Introduction
In this chapter, we study a stochastic version of a simplified model for the dynamics of nematic liquid crystals. The model couples an incompressible fluid velocity field with a director field describing the preferred orientation of the molecules. Our aim is to formulate the system as an abstract stochastic evolution equation and to prove well-posedness results in both a two-dimensional L 2 L^{2} L 2 L p L^{p} L p L q L^{q} L q
Description of the model
In the deterministic setting, the dynamics of liquid crystals are described by the Ericksen–Leslie theory, derived independently by Ericksen and Leslie from conservation laws and nematic modelling; see [6 , 9 ] . In [10 ] , Lin and Liu introduced a simplified version of the Ericksen–Leslie system, which retains the main coupling between the fluid velocity and the molecular orientation while neglecting some stretching and rotational effects. Throughout this chapter, u u u d d d p p p k ∈ N k \in\mathbb{N} k ∈ N k ≥ 1 k \ge1 k ≥ 1 ( a i ) 0 ≤ i ≤ k ∈ R k + 1 (a_i)_{0 \le i \le k} \in\mathbb{R}^{k+1} ( a i ) 0 ≤ i ≤ k ∈ R k + 1
Φ ( d ) : = ( ∑ i = 0 k a i ∣ d ∣ 2 i ) d . (1) \Phi(d) := \left(\sum_{i=0}^{k} a_i |d|^{2i}\right)d. \tag*{(1)} Φ ( d ) := ( i = 0 ∑ k a i ∣ d ∣ 2 i ) d . (1) The deterministic Lin–Liu system considered in [10 ] is then given by
d u = ( Δ u − ( u ⋅ ∇ ) u − ∇ p − ∇ ⋅ ( ∇ d ⊙ ∇ d ) ) d t , d d = ( Δ d − ( u ⋅ ∇ ) d − Φ ( d ) ) d t , ∇ ⋅ u = 0 , \begin{aligned}
\mathrm{d}u &= \left(\Delta u - (u \cdot\nabla)u - \nabla p - \nabla\cdot(\nabla d \odot\nabla d)\right)\,\mathrm{d}t,\\
\mathrm{d}d &= \left(\Delta d - (u \cdot\nabla)d - \Phi(d)\right)\,\mathrm{d}t,\\
\nabla\cdot u &= 0,
\end{aligned} d u d d ∇ ⋅ u = ( Δ u − ( u ⋅ ∇ ) u − ∇ p − ∇ ⋅ ( ∇ d ⊙ ∇ d ) ) d t , = ( Δ d − ( u ⋅ ∇ ) d − Φ ( d ) ) d t , = 0 , where we use the notation
∇ d ⊙ ∇ d : = ∑ i = 1 d ∇ d i ⊗ ∇ d i . (2) \nabla d \odot\nabla d := \sum_{i=1}^{d} \nabla d_i \otimes\nabla d_i. \tag*{(2)} ∇ d ⊙ ∇ d := i = 1 ∑ d ∇ d i ⊗ ∇ d i . (2) This system can be viewed as a simplified Ericksen-Leslie model in which the stretching and rotation effects generated by the director field are neglected. This simplification makes the model closer to the incompressible Navier-Stokes equations and allows one to develop existence and uniqueness results using related techniques, although it also removes some features of the full physical theory; see, for instance, [7 ] .
The stochastic model we study consists in the aforementioned system (Lin-Liu), where transport and multiplicative noises have been included. More explicitly, let U = ℓ 2 U = \ell^{2} U = ℓ 2 W W W U \mathcal{U} U
∑ n ≥ 1 ( b 1 , n ⋅ ∇ ) u d W n and ∑ n ≥ 1 ( b 2 , n ⋅ ∇ ) d d W n , (3) \sum_{n\geq1}(b_{1,n}\cdot\nabla)u\,\mathrm{d}W^{n}\quad\text{and}\quad\sum_{n\geq1}(b_{2,n}\cdot\nabla)d\,\mathrm{d}W^{n}, \tag*{(3)} n ≥ 1 ∑ ( b 1 , n ⋅ ∇ ) u d W n and n ≥ 1 ∑ ( b 2 , n ⋅ ∇ ) d d W n , (3) where ( b i , n ) n ∈ N ∈ ℓ 2 ( R d ) (b_{i,n})_{n\in\mathbb{N}}\in\ell^{2}(\mathbb{R}^{d}) ( b i , n ) n ∈ N ∈ ℓ 2 ( R d ) i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 }
∑ n ≥ 1 g 1 , n ( u , d ) d W n and ∑ n ≥ 1 g 2 , n ( d ) d W n , (4) \sum_{n\geq1}g_{1,n}(u,d)\,\mathrm{d}W^{n}\quad\text{and}\quad\sum_{n\geq1}g_{2,n}(d)\,\mathrm{d}W^{n}, \tag*{(4)} n ≥ 1 ∑ g 1 , n ( u , d ) d W n and n ≥ 1 ∑ g 2 , n ( d ) d W n , (4) with ( g 1 , n ) n ∈ N : R d × R d → ℓ 2 ( R d ) (g_{1,n})_{n\in\mathbb{N}}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\ell^{2}(\mathbb{R}^{d}) ( g 1 , n ) n ∈ N : R d × R d → ℓ 2 ( R d ) ( g 2 , n ) n ∈ N : R d → ℓ 2 ( R d ) (g_{2,n})_{n\in\mathbb{N}}:\mathbb{R}^{d}\to\ell^{2}(\mathbb{R}^{d}) ( g 2 , n ) n ∈ N : R d → ℓ 2 ( R d ) ∑ n ≥ 1 π n d W n \sum_{n\geq1}\pi_{n}\mathrm{d}W^{n} ∑ n ≥ 1 π n d W n
∇ π n = ( I − P ) [ ( b 1 , n ⋅ ∇ ) u + g 1 , n ( u , d ) ] . (5) \nabla\pi_{n}=(I-\mathbb{P})\left[(b_{1,n}\cdot\nabla)u+g_{1,n}(u,d)\right]. \tag*{(5)} ∇ π n = ( I − P ) [ ( b 1 , n ⋅ ∇ ) u + g 1 , n ( u , d ) ] . (5) where P \mathbb{P} P
Moreover, the system is studied on a convex bounded smooth domain O ⊂ R d \mathcal{O}\subset\mathbb{R}^{d} O ⊂ R d u u u d d d
(MLC)
{ d u = ( Δ u − ( u ⋅ ∇ ) u − ∇ p − ∇ ⋅ ( ∇ d ⊙ ∇ d ) ) d t + ∑ n ≥ 1 [ ( b 1 , n ⋅ ∇ ) u + g 1 , n ( u , d ) − ∇ π n ] d W n on O , d d = ( Δ d − ( u ⋅ ∇ ) d − Φ ( d ) ) d t + ∑ n ≥ 1 [ ( b 2 , n ⋅ ∇ ) d + g 2 , n ( d ) ] d W n on O , ∇ ⋅ u = 0 on O , u = 0 and ∂ ν d = 0 on ∂ O , u ∣ t = 0 = u 0 and d ∣ t = 0 = d 0 . \left\{
\begin{aligned}
\mathrm{d}u&=\left(\Delta u-(u\cdot\nabla)u-\nabla p-\nabla\cdot(\nabla d\odot\nabla d)\right)\mathrm{d}t+\sum_{n\geq1}\left[(b_{1,n}\cdot\nabla)u+g_{1,n}(u,d)-\nabla\pi_{n}\right]\mathrm{d}W^{n}\quad\text{on }\mathcal{O},\\
\mathrm{d}d&=\left(\Delta d-(u\cdot\nabla)d-\Phi(d)\right)\mathrm{d}t+\sum_{n\geq1}\left[(b_{2,n}\cdot\nabla)d+g_{2,n}(d)\right]\mathrm{d}W^{n}\quad\text{on }\mathcal{O},\\
\nabla\cdot u&=0\quad\text{on }\mathcal{O},\\
u&=0\ \text{and}\ \partial_{\nu}d=0\quad\text{on }\partial\mathcal{O},\\
u|_{t=0}&=u_{0}\ \text{and}\ d|_{t=0}=d_{0}.
\end{aligned}
\right. ⎩ ⎨ ⎧ d u d d ∇ ⋅ u u u ∣ t = 0 = ( Δ u − ( u ⋅ ∇ ) u − ∇ p − ∇ ⋅ ( ∇ d ⊙ ∇ d ) ) d t + n ≥ 1 ∑ [ ( b 1 , n ⋅ ∇ ) u + g 1 , n ( u , d ) − ∇ π n ] d W n on O , = ( Δ d − ( u ⋅ ∇ ) d − Φ ( d ) ) d t + n ≥ 1 ∑ [ ( b 2 , n ⋅ ∇ ) d + g 2 , n ( d ) ] d W n on O , = 0 on O , = 0 and ∂ ν d = 0 on ∂ O , = u 0 and d ∣ t = 0 = d 0 . The transport noise terms ∑ n ≥ 1 ( b 1 , n ⋅ ∇ ) u d W n \sum_{n\geq1}(b_{1,n}\cdot\nabla)u\,\mathrm{d}W^{n} ∑ n ≥ 1 ( b 1 , n ⋅ ∇ ) u d W n ∑ n ≥ 1 ( b 2 , n ⋅ ∇ ) d d W n \sum_{n\geq1}(b_{2,n}\cdot\nabla)d\,\mathrm{d}W^{n} ∑ n ≥ 1 ( b 2 , n ⋅ ∇ ) d d W n ∑ n ≥ 1 g 1 , n ( u , d ) d W n \sum_{n\geq1}g_{1,n}(u,d)\,\mathrm{d}W^{n} ∑ n ≥ 1 g 1 , n ( u , d ) d W n ∑ n ≥ 1 g 2 , n ( d ) d W n \sum_{n\geq1}g_{2,n}(d)\,\mathrm{d}W^{n} ∑ n ≥ 1 g 2 , n ( d ) d W n ( b i , n ) n ≥ 1 (b_{i,n})_{n\geq1} ( b i , n ) n ≥ 1 i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } [1 , 4 ] for example.
The abstract Cauchy problem
First, we denote by
L σ 2 ( O ) = { u ∈ L 2 ( O , R d ) ∣ div u = 0 , u ⋅ ν = 0 in H − 1 / 2 ( ∂ O ; R ) } , (6) L^{2}_{\sigma}(\mathcal{O})=\{u\in L^{2}(\mathcal{O},\mathbb{R}^{d})\mid\operatorname{div}u=0,\ u\cdot\nu=0\text{ in }H^{-1/2}(\partial\mathcal{O};\mathbb{R})\}, \tag*{(6)} L σ 2 ( O ) = { u ∈ L 2 ( O , R d ) ∣ div u = 0 , u ⋅ ν = 0 in H − 1/2 ( ∂ O ; R )} , (6) H 0 1 ( O ) = H 0 1 ( O , R d ) ∩ L σ 2 ( O ) . (7) H^{1}_{0}(\mathcal{O})=H^{1}_{0}(\mathcal{O},\mathbb{R}^{d})\cap L^{2}_{\sigma}(\mathcal{O}). \tag*{(7)} H 0 1 ( O ) = H 0 1 ( O , R d ) ∩ L σ 2 ( O ) . (7) Then, we consider the spaces
H = L σ 2 ( O ) × H 1 ( O ; R d ) and V = H 0 1 ( O ) × { d ∈ H 2 ( O ; R d ) ∣ ∂ ν d = 0 on ∂ O } , (8) H=L^{2}_{\sigma}(\mathcal{O})\times H^{1}(\mathcal{O};\mathbb{R}^{d})\quad\text{and}\quad V=H^{1}_{0}(\mathcal{O})\times\{d\in H^{2}(\mathcal{O};\mathbb{R}^{d})\mid\partial_{\nu}d=0\text{ on }\partial\mathcal{O}\}, \tag*{(8)} H = L σ 2 ( O ) × H 1 ( O ; R d ) and V = H 0 1 ( O ) × { d ∈ H 2 ( O ; R d ) ∣ ∂ ν d = 0 on ∂ O } , (8) equipped with the following inner products: for all ( u , d ) , ( u ′ , d ′ ) ∈ H (u,d),(u',d')\in H ( u , d ) , ( u ′ , d ′ ) ∈ H
( ( u , d ) , ( u ′ , d ′ ) ) H : = ∫ O ( u ⋅ u ′ + d ⋅ d ′ + ∇ d : ∇ d ′ ) d O , (9) ((u,d),(u',d'))_{H}:=\int_{\mathcal{O}}(u\cdot u'+d\cdot d'+\nabla d:\nabla d')\,\mathrm{d}\mathcal{O}, \tag*{(9)} (( u , d ) , ( u ′ , d ′ ) ) H := ∫ O ( u ⋅ u ′ + d ⋅ d ′ + ∇ d : ∇ d ′ ) d O , (9) and, for all ( u , d ) , ( u ′ , d ′ ) ∈ V (u,d),(u',d')\in V ( u , d ) , ( u ′ , d ′ ) ∈ V
( ( u , d ) , ( u ′ , d ′ ) ) V : = ∫ O ( u ⋅ u ′ + d ⋅ d ′ + ∇ u : ∇ u ′ + ∇ d : ∇ d ′ + Δ d ⋅ Δ d ′ ) d O . (10) ((u,d),(u',d'))_{V}:=\int_{\mathcal{O}}(u\cdot u'+d\cdot d'+\nabla u:\nabla u'+\nabla d:\nabla d'+\Delta d\cdot\Delta d')\,\mathrm{d}\mathcal{O}. \tag*{(10)} (( u , d ) , ( u ′ , d ′ ) ) V := ∫ O ( u ⋅ u ′ + d ⋅ d ′ + ∇ u : ∇ u ′ + ∇ d : ∇ d ′ + Δ d ⋅ Δ d ′ ) d O . (10) Then, we introduce the following duality pairing ⟨ ⋅ , ⋅ ⟩ V × V ′ \langle\cdot,\cdot\rangle_{V\times V'} ⟨ ⋅ , ⋅ ⟩ V × V ′ ( u , d ) ∈ V (u,d)\in V ( u , d ) ∈ V
⟨ ( u , d ) , ( u ′ , d ′ ) ⟩ V × V ′ = ( ( u , d ) , ( u ′ , d ′ ) ) H . \langle(u,d),(u',d')\rangle_{V\times V'}=((u,d),(u',d'))_{H}. ⟨( u , d ) , ( u ′ , d ′ ) ⟩ V × V ′ = (( u , d ) , ( u ′ , d ′ ) ) H . As such, V ⊂ H ⊂ V ′ V\subset H\subset V' V ⊂ H ⊂ V ′
We consider W : = ( W n ) n ≥ 1 W:=(W^{n})_{n\geq1} W := ( W n ) n ≥ 1 ℓ 2 \ell^{2} ℓ 2 Y = ( u , d ) Y=(u,d) Y = ( u , d )
A Y = ( − P Δ u − Δ d ) AY=\begin{pmatrix}-\mathbb{P}\Delta u\\-\Delta d\end{pmatrix} A Y = ( − P Δ u − Δ d ) where P : L 2 ( O ; R d ) → L 2 ( O ) \mathbb{P}: L^2(\mathcal{O};\mathbb{R}^d) \to L^2(\mathcal{O}) P : L 2 ( O ; R d ) → L 2 ( O ) A : V → V ′ A: V \to V' A : V → V ′
⟨ ( u ♯ , d ♯ ) , A ( u , d ) ⟩ V × V ′ = ∫ O ( ∇ u : ∇ u ♯ + ∇ d : ∇ d ♯ + Δ d ⋅ Δ d ♯ ) d O , \langle(u^\sharp,d^\sharp),A(u,d)\rangle_{V\times V'}=\int_{\mathcal{O}}(\nabla u:\nabla u^\sharp+\nabla d:\nabla d^\sharp+\Delta d\cdot\Delta d^\sharp)\,\mathrm{d}\mathcal{O}, ⟨( u ♯ , d ♯ ) , A ( u , d ) ⟩ V × V ′ = ∫ O ( ∇ u : ∇ u ♯ + ∇ d : ∇ d ♯ + Δ d ⋅ Δ d ♯ ) d O , for ( u , d ) , ( u ♯ , d ♯ ) ∈ V (u,d),(u^\sharp,d^\sharp)\in V ( u , d ) , ( u ♯ , d ♯ ) ∈ V V V V V ′ V' V ′ f ∈ L 2 ( O ; R d ) f\in L^2(\mathcal{O};\mathbb{R}^d) f ∈ L 2 ( O ; R d ) ⟨ ( 0 , d ♯ ) , ( 0 , f ) ⟩ = ∫ O f ⋅ ( d ♯ − Δ d ♯ ) d O \langle(0,d^\sharp),(0,f)\rangle=\int_{\mathcal{O}}f\cdot(d^\sharp-\Delta d^\sharp)\,\mathrm{d}\mathcal{O} ⟨( 0 , d ♯ ) , ( 0 , f )⟩ = ∫ O f ⋅ ( d ♯ − Δ d ♯ ) d O f ∈ H 1 f\in H^1 f ∈ H 1 ∂ ν d ♯ = 0 \partial_\nu d^\sharp=0 ∂ ν d ♯ = 0 H 1 H^1 H 1
The model (MLC) can now be written as an abstract Cauchy problem in the following form:
{ d Y ( t ) + A Y ( t ) d t = F ( Y ( t ) ) d t + ( B Y ( t ) + G ( Y ( t ) ) ) d W ( t ) , Y ∣ t = 0 = Y 0 , (11) \left\{
\begin{aligned}
\mathrm{d}Y(t)+AY(t)\,\mathrm{d}t&=F(Y(t))\,\mathrm{d}t+(BY(t)+G(Y(t)))\,\mathrm{d}W(t),\\
Y|_{t=0}&=Y_0,
\end{aligned}
\right. \tag*{(11)} { d Y ( t ) + A Y ( t ) d t Y ∣ t = 0 = F ( Y ( t )) d t + ( B Y ( t ) + G ( Y ( t ))) d W ( t ) , = Y 0 , (11) where Y 0 = ( u 0 , d 0 ) T Y_0=(u_0,d_0)^T Y 0 = ( u 0 , d 0 ) T
F ( Y ) = ( − P div ( u ⊗ u + ( ∇ d ⊙ ∇ d ) ) − ( u ⋅ ∇ ) d − Φ ( d ) ) , B Y = ( P ( b 1 , n ⋅ ∇ ) u ( b 2 , n ⋅ ∇ ) d ) n ≥ 1 , G Y = ( P g 1 , n ( u , d ) g 2 , n ( d ) ) n ≥ 1 . F(Y)=\begin{pmatrix}
-\mathbb{P}\operatorname{div}(u\otimes u+(\nabla d\odot\nabla d))\\
-(u\cdot\nabla)d-\Phi(d)
\end{pmatrix},\qquad
BY=\begin{pmatrix}
\mathbb{P}(b_{1,n}\cdot\nabla)u\\
(b_{2,n}\cdot\nabla)d
\end{pmatrix}_{n\geq1},\qquad
GY=\begin{pmatrix}
\mathbb{P}g_{1,n}(u,d)\\
g_{2,n}(d)
\end{pmatrix}_{n\geq1}. F ( Y ) = ( − P div ( u ⊗ u + ( ∇ d ⊙ ∇ d )) − ( u ⋅ ∇ ) d − Φ ( d ) ) , B Y = ( P ( b 1 , n ⋅ ∇ ) u ( b 2 , n ⋅ ∇ ) d ) n ≥ 1 , G Y = ( P g 1 , n ( u , d ) g 2 , n ( d ) ) n ≥ 1 . In dimension two, the drift is defined on V V V
⟨ ( u ♯ , d ♯ ) , F ( u , d ) ⟩ V × V ′ = ∫ O ( u ⊗ u + ∇ d ⊙ ∇ d ) : ∇ u ♯ d O − ∫ O ( ( u ⋅ ∇ ) d + Φ ( d ) ) ⋅ ( d ♯ − Δ d ♯ ) d O . \langle(u^\sharp,d^\sharp),F(u,d)\rangle_{V\times V'}=\int_{\mathcal{O}}(u\otimes u+\nabla d\odot\nabla d):\nabla u^\sharp\,\mathrm{d}\mathcal{O}-\int_{\mathcal{O}}((u\cdot\nabla)d+\Phi(d))\cdot(d^\sharp-\Delta d^\sharp)\,\mathrm{d}\mathcal{O}. ⟨( u ♯ , d ♯ ) , F ( u , d ) ⟩ V × V ′ = ∫ O ( u ⊗ u + ∇ d ⊙ ∇ d ) : ∇ u ♯ d O − ∫ O (( u ⋅ ∇ ) d + Φ ( d )) ⋅ ( d ♯ − Δ d ♯ ) d O . All these integrals are finite for ( u , d ) , ( u ♯ , d ♯ ) ∈ V (u,d),(u^\sharp,d^\sharp)\in V ( u , d ) , ( u ♯ , d ♯ ) ∈ V
Scaling and criticality
We now investigate the scaling properties of the model (Lin–Liu) and determine the corresponding critical function spaces. We consider smooth solutions on R d \mathbb{R}^d R d
Formal scaling analysis. Let λ > 0 \lambda>0 λ > 0
( u λ , d λ ) ( t , x ) = ( λ α u ( λ t , λ 1 / 2 x ) , λ β d ( λ t , λ 1 / 2 x ) ) . (12) (u^\lambda,d^\lambda)(t,x)=(\lambda^\alpha u(\lambda t,\lambda^{1/2}x),\lambda^\beta d(\lambda t,\lambda^{1/2}x)). \tag*{(12)} ( u λ , d λ ) ( t , x ) = ( λ α u ( λ t , λ 1/2 x ) , λ β d ( λ t , λ 1/2 x )) . (12) where α , β ∈ R \alpha,\beta\in\mathbb{R} α , β ∈ R x ↦ λ 1 / 2 x x\mapsto\lambda^{1/2}x x ↦ λ 1/2 x t ↦ λ t t\mapsto\lambda t t ↦ λ t ∂ t u \partial_tu ∂ t u Δ u \Delta u Δ u λ α + 1 \lambda^{\alpha+1} λ α + 1 ( u ⋅ ∇ ) u (u\cdot\nabla)u ( u ⋅ ∇ ) u λ 2 α + 1 / 2 \lambda^{2\alpha+1/2} λ 2 α + 1/2 ∇ ⋅ ( ∇ d ⊙ ∇ d ) \nabla\cdot(\nabla d\odot\nabla d) ∇ ⋅ ( ∇ d ⊙ ∇ d ) λ 2 β + 3 / 2 \lambda^{2\beta+3/2} λ 2 β + 3/2
α + 1 = 2 α + 1 2 , (13) \alpha+1=2\alpha+\frac{1}{2}, \tag*{(13)} α + 1 = 2 α + 2 1 , (13) which gives α = 1 2 \alpha=\frac{1}{2} α = 2 1 d d d
2 β + 3 2 = α + 1 = 3 2 , (14) 2\beta+\frac{3}{2}=\alpha+1=\frac{3}{2}, \tag*{(14)} 2 β + 2 3 = α + 1 = 2 3 , (14) hence β = 0 \beta=0 β = 0
( α , β ) = ( 1 2 , 0 ) . (\alpha,\beta)=\left(\frac{1}{2},0\right). ( α , β ) = ( 2 1 , 0 ) . With ( α , β ) = ( 1 / 2 , 0 ) (\alpha,\beta)=(1/2,0) ( α , β ) = ( 1/2 , 0 )
∂ t d λ − Δ d λ + ( u λ ⋅ ∇ ) d λ = λ ( ∂ t d − Δ d + ( u ⋅ ∇ ) d ) ( λ t , λ 1 / 2 x ) . \partial_td^\lambda-\Delta d^\lambda+(u^\lambda\cdot\nabla)d^\lambda=\lambda(\partial_td-\Delta d+(u\cdot\nabla)d)(\lambda t,\lambda^{1/2}x). ∂ t d λ − Δ d λ + ( u λ ⋅ ∇ ) d λ = λ ( ∂ t d − Δ d + ( u ⋅ ∇ ) d ) ( λ t , λ 1/2 x ) . Hence the principal deterministic terms are scale invariant under (12) .
Scaling of stochastic perturbations. Define the rescaled processes β t , λ n : = λ − 1 / 2 W λ t n \beta_{t,\lambda}^{n} := \lambda^{-1/2}W_{\lambda t}^{n} β t , λ n := λ − 1/2 W λ t n W n W^n W n α = 1 / 2 \alpha= 1/2 α = 1/2
( b 1 , n ⋅ ∇ ) u λ d β t , λ n = λ ( ( b 1 , n ⋅ ∇ ) u ) ( λ t , λ 1 / 2 x ) d β t , λ n , (b_{1,n}\cdot\nabla)u^\lambda\,d\beta_{t,\lambda}^{n}=\lambda((b_{1,n}\cdot\nabla)u)(\lambda t,\lambda^{1/2}x)\,d\beta_{t,\lambda}^{n}, ( b 1 , n ⋅ ∇ ) u λ d β t , λ n = λ (( b 1 , n ⋅ ∇ ) u ) ( λ t , λ 1/2 x ) d β t , λ n , ( b 2 , n ⋅ ∇ ) d λ d β t , λ n = λ 1 / 2 ( ( b 2 , n ⋅ ∇ ) d ) ( λ t , λ 1 / 2 x ) d β t , λ n . (b_{2,n}\cdot\nabla)d^\lambda\,d\beta_{t,\lambda}^{n}=\lambda^{1/2}((b_{2,n}\cdot\nabla)d)(\lambda t,\lambda^{1/2}x)\,d\beta_{t,\lambda}^{n}. ( b 2 , n ⋅ ∇ ) d λ d β t , λ n = λ 1/2 (( b 2 , n ⋅ ∇ ) d ) ( λ t , λ 1/2 x ) d β t , λ n . Thus, transport noise preserves the principal deterministic scaling under the Brownian time rescaling. Let ( g 1 , n ) n (g_{1,n})_n ( g 1 , n ) n ( g 2 , n ) n (g_{2,n})_n ( g 2 , n ) n β ˙ t , λ n = λ 1 / 2 W ˙ λ t n \dot{\beta}_{t,\lambda}^{n}=\lambda^{1/2}\dot{W}_{\lambda t}^{n} β ˙ t , λ n = λ 1/2 W ˙ λ t n
g 1 ( u λ , d λ ) d β t , λ n = λ 1 / 2 g 1 ( u λ , d λ ) W ˙ λ t n d t , g 2 ( d λ ) d β t , λ n = λ 1 / 2 g 2 ( d λ ) W ˙ λ t n d t . g_1(u^\lambda,d^\lambda)\,d\beta_{t,\lambda}^{n}=\lambda^{1/2}g_1(u^\lambda,d^\lambda)\dot{W}_{\lambda t}^{n}\,dt,\qquad g_2(d^\lambda)\,d\beta_{t,\lambda}^{n}=\lambda^{1/2}g_2(d^\lambda)\dot{W}_{\lambda t}^{n}\,dt. g 1 ( u λ , d λ ) d β t , λ n = λ 1/2 g 1 ( u λ , d λ ) W ˙ λ t n d t , g 2 ( d λ ) d β t , λ n = λ 1/2 g 2 ( d λ ) W ˙ λ t n d t . Assuming a polynomial ansatz
g 1 , n ( u , d ) ∼ ∣ u ∣ δ 1 ∣ d ∣ η 1 , g 2 , n ( d ) ∼ ∣ d ∣ η 2 , g_{1,n}(u,d)\sim|u|^{\delta_1}|d|^{\eta_1},\qquad g_{2,n}(d)\sim|d|^{\eta_2}, g 1 , n ( u , d ) ∼ ∣ u ∣ δ 1 ∣ d ∣ η 1 , g 2 , n ( d ) ∼ ∣ d ∣ η 2 , and substituting u λ = λ 1 / 2 u u^\lambda=\lambda^{1/2}u u λ = λ 1/2 u d λ = d d^\lambda=d d λ = d
g 1 , n ( u λ , d λ ) d β t , λ n ∼ λ ( 1 + δ 1 ) / 2 ∣ u ∣ δ 1 ∣ d ∣ η 1 W ˙ λ t n d t , g 2 , n ( d λ ) d β t , λ n ∼ λ 1 / 2 ∣ d ∣ η 2 W ˙ λ t n d t . g_{1,n}(u^\lambda,d^\lambda)\,d\beta_{t,\lambda}^{n}\sim\lambda^{(1+\delta_1)/2}|u|^{\delta_1}|d|^{\eta_1}\dot{W}_{\lambda t}^{n}\,dt,\qquad g_{2,n}(d^\lambda)\,d\beta_{t,\lambda}^{n}\sim\lambda^{1/2}|d|^{\eta_2}\dot{W}_{\lambda t}^{n}\,dt. g 1 , n ( u λ , d λ ) d β t , λ n ∼ λ ( 1 + δ 1 ) /2 ∣ u ∣ δ 1 ∣ d ∣ η 1 W ˙ λ t n d t , g 2 , n ( d λ ) d β t , λ n ∼ λ 1/2 ∣ d ∣ η 2 W ˙ λ t n d t . Critical or subcritical growth relative to the leading-order terms requires
1 + δ 1 2 ≤ 3 2 , \frac{1+\delta_1}{2}\leq\frac{3}{2}, 2 1 + δ 1 ≤ 2 3 , hence
δ 1 ≤ 2. \delta_1\leq2. δ 1 ≤ 2. For the director multiplicative noise, however, there is no velocity exponent anymore. Since
g 2 , n ( d λ ) d β t , λ n ∼ λ 1 / 2 , g_{2,n}(d^\lambda)\,d\beta_{t,\lambda}^{n}\sim\lambda^{1/2}, g 2 , n ( d λ ) d β t , λ n ∼ λ 1/2 , while the deterministic director equation scales like λ \lambda λ
disappears, because δ 2 \delta_2 δ 2 g 2 , n = g 2 , n ( d ) g_{2,n}=g_{2,n}(d) g 2 , n = g 2 , n ( d ) u u u g 2 , n = g 2 , n ( d ) g_{2,n}=g_{2,n}(d) g 2 , n = g 2 , n ( d ) d d d
Critical spaces. Let ( u 0 , d 0 ) (u_0,d_0) ( u 0 , d 0 )
( u 0 , λ , d 0 , λ ) ( x ) = ( λ 1 / 2 u 0 ( λ 1 / 2 x ) , d 0 ( λ 1 / 2 x ) ) . (u_{0,\lambda},d_{0,\lambda})(x)=(\lambda^{1/2}u_0(\lambda^{1/2}x),d_0(\lambda^{1/2}x)). ( u 0 , λ , d 0 , λ ) ( x ) = ( λ 1/2 u 0 ( λ 1/2 x ) , d 0 ( λ 1/2 x )) . Then for Lebesgue norms we have
∥ u 0 , λ ∥ L q ( R d ) = λ 1 2 − d 2 q ∥ u 0 ∥ L q ( R d ) , ∥ d 0 , λ ∥ L r ( R d ) = λ − d 2 r ∥ d 0 ∥ L r ( R d ) . \|u_{0,\lambda}\|_{L^q(\mathbb{R}^d)}=\lambda^{\frac{1}{2}-\frac{d}{2q}}\|u_0\|_{L^q(\mathbb{R}^d)},\qquad\|d_{0,\lambda}\|_{L^r(\mathbb{R}^d)}=\lambda^{-\frac{d}{2r}}\|d_0\|_{L^r(\mathbb{R}^d)}. ∥ u 0 , λ ∥ L q ( R d ) = λ 2 1 − 2 q d ∥ u 0 ∥ L q ( R d ) , ∥ d 0 , λ ∥ L r ( R d ) = λ − 2 r d ∥ d 0 ∥ L r ( R d ) . Hence scale invariance requires q = d q=d q = d r = ∞ r=\infty r = ∞
L d ( R d ) × L ∞ ( R d ) . L^d(\mathbb{R}^d)\times L^\infty(\mathbb{R}^d). L d ( R d ) × L ∞ ( R d ) . Similarly, for homogeneous Besov norms one obtains
∥ u 0 , λ ∥ B ˙ p , q s ∼ λ 1 + s 2 − d 2 p ∥ u 0 ∥ B ˙ p , q s , ∥ d 0 , λ ∥ B ˙ p , q s ′ ∼ λ s ′ 2 − d 2 p ∥ d 0 ∥ B ˙ p , q s ′ . \|u_{0,\lambda}\|_{\dot{B}^{s}_{p,q}}\sim\lambda^{\frac{1+s}{2}-\frac{d}{2p}}\|u_0\|_{\dot{B}^{s}_{p,q}},\qquad\|d_{0,\lambda}\|_{\dot{B}^{s'}_{p,q}}\sim\lambda^{\frac{s'}{2}-\frac{d}{2p}}\|d_0\|_{\dot{B}^{s'}_{p,q}}. ∥ u 0 , λ ∥ B ˙ p , q s ∼ λ 2 1 + s − 2 p d ∥ u 0 ∥ B ˙ p , q s , ∥ d 0 , λ ∥ B ˙ p , q s ′ ∼ λ 2 s ′ − 2 p d ∥ d 0 ∥ B ˙ p , q s ′ . The invariant Besov pair thus satisfies
1 + s 2 − d 2 p = 0 , s ′ 2 − d 2 p = 0 , that is, s = d p − 1 , s ′ = d p . \frac{1+s}{2}-\frac{d}{2p}=0,\qquad\frac{s'}{2}-\frac{d}{2p}=0,\qquad\text{that is,}\qquad s=\frac{d}{p}-1,\ s'=\frac{d}{p}. 2 1 + s − 2 p d = 0 , 2 s ′ − 2 p d = 0 , that is, s = p d − 1 , s ′ = p d . Therefore, the scaling-invariant function space for the pair ( u , d ) (u,d) ( u , d )
B ˙ p , q d p − 1 ( R d ) × B ˙ p , q d p ( R d ) . \dot{B}^{\frac{d}{p}-1}_{p,q}(\mathbb{R}^d)\times\dot{B}^{\frac{d}{p}}_{p,q}(\mathbb{R}^d). B ˙ p , q p d − 1 ( R d ) × B ˙ p , q p d ( R d ) . Notation
Throughout the paper, we adopt the following notation regarding stopping times. Denote by E E E T > 0 T>0 T > 0 τ \tau τ 0 ≤ τ ≤ T 0\leq\tau\leq T 0 ≤ τ ≤ T P \mathbb{P} P L p ( 0 , τ ; E ) L^p(0,\tau;E) L p ( 0 , τ ; E ) H m , p ( 0 , τ ; E ) H^{m,p}(0,\tau;E) H m , p ( 0 , τ ; E ) ( 0 , τ ( ω ) ) (0,\tau(\omega)) ( 0 , τ ( ω )) τ \tau τ L p L^p L p H m , p ( 0 , τ ; E ) H^{m,p}(0,\tau;E) H m , p ( 0 , τ ; E ) H θ , p ( 0 , τ ; E ) H^{\theta,p}(0,\tau;E) H θ , p ( 0 , τ ; E ) ( 0 , τ ( ω ) ) (0,\tau(\omega)) ( 0 , τ ( ω ))
Definition of solutions and main assumptions
Consider the ground spaces X 0 X_0 X 0 X 1 X_1 X 1 [1 ] , we denote the real interpolation space with parameters ( θ , p ) ∈ ( 0 , 1 ) × [ 1 , ∞ ] (\theta,p)\in(0,1)\times[1,\infty] ( θ , p ) ∈ ( 0 , 1 ) × [ 1 , ∞ ] X θ , p = ( X 0 , X 1 ) θ , p X_{\theta,p}=(X_0,X_1)_{\theta,p} X θ , p = ( X 0 , X 1 ) θ , p X θ = [ X 0 , X 1 ] θ X_\theta=[X_0,X_1]_\theta X θ = [ X 0 , X 1 ] θ θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) X 0 X_0 X 0 X 1 X_1 X 1 ( Ω , F , ( F t ) t ≥ 0 , P ) (\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},\mathbb{P}) ( Ω , F , ( F t ) t ≥ 0 , P ) W W W ( F t ) (\mathcal{F}_t) ( F t ) w κ ( t ) = t κ w_\kappa(t)=t^\kappa w κ ( t ) = t κ L κ p ( 0 , T ; E ) = L p ( ( 0 , T ) , w κ ( t ) d t ; E ) L^p_\kappa(0,T;E)=L^p((0,T),w_\kappa(t)\,\mathrm{d}t;E) L κ p ( 0 , T ; E ) = L p (( 0 , T ) , w κ ( t ) d t ; E )
Definition 1.1. Let p ∈ [ 2 , ∞ ) p\in[2,\infty) p ∈ [ 2 , ∞ ) κ ∈ [ 0 , p − 2 2 ) ∪ { 0 } \kappa\in[0,\frac{p-2}{2})\cup\{0\} κ ∈ [ 0 , 2 p − 2 ) ∪ { 0 } Y 0 Y_0 Y 0 F 0 \mathcal{F}_0 F 0 X 1 − 1 + κ p , p X_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p
(1) A pair ( Y , σ ) (Y,\sigma) ( Y , σ ) L κ p L^p_\kappa L κ p (11) if σ : Ω → [ 0 , ∞ ] \sigma:\Omega\to[0,\infty] σ : Ω → [ 0 , ∞ ] Y : R + × Ω → X 0 Y:\mathbb{R}_+\times\Omega\to X_0 Y : R + × Ω → X 0
Y σ ∧ ⋅ Y_{\sigma\wedge\cdot} Y σ ∧⋅ T > 0 T>0 T > 0
Y ∈ L κ p ( 0 , T ∧ σ ; X 1 ) , Y σ ∧ ⋅ ∈ C ( [ 0 , T ] ; X 1 − 1 + κ p , p ) Y\in L^p_\kappa(0,T\wedge\sigma;X_1),\qquad Y_{\sigma\wedge\cdot}\in C([0,T];X_{1-\frac{1+\kappa}{p},p}) Y ∈ L κ p ( 0 , T ∧ σ ; X 1 ) , Y σ ∧⋅ ∈ C ([ 0 , T ] ; X 1 − p 1 + κ , p )
Y σ ∧ t − Y 0 = ∫ 0 t 1 [ 0 , σ ] ( s ) [ − A Y ( s ) + F ( Y ( s ) ) ] d s + ∫ 0 t 1 [ 0 , σ ] ( s ) [ B Y ( s ) + G ( Y ( s ) ) ] d W s . Y_{\sigma\wedge t}-Y_0=\int_0^t\mathbf{1}_{[0,\sigma]}(s)\left[-AY(s)+F(Y(s))\right]\,\mathrm{d}s+\int_0^t\mathbf{1}_{[0,\sigma]}(s)\left[BY(s)+G(Y(s))\right]\,\mathrm{d}W_s. Y σ ∧ t − Y 0 = ∫ 0 t 1 [ 0 , σ ] ( s ) [ − A Y ( s ) + F ( Y ( s )) ] d s + ∫ 0 t 1 [ 0 , σ ] ( s ) [ B Y ( s ) + G ( Y ( s )) ] d W s . (2) A pair ( Y , σ ) (Y,\sigma) ( Y , σ ) L κ p L^p_\kappa L κ p σ \sigma σ σ n ≤ σ \sigma_n\leq\sigma σ n ≤ σ σ n ↑ σ \sigma_n\uparrow\sigma σ n ↑ σ ( Y , σ n ) (Y,\sigma_n) ( Y , σ n ) L κ p L^p_\kappa L κ p n n n ( σ n ) n ≥ 1 (\sigma_n)_{n\geq1} ( σ n ) n ≥ 1 ( Y ~ , σ ~ ) (\widetilde{Y},\widetilde{\sigma}) ( Y , σ ) Y = Y ~ Y=\widetilde{Y} Y = Y [ 0 , σ ∧ σ ~ ) [0,\sigma\wedge\widetilde{\sigma}) [ 0 , σ ∧ σ ) σ ~ ≤ σ \widetilde{\sigma}\leq\sigma σ ≤ σ L κ p L^p_\kappa L κ p ( Y , σ ) (Y,\sigma) ( Y , σ ) σ = ∞ \sigma=\infty σ = ∞
Throughout the next sections, we will work with the following assumptions.
Assumption 1.2. The following propositions hold:
(1) there exists some ν ∈ ( 0 , 2 ) \nu\in(0,2) ν ∈ ( 0 , 2 ) i = 1 , 2 i=1,2 i = 1 , 2 ( b i , n ) n ∈ N ∈ ℓ 2 ( R d ) (b_{i,n})_{n\in\mathbb{N}}\in\ell^2(\mathbb{R}^d) ( b i , n ) n ∈ N ∈ ℓ 2 ( R d )
∑ n ∈ N ( b i , n ⋅ ξ ) 2 ≤ ν ∣ ξ ∣ 2 for all ξ ∈ R d , \sum_{n\in\mathbb{N}}(b_{i,n}\cdot\xi)^2\leq\nu|\xi|^2\quad\text{for all }\xi\in\mathbb{R}^d, n ∈ N ∑ ( b i , n ⋅ ξ ) 2 ≤ ν ∣ ξ ∣ 2 for all ξ ∈ R d , (parabolicity)
(2) ( g 1 , n ) n ∈ N : R d × R d → ℓ 2 ( R d ) (g_{1,n})_{n\in\mathbb{N}}:\mathbb{R}^d\times\mathbb{R}^d\to\ell^2(\mathbb{R}^d) ( g 1 , n ) n ∈ N : R d × R d → ℓ 2 ( R d ) ( g 2 , n ) n ∈ N : R d → ℓ 2 ( R d ) (g_{2,n})_{n\in\mathbb{N}}:\mathbb{R}^d\to\ell^2(\mathbb{R}^d) ( g 2 , n ) n ∈ N : R d → ℓ 2 ( R d )
(3) The map g 2 : R d → ℓ 2 ( R d ) g_2:\mathbb{R}^d\to\ell^2(\mathbb{R}^d) g 2 : R d → ℓ 2 ( R d ) g 2 ( y ) = ( g 2 , n ( y ) ) n ∈ N g_2(y)=(g_{2,n}(y))_{n\in\mathbb{N}} g 2 ( y ) = ( g 2 , n ( y ) ) n ∈ N D g 2 = ( ∇ g 2 , n ) n ∈ N : R d → ℓ 2 ( R d × d ) Dg_2=(\nabla g_{2,n})_{n\in\mathbb{N}}:\mathbb{R}^d\to\ell^2(\mathbb{R}^{d\times d}) D g 2 = ( ∇ g 2 , n ) n ∈ N : R d → ℓ 2 ( R d × d )
Assumption 1.3. Assumption 1.2 holds, and the coefficient a k a_k a k Φ \Phi Φ (1) ) is positive.
As far as the multiplicative noise coefficients are concerned, Assumption 1.2 is sufficient for local well-posedness in Propositions 2.1 and 3.3. For − 1 < s < 1 -1<s<1 − 1 < s < 1 Φ \Phi Φ L 2 L^2 L 2
L2-setting in 2D
In this section we take X 0 = V ′ X_0=V' X 0 = V ′ X 1 = V X_1=V X 1 = V p = 2 p=2 p = 2 κ = 0 \kappa=0 κ = 0 X 1 / 2 = H X_{1/2}=H X 1/2 = H
X 3 / 4 = [ H , V ] 1 / 2 ↪ H 1 / 2 ( O ; R 2 ) × H 3 / 2 ( O ; R 2 ) . X_{3/4}=[H,V]_{1/2}\hookrightarrow H^{1/2}(\mathcal{O};\mathbb{R}^2)\times H^{3/2}(\mathcal{O};\mathbb{R}^2). X 3/4 = [ H , V ] 1/2 ↪ H 1/2 ( O ; R 2 ) × H 3/2 ( O ; R 2 ) . The aim of this section is to prove the well-posedness of 1.11, that is the following proposition,
Proposition 2.1 (2D well-posedness). Assume d = 2 d=2 d = 2
(1) Assume that Assumption 1.2 holds. For all Y 0 ∈ H Y_0\in H Y 0 ∈ H ( Y , σ ) (Y,\sigma) ( Y , σ ) 1.11, with σ > 0 \sigma>0 σ > 0
P ( σ < ∞ , sup [ 0 , σ ) ∥ Y ∥ H 2 + ∫ 0 σ ∥ Y ∥ V 2 < ∞ ) = 0. \mathbb{P}\left(\sigma<\infty,\sup_{[0,\sigma)}\|Y\|_H^2+\int_0^\sigma\|Y\|_V^2<\infty\right)=0. P ( σ < ∞ , [ 0 , σ ) sup ∥ Y ∥ H 2 + ∫ 0 σ ∥ Y ∥ V 2 < ∞ ) = 0. (2) Assume that Assumption 1.3 holds. Then, the unique maximal solution described above is global.
To prove the result above, we use some theorems from [1 ] . We remind first that the Helmholtz-Leray projector P \mathbb{P} P
Local well-posedness
We prove the first point of Proposition 2.1 by using Theorem 6.2 of [1 ] . We check hereafter Condition 2.2 and Condition 2.3 – see Assumption 6.1 of [1 ] .
First condition of Assumption 6.1
The first condition we prove reads as follows,
Condition 2.2. There exists θ > 0 \theta>0 θ > 0 M ≥ 0 M\geq0 M ≥ 0 ( u , d ) ∈ V (\mathbf{u},\mathbf{d})\in V ( u , d ) ∈ V
⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ − 1 2 ∥ B ( u , d ) ∥ L 2 ( U , H ) 2 ≥ θ ∥ ( u , d ) ∥ V 2 − M ∥ ( u , d ) ∥ H 2 . (15) \left\langle(\mathbf{u},\mathbf{d}),A(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}-\frac{1}{2}\|B(\mathbf{u},\mathbf{d})\|_{L_2(\mathcal{U},H)}^2\geq\theta\|(\mathbf{u},\mathbf{d})\|_V^2-M\|(\mathbf{u},\mathbf{d})\|_H^2. \tag*{(15)} ⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ − 2 1 ∥ B ( u , d ) ∥ L 2 ( U , H ) 2 ≥ θ ∥ ( u , d ) ∥ V 2 − M ∥ ( u , d ) ∥ H 2 . (15) Proof: Let ( u , d ) ∈ V (\mathbf{u},\mathbf{d})\in V ( u , d ) ∈ V A A A
⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ = ∥ ∇ u ∥ L 2 2 + ∥ ∇ d ∥ L 2 2 + ∥ Δ d ∥ L 2 2 ≥ ∥ ( u , d ) ∥ V 2 − ∥ ( u , d ) ∥ H 2 . \left\langle(\mathbf{u},\mathbf{d}),A(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}=\|\nabla\mathbf{u}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2+\|\Delta\mathbf{d}\|_{L^2}^2\geq\|(\mathbf{u},\mathbf{d})\|_V^2-\|(\mathbf{u},\mathbf{d})\|_H^2. ⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ = ∥∇ u ∥ L 2 2 + ∥∇ d ∥ L 2 2 + ∥Δ d ∥ L 2 2 ≥ ∥ ( u , d ) ∥ V 2 − ∥ ( u , d ) ∥ H 2 . Summing over the standard orthonormal basis of U = ℓ 2 \mathcal{U}=\ell^2 U = ℓ 2
∥ B ( u , d ) ∥ L 2 ( U , H ) 2 = ∑ n ≥ 1 ( ∥ P ( b 1 , n ⋅ ∇ ) u ∥ L 2 2 + ∥ ( b 2 , n ⋅ ∇ ) d ∥ H 1 2 ) ≤ ν ( ∥ ∇ u ∥ L 2 2 + ∥ ∇ d ∥ L 2 2 + ∥ ( ∇ ⊗ ∇ ) d ∥ L 2 2 ) ≤ ν ( ∥ ∇ u ∥ L 2 2 + ∥ ∇ d ∥ L 2 2 + ∥ Δ d ∥ L 2 2 ) . \begin{aligned}
\|B(\mathbf{u},\mathbf{d})\|_{L_2(\mathcal{U},H)}^2=\sum_{n\geq1}\left(\|\mathbb{P}(b_{1,n}\cdot\nabla)\mathbf{u}\|_{L^2}^2+\|(b_{2,n}\cdot\nabla)\mathbf{d}\|_{H^1}^2\right) \\
&\leq\nu\left(\|\nabla\mathbf{u}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2+\|(\nabla\otimes\nabla)\mathbf{d}\|_{L^2}^2\right) \\
&\leq\nu\left(\|\nabla\mathbf{u}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2+\|\Delta\mathbf{d}\|_{L^2}^2\right).
\end{aligned} ∥ B ( u , d ) ∥ L 2 ( U , H ) 2 = n ≥ 1 ∑ ( ∥ P ( b 1 , n ⋅ ∇ ) u ∥ L 2 2 + ∥ ( b 2 , n ⋅ ∇ ) d ∥ H 1 2 ) ≤ ν ( ∥∇ u ∥ L 2 2 + ∥∇ d ∥ L 2 2 + ∥ ( ∇ ⊗ ∇ ) d ∥ L 2 2 ) ≤ ν ( ∥∇ u ∥ L 2 2 + ∥∇ d ∥ L 2 2 + ∥Δ d ∥ L 2 2 ) . Here the coefficients b i , n b_{i,n} b i , n P \mathbb{P} P
⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ − 1 2 ∥ B ( u , d ) ∥ L 2 ( U , H ) 2 ≥ ( 1 − ν 2 ) ∥ ( u , d ) ∥ V 2 − C ∥ ( u , d ) ∥ H 2 . (16) \left\langle(\mathbf{u},\mathbf{d}),A(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}-\frac{1}{2}\|B(\mathbf{u},\mathbf{d})\|_{L_2(\mathcal{U},H)}^2\geq\left(1-\frac{\nu}{2}\right)\|(\mathbf{u},\mathbf{d})\|_V^2-C\|(\mathbf{u},\mathbf{d})\|_H^2. \tag*{(16)} ⟨ ( u , d ) , A ( u , d ) ⟩ V × V ′ − 2 1 ∥ B ( u , d ) ∥ L 2 ( U , H ) 2 ≥ ( 1 − 2 ν ) ∥ ( u , d ) ∥ V 2 − C ∥ ( u , d ) ∥ H 2 . (16) Second condition of Assumption 6.1
Moreover, the second condition that we prove is the following,
Condition 2.3. For all m ≥ 1 m\geq1 m ≥ 1 L m ≥ 0 L_m\geq0 L m ≥ 0 r r r ( ( β j , ρ j ) ) 1 ≤ j ≤ r ∈ ( ( 1 / 2 , 1 ) × R + ) r \left((\beta_j,\rho_j)\right)_{1\leq j\leq r}\in\left((1/2,1)\times\mathbb{R}_+\right)^r ( ( β j , ρ j ) ) 1 ≤ j ≤ r ∈ ( ( 1/2 , 1 ) × R + ) r ( 2 β j − 1 ) ( ρ j + 1 ) ≤ 1 (2\beta_j-1)(\rho_j+1)\leq1 ( 2 β j − 1 ) ( ρ j + 1 ) ≤ 1 ∀ j ∈ { 1 , … , r } \forall j\in\{1,\ldots,r\} ∀ j ∈ { 1 , … , r } ( u , d ) , ( u ′ , d ′ ) ∈ V (\mathbf{u},\mathbf{d}),(\mathbf{u}',\mathbf{d}')\in V ( u , d ) , ( u ′ , d ′ ) ∈ V ∥ ( u , d ) ∥ H , ∥ ( u ′ , d ′ ) ∥ H ≤ m \|(\mathbf{u},\mathbf{d})\|_H,\|(\mathbf{u}',\mathbf{d}')\|_H\leq m ∥ ( u , d ) ∥ H , ∥ ( u ′ , d ′ ) ∥ H ≤ m
∥ F ( u , d ) − F ( u ′ , d ′ ) ∥ V ′ + ∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) \|F(\mathbf{u},\mathbf{d})-F(\mathbf{u}',\mathbf{d}')\|_{V'}+\|G(\mathbf{u}',\mathbf{d}')-G(\mathbf{u},\mathbf{d})\|_{L_2(\mathcal{U},H)} ∥ F ( u , d ) − F ( u ′ , d ′ ) ∥ V ′ + ∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) ≤ L m ∑ j = 1 r ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X β j ( 1 + ∥ ( u ′ , d ′ ) ∥ X β j ρ j + ∥ ( u , d ) ∥ X β j ρ j ) . (17) \leq L_m\sum_{j=1}^{r}\|(\mathbf{u}',\mathbf{d}')-(\mathbf{u},\mathbf{d})\|_{X^{\beta_j}}\left(1+\|(\mathbf{u}',\mathbf{d}')\|_{X^{\beta_j}}^{\rho_j}+\|(\mathbf{u},\mathbf{d})\|_{X^{\beta_j}}^{\rho_j}\right). \tag*{(17)} ≤ L m j = 1 ∑ r ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X β j ( 1 + ∥ ( u ′ , d ′ ) ∥ X β j ρ j + ∥ ( u , d ) ∥ X β j ρ j ) . (17) Proof. Let m ≥ 1 m \ge1 m ≥ 1 ( u , d ) , ( u ′ , d ′ ) ∈ V (u,d),(u',d') \in V ( u , d ) , ( u ′ , d ′ ) ∈ V ∥ ( u , d ) ∥ H , ∥ ( u ′ , d ′ ) ∥ H ≤ m \|(u,d)\|_{H},\|(u',d')\|_{H} \le m ∥ ( u , d ) ∥ H , ∥ ( u ′ , d ′ ) ∥ H ≤ m ( u ♯ , d ♯ ) ∈ V (u^\sharp,d^\sharp) \in V ( u ♯ , d ♯ ) ∈ V F F F
⟨ ( u ♯ , d ♯ ) , F ( u , d ) − F ( u ′ , d ′ ) ⟩ V × V ′ = J 1 + J 2 , \langle(u^\sharp,d^\sharp),F(u,d)-F(u',d')\rangle_{V\times V'}=J_1+J_2, ⟨( u ♯ , d ♯ ) , F ( u , d ) − F ( u ′ , d ′ ) ⟩ V × V ′ = J 1 + J 2 , J 1 = ∫ O ( u ⊗ u − u ′ ⊗ u ′ + ∇ d ⊙ ∇ d − ∇ d ′ ⊙ ∇ d ′ ) : ∇ u ♯ d O , J_1=\int_{\mathcal{O}}\left(u\otimes u-u'\otimes u'+\nabla d\odot\nabla d-\nabla d'\odot\nabla d'\right):\nabla u^\sharp\,\mathrm{d}\mathcal{O}, J 1 = ∫ O ( u ⊗ u − u ′ ⊗ u ′ + ∇ d ⊙ ∇ d − ∇ d ′ ⊙ ∇ d ′ ) : ∇ u ♯ d O , J 2 = ∫ O ( ( u ′ ⋅ ∇ ) d ′ − ( u ⋅ ∇ ) d + Φ ( d ′ ) − Φ ( d ) ) ⋅ ( d ♯ − Δ d ♯ ) d O . J_2=\int_{\mathcal{O}}\left((u'\cdot\nabla)d'-(u\cdot\nabla)d+\Phi(d')-\Phi(d)\right)\cdot\left(d^\sharp-\Delta d^\sharp\right)\,\mathrm{d}\mathcal{O}. J 2 = ∫ O ( ( u ′ ⋅ ∇ ) d ′ − ( u ⋅ ∇ ) d + Φ ( d ′ ) − Φ ( d ) ) ⋅ ( d ♯ − Δ d ♯ ) d O . Using u ⊗ u − u ′ ⊗ u ′ = ( u − u ′ ) ⊗ u + u ′ ⊗ ( u − u ′ ) u\otimes u-u'\otimes u'=(u-u')\otimes u+u'\otimes(u-u') u ⊗ u − u ′ ⊗ u ′ = ( u − u ′ ) ⊗ u + u ′ ⊗ ( u − u ′ )
∣ J 1 ∣ ≤ ∥ u ′ − u ∥ L 4 ∥ ∇ u ♯ ∥ L 2 ( ∥ u ′ ∥ L 4 + ∥ u ∥ L 4 ) + ∥ ∇ ( d ′ − d ) ∥ L 4 ∥ ∇ u ♯ ∥ L 2 ( ∥ ∇ d ′ ∥ L 4 + ∥ ∇ d ∥ L 4 ) , \begin{aligned}
|J_1| &\le\|u'-u\|_{L^4}\|\nabla u^\sharp\|_{L^2}\left(\|u'\|_{L^4}+\|u\|_{L^4}\right)\\
&\quad+\|\nabla(d'-d)\|_{L^4}\|\nabla u^\sharp\|_{L^2}\left(\|\nabla d'\|_{L^4}+\|\nabla d\|_{L^4}\right),
\end{aligned} ∣ J 1 ∣ ≤ ∥ u ′ − u ∥ L 4 ∥∇ u ♯ ∥ L 2 ( ∥ u ′ ∥ L 4 + ∥ u ∥ L 4 ) + ∥∇ ( d ′ − d ) ∥ L 4 ∥∇ u ♯ ∥ L 2 ( ∥∇ d ′ ∥ L 4 + ∥∇ d ∥ L 4 ) , ∣ J 2 ∣ ≤ ( ∥ u ′ − u ∥ L 4 ∥ ∇ d ′ ∥ L 4 + ∥ u ∥ L 4 ∥ ∇ ( d ′ − d ) ∥ L 4 + ∥ Φ ( d ′ ) − Φ ( d ) ∥ L 2 ) × ( ∥ d ♯ ∥ L 2 + ∥ Δ d ♯ ∥ L 2 ) . \begin{aligned}
|J_2| &\le\left(\|u'-u\|_{L^4}\|\nabla d'\|_{L^4}+\|u\|_{L^4}\|\nabla(d'-d)\|_{L^4}+\|\Phi(d')-\Phi(d)\|_{L^2}\right)\\
&\quad\times\left(\|d^\sharp\|_{L^2}+\|\Delta d^\sharp\|_{L^2}\right).
\end{aligned} ∣ J 2 ∣ ≤ ( ∥ u ′ − u ∥ L 4 ∥∇ d ′ ∥ L 4 + ∥ u ∥ L 4 ∥∇ ( d ′ − d ) ∥ L 4 + ∥Φ ( d ′ ) − Φ ( d ) ∥ L 2 ) × ( ∥ d ♯ ∥ L 2 + ∥Δ d ♯ ∥ L 2 ) . In addition, the polynomial definition of Φ \Phi Φ
∣ Φ ( d ′ ) − Φ ( d ) ∣ ≤ C ∣ d ′ − d ∣ ( 1 + ∣ d ′ ∣ 2 k + ∣ d ∣ 2 k ) . |\Phi(d')-\Phi(d)|\le C|d'-d|\left(1+|d'|^{2k}+|d|^{2k}\right). ∣Φ ( d ′ ) − Φ ( d ) ∣ ≤ C ∣ d ′ − d ∣ ( 1 + ∣ d ′ ∣ 2 k + ∣ d ∣ 2 k ) . For k ≥ 1 k\ge1 k ≥ 1
∥ Φ ( d ′ ) − Φ ( d ) ∥ L 2 ≤ C ∥ d ′ − d ∥ L 4 ( 1 + ∥ d ′ ∥ L 8 k 2 k + ∥ d ∥ L 8 k 2 k ) . \|\Phi(d')-\Phi(d)\|_{L^2}\le C\|d'-d\|_{L^4}\left(1+\|d'\|_{L^{8k}}^{2k}+\|d\|_{L^{8k}}^{2k}\right). ∥Φ ( d ′ ) − Φ ( d ) ∥ L 2 ≤ C ∥ d ′ − d ∥ L 4 ( 1 + ∥ d ′ ∥ L 8 k 2 k + ∥ d ∥ L 8 k 2 k ) . Gathering the estimates on J 1 J_1 J 1 J 2 J_2 J 2 H 1 / 2 ( O ) ↪ L 4 ( O ) H^{1/2}(\mathcal{O})\hookrightarrow L^4(\mathcal{O}) H 1/2 ( O ) ↪ L 4 ( O ) H 1 ( O ) ↪ L 8 k ( O ) H^1(\mathcal{O})\hookrightarrow L^{8k}(\mathcal{O}) H 1 ( O ) ↪ L 8 k ( O ) k ≥ 1 k\ge1 k ≥ 1 X 3 / 4 X_{3/4} X 3/4
∥ F ( u , d ) − F ( u ′ , d ′ ) ∥ V ′ ≤ C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X 3 / 4 ( 1 + m 2 k + ∥ ( u ′ , d ′ ) ∥ X 3 / 4 + ∥ ( u , d ) ∥ X 3 / 4 ) . (18) \|F(u,d)-F(u',d')\|_{V'}\le C\|(u',d')-(u,d)\|_{X_{3/4}}\left(1+m^{2k}+\|(u',d')\|_{X_{3/4}}+\|(u,d)\|_{X_{3/4}}\right). \tag*{(18)} ∥ F ( u , d ) − F ( u ′ , d ′ ) ∥ V ′ ≤ C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X 3/4 ( 1 + m 2 k + ∥ ( u ′ , d ′ ) ∥ X 3/4 + ∥ ( u , d ) ∥ X 3/4 ) . (18) For the multiplicative noise, the Hilbert–Schmidt square sum and the contractivity of P P P
∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ ∥ ( g 1 , n ( u ′ , d ′ ) − g 1 , n ( u , d ) ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) 2 + ∥ ( g 2 , n ( d ′ ) − g 2 , n ( d ) ) n ≥ 1 ∥ H 1 ( O ; ℓ 2 ) 2 . \|G(u',d')-G(u,d)\|_{L_2(U,H)}^2\le\|(g_{1,n}(u',d')-g_{1,n}(u,d))_{n\ge1}\|_{L^2(\mathcal{O};\ell^2)}^2+\|(g_{2,n}(d')-g_{2,n}(d))_{n\ge1}\|_{H^1(\mathcal{O};\ell^2)}^2. ∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ ∥ ( g 1 , n ( u ′ , d ′ ) − g 1 , n ( u , d ) ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) 2 + ∥ ( g 2 , n ( d ′ ) − g 2 , n ( d ) ) n ≥ 1 ∥ H 1 ( O ; ℓ 2 ) 2 . The L 2 L^2 L 2 C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ H 2 C\|(u',d')-(u,d)\|_H^2 C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ H 2 g 2 g_2 g 2 R 2 \mathbb{R}^2 R 2 ℓ 2 ( R 2 ) \ell^2(\mathbb{R}^2) ℓ 2 ( R 2 ) ∇ g 2 \nabla g_2 ∇ g 2
∥ ( ∇ g 2 , n ( d ′ ) ⋅ ∇ d ′ − ∇ g 2 , n ( d ) ⋅ ∇ d ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) ≤ C ∥ d ′ − d ∥ L 4 ∥ ∇ d ′ ∥ L 4 + C ∥ ∇ ( d ′ − d ) ∥ L 2 ≤ C ∥ d ′ − d ∥ H 1 ( 1 + ∥ d ′ ∥ H 3 / 2 ) . \begin{aligned}
\|(\nabla g_{2,n}(d')\cdot\nabla d'-\nabla g_{2,n}(d)\cdot\nabla d)_{n\ge1}\|_{L^2(\mathcal{O};\ell^2)}
&\le C\|d'-d\|_{L^4}\|\nabla d'\|_{L^4}+C\|\nabla(d'-d)\|_{L^2}\\
&\le C\|d'-d\|_{H^1}\left(1+\|d'\|_{H^{3/2}}\right).
\end{aligned} ∥ ( ∇ g 2 , n ( d ′ ) ⋅ ∇ d ′ − ∇ g 2 , n ( d ) ⋅ ∇ d ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) ≤ C ∥ d ′ − d ∥ L 4 ∥∇ d ′ ∥ L 4 + C ∥∇ ( d ′ − d ) ∥ L 2 ≤ C ∥ d ′ − d ∥ H 1 ( 1 + ∥ d ′ ∥ H 3/2 ) . Consequently,
∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) ≤ C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X 3 / 4 ( 1 + ∥ ( u ′ , d ′ ) ∥ X 3 / 4 + ∥ ( u , d ) ∥ X 3 / 4 ) . (19) \|G(u',d')-G(u,d)\|_{L_2(U,H)}\le C\|(u',d')-(u,d)\|_{X_{3/4}}\left(1+\|(u',d')\|_{X_{3/4}}+\|(u,d)\|_{X_{3/4}}\right). \tag*{(19)} ∥ G ( u ′ , d ′ ) − G ( u , d ) ∥ L 2 ( U , H ) ≤ C ∥ ( u ′ , d ′ ) − ( u , d ) ∥ X 3/4 ( 1 + ∥ ( u ′ , d ′ ) ∥ X 3/4 + ∥ ( u , d ) ∥ X 3/4 ) . (19) Combining (18) and (19) proves Condition 2.3 with L m = C ( 1 + m 2 k ) L_m=C(1+m^{2k}) L m = C ( 1 + m 2 k ) r = 1 r=1 r = 1 β 1 = 3 / 4 \beta_1=3/4 β 1 = 3/4 ρ 1 = 1 \rho_1=1 ρ 1 = 1 ( 2 β 1 − 1 ) ( ρ 1 + 1 ) = 1 (2\beta_1-1)(\rho_1+1)=1 ( 2 β 1 − 1 ) ( ρ 1 + 1 ) = 1 [1 Assumption 6.1] follows from ( V ′ , V ) 3 / 4 , 1 ↪ [ V ′ , V ] 3 / 4 (V',V)_{3/4,1}\hookrightarrow[V',V]_{3/4} ( V ′ , V ) 3/4 , 1 ↪ [ V ′ , V ] 3/4
As such, the conditions of Theorem 6.2 of [1 ] are fulfilled, which shows the existence of a maximal solution to the Cauchy problem.
Global well-posedness
Assume now that Assumption 1.3 holds. To prove the second point of Proposition 2.1 , we use Theorem 6.4 of [1 ] . Since we already proved the two conditions used in Theorem 6.2, we only need to prove the additional following one,
Condition 2.4. There exist θ ′ > 0 \theta' > 0 θ ′ > 0 M ′ ≥ 0 M' \ge0 M ′ ≥ 0 ( u , d ) ∈ V (\mathbf{u},\mathbf{d}) \in V ( u , d ) ∈ V
⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ − 1 2 ∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≥ θ ′ ∥ ( u , d ) ∥ V 2 − M ′ ( ∥ ( u , d ) ∥ H 2 + 1 ) . (20) \left\langle(\mathbf{u},\mathbf{d}), A(\mathbf{u},\mathbf{d})-F(\mathbf{u},\mathbf{d}) \right\rangle_{V\times V'}-\frac{1}{2}\left\|B(\mathbf{u},\mathbf{d})+G(\mathbf{u},\mathbf{d})\right\|_{L_{2}(U,H)}^{2}\geq\theta'\left\|(\mathbf{u},\mathbf{d})\right\|_{V}^{2}-M'\left(\left\|(\mathbf{u},\mathbf{d})\right\|_{H}^{2}+1\right). \tag*{(20)} ⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ − 2 1 ∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≥ θ ′ ∥ ( u , d ) ∥ V 2 − M ′ ( ∥ ( u , d ) ∥ H 2 + 1 ) . (20) Proof. We first perform the following calculation for smooth fields satisfying the boundary conditions. Smooth solenoidal Dirichlet fields are dense in H 0 1 H_{0}^{1} H 0 1 V V V
⟨ ( u , d ) , − F ( u , d ) ⟩ V × V ′ = ( P [ ( u ⋅ ∇ ) u + ∇ ⋅ ( ∇ d ⊙ ∇ d ) ] , u ) L 2 ( O , R 2 ) + ( ( u ⋅ ∇ ) d + Φ ( d ) , d − Δ d ) L 2 ( O , R 2 ) = ∫ O [ ∇ ⋅ ( ∇ d ⊙ ∇ d ) ⋅ u − ( u ⋅ ∇ ) d ⋅ Δ d ] d O + ∫ O ∑ i = 0 k a i ∣ d ∣ 2 i + 2 d O + ∫ O ∑ i = 0 k a i ∣ d ∣ 2 i ∣ ∇ d ∣ 2 d O + ∫ O ∑ i = 1 k 2 i a i ∣ d ∣ 2 i − 2 ∑ ℓ = 1 d ( d ⋅ ∂ ℓ d ) 2 d O . \begin{aligned}
\left\langle(\mathbf{u},\mathbf{d}),-F(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}={}&\left(\mathbb{P}\left[(\mathbf{u}\cdot\nabla)\mathbf{u}+\nabla\cdot(\nabla\mathbf{d}\odot\nabla\mathbf{d})\right],\mathbf{u}\right)_{L_{2}(\mathcal{O},\mathbb{R}^{2})}\\
&+\left((\mathbf{u}\cdot\nabla)\mathbf{d}+\Phi(\mathbf{d}),\mathbf{d}-\Delta\mathbf{d}\right)_{L_{2}(\mathcal{O},\mathbb{R}^{2})}\\
={}&\int_{\mathcal{O}}\left[\nabla\cdot(\nabla\mathbf{d}\odot\nabla\mathbf{d})\cdot\mathbf{u}-(\mathbf{u}\cdot\nabla)\mathbf{d}\cdot\Delta\mathbf{d}\right]\,\mathrm{d}\mathcal{O}+\int_{\mathcal{O}}\sum_{i=0}^{k}a_i|\mathbf{d}|^{2i+2}\,\mathrm{d}\mathcal{O}\\
&+\int_{\mathcal{O}}\sum_{i=0}^{k}a_i|\mathbf{d}|^{2i}|\nabla\mathbf{d}|^{2}\,\mathrm{d}\mathcal{O}+\int_{\mathcal{O}}\sum_{i=1}^{k}2i a_i|\mathbf{d}|^{2i-2}\sum_{\ell=1}^{d}(\mathbf{d}\cdot\partial_{\ell}\mathbf{d})^{2}\,\mathrm{d}\mathcal{O}.
\end{aligned} ⟨ ( u , d ) , − F ( u , d ) ⟩ V × V ′ = = ( P [ ( u ⋅ ∇ ) u + ∇ ⋅ ( ∇ d ⊙ ∇ d ) ] , u ) L 2 ( O , R 2 ) + ( ( u ⋅ ∇ ) d + Φ ( d ) , d − Δ d ) L 2 ( O , R 2 ) ∫ O [ ∇ ⋅ ( ∇ d ⊙ ∇ d ) ⋅ u − ( u ⋅ ∇ ) d ⋅ Δ d ] d O + ∫ O i = 0 ∑ k a i ∣ d ∣ 2 i + 2 d O + ∫ O i = 0 ∑ k a i ∣ d ∣ 2 i ∣∇ d ∣ 2 d O + ∫ O i = 1 ∑ k 2 i a i ∣ d ∣ 2 i − 2 ℓ = 1 ∑ d ( d ⋅ ∂ ℓ d ) 2 d O . In addition, using the identity
∇ ⋅ ( ∇ d ⊙ ∇ d ) = ∑ i 1 2 ∇ ∣ ∇ d i ∣ 2 + ∇ d i Δ d i (21) \nabla\cdot(\nabla\mathbf{d}\odot\nabla\mathbf{d})=\sum_i\frac{1}{2}\nabla|\nabla d_i|^{2}+\nabla d_i\Delta d_i \tag*{(21)} ∇ ⋅ ( ∇ d ⊙ ∇ d ) = i ∑ 2 1 ∇∣∇ d i ∣ 2 + ∇ d i Δ d i (21) we infer that, since u \mathbf{u} u
∫ O [ ∇ ⋅ ( ∇ d ⊙ ∇ d ) ⋅ u − ( u ⋅ ∇ ) d ⋅ Δ d ] d O = ∑ i ∫ O [ ( ∇ d i Δ d i ) ⋅ u − ( u ⋅ ∇ d i ) Δ d i ] d O = 0. (22) \int_{\mathcal{O}}\left[\nabla\cdot(\nabla\mathbf{d}\odot\nabla\mathbf{d})\cdot\mathbf{u}-(\mathbf{u}\cdot\nabla)\mathbf{d}\cdot\Delta\mathbf{d}\right]\,\mathrm{d}\mathcal{O}
=\sum_i\int_{\mathcal{O}}\left[(\nabla d_i\Delta d_i)\cdot\mathbf{u}-(\mathbf{u}\cdot\nabla d_i)\Delta d_i\right]\,\mathrm{d}\mathcal{O}=0. \tag*{(22)} ∫ O [ ∇ ⋅ ( ∇ d ⊙ ∇ d ) ⋅ u − ( u ⋅ ∇ ) d ⋅ Δ d ] d O = i ∑ ∫ O [ ( ∇ d i Δ d i ) ⋅ u − ( u ⋅ ∇ d i ) Δ d i ] d O = 0. (22) Furthermore, since a k > 0 a_k>0 a k > 0 ξ ∈ ( 0 , a k ) \xi\in(0,a_k) ξ ∈ ( 0 , a k ) C ξ > 0 C_{\xi}>0 C ξ > 0
∑ i = 0 k a i ∣ d ∣ 2 i + 2 + ∑ i = 0 k a i ∣ d ∣ 2 i ∣ ∇ d ∣ 2 + ∑ i = 1 k 2 i a i ∣ d ∣ 2 i − 2 ∑ ℓ = 1 d ( d ⋅ ∂ ℓ d ) 2 ≥ ( a k − ξ ) ∣ d ∣ 2 k + 2 + ( a k − ξ ) ∣ d ∣ 2 k ∣ ∇ d ∣ 2 − C ξ ( 1 + ∣ ∇ d ∣ 2 ) . (23) \sum_{i=0}^{k}a_i|\mathbf{d}|^{2i+2}+\sum_{i=0}^{k}a_i|\mathbf{d}|^{2i}|\nabla\mathbf{d}|^{2}+\sum_{i=1}^{k}2i a_i|\mathbf{d}|^{2i-2}\sum_{\ell=1}^{d}(\mathbf{d}\cdot\partial_{\ell}\mathbf{d})^{2}
\geq(a_k-\xi)|\mathbf{d}|^{2k+2}+(a_k-\xi)|\mathbf{d}|^{2k}|\nabla\mathbf{d}|^{2}-C_{\xi}(1+|\nabla\mathbf{d}|^{2}). \tag*{(23)} i = 0 ∑ k a i ∣ d ∣ 2 i + 2 + i = 0 ∑ k a i ∣ d ∣ 2 i ∣∇ d ∣ 2 + i = 1 ∑ k 2 i a i ∣ d ∣ 2 i − 2 ℓ = 1 ∑ d ( d ⋅ ∂ ℓ d ) 2 ≥ ( a k − ξ ) ∣ d ∣ 2 k + 2 + ( a k − ξ ) ∣ d ∣ 2 k ∣∇ d ∣ 2 − C ξ ( 1 + ∣∇ d ∣ 2 ) . (23) Indeed, the leading-order terms are positive, while the lower-order terms are controlled by Young’s inequality. Integrating (23) and using the cancellation in (22) , we obtain
⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ ≥ ∥ ( u , d ) ∥ V 2 − ∥ ( u , d ) ∥ H 2 + ( a k − ξ ) ∫ O ∣ d ∣ 2 k + 2 d O + ( a k − ξ ) ∫ O ∣ d ∣ 2 k ∣ ∇ d ∣ 2 d O − C ξ ( 1 + ∥ ∇ d ∥ L 2 2 ) . \begin{aligned}
\left\langle(\mathbf{u},\mathbf{d}),A(\mathbf{u},\mathbf{d})-F(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}\geq{}&\left\|(\mathbf{u},\mathbf{d})\right\|_{V}^{2}-\left\|(\mathbf{u},\mathbf{d})\right\|_{H}^{2}+(a_k-\xi)\int_{\mathcal{O}}|\mathbf{d}|^{2k+2}\,\mathrm{d}\mathcal{O}\\
&+(a_k-\xi)\int_{\mathcal{O}}|\mathbf{d}|^{2k}|\nabla\mathbf{d}|^{2}\,\mathrm{d}\mathcal{O}-C_{\xi}(1+\|\nabla\mathbf{d}\|_{L^{2}}^{2}).
\end{aligned} ⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ ≥ ∥ ( u , d ) ∥ V 2 − ∥ ( u , d ) ∥ H 2 + ( a k − ξ ) ∫ O ∣ d ∣ 2 k + 2 d O + ( a k − ξ ) ∫ O ∣ d ∣ 2 k ∣∇ d ∣ 2 d O − C ξ ( 1 + ∥∇ d ∥ L 2 2 ) . In particular, discarding the nonnegative leading-order terms gives
⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ ≥ ∥ ( u , d ) ∥ V 2 − C 0 ( 1 + ∥ ( u , d ) ∥ H 2 ) . (24) \left\langle(\mathbf{u},\mathbf{d}),A(\mathbf{u},\mathbf{d})-F(\mathbf{u},\mathbf{d})\right\rangle_{V\times V'}\geq\left\|(\mathbf{u},\mathbf{d})\right\|_{V}^{2}-C_{0}\left(1+\left\|(\mathbf{u},\mathbf{d})\right\|_{H}^{2}\right). \tag*{(24)} ⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ ≥ ∥ ( u , d ) ∥ V 2 − C 0 ( 1 + ∥ ( u , d ) ∥ H 2 ) . (24) for some constant C 0 > 0 C_{0}>0 C 0 > 0 g 1 , g 2 g_{1},g_{2} g 1 , g 2 g 2 g_{2} g 2
∥ G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ C ( 1 + ∥ ( u , d ) ∥ H 2 ) . \left\|G(\mathbf{u},\mathbf{d})\right\|_{L_{2}(U,H)}^{2}\leq C\left(1+\left\|(\mathbf{u},\mathbf{d})\right\|_{H}^{2}\right). ∥ G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ C ( 1 + ∥ ( u , d ) ∥ H 2 ) . Indeed, the L 2 L^{2} L 2 ∥ ( ∇ g 2 , n ( d ) ⋅ ∇ d ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) ≤ C ∥ ∇ d ∥ L 2 \|(\nabla g_{2,n}(\mathbf{d})\cdot\nabla\mathbf{d})_{n\geq1}\|_{L^{2}(\mathcal{O};\ell^{2})}\leq C\|\nabla\mathbf{d}\|_{L^{2}} ∥ ( ∇ g 2 , n ( d ) ⋅ ∇ d ) n ≥ 1 ∥ L 2 ( O ; ℓ 2 ) ≤ C ∥∇ d ∥ L 2 B B B ξ ′ > 0 \xi'>0 ξ ′ > 0 M ξ ′ > 0 M_{\xi'}>0 M ξ ′ > 0
∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ ( ν + ξ ′ ) ∥ ( u , d ) ∥ V 2 + M ξ ′ ( 1 + ∥ ( u , d ) ∥ H 2 ) . \left\|B(\mathbf{u},\mathbf{d})+G(\mathbf{u},\mathbf{d})\right\|_{L_{2}(U,H)}^{2}\leq(\nu+\xi')\left\|(\mathbf{u},\mathbf{d})\right\|_{V}^{2}+M_{\xi'}\left(1+\left\|(\mathbf{u},\mathbf{d})\right\|_{H}^{2}\right). ∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≤ ( ν + ξ ′ ) ∥ ( u , d ) ∥ V 2 + M ξ ′ ( 1 + ∥ ( u , d ) ∥ H 2 ) . Combining this with (24) and choosing ξ ′ = 1 − ν / 2 > 0 \xi' = 1-\nu/2 > 0 ξ ′ = 1 − ν /2 > 0
⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ − 1 2 ∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≥ 2 − ν 4 ∥ ( u , d ) ∥ V 2 − M ′ ( 1 + ∥ ( u , d ) ∥ H 2 ) (25) \left\langle(u,d),A(u,d)-F(u,d)\right\rangle_{V\times V'}-\frac{1}{2}\lVert B(u,d)+G(u,d)\rVert_{L_2(U,H)}^2 \ge\frac{2-\nu}{4}\lVert(u,d)\rVert_V^2-M'(1+\lVert(u,d)\rVert_H^2) \tag*{(25)} ⟨ ( u , d ) , A ( u , d ) − F ( u , d ) ⟩ V × V ′ − 2 1 ∥ B ( u , d ) + G ( u , d ) ∥ L 2 ( U , H ) 2 ≥ 4 2 − ν ∥( u , d ) ∥ V 2 − M ′ ( 1 + ∥( u , d ) ∥ H 2 ) (25) for a constant M ′ ≥ 0 M' \ge0 M ′ ≥ 0 ν < 2 \nu< 2 ν < 2 θ ′ = ( 2 − ν ) / 4 \theta'=(2-\nu)/4 θ ′ = ( 2 − ν ) /4
Consequently, the assumptions of the Theorem 6.4 of [1 ] are met, therefore there exists a unique global solution to the problem 1.11.
L p L^p L p L q L^q L q T d \mathbb{T}^d T d In this section, we study the liquid crystal model (MLC) in an L p L^p L p L q L^q L q d d d O : = T d \mathcal{O}:=\mathbb{T}^d O := T d L 2 L^2 L 2
For the spatial variable, we utilise the following Banach spaces, for s ∈ [ − 1 , 2 ) s\in[-1,2) s ∈ [ − 1 , 2 ) p , q > 2 p,q>2 p , q > 2
X 0 : = H − s , q ( T d ) × H 1 − s , q ( T d ; R d ) , X 1 : = H 2 − s , q ( T d ) × H 3 − s , q ( T d ; R d ) . (26) X_0:=H^{-s,q}(\mathbb{T}^d)\times H^{1-s,q}(\mathbb{T}^d;\mathbb{R}^d),\qquad X_1:=H^{2-s,q}(\mathbb{T}^d)\times H^{3-s,q}(\mathbb{T}^d;\mathbb{R}^d). \tag*{(26)} X 0 := H − s , q ( T d ) × H 1 − s , q ( T d ; R d ) , X 1 := H 2 − s , q ( T d ) × H 3 − s , q ( T d ; R d ) . (26) Here, for δ ∈ R \delta\in\mathbb{R} δ ∈ R q > 1 q>1 q > 1 H δ , q ( T d ) H^{\delta,q}(\mathbb{T}^d) H δ , q ( T d )
H σ δ , q ( T d ) : = { f ∈ H δ , q ( T d ; R d ) : div f = 0 in D ′ ( T d ) } . H_{\sigma}^{\delta,q}(\mathbb{T}^d):=\{f\in H^{\delta,q}(\mathbb{T}^d;\mathbb{R}^d):\operatorname{div}f=0\text{ in }\mathcal{D}'(\mathbb{T}^d)\}. H σ δ , q ( T d ) := { f ∈ H δ , q ( T d ; R d ) : div f = 0 in D ′ ( T d )} . For details, we refer to [3 §2.1] .
For θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 )
X θ = H − s + 2 θ , q ( T d ) × H 1 − s + 2 θ , q ( T d ; R d ) (27) X_\theta=H^{-s+2\theta,q}(\mathbb{T}^d)\times H^{1-s+2\theta,q}(\mathbb{T}^d;\mathbb{R}^d) \tag*{(27)} X θ = H − s + 2 θ , q ( T d ) × H 1 − s + 2 θ , q ( T d ; R d ) (27) and the trace space for κ ≥ 0 \kappa\ge0 κ ≥ 0
X 1 − 1 + κ p = B q p − s + 2 − 2 ( 1 + κ ) p ( T d ) × B q p − s + 3 − 2 ( 1 + κ ) p ( T d ; R d ) . (28) X_{1-\frac{1+\kappa}{p}}=B_{qp}^{-s+2-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d)\times B_{qp}^{-s+3-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d;\mathbb{R}^d). \tag*{(28)} X 1 − p 1 + κ = B q p − s + 2 − p 2 ( 1 + κ ) ( T d ) × B q p − s + 3 − p 2 ( 1 + κ ) ( T d ; R d ) . (28) As before, we study (MLC) as a stochastic evolution equation:
{ d ( u , d ) + A ( u , d ) d t = F ( u , d ) d t + ( B ( u , d ) + G ( u , d ) ) d W , ( u , d ) ( 0 ) = ( u 0 , d 0 ) , \left\{
\begin{aligned}
\mathrm{d}(u,d)+A(u,d)\,\mathrm{d}t&=F(u,d)\,\mathrm{d}t+(B(u,d)+G(u,d))\,\mathrm{d}W,\\
(u,d)(0)&=(u_0,d_0),
\end{aligned}
\right. { d ( u , d ) + A ( u , d ) d t ( u , d ) ( 0 ) = F ( u , d ) d t + ( B ( u , d ) + G ( u , d )) d W , = ( u 0 , d 0 ) , where W : = ( W n ) n ≥ 1 W:=(W^n)_{n\ge1} W := ( W n ) n ≥ 1 ℓ 2 \ell^2 ℓ 2
A ( u , d ) : = ( − P Δ u , − Δ d ) , A(u,d):=(-\mathbb{P}\Delta u,-\Delta d), A ( u , d ) := ( − P Δ u , − Δ d ) , B ( u , d ) : = ( P ( b 1 , n ⋅ ∇ ) u , ( b 2 , n ⋅ ∇ ) d ) n ≥ 1 , B(u,d):=(\mathbb{P}(b_{1,n}\cdot\nabla)u,(b_{2,n}\cdot\nabla)d)_{n\ge1}, B ( u , d ) := ( P ( b 1 , n ⋅ ∇ ) u , ( b 2 , n ⋅ ∇ ) d ) n ≥ 1 , F ( u , d ) : = ( − P ∇ ⋅ ( u ⊗ u + ∇ d ⊙ ∇ d ) , − ( u ⋅ ∇ ) d − Φ ( d ) ) , F(u,d):=(-\mathbb{P}\nabla\cdot(u\otimes u+\nabla d\odot\nabla d),-(u\cdot\nabla)d-\Phi(d)), F ( u , d ) := ( − P ∇ ⋅ ( u ⊗ u + ∇ d ⊙ ∇ d ) , − ( u ⋅ ∇ ) d − Φ ( d )) , G ( u , d ) : = ( P g 1 , n ( u , d ) , g 2 , n ( d ) ) n ≥ 1 . G(u,d):=(\mathbb{P}g_{1,n}(u,d),g_{2,n}(d))_{n\ge1}. G ( u , d ) := ( P g 1 , n ( u , d ) , g 2 , n ( d ) ) n ≥ 1 . In the current setting, P \mathbb{P} P A A A H δ , q ( T d ; R d ) H^{\delta,q}(\mathbb{T}^d;\mathbb{R}^d) H δ , q ( T d ; R d ) H σ δ , q ( T d ) H_{\sigma}^{\delta,q}(\mathbb{T}^d) H σ δ , q ( T d ) [3 §2.1, (3.7)] ). Note that the divergence free condition and the boundary conditions are accounted for by the Helmholtz–Leray projection and the choice of the ground spaces.
Throughout what follows, we will be using the following identity in distributional sense:
if ∇ ⋅ u = 0 , then ( u ⋅ ∇ ) v = ∇ ⋅ ( u ⊗ v ) . (29) \text{if }\nabla\cdot u=0,\text{ then }(u\cdot\nabla)v=\nabla\cdot(u\otimes v). \tag*{(29)} if ∇ ⋅ u = 0 , then ( u ⋅ ∇ ) v = ∇ ⋅ ( u ⊗ v ) . (29) Remark 3.1. The following estimate will be frequently used for the Φ \Phi Φ F F F x , y ∈ R d x,y\in\mathbb{R}^d x , y ∈ R d m ∈ N m\in\mathbb{N} m ∈ N
∣ ∣ x ∣ m x − ∣ y ∣ m y ∣ ≤ C m ∣ x − y ∣ ( ∣ x ∣ m + ∣ y ∣ m ) . \left||x|^m x-|y|^m y\right|\le C_m|x-y|\left(|x|^m+|y|^m\right). ∣ ∣ x ∣ m x − ∣ y ∣ m y ∣ ≤ C m ∣ x − y ∣ ( ∣ x ∣ m + ∣ y ∣ m ) . Therefore,
∣ Φ ( d ) − Φ ( f ) ∣ ≲ ∣ d − f ∣ ∑ i = 0 k ( ∣ d ∣ 2 i + ∣ f ∣ 2 i ) ≲ ∣ d − f ∣ ( 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k ) . |\Phi(d)-\Phi(f)|\lesssim|d-f|\sum_{i=0}^{k}\left(|d|^{2i}+|f|^{2i}\right)\lesssim|d-f|\left(1+|d|^{2k}+|f|^{2k}\right). ∣Φ ( d ) − Φ ( f ) ∣ ≲ ∣ d − f ∣ i = 0 ∑ k ( ∣ d ∣ 2 i + ∣ f ∣ 2 i ) ≲ ∣ d − f ∣ ( 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k ) . Proposition 3.2 (Stochastic maximal regularity of the linear part in (MLC)). Let s ∈ ( − 1 , 2 ) s \in(-1,2) s ∈ ( − 1 , 2 ) q ∈ [ 2 , ∞ ) q \in[2,\infty) q ∈ [ 2 , ∞ ) X 0 X_0 X 0 X 1 X_1 X 1 (26) and let Assumption 1.2 be satisfied. Then for every r ∈ ( 2 , ∞ ) r \in(2,\infty) r ∈ ( 2 , ∞ ) α ∈ [ 0 , r 2 − 1 ) \alpha\in[0,\frac{r}{2}-1) α ∈ [ 0 , 2 r − 1 ) ( A , B ) (A,B) ( A , B )
( A , B ) ∈ S M R r , α ∙ (A,B) \in SMR^{\bullet}_{r,\alpha} ( A , B ) ∈ S M R r , α ∙ in the sense of [1 Definition 3.8] .
Proof. Fix s ∈ ( − 1 , 2 ) s \in(-1,2) s ∈ ( − 1 , 2 ) r ∈ ( 2 , ∞ ) r \in(2,\infty) r ∈ ( 2 , ∞ ) α ∈ [ 0 , r 2 − 1 ) \alpha\in[0,\frac{r}{2}-1) α ∈ [ 0 , 2 r − 1 )
Velocity component. Set δ : = 1 − s \delta:= 1-s δ := 1 − s δ ∈ ( − 1 , 2 ) \delta\in(-1,2) δ ∈ ( − 1 , 2 )
H − 1 + δ , q = H − s , q , H 1 + δ , q = H 2 − s , q . H^{-1+\delta,q}=H^{-s,q}, \qquad H^{1+\delta,q}=H^{2-s,q}. H − 1 + δ , q = H − s , q , H 1 + δ , q = H 2 − s , q . In our setting on T d \mathbb{T}^d T d
A S = − P Δ , ( B S , n u ) n ≥ 1 = ( P ( b 1 , n ⋅ ∇ u ) ) n ≥ 1 , A_S=-P\Delta, \qquad(B_{S,n}u)_{n\geq1}=(P(b_{1,n}\cdot\nabla u))_{n\geq1}, A S = − P Δ , ( B S , n u ) n ≥ 1 = ( P ( b 1 , n ⋅ ∇ u ) ) n ≥ 1 , where P P P ( b 1 , n ) n ≥ 1 (b_{1,n})_{n\geq1} ( b 1 , n ) n ≥ 1 a i j = δ i j a^{ij}=\delta_{ij} a ij = δ ij c = h = 0 c=h=0 c = h = 0 [3 Assumption 3.1(1)–(3)] ; indeed, the stochastic parabolicity bound in Assumption 1.2(1) is equivalent to [3 Assumption 3.1(3)] (with a i j = δ i j a^{ij}=\delta_{ij} a ij = δ ij b 1 , n b_{1,n} b 1 , n [3 Assumption 3.1] are satisfied for the above value of δ \delta δ [3 Theorem 3.2] (with p = r p=r p = r κ = α \kappa=\alpha κ = α δ \delta δ L r L^r L r
X 0 u = H − s , q , X 1 u = H 2 − s , q . X_0^u=H^{-s,q}, \qquad X_1^u=H^{2-s,q}. X 0 u = H − s , q , X 1 u = H 2 − s , q . Director component. Set σ : = 1 − s \sigma:=1-s σ := 1 − s σ ∈ ( − 1 , 2 ) \sigma\in(-1,2) σ ∈ ( − 1 , 2 )
H σ , q = H 1 − s , q , H 2 + σ , q = H 3 − s , q . H^{\sigma,q}=H^{1-s,q}, \qquad H^{2+\sigma,q}=H^{3-s,q}. H σ , q = H 1 − s , q , H 2 + σ , q = H 3 − s , q . Consider the scalar operator pair
A d = − Δ , ( B d , n φ ) n ≥ 1 = ( b 2 , n ⋅ ∇ φ ) n ≥ 1 . A_d=-\Delta, \qquad(B_{d,n}\varphi)_{n\geq1}=(b_{2,n}\cdot\nabla\varphi)_{n\geq1}. A d = − Δ , ( B d , n φ ) n ≥ 1 = ( b 2 , n ⋅ ∇ φ ) n ≥ 1 . By Assumption 1.2(1), and since the coefficients are constant, the conditions in [2 Assumption 5.1] are satisfied (with a i j = δ i j a^{ij}=\delta_{ij} a ij = δ ij b n j = b 2 , n j b_n^j=b_{2,n}^j b n j = b 2 , n j [2 Theorem 5.2] (with p = r p=r p = r κ = α \kappa=\alpha κ = α σ = 1 − s \sigma=1-s σ = 1 − s L r L^r L r ( − Δ , ( b 2 , n ⋅ ∇ ) n ) (-\Delta,(b_{2,n}\cdot\nabla)_n) ( − Δ , ( b 2 , n ⋅ ∇ ) n )
X 0 d = H 1 − s , q , X 1 d = H 3 − s , q . X_0^d=H^{1-s,q}, \qquad X_1^d=H^{3-s,q}. X 0 d = H 1 − s , q , X 1 d = H 3 − s , q . Applying the scalar result componentwise gives the same conclusion for R d \mathbb{R}^d R d ( A , B ) ∈ S M R r , α ∙ (A,B)\in SMR^\bullet_{r,\alpha} ( A , B ) ∈ S M R r , α ∙ [1 Definition 3.8] . □
In the statements of the next three local well-posedness results, we will distinguish between the ranges s ∈ [ 1 , 2 ) s \in[1,2) s ∈ [ 1 , 2 ) s ∈ ( − 1 , 1 ) s \in(-1,1) s ∈ ( − 1 , 1 )
Proposition 3.3 (Local well-posedness, 1 ≤ s < 2 1\leq s<2 1 ≤ s < 2 s ∈ [ 1 , 2 ) s \in[1,2) s ∈ [ 1 , 2 ) p , q ∈ ( 2 , ∞ ) p,q \in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ [ 0 , p 2 − 1 ) \kappa\in[0,\frac{p}{2}-1) κ ∈ [ 0 , 2 p − 1 ) X 0 X_0 X 0 X 1 X_1 X 1 (26) and let Assumption 1.2 be satisfied. Suppose that
1 − 1 + κ p < d − ( s − 1 ) q 4 q + s 2 ≤ 1 − 1 + κ 2 p (30) 1-\frac{1+\kappa}{p}<\frac{d-(s-1)q}{4q}+\frac{s}{2}\leq1-\frac{1+\kappa}{2p} \tag*{(30)} 1 − p 1 + κ < 4 q d − ( s − 1 ) q + 2 s ≤ 1 − 2 p 1 + κ (30) 1 κ > s − 3 + d q + 2 ( 1 + κ p ) , (31) \frac{1}{\kappa}>s-3+\frac{d}{q}+2\left(\frac{1+\kappa}{p}\right), \tag*{(31)} κ 1 > s − 3 + q d + 2 ( p 1 + κ ) , (31) ( s − 1 ) q ≤ d . (32) (s-1)q\leq d. \tag*{(32)} ( s − 1 ) q ≤ d . (32) Then, for all (u_0,d_0)\in L^0_{\mathcal{F}_0}(\Omega;X_1^{-\frac{1+\kappa}{p}}_p) , (MLC) has a unique maximal solution ( u , d , σ ) (u,d,\sigma) ( u , d , σ ) σ > 0 \sigma>0 σ > 0
( u , d ) ∈ H l o c θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ( [ 0 , σ ) ; X 1 − 1 + κ p , p ) for all θ ∈ [ 0 , 1 / 2 ) , (u,d)\in H^\theta_{\mathrm{loc}}(0,\sigma,w_\kappa;X_{1-\theta})\cap C([0,\sigma);X_{1-\frac{1+\kappa}{p},p}) \qquad\text{for all }\theta\in[0,1/2), ( u , d ) ∈ H loc θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ([ 0 , σ ) ; X 1 − p 1 + κ , p ) for all θ ∈ [ 0 , 1/2 ) , where X 1 − θ X_{1-\theta} X 1 − θ X 1 − 1 + κ p , p X_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p (27) and (28) , respectively.
Proposition 3.4 (Local well-posedness, 0 < s < 1 0 < s < 1 0 < s < 1 s ∈ ( 0 , 1 ) s \in(0,1) s ∈ ( 0 , 1 ) p , q ∈ ( 2 , ∞ ) p,q \in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ [ 0 , p 2 − 1 ) \kappa\in[0,\frac{p}{2}-1) κ ∈ [ 0 , 2 p − 1 ) X 0 X_0 X 0 X 1 X_1 X 1 (26) and let Assumption 1.2 be satisfied. Assume additionally that
( g 1 , n ) n ≥ 1 ∈ C 3 ( R d × R d ; ℓ 2 ( R d ) ) , ( g 2 , n ) n ≥ 1 ∈ C 4 ( R d ; ℓ 2 ( R d ) ) , (g_{1,n})_{n\geq1} \in C^3(\mathbb{R}^d \times\mathbb{R}^d;\ell^2(\mathbb{R}^d)), \qquad(g_{2,n})_{n\geq1} \in C^4(\mathbb{R}^d;\ell^2(\mathbb{R}^d)), ( g 1 , n ) n ≥ 1 ∈ C 3 ( R d × R d ; ℓ 2 ( R d )) , ( g 2 , n ) n ≥ 1 ∈ C 4 ( R d ; ℓ 2 ( R d )) , and that
d q < 2 − s − 2 ( 1 + κ ) p . \frac{d}{q} < 2-s-\frac{2(1+\kappa)}{p}. q d < 2 − s − p 2 ( 1 + κ ) . Suppose that
1 + s 2 ≤ 1 − 2 κ 1 + 2 κ 1 + κ p (33) \frac{1+s}{2} \leq1-\frac{2\kappa}{1+2\kappa}\frac{1+\kappa}{p} \tag*{(33)} 2 1 + s ≤ 1 − 1 + 2 κ 2 κ p 1 + κ (33) q ≥ d 3 − s − 8 κ 1 + 2 κ 1 + κ p . (34) q \geq\frac{d}{3-s-\frac{8\kappa}{1+2\kappa}\frac{1+\kappa}{p}}. \tag*{(34)} q ≥ 3 − s − 1 + 2 κ 8 κ p 1 + κ d . (34) Then, for all ( u 0 , d 0 ) ∈ L F 0 0 ( Ω ; X 1 − 1 + κ p ) (u_0,d_0) \in L^0_{\mathcal{F}_0}(\Omega;X_{1-\frac{1+\kappa}{p}}) ( u 0 , d 0 ) ∈ L F 0 0 ( Ω ; X 1 − p 1 + κ ) ( u , d , σ ) (u,d,\sigma) ( u , d , σ ) σ > 0 \sigma>0 σ > 0
( u , d ) ∈ H l o c θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ( [ 0 , σ ) ; X 1 − 1 + κ p ) (u,d) \in H^\theta_{\mathrm{loc}}(0,\sigma,w_\kappa;X_{1-\theta}) \cap C([0,\sigma);X_{1-\frac{1+\kappa}{p}}) ( u , d ) ∈ H loc θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ([ 0 , σ ) ; X 1 − p 1 + κ ) for all θ ∈ [ 0 , 1 / 2 ) \theta\in[0,1/2) θ ∈ [ 0 , 1/2 ) X 1 − θ X_{1-\theta} X 1 − θ X 1 − 1 + κ p X_{1-\frac{1+\kappa}{p}} X 1 − p 1 + κ (27) and (28) , respectively.
Proposition 3.5 (Local well-posedness, − 1 < s ≤ 0 -1 < s \leq0 − 1 < s ≤ 0 s ∈ ( − 1 , 0 ] s \in(-1,0] s ∈ ( − 1 , 0 ] p , q ∈ ( 2 , ∞ ) p,q \in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ [ 0 , p 2 − 1 ) \kappa\in[0,\frac{p}{2}-1) κ ∈ [ 0 , 2 p − 1 ) X 0 X_0 X 0 X 1 X_1 X 1 (26) and let Assumption 1.2 be satisfied. Assume additionally that
( g 1 , n ) n ≥ 1 ∈ C 3 ( R d × R d ; ℓ 2 ( R d ) ) , ( g 2 , n ) n ≥ 1 ∈ C 4 ( R d ; ℓ 2 ( R d ) ) , (g_{1,n})_{n\geq1} \in C^3(\mathbb{R}^d \times\mathbb{R}^d;\ell^2(\mathbb{R}^d)), \qquad(g_{2,n})_{n\geq1} \in C^4(\mathbb{R}^d;\ell^2(\mathbb{R}^d)), ( g 1 , n ) n ≥ 1 ∈ C 3 ( R d × R d ; ℓ 2 ( R d )) , ( g 2 , n ) n ≥ 1 ∈ C 4 ( R d ; ℓ 2 ( R d )) , and that
d q < 2 − s − 2 ( 1 + κ ) p . \frac{d}{q} < 2-s-\frac{2(1+\kappa)}{p}. q d < 2 − s − p 2 ( 1 + κ ) . Suppose that
1 − 1 + κ p ≤ 1 − 2 κ 1 + 2 κ 1 + κ p (35) 1-\frac{1+\kappa}{p} \leq1-\frac{2\kappa}{1+2\kappa}\frac{1+\kappa}{p} \tag*{(35)} 1 − p 1 + κ ≤ 1 − 1 + 2 κ 2 κ p 1 + κ (35) q ≥ d 3 − s − 8 κ 1 + 2 κ 1 + κ p . (36) q \geq\frac{d}{3-s-\frac{8\kappa}{1+2\kappa}\frac{1+\kappa}{p}}. \tag*{(36)} q ≥ 3 − s − 1 + 2 κ 8 κ p 1 + κ d . (36) Then, for all ( u 0 , d 0 ) ∈ L F 0 0 ( Ω ; X 1 − 1 + κ p ) (u_0,d_0) \in L^0_{\mathcal{F}_0}(\Omega;X_{1-\frac{1+\kappa}{p}}) ( u 0 , d 0 ) ∈ L F 0 0 ( Ω ; X 1 − p 1 + κ ) ( u , d , σ ) (u,d,\sigma) ( u , d , σ ) σ > 0 \sigma>0 σ > 0
( u , d ) ∈ H l o c θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ( [ 0 , σ ) ; X 1 − 1 + κ p ) (u,d) \in H^\theta_{\mathrm{loc}}(0,\sigma,w_\kappa;X_{1-\theta}) \cap C([0,\sigma);X_{1-\frac{1+\kappa}{p}}) ( u , d ) ∈ H loc θ ( 0 , σ , w κ ; X 1 − θ ) ∩ C ([ 0 , σ ) ; X 1 − p 1 + κ ) for all θ ∈ [ 0 , 1 / 2 ) \theta\in[0,1/2) θ ∈ [ 0 , 1/2 ) X 1 − θ X_{1-\theta} X 1 − θ X 1 − 1 + κ p X_{1-\frac{1+\kappa}{p}} X 1 − p 1 + κ (27) and (28) , respectively.
We start with the proof of Proposition 3.3 , in which we have s ∈ [ 1 , 2 ) s \in[1,2) s ∈ [ 1 , 2 )
Proof of Proposition 3.3 . We prove that the abstract conditions in [1 Assumption 4.1] are satisfied, so that the result follows from [1, Theorem 4.7.
We note that A ∈ L ( X 1 , X 0 ) A \in\mathcal{L}(X_1,X_0) A ∈ L ( X 1 , X 0 ) B ∈ L ( X 1 , γ ( ℓ 2 , X 1 / 2 ) ) B \in\mathcal{L}(X_1,\gamma(\ell^2,X_{1/2})) B ∈ L ( X 1 , γ ( ℓ 2 , X 1/2 )) λ + A \lambda+A λ + A λ ≥ 0 \lambda\geq0 λ ≥ 0 [3 Theorem 3.2] ) and ( A , B ) ∈ S M R p , κ ∙ (A,B) \in SMR^\bullet_{p,\kappa} ( A , B ) ∈ S M R p , κ ∙ Proposition 3.2 . It remains to prove local Lipschitz estimates for F F F G G G
We estimate F F F ( u , d ) , ( v , f ) ∈ X 1 (u,d),(v,f) \in X_1 ( u , d ) , ( v , f ) ∈ X 1
∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ ( ( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ ( ( u − v ) ⋅ ∇ ) d ∥ L λ + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ L λ ≲ ∥ u − v ∥ L 2 λ ∥ ∇ d ∥ L 2 λ + ∥ v ∥ L 2 λ ∥ ∇ ( d − f ) ∥ L 2 λ ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q , \begin{aligned}
\|(u\cdot\nabla)d-(v\cdot\nabla)f\|_{H^{1-s,q}}
&\leq\|((u-v)\cdot\nabla)d\|_{H^{1-s,q}}+\|(v\cdot\nabla)(d-f)\|_{H^{1-s,q}}\\
&\lesssim\|((u-v)\cdot\nabla)d\|_{L^\lambda}+\|(v\cdot\nabla)(d-f)\|_{L^\lambda}\\
&\lesssim\|u-v\|_{L^{2\lambda}}\|\nabla d\|_{L^{2\lambda}}+\|v\|_{L^{2\lambda}}\|\nabla(d-f)\|_{L^{2\lambda}}\\
&\lesssim\|u-v\|_{H^{-s+2\beta,q}}\|d\|_{H^{1-s+2\beta,q}}+\|v\|_{H^{-s+2\beta,q}}\|d-f\|_{H^{1-s+2\beta,q}},
\end{aligned} ∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ (( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ (( u − v ) ⋅ ∇ ) d ∥ L λ + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ L λ ≲ ∥ u − v ∥ L 2 λ ∥∇ d ∥ L 2 λ + ∥ v ∥ L 2 λ ∥∇ ( d − f ) ∥ L 2 λ ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q , where λ : = d q ( s − 1 ) q + d \lambda:= \frac{dq}{(s-1)q+d} λ := ( s − 1 ) q + d d q β : = d − ( s − 1 ) q 4 q + s 2 ∈ [ 1 2 , 1 ) \beta:= \frac{d-(s-1)q}{4q}+\frac{s}{2}\in\left[\frac{1}{2},1\right) β := 4 q d − ( s − 1 ) q + 2 s ∈ [ 2 1 , 1 )
− d λ = 1 − s − d q , − s + 2 β − d q = − d 2 λ . (37) -\frac{d}{\lambda}=1-s-\frac{d}{q},\qquad-s+2\beta-\frac{d}{q}=-\frac{d}{2\lambda}. \tag*{(37)} − λ d = 1 − s − q d , − s + 2 β − q d = − 2 λ d . (37) Since s ≥ 1 s\geq1 s ≥ 1 1 / λ = 1 / q + ( s − 1 ) / d 1/\lambda=1/q+(s-1)/d 1/ λ = 1/ q + ( s − 1 ) / d L λ ↪ H 1 − s , q L^\lambda\hookrightarrow H^{1-s,q} L λ ↪ H 1 − s , q ( s − 1 ) q ≤ d (s-1)q\leq d ( s − 1 ) q ≤ d λ ≥ q / 2 > 1 \lambda\geq q/2>1 λ ≥ q /2 > 1 β \beta β λ \lambda λ ( s − 1 ) q ≤ d (s-1)q\leq d ( s − 1 ) q ≤ d
− s + 2 β = − d 2 λ + d q = 1 − s 2 + d 2 q ≥ 0 , -s+2\beta=-\frac{d}{2\lambda}+\frac{d}{q}=\frac{1-s}{2}+\frac{d}{2q}\geq0, − s + 2 β = − 2 λ d + q d = 2 1 − s + 2 q d ≥ 0 , thus H − s + 2 β , q ↪ L 2 λ H^{-s+2\beta,q}\hookrightarrow L^{2\lambda} H − s + 2 β , q ↪ L 2 λ
1 − 1 + κ p < β ≤ 1 − 1 + κ 2 p , 1-\frac{1+\kappa}{p}<\beta\leq1-\frac{1+\kappa}{2p}, 1 − p 1 + κ < β ≤ 1 − 2 p 1 + κ , which can be rewritten as (30) using the definition of λ \lambda λ β \beta β β \beta β λ \lambda λ (29) , we find similarly
∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≲ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q \|(u\cdot\nabla)u-(v\cdot\nabla)v\|_{H^{-s,q}}
\lesssim\|(u-v)\otimes u\|_{H^{1-s,q}}+\|v\otimes(u-v)\|_{H^{1-s,q}} ∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≲ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q ≲ ∥ u − v ∥ L 2 λ ∥ u ∥ L 2 λ + ∥ v ∥ L 2 λ ∥ u − v ∥ L 2 λ \lesssim\|u-v\|_{L^{2\lambda}}\|u\|_{L^{2\lambda}}+\|v\|_{L^{2\lambda}}\|u-v\|_{L^{2\lambda}} ≲ ∥ u − v ∥ L 2 λ ∥ u ∥ L 2 λ + ∥ v ∥ L 2 λ ∥ u − v ∥ L 2 λ ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) . \lesssim\|u-v\|_{H^{-s+2\beta,q}}\left(\|u\|_{H^{-s+2\beta,q}}+\|v\|_{H^{-s+2\beta,q}}\right). ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) . and
∥ ∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≲ ∥ ∇ d − ∇ f ∥ L 2 λ ∥ ∇ d ∥ L 2 λ + ∥ ∇ f ∥ L 2 λ ∥ ∇ d − ∇ f ∥ L 2 λ \|\nabla\cdot(\nabla d\odot\nabla d)-\nabla\cdot(\nabla f\odot\nabla f)\|_{H^{-s,q}}
\lesssim\|\nabla d-\nabla f\|_{L^{2\lambda}}\|\nabla d\|_{L^{2\lambda}}+\|\nabla f\|_{L^{2\lambda}}\|\nabla d-\nabla f\|_{L^{2\lambda}} ∥∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≲ ∥∇ d − ∇ f ∥ L 2 λ ∥∇ d ∥ L 2 λ + ∥∇ f ∥ L 2 λ ∥∇ d − ∇ f ∥ L 2 λ ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . \lesssim\|d-f\|_{H^{1-s+2\beta,q}}\left(\|d\|_{H^{1-s+2\beta,q}}+\|f\|_{H^{1-s+2\beta,q}}\right). ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . Lastly, using Remark 3.1, Sobolev embeddings and Hölder’s inequality, we have
∥ Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥ ∣ d − f ∣ ( 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k ) ∥ L λ \|\Phi(d)-\Phi(f)\|_{H^{1-s,q}}\lesssim\left\||d-f|(1+|d|^{2k}+|f|^{2k})\right\|_{L^\lambda} ∥Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∣ d − f ∣ ( 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k ) L λ ≲ ∥ d − f ∥ L λ ( 1 + 2 k ) ( 1 + ∥ d ∥ L λ ( 1 + 2 k ) 2 k + ∥ f ∥ L λ ( 1 + 2 k ) 2 k ) \lesssim\|d-f\|_{L^{\lambda(1+2k)}}\left(1+\|d\|_{L^{\lambda(1+2k)}}^{2k}+\|f\|_{L^{\lambda(1+2k)}}^{2k}\right) ≲ ∥ d − f ∥ L λ ( 1 + 2 k ) ( 1 + ∥ d ∥ L λ ( 1 + 2 k ) 2 k + ∥ f ∥ L λ ( 1 + 2 k ) 2 k ) ≲ ∥ d − f ∥ H 1 − s + 2 β ^ , q ( 1 + ∥ d ∥ H 1 − s + 2 β ^ , q 2 k + ∥ f ∥ H 1 − s + 2 β ^ , q 2 k ) , \lesssim\|d-f\|_{H^{1-s+2\widehat{\beta},q}}\left(1+\|d\|_{H^{1-s+2\widehat{\beta},q}}^{2k}+\|f\|_{H^{1-s+2\widehat{\beta},q}}^{2k}\right), ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q 2 k + ∥ f ∥ H 1 − s + 2 β , q 2 k ) , provided that 1 − s + 2 β ^ − d q ≥ − d λ ( 1 + 2 k ) 1-s+2\widehat{\beta}-\frac{d}{q}\geq-\frac{d}{\lambda(1+2k)} 1 − s + 2 β − q d ≥ − λ ( 1 + 2 k ) d (37) , the latter is equivalent to
β ^ ≥ k 1 + 2 k ( s − 1 + d q ) . (38) \widehat{\beta}\geq\frac{k}{1+2k}\left(s-1+\frac{d}{q}\right). \tag*{(38)} β ≥ 1 + 2 k k ( s − 1 + q d ) . (38) Moreover, the (sub)criticality conditions for the polynomial term (ρ ^ = 2 k \widehat{\rho}=2k ρ = 2 k
1 − 1 + κ p < β ^ ≤ 1 − 2 k 1 + 2 k 1 + κ p . (39) 1-\frac{1+\kappa}{p}<\widehat{\beta}\leq1-\frac{2k}{1+2k}\frac{1+\kappa}{p}. \tag*{(39)} 1 − p 1 + κ < β ≤ 1 − 1 + 2 k 2 k p 1 + κ . (39) The upper bound above is strictly greater than 1 − ( 1 + κ ) / p 1-(1+\kappa)/p 1 − ( 1 + κ ) / p (31) is equivalent to
1 − 2 k 1 + 2 k 1 + κ p > k 1 + 2 k ( s − 1 + d q ) , 1-\frac{2k}{1+2k}\frac{1+\kappa}{p}>\frac{k}{1+2k}\left(s-1+\frac{d}{q}\right), 1 − 1 + 2 k 2 k p 1 + κ > 1 + 2 k k ( s − 1 + q d ) , so there exists a β ^ \widehat{\beta} β (38) and (39) . Combining the estimates above and recalling (27) , we conclude that
∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β + ∥ ( v , f ) ∥ X β ) \|F(u,d)-F(v,f)\|_{X_0}\lesssim\|(u-v,d-f)\|_{X_\beta}\left(1+\|(u,d)\|_{X_\beta}+\|(v,f)\|_{X_\beta}\right) ∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β + ∥ ( v , f ) ∥ X β ) + ∥ ( u − v , d − f ) ∥ X β ^ ( 1 + ∥ ( u , d ) ∥ X β ^ 2 k + ∥ ( v , f ) ∥ X β ^ 2 k ) . \qquad+\|(u-v,d-f)\|_{X_{\widehat{\beta}}}\left(1+\|(u,d)\|_{X_{\widehat{\beta}}}^{2k}+\|(v,f)\|_{X_{\widehat{\beta}}}^{2k}\right). + ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β 2 k + ∥ ( v , f ) ∥ X β 2 k ) . Next, we estimate G G G λ \lambda λ β \beta β
∥ ( P [ g 1 , n ( u , d ) − g 1 , n ( v , f ) ] ) n ≥ 1 ∥ γ ( ℓ 2 , H − s + 1 , q ) ≲ ∥ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ≥ 1 ∥ γ ( ℓ 2 , L λ ) ≲ ∥ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ≥ 1 ∥ L λ ( T d ; ℓ 2 ) ≲ ∥ ( u , d ) − ( v , f ) ∥ L λ ≲ ∥ ( u , d ) − ( v , f ) ∥ X β . \begin{aligned}
\left\|\left(\mathbb{P}[g_{1,n}(\mathbf{u},d)-g_{1,n}(\mathbf{v},f)]\right)_{n\geq1}\right\|_{\gamma(\ell^2,H^{-s+1,q})}
&\lesssim\left\|\left(g_{1,n}(\mathbf{u},d)-g_{1,n}(\mathbf{v},f)\right)_{n\geq1}\right\|_{\gamma(\ell^2,L^\lambda)} \\
&\lesssim\left\|\left(g_{1,n}(\mathbf{u},d)-g_{1,n}(\mathbf{v},f)\right)_{n\geq1}\right\|_{L^\lambda(\mathbb{T}^d;\ell^2)} \\
&\lesssim\|(\mathbf{u},d)-(\mathbf{v},f)\|_{L^\lambda} \\
&\lesssim\|(\mathbf{u},d)-(\mathbf{v},f)\|_{X_\beta}.
\end{aligned} ( P [ g 1 , n ( u , d ) − g 1 , n ( v , f )] ) n ≥ 1 γ ( ℓ 2 , H − s + 1 , q ) ≲ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ≥ 1 γ ( ℓ 2 , L λ ) ≲ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ≥ 1 L λ ( T d ; ℓ 2 ) ≲ ∥ ( u , d ) − ( v , f ) ∥ L λ ≲ ∥ ( u , d ) − ( v , f ) ∥ X β . where we used [8 Proposition 9.3.2] in the second line, Assumption 1.2 (2) in the third line, and in the last line we used that H 1 − s + 2 β , q ↪ H − s + 2 β , q ↪ L 2 λ ↪ L λ H^{1-s+2\beta,q}\hookrightarrow H^{-s+2\beta,q}\hookrightarrow L^{2\lambda}\hookrightarrow L^\lambda H 1 − s + 2 β , q ↪ H − s + 2 β , q ↪ L 2 λ ↪ L λ
Similarly, for the second component of G G G Assumption 1.2 (2)(3):
∥ ( g 2 , n ( d ) − g 2 , n ( f ) ) n ≥ 1 ∥ γ ( ℓ 2 , H − s + 2 , q ) ≲ ∥ ( g 2 , n ( d ) − g 2 , n ( f ) ) n ≥ 1 ∥ H 1 , λ ( T d ; ℓ 2 ) ≲ ∥ d − f ∥ L λ + ∥ ∇ g 2 , n ( d ) ⋅ ∇ d − ∇ g 2 , n ( f ) ⋅ ∇ f ∥ L λ ( T d ; ℓ 2 ) ≲ ∥ d − f ∥ L λ + ∥ ∇ g 2 , n ( d ) − ∇ g 2 , n ( f ) ∥ L 2 λ ( T d ; ℓ 2 ) ∥ ∇ d ∥ L 2 λ + ∥ ∇ g 2 , n ( f ) ∥ L 2 λ ( T d ; ℓ 2 ) ∥ ∇ d − ∇ f ∥ L 2 λ ≲ ∥ d − f ∥ L λ + ∥ d − f ∥ L 2 λ ∥ ∇ d ∥ L 2 λ + ∥ 1 + ∣ f ∣ ∥ L 2 λ ∥ ∇ d − ∇ f ∥ L 2 λ ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . \begin{aligned}
\left\|\left(g_{2,n}(d)-g_{2,n}(f)\right)_{n\geq1}\right\|_{\gamma(\ell^2,H^{-s+2,q})}
&\lesssim\left\|\left(g_{2,n}(d)-g_{2,n}(f)\right)_{n\geq1}\right\|_{H^{1,\lambda}(\mathbb{T}^d;\ell^2)} \\
&\lesssim\|d-f\|_{L^\lambda}+\|\nabla g_{2,n}(d)\cdot\nabla d-\nabla g_{2,n}(f)\cdot\nabla f\|_{L^\lambda(\mathbb{T}^d;\ell^2)} \\
&\lesssim\|d-f\|_{L^\lambda}+\|\nabla g_{2,n}(d)-\nabla g_{2,n}(f)\|_{L^{2\lambda}(\mathbb{T}^d;\ell^2)}\|\nabla d\|_{L^{2\lambda}} \\
&\qquad+\|\nabla g_{2,n}(f)\|_{L^{2\lambda}(\mathbb{T}^d;\ell^2)}\|\nabla d-\nabla f\|_{L^{2\lambda}} \\
&\lesssim\|d-f\|_{L^\lambda}+\|d-f\|_{L^{2\lambda}}\|\nabla d\|_{L^{2\lambda}} \\
&\qquad+\|1+|f|\|_{L^{2\lambda}}\|\nabla d-\nabla f\|_{L^{2\lambda}} \\
&\lesssim\|d-f\|_{H^{1-s+2\beta,q}}\left(1+\|d\|_{H^{1-s+2\beta,q}}+\|f\|_{H^{1-s+2\beta,q}}\right).
\end{aligned} ( g 2 , n ( d ) − g 2 , n ( f ) ) n ≥ 1 γ ( ℓ 2 , H − s + 2 , q ) ≲ ( g 2 , n ( d ) − g 2 , n ( f ) ) n ≥ 1 H 1 , λ ( T d ; ℓ 2 ) ≲ ∥ d − f ∥ L λ + ∥∇ g 2 , n ( d ) ⋅ ∇ d − ∇ g 2 , n ( f ) ⋅ ∇ f ∥ L λ ( T d ; ℓ 2 ) ≲ ∥ d − f ∥ L λ + ∥∇ g 2 , n ( d ) − ∇ g 2 , n ( f ) ∥ L 2 λ ( T d ; ℓ 2 ) ∥∇ d ∥ L 2 λ + ∥∇ g 2 , n ( f ) ∥ L 2 λ ( T d ; ℓ 2 ) ∥∇ d − ∇ f ∥ L 2 λ ≲ ∥ d − f ∥ L λ + ∥ d − f ∥ L 2 λ ∥∇ d ∥ L 2 λ + ∥1 + ∣ f ∣ ∥ L 2 λ ∥∇ d − ∇ f ∥ L 2 λ ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . Recalling (27) , the preceding bounds yield
∥ G ( u , d ) − G ( v , f ) ∥ γ ( ℓ 2 , X 1 / 2 ) ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β + ∥ ( v , f ) ∥ X β ) (40) \|G(\mathbf{u},d)-G(\mathbf{v},f)\|_{\gamma(\ell^2,X_{1/2})}
\lesssim\|(\mathbf{u}-\mathbf{v},d-f)\|_{X_\beta}\left(1+\|(\mathbf{u},d)\|_{X_\beta}+\|(\mathbf{v},f)\|_{X_\beta}\right) \tag*{(40)} ∥ G ( u , d ) − G ( v , f ) ∥ γ ( ℓ 2 , X 1/2 ) ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β + ∥ ( v , f ) ∥ X β ) (40) Finally, note that X β , 1 ↪ X β X_{\beta,1}\hookrightarrow X_\beta X β , 1 ↪ X β X β ^ , 1 ↪ X β ^ X_{\widehat{\beta},1}\hookrightarrow X_{\widehat{\beta}} X β , 1 ↪ X β
Remark 3.6 (Local Lipschitz and growth properties). Under the assumptions of Proposition 3.3 , the termwise Sobolev product estimates in the proof of Proposition 3.3 validate the local Lipschitz estimate of [1 Assumption 4.1] with ( β 1 , ρ 1 ) = ( β , 1 ) (\beta_1,\rho_1)=(\beta,1) ( β 1 , ρ 1 ) = ( β , 1 ) and ( β 2 , ρ 2 ) = ( β ^ , 2 k ) (\beta_2,\rho_2)=(\widehat{\beta},2k) ( β 2 , ρ 2 ) = ( β , 2 k ) where β \beta β and β ^ \widehat{\beta} β are the exponents introduced in that proof. Moreover, let r ∈ ( 2 , ∞ ) r\in(2,\infty) r ∈ ( 2 , ∞ ) α ∈ [ 0 , r / 2 − 1 ) \alpha\in[0,r/2-1) α ∈ [ 0 , r /2 − 1 ) and 1 + α r ≤ 1 + κ p \frac{1+\alpha}{r}\leq\frac{1+\kappa}{p} r 1 + α ≤ p 1 + κ The upper admissibility bounds in [1 (4.3)] remain valid, but the interpolation exponents may have to be increased. For j = 1 , 2 j=1,2 j = 1 , 2 choose β j ′ ≥ β j \beta'_j\geq\beta_j β j ′ ≥ β j such that
1 − 1 + α r < β j ′ ≤ 1 − ρ j 1 + ρ j 1 + α r . 1-\frac{1+\alpha}{r}<\beta'_j\leq1-\frac{\rho_j}{1+\rho_j}\frac{1+\alpha}{r}. 1 − r 1 + α < β j ′ ≤ 1 − 1 + ρ j ρ j r 1 + α . Such choices exist because the right endpoint is no smaller than the original upper bound and is strictly larger than 1 − 1 + α r 1-\frac{1+\alpha}{r} 1 − r 1 + α The embeddings X β j ′ , 1 ↪ X β j , 1 X_{\beta'_j,1}\hookrightarrow X_{\beta_j,1} X β j ′ , 1 ↪ X β j , 1 preserve the nonlinear estimates. Hence [1 Assumption 4.1] holds in the ( r , α ) (r,\alpha) ( r , α ) framework with these exponents. This is relevant for § 5 . Finally, there exists a constant C > 0 C>0 C > 0 such that for all x ∈ X 1 x\in X_1 x ∈ X 1 one has
∥ F ( x ) ∥ X 0 + ∥ G ( x ) ∥ γ ( U , X 1 / 2 ) ≤ C ( 1 + ∥ x ∥ X β , 1 2 + ∥ x ∥ X β ^ , 1 2 k + 1 ) (41) \|F(x)\|_{X_0}+\|G(x)\|_{\gamma(U,X_{1/2})}\leq C\left(1+\|x\|_{X_{\beta,1}}^2+\|x\|_{X_{\widehat{\beta},1}}^{2k+1}\right) \tag*{(41)} ∥ F ( x ) ∥ X 0 + ∥ G ( x ) ∥ γ ( U , X 1/2 ) ≤ C ( 1 + ∥ x ∥ X β , 1 2 + ∥ x ∥ X β , 1 2 k + 1 ) (41) where X θ , 1 = ( X 0 , X 1 ) θ , 1 X_{\theta,1}=(X_0,X_1)_{\theta,1} X θ , 1 = ( X 0 , X 1 ) θ , 1 denotes real interpolation. To prove this, let x = ( u , d ) ∈ X 1 x=(\mathbf{u},d)\in X_1 x = ( u , d ) ∈ X 1 Applying the local Lipschitz estimates from the proof of Proposition 3.3 with ( v , f ) = ( 0 , 0 ) (\mathbf{v},f)=(0,0) ( v , f ) = ( 0 , 0 ) and observing that F ( 0 ) = 0 F(0)=0 F ( 0 ) = 0 and G ( 0 ) ∈ γ ( U , X 1 / 2 ) G(0)\in\gamma(U,X_{1/2}) G ( 0 ) ∈ γ ( U , X 1/2 ) are constant, we obtain (41) .
Proof of Proposition 3.4. Throughout the proof we repeatedly use the fractional Leibniz rule
∥ f g ∥ H 1 − s , q ≲ ∥ f ∥ H 1 − s , a 1 ∥ g ∥ L a 2 + ∥ f ∥ L b 1 ∥ g ∥ H 1 − s , b 2 , 1 q = 1 a 1 + 1 a 2 = 1 b 1 + 1 b 2 , \|fg\|_{H^{1-s,q}}\lesssim\|f\|_{H^{1-s,a_1}}\|g\|_{L^{a_2}}+\|f\|_{L^{b_1}}\|g\|_{H^{1-s,b_2}},\qquad\frac{1}{q}=\frac{1}{a_1}+\frac{1}{a_2}=\frac{1}{b_1}+\frac{1}{b_2}, ∥ f g ∥ H 1 − s , q ≲ ∥ f ∥ H 1 − s , a 1 ∥ g ∥ L a 2 + ∥ f ∥ L b 1 ∥ g ∥ H 1 − s , b 2 , q 1 = a 1 1 + a 2 1 = b 1 1 + b 2 1 , for suitable exponents a 1 , a 2 , b 1 , b 2 ∈ ( 1 , ∞ ) a_1,a_2,b_1,b_2\in(1,\infty) a 1 , a 2 , b 1 , b 2 ∈ ( 1 , ∞ ) F F F q 1 , q 2 ∈ ( 1 , ∞ ) q_1,q_2\in(1,\infty) q 1 , q 2 ∈ ( 1 , ∞ )
1 q = 1 q 1 + 1 q 2 . \frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}. q 1 = q 1 1 + q 2 1 . Then, for all ( u , d ) , ( v , f ) ∈ X 1 (\mathbf{u},d),(\mathbf{v},f) \in X_1 ( u , d ) , ( v , f ) ∈ X 1
∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ ( ( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ u − v ∥ H 1 − s , q 1 ∥ ∇ d ∥ L q 2 + ∥ u − v ∥ L q 2 ∥ ∇ d ∥ H 1 − s , q 1 + ∥ v ∥ H 1 − s , q 1 ∥ ∇ ( d − f ) ∥ L q 2 + ∥ v ∥ L q 2 ∥ ∇ ( d − f ) ∥ H 1 − s , q 1 ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q , \begin{aligned}
\|(\mathbf{u}\cdot\nabla)d-(\mathbf{v}\cdot\nabla)f\|_{H^{1-s,q}}
&\leq\|((\mathbf{u}-\mathbf{v})\cdot\nabla)d\|_{H^{1-s,q}}+\|(\mathbf{v}\cdot\nabla)(d-f)\|_{H^{1-s,q}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_1}}\|\nabla d\|_{L^{q_2}}+\|\mathbf{u}-\mathbf{v}\|_{L^{q_2}}\|\nabla d\|_{H^{1-s,q_1}}\\
&\quad+\|\mathbf{v}\|_{H^{1-s,q_1}}\|\nabla(d-f)\|_{L^{q_2}}+\|\mathbf{v}\|_{L^{q_2}}\|\nabla(d-f)\|_{H^{1-s,q_1}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{-s+2\beta,q}}\|d\|_{H^{1-s+2\beta,q}}+\|\mathbf{v}\|_{H^{-s+2\beta,q}}\|d-f\|_{H^{1-s+2\beta,q}},
\end{aligned} ∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ (( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ u − v ∥ H 1 − s , q 1 ∥∇ d ∥ L q 2 + ∥ u − v ∥ L q 2 ∥∇ d ∥ H 1 − s , q 1 + ∥ v ∥ H 1 − s , q 1 ∥∇ ( d − f ) ∥ L q 2 + ∥ v ∥ L q 2 ∥∇ ( d − f ) ∥ H 1 − s , q 1 ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q , and recalling (29) , we find similarly
∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≤ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q ≲ ∥ u − v ∥ H 1 − s , q 1 ∥ u ∥ L q 2 + ∥ u − v ∥ L q 2 ∥ u ∥ H 1 − s , q 1 + ∥ v ∥ H 1 − s , q 1 ∥ u − v ∥ L q 2 + ∥ v ∥ L q 2 ∥ u − v ∥ H 1 − s , q 1 ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) , \begin{aligned}
\|(\mathbf{u}\cdot\nabla)\mathbf{u}-(\mathbf{v}\cdot\nabla)\mathbf{v}\|_{H^{-s,q}}
&\leq\|(\mathbf{u}-\mathbf{v})\otimes\mathbf{u}\|_{H^{1-s,q}}+\|\mathbf{v}\otimes(\mathbf{u}-\mathbf{v})\|_{H^{1-s,q}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_1}}\|\mathbf{u}\|_{L^{q_2}}+\|\mathbf{u}-\mathbf{v}\|_{L^{q_2}}\|\mathbf{u}\|_{H^{1-s,q_1}}\\
&\quad+\|\mathbf{v}\|_{H^{1-s,q_1}}\|\mathbf{u}-\mathbf{v}\|_{L^{q_2}}+\|\mathbf{v}\|_{L^{q_2}}\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_1}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{-s+2\beta,q}}\left(\|\mathbf{u}\|_{H^{-s+2\beta,q}}+\|\mathbf{v}\|_{H^{-s+2\beta,q}}\right),
\end{aligned} ∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≤ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q ≲ ∥ u − v ∥ H 1 − s , q 1 ∥ u ∥ L q 2 + ∥ u − v ∥ L q 2 ∥ u ∥ H 1 − s , q 1 + ∥ v ∥ H 1 − s , q 1 ∥ u − v ∥ L q 2 + ∥ v ∥ L q 2 ∥ u − v ∥ H 1 − s , q 1 ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) , and
∥ ∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≤ ∥ ( ∇ d − ∇ f ) ⊙ ∇ d ∥ H 1 − s , q + ∥ ∇ f ⊙ ( ∇ d − ∇ f ) ∥ H 1 − s , q ≲ ∥ ∇ d − ∇ f ∥ H 1 − s , q 1 ∥ ∇ d ∥ L q 2 + ∥ ∇ d − ∇ f ∥ L q 2 ∥ ∇ d ∥ H 1 − s , q 1 + ∥ ∇ f ∥ H 1 − s , q 1 ∥ ∇ d − ∇ f ∥ L q 2 + ∥ ∇ f ∥ L q 2 ∥ ∇ d − ∇ f ∥ H 1 − s , q 1 ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . \begin{aligned}
\|\nabla\cdot(\nabla d\odot\nabla d)-\nabla\cdot(\nabla f\odot\nabla f)\|_{H^{-s,q}}
&\leq\|(\nabla d-\nabla f)\odot\nabla d\|_{H^{1-s,q}}+\|\nabla f\odot(\nabla d-\nabla f)\|_{H^{1-s,q}}\\
&\lesssim\|\nabla d-\nabla f\|_{H^{1-s,q_1}}\|\nabla d\|_{L^{q_2}}+\|\nabla d-\nabla f\|_{L^{q_2}}\|\nabla d\|_{H^{1-s,q_1}}\\
&\quad+\|\nabla f\|_{H^{1-s,q_1}}\|\nabla d-\nabla f\|_{L^{q_2}}+\|\nabla f\|_{L^{q_2}}\|\nabla d-\nabla f\|_{H^{1-s,q_1}}\\
&\lesssim\|d-f\|_{H^{1-s+2\beta,q}}\left(\|d\|_{H^{1-s+2\beta,q}}+\|f\|_{H^{1-s+2\beta,q}}\right).
\end{aligned} ∥∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≤ ∥ ( ∇ d − ∇ f ) ⊙ ∇ d ∥ H 1 − s , q + ∥∇ f ⊙ ( ∇ d − ∇ f ) ∥ H 1 − s , q ≲ ∥∇ d − ∇ f ∥ H 1 − s , q 1 ∥∇ d ∥ L q 2 + ∥∇ d − ∇ f ∥ L q 2 ∥∇ d ∥ H 1 − s , q 1 + ∥∇ f ∥ H 1 − s , q 1 ∥∇ d − ∇ f ∥ L q 2 + ∥∇ f ∥ L q 2 ∥∇ d − ∇ f ∥ H 1 − s , q 1 ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . Lastly, using Remark 3.1, define the matrix-valued polynomial
Ψ ( d , f ) : = ∫ 0 1 D Φ ( f + θ ( d − f ) ) d θ . \Psi(d,f):=\int_0^1 D\Phi(f+\theta(d-f))\,d\theta. Ψ ( d , f ) := ∫ 0 1 D Φ ( f + θ ( d − f )) d θ . Then
Φ ( d ) − Φ ( f ) = Ψ ( d , f ) ( d − f ) , ∣ Ψ ( d , f ) ∣ ≲ 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k . \Phi(d)-\Phi(f)=\Psi(d,f)(d-f),\qquad|\Psi(d,f)|\lesssim1+|d|^{2k}+|f|^{2k}. Φ ( d ) − Φ ( f ) = Ψ ( d , f ) ( d − f ) , ∣Ψ ( d , f ) ∣ ≲ 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k . Let p 0 , p 1 ∈ ( 1 , ∞ ) p_0,p_1\in(1,\infty) p 0 , p 1 ∈ ( 1 , ∞ )
q ~ 1 : = p 1 , q ~ 2 : = p 0 2 k , r ~ 1 : = p 0 , 1 r ~ 2 : = 1 p 1 + 2 k − 1 p 0 . \widetilde{q}_1:=p_1,\qquad\widetilde{q}_2:=\frac{p_0}{2k},\qquad\widetilde{r}_1:=p_0,\qquad\frac{1}{\widetilde{r}_2}:=\frac{1}{p_1}+\frac{2k-1}{p_0}. q 1 := p 1 , q 2 := 2 k p 0 , r 1 := p 0 , r 2 1 := p 1 1 + p 0 2 k − 1 . Then
1 q = 1 q ~ 1 + 1 q ~ 2 = 1 r ~ 1 + 1 r ~ 2 \frac{1}{q}=\frac{1}{\widetilde{q}_1}+\frac{1}{\widetilde{q}_2}=\frac{1}{\widetilde{r}_1}+\frac{1}{\widetilde{r}_2} q 1 = q 1 1 + q 2 1 = r 1 1 + r 2 1 and the fractional Leibniz rule yields
∥ Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥ Ψ ( d , f ) ( d − f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , q ~ 1 ∥ Ψ ( d , f ) ∥ L q ~ 2 + ∥ d − f ∥ L r ~ 1 ∥ Ψ ( d , f ) ∥ H 1 − s , r ~ 2 ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 ∥ Ψ ( d , f ) ∥ H 1 − s , r ~ 2 . \begin{aligned}
\|\Phi(d)-\Phi(f)\|_{H^{1-s,q}}
&\lesssim\|\Psi(d,f)(d-f)\|_{H^{1-s,q}}\\
&\lesssim\|d-f\|_{H^{1-s,\widetilde{q}_1}}\|\Psi(d,f)\|_{L^{\widetilde{q}_2}}+\|d-f\|_{L^{\widetilde{r}_1}}\|\Psi(d,f)\|_{H^{1-s,\widetilde{r}_2}}\\
&\lesssim\|d-f\|_{H^{1-s,p_1}}\left(1+\|d\|_{L^{p_0}}^{2k}+\|f\|_{L^{p_0}}^{2k}\right)\\
&\quad+\|d-f\|_{L^{p_0}}\|\Psi(d,f)\|_{H^{1-s,\widetilde{r}_2}}.
\end{aligned} ∥Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥Ψ ( d , f ) ( d − f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , q 1 ∥Ψ ( d , f ) ∥ L q 2 + ∥ d − f ∥ L r 1 ∥Ψ ( d , f ) ∥ H 1 − s , r 2 ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 ∥Ψ ( d , f ) ∥ H 1 − s , r 2 . To estimate the last term, we apply the fractional Leibniz rule repeatedly to each monomial in Ψ ( d , f ) \Psi(d,f) Ψ ( d , f ) H 1 − s , p 1 H^{1-s,p_1} H 1 − s , p 1 L p 0 L^{p_0} L p 0 Ψ \Psi Ψ 2 k 2k 2 k
∥ Ψ ( d , f ) ∥ H 1 − s , r ~ 2 ≲ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) . \|\Psi(d,f)\|_{H^{1-s,\widetilde{r}_2}}\lesssim1+\left(\|d\|_{H^{1-s,p_1}}+\|f\|_{H^{1-s,p_1}}\right)\left(1+\|d\|_{L^{p_0}}^{2k-1}+\|f\|_{L^{p_0}}^{2k-1}\right). ∥Ψ ( d , f ) ∥ H 1 − s , r 2 ≲ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) . Consequently,
∥ Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 [ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) ] ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q 2 k + ∥ f ∥ H 1 − s + 2 β , q 2 k ) . \begin{aligned}
\|\Phi(d)-\Phi(f)\|_{H^{1-s,q}}
&\lesssim\|d-f\|_{H^{1-s,p_1}}\left(1+\|d\|_{L^{p_0}}^{2k}+\|f\|_{L^{p_0}}^{2k}\right)\\
&\quad+\|d-f\|_{L^{p_0}}\left[1+\left(\|d\|_{H^{1-s,p_1}}+\|f\|_{H^{1-s,p_1}}\right)\left(1+\|d\|_{L^{p_0}}^{2k-1}+\|f\|_{L^{p_0}}^{2k-1}\right)\right]\\
&\lesssim\|d-f\|_{H^{1-s+2\beta,q}}\left(1+\|d\|_{H^{1-s+2\beta,q}}^{2k}+\|f\|_{H^{1-s+2\beta,q}}^{2k}\right).
\end{aligned} ∥Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 [ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) ] ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q 2 k + ∥ f ∥ H 1 − s + 2 β , q 2 k ) . We now specify the parameters. Choose β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
max { 1 − 1 + κ p , 1 + s 2 , 1 4 ( 1 + s + d q ) } ≤ β ≤ 1 − 2 k 1 + 2 k 1 + κ p . (42) \max\left\{1-\frac{1+\kappa}{p},\frac{1+s}{2},\frac{1}{4}\left(1+s+\frac{d}{q}\right)\right\}\leq\beta\leq1-\frac{2k}{1+2k}\frac{1+\kappa}{p}. \tag*{(42)} max { 1 − p 1 + κ , 2 1 + s , 4 1 ( 1 + s + q d ) } ≤ β ≤ 1 − 1 + 2 k 2 k p 1 + κ . (42) This is possible by (33) and (34) . We additionally choose β > 1 − 1 + κ p \beta> 1 - \frac{1+\kappa}{p} β > 1 − p 1 + κ
1 − 2 k 1 + 2 k 1 + κ p − ( 1 − 1 + κ p ) = 1 + κ p ( 1 + 2 k ) > 0. 1 - \frac{2k}{1+2k}\frac{1+\kappa}{p} - \left(1-\frac{1+\kappa}{p}\right) = \frac{1+\kappa}{p(1+2k)} > 0. 1 − 1 + 2 k 2 k p 1 + κ − ( 1 − p 1 + κ ) = p ( 1 + 2 k ) 1 + κ > 0. For the bilinear terms, we choose q 1 , q 2 q_1,q_2 q 1 , q 2
1 q = 1 q 1 + 1 q 2 , H − s + 2 β , q ↪ H 1 − s , q 1 , H − s + 2 β , q ↪ L q 2 . \frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}, \qquad H^{-s+2\beta,q} \hookrightarrow H^{1-s,q_1}, \qquad H^{-s+2\beta,q} \hookrightarrow L^{q_2}. q 1 = q 1 1 + q 2 1 , H − s + 2 β , q ↪ H 1 − s , q 1 , H − s + 2 β , q ↪ L q 2 . Such exponents exist precisely when
4 β ≥ 1 + s + d q , (43) 4\beta\ge1+s+\frac{d}{q}, \tag*{(43)} 4 β ≥ 1 + s + q d , (43) that is, whenever
β ≥ 1 4 ( 1 + s + d q ) . \beta\ge\frac{1}{4}\left(1+s+\frac{d}{q}\right). β ≥ 4 1 ( 1 + s + q d ) . Moreover, since β ≥ 1 + s 2 \beta\ge\frac{1+s}{2} β ≥ 2 1 + s
− s + 2 β − 1 ≥ 0 , -s+2\beta-1 \ge0, − s + 2 β − 1 ≥ 0 , so that
∥ ∇ d ∥ L q 2 + ∥ ∇ d ∥ H 1 − s , q 1 ≲ ∥ d ∥ H 1 − s + 2 β , q . \|\nabla d\|_{L^{q_2}}+\|\nabla d\|_{H^{1-s,q_1}} \lesssim\|d\|_{H^{1-s+2\beta,q}}. ∥∇ d ∥ L q 2 + ∥∇ d ∥ H 1 − s , q 1 ≲ ∥ d ∥ H 1 − s + 2 β , q . This justifies the preceding bounds for ( u ⋅ ∇ ) d (u\cdot\nabla)d ( u ⋅ ∇ ) d ( u ⋅ ∇ ) u (u\cdot\nabla)u ( u ⋅ ∇ ) u ∇ ⋅ ( ∇ d ⊙ ∇ d ) \nabla\cdot(\nabla d\odot\nabla d) ∇ ⋅ ( ∇ d ⊙ ∇ d ) p 0 , p 1 p_0,p_1 p 0 , p 1
1 q = 1 p 1 + 2 k p 0 , H 1 − s + 2 β , q ↪ H 1 − s , p 1 , H 1 − s + 2 β , q ↪ L p 0 . \frac{1}{q}=\frac{1}{p_1}+\frac{2k}{p_0}, \qquad H^{1-s+2\beta,q}\hookrightarrow H^{1-s,p_1}, \qquad H^{1-s+2\beta,q}\hookrightarrow L^{p_0}. q 1 = p 1 1 + p 0 2 k , H 1 − s + 2 β , q ↪ H 1 − s , p 1 , H 1 − s + 2 β , q ↪ L p 0 . Such exponents exist whenever
k 2 k + 1 ( s − 1 + d q ) ≤ β . \frac{k}{2k+1}\left(s-1+\frac{d}{q}\right)\le\beta. 2 k + 1 k ( s − 1 + q d ) ≤ β . This condition is weaker than the lower bound
β ≥ 1 4 ( 1 + s + d q ) \beta\ge\frac{1}{4}\left(1+s+\frac{d}{q}\right) β ≥ 4 1 ( 1 + s + q d ) appearing in (42) , and is therefore automatically satisfied. Indeed, (42) implies
1 4 ( 1 + s + d q ) < 1 , \frac{1}{4}\left(1+s+\frac{d}{q}\right)<1, 4 1 ( 1 + s + q d ) < 1 , hence s − 1 + d q < 2 s-1+\frac{d}{q}<2 s − 1 + q d < 2 s − 1 + d q ≤ 0 s-1+\frac{d}{q}\le0 s − 1 + q d ≤ 0 β > 0 \beta>0 β > 0
k 2 k + 1 ( s − 1 + d q ) ≤ 1 2 ( s − 1 + d q ) < 1 2 + 1 4 ( s − 1 + d q ) = 1 4 ( 1 + s + d q ) . \frac{k}{2k+1}\left(s-1+\frac{d}{q}\right)\le\frac{1}{2}\left(s-1+\frac{d}{q}\right)<\frac{1}{2}+\frac{1}{4}\left(s-1+\frac{d}{q}\right)=\frac{1}{4}\left(1+s+\frac{d}{q}\right). 2 k + 1 k ( s − 1 + q d ) ≤ 2 1 ( s − 1 + q d ) < 2 1 + 4 1 ( s − 1 + q d ) = 4 1 ( 1 + s + q d ) . The strict lower bound β > 1 − 1 + κ p \beta>1-\frac{1+\kappa}{p} β > 1 − p 1 + κ
β ≤ 1 − 2 k 1 + 2 k 1 + κ p \beta\le1-\frac{2k}{1+2k}\frac{1+\kappa}{p} β ≤ 1 − 1 + 2 k 2 k p 1 + κ is exactly the (sub)criticality condition for the polynomial nonlinearity Φ \Phi Φ ρ = 2 k \rho=2k ρ = 2 k
Combining the estimates above and recalling (27) , we conclude that
∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β 2 k + ∥ ( v , f ) ∥ X β 2 k ) . (44) \|F(u,d)-F(v,f)\|_{X_0}\lesssim\|(u-v,d-f)\|_{X_\beta}\left(1+\|(u,d)\|_{X_\beta}^{2k}+\|(v,f)\|_{X_\beta}^{2k}\right). \tag*{(44)} ∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β 2 k + ∥ ( v , f ) ∥ X β 2 k ) . (44) For G G G d / q < 2 − s − 2 ( 1 + κ ) / p d/q<2-s-2(1+\kappa)/p d / q < 2 − s − 2 ( 1 + κ ) / p ( 1 + κ ) / p < 1 / 2 (1+\kappa)/p<1/2 ( 1 + κ ) / p < 1/2
B q , p 2 − s − 2 ( 1 + κ ) p ↪ H 1 − s , q ∩ L ∞ ( T d ; R d ) , B_{q,p}^{2-s-\frac{2(1+\kappa)}{p}}\hookrightarrow H^{1-s,q}\cap L^\infty(\mathbb{T}^d;\mathbb{R}^d), B q , p 2 − s − p 2 ( 1 + κ ) ↪ H 1 − s , q ∩ L ∞ ( T d ; R d ) , B q , p 3 − s − 2 ( 1 + κ ) p ↪ H 2 − s , q ( T d ; R d ) ∩ L ∞ ( T d ; R d ) . B_{q,p}^{3-s-\frac{2(1+\kappa)}{p}}\hookrightarrow H^{2-s,q}(\mathbb{T}^d;\mathbb{R}^d)\cap L^\infty(\mathbb{T}^d;\mathbb{R}^d). B q , p 3 − s − p 2 ( 1 + κ ) ↪ H 2 − s , q ( T d ; R d ) ∩ L ∞ ( T d ; R d ) . The Sobolev smoothness gap in both embeddings is 1 − 2 ( 1 + κ ) / p > 0 1-2(1+\kappa)/p>0 1 − 2 ( 1 + κ ) / p > 0 m ≥ 1 m\ge1 m ≥ 1 Y = ( u , d ) Y=(u,d) Y = ( u , d ) Y ~ = ( v , f ) ∈ X 1 \widetilde{Y}=(v,f)\in X_1 Y = ( v , f ) ∈ X 1 m m m C 3 C^3 C 3 C 4 C^4 C 4 ℓ 2 \ell^2 ℓ 2 [3 Lemma 4.3] , give
∥ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ∥ H 1 − s , q ( T d ; ℓ 2 ) + ∥ ( g 2 , n ( d ) − g 2 , n ( f ) ) n ∥ H 2 − s , q ( T d ; ℓ 2 ) ≤ C m ∥ Y − Y ~ ∥ X 1 − 1 + κ p , p . \begin{aligned}
\|(g_{1,n}(\mathbf{u},\mathbf{d})-g_{1,n}(\mathbf{v},\mathbf{f}))_n\|_{H^{1-s,q}(\mathbb{T}^d;\ell^2)}
+ \|(g_{2,n}(\mathbf{d})-g_{2,n}(\mathbf{f}))_n\|_{H^{2-s,q}(\mathbb{T}^d;\ell^2)}
\le C_m\|Y-\widetilde{Y}\|_{X_{1-\frac{1+\kappa}{p},p}}.
\end{aligned} ∥ ( g 1 , n ( u , d ) − g 1 , n ( v , f ) ) n ∥ H 1 − s , q ( T d ; ℓ 2 ) + ∥ ( g 2 , n ( d ) − g 2 , n ( f ) ) n ∥ H 2 − s , q ( T d ; ℓ 2 ) ≤ C m ∥ Y − Y ∥ X 1 − p 1 + κ , p . By the γ \gamma γ P \mathbb{P} P X β , 1 ↪ X 1 − 1 + κ p , p X_{\beta,1}\hookrightarrow X_{1-\frac{1+\kappa}{p},p} X β , 1 ↪ X 1 − p 1 + κ , p
∥ G ( Y ) − G ( Y ~ ) ∥ γ ( ℓ 2 , X 1 / 2 ) ≤ C m ∥ Y − Y ~ ∥ X 1 − 1 + κ p , p ≲ C m ∥ Y − Y ~ ∥ X β , 1 . (45) \|G(Y)-G(\widetilde{Y})\|_{\gamma(\ell^2,X_{1/2})}
\le C_m\|Y-\widetilde{Y}\|_{X_{1-\frac{1+\kappa}{p},p}}
\lesssim C_m\|Y-\widetilde{Y}\|_{X_{\beta,1}}. \tag*{(45)} ∥ G ( Y ) − G ( Y ) ∥ γ ( ℓ 2 , X 1/2 ) ≤ C m ∥ Y − Y ∥ X 1 − p 1 + κ , p ≲ C m ∥ Y − Y ∥ X β , 1 . (45) Moreover, G ( 0 ) ∈ γ ( ℓ 2 , X 1 / 2 ) G(0)\in\gamma(\ell^2,X_{1/2}) G ( 0 ) ∈ γ ( ℓ 2 , X 1/2 ) (44) and X β , 1 ↪ X β X_{\beta,1}\hookrightarrow X_\beta X β , 1 ↪ X β [1 Assumption 4.1] , using the drift pair ( β , 2 k ) (\beta,2k) ( β , 2 k ) ( β , 0 ) (\beta,0) ( β , 0 ) (42) , while the noise pair is admissible since 1 − ( 1 + κ ) / p < β < 1 1-(1+\kappa)/p<\beta<1 1 − ( 1 + κ ) / p < β < 1 [1 Theorem 4.7] now give the unique maximal solution and its stated regularity. □
Proof of Proposition 3.5. Throughout the proof we use the paraproduct estimate on T d \mathbb{T}^d T d
∥ f g ∥ H r , p ( T d ) ≲ ∥ f ∥ L p 1 ( T d ) ∥ g ∥ H r , p 2 ( T d ) + ∥ g ∥ L r 1 ( T d ) ∥ f ∥ H r , r 2 ( T d ) , (46) \|fg\|_{H^{r,p}(\mathbb{T}^d)}
\lesssim\|f\|_{L^{p_1}(\mathbb{T}^d)}\|g\|_{H^{r,p_2}(\mathbb{T}^d)}
+\|g\|_{L^{r_1}(\mathbb{T}^d)}\|f\|_{H^{r,r_2}(\mathbb{T}^d)}, \tag*{(46)} ∥ f g ∥ H r , p ( T d ) ≲ ∥ f ∥ L p 1 ( T d ) ∥ g ∥ H r , p 2 ( T d ) + ∥ g ∥ L r 1 ( T d ) ∥ f ∥ H r , r 2 ( T d ) , (46) valid for every r > 0 r>0 r > 0 p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) p 1 , p 2 , r 1 , r 2 ∈ ( 1 , ∞ ) p_1,p_2,r_1,r_2\in(1,\infty) p 1 , p 2 , r 1 , r 2 ∈ ( 1 , ∞ )
1 p = 1 p 1 + 1 p 2 = 1 r 1 + 1 r 2 , \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r_1}+\frac{1}{r_2}, p 1 = p 1 1 + p 2 1 = r 1 1 + r 2 1 , see [5 ] . We estimate F F F q 1 , q 2 ∈ ( 1 , ∞ ) q_1,q_2\in(1,\infty) q 1 , q 2 ∈ ( 1 , ∞ )
1 q = 1 q 1 + 1 q 2 . \frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}. q 1 = q 1 1 + q 2 1 . Then, for all ( u , d ) , ( v , f ) ∈ X 1 (\mathbf{u},\mathbf{d}),(\mathbf{v},\mathbf{f})\in X_1 ( u , d ) , ( v , f ) ∈ X 1 (46) with r = 1 − s r=1-s r = 1 − s
∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ ( ( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ u − v ∥ L q 1 ∥ ∇ d ∥ H 1 − s , q 2 + ∥ ∇ d ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ v ∥ L q 1 ∥ ∇ ( d − f ) ∥ H 1 − s , q 2 + ∥ ∇ ( d − f ) ∥ L q 1 ∥ v ∥ H 1 − s , q 2 ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q . \begin{aligned}
\|(\mathbf{u}\cdot\nabla)\mathbf{d}-(\mathbf{v}\cdot\nabla)\mathbf{f}\|_{H^{1-s,q}}
&\le\|((\mathbf{u}-\mathbf{v})\cdot\nabla)\mathbf{d}\|_{H^{1-s,q}}
+\|(\mathbf{v}\cdot\nabla)(\mathbf{d}-\mathbf{f})\|_{H^{1-s,q}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{L^{q_1}}\|\nabla\mathbf{d}\|_{H^{1-s,q_2}}
+\|\nabla\mathbf{d}\|_{L^{q_1}}\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_2}}\\
&\quad+\|\mathbf{v}\|_{L^{q_1}}\|\nabla(\mathbf{d}-\mathbf{f})\|_{H^{1-s,q_2}}
+\|\nabla(\mathbf{d}-\mathbf{f})\|_{L^{q_1}}\|\mathbf{v}\|_{H^{1-s,q_2}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{-s+2\beta,q}}\|\mathbf{d}\|_{H^{1-s+2\beta,q}}
+\|\mathbf{v}\|_{H^{-s+2\beta,q}}\|\mathbf{d}-\mathbf{f}\|_{H^{1-s+2\beta,q}}.
\end{aligned} ∥ ( u ⋅ ∇ ) d − ( v ⋅ ∇ ) f ∥ H 1 − s , q ≤ ∥ (( u − v ) ⋅ ∇ ) d ∥ H 1 − s , q + ∥ ( v ⋅ ∇ ) ( d − f ) ∥ H 1 − s , q ≲ ∥ u − v ∥ L q 1 ∥∇ d ∥ H 1 − s , q 2 + ∥∇ d ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ v ∥ L q 1 ∥∇ ( d − f ) ∥ H 1 − s , q 2 + ∥∇ ( d − f ) ∥ L q 1 ∥ v ∥ H 1 − s , q 2 ≲ ∥ u − v ∥ H − s + 2 β , q ∥ d ∥ H 1 − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ∥ d − f ∥ H 1 − s + 2 β , q . Recalling (29) , we find similarly
∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≤ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q ≲ ∥ u − v ∥ L q 1 ∥ u ∥ H 1 − s , q 2 + ∥ u ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ v ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ u − v ∥ L q 1 ∥ v ∥ H 1 − s , q 2 ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) , \begin{aligned}
\|(\mathbf{u}\cdot\nabla)\mathbf{u}-(\mathbf{v}\cdot\nabla)\mathbf{v}\|_{H^{-s,q}}
&\le\|(\mathbf{u}-\mathbf{v})\otimes\mathbf{u}\|_{H^{1-s,q}}
+\|\mathbf{v}\otimes(\mathbf{u}-\mathbf{v})\|_{H^{1-s,q}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{L^{q_1}}\|\mathbf{u}\|_{H^{1-s,q_2}}
+\|\mathbf{u}\|_{L^{q_1}}\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_2}}\\
&\quad+\|\mathbf{v}\|_{L^{q_1}}\|\mathbf{u}-\mathbf{v}\|_{H^{1-s,q_2}}
+\|\mathbf{u}-\mathbf{v}\|_{L^{q_1}}\|\mathbf{v}\|_{H^{1-s,q_2}}\\
&\lesssim\|\mathbf{u}-\mathbf{v}\|_{H^{-s+2\beta,q}}
\left(\|\mathbf{u}\|_{H^{-s+2\beta,q}}+\|\mathbf{v}\|_{H^{-s+2\beta,q}}\right),
\end{aligned} ∥ ( u ⋅ ∇ ) u − ( v ⋅ ∇ ) v ∥ H − s , q ≤ ∥ ( u − v ) ⊗ u ∥ H 1 − s , q + ∥ v ⊗ ( u − v ) ∥ H 1 − s , q ≲ ∥ u − v ∥ L q 1 ∥ u ∥ H 1 − s , q 2 + ∥ u ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ v ∥ L q 1 ∥ u − v ∥ H 1 − s , q 2 + ∥ u − v ∥ L q 1 ∥ v ∥ H 1 − s , q 2 ≲ ∥ u − v ∥ H − s + 2 β , q ( ∥ u ∥ H − s + 2 β , q + ∥ v ∥ H − s + 2 β , q ) , and
∥ ∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≤ ∥ ( ∇ d − ∇ f ) ⊙ ∇ d ∥ H 1 − s , q + ∥ ∇ f ⊙ ( ∇ d − ∇ f ) ∥ H 1 − s , q ≲ ∥ ∇ d − ∇ f ∥ L q 1 ∥ ∇ d ∥ H 1 − s , q 2 + ∥ ∇ d ∥ L q 1 ∥ ∇ d − ∇ f ∥ H 1 − s , q 2 + ∥ ∇ f ∥ L q 1 ∥ ∇ d − ∇ f ∥ H 1 − s , q 2 + ∥ ∇ d − ∇ f ∥ L q 1 ∥ ∇ f ∥ H 1 − s , q 2 ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . \begin{aligned}
\|\nabla\cdot(\nabla\mathbf{d}\odot\nabla\mathbf{d})-\nabla\cdot(\nabla\mathbf{f}\odot\nabla\mathbf{f})\|_{H^{-s,q}}
&\le\|(\nabla\mathbf{d}-\nabla\mathbf{f})\odot\nabla\mathbf{d}\|_{H^{1-s,q}}
+\|\nabla\mathbf{f}\odot(\nabla\mathbf{d}-\nabla\mathbf{f})\|_{H^{1-s,q}}\\
&\lesssim\|\nabla\mathbf{d}-\nabla\mathbf{f}\|_{L^{q_1}}\|\nabla\mathbf{d}\|_{H^{1-s,q_2}}
+\|\nabla\mathbf{d}\|_{L^{q_1}}\|\nabla\mathbf{d}-\nabla\mathbf{f}\|_{H^{1-s,q_2}}\\
&\quad+\|\nabla\mathbf{f}\|_{L^{q_1}}\|\nabla\mathbf{d}-\nabla\mathbf{f}\|_{H^{1-s,q_2}}
+\|\nabla\mathbf{d}-\nabla\mathbf{f}\|_{L^{q_1}}\|\nabla\mathbf{f}\|_{H^{1-s,q_2}}\\
&\lesssim\|\mathbf{d}-\mathbf{f}\|_{H^{1-s+2\beta,q}}
\left(\|\mathbf{d}\|_{H^{1-s+2\beta,q}}+\|\mathbf{f}\|_{H^{1-s+2\beta,q}}\right).
\end{aligned} ∥∇ ⋅ ( ∇ d ⊙ ∇ d ) − ∇ ⋅ ( ∇ f ⊙ ∇ f ) ∥ H − s , q ≤ ∥ ( ∇ d − ∇ f ) ⊙ ∇ d ∥ H 1 − s , q + ∥∇ f ⊙ ( ∇ d − ∇ f ) ∥ H 1 − s , q ≲ ∥∇ d − ∇ f ∥ L q 1 ∥∇ d ∥ H 1 − s , q 2 + ∥∇ d ∥ L q 1 ∥∇ d − ∇ f ∥ H 1 − s , q 2 + ∥∇ f ∥ L q 1 ∥∇ d − ∇ f ∥ H 1 − s , q 2 + ∥∇ d − ∇ f ∥ L q 1 ∥∇ f ∥ H 1 − s , q 2 ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( ∥ d ∥ H 1 − s + 2 β , q + ∥ f ∥ H 1 − s + 2 β , q ) . Lastly, using Remark 3.1, define the matrix-valued polynomial
Ψ ( d , f ) : = ∫ 0 1 D Φ ( f + θ ( d − f ) ) d θ . \Psi(\mathbf{d},\mathbf{f}):=\int_0^1D\Phi(\mathbf{f}+\theta(\mathbf{d}-\mathbf{f}))\,\mathrm{d}\theta. Ψ ( d , f ) := ∫ 0 1 D Φ ( f + θ ( d − f )) d θ . Then
Φ ( d ) − Φ ( f ) = Ψ ( d , f ) ( d − f ) , ∣ Ψ ( d , f ) ∣ ≲ 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k . \Phi(\mathbf{d})-\Phi(\mathbf{f})=\Psi(\mathbf{d},\mathbf{f})(\mathbf{d}-\mathbf{f}),\qquad
|\Psi(\mathbf{d},\mathbf{f})|\lesssim1+|\mathbf{d}|^{2k}+|\mathbf{f}|^{2k}. Φ ( d ) − Φ ( f ) = Ψ ( d , f ) ( d − f ) , ∣Ψ ( d , f ) ∣ ≲ 1 + ∣ d ∣ 2 k + ∣ f ∣ 2 k . Let p 0 , p 1 ∈ ( 1 , ∞ ) p_0,p_1 \in(1,\infty) p 0 , p 1 ∈ ( 1 , ∞ )
1 q = 1 p 1 + 2 k p 0 . \frac{1}{q}=\frac{1}{p_1}+\frac{2k}{p_0}. q 1 = p 1 1 + p 0 2 k . Then, applying (46) with r = 1 − s r=1-s r = 1 − s Ψ \Psi Ψ
∥ Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥ Ψ ( d , f ) ( d − f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , p 1 ∥ Ψ ( d , f ) ∥ L p 0 2 k + ∥ d − f ∥ L p 0 ∥ Ψ ( d , f ) ∥ H 1 − s , p ~ ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 [ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) ] ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q 2 k + ∥ f ∥ H 1 − s + 2 β , q 2 k ) . \begin{aligned}
\|\Phi(d)-\Phi(f)\|_{H^{1-s,q}}&\lesssim\|\Psi(d,f)(d-f)\|_{H^{1-s,q}}\\
&\lesssim\|d-f\|_{H^{1-s,p_1}}\|\Psi(d,f)\|_{L^{\frac{p_0}{2k}}}+\|d-f\|_{L^{p_0}}\|\Psi(d,f)\|_{H^{1-s,\widetilde p}}\\
&\lesssim\|d-f\|_{H^{1-s,p_1}}\left(1+\|d\|_{L^{p_0}}^{2k}+\|f\|_{L^{p_0}}^{2k}\right)\\
&\quad+\|d-f\|_{L^{p_0}}\left[1+\left(\|d\|_{H^{1-s,p_1}}+\|f\|_{H^{1-s,p_1}}\right)\left(1+\|d\|_{L^{p_0}}^{2k-1}+\|f\|_{L^{p_0}}^{2k-1}\right)\right]\\
&\lesssim\|d-f\|_{H^{1-s+2\beta,q}}\left(1+\|d\|_{H^{1-s+2\beta,q}}^{2k}+\|f\|_{H^{1-s+2\beta,q}}^{2k}\right).
\end{aligned} ∥Φ ( d ) − Φ ( f ) ∥ H 1 − s , q ≲ ∥Ψ ( d , f ) ( d − f ) ∥ H 1 − s , q ≲ ∥ d − f ∥ H 1 − s , p 1 ∥Ψ ( d , f ) ∥ L 2 k p 0 + ∥ d − f ∥ L p 0 ∥Ψ ( d , f ) ∥ H 1 − s , p ≲ ∥ d − f ∥ H 1 − s , p 1 ( 1 + ∥ d ∥ L p 0 2 k + ∥ f ∥ L p 0 2 k ) + ∥ d − f ∥ L p 0 [ 1 + ( ∥ d ∥ H 1 − s , p 1 + ∥ f ∥ H 1 − s , p 1 ) ( 1 + ∥ d ∥ L p 0 2 k − 1 + ∥ f ∥ L p 0 2 k − 1 ) ] ≲ ∥ d − f ∥ H 1 − s + 2 β , q ( 1 + ∥ d ∥ H 1 − s + 2 β , q 2 k + ∥ f ∥ H 1 − s + 2 β , q 2 k ) . where p ~ \widetilde p p
1 p ~ = 1 p 1 + 2 k − 1 p 0 . \frac{1}{\widetilde p}=\frac{1}{p_1}+\frac{2k-1}{p_0}. p 1 = p 1 1 + p 0 2 k − 1 . We now specify the parameters. Choose β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
max { 1 − 1 + κ p , 1 4 ( 1 + s + d q ) } ≤ β ≤ 1 − 2 k 1 + 2 k 1 + κ p . (47) \max\left\{1-\frac{1+\kappa}{p},\frac{1}{4}\left(1+s+\frac{d}{q}\right)\right\}\leq\beta\leq1-\frac{2k}{1+2k}\frac{1+\kappa}{p}. \tag*{(47)} max { 1 − p 1 + κ , 4 1 ( 1 + s + q d ) } ≤ β ≤ 1 − 1 + 2 k 2 k p 1 + κ . (47) Choose additionally β > 1 − 1 + κ p \beta>1-\frac{1+\kappa}{p} β > 1 − p 1 + κ (36) , and this strict choice is possible because its upper endpoint exceeds the trace index by 1 + κ p ( 1 + 2 k ) > 0 \frac{1+\kappa}{p(1+2k)}>0 p ( 1 + 2 k ) 1 + κ > 0 q 1 , q 2 q_1,q_2 q 1 , q 2
1 q = 1 q 1 + 1 q 2 , H − s + 2 β , q ↪ L q 1 , H − s + 2 β , q ↪ H 1 − s , q 2 . \frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2},\qquad H^{-s+2\beta,q}\hookrightarrow L^{q_1},\qquad H^{-s+2\beta,q}\hookrightarrow H^{1-s,q_2}. q 1 = q 1 1 + q 2 1 , H − s + 2 β , q ↪ L q 1 , H − s + 2 β , q ↪ H 1 − s , q 2 . These embeddings are possible whenever
2 β ≥ 1 and 4 β ≥ 1 + s + d q . 2\beta\geq1\qquad\text{and}\qquad4\beta\geq1+s+\frac{d}{q}. 2 β ≥ 1 and 4 β ≥ 1 + s + q d . Since β ≥ 1 − 1 + κ p ≥ 1 2 \beta\geq1-\frac{1+\kappa}{p}\geq\frac{1}{2} β ≥ 1 − p 1 + κ ≥ 2 1
β ≥ 1 4 ( 1 + s + d q ) . \beta\geq\frac{1}{4}\left(1+s+\frac{d}{q}\right). β ≥ 4 1 ( 1 + s + q d ) . For the potential term, we choose p 0 , p 1 p_0,p_1 p 0 , p 1
1 q = 1 p 1 + 2 k p 0 , H 1 − s + 2 β , q ↪ H 1 − s , p 1 , H 1 − s + 2 β , q ↪ L p 0 . \frac{1}{q}=\frac{1}{p_1}+\frac{2k}{p_0},\qquad H^{1-s+2\beta,q}\hookrightarrow H^{1-s,p_1},\qquad H^{1-s+2\beta,q}\hookrightarrow L^{p_0}. q 1 = p 1 1 + p 0 2 k , H 1 − s + 2 β , q ↪ H 1 − s , p 1 , H 1 − s + 2 β , q ↪ L p 0 . Such exponents exist whenever
β ≥ k 2 k + 1 ( s − 1 + d q ) . \beta\geq\frac{k}{2k+1}\left(s-1+\frac{d}{q}\right). β ≥ 2 k + 1 k ( s − 1 + q d ) . This condition is weaker than the lower bound
β ≥ 1 4 ( 1 + s + d q ) \beta\geq\frac{1}{4}\left(1+s+\frac{d}{q}\right) β ≥ 4 1 ( 1 + s + q d ) appearing in (47) , and is therefore automatically satisfied. Indeed, if s − 1 + d / q ≤ 0 s-1+d/q\leq0 s − 1 + d / q ≤ 0 0 < s − 1 + d / q < 2 0<s-1+d/q<2 0 < s − 1 + d / q < 2
The strict lower bound β > 1 − 1 + κ p \beta>1-\frac{1+\kappa}{p} β > 1 − p 1 + κ
β ≤ 1 − 2 k 1 + 2 k 1 + κ p \beta\leq1-\frac{2k}{1+2k}\frac{1+\kappa}{p} β ≤ 1 − 1 + 2 k 2 k p 1 + κ is exactly the (sub)criticality condition for the polynomial nonlinearity Φ \Phi Φ ρ = 2 k \rho=2k ρ = 2 k (27) , we conclude that
∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β 2 k + ∥ ( v , f ) ∥ X β 2 k ) . (48) \|F(u,d)-F(v,f)\|_{X_0}\lesssim\|(u-v,d-f)\|_{X_\beta}\left(1+\|(u,d)\|_{X_\beta}^{2k}+\|(v,f)\|_{X_\beta}^{2k}\right). \tag*{(48)} ∥ F ( u , d ) − F ( v , f ) ∥ X 0 ≲ ∥ ( u − v , d − f ) ∥ X β ( 1 + ∥ ( u , d ) ∥ X β 2 k + ∥ ( v , f ) ∥ X β 2 k ) . (48) For G G G d / q < 2 − s − 2 ( 1 + κ ) / p d/q < 2-s-2(1+\kappa)/p d / q < 2 − s − 2 ( 1 + κ ) / p 1 − s ∈ [ 1 , 2 ) 1-s \in[1,2) 1 − s ∈ [ 1 , 2 ) 2 − s ∈ [ 2 , 3 ) 2-s \in[2,3) 2 − s ∈ [ 2 , 3 ) C 3 C^3 C 3 ( g 1 , n ) n (g_{1,n})_n ( g 1 , n ) n C 4 C^4 C 4 ( g 2 , n ) n (g_{2,n})_n ( g 2 , n ) n L ∞ L^\infty L ∞ (45) . Also G ( 0 ) ∈ γ ( ℓ 2 , X 1 / 2 ) G(0) \in\gamma(\ell^2,X_{1/2}) G ( 0 ) ∈ γ ( ℓ 2 , X 1/2 ) (48) and X β , 1 ↪ X β X_{\beta,1} \hookrightarrow X_\beta X β , 1 ↪ X β ( β , 2 κ ) (\beta,2\kappa) ( β , 2 κ ) ( β , 0 ) (\beta,0) ( β , 0 ) (47) , while the noise pair is admissible since 1 − ( 1 + κ ) / p < β < 1 1-(1+\kappa)/p < \beta< 1 1 − ( 1 + κ ) / p < β < 1 [1 Theorem 4.7] now give the unique maximal solution and its stated regularity. □
Blow-up criteria
Throughout this section we work on O = T d \mathcal{O}=\mathbb{T}^d O = T d s ∈ [ 1 , 2 ) s\in[1,2) s ∈ [ 1 , 2 ) p , q ∈ ( 2 , ∞ ) p,q\in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ [ 0 , p 2 − 1 ) \kappa\in\left[0,\frac{p}{2}-1\right) κ ∈ [ 0 , 2 p − 1 ) Y = ( u , d ) Y=(u,d) Y = ( u , d ) ( Y , σ ) (Y,\sigma) ( Y , σ )
Y ∈ H l o c θ , p ( 0 , σ , ω κ ; X 1 − θ ) ∩ C ( [ 0 , σ ) ; X 1 − 1 + κ p , p ) Y\in H_{\mathrm{loc}}^{\theta,p}(0,\sigma,\omega_\kappa;X_{1-\theta})\cap C([0,\sigma);X_{1-\frac{1+\kappa}{p},p}) Y ∈ H loc θ , p ( 0 , σ , ω κ ; X 1 − θ ) ∩ C ([ 0 , σ ) ; X 1 − p 1 + κ , p ) for all θ ∈ [ 0 , 1 / 2 ) \theta\in[0,1/2) θ ∈ [ 0 , 1/2 )
Recall that the trace space is given in (28) , i.e.
X 1 − 1 + κ p , p = B q , p − s + 2 − 2 ( 1 + κ ) p ( T d ; R d ) × B q , p − s + 3 − 2 ( 1 + κ ) p ( T d ; R d ) , X_{1-\frac{1+\kappa}{p},p}=\mathbb{B}_{q,p}^{-s+2-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d;\mathbb{R}^d)\times\mathbb{B}_{q,p}^{-s+3-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d;\mathbb{R}^d), X 1 − p 1 + κ , p = B q , p − s + 2 − p 2 ( 1 + κ ) ( T d ; R d ) × B q , p − s + 3 − p 2 ( 1 + κ ) ( T d ; R d ) , and the complex interpolation spaces are given by (27) , i.e.
X 1 − κ p = H q − s + 2 − 2 κ p ( T d ) × H q 3 − s − 2 κ p ( T d ; R d ) . X_{1-\frac{\kappa}{p}}=H_q^{-s+2-\frac{2\kappa}{p}}(\mathbb{T}^d)\times H_q^{3-s-\frac{2\kappa}{p}}(\mathbb{T}^d;\mathbb{R}^d). X 1 − p κ = H q − s + 2 − p 2 κ ( T d ) × H q 3 − s − p 2 κ ( T d ; R d ) . Theorem 4.1 (Blow-up criteria for (MLC) in the L p L^p L p L q L^q L q ( Y , σ ) (Y,\sigma) ( Y , σ )
(i) (No trace limit at the explosion time.) For every T > 0 T>0 T > 0
P ( σ < T and lim t ↑ σ Y ( t ) exists in X 1 − 1 + κ p , p ) = 0. \mathbb{P}\left(\sigma<T\ \text{and}\ \lim_{t\uparrow\sigma}Y(t)\ \text{exists in}\ X_{1-\frac{1+\kappa}{p},p}\right)=0. P ( σ < T and t ↑ σ lim Y ( t ) exists in X 1 − p 1 + κ , p ) = 0. (ii) (Critical blow-up criterion.) For every T > 0 T>0 T > 0
P ( σ < T and sup t ∈ [ 0 , σ ) ∥ Y ( t ) ∥ X 1 − 1 + κ p , p + ∥ Y ∥ L p ( 0 , σ ; X 1 − κ p ) < ∞ ) = 0. \mathbb{P}\left(\sigma<T\ \text{and}\ \sup_{t\in[0,\sigma)}\|Y(t)\|_{X_{1-\frac{1+\kappa}{p},p}}+\|Y\|_{L^p(0,\sigma;X_{1-\frac{\kappa}{p}})}<\infty\right)=0. P ( σ < T and t ∈ [ 0 , σ ) sup ∥ Y ( t ) ∥ X 1 − p 1 + κ , p + ∥ Y ∥ L p ( 0 , σ ; X 1 − p κ ) < ∞ ) = 0. Equivalently, on { σ < ∞ } \{\sigma<\infty\} { σ < ∞ }
sup t ∈ [ 0 , σ ) ∥ Y ( t ) ∥ X 1 − 1 + κ p , p + ∥ Y ∥ L p ( 0 , σ ; X 1 − κ p ) = + ∞ . \sup_{t\in[0,\sigma)}\|Y(t)\|_{X_{1-\frac{1+\kappa}{p},p}}+\|Y\|_{L^p(0,\sigma;X_{1-\frac{\kappa}{p}})}=+\infty. t ∈ [ 0 , σ ) sup ∥ Y ( t ) ∥ X 1 − p 1 + κ , p + ∥ Y ∥ L p ( 0 , σ ; X 1 − p κ ) = + ∞. (iii) (Subcritical blow-up criterion.) Assume ( p , κ ) (p,\kappa) ( p , κ ) [1 ] holds with strict inequality for every pair ( β j , ρ j ) (\beta_j,\rho_j) ( β j , ρ j ) F F F G G G T > 0 T>0 T > 0
P ( σ < T and sup t ∈ [ 0 , σ ) ∥ Y ( t ) ∥ X 1 − 1 + κ p , p < ∞ ) = 0 , \mathbb{P}\left(\sigma<T\ \text{and}\ \sup_{t\in[0,\sigma)}\|Y(t)\|_{X_{1-\frac{1+\kappa}{p},p}}<\infty\right)=0, P ( σ < T and t ∈ [ 0 , σ ) sup ∥ Y ( t ) ∥ X 1 − p 1 + κ , p < ∞ ) = 0 , i.e. on { σ < ∞ } \{\sigma<\infty\} { σ < ∞ }
sup t ∈ [ 0 , σ ) ∥ Y ( t ) ∥ X 1 − 1 + κ p , p = + ∞ . \sup_{t\in[0,\sigma)}\|Y(t)\|_{X_{1-\frac{1+\kappa}{p},p}}=+\infty. t ∈ [ 0 , σ ) sup ∥ Y ( t ) ∥ X 1 − p 1 + κ , p = + ∞. Proof of Theorem 4.1. This is a direct reformulation, in the present setting, of the abstract blow-up criteria in [1 Theorems 5.1 and Theorem 5.2] . □
Remark 4.2. The criterion in Theorem 4.1 (ii)-(iii) is typically used as follows: if one can prove a priori that on { σ < T } \{\sigma<T\} { σ < T }
sup t ∈ [ 0 , σ ) ∥ Y ( t ) ∥ X 1 − 1 + κ p , p (and in the critical case also ∥ Y ∥ L p ( 0 , σ ; X 1 − κ p ) ) \sup_{t\in[0,\sigma)}\|Y(t)\|_{X_{1-\frac{1+\kappa}{p},p}}
\qquad\text{(and in the critical case also }\|Y\|_{L^p(0,\sigma;X_{1-\frac{\kappa}{p}})}\text{)} t ∈ [ 0 , σ ) sup ∥ Y ( t ) ∥ X 1 − p 1 + κ , p (and in the critical case also ∥ Y ∥ L p ( 0 , σ ; X 1 − p κ ) ) is finite almost surely, then P ( σ < T ) = 0 \mathbb{P}(\sigma<T)=0 P ( σ < T ) = 0 T > 0 T>0 T > 0 σ = ∞ \sigma=\infty σ = ∞
Regularisation in time
In § 4 we proved blow-up criteria for the maximal solution ( Y , σ ) (Y,\sigma) ( Y , σ ) Proposition 3.3 . In particular, Theorem 4.1 (i) states that at the maximal existence time σ \sigma σ Y Y Y X 1 − 1 + κ p , p X_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p instantaneous time regularisation : for every t > 0 t>0 t > 0
Proposition 5.1 (Instantaneous time regularisation for κ > 0 \kappa>0 κ > 0 s ∈ [ 1 , 2 ) s\in[1,2) s ∈ [ 1 , 2 ) p , q ∈ ( 2 , ∞ ) p,q\in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ ( 0 , p 2 − 1 ) \kappa\in\left(0,\frac{p}{2}-1\right) κ ∈ ( 0 , 2 p − 1 ) Proposition 3.3 . Let ( Y , σ ) (Y,\sigma) ( Y , σ ) Proposition 3.3 . Then, almost surely, for every r ∈ ( 2 , ∞ ) r\in(2,\infty) r ∈ ( 2 , ∞ ) θ ∈ [ 0 , p 2 ) \theta\in\left[0,\frac{p}{2}\right) θ ∈ [ 0 , 2 p ) ε ∈ ( 0 , θ ) \varepsilon\in(0,\theta) ε ∈ ( 0 , θ )
Y ∈ H l o c θ , r ( ( 0 , σ ) ; X 1 − θ ) , (49) Y\in H_{\mathrm{loc}}^{\theta,r}((0,\sigma);X_{1-\theta}), \tag*{(49)} Y ∈ H loc θ , r (( 0 , σ ) ; X 1 − θ ) , (49) Y ∈ C l o c θ − ε ( ( 0 , σ ) ; X 1 − θ ) . (50) Y\in C_{\mathrm{loc}}^{\theta-\varepsilon}((0,\sigma);X_{1-\theta}). \tag*{(50)} Y ∈ C loc θ − ε (( 0 , σ ) ; X 1 − θ ) . (50) Proof. This is a direct application of the abstract parabolic regularisation result [1 Theorem 5.6] to the equation (MLC). Indeed, the proof of Proposition 3.3 verifies the hypotheses of [1 Theorem 4.7] in the ( p , κ ) (p,\kappa) ( p , κ ) Proposition 3.2 gives
( A , B ) ∈ S M R r , α ∙ for every r ∈ ( 2 , ∞ ) , α ∈ [ 0 , r 2 − 1 ) , (A,B)\in\mathcal{SMR}^{\bullet}_{r,\alpha}\qquad\text{for every }r\in(2,\infty),\quad\alpha\in\left[0,\frac{r}{2}-1\right), ( A , B ) ∈ SM R r , α ∙ for every r ∈ ( 2 , ∞ ) , α ∈ [ 0 , 2 r − 1 ) , and Remark 3.6 records that the nonlinear estimates required in [1 Assumption 4.1] are satisfied for the present equation and are stable under the change of parameters used in the proof of [1 Theorem 5.6] . Therefore all assumptions of [1 Theorem 5.6] are satisfied, and the conclusion of that theorem gives exactly (5.1) and (5.2). □
Proposition 5.2 (Instantaneous time regularisation for κ = 0 \kappa=0 κ = 0 s ∈ [ 1 , 2 ) s\in[1,2) s ∈ [ 1 , 2 ) p , q ∈ ( 2 , ∞ ) p,q\in(2,\infty) p , q ∈ ( 2 , ∞ ) Proposition 3.3 with κ = 0 \kappa=0 κ = 0 ( Y , σ ) (Y,\sigma) ( Y , σ ) r ∈ ( 2 , ∞ ) r\in(2,\infty) r ∈ ( 2 , ∞ ) θ ∈ [ 0 , 1 2 ) \theta\in\left[0,\frac{1}{2}\right) θ ∈ [ 0 , 2 1 ) ε ∈ ( 0 , θ ) \varepsilon\in(0,\theta) ε ∈ ( 0 , θ )
Y ∈ H l o c θ , r ( ( 0 , σ ) ; X 1 − θ ) , (51) Y\in H_{\mathrm{loc}}^{\theta,r}((0,\sigma);X_{1-\theta}), \tag*{(51)} Y ∈ H loc θ , r (( 0 , σ ) ; X 1 − θ ) , (51) Y ∈ C l o c θ − ε ( ( 0 , σ ) ; X 1 − θ ) . (52) Y\in C_{\mathrm{loc}}^{\theta-\varepsilon}((0,\sigma);X_{1-\theta}). \tag*{(52)} Y ∈ C loc θ − ε (( 0 , σ ) ; X 1 − θ ) . (52) Proof. This is a direct application of [1 Theorem 5.7] . Indeed, Proposition 3.3 verifies the hypotheses of [1 Theorem 4.7] in the ( p , 0 ) (p,0) ( p , 0 ) L 0 p L^p_0 L 0 p ( Y , σ ) (Y,\sigma) ( Y , σ ) Proposition 3.2 gives
( A , B ) ∈ S M R r , α ∙ for all r ∈ ( 2 , ∞ ) , α ∈ [ 0 , r 2 − 1 ) . (A,B)\in\mathcal{SMR}^{\bullet}_{r,\alpha}\qquad\text{for all }r\in(2,\infty),\quad\alpha\in\left[0,\frac{r}{2}-1\right). ( A , B ) ∈ SM R r , α ∙ for all r ∈ ( 2 , ∞ ) , α ∈ [ 0 , 2 r − 1 ) . It remains only to check the additional growth assumption [1 (5.21)] . In the notation of [1 Assumption 4.1] , Remark 3.6 identifies the two pairs
( β 1 , ρ 1 ) = ( β , 1 ) , ( β 2 , ρ 2 ) = ( β ^ , 2 k ) , (\beta_1,\rho_1)=(\beta,1),\qquad(\beta_2,\rho_2)=(\widehat{\beta},2k), ( β 1 , ρ 1 ) = ( β , 1 ) , ( β 2 , ρ 2 ) = ( β , 2 k ) , and proves the estimate
∥ F ( x ) ∥ X 0 + ∥ G ( x ) ∥ γ ( U , X 1 / 2 ) ≤ C ( 1 + ∥ x ∥ X β , 1 + ∥ x ∥ X β ^ , 1 2 k + 1 ) , x ∈ X 1 . \|F(x)\|_{X_0}+\|G(x)\|_{\gamma(U,X_{1/2})}\leq C\left(1+\|x\|_{X_{\beta,1}}+\|x\|_{X_{\widehat{\beta},1}}^{2k+1}\right),\qquad x\in X_1. ∥ F ( x ) ∥ X 0 + ∥ G ( x ) ∥ γ ( U , X 1/2 ) ≤ C ( 1 + ∥ x ∥ X β , 1 + ∥ x ∥ X β , 1 2 k + 1 ) , x ∈ X 1 . This is stronger than [1 (5.21)] : one can choose C n = C C_n=C C n = C n n n ∥ x ∥ X 1 − 1 p , ∞ \|x\|_{X_{1-\frac{1}{p},\infty}} ∥ x ∥ X 1 − p 1 , ∞ [1 Theorem 5.7] are satisfied, and its conclusion gives (5.3) and (5.4). □
Bootstrap in space
In this section we first keep the Sobolev index s s s q q q § 5 . Indeed, for every ε > 0 \varepsilon>0 ε > 0 { σ > ε } \{\sigma>\varepsilon\} { σ > ε } Y ( ε ) Y(\varepsilon) Y ( ε ) L p L^p L p L q ∗ L^{q_*} L q ∗ q ∗ > q q_*>q q ∗ > q
Let ( Y , σ ) (Y,\sigma) ( Y , σ ) Proposition 3.3 , with
Fix q ∗ ∈ ( q , ∞ ) q_*\in(q,\infty) q ∗ ∈ ( q , ∞ )
X 0 ( q ∗ ) : = H − s , q ∗ ( T d ) × H 1 − s , q ∗ ( T d ; R d ) , X 1 ( q ∗ ) : = H 2 − s , q ∗ ( T d ) × H 3 − s , q ∗ ( T d ; R d ) , X_0^{(q_*)}:=H^{-s,q_*}(\mathbb{T}^d)\times H^{1-s,q_*}(\mathbb{T}^d;\mathbb{R}^d),\qquad X_1^{(q_*)}:=H^{2-s,q_*}(\mathbb{T}^d)\times H^{3-s,q_*}(\mathbb{T}^d;\mathbb{R}^d), X 0 ( q ∗ ) := H − s , q ∗ ( T d ) × H 1 − s , q ∗ ( T d ; R d ) , X 1 ( q ∗ ) := H 2 − s , q ∗ ( T d ) × H 3 − s , q ∗ ( T d ; R d ) , and, for θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
X θ ( q ∗ ) : = H − s + 2 θ , q ∗ ( T d ) × H 1 − s + 2 θ , q ∗ ( T d ; R d ) . X_\theta^{(q_*)}:=H^{-s+2\theta,q_*}(\mathbb{T}^d)\times H^{1-s+2\theta,q_*}(\mathbb{T}^d;\mathbb{R}^d). X θ ( q ∗ ) := H − s + 2 θ , q ∗ ( T d ) × H 1 − s + 2 θ , q ∗ ( T d ; R d ) . Recall that the trace space is given in (28) , i.e.
X 1 − 1 + κ p , p ( q ∗ ) = B q ∗ , p − s + 2 − 2 ( 1 + κ ) p ( T d ; R d ) × B q ∗ , p − s + 3 − 2 ( 1 + κ ) p ( T d ; R d ) . X_{1-\frac{1+\kappa}{p},p}^{(q_*)}=\mathbb{B}_{q_*,p}^{-s+2-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d;\mathbb{R}^d)\times\mathbb{B}_{q_*,p}^{-s+3-\frac{2(1+\kappa)}{p}}(\mathbb{T}^d;\mathbb{R}^d). X 1 − p 1 + κ , p ( q ∗ ) = B q ∗ , p − s + 2 − p 2 ( 1 + κ ) ( T d ; R d ) × B q ∗ , p − s + 3 − p 2 ( 1 + κ ) ( T d ; R d ) . Proposition 6.1 (Bootstrap in L q L^q L q s ∈ [ 1 , 2 ) s\in[1,2) s ∈ [ 1 , 2 ) p , q ∈ ( 2 , ∞ ) p,q\in(2,\infty) p , q ∈ ( 2 , ∞ ) κ ∈ [ 0 , p 2 − 1 ) \kappa\in[0,\frac{p}{2}-1) κ ∈ [ 0 , 2 p − 1 ) Proposition 3.3 . Let ( Y , σ ) (Y,\sigma) ( Y , σ ) q ∗ > q q_*>q q ∗ > q Proposition 3.3 remains valid with q q q q ∗ q_* q ∗
1 − 1 + κ p ≤ d − ( s − 1 ) q ∗ 4 q ∗ + s 2 ≤ 1 − 1 + κ 2 p , (53) 1-\frac{1+\kappa}{p}\leq\frac{d-(s-1)q_*}{4q_*}+\frac{s}{2}\leq1-\frac{1+\kappa}{2p}, \tag*{(53)} 1 − p 1 + κ ≤ 4 q ∗ d − ( s − 1 ) q ∗ + 2 s ≤ 1 − 2 p 1 + κ , (53) \frac{1}{\kappa}>s-3+\frac{d}{q_*}+2\left(\frac{1+\kappa}{p}\right),
{#eq:6.2}
q ∗ ≤ d s − 1 , if s > 1. (54) q_*\leq\frac{d}{s-1},\qquad\text{if }s>1. \tag*{(54)} q ∗ ≤ s − 1 d , if s > 1. (54) Assume moreover that there exists β ∈ ( 0 , min { 1 2 , 1 + κ p } ) \beta\in\left(0,\min\left\{\frac{1}{2},\frac{1+\kappa}{p}\right\}\right) β ∈ ( 0 , min { 2 1 , p 1 + κ } )
0 < d q − d q ∗ < 2 ( 1 + κ p − β ) . (55) 0<\frac{d}{q}-\frac{d}{q_*}<2\left(\frac{1+\kappa}{p}-\beta\right). \tag*{(55)} 0 < q d − q ∗ d < 2 ( p 1 + κ − β ) . (55) Then almost surely,
Y ∈ L l o c p ( 0 , σ ; X 1 ( q ∗ ) ) ∩ C ( ( 0 , σ ) ; X 1 − 1 + κ p , p ( q ∗ ) ) . Y\in L^p_{\mathrm{loc}}(0,\sigma;X_1^{(q_*)})\cap C\left((0,\sigma);X_{1-\frac{1+\kappa}{p},p}^{(q_*)}\right). Y ∈ L loc p ( 0 , σ ; X 1 ( q ∗ ) ) ∩ C ( ( 0 , σ ) ; X 1 − p 1 + κ , p ( q ∗ ) ) . In particular,
u ∈ L l o c p ( 0 , σ ; H 2 − s , q ∗ ( T d ) ) , d ∈ L l o c p ( 0 , σ ; H 3 − s , q ∗ ( T d ; R d ) ) . u\in L^p_{\mathrm{loc}}(0,\sigma;H^{2-s,q_*}(\mathbb{T}^d)),\qquad d\in L^p_{\mathrm{loc}}(0,\sigma;H^{3-s,q_*}(\mathbb{T}^d;\mathbb{R}^d)). u ∈ L loc p ( 0 , σ ; H 2 − s , q ∗ ( T d )) , d ∈ L loc p ( 0 , σ ; H 3 − s , q ∗ ( T d ; R d )) . Proof. Fix ε > 0 \varepsilon>0 ε > 0
V ε : = { σ > ε } . V_\varepsilon:=\{\sigma>\varepsilon\}. V ε := { σ > ε } . By Proposition 5.1 if κ > 0 \kappa>0 κ > 0 Proposition 5.2 if κ = 0 \kappa=0 κ = 0 η > 0 \eta>0 η > 0
Y ∈ C l o c β ( ( 0 , σ ) ; X 1 − β ) a.s. Y\in C^\beta_{\mathrm{loc}}((0,\sigma);X_{1-\beta})\quad\text{a.s.} Y ∈ C loc β (( 0 , σ ) ; X 1 − β ) a.s. Hence
1 V ε Y ( ε ) ∈ X 1 − β = H − s + 2 − 2 β , q ( T d ) × H 3 − s − 2 β , q ( T d ; R d ) a.s. \mathbf{1}_{V_\varepsilon}Y(\varepsilon)\in X_{1-\beta}=H^{-s+2-2\beta,q}(\mathbb{T}^d)\times H^{3-s-2\beta,q}(\mathbb{T}^d;\mathbb{R}^d)\quad\text{a.s.} 1 V ε Y ( ε ) ∈ X 1 − β = H − s + 2 − 2 β , q ( T d ) × H 3 − s − 2 β , q ( T d ; R d ) a.s. Next, (55) guarantees the embedding
X 1 − β ↪ X 1 − 1 + κ p , p ( q ∗ ) . (56) X_{1-\beta}\hookrightarrow X_{1-\frac{1+\kappa}{p},p}^{(q_*)}. \tag*{(56)} X 1 − β ↪ X 1 − p 1 + κ , p ( q ∗ ) . (56) Indeed, for the velocity component one has
− s + 2 − 2 β − d q > − s + 2 − 2 ( 1 + κ ) p − d q ∗ , -s+2-2\beta-\frac{d}{q}>-s+2-\frac{2(1+\kappa)}{p}-\frac{d}{q_*}, − s + 2 − 2 β − q d > − s + 2 − p 2 ( 1 + κ ) − q ∗ d , and the director component satisfies the same estimate, shifted by one derivative. Therefore,
1 V ε Y ( ε ) ∈ X 1 − 1 + κ p , p ( q ∗ ) \mathbf{1}_{V_\varepsilon}Y(\varepsilon)\in X^{(q_*)}_{1-\frac{1+\kappa}{p},p} 1 V ε Y ( ε ) ∈ X 1 − p 1 + κ , p ( q ∗ ) a.s.
We now restart (MLC) at time ε \varepsilon ε ( p , κ , q ∗ ) (p,\kappa,q_*) ( p , κ , q ∗ ) Proposition 3.3 applied with q ∗ q_* q ∗ q q q ( Z , τ ∗ ) (Z,\tau_*) ( Z , τ ∗ ) [ ε , ∞ ) [\varepsilon,\infty) [ ε , ∞ )
Z ( ε ) = 1 V ε Y ( ε ) , Z(\varepsilon)=\mathbf{1}_{V_\varepsilon}Y(\varepsilon), Z ( ε ) = 1 V ε Y ( ε ) , such that almost surely
Z ∈ L l o c p ( ε , τ ∗ ; X 1 ( q ∗ ) ) ∩ C ( [ ε , τ ∗ ) ; X 1 − 1 + κ p , p ( q ∗ ) ) . Z\in L^p_{\mathrm{loc}}(\varepsilon,\tau_*;X^{(q_*)}_1)\cap C([\varepsilon,\tau_*);X^{(q_*)}_{1-\frac{1+\kappa}{p},p}). Z ∈ L loc p ( ε , τ ∗ ; X 1 ( q ∗ ) ) ∩ C ([ ε , τ ∗ ) ; X 1 − p 1 + κ , p ( q ∗ ) ) . Since q ∗ > q q_*>q q ∗ > q
X 1 ( q ∗ ) ↪ X 1 , X 1 − 1 + κ p , p ( q ∗ ) ↪ X 1 − 1 + κ p , p . X^{(q_*)}_1\hookrightarrow X_1,\qquad X^{(q_*)}_{1-\frac{1+\kappa}{p},p}\hookrightarrow X_{1-\frac{1+\kappa}{p},p}. X 1 ( q ∗ ) ↪ X 1 , X 1 − p 1 + κ , p ( q ∗ ) ↪ X 1 − p 1 + κ , p . Hence Z Z Z ( p , κ , q ) (p,\kappa,q) ( p , κ , q ) ε \varepsilon ε
Y = Z a.s. on V ε × [ ε , σ ∧ τ ∗ ) . Y=Z\qquad\text{a.s. on }V_\varepsilon\times[\varepsilon,\sigma\wedge\tau_*). Y = Z a.s. on V ε × [ ε , σ ∧ τ ∗ ) . It remains to prove that τ ∗ = σ \tau_*=\sigma τ ∗ = σ V ε V_\varepsilon V ε
First, assume that P ( V ε ∩ { σ < τ ∗ } ) > 0 \mathbb{P}(V_\varepsilon\cap\{\sigma<\tau_*\})>0 P ( V ε ∩ { σ < τ ∗ }) > 0 Z Z Z X 1 − 1 + κ p , p ( q ∗ ) X^{(q_*)}_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p ( q ∗ ) Y Y Y σ \sigma σ X 1 − 1 + κ p , p X_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p Theorem 4.1 (i). Hence σ < τ ∗ \sigma<\tau_* σ < τ ∗ P ( V ε ∩ { τ ∗ < σ } ) > 0 \mathbb{P}(V_\varepsilon\cap\{\tau_*<\sigma\})>0 P ( V ε ∩ { τ ∗ < σ }) > 0 τ ∗ < σ \tau_*<\sigma τ ∗ < σ Proposition 5.1 if κ > 0 \kappa>0 κ > 0 Proposition 5.2 if κ = 0 \kappa=0 κ = 0
Y ( τ ∗ ) ∈ X 1 − β a.s. on V ε ∩ { τ ∗ < σ } . (57) Y(\tau_*)\in X_{1-\beta}\qquad\text{a.s. on }V_\varepsilon\cap\{\tau_*<\sigma\}. \tag*{(57)} Y ( τ ∗ ) ∈ X 1 − β a.s. on V ε ∩ { τ ∗ < σ } . (57) By the embedding in (56) , we have
Y ( τ ∗ ) ∈ X 1 − 1 + κ p , p ( q ∗ ) a.s. on V ε ∩ { τ ∗ < σ } . Y(\tau_*)\in X^{(q_*)}_{1-\frac{1+\kappa}{p},p}\qquad\text{a.s. on }V_\varepsilon\cap\{\tau_*<\sigma\}. Y ( τ ∗ ) ∈ X 1 − p 1 + κ , p ( q ∗ ) a.s. on V ε ∩ { τ ∗ < σ } . Using the continuity of Y Y Y X 1 − β X_{1-\beta} X 1 − β τ ∗ < σ \tau_*<\sigma τ ∗ < σ (56) , and the equality Y = Z Y=Z Y = Z τ ∗ \tau_* τ ∗
lim t ↑ τ ∗ Z ( t ) = Y ( τ ∗ ) in X 1 − 1 + κ p , p ( q ∗ ) \lim_{t\uparrow\tau_*}Z(t)=Y(\tau_*)\quad\text{in }X^{(q_*)}_{1-\frac{1+\kappa}{p},p} t ↑ τ ∗ lim Z ( t ) = Y ( τ ∗ ) in X 1 − p 1 + κ , p ( q ∗ ) a.s. on V ε ∩ { τ ∗ < σ } V_\varepsilon\cap\{\tau_*<\sigma\} V ε ∩ { τ ∗ < σ } Theorem 4.1 (i), applied with q q q q ∗ q_* q ∗ τ ∗ < σ \tau_*<\sigma τ ∗ < σ
Therefore τ ∗ = σ \tau_*=\sigma τ ∗ = σ V ε V_\varepsilon V ε ε > 0 \varepsilon>0 ε > 0
Remark 6.2 (How to choose the exponent q ∗ q_* q ∗ β ∈ ( 0 , min { 1 2 , 1 + κ p } ) \beta\in\left(0,\min\left\{\frac{1}{2},\frac{1+\kappa}{p}\right\}\right) β ∈ ( 0 , min { 2 1 , p 1 + κ } )
1 − 1 + κ p < d − ( s − 1 ) q 4 q + s 2 and q < d s − 1 if s > 1 , 1-\frac{1+\kappa}{p}<\frac{d-(s-1)q}{4q}+\frac{s}{2}
\qquad\text{and}\qquad
q<\frac{d}{s-1}\quad\text{if }s>1, 1 − p 1 + κ < 4 q d − ( s − 1 ) q + 2 s and q < s − 1 d if s > 1 , with no additional upper restriction when s = 1 s=1 s = 1 q ∗ > q q_*>q q ∗ > q q q q
0 < d q − d q ∗ < 2 ( 1 + κ p − β ) . 0<\frac{d}{q}-\frac{d}{q_*}<2\left(\frac{1+\kappa}{p}-\beta\right). 0 < q d − q ∗ d < 2 ( p 1 + κ − β ) . Hence, if the lower inequality in (30) is strict and q < d s − 1 q<\frac{d}{s-1} q < s − 1 d s > 1 s>1 s > 1 q = d s − 1 q=\frac{d}{s-1} q = s − 1 d s > 1 s>1 s > 1 (30) is an equality, the present argument does not give a larger exponent q ∗ q_* q ∗
Remark 6.3. In the case κ = 0 \kappa=0 κ = 0 q ∗ > q q_*>q q ∗ > q q q q
0 < d q − d q ∗ < 2 p . 0<\frac{d}{q}-\frac{d}{q_*}<\frac{2}{p}. 0 < q d − q ∗ d < p 2 . Since β \beta β Proposition 5.2 , this still yields a nontrivial bootstrap step in spatial integrability whenever a choice of q ∗ > q q_* > q q ∗ > q
We now discuss the Sobolev parameter. A genuine gain in Sobolev regularity corresponds to decreasing the parameter s s s s ∗ < s s_* < s s ∗ < s H 2 − s ∗ , q H^{2-s_*,q} H 2 − s ∗ , q H 3 − s ∗ , q H^{3-s_*,q} H 3 − s ∗ , q H 2 − s , q H^{2-s,q} H 2 − s , q H 3 − s , q H^{3-s,q} H 3 − s , q
Fix s ∗ ∈ [ 1 , s ) s_* \in[1,s) s ∗ ∈ [ 1 , s )
X 0 ( s ∗ ) : = H − s ∗ , q ( T d ) × H 1 − s ∗ , q ( T d ; R d ) , X 1 ( s ∗ ) : = H 2 − s ∗ , q ( T d ) × H 3 − s ∗ , q ( T d ; R d ) . X_0^{(s_*)} := H^{-s_*,q}(\mathbb{T}^d) \times H^{1-s_*,q}(\mathbb{T}^d;\mathbb{R}^d), \qquad X_1^{(s_*)} := H^{2-s_*,q}(\mathbb{T}^d) \times H^{3-s_*,q}(\mathbb{T}^d;\mathbb{R}^d). X 0 ( s ∗ ) := H − s ∗ , q ( T d ) × H 1 − s ∗ , q ( T d ; R d ) , X 1 ( s ∗ ) := H 2 − s ∗ , q ( T d ) × H 3 − s ∗ , q ( T d ; R d ) . and, for θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
X θ ( s ∗ ) : = H − s ∗ + 2 θ , q ( T d ) × H 1 − s ∗ + 2 θ , q ( T d ; R d ) . X_\theta^{(s_*)} := H^{-s_*+2\theta,q}(\mathbb{T}^d) \times H^{1-s_*+2\theta,q}(\mathbb{T}^d;\mathbb{R}^d). X θ ( s ∗ ) := H − s ∗ + 2 θ , q ( T d ) × H 1 − s ∗ + 2 θ , q ( T d ; R d ) . From (28) , the corresponding trace space is
X 1 − 1 + κ p ( s ∗ ) = B q , p − s ∗ + 2 − 2 ( 1 + κ ) p ( T d ) × B q , p − s ∗ + 3 − 2 ( 1 + κ ) p ( T d ; R d ) . X_{1-\frac{1+\kappa}{p}}^{(s_*)}
= B^{-s_*+2-\frac{2(1+\kappa)}{p}}_{q,p}(\mathbb{T}^d) \times B^{-s_*+3-\frac{2(1+\kappa)}{p}}_{q,p}(\mathbb{T}^d;\mathbb{R}^d). X 1 − p 1 + κ ( s ∗ ) = B q , p − s ∗ + 2 − p 2 ( 1 + κ ) ( T d ) × B q , p − s ∗ + 3 − p 2 ( 1 + κ ) ( T d ; R d ) . Proposition 6.4 (Bootstrap in Sobolev regularity). Assume the hypotheses of Proposition 3.3 , and assume in addition that κ ∈ [ 0 , p 2 − 1 ) \kappa\in[0,\frac{p}{2}-1) κ ∈ [ 0 , 2 p − 1 ) ( Y , σ ) (Y,\sigma) ( Y , σ ) Proposition 3.3 . Let s ∗ ∈ [ 1 , s ) s_* \in[1,s) s ∗ ∈ [ 1 , s ) Proposition 3.3 remain valid with s s s s ∗ s_* s ∗
1 − 1 + κ p ≤ d − ( s ∗ − 1 ) q 4 q + s ∗ 2 ≤ 1 − 1 + κ 2 p , (58) 1-\frac{1+\kappa}{p} \le\frac{d-(s_*-1)q}{4q}+\frac{s_*}{2} \le1-\frac{1+\kappa}{2p}, \tag*{(58)} 1 − p 1 + κ ≤ 4 q d − ( s ∗ − 1 ) q + 2 s ∗ ≤ 1 − 2 p 1 + κ , (58) 1 κ > s ∗ − 3 + d q + 2 ( 1 + κ p ) , (59) \frac{1}{\kappa} > s_*-3+\frac{d}{q}+2\left(\frac{1+\kappa}{p}\right), \tag*{(59)} κ 1 > s ∗ − 3 + q d + 2 ( p 1 + κ ) , (59) q ≤ d s ∗ − 1 , if s ∗ > 1. (60) q \le\frac{d}{s_*-1}, \qquad\text{if }s_*>1. \tag*{(60)} q ≤ s ∗ − 1 d , if s ∗ > 1. (60) Assume moreover that there exists β ∈ ( 0 , min { 1 2 , 1 + κ p } ) \beta\in(0,\min\{\frac{1}{2},\frac{1+\kappa}{p}\}) β ∈ ( 0 , min { 2 1 , p 1 + κ })
0 < s − s ∗ < 2 ( 1 + κ p − β ) . (61) 0 < s-s_* < 2\left(\frac{1+\kappa}{p}-\beta\right). \tag*{(61)} 0 < s − s ∗ < 2 ( p 1 + κ − β ) . (61) Then almost surely,
Y ∈ L l o c p ( 0 , σ ; X 1 ( s ∗ ) ) ∩ C ( ( 0 , σ ) ; X 1 − 1 + κ p , p ( s ∗ ) ) . Y \in L^p_{\mathrm{loc}}(0,\sigma;X_1^{(s_*)}) \cap C\left((0,\sigma);X_{1-\frac{1+\kappa}{p},p}^{(s_*)}\right). Y ∈ L loc p ( 0 , σ ; X 1 ( s ∗ ) ) ∩ C ( ( 0 , σ ) ; X 1 − p 1 + κ , p ( s ∗ ) ) . Equivalently,
u ∈ L l o c p ( 0 , σ ; H 2 − s ∗ , q ( T d ) ) , d ∈ L l o c p ( 0 , σ ; H 3 − s ∗ , q ( T d ; R d ) ) . u \in L^p_{\mathrm{loc}}(0,\sigma;H^{2-s_*,q}(\mathbb{T}^d)), \qquad d \in L^p_{\mathrm{loc}}(0,\sigma;H^{3-s_*,q}(\mathbb{T}^d;\mathbb{R}^d)). u ∈ L loc p ( 0 , σ ; H 2 − s ∗ , q ( T d )) , d ∈ L loc p ( 0 , σ ; H 3 − s ∗ , q ( T d ; R d )) . Proof. Fix ε > 0 \varepsilon>0 ε > 0 V ε : = { σ > ε } V_\varepsilon:=\{\sigma>\varepsilon\} V ε := { σ > ε } Proposition 5.1 (if κ > 0 \kappa>0 κ > 0 Proposition 5.2 (if κ = 0 \kappa=0 κ = 0 η > 0 \eta>0 η > 0
Y ∈ C l o c β ( ( 0 , σ ) ; X 1 − β ) a.s. Y \in C^\beta_{\mathrm{loc}}((0,\sigma);X_{1-\beta}) \quad\text{a.s.} Y ∈ C loc β (( 0 , σ ) ; X 1 − β ) a.s. Hence
1 V ε Y ( ε ) ∈ X 1 − β = H − s + 2 − 2 β , q ( T d ) × H 3 − s − 2 β , q ( T d ; R d ) a.s. \mathbf{1}_{V_\varepsilon}Y(\varepsilon)\in X_{1-\beta}=H^{-s+2-2\beta,q}(\mathbb{T}^d)\times H^{3-s-2\beta,q}(\mathbb{T}^d;\mathbb{R}^d) \quad\text{a.s.} 1 V ε Y ( ε ) ∈ X 1 − β = H − s + 2 − 2 β , q ( T d ) × H 3 − s − 2 β , q ( T d ; R d ) a.s. Next, (61) ensures the embedding
X 1 − β ↪ X 1 − 1 + κ p ( s ∗ ) . X_{1-\beta}\hookrightarrow X_{1-\frac{1+\kappa}{p}}^{(s_*)}. X 1 − β ↪ X 1 − p 1 + κ ( s ∗ ) . Indeed, for the velocity component one has
− s + 2 − 2 β > − s ∗ + 2 − 2 ( 1 + κ ) p , -s+2-2\beta>-s_*+2-\frac{2(1+\kappa)}{p}, − s + 2 − 2 β > − s ∗ + 2 − p 2 ( 1 + κ ) , and the director component satisfies the same estimate, shifted by one derivative. The embedding H ρ 1 , q ( T d ) ↪ B q , p ρ 2 ( T d ) H^{\rho_1,q}(\mathbb{T}^d)\hookrightarrow B^{\rho_2}_{q,p}(\mathbb{T}^d) H ρ 1 , q ( T d ) ↪ B q , p ρ 2 ( T d ) ρ 1 > ρ 2 \rho_1>\rho_2 ρ 1 > ρ 2 1 < p , q < ∞ 1<p,q<\infty 1 < p , q < ∞
1 V ε Y ( ε ) ∈ X 1 − 1 + κ p ( s ∗ ) a.s. \mathbf{1}_{V_\varepsilon}Y(\varepsilon)\in X_{1-\frac{1+\kappa}{p}}^{(s_*)} \quad\text{a.s.} 1 V ε Y ( ε ) ∈ X 1 − p 1 + κ ( s ∗ ) a.s. We now restart (MLC) at time ε \varepsilon ε ( p , κ , s ∗ ) (p,\kappa,s_*) ( p , κ , s ∗ ) Proposition 3.3 applied with s ∗ s_* s ∗ s s s ( Z , τ ∗ ) (Z,\tau_*) ( Z , τ ∗ ) [ ε , ∞ ) [\varepsilon,\infty) [ ε , ∞ )
Z ( ε ) = 1 V ε Y ( ε ) , Z(\varepsilon)=1_{V_\varepsilon}Y(\varepsilon), Z ( ε ) = 1 V ε Y ( ε ) , such that almost surely
Z ∈ L l o c p ( ε , τ ∗ ; X 1 ( s ∗ ) ) ∩ C ( [ ε , τ ∗ ) ; X 1 − 1 + κ p ( s ∗ ) ) . Z\in L_{\mathrm{loc}}^p(\varepsilon,\tau_*;X_1^{(s_*)})\cap C([\varepsilon,\tau_*);X_{1-\frac{1+\kappa}{p}}^{(s_*)}). Z ∈ L loc p ( ε , τ ∗ ; X 1 ( s ∗ ) ) ∩ C ([ ε , τ ∗ ) ; X 1 − p 1 + κ ( s ∗ ) ) . Since s ∗ < s s_*<s s ∗ < s
X 1 ( s ∗ ) ↪ X 1 , X 1 − 1 + κ p ( s ∗ ) ↪ X 1 − 1 + κ p , p . X_1^{(s_*)}\hookrightarrow X_1,\qquad X_{1-\frac{1+\kappa}{p}}^{(s_*)}\hookrightarrow X_{1-\frac{1+\kappa}{p},p}. X 1 ( s ∗ ) ↪ X 1 , X 1 − p 1 + κ ( s ∗ ) ↪ X 1 − p 1 + κ , p . Hence Z Z Z ( p , κ , s ) (p,\kappa,s) ( p , κ , s ) ε \varepsilon ε Proposition 3.3 ,
Y = Z a.s. on V ε × [ ε , σ ∧ τ ∗ ) . Y=Z\qquad\text{a.s. on }V_\varepsilon\times[\varepsilon,\sigma\wedge\tau_*). Y = Z a.s. on V ε × [ ε , σ ∧ τ ∗ ) . It remains to prove that τ ∗ = σ \tau_*=\sigma τ ∗ = σ V ε V_\varepsilon V ε P ( V ε ∩ { σ < τ ∗ } ) > 0 \mathbb{P}(V_\varepsilon\cap\{\sigma<\tau_*\})>0 P ( V ε ∩ { σ < τ ∗ }) > 0 Z Z Z X 1 − 1 + κ p ( s ∗ ) X_{1-\frac{1+\kappa}{p}}^{(s_*)} X 1 − p 1 + κ ( s ∗ ) Y Y Y σ \sigma σ X 1 − 1 + κ p , p X_{1-\frac{1+\kappa}{p},p} X 1 − p 1 + κ , p Theorem 4.1 (i). Hence σ < τ ∗ \sigma<\tau_* σ < τ ∗
Conversely, assume that P ( V ε ∩ { τ ∗ < σ } ) > 0 \mathbb{P}(V_\varepsilon\cap\{\tau_*<\sigma\})>0 P ( V ε ∩ { τ ∗ < σ }) > 0 τ ∗ < σ \tau_*<\sigma τ ∗ < σ Proposition 5.1 (if κ > 0 \kappa>0 κ > 0 Proposition 5.2 (if κ = 0 \kappa=0 κ = 0
Y ( τ ∗ ) ∈ X 1 − β a.s. on V ε ∩ { τ ∗ < σ } . Y(\tau_*)\in X_{1-\beta}\qquad\text{a.s. on }V_\varepsilon\cap\{\tau_*<\sigma\}. Y ( τ ∗ ) ∈ X 1 − β a.s. on V ε ∩ { τ ∗ < σ } . By the embedding above,
X 1 − β ↪ X 1 − 1 + κ p ( s ∗ ) . (62) X_{1-\beta}\hookrightarrow X_{1-\frac{1+\kappa}{p}}^{(s_*)}. \tag*{(62)} X 1 − β ↪ X 1 − p 1 + κ ( s ∗ ) . (62) and therefore
Y ( τ ∗ ) ∈ X 1 − 1 + κ p ( s ∗ ) a.s. on V ε ∩ { τ ∗ < σ } . Y(\tau_*)\in X_{1-\frac{1+\kappa}{p}}^{(s_*)}\qquad\text{a.s. on }V_\varepsilon\cap\{\tau_*<\sigma\}. Y ( τ ∗ ) ∈ X 1 − p 1 + κ ( s ∗ ) a.s. on V ε ∩ { τ ∗ < σ } . Using
Y = Z a.s. on V ε × [ ε , τ ∗ ) , Y=Z\qquad\text{a.s. on }V_\varepsilon\times[\varepsilon,\tau_*), Y = Z a.s. on V ε × [ ε , τ ∗ ) , together with the continuity of Y Y Y X 1 − β X_{1-\beta} X 1 − β τ ∗ < σ \tau_*<\sigma τ ∗ < σ (62) , we obtain
lim t ↑ τ ∗ Z ( t ) = Y ( τ ∗ ) in X 1 − 1 + κ p ( s ∗ ) \lim_{t\uparrow\tau_*}Z(t)=Y(\tau_*)\quad\text{in }X_{1-\frac{1+\kappa}{p}}^{(s_*)} t ↑ τ ∗ lim Z ( t ) = Y ( τ ∗ ) in X 1 − p 1 + κ ( s ∗ ) a.s. on V ε ∩ { τ ∗ < σ } V_\varepsilon\cap\{\tau_*<\sigma\} V ε ∩ { τ ∗ < σ } Theorem 4.1 (i), applied with s s s s ∗ s_* s ∗ τ ∗ < σ \tau_*<\sigma τ ∗ < σ τ ∗ = σ \tau_*=\sigma τ ∗ = σ V ε V_\varepsilon V ε ε > 0 \varepsilon>0 ε > 0
Remark 6.5 (How to choose the exponent s ∗ s_* s ∗ β ∈ ( 0 , min { 1 2 , 1 + κ p } ) \beta\in(0,\min\{\frac{1}{2},\frac{1+\kappa}{p}\}) β ∈ ( 0 , min { 2 1 , p 1 + κ }) s > 1 s>1 s > 1 (30) is strict, i.e.
1 − 1 + κ p < d − ( s − 1 ) q 4 q + s 2 . 1-\frac{1+\kappa}{p}<\frac{d-(s-1)q}{4q}+\frac{s}{2}. 1 − p 1 + κ < 4 q d − ( s − 1 ) q + 2 s . Then one may choose s ∗ ∈ [ 1 , s ) s_*\in[1,s) s ∗ ∈ [ 1 , s ) s s s (58) –(60) remain valid and
0 < s − s ∗ < 2 ( 1 + κ p − β ) . 0<s-s_*<2\left(\frac{1+\kappa}{p}-\beta\right). 0 < s − s ∗ < 2 ( p 1 + κ − β ) . Thus the argument yields a nontrivial gain in Sobolev regularity whenever there is room to move s s s s = 1 s=1 s = 1 (30) is saturated, this particular argument does not produce such a gain within the regime of Proposition 3.3 .
Kadlec’s formula
We state and prove a Kadlec’s formula for Neumann boundary conditions below. In particular, we use a result from [Chap 5, Taylor Vol. 1], of which Theorem 1.3 announces a similar result with Dirichlet boundary conditions.
Lemma 1 (Kadlec’s formula with Neumann boundary conditions). Let O ⊂ R d \mathcal{O} \subset\mathbb{R}^{d} O ⊂ R d f ∈ { g ∈ H 2 ( O , R ) ∣ ∂ ν g = 0 on ∂ O } f \in\{g \in H^{2}(\mathcal{O},\mathbb{R}) \mid\partial_{\nu}g = 0 \text{ on } \partial\mathcal{O}\} f ∈ { g ∈ H 2 ( O , R ) ∣ ∂ ν g = 0 on ∂ O }
∥ ( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) ≤ ∥ Δ f ∥ L 2 ( O , R ) . \lVert(\nabla\otimes\nabla)f\rVert_{L^{2}(\mathcal{O},\mathbb{R}^{d \times d})} \leq\lVert\Delta f\rVert_{L^{2}(\mathcal{O},\mathbb{R})}. ∥( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) ≤ ∥ Δ f ∥ L 2 ( O , R ) . In particular, this shows that the inner product ( ⋅ , ⋅ ) V (\cdot,\cdot)_{V} ( ⋅ , ⋅ ) V (10) is equivalent to the standard inner product on H 1 ( O ; R d ) × H 2 ( O ; R d ) H^{1}(\mathcal{O};\mathbb{R}^{d}) \times H^{2}(\mathcal{O};\mathbb{R}^{d}) H 1 ( O ; R d ) × H 2 ( O ; R d )
Proof. First we prove that C N ∞ : = { f ∈ C ∞ ( O ‾ , R ) ∣ ∂ ν f = 0 on ∂ O } C^{\infty}_{N} := \{f \in C^{\infty}(\overline{\mathcal{O}},\mathbb{R}) \mid\partial_{\nu}f = 0 \text{ on } \partial\mathcal{O}\} C N ∞ := { f ∈ C ∞ ( O , R ) ∣ ∂ ν f = 0 on ∂ O } H N 2 : = { f ∈ H 2 ( O , R ) ∣ ∂ ν f = 0 on ∂ O } H^{2}_{N} := \{f \in H^{2}(\mathcal{O},\mathbb{R}) \mid\partial_{\nu}f = 0 \text{ on } \partial\mathcal{O}\} H N 2 := { f ∈ H 2 ( O , R ) ∣ ∂ ν f = 0 on ∂ O } ∥ ⋅ ∥ H 2 ( O , R ) \lVert\cdot\rVert_{H^{2}(\mathcal{O},\mathbb{R})} ∥ ⋅ ∥ H 2 ( O , R ) f ∈ H N 2 f \in H^{2}_{N} f ∈ H N 2 C ∞ ( O ‾ , R ) C^{\infty}(\overline{\mathcal{O}},\mathbb{R}) C ∞ ( O , R ) φ n ∈ C ∞ ( O ‾ , R ) \varphi_{n} \in C^{\infty}(\overline{\mathcal{O}},\mathbb{R}) φ n ∈ C ∞ ( O , R ) φ n → f \varphi_{n} \to f φ n → f H 2 ( O , R ) H^{2}(\mathcal{O},\mathbb{R}) H 2 ( O , R ) ∇ φ n → ∇ f \nabla\varphi_{n} \to\nabla f ∇ φ n → ∇ f H 1 ( O , R d ) H^{1}(\mathcal{O},\mathbb{R}^{d}) H 1 ( O , R d ) ∂ ν φ n → ∂ ν f = 0 \partial_{\nu}\varphi_{n} \to\partial_{\nu}f = 0 ∂ ν φ n → ∂ ν f = 0 H 1 / 2 ( ∂ O , R ) H^{1/2}(\partial\mathcal{O},\mathbb{R}) H 1/2 ( ∂ O , R ) φ ~ n \widetilde{\varphi}_{n} φ n
Δ φ ~ n = φ ~ n , ∂ ν φ ~ n = ∂ ν φ n on ∂ O . \Delta\widetilde{\varphi}_{n} = \widetilde{\varphi}_{n}, \qquad\partial_{\nu}\widetilde{\varphi}_{n} = \partial_{\nu}\varphi_{n} \text{ on } \partial\mathcal{O}. Δ φ n = φ n , ∂ ν φ n = ∂ ν φ n on ∂ O . By standard arguments, we infer that ∥ φ ~ n ∥ H 1 ( O , R ) + ∥ Δ φ ~ n ∥ L 2 ( O , R ) → 0 \lVert\widetilde{\varphi}_{n}\rVert_{H^{1}(\mathcal{O},\mathbb{R})} + \lVert\Delta\widetilde{\varphi}_{n}\rVert_{L^{2}(\mathcal{O},\mathbb{R})} \to0 ∥ φ n ∥ H 1 ( O , R ) + ∥ Δ φ n ∥ L 2 ( O , R ) → 0 [11 Proposition 7.5] , we have φ ~ n → 0 \widetilde{\varphi}_{n} \to0 φ n → 0 H 2 ( O , R ) H^{2}(\mathcal{O},\mathbb{R}) H 2 ( O , R ) φ ~ n ∈ C ∞ ( O ‾ , R ) \widetilde{\varphi}_{n} \in C^{\infty}(\overline{\mathcal{O}},\mathbb{R}) φ n ∈ C ∞ ( O , R ) φ ^ n : = φ n − φ ~ n ∈ C N ∞ \widehat{\varphi}_{n} := \varphi_{n} - \widetilde{\varphi}_{n} \in C^{\infty}_{N} φ n := φ n − φ n ∈ C N ∞ φ ^ n → f \widehat{\varphi}_{n} \to f φ n → f H N 2 H^{2}_{N} H N 2
Now, let f ∈ C N ∞ f \in C^{\infty}_{N} f ∈ C N ∞
∥ ( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) 2 = ∫ O ( Δ f ) 2 d O + ∑ i , j = 1 d ∫ ∂ O ν i ∂ j f ∂ i j f d ∂ O , \lVert(\nabla\otimes\nabla)f\rVert_{L^{2}(\mathcal{O},\mathbb{R}^{d \times d})}^{2}
= \int_{\mathcal{O}}(\Delta f)^{2}\,\mathrm{d}\mathcal{O}
+ \sum_{i,j=1}^{d}\int_{\partial\mathcal{O}}\nu_{i}\partial_{j}f\,\partial_{ij}f\,\mathrm{d}\partial\mathcal{O}, ∥( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) 2 = ∫ O ( Δ f ) 2 d O + i , j = 1 ∑ d ∫ ∂ O ν i ∂ j f ∂ ij f d ∂ O , since ∑ i = 1 d ν i ∂ i f = ∂ ν f = 0 \sum_{i=1}^{d}\nu_{i}\partial_{i}f = \partial_{\nu}f = 0 ∑ i = 1 d ν i ∂ i f = ∂ ν f = 0 ∂ O \partial\mathcal{O} ∂ O
∑ i , j = 1 d ∫ ∂ O ν i ∂ j f ∂ i j f d ∂ O = ∑ i , j = 1 d ∫ ∂ O ∂ j f ∂ j ( ν i ∂ i f ) − ( ∂ j f ) ( ∂ i f ) ( ∂ j ν i ) d ∂ O = ∫ ∂ O ∇ f ⋅ ∇ ( ∂ ν f ) − ( ∇ f ) ⋅ J ν ⋅ ( ∇ f ) d ∂ O , \begin{aligned}
\sum_{i,j=1}^{d}\int_{\partial\mathcal{O}}\nu_{i}\partial_{j}f\,\partial_{ij}f\,\mathrm{d}\partial\mathcal{O}
= \sum_{i,j=1}^{d}\int_{\partial\mathcal{O}}\partial_{j}f\,\partial_{j}(\nu_{i}\partial_{i}f)-(\partial_{j}f)(\partial_{i}f)(\partial_{j}\nu_{i})\,\mathrm{d}\partial\mathcal{O} \\
&= \int_{\partial\mathcal{O}}\nabla f\cdot\nabla(\partial_{\nu}f)-(\nabla f)\cdot J_{\nu}\cdot(\nabla f)\,\mathrm{d}\partial\mathcal{O},
\end{aligned} i , j = 1 ∑ d ∫ ∂ O ν i ∂ j f ∂ ij f d ∂ O = i , j = 1 ∑ d ∫ ∂ O ∂ j f ∂ j ( ν i ∂ i f ) − ( ∂ j f ) ( ∂ i f ) ( ∂ j ν i ) d ∂ O = ∫ ∂ O ∇ f ⋅ ∇ ( ∂ ν f ) − ( ∇ f ) ⋅ J ν ⋅ ( ∇ f ) d ∂ O , Here, since the outward unit normal ν \nu ν ∂ O \partial\mathcal{O} ∂ O ν = ∇ ρ \nu= \nabla\rho ν = ∇ ρ ρ \rho ρ ∂ O \partial\mathcal{O} ∂ O O \mathcal{O} O J ν = D ν = D 2 ρ J_{\nu} = D\nu= D^{2}\rho J ν = D ν = D 2 ρ ∂ τ f : = ∇ f − ( ∂ ν f ) ν \partial_{\tau}f := \nabla f - (\partial_{\nu}f)\nu ∂ τ f := ∇ f − ( ∂ ν f ) ν
∫ ∂ O ∇ f ⋅ ∇ ( ∂ ν f ) d ∂ O = ∫ ∂ O ( ∂ ν f ) ∂ ν ( ∂ ν f ) + ∂ τ f ⋅ ∂ τ ( ∂ ν f ) d ∂ O = 0 , \int_{\partial\mathcal{O}}\nabla f\cdot\nabla(\partial_{\nu}f)\,\mathrm{d}\partial\mathcal{O}
= \int_{\partial\mathcal{O}}(\partial_{\nu}f)\partial_{\nu}(\partial_{\nu}f)+\partial_{\tau}f\cdot\partial_{\tau}(\partial_{\nu}f)\,\mathrm{d}\partial\mathcal{O}=0, ∫ ∂ O ∇ f ⋅ ∇ ( ∂ ν f ) d ∂ O = ∫ ∂ O ( ∂ ν f ) ∂ ν ( ∂ ν f ) + ∂ τ f ⋅ ∂ τ ( ∂ ν f ) d ∂ O = 0 , since ∂ ν f = ∂ τ ( ∂ ν f ) = 0 \partial_{\nu}f = \partial_{\tau}(\partial_{\nu}f) = 0 ∂ ν f = ∂ τ ( ∂ ν f ) = 0 ∂ O \partial\mathcal{O} ∂ O
∥ ( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) 2 = ∥ Δ f ∥ L 2 ( O , R ) 2 − ∫ ∂ O ( ∇ f ) ⋅ J ν ⋅ ( ∇ f ) d ∂ O ≤ ∥ Δ f ∥ L 2 ( O , R ) 2 , \lVert(\nabla\otimes\nabla)f\rVert_{L^{2}(\mathcal{O},\mathbb{R}^{d \times d})}^{2}
= \lVert\Delta f\rVert_{L^{2}(\mathcal{O},\mathbb{R})}^{2}
- \int_{\partial\mathcal{O}}(\nabla f)\cdot J_{\nu}\cdot(\nabla f)\,\mathrm{d}\partial\mathcal{O}
\leq\lVert\Delta f\rVert_{L^{2}(\mathcal{O},\mathbb{R})}^{2}, ∥( ∇ ⊗ ∇ ) f ∥ L 2 ( O , R d × d ) 2 = ∥ Δ f ∥ L 2 ( O , R ) 2 − ∫ ∂ O ( ∇ f ) ⋅ J ν ⋅ ( ∇ f ) d ∂ O ≤ ∥ Δ f ∥ L 2 ( O , R ) 2 , because J ν J_{\nu} J ν O \mathcal{O} O f ∈ C N ∞ f \in C^{\infty}_{N} f ∈ C N ∞ H N 2 H^{2}_{N} H N 2 □ \square □
Acknowledgements
Acknowledgements. Gianmarco Del Sarto and Tarek Zöchling acknowledge the support from the DFG project FOR 5528. Esmée Theewis is supported by the VICI subsidy VI.C.212.027 of the Netherlands Organisation for Scientific Research (NWO). Antoine Moneyron acknowledges the support of the ERC project 856408-STUOD.
References
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email address: delsarto@mathematik.tu-darmstadt.de
Univ Rennes, INRIA, IRMAR UMR 6625, Campus de Beaulieu, 263 avenue du Général Leclerc, 35042 Rennes Cedex, France
Email address: antoine.moneyron@inria.fr
Delft University of Technology, Delft Institute of Applied Mathematics, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email address: e.s.theewis@tudelft.nl
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email address: zoechling@mathematik.tu-darmstadt.de
