Introduction
There has been significant recent interest in the construction of Euclidean quantum field theories using Parisi and Wu’s stochastic quantisation procedure [PW81], which is also the main focus of this article. At a formal level, Euclidean quantum field theories (EQFTs) can be interpreted as probability measures of the form
d μ ( φ ) = Z − 1 exp ( − 2 E ( φ ) ) d φ . (1) \mathrm{d}\mu(\varphi) = \mathcal{Z}^{-1}\exp(-2\mathcal{E}(\varphi))\mathrm{d}\varphi. \tag*{(1)} d μ ( φ ) = Z − 1 exp ( − 2 E ( φ )) d φ . (1) Here, φ \varphi φ Z \mathcal{Z} Z E \mathcal{E} E d φ \mathrm{d}\varphi d φ (1) . Indeed, the Lebesgue measure d φ \mathrm{d}\varphi d φ E \mathcal{E} E μ \mu μ Z \mathcal{Z} Z μ \mu μ μ \mu μ μ \mu μ Φ d 4 \Phi^4_d Φ d 4
The aforementioned stochastic quantisation method of Parisi and Wu is a specific approach towards the construction of (1) which has seen growing interest in the singular SPDE community in recent years. Rather than starting directly from (1) , it relies on the Langevin dynamics
∂ t φ = − ∇ E ( φ ) + ξ , (2) \partial_t\varphi=-\nabla\mathcal{E}(\varphi)+\xi, \tag*{(2)} ∂ t φ = − ∇ E ( φ ) + ξ , (2) where both the gradient ∇ \nabla ∇ ξ \xi ξ L 2 L^2 L 2 E \mathcal{E} E (2) is globally well-posed and admits a unique invariant measure, which is then taken as the definition of the EQFT from (1) . The reason why this is sensible is that, at least in finite dimensions, it is easily seen that the right-hand side of (1) is an invariant measure of (2) . The stochastic quantisation method has now been implemented for many different models, including Φ d 4 \Phi^4_d Φ d 4 (2) . One point that we would like to emphasise is that the original motivation for Parisi and Wu to introduce stochastic quantisation was precisely the hope to study/construct gauge theories without having to fix a specific gauge, thanks to the fact that the dynamical perspective allows one to adjust the gauge “on the fly”. This aspect of the stochastic quantisation method will be crucially exploited here in order to obtain global estimates, so the present work can be viewed as vindicating their original vision.
In this article, our primary interest lies in the two-dimensional non-Abelian Yang–Mills–Higgs model (as well as the pure Yang–Mills model as a special case) originally introduced in [EB64, Hig64, GHK64] in order to explain the mechanism giving mass to elementary particles. In order to state (an informal version of) our main theorem, we first formulate our setting where we use the notation End ( W ) \operatorname{End}(W) End ( W ) W → W W \to W W → W
Setting 1.1. Throughout this paper, we work in the following setting.
(i) We fix a connected compact Lie group G G G g \mathfrak{g} g Ad : G → GL ( g ) \operatorname{Ad}: G \to\operatorname{GL}(\mathfrak{g}) Ad : G → GL ( g ) Ad g ( X ) : = g X g − 1 \operatorname{Ad}_{g}(X) := gXg^{-1} Ad g ( X ) := g X g − 1 g ∈ G g \in G g ∈ G X ∈ g X \in\mathfrak{g} X ∈ g ad : g → End ( g ) \operatorname{ad}: \mathfrak{g} \to\operatorname{End}(\mathfrak{g}) ad : g → End ( g ) ad X ( Y ) = [ X , Y ] \operatorname{ad}_{X}(Y) = [X,Y] ad X ( Y ) = [ X , Y ] [ ⋅ , ⋅ ] [\mathord{\cdot},\mathord{\cdot}] [ ⋅ , ⋅ ] g \mathfrak{g} g
(ii) We equip the Lie algebra g \mathfrak{g} g Ad \operatorname{Ad} Ad ⟨ ⋅ , ⋅ ⟩ g \langle\mathord{\cdot},\mathord{\cdot}\rangle_{\mathfrak{g}} ⟨ ⋅ , ⋅ ⟩ g ⟨ Ad g X , Ad g Y ⟩ g = ⟨ X , Y ⟩ g \langle\operatorname{Ad}_{g}X,\operatorname{Ad}_{g}Y\rangle_{\mathfrak{g}} = \langle X,Y\rangle_{\mathfrak{g}} ⟨ Ad g X , Ad g Y ⟩ g = ⟨ X , Y ⟩ g g ∈ G g \in G g ∈ G X , Y ∈ g X,Y \in\mathfrak{g} X , Y ∈ g
(iii) We let V V V ϱ : G → O ( V ) \varrho: G \to O(V) ϱ : G → O ( V ) ϱ ∗ : g → End ( V ) \varrho_{*}: \mathfrak{g} \to\operatorname{End}(V) ϱ ∗ : g → End ( V )
(iv) We fix a symmetric M ∈ End ( V ) M \in\operatorname{End}(V) M ∈ End ( V ) G G G ⟨ ϱ ( g ) Φ , M ϱ ( g ) Φ ⟩ V = ⟨ Φ , M Φ ⟩ V \langle\varrho(g)\Phi,M\varrho(g)\Phi\rangle_{V} = \langle\Phi,M\Phi\rangle_{V} ⟨ ϱ ( g ) Φ , M ϱ ( g ) Φ ⟩ V = ⟨ Φ , M Φ ⟩ V g ∈ G g \in G g ∈ G Φ ∈ V \Phi\in V Φ ∈ V
Remark 1.2. The Ad \operatorname{Ad} Ad g \mathfrak{g} g G G G g \mathfrak{g} g Ad \operatorname{Ad} Ad ⟨ X , Y ⟩ g = − Kil ( X , Y ) = − Tr ( ad X ∘ ad Y ) \langle X,Y\rangle_{\mathfrak{g}} = -\operatorname{Kil}(X,Y) = -\operatorname{Tr}(\operatorname{ad}_{X} \circ\operatorname{ad}_{Y}) ⟨ X , Y ⟩ g = − Kil ( X , Y ) = − Tr ( ad X ∘ ad Y ) g \mathfrak{g} g
In order to formulate Yang–Mills (YM) theory in the above setting, one furthermore needs to fix a principal G G G P P P P = G × T 2 P = G \times\mathbb{T}^{2} P = G × T 2 T 2 : = ( R / 2 π Z ) 2 \mathbb{T}^{2} := (\mathbb{R}/2\pi\mathbb{Z})^{2} T 2 := ( R /2 π Z ) 2 G G G P P P g \mathfrak{g} g A A A T 2 \mathbb{T}^{2} T 2 A ( x ) = A j ( x ) d x j A(x) = A_{j}(x)\,\mathrm{d}x^{j} A ( x ) = A j ( x ) d x j j ∈ { 1 , 2 } j \in\{1,2\} j ∈ { 1 , 2 } A j : T 2 → g A_{j}: \mathbb{T}^{2} \to\mathfrak{g} A j : T 2 → g Φ \Phi Φ V V V Φ : T 2 → V \Phi: \mathbb{T}^{2} \to V Φ : T 2 → V
The Yang–Mills–Higgs (YMH) energy, which depends on the gauge and Higgs fields, is defined as
E Y M H ( A , Φ ) = ∫ T 2 ( 1 2 ∣ F A ∣ g 2 + 1 2 ∣ D A Φ ∣ V 2 + 1 4 ∣ Φ ∣ V 4 + 1 2 ⟨ Φ , M Φ ⟩ V ) d x . (3) \mathcal{E}_{\mathrm{YMH}}(A,\Phi) = \int_{\mathbb{T}^{2}} \left( \frac{1}{2}|F_{A}|_{\mathfrak{g}}^{2} + \frac{1}{2}|D_{A}\Phi|_{V}^{2} + \frac{1}{4}|\Phi|_{V}^{4} + \frac{1}{2}\langle\Phi,M\Phi\rangle_{V} \right)\,\mathrm{d}x . \tag*{(3)} E YMH ( A , Φ ) = ∫ T 2 ( 2 1 ∣ F A ∣ g 2 + 2 1 ∣ D A Φ ∣ V 2 + 4 1 ∣Φ ∣ V 4 + 2 1 ⟨ Φ , M Φ ⟩ V ) d x . (3) Here, F A F_{A} F A D A D_{A} D A A A A E Y M H \mathcal{E}_{\mathrm{YMH}} E YMH
Remark 1.3. The special case V = { 0 } V = \{0\} V = { 0 }
Remark 1.4. One could in principle consider a torus of arbitrary size L L L (3) and the definition (5) of D A D_{A} D A (3) by rescaling the fields as well as the norms ∣ ⋅ ∣ g |\mathord{\cdot}|_{\mathfrak{g}} ∣ ⋅ ∣ g ∣ ⋅ ∣ V |\mathord{\cdot}|_{V} ∣ ⋅ ∣ V
The Langevin equation associated to (3) is then formally given by the system of SPDEs
∂ t A = − ∇ A E Y M H ( A , Φ ) + ξ , ∂ t Φ = − ∇ Φ E Y M H ( A , Φ ) + ζ , (4) \partial_t A=-\nabla_A\mathcal{E}_{\mathrm{YMH}}(A,\Phi)+\xi,\qquad\partial_t\Phi=-\nabla_\Phi\mathcal{E}_{\mathrm{YMH}}(A,\Phi)+\zeta, \tag*{(4)} ∂ t A = − ∇ A E YMH ( A , Φ ) + ξ , ∂ t Φ = − ∇ Φ E YMH ( A , Φ ) + ζ , (4) where ξ \xi ξ ζ \zeta ζ g 2 \mathfrak{g}^2 g 2 V \mathbf{V} V (4) as the stochastic YMH flow. Over both two- and three-dimensional tori, its1
Theorem 1.5 (Informal version). In Setting 1.1 with base manifold T 2 \mathbb{T}^2 T 2
Remark 1.6. The invariant measure from Theorem 1.5 is a natural candidate for the non-Abelian YMH measure, which had not been constructed previously. To the best of our knowledge, Theorem 1.5 yields the first example of a classical Euclidean quantum field theory that has only been constructed via stochastic quantisation, i.e. using stochastic PDE methods.
Remark 1.7. The space of gauge orbits D κ Y M H \mathcal{D}^{\mathrm{YMH}}_\kappa D κ YMH Theorem 1.5 is analogous to the space D 1 − κ \mathcal{D}_{1-\kappa} D 1 − κ C − κ \mathcal{C}^{-\kappa} C − κ not given by D 1 − κ × C − κ \mathcal{D}_{1-\kappa}\times\mathcal{C}^{-\kappa} D 1 − κ × C − κ D α = Ω α 1 / G α \mathcal{D}_\alpha=\Omega^1_\alpha/\mathcal{G}_\alpha D α = Ω α 1 / G α Ω α 1 \Omega^1_\alpha Ω α 1 G α \mathcal{G}_\alpha G α α \alpha α D κ Y M H = ( Ω 1 − κ 1 × C − κ ) / G 1 − κ \mathcal{D}^{\mathrm{YMH}}_\kappa=(\Omega^1_{1-\kappa}\times\mathcal{C}^{-\kappa})/\mathcal{G}_{1-\kappa} D κ YMH = ( Ω 1 − κ 1 × C − κ ) / G 1 − κ
In particular, it admits Wilson loop observables as jointly continuous functions C 2 ( S 1 , T 2 ) × D κ → G / Ad ( G ) \mathcal{C}^2(S^1,\mathbb{T}^2)\times\mathcal{D}_\kappa\to G/\operatorname{Ad}(G) C 2 ( S 1 , T 2 ) × D κ → G / Ad ( G ) ( A , γ ) ↦ [ hol ( A , γ ) ] (A,\gamma)\mapsto[\operatorname{hol}(A,\gamma)] ( A , γ ) ↦ [ hol ( A , γ )] κ \kappa κ κ \kappa κ
In order to define covariant derivatives, we let W W W π \pi π G G G π ∗ \pi_* π ∗ ( W , π ) (W,\pi) ( W , π ) ( g , Ad ) (\mathfrak{g},\operatorname{Ad}) ( g , Ad ) ( V , ϱ ) (V,\varrho) ( V , ϱ ) A A A D A D_A D A W W W
D A , j : = ∂ j + π ∗ ( A j ) . (5) D_{A,j}:=\partial_j+\pi_*(A_j). \tag*{(5)} D A , j := ∂ j + π ∗ ( A j ) . (5) The curvature tensor F A F_A F A g \mathfrak{g} g ( W , π ) (W,\pi) ( W , π )
[ D A , j , D A , k ] = π ∗ ( F A , j k ) . (6) [D_{A,j},D_{A,k}]=\pi_*(F_{A,jk}). \tag*{(6)} [ D A , j , D A , k ] = π ∗ ( F A , j k ) . (6) A direct computation shows that it is explicitly given by
F A , j k = ∂ j A k − ∂ k A j + [ A j , A k ] . F_{A,jk}=\partial_jA_k-\partial_kA_j+[A_j,A_k]. F A , j k = ∂ j A k − ∂ k A j + [ A j , A k ] . We write G \mathcal{G} G P P P G \mathcal{G} G g : T 2 → G g:\mathbb{T}^2\to G g : T 2 → G α ∈ ( 1 2 , 1 ] \alpha\in(\frac{1}{2},1] α ∈ ( 2 1 , 1 ] G α \mathcal{G}_{\alpha} G α G \mathcal{G} G α \alpha α G α \mathcal{G}_{\alpha} G α A A A Φ \Phi Φ
g ∙ A = Ad g A − d g g − 1 and g ∙ Φ = ϱ ( g ) Φ . (7) g \mathbin{\bullet} A = \operatorname{Ad}_{g} A - \mathrm{d}g\,g^{-1}
\qquad\text{and}\qquad
g \mathbin{\bullet} \Phi= \varrho(g)\Phi. \tag*{(7)} g ∙ A = Ad g A − d g g − 1 and g ∙ Φ = ϱ ( g ) Φ. (7) Here, the assumption α > 1 2 \alpha> \frac{1}{2} α > 2 1 d g g − 1 \mathrm{d}g\,g^{-1} d g g − 1 C x α × C x α − 1 → C x α − 1 C^{\alpha}_{x} \times C^{\alpha-1}_{x} \to C^{\alpha-1}_{x} C x α × C x α − 1 → C x α − 1 W W W w w w g ∙ w = π ( g ) w g \mathbin{\bullet} w = \pi(g)w g ∙ w = π ( g ) w A A A B B B g ∈ G 1 − κ g \in\mathcal{G}_{1-\kappa} g ∈ G 1 − κ B = g ∙ A B = g \mathbin{\bullet} A B = g ∙ A Φ \Phi Φ Ψ \Psi Ψ ( A , Φ ) (A,\Phi) ( A , Φ ) ( B , Ψ ) (B,\Psi) ( B , Ψ )
Remark 1.8. Geometrically, these definitions make sense if we think of all functions as being defined on the whole principal bundle P P P G \mathcal{G} G G \mathcal{G} G P P P
Remark 1.9. While we refer to A A A A A A g \mathfrak{g} g ω \omega ω g ∙ ω = Ad g ω g \mathbin{\bullet} \omega= \operatorname{Ad}_{g}\omega g ∙ ω = Ad g ω
The gauge covariance of the covariant derivatives D A D_A D A W W W w w w π ( g ) D A w = D g ∙ A ( π ( g ) w ) \pi(g)D_Aw = D_{g\mathbin{\bullet} A}(\pi(g)w) π ( g ) D A w = D g ∙ A ( π ( g ) w ) g ∙ D A w = D g ∙ A ( g ∙ w ) g \mathbin{\bullet} D_Aw = D_{g\mathbin{\bullet} A}(g \mathbin{\bullet} w) g ∙ D A w = D g ∙ A ( g ∙ w ) F A F_A F A g ∙ F A : = Ad g F A = F g ∙ A g \mathbin{\bullet} F_A := \operatorname{Ad}_{g}F_A = F_{g\mathbin{\bullet} A} g ∙ F A := Ad g F A = F g ∙ A A A A F A F_A F A raison d’être of gauge theories) is that, since both Ad \operatorname{Ad} Ad ϱ \varrho ϱ E Y M H ( g ∙ A , g ∙ Φ ) = E Y M H ( A , Φ ) E_{\mathrm{YMH}}(g \mathbin{\bullet} A,g \mathbin{\bullet} \Phi) = E_{\mathrm{YMH}}(A,\Phi) E YMH ( g ∙ A , g ∙ Φ ) = E YMH ( A , Φ ) E Y M H E_{\mathrm{YMH}} E YMH
Since we discussed the covariant derivatives, curvature tensor, and gauge transformations, we can now turn our attention to the dynamic. While our main result concerns the Langevin dynamics of the YMH energy (3) , the Yang–Mills gradient flow also plays a crucial role in our proof. It is the gradient flow associated with the half squared L 2 L^{2} L 2 (3) , and it is given by solutions of
∂ t B = − D B ∗ F B . (8) \partial_t B = -D_B^{*}F_B . \tag*{(8)} ∂ t B = − D B ∗ F B . (8) The local and global well-posedness of (8) in dimension d = 2 d=2 d = 2 d = 3 d=3 d = 3 (3) is then (at least formally) given by2
∂ t A = − D A ∗ F A + B ( Φ ⊗ D A Φ ) + ξ , ∂ t Φ = Δ A Φ − M Φ − ∣ Φ ∣ 2 Φ + ζ . (9) \begin{aligned}
\partial_t A &= -D_A^{*}F_A + \mathbf{B}(\Phi\otimes D_A\Phi) + \xi,\\
\partial_t \Phi&= \Delta_A\Phi- M\Phi- \lvert\Phi\rvert^2\Phi+ \zeta. \tag*{(9)}
\end{aligned} ∂ t A ∂ t Φ = − D A ∗ F A + B ( Φ ⊗ D A Φ ) + ξ , = Δ A Φ − M Φ − ∣ Φ ∣ 2 Φ + ζ . (9) Here, ξ \xi ξ ζ \zeta ζ g 2 \mathfrak{g}^{2} g 2 V V V Δ A = δ k ℓ D A , k D A , ℓ = D A ℓ D A , ℓ \Delta_A = \delta^{k\ell}D_{A,k}D_{A,\ell} = D_A^{\ell}D_{A,\ell} Δ A = δ k ℓ D A , k D A , ℓ = D A ℓ D A , ℓ B : V ⊗ V → g \mathbf{B}: V \otimes V \to\mathfrak{g} B : V ⊗ V → g
⟨ B ( u ⊗ v ) , h ⟩ g = ⟨ u , ϱ ∗ ( h ) v ⟩ V , (10) \langle\mathbf{B}(u \otimes v),h\rangle_{\mathfrak{g}} = \langle u,\varrho_{*}(h)v\rangle_{V} , \tag*{(10)} ⟨ B ( u ⊗ v ) , h ⟩ g = ⟨ u , ϱ ∗ ( h ) v ⟩ V , (10) see [CCHS24, Eq. 1.5]. Since the representation ϱ \varrho ϱ G \mathcal{G} G V V V ϱ ∗ ( h ) \varrho_{*}(h) ϱ ∗ ( h ) B ( u ⊗ v ) = − B ( v ⊗ u ) \mathbf{B}(u \otimes v) = -\mathbf{B}(v \otimes u) B ( u ⊗ v ) = − B ( v ⊗ u ) (9) in coordinates, we also refer the reader to (353) below.
We recall from above that the Yang–Mills–Higgs energy E Y M H \mathcal{E}_{\mathrm{YMH}} E YMH L 2 L^2 L 2 ∇ E Y M H ( A , Φ ) \nabla\mathcal{E}_{\mathrm{YMH}}(A,\Phi) ∇ E YMH ( A , Φ ) (1.9) cannot possibly be parabolic since elliptic operators on compact manifolds have compact resolvent and therefore a finite-dimensional kernel. Indeed, a direct calculation shows that the principal part of − ( D A ∗ F A ) k -(D_A^*F_A)_k − ( D A ∗ F A ) k Δ A k − ∂ k ( ∂ ℓ A ℓ ) \Delta A_k-\partial_k(\partial^\ell A_\ell) Δ A k − ∂ k ( ∂ ℓ A ℓ ) ξ \xi ξ ζ \zeta ζ (1.9) infeasible. However, one needs to keep in mind that we are not interested in the solution ( A , Φ ) (A,\Phi) ( A , Φ ) (1.9) itself, but only its equivalence class under gauge transformations, i.e., its gauge orbit. By using a time-dependent gauge transformation, we can3 (1.9) into the stochastic Yang–Mills–Higgs equations
∂ t A = − D A ∗ F A − D A D A ∗ ( A − B ) + B ( Φ ⊗ D A Φ ) + ξ , ∂ t Φ = Δ A Φ + ( D A ∗ ( A − B ) ) Φ − M Φ − ∣ Φ ∣ 2 Φ + ζ . (11) \begin{aligned}
\partial_t A &= -D_A^*F_A-D_AD_A^*(A-B)+B(\Phi\otimes D_A\Phi)+\xi,\\
\partial_t\Phi&= \Delta_A\Phi+(D_A^*(A-B))\Phi-M\Phi-|\Phi|^2\Phi+\zeta. \tag*{(11)}
\end{aligned} ∂ t A ∂ t Φ = − D A ∗ F A − D A D A ∗ ( A − B ) + B ( Φ ⊗ D A Φ ) + ξ , = Δ A Φ + ( D A ∗ ( A − B )) Φ − M Φ − ∣Φ ∣ 2 Φ + ζ . (11) (This is all still formal at this stage.) Here, B B B (1.8) with the same initial data as A A A
Remark 1.10. We will show that, when renormalising {(xref)}(1.11) in a natural way, the process one obtains is indeed gauge equivalent to the one considered in [CCHS22, CCHS24] (see Proposition 2.35 ). In particular, when quotienting by the action of the gauge group, we do obtain a representation for the Markov process constructed in [CCHS22] for any choice of smooth B B B B = 0 B=0 B = 0
Remark 1.11. We do not prove the global well-posedness for {(xref)}(1.11) with B B B (1.8), nor do we expect it to hold. For this reason, we only work with {(xref)}(1.11) for short times and then restart the equation, updating the initial data A 0 A_0 A 0 B B B
Related works
At the beginning of this introduction, we briefly mentioned works related to Theorem 1.5 , but we now discuss the literature in more detail. In the following, we focus our discussion on the quantum Yang–Mills–Higgs (or Yang–Mills) model. The literature concerning the classical YM(H) model, such as work on geometric elliptic equations [Uhl82a, TO4], geometric evolution equations [Råd92, Str94, KM95, KT99, Oh15, OT19, Wal91], and topology [AB83, Don83, Hit87], will not be discussed here. The reader interested only in the main ideas of this article, and not in its relationship to the earlier literature, is encouraged to skip ahead to Section 1.2.
Stochastic quantisation and global bounds on singular SPDEs: As already mentioned, this article makes progress on the rigorous implementation of Parisi and Wu’s stochastic quantisation method. This method is best understood in the context of the Φ 3 4 \Phi^4_3 Φ 3 4 Φ 2 4 \Phi^4_2 Φ 2 4 φ : [ 0 , ∞ ) × T d → R \varphi:[0,\infty)\times\mathbb{T}^d\to\mathbb{R} φ : [ 0 , ∞ ) × T d → R
( ∂ t − Δ ) φ = − φ 3 + ∞ ⋅ φ + ξ . (12) (\partial_t-\Delta)\varphi=-\varphi^3+\infty\cdot\varphi+\xi. \tag*{(12)} ( ∂ t − Δ ) φ = − φ 3 + ∞ ⋅ φ + ξ . (12) (Here, given a regularisation procedure depending on a small parameter ε \varepsilon ε ∞ \infty ∞ ε \varepsilon ε ε → 0 \varepsilon\to0 ε → 0 (12) relies heavily on the strong damping term − φ d 3 -\varphi_d^3 − φ d 3
( ∂ t − Δ ) ψ = − ψ 3 . (13) (\partial_t-\Delta)\psi=-\psi^3. \tag*{(13)} ( ∂ t − Δ ) ψ = − ψ 3 . (13) By comparison with the ODE y ′ = − y 3 y'=-y^3 y ′ = − y 3 (13) satisfies ∥ ψ ( t ) ∥ L ∞ ≤ ( ∥ ψ ( 0 ) ∥ L ∞ − 2 + 2 t ) − 1 / 2 \|\psi(t)\|_{L^\infty}\leq(\|\psi(0)\|_{L^\infty}^{-2}+2t)^{-1/2} ∥ ψ ( t ) ∥ L ∞ ≤ ( ∥ ψ ( 0 ) ∥ L ∞ − 2 + 2 t ) − 1/2 L ∞ L^\infty L ∞ ψ ( t ) \psi(t) ψ ( t ) ( 2 t ) − 1 / 2 (2t)^{-1/2} ( 2 t ) − 1/2 (12) . In fact, by treating the Φ d 4 \Phi_d^4 Φ d 4 (13) , similar estimates (depending also on a suitable measure of the “size” of ξ \xi ξ (12) (see e.g. [MW20, Thm 2.1 and Lem. 2.7).
More recently, there has been growing interest in proving global well-posedness and uniform-in-time estimates for singular SPDEs without such strong damping terms. One model of particular interest is the stochastic Navier–Stokes equation in two dimensions, which was studied in [HR24, HZ25]. For a divergence-free vector-valued unknown u : [ 0 , ∞ ) × T 2 → R 2 u\colon[0,\infty)\times\mathbb{T}^2\to\mathbb{R}^2 u : [ 0 , ∞ ) × T 2 → R 2
( ∂ t − Δ ) u + P div ( u ⊗ u ) = η . (14) (\partial_t-\Delta)u+\mathbf{P}\operatorname{div}(u\otimes u)=\eta. \tag*{(14)} ( ∂ t − Δ ) u + P div ( u ⊗ u ) = η . (14) Here, P \mathbf{P} P η \eta η η \eta η Φ d 4 \Phi_d^4 Φ d 4 (14) does not have a strong damping effect since ⟨ u , P div ( u ⊗ u ) ⟩ L x 2 = 0 \langle u,\mathbf{P}\operatorname{div}(u\otimes u)\rangle_{L_x^2}=0 ⟨ u , P div ( u ⊗ u ) ⟩ L x 2 = 0 L x 2 L_x^2 L x 2
In [HR24, HZ25], the global well-posedness of (14) was shown. To be more precise, the solution of (14) is first decomposed into a large, low-frequency component and a small, high-frequency component. The low- and high-frequency components are then estimated using a monotonicity formula for the energy and the stochastic local theory, respectively. The most delicate parts of the argument concern interactions between the low- and high-frequency components, which witness both the large size of the low-frequency component and the roughness of the high-frequency component. In the later article [HZ25], the third and fourth authors not only proved the global well-posedness of (14) , but also uniform-in-time estimates for the solution. Compared to [HR24], the main ingredient was a new trick involving Itô’s formula, which led to more refined estimates of the low-frequency component.
In [BC24], the first two authors studied the massless two-dimensional stochastic Abelian Higgs model. In this model, one considers the Abelian group4 ^{4} 4 G = U ( 1 ) \mathbf{G}=U(1) G = U ( 1 ) V = C \mathbf{V}=\mathbb{C} V = C ϱ ( e i θ ) z = e i θ z \varrho(e^{i\theta})z=e^{i\theta}z ϱ ( e i θ ) z = e i θ z M = 0 \mathbf{M}=0 M = 0 ∂ j A j = 0 \partial_j A^j=0 ∂ j A j = 0 (3) can then be written as
∂ t A = Δ A − P Im ( Φ ‾ D A Φ ) + P ξ , ∂ t Φ = Δ A Φ − ∣ Φ ∣ 2 Φ + ζ , (15) \partial_t A=\Delta A-\mathbf{P}\operatorname{Im}(\overline{\Phi}D_A\Phi)+\mathbf{P}\xi,\qquad
\partial_t\Phi=\Delta_A\Phi-|\Phi|^2\Phi+\zeta, \tag*{(15)} ∂ t A = Δ A − P Im ( Φ D A Φ ) + P ξ , ∂ t Φ = Δ A Φ − ∣Φ ∣ 2 Φ + ζ , (15) where P \mathbf{P} P Φ \Phi Φ (15) contains the strong damping term − ∣ Φ ∣ 2 Φ -|\Phi|^2\Phi − ∣Φ ∣ 2 Φ A A A A A A Φ \Phi Φ Δ A Φ \Delta_A\Phi Δ A Φ Im ( Φ ‾ D A Φ ) \operatorname{Im}(\overline{\Phi}D_A\Phi) Im ( Φ D A Φ ) T > 0 T>0 T > 0 B : ( 0 , T ] × T 2 → R 2 B\colon(0,T]\times\mathbb{T}^2\to\mathbb{R}^2 B : ( 0 , T ] × T 2 → R 2 T B \mathbf{T}_B T B
( ∂ t − Δ B ) T B = ζ . (16) (\partial_t-\Delta_B)\mathbf{T}_B=\zeta. \tag*{(16)} ( ∂ t − Δ B ) T B = ζ . (16) The estimates of the covariant stochastic object itself, as well as the estimates of the derivative nonlinearity Im ( T ‾ B D B T B ) \operatorname{Im}(\overline{\mathbf{T}}_B D_B\mathbf{T}_B) Im ( T B D B T B ) § 3 and the body of this article, our argument will significantly expand on the ideas in [BC24] in many different ways.
Lastly, we mention that the rigorous construction of the EQFTs from (1.1) has also been tackled using the Boué–Dupuis formula [BD98]. This approach was pioneered by Barashkov and Gubinelli in [BG20] and, since then, has had several further applications [Bri22, GOTW22, OST22, OOT24, BG25, TW26]. While it is not based on the Langevin equation (1.2), its implementation often uses many of the same tools as the stochastic quantisation method. Very recently, the variational approach has also been used to construct the two-dimensional Abelian Higgs measure in the Coulomb gauge [BCC+26].
The pure Yang–Mills model using exact solvability: There is a rich literature on the pure YM model on two-dimensional Riemannian manifolds M \mathcal{M} M A A A g \mathfrak{g} g M \mathcal{M} M γ : [ 0 , 1 ] → M \gamma\colon[0,1]\to\mathcal{M} γ : [ 0 , 1 ] → M hol ( A , γ ) : = h ( 1 ) ∈ G \operatorname{hol}(A,\gamma)\mathrel{\mathop:}=h(1)\in G hol ( A , γ ) : = h ( 1 ) ∈ G h : [ 0 , 1 ] → G h\colon[0,1]\to G h : [ 0 , 1 ] → G
d h ( t ) h − 1 ( t ) = − A ( d γ ( t ) ) , h ( 0 ) = id G . (17) \mathrm{d}h(t)h^{-1}(t)=-A(\mathrm{d}\gamma(t)),\qquad h(0)=\operatorname{id}_{G}. \tag*{(17)} d h ( t ) h − 1 ( t ) = − A ( d γ ( t )) , h ( 0 ) = id G . (17) The holonomy hol ( A , γ ) \operatorname{hol}(A,\gamma) hol ( A , γ )
hol ( g ∙ A , γ ) = g ( γ ( 1 ) ) hol ( A , γ ) g ( γ ( 0 ) ) − 1 . (18) \operatorname{hol}(g\mathbin{\bullet} A,\gamma)=g(\gamma(1))\operatorname{hol}(A,\gamma)g(\gamma(0))^{-1}. \tag*{(18)} hol ( g ∙ A , γ ) = g ( γ ( 1 )) hol ( A , γ ) g ( γ ( 0 ) ) − 1 . (18) To obtain (1.18), one only needs to verify that if h ( t ) h(t) h ( t ) g ( γ ( t ) ) h ( t ) g ( γ ( 0 ) ) − 1 g(\gamma(t))h(t)g(\gamma(0))^{-1} g ( γ ( t )) h ( t ) g ( γ ( 0 ) ) − 1 A A A g ∙ A g\mathbin{\bullet} A g ∙ A x 0 ∈ M x_0\in\mathcal{M} x 0 ∈ M γ \gamma γ x 0 x_0 x 0 hol ( g ∙ A , γ ) = g ( x 0 ) hol ( A , γ ) g ( x 0 ) − 1 \operatorname{hol}(g\mathbin{\bullet} A,\gamma)=g(x_0)\operatorname{hol}(A,\gamma)g(x_0)^{-1} hol ( g ∙ A , γ ) = g ( x 0 ) hol ( A , γ ) g ( x 0 ) − 1
( hol ( A , γ ) ) loops γ . (19) \left(\operatorname{hol}(A,\gamma)\right)_{\text{loops }\gamma}. \tag*{(19)} ( hol ( A , γ ) ) loops γ . (19) which is defined modulo simultaneous conjugation by elements of the Lie group G G G
More recently, there has been interest in going beyond the construction of (1.19) as a stochastic process. The goal is then to construct a state space (consisting of gauge orbits of connection one-forms) on which the holonomies along smooth loops are well-defined (modulo the action of G G G M = T 2 \mathcal{M}=\mathbb{T}^2 M = T 2 C x − κ ( T 2 , g 2 ) \mathcal{C}^{-\kappa}_x(\mathbb{T}^2,\mathfrak{g}^2) C x − κ ( T 2 , g 2 ) κ > 0 \kappa>0 κ > 0 Theorem 1.5 ), the axial gauge loses much of its utility. The reason is that while the axial gauge eliminates nonlinear interactions involving only the connection one-form A A A Φ \Phi Φ Φ \Phi Φ A A A T 2 \mathbb{T}^{2} T 2 T 2 \mathbb{T}^{2} T 2 M \mathcal{M} M
In [CS23], Chevyrev and Shen study the Langevin equation of the pure YM model on the two-dimensional torus T 2 \mathbb{T}^{2} T 2 T 2 \mathbb{T}^{2} T 2
Lattice gauge theories: A natural approach towards constructing continuum YM theories involves taking scaling limits of lattice discretisations. Such discretisations are known as lattice gauge theories (alternatively, lattice YM theories), and were first proposed by Wilson [Wil74]. For a more thorough introduction to these models, see the survey [Cha19a]. Brydges, Fröhlich and Seiler [BFS79, BFS80, BFS81] constructed the two-dimensional Abelian Higgs model via this approach. Soon after, Balaban [Bal82a, Bal82b, Bal83a, Bal83b] proved ultraviolet stability for the two- and three-dimensional Abelian Higgs models. (In the 2D case, this was recently strengthened in [CC24].) Later on, King [Kin86a, Kin86b] built on Balaban’s Abelian Higgs work by constructing the continuum theory in three dimensions. In all these cited works on the Abelian Higgs model, the main technique was a renormalisation group analysis to understand how the various theories behave under changes of length scales. Then, by inductively applying this analysis, the authors were able to control the unit length scale behaviour of theories which were originally defined on microscopic length scales.
In higher dimensions, Gross [Gro83] established convergence of the three-dimensional U ( 1 ) U(1) U ( 1 )
Next, we turn to the non-Abelian setting, i.e. the YM model. In the series of works [Bal84a, Bal84b, Bal85a, Bal85b, Bal85c, Bal85d, Bal85e], Balaban studied this model in three dimensions (again using renormalisation group methods). To discuss Balaban’s work, we let T ε 3 \mathbb{T}^{3}_{\varepsilon} T ε 3 T 3 \mathbb{T}^{3} T 3 ε > 0 \varepsilon> 0 ε > 0 G ∈ { U ( N ) , S U ( N ) , O ( N ) } G \in\{U(N), SU(N), O(N)\} G ∈ { U ( N ) , S U ( N ) , O ( N )} U ε : E ε → G U_{\varepsilon}: E_{\varepsilon} \to G U ε : E ε → G E ε E_{\varepsilon} E ε T ε 3 \mathbb{T}^{3}_{\varepsilon} T ε 3 U ε ( x , y ) = U ε ( y , x ) − 1 U_{\varepsilon}(x,y)=U_{\varepsilon}(y,x)^{-1} U ε ( x , y ) = U ε ( y , x ) − 1 ( x , y ) ∈ E ε (x,y) \in E_{\varepsilon} ( x , y ) ∈ E ε
E ε ( U ε ) = E Y M , ε ( U ε ) ≔ 1 g 2 ε ∑ p ∈ P ε ( 1 − Re tr ( U ε ( ∂ p ) ) ) . \mathcal{E}_{\varepsilon}(U_{\varepsilon})=\mathcal{E}_{\mathrm{YM},\varepsilon}(U_{\varepsilon})\coloneqq\frac{1}{g^{2}\varepsilon}\sum_{p\in\mathcal{P}_{\varepsilon}}\left(1-\operatorname{Re}\operatorname{tr}(U_{\varepsilon}(\partial p))\right). E ε ( U ε ) = E YM , ε ( U ε ) : = g 2 ε 1 p ∈ P ε ∑ ( 1 − Re tr ( U ε ( ∂ p )) ) . Here, g g g P ε P_{\varepsilon} P ε T ε 3 \mathbb{T}_{\varepsilon}^{3} T ε 3 U ε ( ∂ p ) U_{\varepsilon}(\partial p) U ε ( ∂ p ) U ε U_{\varepsilon} U ε p p p tr \operatorname{tr} tr E ε ( k ) \mathcal{E}_{\varepsilon}^{(k)} E ε ( k ) T L k ε 3 \mathbb{T}_{L^{k}\varepsilon}^{3} T L k ε 3 L ≥ 2 L \ge2 L ≥ 2 ε → 0 \varepsilon\to0 ε → 0 5
In more recent years, there has been a renewed interest in the mathematical analysis of lattice gauge theory. In [Cha16], Chatterjee computed the leading term for the free energy of the three- and four-dimensional pure U ( N ) U(N) U ( N ) S U ( 2 ) SU(2) S U ( 2 ) d ≥ 2 d \ge2 d ≥ 2
We also mention various works which primarily focus on the long-distance behaviour of lattice gauge theories. The primary questions here are whether there is “quark confinement” and a “mass gap”; see [Cha19a, Sections 4, 5] for an introduction to these properties. Generally speaking, a theory that possesses these properties looks very disordered at long distances, while a theory without these properties exhibits a form of long-range order. Osterwalder and Seiler [OS78] showed that lattice gauge theories with arbitrary gauge group G G G S U ( N ) SU(N) S U ( N ) U ( 1 ) U(1) U ( 1 ) U ( 1 ) U(1) U ( 1 )
One approach towards this problem is to relate lattice gauge theories to certain “string theories”, i.e. surface models, via a “gauge / string duality”. See e.g. Polyakov’s retrospective [Pol08] on this subject from a physics point of view. In mathematics, Chatterjee [Cha19b] established a gauge / string duality for the large-N N N S O ( N ) SO(N) S O ( N ) S U ( N ) SU(N) S U ( N ) N N N N N N N N N N N N
Finally, there has also been a recent line of work initiated by Shen, Smith, R. Zhu, and X. Zhu [SZZ23, SSZ24, SSZ25, SZZ26a], which uses the lattice Langevin dynamics to prove results such as the Makeenko–Migdal equations and a mass gap (in certain strong coupling regimes) for lattice gauge theories. See also [CNS25b], which proves confinement at strong coupling.
Brief discussion of main ideas and techniques
We conclude the introduction with a brief discussion of the main ideas behind the proof of Theorem 1.5 , but refer the reader to § 3 (as well as the body of this article) for more detailed discussions. For most of this brief discussion, we focus on the stochastic Yang–Mills (rather than Yang–Mills–Higgs) equations in the infinitesimal DeTurck gauge (see (11) ), which are (formally) given by
∂ t A = − D A ∗ F A − D A D A ∗ ( A − B ) + ξ . (20) \partial_t A=-\mathrm{D}_A^*F_A-\mathrm{D}_A\mathrm{D}_A^*(A-B)+\xi. \tag*{(20)} ∂ t A = − D A ∗ F A − D A D A ∗ ( A − B ) + ξ . (20) Infinitesimal DeTurck gauge: The idea behind this gauge is to use the DeTurck trick (which would correspond to taking B = 0 B=0 B = 0 (20) ) only at the level of the linearised dynamic, not at the level of the full Langevin dynamics. To make this more precise, take B B B C : = A − B C:=A-B C := A − B Lemma 3.22 below, it can then be shown that C C C
( ∂ t − Δ B ) C k − 2 [ F B k j , C j ] = [ C j , 2 D B j C k − D B k C j ] + [ C j , [ C j , C k ] ] + ξ k . (21) (\partial_t-\Delta_B)C^k-2[F_B^{kj},C_j]=[C_j,2\mathrm{D}_B^jC^k-\mathrm{D}_B^kC^j]+[C_j,[C^j,C^k]]+\xi^k . \tag*{(21)} ( ∂ t − Δ B ) C k − 2 [ F B k j , C j ] = [ C j , 2 D B j C k − D B k C j ] + [ C j , [ C j , C k ]] + ξ k . (21) We emphasise that the connection one-form B B B (21) through D B \mathrm{D}_B D B F B F_B F B B B B (8) . This is important because the Yang–Mills heat flow (with DeTurck term) does not obey suitable long-time estimates. The reader may think of B B B A A A C C C A A A A A A
Covariant heat kernel estimates: Motivated by (21) , we examine the covariant heat kernel q B ( s , y ; t , x ) q_B(s,y;t,x) q B ( s , y ; t , x ) ∂ t − Δ B − 2 ad ( F B ) \partial_t-\Delta_B-2\operatorname{ad}(F_B) ∂ t − Δ B − 2 ad ( F B ) (21) . We emphasise that, while the curvature term 2 ad ( F B ) 2\operatorname{ad}(F_B) 2 ad ( F B ) q B ( s , y ; t , x ) q_B(s,y;t,x) q B ( s , y ; t , x ) q B ( s , y ; t , x ) q_B(s,y;t,x) q B ( s , y ; t , x ) 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T x , y ∈ T 2 x,y\in\mathbb{T}^2 x , y ∈ T 2
∣ D B j 1 ⋯ D B j k q B ( s , y ; t , x ) ∣ ≲ exp ( C ∥ F B ∥ L t ∞ L x ∞ ( [ 0 , T ] ) ) ( 1 + ∥ F B ∥ T , T p e r t ) k ( t − s ) 1 + k 2 exp ( − c ∣ x − y ∣ 2 t − s ) . (22) \left|\mathrm{D}_B^{j_1}\cdots\mathrm{D}_B^{j_k}q_B(s,y;t,x)\right|\lesssim\exp\left(C\|F_B\|_{L_t^\infty L_x^\infty([0,T])}\right)\frac{\left(1+\|F_B\|_{\mathcal{T},T}^{\mathrm{pert}}\right)^k}{(t-s)^{1+\frac{k}{2}}}\exp\left(-c\frac{|x-y|^2}{t-s}\right). \tag*{(22)} D B j 1 ⋯ D B j k q B ( s , y ; t , x ) ≲ exp ( C ∥ F B ∥ L t ∞ L x ∞ ([ 0 , T ]) ) ( t − s ) 1 + 2 k ( 1 + ∥ F B ∥ T , T pert ) k exp ( − c t − s ∣ x − y ∣ 2 ) . (22) Here, k ≥ 1 k\ge1 k ≥ 1 1 ≤ j 1 , … , j k ≤ 2 1\le j_1,\ldots,j_k\le2 1 ≤ j 1 , … , j k ≤ 2 ∥ F B ∥ T , T p e r t \|F_B\|_{\mathcal{T},T}^{\mathrm{pert}} ∥ F B ∥ T , T pert (22) relies on covariant monotonicity formulas, as well as delicate estimates of curvature terms.
Gauge-covariant smooth approximations: In singular SPDEs, local and global well-posedness are studied using limits of smooth approximations. In the earlier works [CCHS22, CCHS24, BC26a], the smooth approximations broke the gauge invariance of the law of the solution, which could only be restored in the limit. In this work, we introduce a gauge covariant smooth approximation of (20) , which is defined as
∂ t A ε = − D A ε ∗ F A ε − D A ε D A ε ∗ ( A ε − B ) + C ρ A ( A ε − B ) + ξ B , ε . (23) \partial_t A_\varepsilon=-\mathrm{D}_{A_\varepsilon}^*F_{A_\varepsilon}-\mathrm{D}_{A_\varepsilon}\mathrm{D}_{A_\varepsilon}^*(A_\varepsilon-B)+C_\rho^{A}(A_\varepsilon-B)+\xi_{B,\varepsilon}. \tag*{(23)} ∂ t A ε = − D A ε ∗ F A ε − D A ε D A ε ∗ ( A ε − B ) + C ρ A ( A ε − B ) + ξ B , ε . (23) Here, C ρ A ( A ε − B ) C_{\rho}^{A}(A_{\varepsilon}-B) C ρ A ( A ε − B )
ξ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (24) \xi_{B,\varepsilon}(t,x)=\int_{0}^{t} ds\int_{\mathbb{T}^{2}}dy\,\rho_{\varepsilon}'(t-s)q_{B}(s,y;t,x)\xi(s,y), \tag*{(24)} ξ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (24) where ρ ε : [ 0 , ∞ ) → [ 0 , 1 ] \rho_{\varepsilon}:[0,\infty)\to[0,1] ρ ε : [ 0 , ∞ ) → [ 0 , 1 ] (23) , however, comes at a price. Unlike for the non-covariant smooth approximations from [CCHS22, CCHS24, BC26a], it is not clear that the resulting stochastic process (at the level of the gauge orbits) is Markov. To establish the Markov property, we will show the gauge equivalence of the limiting solutions from this article and [CCHS22], see Proposition 2.35 2.35 2.35
Covariant local theory: We develop a covariant local theory for (23) . The resulting local time of existence only depends on the size of the curvature F B F_B F B B B B (8) only control the size of F B F_B F B
( ∂ t − Δ B − 2 ad ( F B ) ) Q ˚ B = ξ , Q ˚ B ( 0 ) = 0. (25) (\partial_t-\Delta_B-2\operatorname{ad}(F_B))\mathring{Q}_B=\xi,\qquad\mathring{Q}_B(0)=0. \tag*{(25)} ( ∂ t − Δ B − 2 ad ( F B )) Q ˚ B = ξ , Q ˚ B ( 0 ) = 0. (25) Similar covariant stochastic objects were previously used in [BC24] for the Abelian Higgs model, but our non-Abelian setting prevents us from using many of the estimates in [BC24]. In addition, we introduce a covariant paracontrolled calculus, which comes with covariant commutator estimates, Littlewood–Paley decompositions, paraproduct estimates, and Schauder estimates.
Global iteration argument: In order to prove the global well-posedness in Theorem 1.5 1.5 1.5 ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ τ m + 1 ≔ t m + 1 − t m \tau_{m+1}\coloneqq t_{m+1}-t_m τ m + 1 : = t m + 1 − t m (20) on each interval [ t m , t m + 1 ] [t_m,t_{m+1}] [ t m , t m + 1 ] A A A B B B B B B t = t m + 1 t=t_{m+1} t = t m + 1 6
∥ F B ( t m + 1 ) ∥ L x 2 2 − ∥ F B ( t m ) ∥ L x 2 2 = − 2 ∥ D B ∗ F B ∥ L t , x 2 ( [ t m , t m + 1 ] ) 2 . (26) \|F_B(t_{m+1})\|_{L_x^2}^2-\|F_B(t_m)\|_{L_x^2}^2=-2\|D_B^*F_B\|_{L_{t,x}^2([t_m,t_{m+1}])}^2. \tag*{(26)} ∥ F B ( t m + 1 ) ∥ L x 2 2 − ∥ F B ( t m ) ∥ L x 2 2 = − 2∥ D B ∗ F B ∥ L t , x 2 ([ t m , t m + 1 ]) 2 . (26) Due to the space-time white noise ξ \xi ξ (20) also contains several rough terms. On a single time interval, the rough terms can be controlled using our covariant local theory. However, due to the exponential growth in (22) , the rough terms have to enter into the background B B B Q ˚ B \mathring{Q}_B Q ˚ B B B B L 2 L^2 L 2 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 (26) .
Decay mechanism near Yang–Mills connections: If the background B B B D B ∗ F B D_B^*F_B D B ∗ F B (26) does not provide us with any appreciable decay. In order to obtain the uniform-in-time estimates of Theorem 1.5 1.5 1.5 § 5.5 ). While the unstable directions lead to growth of the linearised dynamic, this growth should lead to a decrease in energy (see Figure 1 as well as the identity (88) below). To be more precise, if B B B
Figure 1. Illustration of Langevin dynamics on a surface equipped with a Morse function.
( ∇ 2 E Y M ( v , v ) ) , v = e ( t m + 1 − t m + 1 / 2 ) ( Δ B + 2 ad ( F B ) ) Q ˚ B ( t m + 1 / 2 ) . (27) (\nabla^2\mathcal{E}_{\mathrm{YM}}(v,v)),\qquad v=e^{(t_{m+1}-t_{m+1/2})(\Delta_B+2\operatorname{ad}(F_B))}\mathring{Q}_B(t_{m+1/2}). \tag*{(27)} ( ∇ 2 E YM ( v , v )) , v = e ( t m + 1 − t m + 1/2 ) ( Δ B + 2 ad ( F B )) Q ˚ B ( t m + 1/2 ) . (27) Here, E Y M \mathcal{E}_{\mathrm{YM}} E YM t m + 1 / 2 t_{m+1/2} t m + 1/2 ( t m , t m + 1 ) (t_m,t_{m+1}) ( t m , t m + 1 ) B ˚ ( t m + 1 / 2 ) \mathring{B}(t_{m+1/2}) B ˚ ( t m + 1/2 ) (27) is the covariant heat flow induced by the time-dependent connection one-form B B B Proposition 2.8 ).
The mechanism here is strongly reminiscent of the situation arising when one considers the projective dynamic for a linear stochastic PDE, see [HR25, HPRY24]. There, each eigenvector of the linear operator yields a saddle point for the deterministic dynamic and one also relies on the presence of the noise to show that the dynamic get “unstuck”.
An even simpler model, illustrated in Figure 1 , is the stochastic gradient flow with respect to a Morse function on a surface. In the figure, the Morse function f f f
Control of the Higgs field: While we have so far focused our discussion on the stochastic Yang–Mills equations (20) , including the Higgs field requires at least one new ingredient. The counterpart of the background connection one-form B B B φ \varphi φ B B B ( B , φ ) (B,\varphi) ( B , φ )
∥ F B ∥ L x 2 2 + ∥ φ ∥ L x γ 1 γ 1 + ∥ φ ∥ L x r γ 2 . (28) \lVert F_B\rVert_{L_x^2}^2+\lVert\varphi\rVert_{L_x^{\gamma_1}}^{\gamma_1}+\lVert\varphi\rVert_{L_x^r}^{\gamma_2}. \tag*{(28)} ∥ F B ∥ L x 2 2 + ∥ φ ∥ L x γ 1 γ 1 + ∥ φ ∥ L x r γ 2 . (28) Here, γ 1 \gamma_1 γ 1 4 4 4 γ 2 \gamma_2 γ 2 3 3 3 r r r L x r L_x^r L x r (28) is useful in our estimates of the B ( φ ⊗ D B φ ) B(\varphi\otimes D_B\varphi) B ( φ ⊗ D B φ ) A A A L x r L_x^r L x r (28) is useful in closing our estimates for the Higgs field itself.
Remnants of exact solvability: We previously emphasised that, unlike the two-dimensional pure Yang–Mills model, the Yang–Mills–Higgs model is not exactly solvable. Nevertheless, since we take as our reference connection in (11) the solution to the (pure) Yang–Mills gradient flow, some remnants of the exact solvability of the pure Yang–Mills model can be seen in parts of our argument. In two dimensions, the four components of the curvature tensor F A F_A F A F A 11 = F A 22 = 0 F_A^{11}=F_A^{22}=0 F A 11 = F A 22 = 0 F A 21 = − F A 12 F_A^{21}=-F_A^{12} F A 21 = − F A 12 [ F A i j , F A k l ] = 0 [F_A^{ij},F_A^{kl}]=0 [ F A ij , F A k l ] = 0 − D A ∗ F A -D_A^*F_A − D A ∗ F A ( − D A ∗ F A ) 1 = − D A , 2 F A 12 (-D_A^*F_A)^1=-D_{A,2}F_A^{12} ( − D A ∗ F A ) 1 = − D A , 2 F A 12 ( − D A ∗ F A ) 2 = D A , 1 F A 12 (-D_A^*F_A)^2=D_{A,1}F_A^{12} ( − D A ∗ F A ) 2 = D A , 1 F A 12 − D A ∗ F A -D_A^*F_A − D A ∗ F A
For any solution B B B (8) , it can then be shown that the component F B 12 F_B^{12} F B 12 ( ∂ t − Δ B ) F B 12 = 0 (\partial_t-\Delta_B)F_B^{12}=0 ( ∂ t − Δ B ) F B 12 = 0 B B B F B 12 F_B^{12} F B 12 F B 12 ( t ) F_B^{12}(t) F B 12 ( t ) F B 12 ( 0 ) F_B^{12}(0) F B 12 ( 0 ) B B B F B F_B F B
Going for the kill: Once we have global-in-time curvature bounds thanks to our control of (28) , the remainder of the argument is by now quite standard. There are, however, a few subtleties that need verifying. First, we need to make sure that the process constructed as the limit ε → 0 \varepsilon\to0 ε → 0 (23) coincides with the process constructed in [CCHS22], modulo gauge transformation (and with the Higgs field added on both sides). Since the approximation used there is quite different, this is actually far from obvious! However, the problems are virtually identical to those arising when trying to show gauge covariance in [CCHS22]. (This is not too surprising since the process (23) is already gauge covariant for fixed ε > 0 \varepsilon> 0 ε > 0 (23) create quite a few more counterterms, so that the computation of the counterterms is more involved.
Once we know that we are looking at the correct process, it remains to show that it admits a unique invariant probability measure. Existence is immediate from the usual tightness argument, while uniqueness follows if we can show that the process is strong Feller and topologically irreducible. The strong Feller property follows as in [CS23]. Regarding irreducibility, it unfortunately appears quite difficult to apply the results of [CF18, HS22]. Instead, we develop a new (and simpler) argument which is more widely applicable and should therefore be of independent interest.
AI disclosure statement
The main mathematical ideas contained in this article as well as the vast majority of its write-up were generated without the assistance of AI. The authors started working on this project in April 2025 and arrived at the main ideas by February 2026, which predates the most recent AI-powered breakthroughs. These ideas and results were presented by the authors at various conferences and seminar talks in early 2026, including at MPI Leipzig (March 10), Stanford University (April 6), and a Simons Collaboration Meeting (April 30).
While AI assistance was not used to generate new ideas, AI tools were used for other research tasks. Different versions of ChatGPT Pro and Claude were used to tweak the exposition, search the literature, and proofread parts of the article. During the final stages of this project, GPT-6 Astra and GPT-5.6 Sol were also used to prove a small number of technical lemmas and propositions, namely, Proposition 3.55, Lemma 3.58, Proposition 3.59, Lemma 5.26, Lemma 11.45, Lemma 11.46 and Lemma 11.55. GPT-6 Astra was also used by WZ to assist with typing some of the formulas in Section 11.3. They were generated one at a time and carefully reviewed before being incorporated into the manuscript.
In addition, during the development of the main ideas in 2025, conversations (for lack of a better word) with ChatGPT about existing tools in differential geometry and geometric analysis were extremely helpful.
Acknowledgements
BB’s research was partially supported by the National Science Foundation under Grant No. DMS-2453126; and he thanks Princeton University for providing a stimulating environment during most stages of this work. SC’s research was partially supported by the National Science Foundation under Grant No. DMS-2303165; he would also like to thank MIT for providing excellent working conditions under which much of this work was completed. MH’s research was partially funded by the Simons Collaboration “Probabilistic Paths to Quantum Field Theory”, as well as by the Swiss National Science Foundation, grant number 10006872. We thank the Courant Institute of Mathematical Sciences, EPFL, the Institute for Mathematical Sciences at NUS, the Hausdorff Research Institute for Mathematics, Peking University, Princeton University, the Simons Foundation, and SLMath for hosting various subsets of the authors during the completion of this work.
Main results
In the previous section, we stated our main theorem (Theorem 1.5 ) and briefly discussed our main ideas, but this was all done at an informal level. In this section, we will be detailed and rigorous. In particular, we introduce rigorous definitions and state several precise theorems capturing the main results from different parts of this article. In Section 2.1, we discuss our covariant heat kernels and their estimates. In Section 2.2, we state our covariant local well-posedness theorem. In Section 2.3, we discuss our global iteration scheme and uniform-in-time estimates. Finally, in Section 2.4, we discuss the Markov process on the space of gauge orbits and its unique invariant measure, namely, the Yang–Mills–Higgs measure.
Covariant heat kernel estimates
Throughout most of this article (except when comparing the process constructed here to the one constructed in [CCHS22, CCHS24], which is required to show that it has the Markov property), we adopt the following central principle: we always work covariantly with respect to a (large) background connection, because we want to treat the background non-perturbatively. This requires us to develop an entire covariant Schauder theory which constitutes the analytic backbone of this article and is the subject of Section 4. In particular, we obtain estimates on covariant heat kernels, which we heavily use in the ensuing covariant local theory (see Section 2.2 for more discussion). We now begin to describe these estimates. Unless otherwise specified, the following discussion applies to a general dimension d ≥ 2 d \ge2 d ≥ 2
Throughout, we fix a time T ∈ ( 0 , 1 ] T \in(0,1] T ∈ ( 0 , 1 ] B : [ 0 , T ] × T d → g d B : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} B : [ 0 , T ] × T d → g d
( ∂ t − Δ B ) ϕ = G , (29) (\partial_t-\Delta_B)\phi=G, \tag*{(29)} ( ∂ t − Δ B ) ϕ = G , (29) ( ∂ t − Δ B ) ψ i = 2 ( F B ) ℓ i ψ ℓ + H i , i ∈ [ d ] . (30) (\partial_t-\Delta_B)\psi_i=2(F_B)^i_{\ell}\psi_\ell+H_i,\qquad i\in[d]. \tag*{(30)} ( ∂ t − Δ B ) ψ i = 2 ( F B ) ℓ i ψ ℓ + H i , i ∈ [ d ] . (30) In the first equation, ϕ , G \phi,G ϕ , G W \mathbf{W} W ψ , H \psi,H ψ , H W d \mathbf{W}^{d} W d 7
( ∂ t − Δ B ) ψ = 2 F B ψ + H , (\partial_t-\Delta_B)\psi=2F_B\psi+H, ( ∂ t − Δ B ) ψ = 2 F B ψ + H , which is in line with the interpretation of F F F W d \mathbf{W}^{d} W d ( F X ) i = F i j X j (FX)_i=F_i^{\,j}X_j ( F X ) i = F i j X j X = ( X i , i ∈ [ d ] ) ∈ W d X=(X_i,i\in[d])\in\mathbf{W}^{d} X = ( X i , i ∈ [ d ]) ∈ W d
Definition 2.1 (Covariant heat kernels). Let p B ( s , y ; t , x ; W ) p_B(s,y;t,x;\mathbf{W}) p B ( s , y ; t , x ; W ) (29) and q B ( s , y ; t , x ; W d ) q_B(s,y;t,x;\mathbf{W}^{d}) q B ( s , y ; t , x ; W d ) (30) , i.e., for ϕ s ∈ C ∞ ( T d , W ) \phi_s\in C^\infty(\mathbb{T}^{d},\mathbf{W}) ϕ s ∈ C ∞ ( T d , W )
ϕ ( t , x ) : = ∫ d y p B ( s , y ; t , x ; W ) ϕ s ( y ) (31) \phi(t,x):=\int\mathrm{d}y\,p_B(s,y;t,x;\mathbf{W})\phi_s(y) \tag*{(31)} ϕ ( t , x ) := ∫ d y p B ( s , y ; t , x ; W ) ϕ s ( y ) (31) is the solution to (29) on [ s , T ] [s,T] [ s , T ] G = 0 G=0 G = 0 ϕ s \phi_s ϕ s ψ s ∈ C ∞ ( T d , W d ) \psi_s\in C^\infty(\mathbb{T}^{d},\mathbf{W}^{d}) ψ s ∈ C ∞ ( T d , W d )
ψ ( t , x ) : = ∫ d y q B ( s , y ; t , x ; W d ) ψ s ( y ) (32) \psi(t,x):=\int\mathrm{d}y\,q_B(s,y;t,x;\mathbf{W}^{d})\psi_s(y) \tag*{(32)} ψ ( t , x ) := ∫ d y q B ( s , y ; t , x ; W d ) ψ s ( y ) (32) is the solution to (30) with H = 0 H=0 H = 0 ψ s \psi_s ψ s ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d (s,y),(t,x)\in[0,T]\times\mathbb{T}^d ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d s < t s<t s < t p B ( s , y ; t , x ; W ) ∈ End ( W ) p_B(s,y;t,x;W)\in\operatorname{End}(W) p B ( s , y ; t , x ; W ) ∈ End ( W ) q B ( s , y ; t , x ; W d ) ∈ End ( W d ) q_B(s,y;t,x;W^d)\in\operatorname{End}(W^d) q B ( s , y ; t , x ; W d ) ∈ End ( W d ) p B ∈ End ( W d ) p_B\in\operatorname{End}(W^d) p B ∈ End ( W d ) p B , q B p_B,q_B p B , q B W , W d W,W^d W , W d p B ( s , y ; t , x ) , q B ( s , y ; t , x ) p_B(s,y;t,x),q_B(s,y;t,x) p B ( s , y ; t , x ) , q B ( s , y ; t , x ) p B ( t , x ) = p B ( 0 , 0 ; t , x ) , q B ( t , x ) = q B ( 0 , 0 ; t , x ) p_B(t,x)=p_B(0,0;t,x),q_B(t,x)=q_B(0,0;t,x) p B ( t , x ) = p B ( 0 , 0 ; t , x ) , q B ( t , x ) = q B ( 0 , 0 ; t , x )
Remark 2.2. The correct geometric setting consists of a vector bundle E → M E\to M E → M M = T d M=\mathbb{T}^d M = T d W W W T ∗ M ⊗ E T^*M\otimes E T ∗ M ⊗ E p B ( s , y ; t , x ) ∈ L ( E y , E x ) p_B(s,y;t,x)\in L(E_y,E_x) p B ( s , y ; t , x ) ∈ L ( E y , E x ) q B ( s , y ; t , x ) ∈ L ( ( T y ∗ M ) ⊗ E y , ( T x ∗ M ) ⊗ E x ) q_B(s,y;t,x)\in L((T_y^*M)\otimes E_y,(T_x^*M)\otimes E_x) q B ( s , y ; t , x ) ∈ L (( T y ∗ M ) ⊗ E y , ( T x ∗ M ) ⊗ E x ) L ( X , Y ) L(X,Y) L ( X , Y ) X X X Y Y Y p B p_B p B q B q_B q B (31) and (32) , where we view ϕ ∈ Γ ( E ) \phi\in\Gamma(E) ϕ ∈ Γ ( E ) ψ ∈ Γ ( T ∗ M ⊗ E ) \psi\in\Gamma(T^*M\otimes E) ψ ∈ Γ ( T ∗ M ⊗ E ) ϕ ( t , x ) ∈ E x \phi(t,x)\in E_x ϕ ( t , x ) ∈ E x ψ ( t , x ) ∈ T x ∗ M ⊗ E x \psi(t,x)\in T_x^*M\otimes E_x ψ ( t , x ) ∈ T x ∗ M ⊗ E x
The more explicit viewpoints of p B ∈ End ( W ) p_B\in\operatorname{End}(W) p B ∈ End ( W ) q B ∈ End ( W d ) q_B\in\operatorname{End}(W^d) q B ∈ End ( W d ) E E E T M = T d TM=\mathbb{T}^d T M = T d E x ≅ W E_x\cong W E x ≅ W T x ∗ M ⊗ E x ≅ W d T_x^*M\otimes E_x\cong W^d T x ∗ M ⊗ E x ≅ W d
In order to obtain covariant Schauder estimates, we need to estimate the solutions ϕ , ψ \phi,\psi ϕ , ψ (29) and (30) . This will require us to assume control on certain quantities involving the curvature F B F_B F B
Definition 2.3 (Curvature norm). Let I = [ s 0 , t 0 ] ⊆ [ 0 , T ] I=[s_0,t_0]\subseteq[0,T] I = [ s 0 , t 0 ] ⊆ [ 0 , T ] k ≥ 0 k\geq0 k ≥ 0
∥ F B ∥ I , k p e r t : = ∑ j = 0 k ( ∥ ( t − s 0 ) ( j + 1 ) / 2 D B ( j ) F B ∥ L t 2 L x ∞ ( I ) + ∥ ( t − s 0 ) j / 2 D B ( j ) F B ∥ L t 1 L x ∞ ( I ) ) . \|F_B\|^{\mathrm{pert}}_{I,k}:=\sum_{j=0}^{k}\left(\|(t-s_0)^{(j+1)/2}D_B^{(j)}F_B\|_{L_t^2L_x^\infty(I)}+\|(t-s_0)^{j/2}D_B^{(j)}F_B\|_{L_t^1L_x^\infty(I)}\right). ∥ F B ∥ I , k pert := j = 0 ∑ k ( ∥ ( t − s 0 ) ( j + 1 ) /2 D B ( j ) F B ∥ L t 2 L x ∞ ( I ) + ∥ ( t − s 0 ) j /2 D B ( j ) F B ∥ L t 1 L x ∞ ( I ) ) . In particular, when k = 0 k=0 k = 0
∥ F B ∥ I , 0 p e r t = ∥ ( t − s 0 ) 1 / 2 F B ∥ L t 2 L x ∞ ( I ) + ∥ F B ∥ L t 1 L x ∞ ( I ) . \|F_B\|^{\mathrm{pert}}_{I,0}=\|(t-s_0)^{1/2}F_B\|_{L_t^2L_x^\infty(I)}+\|F_B\|_{L_t^1L_x^\infty(I)}. ∥ F B ∥ I , 0 pert = ∥ ( t − s 0 ) 1/2 F B ∥ L t 2 L x ∞ ( I ) + ∥ F B ∥ L t 1 L x ∞ ( I ) . If I = [ 0 , t ] I=[0,t] I = [ 0 , t ] 0 0 0 ∥ F B ∥ t , k p e r t \|F_B\|^{\mathrm{pert}}_{t,k} ∥ F B ∥ t , k pert ∥ F B ∥ [ 0 , t ] , k p e r t \|F_B\|^{\mathrm{pert}}_{[0,t],k} ∥ F B ∥ [ 0 , t ] , k pert
We now present one of the main results of § 5 (more precisely, this is a combination of two results, Propositions 4.17 and 4.18 ). This result says that the covariant heat kernels p B , q B p_B,q_B p B , q B ∥ F B ∥ p e r t \|F_B\|^{\mathrm{pert}} ∥ F B ∥ pert p B ( t , x ) = p B ( 0 , 0 ; t , x ) p_B(t,x)=p_B(0,0;t,x) p B ( t , x ) = p B ( 0 , 0 ; t , x ) q B ( t , x ) q_B(t,x) q B ( t , x )
Theorem 2.4 (Pointwise estimates of covariant heat kernels). There is a constant c > 0 c>0 c > 0 ( t 0 , x 0 ) ∈ ( 0 , T ] × T d (t_0,x_0)\in(0,T]\times\mathbb{T}^d ( t 0 , x 0 ) ∈ ( 0 , T ] × T d k ≥ 0 k\geq0 k ≥ 0
∣ D B ( k ) p B ( t 0 , x 0 ) ∣ ≲ k ( 1 + ( ∥ F B ∥ t 0 , ( k − 1 ) ∨ 0 p e r t ) k ) t 0 − k + d 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . |D_B^{(k)}p_B(t_0,x_0)|\lesssim_k\left(1+\left(\|F_B\|^{\mathrm{pert}}_{t_0,(k-1)\vee0}\right)^k\right)t_0^{-\frac{k+d}{2}}\exp\left(-c|x_0|^2/t_0\right). ∣ D B ( k ) p B ( t 0 , x 0 ) ∣ ≲ k ( 1 + ( ∥ F B ∥ t 0 , ( k − 1 ) ∨ 0 pert ) k ) t 0 − 2 k + d exp ( − c ∣ x 0 ∣ 2 / t 0 ) . Additionally, there is a constant C 0 > 0 C_0>0 C 0 > 0 ( t 0 , x 0 ) ∈ ( 0 , T ] × T d (t_0,x_0)\in(0,T]\times\mathbb{T}^d ( t 0 , x 0 ) ∈ ( 0 , T ] × T d k ≥ 0 k\geq0 k ≥ 0 q B q_B q B p B p_B p B
∣ D B ( k ) ( q B − p B ) ( t 0 , x 0 ) ∣ ≲ k exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ) ∥ F B ∥ t 0 , k p e r t ( 1 + ( ∥ F B ∥ t 0 , k p e r t ) ( k − 1 ) ∨ 0 ) t 0 − k + d 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . |D_B^{(k)}(q_B-p_B)(t_0,x_0)|\lesssim_k\exp\left(C_0\|F_B\|_{L_t^1L_x^\infty([0,t_0])}\right)\|F_B\|^{\mathrm{pert}}_{t_0,k}\left(1+\left(\|F_B\|^{\mathrm{pert}}_{t_0,k}\right)^{(k-1)\vee0}\right)t_0^{-\frac{k+d}{2}}\exp\left(-c|x_0|^2/t_0\right). ∣ D B ( k ) ( q B − p B ) ( t 0 , x 0 ) ∣ ≲ k exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ) ∥ F B ∥ t 0 , k pert ( 1 + ( ∥ F B ∥ t 0 , k pert ) ( k − 1 ) ∨ 0 ) t 0 − 2 k + d exp ( − c ∣ x 0 ∣ 2 / t 0 ) . Remark 2.5. We make the following remarks regarding Theorem 2.4 .
(a) For simplicity, we state the theorem with the specific base point ( 0 , 0 ) (0,0) ( 0 , 0 ) p B ( s 0 , y 0 ; t 0 , x 0 ) p_B(s_0,y_0;t_0,x_0) p B ( s 0 , y 0 ; t 0 , x 0 ) q B ( s 0 , y 0 ; t 0 , x 0 ) q_B(s_0,y_0;t_0,x_0) q B ( s 0 , y 0 ; t 0 , x 0 )
(b) In [BC24, Section 3], various covariant heat kernel estimates (in the case G = U ( 1 ) \mathrm{G}=U(1) G = U ( 1 )
Next, we discuss some ideas behind the proof of Theorem 2.4 . In particular, we discuss how to estimate a solution ϕ \phi ϕ (29) . In one sentence, the main technique is to use monotonicity formulas. We begin by computing the evolution of ∣ ϕ ∣ 2 \lvert\phi\rvert^2 ∣ ϕ ∣ 2
1 2 ( ∂ t − Δ ) ( ∣ ϕ ∣ 2 ) + ∣ D B ϕ ∣ 2 = ⟨ ϕ , G ⟩ W . \frac{1}{2}(\partial_t-\Delta)(\lvert\phi\rvert^2)+\lvert D_B\phi\rvert^2=\langle\phi,G\rangle_W. 2 1 ( ∂ t − Δ ) (∣ ϕ ∣ 2 ) + ∣ D B ϕ ∣ 2 = ⟨ ϕ , G ⟩ W . Now let K ( t , x ) K(t,x) K ( t , x ) ( t 0 , x 0 ) (t_0,x_0) ( t 0 , x 0 ) ( ∂ t + Δ ) K = 0 (\partial_t+\Delta)K=0 ( ∂ t + Δ ) K = 0 t < t 0 t<t_0 t < t 0 K ( t 0 , ⋅ ) = δ x 0 K(t_0,\cdot)=\delta_{x_0} K ( t 0 , ⋅ ) = δ x 0 K K K
1 2 ∂ t ( K ∣ ϕ ∣ 2 ) + K ∣ D B ϕ ∣ 2 = 1 2 K Δ ( ∣ ϕ ∣ 2 ) − 1 2 ( Δ K ) ∣ ϕ ∣ 2 + K ⟨ ϕ , G ⟩ W . \frac{1}{2}\partial_t(K\lvert\phi\rvert^2)+K\lvert D_B\phi\rvert^2=\frac{1}{2}K\Delta(\lvert\phi\rvert^2)-\frac{1}{2}(\Delta K)\lvert\phi\rvert^2+K\langle\phi,G\rangle_W. 2 1 ∂ t ( K ∣ ϕ ∣ 2 ) + K ∣ D B ϕ ∣ 2 = 2 1 K Δ (∣ ϕ ∣ 2 ) − 2 1 ( Δ K ) ∣ ϕ ∣ 2 + K ⟨ ϕ , G ⟩ W . Thus upon integrating in space-time (and applying integration by parts in space to cancel the first two terms on the right-hand side above) we obtain
∥ ( K 1 / 2 ϕ ) ( t 0 ) ∥ L x 2 2 + 2 ∥ K 1 / 2 D B ϕ ∥ L t , x 2 ( [ 0 , t 0 ] ) 2 = ∥ ( K 1 / 2 ϕ ) ( 0 ) ∥ L x 2 2 + 2 ∫ 0 t 0 d t ∫ T d d x ( K ⟨ ϕ , G ⟩ W ) ( t , x ) . (33) \begin{aligned}
\lVert(K^{1/2}\phi)(t_0)\rVert_{L_x^2}^2+2\lVert K^{1/2}D_B\phi\rVert_{L^2_{t,x}([0,t_0])}^2
={}&\lVert(K^{1/2}\phi)(0)\rVert_{L_x^2}^2\\
&+2\int_0^{t_0}\mathrm{d}t\int_{\mathbb{T}^d}\mathrm{d}x\,(K\langle\phi,G\rangle_W)(t,x). \tag*{(33)}
\end{aligned} ∥( K 1/2 ϕ ) ( t 0 ) ∥ L x 2 2 + 2 ∥ K 1/2 D B ϕ ∥ L t , x 2 ([ 0 , t 0 ]) 2 = ∥( K 1/2 ϕ ) ( 0 ) ∥ L x 2 2 + 2 ∫ 0 t 0 d t ∫ T d d x ( K ⟨ ϕ , G ⟩ W ) ( t , x ) . (33) In particular, if G = 0 G=0 G = 0 ϕ \phi ϕ
∥ ( K 1 / 2 ϕ ) ( t 0 ) ∥ L x 0 2 ≤ ∥ ( K 1 / 2 ϕ ) ( 0 ) ∥ L x 2 . \lVert(K^{1/2}\phi)(t_0)\rVert_{L_{x_0}^2}\leq\lVert(K^{1/2}\phi)(0)\rVert_{L_x^2}. ∥( K 1/2 ϕ ) ( t 0 ) ∥ L x 0 2 ≤ ∥( K 1/2 ϕ ) ( 0 ) ∥ L x 2 . The heat kernel localisation K K K L x 2 L_x^2 L x 2 L x ∞ L_x^\infty L x ∞ ϕ \phi ϕ K ( t 0 , ⋅ ) = δ x 0 K(t_0,\cdot)=\delta_{x_0} K ( t 0 , ⋅ ) = δ x 0 ∥ ( K 1 / 2 ϕ ) ( t 0 ) ∥ L x 0 2 = ∣ ϕ ( t 0 , x 0 ) ∣ \lVert(K^{1/2}\phi)(t_0)\rVert_{L_{x_0}^2}=\lvert\phi(t_0,x_0)\rvert ∥( K 1/2 ϕ ) ( t 0 ) ∥ L x 0 2 = ∣ ϕ ( t 0 , x 0 )∣ K ( t 0 , ⋅ ) K(t_0,\cdot) K ( t 0 , ⋅ )
∣ ϕ ( t 0 , x 0 ) ∣ ≤ ∥ ( K 1 / 2 ϕ ) ( 0 ) ∥ L x 2 ≤ ∥ K 1 / 2 ( 0 ) ∥ L x ∞ ∥ ϕ ( 0 ) ∥ L x 2 ≲ t 0 − d 4 ∥ ϕ ( 0 ) ∥ L x 2 . \lvert\phi(t_0,x_0)\rvert\leq\lVert(K^{1/2}\phi)(0)\rVert_{L_x^2}\leq\lVert K^{1/2}(0)\rVert_{L_x^\infty}\lVert\phi(0)\rVert_{L_x^2}\lesssim t_0^{-\frac{d}{4}}\lVert\phi(0)\rVert_{L_x^2}. ∣ ϕ ( t 0 , x 0 )∣ ≤ ∥( K 1/2 ϕ ) ( 0 ) ∥ L x 2 ≤ ∥ K 1/2 ( 0 ) ∥ L x ∞ ∥ ϕ ( 0 ) ∥ L x 2 ≲ t 0 − 4 d ∥ ϕ ( 0 ) ∥ L x 2 . By varying x 0 x_0 x 0 L x ∞ L_x^\infty L x ∞
∥ ϕ ( t 0 ) ∥ L x ∞ ≲ t 0 − d 4 ∥ ϕ ( 0 ) ∥ L x 2 . \lVert\phi(t_0)\rVert_{L_x^\infty}\lesssim t_0^{-\frac{d}{4}}\lVert\phi(0)\rVert_{L_x^2}. ∥ ϕ ( t 0 ) ∥ L x ∞ ≲ t 0 − 4 d ∥ ϕ ( 0 ) ∥ L x 2 . Note that this is the same estimate one would obtain if ϕ ( t ) = e t Δ ϕ ( 0 ) \phi(t)=e^{t\Delta}\phi(0) ϕ ( t ) = e t Δ ϕ ( 0 )
Remark 2.6. The heat kernel localisation K K K
For general G G G (33) by
∣ ∫ 0 t 0 d t ∫ T d d x ( K ⟨ ϕ , G ⟩ W ) ( t , x ) ∣ ≤ ∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , t 0 ] ) ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , t 0 ] ) ≤ δ ∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , t 0 ] ) 2 + C δ ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , t 0 ] ) 2 , \begin{aligned}
\left\lvert\int_0^{t_0}\mathrm{d}t\int_{\mathbb{T}^d}\mathrm{d}x\,(K\langle\phi,G\rangle_W)(t,x)\right\rvert\leq\lVert K^{1/2}\phi\rVert_{L_t^\infty L_x^2([0,t_0])}\lVert K^{1/2}G\rVert_{L_t^1L_x^2([0,t_0])} \\
&\le\delta\lVert K^{1/2}\phi\rVert_{L^\infty_tL^2_x([0,t_0])}^2+C_\delta\lVert K^{1/2}G\rVert_{L^1_tL^2_x([0,t_0])}^2,
\end{aligned} ∫ 0 t 0 d t ∫ T d d x ( K ⟨ ϕ , G ⟩ W ) ( t , x ) ≤ ∥ K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , t 0 ]) ∥ K 1/2 G ∥ L t 1 L x 2 ([ 0 , t 0 ]) ≤ δ ∥ K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , t 0 ]) 2 + C δ ∥ K 1/2 G ∥ L t 1 L x 2 ([ 0 , t 0 ]) 2 , for any δ > 0 \delta>0 δ > 0 C δ C_\delta C δ δ \delta δ (33) , we may obtain the further estimate
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , t 0 ] ) + ∥ K 1 / 2 D B ϕ ∥ L t , x 2 ( [ 0 , t 0 ] ) ≲ ∥ ( K 1 / 2 ϕ ) ( 0 ) ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , t 0 ] ) . (34) \lVert K^{1/2}\phi\rVert_{L^\infty_tL^2_x([0,t_0])}+\lVert K^{1/2}D_B\phi\rVert_{L^2_{t,x}([0,t_0])}\lesssim\lVert(K^{1/2}\phi)(0)\rVert_{L^2_x}+\lVert K^{1/2}G\rVert_{L^1_tL^2_x([0,t_0])}. \tag*{(34)} ∥ K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , t 0 ]) + ∥ K 1/2 D B ϕ ∥ L t , x 2 ([ 0 , t 0 ]) ≲ ∥( K 1/2 ϕ ) ( 0 ) ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ([ 0 , t 0 ]) . (34) A suitably time-weighted version of this estimate will allow us to control higher derivatives of ϕ \phi ϕ
( ∂ t − Δ B ) ( D B , i ϕ ) = 2 ( F B ) i ℓ D B , ℓ ϕ + D B , i G , i ∈ [ d ] . (\partial_t-\Delta_B)(D_{B,i}\phi)=2(F_B)_{i}^{\ \ell}D_{B,\ell}\phi+D_{B,i}G,\qquad i\in[d]. ( ∂ t − Δ B ) ( D B , i ϕ ) = 2 ( F B ) i ℓ D B , ℓ ϕ + D B , i G , i ∈ [ d ] . In particular, D B ϕ D_B\phi D B ϕ (30) ). Applying the estimate (34) with ϕ \phi ϕ D B ϕ D_B\phi D B ϕ G G G 2 F B D B ϕ + D B G 2F_BD_B\phi+D_BG 2 F B D B ϕ + D B G L t ∞ L x 2 L^\infty_tL^2_x L t ∞ L x 2 K 1 / 2 D B ϕ K^{1/2}D_B\phi K 1/2 D B ϕ L t 1 L x 2 L^1_tL^2_x L t 1 L x 2 K 1 / 2 F B D B ϕ K^{1/2}F_BD_B\phi K 1/2 F B D B ϕ D B G D_BG D B G (34) also gives an L t , x 2 L^2_{t,x} L t , x 2 K 1 / 2 D B ϕ K^{1/2}D_B\phi K 1/2 D B ϕ
Even higher derivatives of ϕ \phi ϕ D B ( k ) ϕ D_B^{(k)}\phi D B ( k ) ϕ
( ∂ t − Δ B ) ( D B ( k ) ϕ ) = ∑ j 1 + j 2 = k j 2 ≥ 1 O ( ( D B ( j 1 ) F B ) D B ( j 2 ) ϕ ) + D B ( k ) G . (\partial_t-\Delta_B)(D_B^{(k)}\phi)=\sum_{\substack{j_1+j_2=k\\j_2\ge1}}O\bigl((D_B^{(j_1)}F_B)D_B^{(j_2)}\phi\bigr)+D_B^{(k)}G. ( ∂ t − Δ B ) ( D B ( k ) ϕ ) = j 1 + j 2 = k j 2 ≥ 1 ∑ O ( ( D B ( j 1 ) F B ) D B ( j 2 ) ϕ ) + D B ( k ) G . All terms involving ϕ \phi ϕ
Besides the pointwise estimates of Theorem 2.4, § 5 collects various other Schauder estimates that are needed later on. Additionally, the same section contains two different short-time expansions for the covariant heat kernels, which we next discuss. The first is a heat kernel parametrix, which approximates p B p_B p B p τ B p\tau_B p τ B p p p τ B ∈ End ( W ) \tau_B\in\operatorname{End}(W) τ B ∈ End ( W ) B B B § 4.3 ). We state this result as follows.
Proposition 2.7 (Covariant heat kernel parametrix). There is a constant c > 0 c>0 c > 0 t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
N ( B ) ≔ ( t 0 1 / 2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + ∥ F B ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) ) 2 + t 0 1 / 2 ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ( 1 + ∥ F B ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) . \mathcal{N}(B)\coloneqq\left(t_0^{1/2}\left(\lVert\partial_tB\rVert_{L^1_tL^\infty_x([0,t_0])}+\lVert F_B\rVert_{L^2_tL^\infty_x([0,t_0])}\right)\right)^2+t_0^{1/2}\lVert\partial_tB\rVert_{L^1_tL^\infty_x([0,t_0])}\left(1+\lVert F_B\rVert_{L^2_tL^\infty_x([0,t_0])}\right). N ( B ) : = ( t 0 1/2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + ∥ F B ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) ) 2 + t 0 1/2 ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ( 1 + ∥ F B ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) . For any point x 0 x_0 x 0 1 / 10 1/10 1/10 0 0 0
∣ p B − p τ B ∣ ( t 0 , x 0 ) ≲ N ( B ) t 0 − d / 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , \lvert p_B-p\tau_B\rvert(t_0,x_0)\lesssim\mathcal{N}(B)t_0^{-d/2}\exp(-c\lvert x_0\rvert^2/t_0), ∣ p B − p τ B ∣ ( t 0 , x 0 ) ≲ N ( B ) t 0 − d /2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , ∣ D B ( p B − p τ B ) ∣ ( t 0 , x 0 ) ≲ N ( B ) ( 1 + ∥ F B ∥ t 0 , 0 p e r t ) t 0 − ( d + 1 ) / 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . \lvert D_B(p_B-p\tau_B)\rvert(t_0,x_0)\lesssim\mathcal{N}(B)(1+\lVert F_B\rVert_{t_0,0}^{\mathrm{pert}})t_0^{-(d+1)/2}\exp(-c\lvert x_0\rvert^2/t_0). ∣ D B ( p B − p τ B )∣ ( t 0 , x 0 ) ≲ N ( B ) ( 1 + ∥ F B ∥ t 0 , 0 pert ) t 0 − ( d + 1 ) /2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . We prove this proposition in § 5.4 . In this paper, we will generally work on time scales such that the quantity N ( B ) \mathcal{N}(B) N ( B ) p τ B p\tau_B p τ B p B p_B p B § 3.2 ). To prove Proposition 2.7, we begin by deriving the evolution equation of the difference p B − p τ B p_B-p\tau_B p B − p τ B
( ∂ t − Δ B ) ( p B − p τ B ) = perturbative terms , (\partial_t-\Delta_B)(p_B-p\tau_B)=\text{perturbative terms}, ( ∂ t − Δ B ) ( p B − p τ B ) = perturbative terms , and then we proceed to estimate all perturbative terms appearing on the right-hand side. These terms involve covariant derivatives of parallel transport, which we estimate (in § 5.3 ) using the quantity N ( B ) \mathcal{N}(B) N ( B )
By combining Theorem 2.4 and Proposition 2.7 , we also obtain a parametrix for q B q_B q B F B F_B F B t 0 t_0 t 0 τ B \tau_B τ B q B q_B q B B B B q B q_B q B exhibits exponential growth . This exponential growth arises due to instabilities caused by the additional curvature term in the equation (30) and is one of the main inputs needed for our decay mechanism (which ultimately leads to uniform-in-time estimates), to be discussed in more detail in § 3.3 . We now state an informal version of the second short-time expansion. We avoid stating the precise version here, as it takes a fair amount of notational buildup. See the discussion at the beginning of § 5.5 as well as Proposition 4.51 .
Proposition 2.8 (Informal. Covariant heat kernel near Yang–Mills connection). Let d = 2 d=2 d = 2 B B B D B F B D_BF_B D B F B F B F_B F B q B q_B q B τ B s \tau_B\mathfrak{s} τ B s s ( t , x ) ∈ End ( W 2 ) \mathfrak{s}(t,x)\in\operatorname{End}(W^2) s ( t , x ) ∈ End ( W 2 )
( ∂ t − Δ − 2 F B 12 ( 0 , 0 ) − 1 4 ∣ x ∣ 2 F B 12 ( 0 , 0 ) 2 ) s = 0 , s ( 0 ) = δ 0 id W 2 . \left(\partial_t-\Delta-2F_B^{12}(0,0)-\frac{1}{4}|x|^2F_B^{12}(0,0)^2\right)\mathfrak{s}=0,\qquad\mathfrak{s}(0)=\delta_0\operatorname{id}_{W^2}. ( ∂ t − Δ − 2 F B 12 ( 0 , 0 ) − 4 1 ∣ x ∣ 2 F B 12 ( 0 , 0 ) 2 ) s = 0 , s ( 0 ) = δ 0 id W 2 . Here F B 12 ( 0 , 0 ) ∈ g F_B^{12}(0,0)\in\mathfrak{g} F B 12 ( 0 , 0 ) ∈ g π ∗ \pi_* π ∗ F B 12 ( 0 , 0 ) ∈ End ( W ) F_B^{12}(0,0)\in\operatorname{End}(W) F B 12 ( 0 , 0 ) ∈ End ( W ) F B 12 ( 0 , 0 ) F_B^{12}(0,0) F B 12 ( 0 , 0 ) W 2 W^2 W 2 π ∗ ⊕ π ∗ \pi_*\oplus\pi_* π ∗ ⊕ π ∗ F B 12 ( 0 , 0 ) 2 s F_B^{12}(0,0)^2\mathfrak{s} F B 12 ( 0 , 0 ) 2 s F B ( 0 , 0 ) ∈ End ( W 2 ) F_B(0,0)\in\operatorname{End}(W^2) F B ( 0 , 0 ) ∈ End ( W 2 ) ( F B ( 0 , 0 ) X ) i = ( F B ( 0 , 0 ) ) i j X j (F_B(0,0)X)_i=(F_B(0,0))_i{}^jX_j ( F B ( 0 , 0 ) X ) i = ( F B ( 0 , 0 ) ) i j X j X ∈ W 2 X\in W^2 X ∈ W 2 i ∈ [ 2 ] i\in[2] i ∈ [ 2 ]
Additionally, s ( t , x ) ∈ End ( W 2 ) \mathfrak{s}(t,x)\in\operatorname{End}(W^2) s ( t , x ) ∈ End ( W 2 ) F B 12 ( 0 , 0 ) ∈ End ( W ) F_B^{12}(0,0)\in\operatorname{End}(W) F B 12 ( 0 , 0 ) ∈ End ( W ) i λ i\lambda iλ λ ∈ R \lambda\in\mathbb{R} λ ∈ R s ( t , x ) \mathfrak{s}(t,x) s ( t , x )
e 2 λ t 4 π sinh ( λ t ) exp ( − λ ∣ x ∣ 2 4 tanh ( λ t ) ) . \frac{e^{2\lambda t}}{4\pi\sinh(\lambda t)}\exp\left(-\frac{\lambda|x|^2}{4\tanh(\lambda t)}\right). 4 π sinh ( λ t ) e 2 λ t exp ( − 4 tanh ( λ t ) λ ∣ x ∣ 2 ) . (When λ = 0 \lambda=0 λ = 0 λ → 0 \lambda\to0 λ → 0 λ > 0 \lambda>0 λ > 0 x x x 0 0 0 e λ t e^{\lambda t} e λ t F B 12 ( 0 , 0 ) F_B^{12}(0,0) F B 12 ( 0 , 0 ) F B 12 ( 0 , 0 ) F_B^{12}(0,0) F B 12 ( 0 , 0 ) ± i λ \pm i\lambda ± iλ F B 12 ( 0 , 0 ) F_B^{12}(0,0) F B 12 ( 0 , 0 ) i λ i\lambda iλ λ > 0 \lambda>0 λ > 0
Remark 2.9. Unlike the other results of this section, we must take d = 2 d=2 d = 2 Proposition 2.8 . This reflects the fact that two-dimensional Yang–Mills connections exhibit special structure not seen in higher dimensions, as briefly discussed at the end of § 1.2 . As one particular consequence, when working on the whole-plane R 2 \mathbb{R}^2 R 2 B B B τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B s \mathfrak{s} s Proposition 2.8 whose eigenvalues we completely understand. Thus Proposition 2.8 (and our decay mechanism which rests on this proposition) is one of the places in this article where we most heavily use the assumption that d = 2 d=2 d = 2
We also remark that Avramidi [Avr93, Avr95a, Avr95b] previously studied similar questions. He showed that in general dimensions, if one assumes that the curvature is covariantly constant (when d ≥ 3 d\geq3 d ≥ 3 d = 2 d=2 d = 2
We prove Proposition 2.8 (or more precisely, Proposition 4.51) in Section 4.5. To briefly discuss the proof, a natural approach would be to proceed as in the parametrix construction, by deriving the following form for the evolution equation of q B − τ B s q_B-\tau B s q B − τ B s
( ∂ t − Δ B − 2 F B ) ( q B − τ B s ) = perturbative terms . (\partial_t-\Delta_B-2F_B)(q_B-\tau B s)=\text{perturbative terms}. ( ∂ t − Δ B − 2 F B ) ( q B − τ B s ) = perturbative terms . However, the above is not exactly true, as there is still a single term on the right-hand side which is non-perturbative, because it involves the curvature F B F_B F B F B F_B F B B B B F B F_B F B F B F_B F B ( ∂ t − L B ) (\partial_t-\mathcal{L}_B) ( ∂ t − L B ) L B \mathcal{L}_B L B ( ∂ t − Δ B − 2 F B ) (\partial_t-\Delta_B-2F_B) ( ∂ t − Δ B − 2 F B ) τ B − 1 q B − s \tau_B^{-1}q_B-s τ B − 1 q B − s q B − τ B s q_B-\tau_Bs q B − τ B s
Covariant local well-posedness
One of the major goals of this article is to establish a covariant local theory of (1.11). This covariant local theory will heavily rely on the covariant heat kernel estimates discussed in Section 2.1 and will be an important ingredient in our global theory (see Section 2.3 below). As a first step towards our covariant local theory, we discuss the Yang–Mills gradient flow (1.8) with rough initial data. This is not only needed to formulate the infinitesimal DeTurck gauge appearing in (1.11), but will later serve as a warm-up for our full covariant local theory (cf. Section 5 and Sections 6–9).
Proposition 2.10 (Yang–Mills gradient flow with rough initial data). Let κ > 0 \kappa>0 κ > 0 C 0 > 1 C_0>1 C 0 > 1 0 < c 0 ≤ 1 0<c_0\le1 0 < c 0 ≤ 1 B 0 + , B 0 − : T 2 → g 2 B_0^+,B_0^-:\mathbb{T}^2\to\mathfrak{g}^2 B 0 + , B 0 − : T 2 → g 2 B 0 − : T 2 → g 2 B_0^-:\mathbb{T}^2\to\mathfrak{g}^2 B 0 − : T 2 → g 2 B 0 + B_0^+ B 0 + B 0 − ∈ C − κ B_0^-\in\mathcal{C}^{-\kappa} B 0 − ∈ C − κ B 0 # = B 0 + + B 0 − B_0^\#=B_0^++B_0^- B 0 # = B 0 + + B 0 − B + B^+ B + B + ( 0 ) = B 0 + B^{+(0)}=B_0^+ B + ( 0 ) = B 0 + 0 < T ≤ 1 0<T\le1 0 < T ≤ 1 K ≥ 1 K\ge1 K ≥ 1
exp ( C 0 ∥ F B + ∥ L t 1 L x ∞ ( [ 0 , T ] ) ( 1 + ∥ F B + p e r t , 2 ∥ T , 1 2 2 ) T 1 − κ 2 K ) ≤ c 0 , (35) \exp\left(C_0\|F_{B^+}\|_{L^1_tL^\infty_x([0,T])}(1+\|F_{B^+}^{\mathrm{pert},2}\|_{T,\frac{1}{2}}^2)T^{\frac{1-\kappa}{2}}K\right)\le c_0, \tag*{(35)} exp ( C 0 ∥ F B + ∥ L t 1 L x ∞ ([ 0 , T ]) ( 1 + ∥ F B + pert , 2 ∥ T , 2 1 2 ) T 2 1 − κ K ) ≤ c 0 , (35) ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ K , (36) \|B_0^-\|_{\mathcal{C}^{-\kappa}_{B_0^+,F,T,2}}\le K, \tag*{(36)} ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ K , (36) where the last condition involves the covariant Hölder norm at regularity − κ -\kappa − κ B ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 , g 2 ) B\in C_t^0\mathcal{C}_x^{-\kappa}([0,T]\times\mathbb{T}^2,\mathfrak{g}^2) B ∈ C t 0 C x − κ ([ 0 , T ] × T 2 , g 2 ) g ∈ C t 0 G 1 − κ ( [ 0 , T ] ) g\in C_t^0\mathcal{G}_1^{-\kappa}([0,T]) g ∈ C t 0 G 1 − κ ([ 0 , T ]) G 1 − κ \mathcal{G}_{1-\kappa} G 1 − κ ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 B ( 0 ) = g ( 0 ) ⋅ B 0 # B(0)=g(0)\cdot B_0^\# B ( 0 ) = g ( 0 ) ⋅ B 0 # t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ F B ( t ) ∥ L x ∞ ≲ ∥ F B + ( t ) ∥ L x ∞ + C t − 1 + κ 2 K , \|F_{B}(t)\|_{L^\infty_x}\lesssim\|F_{B^+}(t)\|_{L^\infty_x}+Ct^{-\frac{1+\kappa}{2}}K, ∥ F B ( t ) ∥ L x ∞ ≲ ∥ F B + ( t ) ∥ L x ∞ + C t − 2 1 + κ K , ∥ D B ∗ F B ( t ) ∥ L x ∞ ≲ ∥ D B + ∗ F B + ( t ) ∥ L x ∞ + C t − κ 2 ( t − 1 + ∥ F B + ( t ) ∥ L x ∞ ) K . (37) \|D_B^*F_B(t)\|_{L^\infty_x}\lesssim\|D_{B^+}^*F_{B^+}(t)\|_{L^\infty_x}+Ct^{-\frac{\kappa}{2}}\left(t^{-1}+\|F_{B^+}(t)\|_{L^\infty_x}\right)K. \tag*{(37)} ∥ D B ∗ F B ( t ) ∥ L x ∞ ≲ ∥ D B + ∗ F B + ( t ) ∥ L x ∞ + C t − 2 κ ( t − 1 + ∥ F B + ( t ) ∥ L x ∞ ) K . (37) where C C C
For more detailed statements concerning the Yang–Mills gradient flow with low-regularity initial data, we refer the reader to Proposition 5.7 and Lemma 5.9 below. The former result contains additional estimates of the solution, and the latter yields the uniqueness of B B B B ( 0 ) B(0) B ( 0 ) Proposition 2.10 does not construct a Yang–Mills gradient flow for the initial data B 0 ♯ B_0^\sharp B 0 ♯ g ( 0 ) ∙ B 0 ♯ g(0) \mathbin{\bullet} B_0^\sharp g ( 0 ) ∙ B 0 ♯ Theorem 1.5 ) and not in the solution of (1.11) itself. In any case, the aspect of Proposition 2.10 we use the most is the curvature estimate (2.9), which is gauge invariant. In applications of Proposition 2.10 , K K K F B ( t ) F_B(t) F B ( t ) F B + ( t ) F_{B_+}(t) F B + ( t ) L t 1 L x ∞ L_t^1L_x^\infty L t 1 L x ∞ F B F_B F B L t 1 L x ∞ L_t^1L_x^\infty L t 1 L x ∞ F B + F_{B_+} F B +
The proof of Proposition 2.10 , which is the subject of § 6.1 , is rather simple. The main idea is to first show the existence of a unique solution B ♯ : [ 0 , T ] × T 2 → g 2 B^\sharp\colon[0,T] \times\mathbb{T}^2 \to\mathfrak{g}^2 B ♯ : [ 0 , T ] × T 2 → g 2
∂ t B ♯ = − D B ♯ ∗ F B ♯ − D B ♯ D B ♯ ∗ ( B ♯ − B + ) , B ♯ ( 0 ) = B 0 ♯ . (38) \partial_t B^\sharp=-D_{B^\sharp}^{*}F_{B^\sharp}-D_{B^\sharp}D_{B^\sharp}^{*}(B^\sharp-B^+),\qquad B^\sharp(0)=B_0^\sharp. \tag*{(38)} ∂ t B ♯ = − D B ♯ ∗ F B ♯ − D B ♯ D B ♯ ∗ ( B ♯ − B + ) , B ♯ ( 0 ) = B 0 ♯ . (38) That is, we first solve the Yang–Mills gradient flow from (1.8) in the infinitesimal DeTurck gauge. Since the − D B ♯ D B ♯ ∗ ( B ♯ − B + ) -D_{B^\sharp}D_{B^\sharp}^{*}(B^\sharp-B^+) − D B ♯ D B ♯ ∗ ( B ♯ − B + ) § 3.1 and a standard contraction-mapping argument. Once (2.10) has been solved, we then construct a gauge transformation g : [ 0 , T ] × T 2 → G g \colon[0,T] \times\mathbb{T}^2 \to G g : [ 0 , T ] × T 2 → G
( ∂ t g ) g − 1 = − Ad g ( D B ♯ ∗ ( B ♯ − B + ) ) , g ( T ) = id G . (39) (\partial_tg)g^{-1}=-\operatorname{Ad}_g(D_{B^\sharp}^{*}(B^\sharp-B^+)),\qquad g(T)=\operatorname{id}_G. \tag*{(39)} ( ∂ t g ) g − 1 = − Ad g ( D B ♯ ∗ ( B ♯ − B + )) , g ( T ) = id G . (39) The reason for imposing a condition on g g g t = T t=T t = T t = 0 t=0 t = 0 B ♯ B^\sharp B ♯ B B B g ∙ B ♯ g\mathbin{\bullet}B^\sharp g ∙ B ♯ Proposition 2.10 are rather long and our covariant local theory only relies on bounds of the curvature tensor F B F_B F B
J T : = { ( A 0 , Φ 0 ) ∈ C x − κ ( T 2 , g 2 × V ) : ∃ YM gradient flow \mathcal{J}_T:=\{(A_0,\Phi_0)\in C^{-\kappa}_{x}(\mathbb{T}^2,\mathfrak{g}^2\times V):\ \exists\ \text{YM gradient flow} J T := {( A 0 , Φ 0 ) ∈ C x − κ ( T 2 , g 2 × V ) : ∃ YM gradient flow B ∈ C t , x 0 , − κ ( [ 0 , T ] × T 2 , g 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 , g 2 ) with initial data A 0 B\in C^{0,-\kappa}_{t,x}([0,T]\times\mathbb{T}^2,\mathfrak{g}^2)\cap C^\infty_{t,x}((0,T]\times\mathbb{T}^2,\mathfrak{g}^2)\text{ with initial data }A_0 B ∈ C t , x 0 , − κ ([ 0 , T ] × T 2 , g 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 , g 2 ) with initial data A 0 satisfying ∥ t 1 2 + κ 2 F B ∥ L t , x ∞ , ∥ t 1 2 + κ 2 D B ∗ F B ∥ L t , x ∞ < ∞ , ∥ Φ 0 ∥ C A 0 , T − κ < ∞ } . (40) \text{satisfying }\|t^{\frac12+\frac{\kappa}{2}}F_B\|_{L^\infty_{t,x}},\ \|t^{\frac12+\frac{\kappa}{2}}D_{B}^{*}F_B\|_{L^\infty_{t,x}}<\infty,\ \|\Phi_0\|_{C^{-\kappa}_{A_0,T}}<\infty\}. \tag*{(40)} satisfying ∥ t 2 1 + 2 κ F B ∥ L t , x ∞ , ∥ t 2 1 + 2 κ D B ∗ F B ∥ L t , x ∞ < ∞ , ∥ Φ 0 ∥ C A 0 , T − κ < ∞ } . (40) Here, ∥ ⋅ ∥ C A 0 , T − κ \|\cdot\|_{C^{-\kappa}_{A_0,T}} ∥ ⋅ ∥ C A 0 , T − κ
Remark 2.11. The initial-data class J T \mathcal{J}_T J T T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] J T = J 1 \mathcal{J}_T=\mathcal{J}_1 J T = J 1 T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] B B B [ 0 , T ] [0,T] [ 0 , T ] [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ] (0,1] ( 0 , 1 ] ∥ Φ 0 ∥ C A 0 , T − κ < ∞ \|\Phi_0\|_{C^{-\kappa}_{A_0,T}}<\infty ∥ Φ 0 ∥ C A 0 , T − κ < ∞ ∥ Φ 0 ∥ C A 0 , 1 − κ < ∞ \|\Phi_0\|_{C^{-\kappa}_{A_0,1}}<\infty ∥ Φ 0 ∥ C A 0 , 1 − κ < ∞
We nevertheless retain T T T Theorem 2.19 below, we only work with the Yang–Mills gradient flow on [ 0 , T ] [0,T] [ 0 , T ]
After the above discussion of the Yang–Mills gradient flow, we now turn to our gauge covariant smooth approximation of (1.11). To this end, we first let C P C_{\mathcal{P}} C P
P : = { ρ ∈ C ∞ ( [ 0 , ∞ ) → [ 0 , 1 ] ) ∣ ρ ∣ [ 0 , 1 ] = 0 , ρ ′ ∣ [ 1 , 2 ] ≥ 0 , ρ ∣ [ 2 , ∞ ) = 1 , \mathcal{P}:=\{\rho\in C^\infty([0,\infty)\to[0,1])\mid\rho|_{[0,1]}=0,\ \rho'|_{[1,2]}\geq0,\ \rho|_{[2,\infty)}=1, P := { ρ ∈ C ∞ ([ 0 , ∞ ) → [ 0 , 1 ]) ∣ ρ ∣ [ 0 , 1 ] = 0 , ρ ′ ∣ [ 1 , 2 ] ≥ 0 , ρ ∣ [ 2 , ∞ ) = 1 , and ∥ ρ ( k ) ∥ L ∞ ≤ C P for all 0 ≤ k ≤ 20 } . (41) \text{and }\|\rho^{(k)}\|_{L^\infty}\leq C_{\mathcal{P}}\text{ for all }0\leq k\leq20\}. \tag*{(41)} and ∥ ρ ( k ) ∥ L ∞ ≤ C P for all 0 ≤ k ≤ 20 } . (41) For all ρ ∈ P \rho\in\mathcal{P} ρ ∈ P ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0
ρ ε ( t ) ≔ ρ ( ε − 2 t ) . (42) \rho_\varepsilon(t) \coloneqq\rho(\varepsilon^{-2}t). \tag*{(42)} ρ ε ( t ) : = ρ ( ε − 2 t ) . (42) From (41) and (42) , it follows that ρ ε \rho_\varepsilon ρ ε t ≳ ε 2 t \gtrsim\varepsilon^2 t ≳ ε 2 ρ ε ′ \rho_\varepsilon' ρ ε ′ L 1 L^1 L 1 t ∼ ε 2 t \sim\varepsilon^2 t ∼ ε 2 t = 0 t=0 t = 0
Definition 2.12 (Gauge-covariant approximations of ξ \xi ξ ζ \zeta ζ T > 0 T>0 T > 0 B B B (40) , let ρ ∈ P \rho\in\mathcal{P} ρ ∈ P ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0 [ 0 , T ] [0,T] [ 0 , T ]
ξ B , ε ( t , x ) ≔ ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , \xi_{B,\varepsilon}(t,x) \coloneqq\int_0^t \mathrm{d}s \int_{\mathbb{T}^2} \mathrm{d}y\,\rho_\varepsilon'(t-s)q_B(s,y;t,x)\xi(s,y), ξ B , ε ( t , x ) : = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , ζ B , ε ( t , x ) ≔ ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) , (43) \zeta_{B,\varepsilon}(t,x) \coloneqq\int_0^t \mathrm{d}s \int_{\mathbb{T}^2} \mathrm{d}y\,\rho_\varepsilon'(t-s)p_B(s,y;t,x)\zeta(s,y), \tag*{(43)} ζ B , ε ( t , x ) : = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) , (43) where p B p_B p B q B q_B q B
In the earlier works [CCHS22, CCHS24, BC26a], the smooth approximations of space-time white noises were defined using standard mollifiers. In Definition 2.12, however, we use covariant heat kernels, which makes both ξ B , ε \xi_{B,\varepsilon} ξ B , ε ζ B , ε \zeta_{B,\varepsilon} ζ B , ε g ∈ G g \in\mathcal{G} g ∈ G g g g g g g ξ \xi ξ ζ \zeta ζ g ∈ G 1 − κ g \in\mathcal{G}_{1-\kappa} g ∈ G 1 − κ ξ ∈ C t , x − 2 − κ ( [ 0 , T ] × T 2 , g 2 ) \xi\in\mathcal{C}^{-2-\kappa}_{t,x}([0,T]\times\mathbb{T}^2,\mathfrak{g}^2) ξ ∈ C t , x − 2 − κ ([ 0 , T ] × T 2 , g 2 ) ζ ∈ C t , x − 2 − κ ( [ 0 , T ] × T 2 , V ) \zeta\in\mathcal{C}^{-2-\kappa}_{t,x}([0,T]\times\mathbb{T}^2,V) ζ ∈ C t , x − 2 − κ ([ 0 , T ] × T 2 , V ) g ∙ ξ B , ε = ( g ∙ ξ ) g ∙ B , ε g\mathbin{\bullet}\xi_{B,\varepsilon}=(g\mathbin{\bullet}\xi)_{g\mathbin{\bullet}B,\varepsilon} g ∙ ξ B , ε = ( g ∙ ξ ) g ∙ B , ε g ∙ ζ B , ε = ( g ∙ ζ ) g ∙ B , ε g\mathbin{\bullet}\zeta_{B,\varepsilon}=(g\mathbin{\bullet}\zeta)_{g\mathbin{\bullet}B,\varepsilon} g ∙ ζ B , ε = ( g ∙ ζ ) g ∙ B , ε g g g ξ \xi ξ ζ \zeta ζ ϱ \varrho ϱ
In our gauge covariant, smooth approximation of (11) , we also need renormalisation constants C ρ A ∈ End G ( g ) C_\rho^A \in\operatorname{End}_{G}(\mathfrak{g}) C ρ A ∈ End G ( g ) C ρ , ε Φ ∈ End G ( V ) C_{\rho,\varepsilon}^{\Phi} \in\operatorname{End}_{G}(V) C ρ , ε Φ ∈ End G ( V ) End G \operatorname{End}_{G} End G G G G g \mathfrak{g} g V V V ϱ \varrho ϱ C ρ A Ad g = Ad g C ρ A C_\rho^A\operatorname{Ad}_g=\operatorname{Ad}_gC_\rho^A C ρ A Ad g = Ad g C ρ A C ρ , ε Φ ϱ ( g ) = ϱ ( g ) C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi}\varrho(g)=\varrho(g)C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ ϱ ( g ) = ϱ ( g ) C ρ , ε Φ g ∈ G g \in G g ∈ G C ρ A C_\rho^A C ρ A C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ C ρ A C_\rho^A C ρ A C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ
∂ t A ε = − D A ε ∗ F A ε − D A ε D A ε ∗ ( A ε − B ) + B ( Φ ε ⊗ D A ε Φ ε ) + C ρ A ( A ε − B ) + ξ B , ε , ∂ t Φ ε = Δ A ε Φ ε + ( D A ε ∗ ( A ε − B ) ) Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε , A ε ( 0 ) = A 0 , Φ ε ( 0 ) = Φ 0 . (44) \begin{aligned}
\partial_t A_\varepsilon&= -D_{A_\varepsilon}^{*}F_{A_\varepsilon}-D_{A_\varepsilon}D_{A_\varepsilon}^{*}(A_\varepsilon-B)+\mathbf{B}(\Phi_\varepsilon\otimes D_{A_\varepsilon}\Phi_\varepsilon)+C_\rho^A(A_\varepsilon-B)+\xi_{B,\varepsilon},\\
\partial_t\Phi_\varepsilon&= \Delta_{A_\varepsilon}\Phi_\varepsilon+(D_{A_\varepsilon}^{*}(A_\varepsilon-B))\Phi_\varepsilon-\lvert\Phi_\varepsilon\rvert^2\Phi_\varepsilon+C_{\rho,\varepsilon}^{\Phi}(\Phi_\varepsilon)+\zeta_{B,\varepsilon},\\
A_\varepsilon(0)&=A_0,\qquad\Phi_\varepsilon(0)=\Phi_0. \tag*{(44)}
\end{aligned} ∂ t A ε ∂ t Φ ε A ε ( 0 ) = − D A ε ∗ F A ε − D A ε D A ε ∗ ( A ε − B ) + B ( Φ ε ⊗ D A ε Φ ε ) + C ρ A ( A ε − B ) + ξ B , ε , = Δ A ε Φ ε + ( D A ε ∗ ( A ε − B )) Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε , = A 0 , Φ ε ( 0 ) = Φ 0 . (44) (We have absorbed the mass term M Φ ε M\Phi_\varepsilon M Φ ε C ρ , ε Φ Φ ε . ) F o r n o t a t i o n a l r e a s o n s , i t i s c o n v e n i e n t t o d e n o t e t h e u n i q u e s o l u t i o n C_{\rho,\varepsilon}^{\Phi}\Phi_\varepsilon.) For notational reasons, it is convenient to denote the unique solution C ρ , ε Φ Φ ε . ) F or n o t a t i o na l r e a so n s , i t i sco n v e ni e n tt o d e n o t e t h e u ni q u eso l u t i o n (44) on the time interval [ 0 , T ] [0,T] [ 0 , T ]
S T , ρ , ε ( A 0 , Φ 0 , ξ , ζ ; C ρ A , C ρ , ε Φ ) . (45) \mathfrak{S}_{T,\rho,\varepsilon}(A_0,\Phi_0,\xi,\zeta;C_\rho^A,C_{\rho,\varepsilon}^{\Phi}). \tag*{(45)} S T , ρ , ε ( A 0 , Φ 0 , ξ , ζ ; C ρ A , C ρ , ε Φ ) . (45) In the following, we refer to (45) as the data-to-solution map.
Lemma 2.13 (Gauge-covariant approximations). Let T > 0 T>0 T > 0 ρ ∈ P \rho\in\mathcal{P} ρ ∈ P ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0 C ρ A ∈ End G ( g ) C_\rho^A \in\operatorname{End}_{G}(\mathfrak{g}) C ρ A ∈ End G ( g ) C ρ , ε Φ ∈ End G ( V ) C_{\rho,\varepsilon}^{\Phi} \in\operatorname{End}_{G}(V) C ρ , ε Φ ∈ End G ( V ) C ρ A C_\rho^A C ρ A C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ G G G ( A 0 , Φ 0 ) ∈ T T (A_0,\Phi_0)\in\mathcal{T}_T ( A 0 , Φ 0 ) ∈ T T ξ ∈ C t , x θ ( [ 0 , T ] × T 2 , g 2 ) \xi\in\mathcal{C}^{\theta}_{t,x}([0,T]\times\mathbb{T}^2,\mathfrak{g}^2) ξ ∈ C t , x θ ([ 0 , T ] × T 2 , g 2 ) ζ ∈ C t , x θ ( [ 0 , T ] × T 2 , V ) \zeta\in\mathcal{C}^{\theta}_{t,x}([0,T]\times\mathbb{T}^2,V) ζ ∈ C t , x θ ([ 0 , T ] × T 2 , V ) (44) exists on the time interval [ 0 , T ] [0,T] [ 0 , T ] g ∈ G g\in\mathcal{G} g ∈ G
g ∙ S T , ρ , ε ( A 0 , Φ 0 , ξ , ζ ; C ρ A , C ρ , ε Φ ) = S T , ρ , ε ( g ∙ A 0 , g ∙ Φ 0 , g ∙ ξ , g ∙ ζ ; C ρ A , C ρ , ε Φ ) . (46) g\mathbin{\bullet}\mathfrak{S}_{T,\rho,\varepsilon}(A_0,\Phi_0,\xi,\zeta;C_\rho^A,C_{\rho,\varepsilon}^{\Phi})=\mathfrak{S}_{T,\rho,\varepsilon}(g\mathbin{\bullet}A_0,g\mathbin{\bullet}\Phi_0,g\mathbin{\bullet}\xi,g\mathbin{\bullet}\zeta;C_\rho^A,C_{\rho,\varepsilon}^{\Phi}). \tag*{(46)} g ∙ S T , ρ , ε ( A 0 , Φ 0 , ξ , ζ ; C ρ A , C ρ , ε Φ ) = S T , ρ , ε ( g ∙ A 0 , g ∙ Φ 0 , g ∙ ξ , g ∙ ζ ; C ρ A , C ρ , ε Φ ) . (46) In other words, the data-to-solution map is gauge covariant.
Remark 2.14. The gauge covariance statement in Lemma 2.13 is very different from the gauge covariance statements proven in [CCHS22, CCHS24, BC26a]. In [CCHS22, CCHS24, BC26a], the gauge covariance is only shown after taking the limit ε → 0 \varepsilon\to0 ε → 0 C ρ A C^{A}_{\rho} C ρ A C ρ , ε Φ C^{\Phi}_{\rho,\varepsilon} C ρ , ε Φ Lemma 2.13 holds for all values of ε \varepsilon ε (44) may be useful in situations other than just the proof of Theorem 1.5 . In particular, it is possible that the gauge covariant smooth approximation (44) can be compared more easily with lattice dynamics (such as in [She21, CS23]) than the non-covariant smooth approximations from [CCHS22, CCHS24, BC26a].
Since the proof of Lemma 2.13 is rather short, we present it here.
Proof. Let ( A ε , Φ ε ) (A_{\varepsilon}, \Phi_{\varepsilon}) ( A ε , Φ ε ) (44) and, to simplify the notation, let ( A ~ ε , Φ ~ ε ) ≔ g ∙ ( A ε , Φ ε ) (\widetilde{A}_{\varepsilon}, \widetilde{\Phi}_{\varepsilon}) \coloneqq g \mathbin{\bullet} (A_{\varepsilon}, \Phi_{\varepsilon}) ( A ε , Φ ε ) : = g ∙ ( A ε , Φ ε ) ( A ~ 0 , Φ ~ 0 ) ≔ g ∙ ( A 0 , Φ 0 ) (\widetilde{A}_{0}, \widetilde{\Phi}_{0}) \coloneqq g \mathbin{\bullet} (A_{0}, \Phi_{0}) ( A 0 , Φ 0 ) : = g ∙ ( A 0 , Φ 0 ) ( A ~ 0 , Φ ~ 0 ) (\widetilde{A}_{0}, \widetilde{\Phi}_{0}) ( A 0 , Φ 0 ) I T \mathcal{I}_{T} I T B ~ = g ∙ B \widetilde{B} = g \mathbin{\bullet} B B = g ∙ B A ~ 0 \widetilde{A}_{0} A 0 D A ε D_{A_{\varepsilon}} D A ε B B B A ε − B A_{\varepsilon} - B A ε − B
∂ t A ~ ε = − D A ~ ε ∗ F A ~ ε − D A ~ ε ∗ D A ~ ε ∗ ( A ~ ε − B ~ ) + B ( Φ ~ ε ⊗ D A ~ ε Φ ~ ε ) + g ∙ C ρ A ( A ε − B ) + g ∙ ξ B , ε , \partial_t \widetilde{A}_{\varepsilon} = -D_{\widetilde{A}_{\varepsilon}}^{*} F_{\widetilde{A}_{\varepsilon}} - D_{\widetilde{A}_{\varepsilon}}^{*}D_{\widetilde{A}_{\varepsilon}}^{*}(\widetilde{A}_{\varepsilon} - \widetilde{B}) + B(\widetilde{\Phi}_{\varepsilon} \otimes D_{\widetilde{A}_{\varepsilon}}\widetilde{\Phi}_{\varepsilon}) + g \mathbin{\bullet} C_{\rho}^{A}(A_{\varepsilon} - B) + g \mathbin{\bullet} \xi_{B,\varepsilon}, ∂ t A ε = − D A ε ∗ F A ε − D A ε ∗ D A ε ∗ ( A ε − B ) + B ( Φ ε ⊗ D A ε Φ ε ) + g ∙ C ρ A ( A ε − B ) + g ∙ ξ B , ε , ∂ t Φ ~ ε = Δ A ~ ε Φ ~ ε + ( D A ~ ε ∗ ( A ~ ε − B ~ ) ) Φ ~ ε − ∣ Φ ~ ε ∣ 2 Φ ~ ε + g ∙ C ρ , ε Φ Φ ε + g ∙ ζ B , ε , \partial_t \widetilde{\Phi}_{\varepsilon} = \Delta_{\widetilde{A}_{\varepsilon}}\widetilde{\Phi}_{\varepsilon} + (D_{\widetilde{A}_{\varepsilon}}^{*}(\widetilde{A}_{\varepsilon} - \widetilde{B}))\widetilde{\Phi}_{\varepsilon} - |\widetilde{\Phi}_{\varepsilon}|^{2}\widetilde{\Phi}_{\varepsilon} + g \mathbin{\bullet} C_{\rho,\varepsilon}^{\Phi}\Phi_{\varepsilon} + g \mathbin{\bullet} \zeta_{B,\varepsilon}, ∂ t Φ ε = Δ A ε Φ ε + ( D A ε ∗ ( A ε − B )) Φ ε − ∣ Φ ε ∣ 2 Φ ε + g ∙ C ρ , ε Φ Φ ε + g ∙ ζ B , ε , A ~ ε ( 0 ) = A ~ 0 , Φ ~ ε ( 0 ) = Φ ~ 0 . \widetilde{A}_{\varepsilon}(0) = \widetilde{A}_{0}, \qquad\widetilde{\Phi}_{\varepsilon}(0) = \widetilde{\Phi}_{0}. A ε ( 0 ) = A 0 , Φ ε ( 0 ) = Φ 0 . Since both C ρ A C_{\rho}^{A} C ρ A C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ
g ∙ C ρ A ( A ε − B ) = C ρ A ( A ~ ε − B ~ ) and g ∙ C ρ , ε Φ Φ ε = C ρ , ε Φ Φ ~ ε . g \mathbin{\bullet} C_{\rho}^{A}(A_{\varepsilon} - B) = C_{\rho}^{A}(\widetilde{A}_{\varepsilon} - \widetilde{B}) \quad\text{and} \quad g \mathbin{\bullet} C_{\rho,\varepsilon}^{\Phi}\Phi_{\varepsilon} = C_{\rho,\varepsilon}^{\Phi}\widetilde{\Phi}_{\varepsilon}. g ∙ C ρ A ( A ε − B ) = C ρ A ( A ε − B ) and g ∙ C ρ , ε Φ Φ ε = C ρ , ε Φ Φ ε . Furthermore, since the heat kernels p B p_{B} p B q B q_{B} q B Definition 2.1 are gauge covariant, it holds that
g ∙ ξ B , ε = ( g ∙ ξ ) B ~ , ε and g ∙ ζ B , ε = ( g ∙ ζ ) B ~ , ε . g \mathbin{\bullet} \xi_{B,\varepsilon} = (g \mathbin{\bullet} \xi)_{\widetilde{B},\varepsilon} \quad\text{and} \quad g \mathbin{\bullet} \zeta_{B,\varepsilon} = (g \mathbin{\bullet} \zeta)_{\widetilde{B},\varepsilon}. g ∙ ξ B , ε = ( g ∙ ξ ) B , ε and g ∙ ζ B , ε = ( g ∙ ζ ) B , ε . From this, we obtain (46) .
As previously mentioned in the introduction, the gauge covariance of (44) comes at a price. Unlike for the non-covariant smooth approximations from [CCHS22, CCHS24, BC26a], the process on the space of gauge orbits induced by (44) is not necessarily Markov, and establishing the Markov property will require us to choose suitable renormalisation constants (see § 3.4 ). Of course, aside from the Markov property, we will also need to choose suitable renormalisation constants to obtain the local convergence of ( A ε , Φ ε ) (A_{\varepsilon}, \Phi_{\varepsilon}) ( A ε , Φ ε ) ε → 0 \varepsilon\to0 ε → 0
Definition 2.15 (Renormalisation constants). Let ρ ∈ P \rho\in\mathcal{P} ρ ∈ P (41) ) and let ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_{0}} ε ∈ 2 − N 0
σ ρ , ε 2 ≔ ∫ 0 1 d t ρ ε 2 ( t ) 8 π t and C ρ ≔ ∫ 0 ∞ d t ( ∫ 0 t d s ρ ε ( s ) ) ρ ε ′ ( t ) 8 π t . (47) \sigma_{\rho,\varepsilon}^{2} \coloneqq\int_{0}^{1} \mathrm{d}t \frac{\rho_{\varepsilon}^{2}(t)}{8\pi t}
\qquad\text{and} \qquad
C_{\rho} \coloneqq\int_{0}^{\infty} \mathrm{d}t \left(\int_{0}^{t} \mathrm{d}s\,\rho_{\varepsilon}(s)\right)\frac{\rho_{\varepsilon}'(t)}{8\pi t}. \tag*{(47)} σ ρ , ε 2 : = ∫ 0 1 d t 8 π t ρ ε 2 ( t ) and C ρ : = ∫ 0 ∞ d t ( ∫ 0 t d s ρ ε ( s ) ) 8 π t ρ ε ′ ( t ) . (47) Second, we define C ρ A ∈ End ( g ) C_{\rho}^{A} \in\operatorname{End}(\mathfrak{g}) C ρ A ∈ End ( g ) C ρ , ε Φ ∈ End ( V ) C_{\rho,\varepsilon}^{\Phi} \in\operatorname{End}(V) C ρ , ε Φ ∈ End ( V )
C ρ A ≔ C ρ ( Cov g g + Cov V g ) − 1 8 π Cov V g , (48) C_{\rho}^{A} \coloneqq C_{\rho}(\operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}} + \operatorname{Cov}_{V}^{\mathfrak{g}}) - \frac{1}{8\pi}\operatorname{Cov}_{V}^{\mathfrak{g}}, \tag*{(48)} C ρ A : = C ρ ( Cov g g + Cov V g ) − 8 π 1 Cov V g , (48) C ρ , ε Φ ≔ ( − σ ρ , ε 2 + C ρ ) Cov g V + ( dim ( V ) + 2 ) σ ρ , ε 2 Id V − M , C_{\rho,\varepsilon}^{\Phi} \coloneqq(-\sigma_{\rho,\varepsilon}^{2} + C_{\rho})\operatorname{Cov}_{\mathfrak{g}}^{V} + (\dim(V) + 2)\sigma_{\rho,\varepsilon}^{2}\operatorname{Id}_{V} - M, C ρ , ε Φ : = ( − σ ρ , ε 2 + C ρ ) Cov g V + ( dim ( V ) + 2 ) σ ρ , ε 2 Id V − M , where Cov g g \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}} Cov g g Cov V g \operatorname{Cov}_{V}^{\mathfrak{g}} Cov V g Cov g V \operatorname{Cov}_{\mathfrak{g}}^{V} Cov g V
In Lemma 3.42 below, it will be shown that C ρ A ∈ End G ( g ) C_\rho^{A} \in\operatorname{End}_{G}(\mathfrak{g}) C ρ A ∈ End G ( g ) C ρ , ε Φ ∈ End G ( V ) C_{\rho,\varepsilon}^{\Phi} \in\operatorname{End}_{G}(V) C ρ , ε Φ ∈ End G ( V ) C ρ A C_\rho^{A} C ρ A C ρ , ε Φ C_{\rho,\varepsilon}^{\Phi} C ρ , ε Φ G G G
Remark 2.16. While σ ρ , ε 2 \sigma_{\rho,\varepsilon}^{2} σ ρ , ε 2 ε → 0 \varepsilon\to0 ε → 0 C ρ C_\rho C ρ ε \varepsilon ε ρ ε \rho_\varepsilon ρ ε ρ \rho ρ (47) –(48) . For the explicit calculations that led us to (47) –(48) , we refer the reader to Sections 6, 9, and 13 below. In particular, this specific choice for C ρ A C_\rho^{A} C ρ A (44) yields a Markov process in the limit ε → 0 \varepsilon\to0 ε → 0 C ρ C_\rho C ρ ∞ \infty ∞ ε \varepsilon ε C ρ C_\rho C ρ ε → 0 \varepsilon\to0 ε → 0
Remark 2.17. We emphasise that, while most of our estimates will be performed with respect to the background connection one-form B B B B B B p τ B p_{\tau}^{B} p τ B
Remark 2.18. The choice of upper limit 1 in the definition of σ ρ , ε 2 \sigma_{\rho,\varepsilon}^{2} σ ρ , ε 2 M M M
In the following, the renormalisation constants in (44) will always be chosen as in Definition 2.15. Since the renormalisation constants were the last ingredient we needed, we can now state the main result of this subsection, namely, the covariant local well-posedness of the stochastic Yang–Mills–Higgs equations.
Theorem 2.19 (Covariant local well-posedness). Let C 0 > 1 C_0 > 1 C 0 > 1 c 0 > 0 c_0 > 0 c 0 > 0 0 < T ≤ c 0 0 < T \le c_0 0 < T ≤ c 0 ρ ∈ P \rho\in\mathcal{P} ρ ∈ P ( A 0 , Φ 0 ) ∈ I T (A_0,\Phi_0) \in\mathcal{I}_T ( A 0 , Φ 0 ) ∈ I T
T δ exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ( [ 0 , T ] ) ) ( 1 + ∥ F B ∥ T , 100 p e r t + T ( 1 − 2 κ ) / 2 ∥ t − ( 1 − 2 κ ) / 2 F B ∥ T , 100 p e r t ) 100 ≤ 1 , (49) T^\delta\exp\left(C_0\lVert F_B\rVert_{L_t^1L_x^\infty([0,T])}\right)\left(1+\lVert F_B\rVert_{T,100}^{\mathrm{pert}}+T^{(1-2\kappa)/2}\lVert t^{-(1-2\kappa)/2}F_B\rVert_{T,100}^{\mathrm{pert}}\right)^{100} \le1, \tag*{(49)} T δ exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ([ 0 , T ]) ) ( 1 + ∥ F B ∥ T , 100 pert + T ( 1 − 2 κ ) /2 ∥ t − ( 1 − 2 κ ) /2 F B ∥ T , 100 pert ) 100 ≤ 1 , (49) T ∥ Φ 0 ∥ C A 0 , T γ 0 ≤ 1. (50) T\lVert\Phi_0\rVert_{\mathcal{C}_{A_0,T}^{\gamma_0}} \le1. \tag*{(50)} T ∥ Φ 0 ∥ C A 0 , T γ 0 ≤ 1. (50) Here, B B B (40) , δ \delta δ κ \kappa κ (96) , and γ 0 \gamma_0 γ 0 (99) . Then, there exists an event E ( T , A 0 , Φ 0 ) E(T,A_0,\Phi_0) E ( T , A 0 , Φ 0 ) T T T
(i) (High probability) It holds that
P ( E ( T , A 0 , Φ 0 ) ) ≥ 1 − c − 1 exp ( − c T − c ) , \mathbb{P}\left(E(T,A_0,\Phi_0)\right) \ge1-c^{-1}\exp\left(-cT^{-c}\right), P ( E ( T , A 0 , Φ 0 ) ) ≥ 1 − c − 1 exp ( − c T − c ) , where c c c (96) .
(ii) (Local convergence) For each ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0 ( A ε , Φ ε ) (A_\varepsilon,\Phi_\varepsilon) ( A ε , Φ ε ) (44) with the renormalisation constants from Definition 2.15. Then, on the event E ( T , A 0 , Φ 0 ) E(T,A_0,\Phi_0) E ( T , A 0 , Φ 0 )
( A , Φ ) = lim ε → 0 ( A ε , Φ ε ) (A,\Phi)=\lim_{\varepsilon\to0}(A_\varepsilon,\Phi_\varepsilon) ( A , Φ ) = ε → 0 lim ( A ε , Φ ε ) exists in C t 0 C x − κ ( [ 0 , T ] × T 2 , g 2 × V ) C_t^{0}C_x^{-\kappa}([0,T]\times\mathbb{T}^{2},\mathfrak{g}^{2}\times V) C t 0 C x − κ ([ 0 , T ] × T 2 , g 2 × V )
Remark 2.20. Since the C t 0 C x − κ C_t^{0}C_x^{-\kappa} C t 0 C x − κ ( A , Φ ) (A,\Phi) ( A , Φ ) Theorem 2.19 . Our reasons for this are twofold: First, the convergence of the smooth approximations is a qualitative statement. And while we need all of our quantitative estimates, like the conditions in (49) –(50) , to only involve gauge invariant norms, we do not require this in qualitative statements. Second, we could use our solution norms (from Definition 7.2) to state a gauge covariant convergence result. However, this would require notation that has not yet been introduced.
Remark 2.21. The conditions (49) –(50) will later be included in Assumptions 8.1 and 8.2, which will be used throughout all of our local arguments. Since the condition (49) is important for our global iteration scheme, we discuss its main aspects. The polynomial factor in ∥ F B ∥ T , 100 p e r t \lVert F_B\rVert_{T,100}^{\mathrm{pert}} ∥ F B ∥ T , 100 pert T ( 1 − 2 κ ) / 2 ∥ t − ( 1 − 2 κ ) / 2 F B ∥ T , 1 p e r t T^{(1-2\kappa)/2}\lVert t^{-(1-2\kappa)/2}F_B\rVert_{T,1}^{\mathrm{pert}} T ( 1 − 2 κ ) /2 ∥ t − ( 1 − 2 κ ) /2 F B ∥ T , 1 pert (49) is less important than the factors T δ T^\delta T δ exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ( [ 0 , T ] ) ) \exp(C_0\lVert F_B\rVert_{L_t^1L_x^\infty([0,T])}) exp ( C 0 ∥ F B ∥ L t 1 L x ∞ ([ 0 , T ]) ) δ > 0 \delta>0 δ > 0 (49) as
∥ F B ∥ L t 1 L x ∞ ( [ 0 , T ] ) ≪ ∣ log ( T ) ∣ . \lVert F_B\rVert_{L_t^1L_x^\infty([0,T])}\ll\lvert\log(T)\rvert. ∥ F B ∥ L t 1 L x ∞ ([ 0 , T ]) ≪ ∣ log ( T )∣ . If the L x ∞ L_x^\infty L x ∞ 8 F B F_B F B [ 0 , T ] [0,T] [ 0 , T ] (49) as
T ∥ F B ( 0 ) ∥ L x ∞ ≪ ∣ log ( T ) ∣ . T\lVert F_B(0)\rVert_{L_x^\infty}\ll\lvert\log(T)\rvert. T ∥ F B ( 0 ) ∥ L x ∞ ≪ ∣ log ( T )∣ . Hence, T T T F B F_B F B § 1.2 ). The quantitative condition on the size of the local time of existence from (49) is one of the main benefits of our covariant approach over the non-covariant approach from [CCHS22, CCHS24, BC26a]. If we only used the existing non-covariant local theory, we would still be able to work only with gauge invariant norms of the initial data A 0 A_0 A 0 Proposition 2.10 ). However, the cubic term in [CCHS22, (1.7)] means that the resulting local time of existence T T T A 0 A_0 A 0
We emphasise that the precise formulation of Theorem 2.19 was chosen for its relative simplicity. While Theorem 2.19 itself contains little information on the structure of the solution, its proof (as well as Proposition 8.3 ) shows that the solution can be decomposed into covariant stochastic objects, paracontrolled terms, and smooth, nonlinear remainders. The structure of the solution will not only be needed in the proof, but will also be used in our global iteration argument.
In the following, we describe the main ideas behind the proof of Theorem 2.19 . Our discussion will be more detailed than in § 1.2 , but less detailed than in the body of the article (§ 7 –§ 10 ). In our covariant local theory for (44) , we expand around the background connection one-form B B B C ε : = A ε − B C_\varepsilon:=A_\varepsilon-B C ε := A ε − B
( ∂ t − Δ B ) C ε k − 2 [ F B k j , C ε , j ] = [ C ε , j , 2 D B j C ε k − D B k C ε j ] + [ C ε , j , [ C ε j , C ε k ] ] + B ( Φ ε ⊗ D B k Φ ε ) + B ( Φ ε ⊗ C ε k Φ ε ) + C A ρ ( C ε k ) + ξ B , ε k , ( ∂ t − Δ B ) Φ ε = 2 C ε , j D B j Φ ε + C ε j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (51) \begin{aligned}
(\partial_t-\Delta_B)C_\varepsilon^k-2[F_B^{kj},C_{\varepsilon,j}]&=[C_{\varepsilon,j},2D_B^jC_\varepsilon^k-D_B^kC_\varepsilon^j]+[C_{\varepsilon,j},[C_\varepsilon^j,C_\varepsilon^k]]\\
&\quad+\mathbf{B}(\Phi_\varepsilon\otimes D_B^k\Phi_\varepsilon)+\mathbf{B}(\Phi_\varepsilon\otimes C_\varepsilon^k\Phi_\varepsilon)+C_A^\rho(C_\varepsilon^k)+\xi_{B,\varepsilon}^k,\\
(\partial_t-\Delta_B)\Phi_\varepsilon&=2C_{\varepsilon,j}D_B^j\Phi_\varepsilon+C_\varepsilon^jC_{\varepsilon,j}\Phi_\varepsilon-\lvert\Phi_\varepsilon\rvert^2\Phi_\varepsilon+C_{\rho,\varepsilon}^\Phi(\Phi_\varepsilon)+\zeta_{B,\varepsilon}. \tag*{(51)}
\end{aligned} ( ∂ t − Δ B ) C ε k − 2 [ F B k j , C ε , j ] ( ∂ t − Δ B ) Φ ε = [ C ε , j , 2 D B j C ε k − D B k C ε j ] + [ C ε , j , [ C ε j , C ε k ]] + B ( Φ ε ⊗ D B k Φ ε ) + B ( Φ ε ⊗ C ε k Φ ε ) + C A ρ ( C ε k ) + ξ B , ε k , = 2 C ε , j D B j Φ ε + C ε j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (51) For the derivation of (51) , we refer the reader to Lemma 3.22 and § 7 . The first step in our covariant ansatz for ( C ε , Φ ε ) (C_\varepsilon,\Phi_\varepsilon) ( C ε , Φ ε ) S S S φ \varphi φ ξ B , ε \xi_{B,\varepsilon} ξ B , ε ζ B , ε \zeta_{B,\varepsilon} ζ B , ε
( ∂ t − Δ B ) S k − 2 [ F B k j , S j ] = [ S j , 2 D B j S k − D B k S j ] + [ S j , [ S j , S k ] ] + B ( φ ⊗ D B k φ ) + B ( φ ⊗ S k φ ) , (52) (\partial_t-\Delta_B)S^k-2[F_B^{kj},S_j]=[S_j,2D_B^jS^k-D_B^kS^j]+[S_j,[S^j,S^k]]
+\mathbf{B}(\varphi\otimes D_B^k\varphi)+\mathbf{B}(\varphi\otimes S^k\varphi), \tag*{(52)} ( ∂ t − Δ B ) S k − 2 [ F B k j , S j ] = [ S j , 2 D B j S k − D B k S j ] + [ S j , [ S j , S k ]] + B ( φ ⊗ D B k φ ) + B ( φ ⊗ S k φ ) , (52) ( ∂ t − Δ B ) φ = 2 S j D B j φ + S j S j φ − ∣ φ ∣ 2 φ , S ( 0 ) = 0 , φ ( 0 ) = Φ 0 . (53) \begin{aligned}
(\partial_t-\Delta_B)\varphi&=2S_jD_B^j\varphi+S^jS_j\varphi-\lvert\varphi\rvert^2\varphi,\\
S(0)&=0,\qquad\varphi(0)=\Phi_0. \tag*{(53)}
\end{aligned} ( ∂ t − Δ B ) φ S ( 0 ) = 2 S j D B j φ + S j S j φ − ∣ φ ∣ 2 φ , = 0 , φ ( 0 ) = Φ 0 . (53) The equations (52) –(53) are chosen so that ( B + S , φ ) (B+S,\varphi) ( B + S , φ ) B B B S S S φ \varphi φ Proposition 5.15 ), S S S φ \varphi φ S S S φ \varphi φ Φ 0 \Phi_0 Φ 0 S S S φ \varphi φ φ ˚ B , ε \mathring{\varphi}_{B,\varepsilon} φ ˚ B , ε ψ ˚ B , ε \mathring{\psi}_{B,\varepsilon} ψ ˚ B , ε § 1.2 , they are defined as solutions of
( ∂ t − Δ B ) φ ˚ B , ε k − 2 [ F B k j , φ ˚ B , ε , j ] = ξ B , ε k , φ ˚ B , ε k ( 0 ) = 0 , (54) (\partial_t-\Delta_B)\mathring{\varphi}^{k}_{B,\varepsilon}-2[F_B^{kj},\mathring{\varphi}_{B,\varepsilon,j}]=\xi^{k}_{B,\varepsilon},\qquad\mathring{\varphi}^{k}_{B,\varepsilon}(0)=0, \tag*{(54)} ( ∂ t − Δ B ) φ ˚ B , ε k − 2 [ F B k j , φ ˚ B , ε , j ] = ξ B , ε k , φ ˚ B , ε k ( 0 ) = 0 , (54) ( ∂ t − Δ B ) ψ ˚ B , ε = ζ B , ε , ψ ˚ B , ε ( 0 ) = 0. (55) (\partial_t-\Delta_B)\mathring{\psi}_{B,\varepsilon}=\zeta_{B,\varepsilon},\qquad\mathring{\psi}_{B,\varepsilon}(0)=0. \tag*{(55)} ( ∂ t − Δ B ) ψ ˚ B , ε = ζ B , ε , ψ ˚ B , ε ( 0 ) = 0. (55) We also denote the limits of φ ˚ B , ε \mathring{\varphi}_{B,\varepsilon} φ ˚ B , ε ψ ˚ B , ε \mathring{\psi}_{B,\varepsilon} ψ ˚ B , ε ε → 0 \varepsilon\to0 ε → 0 φ ˚ B \mathring{\varphi}_{B} φ ˚ B ψ ˚ \mathring{\psi} ψ ˚
( ∂ t − Δ B ) φ ˚ B , ε ∘ k − 2 [ F B k j , φ ˚ B , ε , j ∘ ] = [ φ ˚ B , ε , j , 2 D B j φ ˚ B , ε k − D B k φ ˚ B , ε , j ] , (56) (\partial_t-\Delta_B)\mathring{\varphi}^{\circ k}_{B,\varepsilon}-2[F_B^{kj},\mathring{\varphi}^{\circ}_{B,\varepsilon,j}]=[\mathring{\varphi}_{B,\varepsilon,j},2D_B^j\mathring{\varphi}^{k}_{B,\varepsilon}-D_B^k\mathring{\varphi}_{B,\varepsilon,j}], \tag*{(56)} ( ∂ t − Δ B ) φ ˚ B , ε ∘ k − 2 [ F B k j , φ ˚ B , ε , j ∘ ] = [ φ ˚ B , ε , j , 2 D B j φ ˚ B , ε k − D B k φ ˚ B , ε , j ] , (56) ( ∂ t − Δ B ) ψ ˚ B , ε ∙ − 2 [ F B k j , ψ ˚ B , ε , j ∙ ] = B ( ψ ˚ B , ε ⊗ D B k ψ ˚ B , ε ) , (57) (\partial_t-\Delta_B)\mathring{\psi}^{\bullet}_{B,\varepsilon}-2[F_B^{kj},\mathring{\psi}^{\bullet}_{B,\varepsilon,j}]=B(\mathring{\psi}_{B,\varepsilon}\otimes D_B^k\mathring{\psi}_{B,\varepsilon}), \tag*{(57)} ( ∂ t − Δ B ) ψ ˚ B , ε ∙ − 2 [ F B k j , ψ ˚ B , ε , j ∙ ] = B ( ψ ˚ B , ε ⊗ D B k ψ ˚ B , ε ) , (57) ( ∂ t − Δ B ) ψ ˚ B , ε ∘ = 2 ψ ˚ B , ε , j D B j ψ ˚ B , ε . (58) (\partial_t-\Delta_B)\mathring{\psi}^{\circ}_{B,\varepsilon}=2\mathring{\psi}_{B,\varepsilon,j}D_B^j\mathring{\psi}_{B,\varepsilon}. \tag*{(58)} ( ∂ t − Δ B ) ψ ˚ B , ε ∘ = 2 ψ ˚ B , ε , j D B j ψ ˚ B , ε . (58) We emphasise that the covariant objects from (2.26)–(2.30) can also be represented using the covariant heat kernels p B p_B p B q B q_B q B
ψ ˚ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . (59) \mathring{\psi}_{B,\varepsilon}(t,x)=\int_0^t ds\int_{\mathbb{T}^2}dy\,\rho_\varepsilon(t-s)q_B(s,y;t,x)\xi(s,y). \tag*{(59)} ψ ˚ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . (59) In our analysis of the covariant stochastic objects, we then heavily rely on covariant heat kernel estimates. While a similar approach was previously used in work of the first and second authors on the Abelian Higgs model [BC24], our implementation of this approach differs significantly from [BC24]. As already mentioned in Remark 2.5, [BC24] relies on operator estimates of the covariant heat kernel and its covariant derivatives, whereas we rely on the pointwise estimates from § 3.1 . This drastically simplifies one of the hardest parts of the proof in [BC24], namely, the estimates of the resonant part of the quadratic stochastic object (cf. [BC24, Proposition 5.29] and Lemmas 9.15 and 9.18 below).
Another important difference between [BC24] and the present article is that the covariant methods in our local theory are not only used for the covariant stochastic objects. Instead, we build an entire covariant functional framework, which encompasses covariant Littlewood–Paley decompositions, function spaces, Schauder estimates, and paraproducts. To motivate this framework, we first briefly recall the standard Littlewood–Paley decompositions. Throughout this discussion only, we let ς : R 2 → [ 0 , ∞ ) \varsigma:\mathbb{R}^2\to[0,\infty) ς : R 2 → [ 0 , ∞ ) ς ∣ B ( 0 , 1 ) ≡ 1 \varsigma|_{B(0,1)}\equiv1 ς ∣ B ( 0 , 1 ) ≡ 1 ς ∣ R 2 ∖ B ( 0 , 2 ) ≡ 0 \varsigma|_{\mathbb{R}^2\setminus B(0,2)}\equiv0 ς ∣ R 2 ∖ B ( 0 , 2 ) ≡ 0 B ( 0 , r ) ⊆ R 2 B(0,r)\subseteq\mathbb{R}^2 B ( 0 , r ) ⊆ R 2 r r r ϕ : T 2 → C \phi:\mathbb{T}^2\to\mathbb{C} ϕ : T 2 → C ( P ≤ N ϕ ) N ∈ 2 N 0 (P_{\leq N}\phi)_{N\in2^{\mathbb{N}_0}} ( P ≤ N ϕ ) N ∈ 2 N 0 ( P N ϕ ) N ∈ 2 N 0 (P_N\phi)_{N\in2^{\mathbb{N}_0}} ( P N ϕ ) N ∈ 2 N 0
P ≤ N ϕ ^ ( n ) = d e f ς ( n / N ) ϕ ^ ( n ) and P N ϕ = d e f ( P ≤ N − P ≤ N / 2 ) ϕ , (60) \widehat{P_{\leq N}\phi}(n)\overset{\mathrm{def}}{=}\varsigma(n/N)\widehat{\phi}(n)
\qquad\text{and}\qquad
P_N\phi\overset{\mathrm{def}}{=}(P_{\leq N}-P_{\leq N/2})\phi, \tag*{(60)} P ≤ N ϕ ( n ) = def ς ( n / N ) ϕ ( n ) and P N ϕ = def ( P ≤ N − P ≤ N /2 ) ϕ , (60) where we use the convention P ≤ 1 / 2 = d e f 0 P_{\leq1/2}\overset{\mathrm{def}}{=}0 P ≤ 1/2 = def 0
Unfortunately, since they do not interact nicely with gauge transformations, the standard Littlewood–Paley projections from (2.32) will not be too useful in this article. In order to introduce our covariant version, recall from (2.31) that the covariant stochastic object φ ˚ B , ε \mathring{\varphi}_{B,\varepsilon} φ ˚ B , ε ρ ε q B \rho_\varepsilon q_B ρ ε q B ξ \xi ξ N ∈ 2 N 0 N\in2^{\mathbb{N}_0} N ∈ 2 N 0
φ ˚ B , ε , N = d e f ∬ d s d y ρ ε ( t − s ) χ N ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . (61) \mathring{\varphi}_{B,\varepsilon,N}\overset{\mathrm{def}}{=}\iint ds\,dy\,\rho_\varepsilon(t-s)\chi_N(t-s)q_B(s,y;t,x)\xi(s,y). \tag*{(61)} φ ˚ B , ε , N = def ∬ d s d y ρ ε ( t − s ) χ N ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . (61) where χ N ( t − s ) \chi_N(t-s) χ N ( t − s ) ( t − s ) ∼ N − 2 (t-s)\sim N^{-2} ( t − s ) ∼ N − 2 (95) . Due to parabolic scaling and the uncertainty principle, the restriction to times N 2 ( t − s ) ∼ 1 N^2(t-s)\sim1 N 2 ( t − s ) ∼ 1 (61) is morally similar to a projection to frequencies of size ∼ N \sim N ∼ N (61) , as well as potential alternatives, we refer the reader to Section 6.2 below. More generally, we introduce localised versions of the Duhamel integral operators of (51) . For example, for any H : [ 0 , T ] × T 2 → g 2 H:[0,T]\times\mathbb{T}^2\to\mathfrak{g}^2 H : [ 0 , T ] × T 2 → g 2
Duh B , N F ( H ) ( t , x ) ≔ ∬ d s d y χ N ( t − s ) q B ( s , y ; t , x ) H ( s , y ) . (62) \operatorname{Duh}^{F}_{B,N}(H)(t,x)\coloneqq\iint\mathrm{d}s\mathrm{d}y\,\chi_N(t-s)q_B(s,y;t,x)H(s,y). \tag*{(62)} Duh B , N F ( H ) ( t , x ) : = ∬ d s d y χ N ( t − s ) q B ( s , y ; t , x ) H ( s , y ) . (62) Definitions similar to those in (61) and (62) can also be made for the Higgs component of our solution by replacing the kernel q B q_B q B p B p_B p B (61) and (62) , we then build an entire covariant functional framework. Compared to the non-covariant paracontrolled calculus of [GIP15], the main difficulty is that our covariant Littlewood–Paley decomposition does not obey the Littlewood–Paley trichotomy; see Section 6.2. Since the full definitions and estimates included in the functional framework are well beyond the scope of this section, we do not include them here, and instead only provide the following brief descriptions and examples:
(i) (Solution spaces) Our solution spaces will be defined in terms of the covariant Littlewood–Paley blocks. The covariant linear stochastic object will be controlled in our solution space at regularity − κ -\kappa − κ
sup N ∈ 2 N 0 N − m − κ ∥ D B , j 1 ⋯ D B , j m B B , ε , N ∥ L t , x ∞ . \sup_{N\in2^{\mathbb{N}_0}}N^{-m-\kappa}\lVert D_{B,j_1}\cdots D_{B,j_m}B_{\mathcal{B},\varepsilon,N}\rVert_{L^\infty_{t,x}}. N ∈ 2 N 0 sup N − m − κ ∥ D B , j 1 ⋯ D B , j m B B , ε , N ∥ L t , x ∞ . Here, 0 ≤ m ≤ 10 0\leq m\leq10 0 ≤ m ≤ 10 1 ≤ j 1 , … , j m ≤ 2 1\leq j_1,\ldots,j_m\leq2 1 ≤ j 1 , … , j m ≤ 2
(ii) (Schauder estimates) Our Schauder estimates will be phrased in terms of the localised Duhamel integrals from (62) . For example, we will prove that
∥ D B , j 1 ⋯ D B , j m Duh B , N F ( H ) ∥ L t , x ∞ ≲ L T ( F B ) N m − 2 ∥ H ∥ L t , x ∞ . \lVert D_{B,j_1}\cdots D_{B,j_m}\operatorname{Duh}^{F}_{B,N}(H)\rVert_{L^\infty_{t,x}}\lesssim L_{T}(F_B)N^{m-2}\lVert H\rVert_{L^\infty_{t,x}}. ∥ D B , j 1 ⋯ D B , j m Duh B , N F ( H ) ∥ L t , x ∞ ≲ L T ( F B ) N m − 2 ∥ H ∥ L t , x ∞ . Here, as before, 0 ≤ m ≤ 10 0\leq m\leq10 0 ≤ m ≤ 10 1 ≤ j 1 , … , j m ≤ 2 1\leq j_1,\ldots,j_m\leq2 1 ≤ j 1 , … , j m ≤ 2 L T ( F B ) L_{T}(F_B) L T ( F B ) (422) captures the F B F_B F B Theorem 2.4 ).
(iii) (Paraproducts) We will define several covariant paraproducts. For example, if ( W M ) M (W_M)_M ( W M ) M g 2 \mathfrak{g}^2 g 2 ( Υ B , ε , N ) N (\Upsilon_{B,\varepsilon,N})_N ( Υ B , ε , N ) N (61) , we define the low× \times ×
[ W i ⋆ D B k Υ B , ε , ⋆ j ] = ∑ M , N ∈ 2 N 0 : M ≤ N 1 − θ [ W M i , D B k Υ B , ε , N j ] . \left[W^i\star D_B^k\Upsilon^j_{B,\varepsilon,\star}\right]=\sum_{\substack{M,N\in2^{\mathbb{N}_0}:\\M\leq N^{1-\theta}}}\left[W^i_M,D_B^k\Upsilon^j_{B,\varepsilon,N}\right]. [ W i ⋆ D B k Υ B , ε , ⋆ j ] = M , N ∈ 2 N 0 : M ≤ N 1 − θ ∑ [ W M i , D B k Υ B , ε , N j ] . Here, 1 ≤ i , j , k ≤ 2 1\leq i,j,k\leq2 1 ≤ i , j , k ≤ 2 ⋆ \star ⋆ θ \theta θ (96) , and it is needed due to the absence of the Littlewood–Paley trichotomy.
(iv) (Commutator estimates) We prove several commutator estimates that are needed to understand the structure of the paracontrolled term. For example, in the same setting as above, we will estimate9
Duh B , κ ( [ W M i , D B k Υ B , ε , N j ] − [ W M i , D B k Duh B , κ ( Υ B , ε , N j ) ] ) . \operatorname{Duh}_{B,\kappa}\left([W^i_M,D_B^k\Upsilon^j_{B,\varepsilon,N}]-[W^i_M,D_B^k\operatorname{Duh}_{B,\kappa}(\Upsilon^j_{B,\varepsilon,N})]\right). Duh B , κ ( [ W M i , D B k Υ B , ε , N j ] − [ W M i , D B k Duh B , κ ( Υ B , ε , N j )] ) . Remark 2.22. As described above, this article relies on a covariant version of the paracontrolled calculus of [GIP15]. In order to make progress towards the global well-posedness of the stochastic Yang–Mills–Higgs model in dimension d = 3 d=3 d = 3 d = 2 d=2 d = 2 (51) . However, the focus of [HS23, Sin25, BSS25] and this article is quite different. In [HS23, Sin25, BSS25], the geometry is fixed once and for all, and the main task is to show that the counterterms only depend on local aspects of the geometry. In this article, the geometry is determined by the background connection one-form B B B § 3.3 ). In particular, one of the main tasks in the proof of Theorem 2.19 is to determine how the size of the covariant stochastic objects depends on the size of F B F_B F B Theorem 2.4 ).
Global well-posedness and uniform-in-time estimates
In this section, we state both soft and hard global well-posedness theorems (Theorems 2.24 and 2.30 ). The statement of the soft result (Theorem 2.24) will be relatively simple and, after combining it with the Krylov–Bogoliubov theorem, it will be sufficient to conclude the existence of the invariant measure from Theorem 1.5 . However, the soft result contains little information on the structure or the decay of the solution to the stochastic Yang–Mills–Higgs equations. The ∥ ⋅ ∥ i n v , − κ \|\cdot\|_{\mathrm{inv},-\kappa} ∥ ⋅ ∥ inv , − κ ∥ ⋅ ∥ i n v , − κ \|\cdot\|_{\mathrm{inv},-\kappa} ∥ ⋅ ∥ inv , − κ Theorem 2.30 ) has a more complicated statement, but it makes up for it by providing more information on the structure of the solution. In particular, the statement of our hard result will be closer to the estimates used in our proof.
In order to state our soft global well-posedness result, we first make the following definition.
Definition 2.23. Let A 0 ∈ C x − κ ( T 2 , g 2 ) A_0 \in\mathcal{C}^{-\kappa}_x(\mathbb{T}^2,\mathfrak{g}^2) A 0 ∈ C x − κ ( T 2 , g 2 ) Φ 0 ∈ C x − κ ( T 2 , V ) \Phi_0 \in\mathcal{C}^{-\kappa}_x(\mathbb{T}^2,V) Φ 0 ∈ C x − κ ( T 2 , V ) T > 0 T>0 T > 0 g ∈ G 1 − κ g \in\mathcal{G}_{1-\kappa} g ∈ G 1 − κ g ⋅ ( A 0 , Φ 0 ) ∈ J T g\cdot(A_0,\Phi_0)\in\mathcal{J}_T g ⋅ ( A 0 , Φ 0 ) ∈ J T J T \mathcal{J}_T J T (40) . Recall from Remark 2.11 that we then also have g ⋅ ( A 0 , Φ 0 ) ∈ J 1 g\cdot(A_0,\Phi_0)\in\mathcal{J}_1 g ⋅ ( A 0 , Φ 0 ) ∈ J 1 B ∈ C 0 ∞ C x − κ ( [ 0 , 1 ] × T 2 , g 2 ) ∩ C 0 , x ∞ ( [ 0 , 1 ] × T 2 , g 2 ) B\in C^\infty_0\mathcal{C}^{-\kappa}_x([0,1]\times\mathbb{T}^2,\mathfrak{g}^2)\cap C^\infty_{0,x}([0,1]\times\mathbb{T}^2,\mathfrak{g}^2) B ∈ C 0 ∞ C x − κ ([ 0 , 1 ] × T 2 , g 2 ) ∩ C 0 , x ∞ ([ 0 , 1 ] × T 2 , g 2 ) B ( 0 ) = g ⋅ A 0 B(0)=g\cdot A_0 B ( 0 ) = g ⋅ A 0 Ψ ( s ) \Psi(s) Ψ ( s ) ( ∂ s − Δ B ( s ) ) Ψ ( s ) = 0 (\partial_s-\Delta_{B(s)})\Psi(s)=0 ( ∂ s − Δ B ( s ) ) Ψ ( s ) = 0 g ⋅ Φ 0 g\cdot\Phi_0 g ⋅ Φ 0
∥ ( A 0 , Φ 0 ) ∥ i n v , − κ ≔ sup s ∈ ( 0 , 1 ] s κ 2 ( 1 s 1 2 ∥ F B ( s ) ∥ L x ∞ + s ∥ D B F B ∥ L x ∞ + ∥ Ψ ( s ) ∥ L x ∞ ) . (63) \|(A_0,\Phi_0)\|_{\mathrm{inv},-\kappa}\coloneqq\sup_{s\in(0,1]}s^{\frac{\kappa}{2}}\left(\frac{1}{s^{\frac{1}{2}}}\|F_{B(s)}\|_{L^\infty_x}+s\|D_BF_B\|_{L^\infty_x}+\|\Psi(s)\|_{L^\infty_x}\right). \tag*{(63)} ∥ ( A 0 , Φ 0 ) ∥ inv , − κ : = s ∈ ( 0 , 1 ] sup s 2 κ ( s 2 1 1 ∥ F B ( s ) ∥ L x ∞ + s ∥ D B F B ∥ L x ∞ + ∥Ψ ( s ) ∥ L x ∞ ) . (63) The fact that this gauge invariant Hölder norm is well-defined follows by Lemma 5.14. We define the norm in this way, because it is not necessarily possible to start a Yang–Mills gradient flow from arbitrary C x − κ ( T 2 , g 2 ) \mathcal{C}^{-\kappa}_x(\mathbb{T}^2,\mathfrak{g}^2) C x − κ ( T 2 , g 2 ) G 1 − κ \mathcal{G}_{1-\kappa} G 1 − κ (63) does not depend on this choice of gauge transformation (and moreover that such a gauge transformation exists).
We emphasise that the gauge invariant Hölder norm in (63) does not require the choice of a background connection one-form, which is not the case for the norms used in our covariant local theory (or the global iteration scheme described below). This allows us to measure the size of the solution to the stochastic Yang–Mills–Higgs equations without any reference to its random structure.
Theorem 2.24 (Soft global well-posedness and uniform-in-time estimates). Let A 0 ∈ C x − κ ( T 2 , g 2 ) A_0\in\mathcal{C}^{-\kappa}_x(\mathbb{T}^2,\mathfrak{g}^2) A 0 ∈ C x − κ ( T 2 , g 2 ) Φ 0 ∈ C x − κ ( T 2 , V ) \Phi_0\in\mathcal{C}^{-\kappa}_x(\mathbb{T}^2,V) Φ 0 ∈ C x − κ ( T 2 , V ) A ∈ L t , l o c ∞ C x − κ ( [ 0 , ∞ ) × T 2 , g 2 ) A\in L^\infty_{t,\mathrm{loc}}\mathcal{C}^{-\kappa}_x([0,\infty)\times\mathbb{T}^2,\mathfrak{g}^2) A ∈ L t , loc ∞ C x − κ ([ 0 , ∞ ) × T 2 , g 2 )
Higgs field Φ ∈ L t , l o c ∞ C x − κ ( [ 0 , ∞ ) × T 2 , V ) \Phi\in L^\infty_{t,\mathrm{loc}}\mathcal{C}^{-\kappa}_x([0,\infty)\times\mathbb{T}^2,\mathbf{V}) Φ ∈ L t , loc ∞ C x − κ ([ 0 , ∞ ) × T 2 , V ) ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ t 0 = 0 t_0=0 t 0 = 0
(i) (Stochastic YMH equations ) On each time interval ( t m , t m + 1 ] (t_m,t_{m+1}] ( t m , t m + 1 ] m ≥ 0 m\geq0 m ≥ 0 ( A , Φ ) (A,\Phi) ( A , Φ ) (11) . To be more precise, it is the limit of the smooth, gauge covariant approximations from (44) in C t 0 C x − κ ( ( t m , t m + 1 ] × T 2 , g 2 × V ) \mathcal{C}^{0}_{t}\mathcal{C}^{-\kappa}_{x}((t_m,t_{m+1}]\times\mathbb{T}^2,\mathfrak{g}^2\times\mathbf{V}) C t 0 C x − κ (( t m , t m + 1 ] × T 2 , g 2 × V )
(ii) (Initial conditions and right-limits ) The initial value ( A ( 0 ) , Φ ( 0 ) ) (A(0),\Phi(0)) ( A ( 0 ) , Φ ( 0 )) ( A 0 , Φ 0 ) (A_0,\Phi_0) ( A 0 , Φ 0 ) m ≥ 0 m\geq0 m ≥ 0 ( A ( t m + ) , Φ ( t m + ) ) (A(t_m+),\Phi(t_m+)) ( A ( t m + ) , Φ ( t m + )) C x − κ ( T 2 , g 2 × V ) \mathcal{C}^{-\kappa}_{x}(\mathbb{T}^2,\mathfrak{g}^2\times\mathbf{V}) C x − κ ( T 2 , g 2 × V ) J t m + 1 − t m \mathcal{J}_{t_{m+1}-t_m} J t m + 1 − t m (40) and is gauge-equivalent to ( A ( t m ) , Φ ( t m ) ) (A(t_m),\Phi(t_m)) ( A ( t m ) , Φ ( t m )) g m + 1 ∈ G 1 − κ g_{m+1}\in\mathcal{G}^{1-\kappa} g m + 1 ∈ G 1 − κ ( A ( t m + ) , Φ ( t m + ) ) = g m + 1 ∙ ( A ( t m ) , Φ ( t m ) ) (A(t_m+),\Phi(t_m+))=g_{m+1}\bullet(A(t_m),\Phi(t_m)) ( A ( t m + ) , Φ ( t m + )) = g m + 1 ∙ ( A ( t m ) , Φ ( t m ))
(iii) (Uniform-in-time estimate ) For all p ≥ 1 p\geq1 p ≥ 1
sup t ∈ [ 0 , ∞ ) E [ ∥ ( A ( t ) , Φ ( t ) ) ∥ i n v , − κ p ] ≤ C ( A 0 , Φ 0 , κ , p ) . \sup_{t\in[0,\infty)}\mathbb{E}\left[\|(A(t),\Phi(t))\|_{\mathrm{inv},-\kappa}^{p}\right]\leq C(A_0,\Phi_0,\kappa,p). t ∈ [ 0 , ∞ ) sup E [ ∥ ( A ( t ) , Φ ( t )) ∥ inv , − κ p ] ≤ C ( A 0 , Φ 0 , κ , p ) . Moreover, as t → ∞ t\to\infty t → ∞ ( A , Φ ) (A,\Phi) ( A , Φ ) ( A 0 , Φ 0 ) (A_0,\Phi_0) ( A 0 , Φ 0 )
lim sup t → ∞ E [ ∥ ( A ( t ) , Φ ( t ) ) ∥ i n v , − κ p ] ≤ C ( κ , p ) . \limsup_{t\to\infty}\mathbb{E}\left[\|(A(t),\Phi(t))\|_{\mathrm{inv},-\kappa}^{p}\right]\leq C(\kappa,p). t → ∞ lim sup E [ ∥ ( A ( t ) , Φ ( t )) ∥ inv , − κ p ] ≤ C ( κ , p ) . We emphasise that, as in Theorem 2.19 , we only use the C t 0 C x − κ \mathcal{C}^{0}_{t}\mathcal{C}^{-\kappa}_{x} C t 0 C x − κ C x − κ \mathcal{C}^{-\kappa}_{x} C x − κ (11) themselves are not solved globally in time. Instead, we only solve (11) on a short time interval, then apply a gauge transformation, and then solve (11) again on the next short time interval. The reason is that, as previously discussed below (21) , the Yang–Mills–Higgs gradient flow with a DeTurck term does not obey suitable long-time estimates. Since we are ultimately interested in the global dynamics of the Markov process on the space of gauge orbits, this use of gauge transformations is allowed.
Remark 2.25. By (i) and (ii), the process ( A , Φ ) (A,\Phi) ( A , Φ ) ( A , Φ ) (A,\Phi) ( A , Φ ) ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ ( t m , t m + 1 ] (t_m,t_{m+1}] ( t m , t m + 1 ] [ t m , t m + 1 ) [t_m,t_{m+1}) [ t m , t m + 1 ) ( A , Φ ) (A,\Phi) ( A , Φ )
As stated at the beginning of this subsection, the soft result from Theorem 2.24 will be the main ingredient needed for the existence of the invariant measure from Theorem 1.5 . For this reason, Theorem 2.24 can perhaps be regarded as the most important statement in the article. However, our soft result will not be proven on its own, but together with our hard result Theorem 2.30 . In the following, we first discuss the main definitions and ideas behind the global iteration scheme that leads to our hard estimates. The statement of our hard global well-posedness result will be included after this discussion.
To start off, we assume that our solution ( A , Φ ) (A,\Phi) ( A , Φ ) t m t_m t m m ≥ 0 m\geq0 m ≥ 0 ( A , Φ ) ( t m ) (A,\Phi)(t_m) ( A , Φ ) ( t m ) ( A + , Φ + ) ( t m ) (A^+,\Phi^+)(t_m) ( A + , Φ + ) ( t m ) ( A − , Φ − ) ( t m ) (A^-,\Phi^-)(t_m) ( A − , Φ − ) ( t m )
E ( A + ( t m ) , Φ + ( t m ) ) ≔ ∥ F A + ( t m ) ∥ L x 2 2 + ∥ Φ + ( t m ) ∥ L x 1 4 + ∥ Φ + ( t m ) ∥ L x 2 2 + 10 , (64) E(A^+(t_m),\Phi^+(t_m))\coloneqq\|F_{A^+}(t_m)\|_{L^2_x}^2+\|\Phi^+(t_m)\|_{L^1_x}^4+\|\Phi^+(t_m)\|_{L^2_x}^2+10, \tag*{(64)} E ( A + ( t m ) , Φ + ( t m )) : = ∥ F A + ( t m ) ∥ L x 2 2 + ∥ Φ + ( t m ) ∥ L x 1 4 + ∥ Φ + ( t m ) ∥ L x 2 2 + 10 , (64) where r ≫ 1 r\gg1 r ≫ 1 (96) and γ 1 > 4 > γ 2 = 3 \gamma_1>4>\gamma_2=3 γ 1 > 4 > γ 2 = 3 (99) . We emphasise that, since γ 1 > γ 2 \gamma_1>\gamma_2 γ 1 > γ 2 4 < r 4<r 4 < r L 4 L^4 L 4 L r L^r L r L 4 L^4 L 4 B ( Φ ⊗ D A Φ ) \mathbf{B}(\Phi\otimes\mathbf{D}_A\Phi) B ( Φ ⊗ D A Φ ) (11) , and that the L r L^r L r (64) as the monotone quantity. The reason is that it decreases under the deterministic dynamics10 τ m + 1 \tau_{m+1} τ m + 1 ( t m , t m + 1 ] (t_m,t_{m+1}] ( t m , t m + 1 ]
Dec m + 1 ≔ 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t , x 2 ( ( t m , t m + 1 ] ) 2 + C γ 2 , r 1 ∥ Φ + ( t m ) ∥ L x r ≥ 1 ∥ Φ + ( t m ) ∥ L x r γ 2 + 2 τ m + 1 + 1 ∥ Φ + ( t m ) ∥ L x 4 ≥ 1 ( γ 1 ∥ φ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 − 4 ∥ ∣ D B m + 1 + S m + 1 φ m + 1 + ∣ ∥ L t , x 2 ( ( t m , t m + 1 ] ) 2 ) ∨ C γ 1 , 4 ∥ Φ + ( t m ) ∥ L x 4 γ 1 + 2 τ m + 1 . (65) \operatorname{Dec}_{m+1} \coloneqq2\lVert D_{B_{m+1}^{+}}^{*}F_{B_{m+1}^{+}}\rVert_{L_{t,x}^{2}((t_m,t_{m+1}])}^{2}
+ C_{\gamma_{2},r}\mathbf{1}_{\lVert\Phi^{+}(t_m)\rVert_{L_x^r}\ge1}\lVert\Phi^{+}(t_m)\rVert_{L_x^r}^{\gamma_{2}+2}\tau_{m+1}
+ \mathbf{1}_{\lVert\Phi^{+}(t_m)\rVert_{L_x^4}\ge1}\left(\gamma_{1}\lVert\varphi_{m+1}^{+}(t_{m+1})\rVert_{L_x^4}^{\gamma_{1}-4}\lVert\lvert D_{B_{m+1}+S_{m+1}}\varphi_{m+1}^{+}\rvert\lVert_{L_{t,x}^{2}((t_m,t_{m+1}])}^{2}\right)
\vee C_{\gamma_{1},4}\lVert\Phi^{+}(t_m)\rVert_{L_x^4}^{\gamma_{1}+2}\tau_{m+1}. \tag*{(65)} Dec m + 1 : = 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t , x 2 (( t m , t m + 1 ]) 2 + C γ 2 , r 1 ∥ Φ + ( t m ) ∥ L x r ≥ 1 ∥ Φ + ( t m ) ∥ L x r γ 2 + 2 τ m + 1 + 1 ∥ Φ + ( t m ) ∥ L x 4 ≥ 1 ( γ 1 ∥ φ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 − 4 ∥∣ D B m + 1 + S m + 1 φ m + 1 + ∣ ∥ L t , x 2 (( t m , t m + 1 ]) 2 ) ∨ C γ 1 , 4 ∥ Φ + ( t m ) ∥ L x 4 γ 1 + 2 τ m + 1 . (65) Here, B m + 1 + B_{m+1}^{+} B m + 1 + A + ( t m ) A^{+}(t_m) A + ( t m ) [ t m , t m + 1 ] [t_m,t_{m+1}] [ t m , t m + 1 ] S m + 1 S_{m+1} S m + 1 (52) and φ m + 1 + \varphi_{m+1}^{+} φ m + 1 + (53) . Finally, C γ 1 , 4 , C γ 2 , r > 0 C_{\gamma_{1},4}, C_{\gamma_{2},r}>0 C γ 1 , 4 , C γ 2 , r > 0 (70) and § 6 . The size of the rough part ( A − , Φ − ) ( t m ) (A^{-},\Phi^{-})(t_m) ( A − , Φ − ) ( t m ) B m + 1 + B_{m+1}^{+} B m + 1 + A + ( t m ) A^{+}(t_m) A + ( t m )
∥ A − ( t m ) ∥ C A + + ( t m ) , F − ≲ 1 and ∥ Φ − ( t m ) ∥ C A + + ( t m ) − ≲ 1. (66) \lVert A^{-}(t_m)\rVert_{C_{A_{+}^{+}(t_m),F}^{-}} \lesssim1
\qquad\text{and}\qquad
\lVert\Phi^{-}(t_m)\rVert_{C_{A_{+}^{+}(t_m)}^{-}} \lesssim1. \tag*{(66)} ∥ A − ( t m ) ∥ C A + + ( t m ) , F − ≲ 1 and ∥ Φ − ( t m ) ∥ C A + + ( t m ) − ≲ 1. (66) For a more rigorous treatment of the rough part, we refer the reader to Definition 11.6 (as well as the rest of Section 11). After the above discussion of the sizes of the smooth and rough parts, we now turn to the choice of the time scale τ m + 1 \tau_{m+1} τ m + 1 t m + 1 ≔ t m + τ m + 1 t_{m+1}\coloneqq t_m+\tau_{m+1} t m + 1 : = t m + τ m + 1 τ m + 1 \tau_{m+1} τ m + 1 ( A + , Φ + ) ( t m ) (A^{+},\Phi^{+})(t_m) ( A + , Φ + ) ( t m )
Definition 2.26 (Time scales). Let A + : T 2 → g 2 A^{+}\colon\mathbb{T}^{2}\to\mathfrak{g}^{2} A + : T 2 → g 2 Φ + : T 2 → V \Phi^{+}\colon\mathbb{T}^{2}\to V Φ + : T 2 → V T 0 T_0 T 0 (98) below and let 0 < ε 0 ≤ 10 − 10 0<\varepsilon_0\le10^{-10} 0 < ε 0 ≤ 1 0 − 10 0 < δ s t e p ≤ T 0 0<\delta_{\mathrm{step}}\le T_0 0 < δ step ≤ T 0
T ( 1 ) ( A + , Φ + ) ≔ min { ε 0 ∥ F A + ∥ L x 2 − 1 log ( ∥ F A + ∥ L x 2 ∨ 2 ) , ∥ Φ + ∥ L x γ 0 − γ 0 , δ s t e p , ε 0 } . (67) T^{(1)}(A^{+},\Phi^{+})\coloneqq\min\left\{\varepsilon_0\lVert F_{A^{+}}\rVert_{L_x^2}^{-1}\log\left(\lVert F_{A^{+}}\rVert_{L_x^2}\vee2\right),\lVert\Phi^{+}\rVert_{L_x^{\gamma_0}}^{-\gamma_0},\delta_{\mathrm{step}},\varepsilon_0\right\}. \tag*{(67)} T ( 1 ) ( A + , Φ + ) : = min { ε 0 ∥ F A + ∥ L x 2 − 1 log ( ∥ F A + ∥ L x 2 ∨ 2 ) , ∥ Φ + ∥ L x γ 0 − γ 0 , δ step , ε 0 } . (67) and
T ( 2 ) ( A + , Φ + ) = T ( 2 ) ( A + ) ≔ inf { t > 0 : 2 ∥ D B + ∗ F B + ∥ L t , x 2 ( [ 0 , t ] ) 2 ≥ ε 0 t − 1 log ( t − 1 ) } . (68) T^{(2)}(A^{+},\Phi^{+})=T^{(2)}(A^{+})\coloneqq\inf\left\{t>0:2\lVert D_{B^{+}}^{*}F_{B^{+}}\rVert_{L_{t,x}^{2}([0,t])}^{2}\ge\varepsilon_0t^{-1}\log(t^{-1})\right\}. \tag*{(68)} T ( 2 ) ( A + , Φ + ) = T ( 2 ) ( A + ) : = inf { t > 0 : 2 ∥ D B + ∗ F B + ∥ L t , x 2 ([ 0 , t ]) 2 ≥ ε 0 t − 1 log ( t − 1 ) } . (68) Here, γ 0 > 2 \gamma_0>2 γ 0 > 2 (99) and B + B^{+} B + A + A^{+} A +
For the rest of this discussion, the reader may assume that the time scale τ m + 1 \tau_{m+1} τ m + 1
τ m + 1 = ( T ( 1 ) ∧ T ( 2 ) ) ( A + ( t m ) , Φ + ( t m ) ) . \tau_{m+1}=\left(T^{(1)}\wedge T^{(2)}\right)(A^{+}(t_m),\Phi^{+}(t_m)). τ m + 1 = ( T ( 1 ) ∧ T ( 2 ) ) ( A + ( t m ) , Φ + ( t m )) . In the full definition of our time scale τ m + 1 \tau_{m+1} τ m + 1 § 3.2 can be used on the time interval ( t m , t m + 1 ] (t_m,t_{m+1}] ( t m , t m + 1 ] Proposition 2.10 and Theorem 2.19 , the most important question for this is whether we can control ∥ F B m + 1 + ∥ L t 1 L x ∞ ( ( t m , t m + 1 ] ) \|F_{B_{m+1}^{+}}\|_{L_t^1L_x^\infty((t_m,t_{m+1}])} ∥ F B m + 1 + ∥ L t 1 L x ∞ (( t m , t m + 1 ]) B m + 1 + B_{m+1}^{+} B m + 1 + B m + 1 + ( t m ) = A + ( t m ) B_{m+1}^{+}(t_m)=A^{+}(t_m) B m + 1 + ( t m ) = A + ( t m ) Lemmas 3.28 , 3.33 , and 3.48 from the body of this article, one has that
∥ F B m + 1 + ∥ L t 1 L x ∞ ( ( t m , t m + 1 ] ) ≲ τ m + 1 1 / 2 ∥ F B m + 1 + ∥ L t 2 L x ∞ ( ( t m , t m + 1 ] ) ≲ τ m + 1 ∥ F B m + 1 + ( t m ) ∥ L x 2 + τ m + 1 1 / 2 ∣ log ( τ m + 1 ) ∣ 1 / 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 ( ( t m , t m + 1 ] ) ≲ ε 0 ∣ log ( τ m + 1 ) ∣ . (69) \begin{aligned}
\|F_{B_{m+1}^{+}}\|_{L_t^1L_x^\infty((t_m,t_{m+1}])}
&\lesssim\tau_{m+1}^{1/2}\|F_{B_{m+1}^{+}}\|_{L_t^2L_x^\infty((t_m,t_{m+1}])}\\
&\lesssim\tau_{m+1}\|F_{B_{m+1}^{+}}(t_m)\|_{L_x^2}+\tau_{m+1}^{1/2}|\log(\tau_{m+1})|^{1/2}\|D_{B_{m+1}^{+}}^{*}F_{B_{m+1}^{+}}\|_{L_t^2L_x^2((t_m,t_{m+1}])}\\
&\lesssim\sqrt{\varepsilon_0}|\log(\tau_{m+1})|. \tag*{(69)}
\end{aligned} ∥ F B m + 1 + ∥ L t 1 L x ∞ (( t m , t m + 1 ]) ≲ τ m + 1 1/2 ∥ F B m + 1 + ∥ L t 2 L x ∞ (( t m , t m + 1 ]) ≲ τ m + 1 ∥ F B m + 1 + ( t m ) ∥ L x 2 + τ m + 1 1/2 ∣ log ( τ m + 1 ) ∣ 1/2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 (( t m , t m + 1 ]) ≲ ε 0 ∣ log ( τ m + 1 ) ∣. (69) Remark 2.27. The log-factors in (67) and (68) will be needed in our global iteration scheme. The reason is that the decay mechanism near Yang–Mills connections (see § 1.2 ) relies on their instability, and only the log-factors allow us to venture slightly into the unstable regime.
Remark 2.28. Since the L t 1 L x ∞ L_t^1L_x^\infty L t 1 L x ∞ Definition 2.26 ). The reason for this is that it appears neither in the monotone quantity (64) nor in its decrease (65) , and including it in the time scales would therefore not fit nicely into our global iteration scheme. In particular, the fact that T ( 2 ) T^{(2)} T ( 2 ) D B + ∗ F B + D_{B^{+}}^{*}F_{B^{+}} D B + ∗ F B + (74) and (75) .
Assuming now that Proposition 2.10 and Theorem 2.19 (as well as their more detailed versions from Propositions 5.7 and 8.3 ) can be applied11 [ t m , t m + 1 ] [t_m,t_{m+1}] [ t m , t m + 1 ] ( A + , A − , Φ + , Φ − ) ( t m ) (A^{+},A^{-},\Phi^{+},\Phi^{-})(t_m) ( A + , A − , Φ + , Φ − ) ( t m )
g m + 1 , A m + 1 , Φ m + 1 , B m + 1 , S m + 1 , φ m + 1 , B ∘ m + 1 , B ∙ m + 1 , R m + 1 , and r m + 1 , (70) g_{m+1},\ A_{m+1},\ \Phi_{m+1},\ B_{m+1},\ S_{m+1},\ \varphi_{m+1},\ \overset{\circ}{B}_{m+1},\ \overset{\bullet}{B}_{m+1},\ R_{m+1},\ \text{and}\ r_{m+1}, \tag*{(70)} g m + 1 , A m + 1 , Φ m + 1 , B m + 1 , S m + 1 , φ m + 1 , B ∘ m + 1 , B ∙ m + 1 , R m + 1 , and r m + 1 , (70) which are all defined on [ t m , t m + 1 ] × T 2 [t_m,t_{m+1}]\times\mathbb{T}^2 [ t m , t m + 1 ] × T 2
(b) ( A m + 1 , Φ m + 1 ) (A_{m+1},\Phi_{m+1}) ( A m + 1 , Φ m + 1 ) (11) with initial data g m + 1 ( t m ) ∙ ( A , Φ ) ( t m ) g_{m+1}(t_m)\mathbin{\bullet}(A,\Phi)(t_m) g m + 1 ( t m ) ∙ ( A , Φ ) ( t m ) Theorem 2.19 . It can be decomposed into A m + 1 = B m + 1 + S m + 1 + B ∘ m + 1 + R m + 1 A_{m+1}=B_{m+1}+S_{m+1}+\overset{\circ}{B}_{m+1}+R_{m+1} A m + 1 = B m + 1 + S m + 1 + B ∘ m + 1 + R m + 1 Φ m + 1 = φ m + 1 + B ∙ m + 1 + r m + 1 \Phi_{m+1}=\varphi_{m+1}+\overset{\bullet}{B}_{m+1}+r_{m+1} Φ m + 1 = φ m + 1 + B ∙ m + 1 + r m + 1
(c) B m + 1 B_{m+1} B m + 1 g m + 1 ( t m ) ∙ A ( t m ) g_{m+1}(t_m)\mathbin{\bullet}A(t_m) g m + 1 ( t m ) ∙ A ( t m ) Proposition 2.10 . Furthermore, it can be written as B m + 1 = g m + 1 ∙ ( B m + 1 + + B m + 1 − ) B_{m+1}=g_{m+1}\mathbin{\bullet}(B_{m+1}^{+}+B_{m+1}^{-}) B m + 1 = g m + 1 ∙ ( B m + 1 + + B m + 1 − ) B m + 1 + B_{m+1}^{+} B m + 1 + A + ( t m ) A^{+}(t_m) A + ( t m ) B m + 1 − B_{m+1}^{-} B m + 1 −
(d) ( S m + 1 , φ m + 1 ) (S_{m+1},\varphi_{m+1}) ( S m + 1 , φ m + 1 ) (52) –(53) with initial data ( 0 , g m + 1 ( t m ) ∙ Φ ( t m ) ) (0,g_{m+1}(t_m)\mathbin{\bullet}\Phi(t_m)) ( 0 , g m + 1 ( t m ) ∙ Φ ( t m )) φ m + 1 = φ m + 1 + + φ m + 1 − \varphi_{m+1}=\varphi_{m+1}^{+}+\varphi_{m+1}^{-} φ m + 1 = φ m + 1 + + φ m + 1 − φ m + 1 + \varphi_{m+1}^{+} φ m + 1 + (53) with initial data g m + 1 ( t m ) ∙ Φ + ( t m ) g_{m+1}(t_m)\mathbin{\bullet}\Phi^{+}(t_m) g m + 1 ( t m ) ∙ Φ + ( t m ) S S S S m + 1 S_{m+1} S m + 1 φ m + 1 − \varphi_{m+1}^{-} φ m + 1 −
The properties in (a)–(f) provide us with detailed information on the structure of our solution. Equipped with this information, we can now decide on the definition of the smooth and rough parts ( A + , Φ + ) (A^+,\Phi^+) ( A + , Φ + ) ( A − , Φ − ) (A^-,\Phi^-) ( A − , Φ − ) (89) below, is to define
A + ( t ) ≔ ( B m + 1 + S m + 1 ) ( t ) , A − ( t ) ≔ ( B ˚ m + 1 + R m + 1 ) ( t ) , Φ + ( t ) ≔ φ m + 1 ( t ) , Φ − ( t ) ≔ ( B ˚ m + 1 + r m + 1 ) ( t ) (71) \begin{aligned}
A^+(t)&\coloneqq(B_{m+1}+S_{m+1})(t), & A^-(t)&\coloneqq(\mathring{B}_{m+1}+R_{m+1})(t),\\
\Phi^+(t)&\coloneqq\varphi_{m+1}(t), & \Phi^-(t)&\coloneqq(\mathring{B}_{m+1}+r_{m+1})(t) \tag*{(71)}
\end{aligned} A + ( t ) Φ + ( t ) : = ( B m + 1 + S m + 1 ) ( t ) , : = φ m + 1 ( t ) , A − ( t ) Φ − ( t ) : = ( B ˚ m + 1 + R m + 1 ) ( t ) , : = ( B ˚ m + 1 + r m + 1 ) ( t ) (71) for all t ∈ ( t m , t m + 1 ] t\in(t_m,t_{m+1}] t ∈ ( t m , t m + 1 ] (70) and (71) is that many of the terms are essentially conditionally independent. To illustrate this in one example, note that the initial data A ( t m ) A(t_m) A ( t m ) F t m \mathcal{F}_{t_m} F t m σ \sigma σ ξ \xi ξ ζ \zeta ζ t m t_m t m B m + 1 B_{m+1} B m + 1 g m + 1 ( t m ) ⋅ A ( t m ) g_{m+1}(t_m)\mathbin{\cdot} A(t_m) g m + 1 ( t m ) ⋅ A ( t m ) B m + 1 B_{m+1} B m + 1 12 ^{12} 12 F t m \mathcal{F}_{t_m} F t m ζ ( ⋅ + t m ) ∣ [ 0 , ∞ ) \zeta(\mathord{\cdot}+t_m)|_{[0,\infty)} ζ ( ⋅ + t m ) ∣ [ 0 , ∞ ) F t m \mathcal{F}_{t_m} F t m B m + 1 B_{m+1} B m + 1 B ˚ m + 1 \mathring{B}_{m+1} B ˚ m + 1
Using the estimates from our covariant local theory, it is relatively easy to control the rough part ( A − , Φ − ) ( t m + 1 ) (A^-,\Phi^-)(t_{m+1}) ( A − , Φ − ) ( t m + 1 )
E ( A + ( t m + 1 ) , Φ + ( t m + 1 ) ) = E ( ( B m + 1 + S m + 1 ) ( t m + 1 ) , φ m + 1 ( t m + 1 ) ) , (72) \mathsf{E}(A^+(t_{m+1}),\Phi^+(t_{m+1}))=\mathsf{E}((B_{m+1}+S_{m+1})(t_{m+1}),\varphi_{m+1}(t_{m+1})), \tag*{(72)} E ( A + ( t m + 1 ) , Φ + ( t m + 1 )) = E (( B m + 1 + S m + 1 ) ( t m + 1 ) , φ m + 1 ( t m + 1 )) , (72) and to hopefully see it decrease from its earlier value E ( A + ( t m ) , Φ + ( t m ) ) \mathsf{E}(A^+(t_m),\Phi^+(t_m)) E ( A + ( t m ) , Φ + ( t m )) ( B m + 1 , φ m + 1 ) (B_{m+1},\varphi_{m+1}) ( B m + 1 , φ m + 1 ) ( A + , Φ + ) ( t m ) (A^+,\Phi^+)(t_m) ( A + , Φ + ) ( t m ) ( A − , Φ − ) ( t m ) (A^-,\Phi^-)(t_m) ( A − , Φ − ) ( t m ) ( A + , Φ + ) ( t m + 1 ) (A^+,\Phi^+)(t_{m+1}) ( A + , Φ + ) ( t m + 1 ) ( A − , Φ − ) ( t m ) (A^-,\Phi^-)(t_m) ( A − , Φ − ) ( t m ) Theorem 2.4 , each term in our solution can only be treated perturbatively for one step of our iteration (or perhaps a bounded number of steps). But all of our non-perturbative arguments can only be used for the smooth part, and hence the rough part has to enter into the smooth part after one step. In order to estimate (72) , it is insufficient to rely only on the covariant Hölder norm of ( A − , Φ − ) ( t m ) (A^-,\Phi^-)(t_m) ( A − , Φ − ) ( t m ) (71) also holds at the earlier step in our iteration, this structure is given by ( A − , Φ − ) ( t m ) = ( B ˚ m , B ˚ m ) ( t m ) + ( R m , r m ) ( t m ) (A^-,\Phi^-)(t_m)=(\mathring{B}_m,\mathring{B}_m)(t_m)+(R_m,r_m)(t_m) ( A − , Φ − ) ( t m ) = ( B ˚ m , B ˚ m ) ( t m ) + ( R m , r m ) ( t m ) (72) . The curvature term can be expanded13 ^{13} 13
∥ F B m + 1 + S m + 1 ( t m + 1 ) ∥ L x 2 2 ≈ ∥ F B m + 1 − ( t m + 1 ) ∥ L x 2 2 (73) \left\|F_{B_{m+1}+S_{m+1}}(t_{m+1})\right\|_{L_x^2}^2
\approx\left\|F_{B_{m+1}^-}(t_{m+1})\right\|_{L_x^2}^2 \tag*{(73)} ∥ F B m + 1 + S m + 1 ( t m + 1 ) ∥ L x 2 2 ≈ F B m + 1 − ( t m + 1 ) L x 2 2 (73) + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , curl B m + 1 + ( t m + 1 ) ( Ad g m + 1 − 1 S m + 1 ) ( t m + 1 ) ⟩ g d x (74) +2\int_{\mathbb{T}^2}\left\langle F_{B_{m+1}^+}^{12}(t_{m+1}),\operatorname{curl}B_{m+1}^+(t_{m+1})(\operatorname{Ad}_{g_{m+1}^{-1}}S_{m+1})(t_{m+1})\right\rangle_{\mathfrak{g}}\,\mathrm{d}x \tag*{(74)} + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , curl B m + 1 + ( t m + 1 ) ( Ad g m + 1 − 1 S m + 1 ) ( t m + 1 ) ⟩ g d x (74) + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ⟩ g d x (75) +2\int_{\mathbb{T}^2}\left\langle F_{B_{m+1}^+}^{12}(t_{m+1}),\operatorname{curl}B_{m+1}^+(t_{m+1})e^{\tau_{m+1}\Delta_{B_{m+1}^+}^{F}(t_m)}\mathring{B}_m(t_m)\right\rangle_{\mathfrak{g}}\,\mathrm{d}x \tag*{(75)} + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ⟩ g d x (75) + ∥ curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ∥ L x 2 2 (76) +\left\|\operatorname{curl}B_{m+1}^+(t_{m+1})e^{\tau_{m+1}\Delta_{B_{m+1}^+}^{F}(t_m)}\mathring{B}_m(t_m)\right\|_{L_x^2}^2 \tag*{(76)} + curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) L x 2 2 (76) + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , [ ( e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ) 1 , ( e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ) 2 ] ⟩ g d x . (77) +2\int_{\mathbb{T}^2}\left\langle F_{B_{m+1}^+}^{12}(t_{m+1}),\left[\left(e^{\tau_{m+1}\Delta_{B_{m+1}^+}^{F}(t_m)}\mathring{B}_m(t_m)\right)^1,\left(e^{\tau_{m+1}\Delta_{B_{m+1}^+}^{F}(t_m)}\mathring{B}_m(t_m)\right)^2\right]\right\rangle_{\mathfrak{g}}\,\mathrm{d}x. \tag*{(77)} + 2 ∫ T 2 ⟨ F B m + 1 + 12 ( t m + 1 ) , [ ( e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ) 1 , ( e τ m + 1 Δ B m + 1 + F ( t m ) B ˚ m ( t m ) ) 2 ] ⟩ g d x . (77) Here, ≈ \approx ≈ e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) e^{\tau_{m+1}\Delta_{B_{m+1}^{+}}^{F_{m+1}}(t_m)} e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) L x 4 L_x^4 L x 4
∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 ≈ ∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 (78) \left\lVert\phi_{m+1}^{+}(t_{m+1})\right\rVert_{L_x^4}^{\gamma_1}
\approx\left\lVert\phi_{m+1}^{+}(t_{m+1})\right\rVert_{L_x^4}^{\gamma_1} \tag*{(78)} ϕ m + 1 + ( t m + 1 ) L x 4 γ 1 ≈ ϕ m + 1 + ( t m + 1 ) L x 4 γ 1 (78) + γ 1 ∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 − 4 ( ∫ T 2 ∣ ϕ m + 1 + ( t m + 1 ) ∣ 2 ⟨ ϕ m + 1 + ( t m + 1 ) , ϕ l i n , m + 1 − ( t m + 1 ) ⟩ V d x ) . (79) +\gamma_1\left\lVert\phi_{m+1}^{+}(t_{m+1})\right\rVert_{L_x^4}^{\gamma_1-4}\left(\int_{\mathbb{T}^2}\left|\phi_{m+1}^{+}(t_{m+1})\right|^2\left\langle\phi_{m+1}^{+}(t_{m+1}),\phi_{\mathrm{lin},m+1}^{-}(t_{m+1})\right\rangle_V\,\mathrm{d}x\right). \tag*{(79)} + γ 1 ϕ m + 1 + ( t m + 1 ) L x 4 γ 1 − 4 ( ∫ T 2 ϕ m + 1 + ( t m + 1 ) 2 ⟨ ϕ m + 1 + ( t m + 1 ) , ϕ lin , m + 1 − ( t m + 1 ) ⟩ V d x ) . (79) Here, ϕ l i n , m + 1 − ( t m + 1 ) \phi_{\mathrm{lin},m+1}^{-}(t_{m+1}) ϕ lin , m + 1 − ( t m + 1 ) g m + 1 ( t m + 1 ) ∙ ( e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) ∙ B m ( t m ) ) g_{m+1}(t_{m+1})\mathbin{\bullet}\left(e^{\tau_{m+1}\Delta_{B_{m+1}^{+}}^{F_{m+1}}(t_m)}\mathbin{\bullet}B_m(t_m)\right) g m + 1 ( t m + 1 ) ∙ ( e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) ∙ B m ( t m ) ) e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) e^{\tau_{m+1}\Delta_{B_{m+1}^{+}}^{F_{m+1}}(t_m)} e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) B m ( t m ) B_m(t_m) B m ( t m ) L x r L_x^r L x r γ 1 \gamma_1 γ 1 γ 2 \gamma_2 γ 2 L x r L_x^r L x r
In the following, we discuss the main ideas behind the estimates of (73) –(79) . We first focus on estimates that are sufficient for global well-posedness, but not for the uniform-in-time estimates. The necessary ingredients for the latter will only be discussed at the end of this subsection. We start with the curl-term from (76) , since it will serve as a benchmark for all remaining terms. To determine the expected size of (76) , let us briefly turn back to the non-covariant setting. For the non-covariant linear stochastic object, a direct calculation (in either frequency space or physical space) yields that E [ ∥ ∇ e τ Δ O ( t ) ∥ L 2 2 ] ≲ τ − 1 \mathbb{E}\left[\left\lVert\nabla e^{\tau\Delta}\mathcal{O}(t)\right\rVert_{L^2}^2\right]\lesssim\tau^{-1} E [ ∇ e τ Δ O ( t ) L 2 2 ] ≲ τ − 1 t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] τ ∈ ( 0 , 1 ] \tau\in(0,1] τ ∈ ( 0 , 1 ] L ∞ L^\infty L ∞ (71) suggest that this second-moment estimate may not only hold in the non-covariant setting, but also in the covariant setting. The best estimate one can hope for is therefore
E [ ∥ curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) ∙ B m ( t m ) ∥ L x 2 2 ] ≲ E [ τ m + 1 − 1 ] . (80) \mathbb{E}\left[\left\lVert\operatorname{curl}B_{m+1}^{+}(t_{m+1})e^{\tau_{m+1}\Delta_{B_{m+1}^{+}}^{F_{m+1}}(t_m)}\mathbin{\bullet}B_m(t_m)\right\rVert_{L_x^2}^2\right]\lesssim\mathbb{E}\left[\tau_{m+1}^{-1}\right]. \tag*{(80)} E [ curl B m + 1 + ( t m + 1 ) e τ m + 1 Δ B m + 1 + F m + 1 ( t m ) ∙ B m ( t m ) L x 2 2 ] ≲ E [ τ m + 1 − 1 ] . (80) The estimate (80) turns out to be true, and it is a consequence of Lemma 10.5 below. However, its proof is more difficult than one may expect. The reason is that we cannot directly use our covariant heat kernel estimates of q B m q_{B_m} q B m q B m + 1 + q_{B_{m+1}^{+}} q B m + 1 + (69) and the exp-factor in Theorem 2.4, this would only yield an upper bound of the form E [ τ m + 1 − ( 1 + C ε 0 ) ] \mathbb{E}[\tau_{m+1}^{-(1+C\sqrt{\varepsilon_0})}] E [ τ m + 1 − ( 1 + C ε 0 ) ] (80) , we need to rely on the fact that (80) only involves the covariant curl, and not a general covariant derivative. As will be shown in Lemma 3.32 below, the geometric evolution equation for the covariant curl primarily involves the covariant Laplacian and the covariant derivatives of the curvature, but not the curvature term itself as in (54) . Together with the control provided by the decrease (65) , this allows us to treat the covariant derivatives of the curvature perturbatively and replace the pointwise estimates for q B m q_{B_m} q B m q B m + 1 + q_{B_{m+1}^{+}} q B m + 1 + p B m p_{B_m} p B m p B m + 1 + p_{B_{m+1}^{+}} p B m + 1 + (80) .
We now discuss why the exponent − 1 -1 − 1 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 (80) is significant for our global iteration scheme. The discussion of this exponent will be entirely heuristic, and making this discussion more precise will occupy much of Section 11. First, we assume14 τ m + 1 \tau_{m+1} τ m + 1 (67) , i.e., that
τ m + 1 = ε 0 ∥ F A + ( t m ) ∥ L x 2 − 1 log ( ∥ F A + ( t m ) ∥ L x 2 ∨ 2 ) \tau_{m+1}=\varepsilon_0\left\lVert F_{A_+}(t_m)\right\rVert_{L_x^2}^{-1}\log\left(\left\lVert F_{A_+}(t_m)\right\rVert_{L_x^2}\vee2\right) τ m + 1 = ε 0 ∥ F A + ( t m ) ∥ L x 2 − 1 log ( ∥ F A + ( t m ) ∥ L x 2 ∨ 2 ) for all m ≥ 0 m\ge0 m ≥ 0 (73) by the squared L x 2 L_x^2 L x 2 F B m + 1 + ( t m ) F_{B_{m+1}^{+}}(t_m) F B m + 1 + ( t m ) (74) , (75) , and (77) , and omitting all expectations, one obtains an estimate of the form
∥ F A + ( t m + 1 ) ∥ L x 2 2 ≤ ∥ F A + ( t m ) ∥ L x 2 2 + C τ m + 1 − 1 ≤ ∥ F A + ( t m ) ∥ L x 2 2 + C ε 0 − 1 ∥ F A + ( t m ) ∥ L x 2 2 log ( ∥ F A + ( t m ) ∥ L x 2 ∨ 2 ) − 1 . (81) \begin{aligned}
\lVert F_{A_+}(t_{m+1})\rVert_{L_x^2}^2 &\leq\lVert F_{A_+}(t_m)\rVert_{L_x^2}^2+C\tau_{m+1}^{-1}\\
&\leq\lVert F_{A_+}(t_m)\rVert_{L_x^2}^2+C\varepsilon_0^{-1}\lVert F_{A_+}(t_m)\rVert_{L_x^2}^2\log\bigl(\lVert F_{A_+}(t_m)\rVert_{L_x^2}\vee2\bigr)^{-1}. \tag*{(81)}
\end{aligned} ∥ F A + ( t m + 1 ) ∥ L x 2 2 ≤ ∥ F A + ( t m ) ∥ L x 2 2 + C τ m + 1 − 1 ≤ ∥ F A + ( t m ) ∥ L x 2 2 + C ε 0 − 1 ∥ F A + ( t m ) ∥ L x 2 2 log ( ∥ F A + ( t m ) ∥ L x 2 ∨ 2 ) − 1 . (81) By iterating (81) until the time t = 1 t=1 t = 1 15
∥ F A + ( 1 ) ∥ L x 2 2 ≤ ∥ F A + ( 0 ) ∥ L x 2 2 + C ( ε 0 − 1 ∥ F A + ( 0 ) ∥ L x 2 log ( ∥ F A + ( 0 ) ∥ L x 2 ∨ 2 ) − 1 ) 2 . (82) \lVert F_{A_+}(1)\rVert_{L_x^2}^2\leq\lVert F_{A_+}(0)\rVert_{L_x^2}^2+C\left(\varepsilon_0^{-1}\lVert F_{A_+}(0)\rVert_{L_x^2}\log\bigl(\lVert F_{A_+}(0)\rVert_{L_x^2}\vee2\bigr)^{-1}\right)^2. \tag*{(82)} ∥ F A + ( 1 ) ∥ L x 2 2 ≤ ∥ F A + ( 0 ) ∥ L x 2 2 + C ( ε 0 − 1 ∥ F A + ( 0 ) ∥ L x 2 log ( ∥ F A + ( 0 ) ∥ L x 2 ∨ 2 ) − 1 ) 2 . (82) The estimate (82) is sufficient for global well-posedness and, upon further iteration, even yields a growth rate of the form exp ( C t 1 / 3 ) \exp(Ct^{1/3}) exp ( C t 1/3 ) ∥ F A + ( 0 ) ∥ L x 2 \lVert F_{A_+}(0)\rVert_{L_x^2} ∥ F A + ( 0 ) ∥ L x 2 (82) would not allow us to rule out finite-time blowup. As a result, we cannot allow for a worse exponent on the right-hand side of (80) , and thus τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 (76) , we now turn to a brief discussion of the remaining terms in (73) –(79) . In this discussion, we only try to convey the main ideas, but will often neglect various technical difficulties.
Estimate of (73) : In order to control the L 2 L^2 L 2 F B m + 1 + ( t m + 1 ) F_{B^+_{m+1}}(t_{m+1}) F B m + 1 + ( t m + 1 )
∥ F B m + 1 + ( t m + 1 ) ∥ L x 2 2 = ∥ F B m + 1 + ( t m ) ∥ L x 2 2 − 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 ( [ t m , t m + 1 ] ) 2 . (83) \lVert F_{B^+_{m+1}}(t_{m+1})\rVert_{L_x^2}^2=\lVert F_{B^+_{m+1}}(t_m)\rVert_{L_x^2}^2-2\lVert D^*_{B^+_{m+1}}F_{B^+_{m+1}}\rVert_{L_t^2L_x^2([t_m,t_{m+1}])}^2. \tag*{(83)} ∥ F B m + 1 + ( t m + 1 ) ∥ L x 2 2 = ∥ F B m + 1 + ( t m ) ∥ L x 2 2 − 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 ([ t m , t m + 1 ]) 2 . (83) The first summand in (83) appears in our monotone quantity (64) at the time t = t m t=t_m t = t m (83) appears in our decrease (65) , and it will be used to absorb several different contributions.
Estimate of (74) : By using integration by parts and the Ad-invariance of the inner product on g \mathfrak{g} g
∣ ( 2.46 ) ∣ ≲ ∥ D B m + 1 + ∗ F B m + 1 + ( t m + 1 ) ∥ L x 2 ∥ S m + 1 ( t m + 1 ) ∥ L x 2 . \lvert(2.46)\rvert\lesssim\lVert D^*_{B^+_{m+1}}F_{B^+_{m+1}}(t_{m+1})\rVert_{L_x^2}\lVert S_{m+1}(t_{m+1})\rVert_{L_x^2}. ∣( 2.46 )∣ ≲ ∥ D B m + 1 + ∗ F B m + 1 + ( t m + 1 ) ∥ L x 2 ∥ S m + 1 ( t m + 1 ) ∥ L x 2 . The first factor can essentially be estimated by τ m + 1 − 1 / 2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 ( [ t m , t m + 1 ] ) \tau_{m+1}^{-1/2}\lVert D^*_{B^+_{m+1}}F_{B^+_{m+1}}\rVert_{L_t^2L_x^2([t_m,t_{m+1}])} τ m + 1 − 1/2 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L t 2 L x 2 ([ t m , t m + 1 ]) (65) . In the equation for S m + 1 S_{m+1} S m + 1 (52) ), the largest term is B ( φ m + 1 + ⊗ D B m + 1 + S m + 1 φ m + 1 + ) \mathbf{B}(\varphi^+_{m+1}\otimes D_{B_{m+1}+S_{m+1}}\varphi^+_{m+1}) B ( φ m + 1 + ⊗ D B m + 1 + S m + 1 φ m + 1 + ) L 2 L^2 L 2 S m + 1 ( t m + 1 ) S_{m+1}(t_{m+1}) S m + 1 ( t m + 1 )
τ m + 1 1 / 2 ∥ B ( φ m + 1 + ⊗ D B m + 1 + S m + 1 φ m + 1 + ) ∥ L t 2 L x 2 ( [ t m , t m + 1 ] ) , \tau_{m+1}^{1/2}\lVert\mathbf{B}(\varphi^+_{m+1}\otimes D_{B_{m+1}+S_{m+1}}\varphi^+_{m+1})\rVert_{L_t^2L_x^2([t_m,t_{m+1}])}, τ m + 1 1/2 ∥ B ( φ m + 1 + ⊗ D B m + 1 + S m + 1 φ m + 1 + ) ∥ L t 2 L x 2 ([ t m , t m + 1 ]) , which involves the decrease of the L 4 L^4 L 4 (65) . Provided16 γ 1 > 4 \gamma_1>4 γ 1 > 4 (74) in the decrease.
Estimate of (75) : As in our estimate of (74) , we first use integration by parts. Then, we use that B m + 1 + ( t m + 1 ) B^+_{m+1}(t_{m+1}) B m + 1 + ( t m + 1 ) ∘ B m ( t m ) {}^{\circ}B_m(t_m) ∘ B m ( t m ) 17 (75) using moment and covariant heat kernel estimates, as well as our previous estimates of D B m + 1 + ∗ F B m + 1 + ( t m + 1 ) D^*_{B^+_{m+1}}F_{B^+_{m+1}}(t_{m+1}) D B m + 1 + ∗ F B m + 1 + ( t m + 1 )
Estimate of (77) : We now turn to the interaction involving F B m + 1 + ( t m + 1 ) F_{B^+_{m+1}}(t_{m+1}) F B m + 1 + ( t m + 1 ) ∘ B m ( t m ) {}^{\circ}B_m(t_m) ∘ B m ( t m ) (73) –(79) . A naïve estimate yields
∣ ( 2.49 ) ∣ ≲ ∥ F B m + 1 + ( t m + 1 ) ∥ L x 2 ∥ e τ m + 1 Δ B m + 1 + ( t m + 1 ) ∘ B m ( t m ) ∥ L x 4 2 . \lvert(2.49)\rvert\lesssim\lVert F_{B^+_{m+1}}(t_{m+1})\rVert_{L_x^2}\lVert e^{\tau_{m+1}\Delta_{B^+_{m+1}(t_{m+1})}}{}^{\circ}B_m(t_m)\rVert_{L_x^4}^2. ∣( 2.49 )∣ ≲ ∥ F B m + 1 + ( t m + 1 ) ∥ L x 2 ∥ e τ m + 1 Δ B m + 1 + ( t m + 1 ) ∘ B m ( t m ) ∥ L x 4 2 . By using the definition of T ( 1 ) T^{(1)} T ( 1 ) (67) , the first factor can be estimated by τ m + 1 − 1 ∣ log ( τ m + 1 ) ∣ \tau_{m+1}^{-1}|\log(\tau_{m+1})| τ m + 1 − 1 ∣ log ( τ m + 1 ) ∣ B ˚ m ( t m ) \mathring{B}_m(t_m) B ˚ m ( t m ) τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 (77) directly, we rely on the geometric identity
2 ∫ T 2 ⟨ F B m + 1 12 ( t m + 1 ) , [ X 1 , X 2 ] ⟩ g d x = ∥ curl B m + 1 + ( t m + 1 ) X ∥ L x 2 2 + ∥ div B m + 1 + ( t m + 1 ) X ∥ L x 2 2 − ∥ D B m + 1 + ( t m + 1 ) X ∥ L x 2 2 (84) 2\int_{\mathbb{T}^2}\langle F_{B_{m+1}^{12}}(t_{m+1}),[X_1,X_2]\rangle_g\,\mathrm{d}x
= \lVert\operatorname{curl}_{B_{m+1}^{+}(t_{m+1})}X\rVert_{L_x^2}^2+\lVert\operatorname{div}_{B_{m+1}^{+}(t_{m+1})}X\rVert_{L_x^2}^2-\lVert D_{B_{m+1}^{+}(t_{m+1})}X\rVert_{L_x^2}^2 \tag*{(84)} 2 ∫ T 2 ⟨ F B m + 1 12 ( t m + 1 ) , [ X 1 , X 2 ] ⟩ g d x = ∥ curl B m + 1 + ( t m + 1 ) X ∥ L x 2 2 + ∥ div B m + 1 + ( t m + 1 ) X ∥ L x 2 2 − ∥ D B m + 1 + ( t m + 1 ) X ∥ L x 2 2 (84) with X = e τ m + 1 Δ B m + 1 + ( t m ) ϕ B m ( t m ) X=e^{\tau_{m+1}\Delta_{B_{m+1}^{+}(t_m)}}\boldsymbol{\phi}_{B_m}(t_m) X = e τ m + 1 Δ B m + 1 + ( t m ) ϕ B m ( t m ) Lemma 3.24 below. The curl-term has already been estimated in (80) . In two dimensions, the curl and divergence of the covariant stochastic object have the same law (see Lemma 9.17), and hence the div-term satisfies the same estimate as the curl-term. Finally, due to the negative sign, the last summand in (84) can be bounded from above by zero. All in all, we obtain an upper bound of the expectation of (77) by τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1
Since this will be important in our proof of uniform-in-time estimates, which will be discussed momentarily, we emphasise that the expectation of (77) cannot be bounded from below by − τ m + 1 − 1 -\tau_{m+1}^{-1} − τ m + 1 − 1 (77) can be rather negative.
Estimate of (78) : For any p ≥ 2 p\ge2 p ≥ 2 L p L^p L p ϕ m + 1 + \phi^+_{m+1} ϕ m + 1 + Proposition 4.4 ). The covariant Laplacian in (53) leads to smoothing, and the cubic nonlinearity in (53) leads to decay. Altogether, we obtain
∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 ≤ ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − c ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 + 2 τ m + 1 − c ∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 − 4 ∥ ∣ ϕ m + 1 + ∣ ∥ D B m + 1 + S m + 1 ϕ m + 1 + ∥ ∥ L t , x 2 ( [ t m , t m + 1 ] ) 2 . \begin{aligned}
\lVert\phi^+_{m+1}(t_{m+1})\rVert_{L_x^4}^{\gamma_1}
&\le\lVert\phi^+_{m+1}(t_m)\rVert_{L_x^4}^{\gamma_1}
-c\lVert\phi^+_{m+1}(t_m)\rVert_{L_x^4}^{\gamma_1+2}\tau_{m+1}\\
&\quad-c\lVert\phi^+_{m+1}(t_{m+1})\rVert_{L_x^4}^{\gamma_1-4}
\big\lVert\lvert\phi^+_{m+1}\rvert\lVert D_{B_{m+1}+S_{m+1}}\phi^+_{m+1}\rVert\big\rVert_{L^2_{t,x}([t_m,t_{m+1}])}^{2}.
\end{aligned} ∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 ≤ ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − c ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 + 2 τ m + 1 − c ∥ ϕ m + 1 + ( t m + 1 ) ∥ L x 4 γ 1 − 4 ∣ ϕ m + 1 + ∣ ∥ D B m + 1 + S m + 1 ϕ m + 1 + ∥ L t , x 2 ([ t m , t m + 1 ]) 2 . We emphasise that the ∣ ϕ m + 1 + ∣ ∥ D B m + 1 + S m + 1 ϕ m + 1 + ∥ \lvert\phi^+_{m+1}\rvert\lVert D_{B_{m+1}+S_{m+1}}\phi^+_{m+1}\rVert ∣ ϕ m + 1 + ∣ ∥ D B m + 1 + S m + 1 ϕ m + 1 + ∥ ∥ ⋅ ∥ L 4 γ 1 \lVert\cdot\rVert_{L^4}^{\gamma_1} ∥ ⋅ ∥ L 4 γ 1 (64) , and this term was already used in our estimate of (74) .
Estimate of (79) : Recall from below (79) that ϕ l i n , m + 1 − \phi^-_{\mathrm{lin},m+1} ϕ lin , m + 1 − ϕ B m ( t m ) \boldsymbol{\phi}_{B_m}(t_m) ϕ B m ( t m ) ϕ B m ( t m ) \boldsymbol{\phi}_{B_m}(t_m) ϕ B m ( t m ) 18 ϕ m + 1 + ( t m + 1 ) \phi^+_{m+1}(t_{m+1}) ϕ m + 1 + ( t m + 1 ) (79) using moment and covariant heat kernel estimates. The first moment of this contribution is then bounded, up to the preceding approximations, by
∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − 4 × τ m 1 / 2 ∥ ( ∣ ϕ m + 1 + ∣ 2 ϕ m + 1 + ) ( t m + 1 ) ∥ L x 2 ≲ τ m 1 / 2 ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − 4 ∥ ϕ m + 1 + ( t m ) ∥ L x 6 3 . (85) \lVert\phi^+_{m+1}(t_m)\rVert_{L_x^4}^{\gamma_1-4}\times\tau_m^{1/2}\big\lVert(\lvert\phi^+_{m+1}\rvert^2\phi^+_{m+1})(t_{m+1})\big\rVert_{L_x^2}\lesssim\tau_m^{1/2}\lVert\phi^+_{m+1}(t_m)\rVert_{L_x^4}^{\gamma_1-4}\lVert\phi^+_{m+1}(t_m)\rVert_{L_x^6}^{3}. \tag*{(85)} ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − 4 × τ m 1/2 (∣ ϕ m + 1 + ∣ 2 ϕ m + 1 + ) ( t m + 1 ) L x 2 ≲ τ m 1/2 ∥ ϕ m + 1 + ( t m ) ∥ L x 4 γ 1 − 4 ∥ ϕ m + 1 + ( t m ) ∥ L x 6 3 . (85) In passing from ϕ m + 1 + ( t m + 1 ) \phi^+_{m+1}(t_{m+1}) ϕ m + 1 + ( t m + 1 ) ϕ m + 1 + ( t m ) \phi^+_{m+1}(t_m) ϕ m + 1 + ( t m ) τ m 1 / 2 \tau_m^{1/2} τ m 1/2 ϕ B m ( t m ) \boldsymbol{\phi}_{B_m}(t_m) ϕ B m ( t m ) H − 1 H^{-1} H − 1 τ m 1 / 2 \tau_m^{1/2} τ m 1/2 L 2 L^2 L 2 (85) involves the previous time increment τ m \tau_m τ m τ m + 1 \tau_{m+1} τ m + 1 (85) rather technical (see Lemma 11.19). However, the desired upper bound by τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 T ( 1 ) T^{(1)} T ( 1 ) (67) involving the Higgs field.
Altogether, one obtains an estimate of the schematic form19
E ( A + ( t m + 1 ) , Φ + ( t m + 1 ) ) ≤ E ( A + ( t m ) , Φ + ( t m ) ) − 1 4 Dec m + 1 + C τ m + 1 − 1 + C . (86) \mathbb{E}\bigl(A^+(t_{m+1}),\Phi^+(t_{m+1})\bigr)\le\mathbb{E}\bigl(A^+(t_m),\Phi^+(t_m)\bigr)-\frac{1}{4}\operatorname{Dec}_{m+1}+C\tau_{m+1}^{-1}+C. \tag*{(86)} E ( A + ( t m + 1 ) , Φ + ( t m + 1 ) ) ≤ E ( A + ( t m ) , Φ + ( t m ) ) − 4 1 Dec m + 1 + C τ m + 1 − 1 + C . (86) For the precise formulation of our estimate of E ( A + ( t m + 1 ) , Φ + ( t m + 1 ) ) \mathbb{E}(A^{+}(t_{m+1}),\Phi^{+}(t_{m+1})) E ( A + ( t m + 1 ) , Φ + ( t m + 1 )) τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 Dec m + 1 \operatorname{Dec}_{m+1} Dec m + 1 Definition 2.26 . The δ s t e p \delta_{\mathrm{step}} δ step ε 0 \varepsilon_{0} ε 0 (67) can be ignored, since they can only be the minimum in (67) when the solution is bounded. When our stopping criterion is triggered by the ∥ Φ + ( t m ) ∥ L r γ \lVert\Phi^{+}(t_m)\rVert_{L^\gamma_r} ∥ Φ + ( t m ) ∥ L r γ (67) , the ∥ Φ + ( t m ) ∥ L r γ 2 + 2 τ m + 1 − 1 \lVert\Phi^{+}(t_m)\rVert_{L^r}^{\gamma_2+2}\tau_{m+1}^{-1} ∥ Φ + ( t m ) ∥ L r γ 2 + 2 τ m + 1 − 1 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 ∥ D B m + 1 + ∗ F B m + 1 + ∥ L 2 \lVert D^*_{B_{m+1}^{+}}F_{B_{m+1}^{+}}\rVert_{L^2} ∥ D B m + 1 + ∗ F B m + 1 + ∥ L 2 (68) , the curvature term in the decrease will allow us to beat τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 ∥ F A + ( t m ) ∥ L 2 \lVert F_{A^{+}(t_m)}\rVert_{L^2} ∥ F A + ( t m ) ∥ L 2 (67) , and this is for a good reason: If the initial connection one-form A + ( t m ) A^{+}(t_m) A + ( t m ) Φ + ( t m ) \Phi^{+}(t_m) Φ + ( t m ) ( A + , Φ + ) ( t m ) (A^{+},\Phi^{+})(t_m) ( A + , Φ + ) ( t m ) τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1
We emphasize that the failure to absorb the τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 Dec m + 1 \operatorname{Dec}_{m+1} Dec m + 1 (86) . Indeed, this obstruction only arises when the time step is determined by the curvature term in (67) , placing us in the regime discussed around (2.53)–(2.54). However, the obstruction shows that (86) , by itself, is insufficient for uniform-in-time estimates.
To obtain uniform-in-time estimates, we need to exploit our decay mechanism near Yang–Mills connections, which was briefly discussed in § 1.2 . To motivate this decay mechanism further, we briefly examine a finite-dimensional toy model. In this toy model, M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n ( λ j ) j = 1 n (\lambda_j)_{j=1}^{n} ( λ j ) j = 1 n W W W R n \mathbb{R}^{n} R n X X X
d X = − M X d t + d W , X ( 0 ) = 0. (87) \mathrm{d}X=-MX\,\mathrm{d}t+\mathrm{d}W,\qquad X(0)=0. \tag*{(87)} d X = − M X d t + d W , X ( 0 ) = 0. (87) We note that (87) is the Langevin equation corresponding to the quadratic energy 1 2 ⟨ M X , X ⟩ \frac{1}{2}\langle MX,X\rangle 2 1 ⟨ M X , X ⟩
E ⟨ M X ( t ) , X ( t ) ⟩ = 1 2 ∑ j = 1 n ( 1 − e − 2 λ j t ) . (88) \mathbb{E}\langle MX(t),X(t)\rangle=\frac{1}{2}\sum_{j=1}^{n}(1-\mathrm{e}^{-2\lambda_jt}). \tag*{(88)} E ⟨ M X ( t ) , X ( t )⟩ = 2 1 j = 1 ∑ n ( 1 − e − 2 λ j t ) . (88) For any negative eigenvalue λ j < 0 \lambda_j<0 λ j < 0 1 − e − 2 λ j t 1-\mathrm{e}^{-2\lambda_jt} 1 − e − 2 λ j t − ∞ -\infty − ∞ M M M (88) eventually becomes negative.
To connect the toy problem back to the Yang–Mills–Higgs model, we note that Yang–Mills connections with large curvature are unstable (see § 5.5 ). Based on our analysis of (88) , we then hope that the combined contribution of (2.48)–(2.49), which comes from the Hessian of the Yang–Mills energy, is negative. Unfortunately, it is difficult to exhibit our decay mechanism directly in (2.48)–(2.49). To see why, recall that we are currently only thinking of B m + 1 + B^{+}_{m+1} B m + 1 + B m + 1 − B^{-}_{m+1} B m + 1 − B ˚ m + 1 \mathring{B}_{m+1} B ˚ m + 1 B m B_m B m B m + 1 B_{m+1} B m + 1 Θ ˚ B m + 1 \mathring{\Theta}_{B_{m+1}} Θ ˚ B m + 1 τ m + 1 − 1 \tau_{m+1}^{-1} τ m + 1 − 1 A + ( t m + 1 ) A^{+}(t_{m+1}) A + ( t m + 1 ) A − ( t m + 1 ) A^{-}(t_{m+1}) A − ( t m + 1 )
A ~ + ( t m + 1 ) ≔ A + ( t m + 1 ) + e ( t m + 1 − t m + 1 / 2 ) Δ B m + 1 ( t m + 1 / 2 ) F Θ ˚ B m + 1 ( t m + 1 / 2 ) , \widetilde{A}^{+}(t_{m+1})\coloneqq A^{+}(t_{m+1})+\mathrm{e}^{(t_{m+1}-t_{m+1/2})\Delta^{F}_{B_{m+1}(t_{m+1/2})}}\mathring{\Theta}_{B_{m+1}}(t_{m+1/2}), A + ( t m + 1 ) : = A + ( t m + 1 ) + e ( t m + 1 − t m + 1/2 ) Δ B m + 1 ( t m + 1/2 ) F Θ ˚ B m + 1 ( t m + 1/2 ) , A ~ − ( t m + 1 ) ≔ A − ( t m + 1 ) − e ( t m + 1 − t m + 1 / 2 ) Δ B m + 1 ( t m + 1 / 2 ) F Θ ˚ B m + 1 ( t m + 1 / 2 ) . (89) \widetilde{A}^{-}(t_{m+1})\coloneqq A^{-}(t_{m+1})-\mathrm{e}^{(t_{m+1}-t_{m+1/2})\Delta^{F}_{B_{m+1}(t_{m+1/2})}}\mathring{\Theta}_{B_{m+1}}(t_{m+1/2}). \tag*{(89)} A − ( t m + 1 ) : = A − ( t m + 1 ) − e ( t m + 1 − t m + 1/2 ) Δ B m + 1 ( t m + 1/2 ) F Θ ˚ B m + 1 ( t m + 1/2 ) . (89) Here, the intermediate time t m + 1 / 2 t_{m+1/2} t m + 1/2 t m t_m t m t m + 1 t_{m+1} t m + 1 e ( t m + 1 − t m + 1 / 2 ) Δ B m + 1 ( t m + 1 / 2 ) F Θ ˚ B m + 1 ( t m + 1 / 2 ) \mathrm{e}^{(t_{m+1}-t_{m+1/2})\Delta^{F}_{B_{m+1}(t_{m+1/2})}}\mathring{\Theta}_{B_{m+1}}(t_{m+1/2}) e ( t m + 1 − t m + 1/2 ) Δ B m + 1 ( t m + 1/2 ) F Θ ˚ B m + 1 ( t m + 1/2 ) Θ ˚ B m + 1 ( t m + 1 ) \mathring{\Theta}_{B_{m+1}}(t_{m+1}) Θ ˚ B m + 1 ( t m + 1 ) 20 (89) has been included, the desired decay can be obtained using the following proposition.
Proposition 2.29 (Decay mechanism near Yang–Mills connections). Let C 1 , C 2 > 0 C_1,C_2>0 C 1 , C 2 > 0 T > 0 T>0 T > 0 C , c > 0 C,c>0 C , c > 0 C 1 C_1 C 1 C 2 C_2 C 2 (96) . Finally, let B : [ 0 , T ] × T 2 → g 2 B:[0,T]\times\mathbb{T}^2\to\mathfrak{g}^2 B : [ 0 , T ] × T 2 → g 2 (49) , as well as the bounds
∥ D B ∗ F B ∥ L t , x 2 ( ( 0 , T ] ) ≤ C 1 T . (90) \lVert D_B^*F_B\rVert_{L^2_{t,x}((0,T])}\leq\frac{C_1}{\sqrt{T}} . \tag*{(90)} ∥ D B ∗ F B ∥ L t , x 2 (( 0 , T ]) ≤ T C 1 . (90) and
inf t ∈ ( 0 , T ] ∥ F B ( t ) ∥ L 2 = ∥ F B ( T ) ∥ L 2 ≥ C 2 ∣ log T ∣ T . (91) \inf_{t\in(0,T]}\lVert F_B(t)\rVert_{L^2}=\lVert F_B(T)\rVert_{L^2}\geq C_2\frac{\lvert\log T\rvert}{T}. \tag*{(91)} t ∈ ( 0 , T ] inf ∥ F B ( t ) ∥ L 2 = ∥ F B ( T ) ∥ L 2 ≥ C 2 T ∣ log T ∣ . (91) Then, we have that
E ∫ T 2 ⟨ F B 12 ( T ) , [ ( e ( T / 2 ) Δ B ( T / 2 ) F ρ B ( T / 2 ) ) 1 , ( e ( T / 2 ) Δ B ( T / 2 ) F ρ B ( T / 2 ) ) 2 ] g ⟩ d x ≤ − C ∣ log T ∣ 1 + c T . (92) \mathbb{E}\int_{\mathbb{T}^2}\left\langle F_B^{12}(T),\left[\left(e^{(T/2)\Delta_{B(T/2)}^F}\rho_B(T/2)\right)^1,\left(e^{(T/2)\Delta_{B(T/2)}^F}\rho_B(T/2)\right)^2\right]_{\mathfrak{g}}\right\rangle\,\mathrm{d}x
\leq-C\frac{\lvert\log T\rvert^{1+c}}{T}. \tag*{(92)} E ∫ T 2 ⟨ F B 12 ( T ) , [ ( e ( T /2 ) Δ B ( T /2 ) F ρ B ( T /2 ) ) 1 , ( e ( T /2 ) Δ B ( T /2 ) F ρ B ( T /2 ) ) 2 ] g ⟩ d x ≤ − C T ∣ log T ∣ 1 + c . (92) Our assumptions in (90) and (91) imply that the average size of D B ∗ F B D_B^*F_B D B ∗ F B [ 0 , T ] [0,T] [ 0 , T ] F B F_B F B D B ∗ F B = 0 D_B^*F_B=0 D B ∗ F B = 0 (90) and (91) as saying that B B B ρ B \rho_B ρ B s \mathfrak{s} s (92) can be obtained using a rather explicit calculation; see Section 10.2.
As we will see in Section 11, a combination of our earlier estimates of (73) –(79) and the decay mechanism from Proposition 2.29 can be used to prove a hard global well-posedness result, which is the subject of the following theorem. In its statement, all explicit and implicit constants may depend on the parameters from (96) and (99) .
Theorem 2.30 (Hard global well-posedness and uniform-in-time estimates). Let A 0 ∈ C x − κ ( T 2 , g 2 ) A_0\in C_x^{-\kappa}(\mathbb{T}^2,\mathfrak{g}^2) A 0 ∈ C x − κ ( T 2 , g 2 ) Φ 0 ∈ C x − κ ( T 2 , V ) \Phi_0\in C_x^{-\kappa}(\mathbb{T}^2,V) Φ 0 ∈ C x − κ ( T 2 , V ) A A A A ~ + \widetilde{A}^+ A + A ~ − \widetilde{A}^- A − Φ \Phi Φ Φ ~ + \widetilde{\Phi}^+ Φ + Φ ~ − \widetilde{\Phi}^- Φ − L t , l o c ∞ C x − κ ( [ 0 , ∞ ) × T 2 , W ) L^\infty_{t,\mathrm{loc}}C_x^{-\kappa}([0,\infty)\times\mathbb{T}^2,W) L t , loc ∞ C x − κ ([ 0 , ∞ ) × T 2 , W ) W = g 2 W=\mathfrak{g}^2 W = g 2 W = V W=V W = V ( t ~ m ) m = 0 ∞ (\widetilde{t}_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ t ~ 0 = 0 \widetilde{t}_0=0 t 0 = 0
(i) (Stochastic YMH equations) On each time interval ( t ~ m , t ~ m + 1 ] (\widetilde{t}_m,\widetilde{t}_{m+1}] ( t m , t m + 1 ] m ≥ 0 m\geq0 m ≥ 0 ( A , Φ ) (A,\Phi) ( A , Φ ) (11) . To be more precise, it is the limit of the smooth, gauge covariant approximations from (44) in C t 0 C x − κ ( ( t ~ m , t ~ m + 1 ] × T 2 × g 2 × V ) C_t^0C_x^{-\kappa}((\widetilde{t}_m,\widetilde{t}_{m+1}]\times\mathbb{T}^2\times\mathfrak{g}^2\times V) C t 0 C x − κ (( t m , t m + 1 ] × T 2 × g 2 × V )
(ii) (Initial conditions and right-limits) The initial value ( A ( 0 ) , Φ ( 0 ) ) (A(0),\Phi(0)) ( A ( 0 ) , Φ ( 0 )) ( A 0 , Φ 0 ) (A_0,\Phi_0) ( A 0 , Φ 0 ) m ≥ 0 m\geq0 m ≥ 0 ( A ( t ~ m + ) , Φ ( t ~ m + ) ) (A(\widetilde{t}_m+),\Phi(\widetilde{t}_m+)) ( A ( t m + ) , Φ ( t m + )) C x − κ ( T 2 , g 2 × V ) C_x^{-\kappa}(\mathbb{T}^2,\mathfrak{g}^2\times V) C x − κ ( T 2 , g 2 × V ) J ~ t ~ m + 1 − t ~ m \widetilde{\mathcal{J}}_{\widetilde{t}_{m+1}-\widetilde{t}_m} J t m + 1 − t m (40) and is gauge-equivalent to ( A ( t ~ m ) , Φ ( t ~ m ) ) (A(\widetilde{t}_m),\Phi(\widetilde{t}_m)) ( A ( t m ) , Φ ( t m )) g m + 1 ∈ G − 1 g_{m+1}\in\mathcal{G}_{-1} g m + 1 ∈ G − 1 ( A ( t ~ m + ) , Φ ( t ~ m + ) ) = g m + 1 ⋅ ( A ( t ~ m ) , Φ ( t ~ m ) ) (A(\widetilde{t}_m+),\Phi(\widetilde{t}_m+))=g_{m+1}\cdot(A(\widetilde{t}_m),\Phi(\widetilde{t}_m)) ( A ( t m + ) , Φ ( t m + )) = g m + 1 ⋅ ( A ( t m ) , Φ ( t m ))
(iii) (Decomposition) For all m ≥ 1 m\geq1 m ≥ 1 A ( t ~ m ) = A ~ + ( t ~ m ) + A ~ − ( t ~ m ) A(\widetilde{t}_m)=\widetilde{A}^+(\widetilde{t}_m)+\widetilde{A}^-(\widetilde{t}_m) A ( t m ) = A + ( t m ) + A − ( t m ) Φ ( t ~ m ) = Φ ~ + ( t ~ m ) + Φ ~ − ( t ~ m ) \Phi(\widetilde{t}_m)=\widetilde{\Phi}^+(\widetilde{t}_m)+\widetilde{\Phi}^-(\widetilde{t}_m) Φ ( t m ) = Φ + ( t m ) + Φ − ( t m )
(iv) (Uniform-in-time estimates for the smooth part ) For all p ≥ 1 p \ge1 p ≥ 1
sup m ≥ 1 E [ E ( A ~ + ( t ~ m ) , Φ ~ + ( t ~ m ) ) p ] ≤ C ( A 0 , Φ 0 , κ , p ) , \sup_{m \ge1} \mathbb{E}\left[\mathcal{E}(\widetilde{A}^{+}(\widetilde{t}_m),\widetilde{\Phi}^{+}(\widetilde{t}_m))^{p}\right] \le C(A_0,\Phi_0,\kappa,p), m ≥ 1 sup E [ E ( A + ( t m ) , Φ + ( t m ) ) p ] ≤ C ( A 0 , Φ 0 , κ , p ) , lim sup m → ∞ E [ E ( A ~ + ( t ~ m ) , Φ ~ + ( t ~ m ) ) p ] ≤ C ( κ , p ) . \limsup_{m \to\infty} \mathbb{E}\left[\mathcal{E}(\widetilde{A}^{+}(\widetilde{t}_m),\widetilde{\Phi}^{+}(\widetilde{t}_m))^{p}\right] \le C(\kappa,p). m → ∞ lim sup E [ E ( A + ( t m ) , Φ + ( t m ) ) p ] ≤ C ( κ , p ) . We emphasise that, in contrast to the bound on the supremum, the bound on the limit superior does not depend on the initial data ( A 0 , Φ 0 ) (A_0,\Phi_0) ( A 0 , Φ 0 )
(v) (Uniform-in-time estimates for the rough part ) For all p ≥ 1 p \ge1 p ≥ 1
sup m ≥ 1 E [ ∥ A ~ − ( t ~ m ) ∥ C A ~ + ( t ~ m ) , F − κ p ] + sup m ≥ 1 E [ ∥ Φ ~ − ( t ~ m ) ∥ C A ~ + ( t ~ m ) − κ p ] ≤ C ( κ , p ) , \sup_{m \ge1} \mathbb{E}\left[\left\|\widetilde{A}^{-}(\widetilde{t}_m)\right\|_{\mathcal{C}^{-\kappa}_{\widetilde{A}^{+}(\widetilde{t}_m),F}}^{p}\right] + \sup_{m \ge1} \mathbb{E}\left[\left\|\widetilde{\Phi}^{-}(\widetilde{t}_m)\right\|_{\mathcal{C}^{-\kappa}_{\widetilde{A}^{+}(\widetilde{t}_m)}}^{p}\right] \le C(\kappa,p), m ≥ 1 sup E [ A − ( t m ) C A + ( t m ) , F − κ p ] + m ≥ 1 sup E [ Φ − ( t m ) C A + ( t m ) − κ p ] ≤ C ( κ , p ) , where ∥ ⋅ ∥ C A ~ + ( t ~ m ) , F − κ \|\cdot\|_{\mathcal{C}^{-\kappa}_{\widetilde{A}^{+}(\widetilde{t}_m),F}} ∥ ⋅ ∥ C A + ( t m ) , F − κ ∥ ⋅ ∥ C A ~ + ( t ~ m ) − κ \|\cdot\|_{\mathcal{C}^{-\kappa}_{\widetilde{A}^{+}(\widetilde{t}_m)}} ∥ ⋅ ∥ C A + ( t m ) − κ
The properties in (i) and (ii) are exactly as in our soft result and the decompositions in (iii) follow directly from our definitions. The main aspects of Theorem 2.30 are therefore contained in (iv) and (v), which imply that the smooth and rough parts are uniformly bounded.
Markov process and Yang–Mills–Higgs measure
Recall that one of the main results of [CCHS22, CS23] was to construct a natural state space D α = Ω α 1 / G α \mathcal{D}_{\alpha} = \Omega^{1}_{\alpha}/\mathcal{G}_{\alpha} D α = Ω α 1 / G α D \mathcal{D} D G α \mathcal{G}_{\alpha} G α α \alpha α Ω α 1 \Omega^{1}_{\alpha} Ω α 1 α \alpha α ∣ ∫ ℓ A ∣ ≲ ∣ ℓ ∣ α \left|\int_{\ell} A\right| \lesssim|\ell|^{\alpha} ∫ ℓ A ≲ ∣ ℓ ∣ α A ∈ Ω α 1 A \in\Omega^{1}_{\alpha} A ∈ Ω α 1 ℓ \ell ℓ
In view of the results of [CCHS24] which considers the 3-dimensional case, it is unsurprising that an analogous Markov process can also be constructed locally in time in the case of 2D Yang–Mills–Higgs on the space D κ Y M H = Ω κ Y M H / G 1 − κ \mathcal{D}^{\mathrm{YMH}}_{\kappa} = \Omega^{\mathrm{YMH}}_{\kappa}/\mathcal{G}_{1-\kappa} D κ YMH = Ω κ YMH / G 1 − κ κ > 0 \kappa> 0 κ > 0 Ω κ Y M H = Ω 1 − κ 1 × C − κ \Omega^{\mathrm{YMH}}_{\kappa} = \Omega^{1}_{1-\kappa} \times\mathcal{C}^{-\kappa} Ω κ YMH = Ω 1 − κ 1 × C − κ Ω 1 − κ 1 \Omega^{1}_{1-\kappa} Ω 1 − κ 1
∂ t Y = ( Δ − 1 ) Y + L ^ Y + B ^ ( Y , D Y ) + T ^ ( Y , Y , Y ) + ξ , (93) \partial_t Y = (\Delta- 1)Y + \widehat{L}Y + \widehat{B}(Y,DY) + \widehat{T}(Y,Y,Y) + \xi, \tag*{(93)} ∂ t Y = ( Δ − 1 ) Y + L Y + B ( Y , D Y ) + T ( Y , Y , Y ) + ξ , (93) for Y = ( A , Φ ) Y = (A,\Phi) Y = ( A , Φ ) ξ = ( ξ , ζ ) \xi= (\xi,\zeta) ξ = ( ξ , ζ ) g 2 ⊕ V \mathfrak{g}^{2} \oplus V g 2 ⊕ V Y Y Y
∂ t Y ε = ( Δ − 1 ) Y ε + ( L ^ + C ρ , ε Y ) Y ε + B ^ ( Y ε , D Y ε ) + T ^ ( Y ε , Y ε , Y ε ) + ξ ε , (94) \partial_t Y_{\varepsilon} = (\Delta- 1)Y_{\varepsilon} + (\widehat{L} + C^{Y}_{\rho,\varepsilon})Y_{\varepsilon} + \widehat{B}(Y_{\varepsilon},DY_{\varepsilon}) + \widehat{T}(Y_{\varepsilon},Y_{\varepsilon},Y_{\varepsilon}) + \xi_{\varepsilon}, \tag*{(94)} ∂ t Y ε = ( Δ − 1 ) Y ε + ( L + C ρ , ε Y ) Y ε + B ( Y ε , D Y ε ) + T ( Y ε , Y ε , Y ε ) + ξ ε , (94) for a suitable choice of C ρ , ε Y ∈ End ( Y ) C^{Y}_{\rho,\varepsilon} \in\operatorname{End}(Y) C ρ , ε Y ∈ End ( Y ) ξ ε = ρ ^ ε ⋆ ξ \xi_{\varepsilon} = \widehat{\rho}_{\varepsilon} \star\xi ξ ε = ρ ε ⋆ ξ ρ ^ ε \widehat{\rho}_{\varepsilon} ρ ε C ρ , ε Y C^{Y}_{\rho,\varepsilon} C ρ , ε Y C ~ ρ , ε Y \widetilde{C}^{Y}_{\rho,\varepsilon} C ρ , ε Y lim ε → 0 ∣ C ρ , ε Y − C ~ ρ , ε Y ∣ = 0 \lim_{\varepsilon\to0} |C^{Y}_{\rho,\varepsilon} - \widetilde{C}^{Y}_{\rho,\varepsilon}| = 0 lim ε → 0 ∣ C ρ , ε Y − C ρ , ε Y ∣ = 0 Y ∈ Ω κ Y M H Y \in\Omega^{\mathrm{YMH}}_{\kappa} Y ∈ Ω κ YMH [ Y ] ∈ D κ Y M H [Y] \in\mathcal{D}^{\mathrm{YMH}}_{\kappa} [ Y ] ∈ D κ YMH D ‾ κ Y M H = D κ Y M H ∪ { ∞ } \overline{\mathcal{D}}^{\mathrm{YMH}}_{\kappa} = \mathcal{D}^{\mathrm{YMH}}_{\kappa} \cup\{\infty\} D κ YMH = D κ YMH ∪ { ∞ } ∞ \infty ∞ ∥ ⋅ ∥ κ \|\cdot\|_{\kappa} ∥ ⋅ ∥ κ
Proposition 2.31. There exists a choice of renormalisation C ρ , ε Y C^{Y}_{\rho,\varepsilon} C ρ , ε Y
(i) For every Y 0 ∈ Ω κ Y M H Y_0 \in\Omega^{\mathrm{YMH}}_{\kappa} Y 0 ∈ Ω κ YMH τ > 0 \tau> 0 τ > 0 Ω κ Y M H \Omega^{\mathrm{YMH}}_{\kappa} Ω κ YMH Y Y Y Y ε Y_{\varepsilon} Y ε (94) with initial condition Y 0 Y_0 Y 0
lim ε → 0 sup t ∈ [ 0 , τ ] ∥ Y ε ( t ) − Y ( t ) ∥ κ = 0 , \lim_{\varepsilon\to0}\sup_{t\in[0,\tau]}\left\|Y_{\varepsilon}(t)-Y(t)\right\|_{\kappa}=0, ε → 0 lim t ∈ [ 0 , τ ] sup ∥ Y ε ( t ) − Y ( t ) ∥ κ = 0 , in probability.
(ii) There exists a Markov process Y ‾ \overline{Y} Y O ‾ κ Y M H \overline{\mathcal{O}}^{\mathrm{YMH}}_{\kappa} O κ YMH Y 0 Y_0 Y 0 Y ‾ ( t ) = [ Y ( t ) ] \overline{Y}(t)=[Y(t)] Y ( t ) = [ Y ( t )] t ∈ [ 0 , τ ] t\in[0,\tau] t ∈ [ 0 , τ ] Y ‾ ( 0 ) = [ Y 0 ] \overline{Y}(0)=[Y_0] Y ( 0 ) = [ Y 0 ]
Remark 2.32. Note that all of these stochastic processes are defined on the same probability space and they are adapted to the natural filtration generated by the noise ξ \xi ξ (93) . (In the case of (94) this statement requires the regularisation ξ ε \xi_{\varepsilon} ξ ε Proposition 2.31 to hold.)
Remark 2.33. While the gauge orbit process Y ‾ \overline{Y} Y ∞ \infty ∞ Y ‾ \overline{Y} Y Y Y Y
Remark 2.34. The fact that the process Y Y Y C ‾ ρ , ε Y \overline{C}^{Y}_{\rho,\varepsilon} C ρ , ε Y lim ε → 0 ( C ρ , ε Y − C ‾ ρ , ε Y ) \lim_{\varepsilon\to0}(C^{Y}_{\rho,\varepsilon}-\overline{C}^{Y}_{\rho,\varepsilon}) lim ε → 0 ( C ρ , ε Y − C ρ , ε Y ) Y ‾ \overline{Y} Y C ρ , ε Y C^{Y}_{\rho,\varepsilon} C ρ , ε Y
In fact, within the class of renormalisation constants that naturally arise in this context, for any other choice of renormalisation C ‾ ρ , ε Y \overline{C}^{Y}_{\rho,\varepsilon} C ρ , ε Y G G G M ∈ End ( V ) M\in\operatorname{End}(V) M ∈ End ( V ) C ρ , ε Y − C ‾ ρ , ε Y C^{Y}_{\rho,\varepsilon}-\overline{C}^{Y}_{\rho,\varepsilon} C ρ , ε Y − C ρ , ε Y 0 ⊕ M 0\oplus M 0 ⊕ M Y = g 2 ⊕ V Y=\mathfrak{g}^{2}\oplus V Y = g 2 ⊕ V M M M
Consider now the process Z = ( A , Φ ) Z=(A,\Phi) Z = ( A , Φ )
Proposition 2.35. The process Z Z Z Ω κ Y M H \Omega^{\mathrm{YMH}}_{\kappa} Ω κ YMH Z ‾ ( t ) = [ Z ( t ) ] \overline{Z}(t)=[Z(t)] Z ( t ) = [ Z ( t )] Y Y Y Z Z Z t > 0 t>0 t > 0 Y ‾ ( t ) = Z ‾ ( t ) \overline{Y}(t)=\overline{Z}(t) Y ( t ) = Z ( t )
(i) The process Y ‾ \overline{Y} Y Y ‾ ( t ) ≠ ∞ \overline{Y}(t)\ne\infty Y ( t ) = ∞
(ii) The process Z ‾ \overline{Z} Z
Remark 2.36. In view of Lemma 2.13 , we know that the process Z Z Z Z ‾ \overline{Z} Z
Results of this type (symmetry properties of solutions to singular SPDEs) have recently been obtained in a number of different contexts and their proofs fall into two categories. The first consists of “hard” estimates where one obtains explicit expressions for the BPHZ renormalisation constants (see [BHZ19, Hai18] for this terminology) and exploits the fact that the BPHZ renormalisations of different approximations to the same equation converge to the same limit. See for example [CCHS22, GH19]. The second consists of “soft” arguments which are more conceptual in nature and were first introduced in [BGHZ22], but then also used in the context of Yang–Mills–Higgs in [CCHS24] where explicit calculations would have been prohibitive.
In our case, Proposition 2.31 could in principle be obtained by the same “soft” argument used in [CCHS24], but it is less clear how to set up such a soft argument to prove Proposition 2.35. Therefore, we decided to go down the “hard” route and we obtain in Section 13.4 precise (up to equivalence) expressions for all the renormalisation constants appearing in the equation.
This then allows us to compare three different processes, as described in Section 13.1. The first one is simply the process Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 ) (94) with some initial data Y 0 Y_0 Y 0 ξ ϵ = ρ ^ ϵ ⋆ ξ \xi_\epsilon= \widehat{\rho}_\epsilon\star\xi ξ ϵ = ρ ϵ ⋆ ξ ρ ^ ϵ ⋆ ( g ∙ ξ ) \widehat{\rho}_\epsilon\star(g \mathbin{\bullet} \xi) ρ ϵ ⋆ ( g ∙ ξ ) G \mathbf{G} G g t g_t g t
Remark 2.37. The crucial observation here is that the law of the process Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 ) g g g
Regarding the second process, we start from the solution Y ϵ B Y_\epsilon^B Y ϵ B (44) which includes the term coming from the “infinitesimal DeTurck gauge” with background connection B B B Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 ) g t ( 2 ) g_t^{(2)} g t ( 2 ) Y ϵ ( 2 ) ( t ) = g t ( 2 ) ∙ Y ϵ B ( t ) Y_\epsilon^{(2)}(t) = g_t^{(2)} \mathbin{\bullet} Y_\epsilon^B(t) Y ϵ ( 2 ) ( t ) = g t ( 2 ) ∙ Y ϵ B ( t ) Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 ) ξ \xi ξ g g g Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 ) Y ϵ ( 2 ) Y_\epsilon^{(2)} Y ϵ ( 2 ) g g g Y ϵ ( 1 ) Y_\epsilon^{(1)} Y ϵ ( 1 )
To construct the third process, we consider the process Y ϵ g 0 Y_\epsilon^{g_0} Y ϵ g 0 (94) (this time driven by the original noise ξ ϵ \xi_\epsilon ξ ϵ g 0 ∙ Y 0 g_0 \mathbin{\bullet} Y_0 g 0 ∙ Y 0 g t ( 3 ) g_t^{(3)} g t ( 3 ) Y 0 Y_0 Y 0 g 0 ∙ Y 0 g_0 \mathbin{\bullet} Y_0 g 0 ∙ Y 0 Y ϵ ( 3 ) Y_\epsilon^{(3)} Y ϵ ( 3 ) Y ϵ ( 3 ) ( t ) = g t ( 3 ) ∙ Y ϵ g 0 ( t ) Y_\epsilon^{(3)}(t) = g_t^{(3)} \mathbin{\bullet} Y_\epsilon^{g_0}(t) Y ϵ ( 3 ) ( t ) = g t ( 3 ) ∙ Y ϵ g 0 ( t )
At fixed ϵ > 0 \epsilon> 0 ϵ > 0 Y ϵ ( i ) Y_\epsilon^{(i)} Y ϵ ( i ) Y ( i ) = lim ϵ → 0 Y ϵ ( i ) Y^{(i)} = \lim_{\epsilon\to0} Y_\epsilon^{(i)} Y ( i ) = lim ϵ → 0 Y ϵ ( i ) Y ( 1 ) = Y ( 2 ) Y^{(1)} = Y^{(2)} Y ( 1 ) = Y ( 2 ) g g g Y = l a w Y ( 1 ) Y \overset{\mathrm{law}}{=} Y^{(1)} Y = law Y ( 1 ) Z = ( g t ( 2 ) ) − 1 ∙ Y ( 2 ) Z = (g_t^{(2)})^{-1} \mathbin{\bullet} Y^{(2)} Z = ( g t ( 2 ) ) − 1 ∙ Y ( 2 ) Y ‾ \overline{Y} Y Z ‾ \overline{Z} Z
As a reality check, we verify that the same choice of renormalisation constants is such that one also has Y ( 1 ) = Y ( 3 ) Y^{(1)} = Y^{(3)} Y ( 1 ) = Y ( 3 ) g g g Y ( 1 ) Y^{(1)} Y ( 1 ) g g g Y = Y ( 1 ) Y = Y^{(1)} Y = Y ( 1 ) Y 0 Y_0 Y 0 g 0 ∙ Y 0 g_0 \mathbin{\bullet} Y_0 g 0 ∙ Y 0 [ Y ] [Y] [ Y ] Y ( 2 ) Y^{(2)} Y ( 2 ) ∥ ⋅ ∥ κ \| \cdot\|_{\kappa} ∥ ⋅ ∥ κ D 1 − κ 1 × C κ − 1 \mathcal{D}_{1-\kappa}^{1} \times\mathcal{C}_{\kappa}^{-1} D 1 − κ 1 × C κ − 1
Theorem 2.38 (Yang–Mills–Higgs measure). The Markov process [ Y ] [Y] [ Y ] μ Y M H \mu_{\mathrm{YMH}} μ YMH D κ Y M H \mathcal{D}_{\kappa}^{\mathrm{YMH}} D κ YMH μ Y M H \mu_{\mathrm{YMH}} μ YMH inf Y ‾ ∈ [ Y ] ∥ Y ‾ ∥ κ \inf_{\overline{Y} \in[Y]} \| \overline{Y} \|_{\kappa} inf Y ∈ [ Y ] ∥ Y ∥ κ
Existence of μ Y M H \mu_{\mathrm{YMH}} μ YMH D κ Y M H \mathcal{D}_{\kappa}^{\mathrm{YMH}} D κ YMH
Preliminaries
We first let χ : [ 0 , ∞ ) → [ 0 , 1 ] \chi\colon[0,\infty) \to[0,1] χ : [ 0 , ∞ ) → [ 0 , 1 ] supp χ ⊆ [ 1 / 16 , 16 ] \operatorname{supp}\chi\subseteq[1/16,16] supp χ ⊆ [ 1/16 , 16 ]
∑ N ∈ 2 N 0 χ ( N 2 t ) = 1 \sum_{N\in2\mathbb{N}_0} \chi(N^2t)=1 N ∈ 2 N 0 ∑ χ ( N 2 t ) = 1 for all t ∈ ( 0 , 1 ] t\in(0,1] t ∈ ( 0 , 1 ] N ∈ 2 N 0 N\in2^{\mathbb{N}_0} N ∈ 2 N 0
χ N ( t ) ≔ χ ( N 2 t ) . (95) \chi_N(t) \coloneqq\chi(N^2t). \tag*{(95)} χ N ( t ) : = χ ( N 2 t ) . (95) We emphasise that the function χ \chi χ ρ \rho ρ (41) can be chosen completely separately.
We now define parameters δ \delta δ δ 1 \delta_1 δ 1 δ 2 \delta_2 δ 2 θ \theta θ θ ′ \theta' θ ′ κ \kappa κ λ \lambda λ λ 1 \lambda_1 λ 1 λ 2 ∈ ( 0 , 1 ) \lambda_2 \in(0,1) λ 2 ∈ ( 0 , 1 ) r ∈ [ 1 , ∞ ) r\in[1,\infty) r ∈ [ 1 , ∞ )
0 < δ ≪ 1 r ≪ κ ≪ δ 1 = δ 2 ≪ θ ≪ θ ′ ≪ λ ≪ λ 1 ≪ λ 2 ≪ 1. (96) 0<\delta\ll\frac{1}{r}\ll\kappa\ll\delta_1=\delta_2\ll\theta\ll\theta'\ll\lambda\ll\lambda_1\ll\lambda_2\ll1. \tag*{(96)} 0 < δ ≪ r 1 ≪ κ ≪ δ 1 = δ 2 ≪ θ ≪ θ ′ ≪ λ ≪ λ 1 ≪ λ 2 ≪ 1. (96) The reason for introducing the two parameters δ 1 \delta_1 δ 1 δ 2 \delta_2 δ 2 § 9 . The parameter δ 1 \delta_1 δ 1 δ 2 \delta_2 δ 2 ϰ > 0 \varkappa>0 ϰ > 0
λ 2 ≔ ϰ , λ 1 ≔ ϰ 2 , λ ≔ ϰ 4 , θ ′ ≔ ϰ 8 , \lambda_2\coloneqq\varkappa,\qquad\lambda_1\coloneqq\varkappa^2,\qquad\lambda\coloneqq\varkappa^4,\qquad\theta'\coloneqq\varkappa^8, λ 2 : = ϰ , λ 1 : = ϰ 2 , λ : = ϰ 4 , θ ′ : = ϰ 8 , θ ≔ ϰ 16 , δ 1 ≔ ϰ 32 , δ 2 ≔ ϰ 32 , κ ≔ ϰ 64 , r ≔ ϰ − 128 , and δ ≔ ϰ 256 . (97) \theta\coloneqq\varkappa^{16},\qquad\delta_1\coloneqq\varkappa^{32},\qquad\delta_2\coloneqq\varkappa^{32},\qquad\kappa\coloneqq\varkappa^{64},\qquad r\coloneqq\varkappa^{-128},\qquad\text{and}\qquad\delta\coloneqq\varkappa^{256}. \tag*{(97)} θ : = ϰ 16 , δ 1 : = ϰ 32 , δ 2 : = ϰ 32 , κ : = ϰ 64 , r : = ϰ − 128 , and δ : = ϰ 256 . (97) Note that, as a consequence of our choices, we also have that
δ 1 θ = δ 2 θ ≪ λ and λ θ ′ ≫ θ . \frac{\delta_1}{\theta}=\frac{\delta_2}{\theta}\ll\lambda\qquad\text{and}\qquad\lambda\theta'\gg\theta. θ δ 1 = θ δ 2 ≪ λ and λ θ ′ ≫ θ . For future use, we also choose
T 0 = T 0 ( ϰ ) (98) T_0=T_0(\varkappa) \tag*{(98)} T 0 = T 0 ( ϰ ) (98) as sufficiently small depending on the parameters appearing in (97) or, equivalently, depending on ϰ \varkappa ϰ α \alpha α γ 0 \gamma_0 γ 0 γ 1 \gamma_1 γ 1 γ 2 \gamma_2 γ 2 ε 0 \varepsilon_0 ε 0
ε 0 ≪ δ , λ 2 ≪ α ≪ γ 1 − 4 ≪ γ 0 − 2 ≪ 1 , and γ 2 = 3. (99) \varepsilon_0\ll\delta,\qquad\lambda_2\ll\alpha\ll\gamma_1-4\ll\gamma_0-2\ll1,\qquad\text{and}\qquad\gamma_2=3. \tag*{(99)} ε 0 ≪ δ , λ 2 ≪ α ≪ γ 1 − 4 ≪ γ 0 − 2 ≪ 1 , and γ 2 = 3. (99) Once all these constants are chosen, choose some δ s t e p \delta_{\mathrm{step}} δ step
Notations
In this section, we set notation and conventions that will be used throughout this paper. We use W \mathsf{W} W G \mathbf{G} G W 1 , W 2 \mathsf{W}_1,\mathsf{W}_2 W 1 , W 2
Notation 3.1 (Dyadic integers). We will often use the capital letters K K K L L L M M M N N N 2 N 0 2\mathbb{N}_0 2 N 0 N N N ∑ N ∈ 2 N 0 \sum_{N\in2\mathbb{N}_0} ∑ N ∈ 2 N 0 ∑ N \sum_N ∑ N
Notation 3.2 (Space-time indices). We use Greek letters such as α \alpha α β \beta β γ \gamma γ μ \mu μ ν \nu ν { 0 } ∪ [ d ] = { 0 , 1 , … , d } \{0\}\cup[d]=\{0,1,\ldots,d\} { 0 } ∪ [ d ] = { 0 , 1 , … , d } i i i j j j k k k ℓ ∈ [ d ] \ell\in[d] ℓ ∈ [ d ] { 0 } ∪ [ d ] \{0\}\cup[d] { 0 } ∪ [ d ] [ d ] [d] [ d ]
Notation 3.3 (Higher derivatives). For k ≥ 0 k\geq0 k ≥ 0 D A ( k ) Φ D_A^{(k)}\Phi D A ( k ) Φ D A i 1 ⋯ D A i k Φ D_A^{i_1}\cdots D_A^{i_k}\Phi D A i 1 ⋯ D A i k Φ i 1 , … , i k ∈ [ d ] i_1,\ldots,i_k\in[d] i 1 , … , i k ∈ [ d ] ∥ ⋅ ∥ \|\mathbin{\cdot}\| ∥ ⋅ ∥
∥ D A ( k ) Φ ∥ = ∑ i 1 , … , i k = 1 d ∥ D A i 1 ⋯ D A i k Φ ∥ . \|D_A^{(k)}\Phi\|=\sum_{i_1,\ldots,i_k=1}^{d}\|D_A^{i_1}\cdots D_A^{i_k}\Phi\|. ∥ D A ( k ) Φ∥ = i 1 , … , i k = 1 ∑ d ∥ D A i 1 ⋯ D A i k Φ∥. i.e. ∥ D A ( k ) Φ ∥ \|D_A^{(k)}\Phi\| ∥ D A ( k ) Φ∥ k k k
Notation 3.4 (Space-time derivatives). Given a multi-index m ∈ N 0 { 0 } ∪ [ d ] \mathbf{m}\in\mathbb{N}_0^{\{0\}\cup[d]} m ∈ N 0 { 0 } ∪ [ d ]
∣ m ∣ s ≔ 2 m 0 + ∑ i = 1 d m i . |\mathbf{m}|_{\mathsf{s}}\mathrel{\mathop{\coloneqq}}2m_0+\sum_{i=1}^{d}m_i. ∣ m ∣ s : = 2 m 0 + i = 1 ∑ d m i . For m ≥ 0 \mathbf{m}\geq0 m ≥ 0 D A , t , x ( m ) D_{A,t,x}^{(\mathbf{m})} D A , t , x ( m ) D A α 1 ⋯ D A α k D_A^{\alpha_1}\cdots D_A^{\alpha_k} D A α 1 ⋯ D A α k k ≥ 0 k\geq0 k ≥ 0 α 1 , … , α k ∈ { 0 } ∪ [ d ] \alpha_1,\ldots,\alpha_k\in\{0\}\cup[d] α 1 , … , α k ∈ { 0 } ∪ [ d ] m \mathbf{m} m m α ∈ N 0 { 0 } ∪ [ d ] \mathbf{m}_\alpha\in\mathbb{N}_0^{\{0\}\cup[d]} m α ∈ N 0 { 0 } ∪ [ d ] 0 , 1 , … , d 0,1,\ldots,d 0 , 1 , … , d α 1 , … , α k \alpha_1,\ldots,\alpha_k α 1 , … , α k α 1 , … , α k ∈ { 0 } ∪ [ d ] \alpha_1,\ldots,\alpha_k\in\{0\}\cup[d] α 1 , … , α k ∈ { 0 } ∪ [ d ] ∣ m α ∣ s = m |\mathbf{m}_\alpha|_{\mathsf{s}}=m ∣ m α ∣ s = m D A 0 = ∂ t D_A^0=\partial_t D A 0 = ∂ t ∥ ⋅ ∥ \|\mathbin{\cdot}\| ∥ ⋅ ∥
∥ D A , t , x ( m ) Φ ∥ = ∑ α 1 , … , α k ∣ m α ∣ s = m ∥ D A α 1 ⋯ D A α k Φ ∥ . \|D_{A,t,x}^{(\mathbf{m})}\Phi\|=\sum_{\substack{\alpha_1,\ldots,\alpha_k\\|\mathbf{m}_\alpha|_{\mathsf{s}}=m}}\|D_A^{\alpha_1}\cdots D_A^{\alpha_k}\Phi\|. ∥ D A , t , x ( m ) Φ∥ = α 1 , … , α k ∣ m α ∣ s = m ∑ ∥ D A α 1 ⋯ D A α k Φ∥. Notation 3.5 (Wick products). If X X X k k k ξ \xi ξ ζ \zeta ζ : X : :X: : X : k k k
Notation 3.6 (Standard and schematic Landau symbols). We write O ( ⋅ ) O(\mathbin{\cdot}) O ( ⋅ ) X = O ( Y ) X=O(Y) X = O ( Y ) C > 0 C>0 C > 0 ∣ X ∣ ≤ C ∣ Y ∣ |X|\leq C|Y| ∣ X ∣ ≤ C ∣ Y ∣ O ( ⋅ ) O(\mathbin{\cdot}) O ( ⋅ ) X 1 , … , X n X_1,\ldots,X_n X 1 , … , X n g \mathfrak{g} g V V V W W W G G G O ( X 1 … X n ) O(X_1\ldots X_n) O ( X 1 … X n ) G G G (21) can be written as
[ K B i j , C j ] = O ( F B C ) , [ C j , 2 D B j C k − D B k C j ] = O ( C D B C ) , [ C j , [ C j , C k ] ] = O ( C C C ) . [K_B^{ij},C_j]=O(F_{BC}),\qquad[C_j,2D_B^jC^k-D_B^kC^j]=O(CD_BC),\qquad[C_j,[C^j,C^k]]=O(CCC). [ K B ij , C j ] = O ( F B C ) , [ C j , 2 D B j C k − D B k C j ] = O ( C D B C ) , [ C j , [ C j , C k ]] = O ( C C C ) . The two most important aspects of the O ( ⋅ ) O(\mathbin{\cdot}) O ( ⋅ ) ∣ O ( X 1 … X n ) ∣ = O ( ∣ X 1 ∣ … ∣ X n ∣ ) |O(X_1\ldots X_n)|=O(|X_1|\ldots|X_n|) ∣ O ( X 1 … X n ) ∣ = O ( ∣ X 1 ∣ … ∣ X n ∣ )
D B , j O ( X 1 … X n ) = ∑ m = 1 n O ( X 1 … D B , j X m … X n ) , D_{B,j}O(X_1\ldots X_n)=\sum_{m=1}^{n}O(X_1\ldots D_{B,j}X_m\ldots X_n), D B , j O ( X 1 … X n ) = m = 1 ∑ n O ( X 1 … D B , j X m … X n ) , which will be further discussed in Lemma 3.17 below.
Notation 3.7 (Euclidean heat kernels). We let p E u c p^{\mathrm{Euc}} p Euc p p p R d \mathbb{R}^{d} R d T d \mathbb{T}^{d} T d
p E u c ( s , y ; t , x ) = p t − s E u c ( x − y ) = 1 ( 4 π ( t − s ) ) d / 2 exp ( − ∣ x − y ∣ 2 4 ( t − s ) ) p^{\mathrm{Euc}}(s,y;t,x)=p^{\mathrm{Euc}}_{t-s}(x-y)=\frac{1}{\left(4\pi(t-s)\right)^{d/2}}\exp\left(-\frac{\lvert x-y\rvert^{2}}{4(t-s)}\right) p Euc ( s , y ; t , x ) = p t − s Euc ( x − y ) = ( 4 π ( t − s ) ) d /2 1 exp ( − 4 ( t − s ) ∣ x − y ∣ 2 ) p ( s , y ; t , x ) = p t − s ( x − y ) = ∑ n ∈ Z d p t − s E u c ( x − y + 2 π n ) . p(s,y;t,x)=p_{t-s}(x-y)=\sum_{n\in\mathbb{Z}^{d}}p^{\mathrm{Euc}}_{t-s}(x-y+2\pi n). p ( s , y ; t , x ) = p t − s ( x − y ) = n ∈ Z d ∑ p t − s Euc ( x − y + 2 π n ) . We refer to both p E u c p^{\mathrm{Euc}} p Euc p p p
Notation 3.8 (Limits of integration). In order to simplify the notation, we will omit the limits of integration whenever they are clear from context. For example, we will often write
∬ d s d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) \iint\mathrm{d}s\mathrm{d}y\rho_{\varepsilon}(t-s)q_{B}(s,y;t,x)\xi(s,y) ∬ d s d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) instead of
∫ 0 t d s ∫ T 2 d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . \int_{0}^{t}\mathrm{d}s\int_{\mathbb{T}^{2}}\mathrm{d}y\rho_{\varepsilon}(t-s)q_{B}(s,y;t,x)\xi(s,y). ∫ 0 t d s ∫ T 2 d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) . More generally, all time variables will be nonnegative, and the second time variable in any heat kernel will always be greater than or equal to the first time variable.
Notation 3.9 (Norms and inner products of differential forms). For W \mathrm{W} W C , D : T d → W d C,D:\mathbb{T}^{d}\to\mathrm{W}^{d} C , D : T d → W d
⟨ C , D ⟩ W : = δ i j ⟨ C i , D j ⟩ W = ⟨ C i , D i ⟩ W , \langle C,D\rangle_{\mathrm{W}}\mathrel{\mathop:}=\delta^{ij}\langle C_{i},D_{j}\rangle_{\mathrm{W}}=\langle C_{i},D^{i}\rangle_{\mathrm{W}}, ⟨ C , D ⟩ W : = δ ij ⟨ C i , D j ⟩ W = ⟨ C i , D i ⟩ W , ∣ C ∣ W : = ⟨ C , C ⟩ W 1 / 2 . \lvert C\rvert_{\mathrm{W}}\mathrel{\mathop:}=\langle C,C\rangle_{\mathrm{W}}^{1/2}. ∣ C ∣ W : = ⟨ C , C ⟩ W 1/2 . Similarly, given W \mathrm{W} W Q , R : T d → W d × d Q,R:\mathbb{T}^{d}\to\mathrm{W}^{d\times d} Q , R : T d → W d × d
⟨ Q , R ⟩ W : = 1 2 δ i k δ j ℓ ⟨ Q i j , R k ℓ ⟩ W = 1 2 ⟨ Q i j , R i j ⟩ W , \langle Q,R\rangle_{\mathrm{W}}\mathrel{\mathop:}=\frac{1}{2}\delta^{ik}\delta^{j\ell}\langle Q_{ij},R_{k\ell}\rangle_{\mathrm{W}}=\frac{1}{2}\langle Q_{ij},R^{ij}\rangle_{\mathrm{W}}, ⟨ Q , R ⟩ W : = 2 1 δ ik δ j ℓ ⟨ Q ij , R k ℓ ⟩ W = 2 1 ⟨ Q ij , R ij ⟩ W , ∣ Q ∣ W : = ⟨ Q , Q ⟩ W 1 / 2 . \lvert Q\rvert_{\mathrm{W}}\mathrel{\mathop:}=\langle Q,Q\rangle_{\mathrm{W}}^{1/2}. ∣ Q ∣ W : = ⟨ Q , Q ⟩ W 1/2 . We will often omit the subscript W \mathrm{W} W ∣ C ∣ , ∣ Q ∣ \lvert C\rvert,\lvert Q\rvert ∣ C ∣ , ∣ Q ∣ L x 2 L^{2}_{x} L x 2
⟨ C , D ⟩ : = ∫ T d ⟨ C ( x ) , D ( x ) ⟩ W d x , \langle C,D\rangle\mathrel{\mathop:}=\int_{\mathbb{T}^{d}}\langle C(x),D(x)\rangle_{\mathrm{W}}\,\mathrm{d}x, ⟨ C , D ⟩ : = ∫ T d ⟨ C ( x ) , D ( x ) ⟩ W d x , ⟨ Q , R ⟩ : = ∫ T d ⟨ Q ( x ) , R ( x ) ⟩ W d x . \langle Q,R\rangle\mathrel{\mathop:}=\int_{\mathbb{T}^{d}}\langle Q(x),R(x)\rangle_{\mathrm{W}}\,\mathrm{d}x. ⟨ Q , R ⟩ : = ∫ T d ⟨ Q ( x ) , R ( x ) ⟩ W d x . For W \mathrm{W} W Φ , Ψ : T d → W \Phi,\Psi:\mathbb{T}^{d}\to\mathrm{W} Φ , Ψ : T d → W ⟨ Φ , Ψ ⟩ \langle\Phi,\Psi\rangle ⟨ Φ , Ψ ⟩ L x 2 L^{2}_{x} L x 2 L 2 L^{2} L 2 ⟨ ⋅ , ⋅ ⟩ L 2 \langle\cdot,\cdot\rangle_{L^{2}} ⟨ ⋅ , ⋅ ⟩ L 2 ⟨ ⋅ , ⋅ ⟩ L x 2 \langle\cdot,\cdot\rangle_{L^{2}_{x}} ⟨ ⋅ , ⋅ ⟩ L x 2
Notation 3.10 (Covariant curl and divergence of a one-form). Let A : T 2 → g 2 A:\mathbb{T}^{2}\to\mathfrak{g}^{2} A : T 2 → g 2 C : T 2 → W 2 C:\mathbb{T}^{2}\to\mathrm{W}^{2} C : T 2 → W 2 W \mathrm{W} W curl A C : T 2 → W \operatorname{curl}_{A}C:\mathbb{T}^{2}\to\mathrm{W} curl A C : T 2 → W
curl A C : = D A , 1 C 2 − D A , 2 C 1 . \operatorname{curl}_{A}C\mathrel{\mathop:}=D_{A,1}C_{2}-D_{A,2}C_{1}. curl A C : = D A , 1 C 2 − D A , 2 C 1 . We remark that this is (up to a possible sign) the Hodge-⋆ \star ⋆ C C C div A C : T 2 → W \operatorname{div}_{A}C:\mathbb{T}^{2}\to\mathrm{W} div A C : T 2 → W
div A C : = D A ℓ C ℓ . \operatorname{div}_{A}C\mathrel{\mathop:}=D_{A}^{\ell}C_{\ell}. div A C : = D A ℓ C ℓ . Notation 3.11 (Covariant divergence of a two-form). Let A : T d → g d A:\mathbb{T}^{d} \to\mathfrak{g}^{d} A : T d → g d F : T d → W d × d F:\mathbb{T}^{d} \to\mathbb{W}^{d \times d} F : T d → W d × d W W W D A ∗ F : T d → W d D_A^*F:\mathbb{T}^{d} \to W^{d} D A ∗ F : T d → W d
( D A ∗ F ) i ≔ − D A j F j i , i ∈ [ d ] . (D_A^*F)_i \coloneqq-D_A^jF_{ji}, \quad i \in[d]. ( D A ∗ F ) i : = − D A j F j i , i ∈ [ d ] . Notation 3.12 (Covariant Hessian). Let A : T d → g d A:\mathbb{T}^{d} \to\mathfrak{g}^{d} A : T d → g d Φ : T d → W \Phi:\mathbb{T}^{d} \to W Φ : T d → W W W W X , Y X,Y X , Y T d \mathbb{T}^{d} T d Φ \Phi Φ X X X
D A , X Φ ≔ X i D A , i Φ , D_{A,X}\Phi\coloneqq X^iD_{A,i}\Phi, D A , X Φ : = X i D A , i Φ , and using this, we define the covariant Hessian
( Hess A Φ ) ( X , Y ) ≔ D A , X D A , Y Φ − D A , ∂ X Y Φ . (\operatorname{Hess}_{A}\Phi)(X,Y) \coloneqq D_{A,X}D_{A,Y}\Phi-D_{A,\partial_XY}\Phi. ( Hess A Φ ) ( X , Y ) : = D A , X D A , Y Φ − D A , ∂ X Y Φ. Here, ∂ X Y \partial_XY ∂ X Y
Geometric identities
In this section, we state various geometric identities which will be used throughout the paper. These identities are all quite classical and perhaps do not require proofs. However, we still prefer to give some of the slightly more involved proofs, since we do not expect the reader to be intimately familiar with the various standard manipulations of geometric analysis. Throughout this section, suppose that all functions which appear are smooth, or more generally, have enough derivatives so that all expressions which appear are classically well-defined.
Covariant derivatives, product formulas, and commutator identities
The following definitions and results will be used to, at least in parts of our argument, treat the connection one-form and Higgs field on equal footing.
Definition 3.13 (G-equivariant maps). Let π 1 \pi_1 π 1 π 2 \pi_2 π 2 G G G W W W Z Z Z U ⊆ W U \subseteq W U ⊆ W π 1 \pi_1 π 1 π 1 ( g ) u ∈ U \pi_1(g)u \in U π 1 ( g ) u ∈ U g ∈ G g \in G g ∈ G u ∈ U u \in U u ∈ U Q : U → Z Q:U \to Z Q : U → Z g ∈ G g \in G g ∈ G u ∈ U u \in U u ∈ U
Q ( π 1 ( g ) u ) = π 2 ( g ) Q ( u ) . Q(\pi_1(g)u)=\pi_2(g)Q(u). Q ( π 1 ( g ) u ) = π 2 ( g ) Q ( u ) . Remark 3.14. The reason for allowing Q Q Q U ⊆ W U \subseteq W U ⊆ W W W W Q Q Q
Lemma 3.15. The Lie bracket [ ∙ , ∙ ] : g × g → g [\bullet,\bullet]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g} [ ∙ , ∙ ] : g × g → g ϱ ∗ : g × V → V \varrho_*:\mathfrak{g}\times V\to V ϱ ∗ : g × V → V B : V ⊗ V → g B:V\otimes V\to\mathfrak{g} B : V ⊗ V → g
Proof. Since the action of G G G g \mathfrak{g} g 21 Ad g \operatorname{Ad}_g Ad g ( X , v ) ↦ ϱ ∗ ( X ) v (X,v)\mapsto\varrho_*(X)v ( X , v ) ↦ ϱ ∗ ( X ) v g ∈ G g\in G g ∈ G X ∈ g X\in\mathfrak{g} X ∈ g v ∈ V v\in V v ∈ V
ϱ ( g ) ϱ ∗ ( X ) v = ϱ ( g ) d d t ( ϱ ( e t X ) v ) ∣ t = 0 = d d t ( ϱ ( g e t X ) v ) ∣ t = 0 \varrho(g)\varrho_*(X)v=\varrho(g)\left.\frac{\mathrm{d}}{\mathrm{d}t}\left(\varrho(e^{tX})v\right)\right|_{t=0}
=\left.\frac{\mathrm{d}}{\mathrm{d}t}\left(\varrho(ge^{tX})v\right)\right|_{t=0} ϱ ( g ) ϱ ∗ ( X ) v = ϱ ( g ) d t d ( ϱ ( e tX ) v ) t = 0 = d t d ( ϱ ( g e tX ) v ) t = 0 = d d t ( ϱ ( g e t X g − 1 ) ϱ ( g ) v ) ∣ t = 0 = ϱ ∗ ( Ad g X ) ϱ ( g ) v . (100) =\left.\frac{\mathrm{d}}{\mathrm{d}t}\left(\varrho(ge^{tX}g^{-1})\varrho(g)v\right)\right|_{t=0}
=\varrho_*(\operatorname{Ad}_gX)\varrho(g)v. \tag*{(100)} = d t d ( ϱ ( g e tX g − 1 ) ϱ ( g ) v ) t = 0 = ϱ ∗ ( Ad g X ) ϱ ( g ) v . (100) as claimed. To show the G \mathbf{G} G B \mathbf{B} B g ∈ G g \in G g ∈ G X ∈ g X \in\mathfrak{g} X ∈ g u , v ∈ V u,v \in V u , v ∈ V Ad \operatorname{Ad} Ad g \mathfrak{g} g B \mathbf{B} B ϱ \varrho ϱ
⟨ Ad g B ( u ⊗ v ) , X ⟩ g = ⟨ B ( u ⊗ v ) , Ad g − 1 X ⟩ g = ⟨ u , ϱ ∗ ( Ad g − 1 X ) v ⟩ V = ⟨ ϱ ( g ) u , ϱ ( g ) ϱ ∗ ( Ad g − 1 X ) v ⟩ V = ⟨ ϱ ( g ) u , ϱ ∗ ( X ) ϱ ( g ) v ⟩ V = ⟨ B ( ϱ ( g ) u ⊗ ϱ ( g ) v ) , X ⟩ g . \begin{aligned}
\langle\operatorname{Ad}_{g}\mathbf{B}(u\otimes v),X\rangle_{\mathfrak{g}}=\langle\mathbf{B}(u\otimes v),\operatorname{Ad}_{g^{-1}}X\rangle_{\mathfrak{g}}=\langle u,\varrho_{*}(\operatorname{Ad}_{g^{-1}}X)v\rangle_{V} \\
&=\langle\varrho(g)u,\varrho(g)\varrho_{*}(\operatorname{Ad}_{g^{-1}}X)v\rangle_{V}=\langle\varrho(g)u,\varrho_{*}(X)\varrho(g)v\rangle_{V} \\
&=\langle\mathbf{B}(\varrho(g)u\otimes\varrho(g)v),X\rangle_{\mathfrak{g}}.
\end{aligned} ⟨ Ad g B ( u ⊗ v ) , X ⟩ g = ⟨ B ( u ⊗ v ) , Ad g − 1 X ⟩ g = ⟨ u , ϱ ∗ ( Ad g − 1 X ) v ⟩ V = ⟨ ϱ ( g ) u , ϱ ( g ) ϱ ∗ ( Ad g − 1 X ) v ⟩ V = ⟨ ϱ ( g ) u , ϱ ∗ ( X ) ϱ ( g ) v ⟩ V = ⟨ B ( ϱ ( g ) u ⊗ ϱ ( g ) v ) , X ⟩ g . Since X X X Ad g B ( u ⊗ v ) = B ( ϱ ( g ) u ⊗ ϱ ( g ) v ) \operatorname{Ad}_{g}\mathbf{B}(u\otimes v)=\mathbf{B}(\varrho(g)u\otimes\varrho(g)v) Ad g B ( u ⊗ v ) = B ( ϱ ( g ) u ⊗ ϱ ( g ) v )
Remark 3.16. Because we always assume that the vector space W \mathbf{W} W ⟨ ⋅ , ⋅ ⟩ W \langle\mathord{\cdot},\mathord{\cdot}\rangle_{\mathbf{W}} ⟨ ⋅ , ⋅ ⟩ W π \pi π G \mathbf{G} G W \mathbf{W} W G \mathbf{G} G W × W → R \mathbf{W}\times\mathbf{W}\to\mathbb{R} W × W → R G \mathbf{G} G R \mathbb{R} R
In the following lemma, we show that G \mathbf{G} G
Lemma 3.17. Let A : T d → g d A:\mathbb{T}^{d}\to\mathfrak{g}^{d} A : T d → g d k ≥ 1 k\geq1 k ≥ 1 1 ≤ j ≤ k 1\leq j\leq k 1 ≤ j ≤ k π j \pi_j π j G \mathbf{G} G W j \mathbf{W}_j W j U j ⊆ W j \mathbf{U}_j\subseteq\mathbf{W}_j U j ⊆ W j π j \pi_j π j u j : T d → U j u_j:\mathbb{T}^{d}\to\mathbf{U}_j u j : T d → U j π \pi π G \mathbf{G} G Z \mathbf{Z} Z Q : U 1 × ⋯ × U k → Z Q:\mathbf{U}_1\times\cdots\times\mathbf{U}_k\to\mathbf{Z} Q : U 1 × ⋯ × U k → Z G \mathbf{G} G
D A ( Q ( u 1 , … , u k ) ) = ∑ j = 1 k ∂ Q ∂ u j ( u 1 , … , u k ) D A u j . (101) D_A(Q(u_1,\ldots,u_k))=\sum_{j=1}^{k}\frac{\partial Q}{\partial u_j}(u_1,\ldots,u_k)D_Au_j. \tag*{(101)} D A ( Q ( u 1 , … , u k )) = j = 1 ∑ k ∂ u j ∂ Q ( u 1 , … , u k ) D A u j . (101) In particular, if Q : W 1 × ⋯ × W k → Z Q:\mathbf{W}_1\times\cdots\times\mathbf{W}_k\to\mathbf{Z} Q : W 1 × ⋯ × W k → Z G \mathbf{G} G w j : T d → W j w_j:\mathbb{T}^{d}\to\mathbf{W}_j w j : T d → W j 1 ≤ j ≤ k 1\leq j\leq k 1 ≤ j ≤ k
D A ( Q ( w 1 , … , w k ) ) = ∑ j = 1 k Q ( w 1 , … , D A w j , … , w k ) . (102) D_A(Q(w_1,\ldots,w_k))=\sum_{j=1}^{k}Q(w_1,\ldots,D_Aw_j,\ldots,w_k). \tag*{(102)} D A ( Q ( w 1 , … , w k )) = j = 1 ∑ k Q ( w 1 , … , D A w j , … , w k ) . (102) We note that, with a slight abuse of notation, we denote the different covariant derivatives corresponding to the different representations of G \mathbf{G} G D A D_A D A
Proof. Since (102) is a special case of (101) , it suffices to prove (101) . Using Definition 3.13 , we obtain for all X ∈ g X\in\mathfrak{g} X ∈ g t ∈ R t\in\mathbb{R} t ∈ R
π ( e t X ) Q ( u 1 , … , u k ) = Q ( π 1 ( e t X ) u 1 , … , π k ( e t X ) u k ) . \pi(\mathrm{e}^{tX})Q(u_1,\ldots,u_k)=Q(\pi_1(\mathrm{e}^{tX})u_1,\ldots,\pi_k(\mathrm{e}^{tX})u_k). π ( e tX ) Q ( u 1 , … , u k ) = Q ( π 1 ( e tX ) u 1 , … , π k ( e tX ) u k ) . By taking the derivative at t = 0 t=0 t = 0
π ∗ ( X ) Q ( u 1 , … , u k ) = ∑ j = 1 k ∂ Q ∂ u j ( u 1 , … , u k ) ( π j ) ∗ ( X ) u j . \pi_{*}(X)Q(u_1,\ldots,u_k)=\sum_{j=1}^{k}\frac{\partial Q}{\partial u_j}(u_1,\ldots,u_k)(\pi_j)_{*}(X)u_j. π ∗ ( X ) Q ( u 1 , … , u k ) = j = 1 ∑ k ∂ u j ∂ Q ( u 1 , … , u k ) ( π j ) ∗ ( X ) u j . Together with the definition of the covariant derivatives D A D_A D A (101) .
Remark 3.18. In particular, for smooth Φ , Ψ : T d → g \Phi,\Psi:\mathbb{T}^{d}\to\mathfrak{g} Φ , Ψ : T d → g
D A j [ Φ , Ψ ] = [ D A j Φ , Ψ ] + [ Φ , D A j Ψ ] . (103) D_A^j[\Phi,\Psi]=[D_A^j\Phi,\Psi]+[\Phi,D_A^j\Psi]. \tag*{(103)} D A j [ Φ , Ψ ] = [ D A j Φ , Ψ ] + [ Φ , D A j Ψ ] . (103) Lemma 3.19 (Commutator between covariant heat operator and covariant derivatives). Let A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d Ψ : [ 0 , T ] × T d → W \Psi: [0,T] \times\mathbb{T}^{d} \to W Ψ : [ 0 , T ] × T d → W W W W j ∈ [ d ] j \in[d] j ∈ [ d ] [ ⋅ , ⋅ ] [\mathord{\cdot},\mathord{\cdot}] [ ⋅ , ⋅ ]
[ Δ A , D A , j ] Ψ : = ( Δ A D A , j − D A , j Δ A ) Ψ = − 2 ( F A ) j ℓ D A ℓ Ψ + ( D A ℓ ( F A ) ℓ j ) Ψ . (104) [\Delta_{A},D_{A,j}]\Psi:= (\Delta_{A}D_{A,j}-D_{A,j}\Delta_{A})\Psi=-2(F_{A})_{j\ell}D_{A}^{\ell}\Psi+(D_{A}^{\ell}(F_{A})_{\ell j})\Psi. \tag*{(104)} [ Δ A , D A , j ] Ψ := ( Δ A D A , j − D A , j Δ A ) Ψ = − 2 ( F A ) j ℓ D A ℓ Ψ + ( D A ℓ ( F A ) ℓ j ) Ψ. (104) [ ∂ t − Δ A , D A , j ] Ψ : = ( ( ∂ t − Δ A ) D A , j − D A , j ( ∂ t − Δ A ) ) Ψ = 2 ( F A ) j ℓ D A ℓ Ψ + ( ∂ t A j − D A ℓ ( F A ) ℓ j ) Ψ . (105) [\partial_{t}-\Delta_{A},D_{A,j}]\Psi:= ((\partial_{t}-\Delta_{A})D_{A,j}-D_{A,j}(\partial_{t}-\Delta_{A}))\Psi
=2(F_{A})_{j\ell}D_{A}^{\ell}\Psi+(\partial_{t}A_{j}-D_{A}^{\ell}(F_{A})_{\ell j})\Psi. \tag*{(105)} [ ∂ t − Δ A , D A , j ] Ψ := (( ∂ t − Δ A ) D A , j − D A , j ( ∂ t − Δ A )) Ψ = 2 ( F A ) j ℓ D A ℓ Ψ + ( ∂ t A j − D A ℓ ( F A ) ℓ j ) Ψ. (105) Remark 3.20. In particular, if A A A ∂ t A j = D A ℓ ( F A ) ℓ j \partial_{t}A_{j}=D_{A}^{\ell}(F_{A})_{\ell j} ∂ t A j = D A ℓ ( F A ) ℓ j (105) drops out. In the Abelian case, this was used in [BC24], see e.g. § 4.4 of that paper. We will also use this extensively (in the non-Abelian case) in § 5 .
Proof of Lemma 3.19. For the first commutation identity (104) , we apply (6) twice and (102) once to obtain
D A ℓ D A , ℓ D A , j Ψ = D A ℓ D A , j D A , ℓ Ψ + D A ℓ ( ( F A ) ℓ j Ψ ) = D A , j D A ℓ D A , ℓ Ψ + ( F A ) j ℓ D A , ℓ Ψ + D A ℓ ( ( F A ) ℓ j Ψ ) = D A , j D A ℓ D A , ℓ Ψ − 2 ( F A ) j ℓ D A ℓ Ψ + ( D A ℓ ( F A ) ℓ j ) Ψ . \begin{aligned}
D_{A}^{\ell}D_{A,\ell}D_{A,j}\Psi=D_{A}^{\ell}D_{A,j}D_{A,\ell}\Psi+D_{A}^{\ell}((F_{A})_{\ell j}\Psi) \\
&=D_{A,j}D_{A}^{\ell}D_{A,\ell}\Psi+(F_{A})_{j}^{\ \ell}D_{A,\ell}\Psi+D_{A}^{\ell}((F_{A})_{\ell j}\Psi) \\
&=D_{A,j}D_{A}^{\ell}D_{A,\ell}\Psi-2(F_{A})_{j\ell}D_{A}^{\ell}\Psi+(D_{A}^{\ell}(F_{A})_{\ell j})\Psi.
\end{aligned} D A ℓ D A , ℓ D A , j Ψ = D A ℓ D A , j D A , ℓ Ψ + D A ℓ (( F A ) ℓ j Ψ ) = D A , j D A ℓ D A , ℓ Ψ + ( F A ) j ℓ D A , ℓ Ψ + D A ℓ (( F A ) ℓ j Ψ ) = D A , j D A ℓ D A , ℓ Ψ − 2 ( F A ) j ℓ D A ℓ Ψ + ( D A ℓ ( F A ) ℓ j ) Ψ. The second identity (105) follows by combining the first identity (104) along with the fact that [ ∂ t , D A , j ] Ψ = π ∗ ( ∂ t A j ) Ψ [\partial_{t},D_{A,j}]\Psi=\pi_{*}(\partial_{t}A_{j})\Psi [ ∂ t , D A , j ] Ψ = π ∗ ( ∂ t A j ) Ψ
Lemma 3.21 (Bochner formulas). Let A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d Ψ : [ 0 , T ] × T d → W \Psi: [0,T] \times\mathbb{T}^{d} \to W Ψ : [ 0 , T ] × T d → W W W W Δ \Delta Δ
1 2 Δ ( ∣ Ψ ∣ W 2 ) = ⟨ Ψ , Δ A Ψ ⟩ W + ∣ D A Ψ ∣ W 2 , (106) \frac{1}{2}\Delta(|\Psi|_{W}^{2})=\langle\Psi,\Delta_{A}\Psi\rangle_{W}+|D_{A}\Psi|_{W}^{2}, \tag*{(106)} 2 1 Δ ( ∣Ψ ∣ W 2 ) = ⟨ Ψ , Δ A Ψ ⟩ W + ∣ D A Ψ ∣ W 2 , (106) 1 2 ( ∂ t − Δ ) ( ∣ Ψ ∣ W 2 ) = ⟨ Ψ , ( ∂ t − Δ A ) Ψ ⟩ W − ∣ D A Ψ ∣ W 2 . (107) \frac{1}{2}(\partial_{t}-\Delta)(|\Psi|_{W}^{2})=\langle\Psi,(\partial_{t}-\Delta_{A})\Psi\rangle_{W}-|D_{A}\Psi|_{W}^{2}. \tag*{(107)} 2 1 ( ∂ t − Δ ) ( ∣Ψ ∣ W 2 ) = ⟨ Ψ , ( ∂ t − Δ A ) Ψ ⟩ W − ∣ D A Ψ ∣ W 2 . (107) Proof. We prove (106) ; the proof for (107) is similar. We compute (recall Remark 3.16 and the product formula (102) )
Δ ⟨ Ψ , Ψ ⟩ W = ∂ ℓ ∂ ℓ ⟨ Ψ , Ψ ⟩ W = 2 ∂ ℓ ⟨ Ψ , D A ℓ Ψ ⟩ W = 2 ⟨ Ψ , D A , ℓ D A ℓ Ψ ⟩ W + 2 ⟨ D A , ℓ Ψ , D A ℓ Ψ ⟩ W . \Delta\langle\Psi,\Psi\rangle_{W}=\partial_{\ell}\partial^{\ell}\langle\Psi,\Psi\rangle_{W}=2\partial_{\ell}\langle\Psi,D_{A}^{\ell}\Psi\rangle_{W}=2\langle\Psi,D_{A,\ell}D_{A}^{\ell}\Psi\rangle_{W}+2\langle D_{A,\ell}\Psi,D_{A}^{\ell}\Psi\rangle_{W}. Δ ⟨ Ψ , Ψ ⟩ W = ∂ ℓ ∂ ℓ ⟨ Ψ , Ψ ⟩ W = 2 ∂ ℓ ⟨ Ψ , D A ℓ Ψ ⟩ W = 2 ⟨ Ψ , D A , ℓ D A ℓ Ψ ⟩ W + 2 ⟨ D A , ℓ Ψ , D A ℓ Ψ ⟩ W . The desired result now follows.
Lemma 3.22 (Linearisation). Let A , B : T d → g d A,B:\mathbb{T}^{d}\to\mathfrak{g}^{d} A , B : T d → g d C : T d → g d C:\mathbb{T}^{d}\to\mathfrak{g}^{d} C : T d → g d A = B + C A=B+C A = B + C Ψ : T d → W \Psi:\mathbb{T}^{d}\to W Ψ : T d → W j , k ∈ [ d ] j,k\in[d] j , k ∈ [ d ]
D A j Ψ = D B j Ψ + C j Ψ , (108) D_{A}^{j}\Psi=D_{B}^{j}\Psi+C^{j}\Psi, \tag*{(108)} D A j Ψ = D B j Ψ + C j Ψ , (108) F A j k = F B j k + D B j C k − D B k C j + [ C j , C k ] , (109) F_{A}^{jk}=F_{B}^{jk}+D_{B}^{j}C^{k}-D_{B}^{k}C^{j}+[C^{j},C^{k}], \tag*{(109)} F A j k = F B j k + D B j C k − D B k C j + [ C j , C k ] , (109) D A , j F A j k = D B , j F B j k + Δ B C k + 2 [ F B k j , C j ] − D B k D B j C j D_{A,j}F_{A}^{jk}=D_{B,j}F_{B}^{jk}+\Delta_{B}C^{k}+2[F_{B}^{kj},C_{j}]-D_{B}^{k}D_{B}^{j}C_{j} D A , j F A j k = D B , j F B j k + Δ B C k + 2 [ F B k j , C j ] − D B k D B j C j + [ D B j C j , C k ] + [ C j , 2 D B j C k − D B k C j ] + [ C j , [ C j , C k ] ] , (110) \qquad+[D_{B}^{j}C_{j},C^{k}]+[C_{j},2D_{B}^{j}C^{k}-D_{B}^{k}C^{j}]+[C_{j},[C^{j},C^{k}]], \tag*{(110)} + [ D B j C j , C k ] + [ C j , 2 D B j C k − D B k C j ] + [ C j , [ C j , C k ]] , (110) D A k D A j C j = D B k D B j C j − [ D B j C j , C k ] , (111) D_{A}^{k}D_{A}^{j}C_{j}=D_{B}^{k}D_{B}^{j}C_{j}-[D_{B}^{j}C_{j},C^{k}], \tag*{(111)} D A k D A j C j = D B k D B j C j − [ D B j C j , C k ] , (111) Δ A Ψ − ( D A j C j ) Ψ = Δ B Ψ + 2 C j D B j Ψ + C j C j Ψ . (112) \Delta_{A}\Psi-(D_{A}^{j}C_{j})\Psi=\Delta_{B}\Psi+2C_{j}D_{B}^{j}\Psi+C^{j}C_{j}\Psi. \tag*{(112)} Δ A Ψ − ( D A j C j ) Ψ = Δ B Ψ + 2 C j D B j Ψ + C j C j Ψ. (112) Proof. The first identity (108) follows directly from the definition of our covariant derivatives. To obtain the second identity (109) , we write
F B + C j k = ∂ j ( B + C ) k − ∂ k ( B + C ) j + [ ( B + C ) j , ( B + C ) k ] = ( ∂ j B k − ∂ k B j + [ B j , B k ] ) + ( ∂ j C k + [ B j , C k ] − ∂ k C j − [ B k , C j ] ) + [ C j , C k ] = F B j k + ( D B C ) j k + [ C j , C k ] . F_{B+C}^{jk} = \partial^j(B+C)^k-\partial^k(B+C)^j+[(B+C)^j,(B+C)^k]
= (\partial^jB^k-\partial^kB^j+[B^j,B^k])+(\partial^jC^k+[B^j,C^k]-\partial^kC^j-[B^k,C^j])+[C^j,C^k]
= F_B^{jk}+(D_BC)^{jk}+[C^j,C^k]. F B + C j k = ∂ j ( B + C ) k − ∂ k ( B + C ) j + [( B + C ) j , ( B + C ) k ] = ( ∂ j B k − ∂ k B j + [ B j , B k ]) + ( ∂ j C k + [ B j , C k ] − ∂ k C j − [ B k , C j ]) + [ C j , C k ] = F B j k + ( D B C ) j k + [ C j , C k ] . To obtain the third identity (110) , we first use (108) and (109) , which yield that
D A , j F A j k = ( D B , j + ad ( C j ) ) ( F B j k + ( D B C ) j k + [ C j , C k ] ) = D B , j F B j k + D B , j ( D B C ) j k + [ C j , F B j k ] D_{A,j}F_A^{jk}=(D_{B,j}+\operatorname{ad}(C_j))(F_B^{jk}+(D_BC)^{jk}+[C^j,C^k])
= D_{B,j}F_B^{jk}+D_{B,j}(D_BC)^{jk}+[C_j,F_B^{jk}] D A , j F A j k = ( D B , j + ad ( C j )) ( F B j k + ( D B C ) j k + [ C j , C k ]) = D B , j F B j k + D B , j ( D B C ) j k + [ C j , F B j k ] + D B j [ C j , C k ] + [ C j , ( D B C ) j k ] + [ C j , [ C j , C k ] ] . (113) \qquad+D_B^j[C_j,C^k]+[C_j,(D_BC)^{jk}]+[C_j,[C^j,C^k]]. \tag*{(113)} + D B j [ C j , C k ] + [ C j , ( D B C ) j k ] + [ C j , [ C j , C k ]] . (113) Using the commutator identity (6) , we see that the second and third summands in (113) can be written as
D B , j ( D B j C k − D B k C j ) + [ C j , F B j k ] = Δ B C k − D B k D B j C j + 2 [ F B j k , C j ] . D_{B,j}(D_B^jC^k-D_B^kC^j)+[C_j,F_B^{jk}]=\Delta_BC^k-D_B^kD_B^jC_j+2[F_B^{jk},C_j]. D B , j ( D B j C k − D B k C j ) + [ C j , F B j k ] = Δ B C k − D B k D B j C j + 2 [ F B j k , C j ] . Furthermore, using the covariant Leibniz rule (103) , we see that the first and second summands in (3.20) can be written as
D B j [ C j , C k ] + [ C j , ( D B C ) j k ] = [ D B j C j , C k ] + [ C j , D B j C k ] + [ C j , D B j C k ] − [ C j , D B k C j ] = [ D B j C j , C k ] + [ C j , 2 D B j C k − D B k C j ] . \begin{aligned}
D_B^j[C_j,C^k]+[C_j,(D_BC)^{jk}]
=[D_B^jC_j,C^k]+[C_j,D_B^jC^k]+[C_j,D_B^jC^k]-[C_j,D_B^kC^j] \\
&=[D_B^jC_j,C^k]+[C_j,2D_B^jC^k-D_B^kC^j].
\end{aligned} D B j [ C j , C k ] + [ C j , ( D B C ) j k ] = [ D B j C j , C k ] + [ C j , D B j C k ] + [ C j , D B j C k ] − [ C j , D B k C j ] = [ D B j C j , C k ] + [ C j , 2 D B j C k − D B k C j ] . Altogether, we therefore obtain (110) . In order to obtain the fourth identity (111) , we first apply (108) to D A j D_A^j D A j
D A k D A j C j = D A k ( D B j C j + [ C j , C j ] ) = D A k D B j C j . D_A^kD_A^jC_j=D_A^k(D_B^jC_j+[C^j,C_j])=D_A^kD_B^jC_j. D A k D A j C j = D A k ( D B j C j + [ C j , C j ]) = D A k D B j C j . After applying (108) to D A k D_A^k D A k (111) . Finally, in order to obtain the last identity (112) , we use Lemma 3.15 and Lemma 3.17 , which yield that
Δ A Ψ − ( D A j C j ) Ψ = ( D B j + C j ) ( D B , j + C j ) Ψ − ( D B j C j + [ C j , C j ] ) Ψ = D B j D B , j Ψ + C j D B , j Ψ + D B j ( C j Ψ ) + C j C j Ψ − ( D B j C j ) Ψ = Δ B Ψ + 2 C j D B j Ψ + C j C j Ψ . \begin{aligned}
\Delta_A\Psi-(D_A^jC_j)\Psi=(D_B^j+C^j)(D_{B,j}+C_j)\Psi-(D_B^jC_j+[C^j,C_j])\Psi\\
&=D_B^jD_{B,j}\Psi+C^jD_{B,j}\Psi+D_B^j(C_j\Psi)+C^jC_j\Psi-(D_B^jC_j)\Psi\\
&=\Delta_B\Psi+2C_jD_B^j\Psi+C^jC_j\Psi.
\end{aligned} Δ A Ψ − ( D A j C j ) Ψ = ( D B j + C j ) ( D B , j + C j ) Ψ − ( D B j C j + [ C j , C j ]) Ψ = D B j D B , j Ψ + C j D B , j Ψ + D B j ( C j Ψ ) + C j C j Ψ − ( D B j C j ) Ψ = Δ B Ψ + 2 C j D B j Ψ + C j C j Ψ. In the following, recall our definitions of inner products of differential forms (Notation 3.9 ), the covariant curl (Notation 3.10 ), and the covariant divergence of a two-form (Notation 3.11).
Lemma 3.23. Let A : T 2 → g 2 A:\mathbb{T}^2\to\mathfrak{g}^2 A : T 2 → g 2 C : T 2 → W 2 C:\mathbb{T}^2\to W^2 C : T 2 → W 2 F : T 2 → W 2 × 2 F:\mathbb{T}^2\to\mathbf{W}^{2\times2} F : T 2 → W 2 × 2
⟨ curl A C , F 12 ⟩ = ⟨ C , D A ∗ F ⟩ . \langle\operatorname{curl}_A C,F_{12}\rangle=\langle C,D_A^*F\rangle. ⟨ curl A C , F 12 ⟩ = ⟨ C , D A ∗ F ⟩ . Proof. We have that
⟨ curl A C , F 12 ⟩ = ∫ T 2 ⟨ curl A C ( x ) , F 12 ( x ) ⟩ W d x = ∫ T 2 ⟨ D A 1 C 2 ( x ) − D A 2 C 1 ( x ) , F 12 ( x ) ⟩ W d x . \langle\operatorname{curl}_A C,F_{12}\rangle=\int_{\mathbb{T}^2}\langle\operatorname{curl}_A C(x),F_{12}(x)\rangle_W\,\mathrm{d}x=\int_{\mathbb{T}^2}\langle D_A^1C^2(x)-D_A^2C^1(x),F_{12}(x)\rangle_W\,\mathrm{d}x. ⟨ curl A C , F 12 ⟩ = ∫ T 2 ⟨ curl A C ( x ) , F 12 ( x ) ⟩ W d x = ∫ T 2 ⟨ D A 1 C 2 ( x ) − D A 2 C 1 ( x ) , F 12 ( x ) ⟩ W d x . To finish, apply integration by parts and use the antisymmetry of the two-form F F F
Lemma 3.24. Let A : T d → g d A:\mathbb{T}^{d}\to\mathfrak{g}^{d} A : T d → g d C : T d → g d C:\mathbb{T}^{d}\to\mathfrak{g}^{d} C : T d → g d g \mathfrak{g} g
2 ∑ 1 ≤ i < j ≤ d ⟨ F A i j , [ C i , C j ] ⟩ = ∑ 1 ≤ i < j ≤ d ∥ D A , i C j − D A , j C i ∥ L 2 2 + ∥ D A j C j ∥ L 2 2 − ∑ i , j = 1 d ∥ D A , i C j ∥ L 2 2 . (114) 2\sum_{1\le i<j\le d}\langle F_{A}^{ij},[C_i,C_j]\rangle
=
\sum_{1\le i<j\le d}\lVert D_{A,i}C_j-D_{A,j}C_i\rVert_{L^2}^{2}
+\lVert D_{A}^{j}C_j\rVert_{L^2}^{2}
-\sum_{i,j=1}^{d}\lVert D_{A,i}C_j\rVert_{L^2}^{2}. \tag*{(114)} 2 1 ≤ i < j ≤ d ∑ ⟨ F A ij , [ C i , C j ]⟩ = 1 ≤ i < j ≤ d ∑ ∥ D A , i C j − D A , j C i ∥ L 2 2 + ∥ D A j C j ∥ L 2 2 − i , j = 1 ∑ d ∥ D A , i C j ∥ L 2 2 . (114) Also, we have that
2 ∑ 1 ≤ i < j ≤ d ⟨ F A i j , [ C i , C j ] ⟩ = − ∑ 1 ≤ i < j ≤ d ∥ D A , i C j − D A , j C i ∥ L 2 2 − ∥ D A j C j ∥ L 2 2 − ∑ i = 1 d ⟨ C i , Δ A C i + 2 [ ( F A ) i j , C j ] ⟩ . (115) 2\sum_{1\le i<j\le d}\langle F_{A}^{ij},[C_i,C_j]\rangle
=
-\sum_{1\le i<j\le d}\lVert D_{A,i}C_j-D_{A,j}C_i\rVert_{L^2}^{2}
-\lVert D_{A}^{j}C_j\rVert_{L^2}^{2}
-\sum_{i=1}^{d}\langle C_i,\Delta_A C_i+2[(F_A)_{ij},C^j]\rangle. \tag*{(115)} 2 1 ≤ i < j ≤ d ∑ ⟨ F A ij , [ C i , C j ]⟩ = − 1 ≤ i < j ≤ d ∑ ∥ D A , i C j − D A , j C i ∥ L 2 2 − ∥ D A j C j ∥ L 2 2 − i = 1 ∑ d ⟨ C i , Δ A C i + 2 [( F A ) ij , C j ]⟩ . (115) Proof. By expanding the squares, the right-hand side of (114) is equal to
2 ∑ 1 ≤ i < j ≤ d ∫ T d ( ⟨ D A , i C i , D A , j C j ⟩ g − ⟨ D A , i C j , D A , j C i ⟩ g ) d x . 2\sum_{1\le i<j\le d}\int_{\mathbb{T}^{d}}\left(\langle D_{A,i}C_i,D_{A,j}C_j\rangle_{\mathfrak{g}}-\langle D_{A,i}C_j,D_{A,j}C_i\rangle_{\mathfrak{g}}\right)\,dx. 2 1 ≤ i < j ≤ d ∑ ∫ T d ( ⟨ D A , i C i , D A , j C j ⟩ g − ⟨ D A , i C j , D A , j C i ⟩ g ) d x . For each i < j i<j i < j (6) give
∫ T d ( ⟨ D A , i C i , D A , j C j ⟩ g − ⟨ D A , i C j , D A , j C i ⟩ g ) d x = ∫ T d ⟨ ( D A , j D A , i − D A , i D A , j ) C j , C i ⟩ g d x = ∫ T d ⟨ F A i j [ C i , C j ] ⟩ g d x . \begin{aligned}
\int_{\mathbb{T}^{d}}\left(\langle D_{A,i}C_i,D_{A,j}C_j\rangle_{\mathfrak{g}}-\langle D_{A,i}C_j,D_{A,j}C_i\rangle_{\mathfrak{g}}\right)\,dx
&=\int_{\mathbb{T}^{d}}\langle(D_{A,j}D_{A,i}-D_{A,i}D_{A,j})C_j,C_i\rangle_{\mathfrak{g}}\,dx\\
&=\int_{\mathbb{T}^{d}}\langle F_{A}^{ij}[C_i,C_j]\rangle_{\mathfrak{g}}\,dx.
\end{aligned} ∫ T d ( ⟨ D A , i C i , D A , j C j ⟩ g − ⟨ D A , i C j , D A , j C i ⟩ g ) d x = ∫ T d ⟨( D A , j D A , i − D A , i D A , j ) C j , C i ⟩ g d x = ∫ T d ⟨ F A ij [ C i , C j ] ⟩ g d x . The equality (114) follows. For (115) , note that
− ∑ i = 1 d ⟨ C i , 2 [ ( F A ) i j , C j ] ⟩ g = 2 ∑ i , j = 1 d ⟨ [ C i , C j ] , ( F A ) i j ⟩ g = 4 ∑ 1 ≤ i < j ≤ d ⟨ [ C i , C j ] , ( F A ) i j ⟩ g -\sum_{i=1}^{d}\langle C_i,2[(F_A)_{ij},C^j]\rangle_{\mathfrak{g}}
=2\sum_{i,j=1}^{d}\langle[C_i,C_j],(F_A)_{ij}\rangle_{\mathfrak{g}}
=4\sum_{1\le i<j\le d}\langle[C_i,C_j],(F_A)_{ij}\rangle_{\mathfrak{g}} − i = 1 ∑ d ⟨ C i , 2 [( F A ) ij , C j ] ⟩ g = 2 i , j = 1 ∑ d ⟨[ C i , C j ] , ( F A ) ij ⟩ g = 4 1 ≤ i < j ≤ d ∑ ⟨[ C i , C j ] , ( F A ) ij ⟩ g and
− ∑ i = 1 d ⟨ C i , Δ A C i ⟩ = ∑ i , j = 1 d ∥ D A , j C i ∥ L 2 2 . -\sum_{i=1}^{d}\langle C_i,\Delta_A C_i\rangle
=\sum_{i,j=1}^{d}\lVert D_{A,j}C_i\rVert_{L^2}^{2}. − i = 1 ∑ d ⟨ C i , Δ A C i ⟩ = i , j = 1 ∑ d ∥ D A , j C i ∥ L 2 2 . Combining this with (114) proves (115) . □
Identities for the Yang–Mills gradient flow Next, we discuss a fully covariant formulation of the Yang–Mills gradient flow, which is described in [Oh14, Oh15]. We find this formulation to be quite elegant, and moreover extremely useful for computations. To this end, we first recall the Yang–Mills gradient flow:
∂ t A i = D A ℓ ( F A ) ℓ i , i ∈ [ d ] . (116) \partial_t A_i=D_A^\ell(F_A)_{\ell i},\quad i\in[d]. \tag*{(116)} ∂ t A i = D A ℓ ( F A ) ℓ i , i ∈ [ d ] . (116) Here, A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d T d \mathbb{T}^{d} T d A 0 : [ 0 , T ] × T d → g A_0 : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g} A 0 : [ 0 , T ] × T d → g ( A α , α ∈ { 0 , … , d } ) (A_\alpha, \alpha\in\{0,\ldots,d\}) ( A α , α ∈ { 0 , … , d }) [ 0 , T ] × T d [0,T] \times\mathbb{T}^{d} [ 0 , T ] × T d
( F A ) 0 i = D A ℓ ( F A ) ℓ i , i ∈ [ d ] , (117) (F_A)_{0i} = D_A^\ell(F_A)_{\ell i}, \quad i \in[d], \tag*{(117)} ( F A ) 0 i = D A ℓ ( F A ) ℓ i , i ∈ [ d ] , (117) A 0 = 0. (118) A_0 = 0. \tag*{(118)} A 0 = 0. (118) Here, the index 0 denotes the time direction, so that ∂ 0 \partial_0 ∂ 0
( F A ) 0 i = ∂ 0 A i − ∂ i A 0 + [ A 0 , A i ] = ∂ 0 A i − D A , i A 0 , (F_A)_{0i} = \partial_0 A_i - \partial_i A_0 + [A_0,A_i] = \partial_0 A_i - D_{A,i}A_0, ( F A ) 0 i = ∂ 0 A i − ∂ i A 0 + [ A 0 , A i ] = ∂ 0 A i − D A , i A 0 , and thus since A 0 = 0 A_0 = 0 A 0 = 0 A 0 A_0 A 0
∂ 0 A i = D A ℓ ( F A ) ℓ i + D A , i A 0 , i ∈ [ d ] , \partial_0 A_i = D_A^\ell(F_A)_{\ell i} + D_{A,i}A_0, \quad i \in[d], ∂ 0 A i = D A ℓ ( F A ) ℓ i + D A , i A 0 , i ∈ [ d ] , and we recall (see e.g. the discussion around [CCHS22, Equation (1.6)]) that the extra pure gradient term D A , i A 0 D_{A,i}A_0 D A , i A 0 A A A A 0 = ∂ ℓ A ℓ A_0 = \partial^\ell A_\ell A 0 = ∂ ℓ A ℓ
The name “fully covariant formulation” arises from the fact that the equation (117) is covariant even under space-time gauge transformations. Under such transformations, the only thing that may change is the gauge condition (118) . In particular, given a space-time gauge transformation g : [ 0 , T ] × T d → G g : [0,T] \times\mathbb{T}^{d} \to G g : [ 0 , T ] × T d → G (117) by Ad g \operatorname{Ad}_g Ad g A ~ = g ∙ A \widetilde{A} = g \mathbin{\bullet} A A = g ∙ A
( F A ~ ) 0 i = Ad g ( F A ) 0 i = Ad g ( D A ℓ ( F A ) ℓ i ) = D A ~ ℓ ( F A ~ ) ℓ i , i ∈ [ d ] , (F_{\widetilde{A}})_{0i} = \operatorname{Ad}_g(F_A)_{0i} = \operatorname{Ad}_g(D_A^\ell(F_A)_{\ell i}) = D_{\widetilde{A}}^\ell(F_{\widetilde{A}})_{\ell i}, \quad i \in[d], ( F A ) 0 i = Ad g ( F A ) 0 i = Ad g ( D A ℓ ( F A ) ℓ i ) = D A ℓ ( F A ) ℓ i , i ∈ [ d ] , A ~ 0 = ( g ∙ A ) 0 = Ad g A 0 − ( ∂ 0 g ) g − 1 . \widetilde{A}_0 = (g \mathbin{\bullet} A)_0 = \operatorname{Ad}_g A_0 - (\partial_0 g)g^{-1}. A 0 = ( g ∙ A ) 0 = Ad g A 0 − ( ∂ 0 g ) g − 1 . As a sample computation, suppose that A ~ \widetilde{A} A
( F A ~ ) 0 i = D A ~ ℓ ( F A ~ ) ℓ i , i ∈ [ d ] (F_{\widetilde{A}})_{0i} = D_{\widetilde{A}}^\ell(F_{\widetilde{A}})_{\ell i}, \quad i \in[d] ( F A ) 0 i = D A ℓ ( F A ) ℓ i , i ∈ [ d ] A ~ 0 = ∂ ℓ A ~ ℓ . \widetilde{A}_0 = \partial^\ell\widetilde{A}_\ell. A 0 = ∂ ℓ A ℓ . Now suppose that we want to solve for a space-time gauge transformation g g g g ∙ A ~ g \mathbin{\bullet} \widetilde{A} g ∙ A g g g ( g ∙ A ~ ) 0 = 0 (g \mathbin{\bullet} \widetilde{A})_0 = 0 ( g ∙ A ) 0 = 0
( ∂ 0 g ) g − 1 = Ad g A ~ 0 = Ad g ( ∂ ℓ A ~ ℓ ) . (\partial_0 g)g^{-1} = \operatorname{Ad}_g\widetilde{A}_0 = \operatorname{Ad}_g(\partial^\ell\widetilde{A}_\ell). ( ∂ 0 g ) g − 1 = Ad g A 0 = Ad g ( ∂ ℓ A ℓ ) . We also remark that the fully covariant formulation is helpful when dealing with equations of the form
( ∂ t − Δ A ) ϕ = G , (\partial_t - \Delta_A)\phi= G, ( ∂ t − Δ A ) ϕ = G , where ϕ \phi ϕ G G G W \mathbf{W} W A A A
( D A , 0 − Δ A ) ϕ = G , (D_{A,0} - \Delta_A)\phi= G, ( D A , 0 − Δ A ) ϕ = G , and given a space-time gauge transformation g g g g g g G G G W \mathbf{W} W g g g
( D g ∙ A , 0 − Δ g ∙ A ) ( g ∙ ϕ ) = g ∙ G . (D_{g \mathbin{\bullet} A,0} - \Delta_{g \mathbin{\bullet} A})(g \mathbin{\bullet} \phi) = g \mathbin{\bullet} G. ( D g ∙ A , 0 − Δ g ∙ A ) ( g ∙ ϕ ) = g ∙ G . This may be converted back into the equation
( ∂ t − Δ g ∙ A ) ( g ∙ ϕ ) = ( ( ∂ t g ) g − 1 ) ( g ∙ ϕ ) + g ∙ G . (119) (\partial_t - \Delta_{g \mathbin{\bullet} A})(g \mathbin{\bullet} \phi) = ((\partial_t g)g^{-1})(g \mathbin{\bullet} \phi) + g \mathbin{\bullet} G. \tag*{(119)} ( ∂ t − Δ g ∙ A ) ( g ∙ ϕ ) = (( ∂ t g ) g − 1 ) ( g ∙ ϕ ) + g ∙ G . (119) Next, we discuss Yang–Mills connections, which are the stationary solutions to the Yang–Mills gradient flow.
Definition 3.25. Let A : T d → g d A:\mathbb{T}^{d}\to\mathfrak{g}^{d} A : T d → g d A A A
D A ℓ ( F A ) ℓ i = 0 , i ∈ [ d ] . D_{A}^{\ell}(F_{A})_{\ell i}=0,\qquad i\in[d]. D A ℓ ( F A ) ℓ i = 0 , i ∈ [ d ] . We refer to the above equations as the Yang–Mills equations.
Having introduced the fully covariant formulation of the Yang–Mills gradient flow, we next turn to deriving various identities associated to this flow. First, we record the Bianchi identity:
D A , α ( F A ) β γ + D A , β ( F A ) γ α + D A , γ ( F A ) α β = 0. (120) D_{A,\alpha}(F_{A})_{\beta\gamma}+D_{A,\beta}(F_{A})_{\gamma\alpha}+D_{A,\gamma}(F_{A})_{\alpha\beta}=0. \tag*{(120)} D A , α ( F A ) β γ + D A , β ( F A ) γ α + D A , γ ( F A ) α β = 0. (120) valid for any values α , β , γ \alpha,\beta,\gamma α , β , γ { 0 } ∪ [ d ] \{0\}\cup[d] { 0 } ∪ [ d ]
Lemma 3.27. Let A : [ 0 , T ] × T d → g d A:[0,T]\times\mathbb{T}^{d}\to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d (116) . Then ∂ t A \partial_tA ∂ t A D A j ∂ t A j = 0 D_{A}^{j}\partial_tA_j=0 D A j ∂ t A j = 0
Proof. We have that D A j ∂ t A j = D A j D A ℓ ( F A ) ℓ j D_{A}^{j}\partial_tA_j=D_{A}^{j}D_{A}^{\ell}(F_A)_{\ell j} D A j ∂ t A j = D A j D A ℓ ( F A ) ℓ j j ↔ ℓ j\leftrightarrow\ell j ↔ ℓ (6) , we obtain
D A j D A ℓ ( F A ) ℓ j = D A ℓ D A j ( F A ) j ℓ = − D A ℓ D A j ( F A ) ℓ j = − D A j D A ℓ ( F A ) ℓ j − [ F A ℓ j , ( F A ) ℓ j ] = − D A j D A ℓ ( F A ) ℓ j , \begin{aligned}
D_{A}^{j}D_{A}^{\ell}(F_A)_{\ell j}=D_{A}^{\ell}D_{A}^{j}(F_A)_{j\ell}=-D_{A}^{\ell}D_{A}^{j}(F_A)_{\ell j} \\
&=-D_{A}^{j}D_{A}^{\ell}(F_A)_{\ell j}-[F_A^{\ell j},(F_A)_{\ell j}]=-D_{A}^{j}D_{A}^{\ell}(F_A)_{\ell j},
\end{aligned} D A j D A ℓ ( F A ) ℓ j = D A ℓ D A j ( F A ) j ℓ = − D A ℓ D A j ( F A ) ℓ j = − D A j D A ℓ ( F A ) ℓ j − [ F A ℓ j , ( F A ) ℓ j ] = − D A j D A ℓ ( F A ) ℓ j , from which the desired result follows.
Lemma 3.28 (Equation for the curvature). Let A : [ 0 , T ] × T d → g d A:[0,T]\times\mathbb{T}^{d}\to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d (116) . Then the curvature F A F_A F A
( ∂ t − Δ A ) ( F A ) i j = 2 [ ( F A ) i ℓ , ( F A ) ℓ j ] . (\partial_t-\Delta_A)(F_A)_{ij}=2[(F_A)_{i}^{\ \ell},(F_A)_{\ell j}]. ( ∂ t − Δ A ) ( F A ) ij = 2 [( F A ) i ℓ , ( F A ) ℓ j ] . In the special case d = 2 d=2 d = 2 ( ∂ t − Δ A ) F A = 0 (\partial_t-\Delta_A)F_A=0 ( ∂ t − Δ A ) F A = 0
Proof. For simplicity, write F = F A F=F_A F = F A D = D A D=D_A D = D A (117) –(118) . By several applications of the Bianchi identity (120) and curvature commutation (6) , we obtain
D 0 F i j = − D i F j 0 − D j F 0 i = D i F 0 j − D j F 0 i = D i D ℓ F ℓ j − D j D ℓ F ℓ i = D ℓ D i F ℓ j − D j D ℓ F ℓ i + [ F i ℓ , F ℓ j ] = − D ℓ D ℓ F j i − D ℓ D j F i ℓ − D j D ℓ F ℓ i − [ F j ℓ , F i ℓ ] + [ F i ℓ , F ℓ j ] = Δ A F i j − D j D ℓ F i ℓ − D j D ℓ F ℓ i − [ F j ℓ , F i ℓ ] + [ F i ℓ , F ℓ j ] = Δ A F i j + 2 [ F i ℓ , F ℓ j ] , \begin{aligned}
D_0F_{ij}=-D_iF_{j0}-D_jF_{0i}=D_iF_{0j}-D_jF_{0i} \\
&=D_iD^{\ell}F_{\ell j}-D_jD^{\ell}F_{\ell i}=D^{\ell}D_iF_{\ell j}-D_jD^{\ell}F_{\ell i}+[F_i^{\ \ell},F_{\ell j}] \\
&=-D^{\ell}D_{\ell}F_{ji}-D^{\ell}D_jF_{i\ell}-D_jD^{\ell}F_{\ell i}-[F_j^{\ \ell},F_{i\ell}]+[F_i^{\ \ell},F_{\ell j}] \\
&=\Delta_AF_{ij}-D_jD^{\ell}F_{i\ell}-D_jD^{\ell}F_{\ell i}-[F_j^{\ \ell},F_{i\ell}]+[F_i^{\ \ell},F_{\ell j}] \\
&=\Delta_AF_{ij}+2[F_i^{\ \ell},F_{\ell j}],
\end{aligned} D 0 F ij = − D i F j 0 − D j F 0 i = D i F 0 j − D j F 0 i = D i D ℓ F ℓ j − D j D ℓ F ℓ i = D ℓ D i F ℓ j − D j D ℓ F ℓ i + [ F i ℓ , F ℓ j ] = − D ℓ D ℓ F j i − D ℓ D j F i ℓ − D j D ℓ F ℓ i − [ F j ℓ , F i ℓ ] + [ F i ℓ , F ℓ j ] = Δ A F ij − D j D ℓ F i ℓ − D j D ℓ F ℓ i − [ F j ℓ , F i ℓ ] + [ F i ℓ , F ℓ j ] = Δ A F ij + 2 [ F i ℓ , F ℓ j ] , as desired (note that D 0 = ∂ 0 D_0=\partial_0 D 0 = ∂ 0 A 0 = 0 A_0=0 A 0 = 0 d = 2 d=2 d = 2 i ≠ j i\ne j i = j F i ℓ , F ℓ j F_i^{\ \ell},F_{\ell j} F i ℓ , F ℓ j F 11 F_{11} F 11 F 22 F_{22} F 22
:::
Lemma 3.29 (Commutators for higher derivatives). Let A : [ 0 , T ] × T d → g d A:[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d A : [ 0 , T ] × T d → g d (116) . Let Ψ : [ 0 , T ] × T d → W \Psi:[0,T]\times\mathbb{T}^d\to W Ψ : [ 0 , T ] × T d → W W W W k ≥ 1 k\geq1 k ≥ 1
[ ( ∂ t − Δ A ) , D A ( k ) ] Ψ = ∑ j 1 + j 2 = k j 1 ≥ 0 , j 2 ≥ 1 O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . [(\partial_t-\Delta_A),D_A^{(k)}]\Psi=\sum_{\substack{j_1+j_2=k\\j_1\geq0,j_2\geq1}}\mathcal{O}\bigl((D_A^{(j_1)}F_A)(D_A^{(j_2)}\Psi)\bigr). [( ∂ t − Δ A ) , D A ( k ) ] Ψ = j 1 + j 2 = k j 1 ≥ 0 , j 2 ≥ 1 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . Here, as in Notation 3.3, D A ( k ) D_A^{(k)} D A ( k ) D A ℓ 1 ⋯ D A ℓ k D_A^{\ell_1}\cdots D_A^{\ell_k} D A ℓ 1 ⋯ D A ℓ k ℓ 1 , … , ℓ k ∈ [ d ] \ell_1,\ldots,\ell_k\in[d] ℓ 1 , … , ℓ k ∈ [ d ] A ~ : [ 0 , T ] × T d → g d \widetilde{A}:[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d A : [ 0 , T ] × T d → g d
∂ t A ~ j = − D A ~ ℓ j ( F A ~ ) ℓ j , j ∈ [ d ] , \partial_t\widetilde{A}_j=-D_{\widetilde{A}}^{\ell_j}(F_{\widetilde{A}})_{\ell_j},\qquad j\in[d], ∂ t A j = − D A ℓ j ( F A ) ℓ j , j ∈ [ d ] , then for any k ≥ 1 k\geq1 k ≥ 1
[ ( ∂ t − Δ A ~ ) , D A ~ ( k ) ] Ψ = ∑ j 1 + j 2 = k j 1 , j 2 ≥ 0 O ( ( D A ~ ( j 1 ) F A ~ ) ( D A ~ ( j 2 ) Ψ ) ) . [(\partial_t-\Delta_{\widetilde{A}}),D_{\widetilde{A}}^{(k)}]\Psi=\sum_{\substack{j_1+j_2=k\\j_1,j_2\geq0}}\mathcal{O}\bigl((D_{\widetilde{A}}^{(j_1)}F_{\widetilde{A}})(D_{\widetilde{A}}^{(j_2)}\Psi)\bigr). [( ∂ t − Δ A ) , D A ( k ) ] Ψ = j 1 + j 2 = k j 1 , j 2 ≥ 0 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . Remark 3.30. In particular, if Ψ \Psi Ψ
( ∂ t − Δ A ) Ψ = G , (\partial_t-\Delta_A)\Psi=G, ( ∂ t − Δ A ) Ψ = G , then Lemma 3.29 implies
( ∂ t − Δ A ) D A ( k ) Ψ = ∑ j 1 + j 2 = k j 1 ≥ 0 j 2 ≥ 1 O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) + D A ( k ) G . (\partial_t-\Delta_A)D_A^{(k)}\Psi=\sum_{\substack{j_1+j_2=k\\j_1\geq0\\j_2\geq1}}\mathcal{O}\bigl((D_A^{(j_1)}F_A)(D_A^{(j_2)}\Psi)\bigr)+D_A^{(k)}G. ( ∂ t − Δ A ) D A ( k ) Ψ = j 1 + j 2 = k j 1 ≥ 0 j 2 ≥ 1 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) + D A ( k ) G . If moreover G G G G = O ( F A Ψ ) G=\mathcal{O}(F_A\Psi) G = O ( F A Ψ )
( ∂ t − Δ A ) D A ( k ) Ψ = ∑ j 1 + j 2 = k j 1 , j 2 ≥ 0 O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . (\partial_t-\Delta_A)D_A^{(k)}\Psi=\sum_{\substack{j_1+j_2=k\\j_1,j_2\geq0}}\mathcal{O}\bigl((D_A^{(j_1)}F_A)(D_A^{(j_2)}\Psi)\bigr). ( ∂ t − Δ A ) D A ( k ) Ψ = j 1 + j 2 = k j 1 , j 2 ≥ 0 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . This follows by inductively applying the product formula (102) to show that D A ( k ) O ( F A Ψ ) D_A^{(k)}\mathcal{O}(F_A\Psi) D A ( k ) O ( F A Ψ ) j 1 , j 2 ≥ 0 j_1,j_2\geq0 j 1 , j 2 ≥ 0 j 1 ≥ 0 , j 2 ≥ 1 j_1\geq0,j_2\geq1 j 1 ≥ 0 , j 2 ≥ 1
( ∂ t − Δ A ~ ) Ψ ~ = O ( F A ~ Ψ ~ ) , (\partial_t-\Delta_{\widetilde{A}})\widetilde{\Psi}=\mathcal{O}(F_{\widetilde{A}}\widetilde{\Psi}), ( ∂ t − Δ A ) Ψ = O ( F A Ψ ) , then we have that
( ∂ t − Δ A ~ ) D A ~ ( k ) Ψ ~ = ∑ j 1 + j 2 = k j 1 , j 2 ≥ 0 O ( ( D A ~ ( j 1 ) F A ~ ) ( D A ~ ( j 2 ) Ψ ~ ) ) . (\partial_t-\Delta_{\widetilde{A}})D_{\widetilde{A}}^{(k)}\widetilde{\Psi}=\sum_{\substack{j_1+j_2=k\\j_1,j_2\geq0}}\mathcal{O}\bigl((D_{\widetilde{A}}^{(j_1)}F_{\widetilde{A}})(D_{\widetilde{A}}^{(j_2)}\widetilde{\Psi})\bigr). ( ∂ t − Δ A ) D A ( k ) Ψ = j 1 + j 2 = k j 1 , j 2 ≥ 0 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) . Proof of Lemma 3.29. To show the first claim, we argue by induction. The base case k = 1 k=1 k = 1 (105) , along with the fact that A A A k ≥ 1 k\geq1 k ≥ 1 (105) as well as the product formula (102) , we obtain
[ ∂ t − Δ A , D A ( k + 1 ) ] Ψ = [ ∂ t − Δ A , D A ] D A ( k ) Ψ + D A ( [ ∂ t − Δ A , D A ( k ) ] Ψ ) = O ( F A D A ( k + 1 ) Ψ ) + D A ∑ j 1 + j 2 = k j 1 ≥ 0 , j 2 ≥ 1 O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) = ∑ j 1 + j 2 = k + 1 j 1 ≥ 0 , j 2 ≥ 1 O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) , (121) \begin{aligned}
[\partial_t-\Delta_A,D_A^{(k+1)}]\Psi=[\partial_t-\Delta_A,D_A]D_A^{(k)}\Psi+D_A\bigl([\partial_t-\Delta_A,D_A^{(k)}]\Psi\bigr) \\
&= \mathcal{O}(F_A D_A^{(k+1)}\Psi)+D_A\sum_{\substack{j_1+j_2=k\\j_1\geq0,\ j_2\geq1}}\mathcal{O}\left((D_A^{(j_1)}F_A)(D_A^{(j_2)}\Psi)\right) \\
&=\sum_{\substack{j_1+j_2=k+1\\j_1\geq0,\ j_2\geq1}}\mathcal{O}\left((D_A^{(j_1)}F_A)(D_A^{(j_2)}\Psi)\right), \tag*{(121)}
\end{aligned} [ ∂ t − Δ A , D A ( k + 1 ) ] Ψ = [ ∂ t − Δ A , D A ] D A ( k ) Ψ + D A ( [ ∂ t − Δ A , D A ( k ) ] Ψ ) = O ( F A D A ( k + 1 ) Ψ ) + D A j 1 + j 2 = k j 1 ≥ 0 , j 2 ≥ 1 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) = j 1 + j 2 = k + 1 j 1 ≥ 0 , j 2 ≥ 1 ∑ O ( ( D A ( j 1 ) F A ) ( D A ( j 2 ) Ψ ) ) , (121) as desired. The second claim follows by a similar argument.
In order to control space-time derivatives, it will sometimes be useful to combine Lemma 3.29 and Remark 3.30 with the following result.
Lemma 3.31. Let A : [ 0 , T ] × T d → g d A:[0,T]\times\mathbb{T}^{d}\to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d (116) . Let Ψ : [ 0 , T ] × T d → W \Psi:[0,T]\times\mathbb{T}^{d}\to W Ψ : [ 0 , T ] × T d → W W W W
( ∂ t − Δ A ) Ψ = O ( F A Ψ ) . (122) (\partial_t-\Delta_A)\Psi=\mathcal{O}(F_A\Psi). \tag*{(122)} ( ∂ t − Δ A ) Ψ = O ( F A Ψ ) . (122) Then, it holds for all k ≥ 0 k\geq0 k ≥ 0
D A , t , x ( k ) Ψ = ∑ ℓ ≥ 0 j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ j 1 , … , j ℓ + 1 ≥ 0 O ( ( D A ( j 1 ) F A ) ⋯ ( D A ( j ℓ ) F A ) ( D A ( j ℓ + 1 ) Ψ ) ) . (123) D_{A,t,x}^{(k)}\Psi=\sum_{\substack{\ell\geq0\\j_1+\cdots+j_{\ell+1}=k-2\ell\\j_1,\ldots,j_{\ell+1}\geq0}}\mathcal{O}\left((D_A^{(j_1)}F_A)\cdots(D_A^{(j_\ell)}F_A)(D_A^{(j_{\ell+1})}\Psi)\right). \tag*{(123)} D A , t , x ( k ) Ψ = ℓ ≥ 0 j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ j 1 , … , j ℓ + 1 ≥ 0 ∑ O ( ( D A ( j 1 ) F A ) ⋯ ( D A ( j ℓ ) F A ) ( D A ( j ℓ + 1 ) Ψ ) ) . (123) Here, as in Notation 3.4, D A , t , x ( k ) Ψ D_{A,t,x}^{(k)}\Psi D A , t , x ( k ) Ψ D A α 1 ⋯ D A α m Ψ D_A^{\alpha_1}\cdots D_A^{\alpha_m}\Psi D A α 1 ⋯ D A α m Ψ m ≥ 0 m\geq0 m ≥ 0 α 1 , … , α m ∈ { 0 } ∪ [ d ] \alpha_1,\ldots,\alpha_m\in\{0\}\cup[d] α 1 , … , α m ∈ { 0 } ∪ [ d ] k k k
We emphasise that, unlike the left-hand side, the right-hand side of (3.29) only contains spatial covariant derivatives.
Proof. Similar to the proof of (105) , we may compute (in an intermediate step we apply Lemma 3.27 to conclude that D ℓ F 0 ℓ = 0 D^\ell F_{0\ell}=0 D ℓ F 0 ℓ = 0
[ ( ∂ t − Δ A ) , D A , 0 ] Ψ = − [ Δ A , D A , 0 ] Ψ = 2 ( F A ) 0 ℓ D A ℓ Ψ = O ( ( D A F A ) ( D A Ψ ) ) . (124) [(\partial_t-\Delta_A),D_{A,0}]\Psi=-[\Delta_A,D_{A,0}]\Psi=2(F_A)_{0\ell}D_A^\ell\Psi=\mathcal{O}((D_AF_A)(D_A\Psi)). \tag*{(124)} [( ∂ t − Δ A ) , D A , 0 ] Ψ = − [ Δ A , D A , 0 ] Ψ = 2 ( F A ) 0 ℓ D A ℓ Ψ = O (( D A F A ) ( D A Ψ )) . (124) From Lemma 3.28 , it follows that ∂ t F A = Δ A F A + O ( F A F A ) \partial_tF_A=\Delta_AF_A+\mathcal{O}(F_AF_A) ∂ t F A = Δ A F A + O ( F A F A ) (122) and (124) , this allows us to eliminate time derivatives acting on Ψ \Psi Ψ F A F_A F A F A F_A F A
Lemma 3.32 (Equation for the covariant exterior derivative). Let A : [ 0 , T ] × T d → g d A:[0,T]\times\mathbb{T}^{d}\to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d (116) . Let Ψ : [ 0 , T ] × T d → W d \Psi:[0,T]\times\mathbb{T}^{d}\to W^{d} Ψ : [ 0 , T ] × T d → W d
( ∂ t − Δ A ) Ψ i = 2 F A i ℓ Ψ ℓ , i ∈ [ d ] . (\partial_t-\Delta_A)\Psi^i=2F_A^{i\ell}\Psi_\ell,\quad i\in[d]. ( ∂ t − Δ A ) Ψ i = 2 F A i ℓ Ψ ℓ , i ∈ [ d ] . Define the covariant exterior derivative d A Ψ d_A\Psi d A Ψ ( d A Ψ ) i j ≔ D A , i Ψ j − D A , j Ψ i (d_A\Psi)_{ij}\coloneqq D_{A,i}\Psi_j-D_{A,j}\Psi_i ( d A Ψ ) ij : = D A , i Ψ j − D A , j Ψ i i , j ∈ [ d ] i,j\in[d] i , j ∈ [ d ] d A Ψ d_A\Psi d A Ψ
( ∂ t − Δ A ) ( d A Ψ ) i j = 2 ( ( F A ) i ℓ D A ℓ Ψ j − ( F A ) j ℓ D A ℓ Ψ i ) + 2 ( D A , i ( ( F A ) j ℓ Ψ ℓ ) − D A , j ( ( F A ) i ℓ Ψ ℓ ) ) = 2 ( F A ) i ℓ ( d A Ψ ) ℓ j − 2 ( F A ) j ℓ ( d A Ψ ) ℓ i − 2 ( D A ℓ ( F A ) i j ) Ψ ℓ . (\partial_t-\Delta_A)(d_A\Psi)_{ij}=2\left((F_A)_{i\ell}D_A^\ell\Psi_j-(F_A)_{j\ell}D_A^\ell\Psi_i\right)
+2\left(D_{A,i}((F_A)_{j\ell}\Psi^\ell)-D_{A,j}((F_A)_{i\ell}\Psi^\ell)\right)
=2(F_A)_i{}^\ell(d_A\Psi)_{\ell j}-2(F_A)_j{}^\ell(d_A\Psi)_{\ell i}-2(D_A^\ell(F_A)_{ij})\Psi_\ell. ( ∂ t − Δ A ) ( d A Ψ ) ij = 2 ( ( F A ) i ℓ D A ℓ Ψ j − ( F A ) j ℓ D A ℓ Ψ i ) + 2 ( D A , i (( F A ) j ℓ Ψ ℓ ) − D A , j (( F A ) i ℓ Ψ ℓ ) ) = 2 ( F A ) i ℓ ( d A Ψ ) ℓ j − 2 ( F A ) j ℓ ( d A Ψ ) ℓ i − 2 ( D A ℓ ( F A ) ij ) Ψ ℓ . In the special case d = 2 d=2 d = 2 curl A Ψ = D A , 1 Ψ 2 − D A , 2 Ψ 1 = ( d A Ψ ) 12 \operatorname{curl}_A\Psi=D_{A,1}\Psi_2-D_{A,2}\Psi_1=(d_A\Psi)_{12} curl A Ψ = D A , 1 Ψ 2 − D A , 2 Ψ 1 = ( d A Ψ ) 12
( ∂ t − Δ A ) curl A Ψ = 2 ( ∂ t A 1 ) Ψ 2 − 2 ( ∂ t A 2 ) Ψ 1 . (\partial_t-\Delta_A)\operatorname{curl}_A\Psi=2(\partial_tA_1)\Psi_2-2(\partial_tA_2)\Psi_1. ( ∂ t − Δ A ) curl A Ψ = 2 ( ∂ t A 1 ) Ψ 2 − 2 ( ∂ t A 2 ) Ψ 1 . Proof. For simplicity, write D = D A D=D_A D = D A F = F A F=F_A F = F A (105) , we have that
( ∂ t − Δ A ) D i Ψ j = 2 F i ℓ D ℓ Ψ j + 2 D i ( F j ℓ Ψ ℓ ) . (\partial_t-\Delta_A)D_i\Psi_j=2F_{i\ell}D^\ell\Psi_j+2D_i(F_{j\ell}\Psi^\ell). ( ∂ t − Δ A ) D i Ψ j = 2 F i ℓ D ℓ Ψ j + 2 D i ( F j ℓ Ψ ℓ ) . The first desired equation for ( d A Ψ ) i j (d_A\Psi)_{ij} ( d A Ψ ) ij D i Ψ j D_i\Psi_j D i Ψ j D j Ψ i D_j\Psi_i D j Ψ i
D i ( F j ℓ Ψ ℓ ) − D j ( F i ℓ Ψ ℓ ) = ( D i F j ℓ − D j F i ℓ ) Ψ ℓ + F j ℓ D i Ψ ℓ − F i ℓ D j Ψ ℓ . D_i(F_{j\ell}\Psi^\ell)-D_j(F_{i\ell}\Psi^\ell)=(D_iF_{j\ell}-D_jF_{i\ell})\Psi^\ell+F_{j\ell}D_i\Psi^\ell-F_{i\ell}D_j\Psi^\ell. D i ( F j ℓ Ψ ℓ ) − D j ( F i ℓ Ψ ℓ ) = ( D i F j ℓ − D j F i ℓ ) Ψ ℓ + F j ℓ D i Ψ ℓ − F i ℓ D j Ψ ℓ . By the antisymmetry of F F F (120) , we have that
D i F j ℓ − D j F i ℓ = D i F j ℓ + D j F ℓ i = − D ℓ F i j . D_iF_{j\ell}-D_jF_{i\ell}=D_iF_{j\ell}+D_jF_{\ell i}=-D_\ell F_{ij}. D i F j ℓ − D j F i ℓ = D i F j ℓ + D j F ℓ i = − D ℓ F ij . Combining the previous few identities, it follows that
F i ℓ D ℓ Ψ j − F j ℓ D ℓ Ψ i + D i ( F j ℓ Ψ ℓ ) − D j ( F i ℓ Ψ ℓ ) = F i ℓ ( D ℓ Ψ j − D j Ψ ℓ ) − F j ℓ ( D ℓ Ψ i − D i Ψ ℓ ) − ( D ℓ F i j ) Ψ ℓ , \begin{aligned}
F_{i\ell}D^\ell\Psi_j-F_{j\ell}D^\ell\Psi_i+D_i(F_{j\ell}\Psi^\ell)-D_j(F_{i\ell}\Psi^\ell)
&=F_{i\ell}(D^\ell\Psi_j-D_j\Psi^\ell)-F_{j\ell}(D^\ell\Psi_i-D_i\Psi^\ell)-(D_\ell F_{ij})\Psi^\ell,
\end{aligned} F i ℓ D ℓ Ψ j − F j ℓ D ℓ Ψ i + D i ( F j ℓ Ψ ℓ ) − D j ( F i ℓ Ψ ℓ ) = F i ℓ ( D ℓ Ψ j − D j Ψ ℓ ) − F j ℓ ( D ℓ Ψ i − D i Ψ ℓ ) − ( D ℓ F ij ) Ψ ℓ , which is the second desired equation.
In the case d = 2 d=2 d = 2 i = 1 i=1 i = 1 j = 2 j=2 j = 2 ( d A Ψ ) 11 = ( d A Ψ ) 22 = F 11 = F 22 = 0 (d_A\Psi)_{11}=(d_A\Psi)_{22}=F_{11}=F_{22}=0 ( d A Ψ ) 11 = ( d A Ψ ) 22 = F 11 = F 22 = 0
( ∂ t − Δ A ) curl A Ψ = − 2 ( D ℓ F 12 ) Ψ ℓ = − 2 ( D 1 F 12 ) Ψ 1 − 2 ( D 2 F 12 ) Ψ 2 . (\partial_t-\Delta_A)\operatorname{curl}_A\Psi=-2(D_\ell F_{12})\Psi^\ell=-2(D_1F_{12})\Psi^1-2(D_2F_{12})\Psi^2. ( ∂ t − Δ A ) curl A Ψ = − 2 ( D ℓ F 12 ) Ψ ℓ = − 2 ( D 1 F 12 ) Ψ 1 − 2 ( D 2 F 12 ) Ψ 2 . The desired equation now follows upon using that D 1 F 12 = ∂ t A 2 D_1F_{12}=\partial_tA_2 D 1 F 12 = ∂ t A 2 D 2 F 12 = − ∂ t A 1 D_2F_{12}=-\partial_tA_1 D 2 F 12 = − ∂ t A 1
Next, we state the following diamagnetic inequalities (which are also variously referred to as Kato’s inequality and the Bochner–Weitzenböck inequality). For proofs, see e.g. [Oh15, Lemmas 4.2 and 4.6].
Lemma 3.33 (Diamagnetic inequalities). Let A : T d → g d A:\mathbb{T}^d\to\mathfrak{g}^d A : T d → g d Ψ : T d → W \Psi:\mathbb{T}^d\to W Ψ : T d → W W W W
∣ ∇ ∣ Ψ ∣ W ∣ ≤ ∣ D A Ψ ∣ W . (125) |\nabla|\Psi|_W|\leq|D_A\Psi|_W. \tag*{(125)} ∣∇∣Ψ ∣ W ∣ ≤ ∣ D A Ψ ∣ W . (125) Let B : [ 0 , T ] × T d → g d B:[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d B : [ 0 , T ] × T d → g d Ψ , G : [ 0 , T ] × T d → W \Psi,G:[0,T]\times\mathbb{T}^d\to W Ψ , G : [ 0 , T ] × T d → W ( ∂ t − Δ B ) Ψ = G (\partial_t-\Delta_B)\Psi=G ( ∂ t − Δ B ) Ψ = G
( ∂ t − Δ ) ∣ Ψ ∣ W ≤ ∣ G ∣ W . (126) (\partial_t-\Delta)|\Psi|_W\leq|G|_W. \tag*{(126)} ( ∂ t − Δ ) ∣Ψ ∣ W ≤ ∣ G ∣ W . (126) Remark 3.34. Combining Lemma 3.28 with the parabolic diamagnetic inequality (126) , we see that in dimension d = 2 d=2 d = 2 A A A (116) , the norm of its curvature ∣ F A 12 ∣ |F_A^{12}| ∣ F A 12 ∣ ( ∂ t − Δ ) ∣ F A 12 ∣ ≤ 0 (\partial_t-\Delta)|F_A^{12}|\leq0 ( ∂ t − Δ ) ∣ F A 12 ∣ ≤ 0
Parallel transport
In this section, we recall and discuss the parallel transport associated to a connection one-form A A A × \times × § 5.5 ) behind the decay mechanism that we uncover.
Throughout this section, fix a vector space W W W π \pi π G G G W W W W = g W=\mathfrak{g} W = g π = Ad \pi=\operatorname{Ad} π = Ad W = V W=V W = V π = ϱ \pi=\varrho π = ϱ
Definition 3.35 (Parallel transport). Let A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d A 0 ≔ 0 A_{0} \coloneqq0 A 0 : = 0 A : [ 0 , T ] × T d → g d + 1 A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d+1} A : [ 0 , T ] × T d → g d + 1 ( 0 , y 0 ) (0,y_{0}) ( 0 , y 0 ) B 1 / 10 ( y 0 ) B_{1/10}(y_{0}) B 1/10 ( y 0 ) 1 / 10 1/10 1/10 y 0 y_{0} y 0 T d \mathbb{T}^{d} T d τ W , A ( 0 , y 0 ; ⋅ , ⋅ ) : [ 0 , T ] × B 1 / 10 ( y 0 ) → π ( G ) \tau_{W,A}(0,y_{0};\mathord\cdot,\mathord\cdot) : [0,T] \times B_{1/10}(y_{0}) \to\pi(G) τ W , A ( 0 , y 0 ; ⋅ , ⋅ ) : [ 0 , T ] × B 1/10 ( y 0 ) → π ( G ) π ( G ) \pi(G) π ( G ) τ W , A ( 0 , y 0 ; 0 , y 0 ) = id W \tau_{W,A}(0,y_{0};0,y_{0}) = \operatorname{id}_{W} τ W , A ( 0 , y 0 ; 0 , y 0 ) = id W ( t , x ) ∈ [ 0 , T ] × B 1 / 10 ( y 0 ) (t,x) \in[0,T] \times B_{1/10}(y_{0}) ( t , x ) ∈ [ 0 , T ] × B 1/10 ( y 0 ) ( t , x ) ≠ ( 0 , y 0 ) (t,x) \ne(0,y_{0}) ( t , x ) = ( 0 , y 0 )
D A , ( t , x − y 0 ) τ W , A ( 0 , y 0 ; t , x ) = 0. D_{A,(t,x-y_{0})}\tau_{W,A}(0,y_{0};t,x)=0. D A , ( t , x − y 0 ) τ W , A ( 0 , y 0 ; t , x ) = 0. These equations may be equivalently viewed as transport equations along rays emanating from the base point ( 0 , y 0 ) (0,y_{0}) ( 0 , y 0 ) v ∈ R 1 + d v \in\mathbb{R}^{1+d} v ∈ R 1 + d γ ( t ) = ( 0 , y 0 ) + t v \gamma(t)=(0,y_{0})+tv γ ( t ) = ( 0 , y 0 ) + t v t t t γ ( t ) ∈ [ 0 , T ] × B 1 / 10 ( y 0 ) \gamma(t) \in[0,T] \times B_{1/10}(y_{0}) γ ( t ) ∈ [ 0 , T ] × B 1/10 ( y 0 ) τ W , A \tau_{W,A} τ W , A
D A , γ ˙ ( t ) τ W , A ( 0 , y 0 ; γ ( t ) ) = 0 , τ W , A ( 0 , y 0 ; γ ( 0 ) ) = id W . (127) D_{A,\dot{\gamma}(t)}\tau_{W,A}(0,y_{0};\gamma(t))=0,\qquad\tau_{W,A}(0,y_{0};\gamma(0))=\operatorname{id}_{W}. \tag*{(127)} D A , γ ˙ ( t ) τ W , A ( 0 , y 0 ; γ ( t )) = 0 , τ W , A ( 0 , y 0 ; γ ( 0 )) = id W . (127) More explicitly, the first equation may be written:
D A α τ W , A ( 0 , y 0 ; γ ( t ) ) γ ˙ α ( t ) = D A 0 τ W , A ( 0 , y 0 ; γ ( t ) ) γ ˙ 0 ( t ) + D A ℓ τ W , A ( 0 , y 0 ; γ ( t ) ) γ ˙ ℓ ( t ) = 0. D_{A}^{\alpha}\tau_{W,A}(0,y_{0};\gamma(t))\dot{\gamma}_{\alpha}(t)=D_{A}^{0}\tau_{W,A}(0,y_{0};\gamma(t))\dot{\gamma}_{0}(t)+D_{A}^{\ell}\tau_{W,A}(0,y_{0};\gamma(t))\dot{\gamma}_{\ell}(t)=0. D A α τ W , A ( 0 , y 0 ; γ ( t )) γ ˙ α ( t ) = D A 0 τ W , A ( 0 , y 0 ; γ ( t )) γ ˙ 0 ( t ) + D A ℓ τ W , A ( 0 , y 0 ; γ ( t )) γ ˙ ℓ ( t ) = 0. Here, to define the covariant differentiation, we think of τ W , A \tau_{W,A} τ W , A End ( W ) \operatorname{End}(W) End ( W ) G G G g ∈ G g \in G g ∈ G g g g M ∈ End ( W ) M \in\operatorname{End}(W) M ∈ End ( W ) π ( g ) M \pi(g)M π ( g ) M
Notation 3.36. For notational brevity, we will often omit the subscript W W W τ A \tau_{A} τ A τ W , A \tau_{W,A} τ W , A
(e) By (d), we have that τ A − 1 = τ A ∗ \tau_A^{-1}=\tau_A^* τ A − 1 = τ A ∗ τ A ∗ \tau_A^* τ A ∗ E ( 0 , y 0 ) s t ⊗ ( E s t ) ∗ E^{\mathrm{st}}_{(0,y_0)}\otimes(E^{\mathrm{st}})^* E ( 0 , y 0 ) st ⊗ ( E st ) ∗ τ A ∗ ∈ End ( W ) \tau_A^*\in\operatorname{End}(\mathbf{W}) τ A ∗ ∈ End ( W ) G \mathbf{G} G g ∈ G g\in\mathbf{G} g ∈ G M ∈ End ( W ) M\in\operatorname{End}(\mathbf{W}) M ∈ End ( W ) M π ( g ) ∗ M\pi(g)^* M π ( g ) ∗
D A ( τ A − 1 ) = ∂ ( τ A − 1 ) + τ A − 1 π ∗ ( A ) ∗ = ∂ ( τ A − 1 ) − τ A − 1 π ∗ ( A ) . (128) D_A(\tau_A^{-1})=\partial(\tau_A^{-1})+\tau_A^{-1}\pi_*(A)^*=\partial(\tau_A^{-1})-\tau_A^{-1}\pi_*(A). \tag*{(128)} D A ( τ A − 1 ) = ∂ ( τ A − 1 ) + τ A − 1 π ∗ ( A ) ∗ = ∂ ( τ A − 1 ) − τ A − 1 π ∗ ( A ) . (128) Now by differentiating the equation id W = τ A τ A − 1 \operatorname{id}_{\mathbf{W}}=\tau_A\tau_A^{-1} id W = τ A τ A − 1 (102) (where the Z \mathbf{Z} Z End ( W ) \operatorname{End}(\mathbf{W}) End ( W ) G \mathbf{G} G Z = End ( W ) \mathbf{Z}=\operatorname{End}(\mathbf{W}) Z = End ( W )
0 = ( D A τ A ) τ A − 1 + τ A D A ( τ A − 1 ) , 0=(D_A\tau_A)\tau_A^{-1}+\tau_A D_A(\tau_A^{-1}), 0 = ( D A τ A ) τ A − 1 + τ A D A ( τ A − 1 ) , which implies that
D A ( τ A − 1 ) = − τ A − 1 ( D A τ A ) τ A − 1 . (129) D_A(\tau_A^{-1})=-\tau_A^{-1}(D_A\tau_A)\tau_A^{-1}. \tag*{(129)} D A ( τ A − 1 ) = − τ A − 1 ( D A τ A ) τ A − 1 . (129) In particular, τ A − 1 \tau_A^{-1} τ A − 1 γ \gamma γ
D A , γ ˙ ( t ) ( τ A ( 0 , y 0 ; γ ( t ) ) − 1 ) = 0. (130) D_{A,\dot{\gamma}(t)}\bigl(\tau_A(0,y_0;\gamma(t))^{-1}\bigr)=0. \tag*{(130)} D A , γ ˙ ( t ) ( τ A ( 0 , y 0 ; γ ( t ) ) − 1 ) = 0. (130) The existence of parallel transport follows from standard ODE arguments. In particular, for a time t 0 t_0 t 0 γ ( t 0 ) \gamma(t_0) γ ( t 0 ) B 1 / 10 ( y 0 ) B_{1/10}(y_0) B 1/10 ( y 0 ) h : [ 0 , t 0 ] → π ( G ) h:[0,t_0]\to\pi(\mathbf{G}) h : [ 0 , t 0 ] → π ( G )
h ˙ ( t ) = − π ∗ ( A ( γ ( t ) ) ) ⋅ γ ˙ ( t ) h ( t ) = − π ∗ ( A α ( γ ( t ) ) ) γ ˙ α ( t ) h ( t ) , h ( 0 ) = id W . (131) \dot{h}(t)=-\pi_*(A(\gamma(t)))\cdot\dot{\gamma}(t)h(t)=-\pi_*(A_\alpha(\gamma(t)))\dot{\gamma}^{\alpha}(t)h(t),\qquad h(0)=\operatorname{id}_{\mathbf{W}}. \tag*{(131)} h ˙ ( t ) = − π ∗ ( A ( γ ( t ))) ⋅ γ ˙ ( t ) h ( t ) = − π ∗ ( A α ( γ ( t ))) γ ˙ α ( t ) h ( t ) , h ( 0 ) = id W . (131) Setting τ A ( 0 , y 0 ; γ ( t ) ) = h ( t ) \tau_A(0,y_0;\gamma(t))=h(t) τ A ( 0 , y 0 ; γ ( t )) = h ( t ) (127) . Towards this end, let t > 0 t>0 t > 0
( ∂ γ ˙ ( t ) τ A ) ( 0 , y 0 ; γ ( t ) ) = d d s τ A ( 0 , y 0 ; γ ( t + s ) ) ∣ s = 0 = d d s h ( t + s ) ∣ s = 0 = h ˙ ( t ) = − π ∗ ( A ( γ ( t ) ) ) ⋅ γ ˙ ( t ) h ( t ) , (\partial_{\dot{\gamma}(t)}\tau_A)(0,y_0;\gamma(t))=\left.\frac{d}{ds}\tau_A(0,y_0;\gamma(t+s))\right|_{s=0}=\left.\frac{d}{ds}h(t+s)\right|_{s=0}=\dot{h}(t)=-\pi_*(A(\gamma(t)))\cdot\dot{\gamma}(t)h(t), ( ∂ γ ˙ ( t ) τ A ) ( 0 , y 0 ; γ ( t )) = d s d τ A ( 0 , y 0 ; γ ( t + s )) s = 0 = d s d h ( t + s ) s = 0 = h ˙ ( t ) = − π ∗ ( A ( γ ( t ))) ⋅ γ ˙ ( t ) h ( t ) , from which (127) follows upon moving the right-hand side to the left-hand side, and recalling the definition of D A D_A D A
We remark that τ A ( 0 , y 0 ; γ ( t ) ) \tau_A(0,y_0;\gamma(t)) τ A ( 0 , y 0 ; γ ( t )) γ ~ ( t ) = γ ( c t ) \widetilde{\gamma}(t)=\gamma(ct) γ ( t ) = γ ( c t ) c > 0 c>0 c > 0 h ~ \widetilde{h} h γ \gamma γ γ ~ \widetilde{\gamma} γ h ~ ( t / c ) = h ( t ) \widetilde{h}(t/c)=h(t) h ( t / c ) = h ( t )
In the following, given a space-time line segment γ ( t ) = ( 0 , y 0 ) + t v \gamma(t)=(0,y_0)+tv γ ( t ) = ( 0 , y 0 ) + t v t ≥ 0 t\geq0 t ≥ 0 t ∣ v x ∣ ≤ 1 / 10 t|v_x|\leq1/10 t ∣ v x ∣ ≤ 1/10 v x v_x v x v ∈ R 1 + d v\in\mathbb{R}^{1+d} v ∈ R 1 + d τ A ( γ ( t ) ) = τ A ( 0 , y 0 ; γ ( t ) ) \tau_A(\gamma(t))=\tau_A(0,y_0;\gamma(t)) τ A ( γ ( t )) = τ A ( 0 , y 0 ; γ ( t )) 0 ≤ s ≤ t 0\leq s\leq t 0 ≤ s ≤ t
τ A ( γ ( s ) ; γ ( t ) ) ≔ τ A ( γ ( t ) ) τ A ( γ ( s ) ) − 1 . \tau_A(\gamma(s);\gamma(t))\coloneqq\tau_A(\gamma(t))\tau_A(\gamma(s))^{-1}. τ A ( γ ( s ) ; γ ( t )) : = τ A ( γ ( t )) τ A ( γ ( s ) ) − 1 . Note in particular that τ A ( 0 , y 0 ; γ ( t ) ) = τ A ( γ ( 0 ) ; γ ( t ) ) \tau_A(0,y_0;\gamma(t))=\tau_A(\gamma(0);\gamma(t)) τ A ( 0 , y 0 ; γ ( t )) = τ A ( γ ( 0 ) ; γ ( t ))
Lemma 3.39 (Transport lemma). Let A : [ 0 , T ] × T d → g d A: [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d Ψ : [ 0 , T ] × T d → W \Psi: [0,T] \times\mathbb{T}^{d} \to W Ψ : [ 0 , T ] × T d → W W W W y 0 ∈ T d y_{0} \in\mathbb{T}^{d} y 0 ∈ T d v = ( v 0 , v x ) ∈ R 1 + d v=(v_{0},v_{x}) \in\mathbb{R}^{1+d} v = ( v 0 , v x ) ∈ R 1 + d v 0 ≥ 0 v_{0} \ge0 v 0 ≥ 0 γ ( t ) = ( 0 , y 0 ) + t v \gamma(t)=(0,y_{0})+tv γ ( t ) = ( 0 , y 0 ) + t v t ≥ 0 t \ge0 t ≥ 0 t v 0 ≤ T tv_{0} \le T t v 0 ≤ T t ∣ v x ∣ ≤ 1 / 10 t|v_{x}| \le1/10 t ∣ v x ∣ ≤ 1/10
Ψ ( γ ( t ) ) = τ A ( 0 , y 0 ; γ ( t ) ) Ψ ( 0 , y 0 ) + ∫ 0 t d s τ A ( γ ( s ) ; γ ( t ) ) D A , γ ˙ ( s ) Ψ ( γ ( s ) ) . \Psi(\gamma(t))=\tau_{A}(0,y_{0};\gamma(t))\Psi(0,y_{0})+\int_{0}^{t} ds\,\tau_{A}(\gamma(s);\gamma(t))D_{A,\dot{\gamma}(s)}\Psi(\gamma(s)). Ψ ( γ ( t )) = τ A ( 0 , y 0 ; γ ( t )) Ψ ( 0 , y 0 ) + ∫ 0 t d s τ A ( γ ( s ) ; γ ( t )) D A , γ ˙ ( s ) Ψ ( γ ( s )) . Remark 3.40. One may view this as a geometric fundamental theorem of calculus, since it expresses the difference between the values of Ψ \Psi Ψ γ \gamma γ Ψ \Psi Ψ Ψ ( 0 , y 0 ) \Psi(0,y_{0}) Ψ ( 0 , y 0 ) τ A ( 0 , y 0 ; γ ( t ) ) Ψ ( 0 , y 0 ) \tau_{A}(0,y_{0};\gamma(t))\Psi(0,y_{0}) τ A ( 0 , y 0 ; γ ( t )) Ψ ( 0 , y 0 )
Ψ ( γ ( t ) ) − Ψ ( 0 , y 0 ) \Psi(\gamma(t))-\Psi(0,y_{0}) Ψ ( γ ( t )) − Ψ ( 0 , y 0 ) does not make geometric sense. Geometrically, Ψ \Psi Ψ
Proof. Recall our notation τ A ( γ ( t ) ) = τ A ( 0 , y 0 ; γ ( t ) ) \tau_{A}(\gamma(t))=\tau_{A}(0,y_{0};\gamma(t)) τ A ( γ ( t )) = τ A ( 0 , y 0 ; γ ( t )) τ A ( γ ( t ) ) − 1 Ψ ( γ ( t ) ) \tau_{A}(\gamma(t))^{-1}\Psi(\gamma(t)) τ A ( γ ( t ) ) − 1 Ψ ( γ ( t )) (102) (with the Z Z Z W W W G G G Z = W Z=W Z = W (130) (recalling that covariant differentiation of τ A − 1 \tau_{A}^{-1} τ A − 1 (128) ), we have that
d d t ( τ A ( γ ( t ) ) − 1 Ψ ( γ ( t ) ) ) = D A , γ ˙ ( t ) ( τ A ( γ ( t ) ) − 1 ) Ψ ( γ ( t ) ) + τ A ( γ ( t ) ) − 1 D A , γ ˙ ( t ) Ψ ( γ ( t ) ) = τ A ( γ ( t ) ) − 1 D A , γ ˙ ( t ) Ψ ( γ ( t ) ) . \begin{aligned}
\frac{d}{dt}\left(\tau_{A}(\gamma(t))^{-1}\Psi(\gamma(t))\right)=D_{A,\dot{\gamma}(t)}(\tau_{A}(\gamma(t))^{-1})\Psi(\gamma(t))+\tau_{A}(\gamma(t))^{-1}D_{A,\dot{\gamma}(t)}\Psi(\gamma(t)) \\
&=\tau_{A}(\gamma(t))^{-1}D_{A,\dot{\gamma}(t)}\Psi(\gamma(t)).
\end{aligned} d t d ( τ A ( γ ( t ) ) − 1 Ψ ( γ ( t )) ) = D A , γ ˙ ( t ) ( τ A ( γ ( t ) ) − 1 ) Ψ ( γ ( t )) + τ A ( γ ( t ) ) − 1 D A , γ ˙ ( t ) Ψ ( γ ( t )) = τ A ( γ ( t ) ) − 1 D A , γ ˙ ( t ) Ψ ( γ ( t )) . Upon integrating, we obtain
τ A ( γ ( t ) ) − 1 Ψ ( γ ( t ) ) − Ψ ( γ ( 0 ) ) = ∫ 0 t d s τ A ( γ ( s ) ) − 1 D A , γ ˙ ( s ) Ψ ( γ ( s ) ) . \tau_{A}(\gamma(t))^{-1}\Psi(\gamma(t))-\Psi(\gamma(0))=\int_{0}^{t} ds\,\tau_{A}(\gamma(s))^{-1}D_{A,\dot{\gamma}(s)}\Psi(\gamma(s)). τ A ( γ ( t ) ) − 1 Ψ ( γ ( t )) − Ψ ( γ ( 0 )) = ∫ 0 t d s τ A ( γ ( s ) ) − 1 D A , γ ˙ ( s ) Ψ ( γ ( s )) . Rearranging and using that τ A ( γ ( t ) ) τ A ( γ ( s ) ) − 1 = τ A ( γ ( s ) ; γ ( t ) ) \tau_{A}(\gamma(t))\tau_{A}(\gamma(s))^{-1}=\tau_{A}(\gamma(s);\gamma(t)) τ A ( γ ( t )) τ A ( γ ( s ) ) − 1 = τ A ( γ ( s ) ; γ ( t ))
We will derive more results about parallel transport in § 5.3 . We do not prove those results here, as this section is intended to be short and introductory.
Covariance and induced linear maps
We recall that the real vector space V V V ϱ : G → End ( V ) \varrho:G \to\operatorname{End}(V) ϱ : G → End ( V ) ϱ ∗ \varrho_{*} ϱ ∗
ϱ ∗ : g → End ( V ) , \varrho_{*}:\mathfrak{g}\to\operatorname{End}(V), ϱ ∗ : g → End ( V ) , which is defined as ϱ ∗ ( A ) = d d ε ϱ ( e ε A ) ∣ ε = 0 \varrho_{*}(A)=\left.\frac{d}{d\varepsilon}\varrho(e^{\varepsilon A})\right|_{\varepsilon=0} ϱ ∗ ( A ) = d ε d ϱ ( e ε A ) ε = 0 ϱ \varrho ϱ ϱ ∗ \varrho_{*} ϱ ∗ g ∈ G g \in G g ∈ G A ∈ g A \in\mathfrak{g} A ∈ g Φ ∈ V \Phi\in V Φ ∈ V
g Φ = ϱ ( g ) Φ and A Φ = ϱ ∗ ( A ) Φ . g\Phi=\varrho(g)\Phi\qquad\text{and}\qquad A\Phi=\varrho_{*}(A)\Phi. g Φ = ϱ ( g ) Φ and A Φ = ϱ ∗ ( A ) Φ. Definition 3.41. We let ( u a ) a (u_{a})_{a} ( u a ) a ( v m ) m (v_{m})_{m} ( v m ) m g \mathfrak{g} g V V V Cov g ∈ g ⊗ g \operatorname{Cov}_{\mathfrak{g}} \in\mathfrak{g}\otimes\mathfrak{g} Cov g ∈ g ⊗ g Cov V ∈ V ⊗ V \operatorname{Cov}_{V} \in V\otimes V Cov V ∈ V ⊗ V
Cov g ≔ u a ⊗ u a and Cov V ≔ v m ⊗ v m . (132) \operatorname{Cov}_{\mathfrak{g}}\coloneqq u_{a}\otimes u^{a}\qquad\text{and}\qquad\operatorname{Cov}_{V}\coloneqq v_{m}\otimes v^{m}. \tag*{(132)} Cov g : = u a ⊗ u a and Cov V : = v m ⊗ v m . (132) (Summation over repeated indices is implicit.) For all A ∈ g A \in\mathfrak{g} A ∈ g Φ ∈ V \Phi\in V Φ ∈ V
Cov g g ( A ) ≔ [ u a , [ u a , A ] ] , Cov V g ( A ) ≔ B ( v m ⊗ ϱ ∗ ( A ) v m ) , (133) \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}}(A) \coloneqq[u^{a},[u_{a},A]], \qquad\operatorname{Cov}_{V}^{\mathfrak{g}}(A) \coloneqq\mathbf{B}(v_{m}\otimes\varrho_{*}(A)v^{m}), \tag*{(133)} Cov g g ( A ) : = [ u a , [ u a , A ]] , Cov V g ( A ) : = B ( v m ⊗ ϱ ∗ ( A ) v m ) , (133) Cov g V ( Φ ) ≔ ϱ ∗ ( u a ) ϱ ∗ ( u a ) Φ , Cov V V ( Φ ) ≔ ⟨ v m , Φ ⟩ V v m = Φ . (134) \operatorname{Cov}_{\mathfrak{g}}^{V}(\Phi) \coloneqq\varrho_{*}(u_{a})\varrho_{*}(u^{a})\Phi, \qquad\operatorname{Cov}_{V}^{V}(\Phi) \coloneqq\langle v_{m},\Phi\rangle_{V}v^{m}=\Phi. \tag*{(134)} Cov g V ( Φ ) : = ϱ ∗ ( u a ) ϱ ∗ ( u a ) Φ , Cov V V ( Φ ) : = ⟨ v m , Φ ⟩ V v m = Φ. (134) In (133) –(134) , the subscript is the space one takes a basis of and the superscript is the space the map acts on. We note that Cov g g \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}} Cov g g (133) –(134) .
In order to apply our gauge covariance result (Lemma 2.13 ) with our choice of the renormalisation constants from Definition 2.15 , we need the following lemma.
Lemma 3.42. The four maps Cov g g \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}} Cov g g Cov V g \operatorname{Cov}_{V}^{\mathfrak{g}} Cov V g Cov g V \operatorname{Cov}_{\mathfrak{g}}^{V} Cov g V Cov V V \operatorname{Cov}_{V}^{V} Cov V V G G G
Proof. Since the proofs for the four different maps are quite similar, we only present the details for Cov g V \operatorname{Cov}_{\mathfrak{g}}^{V} Cov g V g ∈ G g \in G g ∈ G Φ ∈ V \Phi\in V Φ ∈ V (100) , we have that
Cov g V ( ϱ ( g ) Φ ) = ϱ ∗ ( u a ) ϱ ∗ ( u a ) ϱ ( g ) Φ = ϱ ( g ) ϱ ∗ ( Ad g − 1 u a ) ϱ ∗ ( Ad g − 1 u a ) Φ . \operatorname{Cov}_{\mathfrak{g}}^{V}(\varrho(g)\Phi)=\varrho_{*}(u_{a})\varrho_{*}(u^{a})\varrho(g)\Phi=\varrho(g)\varrho_{*}(\operatorname{Ad}_{g^{-1}}u_{a})\varrho_{*}(\operatorname{Ad}_{g^{-1}}u^{a})\Phi. Cov g V ( ϱ ( g ) Φ ) = ϱ ∗ ( u a ) ϱ ∗ ( u a ) ϱ ( g ) Φ = ϱ ( g ) ϱ ∗ ( Ad g − 1 u a ) ϱ ∗ ( Ad g − 1 u a ) Φ. Since the adjoint representation is orthogonal, ( Ad g − 1 u a ) a (\operatorname{Ad}_{g^{-1}}u_{a})_{a} ( Ad g − 1 u a ) a g \mathfrak{g} g ( Ad g − 1 u a ) a (\operatorname{Ad}_{g^{-1}}u_{a})_{a} ( Ad g − 1 u a ) a ( u a ) (u_{a}) ( u a )
ϱ ∗ ( Ad g − 1 u a ) ϱ ∗ ( Ad g − 1 u a ) = ϱ ∗ ( u a ) ϱ ∗ ( u a ) , \varrho_{*}(\operatorname{Ad}_{g^{-1}}u_{a})\varrho_{*}(\operatorname{Ad}_{g^{-1}}u^{a})=\varrho_{*}(u_{a})\varrho_{*}(u^{a}), ϱ ∗ ( Ad g − 1 u a ) ϱ ∗ ( Ad g − 1 u a ) = ϱ ∗ ( u a ) ϱ ∗ ( u a ) , which then implies the desired result. □ \square □
Example 3.43. Let us consider the case G = SU ( N ) G=\operatorname{SU}(N) G = SU ( N ) g = s u ( N ) \mathfrak{g}=\mathfrak{su}(N) g = su ( N ) V = C N V=\mathbb{C}^{N} V = C N ϱ \varrho ϱ s u ( N ) ⊆ C N × N \mathfrak{su}(N)\subseteq\mathbb{C}^{N\times N} su ( N ) ⊆ C N × N C N \mathbb{C}^{N} C N dim R ( s u ( N ) ) = N 2 − 1 \dim_{\mathbb{R}}(\mathfrak{su}(N))=N^{2}-1 dim R ( su ( N )) = N 2 − 1 dim R ( C N ) = 2 N \dim_{\mathbb{R}}(\mathbb{C}^{N})=2N dim R ( C N ) = 2 N
The standard inner products on s u ( N ) \mathfrak{su}(N) su ( N ) C N \mathbb{C}^{N} C N
⟨ A , B ⟩ s u ( N ) = Tr ( A † B ) and ⟨ u , v ⟩ C N = Re ( u † v ) , \langle A,B\rangle_{\mathfrak{su}(N)}=\operatorname{Tr}(A^{\dagger}B) \qquad\text{and}\qquad\langle u,v\rangle_{\mathbb{C}^{N}}=\operatorname{Re}(u^{\dagger}v), ⟨ A , B ⟩ su ( N ) = Tr ( A † B ) and ⟨ u , v ⟩ C N = Re ( u † v ) , where Tr \operatorname{Tr} Tr † \dagger † C N × M → C M × N \mathbb{C}^{N\times M}\to\mathbb{C}^{M\times N} C N × M → C M × N C N \mathbb{C}^{N} C N ( v m ) = { e j } j ≤ N ∪ { i e j } j ≤ N (v_{m})=\{e_{j}\}_{j\leq N}\cup\{ie_{j}\}_{j\leq N} ( v m ) = { e j } j ≤ N ∪ { i e j } j ≤ N j j j e j ∈ C N e_{j}\in\mathbb{C}^{N} e j ∈ C N A ∈ s u ( N ) A\in\mathfrak{su}(N) A ∈ su ( N ) v , w ∈ C N v,w\in\mathbb{C}^{N} v , w ∈ C N
⟨ B ( v ⊗ w ) , A ⟩ s u ( N ) = ⟨ v , A w ⟩ C N = Re ( v † A w ) = 1 2 ( v † A w − w † A v ) = − Tr ( 1 2 ( v w † − w v † ) A ) . \langle\mathbf{B}(v\otimes w),A\rangle_{\mathfrak{su}(N)}=\langle v,Aw\rangle_{\mathbb{C}^{N}}=\operatorname{Re}(v^{\dagger}Aw)=\frac{1}{2}(v^{\dagger}Aw-w^{\dagger}Av)=-\operatorname{Tr}\left(\frac{1}{2}(vw^{\dagger}-wv^{\dagger})A\right). ⟨ B ( v ⊗ w ) , A ⟩ su ( N ) = ⟨ v , A w ⟩ C N = Re ( v † A w ) = 2 1 ( v † A w − w † A v ) = − Tr ( 2 1 ( v w † − w v † ) A ) . We now note that v w † − w v † vw^{\dagger}-wv^{\dagger} v w † − w v †
B ( v ⊗ w ) = 1 2 ( v w † − w v † ) − 1 2 N Tr ( v w † − w v † ) id = P 0 ( 1 2 ( v w † − w v † ) ) , \mathbf{B}(v\otimes w)=\frac{1}{2}(vw^{\dagger}-wv^{\dagger})-\frac{1}{2N}\operatorname{Tr}(vw^{\dagger}-wv^{\dagger})\operatorname{id}=P_{0}\left(\frac{1}{2}(vw^{\dagger}-wv^{\dagger})\right), B ( v ⊗ w ) = 2 1 ( v w † − w v † ) − 2 N 1 Tr ( v w † − w v † ) id = P 0 ( 2 1 ( v w † − w v † ) ) , where P 0 A ≔ A − 1 N Tr ( A ) id P_{0}A\coloneqq A-\frac{1}{N}\operatorname{Tr}(A)\operatorname{id} P 0 A : = A − N 1 Tr ( A ) id Cov g g \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}} Cov g g A , B ∈ s u ( N ) A,B\in\mathfrak{su}(N) A , B ∈ su ( N ) ⟨ Cov g g A , B ⟩ s u ( N ) = Kil ( A , B ) \langle\operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}}A,B\rangle_{\mathfrak{su}(N)}=\operatorname{Kil}(A,B) ⟨ Cov g g A , B ⟩ su ( N ) = Kil ( A , B ) Kil \operatorname{Kil} Kil s u ( N ) \mathfrak{su}(N) su ( N ) s u ( N ) \mathfrak{su}(N) su ( N ) Kil ( A , B ) = 2 N Tr ( A B ) = − 2 N ⟨ A , B ⟩ s u ( N ) \operatorname{Kil}(A,B)=2N\operatorname{Tr}(AB)=-2N\langle A,B\rangle_{\mathfrak{su}(N)} Kil ( A , B ) = 2 N Tr ( A B ) = − 2 N ⟨ A , B ⟩ su ( N )
Cov g g A = − 2 N A . \operatorname{Cov}_{\mathfrak{g}}^{\mathfrak{g}}A=-2NA. Cov g g A = − 2 N A . From a direct computation, one sees that
u a u a = − N 2 − 1 N ⋅ i d and v m v m † = 2 ⋅ i d . (135) u^{a}u_{a}=-\frac{N^{2}-1}{N}\cdot\mathrm{id}\qquad\text{and}\qquad v^{m}v_{m}^{\dagger}=2\cdot\mathrm{id}. \tag*{(135)} u a u a = − N N 2 − 1 ⋅ id and v m v m † = 2 ⋅ id . (135) Indeed, to obtain the first identity, one first shows that u a u a u^{a}u_{a} u a u a U ∈ S U ( N ) U\in\mathrm{SU}(N) U ∈ SU ( N ) ( v m ) = { e j } j ≤ N ∪ { i e j } j ≤ N (v_{m})=\{e_{j}\}_{j\leq N}\cup\{ie_{j}\}_{j\leq N} ( v m ) = { e j } j ≤ N ∪ { i e j } j ≤ N e j e j † e_{j}e_{j}^{\dagger} e j e j † ( i e j ) ( i e j ) † (ie_{j})(ie_{j})^{\dagger} ( i e j ) ( i e j ) † (135) , we obtain for all v ∈ C N v\in\mathbb{C}^{N} v ∈ C N
Cov g V v = u a u a v = − N 2 − 1 N v . \operatorname{Cov}_{\mathfrak{g}}^{V}v=u^{a}u_{a}v=-\frac{N^{2}-1}{N}v. Cov g V v = u a u a v = − N N 2 − 1 v . Similarly, from the second identity in (135) , we obtain that
Cov V g A = B ( v m ⊗ A v m ) = 1 2 P 0 ( v m ( A v m ) † − A v m v m † ) = 1 2 P 0 ( v m ( v m ) † A † − A v m v m † ) = − 2 P 0 ( A ) = − 2 A . \begin{aligned}
\operatorname{Cov}_{V}^{\mathfrak{g}}A=\mathbf{B}(v_{m}\otimes Av^{m})=\frac{1}{2}P_{0}(v_{m}(Av^{m})^{\dagger}-Av^{m}v_{m}^{\dagger}) \\
&=\frac{1}{2}P_{0}(v_{m}(v^{m})^{\dagger}A^{\dagger}-Av^{m}v_{m}^{\dagger})=-2P_{0}(A)=-2A.
\end{aligned} Cov V g A = B ( v m ⊗ A v m ) = 2 1 P 0 ( v m ( A v m ) † − A v m v m † ) = 2 1 P 0 ( v m ( v m ) † A † − A v m v m † ) = − 2 P 0 ( A ) = − 2 A . In the last identity, we used that since A ∈ s u ( N ) A\in\mathfrak{su}(N) A ∈ su ( N ) V = C N V=\mathbb{C}^{N} V = C N V = C N V=\mathbb{C}^{N} V = C N { e j } j ≤ N \{e_{j}\}_{j\leq N} { e j } j ≤ N V = C N V=\mathbb{C}^{N} V = C N R \mathbb{R} R C \mathbb{C} C ( X j ) j = 1 2 N (X_{j})_{j=1}^{2N} ( X j ) j = 1 2 N
∑ j = 1 N X j e j + ∑ j = 1 N X j + N ( i e j ) and ∑ j = 1 N X j + i X j + N 2 e j , \sum_{j=1}^{N}X_{j}e_{j}+\sum_{j=1}^{N}X_{j+N}(ie_{j})
\qquad\text{and}\qquad
\sum_{j=1}^{N}\frac{X_{j}+iX_{j+N}}{\sqrt{2}}e_{j}, j = 1 ∑ N X j e j + j = 1 ∑ N X j + N ( i e j ) and j = 1 ∑ N 2 X j + i X j + N e j , respectively. Thus, the standard Gaussian vectors on V V V 2 \sqrt{2} 2
Lemma 3.44. For every orthonormal basis ( v m ) (v_{m}) ( v m ) V V V Φ ∈ V \Phi\in V Φ ∈ V
B ( v m ⊗ Φ ) v m = Cov g V ( Φ ) . \mathbf{B}(v_{m}\otimes\Phi)v^{m}=\operatorname{Cov}_{\mathfrak{g}}^{V}(\Phi). B ( v m ⊗ Φ ) v m = Cov g V ( Φ ) . Proof. Writing ( u a ) (u_{a}) ( u a ) g \mathfrak{g} g
B ( v k ⊗ Φ ) v k = ⟨ B ( v k ⊗ Φ ) , u a ⟩ g u a v k = u a ⟨ v k , u a Φ ⟩ V v k = u a u a Φ = Cov g V Φ , \mathbf{B}(v^{k}\otimes\Phi)v_{k}=\langle\mathbf{B}(v^{k}\otimes\Phi),u^{a}\rangle_{\mathfrak{g}}u_{a}v_{k}
=u_{a}\langle v^{k},u^{a}\Phi\rangle_{V}v_{k}
=u_{a}u^{a}\Phi=\operatorname{Cov}_{\mathfrak{g}}^{V}\Phi, B ( v k ⊗ Φ ) v k = ⟨ B ( v k ⊗ Φ ) , u a ⟩ g u a v k = u a ⟨ v k , u a Φ ⟩ V v k = u a u a Φ = Cov g V Φ , as claimed. (Here we do not explicitly write ϱ ∗ \varrho_{*} ϱ ∗ g \mathfrak{g} g
Analytic preliminaries
In this section, we collect various analytic preliminaries. We begin by setting our convention for Hölder spaces.
Definition 3.45 (Hölder spaces). For α ∈ R \alpha\in\mathbb{R} α ∈ R ∥ ⋅ ∥ C x α \|\mathord{\cdot}\|_{\mathcal{C}^{\alpha}_{x}} ∥ ⋅ ∥ C x α C x , 0 α \mathcal{C}^{\alpha}_{x,0} C x , 0 α C x ∞ \mathcal{C}^{\infty}_{x} C x ∞
The following gauge covariant Gagliardo–Nirenberg–Sobolev inequality is a direct consequence of the diamagnetic inequality (125) and the classical Gagliardo–Nirenberg–Sobolev inequality (see e.g. [Tao06, Proposition A.3] for the statement of the inequality on R d \mathbb{R}^{d} R d
Lemma 3.46 (Gauge-covariant Gagliardo–Nirenberg–Sobolev inequality). Let B : T 2 → g 2 B:\mathbb{T}^2 \to\mathfrak{g}^2 B : T 2 → g 2 ϕ : T 2 → W \phi:\mathbb{T}^2 \to W ϕ : T 2 → W θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) r ∈ [ 1 , ∞ ) r \in[1,\infty) r ∈ [ 1 , ∞ ) θ ≥ 2 r \theta\ge\frac{2}{r} θ ≥ r 2
∥ ϕ ∥ L x ∞ ≲ θ , r ∥ ϕ ∥ L x r + ∥ ϕ ∥ L x r 1 − θ ∥ D B ϕ ∥ L x r θ . \|\phi\|_{L^\infty_x} \lesssim_{\theta,r} \|\phi\|_{L^r_x} + \|\phi\|_{L^r_x}^{1-\theta}\|D_B\phi\|_{L^r_x}^{\theta}. ∥ ϕ ∥ L x ∞ ≲ θ , r ∥ ϕ ∥ L x r + ∥ ϕ ∥ L x r 1 − θ ∥ D B ϕ ∥ L x r θ . where the implicit constant is uniform in B B B
We next state the following endpoint parabolic Schauder estimate, whose proof is similar to the proof of [Tao08, Lemma 2.5] (which works on R 2 \mathbb{R}^2 R 2
Lemma 3.47 (Endpoint Schauder estimate). Let T ∈ ( 0 , 1 ] T \in(0,1] T ∈ ( 0 , 1 ] u : [ 0 , T ] × T 2 → R u:[0,T]\times\mathbb{T}^2 \to\mathbb{R} u : [ 0 , T ] × T 2 → R
( ∂ t − Δ ) u = 0. (\partial_t-\Delta)u=0. ( ∂ t − Δ ) u = 0. Then we have the following endpoint Schauder estimate:
∥ u ∥ L t 2 L x ∞ ( [ 0 , T ] ) ≲ ∥ u ( 0 ) ∥ L x 2 . \|u\|_{L^2_tL^\infty_x([0,T])} \lesssim\|u(0)\|_{L^2_x}. ∥ u ∥ L t 2 L x ∞ ([ 0 , T ]) ≲ ∥ u ( 0 ) ∥ L x 2 . Proof. We proceed by duality. It suffices to show that for F ∈ L t 2 L x 1 ( [ 0 , T ] ) F \in L^2_tL^1_x([0,T]) F ∈ L t 2 L x 1 ([ 0 , T ])
⟨ u , F ⟩ L t , x 2 ( [ 0 , T ] ) ≲ ∥ u ( 0 ) ∥ L x 2 ∥ F ∥ L t 2 L x 1 ( [ 0 , T ] ) . \langle u,F\rangle_{L^2_{t,x}([0,T])} \lesssim\|u(0)\|_{L^2_x}\|F\|_{L^2_tL^1_x([0,T])}. ⟨ u , F ⟩ L t , x 2 ([ 0 , T ]) ≲ ∥ u ( 0 ) ∥ L x 2 ∥ F ∥ L t 2 L x 1 ([ 0 , T ]) . Towards this end, we may write
⟨ u , F ⟩ L t , x 2 ( [ 0 , T ] ) = ⟨ u ( 0 ) , ∫ 0 T d s e s Δ F ( s ) ⟩ L x 2 ≤ ∥ u ( 0 ) ∥ L x 2 ∥ ∫ 0 T d s e s Δ F ( s ) ∥ L x 2 . \langle u,F\rangle_{L^2_{t,x}([0,T])}=\left\langle u(0),\int_0^T \mathrm{d}s\,e^{s\Delta}F(s)\right\rangle_{L^2_x}\leq\|u(0)\|_{L^2_x}\left\|\int_0^T \mathrm{d}s\,e^{s\Delta}F(s)\right\|_{L^2_x}. ⟨ u , F ⟩ L t , x 2 ([ 0 , T ]) = ⟨ u ( 0 ) , ∫ 0 T d s e s Δ F ( s ) ⟩ L x 2 ≤ ∥ u ( 0 ) ∥ L x 2 ∫ 0 T d s e s Δ F ( s ) L x 2 . Thus to finish, it suffices to show that
∥ ∫ 0 T d s e s Δ F ( s ) ∥ L x 2 2 ≲ ∥ F ∥ L t 2 L x 1 ( [ 0 , T ] ) 2 . \left\|\int_0^T \mathrm{d}s\,e^{s\Delta}F(s)\right\|_{L^2_x}^2\lesssim\|F\|_{L^2_tL^1_x([0,T])}^2. ∫ 0 T d s e s Δ F ( s ) L x 2 2 ≲ ∥ F ∥ L t 2 L x 1 ([ 0 , T ]) 2 . Towards this end, observe that (noting a symmetry between s s s s ′ s' s ′ s > s ′ s>s' s > s ′
∥ ∫ 0 T d s e s Δ F ( s ) ∥ L x 2 2 = 2 ∫ 0 T d s ∫ 0 s d s ′ ⟨ F ( s ) , e ( s + s ′ ) Δ F ( s ′ ) ⟩ L x 2 ≲ ∫ 0 T d s ∥ F ( s ) ∥ L x 1 s − 1 ∫ 0 s d s ′ ∥ F ( s ′ ) ∥ L x 1 = ∫ 0 T ∥ F ( s ′ ) ∥ L x 1 ( ∫ s ′ T ∥ F ( s ) ∥ L x 1 s − 1 d s ) d s ′ ≤ ∫ 0 T ∥ F ( s ′ ) ∥ L x 1 ( ∫ s ′ T ∥ F ( s ) ∥ L x 2 2 s − 1 / 2 d s ) 1 / 2 ( ∫ s ′ T s − 3 / 2 d s ) 1 / 2 d s ′ ≲ ∥ F ∥ L t 2 L x 1 ( [ 0 , T ] ) ( ∫ 0 T ( ∫ s ′ T ∥ F ( s ) ∥ L x 2 2 s − 1 / 2 d s ) ( s ′ ) − 1 d s ′ ) 1 / 2 ≲ ∥ F ∥ L t 2 L x 1 ( [ 0 , T ] ) 2 , \begin{aligned}
\left\|\int_0^T \mathrm{d}s\,e^{s\Delta}F(s)\right\|_{L^2_x}^2
&=2\int_0^T\mathrm{d}s\int_0^s\mathrm{d}s'\,\langle F(s),e^{(s+s')\Delta}F(s')\rangle_{L^2_x}\\
&\lesssim\int_0^T\mathrm{d}s\,\|F(s)\|_{L^1_x}s^{-1}\int_0^s\mathrm{d}s'\,\|F(s')\|_{L^1_x}\\
&=\int_0^T\|F(s')\|_{L^1_x}\left(\int_{s'}^T\|F(s)\|_{L^1_x}s^{-1}\mathrm{d}s\right)\mathrm{d}s'\\
&\leq\int_0^T\|F(s')\|_{L^1_x}\left(\int_{s'}^T\|F(s)\|_{L^2_x}^2s^{-1/2}\mathrm{d}s\right)^{1/2}\left(\int_{s'}^T s^{-3/2}\mathrm{d}s\right)^{1/2}\mathrm{d}s'\\
&\lesssim\|F\|_{L^2_tL^1_x([0,T])}\left(\int_0^T\left(\int_{s'}^T\|F(s)\|_{L^2_x}^2s^{-1/2}\mathrm{d}s\right)(s')^{-1}\mathrm{d}s'\right)^{1/2}\\
&\lesssim\|F\|_{L^2_tL^1_x([0,T])}^2,
\end{aligned} ∫ 0 T d s e s Δ F ( s ) L x 2 2 = 2 ∫ 0 T d s ∫ 0 s d s ′ ⟨ F ( s ) , e ( s + s ′ ) Δ F ( s ′ ) ⟩ L x 2 ≲ ∫ 0 T d s ∥ F ( s ) ∥ L x 1 s − 1 ∫ 0 s d s ′ ∥ F ( s ′ ) ∥ L x 1 = ∫ 0 T ∥ F ( s ′ ) ∥ L x 1 ( ∫ s ′ T ∥ F ( s ) ∥ L x 1 s − 1 d s ) d s ′ ≤ ∫ 0 T ∥ F ( s ′ ) ∥ L x 1 ( ∫ s ′ T ∥ F ( s ) ∥ L x 2 2 s − 1/2 d s ) 1/2 ( ∫ s ′ T s − 3/2 d s ) 1/2 d s ′ ≲ ∥ F ∥ L t 2 L x 1 ([ 0 , T ]) ( ∫ 0 T ( ∫ s ′ T ∥ F ( s ) ∥ L x 2 2 s − 1/2 d s ) ( s ′ ) − 1 d s ′ ) 1/2 ≲ ∥ F ∥ L t 2 L x 1 ([ 0 , T ]) 2 , as desired.
The following lemma will allow us to control certain norms (see Definition 2.3) of the curvature of a given solution to the Yang–Mills gradient flow. This will allow us to obtain covariant heat kernel and Schauder estimates which behave very similarly to the standard heat kernel and Schauder estimates, which in turn will be crucial in our covariant local theory (§ 7 –§ 9 ). In the following, for solutions u : [ 0 , T ] × T 2 → R u:[0,T]\times\mathbb{T}^2\to\mathbb{R} u : [ 0 , T ] × T 2 → R ( ∂ t − Δ ) u = 0 (\partial_t-\Delta)u=0 ( ∂ t − Δ ) u = 0
∥ u ( 0 ) ∥ L x 2 2 − ∥ u ( T ) ∥ L x 2 2 = 2 ∥ ∇ u ∥ L t , x 2 ( [ 0 , T ] ) 2 . (136) \|u(0)\|_{L^2_x}^2-\|u(T)\|_{L^2_x}^2=2\|\nabla u\|_{L^2_{t,x}([0,T])}^2. \tag*{(136)} ∥ u ( 0 ) ∥ L x 2 2 − ∥ u ( T ) ∥ L x 2 2 = 2∥∇ u ∥ L t , x 2 ([ 0 , T ]) 2 . (136) Lemma 3.48. Let T ∈ ( 0 , 1 / 2 ] T \in(0,1/2] T ∈ ( 0 , 1/2 ] f : [ 0 , T ] × T 2 → [ 0 , ∞ ) f : [0,T] \times\mathbb{T}^{2} \to[0,\infty) f : [ 0 , T ] × T 2 → [ 0 , ∞ )
( ∂ t − Δ ) f ≤ 0. (\partial_t-\Delta)f \le0. ( ∂ t − Δ ) f ≤ 0. Let D : [ 0 , T ] → [ 0 , ∞ ) D : [0,T] \to[0,\infty) D : [ 0 , T ] → [ 0 , ∞ ) L x 2 L_x^2 L x 2 f f f
D ( t ) ≔ ∥ f ( 0 ) ∥ L x 2 2 − ∥ f ( t ) ∥ L x 2 2 , t ∈ [ 0 , T ] . D(t) \coloneqq\lVert f(0)\rVert_{L_x^2}^2-\lVert f(t)\rVert_{L_x^2}^2,\qquad t\in[0,T]. D ( t ) : = ∥ f ( 0 ) ∥ L x 2 2 − ∥ f ( t ) ∥ L x 2 2 , t ∈ [ 0 , T ] . Then we have the following estimate:
∥ f ∥ L t 2 L x ∞ ( [ 0 , T ] ) ≲ T 1 / 2 ∥ f ( 0 ) ∥ L x 2 + ∣ log T ∣ 1 / 2 D ( T ) 1 / 2 . \lVert f\rVert_{L_t^2L_x^\infty([0,T])} \lesssim T^{1/2}\lVert f(0)\rVert_{L_x^2}+\lvert\log T\rvert^{1/2}D(T)^{1/2}. ∥ f ∥ L t 2 L x ∞ ([ 0 , T ]) ≲ T 1/2 ∥ f ( 0 ) ∥ L x 2 + ∣ log T ∣ 1/2 D ( T ) 1/2 . Proof. Let g g g f ( 0 ) f(0) f ( 0 ) 0 ≤ f ≤ g 0\le f\le g 0 ≤ f ≤ g g g g N 0 ≔ 10 3 T − 100 N_0 \coloneqq10^3T^{-100} N 0 : = 1 0 3 T − 100 P ≤ N 0 g P_{\le N_0}g P ≤ N 0 g P > N 0 g P_{>N_0}g P > N 0 g t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
∥ P ≤ N 0 g ( t ) ∥ L x ∞ ≤ ∑ N ≤ N 0 ∥ P N g ( t ) ∥ L x ∞ ≲ ∥ g ( t ) ∥ L x 2 + ∑ 1 < N ≤ N 0 ∥ ∇ P N g ( t ) ∥ L x 2 \lVert P_{\le N_0}g(t)\rVert_{L_x^\infty}
\le\sum_{N\le N_0}\lVert P_Ng(t)\rVert_{L_x^\infty}
\lesssim\lVert g(t)\rVert_{L_x^2}+\sum_{1<N\le N_0}\lVert\nabla P_Ng(t)\rVert_{L_x^2} ∥ P ≤ N 0 g ( t ) ∥ L x ∞ ≤ N ≤ N 0 ∑ ∥ P N g ( t ) ∥ L x ∞ ≲ ∥ g ( t ) ∥ L x 2 + 1 < N ≤ N 0 ∑ ∥ ∇ P N g ( t ) ∥ L x 2 ≲ ∥ g ( t ) ∥ L x 2 + ( log N 0 ) 1 / 2 ∥ ∇ P ≤ N 0 g ( t ) ∥ L x 2 , \lesssim\lVert g(t)\rVert_{L_x^2}+(\log N_0)^{1/2}\lVert\nabla P_{\le N_0}g(t)\rVert_{L_x^2}, ≲ ∥ g ( t ) ∥ L x 2 + ( log N 0 ) 1/2 ∥ ∇ P ≤ N 0 g ( t ) ∥ L x 2 , where we applied Cauchy–Schwarz in the final inequality (recalling that the sum over N N N L t 2 L_t^2 L t 2
∥ P ≤ N 0 g ∥ L t 2 L x ∞ ( [ 0 , T ] ) ≲ T 1 / 2 ∥ f ( 0 ) ∥ L x 2 + ∣ log T ∣ 1 / 2 ∥ ∇ P ≤ N 0 g ∥ L t , x 2 ( [ 0 , T ] ) , \lVert P_{\le N_0}g\rVert_{L_t^2L_x^\infty([0,T])}
\lesssim T^{1/2}\lVert f(0)\rVert_{L_x^2}+\lvert\log T\rvert^{1/2}\lVert\nabla P_{\le N_0}g\rVert_{L_{t,x}^2([0,T])}, ∥ P ≤ N 0 g ∥ L t 2 L x ∞ ([ 0 , T ]) ≲ T 1/2 ∥ f ( 0 ) ∥ L x 2 + ∣ log T ∣ 1/2 ∥ ∇ P ≤ N 0 g ∥ L t , x 2 ([ 0 , T ]) , where we used (136) to bound ∥ g ( t ) ∥ L x 2 ≤ ∥ g ( 0 ) ∥ L x 2 = ∥ f ( 0 ) ∥ L x 2 \lVert g(t)\rVert_{L_x^2}\le\lVert g(0)\rVert_{L_x^2}=\lVert f(0)\rVert_{L_x^2} ∥ g ( t ) ∥ L x 2 ≤ ∥ g ( 0 ) ∥ L x 2 = ∥ f ( 0 ) ∥ L x 2 (136) that
∥ ∇ P ≤ N 0 g ∥ L t , x 2 ( [ 0 , T ] ) 2 ≲ ∥ ∇ g ∥ L t , x 2 ( [ 0 , T ] ) 2 ∼ ∥ g ( 0 ) ∥ L x 2 2 − ∥ g ( T ) ∥ L x 2 2 ≤ ∥ f ( 0 ) ∥ L x 2 2 − ∥ f ( T ) ∥ L x 2 2 = D ( T ) , \lVert\nabla P_{\le N_0}g\rVert_{L_{t,x}^2([0,T])}^2
\lesssim\lVert\nabla g\rVert_{L_{t,x}^2([0,T])}^2
\sim\lVert g(0)\rVert_{L_x^2}^2-\lVert g(T)\rVert_{L_x^2}^2
\le\lVert f(0)\rVert_{L_x^2}^2-\lVert f(T)\rVert_{L_x^2}^2=D(T), ∥ ∇ P ≤ N 0 g ∥ L t , x 2 ([ 0 , T ]) 2 ≲ ∥ ∇ g ∥ L t , x 2 ([ 0 , T ]) 2 ∼ ∥ g ( 0 ) ∥ L x 2 2 − ∥ g ( T ) ∥ L x 2 2 ≤ ∥ f ( 0 ) ∥ L x 2 2 − ∥ f ( T ) ∥ L x 2 2 = D ( T ) , where in the second-to-final step, we used the pointwise bound f ≤ g f\le g f ≤ g f ( 0 ) = g ( 0 ) f(0)=g(0) f ( 0 ) = g ( 0 ) P ≤ N 0 g P_{\le N_0}g P ≤ N 0 g
For the high-frequency part, by the endpoint Schauder estimate (Lemma 3.47) and the fact that P > N 0 g P_{>N_0}g P > N 0 g
∥ P > N 0 g ∥ L t 2 L x ∞ ( [ 0 , T ] ) ≲ ∥ P > N 0 g ( 0 ) ∥ L x 2 . \lVert P_{>N_0}g\rVert_{L_t^2L_x^\infty([0,T])}
\lesssim\lVert P_{>N_0}g(0)\rVert_{L_x^2}. ∥ P > N 0 g ∥ L t 2 L x ∞ ([ 0 , T ]) ≲ ∥ P > N 0 g ( 0 ) ∥ L x 2 . By classical heat smoothing estimates (see e.g. [BCD11, Lemma 2.4]), we also have that
∥ P > N 0 g ( T ) ∥ L x 2 ≲ e − c N 0 2 T ∥ P > N 0 g ( 0 ) ∥ L x 2 ≪ ∥ P > N 0 g ( 0 ) ∥ L x 2 , \lVert P_{>N_0}g(T)\rVert_{L_x^2}
\lesssim e^{-cN_0^2T}\lVert P_{>N_0}g(0)\rVert_{L_x^2}
\ll\lVert P_{>N_0}g(0)\rVert_{L_x^2}, ∥ P > N 0 g ( T ) ∥ L x 2 ≲ e − c N 0 2 T ∥ P > N 0 g ( 0 ) ∥ L x 2 ≪ ∥ P > N 0 g ( 0 ) ∥ L x 2 , where the final inequality follows by our choice of N 0 = 10 3 T − 100 N_0=10^3T^{-100} N 0 = 1 0 3 T − 100 (136) , we obtain
∥ ∇ P > N 0 g ∥ L t , x 2 ( [ 0 , T ] ) 2 ∼ ∥ P > N 0 g ( 0 ) ∥ L x 2 2 − ∥ P > N 0 g ( T ) ∥ L x 2 2 ∼ ∥ P > N 0 g ( 0 ) ∥ L x 2 2 , \lVert\nabla P_{>N_0}g\rVert_{L_{t,x}^2([0,T])}^2
\sim\lVert P_{>N_0}g(0)\rVert_{L_x^2}^2-\lVert P_{>N_0}g(T)\rVert_{L_x^2}^2
\sim\lVert P_{>N_0}g(0)\rVert_{L_x^2}^2, ∥ ∇ P > N 0 g ∥ L t , x 2 ([ 0 , T ]) 2 ∼ ∥ P > N 0 g ( 0 ) ∥ L x 2 2 − ∥ P > N 0 g ( T ) ∥ L x 2 2 ∼ ∥ P > N 0 g ( 0 ) ∥ L x 2 2 , and from this we further obtain (again applying (136) in the penultimate estimate)
∥ P > N 0 g ∥ L t 2 L x ∞ ( [ 0 , T ] ) 2 ≲ ∥ ∇ P > N 0 g ∥ L t , x 2 ( [ 0 , T ] ) 2 ≲ ∥ ∇ g ∥ L t , x 2 ( [ 0 , T ] ) 2 \lVert P_{>N_0}g\rVert_{L_t^2L_x^\infty([0,T])}^2
\lesssim\lVert\nabla P_{>N_0}g\rVert_{L_{t,x}^2([0,T])}^2
\lesssim\lVert\nabla g\rVert_{L_{t,x}^2([0,T])}^2 ∥ P > N 0 g ∥ L t 2 L x ∞ ([ 0 , T ]) 2 ≲ ∥ ∇ P > N 0 g ∥ L t , x 2 ([ 0 , T ]) 2 ≲ ∥ ∇ g ∥ L t , x 2 ([ 0 , T ]) 2 ∼ ∥ g ( 0 ) ∥ L x 2 2 − ∥ g ( T ) ∥ L x 2 2 ≲ D ( T ) , \sim\lVert g(0)\rVert_{L_x^2}^2-\lVert g(T)\rVert_{L_x^2}^2\lesssim D(T), ∼ ∥ g ( 0 ) ∥ L x 2 2 − ∥ g ( T ) ∥ L x 2 2 ≲ D ( T ) , where in the final inequality, we again used the pointwise bound f ≤ g f\le g f ≤ g g ( 0 ) = f ( 0 ) g(0)=f(0) g ( 0 ) = f ( 0 )
Lemma 3.49. For t ∈ ( 0 , 1 ] t \in(0,1] t ∈ ( 0 , 1 ] x , y ∈ T d x,y \in\mathbb{T}^d x , y ∈ T d ∣ x − y ∣ ≤ 10 − 1 |x-y| \le10^{-1} ∣ x − y ∣ ≤ 1 0 − 1
∣ ∇ p ( 0 , y ; t , x ) p ( 0 , y ; t , x ) + x − y 2 t ∣ ≲ t 10 . \left|\frac{\nabla p(0,y;t,x)}{p(0,y;t,x)}+\frac{x-y}{2t}\right|\lesssim t^{10}. p ( 0 , y ; t , x ) ∇ p ( 0 , y ; t , x ) + 2 t x − y ≲ t 10 . Proof. We view p p p 2 π 2\pi 2 π R d \mathbb{R}^d R d T d ≅ ( − π , π ] d \mathbb{T}^d \cong(-\pi,\pi]^d T d ≅ ( − π , π ] d p t E u c ( x ) p_t^{\mathrm{Euc}}(x) p t Euc ( x ) R d \mathbb{R}^d R d
p ( 0 , y ; t , x ) = ∑ n ∈ Z d p t E u c ( x − y + 2 π n ) . p(0,y;t,x)=\sum_{n\in\mathbb{Z}^d}p_t^{\mathrm{Euc}}(x-y+2\pi n). p ( 0 , y ; t , x ) = n ∈ Z d ∑ p t Euc ( x − y + 2 π n ) . By translation invariance, we may assume that y = 0 y=0 y = 0 x x x ∣ x ∣ ≤ 10 − 1 |x|\le10^{-1} ∣ x ∣ ≤ 1 0 − 1
∇ p ( 0 , 0 ; t , x ) = ∇ p t E u c ( x ) + ∑ n ≠ 0 ∇ p t E u c ( x + 2 π n ) \nabla p(0,0;t,x)=\nabla p_t^{\mathrm{Euc}}(x)+\sum_{n\ne0}\nabla p_t^{\mathrm{Euc}}(x+2\pi n) ∇ p ( 0 , 0 ; t , x ) = ∇ p t Euc ( x ) + n = 0 ∑ ∇ p t Euc ( x + 2 π n ) = − x 2 t p t E u c ( x ) − ∑ n ≠ 0 x + 2 π n 2 t p t E u c ( x + 2 π n ) =-\frac{x}{2t}p_t^{\mathrm{Euc}}(x)-\sum_{n\ne0}\frac{x+2\pi n}{2t}p_t^{\mathrm{Euc}}(x+2\pi n) = − 2 t x p t Euc ( x ) − n = 0 ∑ 2 t x + 2 π n p t Euc ( x + 2 π n ) = − x 2 t p ( 0 , 0 ; t , x ) − ∑ n ≠ 0 π n t p t E u c ( x + 2 π n ) . =-\frac{x}{2t}p(0,0;t,x)-\sum_{n\ne0}\frac{\pi n}{t}p_t^{\mathrm{Euc}}(x+2\pi n). = − 2 t x p ( 0 , 0 ; t , x ) − n = 0 ∑ t π n p t Euc ( x + 2 π n ) . Since ∣ x ∣ ≤ 10 − 1 |x|\le10^{-1} ∣ x ∣ ≤ 1 0 − 1 n ≠ 0 n\ne0 n = 0
∣ π n p t E u c ( x + 2 π n ) t p t E u c ( x ) ∣ ≲ t − 1 exp ( − c ∣ n ∣ 2 t ) \left|\frac{\pi n p_t^{\mathrm{Euc}}(x+2\pi n)}{t p_t^{\mathrm{Euc}}(x)}\right|\lesssim t^{-1}\exp\left(-c\frac{|n|^2}{t}\right) t p t Euc ( x ) π n p t Euc ( x + 2 π n ) ≲ t − 1 exp ( − c t ∣ n ∣ 2 ) ≤ t − 1 exp ( − c ∣ n ∣ 2 2 t ) exp ( − c ∣ n ∣ 2 2 ) ≲ t 10 exp ( − c ∣ n ∣ 2 2 ) . \le t^{-1}\exp\left(-\frac{c|n|^2}{2t}\right)\exp\left(-\frac{c|n|^2}{2}\right)\lesssim t^{10}\exp\left(-\frac{c|n|^2}{2}\right). ≤ t − 1 exp ( − 2 t c ∣ n ∣ 2 ) exp ( − 2 c ∣ n ∣ 2 ) ≲ t 10 exp ( − 2 c ∣ n ∣ 2 ) . It follows that
∑ n ≠ 0 ∣ π n p t E u c ( x + 2 π n ) t p t E u c ( x ) ∣ ≲ t 10 , \sum_{n\ne0}\left|\frac{\pi n p_t^{\mathrm{Euc}}(x+2\pi n)}{t p_t^{\mathrm{Euc}}(x)}\right|\lesssim t^{10}, n = 0 ∑ t p t Euc ( x ) π n p t Euc ( x + 2 π n ) ≲ t 10 , from which the desired result follows.
Lemma 3.50 (Gronwall). Let f , a , b : [ 0 , T ] → [ 0 , ∞ ) f,a,b:[0,T]\to[0,\infty) f , a , b : [ 0 , T ] → [ 0 , ∞ ) a ∈ L 1 ( [ 0 , T ] ) a\in L^1([0,T]) a ∈ L 1 ([ 0 , T ]) f , b ∈ L ∞ ( [ 0 , T ] ) f,b\in L^\infty([0,T]) f , b ∈ L ∞ ([ 0 , T ])
∥ f ∥ L ∞ ( [ 0 , t ] ) ≤ ∥ a f ∥ L 1 ( [ 0 , t ] ) + ∥ b ∥ L ∞ ( [ 0 , t ] ) , \|f\|_{L^\infty([0,t])}\le\|af\|_{L^1([0,t])}+\|b\|_{L^\infty([0,t])}, ∥ f ∥ L ∞ ([ 0 , t ]) ≤ ∥ a f ∥ L 1 ([ 0 , t ]) + ∥ b ∥ L ∞ ([ 0 , t ]) , uniformly over t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
∥ f ∥ L ∞ ( [ 0 , t ] ) ≤ ∥ b ∥ L ∞ ( [ 0 , t ] ) exp ∥ a ∥ L 1 ( [ 0 , t ] ) , \|f\|_{L^\infty([0,t])}\le\|b\|_{L^\infty([0,t])}\exp\|a\|_{L^1([0,t])}, ∥ f ∥ L ∞ ([ 0 , t ]) ≤ ∥ b ∥ L ∞ ([ 0 , t ]) exp ∥ a ∥ L 1 ([ 0 , t ]) , uniformly over t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
In the following, we define function spaces of negative temporal regularity. They will be used as part of our modified solution spaces, see Definition 7.4 below.
Definition 3.51 (Temporal regularity spaces). Let T > 0 T>0 T > 0 ( B , ∥ ⋅ ∥ B ) (\mathcal{B},\|\cdot\|_{\mathcal{B}}) ( B , ∥ ⋅ ∥ B ) α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] ∥ ⋅ ∥ C t α B \|\cdot\|_{C_t^\alpha\mathcal{B}} ∥ ⋅ ∥ C t α B α \alpha α B \mathcal{B} B
∥ ϕ ∥ C t α B ( [ 0 , T ] ) ≔ ∥ ϕ ∥ C t 0 B ( [ 0 , T ] ) + sup 0 ≤ s < t ≤ T ∥ ϕ ( t ) − ϕ ( s ) ∥ B ∣ t − s ∣ α . \|\phi\|_{C_t^\alpha\mathcal{B}([0,T])}\coloneqq\|\phi\|_{C_t^0\mathcal{B}([0,T])}+\sup_{0\le s<t\le T}\frac{\|\phi(t)-\phi(s)\|_{\mathcal{B}}}{|t-s|^\alpha}. ∥ ϕ ∥ C t α B ([ 0 , T ]) : = ∥ ϕ ∥ C t 0 B ([ 0 , T ]) + 0 ≤ s < t ≤ T sup ∣ t − s ∣ α ∥ ϕ ( t ) − ϕ ( s ) ∥ B . For α ∈ ( − 1 , 0 ) \alpha\in(-1,0) α ∈ ( − 1 , 0 )
∥ ϕ ∥ C t α B ( [ 0 , T ] ) ≔ ∥ ∫ 0 t ϕ ( s ) d s ∥ C t 1 + α B ( [ 0 , T ] ) . \|\phi\|_{\mathcal{C}^{\alpha}_{t}\mathcal{B}([0,T])} \coloneqq\left\|\int_{0}^{t}\phi(s)\,\mathrm{d}s\right\|_{\mathcal{C}^{1+\alpha}_{t}\mathcal{B}([0,T])}. ∥ ϕ ∥ C t α B ([ 0 , T ]) : = ∫ 0 t ϕ ( s ) d s C t 1 + α B ([ 0 , T ]) . In other words, the C t α B \mathcal{C}^{\alpha}_{t}\mathcal{B} C t α B ϕ \phi ϕ C t 1 + α B \mathcal{C}^{1+\alpha}_{t}\mathcal{B} C t 1 + α B t ↦ ∫ 0 t ϕ ( s ) d s t\mapsto\int_{0}^{t}\phi(s)\,\mathrm{d}s t ↦ ∫ 0 t ϕ ( s ) d s
Remark 3.52. It follows directly from the definition that for α ∈ ( − 1 , 0 ) \alpha\in(-1,0) α ∈ ( − 1 , 0 ) ϕ ∈ C t α B \phi\in\mathcal{C}^{\alpha}_{t}\mathcal{B} ϕ ∈ C t α B t ↦ ∫ 0 t ϕ ( s ) d s ∈ C t 1 + α B t\mapsto\int_{0}^{t}\phi(s)\mathrm{d}s\in\mathcal{C}^{1+\alpha}_{t}\mathcal{B} t ↦ ∫ 0 t ϕ ( s ) d s ∈ C t 1 + α B ϕ ∈ C t 1 + α B \phi\in\mathcal{C}^{1+\alpha}_{t}\mathcal{B} ϕ ∈ C t 1 + α B ∂ t ϕ ( t ) ∈ C t α B \partial_{t}\phi(t)\in\mathcal{C}^{\alpha}_{t}\mathcal{B} ∂ t ϕ ( t ) ∈ C t α B
The following version of Young’s integration bound [You36] is an immediate consequence of the sewing lemma [FH20, Lem. 4.2] (just set Ξ s , t = ( ∫ s t ϕ 1 ( r ) d r ) ϕ 2 ( s ) \Xi_{s,t}=\left(\int_{s}^{t}\phi_{1}(r)\,\mathrm{d}r\right)\phi_{2}(s) Ξ s , t = ( ∫ s t ϕ 1 ( r ) d r ) ϕ 2 ( s )
Lemma 3.53 (Product estimate for temporal regularity spaces). Let T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] ( B , ∥ ⋅ ∥ B ) (\mathcal{B},\|\cdot\|_{\mathcal{B}}) ( B , ∥ ⋅ ∥ B ) ( B i , ∥ ⋅ ∥ B i ) (\mathcal{B}_{i},\|\cdot\|_{\mathcal{B}_{i}}) ( B i , ∥ ⋅ ∥ B i ) i = 1 , 2 i=1,2 i = 1 , 2 B 1 × B 2 → B \mathcal{B}_{1}\times\mathcal{B}_{2}\to\mathcal{B} B 1 × B 2 → B α ∈ ( − 1 , 0 ) \alpha\in(-1,0) α ∈ ( − 1 , 0 ) β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) α + β > 0 \alpha+\beta>0 α + β > 0 ϕ i : [ 0 , T ] → B i \phi_{i}:[0,T]\to\mathcal{B}_{i} ϕ i : [ 0 , T ] → B i i = 1 , 2 i=1,2 i = 1 , 2
∥ ϕ 1 ϕ 2 ∥ C t α B ( [ 0 , T ] ) ≲ ∥ ϕ 1 ∥ C t α B 1 ( [ 0 , T ] ) ∥ ϕ 2 ∥ C t β B 2 ( [ 0 , T ] ) . \|\phi_{1}\phi_{2}\|_{\mathcal{C}^{\alpha}_{t}\mathcal{B}([0,T])}\lesssim\|\phi_{1}\|_{\mathcal{C}^{\alpha}_{t}\mathcal{B}_{1}([0,T])}\|\phi_{2}\|_{\mathcal{C}^{\beta}_{t}\mathcal{B}_{2}([0,T])}. ∥ ϕ 1 ϕ 2 ∥ C t α B ([ 0 , T ]) ≲ ∥ ϕ 1 ∥ C t α B 1 ([ 0 , T ]) ∥ ϕ 2 ∥ C t β B 2 ([ 0 , T ]) . The following lemma shows that our temporal regularity spaces are well-behaved under time localisations, which will be used in the proof of our weighted covariant Schauder estimates (see Proposition 7.16).
Lemma 3.54 (Time-localisation). Let 0 < T ≤ 1 0<T\leq1 0 < T ≤ 1 − 1 < α ≤ β ≤ 0 -1<\alpha\leq\beta\leq0 − 1 < α ≤ β ≤ 0 χ ∈ C c ∞ ( ( 0 , ∞ ) ) \chi\in C^{\infty}_{c}((0,\infty)) χ ∈ C c ∞ (( 0 , ∞ )) B \mathcal{B} B ϕ : [ 0 , T ] → B \phi:[0,T]\to\mathcal{B} ϕ : [ 0 , T ] → B N ∈ 2 N 0 N\in2^{\mathbb{N}_{0}} N ∈ 2 N 0
∥ χ ( N 2 t ) ϕ ∥ C t α B ( [ 0 , T ] ) ≲ N 2 ( α − β ) ∥ ϕ ∥ C t β B ( [ 0 , T ] ) . (137) \|\chi(N^{2}t)\phi\|_{\mathcal{C}^{\alpha}_{t}\mathcal{B}([0,T])}\lesssim N^{2(\alpha-\beta)}\|\phi\|_{\mathcal{C}^{\beta}_{t}\mathcal{B}([0,T])}. \tag*{(137)} ∥ χ ( N 2 t ) ϕ ∥ C t α B ([ 0 , T ]) ≲ N 2 ( α − β ) ∥ ϕ ∥ C t β B ([ 0 , T ]) . (137) Proof. We let Φ \Phi Φ ϕ \phi ϕ Φ ( 0 ) = 0 \Phi(0)=0 Φ ( 0 ) = 0 ∥ Φ ∥ C t 1 + β B ≲ ∥ ϕ ∥ C t β B \|\Phi\|_{\mathcal{C}^{1+\beta}_{t}\mathcal{B}}\lesssim\|\phi\|_{\mathcal{C}^{\beta}_{t}\mathcal{B}} ∥Φ ∥ C t 1 + β B ≲ ∥ ϕ ∥ C t β B t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
∥ Φ ( t ) ∥ B = ∥ Φ ( t ) − Φ ( 0 ) ∥ B ≲ t 1 + β ∥ Φ ∥ C t 1 + β B . (138) \|\Phi(t)\|_{\mathcal{B}}=\|\Phi(t)-\Phi(0)\|_{\mathcal{B}}\lesssim t^{1+\beta}\|\Phi\|_{\mathcal{C}^{1+\beta}_{t}\mathcal{B}}. \tag*{(138)} ∥Φ ( t ) ∥ B = ∥Φ ( t ) − Φ ( 0 ) ∥ B ≲ t 1 + β ∥Φ ∥ C t 1 + β B . (138) Since
∫ 0 t χ ( N 2 s ) ϕ ( s ) d s = χ ( N 2 t ) Φ ( t ) − ∫ 0 t N 2 χ ′ ( N 2 s ) Φ ( s ) d s , \int_{0}^{t}\chi(N^{2}s)\phi(s)\,\mathrm{d}s=\chi(N^{2}t)\Phi(t)-\int_{0}^{t}N^{2}\chi'(N^{2}s)\Phi(s)\,\mathrm{d}s, ∫ 0 t χ ( N 2 s ) ϕ ( s ) d s = χ ( N 2 t ) Φ ( t ) − ∫ 0 t N 2 χ ′ ( N 2 s ) Φ ( s ) d s , our desired estimate (137) can be reduced to the two estimates
∥ χ ( N 2 t ) Φ ( t ) ∥ C t α ′ B ≲ N 2 ( α ′ − β ′ ) ∥ Φ ( t ) ∥ C t β ′ B . (139) \|\chi(N^{2}t)\Phi(t)\|_{\mathcal{C}^{\alpha'}_{t}\mathcal{B}}\lesssim N^{2(\alpha'-\beta')}\|\Phi(t)\|_{\mathcal{C}^{\beta'}_{t}\mathcal{B}}. \tag*{(139)} ∥ χ ( N 2 t ) Φ ( t ) ∥ C t α ′ B ≲ N 2 ( α ′ − β ′ ) ∥Φ ( t ) ∥ C t β ′ B . (139) ∥ N 2 ∫ 0 t χ ′ ( N 2 s ) Φ ( s ) d s ∥ C t α ′ B ≲ N 2 ( α ′ − β ′ ) ∥ Φ ( t ) ∥ C t β ′ B . (140) \left\|N^{2}\int_{0}^{t}\chi'(N^{2}s)\Phi(s)\,\mathrm{d}s\right\|_{\mathcal{C}^{\alpha'}_{t}\mathcal{B}}\lesssim N^{2(\alpha'-\beta')}\|\Phi(t)\|_{\mathcal{C}^{\beta'}_{t}\mathcal{B}}. \tag*{(140)} N 2 ∫ 0 t χ ′ ( N 2 s ) Φ ( s ) d s C t α ′ B ≲ N 2 ( α ′ − β ′ ) ∥Φ ( t ) ∥ C t β ′ B . (140) where α ′ : = 1 + α \alpha':=1+\alpha α ′ := 1 + α β ′ : = 1 + β \beta':=1+\beta β ′ := 1 + β (139) and (140) for ( α ′ , β ′ ) = ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) (\alpha',\beta')=(0,0),(0,1),(1,1) ( α ′ , β ′ ) = ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) (138) and standard estimates, and we omit the remaining details.
Supermartingale estimates
In this subsection, we prove uniform-in-time moment bounds for a stochastic process satisfying supermartingale-type conditions. Near the end of the article (see Section 11), they will be used to prove the uniform-in-time bounds in our soft and hard global well-posedness results (Theorems 2.24 and 2.30 ). The reason for presenting our supermartingale-type bounds in a separate subsection, rather than including them in Section 11, is that the arguments are easier to understand without the heavy notation used in Section 11. As previously mentioned in the AI disclosure statement, the proofs in this subsection were suggested to the authors by AI (GPT-6 Astra). The closest arguments in the existing literature, which formed the basis of the AI’s response, are in [PR99, DFMS04].
Proposition 3.55. Let ( G m ) m = 0 ∞ (\mathcal{G}_m)_{m=0}^{\infty} ( G m ) m = 0 ∞ ( X m ) m = 0 ∞ (X_m)_{m=0}^{\infty} ( X m ) m = 0 ∞ ( Y m ) m = 0 ∞ (Y_m)_{m=0}^{\infty} ( Y m ) m = 0 ∞ ( D m ) m = 0 ∞ (D_m)_{m=0}^{\infty} ( D m ) m = 0 ∞ X m ≥ 10 X_m \ge10 X m ≥ 10 m ≥ 0 m \ge0 m ≥ 0 Y 0 ≥ 0 Y_0 \ge0 Y 0 ≥ 0 r ≥ 1 r \ge1 r ≥ 1 p > r + 1 2 p > r + \frac{1}{2} p > r + 2 1 0 ≤ ε < 1 0 \le\varepsilon< 1 0 ≤ ε < 1 c > 0 c > 0 c > 0 K ≥ 1 K \ge1 K ≥ 1
(i) (Recursive estimate) For all m ≥ 0 m \ge0 m ≥ 0
X m + 1 ≤ X m − D m + Y m + 1 + K . (141) X_{m+1} \le X_m - D_m + Y_{m+1} + K. \tag*{(141)} X m + 1 ≤ X m − D m + Y m + 1 + K . (141) (ii) (Decay) For all m ≥ 0 m \ge0 m ≥ 0
D m ≥ c X m log ( X m ) . (142) D_m \ge c\frac{\sqrt{X_m}}{\log(X_m)}. \tag*{(142)} D m ≥ c log ( X m ) X m . (142) (iii) (Fluctuation bounds) For all m ≥ 1 m \ge1 m ≥ 1
E [ ( Y m + 1 − ε D m ) + p ∣ G m − 1 ] ≤ K . (143) \mathbb{E}\left[(Y_{m+1} - \varepsilon D_m)_+^p \mid\mathcal{G}_{m-1}\right] \le K. \tag*{(143)} E [ ( Y m + 1 − ε D m ) + p ∣ G m − 1 ] ≤ K . (143) Furthermore, it holds that
E [ ( Y 1 − ε D 0 ) + p ∣ G 0 ] ≤ Y 0 p . (144) \mathbb{E}\left[(Y_1-\varepsilon D_0)_+^p\mid\mathcal{G}_0\right]\le Y_0^p. \tag*{(144)} E [ ( Y 1 − ε D 0 ) + p ∣ G 0 ] ≤ Y 0 p . (144) Then, there exists a constant C = C ( ε , c , p , r , K ) C=C(\varepsilon,c,p,r,K) C = C ( ε , c , p , r , K )
sup m ≥ 0 E [ X m r ∣ G 0 ] ≤ C ( X 0 r + Y 0 r + 1 ) and lim sup m → ∞ E [ X m r ∣ G 0 ] ≤ C . (145) \sup_{m\ge0}\mathbb{E}[X_m^r\mid\mathcal{G}_0]\le C(X_0^r+Y_0^r+1)
\qquad\text{and}\qquad
\limsup_{m\to\infty}\mathbb{E}[X_m^r\mid\mathcal{G}_0]\le C. \tag*{(145)} m ≥ 0 sup E [ X m r ∣ G 0 ] ≤ C ( X 0 r + Y 0 r + 1 ) and m → ∞ lim sup E [ X m r ∣ G 0 ] ≤ C . (145) Remark 3.56. Since the conclusions are conditional on G 0 \mathcal{G}_0 G 0 X 0 X_0 X 0 Y 0 Y_0 Y 0 Z 0 Z_0 Z 0
Remark 3.57. In our global iteration scheme (see Section 11), X m X_m X m D m D_m D m Y m + 1 Y_{m+1} Y m + 1 G m − 1 \mathcal{G}_{m-1} G m − 1 G m \mathcal{G}_m G m
The reason for treating the initial fluctuation separately, i.e., for separating (143) and (144) , also lies in our global iteration scheme. To be more precise, since the initial data in our global well-posedness theorems (Theorems 2.24 and 2.30 ) is not required to have any random structure, the estimates in the first step of our global iteration scheme sometimes have to differ from the estimates in later steps.
Before we turn to the proof of Proposition 3.55 , we first state and prove the following simpler lemma.
Lemma 3.58. Let ( G m ) m = 0 ∞ (\mathcal{G}_m)_{m=0}^{\infty} ( G m ) m = 0 ∞ ( Z m ) m = 0 ∞ (Z_m)_{m=0}^{\infty} ( Z m ) m = 0 ∞ Z m ≥ 10 Z_m \ge10 Z m ≥ 10 m ≥ 0 m \ge0 m ≥ 0 c > 0 c > 0 c > 0 K ≥ 1 K \ge1 K ≥ 1 p > 3 2 p > \frac{3}{2} p > 2 3 m ≥ 0 m \ge0 m ≥ 0
E [ Z m + 1 p ∣ G m ] 1 p ≤ Z m − c Z m log ( Z m ) + K . (146) \mathbb{E}\left[Z_{m+1}^{p}\mid\mathcal{G}_m\right]^{\frac{1}{p}} \le Z_m-c\frac{\sqrt{Z_m}}{\log(Z_m)}+K. \tag*{(146)} E [ Z m + 1 p ∣ G m ] p 1 ≤ Z m − c log ( Z m ) Z m + K . (146) Then, it holds for all 0 < r < p − 1 2 0 < r < p-\frac{1}{2} 0 < r < p − 2 1 1 2 < α < min ( p − r , 1 ) \frac{1}{2}<\alpha<\min(p-r,1) 2 1 < α < min ( p − r , 1 ) m ≥ 0 m \ge0 m ≥ 0
E [ Z m r ∣ G 0 ] ≤ min ( Z 0 r , C Z 0 p ( m + 1 ) p − r α ) + C , (147) \mathbb{E}[Z_m^r\mid\mathcal{G}_0]\le\min\left(Z_0^r,\frac{CZ_0^p}{(m+1)^{\frac{p-r}{\alpha}}}\right)+C, \tag*{(147)} E [ Z m r ∣ G 0 ] ≤ min ( Z 0 r , ( m + 1 ) α p − r C Z 0 p ) + C , (147) where C = C ( α , c , p , r , K ) C=C(\alpha,c,p,r,K) C = C ( α , c , p , r , K )
Proof of Lemma 3.58. For expository purposes, we split the proof into five different steps.
Step 1: Initial simplification. After choosing R = R ( α , c , p , K ) R=R(\alpha,c,p,K) R = R ( α , c , p , K ) A = A ( α , c , p , K , R ) A=A(\alpha,c,p,K,R) A = A ( α , c , p , K , R ) (146) implies that
E [ Z m + 1 p ∣ G m ] ≤ Z m p − Z m p − α on { Z m > R } , (148) \mathbb{E}[Z_{m+1}^p\mid\mathcal{G}_m]\le Z_m^p-Z_m^{p-\alpha}\qquad\text{on }\{Z_m>R\}, \tag*{(148)} E [ Z m + 1 p ∣ G m ] ≤ Z m p − Z m p − α on { Z m > R } , (148) E [ Z m + 1 p ∣ G m ] ≤ A p on { Z m ≤ R } . (149) \mathbb{E}[Z_{m+1}^p\mid\mathcal{G}_m]\le A^p\qquad\text{on }\{Z_m\le R\}. \tag*{(149)} E [ Z m + 1 p ∣ G m ] ≤ A p on { Z m ≤ R } . (149) Step 2: The supermartingales. To simplify the notation, set β : = α / p ∈ ( 0 , 1 ) \beta\mathrel{\mathop:}=\alpha/p\in(0,1) β : = α / p ∈ ( 0 , 1 ) m ≥ 0 m\ge0 m ≥ 0 H m : [ 0 , ∞ ) → [ 0 , ∞ ) H_m:[0,\infty)\to[0,\infty) H m : [ 0 , ∞ ) → [ 0 , ∞ )
H m ( x ) : = ( x β + β m ) 1 β . H_m(x)\mathrel{\mathop:}=(x^\beta+\beta m)^{\frac{1}{\beta}}. H m ( x ) : = ( x β + β m ) β 1 . From a direct computation of the first and second derivatives of H m H_m H m H m H_m H m 22 ^{22} 22 ν \nu ν
ν : = inf { n ≥ 0 : Z n ≤ R } , \nu\mathrel{\mathop:}=\inf\{n\ge0:Z_n\le R\}, ν : = inf { n ≥ 0 : Z n ≤ R } , i.e., as the first time at which the process is bounded by R R R
( H ν ∧ m ( Z ν ∧ m p ) ) m = 0 ∞ (150) \left(H_{\nu\wedge m}(Z_{\nu\wedge m}^p)\right)_{m=0}^{\infty} \tag*{(150)} ( H ν ∧ m ( Z ν ∧ m p ) ) m = 0 ∞ (150) is a supermartingale. To see this, we first split
E [ H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] = E [ 1 { ν ≤ m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] (151) \mathbb{E}\left[H_{\nu\wedge(m+1)}(Z_{\nu\wedge(m+1)}^p)\mid\mathcal{G}_m\right]
=\mathbb{E}\left[\mathbf{1}_{\{\nu\le m\}}H_{\nu\wedge(m+1)}(Z_{\nu\wedge(m+1)}^p)\mid\mathcal{G}_m\right] \tag*{(151)} E [ H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] = E [ 1 { ν ≤ m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] (151) + E [ 1 { ν > m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] . (152) +\mathbb{E}\left[\mathbf{1}_{\{\nu>m\}}H_{\nu\wedge(m+1)}(Z_{\nu\wedge(m+1)}^p)\mid\mathcal{G}_m\right]. \tag*{(152)} + E [ 1 { ν > m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] . (152) The first summand (151) is clearly equal to 1 { ν ≤ m } H ν ∧ m ( Z ν ∧ m p ) \mathbf{1}_{\{\nu\le m\}}H_{\nu\wedge m}(Z_{\nu\wedge m}^p) 1 { ν ≤ m } H ν ∧ m ( Z ν ∧ m p ) (148) , the second summand can be estimated by
E [ 1 { ν > m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] = 1 { ν > m } E [ H m + 1 ( Z m + 1 p ) ∣ G m ] ≤ 1 { ν > m } H m + 1 ( E [ Z m + 1 p ∣ G m ] ) ≤ 1 { ν > m } H m + 1 ( Z m p − Z m p − α ) . \begin{aligned}
\mathbb{E}\left[\mathbf{1}_{\{\nu>m\}}H_{\nu\wedge(m+1)}(Z_{\nu\wedge(m+1)}^p)\mid\mathcal{G}_m\right]
=\mathbf{1}_{\{\nu>m\}}\mathbb{E}[H_{m+1}(Z_{m+1}^p)\mid\mathcal{G}_m] \\
&\le\mathbf{1}_{\{\nu>m\}}H_{m+1}\left(\mathbb{E}[Z_{m+1}^p\mid\mathcal{G}_m]\right)
\le\mathbf{1}_{\{\nu>m\}}H_{m+1}(Z_m^p-Z_m^{p-\alpha}).
\end{aligned} E [ 1 { ν > m } H ν ∧ ( m + 1 ) ( Z ν ∧ ( m + 1 ) p ) ∣ G m ] = 1 { ν > m } E [ H m + 1 ( Z m + 1 p ) ∣ G m ] ≤ 1 { ν > m } H m + 1 ( E [ Z m + 1 p ∣ G m ] ) ≤ 1 { ν > m } H m + 1 ( Z m p − Z m p − α ) . Since x ∈ [ 0 , ∞ ) ↦ x β x\in[0,\infty)\mapsto x^\beta x ∈ [ 0 , ∞ ) ↦ x β
( Z m p − Z m p − α ) β ≤ ( Z m p ) β − β ( Z m p ) β − 1 Z m p − α = ( Z m p ) β − β , (Z_m^p-Z_m^{p-\alpha})^\beta\le(Z_m^p)^\beta-\beta(Z_m^p)^{\beta-1}Z_m^{p-\alpha}=(Z_m^p)^\beta-\beta, ( Z m p − Z m p − α ) β ≤ ( Z m p ) β − β ( Z m p ) β − 1 Z m p − α = ( Z m p ) β − β , where we used that β = α p \beta= \frac{\alpha}{p} β = p α H m H_m H m H m + 1 H_{m+1} H m + 1
H m + 1 ( Z m p − Z m p − α ) ≤ H m ( Z m p ) . H_{m+1}(Z_m^p-Z_m^{p-\alpha})\le H_m(Z_m^p). H m + 1 ( Z m p − Z m p − α ) ≤ H m ( Z m p ) . Thus, the second summand (152) can be bounded by 1 { ν > m } H m ( Z m p ) = 1 { ν > m } H ν ∧ m ( Z ν ∧ m p ) \mathbb{1}\{\nu>m\}H_m(Z_m^p)=\mathbb{1}\{\nu>m\}H_{\nu\wedge m}(Z_{\nu\wedge m}^p) 1 { ν > m } H m ( Z m p ) = 1 { ν > m } H ν ∧ m ( Z ν ∧ m p ) (150) , one can also show that
( Z ν ∧ m r ) m = 0 ∞ (153) \left(Z_{\nu\wedge m}^r\right)_{m=0}^{\infty} \tag*{(153)} ( Z ν ∧ m r ) m = 0 ∞ (153) is a supermartingale. Indeed, as in (151) –(152) , one first decomposes
E [ Z ν ∧ ( m + 1 ) r ∣ G m ] = E [ 1 { ν ≤ m } Z ν ∧ ( m + 1 ) r ∣ G m ] + E [ 1 { ν > m } Z ν ∧ ( m + 1 ) r ∣ G m ] . (154) \mathbb{E}[Z_{\nu\wedge(m+1)}^r\mid\mathcal{G}_m]=\mathbb{E}[\mathbb{1}\{\nu\le m\}Z_{\nu\wedge(m+1)}^r\mid\mathcal{G}_m]+\mathbb{E}[\mathbb{1}\{\nu>m\}Z_{\nu\wedge(m+1)}^r\mid\mathcal{G}_m]. \tag*{(154)} E [ Z ν ∧ ( m + 1 ) r ∣ G m ] = E [ 1 { ν ≤ m } Z ν ∧ ( m + 1 ) r ∣ G m ] + E [ 1 { ν > m } Z ν ∧ ( m + 1 ) r ∣ G m ] . (154) The first summand in (154) is clearly equal to 1 { ν ≤ m } Z ν ∧ m r \mathbb{1}\{\nu\le m\}Z_{\nu\wedge m}^r 1 { ν ≤ m } Z ν ∧ m r (148) , the second summand in (154) can be bounded by
1 { ν > m } E [ Z m + 1 r ∣ G m ] ≤ 1 { ν > m } ( Z m p − Z m p − α ) r p ≤ 1 { ν > m } Z m r = 1 { ν > m } Z ν ∧ m r , \mathbb{1}\{\nu>m\}\mathbb{E}[Z_{m+1}^r\mid\mathcal{G}_m]\le\mathbb{1}\{\nu>m\}(Z_m^p-Z_m^{p-\alpha})^{\frac{r}{p}}\le\mathbb{1}\{\nu>m\}Z_m^r=\mathbb{1}\{\nu>m\}Z_{\nu\wedge m}^r, 1 { ν > m } E [ Z m + 1 r ∣ G m ] ≤ 1 { ν > m } ( Z m p − Z m p − α ) p r ≤ 1 { ν > m } Z m r = 1 { ν > m } Z ν ∧ m r , which is also as desired.
Step 3: Estimate for an excursion above R R R For all m ≥ 0 m\ge0 m ≥ 0
E [ 1 { ν > m } Z m r ∣ G 0 ] ≤ min ( Z 0 r , B Z 0 p ( m + 1 ) p − r α ) , (155) \mathbb{E}[\mathbb{1}\{\nu>m\}Z_m^r\mid\mathcal{G}_0]\le\min\left(Z_0^r,\frac{BZ_0^p}{(m+1)^{\frac{p-r}{\alpha}}}\right), \tag*{(155)} E [ 1 { ν > m } Z m r ∣ G 0 ] ≤ min ( Z 0 r , ( m + 1 ) α p − r B Z 0 p ) , (155) where B = B ( α , p , r , A ) B=B(\alpha,p,r,A) B = B ( α , p , r , A ) m = 0 m=0 m = 0 m ≥ 1 m\ge1 m ≥ 1 Z 0 r Z_0^r Z 0 r (155) , which is the simpler estimate, we only need to use the supermartingale property for (153) and that 1 { ν > m } Z m r ≤ Z ν ∧ m r \mathbb{1}\{\nu>m\}Z_m^r\le Z_{\nu\wedge m}^r 1 { ν > m } Z m r ≤ Z ν ∧ m r B Z 0 p / ( m + 1 ) p − r α BZ_0^p/(m+1)^{\frac{p-r}{\alpha}} B Z 0 p / ( m + 1 ) α p − r (155) is more difficult to obtain and relies on our supermartingale from (150) . Indeed, using the supermartingale property, it holds that
E [ H ν ∧ m ( Z ν ∧ m p ) ∣ G 0 ] ≤ H 0 ( Z 0 p ) . (156) \mathbb{E}[H_{\nu\wedge m}(Z_{\nu\wedge m}^p)\mid\mathcal{G}_0]\le H_0(Z_0^p). \tag*{(156)} E [ H ν ∧ m ( Z ν ∧ m p ) ∣ G 0 ] ≤ H 0 ( Z 0 p ) . (156) By definition, the right-hand side of (156) is equal to Z 0 p Z_0^p Z 0 p (156) can be bounded below by
E [ H ν ∧ m ( Z ν ∧ m p ) ∣ G 0 ] ≥ E [ 1 { ν > m } H m ( Z m p ) ∣ G 0 ] . \mathbb{E}[H_{\nu\wedge m}(Z_{\nu\wedge m}^p)\mid\mathcal{G}_0]\ge\mathbb{E}[\mathbb{1}\{\nu>m\}H_m(Z_m^p)\mid\mathcal{G}_0]. E [ H ν ∧ m ( Z ν ∧ m p ) ∣ G 0 ] ≥ E [ 1 { ν > m } H m ( Z m p ) ∣ G 0 ] . Since 0 < r < p 0<r<p 0 < r < p
H m ( Z m p ) = ( ( Z m p ) α / p + α p m ) α = ( Z m α + α p m ) p α ( Z m α + α p m ) r α ≥ Z m r ( α p m ) p − r α . H_m(Z_m^p)=\left((Z_m^p)^{\alpha/p}+\frac{\alpha}{p}m\right)^{\alpha}=\left(Z_m^{\alpha}+\frac{\alpha}{p}m\right)^{\frac{p}{\alpha}}\left(Z_m^{\alpha}+\frac{\alpha}{p}m\right)^{\frac{r}{\alpha}}\ge Z_m^r\left(\frac{\alpha}{p}m\right)^{\frac{p-r}{\alpha}}. H m ( Z m p ) = ( ( Z m p ) α / p + p α m ) α = ( Z m α + p α m ) α p ( Z m α + p α m ) α r ≥ Z m r ( p α m ) α p − r . From this, it follows that
E [ 1 { ν > m } Z m r ∣ G 0 ] ≤ ( α p m ) − p − r α Z 0 p . \mathbb{E}[\mathbb{1}\{\nu>m\}Z_m^r\mid\mathcal{G}_0]\le\left(\frac{\alpha}{p}m\right)^{-\frac{p-r}{\alpha}}Z_0^p. E [ 1 { ν > m } Z m r ∣ G 0 ] ≤ ( p α m ) − α p − r Z 0 p . Provided that B B B m m m m + 1 m+1 m + 1 (155) .
Step 4: Estimate for an excursion starting from below R R R In this step, we show for all 0 ≤ k < m 0\le k<m 0 ≤ k < m
E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G k ] ≤ A p B ( m − k ) p − r α . (157) \mathbb{E}[\mathbb{1}\{Z_m>R,\ldots,Z_{k+1}>R,Z_k\le R\}Z_m^r\mid\mathcal{G}_k]\le\frac{A^pB}{(m-k)^{\frac{p-r}{\alpha}}}. \tag*{(157)} E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G k ] ≤ ( m − k ) α p − r A p B . (157) To obtain (157) , we first use the time-shifted version of (155) , which implies that
E [ 1 { Z m > R , … , Z k + 1 > R } Z m r ∣ G k + 1 ] ≤ B Z k + 1 p ( m − k ) p − r α . \mathbb{E}[\mathbb{1}\{Z_m>R,\ldots,Z_{k+1}>R\}Z_m^r\mid\mathcal{G}_{k+1}]\le\frac{BZ_{k+1}^p}{(m-k)^{\frac{p-r}{\alpha}}}. E [ 1 { Z m > R , … , Z k + 1 > R } Z m r ∣ G k + 1 ] ≤ ( m − k ) α p − r B Z k + 1 p . Using (149) , we then obtain that
E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G k ] ≤ E [ 1 { Z k ≤ R } B Z k + 1 p ( m − k ) p − r α | G k ] ≤ A p B ( m − k ) p − r α , \mathbb{E}\left[1\{Z_m>R,\ldots,Z_{k+1}>R,Z_k\le R\}Z_m^r\mid\mathcal{G}_k\right]
\le\mathbb{E}\left[1\{Z_k\le R\}\frac{BZ_{k+1}^p}{(m-k)^{\frac{p-r}{\alpha}}}\,\middle|\,\mathcal{G}_k\right]
\le\frac{A^pB}{(m-k)^{\frac{p-r}{\alpha}}}, E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G k ] ≤ E [ 1 { Z k ≤ R } ( m − k ) α p − r B Z k + 1 p G k ] ≤ ( m − k ) α p − r A p B , which yields the desired estimate (157) .
Step 5: Conclusion. Using (155) and (157) , we obtain that
E [ Z m r ∣ G 0 ] = E [ 1 { Z m > R , … , Z 0 > R } Z m r ∣ G 0 ] \mathbb{E}[Z_m^r\mid\mathcal{G}_0]=\mathbb{E}[1\{Z_m>R,\ldots,Z_0>R\}Z_m^r\mid\mathcal{G}_0] E [ Z m r ∣ G 0 ] = E [ 1 { Z m > R , … , Z 0 > R } Z m r ∣ G 0 ] + ∑ k = 0 m − 1 E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G 0 ] + E [ 1 { Z m ≤ R } Z m r ∣ G 0 ] ≤ min ( Z 0 r , B Z 0 p ( m + 1 ) p − r α ) + A p B ∑ k = 0 m − 1 1 ( m − k ) p − r α + R r ≤ min ( Z 0 r , C Z 0 p ( m + 1 ) p − r α ) + C , \begin{aligned}
\quad+\sum_{k=0}^{m-1}\mathbb{E}[1\{Z_m>R,\ldots,Z_{k+1}>R,Z_k\le R\}Z_m^r\mid\mathcal{G}_0]+\mathbb{E}[1\{Z_m\le R\}Z_m^r\mid\mathcal{G}_0] \\
&\le\min\left(Z_0^r,\frac{BZ_0^p}{(m+1)^{\frac{p-r}{\alpha}}}\right)+A^pB\sum_{k=0}^{m-1}\frac{1}{(m-k)^{\frac{p-r}{\alpha}}}+R^r \\
&\le\min\left(Z_0^r,\frac{CZ_0^p}{(m+1)^{\frac{p-r}{\alpha}}}\right)+C,
\end{aligned} + k = 0 ∑ m − 1 E [ 1 { Z m > R , … , Z k + 1 > R , Z k ≤ R } Z m r ∣ G 0 ] + E [ 1 { Z m ≤ R } Z m r ∣ G 0 ] ≤ min ( Z 0 r , ( m + 1 ) α p − r B Z 0 p ) + A p B k = 0 ∑ m − 1 ( m − k ) α p − r 1 + R r ≤ min ( Z 0 r , ( m + 1 ) α p − r C Z 0 p ) + C , where C = C ( α , p , r , A , B , R ) C=C(\alpha,p,r,A,B,R) C = C ( α , p , r , A , B , R ) ( p − r ) / α > 1 (p-r)/\alpha>1 ( p − r ) / α > 1 (147) .
Proof of Proposition 3.55. To simplify the notation, we set U m + 1 ≔ ( Y m + 1 − ε D m ) + U_{m+1}\coloneqq(Y_{m+1}-\varepsilon D_m)_+ U m + 1 : = ( Y m + 1 − ε D m ) + (141) and the definition of U m + 1 U_{m+1} U m + 1
X m + 1 ≤ X m − ( 1 − ε ) D m + K + U m + 1 ≤ X m − ( 1 − ε ) ( D m ∧ X m ) + K + U m + 1 . (158) \begin{aligned}
X_{m+1}\le X_m-(1-\varepsilon)D_m+K+U_{m+1} \\
&\le X_m-(1-\varepsilon)(D_m\wedge X_m)+K+U_{m+1}. \tag*{(158)}
\end{aligned} X m + 1 ≤ X m − ( 1 − ε ) D m + K + U m + 1 ≤ X m − ( 1 − ε ) ( D m ∧ X m ) + K + U m + 1 . (158) The reason for introducing the minimum of D m D_m D m X m X_m X m (158) is that X m − ( 1 − ε ) ( D m ∧ X m ) + K X_m-(1-\varepsilon)(D_m\wedge X_m)+K X m − ( 1 − ε ) ( D m ∧ X m ) + K
Z m ≔ X m + 2 W m , where W m ≔ E [ U m + 1 p ∣ G m ] 1 p . (159) Z_m\coloneqq X_m+2W_m,\qquad\text{where}\qquad W_m\coloneqq\mathbb{E}[U_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}}. \tag*{(159)} Z m : = X m + 2 W m , where W m : = E [ U m + 1 p ∣ G m ] p 1 . (159) The purpose of the correction 2 W m 2W_m 2 W m (159) is to later absorb a contribution resulting from the U m + 1 U_{m+1} U m + 1 (158) , while leaving the additional decay term − W m -W_m − W m ( W m ) m = 0 ∞ (W_m)_{m=0}^{\infty} ( W m ) m = 0 ∞ ( Z m ) m = 0 ∞ (Z_m)_{m=0}^{\infty} ( Z m ) m = 0 ∞ X m ≤ Z m X_m\le Z_m X m ≤ Z m m ≥ 0 m\ge0 m ≥ 0 X m X_m X m Z m Z_m Z m m ≥ 0 m\ge0 m ≥ 0
E [ Z m + 1 p ∣ G m ] 1 p ≤ Z m − c ′ Z m log ( Z m ) + K ′ , (160) \mathbb{E}[Z_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}}\le Z_m-c'\frac{\sqrt{Z_m}}{\log(Z_m)}+K', \tag*{(160)} E [ Z m + 1 p ∣ G m ] p 1 ≤ Z m − c ′ log ( Z m ) Z m + K ′ , (160) where c ′ > 0 c'>0 c ′ > 0 K ′ ≥ 1 K'\ge1 K ′ ≥ 1 c , ε , p , r c,\varepsilon,p,r c , ε , p , r K K K (143) , it follows that
E [ W m + 1 p ∣ G m ] = E [ E [ U m + 2 p ∣ G m + 1 ] ∣ G m ] = E [ U m + 2 p ∣ G m ] ≤ K . \mathbb{E}[W_{m+1}^p\mid\mathcal{G}_m]=\mathbb{E}\left[\mathbb{E}[U_{m+2}^p\mid\mathcal{G}_{m+1}]\mid\mathcal{G}_m\right]=\mathbb{E}[U_{m+2}^p\mid\mathcal{G}_m]\le K. E [ W m + 1 p ∣ G m ] = E [ E [ U m + 2 p ∣ G m + 1 ] ∣ G m ] = E [ U m + 2 p ∣ G m ] ≤ K . Together with our recursive estimate from (158) , it then follows that
E [ Z m + 1 p ∣ G m ] 1 p ≤ E [ X m + 1 p ∣ G m ] 1 p + 2 E [ W m + 1 p ∣ G m ] 1 p ≤ E [ ( X m − ( 1 − ε ) ( D m ∧ X m ) + K ) p ∣ G m ] 1 p + E [ U m + 1 p ∣ G m ] 1 p + 2 K 1 p = X m − ( 1 − ε ) ( D m ∧ X m ) + K + W m + 2 K 1 p = Z m − ( 1 − ε ) ( D m ∧ X m ) − W m + K + 2 K 1 p . \begin{aligned}
\mathbb{E}[Z_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}}
\le\mathbb{E}[X_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}}+2\mathbb{E}[W_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}} \\
&\le\mathbb{E}[(X_m-(1-\varepsilon)(D_m\wedge X_m)+K)^p\mid\mathcal{G}_m]^{\frac{1}{p}}+\mathbb{E}[U_{m+1}^p\mid\mathcal{G}_m]^{\frac{1}{p}}+2K^{\frac{1}{p}} \\
&=X_m-(1-\varepsilon)(D_m\wedge X_m)+K+W_m+2K^{\frac{1}{p}} \\
&=Z_m-(1-\varepsilon)(D_m\wedge X_m)-W_m+K+2K^{\frac{1}{p}}.
\end{aligned} E [ Z m + 1 p ∣ G m ] p 1 ≤ E [ X m + 1 p ∣ G m ] p 1 + 2 E [ W m + 1 p ∣ G m ] p 1 ≤ E [( X m − ( 1 − ε ) ( D m ∧ X m ) + K ) p ∣ G m ] p 1 + E [ U m + 1 p ∣ G m ] p 1 + 2 K p 1 = X m − ( 1 − ε ) ( D m ∧ X m ) + K + W m + 2 K p 1 = Z m − ( 1 − ε ) ( D m ∧ X m ) − W m + K + 2 K p 1 . In the penultimate identity, we used that X m X_m X m D m D_m D m G m \mathcal{G}_m G m W m W_m W m Z m Z_m Z m (142) , as well as the fact that X m ≥ 10 X_m \ge10 X m ≥ 10 m ≥ 0 m \ge0 m ≥ 0
( 1 − ε ) ( D m ∧ X m ) + W m ≥ ( 1 − ε ) min ( c X m log ( X m ) , X m ) + W m ≥ c ′ Z m log ( Z m ) . (1-\varepsilon)(D_m \wedge X_m)+W_m \ge(1-\varepsilon)\min\left(c\frac{\sqrt{X_m}}{\log(X_m)},X_m\right)+W_m \ge c'\frac{\sqrt{Z_m}}{\log(Z_m)}. ( 1 − ε ) ( D m ∧ X m ) + W m ≥ ( 1 − ε ) min ( c log ( X m ) X m , X m ) + W m ≥ c ′ log ( Z m ) Z m . This completes the proof of (160) . By using Lemma 3.58 for some 1 2 < α < min ( 1 , p − r ) \frac{1}{2}<\alpha<\min(1,p-r) 2 1 < α < min ( 1 , p − r ) m ≥ 0 m\ge0 m ≥ 0
E [ Z m r ∣ G 0 ] ≤ min ( Z 0 r , C Z 0 p ( m + 1 ) p − r α ) + C . \mathbb{E}[Z_m^r\mid\mathcal{G}_0]\le\min\left(Z_0^r,\frac{CZ_0^p}{(m+1)^{\frac{p-r}{\alpha}}}\right)+C. E [ Z m r ∣ G 0 ] ≤ min ( Z 0 r , ( m + 1 ) α p − r C Z 0 p ) + C . Since X m ≤ Z m X_m\le Z_m X m ≤ Z m m ≥ 0 m\ge0 m ≥ 0 (144) implies that W 0 ≤ Y 0 W_0\le Y_0 W 0 ≤ Y 0
E [ X m r ∣ G 0 ] ≤ E [ Z m r ∣ G 0 ] ≤ min ( ( X 0 + 2 Y 0 ) r , C ( X 0 + 2 Y 0 ) p ( m + 1 ) p − r α ) + C . \mathbb{E}[X_m^r\mid\mathcal{G}_0]\le\mathbb{E}[Z_m^r\mid\mathcal{G}_0]\le\min\left((X_0+2Y_0)^r,\frac{C(X_0+2Y_0)^p}{(m+1)^{\frac{p-r}{\alpha}}}\right)+C. E [ X m r ∣ G 0 ] ≤ E [ Z m r ∣ G 0 ] ≤ min ( ( X 0 + 2 Y 0 ) r , ( m + 1 ) α p − r C ( X 0 + 2 Y 0 ) p ) + C . This readily leads to the desired estimates on the supremum and limit superior in (145) . □
In Section 11, Proposition 3.55 will be used to prove the estimate (iv) from our hard global well-posedness result (Theorem 2.30). While the estimate (iv) is uniform in our iteration index m ≥ 1 m\ge1 m ≥ 1 τ ~ m \widetilde{\tau}_m τ m
Proposition 3.59. Let the same assumptions as in Proposition 3.55 be satisfied. In addition, let q 1 > 0 q_1>0 q 1 > 0 0 ≤ q 2 ≤ p 0\le q_2\le p 0 ≤ q 2 ≤ p ( τ m ) m = 1 ∞ (\tau_m)_{m=1}^{\infty} ( τ m ) m = 1 ∞ m ≥ 0 m\ge0 m ≥ 0 τ m + 1 \tau_{m+1} τ m + 1 G m + 1 \mathcal{G}_{m+1} G m + 1 0 < τ m + 1 ≤ 1 0<\tau_{m+1}\le1 0 < τ m + 1 ≤ 1 m ≥ 1 m\ge1 m ≥ 1
E [ τ m + 1 − q 1 ∣ G m − 1 ] ≤ E [ X m q 2 ∣ G m − 1 ] + K . (161) \mathbb{E}[\tau_{m+1}^{-q_1}\mid\mathcal{G}_{m-1}]\le\mathbb{E}[X_m^{q_2}\mid\mathcal{G}_{m-1}]+K. \tag*{(161)} E [ τ m + 1 − q 1 ∣ G m − 1 ] ≤ E [ X m q 2 ∣ G m − 1 ] + K . (161) For each T > 0 T>0 T > 0
M T = d e f inf { m ≥ 1 : ∑ k = 0 m − 1 τ k + 1 ≥ T } . M_T\overset{\mathrm{def}}{=}\inf\left\{m\ge1:\sum_{k=0}^{m-1}\tau_{k+1}\ge T\right\}. M T = def inf { m ≥ 1 : k = 0 ∑ m − 1 τ k + 1 ≥ T } . Then, M T M_T M T
sup T > 0 E [ X M T + 1 r ∣ G 0 ] ≤ C ′ ( X 0 r + Y 0 r + 1 ) and lim sup T → ∞ E [ X M T + 1 r ∣ G 0 ] ≤ C ′ , (162) \sup_{T>0}\mathbb{E}[X_{M_T+1}^r\mid\mathcal{G}_0]\le C'(X_0^r+Y_0^r+1)\quad\text{and}\quad\limsup_{T\to\infty}\mathbb{E}[X_{M_T+1}^r\mid\mathcal{G}_0]\le C', \tag*{(162)} T > 0 sup E [ X M T + 1 r ∣ G 0 ] ≤ C ′ ( X 0 r + Y 0 r + 1 ) and T → ∞ lim sup E [ X M T + 1 r ∣ G 0 ] ≤ C ′ , (162) where C ′ = C ′ ( ε , c , p , q 1 , q 2 , r , K ) C'=C'(\varepsilon,c,p,q_1,q_2,r,K) C ′ = C ′ ( ε , c , p , q 1 , q 2 , r , K )
We emphasise two technical aspects of Proposition 3.59, which will be convenient in the proof of Proposition 11.31 below. First, the conditional moment estimate (161) is only assumed for m ≥ 1 m\ge1 m ≥ 1 m = 0 m=0 m = 0 (162) controls the process X = ( X m ) m = 0 ∞ X=(X_m)_{m=0}^{\infty} X = ( X m ) m = 0 ∞ M T M_T M T M T M_T M T
Throughout the proof, we will refer to the sequence ( τ m ) m = 1 ∞ (\tau_m)_{m=1}^{\infty} ( τ m ) m = 1 ∞ ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ m ∈ N 0 m\in\mathbb{N}_0 m ∈ N 0 t m ∈ [ 0 , ∞ ) t_m\in[0,\infty) t m ∈ [ 0 , ∞ )
Proof of Proposition 3.59 . We use the same notation as in the proofs of Proposition 3.55 and Lemma 3.58 . In particular, Z m = X m + 2 W m Z_m = X_m + 2W_m Z m = X m + 2 W m R R R A A A B B B
Step 1: Further applications of our supermartingale. In this step, we prove variants of the supermartingale bounds from the proof of Lemma 3.58 . To this end, we first recall that (150) and (153) are supermartingales. For any j ≥ 0 j \ge0 j ≥ 0
ν j : = inf { m ≥ j : Z m ≤ R } , \nu_j \mathrel{\mathop:}= \inf\{m \ge j: Z_m \le R\}, ν j : = inf { m ≥ j : Z m ≤ R } , which is the first return to the region below R R R j j j ν j − j \nu_j-j ν j − j ν \nu ν Lemma 3.58 , except that the process ( Z m ) m = 0 ∞ (Z_m)_{m=0}^{\infty} ( Z m ) m = 0 ∞ ( Z j + m ) m = 0 ∞ (Z_{j+m})_{m=0}^{\infty} ( Z j + m ) m = 0 ∞ Lemma 3.58 , one can then show that
( H ( ν j − j ) ∧ m ( Z j + ( ν j − j ) ∧ m p ) ) m = 0 ∞ and ( Z j + ( ν j − j ) ∧ m r ) m = 0 ∞ (163) \left(H_{(\nu_j-j)\wedge m}\left(Z_{j+(\nu_j-j)\wedge m}^{p}\right)\right)_{m=0}^{\infty}
\quad\text{and}\quad
\left(Z_{j+(\nu_j-j)\wedge m}^{r}\right)_{m=0}^{\infty} \tag*{(163)} ( H ( ν j − j ) ∧ m ( Z j + ( ν j − j ) ∧ m p ) ) m = 0 ∞ and ( Z j + ( ν j − j ) ∧ m r ) m = 0 ∞ (163) are supermartingales with respect to the shifted filtration ( G j + m ) m = 0 ∞ (\mathcal{G}_{j+m})_{m=0}^{\infty} ( G j + m ) m = 0 ∞ M M M ( G m ) m = 0 ∞ (\mathcal{G}_m)_{m=0}^{\infty} ( G m ) m = 0 ∞ M ≥ j M \ge j M ≥ j M − j M-j M − j ( G j + m ) m = 0 ∞ (\mathcal{G}_{j+m})_{m=0}^{\infty} ( G j + m ) m = 0 ∞
E [ H ( ν j ∧ M ) − j ( Z ν j ∧ M p ) ∣ G j ] = E [ H ( ν j − j ) ∧ ( M − j ) ( Z j + ( ν j − j ) ∧ ( M − j ) p ) ∣ G j ] ≤ H 0 ( Z j p ) (164) \mathbb{E}\left[H_{(\nu_j\wedge M)-j}\left(Z_{\nu_j\wedge M}^{p}\right)\mid\mathcal{G}_j\right]
=
\mathbb{E}\left[H_{(\nu_j-j)\wedge(M-j)}\left(Z_{j+(\nu_j-j)\wedge(M-j)}^{p}\right)\mid\mathcal{G}_j\right]
\le H_0\left(Z_j^{p}\right) \tag*{(164)} E [ H ( ν j ∧ M ) − j ( Z ν j ∧ M p ) ∣ G j ] = E [ H ( ν j − j ) ∧ ( M − j ) ( Z j + ( ν j − j ) ∧ ( M − j ) p ) ∣ G j ] ≤ H 0 ( Z j p ) (164) and
E [ Z ν j ∧ M r ∣ G j ] = E [ Z j + ( ν j − j ) ∧ ( M − j ) r ∣ G j ] ≤ Z j r . (165) \mathbb{E}\left[Z_{\nu_j\wedge M}^{r}\mid\mathcal{G}_j\right]
=
\mathbb{E}\left[Z_{j+(\nu_j-j)\wedge(M-j)}^{r}\mid\mathcal{G}_j\right]
\le Z_j^{r}. \tag*{(165)} E [ Z ν j ∧ M r ∣ G j ] = E [ Z j + ( ν j − j ) ∧ ( M − j ) r ∣ G j ] ≤ Z j r . (165) By using (164) –(165) and following the same argument as in the derivation of (155) , we further obtain that
E [ 1 { ν j > M } Z M r ∣ G j ] ≤ Z j r , (166) \mathbb{E}\left[\boldsymbol{1}_{\{\nu_j>M\}}Z_M^{r}\mid\mathcal{G}_j\right]\le Z_j^{r}, \tag*{(166)} E [ 1 { ν j > M } Z M r ∣ G j ] ≤ Z j r , (166) E [ 1 { ν j > M } ( M − j + 1 ) p − r α Z M r ∣ G j ] ≤ B Z j p . (167) \mathbb{E}\left[\boldsymbol{1}_{\{\nu_j>M\}}(M-j+1)^{\frac{p-r}{\alpha}}Z_M^{r}\mid\mathcal{G}_j\right]\le BZ_j^{p}. \tag*{(167)} E [ 1 { ν j > M } ( M − j + 1 ) α p − r Z M r ∣ G j ] ≤ B Z j p . (167) We note that (167) with M = j + n M=j+n M = j + n n ≥ 0 n\ge0 n ≥ 0
P ( ν j > j + n ∣ G j ) ≤ B Z j p ( n + 1 ) p − r α . \mathbb{P}(\nu_j>j+n\mid\mathcal{G}_j)\le\frac{BZ_j^{p}}{(n+1)^{\frac{p-r}{\alpha}}}. P ( ν j > j + n ∣ G j ) ≤ ( n + 1 ) α p − r B Z j p . From this, it follows that ν j \nu_j ν j j ≥ 0 j\ge0 j ≥ 0 { ν j < ∞ } \{\nu_j<\infty\} { ν j < ∞ } j ≥ 0 j\ge0 j ≥ 0
P ( Z m ≤ R infinitely often ) = 1. (168) \mathbb{P}(Z_m\le R\text{ infinitely often})=1. \tag*{(168)} P ( Z m ≤ R infinitely often ) = 1. (168) Step 2: Estimates of physical times. In this step, we examine the physical time increments ( τ m ) m = 1 ∞ (\tau_m)_{m=1}^{\infty} ( τ m ) m = 1 ∞ ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞
t 0 : = 0 and t m : = ∑ k = 0 m − 1 τ k + 1 for all m ≥ 1. (169) t_0\mathrel{\mathop:}=0
\quad\text{and}\quad
t_m\mathrel{\mathop:}=\sum_{k=0}^{m-1}\tau_{k+1}\quad\text{for all }m\ge1. \tag*{(169)} t 0 : = 0 and t m : = k = 0 ∑ m − 1 τ k + 1 for all m ≥ 1. (169) We emphasise that, since τ k + 1 \tau_{k+1} τ k + 1 G k + 1 \mathcal{G}_{k+1} G k + 1 k ≥ 0 k\ge0 k ≥ 0 t m t_m t m G m \mathcal{G}_m G m τ k + 2 \tau_{k+2} τ k + 2 { Z k ≤ R } \{Z_k\le R\} { Z k ≤ R } (161) implies that E [ τ k + 2 − q 1 ∣ G k ] ≤ E [ X k + 1 q 2 ∣ G k ] + K \mathbb{E}[\tau_{k+2}^{-q_1}\mid\mathcal{G}_k]\leq\mathbb{E}[X_{k+1}^{q_2}\mid\mathcal{G}_k]+K E [ τ k + 2 − q 1 ∣ G k ] ≤ E [ X k + 1 q 2 ∣ G k ] + K q 2 ≤ p q_2\leq p q 2 ≤ p (149) , the remaining conditional moment can be bounded by
E [ X k + 1 q 2 ∣ G k ] ≤ E [ Z k + 1 q 2 ∣ G k ] ≤ E [ Z k + 1 p ∣ G k ] q 2 p ≤ A q 2 . \mathbb{E}[X_{k+1}^{q_2}\mid\mathcal{G}_k]\leq\mathbb{E}[Z_{k+1}^{q_2}\mid\mathcal{G}_k]\leq\mathbb{E}[Z_{k+1}^{p}\mid\mathcal{G}_k]^{\frac{q_2}{p}}\leq A^{q_2}. E [ X k + 1 q 2 ∣ G k ] ≤ E [ Z k + 1 q 2 ∣ G k ] ≤ E [ Z k + 1 p ∣ G k ] p q 2 ≤ A q 2 . For h : = 2 − 1 q 1 ( A q 2 + K ) − 1 q 1 h\mathrel{\mathop:}=2^{-\frac{1}{q_1}}(A^{q_2}+K)^{-\frac{1}{q_1}} h : = 2 − q 1 1 ( A q 2 + K ) − q 1 1
P ( τ k + 2 ≥ h ∣ G k ) = 1 − P ( τ k + 2 < h ∣ G k ) ≥ 1 − h q 1 ( A q 2 + K ) ≥ 1 2 . (170) \mathbb{P}(\tau_{k+2}\geq h\mid\mathcal{G}_k)=1-\mathbb{P}(\tau_{k+2}<h\mid\mathcal{G}_k)\geq1-h^{q_1}(A^{q_2}+K)\geq\frac{1}{2}. \tag*{(170)} P ( τ k + 2 ≥ h ∣ G k ) = 1 − P ( τ k + 2 < h ∣ G k ) ≥ 1 − h q 1 ( A q 2 + K ) ≥ 2 1 . (170) Next, we show that the process ( Z m ) m = 0 ∞ (Z_m)_{m=0}^{\infty} ( Z m ) m = 0 ∞ R R R ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞ I ⊆ [ 0 , ∞ ) I\subseteq[0,\infty) I ⊆ [ 0 , ∞ )
N ( I ) : = ∑ k ≥ 0 1 { Z k ≤ R , t k ∈ I } . (171) N(I)\mathrel{\mathop:}=\sum_{k\geq0}\mathbf{1}\{Z_k\leq R,\ t_k\in I\}. \tag*{(171)} N ( I ) : = k ≥ 0 ∑ 1 { Z k ≤ R , t k ∈ I } . (171) We note that for each k ≥ 0 k\geq0 k ≥ 0 k k k (171) is G k \mathcal{G}_k G k (170) , it then holds that
E [ N ( I ) ∣ G 0 ] = ∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k ∈ I } ∣ G 0 ] ≤ 2 ∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k ∈ I } P ( τ k + 2 ≥ h ∣ G k ) ∣ G 0 ] = 2 ∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k ∈ I , τ k + 2 ≥ h } ∣ G 0 ] = 2 E [ ∑ k = 0 ∞ 1 { Z k ≤ R , t k ∈ I , τ k + 2 ≥ h } ∣ G 0 ] . \begin{aligned}
\mathbb{E}[N(I)\mid\mathcal{G}_0]
&=\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbf{1}\{Z_k\leq R,\ t_k\in I\}\mid\mathcal{G}_0\right]\\
&\leq2\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbf{1}\{Z_k\leq R,\ t_k\in I\}\mathbb{P}(\tau_{k+2}\geq h\mid\mathcal{G}_k)\mid\mathcal{G}_0\right]\\
&=2\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbf{1}\{Z_k\leq R,\ t_k\in I,\ \tau_{k+2}\geq h\}\mid\mathcal{G}_0\right]\\
&=2\mathbb{E}\left[\sum_{k=0}^{\infty}\mathbf{1}\{Z_k\leq R,\ t_k\in I,\ \tau_{k+2}\geq h\}\mid\mathcal{G}_0\right].
\end{aligned} E [ N ( I ) ∣ G 0 ] = k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k ∈ I } ∣ G 0 ] ≤ 2 k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k ∈ I } P ( τ k + 2 ≥ h ∣ G k ) ∣ G 0 ] = 2 k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k ∈ I , τ k + 2 ≥ h } ∣ G 0 ] = 2 E [ k = 0 ∑ ∞ 1 { Z k ≤ R , t k ∈ I , τ k + 2 ≥ h } ∣ G 0 ] . The intervals [ t k + 1 , t k + 2 ) [t_{k+1},t_{k+2}) [ t k + 1 , t k + 2 ) t k ∈ I t_k\in I t k ∈ I τ k + 2 ≥ h \tau_{k+2}\geq h τ k + 2 ≥ h h h h [ inf ( I ) , sup ( I ) + 2 ] [\inf(I),\sup(I)+2] [ inf ( I ) , sup ( I ) + 2 ] ( 2 + ∣ I ∣ ) / h (2+|I|)/h ( 2 + ∣ I ∣ ) / h
E [ N ( I ) ∣ G 0 ] ≤ 2 ( 2 + ∣ I ∣ ) / h . (172) \mathbb{E}[N(I)\mid\mathcal{G}_0]\leq2(2+|I|)/h. \tag*{(172)} E [ N ( I ) ∣ G 0 ] ≤ 2 ( 2 + ∣ I ∣ ) / h . (172) While our main motivation for proving (172) lies in (177) below, it also implies that t m → ∞ t_m\to\infty t m → ∞ m → ∞ m\to\infty m → ∞ b ≥ 0 b\geq0 b ≥ 0 lim m → ∞ t m ≤ b \lim_{m\to\infty}t_m\leq b lim m → ∞ t m ≤ b (168) , this would imply that N ( [ 0 , b ] ) = ∞ N([0,b])=\infty N ([ 0 , b ]) = ∞ (172) .
Step 3: Properties of M T M_T M T Using our physical times ( t m ) m = 0 ∞ (t_m)_{m=0}^{\infty} ( t m ) m = 0 ∞
M T = inf { m ≥ 1 : t m ≥ T } . M_T=\inf\{m\geq1:t_m\geq T\}. M T = inf { m ≥ 1 : t m ≥ T } . Since t m t_m t m G m \mathcal{G}_m G m { M T ≤ m } = { t m ≥ T } ∈ G m \{M_T\leq m\}=\{t_m\geq T\}\in\mathcal{G}_m { M T ≤ m } = { t m ≥ T } ∈ G m M T M_T M T t m → ∞ t_m\to\infty t m → ∞ m → ∞ m\to\infty m → ∞ M T M_T M T τ k + 1 ≤ 1 \tau_{k+1}\leq1 τ k + 1 ≤ 1 t m ≤ m t_m\leq m t m ≤ m M T M_T M T M T ≥ T M_T\geq T M T ≥ T { M T > m } \{M_T>m\} { M T > m }
T − t m ≤ t M T − t m = ∑ k = m M T − 1 τ k + 1 ≤ M T − m . T-t_m\leq t_{M_T}-t_m=\sum_{k=m}^{M_T-1}\tau_{k+1}\leq M_T-m. T − t m ≤ t M T − t m = k = m ∑ M T − 1 τ k + 1 ≤ M T − m . Step 4: Bounds on excursions. We first show that
E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] ≤ min ( Z 0 r , B Z 0 p ( 1 + T ) p − r α ) . (173) \mathbb{E}[\mathbf{1}\{M_T<\nu_0\}Z_{M_T}^{r}\mid\mathcal{G}_0]\leq\min\left(Z_0^r,\frac{BZ_0^p}{(1+T)^{\frac{p-r}{\alpha}}}\right). \tag*{(173)} E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] ≤ min ( Z 0 r , ( 1 + T ) α p − r B Z 0 p ) . (173) The bound by the first argument in (173) follows directly from (166) with M = M T M=M_T M = M T j = 0 j=0 j = 0 (173) , we use (167) with M = M T M=M_T M = M T j = 0 j=0 j = 0 M T + 1 ≥ 1 + T M_T+1\ge1+T M T + 1 ≥ 1 + T
( 1 + T ) p − r α E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] ≤ E [ 1 { M T < ν 0 } ( M T + 1 ) p − r α Z M T r ∣ G 0 ] ≤ B Z 0 p . (1+T)^{\frac{p-r}{\alpha}}\mathbb{E}[\boldsymbol{1}\{M_T<\nu_0\}Z_{M_T}^r\mid\mathcal{G}_0]\le\mathbb{E}[\boldsymbol{1}\{M_T<\nu_0\}(M_T+1)^{\frac{p-r}{\alpha}}Z_{M_T}^r\mid\mathcal{G}_0]\le BZ_0^p. ( 1 + T ) α p − r E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] ≤ E [ 1 { M T < ν 0 } ( M T + 1 ) α p − r Z M T r ∣ G 0 ] ≤ B Z 0 p . Next, we show for all k ≥ 0 k\ge0 k ≥ 0
E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G k ] ≤ 2 p − r α A p B ( 1 + T − t k ) p − r α 1 { Z k ≤ R , t k < T } . (174) \mathbb{E}\left[\boldsymbol{1}\{Z_k\le R,t_k<T\}\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_k\right]\le\frac{2^{\frac{p-r}{\alpha}}A^pB}{(1+T-t_k)^{\frac{p-r}{\alpha}}}\boldsymbol{1}\{Z_k\le R,t_k<T\}. \tag*{(174)} E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G k ] ≤ ( 1 + T − t k ) α p − r 2 α p − r A p B 1 { Z k ≤ R , t k < T } . (174) On the event { t k < T } \{t_k<T\} { t k < T } M T − k ≥ ( k + 1 ) − k ≥ 1 M_T-k\ge(k+1)-k\ge1 M T − k ≥ ( k + 1 ) − k ≥ 1 M T − k ≥ T − t k M_T-k\ge T-t_k M T − k ≥ T − t k M T − k ≥ 2 − 1 ( 1 + T − t k ) M_T-k\ge2^{-1}(1+T-t_k) M T − k ≥ 2 − 1 ( 1 + T − t k ) (167) for M = M T ∨ ( k + 1 ) M=M_T\vee(k+1) M = M T ∨ ( k + 1 ) j = k + 1 j=k+1 j = k + 1
2 − p − r α ( 1 + T − t k ) p − r α 1 { Z k ≤ R , t k < T } E [ 1 { M T < ν k + 1 } Z M T r ∣ G k + 1 ] ≤ 1 { Z k ≤ R , t k < T } E [ 1 { M T < ν k + 1 } ( M T − k ) p − r α Z M T r ∣ G k + 1 ] ≤ 1 { Z k ≤ R , t k < T } B Z k + 1 p . \begin{aligned}
2^{-\frac{p-r}{\alpha}}(1+T-t_k)^{\frac{p-r}{\alpha}}\boldsymbol{1}\{Z_k\le R,t_k<T\}\mathbb{E}\left[\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_{k+1}\right] \\
&\le\boldsymbol{1}\{Z_k\le R,t_k<T\}\mathbb{E}\left[\boldsymbol{1}\{M_T<\nu_{k+1}\}(M_T-k)^{\frac{p-r}{\alpha}}Z_{M_T}^r\mid\mathcal{G}_{k+1}\right] \\
&\le\boldsymbol{1}\{Z_k\le R,t_k<T\}BZ_{k+1}^p.
\end{aligned} 2 − α p − r ( 1 + T − t k ) α p − r 1 { Z k ≤ R , t k < T } E [ 1 { M T < ν k + 1 } Z M T r ∣ G k + 1 ] ≤ 1 { Z k ≤ R , t k < T } E [ 1 { M T < ν k + 1 } ( M T − k ) α p − r Z M T r ∣ G k + 1 ] ≤ 1 { Z k ≤ R , t k < T } B Z k + 1 p . After conditioning on G k \mathcal{G}_k G k (149) , we then obtain (174) .
Step 5: Conclusion. We first estimate E [ Z M T r ∣ G 0 ] \mathbb{E}[Z_{M_T}^r\mid\mathcal{G}_0] E [ Z M T r ∣ G 0 ] E [ X M T + 1 r ∣ G 0 ] \mathbb{E}[X_{M_T+1}^r\mid\mathcal{G}_0] E [ X M T + 1 r ∣ G 0 ] E [ Z M T r ∣ G 0 ] \mathbb{E}[Z_{M_T}^r\mid\mathcal{G}_0] E [ Z M T r ∣ G 0 ] R R R M T M_T M T R R R M T M_T M T R R R 0 ≤ k < M T 0\le k<M_T 0 ≤ k < M T Z M T ≤ R Z_{M_T}\le R Z M T ≤ R
E [ Z M T r ∣ G 0 ] = E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] + ∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G 0 ] + E [ 1 { Z M T ≤ R } Z M T r ∣ G 0 ] . (175) \begin{aligned}
\mathbb{E}[Z_{M_T}^r\mid\mathcal{G}_0]
={}&\mathbb{E}[\boldsymbol{1}\{M_T<\nu_0\}Z_{M_T}^r\mid\mathcal{G}_0]\\
&+\sum_{k=0}^{\infty}\mathbb{E}[\boldsymbol{1}\{Z_k\le R,t_k<T\}\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_0]\\
&+\mathbb{E}[\boldsymbol{1}\{Z_{M_T}\le R\}Z_{M_T}^r\mid\mathcal{G}_0]. \tag*{(175)}
\end{aligned} E [ Z M T r ∣ G 0 ] = E [ 1 { M T < ν 0 } Z M T r ∣ G 0 ] + k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G 0 ] + E [ 1 { Z M T ≤ R } Z M T r ∣ G 0 ] . (175) The first summand in (175) can be bounded by using (173) , and the third summand in (175) is trivially bounded by R r R^r R r (175) , we group the physical times ( t k ) k = 0 ∞ (t_k)_{k=0}^{\infty} ( t k ) k = 0 ∞ T T T j ≥ 0 j\ge0 j ≥ 0
I j ( T ) : = [ T − j − 1 , T − j ) ∩ [ 0 , ∞ ) . I_j(T)\mathrel{\mathop:}=[T-j-1,T-j)\cap[0,\infty). I j ( T ) : = [ T − j − 1 , T − j ) ∩ [ 0 , ∞ ) . Then, together with (174) , we obtain that
∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G 0 ] = ∑ k = 0 ∞ E [ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G k ] ∣ G 0 ] = ∑ j = 0 ∞ ∑ k = 0 ∞ E [ E [ 1 { Z k ≤ R , t k ∈ I j ( T ) } 1 { M T < ν k + 1 } Z M T r ∣ G k ] ∣ G 0 ] \begin{aligned}
\sum_{k=0}^{\infty}\mathbb{E}\left[\boldsymbol{1}\{Z_k\le R,t_k<T\}\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_0\right]
={}&\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbb{E}\left[\boldsymbol{1}\{Z_k\le R,t_k<T\}\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_k\right]\mid\mathcal{G}_0\right]\\
={}&\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbb{E}\left[\boldsymbol{1}\{Z_k\le R,t_k\in I_j(T)\}\boldsymbol{1}\{M_T<\nu_{k+1}\}Z_{M_T}^r\mid\mathcal{G}_k\right]\mid\mathcal{G}_0\right]
\end{aligned} k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G 0 ] = = k = 0 ∑ ∞ E [ E [ 1 { Z k ≤ R , t k < T } 1 { M T < ν k + 1 } Z M T r ∣ G k ] ∣ G 0 ] j = 0 ∑ ∞ k = 0 ∑ ∞ E [ E [ 1 { Z k ≤ R , t k ∈ I j ( T )} 1 { M T < ν k + 1 } Z M T r ∣ G k ] ∣ G 0 ] ≤ 2 p − r α A p B ∑ j = 0 ∞ ∑ k = 0 ∞ E [ 1 { Z k ≤ R , t k ∈ I j ( T ) } 1 ( 1 + T − t k ) p − r α | G 0 ] (176) \le2^{\frac{p-r}{\alpha}} A^{p}B \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\mathbb{E}\left[\mathbf{1}_{\{Z_k\le R,t_k\in I_j(T)\}}\frac{1}{(1+T-t_k)^{\frac{p-r}{\alpha}}}\,\middle|\,\mathcal{G}_0\right] \tag*{(176)} ≤ 2 α p − r A p B j = 0 ∑ ∞ k = 0 ∑ ∞ E [ 1 { Z k ≤ R , t k ∈ I j ( T )} ( 1 + T − t k ) α p − r 1 G 0 ] (176) On the event { t k ∈ I j ( T ) } \{t_k\in I_j(T)\} { t k ∈ I j ( T )} T − t k > j T-t_k>j T − t k > j N N N (171) and our estimate (172) , we then obtain that
x r e f ‘ ( 3.82 ) ‘ ≤ 2 p − r α A p B ∑ j = 0 ∞ ∑ k = 0 ∞ 1 ( 1 + j ) p − r α E [ 1 { Z k ≤ R , t k ∈ I j ( T ) } | G 0 ] {xref}`(3.82)` \le2^{\frac{p-r}{\alpha}} A^{p}B \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{(1+j)^{\frac{p-r}{\alpha}}}\mathbb{E}\left[\mathbf{1}_{\{Z_k\le R,t_k\in I_j(T)\}}\,\middle|\,\mathcal{G}_0\right] x r e f ‘ ( 3.82 ) ‘ ≤ 2 α p − r A p B j = 0 ∑ ∞ k = 0 ∑ ∞ ( 1 + j ) α p − r 1 E [ 1 { Z k ≤ R , t k ∈ I j ( T )} G 0 ] ≤ 2 p − r α A p B ∑ j = 0 ∞ 1 ( 1 + j ) p − r α E [ N ( I j ( T ) ) | G 0 ] (177) \le2^{\frac{p-r}{\alpha}} A^{p}B \sum_{j=0}^{\infty}\frac{1}{(1+j)^{\frac{p-r}{\alpha}}}\mathbb{E}\left[N(I_j(T))\,\middle|\,\mathcal{G}_0\right] \tag*{(177)} ≤ 2 α p − r A p B j = 0 ∑ ∞ ( 1 + j ) α p − r 1 E [ N ( I j ( T )) ∣ G 0 ] (177) ≤ 2 1 + p − r α ( 3 h − 1 ) A p B ∑ j = 0 ∞ 1 ( 1 + j ) p − r α . \le2^{1+\frac{p-r}{\alpha}}(3h^{-1})A^{p}B\sum_{j=0}^{\infty}\frac{1}{(1+j)^{\frac{p-r}{\alpha}}}. ≤ 2 1 + α p − r ( 3 h − 1 ) A p B j = 0 ∑ ∞ ( 1 + j ) α p − r 1 . As in Lemma 3.58, we have that ( p − r ) / α > 1 (p-r)/\alpha>1 ( p − r ) / α > 1
E [ Z M T r | G 0 ] ≤ min ( Z 0 r , B Z 0 p ( 1 + T ) p − r α ) + C ~ , \mathbb{E}\left[Z_{M_T}^{r}\,\middle|\,\mathcal{G}_0\right]\le\min\left(Z_0^{r},\frac{BZ_0^{p}}{(1+T)^{\frac{p-r}{\alpha}}}\right)+\widetilde{C}, E [ Z M T r G 0 ] ≤ min ( Z 0 r , ( 1 + T ) α p − r B Z 0 p ) + C , where C ~ = C ~ ( ε , c , p , q 1 , q 2 , r , K ) \widetilde{C}=\widetilde{C}(\varepsilon,c,p,q_1,q_2,r,K) C = C ( ε , c , p , q 1 , q 2 , r , K ) Z M T r Z_{M_T}^{r} Z M T r X M T + 1 r X_{M_T+1}^{r} X M T + 1 r M T M_T M T M T + 1 M_T+1 M T + 1 (148) and (149) . Together with Hölder’s inequality, we then obtain that E [ Z m + 1 r ∣ G m ] ≤ Z m r + A r \mathbb{E}[Z_{m+1}^{r}\mid\mathcal{G}_m]\le Z_m^{r}+A^{r} E [ Z m + 1 r ∣ G m ] ≤ Z m r + A r m ≥ 0 m\ge0 m ≥ 0 M T M_T M T
E [ Z M T + 1 r | G 0 ] = ∑ m = 0 ∞ E [ 1 { M T = m } Z m + 1 r | G 0 ] = ∑ m = 0 ∞ E [ 1 { M T = m } E [ Z m + 1 r ∣ G m ] | G 0 ] ≤ ∑ m = 0 ∞ E [ 1 { M T = m } ( Z m r + A r ) | G 0 ] = E [ Z M T r | G 0 ] + A r . \begin{aligned}
\mathbb{E}\left[Z_{M_T+1}^{r}\,\middle|\,\mathcal{G}_0\right]
=\sum_{m=0}^{\infty}\mathbb{E}\left[\mathbf{1}_{\{M_T=m\}}Z_{m+1}^{r}\,\middle|\,\mathcal{G}_0\right]
=\sum_{m=0}^{\infty}\mathbb{E}\left[\mathbf{1}_{\{M_T=m\}}\mathbb{E}[Z_{m+1}^{r}\mid\mathcal{G}_m]\,\middle|\,\mathcal{G}_0\right] \\
&\le\sum_{m=0}^{\infty}\mathbb{E}\left[\mathbf{1}_{\{M_T=m\}}(Z_m^{r}+A^{r})\,\middle|\,\mathcal{G}_0\right]
=\mathbb{E}\left[Z_{M_T}^{r}\,\middle|\,\mathcal{G}_0\right]+A^{r}.
\end{aligned} E [ Z M T + 1 r G 0 ] = m = 0 ∑ ∞ E [ 1 { M T = m } Z m + 1 r G 0 ] = m = 0 ∑ ∞ E [ 1 { M T = m } E [ Z m + 1 r ∣ G m ] G 0 ] ≤ m = 0 ∑ ∞ E [ 1 { M T = m } ( Z m r + A r ) G 0 ] = E [ Z M T r G 0 ] + A r . By combining the above estimates with the fact that X M T + 1 ≤ Z M T + 1 X_{M_T+1}\le Z_{M_T+1} X M T + 1 ≤ Z M T + 1 Z 0 ≤ X 0 + 2 Y 0 Z_0\le X_0+2Y_0 Z 0 ≤ X 0 + 2 Y 0
E [ X M T + 1 r | G 0 ] ≤ min ( ( X 0 + 2 Y 0 ) r , B ( X 0 + 2 Y 0 ) p ( 1 + T ) p − r α ) + C ~ + A r . \mathbb{E}\left[X_{M_T+1}^{r}\,\middle|\,\mathcal{G}_0\right]\le\min\left((X_0+2Y_0)^{r},\frac{B(X_0+2Y_0)^{p}}{(1+T)^{\frac{p-r}{\alpha}}}\right)+\widetilde{C}+A^{r}. E [ X M T + 1 r G 0 ] ≤ min ( ( X 0 + 2 Y 0 ) r , ( 1 + T ) α p − r B ( X 0 + 2 Y 0 ) p ) + C + A r . This readily implies the desired estimates of the supremum and limit superior in (162) . □
Covariant parabolic estimates
In this section, we state and prove various parabolic estimates that will be heavily used in the following sections. Because we will prove quite a few results, many of which are only tangentially connected, we will split this section into further subsections, such that the main results within each subsection will be closely related. If the reader wishes, they may choose to read only the statements of the main results of each subsection, before moving on to later sections. This way, the reader may more quickly get an idea of the analytic input that will be needed in the coming sections, and then later they can come back to fill in the details.
Before diving into the subsections, we first introduce the main players of the present section, and then prove some preliminary lemmas which will be frequently used in the coming subsections. Throughout this section, we work with a general dimension d ≥ 2 d\ge2 d ≥ 2 g \mathfrak{g} g π \pi π π ∗ \pi_* π ∗ π ∗ \pi_* π ∗ g \mathfrak{g} g π \pi π π \pi π π ∗ \pi_* π ∗ g \mathfrak{g} g g \mathfrak{g} g
In the following, let B : [ 0 , T ] × T d → g d B : [0,T] \times\mathbb{T}^d \to\mathfrak{g}^d B : [ 0 , T ] × T d → g d B B B § 4 , we typically no longer explicitly denote the dependence of the covariant derivatives D and the curvature F on the underlying connection B B B § 3.1 . First, consider two natural linear parabolic equations associated to B B B
( ∂ t − Δ B ) ϕ = G , (178) (\partial_t-\Delta_B)\phi=G, \tag*{(178)} ( ∂ t − Δ B ) ϕ = G , (178) ( ∂ t − Δ B ) ψ i = 2 F i ℓ ψ ℓ + H i , i ∈ [ d ] . (179) (\partial_t-\Delta_B)\psi_i=2F_i^{\ \ell}\psi_\ell+H_i,\qquad i\in[d]. \tag*{(179)} ( ∂ t − Δ B ) ψ i = 2 F i ℓ ψ ℓ + H i , i ∈ [ d ] . (179) In the first equation, ϕ \phi ϕ G G G ψ \psi ψ H H H W d \mathrm{W}^d W d
( ∂ t − Δ B ) ψ = 2 F ψ + H , (\partial_t-\Delta_B)\psi=2F\psi+H, ( ∂ t − Δ B ) ψ = 2 F ψ + H , and recall from Definition 2.1 the covariant heat kernels p B p_B p B q B q_B q B
Notation 4.1. In the following, when we write ∣ p B ∣ |p_B| ∣ p B ∣ ∣ q B ∣ |q_B| ∣ q B ∣ p B p_B p B q B q_B q B W d \mathrm{W}^d W d
Next, recall from Definition 2.3 the following norm on the curvature F F F I = [ s 0 , t 0 ] ⊆ [ 0 , ∞ ) I=[s_0,t_0]\subseteq[0,\infty) I = [ s 0 , t 0 ] ⊆ [ 0 , ∞ ) k ≥ 0 k\geq0 k ≥ 0
∥ F ∥ I , k p e r t = ∑ j = 0 k ( ∥ ( t − s 0 ) ( j + 1 ) / 2 D ( j ) F ∥ L t 2 L x ∞ ( I ) + ∥ ( t − s 0 ) j / 2 D ( j ) F ∥ L t 1 L x ∞ ( I ) ) . \|F\|^{\mathrm{pert}}_{I,k}=\sum_{j=0}^{k}\left(\|(t-s_0)^{(j+1)/2}D^{(j)}F\|_{L^2_tL^\infty_x(I)}+\|(t-s_0)^{j/2}D^{(j)}F\|_{L^1_tL^\infty_x(I)}\right). ∥ F ∥ I , k pert = j = 0 ∑ k ( ∥ ( t − s 0 ) ( j + 1 ) /2 D ( j ) F ∥ L t 2 L x ∞ ( I ) + ∥ ( t − s 0 ) j /2 D ( j ) F ∥ L t 1 L x ∞ ( I ) ) . In particular, when k = 0 k=0 k = 0
∥ F ∥ I , 0 p e r t = ∥ ( t − s 0 ) 1 / 2 F ∥ L t 2 L x ∞ ( I ) + ∥ F ∥ L t 1 L x ∞ ( I ) . \|F\|^{\mathrm{pert}}_{I,0}=\|(t-s_0)^{1/2}F\|_{L^2_tL^\infty_x(I)}+\|F\|_{L^1_tL^\infty_x(I)}. ∥ F ∥ I , 0 pert = ∥ ( t − s 0 ) 1/2 F ∥ L t 2 L x ∞ ( I ) + ∥ F ∥ L t 1 L x ∞ ( I ) . In the case I = [ 0 , t ] I=[0,t] I = [ 0 , t ] ∥ F ∥ t , k p e r t \|F\|^{\mathrm{pert}}_{t,k} ∥ F ∥ t , k pert ∥ F ∥ [ 0 , t ] , k p e r t \|F\|^{\mathrm{pert}}_{[0,t],k} ∥ F ∥ [ 0 , t ] , k pert
We will apply Gronwall (Lemma 3.50 ) in many places, which will incur an exponential loss. We next set notation for this loss.
Notation 4.3 (Gronwall loss). Given an interval I ⊆ R + I\subseteq\mathbb{R}^+ I ⊆ R +
Gr I ( F ) ≔ exp ( ∥ F ∥ L t 1 L x ∞ ( I ) ) . \operatorname{Gr}_I(F)\coloneqq\exp\left(\|F\|_{L^1_tL^\infty_x(I)}\right). Gr I ( F ) : = exp ( ∥ F ∥ L t 1 L x ∞ ( I ) ) . When I = [ 0 , t ] I=[0,t] I = [ 0 , t ] Gr t ( F ) \operatorname{Gr}_t(F) Gr t ( F ) Gr [ 0 , t ] ( F ) \operatorname{Gr}_{[0,t]}(F) Gr [ 0 , t ] ( F ) ≲ \lesssim ≲ Gr I ( C F ) \operatorname{Gr}_I(CF) Gr I ( C F ) C C C
Next, we discuss several preliminary results which will form the backbone for the arguments of the coming subsections. For the reader who just wishes to get an idea of the main results of this section, the ensuing results may be safely skipped.
Proposition 4.4 (Covariant monotonicity formulas). Let G : [ 0 , T ] × T d → W G\colon[0,T]\times\mathbb{T}^{d}\to W G : [ 0 , T ] × T d → W ϕ \phi ϕ
( ∂ t − Δ B ) ϕ = G . (\partial_t-\Delta_B)\phi=G. ( ∂ t − Δ B ) ϕ = G . Furthermore, let K : [ 0 , T ] × T d → [ 0 , ∞ ) K\colon[0,T]\times\mathbb{T}^{d}\to[0,\infty) K : [ 0 , T ] × T d → [ 0 , ∞ )
( ∂ t + Δ ) K = 0. (\partial_t+\Delta)K=0. ( ∂ t + Δ ) K = 0. Then, it holds that
∂ t ( K ∣ ϕ ∣ 2 2 ) = − ( Δ K ) ∣ ϕ ∣ 2 2 + K Δ ( ∣ ϕ ∣ 2 2 ) − K ∣ D ϕ ∣ 2 + K ⟨ ϕ , G ⟩ W . (180) \partial_t\left(K\frac{|\phi|^2}{2}\right)=-(\Delta K)\frac{|\phi|^2}{2}+K\Delta\left(\frac{|\phi|^2}{2}\right)-K|D\phi|^2+K\langle\phi,G\rangle_W. \tag*{(180)} ∂ t ( K 2 ∣ ϕ ∣ 2 ) = − ( Δ K ) 2 ∣ ϕ ∣ 2 + K Δ ( 2 ∣ ϕ ∣ 2 ) − K ∣ D ϕ ∣ 2 + K ⟨ ϕ , G ⟩ W . (180) More generally, for any p ≥ 2 p\ge2 p ≥ 2
∂ t ( K ∣ ϕ ∣ p p ) = − ( Δ K ) ∣ ϕ ∣ p p + K Δ ( ∣ ϕ ∣ p p ) − K ( p − 2 4 ∣ ϕ ∣ p − 4 ∣ ∇ ( ∣ ϕ ∣ 2 ) ∣ 2 + ∣ ϕ ∣ p − 2 ∣ D ϕ ∣ 2 − ∣ ϕ ∣ p − 2 ⟨ ϕ , G ⟩ W ) . \begin{aligned}
\partial_t\left(K\frac{|\phi|^p}{p}\right)
&=-(\Delta K)\frac{|\phi|^p}{p}+K\Delta\left(\frac{|\phi|^p}{p}\right)\\
&\quad-K\left(\frac{p-2}{4}|\phi|^{p-4}|\nabla(|\phi|^2)|^2+|\phi|^{p-2}|D\phi|^2-|\phi|^{p-2}\langle\phi,G\rangle_W\right).
\end{aligned} ∂ t ( K p ∣ ϕ ∣ p ) = − ( Δ K ) p ∣ ϕ ∣ p + K Δ ( p ∣ ϕ ∣ p ) − K ( 4 p − 2 ∣ ϕ ∣ p − 4 ∣∇ ( ∣ ϕ ∣ 2 ) ∣ 2 + ∣ ϕ ∣ p − 2 ∣ D ϕ ∣ 2 − ∣ ϕ ∣ p − 2 ⟨ ϕ , G ⟩ W ) . Proof. By the Bochner formula (106) , we have that
∂ t ( ∣ ϕ ∣ 2 2 ) = ⟨ ϕ , ∂ t ϕ ⟩ W = ⟨ ϕ , Δ B ϕ ⟩ W + ⟨ ϕ , G ⟩ W = Δ ( ∣ ϕ ∣ 2 2 ) − ∣ D ϕ ∣ 2 + ⟨ ϕ , G ⟩ W , \partial_t\left(\frac{|\phi|^2}{2}\right)=\langle\phi,\partial_t\phi\rangle_W=\langle\phi,\Delta_B\phi\rangle_W+\langle\phi,G\rangle_W=\Delta\left(\frac{|\phi|^2}{2}\right)-|D\phi|^2+\langle\phi,G\rangle_W, ∂ t ( 2 ∣ ϕ ∣ 2 ) = ⟨ ϕ , ∂ t ϕ ⟩ W = ⟨ ϕ , Δ B ϕ ⟩ W + ⟨ ϕ , G ⟩ W = Δ ( 2 ∣ ϕ ∣ 2 ) − ∣ D ϕ ∣ 2 + ⟨ ϕ , G ⟩ W , and the claim follows from the Leibniz rule (for the calculations for general p ≥ 2 p\ge2 p ≥ 2
Remark 4.5. By integrating (180) in space-time and using integration by parts to cancel the two terms which involve the Laplacian, we have that for any t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ]
∥ K ( t 0 ) 1 / 2 ϕ ( t 0 ) ∥ L x 2 2 + 2 ∥ K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , t 0 ] ) 2 = ∥ K 1 / 2 ( 0 ) ϕ ( 0 ) ∥ L x 2 2 + 2 ∫ 0 t 0 d t ∫ T d d x K ⟨ ϕ , G ⟩ W . (181) \|K(t_0)^{1/2}\phi(t_0)\|_{L_x^2}^2+2\|K^{1/2}D\phi\|_{L_{t,x}^2([0,t_0])}^2
=\|K^{1/2}(0)\phi(0)\|_{L_x^2}^2+2\int_0^{t_0}dt\int_{\mathbb{T}^d}dxK\langle\phi,G\rangle_W. \tag*{(181)} ∥ K ( t 0 ) 1/2 ϕ ( t 0 ) ∥ L x 2 2 + 2∥ K 1/2 D ϕ ∥ L t , x 2 ([ 0 , t 0 ]) 2 = ∥ K 1/2 ( 0 ) ϕ ( 0 ) ∥ L x 2 2 + 2 ∫ 0 t 0 d t ∫ T d d x K ⟨ ϕ , G ⟩ W . (181) Later on, we will need the following special case of the above, where K ≡ 1 K\equiv1 K ≡ 1 G = D k H k G=D^kH_k G = D k H k H : [ 0 , T ] × T d → W d H\colon[0,T]\times\mathbb{T}^d\to W^d H : [ 0 , T ] × T d → W d (181) by parts, and then applying Hölder and Young, we may obtain
∥ ϕ ( t 0 ) ∥ L x 2 2 + ∥ D ϕ ∥ L t , x 2 ( [ 0 , t 0 ] ) 2 ≤ ∥ ϕ ( 0 ) ∥ L x 2 2 + ∥ H ∥ L t , x 2 ( [ 0 , t 0 ] ) 2 . (182) \|\phi(t_0)\|_{L_x^2}^2+\|D\phi\|_{L_{t,x}^2([0,t_0])}^2\le\|\phi(0)\|_{L_x^2}^2+\|H\|_{L_{t,x}^2([0,t_0])}^2. \tag*{(182)} ∥ ϕ ( t 0 ) ∥ L x 2 2 + ∥ D ϕ ∥ L t , x 2 ([ 0 , t 0 ]) 2 ≤ ∥ ϕ ( 0 ) ∥ L x 2 2 + ∥ H ∥ L t , x 2 ([ 0 , t 0 ]) 2 . (182) Here, we absorbed one of the terms arising from Young into the good term ∥ D ϕ ∥ L t , x 2 ( [ 0 , t 0 ] ) 2 \|D\phi\|_{L_{t,x}^2([0,t_0])}^2 ∥ D ϕ ∥ L t , x 2 ([ 0 , t 0 ]) 2 (181) .
The covariant monotonicity formula leads to the following estimates, which will be used shortly.
Lemma 4.6. Let G : [ 0 , T ] × T d → W G\colon[0,T]\times\mathbb{T}^{d}\to W G : [ 0 , T ] × T d → W ϕ 0 : T d → W \phi_0\colon\mathbb{T}^{d}\to W ϕ 0 : T d → W ϕ \phi ϕ
( ∂ t − Δ B ) ϕ = G , ϕ ( 0 ) = ϕ 0 . (\partial_t-\Delta_B)\phi=G,\qquad\phi(0)=\phi_0. ( ∂ t − Δ B ) ϕ = G , ϕ ( 0 ) = ϕ 0 . Furthermore, let K : [ 0 , T ] × T d → ( 0 , ∞ ) K : [0,T] \times\mathbb{T}^{d} \to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ )
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + sup t 0 ∈ [ 0 , T ] ∣ ∫ 0 t 0 ∫ T d d t d x K ⟨ ϕ , G ⟩ W ∣ 1 / 2 . (183) \|K^{1/2}\phi\|_{L^\infty_tL^2_x([0,T])}+\|K^{1/2}D\phi\|_{L^2_{t,x}([0,T])}\lesssim\|K^{1/2}(0)\phi_0\|_{L^2_x}+\sup_{t_0\in[0,T]}\left|\int_0^{t_0}\int_{\mathbb{T}^d}\mathrm{d}t\mathrm{d}x\,K\langle\phi,G\rangle_W\right|^{1/2}. \tag*{(183)} ∥ K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ K 1/2 D ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + t 0 ∈ [ 0 , T ] sup ∫ 0 t 0 ∫ T d d t d x K ⟨ ϕ , G ⟩ W 1/2 . (183) Additionally, for α > 0 \alpha>0 α > 0
∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t α / 2 K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) \|t^{\alpha/2}K^{1/2}\phi\|_{L^\infty_tL^2_x([0,T])}+\|t^{\alpha/2}K^{1/2}D\phi\|_{L^2_{t,x}([0,T])} ∥ t α /2 K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ t α /2 K 1/2 D ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ sup t 0 ∈ [ 0 , T ] ∣ ∫ 0 t 0 ∫ T d d t d x t α K ⟨ ϕ , G ⟩ W ∣ 1 / 2 + α 1 / 2 ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 ( [ 0 , T ] ) . (184) \lesssim\sup_{t_0\in[0,T]}\left|\int_0^{t_0}\int_{\mathbb{T}^d}\mathrm{d}t\mathrm{d}x\,t^\alpha K\langle\phi,G\rangle_W\right|^{1/2}+\alpha^{1/2}\|t^{(\alpha-1)/2}K^{1/2}\phi\|_{L^2_{t,x}([0,T])}. \tag*{(184)} ≲ t 0 ∈ [ 0 , T ] sup ∫ 0 t 0 ∫ T d d t d x t α K ⟨ ϕ , G ⟩ W 1/2 + α 1/2 ∥ t ( α − 1 ) /2 K 1/2 ϕ ∥ L t , x 2 ([ 0 , T ]) . (184) Remark 4.7. In most use cases of (4.6)–(4.7), we will directly estimate the integral by the L t , x 1 L^1_{t,x} L t , x 1
sup t 0 ∈ [ 0 , T ] ∣ ∫ 0 t 0 ∫ T d d t d x K ⟨ ϕ , G ⟩ W ∣ 1 / 2 ≤ ∥ K ⟨ ϕ , G ⟩ W ∥ L t , x 1 ( [ 0 , T ] ) 1 / 2 . \sup_{t_0\in[0,T]}\left|\int_0^{t_0}\int_{\mathbb{T}^d}\mathrm{d}t\mathrm{d}x\,K\langle\phi,G\rangle_W\right|^{1/2}\leq\|K\langle\phi,G\rangle_W\|_{L^1_{t,x}([0,T])}^{1/2}. t 0 ∈ [ 0 , T ] sup ∫ 0 t 0 ∫ T d d t d x K ⟨ ϕ , G ⟩ W 1/2 ≤ ∥ K ⟨ ϕ , G ⟩ W ∥ L t , x 1 ([ 0 , T ]) 1/2 . We will typically insert this estimate without further comment.
Proof. The first estimate follows directly from (4.4), upon taking a sup over t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ] t α K ∣ ϕ ∣ 2 t^\alpha K|\phi|^2 t α K ∣ ϕ ∣ 2
Next, we prove the following very general estimate, which will be repeatedly used in the coming subsections. On a first reading, the reader can choose to set H = 0 H=0 H = 0 H = 0 H=0 H = 0
Lemma 4.8. Let ϕ , G : [ 0 , T ] × T d → W d \phi,G:[0,T]\times\mathbb{T}^{d}\to W^{d} ϕ , G : [ 0 , T ] × T d → W d H : [ 0 , T ] × T d → W H:[0,T]\times\mathbb{T}^{d}\to W H : [ 0 , T ] × T d → W ϕ 0 : T d → W d \phi_0:\mathbb{T}^{d}\to W^{d} ϕ 0 : T d → W d
( ∂ t − Δ B ) ϕ = O ( F ϕ ) + G + D H , ϕ ( 0 ) = ϕ 0 . (185) (\partial_t-\Delta_B)\phi=O(F\phi)+G+DH,\qquad\phi(0)=\phi_0. \tag*{(185)} ( ∂ t − Δ B ) ϕ = O ( F ϕ ) + G + D H , ϕ ( 0 ) = ϕ 0 . (185) Let K : [ 0 , T ] × T d → ( 0 , ∞ ) K:[0,T]\times\mathbb{T}^{d}\to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ ) α > 0 \alpha>0 α > 0
∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t α / 2 K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) \|t^{\alpha/2}K^{1/2}\phi\|_{L^\infty_tL^2_x([0,T])}+\|t^{\alpha/2}K^{1/2}D\phi\|_{L^2_{t,x}([0,T])} ∥ t α /2 K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ t α /2 K 1/2 D ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ ( ∥ t 1 / 2 F ∥ L t 2 L x ∞ ( [ 0 , T ] ) + α 1 / 2 ) ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 ( [ 0 , T ] ) + ∥ t α / 2 K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , T ] ) + N α ( H ) , (186) \lesssim\left(\|t^{1/2}F\|_{L^2_tL^\infty_x([0,T])}+\alpha^{1/2}\right)\|t^{(\alpha-1)/2}K^{1/2}\phi\|_{L^2_{t,x}([0,T])}
+\|t^{\alpha/2}K^{1/2}G\|_{L^1_tL^2_x([0,T])}+N_\alpha(H), \tag*{(186)} ≲ ( ∥ t 1/2 F ∥ L t 2 L x ∞ ([ 0 , T ]) + α 1/2 ) ∥ t ( α − 1 ) /2 K 1/2 ϕ ∥ L t , x 2 ([ 0 , T ]) + ∥ t α /2 K 1/2 G ∥ L t 1 L x 2 ([ 0 , T ]) + N α ( H ) , (186) where
N α ( H ) = ∥ t α / 2 K 1 / 2 H ∥ L t , x 2 ( [ 0 , T ] ) + ∥ t α / 2 ( T − t ) − 1 / 2 K 1 / 2 H ∥ L t 1 L x 2 ( [ 0 , T ] ) + ∥ t α / 2 ∣ Δ K ∣ 1 / 2 H ∥ L t 1 L x 2 ( [ 0 , T ] ) . N_\alpha(H)=\|t^{\alpha/2}K^{1/2}H\|_{L^2_{t,x}([0,T])}+\|t^{\alpha/2}(T-t)^{-1/2}K^{1/2}H\|_{L^1_tL^2_x([0,T])}+\|t^{\alpha/2}|\Delta K|^{1/2}H\|_{L^1_tL^2_x([0,T])}. N α ( H ) = ∥ t α /2 K 1/2 H ∥ L t , x 2 ([ 0 , T ]) + ∥ t α /2 ( T − t ) − 1/2 K 1/2 H ∥ L t 1 L x 2 ([ 0 , T ]) + ∥ t α /2 ∣Δ K ∣ 1/2 H ∥ L t 1 L x 2 ([ 0 , T ]) . If K K K N α ( H ) N_\alpha(H) N α ( H )
Assuming that H = 0 H=0 H = 0 T 1 > T T_1>T T 1 > T
∥ t α / 2 ( T 1 − t ) α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t α / 2 ( T 1 − t ) α / 2 K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) (187) \|t^{\alpha/2}(T_1-t)^{\alpha/2}K^{1/2}\phi\|_{L^\infty_tL^2_x([0,T])}+\|t^{\alpha/2}(T_1-t)^{\alpha/2}K^{1/2}D\phi\|_{L^2_{t,x}([0,T])} \tag*{(187)} ∥ t α /2 ( T 1 − t ) α /2 K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ t α /2 ( T 1 − t ) α /2 K 1/2 D ϕ ∥ L t , x 2 ([ 0 , T ]) (187) + α 1 / 2 ∥ t α / 2 ( T 1 − t ) ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 ( [ 0 , T ] ) +\alpha^{1/2}\|t^{\alpha/2}(T_1-t)^{(\alpha-1)/2}K^{1/2}\phi\|_{L^2_{t,x}([0,T])} + α 1/2 ∥ t α /2 ( T 1 − t ) ( α − 1 ) /2 K 1/2 ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ α 1 / 2 ∥ t ( α − 1 ) / 2 ( T 1 − t ) α / 2 K 1 / 2 ϕ ∥ L t , x 2 ( [ 0 , T ] ) + ∥ ( T 1 − t ) 1 / 2 F ∥ L t 2 L x ∞ ( [ 0 , T ] ) ∥ t α / 2 ( T 1 − t ) ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 ( [ 0 , T ] ) + ∥ t α / 2 ( T 1 − t ) α / 2 K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , T ] ) . \begin{aligned}
&\lesssim\alpha^{1/2}\left\|t^{(\alpha-1)/2}(T_1-t)^{\alpha/2}K^{1/2}\phi\right\|_{L^2_{t,x}([0,T])}\\
&\quad+\left\|(T_1-t)^{1/2}F\right\|_{L^2_tL^\infty_x([0,T])}\left\|t^{\alpha/2}(T_1-t)^{(\alpha-1)/2}K^{1/2}\phi\right\|_{L^2_{t,x}([0,T])}\\
&\quad+\left\|t^{\alpha/2}(T_1-t)^{\alpha/2}K^{1/2}G\right\|_{L^1_tL^2_x([0,T])}.
\end{aligned} ≲ α 1/2 t ( α − 1 ) /2 ( T 1 − t ) α /2 K 1/2 ϕ L t , x 2 ([ 0 , T ]) + ( T 1 − t ) 1/2 F L t 2 L x ∞ ([ 0 , T ]) t α /2 ( T 1 − t ) ( α − 1 ) /2 K 1/2 ϕ L t , x 2 ([ 0 , T ]) + t α /2 ( T 1 − t ) α /2 K 1/2 G L t 1 L x 2 ([ 0 , T ]) . Proof. In the following, all space-time norms are on [ 0 , T ] × T d [0,T]\times\mathbb{T}^d [ 0 , T ] × T d (184) (with the G G G (185) ), we have that
∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 + ∥ t α / 2 K 1 / 2 D ϕ ∥ L t , x 2 ≲ ∥ t α K ∣ F ∣ ∣ ϕ ∣ 2 ∥ L t , x 1 1 / 2 + ∥ t α K ⟨ G , ϕ ⟩ W ∥ L t , x 1 1 / 2 + sup t 0 ∈ [ 0 , T ] ∣ ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W ∣ 1 / 2 + α 1 / 2 ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 . \begin{aligned}
\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}
+\left\|t^{\alpha/2}K^{1/2}D\phi\right\|_{L^2_{t,x}}
\lesssim{}&\left\|t^\alpha K|F||\phi|^2\right\|_{L^1_{t,x}}^{1/2}
+\left\|t^\alpha K\langle G,\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}\\
&+\sup_{t_0\in[0,T]}\left|\int_0^{t_0}\int_{\mathbb{T}^d}dt\,dx\,t^\alpha K\langle DH,\phi\rangle_W\right|^{1/2}\\
&+\alpha^{1/2}\left\|t^{(\alpha-1)/2}K^{1/2}\phi\right\|_{L^2_{t,x}}.
\end{aligned} t α /2 K 1/2 ϕ L t ∞ L x 2 + t α /2 K 1/2 D ϕ L t , x 2 ≲ t α K ∣ F ∣∣ ϕ ∣ 2 L t , x 1 1/2 + ∥ t α K ⟨ G , ϕ ⟩ W ∥ L t , x 1 1/2 + t 0 ∈ [ 0 , T ] sup ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W 1/2 + α 1/2 t ( α − 1 ) /2 K 1/2 ϕ L t , x 2 . By Hölder and Young, we have that
∥ t α K ∣ F ∣ ∣ ϕ ∣ 2 ∥ L t , x 1 1 / 2 ≤ ∥ t 1 / 2 F ∥ L t 2 L x ∞ 1 / 2 ∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 1 / 2 ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 1 / 2 ≤ δ ∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 + C δ ∥ t 1 / 2 F ∥ L t 2 L x ∞ ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 , \begin{aligned}
\left\|t^\alpha K|F||\phi|^2\right\|_{L^1_{t,x}}^{1/2}
&\leq\left\|t^{1/2}F\right\|_{L^2_tL^\infty_x}^{1/2}
\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}^{1/2}
\left\|t^{(\alpha-1)/2}K^{1/2}\phi\right\|_{L^2_{t,x}}^{1/2}\\
&\leq\delta\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}
+C_\delta\left\|t^{1/2}F\right\|_{L^2_tL^\infty_x}
\left\|t^{(\alpha-1)/2}K^{1/2}\phi\right\|_{L^2_{t,x}},
\end{aligned} t α K ∣ F ∣∣ ϕ ∣ 2 L t , x 1 1/2 ≤ t 1/2 F L t 2 L x ∞ 1/2 t α /2 K 1/2 ϕ L t ∞ L x 2 1/2 t ( α − 1 ) /2 K 1/2 ϕ L t , x 2 1/2 ≤ δ t α /2 K 1/2 ϕ L t ∞ L x 2 + C δ t 1/2 F L t 2 L x ∞ t ( α − 1 ) /2 K 1/2 ϕ L t , x 2 , where δ \delta δ
∥ t α K ⟨ G , ϕ ⟩ W ∥ L t , x 1 1 / 2 ≤ δ ∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 + C δ ∥ t α / 2 K 1 / 2 G ∥ L t 1 L x 2 . \left\|t^\alpha K\langle G,\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}
\leq\delta\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}
+C_\delta\left\|t^{\alpha/2}K^{1/2}G\right\|_{L^1_tL^2_x}. ∥ t α K ⟨ G , ϕ ⟩ W ∥ L t , x 1 1/2 ≤ δ t α /2 K 1/2 ϕ L t ∞ L x 2 + C δ t α /2 K 1/2 G L t 1 L x 2 . Combining, we thus obtain the intermediate estimate that for some constant C C C
∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 + ∥ t α / 2 K 1 / 2 D ϕ ∥ L t , x 2 ≤ C ( ∥ t 1 / 2 F ∥ L t 2 L x ∞ + α 1 / 2 ) ∥ t ( α − 1 ) / 2 K 1 / 2 ϕ ∥ L t , x 2 + C ∥ t α / 2 K 1 / 2 G ∥ L t 1 L x 2 + C sup t 0 ∈ [ 0 , T ] ∣ ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W ∣ 1 / 2 . (188) \begin{aligned}
\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}
+\left\|t^{\alpha/2}K^{1/2}D\phi\right\|_{L^2_{t,x}}
\leq{}&C\left(\left\|t^{1/2}F\right\|_{L^2_tL^\infty_x}+\alpha^{1/2}\right)
\left\|t^{(\alpha-1)/2}K^{1/2}\phi\right\|_{L^2_{t,x}}\\
&+C\left\|t^{\alpha/2}K^{1/2}G\right\|_{L^1_tL^2_x}
+C\sup_{t_0\in[0,T]}\left|\int_0^{t_0}\int_{\mathbb{T}^d}dt\,dx\,t^\alpha K\langle DH,\phi\rangle_W\right|^{1/2}. \tag*{(188)}
\end{aligned} t α /2 K 1/2 ϕ L t ∞ L x 2 + t α /2 K 1/2 D ϕ L t , x 2 ≤ C ( t 1/2 F L t 2 L x ∞ + α 1/2 ) t ( α − 1 ) /2 K 1/2 ϕ L t , x 2 + C t α /2 K 1/2 G L t 1 L x 2 + C t 0 ∈ [ 0 , T ] sup ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W 1/2 . (188) To handle the last term, for t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ]
C ∣ ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W ∣ 1 / 2 ≤ C ∥ t α K ⟨ H , D ϕ ⟩ W ∥ L t , x 1 1 / 2 + C ∥ t α ∇ K ⟨ H , ϕ ⟩ W ∥ L t , x 1 1 / 2 . C\left|\int_0^{t_0}\int_{\mathbb{T}^d}dt\,dx\,t^\alpha K\langle DH,\phi\rangle_W\right|^{1/2}
\leq C\left\|t^\alpha K\langle H,D\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}
+C\left\|t^\alpha\nabla K\langle H,\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}. C ∫ 0 t 0 ∫ T d d t d x t α K ⟨ D H , ϕ ⟩ W 1/2 ≤ C ∥ t α K ⟨ H , D ϕ ⟩ W ∥ L t , x 1 1/2 + C ∥ t α ∇ K ⟨ H , ϕ ⟩ W ∥ L t , x 1 1/2 . For the first term on the right-hand side, again by Hölder and Young, we may estimate
C ∥ t α K ⟨ H , D ϕ ⟩ W ∥ L t , x 1 1 / 2 ≤ δ ∥ t α / 2 K 1 / 2 D ϕ ∥ L t , x 2 + C δ ∥ t α / 2 K 1 / 2 H ∥ L t , x 2 . C\left\|t^\alpha K\langle H,D\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}
\leq\delta\left\|t^{\alpha/2}K^{1/2}D\phi\right\|_{L^2_{t,x}}
+C_\delta\left\|t^{\alpha/2}K^{1/2}H\right\|_{L^2_{t,x}}. C ∥ t α K ⟨ H , D ϕ ⟩ W ∥ L t , x 1 1/2 ≤ δ t α /2 K 1/2 D ϕ L t , x 2 + C δ t α /2 K 1/2 H L t , x 2 . Taking δ ≤ 1 / 10 \delta\leq1/10 δ ≤ 1/10 (188) , while the second term is already acceptable. For the other term (the one involving ∇ K \nabla K ∇ K K K K ∇ K = 0 \nabla K=0 ∇ K = 0 T d \mathbb{T}^d T d K K K
∣ ∇ K ∣ 2 K ≤ Δ K + d 2 ( T − t ) K . \frac{|\nabla K|^2}{K}\leq\Delta K+\frac{d}{2(T-t)}K. K ∣∇ K ∣ 2 ≤ Δ K + 2 ( T − t ) d K . We may thus estimate
C ∥ t α ∇ K ⟨ H , ϕ ⟩ W ∥ L t , x 1 1 / 2 ≤ C ∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 1 / 2 ∥ t α / 2 ∣ ∇ K ∣ K 1 / 2 H ∥ L t 1 L x 2 1 / 2 ≤ δ ∥ t α / 2 K 1 / 2 ϕ ∥ L t ∞ L x 2 + C δ ∥ t α / 2 ∣ Δ K ∣ 1 / 2 H ∥ L t 1 L x 2 + C δ ∥ t α / 2 ( T − t ) − 1 / 2 K 1 / 2 H ∥ L t 1 L x 2 . \begin{aligned}
C\left\|t^\alpha\nabla K\langle H,\phi\rangle_W\right\|_{L^1_{t,x}}^{1/2}
\leq C\left\|t^{\alpha/2}K^{1/2}\phi\right\|_{L^\infty_tL^2_x}^{1/2}
\left\|t^{\alpha/2}\frac{|\nabla K|}{K^{1/2}}H\right\|_{L^1_tL^2_x}^{1/2} \\
&\le\delta\lVert t^{\alpha/2}K^{1/2}\phi\rVert_{L^\infty_tL_x^2}+C_\delta\lVert t^{\alpha/2}\lvert\Delta K\rvert^{1/2}H\rVert_{L_t^1L_x^2}+C_\delta\lVert t^{\alpha/2}(T-t)^{-1/2}K^{1/2}H\rVert_{L_t^1L_x^2}.
\end{aligned} C ∥ t α ∇ K ⟨ H , ϕ ⟩ W ∥ L t , x 1 1/2 ≤ C t α /2 K 1/2 ϕ L t ∞ L x 2 1/2 t α /2 K 1/2 ∣∇ K ∣ H L t 1 L x 2 1/2 ≤ δ ∥ t α /2 K 1/2 ϕ ∥ L t ∞ L x 2 + C δ ∥ t α /2 ∣ Δ K ∣ 1/2 H ∥ L t 1 L x 2 + C δ ∥ t α /2 ( T − t ) − 1/2 K 1/2 H ∥ L t 1 L x 2 . Taking δ ≤ 1 / 10 \delta\le1/10 δ ≤ 1/10 (188) , while the last two terms are already acceptable. Combining, the first desired estimate (186) now follows. The second desired estimate (187) follows by a similar argument; we omit the proof.
A small modification of the proof of (186) in Lemma 4.8 also gives the following result. We avoid combining this with the statement of Lemma 4.8, because the statement of that lemma is already long enough as it is. In the following, recall our notation (Notation 4.3) for Gronwall loss.
Lemma 4.9. There is some constant C 0 C_0 C 0 ϕ , G : [ 0 , T ] × T d → W d \phi,G\colon[0,T]\times\mathbb{T}^d\to W^d ϕ , G : [ 0 , T ] × T d → W d H : [ 0 , T ] × T d → W H\colon[0,T]\times\mathbb{T}^d\to W H : [ 0 , T ] × T d → W ϕ 0 : T d → W d \phi_0\colon\mathbb{T}^d\to W^d ϕ 0 : T d → W d (185) . Let K : [ 0 , T ] × T d → ( 0 , ∞ ) K\colon[0,T]\times\mathbb{T}^d\to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ )
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , T ] ) + N ( H ) ) . (189) \lVert K^{1/2}\phi\rVert_{L^\infty_tL_x^2([0,T])}+\lVert K^{1/2}D\phi\rVert_{L^2_{t,x}([0,T])}
\lesssim\operatorname{Gr}_T(C_0F)\left(\lVert K^{1/2}(0)\phi_0\rVert_{L_x^2}+\lVert K^{1/2}G\rVert_{L_t^1L_x^2([0,T])}+N(H)\right). \tag*{(189)} ∥ K 1/2 ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ K 1/2 D ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ([ 0 , T ]) + N ( H ) ) . (189) where
N ( H ) = ∥ K 1 / 2 H ∥ L t , x 2 ( [ 0 , T ] ) + ∥ K 1 / 2 ( T − t ) − 1 / 2 H ∥ L t 1 L x 2 ( [ 0 , T ] ) + ∥ ∣ Δ K ∣ 1 / 2 H ∥ L t 1 L x 2 ( [ 0 , T ] ) . N(H)=\lVert K^{1/2}H\rVert_{L^2_{t,x}([0,T])}+\lVert K^{1/2}(T-t)^{-1/2}H\rVert_{L_t^1L_x^2([0,T])}+\lVert\lvert\Delta K\rvert^{1/2}H\rVert_{L_t^1L_x^2([0,T])}. N ( H ) = ∥ K 1/2 H ∥ L t , x 2 ([ 0 , T ]) + ∥ K 1/2 ( T − t ) − 1/2 H ∥ L t 1 L x 2 ([ 0 , T ]) + ∥∣ Δ K ∣ 1/2 H ∥ L t 1 L x 2 ([ 0 , T ]) . Remark 4.10. The main difference between (189) and (186) is the lack of the ∥ t 1 / 2 F ∥ L t 2 L x ∞ \lVert t^{1/2}F\rVert_{L_t^2L_x^\infty} ∥ t 1/2 F ∥ L t 2 L x ∞ Gr T ( C 0 F ) \operatorname{Gr}_T(C_0F) Gr T ( C 0 F ) (183) , we obtain
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 2 + ∥ K 1 / 2 D ϕ ∥ L t , x 2 2 ≲ ∥ K ∣ F ∣ ∣ ϕ ∣ 2 ∥ L t , x 1 + other terms \lVert K^{1/2}\phi\rVert_{L^\infty_tL_x^2}^2+\lVert K^{1/2}D\phi\rVert_{L^2_{t,x}}^2\lesssim\lVert K\lvert F\rvert\lvert\phi\rvert^2\rVert_{L^1_{t,x}}+\text{other terms} ∥ K 1/2 ϕ ∥ L t ∞ L x 2 2 + ∥ K 1/2 D ϕ ∥ L t , x 2 2 ≲ ∥ K ∣ F ∣ ∣ ϕ ∣ 2 ∥ L t , x 1 + other terms ≲ ∥ F ( t ) ∥ L x ∞ ∥ K 1 / 2 ϕ ( t ) ∥ L x 2 2 ∣ L t 1 + other terms . \lesssim\left.\lVert F(t)\rVert_{L_x^\infty}\lVert K^{1/2}\phi(t)\rVert_{L_x^2}^2\right|_{L_t^1}+\text{other terms}. ≲ ∥ F ( t ) ∥ L x ∞ ∥ K 1/2 ϕ ( t ) ∥ L x 2 2 L t 1 + other terms . (We previously treated the first term on the right-hand side by a different application of Hölder.) At this stage, we may apply Gronwall (Lemma 3.50) and estimate “other terms” exactly the same as before, to arrive at (189) . (There is also the fact that (189) does not include a time-weight, which is not as essential, but does result in a slight modification of the proof.)
Recall that p p p T d \mathbb{T}^d T d
Lemma 4.11. For T > 0 T>0 T > 0 f : [ 0 , T ] × T d → R f\colon[0,T]\times\mathbb{T}^d\to\mathbb{R} f : [ 0 , T ] × T d → R ( t , x ) ∈ ( 0 , T ] × T d (t,x)\in(0,T]\times\mathbb{T}^d ( t , x ) ∈ ( 0 , T ] × T d s 1 ∈ [ 0 , t ) s_1\in[0,t) s 1 ∈ [ 0 , t )
∣ f ( t , x ) ∣ ≤ ∥ p 1 / 2 ( s , y ; t , x ) f ( s ) ∥ L s ∞ L y 2 ( [ s 1 , t ] ) . \lvert f(t,x)\rvert\le\lVert p^{1/2}(s,y;t,x)f(s)\rVert_{L^\infty_sL_y^2([s_1,t])}. ∣ f ( t , x )∣ ≤ ∥ p 1/2 ( s , y ; t , x ) f ( s ) ∥ L s ∞ L y 2 ([ s 1 , t ]) . Remark 4.12. In the following, we will often apply this lemma with f = ∣ ϕ ∣ f=\lvert\phi\rvert f = ∣ ϕ ∣ f = ∣ D ( k ) ϕ ∣ f=\lvert D^{(k)}\phi\rvert f = ∣ D ( k ) ϕ ∣ ϕ : [ 0 , T ] × T d → W \phi\colon[0,T]\times\mathbb{T}^d\to W ϕ : [ 0 , T ] × T d → W ϕ : [ 0 , T ] × T d → W d \phi\colon[0,T]\times\mathbb{T}^d\to W^d ϕ : [ 0 , T ] × T d → W d
Proof. It suffices to note that ∫ d y p ( s , y ; t , x ) ∣ f ( s , y ) ∣ 2 → ∣ f ( t , x ) ∣ 2 \int dy\,p(s,y;t,x)\lvert f(s,y)\rvert^2\to\lvert f(t,x)\rvert^2 ∫ d y p ( s , y ; t , x ) ∣ f ( s , y ) ∣ 2 → ∣ f ( t , x ) ∣ 2 s ↑ t s\uparrow t s ↑ t
In the remainder of the section, unless otherwise specified, we let B : [ 0 , T ] × T d → g d B\colon[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d B : [ 0 , T ] × T d → g d
∂ t B i = D ℓ F ℓ i , i ∈ [ d ] . \partial_tB_i=D^\ell F_{\ell i},\qquad i\in[d]. ∂ t B i = D ℓ F ℓ i , i ∈ [ d ] . Next, we record several estimates involving B B B
Lemma 4.13. Let d = 2 d=2 d = 2 T > 0 T>0 T > 0
∥ t 1 / 2 D F ∥ L t ∞ L x 2 ( [ 0 , T ] ) ≲ ( 1 + ∥ t 1 / 2 F ∥ L t 2 L x ∞ ( [ 0 , T ] ) ) ∥ D F ∥ L 2 ( [ 0 , T ] ) . \|t^{1/2}DF\|_{L^\infty_tL^2_x([0,T])}\lesssim(1+\|t^{1/2}F\|_{L^2_tL^\infty_x([0,T])})\|DF\|_{L^2([0,T])}. ∥ t 1/2 D F ∥ L t ∞ L x 2 ([ 0 , T ]) ≲ ( 1 + ∥ t 1/2 F ∥ L t 2 L x ∞ ([ 0 , T ]) ) ∥ D F ∥ L 2 ([ 0 , T ]) . Proof. By Lemma 3.28 , we have that in dimension d = 2 d=2 d = 2 F F F (105) , we have that ( ∂ t − Δ B − 2 F ) D F = 0 (\partial_t-\Delta_B-2F)DF=0 ( ∂ t − Δ B − 2 F ) D F = 0 (186) (with the α \alpha α 1 1 1 K ≡ 1 K\equiv1 K ≡ 1 G , H ≡ 0 G,H\equiv0 G , H ≡ 0
Remark 4.14. Assume d = 2 d=2 d = 2 ∣ F 12 ∣ \lvert F^{12}\rvert ∣ F 12 ∣ ( ∂ t − Δ ) ∣ F 12 ∣ ≤ 0 (\partial_t-\Delta)\lvert F^{12}\rvert\leq0 ( ∂ t − Δ ) ∣ F 12 ∣ ≤ 0 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞ t ↦ ∥ F ( t ) ∥ L x p t\mapsto\|F(t)\|_{L^p_x} t ↦ ∥ F ( t ) ∥ L x p t t t t > 0 t>0 t > 0
∥ F ( t ) ∥ L x ∞ ≤ 1 t ∫ 0 t ∥ F ( s ) ∥ L x ∞ d s ≤ t − 1 ∥ F ∥ T , 0 p e r t . (190) \|F(t)\|_{L^\infty_x}\leq\frac{1}{t}\int_0^t\|F(s)\|_{L^\infty_x}\,ds\leq t^{-1}\|F\|_{T,0}^{\mathrm{pert}}. \tag*{(190)} ∥ F ( t ) ∥ L x ∞ ≤ t 1 ∫ 0 t ∥ F ( s ) ∥ L x ∞ d s ≤ t − 1 ∥ F ∥ T , 0 pert . (190) Remark 4.15. While ∥ ⋅ ∥ p e r t \|\mathord\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert Corollary 4.41 ) in § 5.4 . With this in mind, we record here the following estimates which will be used in several places later. Let 0 < s 0 < t 0 ≤ T 0<s_0<t_0\leq T 0 < s 0 < t 0 ≤ T α ∈ [ 0 , 1 ] \alpha\in[0,1] α ∈ [ 0 , 1 ]
∥ ∂ t B ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) ≲ s 0 − α / 2 ∥ t α / 2 D F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ≤ s 0 − α / 2 T − ( 1 − α ) / 2 ( T ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t ) , (191) \begin{aligned}
\|\partial_tB\|_{L^1_tL^\infty_x([s_0,t_0])}
&\lesssim s_0^{-\alpha/2}\|t^{\alpha/2}DF\|_{L^1_tL^\infty_x([0,T])}\\
&\leq s_0^{-\alpha/2}T^{-(1-\alpha)/2}\bigl(T^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}\bigr), \tag*{(191)}
\end{aligned} ∥ ∂ t B ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) ≲ s 0 − α /2 ∥ t α /2 D F ∥ L t 1 L x ∞ ([ 0 , T ]) ≤ s 0 − α /2 T − ( 1 − α ) /2 ( T ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ) , (191) ∥ F ∥ L t 2 L x ∞ ( [ s 0 , t 0 ] ) ≤ s 0 − α / 2 ∥ t α / 2 F ∥ L t 2 L x ∞ ( [ 0 , T ] ) ≤ s 0 − α / 2 T − ( 1 − α ) / 2 ( T ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t ) , (192) \begin{aligned}
\|F\|_{L^2_tL^\infty_x([s_0,t_0])}
&\leq s_0^{-\alpha/2}\|t^{\alpha/2}F\|_{L^2_tL^\infty_x([0,T])}\\
&\leq s_0^{-\alpha/2}T^{-(1-\alpha)/2}\bigl(T^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}\bigr), \tag*{(192)}
\end{aligned} ∥ F ∥ L t 2 L x ∞ ([ s 0 , t 0 ]) ≤ s 0 − α /2 ∥ t α /2 F ∥ L t 2 L x ∞ ([ 0 , T ]) ≤ s 0 − α /2 T − ( 1 − α ) /2 ( T ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ) , (192) ∥ F ∥ [ s 0 , t 0 ] , 1 p e r t ≲ ( t 0 − s 0 ) ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t ≤ ( T − 1 ( t 0 − s 0 ) ) ( 1 − α ) / 2 ( T ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t ) . (193) \begin{aligned}
\|F\|_{[s_0,t_0],1}^{\mathrm{pert}}
&\lesssim(t_0-s_0)^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}\\
&\leq\bigl(T^{-1}(t_0-s_0)\bigr)^{(1-\alpha)/2}\bigl(T^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}\bigr). \tag*{(193)}
\end{aligned} ∥ F ∥ [ s 0 , t 0 ] , 1 pert ≲ ( t 0 − s 0 ) ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ≤ ( T − 1 ( t 0 − s 0 ) ) ( 1 − α ) /2 ( T ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ) . (193) In the first two estimates, we use that ∥ t α / 2 D F ∥ L t 1 L x ∞ ( [ 0 , T ] ) \|t^{\alpha/2}DF\|_{L^1_tL^\infty_x([0,T])} ∥ t α /2 D F ∥ L t 1 L x ∞ ([ 0 , T ]) ∥ t α / 2 F ∥ L t 2 L x ∞ ( [ 0 , T ] ) \|t^{\alpha/2}F\|_{L^2_tL^\infty_x([0,T])} ∥ t α /2 F ∥ L t 2 L x ∞ ([ 0 , T ]) ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t \|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}} ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ∥ F ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) \|F\|_{L^1_tL^\infty_x([s_0,t_0])} ∥ F ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) (193) , we split into cases s 0 ≤ t 0 / 2 s_0\leq t_0/2 s 0 ≤ t 0 /2 s 0 > t 0 / 2 s_0>t_0/2 s 0 > t 0 /2
∥ F ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) ≤ t 0 ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) ≲ ( t 0 − s 0 ) ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t , \|F\|_{L^1_tL^\infty_x([s_0,t_0])}\leq t_0^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{L^1_tL^\infty_x([s_0,t_0])}\lesssim(t_0-s_0)^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}, ∥ F ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) ≤ t 0 ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) ≲ ( t 0 − s 0 ) ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert , and in the latter case, we first apply Hölder and then the first inequality in (192) to estimate
∥ F ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) ≤ ( t 0 − s 0 ) 1 / 2 ∥ F ∥ L t 2 L x ∞ ( [ s 0 , t 0 ] ) ≲ ( t 0 − s 0 ) ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t . \|F\|_{L^1_tL^\infty_x([s_0,t_0])}\leq(t_0-s_0)^{1/2}\|F\|_{L^2_tL^\infty_x([s_0,t_0])}\lesssim(t_0-s_0)^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}. ∥ F ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) ≤ ( t 0 − s 0 ) 1/2 ∥ F ∥ L t 2 L x ∞ ([ s 0 , t 0 ]) ≲ ( t 0 − s 0 ) ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert . One should think of the quantity T ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , 1 p e r t T^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}} T ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ T , 1 pert ∥ ⋅ ∥ p e r t \|\mathord\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert
Heat kernel estimates
In this subsection, we prove pointwise bounds on the covariant heat kernels p B , q B p_B,q_B p B , q B ∥ ⋅ ∥ p e r t \|\mathord\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert p p p p B p_B p B q B q_B q B q B − p B q_B-p_B q B − p B B B B
Remark 4.16. Throughout this section, we will fix the base point ( s , y ) = ( 0 , 0 ) (s,y)=(0,0) ( s , y ) = ( 0 , 0 ) p B ( t , x ) = p B ( 0 , 0 ; t , x ) p_B(t,x)=p_B(0,0;t,x) p B ( t , x ) = p B ( 0 , 0 ; t , x ) q B ( t , x ) = q B ( 0 , 0 ; t , x ) q_B(t,x)=q_B(0,0;t,x) q B ( t , x ) = q B ( 0 , 0 ; t , x ) ( s , y ) (s,y) ( s , y ) ∣ x ∣ |x| ∣ x ∣ ∣ x − y ∣ |x-y| ∣ x − y ∣ t t t ( t − s ) (t-s) ( t − s ) ∥ F ∥ t , k p e r t \|F\|^{\mathrm{pert}}_{t,k} ∥ F ∥ t , k pert ∥ F ∥ [ s , t ] , k p e r t \|F\|^{\mathrm{pert}}_{[s,t],k} ∥ F ∥ [ s , t ] , k pert Gr t \operatorname{Gr}_t Gr t Gr [ s , t ] \operatorname{Gr}_{[s,t]} Gr [ s , t ]
Proposition 4.17. There is a constant c > 0 c>0 c > 0 ( t 0 , x 0 ) ∈ ( 0 , 1 ] × T d (t_0,x_0)\in(0,1]\times\mathbb{T}^d ( t 0 , x 0 ) ∈ ( 0 , 1 ] × T d
∣ p B ( t 0 , x 0 ) ∣ ≤ p ( t 0 , x 0 ) . (194) |p_B(t_0,x_0)|\leq p(t_0,x_0). \tag*{(194)} ∣ p B ( t 0 , x 0 ) ∣ ≤ p ( t 0 , x 0 ) . (194) ∣ D ( k ) p B ( t 0 , x 0 ) ∣ ≲ k ( 1 + ( ∥ F ∥ t 0 , k − 1 p e r t ) k ) t 0 − k + d 2 exp ( − c ∣ x 0 ∣ 2 t 0 ) . (195) |\mathrm{D}^{(k)}p_B(t_0,x_0)|\lesssim_k \left(1+\left(\|F\|^{\mathrm{pert}}_{t_0,k-1}\right)^k\right)t_0^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_0|^2}{t_0}\right). \tag*{(195)} ∣ D ( k ) p B ( t 0 , x 0 ) ∣ ≲ k ( 1 + ( ∥ F ∥ t 0 , k − 1 pert ) k ) t 0 − 2 k + d exp ( − c t 0 ∣ x 0 ∣ 2 ) . (195) The second inequality holds for any k ≥ 1 k\geq1 k ≥ 1 c c c k k k
Proposition 4.18. There are constants C 0 , c C_0,c C 0 , c ( t 0 , x 0 ) ∈ ( 0 , 1 ] × T d (t_0,x_0)\in(0,1]\times\mathbb{T}^d ( t 0 , x 0 ) ∈ ( 0 , 1 ] × T d k ≥ 0 k\geq0 k ≥ 0
∣ D ( k ) q B ( t 0 , x 0 ) ∣ ≲ k Gr t 0 ( C 0 F ) ( 1 + ( ∥ F ∥ t 0 , k p e r t ) k ) t 0 − k + d 2 exp ( − c ∣ x 0 ∣ 2 t 0 ) . (196) |\mathrm{D}^{(k)}q_B(t_0,x_0)|\lesssim_k \operatorname{Gr}_{t_0}(C_0F)\left(1+\left(\|F\|^{\mathrm{pert}}_{t_0,k}\right)^k\right)t_0^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_0|^2}{t_0}\right). \tag*{(196)} ∣ D ( k ) q B ( t 0 , x 0 ) ∣ ≲ k Gr t 0 ( C 0 F ) ( 1 + ( ∥ F ∥ t 0 , k pert ) k ) t 0 − 2 k + d exp ( − c t 0 ∣ x 0 ∣ 2 ) . (196) We also have the pointwise estimates on the difference between q B q_B q B p B p_B p B
∣ D ( k ) ( q B ( t 0 , x 0 ) − p B ( t 0 , x 0 ) ) ∣ ≲ k Gr t 0 ( C 0 F ) ∥ F ∥ t 0 , k p e r t ( 1 + ( ∥ F ∥ t 0 , k p e r t ) ( k − 1 ) ∨ 0 ) t 0 − k + d 2 exp ( − c ∣ x 0 ∣ 2 t 0 ) . (197) |\mathrm{D}^{(k)}(q_B(t_0,x_0)-p_B(t_0,x_0))|\lesssim_k \operatorname{Gr}_{t_0}(C_0F)\|F\|^{\mathrm{pert}}_{t_0,k}\left(1+\left(\|F\|^{\mathrm{pert}}_{t_0,k}\right)^{(k-1)\vee0}\right)t_0^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_0|^2}{t_0}\right). \tag*{(197)} ∣ D ( k ) ( q B ( t 0 , x 0 ) − p B ( t 0 , x 0 )) ∣ ≲ k Gr t 0 ( C 0 F ) ∥ F ∥ t 0 , k pert ( 1 + ( ∥ F ∥ t 0 , k pert ) ( k − 1 ) ∨ 0 ) t 0 − 2 k + d exp ( − c t 0 ∣ x 0 ∣ 2 ) . (197) Having stated our main Propositions 4.17 and 4.18, we now proceed to their proofs. First, we prove the following intermediate proposition regarding higher regularity estimates for solutions to the covariant heat equation (178) . Our first main result (Proposition 4.17) will directly follow from this proposition.
Proposition 4.19 (Higher regularity for solutions to the covariant heat equation). Let ϕ , G : [ 0 , T ] × T d → W \phi,G:[0,T]\times\mathbb{T}^d\to W ϕ , G : [ 0 , T ] × T d → W
( ∂ t − Δ B ) ϕ = G , (\partial_t-\Delta_B)\phi=G, ( ∂ t − Δ B ) ϕ = G , and let K : [ 0 , T ] × T d → ( 0 , ∞ ) K:[0,T]\times\mathbb{T}^d\to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ ) k ≥ 0 k\geq0 k ≥ 0
∥ t k / 2 K 1 / 2 D ( k ) ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t k / 2 K 1 / 2 D ( k + 1 ) ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ k ( 1 + ( ∥ F ∥ T , k − 1 p e r t ) k ) ( ∥ K 1 / 2 ( 0 ) ϕ ( 0 ) ∥ L x 2 + ∑ j = 0 k ∥ t j / 2 K 1 / 2 D ( j ) G ∥ L t 1 L x 2 ( [ 0 , T ] ) ) . (198) \left\|t^{k/2}K^{1/2}\mathrm{D}^{(k)}\phi\right\|_{L^\infty_tL^2_x([0,T])}
+\left\|t^{k/2}K^{1/2}\mathrm{D}^{(k+1)}\phi\right\|_{L^2_{t,x}([0,T])}
\lesssim_k \left(1+\left(\|F\|^{\mathrm{pert}}_{T,k-1}\right)^k\right)\left(\left\|K^{1/2}(0)\phi(0)\right\|_{L^2_x}+\sum_{j=0}^{k}\left\|t^{j/2}K^{1/2}\mathrm{D}^{(j)}G\right\|_{L^1_tL^2_x([0,T])}\right). \tag*{(198)} t k /2 K 1/2 D ( k ) ϕ L t ∞ L x 2 ([ 0 , T ]) + t k /2 K 1/2 D ( k + 1 ) ϕ L t , x 2 ([ 0 , T ]) ≲ k ( 1 + ( ∥ F ∥ T , k − 1 pert ) k ) ( K 1/2 ( 0 ) ϕ ( 0 ) L x 2 + j = 0 ∑ k t j /2 K 1/2 D ( j ) G L t 1 L x 2 ([ 0 , T ]) ) . (198) Proof. In the following, all space-time norms are implicitly taken on [ 0 , T ] × T d [0,T]\times\mathbb{T}^d [ 0 , T ] × T d k = 0 k=0 k = 0 (183) , and estimating the term
∥ K ⟨ ϕ , G ⟩ W ∥ L t , x 1 ≤ δ ∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 + C δ ∥ K 1 / 2 G ∥ L t 1 L x 2 , \|K\langle\phi,G\rangle_W\|_{L^1_{t,x}}\leq\delta\|K^{1/2}\phi\|_{L^\infty_tL^2_x}+C_\delta\|K^{1/2}G\|_{L^1_tL^2_x}, ∥ K ⟨ ϕ , G ⟩ W ∥ L t , x 1 ≤ δ ∥ K 1/2 ϕ ∥ L t ∞ L x 2 + C δ ∥ K 1/2 G ∥ L t 1 L x 2 , where δ \delta δ ∥ F ∥ T , − 1 p e r t \|F\|^{\mathrm{pert}}_{T,-1} ∥ F ∥ T , − 1 pert 0 0 0 k 0 ≥ 0 k_0\geq0 k 0 ≥ 0 (198) holds for all 0 ≤ k ≤ k 0 0\leq k\leq k_0 0 ≤ k ≤ k 0 k = k 0 + 1 k=k_0+1 k = k 0 + 1 D ( k 0 + 1 ) ϕ \mathrm{D}^{(k_0+1)}\phi D ( k 0 + 1 ) ϕ
( ∂ t − Δ B ) D ( k 0 + 1 ) ϕ = ∑ j 1 + j 2 = k 0 + 1 j 1 ≥ 0 j 2 ≥ 1 O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) + D ( k 0 + 1 ) G . = O ( F D ( k 0 + 1 ) ϕ ) + R + D ( k 0 + 1 ) G , \begin{aligned}
(\partial_t-\Delta_B)\mathrm{D}^{(k_0+1)}\phi
=
\sum_{\substack{j_1+j_2=k_0+1\\ j_1\geq0\\ j_2\geq1}}
\mathcal{O}\bigl((\mathrm{D}^{(j_1)}F)(\mathrm{D}^{(j_2)}\phi)\bigr)
+\mathrm{D}^{(k_0+1)}G. \\
&= \mathcal{O}(F D^{(k_0+1)}\phi)+R+D^{(k_0+1)}G,
\end{aligned} ( ∂ t − Δ B ) D ( k 0 + 1 ) ϕ = j 1 + j 2 = k 0 + 1 j 1 ≥ 0 j 2 ≥ 1 ∑ O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) + D ( k 0 + 1 ) G . = O ( F D ( k 0 + 1 ) ϕ ) + R + D ( k 0 + 1 ) G , where R R R 1 ≤ j 1 , j 2 ≤ k 0 1\leq j_1,j_2\leq k_0 1 ≤ j 1 , j 2 ≤ k 0 (186) with α = k 0 + 1 \alpha=k_0+1 α = k 0 + 1 H = 0 H=0 H = 0
∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 1 ) ϕ ∥ L t ∞ L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 2 ) ϕ ∥ L t , x 2 ≲ ( ∥ t 1 / 2 F ∥ L t 2 L x ∞ + 1 ) ∥ t k 0 / 2 K 1 / 2 D ( k 0 + 1 ) ϕ ∥ L t , x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 R ∥ L t 1 L x 2 \left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+1)}\phi\right\|_{L^\infty_tL^2_x}
+\left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+2)}\phi\right\|_{L^2_{t,x}}
\lesssim
\left(\left\|t^{1/2}F\right\|_{L^2_tL^\infty_x}+1\right)
\left\|t^{k_0/2}K^{1/2}D^{(k_0+1)}\phi\right\|_{L^2_{t,x}}
+\left\|t^{(k_0+1)/2}K^{1/2}R\right\|_{L^1_tL^2_x} t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 1 ) ϕ L t ∞ L x 2 + t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 2 ) ϕ L t , x 2 ≲ ( t 1/2 F L t 2 L x ∞ + 1 ) t k 0 /2 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 + t ( k 0 + 1 ) /2 K 1/2 R L t 1 L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 1 ) G ∥ L t 1 L x 2 . \qquad
+\left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+1)}G\right\|_{L^1_tL^2_x}. + t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 1 ) G L t 1 L x 2 . The first term on the right-hand side above may be directly handled by our inductive assumption, and the last term is already acceptable. To estimate the second term, first note that for fixed 1 ≤ j 1 , j 2 ≤ k 0 1\leq j_1,j_2\leq k_0 1 ≤ j 1 , j 2 ≤ k 0 j 1 + j 2 = k 0 + 1 j_1+j_2=k_0+1 j 1 + j 2 = k 0 + 1
∥ t ( k 0 + 1 ) / 2 K 1 / 2 ( D ( j 1 ) F ) D ( j 2 ) ϕ ∥ L t 1 L x 2 ≲ ∥ t ( j 1 + 1 ) / 2 D ( j 1 ) F ∥ L t 2 L x ∞ ∥ t ( j 2 − 1 ) / 2 K 1 / 2 D ( j 2 ) ϕ ∥ L t , x 2 \left\|t^{(k_0+1)/2}K^{1/2}(D^{(j_1)}F)D^{(j_2)}\phi\right\|_{L^1_tL^2_x}
\lesssim
\left\|t^{(j_1+1)/2}D^{(j_1)}F\right\|_{L^2_tL^\infty_x}
\left\|t^{(j_2-1)/2}K^{1/2}D^{(j_2)}\phi\right\|_{L^2_{t,x}} t ( k 0 + 1 ) /2 K 1/2 ( D ( j 1 ) F ) D ( j 2 ) ϕ L t 1 L x 2 ≲ t ( j 1 + 1 ) /2 D ( j 1 ) F L t 2 L x ∞ t ( j 2 − 1 ) /2 K 1/2 D ( j 2 ) ϕ L t , x 2 ≲ ( ∥ F ∥ T , k 0 p e r t + ( ∥ F ∥ T , k 0 p e r t ) k 0 + 1 ) ( ∥ K ( 0 ) 1 / 2 ϕ ( 0 ) ∥ L x 2 + ∑ j = 0 k 0 ∥ t j / 2 K 1 / 2 D ( j ) G ∥ L t 1 L x 2 ) . \lesssim
\left(\|F\|_{T,k_0}^{\mathrm{pert}}+\left(\|F\|_{T,k_0}^{\mathrm{pert}}\right)^{k_0+1}\right)
\left(\left\|K^{1/2}_{(0)}\phi(0)\right\|_{L^2_x}
+\sum_{j=0}^{k_0}\left\|t^{j/2}K^{1/2}D^{(j)}G\right\|_{L^1_tL^2_x}\right). ≲ ( ∥ F ∥ T , k 0 pert + ( ∥ F ∥ T , k 0 pert ) k 0 + 1 ) ( K ( 0 ) 1/2 ϕ ( 0 ) L x 2 + j = 0 ∑ k 0 t j /2 K 1/2 D ( j ) G L t 1 L x 2 ) . The estimate of the term involving R R R R R R (198) for k = k 0 + 1 k=k_0+1 k = k 0 + 1
We now use Proposition 4.19 to prove our first main result (Proposition 4.17).
Proof of Proposition 4.17. The first estimate (194) is a consequence of the diamagnetic inequality (126) . Next, we show the higher order estimates (195) . Let k ≥ 1 k\geq1 k ≥ 1 ( t 0 , x 0 ) (t_0,x_0) ( t 0 , x 0 ) K ( t , x ) : = p ( t , x ; t 0 , x 0 ) K(t,x):=p(t,x;t_0,x_0) K ( t , x ) := p ( t , x ; t 0 , x 0 ) I : = [ t 0 2 , t 0 ] I:=\left[\frac{t_0}{2},t_0\right] I := [ 2 t 0 , t 0 ] p B p_B p B I × T d I\times\mathbb{T}^d I × T d G = 0 G=0 G = 0
∥ ( t − t 0 2 ) k / 2 K 1 / 2 D ( k ) p B ∥ L t ∞ L x 2 ( I ) ≲ ( 1 + ∥ F ∥ T , t 0 , k − 1 p e r t k ) ∥ ( K 1 / 2 p B ) ( t 0 2 ) ∥ L x 2 . (199) \left\|(t-\frac{t_0}{2})^{k/2}K^{1/2}D^{(k)}p_B\right\|_{L^\infty_tL^2_x(I)}
\lesssim
\left(1+\|F\|_{T,t_0,k-1}^{\mathrm{pert}\,k}\right)
\left\|(K^{1/2}p_B)(\frac{t_0}{2})\right\|_{L^2_x}. \tag*{(199)} ( t − 2 t 0 ) k /2 K 1/2 D ( k ) p B L t ∞ L x 2 ( I ) ≲ ( 1 + ∥ F ∥ T , t 0 , k − 1 pert k ) ( K 1/2 p B ) ( 2 t 0 ) L x 2 . (199) By Lemma 4.11, the left-hand side of (199) can be bounded below by
∥ ( t − t 0 2 ) k / 2 K 1 / 2 D ( k ) p B ∥ L t ∞ L x 2 ( I ) ≳ t 0 k / 2 ∣ D ( k ) p B ∣ ( t 0 , x 0 ) . \left\|(t-\frac{t_0}{2})^{k/2}K^{1/2}D^{(k)}p_B\right\|_{L^\infty_tL^2_x(I)}
\gtrsim t_0^{k/2}\left|D^{(k)}p_B\right|(t_0,x_0). ( t − 2 t 0 ) k /2 K 1/2 D ( k ) p B L t ∞ L x 2 ( I ) ≳ t 0 k /2 D ( k ) p B ( t 0 , x 0 ) . By using the zeroth order estimate (194) , the estimate p ( t , x ) ≲ t − d / 2 p(t,x)\lesssim t^{-d/2} p ( t , x ) ≲ t − d /2 K 1 / 2 p B K^{1/2}p_B K 1/2 p B (199) can be bounded above by
∥ ( K 1 / 2 p B ) ( t 0 2 ) ∥ L x 2 ≲ ∥ p ( t 0 2 , x ; t 0 , x 0 ) 1 / 2 p ( t 0 2 , x ) ∥ L x 2 \left\|(K^{1/2}p_B)(\frac{t_0}{2})\right\|_{L^2_x}
\lesssim
\left\|p(\frac{t_0}{2},x;t_0,x_0)^{1/2}p(\frac{t_0}{2},x)\right\|_{L^2_x} ( K 1/2 p B ) ( 2 t 0 ) L x 2 ≲ p ( 2 t 0 , x ; t 0 , x 0 ) 1/2 p ( 2 t 0 , x ) L x 2 ≲ t 0 − d / 4 ∥ p ( t 0 2 , x ; t 0 , x 0 ) 1 / 2 p ( t 0 2 , x ) 1 / 2 ∥ L x 2 ≲ t 0 − d / 4 p ( t 0 , x 0 ) 1 / 2 . \lesssim
t_0^{-d/4}\left\|p(\frac{t_0}{2},x;t_0,x_0)^{1/2}p(\frac{t_0}{2},x)^{1/2}\right\|_{L^2_x}
\lesssim
t_0^{-d/4}p(t_0,x_0)^{1/2}. ≲ t 0 − d /4 p ( 2 t 0 , x ; t 0 , x 0 ) 1/2 p ( 2 t 0 , x ) 1/2 L x 2 ≲ t 0 − d /4 p ( t 0 , x 0 ) 1/2 . Combining, we obtain the desired estimate (195) .
In dimension d = 2 d=2 d = 2 F F F (178) (recall Lemma 3.28), we may also apply Proposition 4.19 to obtain higher regularity estimates of F F F F F F ∥ F ∥ T , 0 p e r t \|F\|_{T,0}^{\mathrm{pert}} ∥ F ∥ T , 0 pert
Corollary 4.20. Let d = 2 d=2 d = 2 T ≤ 1 T\leq1 T ≤ 1 k ≥ 0 k\geq0 k ≥ 0
∥ F ∥ T , k p e r t ≲ k ∥ F ∥ T , 0 p e r t + ( ∥ F ∥ T , 0 p e r t ) ( k + 1 ) ! . \|F\|^{\mathrm{pert}}_{T,k}\lesssim_k \|F\|^{\mathrm{pert}}_{T,0}+\left(\|F\|^{\mathrm{pert}}_{T,0}\right)^{(k+1)!}. ∥ F ∥ T , k pert ≲ k ∥ F ∥ T , 0 pert + ( ∥ F ∥ T , 0 pert ) ( k + 1 )! . Remark 4.21. Since we will only ever need this corollary for a bounded range of k k k k ≤ 10 k\leq10 k ≤ 10 k k k
Proof. We proceed by induction. The base case k = 0 k=0 k = 0 k 0 ≥ 0 k_0\geq0 k 0 ≥ 0 0 ≤ k ≤ k 0 0\leq k\leq k_0 0 ≤ k ≤ k 0 T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] k = k 0 + 1 k=k_0+1 k = k 0 + 1 T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] F F F Lemma 3.28 ), we have that
D ( k 0 + 1 ) F ( t 0 ) = ∫ T d d y D ( k 0 + 1 ) p B ( t 0 / 2 , y ; t 0 , ⋅ ) F ( t 0 / 2 , y ) . D^{(k_0+1)}F(t_0)=\int_{\mathbb{T}^d} \mathrm{d}y\,D^{(k_0+1)}p_B(t_0/2,y;t_0,\mathord\cdot)F(t_0/2,y). D ( k 0 + 1 ) F ( t 0 ) = ∫ T d d y D ( k 0 + 1 ) p B ( t 0 /2 , y ; t 0 , ⋅ ) F ( t 0 /2 , y ) . Thus by (195) , we have that (using that ∥ F ∥ [ t 0 / 2 , t 0 ] , k p e r t ≤ ∥ F ∥ T , k p e r t \|F\|^{\mathrm{pert}}_{[t_0/2,t_0],k}\leq\|F\|^{\mathrm{pert}}_{T,k} ∥ F ∥ [ t 0 /2 , t 0 ] , k pert ≤ ∥ F ∥ T , k pert ∥ ⋅ ∥ p e r t \|\mathord\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert
∥ D ( k 0 + 1 ) F ( t 0 ) ∥ L x ∞ ≲ ( 1 + ( ∥ F ∥ T , k 0 p e r t ) k 0 + 1 ) t 0 − ( k 0 + 1 ) / 2 ∥ F ( t 0 / 2 ) ∥ L x ∞ . \|D^{(k_0+1)}F(t_0)\|_{L^\infty_x}\lesssim\left(1+\left(\|F\|^{\mathrm{pert}}_{T,k_0}\right)^{k_0+1}\right)t_0^{-(k_0+1)/2}\|F(t_0/2)\|_{L^\infty_x}. ∥ D ( k 0 + 1 ) F ( t 0 ) ∥ L x ∞ ≲ ( 1 + ( ∥ F ∥ T , k 0 pert ) k 0 + 1 ) t 0 − ( k 0 + 1 ) /2 ∥ F ( t 0 /2 ) ∥ L x ∞ . Since this is true for arbitrary t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
∥ t ( k 0 + 2 ) / 2 D ( k 0 + 1 ) F ∥ L t 2 L x ∞ ( [ 0 , T ] ) + ∥ t ( k 0 + 1 ) / 2 D ( k 0 + 1 ) F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ≲ ( 1 + ( ∥ F ∥ T , k 0 p e r t ) k 0 + 1 ) ∥ F ∥ T , 0 p e r t ≲ ∥ F ∥ T , 0 p e r t + ( ∥ F ∥ T , 0 p e r t ) ( k 0 + 1 ) ! ( k 0 + 1 ) + 1 . \begin{aligned}
&\|t^{(k_0+2)/2}D^{(k_0+1)}F\|_{L^2_tL^\infty_x([0,T])}+\|t^{(k_0+1)/2}D^{(k_0+1)}F\|_{L^1_tL^\infty_x([0,T])}\\
&\qquad\lesssim\left(1+\left(\|F\|^{\mathrm{pert}}_{T,k_0}\right)^{k_0+1}\right)\|F\|^{\mathrm{pert}}_{T,0}\\
&\qquad\lesssim\|F\|^{\mathrm{pert}}_{T,0}+\left(\|F\|^{\mathrm{pert}}_{T,0}\right)^{(k_0+1)!(k_0+1)+1}.
\end{aligned} ∥ t ( k 0 + 2 ) /2 D ( k 0 + 1 ) F ∥ L t 2 L x ∞ ([ 0 , T ]) + ∥ t ( k 0 + 1 ) /2 D ( k 0 + 1 ) F ∥ L t 1 L x ∞ ([ 0 , T ]) ≲ ( 1 + ( ∥ F ∥ T , k 0 pert ) k 0 + 1 ) ∥ F ∥ T , 0 pert ≲ ∥ F ∥ T , 0 pert + ( ∥ F ∥ T , 0 pert ) ( k 0 + 1 )! ( k 0 + 1 ) + 1 . The inductive step now follows upon noting that ( k 0 + 1 ) ! × ( k 0 + 1 ) + 1 ≤ ( k 0 + 2 ) ! (k_0+1)!\times(k_0+1)+1\leq(k_0+2)! ( k 0 + 1 )! × ( k 0 + 1 ) + 1 ≤ ( k 0 + 2 )!
Next, we proceed towards the proof of the second main result (Proposition 4.18 ), which gives estimates for q B q_B q B q B − p B q_B-p_B q B − p B (179) :
( ∂ t − Δ B ) ( q B − p B ) = 2 F q B = 2 F ( q B − p B ) + 2 F p B . (\partial_t-\Delta_B)(q_B-p_B)=2Fq_B=2F(q_B-p_B)+2Fp_B. ( ∂ t − Δ B ) ( q B − p B ) = 2 F q B = 2 F ( q B − p B ) + 2 F p B . Motivated by this, we first prove a higher regularity estimate for solutions to the above equation.
Proposition 4.22 (Higher regularity for covariant heat equation with curvature term). There exists a constant C 0 C_0 C 0 ϕ , ψ , G : [ 0 , T ] × T d → W d \phi,\psi,G:[0,T]\times\mathbb{T}^d\to\mathcal{W}^d ϕ , ψ , G : [ 0 , T ] × T d → W d
( ∂ t − Δ B ) ϕ = G , ϕ ( 0 ) = ϕ 0 , (\partial_t-\Delta_B)\phi=G,\qquad\phi(0)=\phi_0, ( ∂ t − Δ B ) ϕ = G , ϕ ( 0 ) = ϕ 0 , ( ∂ t − Δ B ) ψ = 2 F ( ψ + ϕ ) , ψ ( 0 ) = ψ 0 , (\partial_t-\Delta_B)\psi=2F(\psi+\phi),\qquad\psi(0)=\psi_0, ( ∂ t − Δ B ) ψ = 2 F ( ψ + ϕ ) , ψ ( 0 ) = ψ 0 , and let K : [ 0 , T ] × T d → ( 0 , ∞ ) K:[0,T]\times\mathbb{T}^d\to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ )
∥ K 1 / 2 ψ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ K 1 / 2 D ψ ∥ L t , x 2 ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , T ] ) ) ) . \|K^{1/2}\psi\|_{L^\infty_tL^2_x([0,T])}+\|K^{1/2}D\psi\|_{L^2_{t,x}([0,T])}
\lesssim\operatorname{Gr}_T(C_0F)\left(\|K^{1/2}(0)\psi_0\|_{L^2_x}+\|F\|_{L^1_tL^\infty_x([0,T])}\left(\|K^{1/2}(0)\phi_0\|_{L^2_x}+\|K^{1/2}G\|_{L^1_tL^2_x([0,T])}\right)\right). ∥ K 1/2 ψ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ K 1/2 D ψ ∥ L t , x 2 ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ L t 1 L x ∞ ([ 0 , T ]) ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ([ 0 , T ]) ) ) . Moreover, for k ≥ 1 k\geq1 k ≥ 1
∥ t k / 2 K 1 / 2 D ( k ) ψ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t k / 2 K 1 / 2 D ( k + 1 ) ψ ∥ L t , x 2 ( [ 0 , T ] ) ≲ k Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k p e r t ) k − 1 ) × ( ( 1 + ∥ F ∥ T , k p e r t ) ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ T , k p e r t ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + N k ( G ) ) ) , (200) \begin{aligned}
&\|t^{k/2}K^{1/2}D^{(k)}\psi\|_{L^\infty_tL^2_x([0,T])}+\|t^{k/2}K^{1/2}D^{(k+1)}\psi\|_{L^2_{t,x}([0,T])}\\
&\qquad\lesssim_k \operatorname{Gr}_T(C_0F)\left(1+\left(\|F\|^{\mathrm{pert}}_{T,k}\right)^{k-1}\right)\\
&\qquad\quad\times\left(\left(1+\|F\|^{\mathrm{pert}}_{T,k}\right)\|K^{1/2}(0)\psi_0\|_{L^2_x}+\|F\|^{\mathrm{pert}}_{T,k}\left(\|K^{1/2}(0)\phi_0\|_{L^2_x}+N_k(G)\right)\right), \tag*{(200)}
\end{aligned} ∥ t k /2 K 1/2 D ( k ) ψ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ t k /2 K 1/2 D ( k + 1 ) ψ ∥ L t , x 2 ([ 0 , T ]) ≲ k Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k pert ) k − 1 ) × ( ( 1 + ∥ F ∥ T , k pert ) ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ T , k pert ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + N k ( G ) ) ) , (200) where
N k ( G ) = ∑ j = 0 k − 1 ∥ t j / 2 K 1 / 2 D ( j ) G ∥ L t 1 L x 2 ( [ 0 , T ] ) . N_k(G)=\sum_{j=0}^{k-1}\|t^{j/2}K^{1/2}D^{(j)}G\|_{L^1_tL^2_x([0,T])}. N k ( G ) = j = 0 ∑ k − 1 ∥ t j /2 K 1/2 D ( j ) G ∥ L t 1 L x 2 ([ 0 , T ]) . Proof. In the following, all space-time norms are implicitly taken on [ 0 , T ] × T d [0,T]\times\mathbb{T}^{d} [ 0 , T ] × T d k = 0 k=0 k = 0 (189) , Hölder, and Proposition 4.19, we have that
∥ K 1 / 2 ψ ∥ L t ∞ L x 2 + ∥ K 1 / 2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ K 1 / 2 F ϕ ∥ L t 1 L x 2 ) \lVert K^{1/2}\psi\rVert_{L^\infty_tL^2_x}+\lVert K^{1/2}D\psi\rVert_{L^2_{t,x}}
\lesssim\operatorname{Gr}_T(C_0F)\left(\lVert K^{1/2}(0)\psi_0\rVert_{L^2_x}+\lVert K^{1/2}F\phi\rVert_{L^1_tL^2_x}\right) ∥ K 1/2 ψ ∥ L t ∞ L x 2 + ∥ K 1/2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ K 1/2 F ϕ ∥ L t 1 L x 2 ) ≲ Gr T ( C 0 F ) ( ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ L t 1 L x ∞ ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ) ) . (201) \lesssim\operatorname{Gr}_T(C_0F)\left(\lVert K^{1/2}(0)\psi_0\rVert_{L^2_x}+\lVert F\rVert_{L^1_tL^\infty_x}\left(\lVert K^{1/2}(0)\phi_0\rVert_{L^2_x}+\lVert K^{1/2}G\rVert_{L^1_tL^2_x}\right)\right). \tag*{(201)} ≲ Gr T ( C 0 F ) ( ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ L t 1 L x ∞ ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ) ) . (201) which is acceptable. By possibly increasing the constant C 0 C_0 C 0
∥ K 1 / 2 ψ ∥ L t ∞ L x 2 + ∥ K 1 / 2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ) . (202) \lVert K^{1/2}\psi\rVert_{L^\infty_tL^2_x}+\lVert K^{1/2}D\psi\rVert_{L^2_{t,x}}\lesssim\operatorname{Gr}_T(C_0F)\left(\lVert K^{1/2}(0)\psi_0\rVert_{L^2_x}+\lVert K^{1/2}(0)\phi_0\rVert_{L^2_x}+\lVert K^{1/2}G\rVert_{L^1_tL^2_x}\right). \tag*{(202)} ∥ K 1/2 ψ ∥ L t ∞ L x 2 + ∥ K 1/2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ) . (202) which we will use later.
To prove (200) , we proceed by induction. For the base case k = 1 k=1 k = 1 D ψ D\psi D ψ
( ∂ t − Δ B ) D ψ = O ( F D ψ ) + O ( ( D F ) ψ ) + O ( F D ϕ + ( D F ) ϕ ) . (\partial_t-\Delta_B)D\psi=\mathcal{O}(FD\psi)+\mathcal{O}((DF)\psi)+\mathcal{O}(FD\phi+(DF)\phi). ( ∂ t − Δ B ) D ψ = O ( F D ψ ) + O (( D F ) ψ ) + O ( F D ϕ + ( D F ) ϕ ) . By the time-weighted estimate (186) with α = 1 \alpha=1 α = 1 G G G
∥ t 1 / 2 K 1 / 2 D ψ ∥ L t ∞ L x 2 + ∥ t 1 / 2 K 1 / 2 D ( 2 ) ψ ∥ L t , x 2 ≲ ( ∥ t 1 / 2 F ∥ L t 2 L x ∞ + 1 ) ∥ K 1 / 2 D ψ ∥ L t , x 2 + ∥ t 1 / 2 K 1 / 2 ∣ D F ∣ ∣ ψ ∣ ∥ L t 1 L x 2 + ∥ t 1 / 2 K 1 / 2 ( ∣ D F ∣ ∣ ϕ ∣ + ∣ F ∣ ∣ D ϕ ∣ ) ∥ L t 1 L x 2 . \lVert t^{1/2}K^{1/2}D\psi\rVert_{L^\infty_tL^2_x}+\lVert t^{1/2}K^{1/2}D^{(2)}\psi\rVert_{L^2_{t,x}}\lesssim\left(\lVert t^{1/2}F\rVert_{L^2_tL^\infty_x}+1\right)\lVert K^{1/2}D\psi\rVert_{L^2_{t,x}}+\lVert t^{1/2}K^{1/2}|DF||\psi|\rVert_{L^1_tL^2_x}+\lVert t^{1/2}K^{1/2}(|DF||\phi|+|F||D\phi|)\rVert_{L^1_tL^2_x}. ∥ t 1/2 K 1/2 D ψ ∥ L t ∞ L x 2 + ∥ t 1/2 K 1/2 D ( 2 ) ψ ∥ L t , x 2 ≲ ( ∥ t 1/2 F ∥ L t 2 L x ∞ + 1 ) ∥ K 1/2 D ψ ∥ L t , x 2 + ∥ t 1/2 K 1/2 ∣ D F ∣∣ ψ ∣ ∥ L t 1 L x 2 + ∥ t 1/2 K 1/2 ( ∣ D F ∣∣ ϕ ∣ + ∣ F ∣∣ D ϕ ∣ ) ∥ L t 1 L x 2 . For the first term on the right-hand side above, we combine (201) and (202) to estimate
( ∥ t 1 / 2 F ∥ L t 2 L x ∞ + 1 ) ∥ K 1 / 2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ( 1 + ∥ F ∥ T , 1 p e r t ) ∥ K 1 / 2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ T , 1 p e r t ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ) ) , \left(\lVert t^{1/2}F\rVert_{L^2_tL^\infty_x}+1\right)\lVert K^{1/2}D\psi\rVert_{L^2_{t,x}}\lesssim\operatorname{Gr}_T(C_0F)\left(\left(1+\lVert F\rVert^{\mathrm{pert}}_{T,1}\right)\lVert K^{1/2}(0)\psi_0\rVert_{L^2_x}+\lVert F\rVert^{\mathrm{pert}}_{T,1}\left(\lVert K^{1/2}(0)\phi_0\rVert_{L^2_x}+\lVert K^{1/2}G\rVert_{L^1_tL^2_x}\right)\right), ( ∥ t 1/2 F ∥ L t 2 L x ∞ + 1 ) ∥ K 1/2 D ψ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( ( 1 + ∥ F ∥ T , 1 pert ) ∥ K 1/2 ( 0 ) ψ 0 ∥ L x 2 + ∥ F ∥ T , 1 pert ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + ∥ K 1/2 G ∥ L t 1 L x 2 ) ) , which is acceptable. For the second term, we apply Hölder to estimate
∥ t 1 / 2 K 1 / 2 ∣ D F ∣ ∣ ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t 1 / 2 ∣ D F ∣ ∥ L t 1 L x ∞ ∥ K 1 / 2 ∣ ψ ∣ ∥ L t ∞ L x 2 ≤ ∥ F ∥ T , 1 p e r t ∥ K 1 / 2 ∣ ψ ∣ ∥ L t ∞ L x 2 , \lVert t^{1/2}K^{1/2}|DF||\psi|\rVert_{L^1_tL^2_x}\leq\lVert t^{1/2}|DF|\rVert_{L^1_tL^\infty_x}\lVert K^{1/2}|\psi|\rVert_{L^\infty_tL^2_x}\leq\lVert F\rVert^{\mathrm{pert}}_{T,1}\lVert K^{1/2}|\psi|\rVert_{L^\infty_tL^2_x}, ∥ t 1/2 K 1/2 ∣ D F ∣∣ ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t 1/2 ∣ D F ∣ ∥ L t 1 L x ∞ ∥ K 1/2 ∣ ψ ∣ ∥ L t ∞ L x 2 ≤ ∥ F ∥ T , 1 pert ∥ K 1/2 ∣ ψ ∣ ∥ L t ∞ L x 2 , and then apply (202) to obtain an acceptable contribution. The last term which involves ϕ \phi ϕ D ϕ D\phi D ϕ k = 0 k=0 k = 0 k = 1 k=1 k = 1 (200) .
Next, we suppose that for some k 0 ≥ 1 k_0\geq1 k 0 ≥ 1 (200) is true for all 1 ≤ k ≤ k 0 1\leq k\leq k_0 1 ≤ k ≤ k 0 k = k 0 + 1 k=k_0+1 k = k 0 + 1 D ( k 0 + 1 ) ψ D^{(k_0+1)}\psi D ( k 0 + 1 ) ψ
( ∂ t − Δ B ) D ( k 0 + 1 ) ψ = ∑ ℓ + m = k 0 + 1 ℓ , m ≥ 0 O ( ( D ( ℓ ) F ) ( D ( m ) ψ + D ( m ) ϕ ) ) = O ( F D ( k 0 + 1 ) ψ ) + O ( ( D ( k 0 + 1 ) F ) ψ ) + R 1 + R 2 , (\partial_t-\Delta_B)D^{(k_0+1)}\psi=
\sum_{\substack{\ell+m=k_0+1\\\ell,m\geq0}}\mathcal{O}\left((D^{(\ell)}F)(D^{(m)}\psi+D^{(m)}\phi)\right)
=\mathcal{O}(FD^{(k_0+1)}\psi)+\mathcal{O}((D^{(k_0+1)}F)\psi)+R_1+R_2, ( ∂ t − Δ B ) D ( k 0 + 1 ) ψ = ℓ + m = k 0 + 1 ℓ , m ≥ 0 ∑ O ( ( D ( ℓ ) F ) ( D ( m ) ψ + D ( m ) ϕ ) ) = O ( F D ( k 0 + 1 ) ψ ) + O (( D ( k 0 + 1 ) F ) ψ ) + R 1 + R 2 , where
R 1 : = ∑ ℓ + m = k 0 + 1 ℓ , m ≥ 1 O ( ( D ( ℓ ) F ) ( D ( m ) ψ ) ) , R 2 : = ∑ ℓ + m = k 0 + 1 ℓ , m ≥ 0 O ( ( D ( ℓ ) F ) ( D ( m ) ϕ ) ) . R_1:=\sum_{\substack{\ell+m=k_0+1\\\ell,m\geq1}}\mathcal{O}\left((D^{(\ell)}F)(D^{(m)}\psi)\right),\qquad R_2:=\sum_{\substack{\ell+m=k_0+1\\\ell,m\geq0}}\mathcal{O}\left((D^{(\ell)}F)(D^{(m)}\phi)\right). R 1 := ℓ + m = k 0 + 1 ℓ , m ≥ 1 ∑ O ( ( D ( ℓ ) F ) ( D ( m ) ψ ) ) , R 2 := ℓ + m = k 0 + 1 ℓ , m ≥ 0 ∑ O ( ( D ( ℓ ) F ) ( D ( m ) ϕ ) ) . By the time-weighted estimate (186) with α = k 0 + 1 \alpha=k_0+1 α = k 0 + 1 G = O ( ( D ( k 0 + 1 ) F ) ψ ) + R 1 + R 2 G=O((D^{(k_0+1)}F)\psi)+R_1+R_2 G = O (( D ( k 0 + 1 ) F ) ψ ) + R 1 + R 2 H = 0 H=0 H = 0
∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 1 ) ψ ∥ L t ∞ L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 2 ) ψ ∥ L t , x 2 ≲ ( ∥ t 1 / 2 F ∥ L t 2 L x ∞ + 1 ) ∥ t k 0 / 2 K 1 / 2 D ( k 0 + 1 ) ψ ∥ L t , x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 ∣ D ( k 0 + 1 ) F ∣ ∣ ψ ∣ ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 ∣ R 1 ∣ ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 ∣ R 2 ∣ ∥ L t 1 L x 2 . \begin{aligned}
&\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+1)}\psi\|_{L_t^\infty L_x^2}+\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+2)}\psi\|_{L_{t,x}^2}\\
&\lesssim(\|t^{1/2}F\|_{L_t^2L_x^\infty}+1)\|t^{k_0/2}K^{1/2}D^{(k_0+1)}\psi\|_{L_{t,x}^2}\\
&\quad+\|t^{(k_0+1)/2}K^{1/2}|D^{(k_0+1)}F||\psi|\|_{L_t^1L_x^2}\\
&\quad+\|t^{(k_0+1)/2}K^{1/2}|R_1|\|_{L_t^1L_x^2}+\|t^{(k_0+1)/2}K^{1/2}|R_2|\|_{L_t^1L_x^2}.
\end{aligned} ∥ t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 1 ) ψ ∥ L t ∞ L x 2 + ∥ t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 2 ) ψ ∥ L t , x 2 ≲ ( ∥ t 1/2 F ∥ L t 2 L x ∞ + 1 ) ∥ t k 0 /2 K 1/2 D ( k 0 + 1 ) ψ ∥ L t , x 2 + ∥ t ( k 0 + 1 ) /2 K 1/2 ∣ D ( k 0 + 1 ) F ∣∣ ψ ∣ ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) /2 K 1/2 ∣ R 1 ∣ ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) /2 K 1/2 ∣ R 2 ∣ ∥ L t 1 L x 2 . The first term above may be handled by the inductive assumption. To estimate the second term, we apply Hölder to obtain
∥ t ( k 0 + 1 ) / 2 K 1 / 2 ∣ D ( k 0 + 1 ) F ∣ ∣ ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t ( k 0 + 1 ) / 2 D ( k 0 + 1 ) F ∥ L t 1 L x ∞ ∥ K 1 / 2 ψ ∥ L t ∞ L x 2 , \|t^{(k_0+1)/2}K^{1/2}|D^{(k_0+1)}F||\psi|\|_{L_t^1L_x^2}\leq\|t^{(k_0+1)/2}D^{(k_0+1)}F\|_{L_t^1L_x^\infty}\|K^{1/2}\psi\|_{L_t^\infty L_x^2}, ∥ t ( k 0 + 1 ) /2 K 1/2 ∣ D ( k 0 + 1 ) F ∣∣ ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t ( k 0 + 1 ) /2 D ( k 0 + 1 ) F ∥ L t 1 L x ∞ ∥ K 1/2 ψ ∥ L t ∞ L x 2 , and by (202) , this is acceptable. To estimate the term involving R 1 R_1 R 1 ℓ , m ≥ 1 \ell,m\geq1 ℓ , m ≥ 1 ℓ + m = k 0 + 1 \ell+m=k_0+1 ℓ + m = k 0 + 1
∥ t ( k 0 + 1 ) / 2 K 1 / 2 ∣ D ( ℓ ) F ∣ ∣ D ( m ) ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t ( ℓ + 1 ) / 2 D ( ℓ ) F ∥ L t 2 L x ∞ ∥ t ( m − 1 ) / 2 K 1 / 2 D ( m ) ψ ∥ L t , x 2 \|t^{(k_0+1)/2}K^{1/2}|D^{(\ell)}F||D^{(m)}\psi|\|_{L_t^1L_x^2}\leq\|t^{(\ell+1)/2}D^{(\ell)}F\|_{L_t^2L_x^\infty}\|t^{(m-1)/2}K^{1/2}D^{(m)}\psi\|_{L_{t,x}^2} ∥ t ( k 0 + 1 ) /2 K 1/2 ∣ D ( ℓ ) F ∣∣ D ( m ) ψ ∣ ∥ L t 1 L x 2 ≤ ∥ t ( ℓ + 1 ) /2 D ( ℓ ) F ∥ L t 2 L x ∞ ∥ t ( m − 1 ) /2 K 1/2 D ( m ) ψ ∥ L t , x 2 ≲ ∥ F ∥ T , k 0 + 1 p e r t ∥ t ( m − 1 ) / 2 K 1 / 2 D ( m ) ψ ∥ L t , x 2 . \lesssim\|F\|_{T,k_0+1}^{\mathrm{pert}}\|t^{(m-1)/2}K^{1/2}D^{(m)}\psi\|_{L_{t,x}^2}. ≲ ∥ F ∥ T , k 0 + 1 pert ∥ t ( m − 1 ) /2 K 1/2 D ( m ) ψ ∥ L t , x 2 . Inserting the inductive assumption as well as the zeroth order estimate (note 1 ≤ m ≤ k 0 1\leq m\leq k_0 1 ≤ m ≤ k 0 R 2 R_2 R 2
Using Proposition 4.22, we may now prove our second main result (Proposition 4.18).
Proof of Proposition 4.18. We first show the difference estimate (197) . Fix ( t 0 , x 0 ) (t_0,x_0) ( t 0 , x 0 ) ψ ( t , x ) = ( q B − p B ) ( t , x ) \psi(t,x)=(q_B-p_B)(t,x) ψ ( t , x ) = ( q B − p B ) ( t , x )
( ∂ t − Δ B ) ψ = 2 F q B = 2 F ψ + 2 F p B . (\partial_t-\Delta_B)\psi=2Fq_B=2F\psi+2Fp_B. ( ∂ t − Δ B ) ψ = 2 F q B = 2 F ψ + 2 F p B . Thus, ψ \psi ψ ϕ = p B \phi=p_B ϕ = p B
∣ ψ ( t , x ) ∣ ≲ G r t ( C F ) ∥ F ∥ L t 1 L x ∞ ( [ 0 , t ] ) p ( t , x ) , for all t ∈ ( 0 , t 0 ] , x ∈ T d . (203) |\psi(t,x)|\lesssim G_{\mathrm{rt}}(C_F)\|F\|_{L_t^1L_x^\infty([0,t])}p(t,x),\quad\text{for all }t\in(0,t_0],\ x\in\mathbb{T}^d. \tag*{(203)} ∣ ψ ( t , x ) ∣ ≲ G rt ( C F ) ∥ F ∥ L t 1 L x ∞ ([ 0 , t ]) p ( t , x ) , for all t ∈ ( 0 , t 0 ] , x ∈ T d . (203) For t ∈ ( 0 , t 0 ] t\in(0,t_0] t ∈ ( 0 , t 0 ]
f ( t ) ≔ sup x ∈ T d ∣ ψ ( t , x ) ∣ p ( t , x ) . f(t)\coloneqq\sup_{x\in\mathbb{T}^d}\frac{|\psi(t,x)|}{p(t,x)}. f ( t ) : = x ∈ T d sup p ( t , x ) ∣ ψ ( t , x ) ∣ . For x ∈ T d x\in\mathbb{T}^d x ∈ T d (194) on p B p_B p B
∣ ψ ( t , x ) ∣ ≲ ∫ d s d y p ( s , y ; t , x ) ∣ F ( s , y ) ∣ ∣ ψ ( s , y ) ∣ + ∫ d s d y p ( s , y ; t , x ) ∣ F ( s , y ) ∣ p ( s , y ) ≲ ∫ d s d y p ( s , y ; t , x ) p ( s , y ) f ( s ) ∣ F ( s , y ) ∣ + p ( t , x ) ∥ F ∥ L s 1 L y ∞ ( [ 0 , t ] ) ≲ p ( t , x ) ( ∫ 0 t d s f ( s ) ∥ F ( s ) ∥ L y ∞ + ∥ F ∥ L s 1 L y ∞ ( [ 0 , t ] ) ) . \begin{aligned}
|\psi(t,x)|&\lesssim\int ds\,dy\,p(s,y;t,x)|F(s,y)||\psi(s,y)|+\int ds\,dy\,p(s,y;t,x)|F(s,y)|p(s,y)\\
&\lesssim\int ds\,dy\,p(s,y;t,x)p(s,y)f(s)|F(s,y)|+p(t,x)\|F\|_{L_s^1L_y^\infty([0,t])}\\
&\lesssim p(t,x)\left(\int_0^t ds\,f(s)\|F(s)\|_{L_y^\infty}+\|F\|_{L_s^1L_y^\infty([0,t])}\right).
\end{aligned} ∣ ψ ( t , x ) ∣ ≲ ∫ d s d y p ( s , y ; t , x ) ∣ F ( s , y ) ∣∣ ψ ( s , y ) ∣ + ∫ d s d y p ( s , y ; t , x ) ∣ F ( s , y ) ∣ p ( s , y ) ≲ ∫ d s d y p ( s , y ; t , x ) p ( s , y ) f ( s ) ∣ F ( s , y ) ∣ + p ( t , x ) ∥ F ∥ L s 1 L y ∞ ([ 0 , t ]) ≲ p ( t , x ) ( ∫ 0 t d s f ( s ) ∥ F ( s ) ∥ L y ∞ + ∥ F ∥ L s 1 L y ∞ ([ 0 , t ]) ) . It follows that
f ( t ) ≲ ∫ 0 t d s f ( s ) ∥ F ( s ) ∥ L x ∞ + ∥ F ∥ L t 1 L x ∞ ( [ 0 , t ] ) f(t) \lesssim\int_0^t ds\,f(s)\lVert F(s)\rVert_{L^\infty_x}+\lVert F\rVert_{L^1_tL^\infty_x([0,t])} f ( t ) ≲ ∫ 0 t d s f ( s ) ∥ F ( s ) ∥ L x ∞ + ∥ F ∥ L t 1 L x ∞ ([ 0 , t ]) and the zeroth order estimate (203) now follows by Gronwall (Lemma 3.50 ); we remark that f ∈ L t ∞ ( [ 0 , t 0 ] ) f\in L^\infty_t([0,t_0]) f ∈ L t ∞ ([ 0 , t 0 ]) B B B (203) with our zeroth order estimate (194) on p B p_B p B q B q_B q B k = 0 k=0 k = 0 (196) ).
To obtain the higher order estimates, let k ≥ 1 k\geq1 k ≥ 1 I = [ t 0 2 , t 0 ] I=[\frac{t_0}{2},t_0] I = [ 2 t 0 , t 0 ] x 0 ∈ T d x_0\in\mathbb{T}^d x 0 ∈ T d K ( t , x ) = p ( t , x ; t 0 , x 0 ) K(t,x)=p(t,x;t_0,x_0) K ( t , x ) = p ( t , x ; t 0 , x 0 ) (200) from Proposition 4.22, we have that
∥ ( t − t 0 2 ) k / 2 K 1 / 2 D ( k ) ψ ∥ L t ∞ L x 2 ( I ) ≲ Gr [ t 0 / 2 , t 0 ] ( C F ) ( ( 1 + ∥ F ∥ t 0 , k p e r t ) k ∥ ( K 1 / 2 ψ ) ( t 0 2 ) ∥ L x 2 + ∥ F ∥ t 0 , k p e r t ( 1 + ∥ F ∥ t 0 , k p e r t ) k − 1 ∥ ( K 1 / 2 p B ) ( t 0 2 ) ∥ L x 2 ) \left\lVert(t-\tfrac{t_0}{2})^{k/2}K^{1/2}D^{(k)}\psi\right\rVert_{L^\infty_tL^2_x(I)}
\lesssim\operatorname{Gr}_{[t_0/2,t_0]}(CF)\left((1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^k\lVert(K^{1/2}\psi)(\tfrac{t_0}{2})\rVert_{L^2_x}+\lVert F\rVert_{t_0,k}^{\mathrm{pert}}(1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^{k-1}\lVert(K^{1/2}p_B)(\tfrac{t_0}{2})\rVert_{L^2_x}\right) ( t − 2 t 0 ) k /2 K 1/2 D ( k ) ψ L t ∞ L x 2 ( I ) ≲ Gr [ t 0 /2 , t 0 ] ( C F ) ( ( 1 + ∥ F ∥ t 0 , k pert ) k ∥( K 1/2 ψ ) ( 2 t 0 ) ∥ L x 2 + ∥ F ∥ t 0 , k pert ( 1 + ∥ F ∥ t 0 , k pert ) k − 1 ∥( K 1/2 p B ) ( 2 t 0 ) ∥ L x 2 ) ≲ Gr t 0 ( C F ) ( ( 1 + ∥ F ∥ t 0 , k p e r t ) k ∥ F ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + ∥ F ∥ t 0 , k p e r t ( 1 + ∥ F ∥ t 0 , k p e r t ) k − 1 ) ∥ K 1 / 2 p ( t 0 2 ) ∥ L x 2 , \lesssim\operatorname{Gr}_{t_0}(CF)\left((1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^k\lVert F\rVert_{L^1_tL^\infty_x([0,t_0])}+\lVert F\rVert_{t_0,k}^{\mathrm{pert}}(1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^{k-1}\right)\lVert K^{1/2}p(\tfrac{t_0}{2})\rVert_{L^2_x}, ≲ Gr t 0 ( C F ) ( ( 1 + ∥ F ∥ t 0 , k pert ) k ∥ F ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + ∥ F ∥ t 0 , k pert ( 1 + ∥ F ∥ t 0 , k pert ) k − 1 ) ∥ K 1/2 p ( 2 t 0 ) ∥ L x 2 , where we used the zeroth order estimate (203) as well as the pointwise estimate (194) . To obtain the desired inequality, we split into cases where ∥ F ∥ t 0 , k p e r t > 1 \lVert F\rVert_{t_0,k}^{\mathrm{pert}}>1 ∥ F ∥ t 0 , k pert > 1 ∥ F ∥ t 0 , k p e r t ≤ 1 \lVert F\rVert_{t_0,k}^{\mathrm{pert}}\leq1 ∥ F ∥ t 0 , k pert ≤ 1 ∥ F ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ≤ Gr t 0 ( F ) \lVert F\rVert_{L^1_tL^\infty_x([0,t_0])}\leq\operatorname{Gr}_{t_0}(F) ∥ F ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ≤ Gr t 0 ( F ) x ≤ e x x\leq e^x x ≤ e x Gr t 0 ( F ) \operatorname{Gr}_{t_0}(F) Gr t 0 ( F )
Gr t 0 ( ( C + 1 ) F ) ∥ F ∥ t 0 , k p e r t ( 1 + ∥ F ∥ t 0 , k p e r t ) k − 1 ∥ K 1 / 2 p ( t 0 2 ) ∥ L x 2 , \operatorname{Gr}_{t_0}((C+1)F)\lVert F\rVert_{t_0,k}^{\mathrm{pert}}(1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^{k-1}\lVert K^{1/2}p(\tfrac{t_0}{2})\rVert_{L^2_x}, Gr t 0 (( C + 1 ) F ) ∥ F ∥ t 0 , k pert ( 1 + ∥ F ∥ t 0 , k pert ) k − 1 ∥ K 1/2 p ( 2 t 0 ) ∥ L x 2 , while in the latter case, we may directly obtain the estimate
Gr t 0 ( C F ) ∥ F ∥ t 0 , k p e r t ( 1 + ∥ F ∥ t 0 , k p e r t ) k − 1 ∥ K 1 / 2 p ( t 0 2 ) ∥ L x 2 . \operatorname{Gr}_{t_0}(CF)\lVert F\rVert_{t_0,k}^{\mathrm{pert}}(1+\lVert F\rVert_{t_0,k}^{\mathrm{pert}})^{k-1}\lVert K^{1/2}p(\tfrac{t_0}{2})\rVert_{L^2_x}. Gr t 0 ( C F ) ∥ F ∥ t 0 , k pert ( 1 + ∥ F ∥ t 0 , k pert ) k − 1 ∥ K 1/2 p ( 2 t 0 ) ∥ L x 2 . Having reached this point, the remainder of the argument is similar to the proof of Proposition 4.17; we omit the details. We conclude by noting that the difference estimate (197) combined with our higher order estimates (195) on p B p_B p B k ≥ 1 k\geq1 k ≥ 1 (196) .
Remark 4.23. From the proof of Proposition 4.18, we see that the constant C 0 C_0 C 0 C 0 C_0 C 0
The following result is a direct corollary of Propositions 4.19 and 4.22.
Corollary 4.24. Let C 0 C_0 C 0 ϕ , G : [ 0 , T ] × T d → W d \phi,G:[0,T]\times\mathbb{T}^d\to W^d ϕ , G : [ 0 , T ] × T d → W d
( ∂ t − Δ B − 2 F ) ϕ = G , ϕ ( 0 ) = ϕ 0 . (\partial_t-\Delta_B-2F)\phi=G,\qquad\phi(0)=\phi_0. ( ∂ t − Δ B − 2 F ) ϕ = G , ϕ ( 0 ) = ϕ 0 . Let K : [ 0 , T ] × T d → ( 0 , ∞ ) K:[0,T]\times\mathbb{T}^d\to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ ) k ≥ 0 k\geq0 k ≥ 0
∥ t k / 2 K 1 / 2 D ( k ) ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t k / 2 K 1 / 2 D ( k + 1 ) ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k p e r t ) k ( ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 + ∑ j = 0 k ∥ t j / 2 K 1 / 2 D ( j ) G ∥ L t 1 L x 2 ( [ 0 , T ] ) ) \lVert t^{k/2}K^{1/2}D^{(k)}\phi\rVert_{L^\infty_tL^2_x([0,T])}+\lVert t^{k/2}K^{1/2}D^{(k+1)}\phi\rVert_{L^2_{t,x}([0,T])}
\lesssim\operatorname{Gr}_T(C_0F)\left(1+\lVert F\rVert_{T,k}^{\mathrm{pert}}\right)^k\left(\lVert K^{1/2}(0)\phi_0\rVert_{L^2_x}+\sum_{j=0}^k\lVert t^{j/2}K^{1/2}D^{(j)}G\rVert_{L^1_tL^2_x([0,T])}\right) ∥ t k /2 K 1/2 D ( k ) ϕ ∥ L t ∞ L x 2 ([ 0 , T ]) + ∥ t k /2 K 1/2 D ( k + 1 ) ϕ ∥ L t , x 2 ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k pert ) k ( ∥ K 1/2 ( 0 ) ϕ 0 ∥ L x 2 + j = 0 ∑ k ∥ t j /2 K 1/2 D ( j ) G ∥ L t 1 L x 2 ([ 0 , T ]) ) Proof. We split ϕ = ϕ ~ + ψ \phi= \widetilde{\phi} + \psi ϕ = ϕ + ψ ( ∂ t − Δ B ) ϕ ~ = G (\partial_t - \Delta_B)\widetilde{\phi} = G ( ∂ t − Δ B ) ϕ = G ϕ ~ ( 0 ) = ϕ 0 \widetilde{\phi}(0) = \phi_0 ϕ ( 0 ) = ϕ 0 ϕ ~ \widetilde{\phi} ϕ Proposition 4.19 , and ψ \psi ψ Proposition 4.22 . We remark that in the case k = 0 k = 0 k = 0
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ K 1 / 2 D ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ Gr T ( C F ) ( 1 + ∥ F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ) ( ∥ K ( 0 ) 1 / 2 ϕ 0 ∥ L 2 + ∥ K 1 / 2 G ∥ L t 1 L x 2 ( [ 0 , T ] ) ) . \left\|K^{1/2}\phi\right\|_{L^\infty_tL^2_x([0,T])}+\left\|K^{1/2}D\phi\right\|_{L^2_{t,x}([0,T])}
\lesssim\operatorname{Gr}_T(CF)\left(1+\|F\|_{L^1_tL^\infty_x([0,T])}\right)\left(\left\|K^{1/2}_{(0)}\phi_0\right\|_{L^2}+\left\|K^{1/2}G\right\|_{L^1_tL^2_x([0,T])}\right). K 1/2 ϕ L t ∞ L x 2 ([ 0 , T ]) + K 1/2 D ϕ L t , x 2 ([ 0 , T ]) ≲ Gr T ( C F ) ( 1 + ∥ F ∥ L t 1 L x ∞ ([ 0 , T ]) ) ( K ( 0 ) 1/2 ϕ 0 L 2 + K 1/2 G L t 1 L x 2 ([ 0 , T ]) ) . where C C C Proposition 4.22 . In this case, we estimate ∥ F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ≤ Gr T ( F ) \|F\|_{L^1_tL^\infty_x([0,T])}\leq\operatorname{Gr}_T(F) ∥ F ∥ L t 1 L x ∞ ([ 0 , T ]) ≤ Gr T ( F ) x ≤ e x x\leq e^x x ≤ e x Gr T ( ( C + 1 ) F ) \operatorname{Gr}_T((C+1)F) Gr T (( C + 1 ) F ) □ \square □
We also record the following corollary of Lemma 4.11 and Proposition 4.22 .
Corollary 4.25. Let C 0 C_0 C 0 Proposition 4.22 . Let α ∈ [ 0 , 1 ) \alpha\in[0,1) α ∈ [ 0 , 1 ) ϕ , ψ : [ 0 , T ] × T d → W d \phi,\psi\colon[0,T]\times\mathbb{T}^d\to\mathbb{W}^d ϕ , ψ : [ 0 , T ] × T d → W d ϕ ( 0 ) = ψ ( 0 ) \phi(0)=\psi(0) ϕ ( 0 ) = ψ ( 0 )
( ∂ t − Δ B − 2 F ) ψ = 0 , (\partial_t-\Delta_B-2F)\psi=0, ( ∂ t − Δ B − 2 F ) ψ = 0 , ( ∂ t − Δ B ) ϕ = 0. (\partial_t-\Delta_B)\phi=0. ( ∂ t − Δ B ) ϕ = 0. Then for any k ≥ 0 k\geq0 k ≥ 0
∥ t ( k − ( 1 − α ) ) / 2 D ( k ) ( ψ − ϕ ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ∥ t − ( 1 − α ) / 2 F ∥ T , k p e r t ( 1 + ( ∥ F ∥ T , k p e r t ) ( k − 1 ) ∨ 0 ) ∥ ϕ ( 0 ) ∥ L x ∞ . (204) \left\|t^{(k-(1-\alpha))/2}D^{(k)}(\psi-\phi)\right\|_{L^\infty_{t,x}([0,T])}
\lesssim\operatorname{Gr}_T(C_0F)\left\|t^{-(1-\alpha)/2}F\right\|_{T,k}^{\mathrm{pert}}\left(1+\left(\|F\|_{T,k}^{\mathrm{pert}}\right)^{(k-1)\vee0}\right)\|\phi(0)\|_{L^\infty_x}. \tag*{(204)} t ( k − ( 1 − α )) /2 D ( k ) ( ψ − ϕ ) L t , x ∞ ([ 0 , T ]) ≲ Gr T ( C 0 F ) t − ( 1 − α ) /2 F T , k pert ( 1 + ( ∥ F ∥ T , k pert ) ( k − 1 ) ∨ 0 ) ∥ ϕ ( 0 ) ∥ L x ∞ . (204) Proof. Let t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] Lemma 4.11 with Proposition 4.22 (with the ψ \psi ψ ψ − ϕ \psi-\phi ψ − ϕ G G G 0 0 0
∥ t 0 k / 2 D ( k ) ( ψ − ϕ ) ( t 0 ) ∥ L x ∞ ≲ Gr T ( C 0 F ) ∥ F ∥ t 0 , k p e r t ( 1 + ( ∥ F ∥ T , k p e r t ) ( k − 1 ) ∨ 0 ) ∥ ϕ ( 0 ) ∥ L x ∞ . \left\|t_0^{k/2}D^{(k)}(\psi-\phi)(t_0)\right\|_{L^\infty_x}
\lesssim\operatorname{Gr}_T(C_0F)\|F\|_{t_0,k}^{\mathrm{pert}}\left(1+\left(\|F\|_{T,k}^{\mathrm{pert}}\right)^{(k-1)\vee0}\right)\|\phi(0)\|_{L^\infty_x}. t 0 k /2 D ( k ) ( ψ − ϕ ) ( t 0 ) L x ∞ ≲ Gr T ( C 0 F ) ∥ F ∥ t 0 , k pert ( 1 + ( ∥ F ∥ T , k pert ) ( k − 1 ) ∨ 0 ) ∥ ϕ ( 0 ) ∥ L x ∞ . Next, we insert the following estimate into the above:
∥ F ∥ t 0 , k p e r t ≤ t 0 ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ t 0 , k p e r t ≤ t 0 ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ T , k p e r t , \|F\|_{t_0,k}^{\mathrm{pert}}\leq t_0^{(1-\alpha)/2}\left\|t^{-(1-\alpha)/2}F\right\|_{t_0,k}^{\mathrm{pert}}\leq t_0^{(1-\alpha)/2}\left\|t^{-(1-\alpha)/2}F\right\|_{T,k}^{\mathrm{pert}}, ∥ F ∥ t 0 , k pert ≤ t 0 ( 1 − α ) /2 t − ( 1 − α ) /2 F t 0 , k pert ≤ t 0 ( 1 − α ) /2 t − ( 1 − α ) /2 F T , k pert , and then we finish by taking a sup over t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ] □ \square □
Heat kernel estimates with mixed derivatives
In this section, we extend the covariant heat kernel estimates of Propositions 4.17 and 4.18 to the case of mixed derivatives, i.e. the case where we differentiate in both spatial variables. Such an expression arises naturally when derivatives fall on the forcing term of a covariant heat equation (see (206) –(207) ). We first introduce the following convenient notation.
Notation 4.26. We introduce the covariant derivatives D y † D_y^\dagger D y † Ψ : [ 0 , T ] × T d → End ( W ) \Psi\colon[0,T]\times\mathbb{T}^d\to\operatorname{End}(\mathbb{W}) Ψ : [ 0 , T ] × T d → End ( W ) j ∈ [ d ] j\in[d] j ∈ [ d ]
D y † , j Ψ ( s , y ) ≔ − ∂ y j Ψ ( s , y ) + Ψ ( s , y ) B j ( s , y ) . (205) D_y^{\dagger,j}\Psi(s,y)\coloneqq-\partial_y^j\Psi(s,y)+\Psi(s,y)B^j(s,y). \tag*{(205)} D y † , j Ψ ( s , y ) : = − ∂ y j Ψ ( s , y ) + Ψ ( s , y ) B j ( s , y ) . (205) Here, B j ( s , y ) B^j(s,y) B j ( s , y ) End ( W ) \operatorname{End}(\mathbb{W}) End ( W ) π ∗ \pi_* π ∗ g \mathfrak{g} g End ( W d ) \operatorname{End}(\mathbb{W}^d) End ( W d ) Ψ \Psi Ψ D y † , ( k ) D_y^{\dagger,(k)} D y † , ( k ) k k k D y † , j 1 ⋯ D y † , j k D_y^{\dagger,j_1}\cdots D_y^{\dagger,j_k} D y † , j 1 ⋯ D y † , j k j 1 , … , j k ∈ [ d ] j_1,\ldots,j_k\in[d] j 1 , … , j k ∈ [ d ]
The reason for introducing the symbol † \dagger † D y † D_y^\dagger D y † End ( W ) \operatorname{End}(\mathbb{W}) End ( W )
We remark that D y † \mathrm{D}^{\dagger}_{y} D y † D x \mathrm{D}_{x} D x ∂ t \partial_t ∂ t End ( W ) \operatorname{End}(\mathbf{W}) End ( W ) End ( W d ) \operatorname{End}(\mathbf{W}^{d}) End ( W d ) (205) , it directly follows for all m , k ≥ 0 m,k \ge0 m , k ≥ 0 0 ≤ s < t 0 \le s < t 0 ≤ s < t G : T d → W d G:\mathbb{T}^{d} \to\mathbf{W}^{d} G : T d → W d H : T d → W H:\mathbb{T}^{d} \to\mathbf{W} H : T d → W
∫ d y D x ( k ) q B ( s , y ; t , x ) ( D y ( m ) G ) ( y ) = ∫ d y ( D y † , ( m ) D x ( k ) q B ) ( s , y ; t , x ) G ( y ) , (206) \int\mathrm{d}y\,\mathrm{D}_{x}^{(k)}q_{B}(s,y;t,x)(\mathrm{D}_{y}^{(m)}G)(y)=\int\mathrm{d}y\,(\mathrm{D}_{y}^{\dagger,(m)}\mathrm{D}_{x}^{(k)}q_{B})(s,y;t,x)G(y), \tag*{(206)} ∫ d y D x ( k ) q B ( s , y ; t , x ) ( D y ( m ) G ) ( y ) = ∫ d y ( D y † , ( m ) D x ( k ) q B ) ( s , y ; t , x ) G ( y ) , (206) ∫ d y D x ( k ) p B ( s , y ; t , x ) ( D y ( m ) H ) ( y ) = ∫ d y ( D y † , ( m ) D x ( k ) p B ) ( s , y ; t , x ) H ( y ) . (207) \int\mathrm{d}y\,\mathrm{D}_{x}^{(k)}p_{B}(s,y;t,x)(\mathrm{D}_{y}^{(m)}H)(y)=\int\mathrm{d}y\,(\mathrm{D}_{y}^{\dagger,(m)}\mathrm{D}_{x}^{(k)}p_{B})(s,y;t,x)H(y). \tag*{(207)} ∫ d y D x ( k ) p B ( s , y ; t , x ) ( D y ( m ) H ) ( y ) = ∫ d y ( D y † , ( m ) D x ( k ) p B ) ( s , y ; t , x ) H ( y ) . (207) The main result of this section provides estimates for the kernels appearing in the right-hand sides of (206) –(207) . We state it as follows.
Proposition 4.27. Let C 0 C_{0} C 0 m , k ≥ 0 m,k \ge0 m , k ≥ 0 0 ≤ s 0 < t 0 0 \le s_{0}<t_{0} 0 ≤ s 0 < t 0 t 0 − s 0 ≤ 1 t_{0}-s_{0}\le1 t 0 − s 0 ≤ 1 x 0 , y 0 ∈ T d x_{0},y_{0}\in\mathbb{T}^{d} x 0 , y 0 ∈ T d
∣ D y † , ( m ) D x ( k ) p B ( s 0 , y 0 ; t 0 , x 0 ) ∣ ≲ ( 1 + ( ∥ F ∥ t 0 , m p e r t ) m ) ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k p e r t ) k ) × min ( s 0 , t 0 − s 0 ) − m 2 ( t 0 − s 0 ) − k + d 2 exp ( − c ∣ x 0 − y 0 ∣ 2 t 0 − s 0 ) . \left|\mathrm{D}_{y}^{\dagger,(m)}\mathrm{D}_{x}^{(k)}p_{B}(s_{0},y_{0};t_{0},x_{0})\right|\lesssim\left(1+\left(\|F\|_{t_{0},m}^{\mathrm{pert}}\right)^{m}\right)\left(1+\left(\|F\|_{[s_{0},t_{0}],k}^{\mathrm{pert}}\right)^{k}\right)\times\min(s_{0},t_{0}-s_{0})^{-\frac{m}{2}}(t_{0}-s_{0})^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_{0}-y_{0}|^{2}}{t_{0}-s_{0}}\right). D y † , ( m ) D x ( k ) p B ( s 0 , y 0 ; t 0 , x 0 ) ≲ ( 1 + ( ∥ F ∥ t 0 , m pert ) m ) ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k pert ) k ) × min ( s 0 , t 0 − s 0 ) − 2 m ( t 0 − s 0 ) − 2 k + d exp ( − c t 0 − s 0 ∣ x 0 − y 0 ∣ 2 ) . ∣ D y † , ( m ) D x ( k ) q B ( s 0 , y 0 ; t 0 , x 0 ) ∣ ≲ Gr [ s 0 , t 0 ] ( C 0 F ) ( 1 + ( ∥ F ∥ t 0 , m p e r t ) m ) ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k p e r t ) k ) × min ( s 0 , t 0 − s 0 ) − m 2 ( t 0 − s 0 ) − k + d 2 exp ( − c ∣ x 0 − y 0 ∣ 2 t 0 − s 0 ) . \left|\mathrm{D}_{y}^{\dagger,(m)}\mathrm{D}_{x}^{(k)}q_{B}(s_{0},y_{0};t_{0},x_{0})\right|\lesssim\operatorname{Gr}_{[s_{0},t_{0}]}(C_{0}F)\left(1+\left(\|F\|_{t_{0},m}^{\mathrm{pert}}\right)^{m}\right)\left(1+\left(\|F\|_{[s_{0},t_{0}],k}^{\mathrm{pert}}\right)^{k}\right)\times\min(s_{0},t_{0}-s_{0})^{-\frac{m}{2}}(t_{0}-s_{0})^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_{0}-y_{0}|^{2}}{t_{0}-s_{0}}\right). D y † , ( m ) D x ( k ) q B ( s 0 , y 0 ; t 0 , x 0 ) ≲ Gr [ s 0 , t 0 ] ( C 0 F ) ( 1 + ( ∥ F ∥ t 0 , m pert ) m ) ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k pert ) k ) × min ( s 0 , t 0 − s 0 ) − 2 m ( t 0 − s 0 ) − 2 k + d exp ( − c t 0 − s 0 ∣ x 0 − y 0 ∣ 2 ) . In order to prove this proposition, we will need several intermediate results, which we discuss next. The starting point is to derive an evolution equation for p B , q B p_{B},q_{B} p B , q B s , y s,y s , y t , x t,x t , x p B ∗ , q B ∗ p_{B}^{*},q_{B}^{*} p B ∗ , q B ∗ p B , q B p_{B},q_{B} p B , q B W , W d \mathbf{W},\mathbf{W}^{d} W , W d p B ∗ , q B ∗ p_{B}^{*},q_{B}^{*} p B ∗ , q B ∗ backward covariant heat equations.
Lemma 4.28 (Adjoints and backward heat equations). For all s 0 < t 0 s_{0}<t_{0} s 0 < t 0 x 0 , y 0 ∈ T d x_{0},y_{0}\in\mathbb{T}^{d} x 0 , y 0 ∈ T d
( ∂ s + Δ B , y ) p B ∗ ( s 0 , y 0 ; t 0 , x 0 ) = 0 , (\partial_{s}+\Delta_{B,y})p_{B}^{*}(s_{0},y_{0};t_{0},x_{0})=0, ( ∂ s + Δ B , y ) p B ∗ ( s 0 , y 0 ; t 0 , x 0 ) = 0 , ( ∂ s + Δ B , y + 2 F ) q B ∗ ( s 0 , y 0 ; t 0 , x 0 ) = 0. (\partial_{s}+\Delta_{B,y}+2F)q_{B}^{*}(s_{0},y_{0};t_{0},x_{0})=0. ( ∂ s + Δ B , y + 2 F ) q B ∗ ( s 0 , y 0 ; t 0 , x 0 ) = 0. Here, the subscript y y y Δ B , y \Delta_{B,y} Δ B , y y y y
Proof. By standard considerations, given ψ t 0 : T d → W d \psi_{t_{0}}:\mathbb{T}^{d}\to\mathbf{W}^{d} ψ t 0 : T d → W d
ψ ( s , y ) : = ∫ d x q B ∗ ( s , y ; t 0 , x ) ψ t 0 ( x ) \psi(s,y)\mathrel{\mathop:}=\int\mathrm{d}x\,q_{B}^{*}(s,y;t_{0},x)\psi_{t_{0}}(x) ψ ( s , y ) : = ∫ d x q B ∗ ( s , y ; t 0 , x ) ψ t 0 ( x ) solves the adjoint equation
( ∂ s + L s ∗ ) ψ = 0 , (\partial_{s}+\mathcal{L}_{s}^{*})\psi=0, ( ∂ s + L s ∗ ) ψ = 0 , where L s = Δ B + 2 F \mathcal{L}_{s}=\Delta_{B}+2F L s = Δ B + 2 F Δ B \Delta_{B} Δ B F F F W d \mathbf{W}^{d} W d L 2 L^{2} L 2 T d → W d \mathbb{T}^{d}\to\mathbf{W}^{d} T d → W d X , Y ∈ W d X,Y\in\mathbf{W}^{d} X , Y ∈ W d
⟨ X , F Y ⟩ W d = ⟨ X i , F i j Y j ⟩ W = − ⟨ F i j X i , Y j ⟩ W = ⟨ F j i X i , Y j ⟩ W = ⟨ F X , Y ⟩ W d . \langle X,FY\rangle_{\mathbf{W}^{d}}=\langle X^{i},F_{ij}Y^{j}\rangle_{\mathbf{W}}=-\langle F_{ij}X^{i},Y^{j}\rangle_{\mathbf{W}}=\langle F_{ji}X^{i},Y^{j}\rangle_{\mathbf{W}}=\langle FX,Y\rangle_{\mathbf{W}^{d}}. ⟨ X , F Y ⟩ W d = ⟨ X i , F ij Y j ⟩ W = − ⟨ F ij X i , Y j ⟩ W = ⟨ F j i X i , Y j ⟩ W = ⟨ F X , Y ⟩ W d . Thus L s ∗ = L s \mathcal{L}_{s}^{*}=\mathcal{L}_{s} L s ∗ = L s q B ∗ q_{B}^{*} q B ∗ p B ∗ p_{B}^{*} p B ∗ F F F
Next, we relate the kernels estimated in Proposition 4.27 to the adjoints p B ∗ , q B ∗ p_B^*,q_B^* p B ∗ , q B ∗
Lemma 4.29. For any m , k ≥ 0 m,k\ge0 m , k ≥ 0 0 ≤ s < t 0\le s<t 0 ≤ s < t x , y ∈ T d x,y\in\mathbb{T}^d x , y ∈ T d
( D y † , ( m ) D x ( k ) p B ( s , y ; t , x ) ) ∗ = ( − 1 ) m D y ( m ) ( D x ( k ) p B ( s , y ; t , x ) ) ∗ , \left(D_y^{\dagger,(m)}D_x^{(k)}p_B(s,y;t,x)\right)^*=(-1)^mD_y^{(m)}\left(D_x^{(k)}p_B(s,y;t,x)\right)^*, ( D y † , ( m ) D x ( k ) p B ( s , y ; t , x ) ) ∗ = ( − 1 ) m D y ( m ) ( D x ( k ) p B ( s , y ; t , x ) ) ∗ , ( D y † , ( m ) D x ( k ) q B ( s , y ; t , x ) ) ∗ = ( − 1 ) m D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ . \left(D_y^{\dagger,(m)}D_x^{(k)}q_B(s,y;t,x)\right)^*=(-1)^mD_y^{(m)}\left(D_x^{(k)}q_B(s,y;t,x)\right)^*. ( D y † , ( m ) D x ( k ) q B ( s , y ; t , x ) ) ∗ = ( − 1 ) m D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ . Proof. We show the identity for q B q_B q B p B p_B p B G , H : T d → W d G,H:\mathbb{T}^d\to W^d G , H : T d → W d
∫ d x ∫ d y ⟨ H ( x ) , D x ( k ) q B ( s , y ; t , x ) D y ( m ) G ( y ) ⟩ W d = ( − 1 ) m ∫ d x ∫ d y ⟨ D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ H ( x ) , G ( y ) ⟩ W d = ( − 1 ) m ∫ d x ∫ d y ⟨ H ( x ) , ( D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ ) ∗ G ( y ) ⟩ W d . \begin{aligned}
\int\mathrm{d}x\int\mathrm{d}y\,\langle H(x),D_x^{(k)}q_B(s,y;t,x)D_y^{(m)}G(y)\rangle_{W^d} \\
&=(-1)^m\int\mathrm{d}x\int\mathrm{d}y\,\langle D_y^{(m)}(D_x^{(k)}q_B(s,y;t,x))^*H(x),G(y)\rangle_{W^d} \\
&=(-1)^m\int\mathrm{d}x\int\mathrm{d}y\,\left\langle H(x),\left(D_y^{(m)}(D_x^{(k)}q_B(s,y;t,x))^*\right)^*G(y)\right\rangle_{W^d}.
\end{aligned} ∫ d x ∫ d y ⟨ H ( x ) , D x ( k ) q B ( s , y ; t , x ) D y ( m ) G ( y ) ⟩ W d = ( − 1 ) m ∫ d x ∫ d y ⟨ D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ H ( x ) , G ( y ) ⟩ W d = ( − 1 ) m ∫ d x ∫ d y ⟨ H ( x ) , ( D y ( m ) ( D x ( k ) q B ( s , y ; t , x ) ) ∗ ) ∗ G ( y ) ⟩ W d . Since this is true for arbitrary H H H (206) . □
In summary, so far we have expressed the kernels in Proposition 4.27 in terms of solutions to backward covariant heat equations. We proceed to estimate such solutions. Since the time reversal of a solution to the backward covariant heat equation is a solution to the forward covariant heat equation, a natural approach would be to reduce to the estimates of § 5.1 . However, one needs to be careful here, because the norm ∥ ⋅ ∥ p e r t \|\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert F F F Lemma 3.28 ). However, after time reversal, the curvature will now possibly be singular at the final time, not the initial time. We therefore need to define a new norm which reflects this behaviour.
Definition 4.30 (Norms for time reversals). Let [ s 0 , t 0 ] [s_0,t_0] [ s 0 , t 0 ] t 1 > t 0 t_1>t_0 t 1 > t 0 k ≥ 0 k\ge0 k ≥ 0
∥ F ∥ [ s 0 , t 0 ] , t 1 , k t r e p ≔ ∑ j = 0 k ( ∥ ( t 1 − t ) ( j + 1 ) / 2 D ( j ) F ∥ L t 2 L x ∞ ( [ s 0 , t 0 ] ) + ∥ ( t 1 − t ) j / 2 D ( j ) F ∥ L t 1 L x ∞ ( [ s 0 , t 0 ] ) ) . \|F\|^{\mathrm{trep}}_{[s_0,t_0],t_1,k}\coloneqq\sum_{j=0}^{k}\left(\|(t_1-t)^{(j+1)/2}D^{(j)}F\|_{L_t^2L_x^\infty([s_0,t_0])}+\|(t_1-t)^{j/2}D^{(j)}F\|_{L_t^1L_x^\infty([s_0,t_0])}\right). ∥ F ∥ [ s 0 , t 0 ] , t 1 , k trep : = j = 0 ∑ k ( ∥ ( t 1 − t ) ( j + 1 ) /2 D ( j ) F ∥ L t 2 L x ∞ ([ s 0 , t 0 ]) + ∥ ( t 1 − t ) j /2 D ( j ) F ∥ L t 1 L x ∞ ([ s 0 , t 0 ]) ) . This norm will be needed when dealing with time reversals, hence the superscript “trep”, which is the reversal of “pert”.
Next, we state the main intermediate result towards the proof of Proposition 4.27 , which should be considered the analogue of Propositions 4.19 and 4.22 , except now we use the norm ∥ ⋅ ∥ t r e p \|\cdot\|^{\mathrm{trep}} ∥ ⋅ ∥ trep ∥ ⋅ ∥ p e r t \|\cdot\|^{\mathrm{pert}} ∥ ⋅ ∥ pert
Lemma 4.31. Let C 0 C_0 C 0 Lemma 4.9 . Let B ~ : [ 0 , T ] × T d → g d \widetilde{B}:[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d B : [ 0 , T ] × T d → g d
∂ t B ~ i = − D ~ ℓ F ~ ℓ i , i ∈ [ d ] , \partial_t\widetilde{B}_i=-\widetilde{D}^{\ell}\widetilde{F}_{\ell i},\qquad i\in[d], ∂ t B i = − D ℓ F ℓ i , i ∈ [ d ] , where D ~ = D B ~ \widetilde{D}=D_{\widetilde{B}} D = D B F ~ = F B ~ \widetilde{F}=F_{\widetilde{B}} F = F B ϕ : [ 0 , T ] × T d → W d \phi:[0,T]\times\mathbb{T}^d\to W^d ϕ : [ 0 , T ] × T d → W d ϕ 0 : T d → W d \phi_0:\mathbb{T}^d\to W^d ϕ 0 : T d → W d
( ∂ t − Δ B ~ − 2 F ~ ) ϕ = 0 , ϕ ( 0 ) = ϕ 0 , (\partial_t-\Delta_{\widetilde{B}}-2\widetilde{F})\phi=0,\qquad\phi(0)=\phi_0, ( ∂ t − Δ B − 2 F ) ϕ = 0 , ϕ ( 0 ) = ϕ 0 , and let K : [ 0 , T ] × T d → ( 0 , ∞ ) K : [0,T] \times\mathbb{T}^d \to(0,\infty) K : [ 0 , T ] × T d → ( 0 , ∞ ) T 1 > T T_1 > T T 1 > T k ≥ 0 k \ge0 k ≥ 0
∥ t k / 2 ( T 1 − t ) k / 2 K 1 / 2 D ~ ( k ) ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t k / 2 ( T 1 − t ) k / 2 K 1 / 2 D ~ ( k + 1 ) ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ G T ( C 0 F ~ ) T 1 k / 2 ( 1 + ( ∥ F ~ ∥ [ 0 , T ] , T 1 , k t r e p ) k ) ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 . \left\|t^{k/2}(T_1-t)^{k/2}K^{1/2}\widetilde{\mathcal{D}}^{(k)}\phi\right\|_{L^\infty_tL^2_x([0,T])}+\left\|t^{k/2}(T_1-t)^{k/2}K^{1/2}\widetilde{\mathcal{D}}^{(k+1)}\phi\right\|_{L^2_{t,x}([0,T])}
\lesssim G_{\mathsf{T}}(C_0\widetilde{F})T_1^{k/2}\left(1+\left(\|\widetilde{F}\|^{\mathrm{trep}}_{[0,T],T_1,k}\right)^k\right)\left\|K^{1/2}(0)\phi_0\right\|_{L^2_x}. t k /2 ( T 1 − t ) k /2 K 1/2 D ( k ) ϕ L t ∞ L x 2 ([ 0 , T ]) + t k /2 ( T 1 − t ) k /2 K 1/2 D ( k + 1 ) ϕ L t , x 2 ([ 0 , T ]) ≲ G T ( C 0 F ) T 1 k /2 ( 1 + ( ∥ F ∥ [ 0 , T ] , T 1 , k trep ) k ) K 1/2 ( 0 ) ϕ 0 L x 2 . Similarly, if instead ϕ : [ 0 , T ] × T d → W \phi: [0,T] \times\mathbb{T}^d \to W ϕ : [ 0 , T ] × T d → W ϕ 0 : T d → W \phi_0 : \mathbb{T}^d \to W ϕ 0 : T d → W
( ∂ t − Δ B ~ ) ϕ = 0 , ϕ ( 0 ) = ϕ 0 , (\partial_t-\Delta_{\widetilde{B}})\phi=0,\qquad\phi(0)=\phi_0, ( ∂ t − Δ B ) ϕ = 0 , ϕ ( 0 ) = ϕ 0 , then for any K , T 1 , k K,T_1,k K , T 1 , k
∥ t k / 2 ( T 1 − t ) k / 2 K 1 / 2 D ~ ( k ) ϕ ∥ L t ∞ L x 2 ( [ 0 , T ] ) + ∥ t k / 2 ( T 1 − t ) k / 2 K 1 / 2 D ~ ( k + 1 ) ϕ ∥ L t , x 2 ( [ 0 , T ] ) ≲ T 1 k / 2 ( 1 + ( ∥ F ~ ∥ [ 0 , T ] , T 1 , k t r e p ) k ) ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 . \left\|t^{k/2}(T_1-t)^{k/2}K^{1/2}\widetilde{\mathcal{D}}^{(k)}\phi\right\|_{L^\infty_tL^2_x([0,T])}+\left\|t^{k/2}(T_1-t)^{k/2}K^{1/2}\widetilde{\mathcal{D}}^{(k+1)}\phi\right\|_{L^2_{t,x}([0,T])}
\lesssim T_1^{k/2}\left(1+\left(\|\widetilde{F}\|^{\mathrm{trep}}_{[0,T],T_1,k}\right)^k\right)\left\|K^{1/2}(0)\phi_0\right\|_{L^2_x}. t k /2 ( T 1 − t ) k /2 K 1/2 D ( k ) ϕ L t ∞ L x 2 ([ 0 , T ]) + t k /2 ( T 1 − t ) k /2 K 1/2 D ( k + 1 ) ϕ L t , x 2 ([ 0 , T ]) ≲ T 1 k /2 ( 1 + ( ∥ F ∥ [ 0 , T ] , T 1 , k trep ) k ) K 1/2 ( 0 ) ϕ 0 L x 2 . Proof. We focus on the first estimate, as the proof of the second is quite similar. In the following, all space-time norms are implicitly on [ 0 , T ] × T d [0,T]\times\mathbb{T}^d [ 0 , T ] × T d k = 0 k=0 k = 0 G , H G,H G , H k 0 ≥ 0 k_0\ge0 k 0 ≥ 0 0 ≤ k ≤ k 0 0\le k\le k_0 0 ≤ k ≤ k 0 k 0 + 1 k_0+1 k 0 + 1
( ∂ t − Δ B ~ ) D ~ ( k 0 + 1 ) ϕ = O ( F ~ D ~ ( k 0 + 1 ) ϕ ) + ∑ j 1 + j 2 = k 0 + 1 j 1 ≥ 1 j 2 ≥ 0 O ( ( D ~ ( j 1 ) F ~ ) ( D ~ ( j 2 ) ϕ ) ) . (\partial_t-\Delta_{\widetilde{B}})\widetilde{\mathcal{D}}^{(k_0+1)}\phi
=\mathcal{O}\left(\widetilde{F}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right)
+\sum_{\substack{j_1+j_2=k_0+1\\j_1\ge1\\j_2\ge0}}
\mathcal{O}\left((\widetilde{\mathcal{D}}^{(j_1)}\widetilde{F})(\widetilde{\mathcal{D}}^{(j_2)}\phi)\right). ( ∂ t − Δ B ) D ( k 0 + 1 ) ϕ = O ( F D ( k 0 + 1 ) ϕ ) + j 1 + j 2 = k 0 + 1 j 1 ≥ 1 j 2 ≥ 0 ∑ O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) . For notational convenience, let w k ( t ) = t k / 2 ( T 1 − t ) k / 2 w_k(t)=t^{k/2}(T_1-t)^{k/2} w k ( t ) = t k /2 ( T 1 − t ) k /2 (187) , we have that
∥ w k 0 + 1 K 1 / 2 D ~ ( k 0 + 1 ) ϕ ∥ L t ∞ L x 2 + ∥ w k 0 + 1 K 1 / 2 D ~ ( k 0 + 2 ) ϕ ∥ L t , x 2 ≲ ∥ ( T 1 − t ) 1 / 2 w k 0 K 1 / 2 D ~ ( k 0 + 1 ) ϕ ∥ L t , x 2 + ∥ ( T 1 − t ) 1 / 2 F ~ ∥ L t 2 L x ∞ ∥ t 1 / 2 w k 0 K 1 / 2 D ~ ( k 0 + 1 ) ϕ ∥ L t , x 2 + ∑ j 1 + j 2 = k 0 + 1 j 1 ≥ 1 j 2 ≥ 0 ∥ w k 0 + 1 K 1 / 2 ∣ D ~ ( j 1 ) F ~ ∣ ∣ D ~ ( j 2 ) ϕ ∣ ∥ L t 1 L x 2 . \begin{aligned}
&\left\|w_{k_0+1}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right\|_{L^\infty_tL^2_x}
+\left\|w_{k_0+1}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+2)}\phi\right\|_{L^2_{t,x}}\\
&\lesssim
\left\|(T_1-t)^{1/2}w_{k_0}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right\|_{L^2_{t,x}}
+\left\|(T_1-t)^{1/2}\widetilde{F}\right\|_{L^2_tL^\infty_x}
\left\|t^{1/2}w_{k_0}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right\|_{L^2_{t,x}}\\
&\qquad+
\sum_{\substack{j_1+j_2=k_0+1\\j_1\ge1\\j_2\ge0}}
\left\|w_{k_0+1}K^{1/2}\left|\widetilde{\mathcal{D}}^{(j_1)}\widetilde{F}\right|
\left|\widetilde{\mathcal{D}}^{(j_2)}\phi\right|\right\|_{L^1_tL^2_x}.
\end{aligned} w k 0 + 1 K 1/2 D ( k 0 + 1 ) ϕ L t ∞ L x 2 + w k 0 + 1 K 1/2 D ( k 0 + 2 ) ϕ L t , x 2 ≲ ( T 1 − t ) 1/2 w k 0 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 + ( T 1 − t ) 1/2 F L t 2 L x ∞ t 1/2 w k 0 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 + j 1 + j 2 = k 0 + 1 j 1 ≥ 1 j 2 ≥ 0 ∑ w k 0 + 1 K 1/2 D ( j 1 ) F D ( j 2 ) ϕ L t 1 L x 2 . By the inductive assumption, we may estimate the first term on the right-hand side above by
∥ ( T 1 − t ) 1 / 2 w k 0 K 1 / 2 D ~ ( k 0 + 1 ) ϕ ∥ L t , x 2 ≲ G T ( C 0 F ~ ) T 1 1 / 2 T 1 k 0 / 2 ( 1 + ( ∥ F ~ ∥ [ 0 , T ] , T 1 , k 0 t r e p ) k 0 ) ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 , \left\|(T_1-t)^{1/2}w_{k_0}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right\|_{L^2_{t,x}}
\lesssim G_{\mathsf{T}}(C_0\widetilde{F})T_1^{1/2}T_1^{k_0/2}\left(1+\left(\|\widetilde{F}\|^{\mathrm{trep}}_{[0,T],T_1,k_0}\right)^{k_0}\right)\left\|K^{1/2}(0)\phi_0\right\|_{L^2_x}, ( T 1 − t ) 1/2 w k 0 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 ≲ G T ( C 0 F ) T 1 1/2 T 1 k 0 /2 ( 1 + ( ∥ F ∥ [ 0 , T ] , T 1 , k 0 trep ) k 0 ) K 1/2 ( 0 ) ϕ 0 L x 2 , which is acceptable. For the second term, again applying the inductive assumption, we have that
∥ ( T 1 − t ) 1 / 2 F ~ ∥ L t 2 L x ∞ ∥ t 1 / 2 w k 0 K 1 / 2 D ~ ( k 0 + 1 ) ϕ ∥ L t , x 2 ≲ G T ( C 0 F ~ ) T 1 / 2 T 1 k 0 / 2 ∥ F ~ ∥ [ 0 , T ] , T 1 , 0 t r e p ( 1 + ( ∥ F ~ ∥ [ 0 , T ] , T 1 , k 0 t r e p ) k 0 ) ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 , \begin{aligned}
&\left\|(T_1-t)^{1/2}\widetilde{F}\right\|_{L^2_tL^\infty_x}
\left\|t^{1/2}w_{k_0}K^{1/2}\widetilde{\mathcal{D}}^{(k_0+1)}\phi\right\|_{L^2_{t,x}}\\
&\lesssim G_{\mathsf{T}}(C_0\widetilde{F})T^{1/2}T_1^{k_0/2}
\|\widetilde{F}\|^{\mathrm{trep}}_{[0,T],T_1,0}
\left(1+\left(\|\widetilde{F}\|^{\mathrm{trep}}_{[0,T],T_1,k_0}\right)^{k_0}\right)
\left\|K^{1/2}(0)\phi_0\right\|_{L^2_x},
\end{aligned} ( T 1 − t ) 1/2 F L t 2 L x ∞ t 1/2 w k 0 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 ≲ G T ( C 0 F ) T 1/2 T 1 k 0 /2 ∥ F ∥ [ 0 , T ] , T 1 , 0 trep ( 1 + ( ∥ F ∥ [ 0 , T ] , T 1 , k 0 trep ) k 0 ) K 1/2 ( 0 ) ϕ 0 L x 2 , which is also acceptable. Finally, for the third term, for fixed j 1 ≥ 1 j_1\ge1 j 1 ≥ 1 j 2 ≥ 0 j_2\ge0 j 2 ≥ 0 j 1 + j 2 = k 0 + 1 j_1+j_2=k_0+1 j 1 + j 2 = k 0 + 1
∥ w k 0 + 1 K 1 / 2 ∣ D ~ ( j 1 ) F ~ ∣ ∣ D ~ ( j 2 ) ϕ ∣ ∥ L t 1 L x 2 ≤ ∥ ( T 1 − t ) j 1 / 2 D ~ ( j 1 ) F ~ ∥ L t 1 L x ∞ ∥ t j 2 / 2 w j 2 K 1 / 2 D ~ ( j 2 ) ϕ ∥ L t ∞ L x 2 \begin{aligned}
\left\|w_{k_0+1}K^{1/2}\left|\widetilde{\mathcal{D}}^{(j_1)}\widetilde{F}\right|
\left|\widetilde{\mathcal{D}}^{(j_2)}\phi\right|\right\|_{L^1_tL^2_x} \\
&\le\left\|(T_1-t)^{j_1/2}\widetilde D^{(j_1)}\widetilde F\right\|_{L_t^1L_x^\infty}\left\|t^{j_2/2}w_{j_2}K^{1/2}\widetilde D^{(j_2)}\phi\right\|_{L_t^\infty L_x^2}
\end{aligned} w k 0 + 1 K 1/2 D ( j 1 ) F D ( j 2 ) ϕ L t 1 L x 2 ≤ ( T 1 − t ) j 1 /2 D ( j 1 ) F L t 1 L x ∞ t j 2 /2 w j 2 K 1/2 D ( j 2 ) ϕ L t ∞ L x 2 ≲ Gr T ( C 0 F ~ ) T j 1 / 2 T 1 j 2 / 2 ∥ F ~ ∥ [ 0 , T ] , T 1 , k 0 + 1 trep ( 1 + ∥ F ~ ∥ [ 0 , T ] , T 1 , k 0 trep ) ∥ K 1 / 2 ( 0 ) ϕ 0 ∥ L x 2 , \lesssim\operatorname{Gr}_{T}(C_0\widetilde F)T^{j_1/2}T_1^{j_2/2}\|\widetilde F\|_{[0,T],T_1,k_0+1}^{\operatorname{trep}}\left(1+\|\widetilde F\|_{[0,T],T_1,k_0}^{\operatorname{trep}}\right)\left\|K^{1/2}(0)\phi_0\right\|_{L_x^2}, ≲ Gr T ( C 0 F ) T j 1 /2 T 1 j 2 /2 ∥ F ∥ [ 0 , T ] , T 1 , k 0 + 1 trep ( 1 + ∥ F ∥ [ 0 , T ] , T 1 , k 0 trep ) K 1/2 ( 0 ) ϕ 0 L x 2 , which is also acceptable. Combining, we have thus proven the inductive step.
We now have everything in place to prove Proposition 4.27 .
Proof of Proposition 4.27 . It suffices to assume s 0 > 0 s_0>0 s 0 > 0 m > 0 m>0 m > 0 m = 0 m=0 m = 0 Propositions 4.17 and 4.18 ). We focus on the estimate involving q B q_B q B p B p_B p B Lemma 4.29 , it suffices to estimate D y ( m ) ( D x ( k ) q B ) ∗ D_y^{(m)}(D_x^{(k)}q_B)^* D y ( m ) ( D x ( k ) q B ) ∗ r 0 = ( t 0 + s 0 ) / 2 r_0=(t_0+s_0)/2 r 0 = ( t 0 + s 0 ) /2 s 0 s_0 s 0 t 0 t_0 t 0 s ∈ [ s 0 , r 0 ] s\in[s_0,r_0] s ∈ [ s 0 , r 0 ]
ϕ ( s , y ) = ( D x ( k ) q B ( s , y ; t 0 , x 0 ) ) ∗ . \phi(s,y)=\left(D_x^{(k)}q_B(s,y;t_0,x_0)\right)^*. ϕ ( s , y ) = ( D x ( k ) q B ( s , y ; t 0 , x 0 ) ) ∗ . By the semigroup property of the covariant heat kernel, we have that
ϕ ( s , y ) ∗ = ∫ d z ( D x ( k ) q B ( r 0 , z ; t 0 , x 0 ) ) q B ( s , y ; r 0 , z ) . \phi(s,y)^*=\int dz\,\left(D_x^{(k)}q_B(r_0,z;t_0,x_0)\right)q_B(s,y;r_0,z). ϕ ( s , y ) ∗ = ∫ d z ( D x ( k ) q B ( r 0 , z ; t 0 , x 0 ) ) q B ( s , y ; r 0 , z ) . Then by Lemma 4.28 , we have
( ∂ s + Δ B + 2 F ) ϕ = 0 , ϕ ( r 0 ) = ( D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) ) ∗ . (\partial_s+\Delta_B+2F)\phi=0,\qquad\phi(r_0)=\left(D_x^{(k)}q_B(r_0,\mathord\cdot;t_0,x_0)\right)^*. ( ∂ s + Δ B + 2 F ) ϕ = 0 , ϕ ( r 0 ) = ( D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) ) ∗ . Define the (shifted) time-reversal ϕ ~ : [ 0 , r 0 − s 0 ] × T d → End ( W d ) \widetilde\phi:[0,r_0-s_0]\times\mathbb{T}^d\to\operatorname{End}(W^d) ϕ : [ 0 , r 0 − s 0 ] × T d → End ( W d ) ϕ ~ ( t ) = ϕ ( r 0 − t ) \widetilde\phi(t)=\phi(r_0-t) ϕ ( t ) = ϕ ( r 0 − t ) B ~ ( t ) = B ( r 0 − t ) \widetilde B(t)=B(r_0-t) B ( t ) = B ( r 0 − t ) F ~ ( t ) = F ( r 0 − t ) \widetilde F(t)=F(r_0-t) F ( t ) = F ( r 0 − t ) B B B B ~ \widetilde B B
( ∂ t − Δ B ~ − 2 F ~ ) ϕ ~ = 0 , ϕ ~ ( 0 ) = ( D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) ) ∗ . (\partial_t-\Delta_{\widetilde B}-2\widetilde F)\widetilde\phi=0,\qquad\widetilde\phi(0)=\left(D_x^{(k)}q_B(r_0,\mathord\cdot;t_0,x_0)\right)^*. ( ∂ t − Δ B − 2 F ) ϕ = 0 , ϕ ( 0 ) = ( D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) ) ∗ . In the following, write D ~ = D B ~ \widetilde D=D_{\widetilde B} D = D B
D ~ y ( m ) ϕ ~ ( r 0 − s 0 , y 0 ) = D y ( m ) ( D x ( k ) q B ( s 0 , y 0 ; t 0 , x 0 ) ) ∗ , \widetilde D_y^{(m)}\widetilde\phi(r_0-s_0,y_0)=D_y^{(m)}\left(D_x^{(k)}q_B(s_0,y_0;t_0,x_0)\right)^*, D y ( m ) ϕ ( r 0 − s 0 , y 0 ) = D y ( m ) ( D x ( k ) q B ( s 0 , y 0 ; t 0 , x 0 ) ) ∗ , which is precisely the quantity we want to estimate. By Lemma 4.31 applied with T = r 0 − s 0 T=r_0-s_0 T = r 0 − s 0 T 1 = r 0 T_1=r_0 T 1 = r 0 T 1 > T T_1>T T 1 > T s 0 > 0 s_0>0 s 0 > 0 K : [ 0 , r 0 − s 0 ] × T d → ( 0 , ∞ ) K:[0,r_0-s_0]\times\mathbb{T}^d\to(0,\infty) K : [ 0 , r 0 − s 0 ] × T d → ( 0 , ∞ ) m ≥ 0 m\ge0 m ≥ 0
∥ t m / 2 ( r 0 − t ) m / 2 K 1 / 2 D ~ ( m ) ϕ ~ ∥ L t ∞ L x 2 ( [ 0 , r 0 − s 0 ] ) ≲ Gr r 0 − s 0 ( C 0 F ~ ) r 0 m / 2 ( 1 + ∥ F ~ ∥ [ 0 , r 0 − s 0 ] , r 0 , m trep ) m ∥ K 1 / 2 ( 0 ) ϕ ~ ( 0 ) ∥ L x 2 . \left\|t^{m/2}(r_0-t)^{m/2}K^{1/2}\widetilde D^{(m)}\widetilde\phi\right\|_{L_t^\infty L_x^2([0,r_0-s_0])}
\lesssim\operatorname{Gr}_{r_0-s_0}(C_0\widetilde F)r_0^{m/2}\left(1+\|\widetilde F\|_{[0,r_0-s_0],r_0,m}^{\operatorname{trep}}\right)^m\left\|K^{1/2}(0)\widetilde\phi(0)\right\|_{L_x^2}. t m /2 ( r 0 − t ) m /2 K 1/2 D ( m ) ϕ L t ∞ L x 2 ([ 0 , r 0 − s 0 ]) ≲ Gr r 0 − s 0 ( C 0 F ) r 0 m /2 ( 1 + ∥ F ∥ [ 0 , r 0 − s 0 ] , r 0 , m trep ) m K 1/2 ( 0 ) ϕ ( 0 ) L x 2 . Taking K ( t , x ) = p ( t , x ; r 0 − s 0 , y 0 ) K(t,x)=p(t,x;r_0-s_0,y_0) K ( t , x ) = p ( t , x ; r 0 − s 0 , y 0 ) Proposition 4.18 that
∥ K 1 / 2 ( 0 ) ϕ ~ ( 0 ) ∥ L x 2 ≤ ∥ K 1 / 2 ( 0 ) ∣ D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) ∣ ∥ L x 2 \left\|K^{1/2}(0)\widetilde\phi(0)\right\|_{L_x^2}\le\left\|K^{1/2}(0)\left|D_x^{(k)}q_B(r_0,\mathord\cdot;t_0,x_0)\right|\right\|_{L_x^2} K 1/2 ( 0 ) ϕ ( 0 ) L x 2 ≤ K 1/2 ( 0 ) D x ( k ) q B ( r 0 , ⋅ ; t 0 , x 0 ) L x 2 ≲ Gr [ r 0 , t 0 ] ( C 0 F ) ( 1 + ∥ F ∥ [ r 0 , t 0 ] , k pert ) ( t 0 − r 0 ) − k + d 2 exp ( − c ∣ x 0 − y 0 ∣ 2 t 0 − r 0 ) . \lesssim\operatorname{Gr}_{[r_0,t_0]}(C_0F)\left(1+\|F\|_{[r_0,t_0],k}^{\operatorname{pert}}\right)(t_0-r_0)^{-\frac{k+d}{2}}\exp\left(-c\frac{|x_0-y_0|^2}{t_0-r_0}\right). ≲ Gr [ r 0 , t 0 ] ( C 0 F ) ( 1 + ∥ F ∥ [ r 0 , t 0 ] , k pert ) ( t 0 − r 0 ) − 2 k + d exp ( − c t 0 − r 0 ∣ x 0 − y 0 ∣ 2 ) . Combining this with Lemma 4.11 , we thus obtain (note that t 0 − r 0 ∼ t 0 − s 0 t_0-r_0\sim t_0-s_0 t 0 − r 0 ∼ t 0 − s 0 r 0 r_0 r 0 s 0 , t 0 s_0,t_0 s 0 , t 0 Gr r 0 − s 0 ( C 0 F ~ ) Gr [ r 0 , t 0 ] ( C 0 F ) \operatorname{Gr}_{r_0-s_0}(C_0\widetilde F)\operatorname{Gr}_{[r_0,t_0]}(C_0F) Gr r 0 − s 0 ( C 0 F ) Gr [ r 0 , t 0 ] ( C 0 F ) Gr [ s 0 , t 0 ] ( C 0 F ) \operatorname{Gr}_{[s_0,t_0]}(C_0F) Gr [ s 0 , t 0 ] ( C 0 F )
( r 0 − s 0 ) m / 2 s 0 m / 2 ∣ D ~ ( m ) ϕ ~ ( r 0 − s 0 , y 0 ) ∣ ≲ Gr [ s 0 , t 0 ] ( C 0 F ) r 0 m / 2 ( 1 + ∥ F ~ ∥ [ 0 , r 0 − s 0 ] , r 0 , m trep ) m × ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k p e r t ) k ) ( t 0 − s 0 ) − k + d 2 exp ( − c ∣ x 0 − y 0 ∣ 2 t 0 − s 0 ) . \begin{aligned}
(r_0-s_0)^{m/2}s_0^{m/2}\left|\widetilde D^{(m)}\widetilde\phi(r_0-s_0,y_0)\right|\lesssim\operatorname{Gr}_{[s_0,t_0]}(C_0F)r_0^{m/2}\left(1+\|\widetilde F\|_{[0,r_0-s_0],r_0,m}^{\operatorname{trep}}\right)^m \\
&\times\left(1+\left(\lVert F\rVert^{\mathrm{pert}}_{[s_0,t_0],k}\right)^k\right)(t_0-s_0)^{-\frac{k+d}{2}}\exp\left(-c\frac{\lvert x_0-y_0\rvert^2}{t_0-s_0}\right).
\end{aligned} ( r 0 − s 0 ) m /2 s 0 m /2 D ( m ) ϕ ( r 0 − s 0 , y 0 ) ≲ Gr [ s 0 , t 0 ] ( C 0 F ) r 0 m /2 ( 1 + ∥ F ∥ [ 0 , r 0 − s 0 ] , r 0 , m trep ) m × ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k pert ) k ) ( t 0 − s 0 ) − 2 k + d exp ( − c t 0 − s 0 ∣ x 0 − y 0 ∣ 2 ) . Next, by the fact that B ~ ( t ) = B ( r 0 − t ) \widetilde{B}(t)=B(r_0-t) B ( t ) = B ( r 0 − t ) F ~ ( t ) = F ( r 0 − t ) \widetilde{F}(t)=F(r_0-t) F ( t ) = F ( r 0 − t ) α , β ∈ R \alpha,\beta\in\mathbb{R} α , β ∈ R
∫ 0 r 0 − s 0 d t ( r 0 − t ) α ∥ D B ~ ( t ) ( m ) F ~ ( t ) ∥ L x ∞ β = ∫ s 0 r 0 d t t α ∥ D B ( t ) ( m ) F ( t ) ∥ L x ∞ β , \int_0^{r_0-s_0}\mathrm{d}t(r_0-t)^\alpha\lVert D_{\widetilde{B}(t)}^{(m)}\widetilde{F}(t)\rVert_{L_x^\infty}^\beta=\int_{s_0}^{r_0}\mathrm{d}t\,t^\alpha\lVert D_{B(t)}^{(m)}F(t)\rVert_{L_x^\infty}^\beta, ∫ 0 r 0 − s 0 d t ( r 0 − t ) α ∥ D B ( t ) ( m ) F ( t ) ∥ L x ∞ β = ∫ s 0 r 0 d t t α ∥ D B ( t ) ( m ) F ( t ) ∥ L x ∞ β , from which we obtain
∥ F ~ ∥ [ 0 , r 0 − s 0 ] , r 0 , m t r e p ≤ ∥ F ∥ r 0 , m p e r t . \lVert\widetilde{F}\rVert^{\mathrm{trep}}_{[0,r_0-s_0],r_0,m}\leq\lVert F\rVert^{\mathrm{pert}}_{r_0,m}. ∥ F ∥ [ 0 , r 0 − s 0 ] , r 0 , m trep ≤ ∥ F ∥ r 0 , m pert . We thus further obtain
∣ D ~ ( m ) ϕ ~ ( r 0 − s 0 , y 0 ) ∣ ≲ G [ s 0 , t 0 ] ( C 0 F ) ( r 0 − s 0 ) − m / 2 s 0 − m / 2 r 0 m / 2 ( 1 + ( ∥ F ∥ t 0 , m p e r t ) m ) × ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k p e r t ) k ) ( t 0 − s 0 ) − k + d 2 exp ( − c ∣ x 0 − y 0 ∣ 2 t 0 − s 0 ) . \begin{aligned}
\left\lvert\widetilde{D}^{(m)}\widetilde{\phi}(r_0-s_0,y_0)\right\rvert\lesssim G_{[s_0,t_0]}(C_0F)(r_0-s_0)^{-m/2}s_0^{-m/2}r_0^{m/2}\left(1+\left(\lVert F\rVert^{\mathrm{pert}}_{t_0,m}\right)^m\right) \\
&\times\left(1+\left(\lVert F\rVert^{\mathrm{pert}}_{[s_0,t_0],k}\right)^k\right)(t_0-s_0)^{-\frac{k+d}{2}}\exp\left(-c\frac{\lvert x_0-y_0\rvert^2}{t_0-s_0}\right).
\end{aligned} D ( m ) ϕ ( r 0 − s 0 , y 0 ) ≲ G [ s 0 , t 0 ] ( C 0 F ) ( r 0 − s 0 ) − m /2 s 0 − m /2 r 0 m /2 ( 1 + ( ∥ F ∥ t 0 , m pert ) m ) × ( 1 + ( ∥ F ∥ [ s 0 , t 0 ] , k pert ) k ) ( t 0 − s 0 ) − 2 k + d exp ( − c t 0 − s 0 ∣ x 0 − y 0 ∣ 2 ) . Finally, using that
( r 0 − s 0 ) − m / 2 s 0 − m / 2 r 0 m / 2 ≲ min ( s 0 , t 0 − s 0 ) − m / 2 , (r_0-s_0)^{-m/2}s_0^{-m/2}r_0^{m/2}\lesssim\min(s_0,t_0-s_0)^{-m/2}, ( r 0 − s 0 ) − m /2 s 0 − m /2 r 0 m /2 ≲ min ( s 0 , t 0 − s 0 ) − m /2 , the desired result now follows.
Parallel transport estimates
In this subsection, we derive identities and estimates involving parallel transport (introduced in § 4.3 , see Definition 3.35 ) and its covariant derivatives. With some exceptions, we will typically work with a general time-varying connection one-form A A A A A A (8) . These results will be needed in several places later—see e.g. § 5.4 , § 5.5 , and 9.4. Because their proofs are all quite direct, we will break from the convention of previous sections, and instead give the proofs directly after stating the results. We begin by setting some notation.
Notation 4.32. Let A : [ 0 , T ] × T d → g d A\colon[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d A : [ 0 , T ] × T d → g d F A F_A F A A 0 = 0 A_0=0 A 0 = 0 A A A [ 0 , T ] × T d [0,T]\times\mathbb{T}^d [ 0 , T ] × T d v , w ∈ R 1 + d v,w\in\mathbb{R}^{1+d} v , w ∈ R 1 + d
F A ( v , w ) = ( F A ) α β v α w β . F_A(v,w)=(F_A)_{\alpha\beta}v^\alpha w^\beta. F A ( v , w ) = ( F A ) α β v α w β . Note that this is a function of ( t , x ) (t,x) ( t , x )
F A ( v , w ) ( t , x ) = F A ( t , x ) ( v , w ) = ( F A ( t , x ) ) α β v α w β . F_A(v,w)(t,x)=F_A(t,x)(v,w)=(F_A(t,x))_{\alpha\beta}v^\alpha w^\beta. F A ( v , w ) ( t , x ) = F A ( t , x ) ( v , w ) = ( F A ( t , x ) ) α β v α w β . However, as this gets quite long, we will often refrain from writing out ( t , x ) (t,x) ( t , x ) v , w ∈ R d v,w\in\mathbb{R}^d v , w ∈ R d F A ( v , w ) F_A(v,w) F A ( v , w ) F A ( t , x ) ( v , w ) F_A(t,x)(v,w) F A ( t , x ) ( v , w ) v , w v,w v , w R 1 + d \mathbb{R}^{1+d} R 1 + d u ∈ R 1 + d u\in\mathbb{R}^{1+d} u ∈ R 1 + d
( D A , w F A ) ( v , u ) = ( D A , w ( F A ) α β ) v α u β = ( D A , γ ( F A ) α β ) v α u β w γ . (D_{A,w}F_A)(v,u)=(D_{A,w}(F_A)_{\alpha\beta})v^\alpha u^\beta=(D_{A,\gamma}(F_A)_{\alpha\beta})v^\alpha u^\beta w^\gamma. ( D A , w F A ) ( v , u ) = ( D A , w ( F A ) α β ) v α u β = ( D A , γ ( F A ) α β ) v α u β w γ . We proceed to the first set of main results of this section, which provide identities involving parallel transport. Recall our notation τ A ( γ ( t ) ) = τ A ( 0 , y 0 ; γ ( t ) ) \tau_A(\gamma(t))=\tau_A(0,y_0;\gamma(t)) τ A ( γ ( t )) = τ A ( 0 , y 0 ; γ ( t )) γ \gamma γ γ ( 0 ) = ( 0 , y 0 ) \gamma(0)=(0,y_0) γ ( 0 ) = ( 0 , y 0 )
Remark 4.33. In the following results (Lemmas 4.34 and 4.35 ), one should view F A ( v , w ) F_A(v,w) F A ( v , w ) ( D A , w F A ) ( v , u ) (D_{A,w}F_A)(v,u) ( D A , w F A ) ( v , u ) W W W π ∗ \pi_* π ∗ End ( W ) \operatorname{End}(W) End ( W )
Lemma 4.34 (Covariant derivatives of parallel transport). Let A : [ 0 , T ] × T d → g d A\colon[0,T]\times\mathbb{T}^d\to\mathfrak{g}^d A : [ 0 , T ] × T d → g d y 0 ∈ T d y_0\in\mathbb{T}^d y 0 ∈ T d z = ( z 0 , z x ) ∈ R 1 + d z=(z_0,z_x)\in\mathbb{R}^{1+d} z = ( z 0 , z x ) ∈ R 1 + d z 0 ≥ 0 z_0\geq0 z 0 ≥ 0 γ ( t ) = ( 0 , y 0 ) + t z \gamma(t)=(0,y_0)+tz γ ( t ) = ( 0 , y 0 ) + t z t 0 > 0 t_0>0 t 0 > 0 t 0 z 0 ≤ T t_0z_0\leq T t 0 z 0 ≤ T t 0 ∣ z x ∣ ≤ 1 / 10 t_0|z_x|\leq1/10 t 0 ∣ z x ∣ ≤ 1/10 v , w ∈ R 1 + d v,w\in\mathbb{R}^{1+d} v , w ∈ R 1 + d
( τ A − 1 D A , v τ A ) ( γ ( t 0 ) ) = 1 t 0 ∫ 0 t 0 d t t ( τ A − 1 F A ( z , v ) τ A ) ( γ ( t ) ) . (\tau_A^{-1}D_{A,v}\tau_A)(\gamma(t_0))=\frac{1}{t_0}\int_0^{t_0}dt\,t(\tau_A^{-1}F_A(z,v)\tau_A)(\gamma(t)). ( τ A − 1 D A , v τ A ) ( γ ( t 0 )) = t 0 1 ∫ 0 t 0 d t t ( τ A − 1 F A ( z , v ) τ A ) ( γ ( t )) . ( τ A − 1 D A , w D A , v τ A ) ( γ ( t 0 ) ) = 1 t 0 2 ∫ 0 t 0 d t t 2 ( ( τ A − 1 F A ( z , w ) D A , v τ A ) ( γ ( t ) ) + ( τ A − 1 F A ( z , v ) D A , w τ A ) ( γ ( t ) ) ) + 1 t 0 2 ∫ 0 t 0 d t t 2 ( τ A − 1 ( D A , w F A ) ( z , v ) τ A ) ( γ ( t ) ) + 1 t 0 2 ∫ 0 t 0 d t t ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t ) ) . (\tau_A^{-1}D_{A,w}D_{A,v}\tau_A)(\gamma(t_0))=\frac{1}{t_0^2}\int_0^{t_0}dt\,t^2\left((\tau_A^{-1}F_A(z,w)D_{A,v}\tau_A)(\gamma(t))+(\tau_A^{-1}F_A(z,v)D_{A,w}\tau_A)(\gamma(t))\right)
+\frac{1}{t_0^2}\int_0^{t_0}dt\,t^2(\tau_A^{-1}(D_{A,w}F_A)(z,v)\tau_A)(\gamma(t))+\frac{1}{t_0^2}\int_0^{t_0}dt\,t(\tau_A^{-1}F_A(w,v)\tau_A)(\gamma(t)). ( τ A − 1 D A , w D A , v τ A ) ( γ ( t 0 )) = t 0 2 1 ∫ 0 t 0 d t t 2 ( ( τ A − 1 F A ( z , w ) D A , v τ A ) ( γ ( t )) + ( τ A − 1 F A ( z , v ) D A , w τ A ) ( γ ( t )) ) + t 0 2 1 ∫ 0 t 0 d t t 2 ( τ A − 1 ( D A , w F A ) ( z , v ) τ A ) ( γ ( t )) + t 0 2 1 ∫ 0 t 0 d t t ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t )) . Proof. For notational simplicity, write D = D A D=D_A D = D A F = F A F=F_A F = F A γ ( t , s ) = ( 0 , y 0 ) + t ( z + s v ) \gamma(t,s)=(0,y_0)+t(z+sv) γ ( t , s ) = ( 0 , y 0 ) + t ( z + s v ) γ ( t , 0 ) = γ ( t ) \gamma(t,0)=\gamma(t) γ ( t , 0 ) = γ ( t ) γ \gamma γ [ 0 , t 0 ] × ( − ε , ε ) [0,t_0]\times(-\varepsilon,\varepsilon) [ 0 , t 0 ] × ( − ε , ε ) ε \varepsilon ε γ ( t , s ) \gamma(t,s) γ ( t , s ) τ A \tau_A τ A ( t , s ) ∈ [ 0 , t 0 ] × ( − ε , ε ) (t,s)\in[0,t_0]\times(-\varepsilon,\varepsilon) ( t , s ) ∈ [ 0 , t 0 ] × ( − ε , ε ) γ ˙ = ∂ t γ \dot{\gamma}=\partial_t\gamma γ ˙ = ∂ t γ J = ∂ s γ J=\partial_s\gamma J = ∂ s γ D t = D γ ˙ D_t=D_{\dot{\gamma}} D t = D γ ˙ D s = D J D_s=D_J D s = D J τ A \tau_A τ A D t τ A = 0 D_t\tau_A=0 D t τ A = 0 F ( ∂ t , ∂ s ) = F ( γ ˙ , J ) F(\partial_t,\partial_s)=F(\dot{\gamma},J) F ( ∂ t , ∂ s ) = F ( γ ˙ , J )
D t D s τ A = D s D t τ A + F ( ∂ t , ∂ s ) τ A = F ( γ ˙ , J ) τ A . D_tD_s\tau_A=D_sD_t\tau_A+F(\partial_t,\partial_s)\tau_A=F(\dot{\gamma},J)\tau_A. D t D s τ A = D s D t τ A + F ( ∂ t , ∂ s ) τ A = F ( γ ˙ , J ) τ A . Applying the above identity at s = 0 s=0 s = 0 γ ˙ = z \dot{\gamma}=z γ ˙ = z J = t v J=tv J = t v Lemma 3.39 , we obtain (note that the initial condition is zero since t v = 0 tv=0 t v = 0 t = 0 t=0 t = 0
( τ A − 1 D t 0 v τ A ) ( γ ( t 0 ) ) = ∫ 0 t 0 d t t ( τ A − 1 F ( z , v ) τ A ) ( γ ( t ) ) . (\tau_A^{-1}D_{t_0v}\tau_A)(\gamma(t_0))=\int_0^{t_0}dt\,t(\tau_A^{-1}F(z,v)\tau_A)(\gamma(t)). ( τ A − 1 D t 0 v τ A ) ( γ ( t 0 )) = ∫ 0 t 0 d t t ( τ A − 1 F ( z , v ) τ A ) ( γ ( t )) . Dividing both sides by t 0 t_0 t 0
For the second desired identity, we proceed similarly, except now we define γ ( t , s 1 , s 2 ) = ( 0 , y 0 ) + t ( z + s 1 v + s 2 w ) \gamma(t,s_1,s_2)=(0,y_0)+t(z+s_1v+s_2w) γ ( t , s 1 , s 2 ) = ( 0 , y 0 ) + t ( z + s 1 v + s 2 w ) γ ˙ ( t ) = ∂ t γ \dot{\gamma}(t)=\partial_t\gamma γ ˙ ( t ) = ∂ t γ J 1 = ∂ s 1 γ J_1=\partial_{s_1}\gamma J 1 = ∂ s 1 γ J 2 = ∂ s 2 γ J_2=\partial_{s_2}\gamma J 2 = ∂ s 2 γ γ \gamma γ [ 0 , t 0 ] × ( − ε , ε ) 2 [0,t_0]\times(-\varepsilon,\varepsilon)^2 [ 0 , t 0 ] × ( − ε , ε ) 2 D t = D γ ˙ D_t=D_{\dot{\gamma}} D t = D γ ˙ D s i = D J i D_{s_i}=D_{J_i} D s i = D J i i = 1 , 2 i=1,2 i = 1 , 2
D t D s 1 τ A = F ( ∂ t , ∂ s 1 ) τ A . D_tD_{s_1}\tau_A=F(\partial_t,\partial_{s_1})\tau_A. D t D s 1 τ A = F ( ∂ t , ∂ s 1 ) τ A . We may then obtain (applying the above identity and the product formula in the second identity)
D t D s 2 D s 1 τ A = F ( ∂ t , ∂ s 2 ) D s 1 τ A + D s 2 ( D t D s 1 τ A ) = F ( ∂ t , ∂ s 2 ) D s 1 τ A + F ( ∂ t , ∂ s 1 ) D s 2 τ A + ( D s 2 F ) ( ∂ t , ∂ s 1 ) τ A + ( F ( ∂ s 2 ∂ t , ∂ s 1 ) + F ( ∂ t , ∂ s 2 ∂ s 1 ) ) τ A = F ( γ ˙ , J 2 ) D J 1 τ A + F ( γ ˙ , J 1 ) D J 2 τ A + ( D J 2 F ) ( γ ˙ , J 1 ) τ A + ( F ( ∂ s 2 γ ˙ , J 1 ) + F ( γ ˙ , ∂ s 2 J 1 ) ) τ A . \begin{aligned}
D_tD_{s_2}D_{s_1}\tau_A
&=F(\partial_t,\partial_{s_2})D_{s_1}\tau_A+D_{s_2}(D_tD_{s_1}\tau_A)\\
&=F(\partial_t,\partial_{s_2})D_{s_1}\tau_A+F(\partial_t,\partial_{s_1})D_{s_2}\tau_A+(D_{s_2}F)(\partial_t,\partial_{s_1})\tau_A\\
&\quad+(F(\partial_{s_2}\partial_t,\partial_{s_1})+F(\partial_t,\partial_{s_2}\partial_{s_1}))\tau_A\\
&=F(\dot{\gamma},J_2)D_{J_1}\tau_A+F(\dot{\gamma},J_1)D_{J_2}\tau_A+(D_{J_2}F)(\dot{\gamma},J_1)\tau_A\\
&\quad+(F(\partial_{s_2}\dot{\gamma},J_1)+F(\dot{\gamma},\partial_{s_2}J_1))\tau_A.
\end{aligned} D t D s 2 D s 1 τ A = F ( ∂ t , ∂ s 2 ) D s 1 τ A + D s 2 ( D t D s 1 τ A ) = F ( ∂ t , ∂ s 2 ) D s 1 τ A + F ( ∂ t , ∂ s 1 ) D s 2 τ A + ( D s 2 F ) ( ∂ t , ∂ s 1 ) τ A + ( F ( ∂ s 2 ∂ t , ∂ s 1 ) + F ( ∂ t , ∂ s 2 ∂ s 1 )) τ A = F ( γ ˙ , J 2 ) D J 1 τ A + F ( γ ˙ , J 1 ) D J 2 τ A + ( D J 2 F ) ( γ ˙ , J 1 ) τ A + ( F ( ∂ s 2 γ ˙ , J 1 ) + F ( γ ˙ , ∂ s 2 J 1 )) τ A . Using that ∂ s 2 γ ˙ = w \partial_{s_2}\dot{\gamma}=w ∂ s 2 γ ˙ = w ∂ s 2 J 1 = 0 \partial_{s_2}J_1=0 ∂ s 2 J 1 = 0 s 1 = s 2 = 0 s_1=s_2=0 s 1 = s 2 = 0 γ ˙ = z \dot{\gamma}=z γ ˙ = z J 1 = t v J_1=tv J 1 = t v J 2 = t w J_2=tw J 2 = tw Lemma 3.39 , we may obtain the second desired identity (after multiplying both sides by τ A ( γ ( t 0 ) ) − 1 \tau_A(\gamma(t_0))^{-1} τ A ( γ ( t 0 ) ) − 1 t 0 2 t_0^2 t 0 2
Lemma 4.35 (Commutator identities for parallel transport and curvature). Let A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d y 0 ∈ T d y_{0} \in\mathbb{T}^{d} y 0 ∈ T d u = ( u 0 , u x ) ∈ R 1 + d u=(u_{0},u_{x}) \in\mathbb{R}^{1+d} u = ( u 0 , u x ) ∈ R 1 + d u 0 ≥ 0 u_{0} \ge0 u 0 ≥ 0 γ ( t ) = ( 0 , y 0 ) + t u \gamma(t)=(0,y_{0})+tu γ ( t ) = ( 0 , y 0 ) + t u t ≥ 0 t \ge0 t ≥ 0 t u 0 ≤ T tu_{0} \le T t u 0 ≤ T t ∣ u x ∣ ≤ 1 / 10 t|u_{x}| \le1/10 t ∣ u x ∣ ≤ 1/10 w , v ∈ R 1 + d w,v \in\mathbb{R}^{1+d} w , v ∈ R 1 + d
( τ A − 1 F A ( w , v ) τ A ) ( γ ( t ) ) − F A ( γ ( 0 ) ) ( w , v ) = ∫ 0 t d s ( τ A − 1 ( D A , γ ˙ ( s ) F A ) ( w , v ) τ A ) ( γ ( s ) ) . (\tau_{A}^{-1}F_{A}(w,v)\tau_{A})(\gamma(t))-F_{A}(\gamma(0))(w,v)=\int_{0}^{t} \mathrm{d}s\,(\tau_{A}^{-1}(D_{A,\dot{\gamma}(s)}F_{A})(w,v)\tau_{A})(\gamma(s)). ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t )) − F A ( γ ( 0 )) ( w , v ) = ∫ 0 t d s ( τ A − 1 ( D A , γ ˙ ( s ) F A ) ( w , v ) τ A ) ( γ ( s )) . Consequently, for any t 0 > 0 t_{0}>0 t 0 > 0 t 0 u 0 ≤ T t_{0}u_{0}\le T t 0 u 0 ≤ T t 0 ∣ u x ∣ ≤ 1 / 10 t_{0}|u_{x}|\le1/10 t 0 ∣ u x ∣ ≤ 1/10
( τ A − 1 D A , v τ A ) ( γ ( t 0 ) ) − t 0 2 F A ( γ ( 0 ) ) ( γ ˙ ( 0 ) , v ) = 1 t 0 ∫ 0 t 0 d t ∫ 0 t d s ( τ A − 1 ( D A , γ ˙ ( s ) F A ) ( γ ˙ ( s ) , v ) τ A ) ( γ ( s ) ) . (\tau_{A}^{-1}D_{A,v}\tau_{A})(\gamma(t_{0}))-\frac{t_{0}}{2}F_{A}(\gamma(0))(\dot{\gamma}(0),v)
=\frac{1}{t_{0}}\int_{0}^{t_{0}}\mathrm{d}t\int_{0}^{t}\mathrm{d}s\,(\tau_{A}^{-1}(D_{A,\dot{\gamma}(s)}F_{A})(\dot{\gamma}(s),v)\tau_{A})(\gamma(s)). ( τ A − 1 D A , v τ A ) ( γ ( t 0 )) − 2 t 0 F A ( γ ( 0 )) ( γ ˙ ( 0 ) , v ) = t 0 1 ∫ 0 t 0 d t ∫ 0 t d s ( τ A − 1 ( D A , γ ˙ ( s ) F A ) ( γ ˙ ( s ) , v ) τ A ) ( γ ( s )) . Additionally, if d = 2 d=2 d = 2 A A A t ≥ 0 t\ge0 t ≥ 0 t ∣ u x ∣ ≤ 1 / 10 t|u_{x}|\le1/10 t ∣ u x ∣ ≤ 1/10
( τ A − 1 F A ( w , v ) τ A ) ( γ ( t ) ) = F A ( γ ( 0 ) ) ( w , v ) . (\tau_{A}^{-1}F_{A}(w,v)\tau_{A})(\gamma(t))=F_{A}(\gamma(0))(w,v). ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t )) = F A ( γ ( 0 )) ( w , v ) . Proof. The second identity follows by combining the first identity with Lemma 4.34 (note that γ ˙ ( s ) = u \dot{\gamma}(s)=u γ ˙ ( s ) = u s s s w = u w=u w = u d = 2 d=2 d = 2 A A A F A F_{A} F A Remark 3.26 , which gives that all covariant spatial derivatives are zero, and recall that A A A F A F_{A} F A (127) of parallel transport, we have that
d d t ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t ) ) = ( τ A − 1 ( D A , γ ˙ ( t ) F A ) ( w , v ) τ A ) ( γ ( t ) ) . \frac{\mathrm{d}}{\mathrm{d}t}(\tau_{A}^{-1}F_{A}(w,v)\tau_{A})(\gamma(t))=(\tau_{A}^{-1}(D_{A,\dot{\gamma}(t)}F_{A})(w,v)\tau_{A})(\gamma(t)). d t d ( τ A − 1 F A ( w , v ) τ A ) ( γ ( t )) = ( τ A − 1 ( D A , γ ˙ ( t ) F A ) ( w , v ) τ A ) ( γ ( t )) . whence the first identity follows by integrating both sides.
Next, we turn to the second set of main results of this section, which provide estimates involving parallel transport.
Lemma 4.36 (Estimates on covariant derivatives of parallel transport). Let A : [ 0 , T ] × T d → g d A : [0,T] \times\mathbb{T}^{d} \to\mathfrak{g}^{d} A : [ 0 , T ] × T d → g d ( t 0 , x 0 ) ∈ ( 0 , T ] × T d (t_{0},x_{0}) \in(0,T] \times\mathbb{T}^{d} ( t 0 , x 0 ) ∈ ( 0 , T ] × T d ∣ x 0 ∣ ≤ 10 − 1 |x_{0}| \le10^{-1} ∣ x 0 ∣ ≤ 1 0 − 1
∣ D A τ A ( t 0 , x 0 ) ∣ ≲ ( ∥ ∂ t A ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + ∥ F A ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) ( 1 + ∣ x 0 ∣ t 0 − 1 / 2 ) , (208) |D_{A}\tau_{A}(t_{0},x_{0})| \lesssim\left(\|\partial_{t}A\|_{L_{t}^{1}L_{x}^{\infty}([0,t_{0}])}+\|F_{A}\|_{L_{t}^{2}L_{x}^{\infty}([0,t_{0}])}\right)(1+|x_{0}|t_{0}^{-1/2}), \tag*{(208)} ∣ D A τ A ( t 0 , x 0 ) ∣ ≲ ( ∥ ∂ t A ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + ∥ F A ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) ( 1 + ∣ x 0 ∣ t 0 − 1/2 ) , (208) ∣ ∂ t τ A ( t 0 , x 0 ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ∣ x 0 ∣ t 0 − 1 . (209) |\partial_{t}\tau_{A}(t_{0},x_{0})| \lesssim\|\partial_{t}A\|_{L_{t}^{1}L_{x}^{\infty}([0,t_{0}])}|x_{0}|t_{0}^{-1}. \tag*{(209)} ∣ ∂ t τ A ( t 0 , x 0 ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ∣ x 0 ∣ t 0 − 1 . (209) If, moreover, A A A (8) , then we have that
∣ Δ A τ A ( t 0 , x 0 ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ∣ x 0 ∣ t 0 − 1 + ( ∥ ∂ t A ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + ∥ F A ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) 2 ( 1 + ∣ x 0 ∣ t 0 − 1 / 2 ) 2 . (210) |\Delta_{A}\tau_{A}(t_{0},x_{0})| \lesssim\|\partial_{t}A\|_{L_{t}^{1}L_{x}^{\infty}([0,t_{0}])}|x_{0}|t_{0}^{-1}
+\left(\|\partial_{t}A\|_{L_{t}^{1}L_{x}^{\infty}([0,t_{0}])}+\|F_{A}\|_{L_{t}^{2}L_{x}^{\infty}([0,t_{0}])}\right)^{2}(1+|x_{0}|t_{0}^{-1/2})^{2}. \tag*{(210)} ∣ Δ A τ A ( t 0 , x 0 ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ∣ x 0 ∣ t 0 − 1 + ( ∥ ∂ t A ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + ∥ F A ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) 2 ( 1 + ∣ x 0 ∣ t 0 − 1/2 ) 2 . (210) Proof. In the following, all space-time norms will implicitly be on [ 0 , t 0 ] × T d [0,t_{0}] \times\mathbb{T}^{d} [ 0 , t 0 ] × T d D = D A D=D_{A} D = D A F = F A F=F_{A} F = F A z = ( 1 , x 0 / t 0 ) z=(1,x_{0}/t_{0}) z = ( 1 , x 0 / t 0 ) γ ( t ) = t z \gamma(t)=tz γ ( t ) = t z γ ( 0 ) = ( 0 , 0 ) \gamma(0)=(0,0) γ ( 0 ) = ( 0 , 0 ) γ ( t 0 ) = ( t 0 , x 0 ) \gamma(t_{0})=(t_{0},x_{0}) γ ( t 0 ) = ( t 0 , x 0 ) τ A \tau_{A} τ A Remark 3.37 (d)), it suffices to estimate τ A − 1 D τ A \tau_{A}^{-1}D\tau_{A} τ A − 1 D τ A Lemma 4.34 , we have that for ℓ ∈ [ d ] \ell\in[d] ℓ ∈ [ d ] t ∈ [ 0 , t 0 ] t \in[0,t_{0}] t ∈ [ 0 , t 0 ]
∣ ( τ A − 1 D ℓ τ A ) ( γ ( t ) ) ∣ ≲ 1 t ∫ 0 t d s ∣ F ( z , e ℓ ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ + ∥ F ∥ L t 2 L x ∞ ∣ x 0 ∣ t 0 − 1 / 2 . (211) |(\tau_{A}^{-1}D_{\ell}\tau_{A})(\gamma(t))| \lesssim\frac{1}{t}\int_{0}^{t}\mathrm{d}s\,|F(z,e_{\ell})| \lesssim\|\partial_{t}A\|_{L_{t}^{1}L_{x}^{\infty}}+\|F\|_{L_{t}^{2}L_{x}^{\infty}}|x_{0}|t_{0}^{-1/2}. \tag*{(211)} ∣ ( τ A − 1 D ℓ τ A ) ( γ ( t )) ∣ ≲ t 1 ∫ 0 t d s ∣ F ( z , e ℓ ) ∣ ≲ ∥ ∂ t A ∥ L t 1 L x ∞ + ∥ F ∥ L t 2 L x ∞ ∣ x 0 ∣ t 0 − 1/2 . (211) where we used the identity F ( z , e ℓ ) = ∂ t A ℓ + F i ℓ ( x 0 ) i / t 0 F(z,e^\ell)=\partial_tA^\ell+F^{i\ell}(x_0)i/t_0 F ( z , e ℓ ) = ∂ t A ℓ + F i ℓ ( x 0 ) i / t 0 t = t 0 t=t_0 t = t 0 (208) . The second desired estimate (209) follows by similar considerations (we replace e ℓ e_\ell e ℓ e 0 e_0 e 0 R 1 + d \mathbb{R}^{1+d} R 1 + d (210) , first note that by Lemma 4.34, we have that
( τ A − 1 Δ A τ A ) ( γ ( t 0 ) ) = ( τ A − 1 D ℓ τ A ) ( γ ( t 0 ) ) = 1 t 0 2 ∫ 0 t 0 d t t 2 ( 2 ( τ A − 1 F ( z , e ℓ ) D ℓ τ A ) ( γ ( t ) ) + ( τ A − 1 ( D ℓ F ) ( z , e ℓ ) τ A ) ( γ ( t ) ) ) . \begin{aligned}
(\tau_A^{-1}\Delta_A\tau_A)(\gamma(t_0))=(\tau_A^{-1}D^\ell\tau_A)(\gamma(t_0)) \\
&=\frac{1}{t_0^2}\int_0^{t_0}dt\,t^2\left(2(\tau_A^{-1}F(z,e^\ell)D_\ell\tau_A)(\gamma(t))+(\tau_A^{-1}(D^\ell F)(z,e_\ell)\tau_A)(\gamma(t))\right).
\end{aligned} ( τ A − 1 Δ A τ A ) ( γ ( t 0 )) = ( τ A − 1 D ℓ τ A ) ( γ ( t 0 )) = t 0 2 1 ∫ 0 t 0 d t t 2 ( 2 ( τ A − 1 F ( z , e ℓ ) D ℓ τ A ) ( γ ( t )) + ( τ A − 1 ( D ℓ F ) ( z , e ℓ ) τ A ) ( γ ( t )) ) . Since A A A (8) , we have that (in the last identity we use that ∂ t A \partial_tA ∂ t A
( D ℓ F ) ( z , e ℓ ) = D ℓ ∂ t A ℓ + D ℓ F i ℓ ( x 0 i / t 0 ) = − ∂ t A i ( x 0 i / t 0 ) . (D^\ell F)(z,e_\ell)=D^\ell\partial_tA_\ell+D^\ell F_{i\ell}(x_0^i/t_0)=-\partial_tA_i(x_0^i/t_0). ( D ℓ F ) ( z , e ℓ ) = D ℓ ∂ t A ℓ + D ℓ F i ℓ ( x 0 i / t 0 ) = − ∂ t A i ( x 0 i / t 0 ) . The estimate (210) now follows by inserting this identity and using the estimate (211) .
The following result is a bit unlike the previous results, in that it works with a general base point, and it also assumes control on a slightly different quantity than we have seen before (see (212) ). This will be needed in Section 9.4 to compute the resonance of certain high× \times × B B B (8) , and F = F B F=F_B F = F B
Lemma 4.37 (Approximate semigroup property of parallel transport). Let α ∈ [ 0 , 1 ] \alpha\in[0,1] α ∈ [ 0 , 1 ] L ≥ 1 L\geq1 L ≥ 1
T ( 1 − α ) / 2 ∥ t − ( 1 − α ) / 2 F ∥ F T , 1 p e r t ≤ L . (212) T^{(1-\alpha)/2}\|t^{-(1-\alpha)/2}F\|_{\mathcal{F}^{\mathrm{pert}}_{T,1}}\leq L. \tag*{(212)} T ( 1 − α ) /2 ∥ t − ( 1 − α ) /2 F ∥ F T , 1 pert ≤ L . (212) Let 0 ≤ r 0 < s 0 < t 0 ≤ T 0\leq r_0<s_0<t_0\leq T 0 ≤ r 0 < s 0 < t 0 ≤ T x 0 , y 0 , z 0 ∈ T d x_0,y_0,z_0\in\mathbb{T}^d x 0 , y 0 , z 0 ∈ T d ∣ x 0 − y 0 ∣ , ∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣ ≤ 10 − 1 |x_0-y_0|,|x_0-z_0|,|y_0-z_0|\leq10^{-1} ∣ x 0 − y 0 ∣ , ∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣ ≤ 1 0 − 1 ρ ≔ max ( ∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣ ) \rho\coloneqq\max(|x_0-z_0|,|y_0-z_0|) ρ : = max ( ∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣ )
∣ τ B ( r 0 , z 0 ; t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) τ B ( r 0 , z 0 ; s 0 , y 0 ) ∣ ≲ L T − ( 1 − α ) / 2 r 0 − α / 2 ( ρ log t 0 − r 0 s 0 − r 0 + ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1 / 2 ) ) . \left|\tau_B(r_0,z_0;t_0,x_0)-\tau_B(s_0,y_0;t_0,x_0)\tau_B(r_0,z_0;s_0,y_0)\right|
\lesssim LT^{-(1-\alpha)/2}r_0^{-\alpha/2}\left(\rho\log\frac{t_0-r_0}{s_0-r_0}+|x_0-y_0|\left(1+\rho(t_0-s_0)^{-1/2}\right)\right). ∣ τ B ( r 0 , z 0 ; t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) τ B ( r 0 , z 0 ; s 0 , y 0 ) ∣ ≲ L T − ( 1 − α ) /2 r 0 − α /2 ( ρ log s 0 − r 0 t 0 − r 0 + ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1/2 ) ) . Proof. It suffices to show that for any X ∈ W X\in W X ∈ W ∣ X ∣ W = 1 |X|_W=1 ∣ X ∣ W = 1
∣ ( τ B ( r 0 , z 0 ; t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) τ B ( r 0 , z 0 ; s 0 , y 0 ) ) X ∣ ≲ R , \left|\left(\tau_B(r_0,z_0;t_0,x_0)-\tau_B(s_0,y_0;t_0,x_0)\tau_B(r_0,z_0;s_0,y_0)\right)X\right|\lesssim R, ∣ ( τ B ( r 0 , z 0 ; t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) τ B ( r 0 , z 0 ; s 0 , y 0 ) ) X ∣ ≲ R , where R R R X X X Ψ ( t , x ) ≔ τ B ( r 0 , z 0 ; t , x ) X \Psi(t,x)\coloneqq\tau_B(r_0,z_0;t,x)X Ψ ( t , x ) : = τ B ( r 0 , z 0 ; t , x ) X ∣ x − z 0 ∣ ≤ 1 / 10 |x-z_0|\leq1/10 ∣ x − z 0 ∣ ≤ 1/10 γ : [ 0 , t 0 − s 0 ] → R + × T d \gamma:[0,t_0-s_0]\to\mathbb{R}_+\times\mathbb{T}^d γ : [ 0 , t 0 − s 0 ] → R + × T d ( s 0 , y 0 ) (s_0,y_0) ( s 0 , y 0 ) ( t 0 , x 0 ) (t_0,x_0) ( t 0 , x 0 )
γ ( t ) ≔ ( γ t ( t ) , γ x ( t ) ) ≔ ( s 0 , y 0 ) + t ( 1 , x 0 − y 0 t 0 − s 0 ) . \gamma(t)\coloneqq(\gamma_t(t),\gamma_x(t))\coloneqq(s_0,y_0)+t\left(1,\frac{x_0-y_0}{t_0-s_0}\right). γ ( t ) : = ( γ t ( t ) , γ x ( t )) : = ( s 0 , y 0 ) + t ( 1 , t 0 − s 0 x 0 − y 0 ) . By Lemma 3.39, we have that
∣ Ψ ( t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) Ψ ( s 0 , y 0 ) ∣ ≤ ∫ 0 t 0 − s 0 d t ∣ D γ ˙ ( t ) Ψ ( γ ( t ) ) ∣ |\Psi(t_0,x_0)-\tau_B(s_0,y_0;t_0,x_0)\Psi(s_0,y_0)|
\leq\int_0^{t_0-s_0}dt\,|D_{\dot{\gamma}(t)}\Psi(\gamma(t))| ∣Ψ ( t 0 , x 0 ) − τ B ( s 0 , y 0 ; t 0 , x 0 ) Ψ ( s 0 , y 0 ) ∣ ≤ ∫ 0 t 0 − s 0 d t ∣ D γ ˙ ( t ) Ψ ( γ ( t )) ∣ ≲ ∫ 0 t 0 − s 0 d t ∣ ∂ t τ B ( r 0 , z 0 ; γ ( t ) ) ∣ + ∣ x 0 − y 0 ∣ t 0 − s 0 ∣ D τ B ( r 0 , z 0 ; γ ( t ) ) ∣ . \lesssim\int_0^{t_0-s_0}dt\,|\partial_t\tau_B(r_0,z_0;\gamma(t))|+\frac{|x_0-y_0|}{t_0-s_0}|D\tau_B(r_0,z_0;\gamma(t))|. ≲ ∫ 0 t 0 − s 0 d t ∣ ∂ t τ B ( r 0 , z 0 ; γ ( t )) ∣ + t 0 − s 0 ∣ x 0 − y 0 ∣ ∣ D τ B ( r 0 , z 0 ; γ ( t )) ∣. Applying Lemma 4.36 (in particular (208) and (209) ), we obtain the further upper bound:
∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ( [ r 0 , t + s 0 ] ) ∣ γ x ( t ) − z 0 ∣ t + s 0 − r 0 + ∣ x 0 − y 0 ∣ t 0 − s 0 ∫ 0 t 0 − s 0 d t ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ r 0 , t + s 0 ] ) + ∥ F ∥ L t 2 L x ∞ ( [ r 0 , t + s 0 ] ) ) ( 1 + ∣ γ x ( t ) − z 0 ∣ ( t + s 0 − r 0 ) 1 / 2 ) . \int_{0}^{t_{0}-s_{0}} \mathrm{d}t \, \left\lVert\partial_{t}B \right\rVert_{L^{1}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])} \frac{\lvert\gamma_{x}(t)-z_{0}\rvert}{t+s_{0}-r_{0}}
+
\frac{\lvert x_{0}-y_{0}\rvert}{t_{0}-s_{0}} \int_{0}^{t_{0}-s_{0}} \mathrm{d}t \, \left(\left\lVert\partial_{t}B \right\rVert_{L^{1}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])}+\left\lVert F \right\rVert_{L^{2}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])}\right)\left(1+\frac{\lvert\gamma_{x}(t)-z_{0}\rvert}{(t+s_{0}-r_{0})^{1/2}}\right). ∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ([ r 0 , t + s 0 ]) t + s 0 − r 0 ∣ γ x ( t ) − z 0 ∣ + t 0 − s 0 ∣ x 0 − y 0 ∣ ∫ 0 t 0 − s 0 d t ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ r 0 , t + s 0 ]) + ∥ F ∥ L t 2 L x ∞ ([ r 0 , t + s 0 ]) ) ( 1 + ( t + s 0 − r 0 ) 1/2 ∣ γ x ( t ) − z 0 ∣ ) . To finish, it suffices to show the separate estimates
∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ( [ r 0 , t + s 0 ] ) ∣ γ x ( t ) − z 0 ∣ t + s 0 − r 0 ≲ L T − 1 − α 2 r 0 − α 2 ρ log t 0 − r 0 s 0 − r 0 , (213) \int_{0}^{t_{0}-s_{0}} \mathrm{d}t \, \left\lVert\partial_{t}B \right\rVert_{L^{1}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])} \frac{\lvert\gamma_{x}(t)-z_{0}\rvert}{t+s_{0}-r_{0}}
\lesssim LT^{-\frac{1-\alpha}{2}}r_{0}^{-\frac{\alpha}{2}}\rho\log\frac{t_{0}-r_{0}}{s_{0}-r_{0}}, \tag*{(213)} ∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ([ r 0 , t + s 0 ]) t + s 0 − r 0 ∣ γ x ( t ) − z 0 ∣ ≲ L T − 2 1 − α r 0 − 2 α ρ log s 0 − r 0 t 0 − r 0 , (213) ∣ x 0 − y 0 ∣ t 0 − s 0 ∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ( [ r 0 , t + s 0 ] ) ( 1 + ∣ γ x ( t ) − z 0 ∣ ( t + s 0 − r 0 ) 1 / 2 ) ≲ L T − 1 − α 2 r 0 − α 2 ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1 / 2 ) , (214) \frac{\lvert x_{0}-y_{0}\rvert}{t_{0}-s_{0}}\int_{0}^{t_{0}-s_{0}}\mathrm{d}t \, \left\lVert\partial_{t}B \right\rVert_{L^{1}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])}\left(1+\frac{\lvert\gamma_{x}(t)-z_{0}\rvert}{(t+s_{0}-r_{0})^{1/2}}\right)
\lesssim LT^{-\frac{1-\alpha}{2}}r_{0}^{-\frac{\alpha}{2}}\lvert x_{0}-y_{0}\rvert\left(1+\rho(t_{0}-s_{0})^{-1/2}\right), \tag*{(214)} t 0 − s 0 ∣ x 0 − y 0 ∣ ∫ 0 t 0 − s 0 d t ∥ ∂ t B ∥ L t 1 L x ∞ ([ r 0 , t + s 0 ]) ( 1 + ( t + s 0 − r 0 ) 1/2 ∣ γ x ( t ) − z 0 ∣ ) ≲ L T − 2 1 − α r 0 − 2 α ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1/2 ) , (214) ∣ x 0 − y 0 ∣ t 0 − s 0 ∫ 0 t 0 − s 0 d t ∥ F ∥ L t 2 L x ∞ ( [ r 0 , t + s 0 ] ) ( 1 + ∣ γ x ( t ) − z 0 ∣ ( t + s 0 − r 0 ) 1 / 2 ) ≲ L T − 1 − α 2 r 0 − α 2 ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1 / 2 ) . (215) \frac{\lvert x_{0}-y_{0}\rvert}{t_{0}-s_{0}}\int_{0}^{t_{0}-s_{0}}\mathrm{d}t \left\lVert F \right\rVert_{L^{2}_{t}L^{\infty}_{x}([r_{0},t+s_{0}])}\left(1+\frac{\lvert\gamma_{x}(t)-z_{0}\rvert}{(t+s_{0}-r_{0})^{1/2}}\right)
\lesssim LT^{-\frac{1-\alpha}{2}}r_{0}^{-\frac{\alpha}{2}}\lvert x_{0}-y_{0}\rvert\left(1+\rho(t_{0}-s_{0})^{-1/2}\right). \tag*{(215)} t 0 − s 0 ∣ x 0 − y 0 ∣ ∫ 0 t 0 − s 0 d t ∥ F ∥ L t 2 L x ∞ ([ r 0 , t + s 0 ]) ( 1 + ( t + s 0 − r 0 ) 1/2 ∣ γ x ( t ) − z 0 ∣ ) ≲ L T − 2 1 − α r 0 − 2 α ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1/2 ) . (215) First, note that for any t ∈ [ 0 , t 0 − s 0 ] t \in[0,t_{0}-s_{0}] t ∈ [ 0 , t 0 − s 0 ]
∣ γ x ( t ) − z 0 ∣ ≤ max ( ∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣ ) = ρ , \lvert\gamma_{x}(t)-z_{0}\rvert\leq\max(\lvert x_{0}-z_{0}\rvert,\lvert y_{0}-z_{0}\rvert)=\rho, ∣ γ x ( t ) − z 0 ∣ ≤ max (∣ x 0 − z 0 ∣ , ∣ y 0 − z 0 ∣) = ρ , which follows because ( γ x ( t ) , t ∈ [ 0 , t 0 − s 0 ] ) (\gamma_{x}(t),t \in[0,t_{0}-s_{0}]) ( γ x ( t ) , t ∈ [ 0 , t 0 − s 0 ]) ( y 0 , x 0 ) (y_{0},x_{0}) ( y 0 , x 0 ) x 0 x_{0} x 0 y 0 y_{0} y 0 z 0 z_{0} z 0 (213) , we apply (191) to estimate the left-hand side of (213) by
L T − ( 1 − α ) / 2 r 0 − α / 2 ρ ∫ 0 t 0 − s 0 d t ( t + s 0 − r 0 ) − 1 ≲ L T − ( 1 − α ) / 2 r 0 − α / 2 ρ log t 0 − r 0 s 0 − r 0 , LT^{-(1-\alpha)/2}r_{0}^{-\alpha/2}\rho\int_{0}^{t_{0}-s_{0}}\mathrm{d}t\,(t+s_{0}-r_{0})^{-1}\lesssim LT^{-(1-\alpha)/2}r_{0}^{-\alpha/2}\rho\log\frac{t_{0}-r_{0}}{s_{0}-r_{0}}, L T − ( 1 − α ) /2 r 0 − α /2 ρ ∫ 0 t 0 − s 0 d t ( t + s 0 − r 0 ) − 1 ≲ L T − ( 1 − α ) /2 r 0 − α /2 ρ log s 0 − r 0 t 0 − r 0 , which proves (213) . Next, for (214) , we again apply (191) to estimate the left-hand side of (214) by
L T − ( 1 − α ) / 2 r 0 − α / 2 ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1 / 2 ) , LT^{-(1-\alpha)/2}r_{0}^{-\alpha/2}\lvert x_{0}-y_{0}\rvert\left(1+\rho(t_{0}-s_{0})^{-1/2}\right), L T − ( 1 − α ) /2 r 0 − α /2 ∣ x 0 − y 0 ∣ ( 1 + ρ ( t 0 − s 0 ) − 1/2 ) , as desired. Finally, (215) follows by an argument similar to that for (214) , where we replace the application of (191) by (192) .
Heat kernel parametrix
In this subsection, we obtain more refined short-time expansions for p B p_{B} p B q B q_{B} q B τ B \tau_{B} τ B Definition 3.35 ). While some versions (and extensions) of these results may be found in the textbook [BGV04], we prefer to give self-contained proofs here. We state the main results of this subsection (Proposition 4.39 and Corollary 4.41 ), and then proceed with the proofs. In the following, recall that p p p
Definition 4.38. Let σ : R d → [ 0 , 1 ] \sigma:\mathbb{R}^{d}\to[0,1] σ : R d → [ 0 , 1 ] σ ( x ) = 1 \sigma(x)=1 σ ( x ) = 1 ∣ x ∣ ≤ 1 / 20 \lvert x\rvert\leq1/20 ∣ x ∣ ≤ 1/20 σ ( x ) = 0 \sigma(x)=0 σ ( x ) = 0 ∣ x ∣ > 1 / 10 \lvert x\rvert>1/10 ∣ x ∣ > 1/10 σ \sigma σ T d \mathbb{T}^{d} T d ( s , y ) (s,y) ( s , y ) ( t , x ) (t,x) ( t , x ) σ ( s , y ; t , x ) = σ ( x − y ) \sigma(s,y;t,x)=\sigma(x-y) σ ( s , y ; t , x ) = σ ( x − y ) σ ( t , x ) = σ ( 0 , 0 ; t , x ) = σ ( x ) \sigma(t,x)=\sigma(0,0;t,x)=\sigma(x) σ ( t , x ) = σ ( 0 , 0 ; t , x ) = σ ( x )
A version of this proposition was already stated as Proposition 2.7 , but we choose to reproduce it here for the reader’s convenience (there is also the slight difference that the following proposition includes the spatial cutoff σ \sigma σ
Proposition 4.39. There is a constant c > 0 c>0 c > 0 t 0 ∈ ( 0 , 1 ] t_0\in(0,1] t 0 ∈ ( 0 , 1 ]
N ( B ) ≔ ( t 0 1 / 2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + ∥ F ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) ) 2 + t 0 1 / 2 ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) + t 0 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) ) . \mathcal{N}(B)\coloneqq\left(t_0^{1/2}\left(\|\partial_tB\|_{L_t^1L_x^\infty([0,t_0])}+\|F\|_{L_t^2L_x^\infty([0,t_0])}\right)\right)^2+t_0^{1/2}\|\partial_tB\|_{L_t^1L_x^\infty([0,t_0])}+t_0^{10}\left(1+\|F\|_{L_t^2L_x^\infty([0,t_0])}\right). N ( B ) : = ( t 0 1/2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + ∥ F ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) ) 2 + t 0 1/2 ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) + t 0 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ([ 0 , t 0 ]) ) . For any point x 0 ∈ T d x_0\in\mathbb{T}^d x 0 ∈ T d
∣ ρ B − ρ σ τ B ∣ ( t 0 , x 0 ) ≲ N ( B ) t 0 − d 2 exp ( − c ∣ x 0 ∣ 2 t 0 ) , (216) |\rho_B-\rho_{\sigma\tau B}|(t_0,x_0)\lesssim\mathcal{N}(B)t_0^{-\frac d2}\exp\left(-c\frac{|x_0|^2}{t_0}\right), \tag*{(216)} ∣ ρ B − ρ σ τ B ∣ ( t 0 , x 0 ) ≲ N ( B ) t 0 − 2 d exp ( − c t 0 ∣ x 0 ∣ 2 ) , (216) ∣ D ( ρ B − ρ σ τ B ) ∣ ( t 0 , x 0 ) ≲ N ( B ) ( 1 + ∥ F ∥ [ 0 , t 0 ] , 0 p e r t ) t 0 − d + 1 2 exp ( − c ∣ x 0 ∣ 2 t 0 ) . (217) |D(\rho_B-\rho_{\sigma\tau B})|(t_0,x_0)\lesssim\mathcal{N}(B)\left(1+\|F\|_{[0,t_0],0}^{\mathrm{pert}}\right)t_0^{-\frac{d+1}{2}}\exp\left(-c\frac{|x_0|^2}{t_0}\right). \tag*{(217)} ∣ D ( ρ B − ρ σ τ B ) ∣ ( t 0 , x 0 ) ≲ N ( B ) ( 1 + ∥ F ∥ [ 0 , t 0 ] , 0 pert ) t 0 − 2 d + 1 exp ( − c t 0 ∣ x 0 ∣ 2 ) . (217) Remark 4.40. One might naturally wonder whether higher order pointwise estimates also hold for the difference ρ B − ρ σ τ B \rho_B-\rho_{\sigma\tau B} ρ B − ρ σ τ B ρ B \rho_B ρ B q B q_B q B ρ B − q B \rho_B-q_B ρ B − q B ρ B \rho_B ρ B ρ σ τ B \rho_{\sigma\tau B} ρ σ τ B ρ σ τ B \rho_{\sigma\tau B} ρ σ τ B ρ B \rho_B ρ B ρ B − ρ σ τ B \rho_B-\rho_{\sigma\tau B} ρ B − ρ σ τ B ρ σ Δ B τ B \rho\sigma\Delta_B\tau B ρ σ Δ B τ B (222) for ρ B − ρ σ τ B \rho_B-\rho_{\sigma\tau B} ρ B − ρ σ τ B
We state the following corollary of Proposition 4.39 which will in fact be the result which is applied later (for instance, in § 10.1 ). Unlike the previous result, this corollary is stated for a general pair of space-time points ( s , y ) (s,y) ( s , y ) ( t , x ) (t,x) ( t , x ) t − s t-s t − s (218) (this assumption is similar to the one appearing in (212) ). This norm is stronger because we are placing less of a time weight on F F F D F DF D F
Corollary 4.41. Let T ≤ 1 T\leq1 T ≤ 1 ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d (s,y),(t,x)\in[0,T]\times\mathbb{T}^d ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d s < t s<t s < t α ∈ [ 0 , 1 ] \alpha\in[0,1] α ∈ [ 0 , 1 ] L ≥ 1 L\geq1 L ≥ 1
( T ( 1 − α ) / 2 ∥ u − ( 1 − α ) / 2 F ∥ T , 1 p e r t ) 2 ≤ L . (218) \left(T^{(1-\alpha)/2}\|u^{-(1-\alpha)/2}F\|_{T,1}^{\mathrm{pert}}\right)^2\leq L. \tag*{(218)} ( T ( 1 − α ) /2 ∥ u − ( 1 − α ) /2 F ∥ T , 1 pert ) 2 ≤ L . (218) For notational convenience, let
N ( B , L , s , t ) = L ( T − 1 ( t − s ) ) 1 − α s − α ( t − s ) α + ( ∥ ∂ u B ∥ L u 1 L x ∞ ( [ s , t ] ) + ∥ F ∥ L u 2 L x ∞ ( [ s , t ] ) ) ( t − s ) 1 / 2 . \mathcal{N}(B,L,s,t)=L(T^{-1}(t-s))^{1-\alpha}s^{-\alpha}(t-s)^\alpha+\left(\|\partial_uB\|_{L_u^1L_x^\infty([s,t])}+\|F\|_{L_u^2L_x^\infty([s,t])}\right)(t-s)^{1/2}. N ( B , L , s , t ) = L ( T − 1 ( t − s ) ) 1 − α s − α ( t − s ) α + ( ∥ ∂ u B ∥ L u 1 L x ∞ ([ s , t ]) + ∥ F ∥ L u 2 L x ∞ ([ s , t ]) ) ( t − s ) 1/2 . Then, we have that
∣ ( ρ B − ρ σ τ B ) ( s , y ; t , x ) ∣ ≲ N ( B , L , s , t ) ( t − s ) − d 2 exp ( − c ∣ x − y ∣ 2 t − s ) , (219) |(\rho_B-\rho_{\sigma\tau B})(s,y;t,x)|\lesssim\mathcal{N}(B,L,s,t)(t-s)^{-\frac d2}\exp\left(-c\frac{|x-y|^2}{t-s}\right), \tag*{(219)} ∣ ( ρ B − ρ σ τ B ) ( s , y ; t , x ) ∣ ≲ N ( B , L , s , t ) ( t − s ) − 2 d exp ( − c t − s ∣ x − y ∣ 2 ) , (219) ∣ ( D ρ B − ( ∇ ρ ) σ τ B ) ( s , y ; t , x ) ∣ ≲ ( 1 + ∥ F ∥ [ s , t ] , 0 p e r t ) N ( B , L , s , t ) ( t − s ) − d + 1 2 exp ( − c ∣ x − y ∣ 2 t − s ) . (220) |(D\rho_B-(\nabla\rho)_{\sigma\tau B})(s,y;t,x)|\lesssim\left(1+\|F\|_{[s,t],0}^{\mathrm{pert}}\right)\mathcal{N}(B,L,s,t)(t-s)^{-\frac{d+1}{2}}\exp\left(-c\frac{|x-y|^2}{t-s}\right). \tag*{(220)} ∣ ( D ρ B − ( ∇ ρ ) σ τ B ) ( s , y ; t , x ) ∣ ≲ ( 1 + ∥ F ∥ [ s , t ] , 0 pert ) N ( B , L , s , t ) ( t − s ) − 2 d + 1 exp ( − c t − s ∣ x − y ∣ 2 ) . (220) Furthermore, with C 0 C_0 C 0 0 ≤ k ≤ 1 0\leq k\leq1 0 ≤ k ≤ 1
∣ ( D ( k ) q B − ( ( ∇ ( k ) ρ ) σ τ B ) ) ( s , y ; t , x ) ∣ ≲ G r [ s , t ] ( C 0 F ) ( N ( B , L , s , t ) ( 1 + ∥ F ∥ [ s , t ] , 0 p e r t ) + ∥ F ∥ [ s , t ] , 1 p e r t ) ( t − s ) − d + k 2 exp ( − c ∣ x − y ∣ 2 t − s ) . (221) |(D^{(k)}q_B-((\nabla^{(k)}\rho)_{\sigma\tau B}))(s,y;t,x)|
\lesssim G_{r[s,t]}(C_0F)\left(\mathcal{N}(B,L,s,t)(1+\|F\|_{[s,t],0}^{\mathrm{pert}})+\|F\|_{[s,t],1}^{\mathrm{pert}}\right)(t-s)^{-\frac{d+k}{2}}\exp\left(-c\frac{|x-y|^2}{t-s}\right). \tag*{(221)} ∣ ( D ( k ) q B − (( ∇ ( k ) ρ ) σ τ B )) ( s , y ; t , x ) ∣ ≲ G r [ s , t ] ( C 0 F ) ( N ( B , L , s , t ) ( 1 + ∥ F ∥ [ s , t ] , 0 pert ) + ∥ F ∥ [ s , t ] , 1 pert ) ( t − s ) − 2 d + k exp ( − c t − s ∣ x − y ∣ 2 ) . (221) Next, we proceed to prove the main results. We start with Proposition 4.39. Since the proposition shows that ρ B \rho_B ρ B ρ σ τ B \rho_{\sigma\tau B} ρ σ τ B ρ σ τ B \rho_{\sigma\tau B} ρ σ τ B
( ∂ t − Δ B ) ( ρ σ τ B ) = ( ( ∂ t − Δ ) ( p σ ) τ B + p σ ( ∂ t − Δ B ) τ B − 2 ∂ ℓ ( p σ ) D ℓ τ B = − 2 ∂ ℓ p ( ∂ ℓ σ ) τ B − p ( Δ σ ) τ B + p σ ( ∂ t τ B − 2 ( ∂ ℓ p / p ) D ℓ τ B − Δ B τ B ) − 2 p ∂ ℓ σ D ℓ τ B . \begin{aligned}
(\partial_t-\Delta_B)(\rho_{\sigma\tau B}) \\
&=((\partial_t-\Delta)(p\sigma)\tau_B+p\sigma(\partial_t-\Delta_B)\tau_B-2\partial^\ell(p\sigma)D_\ell\tau_B \\
&=-2\partial^\ell p(\partial_\ell\sigma)\tau_B-p(\Delta\sigma)\tau_B+p\sigma\left(\partial_t\tau_B-2(\partial^\ell p/p)D_\ell\tau_B-\Delta_B\tau_B\right)-2p\partial^\ell\sigma D_\ell\tau_B.
\end{aligned} ( ∂ t − Δ B ) ( ρ σ τ B ) = (( ∂ t − Δ ) ( p σ ) τ B + p σ ( ∂ t − Δ B ) τ B − 2 ∂ ℓ ( p σ ) D ℓ τ B = − 2 ∂ ℓ p ( ∂ ℓ σ ) τ B − p ( Δ σ ) τ B + p σ ( ∂ t τ B − 2 ( ∂ ℓ p / p ) D ℓ τ B − Δ B τ B ) − 2 p ∂ ℓ σ D ℓ τ B . Now observe that
∂ t τ B − 2 ( ∂ ℓ p / p ) D ℓ τ B = ∂ t τ B + ( x ℓ / t ) D ℓ τ B − ( 2 ∂ ℓ p / p + x ℓ / t ) D ℓ τ B = D γ ˙ ( t ) τ B ( γ ( t ) ) − ( 2 ∂ ℓ p / p + x ℓ / t ) D ℓ τ B . \begin{aligned}
\partial_t\tau_B-2(\partial^\ell p/p)D_\ell\tau_B=\partial_t\tau_B+(x^\ell/t)D_\ell\tau_B-(2\partial^\ell p/p+x^\ell/t)D_\ell\tau_B \\
&=D_{\dot\gamma(t)}\tau_B(\gamma(t))-(2\partial^\ell p/p+x^\ell/t)D_\ell\tau_B.
\end{aligned} ∂ t τ B − 2 ( ∂ ℓ p / p ) D ℓ τ B = ∂ t τ B + ( x ℓ / t ) D ℓ τ B − ( 2 ∂ ℓ p / p + x ℓ / t ) D ℓ τ B = D γ ˙ ( t ) τ B ( γ ( t )) − ( 2 ∂ ℓ p / p + x ℓ / t ) D ℓ τ B . where for fixed x , t x,t x , t γ \gamma γ γ ( s ) = s ( 1 , x / t ) \gamma(s)=s(1,x/t) γ ( s ) = s ( 1 , x / t ) σ ≠ 0 \sigma\ne0 σ = 0 x x x
R : = 2 ∇ p / p + x / t . R\mathrel{\mathop:}=2\nabla p/p+x/t. R : = 2∇ p / p + x / t . We thus obtain
( ∂ t − Δ B ) ( p σ τ B ) = − 2 ∂ ℓ p ( ∂ ℓ σ ) τ B − p σ Δ B τ B − p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B − p ( Δ σ ) τ B , (\partial_t-\Delta_B)(p\sigma\tau_B)=-2\partial^\ell p(\partial_\ell\sigma)\tau_B-p\sigma\Delta_B\tau_B-p(2\partial^\ell\sigma+\sigma R^\ell)D_\ell\tau_B-p(\Delta\sigma)\tau_B, ( ∂ t − Δ B ) ( p σ τ B ) = − 2 ∂ ℓ p ( ∂ ℓ σ ) τ B − p σ Δ B τ B − p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B − p ( Δ σ ) τ B , and so the difference p B − p σ τ B p_B-p\sigma\tau_B p B − p σ τ B
( ∂ t − Δ B ) ( p B − p σ τ B ) = 2 ∂ ℓ p ( ∂ ℓ σ ) τ B + p σ Δ B τ B + p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B + p ( Δ σ ) τ B . (222) (\partial_t-\Delta_B)(p_B-p\sigma\tau_B)=2\partial^\ell p(\partial_\ell\sigma)\tau_B+p\sigma\Delta_B\tau_B+p(2\partial^\ell\sigma+\sigma R^\ell)D_\ell\tau_B+p(\Delta\sigma)\tau_B. \tag*{(222)} ( ∂ t − Δ B ) ( p B − p σ τ B ) = 2 ∂ ℓ p ( ∂ ℓ σ ) τ B + p σ Δ B τ B + p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B + p ( Δ σ ) τ B . (222) ( p B − p σ τ B ) ( 0 ) = 0. (p_B-p\sigma\tau_B)(0)=0. ( p B − p σ τ B ) ( 0 ) = 0. Thus in order to estimate the difference p B − p σ τ B p_B-p\sigma\tau_B p B − p σ τ B D τ B D\tau_B D τ B Δ B τ B \Delta_B\tau_B Δ B τ B
Proof of Proposition 4.39. By (222) , we have that
( p B − p σ τ B ) ( t 0 , x 0 ) = I m + I r , (p_B-p\sigma\tau_B)(t_0,x_0)=I_m+I_r, ( p B − p σ τ B ) ( t 0 , x 0 ) = I m + I r , where (here ‘m’ stands for main, and ‘r’ stands for remainder)
I m : = ∫ d t d x p B ( t , x ; t 0 , x 0 ) ( p σ Δ B τ B ) ( t , x ) , I_m\mathrel{\mathop:}=\int dt\,dx\,p_B(t,x;t_0,x_0)(p\sigma\Delta_B\tau_B)(t,x), I m : = ∫ d t d x p B ( t , x ; t 0 , x 0 ) ( p σ Δ B τ B ) ( t , x ) , I r : = ∫ d t d x p B ( t , x ; t 0 , x 0 ) ( 2 ∂ ℓ p ( ∂ ℓ σ ) τ B + p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B + p ( Δ σ ) τ B ) ( t , x ) . I_r\mathrel{\mathop:}=\int dt\,dx\,p_B(t,x;t_0,x_0)\left(2\partial^\ell p(\partial_\ell\sigma)\tau_B+p(2\partial^\ell\sigma+\sigma R^\ell)D_\ell\tau_B+p(\Delta\sigma)\tau_B\right)(t,x). I r : = ∫ d t d x p B ( t , x ; t 0 , x 0 ) ( 2 ∂ ℓ p ( ∂ ℓ σ ) τ B + p ( 2 ∂ ℓ σ + σ R ℓ ) D ℓ τ B + p ( Δ σ ) τ B ) ( t , x ) . Applying the diamagnetic inequality (4.17) and inserting our pointwise estimate of Δ B τ B \Delta_B\tau_B Δ B τ B
∣ I m ∣ ≲ ∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) J m ( t , x ) , |I_m|\lesssim\int dt\,dx\,p(t,x;t_0,x_0)p(t,x)J_m(t,x), ∣ I m ∣ ≲ ∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) J m ( t , x ) , where
J m ( t , x ) : = ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ 0 , t ] ) ) 2 ( 1 + ∣ x ∣ t − 1 / 2 ) 2 + ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t ] ) ∣ x ∣ t − 1 . J_m(t,x)\mathrel{\mathop:}=\left(\|\partial_tB\|_{L^1_tL^\infty_x([0,t])}+\|F\|_{L^2_tL^\infty_x([0,t])}\right)^2(1+|x|t^{-1/2})^2+\|\partial_tB\|_{L^1_tL^\infty_x([0,t])}|x|t^{-1}. J m ( t , x ) : = ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t ]) + ∥ F ∥ L t 2 L x ∞ ([ 0 , t ]) ) 2 ( 1 + ∣ x ∣ t − 1/2 ) 2 + ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t ]) ∣ x ∣ t − 1 . Combining this with the estimates
∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) ∣ x ∣ t − 1 ≲ t 0 1 / 2 t 0 − d / 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , \int dt\,dx\,p(t,x;t_0,x_0)p(t,x)|x|t^{-1}\lesssim t_0^{1/2}t_0^{-d/2}\exp(-c|x_0|^2/t_0), ∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) ∣ x ∣ t − 1 ≲ t 0 1/2 t 0 − d /2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , ∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) ( 1 + ∣ x ∣ t − 1 / 2 ) 2 ≲ t 0 t 0 − d / 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , \int dt\,dx\,p(t,x;t_0,x_0)p(t,x)(1+|x|t^{-1/2})^2\lesssim t_0t_0^{-d/2}\exp(-c|x_0|^2/t_0), ∫ d t d x p ( t , x ; t 0 , x 0 ) p ( t , x ) ( 1 + ∣ x ∣ t − 1/2 ) 2 ≲ t 0 t 0 − d /2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) , we may obtain an acceptable bound on I m I_m I m I r I_r I r τ B \tau_B τ B (194) combined with (208) and Lemma 3.49, we may estimate
∣ I r ∣ ≲ ∫ d t d x p ( t , x ; t 0 , x 0 ) ( ∣ ∂ p ∣ ∣ ∂ σ ∣ + p ( ∣ ∂ σ ∣ + t 10 ) J r + p ∣ Δ σ ∣ ) ( t , x ) . |I_r| \lesssim\int\mathrm{d}t\mathrm{d}x\ p(t,x;t_0,x_0)(|\partial p||\partial\sigma|+p(|\partial\sigma|+t^{10})J_r+p|\Delta\sigma|)(t,x). ∣ I r ∣ ≲ ∫ d t d x p ( t , x ; t 0 , x 0 ) ( ∣ ∂ p ∣∣ ∂ σ ∣ + p ( ∣ ∂ σ ∣ + t 10 ) J r + p ∣Δ σ ∣ ) ( t , x ) . where
J r ( t , x ) ≔ ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ 0 , t ] ) ) ( 1 + ∣ x ∣ t − 1 / 2 ) . J_r(t,x)\coloneqq\left(\|\partial_tB\|_{L_t^1L_x^\infty([0,t])}+\|F\|_{L_t^2L_x^\infty([0,t])}\right)(1+|x|t^{-1/2}). J r ( t , x ) : = ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t ]) + ∥ F ∥ L t 2 L x ∞ ([ 0 , t ]) ) ( 1 + ∣ x ∣ t − 1/2 ) . Note that ∂ σ \partial\sigma ∂ σ Δ σ \Delta\sigma Δ σ ∣ x ∣ ∼ 1 |x|\sim1 ∣ x ∣ ∼ 1
( ∣ ∂ p ∣ ∣ ∂ σ ∣ ) ( t , x ) , ( p ∣ ∂ σ ∣ ) ( t , x ) , ( p ∣ Δ σ ∣ ) ( t , x ) ≲ t 10 exp ( − c ∣ x ∣ 2 / t ) . (|\partial p||\partial\sigma|)(t,x),\quad(p|\partial\sigma|)(t,x),\quad(p|\Delta\sigma|)(t,x)\lesssim t^{10}\exp(-c|x|^2/t). ( ∣ ∂ p ∣∣ ∂ σ ∣ ) ( t , x ) , ( p ∣ ∂ σ ∣ ) ( t , x ) , ( p ∣Δ σ ∣ ) ( t , x ) ≲ t 10 exp ( − c ∣ x ∣ 2 / t ) . Combining, we may obtain the further estimate
∣ I r ∣ ≲ t 0 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ( [ 0 , t 0 ] ) + ∥ ∂ t B ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ) t 0 − d / 2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . |I_r|\lesssim t_0^{10}(1+\|F\|_{L_t^2L_x^\infty([0,t_0])}+\|\partial_tB\|_{L_t^1L_x^\infty([0,t_0])})t_0^{-d/2}\exp(-c|x_0|^2/t_0). ∣ I r ∣ ≲ t 0 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ([ 0 , t 0 ]) + ∥ ∂ t B ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ) t 0 − d /2 exp ( − c ∣ x 0 ∣ 2 / t 0 ) . which is acceptable. Combining our estimates of I m I_m I m I r I_r I r (216) . For the first order estimate (217) , we proceed similarly, where instead of applying the diamagnetic inequality ∣ p B ( t , x ; t 0 , x 0 ) ∣ ≤ p ( t , x ; t 0 , x 0 ) |p_B(t,x;t_0,x_0)|\leq p(t,x;t_0,x_0) ∣ p B ( t , x ; t 0 , x 0 ) ∣ ≤ p ( t , x ; t 0 , x 0 ) (195) to bound
∣ D p B ( t , x ; t 0 , x 0 ) ∣ ≲ ( 1 + ∥ F ∥ [ t , t 0 ] , 0 p e r t ) ( t 0 − t ) − ( d + 1 ) / 2 exp ( − c ∣ x 0 − x ∣ 2 t 0 − t ) . |D p_B(t,x;t_0,x_0)|\lesssim(1+\|F\|^{\mathrm{pert}}_{[t,t_0],0})(t_0-t)^{-(d+1)/2}\exp\left(-c\frac{|x_0-x|^2}{t_0-t}\right). ∣ D p B ( t , x ; t 0 , x 0 ) ∣ ≲ ( 1 + ∥ F ∥ [ t , t 0 ] , 0 pert ) ( t 0 − t ) − ( d + 1 ) /2 exp ( − c t 0 − t ∣ x 0 − x ∣ 2 ) . The remainder of the proof proceeds analogously; we omit the details.
Having proven Proposition 4.39, we next show how Corollary 4.41 follows from Proposition 4.39, Proposition 4.18, and Lemma 4.36.
Proof of Corollary 4.41. By Proposition 4.39, we have that for k = 0 , 1 k=0,1 k = 0 , 1
∣ D ( k ) ( p B − p σ τ B ) ( s , y ; t , x ) ∣ ≲ ( 1 + ∥ F ∥ [ s , t ] , 0 p e r t ) k N ( B ) × ( t − s ) − ( d + k ) / 2 exp ( − c ∣ x − y ∣ 2 t − s ) . |D^{(k)}(p_B-p\sigma\tau_B)(s,y;t,x)|\lesssim(1+\|F\|^{\mathrm{pert}}_{[s,t],0})^k\mathcal{N}(B)\times(t-s)^{-(d+k)/2}\exp\left(-c\frac{|x-y|^2}{t-s}\right). ∣ D ( k ) ( p B − p σ τ B ) ( s , y ; t , x ) ∣ ≲ ( 1 + ∥ F ∥ [ s , t ] , 0 pert ) k N ( B ) × ( t − s ) − ( d + k ) /2 exp ( − c t − s ∣ x − y ∣ 2 ) . where
N ( B ) = ( ( t − s ) 1 / 2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ s , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ s , t ] ) ) ) 2 + ( t − s ) 1 / 2 ∥ ∂ t B ∥ L t 1 L x ∞ ( [ s , t ] ) + ( t − s ) 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ( [ s , t ] ) ) . \mathcal{N}(B)=\left((t-s)^{1/2}(\|\partial_tB\|_{L_t^1L_x^\infty([s,t])}+\|F\|_{L_t^2L_x^\infty([s,t])})\right)^2
+(t-s)^{1/2}\|\partial_tB\|_{L_t^1L_x^\infty([s,t])}+(t-s)^{10}(1+\|F\|_{L_t^2L_x^\infty([s,t])}). N ( B ) = ( ( t − s ) 1/2 ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ s , t ]) + ∥ F ∥ L t 2 L x ∞ ([ s , t ]) ) ) 2 + ( t − s ) 1/2 ∥ ∂ t B ∥ L t 1 L x ∞ ([ s , t ]) + ( t − s ) 10 ( 1 + ∥ F ∥ L t 2 L x ∞ ([ s , t ]) ) . Combining (191) and (192) , we may estimate
N ( B ) ≲ L T − ( 1 − α ) ( t − s ) s − α + ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ s , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ s , t ] ) ) ( t − s ) 1 / 2 . \mathcal{N}(B)\lesssim LT^{-(1-\alpha)}(t-s)s^{-\alpha}+(\|\partial_tB\|_{L_t^1L_x^\infty([s,t])}+\|F\|_{L_t^2L_x^\infty([s,t])})(t-s)^{1/2}. N ( B ) ≲ L T − ( 1 − α ) ( t − s ) s − α + ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ s , t ]) + ∥ F ∥ L t 2 L x ∞ ([ s , t ]) ) ( t − s ) 1/2 . Combining, we obtain the estimate (219) . For (220) , we additionally estimate (applying (208) , (191) and (192) )
∣ ( D ( p σ τ B ) − ( ∇ p ) σ τ B ) ( s , y ; t , x ) ∣ ≤ ∣ ( p σ D τ B ) ( s , y ; t , x ) ∣ + ∣ ( p ( ∇ σ ) τ B ) ( s , y ; t , x ) ∣ ≲ ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ s , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ s , t ] ) ) ( 1 + ∣ x − y ∣ ( t − s ) 1 / 2 ) p ( s , y ; t , x ) + ∣ ∇ σ ( x − y ) ∣ p ( s , y ; t , x ) \begin{aligned}
\left|\left(D(p\sigma\tau_B)-(\nabla p)\sigma\tau_B\right)(s,y;t,x)\right|
&\leq|(p\sigma D\tau_B)(s,y;t,x)|+|(p(\nabla\sigma)\tau_B)(s,y;t,x)|\\
&\lesssim\left(\|\partial_tB\|_{L_t^1L_x^\infty([s,t])}+\|F\|_{L_t^2L_x^\infty([s,t])}\right)
\left(1+\frac{|x-y|}{(t-s)^{1/2}}\right)p(s,y;t,x)+|\nabla\sigma(x-y)|p(s,y;t,x)
\end{aligned} ∣ ( D ( p σ τ B ) − ( ∇ p ) σ τ B ) ( s , y ; t , x ) ∣ ≤ ∣ ( p σ D τ B ) ( s , y ; t , x ) ∣ + ∣ ( p ( ∇ σ ) τ B ) ( s , y ; t , x ) ∣ ≲ ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ s , t ]) + ∥ F ∥ L t 2 L x ∞ ([ s , t ]) ) ( 1 + ( t − s ) 1/2 ∣ x − y ∣ ) p ( s , y ; t , x ) + ∣∇ σ ( x − y ) ∣ p ( s , y ; t , x ) ≲ ( ∥ ∂ t B ∥ L t 1 L x ∞ ( [ s , t ] ) + ∥ F ∥ L t 2 L x ∞ ( [ s , t ] ) + ( t − s ) 10 ) ( t − s ) 1 / 2 ( t − s ) − d + 1 2 exp ( − c ∣ x − y ∣ 2 t − s ) , \lesssim\left(\|\partial_tB\|_{L_t^1L_x^\infty([s,t])}+\|F\|_{L_t^2L_x^\infty([s,t])}+(t-s)^{10}\right)(t-s)^{1/2}(t-s)^{-\frac{d+1}{2}}\exp\left(-c\frac{|x-y|^2}{t-s}\right), ≲ ( ∥ ∂ t B ∥ L t 1 L x ∞ ([ s , t ]) + ∥ F ∥ L t 2 L x ∞ ([ s , t ]) + ( t − s ) 10 ) ( t − s ) 1/2 ( t − s ) − 2 d + 1 exp ( − c t − s ∣ x − y ∣ 2 ) , where we also used that ∇ σ ( x − y ) \nabla\sigma(x-y) ∇ σ ( x − y ) ∣ x − y ∣ ∼ 1 |x-y|\sim1 ∣ x − y ∣ ∼ 1 x , y x,y x , y p ( s , y ; t , x ) ≲ ( t − s ) 10 ( t − s ) − d / 2 exp ( − c ∣ x − y ∣ 2 / ( t − s ) ) p(s,y;t,x)\lesssim(t-s)^{10}(t-s)^{-d/2}\exp(-c|x-y|^2/(t-s)) p ( s , y ; t , x ) ≲ ( t − s ) 10 ( t − s ) − d /2 exp ( − c ∣ x − y ∣ 2 / ( t − s )) (220) now follows by combining our previous few estimates.
Finally, for the estimate (221) , by (197) , we have that
∣ D ( k ) ( q B − p B ) ( s , y ; t , x ) ∣ ≲ Gr [ s , t ] ( C 0 F ) ∥ F ∥ [ s , t ] , 1 p e r t ( t − s ) − d + k 2 exp ( − c ∣ x − y ∣ 2 t − s ) , \left|D^{(k)}(q_B-p_B)(s,y;t,x)\right|\lesssim\operatorname{Gr}_{[s,t]}(C_0F)\|F\|^{\mathrm{pert}}_{[s,t],1}(t-s)^{-\frac{d+k}{2}}\exp\left(-c\frac{|x-y|^2}{t-s}\right), D ( k ) ( q B − p B ) ( s , y ; t , x ) ≲ Gr [ s , t ] ( C 0 F ) ∥ F ∥ [ s , t ] , 1 pert ( t − s ) − 2 d + k exp ( − c t − s ∣ x − y ∣ 2 ) , which is acceptable. Combining this with (219) and (220) , we obtain the desired result.
Perturbing around a Yang–Mills connection
In this section, we discuss a different approach towards approximating the covariant heat kernel q B q_B q B B B B
As a starting point, it turns out that when B B B R 2 \mathbb{R}^2 R 2 q B q_B q B q B q_B q B B B B
As in § 5.1 and § 5.4 , throughout we fix the base point ( s , y ) = ( 0 , 0 ) (s,y)=(0,0) ( s , y ) = ( 0 , 0 ) T 2 ≅ ( − π , π ] 2 ⊆ R 2 \mathbb{T}^2\cong(-\pi,\pi]^2\subseteq\mathbb{R}^2 T 2 ≅ ( − π , π ] 2 ⊆ R 2 0 ∈ R 2 0\in\mathbb{R}^2 0 ∈ R 2
Notation 4.42. Define the angular and radial vector fields e θ e_\theta e θ e r e_r e r R 2 ∖ { 0 } \mathbb{R}^2\setminus\{0\} R 2 ∖ { 0 }
e θ ( x ) ≔ ( − x 2 , x 1 ) , e r ( x ) ≔ x , e_\theta(x)\coloneqq(-x_2,x_1),\qquad e_r(x)\coloneqq x, e θ ( x ) : = ( − x 2 , x 1 ) , e r ( x ) : = x , as well as the function r ( x ) ≔ ∣ x ∣ r(x)\coloneqq|x| r ( x ) : = ∣ x ∣ e θ e_\theta e θ e r e_r e r 0 0 0 T 2 \mathbb{T}^2 T 2 ∣ x ∣ ≤ 1 / 10 |x|\leq1/10 ∣ x ∣ ≤ 1/10 ∂ θ \partial_\theta ∂ θ ∂ e θ \partial_{e_\theta} ∂ e θ e θ e_\theta e θ ∂ r \partial_r ∂ r ∂ r − 1 e r \partial_{r^{-1}e_r} ∂ r − 1 e r ∂ θ \partial_\theta ∂ θ ∂ r \partial_r ∂ r ∂ e r \partial_{e_r} ∂ e r
For the moment, we will work on R 2 \mathbb{R}^2 R 2 λ ≠ 0 \lambda\neq0 λ = 0 m λ : ( 0 , ∞ ) × R 2 → [ 0 , ∞ ) m_\lambda:(0,\infty)\times\mathbb{R}^2\to[0,\infty) m λ : ( 0 , ∞ ) × R 2 → [ 0 , ∞ )
m λ ( t , x ) = λ 4 π sinh ( λ t ) exp ( − λ ∣ x ∣ 2 4 tanh ( λ t ) ) . (223) m_\lambda(t,x)=\frac{\lambda}{4\pi\sinh(\lambda t)}\exp\left(-\frac{\lambda|x|^2}{4\tanh(\lambda t)}\right). \tag*{(223)} m λ ( t , x ) = 4 π sinh ( λ t ) λ exp ( − 4 tanh ( λ t ) λ ∣ x ∣ 2 ) . (223) When λ = 0 \lambda=0 λ = 0 m 0 = p E u c m_0=p^{\mathrm{Euc}} m 0 = p Euc R 2 \mathbb{R}^2 R 2 m λ → m 0 m_\lambda\to m_0 m λ → m 0 λ → 0 \lambda\to0 λ → 0
( ∂ t − Δ + λ 2 4 ∣ x ∣ 2 ) m λ = 0 , m λ ( 0 ) = δ 0 . (224) \left(\partial_t-\Delta+\frac{\lambda^2}{4}|x|^2\right)m_\lambda=0,\qquad m_\lambda(0)=\delta_0. \tag*{(224)} ( ∂ t − Δ + 4 λ 2 ∣ x ∣ 2 ) m λ = 0 , m λ ( 0 ) = δ 0 . (224) Because it will be needed later (in the proof of the main result of this section, Proposition 4.51), for ( s , y ) ∈ [ 0 , ∞ ) × R 2 (s,y)\in[0,\infty)\times\mathbb{R}^2 ( s , y ) ∈ [ 0 , ∞ ) × R 2 m λ 0 ( s , y ; ⋅ , ⋅ ) : ( s , ∞ ) × R 2 → C m_\lambda^0(s,y;\cdot,\cdot):(s,\infty)\times\mathbb{R}^2\to\mathbb{C} m λ 0 ( s , y ; ⋅ , ⋅ ) : ( s , ∞ ) × R 2 → C
m λ θ ( s , y ; t , x ) = { λ 4 π sinh ( λ ( t − s ) ) exp ( − λ ∣ x − y ∣ 2 4 tanh ( λ ( t − s ) ) ) exp ( i λ 2 ( x 1 y 2 − x 2 y 1 ) ) λ ≠ 0 , μ E u c ( s , y ; t , x ) λ = 0. m^\theta_\lambda(s,y;t,x)=\begin{cases}
\frac{\lambda}{4\pi\sinh(\lambda(t-s))}\exp(-\frac{\lambda|x-y|^2}{4\tanh(\lambda(t-s))})\exp(\mathrm{i}\frac{\lambda}{2}(x_1y_2-x_2y_1)) & \lambda\ne0,\\
\mu^{\mathrm{Euc}}(s,y;t,x) & \lambda=0.
\end{cases} m λ θ ( s , y ; t , x ) = { 4 π s i n h ( λ ( t − s )) λ exp ( − 4 t a n h ( λ ( t − s )) λ ∣ x − y ∣ 2 ) exp ( i 2 λ ( x 1 y 2 − x 2 y 1 )) μ Euc ( s , y ; t , x ) λ = 0 , λ = 0. When ( s , y ) = ( 0 , 0 ) (s,y)=(0,0) ( s , y ) = ( 0 , 0 ) m λ θ ( 0 , 0 ; t , x ) = m λ ( t , x ) m^\theta_\lambda(0,0;t,x)=m_\lambda(t,x) m λ θ ( 0 , 0 ; t , x ) = m λ ( t , x ) m λ m_\lambda m λ m λ θ ( s , y ; t , x ) m^\theta_\lambda(s,y;t,x) m λ θ ( s , y ; t , x ) ( s , y ) (s,y) ( s , y ) m λ θ m^\theta_\lambda m λ θ m λ θ ( s , y ; ⋅ , ⋅ ) m^\theta_\lambda(s,y;\mathord\cdot,\mathord\cdot) m λ θ ( s , y ; ⋅ , ⋅ )
( ∂ t − Δ + λ 2 4 ∣ x ∣ 2 − i λ ∂ θ ) m λ θ ( s , y ; ⋅ , ⋅ ) = 0 , m λ θ ( s , y ; s , ⋅ ) = δ y ( ⋅ ) . (225) \left(\partial_t-\Delta+\frac{\lambda^2}{4}|x|^2-\mathrm{i}\lambda\partial_\theta\right)m^\theta_\lambda(s,y;\mathord\cdot,\mathord\cdot)=0,\qquad m^\theta_\lambda(s,y;s,\mathord\cdot)=\delta_y(\mathord\cdot). \tag*{(225)} ( ∂ t − Δ + 4 λ 2 ∣ x ∣ 2 − i λ ∂ θ ) m λ θ ( s , y ; ⋅ , ⋅ ) = 0 , m λ θ ( s , y ; s , ⋅ ) = δ y ( ⋅ ) . (225) To describe q B q_B q B (224) and (225) . For ease of notation, we make the following definition regarding operators and kernels that will frequently appear.
Definition 4.43. Let X ∈ g X\in\mathfrak{g} X ∈ g
L X ≔ Δ + 1 4 ∣ x ∣ 2 π ∗ ( X ) 2 , L X θ ≔ Δ + 1 4 ∣ x ∣ 2 π ∗ ( X ) 2 + π ∗ ( X ) ∂ θ , \mathcal{L}_X\coloneqq\Delta+\frac{1}{4}|x|^2\pi_*(X)^2,\qquad\mathcal{L}^\theta_X\coloneqq\Delta+\frac{1}{4}|x|^2\pi_*(X)^2+\pi_*(X)\partial_\theta, L X : = Δ + 4 1 ∣ x ∣ 2 π ∗ ( X ) 2 , L X θ : = Δ + 4 1 ∣ x ∣ 2 π ∗ ( X ) 2 + π ∗ ( X ) ∂ θ , acting on functions ϕ : T 2 → W \phi:\mathbb{T}^2\to W ϕ : T 2 → W ϕ : R 2 → W \phi:\mathbb{R}^2\to W ϕ : R 2 → W ϕ \phi ϕ 0 0 0 W 2 W^2 W 2 Rot ( X ) ∈ End ( W 2 ) \operatorname{Rot}(X)\in\operatorname{End}(W^2) Rot ( X ) ∈ End ( W 2 ) Rot ( X ) ( u , v ) = ( π ∗ ( X ) v , − π ∗ ( X ) u ) \operatorname{Rot}(X)(u,v)=(\pi_*(X)v,-\pi_*(X)u) Rot ( X ) ( u , v ) = ( π ∗ ( X ) v , − π ∗ ( X ) u ) u , v ∈ W u,v\in W u , v ∈ W Rot ( X ) \operatorname{Rot}(X) Rot ( X )
L X F ≔ L X + 2 Rot ( X ) = Δ + 2 Rot ( X ) + 1 4 ∣ x ∣ 2 π ∗ ( X ) 2 , \mathcal{L}^F_X\coloneqq\mathcal{L}_X+2\operatorname{Rot}(X)=\Delta+2\operatorname{Rot}(X)+\frac{1}{4}|x|^2\pi_*(X)^2, L X F : = L X + 2 Rot ( X ) = Δ + 2 Rot ( X ) + 4 1 ∣ x ∣ 2 π ∗ ( X ) 2 , L X F , θ ≔ L X θ + 2 Rot ( X ) = Δ + 2 Rot ( X ) + 1 4 ∣ x ∣ 2 π ∗ ( X ) 2 + π ∗ ( X ) ∂ θ . \mathcal{L}^{F,\theta}_X\coloneqq\mathcal{L}^\theta_X+2\operatorname{Rot}(X)=\Delta+2\operatorname{Rot}(X)+\frac{1}{4}|x|^2\pi_*(X)^2+\pi_*(X)\partial_\theta. L X F , θ : = L X θ + 2 Rot ( X ) = Δ + 2 Rot ( X ) + 4 1 ∣ x ∣ 2 π ∗ ( X ) 2 + π ∗ ( X ) ∂ θ . In the above, we abuse notation and also view π ∗ ( X ) ∈ End ( W 2 ) \pi_*(X)\in\operatorname{End}(W^2) π ∗ ( X ) ∈ End ( W 2 ) ( u , v ) ↦ ( π ∗ ( X ) u , π ∗ ( X ) v ) (u,v)\mapsto(\pi_*(X)u,\pi_*(X)v) ( u , v ) ↦ ( π ∗ ( X ) u , π ∗ ( X ) v )
In the following, we will often abuse notation and write X X X π ∗ ( X ) \pi_*(X) π ∗ ( X )
Remark 4.44. Note that in two dimensions, the curvature F B F_B F B W 2 W^2 W 2 F B 1 = − F B 2 F^1_B=-F^2_B F B 1 = − F B 2 Rot ( F B 1 ) = F B \operatorname{Rot}(F^1_B)=F_B Rot ( F B 1 ) = F B X ∈ g X\in\mathfrak{g} X ∈ g Rot ( X ) \operatorname{Rot}(X) Rot ( X ) W 2 W^2 W 2
As we will see later on, the action of the operators L X , L X θ \mathcal{L}_X,\mathcal{L}^\theta_X L X , L X θ X X X W W W X X X (224) and (225) . This will provide the link between q B q_B q B
Definition 4.45. Let X ∈ g X\in\mathfrak{g} X ∈ g s X ( ⋅ , ⋅ ) : [ 0 , ∞ ) × R 2 → End ( W ) s_X(\mathord\cdot,\mathord\cdot):[0,\infty)\times\mathbb{R}^2\to\operatorname{End}(W) s X ( ⋅ , ⋅ ) : [ 0 , ∞ ) × R 2 → End ( W ) s X F ( ⋅ , ⋅ ) : [ 0 , ∞ ) × R 2 → End ( W 2 ) s^F_X(\mathord\cdot,\mathord\cdot):[0,\infty)\times\mathbb{R}^2\to\operatorname{End}(W^2) s X F ( ⋅ , ⋅ ) : [ 0 , ∞ ) × R 2 → End ( W 2 )
( ∂ t − L X ) s X = 0 , s X ( 0 ) = δ 0 id W , (\partial_t-\mathcal{L}_X)s_X=0,\qquad s_X(0)=\delta_0\operatorname{id}_W, ( ∂ t − L X ) s X = 0 , s X ( 0 ) = δ 0 id W , ( ∂ t − L X F ) s X F = 0 , s X F ( 0 ) = δ 0 id W 2 , (\partial_t-\mathcal{L}^F_X)s^F_X=0,\qquad s^F_X(0)=\delta_0\operatorname{id}_{W^2}, ( ∂ t − L X F ) s X F = 0 , s X F ( 0 ) = δ 0 id W 2 , where id W \operatorname{id}_{W} id W id W 2 \operatorname{id}_{W^{2}} id W 2 W W W W 2 W^{2} W 2 L X θ L_{X}^{\theta} L X θ L X F , θ L_{X}^{F,\theta} L X F , θ s X θ \mathfrak{s}_{X}^{\theta} s X θ s X F , θ \mathfrak{s}_{X}^{F,\theta} s X F , θ W W W W 2 W^{2} W 2 s X θ ( ⋅ , ⋅ ; ⋅ , ⋅ ) ∈ End ( W C ) \mathfrak{s}_{X}^{\theta}(\mathord{\cdot},\mathord{\cdot};\mathord{\cdot},\mathord{\cdot}) \in\operatorname{End}(W_{\mathbb{C}}) s X θ ( ⋅ , ⋅ ; ⋅ , ⋅ ) ∈ End ( W C ) s X F , θ ( ⋅ , ⋅ ; ⋅ , ⋅ ) ∈ End ( W C 2 ) \mathfrak{s}_{X}^{F,\theta}(\mathord{\cdot},\mathord{\cdot};\mathord{\cdot},\mathord{\cdot}) \in\operatorname{End}(W_{\mathbb{C}}^{2}) s X F , θ ( ⋅ , ⋅ ; ⋅ , ⋅ ) ∈ End ( W C 2 ) W C W_{\mathbb{C}} W C W W W
( ∂ t − L X θ ) s X θ ( s , y ; ⋅ , ⋅ ) = 0 , s X θ ( s , y ; s , ⋅ ) = δ y id W C , ( ∂ t − L X F , θ ) s X F , θ ( s , y ; ⋅ , ⋅ ) = 0 , s X F , θ ( s , y ; s , ⋅ ) = δ y id W C 2 . (226) \begin{aligned}
(\partial_t-L_{X}^{\theta})\mathfrak{s}_{X}^{\theta}(s,y;\mathord{\cdot},\mathord{\cdot})&=0, & \mathfrak{s}_{X}^{\theta}(s,y;s,\mathord{\cdot})&=\delta_y\operatorname{id}_{W_{\mathbb{C}}},\\
(\partial_t-L_{X}^{F,\theta})\mathfrak{s}_{X}^{F,\theta}(s,y;\mathord{\cdot},\mathord{\cdot})&=0, & \mathfrak{s}_{X}^{F,\theta}(s,y;s,\mathord{\cdot})&=\delta_y\operatorname{id}_{W_{\mathbb{C}}^{2}}. \tag*{(226)}
\end{aligned} ( ∂ t − L X θ ) s X θ ( s , y ; ⋅ , ⋅ ) ( ∂ t − L X F , θ ) s X F , θ ( s , y ; ⋅ , ⋅ ) = 0 , = 0 , s X θ ( s , y ; s , ⋅ ) s X F , θ ( s , y ; s , ⋅ ) = δ y id W C , = δ y id W C 2 . (226) for any s ≥ 0 s \ge0 s ≥ 0 y ∈ R 2 y \in\mathbb{R}^{2} y ∈ R 2
We claim that both s X \mathfrak{s}_{X} s X s X F \mathfrak{s}_{X}^{F} s X F m λ m_{\lambda} m λ s X θ \mathfrak{s}_{X}^{\theta} s X θ s X F , θ \mathfrak{s}_{X}^{F,\theta} s X F , θ m λ θ m_{\lambda}^{\theta} m λ θ m λ θ m_{\lambda}^{\theta} m λ θ W C W_{\mathbb{C}} W C W C 2 W_{\mathbb{C}}^{2} W C 2
In the following discussion, we focus primarily on s X \mathfrak{s}_{X} s X s X F \mathfrak{s}_{X}^{F} s X F s X θ \mathfrak{s}_{X}^{\theta} s X θ s X F , θ \mathfrak{s}_{X}^{F,\theta} s X F , θ X ∈ End ( W ) X \in\operatorname{End}(W) X ∈ End ( W ) W C W_{\mathbb{C}} W C i λ i\lambda iλ − i λ -i\lambda − iλ i λ i\lambda iλ u , v ∈ W u,v \in W u , v ∈ W i λ i\lambda iλ X u = − λ v Xu=-\lambda v X u = − λ v X v = λ u Xv=\lambda u X v = λ u X X X X 2 X^{2} X 2 − λ 2 -\lambda^{2} − λ 2 λ \lambda λ i λ i\lambda iλ X X X
From this, it follows that on each − λ 2 -\lambda^{2} − λ 2 X 2 X^{2} X 2 s X \mathfrak{s}_{X} s X m λ m_{\lambda} m λ
X 2 = O D O T = O ( − λ 1 2 0 ⋯ 0 0 − λ 2 2 ⋯ 0 0 0 ⋯ − λ n 2 ) O T , X^{2}=ODO^{T}=O
\begin{pmatrix}
-\lambda_{1}^{2} & 0 & \cdots& 0\\
0 & -\lambda_{2}^{2} & \cdots& 0\\
0 & 0 & \cdots& -\lambda_{n}^{2}
\end{pmatrix}
O^{T}, X 2 = O D O T = O − λ 1 2 0 0 0 − λ 2 2 0 ⋯ ⋯ ⋯ 0 0 − λ n 2 O T , is a diagonalisation of X 2 X^{2} X 2
s X = O ( m λ 1 0 ⋯ 0 0 m λ 2 ⋯ 0 0 0 ⋯ m λ n ) O T . \mathfrak{s}_{X}=O
\begin{pmatrix}
m_{\lambda_{1}} & 0 & \cdots& 0\\
0 & m_{\lambda_{2}} & \cdots& 0\\
0 & 0 & \cdots& m_{\lambda_{n}}
\end{pmatrix}
O^{T}. s X = O m λ 1 0 0 0 m λ 2 0 ⋯ ⋯ ⋯ 0 0 m λ n O T . Next, recall that Rot ( X ) \operatorname{Rot}(X) Rot ( X ) W 2 W^{2} W 2
Rot ( X ) = ( 0 X − X 0 ) . \operatorname{Rot}(X)=
\begin{pmatrix}
0 & X\\
-X & 0
\end{pmatrix}. Rot ( X ) = ( 0 − X X 0 ) . To discuss s X F \mathfrak{s}_{X}^{F} s X F Rot ( X ) \operatorname{Rot}(X) Rot ( X ) X X X Rot ( X ) \operatorname{Rot}(X) Rot ( X ) v ∈ W v \in W v ∈ W X ∈ End ( W ) X \in\operatorname{End}(W) X ∈ End ( W ) ( v , 0 ) (v,0) ( v , 0 ) ( 0 , v ) (0,v) ( 0 , v ) Rot ( X ) \operatorname{Rot}(X) Rot ( X ) u , v ∈ W u,v \in W u , v ∈ W X u = − λ v Xu=-\lambda v X u = − λ v X v = λ u Xv=\lambda u X v = λ u λ \lambda λ ( u , v ) (u,v) ( u , v ) ( v , − u ) (v,-u) ( v , − u ) − λ -\lambda − λ ( v , u ) (v,u) ( v , u ) ( u , − v ) (u,-v) ( u , − v )
From the linear independence of u , v u,v u , v Rot ( X ) \operatorname{Rot}(X) Rot ( X ) Rot ( X ) \operatorname{Rot}(X) Rot ( X ) 2 dim ( W ) 2\dim(W) 2 dim ( W ) X X X dim ( W ) \dim(W) dim ( W ) X X X W 2 W^2 W 2 2 dim ( ker ( X ) ) 2\dim(\ker(X)) 2 dim ( ker ( X )) W W W u , v u,v u , v X X X W 2 W^2 W 2 Rot ( X ) \operatorname{Rot}(X) Rot ( X ) 2 dim ( W ) 2\dim(W) 2 dim ( W ) Rot ( X ) \operatorname{Rot}(X) Rot ( X )
From our previous discussion, this diagonalisation of Rot ( X ) ∈ End ( W 2 ) \operatorname{Rot}(X) \in\operatorname{End}(W^2) Rot ( X ) ∈ End ( W 2 ) Rot ( X ) \operatorname{Rot}(X) Rot ( X ) λ \lambda λ − λ -\lambda − λ s X F \mathfrak{s}_X^F s X F
By a short calculation, one may show that s X F = exp ( 2 t Rot ( X ) ) s X \mathfrak{s}_X^F = \exp(2t\operatorname{Rot}(X))\mathfrak{s}_X s X F = exp ( 2 t Rot ( X )) s X
( ∂ t − L X F ) ( exp ( 2 t Rot ( X ) ) s X ) = 0. (\partial_t-\mathcal{L}_X^F)(\exp(2t\operatorname{Rot}(X))\mathfrak{s}_X)=0. ( ∂ t − L X F ) ( exp ( 2 t Rot ( X )) s X ) = 0. Here, similar to before, we abuse notation and view s X ∈ End ( W 2 ) \mathfrak{s}_X \in\operatorname{End}(W^2) s X ∈ End ( W 2 ) s X ∈ End ( W ) \mathfrak{s}_X \in\operatorname{End}(W) s X ∈ End ( W ) λ ≠ 0 \lambda\ne0 λ = 0 u , v ∈ W u,v \in W u , v ∈ W X u = − λ v Xu=-\lambda v X u = − λ v X v = λ u Xv=\lambda u X v = λ u u , v u,v u , v X 2 X^2 X 2 − λ 2 -\lambda^2 − λ 2 u , v u,v u , v s X \mathfrak{s}_X s X m λ m_\lambda m λ W 2 W^2 W 2 u , v u,v u , v s X \mathfrak{s}_X s X W 2 W^2 W 2 m λ m_\lambda m λ s X F \mathfrak{s}_X^F s X F e ± 2 λ t m λ e^{\pm2\lambda t}m_\lambda e ± 2 λ t m λ ± λ \pm\lambda ± λ u , v u,v u , v Rot ( X ) \operatorname{Rot}(X) Rot ( X ) s X , s X F \mathfrak{s}_X,\mathfrak{s}_X^F s X , s X F p p p
To summarise the discussion, we have described explicit formulas for s X , s X F \mathfrak{s}_X,\mathfrak{s}_X^F s X , s X F X ∈ End ( W ) X \in\operatorname{End}(W) X ∈ End ( W ) X X X W W W s X θ , s X F , θ \mathfrak{s}_X^\theta,\mathfrak{s}_X^{F,\theta} s X θ , s X F , θ s X F , θ ( s , y ; t , x ) = exp ( 2 ( t − s ) Rot ( X ) ) s X θ ( s , y ; t , x ) \mathfrak{s}_X^{F,\theta}(s,y;t,x)=\exp(2(t-s)\operatorname{Rot}(X))\mathfrak{s}_X^\theta(s,y;t,x) s X F , θ ( s , y ; t , x ) = exp ( 2 ( t − s ) Rot ( X )) s X θ ( s , y ; t , x ) λ \lambda λ i λ i\lambda iλ X X X m λ θ m_\lambda^\theta m λ θ s X θ \mathfrak{s}_X^\theta s X θ
∣ s X F , θ ( s , y ; t , x ) ∣ ≲ exp ( C ( t − s ) ∣ X ∣ ) p ( s , y ; t , x ) , (227) \left|\mathfrak{s}_X^{F,\theta}(s,y;t,x)\right|\lesssim\exp(C(t-s)|X|)p(s,y;t,x), \tag*{(227)} s X F , θ ( s , y ; t , x ) ≲ exp ( C ( t − s ) ∣ X ∣ ) p ( s , y ; t , x ) , (227) which we will use later.
Remark 4.46 (Exponential growth). It follows from this discussion that s X F \mathfrak{s}_X^F s X F X X X W W W s X F \mathfrak{s}_X^F s X F e 2 λ t m λ e^{2\lambda t}m_\lambda e 2 λ t m λ λ > 0 \lambda>0 λ > 0 (223) for m λ m_\lambda m λ ( sinh ( λ t ) ) − 1 (\sinh(\lambda t))^{-1} ( sinh ( λ t ) ) − 1 m λ m_\lambda m λ ( sinh ( λ t ) ) − 1 (\sinh(\lambda t))^{-1} ( sinh ( λ t ) ) − 1 e − λ t e^{-\lambda t} e − λ t e 2 λ t m λ e^{2\lambda t}m_\lambda e 2 λ t m λ e λ t e^{\lambda t} e λ t
Remark 4.47. Recall we previously observed that m λ θ ( 0 , 0 ; t , x ) = m λ ( t , x ) m_\lambda^\theta(0,0;t,x)=m_\lambda(t,x) m λ θ ( 0 , 0 ; t , x ) = m λ ( t , x ) s X F , s X F , θ \mathfrak{s}_X^F,\mathfrak{s}_X^{F,\theta} s X F , s X F , θ m λ , m λ θ m_\lambda,m_\lambda^\theta m λ , m λ θ s X F , θ ( 0 , 0 ; t , x ) = s X F ( t , x ) \mathfrak{s}_X^{F,\theta}(0,0;t,x)=\mathfrak{s}_X^F(t,x) s X F , θ ( 0 , 0 ; t , x ) = s X F ( t , x )
( ∂ t − L X F , θ ) s X F = 0. (\partial_t-\mathcal{L}_X^{F,\theta})\mathfrak{s}_X^F=0. ( ∂ t − L X F , θ ) s X F = 0. We may now return to the claim made at the beginning of the section, which is if B B B q B q_B q B q B q_B q B s X F \mathfrak{s}_X^F s X F B B B q B q_B q B q B q_B q B Proposition 4.51 is needed).
Lemma 4.48. If B : R 2 → g 2 B:\mathbb{R}^2 \to\mathfrak{g}^2 B : R 2 → g 2 X = F B 12 ( 0 ) X=F_B^{12}(0) X = F B 12 ( 0 )
τ B − 1 q B = σ X F . \tau_B^{-1}q_B=\sigma_X^F. τ B − 1 q B = σ X F . Remark 4.49. In the setting of Lemma 4.48 , it is also true that τ B − 1 p B = σ X \tau_B^{-1}p_B=\sigma_X τ B − 1 p B = σ X τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B q B q_B q B τ B \tau_B τ B E → R 2 E\to\mathbb{R}^2 E → R 2 W W W τ B − 1 \tau_B^{-1} τ B − 1 ( τ B − 1 q B ) ( t , x ) (\tau_B^{-1}q_B)(t,x) ( τ B − 1 q B ) ( t , x ) End ( E y ) \operatorname{End}(E_y) End ( E y ) q B ( t , x ) q_B(t,x) q B ( t , x ) L ( E y , E x ) L(E_y,E_x) L ( E y , E x ) τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B Lemma 4.48 provides exact formulas for the eigenvalues in the special case where B B B
With this result in mind, it is then natural to try to approximate τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B σ X F \sigma_X^F σ X F X = F B 12 ( 0 , 0 ) X=F_B^{12}(0,0) X = F B 12 ( 0 , 0 ) B B B Proposition 4.51 ). First, we define a quantity which captures how far away B B B
Definition 4.50. Let I ⊆ [ 0 , ∞ ) I\subseteq[0,\infty) I ⊆ [ 0 , ∞ ) B : I × T 2 → g 2 B:I\times\mathbb{T}^2\to\mathfrak{g}^2 B : I × T 2 → g 2
∥ B ∥ I Y M ≔ ∣ I ∣ 3 / 2 ∥ ∂ t 2 B ∥ L t , x ∞ ( I ) + ∣ I ∣ ( ∥ ∂ t F B ∥ L t , x ∞ ( I ) + ∥ D ∂ t B ∥ L t , x ∞ ( I ) ) + ∣ I ∣ 1 / 2 ( ∥ D F B ∥ L t , x ∞ ( I ) + ∥ ∂ t B ∥ L t , x ∞ ( I ) ) . (228) \lVert B\rVert_I^{\mathrm{YM}} \coloneqq|I|^{3/2}\lVert\partial_t^2B\rVert_{L^\infty_{t,x}(I)}+|I|\left(\lVert\partial_tF_B\rVert_{L^\infty_{t,x}(I)}+\lVert D\partial_tB\rVert_{L^\infty_{t,x}(I)}\right)+|I|^{1/2}\left(\lVert DF_B\rVert_{L^\infty_{t,x}(I)}+\lVert\partial_tB\rVert_{L^\infty_{t,x}(I)}\right). \tag*{(228)} ∥ B ∥ I YM : = ∣ I ∣ 3/2 ∥ ∂ t 2 B ∥ L t , x ∞ ( I ) + ∣ I ∣ ( ∥ ∂ t F B ∥ L t , x ∞ ( I ) + ∥ D ∂ t B ∥ L t , x ∞ ( I ) ) + ∣ I ∣ 1/2 ( ∥ D F B ∥ L t , x ∞ ( I ) + ∥ ∂ t B ∥ L t , x ∞ ( I ) ) . (228) When I = [ 0 , t 0 ] I=[0,t_0] I = [ 0 , t 0 ] ∥ B ∥ t 0 Y M = ∥ B ∥ I Y M \lVert B\rVert_{t_0}^{\mathrm{YM}}=\lVert B\rVert_I^{\mathrm{YM}} ∥ B ∥ t 0 YM = ∥ B ∥ I YM
In two dimensions, Yang–Mills connections B B B D F B = 0 DF_B=0 D F B = 0 ∥ B ∥ I Y M = 0 \lVert B\rVert_I^{\mathrm{YM}}=0 ∥ B ∥ I YM = 0 B B B σ \sigma σ T 2 \mathbb{T}^2 T 2 R 2 \mathbb{R}^2 R 2 Proposition 2.8 .
Proposition 4.51. There is a constant C 0 C_0 C 0 t 0 ≤ 1 t_0\leq1 t 0 ≤ 1 B : [ 0 , t 0 ] × T 2 → g 2 B:[0,t_0]\times\mathbb{T}^2\to\mathfrak{g}^2 B : [ 0 , t 0 ] × T 2 → g 2 c 0 ≥ 0 c_0\geq0 c 0 ≥ 0
t 0 ( 1 + t 0 ∥ F B ∥ L t , x ∞ ( [ 0 , t 0 ] ) 5 ) ∥ B ∥ t 0 Y M 5 ( 1 + t 0 ∥ B ∥ t 0 Y M 5 ) ≤ c 0 . (229) t_0\left(1+t_0\lVert F_B\rVert_{L^\infty_{t,x}([0,t_0])}^5\right)\lVert B\rVert_{t_0}^{\mathrm{YM}\,5}\left(1+t_0\lVert B\rVert_{t_0}^{\mathrm{YM}\,5}\right)\leq c_0. \tag*{(229)} t 0 ( 1 + t 0 ∥ F B ∥ L t , x ∞ ([ 0 , t 0 ]) 5 ) ∥ B ∥ t 0 YM 5 ( 1 + t 0 ∥ B ∥ t 0 YM 5 ) ≤ c 0 . (229) Take X = F B 12 ( 0 , 0 ) X=F_B^{12}(0,0) X = F B 12 ( 0 , 0 ) ( t , x ) ∈ ( 0 , t 0 ] × T 2 (t,x)\in(0,t_0]\times\mathbb{T}^2 ( t , x ) ∈ ( 0 , t 0 ] × T 2
∣ σ ( τ B − 1 q B − σ X F ) ( t , x ) ∣ ≲ ( c 0 + t 10 ) exp ( C c 0 ∥ F B ∥ L t , x ∞ ( [ 0 , t 0 ] ) t ) t − 1 exp ( − ∣ x ∣ 2 C t ) . \left|\sigma\left(\tau_B^{-1}q_B-\sigma_X^F\right)(t,x)\right|\lesssim(c_0+t^{10})\exp\left(Cc_0\lVert F_B\rVert_{L^\infty_{t,x}([0,t_0])}t\right)t^{-1}\exp\left(-\frac{|x|^2}{C t}\right). σ ( τ B − 1 q B − σ X F ) ( t , x ) ≲ ( c 0 + t 10 ) exp ( C c 0 ∥ F B ∥ L t , x ∞ ([ 0 , t 0 ]) t ) t − 1 exp ( − C t ∣ x ∣ 2 ) . Remark 4.52. While reading the arguments in this section, it may be good to keep in mind the following model scenario (see also Lemma 10.7): B B B N ≫ 1 N\gg1 N ≫ 1 c ≪ 1 c\ll1 c ≪ 1 t 0 = c N − 1 log log N t_0=cN^{-1}\log\log N t 0 = c N − 1 log log N
∥ F B ∥ L t , x ∞ ( [ 0 , t 0 ] ) ∼ N , ∥ D ( j ) F B ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ N 1 + j / 2 log N for 1 ≤ j ≤ 3. \lVert F_B\rVert_{L^\infty_{t,x}([0,t_0])}\sim N,\qquad\lVert D^{(j)}F_B\rVert_{L^\infty_{t,x}([0,t_0])}\lesssim\frac{N^{1+j/2}}{\log N}\quad\text{for }1\leq j\leq3. ∥ F B ∥ L t , x ∞ ([ 0 , t 0 ]) ∼ N , ∥ D ( j ) F B ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ log N N 1 + j /2 for 1 ≤ j ≤ 3. One can heuristically think of D F B D F_B D F B ≲ N 1 / 2 \lesssim N^{1/2} ≲ N 1/2 N 1 / 2 N^{1/2} N 1/2 log N \log N log N F B F_B F B B B B B B B ∂ t F B = Δ B F B \partial_t F_B=\Delta_B F_B ∂ t F B = Δ B F B ∂ t D F B = D ( 3 ) F B + O ( [ F B , D F B ] ) \partial_t D F_B=D^{(3)}F_B+O([F_B,D F_B]) ∂ t D F B = D ( 3 ) F B + O ([ F B , D F B ]) ∥ ∂ t D F B ∥ L t , x ∞ ≲ N 5 / 2 / log N \|\partial_t D F_B\|_{L^\infty_{t,x}}\lesssim N^{5/2}/\log N ∥ ∂ t D F B ∥ L t , x ∞ ≲ N 5/2 / log N
t 0 ∥ B ∥ t 0 Y M ≲ c ( log log N ) 5 / 2 log N ≪ 1. t_0\|B\|_{t_0}^{\mathrm{YM}}\lesssim c\frac{(\log\log N)^{5/2}}{\log N}\ll1. t 0 ∥ B ∥ t 0 YM ≲ c log N ( log log N ) 5/2 ≪ 1. In the context of Proposition 4.51 , we may take c 0 ∼ c ( log log N ) 100 / log N c_0\sim c(\log\log N)^{100}/\log N c 0 ∼ c ( log log N ) 100 / log N Proposition 4.51 may then be estimated (assuming that c ≪ 1 c\ll1 c ≪ 1 C 0 C_0 C 0
( c 0 + t 0 10 ) exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ( [ 0 , t 0 ] ) ) ≲ c ( log log N ) 100 log N exp ( C 0 c log log N ) ≲ 1 ( log N ) 1 − c 1 , (c_0+t_0^{10})\exp(C_0t_0\|F_B\|_{L^\infty_{t,x}([0,t_0])})\lesssim c\frac{(\log\log N)^{100}}{\log N}\exp(C_0c\log\log N)\lesssim\frac{1}{(\log N)^{1-c_1}}, ( c 0 + t 0 10 ) exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ([ 0 , t 0 ]) ) ≲ c log N ( log log N ) 100 exp ( C 0 c log log N ) ≲ ( log N ) 1 − c 1 1 , where c 1 > 0 c_1>0 c 1 > 0 c c c τ B − 1 q B − ξ X F \tau_B^{-1}q_B-\xi_X^F τ B − 1 q B − ξ X F
We now proceed to prove Lemma 4.48 and Proposition 4.51 . While the lemma is not strictly needed in the following, its proof is a nice warm-up towards the proof of the proposition. To begin, recall that q B q_B q B
( ∂ t − Δ B − 2 F B ) q B = 0 , q B ( 0 ) = δ 0 i d W 2 . (\partial_t-\Delta_B-2F_B)q_B=0,\qquad q_B(0)=\delta_{0\mathrm{id}_{W_2}}. ( ∂ t − Δ B − 2 F B ) q B = 0 , q B ( 0 ) = δ 0 id W 2 . As an intermediate step, we derive the equation satisfied by τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B Proposition 4.51 . The following derivation applies both when q B q_B q B
Lemma 4.53. We have that
( ∂ t − Δ ) ( τ B − 1 q B ) = − τ B − 1 ( ( ∂ t − Δ B ) τ B ) τ B − 1 q B + 2 ( τ B − 1 F B τ B ) τ B − 1 q B + 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) + 2 ( τ B − 1 D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) . (\partial_t-\Delta)(\tau_B^{-1}q_B)=-\tau_B^{-1}((\partial_t-\Delta_B)\tau_B)\tau_B^{-1}q_B+2(\tau_B^{-1}F_B\tau_B)\tau_B^{-1}q_B
+2(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)\partial_{r^{-1}e_\theta}(\tau_B^{-1}q_B)+2(\tau_B^{-1}D_{r^{-1}e_r}\tau_B)\partial_{r^{-1}e_r}(\tau_B^{-1}q_B). ( ∂ t − Δ ) ( τ B − 1 q B ) = − τ B − 1 (( ∂ t − Δ B ) τ B ) τ B − 1 q B + 2 ( τ B − 1 F B τ B ) τ B − 1 q B + 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) + 2 ( τ B − 1 D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) . Proof. To begin, we compute by the product rule
( ∂ t − Δ ) ( τ B − 1 q B ) = ( ( ∂ t − Δ B ) ( τ B − 1 ) ) q B + τ B − 1 ( ∂ t − Δ B ) q B − 2 D ℓ ( τ B − 1 ) D ℓ q B . (\partial_t-\Delta)(\tau_B^{-1}q_B)=((\partial_t-\Delta_B)(\tau_B^{-1}))q_B+\tau_B^{-1}(\partial_t-\Delta_B)q_B-2D_\ell(\tau_B^{-1})D^\ell q_B. ( ∂ t − Δ ) ( τ B − 1 q B ) = (( ∂ t − Δ B ) ( τ B − 1 )) q B + τ B − 1 ( ∂ t − Δ B ) q B − 2 D ℓ ( τ B − 1 ) D ℓ q B . We will need the intermediate computations (recall Remark 3.37(e), in particular (129) ):
∂ t ( τ B − 1 ) = − τ B − 1 ( ∂ t τ B ) τ B − 1 , \partial_t(\tau_B^{-1})=-\tau_B^{-1}(\partial_t\tau_B)\tau_B^{-1}, ∂ t ( τ B − 1 ) = − τ B − 1 ( ∂ t τ B ) τ B − 1 , D ℓ ( τ B − 1 ) = − τ B − 1 ( D ℓ τ B ) τ B − 1 , D_\ell(\tau_B^{-1})=-\tau_B^{-1}(D_\ell\tau_B)\tau_B^{-1}, D ℓ ( τ B − 1 ) = − τ B − 1 ( D ℓ τ B ) τ B − 1 , Δ B ( τ B − 1 ) = − D ℓ ( τ B − 1 ( D ℓ τ B ) τ B − 1 ) = − τ B − 1 ( Δ B τ B ) τ B − 1 + 2 τ B − 1 ( D ℓ τ B ) τ B − 1 ( D ℓ τ B ) τ B − 1 , \Delta_B(\tau_B^{-1})=-D^\ell(\tau_B^{-1}(D_\ell\tau_B)\tau_B^{-1})=-\tau_B^{-1}(\Delta_B\tau_B)\tau_B^{-1}+2\tau_B^{-1}(D_\ell\tau_B)\tau_B^{-1}(D^\ell\tau_B)\tau_B^{-1}, Δ B ( τ B − 1 ) = − D ℓ ( τ B − 1 ( D ℓ τ B ) τ B − 1 ) = − τ B − 1 ( Δ B τ B ) τ B − 1 + 2 τ B − 1 ( D ℓ τ B ) τ B − 1 ( D ℓ τ B ) τ B − 1 , D ℓ ( q B ) = D ℓ ( τ B − 1 τ B q B ) = ( D ℓ τ B ) τ B − 1 q B + τ B − 1 D ℓ ( τ B q B ) . D^\ell(q_B)=D^\ell(\tau_B^{-1}\tau_Bq_B)=(D^\ell\tau_B)\tau_B^{-1}q_B+\tau_B^{-1}D^\ell(\tau_Bq_B). D ℓ ( q B ) = D ℓ ( τ B − 1 τ B q B ) = ( D ℓ τ B ) τ B − 1 q B + τ B − 1 D ℓ ( τ B q B ) . Inserting these intermediate computations, we may further obtain
( ∂ t − Δ ) ( τ B − 1 q B ) = − τ B − 1 ( ( ∂ t − Δ B ) τ B ) τ B − 1 q B + 2 τ B − 1 F B q B + 2 τ B − 1 ( D ℓ τ B ) D ℓ ( τ B − 1 q B ) . (\partial_t-\Delta)(\tau_B^{-1}q_B)=-\tau_B^{-1}((\partial_t-\Delta_B)\tau_B)\tau_B^{-1}q_B+2\tau_B^{-1}F_Bq_B+2\tau_B^{-1}(D_\ell\tau_B)D^\ell(\tau_B^{-1}q_B). ( ∂ t − Δ ) ( τ B − 1 q B ) = − τ B − 1 (( ∂ t − Δ B ) τ B ) τ B − 1 q B + 2 τ B − 1 F B q B + 2 τ B − 1 ( D ℓ τ B ) D ℓ ( τ B − 1 q B ) . To finish, we use that r − 1 e r , r − 1 e θ r^{-1}e_r,r^{-1}e_\theta r − 1 e r , r − 1 e θ
( D ℓ τ B ) D ℓ ( τ B − 1 q B ) = ( D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) + ( D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) . (D_\ell\tau_B)D^\ell(\tau_B^{-1}q_B)=(D_{r^{-1}e_r}\tau_B)\partial_{r^{-1}e_r}(\tau_B^{-1}q_B)+(D_{r^{-1}e_\theta}\tau_B)\partial_{r^{-1}e_\theta}(\tau_B^{-1}q_B). ( D ℓ τ B ) D ℓ ( τ B − 1 q B ) = ( D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) + ( D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) . The desired result now follows by combining the previous few identities.
From Lemma 4.53 , we see that the Laplacian Δ B τ B \Delta_B\tau_B Δ B τ B τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B ( e θ , e r ) (e_\theta,e_r) ( e θ , e r ) Notation 4.42 . As before, the following derivation holds both on the plane as well as on a neighbourhood around 0 0 0
Lemma 4.54 (Polar coordinates). We have that
( Hess B Φ ) ( r − 1 e r , r − 1 e r ) = D r − 1 e r D r − 1 e r Φ , (\operatorname{Hess}_B\Phi)(r^{-1}e_r,r^{-1}e_r)=D_{r^{-1}e_r}D_{r^{-1}e_r}\Phi, ( Hess B Φ ) ( r − 1 e r , r − 1 e r ) = D r − 1 e r D r − 1 e r Φ , ( Hess B Φ ) ( r − 1 e θ , r − 1 e θ ) = ( D r − 1 e θ D r − 1 e θ + r − 1 D r − 1 e r ) Φ . (\operatorname{Hess}_B\Phi)(r^{-1}e_\theta,r^{-1}e_\theta)=\left(D_{r^{-1}e_\theta}D_{r^{-1}e_\theta}+r^{-1}D_{r^{-1}e_r}\right)\Phi. ( Hess B Φ ) ( r − 1 e θ , r − 1 e θ ) = ( D r − 1 e θ D r − 1 e θ + r − 1 D r − 1 e r ) Φ. Consequently, we have that
Δ B Φ = ( D r − 1 e r D r − 1 e r + r − 1 D r − 1 e r + D r − 1 e θ D r − 1 e θ ) Φ . (230) \Delta_B\Phi=\left(D_{r^{-1}e_r}D_{r^{-1}e_r}+r^{-1}D_{r^{-1}e_r}+D_{r^{-1}e_\theta}D_{r^{-1}e_\theta}\right)\Phi. \tag*{(230)} Δ B Φ = ( D r − 1 e r D r − 1 e r + r − 1 D r − 1 e r + D r − 1 e θ D r − 1 e θ ) Φ. (230) Proof. We compute
D r − 1 e r D r − 1 e r Φ = ( Hess B Φ ) ( r − 1 e r , r − 1 e r ) + D ∂ r − 1 e r ( r − 1 e r ) Φ = ( Hess B Φ ) ( r − 1 e r , r − 1 e r ) , D_{r^{-1}e_r}D_{r^{-1}e_r}\Phi=(\operatorname{Hess}_B\Phi)(r^{-1}e_r,r^{-1}e_r)+D_{\partial_{r^{-1}e_r}}(r^{-1}e_r)\Phi=(\operatorname{Hess}_B\Phi)(r^{-1}e_r,r^{-1}e_r), D r − 1 e r D r − 1 e r Φ = ( Hess B Φ ) ( r − 1 e r , r − 1 e r ) + D ∂ r − 1 e r ( r − 1 e r ) Φ = ( Hess B Φ ) ( r − 1 e r , r − 1 e r ) , where we used that ∂ r − 1 e r ( r − 1 e r ) = 0 \partial_{r^{-1}e_r}(r^{-1}e_r)=0 ∂ r − 1 e r ( r − 1 e r ) = 0
D r − 1 e θ D r − 1 e θ Φ = ( Hess B Φ ) ( r − 1 e θ , r − 1 e θ ) + D ∂ r − 1 e θ ( r − 1 e θ ) Φ . D_{r^{-1}e_\theta}D_{r^{-1}e_\theta}\Phi=(\operatorname{Hess}_B\Phi)(r^{-1}e_\theta,r^{-1}e_\theta)+D_{\partial_{r^{-1}e_\theta}}(r^{-1}e_\theta)\Phi. D r − 1 e θ D r − 1 e θ Φ = ( Hess B Φ ) ( r − 1 e θ , r − 1 e θ ) + D ∂ r − 1 e θ ( r − 1 e θ ) Φ. We have that
∂ r − 1 e θ ( r − 1 e θ ) = r − 1 ∂ r − 1 e θ e θ = r − 2 ∂ e θ e θ = − r − 2 e r . (231) \partial_{r^{-1}e_\theta}(r^{-1}e_\theta)=r^{-1}\partial_{r^{-1}e_\theta}e_\theta=r^{-2}\partial_{e_\theta}e_\theta=-r^{-2}e_r. \tag*{(231)} ∂ r − 1 e θ ( r − 1 e θ ) = r − 1 ∂ r − 1 e θ e θ = r − 2 ∂ e θ e θ = − r − 2 e r . (231) Since r − 1 e θ r^{-1}e_\theta r − 1 e θ r − 1 e r r^{-1}e_r r − 1 e r R 2 \mathbb{R}^2 R 2
The remaining ingredient in the proof of Lemma 4.48 is an exact computation of covariant derivatives of parallel transport in the case where B B B Remark 3.37 (c), which says that if B B B τ B \tau_B τ B
Lemma 4.55. Let B : R 2 → g 2 B:\mathbb{R}^2\to\mathfrak{g}^2 B : R 2 → g 2 x ∈ R 2 x\in\mathbb{R}^2 x ∈ R 2 v ∈ R 2 v\in\mathbb{R}^2 v ∈ R 2
( τ B − 1 D v τ B ) ( x ) = 1 2 F B 12 ( 0 ) ( x , v ) , (\tau_B^{-1}D_v\tau_B)(x)=\frac{1}{2}F_B^{12}(0)(x,v), ( τ B − 1 D v τ B ) ( x ) = 2 1 F B 12 ( 0 ) ( x , v ) , ( τ B − 1 D r − 1 e θ τ B ) ( x ) = 1 2 ∣ x ∣ F B 12 ( 0 ) , (\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(x)=\frac{1}{2}|x|F_B^{12}(0), ( τ B − 1 D r − 1 e θ τ B ) ( x ) = 2 1 ∣ x ∣ F B 12 ( 0 ) , ( τ B − 1 Δ B τ B ) ( x ) = 1 4 ∣ x ∣ 2 ( F B 12 ( 0 ) ) 2 . (\tau_B^{-1}\Delta_B\tau_B)(x)=\frac{1}{4}|x|^2(F_B^{12}(0))^2. ( τ B − 1 Δ B τ B ) ( x ) = 4 1 ∣ x ∣ 2 ( F B 12 ( 0 ) ) 2 . Proof. By Lemmas 4.34 and 4.35 (or more precisely, analogues of these lemmas on the plane for constant-in-time connections), we have that (with γ ( t ) = t x \gamma(t)=tx γ ( t ) = t x
( τ B − 1 D v τ B ) ( x ) = ∫ 0 1 d t ⋅ ( τ B − 1 F B ( x , v ) τ B ) ( γ ( t ) ) = ∫ 0 1 d t ⋅ F B 12 ( 0 ) ( x , v ) = 1 2 F B 12 ( 0 ) ( x , v ) , (232) \begin{aligned}
(\tau_B^{-1}D_v\tau_B)(x)=\int_0^1\mathrm{d}t\cdot(\tau_B^{-1}F_B(x,v)\tau_B)(\gamma(t)) \\
&=\int_0^1\mathrm{d}t\cdot F_B^{12}(0)(x,v)=\frac{1}{2}F_B^{12}(0)(x,v), \tag*{(232)}
\end{aligned} ( τ B − 1 D v τ B ) ( x ) = ∫ 0 1 d t ⋅ ( τ B − 1 F B ( x , v ) τ B ) ( γ ( t )) = ∫ 0 1 d t ⋅ F B 12 ( 0 ) ( x , v ) = 2 1 F B 12 ( 0 ) ( x , v ) , (232) which gives the first desired identity.
For the second identity, we apply the first identity to obtain
( τ B − 1 D r − 1 e θ τ B ) ( x ) = 1 2 F B ( 0 ) ( x , r − 1 e θ ) = 1 2 ( r − 1 F B 12 ( 0 ) x 1 2 + r − 1 F B 21 ( 0 ) x 2 ( − x 2 ) ) = 1 2 ∣ x ∣ F B 12 ( 0 ) , (\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(x)=\frac{1}{2}F_B(0)(x,r^{-1}e_\theta)
=\frac{1}{2}\left(r^{-1}F_B^{12}(0)x_1^2+r^{-1}F_B^{21}(0)x_2(-x_2)\right)=\frac{1}{2}|x|F_B^{12}(0), ( τ B − 1 D r − 1 e θ τ B ) ( x ) = 2 1 F B ( 0 ) ( x , r − 1 e θ ) = 2 1 ( r − 1 F B 12 ( 0 ) x 1 2 + r − 1 F B 21 ( 0 ) x 2 ( − x 2 ) ) = 2 1 ∣ x ∣ F B 12 ( 0 ) , as desired.
Finally, for the last identity, we differentiate both sides above by ∂ r − 1 e θ \partial_{r^{-1}e_\theta} ∂ r − 1 e θ
τ B − 1 D r − 1 e θ D r − 1 e θ τ B − τ B − 1 ( D r − 1 e θ τ B ) τ B − 1 D r − 1 e θ τ B = ∂ r − 1 e θ ( τ B − 1 D r − 1 e θ τ B ) = 1 2 ( ∂ r − 1 e θ r − 1 ) F B 12 ( 0 ) = 0 , \tau_B^{-1}D_{r^{-1}e_\theta}D_{r^{-1}e_\theta}\tau_B-\tau_B^{-1}(D_{r^{-1}e_\theta}\tau_B)\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B=\partial_{r^{-1}e_\theta}\left(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B\right)
=\frac{1}{2}(\partial_{r^{-1}e_\theta}r^{-1})F_B^{12}(0)=0, τ B − 1 D r − 1 e θ D r − 1 e θ τ B − τ B − 1 ( D r − 1 e θ τ B ) τ B − 1 D r − 1 e θ τ B = ∂ r − 1 e θ ( τ B − 1 D r − 1 e θ τ B ) = 2 1 ( ∂ r − 1 e θ r − 1 ) F B 12 ( 0 ) = 0 , where we used that r = ∣ x ∣ r=|x| r = ∣ x ∣ D e r τ B = 0 D_{e_r}\tau_B=0 D e r τ B = 0 (230) , we thus obtain
( τ B − 1 Δ B τ B ) ( x ) = ( τ B − 1 D r − 1 e θ τ B ) ( x ) ( τ B − 1 D r − 1 e θ τ B ) ( x ) = 1 4 ∣ x ∣ 2 ( F B 12 ( 0 ) ) 2 , (\tau_B^{-1}\Delta_B\tau_B)(x)=(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(x)(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(x)=\frac{1}{4}|x|^2(F_B^{12}(0))^2, ( τ B − 1 Δ B τ B ) ( x ) = ( τ B − 1 D r − 1 e θ τ B ) ( x ) ( τ B − 1 D r − 1 e θ τ B ) ( x ) = 4 1 ∣ x ∣ 2 ( F B 12 ( 0 ) ) 2 , as desired. □
Because it will be useful later, we record the following calculation from the proof of Lemma 4.55 :
F B ( x , e θ ) = F B 12 x 1 2 + F B 21 x 2 ( − x 2 ) = F B 12 ∣ x ∣ 2 . (233) F_B(x,e_\theta)=F_B^{12}x_1^2+F_B^{21}x_2(-x_2)=F_B^{12}|x|^2. \tag*{(233)} F B ( x , e θ ) = F B 12 x 1 2 + F B 21 x 2 ( − x 2 ) = F B 12 ∣ x ∣ 2 . (233) We are now ready to prove Lemma 4.48 .
Proof of Lemma 4.48 . By Lemma 4.53 combined with the facts that when B B B τ B \tau_B τ B D e r τ B = 0 D_{e_r}\tau_B=0 D e r τ B = 0
( ∂ t − Δ ) ( τ B − 1 q B ) = 2 ( τ B − 1 F B τ B ) ( τ B − 1 q B ) + ( τ B − 1 Δ B τ B ) ( τ B − 1 q B ) + 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) . (\partial_t-\Delta)(\tau_B^{-1}q_B)=2(\tau_B^{-1}F_B\tau_B)(\tau_B^{-1}q_B)+(\tau_B^{-1}\Delta_B\tau_B)(\tau_B^{-1}q_B)+2(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)\partial_{r^{-1}e_\theta}(\tau_B^{-1}q_B). ( ∂ t − Δ ) ( τ B − 1 q B ) = 2 ( τ B − 1 F B τ B ) ( τ B − 1 q B ) + ( τ B − 1 Δ B τ B ) ( τ B − 1 q B ) + 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) . Next, by Lemmas 4.35 and 4.55 (the first of which implies that τ B − 1 F B τ B = F B ( 0 ) \tau_B^{-1}F_B\tau_B=F_B(0) τ B − 1 F B τ B = F B ( 0 )
( ∂ t − L X F ) ( τ B − 1 q B ) = 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) , (\partial_t-\mathcal{L}_X^F)(\tau_B^{-1}q_B)=2(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)\partial_{r^{-1}e_\theta}(\tau_B^{-1}q_B), ( ∂ t − L X F ) ( τ B − 1 q B ) = 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ ( τ B − 1 q B ) , where L X F \mathcal{L}_X^F L X F Definition 4.43 . To finish, observe that s X F s_X^F s X F ∂ e θ s X F = 0 \partial_{e_\theta}s_X^F=0 ∂ e θ s X F = 0
( ∂ t − L X F ) s X F = 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ s X F . (\partial_t-\mathcal{L}_X^F)s_X^F=2(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)\partial_{r^{-1}e_\theta}s_X^F. ( ∂ t − L X F ) s X F = 2 ( τ B − 1 D r − 1 e θ τ B ) ∂ r − 1 e θ s X F . Since s X F ( 0 ) = δ 0 id W 2 = ( τ B − 1 q B ) ( 0 ) s_X^F(0)=\delta_{0}\operatorname{id}_{W^2}=(\tau_B^{-1}q_B)(0) s X F ( 0 ) = δ 0 id W 2 = ( τ B − 1 q B ) ( 0 ) s X F = τ B − 1 q B s_X^F=\tau_B^{-1}q_B s X F = τ B − 1 q B
The proof of Lemma 4.48 also suggests how to go about proving Proposition 4.51 : we should derive the equation satisfied by τ B − 1 q B − s X F \tau_B^{-1}q_B-s_X^F τ B − 1 q B − s X F ( ∂ t − L X F , θ ) ( τ B − 1 q B − s X F ) = G B (\partial_t-\mathcal{L}_X^{F,\theta})(\tau_B^{-1}q_B-s_X^F)=G_B ( ∂ t − L X F , θ ) ( τ B − 1 q B − s X F ) = G B G B = 0 G_B=0 G B = 0 B B B G B G_B G B B B B X = F B 12 ( 0 , 0 ) X=F_B^{12}(0,0) X = F B 12 ( 0 , 0 )
r B F ≔ σ ( τ B − 1 q B − s X F ) : [ 0 , ∞ ) × R 2 → End ( W 2 ) . r_B^F\mathrel{\mathop{\coloneqq}}\sigma(\tau_B^{-1}q_B-s_X^F):[0,\infty)\times\mathbb{R}^2\to\operatorname{End}(W^2). r B F : = σ ( τ B − 1 q B − s X F ) : [ 0 , ∞ ) × R 2 → End ( W 2 ) . Here, while τ B − 1 q B \tau_B^{-1}q_B τ B − 1 q B σ \sigma σ σ τ B − 1 q B \sigma\tau_B^{-1}q_B σ τ B − 1 q B r B F r_B^F r B F Proposition 4.51 .
Combining Remark 4.47 (which gives that ( ∂ t − L X F , θ ) F X F = 0 (\partial_t-\mathcal{L}^{F,\theta}_X)\mathcal{F}^F_X=0 ( ∂ t − L X F , θ ) F X F = 0 r B F r^F_B r B F
( ∂ t − L X F , θ ) r B F = 2 ( τ B − 1 F B τ B − Rot ( X ) ) ( σ τ B − 1 q B ) + ( τ B − 1 Δ B τ B − r 2 4 X 2 ) ( σ τ B − 1 q B ) + 2 σ ( τ B − 1 D r − 1 e θ τ B − r 2 X ) ∂ r − 1 e θ ( τ B − 1 q B ) − ( τ B − 1 ∂ t τ B ) ( σ τ B − 1 q B ) + 2 σ ( τ B − 1 D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) − r X ( ∂ r − 1 e θ σ ) ( τ B − 1 q B − F X F ) − 2 ( ∂ ℓ σ ) ∂ ℓ ( τ B − 1 q B − F X F ) − ( Δ σ ) ( τ B − 1 q B − F X F ) . (234) (\partial_t-\mathcal{L}^{F,\theta}_X)r^F_B
=2(\tau_B^{-1}F_{B\tau_B}-\operatorname{Rot}(X))(\sigma\tau_B^{-1}q_B)
+\left(\tau_B^{-1}\Delta_B\tau_B-\frac{r^2}{4}X^2\right)(\sigma\tau_B^{-1}q_B)
+2\sigma\left(\tau_B^{-1}D_{r-1}e_{\theta}\tau_B-\frac{r}{2}X\right)\partial_{r-1}e_{\theta}(\tau_B^{-1}q_B)
-(\tau_B^{-1}\partial_t\tau_B)(\sigma\tau_B^{-1}q_B)+2\sigma(\tau_B^{-1}D_{r-1}e_r\tau_B)\partial_{r-1}e_r(\tau_B^{-1}q_B)
-rX(\partial_{r-1}e_{\theta}\sigma)(\tau_B^{-1}q_B-\mathcal{F}^F_X)
-2(\partial^\ell\sigma)\partial_\ell(\tau_B^{-1}q_B-\mathcal{F}^F_X)-(\Delta\sigma)(\tau_B^{-1}q_B-\mathcal{F}^F_X). \tag*{(234)} ( ∂ t − L X F , θ ) r B F = 2 ( τ B − 1 F B τ B − Rot ( X )) ( σ τ B − 1 q B ) + ( τ B − 1 Δ B τ B − 4 r 2 X 2 ) ( σ τ B − 1 q B ) + 2 σ ( τ B − 1 D r − 1 e θ τ B − 2 r X ) ∂ r − 1 e θ ( τ B − 1 q B ) − ( τ B − 1 ∂ t τ B ) ( σ τ B − 1 q B ) + 2 σ ( τ B − 1 D r − 1 e r τ B ) ∂ r − 1 e r ( τ B − 1 q B ) − r X ( ∂ r − 1 e θ σ ) ( τ B − 1 q B − F X F ) − 2 ( ∂ ℓ σ ) ∂ ℓ ( τ B − 1 q B − F X F ) − ( Δ σ ) ( τ B − 1 q B − F X F ) . (234) The main step towards Proposition 4.51 is to estimate all terms on the right-hand side of (234) in terms of the quantity ∥ B ∥ t 0 Y M \|B\|^{\mathrm{YM}}_{t_0} ∥ B ∥ t 0 YM B B B
Lemma 4.56 (Commutator estimates of parallel transport and curvature). Let B : [ 0 , t 0 ] × T 2 → g 2 B:[0,t_0]\times\mathbb{T}^2\to\mathfrak{g}^2 B : [ 0 , t 0 ] × T 2 → g 2 u , w ∈ R 2 u,w\in\mathbb{R}^2 u , w ∈ R 2 γ ( t ) = t ( 1 , w ) \gamma(t)=t(1,w) γ ( t ) = t ( 1 , w ) t ∈ [ 0 , t 0 ] t\in[0,t_0] t ∈ [ 0 , t 0 ] t ∣ w ∣ ≤ 1 / 10 t|w|\le1/10 t ∣ w ∣ ≤ 1/10 v ∈ R 2 v\in\mathbb{R}^2 v ∈ R 2
∣ ( τ B − 1 F B ( u , v ) τ B ) ( γ ( t ) ) − F B ( γ ( 0 ) ) ( u , v ) ∣ ≲ t ∣ u ∣ ∣ v ∣ ( ∥ ∂ t F B ∥ L t , x ∞ ( [ 0 , t ] ) + ∥ D F B ∥ L t , x ∞ ( [ 0 , t ] ) ∣ w ∣ ) . (235) \left|(\tau_B^{-1}F_B(u,v)\tau_B)(\gamma(t))-F_B(\gamma(0))(u,v)\right|
\lesssim t|u||v|\left(\|\partial_tF_B\|_{L^\infty_{t,x}([0,t])}+\|DF_B\|_{L^\infty_{t,x}([0,t])}|w|\right). \tag*{(235)} ( τ B − 1 F B ( u , v ) τ B ) ( γ ( t )) − F B ( γ ( 0 )) ( u , v ) ≲ t ∣ u ∣∣ v ∣ ( ∥ ∂ t F B ∥ L t , x ∞ ([ 0 , t ]) + ∥ D F B ∥ L t , x ∞ ([ 0 , t ]) ∣ w ∣ ) . (235) If, more generally, v ∈ R 1 + 2 v\in\mathbb{R}^{1+2} v ∈ R 1 + 2
∣ ( τ B − 1 F B ( γ ˙ ( t ) , v ) τ B ) ( γ ( t ) ) − F B ( γ ( 0 ) ) ( γ ˙ ( 0 ) , v ) ∣ ≲ t ( ∥ ∂ t 2 B ∥ L t , x ∞ ( [ 0 , t ] ) ( ∣ v x ∣ + ∣ w ∣ ∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ( [ 0 , t ] ) ∣ w ∣ ∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ( [ 0 , t ] ) ( ∣ w ∣ ∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ( [ 0 , t ] ) ∣ w ∣ 2 ∣ v x ∣ ) . (236) \left|(\tau_B^{-1}F_B(\dot{\gamma}(t),v)\tau_B)(\gamma(t))-F_B(\gamma(0))(\dot{\gamma}(0),v)\right|
\lesssim t\left(\|\partial_t^2B\|_{L^\infty_{t,x}([0,t])}(|v_x|+|w||v_0|)+\|\partial_tF_B\|_{L^\infty_{t,x}([0,t])}|w||v_x|
+\|D\partial_tB\|_{L^\infty_{t,x}([0,t])}(|w||v_x|+|w|^2|v_0|)
+\|DF_B\|_{L^\infty_{t,x}([0,t])}|w|^2|v_x|\right). \tag*{(236)} ( τ B − 1 F B ( γ ˙ ( t ) , v ) τ B ) ( γ ( t )) − F B ( γ ( 0 )) ( γ ˙ ( 0 ) , v ) ≲ t ( ∥ ∂ t 2 B ∥ L t , x ∞ ([ 0 , t ]) ( ∣ v x ∣ + ∣ w ∣∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ([ 0 , t ]) ∣ w ∣∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ([ 0 , t ]) ( ∣ w ∣∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ([ 0 , t ]) ∣ w ∣ 2 ∣ v x ∣ ) . (236) ∣ ( τ B − 1 D v τ B ) ( γ ( t ) ) − t 2 F B ( γ ( 0 ) ) ( γ ˙ ( 0 ) , v ) ∣ ≲ t 2 ( ∥ ∂ t 2 B ∥ L t , x ∞ ( [ 0 , t ] ) ( ∣ v x ∣ + ∣ w ∣ ∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ( [ 0 , t ] ) ∣ w ∣ ∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ( [ 0 , t ] ) ( ∣ w ∣ ∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ( [ 0 , t ] ) ∣ w ∣ 2 ∣ v x ∣ ) . (237) \left|(\tau_B^{-1}D_v\tau_B)(\gamma(t))-\frac{t}{2}F_B(\gamma(0))(\dot{\gamma}(0),v)\right|
\lesssim t^2\left(\|\partial_t^2B\|_{L^\infty_{t,x}([0,t])}(|v_x|+|w||v_0|)+\|\partial_tF_B\|_{L^\infty_{t,x}([0,t])}|w||v_x|
+\|D\partial_tB\|_{L^\infty_{t,x}([0,t])}(|w||v_x|+|w|^2|v_0|)
+\|DF_B\|_{L^\infty_{t,x}([0,t])}|w|^2|v_x|\right). \tag*{(237)} ( τ B − 1 D v τ B ) ( γ ( t )) − 2 t F B ( γ ( 0 )) ( γ ˙ ( 0 ) , v ) ≲ t 2 ( ∥ ∂ t 2 B ∥ L t , x ∞ ([ 0 , t ]) ( ∣ v x ∣ + ∣ w ∣∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ([ 0 , t ]) ∣ w ∣∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ([ 0 , t ]) ( ∣ w ∣∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ([ 0 , t ]) ∣ w ∣ 2 ∣ v x ∣ ) . (237) Proof. By Lemma 4.35, we have that
( τ B − 1 F B ( u , v ) τ B ) ( γ ( t ) ) − F B ( γ ( 0 ) ) ( u , v ) = ∫ 0 t d s ( τ B − 1 ( D γ ˙ ( s ) F B ) ( u , v ) τ B ) ( γ ( s ) ) . (\tau_B^{-1}F_B(u,v)\tau_B)(\gamma(t))-F_B(\gamma(0))(u,v)
=\int_0^t ds\,(\tau_B^{-1}(D_{\dot{\gamma}(s)}F_B)(u,v)\tau_B)(\gamma(s)). ( τ B − 1 F B ( u , v ) τ B ) ( γ ( t )) − F B ( γ ( 0 )) ( u , v ) = ∫ 0 t d s ( τ B − 1 ( D γ ˙ ( s ) F B ) ( u , v ) τ B ) ( γ ( s )) . The estimate (235) now follows by integrating the pointwise estimate (note that D γ ˙ ( s ) F B = ∂ t F B + D w F B D_{\dot{\gamma}(s)}F_B=\partial_tF_B+D_wF_B D γ ˙ ( s ) F B = ∂ t F B + D w F B
∣ ( τ B − 1 ( D γ ˙ ( s ) F B ) ( u , v ) τ B ) ( γ ( s ) ) ∣ ≲ ∥ ∂ t F B ∥ L t , x ∞ ∣ u ∣ ∣ v ∣ + ∥ D F B ∥ L t , x ∞ ∣ u ∣ ∣ v ∣ ∣ w ∣ . |(\tau_B^{-1}(D_{\dot{\gamma}(s)}F_B)(u,v)\tau_B)(\gamma(s))|
\lesssim\|\partial_tF_B\|_{L^\infty_{t,x}}|u||v|+\|DF_B\|_{L^\infty_{t,x}}|u||v||w|. ∣ ( τ B − 1 ( D γ ˙ ( s ) F B ) ( u , v ) τ B ) ( γ ( s )) ∣ ≲ ∥ ∂ t F B ∥ L t , x ∞ ∣ u ∣∣ v ∣ + ∥ D F B ∥ L t , x ∞ ∣ u ∣∣ v ∣∣ w ∣. Here, we used the assumption that v ∈ R 2 v\in\mathbb{R}^2 v ∈ R 2 F B F_B F B
Next, again by Lemma 4.35, we have that
( τ B − 1 F B ( γ ˙ ( t ) , v ) τ B ) ( γ ( t ) ) − F B ( γ ( 0 ) ) ( γ ˙ ( 0 ) , v ) = ∫ 0 t d s ( τ B − 1 ( D γ ˙ ( s ) F B ) ( γ ˙ ( s ) , v ) τ B ) ( γ ( s ) ) . (\tau_B^{-1}F_B(\dot{\gamma}(t),v)\tau_B)(\gamma(t))-F_B(\gamma(0))(\dot{\gamma}(0),v)
=\int_0^t ds\,(\tau_B^{-1}(D_{\dot{\gamma}(s)}F_B)(\dot{\gamma}(s),v)\tau_B)(\gamma(s)). ( τ B − 1 F B ( γ ˙ ( t ) , v ) τ B ) ( γ ( t )) − F B ( γ ( 0 )) ( γ ˙ ( 0 ) , v ) = ∫ 0 t d s ( τ B − 1 ( D γ ˙ ( s ) F B ) ( γ ˙ ( s ) , v ) τ B ) ( γ ( s )) . Note that γ ˙ ( s ) = ( 1 , w ) \dot{\gamma}(s)=(1,w) γ ˙ ( s ) = ( 1 , w )
( D γ ˙ ( s ) F ) ( γ ˙ ( s ) , v ) = ( D ( 1 , w ) F ) ( ( 1 , w ) , v ) = D 0 F 0 j v j + D 0 F i j w i v j + D 0 F i 0 w i v 0 + w i D i F 0 j v j + w i D i F j k w j v k + w i D i F j 0 w j v 0 . \begin{aligned}
(D_{\dot{\gamma}(s)}F)(\dot{\gamma}(s),v)
&=(D_{(1,w)}F)((1,w),v)\\
&=D_0F_{0j}v^j+D_0F_{ij}w^iv^j+D_0F_{i0}w^iv_0+w^iD_iF_{0j}v^j+w^iD_iF_{jk}w^jv^k+w^iD_iF_{j0}w^jv^0.
\end{aligned} ( D γ ˙ ( s ) F ) ( γ ˙ ( s ) , v ) = ( D ( 1 , w ) F ) (( 1 , w ) , v ) = D 0 F 0 j v j + D 0 F ij w i v j + D 0 F i 0 w i v 0 + w i D i F 0 j v j + w i D i F j k w j v k + w i D i F j 0 w j v 0 . Noting that F 0 i = ∂ t B i F_{0i}=\partial_tB_i F 0 i = ∂ t B i D 0 = ∂ t D_0=\partial_t D 0 = ∂ t
∣ ( τ B − 1 ( D γ ˙ ( s ) F B ) ( γ ˙ ( s ) , v ) τ B ) ( γ ( s ) ) ∣ ≲ ∥ ∂ t 2 B ∥ L t , x ∞ ( ∣ v x ∣ + ∣ w ∣ ∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ∣ w ∣ ∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ( ∣ w ∣ ∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ∣ w ∣ 2 ∣ v x ∣ . \left|(\tau_B^{-1}(D_{\dot{\gamma}(s)}F_B)(\dot{\gamma}(s),v)\tau_B)(\gamma(s))\right|
\lesssim\|\partial_t^2B\|_{L^\infty_{t,x}}(|v_x|+|w||v_0|)+\|\partial_tF_B\|_{L^\infty_{t,x}}|w||v_x|
+\|D\partial_tB\|_{L^\infty_{t,x}}(|w||v_x|+|w|^2|v_0|)+\|DF_B\|_{L^\infty_{t,x}}|w|^2|v_x|. ( τ B − 1 ( D γ ˙ ( s ) F B ) ( γ ˙ ( s ) , v ) τ B ) ( γ ( s )) ≲ ∥ ∂ t 2 B ∥ L t , x ∞ ( ∣ v x ∣ + ∣ w ∣∣ v 0 ∣ ) + ∥ ∂ t F B ∥ L t , x ∞ ∣ w ∣∣ v x ∣ + ∥ D ∂ t B ∥ L t , x ∞ ( ∣ w ∣∣ v x ∣ + ∣ w ∣ 2 ∣ v 0 ∣ ) + ∥ D F B ∥ L t , x ∞ ∣ w ∣ 2 ∣ v x ∣. Upon integrating, we obtain (236) . For (237) , we integrate once more and again apply Lemma 4.35.
Lemma 4.57. Let B : [ 0 , t 0 ] × T 2 → g 2 B:[0,t_0]\times\mathbb{T}^2\to\mathfrak{g}^2 B : [ 0 , t 0 ] × T 2 → g 2 x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2 ∣ x ∣ ≤ 1 / 10 |x|\leq1/10 ∣ x ∣ ≤ 1/10 t ∈ ( 0 , t 0 ] t\in(0,t_0] t ∈ ( 0 , t 0 ]
∣ ( t / 2 ) F B ( ( 1 , x / t ) , r − 1 e θ ) − ( r / 2 ) F B 12 ( 0 , 0 ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M , (238) \left|(t/2)F_B((1,x/t),r^{-1}e_\theta)-(r/2)F_B^{12}(0,0)\right|\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}, \tag*{(238)} ( t /2 ) F B (( 1 , x / t ) , r − 1 e θ ) − ( r /2 ) F B 12 ( 0 , 0 ) ≲ t 1/2 ∥ B ∥ t 0 YM , (238) ∣ t F B ( ( 1 , x / t ) , r − 1 e r ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M , (239) |tF_B((1,x/t),r^{-1}e_r)|\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}, \tag*{(239)} ∣ t F B (( 1 , x / t ) , r − 1 e r ) ∣ ≲ t 1/2 ∥ B ∥ t 0 YM , (239) ∣ t F B ( ( 1 , x / t ) , ( 1 , 0 ) ) ∣ ≲ ∥ B ∥ t 0 Y M ( r t − 1 / 2 ) . (240) |tF_B((1,x/t),(1,0))|\lesssim\|B\|_{t_0}^{\mathrm{YM}}(rt^{-1/2}). \tag*{(240)} ∣ t F B (( 1 , x / t ) , ( 1 , 0 )) ∣ ≲ ∥ B ∥ t 0 YM ( r t − 1/2 ) . (240) To be clear, in the above F B F_B F B ( 0 , 0 ) (0,0) ( 0 , 0 )
Proof. To show (238) , we first expand (and use (4.55))
( t / 2 ) F B ( ( 1 , x / t ) , r − 1 e θ ) = ( t / 2 ) F B ( x / t , r − 1 e θ ) + ( t / 2 ) ∂ t B ⋅ ( r − 1 e θ ) = ( r / 2 ) F B 12 ( 0 , 0 ) + ( t / 2 ) ∂ t B ⋅ ( r − 1 e θ ) , \begin{aligned}
(t/2)F_B((1,x/t),r^{-1}e_\theta)
&=(t/2)F_B(x/t,r^{-1}e_\theta)+(t/2)\partial_tB\cdot(r^{-1}e_\theta)\\
&=(r/2)F_B^{12}(0,0)+(t/2)\partial_tB\cdot(r^{-1}e_\theta),
\end{aligned} ( t /2 ) F B (( 1 , x / t ) , r − 1 e θ ) = ( t /2 ) F B ( x / t , r − 1 e θ ) + ( t /2 ) ∂ t B ⋅ ( r − 1 e θ ) = ( r /2 ) F B 12 ( 0 , 0 ) + ( t /2 ) ∂ t B ⋅ ( r − 1 e θ ) , and to finish, we estimate
t ∣ ∂ t B ⋅ ( r − 1 e θ ) ∣ ≲ t ∥ ∂ t B ∥ L t , x ∞ ≲ t 1 / 2 ∥ B ∥ t 0 Y M . t|\partial_tB\cdot(r^{-1}e_\theta)|\lesssim t\|\partial_tB\|_{L^\infty_{t,x}}\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}. t ∣ ∂ t B ⋅ ( r − 1 e θ ) ∣ ≲ t ∥ ∂ t B ∥ L t , x ∞ ≲ t 1/2 ∥ B ∥ t 0 YM . To show (239) , we first note that F B ( ( 1 , x / t ) , r − 1 e r ) = ( ∂ t B ) ⋅ ( r − 1 e r ) F_B((1,x/t),r^{-1}e_r)=(\partial_tB)\cdot(r^{-1}e_r) F B (( 1 , x / t ) , r − 1 e r ) = ( ∂ t B ) ⋅ ( r − 1 e r ) (240) , we estimate
∣ t F B ( ( 1 , x / t ) , ( 1 , 0 ) ) ∣ = ∣ t ( ∂ t B ) ⋅ ( x / t ) ∣ ≲ ∥ B ∥ t 0 Y M ∣ x ∣ t − 1 / 2 = ∥ B ∥ t 0 Y M ( r t − 1 / 2 ) . |tF_B((1,x/t),(1,0))|=|t(\partial_tB)\cdot(x/t)|\lesssim\|B\|_{t_0}^{\mathrm{YM}}|x|t^{-1/2}=\|B\|_{t_0}^{\mathrm{YM}}(rt^{-1/2}). ∣ t F B (( 1 , x / t ) , ( 1 , 0 )) ∣ = ∣ t ( ∂ t B ) ⋅ ( x / t ) ∣ ≲ ∥ B ∥ t 0 YM ∣ x ∣ t − 1/2 = ∥ B ∥ t 0 YM ( r t − 1/2 ) . Using Lemmas 4.56 and 4.57, we may now estimate all error terms appearing on the right-hand side of (234) .
Lemma 4.58. Let B : [ 0 , t 0 ] × T 2 → g 2 B:[0,t_0]\times\mathbb{T}^2\to\mathfrak{g}^2 B : [ 0 , t 0 ] × T 2 → g 2 x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2 ∣ x ∣ ≤ 1 / 10 |x|\leq1/10 ∣ x ∣ ≤ 1/10 t ∈ ( 0 , t 0 ] t\in(0,t_0] t ∈ ( 0 , t 0 ]
∣ ( τ B − 1 F B τ B ) ( t , x ) − F B ( 0 , 0 ) ∣ ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) ) , (241) \left|(\tau_B^{-1}F_B\tau_B)(t,x)-F_B(0,0)\right|\lesssim\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})), \tag*{(241)} ( τ B − 1 F B τ B ) ( t , x ) − F B ( 0 , 0 ) ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 )) , (241) ∣ ( τ B − 1 D r − 1 e θ τ B ) ( t , x ) − r 2 F B 12 ( 0 , 0 ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) , (242) \left|(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(t,x)-\frac{r}{2}F_B^{12}(0,0)\right|\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2), \tag*{(242)} ( τ B − 1 D r − 1 e θ τ B ) ( t , x ) − 2 r F B 12 ( 0 , 0 ) ≲ t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) , (242) ∣ τ B − 1 D r − 1 e r τ B ( t , x ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) , (243) |\tau_B^{-1}D_{r^{-1}e_r}\tau_B(t,x)|\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2), \tag*{(243)} ∣ τ B − 1 D r − 1 e r τ B ( t , x ) ∣ ≲ t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) , (243) ∣ ( τ B − 1 ∂ t τ B ) ( t , x ) ∣ ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) , (244) |(\tau_B^{-1}\partial_t\tau_B)(t,x)|\lesssim\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2), \tag*{(244)} ∣ ( τ B − 1 ∂ t τ B ) ( t , x ) ∣ ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) , (244) ∣ ( τ B − 1 Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) − r 2 4 ( F B 12 ( 0 , 0 ) ) 2 ∣ ≲ ( 1 + t ∥ F ∥ L t , x ∞ ( [ 0 , t 0 ] ) + t ∥ B ∥ t 0 Y M ) × ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 5 ) . (245) \left|(\tau_B^{-1}\operatorname{Hess}_B\tau_B)(r^{-1}e_\theta,r^{-1}e_\theta)(t,x)-\frac{r^2}{4}(F_B^{12}(0,0))^2\right|
\lesssim\left(1+t\|F\|_{L^\infty_{t,x}([0,t_0])}+t\|B\|_{t_0}^{\mathrm{YM}}\right)
\times\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^5). \tag*{(245)} ( τ B − 1 Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) − 4 r 2 ( F B 12 ( 0 , 0 ) ) 2 ≲ ( 1 + t ∥ F ∥ L t , x ∞ ([ 0 , t 0 ]) + t ∥ B ∥ t 0 YM ) × ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 5 ) . (245) ∣ ( τ B − 1 Hess B τ B ) ( r − 1 e r , r − 1 e r ) ( t , x ) ∣ ≲ ( 1 + t ∥ B ∥ t 0 Y M ) × ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) (246) \left|(\tau_B^{-1}\operatorname{Hess}_B\tau_B)(r^{-1}e_r,r^{-1}e_r)(t,x)\right| \lesssim(1+t\|B\|_{t_0}^{\mathrm{YM}})\times\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right) \tag*{(246)} ( τ B − 1 Hess B τ B ) ( r − 1 e r , r − 1 e r ) ( t , x ) ≲ ( 1 + t ∥ B ∥ t 0 YM ) × ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) (246) ∣ ( τ B − 1 Δ B τ B ) ( t , x ) − r 2 4 ( F B 12 ( 0 , 0 ) ) 2 ∣ ≲ ( 1 + t ∥ F ∥ L t , x ∞ ( [ 0 , t 0 ] ) + t ∥ B ∥ t 0 Y M ) × ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 5 ) . (247) \left|(\tau_B^{-1}\Delta_B\tau_B)(t,x)-\frac{r^2}{4}(F_B^{12}(0,0))^2\right| \lesssim\left(1+t\|F\|_{L^\infty_{t,x}([0,t_0])}+t\|B\|_{t_0}^{\mathrm{YM}}\right)\times\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^5\right). \tag*{(247)} ( τ B − 1 Δ B τ B ) ( t , x ) − 4 r 2 ( F B 12 ( 0 , 0 ) ) 2 ≲ ( 1 + t ∥ F ∥ L t , x ∞ ([ 0 , t 0 ]) + t ∥ B ∥ t 0 YM ) × ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 5 ) . (247) Proof. We proceed with the estimates one-by-one.
Proof of (4.63). We apply (235) with w = x / t w=x/t w = x / t u , v ∈ R 2 u,v\in\mathbb{R}^2 u , v ∈ R 2 ∣ u ∣ = ∣ v ∣ = 1 |u|=|v|=1 ∣ u ∣ = ∣ v ∣ = 1
∣ ( τ B − 1 F B τ B ) ( t , x ) − F B ( 0 , 0 ) ∣ ≲ ∥ ∂ t F B ∥ L t , x ∞ t + ∥ D F B ∥ L t , x ∞ ∣ x ∣ ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) ) . \left|(\tau_B^{-1}F_{B\tau_B})(t,x)-F_B(0,0)\right| \lesssim\|\partial_tF_B\|_{L^\infty_{t,x}}t+\|DF_B\|_{L^\infty_{t,x}}|x| \lesssim\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})\right). ( τ B − 1 F B τ B ) ( t , x ) − F B ( 0 , 0 ) ≲ ∥ ∂ t F B ∥ L t , x ∞ t + ∥ D F B ∥ L t , x ∞ ∣ x ∣ ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) ) . Proofs of (4.64) and (4.65). We apply (237) with v = ( 0 , r − 1 e θ ) v=(0,r^{-1}e_\theta) v = ( 0 , r − 1 e θ ) w = x / t w=x/t w = x / t
∣ ( τ B − 1 D r − 1 e θ τ B ) ( t , x ) − t 2 F B ( ( 1 , w ) , r − 1 e θ ) ∣ ≲ t 2 ( ∥ ∂ t B ∥ L t , x ∞ + ∥ ∂ t F B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D F B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) \left|(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(t,x)-\frac{t}{2}F_B((1,w),r^{-1}e_\theta)\right| \lesssim t^2\left(\|\partial_tB\|_{L^\infty_{t,x}}+\|\partial_tF_B\|_{L^\infty_{t,x}}|x|t^{-1}+\|D\partial_tB\|_{L^\infty_{t,x}}|x|t^{-1}+\|DF_B\|_{L^\infty_{t,x}}|x|^2t^{-2}\right) ( τ B − 1 D r − 1 e θ τ B ) ( t , x ) − 2 t F B (( 1 , w ) , r − 1 e θ ) ≲ t 2 ( ∥ ∂ t B ∥ L t , x ∞ + ∥ ∂ t F B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D F B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) ≲ t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) . \lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right). ≲ t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) . Combining this with (238) , we have thus shown (4.64). The proof of (4.65) is similar, except we take v = ( 0 , r − 1 e r ) v=(0,r^{-1}e_r) v = ( 0 , r − 1 e r ) (239) in place of (238) .
Proof of (4.66). We apply (237) with v = ( 1 , 0 ) v=(1,0) v = ( 1 , 0 ) w = x / t w=x/t w = x / t
∣ ( τ B − 1 ∂ t τ B ) ( t , x ) − t 2 F B ( ( 1 , w ) , ( 1 , 0 ) ) ∣ ≲ t 2 ( ∥ ∂ t 2 B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) \left|(\tau_B^{-1}\partial_t\tau_B)(t,x)-\frac{t}{2}F_B((1,w),(1,0))\right| \lesssim t^2\left(\|\partial_t^2B\|_{L^\infty_{t,x}}|x|t^{-1}+\|D\partial_tB\|_{L^\infty_{t,x}}|x|^2t^{-2}\right) ( τ B − 1 ∂ t τ B ) ( t , x ) − 2 t F B (( 1 , w ) , ( 1 , 0 )) ≲ t 2 ( ∥ ∂ t 2 B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) . \lesssim\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right). ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) . Combining this with (240) , we may thus obtain (4.66).
Proof of (4.67). We want to apply Lemma 4.34, but we need to be careful because r − 1 e θ r^{-1}e_\theta r − 1 e θ ∂ r − 1 e θ ( r − 1 e θ ) = − r − 1 ( r − 1 e r ) ≠ 0 \partial_{r^{-1}e_\theta}(r^{-1}e_\theta)=-r^{-1}(r^{-1}e_r)\ne0 ∂ r − 1 e θ ( r − 1 e θ ) = − r − 1 ( r − 1 e r ) = 0 (231) ). Towards this end, let v 1 = v 2 = ( r − 1 e θ ) ( x ) v_1=v_2=(r^{-1}e_\theta)(x) v 1 = v 2 = ( r − 1 e θ ) ( x ) ( Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) = ( D v 1 D v 2 τ B ) ( t , x ) (\operatorname{Hess}_B\tau_B)(r^{-1}e_\theta,r^{-1}e_\theta)(t,x)=(D_{v_1}D_{v_2}\tau_B)(t,x) ( Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) = ( D v 1 D v 2 τ B ) ( t , x ) v 1 , v 2 v_1,v_2 v 1 , v 2 r − 1 e θ r^{-1}e_\theta r − 1 e θ γ ( s ) = s ( 1 , x / t ) \gamma(s)=s(1,x/t) γ ( s ) = s ( 1 , x / t )
( τ B − 1 Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) = ( τ B − 1 D v 1 D v 2 τ B ) ( t , x ) = 2 t 2 ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s ) ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) + 1 t 2 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e θ F B ) ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s ) ) . \begin{aligned}
(\tau_B^{-1}\operatorname{Hess}_B\tau_B)(r^{-1}e_\theta,r^{-1}e_\theta)(t,x) \\
&=(\tau_B^{-1}D_{v_1}D_{v_2}\tau_B)(t,x) \\
&=\frac{2}{t^2}\int_0^t ds\,s^2(\tau_B^{-1}F_B(\dot{\gamma}(s),r^{-1}e_\theta)\tau_B)(\gamma(s))(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(\gamma(s)) \\
&+\frac{1}{t^2}\int_0^t ds\,s^2(\tau_B^{-1}(D_{r^{-1}e_\theta}F_B)(\dot{\gamma}(s),r^{-1}e_\theta)\tau_B)(\gamma(s)).
\end{aligned} ( τ B − 1 Hess B τ B ) ( r − 1 e θ , r − 1 e θ ) ( t , x ) = ( τ B − 1 D v 1 D v 2 τ B ) ( t , x ) = t 2 2 ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s )) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) + t 2 1 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e θ F B ) ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s )) . The second term above is a perturbation, as we may estimate
1 t 2 ∫ 0 t d s s 2 ∣ ( τ B − 1 ( D r − 1 e θ F B ) ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s ) ) ∣ ≲ ∫ 0 t d s ( ∥ D F B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ) \frac{1}{t^2}\int_0^t ds\,s^2\left|(\tau_B^{-1}(D_{r^{-1}e_\theta}F_B)(\dot{\gamma}(s),r^{-1}e_\theta)\tau_B)(\gamma(s))\right| \lesssim\int_0^t ds\left(\|DF_B\|_{L^\infty_{t,x}}|x|t^{-1}+\|D\partial_tB\|_{L^\infty_{t,x}}\right) t 2 1 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e θ F B ) ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s )) ≲ ∫ 0 t d s ( ∥ D F B ∥ L t , x ∞ ∣ x ∣ t − 1 + ∥ D ∂ t B ∥ L t , x ∞ ) ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) ) , \lesssim\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})\right), ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) ) , which is acceptable. To handle the leading term, first, for notational brevity, let
R 1 ( s ) = ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s ) ) − r t F B 12 ( 0 , 0 ) , R_1(s)=\bigl(\tau_B^{-1}F_B(\dot{\gamma}(s),r^{-1}e_\theta)\tau_B\bigr)(\gamma(s))-\frac{r}{t}F_B^{12}(0,0), R 1 ( s ) = ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s )) − t r F B 12 ( 0 , 0 ) , R 2 ( s ) = ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) − s 2 t r F B 12 ( 0 , 0 ) . R_2(s)=\bigl(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B\bigr)(\gamma(s))-\frac{s}{2t}rF_B^{12}(0,0). R 2 ( s ) = ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) − 2 t s r F B 12 ( 0 , 0 ) . Note that (recalling (4.55)) F B ( γ ˙ ( 0 ) , r − 1 e θ ) = F B 12 ( 0 , 0 ) r t − 1 + ( ∂ t B ) ⋅ r − 1 e θ F_B(\dot{\gamma}(0),r^{-1}e_\theta)=F_B^{12}(0,0)rt^{-1}+(\partial_tB)\cdot r^{-1}e_\theta F B ( γ ˙ ( 0 ) , r − 1 e θ ) = F B 12 ( 0 , 0 ) r t − 1 + ( ∂ t B ) ⋅ r − 1 e θ (236) , we may estimate
∣ R 1 ( s ) ∣ ≲ s ( ∥ ∂ t 2 B ∥ L t , x ∞ + ( ∥ ∂ t F B ∥ L t , x ∞ + ∥ D ∂ t B ∥ L t , x ∞ ) ∣ x ∣ t − 1 + ∥ D F B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) + ∥ ∂ t B ∥ L t , x ∞ ≲ t − 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) . \begin{aligned}
|R_1(s)|&\lesssim s\bigl(\|\partial_t^2B\|_{L^\infty_{t,x}}+(\|\partial_tF_B\|_{L^\infty_{t,x}}+\|D\partial_tB\|_{L^\infty_{t,x}})|x|t^{-1}+\|DF_B\|_{L^\infty_{t,x}}|x|^2t^{-2}\bigr)+\|\partial_tB\|_{L^\infty_{t,x}}\\
&\lesssim t^{-1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2).
\end{aligned} ∣ R 1 ( s ) ∣ ≲ s ( ∥ ∂ t 2 B ∥ L t , x ∞ + ( ∥ ∂ t F B ∥ L t , x ∞ + ∥ D ∂ t B ∥ L t , x ∞ ) ∣ x ∣ t − 1 + ∥ D F B ∥ L t , x ∞ ∣ x ∣ 2 t − 2 ) + ∥ ∂ t B ∥ L t , x ∞ ≲ t − 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) . We also apply (242) to estimate (here, note that ∣ x ∣ − 1 e θ ( x ) = ∣ ( s / t ) x ∣ − 1 e θ ( γ ( s ) ) |x|^{-1}e_\theta(x)=|(s/t)x|^{-1}e_\theta(\gamma(s)) ∣ x ∣ − 1 e θ ( x ) = ∣ ( s / t ) x ∣ − 1 e θ ( γ ( s )) (242) with ( t , x ) (t,x) ( t , x ) γ ( s ) = ( s , ( s / t ) x ) \gamma(s)=(s,(s/t)x) γ ( s ) = ( s , ( s / t ) x )
∣ R 2 ( s ) ∣ ≲ s 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) . |R_2(s)|\lesssim s^{1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2). ∣ R 2 ( s ) ∣ ≲ s 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) . Inserting these approximations, it follows that
1 t 2 ∣ ∫ 0 t d s s 2 ( 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s ) ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) − s t ∣ x ∣ 2 ( F B 12 ( 0 , 0 ) ) 2 ) ∣ ≲ 1 t 2 ∫ 0 t d s s 2 ∣ R 1 ( s ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) ∣ + 1 t 2 ∫ 0 t d s s 2 s t ∣ F B 12 ( 0 , 0 ) R 2 ( s ) ∣ . \begin{aligned}
\frac{1}{t^2}\left|\int_0^t ds\,s^2\left(2\bigl(\tau_B^{-1}F_B(\dot{\gamma}(s),r^{-1}e_\theta)\tau_B\bigr)(\gamma(s))\bigl(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B\bigr)(\gamma(s))-\frac{s}{t}|x|^2(F_B^{12}(0,0))^2\right)\right|\\
\lesssim\frac{1}{t^2}\int_0^t ds\,s^2|R_1(s)(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(\gamma(s))|+\frac{1}{t^2}\int_0^t ds\,s^2\frac{s}{t}|F_B^{12}(0,0)R_2(s)|.
\end{aligned} t 2 1 ∫ 0 t d s s 2 ( 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e θ ) τ B ) ( γ ( s )) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) − t s ∣ x ∣ 2 ( F B 12 ( 0 , 0 ) ) 2 ) ≲ t 2 1 ∫ 0 t d s s 2 ∣ R 1 ( s ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) ∣ + t 2 1 ∫ 0 t d s s 2 t s ∣ F B 12 ( 0 , 0 ) R 2 ( s ) ∣. We may combine our estimates on R 1 R_1 R 1 R 2 R_2 R 2
r ∣ F B 12 ( 0 , 0 ) ∣ ≲ ∥ F B ∥ L t , x ∞ r , r|F_B^{12}(0,0)|\lesssim\|F_B\|_{L^\infty_{t,x}}r, r ∣ F B 12 ( 0 , 0 ) ∣ ≲ ∥ F B ∥ L t , x ∞ r , ∣ ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) ∣ ≲ ∣ ( s / t ) r F B 12 ( 0 , 0 ) ∣ + ∣ R 2 ( s ) ∣ ≲ ∥ F B ∥ L t , x ∞ r + ∣ R 2 ( s ) ∣ , |(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(\gamma(s))|\lesssim|(s/t)rF_B^{12}(0,0)|+|R_2(s)|\lesssim\|F_B\|_{L^\infty_{t,x}}r+|R_2(s)|, ∣ ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) ∣ ≲ ∣ ( s / t ) r F B 12 ( 0 , 0 ) ∣ + ∣ R 2 ( s ) ∣ ≲ ∥ F B ∥ L t , x ∞ r + ∣ R 2 ( s ) ∣ , in order to estimate
∫ 0 t d s ∣ R 1 ( s ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s ) ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) ( ∥ F B ∥ L t , x ∞ r + t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) ) ≲ t ( ∥ F B ∥ L t , x ∞ + ∥ B ∥ t 0 Y M ) ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 5 ) , \begin{aligned}
\int_0^t ds\,|R_1(s)(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B)(\gamma(s))|
&\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2)(\|F_B\|_{L^\infty_{t,x}}r+t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^2))\\
&\lesssim t(\|F_B\|_{L^\infty_{t,x}}+\|B\|_{t_0}^{\mathrm{YM}})\|B\|_{t_0}^{\mathrm{YM}}(1+(rt^{-1/2})^5),
\end{aligned} ∫ 0 t d s ∣ R 1 ( s ) ( τ B − 1 D r − 1 e θ τ B ) ( γ ( s )) ∣ ≲ t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) ( ∥ F B ∥ L t , x ∞ r + t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 )) ≲ t ( ∥ F B ∥ L t , x ∞ + ∥ B ∥ t 0 YM ) ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 5 ) , and similarly for the second term. To finish, we simply compute
1 t 2 ∫ 0 t d s s 3 t 2 ∣ x ∣ 2 ( F B 12 ( 0 , 0 ) ) 2 = r 2 4 ( F B 12 ( 0 , 0 ) ) 2 . \frac{1}{t^2}\int_0^t ds\,\frac{s^3}{t^2}|x|^2(F_B^{12}(0,0))^2=\frac{r^2}{4}(F_B^{12}(0,0))^2. t 2 1 ∫ 0 t d s t 2 s 3 ∣ x ∣ 2 ( F B 12 ( 0 , 0 ) ) 2 = 4 r 2 ( F B 12 ( 0 , 0 ) ) 2 . The desired estimate (245) now follows.
Proof of (246) . We start as before, except now since ∂ r − 1 e r ( r − 1 e r ) = 0 \partial_{r^{-1}e_r}(r^{-1}e_r)=0 ∂ r − 1 e r ( r − 1 e r ) = 0 D r − 1 e r D r − 1 e r τ B = ( Hess B τ B ) ( r − 1 e r , r − 1 e r ) D_{r^{-1}e_r}D_{r^{-1}e_r}\tau_B=(\operatorname{Hess}_B\tau_B)(r^{-1}e_r,r^{-1}e_r) D r − 1 e r D r − 1 e r τ B = ( Hess B τ B ) ( r − 1 e r , r − 1 e r ) γ ( s ) = s ( 1 , x / t ) \gamma(s)=s(1,x/t) γ ( s ) = s ( 1 , x / t )
( τ B − 1 Hess B τ B ) ( r − 1 e r , r − 1 e r ) ( t , x ) = 2 t 2 ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s ) ) ( τ B − 1 D r − 1 e r τ B ) ( γ ( s ) ) + 1 t 2 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e r F B ) ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s ) ) . \begin{aligned}
(\tau_B^{-1}\operatorname{Hess}_B\tau_B)(r^{-1}e_r,r^{-1}e_r)(t,x)
=\frac{2}{t^2}\int_0^t ds\,s^2\bigl(\tau_B^{-1}F_B(\dot{\gamma}(s),r^{-1}e_r)\tau_B\bigr)(\gamma(s))\bigl(\tau_B^{-1}D_{r^{-1}e_r}\tau_B\bigr)(\gamma(s)) \\
&+\frac{1}{t^2}\int_0^t ds\,s^2\left(\tau_B^{-1}(D_{r^{-1}e_r}F_B)(\dot{\gamma}(s),r^{-1}e_r)\tau_B\right)(\gamma(s)).
\end{aligned} ( τ B − 1 Hess B τ B ) ( r − 1 e r , r − 1 e r ) ( t , x ) = t 2 2 ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s )) ( τ B − 1 D r − 1 e r τ B ) ( γ ( s )) + t 2 1 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e r F B ) ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s )) . The second term may be estimated as before:
1 t 2 ∫ 0 t d s s 2 ∣ ( τ B − 1 ( D r − 1 e r F B ) ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s ) ) ∣ ≲ ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) ) . \frac{1}{t^2}\int_0^t ds\,s^2\left|\left(\tau_B^{-1}(D_{r^{-1}e_r}F_B)(\dot{\gamma}(s),r^{-1}e_r)\tau_B\right)(\gamma(s))\right|\lesssim\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})\right). t 2 1 ∫ 0 t d s s 2 ( τ B − 1 ( D r − 1 e r F B ) ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s )) ≲ ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) ) . The first term is now also a perturbation. To see this, first note that
F B ( γ ˙ ( s ) , r − 1 e r ) = F B ( ( 1 , x / t ) , r − 1 e r ) = ( ∂ t B ) ⋅ ( r − 1 e r ) . F_B(\dot{\gamma}(s),r^{-1}e_r)=F_B((1,x/t),r^{-1}e_r)=(\partial_tB)\cdot(r^{-1}e_r). F B ( γ ˙ ( s ) , r − 1 e r ) = F B (( 1 , x / t ) , r − 1 e r ) = ( ∂ t B ) ⋅ ( r − 1 e r ) . Combining this with (243) , we may estimate
1 t 2 ∣ ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s ) ) ( τ B − 1 D r − 1 e r τ B ) ( γ ( s ) ) ∣ \frac{1}{t^2}\left|\int_0^t ds\,s^2\left(\tau_B^{-1}F_B(\dot{\gamma}(s),r^{-1}e_r)\tau_B\right)(\gamma(s))\left(\tau_B^{-1}D_{r^{-1}e_r}\tau_B\right)(\gamma(s))\right| t 2 1 ∫ 0 t d s s 2 ( τ B − 1 F B ( γ ˙ ( s ) , r − 1 e r ) τ B ) ( γ ( s )) ( τ B − 1 D r − 1 e r τ B ) ( γ ( s )) ≲ ∫ 0 t d s ∥ ∂ t B ∥ L t , x ∞ s 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) \lesssim\int_0^t ds\,\|\partial_tB\|_{L^\infty_{t,x}}s^{1/2}\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right) ≲ ∫ 0 t d s ∥ ∂ t B ∥ L t , x ∞ s 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) ≲ t ( ∥ B ∥ t 0 Y M ) 2 ( 1 + ( r t − 1 / 2 ) 2 ) , \lesssim t\left(\|B\|_{t_0}^{\mathrm{YM}}\right)^2\left(1+(rt^{-1/2})^2\right), ≲ t ( ∥ B ∥ t 0 YM ) 2 ( 1 + ( r t − 1/2 ) 2 ) , which is acceptable. Combining, we may thus obtain (246) .
Proof of (247) . We combine (245) and (246) .
Remark 4.59. In the estimates of Lemmas 4.56, 4.57, and 4.58, we did not assume that B B B (196) of q B q_B q B
With the estimates from Lemma 4.58 in hand, we may now prove Proposition 4.51.
Proof of Proposition 4.51. Recall that r B F r_B^F r B F (234) . We distribute
∂ r − 1 e θ ( τ B − 1 q B ) = τ B − 1 D r − 1 e θ q B − τ B − 1 ( D r − 1 e θ τ B ) τ B − 1 q B , \partial_{r^{-1}e_\theta}(\tau_B^{-1}q_B)=\tau_B^{-1}D_{r^{-1}e_\theta}q_B-\tau_B^{-1}(D_{r^{-1}e_\theta}\tau_B)\tau_B^{-1}q_B, ∂ r − 1 e θ ( τ B − 1 q B ) = τ B − 1 D r − 1 e θ q B − τ B − 1 ( D r − 1 e θ τ B ) τ B − 1 q B , and similarly for r − 1 e r r^{-1}e_r r − 1 e r (234) , we further obtain
( ∂ t − L X F B , θ ) r B F = R 1 ( σ τ B − 1 q B ) + 2 σ R 2 τ B − 1 D r − 1 e θ q B + 2 σ R 3 τ B − 1 D r − 1 e r q B + R 4 , (\partial_t-\mathcal{L}_X^{F_B,\theta})r_B^F=R_1(\sigma\tau_B^{-1}q_B)+2\sigma R_2\tau_B^{-1}D_{r^{-1}e_\theta}q_B+2\sigma R_3\tau_B^{-1}D_{r^{-1}e_r}q_B+R_4, ( ∂ t − L X F B , θ ) r B F = R 1 ( σ τ B − 1 q B ) + 2 σ R 2 τ B − 1 D r − 1 e θ q B + 2 σ R 3 τ B − 1 D r − 1 e r q B + R 4 , where (recall X = F B 12 ( 0 , 0 ) X=F_B^{12}(0,0) X = F B 12 ( 0 , 0 )
R 1 = 2 ( τ B − 1 F B τ B − Rot ( X ) ) + ( τ B − 1 Δ B τ B − r 2 4 X 2 ) − 2 ( τ B − 1 D r − 1 e θ τ B − r 2 X ) τ B − 1 D r − 1 e θ τ B − τ B − 1 ∂ t τ B − 2 ( τ B − 1 D r − 1 e r τ B ) ( τ B − 1 D r − 1 e r τ B ) , R 2 = τ B − 1 D r − 1 e θ τ B − r 2 X , R 3 = τ B − 1 D r − 1 e r τ B , R 4 = − r X ( ∂ r − 1 e θ σ ) ( τ B − 1 q B − s X F ) − 2 ( ∂ ℓ σ ) ∂ ℓ ( τ B − 1 q B − s X F ) − ( Δ σ ) ( τ B − 1 q B − s X F ) . \begin{aligned}
R_1={}&2(\tau_B^{-1}F_B\tau_B-\operatorname{Rot}(X))+\left(\tau_B^{-1}\Delta_B\tau_B-\frac{r^2}{4}X^2\right)\\
&-2\left(\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B-\frac{r}{2}X\right)\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B\\
&-\tau_B^{-1}\partial_t\tau_B-2(\tau_B^{-1}D_{r^{-1}e_r}\tau_B)(\tau_B^{-1}D_{r^{-1}e_r}\tau_B),\\
R_2={}&\tau_B^{-1}D_{r^{-1}e_\theta}\tau_B-\frac{r}{2}X,\\
R_3={}&\tau_B^{-1}D_{r^{-1}e_r}\tau_B,\\
R_4={}&-rX(\partial_{r^{-1}e_\theta}\sigma)(\tau_B^{-1}q_B-s_X^F)-2(\partial^\ell\sigma)\partial_\ell(\tau_B^{-1}q_B-s_X^F)-(\Delta\sigma)(\tau_B^{-1}q_B-s_X^F).
\end{aligned} R 1 = R 2 = R 3 = R 4 = 2 ( τ B − 1 F B τ B − Rot ( X )) + ( τ B − 1 Δ B τ B − 4 r 2 X 2 ) − 2 ( τ B − 1 D r − 1 e θ τ B − 2 r X ) τ B − 1 D r − 1 e θ τ B − τ B − 1 ∂ t τ B − 2 ( τ B − 1 D r − 1 e r τ B ) ( τ B − 1 D r − 1 e r τ B ) , τ B − 1 D r − 1 e θ τ B − 2 r X , τ B − 1 D r − 1 e r τ B , − r X ( ∂ r − 1 e θ σ ) ( τ B − 1 q B − s X F ) − 2 ( ∂ ℓ σ ) ∂ ℓ ( τ B − 1 q B − s X F ) − ( Δ σ ) ( τ B − 1 q B − s X F ) . By Lemma 4.58, we have the following pointwise bounds on R 1 R_1 R 1 R 2 R_2 R 2 R 3 R_3 R 3 σ \sigma σ x x x ∣ x ∣ ≤ 1 / 10 |x|\le1/10 ∣ x ∣ ≤ 1/10
∣ R 1 ( t , x ) ∣ ≲ ( 1 + t ∥ F B ∥ L t , x ∞ + t ∥ B ∥ t 0 Y M ) ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 5 ) , |R_1(t,x)|\lesssim\left(1+t\|F_B\|_{L^\infty_{t,x}}+t\|B\|_{t_0}^{\mathrm{YM}}\right)\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^5\right), ∣ R 1 ( t , x ) ∣ ≲ ( 1 + t ∥ F B ∥ L t , x ∞ + t ∥ B ∥ t 0 YM ) ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 5 ) , ∣ R 2 ( t , x ) ∣ + ∣ R 3 ( t , x ) ∣ ≲ t 1 / 2 ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) . |R_2(t,x)|+|R_3(t,x)|\lesssim t^{1/2}\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right). ∣ R 2 ( t , x ) ∣ + ∣ R 3 ( t , x ) ∣ ≲ t 1/2 ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) . To estimate R 4 R_4 R 4 σ \sigma σ R 4 R_4 R 4 x x x ∣ x ∣ ≳ 1 |x|\gtrsim1 ∣ x ∣ ≳ 1 q B q_B q B s X F , θ \mathfrak{s}^{F,\theta}_X s X F , θ
∣ R 4 ( t , x ) ∣ ≲ t 10 ( 1 + ∥ F ∥ L t , x ∞ ( [ 0 , t 0 ] ) + ∥ B ∥ t 0 Y M ) exp ( C t ∥ F ∥ L t , x ∞ ( [ 0 , t 0 ] ) ) t − 1 exp ( − c ∣ x ∣ 2 / t ) . |R_4(t,x)|\lesssim t^{10}\left(1+\|F\|_{L^\infty_{t,x}([0,t_0])}+\|B\|_{t_0}^{\mathrm{YM}}\right)\exp\left(Ct\|F\|_{L^\infty_{t,x}([0,t_0])}\right)t^{-1}\exp\left(-c|x|^2/t\right). ∣ R 4 ( t , x ) ∣ ≲ t 10 ( 1 + ∥ F ∥ L t , x ∞ ([ 0 , t 0 ]) + ∥ B ∥ t 0 YM ) exp ( C t ∥ F ∥ L t , x ∞ ([ 0 , t 0 ]) ) t − 1 exp ( − c ∣ x ∣ 2 / t ) . Here, the power 10 could have been any number. The point is that R 4 R_4 R 4
From the defining equation (226) for s X F , θ \mathfrak{s}^{F,\theta}_X s X F , θ s X F , θ \mathfrak{s}^{F,\theta}_X s X F , θ ( ∂ t − L X F , θ ) − 1 (\partial_t-L_X^{F,\theta})^{-1} ( ∂ t − L X F , θ ) − 1 s X F , θ \mathfrak{s}^{F,\theta}_X s X F , θ (227) ) as well as q B q_B q B D q B Dq_B D q B ( t 1 , x 1 ) ∈ ( 0 , t 0 ] × T 2 (t_1,x_1)\in(0,t_0]\times\mathbb{T}^2 ( t 1 , x 1 ) ∈ ( 0 , t 0 ] × T 2
∣ r B F ( t 1 , x 1 ) ∣ ≲ ∫ d t d x ∣ s X F , θ ( t , x ; t 1 , x 1 ) ∣ ( ∣ R 1 ( t , x ) ∣ ∣ q B ( t , x ) ∣ + ( ∣ R 2 ( t , x ) ∣ + ∣ R 3 ( t , x ) ∣ ) ∣ D q B ( t , x ) ∣ + ∣ R 4 ( t , x ) ∣ ) ≲ exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ) ∫ d t d x p ( t , x ; t 1 , x 1 ) t − 1 exp ( − c ∣ x ∣ 2 / t ) × ( t 10 ( 1 + ∥ B ∥ t 0 Y M ) + ( 1 + t ∥ F B ∥ L t , x ∞ + t ∥ B ∥ t 0 Y M ) ∥ B ∥ t 0 Y M ( 1 + ( r t − 1 / 2 ) 2 ) ) ≲ ( c 0 + t 10 ) exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ) t 1 − 1 exp ( − c ∣ x 1 ∣ 2 / t 1 ) , \begin{aligned}
|\mathfrak{r}^{F}_B(t_1,x_1)|&\lesssim\int dt\,dx\,|\mathfrak{s}^{F,\theta}_X(t,x;t_1,x_1)|\bigl(|R_1(t,x)||q_B(t,x)|\\
&\qquad+\left(|R_2(t,x)|+|R_3(t,x)|\right)|Dq_B(t,x)|+|R_4(t,x)|\bigr)\\
&\lesssim\exp\left(C_0t_0\|F_B\|_{L^\infty_{t,x}}\right)\int dt\,dx\,p(t,x;t_1,x_1)t^{-1}\exp\left(-c|x|^2/t\right)\\
&\qquad\times\left(t^{10}\left(1+\|B\|_{t_0}^{\mathrm{YM}}\right)+\left(1+t\|F_B\|_{L^\infty_{t,x}}+t\|B\|_{t_0}^{\mathrm{YM}}\right)\|B\|_{t_0}^{\mathrm{YM}}\left(1+(rt^{-1/2})^2\right)\right)\\
&\lesssim\left(c_0+t^{10}\right)\exp\left(C_0t_0\|F_B\|_{L^\infty_{t,x}}\right)t_1^{-1}\exp\left(-c|x_1|^2/t_1\right),
\end{aligned} ∣ r B F ( t 1 , x 1 ) ∣ ≲ ∫ d t d x ∣ s X F , θ ( t , x ; t 1 , x 1 ) ∣ ( ∣ R 1 ( t , x ) ∣∣ q B ( t , x ) ∣ + ( ∣ R 2 ( t , x ) ∣ + ∣ R 3 ( t , x ) ∣ ) ∣ D q B ( t , x ) ∣ + ∣ R 4 ( t , x ) ∣ ) ≲ exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ) ∫ d t d x p ( t , x ; t 1 , x 1 ) t − 1 exp ( − c ∣ x ∣ 2 / t ) × ( t 10 ( 1 + ∥ B ∥ t 0 YM ) + ( 1 + t ∥ F B ∥ L t , x ∞ + t ∥ B ∥ t 0 YM ) ∥ B ∥ t 0 YM ( 1 + ( r t − 1/2 ) 2 ) ) ≲ ( c 0 + t 10 ) exp ( C 0 t 0 ∥ F B ∥ L t , x ∞ ) t 1 − 1 exp ( − c ∣ x 1 ∣ 2 / t 1 ) , as desired (in the last step we applied the smallness assumption (229) ).
Schauder estimates
In this section, we collect several Schauder estimates which will be needed when carrying out perturbative arguments in the coming sections. These results will be obtained as consequences of the material in Sections 4.1–4.2. We begin by setting notation regarding covariant Duhamel operators.
Definition 4.60. For H : [ 0 , T ] × T d → W H:[0,T]\times\mathbb{T}^d\to W H : [ 0 , T ] × T d → W
Duh B ( H ) ( t , x ) ≔ ∫ 0 t d s ∫ T d d y p B ( s , y ; t , x ) H ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d . \operatorname{Duh}_B(H)(t,x)\coloneqq\int_0^t ds\int_{\mathbb{T}^d}dy\,p_B(s,y;t,x)H(s,y),\qquad(t,x)\in[0,T]\times\mathbb{T}^d. Duh B ( H ) ( t , x ) : = ∫ 0 t d s ∫ T d d y p B ( s , y ; t , x ) H ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d . Similarly, for G : [ 0 , T ] × T d → W d G:[0,T]\times\mathbb{T}^d\to W^d G : [ 0 , T ] × T d → W d
Duh B F ( G ) ( t , x ) ≔ ∫ 0 t d s ∫ T d d y q B ( s , y ; t , x ) G ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d . \operatorname{Duh}^{F}_B(G)(t,x)\coloneqq\int_0^t ds\int_{\mathbb{T}^d}dy\,q_B(s,y;t,x)G(s,y),\qquad(t,x)\in[0,T]\times\mathbb{T}^d. Duh B F ( G ) ( t , x ) : = ∫ 0 t d s ∫ T d d y q B ( s , y ; t , x ) G ( s , y ) , ( t , x ) ∈ [ 0 , T ] × T d . Note that Duh B ( H ) , Duh B F ( G ) \operatorname{Duh}_B(H),\operatorname{Duh}^{F}_B(G) Duh B ( H ) , Duh B F ( G )
( ∂ t − Δ B ) Duh B ( H ) = H , Duh B ( H ) ( 0 ) = 0 , (\partial_t-\Delta_B)\operatorname{Duh}_B(H)=H,\qquad\operatorname{Duh}_B(H)(0)=0, ( ∂ t − Δ B ) Duh B ( H ) = H , Duh B ( H ) ( 0 ) = 0 , ( ∂ t − Δ B − 2 F ) Duh B F ( G ) = G , Duh B F ( G ) ( 0 ) = 0. (\partial_t-\Delta_B-2F)\operatorname{Duh}^{F}_B(G)=G,\qquad\operatorname{Duh}^{F}_B(G)(0)=0. ( ∂ t − Δ B − 2 F ) Duh B F ( G ) = G , Duh B F ( G ) ( 0 ) = 0. We proceed with the main results of this section.
Lemma 4.61 (Covariant Schauder estimates). Let C 0 C_0 C 0 T ≤ 1 T\leq1 T ≤ 1 G : [ 0 , T ] × T d → W d G:[0,T]\times\mathbb{T}^d\to W^d G : [ 0 , T ] × T d → W d 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞
∥ Duh B F ( G ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ G r T ( C 0 F ) ∥ G ∥ L t 1 L x p ( [ 0 , T ] ) , \left\|\operatorname{Duh}^{F}_B(G)\right\|_{L^\infty_tL^p_x([0,T])}\lesssim G_{rT}(C_0F)\|G\|_{L^1_tL^p_x([0,T])}, Duh B F ( G ) L t ∞ L x p ([ 0 , T ]) ≲ G r T ( C 0 F ) ∥ G ∥ L t 1 L x p ([ 0 , T ]) , ∥ D B Duh B F ( G ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , 1 p e r t ) sup t 0 ∈ ( 0 , T ] ∥ ( t 0 − t ) − 1 / 2 G ∥ L t 1 L x p ( [ 0 , t 0 ] ) , \|D_B\operatorname{Duh}_B^F(G)\|_{L_t^\infty L_x^p([0,T])} \lesssim\operatorname{Gr}_T(C_0F)(1+\|F\|_{T,1}^{\mathrm{pert}})\sup_{t_0\in(0,T]}\|(t_0-t)^{-1/2}G\|_{L_t^1L_x^p([0,t_0])}, ∥ D B Duh B F ( G ) ∥ L t ∞ L x p ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , 1 pert ) t 0 ∈ ( 0 , T ] sup ∥ ( t 0 − t ) − 1/2 G ∥ L t 1 L x p ([ 0 , t 0 ]) , ∥ Duh B F ( D B G ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , 1 p e r t ) sup t 0 ∈ ( 0 , T ] ∥ min ( t , ( t 0 − t ) ) − 1 / 2 G ∥ L t 1 L x p ( [ 0 , t 0 ] ) . \|\operatorname{Duh}_B^F(D_BG)\|_{L_t^\infty L_x^p([0,T])} \lesssim\operatorname{Gr}_T(C_0F)(1+\|F\|_{T,1}^{\mathrm{pert}})\sup_{t_0\in(0,T]}\|\min(t,(t_0-t))^{-1/2}G\|_{L_t^1L_x^p([0,t_0])}. ∥ Duh B F ( D B G ) ∥ L t ∞ L x p ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , 1 pert ) t 0 ∈ ( 0 , T ] sup ∥ min ( t , ( t 0 − t ) ) − 1/2 G ∥ L t 1 L x p ([ 0 , t 0 ]) . Similarly, for H : [ 0 , T ] × T d → W H:[0,T]\times\mathbb{T}^d\to W H : [ 0 , T ] × T d → W
∥ Duh B ( H ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ ∥ H ∥ L t 1 L x p ( [ 0 , T ] ) , \|\operatorname{Duh}_B(H)\|_{L_t^\infty L_x^p([0,T])}\lesssim\|H\|_{L_t^1L_x^p([0,T])}, ∥ Duh B ( H ) ∥ L t ∞ L x p ([ 0 , T ]) ≲ ∥ H ∥ L t 1 L x p ([ 0 , T ]) , ∥ D B Duh B ( H ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) sup t 0 ∈ ( 0 , T ] ∥ ( t 0 − t ) − 1 / 2 H ∥ L t 1 L x p ( [ 0 , t 0 ] ) , \|D_B\operatorname{Duh}_B(H)\|_{L_t^\infty L_x^p([0,T])}\lesssim(1+\|F\|_{T,0}^{\mathrm{pert}})\sup_{t_0\in(0,T]}\|(t_0-t)^{-1/2}H\|_{L_t^1L_x^p([0,t_0])}, ∥ D B Duh B ( H ) ∥ L t ∞ L x p ([ 0 , T ]) ≲ ( 1 + ∥ F ∥ T , 0 pert ) t 0 ∈ ( 0 , T ] sup ∥ ( t 0 − t ) − 1/2 H ∥ L t 1 L x p ([ 0 , t 0 ]) , ∥ Duh B ( D B H ) ∥ L t ∞ L x p ( [ 0 , T ] ) ≲ ( 1 + ∥ F ∥ T , 1 p e r t ) sup t 0 ∈ ( 0 , T ] ∥ min ( t , ( t 0 − t ) ) − 1 / 2 H ∥ L t 1 L x p ( [ 0 , t 0 ] ) . \|\operatorname{Duh}_B(D_BH)\|_{L_t^\infty L_x^p([0,T])}\lesssim(1+\|F\|_{T,1}^{\mathrm{pert}})\sup_{t_0\in(0,T]}\|\min(t,(t_0-t))^{-1/2}H\|_{L_t^1L_x^p([0,t_0])}. ∥ Duh B ( D B H ) ∥ L t ∞ L x p ([ 0 , T ]) ≲ ( 1 + ∥ F ∥ T , 1 pert ) t 0 ∈ ( 0 , T ] sup ∥ min ( t , ( t 0 − t ) ) − 1/2 H ∥ L t 1 L x p ([ 0 , t 0 ]) . Proof. We focus on the estimates for Duh B F \operatorname{Duh}_B^F Duh B F Duh B \operatorname{Duh}_B Duh B Proposition 4.17 applied in place of Proposition 4.18 ). Let t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ] Proposition 4.18 (in particular (196) with k = 0 k=0 k = 0
∥ Duh B F ( G ) ( t 0 ) ∥ L x p ≤ ∫ d s ∥ ∫ d y q B ( s , y ; t 0 , x ) G ( s , y ) ∥ L x p \|\operatorname{Duh}_B^F(G)(t_0)\|_{L_x^p}\leq\int ds\left\|\int dy\,q_B(s,y;t_0,x)G(s,y)\right\|_{L_x^p} ∥ Duh B F ( G ) ( t 0 ) ∥ L x p ≤ ∫ d s ∫ d y q B ( s , y ; t 0 , x ) G ( s , y ) L x p ≲ Gr T ( C 0 F ) ∫ d s ∥ ∫ d y ( t 0 − s ) − d / 2 exp ( − c ∣ x − y ∣ 2 t 0 − s ) ∣ G ( s , y ) ∣ ∥ L x p \lesssim\operatorname{Gr}_T(C_0F)\int ds\left\|\int dy\,(t_0-s)^{-d/2}\exp\left(-c\frac{|x-y|^2}{t_0-s}\right)|G(s,y)|\right\|_{L_x^p} ≲ Gr T ( C 0 F ) ∫ d s ∫ d y ( t 0 − s ) − d /2 exp ( − c t 0 − s ∣ x − y ∣ 2 ) ∣ G ( s , y ) ∣ L x p ≲ Gr T ( C 0 F ) ∫ d s ∥ G ( s ) ∥ L x p ≲ Gr T ( C 0 F ) ∥ G ∥ L t 1 L x p ( [ 0 , T ] ) , \lesssim\operatorname{Gr}_T(C_0F)\int ds\,\|G(s)\|_{L_x^p}\lesssim\operatorname{Gr}_T(C_0F)\|G\|_{L_t^1L_x^p([0,T])}, ≲ Gr T ( C 0 F ) ∫ d s ∥ G ( s ) ∥ L x p ≲ Gr T ( C 0 F ) ∥ G ∥ L t 1 L x p ([ 0 , T ]) , which shows the first estimate. The second estimate may be proven similarly, where we apply (196) with k = 1 k=1 k = 1 (206) to first move the derivative off of G G G q B q_B q B Proposition 4.27 in place of the estimate (196) . □
Lemma 4.62. Let C 0 C_0 C 0 Proposition 4.18 . Let G : [ 0 , T ] × T d → W d G:[0,T]\times\mathbb{T}^d\to W^d G : [ 0 , T ] × T d → W d k ≥ 0 k\geq0 k ≥ 0
∥ t k / 2 D ( k ) Duh B F ( G ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k p e r t ) k ) N k ( G ) , \|t^{k/2}D^{(k)}\operatorname{Duh}_B^F(G)\|_{L_{t,x}^\infty([0,T])}\lesssim\operatorname{Gr}_T(C_0F)(1+(\|F\|_{T,k}^{\mathrm{pert}})^k)N_k(G), ∥ t k /2 D ( k ) Duh B F ( G ) ∥ L t , x ∞ ([ 0 , T ]) ≲ Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k pert ) k ) N k ( G ) , where
N k ( G ) = ∥ G ∥ L t 1 L x ∞ ( [ 0 , T ] ) + ∑ j = 0 k − 1 ( ∥ t ( j + 1 ) / 2 D ( j ) G ∥ L t 2 L x ∞ ( [ 0 , T ] ) + sup u ∈ [ 0 , T ] ∥ t ( j + 1 ) / 2 ( u − t ) − 1 / 2 D ( j ) G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . N_k(G)=\|G\|_{L_t^1L_x^\infty([0,T])}+\sum_{j=0}^{k-1}\left(\|t^{(j+1)/2}D^{(j)}G\|_{L_t^2L_x^\infty([0,T])}+\sup_{u\in[0,T]}\|t^{(j+1)/2}(u-t)^{-1/2}D^{(j)}G\|_{L_t^1L_x^\infty([0,u])}\right). N k ( G ) = ∥ G ∥ L t 1 L x ∞ ([ 0 , T ]) + j = 0 ∑ k − 1 ( ∥ t ( j + 1 ) /2 D ( j ) G ∥ L t 2 L x ∞ ([ 0 , T ]) + u ∈ [ 0 , T ] sup ∥ t ( j + 1 ) /2 ( u − t ) − 1/2 D ( j ) G ∥ L t 1 L x ∞ ([ 0 , u ]) ) . In particular, when k = 0 k=0 k = 0 N 0 ( G ) = ∥ G ∥ L t 1 L x ∞ ( [ 0 , T ] ) N_0(G)=\|G\|_{L_t^1L_x^\infty([0,T])} N 0 ( G ) = ∥ G ∥ L t 1 L x ∞ ([ 0 , T ]) H : [ 0 , T ] × T d → W H:[0,T]\times\mathbb{T}^d\to W H : [ 0 , T ] × T d → W k ≥ 0 k\geq0 k ≥ 0
∥ t k / 2 D ( k ) Duh B ( H ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ( 1 + ( ∥ F ∥ T , k − 1 p e r t ) k ) N k ( H ) . \|t^{k/2}D^{(k)}\operatorname{Duh}_B(H)\|_{L_{t,x}^\infty([0,T])}\lesssim(1+(\|F\|_{T,k-1}^{\mathrm{pert}})^k)N_k(H). ∥ t k /2 D ( k ) Duh B ( H ) ∥ L t , x ∞ ([ 0 , T ]) ≲ ( 1 + ( ∥ F ∥ T , k − 1 pert ) k ) N k ( H ) . Proof. We focus on the estimate of Duh B F ( G ) \operatorname{Duh}_B^F(G) Duh B F ( G ) Duh B ( H ) \operatorname{Duh}_B(H) Duh B ( H ) Corollary 4.24 by Proposition 4.19 in the base case of the ensuing inductive argument). Let ϕ = Duh B F ( G ) \phi=\operatorname{Duh}_B^F(G) ϕ = Duh B F ( G ) ( t 0 , x 0 ) ∈ [ 0 , T ] × T d (t_0,x_0)\in[0,T]\times\mathbb{T}^d ( t 0 , x 0 ) ∈ [ 0 , T ] × T d K ( t , x ) = p ( t , x ; t 0 , x 0 ) K(t,x)=p(t,x;t_0,x_0) K ( t , x ) = p ( t , x ; t 0 , x 0 ) [ 0 , t 0 ] × T d [0,t_0]\times\mathbb{T}^d [ 0 , t 0 ] × T d Lemma 4.11 , it suffices to show that for all k ≥ 0 k\geq0 k ≥ 0
∥ t k / 2 K 1 / 2 D ( k ) ϕ ∥ L t ∞ L x 2 + ∥ t k / 2 K 1 / 2 D ( k + 1 ) ϕ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k p e r t ) k ) N k ( G ) . (248) \|t^{k/2}K^{1/2}D^{(k)}\phi\|_{L_t^\infty L_x^2}+\|t^{k/2}K^{1/2}D^{(k+1)}\phi\|_{L_{t,x}^2}\lesssim\operatorname{Gr}_T(C_0F)(1+(\|F\|_{T,k}^{\mathrm{pert}})^k)N_k(G). \tag*{(248)} ∥ t k /2 K 1/2 D ( k ) ϕ ∥ L t ∞ L x 2 + ∥ t k /2 K 1/2 D ( k + 1 ) ϕ ∥ L t , x 2 ≲ Gr T ( C 0 F ) ( 1 + ( ∥ F ∥ T , k pert ) k ) N k ( G ) . (248) We proceed by induction. For the base case k = 0 k=0 k = 0 Corollary 4.24 , we have that
∥ K 1 / 2 ϕ ∥ L t ∞ L x 2 + ∥ K 1 / 2 D ϕ ∥ L t , x 2 ≲ G T ( C 0 F ) ∥ K 1 / 2 G ∥ L t 1 L x 2 ≲ G T ( C 0 F ) ∥ G ∥ L t 1 L x ∞ . (249) \left\|K^{1/2}\phi\right\|_{L^\infty_tL^2_x}+\left\|K^{1/2}D\phi\right\|_{L^2_{t,x}}\lesssim G_T(C_0F)\left\|K^{1/2}G\right\|_{L^1_tL^2_x}\lesssim G_T(C_0F)\left\|G\right\|_{L^1_tL^\infty_x}. \tag*{(249)} K 1/2 ϕ L t ∞ L x 2 + K 1/2 D ϕ L t , x 2 ≲ G T ( C 0 F ) K 1/2 G L t 1 L x 2 ≲ G T ( C 0 F ) ∥ G ∥ L t 1 L x ∞ . (249) where in the last inequality, we used Hölder and the fact that ∥ K 1 / 2 ∥ L t ∞ L x 2 = 1 \left\|K^{1/2}\right\|_{L^\infty_tL^2_x}=1 K 1/2 L t ∞ L x 2 = 1
Next, suppose that for some k 0 ≥ 0 k_0\ge0 k 0 ≥ 0 (248) has been proven for all 0 ≤ k ≤ k 0 0\le k\le k_0 0 ≤ k ≤ k 0 k 0 + 1 k_0+1 k 0 + 1 Lemma 3.29 and Remark 3.30 that D ( k 0 + 1 ) ϕ D^{(k_0+1)}\phi D ( k 0 + 1 ) ϕ
( ∂ t − Δ B ) D ( k 0 + 1 ) ϕ = ∑ j 1 + j 2 = k 0 + 1 j 1 , j 2 ≥ 0 O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) + D ( k 0 + 1 ) G = O ( F D ( k 0 + 1 ) ϕ ) + R + D ( k 0 + 1 ) G , (\partial_t-\Delta_B)D^{(k_0+1)}\phi=\sum_{\substack{j_1+j_2=k_0+1\\j_1,j_2\ge0}}\mathcal{O}\bigl((D^{(j_1)}F)(D^{(j_2)}\phi)\bigr)+D^{(k_0+1)}G
=\mathcal{O}\bigl(FD^{(k_0+1)}\phi\bigr)+R+D^{(k_0+1)}G, ( ∂ t − Δ B ) D ( k 0 + 1 ) ϕ = j 1 + j 2 = k 0 + 1 j 1 , j 2 ≥ 0 ∑ O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) + D ( k 0 + 1 ) G = O ( F D ( k 0 + 1 ) ϕ ) + R + D ( k 0 + 1 ) G , where
R : = ∑ j 1 + j 2 = k 0 + 1 j 1 ≥ 1 , j 2 ≥ 0 O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) . R:=\sum_{\substack{j_1+j_2=k_0+1\\j_1\ge1,j_2\ge0}}\mathcal{O}\bigl((D^{(j_1)}F)(D^{(j_2)}\phi)\bigr). R := j 1 + j 2 = k 0 + 1 j 1 ≥ 1 , j 2 ≥ 0 ∑ O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) . By the time-weighted estimate (186) with α = k 0 + 1 \alpha=k_0+1 α = k 0 + 1 G , H G,H G , H R R R D ( k 0 ) G D^{(k_0)}G D ( k 0 ) G
∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 1 ) ϕ ∥ L t ∞ L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 + 2 ) ϕ ∥ L t , x 2 ≲ ( 1 + ∥ F ∥ t 0 , 0 p e r t ) ∥ t k 0 / 2 K 1 / 2 D ( k 0 + 1 ) ϕ ∥ L t , x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 R ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 ) G ∥ L t , x 2 + ∥ t ( k 0 + 1 ) / 2 ( t 0 − t ) − 1 / 2 K 1 / 2 D ( k 0 ) G ∥ L t 1 L x 2 + ∥ t ( k 0 + 1 ) / 2 ∣ Δ K ∣ 1 / 2 D ( k 0 ) G ∥ L t 1 L x 2 . \begin{aligned}
&\left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+1)}\phi\right\|_{L^\infty_tL^2_x}
+\left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0+2)}\phi\right\|_{L^2_{t,x}} \\
&\lesssim\left(1+\left\|F\right\|^{\mathrm{pert}}_{t_0,0}\right)\left\|t^{k_0/2}K^{1/2}D^{(k_0+1)}\phi\right\|_{L^2_{t,x}} \\
&\quad+\left\|t^{(k_0+1)/2}K^{1/2}R\right\|_{L^1_tL^2_x}
+\left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0)}G\right\|_{L^2_{t,x}} \\
&\quad+\left\|t^{(k_0+1)/2}(t_0-t)^{-1/2}K^{1/2}D^{(k_0)}G\right\|_{L^1_tL^2_x}
+\left\|t^{(k_0+1)/2}|\Delta K|^{1/2}D^{(k_0)}G\right\|_{L^1_tL^2_x}.
\end{aligned} t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 1 ) ϕ L t ∞ L x 2 + t ( k 0 + 1 ) /2 K 1/2 D ( k 0 + 2 ) ϕ L t , x 2 ≲ ( 1 + ∥ F ∥ t 0 , 0 pert ) t k 0 /2 K 1/2 D ( k 0 + 1 ) ϕ L t , x 2 + t ( k 0 + 1 ) /2 K 1/2 R L t 1 L x 2 + t ( k 0 + 1 ) /2 K 1/2 D ( k 0 ) G L t , x 2 + t ( k 0 + 1 ) /2 ( t 0 − t ) − 1/2 K 1/2 D ( k 0 ) G L t 1 L x 2 + t ( k 0 + 1 ) /2 ∣Δ K ∣ 1/2 D ( k 0 ) G L t 1 L x 2 . By our inductive assumption, the first term on the right-hand side is acceptable. To estimate the last three terms, we apply Hölder to obtain
∥ t ( k 0 + 1 ) / 2 K 1 / 2 D ( k 0 ) G ∥ L t , x 2 ≤ ∥ K 1 / 2 ∥ L t ∞ L x 2 ∥ t ( k 0 + 1 ) / 2 D ( k 0 ) G ∥ L t 2 L x ∞ ≤ ∥ t ( k 0 + 1 ) / 2 D ( k 0 ) G ∥ L t 2 L x ∞ , \left\|t^{(k_0+1)/2}K^{1/2}D^{(k_0)}G\right\|_{L^2_{t,x}}
\le\left\|K^{1/2}\right\|_{L^\infty_tL^2_x}\left\|t^{(k_0+1)/2}D^{(k_0)}G\right\|_{L^2_tL^\infty_x}
\le\left\|t^{(k_0+1)/2}D^{(k_0)}G\right\|_{L^2_tL^\infty_x}, t ( k 0 + 1 ) /2 K 1/2 D ( k 0 ) G L t , x 2 ≤ K 1/2 L t ∞ L x 2 t ( k 0 + 1 ) /2 D ( k 0 ) G L t 2 L x ∞ ≤ t ( k 0 + 1 ) /2 D ( k 0 ) G L t 2 L x ∞ , ∥ t ( k 0 + 1 ) / 2 K 1 / 2 ( t 0 − t ) − 1 / 2 D ( k 0 ) G ∥ L t 1 L x 2 ≤ ∥ t ( k 0 + 1 ) / 2 ( t 0 − t ) − 1 / 2 D ( k 0 ) G ∥ L t 1 L x ∞ , \left\|t^{(k_0+1)/2}K^{1/2}(t_0-t)^{-1/2}D^{(k_0)}G\right\|_{L^1_tL^2_x}
\le\left\|t^{(k_0+1)/2}(t_0-t)^{-1/2}D^{(k_0)}G\right\|_{L^1_tL^\infty_x}, t ( k 0 + 1 ) /2 K 1/2 ( t 0 − t ) − 1/2 D ( k 0 ) G L t 1 L x 2 ≤ t ( k 0 + 1 ) /2 ( t 0 − t ) − 1/2 D ( k 0 ) G L t 1 L x ∞ , ∥ t ( k 0 + 1 ) / 2 ∣ Δ K ∣ 1 / 2 D ( k 0 ) G ∥ L t 1 L x 2 ≤ ∥ ( t 0 − t ) 1 / 2 ∣ Δ K ∣ 1 / 2 ∥ L t ∞ L x 2 ∥ t ( k 0 + 1 ) / 2 ( t 0 − t ) − 1 / 2 D ( k 0 ) G ∥ L t 1 L x ∞ ≲ ∥ t ( k 0 + 1 ) / 2 ( t 0 − t ) − 1 / 2 D ( k 0 ) G ∥ L t 1 L x ∞ . \begin{aligned}
\left\|t^{(k_0+1)/2}|\Delta K|^{1/2}D^{(k_0)}G\right\|_{L^1_tL^2_x}
&\le\left\|(t_0-t)^{1/2}|\Delta K|^{1/2}\right\|_{L^\infty_tL^2_x}
\left\|t^{(k_0+1)/2}(t_0-t)^{-1/2}D^{(k_0)}G\right\|_{L^1_tL^\infty_x} \\
&\lesssim\left\|t^{(k_0+1)/2}(t_0-t)^{-1/2}D^{(k_0)}G\right\|_{L^1_tL^\infty_x}.
\end{aligned} t ( k 0 + 1 ) /2 ∣Δ K ∣ 1/2 D ( k 0 ) G L t 1 L x 2 ≤ ( t 0 − t ) 1/2 ∣Δ K ∣ 1/2 L t ∞ L x 2 t ( k 0 + 1 ) /2 ( t 0 − t ) − 1/2 D ( k 0 ) G L t 1 L x ∞ ≲ t ( k 0 + 1 ) /2 ( t 0 − t ) − 1/2 D ( k 0 ) G L t 1 L x ∞ . By the definition of N k ( G ) N_k(G) N k ( G ) R R R R R R O ( ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ) \mathcal{O}((D^{(j_1)}F)(D^{(j_2)}\phi)) O (( D ( j 1 ) F ) ( D ( j 2 ) ϕ )) j 1 + j 2 = k 0 + 1 j_1+j_2=k_0+1 j 1 + j 2 = k 0 + 1 j 1 ≥ 1 j_1\ge1 j 1 ≥ 1 j 2 ≥ 0 j_2\ge0 j 2 ≥ 0 j 2 ≤ k 0 j_2\le k_0 j 2 ≤ k 0
∥ t ( k 0 + 1 ) / 2 K 1 / 2 ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) ∥ L t 1 L x 2 ≤ ∥ t j 1 / 2 D ( j 1 ) F ∥ L t 1 L x ∞ ∥ t j 2 / 2 K 1 / 2 D ( j 2 ) ϕ ∥ L t ∞ L x 2 ≲ G T ( C 0 F ) ∥ F ∥ t 0 , k 0 + 1 p e r t ( 1 + ( ∥ F ∥ T , k 0 p e r t ) k 0 ) N k 0 ( G ) , \begin{aligned}
\left\|t^{(k_0+1)/2}K^{1/2}(D^{(j_1)}F)(D^{(j_2)}\phi)\right\|_{L^1_tL^2_x}
&\le\left\|t^{j_1/2}D^{(j_1)}F\right\|_{L^1_tL^\infty_x}
\left\|t^{j_2/2}K^{1/2}D^{(j_2)}\phi\right\|_{L^\infty_tL^2_x} \\
&\lesssim G_T(C_0F)\left\|F\right\|^{\mathrm{pert}}_{t_0,k_0+1}
\left(1+\left(\left\|F\right\|^{\mathrm{pert}}_{T,k_0}\right)^{k_0}\right)N_{k_0}(G),
\end{aligned} t ( k 0 + 1 ) /2 K 1/2 ( D ( j 1 ) F ) ( D ( j 2 ) ϕ ) L t 1 L x 2 ≤ t j 1 /2 D ( j 1 ) F L t 1 L x ∞ t j 2 /2 K 1/2 D ( j 2 ) ϕ L t ∞ L x 2 ≲ G T ( C 0 F ) ∥ F ∥ t 0 , k 0 + 1 pert ( 1 + ( ∥ F ∥ T , k 0 pert ) k 0 ) N k 0 ( G ) , which is acceptable.
Analysis of deterministic flows
In this section, we collect many of the deterministic estimates that will be needed later on. In Sections 5.1 and 5.2 we estimate the various deterministic pieces that will appear in our decomposition of the solution ( A , Φ ) (A,\Phi) ( A , Φ ) (11) (recall that various aspects of the decomposition were already informally discussed in Section 1). Many of these estimates involve analysing (small) rough perturbations of (large) smooth backgrounds. They will be needed in both the local theory (see e.g. Section 8.2) as well as in the global theory. For the latter, in Section 5.3, we consider a single step of our global iteration scheme and treat all terms which may be estimated deterministically (by applying the estimates from Sections 5.1 and 5.2).
We begin with the following lemma, which estimates solutions to the Yang–Mills gradient flow on an interval [ 0 , T ] [0,T] [ 0 , T ] T T T (253) is quite crude, and will only be needed in Section 13.6 to establish a soft statement, thus the reader may choose to skip its proof on a first reading.
Lemma 5.1. Let 0 < T ≤ 1 0<T\leq1 0 < T ≤ 1 B ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) B\in C_t^0C_x^{-\kappa}([0,T]\times\mathbb{T}^2)\cap C_{t,x}^{\infty}((0,T]\times\mathbb{T}^2) B ∈ C t 0 C x − κ ([ 0 , T ] × T 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 )
M 0 ≔ ∥ t 1 + κ 2 F B ∥ L t , x ∞ ( [ 0 , T ] ) , M 1 ≔ ∥ t 1 + κ 2 D B F B ∥ L t , x ∞ ( [ 0 , T ] ) . M_0\coloneqq\lVert t^{\frac{1+\kappa}{2} }F_B\rVert_{L_{t,x}^{\infty}([0,T])},\qquad M_1\coloneqq\lVert t^{\frac{1+\kappa}{2}}D_BF_B\rVert_{L_{t,x}^{\infty}([0,T])}. M 0 : = ∥ t 2 1 + κ F B ∥ L t , x ∞ ([ 0 , T ]) , M 1 : = ∥ t 2 1 + κ D B F B ∥ L t , x ∞ ([ 0 , T ]) . (These quantities may of course be infinite, in which case some of the following estimates are trivially true.) Then we have the bounds
∥ t κ 2 B ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ∥ B ( T ) ∥ L x ∞ + M 1 , (250) \lVert t^{\frac{\kappa}{2}}B\rVert_{L_{t,x}^{\infty}([0,T])}\lesssim\lVert B(T)\rVert_{L_x^\infty}+M_1, \tag*{(250)} ∥ t 2 κ B ∥ L t , x ∞ ([ 0 , T ]) ≲ ∥ B ( T ) ∥ L x ∞ + M 1 , (250) ∥ B ∥ C t 0 C x − κ ( [ 0 , T ] ) ≲ ( M 0 + 1 ) ( ∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) , (251) \lVert B\rVert_{C_t^0C_x^{-\kappa}([0,T])}\lesssim(M_0+1)(\lVert B(T)\rVert_{L_x^\infty}+M_1+1), \tag*{(251)} ∥ B ∥ C t 0 C x − κ ([ 0 , T ]) ≲ ( M 0 + 1 ) (∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) , (251) ∥ t κ ∂ ℓ B ℓ ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ∥ ∂ ℓ B ℓ ( T ) ∥ L x ∞ + ( ∥ B ( T ) ∥ L x ∞ + M 1 ) M 1 , (252) \lVert t^\kappa\partial^\ell B_\ell\rVert_{L_{t,x}^{\infty}([0,T])}\lesssim\lVert\partial^\ell B_\ell(T)\rVert_{L_x^\infty}+(\lVert B(T)\rVert_{L_x^\infty}+M_1)M_1, \tag*{(252)} ∥ t κ ∂ ℓ B ℓ ∥ L t , x ∞ ([ 0 , T ]) ≲ ∥ ∂ ℓ B ℓ ( T ) ∥ L x ∞ + (∥ B ( T ) ∥ L x ∞ + M 1 ) M 1 , (252) ∥ t 1 + κ 2 ∂ ( ∂ ℓ B ℓ ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ∥ ∂ ( ∂ ℓ B ℓ ) ( T ) ∥ L x ∞ + ∥ ∂ B ( T ) ∥ L x ∞ ( M 1 + 1 ) + ( ∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) 2 ( M 0 + M 1 + 1 ) 5 . (253) \begin{aligned}
\lVert t^{\frac{1+\kappa}{2}}\partial(\partial^\ell B_\ell)\rVert_{L_{t,x}^{\infty}([0,T])}
&\lesssim\lVert\partial(\partial^\ell B_\ell)(T)\rVert_{L_x^\infty}
+\lVert\partial B(T)\rVert_{L_x^\infty}(M_1+1)\\
&\quad+(\lVert B(T)\rVert_{L_x^\infty}+M_1+1)^2(M_0+M_1+1)^5. \tag*{(253)}
\end{aligned} ∥ t 2 1 + κ ∂ ( ∂ ℓ B ℓ ) ∥ L t , x ∞ ([ 0 , T ]) ≲ ∥ ∂ ( ∂ ℓ B ℓ ) ( T ) ∥ L x ∞ + ∥ ∂ B ( T ) ∥ L x ∞ ( M 1 + 1 ) + (∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) 2 ( M 0 + M 1 + 1 ) 5 . (253) Proof. Since B B B (8) , we have that for t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] i ∈ [ 2 ] i\in[2] i ∈ [ 2 ]
B i ( t 0 ) = B i ( T ) − ∫ t 0 T d t D B ℓ ( F B ) ℓ i ( t ) . (254) B_i(t_0)=B_i(T)-\int_{t_0}^{T}\mathrm{d}t\,D_B^\ell(F_B)_{\ell i}(t). \tag*{(254)} B i ( t 0 ) = B i ( T ) − ∫ t 0 T d t D B ℓ ( F B ) ℓ i ( t ) . (254) The estimate (250) now directly follows from the definition of M 1 M_1 M 1
For the estimate (251) , let ϕ x λ \phi_x^\lambda ϕ x λ g \mathfrak{g} g L x 1 L_x^1 L x 1 x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2 λ ≪ 1 \lambda\ll1 λ ≪ 1 B ∈ C t 0 C x − κ ( [ 0 , T ] ) B\in C_t^0C_x^{-\kappa}([0,T]) B ∈ C t 0 C x − κ ([ 0 , T ]) x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2 λ ≪ 1 \lambda\ll1 λ ≪ 1
max i ∈ [ 2 ] sup t 0 ∈ ( 0 , T ] ∣ ⟨ B i ( t 0 ) , ϕ x λ ⟩ ∣ ≲ λ − κ ( M 0 + 1 ) ( ∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) . (255) \max_{i\in[2]}\sup_{t_0\in(0,T]}\bigl|\langle B_i(t_0),\phi_x^\lambda\rangle\bigr|\lesssim\lambda^{-\kappa}(M_0+1)(\lVert B(T)\rVert_{L_x^\infty}+M_1+1). \tag*{(255)} i ∈ [ 2 ] max t 0 ∈ ( 0 , T ] sup ⟨ B i ( t 0 ) , ϕ x λ ⟩ ≲ λ − κ ( M 0 + 1 ) (∥ B ( T ) ∥ L x ∞ + M 1 + 1 ) . (255) Towards this end, if t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] t 0 ≥ λ 2 t_0\geq\lambda^2 t 0 ≥ λ 2 (250) we have that
∣ ⟨ B i ( t 0 ) , ϕ x λ ⟩ ∣ ≲ t 0 − κ 2 ( ∥ B ( T ) ∥ L x ∞ + M 1 ) ≲ λ − κ ( ∥ B ( T ) ∥ L x ∞ + M 1 ) , \bigl|\langle B_i(t_0),\phi_x^\lambda\rangle\bigr|\lesssim t_0^{-\frac{\kappa}{2}}(\lVert B(T)\rVert_{L_x^\infty}+M_1)\lesssim\lambda^{-\kappa}(\lVert B(T)\rVert_{L_x^\infty}+M_1), ⟨ B i ( t 0 ) , ϕ x λ ⟩ ≲ t 0 − 2 κ (∥ B ( T ) ∥ L x ∞ + M 1 ) ≲ λ − κ (∥ B ( T ) ∥ L x ∞ + M 1 ) , which is acceptable. Therefore, we may assume that t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] t 0 ≤ λ 2 t_0\leq\lambda^2 t 0 ≤ λ 2 ∂ t B i = D B ℓ ( F B ) ℓ i \partial_tB_i=D_B^\ell(F_B)_{\ell i} ∂ t B i = D B ℓ ( F B ) ℓ i t 0 t_0 t 0 λ 2 ∧ T \lambda^2\wedge T λ 2 ∧ T
⟨ B i ( t 0 ) , ϕ x λ ⟩ = ⟨ B i ( λ 2 ∧ T ) , ϕ x λ ⟩ − ∫ t 0 λ 2 ∧ T d t ⟨ D B ℓ ( F B ) ℓ i ( t ) , ϕ x λ ⟩ . \langle B_i(t_0),\phi_x^\lambda\rangle=\langle B_i(\lambda^2\wedge T),\phi_x^\lambda\rangle-\int_{t_0}^{\lambda^2\wedge T}\mathrm{d}t\langle D_B^\ell(F_B)_{\ell i}(t),\phi_x^\lambda\rangle. ⟨ B i ( t 0 ) , ϕ x λ ⟩ = ⟨ B i ( λ 2 ∧ T ) , ϕ x λ ⟩ − ∫ t 0 λ 2 ∧ T d t ⟨ D B ℓ ( F B ) ℓ i ( t ) , ϕ x λ ⟩ . If λ 2 ≥ T \lambda^{2} \ge T λ 2 ≥ T ∥ B ( T ) ∥ L x ∞ \|B(T)\|_{L^\infty_x} ∥ B ( T ) ∥ L x ∞ λ 2 < T \lambda^{2}<T λ 2 < T D B ℓ = ∂ ℓ + ad ( B ℓ ) D_B^\ell=\partial^\ell+\operatorname{ad}(B^\ell) D B ℓ = ∂ ℓ + ad ( B ℓ )
∫ t 0 λ 2 ∧ T d t ⟨ D B ℓ ( F B ) ℓ i ( t ) , ϕ x λ ⟩ = − ∫ t 0 λ 2 ∧ T d t ⟨ ( F B ) ℓ i ( t ) , ∂ ℓ ϕ x λ ⟩ + ∫ t 0 λ 2 ∧ T d t ⟨ [ B ℓ , ( F B ) ℓ i ] , ϕ x λ ⟩ . \int_{t_0}^{\lambda^{2}\wedge T}dt\,\langle D_B^\ell(F_B)_{\ell i}(t),\phi_x^\lambda\rangle=-\int_{t_0}^{\lambda^{2}\wedge T}dt\,\langle(F_B)_{\ell i}(t),\partial^\ell\phi_x^\lambda\rangle+\int_{t_0}^{\lambda^{2}\wedge T}dt\,\langle[B^\ell,(F_B)_{\ell i}],\phi_x^\lambda\rangle. ∫ t 0 λ 2 ∧ T d t ⟨ D B ℓ ( F B ) ℓ i ( t ) , ϕ x λ ⟩ = − ∫ t 0 λ 2 ∧ T d t ⟨( F B ) ℓ i ( t ) , ∂ ℓ ϕ x λ ⟩ + ∫ t 0 λ 2 ∧ T d t ⟨[ B ℓ , ( F B ) ℓ i ] , ϕ x λ ⟩ . Combining the definition of M 0 M_0 M 0 ∥ ∂ ℓ ϕ x λ ∥ L x 1 ≲ λ − 1 \|\partial^\ell\phi_x^\lambda\|_{L^1_x}\lesssim\lambda^{-1} ∥ ∂ ℓ ϕ x λ ∥ L x 1 ≲ λ − 1 (250) , we may estimate both terms on the right-hand side above by acceptable contributions. Combining, we thus obtain (5.6}.
For the estimate (5.3}, recall from Lemma 3.27 that ∂ t B \partial_tB ∂ t B D B i ∂ t B i = 0 D_B^i\partial_tB_i=0 D B i ∂ t B i = 0 ∂ t B i = D B ℓ ( F B ) ℓ i \partial_tB_i=D_B^\ell(F_B)_{\ell i} ∂ t B i = D B ℓ ( F B ) ℓ i D B i = ∂ i + [ B i , ⋅ ] D_B^i=\partial^i+[B^i,\mathord\cdot] D B i = ∂ i + [ B i , ⋅ ]
∂ t ∂ i B i = − [ B i , D B ℓ ( F B ) ℓ i ] . (256) \partial_t\partial^iB_i=-[B^i,D_B^\ell(F_B)_{\ell i}]. \tag*{(256)} ∂ t ∂ i B i = − [ B i , D B ℓ ( F B ) ℓ i ] . (256) The estimate (5.3} now follows by integrating this identity back from the end time T T T B B B M 1 M_1 M 1 ∥ ∂ ℓ B ℓ ( T ) ∥ L x ∞ \|\partial^\ell B_\ell(T)\|_{L^\infty_x} ∥ ∂ ℓ B ℓ ( T ) ∥ L x ∞
We move on to the final estimate (5.4}. We will be quite wasteful in the following bounds (as this will be sufficient for our purposes). To start, we hit both sides of (5.7} by a derivative to obtain
∂ t ( ∂ ( ∂ i B i ) ) = − ∂ [ B i , D B ℓ ( F B ) ℓ i ] = − [ B i , D B D B ℓ ( F B ) ℓ i ] + [ B i , [ B , D B ℓ ( F B ) ℓ i ] ] − [ ∂ B i , D B ℓ ( F B ) ℓ i ] . (257) \begin{aligned}
\partial_t(\partial(\partial^iB_i))&=-\partial[B^i,D_B^\ell(F_B)_{\ell i}]\\
&=-[B^i,D_BD_B^\ell(F_B)_{\ell i}]+[B^i,[B,D_B^\ell(F_B)_{\ell i}]]-[\partial B^i,D_B^\ell(F_B)_{\ell i}]. \tag*{(257)}
\end{aligned} ∂ t ( ∂ ( ∂ i B i )) = − ∂ [ B i , D B ℓ ( F B ) ℓ i ] = − [ B i , D B D B ℓ ( F B ) ℓ i ] + [ B i , [ B , D B ℓ ( F B ) ℓ i ]] − [ ∂ B i , D B ℓ ( F B ) ℓ i ] . (257) We thus see that we need additional estimates on D B ( 2 ) F B D_B^{(2)}F_B D B ( 2 ) F B ∂ B \partial B ∂ B M 0 M_0 M 0 M 1 M_1 M 1 ∥ F B ∥ T , 1 p e r t \|F_B\|_{T,1}^{\mathrm{pert}} ∥ F B ∥ T , 1 pert ∥ F B ∥ T , 1 p e r t ≲ M 0 + M 1 \|F_B\|_{T,1}^{\mathrm{pert}}\lesssim M_0+M_1 ∥ F B ∥ T , 1 pert ≲ M 0 + M 1 F B F_B F B Lemma 3.28 ), we obtain by our heat kernel estimate (195) that
∥ t 3 + κ 2 D B ( 2 ) F B ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ( 1 + M 0 + M 1 ) 2 M 0 . (258) \|t^{\frac{3+\kappa}{2}}D_B^{(2)}F_B\|_{L^\infty_{t,x}([0,T])}\lesssim(1+M_0+M_1)^2M_0. \tag*{(258)} ∥ t 2 3 + κ D B ( 2 ) F B ∥ L t , x ∞ ([ 0 , T ]) ≲ ( 1 + M 0 + M 1 ) 2 M 0 . (258) For an estimate on ∂ B \partial B ∂ B
∂ B i ( t 0 ) = ∂ B i ( T ) − ∫ t 0 T d t ( D B D B ℓ ( F B ) ℓ i ( t ) − [ B , D B ℓ ( F B ) ℓ i ] ) , \partial B_i(t_0)=\partial B_i(T)-\int_{t_0}^{T}dt\left(D_BD_B^\ell(F_B)_{\ell i}(t)-[B,D_B^\ell(F_B)_{\ell i}]\right), ∂ B i ( t 0 ) = ∂ B i ( T ) − ∫ t 0 T d t ( D B D B ℓ ( F B ) ℓ i ( t ) − [ B , D B ℓ ( F B ) ℓ i ] ) , and thus (applying (258) to estimate the first term in the integral, and (250) to estimate the second)
t 0 1 + κ 2 ∥ ∂ B ( t 0 ) ∥ L x ∞ ≲ ∥ ∂ B ( T ) ∥ L x ∞ + ( 1 + M 0 + M 1 ) 2 M 0 + ( ∥ B ( T ) ∥ L x ∞ + M 1 ) M 1 . t_0^{\frac{1+\kappa}{2}}\|\partial B(t_0)\|_{L^\infty_x}\lesssim\|\partial B(T)\|_{L^\infty_x}+(1+M_0+M_1)^2M_0+(\|B(T)\|_{L^\infty_x}+M_1)M_1. t 0 2 1 + κ ∥ ∂ B ( t 0 ) ∥ L x ∞ ≲ ∥ ∂ B ( T ) ∥ L x ∞ + ( 1 + M 0 + M 1 ) 2 M 0 + ( ∥ B ( T ) ∥ L x ∞ + M 1 ) M 1 . Combining our estimates with the evolution (257) for ∂ ( ∂ ℓ B ℓ ) \partial(\partial^\ell B_\ell) ∂ ( ∂ ℓ B ℓ ) (253) . □
Next, we define gauge invariant Hölder norms which will appear in the various well-posedness results of the coming sections. First, we set some notation which will be used in the definition, as well as in many places later on.
Notation 5.2. Let A ∈ C x − κ A\in\mathcal{C}^{-\kappa}_x A ∈ C x − κ T > 0 T>0 T > 0 B ∈ C t , x ∞ ( ( 0 , T ] × T 2 ) B\in\mathcal{C}^{\infty}_{t,x}((0,T]\times\mathbb{T}^{2}) B ∈ C t , x ∞ (( 0 , T ] × T 2 ) B ( t ) → A B(t)\to A B ( t ) → A C x − κ \mathcal{C}^{-\kappa}_x C x − κ t ↓ 0 t\downarrow0 t ↓ 0 u 0 u_0 u 0 u u u
( ∂ t − Δ B − 2 F B ) u = 0 , u ( 0 ) = u 0 . (\partial_t-\Delta_B-2F_B)u=0,\qquad u(0)=u_0. ( ∂ t − Δ B − 2 F B ) u = 0 , u ( 0 ) = u 0 . Then for s ∈ ( 0 , T ] s \in(0,T] s ∈ ( 0 , T ] e s Δ A F u 0 : = u ( s ) e^{s\Delta_A^F}u_0 \mathrel{\mathop:}=u(s) e s Δ A F u 0 : = u ( s )
Similarly, given a W-valued distribution v 0 v_0 v 0
( ∂ t − Δ B ) v = 0 , v ( 0 ) = v 0 . (\partial_t-\Delta_B)v=0,\qquad v(0)=v_0. ( ∂ t − Δ B ) v = 0 , v ( 0 ) = v 0 . Then for s ∈ ( 0 , T ] s \in(0,T] s ∈ ( 0 , T ] e s Δ A v 0 : = v ( s ) e^{s\Delta_A}v_0 \mathrel{\mathop:}=v(s) e s Δ A v 0 : = v ( s )
Remark 5.3. If A A A B B B [ 0 , ∞ ) × T 2 [0,\infty)\times\mathbb{T}^2 [ 0 , ∞ ) × T 2 α ∈ R \alpha\in\mathbb{R} α ∈ R u 0 , v 0 ∈ C x α u_0,v_0\in\mathcal{C}^{\alpha}_x u 0 , v 0 ∈ C x α s ≥ 0 s\geq0 s ≥ 0 e s Δ A F u 0 , e s Δ A v 0 e^{s\Delta_A^F}u_0,e^{s\Delta_A}v_0 e s Δ A F u 0 , e s Δ A v 0 A ∈ C x − κ A\in\mathcal{C}^{-\kappa}_x A ∈ C x − κ A A A B B B M 0 , M 1 M_0,M_1 M 0 , M 1 B B B L t , x ∞ L^\infty_{t,x} L t , x ∞ (250) . Using this, one may show using standard parabolic theory that if u 0 , v 0 ∈ C x − κ u_0,v_0\in\mathcal{C}^{-\kappa}_x u 0 , v 0 ∈ C x − κ e s Δ A F u 0 , e s Δ A v 0 e^{s\Delta_A^F}u_0,e^{s\Delta_A}v_0 e s Δ A F u 0 , e s Δ A v 0
Definition 5.4 (Gauge invariant Hölder norms). Let A , B A,B A , B α < 1 \alpha<1 α < 1 T > 0 T>0 T > 0 u 0 u_0 u 0 e s Δ A F u 0 e^{s\Delta_A^F}u_0 e s Δ A F u 0 s ∈ ( 0 , T ] s\in(0,T] s ∈ ( 0 , T ]
∥ u 0 ∥ C A , F , T α : = max 0 ≤ m ≤ 1 sup s ∈ ( 0 , T ] ( s − α + m 2 ∧ 1 ) ∥ D B ( s ) ( m ) e s Δ A F u 0 ∥ L x ∞ . \|u_0\|_{\mathcal{C}^{\alpha}_{A,F,T}}\mathrel{\mathop:}=\max_{0\leq m\leq1}\sup_{s\in(0,T]}\left(s^{\frac{-\alpha+m}{2}}\wedge1\right)\left\|D_{B(s)}^{(m)}e^{s\Delta_A^F}u_0\right\|_{L^\infty_x}. ∥ u 0 ∥ C A , F , T α : = 0 ≤ m ≤ 1 max s ∈ ( 0 , T ] sup ( s 2 − α + m ∧ 1 ) D B ( s ) ( m ) e s Δ A F u 0 L x ∞ . Similarly, given a W-valued distribution v 0 v_0 v 0 e s Δ A v 0 e^{s\Delta_A}v_0 e s Δ A v 0 s ∈ ( 0 , T ] s\in(0,T] s ∈ ( 0 , T ]
∥ v 0 ∥ C A , T α : = max 0 ≤ m ≤ 1 sup s ∈ ( 0 , T ] ( s − α + m 2 ∧ 1 ) ∥ D B ( s ) ( m ) e s Δ A v 0 ∥ L x ∞ . \|v_0\|_{\mathcal{C}^{\alpha}_{A,T}}\mathrel{\mathop:}=\max_{0\leq m\leq1}\sup_{s\in(0,T]}\left(s^{\frac{-\alpha+m}{2}}\wedge1\right)\left\|D_{B(s)}^{(m)}e^{s\Delta_A}v_0\right\|_{L^\infty_x}. ∥ v 0 ∥ C A , T α : = 0 ≤ m ≤ 1 max s ∈ ( 0 , T ] sup ( s 2 − α + m ∧ 1 ) D B ( s ) ( m ) e s Δ A v 0 L x ∞ . More generally, for k ≥ 1 k\geq1 k ≥ 1
∥ u 0 ∥ C A , F , T , k α : = max 0 ≤ m ≤ k sup s ∈ ( 0 , T ] ( s − α + m 2 ∧ 1 ) ∥ D B ( s ) ( m ) e s Δ A F u 0 ∥ L x ∞ , \|u_0\|_{\mathcal{C}^{\alpha}_{A,F,T,k}}\mathrel{\mathop:}=\max_{0\leq m\leq k}\sup_{s\in(0,T]}\left(s^{\frac{-\alpha+m}{2}}\wedge1\right)\left\|D_{B(s)}^{(m)}e^{s\Delta_A^F}u_0\right\|_{L^\infty_x}, ∥ u 0 ∥ C A , F , T , k α : = 0 ≤ m ≤ k max s ∈ ( 0 , T ] sup ( s 2 − α + m ∧ 1 ) D B ( s ) ( m ) e s Δ A F u 0 L x ∞ , and similarly for ∥ v 0 ∥ C A , T , k α \|v_0\|_{\mathcal{C}^{\alpha}_{A,T,k}} ∥ v 0 ∥ C A , T , k α
Remark 5.5. We refer to these as gauge invariant Hölder norms because given a smooth gauge transformation g ∈ G g\in\mathcal{G} g ∈ G
∥ g ∙ u 0 ∥ C g ∙ A , F , T , k α = ∥ u 0 ∥ C A , F , T , k α , \|g\mathbin{\bullet}u_0\|_{\mathcal{C}^{\alpha}_{g\mathbin{\bullet}A,F,T,k}}=\|u_0\|_{\mathcal{C}^{\alpha}_{A,F,T,k}}, ∥ g ∙ u 0 ∥ C g ∙ A , F , T , k α = ∥ u 0 ∥ C A , F , T , k α , and similarly for ∥ ∙ ∥ C A , T , k α \|\mathbin{\bullet}\|_{\mathcal{C}^{\alpha}_{A,T,k}} ∥ ∙ ∥ C A , T , k α
Before proceeding, we set some notation that will simplify many of the ensuing estimates.
Notation 5.6. Let T > 0 T>0 T > 0 B B B ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 C 0 C_0 C 0 k ≥ 0 k\geq0 k ≥ 0
M T ( B , k ) : = Gr T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 2 p e r t ) k . (259) M_T(B,k)\mathrel{\mathop:}=\operatorname{Gr}_T(C_0F_B)(1+\|F_B\|_{T,2}^{\mathrm{pert}})^k. \tag*{(259)} M T ( B , k ) : = Gr T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 2 pert ) k . (259) YM gradient flow with rough initial data
In this section, we prove estimates on solutions to the Yang–Mills gradient flow with rough initial data. We will later apply these estimates in a setting where the initial data is made of two pieces: a large smooth piece and a smaller rough piece. Then in order to obtain estimates, it is natural to treat the smooth piece non-perturbatively and the rough piece perturbatively. We accomplish this by expanding around the background arising from the smooth piece. This is the general method of proof behind the following proposition, which shows that solutions to the Yang–Mills gradient flow exist starting from any C x − κ \mathcal{C}^{-\kappa}_x C x − κ
Proposition 5.7 (Yang–Mills gradient flow with C x − κ \mathcal{C}^{-\kappa}_x C x − κ c 0 ≤ 1 c_0 \le1 c 0 ≤ 1 B 0 # = B 0 + + B 0 − B_0^\# = B_0^+ + B_0^- B 0 # = B 0 + + B 0 − B 0 + B_0^+ B 0 + B 0 − ∈ C x − κ B_0^- \in\mathcal{C}^{-\kappa}_x B 0 − ∈ C x − κ B + B^+ B + B + ( 0 ) = B 0 + B^+(0) = B_0^+ B + ( 0 ) = B 0 + 0 < T ≤ 1 0 < T \le1 0 < T ≤ 1 K ≥ 1 K \ge1 K ≥ 1
M T ( B + , 2 T 1 − κ 2 ) K ≤ c 0 , (260) M_T(B^+,2T^{\frac{1-\kappa}{2}})K \le c_0 , \tag*{(260)} M T ( B + , 2 T 2 1 − κ ) K ≤ c 0 , (260) ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ K . (261) \lVert B_0^- \rVert_{\mathcal{C}^{-\kappa}_{B_0^+,F,T,2}} \le K . \tag*{(261)} ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ K . (261) Then there exist a gauge transformation g 0 ∈ G 1 − κ g_0 \in\mathcal{G}_{1-\kappa} g 0 ∈ G 1 − κ B ∈ C t , x ∞ ( ( 0 , T ] × T 2 ) B \in C^\infty_{t,x}((0,T]\times\mathbb{T}^2) B ∈ C t , x ∞ (( 0 , T ] × T 2 ) (8) such that B ( t ) → g 0 ⋅ B 0 # B(t) \to g_0 \cdot B_0^\# B ( t ) → g 0 ⋅ B 0 # C x − κ \mathcal{C}^{-\kappa}_x C x − κ t ↓ 0 t \downarrow0 t ↓ 0 g ∈ C t 0 G 1 − κ ( [ 0 , T ] ) g \in C^0_t\mathcal{G}_{1-\kappa}([0,T]) g ∈ C t 0 G 1 − κ ([ 0 , T ]) g g g g ( 0 ) = g 0 g(0)=g_0 g ( 0 ) = g 0 g − 1 ∙ B = B + + B − g^{-1}\bullet B=B^++B^- g − 1 ∙ B = B + + B − B + B^+ B + B + ( 0 ) = B 0 + B^+(0)=B_0^+ B + ( 0 ) = B 0 + B − B^- B − B l i n − ( t ) = e t Δ B 0 + B 0 − B^-_{\mathrm{lin}}(t)=e^{t\Delta_{B_0^+}}B_0^- B lin − ( t ) = e t Δ B 0 + B 0 − t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
max 0 ≤ j ≤ 2 ∥ t j + κ 2 D B + ( j ) ( B − − B l i n − ) ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ M T ( B + , 2 ) K 2 t 0 1 − κ 2 . (262) \max_{0\le j\le2}\left\|t^{\frac{j+\kappa}{2}}D_{B^+}^{(j)}(B^- - B^-_{\mathrm{lin}})\right\|_{L^\infty_{t,x}([0,t_0])}
\lesssim M_T(B^+,2)K^2t_0^{\frac{1-\kappa}{2}} . \tag*{(262)} 0 ≤ j ≤ 2 max t 2 j + κ D B + ( j ) ( B − − B lin − ) L t , x ∞ ([ 0 , t 0 ]) ≲ M T ( B + , 2 ) K 2 t 0 2 1 − κ . (262) Moreover, we have the following L x ∞ L^\infty_x L x ∞ L x 2 L^2_x L x 2
∥ D B ( j ) F B ( t ) ∥ L x ∞ ≤ ∥ D B + ( j ) F B + ( t ) ∥ L x ∞ + O ( t − κ 2 ( t − 1 + j 2 + ∥ F B + ( t ) ∥ L x ∞ j ) K ) , j = 0 , 1 , (263) \left\|D_B^{(j)}F_B(t)\right\|_{L^\infty_x}
\le\left\|D_{B^+}^{(j)}F_{B^+}(t)\right\|_{L^\infty_x}
+O\left(t^{-\frac{\kappa}{2}}\left(t^{-\frac{1+j}{2}}+\left\|F_{B^+}(t)\right\|_{L^\infty_x}^{j}\right)K\right),\qquad j=0,1, \tag*{(263)} D B ( j ) F B ( t ) L x ∞ ≤ D B + ( j ) F B + ( t ) L x ∞ + O ( t − 2 κ ( t − 2 1 + j + ∥ F B + ( t ) ∥ L x ∞ j ) K ) , j = 0 , 1 , (263) ∥ D B ( j ) F B ( t ) ∥ L x 2 ≤ ∥ D B + ( j ) F B + ( t ) ∥ L x 2 + O ( t − κ 2 ( t − 1 + j 2 + ∥ F B + ( t ) ∥ L x 2 j ) K ) , j = 0 , 1 , (264) \left\|D_B^{(j)}F_B(t)\right\|_{L^2_x}
\le\left\|D_{B^+}^{(j)}F_{B^+}(t)\right\|_{L^2_x}
+O\left(t^{-\frac{\kappa}{2}}\left(t^{-\frac{1+j}{2}}+\left\|F_{B^+}(t)\right\|_{L^2_x}^{j}\right)K\right),\qquad j=0,1, \tag*{(264)} D B ( j ) F B ( t ) L x 2 ≤ D B + ( j ) F B + ( t ) L x 2 + O ( t − 2 κ ( t − 2 1 + j + ∥ F B + ( t ) ∥ L x 2 j ) K ) , j = 0 , 1 , (264) and for all t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ F B ( t ) ∥ L x 2 2 = ∥ F B + ( t ) ∥ L x 2 2 + ∥ curl B + ( t ) B l i n − ( t ) ∥ L x 2 2 + 2 ⟨ F B + ( t ) , ( curl B + B l i n − + [ B l i n − , B 2 , l i n − ] ) ( t ) ⟩ + M T ( B + , 3 ) O ( K 2 t − κ ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) + K 3 t − 1 2 − 3 κ 2 + ∥ F B + ( t ) ∥ L x 2 K 3 t 1 2 − 3 κ 2 ) . (265) \begin{aligned}
\left\|F_B(t)\right\|_{L^2_x}^2
={}&\left\|F_{B^+}(t)\right\|_{L^2_x}^2
+\left\|\operatorname{curl}_{B^+(t)}B^-_{\mathrm{lin}}(t)\right\|_{L^2_x}^2 \\
&+2\left\langle F_{B^+}(t),\left(\operatorname{curl}_{B^+}B^-_{\mathrm{lin}}+[B^-_{\mathrm{lin}},B^-_{2,\mathrm{lin}}]\right)(t)\right\rangle\\
&+M_T(B^+,3)O\left(K^2t^{-\kappa}\left\|D_{B^+}F_{B^+}\right\|_{L^2_{t,x}([0,t])}+K^3t^{-\frac12-\frac{3\kappa}{2}}+\left\|F_{B^+}(t)\right\|_{L^2_x}K^3t^{\frac12-\frac{3\kappa}{2}}\right). \tag*{(265)}
\end{aligned} ∥ F B ( t ) ∥ L x 2 2 = ∥ F B + ( t ) ∥ L x 2 2 + curl B + ( t ) B lin − ( t ) L x 2 2 + 2 ⟨ F B + ( t ) , ( curl B + B lin − + [ B lin − , B 2 , lin − ] ) ( t ) ⟩ + M T ( B + , 3 ) O ( K 2 t − κ ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) + K 3 t − 2 1 − 2 3 κ + ∥ F B + ( t ) ∥ L x 2 K 3 t 2 1 − 2 3 κ ) . (265) Remark 5.8. In principle, one could always take B 0 + = 0 B_0^+=0 B 0 + = 0 K = ∥ B 0 # ∥ C 0 , F , T , 2 − κ K=\left\|B_0^\#\right\|_{\mathcal{C}^{-\kappa}_{0,F,T,2}} K = B 0 # C 0 , F , T , 2 − κ K ≪ ∥ B 0 # ∥ C 0 , F , T , 2 − κ K\ll\left\|B_0^\#\right\|_{\mathcal{C}^{-\kappa}_{0,F,T,2}} K ≪ B 0 # C 0 , F , T , 2 − κ T T T
It is natural to wonder in what sense the solution to the Yang–Mills gradient flow B B B B B B
Lemma 5.9 (Low-regularity uniqueness of the Yang–Mills gradient flow). Let A 0 ∈ C x − κ A_0\in\mathcal{C}^{-\kappa}_x A 0 ∈ C x − κ T > 0 T>0 T > 0 B , B ^ ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) B,\widehat{B}\in C^0_t\mathcal{C}^{-\kappa}_x([0,T]\times\mathbb{T}^2)\cap C^\infty_{t,x}((0,T]\times\mathbb{T}^2) B , B ∈ C t 0 C x − κ ([ 0 , T ] × T 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 ) B ( t ) , B ^ ( t ) → A 0 B(t),\widehat{B}(t)\to A_0 B ( t ) , B ( t ) → A 0 C x − κ \mathcal{C}^{-\kappa}_x C x − κ t ↓ 0 t\downarrow0 t ↓ 0 B , B ^ B,\widehat{B} B , B
∥ t 1 + κ 2 D B F B ∥ L t , x ∞ ( ( 0 , T ] ) , ∥ t 1 + κ 2 D B ^ F B ^ ∥ L t , x ∞ ( ( 0 , T ] ) < ∞ . \left\|t^{1+\frac{\kappa}{2}}D_BF_B\right\|_{L^\infty_{t,x}((0,T])},\left\|t^{1+\frac{\kappa}{2}}D_{\widehat{B}}F_{\widehat{B}}\right\|_{L^\infty_{t,x}((0,T])}<\infty. t 1 + 2 κ D B F B L t , x ∞ (( 0 , T ]) , t 1 + 2 κ D B F B L t , x ∞ (( 0 , T ]) < ∞. Then B = B ^ B=\widehat{B} B = B
Remark 5.10. A more intrinsic formulation of Lemma 5.9 would be to prove a low-regularity well-posedness result for C x − κ \mathcal{C}^{-\kappa}_x C x − κ A 0 A_0 A 0 ∂ ℓ ( A 0 ) ℓ \partial^\ell(A_0)_\ell ∂ ℓ ( A 0 ) ℓ C x − 1 + ε \mathcal{C}^{-1+\varepsilon}_x C x − 1 + ε ε > 0 \varepsilon>0 ε > 0
We expect that the more intrinsic result may be obtained by analysing a system for the divergence and curl of B B B
Before proving Proposition 5.7, we state the following lemma which compares solutions to covariant heat equations with operator Δ B \Delta_B Δ B Δ B + \Delta_{B_+} Δ B + B B B B + B_+ B + § 6.3 . We defer the proof of the lemma until after the proof of the proposition.
Lemma 5.11. Let all notation and assumptions be as in Proposition 5.7. Upon possibly taking the constant c 0 c_0 c 0 (260) smaller, the following holds. Let g : [ 0 , T ] × T 2 → G g : [0,T] \times\mathbb{T}^2 \to G g : [ 0 , T ] × T 2 → G ϕ 1 , ϕ 2 : [ 0 , T ] × T 2 → W \phi_1,\phi_2 : [0,T] \times\mathbb{T}^2 \to W ϕ 1 , ϕ 2 : [ 0 , T ] × T 2 → W ( g − 1 ∙ ϕ 1 ) ( 0 ) = ϕ 2 ( 0 ) (g^{-1} \mathbin{\bullet} \phi_1)(0) = \phi_2(0) ( g − 1 ∙ ϕ 1 ) ( 0 ) = ϕ 2 ( 0 ) ϕ 1 , ϕ 2 \phi_1,\phi_2 ϕ 1 , ϕ 2
( ∂ t − Δ B ) ϕ 1 = 0 , ( ∂ t − Δ B + ) ϕ 2 = 0. (\partial_t-\Delta_B)\phi_1=0,\qquad(\partial_t-\Delta_{B_+})\phi_2=0. ( ∂ t − Δ B ) ϕ 1 = 0 , ( ∂ t − Δ B + ) ϕ 2 = 0. Then we have that for any t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] α ∈ [ 0 , 1 / 2 ] \alpha\in[0,1/2] α ∈ [ 0 , 1/2 ] 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞
∥ t α 2 ( g − 1 ∙ ϕ 1 − ϕ 2 ) ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 p e r t t 0 1 − κ 2 K ) ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . \lVert t^{\frac{\alpha}{2}}(g^{-1}\mathbin{\bullet}\phi_1-\phi_2)\rVert_{L_t^\infty L_x^p([0,t_0])}\lesssim(1+\lVert F_{B_+}\rVert_{t_0,1}^{\mathrm{pert}}t_0^{\frac{1-\kappa}{2}}K)\lVert t^{\frac{\alpha}{2}}\phi_2\rVert_{L_t^\infty L_x^p([0,t_0])}. ∥ t 2 α ( g − 1 ∙ ϕ 1 − ϕ 2 ) ∥ L t ∞ L x p ([ 0 , t 0 ]) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 pert t 0 2 1 − κ K ) ∥ t 2 α ϕ 2 ∥ L t ∞ L x p ([ 0 , t 0 ]) . Additionally, we have that
∥ t α 2 ϕ 1 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) + ∥ t 1 + α 2 D B ϕ 1 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 p e r t ) 2 ( 1 + t 0 1 − κ 2 K ) ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . \lVert t^{\frac{\alpha}{2}}\phi_1\rVert_{L_t^\infty L_x^p([0,t_0])}+\lVert t^{\frac{1+\alpha}{2}}D_B\phi_1\rVert_{L_t^\infty L_x^p([0,t_0])}\lesssim(1+\lVert F_{B_+}\rVert_{t_0,1}^{\mathrm{pert}})^2(1+t_0^{1-\frac{\kappa}{2}}K)\lVert t^{\frac{\alpha}{2}}\phi_2\rVert_{L_t^\infty L_x^p([0,t_0])}. ∥ t 2 α ϕ 1 ∥ L t ∞ L x p ([ 0 , t 0 ]) + ∥ t 2 1 + α D B ϕ 1 ∥ L t ∞ L x p ([ 0 , t 0 ]) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 pert ) 2 ( 1 + t 0 1 − 2 κ K ) ∥ t 2 α ϕ 2 ∥ L t ∞ L x p ([ 0 , t 0 ]) . Similarly, if ψ 1 , ψ 2 : [ 0 , T ] × T 2 → W 2 \psi_1,\psi_2 : [0,T]\times\mathbb{T}^2\to W^2 ψ 1 , ψ 2 : [ 0 , T ] × T 2 → W 2 ( g − 1 ∙ ψ 1 ) ( 0 ) = ψ 2 ( 0 ) (g^{-1}\mathbin{\bullet}\psi_1)(0)=\psi_2(0) ( g − 1 ∙ ψ 1 ) ( 0 ) = ψ 2 ( 0 )
( ∂ t − Δ B − 2 F B ) ψ 1 = 0 , ( ∂ t − Δ B + − 2 F B + ) ψ 2 = 0 , (\partial_t-\Delta_B-2F_B)\psi_1=0,\qquad(\partial_t-\Delta_{B_+}-2F_{B_+})\psi_2=0, ( ∂ t − Δ B − 2 F B ) ψ 1 = 0 , ( ∂ t − Δ B + − 2 F B + ) ψ 2 = 0 , then for any t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] α ∈ [ 0 , 1 / 2 ] \alpha\in[0,1/2] α ∈ [ 0 , 1/2 ] 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞
∥ t α 2 ( g − 1 ∙ ψ 1 − ψ 2 ) ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≲ M T ( B + , 1 ) t 0 1 − κ 2 K ∥ t α 2 ψ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . \lVert t^{\frac{\alpha}{2}}(g^{-1}\mathbin{\bullet}\psi_1-\psi_2)\rVert_{L_t^\infty L_x^p([0,t_0])}\lesssim M_T(B^+,1)t_0^{\frac{1-\kappa}{2}}K\lVert t^{\frac{\alpha}{2}}\psi_2\rVert_{L_t^\infty L_x^p([0,t_0])}. ∥ t 2 α ( g − 1 ∙ ψ 1 − ψ 2 ) ∥ L t ∞ L x p ([ 0 , t 0 ]) ≲ M T ( B + , 1 ) t 0 2 1 − κ K ∥ t 2 α ψ 2 ∥ L t ∞ L x p ([ 0 , t 0 ]) . We now begin to prove Proposition 5.7. Towards this end, we will implement the classical DeTurck trick. In particular, we first prove local well-posedness in the DeTurck gauge (with appropriate choice of background connection).
Lemma 5.12 (Local solution theory for Yang–Mills heat flow in the infinitesimal DeTurck gauge). Under the assumptions of Proposition 5.7, there exists a unique B ♯ ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 ) B^\sharp\in C_t^0C_x^{-\kappa}([0,T]\times\mathbb{T}^2) B ♯ ∈ C t 0 C x − κ ([ 0 , T ] × T 2 ) ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 B ♯ B^\sharp B ♯
∂ t B k ♯ = D B ♯ j ( F B ♯ ) j k + D B ♯ , k D B + j ( B j ♯ − B j + ) , k ∈ [ 2 ] . (266) \partial_tB_k^\sharp=D_{B^\sharp}^j(F_{B^\sharp})_{jk}+D_{B^\sharp,k}D_{B_+}^j(B_j^\sharp-B_j^+),\qquad k\in[2]. \tag*{(266)} ∂ t B k ♯ = D B ♯ j ( F B ♯ ) j k + D B ♯ , k D B + j ( B j ♯ − B j + ) , k ∈ [ 2 ] . (266) and moreover B # ( t ) → B 0 # B^{\#}(t) \to B_{0}^{\#} B # ( t ) → B 0 # C x − κ \mathcal{C}_{x}^{-\kappa} C x − κ t ↓ 0 t \downarrow0 t ↓ 0 B l i n − ( t ) = e t Δ B + F B 0 − B_{\mathrm{lin}}^{-}(t) = e^{t\Delta_{B_{+}}^{F}}B_{0}^{-} B lin − ( t ) = e t Δ B + F B 0 − t 0 ∈ ( 0 , T ] t_{0} \in(0,T] t 0 ∈ ( 0 , T ] (259) for the definition of M T ( B + , 2 ) M_{T}(B^{+},2) M T ( B + , 2 )
max 0 ≤ j ≤ 2 ∥ t j + κ 2 D B + ( j ) ( B # − B + − B l i n − ) ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ M T ( B + , 2 ) K 2 t 0 1 − κ 2 . (267) \max_{0\leq j\leq2}\left\|t^{\frac{j+\kappa}{2}}D_{B_{+}}^{(j)}\left(B^{\#}-B^{+}-B_{\mathrm{lin}}^{-}\right)\right\|_{L_{t,x}^{\infty}([0,t_{0}])}\lesssim M_{T}(B^{+},2)K^{2}t_{0}^{1-\frac{\kappa}{2}}. \tag*{(267)} 0 ≤ j ≤ 2 max t 2 j + κ D B + ( j ) ( B # − B + − B lin − ) L t , x ∞ ([ 0 , t 0 ]) ≲ M T ( B + , 2 ) K 2 t 0 1 − 2 κ . (267) Finally, there exists a gauge transformation
g ∈ C t 0 G 1 − κ ( [ 0 , T ] ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) , (268) g\in C_{t}^{0}\mathcal{G}_{1-\kappa}([0,T])\cap C_{t,x}^{\infty}((0,T]\times\mathbb{T}^{2}), \tag*{(268)} g ∈ C t 0 G 1 − κ ([ 0 , T ]) ∩ C t , x ∞ (( 0 , T ] × T 2 ) , (268) such that for each x ∈ T 2 x\in\mathbb{T}^{2} x ∈ T 2 g ( ⋅ , x ) g(\mathord{\cdot},x) g ( ⋅ , x )
( ∂ t g g − 1 ) ( t , x ) = Ad g ( D B + ℓ ( B ℓ # − B ℓ + ) ) ( t , x ) , t ∈ ( 0 , T ] , (269) (\partial_{t}gg^{-1})(t,x)=\operatorname{Ad}_{g}\left(D_{B_{+}}^{\ell}(B_{\ell}^{\#}-B_{\ell}^{+})\right)(t,x),\quad t\in(0,T], \tag*{(269)} ( ∂ t g g − 1 ) ( t , x ) = Ad g ( D B + ℓ ( B ℓ # − B ℓ + ) ) ( t , x ) , t ∈ ( 0 , T ] , (269) g ( T , x ) = id G , g(T,x)=\operatorname{id}_{G}, g ( T , x ) = id G , and such that B = g ∙ B # B=g\mathbin{\bullet}B^{\#} B = g ∙ B # ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^{2} ( 0 , T ] × T 2 B ( t ) → g ( 0 ) ∙ B 0 # B(t)\to g(0)\mathbin{\bullet}B_{0}^{\#} B ( t ) → g ( 0 ) ∙ B 0 # C x − κ \mathcal{C}_{x}^{-\kappa} C x − κ t ↓ 0 t\downarrow0 t ↓ 0
Proof. To solve (266) , we make the ansatz B # = B + + B − B^{\#}=B^{+}+B^{-} B # = B + + B − B + B^{+} B + B 0 + B_{0}^{+} B 0 + (266) for B # B^{\#} B # (109) ), it suffices to solve for B − B^{-} B −
( ∂ t − Δ B + − 2 ad ( F B + ) ) B − = O ( B − D B + B − ) + O ( ( B − ) 3 ) , B − ( 0 ) = B 0 − . (270) (\partial_{t}-\Delta_{B_{+}}-2\operatorname{ad}(F_{B_{+}}))B^{-}=\mathcal{O}(B^{-}D_{B_{+}}B^{-})+\mathcal{O}((B^{-})^{3}),\qquad B^{-}(0)=B_{0}^{-}. \tag*{(270)} ( ∂ t − Δ B + − 2 ad ( F B + )) B − = O ( B − D B + B − ) + O (( B − ) 3 ) , B − ( 0 ) = B 0 − . (270) This equation is a non-degenerate semilinear parabolic PDE, so local well-posedness in C x − κ \mathcal{C}_{x}^{-\kappa} C x − κ κ < 1 2 \kappa<\frac{1}{2} κ < 2 1 B − B^{-} B − B r e m = B − − B l i n − B_{\mathrm{rem}}=B^{-}-B_{\mathrm{lin}}^{-} B rem = B − − B lin − B 0 − B_{0}^{-} B 0 − B − B^{-} B − B r e m B_{\mathrm{rem}} B rem
( ∂ t − Δ B + − 2 ad ( F B + ) ) B r e m = O ( B − D B + B − ) + O ( ( B − ) 3 ) , B r e m ( 0 ) = 0. (\partial_{t}-\Delta_{B_{+}}-2\operatorname{ad}(F_{B_{+}}))B_{\mathrm{rem}}=\mathcal{O}(B^{-}D_{B_{+}}B^{-})+\mathcal{O}((B^{-})^{3}),\qquad B_{\mathrm{rem}}(0)=0. ( ∂ t − Δ B + − 2 ad ( F B + )) B rem = O ( B − D B + B − ) + O (( B − ) 3 ) , B rem ( 0 ) = 0. Let G G G t 0 ∈ ( 0 , T ] t_{0}\in(0,T] t 0 ∈ ( 0 , T ]
∥ B − ∥ X ( t 0 ) = d e f max 0 ≤ j ≤ 2 ∥ t j + κ 2 D B + ( j ) B − ∥ L t , x ∞ ( [ 0 , t 0 ] ) . \|B^{-}\|_{\mathcal{X}(t_{0})}\overset{\mathrm{def}}{=}\max_{0\leq j\leq2}\left\|t^{\frac{j+\kappa}{2}}D_{B_{+}}^{(j)}B^{-}\right\|_{L_{t,x}^{\infty}([0,t_{0}])}. ∥ B − ∥ X ( t 0 ) = def 0 ≤ j ≤ 2 max t 2 j + κ D B + ( j ) B − L t , x ∞ ([ 0 , t 0 ]) . For each t 0 ∈ ( 0 , T ] t_{0}\in(0,T] t 0 ∈ ( 0 , T ] Lemma 4.62 (with k ∈ { 0 , 1 , 2 } k\in\{0,1,2\} k ∈ { 0 , 1 , 2 }
max 0 ≤ j ≤ 2 t 0 j + κ 2 ∥ D B + ( j ) B r e m ( t 0 ) ∥ L x ∞ ≲ t 0 κ 2 M T ( B + , 2 ) N 2 ( G ) , \max_{0\leq j\leq2}t_{0}^{\frac{j+\kappa}{2}}\left\|D_{B_{+}}^{(j)}B_{\mathrm{rem}}(t_{0})\right\|_{L_{x}^{\infty}}\lesssim t_{0}^{\frac{\kappa}{2}}M_{T}(B^{+},2)N_{2}(G), 0 ≤ j ≤ 2 max t 0 2 j + κ D B + ( j ) B rem ( t 0 ) L x ∞ ≲ t 0 2 κ M T ( B + , 2 ) N 2 ( G ) , where
N 2 ( G ) = ∥ G ∥ L t 1 L x ∞ ( [ 0 , t 0 ) + ∑ j = 0 1 ( ∥ t j + 1 2 D B + ( j ) G ∥ L t 2 L x ∞ ( [ 0 , t 0 ) + sup u ∈ [ 0 , t 0 ] ∥ t j + 1 2 ( u − t ) − 1 / 2 D B + ( j ) G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . N_{2}(G)=\|G\|_{L_{t}^{1}L_{x}^{\infty}([0,t_{0})}
+\sum_{j=0}^{1}\left(\left\|t^{\frac{j+1}{2}}D_{B_{+}}^{(j)}G\right\|_{L_{t}^{2}L_{x}^{\infty}([0,t_{0})}
+\sup_{u\in[0,t_{0}]}\left\|t^{\frac{j+1}{2}}(u-t)^{-1/2}D_{B_{+}}^{(j)}G\right\|_{L_{t}^{1}L_{x}^{\infty}([0,u])}\right). N 2 ( G ) = ∥ G ∥ L t 1 L x ∞ ([ 0 , t 0 ) + j = 0 ∑ 1 ( t 2 j + 1 D B + ( j ) G L t 2 L x ∞ ([ 0 , t 0 ) + u ∈ [ 0 , t 0 ] sup t 2 j + 1 ( u − t ) − 1/2 D B + ( j ) G L t 1 L x ∞ ([ 0 , u ]) ) . We now estimate the norms on G G G G G G
sup u ∈ ( 0 , t 0 ] ∥ t 1 / 2 ( u − t ) − 1 / 2 G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ≲ t 0 1 2 − κ ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − 3 κ 2 ∥ B − ∥ X ( t 0 ) 3 . \sup_{u\in(0,t_{0}]}\left\|t^{1/2}(u-t)^{-1/2}G\right\|_{L_{t}^{1}L_{x}^{\infty}([0,u])}\lesssim t_{0}^{\frac{1}{2}-\kappa}\|B^{-}\|_{\mathcal{X}(t_{0})}^{2}+t_{0}^{1-\frac{3\kappa}{2}}\|B^{-}\|_{\mathcal{X}(t_{0})}^{3}. u ∈ ( 0 , t 0 ] sup t 1/2 ( u − t ) − 1/2 G L t 1 L x ∞ ([ 0 , u ]) ≲ t 0 2 1 − κ ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − 2 3 κ ∥ B − ∥ X ( t 0 ) 3 . sup u ∈ ( 0 , t 0 ] ∥ t ( u − t ) − 1 / 2 D B + G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ≲ t 0 1 − κ 2 ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − 3 κ 2 ∥ B − ∥ X ( t 0 ) 3 \sup_{u\in(0,t_0]} \left\|t(u-t)^{-1/2}D_{B^+}G\right\|_{L^1_tL^\infty_x([0,u])} \lesssim t_0^{\frac{1-\kappa}{2}}\|B^-\|_{\mathcal{X}(t_0)}^2+t_0^{1-\frac{3\kappa}{2}}\|B^-\|_{\mathcal{X}(t_0)}^3 u ∈ ( 0 , t 0 ] sup t ( u − t ) − 1/2 D B + G L t 1 L x ∞ ([ 0 , u ]) ≲ t 0 2 1 − κ ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − 2 3 κ ∥ B − ∥ X ( t 0 ) 3 All the other norms appearing in N 2 ( G ) N_2(G) N 2 ( G ) C C C t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ] (261) which implies that ∥ B l i n − ∥ X ( t 0 ) ≤ K \|B^-_{\mathrm{lin}}\|_{\mathcal{X}(t_0)}\leq K ∥ B lin − ∥ X ( t 0 ) ≤ K B − = B l i n − + B r e m B^-=B^-_{\mathrm{lin}}+B_{\mathrm{rem}} B − = B lin − + B rem
∥ B r e m ∥ X ( t 0 ) ≤ C M T ( B + , 2 ) ( t 0 1 − κ 2 ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − κ ∥ B − ∥ X ( t 0 ) 3 ) . (271) \|B_{\mathrm{rem}}\|_{\mathcal{X}(t_0)}\leq CM_T(B^+,2)\left(t_0^{\frac{1-\kappa}{2}}\|B^-\|_{\mathcal{X}(t_0)}^2+t_0^{1-\kappa}\|B^-\|_{\mathcal{X}(t_0)}^3\right). \tag*{(271)} ∥ B rem ∥ X ( t 0 ) ≤ C M T ( B + , 2 ) ( t 0 2 1 − κ ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − κ ∥ B − ∥ X ( t 0 ) 3 ) . (271) ∥ B − ∥ X ( t 0 ) ≤ K + C M T ( B + , 2 ) ( t 0 1 − κ 2 ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − κ ∥ B − ∥ X ( t 0 ) 3 ) . (272) \|B^-\|_{\mathcal{X}(t_0)}\leq K+CM_T(B^+,2)\left(t_0^{\frac{1-\kappa}{2}}\|B^-\|_{\mathcal{X}(t_0)}^2+t_0^{1-\kappa}\|B^-\|_{\mathcal{X}(t_0)}^3\right). \tag*{(272)} ∥ B − ∥ X ( t 0 ) ≤ K + C M T ( B + , 2 ) ( t 0 2 1 − κ ∥ B − ∥ X ( t 0 ) 2 + t 0 1 − κ ∥ B − ∥ X ( t 0 ) 3 ) . (272) We now take the constant c 0 c_0 c 0 (260) to be small enough such that 10 4 C c 0 ≤ 1 10^4Cc_0\leq1 1 0 4 C c 0 ≤ 1 C C C (272) . We proceed by a standard continuity argument. Let
I = { t 0 ∈ ( 0 , T ] : ∥ B − ∥ X ( t 0 ) ≤ 10 K } , I=\{t_0\in(0,T]:\|B^-\|_{\mathcal{X}(t_0)}\leq10K\}, I = { t 0 ∈ ( 0 , T ] : ∥ B − ∥ X ( t 0 ) ≤ 10 K } , where K K K (272) , I I I t 0 > 0 t_0>0 t 0 > 0 B 0 − B^-_0 B 0 − t ↦ ∥ B − ∥ X ( t ) t\mapsto\|B^-\|_{\mathcal{X}(t)} t ↦ ∥ B − ∥ X ( t ) t t t I I I ( 0 , T ] (0,T] ( 0 , T ] I I I ( 0 , T ] (0,T] ( 0 , T ] (261) , our choice of c 0 c_0 c 0 I I I t 0 ∈ I t_0\in I t 0 ∈ I
C M T ( B + , 2 ) t 0 1 − κ 2 ∥ B − ∥ X ( t 0 ) ≤ 10 2 C M T ( B + , 2 ) t 0 1 − κ 2 K ≤ 10 2 C c 0 ≤ 1 100 . CM_T(B^+,2)t_0^{\frac{1-\kappa}{2}}\|B^-\|_{\mathcal{X}(t_0)}
\leq10^2CM_T(B^+,2)t_0^{\frac{1-\kappa}{2}}K
\leq10^2Cc_0\leq\frac{1}{100}. C M T ( B + , 2 ) t 0 2 1 − κ ∥ B − ∥ X ( t 0 ) ≤ 1 0 2 C M T ( B + , 2 ) t 0 2 1 − κ K ≤ 1 0 2 C c 0 ≤ 100 1 . Thus by (272) , it follows that for all t 0 ∈ I t_0\in I t 0 ∈ I
∥ B − ∥ X ( t 0 ) ≤ K + 1 2 ∥ B − ∥ X ( t 0 ) , \|B^-\|_{\mathcal{X}(t_0)}\leq K+\frac{1}{2}\|B^-\|_{\mathcal{X}(t_0)}, ∥ B − ∥ X ( t 0 ) ≤ K + 2 1 ∥ B − ∥ X ( t 0 ) , from which we obtain
∥ B − ∥ X ( t 0 ) < 10 K , \|B^-\|_{\mathcal{X}(t_0)}<10K, ∥ B − ∥ X ( t 0 ) < 10 K , and thus I I I ( 0 , T ] (0,T] ( 0 , T ] I = ( 0 , T ] I=(0,T] I = ( 0 , T ] (267) is then a direct consequence of the smallness condition (260) , the estimate (271) , and the estimate ∥ B − ∥ X ( t 0 ) ≤ 10 K \|B^-\|_{\mathcal{X}(t_0)}\leq10K ∥ B − ∥ X ( t 0 ) ≤ 10 K t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
It remains to prove the claimed results about the gauge transformation g g g (268) holds, one may show by direct calculation (using for example (269) and the fully covariant formulation introduced in Section 3.2.2) that B = g ∙ B # B=g\mathbin{\bullet}B^\# B = g ∙ B # ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 B ( t ) → g ( 0 ) ∙ B 0 − B(t)\to g(0)\mathbin{\bullet}B_0^- B ( t ) → g ( 0 ) ∙ B 0 − C x − κ C_x^{-\kappa} C x − κ t ↓ 0 t\downarrow0 t ↓ 0 g g g (269) indeed exists and satisfies the regularity conditions specified by (268) .
Since B + B^+ B + B # B^\# B # ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 g g g (269) on ( 0 , T ] × T 2 (0,T]\times\mathbb{T}^2 ( 0 , T ] × T 2 ∥ B − ∥ X ( T ) ≤ 10 K \|B^-\|_{\mathcal{X}(T)}\leq10K ∥ B − ∥ X ( T ) ≤ 10 K
∥ t 1 + κ 2 ∂ t g g − 1 ∥ L t ∞ L x ∞ ( [ 0 , T ] ) ≲ K . (273) \left\|t^{\frac{1+\kappa}{2}}\partial_tgg^{-1}\right\|_{L^\infty_tL^\infty_x([0,T])}\lesssim K. \tag*{(273)} t 2 1 + κ ∂ t g g − 1 L t ∞ L x ∞ ([ 0 , T ]) ≲ K . (273) From this, it follows that g g g [ 0 , T ] × T 2 [0,T]\times\mathbb{T}^2 [ 0 , T ] × T 2
g ( 0 , x ) = id G − ∫ 0 T d t ( ∂ t g g − 1 ) ( t , x ) g ( t , x ) , x ∈ T 2 . g(0,x)=\operatorname{id}_G-\int_0^Tdt\,(\partial_tgg^{-1})(t,x)g(t,x),\qquad x\in\mathbb{T}^2. g ( 0 , x ) = id G − ∫ 0 T d t ( ∂ t g g − 1 ) ( t , x ) g ( t , x ) , x ∈ T 2 . It remains to show that g ∈ C t 0 G 1 1 − κ ( [ 0 , T ] ) g\in C_t^0G_1^{1-\kappa}([0,T]) g ∈ C t 0 G 1 1 − κ ([ 0 , T ]) C t 0 C x 1 − κ ( [ 0 , T ] × T 2 ) C_t^0C_x^{1-\kappa}([0,T]\times\mathbb{T}^2) C t 0 C x 1 − κ ([ 0 , T ] × T 2 ) g g g L t ∞ C x 1 − κ ( ( 0 , T ] × T 2 ) L_t^\infty C_x^{1-\kappa}((0,T]\times\mathbb{T}^2) L t ∞ C x 1 − κ (( 0 , T ] × T 2 ) t = 0 t=0 t = 0 ( d g g − 1 ) ( t 0 ) (dgg^{-1})(t_0) ( d g g − 1 ) ( t 0 ) t 0 > 0 t_0>0 t 0 > 0
∥ t κ 2 d g g − 1 ∥ L t , x ∞ ( [ 0 , T ] ) ≲ 1 , (274) \left\|t^{\frac{\kappa}{2}}dgg^{-1}\right\|_{L^\infty_{t,x}([0,T])}\lesssim1, \tag*{(274)} t 2 κ d g g − 1 L t , x ∞ ([ 0 , T ]) ≲ 1 , (274) where the implicit constant may depend on B + B^+ B + K K K t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] x , y ∈ T 2 x,y\in\mathbb{T}^2 x , y ∈ T 2 ∣ x − y ∣ 2 ≤ T |x-y|^2\leq T ∣ x − y ∣ 2 ≤ T t 0 = ∣ x − y ∣ 2 ∨ t t_0=|x-y|^2\vee t t 0 = ∣ x − y ∣ 2 ∨ t
∣ g ( t , x ) − g ( t , y ) ∣ ≤ 1 ( t < t 0 ) ( ∣ g ( t , x ) − g ( t 0 , x ) ∣ + ∣ g ( t 0 , y ) − g ( t , y ) ∣ ) + ∣ g ( t 0 , x ) − g ( t 0 , y ) ∣ ≲ 1 ( t < t 0 ) t 0 1 − κ 2 + ∣ x − y ∣ t 0 − κ 2 ≲ ∣ x − y ∣ 1 − κ . \begin{aligned}
|g(t,x)-g(t,y)| &\leq\mathbf{1}(t<t_0)(|g(t,x)-g(t_0,x)|+|g(t_0,y)-g(t,y)|)+|g(t_0,x)-g(t_0,y)|\\
&\lesssim\mathbf{1}(t<t_0)t_0^{\frac{1-\kappa}{2}}+|x-y|t_0^{-\frac{\kappa}{2}}\lesssim|x-y|^{1-\kappa}.
\end{aligned} ∣ g ( t , x ) − g ( t , y ) ∣ ≤ 1 ( t < t 0 ) ( ∣ g ( t , x ) − g ( t 0 , x ) ∣ + ∣ g ( t 0 , y ) − g ( t , y ) ∣ ) + ∣ g ( t 0 , x ) − g ( t 0 , y ) ∣ ≲ 1 ( t < t 0 ) t 0 2 1 − κ + ∣ x − y ∣ t 0 − 2 κ ≲ ∣ x − y ∣ 1 − κ . Here, we estimated the first two terms (in the case t < t 0 t<t_0 t < t 0 t = t 0 t=t_0 t = t 0 t 0 1 − κ 2 t_0^{\frac{1-\kappa}{2}} t 0 2 1 − κ (273) ), and we estimated the last term by ∣ x − y ∣ t 0 − κ 2 |x-y|t_0^{-\frac{\kappa}{2}} ∣ x − y ∣ t 0 − 2 κ (274) ). Since the implicit constants are uniform in t , x , y t,x,y t , x , y g ∈ L t ∞ C x 1 − κ ( ( 0 , T ] × T 2 ) g\in L^\infty_t\mathcal{C}^{1-\kappa}_x((0,T]\times\mathbb{T}^2) g ∈ L t ∞ C x 1 − κ (( 0 , T ] × T 2 ) g ( 0 ) g(0) g ( 0 ) ( 1 − κ ) (1-\kappa) ( 1 − κ ) g ( 0 ) g(0) g ( 0 ) C x 1 − κ \mathcal{C}^{1-\kappa}_x C x 1 − κ g ( 0 ) ∈ G 1 − κ g(0)\in\mathcal{G}_{1-\kappa} g ( 0 ) ∈ G 1 − κ
To complete the proof of the L t ∞ C x 1 − κ ( ( 0 , T ] × T 2 ) L^\infty_t\mathcal{C}^{1-\kappa}_x((0,T]\times\mathbb{T}^2) L t ∞ C x 1 − κ (( 0 , T ] × T 2 ) (274) . We let h α = − ( ∂ α g ) g − 1 h_\alpha=-(\partial_\alpha g)g^{-1} h α = − ( ∂ α g ) g − 1 α ∈ { 0 } ∪ [ 2 ] \alpha\in\{0\}\cup[2] α ∈ { 0 } ∪ [ 2 ] Notation 3.2 ), so that h = g ∙ 0 h=g\bullet0 h = g ∙ 0 h h h i ∈ [ 2 ] i\in[2] i ∈ [ 2 ] 0 = ( F h ) 0 i = ∂ 0 h i − D h , i h 0 0=(F_h)_{0i}=\partial_0h_i-D_{h,i}h_0 0 = ( F h ) 0 i = ∂ 0 h i − D h , i h 0
∂ t h i = D g ∙ 0 , i ( − ∂ t g g − 1 ) = − Ad g ∂ i ( Ad g − 1 ( ∂ t g g − 1 ) ) . (275) \partial_t h_i=D_{g\bullet0,i}(-\partial_tgg^{-1})=-\operatorname{Ad}_g\partial_i\left(\operatorname{Ad}_{g^{-1}}(\partial_tgg^{-1})\right). \tag*{(275)} ∂ t h i = D g ∙ 0 , i ( − ∂ t g g − 1 ) = − Ad g ∂ i ( Ad g − 1 ( ∂ t g g − 1 ) ) . (275) Integrating this on [ t 0 , T ] [t_0,T] [ t 0 , T ] g ( T ) ≡ id G g(T)\equiv\operatorname{id}_G g ( T ) ≡ id G ( d g g − 1 ) ( T ) = 0 (dgg^{-1})(T)=0 ( d g g − 1 ) ( T ) = 0
( ∂ i g g − 1 ) ( t 0 ) = − ∫ t 0 T d t ( Ad g ∂ i D B + j B j − ) ( t ) = − ∫ t 0 T d t ( Ad g D B + , i D B + j B j − ) ( t ) + ∫ t 0 T d t ( Ad g [ B i + , D B + j B j − ] ) ( t ) . \begin{aligned}
(\partial_i gg^{-1})(t_0)=-\int_{t_0}^{T}dt\,(\operatorname{Ad}_g\partial_iD_{B^+}^{j}B_j^-)(t) \\
&=-\int_{t_0}^{T}dt\,(\operatorname{Ad}_gD_{B^+,i}D_{B^+}^{j}B_j^-)(t)+\int_{t_0}^{T}dt\,(\operatorname{Ad}_g[B_i^+,D_{B^+}^{j}B_j^-])(t).
\end{aligned} ( ∂ i g g − 1 ) ( t 0 ) = − ∫ t 0 T d t ( Ad g ∂ i D B + j B j − ) ( t ) = − ∫ t 0 T d t ( Ad g D B + , i D B + j B j − ) ( t ) + ∫ t 0 T d t ( Ad g [ B i + , D B + j B j − ]) ( t ) . Since B + B^+ B + ∥ B − ∥ X ( T ) ≲ K \|B^-\|_{X(T)}\lesssim K ∥ B − ∥ X ( T ) ≲ K
∥ t 1 + κ 2 D B + ( 2 ) B − ∥ L t , x ∞ ( [ 0 , T ] ) ≲ 1 , \left\|t^{1+\frac{\kappa}{2}}D_{B^+}^{(2)}B^-\right\|_{L^\infty_{t,x}([0,T])}\lesssim1, t 1 + 2 κ D B + ( 2 ) B − L t , x ∞ ([ 0 , T ]) ≲ 1 , where the implicit constant may depend on B + B^+ B + K K K B − = B l i n − + B r e m B^-=B^-_{\mathrm{lin}}+B_{\mathrm{rem}} B − = B lin − + B rem (261) and the j = 2 j=2 j = 2 (267) .
Finally, we discuss the continuity of g ( t ) g(t) g ( t ) t = 0 t=0 t = 0 C x 1 − κ \mathcal{C}^{1-\kappa}_x C x 1 − κ g ( 0 ) ∈ G 1 − κ g(0)\in\mathcal{G}_{1-\kappa} g ( 0 ) ∈ G 1 − κ t ↓ 0 t\downarrow0 t ↓ 0 ( ( g − 1 d g ) ( t ) ) t > 0 ((g^{-1}dg)(t))_{t>0} (( g − 1 d g ) ( t ) ) t > 0 C x − κ \mathcal{C}^{-\kappa}_x C x − κ h ~ = g − 1 d g = − Ad g − 1 h \tilde h=g^{-1}dg=-\operatorname{Ad}_{g}^{-1}h h ~ = g − 1 d g = − Ad g − 1 h h ~ \tilde h h ~ h h h (275) , we may obtain
∂ t h ~ i = ∂ i D B + ℓ B ℓ − + Ad g − 1 [ ∂ t g g − 1 , h i ] . \partial_t\tilde h_i=\partial_iD_{B^+}^{\ell}B_\ell^-+\operatorname{Ad}_{g}^{-1}[\partial_tgg^{-1},h_i]. ∂ t h ~ i = ∂ i D B + ℓ B ℓ − + Ad g − 1 [ ∂ t g g − 1 , h i ] . Thus for 0 < s < t 0<s<t 0 < s < t
h ~ i ( t ) − h ~ i ( s ) = ∫ s t d u ( ∂ i D B + ℓ B ℓ − + Ad g − 1 [ ∂ t g g − 1 , h i ] ) . \tilde h_i(t)-\tilde h_i(s)=\int_s^tdu\,\left(\partial_iD_{B^+}^{\ell}B_\ell^-+\operatorname{Ad}_{g}^{-1}[\partial_tgg^{-1},h_i]\right). h ~ i ( t ) − h ~ i ( s ) = ∫ s t d u ( ∂ i D B + ℓ B ℓ − + Ad g − 1 [ ∂ t g g − 1 , h i ] ) . By the estimates (273) and (274) , the second term above may be directly estimated in L x ∞ L^\infty_x L x ∞ B − = B l i n − + B r e m B^-=B^-_{\mathrm{lin}}+B_{\mathrm{rem}} B − = B lin − + B rem B l i n − B^-_{\mathrm{lin}} B lin − e t Δ B 0 − e^{t\Delta}B^{-}_{0} e t Δ B 0 − L x ∞ L^\infty_x L x ∞ L x ∞ L^\infty_x L x ∞
∫ s t d u ∂ i ∂ ℓ e u Δ B 0 , ℓ − = ( − Δ ) − 1 ∂ i ∂ ℓ ( e s Δ B 0 , ℓ − − e t Δ B 0 , ℓ − ) . \int_s^t \mathrm{d}u\,\partial_i\partial^\ell e^{u\Delta}B^{-}_{0,\ell}=(-\Delta)^{-1}\partial_i\partial^\ell\left(e^{s\Delta}B^{-}_{0,\ell}-e^{t\Delta}B^{-}_{0,\ell}\right). ∫ s t d u ∂ i ∂ ℓ e u Δ B 0 , ℓ − = ( − Δ ) − 1 ∂ i ∂ ℓ ( e s Δ B 0 , ℓ − − e t Δ B 0 , ℓ − ) . To finish, we use that
∥ ( − Δ ) − 1 ∂ i ∂ ℓ ( e s Δ B 0 , ℓ − − e t Δ B 0 , ℓ − ) ∥ C x − κ ≲ ∥ e s Δ B 0 − − e t Δ B 0 − ∥ C x − κ , \left\|(-\Delta)^{-1}\partial_i\partial^\ell\left(e^{s\Delta}B^{-}_{0,\ell}-e^{t\Delta}B^{-}_{0,\ell}\right)\right\|_{\mathcal{C}^{-\kappa}_x}\lesssim\left\|e^{s\Delta}B^{-}_{0}-e^{t\Delta}B^{-}_{0}\right\|_{\mathcal{C}^{-\kappa}_x}, ( − Δ ) − 1 ∂ i ∂ ℓ ( e s Δ B 0 , ℓ − − e t Δ B 0 , ℓ − ) C x − κ ≲ e s Δ B 0 − − e t Δ B 0 − C x − κ , combined with the fact that e t Δ B 0 − e^{t\Delta}B^{-}_{0} e t Δ B 0 − C x − κ \mathcal{C}^{-\kappa}_x C x − κ t ↓ 0 t\downarrow0 t ↓ 0 B 0 − B^{-}_{0} B 0 − C x − κ \mathcal{C}^{-\kappa}_x C x − κ C x ∞ C^\infty_x C x ∞
Having shown local well-posedness of the DeTurck Yang–Mills heat flow, we next prove the following curvature estimates. These estimates will be needed to show the curvature estimates claimed in Proposition 5.7.
Lemma 5.13. In the setting of Lemma 5.12, we have for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ] x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2
∣ D B ♯ ( j ) F B ♯ ( t , x ) ∣ ≤ ∣ D B + ( j ) F B + ( t , x ) ∣ + O ( t − κ 2 ( t − 1 + j 2 + ∣ F B + ( t , x ) ∣ j ) K ) , j ∈ { 0 , 1 } , (276) \left|D_{B^\sharp}^{(j)}F_{B^\sharp}(t,x)\right|\leq\left|D_{B^+}^{(j)}F_{B^+}(t,x)\right|+O\left(t^{-\frac{\kappa}{2}}\left(t^{-\frac{1+j}{2}}+\left|F_{B^+}(t,x)\right|^j\right)K\right),\qquad j\in\{0,1\}, \tag*{(276)} D B ♯ ( j ) F B ♯ ( t , x ) ≤ D B + ( j ) F B + ( t , x ) + O ( t − 2 κ ( t − 2 1 + j + ∣ F B + ( t , x ) ∣ j ) K ) , j ∈ { 0 , 1 } , (276) as well as the L x 2 L^2_x L x 2 F B F_B F B F B ♯ F_{B^\sharp} F B ♯
Proof. Let B r e m = B ♯ − B + − B l i n − B_{\mathrm{rem}}=B^\sharp-B^+-B^-_{\mathrm{lin}} B rem = B ♯ − B + − B lin − t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ] M T ( B + , 2 ) M_T(B^+,2) M T ( B + , 2 )
∥ B r e m ( t ) ∥ L x ∞ ≲ M T ( B + , 2 ) K 2 t 1 − κ ≲ c 0 t − κ 2 K , (277) \|B_{\mathrm{rem}}(t)\|_{L^\infty_x}\lesssim M_T(B^+,2)K^2t^{1-\kappa}\lesssim c_0t^{-\frac{\kappa}{2}}K, \tag*{(277)} ∥ B rem ( t ) ∥ L x ∞ ≲ M T ( B + , 2 ) K 2 t 1 − κ ≲ c 0 t − 2 κ K , (277) ∥ ( D B + B r e m ) ( t ) ∥ L x ∞ ≲ M T ( B + , 2 ) K 2 t − κ ≲ c 0 t − 1 + κ 2 K , (278) \|(D_{B^+}B_{\mathrm{rem}})(t)\|_{L^\infty_x}\lesssim M_T(B^+,2)K^2t^{-\kappa}\lesssim c_0t^{-\frac{1+\kappa}{2}}K, \tag*{(278)} ∥ ( D B + B rem ) ( t ) ∥ L x ∞ ≲ M T ( B + , 2 ) K 2 t − κ ≲ c 0 t − 2 1 + κ K , (278) so that B r e m B_{\mathrm{rem}} B rem B l i n − B^-_{\mathrm{lin}} B lin −
F B ♯ 12 = F B + 12 + curl B + ( B l i n − + B r e m ) + [ ( B l i n − + B r e m ) 1 , ( B l i n − + B r e m ) 2 ] . F_{B^\sharp}^{12}=F_{B^+}^{12}+\operatorname{curl}_{B^+}(B^-_{\mathrm{lin}}+B_{\mathrm{rem}})+[(B^-_{\mathrm{lin}}+B_{\mathrm{rem}})^1,(B^-_{\mathrm{lin}}+B_{\mathrm{rem}})^2]. F B ♯ 12 = F B + 12 + curl B + ( B lin − + B rem ) + [( B lin − + B rem ) 1 , ( B lin − + B rem ) 2 ] . The pointwise estimate (5.27) in the case j = 0 j=0 j = 0 B l i n − B^-_{\mathrm{lin}} B lin − B r e m B_{\mathrm{rem}} B rem t − κ K 2 ≪ t − 1 + κ 2 K t^{-\kappa}K^2\ll t^{-\frac{1+\kappa}{2}}K t − κ K 2 ≪ t − 2 1 + κ K j = 1 j=1 j = 1 D B ♯ D_{B^\sharp} D B ♯ B − = B l i n − + B r e m B^-=B^-_{\mathrm{lin}}+B_{\mathrm{rem}} B − = B lin − + B rem D B ♯ = D B + + [ B − , ⋅ ] D_{B^\sharp}=D_{B^+}+[B^-,\mathord\cdot] D B ♯ = D B + + [ B − , ⋅ ] F B ♯ = F B + + O ( D B + B − + ( B − ) 2 ) F_{B^\sharp}=F_{B^+}+O(D_{B^+}B^-+(B^-)^2) F B ♯ = F B + + O ( D B + B − + ( B − ) 2 )
D B ♯ F B ♯ 12 = D B + F B + 12 + [ B − , F B + ] + O ( D B + ( 2 ) B − + B − D B + B − + ( B − ) 3 ) . D_{B^\sharp}F_{B^\sharp}^{12}=D_{B^+}F_{B^+}^{12}+[B^-,F_{B^+}]+O\left(D_{B^+}^{(2)}B^-+B^-D_{B^+}B^-+(B^-)^3\right). D B ♯ F B ♯ 12 = D B + F B + 12 + [ B − , F B + ] + O ( D B + ( 2 ) B − + B − D B + B − + ( B − ) 3 ) . The second and third terms on the right-hand side may be directly estimated by combining the assumed bound on B l i n − B^-_{\mathrm{lin}} B lin − B r e m B_{\mathrm{rem}} B rem
To show the L x 2 L^2_x L x 2
∥ F B ♯ ( t ) ∥ L x 2 2 = ∥ F B + ( t ) ∥ L x 2 2 + ∥ curl B + ( t ) B l i n − ( t ) ∥ L x 2 2 + 2 ⟨ F B + 12 ( t ) , ( curl B + B l i n − ) ( t ) + [ B l i n − 1 , B l i n − 2 ] ( t ) ⟩ + 2 ⟨ curl B + B l i n − ( t ) , [ B l i n − 1 , B l i n − 2 ] ( t ) ⟩ + ∥ [ B l i n − 1 , B l i n − 2 ] ( t ) ∥ L x 2 2 \begin{aligned}
\|F_{B^\sharp}(t)\|_{L^2_x}^2={}&\|F_{B^+}(t)\|_{L^2_x}^2+\|\operatorname{curl}_{B^+}(t)B^-_{\mathrm{lin}}(t)\|_{L^2_x}^2\\
&+2\langle F_{B^+}^{12}(t),(\operatorname{curl}_{B^+}B^-_{\mathrm{lin}})(t)+[B^{-1}_{\mathrm{lin}},B^{-2}_{\mathrm{lin}}](t)\rangle\\
&+2\langle\operatorname{curl}_{B^+}B^-_{\mathrm{lin}}(t),[B^{-1}_{\mathrm{lin}},B^{-2}_{\mathrm{lin}}](t)\rangle+\|[B^{-1}_{\mathrm{lin}},B^{-2}_{\mathrm{lin}}](t)\|_{L^2_x}^2
\end{aligned} ∥ F B ♯ ( t ) ∥ L x 2 2 = ∥ F B + ( t ) ∥ L x 2 2 + ∥ curl B + ( t ) B lin − ( t ) ∥ L x 2 2 + 2 ⟨ F B + 12 ( t ) , ( curl B + B lin − ) ( t ) + [ B lin − 1 , B lin − 2 ] ( t )⟩ + 2 ⟨ curl B + B lin − ( t ) , [ B lin − 1 , B lin − 2 ] ( t )⟩ + ∥ [ B lin − 1 , B lin − 2 ] ( t ) ∥ L x 2 2 where “perturbative terms” collects all terms with at least one factor of B r e m B_{\mathrm{rem}} B rem (261) on the size of B 0 − B_{0}^{-} B 0 −
2 ∣ ⟨ curl B + B l i n − , [ B l i n − 1 , B l i n − 2 ] ( t ) ⟩ ∣ + ∥ [ B l i n − 1 , B l i n − 2 ] ( t ) ∥ L x 2 2 ≲ t − 1 + 3 κ 2 K 3 + t − 4 κ 2 K 4 ≲ t − 1 + 3 κ 2 K 3 , 2\left|\left\langle\operatorname{curl}_{B^{+}}B_{\mathrm{lin}}^{-},[B_{\mathrm{lin}}^{-1},B_{\mathrm{lin}}^{-2}](t)\right\rangle\right|+\left\|[B_{\mathrm{lin}}^{-1},B_{\mathrm{lin}}^{-2}](t)\right\|_{L_{x}^{2}}^{2}\lesssim t^{-\frac{1+3\kappa}{2}}K^{3}+t^{-\frac{4\kappa}{2}}K^{4}\lesssim t^{-\frac{1+3\kappa}{2}}K^{3}, 2 ⟨ curl B + B lin − , [ B lin − 1 , B lin − 2 ] ( t ) ⟩ + [ B lin − 1 , B lin − 2 ] ( t ) L x 2 2 ≲ t − 2 1 + 3 κ K 3 + t − 2 4 κ K 4 ≲ t − 2 1 + 3 κ K 3 , where we used that t 1 − κ 2 K ≤ T 1 − κ 2 K ≤ 1 t^{\frac{1-\kappa}{2}}K\leq T^{\frac{1-\kappa}{2}}K\leq1 t 2 1 − κ K ≤ T 2 1 − κ K ≤ 1 (260) ). This contribution is acceptable.
Next, we estimate all perturbative terms by an acceptable contribution. For the most delicate perturbative term (which requires that we first integrate by parts), we first apply Lemma 4.13 and the definition of ∥ F B + ∥ p e r t \|F_{B^{+}}\|_{\mathrm{pert}} ∥ F B + ∥ pert
∥ t 1 / 2 D B + F B + ∥ L t ∞ L x 2 ( [ 0 , t ] ) ≲ ( 1 + ∥ F B + ∥ t , 0 p e r t ) ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) . (279) \|t^{1/2}D_{B^{+}}F_{B^{+}}\|_{L_{t}^{\infty}L_{x}^{2}([0,t])}\lesssim(1+\|F_{B^{+}}\|_{t,0}^{\mathrm{pert}})\|D_{B^{+}}F_{B^{+}}\|_{L_{t,x}^{2}([0,t])}. \tag*{(279)} ∥ t 1/2 D B + F B + ∥ L t ∞ L x 2 ([ 0 , t ]) ≲ ( 1 + ∥ F B + ∥ t , 0 pert ) ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) . (279) We now estimate the most delicate perturbative term by (here we apply integration by parts (Lemma 3.23) in the first step and (279) in the final step):
∣ ⟨ F B + 12 ( t ) , ( curl B + B r e m ) ( t ) ⟩ ∣ = ∣ ⟨ D B + ∗ F B + ( t ) , B r e m ( t ) ⟩ ∣ ≲ M T ( B + , 2 ) ∥ D B + F B + ( t ) ∥ L x 2 K 2 t 1 2 − κ ≲ M T ( B + , 3 ) ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) K 2 t − κ . \left|\left\langle F_{B^{+}}^{12}(t),(\operatorname{curl}_{B^{+}}B_{\mathrm{rem}})(t)\right\rangle\right|=\left|\left\langle D_{B^{+}}^{*}F_{B^{+}}(t),B_{\mathrm{rem}}(t)\right\rangle\right|
\lesssim M_{T}(B^{+},2)\|D_{B^{+}}F_{B^{+}}(t)\|_{L_{x}^{2}}K^{2}t^{\frac{1}{2}-\kappa}
\lesssim M_{T}(B^{+},3)\|D_{B^{+}}F_{B^{+}}\|_{L_{t,x}^{2}([0,t])}K^{2}t^{-\kappa}. ⟨ F B + 12 ( t ) , ( curl B + B rem ) ( t ) ⟩ = ∣ ⟨ D B + ∗ F B + ( t ) , B rem ( t ) ⟩ ∣ ≲ M T ( B + , 2 ) ∥ D B + F B + ( t ) ∥ L x 2 K 2 t 2 1 − κ ≲ M T ( B + , 3 ) ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) K 2 t − κ . All other perturbative terms are less delicate, as they may be directly estimated (i.e. without first integrating by parts). For instance, one of the perturbative terms is (here we apply (277) )
∣ ⟨ F B + 12 ( t ) , [ B r e m 1 , ( B l i n − + B r e m ) 2 ] ( t ) ⟩ ∣ ≲ ∥ F B + ( t ) ∥ L x 2 ∥ B r e m ∥ L x ∞ ∥ B l i n − + B r e m ∥ L x ∞ ≲ M T ( B + , 2 ) ∥ F B + ( t ) ∥ L x 2 K 3 t 1 2 − 3 κ 2 . \left|\left\langle F_{B^{+}}^{12}(t),[B_{\mathrm{rem}}^{1},(B_{\mathrm{lin}}^{-}+B_{\mathrm{rem}})^{2}](t)\right\rangle\right|\lesssim\|F_{B^{+}}(t)\|_{L_{x}^{2}}\|B_{\mathrm{rem}}\|_{L_{x}^{\infty}}\|B_{\mathrm{lin}}^{-}+B_{\mathrm{rem}}\|_{L_{x}^{\infty}}
\lesssim M_{T}(B^{+},2)\|F_{B^{+}}(t)\|_{L_{x}^{2}}K^{3}t^{\frac{1}{2}-\frac{3\kappa}{2}}. ⟨ F B + 12 ( t ) , [ B rem 1 , ( B lin − + B rem ) 2 ] ( t ) ⟩ ≲ ∥ F B + ( t ) ∥ L x 2 ∥ B rem ∥ L x ∞ ∥ B lin − + B rem ∥ L x ∞ ≲ M T ( B + , 2 ) ∥ F B + ( t ) ∥ L x 2 K 3 t 2 1 − 2 3 κ . Another perturbative term is (here we apply (278) )
∣ ⟨ ( curl B + B r e m ) ( t ) , ( curl B + ( B l i n − + B r e m ) ) ( t ) ⟩ ∣ ≲ M T ( B + , 2 ) K 3 t − 1 2 − 3 κ 2 . \left|\left\langle(\operatorname{curl}_{B^{+}}B_{\mathrm{rem}})(t),(\operatorname{curl}_{B^{+}}(B_{\mathrm{lin}}^{-}+B_{\mathrm{rem}}))(t)\right\rangle\right|\lesssim M_{T}(B^{+},2)K^{3}t^{-\frac{1}{2}-\frac{3\kappa}{2}}. ⟨ ( curl B + B rem ) ( t ) , ( curl B + ( B lin − + B rem )) ( t ) ⟩ ≲ M T ( B + , 2 ) K 3 t − 2 1 − 2 3 κ . All other terms may be estimated by similar arguments.
Combining the previous few intermediate results, we may now prove Proposition 5.7.
Proof of Proposition 5.7. We take the constant c 0 c_{0} c 0 B ♯ , g B^{\sharp},g B ♯ , g B = g ∙ B ♯ B=g\mathbin{\bullet}B^{\sharp} B = g ∙ B ♯
Next, note that g − 1 ∙ B = B + + B − g^{-1}\mathbin{\bullet}B=B^{+}+B^{-} g − 1 ∙ B = B + + B − (262) on B − B^{-} B − (267) . Finally, for the curvature estimates, we use that D B ( j ) F B = Ad g D B ♯ ( j ) F B ♯ D_{B}^{(j)}F_{B}=\operatorname{Ad}_{g}D_{B^{\sharp}}^{(j)}F_{B^{\sharp}} D B ( j ) F B = Ad g D B ♯ ( j ) F B ♯ (276) directly implies (263) and (264) , and the second estimate (265) follows directly from the corresponding estimate given by Lemma 5.13.
Next, we prove the uniqueness result Lemma 5.9. We reiterate that this may be skipped on a first reading.
Proof of Lemma 5.9. Without loss of generality, we may take T ≤ 1 T \le1 T ≤ 1 B B B h h h B B B C x − κ C_x^{-\kappa} C x − κ h ∈ C t 0 C x 2 − 2 κ ( [ 0 , T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) h \in C_t^0C_x^{2-2\kappa}([0,T] \times\mathbb{T}^2) \cap C_{t,x}^{\infty}((0,T] \times\mathbb{T}^2) h ∈ C t 0 C x 2 − 2 κ ([ 0 , T ] × T 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 )
( ∂ t h ) h − 1 = ∂ ℓ ( ( ∂ ℓ h ) h − 1 ) − ∂ ℓ ( Ad h B ℓ ) , h ( 0 ) = id G , (\partial_t h)h^{-1} = \partial^\ell((\partial_\ell h)h^{-1}) - \partial^\ell(\operatorname{Ad}_h B_\ell), \qquad h(0) = \operatorname{id}_G, ( ∂ t h ) h − 1 = ∂ ℓ (( ∂ ℓ h ) h − 1 ) − ∂ ℓ ( Ad h B ℓ ) , h ( 0 ) = id G , then B ~ = h ∙ B \widetilde{B} = h \bullet B B = h ∙ B B ~ ( t ) → A 0 \widetilde{B}(t) \to A_0 B ( t ) → A 0 C x − κ C_x^{-\kappa} C x − κ h h h h h h
( ∂ t h ) h − 1 = ( Δ h ) h − 1 − ( ∂ ℓ h ) h − 1 ( ∂ ℓ h ) h − 1 − ad ( ( ∂ ℓ h ) h − 1 ) Ad h B ℓ − Ad h ( ∂ ℓ B ℓ ) . (\partial_t h)h^{-1} = (\Delta h)h^{-1} - (\partial_\ell h)h^{-1}(\partial^\ell h)h^{-1} - \operatorname{ad}((\partial^\ell h)h^{-1})\operatorname{Ad}_h B_\ell- \operatorname{Ad}_h(\partial^\ell B_\ell). ( ∂ t h ) h − 1 = ( Δ h ) h − 1 − ( ∂ ℓ h ) h − 1 ( ∂ ℓ h ) h − 1 − ad (( ∂ ℓ h ) h − 1 ) Ad h B ℓ − Ad h ( ∂ ℓ B ℓ ) . Recall by Lemma 5.1 (here we use the assumed bound on D B F B D_BF_B D B F B T ≤ 1 T \le1 T ≤ 1
∥ t κ 2 B ∥ L t , x ∞ ( [ 0 , T ] ) , ∥ t κ 2 ∂ ℓ B ℓ ∥ L t , x ∞ ( [ 0 , T ] ) < ∞ . \left\|t^{\frac{\kappa}{2}}B\right\|_{L^\infty_{t,x}([0,T])},\ \left\|t^{\frac{\kappa}{2}}\partial^\ell B_\ell\right\|_{L^\infty_{t,x}([0,T])} < \infty. t 2 κ B L t , x ∞ ([ 0 , T ]) , t 2 κ ∂ ℓ B ℓ L t , x ∞ ([ 0 , T ]) < ∞. Therefore by a standard Picard iteration argument, for some possibly smaller time 0 < T 1 ≤ T 0 < T_1 \le T 0 < T 1 ≤ T h ∈ C t 0 C x 2 − 2 κ ( [ 0 , T 1 ] × T 2 ) h \in C_t^0C_x^{2-2\kappa}([0,T_1] \times\mathbb{T}^2) h ∈ C t 0 C x 2 − 2 κ ([ 0 , T 1 ] × T 2 ) B B B h ∈ C t , x ∞ ( ( 0 , T 1 ] × T 2 ) h \in C_{t,x}^{\infty}((0,T_1] \times\mathbb{T}^2) h ∈ C t , x ∞ (( 0 , T 1 ] × T 2 ) B ~ = h ∙ B \widetilde{B} = h \bullet B B = h ∙ B ( 0 , T 1 ] (0,T_1] ( 0 , T 1 ] B ~ ( t ) → A 0 \widetilde{B}(t) \to A_0 B ( t ) → A 0 C x − κ C_x^{-\kappa} C x − κ t ↓ 0 t \downarrow0 t ↓ 0 B ~ \widetilde{B} B A 0 ∈ C x − κ A_0 \in C_x^{-\kappa} A 0 ∈ C x − κ h h h B ~ \widetilde{B} B x ∈ T 2 x \in\mathbb{T}^2 x ∈ T 2
( ∂ t h ) h − 1 = − ∂ ℓ B ~ ℓ , h ( 0 ) = id G . (\partial_t h)h^{-1} = -\partial^\ell\widetilde{B}_\ell, \qquad h(0) = \operatorname{id}_G. ( ∂ t h ) h − 1 = − ∂ ℓ B ℓ , h ( 0 ) = id G . (We remark that these ODEs may be integrated up from time zero, since by standard parabolic theory, one may show that ∥ t 1 + κ 2 ∂ ℓ B ~ ℓ ∥ L t , x ∞ ( [ 0 , T 1 ] ) < ∞ \left\|t^{\frac{1+\kappa}{2}}\partial^\ell\widetilde{B}_\ell\right\|_{L^\infty_{t,x}([0,T_1])} < \infty t 2 1 + κ ∂ ℓ B ℓ L t , x ∞ ([ 0 , T 1 ]) < ∞ B ~ \widetilde{B} B h h h B = h − 1 ∙ B ~ B = h^{-1} \bullet\widetilde{B} B = h − 1 ∙ B [ 0 , T 1 ] [0,T_1] [ 0 , T 1 ] [ 0 , T ] [0,T] [ 0 , T ] □ \square □
Using the same ideas as in the proof of Lemma 5.9, we can show that the ∥ ⋅ ∥ i n v , − κ \|\cdot\|_{\mathrm{inv},-\kappa} ∥ ⋅ ∥ inv , − κ Definition 2.23 is well-defined. We quickly do this next.
Lemma 5.14. Let A 0 , Φ 0 A_0, \Phi_0 A 0 , Φ 0 Definition 2.23 , i.e. let A 0 , Φ 0 ∈ C x − κ A_0,\Phi_0 \in\mathcal{C}_x^{-\kappa} A 0 , Φ 0 ∈ C x − κ T > 0 T > 0 T > 0 g ∈ G 1 − κ g \in\mathcal{G}_{1-\kappa} g ∈ G 1 − κ g ⋅ ( A 0 , Φ 0 ) ∈ T T g \cdot(A_0,\Phi_0) \in\mathcal{T}_T g ⋅ ( A 0 , Φ 0 ) ∈ T T (40) ), as required in the definition. Moreover, if T 1 , T 2 > 0 T_1,T_2 > 0 T 1 , T 2 > 0 g 1 , g 2 ∈ G 1 − κ g_1,g_2 \in\mathcal{G}_{1-\kappa} g 1 , g 2 ∈ G 1 − κ g 1 ⋅ ( A 0 , Φ 0 ) ∈ T T 1 g_1 \cdot(A_0,\Phi_0) \in\mathcal{T}_{T_1} g 1 ⋅ ( A 0 , Φ 0 ) ∈ T T 1 g 2 ⋅ ( A 0 , Φ 0 ) ∈ T T 2 g_2 \cdot(A_0,\Phi_0) \in\mathcal{T}_{T_2} g 2 ⋅ ( A 0 , Φ 0 ) ∈ T T 2 g 1 g 2 − 1 g_1g_2^{-1} g 1 g 2 − 1 B 1 , B 2 B_1,B_2 B 1 , B 2 B 1 = ( g 1 g 2 − 1 ) ∙ B 2 B_1 = (g_1g_2^{-1}) \bullet B_2 B 1 = ( g 1 g 2 − 1 ) ∙ B 2 Ψ 1 , Ψ 2 \Psi_1,\Psi_2 Ψ 1 , Ψ 2 (63) is the same for ( B 1 , Ψ 1 ) (B_1,\Psi_1) ( B 1 , Ψ 1 ) ( B 2 , Ψ 2 ) (B_2,\Psi_2) ( B 2 , Ψ 2 ) ∥ ⋅ ∥ i n v , − κ \|\cdot\|_{\mathrm{inv},-\kappa} ∥ ⋅ ∥ inv , − κ
Proof. The existence of T > 0 T > 0 T > 0 g ∈ G 1 − κ g \in\mathcal{G}_{1-\kappa} g ∈ G 1 − κ B B B g ⋅ A 0 g \cdot A_0 g ⋅ A 0 B 0 + B_0^+ B 0 + B 0 − B_0^- B 0 − A 0 A_0 A 0 T > 0 T > 0 T > 0 K ≥ 1 K \ge1 K ≥ 1 (260) and (261) are satisfied). This proposition also provides the required bounds on F B F_B F B D B ∗ F B D_B^*F_B D B ∗ F B (40) (from which it follows by Lemma 5.9 that B B B g ⋅ Φ 0 g\mathbin{\cdot}\Phi_0 g ⋅ Φ 0 Lemma 5.1 to obtain bounds on B B B ∂ ℓ B ℓ \partial^\ell B_\ell ∂ ℓ B ℓ g ⋅ Φ 0 g\mathbin{\cdot}\Phi_0 g ⋅ Φ 0
Next, let T 1 , T 2 > 0 T_1,T_2>0 T 1 , T 2 > 0 g 1 , g 2 ∈ G 1 − κ g_1,g_2\in\mathcal{G}_{1-\kappa} g 1 , g 2 ∈ G 1 − κ B 1 , B 2 B_1,B_2 B 1 , B 2 Ψ 1 , Ψ 2 \Psi_1,\Psi_2 Ψ 1 , Ψ 2 Lemma 5.9 , there exist 0 < T ′ ≤ T 1 ∧ T 2 0<T'\leq T_1\wedge T_2 0 < T ′ ≤ T 1 ∧ T 2 h 1 , h 2 ∈ C t 2 C x 2 − 2 κ ( [ 0 , T ′ ] ) ∩ C t , x ∞ ( ( 0 , T ′ ] ) h_1,h_2\in C^2_tC_x^{2-2\kappa}([0,T'])\cap C^\infty_{t,x}((0,T']) h 1 , h 2 ∈ C t 2 C x 2 − 2 κ ([ 0 , T ′ ]) ∩ C t , x ∞ (( 0 , T ′ ]) h 1 ⋅ B 1 , h 2 ⋅ B 2 h_1\mathbin{\cdot}B_1,h_2\mathbin{\cdot}B_2 h 1 ⋅ B 1 , h 2 ⋅ B 2 h 1 ( 0 ) = h 2 ( 0 ) ≡ id G h_1(0)=h_2(0)\equiv\operatorname{id}_G h 1 ( 0 ) = h 2 ( 0 ) ≡ id G
Having reached this point, we finish by a statement of smooth gauge equivalence at positive times between DeTurck Yang–Mills flows whose initial data are related by a G 1 − κ \mathcal{G}_{1-\kappa} G 1 − κ Lemma 5.9 (except now using weighted spaces), it is possible to solve for a gauge transformation h h h h 2 ⋅ B 2 h_2\mathbin{\cdot}B_2 h 2 ⋅ B 2 h 1 ⋅ B 1 h_1\mathbin{\cdot}B_1 h 1 ⋅ B 1 h h h h ⋅ ( h 2 ⋅ B 2 ) = h 1 ⋅ B 1 h\mathbin{\cdot}(h_2\mathbin{\cdot}B_2)=h_1\mathbin{\cdot}B_1 h ⋅ ( h 2 ⋅ B 2 ) = h 1 ⋅ B 1 h ( 0 ) = g 1 g 2 − 1 h(0)=g_1g_2^{-1} h ( 0 ) = g 1 g 2 − 1
From this, letting h ~ = h 1 − 1 h h 2 \widetilde h=h_1^{-1}hh_2 h = h 1 − 1 h h 2 B 1 = h ~ ⋅ B 2 B_1=\widetilde h\mathbin{\cdot}B_2 B 1 = h ⋅ B 2 h ~ \widetilde h h
( ( h 1 − 1 h h 2 ) ⋅ B 2 ) 0 = ( h 1 − 1 ⋅ ( ( h h 2 ) ⋅ B 2 ) ) 0 = ( h 1 − 1 ⋅ ( h 1 ⋅ B 1 ) ) 0 = ( B 1 ) 0 . \bigl((h_1^{-1}hh_2)\mathbin{\cdot}B_2\bigr)_0
=\bigl(h_1^{-1}\mathbin{\cdot}((hh_2)\mathbin{\cdot}B_2)\bigr)_0
=\bigl(h_1^{-1}\mathbin{\cdot}(h_1\mathbin{\cdot}B_1)\bigr)_0
=(B_1)_0. ( ( h 1 − 1 h h 2 ) ⋅ B 2 ) 0 = ( h 1 − 1 ⋅ (( h h 2 ) ⋅ B 2 ) ) 0 = ( h 1 − 1 ⋅ ( h 1 ⋅ B 1 ) ) 0 = ( B 1 ) 0 . Since B 1 , B 2 B_1,B_2 B 1 , B 2 ( B 1 ) 0 = ( B 2 ) 0 = 0 (B_1)_0=(B_2)_0=0 ( B 1 ) 0 = ( B 2 ) 0 = 0 ( ∂ t h ~ ) h ~ − 1 = 0 (\partial_t\widetilde h)\widetilde h^{-1}=0 ( ∂ t h ) h − 1 = 0 h ~ \widetilde h h h ~ ( 0 ) = h 1 − 1 ( 0 ) h ( 0 ) h 2 ( 0 ) = g 1 g 2 − 1 \widetilde h(0)=h_1^{-1}(0)h(0)h_2(0)=g_1g_2^{-1} h ( 0 ) = h 1 − 1 ( 0 ) h ( 0 ) h 2 ( 0 ) = g 1 g 2 − 1 g 1 g 2 − 1 = h ~ g_1g_2^{-1}=\widetilde h g 1 g 2 − 1 = h B 1 = ( g 1 g 2 − 1 ) ⋅ B 2 B_1=(g_1g_2^{-1})\mathbin{\cdot}B_2 B 1 = ( g 1 g 2 − 1 ) ⋅ B 2 (119) ) h ~ ⋅ Ψ 2 \widetilde h\mathbin{\cdot}\Psi_2 h ⋅ Ψ 2 h ~ ⋅ B 2 = B 1 \widetilde h\mathbin{\cdot}B_2=B_1 h ⋅ B 2 = B 1 ( h ~ ⋅ Ψ 2 ) ( 0 ) = ( g 1 g 2 − 1 ) ⋅ ( g 2 ⋅ Φ 0 ) = g 1 ⋅ Φ 0 (\widetilde h\mathbin{\cdot}\Psi_2)(0)=(g_1g_2^{-1})\mathbin{\cdot}(g_2\mathbin{\cdot}\Phi_0)=g_1\mathbin{\cdot}\Phi_0 ( h ⋅ Ψ 2 ) ( 0 ) = ( g 1 g 2 − 1 ) ⋅ ( g 2 ⋅ Φ 0 ) = g 1 ⋅ Φ 0 h ~ ⋅ Ψ 2 = Ψ 1 \widetilde h\mathbin{\cdot}\Psi_2=\Psi_1 h ⋅ Ψ 2 = Ψ 1
To finish off this section, we prove Lemma 5.11 .
Proof of Lemma 5.11. Let ϕ ~ 1 = g − 1 ⋅ ϕ 1 \widetilde\phi_1=g^{-1}\mathbin{\cdot}\phi_1 ϕ 1 = g − 1 ⋅ ϕ 1 g − 1 ⋅ B = B ♯ = B + + B − g^{-1}\mathbin{\cdot}B=B^\sharp=B^++B^- g − 1 ⋅ B = B ♯ = B + + B − g g g (269) . Then we have that (recall the fully covariant discussion in Section 3.2.2)
( ∂ t − Δ B ♯ ) ϕ ~ 1 = − ( g − 1 ∂ t g ) ϕ ~ 1 = − ( D B + j + B j − ) ϕ ~ 1 . (\partial_t-\Delta_{B^\sharp})\widetilde\phi_1=-(g^{-1}\partial_tg)\widetilde\phi_1=-(D_{B^+}^j+B_j^-)\widetilde\phi_1. ( ∂ t − Δ B ♯ ) ϕ 1 = − ( g − 1 ∂ t g ) ϕ 1 = − ( D B + j + B j − ) ϕ 1 . Let ψ = ϕ ~ 1 − ϕ 2 \psi=\widetilde\phi_1-\phi_2 ψ = ϕ 1 − ϕ 2 (112) to expand Δ B ♯ \Delta_{B^\sharp} Δ B ♯ Δ B + \Delta_{B^+} Δ B +
( ∂ t − Δ B + ) ψ = 2 D B + j ( B j − ( ψ + ϕ 2 ) ) − 2 ( D B + j + B j − ) ( ψ + ϕ 2 ) + ( B − ) j B j − ( ψ + ϕ 2 ) , ψ ( 0 ) = 0. (\partial_t-\Delta_{B^+})\psi
=2D_{B^+}^j(B_j^-(\psi+\phi_2))
-2(D_{B^+}^j+B_j^-)(\psi+\phi_2)
+(B^-)^jB_j^-(\psi+\phi_2),
\qquad
\psi(0)=0. ( ∂ t − Δ B + ) ψ = 2 D B + j ( B j − ( ψ + ϕ 2 )) − 2 ( D B + j + B j − ) ( ψ + ϕ 2 ) + ( B − ) j B j − ( ψ + ϕ 2 ) , ψ ( 0 ) = 0. By (260) , (261) , and (262) , we have that
∥ t κ 2 B − ∥ L t , x ∞ ( [ 0 , T ] ) + ∥ t 1 + κ 2 D B + B − ∥ L t , x ∞ ( [ 0 , T ] ) ≲ K . \lVert t^{\frac{\kappa}{2}}B^-\rVert_{L^\infty_{t,x}([0,T])}
+\lVert t^{\frac{1+\kappa}{2}}D_{B^+}B^-\rVert_{L^\infty_{t,x}([0,T])}
\lesssim K. ∥ t 2 κ B − ∥ L t , x ∞ ([ 0 , T ]) + ∥ t 2 1 + κ D B + B − ∥ L t , x ∞ ([ 0 , T ]) ≲ K . Combining this with the Schauder estimates of Lemma 4.61 , we may obtain that for any t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
t 0 α 2 ∥ ψ ( t 0 ) ∥ L x p ≲ ( 1 + ∥ F B + ∥ L t 0 , 1 ∞ ( t 0 1 − κ 2 K + ( t 0 1 − κ 2 K ) 2 ) ) ( ∥ t α 2 ψ ∥ L t ∞ L x p ( [ 0 , t 0 ] ) + ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ) . t_0^{\frac{\alpha}{2}}\lVert\psi(t_0)\rVert_{L_x^p}
\lesssim
\left(1+\lVert F_{B^+}\rVert_{L^\infty_{t_0,1}}\left(t_0^{\frac{1-\kappa}{2}}K+\left(t_0^{\frac{1-\kappa}{2}}K\right)^2\right)\right)
\left(\lVert t^{\frac{\alpha}{2}}\psi\rVert_{L_t^\infty L_x^p([0,t_0])}
+\lVert t^{\frac{\alpha}{2}}\phi_2\rVert_{L_t^\infty L_x^p([0,t_0])}\right). t 0 2 α ∥ ψ ( t 0 ) ∥ L x p ≲ ( 1 + ∥ F B + ∥ L t 0 , 1 ∞ ( t 0 2 1 − κ K + ( t 0 2 1 − κ K ) 2 ) ) ( ∥ t 2 α ψ ∥ L t ∞ L x p ([ 0 , t 0 ]) + ∥ t 2 α ϕ 2 ∥ L t ∞ L x p ([ 0 , t 0 ]) ) . (Here, we used that α ∈ [ 0 , 1 / 2 ] \alpha\in[0,1/2] α ∈ [ 0 , 1/2 ] α + κ < 1 \alpha+\kappa<1 α + κ < 1 (260) , we thus obtain that for some constant C C C t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
∥ t α 2 ψ ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≲ C ( c 0 + c 0 2 ) ∥ t α 2 ψ ∥ L t ∞ L x p ( [ 0 , t 0 ] ) + C ( 1 + ∥ F B + ∥ t 0 , 1 p e r t ) t 0 1 − κ 2 K ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . \left\|t^{\frac{\alpha}{2}}\psi\right\|_{L_t^\infty L_x^p([0,t_0])}\lesssim C(c_0+c_0^2)\left\|t^{\frac{\alpha}{2}}\psi\right\|_{L_t^\infty L_x^p([0,t_0])}+C\left(1+\left\|F_{B+}\right\|_{t_0,1}^{\mathrm{pert}}\right)t_0^{\frac{1-\kappa}{2}}K\left\|t^{\frac{\alpha}{2}}\phi_2\right\|_{L_t^\infty L_x^p([0,t_0])}. t 2 α ψ L t ∞ L x p ([ 0 , t 0 ]) ≲ C ( c 0 + c 0 2 ) t 2 α ψ L t ∞ L x p ([ 0 , t 0 ]) + C ( 1 + ∥ F B + ∥ t 0 , 1 pert ) t 0 2 1 − κ K t 2 α ϕ 2 L t ∞ L x p ([ 0 , t 0 ]) . Thus upon possibly taking c 0 c_0 c 0
∥ t α 2 ψ ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 p e r t t 0 1 − κ 2 K ) ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . \left\|t^{\frac{\alpha}{2}}\psi\right\|_{L_t^\infty L_x^p([0,t_0])}\lesssim\left(1+\left\|F_{B+}\right\|_{t_0,1}^{\mathrm{pert}}t_0^{\frac{1-\kappa}{2}}K\right)\left\|t^{\frac{\alpha}{2}}\phi_2\right\|_{L_t^\infty L_x^p([0,t_0])}. t 2 α ψ L t ∞ L x p ([ 0 , t 0 ]) ≲ ( 1 + ∥ F B + ∥ t 0 , 1 pert t 0 2 1 − κ K ) t 2 α ϕ 2 L t ∞ L x p ([ 0 , t 0 ]) . which is the first desired estimate.
For the second estimate, first note that ∣ ϕ 1 ∣ = ∣ g − 1 ∙ ϕ 1 ∣ |\phi_1|=|g^{-1}\mathbin{\bullet}\phi_1| ∣ ϕ 1 ∣ = ∣ g − 1 ∙ ϕ 1 ∣ C C C
∥ t α 2 ϕ 1 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≤ ∥ t α 2 ( g − 1 ∙ ϕ 1 − ϕ 2 ) ∥ L t ∞ L x p ( [ 0 , t 0 ] ) + ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≤ ( 1 + C ( 1 + ∥ F B + ∥ t 0 , 1 p e r t ) t 0 1 − κ 2 K ) ∥ t α 2 ϕ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) . (280) \begin{aligned}
\left\|t^{\frac{\alpha}{2}}\phi_1\right\|_{L_t^\infty L_x^p([0,t_0])}\leq\left\|t^{\frac{\alpha}{2}}(g^{-1}\mathbin{\bullet}\phi_1-\phi_2)\right\|_{L_t^\infty L_x^p([0,t_0])}+\left\|t^{\frac{\alpha}{2}}\phi_2\right\|_{L_t^\infty L_x^p([0,t_0])} \\
&\leq\left(1+C\left(1+\left\|F_{B+}\right\|_{t_0,1}^{\mathrm{pert}}\right)t_0^{\frac{1-\kappa}{2}}K\right)\left\|t^{\frac{\alpha}{2}}\phi_2\right\|_{L_t^\infty L_x^p([0,t_0])}. \tag*{(280)}
\end{aligned} t 2 α ϕ 1 L t ∞ L x p ([ 0 , t 0 ]) ≤ t 2 α ( g − 1 ∙ ϕ 1 − ϕ 2 ) L t ∞ L x p ([ 0 , t 0 ]) + t 2 α ϕ 2 L t ∞ L x p ([ 0 , t 0 ]) ≤ ( 1 + C ( 1 + ∥ F B + ∥ t 0 , 1 pert ) t 0 2 1 − κ K ) t 2 α ϕ 2 L t ∞ L x p ([ 0 , t 0 ]) . (280) To finish, we estimate D B ϕ 1 D_B\phi_1 D B ϕ 1 p B p_B p B t ∈ ( 0 , t 0 ] t\in(0,t_0] t ∈ ( 0 , t 0 ]
∥ D B ϕ 1 ( t ) ∥ L x p ≲ ( 1 + ∥ F B ∥ t 0 , 0 p e r t ) t − 1 / 2 ∥ ϕ 1 ( t / 2 ) ∥ L x p . \left\|D_B\phi_1(t)\right\|_{L_x^p}\lesssim\left(1+\left\|F_B\right\|_{t_0,0}^{\mathrm{pert}}\right)t^{-1/2}\left\|\phi_1(t/2)\right\|_{L_x^p}. ∥ D B ϕ 1 ( t ) ∥ L x p ≲ ( 1 + ∥ F B ∥ t 0 , 0 pert ) t − 1/2 ∥ ϕ 1 ( t /2 ) ∥ L x p . Recalling the definition of ∥ ∙ ∥ p e r t \|\bullet\|^{\mathrm{pert}} ∥ ∙ ∥ pert j = 0 j=0 j = 0 (263) directly implies that
∥ F B ∥ t 0 , 0 p e r t ≤ ∥ F B + ∥ t 0 , 0 p e r t + C t 0 1 − κ 2 K ≲ ∥ F B + ∥ t 0 , 0 p e r t + 1. \left\|F_B\right\|_{t_0,0}^{\mathrm{pert}}\leq\left\|F_{B+}\right\|_{t_0,0}^{\mathrm{pert}}+Ct_0^{\frac{1-\kappa}{2}}K\lesssim\left\|F_{B+}\right\|_{t_0,0}^{\mathrm{pert}}+1. ∥ F B ∥ t 0 , 0 pert ≤ ∥ F B + ∥ t 0 , 0 pert + C t 0 2 1 − κ K ≲ ∥ F B + ∥ t 0 , 0 pert + 1. Inserting this into the previous math display and then multiplying both sides of that display by t 1 + α 2 t^{\frac{1+\alpha}{2}} t 2 1 + α t ∈ ( 0 , t 0 ] t\in(0,t_0] t ∈ ( 0 , t 0 ] D B ϕ 1 D_B\phi_1 D B ϕ 1 (280) .
Finally, for the third estimate, we proceed similarly to the proof for the first estimate. We now let ψ = g − 1 ∙ ψ 1 − ψ 2 \psi=g^{-1}\mathbin{\bullet}\psi_1-\psi_2 ψ = g − 1 ∙ ψ 1 − ψ 2 (112) as well as the curvature expansion (109) )
( ∂ t − Δ B + − 2 F B + ) ψ = O ( D B + ( B − ( ψ + ψ 2 ) ) ) + O ( ( D B + B − + ( B − ) 2 ) ( ψ + ψ 2 ) ) . (\partial_t-\Delta_{B+}-2F_{B+})\psi=O(D_{B+}(B^-(\psi+\psi_2)))+O((D_{B+}B^-+(B^-)^2)(\psi+\psi_2)). ( ∂ t − Δ B + − 2 F B + ) ψ = O ( D B + ( B − ( ψ + ψ 2 ))) + O (( D B + B − + ( B − ) 2 ) ( ψ + ψ 2 )) . Note also that ψ ( 0 ) = 0 \psi(0)=0 ψ ( 0 ) = 0 t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
t 0 α 2 ∥ ψ ( t 0 ) ∥ L x p ≲ Gr T ( C 0 F B + ) ( 1 + ∥ F B + ∥ t 0 , 1 p e r t ) ( t 0 1 − κ 2 K + ( t 0 1 − κ 2 K ) 2 ) ( ∥ t α 2 ψ ∥ L t ∞ L x p ( [ 0 , t 0 ] ) + ∥ t α 2 ψ 2 ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ) . t_0^{\frac{\alpha}{2}}\|\psi(t_0)\|_{L_x^p}
\lesssim\operatorname{Gr}_T(C_0F_{B+})\left(1+\left\|F_{B+}\right\|_{t_0,1}^{\mathrm{pert}}\right)\left(t_0^{\frac{1-\kappa}{2}}K+\left(t_0^{\frac{1-\kappa}{2}}K\right)^2\right)\left(\left\|t^{\frac{\alpha}{2}}\psi\right\|_{L_t^\infty L_x^p([0,t_0])}+\left\|t^{\frac{\alpha}{2}}\psi_2\right\|_{L_t^\infty L_x^p([0,t_0])}\right). t 0 2 α ∥ ψ ( t 0 ) ∥ L x p ≲ Gr T ( C 0 F B + ) ( 1 + ∥ F B + ∥ t 0 , 1 pert ) ( t 0 2 1 − κ K + ( t 0 2 1 − κ K ) 2 ) ( t 2 α ψ L t ∞ L x p ([ 0 , t 0 ]) + t 2 α ψ 2 L t ∞ L x p ([ 0 , t 0 ]) ) . The rest of the argument proceeds similarly to before; we omit the details.
Estimates of deterministic pieces arising from the Higgs field
In this section, we derive estimates on the pair ( S , ϕ ) (S,\phi) ( S , ϕ ) § 6.3 ). Throughout this section, we take B ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) B\in C_t^0C_x^{-\kappa}([0,T]\times\mathbb{T}^2)\cap C_{t,x}^\infty((0,T]\times\mathbb{T}^2) B ∈ C t 0 C x − κ ([ 0 , T ] × T 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 ) M 0 M_0 M 0 M 1 M_1 M 1 B B B B B B
Recall that the pair ( S , φ ) (\mathcal{S},\varphi) ( S , φ )
( ∂ t − Δ B − 2 ad ( F B ) ) S = [ S j , 2 D B , j S − D B S j ] + [ S j , [ S j , S ] ] + B ( φ ⊗ D B + S φ ) (281) (\partial_t-\Delta_B-2\operatorname{ad}(F_B))\mathcal{S}=[S^j,2D_{B,j}\mathcal{S}-D_BS_j]+[S^j,[S_j,\mathcal{S}]]+\mathbf{B}(\varphi\otimes D_{B+S}\varphi) \tag*{(281)} ( ∂ t − Δ B − 2 ad ( F B )) S = [ S j , 2 D B , j S − D B S j ] + [ S j , [ S j , S ]] + B ( φ ⊗ D B + S φ ) (281) S ( 0 ) = 0 , \mathcal{S}(0)=0, S ( 0 ) = 0 , ( ∂ t − Δ B + S ) φ = − ( D B j S j ) φ − ∣ φ ∣ 2 φ , (282) (\partial_t-\Delta_{B+S})\varphi=-(D_B^jS_j)\varphi-|\varphi|^2\varphi, \tag*{(282)} ( ∂ t − Δ B + S ) φ = − ( D B j S j ) φ − ∣ φ ∣ 2 φ , (282) φ ( 0 ) = φ 0 + + φ 0 − . \varphi(0)=\varphi_0^++\varphi_0^-. φ ( 0 ) = φ 0 + + φ 0 − . To estimate φ \varphi φ φ = φ l i n + φ r e g \varphi=\varphi_{\mathrm{lin}}+\varphi_{\mathrm{reg}} φ = φ lin + φ reg
( ∂ t − Δ B ) φ l i n − = 0 , φ l i n − ( 0 ) = φ 0 − , (283) (\partial_t-\Delta_B)\varphi_{\mathrm{lin}}^-=0,\qquad\varphi_{\mathrm{lin}}^-(0)=\varphi_0^-, \tag*{(283)} ( ∂ t − Δ B ) φ lin − = 0 , φ lin − ( 0 ) = φ 0 − , (283) ( ∂ t − Δ B + S ) φ r e g = 2 S j D B j φ l i n − + S j S j φ l i n − − ( D B j S j ) φ r e g − ∣ φ ∣ 2 φ , φ r e g ( 0 ) = φ 0 + . (284) (\partial_t-\Delta_{B+S})\varphi_{\mathrm{reg}}=2S_jD_B^j\varphi_{\mathrm{lin}}^-+S_jS^j\varphi_{\mathrm{lin}}^--(D_B^jS_j)\varphi_{\mathrm{reg}}-|\varphi|^2\varphi,\qquad\varphi_{\mathrm{reg}}(0)=\varphi_0^+. \tag*{(284)} ( ∂ t − Δ B + S ) φ reg = 2 S j D B j φ lin − + S j S j φ lin − − ( D B j S j ) φ reg − ∣ φ ∣ 2 φ , φ reg ( 0 ) = φ 0 + . (284) Here, we used that (recall (112) )
( Δ B + S − Δ B ) φ l i n − − ( D B j S j ) φ l i n − = 2 S j D B j φ l i n − + S j S j φ l i n − . (\Delta_{B+S}-\Delta_B)\varphi_{\mathrm{lin}}^--(D_B^jS_j)\varphi_{\mathrm{lin}}^-=2S_jD_B^j\varphi_{\mathrm{lin}}^-+S_jS^j\varphi_{\mathrm{lin}}^-. ( Δ B + S − Δ B ) φ lin − − ( D B j S j ) φ lin − = 2 S j D B j φ lin − + S j S j φ lin − . We state the first main result of this section. Recall the covariant Hölder norm introduced in Definition 5.4 as well as the definition of M T ( B , k ) M_T(B,k) M T ( B , k )
Proposition 5.15 (Estimates of S S S φ \varphi φ C 0 C_0 C 0 c 1 ≤ 1 c_1\leq1 c 1 ≤ 1 T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ]
M T ( B , 10 ) ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + T 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ c 1 . (285) M_T(B,10)\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)\leq c_1. \tag*{(285)} M T ( B , 10 ) ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + T 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ c 1 . (285) Then S S S φ \varphi φ [ 0 , T ] [0,T] [ 0 , T ]
max 0 ≤ j ≤ 2 ∥ t j 2 D B ( j ) S ∥ L t , x ∞ ( [ 0 , T ] ) ≲ M T ( B , 10 ) ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + T 1 2 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) , (286) \max_{0\leq j\leq2}\|t^{\frac{j}{2}}D_B^{(j)}S\|_{L^\infty_{t,x}([0,T])}\lesssim M_T(B,10)\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{1}{2}-\frac{\kappa}{2}}\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}\right), \tag*{(286)} 0 ≤ j ≤ 2 max ∥ t 2 j D B ( j ) S ∥ L t , x ∞ ([ 0 , T ]) ≲ M T ( B , 10 ) ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + T 2 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) , (286) max 0 ≤ j ≤ 2 ∥ t j + κ 2 D B ( j ) φ ∥ L t , x ∞ ( [ 0 , T ] ) ≲ ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) , (287) \max_{0\leq j\leq2}\|t^{\frac{j+\kappa}{2}}D_B^{(j)}\varphi\|_{L^\infty_{t,x}([0,T])}\lesssim\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)^2\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}\right), \tag*{(287)} 0 ≤ j ≤ 2 max ∥ t 2 j + κ D B ( j ) φ ∥ L t , x ∞ ([ 0 , T ]) ≲ ( 1 + ∥ F B ∥ T , 1 pert ) 2 ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) , (287) max 0 ≤ j ≤ 1 ∥ t j + κ 2 D B ( j ) S ∥ L t ∞ L x 2 ( [ 0 , T ] ) ≲ G r T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 0 p e r t ) \max_{0\leq j\leq1}\|t^{\frac{j+\kappa}{2}}D_B^{(j)}S\|_{L^\infty_tL_x^2([0,T])}\lesssim Gr_T(C_0F_B)(1+\|F_B\|_{T,0}^{\mathrm{pert}}) 0 ≤ j ≤ 1 max ∥ t 2 j + κ D B ( j ) S ∥ L t ∞ L x 2 ([ 0 , T ]) ≲ G r T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 0 pert ) × ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ( [ 0 , T ] ) + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , T ] ) ) . (288) {}\times\left(\|\mathbf{B}(\varphi\otimes D_{B+S}\varphi)\|_{L_t^1L_x^2([0,T])}+\|t^{1/2}\mathbf{B}(\varphi\otimes D_{B+S}\varphi)\|_{L_{t,x}^2([0,T])}\right). \tag*{(288)} × ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ([ 0 , T ]) + ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ([ 0 , T ]) ) . (288) We remark that unlike the first two estimates, the last estimate only concerns zeroth and first order derivatives.
Remark 5.16 (Soft smoothness assumption). In the course of the proof of Proposition 5.15, we will assume that φ 0 + \varphi_0^+ φ 0 + φ 0 − \varphi_0^- φ 0 − φ ( 0 ) \varphi(0) φ ( 0 ) ∥ φ 0 + ∥ L x r \|\varphi_0^+\|_{L_x^r} ∥ φ 0 + ∥ L x r ∥ φ 0 − ∥ C B ( 0 ) , T − κ \|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}} ∥ φ 0 − ∥ C B ( 0 ) , T − κ
The proof of Proposition 5.15 is deferred until later. We proceed to state several more results regarding φ \varphi φ § 6.3 ). To state these results, we first further decompose φ r e g \varphi_{\mathrm{reg}} φ reg (284) ) into φ r e g = φ + + φ r e m \varphi_{\mathrm{reg}}=\varphi^++\varphi_{\mathrm{rem}} φ reg = φ + + φ rem
( ∂ t − Δ B + S ) φ + = − ( D B j S j ) φ + − ∣ φ + ∣ 2 φ + , (289) (\partial_t-\Delta_{B+S})\varphi^+=-(D_B^jS_j)\varphi^+-|\varphi^+|^2\varphi^+, \tag*{(289)} ( ∂ t − Δ B + S ) φ + = − ( D B j S j ) φ + − ∣ φ + ∣ 2 φ + , (289) φ + ( 0 ) = φ 0 + , \varphi^+(0)=\varphi_0^+, φ + ( 0 ) = φ 0 + , ( ∂ t − Δ B + S ) φ r e m = 2 S j D B j φ l i n − + S j S j φ l i n − − ( D B j S j ) φ r e m − ( ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + ) , (290) (\partial_t-\Delta_{B+S})\varphi_{\mathrm{rem}}=2S_jD_B^j\varphi_{\mathrm{lin}}^-+S_jS^j\varphi_{\mathrm{lin}}^--(D_B^jS_j)\varphi_{\mathrm{rem}}-\left(|\varphi|^2\varphi-|\varphi^+|^2\varphi^+\right), \tag*{(290)} ( ∂ t − Δ B + S ) φ rem = 2 S j D B j φ lin − + S j S j φ lin − − ( D B j S j ) φ rem − ( ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + ) , (290) φ r e m ( 0 ) = 0 \varphi_{\mathrm{rem}}(0)=0 φ rem ( 0 ) = 0
We record the following identity which will be useful in the coming proofs: by the Bochner formula (107) , we have that
1 2 ( ∂ t − Δ ) ( ∣ φ + ∣ 2 ) + ∣ D B + S φ + ∣ 2 = − ⟨ φ + , ( D B j S j ) φ + ⟩ − ∣ φ + ∣ 4 = − ∣ φ + ∣ 4 , (291) \frac{1}{2}(\partial_t-\Delta)(|\varphi^+|^2)+|D_{B+S}\varphi^+|^2=-\langle\varphi^+,(D_B^jS_j)\varphi^+\rangle-|\varphi^+|^4=-|\varphi^+|^4, \tag*{(291)} 2 1 ( ∂ t − Δ ) ( ∣ φ + ∣ 2 ) + ∣ D B + S φ + ∣ 2 = − ⟨ φ + , ( D B j S j ) φ + ⟩ − ∣ φ + ∣ 4 = − ∣ φ + ∣ 4 , (291) where we used that ⟨ u , X u ⟩ = 0 \langle u,Xu\rangle=0 ⟨ u , X u ⟩ = 0 u ∈ V u\in V u ∈ V X ∈ g X\in\mathfrak{g} X ∈ g g \mathfrak{g} g V V V
Our next result gives a lower bound on the decrease of φ + \varphi^+ φ + § 6.3 , this decrease will be used to absorb various bad terms in our estimates. (One of the main bad terms is the potential increase in the L x 2 L_x^2 L x 2 S S S B B B B + S B+S B + S
Proposition 5.17 (Decrease estimate for φ + \varphi^+ φ + T T T γ > 0 \gamma>0 γ > 0 t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] p ∈ [ 2 , r ] p\in[2,r] p ∈ [ 2 , r ] ∥ φ 0 + ∥ L x r ≥ 1 \|\varphi_0^+\|_{L_x^r}\geq1 ∥ φ 0 + ∥ L x r ≥ 1
∥ φ 0 + ∥ L x p γ − ∥ φ + ( t ) ∥ L x p γ ≳ γ ∥ φ 0 + ∥ L x p γ + 2 t . (292) \|\varphi_0^+\|_{L_x^p}^{\gamma}-\|\varphi^+(t)\|_{L_x^p}^{\gamma}\gtrsim_\gamma\|\varphi_0^+\|_{L_x^p}^{\gamma+2}t. \tag*{(292)} ∥ φ 0 + ∥ L x p γ − ∥ φ + ( t ) ∥ L x p γ ≳ γ ∥ φ 0 + ∥ L x p γ + 2 t . (292) Moreover, for any γ ≥ 4 \gamma\geq4 γ ≥ 4
∥ φ 0 + ∥ L x 4 γ − ∥ φ + ( t ) ∥ L x 4 γ ≥ γ ∥ φ + ( t ) ∥ L x 4 γ − 4 ∥ ∣ D B + S φ + ∣ ∣ φ + ∣ ∥ L t , x 2 ( [ 0 , t ] ) 2 . (293) \|\varphi_0^+\|_{L_x^4}^{\gamma}-\|\varphi^+(t)\|_{L_x^4}^{\gamma}\geq\gamma\|\varphi^+(t)\|_{L_x^4}^{\gamma-4}\|\,|D_{B+S}\varphi^+||\varphi^+|\,\|_{L_{t,x}^2([0,t])}^2. \tag*{(293)} ∥ φ 0 + ∥ L x 4 γ − ∥ φ + ( t ) ∥ L x 4 γ ≥ γ ∥ φ + ( t ) ∥ L x 4 γ − 4 ∥ ∣ D B + S φ + ∣∣ φ + ∣ ∥ L t , x 2 ([ 0 , t ]) 2 . (293) One should think of φ + \varphi^+ φ + φ = φ + + φ l i n + φ r e m \varphi=\varphi^++\varphi_{\mathrm{lin}}+\varphi_{\mathrm{rem}} φ = φ + + φ lin + φ rem φ l i n \varphi_{\mathrm{lin}} φ lin φ r e m \varphi_{\mathrm{rem}} φ rem φ r e m \varphi_{\mathrm{rem}} φ rem φ l i n \varphi_{\mathrm{lin}} φ lin
Lemma 5.18 (Smallness of φ r e m \varphi_{\mathrm{rem}} φ rem T ≤ 1 T\leq1 T ≤ 1
c ( T ) ≔ M T ( B , 10 ) ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + T 1 2 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . c(T)\coloneqq M_T(B,10)\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{1}{2}-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right). c ( T ) : = M T ( B , 10 ) ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + T 2 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . There is some constant c 1 ≤ 1 c_1\leq1 c 1 ≤ 1 c ( T ) ≤ c 1 c(T)\leq c_1 c ( T ) ≤ c 1
∥ φ r e m ∥ L t , x ∞ ( [ 0 , T ] ) + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ( [ 0 , T ] ) ≲ c ( T ) 2 T − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ . \|\varphi_{\mathrm{rem}}\|_{L_{t,x}^{\infty}([0,T])}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^{\infty}([0,T])}\lesssim c(T)^2T^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}. ∥ φ rem ∥ L t , x ∞ ([ 0 , T ]) + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ([ 0 , T ]) ≲ c ( T ) 2 T − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ . From the estimate (288) in Proposition 5.15, we see that the L x 2 L_x^2 L x 2 S , D B S S,D_BS S , D B S L t 1 L x 2 L_t^1L_x^2 L t 1 L x 2 B ( φ ⊗ D B + S φ ) \mathbf{B}(\varphi\otimes D_{B+S}\varphi) B ( φ ⊗ D B + S φ ) L t , x 2 L_{t,x}^2 L t , x 2 φ \varphi φ φ + \varphi^+ φ +
Lemma 5.19. There is some constant C C C T > 0 T>0 T > 0 (285) and suppose K ≥ 0 K\geq0 K ≥ 0
∥ φ r e m ∥ L t , x ∞ ( [ 0 , T ] ) + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ( [ 0 , T ] ) ≤ K . (294) \|\varphi_{\mathrm{rem}}\|_{L_{t,x}^{\infty}([0,T])}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^{\infty}([0,T])}\leq K. \tag*{(294)} ∥ φ rem ∥ L t , x ∞ ([ 0 , T ]) + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ([ 0 , T ]) ≤ K . (294) Let
K ′ = C ( 1 + ( ∥ F B ∥ T , 1 / 2 p e r t ) 2 ) ( T − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + K ) ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + T 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . K'=C\left(1+(\|F_B\|_{T,1/2}^{\mathrm{pert}})^2\right)\left(T^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}+K\right)\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right). K ′ = C ( 1 + ( ∥ F B ∥ T , 1/2 pert ) 2 ) ( T − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ + K ) ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + T 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . Then
∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ( [ 0 , T ] ) ≤ ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t 1 L x 2 ( [ 0 , T ] ) + K ′ , \|\mathbf{B}(\varphi\otimes D_{B+S}\varphi)\|_{L_t^1L_x^2([0,T])}\leq\|\mathbf{B}(\varphi^+\otimes D_{B+S}\varphi^+)\|_{L_t^1L_x^2([0,T])}+K', ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ([ 0 , T ]) ≤ ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t 1 L x 2 ([ 0 , T ]) + K ′ , ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , T ] ) ≤ ∥ t 1 / 2 B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , T ] ) + K ′ . \|t^{1/2}\mathbf{B}(\varphi\otimes D_{B+S}\varphi)\|_{L_{t,x}^2([0,T])}\leq\|t^{1/2}\mathbf{B}(\varphi^+\otimes D_{B+S}\varphi^+)\|_{L_{t,x}^2([0,T])}+K'. ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ([ 0 , T ]) ≤ ∥ t 1/2 B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , T ]) + K ′ . The next main result approximates φ + \varphi^{+} φ + φ l i n + \varphi_{\mathrm{lin}}^{+} φ lin +
( ∂ t − Δ B ) φ l i n + = 0 , φ l i n + ( 0 ) = φ 0 + . (295) (\partial_t-\Delta_B)\varphi_{\mathrm{lin}}^{+}=0,\quad\varphi_{\mathrm{lin}}^{+}(0)=\varphi_0^{+}. \tag*{(295)} ( ∂ t − Δ B ) φ lin + = 0 , φ lin + ( 0 ) = φ 0 + . (295) Lemma 5.20. Let C 0 , T C_0,T C 0 , T Proposition 5.15 . Then we have that
∥ φ + − φ l i n + ∥ L t ∞ L x 4 ( [ 0 , T ] ) ≲ G r T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 2 p e r t ) 15 ( T 1 2 − κ 2 ∥ φ 0 + ∥ L x r + T 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ , 2 ) 2 ∥ φ 0 + ∥ L x 4 . \|\varphi^{+}-\varphi_{\mathrm{lin}}^{+}\|_{L_t^\infty L_x^4([0,T])}\lesssim G r_T(C_0F_B)(1+\|F_B\|_{T,2}^{\mathrm{pert}})^{15}\left(T^{\frac12-\frac{\kappa}{2}}\|\varphi_0^{+}\|_{L_x^r}+T^{\frac{1-\kappa}{2}}\|\varphi_0^{-}\|_{\mathcal{C}_{B(0),T}^{-\kappa,2}}\right)^2\|\varphi_0^{+}\|_{L_x^4}. ∥ φ + − φ lin + ∥ L t ∞ L x 4 ([ 0 , T ]) ≲ G r T ( C 0 F B ) ( 1 + ∥ F B ∥ T , 2 pert ) 15 ( T 2 1 − 2 κ ∥ φ 0 + ∥ L x r + T 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ , 2 ) 2 ∥ φ 0 + ∥ L x 4 . We proceed to prove all the main results that we have stated. We start with Proposition 5.15 . For notational convenience, we make the following definition.
Definition 5.21. For t 0 ≥ 0 t_0\geq0 t 0 ≥ 0 ( S , φ ) (S,\varphi) ( S , φ ) [ 0 , t 0 ] [0,t_0] [ 0 , t 0 ]
N S ( t 0 ) ≔ max 0 ≤ j ≤ 2 ∥ t 0 j 2 D B ( j ) S ∥ L t , x ∞ ( [ 0 , t 0 ] ) , (296) N_S(t_0)\coloneqq\max_{0\leq j\leq2}\left\|t_0^{\frac{j}{2}}D_B^{(j)}S\right\|_{L_{t,x}^\infty([0,t_0])}, \tag*{(296)} N S ( t 0 ) : = 0 ≤ j ≤ 2 max t 0 2 j D B ( j ) S L t , x ∞ ([ 0 , t 0 ]) , (296) N φ ( t 0 ; κ ) ≔ max 0 ≤ j ≤ 2 ∥ t 0 j + κ 2 D B ( j ) φ ∥ L t , x ∞ ( [ 0 , t 0 ] ) . N_\varphi(t_0;\kappa)\coloneqq\max_{0\leq j\leq2}\left\|t_0^{\frac{j+\kappa}{2}}D_B^{(j)}\varphi\right\|_{L_{t,x}^\infty([0,t_0])}. N φ ( t 0 ; κ ) : = 0 ≤ j ≤ 2 max t 0 2 j + κ D B ( j ) φ L t , x ∞ ([ 0 , t 0 ]) . We will also use N ψ ( t 0 ; κ ) N_\psi(t_0;\kappa) N ψ ( t 0 ; κ ) ψ \psi ψ ψ = φ r e m \psi=\varphi_{\mathrm{rem}} ψ = φ rem ψ = φ l i n \psi=\varphi_{\mathrm{lin}} ψ = φ lin
By the definition of ∥ ⋅ ∥ C B ( 0 ) , t 0 − κ \|\cdot\|_{\mathcal{C}_{B(0),t_0}^{-\kappa}} ∥ ⋅ ∥ C B ( 0 ) , t 0 − κ (195) , we have that
N φ l i n ( t 0 ; κ ) ≲ ( 1 + ( ∥ F B ∥ t 0 , 1 p e r t ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , t 0 − κ . (297) N_{\varphi_{\mathrm{lin}}}(t_0;\kappa)\lesssim\left(1+\left(\|F_B\|_{t_0,1}^{\mathrm{pert}}\right)^2\right)\|\varphi_0^{-}\|_{\mathcal{C}_{B(0),t_0}^{-\kappa}}. \tag*{(297)} N φ lin ( t 0 ; κ ) ≲ ( 1 + ( ∥ F B ∥ t 0 , 1 pert ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , t 0 − κ . (297) (The heat kernel estimate is used to estimate the second derivatives which appear in Definition 5.21.) It also follows from Definition 5.21 and the above that there is some constant C > 0 C>0 C > 0
N φ ( t 0 ; κ ) ≤ N φ r e g ( t 0 ; κ ) + C ( 1 + ( ∥ F B ∥ t 0 , 1 p e r t ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , t 0 − κ . (298) N_\varphi(t_0;\kappa)\leq N_{\varphi_{\mathrm{reg}}}(t_0;\kappa)+C\left(1+\left(\|F_B\|_{t_0,1}^{\mathrm{pert}}\right)^2\right)\|\varphi_0^{-}\|_{\mathcal{C}_{B(0),t_0}^{-\kappa}}. \tag*{(298)} N φ ( t 0 ; κ ) ≤ N φ reg ( t 0 ; κ ) + C ( 1 + ( ∥ F B ∥ t 0 , 1 pert ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , t 0 − κ . (298) Next, we obtain the following perturbative estimates on S S S Lemma 5.22 ) and φ r e g \varphi_{\mathrm{reg}} φ reg Lemma 5.23 ), which will serve as the main steps towards Proposition 5.15 . In the following, to help track which terms are the main terms and which terms are perturbations, it may help to think of t 0 1 / 2 N S ( t 0 ) ≪ 1 t_0^{1/2}N_S(t_0)\ll1 t 0 1/2 N S ( t 0 ) ≪ 1 t 0 ( 1 − κ ) / 2 N φ ( t 0 ; κ ) ≪ 1 t_0^{(1-\kappa)/2}N_\varphi(t_0;\kappa)\ll1 t 0 ( 1 − κ ) /2 N φ ( t 0 ; κ ) ≪ 1
Lemma 5.22 (Perturbative estimates of S S S Let t 0 ∈ ( 0 , 1 ] t_0\in(0,1] t 0 ∈ ( 0 , 1 ] ( S , φ ) (S,\varphi) ( S , φ ) [ 0 , t 0 ] [0,t_0] [ 0 , t 0 ]
N S ( t 0 ) ≲ M t 0 ( B , 2 ) ( t 0 1 2 − κ 2 N φ ( t 0 ; κ ) 2 ( 1 + t 0 1 2 N S ( t 0 ) ) + t 0 1 2 N S ( t 0 ) 2 + t 0 N S ( t 0 ) 3 ) . N_S(t_0)\lesssim M_{t_0}(B,2)\left(t_0^{\frac12-\frac{\kappa}{2}}N_\varphi(t_0;\kappa)^2(1+t_0^{\frac12}N_S(t_0))+t_0^{\frac12}N_S(t_0)^2+t_0N_S(t_0)^3\right). N S ( t 0 ) ≲ M t 0 ( B , 2 ) ( t 0 2 1 − 2 κ N φ ( t 0 ; κ ) 2 ( 1 + t 0 2 1 N S ( t 0 )) + t 0 2 1 N S ( t 0 ) 2 + t 0 N S ( t 0 ) 3 ) . Proof. Let G = O ( S D B S ) + O ( S 3 ) + B ( φ ⊗ D B + S φ ) G=\mathcal{O}(SD_BS)+\mathcal{O}(S^3)+B(\varphi\otimes D_B+S\varphi) G = O ( S D B S ) + O ( S 3 ) + B ( φ ⊗ D B + S φ ) (281) )
( ∂ t − Δ B − 2 ad ( F B ) ) S = G . (\partial_t-\Delta_B-2\operatorname{ad}(F_B))S=G. ( ∂ t − Δ B − 2 ad ( F B )) S = G . In the following, all space-time norms are implicitly on [ 0 , t 0 ] × T d [0,t_0]\times\mathbb{T}^d [ 0 , t 0 ] × T d Lemma 4.62 (with the k k k 0 , 1 , 2 0,1,2 0 , 1 , 2
N S ( t 0 ) ≲ M t 0 ( B , 2 ) N 2 ( G ) , N_S(t_0)\lesssim M_{t_0}(B,2)N_2(G), N S ( t 0 ) ≲ M t 0 ( B , 2 ) N 2 ( G ) , where
N 2 ( G ) = ∥ G ∥ L t 1 L x ∞ + ∑ j = 0 1 ( ∥ t ( j + 1 ) / 2 D B ( j ) G ∥ L t 2 L x ∞ + sup u ∈ [ 0 , t 0 ] ∥ t ( j + 1 ) / 2 ( u − t ) − 1 / 2 D B ( j ) G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . N_2(G)=\|G\|_{L_t^1L_x^\infty}+\sum_{j=0}^{1}\left(\|t^{(j+1)/2}D_B^{(j)}G\|_{L_t^2L_x^\infty}+\sup_{u\in[0,t_0]}\|t^{(j+1)/2}(u-t)^{-1/2}D_B^{(j)}G\|_{L_t^1L_x^\infty([0,u])}\right). N 2 ( G ) = ∥ G ∥ L t 1 L x ∞ + j = 0 ∑ 1 ( ∥ t ( j + 1 ) /2 D B ( j ) G ∥ L t 2 L x ∞ + u ∈ [ 0 , t 0 ] sup ∥ t ( j + 1 ) /2 ( u − t ) − 1/2 D B ( j ) G ∥ L t 1 L x ∞ ([ 0 , u ]) ) . We proceed to estimate the terms on the right-hand side above. We will just give the argument for the term ∥ G ∥ L t 1 L x ∞ \lVert G\rVert_{L^1_tL^\infty_x} ∥ G ∥ L t 1 L x ∞ D B + S φ = D B φ + S φ D_{B+S}\varphi=D_B\varphi+S\varphi D B + S φ = D B φ + S φ
∥ G ∥ L t 1 L x ∞ ≲ ∥ O ( S D B S ) ∥ L t 1 L x ∞ + ∥ O ( S 3 ) ∥ L t 1 L x ∞ + ∥ D B + S φ ⊗ φ ∥ L t 1 L x ∞ ≲ ∥ t − 1 / 2 ∥ L t 1 N S ( t 0 ) 2 + ∥ 1 ∥ L t 1 N S ( t 0 ) 3 + ∥ D B φ ⊗ φ ∥ L t 1 L x ∞ + ∥ S φ ⊗ φ ∥ L t 1 L x ∞ ≲ t 0 1 / 2 N S ( t 0 ) 2 + t 0 N S ( t 0 ) 3 + t 0 1 2 − κ N φ ( t 0 ; κ ) 2 + t 0 1 − κ N S ( t 0 ) N φ ( t 0 ; κ ) 2 , \begin{aligned}
\lVert G\rVert_{L^1_tL^\infty_x}
&\lesssim\lVert\mathcal{O}(SD_BS)\rVert_{L^1_tL^\infty_x}+\lVert\mathcal{O}(S^3)\rVert_{L^1_tL^\infty_x}+\lVert D_{B+S}\varphi\otimes\varphi\rVert_{L^1_tL^\infty_x}\\
&\lesssim\lVert t^{-1/2}\rVert_{L^1_t}N_S(t_0)^2+\lVert1\rVert_{L^1_t}N_S(t_0)^3+\lVert D_B\varphi\otimes\varphi\rVert_{L^1_tL^\infty_x}+\lVert S\varphi\otimes\varphi\rVert_{L^1_tL^\infty_x}\\
&\lesssim t_0^{1/2}N_S(t_0)^2+t_0N_S(t_0)^3+t_0^{\frac12-\kappa}N_\varphi(t_0;\kappa)^2+t_0^{1-\kappa}N_S(t_0)N_\varphi(t_0;\kappa)^2,
\end{aligned} ∥ G ∥ L t 1 L x ∞ ≲ ∥ O ( S D B S ) ∥ L t 1 L x ∞ + ∥ O ( S 3 ) ∥ L t 1 L x ∞ + ∥ D B + S φ ⊗ φ ∥ L t 1 L x ∞ ≲ ∥ t − 1/2 ∥ L t 1 N S ( t 0 ) 2 + ∥ 1 ∥ L t 1 N S ( t 0 ) 3 + ∥ D B φ ⊗ φ ∥ L t 1 L x ∞ + ∥ S φ ⊗ φ ∥ L t 1 L x ∞ ≲ t 0 1/2 N S ( t 0 ) 2 + t 0 N S ( t 0 ) 3 + t 0 2 1 − κ N φ ( t 0 ; κ ) 2 + t 0 1 − κ N S ( t 0 ) N φ ( t 0 ; κ ) 2 , which is acceptable. The desired result now follows.
Lemma 5.23 (Perturbative estimates of φ r e g \varphi_{\mathrm{reg}} φ reg Let t 0 ∈ ( 0 , 1 ] t_0\in(0,1] t 0 ∈ ( 0 , 1 ] ( S , φ ) (S,\varphi) ( S , φ ) [ 0 , t 0 ] [0,t_0] [ 0 , t 0 ]
N φ r e g ( t 0 ; κ ) ≲ ( 1 + ∥ F B ∥ t 0 , 1 p e r t ) 2 ( t 0 κ 2 − 1 r ∥ φ 0 + ∥ L x r + N φ ( t 0 ; κ ) t 0 1 / 2 N S ( t 0 ) ( 1 + t 0 1 / 2 N S ( t 0 ) ) + t 0 1 − κ N φ ( t 0 ; κ ) 3 ) . N_{\varphi_{\mathrm{reg}}}(t_0;\kappa)\lesssim\left(1+\lVert F_B\rVert_{t_0,1}^{\mathrm{pert}}\right)^2\left(t_0^{\frac{\kappa}{2}-\frac{1}{r}}\lVert\varphi_0^+\rVert_{L^r_x}+N_\varphi(t_0;\kappa)t_0^{1/2}N_S(t_0)\left(1+t_0^{1/2}N_S(t_0)\right)+t_0^{1-\kappa}N_\varphi(t_0;\kappa)^3\right). N φ reg ( t 0 ; κ ) ≲ ( 1 + ∥ F B ∥ t 0 , 1 pert ) 2 ( t 0 2 κ − r 1 ∥ φ 0 + ∥ L x r + N φ ( t 0 ; κ ) t 0 1/2 N S ( t 0 ) ( 1 + t 0 1/2 N S ( t 0 ) ) + t 0 1 − κ N φ ( t 0 ; κ ) 3 ) . Proof. In the following, all space-time norms are implicitly on [ 0 , t 0 ] × T d [0,t_0]\times\mathbb{T}^d [ 0 , t 0 ] × T d (284) in the following form (by using (112) to expand Δ B + S \Delta_{B+S} Δ B + S Δ B \Delta_B Δ B
( ∂ t − Δ B ) φ r e g = 2 S j D B j φ + S j S j φ − ∣ φ ∣ 2 φ = d e f H , φ r e g ( 0 ) = φ 0 + . (299) (\partial_t-\Delta_B)\varphi_{\mathrm{reg}}=2S_jD_B^j\varphi+S^jS_j\varphi-\lvert\varphi\rvert^2\varphi\mathrel{\mathop{=}^{\mathrm{def}}}H,\qquad\varphi_{\mathrm{reg}}(0)=\varphi_0^+. \tag*{(299)} ( ∂ t − Δ B ) φ reg = 2 S j D B j φ + S j S j φ − ∣ φ ∣ 2 φ = def H , φ reg ( 0 ) = φ 0 + . (299) We split φ r e g = φ l i n + + φ ~ \varphi_{\mathrm{reg}}=\varphi_{\mathrm{lin}}^++\widetilde{\varphi} φ reg = φ lin + + φ φ l i n + \varphi_{\mathrm{lin}}^+ φ lin + (295) (i.e. it solves the linear equation with initial data φ 0 + \varphi_0^+ φ 0 + φ ~ \widetilde{\varphi} φ
( ∂ t − Δ B ) φ ~ = H , φ ~ ( 0 ) = 0. (\partial_t-\Delta_B)\widetilde{\varphi}=H,\qquad\widetilde{\varphi}(0)=0. ( ∂ t − Δ B ) φ = H , φ ( 0 ) = 0. Since φ r e g = φ l i n + + φ ~ \varphi_{\mathrm{reg}}=\varphi_{\mathrm{lin}}^++\widetilde{\varphi} φ reg = φ lin + + φ N φ l i n + ( t 0 ; κ ) N_{\varphi_{\mathrm{lin}}^+}(t_0;\kappa) N φ lin + ( t 0 ; κ ) N φ ~ ( t 0 ; κ ) N_{\widetilde{\varphi}}(t_0;\kappa) N φ ( t 0 ; κ ) (194) –(195) , it follows that
N φ l i n + ( t 0 ; κ ) ≲ ( 1 + ∥ F B ∥ t 0 , 1 p e r t ) 2 t 0 κ 2 − 1 r ∥ φ 0 + ∥ L x r , N_{\varphi_{\mathrm{lin}}^+}(t_0;\kappa)\lesssim\left(1+\lVert F_B\rVert_{t_0,1}^{\mathrm{pert}}\right)^2t_0^{\frac{\kappa}{2}-\frac{1}{r}}\lVert\varphi_0^+\rVert_{L^r_x}, N φ lin + ( t 0 ; κ ) ≲ ( 1 + ∥ F B ∥ t 0 , 1 pert ) 2 t 0 2 κ − r 1 ∥ φ 0 + ∥ L x r , which is acceptable. Next, by Lemma 4.62 (applied with k = 0 , 1 , 2 k=0,1,2 k = 0 , 1 , 2 t 1 ∈ ( 0 , t 0 ] t_1\in(0,t_0] t 1 ∈ ( 0 , t 0 ]
max 0 ≤ j ≤ 2 t 1 j / 2 ∥ D B ( j ) φ ~ ( t 1 ) ∥ L x ∞ ≲ ( 1 + ∥ F B ∥ t 0 , 1 p e r t ) 2 N 2 ( H ) , (300) \max_{0\le j\le2}t_1^{j/2}\lVert D_B^{(j)}\widetilde{\varphi}(t_1)\rVert_{L^\infty_x}\lesssim\left(1+\lVert F_B\rVert_{t_0,1}^{\mathrm{pert}}\right)^2N_2(H), \tag*{(300)} 0 ≤ j ≤ 2 max t 1 j /2 ∥ D B ( j ) φ ( t 1 ) ∥ L x ∞ ≲ ( 1 + ∥ F B ∥ t 0 , 1 pert ) 2 N 2 ( H ) , (300) where H H H (299) , and (in the following, all space-time norms are on [ 0 , t 1 ] × T 2 [0,t_1]\times\mathbb{T}^2 [ 0 , t 1 ] × T 2
N 2 ( H ) = ∥ H ∥ L t 1 L x ∞ + ∑ j = 0 1 ( ∥ t ( j + 1 ) / 2 D B ( j ) H ∥ L t 2 L x ∞ + sup u ∈ [ 0 , t 1 ] ∥ t ( j + 1 ) / 2 ( u − t ) − 1 / 2 D B ( j ) H ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . N_2(H)=\lVert H\rVert_{L^1_tL^\infty_x}+\sum_{j=0}^{1}\left(\lVert t^{(j+1)/2}D_B^{(j)}H\rVert_{L^2_tL^\infty_x}+\sup_{u\in[0,t_1]}\lVert t^{(j+1)/2}(u-t)^{-1/2}D_B^{(j)}H\rVert_{L^1_tL^\infty_x([0,u])}\right). N 2 ( H ) = ∥ H ∥ L t 1 L x ∞ + j = 0 ∑ 1 ( ∥ t ( j + 1 ) /2 D B ( j ) H ∥ L t 2 L x ∞ + u ∈ [ 0 , t 1 ] sup ∥ t ( j + 1 ) /2 ( u − t ) − 1/2 D B ( j ) H ∥ L t 1 L x ∞ ([ 0 , u ]) ) . We will focus on estimating the L t 1 L x ∞ L^1_tL^\infty_x L t 1 L x ∞
∥ H ∥ L t 1 L x ∞ ≲ ∥ S j D B j φ ∥ L t 1 L x ∞ + ∥ S j S j φ ∥ L t 1 L x ∞ + ∥ ∣ φ ∣ 2 φ ∥ L t 1 L x ∞ ≲ t 0 1 − κ 2 N S ( t 0 ) N φ ( t 0 ; κ ) + t 0 1 − κ 2 N S ( t 0 ) 2 N φ ( t 0 ; κ ) + t 0 1 − 3 κ 2 N φ ( t 0 ; κ ) 3 . \begin{aligned}
\lVert H\rVert_{L^1_tL^\infty_x}
&\lesssim\lVert S_jD_B^j\varphi\rVert_{L^1_tL^\infty_x}+\lVert S^jS_j\varphi\rVert_{L^1_tL^\infty_x}+\lVert\lvert\varphi\rvert^2\varphi\rVert_{L^1_tL^\infty_x}\\
&\lesssim t_0^{1-\frac{\kappa}{2}}N_S(t_0)N_\varphi(t_0;\kappa)+t_0^{1-\frac{\kappa}{2}}N_S(t_0)^2N_\varphi(t_0;\kappa)+t_0^{1-\frac{3\kappa}{2}}N_\varphi(t_0;\kappa)^3.
\end{aligned} ∥ H ∥ L t 1 L x ∞ ≲ ∥ S j D B j φ ∥ L t 1 L x ∞ + ∥ S j S j φ ∥ L t 1 L x ∞ + ∥∣ φ ∣ 2 φ ∥ L t 1 L x ∞ ≲ t 0 1 − 2 κ N S ( t 0 ) N φ ( t 0 ; κ ) + t 0 1 − 2 κ N S ( t 0 ) 2 N φ ( t 0 ; κ ) + t 0 1 − 2 3 κ N φ ( t 0 ; κ ) 3 . Multiplying both sides of (300) by t 1 κ / 2 t_1^{\kappa/2} t 1 κ /2 t 1 ∈ [ 0 , t 0 ] t_1\in[0,t_0] t 1 ∈ [ 0 , t 0 ]
We are now able to show Proposition 5.15.
Proof of Proposition 5.15. First, we prove (286) and (287) . We focus on deriving an a priori bound, as the local well-posedness on the time interval [ 0 , T ] [0,T] [ 0 , T ] ( S , φ ) (S,\varphi) ( S , φ ) [ 0 , T ] [0,T] [ 0 , T ] N S N_S N S N φ ( ⋅ ; κ ) N_\varphi(\cdot;\kappa) N φ ( ⋅ ; κ ) N φ r e g ( ⋅ ; κ ) N_{\varphi_{\mathrm{reg}}}(\cdot;\kappa) N φ reg ( ⋅ ; κ ) t t t
By combining Lemmas 5.22 and 5.23 with (298) there is some constant C ≥ 1 C\geq1 C ≥ 1 T T T t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
N S ( t ) ≤ C M T ( B , 2 ) ( t 1 2 − κ N φ ( t ; κ ) 2 ( 1 + t 1 2 N S ( t ) ) + t 1 2 N S ( t ) 2 + t N S ( t ) 3 ) , (301) N_S(t)\leq CM_T(B,2)\left(t^{\frac{1}{2}-\kappa}N_\varphi(t;\kappa)^2(1+t^{\frac{1}{2}}N_S(t))+t^{\frac{1}{2}}N_S(t)^2+tN_S(t)^3\right), \tag*{(301)} N S ( t ) ≤ C M T ( B , 2 ) ( t 2 1 − κ N φ ( t ; κ ) 2 ( 1 + t 2 1 N S ( t )) + t 2 1 N S ( t ) 2 + t N S ( t ) 3 ) , (301) N φ r e g ( t ; κ ) ≤ C ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ( t κ 2 − 1 r ∥ φ 0 + ∥ L x r N_{\varphi_{\mathrm{reg}}}(t;\kappa)\leq C(1+\|F_B\|_{T,1}^{\mathrm{pert}})^2\left(t^{\frac{\kappa}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}
\right. N φ reg ( t ; κ ) ≤ C ( 1 + ∥ F B ∥ T , 1 pert ) 2 ( t 2 κ − r 1 ∥ φ 0 + ∥ L x r + ( C ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + N φ r e g ( t ; κ ) ) t 1 2 N S ( t ) ( 1 + t 1 2 N S ( t ) ) + t 1 − κ N φ ( t ; κ ) 3 ) , (302) \left.
+\left(C(1+\|F_B\|_{T,1}^{\mathrm{pert}})^2\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}+N_{\varphi_{\mathrm{reg}}}(t;\kappa)\right)t^{\frac{1}{2}}N_S(t)(1+t^{\frac{1}{2}}N_S(t))+t^{1-\kappa}N_\varphi(t;\kappa)^3\right), \tag*{(302)} + ( C ( 1 + ∥ F B ∥ T , 1 pert ) 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + N φ reg ( t ; κ ) ) t 2 1 N S ( t ) ( 1 + t 2 1 N S ( t )) + t 1 − κ N φ ( t ; κ ) 3 ) , (302) N φ ( t ; κ ) ≤ N φ r e g ( t ; κ ) + C ( 1 + ( ∥ F B ∥ T , 1 p e r t ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , T − κ . N_\varphi(t;\kappa)\leq N_{\varphi_{\mathrm{reg}}}(t;\kappa)+C(1+(\|F_B\|_{T,1}^{\mathrm{pert}})^2)\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}. N φ ( t ; κ ) ≤ N φ reg ( t ; κ ) + C ( 1 + ( ∥ F B ∥ T , 1 pert ) 2 ) ∥ φ 0 − ∥ C B ( 0 ) , T − κ . In the following, let the constant c 1 c_1 c 1 (285) be chosen so that
c 1 : = ( 10 C ) − 100 . (303) c_1\mathrel{\mathop:}= (10C)^{-100}. \tag*{(303)} c 1 : = ( 10 C ) − 100 . (303) Now let I ⊆ [ 0 , T ] I\subseteq[0,T] I ⊆ [ 0 , T ] t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
N S ( t ) ≤ ( 10 C ) 10 M T ( B , 10 ) ( t 1 2 − 2 r ∥ φ 0 + ∥ L x r 2 + t 1 2 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) , (304) N_S(t)\leq(10C)^{10}M_T(B,10)\left(t^{\frac{1}{2}-\frac{2}{r}}\|\varphi_0^+\|_{L_x^r}^2+t^{\frac{1}{2}-\kappa}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}^2\right), \tag*{(304)} N S ( t ) ≤ ( 10 C ) 10 M T ( B , 10 ) ( t 2 1 − r 2 ∥ φ 0 + ∥ L x r 2 + t 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) , (304) N φ r e g ( t ; κ ) ≤ 10 C ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ( t κ 2 − 1 r ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . (305) N_{\varphi_{\mathrm{reg}}}(t;\kappa)\leq10C\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)^2\left(t^{\frac{\kappa}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}\right). \tag*{(305)} N φ reg ( t ; κ ) ≤ 10 C ( 1 + ∥ F B ∥ T , 1 pert ) 2 ( t 2 κ − r 1 ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . (305) Note that 0 ∈ I 0\in I 0 ∈ I I I I c 2 = ( 10 C ) 10 M T ( B , 8 ) − 1 c 1 c_2=(10C)^{10}M_T(B,8)^{-1}c_1 c 2 = ( 10 C ) 10 M T ( B , 8 ) − 1 c 1 t ∈ I t\in I t ∈ I (285) that
( 1 + ∥ F B ∥ T , 1 p e r t ) 2 t 1 2 N S ( t ) ≤ ( 10 C ) 10 M T ( B , 12 ) ( t 1 2 − 1 r ∥ φ 0 + ∥ L x r ) 2 + ( t 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≤ ( 10 C ) 10 M T ( B , 8 ) − 1 c 1 2 ≤ c 2 , (306) \begin{aligned}
(1+\|F_B\|_{T,1}^{\mathrm{pert}})^2t^{\frac{1}{2}}N_S(t)\leq(10C)^{10}M_T(B,12)\left(t^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}\right)^2+\left(t^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}\right)^2 \\
&\leq(10C)^{10}M_T(B,8)^{-1}c_1^2\leq c_2, \tag*{(306)}
\end{aligned} ( 1 + ∥ F B ∥ T , 1 pert ) 2 t 2 1 N S ( t ) ≤ ( 10 C ) 10 M T ( B , 12 ) ( t 2 1 − r 1 ∥ φ 0 + ∥ L x r ) 2 + ( t 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≤ ( 10 C ) 10 M T ( B , 8 ) − 1 c 1 2 ≤ c 2 , (306) t 1 − κ 2 N φ r e g ( t ; κ ) ≤ 10 C M T ( B , 8 ) − 1 c 1 ≤ c 2 , (307) t^{\frac{1-\kappa}{2}}N_{\varphi_{\mathrm{reg}}}(t;\kappa)\leq10CM_T(B,8)^{-1}c_1\leq c_2, \tag*{(307)} t 2 1 − κ N φ reg ( t ; κ ) ≤ 10 C M T ( B , 8 ) − 1 c 1 ≤ c 2 , (307) t 1 − κ 2 N φ ( t ; κ ) ≤ t 1 − κ 2 N φ r e g ( t ; κ ) + C ( 1 + ( ∥ F B ∥ T , 1 p e r t ) 2 ) t 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ c 2 . t^{\frac{1-\kappa}{2}}N_\varphi(t;\kappa)\leq t^{\frac{1-\kappa}{2}}N_{\varphi_{\mathrm{reg}}}(t;\kappa)+C(1+(\|F_B\|_{T,1}^{\mathrm{pert}})^2)t^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}\leq c_2. t 2 1 − κ N φ ( t ; κ ) ≤ t 2 1 − κ N φ reg ( t ; κ ) + C ( 1 + ( ∥ F B ∥ T , 1 pert ) 2 ) t 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ c 2 . Inserting these estimates as well as (304) , (305) back into (301) , we obtain
N S ( t ) ≤ C M T ( B , 2 ) × ( t 1 2 − κ ( 11 C ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ( t κ 2 − 1 r ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ) 2 ( 1 + c 2 ) + c 2 N S ( t ) + c 2 2 N S ( t ) ) ≤ ( 10 C ) 5 M T ( B , 6 ) ( t 1 2 − 2 r ∥ φ 0 + ∥ L x r 2 + t 1 2 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) + 1 2 N S ( t ) , \begin{aligned}
N_S(t)&\leq CM_T(B,2)\\
&\quad\times\left(t^{\frac{1}{2}-\kappa}\left(11C(1+\|F_B\|_{T,1}^{\mathrm{pert}})^2\left(t^{\frac{\kappa}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}\right)\right)^2(1+c_2)+c_2N_S(t)+c_2^2N_S(t)\right)\\
&\leq(10C)^5M_T(B,6)\left(t^{\frac{1}{2}-\frac{2}{r}}\|\varphi_0^+\|_{L_x^r}^2+t^{\frac{1}{2}-\kappa}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}^2\right)+\frac{1}{2}N_S(t),
\end{aligned} N S ( t ) ≤ C M T ( B , 2 ) × ( t 2 1 − κ ( 11 C ( 1 + ∥ F B ∥ T , 1 pert ) 2 ( t 2 κ − r 1 ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ) 2 ( 1 + c 2 ) + c 2 N S ( t ) + c 2 2 N S ( t ) ) ≤ ( 10 C ) 5 M T ( B , 6 ) ( t 2 1 − r 2 ∥ φ 0 + ∥ L x r 2 + t 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) + 2 1 N S ( t ) , where we used the definition of c 2 c_2 c 2 (303) of c 1 c_1 c 1 C C C t ∈ I t\in I t ∈ I
N S ( t ) ≤ 2 ⋅ ( 10 C ) 5 M T ( B , 10 ) ( t 1 2 − 2 r ∥ φ 0 + ∥ L x r 2 + t 1 2 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) , N_S(t)\leq2\cdot(10C)^5M_T(B,10)\left(t^{\frac{1}{2}-\frac{2}{r}}\|\varphi_0^+\|_{L_x^r}^2+t^{\frac{1}{2}-\kappa}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}^2\right), N S ( t ) ≤ 2 ⋅ ( 10 C ) 5 M T ( B , 10 ) ( t 2 1 − r 2 ∥ φ 0 + ∥ L x r 2 + t 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ 2 ) , which is strictly better than (304) . To finish, we estimate N φ r e g N_{\varphi_{\mathrm{reg}}} N φ reg (302) to obtain
N φ r e g ( t ; κ ) ≤ C ( 1 + ∥ F B ∥ T , 1 p e r t ) 2 ( t κ 2 − 1 r ∥ φ 0 + ∥ L x r + ( C c 2 2 ( 1 + c 2 ) + C ( 1 + ∥ F B ∥ T , 1 p e r t , 2 c 2 2 ) ∥ φ 0 ∥ C B ( 0 ) , T − r ) + 1 2 N φ r e g ( t ; κ ) , \begin{aligned}
N_{\varphi_{\mathrm{reg}}}(t;\kappa)\leq C(1+\|F_B\|_{T,1}^{\mathrm{pert}})^2(t^{\frac{\kappa}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r} \\
&+ \left(Cc_2^2(1+c_2)+C\left(1+\lVert F_B\rVert_{T,1}^{\mathrm{pert},2}c_2^2\right)\lVert\varphi_0\rVert_{C^{-r}_{B(0),T}}\right)+\frac{1}{2}\mathcal{N}_{\varphi_{\mathrm{reg}}}(t;\kappa),
\end{aligned} N φ reg ( t ; κ ) ≤ C ( 1 + ∥ F B ∥ T , 1 pert ) 2 ( t 2 κ − r 1 ∥ φ 0 + ∥ L x r + ( C c 2 2 ( 1 + c 2 ) + C ( 1 + ∥ F B ∥ T , 1 pert , 2 c 2 2 ) ∥ φ 0 ∥ C B ( 0 ) , T − r ) + 2 1 N φ reg ( t ; κ ) , from which we further obtain
N φ r e g ( t ; κ ) ≤ 2 C ( 1 + ∥ F B ∥ T , 1 p e r t , 2 ) ( t κ 2 − 1 r ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − r ) . \mathcal{N}_{\varphi_{\mathrm{reg}}}(t;\kappa)\le2C\left(1+\lVert F_B\rVert_{T,1}^{\mathrm{pert},2}\right)\left(t^{\frac{\kappa}{2}-\frac{1}{r}}\lVert\varphi_0\rVert_{L_x^r}+\lVert\varphi_0\rVert_{C^{-r}_{B(0),T}}\right). N φ reg ( t ; κ ) ≤ 2 C ( 1 + ∥ F B ∥ T , 1 pert , 2 ) ( t 2 κ − r 1 ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − r ) . which is strictly better than (305) . Combining our previous estimates (and using the fact that N N N I I I [ 0 , T ] [0,T] [ 0 , T ] I I I I = [ 0 , T ] I=[0,T] I = [ 0 , T ] (298) ).
Next, we prove the L x 2 L_x^2 L x 2 (288) . By Corollary 4.24 (applied with K ≡ 1 K\equiv1 K ≡ 1
∥ S ∥ L t ∞ L x 2 + ∥ D B S ∥ L t , x 2 (308) \lVert S\rVert_{L_t^\infty L_x^2}+\lVert D_BS\rVert_{L_{t,x}^2} \tag*{(308)} ∥ S ∥ L t ∞ L x 2 + ∥ D B S ∥ L t , x 2 (308) ≲ G r T ( C 0 F B ) ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 + ∥ O ( S D B S ) ∥ L t 1 L x 2 + ∥ O ( S 3 ) ∥ L t 1 L x 2 ) \lesssim\mathrm{Gr}_T(C_0F_B)\left(\lVert B(\varphi\otimes D_B+S\varphi)\rVert_{L_t^1L_x^2}+\lVert O(SD_BS)\rVert_{L_t^1L_x^2}+\lVert O(S^3)\rVert_{L_t^1L_x^2}\right) ≲ Gr T ( C 0 F B ) ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 + ∥ O ( S D B S ) ∥ L t 1 L x 2 + ∥ O ( S 3 ) ∥ L t 1 L x 2 ) ≲ G r T ( C 0 F B ) ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 + ( T 1 / 2 N S ( T ) + T N S ( T ) 2 ) ∥ S ∥ L t ∞ L x 2 ) . \lesssim\mathrm{Gr}_T(C_0F_B)\left(\lVert B(\varphi\otimes D_B+S\varphi)\rVert_{L_t^1L_x^2}+\left(T^{1/2}N_S(T)+TN_S(T)^2\right)\lVert S\rVert_{L_t^\infty L_x^2}\right). ≲ Gr T ( C 0 F B ) ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 + ( T 1/2 N S ( T ) + T N S ( T ) 2 ) ∥ S ∥ L t ∞ L x 2 ) . Upon possibly taking the constant c 1 c_1 c 1 (285) on T T T (306) ) the second term on the right-hand side back into the left-hand side, and thus we obtain
∥ S ∥ L t ∞ L x 2 + ∥ D B S ∥ L t , x 2 ≲ G r T ( C 0 F B ) ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 , (309) \lVert S\rVert_{L_t^\infty L_x^2}+\lVert D_BS\rVert_{L_{t,x}^2}\lesssim\mathrm{Gr}_T(C_0F_B)\lVert B(\varphi\otimes D_B+S\varphi)\rVert_{L_t^1L_x^2}, \tag*{(309)} ∥ S ∥ L t ∞ L x 2 + ∥ D B S ∥ L t , x 2 ≲ Gr T ( C 0 F B ) ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 , (309) which gives an acceptable estimate for the first term on the left-hand side of (288) . To complete the proof of (288) , we proceed to estimate D B S D_BS D B S
( ∂ t − Δ B ) D B S = O ( ad ( F B ) D B S ) + D B H , (\partial_t-\Delta_B)D_BS=O(\operatorname{ad}(F_B)D_BS)+D_BH, ( ∂ t − Δ B ) D B S = O ( ad ( F B ) D B S ) + D B H , where H = 2 ad ( F B ) S + O ( S D B S ) + O ( S 3 ) + B ( φ ⊗ D B + S φ ) H=2\operatorname{ad}(F_B)S+O(SD_BS)+O(S^3)+B(\varphi\otimes D_B+S\varphi) H = 2 ad ( F B ) S + O ( S D B S ) + O ( S 3 ) + B ( φ ⊗ D B + S φ ) (186) (with K ≡ 1 K\equiv1 K ≡ 1 α = 1 \alpha=1 α = 1 G = 0 G=0 G = 0
∥ t 1 / 2 D B S ∥ L t ∞ L x 2 + ∥ t 1 / 2 D B ( 2 ) S ∥ L t , x 2 ≲ ( 1 + ∥ F B ∥ T , 0 p e r t ) ∥ D B S ∥ L t , x 2 + ∥ t 1 / 2 H ∥ L t , x 2 . \lVert t^{1/2}D_BS\rVert_{L_t^\infty L_x^2}+\lVert t^{1/2}D_B^{(2)}S\rVert_{L_{t,x}^2}
\lesssim\left(1+\lVert F_B\rVert_{T,0}^{\mathrm{pert}}\right)\lVert D_BS\rVert_{L_{t,x}^2}+\lVert t^{1/2}H\rVert_{L_{t,x}^2}. ∥ t 1/2 D B S ∥ L t ∞ L x 2 + ∥ t 1/2 D B ( 2 ) S ∥ L t , x 2 ≲ ( 1 + ∥ F B ∥ T , 0 pert ) ∥ D B S ∥ L t , x 2 + ∥ t 1/2 H ∥ L t , x 2 . The first term on the right-hand side may be directly estimated by (309) . For the second term, we estimate
∥ t 1 / 2 H ∥ L t , x 2 ≲ ( ∥ F B ∥ T , 0 p e r t + T 1 / 2 N S ( T ) ( 1 + T 1 / 2 N S ( T ) ) ) ∥ S ∥ L t ∞ L x 2 + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 . \lVert t^{1/2}H\rVert_{L_{t,x}^2}\lesssim\left(\lVert F_B\rVert_{T,0}^{\mathrm{pert}}+T^{1/2}N_S(T)\left(1+T^{1/2}N_S(T)\right)\right)\lVert S\rVert_{L_t^\infty L_x^2}+\lVert t^{1/2}B(\varphi\otimes D_B+S\varphi)\rVert_{L_{t,x}^2}. ∥ t 1/2 H ∥ L t , x 2 ≲ ( ∥ F B ∥ T , 0 pert + T 1/2 N S ( T ) ( 1 + T 1/2 N S ( T ) ) ) ∥ S ∥ L t ∞ L x 2 + ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 . To finish, we combine (306) which gives that T 1 / 2 N S ( T ) ≪ 1 T^{1/2}N_S(T)\ll1 T 1/2 N S ( T ) ≪ 1 (309) . The desired estimate (288) now follows.
Next, we show Proposition 5.17.
Proof of Proposition 5.17. By (291) and the Leibniz rule, it follows that for some constant c > 0 c>0 c > 0
1 p ∂ t ∥ φ + ∥ L x p p ≤ − c ∥ φ + ∥ L x p + 2 p + 2 . \frac{1}{p}\partial_t\lVert\varphi^+\rVert_{L_x^p}^p\le-c\lVert\varphi^+\rVert_{L_x^{p+2}}^{p+2}. p 1 ∂ t ∥ φ + ∥ L x p p ≤ − c ∥ φ + ∥ L x p + 2 p + 2 . In particular, the L x p L_x^p L x p φ + \varphi^+ φ + t 1 ∈ [ 0 , T ] t_1\in[0,T] t 1 ∈ [ 0 , T ]
∥ φ + ( t 1 ) ∥ L x 2 ≤ 1 2 ∥ φ 0 + ∥ L x p , \lVert\varphi^+(t_1)\rVert_{L_x^2}\le\frac{1}{2}\lVert\varphi_0^+\rVert_{L_x^p}, ∥ φ + ( t 1 ) ∥ L x 2 ≤ 2 1 ∥ φ 0 + ∥ L x p , then the decrease
∥ φ 0 + ∥ L x p p − ∥ φ + ( t 1 ) ∥ L x p p ≳ ∥ φ 0 + ∥ L x p p + 2 ≳ ∥ φ 0 + ∥ L x p p + 2 t 1 , \lVert\varphi_0^+\rVert_{L_x^p}^p-\lVert\varphi^+(t_1)\rVert_{L_x^p}^p\gtrsim\lVert\varphi_0^+\rVert_{L_x^p}^{p+2}\gtrsim\lVert\varphi_0^+\rVert_{L_x^p}^{p+2}t_1, ∥ φ 0 + ∥ L x p p − ∥ φ + ( t 1 ) ∥ L x p p ≳ ∥ φ 0 + ∥ L x p p + 2 ≳ ∥ φ 0 + ∥ L x p p + 2 t 1 , where the final inequality follows because t 1 ≤ T t_1 \le T t 1 ≤ T T ≤ ∥ φ 0 + ∥ L x r − 2 ≤ ∥ φ 0 + ∥ L x p − 2 T \le\|\varphi_0^+\|_{L_x^r}^{-2} \le\|\varphi_0^+\|_{L_x^p}^{-2} T ≤ ∥ φ 0 + ∥ L x r − 2 ≤ ∥ φ 0 + ∥ L x p − 2 (285) in Proposition 5.15 and the assumption that p ≤ r p \le r p ≤ r
∥ φ + ( t 1 ) ∥ L x p p ≥ 1 2 ∥ φ 0 + ∥ L x p p . (310) \|\varphi^+(t_1)\|_{L_x^p}^p \ge\frac{1}{2}\|\varphi_0^+\|_{L_x^p}^p. \tag*{(310)} ∥ φ + ( t 1 ) ∥ L x p p ≥ 2 1 ∥ φ 0 + ∥ L x p p . (310) Assume that this is the case. By the chain rule, we may obtain
1 γ ∂ t ∥ φ + ∥ L x γ γ ≲ − ∥ φ + ∥ L x γ + 2 γ + 2 . \frac{1}{\gamma}\partial_t\|\varphi^+\|_{L_x^\gamma}^\gamma\lesssim-\|\varphi^+\|_{L_x^{\gamma+2}}^{\gamma+2}. γ 1 ∂ t ∥ φ + ∥ L x γ γ ≲ − ∥ φ + ∥ L x γ + 2 γ + 2 . Integrating this between t ∈ [ 0 , t 1 ] t \in[0,t_1] t ∈ [ 0 , t 1 ] L x p L_x^p L x p φ + \varphi^+ φ + (310) , we obtain
∥ φ 0 + ∥ L x p γ − ∥ φ + ( t 1 ) ∥ L x p γ ≳ ∥ φ + ( t 1 ) ∥ L x p γ + 2 t 1 ≳ ∥ φ 0 + ∥ L x p γ + 2 t 1 , \|\varphi_0^+\|_{L_x^p}^\gamma-\|\varphi^+(t_1)\|_{L_x^p}^\gamma\gtrsim\|\varphi^+(t_1)\|_{L_x^p}^{\gamma+2}t_1 \gtrsim\|\varphi_0^+\|_{L_x^p}^{\gamma+2}t_1, ∥ φ 0 + ∥ L x p γ − ∥ φ + ( t 1 ) ∥ L x p γ ≳ ∥ φ + ( t 1 ) ∥ L x p γ + 2 t 1 ≳ ∥ φ 0 + ∥ L x p γ + 2 t 1 , as desired.
For (293) , we use that ∥ φ + ( t ) ∥ L x 4 \|\varphi^+(t)\|_{L_x^4} ∥ φ + ( t ) ∥ L x 4 t t t
1 γ ∂ t ∥ φ + ∥ L x γ γ = 1 4 ∥ φ + ∥ L x 4 γ − 4 ( ∂ t ∥ φ + ∥ L x 4 4 ) ≤ − ∥ φ + ∥ L x 4 γ − 4 ∥ D B + S φ + ∥ L x 2 2 . \frac{1}{\gamma}\partial_t\|\varphi^+\|_{L_x^\gamma}^\gamma=\frac{1}{4}\|\varphi^+\|_{L_x^4}^{\gamma-4}\left(\partial_t\|\varphi^+\|_{L_x^4}^4\right)\le-\|\varphi^+\|_{L_x^4}^{\gamma-4}\|D_{B+S}\varphi^+\|_{L_x^2}^2. γ 1 ∂ t ∥ φ + ∥ L x γ γ = 4 1 ∥ φ + ∥ L x 4 γ − 4 ( ∂ t ∥ φ + ∥ L x 4 4 ) ≤ − ∥ φ + ∥ L x 4 γ − 4 ∥ D B + S φ + ∥ L x 2 2 . The bound then follows by integrating in time.
Next, we proceed to prove Lemma 5.18. Towards this end, we first need the following additional estimates on φ + \varphi^+ φ +
Lemma 5.24 (A priori estimates of φ + \varphi^+ φ + t 0 ∈ ( 0 , 1 ] t_0 \in(0,1] t 0 ∈ ( 0 , 1 ] ( S , φ + ) (S,\varphi^+) ( S , φ + ) [ 0 , t 0 ] [0,t_0] [ 0 , t 0 ] 1 ≤ p ≤ ∞ 1 \le p \le\infty 1 ≤ p ≤ ∞
∥ φ + ∥ L t ∞ L x p ( [ 0 , t 0 ] ) ≤ ∥ φ 0 + ∥ L x p , \|\varphi^+\|_{L_t^\infty L_x^p([0,t_0])}\le\|\varphi_0^+\|_{L_x^p}, ∥ φ + ∥ L t ∞ L x p ([ 0 , t 0 ]) ≤ ∥ φ 0 + ∥ L x p , as well as the estimates
∥ t 1 / r φ + ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ ∥ φ 0 + ∥ L x r , ∥ D B + S φ + ∥ L t , x 2 ( [ 0 , t 0 ] ) ≲ ∥ φ 0 + ∥ L x 2 . \|t^{1/r}\varphi^+\|_{L_{t,x}^\infty([0,t_0])}\lesssim\|\varphi_0^+\|_{L_x^r},\qquad\|D_{B+S}\varphi^+\|_{L_{t,x}^2([0,t_0])}\lesssim\|\varphi_0^+\|_{L_x^2}. ∥ t 1/ r φ + ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ ∥ φ 0 + ∥ L x r , ∥ D B + S φ + ∥ L t , x 2 ([ 0 , t 0 ]) ≲ ∥ φ 0 + ∥ L x 2 . Proof. From (291) and the Leibniz rule, it follows that ∣ φ + ∣ |\varphi^+| ∣ φ + ∣ ( ∂ t − Δ ) ∣ φ + ∣ ≤ 0 (\partial_t-\Delta)|\varphi^+|\le0 ( ∂ t − Δ ) ∣ φ + ∣ ≤ 0 (291) in space and time.
Proof of Lemma 5.18. We may assume that c 1 c_1 c 1 c 1 c_1 c 1 c ( T ) ≤ c 1 c(T)\le c_1 c ( T ) ≤ c 1 (285) is satisfied. (We will take c 1 c_1 c 1 t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
t 1 / 2 N S ( t ) ≲ c ( T ) 2 ( 1 + ∥ F B ∥ T , 1 p e r t ) − 10 . (311) t^{1/2}N_S(t)\lesssim c(T)^2(1+\|F_B\|_{T,1}^{\mathrm{pert}})^{-10}. \tag*{(311)} t 1/2 N S ( t ) ≲ c ( T ) 2 ( 1 + ∥ F B ∥ T , 1 pert ) − 10 . (311) Fix t 1 ∈ [ 0 , T ] t_1\in[0,T] t 1 ∈ [ 0 , T ] [ 0 , t 1 ] × T 2 [0,t_1]\times\mathbb{T}^2 [ 0 , t 1 ] × T 2 (290) in the form:
( ∂ t − Δ B ) φ r e m = 2 S j D B j ( φ l i n + φ r e m ) + S j S j ( φ l i n + φ r e m ) − ( ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + ) : = G , (\partial_t-\Delta_B)\varphi_{\mathrm{rem}}=2S_jD_B^j(\varphi_{\mathrm{lin}}+\varphi_{\mathrm{rem}})+S_jS^j(\varphi_{\mathrm{lin}}+\varphi_{\mathrm{rem}})-(|\varphi|^2\varphi-|\varphi^+|^2\varphi^+)\mathrel{\mathop:}=G, ( ∂ t − Δ B ) φ rem = 2 S j D B j ( φ lin + φ rem ) + S j S j ( φ lin + φ rem ) − ( ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + ) : = G , φ r e m ( 0 ) = 0. \varphi_{\mathrm{rem}}(0)=0. φ rem ( 0 ) = 0. By Lemma 4.62 (with the k k k 0 , 1 0,1 0 , 1
∥ φ r e m ∥ L t , x ∞ + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ (312) \|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty} \tag*{(312)} ∥ φ rem ∥ L t , x ∞ + ∥ t 1/2 D B φ rem ∥ L t , x ∞ (312) ≲ ( 1 + ∥ F B ∥ T , 1 p e r t ) ( ∥ G ∥ L t 1 L x ∞ + ∥ t 1 / 2 G ∥ L t 2 L x ∞ + sup u ∈ [ 0 , t 1 ] ∥ t 1 / 2 ( u − t ) − 1 / 2 G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . \lesssim\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)\left(\|G\|_{L_t^1L_x^\infty}+\|t^{1/2}G\|_{L_t^2L_x^\infty}+\sup_{u\in[0,t_1]}\|t^{1/2}(u-t)^{-1/2}G\|_{L_t^1L_x^\infty([0,u])}\right). ≲ ( 1 + ∥ F B ∥ T , 1 pert ) ( ∥ G ∥ L t 1 L x ∞ + ∥ t 1/2 G ∥ L t 2 L x ∞ + u ∈ [ 0 , t 1 ] sup ∥ t 1/2 ( u − t ) − 1/2 G ∥ L t 1 L x ∞ ([ 0 , u ]) ) . We proceed to estimate the L t 1 L x ∞ L_t^1L_x^\infty L t 1 L x ∞ G G G (311) and the definition of the covariant Hölder norm (Definition 5.4 5.4 5.4
∥ S j D B j φ ˉ l i n ∥ L t 1 L x ∞ ≲ T 1 − κ 2 N S ( T ) ∥ φ 0 ∥ C B ( 0 ) , T − κ ≲ ( 1 + ∥ F B ∥ T , 1 p e r t ) − 10 c ( T ) 2 T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ , \|S_jD_B^j\bar{\varphi}_{\mathrm{lin}}\|_{L_t^1L_x^\infty}\lesssim T^{1-\frac{\kappa}{2}}N_S(T)\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}\lesssim\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)^{-10}c(T)^2T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}, ∥ S j D B j φ ˉ lin ∥ L t 1 L x ∞ ≲ T 1 − 2 κ N S ( T ) ∥ φ 0 ∥ C B ( 0 ) , T − κ ≲ ( 1 + ∥ F B ∥ T , 1 pert ) − 10 c ( T ) 2 T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ , ∥ S j S j φ l i n ∥ L t 1 L x ∞ ≲ T 1 − κ 2 N S ( T ) 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ≲ ( 1 + ∥ F B ∥ T , 1 p e r t ) − 10 c ( T ) 2 T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ . \|S_jS^j\varphi_{\mathrm{lin}}\|_{L_t^1L_x^\infty}\lesssim T^{1-\frac{\kappa}{2}}N_S(T)^2\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}\lesssim\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)^{-10}c(T)^2T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}. ∥ S j S j φ lin ∥ L t 1 L x ∞ ≲ T 1 − 2 κ N S ( T ) 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ≲ ( 1 + ∥ F B ∥ T , 1 pert ) − 10 c ( T ) 2 T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ . where in the final inequality we used that c ( T ) ≤ c 1 ≤ 1 c(T)\leq c_1\leq1 c ( T ) ≤ c 1 ≤ 1 c ( T ) 4 ≤ c ( T ) 2 c(T)^4\leq c(T)^2 c ( T ) 4 ≤ c ( T ) 2
∥ S j D B j φ r e m ∥ L t 1 L x ∞ + ∥ S j S j φ r e m ∥ L t 1 L x ∞ ≲ ( T 1 / 2 N S ( T ) + ( T 1 / 2 N S ( T ) ) 2 ) ( ∥ φ r e m ∥ L t , x ∞ + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ) ≤ 1 10 ( 1 + ∥ F B ∥ T , 1 p e r t ) − 10 ( ∥ φ r e m ∥ L t , x ∞ + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ) , \begin{aligned}
\|S_jD_B^j\varphi_{\mathrm{rem}}\|_{L_t^1L_x^\infty}+\|S_jS^j\varphi_{\mathrm{rem}}\|_{L_t^1L_x^\infty}
\lesssim\left(T^{1/2}N_S(T)+(T^{1/2}N_S(T))^2\right)\left(\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}\right) \\
&\leq\frac{1}{10}\left(1+\|F_B\|_{T,1}^{\mathrm{pert}}\right)^{-10}\left(\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}\right),
\end{aligned} ∥ S j D B j φ rem ∥ L t 1 L x ∞ + ∥ S j S j φ rem ∥ L t 1 L x ∞ ≲ ( T 1/2 N S ( T ) + ( T 1/2 N S ( T ) ) 2 ) ( ∥ φ rem ∥ L t , x ∞ + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ) ≤ 10 1 ( 1 + ∥ F B ∥ T , 1 pert ) − 10 ( ∥ φ rem ∥ L t , x ∞ + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ) , where the final inequality follows by (311) and the fact that c ( T ) ≤ c 1 c(T)\leq c_1 c ( T ) ≤ c 1 c 1 ≪ 1 c_1\ll1 c 1 ≪ 1 G G G
∥ ∣ φ r e m ∣ 2 φ r e m ∥ L t 1 L x ∞ ≤ t 1 ∥ φ r e m ∥ L t , x ∞ 3 , \||\varphi_{\mathrm{rem}}|^2\varphi_{\mathrm{rem}}\|_{L_t^1L_x^\infty}\leq t_1\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}^3, ∥∣ φ rem ∣ 2 φ rem ∥ L t 1 L x ∞ ≤ t 1 ∥ φ rem ∥ L t , x ∞ 3 , and then we expand the polynomial around this cubic term:
∣ ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + − ∣ φ r e m ∣ 2 φ r e m ∣ ≲ ( ∣ φ + ∣ 2 ∣ φ l i n ∣ + ∣ φ + ∣ ∣ φ l i n ∣ 2 + ∣ φ l i n ∣ 3 ) + ( ∣ φ + ∣ 2 + ∣ φ l i n ∣ 2 ) ∣ φ r e m ∣ + ( ∣ φ + ∣ + ∣ φ l i n ∣ ) ∣ φ r e m ∣ 2 , \left||\varphi|^2\varphi-|\varphi_+|^2\varphi_+-|\varphi_{\mathrm{rem}}|^2\varphi_{\mathrm{rem}}\right|
\lesssim\left(|\varphi_+|^2|\varphi_{\mathrm{lin}}|+|\varphi_+||\varphi_{\mathrm{lin}}|^2+|\varphi_{\mathrm{lin}}|^3\right)
+\left(|\varphi_+|^2+|\varphi_{\mathrm{lin}}|^2\right)|\varphi_{\mathrm{rem}}|
+\left(|\varphi_+|+|\varphi_{\mathrm{lin}}|\right)|\varphi_{\mathrm{rem}}|^2, ∣ φ ∣ 2 φ − ∣ φ + ∣ 2 φ + − ∣ φ rem ∣ 2 φ rem ≲ ( ∣ φ + ∣ 2 ∣ φ lin ∣ + ∣ φ + ∣∣ φ lin ∣ 2 + ∣ φ lin ∣ 3 ) + ( ∣ φ + ∣ 2 + ∣ φ lin ∣ 2 ) ∣ φ rem ∣ + ( ∣ φ + ∣ + ∣ φ lin ∣ ) ∣ φ rem ∣ 2 , where the terms are grouped by the number of appearances of φ r e m \varphi_{\mathrm{rem}} φ rem
A = T ( ( T − 1 r ∥ φ 0 ∥ L x r ) 2 T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ + T − 1 r ∥ φ 0 ∥ L x r ( T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 + ( T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 3 ) . (313) A=T\left((T^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r})^2T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}+T^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r}(T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}})^2+(T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}})^3\right). \tag*{(313)} A = T ( ( T − r 1 ∥ φ 0 ∥ L x r ) 2 T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ + T − r 1 ∥ φ 0 ∥ L x r ( T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 + ( T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 3 ) . (313) We may estimate (recall Lemma 5.24 5.24 5.24
∥ ( ∣ φ + ∣ 2 ∣ φ l i n ∣ + ∣ φ + ∣ ∣ φ l i n ∣ 2 + ∣ φ l i n ∣ 3 ) ∥ L t 1 L x ∞ ≲ t 1 ( ( t 1 − 1 r ∥ φ 0 ∥ L x r ) 2 t 1 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ + t 1 − 1 r ∥ φ 0 ∥ L x r ( t 1 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 + ( t 1 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 3 ) ≤ A . \left\|\left(|\varphi_+|^2|\varphi_{\mathrm{lin}}|+|\varphi_+||\varphi_{\mathrm{lin}}|^2+|\varphi_{\mathrm{lin}}|^3\right)\right\|_{L_t^1L_x^\infty}
\lesssim t_1\left((t_1^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r})^2t_1^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}+t_1^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r}(t_1^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}})^2+(t_1^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}})^3\right)\leq A. ( ∣ φ + ∣ 2 ∣ φ lin ∣ + ∣ φ + ∣∣ φ lin ∣ 2 + ∣ φ lin ∣ 3 ) L t 1 L x ∞ ≲ t 1 ( ( t 1 − r 1 ∥ φ 0 ∥ L x r ) 2 t 1 − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ + t 1 − r 1 ∥ φ 0 ∥ L x r ( t 1 − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 + ( t 1 − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 3 ) ≤ A . Similarly, we may estimate
∥ ( ∣ φ + ∣ 2 + ∣ φ l i n ∣ 2 ) ∣ φ r e m ∣ ∥ L t 1 L x ∞ ≲ t 1 ( ( t 1 − 1 r ∥ φ 0 ∥ L x r ) 2 + ( t 1 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 ) ∥ φ r e m ∥ L t , x ∞ , \left\|\left(|\varphi_+|^2+|\varphi_{\mathrm{lin}}|^2\right)|\varphi_{\mathrm{rem}}|\right\|_{L_t^1L_x^\infty}
\lesssim t_1\left((t_1^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r})^2+(t_1^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}})^2\right)\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}, ( ∣ φ + ∣ 2 + ∣ φ lin ∣ 2 ) ∣ φ rem ∣ L t 1 L x ∞ ≲ t 1 ( ( t 1 − r 1 ∥ φ 0 ∥ L x r ) 2 + ( t 1 − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) 2 ) ∥ φ rem ∥ L t , x ∞ , ∥ ( ∣ φ + ∣ + ∣ φ l i n ∣ ) ∣ φ r e m ∣ 2 ∥ L t 1 L x ∞ ≲ t 1 ( t 1 − 1 r ∥ φ 0 ∥ L x r + t 1 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ ) ∥ φ r e m ∥ L t , x ∞ 2 . \left\|\left(|\varphi_+|+|\varphi_{\mathrm{lin}}|\right)|\varphi_{\mathrm{rem}}|^2\right\|_{L_t^1L_x^\infty}
\lesssim t_1\left(t_1^{-\frac{1}{r}}\|\varphi_0\|_{L_x^r}+t_1^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}}\right)\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty}^2. ( ∣ φ + ∣ + ∣ φ lin ∣ ) ∣ φ rem ∣ 2 L t 1 L x ∞ ≲ t 1 ( t 1 − r 1 ∥ φ 0 ∥ L x r + t 1 − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ ) ∥ φ rem ∥ L t , x ∞ 2 . Combining, we have estimated the L t 1 L x ∞ L_t^1L_x^\infty L t 1 L x ∞ G G G G G G (312) may be estimated by the same quantities by similar arguments. There is therefore some constant C C C t 1 ∈ [ 0 , T ] t_1\in[0,T] t 1 ∈ [ 0 , T ]
∥ φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) ≤ 1 10 ( ∥ φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) ) + C c ( T ) 2 T − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ + C ( 1 + ∥ F B ∥ T , 1 p e r t ) ( A + t 1 ( ( t 1 − 1 r ∥ φ 0 + ∥ L x r ) 2 + ( t 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) 2 ) ∥ φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) + t 1 ( t 1 − 1 r ∥ φ 0 + ∥ L x r + t 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ∥ φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) 2 + t 1 ∥ φ r e m ∥ L t , x ∞ ( [ 0 , t 1 ] ) 3 ) . (314) \begin{aligned}
\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty([0,t_1])}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty([0,t_1])} \\
&\leq\frac{1}{10}\left(\|\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty([0,t_1])}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L_{t,x}^\infty([0,t_1])}\right)+Cc(T)^2T^{-\frac{\kappa}{2}}\|\varphi_0\|_{\mathcal C_{B(0),T}^{-\kappa}} \\
&+ C(1+\|F_B\|^{\mathrm{pert}}_{T,1})(A+t_1((t_1^{-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r})^2+(t_1^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}})^2)\|\varphi_{\mathrm{rem}}\|_{L^\infty_{t,x}([0,t_1])} \\
&+t_1(t_1^{-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+t_1^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}})\|\varphi_{\mathrm{rem}}\|^2_{L^\infty_{t,x}([0,t_1])}+t_1\|\varphi_{\mathrm{rem}}\|^3_{L^\infty_{t,x}([0,t_1])}). \tag*{(314)}
\end{aligned} ∥ φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) ≤ 10 1 ( ∥ φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) ) + C c ( T ) 2 T − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ + C ( 1 + ∥ F B ∥ T , 1 pert ) ( A + t 1 (( t 1 − r 1 ∥ φ 0 + ∥ L x r ) 2 + ( t 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) 2 ) ∥ φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) + t 1 ( t 1 − r 1 ∥ φ 0 + ∥ L x r + t 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ∥ φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) 2 + t 1 ∥ φ rem ∥ L t , x ∞ ([ 0 , t 1 ]) 3 ) . (314) To finish, we may proceed by a continuity argument similar to that in the proof of Proposition 5.15 , to obtain that for all t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
∥ φ r e m ∥ L t , x ∞ ( [ 0 , t ] ) + ∥ t 1 / 2 D B φ r e m ∥ L t , x ∞ ( [ 0 , t ] ) ≤ 10 C ( c ( T ) 2 T − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ( 1 + ∥ F B ∥ T , 1 p e r t A ) ) . \|\varphi_{\mathrm{rem}}\|_{L^\infty_{t,x}([0,t])}+\|t^{1/2}D_B\varphi_{\mathrm{rem}}\|_{L^\infty_{t,x}([0,t])}\leq10C\left(c(T)2T^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}+(1+\|F_B\|^{\mathrm{pert}}_{T,1}A)\right). ∥ φ rem ∥ L t , x ∞ ([ 0 , t ]) + ∥ t 1/2 D B φ rem ∥ L t , x ∞ ([ 0 , t ]) ≤ 10 C ( c ( T ) 2 T − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ( 1 + ∥ F B ∥ T , 1 pert A ) ) . Finally, we simply note that from the definition of A A A (313) ) and c ( T ) c(T) c ( T ) Lemma 5.18 ), we have that
( 1 + ∥ F B ∥ T , 1 p e r t ) A ≲ c ( T ) 2 T − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ . (1+\|F_B\|^{\mathrm{pert}}_{T,1})A\lesssim c(T)2T^{-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0),T}}. ( 1 + ∥ F B ∥ T , 1 pert ) A ≲ c ( T ) 2 T − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ . Next, we prove Lemma 5.19 , which basically follows from the triangle inequality and our previous estimates.
Proof of Lemma 5.19 . We may expand
B ( φ ⊗ D B + S φ ) = B ( φ + ⊗ D B + S φ + ) + B ( ( φ l i n − + φ r e m ) ⊗ D B + S φ + ) + B ( φ ⊗ D B + S ( φ l i n − + φ r e m ) ) . \mathbf{B}(\varphi\otimes D_{B+S}\varphi)=\mathbf{B}(\varphi^+\otimes D_{B+S}\varphi^+)+\mathbf{B}((\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}})\otimes D_{B+S}\varphi^+)+\mathbf{B}(\varphi\otimes D_{B+S}(\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}})). B ( φ ⊗ D B + S φ ) = B ( φ + ⊗ D B + S φ + ) + B (( φ lin − + φ rem ) ⊗ D B + S φ + ) + B ( φ ⊗ D B + S ( φ lin − + φ rem )) . We first estimate the L t 1 L x 2 L_t^1L_x^2 L t 1 L x 2 L t 1 L x 2 L_t^1L_x^2 L t 1 L x 2 (291) , the definition of C B ( 0 ) , T − \mathcal{C}_{B(0),T}^- C B ( 0 ) , T − (294) )
∥ B ( ( φ l i n − + φ r e m ) ⊗ D B + S φ + ) ∥ L t 1 L x 2 ≲ ∥ D B + S φ + ∥ L t 2 L x 2 ( ∥ φ l i n − ∥ L t 2 L x ∞ + ∥ φ r e m ∥ L t 2 L x ∞ ) \|\mathbf{B}((\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}})\otimes D_{B+S}\varphi^+)\|_{L_t^1L_x^2}\lesssim\|D_{B+S}\varphi^+\|_{L_t^2L_x^2}\left(\|\varphi_{\mathrm{lin}}^-\|_{L_t^2L_x^\infty}+\|\varphi_{\mathrm{rem}}\|_{L_t^2L_x^\infty}\right) ∥ B (( φ lin − + φ rem ) ⊗ D B + S φ + ) ∥ L t 1 L x 2 ≲ ∥ D B + S φ + ∥ L t 2 L x 2 ( ∥ φ lin − ∥ L t 2 L x ∞ + ∥ φ rem ∥ L t 2 L x ∞ ) ≲ ∥ φ 0 + ∥ L x 2 ( T 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 / 2 K ) , \lesssim\|\varphi_0^+\|_{L_x^2}\left(T^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}+T^{1/2}K\right), ≲ ∥ φ 0 + ∥ L x 2 ( T 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1/2 K ) , which is acceptable. Next, we estimate
∥ B ( φ ⊗ D B + S ( φ l i n − + φ r e m ) ) ∥ L t 1 L x 2 ≤ ∥ B ( φ ⊗ D B ( φ l i n − + φ r e m ) ) ∥ L t 1 L x 2 + ∥ B ( φ ⊗ S ( φ l i n − + φ r e m ) ) ∥ L t 1 L x 2 \begin{aligned}
\|\mathbf{B}(\varphi\otimes D_{B+S}(\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}}))\|_{L_t^1L_x^2} \\
&\leq\|\mathbf{B}(\varphi\otimes D_B(\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}}))\|_{L_t^1L_x^2}+\|\mathbf{B}(\varphi\otimes S(\varphi_{\mathrm{lin}}^-+\varphi_{\mathrm{rem}}))\|_{L_t^1L_x^2}
\end{aligned} ∥ B ( φ ⊗ D B + S ( φ lin − + φ rem )) ∥ L t 1 L x 2 ≤ ∥ B ( φ ⊗ D B ( φ lin − + φ rem )) ∥ L t 1 L x 2 + ∥ B ( φ ⊗ S ( φ lin − + φ rem )) ∥ L t 1 L x 2 ≲ ( T 1 2 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 − κ 2 K + T 1 − κ N S ( T ) ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 − κ 2 N S ( T ) K ) ∥ t κ 2 φ ∥ L t , x ∞ . \lesssim\left(T^{\frac{1}{2}-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}+T^{\frac{1-\kappa}{2}}K+T^{1-\kappa}N_S(T)\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}+T^{1-\frac{\kappa}{2}}N_S(T)K\right)\|t^{\frac{\kappa}{2}}\varphi\|_{L^\infty_{t,x}}. ≲ ( T 2 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 − κ K + T 1 − κ N S ( T ) ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 − 2 κ N S ( T ) K ) ∥ t 2 κ φ ∥ L t , x ∞ . To finish, we apply (287) and use that Proposition 5.15 implies that T 1 / 2 N S ( T ) ≪ 1 T^{1/2}N_S(T)\ll1 T 1/2 N S ( T ) ≪ 1 L t , x 2 L_{t,x}^2 L t , x 2
Next, we quickly show Lemma 5.20.
Proof of Lemma 5.20. From the equations (289) , (295) for φ + \varphi^+ φ + φ l i n + \varphi_{\mathrm{lin}}^+ φ lin + (112) to expand Δ B + S \Delta_{B+S} Δ B + S Δ B \Delta_B Δ B
( ∂ t − Δ B ) ( φ + − φ l i n + ) = 2 D B j ( S j φ + ) − 2 ( D B j S j ) φ + + S j S j φ + − ∣ φ + ∣ 2 φ + , ( φ + − φ l i n + ) ( 0 ) = 0. (\partial_t-\Delta_B)(\varphi^+-\varphi_{\mathrm{lin}}^+)=2D_B^j(S_j\varphi^+)-2(D_B^jS_j)\varphi^++S_jS^j\varphi^+-|\varphi^+|^2\varphi^+,\qquad(\varphi^+-\varphi_{\mathrm{lin}}^+)(0)=0. ( ∂ t − Δ B ) ( φ + − φ lin + ) = 2 D B j ( S j φ + ) − 2 ( D B j S j ) φ + + S j S j φ + − ∣ φ + ∣ 2 φ + , ( φ + − φ lin + ) ( 0 ) = 0. Recalling the definition (Definition 4.60) of Duh B \operatorname{Duh}_B Duh B
φ + − φ l i n + = Duh B ( 2 D B j ( S j φ + ) − 2 ( D B j S j ) φ + + S j S j φ + − ∣ φ + ∣ 2 φ + ) . \varphi^+-\varphi_{\mathrm{lin}}^+=\operatorname{Duh}_B(2D_B^j(S_j\varphi^+)-2(D_B^jS_j)\varphi^++S_jS^j\varphi^+-|\varphi^+|^2\varphi^+). φ + − φ lin + = Duh B ( 2 D B j ( S j φ + ) − 2 ( D B j S j ) φ + + S j S j φ + − ∣ φ + ∣ 2 φ + ) . In the following, all space-time norms are on [ 0 , T ] × T d [0,T]\times\mathbb{T}^d [ 0 , T ] × T d Lemma 4.61 , we may estimate
∥ φ + − φ l i n + ∥ L t ∞ L x 4 ≲ ( 1 + ∥ F B ∥ T , 1 p e r t ) sup u ∈ ( 0 , T ] ∥ min ( t , ( u − t ) ) − 1 / 2 S φ + ∥ L t 1 L x 4 ( [ 0 , u ] ) + ∥ ( D B j S j ) φ + ∥ L t 1 L x 4 + ∥ S j S j φ + ∥ L t 1 L x 4 + ∥ ∣ φ + ∣ 2 φ + ∥ L t 1 L x 4 . \begin{aligned}
\|\varphi^+-\varphi_{\mathrm{lin}}^+\|_{L_t^\infty L_x^4}\lesssim(1+\|F_B\|^{\mathrm{pert}}_{T,1})\sup_{u\in(0,T]}\|\min(t,(u-t))^{-1/2}S\varphi^+\|_{L_t^1L_x^4([0,u])} \\
&+ \lVert(D^j_{B}S_j)\varphi^+ \rVert_{L^1_tL^4_x} + \lVert S_jS^j\varphi^+ \rVert_{L^1_tL^4_x} + \lVert\lvert\varphi^+\rvert^2\varphi^+ \rVert_{L^1_tL^4_x}.
\end{aligned} ∥ φ + − φ lin + ∥ L t ∞ L x 4 ≲ ( 1 + ∥ F B ∥ T , 1 pert ) u ∈ ( 0 , T ] sup ∥ min ( t , ( u − t ) ) − 1/2 S φ + ∥ L t 1 L x 4 ([ 0 , u ]) + ∥( D B j S j ) φ + ∥ L t 1 L x 4 + ∥ S j S j φ + ∥ L t 1 L x 4 + ∥∣ φ + ∣ 2 φ + ∥ L t 1 L x 4 . By Lemma 5.24, we may further estimate the right-hand side above by
( 1 + ∥ F B ∥ T , 1 p e r t ( T 1 / 2 N S ( T ) + T N S ( T ) 2 + ( T 1 / 2 − 1 r ∥ φ 0 + ∥ L x r ) 2 ) ) ∥ φ 0 + ∥ L x 4 . \left(1+\lVert F_B\rVert_{T,1}^{\mathrm{pert}}\left(T^{1/2}N_S(T)+TN_S(T)^2+\left(T^{1/2}-\frac{1}{r}\lVert\varphi_0^+\rVert_{L_x^r}\right)^2\right)\right)\lVert\varphi_0^+\rVert_{L^4_x}. ( 1 + ∥ F B ∥ T , 1 pert ( T 1/2 N S ( T ) + T N S ( T ) 2 + ( T 1/2 − r 1 ∥ φ 0 + ∥ L x r ) 2 ) ) ∥ φ 0 + ∥ L x 4 . By Proposition 5.15, the terms T 1 / 2 N S ( T ) T^{1/2}N_S(T) T 1/2 N S ( T ) T N S ( T ) 2 TN_S(T)^2 T N S ( T ) 2
We finish this section off with two results of a more auxiliary nature. The first result is a soft statement that will be needed in the proof of Theorem 2.19 at the end of § 9.5 .
Lemma 5.25. In the setting of Proposition 5.15, suppose we further have that φ 0 − ∈ C x − κ \varphi_0^- \in\mathcal{C}^{-\kappa}_x φ 0 − ∈ C x − κ
∥ t κ 2 B ∥ L t , x ∞ ( [ 0 , T ] ) + ∥ t κ ∂ ℓ B ℓ ∥ L t , x ∞ ( [ 0 , T ] ) < ∞ . (315) \lVert t^{\frac{\kappa}{2}}B\rVert_{L^\infty_{t,x}([0,T])}+\lVert t^\kappa\partial^\ell B_\ell\rVert_{L^\infty_{t,x}([0,T])}<\infty. \tag*{(315)} ∥ t 2 κ B ∥ L t , x ∞ ([ 0 , T ]) + ∥ t κ ∂ ℓ B ℓ ∥ L t , x ∞ ([ 0 , T ]) < ∞. (315) Then φ ∈ C t 0 C x − κ ( [ 0 , T ] ) \varphi\in C^0_t\mathcal{C}^{-\kappa}_x([0,T]) φ ∈ C t 0 C x − κ ([ 0 , T ])
Proof. Recall the equation (282) for φ \varphi φ Δ B + S \Delta_{B+S} Δ B + S Δ \Delta Δ
( ∂ t − Δ ) φ = 2 ∂ j ( ( B + S ) j φ ) − ( ∂ j ( B + S ) j ) φ + ( B + S ) j ( B + S ) j φ − ( D B j S j ) φ − ∣ φ ∣ 2 φ , (\partial_t-\Delta)\varphi=2\partial^j((B+S)_j\varphi)-(\partial^j(B+S)_j)\varphi+(B+S)^j(B+S)_j\varphi-(D^j_BS_j)\varphi-\lvert\varphi\rvert^2\varphi, ( ∂ t − Δ ) φ = 2 ∂ j (( B + S ) j φ ) − ( ∂ j ( B + S ) j ) φ + ( B + S ) j ( B + S ) j φ − ( D B j S j ) φ − ∣ φ ∣ 2 φ , φ ( 0 ) = φ 0 + + φ 0 − . \varphi(0)=\varphi_0^++\varphi_0^-. φ ( 0 ) = φ 0 + + φ 0 − . Let G G G φ \varphi φ (286) , (287) with the assumed estimates on B B B Duh ( G ) ∈ C t 0 C x − κ ( [ 0 , T ] ) \operatorname{Duh}(G)\in C^0_t\mathcal{C}^{-\kappa}_x([0,T]) Duh ( G ) ∈ C t 0 C x − κ ([ 0 , T ]) ( ∂ t − Δ ) (\partial_t-\Delta) ( ∂ t − Δ ) φ = e t Δ φ 0 + + e t Δ φ 0 − + Duh ( G ) \varphi=e^{t\Delta}\varphi_0^++e^{t\Delta}\varphi_0^-+\operatorname{Duh}(G) φ = e t Δ φ 0 + + e t Δ φ 0 − + Duh ( G ) e t Δ φ 0 + e^{t\Delta}\varphi_0^+ e t Δ φ 0 + e t Δ φ 0 − ∈ C t 0 C x − κ ( [ 0 , T ] ) e^{t\Delta}\varphi_0^-\in C^0_t\mathcal{C}^{-\kappa}_x([0,T]) e t Δ φ 0 − ∈ C t 0 C x − κ ([ 0 , T ]) e t Δ φ 0 + e^{t\Delta}\varphi_0^+ e t Δ φ 0 + φ 0 + ∈ L x r \varphi_0^+\in L_x^r φ 0 + ∈ L x r 1 r ≪ κ \frac{1}{r}\ll\kappa r 1 ≪ κ (96) ), combined with classical heat smoothing estimates. For e t Δ φ 0 − e^{t\Delta}\varphi_0^- e t Δ φ 0 − φ 0 − ∈ C x − κ \varphi_0^-\in\mathcal{C}^{-\kappa}_x φ 0 − ∈ C x − κ t = 0 t=0 t = 0 C x − κ \mathcal{C}^{-\kappa}_x C x − κ
Next, using our bounds for S S S S S S
Lemma 5.26. Recall the constants specified in (97) and (99) . Let T 0 T_0 T 0 T ∈ ( 0 , T 0 ] T\in(0,T_0] T ∈ ( 0 , T 0 ] B ∈ C t 0 C x − κ ( [ 0 , 2 T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , 2 T ] × T 2 ) B\in C^0_t\mathcal{C}^{-\kappa}_x([0,2T]\times\mathbb{T}^2)\cap C^\infty_{t,x}((0,2T]\times\mathbb{T}^2) B ∈ C t 0 C x − κ ([ 0 , 2 T ] × T 2 ) ∩ C t , x ∞ (( 0 , 2 T ] × T 2 )
M 2 T ( B , 100 ) ≤ T − δ . M_{2T}(B,100)\leq T^{-\delta}. M 2 T ( B , 100 ) ≤ T − δ . Let S , φ S,\varphi S , φ (281) , (282) with initial data satisfying
∥ φ 0 + ∥ L x r ≤ T − 1 / γ 0 , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − 2 α . \lVert\varphi_0^+\rVert_{L_x^r}\leq T^{-1/\gamma_0},\qquad\lVert\varphi_0^-\rVert_{\mathcal{C}^{-\kappa}_{B(0),T}}\leq T^{-2\alpha}. ∥ φ 0 + ∥ L x r ≤ T − 1/ γ 0 , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − 2 α . Then, for any t , t 0 ∈ ( 0 , T ] t,t_0\in(0,T] t , t 0 ∈ ( 0 , T ] u 0 u_0 u 0
∥ u 0 ∥ C ( B + S ) ( t ) , F , t 0 − κ ≲ T − δ ∥ u 0 ∥ C B ( t ) , F , t 0 − κ . (316) \lVert u_0\rVert_{\mathcal{C}^{-\kappa}_{(B+S)(t),F,t_0}}\lesssim T^{-\delta}\lVert u_0\rVert_{\mathcal{C}^{-\kappa}_{B(t),F,t_0}.} \tag*{(316)} ∥ u 0 ∥ C ( B + S ) ( t ) , F , t 0 − κ ≲ T − δ ∥ u 0 ∥ C B ( t ) , F , t 0 − κ . (316) The analogous result also holds for the C ( B + S ) ( t ) , t 0 − κ C^{-\kappa}_{(B+S)(t),t_0} C ( B + S ) ( t ) , t 0 − κ T ′ ∈ ( 0 , 1 ] T' \in(0,1] T ′ ∈ ( 0 , 1 ]
G T ′ ( C 0 F Ψ ∙ ( ( B + S ) ( t ) ) ) ( 1 + ∥ F Ψ ∙ ( ( B + S ) ( t ) ) ∥ T ′ , 1 p e r t ) ≤ ( T ′ ) − δ , (317) G_{T'}\bigl(C_0F_{\Psi\bullet((B+S)(t))}\bigr)\left(1+\left\|F_{\Psi\bullet((B+S)(t))}\right\|^{\mathrm{pert}}_{T',1}\right)\leq(T')^{-\delta}, \tag*{(317)} G T ′ ( C 0 F Ψ ∙ (( B + S ) ( t )) ) ( 1 + F Ψ ∙ (( B + S ) ( t )) T ′ , 1 pert ) ≤ ( T ′ ) − δ , (317) where C 0 C_0 C 0 Ψ ∙ ( ( B + S ) ( t ) ) \Psi_\bullet((B+S)(t)) Ψ ∙ (( B + S ) ( t )) ( B + S ) ( t ) (B+S)(t) ( B + S ) ( t )
∥ u 0 ∥ C ( B + S ) ( t ) , F , T ′ − κ ≲ T − κ 2 − δ ∥ u 0 ∥ C B ( t ) , F , T − κ . (318) \|u_0\|_{C^{-\kappa}_{(B+S)(t),F,T'}}\lesssim T^{-\frac{\kappa}{2}-\delta}\|u_0\|_{C^{-\kappa}_{B(t),F,T}}. \tag*{(318)} ∥ u 0 ∥ C ( B + S ) ( t ) , F , T ′ − κ ≲ T − 2 κ − δ ∥ u 0 ∥ C B ( t ) , F , T − κ . (318) The analogous result also holds for the C ( B + S ) ( t ) , T ′ − κ C^{-\kappa}_{(B+S)(t),T'} C ( B + S ) ( t ) , T ′ − κ
Proof. First, applying (286) on [ 0 , t ] [0,t] [ 0 , t ] ∥ S ( t ) ∥ L ∞ ≲ T 1 2 − 2 r − 2 γ 0 − δ \|S(t)\|_{L^\infty}\lesssim T^{\frac{1}{2}-\frac{2}{r}-\frac{2}{\gamma_0}-\delta} ∥ S ( t ) ∥ L ∞ ≲ T 2 1 − r 2 − γ 0 2 − δ (196) yields
∥ S ( t ) ∥ C B ( t ) , F , T , 2 − κ ≲ T − δ ∥ S ( t ) ∥ L x ∞ ≲ T 1 2 − 2 r − 2 γ 0 − 2 δ . \|S(t)\|_{C^{-\kappa}_{B(t),F,T,2}}\lesssim T^{-\delta}\|S(t)\|_{L^\infty_x}\lesssim T^{\frac{1}{2}-\frac{2}{r}-\frac{2}{\gamma_0}-2\delta}. ∥ S ( t ) ∥ C B ( t ) , F , T , 2 − κ ≲ T − δ ∥ S ( t ) ∥ L x ∞ ≲ T 2 1 − r 2 − γ 0 2 − 2 δ . This estimate is enough to treat S S S K = T 1 2 − 2 r − 2 γ 0 − 2 δ K=T^{\frac{1}{2}-\frac{2}{r}-\frac{2}{\gamma_0}-2\delta} K = T 2 1 − r 2 − γ 0 2 − 2 δ B 0 + = B ( t ) B_0^+=B(t) B 0 + = B ( t ) B 0 − = S ( t ) B_0^-=S(t) B 0 − = S ( t ) (97) and (99) , we indeed have that M T ( Ψ ∙ ( B ( t ) ) , 2 ) T 1 − κ 2 K ≪ 1 M_T(\Psi_\bullet(B(t)),2)T^{\frac{1-\kappa}{2}}K\ll1 M T ( Ψ ∙ ( B ( t )) , 2 ) T 2 1 − κ K ≪ 1 T 0 T_0 T 0 (260) in the proposition is satisfied. Therefore, by Proposition 5.7 and Lemma 5.11 with α = κ \alpha=\kappa α = κ p = ∞ p=\infty p = ∞ t 0 ∈ ( 0 , T ] t_0\in(0,T] t 0 ∈ ( 0 , T ]
sup 0 < s ≤ t 0 s κ 2 ∥ e s Δ B + S ( t ) F u 0 ∥ L x ∞ ≲ sup 0 < s ≤ t 0 s κ 2 ∥ e s Δ B ( t ) F u 0 ∥ L x ∞ ≤ ∥ u 0 ∥ C B ( t ) , F , t 0 − κ . (319) \sup_{0<s\leq t_0}s^{\frac{\kappa}{2}}\left\|e^{s\Delta^F_{B+S(t)} }u_0\right\|_{L^\infty_x}\lesssim\sup_{0<s\leq t_0}s^{\frac{\kappa}{2}}\left\|e^{s\Delta^F_{B(t)}}u_0\right\|_{L^\infty_x}\leq\|u_0\|_{C^{-\kappa}_{B(t),F,t_0}}. \tag*{(319)} 0 < s ≤ t 0 sup s 2 κ e s Δ B + S ( t ) F u 0 L x ∞ ≲ 0 < s ≤ t 0 sup s 2 κ e s Δ B ( t ) F u 0 L x ∞ ≤ ∥ u 0 ∥ C B ( t ) , F , t 0 − κ . (319) Here, we are actually applying Lemma 5.11 to g 0 ∙ u 0 g_0\bullet u_0 g 0 ∙ u 0 Ψ ∙ ( g 0 ∙ ( B + S ) ( t ) ) \Psi_\bullet(g_0\bullet(B+S)(t)) Ψ ∙ ( g 0 ∙ ( B + S ) ( t )) g 0 g_0 g 0
∥ e s Δ g 0 ∙ ( B + S ) ( t ) F ( g 0 ∙ u 0 ) ∥ L x ∞ = ∥ e s Δ ( B + S ) ( t ) F u 0 ∥ L x ∞ \left\|e^{s\Delta^F_{g_0\bullet(B+S)(t)}}(g_0\bullet u_0)\right\|_{L^\infty_x}=\left\|e^{s\Delta^F_{(B+S)(t)}}u_0\right\|_{L^\infty_x} e s Δ g 0 ∙ ( B + S ) ( t ) F ( g 0 ∙ u 0 ) L x ∞ = e s Δ ( B + S ) ( t ) F u 0 L x ∞ to get (319) . The desired bounds then follow from (319) and (196) . We remark that when proving (318) , a direct application would give a factor ( T ′ ) − δ (T')^{-\delta} ( T ′ ) − δ T ′ > T T'>T T ′ > T T ′ ≤ T T'\leq T T ′ ≤ T (318) is the same as (316) .
Monotone quantity in global iteration
Recall the energy (64) that is used to measure the size of the smooth part of the solution. In Section 11, we will perform a careful analysis on the iteration of E E E E E E
Proposition 5.27. Let T 0 T_0 T 0 (98) , chosen sufficiently small. Recall the constants specified as in (99) . There exists some constant C C C B 0 ± = B 0 + + B 0 − B_0^\pm=B_0^++B_0^- B 0 ± = B 0 + + B 0 − T ∈ ( 0 , T 0 ] T\in(0,T_0] T ∈ ( 0 , T 0 ]
M T ( B + , 100 ) ≤ T − a , ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ T − 2 a . (320) M_T(B^+,100)\leq T^{-a},\qquad\|B_0^-\|_{C^{-\kappa}_{B_0^+,F,T,2}}\leq T^{-2a}. \tag*{(320)} M T ( B + , 100 ) ≤ T − a , ∥ B 0 − ∥ C B 0 + , F , T , 2 − κ ≤ T − 2 a . (320) In the following, we will use e t Δ F e^{t\Delta^F} e t Δ F e t Δ + e^{t\Delta^+} e t Δ + e t Δ B 0 + F e^{t\Delta^F_{B_0^+}} e t Δ B 0 + F e t Δ B 0 + + e^{t\Delta^+_{B_0^+}} e t Δ B 0 + + B , B + , g B,B^+,g B , B + , g K K K T − 2 a T^{-2a} T − 2 a φ 0 = φ 0 + + φ 0 − \varphi_0=\varphi_0^++\varphi_0^- φ 0 = φ 0 + + φ 0 −
T 1 2 ∥ φ 0 + ∥ L x r ≤ T γ 0 − 2 100 , ∥ t κ 2 e t Δ + ( g 0 − 1 ∙ φ 0 ) ∥ L t ∞ L x 4 ( [ 0 , T ] ) ≤ T − a , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − 2 a . (321) T^{\frac{1}{2}}\|\varphi_0^+\|_{L^r_x}\leq T^{\frac{\gamma_0-2}{100}},\qquad\|t^{\frac{\kappa}{2}}e^{t\Delta^+}(g_0^{-1}\bullet\varphi_0)\|_{L^\infty_tL^4_x([0,T])}\leq T^{-a},\qquad\|\varphi_0^-\|_{C^{-\kappa}_{B(0),T}}\leq T^{-2a}. \tag*{(321)} T 2 1 ∥ φ 0 + ∥ L x r ≤ T 100 γ 0 − 2 , ∥ t 2 κ e t Δ + ( g 0 − 1 ∙ φ 0 ) ∥ L t ∞ L x 4 ([ 0 , T ]) ≤ T − a , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − 2 a . (321) Let S , φ S,\varphi S , φ (281) –(282) . Then, for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
E ( ( B + S ) ( t ) , φ ( t ) ) ≤ E ( B 0 + , φ 0 + ) − 1 2 Dec t ( B 0 + , φ 0 + ) + C t − 1 + κ + ∥ curl B + ( t ) e t Δ + F B 0 − ∥ L 2 2 + 2 ∫ T 2 ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ + F B 0 − ⟩ g d x + 2 ∫ T 2 ⟨ F B + 12 ( t ) , [ ( e t Δ + F B 0 − ) 1 , ( e t Δ + F B 0 − ) 2 ] ⟩ g d x + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) ∣ 2 + ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) , e t Δ + ( g 0 − 1 ⋅ φ 0 − ) ⟩ g d x . (322) \begin{aligned}
\mathcal{E}((B+S)(t),\varphi(t)) &\leq\mathcal{E}(B_0^+,\varphi_0^+) - \frac{1}{2}\operatorname{Dec}_t(B_0^+,\varphi_0^+) + Ct^{-1+\kappa} \\
&\quad+ \left\|\operatorname{curl}_{B_+(t)}e^{t\Delta_+^F}B_0^-\right\|_{L^2}^2 \\
&\quad+ 2\int_{T^2}\left\langle F_{B_+}^{12}(t),\operatorname{curl}_{B_+(t)}e^{t\Delta_+^F}B_0^-\right\rangle_g\,dx \\
&\quad+ 2\int_{T^2}\left\langle F_{B_+}^{12}(t),\left[(e^{t\Delta_+^F}B_0^-)^1,(e^{t\Delta_+^F}B_0^-)^2\right]\right\rangle_g\,dx \\
&\quad+ C\|\varphi_0^+\|_{L^4}^{\gamma_1-4}\int_{T^2}\left|e^{t\Delta_+}(g_0^{-1}\cdot\varphi_0^+)\right|^2 \\
&\qquad\qquad\qquad\qquad\qquad\qquad+\left\langle e^{t\Delta_+}(g_0^{-1}\cdot\varphi_0^+),e^{t\Delta_+}(g_0^{-1}\cdot\varphi_0^-)\right\rangle_g\,dx . \tag*{(322)}
\end{aligned} E (( B + S ) ( t ) , φ ( t )) ≤ E ( B 0 + , φ 0 + ) − 2 1 Dec t ( B 0 + , φ 0 + ) + C t − 1 + κ + curl B + ( t ) e t Δ + F B 0 − L 2 2 + 2 ∫ T 2 ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ + F B 0 − ⟩ g d x + 2 ∫ T 2 ⟨ F B + 12 ( t ) , [ ( e t Δ + F B 0 − ) 1 , ( e t Δ + F B 0 − ) 2 ] ⟩ g d x + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 e t Δ + ( g 0 − 1 ⋅ φ 0 + ) 2 + ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) , e t Δ + ( g 0 − 1 ⋅ φ 0 − ) ⟩ g d x . (322) where we used the notation
Dec t ( B 0 + , φ 0 + ) = 2 ∥ D B + ∗ F B + ∥ L t , x 2 ( [ 0 , t ] ) 2 + C ( 5.43 ) , γ 2 , r 1 ∥ φ 0 + ∥ L r 1 ≥ 1 γ 2 + 2 t + 1 ∥ φ 0 ∥ L 4 ≥ 1 ( ( γ 1 ∥ φ + ( t ) ∥ L x 4 γ 1 − 4 ∥ D B + + S φ + ∥ L t , x 2 ( [ 0 , t ] ) 2 ) ∨ ( C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t ) ) . (323) \begin{aligned}
\operatorname{Dec}_t(B_0^+,\varphi_0^+)=2\|D_{B_+}^*F_{B_+}\|_{L^2_{t,x}([0,t])}^2+C_{(5.43),\gamma_2,r_1}\|\varphi_0^+\|_{L^{r_1}\geq1}^{\gamma_2+2}t \\
&+ \mathbf{1}_{\|\varphi_0\|_{L^4}\geq1}\left(\left(\gamma_1\|\varphi^+(t)\|_{L_x^4}^{\gamma_1-4}\|D_{B_++S}\varphi^+\|_{L^2_{t,x}([0,t])}^2\right)\vee\left(C_{(5.43),\gamma_1,4}\|\varphi_0^+\|_{L^4}^{\gamma_1+2}t\right)\right). \tag*{(323)}
\end{aligned} Dec t ( B 0 + , φ 0 + ) = 2∥ D B + ∗ F B + ∥ L t , x 2 ([ 0 , t ]) 2 + C ( 5.43 ) , γ 2 , r 1 ∥ φ 0 + ∥ L r 1 ≥ 1 γ 2 + 2 t + 1 ∥ φ 0 ∥ L 4 ≥ 1 ( ( γ 1 ∥ φ + ( t ) ∥ L x 4 γ 1 − 4 ∥ D B + + S φ + ∥ L t , x 2 ([ 0 , t ]) 2 ) ∨ ( C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t ) ) . (323) This also gives the following crude bound, which sometimes will be convenient. For t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
E ( ( B + S ) ( t ) , φ ( t ) ) ≤ E ( B 0 + , φ 0 + ) + C t − 7 4 . (324) \mathcal{E}((B+S)(t),\varphi(t))\leq\mathcal{E}(B_0^+,\varphi_0^+)+Ct^{-\frac{7}{4}}. \tag*{(324)} E (( B + S ) ( t ) , φ ( t )) ≤ E ( B 0 + , φ 0 + ) + C t − 4 7 . (324) Remark 5.28. By Propositions 4.4 and 5.17 we have
Dec t ( B 0 + , φ 0 + ) ≤ E ( B + ( 0 ) , φ + ( 0 ) ) − E ( B + ( t ) , φ + ( t ) ) . (325) \operatorname{Dec}_t(B_0^+,\varphi_0^+)\leq\mathcal{E}(B^+(0),\varphi^+(0))-\mathcal{E}(B^+(t),\varphi^+(t)). \tag*{(325)} Dec t ( B 0 + , φ 0 + ) ≤ E ( B + ( 0 ) , φ + ( 0 )) − E ( B + ( t ) , φ + ( t )) . (325) This is the reason for using the notation Dec, which stands for “decrease”. Additionally, recalling the discussion on remnants of exact solvability near the end of § 1.2 , we have that in two dimensions, the covariant divergence D B + ∗ F B + D_{B_+}^*F_{B_+} D B + ∗ F B + D B + F B + D_{B_+}F_{B_+} D B + F B + (323) to absorb terms involving D B + F B + D_{B_+}F_{B_+} D B + F B +
Proof. We begin by making several preliminary observations. Note that when (320) holds, by (263) we have that ∥ F B ∥ T , 0 p e r t ≤ ∥ F B + ∥ T , 0 p e r t + O ( 1 ) \|F_B\|_{T,0}^{\mathrm{pert}}\leq\|F_{B_+}\|_{T,0}^{\mathrm{pert}}+O(1) ∥ F B ∥ T , 0 pert ≤ ∥ F B + ∥ T , 0 pert + O ( 1 )
Gr T ( C 0 F B ) ≲ Gr T ( C 0 F B + ) ≤ T − a , ∥ F B ∥ T , 2 p e r t ≲ ( 1 + ∥ F B + ∥ T , 0 p e r t ) 6 ≤ T − a 10 . (326) \operatorname{Gr}_T(C_0F_B)\lesssim\operatorname{Gr}_T(C_0F_{B_+})\leq T^{-a},\qquad\|F_B\|_{T,2}^{\mathrm{pert}}\lesssim(1+\|F_{B_+}\|_{T,0}^{\mathrm{pert}})^6\leq T^{-\frac{a}{10}}. \tag*{(326)} Gr T ( C 0 F B ) ≲ Gr T ( C 0 F B + ) ≤ T − a , ∥ F B ∥ T , 2 pert ≲ ( 1 + ∥ F B + ∥ T , 0 pert ) 6 ≤ T − 10 a . (326) By Remark 4.14, this implies that under (320) , for any t > 0 t>0 t > 0
∥ F B + ( t ) ∥ L ∞ + ∥ F B ( t ) ∥ L ∞ ≲ t − 1 − a 10 . (327) \|F_{B_+}(t)\|_{L^\infty}+\|F_B(t)\|_{L^\infty}\lesssim t^{-1-\frac{a}{10}}. \tag*{(327)} ∥ F B + ( t ) ∥ L ∞ + ∥ F B ( t ) ∥ L ∞ ≲ t − 1 − 10 a . (327) Also, combining (326) with (321) , for any t ≤ T t\leq T t ≤ T
M t ( B , 100 ) ( t 1 2 − 1 r ∥ φ 0 + ∥ L x r + t 1 2 − κ 2 ∥ φ 0 − ∥ C B ( 0 , T ) ) ≲ t γ 1 − 4 . (328) M_t(B,100)\left(t^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+t^{\frac{1}{2}-\frac{\kappa}{2}}\|\varphi_0^-\|_{C_B(0,T)}\right)\lesssim t^{\gamma_1-4}. \tag*{(328)} M t ( B , 100 ) ( t 2 1 − r 1 ∥ φ 0 + ∥ L x r + t 2 1 − 2 κ ∥ φ 0 − ∥ C B ( 0 , T ) ) ≲ t γ 1 − 4 . (328) Here we used that γ 1 − 4 ≪ γ 0 − 2 \gamma_1-4\ll\gamma_0-2 γ 1 − 4 ≪ γ 0 − 2 (285) is satisfied when T T T S , φ S,\varphi S , φ T T T
In the following, recall φ l i n \varphi_{\mathrm{lin}} φ lin (283) ) and φ r e m \varphi_{\mathrm{rem}} φ rem φ = φ + + φ l i n + φ r e m \varphi=\varphi^++\varphi_{\mathrm{lin}}+\varphi_{\mathrm{rem}} φ = φ + + φ lin + φ rem φ + \varphi^+ φ + (289) ). By (321) and Lemma 5.18 we get that for any t ≤ T t\leq T t ≤ T
∥ φ l i n ( t ) ∥ L x ∞ ≤ t − 2 a − κ 2 , ∥ φ r e m ( t ) ∥ L x ∞ + ∥ t 1 2 D B φ r e m ( t ) ∥ L x ∞ ≲ t γ 1 − 4 . \|\varphi_{\mathrm{lin}}(t)\|_{L_x^\infty}\leq t^{-2a-\frac{\kappa}{2}},\qquad\|\varphi_{\mathrm{rem}}(t)\|_{L_x^\infty}+\|t^{\frac{1}{2}}D_B\varphi_{\mathrm{rem}}(t)\|_{L_x^\infty}\lesssim t^{\gamma_1-4}. ∥ φ lin ( t ) ∥ L x ∞ ≤ t − 2 a − 2 κ , ∥ φ rem ( t ) ∥ L x ∞ + ∥ t 2 1 D B φ rem ( t ) ∥ L x ∞ ≲ t γ 1 − 4 . We divide the proof of (322) into Lemmas 5.29–5.35 below. Based on the discussion above, one can verify all assumptions in Lemmas 5.29–5.35.
For (324) , plugging (320) and (321) into the right-hand side of (322) and applying (327) , it follows that for any t ≤ T t \le T t ≤ T
E ( ( B + S ) ( t ) , φ ( t ) ) ≤ E ( B 0 + , φ 0 + ) + C t − 1 + κ + C ( t − 1 − α T − 4 α + t − 1 − α ⋅ t − 1 + κ 2 T − 2 α + t − 1 − α ⋅ t − κ T − 4 α + T − ( 1 2 − γ 0 − 2 100 ) ( γ 1 − 1 ) t − κ 2 T − 2 α ) ≤ E ( B 0 + , φ 0 + ) + C t − 7 4 . \begin{aligned}
\mathrm{E}((B+S)(t),\varphi(t)) &\le\mathrm{E}(B_0^+,\varphi_0^+) + Ct^{-1+\kappa} + C\left(t^{-1-\alpha}T^{-4\alpha} + t^{-1-\alpha}\cdot t^{-\frac{1+\kappa}{2}}T^{-2\alpha}\right.\\
&\qquad\left.{}+t^{-1-\alpha}\cdot t^{-\kappa}T^{-4\alpha}+T^{-\left(\frac{1}{2}-\frac{\gamma_0-2}{100}\right)(\gamma_1-1)}t^{-\frac{\kappa}{2}}T^{-2\alpha}\right)\\
&\le\mathrm{E}(B_0^+,\varphi_0^+) + Ct^{-\frac{7}{4}}.
\end{aligned} E (( B + S ) ( t ) , φ ( t )) ≤ E ( B 0 + , φ 0 + ) + C t − 1 + κ + C ( t − 1 − α T − 4 α + t − 1 − α ⋅ t − 2 1 + κ T − 2 α + t − 1 − α ⋅ t − κ T − 4 α + T − ( 2 1 − 100 γ 0 − 2 ) ( γ 1 − 1 ) t − 2 κ T − 2 α ) ≤ E ( B 0 + , φ 0 + ) + C t − 4 7 . This concludes the proof.
We record several preliminary estimates which hold in the setting of Proposition 5.27, and which will be used several times in the following. Let T T T t ∈ ( 0 , T ] t \in(0,T] t ∈ ( 0 , T ] [ 0 , t ) [0,t) [ 0 , t ) (326) , and (321) , we have that
max 0 ≤ m ≤ 1 t m 2 ∥ D B ( m ) S ( t ) ∥ L x 2 ≲ t − ( 1.1 ) α ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ( [ 0 , t ] ) + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , t ] ) ) . (329) \max_{0\le m\le1} t^{\frac{m}{2}}\left\|D_B^{(m)}S(t)\right\|_{L_x^2}\lesssim t^{-(1.1)\alpha}\left(\left\|B(\varphi\otimes D_{B+S}\varphi)\right\|_{L_t^1L_x^2([0,t])}+\left\|t^{1/2}B(\varphi\otimes D_{B+S}\varphi)\right\|_{L_{t,x}^2([0,t])}\right). \tag*{(329)} 0 ≤ m ≤ 1 max t 2 m D B ( m ) S ( t ) L x 2 ≲ t − ( 1.1 ) α ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ([ 0 , t ]) + t 1/2 B ( φ ⊗ D B + S φ ) L t , x 2 ([ 0 , t ]) ) . (329) max 0 ≤ m ≤ 2 t m 2 ∥ D B ( m ) S ( t ) ∥ L x ∞ ≲ t − 2 α ( t − 1 2 + γ 0 − 2 50 − κ 2 + t 1 2 − κ − 4 α ) . (330) \max_{0\le m\le2} t^{\frac{m}{2}}\left\|D_B^{(m)}S(t)\right\|_{L_x^\infty}\lesssim t^{-2\alpha}\left(t^{-\frac{1}{2}+\frac{\gamma_0-2}{50}-\frac{\kappa}{2}}+t^{\frac{1}{2}-\kappa-4\alpha}\right). \tag*{(330)} 0 ≤ m ≤ 2 max t 2 m D B ( m ) S ( t ) L x ∞ ≲ t − 2 α ( t − 2 1 + 50 γ 0 − 2 − 2 κ + t 2 1 − κ − 4 α ) . (330) Additionally, recall that in the setting of Proposition 5.27, we apply Proposition 5.7 with the choice K = T − 2 α K=T^{-2\alpha} K = T − 2 α (261) , (262) , we have that
max 0 ≤ m ≤ 1 t m 2 ∥ D B + ( m ) B − ( t ) ∥ L x ∞ ≲ t − 2 α − κ 2 . (331) \max_{0\le m\le1}t^{\frac{m}{2}}\left\|D_{B^+}^{(m)}B^-(t)\right\|_{L_x^\infty}\lesssim t^{-2\alpha-\frac{\kappa}{2}}. \tag*{(331)} 0 ≤ m ≤ 1 max t 2 m D B + ( m ) B − ( t ) L x ∞ ≲ t − 2 α − 2 κ . (331) We proceed to prove the various intermediate estimates on which the proof of Proposition 5.27 relies.
Lemma 5.29. There exists some constant C C C t ∈ ( 0 , T ] t \in(0,T] t ∈ ( 0 , T ]
∥ F B + S ( t ) ∥ L 2 2 − ∥ F B + ( t ) ∥ L 2 2 ≤ 1 10 ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) 2 + C t − 1 + κ + C t − 1 − 10 α ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ( [ 0 , t ] ) 2 + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ) + ∥ curl B + ( t ) e t Δ F + B 0 − ∥ L 2 2 + 2 ∫ T 2 t ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ F + B 0 − ⟩ g d x + 2 ∫ T 2 t ⟨ F B + 12 ( t ) , [ ( e t Δ F + B 0 − ) − 1 , ( e t Δ F + B 0 − ) 2 ] ⟩ g d x . (332) \begin{aligned}
\left\|F_{B+S}(t)\right\|_{L^2}^2-\left\|F_{B+}(t)\right\|_{L^2}^2
&\le\frac{1}{10}\left\|D_{B+}F_{B+}\right\|_{L_{t,x}^2([0,t])}^2+Ct^{-1+\kappa}\\
&\quad+Ct^{-1-10\alpha}\left(\left\|B(\varphi\otimes D_{B+S}\varphi)\right\|_{L_t^1L_x^2([0,t])}^2+\left\|t^{1/2}B(\varphi\otimes D_{B+S}\varphi)\right\|_{L_{t,x}^2([0,t])}^2\right)\\
&\quad+\left\|\operatorname{curl}_{B+}(t)e^{t\Delta_F^+}B_0^-\right\|_{L^2}^2\\
&\quad+2\int_{T^2}^{t}\left\langle F_{B+}^{12}(t),\operatorname{curl}_{B+}(t)e^{t\Delta_F^+}B_0^-\right\rangle_g\,dx\\
&\quad+2\int_{T^2}^{t}\left\langle F_{B+}^{12}(t),\left[(e^{t\Delta_F^+}B_0^-)^{-1},(e^{t\Delta_F^+}B_0^-)^2\right]\right\rangle_g\,dx. \tag*{(332)}
\end{aligned} ∥ F B + S ( t ) ∥ L 2 2 − ∥ F B + ( t ) ∥ L 2 2 ≤ 10 1 ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) 2 + C t − 1 + κ + C t − 1 − 10 α ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ([ 0 , t ]) 2 + t 1/2 B ( φ ⊗ D B + S φ ) L t , x 2 ([ 0 , t ]) 2 ) + curl B + ( t ) e t Δ F + B 0 − L 2 2 + 2 ∫ T 2 t ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ F + B 0 − ⟩ g d x + 2 ∫ T 2 t ⟨ F B + 12 ( t ) , [ ( e t Δ F + B 0 − ) − 1 , ( e t Δ F + B 0 − ) 2 ] ⟩ g d x . (332) Remark 5.30. The right-hand side is already in the desired form, except for the term on the second line. This term will be absorbed by the dissipation coming from the decrease of the L 4 L^4 L 4 φ \varphi φ L 4 L^4 L 4 φ \varphi φ
Proof. We first expand
∥ F B + S ∥ L 2 2 = ∥ F B ∥ L 2 2 + 2 ⟨ F B 1 , curl B S ⟩ + 2 ⟨ F B 1 , [ S 1 , S 2 ] ⟩ + ∥ curl B S + [ S 1 , S 2 ] ∥ L 2 2 . (333) \left\|F_{B+S}\right\|_{L^2}^2=\left\|F_B\right\|_{L^2}^2+2\left\langle F_B^1,\operatorname{curl}_B S\right\rangle+2\left\langle F_B^1,[S^1,S^2]\right\rangle+\left\|\operatorname{curl}_B S+[S^1,S^2]\right\|_{L^2}^2. \tag*{(333)} ∥ F B + S ∥ L 2 2 = ∥ F B ∥ L 2 2 + 2 ⟨ F B 1 , curl B S ⟩ + 2 ⟨ F B 1 , [ S 1 , S 2 ] ⟩ + curl B S + [ S 1 , S 2 ] L 2 2 . (333) For the first term in (333) , by (265) , we have (here we also apply (327) to estimate the final term in (265) )
∥ F B ( t ) ∥ L 2 2 = ∥ F B + ( t ) ∥ L 2 2 + ∥ curl B + ( t ) e t Δ F B 0 − ∥ L 2 2 + 2 ∫ T 2 ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ F B 0 − + ( e t Δ F B 0 − ) 1 , ( e t Δ F B 0 − ) 2 ⟩ g d x + O ( t − 10 a − κ ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) + t − 1 + κ ) . \begin{aligned}
\|F_B(t)\|_{L^2}^2
&= \|F_{B+}(t)\|_{L^2}^2
+ \left\|\operatorname{curl}_{B+}(t)e^{t\Delta_F}B_0^-\right\|_{L^2}^2 \\
&\quad+ 2\int_{\mathbb{T}^2}\left\langle F_{B+}^{12}(t),\operatorname{curl}_{B+}(t)e^{t\Delta_F}B_0^-+\left(e^{t\Delta_F}B_0^-\right)^1,\left(e^{t\Delta_F}B_0^-\right)^2\right\rangle_g\,dx \\
&\quad+ O\left(t^{-10a-\kappa}\|D_{B+}F_{B+}\|_{L^2_{t,x}([0,t])}+t^{-1+\kappa}\right).
\end{aligned} ∥ F B ( t ) ∥ L 2 2 = ∥ F B + ( t ) ∥ L 2 2 + curl B + ( t ) e t Δ F B 0 − L 2 2 + 2 ∫ T 2 ⟨ F B + 12 ( t ) , curl B + ( t ) e t Δ F B 0 − + ( e t Δ F B 0 − ) 1 , ( e t Δ F B 0 − ) 2 ⟩ g d x + O ( t − 10 a − κ ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) + t − 1 + κ ) . Applying Young’s inequality to the term t − 10 a − κ ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) t^{-10a-\kappa}\|D_{B+}F_{B+}\|_{L^2_{t,x}([0,t])} t − 10 a − κ ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ])
For the second term in (333) , first recall that in Proposition 5.7, g g g B − B^- B − g − 1 ⋅ B = B + + B − g^{-1}\cdot B=B^++B^- g − 1 ⋅ B = B + + B −
⟨ F B 12 , curl B S ⟩ = ⟨ Ad g − 1 F B 12 , Ad g − 1 curl B S ⟩ = ⟨ F B + 12 , curl B + + B − Ad g − 1 S ⟩ (334) \begin{aligned}
\left\langle F_B^{12},\operatorname{curl}_B S\right\rangle
=\left\langle\operatorname{Ad}_{g^{-1}}F_B^{12},\operatorname{Ad}_{g^{-1}}\operatorname{curl}_B S\right\rangle\\
&=\left\langle F_{B+}^{12},\operatorname{curl}_{B++B^-}\operatorname{Ad}_{g^{-1}}S\right\rangle \tag*{(334)}
\end{aligned} ⟨ F B 12 , curl B S ⟩ = ⟨ Ad g − 1 F B 12 , Ad g − 1 curl B S ⟩ = ⟨ F B + 12 , curl B ++ B − Ad g − 1 S ⟩ (334) + ⟨ curl B + B − + [ B 1 − , B 2 − ] , Ad g − 1 curl B S ⟩ . (335) +\left\langle\operatorname{curl}_{B+}B^-+[B_1^-,B_2^-],\operatorname{Ad}_{g^{-1}}\operatorname{curl}_B S\right\rangle. \tag*{(335)} + ⟨ curl B + B − + [ B 1 − , B 2 − ] , Ad g − 1 curl B S ⟩ . (335) For notational brevity, let
J = ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ( [ 0 , t ] ) + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , t ] ) . J=\|\mathbf{B}(\varphi\otimes D_{B+}S\varphi)\|_{L_t^1L_x^2([0,t])}
+\|t^{1/2}\mathbf{B}(\varphi\otimes D_{B+}S\varphi)\|_{L_{t,x}^2([0,t])}. J = ∥ B ( φ ⊗ D B + S φ ) ∥ L t 1 L x 2 ([ 0 , t ]) + ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ([ 0 , t ]) . Now by integration by parts, Lemma 4.13, (326) , (329) , and (331) , the term (334) is bounded (up to constants) by
∥ D B + F B + ( t ) ∥ L 2 ∥ S ( t ) ∥ L 2 + ∥ F B + ( t ) ∥ L 2 ∥ B − ( t ) ∥ L ∞ ∥ S ( t ) ∥ L 2 ≲ t − 1 2 − a ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) ⋅ t − ( 1.1 ) a J + t − 1 − a 10 ⋅ t − 2 a t − κ 2 ⋅ t − ( 1.1 ) a J , \|D_{B+}F_{B+}(t)\|_{L^2}\|S(t)\|_{L^2}
+\|F_{B+}(t)\|_{L^2}\|B^-(t)\|_{L^\infty}\|S(t)\|_{L^2}
\lesssim t^{-\frac12-a}\|D_{B+}F_{B+}\|_{L^2_{t,x}([0,t])}\cdot t^{-(1.1)a}J+t^{-1-\frac{a}{10}}\cdot t^{-2a}t^{-\frac{\kappa}{2}}\cdot t^{-(1.1)a}J, ∥ D B + F B + ( t ) ∥ L 2 ∥ S ( t ) ∥ L 2 + ∥ F B + ( t ) ∥ L 2 ∥ B − ( t ) ∥ L ∞ ∥ S ( t ) ∥ L 2 ≲ t − 2 1 − a ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) ⋅ t − ( 1.1 ) a J + t − 1 − 10 a ⋅ t − 2 a t − 2 κ ⋅ t − ( 1.1 ) a J , which gives the desired bound by Young’s inequality. For (335) , we apply (329) and (331) to estimate this term by
( t − 2 a t − 1 2 − κ 2 + t − 4 a t − κ ) t − ( 1.1 ) a − 1 2 J , \left(t^{-2a}t^{-\frac12-\frac{\kappa}{2}}+t^{-4a}t^{-\kappa}\right)t^{-(1.1)a-\frac12}J, ( t − 2 a t − 2 1 − 2 κ + t − 4 a t − κ ) t − ( 1.1 ) a − 2 1 J , which gives the desired bound by Young’s inequality.
For the third term in (333) , we use Hölder’s inequality, (329) , (330) , and (327) to estimate
⟨ F B 12 ( t ) , [ S 1 ( t ) , S 2 ( t ) ] ⟩ ≤ 2 ∥ F B ( t ) ∥ L x ∞ ∥ S ( t ) ∥ L x 2 2 ≤ 2 t − 1 − a 10 ⋅ t − ( 2.2 ) a J 2 , \left\langle F_B^{12}(t),[S^1(t),S^2(t)]\right\rangle
\leq2\|F_B(t)\|_{L_x^\infty}\|S(t)\|_{L_x^2}^2
\leq2t^{-1-\frac{a}{10}}\cdot t^{-(2.2)a}J^2, ⟨ F B 12 ( t ) , [ S 1 ( t ) , S 2 ( t )] ⟩ ≤ 2∥ F B ( t ) ∥ L x ∞ ∥ S ( t ) ∥ L x 2 2 ≤ 2 t − 1 − 10 a ⋅ t − ( 2.2 ) a J 2 , which gives the desired bound.
For the last term in (333) , we apply (329) and (330) to estimate
2 ∥ D B S ( t ) ∥ L 2 2 + 2 ∥ S ( t ) ∥ L ∞ ∥ S ( t ) ∥ L 2 2 2\|D_BS(t)\|_{L^2}^2+2\|S(t)\|_{L^\infty}\|S(t)\|_{L^2}^2 2∥ D B S ( t ) ∥ L 2 2 + 2∥ S ( t ) ∥ L ∞ ∥ S ( t ) ∥ L 2 2 ≲ ( t − ( 1.1 ) a − 1 2 J + ( t − 2 a t − 1 2 + γ 0 50 − 2 r + t − 2 a t 1 2 − κ t − 4 a ) t − ( 1.1 ) a J ) 2 ≲ t − 1 − 8 a J 2 . \lesssim\left(t^{-(1.1)a-\frac12}J+\left(t^{-2a}t^{-\frac12+\frac{\gamma_0}{50}-\frac{2}{r}}+t^{-2a}t^{\frac12-\kappa}t^{-4a}\right)t^{-(1.1)a}J\right)^2\lesssim t^{-1-8a}J^2. ≲ ( t − ( 1.1 ) a − 2 1 J + ( t − 2 a t − 2 1 + 50 γ 0 − r 2 + t − 2 a t 2 1 − κ t − 4 a ) t − ( 1.1 ) a J ) 2 ≲ t − 1 − 8 a J 2 . The desired estimate follows.
Lemma 5.31. There exists some constant C C C T ∈ ( 0 , 1 ] T\in(0,1] T ∈ ( 0 , 1 ] B ∈ C t 0 C x − κ ( [ 0 , T ] × T 2 ) ∩ C t , x ∞ ( ( 0 , T ] × T 2 ) B\in C_t^0\mathcal{C}_x^{-\kappa}([0,T]\times\mathbb{T}^2)\cap C_{t,x}^{\infty}((0,T]\times\mathbb{T}^2) B ∈ C t 0 C x − κ ([ 0 , T ] × T 2 ) ∩ C t , x ∞ (( 0 , T ] × T 2 ) T T T (285) of Proposition 5.15 . Let
S S S φ \varphi φ (281) –(282) (which is provided by Proposition 5.15 ). Let φ + \varphi^{+} φ + (289) . If φ \varphi φ φ = φ + + φ − = φ + + φ l i n − + φ r e m \varphi=\varphi^{+}+\varphi^{-}=\varphi^{+}+\varphi_{\mathrm{lin}}^{-}+\varphi_{\mathrm{rem}} φ = φ + + φ − = φ + + φ lin − + φ rem
∥ t 2 + 2 a φ l i n − ( t ) ∥ L t , x ∞ ( [ 0 , T ] ) ≤ 1 , ∥ t − ( γ 1 − 4 ) φ r e m ( t ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ 1 , (336) \left\|t^{2+2a}\varphi_{\mathrm{lin}}^{-}(t)\right\|_{L^\infty_{t,x}([0,T])}\leq1,\qquad\left\|t^{-(\gamma_1-4)}\varphi_{\mathrm{rem}}(t)\right\|_{L^\infty_{t,x}([0,T])}\lesssim1, \tag*{(336)} t 2 + 2 a φ lin − ( t ) L t , x ∞ ([ 0 , T ]) ≤ 1 , t − ( γ 1 − 4 ) φ rem ( t ) L t , x ∞ ([ 0 , T ]) ≲ 1 , (336) then for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ φ ( t ) ∥ L 4 γ 1 ≤ ∥ φ 0 + ∥ L 4 γ 1 − 3 4 ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ (337) \|\varphi(t)\|_{L^4}^{\gamma_1}\leq\|\varphi_0^{+}\|_{L^4}^{\gamma_1}-\frac{3}{4}\left(\|\varphi_0^{+}\|_{L^4}^{\gamma_1}-\|\varphi^{+}(t)\|_{L^4}^{\gamma_1}\right)+Ct^{-1+\kappa} \tag*{(337)} ∥ φ ( t ) ∥ L 4 γ 1 ≤ ∥ φ 0 + ∥ L 4 γ 1 − 4 3 ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ (337) + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ l i n − ( t ) ⟩ V d x ∣ . (338) +C\|\varphi_0^{+}\|_{L^4}^{\gamma_1-4}\left|\int_{\mathbb{T}^2}|\varphi^{+}(t)|^2\langle\varphi^{+}(t),\varphi_{\mathrm{lin}}^{-}(t)\rangle_V\,\mathrm{d}x\right|. \tag*{(338)} + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ lin − ( t ) ⟩ V d x . (338) Remark 5.32. The term (338) will be bounded in Lemma 5.35.
Proof. First we expand the L 4 L^4 L 4 t t t (336) )
∥ φ + + φ − ∥ L 4 4 ≤ ∥ φ + ∥ L 4 4 + 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ − ⟩ V d x + C ( 1 + ∥ φ + ∥ L 4 2 ) ⋅ t − 8 a t − 2 κ . \|\varphi^{+}+\varphi^{-}\|_{L^4}^{4}\leq\|\varphi^{+}\|_{L^4}^{4}+4\int_{\mathbb{T}^2}|\varphi^{+}|^2\langle\varphi^{+},\varphi^{-}\rangle_V\,\mathrm{d}x+C(1+\|\varphi^{+}\|_{L^4}^{2})\cdot t^{-8a}t^{-2\kappa}. ∥ φ + + φ − ∥ L 4 4 ≤ ∥ φ + ∥ L 4 4 + 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ − ⟩ V d x + C ( 1 + ∥ φ + ∥ L 4 2 ) ⋅ t − 8 a t − 2 κ . If ∥ φ + ( t ) ∥ L 4 ≤ C ⋆ t − 1 5 \|\varphi^{+}(t)\|_{L^4}\leq C_{\star}t^{-\frac{1}{5}} ∥ φ + ( t ) ∥ L 4 ≤ C ⋆ t − 5 1 C ⋆ C_{\star} C ⋆ (337) directly follows from this expansion and the fact that γ 1 < 5 \gamma_1<5 γ 1 < 5 C ⋆ C_{\star} C ⋆
∥ φ + ( t ) ∥ L 4 ≥ C ⋆ t − 1 5 ≥ ∥ φ − ( t ) ∥ L 4 . \|\varphi^{+}(t)\|_{L^4}\geq C_{\star}t^{-\frac{1}{5}}\geq\|\varphi^{-}(t)\|_{L^4}. ∥ φ + ( t ) ∥ L 4 ≥ C ⋆ t − 5 1 ≥ ∥ φ − ( t ) ∥ L 4 . Since from this we get ∥ φ + ( 0 ) ∥ L 4 ≥ ∥ φ + ( t ) ∥ L 4 ≥ 1 \|\varphi^{+}(0)\|_{L^4}\geq\|\varphi^{+}(t)\|_{L^4}\geq1 ∥ φ + ( 0 ) ∥ L 4 ≥ ∥ φ + ( t ) ∥ L 4 ≥ 1 Proposition 5.17 we get
∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ≥ C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t . (339) \|\varphi_0^{+}\|_{L^4}^{\gamma_1}-\|\varphi^{+}(t)\|_{L^4}^{\gamma_1}\geq C_{(5.43),\gamma_1,4}\|\varphi_0^{+}\|_{L^4}^{\gamma_1+2}t. \tag*{(339)} ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ≥ C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t . (339) For γ 1 ≥ 4 \gamma_1\geq4 γ 1 ≥ 4 ∣ ( 1 + x ) γ 1 / 4 − 1 ∣ ≤ C γ 1 ∣ x ∣ |(1+x)^{\gamma_1/4}-1|\leq C_{\gamma_1}|x| ∣ ( 1 + x ) γ 1 /4 − 1∣ ≤ C γ 1 ∣ x ∣ ∣ x ∣ ≤ 1 |x|\leq1 ∣ x ∣ ≤ 1 t t t
∥ φ + + φ − ∥ L 4 γ 1 = ∥ φ + ∥ L 4 γ 1 ( 1 + ∥ φ + ∥ L 4 − 4 ( 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ − ⟩ V d x + O ( ∥ φ + ∥ L 4 2 ⋅ t − 8 a t − 2 κ ) ) ) γ 1 4 ≤ ∥ φ + ∥ L 4 γ 1 + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ l i n − ⟩ V d x ∣ + C ∥ φ + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ r e m ⟩ V d x ∣ (340) \begin{aligned}
\|\varphi^{+}+\varphi^{-}\|_{L^4}^{\gamma_1}
=\|\varphi^{+}\|_{L^4}^{\gamma_1}\left(1+\|\varphi^{+}\|_{L^4}^{-4}\left(4\int_{\mathbb{T}^2}|\varphi^{+}|^2\langle\varphi^{+},\varphi^{-}\rangle_V\,\mathrm{d}x+O(\|\varphi^{+}\|_{L^4}^{2}\cdot t^{-8a}t^{-2\kappa})\right)\right)^{\frac{\gamma_1}{4}} \\
&\leq\|\varphi^{+}\|_{L^4}^{\gamma_1}+C\|\varphi_0^{+}\|_{L^4}^{\gamma_1-4}\left|\int_{\mathbb{T}^2}|\varphi^{+}|^2\langle\varphi^{+},\varphi_{\mathrm{lin}}^{-}\rangle_V\,\mathrm{d}x\right| \\
&+C\|\varphi^{+}\|_{L^4}^{\gamma_1-4}\left|\int_{\mathbb{T}^2}|\varphi^{+}|^2\langle\varphi^{+},\varphi_{\mathrm{rem}}\rangle_V\,\mathrm{d}x\right| \tag*{(340)}
\end{aligned} ∥ φ + + φ − ∥ L 4 γ 1 = ∥ φ + ∥ L 4 γ 1 ( 1 + ∥ φ + ∥ L 4 − 4 ( 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ − ⟩ V d x + O ( ∥ φ + ∥ L 4 2 ⋅ t − 8 a t − 2 κ ) ) ) 4 γ 1 ≤ ∥ φ + ∥ L 4 γ 1 + C ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ lin − ⟩ V d x + C ∥ φ + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ φ + ∣ 2 ⟨ φ + , φ rem ⟩ V d x (340) + C ∥ φ + ∥ L 4 γ 1 − 2 t − 8 a ⋅ t − 2 κ , (341) +C\|\varphi^{+}\|_{L^4}^{\gamma_1-2}t^{-8a}\cdot t^{-2\kappa}, \tag*{(341)} + C ∥ φ + ∥ L 4 γ 1 − 2 t − 8 a ⋅ t − 2 κ , (341) where we chose C ⋆ C_{\star} C ⋆ 1 1 1 (340) + (341) by 1 4 ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ \frac{1}{4}(\|\varphi_0^{+}\|_{L^4}^{\gamma_1}-\|\varphi^{+}(t)\|_{L^4}^{\gamma_1})+Ct^{-1+\kappa} 4 1 ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ (340) , by (336) , (339) and Young’s inequality, we have
∥ φ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ r e m ( t ) ⟩ V d x ∣ ≲ ∥ φ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 = ( ∥ φ 0 + ∥ L 4 γ 1 + 2 t ) γ 1 − 1 γ 1 + 2 t γ 1 − 4 − γ 1 − 1 γ 1 + 2 ≤ 1 8 C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( γ 1 − 4 − γ 1 − 1 γ 1 + 2 ) γ 1 + 2 3 . (342) \begin{aligned}
\|\varphi_0^{+}\|_{L^4}^{\gamma_1-4}\left|\int_{\mathbb{T}^2}|\varphi^{+}(t)|^2\langle\varphi^{+}(t),\varphi_{\mathrm{rem}}(t)\rangle_V\,\mathrm{d}x\right|\lesssim\|\varphi_0^{+}\|_{L^4}^{\gamma_1-1}t^{\gamma_1-4} \\
&=\left(\|\varphi_0^{+}\|_{L^4}^{\gamma_1+2}t\right)^{\frac{\gamma_1-1}{\gamma_1+2}}t^{\gamma_1-4-\frac{\gamma_1-1}{\gamma_1+2}} \\
&\leq\frac{1}{8}C_{(5.43),\gamma_1,4}\|\varphi_0^{+}\|_{L^4}^{\gamma_1+2}t+Ct^{(\gamma_1-4-\frac{\gamma_1-1}{\gamma_1+2})\frac{\gamma_1+2}{3}}. \tag*{(342)}
\end{aligned} ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ rem ( t ) ⟩ V d x ≲ ∥ φ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 = ( ∥ φ 0 + ∥ L 4 γ 1 + 2 t ) γ 1 + 2 γ 1 − 1 t γ 1 − 4 − γ 1 + 2 γ 1 − 1 ≤ 8 1 C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( γ 1 − 4 − γ 1 + 2 γ 1 − 1 ) 3 γ 1 + 2 . (342) The exponent of t t t
− γ 1 − 1 3 + ( γ 1 − 4 ) γ 1 + 2 3 = − 1 − γ 1 − 4 3 + ( γ 1 − 4 ) γ 1 + 2 3 > − 1 + κ , -\frac{\gamma_1-1}{3}+(\gamma_1-4)\frac{\gamma_1+2}{3}=-1-\frac{\gamma_1-4}{3}+(\gamma_1-4)\frac{\gamma_1+2}{3}>-1+\kappa, − 3 γ 1 − 1 + ( γ 1 − 4 ) 3 γ 1 + 2 = − 1 − 3 γ 1 − 4 + ( γ 1 − 4 ) 3 γ 1 + 2 > − 1 + κ , so this term is acceptable. For (341) , we proceed similarly and get
∥ φ 0 + ∥ L 4 γ 1 − 2 t − 2 κ − 8 a ≤ 1 8 C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( − 2 κ − 8 a − γ 1 − 2 γ 1 + 2 ) γ 1 + 2 4 , (343) \|\varphi_0^+\|_{L^4}^{\gamma_1-2}t^{-2\kappa-8a}\le\frac{1}{8}C_{(5.43),\gamma_1,4}\|\varphi_0^+\|_{L^4}^{\gamma_1+2}t+Ct^{\left(-2\kappa-8a-\frac{\gamma_1-2}{\gamma_1+2}\right)\frac{\gamma_1+2}{4}}, \tag*{(343)} ∥ φ 0 + ∥ L 4 γ 1 − 2 t − 2 κ − 8 a ≤ 8 1 C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( − 2 κ − 8 a − γ 1 + 2 γ 1 − 2 ) 4 γ 1 + 2 , (343) which is acceptable since γ 1 ∈ ( 4 , 5 ) \gamma_1\in(4,5) γ 1 ∈ ( 4 , 5 )
Lemma 5.33. In the same setting as Lemma 5.31, for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ φ ( t ) ∥ L r γ 2 ≤ ∥ φ 0 + ∥ L r γ 2 − 3 4 ( ∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ) + C t − 1 + κ . (344) \|\varphi(t)\|_{L^r}^{\gamma_2}\le\|\varphi_0^+\|_{L^r}^{\gamma_2}-\frac{3}{4}\left(\|\varphi_0^+\|_{L^r}^{\gamma_2}-\|\varphi^+(t)\|_{L^r}^{\gamma_2}\right)+Ct^{-1+\kappa}. \tag*{(344)} ∥ φ ( t ) ∥ L r γ 2 ≤ ∥ φ 0 + ∥ L r γ 2 − 4 3 ( ∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ) + C t − 1 + κ . (344) Proof. If ∥ φ + ( t ) ∥ L r ≤ 100 t − κ / 2 − 2 a \|\varphi^+(t)\|_{L^r}\le100t^{-\kappa/2-2a} ∥ φ + ( t ) ∥ L r ≤ 100 t − κ /2 − 2 a (344) immediately holds by the triangle inequality and (336) . Otherwise by (336) we have
∥ φ + ( t ) ∥ L r ≥ 100 t − κ / 2 − 2 a ≥ ∥ φ − ( t ) ∥ L r . \|\varphi^+(t)\|_{L^r}\ge100t^{-\kappa/2-2a}\ge\|\varphi^-(t)\|_{L^r}. ∥ φ + ( t ) ∥ L r ≥ 100 t − κ /2 − 2 a ≥ ∥ φ − ( t ) ∥ L r . Also from this we get ∥ φ + ( 0 ) ∥ L r ≥ ∥ φ + ( t ) ∥ L r ≥ 1 \|\varphi^+(0)\|_{L^r}\ge\|\varphi^+(t)\|_{L^r}\ge1 ∥ φ + ( 0 ) ∥ L r ≥ ∥ φ + ( t ) ∥ L r ≥ 1
∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ≥ C ( 5.43 ) , γ 2 , r ∥ φ 0 + ∥ L r γ 2 + 2 t . \|\varphi_0^+\|_{L^r}^{\gamma_2}-\|\varphi^+(t)\|_{L^r}^{\gamma_2}\ge C_{(5.43),\gamma_2,r}\|\varphi_0^+\|_{L^r}^{\gamma_2+2}t. ∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ≥ C ( 5.43 ) , γ 2 , r ∥ φ 0 + ∥ L r γ 2 + 2 t . Then, we use the triangle inequality to get
∥ φ ( t ) ∥ L r γ 2 ≤ ∥ φ + ( t ) ∥ L r γ 2 ( 1 + ∥ φ − ( t ) ∥ L r ∥ φ + ( t ) ∥ L r ) γ 2 ≤ ∥ φ + ( t ) ∥ L r γ 2 + C ∥ φ + ( t ) ∥ L r γ 2 − 1 ∥ φ − ( t ) ∥ L r ≤ ∥ φ 0 + ∥ L r γ 2 − ( ∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ) + C ∥ φ 0 + ∥ L r γ 2 − 1 t − κ / 2 − 2 a . \begin{aligned}
\|\varphi(t)\|_{L^r}^{\gamma_2}
&\le\|\varphi^+(t)\|_{L^r}^{\gamma_2}\left(1+\frac{\|\varphi^-(t)\|_{L^r}}{\|\varphi^+(t)\|_{L^r}}\right)^{\gamma_2}\\
&\le\|\varphi^+(t)\|_{L^r}^{\gamma_2}+C\|\varphi^+(t)\|_{L^r}^{\gamma_2-1}\|\varphi^-(t)\|_{L^r}\\
&\le\|\varphi_0^+\|_{L^r}^{\gamma_2}-\left(\|\varphi_0^+\|_{L^r}^{\gamma_2}-\|\varphi^+(t)\|_{L^r}^{\gamma_2}\right)+C\|\varphi_0^+\|_{L^r}^{\gamma_2-1}t^{-\kappa/2-2a}.
\end{aligned} ∥ φ ( t ) ∥ L r γ 2 ≤ ∥ φ + ( t ) ∥ L r γ 2 ( 1 + ∥ φ + ( t ) ∥ L r ∥ φ − ( t ) ∥ L r ) γ 2 ≤ ∥ φ + ( t ) ∥ L r γ 2 + C ∥ φ + ( t ) ∥ L r γ 2 − 1 ∥ φ − ( t ) ∥ L r ≤ ∥ φ 0 + ∥ L r γ 2 − ( ∥ φ 0 + ∥ L r γ 2 − ∥ φ + ( t ) ∥ L r γ 2 ) + C ∥ φ 0 + ∥ L r γ 2 − 1 t − κ /2 − 2 a . The last term is treated using Young’s inequality as follows:
C ∥ φ 0 + ∥ L r γ 2 − 1 t − κ / 2 − 2 a ≤ 1 4 C ( 5.43 ) , γ 2 , r ∥ φ 0 + ∥ L r γ 2 + 2 t + C t ( − κ 2 − 2 a − γ 2 − 1 γ 2 + 2 ) γ 2 + 2 3 . C\|\varphi_0^+\|_{L^r}^{\gamma_2-1}t^{-\kappa/2-2a}\le\frac{1}{4}C_{(5.43),\gamma_2,r}\|\varphi_0^+\|_{L^r}^{\gamma_2+2}t+Ct^{\left(-\frac{\kappa}{2}-2a-\frac{\gamma_2-1}{\gamma_2+2}\right)\frac{\gamma_2+2}{3}}. C ∥ φ 0 + ∥ L r γ 2 − 1 t − κ /2 − 2 a ≤ 4 1 C ( 5.43 ) , γ 2 , r ∥ φ 0 + ∥ L r γ 2 + 2 t + C t ( − 2 κ − 2 a − γ 2 + 2 γ 2 − 1 ) 3 γ 2 + 2 . From the choice of our γ 2 \gamma_2 γ 2 (98) , the exponent in t t t t − 1 + κ t^{-1+\kappa} t − 1 + κ
Lemma 5.34. Let T 0 T_0 T 0 ε > 0 \varepsilon>0 ε > 0 C = C ε C=C_\varepsilon C = C ε B B B ( 0 , T ] (0,T] ( 0 , T ] T ∈ ( 0 , T 0 ] T\in(0,T_0] T ∈ ( 0 , T 0 ] M T ( B , 10 ) ≤ T − 5 a M_T(B,10)\le T^{-5a} M T ( B , 10 ) ≤ T − 5 a φ 0 = φ 0 + + φ 0 − \varphi_0=\varphi_0^++\varphi_0^- φ 0 = φ 0 + + φ 0 − (321) hold. Let S , φ S,\varphi S , φ (281) –(282) and let φ + \varphi^+ φ + (289) . If φ \varphi φ φ = φ + + φ l i n + φ r e m \varphi=\varphi^++\varphi_{\mathrm{lin}}+\varphi_{\mathrm{rem}} φ = φ + + φ lin + φ rem
∥ t κ 2 + 2 a φ l i n ( t ) ∥ L t , x ∞ ( [ 0 , T ] ) ≤ 1 , ∥ t − ( γ 1 − 4 ) φ r e m ( t ) ∥ L t , x ∞ ( [ 0 , T ] ) + ∥ t − ( γ 1 − 4 ) 1 2 D B φ r e m ( t ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ 1 , \|t^{\frac{\kappa}{2}+2a}\varphi_{\mathrm{lin}}(t)\|_{L^\infty_{t,x}([0,T])}\le1,\qquad
\|t^{-(\gamma_1-4)}\varphi_{\mathrm{rem}}(t)\|_{L^\infty_{t,x}([0,T])}+\|t^{-(\gamma_1-4)}\tfrac{1}{2}D_B\varphi_{\mathrm{rem}}(t)\|_{L^\infty_{t,x}([0,T])}\lesssim1, ∥ t 2 κ + 2 a φ lin ( t ) ∥ L t , x ∞ ([ 0 , T ]) ≤ 1 , ∥ t − ( γ 1 − 4 ) φ rem ( t ) ∥ L t , x ∞ ([ 0 , T ]) + ∥ t − ( γ 1 − 4 ) 2 1 D B φ rem ( t ) ∥ L t , x ∞ ([ 0 , T ]) ≲ 1 , then for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
t − 1 − 10 a ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 2 L x 2 ( [ 0 , t ] ) 2 + ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ) ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ . (345) t^{-1-10a}\left(\|\mathbb{B}(\varphi\otimes D_B+S\varphi)\|_{L_t^2L_x^2([0,t])}^2+\|t^{1/2}\mathbb{B}(\varphi\otimes D_B+S\varphi)\|_{L_{t,x}^2([0,t])}^2\right)
\le\varepsilon\left(\|\varphi_0^+\|_{L^4}^{\gamma_1}-\|\varphi^+(t)\|_{L^4}^{\gamma_1}\right)+Ct^{-1+\kappa}. \tag*{(345)} t − 1 − 10 a ( ∥ B ( φ ⊗ D B + S φ ) ∥ L t 2 L x 2 ([ 0 , t ]) 2 + ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ([ 0 , t ]) 2 ) ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ . (345) Proof. By Lemma 5.19 and the last estimate of (321) , we have
∥ B ( φ ⊗ D B + S φ ) ∥ L t 2 L x 2 ( [ 0 , t ] ) 2 ≤ 2 ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t 2 L x 2 ( [ 0 , t ] ) 2 + C t γ 1 − 4 , \|\mathbb{B}(\varphi\otimes D_B+S\varphi)\|_{L_t^2L_x^2([0,t])}^2\le2\|\mathbb{B}(\varphi^+\otimes D_B+S\varphi^+)\|_{L_t^2L_x^2([0,t])}^2+Ct^{\gamma_1-4}, ∥ B ( φ ⊗ D B + S φ ) ∥ L t 2 L x 2 ([ 0 , t ]) 2 ≤ 2∥ B ( φ + ⊗ D B + S φ + ) ∥ L t 2 L x 2 ([ 0 , t ]) 2 + C t γ 1 − 4 , ∥ t 1 / 2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ≤ 2 ∥ t 1 / 2 B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , t ] ) 2 + C t γ 1 − 4 . \|t^{1/2}\mathbb{B}(\varphi\otimes D_B+S\varphi)\|_{L_{t,x}^2([0,t])}^2\le2\|t^{1/2}\mathbb{B}(\varphi^+\otimes D_B+S\varphi^+)\|_{L_{t,x}^2([0,t])}^2+Ct^{\gamma_1-4}. ∥ t 1/2 B ( φ ⊗ D B + S φ ) ∥ L t , x 2 ([ 0 , t ]) 2 ≤ 2∥ t 1/2 B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , t ]) 2 + C t γ 1 − 4 . which means the left-hand side of (345) is bounded by (here we also apply Hölder’s inequality in time)
4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , t ] ) 2 + C t − 1 + κ . 4t^{-10\alpha}\lVert B(\varphi^{+}\otimes D_{B+S}\varphi^{+})\rVert^{2}_{L^{2}_{t,x}([0,t])}+Ct^{-1+\kappa}. 4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , t ]) 2 + C t − 1 + κ . By (293) with γ = 4 \gamma=4 γ = 4 γ = γ 1 \gamma=\gamma_{1} γ = γ 1
4 ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ≲ ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ∥ φ + ( t ) ∥ L 4 γ 1 − 4 ∧ ∥ φ 0 + ∥ L 4 4 . (346) 4\lVert B(\varphi^{+}\otimes D_{B+S}\varphi^{+})\rVert^{2}_{L^{2}_{t,x}([0,t])}\lesssim\frac{\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}}{\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}-4}}\wedge\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{4}. \tag*{(346)} 4 ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , t ]) 2 ≲ ∥ φ + ( t ) ∥ L 4 γ 1 − 4 ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ∧ ∥ φ 0 + ∥ L 4 4 . (346) We discuss the following cases, with C ⋆ C_{\star} C ⋆
If ∥ φ 0 + ∥ L 4 4 ≤ C ⋆ t − 4 5 \lVert\varphi_{0}^{+}\rVert_{L^{4}}^{4}\le C_{\star}t^{-\frac{4}{5}} ∥ φ 0 + ∥ L 4 4 ≤ C ⋆ t − 5 4 (345) directly follows from the above discussion.
If ∥ φ 0 + ∥ L 4 4 ≥ C ⋆ t − 4 5 \lVert\varphi_{0}^{+}\rVert_{L^{4}}^{4}\ge C_{\star}t^{-\frac{4}{5}} ∥ φ 0 + ∥ L 4 4 ≥ C ⋆ t − 5 4 ∥ φ + ( t ) ∥ L 4 ≤ 1 2 ∥ φ 0 + ∥ L 4 \lVert\varphi^{+}(t)\rVert_{L^{4}}\le\frac{1}{2}\lVert\varphi_{0}^{+}\rVert_{L^{4}} ∥ φ + ( t ) ∥ L 4 ≤ 2 1 ∥ φ 0 + ∥ L 4 (346) we have
4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ≤ C t − 10 α ∥ φ 0 + ∥ L 4 4 ≤ C C ⋆ − 5 α 4 ∥ φ 0 + ∥ L 4 γ 1 ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) , \begin{aligned}
4t^{-10\alpha}\lVert B(\varphi^{+}\otimes D_{B+S}\varphi^{+})\rVert^{2}_{L^{2}_{t,x}([0,t])}\le Ct^{-10\alpha}\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{4} \\
&\le CC_{\star}^{-\frac{5\alpha}{4}}\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}\le\varepsilon\bigl(\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}\bigr),
\end{aligned} 4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , t ]) 2 ≤ C t − 10 α ∥ φ 0 + ∥ L 4 4 ≤ C C ⋆ − 4 5 α ∥ φ 0 + ∥ L 4 γ 1 ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) , provided that we choose C ⋆ C_{\star} C ⋆
If ∥ φ 0 + ∥ L 4 4 ≥ C ⋆ t − 4 5 \lVert\varphi_{0}^{+}\rVert_{L^{4}}^{4}\ge C_{\star}t^{-\frac{4}{5}} ∥ φ 0 + ∥ L 4 4 ≥ C ⋆ t − 5 4 ∥ φ + ( t ) ∥ L 4 ≥ 1 2 ∥ φ 0 + ∥ L 4 \lVert\varphi^{+}(t)\rVert_{L^{4}}\ge\frac{1}{2}\lVert\varphi_{0}^{+}\rVert_{L^{4}} ∥ φ + ( t ) ∥ L 4 ≥ 2 1 ∥ φ 0 + ∥ L 4 (346) ,
4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ( [ 0 , t ] ) 2 ≤ C t − 10 α ∥ φ 0 + ∥ L 4 − ( γ 1 − 4 ) ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) ≤ C C ⋆ − γ 1 − 4 4 t − 10 α + 1 5 ( γ 1 − 4 ) ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) , \begin{aligned}
4t^{-10\alpha}\lVert B(\varphi^{+}\otimes D_{B+S}\varphi^{+})\rVert^{2}_{L^{2}_{t,x}([0,t])}\le Ct^{-10\alpha}\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{-(\gamma_{1}-4)}\bigl(\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}\bigr) \\
&\le CC_{\star}^{-\frac{\gamma_{1}-4}{4}}t^{-10\alpha+\frac{1}{5}(\gamma_{1}-4)}\bigl(\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}\bigr) \\
&\le\varepsilon\bigl(\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}\bigr),
\end{aligned} 4 t − 10 α ∥ B ( φ + ⊗ D B + S φ + ) ∥ L t , x 2 ([ 0 , t ]) 2 ≤ C t − 10 α ∥ φ 0 + ∥ L 4 − ( γ 1 − 4 ) ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) ≤ C C ⋆ − 4 γ 1 − 4 t − 10 α + 5 1 ( γ 1 − 4 ) ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) , provided that we choose C ⋆ C_{\star} C ⋆
The bound (345) now follows by combining these three cases.
Lemma 5.35. Let T 0 T_{0} T 0 ε > 0 \varepsilon>0 ε > 0 C = C ε C=C_{\varepsilon} C = C ε B 0 ♯ = B 0 + + B 0 − B_{0}^{\sharp}=B_{0}^{+}+B_{0}^{-} B 0 ♯ = B 0 + + B 0 − T ∈ ( 0 , T 0 ] T\in(0,T_{0}] T ∈ ( 0 , T 0 ] (320) holds. Let B B B B + B^{+} B + g g g φ 0 = φ 0 + + φ 0 − \varphi_{0}=\varphi_{0}^{+}+\varphi_{0}^{-} φ 0 = φ 0 + + φ 0 − (321) holds. Let S , φ S,\varphi S , φ (281) –(282) and let φ + \varphi^{+} φ + (289) . Then for any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ φ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ l i n + ( t ) ⟩ V d x ∣ ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ \lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}-4}\left|\int_{\mathbb{T}^{2}}\lvert\varphi^{+}(t)\rvert^{2}\langle\varphi^{+}(t),\varphi_{\mathrm{lin}}^{+}(t)\rangle_{V}\,dx\right|\le\varepsilon\bigl(\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}}-\lVert\varphi^{+}(t)\rVert_{L^{4}}^{\gamma_{1}}\bigr)+Ct^{-1+\kappa} ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ φ + ( t ) ∣ 2 ⟨ φ + ( t ) , φ lin + ( t ) ⟩ V d x ≤ ε ( ∥ φ 0 + ∥ L 4 γ 1 − ∥ φ + ( t ) ∥ L 4 γ 1 ) + C t − 1 + κ + ∥ φ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) ∣ 2 ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) , e t Δ + ( g 0 − 1 ⋅ φ 0 − ) ⟩ V d x ∣ . (347) +\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}-4}\left|\int_{\mathbb{T}^{2}}\langle e^{t\Delta_{+}}(g_{0}^{-1}\cdot\varphi_{0}^{+})\rvert^{2}\langle e^{t\Delta_{+}}(g_{0}^{-1}\cdot\varphi_{0}^{+}),e^{t\Delta_{+}}(g_{0}^{-1}\cdot\varphi_{0}^{-})\rangle_{V}\,dx\right|. \tag*{(347)} + ∥ φ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) ∣ 2 ⟨ e t Δ + ( g 0 − 1 ⋅ φ 0 + ) , e t Δ + ( g 0 − 1 ⋅ φ 0 − ) ⟩ V d x . (347) Proof. First we replace φ + \varphi^{+} φ + φ l i n + = e t Δ B ( 0 ) φ 0 + \varphi_{\mathrm{lin}}^{+}=e^{t\Delta_{B(0)}}\varphi_{0}^{+} φ lin + = e t Δ B ( 0 ) φ 0 + (328) , we get
∥ φ + − φ l i n + ∥ L t ∞ L x 4 ( [ 0 , t ] ) ≲ t γ 1 − 4 ∥ φ 0 + ∥ L 4 . \lVert\varphi^{+}-\varphi_{\mathrm{lin}}^{+}\rVert_{L^{\infty}_{t}L^{4}_{x}([0,t])}\lesssim t^{\gamma_{1}-4}\lVert\varphi_{0}^{+}\rVert_{L^{4}}. ∥ φ + − φ lin + ∥ L t ∞ L x 4 ([ 0 , t ]) ≲ t γ 1 − 4 ∥ φ 0 + ∥ L 4 . Therefore, the error when replacing φ + \varphi^{+} φ + φ l i n + \varphi_{\mathrm{lin}}^{+} φ lin + (347) is bounded by
∥ φ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 t − γ 1 2 − 2 α ≤ ∥ φ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 2 ≤ ε C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( γ 1 − 4 γ 1 + 2 − γ 1 − 1 γ 1 + 2 ) γ 1 + 2 3 , \begin{aligned}
\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}-1}t^{\gamma_{1}-4}t^{-\frac{\gamma_{1}}{2}-2\alpha} \\
&\le\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}-1}t^{\frac{\gamma_{1}-4}{2}}\le\varepsilon C_{(5.43),\gamma_{1},4}\lVert\varphi_{0}^{+}\rVert_{L^{4}}^{\gamma_{1}+2}t+C t^{\left(\frac{\gamma_{1}-4}{\gamma_{1}+2}-\frac{\gamma_{1}-1}{\gamma_{1}+2}\right)\frac{\gamma_{1}+2}{3}},
\end{aligned} ∥ φ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 t − 2 γ 1 − 2 α ≤ ∥ φ 0 + ∥ L 4 γ 1 − 1 t 2 γ 1 − 4 ≤ ε C ( 5.43 ) , γ 1 , 4 ∥ φ 0 + ∥ L 4 γ 1 + 2 t + C t ( γ 1 + 2 γ 1 − 4 − γ 1 + 2 γ 1 − 1 ) 3 γ 1 + 2 , which is acceptable by the same argument as for (342) . Now let g g g B B B B + B^{+} B + B − B^{-} B − g − 1 ⋅ B = B + + B − g^{-1}\mathbin{\cdot}B=B^{+}+B^{-} g − 1 ⋅ B = B + + B −
Note that
∣ ∫ T 2 ∣ ϕ l i n + ∣ 2 ⟨ ϕ l i n + , ϕ l i n − ⟩ V d x ∣ = ∣ ∫ T 2 ∣ g − 1 ⋅ ϕ l i n + ∣ 2 ⟨ g − 1 ⋅ ϕ l i n + , g − 1 ⋅ ϕ l i n − ⟩ V d x ∣ . \left|\int_{\mathbb{T}^{2}}|\phi_{\mathrm{lin}}^{+}|^{2}\langle\phi_{\mathrm{lin}}^{+},\phi_{\mathrm{lin}}^{-}\rangle_{V}\,\mathrm{d}x\right|=\left|\int_{\mathbb{T}^{2}}|g^{-1}\mathbin{\cdot}\phi_{\mathrm{lin}}^{+}|^{2}\langle g^{-1}\mathbin{\cdot}\phi_{\mathrm{lin}}^{+},g^{-1}\mathbin{\cdot}\phi_{\mathrm{lin}}^{-}\rangle_{V}\,\mathrm{d}x\right|. ∫ T 2 ∣ ϕ lin + ∣ 2 ⟨ ϕ lin + , ϕ lin − ⟩ V d x = ∫ T 2 ∣ g − 1 ⋅ ϕ lin + ∣ 2 ⟨ g − 1 ⋅ ϕ lin + , g − 1 ⋅ ϕ lin − ⟩ V d x . By Lemma 5.11 (applied with t 0 t_{0} t 0 t t t
∥ ( g − 1 ⋅ ϕ l i n + ) ( t ) − e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) ∥ L 4 ≲ t 1 3 ∥ ϕ 0 + ∥ L 4 , \left\|(g^{-1}\mathbin{\cdot}\phi_{\mathrm{lin}}^{+})(t)-e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{+})\right\|_{L^{4}}\lesssim t^{\frac{1}{3}}\|\phi_{0}^{+}\|_{L^{4}}, ( g − 1 ⋅ ϕ lin + ) ( t ) − e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) L 4 ≲ t 3 1 ∥ ϕ 0 + ∥ L 4 , ∥ t κ 2 ( ( g − 1 ⋅ ϕ l i n − ) ( t ) − e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ) ∥ L 4 ≲ t 1 3 ∥ t κ 2 e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ∥ L t ∞ L x 4 . \left\|t^{\frac{\kappa}{2}}\left((g^{-1}\mathbin{\cdot}\phi_{\mathrm{lin}}^{-})(t)-e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{-})\right)\right\|_{L^{4}}\lesssim t^{\frac{1}{3}}\left\|t^{\frac{\kappa}{2}}e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{-})\right\|_{L_{t}^{\infty}L_{x}^{4}}. t 2 κ ( ( g − 1 ⋅ ϕ lin − ) ( t ) − e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ) L 4 ≲ t 3 1 t 2 κ e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) L t ∞ L x 4 . Therefore, we get (applying (321) )
∥ ϕ 0 + ∥ L 4 γ 1 − 4 ∣ ∫ T 2 ∣ ϕ l i n + ( t ) ∣ 2 ⟨ ϕ l i n + ( t ) , ϕ l i n − ( t ) ⟩ V d x − ∫ T 2 ∣ e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) ∣ 2 ⟨ e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) , e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ⟩ V d x ∣ ≲ t 1 3 − κ 2 ∥ ϕ 0 + ∥ L 4 γ 1 − 1 ∥ t κ 2 e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ∥ L t ∞ L x 4 ≲ ∥ ϕ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 , \begin{aligned}
\|\phi_{0}^{+}\|_{L^{4}}^{\gamma_{1}-4}\bigg|&
\int_{\mathbb{T}^{2}}|\phi_{\mathrm{lin}}^{+}(t)|^{2}\langle\phi_{\mathrm{lin}}^{+}(t),\phi_{\mathrm{lin}}^{-}(t)\rangle_{V}\,\mathrm{d}x\\
&-\int_{\mathbb{T}^{2}}\left|e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{+})\right|^{2}
\left\langle e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{+}),e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{-})\right\rangle_{V}\,\mathrm{d}x\bigg|\\
&\lesssim t^{\frac{1}{3}-\frac{\kappa}{2}}\|\phi_{0}^{+}\|_{L^{4}}^{\gamma_{1}-1}
\left\|t^{\frac{\kappa}{2}}e^{t\Delta_{+}}(g_{0}^{-1}\mathbin{\cdot}\phi_{0}^{-})\right\|_{L_{t}^{\infty}L_{x}^{4}}
\lesssim\|\phi_{0}^{+}\|_{L^{4}}^{\gamma_{1}-1}t^{\gamma_{1}-4},
\end{aligned} ∥ ϕ 0 + ∥ L 4 γ 1 − 4 ∫ T 2 ∣ ϕ lin + ( t ) ∣ 2 ⟨ ϕ lin + ( t ) , ϕ lin − ( t ) ⟩ V d x − ∫ T 2 e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) 2 ⟨ e t Δ + ( g 0 − 1 ⋅ ϕ 0 + ) , e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) ⟩ V d x ≲ t 3 1 − 2 κ ∥ ϕ 0 + ∥ L 4 γ 1 − 1 t 2 κ e t Δ + ( g 0 − 1 ⋅ ϕ 0 − ) L t ∞ L x 4 ≲ ∥ ϕ 0 + ∥ L 4 γ 1 − 1 t γ 1 − 4 , which can then be estimated as in (342) (with ε \varepsilon ε 1 / 8 1/8 1/8 (347) now follows.
We also record the following perturbative result on D B + S F B + S D_{B+S}F_{B+S} D B + S F B + S
Lemma 5.36. There exists some constant C C C t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ D B + S F B + S ( t ) ∥ L 2 ≤ C t − 1 2 ( 1 + ∥ t 1 2 F B + ∥ L t 2 L x ∞ ( [ 0 , T ] ) ) ∥ D B + F B + ∥ L t , x 2 ( [ 0 , T ] ) + C t − 3 2 + γ 0 − 2 50 − 2 a − 2 r + C t − 1 2 + γ 0 − 2 50 − 2 a − 2 r ( ∥ F B + ( t ) ∥ L 2 + C ∥ ϕ 0 + ∥ L 4 γ 1 − 4 ∥ ϕ + ( t ) ∥ L 4 γ 1 ) . (348) \begin{aligned}
\|D_{B+S}F_{B+S}(t)\|_{L^{2}}\leq{}&
Ct^{-\frac{1}{2}}\left(1+\|t^{\frac{1}{2}}F_{B+}\|_{L_{t}^{2}L_{x}^{\infty}([0,T])}\right)\|D_{B+}F_{B+}\|_{L_{t,x}^{2}([0,T])}\\
&+Ct^{-\frac{3}{2}+\frac{\gamma_{0}-2}{50}-2a-\frac{2}{r}}\\
&+Ct^{-\frac{1}{2}+\frac{\gamma_{0}-2}{50}-2a-\frac{2}{r}}
\left(\|F_{B+}(t)\|_{L^{2}}+C\sqrt{\|\phi_{0}^{+}\|_{L^{4}}^{\gamma_{1}-4}\|\phi_{+}(t)\|_{L^{4}}^{\gamma_{1}}}\right). \tag*{(348)}
\end{aligned} ∥ D B + S F B + S ( t ) ∥ L 2 ≤ C t − 2 1 ( 1 + ∥ t 2 1 F B + ∥ L t 2 L x ∞ ([ 0 , T ]) ) ∥ D B + F B + ∥ L t , x 2 ([ 0 , T ]) + C t − 2 3 + 50 γ 0 − 2 − 2 a − r 2 + C t − 2 1 + 50 γ 0 − 2 − 2 a − r 2 ( ∥ F B + ( t ) ∥ L 2 + C ∥ ϕ 0 + ∥ L 4 γ 1 − 4 ∥ ϕ + ( t ) ∥ L 4 γ 1 ) . (348) Proof. Expanding D B + S F B + S D_{B+S}F_{B+S} D B + S F B + S
∥ D B + S F B + S ∥ L 2 ≤ ∥ D B F B ∥ L 2 + ∥ D B ( curl B S + [ S 1 , S 2 ] ) ∥ L 2 + ∥ S ∥ L ∞ ∥ F B + S ∥ L 2 . \|D_{B+S}F_{B+S}\|_{L^{2}}\leq\|D_{B}F_{B}\|_{L^{2}}+\|D_{B}(\operatorname{curl}_{B}S+[S^{1},S^{2}])\|_{L^{2}}+\|S\|_{L^{\infty}}\|F_{B+S}\|_{L^{2}}. ∥ D B + S F B + S ∥ L 2 ≤ ∥ D B F B ∥ L 2 + ∥ D B ( curl B S + [ S 1 , S 2 ]) ∥ L 2 + ∥ S ∥ L ∞ ∥ F B + S ∥ L 2 . By Proposition 5.7 (recall that in the setting of Proposition 5.27, we apply this proposition with K = T − 2 a K=T^{-2a} K = T − 2 a L ∞ L^{\infty} L ∞ (327) on F B + ( t ) F_{B+}(t) F B + ( t )
∥ D B F B ( t ) ∥ L 2 ≤ ∥ D B + F B + ( t ) ∥ L 2 + C t − 1 − κ 2 − 3 a ≲ t − 1 2 ( 1 + ∥ t 1 2 F B + ∥ L t 2 L x ∞ ( [ 0 , T ] ) ) ∥ D B + F B + ∥ L t , x 2 ( [ 0 , T ] ) + t − 1 − κ 2 − 3 a . \begin{aligned}
\|D_{B}F_{B}(t)\|_{L^{2}}&\leq\|D_{B+}F_{B+}(t)\|_{L^{2}}+Ct^{-1-\frac{\kappa}{2}-3a}\\
&\lesssim t^{-\frac{1}{2}}\left(1+\|t^{\frac{1}{2}}F_{B+}\|_{L_{t}^{2}L_{x}^{\infty}([0,T])}\right)\|D_{B+}F_{B+}\|_{L_{t,x}^{2}([0,T])}+t^{-1-\frac{\kappa}{2}-3a}.
\end{aligned} ∥ D B F B ( t ) ∥ L 2 ≤ ∥ D B + F B + ( t ) ∥ L 2 + C t − 1 − 2 κ − 3 a ≲ t − 2 1 ( 1 + ∥ t 2 1 F B + ∥ L t 2 L x ∞ ([ 0 , T ]) ) ∥ D B + F B + ∥ L t , x 2 ([ 0 , T ]) + t − 1 − 2 κ − 3 a . By (330) , we have
∥ D B ( curl B S + [ S 1 , S 2 ] ) ∥ L 2 ≲ t − 3 2 + γ 0 − 2 50 − 2 a − 2 r . \|D_{B}(\operatorname{curl}_{B}S+[S^{1},S^{2}])\|_{L^{2}}\lesssim t^{-\frac{3}{2}+\frac{\gamma_{0}-2}{50}-2a-\frac{2}{r}}. ∥ D B ( curl B S + [ S 1 , S 2 ]) ∥ L 2 ≲ t − 2 3 + 50 γ 0 − 2 − 2 a − r 2 . By (330) and (332) , we have
∥ S ∥ L ∞ ∥ F B + S ∥ L 2 ≲ t − 1 2 + γ 0 − 2 50 − 2 a − 2 r ⋅ ( ∥ F B + ( t ) ∥ L 2 + ∥ D B + F B + ∥ L t , x 2 ( [ 0 , t ] ) + t − 1 + κ 2 + t − 1 2 − 5 a ( ∥ B ( ϕ ⊗ D B + S ϕ ) ∥ L t 1 L x 2 ( [ 0 , t ] ) + ∥ t 1 2 B ( ϕ ⊗ D B + S ϕ ) ∥ L t , x 2 ( [ 0 , t ] ) ) + ∥ D B + ( t ) e t Δ F B 0 − ∥ L ∞ + ∥ e t Δ F B 0 − ∥ L ∞ 2 ) . \begin{aligned}
\|S\|_{L^{\infty}}\|F_{B+S}\|_{L^{2}}\lesssim{}&
t^{-\frac{1}{2}+\frac{\gamma_{0}-2}{50}-2a-\frac{2}{r}}\cdot\bigg(
\|F_{B+}(t)\|_{L^{2}}+\|D_{B+}F_{B+}\|_{L_{t,x}^{2}([0,t])}+t^{-\frac{1+\kappa}{2}}\\
&+t^{-\frac{1}{2}-5a}\left(\|\mathcal{B}(\phi\otimes D_{B+S}\phi)\|_{L_{t}^{1}L_{x}^{2}([0,t])}
+\|t^{\frac{1}{2}}\mathcal{B}(\phi\otimes D_{B+S}\phi)\|_{L_{t,x}^{2}([0,t])}\right)\\
&+\|D_{B+}(t)e^{t\Delta_{F}}B_{0}^{-}\|_{L^{\infty}}
+\|e^{t\Delta_{F}}B_{0}^{-}\|_{L^{\infty}}^{2}\bigg).
\end{aligned} ∥ S ∥ L ∞ ∥ F B + S ∥ L 2 ≲ t − 2 1 + 50 γ 0 − 2 − 2 a − r 2 ⋅ ( ∥ F B + ( t ) ∥ L 2 + ∥ D B + F B + ∥ L t , x 2 ([ 0 , t ]) + t − 2 1 + κ + t − 2 1 − 5 a ( ∥ B ( ϕ ⊗ D B + S ϕ ) ∥ L t 1 L x 2 ([ 0 , t ]) + ∥ t 2 1 B ( ϕ ⊗ D B + S ϕ ) ∥ L t , x 2 ([ 0 , t ]) ) + ∥ D B + ( t ) e t Δ F B 0 − ∥ L ∞ + ∥ e t Δ F B 0 − ∥ L ∞ 2 ) . which is acceptable by the assumption (320) and Lemma 5.34. The lemma now follows by combining the previous estimates.
Covariant ansatz
We recall from (11) that we are interested in the stochastic Yang–Mills–Higgs equations in the infinitesimal DeTurck gauge. They are formally given by
∂ t A = − D A ∗ F A − D A D A ∗ ( A − B ) + B ( Φ ⊗ D A Φ ) + ξ , A ( 0 ) = A 0 , ∂ t Φ = Δ A Φ + ( D A ∗ ( A − B ) ) Φ − ∣ Φ ∣ 2 Φ − M Φ + ζ , Φ ( 0 ) = Φ 0 . (349) \begin{aligned}
\partial_t A &= -D_A^*F_A-D_A D_A^*(A-B)+B(\Phi\otimes D_A\Phi)+\xi,\qquad A(0)=A_0,\\
\partial_t\Phi&= \Delta_A\Phi+(D_A^*(A-B))\Phi-|\Phi|^2\Phi-M\Phi+\zeta,\qquad\Phi(0)=\Phi_0. \tag*{(349)}
\end{aligned} ∂ t A ∂ t Φ = − D A ∗ F A − D A D A ∗ ( A − B ) + B ( Φ ⊗ D A Φ ) + ξ , A ( 0 ) = A 0 , = Δ A Φ + ( D A ∗ ( A − B )) Φ − ∣Φ ∣ 2 Φ − M Φ + ζ , Φ ( 0 ) = Φ 0 . (349) where B B B
∂ t B = − D B ∗ F B , B ( 0 ) = A 0 . (350) \partial_t B=-D_B^*F_B,\qquad B(0)=A_0. \tag*{(350)} ∂ t B = − D B ∗ F B , B ( 0 ) = A 0 . (350) For suitable initial data A 0 A_0 A 0 (350) has already been established in § 6 . The local well-posedness of (349) , however, is much more challenging. It can be established using methods similar to those used in [CCHS22, BC26a], which proved the local well-posedness of the stochastic Yang–Mills equations in the original DeTurck gauge. For our purposes, however, the resulting local well-posedness theorem for (349) would be insufficient. As already discussed in Remark 2.21, the reason for this is that the methods in [CCHS22, BC26a] are non-covariant. Used directly, the arguments therefore lead to local results for (349) that involve gauge dependent norms of the initial data, whereas our global iteration scheme only uses gauge invariant norms23
Instead of using the local theory from [CCHS22, BC26a], we develop a covariant local theory of (349) , which will be distributed over § 7 –§ 10 . In this section, we derive a covariant paracontrolled ansatz, which is the starting point of our covariant local theory. Some aspects of this were already described in § 3.2 , i.e., in the statement and discussion of our covariant local well-posedness result. However, given the sheer length of this article, we believe that occasional repetitions are helpful to the reader, and we will therefore start the discussion of our covariant local theory anew (and go into much more detail).
Smooth approximation and stochastic objects
In order to discuss (349) rigorously, we need to first replace the space-time white noises ξ \xi ξ ζ \zeta ζ (349) , we work with different, covariant smooth approximations. They were previously defined in Definition 2.12 and are given by
ξ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (351) \xi_{B,\varepsilon}(t,x)=\int_0^t\!ds\int_{\mathbb{T}^2}\!dy\,\rho_\varepsilon'(t-s)q_B(s,y;t,x)\xi(s,y), \tag*{(351)} ξ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (351) ζ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) . \zeta_{B,\varepsilon}(t,x)=\int_0^t\!ds\int_{\mathbb{T}^2}\!dy\,\rho_\varepsilon'(t-s)p_B(s,y;t,x)\zeta(s,y). ζ B , ε ( t , x ) = ∫ 0 t d s ∫ T 2 d y ρ ε ′ ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) . Here, ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0 ρ ε \rho_\varepsilon ρ ε (42) , ρ ε ′ \rho_\varepsilon' ρ ε ′ p B p_B p B q B q_B q B Definition 2.1 .
The precise form of the covariant smooth approximations from (351) will be useful for many different reasons. For now, we only briefly discuss two aspects of (351) , for which we focus on ξ B , ε \xi_{B,\varepsilon} ξ B , ε ξ B , ε \xi_{B,\varepsilon} ξ B , ε ξ \xi ξ ε → 0 \varepsilon\to0 ε → 0 ξ \xi ξ ρ ε ′ ( t − s ) \rho_\varepsilon'(t-s) ρ ε ′ ( t − s ) δ ( t − s ) \delta(t-s) δ ( t − s ) ε → 0 \varepsilon\to0 ε → 0 q B ( s , y ; t , x ) q_B(s,y;t,x) q B ( s , y ; t , x ) δ ( x − y ) id g ⊕ g \delta(x-y)\operatorname{id}_{\mathfrak{g}\oplus\mathfrak{g}} δ ( x − y ) id g ⊕ g s → t s\to t s → t ε → 0 \varepsilon\to0 ε → 0 ξ B , ε \xi_{B,\varepsilon} ξ B , ε g ∈ G g\in\mathcal{G} g ∈ G q B q_B q B
Ad g ( x ) ξ B , ε ( t , x ) = ∬ d s d y ρ ε ′ ( t − s ) q g ∙ B ( s , y ; t , x ) ( Ad g ξ ) ( s , y ) = l a w ξ g ∙ B , ε ( t , x ) . \operatorname{Ad}_{g(x)}\xi_{B,\varepsilon}(t,x)=\iint\mathrm{d}s\mathrm{d}y\,\rho'_{\varepsilon}(t-s)q_{g\mathbin{\bullet}B}(s,y;t,x)(\operatorname{Ad}_{g}\xi)(s,y)\mathrel{\mathop=^{\mathrm{law}}}\xi_{g\mathbin{\bullet}B,\varepsilon}(t,x). Ad g ( x ) ξ B , ε ( t , x ) = ∬ d s d y ρ ε ′ ( t − s ) q g ∙ B ( s , y ; t , x ) ( Ad g ξ ) ( s , y ) = law ξ g ∙ B , ε ( t , x ) . where the equality in law holds as processes in ( t , x ) (t,x) ( t , x ) (349) , we need to make use of the renormalisation terms previously introduced in Definition 2.15. Recall that we defined
C ρ A = C ρ ( Cov g g + Cov v g ) − 1 8 π Cov v g , C_{\rho}^{A}=C_{\rho}\left(\operatorname{Cov}_{\mathfrak{g}}^{g}+\operatorname{Cov}_{\mathfrak{v}}^{g}\right)-\frac{1}{8\pi}\operatorname{Cov}_{\mathfrak{v}}^{g}, C ρ A = C ρ ( Cov g g + Cov v g ) − 8 π 1 Cov v g , C ρ , ε Φ = ( − σ ρ , ε 2 + C ρ ) Cov v g + ( dim ( V ) + 2 ) σ ρ , ε 2 id V − M , (352) C_{\rho,\varepsilon}^{\Phi}=\left(-\sigma_{\rho,\varepsilon}^{2}+C_{\rho}\right)\operatorname{Cov}_{\mathfrak{v}}^{g}+\left(\dim(\mathbb{V})+2\right)\sigma_{\rho,\varepsilon}^{2}\operatorname{id}_{\mathbb{V}}-M, \tag*{(352)} C ρ , ε Φ = ( − σ ρ , ε 2 + C ρ ) Cov v g + ( dim ( V ) + 2 ) σ ρ , ε 2 id V − M , (352) where Cov g g \operatorname{Cov}_{\mathfrak{g}}^{g} Cov g g Cov v g \operatorname{Cov}_{\mathfrak{v}}^{g} Cov v g Cov V g \operatorname{Cov}_{\mathbb{V}}^{g} Cov V g
The counterterms in (352) , including the non-divergent C ρ C_{\rho} C ρ 1 8 π \frac{1}{8\pi} 8 π 1 ρ ∈ P \rho\in\mathcal{P} ρ ∈ P M M M Φ \Phi Φ (349) , is included in (352) for notational convenience, and could have been kept separate. Equipped with our covariant smooth approximations of ξ \xi ξ ζ \zeta ζ (44) , i.e., our covariant smooth approximation of the stochastic YMH equations (349) . In coordinates, it can be written as
∂ t A ε k = D A ε , j F A ε j k + D A ε k D A ε j ( A ε − B ) j + B ( Φ ε ⊗ D A ε k Φ ε ) + C ρ A ( A ε − B ) k + ξ B , ε k , \partial_t A_{\varepsilon}^{k}=\mathbf{D}_{A_{\varepsilon},j}F_{A_{\varepsilon}}^{jk}+\mathbf{D}_{A_{\varepsilon}}^{k}\mathbf{D}_{A_{\varepsilon}}^{j}(A_{\varepsilon}-B)_{j}+\mathbf{B}(\Phi_{\varepsilon}\otimes\mathbf{D}_{A_{\varepsilon}}^{k}\Phi_{\varepsilon})+C_{\rho}^{A}(A_{\varepsilon}-B)^{k}+\xi_{B,\varepsilon}^{k}, ∂ t A ε k = D A ε , j F A ε j k + D A ε k D A ε j ( A ε − B ) j + B ( Φ ε ⊗ D A ε k Φ ε ) + C ρ A ( A ε − B ) k + ξ B , ε k , ∂ t Φ ε = Δ A ε Φ ε − ( D A ε j ( A ε − B ) j ) Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε , \partial_t\Phi_{\varepsilon}=\Delta_{A_{\varepsilon}}\Phi_{\varepsilon}-(\mathbf{D}_{A_{\varepsilon}}^{j}(A_{\varepsilon}-B)_{j})\Phi_{\varepsilon}-\lvert\Phi_{\varepsilon}\rvert^{2}\Phi_{\varepsilon}+C_{\rho,\varepsilon}^{\Phi}(\Phi_{\varepsilon})+\zeta_{B,\varepsilon}, ∂ t Φ ε = Δ A ε Φ ε − ( D A ε j ( A ε − B ) j ) Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε , A ε ( 0 ) = A 0 , Φ ε ( 0 ) = Φ 0 . (353) A_{\varepsilon}(0)=A_{0},\qquad\Phi_{\varepsilon}(0)=\Phi_{0}. \tag*{(353)} A ε ( 0 ) = A 0 , Φ ε ( 0 ) = Φ 0 . (353) We emphasise that B B B (350) and, in particular, does not depend on ε \varepsilon ε (353) in terms of the difference C ε : = A ε − B C_{\varepsilon}:=A_{\varepsilon}-B C ε := A ε − B
( ∂ t − Δ B ) C ε k − 2 [ F B k j , C ε , j ] = [ C ε , j , 2 D B j C ε k − D B k C ε j ] + [ C ε , j , [ C ε j , C ε k ] ] (\partial_t-\Delta_{B})C_{\varepsilon}^{k}-2[F_{B}^{kj},C_{\varepsilon,j}]=[C_{\varepsilon,j},2\mathbf{D}_{B}^{j}C_{\varepsilon}^{k}-\mathbf{D}_{B}^{k}C_{\varepsilon}^{j}]+[C_{\varepsilon,j},[C_{\varepsilon}^{j},C_{\varepsilon}^{k}]] ( ∂ t − Δ B ) C ε k − 2 [ F B k j , C ε , j ] = [ C ε , j , 2 D B j C ε k − D B k C ε j ] + [ C ε , j , [ C ε j , C ε k ]] + B ( Φ ε ⊗ D B k Φ ε ) + B ( Φ ε ⊗ C ε k Φ ε ) + C ρ A ( C ε k ) + ξ B , ε k , \qquad+\mathbf{B}(\Phi_{\varepsilon}\otimes\mathbf{D}_{B}^{k}\Phi_{\varepsilon})+\mathbf{B}(\Phi_{\varepsilon}\otimes C_{\varepsilon}^{k}\Phi_{\varepsilon})+C_{\rho}^{A}(C_{\varepsilon}^{k})+\xi_{B,\varepsilon}^{k}, + B ( Φ ε ⊗ D B k Φ ε ) + B ( Φ ε ⊗ C ε k Φ ε ) + C ρ A ( C ε k ) + ξ B , ε k , ( ∂ t − Δ B ) Φ ε = 2 C ε , j D B j Φ ε + C ε j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (354) (\partial_t-\Delta_{B})\Phi_{\varepsilon}=2C_{\varepsilon,j}\mathbf{D}_{B}^{j}\Phi_{\varepsilon}+C_{\varepsilon}^{j}C_{\varepsilon,j}\Phi_{\varepsilon}-\lvert\Phi_{\varepsilon}\rvert^{2}\Phi_{\varepsilon}+C_{\rho,\varepsilon}^{\Phi}(\Phi_{\varepsilon})+\zeta_{B,\varepsilon}. \tag*{(354)} ( ∂ t − Δ B ) Φ ε = 2 C ε , j D B j Φ ε + C ε j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (354) We note that in our formulation of (354) , we used that there are two cancellations between (110) and (111) . We also emphasise that the term 2 [ F B k j , C ε , j ] 2[F_{B}^{kj},C_{\varepsilon,j}] 2 [ F B k j , C ε , j ] (354) . The reason is that the dependence of the local time of existence on the size of the curvature tensor F B F_B F B Δ B C ε k + 2 [ F B k j , C ε , j ] \Delta_B C_{\varepsilon}^{k}+2[F_{B}^{kj},C_{\varepsilon,j}] Δ B C ε k + 2 [ F B k j , C ε , j ] Δ B C ε k \Delta_B C_{\varepsilon}^{k} Δ B C ε k (354) , we will work covariantly with respect to the connection one-form B B B
Notation 6.1. Given a solution B B B
D = D B and F = F B . D=D_B\qquad\text{and}\qquad F=F_B. D = D B and F = F B . That is, whenever the connection one-form is omitted in covariant derivatives or curvature tensors, it is given by B B B B B B Δ B \Delta_B Δ B
( ad ( F ) C ε ) k = ad ( F k j ) C ε , j = [ F k j , C ε , j ] . (\operatorname{ad}(F)C_{\varepsilon})^{k}=\operatorname{ad}(F^{kj})C_{\varepsilon,j}=[F^{kj},C_{\varepsilon,j}]. ( ad ( F ) C ε ) k = ad ( F k j ) C ε , j = [ F k j , C ε , j ] . That is, we interpret ad ( F ) \operatorname{ad}(F) ad ( F ) g 2 \mathfrak{g}^{2} g 2 g 2 \mathfrak{g}^{2} g 2 k k k (354) and related equations. In other words, unless the components are contracted, we no longer write out the components of C ε C_{\varepsilon} C ε ad ( F ) C ε \operatorname{ad}(F)C_{\varepsilon} ad ( F ) C ε D C ε j DC_{\varepsilon}^{j} D C ε j g 2 \mathfrak{g}^{2} g 2
In our simplified notation, (354) then takes the form
( ∂ t − Δ B − 2 ad ( F ) ) C ε = [ C ε , j , 2 D j C ε − D C j ] + [ C ε , j , [ C j , C ε ] ] + B ( Φ ε ⊗ D Φ ε ) + B ( Φ ε ⊗ C ε Φ ε ) + C ρ A ( C ε ) + ξ B , ε , ( ∂ t − Δ B ) Φ ε = 2 C ε , j D j Φ ε + C ε , j j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (355) \begin{aligned}
(\partial_t-\Delta_B-2\operatorname{ad}(F))C_\varepsilon
&=[C_{\varepsilon,j},2D^jC_\varepsilon-DC^j]+[C_{\varepsilon,j},[C^j,C_\varepsilon]]\\
&\quad+\mathbf{B}(\Phi_\varepsilon\otimes D\Phi_\varepsilon)+\mathbf{B}(\Phi_\varepsilon\otimes C_\varepsilon\Phi_\varepsilon)+C_\rho^A(C_\varepsilon)+\xi_{B,\varepsilon},\\
(\partial_t-\Delta_B)\Phi_\varepsilon
&=2C_{\varepsilon,j}D^j\Phi_\varepsilon+C_{\varepsilon,j}^jC_{\varepsilon,j}\Phi_\varepsilon-\lvert\Phi_\varepsilon\rvert^2\Phi_\varepsilon+C_{\rho,\varepsilon}^{\Phi}(\Phi_\varepsilon)+\zeta_{B,\varepsilon}. \tag*{(355)}
\end{aligned} ( ∂ t − Δ B − 2 ad ( F )) C ε ( ∂ t − Δ B ) Φ ε = [ C ε , j , 2 D j C ε − D C j ] + [ C ε , j , [ C j , C ε ]] + B ( Φ ε ⊗ D Φ ε ) + B ( Φ ε ⊗ C ε Φ ε ) + C ρ A ( C ε ) + ξ B , ε , = 2 C ε , j D j Φ ε + C ε , j j C ε , j Φ ε − ∣ Φ ε ∣ 2 Φ ε + C ρ , ε Φ ( Φ ε ) + ζ B , ε . (355) We now recall (2.24)–(2.30), i.e., the definitions of the deterministic and random terms in our covariant ansatz. The deterministic terms S S S φ \varphi φ
( ∂ t − Δ B − 2 ad ( F ) ) S = [ S j , 2 D j S − D S j ] + [ S j , [ S j , S ] ] + B ( φ ⊗ D φ ) + B ( φ ⊗ S φ ) , (356) (\partial_t-\Delta_B-2\operatorname{ad}(F))S=[S_j,2D^jS-DS^j]+[S_j,[S^j,S]]+\mathbf{B}(\varphi\otimes D\varphi)+\mathbf{B}(\varphi\otimes S\varphi), \tag*{(356)} ( ∂ t − Δ B − 2 ad ( F )) S = [ S j , 2 D j S − D S j ] + [ S j , [ S j , S ]] + B ( φ ⊗ D φ ) + B ( φ ⊗ S φ ) , (356) ( ∂ t − Δ B ) φ = 2 S j D j φ + S j j S j φ − ∣ φ ∣ 2 φ , (357) (\partial_t-\Delta_B)\varphi=2S_jD^j\varphi+S_j^jS_j\varphi-\lvert\varphi\rvert^2\varphi, \tag*{(357)} ( ∂ t − Δ B ) φ = 2 S j D j φ + S j j S j φ − ∣ φ ∣ 2 φ , (357) S ( 0 ) = 0 , φ ( 0 ) = φ 0 . S(0)=0,\qquad\varphi(0)=\varphi_0. S ( 0 ) = 0 , φ ( 0 ) = φ 0 . Here, in keeping with the notation from § 6 , we denote the initial data of the Higgs field by φ 0 = d e f Φ 0 \varphi_0\overset{\mathrm{def}}{=}\Phi_0 φ 0 = def Φ 0 φ 0 \varphi_0 φ 0 φ 0 + \varphi_0^+ φ 0 + φ 0 − \varphi_0^- φ 0 − § 6 . The covariant linear stochastic objects are the solutions of the inhomogeneous, linear part of (355) , i.e.,
( ∂ t − Δ B − 2 ad ( F ) ) A ˚ B , ε = ξ B , ε , A ˚ B , ε ( 0 ) = 0 , ( ∂ t − Δ B ) Φ ˚ B , ε = ζ B , ε , Φ ˚ B , ε ( 0 ) = 0. (358) \begin{aligned}
(\partial_t-\Delta_B-2\operatorname{ad}(F))\mathring{\mathcal A}_{B,\varepsilon}&=\xi_{B,\varepsilon},\qquad\mathring{\mathcal A}_{B,\varepsilon}(0)=0,\\
(\partial_t-\Delta_B)\mathring{\Phi}_{B,\varepsilon}&=\zeta_{B,\varepsilon},\qquad\mathring{\Phi}_{B,\varepsilon}(0)=0. \tag*{(358)}
\end{aligned} ( ∂ t − Δ B − 2 ad ( F )) A ˚ B , ε ( ∂ t − Δ B ) Φ ˚ B , ε = ξ B , ε , A ˚ B , ε ( 0 ) = 0 , = ζ B , ε , Φ ˚ B , ε ( 0 ) = 0. (358) From the definitions of p B p_B p B q B q_B q B ξ B , ε \xi_{B,\varepsilon} ξ B , ε ζ B , ε \zeta_{B,\varepsilon} ζ B , ε
A ˚ B , ε ( t , x ) = ∬ d s d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (359) \mathring{\mathcal A}_{B,\varepsilon}(t,x)=\iint ds\,dy\,\rho_\varepsilon(t-s)q_B(s,y;t,x)\xi(s,y), \tag*{(359)} A ˚ B , ε ( t , x ) = ∬ d s d y ρ ε ( t − s ) q B ( s , y ; t , x ) ξ ( s , y ) , (359) Φ ˚ B , ε ( t , x ) = ∬ d s d y ρ ε ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) . \mathring{\Phi}_{B,\varepsilon}(t,x)=\iint ds\,dy\,\rho_\varepsilon(t-s)p_B(s,y;t,x)\zeta(s,y). Φ ˚ B , ε ( t , x ) = ∬ d s d y ρ ε ( t − s ) p B ( s , y ; t , x ) ζ ( s , y ) . This can be seen by acting on the right-hand sides of (359) with ∂ t − Δ B − 2 ad ( F ) \partial_t-\Delta_B-2\operatorname{ad}(F) ∂ t − Δ B − 2 ad ( F ) ∂ t − Δ B \partial_t-\Delta_B ∂ t − Δ B q B q_B q B p B p_B p B
( ∂ t − Δ B − 2 ad ( F ) ) A ˚ B , ε , j ∘ = [ A ˚ B , ε , j ∙ , 2 D j A ˚ B , ε − D A ˚ B , ε j ] , (360) (\partial_t-\Delta_B-2\operatorname{ad}(F))\mathring{\mathcal A}_{B,\varepsilon,j}^{\circ}=[\mathring{\mathcal A}_{B,\varepsilon,j}^{\bullet},2D^j\mathring{\mathcal A}_{B,\varepsilon}-D\mathring{\mathcal A}_{B,\varepsilon}^j], \tag*{(360)} ( ∂ t − Δ B − 2 ad ( F )) A ˚ B , ε , j ∘ = [ A ˚ B , ε , j ∙ , 2 D j A ˚ B , ε − D A ˚ B , ε j ] , (360) ( ∂ t − Δ B − 2 ad ( F ) ) A ˚ B , ε ∙ = B ( Φ ˚ B , ε ∙ ⊗ D Φ ˚ B , ε ) , (361) (\partial_t-\Delta_B-2\operatorname{ad}(F))\mathring{\mathcal A}_{B,\varepsilon}^{\bullet}=\mathbf{B}(\mathring{\Phi}_{B,\varepsilon}^{\bullet}\otimes D\mathring{\Phi}_{B,\varepsilon}), \tag*{(361)} ( ∂ t − Δ B − 2 ad ( F )) A ˚ B , ε ∙ = B ( Φ ˚ B , ε ∙ ⊗ D Φ ˚ B , ε ) , (361) ( ∂ t − Δ B ) Φ ˚ B , ε ∘ = 2 Φ ˚ B , ε ∙ D j Φ ˚ B , ε . (362) (\partial_t-\Delta_B)\mathring{\Phi}_{B,\varepsilon}^{\circ}=2\mathring{\Phi}_{B,\varepsilon}^{\bullet}D_j\mathring{\Phi}_{B,\varepsilon}. \tag*{(362)} ( ∂ t − Δ B ) Φ ˚ B , ε ∘ = 2 Φ ˚ B , ε ∙ D j Φ ˚ B , ε . (362) After this recap of (2.24)–(2.30), we now introduce nonlinear remainders, which were not defined in § 3.2 . The nonlinear remainders are given by
W ε : = A ε − B − A ˚ B , ε , Z ε : = A ε − B − S − A ˚ B , ε ∘ − A ˚ B , ε ∙ , w ε : = Φ ε − Φ ˚ B , ε , z ε : = Φ ε − φ − Φ ˚ B , ε − Φ ˚ B , ε ∘ . (363) \begin{aligned}
W_\varepsilon&:=A_\varepsilon-B-\mathring{\mathcal A}_{B,\varepsilon},&
Z_\varepsilon&:=A_\varepsilon-B-S-\mathring{\mathcal A}_{B,\varepsilon}^{\circ}-\mathring{\mathcal A}_{B,\varepsilon}^{\bullet},\\
w_\varepsilon&:=\Phi_\varepsilon-\mathring{\Phi}_{B,\varepsilon},&
z_\varepsilon&:=\Phi_\varepsilon-\varphi-\mathring{\Phi}_{B,\varepsilon}-\mathring{\Phi}_{B,\varepsilon}^{\circ}. \tag*{(363)}
\end{aligned} W ε w ε := A ε − B − A ˚ B , ε , := Φ ε − Φ ˚ B , ε , Z ε z ε := A ε − B − S − A ˚ B , ε ∘ − A ˚ B , ε ∙ , := Φ ε − φ − Φ ˚ B , ε − Φ ˚ B , ε ∘ . (363) The nonlinear remainders W ε W_\varepsilon W ε w ε w_\varepsilon w ε Z ε Z_\varepsilon Z ε z ε z_\varepsilon z ε
( ∂ t − Δ B − 2 ad ( F ) ) Z ε (\partial_t-\Delta_B-2\operatorname{ad}(F))Z_\varepsilon ( ∂ t − Δ B − 2 ad ( F )) Z ε = [ W ε , j , 2 D j B ˚ ε − D B ˚ ε , j ] + [ B ˚ ε , j , 2 D j W ε − D W ε j ] + [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] + [ B ˚ ε , j + W ε , j , [ B ˚ ε j + W ε j , B ˚ ε + W ε ] ] − [ S j , [ S j , S ] ] + B ( w ε ⊗ D B ˚ ε ) + B ( B ˚ ε ⊗ D w ε ) + B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) + B ( ( B ˚ ε + w ε ) ⊗ ( B ˚ ε + W ε ) ( B ˚ ε + w ε ) ) − B ( φ ⊗ S φ ) + C ρ A ( B ˚ ε + W ε ) , (364) \begin{aligned}
={}&[W_{\varepsilon,j},2D^j\mathring{B}_{\varepsilon}-D\mathring{B}_{\varepsilon,j}]+[\mathring{B}_{\varepsilon,j},2D^jW_{\varepsilon}-DW_{\varepsilon}^{j}]\\
&+[W_{\varepsilon,j},2D^jW_{\varepsilon}-DW_{\varepsilon}^{j}]-[S_j,2D^jS-DS^j]\\
&+[\mathring{B}_{\varepsilon,j}+W_{\varepsilon,j},[\mathring{B}_{\varepsilon}^{j}+W_{\varepsilon}^{j},\mathring{B}_{\varepsilon}+W_{\varepsilon}]]-[S_j,[S^j,S]]\\
&+\mathbf{B}(w_{\varepsilon}\otimes D\mathring{B}_{\varepsilon})+\mathbf{B}(\mathring{B}_{\varepsilon}\otimes Dw_{\varepsilon})\\
&+\mathbf{B}(w_{\varepsilon}\otimes Dw_{\varepsilon})-\mathbf{B}(\varphi\otimes D\varphi)\\
&+\mathbf{B}((\mathring{B}_{\varepsilon}+w_{\varepsilon})\otimes(\mathring{B}_{\varepsilon}+W_{\varepsilon})(\mathring{B}_{\varepsilon}+w_{\varepsilon}))-\mathbf{B}(\varphi\otimes S\varphi)\\
&+C_{\rho}^{A}(\mathring{B}_{\varepsilon}+W_{\varepsilon}), \tag*{(364)}
\end{aligned} = [ W ε , j , 2 D j B ˚ ε − D B ˚ ε , j ] + [ B ˚ ε , j , 2 D j W ε − D W ε j ] + [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] + [ B ˚ ε , j + W ε , j , [ B ˚ ε j + W ε j , B ˚ ε + W ε ]] − [ S j , [ S j , S ]] + B ( w ε ⊗ D B ˚ ε ) + B ( B ˚ ε ⊗ D w ε ) + B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) + B (( B ˚ ε + w ε ) ⊗ ( B ˚ ε + W ε ) ( B ˚ ε + w ε )) − B ( φ ⊗ S φ ) + C ρ A ( B ˚ ε + W ε ) , (364) and
( ∂ t − Δ B ) z ε = 2 W ε j D j B ˚ ε + 2 B ˚ ε j D j w ε + 2 ( W ε j D j w ε − S j D j φ ) + ( B ˚ ε + W ε ) j ( B ˚ ε + W ε ) j ( B ˚ ε + w ε ) − S j S j φ − ∣ B ˚ ε + w ε ∣ 2 ( B ˚ ε + w ε ) + ∣ φ ∣ 2 φ + C ρ , ε Φ ( B ˚ ε + w ε ) . (365) \begin{aligned}
(\partial_t-\Delta_B)z_{\varepsilon}&=2W_{\varepsilon}^{j}D_j\mathring{B}_{\varepsilon}+2\mathring{B}_{\varepsilon}^{j}D_jw_{\varepsilon}\\
&\quad+2(W_{\varepsilon}^{j}D_jw_{\varepsilon}-S^{j}D_j\varphi)\\
&\quad+(\mathring{B}_{\varepsilon}+W_{\varepsilon})^{j}(\mathring{B}_{\varepsilon}+W_{\varepsilon})_j(\mathring{B}_{\varepsilon}+w_{\varepsilon})-S^{j}S_j\varphi\\
&\quad-|\mathring{B}_{\varepsilon}+w_{\varepsilon}|^2(\mathring{B}_{\varepsilon}+w_{\varepsilon})+|\varphi|^2\varphi+C_{\rho,\varepsilon}^{\Phi}(\mathring{B}_{\varepsilon}+w_{\varepsilon}). \tag*{(365)}
\end{aligned} ( ∂ t − Δ B ) z ε = 2 W ε j D j B ˚ ε + 2 B ˚ ε j D j w ε + 2 ( W ε j D j w ε − S j D j φ ) + ( B ˚ ε + W ε ) j ( B ˚ ε + W ε ) j ( B ˚ ε + w ε ) − S j S j φ − ∣ B ˚ ε + w ε ∣ 2 ( B ˚ ε + w ε ) + ∣ φ ∣ 2 φ + C ρ , ε Φ ( B ˚ ε + w ε ) . (365) As in the local theory of [CCHS22, BC26a], the evolution equations for Z ε Z_{\varepsilon} Z ε z ε z_{\varepsilon} z ε D B ˚ ε D\mathring{B}_{\varepsilon} D B ˚ ε (364) only have regularity − 1 − -1- − 1 − Z ε Z_{\varepsilon} Z ε 1 − 1- 1 − B ˚ ε \mathring{B}_{\varepsilon} B ˚ ε 0 − 0- 0 − [ Z ε , D B ˚ ε ] [Z_{\varepsilon},D\mathring{B}_{\varepsilon}] [ Z ε , D B ˚ ε ] [ D Z ε , B ˚ ε ] [DZ_{\varepsilon},\mathring{B}_{\varepsilon}] [ D Z ε , B ˚ ε ] Z ε Z_{\varepsilon} Z ε z ε z_{\varepsilon} z ε B B B
Remark 6.2. In (6.17), (6.18), (6.20), (6.21), and (6.24)–(6.26), we subtract different terms involving S S S φ \varphi φ Z ε Z_{\varepsilon} Z ε z ε z_{\varepsilon} z ε
A Littlewood–Paley decomposition
In Section 2.2, we previously explained why the standard Littlewood–Paley operators from (60) will not play an important role in this article, and we also introduced our covariant Littlewood–Paley blocks in (61) . In this subsection, we now discuss our motivation behind (61) in more detail. To this end, we first discuss two alternatives to (61) , both of which encounter different issues in our setting. It is tempting to try to define a covariant Littlewood–Paley decomposition using the covariant heat operator e s Δ B ( t ) e^{s\Delta_{B(t)}} e s Δ B ( t ) s 0 > 0 s_0>0 s 0 > 0
B ˚ ε ( t , x ) = − ∫ 0 s 0 d s ∂ s ( e s Δ B ( t ) B ˚ ε ) ( t , x ) + e s 0 Δ B ( t ) B ˚ ε ( t , x ) . (366) \mathring{B}_{\varepsilon}(t,x)=-\int_{0}^{s_0}ds\,\partial_s\left(e^{s\Delta_{B(t)}}\mathring{B}_{\varepsilon}\right)(t,x)+e^{s_0\Delta_{B(t)}}\mathring{B}_{\varepsilon}(t,x). \tag*{(366)} B ˚ ε ( t , x ) = − ∫ 0 s 0 d s ∂ s ( e s Δ B ( t ) B ˚ ε ) ( t , x ) + e s 0 Δ B ( t ) B ˚ ε ( t , x ) . (366) The e s 0 Δ B ( t ) e^{s_0\Delta_{B(t)}} e s 0 Δ B ( t ) (366) should be interpreted as a smooth remainder, and will not be too important in this discussion. One may then interpret
− ∫ 0 s 0 d s χ ( N 2 s ) ∂ s ( e s Δ B ( t ) B ˚ ε ) ( t , x ) , -\int_{0}^{s_0}ds\,\chi(N^2s)\partial_s\left(e^{s\Delta_{B(t)}}\mathring{B}_{\varepsilon}\right)(t,x), − ∫ 0 s 0 d s χ ( N 2 s ) ∂ s ( e s Δ B ( t ) B ˚ ε ) ( t , x ) , where χ \chi χ § 4 ) and N ∈ 2 N 0 N \in2^{\mathbb{N}_0} N ∈ 2 N 0 N N N
n ↦ − ∫ 0 ∞ d s χ ( N 2 s ) ∂ s ( e − s ∣ n ∣ 2 ) . n \mapsto-\int_0^\infty\mathrm{d}s\,\chi(N^2s)\partial_s\left(e^{-s|n|^2}\right). n ↦ − ∫ 0 ∞ d s χ ( N 2 s ) ∂ s ( e − s ∣ n ∣ 2 ) . This symbol behaves like N − 2 ∣ n ∣ 2 e − c N − 2 ∣ n ∣ 2 N^{-2}|n|^2e^{-cN^{-2}|n|^2} N − 2 ∣ n ∣ 2 e − c N − 2 ∣ n ∣ 2 ∣ n ∣ ∼ N |n| \sim N ∣ n ∣ ∼ N (366) , one can easily write down paraproduct decompositions of multilinear interactions. For example, one can decompose
[ Z ˚ ε , j ( t , x ) , D B ( t ) j ϕ ˚ B , ε ( t , x ) ] = ∬ [ 0 , s 0 ] 2 d s d s ′ [ ∂ s ( e s Δ B ( t ) Z ˚ ε , j ) ( t , x ) , D B ( t ) j ∂ s ′ ( e s ′ Δ B ( t ) ϕ ˚ B , ε ) ( t , x ) ] + { ⋯ } , (367) \left[\mathring{\mathcal{Z}}_{\varepsilon,j}(t,x),D^j_{B(t)}\mathring{\phi}_{B,\varepsilon}(t,x)\right]
=
\iint_{[0,s_0]^2}\mathrm{d}s\mathrm{d}s'\left[\partial_s\left(e^{s\Delta_{B(t)}}\mathring{\mathcal{Z}}_{\varepsilon,j}\right)(t,x),D^j_{B(t)}\partial_{s'}\left(e^{s'\Delta_{B(t)}}\mathring{\phi}_{B,\varepsilon}\right)(t,x)\right]+\{\cdots\}, \tag*{(367)} [ Z ˚ ε , j ( t , x ) , D B ( t ) j ϕ ˚ B , ε ( t , x ) ] = ∬ [ 0 , s 0 ] 2 d s d s ′ [ ∂ s ( e s Δ B ( t ) Z ˚ ε , j ) ( t , x ) , D B ( t ) j ∂ s ′ ( e s ′ Δ B ( t ) ϕ ˚ B , ε ) ( t , x ) ] + { ⋯ } , (367) where we omitted terms involving e s 0 Δ B ( t ) e^{s_0\Delta_{B(t)}} e s 0 Δ B ( t ) × \times × × \times × × \times × s ≫ s ′ s \gg s' s ≫ s ′ s ∼ s ′ s \sim s' s ∼ s ′ s ≪ s ′ s \ll s' s ≪ s ′ (366) and (367) : We do not have suitable estimates of the kernel of the heat propagator e s Δ B ( t ) e^{s\Delta_{B(t)}} e s Δ B ( t ) B ( t ) B(t) B ( t ) t t t L ∞ L^\infty L ∞ F B ( t ) F_{B(t)} F B ( t ) e s Δ B ( t ) ϕ ˚ B , ε e^{s\Delta_{B(t)}}\mathring{\phi}_{B,\varepsilon} e s Δ B ( t ) ϕ ˚ B , ε ϕ ˚ B , ε ( t , x ; s ) \mathring{\phi}_{B,\varepsilon}(t,x;s) ϕ ˚ B , ε ( t , x ; s )
( ∂ s − Δ B ( t + s ) ) ϕ ˚ B , ε ( t , x ; s ) = 0 , ϕ ˚ B , ε ( t , x ; 0 ) = ϕ ˚ B , ε ( t , x ) . (368) (\partial_s-\Delta_{B(t+s)})\mathring{\phi}_{B,\varepsilon}(t,x;s)=0,\qquad\mathring{\phi}_{B,\varepsilon}(t,x;0)=\mathring{\phi}_{B,\varepsilon}(t,x). \tag*{(368)} ( ∂ s − Δ B ( t + s ) ) ϕ ˚ B , ε ( t , x ; s ) = 0 , ϕ ˚ B , ε ( t , x ; 0 ) = ϕ ˚ B , ε ( t , x ) . (368) The solution of (368) can be explicitly represented as
ϕ ˚ B , ε ( t , x ; s ) = ∫ d y p B ( t , y ; t + s , x ) ϕ ˚ B , ε ( t , y ) . (369) \mathring{\phi}_{B,\varepsilon}(t,x;s)=\int\mathrm{d}y\,p_{B}(t,y;t+s,x)\mathring{\phi}_{B,\varepsilon}(t,y). \tag*{(369)} ϕ ˚ B , ε ( t , x ; s ) = ∫ d y p B ( t , y ; t + s , x ) ϕ ˚ B , ε ( t , y ) . (369) By the results from § 5 , the covariant heat kernel appearing in (369) obeys several useful estimates. However, there is an issue with (369) , which appears once we examine the counterpart of (367) . It is now of the form
[ Z ˚ ε , j ( t , x ) , D B ( t ) j ϕ ˚ B , ε ( t , x ) ] = ∬ d y d y ′ ∬ [ 0 , s 0 ] 2 d s d s ′ [ ∂ s ( p B ( t , y ; t + s , x ) Z ˚ ε , j ( t , y ) ) , D B ( t ) j ∂ s ′ ( p B ( t , y ′ ; t + s ′ , x ) ϕ ˚ B , ε ( t , y ′ ) ) ] + { ⋯ } . (370) \left[\mathring{\mathcal{Z}}_{\varepsilon,j}(t,x),D^j_{B(t)}\mathring{\phi}_{B,\varepsilon}(t,x)\right]
=
\iint\mathrm{d}y\mathrm{d}y'\iint_{[0,s_0]^2}\mathrm{d}s\mathrm{d}s'\left[\partial_s\left(p_B(t,y;t+s,x)\mathring{\mathcal{Z}}_{\varepsilon,j}(t,y)\right),D^j_{B(t)}\partial_{s'}\left(p_B(t,y';t+s',x)\mathring{\phi}_{B,\varepsilon}(t,y')\right)\right]+\{\cdots\}. \tag*{(370)} [ Z ˚ ε , j ( t , x ) , D B ( t ) j ϕ ˚ B , ε ( t , x ) ] = ∬ d y d y ′ ∬ [ 0 , s 0 ] 2 d s d s ′ [ ∂ s ( p B ( t , y ; t + s , x ) Z ˚ ε , j ( t , y ) ) , D B ( t ) j ∂ s ′ ( p B ( t , y ′ ; t + s ′ , x ) ϕ ˚ B , ε ( t , y ′ ) ) ] + { ⋯ } . (370) In (370) , the covariant derivative D B ( t ) D_{B(t)} D B ( t ) B B B t t t t + s t+s t + s t + s ′ t+s' t + s ′ D B ( t ) p B ( t , y ′ ; t + s ′ , x ) D_{B(t)}p_B(t,y';t+s',x) D B ( t ) p B ( t , y ′ ; t + s ′ , x ) D B ( t ) p B ( t , y ; t + s , x ) D_{B(t)}p_B(t,y;t+s,x) D B ( t ) p B ( t , y ; t + s , x ) B ( t + s ′ ) − B ( t ) B(t+s')-B(t) B ( t + s ′ ) − B ( t ) B ( t + s ) − B ( t ) B(t+s)-B(t) B ( t + s ) − B ( t ) F B F_B F B B B B
Instead, our covariant Littlewood–Paley decomposition uses the heat kernels that already appear in the Duhamel integral formulations of (358) , (360) –(362) , and (364) –(6.26). We start with the covariant Littlewood–Paley decomposition of ϕ ˚ B , ε \mathring{\phi}_{B,\varepsilon} ϕ ˚ B , ε χ N \chi_N χ N (95) and the integral representation of ϕ ˚ B , ε \mathring{\phi}_{B,\varepsilon} ϕ ˚ B , ε (359) , we write
ϕ ˚ B , ε = ∑ N ϕ ˚ B , ε , N , ϕ ˚ B , ε , N ( t , x ) ≔ ∬ d s d y q B , ε , N ( s , y ; t , x ) χ ( s , y ) . (371) \mathring{\phi}_{B,\varepsilon}=\sum_N\mathring{\phi}_{B,\varepsilon,N},\qquad
\mathring{\phi}_{B,\varepsilon,N}(t,x)\coloneqq\iint\mathrm{d}s\mathrm{d}y\,q_{B,\varepsilon,N}(s,y;t,x)\chi(s,y). \tag*{(371)} ϕ ˚ B , ε = N ∑ ϕ ˚ B , ε , N , ϕ ˚ B , ε , N ( t , x ) : = ∬ d s d y q B , ε , N ( s , y ; t , x ) χ ( s , y ) . (371) where
q ˚ B , ε , N ( s , y ; t , x ) ≔ ρ ε ( t − s ) χ N ( t − s ) q B ( s , y ; t , x ) . (372) \mathring{q}_{B,\varepsilon,N}(s,y;t,x) \coloneqq\rho_{\varepsilon}(t-s)\chi_N(t-s)q_B(s,y;t,x). \tag*{(372)} q ˚ B , ε , N ( s , y ; t , x ) : = ρ ε ( t − s ) χ N ( t − s ) q B ( s , y ; t , x ) . (372) Here, as discussed in Notation 3.1, we use the convention that sums over N N N N ∈ 2 N 0 N \in2^{\mathbb{N}_0} N ∈ 2 N 0 N ∈ N 0 N \in\mathbb{N}_0 N ∈ N 0
q B , N ( s , y ; t , x ) ≔ χ N ( t − s ) q B ( s , y ; t , x ) . (373) q_{B,N}(s,y;t,x) \coloneqq\chi_N(t-s)q_B(s,y;t,x). \tag*{(373)} q B , N ( s , y ; t , x ) : = χ N ( t − s ) q B ( s , y ; t , x ) . (373) Due to the support properties of ρ ε \rho_{\varepsilon} ρ ε χ N \chi_N χ N q ˚ B , ε , N \mathring{q}_{B,\varepsilon,N} q ˚ B , ε , N N ≲ ε − 1 N \lesssim\varepsilon^{-1} N ≲ ε − 1 (371) is finite. Since q B , ε , N q_{B,\varepsilon,N} q B , ε , N q B , N q_{B,N} q B , N q B q_B q B t − s ∼ N − 2 t-s \sim N^{-2} t − s ∼ N − 2 q ˚ B , ε , N \mathring{q}_{B,\varepsilon,N} q ˚ B , ε , N q ˚ B , ε \mathring{q}_{B,\varepsilon} q ˚ B , ε ∼ N \sim N ∼ N q ˚ B , ε , N \mathring{q}_{B,\varepsilon,N} q ˚ B , ε , N ∼ N \sim N ∼ N ∼ N 2 \sim N^2 ∼ N 2
Furthermore, also due to the aforementioned support properties, q ˚ B , ε , N \mathring{q}_{B,\varepsilon,N} q ˚ B , ε , N ρ ε \rho_{\varepsilon} ρ ε N ∼ ε − 1 N \sim\varepsilon^{-1} N ∼ ε − 1 N ∼ ε − 1 N \sim\varepsilon^{-1} N ∼ ε − 1 ρ ε ( t − s ) \rho_{\varepsilon}(t-s) ρ ε ( t − s ) (372) is just equal to zero or one on the support of χ N ( t − s ) \chi_N(t-s) χ N ( t − s ) (353) does not depend on the precise choice of ρ \rho ρ q ˚ B , ε \mathring{q}_{B,\varepsilon} q ˚ B , ε p ˚ B , ε \mathring{p}_{B,\varepsilon} p ˚ B , ε
p ˚ B , ε , N ( t , x ) ≔ ∬ d s d y p B , ε , N ( s , y ; t , x ) ζ ( s , y ) , (374) \mathring{p}_{B,\varepsilon,N}(t,x) \coloneqq\iint ds\,dy\,p_{B,\varepsilon,N}(s,y;t,x)\zeta(s,y), \tag*{(374)} p ˚ B , ε , N ( t , x ) : = ∬ d s d y p B , ε , N ( s , y ; t , x ) ζ ( s , y ) , (374) where
p B , ε , N ( s , y ; t , x ) ≔ ρ ε ( t − s ) χ N ( t − s ) p B ( s , y ; t , x ) . p_{B,\varepsilon,N}(s,y;t,x) \coloneqq\rho_{\varepsilon}(t-s)\chi_N(t-s)p_B(s,y;t,x). p B , ε , N ( s , y ; t , x ) : = ρ ε ( t − s ) χ N ( t − s ) p B ( s , y ; t , x ) . As for q B , N q_{B,N} q B , N
p B , N ( s , y ; t , x ) ≔ χ N ( t − s ) p B ( s , y ; t , x ) . (375) p_{B,N}(s,y;t,x) \coloneqq\chi_N(t-s)p_B(s,y;t,x). \tag*{(375)} p B , N ( s , y ; t , x ) : = χ N ( t − s ) p B ( s , y ; t , x ) . (375) In order to define the covariant Littlewood–Paley decompositions of the remaining terms, we start with a decomposition of the Duhamel integrals of ∂ t − Δ B − 2 ad ( F ) \partial_t-\Delta_B-2\operatorname{ad}(F) ∂ t − Δ B − 2 ad ( F ) ∂ t − Δ B \partial_t-\Delta_B ∂ t − Δ B
Duh B F ( Ψ ) ( t , x ) = ∬ d s d y q B ( s , y ; t , x ) Ψ ( s , y ) , \operatorname{Duh}^{F}_B(\Psi)(t,x)=\iint ds\,dy\,q_B(s,y;t,x)\Psi(s,y), Duh B F ( Ψ ) ( t , x ) = ∬ d s d y q B ( s , y ; t , x ) Ψ ( s , y ) , Duh B ( Ψ ) ( t , x ) = ∬ d s d y p B ( s , y ; t , x ) Ψ ( s , y ) . \operatorname{Duh}_B(\Psi)(t,x)=\iint ds\,dy\,p_B(s,y;t,x)\Psi(s,y). Duh B ( Ψ ) ( t , x ) = ∬ d s d y p B ( s , y ; t , x ) Ψ ( s , y ) . Here, the map Ψ \Psi Ψ g 2 \mathfrak{g}^2 g 2 V V V F F F Duh B F \operatorname{Duh}^{F}_B Duh B F ad ( F ) \operatorname{ad}(F) ad ( F ) (355) , have been accounted for through q B q_B q B
Duh B , N F ( Ψ ) ( t , x ) ≔ ∬ d s d y q B , N ( s , y ; t , x ) Ψ ( s , y ) , \operatorname{Duh}^{F}_{B,N}(\Psi)(t,x) \coloneqq\iint ds\,dy\,q_{B,N}(s,y;t,x)\Psi(s,y), Duh B , N F ( Ψ ) ( t , x ) : = ∬ d s d y q B , N ( s , y ; t , x ) Ψ ( s , y ) , Duh B , N ( Ψ ) ( t , x ) ≔ ∬ d s d y p B , N ( s , y ; t , x ) Ψ ( s , y ) . (376) \operatorname{Duh}_{B,N}(\Psi)(t,x) \coloneqq\iint ds\,dy\,p_{B,N}(s,y;t,x)\Psi(s,y). \tag*{(376)} Duh B , N ( Ψ ) ( t , x ) : = ∬ d s d y p B , N ( s , y ; t , x ) Ψ ( s , y ) . (376) where N ∈ 2 N 0 N \in2^{\mathbb{N}_0} N ∈ 2 N 0 q B , N q_{B,N} q B , N (373) , and p B , N p_{B,N} p B , N (375) . From now on, we work on the time interval [ 0 , T ] [0,T] [ 0 , T ] 0 < T ≤ 1 128 0<T\leq\frac{1}{128} 0 < T ≤ 128 1 [ 0 , T ] [0,T] [ 0 , T ] Ψ \Psi Ψ
Duh B = ∑ N ≥ N l o Duh B , N and Duh B F = ∑ N ≥ N l o Duh B , N F . (377) \operatorname{Duh}_B=\sum_{N\geq N_{\mathrm{lo}}}\operatorname{Duh}_{B,N}
\qquad\text{and}\qquad
\operatorname{Duh}^{F}_B=\sum_{N\geq N_{\mathrm{lo}}}\operatorname{Duh}^{F}_{B,N}. \tag*{(377)} Duh B = N ≥ N lo ∑ Duh B , N and Duh B F = N ≥ N lo ∑ Duh B , N F . (377) where N l o N_{\mathrm{lo}} N lo ( 32 T ) − 1 2 (32T)^{-\frac{1}{2}} ( 32 T ) − 2 1
N l o : = max { N ∈ 2 N 0 : N ≤ ( 32 T ) − 1 2 } . (378) N_{\mathrm{lo}} := \max\{N \in2^{\mathbb{N}_0}: N \le(32T)^{-\frac{1}{2}}\}. \tag*{(378)} N lo := max { N ∈ 2 N 0 : N ≤ ( 32 T ) − 2 1 } . (378) Using the time-localised Duhamel integrals, we now define the covariant Littlewood–Paley blocks of the remaining terms by acting with Duh B , N F \operatorname{Duh}^{F}_{B,N} Duh B , N F Duh B , N \operatorname{Duh}_{B,N} Duh B , N S S S
S N : = Duh B , N F ( R H S o f x r e f ‘ ( 6.8 ‘ ) ) . (379) S_N := \operatorname{Duh}^{F}_{B,N}(\mathrm{RHS\ of\ }{xref}`(6.8`)). \tag*{(379)} S N := Duh B , N F ( RHS of x r e f ‘ ( 6.8‘ )) . (379) Similarly, we define the covariant Littlewood–Paley blocks of φ \varphi φ
φ N = φ l i n , N + Duh B , N ( R H S o f x r e f ‘ ( 6.9 ‘ ) ) , (380) \varphi_N = \varphi_{\mathrm{lin},N} + \operatorname{Duh}_{B,N}(\mathrm{RHS\ of\ }{xref}`(6.9`)), \tag*{(380)} φ N = φ lin , N + Duh B , N ( RHS of x r e f ‘ ( 6.9‘ )) , (380) where the covariant Littlewood–Paley block of the linear evolution is defined as
φ l i n , N ( t , x ) ≔ ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) . \varphi_{\mathrm{lin},N}(t,x) \coloneqq\int\mathrm{d}y\,p_{B,N}(0,y;t,x)\varphi_0(y). φ lin , N ( t , x ) : = ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) . Finally, we define the Littlewood–Paley blocks of the covariant quadratic stochastic objects by
B ∘ ε , N : = Duh B , N F ( R H S o f x r e f ‘ ( 6.12 ) ‘ ) , B ∙ ε , N : = Duh B , N F ( R H S o f x r e f ‘ ( 6.13 ) ‘ ) , B ⊥ ε , N : = Duh B , N ( R H S o f x r e f ‘ ( 6.14 ) ‘ ) . (381) \begin{aligned}
\overset{\circ}{B}_{\varepsilon,N} &:= \operatorname{Duh}^{F}_{B,N}(\mathrm{RHS\ of\ }{xref}`(6.12)`), \qquad
\overset{\bullet}{B}_{\varepsilon,N} := \operatorname{Duh}^{F}_{B,N}(\mathrm{RHS\ of\ }{xref}`(6.13)`),\\
\overset{\mathord{\perp}}{B}_{\varepsilon,N} &:= \operatorname{Duh}_{B,N}(\mathrm{RHS\ of\ }{xref}`(6.14)`). \tag*{(381)}
\end{aligned} B ∘ ε , N B ⊥ ε , N := Duh B , N F ( RHS of x r e f ‘ ( 6.12 ) ‘ ) , B ∙ ε , N := Duh B , N F ( RHS of x r e f ‘ ( 6.13 ) ‘ ) , := Duh B , N ( RHS of x r e f ‘ ( 6.14 ) ‘ ) . (381) and, similarly, we define the Littlewood–Paley blocks of the nonlinear remainders by
Z ε , N : = Duh B , N F ( R H S o f x r e f ‘ ( 6.16 ) ‘ − x r e f ‘ ( 6.22 ) ‘ ) , Z_{\varepsilon,N} := \operatorname{Duh}^{F}_{B,N}\bigl(\mathrm{RHS\ of\ }{xref}`(6.16)`-{xref}`(6.22)`\bigr), Z ε , N := Duh B , N F ( RHS of x r e f ‘ ( 6.16 ) ‘ − x r e f ‘ ( 6.22 ) ‘ ) , z ε , N : = Duh B , N ( R H S o f x r e f ‘ ( 6.23 ) ‘ − x r e f ‘ ( 6.26 ) ‘ ) . (382) z_{\varepsilon,N} := \operatorname{Duh}_{B,N}\bigl(\mathrm{RHS\ of\ }{xref}`(6.23)`-{xref}`(6.26)`\bigr). \tag*{(382)} z ε , N := Duh B , N ( RHS of x r e f ‘ ( 6.23 ) ‘ − x r e f ‘ ( 6.26 ) ‘ ) . (382) Before we continue with our definition of the covariant paraproducts, note that our covariant Littlewood–Paley decomposition does not come with a collection of “projection operators” P N P_N P N A A A A = ∑ N P N A A = \sum_N P_NA A = ∑ N P N A ( A N ) N ≥ N l o (A_N)_{N\ge N_{\mathrm{lo}}} ( A N ) N ≥ N lo A ⋆ A_\star A ⋆ N N N ⋆ \star ⋆ 2 N 0 2^{\mathbb{N}_0} 2 N 0 A ⋆ = ( A N ) N ≥ N l o A_\star= (A_N)_{N\ge N_{\mathrm{lo}}} A ⋆ = ( A N ) N ≥ N lo A A A A = ∑ N ≥ N l o A N A = \sum_{N\ge N_{\mathrm{lo}}} A_N A = ∑ N ≥ N lo A N
Let θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) (97) . For two dyadic scales M , N ∈ 2 N 0 M,N \in2^{\mathbb{N}_0} M , N ∈ 2 N 0
M ≪ θ N ⟺ M ≤ N 1 − θ . (383) M \ll_\theta N \quad\Longleftrightarrow\quad M \le N^{1-\theta}. \tag*{(383)} M ≪ θ N ⟺ M ≤ N 1 − θ . (383) Furthermore, we write M ≫ θ N M \gg_\theta N M ≫ θ N N ≪ θ M N \ll_\theta M N ≪ θ M M ∼ θ N M \sim_\theta N M ∼ θ N M ≪ θ N M \ll_\theta N M ≪ θ N M ≫ θ N M \gg_\theta N M ≫ θ N M ∼ θ N M \sim_\theta N M ∼ θ N min ( M , N ) > max ( M , N ) 1 − θ \min(M,N) > \max(M,N)^{1-\theta} min ( M , N ) > max ( M , N ) 1 − θ
Definition 6.3 (Covariant paraproducts). Let 0 < T ≤ 1 128 0<T\le\frac{1}{128} 0 < T ≤ 128 1 W 1 \mathbf{W}_1 W 1 W 2 \mathbf{W}_2 W 2 W 3 \mathbf{W}_3 W 3 Q : W 1 × W 2 → W 3 Q:\mathbf{W}_1\times\mathbf{W}_2\to\mathbf{W}_3 Q : W 1 × W 2 → W 3 U ⋆ = ( U M ) M ≥ N l o U_\star=(U_M)_{M\ge N_{\mathrm{lo}}} U ⋆ = ( U M ) M ≥ N lo V ⋆ = ( V N ) N ≥ N l o V_\star=(V_N)_{N\ge N_{\mathrm{lo}}} V ⋆ = ( V N ) N ≥ N lo U M : [ 0 , T ] × T 2 → W 1 U_M:[0,T]\times\mathbb{T}^2\to\mathbf{W}_1 U M : [ 0 , T ] × T 2 → W 1 V N : [ 0 , T ] × T 2 → W 2 V_N:[0,T]\times\mathbb{T}^2\to\mathbf{W}_2 V N : [ 0 , T ] × T 2 → W 2
Q ( U ⋆ ⊛ V ⋆ ) : = ∑ M , N ≥ N l o : M ≪ θ N Q ( U M , V N ) . (384) Q(U_\star\circledast V_\star) := \sum_{\substack{M,N\ge N_{\mathrm{lo}}:\\ M\ll_\theta N}} Q(U_M,V_N). \tag*{(384)} Q ( U ⋆ ⊛ V ⋆ ) := M , N ≥ N lo : M ≪ θ N ∑ Q ( U M , V N ) . (384) Similarly, we define Q ( U ⋆ ⊛ V ⋆ ) Q(U_\star\circledast V_\star) Q ( U ⋆ ⊛ V ⋆ ) Q ( U ⋆ ⊛ V ⋆ ) Q(U_\star\circledast V_\star) Q ( U ⋆ ⊛ V ⋆ ) Q ( U ⋆ ⊛ V ⋆ ) Q(U_\star\circledast V_\star) Q ( U ⋆ ⊛ V ⋆ ) M ≪ θ N M\ll_\theta N M ≪ θ N (384) by “M ≫ θ N M\gg_\theta N M ≫ θ N M ∼ θ N M\sim_\theta N M ∼ θ N M ̸ ∼ θ N M\not\sim_\theta N M ∼ θ N (384) , we will also write
Q ( U ⋆ , V ⋆ ) : = ∑ M , N ≥ N l o Q ( U M , V N ) . Q(U_\star,V_\star) := \sum_{M,N\ge N_{\mathrm{lo}}} Q(U_M,V_N). Q ( U ⋆ , V ⋆ ) := M , N ≥ N lo ∑ Q ( U M , V N ) . Remark 6.4. While these definitions apply to general U ⋆ U_\star U ⋆ V ⋆ V_\star V ⋆
We emphasise that the parameter θ \theta θ f , g : T 2 → R f,g:\mathbb{T}^2\to\mathbb{R} f , g : T 2 → R K , M , N ∈ 2 N 0 K,M,N\in2^{\mathbb{N}_0} K , M , N ∈ 2 N 0 ( P L ) L ∈ 2 N 0 (P_L)_{L\in2^{\mathbb{N}_0}} ( P L ) L ∈ 2 N 0
P K ( P M f ⋅ P N g ) = 0 P_K(P_Mf\cdot P_Ng)=0 P K ( P M f ⋅ P N g ) = 0 unless
K ∼ M ≫ N , K ∼ N ≫ M , or K ≲ M ∼ N . (385) K\sim M\gg N,\qquad K\sim N\gg M,\qquad\text{or}\qquad K\lesssim M\sim N. \tag*{(385)} K ∼ M ≫ N , K ∼ N ≫ M , or K ≲ M ∼ N . (385) The conditions in (385) are often called the Littlewood–Paley trichotomy, and the above fact is heavily used in the paracontrolled calculus of [GIP15]. In our covariant setting, this has no direct counterpart. For example, consider one of the terms coming from (364) and (382) , which can be decomposed into terms of the form
Duh B , K F ( [ Z ε , M , j , D j ℓ ˚ B , ε , N ] ) . (386) \operatorname{Duh}^{F}_{B,K}\left([Z_{\varepsilon,M,j},D^{j}\mathring{\ell}_{B,\varepsilon,N}]\right). \tag*{(386)} Duh B , K F ( [ Z ε , M , j , D j ℓ ˚ B , ε , N ] ) . (386) Unlike in the non-covariant paracontrolled calculus of [GIP15], there is no reason for this term to vanish if, for example, M ≪ N M\ll N M ≪ N K ≪ N K\ll N K ≪ N K ∼ 1 K\sim1 K ∼ 1 M ∼ N 1 − θ M\sim N^{1-\theta} M ∼ N 1 − θ × \times × (386) is still only well-defined under certain θ \theta θ Z ε , ⋆ Z_{\varepsilon,\star} Z ε , ⋆ K ∼ N 1 − θ K\sim N^{1-\theta} K ∼ N 1 − θ M ∼ 1 M\sim1 M ∼ 1 K / N ≲ N − θ K/N\lesssim N^{-\theta} K / N ≲ N − θ θ \theta θ
Paracontrolled ansatz
Equipped with our covariant Littlewood–Paley decomposition and paraproducts, we are now able to derive our covariant paracontrolled ansatz. This derivation will closely follow [BC26a, Section 3], except that we include a Higgs field and replace all derivatives and paraproducts by their covariant counterparts. Before turning to the derivation, however, we introduce the following notation.
Notation 6.5 (Omitting the subscript B B B B B B B B B
ℓ ˚ ε , ⋆ = ℓ ˚ B , ε , ⋆ , ℓ ε , ⋆ = ℓ B , ε , ⋆ , ϑ ˚ ε , ⋆ = ϑ ˚ B , ε , ⋆ , \mathring{\ell}_{\varepsilon,\star}=\mathring{\ell}_{B,\varepsilon,\star},\qquad
\ell_{\varepsilon,\star}=\ell_{B,\varepsilon,\star},\qquad
\mathring{\vartheta}_{\varepsilon,\star}=\mathring{\vartheta}_{B,\varepsilon,\star}, ℓ ˚ ε , ⋆ = ℓ ˚ B , ε , ⋆ , ℓ ε , ⋆ = ℓ B , ε , ⋆ , ϑ ˚ ε , ⋆ = ϑ ˚ B , ε , ⋆ , etc. Since the non-covariant stochastic estimates only appear in Section 13, which does not use the graphical symbols ℓ ˚ ε , ⋆ \mathring{\ell}_{\varepsilon,\star} ℓ ˚ ε , ⋆ ℓ ε , ⋆ \ell_{\varepsilon,\star} ℓ ε , ⋆ ϑ ˚ ε , ⋆ \mathring{\vartheta}_{\varepsilon,\star} ϑ ˚ ε , ⋆
We now make the ansatz
Z ε , ⋆ : = X ε , ⋆ + Y ε , ⋆ and z ε , ⋆ : = x ε , ⋆ + y ε , ⋆ , (387) Z_{\varepsilon,\star}:=X_{\varepsilon,\star}+Y_{\varepsilon,\star}
\qquad\text{and}\qquad
z_{\varepsilon,\star}:=x_{\varepsilon,\star}+y_{\varepsilon,\star}, \tag*{(387)} Z ε , ⋆ := X ε , ⋆ + Y ε , ⋆ and z ε , ⋆ := x ε , ⋆ + y ε , ⋆ , (387) where X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
A ε = B + S + ℓ ˚ ε + ϑ ˚ ε + ϑ ε + X ε + Y ε , A_\varepsilon=B+S+\mathring{\ell}_\varepsilon+\mathring{\vartheta}_\varepsilon+\vartheta_\varepsilon+X_\varepsilon+Y_\varepsilon, A ε = B + S + ℓ ˚ ε + ϑ ˚ ε + ϑ ε + X ε + Y ε , Φ ε = φ + ℓ ε + ϑ ˚ ε + x ε + y ε . \Phi_\varepsilon=\varphi+\ell_\varepsilon+\mathring{\vartheta}_\varepsilon+x_\varepsilon+y_\varepsilon. Φ ε = φ + ℓ ε + ϑ ˚ ε + x ε + y ε . Covariant ansatz
The terms in our ansatz for Z ε , ⋆ Z_{\varepsilon,\star} Z ε , ⋆ z ε , ⋆ z_{\varepsilon,\star} z ε , ⋆ 24 X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ X ε , ⋆ = X ε , ⋆ ∘ + X ε , ⋆ ∙ X_{\varepsilon,\star}=X_{\varepsilon,\star}^{\circ}+X_{\varepsilon,\star}^{\bullet} X ε , ⋆ = X ε , ⋆ ∘ + X ε , ⋆ ∙
X ε , K ∘ = Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] ) , (388) X_{\varepsilon,K}^{\circ}=\operatorname{Duh}_{B,K}^{F}\left(\left[W_{\varepsilon,\star,j}\otimes\left(2D^{j}\mathfrak{o}_{\varepsilon,\star}-D\mathfrak{o}_{\varepsilon,\star}^{j}\right)\right]\right), \tag*{(388)} X ε , K ∘ = Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] ) , (388) X ε , K ∙ = Duh B , K F ( B ( w ε , ⋆ ⊗ D o ε , ⋆ ∙ ) ) . (389) X_{\varepsilon,K}^{\bullet}=\operatorname{Duh}_{B,K}^{F}\left(\mathbf{B}\left(w_{\varepsilon,\star}\otimes D\mathfrak{o}_{\varepsilon,\star}^{\bullet}\right)\right). \tag*{(389)} X ε , K ∙ = Duh B , K F ( B ( w ε , ⋆ ⊗ D o ε , ⋆ ∙ ) ) . (389) The paracontrolled component x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆
x ε , K = 2 Duh B , K ( W ε , ⋆ W j ⊗ D j o ε , ⋆ ∙ ) . (390) x_{\varepsilon,K}=2\operatorname{Duh}_{B,K}\left(W_{\varepsilon,\star}^{Wj}\otimes D_{j}\mathfrak{o}_{\varepsilon,\star}^{\bullet}\right). \tag*{(390)} x ε , K = 2 Duh B , K ( W ε , ⋆ W j ⊗ D j o ε , ⋆ ∙ ) . (390) The smooth nonlinear remainder Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ 25
Y ε , K = Duh B , K F ( [ W ε , ⋆ , j ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] − [ W ε , ⋆ , j ⊗ ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] ) (391) Y_{\varepsilon,K}=\operatorname{Duh}_{B,K}^{F}\left(\left[W_{\varepsilon,\star,j}\left(2D^{j}\mathfrak{o}_{\varepsilon,\star}-D\mathfrak{o}_{\varepsilon,\star}^{j}\right)\right]-\left[W_{\varepsilon,\star,j}\otimes\left(2D^{j}\mathfrak{o}_{\varepsilon,\star}-D\mathfrak{o}_{\varepsilon,\star}^{j}\right)\right]\right) \tag*{(391)} Y ε , K = Duh B , K F ( [ W ε , ⋆ , j ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] − [ W ε , ⋆ , j ⊗ ( 2 D j o ε , ⋆ − D o ε , ⋆ j ) ] ) (391) + [ o ε , j , 2 D j W ε − D W ε j ] (392) {}+\left[\mathfrak{o}_{\varepsilon,j},2D^{j}W_{\varepsilon}-DW_{\varepsilon}^{j}\right] \tag*{(392)} + [ o ε , j , 2 D j W ε − D W ε j ] (392) + B ( w ε , ⋆ ⊗ D o ε , ⋆ ∙ ) − B ( w ε , ⋆ ⊗ D o ε , ⋆ ) + B ( o ε ⊗ D w ε ) (393) {}+\mathbf{B}\left(w_{\varepsilon,\star}\otimes D\mathfrak{o}_{\varepsilon,\star}^{\bullet}\right)-\mathbf{B}\left(w_{\varepsilon,\star}\otimes D\mathfrak{o}_{\varepsilon,\star}\right)+\mathbf{B}\left(\mathfrak{o}_{\varepsilon}\otimes Dw_{\varepsilon}\right) \tag*{(393)} + B ( w ε , ⋆ ⊗ D o ε , ⋆ ∙ ) − B ( w ε , ⋆ ⊗ D o ε , ⋆ ) + B ( o ε ⊗ D w ε ) (393) + ( 6.17 ) + ( 6.18 ) + ( 6.20 ) + ( 6.21 ) + ( 6.22 ) . (394) {}+(6.17)+(6.18)+(6.20)+(6.21)+(6.22). \tag*{(394)} + ( 6.17 ) + ( 6.18 ) + ( 6.20 ) + ( 6.21 ) + ( 6.22 ) . (394) And finally, the smooth nonlinear remainder y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
y ε , K = Duh B , K ( 2 W ε , ⋆ W j D j o ε , ⋆ ∙ − 2 W ε , ⋆ W j ⊗ D j o ε , ⋆ ∙ + 2 o ε ∘ j D w ε ) (395) y_{\varepsilon,K}=\operatorname{Duh}_{B,K}\left(2W_{\varepsilon,\star}^{Wj}D_{j}\mathfrak{o}_{\varepsilon,\star}^{\bullet}-2W_{\varepsilon,\star}^{Wj}\otimes D_{j}\mathfrak{o}_{\varepsilon,\star}^{\bullet}+2\mathfrak{o}_{\varepsilon}^{\mathord{\circ}j}Dw_{\varepsilon}\right) \tag*{(395)} y ε , K = Duh B , K ( 2 W ε , ⋆ W j D j o ε , ⋆ ∙ − 2 W ε , ⋆ W j ⊗ D j o ε , ⋆ ∙ + 2 o ε ∘ j D w ε ) (395) + ( 6.24 ) + ( 6.25 ) + ( 6.26 ) . (396) {}+(6.24)+(6.25)+(6.26). \tag*{(396)} + ( 6.24 ) + ( 6.25 ) + ( 6.26 ) . (396) In order to use the paracontrolled ansatz in our estimates of Y ε , K Y_{\varepsilon,K} Y ε , K y ε , K y_{\varepsilon,K} y ε , K (391) , (392) , (393) , and (395) . For this, we first prove the following lemma.
Lemma 6.6. For i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3 W i W_i W i G \mathrm{G} G j = 1 , 2 j=1,2 j = 1 , 2 U ⋆ = ( U M ) M ≥ N l o U_{\star}=(U_M)_{M\ge N_{\mathrm{lo}}} U ⋆ = ( U M ) M ≥ N lo V ⋆ = ( V N ) N ≥ N l o V_{\star}=(V_N)_{N\ge N_{\mathrm{lo}}} V ⋆ = ( V N ) N ≥ N lo M , N ≥ N l o M,N\ge N_{\mathrm{lo}} M , N ≥ N lo U M : [ 0 , T ] × T 2 → W 1 U_M:[0,T]\times\mathbb{T}^{2}\to W_1 U M : [ 0 , T ] × T 2 → W 1 V N : [ 0 , T ] × T 2 → W 2 V_N:[0,T]\times\mathbb{T}^{2}\to W_2 V N : [ 0 , T ] × T 2 → W 2 Q : W 1 × W 2 → W 3 Q:W_1\times W_2\to W_3 Q : W 1 × W 2 → W 3 G \mathrm{G} G
Q ( U ⋆ , D j V ⋆ ) = − Q ( D j U ⋆ ⊗ V ⋆ ) + D j Q ( U ⋆ ⊗ V ⋆ ) + Q ( U ⋆ ⊗ D j V ⋆ ) . Q(U_{\star},D^{j}V_{\star})=-Q(D^{j}U_{\star}\otimes V_{\star})+D^{j}Q(U_{\star}\otimes V_{\star})+Q(U_{\star}\otimes D^{j}V_{\star}). Q ( U ⋆ , D j V ⋆ ) = − Q ( D j U ⋆ ⊗ V ⋆ ) + D j Q ( U ⋆ ⊗ V ⋆ ) + Q ( U ⋆ ⊗ D j V ⋆ ) . Proof. To obtain this identity, we first use the definition of our paraproducts and then use the covariant product rule for G \mathrm{G} G Lemma 3.17 ).
Equipped with Lemma 6.6 , we can now rewrite the terms in (391) , (392) , (393) , and (395) . First, using Lemma 6.6 and the skew-symmetry of the Lie bracket, we see that (391) –(392) can be written as
[ W ε , ⋆ , j , ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] − [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] + [ ∂ ε , j , 2 D j W ε − D W ε j ] = [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] + [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] − 2 [ D j ∂ ε , ⋆ , j ⊗ W ε , ⋆ ] + 2 D j [ ∂ ε , ⋆ , j ⊗ W ε , ⋆ ] + 2 [ ∂ ε , ⋆ , j ⊗ D j W ε , ⋆ ] + [ D ε , ⋆ , j ∘ ⊗ W ε j ] − D [ ∂ ε , ⋆ , j ⊗ W ε j ] − [ ∂ ε , ⋆ , j ⊗ D W ε j ] = 2 [ W ε , ⋆ , j ⊗ ( D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] + 2 [ W ε , ⋆ ⊗ D j ∂ ε , ⋆ , j ] − 2 D j [ W ε , ⋆ ⊗ ∂ ε , ⋆ , j ] + D [ W ε , ⋆ j ⊗ ∂ ε , ⋆ , j ] + [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] + [ ∂ ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j ) ] . (397) \begin{aligned}
&[W_{\varepsilon,\star,j},(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]-[W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]+[\partial_{\varepsilon,j},2D^{j}W_{\varepsilon}-DW_{\varepsilon}^{j}] \\
={}&[W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]+[W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]\\
&-2[D^{j}\partial_{\varepsilon,\star,j}\otimes W_{\varepsilon,\star}]+2D^{j}[\partial_{\varepsilon,\star,j}\otimes W_{\varepsilon,\star}]+2[\partial_{\varepsilon,\star,j}\otimes D^{j}W_{\varepsilon,\star}]\\
&+[D_{\varepsilon,\star,j}^{\circ}\otimes W_{\varepsilon}^{j}]-D[\partial_{\varepsilon,\star,j}\otimes W_{\varepsilon}^{j}]-[\partial_{\varepsilon,\star,j}\otimes DW_{\varepsilon}^{j}]\\
={}&2[W_{\varepsilon,\star,j}\otimes(D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]+2[W_{\varepsilon,\star}\otimes D^{j}\partial_{\varepsilon,\star,j}]\\
&-2D^{j}[W_{\varepsilon,\star}\otimes\partial_{\varepsilon,\star,j}]+D[W_{\varepsilon,\star}^{j}\otimes\partial_{\varepsilon,\star,j}]\\
&+[W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]+[\partial_{\varepsilon,\star,j}\otimes(2D^{j}W_{\varepsilon,\star}-DW_{\varepsilon,\star}^{j})]. \tag*{(397)}
\end{aligned} = = [ W ε , ⋆ , j , ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] − [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] + [ ∂ ε , j , 2 D j W ε − D W ε j ] [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] + [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] − 2 [ D j ∂ ε , ⋆ , j ⊗ W ε , ⋆ ] + 2 D j [ ∂ ε , ⋆ , j ⊗ W ε , ⋆ ] + 2 [ ∂ ε , ⋆ , j ⊗ D j W ε , ⋆ ] + [ D ε , ⋆ , j ∘ ⊗ W ε j ] − D [ ∂ ε , ⋆ , j ⊗ W ε j ] − [ ∂ ε , ⋆ , j ⊗ D W ε j ] 2 [ W ε , ⋆ , j ⊗ ( D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] + 2 [ W ε , ⋆ ⊗ D j ∂ ε , ⋆ , j ] − 2 D j [ W ε , ⋆ ⊗ ∂ ε , ⋆ , j ] + D [ W ε , ⋆ j ⊗ ∂ ε , ⋆ , j ] + [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] + [ ∂ ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j )] . (397) The exact expression in (397) plays an important role in our argument, since it will need to be renormalised. Since it appears repeatedly below, it is convenient to define
Lb ( W ε , ⋆ , D ∂ ε , ⋆ ) : = 2 [ W ε , ⋆ , j ⊗ ( D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] + 2 [ W ε , ⋆ ⊗ D j ∂ ε , ⋆ , j ] . (398) \operatorname{Lb}(W_{\varepsilon,\star},D\partial_{\varepsilon,\star}) := 2[W_{\varepsilon,\star,j}\otimes(D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]+2[W_{\varepsilon,\star}\otimes D^{j}\partial_{\varepsilon,\star,j}]. \tag*{(398)} Lb ( W ε , ⋆ , D ∂ ε , ⋆ ) := 2 [ W ε , ⋆ , j ⊗ ( D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] + 2 [ W ε , ⋆ ⊗ D j ∂ ε , ⋆ , j ] . (398) We now turn to the terms in (393) . Using Lemma 6.6 and the skew-symmetry of B, we obtain
B ( w ε , ⋆ ⊗ D ε , ⋆ ) − B ( w ε , ⋆ ⊗ D ε , ⋆ ) + B ( ∂ ε ⊗ D w ε ) = 2 B ( w ε , ⋆ ⊗ D ε , ⋆ ) (399) \begin{aligned}
B(w_{\varepsilon,\star}\otimes D_{\varepsilon,\star})-B(w_{\varepsilon,\star}\otimes D_{\varepsilon,\star})+B(\partial_{\varepsilon}\otimes Dw_{\varepsilon})
&=2B(w_{\varepsilon,\star}\otimes D_{\varepsilon,\star}) \tag*{(399)}
\end{aligned} B ( w ε , ⋆ ⊗ D ε , ⋆ ) − B ( w ε , ⋆ ⊗ D ε , ⋆ ) + B ( ∂ ε ⊗ D w ε ) = 2 B ( w ε , ⋆ ⊗ D ε , ⋆ ) (399) − D ( B ( w ε , ⋆ ⊗ ∂ ε , ⋆ ) ) + B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) + B ( ∂ ε , ⋆ ⊗ D w ε , ⋆ ) . (400) -D\bigl(B(w_{\varepsilon,\star}\otimes\partial_{\varepsilon,\star})\bigr)+B(w_{\varepsilon,\star}\otimes D\partial_{\varepsilon,\star})+B(\partial_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}). \tag*{(400)} − D ( B ( w ε , ⋆ ⊗ ∂ ε , ⋆ ) ) + B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) + B ( ∂ ε , ⋆ ⊗ D w ε , ⋆ ) . (400) Finally, we turn to the terms in (395) . Using Lemma 6.6 again, we decompose
2 W ε , ⋆ j D j ∂ ε , ⋆ − 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ = 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ + 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ . (401) 2W_{\varepsilon,\star}^{j}D_{j}\partial_{\varepsilon,\star}-2W_{\varepsilon,\star}^{j}\otimes D_{j}\partial_{\varepsilon,\star}=2W_{\varepsilon,\star}^{j}\otimes D_{j}\partial_{\varepsilon,\star}+2W_{\varepsilon,\star}^{j}\otimes D_{j}\partial_{\varepsilon,\star}. \tag*{(401)} 2 W ε , ⋆ j D j ∂ ε , ⋆ − 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ = 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ + 2 W ε , ⋆ j ⊗ D j ∂ ε , ⋆ . (401) and
2 ∂ ε j D j w ε = − 2 ( D j ∂ ε , ⋆ j ) ⊗ w ε , ⋆ + 2 D j ( ∂ ε , ⋆ j ⊗ w ε , ⋆ ) + 2 ∂ ε , ⋆ j ⊗ D j w ε , ⋆ . (402) 2\partial_{\varepsilon}^{j}D_{j}w_{\varepsilon}=-2(D_{j}\partial_{\varepsilon,\star}^{j})\otimes w_{\varepsilon,\star}+2D_{j}(\partial_{\varepsilon,\star}^{j}\otimes w_{\varepsilon,\star})+2\partial_{\varepsilon,\star}^{j}\otimes D_{j}w_{\varepsilon,\star}. \tag*{(402)} 2 ∂ ε j D j w ε = − 2 ( D j ∂ ε , ⋆ j ) ⊗ w ε , ⋆ + 2 D j ( ∂ ε , ⋆ j ⊗ w ε , ⋆ ) + 2 ∂ ε , ⋆ j ⊗ D j w ε , ⋆ . (402) In order to finish our reformulation of (391) –(396) , it only remains to complete two steps: First, we insert our paracontrolled ansatz into the high× \times × (397) , (399) , the first summand in (401) , and the first summand in (402) , which is done using the double Duhamel trick. For this, we need to introduce operators corresponding to the right-hand sides of (388) , (389) , and (390) , which are defined as
X ε , K ∘ ( W ε , ⋆ ) ≔ Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] ) , (403) X_{\varepsilon,K}^{\circ}(W_{\varepsilon,\star})\coloneqq\operatorname{Duh}_{B,K}^{F}\bigl([W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]\bigr), \tag*{(403)} X ε , K ∘ ( W ε , ⋆ ) : = Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] ) , (403) X ε , K ∙ ( w ε , ⋆ ) ≔ Duh B , K F ( B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) ) , (404) X_{\varepsilon,K}^{\bullet}(w_{\varepsilon,\star})\coloneqq\operatorname{Duh}_{B,K}^{F}\bigl(B(w_{\varepsilon,\star}\otimes D\partial_{\varepsilon,\star})\bigr), \tag*{(404)} X ε , K ∙ ( w ε , ⋆ ) : = Duh B , K F ( B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) ) , (404) x ε , K ( W ε , ⋆ ) ≔ 2 Duh B , K ( W ε , ⋆ j ⊗ D j ∂ ε , ⋆ ) . (405) x_{\varepsilon,K}(W_{\varepsilon,\star})\coloneqq2\operatorname{Duh}_{B,K}\bigl(W_{\varepsilon,\star}^{j}\otimes D_{j}\partial_{\varepsilon,\star}\bigr). \tag*{(405)} x ε , K ( W ε , ⋆ ) : = 2 Duh B , K ( W ε , ⋆ j ⊗ D j ∂ ε , ⋆ ) . (405) In line with our earlier decomposition of X ε , K X_{\varepsilon,K} X ε , K X ε , K ∘ X_{\varepsilon,K}^{\circ} X ε , K ∘ X ε , K ∙ X_{\varepsilon,K}^{\bullet} X ε , K ∙
X ε , K ( W ε , ⋆ , w ε , ⋆ ) ≔ X ε , K ∘ ( W ε , ⋆ ) + X ε , K ∙ ( w ε , ⋆ ) . X_{\varepsilon,K}(W_{\varepsilon,\star},w_{\varepsilon,\star})\coloneqq X_{\varepsilon,K}^{\circ}(W_{\varepsilon,\star})+X_{\varepsilon,K}^{\bullet}(w_{\varepsilon,\star}). X ε , K ( W ε , ⋆ , w ε , ⋆ ) : = X ε , K ∘ ( W ε , ⋆ ) + X ε , K ∙ ( w ε , ⋆ ) . Second, we renormalise the individual terms in the nonlinearity, and the counterterms will involve the constants σ ρ , ε 2 \sigma_{\rho,\varepsilon}^{2} σ ρ , ε 2 C ρ C_{\rho} C ρ (388) –(390) (or as on the right-hand sides of (403) –(405) ), the equations for the paracontrolled components, which we repeat for convenience, are
X ε , K ∘ = Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j ) ] ) , (406) X_{\varepsilon,K}^{\circ}=\operatorname{Duh}_{B,K}^{F}\bigl([W_{\varepsilon,\star,j}\otimes(2D^{j}\partial_{\varepsilon,\star}-D_{\varepsilon,\star}^{\circ j})]\bigr), \tag*{(406)} X ε , K ∘ = Duh B , K F ( [ W ε , ⋆ , j ⊗ ( 2 D j ∂ ε , ⋆ − D ε , ⋆ ∘ j )] ) , (406) X ε , K ∙ = Duh B , K F ( B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) ) , (407) X_{\varepsilon,K}^{\bullet}=\operatorname{Duh}_{B,K}^{F}\bigl(B(w_{\varepsilon,\star}\otimes D\partial_{\varepsilon,\star})\bigr), \tag*{(407)} X ε , K ∙ = Duh B , K F ( B ( w ε , ⋆ ⊗ D ∂ ε , ⋆ ) ) , (407) x ε , K = 2 Duh B , K ( W ε , ⋆ j ⊗ D j ∂ ε , ⋆ ) . (408) x_{\varepsilon,K}=2\operatorname{Duh}_{B,K}\bigl(W_{\varepsilon,\star}^{j}\otimes D_{j}\partial_{\varepsilon,\star}\bigr). \tag*{(408)} x ε , K = 2 Duh B , K ( W ε , ⋆ j ⊗ D j ∂ ε , ⋆ ) . (408) Covariant ansatz
For the smooth nonlinear remainder Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆
Y ε , κ = Duh B , κ F ( Lb ( ϱ ˚ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) + Lb ( ϱ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B ( ϱ ˚ ˚ ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ ϑ ( ϱ ˚ ε ) + Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ϱ ˚ ε , ⋆ ) + 2 B ( x ε , ⋆ ( W ε , ⋄ ) ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) ( Cov g ϑ + Cov ϑ ϑ ) ( W ε ) + Lb ( S ⋆ + Y ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B ( ( φ ⋆ + y ε , ⋆ ) ⊗ D ϱ ˚ ε , ⋆ ) − 2 D j [ W ε , ⋆ ⊗ ϱ ˚ ε , ⋆ , j ] + D [ W ε , ⋆ j ⊗ ϱ ˚ ε , ⋆ , j − D ( B ( w ε , ⋆ ⊗ ϱ ˚ ε , ⋆ ) ) ] + [ W ε , ⋆ , j ⊗ ( 2 D j ϱ ˚ ε , ⋆ − D ϱ ˚ ε , ⋆ j ) ] + B ( w ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + [ ϱ ˚ ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j ) ] + B ( ϱ ˚ ε , ⋆ ⊗ D w ε , ⋆ ) + [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] + B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) + [ ϱ ˚ ε , j + W ε , j , [ ϱ ˚ ε j + W ε j , ϱ ˚ ε + W ε ] ] − [ S j , [ S j , S ] ] − σ ρ , ε 2 Cov g ϑ ( ϱ ˚ ε + W ε ) + B ( ( ϱ ˚ ε + w ε ) ⊗ ( ϱ ˚ ε + W ε ) ( ϱ ˚ ε + w ε ) ) − B ( φ ⊗ S φ ) − σ ρ , ε 2 Cov ϑ ϑ ( ϱ ˚ ε + W ε ) − 1 8 π Cov ϑ ϑ ( ϱ ˚ ε + W ε ) ) . (409) \begin{aligned}
Y_{\varepsilon,\kappa}={}&\operatorname{Duh}^{F}_{B,\kappa}\Bigl(
\operatorname{Lb}(\mathring{\mathring{\varrho}}_{\varepsilon,\star},\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{g}^{\vartheta}(\mathring{\varrho}_{\varepsilon})
+\operatorname{Lb}(\mathring{\varrho}_{\varepsilon,\star},\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})\\
&+2\operatorname{B}(\mathring{\mathring{\varrho}}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{\vartheta}^{\vartheta}(\mathring{\varrho}_{\varepsilon})\\
&+\operatorname{Lb}(X_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond}),\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+2\operatorname{B}(x_{\varepsilon,\star}(W_{\varepsilon,\diamond})\mathbin{\otimes}\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})\\
&+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})(\operatorname{Cov}_{g}^{\vartheta}+\operatorname{Cov}_{\vartheta}^{\vartheta})(W_{\varepsilon})\\
&+\operatorname{Lb}(S_{\star}+Y_{\varepsilon,\star},\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+2\operatorname{B}((\varphi_{\star}+y_{\varepsilon,\star})\mathbin{\otimes}\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})\\
&-2\mathrm{D}^{j}[W_{\varepsilon,\star}\mathbin{\otimes}\mathring{\varrho}_{\varepsilon,\star,j}]
+\mathrm{D}[W_{\varepsilon,\star}^{j}\mathbin{\otimes}\mathring{\varrho}_{\varepsilon,\star,j}
-\mathrm{D}(\operatorname{B}(w_{\varepsilon,\star}\mathbin{\otimes}\mathring{\varrho}_{\varepsilon,\star}))]\\
&+[W_{\varepsilon,\star,j}\mathbin{\otimes}(2\mathrm{D}^{j}\mathring{\varrho}_{\varepsilon,\star}-\mathrm{D}\mathring{\varrho}_{\varepsilon,\star}^{j})]
+\operatorname{B}(w_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})\\
&+[\mathring{\varrho}_{\varepsilon,\star,j}\mathbin{\otimes}(2\mathrm{D}^{j}W_{\varepsilon,\star}-\mathrm{D}W_{\varepsilon,\star}^{j})]
+\operatorname{B}(\mathring{\varrho}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}w_{\varepsilon,\star})\\
&+[W_{\varepsilon,j},2\mathrm{D}^{j}W_{\varepsilon}-\mathrm{D}W_{\varepsilon}^{j}]
-[S_{j},2\mathrm{D}^{j}S-\mathrm{D}S^{j}]
+\operatorname{B}(w_{\varepsilon}\mathbin{\otimes}\mathrm{D}w_{\varepsilon})
-\operatorname{B}(\varphi\mathbin{\otimes}\mathrm{D}\varphi)\\
&+[\mathring{\varrho}_{\varepsilon,j}+W_{\varepsilon,j},
[\mathring{\varrho}_{\varepsilon}^{j}+W_{\varepsilon}^{j},\mathring{\varrho}_{\varepsilon}+W_{\varepsilon}]]
-[S_{j},[S^{j},S]]
-\sigma_{\rho,\varepsilon}^{2}\operatorname{Cov}_{g}^{\vartheta}(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})\\
&+\operatorname{B}((\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})\mathbin{\otimes}
(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon}))
-\operatorname{B}(\varphi\mathbin{\otimes}S\varphi)
-\sigma_{\rho,\varepsilon}^{2}\operatorname{Cov}_{\vartheta}^{\vartheta}(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})\\
&-\frac{1}{8\pi}\operatorname{Cov}_{\vartheta}^{\vartheta}(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})
\Bigr). \tag*{(409)}
\end{aligned} Y ε , κ = Duh B , κ F ( Lb ( ϱ ˚ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) + Lb ( ϱ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B ( ϱ ˚ ˚ ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ ϑ ( ϱ ˚ ε ) + Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ϱ ˚ ε , ⋆ ) + 2 B ( x ε , ⋆ ( W ε , ⋄ ) ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) ( Cov g ϑ + Cov ϑ ϑ ) ( W ε ) + Lb ( S ⋆ + Y ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B (( φ ⋆ + y ε , ⋆ ) ⊗ D ϱ ˚ ε , ⋆ ) − 2 D j [ W ε , ⋆ ⊗ ϱ ˚ ε , ⋆ , j ] + D [ W ε , ⋆ j ⊗ ϱ ˚ ε , ⋆ , j − D ( B ( w ε , ⋆ ⊗ ϱ ˚ ε , ⋆ ))] + [ W ε , ⋆ , j ⊗ ( 2 D j ϱ ˚ ε , ⋆ − D ϱ ˚ ε , ⋆ j )] + B ( w ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + [ ϱ ˚ ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j )] + B ( ϱ ˚ ε , ⋆ ⊗ D w ε , ⋆ ) + [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] + B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) + [ ϱ ˚ ε , j + W ε , j , [ ϱ ˚ ε j + W ε j , ϱ ˚ ε + W ε ]] − [ S j , [ S j , S ]] − σ ρ , ε 2 Cov g ϑ ( ϱ ˚ ε + W ε ) + B (( ϱ ˚ ε + w ε ) ⊗ ( ϱ ˚ ε + W ε ) ( ϱ ˚ ε + w ε )) − B ( φ ⊗ S φ ) − σ ρ , ε 2 Cov ϑ ϑ ( ϱ ˚ ε + W ε ) − 8 π 1 Cov ϑ ϑ ( ϱ ˚ ε + W ε ) ) . (409) Lb ( ϱ ˚ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) + Lb ( ϱ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B ( ϱ ˚ ˚ ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ ϑ ( ϱ ˚ ε ) (410) \begin{aligned}
&\operatorname{Lb}(\mathring{\mathring{\varrho}}_{\varepsilon,\star},\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{g}^{\vartheta}(\mathring{\varrho}_{\varepsilon})
+\operatorname{Lb}(\mathring{\varrho}_{\varepsilon,\star},\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})\\
&\qquad+2\operatorname{B}(\mathring{\mathring{\varrho}}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}\mathring{\varrho}_{\varepsilon,\star})
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{\vartheta}^{\vartheta}(\mathring{\varrho}_{\varepsilon}) \tag*{(410)}
\end{aligned} Lb ( ϱ ˚ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) + Lb ( ϱ ˚ ε , ⋆ , D ϱ ˚ ε , ⋆ ) + 2 B ( ϱ ˚ ˚ ε , ⋆ ⊗ D ϱ ˚ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ ϑ ( ϱ ˚ ε ) (410) where we have highlighted the counterterms in light grey. In (6.74), we used the symbols ⋄ \diamond ⋄ ⋆ \star ⋆ X X X x x x (409) –(6.81) are similar to each other. Finally, for the smooth remainder y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
y ε , κ = Duh B , κ ( 2 ( ϱ ˚ ε , ⋆ j + ϱ ˚ ε , ⋆ j ) ⊗ D j ϱ ˚ ε , ⋆ + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ ϱ ˚ ε , ⋆ + 2 X ε , ⋆ j ( W ε , ⋄ , w ε , ⋄ ) ⊗ D j ϱ ˚ ε , ⋆ + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( w ε ) − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ x ε , ⋆ ( W ε , ⋄ ) + 2 ( S ⋆ + Y ε , ⋆ ) j ⊗ D j ϱ ˚ ε , ⋆ − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ ( φ ⋆ + y ε , ⋆ ) + 2 D j ( ϱ ˚ ε , ⋆ j ⊗ w ε , ⋆ ) + 2 W ε , ⋆ j ⊗ D j ϱ ˚ ε , ⋆ + 2 ϱ ˚ ε , ⋆ j ⊗ D j w ε , ⋆ + 2 ( W ε j D j w ε − S j D j φ ) + ( ϱ ˚ ε + W ε ) j ( ϱ ˚ ε + W ε ) j ( ϱ ˚ ε + w ε ) − S j S j φ − 2 σ ρ , ε 2 Cov g ϑ ( ϱ ˚ ε + w ε ) − ∣ ϱ ˚ ε + w ε ∣ 2 ( ϱ ˚ ε + w ε ) + ( dim ( V ) + 2 ) σ ρ , ε 2 ( ϱ ˚ ε + w ε ) + ∣ φ ∣ 2 φ − M ( ϱ ˚ ε + w ε ) ) . (411) \begin{aligned}
y_{\varepsilon,\kappa}={}&\operatorname{Duh}_{B,\kappa}\Bigl(
2(\mathring{\varrho}_{\varepsilon,\star}^{j}+\mathring{\varrho}_{\varepsilon,\star}^{j})\otimes\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{g}^{\vartheta}(\mathring{\varrho}_{\varepsilon})
-2(\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}^{j})\otimes\mathring{\varrho}_{\varepsilon,\star}\\
&+2X_{\varepsilon,\star}^{j}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond})\otimes\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}
+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}_{g}^{\vartheta}(w_{\varepsilon})
-2(\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}^{j})\otimes x_{\varepsilon,\star}(W_{\varepsilon,\diamond})\\
&+2(S_{\star}+Y_{\varepsilon,\star})^{j}\otimes\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}
-2(\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}^{j})\otimes(\varphi_{\star}+y_{\varepsilon,\star})\\
&+2\mathrm{D}_{j}(\mathring{\varrho}_{\varepsilon,\star}^{j}\otimes w_{\varepsilon,\star})
+2W_{\varepsilon,\star}^{j}\otimes\mathrm{D}_{j}\mathring{\varrho}_{\varepsilon,\star}
+2\mathring{\varrho}_{\varepsilon,\star}^{j}\otimes\mathrm{D}_{j}w_{\varepsilon,\star}\\
&+2(W_{\varepsilon}^{j}\mathrm{D}_{j}w_{\varepsilon}-S^{j}\mathrm{D}_{j}\varphi)\\
&+(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})^{j}
(\mathring{\varrho}_{\varepsilon}+W_{\varepsilon})_{j}
(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})
-S^{j}S_{j}\varphi
-2\sigma_{\rho,\varepsilon}^{2}\operatorname{Cov}_{g}^{\vartheta}(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})\\
&-|\mathring{\varrho}_{\varepsilon}+w_{\varepsilon}|^{2}
(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})
+(\dim(V)+2)\sigma_{\rho,\varepsilon}^{2}(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})
+|\varphi|^{2}\varphi\\
&-\operatorname{M}(\mathring{\varrho}_{\varepsilon}+w_{\varepsilon})
\Bigr). \tag*{(411)}
\end{aligned} y ε , κ = Duh B , κ ( 2 ( ϱ ˚ ε , ⋆ j + ϱ ˚ ε , ⋆ j ) ⊗ D j ϱ ˚ ε , ⋆ + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( ϱ ˚ ε ) − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ ϱ ˚ ε , ⋆ + 2 X ε , ⋆ j ( W ε , ⋄ , w ε , ⋄ ) ⊗ D j ϱ ˚ ε , ⋆ + ( σ ρ , ε 2 + C ρ ) Cov g ϑ ( w ε ) − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ x ε , ⋆ ( W ε , ⋄ ) + 2 ( S ⋆ + Y ε , ⋆ ) j ⊗ D j ϱ ˚ ε , ⋆ − 2 ( D j ϱ ˚ ε , ⋆ j ) ⊗ ( φ ⋆ + y ε , ⋆ ) + 2 D j ( ϱ ˚ ε , ⋆ j ⊗ w ε , ⋆ ) + 2 W ε , ⋆ j ⊗ D j ϱ ˚ ε , ⋆ + 2 ϱ ˚ ε , ⋆ j ⊗ D j w ε , ⋆ + 2 ( W ε j D j w ε − S j D j φ ) + ( ϱ ˚ ε + W ε ) j ( ϱ ˚ ε + W ε ) j ( ϱ ˚ ε + w ε ) − S j S j φ − 2 σ ρ , ε 2 Cov g ϑ ( ϱ ˚ ε + w ε ) − ∣ ϱ ˚ ε + w ε ∣ 2 ( ϱ ˚ ε + w ε ) + ( dim ( V ) + 2 ) σ ρ , ε 2 ( ϱ ˚ ε + w ε ) + ∣ φ ∣ 2 φ − M ( ϱ ˚ ε + w ε ) ) . (411) We note that, by Definition 2.15, adding up the counterterms in (409) –(6.82) gives exactly the counterterm in (6.22), and adding up the counterterms in (411) –(6.90) gives exactly the counterterm in (6.26).
Remark 6.7. By writing out the integral equations (409) –(6.90), we arrived at rather long expressions. The precise form of most terms is irrelevant, i.e., the precise indices, prefactors, or signs are not too important. However, in some terms such as (409) –(6.75), the precise form is crucial for our argument. For this reason, we believe it is best to write out the integral equations for Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
Remark 6.8. In (6.82), we left the 1 8 π \frac{1}{8\pi} 8 π 1 C ρ C_\rho C ρ § 7 –§ 10 , one could leave the C ρ C_\rho C ρ § 7 –§ 10 could be used to prove that the limiting solution does not depend on ρ \rho ρ C ρ C_\rho C ρ
Solution spaces, Schauder estimates, and paraproduct estimates
In this section, we introduce the covariant functional framework that will be needed in the proof of our covariant local well-posedness result (Theorem 2.19 ). Throughout the section, we let 0 < T ≤ 1 128 0<T\leq\frac{1}{128} 0 < T ≤ 128 1 N l o N_{\mathrm{lo}} N lo (378) , and let B B B (350) . Furthermore, as in Notation 6.1, we simply write F F F F B F_B F B
Solution spaces
In this subsection, we introduce the functional framework that will be used to solve (6.69)–(6.90). In order to treat the connection one-form and Higgs field simultaneously, we introduce the following notation.
Notation 7.1. First, we introduce Y ≔ g 2 ⊕ V = g ⊕ g ⊕ V Y \coloneqq\mathfrak{g}^{2}\oplus V=\mathfrak{g}\oplus\mathfrak{g}\oplus V Y : = g 2 ⊕ V = g ⊕ g ⊕ V G G G ϱ Y = Ad ⊕ Ad ⊕ ϱ \varrho_Y=\operatorname{Ad}\oplus\operatorname{Ad}\oplus\varrho ϱ Y = Ad ⊕ Ad ⊕ ϱ Y Y Y
Y ( 1 ) = Y ( 2 ) ≔ g and Y ( 3 ) ≔ V . Y^{(1)}=Y^{(2)}\coloneqq\mathfrak{g}
\qquad\text{and}\qquad
Y^{(3)}\coloneqq V. Y ( 1 ) = Y ( 2 ) : = g and Y ( 3 ) : = V . For each N ∈ 2 N 0 N\in2^{\mathbb{N}_0} N ∈ 2 N 0 End ( Y ) \operatorname{End}(Y) End ( Y )
q Y , B ( ⋅ , ⋅ ) ≔ q B ( ⋅ , ⋅ ; g 2 ) ⊕ p B ( ⋅ , ⋅ ; V ) , q Y , B , ε , N ( ⋅ , ⋅ ) ≔ q B , ε , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , ε , N ( ⋅ , ⋅ ; V ) , q Y , B , N ( ⋅ , ⋅ ) ≔ q B , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , N ( ⋅ , ⋅ ; V ) . (412) \begin{aligned}
q_{Y,B}(\mathord{\cdot},\mathord{\cdot})&\coloneqq q_B(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_B(\mathord{\cdot},\mathord{\cdot};V),\\
q_{Y,B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot})&\coloneqq q_{B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_{B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot};V),\\
q_{Y,B,N}(\mathord{\cdot},\mathord{\cdot})&\coloneqq q_{B,N}(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_{B,N}(\mathord{\cdot},\mathord{\cdot};V). \tag*{(412)}
\end{aligned} q Y , B ( ⋅ , ⋅ ) q Y , B , ε , N ( ⋅ , ⋅ ) q Y , B , N ( ⋅ , ⋅ ) : = q B ( ⋅ , ⋅ ; g 2 ) ⊕ p B ( ⋅ , ⋅ ; V ) , : = q B , ε , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , ε , N ( ⋅ , ⋅ ; V ) , : = q B , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , N ( ⋅ , ⋅ ; V ) . (412) where q B ( ⋅ , ⋅ ; g 2 ) q_B(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2}) q B ( ⋅ , ⋅ ; g 2 ) p B ( ⋅ , ⋅ ; V ) p_B(\mathord{\cdot},\mathord{\cdot};V) p B ( ⋅ , ⋅ ; V ) End ( g 2 ) \operatorname{End}(\mathfrak{g}^{2}) End ( g 2 ) End ( V ) \operatorname{End}(V) End ( V ) Definition 2.1 and their decorated versions are as in (372) and (373) . We emphasise that the second kernel in (412) is a p B p_B p B q B q_B q B Φ \Phi Φ (354) does not contain a curvature term. We will sometimes also use the End ( Y ) \operatorname{End}(Y) End ( Y )
p Y , B ( ⋅ , ⋅ ) ≔ p B ( ⋅ , ⋅ ; g 2 ) ⊕ p B ( ⋅ , ⋅ ; V ) , p Y , B , ε , N ( ⋅ , ⋅ ) ≔ p B , ε , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , ε , N ( ⋅ , ⋅ ; V ) , p Y , B , N ( ⋅ , ⋅ ) ≔ p B , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , N ( ⋅ , ⋅ ; V ) . (413) \begin{aligned}
p_{Y,B}(\mathord{\cdot},\mathord{\cdot})&\coloneqq p_B(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_B(\mathord{\cdot},\mathord{\cdot};V),\\
p_{Y,B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot})&\coloneqq p_{B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_{B,\varepsilon,N}(\mathord{\cdot},\mathord{\cdot};V),\\
p_{Y,B,N}(\mathord{\cdot},\mathord{\cdot})&\coloneqq p_{B,N}(\mathord{\cdot},\mathord{\cdot};\mathfrak{g}^{2})\oplus p_{B,N}(\mathord{\cdot},\mathord{\cdot};V). \tag*{(413)}
\end{aligned} p Y , B ( ⋅ , ⋅ ) p Y , B , ε , N ( ⋅ , ⋅ ) p Y , B , N ( ⋅ , ⋅ ) : = p B ( ⋅ , ⋅ ; g 2 ) ⊕ p B ( ⋅ , ⋅ ; V ) , : = p B , ε , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , ε , N ( ⋅ , ⋅ ; V ) , : = p B , N ( ⋅ , ⋅ ; g 2 ) ⊕ p B , N ( ⋅ , ⋅ ; V ) . (413) where the first kernel on each right-hand side is End ( g 2 ) \operatorname{End}(\mathfrak{g}^{2}) End ( g 2 ) End ( V ) \operatorname{End}(V) End ( V ) (412) and (413) , we further write
Duh Y , B , N F ( Ψ ) ( t , x ) ≔ ∬ d s d y q Y , B , N ( s , y ; t , x ) Ψ ( s , y ) , \operatorname{Duh}^{F}_{Y,B,N}(\Psi)(t,x)\coloneqq\iint ds\,dy\,q_{Y,B,N}(s,y;t,x)\Psi(s,y), Duh Y , B , N F ( Ψ ) ( t , x ) : = ∬ d s d y q Y , B , N ( s , y ; t , x ) Ψ ( s , y ) , Duh Y , B , N ( Ψ ) ( t , x ) ≔ ∬ d s d y p Y , B , N ( s , y ; t , x ) Ψ ( s , y ) . (414) \operatorname{Duh}_{Y,B,N}(\Psi)(t,x)\coloneqq\iint ds\,dy\,p_{Y,B,N}(s,y;t,x)\Psi(s,y). \tag*{(414)} Duh Y , B , N ( Ψ ) ( t , x ) : = ∬ d s d y p Y , B , N ( s , y ; t , x ) Ψ ( s , y ) . (414) where Ψ : [ 0 , T ] × T 2 → Y \Psi\colon[0,T]\times\mathbb{T}^{2}\to Y Ψ : [ 0 , T ] × T 2 → Y
We note that, while Notation 7.1 will already be useful in this section, it will be most useful in our treatment of stochastic objects, see e.g. Sections 8.1 and 9. We proceed to introduce our solution spaces. In the following, recall Notation 3.4 regarding space-time derivatives.
Definition 7.2 (Solution spaces). Let α ∈ R \alpha\in\mathbb{R} α ∈ R η ≤ 0 \eta\le0 η ≤ 0 M ∈ 2 N 0 M \in2^{\mathbb{N}_0} M ∈ 2 N 0 M ≥ N l o M \ge N_{\mathrm{lo}} M ≥ N lo ϕ M : [ 0 , T ] × T 2 → Y \phi_M : [0,T] \times\mathbb{T}^2 \to Y ϕ M : [ 0 , T ] × T 2 → Y
∥ ϕ M ∥ s o l M α , η = ∥ ϕ M ∥ s o l M α , η ( [ 0 , T ] ) : = max 0 ≤ m ≤ 10 M α − m ∥ t − η / 2 D t , x ( m ) ϕ M ∥ L t , x ∞ ( [ 0 , T ] × T 2 ) . (415) \lVert\phi_M \rVert_{\mathrm{sol}^{\alpha,\eta}_M} = \lVert\phi_M \rVert_{\mathrm{sol}^{\alpha,\eta}_M([0,T])} := \max_{0 \le m \le10} M^{\alpha-m} \lVert t^{-\eta/2} D_{t,x}^{(m)} \phi_M \rVert_{L^\infty_{t,x}([0,T]\times\mathbb{T}^2)}. \tag*{(415)} ∥ ϕ M ∥ sol M α , η = ∥ ϕ M ∥ sol M α , η ([ 0 , T ]) := 0 ≤ m ≤ 10 max M α − m ∥ t − η /2 D t , x ( m ) ϕ M ∥ L t , x ∞ ([ 0 , T ] × T 2 ) . (415) Furthermore, let ϕ ⋆ = ( ϕ N ) N ≥ N l o \phi_\star= (\phi_N)_{N \ge N_{\mathrm{lo}}} ϕ ⋆ = ( ϕ N ) N ≥ N lo N ≥ N l o N \ge N_{\mathrm{lo}} N ≥ N lo ϕ N : [ 0 , T ] × T 2 → Y \phi_N : [0,T] \times\mathbb{T}^2 \to Y ϕ N : [ 0 , T ] × T 2 → Y
∥ ϕ ⋆ ∥ s o l α , η = ∥ ϕ ⋆ ∥ s o l α , η ( [ 0 , T ] ) : = sup N ≥ N l o ∥ ϕ N ∥ s o l N α , η ( [ 0 , T ] ) . (416) \lVert\phi_\star\rVert_{\mathrm{sol}^{\alpha,\eta}} = \lVert\phi_\star\rVert_{\mathrm{sol}^{\alpha,\eta}([0,T])} := \sup_{N \ge N_{\mathrm{lo}}} \lVert\phi_N \rVert_{\mathrm{sol}^{\alpha,\eta}_N([0,T])}. \tag*{(416)} ∥ ϕ ⋆ ∥ sol α , η = ∥ ϕ ⋆ ∥ sol α , η ([ 0 , T ]) := N ≥ N lo sup ∥ ϕ N ∥ sol N α , η ([ 0 , T ]) . (416) We also write ∥ ϕ M ∥ s o l M α : = ∥ ϕ M ∥ s o l M α , 0 \lVert\phi_M \rVert_{\mathrm{sol}^{\alpha}_M} := \lVert\phi_M \rVert_{\mathrm{sol}^{\alpha,0}_M} ∥ ϕ M ∥ sol M α := ∥ ϕ M ∥ sol M α , 0 ∥ ϕ ⋆ ∥ s o l α : = ∥ ϕ ⋆ ∥ s o l α , 0 \lVert\phi_\star\rVert_{\mathrm{sol}^{\alpha}} := \lVert\phi_\star\rVert_{\mathrm{sol}^{\alpha,0}} ∥ ϕ ⋆ ∥ sol α := ∥ ϕ ⋆ ∥ sol α , 0 Y ( a ) Y^{(a)} Y ( a ) Y Y Y a ∈ [ 3 ] a \in[3] a ∈ [ 3 ] (415) and (416) are defined similarly.
Remark 7.3. In Definition 7.2, it is useful to think of α + η \alpha+\eta α + η B ≡ 0 B \equiv0 B ≡ 0 ∥ ϕ ⋆ ∥ s o l α , η \lVert\phi_\star\rVert_{\mathrm{sol}^{\alpha,\eta}} ∥ ϕ ⋆ ∥ sol α , η ∥ ϕ ( 0 ) ∥ C x α + η \lVert\phi^{(0)} \rVert_{\mathcal{C}^{\alpha+\eta}_x} ∥ ϕ ( 0 ) ∥ C x α + η T T T
The solution spaces s o l M α , η \mathrm{sol}^{\alpha,\eta}_M sol M α , η α \alpha α M M M α \alpha α (415) and (416) . The reason is that, when α \alpha α M M M (415) is always negative, and hence (415) does not allow us to gain powers of M M M
Definition 7.4 (Modified solution spaces). Let α ∈ ( − 1 , 0 ) \alpha\in(-1,0) α ∈ ( − 1 , 0 ) M ∈ 2 N 0 M \in2^{\mathbb{N}_0} M ∈ 2 N 0 M ≥ N l o M \ge N_{\mathrm{lo}} M ≥ N lo ϕ M : [ 0 , T ] × T 2 → Y \phi_M : [0,T] \times\mathbb{T}^2 \to Y ϕ M : [ 0 , T ] × T 2 → Y
∥ ϕ M ∥ m s o l M α = ∥ ϕ M ∥ m s o l M α ( [ 0 , T ] ) : = ∥ ϕ M ∥ s o l M α ( [ 0 , T ] ) + sup − 1 ≤ α ′ ≤ α max 0 ≤ m ≤ 10 M − α ′ + α − m ∥ D t , x ( m ) ϕ M ∥ C t α ′ / 2 L x r . (417) \begin{aligned}
\lVert\phi_M \rVert_{\mathrm{msol}^{\alpha}_M}
&= \lVert\phi_M \rVert_{\mathrm{msol}^{\alpha}_M([0,T])} \\
&:= \lVert\phi_M \rVert_{\mathrm{sol}^{\alpha}_M([0,T])}
+ \sup_{-1 \le\alpha' \le\alpha}\max_{0 \le m \le10}
M^{-\alpha' + \alpha- m}\lVert D_{t,x}^{(m)}\phi_M\rVert_{\mathcal{C}^{\alpha'/2}_t L^r_x}. \tag*{(417)}
\end{aligned} ∥ ϕ M ∥ msol M α = ∥ ϕ M ∥ msol M α ([ 0 , T ]) := ∥ ϕ M ∥ sol M α ([ 0 , T ]) + − 1 ≤ α ′ ≤ α sup 0 ≤ m ≤ 10 max M − α ′ + α − m ∥ D t , x ( m ) ϕ M ∥ C t α ′ /2 L x r . (417) where r r r (96) . Furthermore, let ϕ ⋆ = ( ϕ N ) N ≥ N l o \phi_\star= (\phi_N)_{N \ge N_{\mathrm{lo}}} ϕ ⋆ = ( ϕ N ) N ≥ N lo N ≥ N l o N \ge N_{\mathrm{lo}} N ≥ N lo ϕ N : [ 0 , T ] × T 2 → Y \phi_N : [0,T] \times\mathbb{T}^2 \to Y ϕ N : [ 0 , T ] × T 2 → Y
∥ ϕ ⋆ ∥ m s o l α = ∥ ϕ ⋆ ∥ m s o l α ( [ 0 , T ] ) : = sup N ≥ N l o ∥ ϕ N ∥ m s o l N α ( [ 0 , T ] ) . (418) \lVert\phi_\star\rVert_{\mathrm{msol}^{\alpha}} = \lVert\phi_\star\rVert_{\mathrm{msol}^{\alpha}([0,T])} := \sup_{N \ge N_{\mathrm{lo}}} \lVert\phi_N \rVert_{\mathrm{msol}^{\alpha}_N([0,T])}. \tag*{(418)} ∥ ϕ ⋆ ∥ msol α = ∥ ϕ ⋆ ∥ msol α ([ 0 , T ]) := N ≥ N lo sup ∥ ϕ N ∥ msol N α ([ 0 , T ]) . (418) If the maps take values in the component Y ( a ) Y^{(a)} Y ( a ) Y Y Y a ∈ [ 3 ] a \in[3] a ∈ [ 3 ]
We note that the exponent of M M M (417) is positive when α ′ < α \alpha' < \alpha α ′ < α m = 0 m=0 m = 0
Remark 7.5. As mentioned before, the most important difference between our s o l α , η \mathrm{sol}^{\alpha,\eta} sol α , η m s o l α \mathrm{msol}^{\alpha} msol α m s o l α \mathrm{msol}^{\alpha} msol α m s o l α \mathrm{msol}^{\alpha} msol α m s o l α \mathrm{msol}^{\alpha} msol α L x r L^r_x L x r s o l α , η \mathrm{sol}^{\alpha,\eta} sol α , η L x ∞ L^\infty_x L x ∞ m s o l α \mathrm{msol}^{\alpha} msol α r r r L x r L^r_x L x r
In the following, we state and prove two lemmas related to properties of our solution spaces. In the first lemma, we prove a Hölder estimate in time.
Lemma 7.6 (Hölder estimate in time). Let α ∈ R \alpha\in\mathbb{R} α ∈ R η ∈ ( − 2 , 0 ] \eta\in(-2,0] η ∈ ( − 2 , 0 ] 0 ≤ β ≤ 2 0 \le\beta\le2 0 ≤ β ≤ 2 0 ≤ m ≤ 8 0 \le m \le8 0 ≤ m ≤ 8 M ∈ 2 N 0 M \in2^{\mathbb{N}_0} M ∈ 2 N 0 M ≥ N l o M \ge N_{\mathrm{lo}} M ≥ N lo ϕ M : [ 0 , T ] × T 2 → Y \phi_M \colon[0,T] \times\mathbb{T}^2 \to Y ϕ M : [ 0 , T ] × T 2 → Y ϕ M ( t ) = 0 \phi_M(t)=0 ϕ M ( t ) = 0 t ≪ M − 2 t \ll M^{-2} t ≪ M − 2
∥ t − η / 2 D t , x ( m ) ϕ M ∥ C t β / 2 L x ∞ ≲ M − α + β + m ∥ ϕ M ∥ s o l M α , η . (419) \left\|t^{-\eta/2}D_{t,x}^{(m)}\phi_M\right\|_{\mathcal{C}^{\beta/2}_tL^\infty_x}
\lesssim M^{-\alpha+\beta+m}\|\phi_M\|_{\mathrm{sol}^{\alpha,\eta}_M}. \tag*{(419)} t − η /2 D t , x ( m ) ϕ M C t β /2 L x ∞ ≲ M − α + β + m ∥ ϕ M ∥ sol M α , η . (419) Proof. By interpolation, it suffices to treat the cases β = 0 \beta=0 β = 0 β = 2 \beta=2 β = 2 β = 0 \beta=0 β = 0 β = 2 \beta=2 β = 2
∥ ∂ t ( t − η / 2 D t , x ( m ) ϕ M ) ∥ L t , x ∞ ≲ ∥ t − η / 2 − 1 D t , x ( m ) ϕ M ∥ L t , x ∞ + ∥ t − η / 2 D t , x ( m + 2 ) ϕ M ∥ L t , x ∞ \left\|\partial_t\left(t^{-\eta/2}D_{t,x}^{(m)}\phi_M\right)\right\|_{L^\infty_{t,x}}
\lesssim\left\|t^{-\eta/2-1}D_{t,x}^{(m)}\phi_M\right\|_{L^\infty_{t,x}}+\left\|t^{-\eta/2}D_{t,x}^{(m+2)}\phi_M\right\|_{L^\infty_{t,x}} ∂ t ( t − η /2 D t , x ( m ) ϕ M ) L t , x ∞ ≲ t − η /2 − 1 D t , x ( m ) ϕ M L t , x ∞ + t − η /2 D t , x ( m + 2 ) ϕ M L t , x ∞ ≲ M 2 ∥ t − η / 2 D t , x ( m ) ϕ M ∥ L t , x ∞ + ∥ t − η / 2 D t , x ( m + 2 ) ϕ M ∥ L t , x ∞ \lesssim M^2\left\|t^{-\eta/2}D_{t,x}^{(m)}\phi_M\right\|_{L^\infty_{t,x}}+\left\|t^{-\eta/2}D_{t,x}^{(m+2)}\phi_M\right\|_{L^\infty_{t,x}} ≲ M 2 t − η /2 D t , x ( m ) ϕ M L t , x ∞ + t − η /2 D t , x ( m + 2 ) ϕ M L t , x ∞ ≲ M 2 − α + m ∥ ϕ M ∥ s o l M α , η , \lesssim M^{2-\alpha+m}\|\phi_M\|_{\mathrm{sol}^{\alpha,\eta}_M}, ≲ M 2 − α + m ∥ ϕ M ∥ sol M α , η , where we used that t − 1 ≲ M 2 t^{-1}\lesssim M^2 t − 1 ≲ M 2 ϕ M \phi_M ϕ M
In the second lemma, we prove an estimate that relates solution spaces with different regularities and singular weights.
Lemma 7.7. Let α 1 , α 2 ∈ R \alpha_1,\alpha_2 \in\mathbb{R} α 1 , α 2 ∈ R η 1 , η 2 ≤ 0 \eta_1,\eta_2 \le0 η 1 , η 2 ≤ 0 α 1 + η 1 ≤ α 2 + η 2 \alpha_1+\eta_1 \le\alpha_2+\eta_2 α 1 + η 1 ≤ α 2 + η 2 η 1 ≥ η 2 \eta_1 \ge\eta_2 η 1 ≥ η 2 N ≥ N l o N \ge N_{\mathrm{lo}} N ≥ N lo ϕ N : [ 0 , T ] × T 2 → Y \phi_N \colon[0,T] \times\mathbb{T}^2 \to Y ϕ N : [ 0 , T ] × T 2 → Y t ≳ N − 2 t \gtrsim N^{-2} t ≳ N − 2
∥ ϕ N ∥ s o l N α 1 , η 1 ≲ T α 2 + η 2 − ( α 1 + η 1 ) 2 ∥ ϕ N ∥ s o l N α 2 , η 2 , \|\phi_N\|_{\mathrm{sol}^{\alpha_1,\eta_1}_N}
\lesssim T^{\frac{\alpha_2+\eta_2-(\alpha_1+\eta_1)}{2}}\|\phi_N\|_{\mathrm{sol}^{\alpha_2,\eta_2}_N}, ∥ ϕ N ∥ sol N α 1 , η 1 ≲ T 2 α 2 + η 2 − ( α 1 + η 1 ) ∥ ϕ N ∥ sol N α 2 , η 2 , where the implicit constant is independent of T T T N N N
Roughly speaking, this shows that, at the cost of increasing the regularity parameter, we can make the weight more singular.
Proof. On the support of ϕ N \phi_N ϕ N
N α 1 t − η 1 2 = N α 1 t η 2 − η 1 2 t − η 2 2 ≲ N α 1 N − ( η 2 − η 1 ) t − η 2 2 = N α 2 N α 1 + η 1 − α 2 − η 2 t − η 2 2 . N^{\alpha_1}t^{-\frac{\eta_1}{2}}=N^{\alpha_1}t^{\frac{\eta_2-\eta_1}{2}}t^{-\frac{\eta_2}{2}}\lesssim N^{\alpha_1}N^{-(\eta_2-\eta_1)}t^{-\frac{\eta_2}{2}}=N^{\alpha_2}N^{\alpha_1+\eta_1-\alpha_2-\eta_2}t^{-\frac{\eta_2}{2}}. N α 1 t − 2 η 1 = N α 1 t 2 η 2 − η 1 t − 2 η 2 ≲ N α 1 N − ( η 2 − η 1 ) t − 2 η 2 = N α 2 N α 1 + η 1 − α 2 − η 2 t − 2 η 2 . Since α 1 + η 1 − α 2 − η 2 ≤ 0 \alpha_1+\eta_1-\alpha_2-\eta_2\le0 α 1 + η 1 − α 2 − η 2 ≤ 0 N ≳ N l o ∼ T − 1 / 2 N\gtrsim N_{\mathrm{lo}}\sim T^{-1/2} N ≳ N lo ∼ T − 1/2
After this discussion of the solution spaces themselves, we will soon turn to the corresponding Schauder estimates. To simplify the notation below, we first define the following norm, which will be used for the nonlinearities in (406) –(6.90).
Definition 7.8. Let α ∈ R \alpha\in\mathbb{R} α ∈ R ν ≤ 0 \nu\le0 ν ≤ 0 M ∈ 2 N 0 M \in2^{\mathbb{N}_0} M ∈ 2 N 0 M ≥ N l o M \ge N_{\mathrm{lo}} M ≥ N lo ϕ M : [ 0 , T ] × T 2 → Y \phi_M \colon[0,T] \times\mathbb{T}^2 \to Y ϕ M : [ 0 , T ] × T 2 → Y
∥ ϕ M ∥ n l M α , ν = ∥ ϕ M ∥ n l M α , ν ( [ 0 , T ] ) : = ∥ ∬ d s d y q Y , B , M ( s , y ; t , x ) ϕ M ( s , y ) ∥ s o l M α + 2 , ν ( [ 0 , T ] ) . (420) \|\phi_M\|_{\mathrm{nl}^{\alpha,\nu}_M}=\|\phi_M\|_{\mathrm{nl}^{\alpha,\nu}_M([0,T])}:=\left\|\iint\mathrm{d}s\mathrm{d}y\,q_{Y,B,M}(s,y;t,x)\phi_M(s,y)\right\|_{\mathrm{sol}^{\alpha+2,\nu}_M([0,T])}. \tag*{(420)} ∥ ϕ M ∥ nl M α , ν = ∥ ϕ M ∥ nl M α , ν ([ 0 , T ]) := ∬ d s d y q Y , B , M ( s , y ; t , x ) ϕ M ( s , y ) sol M α + 2 , ν ([ 0 , T ]) . (420) Furthermore, let ϕ : [ 0 , T ] × T 2 → Y \phi\colon[0,T] \times\mathbb{T}^2 \to Y ϕ : [ 0 , T ] × T 2 → Y
∥ ϕ ∥ n l α , ν = ∥ ϕ ∥ n l α , ν ( [ 0 , T ] ) : = sup N ≥ N l o ∥ ϕ ∥ n l N α , ν ( [ 0 , T ] ) . (421) \|\phi\|_{\mathrm{nl}^{\alpha,\nu}}=\|\phi\|_{\mathrm{nl}^{\alpha,\nu}([0,T])}:=\sup_{N\ge N_{\mathrm{lo}}}\|\phi\|_{\mathrm{nl}^{\alpha,\nu}_N([0,T])}. \tag*{(421)} ∥ ϕ ∥ nl α , ν = ∥ ϕ ∥ nl α , ν ([ 0 , T ]) := N ≥ N lo sup ∥ ϕ ∥ nl N α , ν ([ 0 , T ]) . (421) As for our solution spaces, we write n l α \mathrm{nl}^{\alpha} nl α n l M α \mathrm{nl}^{\alpha}_M nl M α ν = 0 \nu=0 ν = 0 ϕ M \phi_M ϕ M ϕ \phi ϕ g 2 \mathfrak{g}^2 g 2 V V V Y Y Y ∥ ϕ M ∥ n l M α , ν \|\phi_M\|_{\mathrm{nl}^{\alpha,\nu}_M} ∥ ϕ M ∥ nl M α , ν ∥ ϕ ∥ n l α , ν \|\phi\|_{\mathrm{nl}^{\alpha,\nu}} ∥ ϕ ∥ nl α , ν g 2 ↪ Y \mathfrak{g}^2 \hookrightarrow Y g 2 ↪ Y V ↪ Y V \hookrightarrow Y V ↪ Y Y Y Y
We also remark that it is natural that the map ϕ \phi ϕ (421) does not depend on N N N (406) –(6.90).
Schauder and commutator estimates
In this subsection, we prove (covariant) Schauder estimates in our solution spaces, i.e., in the spaces from Definitions 7.2 and 7.8 . We also prove commutator estimates (which are stated in L t , x ∞ L^\infty_{t,x} L t , x ∞ § 9.3 , whereas the commutator estimates will be needed in uncovering the structure of the paracontrolled terms — see § 9.4 . We first define the following quantity, which will absorb all losses in the size of F F F Propositions 4.17 , 4.18 , and 4.27 . In the following, let C 0 C_0 C 0 C 0 C_0 C 0 Lemma 4.9 , Proposition 4.18 , and Proposition 4.22 .
Notation 7.9. In analogy with Notation 4.3 , we define
L S T ( F ) ≔ exp ( C 0 ∥ F ∥ L t 1 L x ∞ ( [ 0 , T ] ) ) ( 1 + ∥ F ∥ T , 100 p e r t + T 1 − 2 κ 2 ∥ t − 1 − 2 κ 2 F ∥ T , 1 p e r t ) 100 . (422) L_{S_T}(F) \coloneqq\exp\left(C_0\lVert F\rVert_{L^1_tL^\infty_x([0,T])}\right)\left(1+\lVert F\rVert^{\mathrm{pert}}_{T,100}+T^{\frac{1-2\kappa}{2}}\left\lVert t^{-\frac{1-2\kappa}{2}}F\right\rVert^{\mathrm{pert}}_{T,1}\right)^{100}. \tag*{(422)} L S T ( F ) : = exp ( C 0 ∥ F ∥ L t 1 L x ∞ ([ 0 , T ]) ) ( 1 + ∥ F ∥ T , 100 pert + T 2 1 − 2 κ t − 2 1 − 2 κ F T , 1 pert ) 100 . (422) In contrast to Notation 4.3 , we have also included the constant C 0 C_0 C 0
In the following lemma, we state our unweighted Schauder estimates. While we will later prove weighted Schauder estimates as well, the unweighted Schauder estimates are much less technical. For expository purposes, we therefore split the unweighted and weighted Schauder estimates into Lemma 7.10 and Proposition 7.16, respectively.
Lemma 7.10 (Covariant Schauder estimates). Let N ≥ N l o N \ge N_{\mathrm{lo}} N ≥ N lo α ∈ R \alpha\in\mathbb{R} α ∈ R β ∈ ( − 2 , 0 ] \beta\in(-2,0] β ∈ ( − 2 , 0 ] 1 ≤ r ≤ ∞ 1 \le r \le\infty 1 ≤ r ≤ ∞ Q : Y × Y → Y Q:\mathbb{Y}\times\mathbb{Y}\to\mathbb{Y} Q : Y × Y → Y G \mathbb{G} G ϕ : [ 0 , T ] × T 2 → Y \phi:[0,T]\times\mathbb{T}^2\to\mathbb{Y} ϕ : [ 0 , T ] × T 2 → Y ψ : [ 0 , T ] × T 2 → Y \psi:[0,T]\times\mathbb{T}^2\to\mathbb{Y} ψ : [ 0 , T ] × T 2 → Y
max m = 0 , 1 N − α − m ∥ D ( m ) ϕ ∥ n l N α ≲ L S T ( F ) ∥ ϕ ∥ L t , x ∞ , (423) \max_{m=0,1}N^{-\alpha-m}\lVert D^{(m)}\phi\rVert_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{S_T}(F)\lVert\phi\rVert_{L^\infty_{t,x}}, \tag*{(423)} m = 0 , 1 max N − α − m ∥ D ( m ) ϕ ∥ nl N α ≲ L S T ( F ) ∥ ϕ ∥ L t , x ∞ , (423) max m = 0 , 1 m < 2 + β N − α − m ∥ D ( m ) ϕ ∥ n l N α ≲ L S T ( F ) N − β + 2 r ∥ ϕ ∥ L t β L x r , (424) \max_{\substack{m=0,1\\m<2+\beta}}N^{-\alpha-m}\lVert D^{(m)}\phi\rVert_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{S_T}(F)N^{-\beta+\frac{2}{r}}\lVert\phi\rVert_{L^\beta_tL^r_x}, \tag*{(424)} m = 0 , 1 m < 2 + β max N − α − m ∥ D ( m ) ϕ ∥ nl N α ≲ L S T ( F ) N − β + r 2 ∥ ϕ ∥ L t β L x r , (424) N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α ≲ L S T ( F ) ∥ ϕ ∥ L t , x ∞ ∥ D ψ ∥ L t , x ∞ , (425) N^{-\alpha}\lVert Q(\phi,D\psi)\rVert_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{S_T}(F)\lVert\phi\rVert_{L^\infty_{t,x}}\lVert D\psi\rVert_{L^\infty_{t,x}}, \tag*{(425)} N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α ≲ L S T ( F ) ∥ ϕ ∥ L t , x ∞ ∥ D ψ ∥ L t , x ∞ , (425) N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α ≲ L S T ( F ) ( N ∥ ϕ ∥ L t , x ∞ + ∥ D ϕ ∥ L t , x ∞ ) ∥ ψ ∥ L t , x ∞ . (426) N^{-\alpha}\lVert Q(\phi,D\psi)\rVert_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{S_T}(F)\left(N\lVert\phi\rVert_{L^\infty_{t,x}}+\lVert D\phi\rVert_{L^\infty_{t,x}}\right)\lVert\psi\rVert_{L^\infty_{t,x}}. \tag*{(426)} N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α ≲ L S T ( F ) ( N ∥ ϕ ∥ L t , x ∞ + ∥ D ϕ ∥ L t , x ∞ ) ∥ ψ ∥ L t , x ∞ . (426) Remark 7.11. We note that (423) is the special case of (424) in which ( β , r ) = ( 0 , ∞ ) (\beta,r)=(0,\infty) ( β , r ) = ( 0 , ∞ ) (423) will be used repeatedly, we state it as a separate estimate.
Remark 7.12. We observe that the Schauder estimates (7.12)–(7.15) also hold if the operator D u h Y , B , N F Duh^{F}_{\mathbb{Y},B,N} D u h Y , B , N F Definition 7.8 is replaced by D u h Y , B , N Duh_{\mathbb{Y},B,N} D u h Y , B , N q Y , B , N q_{\mathbb{Y},B,N} q Y , B , N p Y , B , N p_{\mathbb{Y},B,N} p Y , B , N
Remark 7.13. Let a , b , c ∈ [ 3 ] a,b,c\in[3] a , b , c ∈ [ 3 ] Q : Y ( a ) × Y ( b ) → Y ( c ) Q:\mathbb{Y}^{(a)}\times\mathbb{Y}^{(b)}\to\mathbb{Y}^{(c)} Q : Y ( a ) × Y ( b ) → Y ( c ) G \mathbb{G} G G \mathbb{G} G Q ~ : Y × Y → Y \widetilde{Q}:\mathbb{Y}\times\mathbb{Y}\to\mathbb{Y} Q : Y × Y → Y Q ~ ( y 1 , y 2 ) = ι c Q ( π a y 1 , π b y 2 ) \widetilde{Q}(y_1,y_2)=\iota_cQ(\pi_a y_1,\pi_b y_2) Q ( y 1 , y 2 ) = ι c Q ( π a y 1 , π b y 2 ) y 1 , y 2 ∈ Y y_1,y_2\in\mathbb{Y} y 1 , y 2 ∈ Y π a : Y → Y ( a ) \pi_a:\mathbb{Y}\to\mathbb{Y}^{(a)} π a : Y → Y ( a ) π b : Y → Y ( b ) \pi_b:\mathbb{Y}\to\mathbb{Y}^{(b)} π b : Y → Y ( b ) ι c : Y ( c ) → Y \iota_c:\mathbb{Y}^{(c)}\to\mathbb{Y} ι c : Y ( c ) → Y Lemma 7.10 also hold for G \mathbb{G} G V \mathbb{V} V § 8.3 .
Before we turn to the proof of Lemma 7.10 , we need to make additional preparations. First, we prove the following pointwise estimate of the curvature tensor F F F
Lemma 7.14. For any t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ] k ≥ 0 k\ge0 k ≥ 0
∥ D x ( k ) F ( t ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 p e r t ( 1 + ∥ F ∥ T , ( k − 1 ) ∨ 0 p e r t ) k t − ( 1 + k 2 ) . \lVert D_x^{(k)}F(t)\rVert_{L^\infty_x}\lesssim\lVert F\rVert^{\mathrm{pert}}_{T,0}\left(1+\lVert F\rVert^{\mathrm{pert}}_{T,(k-1)\vee0}\right)^k t^{-\left(1+\frac{k}{2}\right)}. ∥ D x ( k ) F ( t ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 pert ( 1 + ∥ F ∥ T , ( k − 1 ) ∨ 0 pert ) k t − ( 1 + 2 k ) . Proof. By (190) , we have that for any t ∈ ( 0 , T ] t \in(0,T] t ∈ ( 0 , T ]
∥ F ( t ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 p e r t t − 1 . \lVert F(t)\rVert_{L^\infty_x} \lesssim\lVert F\rVert^{\mathrm{pert}}_{T,0}t^{-1}. ∥ F ( t ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 pert t − 1 . For k ≥ 1 k \ge1 k ≥ 1 (195) and the fact that F B 12 F_B^{12} F B 12 ( ∂ t − Δ B ) F B 12 = 0 (\partial_t-\Delta_B)F_B^{12}=0 ( ∂ t − Δ B ) F B 12 = 0 Lemma 3.28 ), we have that
t k / 2 ∥ D x ( k ) F ( t ) ∥ L x ∞ ≲ ( 1 + ∥ F ∥ T , k − 1 p e r t ) k ∥ F ( t / 2 ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 p e r t ( 1 + ∥ F ∥ T , k − 1 p e r t ) k t − 1 , t^{k/2}\lVert\mathrm{D}_x^{(k)}F(t)\rVert_{L^\infty_x}\lesssim(1+\lVert F\rVert^{\mathrm{pert}}_{T,k-1})^k\lVert F(t/2)\rVert_{L^\infty_x}\lesssim\lVert F\rVert^{\mathrm{pert}}_{T,0}(1+\lVert F\rVert^{\mathrm{pert}}_{T,k-1})^k t^{-1}, t k /2 ∥ D x ( k ) F ( t ) ∥ L x ∞ ≲ ( 1 + ∥ F ∥ T , k − 1 pert ) k ∥ F ( t /2 ) ∥ L x ∞ ≲ ∥ F ∥ T , 0 pert ( 1 + ∥ F ∥ T , k − 1 pert ) k t − 1 , which implies the desired estimate.
Recall from Notation 6.1 that we omit the subscript B B B D y † , ( m ) \mathrm{D}_y^{\dagger,(m)} D y † , ( m ) x x x D x ( k ) \mathrm{D}_x^{(k)} D x ( k ) Δ B , y † = D y † , j D y , j † \Delta^\dagger_{B,y}=\mathrm{D}^{\dagger,j}_y\mathrm{D}^\dagger_{y,j} Δ B , y † = D y † , j D y , j †
( Δ B , y q Y , B ∗ ) ∗ = Δ B , y † q Y , B . (427) (\Delta_{B,y}q_{Y,B}^{*})^{*}=\Delta^\dagger_{B,y}q_{Y,B}. \tag*{(427)} ( Δ B , y q Y , B ∗ ) ∗ = Δ B , y † q Y , B . (427) Lemma 7.15. Let N ≥ N l o N\ge N_{\mathrm{lo}} N ≥ N lo 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T 64 − 1 N − 2 ≤ t − s ≤ 64 N − 2 64^{-1}N^{-2}\le t-s\le64N^{-2} 6 4 − 1 N − 2 ≤ t − s ≤ 64 N − 2 k , m ≥ 0 k,m\ge0 k , m ≥ 0
∣ D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ∣ ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ m p e r t ) k + m + 2 min ( s , N − 2 ) − m / 2 N k + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) (428) \left|\mathrm{D}_y^{\dagger,(m)}\mathrm{D}_{t,x}^{(k)}q_{Y,B}(s,y;t,x)\right|
\lesssim\operatorname{Gr}_T(C_0F)(1+\lVert F\rVert^{\mathrm{pert}}_{T,k\vee m})^{k+m+2}\min(s,N^{-2})^{-m/2}N^{k+2}\exp(-cN^2|x-y|^2) \tag*{(428)} D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ m pert ) k + m + 2 min ( s , N − 2 ) − m /2 N k + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) (428) and
∣ ∂ s D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ∣ ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ ( m + 2 ) p e r t ) k + m + 4 min ( s , N − 2 ) − ( m + 2 ) / 2 N k + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) . (429) \left|\partial_s\mathrm{D}_y^{\dagger,(m)}\mathrm{D}_{t,x}^{(k)}q_{Y,B}(s,y;t,x)\right|
\lesssim\operatorname{Gr}_T(C_0F)(1+\lVert F\rVert^{\mathrm{pert}}_{T,k\vee(m+2)})^{k+m+4}\min(s,N^{-2})^{-(m+2)/2}N^{k+2}\exp(-cN^2|x-y|^2). \tag*{(429)} ∂ s D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ≲ Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ ( m + 2 ) pert ) k + m + 4 min ( s , N − 2 ) − ( m + 2 ) /2 N k + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) . (429) The same estimates also hold with q Y , B q_{Y,B} q Y , B q Y , B , N q_{Y,B,N} q Y , B , N q Y , B , ε , N q_{Y,B,\varepsilon,N} q Y , B , ε , N ε ∈ 2 − N l o \varepsilon\in2^{-N_{\mathrm{lo}}} ε ∈ 2 − N lo
Proof. We first prove the estimate (428) . Using Lemma 3.31, we have that
D t , x ( k ) q Y , B ( s , y ; t , x ) = ∑ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 O ( ( D x ( j 1 ) F ( t ) ) ⋯ ( D x ( j ℓ ) F ( t ) ) ( D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x ) ) ) . \mathrm{D}_{t,x}^{(k)}q_{Y,B}(s,y;t,x)=\sum_{\ell\ge0}\ \sum_{\substack{j_1+\cdots+j_{\ell+1}=k-2\ell,\\ j_1,\ldots,j_{\ell+1}\ge0}}\mathcal{O}\left((\mathrm{D}_x^{(j_1)}F(t))\cdots(\mathrm{D}_x^{(j_\ell)}F(t))(\mathrm{D}_x^{(j_{\ell+1})}q_{Y,B}(s,y;t,x))\right). D t , x ( k ) q Y , B ( s , y ; t , x ) = ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ∑ O ( ( D x ( j 1 ) F ( t )) ⋯ ( D x ( j ℓ ) F ( t )) ( D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x )) ) . By applying D y † , ( m ) \mathrm{D}_y^{\dagger,(m)} D y † , ( m )
D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) = ∑ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 O ( ( D x ( j 1 ) F ( t ) ) ⋯ ( D x ( j ℓ ) F ( t ) ) ( D y † , ( m ) D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x ) ) ) . \mathrm{D}_y^{\dagger,(m)}\mathrm{D}_{t,x}^{(k)}q_{Y,B}(s,y;t,x)
=\sum_{\ell\ge0}\ \sum_{\substack{j_1+\cdots+j_{\ell+1}=k-2\ell,\\ j_1,\ldots,j_{\ell+1}\ge0}}\mathcal{O}\left((\mathrm{D}_x^{(j_1)}F(t))\cdots(\mathrm{D}_x^{(j_\ell)}F(t))(\mathrm{D}_y^{\dagger,(m)}\mathrm{D}_x^{(j_{\ell+1})}q_{Y,B}(s,y;t,x))\right). D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) = ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ∑ O ( ( D x ( j 1 ) F ( t )) ⋯ ( D x ( j ℓ ) F ( t )) ( D y † , ( m ) D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x )) ) . Using Proposition 4.27 and Lemma 7.14, it then follows that
∣ D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ∣ ≲ ∑ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ∣ D x ( j 1 ) F ( t ) ∣ ⋯ ∣ D x ( j ℓ ) F ( t ) ∣ × ∣ D y † , ( m ) D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x ) ∣ ≲ ∑ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ( 1 + ∥ F ∥ T , k p e r t ) j 1 + ⋯ + j ℓ + ℓ − ( ℓ + j 1 + ⋯ + j ℓ 2 ) × Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ m p e r t ) m + j ℓ + 1 + 2 min ( s , N − 2 ) − m / 2 N j ℓ + 1 + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) . \begin{aligned}
\left|\mathrm{D}_y^{\dagger,(m)}\mathrm{D}_{t,x}^{(k)}q_{Y,B}(s,y;t,x)\right|
&\lesssim\sum_{\ell\ge0}\ \sum_{\substack{j_1+\cdots+j_{\ell+1}=k-2\ell,\\ j_1,\ldots,j_{\ell+1}\ge0}}
\left|\mathrm{D}_x^{(j_1)}F(t)\right|\cdots\left|\mathrm{D}_x^{(j_\ell)}F(t)\right|\\
&\qquad\qquad\times\left|\mathrm{D}_y^{\dagger,(m)}\mathrm{D}_x^{(j_{\ell+1})}q_{Y,B}(s,y;t,x)\right|\\
&\lesssim\sum_{\ell\ge0}\ \sum_{\substack{j_1+\cdots+j_{\ell+1}=k-2\ell,\\ j_1,\ldots,j_{\ell+1}\ge0}}
(1+\lVert F\rVert^{\mathrm{pert}}_{T,k})^{j_1+\cdots+j_\ell+\ell-\left(\ell+\frac{j_1+\cdots+j_\ell}{2}\right)}\\
&\qquad\qquad\times\operatorname{Gr}_T(C_0F)(1+\lVert F\rVert^{\mathrm{pert}}_{T,k\vee m})^{m+j_{\ell+1}+2}\min(s,N^{-2})^{-m/2}N^{j_{\ell+1}+2}\exp(-cN^2|x-y|^2).
\end{aligned} D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) ≲ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ∑ D x ( j 1 ) F ( t ) ⋯ D x ( j ℓ ) F ( t ) × D y † , ( m ) D x ( j ℓ + 1 ) q Y , B ( s , y ; t , x ) ≲ ℓ ≥ 0 ∑ j 1 + ⋯ + j ℓ + 1 = k − 2 ℓ , j 1 , … , j ℓ + 1 ≥ 0 ∑ ( 1 + ∥ F ∥ T , k pert ) j 1 + ⋯ + j ℓ + ℓ − ( ℓ + 2 j 1 + ⋯ + j ℓ ) × Gr T ( C 0 F ) ( 1 + ∥ F ∥ T , k ∨ m pert ) m + j ℓ + 1 + 2 min ( s , N − 2 ) − m /2 N j ℓ + 1 + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) . Since t ≥ t − s ≥ 64 − 1 N − 2 t \ge t-s \ge64^{-1}N^{-2} t ≥ t − s ≥ 6 4 − 1 N − 2
t − ( ℓ + j 1 + ⋯ + j ℓ 2 ) N j ℓ + 1 + 2 ≲ N j 1 + ⋯ + j ℓ + j ℓ + 1 + 2 ℓ + 2 = N k + 2 . t^{-(\ell+\frac{j_{1}+\cdots+j_{\ell}}{2})}N^{j_{\ell+1}+2}\lesssim N^{j_{1}+\cdots+j_{\ell}+j_{\ell+1}+2\ell+2}=N^{k+2}. t − ( ℓ + 2 j 1 + ⋯ + j ℓ ) N j ℓ + 1 + 2 ≲ N j 1 + ⋯ + j ℓ + j ℓ + 1 + 2 ℓ + 2 = N k + 2 . Furthermore, using that j 1 + ⋯ + j ℓ + 1 + ℓ = k − ℓ ≤ k j_{1}+\cdots+j_{\ell+1}+\ell=k-\ell\le k j 1 + ⋯ + j ℓ + 1 + ℓ = k − ℓ ≤ k
( 1 + ∥ F ∥ T , k p e r t ) j 1 + ⋯ + j ℓ + ℓ ( 1 + ∥ F ∥ T , k ∨ m p e r t ) m + j ℓ + 1 + 2 ≲ ( 1 + ∥ F ∥ T , k ∨ m p e r t ) k + m + 2 . (1+\|F\|_{T,k}^{\mathrm{pert}})^{j_{1}+\cdots+j_{\ell}+\ell}(1+\|F\|_{T,k\vee m}^{\mathrm{pert}})^{m+j_{\ell+1}+2}\lesssim(1+\|F\|_{T,k\vee m}^{\mathrm{pert}})^{k+m+2}. ( 1 + ∥ F ∥ T , k pert ) j 1 + ⋯ + j ℓ + ℓ ( 1 + ∥ F ∥ T , k ∨ m pert ) m + j ℓ + 1 + 2 ≲ ( 1 + ∥ F ∥ T , k ∨ m pert ) k + m + 2 . By combining our previous three estimates, we obtain our desired estimate (428) . We now turn to the proof of (429) , i.e., the estimate for the ∂ s \partial_s ∂ s (427) ), it follows that
( ∂ s + Δ B , y † ) q Y , B ( s , y ; t , x ) = O ( q Y , B ( s , y ; t , x ) F ( s ) ) . (\partial_s+\Delta^\dagger_{B,y})q_{Y,B}(s,y;t,x)=\mathcal{O}(q_{Y,B}(s,y;t,x)F(s)). ( ∂ s + Δ B , y † ) q Y , B ( s , y ; t , x ) = O ( q Y , B ( s , y ; t , x ) F ( s )) . Since D t , x ( k ) D^{(k)}_{t,x} D t , x ( k ) ( ∂ s + Δ B , y † ) (\partial_s+\Delta^\dagger_{B,y}) ( ∂ s + Δ B , y † ) F ( s ) F(s) F ( s )
( ∂ s + Δ B , y † ) D t , x ( k ) q Y , B ( s , y ; t , x ) = O ( ( D t , x ( k ) q Y , B ( s , y ; t , x ) ) F ( s ) ) . (\partial_s+\Delta^\dagger_{B,y})D^{(k)}_{t,x}q_{Y,B}(s,y;t,x)=\mathcal{O}((D^{(k)}_{t,x}q_{Y,B}(s,y;t,x))F(s)). ( ∂ s + Δ B , y † ) D t , x ( k ) q Y , B ( s , y ; t , x ) = O (( D t , x ( k ) q Y , B ( s , y ; t , x )) F ( s )) . By using Remark 3.30 or, more precisely, its analogue for ( ∂ s + Δ B , y † ) (\partial_s+\Delta^\dagger_{B,y}) ( ∂ s + Δ B , y † )
( ∂ s + Δ B , y † ) D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) = ∑ j 1 + j 2 = m j 1 , j 2 ≥ 0 O ( ( D y † , ( j 1 ) D t , x ( k ) q Y , B ( s , y ; t , x ) ) ( D y ( j 2 ) F ( s ) ) ) . (\partial_s+\Delta^\dagger_{B,y})D^{\dagger,(m)}_{y}D^{(k)}_{t,x}q_{Y,B}(s,y;t,x)=\sum_{\substack{j_1+j_2=m\\j_1,j_2\ge0}}\mathcal{O}\left((D^{\dagger,(j_1)}_{y}D^{(k)}_{t,x}q_{Y,B}(s,y;t,x))(D^{(j_2)}_{y}F(s))\right). ( ∂ s + Δ B , y † ) D y † , ( m ) D t , x ( k ) q Y , B ( s , y ; t , x ) = j 1 + j 2 = m j 1 , j 2 ≥ 0 ∑ O ( ( D y † , ( j 1 ) D t , x ( k ) q Y , B ( s , y ; t , x )) ( D y ( j 2 ) F ( s )) ) . The desired estimate (429) can now be obtained by first bringing the Δ B , y † \Delta^\dagger_{B,y} Δ B , y † (428) . Finally, in order to see that the estimates remain valid after replacing q Y , B q_{Y,B} q Y , B q Y , B , N q_{Y,B,N} q Y , B , N q Y , B , ε , N q_{Y,B,\varepsilon,N} q Y , B , ε , N (372) , and (373) . □
Equipped with Lemma 7.15, we can now prove our unweighted covariant Schauder estimates.
Proof of Lemma 7.10. We first show that (425) and (426) can be derived from (423) and the product formula (Lemma 3.17). Indeed, (425) follows from
N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α ≲ L T ( F ) ∥ Q ( ϕ , D ψ ) ∥ L t , x ∞ ≲ L T ( F ) ∥ ϕ ∥ L t , x ∞ ∥ D ψ ∥ L t , x ∞ . N^{-\alpha}\|Q(\phi,D\psi)\|_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{T}(F)\|Q(\phi,D\psi)\|_{L^\infty_{t,x}}\lesssim L_{T}(F)\|\phi\|_{L^\infty_{t,x}}\|D\psi\|_{L^\infty_{t,x}}. N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α ≲ L T ( F ) ∥ Q ( ϕ , D ψ ) ∥ L t , x ∞ ≲ L T ( F ) ∥ ϕ ∥ L t , x ∞ ∥ D ψ ∥ L t , x ∞ . Similarly, (426) follows from
N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α ≲ N − α ∥ D Q ( ϕ , ψ ) ∥ n l N α + N − α ∥ Q ( D ϕ , ψ ) ∥ n l N α ≲ L T ( F ) ( N ∥ Q ( ϕ , ψ ) ∥ L t , x ∞ + ∥ Q ( D ϕ , ψ ) ∥ L t , x ∞ ) ≲ L T ( F ) ( N ∥ ϕ ∥ L t , x ∞ + ∥ D ϕ ∥ L t , x ∞ ) ∥ ψ ∥ L t , x ∞ . \begin{aligned}
N^{-\alpha}\|Q(\phi,D\psi)\|_{\mathrm{nl}^{\alpha}_{N}}&\lesssim N^{-\alpha}\|DQ(\phi,\psi)\|_{\mathrm{nl}^{\alpha}_{N}}+N^{-\alpha}\|Q(D\phi,\psi)\|_{\mathrm{nl}^{\alpha}_{N}}\\
&\lesssim L_{T}(F)(N\|Q(\phi,\psi)\|_{L^\infty_{t,x}}+\|Q(D\phi,\psi)\|_{L^\infty_{t,x}})\\
&\lesssim L_{T}(F)(N\|\phi\|_{L^\infty_{t,x}}+\|D\phi\|_{L^\infty_{t,x}})\|\psi\|_{L^\infty_{t,x}}.
\end{aligned} N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α ≲ N − α ∥ D Q ( ϕ , ψ ) ∥ nl N α + N − α ∥ Q ( D ϕ , ψ ) ∥ nl N α ≲ L T ( F ) ( N ∥ Q ( ϕ , ψ ) ∥ L t , x ∞ + ∥ Q ( D ϕ , ψ ) ∥ L t , x ∞ ) ≲ L T ( F ) ( N ∥ ϕ ∥ L t , x ∞ + ∥ D ϕ ∥ L t , x ∞ ) ∥ ψ ∥ L t , x ∞ . We now turn to the proof of (423) . In fact, we prove the more general estimate
∥ D ( m ) ϕ ∥ n l N α ≲ L T ( F ) N α + m + 2 r ∥ ϕ ∥ L t ∞ L x r , (430) \|D^{(m)}\phi\|_{\mathrm{nl}^{\alpha}_{N}}\lesssim L_{T}(F)N^{\alpha+m+\frac{2}{r}}\|\phi\|_{L^\infty_tL^r_x}, \tag*{(430)} ∥ D ( m ) ϕ ∥ nl N α ≲ L T ( F ) N α + m + r 2 ∥ ϕ ∥ L t ∞ L x r , (430) which also covers the case β = 0 \beta=0 β = 0 (424) . To this end, let 0 ≤ k ≤ 10 0\le k\le10 0 ≤ k ≤ 10 0 ≤ m ≤ 1 0\le m\le1 0 ≤ m ≤ 1
( D t , x ( k ) Duh Y , B , N F ( D ( m ) ϕ ) ) ( t , x ) = ∬ d s d y D t , x ( k ) q Y , B , N ( s , y ; t , x ) D y ( m ) ϕ ( s , y ) = ∬ d s d y ( D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ϕ ( s , y ) . (431) \begin{aligned}
(D^{(k)}_{t,x}\operatorname{Duh}^{F}_{Y,B,N}(D^{(m)}\phi))(t,x)
&=\iint dsdy\,D^{(k)}_{t,x}q_{Y,B,N}(s,y;t,x)D^{(m)}_{y}\phi(s,y)\\
&=\iint dsdy\,(D^{\dagger,(m)}_{y}D^{(k)}_{t,x}q_{Y,B,N}(s,y;t,x))\phi(s,y). \tag*{(431)}
\end{aligned} ( D t , x ( k ) Duh Y , B , N F ( D ( m ) ϕ )) ( t , x ) = ∬ d s d y D t , x ( k ) q Y , B , N ( s , y ; t , x ) D y ( m ) ϕ ( s , y ) = ∬ d s d y ( D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x )) ϕ ( s , y ) . (431) Using Lemma 7.15, we then estimate
∣ ( 7.20 ) ∣ ≲ L S T ( F ) N k + 2 ∬ d s d y 1 { t − s ∼ N − 2 } min ( s , N − 2 ) − m 2 exp ( − c N 2 ∣ x − y ∣ 2 ) ∣ ϕ ( s , y ) ∣ ≲ L S T ( F ) N k + 2 ∥ 1 { t − s ∼ N − 2 } min ( s , N − 2 ) − m 2 exp ( − c N 2 ∣ x − y ∣ 2 ) ∥ L s 1 L y r ′ ∥ ϕ ∥ L s ∞ L y r ≲ L S T ( F ) N k + m 2 + 2 r − 2 ∥ ϕ ∥ L s ∞ L y r . (432) \begin{aligned}
|(7.20)| &\lesssim L_{ST}(F)N^{k+2}\iint ds\,dy\,\mathbf{1}_{\{t-s\sim N^{-2}\}}\min(s,N^{-2})^{-\frac{m}{2}}\exp\left(-cN^2|x-y|^2\right)|\phi(s,y)| \\
&\lesssim L_{ST}(F)N^{k+2}\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}\min(s,N^{-2})^{-\frac{m}{2}}\exp\left(-cN^2|x-y|^2\right)\right\|_{L^1_sL^{r'}_y}\|\phi\|_{L^\infty_sL^r_y} \\
&\lesssim L_{ST}(F)N^{k+\frac{m}{2}+\frac{2}{r}-2}\|\phi\|_{L^\infty_sL^r_y}. \tag*{(432)}
\end{aligned} ∣ ( 7.20 ) ∣ ≲ L S T ( F ) N k + 2 ∬ d s d y 1 { t − s ∼ N − 2 } min ( s , N − 2 ) − 2 m exp ( − c N 2 ∣ x − y ∣ 2 ) ∣ ϕ ( s , y ) ∣ ≲ L S T ( F ) N k + 2 1 { t − s ∼ N − 2 } min ( s , N − 2 ) − 2 m exp ( − c N 2 ∣ x − y ∣ 2 ) L s 1 L y r ′ ∥ ϕ ∥ L s ∞ L y r ≲ L S T ( F ) N k + 2 m + r 2 − 2 ∥ ϕ ∥ L s ∞ L y r . (432) In the last line, we used that 0 ≤ m ≤ 1 0\le m\le1 0 ≤ m ≤ 1 min ( s , N − 2 ) − m 2 \min(s,N^{-2})^{-\frac{m}{2}} min ( s , N − 2 ) − 2 m (430) .
It remains to prove (424) . By (430) , we may assume that β < 0 \beta<0 β < 0 ϕ ~ : R × T 2 → Y \widetilde{\phi}\colon\mathbb{R}\times\mathbb{T}^2\to Y ϕ : R × T 2 → Y ∂ t ϕ ~ ( t ) = ϕ ( t ) \partial_t\widetilde{\phi}(t)=\phi(t) ∂ t ϕ ( t ) = ϕ ( t ) t ∈ ( 0 , T ) t\in(0,T) t ∈ ( 0 , T )
∥ ϕ ~ ∥ C t 1 + β / 2 L x r ( R × T 2 ) ≲ ∥ ϕ ∥ C t β / 2 L x r ( [ 0 , T ] × T 2 ) . \|\widetilde{\phi}\|_{C_t^{1+\beta/2}L_x^r(\mathbb{R}\times\mathbb{T}^2)}
\lesssim
\|\phi\|_{C_t^{\beta/2}L_x^r([0,T]\times\mathbb{T}^2)}. ∥ ϕ ∥ C t 1 + β /2 L x r ( R × T 2 ) ≲ ∥ ϕ ∥ C t β /2 L x r ([ 0 , T ] × T 2 ) . For a given t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ] s 0 : = 0 s_0:=0 s 0 := 0 t ≤ 64 N − 2 t\le64N^{-2} t ≤ 64 N − 2 s 0 : = t s_0:=t s 0 := t t > 64 N − 2 t>64N^{-2} t > 64 N − 2 s ∈ [ 0 , T ] s\in[0,T] s ∈ [ 0 , T ] χ N ( t − s ) \chi_N(t-s) χ N ( t − s ) ∣ s − s 0 ∣ ≲ min ( s , t − s ) |s-s_0|\lesssim\min(s,t-s) ∣ s − s 0 ∣ ≲ min ( s , t − s ) ( P L ) L ≥ 1 (P_L)_{L\ge1} ( P L ) L ≥ 1 ∼ L 2 \sim L^2 ∼ L 2 L 2 L^2 L 2 L L L
∥ P L ϕ ~ ( s ) − P L ϕ ~ ( s 0 ) ∥ L x r ≲ min ( L 2 ∣ s − s 0 ∣ , 1 ) L − 2 − β ∥ ϕ ~ ∥ C t 1 + β / 2 L x r ≲ min ( L 2 s , L 2 ( t − s ) , 1 ) L − 2 − β ∥ ϕ ~ ∥ C t 1 + β / 2 L x r . (433) \begin{aligned}
\|P_L\widetilde{\phi}(s)-P_L\widetilde{\phi}(s_0)\|_{L_x^r}
&\lesssim\min(L^2|s-s_0|,1)L^{-2-\beta}\|\widetilde{\phi}\|_{C_t^{1+\beta/2}L_x^r} \\
&\lesssim\min(L^2s,L^2(t-s),1)L^{-2-\beta}\|\widetilde{\phi}\|_{C_t^{1+\beta/2}L_x^r}. \tag*{(433)}
\end{aligned} ∥ P L ϕ ( s ) − P L ϕ ( s 0 ) ∥ L x r ≲ min ( L 2 ∣ s − s 0 ∣ , 1 ) L − 2 − β ∥ ϕ ∥ C t 1 + β /2 L x r ≲ min ( L 2 s , L 2 ( t − s ) , 1 ) L − 2 − β ∥ ϕ ∥ C t 1 + β /2 L x r . (433) Starting from (7.20), we then write
( D t , x ( k ) D u h Y F , B , N ( D ( m ) ϕ ) ) ( t , x ) = ∬ d s d y ( D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ∂ s ( ϕ ~ ( s , y ) − ϕ ~ ( s 0 , y ) ) = − ∑ L ≥ 1 ∬ d s d y ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ( P L ϕ ~ ( s , y ) − P L ϕ ~ ( s 0 , y ) ) . (434) \begin{aligned}
\left(D_{t,x}^{(k)}\mathop{\mathrm{Duh}}{}_Y^{F,B,N}(D^{(m)}\phi)\right)(t,x)
&=\iint ds\,dy\,\left(D_y^{\dagger,(m)}D_{t,x}^{(k)}q_{Y,B,N}(s,y;t,x)\right)\partial_s\left(\widetilde{\phi}(s,y)-\widetilde{\phi}(s_0,y)\right) \\
&=-\sum_{L\ge1}\iint ds\,dy\,\left(\partial_sD_y^{\dagger,(m)}D_{t,x}^{(k)}q_{Y,B,N}(s,y;t,x)\right)\left(P_L\widetilde{\phi}(s,y)-P_L\widetilde{\phi}(s_0,y)\right). \tag*{(434)}
\end{aligned} ( D t , x ( k ) Duh Y F , B , N ( D ( m ) ϕ ) ) ( t , x ) = ∬ d s d y ( D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ∂ s ( ϕ ( s , y ) − ϕ ( s 0 , y ) ) = − L ≥ 1 ∑ ∬ d s d y ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ( P L ϕ ( s , y ) − P L ϕ ( s 0 , y ) ) . (434) We emphasise that there is no boundary term in (434) since, if s = 0 s=0 s = 0 q Y , B , N ( s , y ; t , x ) q_{Y,B,N}(s,y;t,x) q Y , B , N ( s , y ; t , x ) s 0 = 0 s_0=0 s 0 = 0 (433) and Hölder’s inequality, each summand in (434) can be estimated by
∣ ∬ d s d y ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ( P L ϕ ~ ( s , y ) − P L ϕ ~ ( s 0 , y ) ) ∣ ≲ L − 2 − β ∥ min ( L 2 s , L 2 ( t − s ) , 1 ) ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ∥ L s 1 L y r ′ ∥ ϕ ~ ∥ C t 1 + β / 2 L x r . (435) \begin{aligned}
\left|\iint ds\,dy\,\left(\partial_sD_y^{\dagger,(m)}D_{t,x}^{(k)}q_{Y,B,N}(s,y;t,x)\right)\left(P_L\widetilde{\phi}(s,y)-P_L\widetilde{\phi}(s_0,y)\right)\right| \\
\lesssim L^{-2-\beta}\left\|\min(L^2s,L^2(t-s),1)\left(\partial_sD_y^{\dagger,(m)}D_{t,x}^{(k)}q_{Y,B,N}(s,y;t,x)\right)\right\|_{L_s^1L_y^{r'}}\|\widetilde{\phi}\|_{C_t^{1+\beta/2}L_x^r}. \tag*{(435)}
\end{aligned} ∬ d s d y ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ( P L ϕ ( s , y ) − P L ϕ ( s 0 , y ) ) ≲ L − 2 − β min ( L 2 s , L 2 ( t − s ) , 1 ) ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) L s 1 L y r ′ ∥ ϕ ∥ C t 1 + β /2 L x r . (435) Using Lemma 7.15, we can estimate
∥ min ( L 2 s , L 2 ( t − s ) , 1 ) ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) ∥ L s 1 L y r ′ ≲ L S T ( F ) N k + 2 r ∥ 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) min ( s , N − 2 ) − m + 2 2 exp ( − c N 2 ∣ x − y ∣ 2 ) ∥ L s 1 L y r ′ ≲ L S T ( F ) N k + 2 r ∥ 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) min ( s , N − 2 ) − m + 2 2 ∥ L s 1 . \begin{aligned}
&\left\|\min(L^2s,L^2(t-s),1)\left(\partial_sD_y^{\dagger,(m)}D_{t,x}^{(k)}q_{Y,B,N}(s,y;t,x)\right)\right\|_{L_s^1L_y^{r'}} \\
&\lesssim L_{ST}(F)N^{k+\frac{2}{r}}\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}\min(L^2s,L^2(t-s),1)\min(s,N^{-2})^{-\frac{m+2}{2}}\exp\left(-cN^2|x-y|^2\right)\right\|_{L_s^1L_y^{r'}} \\
&\lesssim L_{ST}(F)N^{k+\frac{2}{r}}\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}\min(L^2s,L^2(t-s),1)\min(s,N^{-2})^{-\frac{m+2}{2}}\right\|_{L_s^1}.
\end{aligned} min ( L 2 s , L 2 ( t − s ) , 1 ) ( ∂ s D y † , ( m ) D t , x ( k ) q Y , B , N ( s , y ; t , x ) ) L s 1 L y r ′ ≲ L S T ( F ) N k + r 2 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) min ( s , N − 2 ) − 2 m + 2 exp ( − c N 2 ∣ x − y ∣ 2 ) L s 1 L y r ′ ≲ L S T ( F ) N k + r 2 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) min ( s , N − 2 ) − 2 m + 2 L s 1 . In the L s 1 L^1_{s} L s 1 s − m + 2 2 s^{-\frac{m+2}{2}} s − 2 m + 2 N m + 2 N^{m+2} N m + 2 s − m + 2 2 s^{-\frac{m+2}{2}} s − 2 m + 2
∥ 1 { s ≥ L − 2 } 1 { t − s ∼ N − 2 } s − m + 2 2 ∥ L s 1 + L 2 ∥ 1 { s ≤ L − 2 } 1 { t − s ∼ N − 2 } s 1 − m + 2 2 ∥ L s 1 \left\|\mathbf{1}_{\{s\geq L^{-2}\}}\mathbf{1}_{\{t-s\sim N^{-2}\}}s^{-\frac{m+2}{2}}\right\|_{L^1_s}+L^2\left\|\mathbf{1}_{\{s\leq L^{-2}\}}\mathbf{1}_{\{t-s\sim N^{-2}\}}s^{1-\frac{m+2}{2}}\right\|_{L^1_s} 1 { s ≥ L − 2 } 1 { t − s ∼ N − 2 } s − 2 m + 2 L s 1 + L 2 1 { s ≤ L − 2 } 1 { t − s ∼ N − 2 } s 1 − 2 m + 2 L s 1 ≲ min ( L m + 2 N − 2 , L m log ( 2 + L / N ) ) + min ( L 2 N m − 2 , L m ) \lesssim\min\left(L^{m+2}N^{-2},L^m\log(2+L/N)\right)+\min\left(L^2N^{m-2},L^m\right) ≲ min ( L m + 2 N − 2 , L m log ( 2 + L / N ) ) + min ( L 2 N m − 2 , L m ) ≲ min ( L 2 N m − 2 , L m ) log ( 2 + L / N ) . \lesssim\min\left(L^2N^{m-2},L^m\right)\log(2+L/N). ≲ min ( L 2 N m − 2 , L m ) log ( 2 + L / N ) . where we used that m = 0 , 1 m=0,1 m = 0 , 1 N m + 2 N^{m+2} N m + 2
N m + 2 ∥ 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) ∥ L s 1 N^{m+2}\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}\min\left(L^2s,L^2(t-s),1\right)\right\|_{L^1_s} N m + 2 1 { t − s ∼ N − 2 } min ( L 2 s , L 2 ( t − s ) , 1 ) L s 1 ≲ N m + 2 min ( ∥ 1 { t − s ∼ N − 2 } ∥ L s 1 , L 2 ∥ 1 { t − s ∼ N − 2 } ( t − s ) ∥ L s 1 ) ≲ min ( N m , N m − 2 L 2 ) . \lesssim N^{m+2}\min\left(\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}\right\|_{L^1_s},L^2\left\|\mathbf{1}_{\{t-s\sim N^{-2}\}}(t-s)\right\|_{L^1_s}\right)
\lesssim\min\left(N^m,N^{m-2}L^2\right). ≲ N m + 2 min ( 1 { t − s ∼ N − 2 } L s 1 , L 2 1 { t − s ∼ N − 2 } ( t − s ) L s 1 ) ≲ min ( N m , N m − 2 L 2 ) . Since β < 0 \beta<0 β < 0 m < 2 + β m<2+\beta m < 2 + β
∑ L ≥ 1 L − 2 − β ( min ( L 2 N m − 2 , L m ) log ( 2 + L / N ) + min ( N m , N m − 2 L 2 ) ) ≲ N m − β − 2 , \sum_{L\geq1}L^{-2-\beta}\left(\min\left(L^2N^{m-2},L^m\right)\log(2+L/N)+\min\left(N^m,N^{m-2}L^2\right)\right)\lesssim N^{m-\beta-2}, L ≥ 1 ∑ L − 2 − β ( min ( L 2 N m − 2 , L m ) log ( 2 + L / N ) + min ( N m , N m − 2 L 2 ) ) ≲ N m − β − 2 , which yields the desired estimate. □
So far, we have stated and proved the unweighted Schauder estimates in our solution spaces (Lemma 7.10 ). We now turn to the weighted Schauder estimates, which are the subject of the following proposition.
Proposition 7.16 (Weighted covariant Schauder estimates). Let N ≥ N l o N\geq N_{\mathrm{lo}} N ≥ N lo α ∈ R \alpha\in\mathbb{R} α ∈ R β ∈ ( − 2 , 0 ] \beta\in(-2,0] β ∈ ( − 2 , 0 ] − 2 < η , η 1 , η 2 , η ~ 1 , η ~ 2 , ν ≤ 0 -2<\eta,\eta_1,\eta_2,\widetilde{\eta}_1,\widetilde{\eta}_2,\nu\leq0 − 2 < η , η 1 , η 2 , η 1 , η 2 , ν ≤ 0 1 ≤ r ≤ ∞ 1\leq r\leq\infty 1 ≤ r ≤ ∞
η 1 + η 2 > − 2 and η ~ 1 + η ~ 2 > − 1. \eta_1+\eta_2>-2\qquad\text{and}\qquad\widetilde{\eta}_1+\widetilde{\eta}_2>-1. η 1 + η 2 > − 2 and η 1 + η 2 > − 1. To simplify the notation, let η 12 : = η 1 + η 2 \eta_{12}:=\eta_1+\eta_2 η 12 := η 1 + η 2 η ~ 12 : = η ~ 1 + η ~ 2 \widetilde{\eta}_{12}:=\widetilde{\eta}_1+\widetilde{\eta}_2 η 12 := η 1 + η 2 Q : Y × Y → Y Q:Y\times Y\to Y Q : Y × Y → Y G G G ϕ : [ 0 , T ] × T 2 → Y \phi:[0,T]\times\mathbb{T}^2\to Y ϕ : [ 0 , T ] × T 2 → Y ψ : [ 0 , T ] × T 2 → Y \psi:[0,T]\times\mathbb{T}^2\to Y ψ : [ 0 , T ] × T 2 → Y
max m = 0 , 1 : m < 2 + η N − α − m ∥ D ( m ) ϕ ∥ n l N α , ν ≲ L S T ( F ) max ( N ν − η , N l o ν − η ) ∥ t − η / 2 ϕ ∥ L t , x ∞ , (436) \max_{\substack{m=0,1:\\m<2+\eta}}N^{-\alpha-m}\left\|D^{(m)}\phi\right\|_{\mathrm{nl}^{\alpha,\nu}_N}
\lesssim L_{S_T}(F)\max\left(N^{\nu-\eta},N_{\mathrm{lo}}^{\nu-\eta}\right)\left\|t^{-\eta/2}\phi\right\|_{L^\infty_{t,x}}, \tag*{(436)} m = 0 , 1 : m < 2 + η max N − α − m D ( m ) ϕ nl N α , ν ≲ L S T ( F ) max ( N ν − η , N lo ν − η ) t − η /2 ϕ L t , x ∞ , (436) max m = 0 , 1 : m < 2 + β + η N − α − m ∥ D ( m ) ϕ ∥ n l N α , ν ≲ L S T ( F ) max ( N ν − η , N l o ν − η ) N − β + 2 r ∥ t − η / 2 ϕ ∥ L t β 2 L x r , (437) \max_{\substack{m=0,1:\\m<2+\beta+\eta}}N^{-\alpha-m}\left\|D^{(m)}\phi\right\|_{\mathrm{nl}^{\alpha,\nu}_N}
\lesssim L_{S_T}(F)\max\left(N^{\nu-\eta},N_{\mathrm{lo}}^{\nu-\eta}\right)N^{-\beta+\frac{2}{r}}\left\|t^{-\eta/2}\phi\right\|_{L^{{\beta\over2}}_tL^r_x}, \tag*{(437)} m = 0 , 1 : m < 2 + β + η max N − α − m D ( m ) ϕ nl N α , ν ≲ L S T ( F ) max ( N ν − η , N lo ν − η ) N − β + r 2 t − η /2 ϕ L t 2 β L x r , (437) as well as the estimates
N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α , ν ≲ L S T ( F ) max ( N ν − η 12 , N l o ν − η 12 ) ∥ t − η 1 / 2 ϕ ∥ L t , x ∞ ∥ t − η 2 / 2 D ψ ∥ L t , x ∞ , (438) N^{-\alpha}\left\|Q(\phi,D\psi)\right\|_{\mathrm{nl}^{\alpha,\nu}_N}
\lesssim L_{S_T}(F)\max\left(N^{\nu-\eta_{12}},N_{\mathrm{lo}}^{\nu-\eta_{12}}\right)\left\|t^{-\eta_1/2}\phi\right\|_{L^\infty_{t,x}}\left\|t^{-\eta_2/2}D\psi\right\|_{L^\infty_{t,x}}, \tag*{(438)} N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α , ν ≲ L S T ( F ) max ( N ν − η 12 , N lo ν − η 12 ) t − η 1 /2 ϕ L t , x ∞ t − η 2 /2 D ψ L t , x ∞ , (438) N − α ∥ Q ( ϕ , D ψ ) ∥ n l N α , ν ≲ L S T ( F ) max ( N ν − η 12 , N l o ν − η 12 ) ∥ t − η 1 / 2 D ϕ ∥ L t , x ∞ ∥ t − η 2 / 2 ψ ∥ L t , x ∞ + L S T ( F ) max ( N ν − η ~ 12 , N l o ν − η ~ 12 ) N ∥ t − η ~ 1 / 2 ϕ ∥ L t , x ∞ ∥ t − η ~ 2 / 2 ψ ∥ L t , x ∞ . (439) \begin{aligned}
N^{-\alpha}\left\|Q(\phi,D\psi)\right\|_{\mathrm{nl}^{\alpha,\nu}_N}
&\lesssim L_{S_T}(F)\max\left(N^{\nu-\eta_{12}},N_{\mathrm{lo}}^{\nu-\eta_{12}}\right)\left\|t^{-\eta_1/2}D\phi\right\|_{L^\infty_{t,x}}\left\|t^{-\eta_2/2}\psi\right\|_{L^\infty_{t,x}}\\
&\quad+L_{S_T}(F)\max\left(N^{\nu-\widetilde{\eta}_{12}},N_{\mathrm{lo}}^{\nu-\widetilde{\eta}_{12}}\right)N\left\|t^{-\widetilde{\eta}_1/2}\phi\right\|_{L^\infty_{t,x}}\left\|t^{-\widetilde{\eta}_2/2}\psi\right\|_{L^\infty_{t,x}}. \tag*{(439)}
\end{aligned} N − α ∥ Q ( ϕ , D ψ ) ∥ nl N α , ν ≲ L S T ( F ) max ( N ν − η 12 , N lo ν − η 12 ) t − η 1 /2 D ϕ L t , x ∞ t − η 2 /2 ψ L t , x ∞ + L S T ( F ) max ( N ν − η 12 , N lo ν − η 12 ) N t − η 1 /2 ϕ L t , x ∞ t − η 2 /2 ψ L t , x ∞ . (439) Remark 7.17. In applications of Proposition 7.16, we will mostly take ν ≥ η , η 12 , η ~ 12 \nu\geq\eta,\eta_{12},\widetilde{\eta}_{12} ν ≥ η , η 12 , η 12 N l o N_{\mathrm{lo}} N lo
Proof. Similarly to the proof of Lemma 7.10 , (7.26) and (7.27) can be derived from (7.24) and the product formula. Furthermore, (7.24) is the special case of (7.25) in which ( β , r ) = ( 0 , ∞ ) (\beta,r)=(0,\infty) ( β , r ) = ( 0 , ∞ )
The idea behind the proof of (437) is to combine our earlier, unweighted Schauder estimate (424) with a dyadic decomposition in time. For any 0 ≤ k ≤ 10 0 \le k \le10 0 ≤ k ≤ 10
D t , x ( k ) Duh Y , B , N F ( D x ( m ) ϕ ) = ∑ K , M ≥ N l o χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) . (440) D_{t,x}^{(k)}\operatorname{Duh}_{Y,B,N}^{F}\left(D_x^{(m)}\phi\right)
=
\sum_{K,M \ge N_{\mathrm{lo}}}
\chi_K D_{t,x}^{(k)}\operatorname{Duh}_{Y,B,N}^{F}\left(\chi_M D_x^{(m)}\phi\right). \tag*{(440)} D t , x ( k ) Duh Y , B , N F ( D x ( m ) ϕ ) = K , M ≥ N lo ∑ χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) . (440) Since the supports of ( χ K ) K ≥ N l o (\chi_K)_{K \ge N_{\mathrm{lo}}} ( χ K ) K ≥ N lo
∥ t − k 2 D t , x ( k ) Duh Y , B , N F ( D x ( m ) ϕ ) ∥ L t , x ∞ ≲ sup K ≥ N l o ∑ M ≥ N l o K ν ∥ χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) ∥ L t , x ∞ . (441) \left\|t^{-\frac{k}{2}}D_{t,x}^{(k)}\operatorname{Duh}_{Y,B,N}^{F}\left(D_x^{(m)}\phi\right)\right\|_{L^\infty_{t,x}}
\lesssim
\sup_{K \ge N_{\mathrm{lo}}}
\sum_{M \ge N_{\mathrm{lo}}}
K^\nu
\left\|\chi_K D_{t,x}^{(k)}\operatorname{Duh}_{Y,B,N}^{F}\left(\chi_M D_x^{(m)}\phi\right)\right\|_{L^\infty_{t,x}}. \tag*{(441)} t − 2 k D t , x ( k ) Duh Y , B , N F ( D x ( m ) ϕ ) L t , x ∞ ≲ K ≥ N lo sup M ≥ N lo ∑ K ν χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) L t , x ∞ . (441) We note that, since D x ( m ) D_x^{(m)} D x ( m ) χ M \chi_M χ M D x ( m ) D_x^{(m)} D x ( m ) (440) are such that t ∼ K − 2 t \sim K^{-2} t ∼ K − 2 t − s ∼ N − 2 t-s \sim N^{-2} t − s ∼ N − 2 s ∼ M − 2 s \sim M^{-2} s ∼ M − 2 s s s t = t − s + s t=t-s+s t = t − s + s K ∼ M ≲ N K \sim M \lesssim N K ∼ M ≲ N K ∼ N ≲ M K \sim N \lesssim M K ∼ N ≲ M β ′ ∈ ( − 2 , 0 ] \beta' \in(-2,0] β ′ ∈ ( − 2 , 0 ] β ′ ≤ β \beta' \le\beta β ′ ≤ β β ′ > − 2 + m \beta' > -2+m β ′ > − 2 + m m < 2 + β + η m<2+\beta+\eta m < 2 + β + η η ≤ 0 \eta\le0 η ≤ 0 χ M ( t ) ( M 2 t ) 1 2 \chi_M(t)(M^2t)^{\frac{1}{2}} χ M ( t ) ( M 2 t ) 2 1 (424) , we obtain that
K ν ∥ χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) ∥ L t , x ∞ ≲ L S T ( F ) K ν N − 2 + k + m − β ′ + 2 r ∥ χ M ϕ ∥ L t r L x r K^\nu\left\|\chi_KD_{t,x}^{(k)}\operatorname{Duh}_{Y,B,N}^{F}\left(\chi_MD_x^{(m)}\phi\right)\right\|_{L^\infty_{t,x}}
\lesssim L_{S_T}(F)K^\nu N^{-2+k+m-\beta'+\frac{2}{r}}
\left\|\chi_M\phi\right\|_{L^r_tL^r_x} K ν χ K D t , x ( k ) Duh Y , B , N F ( χ M D x ( m ) ϕ ) L t , x ∞ ≲ L S T ( F ) K ν N − 2 + k + m − β ′ + r 2 ∥ χ M ϕ ∥ L t r L x r ≲ L S T ( F ) K ν N − 2 + k + m − β ′ + 2 r M β ′ − β − η ∥ t − η 2 ϕ ∥ L t r L x r . (442) \lesssim L_{S_T}(F)K^\nu N^{-2+k+m-\beta'+\frac{2}{r}}
M^{\beta'-\beta-\eta}
\left\|t^{-\frac{\eta}{2}}\phi\right\|_{L^r_tL^r_x}. \tag*{(442)} ≲ L S T ( F ) K ν N − 2 + k + m − β ′ + r 2 M β ′ − β − η t − 2 η ϕ L t r L x r . (442) In the case M ∼ K ≲ N M \sim K \lesssim N M ∼ K ≲ N β ′ = β \beta'=\beta β ′ = β K ν − η N − 2 + k + m − β + 2 r K^{\nu-\eta}N^{-2+k+m-\beta+\frac{2}{r}} K ν − η N − 2 + k + m − β + r 2 (442) . Since the sum over M M M (441) then only contains a bounded number of terms, and since N l o ≤ K ≲ N N_{\mathrm{lo}}\le K\lesssim N N lo ≤ K ≲ N K ∼ N ≲ M K\sim N\lesssim M K ∼ N ≲ M β ′ = − 2 + m + ε \beta'=-2+m+\varepsilon β ′ = − 2 + m + ε ε > 0 \varepsilon>0 ε > 0 (442) becomes
K ν N k + 2 r − ε M − 2 + m − β − η + ε . K^\nu N^{k+\frac{2}{r}-\varepsilon}M^{-2+m-\beta-\eta+\varepsilon}. K ν N k + r 2 − ε M − 2 + m − β − η + ε . Due to our assumption m < 2 + β + η m<2+\beta+\eta m < 2 + β + η M M M K ∼ N ≲ M K\sim N\lesssim M K ∼ N ≲ M (441) can be bounded by
1 { K ∼ N } K ν N k + 2 r − ε ∑ M ≥ N M − 2 + m − β − η + ε ≲ 1 { K ∼ N } K ν N − 2 + k + m − β − η + 2 r ≲ N − 2 + k + m − β + ν − η + 2 r , \mathbf{1}_{\{K\sim N\}}K^\nu N^{k+\frac{2}{r}-\varepsilon}
\sum_{M\ge N}M^{-2+m-\beta-\eta+\varepsilon}
\lesssim
\mathbf{1}_{\{K\sim N\}}K^\nu N^{-2+k+m-\beta-\eta+\frac{2}{r}}
\lesssim
N^{-2+k+m-\beta+\nu-\eta+\frac{2}{r}}, 1 { K ∼ N } K ν N k + r 2 − ε M ≥ N ∑ M − 2 + m − β − η + ε ≲ 1 { K ∼ N } K ν N − 2 + k + m − β − η + r 2 ≲ N − 2 + k + m − β + ν − η + r 2 , which is acceptable.
In addition to the Schauder estimates above, we also control covariant linear evolutions in our solution spaces. This will later be needed in the proof of Proposition 8.8.
Lemma 7.18. Let φ 0 : T 2 → V \varphi_0:\mathbb{T}^2\to V φ 0 : T 2 → V α ∈ R \alpha\in\mathbb{R} α ∈ R η ≤ 0 \eta\le0 η ≤ 0 α + η ≤ − 2 r \alpha+\eta\le-\frac{2}{r} α + η ≤ − r 2 r r r (96) . Then, it holds for all N ≥ N l o N\ge N_{\mathrm{lo}} N ≥ N lo
∥ ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) ∥ s o l N α , η ≲ L S T ( F ) T − α + η 2 − 1 r ∥ φ 0 ∥ L x r . (443) \left\|\int dy\,p_{B,N}(0,y;t,x)\varphi_0(y)\right\|_{\mathrm{sol}^{\alpha,\eta}_N}
\lesssim
L_{S_T}(F)T^{-\frac{\alpha+\eta}{2}-\frac{1}{r}}\|\varphi_0\|_{L^r_x}. \tag*{(443)} ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) sol N α , η ≲ L S T ( F ) T − 2 α + η − r 1 ∥ φ 0 ∥ L x r . (443) Alternatively, if α + η ≤ − κ \alpha+\eta\le-\kappa α + η ≤ − κ
∥ ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) ∥ s o l N α , η ≲ L S T ( F ) T − α + η 2 − κ 2 ∥ φ 0 ∥ C B ( 0 ) , T − κ . (444) \left\|\int dy\,p_{B,N}(0,y;t,x)\varphi_0(y)\right\|_{\mathrm{sol}^{\alpha,\eta}_N}
\lesssim
L_{S_T}(F)T^{-\frac{\alpha+\eta}{2}-\frac{\kappa}{2}}\|\varphi_0\|_{C^{-\kappa}_{B(0),T}}. \tag*{(444)} ∫ d y p B , N ( 0 , y ; t , x ) φ 0 ( y ) sol N α , η ≲ L S T ( F ) T − 2 α + η − 2 κ ∥ φ 0 ∥ C B ( 0 ) , T − κ . (444) Proof. We first note that, on the support of the integrals appearing in (443) and (444) , it necessarily holds that 1 16 N − 2 ≤ t ≤ 16 N − 2 \frac{1}{16}N^{-2}\leq t\leq16N^{-2} 16 1 N − 2 ≤ t ≤ 16 N − 2
∥ ∫ d y P B , N ( 0 , y ; t , x ) φ 0 ( y ) ∥ s o l N α , η ∼ max 0 ≤ m ≤ 10 N α + η − m ∥ ∫ d y ( D t , x ( m ) P B , N ) ( 0 , y ; t , x ) φ 0 ( y ) ∥ L t , x ∞ . (445) \left\|\int\mathrm{d}y\,\mathbb{P}_{B,N}(0,y;t,x)\varphi_0(y)\right\|_{\mathrm{sol}^{\alpha,\eta}_N}
\sim\max_{0\leq m\leq10}N^{\alpha+\eta-m}\left\|\int\mathrm{d}y\,(D_{t,x}^{(m)}\mathbb{P}_{B,N})(0,y;t,x)\varphi_0(y)\right\|_{L^\infty_{t,x}}. \tag*{(445)} ∫ d y P B , N ( 0 , y ; t , x ) φ 0 ( y ) sol N α , η ∼ 0 ≤ m ≤ 10 max N α + η − m ∫ d y ( D t , x ( m ) P B , N ) ( 0 , y ; t , x ) φ 0 ( y ) L t , x ∞ . (445) We now prove (443) and (444) separately. To obtain (443) , we use our pointwise estimate (Lemma 7.15) and Young’s convolution inequality, which imply that
N α + η − m ∥ ∫ d y ( D t , x ( m ) P B , N ) ( 0 , y ; t , x ) φ 0 ( y ) ∥ L t , x ∞ ≲ L S T ( F ) N α + η + 2 ∥ ∫ exp ( − c N 2 ∣ x − y ∣ 2 ) ∣ φ 0 ( y ) ∣ d y ∥ L t , x ∞ ≲ L S T ( F ) N α + η + 2 r ∥ φ 0 ∥ L x r . N^{\alpha+\eta-m}\left\|\int\mathrm{d}y\,(D_{t,x}^{(m)}\mathbb{P}_{B,N})(0,y;t,x)\varphi_0(y)\right\|_{L^\infty_{t,x}}
\lesssim L_{S_T}(F)N^{\alpha+\eta+2}\left\|\int\exp\left(-cN^2|x-y|^2\right)|\varphi_0(y)|\mathrm{d}y\right\|_{L^\infty_{t,x}}
\lesssim L_{S_T}(F)N^{\alpha+\eta+\frac{2}{r}}\|\varphi_0\|_{L^r_x}. N α + η − m ∫ d y ( D t , x ( m ) P B , N ) ( 0 , y ; t , x ) φ 0 ( y ) L t , x ∞ ≲ L S T ( F ) N α + η + 2 ∫ exp ( − c N 2 ∣ x − y ∣ 2 ) ∣ φ 0 ( y ) ∣ d y L t , x ∞ ≲ L S T ( F ) N α + η + r 2 ∥ φ 0 ∥ L x r . Due to our condition α + η ≤ − 2 r \alpha+\eta\leq-\frac{2}{r} α + η ≤ − r 2 N N N N ≥ N l o ∼ T − 1 2 N\geq N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} N ≥ N lo ∼ T − 2 1 (443) . It remains to prove the second estimate (444) . To this end, we recall that 1 16 N − 2 ≤ t ≤ 16 N − 2 \frac{1}{16}N^{-2}\leq t\leq16N^{-2} 16 1 N − 2 ≤ t ≤ 16 N − 2 t ′ = 1 32 N − 2 t'=\frac{1}{32}N^{-2} t ′ = 32 1 N − 2
∫ d y D t , x ( m ) P B , N ( 0 , y ; t , x ) φ 0 ( y ) = ∫ d y D t , x ( m ) ( χ N ( t ) P B ( 0 , y ; t , x ) ) φ 0 ( y ) = ∫ d x ′ D t , x ( m ) ( χ N ( t ) P B ( t ′ , x ′ ; t , x ) ) ( ∫ d y P B ( 0 , y ; t ′ , x ′ ) φ 0 ( y ) ) . \begin{aligned}
\int\mathrm{d}y\,D_{t,x}^{(m)}\mathbb{P}_{B,N}(0,y;t,x)\varphi_0(y)
=\int\mathrm{d}y\,D_{t,x}^{(m)}\bigl(\chi_N(t)\mathbb{P}_B(0,y;t,x)\bigr)\varphi_0(y) \\
&=\int\mathrm{d}x'\,D_{t,x}^{(m)}\bigl(\chi_N(t)\mathbb{P}_B(t',x';t,x)\bigr)\left(\int\mathrm{d}y\,\mathbb{P}_B(0,y;t',x')\varphi_0(y)\right).
\end{aligned} ∫ d y D t , x ( m ) P B , N ( 0 , y ; t , x ) φ 0 ( y ) = ∫ d y D t , x ( m ) ( χ N ( t ) P B ( 0 , y ; t , x ) ) φ 0 ( y ) = ∫ d x ′ D t , x ( m ) ( χ N ( t ) P B ( t ′ , x ′ ; t , x ) ) ( ∫ d y P B ( 0 , y ; t ′ , x ′ ) φ 0 ( y ) ) . It therefore follows that
∥ ∫ d y D t , x ( m ) P B , N ( 0 , y ; t , x ) φ 0 ( y ) ∥ L t , x ∞ ≲ ∥ D t , x ( m ) ( χ N ( t ) P B ( t ′ , x ′ ; t , x ) ) ∥ L t , x ∞ L x ′ 1 ∥ ∫ d y P B ( 0 , y ; t ′ , x ′ ) φ 0 ( y ) ∥ L x ′ ∞ . \left\|\int\mathrm{d}y\,D_{t,x}^{(m)}\mathbb{P}_{B,N}(0,y;t,x)\varphi_0(y)\right\|_{L^\infty_{t,x}}
\lesssim\left\|D_{t,x}^{(m)}\bigl(\chi_N(t)\mathbb{P}_B(t',x';t,x)\bigr)\right\|_{L^\infty_{t,x}L^1_{x'}}\left\|\int\mathrm{d}y\,\mathbb{P}_B(0,y;t',x')\varphi_0(y)\right\|_{L^\infty_{x'}}. ∫ d y D t , x ( m ) P B , N ( 0 , y ; t , x ) φ 0 ( y ) L t , x ∞ ≲ D t , x ( m ) ( χ N ( t ) P B ( t ′ , x ′ ; t , x ) ) L t , x ∞ L x ′ 1 ∫ d y P B ( 0 , y ; t ′ , x ′ ) φ 0 ( y ) L x ′ ∞ . Using our pointwise estimates (Lemma 7.15), the first factor can be bounded by L S T ( F ) N m L_{S_T}(F)N^m L S T ( F ) N m ∥ ⋅ ∥ C B ( 0 ) , T κ \|\mathord{\cdot}\|_{\mathcal{C}^{\kappa}_{B(0),T}} ∥ ⋅ ∥ C B ( 0 ) , T κ N κ ∥ φ 0 ∥ C B ( 0 ) , T κ N^\kappa\|\varphi_0\|_{\mathcal{C}^{\kappa}_{B(0),T}} N κ ∥ φ 0 ∥ C B ( 0 ) , T κ (445) and using the condition α + η ≤ − κ \alpha+\eta\leq-\kappa α + η ≤ − κ (444) . □ \square □
Next, we turn to our commutator estimates, which will be needed when uncovering the structure of the paracontrolled term in our local solution ansatz—see Section 8.4.
Lemma 7.19 (Commutator estimate). Let W \mathbf{W} W W 1 \mathbf{W}_1 W 1 W 2 \mathbf{W}_2 W 2 G G G Q : W 1 × W 2 → W Q:\mathbf{W}_1\times\mathbf{W}_2\to\mathbf{W} Q : W 1 × W 2 → W G G G G : [ 0 , T ] × T 2 → W 1 G:[0,T]\times\mathbb{T}^2\to\mathbf{W}_1 G : [ 0 , T ] × T 2 → W 1 H : [ 0 , T ] × T 2 → W 2 H:[0,T]\times\mathbb{T}^2\to\mathbf{W}_2 H : [ 0 , T ] × T 2 → W 2 M ∈ 2 N 0 M\in2^{\mathbb{N}_0} M ∈ 2 N 0 − 2 < η ≤ 0 -2<\eta\leq0 − 2 < η ≤ 0
∥ t − η 2 ( Duh B , M ( Q ( G , H ) ) − Q ( G , Duh B , M ( H ) ) ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ L S T ( F ) ( M − 4 ∥ t − η 2 ( ∂ t − Δ B ) G ∥ L t , x ∞ ( [ 0 , T ] ) + M − 3 ∥ t − η 2 D G ∥ L t , x ∞ ( [ 0 , T ] ) ) ∥ H ∥ L t , x ∞ ( [ 0 , T ] ) . \left\|t^{-\frac{\eta}{2}}\bigl(\operatorname{Duh}_{B,M}(Q(G,H))-Q(G,\operatorname{Duh}_{B,M}(H))\bigr)\right\|_{L^\infty_{t,x}([0,T])}
\lesssim L_{S_T}(F)\left(M^{-4}\left\|t^{-\frac{\eta}{2}}(\partial_t-\Delta_B)G\right\|_{L^\infty_{t,x}([0,T])}+M^{-3}\left\|t^{-\frac{\eta}{2}}DG\right\|_{L^\infty_{t,x}([0,T])}\right)\|H\|_{L^\infty_{t,x}([0,T])}. t − 2 η ( Duh B , M ( Q ( G , H )) − Q ( G , Duh B , M ( H )) ) L t , x ∞ ([ 0 , T ]) ≲ L S T ( F ) ( M − 4 t − 2 η ( ∂ t − Δ B ) G L t , x ∞ ([ 0 , T ]) + M − 3 t − 2 η D G L t , x ∞ ([ 0 , T ]) ) ∥ H ∥ L t , x ∞ ([ 0 , T ]) . Proof. Fix ( t 0 , x 0 ) ∈ [ 0 , T ] × T 2 (t_0,x_0)\in[0,T]\times\mathbb{T}^2 ( t 0 , x 0 ) ∈ [ 0 , T ] × T 2 ϕ : [ 0 , t 0 ] × T 2 → W \phi:[0,t_0]\times\mathbb{T}^2\to\mathbf{W} ϕ : [ 0 , t 0 ] × T 2 → W ψ : [ 0 , t 0 ] × T 2 → W 2 \psi:[0,t_0]\times\mathbb{T}^2\to\mathbf{W}_2 ψ : [ 0 , t 0 ] × T 2 → W 2
ϕ = Duh B ( χ M ( t 0 − ⋅ ) Q ( G , H ) ) , ψ = Duh B ( χ M ( t 0 − ⋅ ) H ) . \phi=\operatorname{Duh}_B\bigl(\chi_M(t_0-\mathord{\cdot})Q(G,H)\bigr),\qquad\psi=\operatorname{Duh}_B\bigl(\chi_M(t_0-\mathord{\cdot})H\bigr). ϕ = Duh B ( χ M ( t 0 − ⋅ ) Q ( G , H ) ) , ψ = Duh B ( χ M ( t 0 − ⋅ ) H ) . Using this notation, we can write
( Duh B , M ( Q ( G , H ) ) − Q ( G , Duh B , M ( H ) ) ) ( t 0 , x 0 ) = ( ϕ − Q ( G , ψ ) ) ( t 0 , x 0 ) . \left(\operatorname{Duh}_{B,M}(\mathcal{Q}(G,H))-\mathcal{Q}(G,\operatorname{Duh}_{B,M}(H))\right)(t_0,x_0)=(\phi-\mathcal{Q}(G,\psi))(t_0,x_0). ( Duh B , M ( Q ( G , H )) − Q ( G , Duh B , M ( H )) ) ( t 0 , x 0 ) = ( ϕ − Q ( G , ψ )) ( t 0 , x 0 ) . Additionally, observe that ϕ ( t ) = ψ ( t ) = 0 \phi(t)=\psi(t)=0 ϕ ( t ) = ψ ( t ) = 0 ∣ t − t 0 ∣ ≲ M − 2 |t-t_0|\lesssim M^{-2} ∣ t − t 0 ∣ ≲ M − 2 I : = [ 0 ∨ ( t 0 − C M − 2 ) , t 0 ] I:=[0\vee(t_0-CM^{-2}),t_0] I := [ 0 ∨ ( t 0 − C M − 2 ) , t 0 ] ϕ \phi ϕ ψ \psi ψ f : I × T 2 → R f:I\times\mathbb{T}^2\to\mathbb{R} f : I × T 2 → R
t 0 − η 2 ∥ f ∥ L t 1 L x ∞ ( I ) ≲ M − 2 ∥ t − η 2 f ∥ L t ∞ L x ∞ ( I ) . (446) t_0^{-\frac{\eta}{2}}\|f\|_{L_t^1L_x^\infty(I)}\lesssim M^{-2}\|t^{-\frac{\eta}{2}}f\|_{L_t^\infty L_x^\infty(I)}. \tag*{(446)} t 0 − 2 η ∥ f ∥ L t 1 L x ∞ ( I ) ≲ M − 2 ∥ t − 2 η f ∥ L t ∞ L x ∞ ( I ) . (446) Indeed, this can be obtained easily by distinguishing the cases t 0 ≫ M − 2 t_0\gg M^{-2} t 0 ≫ M − 2 t 0 ≲ M − 2 t_0\lesssim M^{-2} t 0 ≲ M − 2 t − η / 2 ∼ t 0 − η / 2 t^{-\eta/2}\sim t_0^{-\eta/2} t − η /2 ∼ t 0 − η /2 I I I ∥ t η / 2 ∥ L t 1 ( [ 0 , t 0 ] ) ≲ t 0 1 + η / 2 ≲ M − 2 t 0 η / 2 \|t^{\eta/2}\|_{L_t^1([0,t_0])}\lesssim t_0^{1+\eta/2}\lesssim M^{-2}t_0^{\eta/2} ∥ t η /2 ∥ L t 1 ([ 0 , t 0 ]) ≲ t 0 1 + η /2 ≲ M − 2 t 0 η /2
∥ ψ ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ ∥ χ M ( t 0 − ⋅ ) H ∥ L t 1 L x ∞ ( [ 0 , t 0 ] ) ≲ M − 2 ∥ H ∥ L t , x ∞ ( [ 0 , t 0 ] ) , (447) \|\psi\|_{L_{t,x}^\infty([0,t_0])}\lesssim\|\chi_M(t_0-\mathord\cdot)H\|_{L_t^1L_x^\infty([0,t_0])}\lesssim M^{-2}\|H\|_{L_{t,x}^\infty([0,t_0])}, \tag*{(447)} ∥ ψ ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ ∥ χ M ( t 0 − ⋅ ) H ∥ L t 1 L x ∞ ([ 0 , t 0 ]) ≲ M − 2 ∥ H ∥ L t , x ∞ ([ 0 , t 0 ]) , (447) ∥ D ψ ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) sup t ∈ ( 0 , t 0 ] ∥ ( t − s ) − 1 / 2 χ M ( t 0 − s ) H ∥ L s 1 L x ∞ ( [ 0 , t ] ) ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) M − 1 ∥ H ∥ L t , x ∞ ( [ 0 , t 0 ] ) . (448) \|D\psi\|_{L_{t,x}^\infty([0,t_0])}\lesssim(1+\|F\|_{T,0}^{\mathrm{pert}})\sup_{t\in(0,t_0]}\|(t-s)^{-1/2}\chi_M(t_0-s)H\|_{L_s^1L_x^\infty([0,t])}
\lesssim(1+\|F\|_{T,0}^{\mathrm{pert}})M^{-1}\|H\|_{L_{t,x}^\infty([0,t_0])}. \tag*{(448)} ∥ D ψ ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ ( 1 + ∥ F ∥ T , 0 pert ) t ∈ ( 0 , t 0 ] sup ∥ ( t − s ) − 1/2 χ M ( t 0 − s ) H ∥ L s 1 L x ∞ ([ 0 , t ]) ≲ ( 1 + ∥ F ∥ T , 0 pert ) M − 1 ∥ H ∥ L t , x ∞ ([ 0 , t 0 ]) . (448) Next, we compute
( ∂ t − Δ B ) ( ϕ − Q ( G , ψ ) ) = ( ∂ t − Δ B ) ϕ − Q ( ( ∂ t − Δ B ) G , ψ ) + 2 Q ( D ℓ G , D ℓ ψ ) − Q ( G , ( ∂ t − Δ B ) ψ ) = − Q ( ( ∂ t − Δ B ) G , ψ ) + 2 Q ( D ℓ G , D ℓ ψ ) . \begin{aligned}
(\partial_t-\Delta_B)(\phi-\mathcal{Q}(G,\psi))
&=(\partial_t-\Delta_B)\phi-\mathcal{Q}((\partial_t-\Delta_B)G,\psi)+2\mathcal{Q}(D^\ell G,D_\ell\psi)-\mathcal{Q}(G,(\partial_t-\Delta_B)\psi)\\
&=-\mathcal{Q}((\partial_t-\Delta_B)G,\psi)+2\mathcal{Q}(D^\ell G,D_\ell\psi).
\end{aligned} ( ∂ t − Δ B ) ( ϕ − Q ( G , ψ )) = ( ∂ t − Δ B ) ϕ − Q (( ∂ t − Δ B ) G , ψ ) + 2 Q ( D ℓ G , D ℓ ψ ) − Q ( G , ( ∂ t − Δ B ) ψ ) = − Q (( ∂ t − Δ B ) G , ψ ) + 2 Q ( D ℓ G , D ℓ ψ ) . Here, we used that the ( ∂ t − Δ B ) ϕ (\partial_t-\Delta_B)\phi ( ∂ t − Δ B ) ϕ Q ( G , ( ∂ t − Δ B ) ψ ) \mathcal{Q}(G,(\partial_t-\Delta_B)\psi) Q ( G , ( ∂ t − Δ B ) ψ )
ϕ − Q ( G , ψ ) = − Duh B ( Q ( ( ∂ t − Δ B ) G , ψ ) ) + 2 Duh B ( Q ( D ℓ G , D ℓ ψ ) ) . \phi-\mathcal{Q}(G,\psi)=-\operatorname{Duh}_B(\mathcal{Q}((\partial_t-\Delta_B)G,\psi))+2\operatorname{Duh}_B(\mathcal{Q}(D^\ell G,D_\ell\psi)). ϕ − Q ( G , ψ ) = − Duh B ( Q (( ∂ t − Δ B ) G , ψ )) + 2 Duh B ( Q ( D ℓ G , D ℓ ψ )) . Each of the terms above may be estimated by the Schauder estimates of Lemma 4.61. In particular, we estimate the first term by
t 0 − η 2 ∥ Duh B ( Q ( ( ∂ t − Δ B ) G , ψ ) ) ( t 0 ) ∥ L x ∞ ≲ t 0 − η 2 ∥ Q ( ( ∂ t − Δ B ) G , ψ ) ∥ L t 1 L x ∞ ( I ) ≲ t 0 − η 2 ∥ ( ∂ t − Δ B ) G ∥ L t 1 L x ∞ ( I ) ∥ ψ ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ M − 4 ∥ t − η 2 ( ∂ t − Δ B ) G ∥ L t , x ∞ ( [ 0 , t 0 ] ) ∥ H ∥ L t , x ∞ ( [ 0 , t 0 ] ) . \begin{aligned}
t_0^{-\frac{\eta}{2}}\|\operatorname{Duh}_B(\mathcal{Q}((\partial_t-\Delta_B)G,\psi))(t_0)\|_{L_x^\infty}
&\lesssim t_0^{-\frac{\eta}{2}}\|\mathcal{Q}((\partial_t-\Delta_B)G,\psi)\|_{L_t^1L_x^\infty(I)}\\
&\lesssim t_0^{-\frac{\eta}{2}}\|(\partial_t-\Delta_B)G\|_{L_t^1L_x^\infty(I)}\|\psi\|_{L_{t,x}^\infty([0,t_0])}\\
&\lesssim M^{-4}\|t^{-\frac{\eta}{2}}(\partial_t-\Delta_B)G\|_{L_{t,x}^\infty([0,t_0])}\|H\|_{L_{t,x}^\infty([0,t_0])}.
\end{aligned} t 0 − 2 η ∥ Duh B ( Q (( ∂ t − Δ B ) G , ψ )) ( t 0 ) ∥ L x ∞ ≲ t 0 − 2 η ∥ Q (( ∂ t − Δ B ) G , ψ ) ∥ L t 1 L x ∞ ( I ) ≲ t 0 − 2 η ∥ ( ∂ t − Δ B ) G ∥ L t 1 L x ∞ ( I ) ∥ ψ ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ M − 4 ∥ t − 2 η ( ∂ t − Δ B ) G ∥ L t , x ∞ ([ 0 , t 0 ]) ∥ H ∥ L t , x ∞ ([ 0 , t 0 ]) . In the first line, we used that the support of ψ \psi ψ I I I
t 0 − η 2 ∥ Duh B ( Q ( D ℓ G , D ℓ ψ ) ) ( t 0 ) ∥ L x ∞ ≲ t 0 − η 2 ∥ Q ( D ℓ G , D ℓ ψ ) ∥ L t 1 L x ∞ ( I ) ≲ t 0 − η 2 ∥ D G ∥ L t 1 L x ∞ ( I ) ∥ D ψ ∥ L t , x ∞ ( [ 0 , t 0 ] ) ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) M − 3 ∥ t − η 2 D G ∥ L t , x ∞ ( [ 0 , t 0 ] ) ∥ H ∥ L t , x ∞ ( [ 0 , t 0 ] ) , \begin{aligned}
t_0^{-\frac{\eta}{2}}\|\operatorname{Duh}_B(\mathcal{Q}(D^\ell G,D_\ell\psi))(t_0)\|_{L_x^\infty}
&\lesssim t_0^{-\frac{\eta}{2}}\|\mathcal{Q}(D^\ell G,D_\ell\psi)\|_{L_t^1L_x^\infty(I)}\\
&\lesssim t_0^{-\frac{\eta}{2}}\|DG\|_{L_t^1L_x^\infty(I)}\|D\psi\|_{L_{t,x}^\infty([0,t_0])}\\
&\lesssim(1+\|F\|_{T,0}^{\mathrm{pert}})M^{-3}\|t^{-\frac{\eta}{2}}DG\|_{L_{t,x}^\infty([0,t_0])}\|H\|_{L_{t,x}^\infty([0,t_0])},
\end{aligned} t 0 − 2 η ∥ Duh B ( Q ( D ℓ G , D ℓ ψ )) ( t 0 ) ∥ L x ∞ ≲ t 0 − 2 η ∥ Q ( D ℓ G , D ℓ ψ ) ∥ L t 1 L x ∞ ( I ) ≲ t 0 − 2 η ∥ D G ∥ L t 1 L x ∞ ( I ) ∥ D ψ ∥ L t , x ∞ ([ 0 , t 0 ]) ≲ ( 1 + ∥ F ∥ T , 0 pert ) M − 3 ∥ t − 2 η D G ∥ L t , x ∞ ([ 0 , t 0 ]) ∥ H ∥ L t , x ∞ ([ 0 , t 0 ]) , which completes the proof.
Lemma 7.20 (Derivative commutator estimate). Let G : [ 0 , T ] × T 2 → W G\colon[0,T]\times\mathbb{T}^{2}\to\mathbf{W} G : [ 0 , T ] × T 2 → W M ≥ N l o M\ge N_{\mathrm{lo}} M ≥ N lo i ∈ [ 2 ] i\in[2] i ∈ [ 2 ]
∥ D i Duh B , M ( G ) − Duh B , M ( D i G ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ L S T ( F ) T − ( 1 − 2 κ ) / 2 M − ( 2 − 2 κ ) ∥ G ∥ L t , x ∞ ( [ 0 , T ] ) . \lVert D_i\operatorname{Duh}_{B,M}(G)-\operatorname{Duh}_{B,M}(D_iG)\rVert_{L^\infty_{t,x}([0,T])}\lesssim L_{S_T}(F)T^{-(1-2\kappa)/2}M^{-(2-2\kappa)}\lVert G\rVert_{L^\infty_{t,x}([0,T])}. ∥ D i Duh B , M ( G ) − Duh B , M ( D i G ) ∥ L t , x ∞ ([ 0 , T ]) ≲ L S T ( F ) T − ( 1 − 2 κ ) /2 M − ( 2 − 2 κ ) ∥ G ∥ L t , x ∞ ([ 0 , T ]) . Proof. Fix t 0 ∈ [ 0 , T ] t_0\in[0,T] t 0 ∈ [ 0 , T ]
( D i Duh B , M ( G ) − Duh B , M ( D i G ) ) ( t 0 ) = ( D i ϕ − ψ ) ( t 0 ) , (D_i\operatorname{Duh}_{B,M}(G)-\operatorname{Duh}_{B,M}(D_iG))(t_0)=(D_i\phi-\psi)(t_0), ( D i Duh B , M ( G ) − Duh B , M ( D i G )) ( t 0 ) = ( D i ϕ − ψ ) ( t 0 ) , where ϕ , ψ : [ 0 , t 0 ] × T 2 → W \phi,\psi\colon[0,t_0]\times\mathbb{T}^{2}\to\mathbf{W} ϕ , ψ : [ 0 , t 0 ] × T 2 → W
( ∂ t − Δ B ) ϕ = χ M ( t 0 − ⋅ ) G , ϕ ( 0 ) = 0 , (\partial_t-\Delta_B)\phi=\chi_M(t_0-\mathord\cdot)G,\qquad\phi(0)=0, ( ∂ t − Δ B ) ϕ = χ M ( t 0 − ⋅ ) G , ϕ ( 0 ) = 0 , ( ∂ t − Δ B ) ψ = χ M ( t 0 − ⋅ ) D i G , ψ ( 0 ) = 0. (\partial_t-\Delta_B)\psi=\chi_M(t_0-\mathord\cdot)D_iG,\qquad\psi(0)=0. ( ∂ t − Δ B ) ψ = χ M ( t 0 − ⋅ ) D i G , ψ ( 0 ) = 0. Note that ϕ ( t ) = 0 \phi(t)=0 ϕ ( t ) = 0 ∣ t 0 − t ∣ ≲ M − 2 \lvert t_0-t\rvert\lesssim M^{-2} ∣ t 0 − t ∣ ≲ M − 2 (105) and Remark 3.20)
( ∂ t − Δ B ) ( D i ϕ − ψ ) = 2 F i ℓ D ℓ ϕ . (\partial_t-\Delta_B)(D_i\phi-\psi)=2F_i^\ell D_\ell\phi. ( ∂ t − Δ B ) ( D i ϕ − ψ ) = 2 F i ℓ D ℓ ϕ . Thus, by Lemma 4.61 (in the following, all space-time norms are on [ 0 , t 0 ] × T 2 [0,t_0]\times\mathbb{T}^{2} [ 0 , t 0 ] × T 2
∥ D i ϕ − ψ ∥ L t , x ∞ ≲ ∥ F D ϕ ∥ L t 1 L x ∞ ≲ ∥ t κ / 2 F ∥ L t 2 L x ∞ ∥ t − 2 κ / 2 D ϕ ∥ L t 2 L x ∞ \lVert D_i\phi-\psi\rVert_{L^\infty_{t,x}}\lesssim\lVert F D\phi\rVert_{L^1_tL^\infty_x}\lesssim\lVert t^{\kappa/2}F\rVert_{L^2_tL^\infty_x}\lVert t^{-2\kappa/2}D\phi\rVert_{L^2_tL^\infty_x} ∥ D i ϕ − ψ ∥ L t , x ∞ ≲ ∥ F D ϕ ∥ L t 1 L x ∞ ≲ ∥ t κ /2 F ∥ L t 2 L x ∞ ∥ t − 2 κ /2 D ϕ ∥ L t 2 L x ∞ ≲ M − 1 ∥ t − ( 1 − 2 κ ) / 2 F ∥ T , 0 p e r t ∥ t − 2 κ / 2 D ϕ ∥ L t , x ∞ \lesssim M^{-1}\lVert t^{-(1-2\kappa)/2}F\rVert_{T,0}^{\mathrm{pert}}\lVert t^{-2\kappa/2}D\phi\rVert_{L^\infty_{t,x}} ≲ M − 1 ∥ t − ( 1 − 2 κ ) /2 F ∥ T , 0 pert ∥ t − 2 κ /2 D ϕ ∥ L t , x ∞ ≲ L S T ( F ) 1 / 100 M − 1 T − ( 1 − 2 κ ) / 2 ∥ t − 2 κ / 2 D ϕ ∥ L t , x ∞ , \lesssim L_{S_T}(F)^{1/100}M^{-1}T^{-(1-2\kappa)/2}\lVert t^{-2\kappa/2}D\phi\rVert_{L^\infty_{t,x}}, ≲ L S T ( F ) 1/100 M − 1 T − ( 1 − 2 κ ) /2 ∥ t − 2 κ /2 D ϕ ∥ L t , x ∞ , where in the third inequality, we used that ϕ ( t ) = 0 \phi(t)=0 ϕ ( t ) = 0 ∣ t 0 − t ∣ ≲ M − 2 \lvert t_0-t\rvert\lesssim M^{-2} ∣ t 0 − t ∣ ≲ M − 2 L S T ( F ) L_{S_T}(F) L S T ( F )
∥ t − 2 κ / 2 D ϕ ∥ L t , x ∞ ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) M − ( 1 − 2 κ ) ∥ G ∥ L t , x ∞ . (449) \lVert t^{-2\kappa/2}D\phi\rVert_{L^\infty_{t,x}}\lesssim(1+\lVert F\rVert_{T,0}^{\mathrm{pert}})M^{-(1-2\kappa)}\lVert G\rVert_{L^\infty_{t,x}}. \tag*{(449)} ∥ t − 2 κ /2 D ϕ ∥ L t , x ∞ ≲ ( 1 + ∥ F ∥ T , 0 pert ) M − ( 1 − 2 κ ) ∥ G ∥ L t , x ∞ . (449) (Again by (7.11), we may absorb 1 + ∥ F ∥ T , 0 p e r t ≤ L S T ( F ) 1 / 100 1+\lVert F\rVert_{T,0}^{\mathrm{pert}}\le L_{S_T}(F)^{1/100} 1 + ∥ F ∥ T , 0 pert ≤ L S T ( F ) 1/100 t 1 ∈ [ 0 , t 0 ] t_1\in[0,t_0] t 1 ∈ [ 0 , t 0 ]
∥ t 1 / 2 D ϕ ∥ L t , x ∞ ( [ 0 , t 1 ] ) ≲ ( 1 + ∥ F ∥ T , 0 p e r t ) ( ∥ χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ( [ 0 , t 1 ] ) + ∥ t 1 / 2 χ M ( t 0 − ⋅ ) G ∥ L t 2 L x ∞ ( [ 0 , t 1 ] ) + sup u ∈ [ 0 , t 1 ] ∥ t 1 / 2 ( u − t ) − 1 / 2 χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ) . \lVert t^{1/2}D\phi\rVert_{L^\infty_{t,x}([0,t_1])}\lesssim(1+\lVert F\rVert_{T,0}^{\mathrm{pert}})\left(\lVert\chi_M(t_0-\mathord\cdot)G\rVert_{L^1_tL^\infty_x([0,t_1])}+\lVert t^{1/2}\chi_M(t_0-\mathord\cdot)G\rVert_{L^2_tL^\infty_x([0,t_1])}+\sup_{u\in[0,t_1]}\lVert t^{1/2}(u-t)^{-1/2}\chi_M(t_0-\mathord\cdot)G\rVert_{L^1_tL^\infty_x([0,u])}\right). ∥ t 1/2 D ϕ ∥ L t , x ∞ ([ 0 , t 1 ]) ≲ ( 1 + ∥ F ∥ T , 0 pert ) ( ∥ χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ([ 0 , t 1 ]) + ∥ t 1/2 χ M ( t 0 − ⋅ ) G ∥ L t 2 L x ∞ ([ 0 , t 1 ]) + u ∈ [ 0 , t 1 ] sup ∥ t 1/2 ( u − t ) − 1/2 χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ([ 0 , u ]) ) . We may estimate
∥ χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ( [ 0 , t 1 ] ) ≲ min ( t 1 , M − 2 ) ∥ G ∥ L t , x ∞ , \lVert\chi_M(t_0-\mathord\cdot)G\rVert_{L^1_tL^\infty_x([0,t_1])}\lesssim\min(t_1,M^{-2})\lVert G\rVert_{L^\infty_{t,x}}, ∥ χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ([ 0 , t 1 ]) ≲ min ( t 1 , M − 2 ) ∥ G ∥ L t , x ∞ , ∥ t 1 / 2 χ M ( t 0 − ⋅ ) G ∥ L t 2 L x ∞ ( [ 0 , t 1 ] ) ≲ t 1 1 / 2 min ( t 1 1 / 2 , M − 1 ) ∥ G ∥ L t , x ∞ , \lVert t^{1/2}\chi_M(t_0-\mathord\cdot)G\rVert_{L^2_tL^\infty_x([0,t_1])}\lesssim t_1^{1/2}\min(t_1^{1/2},M^{-1})\lVert G\rVert_{L^\infty_{t,x}}, ∥ t 1/2 χ M ( t 0 − ⋅ ) G ∥ L t 2 L x ∞ ([ 0 , t 1 ]) ≲ t 1 1/2 min ( t 1 1/2 , M − 1 ) ∥ G ∥ L t , x ∞ , sup u ∈ [ 0 , t 1 ] ∥ t 1 / 2 ( u − t ) − 1 / 2 χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ( [ 0 , u ] ) ≲ t 1 1 / 2 min ( t 1 1 / 2 , M − 1 ) ∥ G ∥ L t , x ∞ . \sup_{u\in[0,t_1]}\lVert t^{1/2}(u-t)^{-1/2}\chi_M(t_0-\mathord\cdot)G\rVert_{L^1_tL^\infty_x([0,u])}\lesssim t_1^{1/2}\min(t_1^{1/2},M^{-1})\lVert G\rVert_{L^\infty_{t,x}}. u ∈ [ 0 , t 1 ] sup ∥ t 1/2 ( u − t ) − 1/2 χ M ( t 0 − ⋅ ) G ∥ L t 1 L x ∞ ([ 0 , u ]) ≲ t 1 1/2 min ( t 1 1/2 , M − 1 ) ∥ G ∥ L t , x ∞ . Combining, we thus obtain that for all t 1 ∈ [ 0 , t 0 ] t_1\in[0,t_0] t 1 ∈ [ 0 , t 0 ]
∥ ( D ϕ ) ( t 1 ) ∥ L x ∞ ≲ t 1 2 κ / 2 M − ( 1 − 2 κ ) ( 1 + ∥ F ∥ T , 0 p e r t ) ∥ G ∥ L t , x ∞ , \lVert(D\phi)(t_1)\rVert_{L^\infty_x}\lesssim t_1^{2\kappa/2}M^{-(1-2\kappa)}(1+\lVert F\rVert_{T,0}^{\mathrm{pert}})\lVert G\rVert_{L^\infty_{t,x}}, ∥( D ϕ ) ( t 1 ) ∥ L x ∞ ≲ t 1 2 κ /2 M − ( 1 − 2 κ ) ( 1 + ∥ F ∥ T , 0 pert ) ∥ G ∥ L t , x ∞ , from which the desired estimate (449) follows.
Product and paraproduct estimates
In this subsection, we state and prove different covariant product and paraproduct estimates.
Assumption 7.21 (Setup). We make the following assumptions:
(i) The indices a , b , c ∈ [ 3 ] a,b,c \in[3] a , b , c ∈ [ 3 ]
(ii) The bilinear map Q : Y ( a ) × Y ( b ) → Y ( c ) Q: Y^{(a)} \times Y^{(b)} \to Y^{(c)} Q : Y ( a ) × Y ( b ) → Y ( c )
(iii) We let U ⋆ = ( U M ) M ≥ N l o U_\star= (U_M)_{M \ge N_{\mathrm{lo}}} U ⋆ = ( U M ) M ≥ N lo M ≥ N l o M \ge N_{\mathrm{lo}} M ≥ N lo U M : [ 0 , T ] × T 2 → Y ( a ) U_M : [0,T] \times\mathbb{T}^2 \to Y^{(a)} U M : [ 0 , T ] × T 2 → Y ( a ) U M U_M U M t ≳ M − 2 t \gtrsim M^{-2} t ≳ M − 2
(iv) We let V ⋆ = ( V N ) N ≥ N l o V_\star= (V_N)_{N \ge N_{\mathrm{lo}}} V ⋆ = ( V N ) N ≥ N lo N ≥ N l o N \ge N_{\mathrm{lo}} N ≥ N lo V N : [ 0 , T ] × T 2 → Y ( b ) V_N : [0,T] \times\mathbb{T}^2 \to Y^{(b)} V N : [ 0 , T ] × T 2 → Y ( b ) V N V_N V N t ≳ N − 2 t \gtrsim N^{-2} t ≳ N − 2
Our first lemma will later be used to control (6.79) and (6.87). While these are among the simplest terms in (406) –(6.90), this nevertheless illustrates many of the ideas used to control other terms in (406) –(6.90).
Lemma 7.22 (Product estimate). Let Assumption 7.21 be satisfied. In addition, let α , β , γ ∈ R \alpha,\beta,\gamma\in\mathbb{R} α , β , γ ∈ R η 1 , η 2 , ν ∈ ( − ∞ , 0 ] \eta_1,\eta_2,\nu\in(-\infty,0] η 1 , η 2 , ν ∈ ( − ∞ , 0 ]
0 < α , β < 1 0 < \alpha,\beta< 1 0 < α , β < 1 α + β > 1 \alpha+\beta> 1 α + β > 1 η 1 , η 2 > − 1 \eta_1,\eta_2 > -1 η 1 , η 2 > − 1 η 2 ≤ ν \eta_2 \le\nu η 2 ≤ ν γ + ν − η 1 − η 2 < β − 1 \gamma+\nu-\eta_1-\eta_2 < \beta-1 γ + ν − η 1 − η 2 < β − 1
Then, it holds that
∥ Q ( U ⋆ , D V ⋆ ) ∥ n l K γ , ν ≲ L S T ( F ) T α + β − γ − ν + η 1 + η 2 − 1 2 ∥ U ⋆ ∥ s o l α , η 1 ∥ V ⋆ ∥ s o l β , η 2 . (450) \lVert Q(U_\star,DV_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim L_{\mathrm{ST}}(F)T^{\frac{\alpha+\beta-\gamma-\nu+\eta_1+\eta_2-1}{2}}\lVert U_\star\rVert_{\mathrm{sol}^{\alpha,\eta_1}}\lVert V_\star\rVert_{\mathrm{sol}^{\beta,\eta_2}}. \tag*{(450)} ∥ Q ( U ⋆ , D V ⋆ ) ∥ nl K γ , ν ≲ L ST ( F ) T 2 α + β − γ − ν + η 1 + η 2 − 1 ∥ U ⋆ ∥ sol α , η 1 ∥ V ⋆ ∥ sol β , η 2 . (450) Proof. We emphasise that, as in Assumption 7.21, all dyadic frequency scales in the proof will be bounded below by N l o ∼ T − 1 / 2 N_{\mathrm{lo}} \sim T^{-1/2} N lo ∼ T − 1/2 η : = η 1 + η 2 \eta:= \eta_1+\eta_2 η := η 1 + η 2 (438) and (439) , where we choose ( η ~ 1 , η ~ 2 ) = ( 0 , η 2 ) (\widetilde{\eta}_1,\widetilde{\eta}_2)=(0,\eta_2) ( η 1 , η 2 ) = ( 0 , η 2 )
∥ Q ( U M , D V N ) ∥ n l K γ , ν ≲ L S T ( F ) min ( K γ + ν − η ∥ t − η 1 / 2 U M ∥ L t , x ∞ ∥ t − η 2 / 2 D V N ∥ L t , x ∞ , K γ + ν − η 2 K ∥ U M ∥ L t , x ∞ ∥ t − η 2 / 2 V N ∥ L t , x ∞ + K γ + ν − η ∥ t − η 1 / 2 D U M ∥ L t , x ∞ ∥ t − η 2 / 2 V N ∥ L t , x ∞ ) . \begin{aligned}
\lVert Q(U_M,DV_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim{}& L_{\mathrm{ST}}(F)\min\bigl(K^{\gamma+\nu-\eta}\lVert t^{-\eta_1/2}U_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta_2/2}DV_N\rVert_{L^\infty_{t,x}},\\
&\qquad K^{\gamma+\nu-\eta_2}K\lVert U_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta_2/2}V_N\rVert_{L^\infty_{t,x}}\\
&\qquad+K^{\gamma+\nu-\eta}\lVert t^{-\eta_1/2}DU_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta_2/2}V_N\rVert_{L^\infty_{t,x}}\bigr).
\end{aligned} ∥ Q ( U M , D V N ) ∥ nl K γ , ν ≲ L ST ( F ) min ( K γ + ν − η ∥ t − η 1 /2 U M ∥ L t , x ∞ ∥ t − η 2 /2 D V N ∥ L t , x ∞ , K γ + ν − η 2 K ∥ U M ∥ L t , x ∞ ∥ t − η 2 /2 V N ∥ L t , x ∞ + K γ + ν − η ∥ t − η 1 /2 D U M ∥ L t , x ∞ ∥ t − η 2 /2 V N ∥ L t , x ∞ ) . Using that U M U_M U M t ≳ M − 2 t \gtrsim M^{-2} t ≳ M − 2
K − η 2 ∥ U M ∥ L t , x ∞ ≲ K − η 2 M − η 1 ∥ t − η 1 / 2 U M ∥ L t , x ∞ = K − η + η 1 M − η 1 ∥ t − η 1 / 2 U M ∥ L t , x ∞ . K^{-\eta_2}\lVert U_M\rVert_{L^\infty_{t,x}}
\lesssim K^{-\eta_2}M^{-\eta_1}\lVert t^{-\eta_1/2}U_M\rVert_{L^\infty_{t,x}}
=K^{-\eta+\eta_1}M^{-\eta_1}\lVert t^{-\eta_1/2}U_M\rVert_{L^\infty_{t,x}}. K − η 2 ∥ U M ∥ L t , x ∞ ≲ K − η 2 M − η 1 ∥ t − η 1 /2 U M ∥ L t , x ∞ = K − η + η 1 M − η 1 ∥ t − η 1 /2 U M ∥ L t , x ∞ . Using the definition of our solution spaces, we therefore obtain that
∥ Q ( U M , D V N ) ∥ n l K γ , ν ≲ L S T ( F ) min ( N , max ( K 1 + η 1 M − η 1 , M ) ) K γ + ν − η M − α N − β × ∥ U M ∥ s o l M α , η 1 ∥ V N ∥ s o l N β , η 2 . \begin{aligned}
\lVert Q(U_M,DV_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim{}& L_{\mathrm{ST}}(F)\min\bigl(N,\max(K^{1+\eta_1}M^{-\eta_1},M)\bigr)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}\\
&\qquad\times\lVert U_M\rVert_{\mathrm{sol}^{\alpha,\eta_1}_M}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta_2}_N}.
\end{aligned} ∥ Q ( U M , D V N ) ∥ nl K γ , ν ≲ L ST ( F ) min ( N , max ( K 1 + η 1 M − η 1 , M ) ) K γ + ν − η M − α N − β × ∥ U M ∥ sol M α , η 1 ∥ V N ∥ sol N β , η 2 . Since η 1 ∈ ( − 1 , 0 ] \eta_1 \in(-1,0] η 1 ∈ ( − 1 , 0 ]
min ( N , max ( K 1 + η 1 M − η 1 , M ) ) ≲ min ( N , max ( K , M ) ) . \min\bigl(N,\max(K^{1+\eta_1}M^{-\eta_1},M)\bigr)
\lesssim\min(N,\max(K,M)). min ( N , max ( K 1 + η 1 M − η 1 , M ) ) ≲ min ( N , max ( K , M )) . Due to this, it remains to prove that
sup K ∑ M , N min ( N , max ( K , M ) ) K γ + ν − η M − α N − β ≲ N l o 1 + γ + ν − η − α − β . (451) \sup_K\sum_{M,N}\min(N,\max(K,M))K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim N_{\mathrm{lo}}^{1+\gamma+\nu-\eta-\alpha-\beta}. \tag*{(451)} K sup M , N ∑ min ( N , max ( K , M )) K γ + ν − η M − α N − β ≲ N lo 1 + γ + ν − η − α − β . (451) To obtain (451) , we first use that β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) N N N
∑ N min ( N , max ( K , M ) ) K γ + ν − η M − α N − β ≲ max ( K , M ) 1 − β K γ + ν − η M − α . \sum_N \min(N,\max(K,M))K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim\max(K,M)^{1-\beta}K^{\gamma+\nu-\eta}M^{-\alpha}. N ∑ min ( N , max ( K , M )) K γ + ν − η M − α N − β ≲ max ( K , M ) 1 − β K γ + ν − η M − α . Using α > 0 \alpha> 0 α > 0 α + β > 1 \alpha+\beta> 1 α + β > 1 M M M
∑ M max ( K , M ) 1 − β K γ + ν − η M − α ≲ K 1 + γ + ν − η − β ∑ M M − α + K γ + ν − η ∑ M M 1 − α − β ≲ K 1 + γ + ν − η − β N l o − α + K γ + ν − η N l o 1 − α − β . \begin{aligned}
\sum_M \max(K,M)^{1-\beta}K^{\gamma+\nu-\eta}M^{-\alpha}
&\lesssim K^{1+\gamma+\nu-\eta-\beta}\sum_M M^{-\alpha}
+K^{\gamma+\nu-\eta}\sum_M M^{1-\alpha-\beta} \\
&\lesssim K^{1+\gamma+\nu-\eta-\beta}N_{\mathrm{lo}}^{-\alpha}
+K^{\gamma+\nu-\eta}N_{\mathrm{lo}}^{1-\alpha-\beta}.
\end{aligned} M ∑ max ( K , M ) 1 − β K γ + ν − η M − α ≲ K 1 + γ + ν − η − β M ∑ M − α + K γ + ν − η M ∑ M 1 − α − β ≲ K 1 + γ + ν − η − β N lo − α + K γ + ν − η N lo 1 − α − β . Since both 1 + γ + ν − η − β 1+\gamma+\nu-\eta-\beta 1 + γ + ν − η − β γ + ν − η \gamma+\nu-\eta γ + ν − η
For the rest of this subsection, U ⋆ U_\star U ⋆ V ⋆ V_\star V ⋆ U ⋆ U_\star U ⋆ ε ℓ , ⋆ \varepsilon_{\ell,\star} ε ℓ , ⋆ V ⋆ V_\star V ⋆ W ε , ⋆ W_{\varepsilon,\star} W ε , ⋆ Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ (406) –(408) .
Lemma 7.23 (Product estimate at negative regularity). Let Assumption 7.21 be satisfied. In addition, let α ∈ ( − 1 , 0 ) \alpha\in(-1,0) α ∈ ( − 1 , 0 ) β , γ ∈ R \beta,\gamma\in\mathbb{R} β , γ ∈ R η , ν ∈ ( − ∞ , 0 ] \eta,\nu\in(-\infty,0] η , ν ∈ ( − ∞ , 0 ]
− 1 < α + η < 0 , β > 0 , α + β > 0 , − 1 < η ≤ ν , and γ + ν − η < − 1 + α − 2 r . (452) -1 < \alpha+\eta< 0,\qquad\beta> 0,\qquad\alpha+\beta> 0,\qquad-1 < \eta\le\nu,\qquad\text{and}\qquad\gamma+\nu-\eta< -1+\alpha-\frac{2}{r}. \tag*{(452)} − 1 < α + η < 0 , β > 0 , α + β > 0 , − 1 < η ≤ ν , and γ + ν − η < − 1 + α − r 2 . (452) Then, it holds for all ε > 0 \varepsilon> 0 ε > 0
∥ D Q ( U ⋆ , V ⋆ ) ∥ n l K γ , ν ≲ ε L S T ( F ) T 2 α + β − γ − ν + η − 1 2 T − 1 r − ε ∥ U ⋆ ∥ m s o l α ∥ V ⋆ ∥ s o l β , η . (453) \left\| \mathsf{D}Q(U_\star,V_\star)\right\|_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim_{\varepsilon} L_{\mathrm{ST}}(F)T^{\frac{2\alpha+\beta-\gamma-\nu+\eta-1}{2}}T^{-\frac{1}{r}-\varepsilon}\left\|U_\star\right\|_{\mathrm{msol}^{\alpha}}\left\|V_\star\right\|_{\mathrm{sol}^{\beta,\eta}}. \tag*{(453)} ∥ D Q ( U ⋆ , V ⋆ ) ∥ nl K γ , ν ≲ ε L ST ( F ) T 2 2 α + β − γ − ν + η − 1 T − r 1 − ε ∥ U ⋆ ∥ msol α ∥ V ⋆ ∥ sol β , η . (453) In addition to (453) , we also have the same estimate but with Q ( U ⋆ , V ⋆ ) Q(U_\star,V_\star) Q ( U ⋆ , V ⋆ ) Q ( U ⋆ ⊗ V ⋆ ) Q(U_\star\otimes V_\star) Q ( U ⋆ ⊗ V ⋆ ) Q ( U ⋆ ⊘ V ⋆ ) Q(U_\star\oslash V_\star) Q ( U ⋆ ⊘ V ⋆ ) Q ( U ⋆ ⊙ V ⋆ ) Q(U_\star\odot V_\star) Q ( U ⋆ ⊙ V ⋆ )
Remark 7.24. We emphasise that the estimate (453) cannot be obtained this way if the m s o l α \mathrm{msol}^{\alpha} msol α s o l α \mathrm{sol}^{\alpha} sol α α \alpha α s o l α \mathrm{sol}^{\alpha} sol α U ⋆ U_\star U ⋆
Proof of Lemma 7.23. Without loss of generality, we may assume that ε \varepsilon ε (452) . Using our weighted Schauder estimate (437) and Lemma 7.6, we obtain
∥ D Q ( U M , V N ) ∥ n l K γ , ν ≲ L S T ( F ) K γ + ν − η + 1 − α + 2 r + ε ∥ t − η / 2 Q ( U M , V N ) ∥ L x r ≲ L S T ( F ) K γ + ν − η + 1 − α + 2 r + ε ∥ U M ∥ C t α − ε 2 L x r ∥ t − η / 2 V N ∥ C t α − 2 ε 2 L x ∞ ≲ L S T ( F ) K γ + ν − η + 1 − α + 2 r + ε M − ε N − α − β + 2 ε ∥ U M ∥ m s o l M α ∥ V N ∥ s o l N β , η . \begin{aligned}
\left\|\mathsf{D}Q(U_M,V_N)\right\|_{\mathrm{nl}^{\gamma,\nu}_K}
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta+1-\alpha+\frac{2}{r}+\varepsilon}
\left\|t^{-\eta/2}Q(U_M,V_N)\right\|_{L^r_x} \\
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta+1-\alpha+\frac{2}{r}+\varepsilon}
\left\|U_M\right\|_{\mathcal{C}^{\frac{\alpha-\varepsilon}{2}}_tL^r_x}
\left\|t^{-\eta/2}V_N\right\|_{\mathcal{C}^{\frac{\alpha-2\varepsilon}{2}}_tL^\infty_x} \\
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta+1-\alpha+\frac{2}{r}+\varepsilon}
M^{-\varepsilon}N^{-\alpha-\beta+2\varepsilon}
\left\|U_M\right\|_{\mathrm{msol}^{\alpha}_M}
\left\|V_N\right\|_{\mathrm{sol}^{\beta,\eta}_N}.
\end{aligned} ∥ D Q ( U M , V N ) ∥ nl K γ , ν ≲ L ST ( F ) K γ + ν − η + 1 − α + r 2 + ε t − η /2 Q ( U M , V N ) L x r ≲ L ST ( F ) K γ + ν − η + 1 − α + r 2 + ε ∥ U M ∥ C t 2 α − ε L x r t − η /2 V N C t 2 α − 2 ε L x ∞ ≲ L ST ( F ) K γ + ν − η + 1 − α + r 2 + ε M − ε N − α − β + 2 ε ∥ U M ∥ msol M α ∥ V N ∥ sol N β , η . Due to our conditions from (452) , the exponents of K K K M M M N N N − 2 α − β + γ + ν − η + 1 + 2 r + 2 ε -2\alpha-\beta+\gamma+\nu-\eta+1+\frac{2}{r}+2\varepsilon − 2 α − β + γ + ν − η + 1 + r 2 + 2 ε (453) .
We now turn to low× \times ×
Lemma 7.25 (Low× \times × α , β , γ ∈ R \alpha,\beta,\gamma\in\mathbb{R} α , β , γ ∈ R η , ν ∈ ( − ∞ , 0 ] \eta,\nu\in(-\infty,0] η , ν ∈ ( − ∞ , 0 ]
α < 0 , 0 < β < 1 , γ + ν − η < 0 , − 2 < η ≤ ν , and ( 1 − θ ) α + β > ( 1 − θ ) (454) \alpha< 0,\qquad0 < \beta< 1,\qquad\gamma+\nu-\eta< 0,\qquad-2 < \eta\le\nu,\qquad\text{and}\qquad(1-\theta)\alpha+\beta> (1-\theta) \tag*{(454)} α < 0 , 0 < β < 1 , γ + ν − η < 0 , − 2 < η ≤ ν , and ( 1 − θ ) α + β > ( 1 − θ ) (454) are satisfied. Then, it holds that
∥ Q ( D U ⋆ ⊗ V ⋆ ) ∥ n l γ , ν ≲ L T ( F ) T ( 1 − θ ) α + β − γ − ν + η + θ − 1 2 ∥ U ⋆ ∥ s o l α ∥ V ⋆ ∥ s o l β , η . (455) \lVert Q(DU_\star\otimes V_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}}
\lesssim L_{T}(F)T^{\frac{(1-\theta)\alpha+\beta-\gamma-\nu+\eta+\theta-1}{2}}
\lVert U_\star\rVert_{\mathrm{sol}^{\alpha}}\lVert V_\star\rVert_{\mathrm{sol}^{\beta,\eta}}. \tag*{(455)} ∥ Q ( D U ⋆ ⊗ V ⋆ ) ∥ nl γ , ν ≲ L T ( F ) T 2 ( 1 − θ ) α + β − γ − ν + η + θ − 1 ∥ U ⋆ ∥ sol α ∥ V ⋆ ∥ sol β , η . (455) If, in addition to (454) , the conditions
η > − 1 and ( 1 − θ ) α + β > 1 + γ + ν − η (456) \eta> -1\qquad\text{and}\qquad(1-\theta)\alpha+\beta> 1+\gamma+\nu-\eta \tag*{(456)} η > − 1 and ( 1 − θ ) α + β > 1 + γ + ν − η (456) are satisfied, it also holds that
∥ Q ( U ⋆ D V ⋆ ) ∥ n l γ , ν ≲ L T ( F ) T ( 1 − θ ) α + β − γ − ν + η − 1 2 ∥ U ⋆ ∥ s o l α ∥ V ⋆ ∥ s o l β , η . (457) \lVert Q(U_\star D V_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}}
\lesssim L_{T}(F)T^{\frac{(1-\theta)\alpha+\beta-\gamma-\nu+\eta-1}{2}}
\lVert U_\star\rVert_{\mathrm{sol}^{\alpha}}\lVert V_\star\rVert_{\mathrm{sol}^{\beta,\eta}}. \tag*{(457)} ∥ Q ( U ⋆ D V ⋆ ) ∥ nl γ , ν ≲ L T ( F ) T 2 ( 1 − θ ) α + β − γ − ν + η − 1 ∥ U ⋆ ∥ sol α ∥ V ⋆ ∥ sol β , η . (457) Remark 7.26. The last condition in (454) is one of the reasons for the parameter θ \theta θ θ \theta θ (454) , the condition would become α + β > 1 \alpha+\beta>1 α + β > 1 × \times × D B ∘ \mathrm{D}_{\mathrm{B}}^{\circ} D B ∘ Z Z Z (454) has no counterpart in the usual, non-covariant low× \times × (385) ). But, as already discussed in § 7.2 , the Littlewood–Paley trichotomy is not available in our setting.
Proof. We first prove the estimate (455) . Using Definition 7.2 and our weighted Schauder estimate (438) , we obtain that
∥ Q ( D U M , V N ) ∥ n l K γ , ν ≲ L T ( F ) K γ + ν − η M 1 − α N − β ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \lVert Q(DU_M,V_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim L_T(F)K^{\gamma+\nu-\eta}M^{1-\alpha}N^{-\beta}
\lVert U_M\rVert_{\mathrm{sol}^{\alpha}_M}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta}_N}. ∥ Q ( D U M , V N ) ∥ nl K γ , ν ≲ L T ( F ) K γ + ν − η M 1 − α N − β ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . It then remains to show that, under our assumption (454) ,
sup K ∑ M , N : M ≤ N 1 − θ K γ + ν − η M 1 − α N − β ≲ N l o γ + ν − η + ( 1 − θ ) ( 1 − α ) − β . (458) \sup_K\sum_{\substack{M,N:\\M\le N^{1-\theta}}}
K^{\gamma+\nu-\eta}M^{1-\alpha}N^{-\beta}
\lesssim N_{\mathrm{lo}}^{\gamma+\nu-\eta+(1-\theta)(1-\alpha)-\beta}. \tag*{(458)} K sup M , N : M ≤ N 1 − θ ∑ K γ + ν − η M 1 − α N − β ≲ N lo γ + ν − η + ( 1 − θ ) ( 1 − α ) − β . (458) This can be easily obtained by first summing in M M M N N N (457) . Using Definition 7.2 and our Schauder estimates (438) and (439) , we obtain that
∥ Q ( U M , D V N ) ∥ n l K γ , ν ≲ L T ( F ) min ( max ( K , M ) , N ) K γ + ν − η M − α N − β ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \lVert Q(U_M,DV_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim L_T(F)\min(\max(K,M),N)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lVert U_M\rVert_{\mathrm{sol}^{\alpha}_M}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta}_N}. ∥ Q ( U M , D V N ) ∥ nl K γ , ν ≲ L T ( F ) min ( max ( K , M ) , N ) K γ + ν − η M − α N − β ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . Since the conditions in (454) are still satisfied, the region in which min ( max ( K , M ) , N ) \min(\max(K,M),N) min ( max ( K , M ) , N ) M M M (455) . It then remains to show that, under our assumptions in (454) and (456) ,
sup K ∑ M , N : M ≤ N 1 − θ min ( K , N ) K γ + ν − η M − α N − β ≲ N l o 1 + γ + ν − η − ( 1 − θ ) α − β . (459) \sup_K\sum_{\substack{M,N:\\M\le N^{1-\theta}}}
\min(K,N)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim N_{\mathrm{lo}}^{1+\gamma+\nu-\eta-(1-\theta)\alpha-\beta}. \tag*{(459)} K sup M , N : M ≤ N 1 − θ ∑ min ( K , N ) K γ + ν − η M − α N − β ≲ N lo 1 + γ + ν − η − ( 1 − θ ) α − β . (459) We note that, since θ > 0 \theta>0 θ > 0 (459) is worse than in (458) . In order to obtain (459) , we estimate
∑ M , N : M ≤ N 1 − θ min ( K , N ) K γ + ν − η M − α N − β ≲ ∑ N min ( K , N ) K γ + ν − η N − ( 1 − θ ) α − β \sum_{\substack{M,N:\\M\le N^{1-\theta}}}
\min(K,N)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim
\sum_N\min(K,N)K^{\gamma+\nu-\eta}N^{-(1-\theta)\alpha-\beta} M , N : M ≤ N 1 − θ ∑ min ( K , N ) K γ + ν − η M − α N − β ≲ N ∑ min ( K , N ) K γ + ν − η N − ( 1 − θ ) α − β ≲ ∑ N ≤ K K γ + ν − η N 1 − ( 1 − θ ) α − β + ∑ N > K K 1 + γ + ν − η N − ( 1 − θ ) α − β ≲ K 1 + γ + ν − η − ( 1 − θ ) α − β . \lesssim
\sum_{N\le K}K^{\gamma+\nu-\eta}N^{1-(1-\theta)\alpha-\beta}
+
\sum_{N>K}K^{1+\gamma+\nu-\eta}N^{-(1-\theta)\alpha-\beta}
\lesssim K^{1+\gamma+\nu-\eta-(1-\theta)\alpha-\beta}. ≲ N ≤ K ∑ K γ + ν − η N 1 − ( 1 − θ ) α − β + N > K ∑ K 1 + γ + ν − η N − ( 1 − θ ) α − β ≲ K 1 + γ + ν − η − ( 1 − θ ) α − β . In the estimate of the first sum, we used that ( 1 − θ ) α + β < 1 (1-\theta)\alpha+\beta<1 ( 1 − θ ) α + β < 1 α < 0 \alpha<0 α < 0 β < 1 \beta<1 β < 1 ( 1 − θ ) α + β > 1 − θ > 0 (1-\theta)\alpha+\beta>1-\theta>0 ( 1 − θ ) α + β > 1 − θ > 0 1 + γ + ν − η − ( 1 − θ ) α − β 1+\gamma+\nu-\eta-(1-\theta)\alpha-\beta 1 + γ + ν − η − ( 1 − θ ) α − β (459) .
In our next lemma, we prove different high× \times ×
Lemma 7.27 (High× \times × α , β , γ ∈ R \alpha,\beta,\gamma\in\mathbb{R} α , β , γ ∈ R η , ν ∈ ( − ∞ , 0 ] \eta,\nu\in(-\infty,0] η , ν ∈ ( − ∞ , 0 ]
α < 0 , β > 0 , γ + ν − η < 0 , η > − 2 , ν > − 1 , ν ≥ η , α + ( 1 − θ ) β > 1 + γ + ν − η , and α + ( 1 − θ ) ( β − 1 ) > 0 (460) \begin{aligned}
\alpha&<0,\qquad\beta>0,\qquad\gamma+\nu-\eta<0,\qquad\eta>-2,\qquad\nu>-1,\qquad\nu\geq\eta,\\
\alpha+(1-\theta)\beta&>1+\gamma+\nu-\eta,\qquad\text{and}\qquad\alpha+(1-\theta)(\beta-1)>0 \tag*{(460)}
\end{aligned} α α + ( 1 − θ ) β < 0 , β > 0 , γ + ν − η < 0 , η > − 2 , ν > − 1 , ν ≥ η , > 1 + γ + ν − η , and α + ( 1 − θ ) ( β − 1 ) > 0 (460) are satisfied. Then, it holds that
∥ Q ( U ⋆ ⋄ D V ⋆ ) ∥ n l γ , ν + ∥ Q ( D U ⋆ ⋄ V ⋆ ) ∥ n l γ , ν ≲ L S T ( F ) T α + ( 1 − θ ) β − γ − ν + η − 1 2 ∥ U ⋆ ∥ s o l α ∥ V ⋆ ∥ s o l β , η . (461) \lVert Q(U_\star\diamond D V_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}}+\lVert Q(DU_\star\diamond V_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}}
\lesssim L_{\mathrm{ST}}(F)T^{\frac{\alpha+(1-\theta)\beta-\gamma-\nu+\eta-1}{2}}\lVert U_\star\rVert_{\mathrm{sol}^{\alpha}}\lVert V_\star\rVert_{\mathrm{sol}^{\beta,\eta}}. \tag*{(461)} ∥ Q ( U ⋆ ⋄ D V ⋆ ) ∥ nl γ , ν + ∥ Q ( D U ⋆ ⋄ V ⋆ ) ∥ nl γ , ν ≲ L ST ( F ) T 2 α + ( 1 − θ ) β − γ − ν + η − 1 ∥ U ⋆ ∥ sol α ∥ V ⋆ ∥ sol β , η . (461) Alternatively, assume that
α < 0 , 0 < β < 1 , ν ≥ η > − 1 , and α + ( 1 − θ ) β > max ( 0 , γ + ν − η + 1 ) . (462) \alpha<0,\qquad0<\beta<1,\qquad\nu\geq\eta>-1,\qquad\text{and}\qquad\alpha+(1-\theta)\beta>\max(0,\gamma+\nu-\eta+1). \tag*{(462)} α < 0 , 0 < β < 1 , ν ≥ η > − 1 , and α + ( 1 − θ ) β > max ( 0 , γ + ν − η + 1 ) . (462) Then, it holds that
∥ D Q ( U ⋆ ⋄ V ⋆ ) ∥ n l γ , ν ≲ L S T ( F ) T α + ( 1 − θ ) β − γ − ν + η − 1 2 ∥ U ⋆ ∥ s o l α ∥ V ⋆ ∥ s o l β , η . (463) \lVert DQ(U_\star\diamond V_\star)\rVert_{\mathrm{nl}^{\gamma,\nu}}
\lesssim L_{\mathrm{ST}}(F)T^{\frac{\alpha+(1-\theta)\beta-\gamma-\nu+\eta-1}{2}}\lVert U_\star\rVert_{\mathrm{sol}^{\alpha}}\lVert V_\star\rVert_{\mathrm{sol}^{\beta,\eta}}. \tag*{(463)} ∥ D Q ( U ⋆ ⋄ V ⋆ ) ∥ nl γ , ν ≲ L ST ( F ) T 2 α + ( 1 − θ ) β − γ − ν + η − 1 ∥ U ⋆ ∥ sol α ∥ V ⋆ ∥ sol β , η . (463) Remark. 7.28. In the proof of Lemma 8.14 below, in which V ⋆ V_\star V ⋆ φ ⋆ \varphi_\star φ ⋆ (460) only imposes the lower bound η > − 2 \eta>-2 η > − 2 η > − 1 \eta>-1 η > − 1
Proof. We begin with our proof of (461) . Using our Schauder estimate (438) , we obtain that
∥ Q ( U M , D V N ) ∥ n l K γ , ν ≲ L S T ( F ) K γ + ν − η ∥ U M ∥ L t , x ∞ ∥ t − η / 2 D V N ∥ L t , x ∞ ≲ L S T ( F ) K γ + ν − η M − α N − β ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \begin{aligned}
\lVert Q(U_M,DV_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_{K}}
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta}\lVert U_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta/2}DV_N\rVert_{L^\infty_{t,x}}\\
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}\lVert U_M\rVert_{\mathrm{sol}^{\alpha}_M}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta}_N}.
\end{aligned} ∥ Q ( U M , D V N ) ∥ nl K γ , ν ≲ L ST ( F ) K γ + ν − η ∥ U M ∥ L t , x ∞ ∥ t − η /2 D V N ∥ L t , x ∞ ≲ L ST ( F ) K γ + ν − η M − α N − β ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . Since both Q ( U ⋆ ⋄ D V ⋆ ) Q(U_\star\diamond DV_\star) Q ( U ⋆ ⋄ D V ⋆ ) Q ( D U ⋆ ⋄ V ⋆ ) Q(DU_\star\diamond V_\star) Q ( D U ⋆ ⋄ V ⋆ ) (461) , we will also need to estimate Q ( D U M , V N ) Q(DU_M,V_N) Q ( D U M , V N ) U M U_M U M (439) , which was proven using integration by parts. However, since our parameter η \eta η − 1 -1 − 1 η ~ \widetilde{\eta} η η ~ > − 1 \widetilde{\eta}>-1 η > − 1 η ~ ≥ η \widetilde{\eta}\geq\eta η ≥ η η ~ ≤ min ( ν , η + 1 ) \widetilde{\eta}\leq\min(\nu,\eta+1) η ≤ min ( ν , η + 1 ) η \eta η ν \nu ν V N V_N V N t ≳ N − 2 t\gtrsim N^{-2} t ≳ N − 2
∥ t − η ~ / 2 V N ∥ L t , x ∞ ≲ N η ~ − η N − β ∥ V N ∥ s o l N β , η . \lVert t^{-\widetilde{\eta}/2}V_N\rVert_{L^\infty_{t,x}}
\lesssim N^{\widetilde{\eta}-\eta}N^{-\beta}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta}_N}. ∥ t − η /2 V N ∥ L t , x ∞ ≲ N η − η N − β ∥ V N ∥ sol N β , η . Together with our weighted Schauder estimate (439) , we then have that
∥ Q ( D U M , V N ) ∥ n l K γ , ν ≲ L S T ( F ) K γ min ( K ν − η ∥ D U M ∥ L t , x ∞ ∥ t − η / 2 V N ∥ L t , x ∞ , K ν − η ∥ U M ∥ L t , x ∞ ∥ t − η / 2 D V N ∥ L t , x ∞ + K ν − η ~ + 1 ∥ U M ∥ L t , x ∞ ∥ t − η ~ / 2 V N ∥ L t , x ∞ ) ≲ L S T ( F ) K γ + ν − η M − α N − β min ( M , max ( N , K 1 + η − η ~ N η ~ − η ) ) ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \begin{aligned}
\lVert Q(DU_M,V_N)\rVert_{\mathrm{nl}^{\gamma,\nu}_{K}}
&\lesssim L_{\mathrm{ST}}(F)K^\gamma\min\Bigl(
K^{\nu-\eta}\lVert DU_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta/2}V_N\rVert_{L^\infty_{t,x}},\\
&\qquad\qquad K^{\nu-\eta}\lVert U_M\rVert_{L^\infty_{t,x}}\lVert t^{-\eta/2}DV_N\rVert_{L^\infty_{t,x}}
+K^{\nu-\widetilde{\eta}+1}\lVert U_M\rVert_{L^\infty_{t,x}}\lVert t^{-\widetilde{\eta}/2}V_N\rVert_{L^\infty_{t,x}}\Bigr)\\
&\lesssim L_{\mathrm{ST}}(F)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\min\Bigl(M,\max\bigl(N,K^{1+\eta-\widetilde{\eta}}N^{\widetilde{\eta}-\eta}\bigr)\Bigr)
\lVert U_M\rVert_{\mathrm{sol}^{\alpha}_M}\lVert V_N\rVert_{\mathrm{sol}^{\beta,\eta}_N}.
\end{aligned} ∥ Q ( D U M , V N ) ∥ nl K γ , ν ≲ L ST ( F ) K γ min ( K ν − η ∥ D U M ∥ L t , x ∞ ∥ t − η /2 V N ∥ L t , x ∞ , K ν − η ∥ U M ∥ L t , x ∞ ∥ t − η /2 D V N ∥ L t , x ∞ + K ν − η + 1 ∥ U M ∥ L t , x ∞ ∥ t − η /2 V N ∥ L t , x ∞ ) ≲ L ST ( F ) K γ + ν − η M − α N − β min ( M , max ( N , K 1 + η − η N η − η ) ) ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . Since 0 ≤ η ~ − η ≤ 1 0 \leq\widetilde{\eta}-\eta\leq1 0 ≤ η − η ≤ 1
min ( M , max ( N , K 1 + η − η ~ N η ~ − η ) ) ≤ min ( M , max ( K , N ) ) ≤ min ( K , M ) + N . \min\left(M,\max\left(N,K^{1+\eta-\widetilde{\eta}}N^{\widetilde{\eta}-\eta}\right)\right) \leq\min\left(M,\max(K,N)\right) \leq\min(K,M)+N. min ( M , max ( N , K 1 + η − η N η − η ) ) ≤ min ( M , max ( K , N ) ) ≤ min ( K , M ) + N . After combining our estimates of both Q ( U M , D V N ) \mathcal{Q}(U_M,DV_N) Q ( U M , D V N ) Q ( D U M , V N ) \mathcal{Q}(DU_M,V_N) Q ( D U M , V N )
K γ + ν − η ∑ M , N : M ∼ θ N min ( K , M ) M − α N − β + K γ + ν − η ∑ M , N : M ∼ θ N M − α N 1 − β ≲ N l o 1 + γ + ν − η − α − ( 1 − θ ) β . (464) K^{\gamma+\nu-\eta}\sum_{\substack{M,N:\\M\sim_\theta N}}\min(K,M)M^{-\alpha}N^{-\beta}
+K^{\gamma+\nu-\eta}\sum_{\substack{M,N:\\M\sim_\theta N}}M^{-\alpha}N^{1-\beta}
\lesssim N_{\mathrm{lo}}^{1+\gamma+\nu-\eta-\alpha-(1-\theta)\beta}. \tag*{(464)} K γ + ν − η M , N : M ∼ θ N ∑ min ( K , M ) M − α N − β + K γ + ν − η M , N : M ∼ θ N ∑ M − α N 1 − β ≲ N lo 1 + γ + ν − η − α − ( 1 − θ ) β . (464) We first estimate the first sum in (464) . By using that β > 0 \beta>0 β > 0 N N N
K γ + ν − η ∑ M , N : M ∼ θ N min ( K , M ) M − α N − β ≲ K γ + ν − η ∑ M ≲ K M 1 − α − ( 1 − θ ) β + K 1 + γ + ν − η ∑ M ≳ K M − α − ( 1 − θ ) β . (465) K^{\gamma+\nu-\eta}\sum_{\substack{M,N:\\M\sim_\theta N}}\min(K,M)M^{-\alpha}N^{-\beta}
\lesssim K^{\gamma+\nu-\eta}\sum_{M\lesssim K}M^{1-\alpha-(1-\theta)\beta}
+K^{1+\gamma+\nu-\eta}\sum_{M\gtrsim K}M^{-\alpha-(1-\theta)\beta}. \tag*{(465)} K γ + ν − η M , N : M ∼ θ N ∑ min ( K , M ) M − α N − β ≲ K γ + ν − η M ≲ K ∑ M 1 − α − ( 1 − θ ) β + K 1 + γ + ν − η M ≳ K ∑ M − α − ( 1 − θ ) β . (465) Due to the third and seventh conditions in (460) , the first sum in (7.53) can be bounded by first estimating $K^{\gamma+\nu-\eta}\lesssim M^{\gamma+\nu-\eta}$ and then summing over $M\gtrsim N_{\mathrm{lo}}$. Due to the seventh and eighth conditions in (7.48), the second sum in {xref}(7.53) can be estimated by first summing over M M M K ≳ N l o K\gtrsim N_{\mathrm{lo}} K ≳ N lo
We now turn to the second sum in (464) . To this end, we first note that the first and last conditions in (460) imply that β > 1 \beta>1 β > 1 β > 1 \beta>1 β > 1 α + ( 1 − θ ) ( β − 1 ) > 0 \alpha+(1-\theta)(\beta-1)>0 α + ( 1 − θ ) ( β − 1 ) > 0 γ + ν − η < 0 \gamma+\nu-\eta<0 γ + ν − η < 0 (464) can be estimated by
K γ + ν − η ∑ M , N : M ∼ θ N M − α N 1 − β ≲ K γ + ν − η ∑ M M − α − ( 1 − θ ) ( β − 1 ) K^{\gamma+\nu-\eta}\sum_{\substack{M,N:\\M\sim_\theta N}}M^{-\alpha}N^{1-\beta}
\lesssim K^{\gamma+\nu-\eta}\sum_M M^{-\alpha-(1-\theta)(\beta-1)} K γ + ν − η M , N : M ∼ θ N ∑ M − α N 1 − β ≲ K γ + ν − η M ∑ M − α − ( 1 − θ ) ( β − 1 ) ≲ K γ + ν − η N l o − α − ( 1 − θ ) ( β − 1 ) ≲ N l o γ + ν − η − α − ( 1 − θ ) ( β − 1 ) , \lesssim K^{\gamma+\nu-\eta}N_{\mathrm{lo}}^{-\alpha-(1-\theta)(\beta-1)}
\lesssim N_{\mathrm{lo}}^{\gamma+\nu-\eta-\alpha-(1-\theta)(\beta-1)}, ≲ K γ + ν − η N lo − α − ( 1 − θ ) ( β − 1 ) ≲ N lo γ + ν − η − α − ( 1 − θ ) ( β − 1 ) , which is better than the right-hand side of (464) . This completes the proof of (461) , and it now remains to prove (463) . Using Definition 7.2 and our weighted Schauder estimate (436) with m = 1 m=1 m = 1
∥ D Q ( U M , V N ) ∥ n l K γ , ν ≲ L S T ( F ) K 1 + γ + ν − η M − α N − β ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \left\|D\mathcal{Q}(U_M,V_N)\right\|_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim L_{\mathrm{S}_T}(F)K^{1+\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}\left\|U_M\right\|_{\mathrm{sol}^{\alpha}_M}\left\|V_N\right\|_{\mathrm{sol}^{\beta,\eta}_N}. ∥ D Q ( U M , V N ) ∥ nl K γ , ν ≲ L S T ( F ) K 1 + γ + ν − η M − α N − β ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . Alternatively, using Definition 7.2, our weighted Schauder estimate (436) with m = 0 m=0 m = 0
D Q ( U M , V N ) = Q ( D U M , V N ) + Q ( U M , D V N ) , D\mathcal{Q}(U_M,V_N)=\mathcal{Q}(DU_M,V_N)+\mathcal{Q}(U_M,DV_N), D Q ( U M , V N ) = Q ( D U M , V N ) + Q ( U M , D V N ) , we obtain that
∥ D Q ( U M , V N ) ∥ n l K γ , ν ≲ L S T ( F ) max ( M , N ) K γ + ν − η M − α N − β ∥ U M ∥ s o l M α ∥ V N ∥ s o l N β , η . \left\|D\mathcal{Q}(U_M,V_N)\right\|_{\mathrm{nl}^{\gamma,\nu}_K}
\lesssim L_{\mathrm{S}_T}(F)\max(M,N)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}\left\|U_M\right\|_{\mathrm{sol}^{\alpha}_M}\left\|V_N\right\|_{\mathrm{sol}^{\beta,\eta}_N}. ∥ D Q ( U M , V N ) ∥ nl K γ , ν ≲ L S T ( F ) max ( M , N ) K γ + ν − η M − α N − β ∥ U M ∥ sol M α ∥ V N ∥ sol N β , η . It then remains to show that, under our assumption (462) ,
∑ M , N : M ∼ θ N min ( K , max ( M , N ) ) K γ + ν − η M − α N − β ≲ N l o 1 + γ + ν − η − α − ( 1 − θ ) β . (466) \sum_{\substack{M,N:\\M\sim_\theta N}}\min(K,\max(M,N))K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim N_{\mathrm{lo}}^{1+\gamma+\nu-\eta-\alpha-(1-\theta)\beta}. \tag*{(466)} M , N : M ∼ θ N ∑ min ( K , max ( M , N )) K γ + ν − η M − α N − β ≲ N lo 1 + γ + ν − η − α − ( 1 − θ ) β . (466) Since β \beta β α \alpha α M ∼ θ N M \sim_{\theta} N M ∼ θ N max ( M , N ) \max(M,N) max ( M , N ) M M M N N N M ≥ N M \ge N M ≥ N (466) . Then, we estimate
∑ M , N : M 1 − θ ≤ N ≤ M min ( K , M ) K γ + ν − η M − α N − β ≲ ∑ M min ( K , M ) K γ + ν − η M − α − ( 1 − θ ) β ≲ K 1 + γ + ν − η − α − ( 1 − θ ) β . \sum_{\substack{M,N:\\ M^{1-\theta}\leq N\leq M}} \min(K,M)K^{\gamma+\nu-\eta}M^{-\alpha}N^{-\beta}
\lesssim\sum_M \min(K,M)K^{\gamma+\nu-\eta}M^{-\alpha-(1-\theta)\beta}
\lesssim K^{1+\gamma+\nu-\eta-\alpha-(1-\theta)\beta}. M , N : M 1 − θ ≤ N ≤ M ∑ min ( K , M ) K γ + ν − η M − α N − β ≲ M ∑ min ( K , M ) K γ + ν − η M − α − ( 1 − θ ) β ≲ K 1 + γ + ν − η − α − ( 1 − θ ) β . The last inequality above is obtained by splitting the sum into the two regions M ≤ K M \le K M ≤ K M > K M > K M > K 1 − α − ( 1 − θ ) β > 0 1-\alpha-(1-\theta)\beta>0 1 − α − ( 1 − θ ) β > 0 − α − ( 1 − θ ) β < 0 -\alpha-(1-\theta)\beta<0 − α − ( 1 − θ ) β < 0 (462) . Furthermore, due to (462) , 1 + γ + ν − η − α − ( 1 − θ ) β 1+\gamma+\nu-\eta-\alpha-(1-\theta)\beta 1 + γ + ν − η − α − ( 1 − θ ) β (466) .
In our last lemma, we prove a high× \times ×
Lemma 7.29 (High× \times × α , β , γ ∈ R \alpha,\beta,\gamma\in\mathbb{R} α , β , γ ∈ R η , ν ∈ ( − ∞ , 0 ] \eta,\nu\in(-\infty,0] η , ν ∈ ( − ∞ , 0 ]
− 1 − η < α < 0 , 0 < β < 1 , − 1 < η ≤ ν , (467) -1-\eta<\alpha<0,\qquad0<\beta<1,\qquad-1<\eta\leq\nu, \tag*{(467)} − 1 − η < α < 0 , 0 < β < 1 , − 1 < η ≤ ν , (467) θ ( 1 + α ) > ( 1 − θ ) ( 1 − α − β ) , γ + ν − η < α − 1 − θ θ ( 1 − α − β ) − 2 r . (468) \theta(1+\alpha)>(1-\theta)(1-\alpha-\beta),\qquad\gamma+\nu-\eta<\alpha-\frac{1-\theta}{\theta}(1-\alpha-\beta)-\frac{2}{r}. \tag*{(468)} θ ( 1 + α ) > ( 1 − θ ) ( 1 − α − β ) , γ + ν − η < α − θ 1 − θ ( 1 − α − β ) − r 2 . (468) Then, it holds for all ε > 0 \varepsilon>0 ε > 0
∥ Q ( U ⋆ ⊡ D V ⋆ ) ∥ n l κ γ , ν ≲ ε L s T ( F ) T − γ − ν + η + α 2 − 1 − θ 2 θ ( 1 − α − β ) T − 1 r − ε ∥ U ⋆ ∥ m s o l α ∥ V ⋆ ∥ s o l β , η . (469) \left\|Q(U_\star\mathbin{\boxdot}DV_\star)\right\|_{\mathrm{nl}^{\gamma,\nu}_\kappa}
\lesssim_{\varepsilon} L_{s_T}(F)T^{\frac{-\gamma-\nu+\eta+\alpha}{2}-\frac{1-\theta}{2\theta}(1-\alpha-\beta)}T^{-\frac{1}{r}-\varepsilon}\left\|U_\star\right\|_{\mathrm{msol}^{\alpha}}\left\|V_\star\right\|_{\mathrm{sol}^{\beta,\eta}}. \tag*{(469)} ∥ Q ( U ⋆ ⊡ D V ⋆ ) ∥ nl κ γ , ν ≲ ε L s T ( F ) T 2 − γ − ν + η + α − 2 θ 1 − θ ( 1 − α − β ) T − r 1 − ε ∥ U ⋆ ∥ msol α ∥ V ⋆ ∥ sol β , η . (469) Remark 7.30. In order to control the high× \times × φ B \varphi_B φ B D W DW D W α = − κ \alpha=-\kappa α = − κ β = 1 − δ 1 \beta=1-\delta_1 β = 1 − δ 1 γ = − 1 + δ 2 \gamma=-1+\delta_2 γ = − 1 + δ 2 (468) are satisfied only because δ 1 / θ \delta_1/\theta δ 1 / θ θ \theta θ
Proof. Since our desired estimate (469) becomes worse as ε \varepsilon ε ε > 0 \varepsilon>0 ε > 0 (467) –(468) . For technical reasons, we let ε ′ > 0 \varepsilon'>0 ε ′ > 0 (467) –(468) and ε \varepsilon ε α ′ ∈ [ − 1 , α ] \alpha'\in[-1,\alpha] α ′ ∈ [ − 1 , α ] (437) , we first estimate
∥ Q ( U M , D V N ) ∥ n l κ γ , ν ≲ L s T ( F ) K γ + ν − η − α ′ + 2 r ∥ t − η / 2 Q ( U M , D V N ) ∥ L t α ′ / 2 L x r ≲ L s T ( F ) K γ + ν − η − α ′ + 2 r ∥ U M ∥ L t α ′ / 2 L x r ∥ t − η / 2 D V N ∥ L t − ( α ′ − ε ′ ) / 2 L x ∞ ≲ L s T ( F ) K γ + ν − η − α ′ + 2 r M α ′ − α N 1 − α ′ − β + ε ′ ∥ U ⋆ ∥ m s o l α ∥ V ⋆ ∥ s o l β , η . \begin{aligned}
\left\|Q(U_M,DV_N)\right\|_{\mathrm{nl}^{\gamma,\nu}_\kappa}
&\lesssim L_{s_T}(F)K^{\gamma+\nu-\eta-\alpha'+\frac{2}{r}}\left\|t^{-\eta/2}Q(U_M,DV_N)\right\|_{L_t^{\alpha'/2}L_x^r}\\
&\lesssim L_{s_T}(F)K^{\gamma+\nu-\eta-\alpha'+\frac{2}{r}}\left\|U_M\right\|_{L_t^{\alpha'/2}L_x^r}\left\|t^{-\eta/2}DV_N\right\|_{L_t^{-(\alpha'-\varepsilon')/2}L_x^\infty}\\
&\lesssim L_{s_T}(F)K^{\gamma+\nu-\eta-\alpha'+\frac{2}{r}}M^{\alpha'-\alpha}N^{1-\alpha'-\beta+\varepsilon'}\left\|U_\star\right\|_{\mathrm{msol}^{\alpha}}\left\|V_\star\right\|_{\mathrm{sol}^{\beta,\eta}}.
\end{aligned} ∥ Q ( U M , D V N ) ∥ nl κ γ , ν ≲ L s T ( F ) K γ + ν − η − α ′ + r 2 t − η /2 Q ( U M , D V N ) L t α ′ /2 L x r ≲ L s T ( F ) K γ + ν − η − α ′ + r 2 ∥ U M ∥ L t α ′ /2 L x r t − η /2 D V N L t − ( α ′ − ε ′ ) /2 L x ∞ ≲ L s T ( F ) K γ + ν − η − α ′ + r 2 M α ′ − α N 1 − α ′ − β + ε ′ ∥ U ⋆ ∥ msol α ∥ V ⋆ ∥ sol β , η . Since 1 − β 1-\beta 1 − β − α ′ -\alpha' − α ′ ε ′ \varepsilon' ε ′
M α ′ − α ∑ N ≤ M 1 − θ N 1 − α ′ − β + ε ′ ≲ M α ′ − α M ( 1 − θ ) ( 1 − α ′ − β + ε ′ ) = M ( 1 − θ ) ( 1 − α − β ) + ( 1 − θ ) ε ′ M − θ ( α − α ′ ) . M^{\alpha'-\alpha}\sum_{N\leq M^{1-\theta}}N^{1-\alpha'-\beta+\varepsilon'}
\lesssim M^{\alpha'-\alpha}M^{(1-\theta)(1-\alpha'-\beta+\varepsilon')}
=M^{(1-\theta)(1-\alpha-\beta)+(1-\theta)\varepsilon'}M^{-\theta(\alpha-\alpha')}. M α ′ − α N ≤ M 1 − θ ∑ N 1 − α ′ − β + ε ′ ≲ M α ′ − α M ( 1 − θ ) ( 1 − α ′ − β + ε ′ ) = M ( 1 − θ ) ( 1 − α − β ) + ( 1 − θ ) ε ′ M − θ ( α − α ′ ) . In order to obtain our desired estimate (469) , it then suffices to show that
∑ M K γ + ν − η − α + 2 r M ( 1 − θ ) ( 1 − α − β ) + ( 1 − θ ) ε ′ min α ′ = − 1 , α ( K M − θ ) α ′ − α ≲ K γ + ν − η − α + 2 r + 1 − θ θ ( 1 − α − β ) + 2 ε . (470) \sum_M K^{\gamma+\nu-\eta-\alpha+\frac{2}{r}}M^{(1-\theta)(1-\alpha-\beta)+(1-\theta)\varepsilon'}\min_{\alpha'=-1,\alpha}(KM^{-\theta})^{\alpha'-\alpha}
\lesssim K^{\gamma+\nu-\eta-\alpha+\frac{2}{r}+\frac{1-\theta}{\theta}(1-\alpha-\beta)+2\varepsilon}. \tag*{(470)} M ∑ K γ + ν − η − α + r 2 M ( 1 − θ ) ( 1 − α − β ) + ( 1 − θ ) ε ′ α ′ = − 1 , α min ( K M − θ ) α ′ − α ≲ K γ + ν − η − α + r 2 + θ 1 − θ ( 1 − α − β ) + 2 ε . (470) The reason is that, due to the second condition in (468) and since ε > 0 \varepsilon>0 ε > 0 (470) is negative. In order to prove (470) , we first note that the minimum in (470) is given by min ( 1 , ( K M − θ ) 1 + α ) \min(1,(KM^{-\theta})^{1+\alpha}) min ( 1 , ( K M − θ ) 1 + α ) (467) , the exponent of the first M M M (470) is positive. Furthermore, due to the first condition in (468) , the combined exponent of the first M M M M M M (470) is negative. From this, it follows that the sum in (470) is comparable to its value at M = K 1 / θ M=K^{1/\theta} M = K 1/ θ
K γ + ν − η − α + 2 r × K 1 − θ θ ( 1 − α − β ) + 1 − θ θ ε ′ . K^{\gamma+\nu-\eta-\alpha+\frac{2}{r}}\times K^{\frac{1-\theta}{\theta}(1-\alpha-\beta)+\frac{1-\theta}{\theta}\varepsilon'}. K γ + ν − η − α + r 2 × K θ 1 − θ ( 1 − α − β ) + θ 1 − θ ε ′ . Since ε ′ \varepsilon' ε ′ (470) .
Covariant local well-posedness
In this section, we prove a covariant local well-posedness result for (353) . It will be a more detailed version of Theorem 2.19 , which is one of the main results of this article. For expository purposes, we first make the following assumption on the time T T T B B B
Assumption 8.1. First, we assume that 0 < T ≤ T 0 0<T\le T_0 0 < T ≤ T 0 T 0 T_0 T 0 (98) . Second, we assume that the background connection one-form B B B (350) on the time interval [ 0 , T ] [0,T] [ 0 , T ]
L s T ( F B ) ≤ T − δ , (471) \mathrm{L}_{sT}(F_B)\le T^{-\delta}, \tag*{(471)} L s T ( F B ) ≤ T − δ , (471) where δ \delta δ (97) and L s T ( F ) \mathrm{L}_{sT}(F) L s T ( F ) (422) .
The reasons behind these assumptions on B B B (350) on each time interval appearing in our global iteration scheme. Second, they are sufficient to obtain our Schauder estimates (from Section 7) and our estimates of the covariant stochastic objects (in Section 9 below). In addition to Assumption 8.1, we also make the following assumption on the initial Higgs fields.
Assumption 8.2. We assume that the initial Higgs fields φ 0 + \varphi_0^+ φ 0 + φ 0 − \varphi_0^- φ 0 −
max ( ∥ φ 0 + ∥ L x r , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ T − 1 γ 0 , (472) \max\left(\|\varphi_0^+\|_{L_x^r},\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right)\le T^{-\frac{1}{\gamma_0}}, \tag*{(472)} max ( ∥ φ 0 + ∥ L x r , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ T − γ 0 1 , (472) where γ 0 \gamma_0 γ 0 (99) .
The reason for distributing (471) and (472) over two different assumptions, rather than collecting them in a single assumption, is that many of our estimates only involve the background connection one-form B B B (472) is well-defined.
We also mention that Assumptions 8.1 and 8.2 imply the smallness condition (285) from Proposition 5.15, i.e., the smallness condition needed for our earlier estimates of S S S Φ \Phi Φ M T ( B , 10 ) M_T(B,10) M T ( B , 10 ) L s T ( F ) \mathrm{L}_{sT}(F) L s T ( F ) (259) and (422) , it directly follows that M T ( B , 10 ) ≤ L s T ( F ) M_T(B,10)\le\mathrm{L}_{sT}(F) M T ( B , 10 ) ≤ L s T ( F )
By using Assumptions 8.1 and 8.2, the left-hand side of (285) can then be bounded by
M T ( B , 10 ) ( T 1 2 − 1 r ∥ φ 0 + ∥ L x r + T 1 − κ 2 ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ T − δ ( T 1 2 − 1 r T − 1 γ 0 + T 1 − κ 2 T − 1 γ 0 ) . M_T(B,10)\left(T^{\frac{1}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{1-\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right)\le T^{-\delta}\left(T^{\frac{1}{2}-\frac{1}{r}}T^{-\frac{1}{\gamma_0}}+T^{1-\frac{\kappa}{2}}T^{-\frac{1}{\gamma_0}}\right). M T ( B , 10 ) ( T 2 1 − r 1 ∥ φ 0 + ∥ L x r + T 2 1 − κ ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) ≤ T − δ ( T 2 1 − r 1 T − γ 0 1 + T 1 − 2 κ T − γ 0 1 ) . Since γ 0 − 2 \gamma_0-2 γ 0 − 2 δ \delta δ κ \kappa κ T 0 T_0 T 0 (285) .
Equipped with Assumptions 8.1 and 8.2, we are now ready to state the main result of this section.
Proposition 8.3 (Covariant local well-posedness). Let Assumptions 8.1 and 8.2 be satisfied and let the covariant enhanced data sets (see Definitions 8.6 and 8.7 below) satisfy
sup ε ∈ 2 − N 0 ∥ Ξ B , ε ∥ ≤ T − λ 1 and lim ε , ε ′ → 0 ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0. (473) \sup_{\varepsilon\in2^{-\mathbb{N}_0}}\|\Xi_{B,\varepsilon}\| \le T^{-\lambda_1}
\qquad\text{and}\qquad
\lim_{\varepsilon,\varepsilon'\to0}\|\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\|=0. \tag*{(473)} ε ∈ 2 − N 0 sup ∥ Ξ B , ε ∥ ≤ T − λ 1 and ε , ε ′ → 0 lim ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0. (473) Then, for all ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_0} ε ∈ 2 − N 0 ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) (X_{\varepsilon,\star},x_{\varepsilon,\star},Y_{\varepsilon,\star},y_{\varepsilon,\star}) ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) (406) –(6.90) exist and, as ε → 0 \varepsilon\to0 ε → 0 ( X ⋆ , x ⋆ , Y ⋆ , y ⋆ ) (X_\star,x_\star,Y_\star,y_\star) ( X ⋆ , x ⋆ , Y ⋆ , y ⋆ )
( sol 1 − δ 1 , 0 × sol 1 − δ 1 , 0 × sol 1 + δ 2 , − λ × sol 1 + δ 2 , − λ ) ( [ 0 , T ] ) . \left(\operatorname{sol}^{1-\delta_1,0}\times\operatorname{sol}^{1-\delta_1,0}\times\operatorname{sol}^{1+\delta_2,-\lambda}\times\operatorname{sol}^{1+\delta_2,-\lambda}\right)([0,T]). ( sol 1 − δ 1 , 0 × sol 1 − δ 1 , 0 × sol 1 + δ 2 , − λ × sol 1 + δ 2 , − λ ) ([ 0 , T ]) . Finally, the remainders R : = ℘ B + + ℘ B + + X + Y R:=\wp_B^+ + \wp_B^+ + X+Y R := ℘ B + + ℘ B + + X + Y r : = ℘ B + + x + y r:=\wp_B^+ +x+y r := ℘ B + + x + y
∥ ( R , r ) ∥ L t , x ∞ ( [ 0 , T ] ) ≲ T 1 − λ 2 − 1 γ 0 − λ 2 and ∥ ( R ⋆ , r ⋆ ) ∥ sol 1 − λ , 0 ( [ 0 , T ] ) ≲ T − 1 γ 0 − λ 2 . (474) \|(R,r)\|_{L^\infty_{t,x}([0,T])}\lesssim T^{\frac{1-\lambda}{2}-\frac{1}{\gamma_0}-\lambda_2}
\qquad\text{and}\qquad
\|(R_\star,r_\star)\|_{\operatorname{sol}^{1-\lambda,0}([0,T])}\lesssim T^{-\frac{1}{\gamma_0}-\lambda_2}. \tag*{(474)} ∥ ( R , r ) ∥ L t , x ∞ ([ 0 , T ]) ≲ T 2 1 − λ − γ 0 1 − λ 2 and ∥ ( R ⋆ , r ⋆ ) ∥ sol 1 − λ , 0 ([ 0 , T ]) ≲ T − γ 0 1 − λ 2 . (474) Remark 8.4. From the convergence of the covariant enhanced data set Ξ B , ε \Xi_{B,\varepsilon} Ξ B , ε ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) (X_{\varepsilon,\star},x_{\varepsilon,\star},Y_{\varepsilon,\star},y_{\varepsilon,\star}) ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) ( ( A ε − B ) ⋆ , Φ ε , ⋆ ) ((A_\varepsilon-B)_\star,\Phi_{\varepsilon,\star}) (( A ε − B ) ⋆ , Φ ε , ⋆ ) sol − κ ( [ 0 , T ] ) \operatorname{sol}^{-\kappa}([0,T]) sol − κ ([ 0 , T ])
As in Section 6, all covariant stochastic objects used in this section will depend on the same connection one-form B B B ℘ ε , ε \wp_{\varepsilon,\varepsilon} ℘ ε , ε ℘ ε , ε \wp_{\varepsilon,\varepsilon} ℘ ε , ε ℘ ε , ε \wp_{\varepsilon,\varepsilon} ℘ ε , ε
Previously, we set up the covariant ansatz (Section 6) and a covariant functional framework (Section 7). In order to complete the proof of Proposition 8.3, we will now proceed as follows: In Section 8.1, we introduce the covariant enhanced data set and its norm, which will be estimated in Section 9. In Section 8.2, we control the deterministic terms S S S φ \varphi φ (406) –(6.90), except for the terms that rely on the paracontrolled structure of X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆
Enhanced data set
In this subsection, we discuss the bounds on the covariant stochastic objects that are needed to solve (406) –(6.90). In order to capture these bounds in a single proposition, we will introduce the covariant enhanced data set and its size. Our definitions will be similar to those in [BC26a, Section 5], except that all stochastic objects will be covariant and that we account for the Higgs field.
In order to unify the treatment of the connection one-form and Higgs field, we introduce the following notation, which is in the spirit of Notation 7.1. To this end, recall that Y = g 2 ⊕ V Y=\mathfrak{g}^2\oplus V Y = g 2 ⊕ V
Notation 8.5. We define the Y Y Y
ξ = ξ ⊕ ζ and ℘ ε , ⋆ = ℘ ε , ⋆ ⊕ ℘ ε , ⋆ . (475) \xi=\xi\oplus\zeta
\qquad\text{and}\qquad
\wp_{\varepsilon,\star}=\wp_{\varepsilon,\star}\oplus\wp_{\varepsilon,\star}. \tag*{(475)} ξ = ξ ⊕ ζ and ℘ ε , ⋆ = ℘ ε , ⋆ ⊕ ℘ ε , ⋆ . (475) In addition, we define the covariance Cov Y ∈ Y ⊗ Y \operatorname{Cov}_Y\in Y\otimes Y Cov Y ∈ Y ⊗ Y Cov Y = δ m n u m ⊗ u n \operatorname{Cov}_Y=\delta^{mn}u_m\otimes u_n Cov Y = δ mn u m ⊗ u n ( u m ) m (u_m)_m ( u m ) m Y Y Y
In order to state the next definition, we recall the linear maps Cov g g \operatorname{Cov}_{\mathfrak{g}}^g Cov g g Cov g g \operatorname{Cov}_{\mathfrak{g}}^g Cov g g Cov g V \operatorname{Cov}_{\mathfrak{g}}^V Cov g V σ ρ , ε 2 \sigma_{\rho,\varepsilon}^2 σ ρ , ε 2 C ρ C_\rho C ρ
Definition 8.6 (Covariant enhanced data set). Let 0 < T ≤ 1 128 0<T\leq\frac{1}{128} 0 < T ≤ 128 1 B : [ 0 , T ] × T 2 → g 2 B:[0,T]\times\mathbb{T}^{2}\to\mathfrak{g}^{2} B : [ 0 , T ] × T 2 → g 2 (350) , and let ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_{0}} ε ∈ 2 − N 0
Ξ B , ε ≔ ( Ξ B , ε a n s , Ξ B , ε d e r , Ξ B , ε p o w , Ξ B , ε p a r a ) . (476) \Xi_{B,\varepsilon}\coloneqq\left(\Xi^{\mathrm{ans}}_{B,\varepsilon},\Xi^{\mathrm{der}}_{B,\varepsilon},\Xi^{\mathrm{pow}}_{B,\varepsilon},\Xi^{\mathrm{para}}_{B,\varepsilon}\right). \tag*{(476)} Ξ B , ε : = ( Ξ B , ε ans , Ξ B , ε der , Ξ B , ε pow , Ξ B , ε para ) . (476) The superscripts stand for “ansatz”, “derivative nonlinearity”, “power-type nonlinearity”, and “paracontrolled”, respectively, and the corresponding components of the covariant data set are defined as follows: First, we define
Ξ B , ε a n s ≔ ( ε ∘ ε , ⋆ , ε ∙ ε , ⋆ , ε ∘ ε , ⋆ , ε ∙ ε , ⋆ , ε ∘ ε , ⋆ ) , (477) \Xi^{\mathrm{ans}}_{B,\varepsilon}\coloneqq\left(\overset{\circ}{\varepsilon}_{\varepsilon,\star},\overset{\bullet}{\varepsilon}_{\varepsilon,\star},\overset{\circ}{\varepsilon}_{\varepsilon,\star},\overset{\bullet}{\varepsilon}_{\varepsilon,\star},\overset{\circ}{\varepsilon}_{\varepsilon,\star}\right), \tag*{(477)} Ξ B , ε ans : = ( ε ∘ ε , ⋆ , ε ∙ ε , ⋆ , ε ∘ ε , ⋆ , ε ∙ ε , ⋆ , ε ∘ ε , ⋆ ) , (477) where the stochastic objects are as in (371) , (374) , and (381) . Second, we define
Ξ B , ε d e r ≔ ( ( Lb ( ε ∘ ε , ⋆ ) ⊗ D ε ∘ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , Lb ( ε ∙ ε , ⋆ , D ε ∘ ε , ⋆ ) , (478) \Xi^{\mathrm{der}}_{B,\varepsilon}\coloneqq(
(\operatorname{Lb}(\overset{\circ}{\varepsilon}_{\varepsilon,\star})\mathbin{\otimes}\mathrm{D}\overset{\circ}{\varepsilon}_{\varepsilon,\star})+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathfrak{g}}(\overset{\circ}{\varepsilon}),
\quad\operatorname{Lb}(\overset{\bullet}{\varepsilon}_{\varepsilon,\star},\mathrm{D}\overset{\circ}{\varepsilon}_{\varepsilon,\star}), \tag*{(478)} Ξ B , ε der : = (( Lb ( ε ∘ ε , ⋆ ) ⊗ D ε ∘ ε , ⋆ ) + ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , Lb ( ε ∙ ε , ⋆ , D ε ∘ ε , ⋆ ) , (478) B ( ε ∘ ε , ⋆ ⊗ D ε ∘ ε , ⋆ ) + 1 2 ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , ε ∘ ε , ⋆ ⊗ D ℓ ε ∘ ε , ⋆ , (479) \mathbf{B}\left(\overset{\circ}{\varepsilon}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}\overset{\circ}{\varepsilon}_{\varepsilon,\star}\right)+\frac{1}{2}\left(\sigma_{\rho,\varepsilon}^{2}+C_{\rho}\right)\operatorname{Cov}^{\mathfrak{g}}\left(\overset{\circ}{\varepsilon}\right),
\quad\overset{\circ}{\varepsilon}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}_{\ell}\overset{\circ}{\varepsilon}_{\varepsilon,\star}, \tag*{(479)} B ( ε ∘ ε , ⋆ ⊗ D ε ∘ ε , ⋆ ) + 2 1 ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , ε ∘ ε , ⋆ ⊗ D ℓ ε ∘ ε , ⋆ , (479) ε ∙ ε , ⋆ ⊗ D ℓ ε ∘ ε , ⋆ + 1 2 ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , ( D ℓ ε ∘ ε , ⋆ ) ⊗ ε ∙ ε , ⋆ ) , (480) \overset{\bullet}{\varepsilon}_{\varepsilon,\star}\mathbin{\otimes}\mathrm{D}_{\ell}\overset{\circ}{\varepsilon}_{\varepsilon,\star}+\frac{1}{2}(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathfrak{g}}(\overset{\circ}{\varepsilon}),
\quad(\mathrm{D}_{\ell}\overset{\circ}{\varepsilon}_{\varepsilon,\star})\mathbin{\otimes}\overset{\bullet}{\varepsilon}_{\varepsilon,\star}
), \tag*{(480)} ε ∙ ε , ⋆ ⊗ D ℓ ε ∘ ε , ⋆ + 2 1 ( σ ρ , ε 2 + C ρ ) Cov g ( ε ∘ ) , ( D ℓ ε ∘ ε , ⋆ ) ⊗ ε ∙ ε , ⋆ ) , (480) where B \mathbf{B} B Lb \operatorname{Lb} Lb (10) and (398) . Third, we define26
Ξ B , ε p a r a ≔ ( ∑ K , L : K ∼ θ L D B k Duh Y , B , K ( ε ∘ ) ⊗ D B ℓ ε ∘ ε , L − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y ) k , ℓ ∈ [ 2 ] . (481) \Xi^{\mathrm{para}}_{B,\varepsilon}\coloneqq\left(\sum_{\substack{K,L:\\K\sim_{\theta}L}}\mathrm{D}_{B}^{k}\operatorname{Duh}_{Y,B,K}\left(\overset{\circ}{\varepsilon}\right)\mathbin{\otimes}\mathrm{D}_{B}^{\ell}\overset{\circ}{\varepsilon}_{\varepsilon,L}-\frac{1}{4}\delta^{k\ell}\left(\sigma_{\rho,\varepsilon}^{2}+C_{\rho}\right)\operatorname{Cov}_{Y}\right)_{k,\ell\in[2]}. \tag*{(481)} Ξ B , ε para : = K , L : K ∼ θ L ∑ D B k Duh Y , B , K ( ε ∘ ) ⊗ D B ℓ ε ∘ ε , L − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y k , ℓ ∈ [ 2 ] . (481) Finally, we define
Ξ B , ε p o w ≔ ( ε ∘ ⊗ ε ∘ − σ ρ , ε 2 Cov Y , (482) \Xi^{\mathrm{pow}}_{B,\varepsilon}\coloneqq(\overset{\circ}{\varepsilon}\mathbin{\otimes}\overset{\circ}{\varepsilon}-\sigma_{\rho,\varepsilon}^{2}\operatorname{Cov}_{Y}, \tag*{(482)} Ξ B , ε pow : = ( ε ∘ ⊗ ε ∘ − σ ρ , ε 2 Cov Y , (482) ε ∘ ⊗ ε ∘ ⊗ ε ∘ − σ ρ , ε 2 ( Cov Y ⊗ ε ∘ + u m ⊗ ε ∘ ⊗ u m + ε ∘ ⊗ Cov Y ) ) . (483) \overset{\circ}{\varepsilon}\mathbin{\otimes}\overset{\circ}{\varepsilon}\mathbin{\otimes}\overset{\circ}{\varepsilon}-\sigma_{\rho,\varepsilon}^{2}(\operatorname{Cov}_{Y}\mathbin{\otimes}\overset{\circ}{\varepsilon}+\mathbf{u}^{m}\mathbin{\otimes}\overset{\circ}{\varepsilon}\mathbin{\otimes}\mathbf{u}_{m}+\overset{\circ}{\varepsilon}\mathbin{\otimes}\operatorname{Cov}_{Y})). \tag*{(483)} ε ∘ ⊗ ε ∘ ⊗ ε ∘ − σ ρ , ε 2 ( Cov Y ⊗ ε ∘ + u m ⊗ ε ∘ ⊗ u m + ε ∘ ⊗ Cov Y )) . (483) Here, ( u m ) m (\mathbf{u}_{m})_{m} ( u m ) m Y Y Y
In addition to the covariant enhanced data set itself, we also define the following “norm”.
Definition 8.7 (Size of covariant enhanced data set). Let 0 < T ≤ 1 128 0<T\leq\frac{1}{128} 0 < T ≤ 128 1 B : [ 0 , T ] × T 2 → g 2 B:[0,T]\times\mathbb{T}^{2}\to\mathfrak{g}^{2} B : [ 0 , T ] × T 2 → g 2 (350) , and let ε ∈ 2 − N 0 \varepsilon\in2^{-\mathbb{N}_{0}} ε ∈ 2 − N 0
∥ Ξ B , ε ∥ ≔ ∥ ε ∘ ε , ⋆ ∥ s o l − κ + ∥ ε ∘ ε , ⋆ ∥ 1 2 s o l − 2 κ + ∥ ε ∙ ε , ⋆ ∥ 1 2 s o l − 2 κ + ∥ ε ∘ ε , ⋆ ∥ 1 2 s o l − 2 κ \left\lVert\Xi_{B,\varepsilon}\right\rVert\coloneqq\left\lVert\overset{\circ}{\varepsilon}_{\varepsilon,\star}\right\rVert_{\mathsf{sol}-\kappa}+\left\lVert\overset{\circ}{\varepsilon}_{\varepsilon,\star}\right\rVert_{\frac{1}{2}\mathsf{sol}-2\kappa}+\left\lVert\overset{\bullet}{\varepsilon}_{\varepsilon,\star}\right\rVert_{\frac{1}{2}\mathsf{sol}-2\kappa}+\left\lVert\overset{\circ}{\varepsilon}_{\varepsilon,\star}\right\rVert_{\frac{1}{2}\mathsf{sol}-2\kappa} ∥ Ξ B , ε ∥ : = ε ∘ ε , ⋆ sol − κ + ε ∘ ε , ⋆ 2 1 sol − 2 κ + ε ∙ ε , ⋆ 2 1 sol − 2 κ + ε ∘ ε , ⋆ 2 1 sol − 2 κ + ∥ Terms in (8.8)–(8.10) ∥ 1 3 E t θ L x r + ∥ (8.11) ∥ 1 2 E t θ L x r +\left\lVert\text{Terms in (8.8)--(8.10)}\right\rVert_{\frac{1}{3}\mathcal{E}_{t}^{\theta}L_{x}^{r}}+\left\lVert\text{(8.11)}\right\rVert_{\frac{1}{2}\mathcal{E}_{t}^{\theta}L_{x}^{r}} + ∥ Terms in (8.8)–(8.10) ∥ 3 1 E t θ L x r + ∥ (8.11) ∥ 2 1 E t θ L x r + ∥ (8.12) ∥ 1 2 E t θ L x r + ∥ (8.13) ∥ 1 3 E t θ − 3 κ 2 L x r . (484) +\left\lVert\text{(8.12)}\right\rVert_{\frac{1}{2}\mathcal{E}_{t}^{\theta}L_{x}^{r}}+\left\lVert\text{(8.13)}\right\rVert_{\frac{1}{3}\mathcal{E}_{t}^{\theta-\frac{3\kappa}{2}}L_{x}^{r}}. \tag*{(484)} + ∥ (8.12) ∥ 2 1 E t θ L x r + ∥ (8.13) ∥ 3 1 E t θ − 2 3 κ L x r . (484) Furthermore, if ε ′ ∈ 2 − N 0 \varepsilon'\in2^{-\mathbb{N}_{0}} ε ′ ∈ 2 − N 0
∥ Ξ B , ε ; Ξ B , ε ′ ∥ \left\lVert\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\right\rVert ∥ Ξ B , ε ; Ξ B , ε ′ ∥ as in (484) , but with all stochastic objects replaced by the corresponding difference of stochastic objects. For example, the first summand in (484) is replaced with ∥ ε ∘ ε , ⋆ − ε ∘ ε ′ , ⋆ ∥ s o l − κ \left\lVert\overset{\circ}{\varepsilon}_{\varepsilon,\star}-\overset{\circ}{\varepsilon}_{\varepsilon',\star}\right\rVert_{\mathsf{sol}-\kappa} ε ∘ ε , ⋆ − ε ∘ ε ′ , ⋆ sol − κ
The estimates of both ∥ Ξ B , ε ∥ \left\lVert\Xi_{B,\varepsilon}\right\rVert ∥ Ξ B , ε ∥ ∥ Ξ B , ε ; Ξ B , ε ′ ∥ \left\lVert\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\right\rVert ∥ Ξ B , ε ; Ξ B , ε ′ ∥ ∥ Ξ B , ε ∥ \left\lVert\Xi_{B,\varepsilon}\right\rVert ∥ Ξ B , ε ∥ ∥ Ξ B , ε ; Ξ B , ε ′ ∥ \left\lVert\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\right\rVert ∥ Ξ B , ε ; Ξ B , ε ′ ∥
Deterministic estimates of S S S φ \varphi φ
In Proposition 5.15 above, we estimated S S S φ \varphi φ L t , x ∞ L^\infty_{t,x} L t , x ∞ Definition 7.2 . This is needed in our covariant local theory, but will not enter directly into our global iteration argument. To this end, we recall that the covariant Littlewood–Paley blocks of S S S φ \varphi φ (379) and (380) , respectively.
Proposition 8.8 (Bounds for φ ⋆ \varphi_\star φ ⋆ S ⋆ S_\star S ⋆ 0 ≤ α ≤ 2 0 \leq\alpha\leq2 0 ≤ α ≤ 2 η ≤ 0 \eta\leq0 η ≤ 0 α + η ≤ − κ \alpha+\eta\leq-\kappa α + η ≤ − κ
∥ φ ⋆ ∥ s o l α , η ≲ T − δ T α + η + κ 2 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . (485) \|\varphi_\star\|_{\mathrm{sol}^{\alpha,\eta}} \lesssim T^{-\delta}T^{\frac{\alpha+\eta+\kappa}{2}}\left(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right). \tag*{(485)} ∥ φ ⋆ ∥ sol α , η ≲ T − δ T 2 α + η + κ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . (485) Furthermore, under the condition α + η ≤ 1 − 2 κ \alpha+\eta\leq1-2\kappa α + η ≤ 1 − 2 κ
∥ S ⋆ ∥ s o l α , η ≲ T − 3 δ T 1 − α − η − 2 κ 2 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) 2 . (486) \|S_\star\|_{\mathrm{sol}^{\alpha,\eta}} \lesssim T^{-3\delta}T^{\frac{1-\alpha-\eta-2\kappa}{2}}\left(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right)^2. \tag*{(486)} ∥ S ⋆ ∥ sol α , η ≲ T − 3 δ T 2 1 − α − η − 2 κ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) 2 . (486) We note that, due to our upper bound on T T T (486) is smaller than the right-hand side of (485) .
Proof of Proposition 8.8. The proof does not rely on a contraction-mapping argument. Instead, we just use our earlier estimates of S S S φ \varphi φ Proposition 5.15 , and insert them into our Schauder estimates. In particular, we will often use that
T 1 2 − δ ( ∥ S ∥ L t , x ∞ + ∥ t 1 2 D S ∥ L t , x ∞ ) ≲ T 1 − 2 δ − κ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) 2 ≲ T 1 − 2 δ − κ T − 2 γ 0 ≲ T δ . T^{\frac{1}{2}-\delta}\left(\|S\|_{L^\infty_{t,x}}+\|t^{\frac{1}{2}}DS\|_{L^\infty_{t,x}}\right)\lesssim T^{1-2\delta-\kappa}\left(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right)^2\lesssim T^{1-2\delta-\kappa}T^{-\frac{2}{\gamma_0}}\lesssim T^\delta. T 2 1 − δ ( ∥ S ∥ L t , x ∞ + ∥ t 2 1 D S ∥ L t , x ∞ ) ≲ T 1 − 2 δ − κ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) 2 ≲ T 1 − 2 δ − κ T − γ 0 2 ≲ T δ . In the last estimate, we used that γ 0 − 2 \gamma_0-2 γ 0 − 2 δ \delta δ κ \kappa κ ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ≥ 1 \|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\geq1 ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ≥ 1
Step 1: Estimates of φ ⋆ \varphi_\star φ ⋆ Using the trivial bound t ≤ T t\leq T t ≤ T φ N \varphi_N φ N t ≳ N − 2 t\gtrsim N^{-2} t ≳ N − 2
∥ φ ⋆ ∥ s o l α , η ≲ T α + η + κ 2 ∥ φ ⋆ ∥ s o l α , − α − κ ≲ T α + η + κ 2 ∥ φ ⋆ ∥ s o l 2 , − 2 − κ . \|\varphi_\star\|_{\mathrm{sol}^{\alpha,\eta}}\lesssim T^{\frac{\alpha+\eta+\kappa}{2}}\|\varphi_\star\|_{\mathrm{sol}^{\alpha,-\alpha-\kappa}}\lesssim T^{\frac{\alpha+\eta+\kappa}{2}}\|\varphi_\star\|_{\mathrm{sol}^{2,-2-\kappa}}. ∥ φ ⋆ ∥ sol α , η ≲ T 2 α + η + κ ∥ φ ⋆ ∥ sol α , − α − κ ≲ T 2 α + η + κ ∥ φ ⋆ ∥ sol 2 , − 2 − κ . Due to this, it suffices to estimate the s o l 2 , − 2 − κ \mathrm{sol}^{2,-2-\kappa} sol 2 , − 2 − κ (485) for ( α , η ) = ( 2 , − 2 − κ ) (\alpha,\eta)=(2,-2-\kappa) ( α , η ) = ( 2 , − 2 − κ ) (380) that
φ N = φ l i n , N + Duh B , N ( 2 S j D j φ + S j S j φ − ∣ φ ∣ 2 φ ) . (487) \varphi_N=\varphi_{\mathrm{lin},N}+\operatorname{Duh}_{B,N}\left(2S_jD^j\varphi+S^jS_j\varphi-|\varphi|^2\varphi\right). \tag*{(487)} φ N = φ lin , N + Duh B , N ( 2 S j D j φ + S j S j φ − ∣ φ ∣ 2 φ ) . (487) Using Lemma 7.18, the linear evolution in (487) can be bounded by
∥ φ l i n , ⋆ ∥ s o l 2 , − 2 − κ ≲ T − δ ( T κ 2 − 1 r ∥ φ 0 + ∥ L x r + T κ 2 − κ 2 ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) ≲ T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . \|\varphi_{\mathrm{lin},\star}\|_{\mathrm{sol}^{2,-2-\kappa}}\lesssim T^{-\delta}\left(T^{\frac{\kappa}{2}-\frac{1}{r}}\|\varphi_0^+\|_{L_x^r}+T^{\frac{\kappa}{2}-\frac{\kappa}{2}}\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right)\lesssim T^{-\delta}\left(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right). ∥ φ lin , ⋆ ∥ sol 2 , − 2 − κ ≲ T − δ ( T 2 κ − r 1 ∥ φ 0 + ∥ L x r + T 2 κ − 2 κ ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) ≲ T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . In the last inequality, we used that 1 / r 1/r 1/ r κ \kappa κ ( α , η ) = ( 2 , − 2 − κ ) (\alpha,\eta)=(2,-2-\kappa) ( α , η ) = ( 2 , − 2 − κ ) (485) . The second term in (487) can be estimated by
∥ Duh B , N ( S j D j φ ) ∥ s o l 2 , − 2 − κ ≲ T 1 2 ∥ Duh B , N ( S j D j φ ) ∥ s o l 2 , − 1 − κ ≲ T 1 2 − δ ∥ t 1 2 + κ 2 S j D j φ ∥ L t , x ∞ ≲ T 1 2 − δ ∥ S ∥ L t , x ∞ ∥ t 1 2 + κ 2 D φ ∥ L t , x ∞ ≲ T δ × T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . (488) \begin{aligned}
\|\operatorname{Duh}_{B,N}(S_jD^j\varphi)\|_{\mathrm{sol}^{2,-2-\kappa}}
&\lesssim T^{\frac{1}{2}}\|\operatorname{Duh}_{B,N}(S_jD^j\varphi)\|_{\mathrm{sol}^{2,-1-\kappa}}\\
&\lesssim T^{\frac{1}{2}-\delta}\|t^{\frac{1}{2}+\frac{\kappa}{2}}S_jD^j\varphi\|_{L^\infty_{t,x}}\\
&\lesssim T^{\frac{1}{2}-\delta}\|S\|_{L^\infty_{t,x}}\|t^{\frac{1}{2}+\frac{\kappa}{2}}D\varphi\|_{L^\infty_{t,x}}\\
&\lesssim T^\delta\times T^{-\delta}\left(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{\mathcal{C}^{-\kappa}_{B(0,T)}}\right). \tag*{(488)}
\end{aligned} ∥ Duh B , N ( S j D j φ ) ∥ sol 2 , − 2 − κ ≲ T 2 1 ∥ Duh B , N ( S j D j φ ) ∥ sol 2 , − 1 − κ ≲ T 2 1 − δ ∥ t 2 1 + 2 κ S j D j φ ∥ L t , x ∞ ≲ T 2 1 − δ ∥ S ∥ L t , x ∞ ∥ t 2 1 + 2 κ D φ ∥ L t , x ∞ ≲ T δ × T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 , T ) − κ ) . (488) which is acceptable. The third term in (487) can be bounded by
∥ Duh B , N ( S j S j ϕ ) ∥ s o l N 2 , − 2 − κ ≲ T ∥ Duh B , N ( S j S j ϕ ) ∥ s o l N 2 , − κ ≲ T 1 − δ ∥ t κ 2 S j S j ϕ ∥ L t , x ∞ ≲ T 1 − δ ∥ S ∥ L t , x ∞ 2 ∥ t κ 2 ϕ ∥ L t , x ∞ ≲ T 2 δ × T − δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) . (489) \begin{aligned}
\|\operatorname{Duh}_{B,N}(S^jS_j\phi)\|_{\mathrm{sol}^{2,-2-\kappa}_N}
&\lesssim T\|\operatorname{Duh}_{B,N}(S^jS_j\phi)\|_{\mathrm{sol}^{2,-\kappa}_N}
\lesssim T^{1-\delta}\|t^{\frac{\kappa}{2}}S^jS_j\phi\|_{L^\infty_{t,x}}\\
&\lesssim T^{1-\delta}\|S\|_{L^\infty_{t,x}}^2\|t^{\frac{\kappa}{2}}\phi\|_{L^\infty_{t,x}}
\lesssim T^{2\delta}\times T^{-\delta}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right). \tag*{(489)}
\end{aligned} ∥ Duh B , N ( S j S j ϕ ) ∥ sol N 2 , − 2 − κ ≲ T ∥ Duh B , N ( S j S j ϕ ) ∥ sol N 2 , − κ ≲ T 1 − δ ∥ t 2 κ S j S j ϕ ∥ L t , x ∞ ≲ T 1 − δ ∥ S ∥ L t , x ∞ 2 ∥ t 2 κ ϕ ∥ L t , x ∞ ≲ T 2 δ × T − δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) . (489) which is acceptable. Finally, the fourth term in (487) can be bounded by
∥ Duh B , N ( ∣ ϕ ∣ 2 ϕ ) ∥ s o l N 2 , − 2 − κ ≲ T 1 − κ ∥ Duh B , N ( ∣ ϕ ∣ 2 ϕ ) ∥ s o l N 2 , − 3 κ ≲ T 1 − δ − κ ∥ t 3 κ 2 ∣ ϕ ∣ 2 ϕ ∥ L t , x ∞ ≲ T 1 − 4 δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 3 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 3 − ( 1 − 4 δ − κ ) γ 0 . (490) \begin{aligned}
\|\operatorname{Duh}_{B,N}(|\phi|^2\phi)\|_{\mathrm{sol}^{2,-2-\kappa}_N}
&\lesssim T^{1-\kappa}\|\operatorname{Duh}_{B,N}(|\phi|^2\phi)\|_{\mathrm{sol}^{2,-3\kappa}_N}\\
&\lesssim T^{1-\delta-\kappa}\|t^{\frac{3\kappa}{2}}|\phi|^2\phi\|_{L^\infty_{t,x}}
\lesssim T^{1-4\delta-\kappa}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^3\\
&\lesssim\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^{3-(1-4\delta-\kappa)\gamma_0}. \tag*{(490)}
\end{aligned} ∥ Duh B , N ( ∣ ϕ ∣ 2 ϕ ) ∥ sol N 2 , − 2 − κ ≲ T 1 − κ ∥ Duh B , N ( ∣ ϕ ∣ 2 ϕ ) ∥ sol N 2 , − 3 κ ≲ T 1 − δ − κ ∥ t 2 3 κ ∣ ϕ ∣ 2 ϕ ∥ L t , x ∞ ≲ T 1 − 4 δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 3 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 3 − ( 1 − 4 δ − κ ) γ 0 . (490) Since γ 0 − 2 \gamma_0-2 γ 0 − 2 δ \delta δ κ \kappa κ
Step 2: Estimates of S ⋆ S_\star S ⋆ As in our estimates of ϕ ⋆ \phi_\star ϕ ⋆ t ≤ T t\leq T t ≤ T s o l β , ν \mathrm{sol}^{\beta,\nu} sol β , ν ν ≤ 0 \nu\leq0 ν ≤ 0 α ≥ 1 − 2 κ \alpha\geq1-2\kappa α ≥ 1 − 2 κ α ≤ 1 − 2 κ \alpha\leq1-2\kappa α ≤ 1 − 2 κ
∥ S ⋆ ∥ s o l α , η ≲ T 1 − 2 κ − α − η 2 ∥ S ⋆ ∥ s o l α , 1 − 2 κ − α ≲ T 1 − 2 κ − α − η 2 ∥ S ⋆ ∥ s o l 2 , − 1 − 2 κ . \|S_\star\|_{\mathrm{sol}^{\alpha,\eta}}
\lesssim T^{\frac{1-2\kappa-\alpha-\eta}{2}}\|S_\star\|_{\mathrm{sol}^{\alpha,1-2\kappa-\alpha}}
\lesssim T^{\frac{1-2\kappa-\alpha-\eta}{2}}\|S_\star\|_{\mathrm{sol}^{2,-1-2\kappa}}. ∥ S ⋆ ∥ sol α , η ≲ T 2 1 − 2 κ − α − η ∥ S ⋆ ∥ sol α , 1 − 2 κ − α ≲ T 2 1 − 2 κ − α − η ∥ S ⋆ ∥ sol 2 , − 1 − 2 κ . In the second case, we estimate
∥ S ⋆ ∥ s o l α , η ≲ T − η 2 ∥ S ⋆ ∥ s o l α , 0 ≲ T 1 − 2 κ − α 2 T − η 2 ∥ S ⋆ ∥ s o l 2 , − 1 − 2 κ . \|S_\star\|_{\mathrm{sol}^{\alpha,\eta}}
\lesssim T^{-\frac{\eta}{2}}\|S_\star\|_{\mathrm{sol}^{\alpha,0}}
\lesssim T^{\frac{1-2\kappa-\alpha}{2}}T^{-\frac{\eta}{2}}\|S_\star\|_{\mathrm{sol}^{2,-1-2\kappa}}. ∥ S ⋆ ∥ sol α , η ≲ T − 2 η ∥ S ⋆ ∥ sol α , 0 ≲ T 2 1 − 2 κ − α T − 2 η ∥ S ⋆ ∥ sol 2 , − 1 − 2 κ . Thus, it suffices to treat the case ( α , η ) = ( 2 , − 1 − 2 κ ) (\alpha,\eta)=(2,-1-2\kappa) ( α , η ) = ( 2 , − 1 − 2 κ ) (379) that
S N = Duh B , N F ( [ S j , 2 D j S − D S j ] + [ S j , [ S j , S ] ] + B ( ϕ ⊗ D ϕ ) + B ( ϕ ⊗ S ϕ ) ) . (491) S_N=\operatorname{Duh}^{F}_{B,N}\left([S_j,2D^jS-DS^j]+[S_j,[S^j,S]]+\mathbf{B}(\phi\otimes D\phi)+\mathbf{B}(\phi\otimes S\phi)\right). \tag*{(491)} S N = Duh B , N F ( [ S j , 2 D j S − D S j ] + [ S j , [ S j , S ]] + B ( ϕ ⊗ D ϕ ) + B ( ϕ ⊗ S ϕ ) ) . (491) The first term in (491) can be bounded by
∥ Duh B , N F ( [ S j , 2 D j S − D S j ] ) ∥ s o l N 2 , − 1 − 2 κ ≲ T κ − δ ∥ t 1 2 [ S j , 2 D j S − D S j ] ∥ L t , x ∞ ≲ T κ − δ ∥ S ∥ L t , x ∞ ∥ t 1 2 D S ∥ L t , x ∞ ≲ T 1 − 3 δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 − ( 1 − 3 δ − κ ) γ 0 . (492) \begin{aligned}
\|\operatorname{Duh}^{F}_{B,N}([S_j,2D^jS-DS^j])\|_{\mathrm{sol}^{2,-1-2\kappa}_N}
&\lesssim T^{\kappa-\delta}\|t^{\frac12}[S_j,2D^jS-DS^j]\|_{L^\infty_{t,x}}\\
&\lesssim T^{\kappa-\delta}\|S\|_{L^\infty_{t,x}}\|t^{\frac12}DS\|_{L^\infty_{t,x}}\\
&\lesssim T^{1-3\delta-\kappa}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^4\\
&\lesssim\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^{4-(1-3\delta-\kappa)\gamma_0}. \tag*{(492)}
\end{aligned} ∥ Duh B , N F ([ S j , 2 D j S − D S j ]) ∥ sol N 2 , − 1 − 2 κ ≲ T κ − δ ∥ t 2 1 [ S j , 2 D j S − D S j ] ∥ L t , x ∞ ≲ T κ − δ ∥ S ∥ L t , x ∞ ∥ t 2 1 D S ∥ L t , x ∞ ≲ T 1 − 3 δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 − ( 1 − 3 δ − κ ) γ 0 . (492) The second term in (491) can be bounded by
∥ Duh B , N F ( [ S j , [ S j , S ] ] ) ∥ s o l N 2 , − 1 − 2 κ ≲ T 1 2 + κ ∥ Duh B , N F ( [ S j , [ S j , S ] ] ) ∥ s o l N 2 , 0 ≲ T 1 2 + κ − δ ∥ [ S j , [ S j , S ] ] ∥ L t , x ∞ ≲ T 1 2 + κ − δ ( T 1 2 − δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ) 3 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 6 − ( 2 − 4 δ − 2 κ ) γ 0 . (493) \begin{aligned}
\|\operatorname{Duh}^{F}_{B,N}([S_j,[S^j,S]])\|_{\mathrm{sol}^{2,-1-2\kappa}_N}
&\lesssim T^{\frac12+\kappa}\|\operatorname{Duh}^{F}_{B,N}([S_j,[S^j,S]])\|_{\mathrm{sol}^{2,0}_N}\\
&\lesssim T^{\frac12+\kappa-\delta}\|[S_j,[S^j,S]]\|_{L^\infty_{t,x}}\\
&\lesssim T^{\frac12+\kappa-\delta}\left(T^{\frac12-\delta-\kappa}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^2\right)^3\\
&\lesssim\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^{6-(2-4\delta-2\kappa)\gamma_0}. \tag*{(493)}
\end{aligned} ∥ Duh B , N F ([ S j , [ S j , S ]]) ∥ sol N 2 , − 1 − 2 κ ≲ T 2 1 + κ ∥ Duh B , N F ([ S j , [ S j , S ]]) ∥ sol N 2 , 0 ≲ T 2 1 + κ − δ ∥ [ S j , [ S j , S ]] ∥ L t , x ∞ ≲ T 2 1 + κ − δ ( T 2 1 − δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ) 3 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 6 − ( 2 − 4 δ − 2 κ ) γ 0 . (493) The third term in (491) can be bounded by
∥ Duh B , N F ( B ( ϕ ⊗ D ϕ ) ) ∥ s o l N 2 , − 1 − 2 κ ≲ T − δ ∥ t 1 2 + κ B ( ϕ ⊗ D ϕ ) ∥ L t , x ∞ ≲ T − δ ∥ t κ 2 ϕ ∥ L t , x ∞ ∥ t 1 2 + κ 2 D ϕ ∥ L t , x ∞ ≲ T − 3 δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 . (494) \begin{aligned}
\|\operatorname{Duh}^{F}_{B,N}(\mathbf{B}(\phi\otimes D\phi))\|_{\mathrm{sol}^{2,-1-2\kappa}_N}
&\lesssim T^{-\delta}\|t^{\frac12+\kappa}\mathbf{B}(\phi\otimes D\phi)\|_{L^\infty_{t,x}}\\
&\lesssim T^{-\delta}\|t^{\frac{\kappa}{2}}\phi\|_{L^\infty_{t,x}}\|t^{\frac12+\frac{\kappa}{2}}D\phi\|_{L^\infty_{t,x}}\\
&\lesssim T^{-3\delta}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^2. \tag*{(494)}
\end{aligned} ∥ Duh B , N F ( B ( ϕ ⊗ D ϕ )) ∥ sol N 2 , − 1 − 2 κ ≲ T − δ ∥ t 2 1 + κ B ( ϕ ⊗ D ϕ ) ∥ L t , x ∞ ≲ T − δ ∥ t 2 κ ϕ ∥ L t , x ∞ ∥ t 2 1 + 2 κ D ϕ ∥ L t , x ∞ ≲ T − 3 δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 . (494) Finally, the fourth term in (491) can be bounded by
∥ Duh B , N F ( B ( ϕ ⊗ S ϕ ) ) ∥ s o l N 2 , − 1 − 2 κ ≲ T 1 2 ∥ Duh B , N F ( B ( ϕ ⊗ S ϕ ) ) ∥ s o l N 2 , − 2 κ ≲ T 1 2 − δ ∥ t κ B ( ϕ ⊗ S ϕ ) ∥ L t , x ∞ ≲ T 1 2 − δ ∥ S ∥ L t , x ∞ ∥ t κ 2 ϕ ∥ L t , x ∞ ≲ T 1 2 − δ × T 1 2 − δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 × T − 2 δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 − ( 1 − 4 δ − κ ) γ 0 . (495) \begin{aligned}
\|\operatorname{Duh}^{F}_{B,N}(\mathbf{B}(\phi\otimes S\phi))\|_{\mathrm{sol}^{2,-1-2\kappa}_N}
&\lesssim T^{\frac12}\|\operatorname{Duh}^{F}_{B,N}(\mathbf{B}(\phi\otimes S\phi))\|_{\mathrm{sol}^{2,-2\kappa}_N}\\
&\lesssim T^{\frac12-\delta}\|t^\kappa\mathbf{B}(\phi\otimes S\phi)\|_{L^\infty_{t,x}}
\lesssim T^{\frac12-\delta}\|S\|_{L^\infty_{t,x}}\|t^{\frac{\kappa}{2}}\phi\|_{L^\infty_{t,x}}\\
&\lesssim T^{\frac12-\delta}\times T^{\frac12-\delta-\kappa}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^2\\
&\qquad\times T^{-2\delta}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^2\\
&\lesssim\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{C_{B(0),T}^{-\kappa}}\right)^{4-(1-4\delta-\kappa)\gamma_0}. \tag*{(495)}
\end{aligned} ∥ Duh B , N F ( B ( ϕ ⊗ S ϕ )) ∥ sol N 2 , − 1 − 2 κ ≲ T 2 1 ∥ Duh B , N F ( B ( ϕ ⊗ S ϕ )) ∥ sol N 2 , − 2 κ ≲ T 2 1 − δ ∥ t κ B ( ϕ ⊗ S ϕ ) ∥ L t , x ∞ ≲ T 2 1 − δ ∥ S ∥ L t , x ∞ ∥ t 2 κ ϕ ∥ L t , x ∞ ≲ T 2 1 − δ × T 2 1 − δ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 × T − 2 δ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≲ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 4 − ( 1 − 4 δ − κ ) γ 0 . (495) Since γ 0 − 2 \gamma_0-2 γ 0 − 2 δ \delta δ κ \kappa κ (492) –(495) are acceptable.
Nonlinear estimates
In this subsection, we estimate all terms in (406) –(6.90), except the terms that can be controlled using only ∥ ∥ ∥ Ξ B , ε ∥ ∥ ∥ \lVert\!\lVert\!\lVert\Xi_{B,\varepsilon}\rVert\!\rVert\!\rVert ∥ ∥ ∥ Ξ B , ε ∥ ∥ ∥ X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ (409) , (410) , (6.74), (411) , and (6.84). In most cases, the estimates follow directly from our general estimates in § 8.3 , and the only difficulty lies in choosing the correct regularity parameters.
In the derivation of our ansatz (§ 7 ), we mostly viewed S S S φ \varphi φ W ε W_\varepsilon W ε w ε w_\varepsilon w ε S S S φ \varphi φ S S S φ \varphi φ S S S φ \varphi φ S S S φ \varphi φ Proposition 8.3 . To this end, we introduce the variables
R ε : = W ε − S = Y ˚ ε + Y ˙ ε + X ε + Y ε , (496) R_\varepsilon:= W_\varepsilon-S=\mathring{Y}_\varepsilon+\dot{Y}_\varepsilon+X_\varepsilon+Y_\varepsilon, \tag*{(496)} R ε := W ε − S = Y ˚ ε + Y ˙ ε + X ε + Y ε , (496) r ε : = w ε − φ = y ˚ ε + x ε + y ε . (497) r_\varepsilon:= w_\varepsilon-\varphi=\mathring{y}_\varepsilon+x_\varepsilon+y_\varepsilon. \tag*{(497)} r ε := w ε − φ = y ˚ ε + x ε + y ε . (497) Throughout this subsection, we rely on the following assumption.
Assumption 8.9 (Setup). We assume that
X ε , ⋆ ∈ s o l 1 − δ 1 , − λ ( [ 0 , T ] × T 2 → g 2 ) , Y ε , ⋆ ∈ s o l 1 + δ 2 , − λ ( [ 0 , T ] × T 2 → g 2 ) , X_{\varepsilon,\star}\in\mathrm{sol}^{1-\delta_1,-\lambda}([0,T]\times\mathbb{T}^2\to\mathfrak{g}^2),\qquad
Y_{\varepsilon,\star}\in\mathrm{sol}^{1+\delta_2,-\lambda}([0,T]\times\mathbb{T}^2\to\mathfrak{g}^2), X ε , ⋆ ∈ sol 1 − δ 1 , − λ ([ 0 , T ] × T 2 → g 2 ) , Y ε , ⋆ ∈ sol 1 + δ 2 , − λ ([ 0 , T ] × T 2 → g 2 ) , x ε , ⋆ ∈ s o l 1 − δ 1 , − λ ( [ 0 , T ] × T 2 → V ) , y ε , ⋆ ∈ s o l 1 + δ 2 , − λ ( [ 0 , T ] × T 2 → V ) . x_{\varepsilon,\star}\in\mathrm{sol}^{1-\delta_1,-\lambda}([0,T]\times\mathbb{T}^2\to V),\qquad
y_{\varepsilon,\star}\in\mathrm{sol}^{1+\delta_2,-\lambda}([0,T]\times\mathbb{T}^2\to V). x ε , ⋆ ∈ sol 1 − δ 1 , − λ ([ 0 , T ] × T 2 → V ) , y ε , ⋆ ∈ sol 1 + δ 2 , − λ ([ 0 , T ] × T 2 → V ) . Furthermore, we assume that each covariant Littlewood–Paley block X ε , N X_{\varepsilon,N} X ε , N x ε , N x_{\varepsilon,N} x ε , N Y ε , N Y_{\varepsilon,N} Y ε , N y ε , N y_{\varepsilon,N} y ε , N 0 ≤ t ≤ 1 16 N − 2 0\leq t\leq\frac{1}{16}N^{-2} 0 ≤ t ≤ 16 1 N − 2 X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆ 27 (376) .
Furthermore, throughout this subsection and the next, we repeatedly apply estimates involving L T s ( F ) L_T^s(F) L T s ( F ) Lemma 7.10 and Proposition 7.16. In each such application, we use the bound L T s ( F ) ≤ T − δ L_T^s(F)\leq T^{-\delta} L T s ( F ) ≤ T − δ Assumption 8.1 .
In the first lemma of this subsection, we control the terms appearing in the definitions of the paracontrolled terms X ε ∘ X_{\varepsilon}^{\circ} X ε ∘ X ε ∙ X_{\varepsilon}^{\bullet} X ε ∙ x ε x_\varepsilon x ε (406) , (407) , and (408) .
Lemma 8.10. Let Assumptions 8.1 , 8.2 , and 8.9 be satisfied. Then, it holds that
∥ [ W ε , ⋆ , j ⊛ ( 2 D ε , ⋆ j − D ε , ⋆ j ) ] ∥ n l − 1 − δ 1 , 0 + ∥ B ( w ε , ⋆ ⊛ D ε , ⋆ ) ∥ n l − 1 − δ 1 , 0 + ∥ W ε , ⋆ j ⊛ D j D ε , ⋆ ∥ n l − 1 − δ 1 , 0 ≲ T − λ 1 ∥ ∥ ∥ Ξ B , ε ∥ ∥ ∥ ( ∥ φ 0 x ∥ L T x + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) . (498) \begin{aligned}
&\lVert[W_{\varepsilon,\star,j}\circledast(2D^j_{\varepsilon,\star}-D^j_{\varepsilon,\star})]\rVert_{\mathrm{nl}^{-1-\delta_1,0}}
+\lVert B(w_{\varepsilon,\star}\circledast D_{\varepsilon,\star})\rVert_{\mathrm{nl}^{-1-\delta_1,0}}\\
&\qquad+\lVert W^j_{\varepsilon,\star}\circledast D_jD_{\varepsilon,\star}\rVert_{\mathrm{nl}^{-1-\delta_1,0}}\\
&\lesssim T^{-\lambda_1}\lVert\!\lVert\!\lVert\Xi_{B,\varepsilon}\rVert\!\rVert\!\rVert
\left(\lVert\varphi_0^x\rVert_{L_T^x}+\lVert\varphi_0\rVert_{\mathcal{C}^{-\kappa}_{B(0),T}}
+T^{\frac12}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}\right). \tag*{(498)}
\end{aligned} ∥[ W ε , ⋆ , j ⊛ ( 2 D ε , ⋆ j − D ε , ⋆ j )] ∥ nl − 1 − δ 1 , 0 + ∥ B ( w ε , ⋆ ⊛ D ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 + ∥ W ε , ⋆ j ⊛ D j D ε , ⋆ ∥ nl − 1 − δ 1 , 0 ≲ T − λ 1 ∥ ∥ ∥ Ξ B , ε ∥ ∥ ∥ ( ∥ φ 0 x ∥ L T x + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) . (498) In (498) , we use the convenient notation
∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ : = ∥ R ε , ⋆ ∥ s o l 1 − δ 1 , − λ + ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ , \lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}
\mathrel{\mathop:}= \lVert R_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}+\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}, ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ : = ∥ R ε , ⋆ ∥ sol 1 − δ 1 , − λ + ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ , which allows us to shorten several expressions.
Remark 8.11. We want to use (498) to explain one way of looking at the estimates in this subsection. In (498) , we encounter a loss of the form T − λ 1 T^{-\lambda_1} T − λ 1 γ 0 \gamma_0 γ 0 γ 0 − 2 \gamma_0-2 γ 0 − 2 λ 1 \lambda_1 λ 1 (97) , λ 1 \lambda_1 λ 1 δ \delta δ δ 2 \delta_2 δ 2 λ \lambda λ δ 2 / θ \delta_2/\theta δ 2 / θ T − δ T^{-\delta} T − δ T − δ 2 T^{-\delta_2} T − δ 2 T − λ T^{-\lambda} T − λ T − δ 2 / θ T^{-\delta_2/\theta} T − δ 2 / θ
easily be bounded by T − λ 1 T^{-\lambda_1} T − λ 1 T T T 1 / 2 1/2 1/2 1 1 1
As discussed in Remark 7.3, the scaling exponent associated with the sol α , η \operatorname{sol}^{\alpha,\eta} sol α , η α + η \alpha+\eta α + η ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ \lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\operatorname{sol}^{1-\delta_1,-\lambda}} ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ 1 − δ 1 − λ ≈ 1 1-\delta_1-\lambda\approx1 1 − δ 1 − λ ≈ 1 ∥ φ 0 + ∥ L x r \lVert\varphi_0^+\rVert_{L_x^r} ∥ φ 0 + ∥ L x r ∥ φ 0 − ∥ C B ( 0 ) , T − κ \lVert\varphi_0^-\rVert_{C_{B(0),T}^{-\kappa}} ∥ φ 0 − ∥ C B ( 0 ) , T − κ − 2 r ≈ 0 -\frac{2}{r}\approx0 − r 2 ≈ 0 − κ ≈ 0 -\kappa\approx0 − κ ≈ 0 ∥ φ 0 + ∥ L x r \lVert\varphi_0^+\rVert_{L_x^r} ∥ φ 0 + ∥ L x r ∥ φ 0 − ∥ C B ( 0 ) , T − κ \lVert\varphi_0^-\rVert_{C_{B(0),T}^{-\kappa}} ∥ φ 0 − ∥ C B ( 0 ) , T − κ T T T ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ \lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\operatorname{sol}^{1-\delta_1,-\lambda}} ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ T 1 2 T^{\frac12} T 2 1
Proof. We only control the second summand in (8.28), since the first and third summands can be treated using similar arguments. The only difference is that, instead of using (8.15) as below, one has to use (8.16). Using the G \mathbf{G} G B \mathbf{B} B Lemma 3.15 ) and the product rule (Lemma 3.17 ), we first write
B ( w ε , ⋆ ⊗ D ℓ ε , ⋆ ) = D B ( w ε , ⋆ ⊗ ℓ ε , ⋆ ) − B ( D w ε , ⋆ ⊗ ℓ ε , ⋆ ) . (499) \mathbf{B}(w_{\varepsilon,\star}\otimes D\ell_{\varepsilon,\star})=D\mathbf{B}(w_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star})-\mathbf{B}(Dw_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star}). \tag*{(499)} B ( w ε , ⋆ ⊗ D ℓ ε , ⋆ ) = D B ( w ε , ⋆ ⊗ ℓ ε , ⋆ ) − B ( D w ε , ⋆ ⊗ ℓ ε , ⋆ ) . (499) Using Lemma 7.23 with
( α , β , γ , ν , η ) : = ( − κ , 2 κ , − 1 − δ 1 , 0 , − 3 κ ) , (500) (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,2\kappa,-1-\delta_1,0,-3\kappa), \tag*{(500)} ( α , β , γ , ν , η ) := ( − κ , 2 κ , − 1 − δ 1 , 0 , − 3 κ ) , (500) the first summand in (8.29) can be bounded by
∥ D B ( w ε , ⋆ ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T δ 2 − O ( κ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ w ε , ⋆ ∥ sol 2 κ , − 3 κ . \lVert D\mathbf{B}(w_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star})\rVert_{\operatorname{nl}^{-1-\delta_1,0}}
\lesssim T^{-\delta}T^{\frac{\delta}{2}-O(\kappa)}\lVert\ell_{\varepsilon,\star}\rVert_{\operatorname{msol}^{-\kappa}}\lVert w_{\varepsilon,\star}\rVert_{\operatorname{sol}^{2\kappa,-3\kappa}}. ∥ D B ( w ε , ⋆ ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T 2 δ − O ( κ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ w ε , ⋆ ∥ sol 2 κ , − 3 κ . Using Lemma 7.7 and Proposition 8.8 , we can further estimate
∥ w ε , ⋆ ∥ sol 2 κ , − 3 κ ≤ ∥ φ ⋆ ∥ sol 2 κ , − 3 κ + ∥ r ε , ⋆ ∥ sol 2 κ , − 3 κ \lVert w_{\varepsilon,\star}\rVert_{\operatorname{sol}^{2\kappa,-3\kappa}}\leq\lVert\varphi_\star\rVert_{\operatorname{sol}^{2\kappa,-3\kappa}}+\lVert r_{\varepsilon,\star}\rVert_{\operatorname{sol}^{2\kappa,-3\kappa}} ∥ w ε , ⋆ ∥ sol 2 κ , − 3 κ ≤ ∥ φ ⋆ ∥ sol 2 κ , − 3 κ + ∥ r ε , ⋆ ∥ sol 2 κ , − 3 κ ≲ T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 1 − λ − δ 1 + κ 2 ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ , \lesssim T^{-\delta}\left(\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C_{B(0),T}^{-\kappa}}\right)+T^{\frac{1-\lambda-\delta_1+\kappa}{2}}\lVert r_{\varepsilon,\star}\rVert_{\operatorname{sol}^{1-\delta_1,-\lambda}}, ≲ T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 2 1 − λ − δ 1 + κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ , which then leads to an acceptable contribution to (8.28). In order to estimate the second summand in (8.29), we further split
B ( D w ε , ⋆ ⊗ ℓ ε , ⋆ ) = B ( D φ ⋆ , ε ⊗ ℓ ε , ⋆ ) + B ( D r ε , ⋆ ⊗ ℓ ε , ⋆ ) . (501) \mathbf{B}(Dw_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star})=\mathbf{B}(D\varphi_{\star,\varepsilon}\otimes\ell_{\varepsilon,\star})+\mathbf{B}(Dr_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star}). \tag*{(501)} B ( D w ε , ⋆ ⊗ ℓ ε , ⋆ ) = B ( D φ ⋆ , ε ⊗ ℓ ε , ⋆ ) + B ( D r ε , ⋆ ⊗ ℓ ε , ⋆ ) . (501) Using Lemma 7.29 with
( α , β , γ , ν , η ) : = ( − κ , 1 − 3 κ , − 1 − δ 1 , 0 , − 1 + 2 κ ) (502) (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1-3\kappa,-1-\delta_1,0,-1+2\kappa) \tag*{(502)} ( α , β , γ , ν , η ) := ( − κ , 1 − 3 κ , − 1 − δ 1 , 0 , − 1 + 2 κ ) (502) and Proposition 8.8 , we obtain that
∥ B ( D φ ⋆ , ε ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T δ 2 − O ( κ / θ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ φ ⋆ ∥ sol 1 − 3 κ , − 1 + 2 κ \lVert\mathbf{B}(D\varphi_{\star,\varepsilon}\otimes\ell_{\varepsilon,\star})\rVert_{\operatorname{nl}^{-1-\delta_1,0}}
\lesssim T^{-\delta}T^{\frac{\delta}{2}-O(\kappa/\theta)}\lVert\ell_{\varepsilon,\star}\rVert_{\operatorname{msol}^{-\kappa}}\lVert\varphi_\star\rVert_{\operatorname{sol}^{1-3\kappa,-1+2\kappa}} ∥ B ( D φ ⋆ , ε ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T 2 δ − O ( κ / θ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ φ ⋆ ∥ sol 1 − 3 κ , − 1 + 2 κ ≲ T − δ T δ 2 − O ( κ / θ ) ∥ ℓ ε , ⋆ ∥ msol − κ × T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . \lesssim T^{-\delta}T^{\frac{\delta}{2}-O(\kappa/\theta)}\lVert\ell_{\varepsilon,\star}\rVert_{\operatorname{msol}^{-\kappa}}\times T^{-\delta}\left(\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C_{B(0),T}^{-\kappa}}\right). ≲ T − δ T 2 δ − O ( κ / θ ) ∥ ℓ ε , ⋆ ∥ msol − κ × T − δ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) . Due to (3.3), κ / θ \kappa/\theta κ / θ λ 1 \lambda_1 λ 1 Lemma 7.29 with
( α , β , γ , ν , η ) : = ( − κ , 1 − δ 1 , − 1 − δ 1 , 0 , − λ ) , (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1-\delta_1,-1-\delta_1,0,-\lambda), ( α , β , γ , ν , η ) := ( − κ , 1 − δ 1 , − 1 − δ 1 , 0 , − λ ) , we also obtain that
∥ B ( D r ε , ⋆ ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T δ 2 − O ( λ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ , \lVert\mathbf{B}(Dr_{\varepsilon,\star}\otimes\ell_{\varepsilon,\star})\rVert_{\operatorname{nl}^{-1-\delta_1,0}}
\lesssim T^{-\delta}T^{\frac{\delta}{2}-O(\lambda)}\lVert\ell_{\varepsilon,\star}\rVert_{\operatorname{msol}^{-\kappa}}\lVert r_{\varepsilon,\star}\rVert_{\operatorname{sol}^{1-\delta_1,-\lambda}}, ∥ B ( D r ε , ⋆ ⊗ ℓ ε , ⋆ ) ∥ nl − 1 − δ 1 , 0 ≲ T − δ T 2 δ − O ( λ ) ∥ ℓ ε , ⋆ ∥ msol − κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ , which is acceptable as well.
Remark 8.12. When verifying that the parameters in (500) and (502) satisfy the conditions in Lemma 7.23 and Lemma 7.29, we strongly advise the reader to keep (97) in mind. Due to (97) , most (but not all) of the parameters among κ \kappa κ δ 1 \delta_1 δ 1 δ 2 \delta_2 δ 2 θ \theta θ λ \lambda λ r r r
Remark 8.13. In the proof of Lemma 8.10, we always wrote out the T − δ T^{-\delta} T − δ O ( … ) O(\ldots) O ( … ) L s T ( F ) L_{s_T}(F) L s T ( F ) δ \delta δ O ( … ) O(\ldots) O ( … )
In the next lemma, we estimate high× \times × S S S φ \varphi φ Y ε Y_\varepsilon Y ε y ε y_\varepsilon y ε
Lemma 8.14. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ L b ( S ⋆ + Y ε , ⋆ , D ∘ I ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ + ∥ ( S ⋆ + Y ε , ⋆ ) j ∘ D j I ε , ⋆ ∥ n l − 1 + δ 2 , − λ + ∥ B ( ( φ ⋆ + y ε , ⋆ ) ∘ D I ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ + ∥ ( D j I ε , ⋆ ) ∘ ( φ ⋆ + y ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T − λ ∥ Ξ B , ε ∥ ( ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( Y ε , ⋆ , y ε , ⋆ ) ∥ s o l 1 + δ 2 , − λ ) . (503) \begin{aligned}
&\lVert\mathsf{L}b(S_{\star}+Y_{\varepsilon,\star},D\mathord{\circ}_{\!I_\varepsilon,\star})\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}+\lVert(S_{\star}+Y_{\varepsilon,\star})^{j}\mathord{\circ}_{\!D_j I_\varepsilon,\star}\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\quad+\lVert\mathsf{B}((\varphi_{\star}+y_{\varepsilon,\star})\mathord{\circ}_{\!D I_\varepsilon,\star})\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}+\lVert(D_j I_\varepsilon,\star)\mathord{\circ}(\varphi_{\star}+y_{\varepsilon,\star})\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\qquad\lesssim T^{-\lambda}\lVert\Xi_{B,\varepsilon}\rVert\left(\lVert\varphi_0\rVert_{L_x^r}+\lVert\varphi_0\rVert_{C_{B(0),T}^{-\kappa}}+T^{\frac12}\lVert(Y_{\varepsilon,\star},y_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1+\delta_2,-\lambda}}\right). \tag*{(503)}
\end{aligned} ∥ L b ( S ⋆ + Y ε , ⋆ , D ∘ I ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ + ∥( S ⋆ + Y ε , ⋆ ) j ∘ D j I ε , ⋆ ∥ nl − 1 + δ 2 , − λ + ∥ B (( φ ⋆ + y ε , ⋆ ) ∘ D I ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ + ∥( D j I ε , ⋆ ) ∘ ( φ ⋆ + y ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T − λ ∥ Ξ B , ε ∥ ( ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 2 1 ∥( Y ε , ⋆ , y ε , ⋆ ) ∥ sol 1 + δ 2 , − λ ) . (503) Proof. We only estimate the third summand in (503) , since the estimates for the remaining three terms are similar. First, we use (461) from Lemma 7.27 with
( α , β , γ , ν , η ) : = ( − κ , 1 + 2 κ , − 1 + δ 2 , − λ , − 1 − 3 κ ) (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1+2\kappa,-1+\delta_2,-\lambda,-1-3\kappa) ( α , β , γ , ν , η ) := ( − κ , 1 + 2 κ , − 1 + δ 2 , − λ , − 1 − 3 κ ) and Proposition 8.8, which imply that
∥ B ( φ ⋆ ∘ D I ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ I ε , ⋆ ∥ s o l − κ ∥ φ ⋆ ∥ s o l 1 + 2 κ , − 1 − 3 κ \lVert\mathsf{B}(\varphi_{\star}\mathord{\circ}_{\!D I_\varepsilon,\star})\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{-O(\lambda)}\lVert I_\varepsilon,\star\rVert_{\mathrm{sol}^{-\kappa}}\lVert\varphi_{\star}\rVert_{\mathrm{sol}^{1+2\kappa,-1-3\kappa}} ∥ B ( φ ⋆ ∘ D I ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ I ε , ⋆ ∥ sol − κ ∥ φ ⋆ ∥ sol 1 + 2 κ , − 1 − 3 κ ≲ T − O ( λ ) ∥ I ε , ⋆ ∥ s o l − κ ( ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − κ ) . \lesssim T^{-O(\lambda)}\lVert I_\varepsilon,\star\rVert_{\mathrm{sol}^{-\kappa}}\left(\lVert\varphi_0\rVert_{L_x^r}+\lVert\varphi_0\rVert_{C_{B(0),T}^{-\kappa}}\right). ≲ T − O ( λ ) ∥ I ε , ⋆ ∥ sol − κ ( ∥ φ 0 ∥ L x r + ∥ φ 0 ∥ C B ( 0 ) , T − κ ) . Second, we use (461) from Lemma 7.27 with
( α , β , γ , ν , η ) : = ( − κ , 1 + δ 2 , − 1 + δ 2 , − λ , − λ ) , (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1+\delta_2,-1+\delta_2,-\lambda,-\lambda), ( α , β , γ , ν , η ) := ( − κ , 1 + δ 2 , − 1 + δ 2 , − λ , − λ ) , which implies that
∥ B ( y ε , ⋆ ∘ D I ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − O ( λ ) ∥ I ε , ⋆ ∥ s o l − κ ∥ y ε , ⋆ ∥ s o l 1 + δ 2 , − λ . \lVert\mathsf{B}(y_{\varepsilon,\star}\mathord{\circ}_{\!D I_\varepsilon,\star})\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{\frac12-O(\lambda)}\lVert I_\varepsilon,\star\rVert_{\mathrm{sol}^{-\kappa}}\lVert y_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1+\delta_2,-\lambda}}. ∥ B ( y ε , ⋆ ∘ D I ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T 2 1 − O ( λ ) ∥ I ε , ⋆ ∥ sol − κ ∥ y ε , ⋆ ∥ sol 1 + δ 2 , − λ . Thus, both terms make an acceptable contribution to (503) .
Remark 8.15. The estimate of B ( φ ⋆ ∘ D I ε , ⋆ ) \mathsf{B}(\varphi_{\star}\mathord{\circ}_{\!D I_\varepsilon,\star}) B ( φ ⋆ ∘ D I ε , ⋆ ) Y ε Y_\varepsilon Y ε s o l 1 + δ 2 , − λ \mathrm{sol}^{1+\delta_2,-\lambda} sol 1 + δ 2 , − λ s o l 1 + δ 2 , 0 \mathrm{sol}^{1+\delta_2,0} sol 1 + δ 2 , 0 β ∈ ( 0 , 2 ] \beta\in(0,2] β ∈ ( 0 , 2 ] η ≤ 0 \eta\leq0 η ≤ 0 ∥ φ ⋆ ∥ s o l β , η \lVert\varphi_{\star}\rVert_{\mathrm{sol}^{\beta,\eta}} ∥ φ ⋆ ∥ sol β , η β + η ≤ − κ \beta+\eta\leq-\kappa β + η ≤ − κ ( α , γ ) = ( − κ , − 1 + δ 2 ) (\alpha,\gamma)=(-\kappa,-1+\delta_2) ( α , γ ) = ( − κ , − 1 + δ 2 ) (460) then implies that
ν < α + ( 1 − θ ) β − 1 − γ + η ≤ − κ + ( 1 − θ ) β − δ 2 − β − κ = − δ 2 − κ − κ − θ β . \nu<\alpha+(1-\theta)\beta-1-\gamma+\eta\leq-\kappa+(1-\theta)\beta-\delta_2-\beta-\kappa=-\delta_2-\kappa-\kappa-\theta\beta. ν < α + ( 1 − θ ) β − 1 − γ + η ≤ − κ + ( 1 − θ ) β − δ 2 − β − κ = − δ 2 − κ − κ − θ β . In particular, ν \nu ν (460) , one can also see that ν \nu ν 28
In the next lemma, we control the terms involving both a total derivative and high× \times ×
Lemma 8.16. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ D j [ W ε , ⋆ ⊛ ϱ ε , ⋆ , j ] ∥ n l − 1 + δ 2 , − λ + ∥ D [ W ε , ⋆ j ⊛ ϱ ε , ⋆ , j ] ∥ n l − 1 + δ 2 , − λ + ∥ D ( B ( w ε , ⋆ ⊛ ϱ ε , ⋆ ) ) ∥ n l − 1 + δ 2 , − λ + ∥ D j ( ϱ ε , ⋆ j ⊛ w ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T − λ − 1 ∥ Ξ B , ε ∥ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) . (504) \begin{aligned}
&\lVert\mathrm{D}^{j}[W_{\varepsilon,\star}\circledast\varrho_{\varepsilon,\star,j}]\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\lVert\mathrm{D}[W^{j}_{\varepsilon,\star}\circledast\varrho_{\varepsilon,\star,j}]\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\quad+\lVert\mathrm{D}(\mathrm{B}(w_{\varepsilon,\star}\circledast\varrho_{\varepsilon,\star}))\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\lVert\mathrm{D}_{j}(\varrho^{j}_{\varepsilon,\star}\circledast w_{\varepsilon,\star})\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\lesssim T^{-\lambda-1}\lVert\Xi_{\mathrm{B},\varepsilon}\rVert
\left(\lVert\varphi_{0}^{+}\rVert_{L^{x}_{r}}+\lVert\varphi_{0}^{-}\rVert_{\mathcal{C}^{-\kappa}_{\mathrm{B}(0),T}}+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right). \tag*{(504)}
\end{aligned} ∥ D j [ W ε , ⋆ ⊛ ϱ ε , ⋆ , j ] ∥ nl − 1 + δ 2 , − λ + ∥ D [ W ε , ⋆ j ⊛ ϱ ε , ⋆ , j ] ∥ nl − 1 + δ 2 , − λ + ∥ D ( B ( w ε , ⋆ ⊛ ϱ ε , ⋆ )) ∥ nl − 1 + δ 2 , − λ + ∥ D j ( ϱ ε , ⋆ j ⊛ w ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T − λ − 1 ∥ Ξ B , ε ∥ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) . (504) Proof. We only control the third summand in (8.34), since the remaining three terms can be controlled using similar arguments. Using (7.51) from Lemma 7.27 with
( α , β , γ , ν , η ) : = ( − κ , λ − κ , − 1 + δ 2 , − λ , − λ ) , (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,\lambda-\kappa,-1+\delta_{2},-\lambda,-\lambda), ( α , β , γ , ν , η ) := ( − κ , λ − κ , − 1 + δ 2 , − λ , − λ ) , we obtain that
∥ D ( B ( w ε , ⋆ ⊛ ϱ ε , ⋆ ) ) ∥ n l − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ϱ ε , ⋆ ∥ s o l − κ ∥ w ε , ⋆ ∥ s o l λ − κ , − λ . \lVert\mathrm{D}(\mathrm{B}(w_{\varepsilon,\star}\circledast\varrho_{\varepsilon,\star}))\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}
\lesssim T^{-O(\lambda)}\lVert\varrho_{\varepsilon,\star}\rVert_{\mathrm{sol}^{-\kappa}}\lVert w_{\varepsilon,\star}\rVert_{\mathrm{sol}^{\lambda-\kappa,-\lambda}}. ∥ D ( B ( w ε , ⋆ ⊛ ϱ ε , ⋆ )) ∥ nl − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ϱ ε , ⋆ ∥ sol − κ ∥ w ε , ⋆ ∥ sol λ − κ , − λ . Using Proposition 8.8 and the trivial estimate t ≤ T t\leq T t ≤ T
∥ w ε , ⋆ ∥ s o l λ − κ , − λ ≤ ∥ φ ⋆ ∥ s o l λ − κ , − λ + ∥ r ε , ⋆ ∥ s o l λ − κ , − λ \lVert w_{\varepsilon,\star}\rVert_{\mathrm{sol}^{\lambda-\kappa,-\lambda}}
\leq\lVert\varphi_{\star}\rVert_{\mathrm{sol}^{\lambda-\kappa,-\lambda}}+\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{\lambda-\kappa,-\lambda}} ∥ w ε , ⋆ ∥ sol λ − κ , − λ ≤ ∥ φ ⋆ ∥ sol λ − κ , − λ + ∥ r ε , ⋆ ∥ sol λ − κ , − λ ≲ T − δ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 1 − λ − δ 1 + κ 2 ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ . \lesssim T^{-\delta}\left(\lVert\varphi_{0}^{+}\rVert_{L^{x}_{r}}+\lVert\varphi_{0}^{-}\rVert_{\mathcal{C}^{-\kappa}_{\mathrm{B}(0),T}}\right)+T^{\frac{1-\lambda-\delta_{1}+\kappa}{2}}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}. ≲ T − δ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 2 1 − λ − δ 1 + κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ . After combining both estimates, this finishes the proof.
In the next lemma, we control terms involving D ϱ ε j \mathrm{D}\varrho_{\varepsilon}^{j} D ϱ ε j D ϱ ε j \mathrm{D}\varrho_{\varepsilon}^{j} D ϱ ε j
Lemma 8.17. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ [ W ε , ⋆ , j ⊛ ( 2 D ϱ ε , ⋆ j − D ϱ ε , ⋆ j ) ] ∥ n l − 1 + δ 2 , − λ + ∥ W ε , ⋆ j ⊛ D j ϱ ε , ⋆ ∥ n l − 1 + δ 2 , − λ + ∥ B ( w ε , ⋆ ⊛ D ϱ ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T − λ − 1 ∥ Ξ B , ε ∥ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) . (505) \begin{aligned}
&\lVert[W_{\varepsilon,\star,j}\circledast(2\mathrm{D}\varrho^{j}_{\varepsilon,\star}-\mathrm{D}\varrho^{j}_{\varepsilon,\star})]\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\lVert W^{j}_{\varepsilon,\star}\circledast\mathrm{D}_{j}\varrho_{\varepsilon,\star}\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\quad+\lVert\mathrm{B}(w_{\varepsilon,\star}\circledast\mathrm{D}\varrho_{\varepsilon,\star})\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\lesssim T^{-\lambda-1}\lVert\Xi_{\mathrm{B},\varepsilon}\rVert
\left(\lVert\varphi_{0}^{+}\rVert_{L^{x}_{r}}+\lVert\varphi_{0}^{-}\rVert_{\mathcal{C}^{-\kappa}_{\mathrm{B}(0),T}}+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right). \tag*{(505)}
\end{aligned} ∥[ W ε , ⋆ , j ⊛ ( 2 D ϱ ε , ⋆ j − D ϱ ε , ⋆ j )] ∥ nl − 1 + δ 2 , − λ + ∥ W ε , ⋆ j ⊛ D j ϱ ε , ⋆ ∥ nl − 1 + δ 2 , − λ + ∥ B ( w ε , ⋆ ⊛ D ϱ ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T − λ − 1 ∥ Ξ B , ε ∥ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) . (505) Proof. We only estimate the third summand in (8.35), since the remaining two terms can be treated using similar arguments. We use (7.43) from Lemma 7.25 with
( α , β , γ , ν , η ) : = ( − κ , 1 − δ 1 , − 1 + δ 2 , − λ , − 1 + δ 1 − κ ) , (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1-\delta_{1},-1+\delta_{2},-\lambda,-1+\delta_{1}-\kappa), ( α , β , γ , ν , η ) := ( − κ , 1 − δ 1 , − 1 + δ 2 , − λ , − 1 + δ 1 − κ ) , which implies that
∥ B ( w ε , ⋆ ⊛ D ϱ ε , ⋆ ) ∥ n l − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ϱ ε , ⋆ ∥ s o l − κ ∥ w ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ . \lVert\mathrm{B}(w_{\varepsilon,\star}\circledast\mathrm{D}\varrho_{\varepsilon,\star})\rVert_{\mathrm{nl}-1+\delta_{2},-\lambda}
\lesssim T^{-O(\lambda)}\lVert\varrho_{\varepsilon,\star}\rVert_{\mathrm{sol}^{-\kappa}}\lVert w_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}. ∥ B ( w ε , ⋆ ⊛ D ϱ ε , ⋆ ) ∥ nl − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ϱ ε , ⋆ ∥ sol − κ ∥ w ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ . Using Proposition 8.8, we also have that
∥ w ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ≤ ∥ φ ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ + ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ \lVert w_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}
\leq\lVert\varphi_{\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}+\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}} ∥ w ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≤ ∥ φ ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ + ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≲ T − δ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 1 − λ − δ 1 + κ 2 ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ . \lesssim T^{-\delta}\left(\lVert\varphi_{0}^{+}\rVert_{L^{x}_{r}}+\lVert\varphi_{0}^{-}\rVert_{\mathcal{C}^{-\kappa}_{\mathrm{B}(0),T}}\right)+T^{\frac{1-\lambda-\delta_{1}+\kappa}{2}}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}. ≲ T − δ ( ∥ φ 0 + ∥ L r x + ∥ φ 0 − ∥ C B ( 0 ) , T − κ ) + T 2 1 − λ − δ 1 + κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ . After combining both estimates, this completes the proof.
In the next lemma, we control terms involving our covariant linear stochastic objects, covariant derivatives of W ε W_\varepsilon W ε w ε w_\varepsilon w ε ⊗ \otimes ⊗
Lemma 8.18. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ ∥ E ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j ) ∥ ∥ n l − 1 + δ 2 , − λ + ∥ ∥ B ( E ε , ⋆ ⊗ D W ε , ⋆ ) ∥ ∥ n l − 1 + δ 2 , − λ + ∥ ∥ E ε , ⋆ j ⊗ D j w ε , ⋆ ∥ ∥ n l − 1 + δ 2 , − λ ≲ T − λ 1 ∥ ∣ E B , ε ∥ ∣ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ s o l 1 − δ 1 , − λ ) . (506) \begin{aligned}
&\left\Vert\left\Vert\mathcal{E}_{\varepsilon,\star,j}\otimes\left(2D^{j}W_{\varepsilon,\star}-DW^{j}_{\varepsilon,\star}\right)\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\left\Vert\left\Vert B\left(\mathcal{E}_{\varepsilon,\star}\otimes DW_{\varepsilon,\star}\right)\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\qquad+\left\Vert\left\Vert\mathcal{E}^{j}_{\varepsilon,\star}\otimes D_{j}w_{\varepsilon,\star}\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\lesssim T^{-\lambda_{1}}\|\!|\mathcal{E}_{B,\varepsilon}\|\!|\left(\|\varphi_{0}\|_{L^{r}}+\|\varphi_{0}\|_{C_{B(0),T}^{-\kappa}}+T^{\frac12}\|\!(R_{\varepsilon,\star},r_{\varepsilon,\star})\|\!|_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right). \tag*{(506)}
\end{aligned} E ε , ⋆ , j ⊗ ( 2 D j W ε , ⋆ − D W ε , ⋆ j ) nl − 1 + δ 2 , − λ + ∥ ∥ B ( E ε , ⋆ ⊗ D W ε , ⋆ ) ∥ ∥ nl − 1 + δ 2 , − λ + E ε , ⋆ j ⊗ D j w ε , ⋆ nl − 1 + δ 2 , − λ ≲ T − λ 1 ∥ ∣ E B , ε ∥ ∣ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 2 1 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ sol 1 − δ 1 , − λ ) . (506) Proof. We only control the second summand in (506) , since the remaining two terms can be treated using similar arguments. To this end, we first decompose
B ( E ε , ⋆ ⊗ D w ε , ⋆ ) = B ( E ε , ⋆ ⊗ D w ε , ⋆ ) + B ( E ε , ⋆ ⊗ D w ε , ⋆ ) . B\left(\mathcal{E}_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}\right)
=B\left(\mathcal{E}_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}\right)
+B\left(\mathcal{E}_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}\right). B ( E ε , ⋆ ⊗ D w ε , ⋆ ) = B ( E ε , ⋆ ⊗ D w ε , ⋆ ) + B ( E ε , ⋆ ⊗ D w ε , ⋆ ) . In addition, we choose the parameters
( α , β , γ , ν , η ) : = ( − κ , 1 − δ 1 , − 1 + δ 2 , − λ , − 1 + δ 1 − κ ) . (\alpha,\beta,\gamma,\nu,\eta):=(-\kappa,1-\delta_{1},-1+\delta_{2},-\lambda,-1+\delta_{1}-\kappa). ( α , β , γ , ν , η ) := ( − κ , 1 − δ 1 , − 1 + δ 2 , − λ , − 1 + δ 1 − κ ) . Using (457) from Lemma 7.25 and Lemma 7.29, we then obtain that
∥ ∥ B ( E ε , ⋆ ⊗ D w ε , ⋆ ) ∥ ∥ n l − 1 + δ 2 , − λ + ∥ ∥ B ( E ε , ⋆ ⊗ D w ε , ⋆ ) ∥ ∥ n l − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ∣ E ε , ⋆ ∥ ∣ m s o l − κ ∥ w ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ . \begin{aligned}
&\left\Vert\left\Vert B\left(\mathcal{E}_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}\right)\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\left\Vert\left\Vert B\left(\mathcal{E}_{\varepsilon,\star}\otimes Dw_{\varepsilon,\star}\right)\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\qquad\lesssim T^{-O(\lambda)}\|\!|\mathcal{E}_{\varepsilon,\star}\|\!|_{\mathrm{msol}^{-\kappa}}\|w_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}.
\end{aligned} ∥ ∥ B ( E ε , ⋆ ⊗ D w ε , ⋆ ) ∥ ∥ nl − 1 + δ 2 , − λ + ∥ ∥ B ( E ε , ⋆ ⊗ D w ε , ⋆ ) ∥ ∥ nl − 1 + δ 2 , − λ ≲ T − O ( λ ) ∥ ∣ E ε , ⋆ ∥ ∣ msol − κ ∥ w ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ . Using Proposition 8.8, we have that
∥ w ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ≤ ∥ φ ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ + ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ \|w_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}
\leq\|\varphi_{\star}\|_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}
+\|r_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}} ∥ w ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≤ ∥ φ ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ + ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≲ T − δ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ ) + T 1 − λ − δ 1 + κ 2 ∥ ∣ r ε , ⋆ ∥ ∣ s o l 1 − δ 1 , − λ . \lesssim T^{-\delta}\left(\|\varphi_{0}\|_{L^{r}}+\|\varphi_{0}\|_{C_{B(0),T}^{-\kappa}}\right)
+T^{\frac{1-\lambda-\delta_{1}+\kappa}{2}}\|\!|r_{\varepsilon,\star}\|\!|_{\mathrm{sol}^{1-\delta_{1},-\lambda}}. ≲ T − δ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ ) + T 2 1 − λ − δ 1 + κ ∥ ∣ r ε , ⋆ ∥ ∣ sol 1 − δ 1 , − λ . By combining both estimates, we arrive at the desired estimate of the second term in (506) .
In the next lemma, we control the parts of the derivative nonlinearities containing no covariant linear stochastic objects, i.e., the terms in (6.79) and (6.87). While these terms have better regularity properties than the terms from Lemmas 8.10–8.18 above, we need to be mindful of their size.
Lemma 8.19. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ ∥ [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] ∥ ∥ n l − 1 + δ 2 , − λ + ∥ ∥ B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) ∥ ∥ n l − 1 + δ 2 , − λ + ∥ ∥ W ε j D j w ε − S j D j φ ∥ ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − λ 1 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ s o l 1 − δ 1 , − λ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ s o l 1 − δ 1 , − λ ) . (507) \begin{aligned}
&\left\Vert\left\Vert[W_{\varepsilon,j},2D^{j}W_{\varepsilon}-DW^{j}_{\varepsilon}]-[S_{j},2D^{j}S-DS^{j}]\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\quad+\left\Vert\left\Vert B(w_{\varepsilon}\otimes Dw_{\varepsilon})-B(\varphi\otimes D\varphi)\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}
+\left\Vert\left\Vert W^{j}_{\varepsilon}D_{j}w_{\varepsilon}-S^{j}D_{j}\varphi\right\Vert\right\Vert_{\mathrm{nl}-1+\delta_{2},-\lambda}\\
&\lesssim T^{\frac12-\lambda_{1}}\|\!(R_{\varepsilon,\star},r_{\varepsilon,\star})\|\!|_{\mathrm{sol}^{1-\delta_{1},-\lambda}}
\left(\|\varphi_{0}\|_{L^{r}}+\|\varphi_{0}\|_{C_{B(0),T}^{-\kappa}}+T^{\frac12}\|\!(R_{\varepsilon,\star},r_{\varepsilon,\star})\|\!|_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right). \tag*{(507)}
\end{aligned} [ W ε , j , 2 D j W ε − D W ε j ] − [ S j , 2 D j S − D S j ] nl − 1 + δ 2 , − λ + ∥ ∥ B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) ∥ ∥ nl − 1 + δ 2 , − λ + W ε j D j w ε − S j D j φ nl − 1 + δ 2 , − λ ≲ T 2 1 − λ 1 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ sol 1 − δ 1 , − λ ( ∥ φ 0 ∥ L r + ∥ φ 0 ∥ C B ( 0 ) , T − κ + T 2 1 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ ∣ sol 1 − δ 1 , − λ ) . (507) Proof. We only control the second summand in (507) , since the first and third summands can be treated using similar arguments. To this end, we decompose
B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) = B ( r ε ⊗ D φ ) + B ( φ ⊗ D r ε ) + B ( r ε ⊗ D r ε ) . (508) \begin{aligned}
B(w_{\varepsilon}\otimes Dw_{\varepsilon})-B(\varphi\otimes D\varphi)
&=B(r_{\varepsilon}\otimes D\varphi)+B(\varphi\otimes Dr_{\varepsilon})\\
&\quad+B(r_{\varepsilon}\otimes Dr_{\varepsilon}). \tag*{(508)}
\end{aligned} B ( w ε ⊗ D w ε ) − B ( φ ⊗ D φ ) = B ( r ε ⊗ D φ ) + B ( φ ⊗ D r ε ) + B ( r ε ⊗ D r ε ) . (508) We now estimate all three terms in (508) using Lemma 7.22. We will always choose ( γ , ν ) = ( − 1 + δ 2 , − λ ) (\gamma,\nu)=(-1+\delta_2,-\lambda) ( γ , ν ) = ( − 1 + δ 2 , − λ ) ( α , β , η 1 , η 2 ) (\alpha,\beta,\eta_1,\eta_2) ( α , β , η 1 , η 2 )
( α , β , η 1 , η 2 ) : = ( 1 − δ 1 − λ , 1 − δ 1 , 0 , − 1 + δ 1 − κ ) . (\alpha,\beta,\eta_1,\eta_2):=(1-\delta_1-\lambda,1-\delta_1,0,-1+\delta_1-\kappa). ( α , β , η 1 , η 2 ) := ( 1 − δ 1 − λ , 1 − δ 1 , 0 , − 1 + δ 1 − κ ) . Together with Lemma 7.7 and Proposition 8.8, we then obtain that
∥ B ( r ε ⊗ D φ ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − O ( λ ) ∥ r ε , ⋆ ∥ s o l 1 − δ 1 − λ , 0 ∥ φ ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ≲ T 1 2 − O ( λ ) ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) . \begin{aligned}
\lVert B(r_\varepsilon\otimes D\varphi)\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}} &\lesssim T^{\frac{1}{2}-O(\lambda)}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1-\lambda,0}}\lVert\varphi_\star\rVert_{\mathrm{sol}^{1-\delta_1,-1+\delta_1-\kappa}} \\
&\lesssim T^{\frac{1}{2}-O(\lambda)}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}\left(\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C^{-k}_{B(0),T}}\right).
\end{aligned} ∥ B ( r ε ⊗ D φ ) ∥ nl − 1 + δ 2 , − λ ≲ T 2 1 − O ( λ ) ∥ r ε , ⋆ ∥ sol 1 − δ 1 − λ , 0 ∥ φ ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≲ T 2 1 − O ( λ ) ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) . For the second term, we choose
( α , β , η 1 , η 2 ) : = ( δ 1 + κ , 1 − δ 1 , − δ 1 − 2 κ , − λ ) . (\alpha,\beta,\eta_1,\eta_2):=(\delta_1+\kappa,1-\delta_1,-\delta_1-2\kappa,-\lambda). ( α , β , η 1 , η 2 ) := ( δ 1 + κ , 1 − δ 1 , − δ 1 − 2 κ , − λ ) . Together with Proposition 8.8, we obtain
∥ B ( φ ⊗ D r ε ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − O ( λ ) ∥ φ ⋆ ∥ s o l δ 1 + κ , − δ 1 − 2 κ ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ ≲ T 1 2 − O ( λ ) ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ . \begin{aligned}
\lVert B(\varphi\otimes Dr_\varepsilon)\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}} &\lesssim T^{\frac{1}{2}-O(\lambda)}\lVert\varphi_\star\rVert_{\mathrm{sol}^{\delta_1+\kappa,-\delta_1-2\kappa}}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}} \\
&\lesssim T^{\frac{1}{2}-O(\lambda)}\left(\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C^{-k}_{B(0),T}}\right)\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}.
\end{aligned} ∥ B ( φ ⊗ D r ε ) ∥ nl − 1 + δ 2 , − λ ≲ T 2 1 − O ( λ ) ∥ φ ⋆ ∥ sol δ 1 + κ , − δ 1 − 2 κ ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ ≲ T 2 1 − O ( λ ) ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ . Finally, for the third term, we choose
( α , β , η 1 , η 2 ) : = ( 1 − δ 1 , 1 − δ 1 , − λ , − λ ) . (\alpha,\beta,\eta_1,\eta_2):=(1-\delta_1,1-\delta_1,-\lambda,-\lambda). ( α , β , η 1 , η 2 ) := ( 1 − δ 1 , 1 − δ 1 , − λ , − λ ) . We then obtain that
∥ B ( r ε ⊗ D r ε ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 − O ( λ ) ∥ r ε , ⋆ ∥ s o l 1 − δ 1 , − λ 2 . \lVert B(r_\varepsilon\otimes Dr_\varepsilon)\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\lesssim T^{1-O(\lambda)}\lVert r_{\varepsilon,\star}\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}^2. ∥ B ( r ε ⊗ D r ε ) ∥ nl − 1 + δ 2 , − λ ≲ T 1 − O ( λ ) ∥ r ε , ⋆ ∥ sol 1 − δ 1 , − λ 2 . By combining our estimates for the first, second, and third terms in (508) , we complete the proof. □
In the next lemma, we estimate (6.80), (6.81), (6.88), and (6.89), i.e., the power-type nonlinearity.
Lemma 8.21. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ [ φ ε , j + W ε , j , [ φ ε j + W ε j , φ ε + W ε ] ] − [ S j , [ S j , S ] ] − σ ρ , ε 2 Cov g g ( φ ε + W ε ) ∥ n l − 1 + δ 2 , − λ + ∥ B ( ( φ ε + W ε ) ⊗ ( φ ε + W ε ) ( φ ε + w ε ) ) − B ( φ ⊗ S φ ) − σ ρ , ε 2 Cov g φ ( φ ε + W ε ) ∥ n l − 1 + δ 2 , − λ + ∥ ( φ ε + W ε ) j ( φ ε + W ε ) j ( φ ε + w ε ) − S j S j φ − 2 σ ρ , ε 2 Cov g V g ( φ ε + w ε ) ∥ n l − 1 + δ 2 , − λ + ∥ − ∣ φ ε + w ε ∣ 2 ( φ ε + w ε ) + ( dim ( V ) + 2 ) σ ρ , ε 2 ( φ ε + w ε ) + ∣ φ ∣ 2 φ ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − λ 1 ( ∥ Ξ B , ε ∥ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) × ( ∥ Ξ B , ε ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) 2 . (509) \begin{aligned}
&\big\lVert[\varphi_{\varepsilon,j}+W_{\varepsilon,j},[\varphi_\varepsilon^j+W_\varepsilon^j,\varphi_\varepsilon+W_\varepsilon]]-[S_j,[S^j,S]]-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{g}^{g}(\varphi_\varepsilon+W_\varepsilon)\big\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\quad+\big\lVert B((\varphi_\varepsilon+W_\varepsilon)\otimes(\varphi_\varepsilon+W_\varepsilon)(\varphi_\varepsilon+w_\varepsilon))-B(\varphi\otimes S\varphi)-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{g}^{\varphi}(\varphi_\varepsilon+W_\varepsilon)\big\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\quad+\big\lVert(\varphi_\varepsilon+W_\varepsilon)^j(\varphi_\varepsilon+W_\varepsilon)_j(\varphi_\varepsilon+w_\varepsilon)-S^jS_j\varphi-2\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{g}^{Vg}(\varphi_\varepsilon+w_\varepsilon)\big\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\quad+\big\lVert-|\varphi_\varepsilon+w_\varepsilon|^2(\varphi_\varepsilon+w_\varepsilon)+(\dim(V)+2)\sigma_{\rho,\varepsilon}^2(\varphi_\varepsilon+w_\varepsilon)+|\varphi|^2\varphi\big\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\\
&\lesssim T^{\frac{1}{2}-\lambda_1}\left(\lVert\Xi_{B,\varepsilon}\rVert+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}\right)\\
&\qquad\times\left(\lVert\Xi_{B,\varepsilon}\rVert+\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C^{-k}_{B(0),T}}+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_1,-\lambda}}\right)^2. \tag*{(509)}
\end{aligned} [ φ ε , j + W ε , j , [ φ ε j + W ε j , φ ε + W ε ]] − [ S j , [ S j , S ]] − σ ρ , ε 2 Cov g g ( φ ε + W ε ) nl − 1 + δ 2 , − λ + B (( φ ε + W ε ) ⊗ ( φ ε + W ε ) ( φ ε + w ε )) − B ( φ ⊗ S φ ) − σ ρ , ε 2 Cov g φ ( φ ε + W ε ) nl − 1 + δ 2 , − λ + ( φ ε + W ε ) j ( φ ε + W ε ) j ( φ ε + w ε ) − S j S j φ − 2 σ ρ , ε 2 Cov g V g ( φ ε + w ε ) nl − 1 + δ 2 , − λ + − ∣ φ ε + w ε ∣ 2 ( φ ε + w ε ) + ( dim ( V ) + 2 ) σ ρ , ε 2 ( φ ε + w ε ) + ∣ φ ∣ 2 φ nl − 1 + δ 2 , − λ ≲ T 2 1 − λ 1 ( ∥ Ξ B , ε ∥ + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) × ( ∥ Ξ B , ε ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) 2 . (509) Remark 8.22. As in Lemma 8.19 and Remark 8.20, it is important that the right-hand side of (509) contains no term of the form T 1 2 − λ 1 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) 3 T^{\frac{1}{2}-\lambda_1}(\lVert\varphi_0^+\rVert_{L_x^r}+\lVert\varphi_0^-\rVert_{C^{-k}_{B(0),T}})^3 T 2 1 − λ 1 (∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − k ) 3 [ S j , [ S j , S ] ] [S_j,[S^j,S]] [ S j , [ S j , S ]] B ( φ ⊗ S φ ) B(\varphi\otimes S\varphi) B ( φ ⊗ S φ ) S j S j φ S^jS_j\varphi S j S j φ − ∣ φ ∣ 2 φ -|\varphi|^2\varphi − ∣ φ ∣ 2 φ (509) .
Proof. We only control the first summand in (509) , since the remaining three summands can be controlled using similar arguments. To briefly simplify the notation, set
Pow ≔ [ φ ε , j + W ε , j , [ φ ε j + W ε j , φ ε + W ε ] ] − [ S j , [ S j , S ] ] − σ ρ , ε 2 Cov g g ( φ ε + W ε ) . \operatorname{Pow}\coloneqq[\varphi_{\varepsilon,j}+W_{\varepsilon,j},[\varphi_\varepsilon^j+W_\varepsilon^j,\varphi_\varepsilon+W_\varepsilon]]-[S_j,[S^j,S]]-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{g}^{g}(\varphi_\varepsilon+W_\varepsilon). Pow : = [ φ ε , j + W ε , j , [ φ ε j + W ε j , φ ε + W ε ]] − [ S j , [ S j , S ]] − σ ρ , ε 2 Cov g g ( φ ε + W ε ) . We now recall our weighted Schauder estimate (437) , which, together with Lemma 7.7 and Definition 7.8 , implies that
∥ Pow ∥ n l − 1 + δ 2 , − λ ≲ ∥ Pow ∥ n l − 1 + δ 2 + 2 λ , − 3 λ ≲ T 1 2 − O ( λ ) ∥ t 3 κ 2 Pow ∥ C t − 3 κ 2 L x r . \|\operatorname{Pow}\|_{\mathrm{nl}-1+\delta_2,-\lambda}\lesssim\|\operatorname{Pow}\|_{\mathrm{nl}-1+\delta_2+2\lambda,-3\lambda}\lesssim T^{\frac12-O(\lambda)}\left\|t^{\frac{3\kappa}{2}}\operatorname{Pow}\right\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}. ∥ Pow ∥ nl − 1 + δ 2 , − λ ≲ ∥ Pow ∥ nl − 1 + δ 2 + 2 λ , − 3 λ ≲ T 2 1 − O ( λ ) t 2 3 κ Pow C t − 2 3 κ L x r . We also note that, due to Proposition 8.8 and Lemma 7.7 ,
∥ S ⋆ ∥ s o l 5 κ , − λ ≲ T − 3 δ T 1 − 5 κ + λ 2 − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≲ T − O ( λ ) ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) , \|S_\star\|_{\mathrm{sol}^{5\kappa},-\lambda}\lesssim T^{-3\delta}T^{\frac{1-5\kappa+\lambda}{2}-\kappa}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right)^2\lesssim T^{-O(\lambda)}\left(\|\phi_0^+\|_{L_x^r}+\|\phi_0^-\|_{\mathcal{C}_{B(0),T}^{-\kappa}}\right), ∥ S ⋆ ∥ sol 5 κ , − λ ≲ T − 3 δ T 2 1 − 5 κ + λ − κ ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) 2 ≲ T − O ( λ ) ( ∥ ϕ 0 + ∥ L x r + ∥ ϕ 0 − ∥ C B ( 0 ) , T − κ ) , ∥ R ε , ⋆ ∥ s o l 5 κ , − λ ≲ T 1 − 5 κ − δ 1 2 ∥ R ε , ⋆ ∥ s o l 1 − δ 1 , − λ ≲ T 1 2 − O ( λ ) ∥ R ε , ⋆ ∥ s o l 1 − δ 1 , − λ . \|R_{\varepsilon,\star}\|_{\mathrm{sol}^{5\kappa},-\lambda}\lesssim T^{\frac{1-5\kappa-\delta_1}{2}}\|R_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_1},-\lambda}\lesssim T^{\frac12-O(\lambda)}\|R_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_1},-\lambda}. ∥ R ε , ⋆ ∥ sol 5 κ , − λ ≲ T 2 1 − 5 κ − δ 1 ∥ R ε , ⋆ ∥ sol 1 − δ 1 , − λ ≲ T 2 1 − O ( λ ) ∥ R ε , ⋆ ∥ sol 1 − δ 1 , − λ . By using the above three estimates, as well as the identity W ε = S + R ε W_\varepsilon=S+R_\varepsilon W ε = S + R ε (509) to the estimate
∥ t 3 κ 2 ( [ ε ˚ ε , j + W ε , j , [ ε ˚ j + W ε j , ε ˚ + W ε ] ] − [ S j , [ S j , S ] ] − σ ρ , ε 2 Cov g g ( ε ˚ + W ε ) ) ∥ C t − 3 κ 2 L x r \left\|t^{\frac{3\kappa}{2}}\left([\mathring{\varepsilon}_{\varepsilon,j}+W_{\varepsilon,j},[\mathring{\varepsilon}^{\,j}+W^j_{\varepsilon},\mathring{\varepsilon}+W_\varepsilon]]-[S_j,[S^j,S]]-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}+W_\varepsilon)\right)\right\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r} t 2 3 κ ( [ ε ˚ ε , j + W ε , j , [ ε ˚ j + W ε j , ε ˚ + W ε ]] − [ S j , [ S j , S ]] − σ ρ , ε 2 Cov g g ( ε ˚ + W ε ) ) C t − 2 3 κ L x r ≲ ∥ Ξ B , ε ∥ ( ∥ Ξ B , ε ∥ + ∥ W ε , ⋆ ∥ s o l 5 κ , − λ ) 2 + ∥ R ε , ⋆ ∥ s o l 5 κ , − λ ( ∥ S ⋆ ∥ s o l 5 κ , − λ + ∥ R ε , ⋆ ∥ s o l 5 κ , − λ ) 2 . (510) \lesssim\|\Xi_{B,\varepsilon}\|\left(\|\Xi_{B,\varepsilon}\|+\|W_{\varepsilon,\star}\|_{\mathrm{sol}^{5\kappa},-\lambda}\right)^2+\|R_{\varepsilon,\star}\|_{\mathrm{sol}^{5\kappa},-\lambda}\left(\|S_\star\|_{\mathrm{sol}^{5\kappa},-\lambda}+\|R_{\varepsilon,\star}\|_{\mathrm{sol}^{5\kappa},-\lambda}\right)^2. \tag*{(510)} ≲ ∥ Ξ B , ε ∥ ( ∥ Ξ B , ε ∥ + ∥ W ε , ⋆ ∥ sol 5 κ , − λ ) 2 + ∥ R ε , ⋆ ∥ sol 5 κ , − λ ( ∥ S ⋆ ∥ sol 5 κ , − λ + ∥ R ε , ⋆ ∥ sol 5 κ , − λ ) 2 . (510) In order to prove (510) , we decompose the argument into
[ ε ˚ ε , j + W ε , j , [ ε ˚ j + W ε j , ε ˚ k + W ε k ] ] − σ ρ , ε 2 Cov g g ( ε ˚ k + W ε k ) − [ S j , [ S j , S k ] ] = [ ε ˚ ε , j , [ ε ˚ j , ε ˚ k ] ] − σ ρ , ε 2 Cov g g ( ε ˚ k ) (511) \begin{aligned}
[\mathring{\varepsilon}_{\varepsilon,j}+W_{\varepsilon,j},[\mathring{\varepsilon}^{\,j}+W^j_{\varepsilon},\mathring{\varepsilon}^{\,k}+W^k_{\varepsilon}]]-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}^{\,k}+W^k_{\varepsilon})-[S_j,[S^j,S^k]] \\
&=[\mathring{\varepsilon}_{\varepsilon,j},[\mathring{\varepsilon}^{\,j},\mathring{\varepsilon}^{\,k}]]-\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}^{\,k}) \tag*{(511)}
\end{aligned} [ ε ˚ ε , j + W ε , j , [ ε ˚ j + W ε j , ε ˚ k + W ε k ]] − σ ρ , ε 2 Cov g g ( ε ˚ k + W ε k ) − [ S j , [ S j , S k ]] = [ ε ˚ ε , j , [ ε ˚ j , ε ˚ k ]] − σ ρ , ε 2 Cov g g ( ε ˚ k ) (511) + [ W ε , j , [ ε ˚ j , ε ˚ k ] ] (512) +[W_{\varepsilon,j},[\mathring{\varepsilon}^{\,j},\mathring{\varepsilon}^{\,k}]] \tag*{(512)} + [ W ε , j , [ ε ˚ j , ε ˚ k ]] (512) + [ ε ˚ ε , j , [ W ε j , ε ˚ k ] ] + σ ρ , ε 2 Cov g g ( W ε k ) (513) +[\mathring{\varepsilon}_{\varepsilon,j},[W^j_{\varepsilon},\mathring{\varepsilon}^{\,k}]]+\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(W^k_{\varepsilon}) \tag*{(513)} + [ ε ˚ ε , j , [ W ε j , ε ˚ k ]] + σ ρ , ε 2 Cov g g ( W ε k ) (513) + [ ε ˚ ε , j , [ ε ˚ j , W ε k ] ] − 2 σ ρ , ε 2 Cov g g ( W ε k ) (514) +[\mathring{\varepsilon}_{\varepsilon,j},[\mathring{\varepsilon}^{\,j},W^k_{\varepsilon}]]-2\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(W^k_{\varepsilon}) \tag*{(514)} + [ ε ˚ ε , j , [ ε ˚ j , W ε k ]] − 2 σ ρ , ε 2 Cov g g ( W ε k ) (514) + [ ε ˚ ε , j , [ W ε j , W ε k ] ] + [ W ε , j , [ ε ˚ j , W ε k ] ] + [ W ε , j , [ W ε j , ε ˚ k ] ] (515) +[\mathring{\varepsilon}_{\varepsilon,j},[W^j_{\varepsilon},W^k_{\varepsilon}]]+[W_{\varepsilon,j},[\mathring{\varepsilon}^{\,j},W^k_{\varepsilon}]]+[W_{\varepsilon,j},[W^j_{\varepsilon},\mathring{\varepsilon}^{\,k}]] \tag*{(515)} + [ ε ˚ ε , j , [ W ε j , W ε k ]] + [ W ε , j , [ ε ˚ j , W ε k ]] + [ W ε , j , [ W ε j , ε ˚ k ]] (515) + [ W ε , j , [ W ε j , W ε k ] ] − [ S j , [ S j , S k ] ] . (516) +[W_{\varepsilon,j},[W^j_{\varepsilon},W^k_{\varepsilon}]]-[S_j,[S^j,S^k]]. \tag*{(516)} + [ W ε , j , [ W ε j , W ε k ]] − [ S j , [ S j , S k ]] . (516) To estimate the first term (511) , we observe that the argument can be written as the image of
ε ˚ ε , j ⊗ ε ˚ j ⊗ ε ˚ k − δ j j σ ρ , ε 2 ( u a ⊗ u a ⊗ ε ˚ k ) − δ k k σ ρ , ε 2 ( u a ⊗ ε ˚ j ⊗ u a ) − δ j k σ ρ , ε 2 ( ε ˚ ε , j ⊗ u a ⊗ u a ) , \mathring{\varepsilon}_{\varepsilon,j}\otimes\mathring{\varepsilon}^{\,j}\otimes\mathring{\varepsilon}^{\,k}
-\delta_j^j\sigma_{\rho,\varepsilon}^2(u_a\otimes u^a\otimes\mathring{\varepsilon}^{\,k})
-\delta_k^k\sigma_{\rho,\varepsilon}^2(u_a\otimes\mathring{\varepsilon}^{\,j}\otimes u^a)
-\delta_j^k\sigma_{\rho,\varepsilon}^2(\mathring{\varepsilon}_{\varepsilon,j}\otimes u_a\otimes u^a), ε ˚ ε , j ⊗ ε ˚ j ⊗ ε ˚ k − δ j j σ ρ , ε 2 ( u a ⊗ u a ⊗ ε ˚ k ) − δ k k σ ρ , ε 2 ( u a ⊗ ε ˚ j ⊗ u a ) − δ j k σ ρ , ε 2 ( ε ˚ ε , j ⊗ u a ⊗ u a ) , where ( u a ) a (u_a)_a ( u a ) a
L ( X ⊗ Y ⊗ Z ) : = [ X , [ Y , Z ] ] . L(X\otimes Y\otimes Z)\mathrel{\mathop:}= [X,[Y,Z]]. L ( X ⊗ Y ⊗ Z ) : = [ X , [ Y , Z ]] . Indeed, in order to see that the resulting counterterm is as in (511) , we use that
L ( δ j j ( u a ⊗ u a ⊗ ε ˚ k ) + δ k j ( u a ⊗ ε ˚ j ⊗ u a ) + δ k j ( ε ˚ ε , j ⊗ u a ⊗ u a ) ) = 2 Cov g g ( ε ˚ k ) − Cov g g ( ε ˚ k ) + 0 = Cov g g ( ε ˚ k ) . \begin{aligned}
L\left(\delta_j^j(u_a\otimes u^a\otimes\mathring{\varepsilon}^{\,k})+\delta_k^j(u_a\otimes\mathring{\varepsilon}^{\,j}\otimes u^a)+\delta^{kj}(\mathring{\varepsilon}_{\varepsilon,j}\otimes u_a\otimes u^a)\right) \\
&=2\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}^{\,k})-\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}^{\,k})+0=\operatorname{Cov}_{\mathfrak g}^{\mathfrak g}(\mathring{\varepsilon}^{\,k}).
\end{aligned} L ( δ j j ( u a ⊗ u a ⊗ ε ˚ k ) + δ k j ( u a ⊗ ε ˚ j ⊗ u a ) + δ k j ( ε ˚ ε , j ⊗ u a ⊗ u a ) ) = 2 Cov g g ( ε ˚ k ) − Cov g g ( ε ˚ k ) + 0 = Cov g g ( ε ˚ k ) . Using Definition 8.7, we then obtain that
∥ t 3 κ 2 ( 8.41 ) ∥ C t − 3 κ 2 L x r ≲ ∥ ( 8.41 ) ∥ C t − 3 κ 2 L x r ≲ ∥ Ξ B , ε ∥ 3 . \left\|t^{\frac{3\kappa}{2}}(8.41)\right\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}\lesssim\|(8.41)\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}\lesssim\|\Xi_{B,\varepsilon}\|^3. t 2 3 κ ( 8.41 ) C t − 2 3 κ L x r ≲ ∥ ( 8.41 ) ∥ C t − 2 3 κ L x r ≲ ∥ Ξ B , ε ∥ 3 . To estimate the terms in (512) , (513) , and (514) , we note that all three of them can be written as linear combinations of images of
( ε ˚ e ⊗ ε ˚ f − δ e f σ ρ , ε 2 Cov g ) ⊗ W ε h , (\mathring{\varepsilon}^{\,e}\otimes\mathring{\varepsilon}^{\,f}-\delta^{ef}\sigma_{\rho,\varepsilon}^2\operatorname{Cov}_{\mathfrak g})\otimes W^h_{\varepsilon}, ( ε ˚ e ⊗ ε ˚ f − δ e f σ ρ , ε 2 Cov g ) ⊗ W ε h , where e , f , h ∈ [ 2 ] e,f,h \in[2] e , f , h ∈ [ 2 ] L L L (514) can be written as
δ e i δ f j δ h k L ( ( φ ϵ e ⊗ φ ϵ f − δ e f σ ρ , ϵ 2 Cov g ) ⊗ W ϵ h ) . \delta_e^i\delta_f^j\delta_h^k L\left(\left(\varphi_\epsilon^e\otimes\varphi_\epsilon^f-\delta^{ef}\sigma_{\rho,\epsilon}^2\operatorname{Cov}_g\right)\otimes W_\epsilon^h\right). δ e i δ f j δ h k L ( ( φ ϵ e ⊗ φ ϵ f − δ e f σ ρ , ϵ 2 Cov g ) ⊗ W ϵ h ) . Using Lemma 3.53, Lemma 7.6, and Definition 8.7, we then obtain that
∥ t 3 λ 2 ( 8.42 ) ∥ C t − 3 κ 2 L x r + ∥ t 3 λ 2 ( 8.43 ) ∥ C t − 3 κ 2 L x r + ∥ t 3 λ 2 ( 8.44 ) ∥ C t − 3 κ 2 L x r ≲ max e , f ∈ [ 2 ] ∥ ∥ φ ϵ e ⊗ φ ϵ f − δ e f σ ρ , ϵ 2 Cov g ∥ C t − κ 2 L x r ∥ t 3 λ 2 W ϵ ∥ C t 2 κ L x ∞ ≲ ∥ Ξ B , ϵ ∥ ∥ W ϵ , ⋆ ∥ s o l 5 κ , − 3 λ . \|t^{\frac{3\lambda}{2}}(8.42)\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}+\|t^{\frac{3\lambda}{2}}(8.43)\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}+\|t^{\frac{3\lambda}{2}}(8.44)\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}
\lesssim\max_{e,f\in[2]}\|\|\varphi_\epsilon^e\otimes\varphi_\epsilon^f-\delta^{ef}\sigma_{\rho,\epsilon}^2\operatorname{Cov}_g\|_{\mathcal{C}_t^{-\frac{\kappa}{2}}L_x^r}\|t^{\frac{3\lambda}{2}}W_\epsilon\|_{\mathcal{C}_t^{2\kappa}L_x^\infty}\lesssim\|\Xi_{B,\epsilon}\|\,\|W_{\epsilon,\star}\|_{\mathrm{sol}^{5\kappa,-3\lambda}}. ∥ t 2 3 λ ( 8.42 ) ∥ C t − 2 3 κ L x r + ∥ t 2 3 λ ( 8.43 ) ∥ C t − 2 3 κ L x r + ∥ t 2 3 λ ( 8.44 ) ∥ C t − 2 3 κ L x r ≲ e , f ∈ [ 2 ] max ∥∥ φ ϵ e ⊗ φ ϵ f − δ e f σ ρ , ϵ 2 Cov g ∥ C t − 2 κ L x r ∥ t 2 3 λ W ϵ ∥ C t 2 κ L x ∞ ≲ ∥ Ξ B , ϵ ∥ ∥ W ϵ , ⋆ ∥ sol 5 κ , − 3 λ . To estimate the terms in (515) , we use Lemma 3.53 and Lemma 7.6, which yield that
∥ t 3 λ 2 ( 8.45 ) ∥ C t − 3 κ 2 L x r ≲ ∥ φ ϵ ∥ C t − 3 κ 4 L x r ∥ t 3 λ 2 ( W ϵ ⊗ W ϵ ) ∥ C t κ L x ∞ \left\|t^{\frac{3\lambda}{2}}(8.45)\right\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}\lesssim\|\varphi_\epsilon\|_{\mathcal{C}_t^{-\frac{3\kappa}{4}}L_x^r}\left\|t^{\frac{3\lambda}{2}}(W_\epsilon\otimes W_\epsilon)\right\|_{\mathcal{C}_t^{\kappa}L_x^\infty} t 2 3 λ ( 8.45 ) C t − 2 3 κ L x r ≲ ∥ φ ϵ ∥ C t − 4 3 κ L x r t 2 3 λ ( W ϵ ⊗ W ϵ ) C t κ L x ∞ ≲ ∥ φ ϵ ∥ C t − 3 κ 4 L x r ∥ t 3 λ 4 W ϵ ∥ C t κ 2 L x ∞ 2 ≲ ∥ Ξ B , ϵ ∥ ∥ W ϵ , ⋆ ∥ s o l 3 κ , − 3 λ 2 2 . \lesssim\|\varphi_\epsilon\|_{\mathcal{C}_t^{-\frac{3\kappa}{4}}L_x^r}\left\|t^{\frac{3\lambda}{4}}W_\epsilon\right\|_{\mathcal{C}_t^{\frac{\kappa}{2}}L_x^\infty}^2\lesssim\|\Xi_{B,\epsilon}\|\,\|W_{\epsilon,\star}\|_{\mathrm{sol}^{3\kappa,-\frac{3\lambda}{2}}}^{2}. ≲ ∥ φ ϵ ∥ C t − 4 3 κ L x r t 4 3 λ W ϵ C t 2 κ L x ∞ 2 ≲ ∥ Ξ B , ϵ ∥ ∥ W ϵ , ⋆ ∥ sol 3 κ , − 2 3 λ 2 . Finally, it remains to estimate the terms in (516) . To this end, we insert the identity W ϵ = S + R ϵ W_\epsilon=S+R_\epsilon W ϵ = S + R ϵ (516) . The cubic term in S S S R ϵ R_\epsilon R ϵ
∥ t 3 λ 2 ( 8.46 ) ∥ C t − 3 κ 2 L x r ≲ ∥ t 3 λ 2 ( 8.46 ) ∥ L t ∞ L x ∞ \left\|t^{\frac{3\lambda}{2}}(8.46)\right\|_{\mathcal{C}_t^{-\frac{3\kappa}{2}}L_x^r}\lesssim\left\|t^{\frac{3\lambda}{2}}(8.46)\right\|_{L_t^\infty L_x^\infty} t 2 3 λ ( 8.46 ) C t − 2 3 κ L x r ≲ t 2 3 λ ( 8.46 ) L t ∞ L x ∞ ≲ ∥ t λ 2 R ϵ ∥ L t ∞ L x ∞ ( ∥ t λ 2 S ∥ L t ∞ L x ∞ + ∥ t λ 2 R ϵ ∥ L t ∞ L x ∞ ) 2 \lesssim\left\|t^{\frac{\lambda}{2}}R_\epsilon\right\|_{L_t^\infty L_x^\infty}\left(\left\|t^{\frac{\lambda}{2}}S\right\|_{L_t^\infty L_x^\infty}+\left\|t^{\frac{\lambda}{2}}R_\epsilon\right\|_{L_t^\infty L_x^\infty}\right)^2 ≲ t 2 λ R ϵ L t ∞ L x ∞ ( t 2 λ S L t ∞ L x ∞ + t 2 λ R ϵ L t ∞ L x ∞ ) 2 ≲ ∥ R ϵ , ⋆ ∥ s o l κ , − λ ( ∥ S ⋆ ∥ s o l κ , − λ + ∥ R ϵ , ⋆ ∥ s o l κ , − λ ) 2 . \lesssim\|R_{\epsilon,\star}\|_{\mathrm{sol}^{\kappa,-\lambda}}\left(\|S_\star\|_{\mathrm{sol}^{\kappa,-\lambda}}+\|R_{\epsilon,\star}\|_{\mathrm{sol}^{\kappa,-\lambda}}\right)^2. ≲ ∥ R ϵ , ⋆ ∥ sol κ , − λ ( ∥ S ⋆ ∥ sol κ , − λ + ∥ R ϵ , ⋆ ∥ sol κ , − λ ) 2 . This completes our last estimate of the terms in (511) –(516) , and hence the proof of (510) .
In the last lemma of this subsection, we estimate the 1 8 π \frac{1}{8\pi} 8 π 1
Lemma 8.23. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
∥ 1 8 π Cov g ( I ϵ + W ϵ ) ∥ n l − 1 + δ 2 , − λ + ∥ M ( I ϵ + w ϵ ) ∥ n l − 1 + δ 2 , − λ ≲ T λ 2 2 − λ 1 ( ∥ Ξ B , ϵ ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 + ∥ C B ( 0 ) , T − κ + T λ 2 ∥ ( R ϵ , ⋆ , r ϵ , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) . \left\|\frac{1}{8\pi}\operatorname{Cov}_g(\mathfrak{I}_\epsilon+W_\epsilon)\right\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}+\left\|M(\mathfrak{I}_\epsilon+w_\epsilon)\right\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{\frac{\lambda_2}{2}-\lambda_1}\left(\|\Xi_{B,\epsilon}\|+\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^+\|_{\mathcal{C}^{-\kappa}_{B(0),T}}+T^{\frac{\lambda}{2}}\|(R_{\epsilon,\star},r_{\epsilon,\star})\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\right). 8 π 1 Cov g ( I ϵ + W ϵ ) nl − 1 + δ 2 , − λ + ∥ M ( I ϵ + w ϵ ) ∥ nl − 1 + δ 2 , − λ ≲ T 2 λ 2 − λ 1 ( ∥ Ξ B , ϵ ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 + ∥ C B ( 0 ) , T − κ + T 2 λ ∥ ( R ϵ , ⋆ , r ϵ , ⋆ ) ∥ sol 1 − δ 1 , − λ ) . Proof. Recall that W ϵ = S + R ϵ W_\epsilon=S+R_\epsilon W ϵ = S + R ϵ (436) and (437) , we obtain that
∥ I ϵ ∥ n l − 1 + δ 2 , − λ + ∥ S ∥ n l − 1 + δ 2 , − λ + ∥ R ϵ ∥ n l − 1 + δ 2 , − λ ≲ T 1 2 − O ( λ ) ( ∥ I ϵ ∥ C t − κ L x r + ∥ S ∥ L t , x ∞ + ∥ t λ 2 R ϵ ∥ L t , x ∞ ) \|\mathfrak{I}_\epsilon\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}+\|S\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}+\|R_\epsilon\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{\frac{1}{2}-O(\lambda)}\left(\|\mathfrak{I}_\epsilon\|_{\mathcal{C}_t^{-\kappa}L_x^r}+\|S\|_{L_{t,x}^\infty}+\|t^{\frac{\lambda}{2}}R_\epsilon\|_{L_{t,x}^\infty}\right) ∥ I ϵ ∥ nl − 1 + δ 2 , − λ + ∥ S ∥ nl − 1 + δ 2 , − λ + ∥ R ϵ ∥ nl − 1 + δ 2 , − λ ≲ T 2 1 − O ( λ ) ( ∥ I ϵ ∥ C t − κ L x r + ∥ S ∥ L t , x ∞ + ∥ t 2 λ R ϵ ∥ L t , x ∞ ) ≲ T 1 2 − O ( λ ) ( ∥ Ξ B , ϵ ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 + ∥ C B ( 0 ) , T − κ + T λ 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) . \lesssim T^{\frac{1}{2}-O(\lambda)}\left(\|\Xi_{B,\epsilon}\|+\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^+\|_{\mathcal{C}^{-\kappa}_{B(0),T}}+T^{\frac{\lambda}{2}}\|R_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\right). ≲ T 2 1 − O ( λ ) ( ∥ Ξ B , ϵ ∥ + ∥ φ 0 + ∥ L x r + ∥ φ 0 + ∥ C B ( 0 ) , T − κ + T 2 λ ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) . In the last line, we also used the bound from Assumption 8.2 and Proposition 5.15. The estimate of M ( I ϵ + w ϵ ) M(\mathfrak{I}_\epsilon+w_\epsilon) M ( I ϵ + w ϵ )
Paracontrolled estimates
We now estimate the terms in (6.74) and (6.84), i.e., the high×high-interactions involving the paracontrolled terms X ε X_\varepsilon X ε x ε x_\varepsilon x ε Corollary 8.33 ). To state it, it is notationally convenient to extend our n l α , ν \mathrm{nl}^{\alpha,\nu} nl α , ν Definition 7.8 ) for Y \mathbf{Y} Y ( Y ⊗ Y ) (\mathbf{Y}\otimes\mathbf{Y}) ( Y ⊗ Y ) ( Q i ) i (Q_i)_i ( Q i ) i G \mathbf{G} G Y ⊗ Y \mathbf{Y}\otimes\mathbf{Y} Y ⊗ Y Y \mathbf{Y} Y
∥ ϕ ∥ n l α , ν ≔ max i ∥ Q i ( ϕ ) ∥ n l α , ν , \lVert\phi\rVert_{\mathrm{nl}^{\alpha,\nu}} \coloneqq\max_i \lVert Q_i(\phi)\rVert_{\mathrm{nl}^{\alpha,\nu}}, ∥ ϕ ∥ nl α , ν : = i max ∥ Q i ( ϕ ) ∥ nl α , ν , where ϕ : [ 0 , T ] × T 2 → Y ⊗ Y \phi\colon[0,T]\times\mathbb{T}^2\to\mathbf{Y}\otimes\mathbf{Y} ϕ : [ 0 , T ] × T 2 → Y ⊗ Y
Proposition 8.24 (Paracontrolled estimate). Let Assumption 8.1 be satisfied. Furthermore, let k , ℓ ∈ [ 2 ] k,\ell\in[2] k , ℓ ∈ [ 2 ] Q : Y × Y → Y Q\colon\mathbf{Y}\times\mathbf{Y}\to\mathbf{Y} Q : Y × Y → Y G \mathbf{G} G K : Y → Y ⊗ Y \mathcal{K}\colon\mathbf{Y}\to\mathbf{Y}\otimes\mathbf{Y} K : Y → Y ⊗ Y K ( W ) : = Q ( W , u m ) ⊗ u m \mathcal{K}(W):=Q(W,u_m)\otimes u^m K ( W ) := Q ( W , u m ) ⊗ u m ( u m ) m (u_m)_m ( u m ) m Y \mathbf{Y} Y W ⋆ = ( W M ) M ≥ N l o W_\star=(W_M)_{M\geq N_{\mathrm{lo}}} W ⋆ = ( W M ) M ≥ N lo [ 0 , T ] × T 2 [0,T]\times\mathbb{T}^2 [ 0 , T ] × T 2 Y \mathbf{Y} Y W M W_M W M t ≳ M − 2 t\gtrsim M^{-2} t ≳ M − 2 29
∥ D u h Y , B , ⋆ F ( Q ( W ∘ ⊗ D ε , ∘ k ) ⊗ D ε , ⋆ ℓ − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 − λ 2 ∥ W ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ∥ ∥ Ξ B , ε ∥ ∥ 2 . (517) \left\lVert\mathrm{Duh}^{F}_{\mathbf{Y},B,\star}\left(Q(W_\circ\otimes D^{k}_{\varepsilon,\circ})\otimes D^{\ell}_{\varepsilon,\star}-\frac{1}{4}\delta^{k\ell}(\sigma^2_{\rho,\varepsilon}+C_\rho)\mathcal{K}(W)\right)\right\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{\frac{1-\lambda}{2}}\lVert W_\star\rVert_{\mathrm{sol}^{1-\delta_1,-1+\delta_1-\kappa}}\lVert\!\lVert\Xi_{B,\varepsilon}\rVert\!\rVert^2. \tag*{(517)} Duh Y , B , ⋆ F ( Q ( W ∘ ⊗ D ε , ∘ k ) ⊗ D ε , ⋆ ℓ − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) ) nl − 1 + δ 2 , − λ ≲ T 2 1 − λ ∥ W ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ∥ ∥ Ξ B , ε ∥ ∥ 2 . (517) Remark 8.25. The reason for using the ∥ W ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ \lVert W_\star\rVert_{\mathrm{sol}^{1-\delta_1,-1+\delta_1-\kappa}} ∥ W ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ S ⋆ S_\star S ⋆ ϕ ⋆ \phi_\star ϕ ⋆ X ε , ⋆ X_{\varepsilon,\star} X ε , ⋆ x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
Remark 8.26. A similar statement as in Proposition 8.24 also holds if the tensor product in Y \mathbf{Y} Y G \mathbf{G} G Y \mathbf{Y} Y a , b , c , d , e , f ∈ [ 3 ] a,b,c,d,e,f\in[3] a , b , c , d , e , f ∈ [ 3 ] Q 1 : Y ( a ) × Y ( b ) → Y ( c ) Q_1\colon\mathbf{Y}^{(a)}\times\mathbf{Y}^{(b)}\to\mathbf{Y}^{(c)} Q 1 : Y ( a ) × Y ( b ) → Y ( c ) Q 2 : Y ( d ) × Y ( e ) → Y ( f ) Q_2\colon\mathbf{Y}^{(d)}\times\mathbf{Y}^{(e)}\to\mathbf{Y}^{(f)} Q 2 : Y ( d ) × Y ( e ) → Y ( f ) G \mathbf{G} G
∥ Q 2 ( D u h Y , B , ⋆ F , d c ( Q 1 ( W ∘ a ⊗ D ε , ∘ k ) ⊗ D ε , ⋆ ℓ ) ⊗ D ε , ⋆ ℓ − 1 4 δ d c δ b e δ k ℓ ( σ ρ , ε 2 + C ρ ) K ~ ( W ∘ a ) ) ∥ n l − 1 + δ 2 , − λ ≲ T 1 − λ 2 ∥ W ⋆ ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ∥ ∥ Ξ B , ε ∥ ∥ 2 . (518) \left\lVert Q_2\left(\mathrm{Duh}^{F,dc}_{\mathbf{Y},B,\star}\left(Q_1(W^a_\circ\otimes D^{k}_{\varepsilon,\circ})\otimes D^{\ell}_{\varepsilon,\star}\right)\otimes D^{\ell}_{\varepsilon,\star}-\frac{1}{4}\delta^{dc}\delta^{be}\delta^{k\ell}(\sigma^2_{\rho,\varepsilon}+C_\rho)\widetilde{\mathcal{K}}(W^a_\circ)\right)\right\rVert_{\mathrm{nl}^{-1+\delta_2,-\lambda}}
\lesssim T^{\frac{1-\lambda}{2}}\lVert W_\star\rVert_{\mathrm{sol}^{1-\delta_1,-1+\delta_1-\kappa}}\lVert\!\lVert\Xi_{B,\varepsilon}\rVert\!\rVert^2. \tag*{(518)} Q 2 ( Duh Y , B , ⋆ F , d c ( Q 1 ( W ∘ a ⊗ D ε , ∘ k ) ⊗ D ε , ⋆ ℓ ) ⊗ D ε , ⋆ ℓ − 4 1 δ d c δ b e δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ∘ a ) ) nl − 1 + δ 2 , − λ ≲ T 2 1 − λ ∥ W ⋆ ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ∥ ∥ Ξ B , ε ∥ ∥ 2 . (518) Here, D u h Y , B , ⋆ F , d c \mathrm{Duh}^{F,dc}_{\mathbf{Y},B,\star} Duh Y , B , ⋆ F , d c D u h Y , B , ⋆ F \mathrm{Duh}^{F}_{\mathbf{Y},B,\star} Duh Y , B , ⋆ F d d d c c c K ~ : Y ( a ) → Y ( f ) \widetilde{\mathcal{K}}\colon\mathbf{Y}^{(a)}\to\mathbf{Y}^{(f)} K : Y ( a ) → Y ( f )
K ~ ( W ) ≔ { Q 2 ( Q 1 ( W , u m ( b ) ) , u ( b ) , m ) if b = e and c = d , 0 else . \widetilde{\mathcal{K}}(W)\coloneqq
\begin{cases}
Q_2(Q_1(W,u_m^{(b)}),u^{(b),m}) & \text{if } b=e\text{ and }c=d,\\
0 & \text{else}.
\end{cases} K ( W ) : = { Q 2 ( Q 1 ( W , u m ( b ) ) , u ( b ) , m ) 0 if b = e and c = d , else . where ( u m ( b ) ) m (u_m^{(b)})_m ( u m ( b ) ) m Y ( b ) \mathbf{Y}^{(b)} Y ( b ) Y \mathbf{Y} Y
Remark 8.27. The level of generality in Proposition 8.24, which allows for different G \mathbf{G} G
4 B ( D u h B , ⋆ ( W ε , ∘ j ⊗ D ε , ∘ j ) ⊗ D ε , ⋆ k ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ k ( W ε k ) . 4\mathbf{B}\left(\mathrm{Duh}_{B,\star}(W^j_{\varepsilon,\circ}\otimes D^j_{\varepsilon,\circ})\otimes D^k_{\varepsilon,\star}\right)+(\sigma^2_{\rho,\varepsilon}+C_\rho)\operatorname{Cov}^{k}_{\vartheta}(W^k_\varepsilon). 4 B ( Duh B , ⋆ ( W ε , ∘ j ⊗ D ε , ∘ j ) ⊗ D ε , ⋆ k ) + ( σ ρ , ε 2 + C ρ ) Cov ϑ k ( W ε k ) . The proof of Proposition 8.24 will be postponed until the end of the subsection. In the next lemma, we first prove an approximate version of the Littlewood–Paley trichotomy, which will be used to better understand the structure of the paracontrolled term X X X
Lemma 8.28 (Approximate Littlewood–Paley trichotomy). Let Assumption 8.1 be satisfied. In addition, let L : Y → Y \mathcal{L}: Y \to Y L : Y → Y G G G Q : Y × Y → Y Q: Y \times Y \to Y Q : Y × Y → Y G G G K , M , N ≥ N l o K, M, N \ge N_{\mathrm{lo}} K , M , N ≥ N lo M ≪ θ N M \ll_{\theta} N M ≪ θ N K ̸ ∼ θ ′ N K \not\sim_{\theta'} N K ∼ θ ′ N θ ′ \theta' θ ′ W M : [ 0 , T ] × T 2 → Y W_M : [0,T] \times\mathbb{T}^{2} \to Y W M : [ 0 , T ] × T 2 → Y t ≳ M − 2 t \gtrsim M^{-2} t ≳ M − 2 − 1 + δ 1 − κ ≤ η 1 ≤ 0 -1+\delta_1-\kappa\le\eta_1 \le0 − 1 + δ 1 − κ ≤ η 1 ≤ 0
∥ Duh Y , B , K F ( L ( D B ε ℓ , N ) ) ∥ L t , x ∞ ≲ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ m s o l N − κ . (519) \left\| \operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{L}(D_B \varepsilon_{\ell,N})\right)\right\|_{L^\infty_{t,x}} \lesssim K^{-1-2\theta}N^{-\kappa}\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}. \tag*{(519)} Duh Y , B , K F ( L ( D B ε ℓ , N ) ) L t , x ∞ ≲ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ msol N − κ . (519) ∥ t − η 1 2 Duh Y , B , K F ( Q ( W M , D B ε ℓ , N ) ) ∥ L t , x ∞ ≲ T 1 2 − λ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ m s o l N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 . (520) \left\|t^{-\frac{\eta_1}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(Q(W_M,D_B\varepsilon_{\ell,N})\right)\right\|_{L^\infty_{t,x}} \lesssim T^{\frac{1}{2}-\lambda}K^{-1-2\theta}N^{-\kappa}\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}\|W_M\|_{\mathrm{sol}^{1-\delta_1,\eta_1}_M}. \tag*{(520)} t − 2 η 1 Duh Y , B , K F ( Q ( W M , D B ε ℓ , N ) ) L t , x ∞ ≲ T 2 1 − λ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . (520) The estimates (8.49) and (8.50) also hold if Duh Y , B , K F \operatorname{Duh}^{F}_{Y,B,K} Duh Y , B , K F Duh Y , B , K \operatorname{Duh}_{Y,B,K} Duh Y , B , K
∥ D B Duh Y , B , K ( L ( ε ℓ , N ) ) ∥ L t , x ∞ ≲ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ m s o l N − κ . (521) \left\|D_B\operatorname{Duh}_{Y,B,K}\left(\mathcal{L}(\varepsilon_{\ell,N})\right)\right\|_{L^\infty_{t,x}} \lesssim K^{-1-2\theta}N^{-\kappa}\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}. \tag*{(521)} ∥ D B Duh Y , B , K ( L ( ε ℓ , N ) ) ∥ L t , x ∞ ≲ K − 1 − 2 θ N − κ ∥ ε ℓ , N ∥ msol N − κ . (521) Remark 8.29. In our proof of Proposition 8.24, we will only need the case η 1 = − 1 + δ 1 − κ \eta_1=-1+\delta_1-\kappa η 1 = − 1 + δ 1 − κ Lemma 8.28 . However, in the proof of Proposition 9.25 below, the case η 1 = 0 \eta_1=0 η 1 = 0 Duh Y , B , K \operatorname{Duh}_{Y,B,K} Duh Y , B , K Duh Y , B , K F \operatorname{Duh}^{F}_{Y,B,K} Duh Y , B , K F Lemma 7.20 ) is only available for Duh Y , B , K \operatorname{Duh}_{Y,B,K} Duh Y , B , K Duh Y , B , K F \operatorname{Duh}^{F}_{Y,B,K} Duh Y , B , K F
We emphasise that the N − κ N^{-\kappa} N − κ N N N M ≪ θ N M \ll_{\theta} N M ≪ θ N M M M
Proof of Lemma 8.28. Since Duh Y , B , K \operatorname{Duh}_{Y,B,K} Duh Y , B , K Duh Y , B , K F \operatorname{Duh}^{F}_{Y,B,K} Duh Y , B , K F
∥ Duh Y , B , K F ( L ( D B ε ℓ , N ) ) ∥ L t , x ∞ ≲ T − δ K − 1 N 3 κ min ( N K , K N ) ∥ ε ℓ , N ∥ m s o l N − κ . (522) \left\|\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{L}(D_B\varepsilon_{\ell,N})\right)\right\|_{L^\infty_{t,x}} \lesssim T^{-\delta}K^{-1}N^{3\kappa}\min\left(\frac{N}{K},\frac{K}{N}\right)\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}. \tag*{(522)} Duh Y , B , K F ( L ( D B ε ℓ , N ) ) L t , x ∞ ≲ T − δ K − 1 N 3 κ min ( K N , N K ) ∥ ε ℓ , N ∥ msol N − κ . (522) Indeed, once (8.52) has been proven, the assumption K ̸ ∼ θ ′ N K \not\sim_{\theta'} N K ∼ θ ′ N
∥ Duh Y , B , K F ( L ( D B ε ℓ , N ) ) ∥ L t , x ∞ ≲ T − δ K − 2 ∥ D B ε ℓ , N ∥ L t , x ∞ \left\|\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{L}(D_B\varepsilon_{\ell,N})\right)\right\|_{L^\infty_{t,x}} \lesssim T^{-\delta}K^{-2}\|D_B\varepsilon_{\ell,N}\|_{L^\infty_{t,x}} Duh Y , B , K F ( L ( D B ε ℓ , N ) ) L t , x ∞ ≲ T − δ K − 2 ∥ D B ε ℓ , N ∥ L t , x ∞ ≲ T − δ K − 2 N 1 + κ ∥ ε ℓ , N ∥ m s o l N − κ = T − δ K − 1 N κ × N K × ∥ ε ℓ , N ∥ m s o l N − κ . (523) \lesssim T^{-\delta}K^{-2}N^{1+\kappa}\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}=T^{-\delta}K^{-1}N^\kappa\times\frac{N}{K}\times\|\varepsilon_{\ell,N}\|_{\mathrm{msol}^{-\kappa}_N}. \tag*{(523)} ≲ T − δ K − 2 N 1 + κ ∥ ε ℓ , N ∥ msol N − κ = T − δ K − 1 N κ × K N × ∥ ε ℓ , N ∥ msol N − κ . (523) Second, we use the G G G L \mathcal{L} L ( α , β , m ) = ( − 2 , − 1 + 2 κ , 1 ) (\alpha,\beta,m)=(-2,-1+2\kappa,1) ( α , β , m ) = ( − 2 , − 1 + 2 κ , 1 )
∥ D u h Y , B , K F ( L ( D B φ ε , N ) ) ∥ L t , x ∞ = ∥ D u h Y , B , K F ( D B L ( φ ε , N ) ) ∥ L t , x ∞ ≲ T − δ K 2 r − 2 κ ∥ φ ε , N ∥ C t − 1 2 + κ L x r ≲ T − δ K 2 r − 2 κ N − 1 + 3 κ ∥ φ ε , N ∥ m s o l N − κ = T − δ K − 1 K 2 r − 2 κ N 3 κ × K N ∥ φ ε , N ∥ m s o l N − κ . (524) \begin{aligned}
\left\|\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{L}(\mathrm{D}_{B}\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}
&=\left\|\mathrm{Duh}^{F}_{Y,B,K}\left(\mathrm{D}_{B}\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}\\
&\lesssim T^{-\delta}K^{\frac{2}{r}-2\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{C_t^{-\frac{1}{2}+\kappa}L_x^r}
\lesssim T^{-\delta}K^{\frac{2}{r}-2\kappa}N^{-1+3\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}\\
&=T^{-\delta}K^{-1}K^{\frac{2}{r}-2\kappa}N^{3\kappa}\times\frac{K}{N}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}. \tag*{(524)}
\end{aligned} Duh Y , B , K F ( L ( D B φ ε , N ) ) L t , x ∞ = Duh Y , B , K F ( D B L ( φ ε , N ) ) L t , x ∞ ≲ T − δ K r 2 − 2 κ ∥ φ ε , N ∥ C t − 2 1 + κ L x r ≲ T − δ K r 2 − 2 κ N − 1 + 3 κ ∥ φ ε , N ∥ msol N − κ = T − δ K − 1 K r 2 − 2 κ N 3 κ × N K ∥ φ ε , N ∥ msol N − κ . (524) Since 1 r \frac{1}{r} r 1 κ \kappa κ (522) , and hence the proof of (519) .
Next, we prove the estimate (521) . For this, it is sufficient to prove that
∥ D B D u h Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≲ T − δ K − 1 N 3 κ min ( N K ′ , K N ) 1 − 2 κ ∥ φ ε , N ∥ m s o l N − κ . (525) \left\|\mathrm{D}_{B}\mathrm{Duh}_{Y,B,K}\left(\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}
\lesssim T^{-\delta}K^{-1}N^{3\kappa}\min\left(\frac{N}{K'},\frac{K}{N}\right)^{1-2\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}. \tag*{(525)} ∥ D B Duh Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≲ T − δ K − 1 N 3 κ min ( K ′ N , N K ) 1 − 2 κ ∥ φ ε , N ∥ msol N − κ . (525) In the case K ≫ θ ′ N K\gg_{\theta'}N K ≫ θ ′ N
∥ D B D u h Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≤ ∥ ( D B D u h Y , B , K − D u h Y , B , K D B ) ( L ( φ ε , N ) ) ∥ L t , x ∞ (526) \left\|\mathrm{D}_{B}\mathrm{Duh}_{Y,B,K}\left(\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}
\leq\left\|(\mathrm{D}_{B}\mathrm{Duh}_{Y,B,K}-\mathrm{Duh}_{Y,B,K}\mathrm{D}_{B})\left(\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}} \tag*{(526)} ∥ D B Duh Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≤ ∥ ( D B Duh Y , B , K − Duh Y , B , K D B ) ( L ( φ ε , N ) ) ∥ L t , x ∞ (526) + ∥ D u h Y , B , K ( D B L ( φ ε , N ) ) ∥ L t , x ∞ . (527) +\left\|\mathrm{Duh}_{Y,B,K}\left(\mathrm{D}_{B}\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}. \tag*{(527)} + ∥ Duh Y , B , K ( D B L ( φ ε , N ) ) ∥ L t , x ∞ . (527) Using our commutator estimate (Lemma 7.20) and N ≥ N l o ∼ T − 1 2 N\geq N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} N ≥ N lo ∼ T − 2 1 (526) can be bounded by
( 8.56 ) ≲ T − δ T − 1 − 2 κ 2 K − ( 2 − 2 κ ) ∥ φ ε , N ∥ L t , x ∞ ≲ T − δ N 1 − 2 κ K − ( 2 − 2 κ ) N κ ∥ φ ε , N ∥ m s o l N − κ . {(8.56)}\lesssim T^{-\delta}T^{-\frac{1-2\kappa}{2}}K^{-(2-2\kappa)}\left\|\varphi_{\varepsilon,N}\right\|_{L^\infty_{t,x}}
\lesssim T^{-\delta}N^{1-2\kappa}K^{-(2-2\kappa)}N^\kappa\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}. ( 8.56 ) ≲ T − δ T − 2 1 − 2 κ K − ( 2 − 2 κ ) ∥ φ ε , N ∥ L t , x ∞ ≲ T − δ N 1 − 2 κ K − ( 2 − 2 κ ) N κ ∥ φ ε , N ∥ msol N − κ . Using our earlier estimate (523) , except that D u h Y , B , K F \mathrm{Duh}^{F}_{Y,B,K} Duh Y , B , K F D u h Y , B , K \mathrm{Duh}_{Y,B,K} Duh Y , B , K (527) by T − δ K − 2 N 1 + κ ∥ φ ε , N ∥ m s o l N − κ T^{-\delta}K^{-2}N^{1+\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}} T − δ K − 2 N 1 + κ ∥ φ ε , N ∥ msol N − κ (526) and (527) , we obtain (525) .
In the case K ≪ θ ′ N K\ll_{\theta'}N K ≪ θ ′ N (424) with ( α , β , m ) = ( − 1 , − 1 + 2 κ , 0 ) (\alpha,\beta,m)=(-1,-1+2\kappa,0) ( α , β , m ) = ( − 1 , − 1 + 2 κ , 0 )
∥ D B D u h Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≤ ∥ D u h Y , B , K ( L ( φ ε , N ) ) ∥ s o l K 1 ≲ T − δ K 2 r − 2 κ ∥ φ ε , N ∥ C t − 1 2 + κ L x r . \left\|\mathrm{D}_{B}\mathrm{Duh}_{Y,B,K}\left(\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}}
\leq\left\|\mathrm{Duh}_{Y,B,K}\left(\mathcal{L}(\varphi_{\varepsilon,N})\right)\right\|_{\mathrm{sol}_{K}^{1}}
\lesssim T^{-\delta}K^{\frac{2}{r}-2\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{C_t^{-\frac{1}{2}+\kappa}L_x^r}. ∥ D B Duh Y , B , K ( L ( φ ε , N ) ) ∥ L t , x ∞ ≤ ∥ Duh Y , B , K ( L ( φ ε , N ) ) ∥ sol K 1 ≲ T − δ K r 2 − 2 κ ∥ φ ε , N ∥ C t − 2 1 + κ L x r . After proceeding exactly as in (524) , we then obtain (525) .
Finally, we prove the estimate (520) . For this, we first use a dyadic decomposition to reduce to the case η 1 = − 1 + δ 1 − κ \eta_1=-1+\delta_1-\kappa η 1 = − 1 + δ 1 − κ (520) holds for η 1 = − 1 + δ 1 − κ \eta_1=-1+\delta_1-\kappa η 1 = − 1 + δ 1 − κ η 1 ′ ∈ [ η 1 , 0 ] \eta'_1\in[\eta_1,0] η 1 ′ ∈ [ η 1 , 0 ]
∥ t − η 1 ′ 2 D u h Y , B , K F ( Q ( W M , D B φ ε , N ) ) ∥ L t , x ∞ ( [ 0 , T ] ) = sup k ≥ 0 ∥ t − η 1 ′ 2 D u h Y , B , K F ( Q ( W M , D B φ ε , N ) ) ∥ L t , x ∞ ( [ 2 − ( k + 1 ) T , 2 − k T ] ) \left\|t^{-\frac{\eta'_1}{2}}\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,\mathrm{D}_{B}\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}([0,T])}
=\sup_{k\geq0}\left\|t^{-\frac{\eta'_1}{2}}\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,\mathrm{D}_{B}\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}([2^{-(k+1)}T,2^{-k}T])} t − 2 η 1 ′ Duh Y , B , K F ( Q ( W M , D B φ ε , N ) ) L t , x ∞ ([ 0 , T ]) = k ≥ 0 sup t − 2 η 1 ′ Duh Y , B , K F ( Q ( W M , D B φ ε , N ) ) L t , x ∞ ([ 2 − ( k + 1 ) T , 2 − k T ]) ≲ sup k ≥ 0 ( 2 − k T ) η 1 − η 1 ′ 2 ∥ t − η 1 2 D u h Y , B , K F ( Q ( W M , D B φ ε , N ) ) ∥ L t , x ∞ ( [ 0 , 2 − k T ] ) \lesssim\sup_{k\geq0}(2^{-k}T)^{\frac{\eta_1-\eta'_1}{2}}\left\|t^{-\frac{\eta_1}{2}}\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,\mathrm{D}_{B}\varphi_{\varepsilon,N})\right)\right\|_{L^\infty_{t,x}([0,2^{-k}T])} ≲ k ≥ 0 sup ( 2 − k T ) 2 η 1 − η 1 ′ t − 2 η 1 Duh Y , B , K F ( Q ( W M , D B φ ε , N ) ) L t , x ∞ ([ 0 , 2 − k T ]) ≲ sup k ≥ 0 ( ( 2 − k T ) η 1 − η 1 ′ 2 × T 1 2 − λ K − 1 − 2 θ N − κ ∥ φ ε , N ∥ m s o l N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ( [ 0 , 2 − k T ] ) ) \lesssim\sup_{k\geq0}\left((2^{-k}T)^{\frac{\eta_1-\eta'_1}{2}}\times T^{\frac{1}{2}-\lambda}K^{-1-2\theta}N^{-\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}\left\|W_M\right\|_{\mathrm{sol}_{M}^{1-\delta_1,\eta_1}([0,2^{-k}T])}\right) ≲ k ≥ 0 sup ( ( 2 − k T ) 2 η 1 − η 1 ′ × T 2 1 − λ K − 1 − 2 θ N − κ ∥ φ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ([ 0 , 2 − k T ]) ) ≲ T 1 2 − λ K − 1 − 2 θ N − κ ∥ φ ε , N ∥ m s o l N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ′ ( [ 0 , T ] ) . \lesssim T^{\frac{1}{2}-\lambda}K^{-1-2\theta}N^{-\kappa}\left\|\varphi_{\varepsilon,N}\right\|_{\mathrm{msol}_N^{-\kappa}}\left\|W_M\right\|_{\mathrm{sol}_{M}^{1-\delta_1,\eta'_1}([0,T])}. ≲ T 2 1 − λ K − 1 − 2 θ N − κ ∥ φ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ′ ([ 0 , T ]) . In the last line, we used that η 1 ′ ≥ η 1 \eta'_1\geq\eta_1 η 1 ′ ≥ η 1
∥ W M ∥ s o l M 1 − δ 1 , η 1 ( [ 0 , 2 − k T ] ) ≲ ( 2 − k T ) η 1 ′ − η 1 2 ∥ W M ∥ s o l M 1 − δ 1 , η 1 ′ ( [ 0 , T ] ) . \left\|W_M\right\|_{\mathrm{sol}_{M}^{1-\delta_1,\eta_1}([0,2^{-k}T])}
\lesssim(2^{-k}T)^{\frac{\eta'_1-\eta_1}{2}}\left\|W_M\right\|_{\mathrm{sol}_{M}^{1-\delta_1,\eta'_1}([0,T])}. ∥ W M ∥ sol M 1 − δ 1 , η 1 ([ 0 , 2 − k T ]) ≲ ( 2 − k T ) 2 η 1 ′ − η 1 ∥ W M ∥ sol M 1 − δ 1 , η 1 ′ ([ 0 , T ]) . For the rest of this proof, we now take η 1 = − 1 + δ 1 − κ \eta_1=-1+\delta_1-\kappa η 1 = − 1 + δ 1 − κ (519) , (520) can be reduced to the dyadic estimate
∥ t − η 1 2 Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) ∥ L t , x ∞ ≲ T 1 2 − λ K − 1 N − κ min ( N 2 κ N K , ( K N ) 2 κ ( K N ) λ ) + K − λ ∥ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . (528) \left\|t^{-\frac{\eta_1}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,D_B\circ\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}
\lesssim T^{\frac{1}{2}-\lambda}K^{-1}N^{-\kappa}\min\left(N^{2\kappa}\frac{N}{K},(KN)^{2\kappa}\left(\frac{K}{N}\right)^\lambda\right)+K^{-\lambda}\left\|\varepsilon_{,N}\right\|_{\operatorname{msol}_N^{-\kappa}}\left\|W_M\right\|_{\operatorname{sol}_M^{1-\delta_1,\eta_1}}. \tag*{(528)} t − 2 η 1 Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) L t , x ∞ ≲ T 2 1 − λ K − 1 N − κ min ( N 2 κ K N , ( K N ) 2 κ ( N K ) λ ) + K − λ ∥ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . (528) Indeed, if N ≪ θ ′ K N\ll_{\theta'}K N ≪ θ ′ K (528) and, if K ≪ θ ′ N K\ll_{\theta'}N K ≪ θ ′ N λ θ ′ \lambda\theta' λ θ ′ θ \theta θ (528) , we first use our weighted Schauder estimate (436) , which yields
∥ t − η 1 2 Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) ∥ L t , x ∞ ≲ T − δ K − 2 ∥ t − η 1 2 Q ( W M , D B ∘ ε , N ) ∥ L t , x ∞ \left\|t^{-\frac{\eta_1}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,D_B\circ\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}\lesssim T^{-\delta}K^{-2}\left\|t^{-\frac{\eta_1}{2}}\mathcal{Q}(W_M,D_B\circ\varepsilon_{,N})\right\|_{L^\infty_{t,x}} t − 2 η 1 Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) L t , x ∞ ≲ T − δ K − 2 t − 2 η 1 Q ( W M , D B ∘ ε , N ) L t , x ∞ ≲ T − δ K − 2 M − 1 + δ 1 N 1 + κ ∥ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . \lesssim T^{-\delta}K^{-2}M^{-1+\delta_1}N^{1+\kappa}\left\|\varepsilon_{,N}\right\|_{\operatorname{msol}_N^{-\kappa}}\left\|W_M\right\|_{\operatorname{sol}_M^{1-\delta_1,\eta_1}}. ≲ T − δ K − 2 M − 1 + δ 1 N 1 + κ ∥ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . Since M ≥ N l o ∼ T − 1 2 M\geq N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} M ≥ N lo ∼ T − 2 1 (528) . In order to get the second bound, we split
Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) = Duh Y , B , K F ( D B Q ( W M , ε , N ) ) − Duh Y , B , K F ( Q ( D B W M , ε , N ) ) . (529) \operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,D_B\circ\varepsilon_{,N})\right)
=\operatorname{Duh}^{F}_{Y,B,K}\left(D_B\mathcal{Q}(W_M,\varepsilon_{,N})\right)-\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(D_BW_M,\varepsilon_{,N})\right). \tag*{(529)} Duh Y , B , K F ( Q ( W M , D B ∘ ε , N ) ) = Duh Y , B , K F ( D B Q ( W M , ε , N ) ) − Duh Y , B , K F ( Q ( D B W M , ε , N ) ) . (529) To estimate the first summand in (529) , we let α ≤ − κ \alpha\leq-\kappa α ≤ − κ α ≥ η 1 + 2 κ \alpha\geq\eta_1+2\kappa α ≥ η 1 + 2 κ
∥ t − η 1 2 Duh Y , B , K F ( D B Q ( W M , ε , N ) ) ∥ L t , x ∞ ≲ T α − 2 κ − η 1 2 ∥ t − α − 2 κ 2 Duh Y , B , K F ( D B Q ( W M , ε , N ) ) ∥ L t , x ∞ . \left\|t^{-\frac{\eta_1}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(D_B\mathcal{Q}(W_M,\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}
\lesssim T^{\frac{\alpha-2\kappa-\eta_1}{2}}\left\|t^{-\frac{\alpha-2\kappa}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(D_B\mathcal{Q}(W_M,\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}. t − 2 η 1 Duh Y , B , K F ( D B Q ( W M , ε , N ) ) L t , x ∞ ≲ T 2 α − 2 κ − η 1 t − 2 α − 2 κ Duh Y , B , K F ( D B Q ( W M , ε , N ) ) L t , x ∞ . Provided that ( α − 2 κ ) + α > − 1 (\alpha-2\kappa)+\alpha>-1 ( α − 2 κ ) + α > − 1 t − η 1 2 t^{-\frac{\eta_1}{2}} t − 2 η 1 (437) that
T α − 2 κ − η 1 2 ∥ t − α − 2 κ 2 Duh Y , B , K F ( D B Q ( W M , ε , N ) ) ∥ L t , x ∞ ≲ T α − 2 κ − η 1 2 T − δ K − 1 − α + δ 2 ∥ t − α − 2 κ 2 Q ( W M , ε , N ) ∥ L t 2 L x r . T^{\frac{\alpha-2\kappa-\eta_1}{2}}\left\|t^{-\frac{\alpha-2\kappa}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(D_B\mathcal{Q}(W_M,\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}
\lesssim T^{\frac{\alpha-2\kappa-\eta_1}{2}}T^{-\delta}K^{-1-\alpha+\frac{\delta}{2}}\left\|t^{-\frac{\alpha-2\kappa}{2}}\mathcal{Q}(W_M,\varepsilon_{,N})\right\|_{L^2_tL^r_x}. T 2 α − 2 κ − η 1 t − 2 α − 2 κ Duh Y , B , K F ( D B Q ( W M , ε , N ) ) L t , x ∞ ≲ T 2 α − 2 κ − η 1 T − δ K − 1 − α + 2 δ t − 2 α − 2 κ Q ( W M , ε , N ) L t 2 L x r . From Lemma 3.53, Lemma 7.6, and Lemma 7.7, it then follows that
∥ t − α − 2 κ 2 Q ( W M , ε , N ) ∥ L t 2 L x r ≲ ∥ t − α − 2 κ 2 W M ∥ L t ∞ L − α + κ 2 x ∥ ε , N ∥ L t 2 L x r \left\|t^{-\frac{\alpha-2\kappa}{2}}\mathcal{Q}(W_M,\varepsilon_{,N})\right\|_{L^2_tL^r_x}
\lesssim\left\|t^{-\frac{\alpha-2\kappa}{2}}W_M\right\|_{L^\infty_tL^x_{-\frac{\alpha+\kappa}{2}}}\left\|\varepsilon_{,N}\right\|_{L^2_tL^r_x} t − 2 α − 2 κ Q ( W M , ε , N ) L t 2 L x r ≲ t − 2 α − 2 κ W M L t ∞ L − 2 α + κ x ∥ ε , N ∥ L t 2 L x r ≲ N α + κ ∥ W M ∥ sol M − α + κ , α − 2 κ ∥ ε , N ∥ msol N − κ ≲ N α + κ ∥ W M ∥ sol M 1 − δ 1 , − 1 + δ 1 − κ ∥ ε , N ∥ msol N − κ . \lesssim N^{\alpha+\kappa}\left\|W_M\right\|_{\operatorname{sol}_M^{-\alpha+\kappa,\alpha-2\kappa}}\left\|\varepsilon_{,N}\right\|_{\operatorname{msol}_N^{-\kappa}}
\lesssim N^{\alpha+\kappa}\left\|W_M\right\|_{\operatorname{sol}_M^{1-\delta_1,-1+\delta_1-\kappa}}\left\|\varepsilon_{,N}\right\|_{\operatorname{msol}_N^{-\kappa}}. ≲ N α + κ ∥ W M ∥ sol M − α + κ , α − 2 κ ∥ ε , N ∥ msol N − κ ≲ N α + κ ∥ W M ∥ sol M 1 − δ 1 , − 1 + δ 1 − κ ∥ ε , N ∥ msol N − κ . In total, using the definition of η 1 \eta_1 η 1
T − δ T 1 + α − δ 1 − κ 2 K − 1 N − κ × K δ 2 N 2 κ ( K N ) − α × ∥ W M ∥ sol M 1 − δ 1 , − 1 + δ 1 − κ ∥ ε , N ∥ msol N − κ . T^{-\delta}T^{\frac{1+\alpha-\delta_1-\kappa}{2}}K^{-1}N^{-\kappa}\times K^{\frac{\delta}{2}}N^{2\kappa}\left(\frac{K}{N}\right)^{-\alpha}\times\left\|W_M\right\|_{\operatorname{sol}_M^{1-\delta_1,-1+\delta_1-\kappa}}\left\|\varepsilon_{,N}\right\|_{\operatorname{msol}_N^{-\kappa}}. T − δ T 2 1 + α − δ 1 − κ K − 1 N − κ × K 2 δ N 2 κ ( N K ) − α × ∥ W M ∥ sol M 1 − δ 1 , − 1 + δ 1 − κ ∥ ε , N ∥ msol N − κ . By choosing α = − λ \alpha=-\lambda α = − λ (528) . To estimate the second summand in (529) , we use Lemma 7.29 with ( α , β , γ , ν , η ) = ( − 2 κ , 1 − δ 1 , − 2 δ 1 θ , η 1 , η 1 ) (\alpha,\beta,\gamma,\nu,\eta)=(-2\kappa,1-\delta_1,-\frac{2\delta_1}{\theta},\eta_1,\eta_1) ( α , β , γ , ν , η ) = ( − 2 κ , 1 − δ 1 , − θ 2 δ 1 , η 1 , η 1 )
From this, it follows that30
∥ t − η 2 Duh Y , B , K F ( Q ( D B W M , ε , N ) ) ∥ L t , x ∞ ≲ K − 2 + 2 δ 1 θ ∥ Q ( D B W M , ε , N ) ∥ N K − 2 δ 1 / θ , η 1 \left\|t^{-\frac{\eta}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(Q(D_{B}W_{M},\varepsilon_{,N})\right)\right\|_{L^\infty_{t,x}}
\lesssim K^{-2+\frac{2\delta_{1}}{\theta}}\left\|Q(D_{B}W_{M},\varepsilon_{,N})\right\|_{\mathcal{N}_{K}^{-2\delta_{1}/\theta,\eta_{1}}} t − 2 η Duh Y , B , K F ( Q ( D B W M , ε , N ) ) L t , x ∞ ≲ K − 2 + θ 2 δ 1 ∥ Q ( D B W M , ε , N ) ∥ N K − 2 δ 1 / θ , η 1 ≲ T − δ K δ 1 θ − 1 − θ 2 θ ( δ 1 + 2 κ ) T − 1 / r − 2 κ K − 2 + 2 δ 1 θ ∥ ε , N ∥ m s o l N − 2 κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 \lesssim T^{-\delta K^{\frac{\delta_{1}}{\theta}-\frac{1-\theta}{2\theta}(\delta_{1}+2\kappa)}T^{-1/r-2\kappa}K^{-2+\frac{2\delta_{1}}{\theta}}\|\varepsilon_{,N}\|_{\mathrm{msol}^{-2\kappa}_{N}}\|W_{M}\|_{\mathrm{sol}^{1-\delta_{1},\eta_{1}}_{M}}} ≲ T − δ K θ δ 1 − 2 θ 1 − θ ( δ 1 + 2 κ ) T − 1/ r − 2 κ K − 2 + θ 2 δ 1 ∥ ε , N ∥ msol N − 2 κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ≲ T − O ( δ 1 / θ ) K − 2 + 2 δ 1 θ N − κ ∥ ε , N ∥ m s o l N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 . \lesssim T^{-O(\delta_{1}/\theta)}K^{-2+\frac{2\delta_{1}}{\theta}}N^{-\kappa}\|\varepsilon_{,N}\|_{\mathrm{msol}^{-\kappa}_{N}}\|W_{M}\|_{\mathrm{sol}^{1-\delta_{1},\eta_{1}}_{M}}. ≲ T − O ( δ 1 / θ ) K − 2 + θ 2 δ 1 N − κ ∥ ε , N ∥ msol N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 . Since K ≥ N l o ∼ T − 1 2 K\geq N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} K ≥ N lo ∼ T − 2 1
K − 2 + 2 δ 1 θ = K − 1 − λ K − 1 + λ + 2 δ 1 θ ≲ K − 1 − λ T 1 − λ 2 − δ 1 θ . K^{-2+\frac{2\delta_{1}}{\theta}}=K^{-1-\lambda}K^{-1+\lambda+\frac{2\delta_{1}}{\theta}}\lesssim K^{-1-\lambda}T^{\frac{1-\lambda}{2}-\frac{\delta_{1}}{\theta}}. K − 2 + θ 2 δ 1 = K − 1 − λ K − 1 + λ + θ 2 δ 1 ≲ K − 1 − λ T 2 1 − λ − θ δ 1 . Thus, since λ \lambda λ δ 1 / θ \delta_{1}/\theta δ 1 / θ (528) .
In our next lemma, we show that for large values of K K K Duh Y , B , K F \operatorname{Duh}^{F}_{Y,B,K} Duh Y , B , K F Duh Y , B , K \operatorname{Duh}_{Y,B,K} Duh Y , B , K Ξ B , ε p a r a \Xi^{\mathrm{para}}_{B,\varepsilon} Ξ B , ε para (481) .
Lemma 8.30. Let Assumption 8.1 be satisfied, let K ≥ N l o K\geq N_{\mathrm{lo}} K ≥ N lo − 2 < η ≤ 0 -2<\eta\leq0 − 2 < η ≤ 0 ϕ : [ 0 , T ] × T 2 → Y \phi:[0,T]\times\mathbb{T}^{2}\to Y ϕ : [ 0 , T ] × T 2 → Y
∥ t − η 2 ( Duh Y , B , K F − Duh Y , B , K ) ( ϕ ) ∥ L t , x ∞ ≲ T − 1 − 2 κ 2 − δ K − 3 + 2 κ ∥ t − η 2 ϕ ∥ L t , x ∞ . (530) \left\|t^{-\frac{\eta}{2}}\left(\operatorname{Duh}^{F}_{Y,B,K}-\operatorname{Duh}_{Y,B,K}\right)(\phi)\right\|_{L^\infty_{t,x}}
\lesssim T^{-\frac{1-2\kappa}{2}-\delta}K^{-3+2\kappa}\left\|t^{-\frac{\eta}{2}}\phi\right\|_{L^\infty_{t,x}}. \tag*{(530)} t − 2 η ( Duh Y , B , K F − Duh Y , B , K ) ( ϕ ) L t , x ∞ ≲ T − 2 1 − 2 κ − δ K − 3 + 2 κ t − 2 η ϕ L t , x ∞ . (530) Proof. Let 0 ≤ s ≤ t ≤ T 0\leq s\leq t\leq T 0 ≤ s ≤ t ≤ T L T s ( F ) L^{s}_{T}(F) L T s ( F ) (422) , we obtain
∣ q Y , B ( s , y ; t , x ) − p Y , B ( s , y ; t , x ) ∣ ≲ Gr T ( C 0 F ) ∥ F ∥ [ s , t ] , 0 p e r t ( t − s ) − 1 exp ( − c ∣ x − y ∣ 2 t − s ) \left|q_{Y,B}(s,y;t,x)-p_{Y,B}(s,y;t,x)\right|
\lesssim\operatorname{Gr}_{T}(C_{0}F)\|F\|^{\mathrm{pert}}_{[s,t],0}(t-s)^{-1}\exp\left(-c\frac{|x-y|^{2}}{t-s}\right) ∣ q Y , B ( s , y ; t , x ) − p Y , B ( s , y ; t , x ) ∣ ≲ Gr T ( C 0 F ) ∥ F ∥ [ s , t ] , 0 pert ( t − s ) − 1 exp ( − c t − s ∣ x − y ∣ 2 ) ≲ Gr T ( C 0 F ) ( T 1 − 2 κ 2 ∥ u − 1 − 2 κ 2 F ( u ) ∥ [ 0 , T ] , 1 p e r t ) ( T − 1 ( t − s ) ) 1 − 2 κ 2 ( t − s ) − 1 exp ( − c ∣ x − y ∣ 2 t − s ) \lesssim\operatorname{Gr}_{T}(C_{0}F)\left(T^{\frac{1-2\kappa}{2}}\|u^{-\frac{1-2\kappa}{2}}F(u)\|^{\mathrm{pert}}_{[0,T],1}\right)(T^{-1}(t-s))^{\frac{1-2\kappa}{2}}(t-s)^{-1}\exp\left(-c\frac{|x-y|^{2}}{t-s}\right) ≲ Gr T ( C 0 F ) ( T 2 1 − 2 κ ∥ u − 2 1 − 2 κ F ( u ) ∥ [ 0 , T ] , 1 pert ) ( T − 1 ( t − s ) ) 2 1 − 2 κ ( t − s ) − 1 exp ( − c t − s ∣ x − y ∣ 2 ) ≲ L T s ( F ) ( T − 1 ( t − s ) ) 1 − 2 κ 2 ( t − s ) − 1 exp ( − c ∣ x − y ∣ 2 t − s ) . \lesssim L^{s}_{T}(F)\left(T^{-1}(t-s)\right)^{\frac{1-2\kappa}{2}}(t-s)^{-1}\exp\left(-c\frac{|x-y|^{2}}{t-s}\right). ≲ L T s ( F ) ( T − 1 ( t − s ) ) 2 1 − 2 κ ( t − s ) − 1 exp ( − c t − s ∣ x − y ∣ 2 ) . By distinguishing the cases t ≫ K − 2 t\gg K^{-2} t ≫ K − 2 t ≲ K − 2 t\lesssim K^{-2} t ≲ K − 2
∥ 1 { ∣ t − s ∣ ∼ K − 2 } t − η 2 s η 2 ( q Y , B ( s , y ; t , x ) − p Y , B ( s , y ; t , x ) ) ∥ L s 1 ≲ L T s ( F ) T − 1 − 2 κ 2 K − 3 + 2 κ . \left\|\mathbf{1}_{\{|t-s|\sim K^{-2}\}}t^{-\frac{\eta}{2}}s^{\frac{\eta}{2}}\left(q_{Y,B}(s,y;t,x)-p_{Y,B}(s,y;t,x)\right)\right\|_{L^{1}_{s}}
\lesssim L^{s}_{T}(F)T^{-\frac{1-2\kappa}{2}}K^{-3+2\kappa}. 1 { ∣ t − s ∣ ∼ K − 2 } t − 2 η s 2 η ( q Y , B ( s , y ; t , x ) − p Y , B ( s , y ; t , x ) ) L s 1 ≲ L T s ( F ) T − 2 1 − 2 κ K − 3 + 2 κ . Together with the bound from Assumption 8.1, this implies the desired result.
Equipped with Lemma 8.28 and Lemma 8.30, we now prove the following lemma, which yields information on the structure of the paracontrolled term X X X
Lemma 8.31 (Structure of the paracontrolled term). Let Assumption 8.1 be satisfied. Furthermore, let Q : Y × Y → Y Q:Y\times Y\to Y Q : Y × Y → Y k ∈ [ 2 ] k\in[2] k ∈ [ 2 ] K ≥ N l o K\geq N_{\mathrm{lo}} K ≥ N lo W ⋆ = ( W M ) M ≥ N l o W_{\star}=(W_{M})_{M\geq N_{\mathrm{lo}}} W ⋆ = ( W M ) M ≥ N lo W M : [ 0 , T ] × T 2 → Y W_{M}:[0,T]\times\mathbb{T}^{2}\to Y W M : [ 0 , T ] × T 2 → Y t ≳ M − 2 t\gtrsim M^{-2} t ≳ M − 2 − 1 + δ 1 − κ ≤ η 1 ≤ 0 -1+\delta_{1}-\kappa\leq\eta_{1}\leq0 − 1 + δ 1 − κ ≤ η 1 ≤ 0
∥ t − η 1 2 ( Duh Y , B , K F ( Q ( W ⋆ ⊗ D k ε ⋆ ) ) − Q ( W , D k Duh Y , B , K ( ε ⋆ ) ) ) ∥ L t , x ∞ ≲ T 1 − λ 1 2 K − 1 − 2 θ ∥ W ⋆ ∥ s o l 1 − δ 1 , η 1 ∥ ε ⋆ ∥ m s o l − κ . (531) \left\|t^{-\frac{\eta_{1}}{2}}\left(\operatorname{Duh}^{F}_{Y,B,K}\left(Q(W_{\star}\otimes D^{k}\varepsilon_{\star})\right)-Q\left(W,D^{k}\operatorname{Duh}_{Y,B,K}(\varepsilon_{\star})\right)\right)\right\|_{L^\infty_{t,x}}
\lesssim T^{\frac{1-\lambda_{1}}{2}}K^{-1-2\theta}\|W_{\star}\|_{\mathrm{sol}^{1-\delta_{1},\eta_{1}}}\|\varepsilon_{\star}\|_{\mathrm{msol}^{-\kappa}}. \tag*{(531)} t − 2 η 1 ( Duh Y , B , K F ( Q ( W ⋆ ⊗ D k ε ⋆ ) ) − Q ( W , D k Duh Y , B , K ( ε ⋆ ) ) ) L t , x ∞ ≲ T 2 1 − λ 1 K − 1 − 2 θ ∥ W ⋆ ∥ sol 1 − δ 1 , η 1 ∥ ε ⋆ ∥ msol − κ . (531) In addition, for all M , N ≥ N 0 M,N\geq N_{0} M , N ≥ N 0 M ≪ θ N M\ll_{\theta}N M ≪ θ N
∥ t − η 1 2 ( Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) ) ∥ L t , x ∞ ≲ T 1 − λ 1 2 K − 1 − 2 θ N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ ε , N ˚ ∥ m s o l N − κ . (532) \left\|t^{-\frac{\eta_{1}}{2}}\left(\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}\left(W_{M},D^{k}\mathring{\varepsilon,N}\right)\right)-\mathcal{Q}\left(W_{M},D^{k}\operatorname{Duh}_{Y,B,K}\left(\mathring{\varepsilon,N}\right)\right)\right)\right\|_{L^{\infty}_{t,x}}
\lesssim T^{\frac{1-\lambda_{1}}{2}}K^{-1-2\theta}N^{-\kappa}\left\|W_{M}\right\|_{\mathrm{sol}_{M}^{1-\delta_{1},\eta_{1}}}\left\|\mathring{\varepsilon,N}\right\|_{\mathrm{msol}_{N}^{-\kappa}}. \tag*{(532)} t − 2 η 1 ( Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) ) L t , x ∞ ≲ T 2 1 − λ 1 K − 1 − 2 θ N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ε , N ˚ msol N − κ . (532) Remark 8.32. In this subsection, we will only need the estimate (531) . However, the frequency-localised estimate (532) will be useful in the proof of Lemma 9.35 below.
Proof. Throughout the proof, when using estimates involving the loss-factor L T ( F ) L_{T}(F) L T ( F ) L T ( F ) ≤ T − δ L_{T}(F)\leq T^{-\delta} L T ( F ) ≤ T − δ (532) , for which we distinguish the cases K ∼ θ ′ N K\sim_{\theta'}N K ∼ θ ′ N K ̸ ∼ θ ′ N K\not\sim_{\theta'}N K ∼ θ ′ N (532) , as well as some easier estimates, to prove (531) .
The case K ∼ θ ′ N K\sim_{\theta'}N K ∼ θ ′ N (532) : In this case, we split
Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) = ( Duh Y , B , K F − Duh Y , B , K ) ( Q ( W M , D k ε , N ˚ ) ) + Duh Y , B , K ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , Duh Y , B , K ( D k ε , N ˚ ) ) + Q ( W M , Duh Y , B , K ( D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) . (533) \begin{aligned}
\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}\left(W_{M},D^{k}\mathring{\varepsilon,N}\right)\right)-\mathcal{Q}\left(W_{M},D^{k}\operatorname{Duh}_{Y,B,K}\left(\mathring{\varepsilon,N}\right)\right)
={}&\left(\operatorname{Duh}^{F}_{Y,B,K}-\operatorname{Duh}_{Y,B,K}\right)\left(\mathcal{Q}\left(W_{M},D^{k}\mathring{\varepsilon,N}\right)\right) \\
&+\operatorname{Duh}_{Y,B,K}\left(\mathcal{Q}\left(W_{M},D^{k}\mathring{\varepsilon,N}\right)\right)-\mathcal{Q}\left(W_{M},\operatorname{Duh}_{Y,B,K}\left(D^{k}\mathring{\varepsilon,N}\right)\right) \\
&+\mathcal{Q}\left(W_{M},\operatorname{Duh}_{Y,B,K}\left(D^{k}\mathring{\varepsilon,N}\right)\right)-\mathcal{Q}\left(W_{M},D^{k}\operatorname{Duh}_{Y,B,K}\left(\mathring{\varepsilon,N}\right)\right). \tag*{(533)}
\end{aligned} Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) = ( Duh Y , B , K F − Duh Y , B , K ) ( Q ( W M , D k ε , N ˚ ) ) + Duh Y , B , K ( Q ( W M , D k ε , N ˚ ) ) − Q ( W M , Duh Y , B , K ( D k ε , N ˚ ) ) + Q ( W M , Duh Y , B , K ( D k ε , N ˚ ) ) − Q ( W M , D k Duh Y , B , K ( ε , N ˚ ) ) . (533) ∥ t − η 1 2 (8.63) ∥ L t , x ∞ ≲ T − δ T − 1 2 K − 3 + O ( κ ) ∥ t − η 1 2 W M ∥ L t , x ∞ ∥ D ε , N ˚ ∥ L t , x ∞ \left\|t^{-\frac{\eta_{1}}{2}}\text{(8.63)}\right\|_{L^{\infty}_{t,x}}
\lesssim T^{-\delta}T^{-\frac{1}{2}}K^{-3+O(\kappa)}\left\|t^{-\frac{\eta_{1}}{2}}W_{M}\right\|_{L^{\infty}_{t,x}}\left\|D\mathring{\varepsilon,N}\right\|_{L^{\infty}_{t,x}} t − 2 η 1 (8.63) L t , x ∞ ≲ T − δ T − 2 1 K − 3 + O ( κ ) t − 2 η 1 W M L t , x ∞ D ε , N ˚ L t , x ∞ ≲ T − 1 2 − δ K − 3 M − 1 N ( K M N ) O ( δ 1 ) ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ ε , N ˚ ∥ m s o l N − κ . \lesssim T^{-\frac{1}{2}-\delta}K^{-3}M^{-1}N(KMN)^{O(\delta_{1})}\left\|W_{M}\right\|_{\mathrm{sol}_{M}^{1-\delta_{1},\eta_{1}}}\left\|\mathring{\varepsilon,N}\right\|_{\mathrm{msol}_{N}^{-\kappa}}. ≲ T − 2 1 − δ K − 3 M − 1 N ( K M N ) O ( δ 1 ) ∥ W M ∥ sol M 1 − δ 1 , η 1 ε , N ˚ msol N − κ . Using Lemma 8.30, the first term can be estimated by
Since M ≳ T − 1 2 M\gtrsim T^{-\frac{1}{2}} M ≳ T − 2 1 K ∼ θ ′ N K\sim_{\theta'}N K ∼ θ ′ N K − 2 + O ( θ ′ ) N − κ K^{-2+O(\theta')}N^{-\kappa} K − 2 + O ( θ ′ ) N − κ θ ′ ≪ λ 1 \theta'\ll\lambda_{1} θ ′ ≪ λ 1 K ≳ T − 1 2 K\gtrsim T^{-\frac{1}{2}} K ≳ T − 2 1
∥ t − η 1 2 (8.64) ∥ L t , x ∞ ≲ T − δ ( K − 4 ∥ t − η 1 2 ( ∂ t − Δ B ) W M ∥ L t , x ∞ + K − 3 ∥ t − η 1 2 D W M ∥ L t , x ∞ ) ∥ D ε , N ˚ ∥ L t , x ∞ \left\|t^{-\frac{\eta_{1}}{2}}\text{(8.64)}\right\|_{L^{\infty}_{t,x}}
\lesssim T^{-\delta}\left(K^{-4}\left\|t^{-\frac{\eta_{1}}{2}}(\partial_{t}-\Delta_{B})W_{M}\right\|_{L^{\infty}_{t,x}}+K^{-3}\left\|t^{-\frac{\eta_{1}}{2}}DW_{M}\right\|_{L^{\infty}_{t,x}}\right)\left\|D\mathring{\varepsilon,N}\right\|_{L^{\infty}_{t,x}} t − 2 η 1 (8.64) L t , x ∞ ≲ T − δ ( K − 4 t − 2 η 1 ( ∂ t − Δ B ) W M L t , x ∞ + K − 3 t − 2 η 1 D W M L t , x ∞ ) D ε , N ˚ L t , x ∞ ≲ T − δ ( K − 4 M 1 + δ 1 + K − 3 M δ 1 ) N 1 + κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ ε , N ˚ ∥ m s o l N − κ . \lesssim T^{-\delta}\left(K^{-4}M^{1+\delta_{1}}+K^{-3}M^{\delta_{1}}\right)N^{1+\kappa}\left\|W_{M}\right\|_{\mathrm{sol}_{M}^{1-\delta_{1},\eta_{1}}}\left\|\mathring{\varepsilon,N}\right\|_{\mathrm{msol}_{N}^{-\kappa}}. ≲ T − δ ( K − 4 M 1 + δ 1 + K − 3 M δ 1 ) N 1 + κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ε , N ˚ msol N − κ . Due to our assumptions M ≪ θ N M\ll_{\theta}N M ≪ θ N K ∼ θ ′ N K\sim_{\theta'}N K ∼ θ ′ N M ≤ N ≲ K 1 + O ( θ ′ ) M\leq N\lesssim K^{1+O(\theta')} M ≤ N ≲ K 1 + O ( θ ′ ) K − 2 + O ( θ ′ ) N − κ K^{-2+O(\theta')}N^{-\kappa} K − 2 + O ( θ ′ ) N − κ
∥ t − η 1 2 (8.65) ∥ L t , x ∞ ≲ ∥ t − η 1 2 W M ∥ L t , x ∞ ∥ Duh Y , B , K ( D k ε , N ˚ ) − D k Duh Y , B , K ( ε , N ˚ ) ∥ L t , x ∞ \left\|t^{-\frac{\eta_{1}}{2}}\text{(8.65)}\right\|_{L^{\infty}_{t,x}}
\lesssim\left\|t^{-\frac{\eta_{1}}{2}}W_{M}\right\|_{L^{\infty}_{t,x}}\left\|\operatorname{Duh}_{Y,B,K}\left(D^{k}\mathring{\varepsilon,N}\right)-D^{k}\operatorname{Duh}_{Y,B,K}\left(\mathring{\varepsilon,N}\right)\right\|_{L^{\infty}_{t,x}} t − 2 η 1 (8.65) L t , x ∞ ≲ t − 2 η 1 W M L t , x ∞ Duh Y , B , K ( D k ε , N ˚ ) − D k Duh Y , B , K ( ε , N ˚ ) L t , x ∞ ≲ T − δ T − 1 2 K − 2 + 2 κ ∥ t − η 1 2 W M ∥ L t , x ∞ ∥ ε , N ˚ ∥ L t , x ∞ \lesssim T^{-\delta}T^{-\frac{1}{2}}K^{-2+2\kappa}\left\|t^{-\frac{\eta_{1}}{2}}W_{M}\right\|_{L^{\infty}_{t,x}}\left\|\mathring{\varepsilon,N}\right\|_{L^{\infty}_{t,x}} ≲ T − δ T − 2 1 K − 2 + 2 κ t − 2 η 1 W M L t , x ∞ ε , N ˚ L t , x ∞ ≲ T − δ T − 1 2 K − 2 + 2 κ M − 1 + δ 1 N κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ ε , N ˚ ∥ m s o l N − κ . \lesssim T^{-\delta}T^{-\frac{1}{2}}K^{-2+2\kappa}M^{-1+\delta_{1}}N^{\kappa}\left\|W_{M}\right\|_{\mathrm{sol}_{M}^{1-\delta_{1},\eta_{1}}}\left\|\mathring{\varepsilon,N}\right\|_{\mathrm{msol}_{N}^{-\kappa}}. ≲ T − δ T − 2 1 K − 2 + 2 κ M − 1 + δ 1 N κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ε , N ˚ msol N − κ . Since, as before, M ≤ N ≤ K 1 + O ( θ ′ ) M\leq N\leq K^{1+O(\theta')} M ≤ N ≤ K 1 + O ( θ ′ ) M ≳ T − 1 2 M\gtrsim T^{-\frac{1}{2}} M ≳ T − 2 1 K − 2 + O ( θ ′ ) N − κ K^{-2+O(\theta')}N^{-\kappa} K − 2 + O ( θ ′ ) N − κ
The case K ̸ ∼ θ ′ N K\not\sim_{\theta'}N K ∼ θ ′ N (532) : In this case, we primarily rely on Lemma 8.28. It implies that
∥ t − η 1 2 Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) ∥ L t , x ∞ ≲ T 1 2 − λ K − 1 − 2 θ N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ ε , N ˚ ∥ m s o l N − κ . \left\|t^{-\frac{\eta_{1}}{2}}\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}\left(W_{M},D^{k}\mathring{\varepsilon,N}\right)\right)\right\|_{L^{\infty}_{t,x}}
\lesssim T^{\frac{1}{2}-\lambda}K^{-1-2\theta}N^{-\kappa}\left\|W_{M}\right\|_{\mathrm{sol}_{M}^{1-\delta_{1},\eta_{1}}}\left\|\mathring{\varepsilon,N}\right\|_{\mathrm{msol}_{N}^{-\kappa}}. t − 2 η 1 Duh Y , B , K F ( Q ( W M , D k ε , N ˚ ) ) L t , x ∞ ≲ T 2 1 − λ K − 1 − 2 θ N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ε , N ˚ msol N − κ . as well as
∥ t − η 1 2 Q ( W M , D k D u h Y , B , K ( I ε , N ) ) ∥ L t , x ∞ ≲ ∥ t − η 1 2 W M ∥ L t , x ∞ ∥ D k D u h Y , B , K ( I ε , N ) ∥ L t , x ∞ (534) \left\|t^{-\frac{\eta_1}{2}}\mathcal{Q}(W_M,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N}))\right\|_{L^\infty_{t,x}}
\lesssim\left\|t^{-\frac{\eta_1}{2}}W_M\right\|_{L^\infty_{t,x}}\left\|D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N})\right\|_{L^\infty_{t,x}} \tag*{(534)} t − 2 η 1 Q ( W M , D k Duh Y , B , K ( I ε , N )) L t , x ∞ ≲ t − 2 η 1 W M L t , x ∞ D k Duh Y , B , K ( I ε , N ) L t , x ∞ (534) ≲ M − ( 1 − δ 1 ) K − 1 − 2 θ N − κ ∥ W M ∥ s o l M 1 − δ 1 , η 1 ∥ I ε , N ∥ m s o l N − κ . \lesssim M^{-(1-\delta_1)}K^{-1-2\theta}N^{-\kappa}\|W_M\|_{\mathrm{sol}^{1-\delta_1,\eta_1}_M}\|\mathfrak{I}_{\varepsilon,N}\|_{\mathrm{msol}^{-\kappa}_N}. ≲ M − ( 1 − δ 1 ) K − 1 − 2 θ N − κ ∥ W M ∥ sol M 1 − δ 1 , η 1 ∥ I ε , N ∥ msol N − κ . Since M ≥ N l o ∼ T − 1 2 M\ge N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} M ≥ N lo ∼ T − 2 1
The estimate (531) : In order to prove (531) , we decompose
D u h Y , B , K F ( Q ( W ⋆ D k I ε , ⋆ ) ) − Q ( W , D k D u h Y , B , K ( I ε ) ) = ∑ M , N : M ≪ θ N ( D u h Y , B , K F ( Q ( W M , D k I ε , N ) ) − Q ( W M , D k D u h Y , B , K ( I ε , N ) ) ) (535) \begin{aligned}
\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W\star D^k\mathfrak{I}_{\varepsilon,\star})\right)-\mathcal{Q}\left(W,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon})\right) \\
&= \sum_{\substack{M,N:\\ M\ll_{\theta}N}}\left(\mathrm{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}(W_M,D^k\mathfrak{I}_{\varepsilon,N})\right)-\mathcal{Q}\left(W_M,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N})\right)\right) \tag*{(535)}
\end{aligned} Duh Y , B , K F ( Q ( W ⋆ D k I ε , ⋆ ) ) − Q ( W , D k Duh Y , B , K ( I ε ) ) = M , N : M ≪ θ N ∑ ( Duh Y , B , K F ( Q ( W M , D k I ε , N ) ) − Q ( W M , D k Duh Y , B , K ( I ε , N ) ) ) (535) − ∑ M , N : M ≳ θ N , K ∼ θ ′ N Q ( W M , D k D u h Y , B , K ( I ε , N ) ) (536) -\sum_{\substack{M,N:\\ M\gtrsim_{\theta}N,\\ K\sim_{\theta'}N}}\mathcal{Q}\left(W_M,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N})\right) \tag*{(536)} − M , N : M ≳ θ N , K ∼ θ ′ N ∑ Q ( W M , D k Duh Y , B , K ( I ε , N ) ) (536) − ∑ M , N : M ≳ θ N , K ̸ ∼ θ ′ N Q ( W M , D k D u h Y , B , K ( I ε , N ) ) . (537) -\sum_{\substack{M,N:\\ M\gtrsim_{\theta}N,\\ K\not\sim_{\theta'}N}}\mathcal{Q}\left(W_M,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N})\right). \tag*{(537)} − M , N : M ≳ θ N , K ∼ θ ′ N ∑ Q ( W M , D k Duh Y , B , K ( I ε , N ) ) . (537) We remark that M ≳ θ N M\gtrsim_{\theta}N M ≳ θ N (383) , simply means that M ≫ θ N M\gg_{\theta}N M ≫ θ N M ∼ θ N M\sim_{\theta}N M ∼ θ N (535) can easily be bounded using our frequency-localised estimate (532) . Using our Schauder estimate (423) , the summands in the second term (536) can be bounded by
∥ t − η 1 2 Q ( W M , D k D u h Y , B , K ( I ε , N ) ) ∥ L t , x ∞ ≲ ∥ t − η 1 2 W M ∥ L t , x ∞ ∥ D k D u h Y , B , K ( I ε , N ) ∥ L t , x ∞ \left\|t^{-\frac{\eta_1}{2}}\mathcal{Q}(W_M,D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N}))\right\|_{L^\infty_{t,x}}
\lesssim\left\|t^{-\frac{\eta_1}{2}}W_M\right\|_{L^\infty_{t,x}}\left\|D^k\mathrm{Duh}_{Y,B,K}(\mathfrak{I}_{\varepsilon,N})\right\|_{L^\infty_{t,x}} t − 2 η 1 Q ( W M , D k Duh Y , B , K ( I ε , N )) L t , x ∞ ≲ t − 2 η 1 W M L t , x ∞ D k Duh Y , B , K ( I ε , N ) L t , x ∞ ≲ ∥ t − η 1 2 W M ∥ L t , x ∞ T − δ K − 1 ∥ I ε , N ∥ L t , x ∞ ≲ T − δ M − ( 1 − δ 1 ) K − 1 N κ ∥ W ⋆ ∥ s o l 1 − δ 1 , η 1 ∥ I ε , ⋆ ∥ m s o l − κ . \lesssim\left\|t^{-\frac{\eta_1}{2}}W_M\right\|_{L^\infty_{t,x}}T^{-\delta}K^{-1}\|\mathfrak{I}_{\varepsilon,N}\|_{L^\infty_{t,x}}
\lesssim T^{-\delta}M^{-(1-\delta_1)}K^{-1}N^\kappa\|W_\star\|_{\mathrm{sol}^{1-\delta_1,\eta_1}}\|\mathfrak{I}_{\varepsilon,\star}\|_{\mathrm{msol}^{-\kappa}}. ≲ t − 2 η 1 W M L t , x ∞ T − δ K − 1 ∥ I ε , N ∥ L t , x ∞ ≲ T − δ M − ( 1 − δ 1 ) K − 1 N κ ∥ W ⋆ ∥ sol 1 − δ 1 , η 1 ∥ I ε , ⋆ ∥ msol − κ . Since K ≥ N l o ∼ T − 1 2 K\ge N_{\mathrm{lo}}\sim T^{-\frac{1}{2}} K ≥ N lo ∼ T − 2 1
∑ M , N : M ≳ θ N , K ∼ θ ′ N M − ( 1 − δ 1 ) K − 1 N κ ≲ ∑ K ∼ θ ′ N N − ( 1 − θ ) ( 1 − δ 1 ) + κ K − 1 ≲ K − ( 1 − θ ′ ) ( ( 1 − θ ) ( 1 − δ 1 ) − κ ) K − 1 , \sum_{\substack{M,N:\\ M\gtrsim_{\theta}N,\\ K\sim_{\theta'}N}}M^{-(1-\delta_1)}K^{-1}N^\kappa
\lesssim\sum_{K\sim_{\theta'}N}N^{-(1-\theta)(1-\delta_1)+\kappa}K^{-1}
\lesssim K^{-(1-\theta')((1-\theta)(1-\delta_1)-\kappa)}K^{-1}, M , N : M ≳ θ N , K ∼ θ ′ N ∑ M − ( 1 − δ 1 ) K − 1 N κ ≲ K ∼ θ ′ N ∑ N − ( 1 − θ ) ( 1 − δ 1 ) + κ K − 1 ≲ K − ( 1 − θ ′ ) (( 1 − θ ) ( 1 − δ 1 ) − κ ) K − 1 , this shows that the contribution of (536) is of size ∼ K − 2 + O ( θ ′ ) \sim K^{-2+O(\theta')} ∼ K − 2 + O ( θ ′ ) (534) , the third summand (537) can be bounded by
∥ t − η 1 2 ( 8.69 ) ∥ L t , x ∞ ≲ ∑ M , N : M ≳ θ N , K ̸ ∼ θ ′ N M − ( 1 − δ 1 ) K − 1 − 2 θ N − κ ∥ W M ∥ s o l 1 − δ 1 , η 1 ∥ I ε , N ∥ m s o l − κ \left\|t^{-\frac{\eta_1}{2}}(8.69)\right\|_{L^\infty_{t,x}}
\lesssim\sum_{\substack{M,N:\\ M\gtrsim_{\theta}N,\\ K\not\sim_{\theta'}N}}M^{-(1-\delta_1)}K^{-1-2\theta}N^{-\kappa}\|W_M\|_{\mathrm{sol}^{1-\delta_1,\eta_1}}\|\mathfrak{I}_{\varepsilon,N}\|_{\mathrm{msol}^{-\kappa}} t − 2 η 1 ( 8.69 ) L t , x ∞ ≲ M , N : M ≳ θ N , K ∼ θ ′ N ∑ M − ( 1 − δ 1 ) K − 1 − 2 θ N − κ ∥ W M ∥ sol 1 − δ 1 , η 1 ∥ I ε , N ∥ msol − κ ≲ T 1 − δ 1 2 K − 1 − 2 θ ∥ W ⋆ ∥ s o l 1 − δ 1 , η 1 ∥ I ε , ⋆ ∥ m s o l − κ , \lesssim T^{\frac{1-\delta_1}{2}}K^{-1-2\theta}\|W_\star\|_{\mathrm{sol}^{1-\delta_1,\eta_1}}\|\mathfrak{I}_{\varepsilon,\star}\|_{\mathrm{msol}^{-\kappa}}, ≲ T 2 1 − δ 1 K − 1 − 2 θ ∥ W ⋆ ∥ sol 1 − δ 1 , η 1 ∥ I ε , ⋆ ∥ msol − κ , which is clearly acceptable.
Equipped with Lemma 8.31, we are now ready to prove the main estimates of this subsection.
Proof of Proposition 8.24. To simplify the notation, we set η 1 : = − 1 + δ 1 − κ \eta_1 := -1 + \delta_1 - \kappa η 1 := − 1 + δ 1 − κ × \times × Lemma 8.31 , we can rewrite the argument in (517) as
Duh Y , B , ⋆ F ( Q ( W ⊗ D ε , ⋄ k ) ) ⊗ D ε , ⋆ ℓ − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) = ∑ K , L : K ∼ θ L ( Duh Y , B , K F ( Q ( W ⊗ D ε , ⋄ k ) ) − Q ( W , D k Duh Y , B , K ( I ε ) ) ) ⊗ D ε , L ℓ (538) \begin{aligned}
\operatorname{Duh}^{F}_{Y,B,\star}\left(\mathcal{Q}\left(W\otimes\mathbb{D}^{k}_{\varepsilon,\diamond}\right)\right)\otimes\mathbb{D}^{\ell}_{\varepsilon,\star}
-\frac{1}{4}\delta^{k\ell}\left(\sigma^{2}_{\rho,\varepsilon}+C_{\rho}\right)\mathcal{K}(W)
={}&\sum_{\substack{K,L:\\K\sim_{\theta}L}}\left(\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}\left(W\otimes\mathbb{D}^{k}_{\varepsilon,\diamond}\right)\right)-\mathcal{Q}\left(W,\mathbb{D}^{k}\operatorname{Duh}_{Y,B,K}\left(\mathcal{I}_{\varepsilon}\right)\right)\right)\otimes\mathbb{D}^{\ell}_{\varepsilon,L} \tag*{(538)}
\end{aligned} Duh Y , B , ⋆ F ( Q ( W ⊗ D ε , ⋄ k ) ) ⊗ D ε , ⋆ ℓ − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) = K , L : K ∼ θ L ∑ ( Duh Y , B , K F ( Q ( W ⊗ D ε , ⋄ k ) ) − Q ( W , D k Duh Y , B , K ( I ε ) ) ) ⊗ D ε , L ℓ (538) + ∑ K , L : K ∼ θ L Q ( W , D k Duh Y , B , K ( I ε ) ) ⊗ D ε , L ℓ − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) . (539) +\sum_{\substack{K,L:\\K\sim_{\theta}L}}\mathcal{Q}\left(W,\mathbb{D}^{k}\operatorname{Duh}_{Y,B,K}\left(\mathcal{I}_{\varepsilon}\right)\right)\otimes\mathbb{D}^{\ell}_{\varepsilon,L}
-\frac{1}{4}\delta^{k\ell}\left(\sigma^{2}_{\rho,\varepsilon}+C_{\rho}\right)\mathcal{K}(W). \tag*{(539)} + K , L : K ∼ θ L ∑ Q ( W , D k Duh Y , B , K ( I ε ) ) ⊗ D ε , L ℓ − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) K ( W ) . (539) Using Lemma 7.7 and our weighted Schauder estimate (436) , as well as the parameter identity δ 1 = δ 2 \delta_1=\delta_2 δ 1 = δ 2
∥ ( 8.70 ) ∥ n l − 1 + δ 2 , − λ ≲ ∥ ( 8.70 ) ∥ n l 0 , − 1 + δ 1 − λ ≲ T λ − κ 2 − δ ∥ ( 8.70 ) ∥ n l 0 , η 1 \left\|(8.70)\right\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\lesssim\left\|(8.70)\right\|_{\mathrm{nl}^{0,-1+\delta_1-\lambda}}\lesssim T^{\frac{\lambda-\kappa}{2}-\delta}\left\|(8.70)\right\|_{\mathrm{nl}^{0,\eta_1}} ∥ ( 8.70 ) ∥ nl − 1 + δ 2 , − λ ≲ ∥ ( 8.70 ) ∥ nl 0 , − 1 + δ 1 − λ ≲ T 2 λ − κ − δ ∥ ( 8.70 ) ∥ nl 0 , η 1 ≲ T λ − κ 2 − δ ∑ K , L : K ∼ θ L ∥ D ε , L ℓ ∥ L t , x ∞ ∥ t − η 1 2 ( Duh Y , B , K F ( Q ( W ⊗ D ε , ⋄ k ) ) − Q ( W , D k Duh Y , B , K ( I ε ) ) ) ∥ L t , x ∞ . \lesssim T^{\frac{\lambda-\kappa}{2}-\delta}\sum_{\substack{K,L:\\K\sim_{\theta}L}}\left\|\mathbb{D}^{\ell}_{\varepsilon,L}\right\|_{L^\infty_{t,x}}\left\|t^{-\frac{\eta_1}{2}}\left(\operatorname{Duh}^{F}_{Y,B,K}\left(\mathcal{Q}\left(W\otimes\mathbb{D}^{k}_{\varepsilon,\diamond}\right)\right)-\mathcal{Q}\left(W,\mathbb{D}^{k}\operatorname{Duh}_{Y,B,K}\left(\mathcal{I}_{\varepsilon}\right)\right)\right)\right\|_{L^\infty_{t,x}}. ≲ T 2 λ − κ − δ K , L : K ∼ θ L ∑ D ε , L ℓ L t , x ∞ t − 2 η 1 ( Duh Y , B , K F ( Q ( W ⊗ D ε , ⋄ k ) ) − Q ( W , D k Duh Y , B , K ( I ε ) ) ) L t , x ∞ . We can now estimate the commutator term using Lemma 8.31 which, together with the definition of ∥ Ξ B , ε ∥ \lVert\Xi_{B,\varepsilon}\rVert ∥ Ξ B , ε ∥
T λ − κ 2 − δ T 1 − λ 2 ∥ Ξ B , ε ∥ 2 ∥ W ⋆ ∥ s o l 1 − δ 1 , η 1 ∑ K , L : K ∼ θ L K − 1 − 2 θ L 1 + κ . T^{\frac{\lambda-\kappa}{2}-\delta}T^{\frac{1-\lambda}{2}}\lVert\Xi_{B,\varepsilon}\rVert^{2}\lVert W_{\star}\rVert_{\mathrm{sol}^{1-\delta_1,\eta_1}}\sum_{\substack{K,L:\\K\sim_{\theta}L}}K^{-1-2\theta}L^{1+\kappa}. T 2 λ − κ − δ T 2 1 − λ ∥ Ξ B , ε ∥ 2 ∥ W ⋆ ∥ sol 1 − δ 1 , η 1 K , L : K ∼ θ L ∑ K − 1 − 2 θ L 1 + κ . Since the exponent of K K K − 1 − 2 θ -1-2\theta − 1 − 2 θ − 1 − θ -1-\theta − 1 − θ κ \kappa κ θ \theta θ (539) as
L ( W ⊗ ( ∑ K , L : K ∼ θ L D k Duh Y , B , K ( I ε ) ⊗ D ε , L ℓ − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y ) ) . \mathcal{L}\left(W\otimes\left(\sum_{\substack{K,L:\\K\sim_{\theta}L}}\mathbb{D}^{k}\operatorname{Duh}_{Y,B,K}\left(\mathcal{I}_{\varepsilon}\right)\otimes\mathbb{D}^{\ell}_{\varepsilon,L}-\frac{1}{4}\delta^{k\ell}\left(\sigma^{2}_{\rho,\varepsilon}+C_{\rho}\right)\operatorname{Cov}_{Y}\right)\right). L W ⊗ K , L : K ∼ θ L ∑ D k Duh Y , B , K ( I ε ) ⊗ D ε , L ℓ − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y . Here, L : Y ⊗ Y ⊗ Y → Y ⊗ Y \mathcal{L}:Y\otimes Y\otimes Y\to Y\otimes Y L : Y ⊗ Y ⊗ Y → Y ⊗ Y L ( X ⊗ Y ⊗ Z ) : = Q ( X , Y ) ⊗ Z \mathcal{L}(X\otimes Y\otimes Z):=\mathcal{Q}(X,Y)\otimes Z L ( X ⊗ Y ⊗ Z ) := Q ( X , Y ) ⊗ Z (539) , we first use Lemma 7.7 and then use our weighted covariant Schauder estimate (437) with
( α , β , ν , η ) : = ( − 3 θ , − 2 θ , − 1 + δ 2 + 3 θ − λ , η 1 ) . (\alpha,\beta,\nu,\eta):=(-3\theta,-2\theta,-1+\delta_2+3\theta-\lambda,\eta_1). ( α , β , ν , η ) := ( − 3 θ , − 2 θ , − 1 + δ 2 + 3 θ − λ , η 1 ) . Since θ \theta θ 1 r \frac{1}{r} r 1 N α − β + 2 r N^{\alpha-\beta+\frac{2}{r}} N α − β + r 2 (437) is then N − 3 θ + 2 θ + 2 r ≲ 1 N^{-3\theta+2\theta+\frac{2}{r}}\lesssim1 N − 3 θ + 2 θ + r 2 ≲ 1 λ \lambda λ δ 2 \delta_2 δ 2 κ \kappa κ θ \theta θ ν \nu ν η \eta η max ( N l o ν − η , N ν − η ) \max(N_{\mathrm{lo}}^{\nu-\eta},N^{\nu-\eta}) max ( N lo ν − η , N ν − η ) (437) contributes T ν − η 2 = T λ 2 − O ( θ ) T^{\frac{\nu-\eta}{2}}=T^{\frac{\lambda}{2}-O(\theta)} T 2 ν − η = T 2 λ − O ( θ )
∥ ( 8.71 ) ∥ n l − 1 + δ 2 , − λ ≲ ∥ ( 8.71 ) ∥ n l − 3 θ , − 1 + δ 2 + 3 θ − λ ≲ T λ 2 − O ( θ ) ∥ t − η 1 2 ( 8.71 ) ∥ C θ − θ L x r . \left\|(8.71)\right\|_{\mathrm{nl}^{-1+\delta_2,-\lambda}}\lesssim\left\|(8.71)\right\|_{\mathrm{nl}^{-3\theta,-1+\delta_2+3\theta-\lambda}}\lesssim T^{\frac{\lambda}{2}-O(\theta)}\left\|t^{-\frac{\eta_1}{2}}(8.71)\right\|_{\mathcal{C}^{\theta-\theta}L^r_x}. ∥ ( 8.71 ) ∥ nl − 1 + δ 2 , − λ ≲ ∥ ( 8.71 ) ∥ nl − 3 θ , − 1 + δ 2 + 3 θ − λ ≲ T 2 λ − O ( θ ) t − 2 η 1 ( 8.71 ) C θ − θ L x r . Using product estimates (Lemma 3.53), our Hölder estimate (Lemma 7.6), and our bounds on Ξ B , ε \Xi_{B,\varepsilon} Ξ B , ε
T λ 2 − O ( θ ) ∥ t − η 1 2 ( 8.71 ) ∥ C θ − θ L x r T^{\frac{\lambda}{2}-O(\theta)}\left\|t^{-\frac{\eta_1}{2}}(8.71)\right\|_{\mathcal{C}^{\theta-\theta}L^r_x} T 2 λ − O ( θ ) t − 2 η 1 ( 8.71 ) C θ − θ L x r ≲ T λ 2 − O ( θ ) ∥ t − η 1 2 W ∥ C t 2 θ L x ∞ ∥ ∑ K , L : K ∼ θ L D k Duh Y , B , K ( I ε ) ⊗ D ε , L ℓ − 1 4 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y ∥ C θ − θ L x r \lesssim T^{\frac{\lambda}{2}-O(\theta)}\left\|t^{-\frac{\eta_1}{2}}W\right\|_{\mathcal{C}^{2\theta}_{t}L^\infty_x}
\left\|\sum_{\substack{K,L:\\K\sim_{\theta}L}}\mathbb{D}^{k}\operatorname{Duh}_{Y,B,K}\left(\mathcal{I}_{\varepsilon}\right)\otimes\mathbb{D}^{\ell}_{\varepsilon,L}
-\frac{1}{4}\delta^{k\ell}\left(\sigma^{2}_{\rho,\varepsilon}+C_{\rho}\right)\operatorname{Cov}_{Y}\right\|_{\mathcal{C}^{\theta-\theta}L^r_x} ≲ T 2 λ − O ( θ ) t − 2 η 1 W C t 2 θ L x ∞ K , L : K ∼ θ L ∑ D k Duh Y , B , K ( I ε ) ⊗ D ε , L ℓ − 4 1 δ k ℓ ( σ ρ , ε 2 + C ρ ) Cov Y C θ − θ L x r ≲ T λ 2 − O ( θ ) ∥ W ⋆ ∥ s o l 5 θ , η 1 ∥ Ξ B , ε ∥ 2 ≲ T 1 + λ 2 − O ( θ ) ∥ W ⋆ ∥ s o l 1 − δ 1 , η 1 ∥ Ξ B , ε ∥ 2 . \lesssim T^{\frac{\lambda}{2}-O(\theta)}\lVert W_{\star}\rVert_{\mathrm{sol}^{5\theta,\eta_1}}\lVert\Xi_{B,\varepsilon}\rVert^{2}
\lesssim T^{\frac{1+\lambda}{2}-O(\theta)}\lVert W_{\star}\rVert_{\mathrm{sol}^{1-\delta_1,\eta_1}}\lVert\Xi_{B,\varepsilon}\rVert^{2}. ≲ T 2 λ − O ( θ ) ∥ W ⋆ ∥ sol 5 θ , η 1 ∥ Ξ B , ε ∥ 2 ≲ T 2 1 + λ − O ( θ ) ∥ W ⋆ ∥ sol 1 − δ 1 , η 1 ∥ Ξ B , ε ∥ 2 . Thus, the contributions of both (538) and (539) to (517) are acceptable.
In the following corollary, we use Proposition 8.24 to control the specific terms appearing in our evolution equations for Y ε , ⋆ Y_{\varepsilon,\star} Y ε , ⋆ y ε , ⋆ y_{\varepsilon,\star} y ε , ⋆
Corollary 8.33. Let Assumptions 8.1, 8.2, and 8.9 be satisfied. Then, it holds that
\begin{aligned}
&\left\lVert\operatorname{Lb}(X_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathsf{g}}_{g}(W_{\varepsilon})\right\rVert_{n\ell-1+\delta_{2},-\lambda}\\
&\quad+\left\lVert2\mathsf{B}(x_{\varepsilon,\star}(W_{\varepsilon,\diamond})\mathbin{\mathop{\otimes}}D^\circ_{\varepsilon,\star})+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathsf{V}}_{g}(W_{\varepsilon})\right\rVert_{n\ell-1+\delta_{2},-\lambda}\\
&\quad+\left\lVert2X^j_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond})\mathbin{\mathop{\otimes}}D^\circ_{j\mid\varepsilon,\star}+(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathsf{V}}_{g}(w_{\varepsilon})\right\rVert_{n\ell-1+\delta_{2},-\lambda}\\
&\quad+\left\lVert-2(D^\circ_{j\mid\varepsilon,\star}^{\,j})\mathbin{\mathop{\otimes}}x_{\varepsilon,\star}(W_{\varepsilon,\diamond})\right\rVert_{n\ell-1+\delta_{2},-\lambda}\\
&\lesssim T^{\frac{1}{2}-\lambda_{1}}\left(\lVert\varphi_{0}^{\circ}\rVert_{L^{r}_{x}}+\lVert\varphi_{0}^{\circ}\rVert_{C^{-\kappa}_{B(0),T}}+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right)\lVert\Xi_{B,\varepsilon}\rVert^{2}. \tag*{(540)}
\end{aligned}
Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ε , ⋆ ∘ ) = Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) + Lb ( X ε , ⋆ ∘ ( w ε , ⋄ ) , D ε , ⋆ ∘ ) . \begin{aligned}
&\operatorname{Lb}(X_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})\\
&\qquad=\operatorname{Lb}(X^\circ_{\varepsilon,\star}(W_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})+\operatorname{Lb}(X^\circ_{\varepsilon,\star}(w_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star}).
\end{aligned} Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ε , ⋆ ∘ ) = Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) + Lb ( X ε , ⋆ ∘ ( w ε , ⋄ ) , D ε , ⋆ ∘ ) . Proof. We first observe that the terms in (540) –(8.75) are all of the same form as the abstract term in Proposition 8.24. Using Lemma 7.7 and Proposition 8.8 , we have that
∥ ( W ε , ⋆ , w ε , ⋆ ) ∥ s o l 1 − δ 1 , − 1 + δ 1 − κ ≲ T − O ( λ ) ( ∥ φ 0 ∘ ∥ L x r + ∥ φ 0 ∘ ∥ C B ( 0 ) , T − κ + T 1 2 ∥ ( R ε , ⋆ , r ε , ⋆ ) ∥ s o l 1 − δ 1 , − λ ) . \lVert(W_{\varepsilon,\star},w_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_{1},-1+\delta_{1}-\kappa}}\lesssim T^{-O(\lambda)}\left(\lVert\varphi_{0}^{\circ}\rVert_{L^{r}_{x}}+\lVert\varphi_{0}^{\circ}\rVert_{C^{-\kappa}_{B(0),T}}+T^{\frac{1}{2}}\lVert(R_{\varepsilon,\star},r_{\varepsilon,\star})\rVert_{\mathrm{sol}^{1-\delta_{1},-\lambda}}\right). ∥( W ε , ⋆ , w ε , ⋆ ) ∥ sol 1 − δ 1 , − 1 + δ 1 − κ ≲ T − O ( λ ) ( ∥ φ 0 ∘ ∥ L x r + ∥ φ 0 ∘ ∥ C B ( 0 ) , T − κ + T 2 1 ∥( R ε , ⋆ , r ε , ⋆ ) ∥ sol 1 − δ 1 , − λ ) . Thus, due to Proposition 8.24, it only remains to verify that the renormalisation terms in (540) –(8.75) agree with the renormalisation terms resulting from the corresponding applications of (517) . For this, we discuss each term separately.
The term in (540) : We first split
Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ε , ⋆ ∘ ) = Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) + Lb ( X ε , ⋆ ∘ ( w ε , ⋄ ) , D ε , ⋆ ∘ ) . \operatorname{Lb}(X_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})=\operatorname{Lb}(X^\circ_{\varepsilon,\star}(W_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})+\operatorname{Lb}(X^\circ_{\varepsilon,\star}(w_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star}). Lb ( X ε , ⋆ ( W ε , ⋄ , w ε , ⋄ ) , D ε , ⋆ ∘ ) = Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) + Lb ( X ε , ⋆ ∘ ( w ε , ⋄ ) , D ε , ⋆ ∘ ) . Since X ε , ⋆ ∙ X^\bullet_{\varepsilon,\star} X ε , ⋆ ∙ ϕ ε , ⋆ \boldsymbol{\phi}_{\varepsilon,\star} ϕ ε , ⋆ (398) and (403) , we write the first summand above as
Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) = 2 [ X ε , ⋆ , j ∘ ( W ε , ⋄ ) ⊗ ( D ε , ⋆ j − D ε , ⋆ ∘ j ) ] + 2 [ X ε , ⋆ ∘ ( W ε , ⋄ ) ⊗ D ε , ⋆ , j ∘ j ] = 2 [ Duh B , ⋆ , j k F [ W ε , ⋄ , ℓ ⊗ ( 2 D ε , ⋄ ℓ k − D ε , ⋄ ∘ ℓ k ) ] ⊗ ( D ε , ⋆ ∘ j − D ε , ⋆ ∘ j ) ] + 2 [ Duh B , ⋆ F , i k [ W ε , ⋄ , ℓ ⊗ ( 2 D ε , ⋄ ℓ k − D ε , ⋄ ∘ ℓ k ) ] ⊗ D ε , ⋆ , j ∘ j ] . \begin{aligned}
\operatorname{Lb}(X^\circ_{\varepsilon,\star}(W_{\varepsilon,\diamond}),D^\circ_{\varepsilon,\star})&=2[X^\circ_{\varepsilon,\star,j}(W_{\varepsilon,\diamond})\mathbin{\mathop{\otimes}}(D^j_{\varepsilon,\star}-D^{\circ j}_{\varepsilon,\star})]+2[X^\circ_{\varepsilon,\star}(W_{\varepsilon,\diamond})\mathbin{\mathop{\otimes}}D^{\circ j}_{\varepsilon,\star,j}]\\
&=2\left[\operatorname{Duh}^{F}_{B,\star,jk}\left[W_{\varepsilon,\diamond,\ell}\mathbin{\mathop{\otimes}}(2D^{\ell k}_{\varepsilon,\diamond}-D^{\circ\ell k}_{\varepsilon,\diamond})\right]\mathbin{\mathop{\otimes}}(D^{\circ j}_{\varepsilon,\star}-D^{\circ j}_{\varepsilon,\star})\right]\\
&\quad+2\left[\operatorname{Duh}^{F,i k}_{B,\star}\left[W_{\varepsilon,\diamond,\ell}\mathbin{\mathop{\otimes}}(2D^{\ell k}_{\varepsilon,\diamond}-D^{\circ\ell k}_{\varepsilon,\diamond})\right]\mathbin{\mathop{\otimes}}D^{\circ j}_{\varepsilon,\star,j}\right].
\end{aligned} Lb ( X ε , ⋆ ∘ ( W ε , ⋄ ) , D ε , ⋆ ∘ ) = 2 [ X ε , ⋆ , j ∘ ( W ε , ⋄ ) ⊗ ( D ε , ⋆ j − D ε , ⋆ ∘ j )] + 2 [ X ε , ⋆ ∘ ( W ε , ⋄ ) ⊗ D ε , ⋆ , j ∘ j ] = 2 [ Duh B , ⋆ , j k F [ W ε , ⋄ , ℓ ⊗ ( 2 D ε , ⋄ ℓ k − D ε , ⋄ ∘ ℓ k ) ] ⊗ ( D ε , ⋆ ∘ j − D ε , ⋆ ∘ j ) ] + 2 [ Duh B , ⋆ F , ik [ W ε , ⋄ , ℓ ⊗ ( 2 D ε , ⋄ ℓ k − D ε , ⋄ ∘ ℓ k ) ] ⊗ D ε , ⋆ , j ∘ j ] . Here, as in Remark 8.26, we are using Duh B , ⋆ , j k F \operatorname{Duh}^{F}_{B,\star,jk} Duh B , ⋆ , j k F Duh B , ⋆ F \operatorname{Duh}^{F}_{B,\star} Duh B , ⋆ F j j j k k k
2 × 1 4 ( σ ρ , ε 2 + C ρ ) ( δ j k ( 2 δ j ℓ δ i k − 2 δ ℓ i δ k j − δ k j δ ℓ i + δ i k δ j ℓ ) + δ i k ( 2 δ ℓ j δ k j − δ k j δ j ℓ ) ) Cov g g ( W ε , ℓ ) 2\times\frac{1}{4}(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\left(\delta_{jk}(2\delta^{j\ell}\delta^{ik}-2\delta^{\ell i}\delta^{kj}-\delta^{kj}\delta^{\ell i}+\delta^{ik}\delta^{j\ell})+\delta^{ik}(2\delta^{\ell j}\delta_{kj}-\delta^{j}_{k}\delta^{\ell}_{j})\right)\operatorname{Cov}^{\mathsf{g}}_{g}(W_{\varepsilon,\ell}) 2 × 4 1 ( σ ρ , ε 2 + C ρ ) ( δ j k ( 2 δ j ℓ δ ik − 2 δ ℓ i δ k j − δ k j δ ℓ i + δ ik δ j ℓ ) + δ ik ( 2 δ ℓ j δ k j − δ k j δ j ℓ ) ) Cov g g ( W ε , ℓ ) = − ( σ ρ , ε 2 + C ρ ) Cov g g ( W ε ∘ ) , =-(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\operatorname{Cov}^{\mathsf{g}}_{g}(W^\circ_{\varepsilon}), = − ( σ ρ , ε 2 + C ρ ) Cov g g ( W ε ∘ ) , which is consistent with (540) .
The term in (8.73): Using (405) , we have that
2 B ( x ε , ⋆ ( W ε , ⋄ ) ⊗ D ε , ⋆ k ) = 4 B ( Duh B , ⋆ ( W ε , ⋄ ℓ ⊗ D ℓ ∣ ε , ⋄ ) ⊗ D ε , ⋆ k ) . 2\mathsf{B}(x_{\varepsilon,\star}(W_{\varepsilon,\diamond})\mathbin{\mathop{\otimes}}D^k_{\varepsilon,\star})=4\mathsf{B}\left(\operatorname{Duh}_{B,\star}(W^\ell_{\varepsilon,\diamond}\mathbin{\mathop{\otimes}}D_{\ell\mid\varepsilon,\diamond})\mathbin{\mathop{\otimes}}D^k_{\varepsilon,\star}\right). 2 B ( x ε , ⋆ ( W ε , ⋄ ) ⊗ D ε , ⋆ k ) = 4 B ( Duh B , ⋆ ( W ε , ⋄ ℓ ⊗ D ℓ ∣ ε , ⋄ ) ⊗ D ε , ⋆ k ) . Due to Proposition 8.24, we need to subtract the renormalisation term
4 × 1 4 δ ℓ k ( σ ρ , ε 2 + C ρ ) B ( W ε ℓ ⊗ v m ⊗ v m ) , 4\times\frac{1}{4}\delta^{k}_{\ell}(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\mathsf{B}(W^\ell_{\varepsilon}\mathbin{\mathop{\otimes}}v_{m}\mathbin{\mathop{\otimes}}v^{m}), 4 × 4 1 δ ℓ k ( σ ρ , ε 2 + C ρ ) B ( W ε ℓ ⊗ v m ⊗ v m ) , where ( v m ) m (v_m)_m ( v m ) m V \mathbf{V} V B \mathbf{B} B Cov V g \operatorname{Cov}^{g}_{\mathbf{V}} Cov V g
The term in (541) : We first decompose
2 X ε , ⋆ ∘ , j ( W ε , ⋄ , w ε , ⋄ ) ⊗ D j e ε , ⋆ = 2 X ε , ⋆ ∘ , j ( W ε , ⋄ ) ⊗ D j e ε , ⋆ + 2 X ε , ⋆ ∘ , j ( w ε , ⋄ ) ⊗ D j e ε , ⋆ . (541) 2X^{\circ,j}_{\varepsilon,\star}(W_{\varepsilon,\diamond},w_{\varepsilon,\diamond})\mathbin{\otimes}D_j\mathfrak{e}_{\varepsilon,\star}=2X^{\circ,j}_{\varepsilon,\star}(W_{\varepsilon,\diamond})\mathbin{\otimes}D_j\mathfrak{e}_{\varepsilon,\star}+2X^{\circ,j}_{\varepsilon,\star}(w_{\varepsilon,\diamond})\mathbin{\otimes}D_j\mathfrak{e}_{\varepsilon,\star}. \tag*{(541)} 2 X ε , ⋆ ∘ , j ( W ε , ⋄ , w ε , ⋄ ) ⊗ D j e ε , ⋆ = 2 X ε , ⋆ ∘ , j ( W ε , ⋄ ) ⊗ D j e ε , ⋆ + 2 X ε , ⋆ ∘ , j ( w ε , ⋄ ) ⊗ D j e ε , ⋆ . (541) Since X ε , ⋆ ∘ X^{\circ}_{\varepsilon,\star} X ε , ⋆ ∘ e ε , ⋆ \mathfrak{e}_{\varepsilon,\star} e ε , ⋆ (404) , the second summand above can be written as
2 ( Duh B , ⋆ F ( B ( w ε , ⋄ ⊗ D e ε , ⋄ ) ) ) j ⊗ D j e ε , ⋆ . 2\left(\operatorname{Duh}^{F}_{\mathbf{B},\star}\left(\mathbf{B}(w_{\varepsilon,\diamond}\mathbin{\otimes}D\mathfrak{e}_{\varepsilon,\diamond})\right)\right)^{j}\mathbin{\otimes}D_j\mathfrak{e}_{\varepsilon,\star}. 2 ( Duh B , ⋆ F ( B ( w ε , ⋄ ⊗ D e ε , ⋄ ) ) ) j ⊗ D j e ε , ⋆ . Due to Proposition 8.24 (see also Remark 8.26), we need to subtract the renormalisation term
2 × 1 4 δ i j ( σ ρ , ε 2 + C ρ ) B ( w ε ⊗ v m ) v m . 2\times\frac{1}{4}\delta^{ij}(\sigma_{\rho,\varepsilon}^{2}+C_{\rho})\mathbf{B}(w_{\varepsilon}\otimes v_m)v^m. 2 × 4 1 δ ij ( σ ρ , ε 2 + C ρ ) B ( w ε ⊗ v m ) v m . After using Lemma 3.44, we see that this is consistent with (541) .
The term in (8.75): Since x ε , ⋆ x_{\varepsilon,\star} x ε , ⋆ e ε , ⋆ \mathfrak{e}_{\varepsilon,\star} e ε , ⋆ − 2 ( D j e ε , ⋆ φ ) ⊗ x ε , ⋆ ( W ε , ⋄ ) -2(D_j\mathfrak{e}^{\varphi}_{\varepsilon,\star})\mathbin{\otimes}x_{\varepsilon,\star}(W_{\varepsilon,\diamond}) − 2 ( D j e ε , ⋆ φ ) ⊗ x ε , ⋆ ( W ε , ⋄ )
Proof of covariant local well-posedness
Equipped with the estimates from Sections 8.2–8.4, we are now ready to first prove the main result of this section (Proposition 8.3), and then our covariant local well-posedness theorem (Theorem 2.19).
Proof of Proposition 8.3. The proof is based on a contraction-mapping argument. For this, we first define
B : = { ( X ε , ⋆ ∘ , X ε , ⋆ ∙ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) : max ( ∥ X ε , ⋆ ∘ ∥ s o l 1 − δ 1 , 0 , ∥ X ε , ⋆ ∙ ∥ s o l 1 − δ 1 , 0 , \mathbb{B}:=\{(X^{\circ}_{\varepsilon,\star},X^{\bullet}_{\varepsilon,\star},x_{\varepsilon,\star},Y_{\varepsilon,\star},y_{\varepsilon,\star}):\max(\|X^{\circ}_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,0}},\|X^{\bullet}_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,0}}, B := {( X ε , ⋆ ∘ , X ε , ⋆ ∙ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) : max ( ∥ X ε , ⋆ ∘ ∥ sol 1 − δ 1 , 0 , ∥ X ε , ⋆ ∙ ∥ sol 1 − δ 1 , 0 , ∥ x ε , ⋆ ∥ s o l 1 − δ 1 , 0 , ∥ Y ε , ⋆ ∥ s o l 1 + δ 2 , − λ , ∥ y ε , ⋆ ∥ s o l 1 + δ 2 , − λ ) ≤ T − 1 γ 0 − λ 2 \left.
\|x_{\varepsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,0}},\|Y_{\varepsilon,\star}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}},\|y_{\varepsilon,\star}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}}\right)\leq T^{-\frac{1}{\gamma_0}-\lambda_2} ∥ x ε , ⋆ ∥ sol 1 − δ 1 , 0 , ∥ Y ε , ⋆ ∥ sol 1 + δ 2 , − λ , ∥ y ε , ⋆ ∥ sol 1 + δ 2 , − λ ) ≤ T − γ 0 1 − λ 2 and ( X ε , N ∘ , X ε , N ∙ , x ε , N , Y ε , N , y ε , N ) ( t ) = 0 for all N ≥ N l o and 0 ≤ t ≤ 1 16 N − 2 } . (542) \text{and }(X^{\circ}_{\varepsilon,N},X^{\bullet}_{\varepsilon,N},x_{\varepsilon,N},Y_{\varepsilon,N},y_{\varepsilon,N})(t)=0\text{ for all }N\geq N_{\mathrm{lo}}\text{ and }0\leq t\leq\frac{1}{16}N^{-2}\}. \tag*{(542)} and ( X ε , N ∘ , X ε , N ∙ , x ε , N , Y ε , N , y ε , N ) ( t ) = 0 for all N ≥ N lo and 0 ≤ t ≤ 16 1 N − 2 } . (542) For notational convenience, we also write X ε , ⋆ : = ( X ε , ⋆ ∘ , X ε , ⋆ ∙ , x ε , ⋆ ) \mathcal{X}_{\varepsilon,\star}\mathrel{\mathop:}= (X^{\circ}_{\varepsilon,\star},X^{\bullet}_{\varepsilon,\star},x_{\varepsilon,\star}) X ε , ⋆ : = ( X ε , ⋆ ∘ , X ε , ⋆ ∙ , x ε , ⋆ ) Y ε , ⋆ : = ( Y ε , ⋆ , y ε , ⋆ ) \mathcal{Y}_{\varepsilon,\star}\mathrel{\mathop:}= (Y_{\varepsilon,\star},y_{\varepsilon,\star}) Y ε , ⋆ : = ( Y ε , ⋆ , y ε , ⋆ )
We further define a map Γ ε ( X ε , ⋆ , Y ε , ⋆ ) = ( Γ ε , X , Γ ε , Y ) ( X ε , ⋆ , Y ε , ⋆ ) \Gamma_{\varepsilon}(\mathcal{X}_{\varepsilon,\star},\mathcal{Y}_{\varepsilon,\star})=(\Gamma_{\varepsilon,X},\Gamma_{\varepsilon,Y})(\mathcal{X}_{\varepsilon,\star},\mathcal{Y}_{\varepsilon,\star}) Γ ε ( X ε , ⋆ , Y ε , ⋆ ) = ( Γ ε , X , Γ ε , Y ) ( X ε , ⋆ , Y ε , ⋆ )
Γ ε , X : = ( ( 6.69 ) , ( 6.70 ) , ( 6.71 ) ) and Γ ε , Y : = ( ( 6.72 ) − – ( 6.82 ) , ( 6.83 ) − – ( 6.90 ) ) . \Gamma_{\varepsilon,X}\mathrel{\mathop:}=\left((6.69),(6.70),(6.71)\right)
\quad\text{and}\quad
\Gamma_{\varepsilon,Y}\mathrel{\mathop:}=\left((6.72)-\text{--}(6.82),(6.83)-\text{--}(6.90)\right). Γ ε , X : = ( ( 6.69 ) , ( 6.70 ) , ( 6.71 ) ) and Γ ε , Y : = ( ( 6.72 ) − – ( 6.82 ) , ( 6.83 ) − – ( 6.90 ) ) . In the individual terms in (406) –(6.90) and the estimates from Sections 8.2–8.4, which will be used momentarily, we use that
W ε , ⋆ = S ⋆ + e ε , ⋆ ∘ + e ε , ⋆ ∙ + X ε , ⋆ ∘ + X ε , ⋆ ∙ + Y ε , ⋆ , w ε , ⋆ = φ ⋆ + e ε , ⋆ ∘ + x ε , ⋆ + y ε , ⋆ , W_{\varepsilon,\star}=S_{\star}+\mathfrak{e}_{\varepsilon,\star}^{\circ}+\mathfrak{e}_{\varepsilon,\star}^{\bullet}+X^{\circ}_{\varepsilon,\star}+X^{\bullet}_{\varepsilon,\star}+Y_{\varepsilon,\star},
\qquad
w_{\varepsilon,\star}=\varphi_{\star}+\mathfrak{e}_{\varepsilon,\star}^{\circ}+x_{\varepsilon,\star}+y_{\varepsilon,\star}, W ε , ⋆ = S ⋆ + e ε , ⋆ ∘ + e ε , ⋆ ∙ + X ε , ⋆ ∘ + X ε , ⋆ ∙ + Y ε , ⋆ , w ε , ⋆ = φ ⋆ + e ε , ⋆ ∘ + x ε , ⋆ + y ε , ⋆ , R ε , ⋆ = e ε , ⋆ ∘ + e ε , ⋆ ∙ + X ε , ⋆ ∘ + X ε , ⋆ ∙ + Y ε , ⋆ , r ε , ⋆ = e ε , ⋆ ∘ + x ε , ⋆ + y ε , ⋆ . R_{\varepsilon,\star}=\mathfrak{e}_{\varepsilon,\star}^{\circ}+\mathfrak{e}_{\varepsilon,\star}^{\bullet}+X^{\circ}_{\varepsilon,\star}+X^{\bullet}_{\varepsilon,\star}+Y_{\varepsilon,\star},
\qquad
r_{\varepsilon,\star}=\mathfrak{e}_{\varepsilon,\star}^{\circ}+x_{\varepsilon,\star}+y_{\varepsilon,\star}. R ε , ⋆ = e ε , ⋆ ∘ + e ε , ⋆ ∙ + X ε , ⋆ ∘ + X ε , ⋆ ∙ + Y ε , ⋆ , r ε , ⋆ = e ε , ⋆ ∘ + x ε , ⋆ + y ε , ⋆ . That is, we view every term as a map depending on the given objects S S S φ \varphi φ Ξ B , ε \Xi_{\mathbf{B},\varepsilon} Ξ B , ε (542) . As above, we write R ε , ⋆ = ( R ε , ⋆ , r ε , ⋆ ) R_{\varepsilon,\star}=(R_{\varepsilon,\star},r_{\varepsilon,\star}) R ε , ⋆ = ( R ε , ⋆ , r ε , ⋆ ) Γ ε \Gamma_{\varepsilon} Γ ε Γ ε \Gamma_{\varepsilon} Γ ε B \mathbb{B} B B \mathbb{B} B Γ ε \Gamma_{\varepsilon} Γ ε ∥ Ξ B , ε ∥ ≤ T − λ 1 \|\Xi_{\mathbf{B},\varepsilon}\|\leq T^{-\lambda_1} ∥ Ξ B , ε ∥ ≤ T − λ 1 (473) is satisfied, Γ ϵ \Gamma_{\epsilon} Γ ϵ Ξ ‾ B , ϵ \overline{\Xi}_{B,\epsilon} Ξ B , ϵ Γ ϵ \Gamma_{\epsilon} Γ ϵ
max ( ∥ Γ ϵ , X ∥ s o l 1 − δ 1 , 0 , ∥ Γ ϵ , Y ∥ s o l 1 + δ 2 , − λ ) ≲ T 1 2 λ 1 ∥ Ξ ‾ B , ϵ ∥ 3 + T − λ 1 ∥ Ξ ‾ B , ϵ ∥ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ + T 1 2 ∥ Y ϵ , ⋆ ∥ s o l 1 + δ 2 , − λ ) + T 1 2 λ 2 − λ 1 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) + T 1 2 λ 2 − λ 1 ( ∥ Ξ ‾ B , ϵ ∥ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ∥ Ξ ‾ B , ϵ ∥ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) 2 + T 1 2 λ 2 − λ 1 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ∥ Ξ ‾ B , ϵ ∥ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) + T 1 2 λ 2 − λ 1 ∥ Ξ ‾ B , ϵ ∥ 2 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 1 2 ∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ) . (543) \begin{aligned}
\max\bigl(&\|\Gamma_{\epsilon,X}\|_{\mathrm{sol}^{1-\delta_1,0}},\|\Gamma_{\epsilon,Y}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}}\bigr) \\
&\lesssim T^{\frac{1}{2}\lambda_1}\|\overline{\Xi}_{B,\epsilon}\|^3 \\
&\quad+T^{-\lambda_1}\|\overline{\Xi}_{B,\epsilon}\|\bigl(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}+T^{\frac{1}{2}}\|\mathcal{Y}_{\epsilon,\star}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}}\bigr) \\
&\quad+T^{\frac{1}{2}\lambda_2-\lambda_1}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigl(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigr) \\
&\quad+T^{\frac{1}{2}\lambda_2-\lambda_1}\bigl(\|\overline{\Xi}_{B,\epsilon}\|+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigr)\bigl(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}+\|\overline{\Xi}_{B,\epsilon}\|+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigr)^2 \\
&\quad+T^{\frac{1}{2}\lambda_2-\lambda_1}\bigl(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}+\|\overline{\Xi}_{B,\epsilon}\|+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigr) \\
&\quad+T^{\frac{1}{2}\lambda_2-\lambda_1}\|\overline{\Xi}_{B,\epsilon}\|^2\bigl(\|\varphi_0^+\|_{L_x^r}+\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}+T^{\frac{1}{2}}\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\bigr). \tag*{(543)}
\end{aligned} max ( ∥ Γ ϵ , X ∥ sol 1 − δ 1 , 0 , ∥ Γ ϵ , Y ∥ sol 1 + δ 2 , − λ ) ≲ T 2 1 λ 1 ∥ Ξ B , ϵ ∥ 3 + T − λ 1 ∥ Ξ B , ϵ ∥ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ + T 2 1 ∥ Y ϵ , ⋆ ∥ sol 1 + δ 2 , − λ ) + T 2 1 λ 2 − λ 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) + T 2 1 λ 2 − λ 1 ( ∥ Ξ B , ϵ ∥ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ∥ Ξ B , ϵ ∥ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) 2 + T 2 1 λ 2 − λ 1 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + ∥ Ξ B , ϵ ∥ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) + T 2 1 λ 2 − λ 1 ∥ Ξ B , ϵ ∥ 2 ( ∥ φ 0 + ∥ L x r + ∥ φ 0 − ∥ C B ( 0 ) , T − κ + T 2 1 ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ) . (543) Indeed, once (543) has been proven, the fact that Γ ϵ \Gamma_{\epsilon} Γ ϵ B \mathbf{B} B B \mathbf{B} B γ 0 − 2 \gamma_0-2 γ 0 − 2 λ 2 \lambda_2 λ 2 ∥ φ 0 + ∥ L x r , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − 1 γ 0 \|\varphi_0^+\|_{L_x^r},\|\varphi_0^-\|_{C_{B(0),T}^{-\kappa}}\leq T^{-\frac{1}{\gamma_0}} ∥ φ 0 + ∥ L x r , ∥ φ 0 − ∥ C B ( 0 ) , T − κ ≤ T − γ 0 1 ∥ Ξ ‾ B , ϵ ∥ ≤ T − λ 1 \|\overline{\Xi}_{B,\epsilon}\|\leq T^{-\lambda_1} ∥ Ξ B , ϵ ∥ ≤ T − λ 1
∥ R ϵ , ⋆ ∥ s o l 1 − δ 1 , − λ ≲ ∥ Ξ ‾ B , ϵ ∥ 2 + ∥ X ϵ , ⋆ ∥ s o l 1 − δ 1 , 0 + ∥ Y ϵ , ⋆ ∥ s o l 1 + δ 2 , − λ ≲ T − 2 λ 1 + T − 1 γ 0 − λ 2 ∼ T − 1 γ 0 − λ 2 . \|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,-\lambda}}\lesssim\|\overline{\Xi}_{B,\epsilon}\|^2+\|\mathcal{X}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,0}}+\|\mathcal{Y}_{\epsilon,\star}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}}\lesssim T^{-2\lambda_1}+T^{-\frac{1}{\gamma_0}-\lambda_2}\sim T^{-\frac{1}{\gamma_0}-\lambda_2}. ∥ R ϵ , ⋆ ∥ sol 1 − δ 1 , − λ ≲ ∥ Ξ B , ϵ ∥ 2 + ∥ X ϵ , ⋆ ∥ sol 1 − δ 1 , 0 + ∥ Y ϵ , ⋆ ∥ sol 1 + δ 2 , − λ ≲ T − 2 λ 1 + T − γ 0 1 − λ 2 ∼ T − γ 0 1 − λ 2 . Note that, in each bracket in (543) involving the L x r L_x^r L x r φ 0 + \varphi_0^+ φ 0 + T 1 2 T^{\frac{1}{2}} T 2 1 (543) , we estimate the terms in (406) –(6.90) as follows:
(406) , (407) , and (408) are estimated using Lemma 8.10.
(409) , (410) , and (411) are estimated using our weighted Schauder estimate (437) and ∥ Ξ ‾ B , ϵ ∥ \|\overline{\Xi}_{B,\epsilon}\| ∥ Ξ B , ϵ ∥
(6.74) and (6.84) are estimated using Corollary 8.33.
(6.75) and (6.85) are estimated using Lemma 8.14.
(6.76) and the first term in (6.86) are estimated using Lemma 8.16.
(6.77) and the second term in (6.86) are estimated using Lemma 8.17.
(6.78) and the third term in (6.86) are estimated using Lemma 8.18.
(6.79) and (6.87) are estimated using Lemma 8.19.
(6.80), (6.81), (6.88), and (6.89) are estimated using Lemma 8.21.
(6.82) and (6.90) are estimated using Lemma 8.23.
Thus, all terms in the definition of Γ ϵ \Gamma_{\epsilon} Γ ϵ (543) . It now only remains to prove the estimates in (474) . Using the support properties of our time-localised Duhamel integral operators together with Lemma 7.7, as well as the definition of ∥ Ξ ‾ B , ϵ ∥ \|\overline{\Xi}_{B,\epsilon}\| ∥ Ξ B , ϵ ∥
∥ R ϵ , ⋆ ∥ s o l 1 − λ , 0 ≲ ∥ ( Q ϵ , ⋆ ∘ , X ϵ , ⋆ ∘ , Y ϵ , ⋆ ∘ ) ∥ s o l 1 − λ , 0 + ∥ X ϵ , ⋆ ∥ s o l 1 − λ , 0 + ∥ Y ϵ , ⋆ ∥ s o l 1 − λ , 0 ≲ ∥ Ξ ‾ B , ϵ ∥ 2 + ∥ X ϵ , ⋆ ∥ s o l 1 − δ 1 , 0 + ∥ Y ϵ , ⋆ ∥ s o l 1 + δ 2 , − λ ≲ ( T − λ 1 ) 2 + T − 1 γ 0 − λ 2 ≲ T − 1 γ 0 − λ 2 . \begin{aligned}
\|\mathcal{R}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\lambda,0}}
&\lesssim\|(\mathcal{Q}_{\epsilon,\star}^{\circ},\mathcal{X}_{\epsilon,\star}^{\circ},\mathcal{Y}_{\epsilon,\star}^{\circ})\|_{\mathrm{sol}^{1-\lambda,0}}+\|\mathcal{X}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\lambda,0}}+\|\mathcal{Y}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\lambda,0}} \\
&\lesssim\|\overline{\Xi}_{B,\epsilon}\|^2+\|\mathcal{X}_{\epsilon,\star}\|_{\mathrm{sol}^{1-\delta_1,0}}+\|\mathcal{Y}_{\epsilon,\star}\|_{\mathrm{sol}^{1+\delta_2,-\lambda}} \\
&\lesssim(T^{-\lambda_1})^2+T^{-\frac{1}{\gamma_0}-\lambda_2}\lesssim T^{-\frac{1}{\gamma_0}-\lambda_2}.
\end{aligned} ∥ R ϵ , ⋆ ∥ sol 1 − λ , 0 ≲ ∥ ( Q ϵ , ⋆ ∘ , X ϵ , ⋆ ∘ , Y ϵ , ⋆ ∘ ) ∥ sol 1 − λ , 0 + ∥ X ϵ , ⋆ ∥ sol 1 − λ , 0 + ∥ Y ϵ , ⋆ ∥ sol 1 − λ , 0 ≲ ∥ Ξ B , ϵ ∥ 2 + ∥ X ϵ , ⋆ ∥ sol 1 − δ 1 , 0 + ∥ Y ϵ , ⋆ ∥ sol 1 + δ 2 , − λ ≲ ( T − λ 1 ) 2 + T − γ 0 1 − λ 2 ≲ T − γ 0 1 − λ 2 . Furthermore, using the definition of the s o l 1 − λ , 0 \mathrm{sol}^{1-\lambda,0} sol 1 − λ , 0
∥ R ϵ ∥ L t , x ∞ ≲ ∑ N ≥ N l o N − ( 1 − λ ) ∥ R ϵ , N ∥ s o l 1 − λ , 0 ≲ T 1 2 λ 1 − 1 γ 0 − λ 2 . \|\mathcal{R}_{\epsilon}\|_{L^\infty_{t,x}}\lesssim\sum_{N\geq N_{\mathrm{lo}}}N^{-(1-\lambda)}\|\mathcal{R}_{\epsilon,N}\|_{\mathrm{sol}^{1-\lambda,0}}\lesssim T^{\frac{1}{2}\lambda_1-\frac{1}{\gamma_0}-\lambda_2}. ∥ R ϵ ∥ L t , x ∞ ≲ N ≥ N lo ∑ N − ( 1 − λ ) ∥ R ϵ , N ∥ sol 1 − λ , 0 ≲ T 2 1 λ 1 − γ 0 1 − λ 2 . By taking the limit ϵ → 0 \epsilon\to0 ϵ → 0 (474) .
Proof of Theorem 2.19 . Let ( A 0 , Φ 0 ) (A_0,\Phi_0) ( A 0 , Φ 0 ) B B B A 0 A_0 A 0 φ 0 + : = 0 \varphi_0^+ := 0 φ 0 + := 0 φ 0 − : = Φ 0 \varphi_0^- := \Phi_0 φ 0 − := Φ 0 J T \mathcal{J}_T J T B ∈ C t 0 C x − κ ( [ 0 , T ] ) ∩ C t , x ∞ ( ( 0 , T ] ) B \in C_t^0\mathcal{C}_x^{-\kappa}([0,T]) \cap C_{t,x}^{\infty}((0,T]) B ∈ C t 0 C x − κ ([ 0 , T ]) ∩ C t , x ∞ (( 0 , T ]) 31
∥ t 1 + κ 2 F B ∥ L t , x ∞ ( [ 0 , T ] ) , ∥ t 1 + κ 2 D B F B ∥ L t , x ∞ ( [ 0 , T ] ) < ∞ , (544) \left\|t^{\frac{1+\kappa}{2}}F_B\right\|_{L^\infty_{t,x}([0,T])},\left\|t^{1+\frac{\kappa}{2}}D_BF_B\right\|_{L^\infty_{t,x}([0,T])}<\infty, \tag*{(544)} t 2 1 + κ F B L t , x ∞ ([ 0 , T ]) , t 1 + 2 κ D B F B L t , x ∞ ([ 0 , T ]) < ∞ , (544) where we also used the equivalence of norms of D B F B D_BF_B D B F B D B ∗ F B D_B^*F_B D B ∗ F B
∥ t κ 2 B ∥ L t , x ∞ ( [ 0 , T ] ) , ∥ t κ ∂ t B ∥ L t , x ∞ ( [ 0 , T ] ) < ∞ , (545) \left\|t^{\frac{\kappa}{2}}B\right\|_{L^\infty_{t,x}([0,T])},\left\|t^\kappa\partial_tB\right\|_{L^\infty_{t,x}([0,T])}<\infty, \tag*{(545)} t 2 κ B L t , x ∞ ([ 0 , T ]) , ∥ t κ ∂ t B ∥ L t , x ∞ ([ 0 , T ]) < ∞ , (545) which will be needed momentarily. We now define the event appearing in Theorem 2.19 as
E ( T , A 0 , Φ 0 ) ≔ { sup ε ∈ 2 − N 0 ∥ Ξ B , ε ∥ ≤ T − λ 1 / 2 } ∩ { lim ε , ε ′ → 0 ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0 } E(T,A_0,\Phi_0)\coloneqq\left\{\sup_{\varepsilon\in2^{-\mathbb{N}_0}}\|\Xi_{B,\varepsilon}\|\leq T^{-\lambda_1/2}\right\}\cap\left\{\lim_{\varepsilon,\varepsilon'\to0}\|\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\|=0\right\} E ( T , A 0 , Φ 0 ) : = { ε ∈ 2 − N 0 sup ∥ Ξ B , ε ∥ ≤ T − λ 1 /2 } ∩ { ε , ε ′ → 0 lim ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0 } ∩ { lim ε , ε ′ → 0 ∥ φ B , ε − φ B , ε ′ ∥ C t 0 C x − κ = 0 } . \cap\left\{\lim_{\varepsilon,\varepsilon'\to0}\|\varphi_{B,\varepsilon}-\varphi_{B,\varepsilon'}\|_{C_t^0\mathcal{C}_x^{-\kappa}}=0\right\}. ∩ { ε , ε ′ → 0 lim ∥ φ B , ε − φ B , ε ′ ∥ C t 0 C x − κ = 0 } . From Proposition 9.1 with R = T − λ 1 / 2 R=T^{-\lambda_1/2} R = T − λ 1 /2 c > 0 c>0 c > 0
P ( E ( T , A 0 , Φ 0 ) ) ≥ 1 − c − 1 exp ( − c T − λ 1 ) . \mathbb{P}(E(T,A_0,\Phi_0))\geq1-c^{-1}\exp(-cT^{-\lambda_1}). P ( E ( T , A 0 , Φ 0 )) ≥ 1 − c − 1 exp ( − c T − λ 1 ) . After further decreasing the value of c c c E ( T , A 0 , Φ 0 ) E(T,A_0,\Phi_0) E ( T , A 0 , Φ 0 ) E ( T , A 0 , Φ 0 ) E(T,A_0,\Phi_0) E ( T , A 0 , Φ 0 ) L ∞ ↪ C − κ L^\infty\hookrightarrow\mathcal{C}^{-\kappa} L ∞ ↪ C − κ ( A ε , Φ ε ) (A_\varepsilon,\Phi_\varepsilon) ( A ε , Φ ε )
(a) B ∈ C t 0 C x − κ ( [ 0 , T ] ) B\in C_t^0\mathcal{C}_x^{-\kappa}([0,T]) B ∈ C t 0 C x − κ ([ 0 , T ]) J T \mathcal{J}_T J T
(b) S ⋆ ∈ sol 1 − 2 κ ( [ 0 , T ] ) S_\star\in\operatorname{sol}^{1-2\kappa}([0,T]) S ⋆ ∈ sol 1 − 2 κ ([ 0 , T ])
(c) φ ∈ C t 0 C x − κ ( [ 0 , T ] ) \varphi\in C_t^0\mathcal{C}_x^{-\kappa}([0,T]) φ ∈ C t 0 C x − κ ([ 0 , T ]) φ 0 − ∈ C x − κ \varphi_0^-\in\mathcal{C}_x^{-\kappa} φ 0 − ∈ C x − κ J T \mathcal{J}_T J T
(d) The convergence of φ B , ε \varphi_{B,\varepsilon} φ B , ε ϕ B , ε \phi_{B,\varepsilon} ϕ B , ε C t 0 C x − κ ( [ 0 , T ] ) C_t^0\mathcal{C}_x^{-\kappa}([0,T]) C t 0 C x − κ ([ 0 , T ]) E ( T , A 0 , Φ 0 ) E(T,A_0,\Phi_0) E ( T , A 0 , Φ 0 )
(e) The convergence of ∘ ∘ B , ε , ⋆ \mathord{\circ\!\!\circ}_{B,\varepsilon,\star} ∘ ∘ B , ε , ⋆ ∙ ∙ B , ε , ⋆ \mathord{\bullet\!\!\bullet}_{B,\varepsilon,\star} ∙ ∙ B , ε , ⋆ ∘ ∙ B , ε , ⋆ \mathord{\circ\!\!\bullet}_{B,\varepsilon,\star} ∘ ∙ B , ε , ⋆ sol 1 − 2 κ ( [ 0 , T ] ) \operatorname{sol}^{1-2\kappa}([0,T]) sol 1 − 2 κ ([ 0 , T ])
(f) The convergence of ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) (X_{\varepsilon,\star},x_{\varepsilon,\star},Y_{\varepsilon,\star},y_{\varepsilon,\star}) ( X ε , ⋆ , x ε , ⋆ , Y ε , ⋆ , y ε , ⋆ ) sol 1 − δ 1 − λ 1 ( [ 0 , T ] ) \operatorname{sol}^{1-\delta_1-\lambda_1}([0,T]) sol 1 − δ 1 − λ 1 ([ 0 , T ])
Stochastic estimates
In this section, we prove that the covariant enhanced data set (see Definition 8.6) satisfies the estimates needed in our local theory (see Proposition 8.3) with high probability. The main ingredients in the proof will be the covariant heat kernel estimates from Section 4 and Wiener-chaos decompositions. In Section 10 below, we will also prove more refined stochastic estimates, which will later be needed in our global theory (see Section 11). Roughly speaking, the difference between the “unrefined” and “refined” stochastic estimates is that log-losses are tolerated in the unrefined estimates, but not in the refined estimates.
In the following proposition, we collect the main result of this section.
Proposition 9.1 (Control of covariant enhanced data set). Let Assumption 8.1 be satisfied and let c > 0 c > 0 c > 0 R ≥ 1 R \ge1 R ≥ 1
P ( sup ε ∈ 2 − N 0 ∥ Ξ B , ε ∥ ≥ T − λ 1 / 2 R ) ≤ c − 1 exp ( − c R 2 ) . \mathbb{P}\left(\sup_{\varepsilon\in2^{-\mathbb{N}_0}}\left\lVert\Xi_{B,\varepsilon}\right\rVert\ge T^{-\lambda_1/2}R\right) \le c^{-1}\exp(-cR^2). P ( ε ∈ 2 − N 0 sup ∥ Ξ B , ε ∥ ≥ T − λ 1 /2 R ) ≤ c − 1 exp ( − c R 2 ) . Furthermore, it holds that
lim ε , ε ′ → 0 ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0 P -a.s. \lim_{\varepsilon,\varepsilon'\to0}\left\lVert\Xi_{B,\varepsilon};\Xi_{B,\varepsilon'}\right\rVert= 0 \quad\mathbb{P}\text{-a.s.} ε , ε ′ → 0 lim ∥ Ξ B , ε ; Ξ B , ε ′ ∥ = 0 P -a.s. As in § 7 and § 9 , all covariant stochastic objects will depend on the same connection one-form B B B B B B
Non-resonant estimates
In this subsection, we estimate moments of the non-resonant parts of the objects in the covariant enhanced data set (Definition 8.6). (We also estimate an object which does not appear in the data set—see Lemma 9.9.) As usual, due to various standard reductions (which we defer to Section 9.3), the estimates of the stochastic objects in the appropriate spaces essentially reduce to these moment estimates. We begin with an abstract lemma which will be used for many of the moment estimates.
Lemma 9.2. Let C 0 > 1 C_0 > 1 C 0 > 1 C 1 C_1 C 1 C 0 C_0 C 0 T ∈ [ 0 , 1 ] T \in[0,1] T ∈ [ 0 , 1 ] m ∈ N m \in\mathbb{N} m ∈ N N 1 , … , N m ∈ 2 N 0 N_1,\ldots,N_m \in2^{\mathbb{N}_0} N 1 , … , N m ∈ 2 N 0 k : ( [ 0 , T ] × T 2 ) m + 1 → [ 0 , ∞ ) \mathfrak{k}: ([0,T]\times\mathbb{T}^2)^{m+1}\to[0,\infty) k : ([ 0 , T ] × T 2 ) m + 1 → [ 0 , ∞ ) k \mathfrak{k} k
{ ( s 1 , y 1 , … , s m , y m , t , x ) ∈ ( [ 0 , T ] × T 2 ) m + 1 : ∣ t − s j ∣ ∈ [ C 0 − 1 N j − 2 , C 0 N j − 2 ] for all j = 1 , … , m } . \left\{(s_1,y_1,\ldots,s_m,y_m,t,x)\in([0,T]\times\mathbb{T}^2)^{m+1}: |t-s_j|\in[C_0^{-1}N_j^{-2},C_0N_j^{-2}]\ \text{for all }j=1,\ldots,m\right\}. { ( s 1 , y 1 , … , s m , y m , t , x ) ∈ ([ 0 , T ] × T 2 ) m + 1 : ∣ t − s j ∣ ∈ [ C 0 − 1 N j − 2 , C 0 N j − 2 ] for all j = 1 , … , m } . Let N max = max ( N 1 , … , N m ) N_{\max}=\max(N_1,\ldots,N_m) N m a x = max ( N 1 , … , N m ) t a , t b ∈ [ 0 , T ] t_a,t_b\in[0,T] t a , t b ∈ [ 0 , T ] t a < t b t_a<t_b t a < t b x ∈ T 2 x\in\mathbb{T}^2 x ∈ T 2
∫ d s 1 d y 1 ⋯ d s m d y m ( ∫ t a t b d t k ( s 1 , y 1 , … , s m , y m , t , x ) ) 2 ≤ C 1 ∣ t b − t a ∣ min ( ∣ t b − t a ∣ , N max − 2 ) sup t ∈ [ 0 , T ] ∫ d s 1 d y 1 ⋯ d s m d y m k ( s 1 , y 1 , … , s m , y m , t , x ) 2 . \int\mathrm{d}s_1\mathrm{d}y_1\cdots\mathrm{d}s_m\mathrm{d}y_m\left(\int_{t_a}^{t_b}\mathrm{d}t\,\mathfrak{k}(s_1,y_1,\ldots,s_m,y_m,t,x)\right)^2
\le C_1|t_b-t_a|\min\left(|t_b-t_a|,N_{\max}^{-2}\right)\sup_{t\in[0,T]}\int\mathrm{d}s_1\mathrm{d}y_1\cdots\mathrm{d}s_m\mathrm{d}y_m\,\mathfrak{k}(s_1,y_1,\ldots,s_m,y_m,t,x)^2. ∫ d s 1 d y 1 ⋯ d s m d y m ( ∫ t a t b d t k ( s 1 , y 1 , … , s m , y m , t , x ) ) 2 ≤ C 1 ∣ t b − t a ∣ min ( ∣ t b − t a ∣ , N m a x − 2 ) t ∈ [ 0 , T ] sup ∫ d s 1 d y 1 ⋯ d s m d y m k ( s 1 , y 1 , … , s m , y m , t , x ) 2 . Proof. This follows directly from the Cauchy–Schwarz inequality and Tonelli’s theorem. □
Next, we introduce some convenient notation that will appear in intermediate steps of many of the estimates.
Notation 9.3. In the following, let r ( s , y ; t , x ) r(s,y;t,x) r ( s , y ; t , x )
r ( s , y ; t , x ) ≲ ( t − s ) − 1 exp ( − c ∣ x − y ∣ 2 t − s ) , r(s,y;t,x)\lesssim(t-s)^{-1}\exp\left(-c\frac{|x-y|^2}{t-s}\right), r ( s , y ; t , x ) ≲ ( t − s ) − 1 exp ( − c t − s ∣ x − y ∣ 2 ) , for some constant c c c r r r c c c r N ( s , y ; t , x ) r_N(s,y;t,x) r N ( s , y ; t , x ) r r r r N ( s , y ; t , x ) ≲ 1 ∣ t − s ∣ ∼ N − 2 r ( s , y ; t , x ) r_N(s,y;t,x)\lesssim\mathbf{1}_{|t-s|\sim N^{-2}}r(s,y;t,x) r N ( s , y ; t , x ) ≲ 1 ∣ t − s ∣ ∼ N − 2 r ( s , y ; t , x )
