Introduction
The inhomogeneous incompressible Navier–Stokes equations describe a viscous fluid whose density is transported by the flow. For densities that are bounded and bounded away from zero the Cauchy problem is well understood. Weak solutions with finite energy exist [29 , 39 , 33 ] , strong solutions are unique and exist globally under smallness or two-dimensionality assumptions [31 , 20 , 13 , 1 , 14 , 38 , 32 , 15 ] , and Hao, Shao, Wei and Zhang [25 ] recently proved uniqueness of Lions’ weak solutions in the two-dimensional whole space with L 2 L^2 L 2 H ˙ 1 / 2 ( R 3 ) \dot{H}^{1/2}(\mathbb{R}^3) H ˙ 1/2 ( R 3 )
The question addressed here is how much of this uniqueness survives when the density is only integrable to a power close to one and the equations are imposed in the sense of distributions. We show that in this regime uniqueness fails completely. For every 0 < s < 3 / 2 0 < s < 3/2 0 < s < 3/2 C t L p C_tL^p C t L p p > 1 p > 1 p > 1 C t H s C_tH^s C t H s
The two regimes leave a gap. Uniqueness is known for bounded densities in the energy class, and nonuniqueness is shown here for densities in L p L^p L p p p p L q L^q L q q q q [7 , 10 , 11 ] , and within the Leray class it is known in the presence of a forcing term [2 ] ; here the mechanism is the density rather than a force.
We consider the equations on T 3 = R 3 / Z 3 \mathbb{T}^3 = \mathbb{R}^3/\mathbb{Z}^3 T 3 = R 3 / Z 3
{ ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ Π − Δ u = 0 , div u = 0. (1) \begin{cases}
\partial_t \rho+ \operatorname{div}(\rho u) = 0,\\
\partial_t(\rho u) + \operatorname{div}(\rho u \otimes u) + \nabla\Pi- \Delta u = 0,\\
\operatorname{div} u = 0. \tag*{(1)}
\end{cases} ⎩ ⎨ ⎧ ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇Π − Δ u = 0 , div u = 0. (1) Here ρ \rho ρ u u u R 3 \mathbb{R}^3 R 3 Π \Pi Π
Definition 1.1 (Weak solutions). Let 0 < s < 3 / 2 0 < s < 3/2 0 < s < 3/2 1 < p < ∞ 1 < p < \infty 1 < p < ∞
ρ ∈ C t L p , u ∈ C t H s , Π ∈ D ′ ( ( 0 , 1 ) × T 3 ) , \rho\in C_tL^p,\qquad u \in C_tH^s,\qquad\Pi\in\mathcal{D}'((0,1) \times\mathbb{T}^3), ρ ∈ C t L p , u ∈ C t H s , Π ∈ D ′ (( 0 , 1 ) × T 3 ) , with ρ ≥ ρ ‾ \rho\geq\underline{\rho} ρ ≥ ρ ρ ‾ > 0 \underline{\rho} > 0 ρ > 0
ρ u ∈ C t L 1 , ρ u ⊗ u ∈ C t L 1 , \rho u \in C_tL^1,\qquad\rho u \otimes u \in C_tL^1, ρ u ∈ C t L 1 , ρ u ⊗ u ∈ C t L 1 , such that (1.1) holds in the sense of distributions on ( 0 , 1 ) × T 3 (0,1) \times\mathbb{T}^3 ( 0 , 1 ) × T 3
Main results
Our first result approximates prescribed smooth density and velocity fields inside an arbitrary cylinder.
Throughout, B = B ( x 0 , r B ) ⊂ T 3 B = B(x_0,r_B) \subset\mathbb{T}^3 B = B ( x 0 , r B ) ⊂ T 3 r B < 1 / 4 r_B < 1/4 r B < 1/4 B B B ∫ B ( x − x 0 ) f d x \int_B (x-x_0)f\,\mathrm{d}x ∫ B ( x − x 0 ) f d x B B B
Theorem 1.1 (Localized approximation). For every 0 < s < 3 / 2 0 < s < 3/2 0 < s < 3/2 p ∗ ( s ) > 1 p_*(s) > 1 p ∗ ( s ) > 1 1 < p < p ∗ ( s ) 1 < p < p_*(s) 1 < p < p ∗ ( s ) Q ⋐ ( 0 , 1 ) × T 3 Q \Subset(0,1) \times\mathbb{T}^3 Q ⋐ ( 0 , 1 ) × T 3 B B B 1 / 4 1/4 1/4
Let ( ρ ˉ , u ˉ , Π ˉ ) (\bar{\rho},\bar{u},\bar{\Pi}) ( ρ ˉ , u ˉ , Π ˉ ) ( ρ ∗ , u ∗ ) (\rho_*,u_*) ( ρ ∗ , u ∗ ) [ 0 , 1 ] × T 3 [0,1] \times\mathbb{T}^3 [ 0 , 1 ] × T 3 ρ ˉ , ρ ∗ > 0 \bar{\rho},\rho_* > 0 ρ ˉ , ρ ∗ > 0 div u ∗ = 0 \operatorname{div}u_* = 0 div u ∗ = 0
supp ( ρ ∗ − ρ ˉ , u ∗ − u ˉ ) ⋐ Q , \operatorname{supp}(\rho_*-\bar{\rho},u_*-\bar{u}) \Subset Q, supp ( ρ ∗ − ρ ˉ , u ∗ − u ˉ ) ⋐ Q , ∫ B ( ρ ∗ − ρ ˉ ) ( t , x ) d x = 0 , ∫ B ( x − x 0 ) ( ρ ∗ − ρ ˉ ) ( t , x ) d x = 0 ( 0 ≤ t ≤ 1 ) . (2) \int_B(\rho_*-\bar{\rho})(t,x)\,\mathrm{d}x=0,\qquad
\int_B(x-x_0)(\rho_*-\bar{\rho})(t,x)\,\mathrm{d}x=0\qquad(0\leq t\leq1). \tag*{(2)} ∫ B ( ρ ∗ − ρ ˉ ) ( t , x ) d x = 0 , ∫ B ( x − x 0 ) ( ρ ∗ − ρ ˉ ) ( t , x ) d x = 0 ( 0 ≤ t ≤ 1 ) . (2) Then, for every δ > 0 \delta> 0 δ > 0 ( ρ , u , Π ) (\rho,u,\Pi) ( ρ , u , Π ) Π ∈ L t , x 1 \Pi\in L^1_{t,x} Π ∈ L t , x 1
∥ ρ − ρ ∗ ∥ C t L p + ∥ u − u ∗ ∥ C t H s < δ , ρ ≥ ρ ∗ − δ almost everywhere . \|\rho-\rho_*\|_{C_tL^p}+\|u-u_*\|_{C_tH^s}<\delta,\qquad
\rho\geq\rho_*-\delta\quad\text{almost everywhere}. ∥ ρ − ρ ∗ ∥ C t L p + ∥ u − u ∗ ∥ C t H s < δ , ρ ≥ ρ ∗ − δ almost everywhere . Moreover, ( ρ ˉ , u ˉ , Π ˉ ) = ( ρ , u , Π ) (\bar{\rho},\bar{u},\bar{\Pi})=(\rho,u,\Pi) ( ρ ˉ , u ˉ , Π ˉ ) = ( ρ , u , Π ) Q Q Q
The exponent p ∗ ( s ) p_*(s) p ∗ ( s ) s ↑ 3 / 2 s \uparrow3/2 s ↑ 3/2 H s H^s H s s s s r B < 1 / 4 r_B < 1/4 r B < 1/4 B B B
The prescribed fields need not satisfy either evolution equation inside Q Q Q
Corollary 1.1 (Compactly supported nonuniqueness). Let s , p , ( ρ ˉ , u ˉ , Π ˉ ) s,p,(\bar{\rho},\bar{u},\bar{\Pi}) s , p , ( ρ ˉ , u ˉ , Π ˉ ) Q Q Q ( ρ ˉ , u ˉ , Π ˉ ) (\bar{\rho},\bar{u},\bar{\Pi}) ( ρ ˉ , u ˉ , Π ˉ ) Q Q Q ( ρ ˉ , u ˉ ) (\bar{\rho},\bar{u}) ( ρ ˉ , u ˉ ) C t L p × C t H s C_tL^p \times C_tH^s C t L p × C t H s ρ ≥ 1 2 min ρ ˉ \rho\geq\frac{1}{2}\min\bar{\rho} ρ ≥ 2 1 min ρ ˉ ρ − 1 \rho-1 ρ − 1 u u u Π \Pi Π Q Q Q
Remark 1.1 (Initial data and other domains). The solutions of Corollary 1.1 coincide with ( ρ ˉ , u ˉ , Π ˉ ) (\bar{\rho},\bar{u},\bar{\Pi}) ( ρ ˉ , u ˉ , Π ˉ ) Q Q Q ( ρ ˉ ( 0 ) , u ˉ ( 0 ) ) (\bar{\rho}(0),\bar{u}(0)) ( ρ ˉ ( 0 ) , u ˉ ( 0 )) ρ 0 = 1 \rho_0=1 ρ 0 = 1 u 0 = 0 u_0=0 u 0 = 0 Q Q Q Ω ⊃ B \Omega\supset B Ω ⊃ B R 3 \mathbb{R}^3 R 3
Remark 1.2 (Failure of renormalization). Let ( ρ , u , Π ) (\rho,u,\Pi) ( ρ , u , Π ) Φ ( z ) = 1 + ( z − 1 ) 2 − 1 \Phi(z)=\sqrt{1+(z-1)^2}-1 Φ ( z ) = 1 + ( z − 1 ) 2 − 1 Φ ≥ 0 \Phi\geq0 Φ ≥ 0 Φ ( z ) = 0 \Phi(z)=0 Φ ( z ) = 0 z = 1 z=1 z = 1 ∣ Φ ′ ∣ ≤ 1 |\Phi'|\leq1 ∣ Φ ′ ∣ ≤ 1 Φ ( z ) ≤ ∣ z − 1 ∣ \Phi(z)\leq|z-1| Φ ( z ) ≤ ∣ z − 1∣ Φ ( ρ ) ∈ C t L 1 \Phi(\rho)\in C_tL^1 Φ ( ρ ) ∈ C t L 1 ∣ Φ ( ρ ) u ∣ ≤ ( ρ + 1 ) ∣ u ∣ ∈ L t , x 1 |\Phi(\rho)u|\leq(\rho+1)|u|\in L^1_{t,x} ∣Φ ( ρ ) u ∣ ≤ ( ρ + 1 ) ∣ u ∣ ∈ L t , x 1 ρ ∈ C t L 1 \rho\in C_tL^1 ρ ∈ C t L 1 u ∈ C t L 2 u\in C_tL^2 u ∈ C t L 2 ∂ t Φ ( ρ ) + div ( Φ ( ρ ) u ) = 0 \partial_t\Phi(\rho)+\operatorname{div}(\Phi(\rho)u)=0 ∂ t Φ ( ρ ) + div ( Φ ( ρ ) u ) = 0 D ′ ( ( 0 , 1 ) × T 3 ) \mathcal{D}'((0,1)\times\mathbb{T}^3) D ′ (( 0 , 1 ) × T 3 ) t ↦ ∫ Φ ( ρ ( t ) ) d x t\mapsto\int\Phi(\rho(t))\,\mathrm{d}x t ↦ ∫ Φ ( ρ ( t )) d x Q Q Q ρ = 1 \rho=1 ρ = 1 Φ ( ρ ( t ) ) = 0 \Phi(\rho(t))=0 Φ ( ρ ( t )) = 0 ρ ( t ) = 1 \rho(t)=1 ρ ( t ) = 1 t t t ρ \rho ρ [21 , 3 ] , which guarantees renormalization for ρ ∈ L p \rho\in L^p ρ ∈ L p u ∈ W 1 , p ′ u\in W^{1,p'} u ∈ W 1 , p ′ 1 / p + 1 / p ′ = 1 1/p+1/p'=1 1/ p + 1/ p ′ = 1 u u u L 2 L^2 L 2 p ′ p' p ′ p p p [35 , 36 ] . The point of the present result is that positive density, the momentum equation and viscosity are compatible with this phenomenon.
Remark 1.3 (The spatial moment conditions). Conditions (1.2) from Theorem 1.1 cannot be dropped. Let ( ρ , u , Π ) (\rho,u,\Pi) ( ρ , u , Π ) ( ρ , u , Π ) = ( ρ ˉ , u ˉ , Π ˉ ) (\rho,u,\Pi)=(\bar{\rho},\bar{u},\bar{\Pi}) ( ρ , u , Π ) = ( ρ ˉ , u ˉ , Π ˉ ) K ⋐ Q K\Subset Q K ⋐ Q χ B ∈ C c ∞ ( B ) \chi_B\in C^\infty_c(B) χ B ∈ C c ∞ ( B ) K K K ψ ( t ) χ B \psi(t)\chi_B ψ ( t ) χ B ψ ( t ) ( x − x 0 ) χ B \psi(t)(x-x_0)\chi_B ψ ( t ) ( x − x 0 ) χ B ψ ( t ) e i χ B \psi(t)e_i\chi_B ψ ( t ) e i χ B ψ ∈ C c ∞ ( ( 0 , 1 ) ) \psi\in C^\infty_c((0,1)) ψ ∈ C c ∞ (( 0 , 1 )) ∇ χ B \nabla\chi_B ∇ χ B D ′ ( ( 0 , 1 ) ) \mathcal{D}'((0,1)) D ′ (( 0 , 1 ))
d d t ∫ ( ρ − ρ ˉ ) χ B = 0 , \frac{\mathrm{d}}{\mathrm{d}t}\int(\rho-\bar{\rho})\chi_B=0, d t d ∫ ( ρ − ρ ˉ ) χ B = 0 , d d t ∫ ( x − x 0 ) ( ρ − ρ ˉ ) χ B = ∫ ( ρ u − ρ ˉ u ˉ ) χ B , \frac{\mathrm{d}}{\mathrm{d}t}\int(x-x_0)(\rho-\bar{\rho})\chi_B
=\int(\rho u-\bar{\rho}\bar{u})\chi_B, d t d ∫ ( x − x 0 ) ( ρ − ρ ˉ ) χ B = ∫ ( ρ u − ρ ˉ u ˉ ) χ B , d d t ∫ ( ρ u − ρ ˉ u ˉ ) χ B = 0. \frac{\mathrm{d}}{\mathrm{d}t}\int(\rho u-\bar{\rho}\bar{u})\chi_B=0. d t d ∫ ( ρ u − ρ ˉ u ˉ ) χ B = 0. The three quantities are continuous in time and vanish for t t t Q Q Q ρ − ρ ˉ \rho-\bar{\rho} ρ − ρ ˉ B B B t t t C t L 1 C_tL^1 C t L 1 ρ \rho ρ ( ρ ∗ , u ∗ ) (\rho_*,u_*) ( ρ ∗ , u ∗ ) C t L 1 C_tL^1 C t L 1
Remark 1.4 (Localization in time). If B B B T 3 \mathbb{T}^3 T 3 R 0 \mathcal{R}_0 R 0 R \mathcal{R} R § 5.4 . The correction θ c \theta_c θ c
Main ideas of the proof
We construct smooth approximate solutions with a continuity defect R R R T T T
{ ∂ t ρ + div ( ρ u ) = div R , ∂ t ( ρ u − R ) + div ( ρ u ⊗ u ) + ∇ Π − Δ u = div T , div u = 0. \begin{cases}
\partial_t\rho+\operatorname{div}(\rho u)=\operatorname{div}R,\\
\partial_t(\rho u-R)+\operatorname{div}(\rho u\otimes u)+\nabla\Pi-\Delta u=\operatorname{div}T,\\
\operatorname{div}u=0.
\end{cases} ⎩ ⎨ ⎧ ∂ t ρ + div ( ρ u ) = div R , ∂ t ( ρ u − R ) + div ( ρ u ⊗ u ) + ∇Π − Δ u = div T , div u = 0. At each step we reduce R , T R,T R , T − R -R − R
The principal perturbations
Let Θ k ≥ 0 \Theta_k\geq0 Θ k ≥ 0 V k = ω k V_k=\omega k V k = ω k w w w
∂ t Θ k + V k ⋅ ∇ Θ k = 0 , u + w = V k on supp ( a k Θ k ) . \partial_t\Theta_k+V_k\cdot\nabla\Theta_k=0,\qquad u+w=V_k\quad\text{on }\operatorname{supp}(a_k\Theta_k). ∂ t Θ k + V k ⋅ ∇ Θ k = 0 , u + w = V k on supp ( a k Θ k ) . The principal density perturbation is
θ p = ∑ k a k ( Θ k − 1 ) , a k ≥ 0. \theta_p=\sum_k a_k(\Theta_k-1),\qquad a_k\geq0. θ p = k ∑ a k ( Θ k − 1 ) , a k ≥ 0. The transport of each profile is exact, so its remaining error contains only derivatives of the coefficient:
∂ t [ a k ( Θ k − 1 ) ] + div [ a k ( Θ k − 1 ) V k ] = ( Θ k − 1 ) ( ∂ t + V k ⋅ ∇ ) a k . \partial_t[a_k(\Theta_k-1)]+\operatorname{div}[a_k(\Theta_k-1)V_k]
=(\Theta_k-1)(\partial_t+V_k\cdot\nabla)a_k. ∂ t [ a k ( Θ k − 1 )] + div [ a k ( Θ k − 1 ) V k ] = ( Θ k − 1 ) ( ∂ t + V k ⋅ ∇ ) a k . This is the reason for moving the profiles. A stationary bump would leave the full time derivative of a k Θ k a_k\Theta_k a k Θ k V k V_k V k u + w = V k u+w=V_k u + w = V k
We choose the coefficients to satisfy
∑ k a k ( V k − u ) = − R , \sum_k a_k(V_k-u)=-R, k ∑ a k ( V k − u ) = − R , ∑ k a k ( V k ⊗ V k − u ⊗ u ) = π Id − T . (3) \sum_k a_k(V_k\otimes V_k-u\otimes u)=\pi\operatorname{Id}-T. \tag*{(3)} k ∑ a k ( V k ⊗ V k − u ⊗ u ) = π Id − T . (3) The first identity cancels the continuity defect; the second cancels the stress after a pressure correction. Opposite coordinate directions give the first identity, while the geometric lemma supplies the second. Taking ω \omega ω
Concentration and the regularity range
At frequency λ \lambda λ μ \mu μ
∥ Θ k ∥ L p ≲ μ 3 − 3 / p , ∥ w ∥ H s ≲ ω λ s μ − 3 / 2 . \|\Theta_k\|_{L^p}\lesssim\mu^{3-3/p},\qquad\|w\|_{H^s}\lesssim\omega\lambda^s\mu^{-3/2}. ∥ Θ k ∥ L p ≲ μ 3 − 3/ p , ∥ w ∥ H s ≲ ω λ s μ − 3/2 . Choosing μ = λ N \mu=\lambda^N μ = λ N N ( 3 / 2 − s ) > s + 2 N(3/2-s)>s+2 N ( 3/2 − s ) > s + 2 s < 3 / 2 s<3/2 s < 3/2 3 − 3 / p → 0 3-3/p\to0 3 − 3/ p → 0 p ↓ 1 p\downarrow1 p ↓ 1 p p p θ p \theta_p θ p p p p s s s
The density lower bound
Proposition 2.1 requires a positive density lower bound throughout the iteration. The principal perturbation can decrease the density wherever its positive bumps vanish. With ω = δ n − 4 \omega= \delta_n^{-4} ω = δ n − 4
θ p ≥ − ∑ k a k , ∑ k a k ≤ δ n 2 ∣ R ∣ + δ n 4 ∣ T ∣ + C δ n 12 . \theta_p \ge-\sum_k a_k,\qquad\sum_k a_k \le\delta_n^2|R|+\delta_n^4|T|+C\delta_n^{12}. θ p ≥ − k ∑ a k , k ∑ a k ≤ δ n 2 ∣ R ∣ + δ n 4 ∣ T ∣ + C δ n 12 . Here δ n \delta_n δ n n n n δ n 2 , δ n 4 \delta_n^2,\delta_n^4 δ n 2 , δ n 4 R , T R,T R , T
At the next stage the same subtraction is determined by R + , T + R^+,T^+ R + , T + T + T^+ T + − ( ∇ w + ∇ w T ) -(\nabla w+\nabla w^{T}) − ( ∇ w + ∇ w T ) L p L^p L p
We add a stationary density corrector θ d \theta_d θ d L p L^p L p θ = θ p + θ c + θ d \theta=\theta_p+\theta_c+\theta_d θ = θ p + θ c + θ d θ c \theta_c θ c
θ ≥ − ∑ k a k + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ − C δ n + 1 16 . (4) \theta\ge-\sum_k a_k+\delta_{n+1}^2|R^+|+\delta_{n+1}^4|T^+|-C\delta_{n+1}^{16}. \tag*{(4)} θ ≥ − k ∑ a k + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ − C δ n + 1 16 . (4) The positive defect terms cover the subtraction when R + , T + R^+,T^+ R + , T +
The profile is stationary so that its time derivative falls only on the time cutoff. The resulting error R 0 ∂ t θ d \mathcal{R}_0\partial_t\theta_d R 0 ∂ t θ d
Compact support
For a symmetric solution of div S = f \operatorname{div} S=f div S = f B B B
∫ f = 0 , ∫ x × f = 0. \int f=0,\qquad\int x\times f=0. ∫ f = 0 , ∫ x × f = 0. These are the vanishing linear and angular momentum conditions used in localized convex integration; see Isett–Oh [26 ] and Zhao [40 ] . We arrange them before applying the inverse divergence operators. The density increment θ \theta θ
∫ θ = 0 , ∫ x θ = 0. \int\theta=0,\qquad\int x\theta=0. ∫ θ = 0 , ∫ x θ = 0. Writing m = ρ u − R m=\rho u-R m = ρ u − R ∂ t θ + div ( m + − m ) = 0 \partial_t\theta+\operatorname{div}(m^+-m)=0 ∂ t θ + div ( m + − m ) = 0
∫ ( m + − m ) = − ∫ x div ( m + − m ) = ∂ t ∫ x θ = 0. \int(m^+-m)=-\int x\operatorname{div}(m^+-m)=\partial_t\int x\theta=0. ∫ ( m + − m ) = − ∫ x div ( m + − m ) = ∂ t ∫ x θ = 0. A compactly supported curl removes its angular momentum without changing this divergence. The resulting momentum residual then has a compactly supported symmetric inverse divergence.
Finally, the approximate equations are linear in ρ \rho ρ u u u Π \Pi Π ρ u − R \rho u-R ρ u − R ρ u ⊗ u − T \rho u\otimes u-T ρ u ⊗ u − T
Related work
Convex integration for fluid equations goes back to the differential-inclusion formulation of De Lellis and Székelyhidi [17 ] , which builds on the work of Nash, Kuiper and Gromov [37 , 30 , 24 ] ; see the surveys [19 , 8 ] . Continuous dissipative Euler flows [18 ] , the Mikado flows of Daneri and Székelyhidi [16 ] and Isett’s proof of Onsager’s conjecture [28 ] established the geometric framework used below. Compact support in time was obtained by Isett [27 ] , and compact support in space, subject to the linear and angular momentum conditions that reappear in § 5.4 , by Isett and Oh [26 ] , see also [40 ] .
For the Navier–Stokes equations, Buckmaster and Vicol [7 ] introduced intermittent building blocks whose concentration overcomes viscosity. Sharp Sobolev nonuniqueness and temporal intermittency were obtained by Cheskidov and Luo [10 , 11 ] , and partial regularity in time in [6 ] . These solutions lie outside the Leray class; within it, nonuniqueness is known for the forced equations [2 ] . Our solutions have velocity in H s H^s H s s < 3 / 2 s<3/2 s < 3/2
For the transport equation with Sobolev velocity, nonuniqueness was established by Modena and Székelyhidi [35 , 36 ] in the range complementary to the DiPerna–Lions theory [21 , 3 ] . Moving profiles with full dimensional concentration appear in [34 ] , positive profiles and their characteristic geometry in [5 ] , critical constructions in [9 ] , and temporal intermittency in [12 ] . Convex integration for the inhomogeneous incompressible Euler equations was carried out by Giri and Koley [23 ] .
Here the density and velocity are constructed together. The identities (3) cancel the continuity defect and the stress using the same nonnegative coefficients, and the stationary density corrector gives (4) . For spatial localization, we use the Bogovskii operator [4 , 22 ] and solve the symmetric divergence equation subject to the linear and angular momentum conditions, as in [26 ] .
Organization of the paper
Section 2 formulates the iteration and deduces the main results. Section 3 collects the auxiliary estimates and constructs the stationary profile. We construct one step in Section 4 and prove its estimates in Section 5. Section 6 completes the iteration. Only the elementary cutoff, calculus, and kernel arguments are collected in Appendix A.
Notation
All spatial norms and integrals are over T 3 \mathbb{T}^{3} T 3 C t X = C ( [ 0 , 1 ] ; X ) C_{t}X = C([0,1];X) C t X = C ([ 0 , 1 ] ; X ) m m m
∥ f ∥ C t , x m = max j + ∣ α ∣ ≤ m ∥ ∂ t j ∂ x α f ∥ ∞ . \lVert f\rVert_{C_{t,x}^{m}}=\max_{j+|\alpha|\le m}\lVert\partial_{t}^{j}\partial_{x}^{\alpha}f\rVert_{\infty}. ∥ f ∥ C t , x m = j + ∣ α ∣ ≤ m max ∥ ∂ t j ∂ x α f ∥ ∞ . We use the same notation for scalar, vector, and matrix fields, with the Euclidean norm in the range; a norm of a tuple is the sum of the norms of its entries. The letter p p p q q q
We denote the coordinate unit vectors by e 1 , e 2 , e 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3} e 1 , e 2 , e 3 3 × 3 3\times3 3 × 3 Sym 3 \operatorname{Sym}_{3} Sym 3
( v ⊗ w ) i j = v i w j , ( ∇ v ) i j = ∂ j v i , ( div S ) i = ∑ j ∂ j S i j . (v\otimes w)_{ij}=v_{i}w_{j},\qquad(\nabla v)_{ij}=\partial_{j}v_{i},\qquad(\operatorname{div}S)_{i}=\sum_{j}\partial_{j}S_{ij}. ( v ⊗ w ) ij = v i w j , ( ∇ v ) ij = ∂ j v i , ( div S ) i = j ∑ ∂ j S ij . We also write ( S ∇ χ ) i = ∑ j S i j ∂ j χ (S\nabla\chi)_{i}=\sum_{j}S_{ij}\partial_{j}\chi ( S ∇ χ ) i = ∑ j S ij ∂ j χ ε i j k \varepsilon_{ijk} ε ij k ∇ \nabla ∇ Δ \Delta Δ div \operatorname{div} div curl \operatorname{curl} curl ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣
\fint f=\int_{\mathbb{T}^{3}}f\,\mathrm{d}x,\qquad\mathbb{P}_{\ne0}f=f-\fint f,
and inverse Laplacians act on mean-zero functions.
Acknowledgements
XL would like to thank Xiaoyutao Luo for fruitful discussions and for his invitation to visit the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. XL is supported by PRIN 2022 “Turbulent effects vs Stability in Equations from Oceanography” (TESEO), project number 2022HSSYPN.
During the preparation of this work the author used Chatgpt 6 and Claude as research assistants. The strategy for spatial localization in Section 3.4 was organized by Claude on the basis of the ideas in [40 ] . The models were also used to check algebraic identities and to improve the exposition.
All mathematical arguments and conclusions in that lemma were independently checked and verified by the author, who assumes full responsibility for its content.
The iteration and proof of the main results
We first extend the prescribed fields to a smooth approximate solution with compactly supported errors. Proposition 2.1 reduces these errors to zero with summable increments, while preserving positivity and a common compact support. We then deduce the main results from this proposition.
The approximate system
We introduce defects in both evolution equations. The continuity defect is included in the momentum variable, so that both equations remain linear in the density, modified momentum and modified flux. This form will be used both for the principal cancellation and for regularization.
Definition 2.1 (Approximate solutions). A smooth solution of the approximate system is a quintuple Z = ( ρ , u , Π , R , T ) Z=(\rho,u,\Pi,R,T) Z = ( ρ , u , Π , R , T ) [ 0 , 1 ] × T 3 [0,1]\times\mathbb{T}^3 [ 0 , 1 ] × T 3 ρ , Π \rho,\Pi ρ , Π u , R u,R u , R R 3 \mathbb{R}^3 R 3 T T T Sym 3 \operatorname{Sym}_3 Sym 3
{ ∂ t ρ + div ( ρ u ) = div R , ∂ t ( ρ u − R ) + div ( ρ u ⊗ u ) + ∇ Π − Δ u = div T , div u = 0. (5) \begin{cases}
\partial_t\rho+\operatorname{div}(\rho u)=\operatorname{div}R,\\
\partial_t(\rho u-R)+\operatorname{div}(\rho u\otimes u)+\nabla\Pi-\Delta u=\operatorname{div}T,\\
\operatorname{div}u=0. \tag*{(5)}
\end{cases} ⎩ ⎨ ⎧ ∂ t ρ + div ( ρ u ) = div R , ∂ t ( ρ u − R ) + div ( ρ u ⊗ u ) + ∇Π − Δ u = div T , div u = 0. (5) Define the smooth vector field m m m Φ \Phi Φ
m ( t , x ) = ρ ( t , x ) u ( t , x ) − R ( t , x ) , m(t,x)=\rho(t,x)u(t,x)-R(t,x), m ( t , x ) = ρ ( t , x ) u ( t , x ) − R ( t , x ) , Φ ( t , x ) = ρ ( t , x ) u ( t , x ) ⊗ u ( t , x ) − T ( t , x ) . \Phi(t,x)=\rho(t,x)u(t,x)\otimes u(t,x)-T(t,x). Φ ( t , x ) = ρ ( t , x ) u ( t , x ) ⊗ u ( t , x ) − T ( t , x ) . Then (5) becomes
{ ∂ t ρ + div m = 0 , ∂ t m + div Φ + ∇ Π − Δ u = 0 , div u = 0. \begin{cases}
\partial_t\rho+\operatorname{div}m=0,\\
\partial_t m+\operatorname{div}\Phi+\nabla\Pi-\Delta u=0,\\
\operatorname{div}u=0.
\end{cases} ⎩ ⎨ ⎧ ∂ t ρ + div m = 0 , ∂ t m + div Φ + ∇Π − Δ u = 0 , div u = 0. Parameters
Given 0 < s < 3 / 2 0<s<3/2 0 < s < 3/2 0 < γ < 10 − 3 0<\gamma<10^{-3} 0 < γ < 1 0 − 3 N N N
N ≥ γ − 1 , N ( 3 / 2 − s ) > s + 2. N\geq\gamma^{-1},\qquad N(3/2-s)>s+2. N ≥ γ − 1 , N ( 3/2 − s ) > s + 2. Set b = N 8 b=N^8 b = N 8
1 < p < p ∗ ( s ) , p ∗ ( s ) = 1 + γ N 10 . 1<p<p_*(s),\qquad p_*(s)=1+\frac{\gamma}{N^{10}}. 1 < p < p ∗ ( s ) , p ∗ ( s ) = 1 + N 10 γ . For an integer a ≥ 2 a\geq2 a ≥ 2
λ n = a b n , δ n = λ n − γ ( n ≥ 0 ) . \lambda_n=a^{b^n},\qquad\delta_n=\lambda_n^{-\gamma}\qquad(n\geq0). λ n = a b n , δ n = λ n − γ ( n ≥ 0 ) . The exponents γ , N , b , p \gamma,N,b,p γ , N , b , p a > 0 a>0 a > 0 a a a
Initial defects
Write Q = J × B Q=J\times B Q = J × B B = B ( x 0 , r B ) B=B(x_0,r_B) B = B ( x 0 , r B ) J ⋐ ( 0 , 1 ) J\Subset(0,1) J ⋐ ( 0 , 1 ) r B < 1 / 4 r_B<1/4 r B < 1/4
Lemma 2.1 (Local extension). Under the hypotheses of Theorem 1.1 there are smooth fields R ∗ R_* R ∗ T ∗ = T ∗ T T_*=T_*^{T} T ∗ = T ∗ T K ∗ ⊂ Q K_*\subset Q K ∗ ⊂ Q Z ∗ = ( ρ ∗ , u ∗ , Π ‾ , R ∗ , T ∗ ) Z_*=(\rho_*,u_*,\overline{\Pi},R_*,T_*) Z ∗ = ( ρ ∗ , u ∗ , Π , R ∗ , T ∗ ) Z ∗ = ( ρ ‾ , u ‾ , Π ‾ , 0 , 0 ) Z_*=(\overline{\rho},\overline{u},\overline{\Pi},0,0) Z ∗ = ( ρ , u , Π , 0 , 0 ) K ∗ K_* K ∗
Proof. Let K 0 ⊂ Q K_0\subset Q K 0 ⊂ Q ( ρ ∗ , u ∗ ) = ( ρ ‾ , u ‾ ) (\rho_*,u_*)=(\overline{\rho},\overline{u}) ( ρ ∗ , u ∗ ) = ( ρ , u ) K 0 K_0 K 0 J 0 ⋐ J J_0\Subset J J 0 ⋐ J r ′ < r B r'<r_B r ′ < r B K 0 ⊂ J 0 × B ( x 0 , r ′ ) K_0\subset J_0\times B(x_0,r') K 0 ⊂ J 0 × B ( x 0 , r ′ ) U ′ = B ( x 0 , r ′ ) U'=B(x_0,r') U ′ = B ( x 0 , r ′ ) ϕ ∗ ∈ C c ∞ ( U ′ ) \phi_*\in C^\infty_c(U') ϕ ∗ ∈ C c ∞ ( U ′ ) ∫ ϕ ∗ = 1 \int\phi_*=1 ∫ ϕ ∗ = 1
Step 1: The continuity defect.
Since ρ ‾ \overline{\rho} ρ
f : = ∂ t ρ ∗ + div ( ρ ∗ u ∗ ) = ∂ t ( ρ ∗ − ρ ‾ ) + div ( ρ ∗ u ∗ − ρ ‾ u ‾ ) f:=\partial_t\rho_*+\operatorname{div}(\rho_*u_*)=\partial_t(\rho_*-\overline{\rho})+\operatorname{div}(\rho_*u_*-\overline{\rho}\,\overline{u}) f := ∂ t ρ ∗ + div ( ρ ∗ u ∗ ) = ∂ t ( ρ ∗ − ρ ) + div ( ρ ∗ u ∗ − ρ u ) is supported in K 0 K_0 K 0
∫ f d x = d d t ∫ B ( ρ ∗ − ρ ‾ ) d x = 0 \int f\,\mathrm{d}x=\frac{\mathrm{d}}{\mathrm{d}t}\int_{B}(\rho_*-\overline{\rho})\,\mathrm{d}x=0 ∫ f d x = d t d ∫ B ( ρ ∗ − ρ ) d x = 0 by Theorem 1.1. Lemma 3.7 gives the vector field
R b : = B U ′ f ∈ C c ∞ ( U ′ ; R 3 ) R^{\mathrm b}:=\mathcal{B}_{U'}f\in C^\infty_c(U';\mathbb{R}^3) R b := B U ′ f ∈ C c ∞ ( U ′ ; R 3 ) with div R b = f \operatorname{div}R^{\mathrm b}=f div R b = f t t t t ∉ J 0 t\notin J_0 t ∈ / J 0
F 0 : = ρ ∗ u ∗ − ρ ‾ u ‾ − R b , F^0:=\rho_*u_*-\overline{\rho}\,\overline{u}-R^{\mathrm b}, F 0 := ρ ∗ u ∗ − ρ u − R b , D ∗ ( t , x ) : = 1 2 curl x [ ϕ ∗ ( x ) ∫ ( y − x 0 ) × F 0 ( t , y ) d y ] , D_*(t,x):=\frac{1}{2}\operatorname{curl}_x\left[\phi_*(x)\int(y-x_0)\times F^0(t,y)\,\mathrm{d}y\right], D ∗ ( t , x ) := 2 1 curl x [ ϕ ∗ ( x ) ∫ ( y − x 0 ) × F 0 ( t , y ) d y ] , F ∗ : = F 0 − D ∗ . F_*:=F^0-D_*. F ∗ := F 0 − D ∗ . These fields are smooth and supported in J 0 × U ′ J_0 \times U' J 0 × U ′ div F ∗ = div ( ρ ∗ u ∗ − ρ ˉ u ˉ ) − f = − ∂ t ( ρ ∗ − ρ ˉ ) \operatorname{div} F_* = \operatorname{div}(\rho_*u_* - \bar{\rho}\bar{u}) - f = -\partial_t(\rho_* - \bar{\rho}) div F ∗ = div ( ρ ∗ u ∗ − ρ ˉ u ˉ ) − f = − ∂ t ( ρ ∗ − ρ ˉ )
∫ F ∗ d x = − ∫ ( x − x 0 ) div F ∗ d x = d d t ∫ ( x − x 0 ) ( ρ ∗ − ρ ˉ ) d x = 0. \int F_*\,\mathrm{d}x = -\int(x-x_0)\operatorname{div} F_*\,\mathrm{d}x
= \frac{\mathrm{d}}{\mathrm{d}t}\int(x-x_0)(\rho_*-\bar{\rho})\,\mathrm{d}x = 0. ∫ F ∗ d x = − ∫ ( x − x 0 ) div F ∗ d x = d t d ∫ ( x − x 0 ) ( ρ ∗ − ρ ˉ ) d x = 0. The curl correction also gives
∫ ( x − x 0 ) × F ∗ d x = ( 1 − ∫ ϕ ∗ d x ) ∫ ( x − x 0 ) × F 0 d x = 0. \int(x-x_0)\times F_*\,\mathrm{d}x = \left(1-\int\phi_*\,\mathrm{d}x\right)\int(x-x_0)\times F^0\,\mathrm{d}x = 0. ∫ ( x − x 0 ) × F ∗ d x = ( 1 − ∫ ϕ ∗ d x ) ∫ ( x − x 0 ) × F 0 d x = 0. Step 2: The momentum defect.
Let
g : = ∂ t F ∗ + div ( ρ ∗ u ∗ ⊗ u ∗ − ρ ˉ u ˉ ⊗ u ˉ ) − Δ ( u ∗ − u ˉ ) . g := \partial_t F_* + \operatorname{div}(\rho_*u_* \otimes u_*-\bar{\rho}\bar{u}\otimes\bar{u})-\Delta(u_*-\bar{u}). g := ∂ t F ∗ + div ( ρ ∗ u ∗ ⊗ u ∗ − ρ ˉ u ˉ ⊗ u ˉ ) − Δ ( u ∗ − u ˉ ) . It is smooth and supported in J 0 × U ′ J_0 \times U' J 0 × U ′ Δ v = div ( ∇ v + ∇ v T ) \Delta v = \operatorname{div}(\nabla v+\nabla v^T) Δ v = div ( ∇ v + ∇ v T ) v v v
∫ g = ∂ t ∫ F ∗ = 0 , ∫ ( x − x 0 ) × g = ∂ t ∫ ( x − x 0 ) × F ∗ = 0. \int g = \partial_t\int F_* = 0,\qquad
\int(x-x_0)\times g = \partial_t\int(x-x_0)\times F_* = 0. ∫ g = ∂ t ∫ F ∗ = 0 , ∫ ( x − x 0 ) × g = ∂ t ∫ ( x − x 0 ) × F ∗ = 0. Set
R ∗ : = R b + D ∗ , T ∗ : = S U ′ g , R_* := R^{\mathrm b}+D_*,\qquad T_* := S_{U'}g, R ∗ := R b + D ∗ , T ∗ := S U ′ g , with S U ′ S_{U'} S U ′ div R ∗ = f \operatorname{div}R_* = f div R ∗ = f ρ ∗ u ∗ − R ∗ = ρ ˉ u ˉ + F ∗ \rho_*u_*-R_*=\bar{\rho}\bar{u}+F_* ρ ∗ u ∗ − R ∗ = ρ ˉ u ˉ + F ∗ ( ρ ˉ , u ˉ , Π ˉ ) (\bar{\rho},\bar{u},\bar{\Pi}) ( ρ ˉ , u ˉ , Π ˉ )
∂ t ( ρ ∗ u ∗ − R ∗ ) + div ( ρ ∗ u ∗ ⊗ u ∗ ) + ∇ Π ˉ − Δ u ∗ = g = div T ∗ . \partial_t(\rho_*u_*-R_*)+\operatorname{div}(\rho_*u_*\otimes u_*)+\nabla\bar{\Pi}-\Delta u_* = g = \operatorname{div}T_*. ∂ t ( ρ ∗ u ∗ − R ∗ ) + div ( ρ ∗ u ∗ ⊗ u ∗ ) + ∇ Π ˉ − Δ u ∗ = g = div T ∗ . The fields R ∗ , T ∗ R_*,T_* R ∗ , T ∗ K ∗ : = J 0 × U ′ ⊂ Q K_*:=J_0\times U'\subset Q K ∗ := J 0 × U ′ ⊂ Q
From now on we fix Z ∗ Z_* Z ∗ K ∗ K_* K ∗ r 0 ∈ [ r B / 2 , r B ) r_0\in[r_B/2,r_B) r 0 ∈ [ r B /2 , r B ) K ∗ ⊂ J × B ( x 0 , r 0 ) K_*\subset J\times B(x_0,r_0) K ∗ ⊂ J × B ( x 0 , r 0 ) C ∗ C_* C ∗ Z ∗ Z_* Z ∗
The iteration
The iteration must do more than make the two defects vanish. It must also preserve a common compact support, a positive lower bound for the density, and strong convergence of the nonlinear terms. The next proposition records precisely these conclusions.
Proposition 2.1 (Iteration). Let Z ∗ Z_* Z ∗ K ∗ K_* K ∗ a > 0 a>0 a > 0 K ∞ ⊂ Q K_\infty\subset Q K ∞ ⊂ Q K ∗ K_* K ∗
Z n = ( ρ n , u n , Π n , R n , T n ) , n ≥ 0 , Z_n=(\rho_n,u_n,\Pi_n,R_n,T_n),\qquad n\geq0, Z n = ( ρ n , u n , Π n , R n , T n ) , n ≥ 0 , of (2.1) from Definition 2.1, with Z 0 = Z ∗ Z_0=Z_* Z 0 = Z ∗ n n n Z n − Z ∗ Z_n-Z_* Z n − Z ∗ ( R n , T n ) (R_n,T_n) ( R n , T n ) K ∞ K_\infty K ∞
The density and velocity increments satisfy
∥ ρ n + 1 − ρ n ∥ C t L p + ∥ u n + 1 − u n ∥ C t H s ≤ δ n , (6) \lVert\rho_{n+1}-\rho_n\rVert_{C_tL^p}+\lVert u_{n+1}-u_n\rVert_{C_tH^s}\leq\delta_n, \tag*{(6)} ∥ ρ n + 1 − ρ n ∥ C t L p + ∥ u n + 1 − u n ∥ C t H s ≤ δ n , (6) and the defects tend to zero according to
∥ ( R n , T n ) ∥ C t L p ≤ δ n 8 ( n ≥ 1 ) . (7) \lVert(R_n,T_n)\rVert_{C_tL^p}\leq\delta_n^8\qquad(n\geq1). \tag*{(7)} ∥( R n , T n ) ∥ C t L p ≤ δ n 8 ( n ≥ 1 ) . (7) The nonlinear quantities and pressure have summable increments: for n ≥ 1 n\geq1 n ≥ 1
∥ ρ n + 1 u n + 1 − ρ n u n ∥ C t L 1 + ∥ ρ n + 1 u n + 1 ⊗ u n + 1 − ρ n u n ⊗ u n ∥ C t L 1 + ∥ Π n + 1 − Π n ∥ L t 1 L x ∞ ≤ δ n . (8) \begin{aligned}
&\lVert\rho_{n+1}u_{n+1}-\rho_nu_n\rVert_{C_tL^1}\\
&\quad+\lVert\rho_{n+1}u_{n+1}\otimes u_{n+1}-\rho_nu_n\otimes u_n\rVert_{C_tL^1}\\
&\quad+\lVert\Pi_{n+1}-\Pi_n\rVert_{L_t^1L_x^\infty}\leq\delta_n. \tag*{(8)}
\end{aligned} ∥ ρ n + 1 u n + 1 − ρ n u n ∥ C t L 1 + ∥ ρ n + 1 u n + 1 ⊗ u n + 1 − ρ n u n ⊗ u n ∥ C t L 1 + ∥ Π n + 1 − Π n ∥ L t 1 L x ∞ ≤ δ n . (8) Finally, the density retains the pointwise lower bound
ρ n ≥ ρ ∗ − δ 0 . (9) \rho_n\geq\rho_*-\delta_0. \tag*{(9)} ρ n ≥ ρ ∗ − δ 0 . (9) The proposition is proved in Section 6, using the one-step Proposition 4.1.
Proof of Theorem 1.1
Proof of Theorem 1.1. Let Z ∗ Z_* Z ∗ a a a 2 δ 0 < min { δ , 1 2 min ρ ∗ } 2\delta_0 < \min\{\delta, \frac{1}{2}\min\rho_*\} 2 δ 0 < min { δ , 2 1 min ρ ∗ } (6) , ρ n → ρ \rho_n \to\rho ρ n → ρ C t L p C_tL^p C t L p u n → u u_n \to u u n → u C t H s C_tH^s C t H s
∥ ρ − ρ ∗ ∥ C t L p + ∥ u − u ∗ ∥ C t H s ≤ ∑ n ≥ 0 δ n ≤ 2 δ 0 < δ . \|\rho-\rho_*\|_{C_tL^p}+\|u-u_*\|_{C_tH^s}\leq\sum_{n\geq0}\delta_n\leq2\delta_0<\delta. ∥ ρ − ρ ∗ ∥ C t L p + ∥ u − u ∗ ∥ C t H s ≤ n ≥ 0 ∑ δ n ≤ 2 δ 0 < δ . By (9) , ρ ≥ ρ ∗ − δ 0 ≥ ρ ∗ − δ \rho\geq\rho_*-\delta_0\geq\rho_*-\delta ρ ≥ ρ ∗ − δ 0 ≥ ρ ∗ − δ ρ n , ρ ≥ 3 4 min ρ ∗ \rho_n,\rho\geq\frac{3}{4}\min\rho_* ρ n , ρ ≥ 4 3 min ρ ∗ Z 0 Z_0 Z 0 Z 1 Z_1 Z 1 (8) makes all later increments summable. Hence ρ n u n \rho_nu_n ρ n u n ρ n u n ⊗ u n \rho_nu_n\otimes u_n ρ n u n ⊗ u n C t L 1 C_tL^1 C t L 1 m m m Φ \Phi Φ Π n → Π \Pi_n\to\Pi Π n → Π L t , x 1 L^1_{t,x} L t , x 1 (7) , R n , T n → 0 R_n,T_n\to0 R n , T n → 0 C t L 1 C_tL^1 C t L 1
Fix t t t ρ n ( t ) → ρ ( t ) \rho_n(t)\to\rho(t) ρ n ( t ) → ρ ( t ) u n ( t ) → u ( t ) u_n(t)\to u(t) u n ( t ) → u ( t ) ρ n u n ( t ) → m ( t ) \rho_nu_n(t)\to m(t) ρ n u n ( t ) → m ( t ) ρ n u n ⊗ u n ( t ) → Φ ( t ) \rho_nu_n\otimes u_n(t)\to\Phi(t) ρ n u n ⊗ u n ( t ) → Φ ( t ) m ( t ) = ρ u ( t ) m(t)=\rho u(t) m ( t ) = ρ u ( t ) Φ ( t ) = ρ u ⊗ u ( t ) \Phi(t)=\rho u\otimes u(t) Φ ( t ) = ρ u ⊗ u ( t ) t t t ρ u ∈ C t L 1 \rho u\in C_tL^1 ρ u ∈ C t L 1 ρ u ⊗ u ∈ C t L 1 \rho u\otimes u\in C_tL^1 ρ u ⊗ u ∈ C t L 1 C t L 1 C_tL^1 C t L 1
For φ ∈ C c ∞ ( ( 0 , 1 ) × T 3 ) \varphi\in C^\infty_c((0,1)\times\mathbb{T}^3) φ ∈ C c ∞ (( 0 , 1 ) × T 3 ) ψ ∈ C c ∞ ( ( 0 , 1 ) × T 3 ; R 3 ) \psi\in C^\infty_c((0,1)\times\mathbb{T}^3;\mathbb{R}^3) ψ ∈ C c ∞ (( 0 , 1 ) × T 3 ; R 3 ) (5) gives
∬ ρ n ∂ t φ + ( ρ n u n − R n ) ⋅ ∇ φ d x d t = 0 , \iint\rho_n\partial_t\varphi+(\rho_nu_n-R_n)\cdot\nabla\varphi\,\mathrm{d}x\,\mathrm{d}t=0, ∬ ρ n ∂ t φ + ( ρ n u n − R n ) ⋅ ∇ φ d x d t = 0 , ∬ ( ρ n u n − R n ) ⋅ ∂ t ψ + ( ρ n u n ⊗ u n − T n ) : ∇ ψ + Π n div ψ + u n ⋅ Δ ψ d x d t = 0. \iint(\rho_nu_n-R_n)\cdot\partial_t\psi+(\rho_nu_n\otimes u_n-T_n):\nabla\psi+\Pi_n\operatorname{div}\psi+u_n\cdot\Delta\psi\,\mathrm{d}x\,\mathrm{d}t=0. ∬ ( ρ n u n − R n ) ⋅ ∂ t ψ + ( ρ n u n ⊗ u n − T n ) : ∇ ψ + Π n div ψ + u n ⋅ Δ ψ d x d t = 0. All terms converge, so ( ρ , u , Π ) (\rho,u,\Pi) ( ρ , u , Π ) ( ρ n , u n , Π n ) = ( ρ ˉ , u ˉ , Π ˉ ) (\rho_n,u_n,\Pi_n)=(\bar\rho,\bar u,\bar\Pi) ( ρ n , u n , Π n ) = ( ρ ˉ , u ˉ , Π ˉ ) K ‾ ∞ \overline{K}_\infty K ∞ n n n □ \square □
Proof of Corollary 1.1
Proof. We distinguish the solutions by a linear observable of the density. Choose t ∗ ∈ J t_*\in J t ∗ ∈ J η 0 ∈ C ∞ ( J ; [ 0 , 1 ] ) \eta_0\in C^\infty(J;[0,1]) η 0 ∈ C ∞ ( J ; [ 0 , 1 ]) η 0 ( t ∗ ) = 1 \eta_0(t_*)=1 η 0 ( t ∗ ) = 1 g ∈ C c ∞ ( B ) g\in C^\infty_c(B) g ∈ C c ∞ ( B ) Δ g ≢ 0 \Delta g\not\equiv0 Δ g ≡ 0 f = Δ g / ∥ Δ g ∥ ∞ f=\Delta g/\|\Delta g\|_\infty f = Δ g /∥Δ g ∥ ∞ ∥ f ∥ ∞ = 1 \|f\|_\infty=1 ∥ f ∥ ∞ = 1 ∫ f = 0 \int f=0 ∫ f = 0 ∫ ( x − x 0 ) f = 0 \int(x-x_0)f=0 ∫ ( x − x 0 ) f = 0 c f = ∥ f ∥ L 2 ∈ ( 0 , 1 ) c_f=\|f\|_{L^2}\in(0,1) c f = ∥ f ∥ L 2 ∈ ( 0 , 1 ) α j = 2 − j − 4 min { 1 , min ρ ˉ } \alpha_j=2^{-j-4}\min\{1,\min\bar\rho\} α j = 2 − j − 4 min { 1 , min ρ ˉ } j ≥ 1 j\geq1 j ≥ 1
ρ ∗ , j = ρ ˉ + α j η 0 ( t ) f ( x ) , u ∗ , j = u ˉ \rho_{*,j}=\bar\rho+\alpha_j\eta_0(t)f(x),\qquad u_{*,j}=\bar u ρ ∗ , j = ρ ˉ + α j η 0 ( t ) f ( x ) , u ∗ , j = u ˉ satisfy the hypotheses of Theorem 1.1. Apply it with δ j = α j c f / 4 \delta_j=\alpha_jc_f/4 δ j = α j c f /4 L 1 L^1 L 1 T 3 \mathbb{T}^3 T 3 L p L^p L p
∫ ( ρ j ( t ∗ ) − ρ ˉ ( t ∗ ) ) f d x ∈ [ α j c f − δ j , α j c f + δ j ] = α j c f [ 3 4 , 5 4 ] . \int(\rho_j(t_*)-\bar\rho(t_*))f\,\mathrm{d}x\in[\alpha_jc_f-\delta_j,\alpha_jc_f+\delta_j]=\alpha_jc_f\left[\frac{3}{4},\frac{5}{4}\right]. ∫ ( ρ j ( t ∗ ) − ρ ˉ ( t ∗ )) f d x ∈ [ α j c f − δ j , α j c f + δ j ] = α j c f [ 4 3 , 4 5 ] . These intervals are pairwise disjoint because α j + 1 = α j / 2 \alpha_{j+1}=\alpha_j/2 α j + 1 = α j /2 ρ j ≥ ρ ˉ − α j − δ j ≥ 1 2 min ρ ˉ \rho_j\geq\bar\rho-\alpha_j-\delta_j\geq\frac{1}{2}\min\bar\rho ρ j ≥ ρ ˉ − α j − δ j ≥ 2 1 min ρ ˉ ∥ ρ j − ρ ˉ ∥ C t L p + ∥ u j − u ˉ ∥ C t H s ≤ α j + δ j → 0 \|\rho_j-\bar\rho\|_{C_tL^p}+\|u_j-\bar u\|_{C_tH^s}\leq\alpha_j+\delta_j\to0 ∥ ρ j − ρ ˉ ∥ C t L p + ∥ u j − u ˉ ∥ C t H s ≤ α j + δ j → 0 ( ρ ˉ , u ˉ , Π ˉ ) = ( 1 , 0 , 0 ) (\bar\rho,\bar u,\bar\Pi)=(1,0,0) ( ρ ˉ , u ˉ , Π ˉ ) = ( 1 , 0 , 0 ) ρ j \rho_j ρ j ρ j − 1 \rho_j-1 ρ j − 1 u j u_j u j Π j \Pi_j Π j Q Q Q □ \square □
Preliminaries
We collect the inverse divergence estimates, the geometric lemma, and the stationary density profile used in the perturbation. The periodic operators give a gain in frequency; their localized versions preserve compact support under the appropriate moment conditions. Constants depend only on the fixed parameters and the quantities indicated in each statement.
Time cutoffs and calculus inequalities
We use three nested time cutoffs. The first supports the density weights, the second the velocity perturbation, and the third the stationary density correction. Their separation makes the last cutoff equal to one wherever the principal errors are present.
Lemma 3.1 (Time cutoffs). Let I = [ t − , t + ] ⊂ J ′ ⊂ ( 0 , 1 ) I=[t_-,t_+]\subset J'\subset(0,1) I = [ t − , t + ] ⊂ J ′ ⊂ ( 0 , 1 ) J ′ J' J ′ τ = dist ( I , ∂ J ′ ) \tau=\operatorname{dist}(I,\partial J') τ = dist ( I , ∂ J ′ ) η , η u , η d ∈ C c ∞ ( J ′ ; [ 0 , 1 ] ) \eta,\eta_u,\eta_d\in C^\infty_c(J';[0,1]) η , η u , η d ∈ C c ∞ ( J ′ ; [ 0 , 1 ]) η = 1 \eta=1 η = 1 I I I η u = 1 \eta_u=1 η u = 1 supp η \operatorname{supp}\eta supp η η d = 1 \eta_d=1 η d = 1 supp η u \operatorname{supp}\eta_u supp η u j ≥ 0 j\geq0 j ≥ 0
∥ η ∥ C j + ∥ η u ∥ C j + ∥ η d ∥ C j ≤ C j τ − j , \|\eta\|_{C^j}+\|\eta_u\|_{C^j}+\|\eta_d\|_{C^j}\leq C_j\tau^{-j}, ∥ η ∥ C j + ∥ η u ∥ C j + ∥ η d ∥ C j ≤ C j τ − j , ∥ η d ′ ∥ ≤ C N τ − 1 η d 1 − 1 / ( 16 N ) . \|\eta_d'\| \leq C_N\tau^{-1}\eta_d^{1-1/(16N)}. ∥ η d ′ ∥ ≤ C N τ − 1 η d 1 − 1/ ( 16 N ) . The proof is given in Appendix A.
Lemma 3.2 (Calculus inequalities). Let j ≥ 0 j \ge0 j ≥ 0 f , g f,g f , g [ 0 , 1 ] × T 3 [0,1]\times\mathbb{T}^3 [ 0 , 1 ] × T 3
∥ f g ∥ C t , x j ≤ 2 j ∥ f ∥ C t , x j ∥ g ∥ C t , x j . \|fg\|_{C^j_{t,x}} \le2^j\|f\|_{C^j_{t,x}}\|g\|_{C^j_{t,x}}. ∥ f g ∥ C t , x j ≤ 2 j ∥ f ∥ C t , x j ∥ g ∥ C t , x j . Let K ⊂ R m \mathcal{K}\subset\mathbb{R}^m K ⊂ R m Γ \Gamma Γ K \mathcal{K} K v : [ 0 , 1 ] × T 3 → K v:[0,1]\times\mathbb{T}^3\to\mathcal{K} v : [ 0 , 1 ] × T 3 → K
∥ Γ ∘ v ∥ C t , x j ≤ C ( j , m , ∥ Γ ∥ C j ( K ) ) ( 1 + ∥ v ∥ C t , x j ) j . \|\Gamma\circ v\|_{C^j_{t,x}}\le C(j,m,\|\Gamma\|_{C^j(\mathcal{K})})(1+\|v\|_{C^j_{t,x}})^j. ∥Γ ∘ v ∥ C t , x j ≤ C ( j , m , ∥Γ ∥ C j ( K ) ) ( 1 + ∥ v ∥ C t , x j ) j . If Γ \Gamma Γ R m \mathbb{R}^m R m 1 , … , j 1,\ldots,j 1 , … , j 1 , … , j 1,\ldots,j 1 , … , j Γ ∘ v \Gamma\circ v Γ ∘ v ∥ Γ ∥ C j ( K ) \|\Gamma\|_{C^j(\mathcal{K})} ∥Γ ∥ C j ( K ) max 1 ≤ l ≤ j ∥ ∇ l Γ ∥ L ∞ ( R m ) \max_{1\le l\le j}\|\nabla^l\Gamma\|_{L^\infty(\mathbb{R}^m)} max 1 ≤ l ≤ j ∥ ∇ l Γ ∥ L ∞ ( R m )
The proof is given in Appendix A.
Periodic operators
The periodic operators give a gain in frequency on mean-zero profiles. We later localize them with a cutoff and a Bogovskii corrector.
Lemma 3.3 (Periodic antidivergence operators). For f ∈ C ∞ ( T 3 ) f\in C^\infty(\mathbb{T}^3) f ∈ C ∞ ( T 3 ) F ∈ C ∞ ( T 3 ; R 3 ) F\in C^\infty(\mathbb{T}^3;\mathbb{R}^3) F ∈ C ∞ ( T 3 ; R 3 )
R 0 f = ∇ Δ − 1 P ≠ 0 f , R F = ∇ Δ − 1 P ≠ 0 F + ( ∇ Δ − 1 P ≠ 0 F ) T − ∇ 2 Δ − 2 div F . (10) \mathcal{R}_0f=\nabla\Delta^{-1}\mathbb{P}_{\ne0}f,\qquad
\mathcal{R}F=\nabla\Delta^{-1}\mathbb{P}_{\ne0}F+\left(\nabla\Delta^{-1}\mathbb{P}_{\ne0}F\right)^T-\nabla^2\Delta^{-2}\operatorname{div}F. \tag*{(10)} R 0 f = ∇ Δ − 1 P = 0 f , R F = ∇ Δ − 1 P = 0 F + ( ∇ Δ − 1 P = 0 F ) T − ∇ 2 Δ − 2 div F . (10) Then div R 0 f = P ≠ 0 f \operatorname{div}\mathcal{R}_0f=\mathbb{P}_{\ne0}f div R 0 f = P = 0 f R F \mathcal{R}F R F div R F = P ≠ 0 F + ∇ Δ − 1 div F − ∇ Δ − 1 div F = P ≠ 0 F \operatorname{div}\mathcal{R}F=\mathbb{P}_{\ne0}F+\nabla\Delta^{-1}\operatorname{div}F-\nabla\Delta^{-1}\operatorname{div}F=\mathbb{P}_{\ne0}F div R F = P = 0 F + ∇ Δ − 1 div F − ∇ Δ − 1 div F = P = 0 F
Let G ∈ D ′ ( T 3 ) G\in\mathcal{D}'(\mathbb{T}^3) G ∈ D ′ ( T 3 ) Δ G = δ 0 − 1 \Delta G=\delta_0-1 Δ G = δ 0 − 1 δ 0 \delta_0 δ 0 G 2 = G ∗ G G_2=G*G G 2 = G ∗ G α \alpha α z ∈ [ − 1 2 , 1 2 ] 3 ∖ { 0 } z\in[-\frac12,\frac12]^3\setminus\{0\} z ∈ [ − 2 1 , 2 1 ] 3 ∖ { 0 }
∣ ∂ α G ( z ) ∣ ≤ C α ( 1 + ∣ z ∣ − 1 − ∣ α ∣ ) , |\partial^\alpha G(z)|\le C_\alpha(1+|z|^{-1-|\alpha|}), ∣ ∂ α G ( z ) ∣ ≤ C α ( 1 + ∣ z ∣ − 1 − ∣ α ∣ ) , ∣ ∂ α G 2 ( z ) ∣ ≤ C α ( 1 + ∣ z ∣ 1 − ∣ α ∣ ) . |\partial^\alpha G_2(z)|\le C_\alpha(1+|z|^{1-|\alpha|}). ∣ ∂ α G 2 ( z ) ∣ ≤ C α ( 1 + ∣ z ∣ 1 − ∣ α ∣ ) . In particular, R 0 \mathcal{R}_0 R 0 R \mathcal{R} R Δ − 1 P ≠ 0 \Delta^{-1}\mathbb{P}_{\ne0} Δ − 1 P = 0 K K K j ≥ 0 j\ge0 j ≥ 0
∣ ∇ j K ( z ) ∣ ≤ C j ( 1 + dist ( z , 0 ) − 2 − j ) . (11) |\nabla^jK(z)|\le C_j(1+\operatorname{dist}(z,0)^{-2-j}). \tag*{(11)} ∣ ∇ j K ( z ) ∣ ≤ C j ( 1 + dist ( z , 0 ) − 2 − j ) . (11) They are bounded on L q L^q L q 1 ≤ q ≤ ∞ 1\le q\le\infty 1 ≤ q ≤ ∞ C t , x j C^j_{t,x} C t , x j λ ≥ 1 \lambda\ge1 λ ≥ 1
L [ g ( λ ⋅ ) ] = λ − 1 ( L g ) ( λ ⋅ ) , L ∈ { R 0 , R } , \mathcal{L}[g(\lambda\cdot)]=\lambda^{-1}(\mathcal{L}g)(\lambda\cdot),\qquad\mathcal{L}\in\{\mathcal{R}_0,\mathcal{R}\}, L [ g ( λ ⋅ )] = λ − 1 ( L g ) ( λ ⋅ ) , L ∈ { R 0 , R } , Δ − 1 P ≠ 0 [ g ( λ ⋅ ) ] = λ − 2 ( Δ − 1 P ≠ 0 g ) ( λ ⋅ ) . (12) \Delta^{-1}\mathbb{P}_{\ne0}[g(\lambda\cdot)]=\lambda^{-2}(\Delta^{-1}\mathbb{P}_{\ne0}g)(\lambda\cdot). \tag*{(12)} Δ − 1 P = 0 [ g ( λ ⋅ )] = λ − 2 ( Δ − 1 P = 0 g ) ( λ ⋅ ) . (12) Finally,
∥ Δ − 2 ∂ α P ≠ 0 g ∥ L 1 ≤ C α ∥ g ∥ L 1 if ∣ α ∣ ≤ 3. (13) \|\Delta^{-2}\partial^\alpha\mathbb{P}_{\ne0}g\|_{L^1}\le C_\alpha\|g\|_{L^1}\qquad\text{if }|\alpha|\le3. \tag*{(13)} ∥ Δ − 2 ∂ α P = 0 g ∥ L 1 ≤ C α ∥ g ∥ L 1 if ∣ α ∣ ≤ 3. (13) The kernel estimates and their consequences are proved in Appendix A.
Lemma 3.4 (Products with fast periodic functions). Let λ ∈ N \lambda\in\mathbb{N} λ ∈ N f ∈ C 1 ( T 3 ) f\in C^1(\mathbb{T}^3) f ∈ C 1 ( T 3 ) g ∈ C ∞ ( T 3 ) g\in C^\infty(\mathbb{T}^3) g ∈ C ∞ ( T 3 ) 1 ≤ q < ∞ 1\le q<\infty 1 ≤ q < ∞
∥ f g ( λ ⋅ ) ∥ L q ≤ ∥ g ∥ L q ( ∥ f ∥ L q + 3 λ − 1 ∥ ∇ f ∥ ∞ ) . (14) \|fg(\lambda\cdot)\|_{L^q}\le\|g\|_{L^q}\left(\|f\|_{L^q}+\sqrt{3}\lambda^{-1}\|\nabla f\|_\infty\right). \tag*{(14)} ∥ f g ( λ ⋅ ) ∥ L q ≤ ∥ g ∥ L q ( ∥ f ∥ L q + 3 λ − 1 ∥∇ f ∥ ∞ ) . (14) If ∫ f g = 0 \int fg=0 ∫ f g = 0
∣ ∫ f g ( λ ⋅ ) ∣ ≤ 3 λ − 1 ∥ ∇ f ∥ ∞ ∥ g ∥ L 1 . (15) \left|\int fg(\lambda\cdot)\right|\le\sqrt{3}\lambda^{-1}\|\nabla f\|_\infty\|g\|_{L^1}. \tag*{(15)} ∫ f g ( λ ⋅ ) ≤ 3 λ − 1 ∥∇ f ∥ ∞ ∥ g ∥ L 1 . (15) Let K 0 ( z ) = 1 + dist ( z , 0 ) − 2 K_0(z)=1+\operatorname{dist}(z,0)^{-2} K 0 ( z ) = 1 + dist ( z , 0 ) − 2 ∫ f g = 0 \int fg=0 ∫ f g = 0 L \mathcal{L} L
∣ L [ f g ( λ ⋅ ) ] − f L [ g ( λ ⋅ ) ] ∣ ( x ) ≤ C λ − 1 ∥ f ∥ C 1 ( K 0 ∗ ∣ ( R 0 g ) ( λ ⋅ ) ∣ ) ( x ) . (16) |\mathcal{L}[fg(\lambda\cdot)]-f\mathcal{L}[g(\lambda\cdot)]|(x)
\le C\lambda^{-1}\|f\|_{C^1}\left(K_0*|(\mathcal{R}_0g)(\lambda\cdot)|\right)(x). \tag*{(16)} ∣ L [ f g ( λ ⋅ )] − f L [ g ( λ ⋅ )] ∣ ( x ) ≤ C λ − 1 ∥ f ∥ C 1 ( K 0 ∗ ∣ ( R 0 g ) ( λ ⋅ ) ∣ ) ( x ) . (16) Consequently,
∥ L [ f g ( λ ⋅ ) ] ∥ L 1 ≤ C λ − 1 ∥ f ∥ C 1 ∥ g ∥ L 1 . (17) \|\mathcal{L}[fg(\lambda\cdot)]\|_{L^1}\le C\lambda^{-1}\|f\|_{C^1}\|g\|_{L^1}. \tag*{(17)} ∥ L [ f g ( λ ⋅ )] ∥ L 1 ≤ C λ − 1 ∥ f ∥ C 1 ∥ g ∥ L 1 . (17) All bounds are uniform under translations of g g g L \mathcal{L} L
Proof. The first two estimates follow by averaging over the periodic cells. Divide T 3 \mathbb{T}^{3} T 3 λ 3 \lambda^{3} λ 3 Q i Q_i Q i 1 / λ 1/\lambda 1/ λ x i ∈ Q i x_i \in Q_i x i ∈ Q i ∣ f ∣ |f| ∣ f ∣
∣ f ( x ) ∣ ≤ ∣ f ( x i ) ∣ + 3 λ − 1 ∥ ∇ f ∥ ∞ . |f(x)| \le|f(x_i)|+\sqrt{3}\lambda^{-1}\|\nabla f\|_{\infty}. ∣ f ( x ) ∣ ≤ ∣ f ( x i ) ∣ + 3 λ − 1 ∥∇ f ∥ ∞ . while periodicity gives
∫ Q i ∣ g ( λ x ) ∣ q d x = λ − 3 ∥ g ∥ L q q . \int_{Q_i}|g(\lambda x)|^{q}\,dx=\lambda^{-3}\|g\|_{L^{q}}^{q}. ∫ Q i ∣ g ( λ x ) ∣ q d x = λ − 3 ∥ g ∥ L q q . Minkowski’s inequality in ℓ q \ell^{q} ℓ q λ − 3 ∣ f ( x i ) ∣ q ≤ ∫ Q i ∣ f ∣ q \lambda^{-3}|f(x_i)|^{q}\le\int_{Q_i}|f|^{q} λ − 3 ∣ f ( x i ) ∣ q ≤ ∫ Q i ∣ f ∣ q (14) . When g g g f ( x i ) f(x_i) f ( x i ) (15) .
For the commutator, let K K K L \mathcal{L} L (10) and f g = 0 f g=0 f g = 0 div R 0 g = g \operatorname{div}\mathcal{R}_{0}g=g div R 0 g = g
g ( λ y ) = λ − 1 div y [ ( R 0 g ) ( λ y ) ] . g(\lambda y)=\lambda^{-1}\operatorname{div}_{y}[(\mathcal{R}_{0}g)(\lambda y)]. g ( λ y ) = λ − 1 div y [( R 0 g ) ( λ y )] . An integration by parts therefore yields
L [ f g ( λ ⋅ ) ] ( x ) − f ( x ) L [ g ( λ ⋅ ) ] ( x ) = − λ − 1 ∫ ∇ y [ K ( x − y ) ( f ( y ) − f ( x ) ) ] ⋅ ( R 0 g ) ( λ y ) d y . \mathcal{L}[fg(\lambda\cdot)](x)-f(x)\mathcal{L}[g(\lambda\cdot)](x)
=-\lambda^{-1}\int\nabla_{y}[K(x-y)(f(y)-f(x))]\cdot(\mathcal{R}_{0}g)(\lambda y)\,dy. L [ f g ( λ ⋅ )] ( x ) − f ( x ) L [ g ( λ ⋅ )] ( x ) = − λ − 1 ∫ ∇ y [ K ( x − y ) ( f ( y ) − f ( x ))] ⋅ ( R 0 g ) ( λ y ) d y . To justify this identity, first integrate outside B ( x , h ) B(x,h) B ( x , h )
C h 2 h − 2 h ∥ f ∥ C 1 ∥ R 0 g ∥ ∞ , Ch^{2}h^{-2}h\|f\|_{C^{1}}\|\mathcal{R}_{0}g\|_{\infty}, C h 2 h − 2 h ∥ f ∥ C 1 ∥ R 0 g ∥ ∞ , which tends to zero. By (11) ,
∣ ∇ y [ K ( x − y ) ( f ( y ) − f ( x ) ) ] ∣ ≤ C ∥ f ∥ C 1 K 0 ( x − y ) . |\nabla_{y}[K(x-y)(f(y)-f(x))]|\le C\|f\|_{C^{1}}K_{0}(x-y). ∣ ∇ y [ K ( x − y ) ( f ( y ) − f ( x ))] ∣ ≤ C ∥ f ∥ C 1 K 0 ( x − y ) . This proves (16) . Its right side has L 1 L^{1} L 1 C λ − 1 ∥ f ∥ C 1 ∥ g ∥ L 1 C\lambda^{-1}\|f\|_{C^{1}}\|g\|_{L^{1}} C λ − 1 ∥ f ∥ C 1 ∥ g ∥ L 1 K 0 K_{0} K 0 R 0 \mathcal{R}_{0} R 0 L 1 L^{1} L 1 Lemma 3.3 . The remaining term has the same bound by (12) , proving (17) .
The far field
Localization errors are evaluated away from the support of their source. There the kernel is smooth, and repeated integration by parts gives arbitrarily large gains in the frequency. The resulting bound uses an integral norm of the profile and therefore does not lose a power of its concentration.
Lemma 3.5 (Far field of oscillatory antidivergences). Let L \mathcal{L} L Lemma 3.3 , let g ∈ C ∞ ( T 3 ) g\in C^{\infty}(\mathbb{T}^{3}) g ∈ C ∞ ( T 3 ) f f f L \mathcal{L} L λ ∈ N \lambda\in\mathbb{N} λ ∈ N 0 < d < 1 / 4 0<d<1/4 0 < d < 1/4 dist ( x , supp f ) ≥ d \operatorname{dist}(x,\operatorname{supp}f)\ge d dist ( x , supp f ) ≥ d m ≥ 0 m\ge0 m ≥ 0
∣ ∇ m L [ f g ( λ ⋅ ) ] ( x ) ∣ ≤ C m d − ( m + 6 ) λ − 4 ∥ f ∥ C 4 ∥ Δ − 2 g ∥ L 1 , |\nabla^{m}\mathcal{L}[fg(\lambda\cdot)](x)|\le C_{m}d^{-(m+6)}\lambda^{-4}\|f\|_{C^{4}}\|\Delta^{-2}g\|_{L^{1}}, ∣ ∇ m L [ f g ( λ ⋅ )] ( x ) ∣ ≤ C m d − ( m + 6 ) λ − 4 ∥ f ∥ C 4 ∥ Δ − 2 g ∥ L 1 , uniformly under translations of g g g e e e 0 ≤ j ≤ 3 0\le j\le3 0 ≤ j ≤ 3 ∥ Δ − 2 ( e ⋅ ∇ ) j g ∥ L 1 ≤ C ∥ g ∥ L 1 \|\Delta^{-2}(e\cdot\nabla)^{j}g\|_{L^{1}}\le C\|g\|_{L^{1}} ∥ Δ − 2 ( e ⋅ ∇ ) j g ∥ L 1 ≤ C ∥ g ∥ L 1 (13) from Lemma 3.3 .
If f = f ( t , x ) f=f(t,x) f = f ( t , x ) 0 ≤ j ≤ 3 0\le j\le3 0 ≤ j ≤ 3
∂ t j [ f g ( λ x − λ ω t e ) ] = ∑ i = 0 j ( j i ) ( ∂ t j − i f ) ( − λ ω ) i ( ( e ⋅ ∇ ) i g ) ( λ x − λ ω t e ) . \partial_{t}^{j}[fg(\lambda x-\lambda\omega t e)]=\sum_{i=0}^{j}\binom{j}{i}(\partial_{t}^{j-i}f)(-\lambda\omega)^{i}((e\cdot\nabla)^{i}g)(\lambda x-\lambda\omega t e). ∂ t j [ f g ( λ x − λω t e )] = i = 0 ∑ j ( i j ) ( ∂ t j − i f ) ( − λω ) i (( e ⋅ ∇ ) i g ) ( λ x − λω t e ) . Each differentiated profile satisfies the preceding integral bound, uniformly under translations.
Proof. Since g g g
g ( λ y ) = λ − 4 Δ y 2 [ ( Δ − 2 g ) ( λ y ) ] . g(\lambda y)=\lambda^{-4}\Delta_{y}^{2}[(\Delta^{-2}g)(\lambda y)]. g ( λ y ) = λ − 4 Δ y 2 [( Δ − 2 g ) ( λ y )] . Let K K K L \mathcal{L} L f f f B ( x , d ) B(x,d) B ( x , d ) y ↦ ∇ m K ( x − y ) f ( y ) y\mapsto\nabla^{m}K(x-y)f(y) y ↦ ∇ m K ( x − y ) f ( y )
∇ m L [ f g ( λ ⋅ ) ] ( x ) = λ − 4 ∫ Δ y 2 [ ∇ m K ( x − y ) f ( y ) ] ( Δ − 2 g ) ( λ y ) d y . \nabla^{m}\mathcal{L}[fg(\lambda\cdot)](x)=\lambda^{-4}\int\Delta_{y}^{2}[\nabla^{m}K(x-y)f(y)](\Delta^{-2}g)(\lambda y)\,dy. ∇ m L [ f g ( λ ⋅ )] ( x ) = λ − 4 ∫ Δ y 2 [ ∇ m K ( x − y ) f ( y )] ( Δ − 2 g ) ( λ y ) d y . By (11) , the bracket is bounded by C d − ( m + 6 ) ∥ f ∥ C 4 Cd^{-(m+6)}\|f\|_{C^{4}} C d − ( m + 6 ) ∥ f ∥ C 4 L 1 L^{1} L 1 Δ − 2 g \Delta^{-2}g Δ − 2 g λ ω \lambda\omega λω f f f
Compactly supported antidivergences
Throughout this subsection U = B ( x 0 , r U ) U = B(x_0,r_U) U = B ( x 0 , r U ) 0 < r U ≤ r B < 1 / 4 0 < r_U \le r_B < 1/4 0 < r U ≤ r B < 1/4 U U U r B r_B r B r U r_U r U r ′ r' r ′ r U ≥ r 0 ≥ r B / 2 r_U \ge r_0 \ge r_B/2 r U ≥ r 0 ≥ r B /2 r 0 r_0 r 0 r B r_B r B
The range of a compactly supported divergence is constrained by integration by parts. For a symmetric matrix field, symmetry also forces the angular momentum of its divergence to vanish. These are exactly the compatibility conditions used in the construction below.
Lemma 3.6 (Moment identities). Let F , a ∈ C c 1 ( U ; R 3 ) F,a \in C^1_c(U;\mathbb{R}^3) F , a ∈ C c 1 ( U ; R 3 ) q ∈ C c 1 ( U ) q \in C^1_c(U) q ∈ C c 1 ( U ) M ∈ C c 1 ( U ; Sym 3 ) M \in C^1_c(U;\operatorname{Sym}_3) M ∈ C c 1 ( U ; Sym 3 )
∫ F = − ∫ ( x − x 0 ) div F , ∫ ∇ q = ∫ div M = ∫ curl a = 0 , \int F=-\int(x-x_0)\operatorname{div}F,\qquad\int\nabla q=\int\operatorname{div}M=\int\operatorname{curl}a=0, ∫ F = − ∫ ( x − x 0 ) div F , ∫ ∇ q = ∫ div M = ∫ curl a = 0 , ∫ ( x − x 0 ) × ∇ q = ∫ ( x − x 0 ) × div M = 0 , \int(x-x_0)\times\nabla q=\int(x-x_0)\times\operatorname{div}M=0, ∫ ( x − x 0 ) × ∇ q = ∫ ( x − x 0 ) × div M = 0 , ∫ ( x − x 0 ) × curl a = 2 ∫ a . \int(x-x_0)\times\operatorname{curl}a=2\int a. ∫ ( x − x 0 ) × curl a = 2 ∫ a . Moreover Δ v = div ( ∇ v + ∇ v T ) \Delta v=\operatorname{div}(\nabla v+\nabla v^{T}) Δ v = div ( ∇ v + ∇ v T ) v ∈ C 2 v\in C^2 v ∈ C 2
Proof. All identities follow by integration by parts. In particular, symmetry gives
∫ [ ( x − x 0 ) × div M ] l = ∫ ε l a b ( x − x 0 ) a ∂ j M b j = − ε l a b ∫ M b a = 0. \begin{aligned}
\int[(x-x_0)\times\operatorname{div}M]_l=\int\varepsilon_{lab}(x-x_0)_a\partial_jM_{bj} \\
&=-\varepsilon_{lab}\int M_{ba}=0.
\end{aligned} ∫ [( x − x 0 ) × div M ] l = ∫ ε l ab ( x − x 0 ) a ∂ j M bj = − ε l ab ∫ M ba = 0. The gradient case follows from ∇ q = div ( q I d ) \nabla q=\operatorname{div}(q\,\mathrm{Id}) ∇ q = div ( q Id )
∫ [ ( x − x 0 ) × curl a ] l = − ε l a b ε b a d ∫ a d = 2 ∫ a l , \int[(x-x_0)\times\operatorname{curl}a]_l=-\varepsilon_{lab}\varepsilon_{bad}\int a_d=2\int a_l, ∫ [( x − x 0 ) × curl a ] l = − ε l ab ε ba d ∫ a d = 2 ∫ a l , using ε l a b ε b a d = − 2 δ l d \varepsilon_{lab}\varepsilon_{bad}=-2\delta_{ld} ε l ab ε ba d = − 2 δ l d
Lemma 3.7 (Bogovskii operator). There is a linear operator
B U : { h ∈ C c ∞ ( U ) : ∫ h = 0 } ⟶ C c ∞ ( U ; R 3 ) B_U:\left\{h\in C^\infty_c(U):\int h=0\right\}\longrightarrow C^\infty_c(U;\mathbb{R}^3) B U : { h ∈ C c ∞ ( U ) : ∫ h = 0 } ⟶ C c ∞ ( U ; R 3 ) such that
div B U h = h , ∫ ( x − x 0 ) × B U h = 0 , \operatorname{div}B_Uh=h,\qquad\int(x-x_0)\times B_Uh=0, div B U h = h , ∫ ( x − x 0 ) × B U h = 0 , ∥ B U h ∥ C k ≤ C k ∥ h ∥ C k ( k ≥ 0 ) . \lVert B_Uh\rVert_{C^k}\le C_k\lVert h\rVert_{C^k}\qquad(k\ge0). ∥ B U h ∥ C k ≤ C k ∥ h ∥ C k ( k ≥ 0 ) . It preserves smooth dependence on time and commutes with time derivatives.
Proof. Let b U ( x ) = r U − 3 b 1 ( ( x − x 0 ) / r U ) b_U(x)=r_U^{-3}b_1((x-x_0)/r_U) b U ( x ) = r U − 3 b 1 (( x − x 0 ) / r U ) b 1 ∈ C c ∞ ( B ( 0 , 1 / 2 ) ) b_1\in C^\infty_c(B(0,1/2)) b 1 ∈ C c ∞ ( B ( 0 , 1/2 )) ∫ b 1 = 1 \int b_1=1 ∫ b 1 = 1 U U U B ( x 0 , r U / 2 ) ⊃ supp b U B(x_0,r_U/2)\supset\operatorname{supp}b_U B ( x 0 , r U /2 ) ⊃ supp b U [4 ] (see also [22 Section III.3] ) is
B U h ( x ) = ∫ U h ( y ) x − y ∣ x − y ∣ 3 ∫ ∣ x − y ∣ ∞ b U ( y + r x − y ∣ x − y ∣ ) r 2 d r d y . B_Uh(x)=\int_U h(y)\frac{x-y}{|x-y|^3}\int_{|x-y|}^{\infty}b_U\left(y+r\frac{x-y}{|x-y|}\right)r^2\,dr\,dy. B U h ( x ) = ∫ U h ( y ) ∣ x − y ∣ 3 x − y ∫ ∣ x − y ∣ ∞ b U ( y + r ∣ x − y ∣ x − y ) r 2 d r d y . For h ∈ C c ∞ ( U ) h\in C^\infty_c(U) h ∈ C c ∞ ( U ) ∫ h = 0 \int h=0 ∫ h = 0 B U h ∈ C c ∞ ( U ) B_Uh\in C^\infty_c(U) B U h ∈ C c ∞ ( U ) div B U h = h \operatorname{div}B_Uh=h div B U h = h [4 ] , [22 Section III.3] . The support property is also seen directly: B U h ( x ) ≠ 0 B_Uh(x)\ne0 B U h ( x ) = 0 x x x supp h \operatorname{supp}h supp h supp b U \operatorname{supp}b_U supp b U U U U
To compute the angular momentum, write the inner integral using r ≥ 1 r\ge1 r ≥ 1 z = y + r ( x − y ) z=y+r(x-y) z = y + r ( x − y ) ∫ 1 ∞ r − 2 d r = 1 \int_1^\infty r^{-2}\,dr=1 ∫ 1 ∞ r − 2 d r = 1
∫ ( x − x 0 ) × B U h = ( ∫ ( y − x 0 ) h ( y ) d y ) × ( ∫ ( z − x 0 ) b U ( z ) d z ) = 0. \int(x-x_0)\times B_Uh=\left(\int(y-x_0)h(y)\,dy\right)\times\left(\int(z-x_0)b_U(z)\,dz\right)=0. ∫ ( x − x 0 ) × B U h = ( ∫ ( y − x 0 ) h ( y ) d y ) × ( ∫ ( z − x 0 ) b U ( z ) d z ) = 0. The last equality follows from the radial symmetry of b U b_U b U
For the C k C^k C k y = x − z y=x-z y = x − z
B U h ( x ) = ∫ h ( x − z ) K U ( x , z ) d z , K U ( x , z ) = z ∣ z ∣ 3 ∫ ∣ z ∣ 2 r U b U ( x − z + r z ∣ z ∣ ) r 2 d r , B_Uh(x)=\int h(x-z)K_U(x,z)\,dz,\qquad K_U(x,z)=\frac{z}{|z|^3}\int_{|z|}^{2r_U}b_U\left(x-z+r\frac{z}{|z|}\right)r^2\,dr, B U h ( x ) = ∫ h ( x − z ) K U ( x , z ) d z , K U ( x , z ) = ∣ z ∣ 3 z ∫ ∣ z ∣ 2 r U b U ( x − z + r ∣ z ∣ z ) r 2 d r , where we used that the integrand vanishes for r > 2 r U r > 2r_U r > 2 r U x − z ∈ U x-z \in U x − z ∈ U x x x h h h b U b_U b U ∣ ∂ x α K U ( x , z ) ∣ ≤ C α ∥ b U ∥ C ∣ α ∣ r U 3 ∣ z ∣ − 2 |\partial_x^\alpha K_U(x,z)| \le C_\alpha\|b_U\|_{C^{|\alpha|}}r_U^3|z|^{-2} ∣ ∂ x α K U ( x , z ) ∣ ≤ C α ∥ b U ∥ C ∣ α ∣ r U 3 ∣ z ∣ − 2 ∣ z ∣ − 2 |z|^{-2} ∣ z ∣ − 2 B ( 0 , 2 r U ) B(0,2r_U) B ( 0 , 2 r U ) ∣ ∂ α B U h ( x ) ∣ ≤ C α ∥ h ∥ C ∣ α ∣ |\partial^\alpha B_Uh(x)| \le C_\alpha\|h\|_{C^{|\alpha|}} ∣ ∂ α B U h ( x ) ∣ ≤ C α ∥ h ∥ C ∣ α ∣ C α C_\alpha C α b 1 b_1 b 1 α \alpha α r U , r U − 1 r_U,r_U^{-1} r U , r U − 1 t t t
Lemma 3.8 (Compactly supported symmetric antidivergence). There is a linear operator S U S_U S U f ∈ C c ∞ ( U ; R 3 ) f \in C^\infty_c(U;\mathbb{R}^3) f ∈ C c ∞ ( U ; R 3 )
∫ f = 0 , ∫ ( x − x 0 ) × f = 0 , \int f = 0,\qquad\int(x-x_0)\times f = 0, ∫ f = 0 , ∫ ( x − x 0 ) × f = 0 , the field S U f S_Uf S U f C c ∞ ( U ; Sym 3 ) C^\infty_c(U;\operatorname{Sym}_3) C c ∞ ( U ; Sym 3 )
div S U f = f , ∥ S U f ∥ C k ≤ C k ∥ f ∥ C k + 1 ( k ≥ 0 ) . \operatorname{div} S_Uf=f,\qquad\|S_Uf\|_{C^k}\le C_k\|f\|_{C^{k+1}}\quad(k\ge0). div S U f = f , ∥ S U f ∥ C k ≤ C k ∥ f ∥ C k + 1 ( k ≥ 0 ) . The operator commutes with ∂ t \partial_t ∂ t
Proof. Since ∫ f i = 0 \int f_i=0 ∫ f i = 0 M i j ( x ) = ( B U f i ) j ( x ) M_{ij}(x)=(B_Uf_i)_j(x) M ij ( x ) = ( B U f i ) j ( x ) ∂ j M i j = f i \partial_jM_{ij}=f_i ∂ j M ij = f i M = M s + A M=M^{\mathrm{s}}+A M = M s + A n k = 1 2 ε k i j A i j n_k=\frac{1}{2}\varepsilon_{kij}A_{ij} n k = 2 1 ε k ij A ij A i j = ε i j k n k A_{ij}=\varepsilon_{ijk}n_k A ij = ε ij k n k div A = curl n \operatorname{div}A=\operatorname{curl}n div A = curl n
0 = ∫ ( x − x 0 ) × f = ∫ ( x − x 0 ) × curl n = 2 ∫ n . 0=\int(x-x_0)\times f=\int(x-x_0)\times\operatorname{curl}n=2\int n. 0 = ∫ ( x − x 0 ) × f = ∫ ( x − x 0 ) × curl n = 2 ∫ n . Let Y Y Y Y l ⋅ = B U n l Y_{l\cdot}=B_Un_l Y l ⋅ = B U n l ∂ j Y l j = n l \partial_jY_{lj}=n_l ∂ j Y l j = n l
( S U f ) i j = M i j s + ε i k l ∂ k Y l j + ε j k l ∂ k Y l i . (S_Uf)_{ij}=M^{\mathrm{s}}_{ij}+\varepsilon_{ikl}\partial_kY_{lj}+\varepsilon_{jkl}\partial_kY_{li}. ( S U f ) ij = M ij s + ε ik l ∂ k Y l j + ε j k l ∂ k Y l i . This field is symmetric and satisfies div S U f = f − curl n + curl n = f \operatorname{div}S_Uf=f-\operatorname{curl}n+\operatorname{curl}n=f div S U f = f − curl n + curl n = f
∥ S U f ∥ C k ≤ C ( ∥ M ∥ C k + ∥ Y ∥ C k + 1 ) ≤ C ∥ f ∥ C k + 1 . \|S_Uf\|_{C^k}\le C(\|M\|_{C^k}+\|Y\|_{C^{k+1}})\le C\|f\|_{C^{k+1}}. ∥ S U f ∥ C k ≤ C ( ∥ M ∥ C k + ∥ Y ∥ C k + 1 ) ≤ C ∥ f ∥ C k + 1 . Directions and lines
To cancel the symmetric defect with nonnegative weights, we need a positive decomposition into rank-one matrices. The following explicit formula uses the two diagonals in each coordinate plane.
Lemma 3.9 (Geometric lemma). For 1 ≤ i < j ≤ 3 1\le i<j\le3 1 ≤ i < j ≤ 3 ξ i j ± = ( e i ± e j ) / 2 \xi^\pm_{ij}=(e_i\pm e_j)/\sqrt{2} ξ ij ± = ( e i ± e j ) / 2 S ∈ Sym 3 S\in\operatorname{Sym}_3 S ∈ Sym 3 ∣ S − Id ∣ < 1 / 5 |S-\operatorname{Id}|<1/5 ∣ S − Id ∣ < 1/5
S = ∑ 1 ≤ i < j ≤ 3 ∑ ± Γ i j ± ( S ) ξ i j ± ⊗ ξ i j ± , (18) S=\sum_{1\le i<j\le3}\sum_{\pm}\Gamma^\pm_{ij}(S)\,\xi^\pm_{ij}\otimes\xi^\pm_{ij}, \tag*{(18)} S = 1 ≤ i < j ≤ 3 ∑ ± ∑ Γ ij ± ( S ) ξ ij ± ⊗ ξ ij ± , (18) where the affine weights are
Γ i j ± ( S ) = S i i + S j j − 1 2 Tr S ± S i j ≥ 1 2 − 5 2 ∣ S − Id ∣ > 0. \Gamma^\pm_{ij}(S)=S_{ii}+S_{jj}-\frac{1}{2}\operatorname{Tr}S\pm S_{ij}\ge\frac{1}{2}-\frac{5}{2}|S-\operatorname{Id}|>0. Γ ij ± ( S ) = S ii + S j j − 2 1 Tr S ± S ij ≥ 2 1 − 2 5 ∣ S − Id ∣ > 0. Proof. The i j ij ij (18) is 1 2 ( Γ i j + − Γ i j − ) = S i j \frac{1}{2}(\Gamma^+_{ij}-\Gamma^-_{ij})=S_{ij} 2 1 ( Γ ij + − Γ ij − ) = S ij
1 2 ( Γ 12 + + Γ 12 − + Γ 13 + + Γ 13 − ) = 2 S 11 + S 22 + S 33 − Tr S = S 11 , \frac{1}{2}(\Gamma^+_{12}+\Gamma^-_{12}+\Gamma^+_{13}+\Gamma^-_{13})=2S_{11}+S_{22}+S_{33}-\operatorname{Tr}S=S_{11}, 2 1 ( Γ 12 + + Γ 12 − + Γ 13 + + Γ 13 − ) = 2 S 11 + S 22 + S 33 − Tr S = S 11 , and the other diagonal entries follow in the same way. At S = Id S=\operatorname{Id} S = Id 1 / 2 1/2 1/2 Tr S = S 11 + S 22 + S 33 \operatorname{Tr}S=S_{11}+S_{22}+S_{33} Tr S = S 11 + S 22 + S 33 5 2 ∣ S − Id ∣ \frac{5}{2}|S-\operatorname{Id}| 2 5 ∣ S − Id ∣
This is an explicit instance of the geometric lemma of [16 Lemma 2.4] . We use the finite sets of unit directions
Λ R = { ± e 1 , ± e 2 , ± e 3 } , \Lambda_R=\{\pm e_1,\pm e_2,\pm e_3\}, Λ R = { ± e 1 , ± e 2 , ± e 3 } , Λ T = { ± ξ i j + , ± ξ i j − : 1 ≤ i < j ≤ 3 } , Λ = Λ R ∪ Λ T . \Lambda_T=\{\pm\xi^+_{ij},\pm\xi^-_{ij}:1\le i<j\le3\},\qquad\Lambda=\Lambda_R\cup\Lambda_T. Λ T = { ± ξ ij + , ± ξ ij − : 1 ≤ i < j ≤ 3 } , Λ = Λ R ∪ Λ T . For the opposite directions, set Γ ± ξ i j + = Γ i j + \Gamma^{\pm\xi^+_{ij}}=\Gamma^+_{ij} Γ ± ξ ij + = Γ ij + Γ ± ξ i j − = Γ i j − \Gamma^{\pm\xi^-_{ij}}=\Gamma^-_{ij} Γ ± ξ ij − = Γ ij − k ∈ Λ k\in\Lambda k ∈ Λ y + R k y+\mathbb{R}k y + R k T 3 \mathbb{T}^3 T 3
Lemma 3.10 (Disjoint lines). There are points y k ∈ T 3 y_k\in\mathbb{T}^3 y k ∈ T 3 k ∈ Λ k\in\Lambda k ∈ Λ L k = y k + R k L_k=y_k+\mathbb{R}k L k = y k + R k s ∗ > 0 s_*>0 s ∗ > 0
Proof. Choose the points successively. Given finitely many closed lines L 1 , … , L m L_1,\ldots,L_m L 1 , … , L m y + R k y+\mathbb{R}k y + R k L i L_i L i y ∈ L i + R k y\in L_i+\mathbb{R}k y ∈ L i + R k y y y
The stationary profile
We construct a stationary weight which is large along the lines L k L_k L k L p L^p L p
For μ ≥ 1 \mu\ge1 μ ≥ 1 Ψ μ \Psi_\mu Ψ μ T 3 \mathbb{T}^3 T 3 k ∈ Λ k \in\Lambda k ∈ Λ n k , 1 , n k , 2 ∈ Z 3 n_{k,1}, n_{k,2} \in\mathbb{Z}^3 n k , 1 , n k , 2 ∈ Z 3 k k k
F k ( x ) = ∑ j = 1 2 sin 2 ( π n k , j ⋅ ( x − y k ) ) , F_k(x) = \sum_{j=1}^{2} \sin^2\bigl(\pi n_{k,j} \cdot(x-y_k)\bigr), F k ( x ) = j = 1 ∑ 2 sin 2 ( π n k , j ⋅ ( x − y k ) ) , Ψ μ ( x ) = 1 + ∑ k ∈ Λ ( μ − 2 + F k ( x ) ) − 1 + γ / ( 2 N ) . \Psi_\mu(x) = 1 + \sum_{k\in\Lambda}\bigl(\mu^{-2} + F_k(x)\bigr)^{-1+\gamma/(2N)}. Ψ μ ( x ) = 1 + k ∈ Λ ∑ ( μ − 2 + F k ( x ) ) − 1 + γ / ( 2 N ) . The zero set Z k Z_k Z k F k F_k F k k k k L k L_k L k z ↦ ( n k , 1 ⋅ z , n k , 2 ⋅ z ) z \mapsto(n_{k,1}\cdot z,n_{k,2}\cdot z) z ↦ ( n k , 1 ⋅ z , n k , 2 ⋅ z ) k ⊥ k^\perp k ⊥ R 2 \mathbb{R}^2 R 2 4 dist ( s , Z ) 2 ≤ sin 2 ( π s ) ≤ π 2 dist ( s , Z ) 2 4\operatorname{dist}(s,Z)^2 \le\sin^2(\pi s) \le\pi^2\operatorname{dist}(s,Z)^2 4 dist ( s , Z ) 2 ≤ sin 2 ( π s ) ≤ π 2 dist ( s , Z ) 2
c dist ( x , Z k ) 2 ≤ F k ( x ) ≤ C dist ( x , Z k ) 2 , ∣ ∇ F k ∣ ≤ C F k 1 / 2 . (19) c\operatorname{dist}(x,Z_k)^2 \le F_k(x) \le C\operatorname{dist}(x,Z_k)^2,\qquad\lvert\nabla F_k\rvert\le C F_k^{1/2}. \tag*{(19)} c dist ( x , Z k ) 2 ≤ F k ( x ) ≤ C dist ( x , Z k ) 2 , ∣ ∇ F k ∣ ≤ C F k 1/2 . (19) the last bound because F k F_k F k s 0 > 0 s_0>0 s 0 > 0 ⋃ k Z k \bigcup_k Z_k ⋃ k Z k
For the moving profiles, fix scalar functions ζ ∈ C c ∞ ( B ( 0 , 1 / 16 ) ; [ 0 , ∞ ) ) \zeta\in C^\infty_c(B(0,1/16);[0,\infty)) ζ ∈ C c ∞ ( B ( 0 , 1/16 ) ; [ 0 , ∞ )) χ ∗ ∈ C c ∞ ( B ( 0 , 1 / 8 ) ; [ 0 , 1 ] ) \chi_*\in C^\infty_c(B(0,1/8);[0,1]) χ ∗ ∈ C c ∞ ( B ( 0 , 1/8 ) ; [ 0 , 1 ]) R 3 \mathbb{R}^3 R 3 ∫ R 3 ζ = 1 \int_{\mathbb{R}^3}\zeta=1 ∫ R 3 ζ = 1 χ ∗ = 1 \chi_*=1 χ ∗ = 1 B ( 0 , 1 / 16 ) B(0,1/16) B ( 0 , 1/16 )
ζ μ ( x ) = ∑ j ∈ Z 3 μ 3 ζ ( μ ( x − j ) ) . \zeta_\mu(x)=\sum_{j\in\mathbb{Z}^3}\mu^3\zeta\bigl(\mu(x-j)\bigr). ζ μ ( x ) = j ∈ Z 3 ∑ μ 3 ζ ( μ ( x − j ) ) . Thus ζ μ \zeta_\mu ζ μ
Lemma 3.11 (Stationary profile). The profile Ψ μ \Psi_\mu Ψ μ j ≥ 0 j\ge0 j ≥ 0
∥ Ψ μ ∥ L p + ∥ Ψ μ ∥ L 1 ≤ C , ∥ ∇ j Ψ μ ∥ ∞ ≤ C j μ 2 + j . (20) \|\Psi_\mu\|_{L^p}+\|\Psi_\mu\|_{L^1}\le C,\qquad\|\nabla^j\Psi_\mu\|_\infty\le C_j\mu^{2+j}. \tag*{(20)} ∥ Ψ μ ∥ L p + ∥ Ψ μ ∥ L 1 ≤ C , ∥ ∇ j Ψ μ ∥ ∞ ≤ C j μ 2 + j . (20) Its ball averages are controlled by its value at the center:
\fint_{B(x,h)}\Psi_\mu(\lambda y)\,dy\le C\Psi_\mu(\lambda x). \tag*{(21)}
Consequently, for K 0 ( z ) = 1 + dist ( z , 0 ) − 2 K_0(z)=1+\operatorname{dist}(z,0)^{-2} K 0 ( z ) = 1 + dist ( z , 0 ) − 2
K 0 ∗ Ψ μ ( λ ⋅ ) ≤ C Ψ μ ( λ ⋅ ) . (22) K_0 * \Psi_\mu(\lambda\cdot)\le C\Psi_\mu(\lambda\cdot). \tag*{(22)} K 0 ∗ Ψ μ ( λ ⋅ ) ≤ C Ψ μ ( λ ⋅ ) . (22) For every k ∈ Λ k\in\Lambda k ∈ Λ
Ψ μ ( x ) ≥ c μ 2 − γ / N if dist ( x , L k ) ≤ 1 8 μ . (23) \Psi_\mu(x)\ge c\mu^{2-\gamma/N}\quad\text{if}\quad\operatorname{dist}(x,L_k)\le\frac{1}{8\mu}. \tag*{(23)} Ψ μ ( x ) ≥ c μ 2 − γ / N if dist ( x , L k ) ≤ 8 μ 1 . (23) Let f f f T 3 \mathbb{T}^3 T 3 L = R \mathcal{L}=\mathbb{R} L = R z ∈ L k z\in L_k z ∈ L k g = ζ μ ( ⋅ − z ) − 1 g=\zeta_\mu(\cdot-z)-1 g = ζ μ ( ⋅ − z ) − 1 L ∈ { R 0 , R } \mathcal{L}\in\{\mathcal{R}_0,\mathcal{R}\} L ∈ { R 0 , R }
∣ L [ f g ( λ ⋅ ) ] ( x ) ∣ ≤ C λ − 1 ∥ f ∥ C 1 μ γ / N Ψ μ ( λ x ) . (24) \lvert\mathcal{L}[fg(\lambda\cdot)](x)\rvert\le C\lambda^{-1}\|f\|_{C^1}\mu^{\gamma/N}\Psi_\mu(\lambda x). \tag*{(24)} ∣ L [ f g ( λ ⋅ )] ( x )∣ ≤ C λ − 1 ∥ f ∥ C 1 μ γ / N Ψ μ ( λ x ) . (24) If instead g=\Psi_\mu-\fint\Psi_\mu , then
∣ R 0 [ f g ( λ ⋅ ) ] ( x ) ∣ ≤ C λ − 1 ∥ f ∥ C 1 Ψ μ ( λ x ) . (25) \lvert\mathcal{R}_0[fg(\lambda\cdot)](x)\rvert\le C\lambda^{-1}\|f\|_{C^1}\Psi_\mu(\lambda x). \tag*{(25)} ∣ R 0 [ f g ( λ ⋅ )] ( x )∣ ≤ C λ − 1 ∥ f ∥ C 1 Ψ μ ( λ x ) . (25) All constants are uniform in μ ≥ 1 \mu\ge1 μ ≥ 1 λ ≥ 1 \lambda\ge1 λ ≥ 1 x ∈ R 3 x\in\mathbb{R}^3 x ∈ R 3 h > 0 h>0 h > 0 γ , N , p \gamma,N,p γ , N , p
Proof. The estimates follow from the two-dimensional transverse geometry of the lines. Put
ψ k ( x ) = ( μ − 2 + dist ( x , Z k ) 2 ) − 1 + γ / ( 2 N ) . \psi_k(x)=\bigl(\mu^{-2}+\operatorname{dist}(x,Z_k)^2\bigr)^{-1+\gamma/(2N)}. ψ k ( x ) = ( μ − 2 + dist ( x , Z k ) 2 ) − 1 + γ / ( 2 N ) . By (19) , Ψ μ \Psi_\mu Ψ μ 1 + ∑ k ψ k 1+\sum_k\psi_k 1 + ∑ k ψ k ψ k \psi_k ψ k
Step 1: Integral bounds and ball averages.
Since p − 1 < γ / N 10 p-1<\gamma/N^{10} p − 1 < γ / N 10 ( 2 − γ / N ) q < 2 − γ / ( 2 N ) (2-\gamma/N)q<2-\gamma/(2N) ( 2 − γ / N ) q < 2 − γ / ( 2 N ) q = 1 , p q=1,p q = 1 , p Z k Z_k Z k
∫ ψ k q ≤ C ∫ 0 s 0 ( μ − 2 + t 2 ) − ( 1 − γ / ( 2 N ) ) q t d t + C ≤ C N γ . \int\psi_k^q \le C\int_0^{s_0}(\mu^{-2}+t^2)^{-(1-\gamma/(2N))q}t\,\mathrm{d}t+C\le\frac{CN}{\gamma}. ∫ ψ k q ≤ C ∫ 0 s 0 ( μ − 2 + t 2 ) − ( 1 − γ / ( 2 N )) q t d t + C ≤ γ C N . Away from these lines the profile is uniformly bounded. For its derivatives, the chain rule and ∣ ∇ F k ∣ ≤ C F k 1 / 2 |\nabla F_k|\le CF_k^{1/2} ∣∇ F k ∣ ≤ C F k 1/2
∣ ∇ j ( μ − 2 + F k ) − 1 + γ / ( 2 N ) ∣ ≤ C j ( μ − 2 + F k ) − 1 + γ / ( 2 N ) − j / 2 ≤ C j μ 2 + j . \left|\nabla^j(\mu^{-2}+F_k)^{-1+\gamma/(2N)}\right|\le C_j(\mu^{-2}+F_k)^{-1+\gamma/(2N)-j/2}\le C_j\mu^{2+j}. ∇ j ( μ − 2 + F k ) − 1 + γ / ( 2 N ) ≤ C j ( μ − 2 + F k ) − 1 + γ / ( 2 N ) − j /2 ≤ C j μ 2 + j . Indeed, first derivatives of F k F_k F k F k 1 / 2 F_k^{1/2} F k 1/2
To prove (3.12), a change of variables reduces the claim to λ = 1 \lambda=1 λ = 1 D = dist ( x , Z k ) D=\operatorname{dist}(x,Z_k) D = dist ( x , Z k ) h ≤ D / 2 h\le D/2 h ≤ D /2 Z k Z_k Z k D / 2 D/2 D /2 B ( x , h ) B(x,h) B ( x , h ) ψ k ( y ) ≤ C ψ k ( x ) \psi_k(y)\le C\psi_k(x) ψ k ( y ) ≤ C ψ k ( x ) D / 2 < h ≤ s 0 / 6 D/2<h\le s_0/6 D /2 < h ≤ s 0 /6 3 h 3h 3 h Z k Z_k Z k
∫ B ( x , h ) ψ k ≤ C h − 2 ∫ 0 3 h ( μ − 2 + t 2 ) − 1 + γ / ( 2 N ) t d t ≤ C ( μ − 2 + h 2 ) − 1 + γ / ( 2 N ) ≤ C ψ k ( x ) . \begin{aligned}
\int_{B(x,h)}\psi_k\le Ch^{-2}\int_0^{3h}(\mu^{-2}+t^2)^{-1+\gamma/(2N)}t\,\mathrm{d}t \\
&\le C(\mu^{-2}+h^2)^{-1+\gamma/(2N)}\le C\psi_k(x).
\end{aligned} ∫ B ( x , h ) ψ k ≤ C h − 2 ∫ 0 3 h ( μ − 2 + t 2 ) − 1 + γ / ( 2 N ) t d t ≤ C ( μ − 2 + h 2 ) − 1 + γ / ( 2 N ) ≤ C ψ k ( x ) . The second inequality follows by considering h ≤ μ − 1 h\le\mu^{-1} h ≤ μ − 1 h ≥ μ − 1 h\ge\mu^{-1} h ≥ μ − 1 2 − γ / N < 2 2-\gamma/N<2 2 − γ / N < 2 h > s 0 / 6 h>s_0/6 h > s 0 /6 B ( x , h ) B(x,h) B ( x , h ) C ( 1 + h ) 3 C(1+h)^3 C ( 1 + h ) 3 ψ k ( x ) \psi_k(x) ψ k ( x ) k k k
Step 2: Convolution and the lower bound on the tubes.
On the fundamental domain x + [ − 1 / 2 , 1 / 2 ) 3 x+[-1/2,1/2)^3 x + [ − 1/2 , 1/2 ) 3 K 0 K_0 K 0 C ∣ z ∣ − 2 C|z|^{-2} C ∣ z ∣ − 2 2 − i − 1 < ∣ z ∣ ≤ 2 − i 2^{-i-1}<|z|\le2^{-i} 2 − i − 1 < ∣ z ∣ ≤ 2 − i
C 2 2 i 2 − 3 i Ψ μ ( λ x ) . C2^{2i}2^{-3i}\Psi_\mu(\lambda x). C 2 2 i 2 − 3 i Ψ μ ( λ x ) . Summing over i ≥ 0 i\ge0 i ≥ 0 dist ( x , L k ) ≤ 1 / ( 8 μ ) \operatorname{dist}(x,L_k)\le1/(8\mu) dist ( x , L k ) ≤ 1/ ( 8 μ ) F k ( x ) ≤ C μ − 2 F_k(x)\le C\mu^{-2} F k ( x ) ≤ C μ − 2
Step 3: Pointwise antidivergence estimates.
Let g = ζ μ ( ⋅ − z ) − 1 g=\zeta_\mu(\cdot-z)-1 g = ζ μ ( ⋅ − z ) − 1 z ∈ L k z\in L_k z ∈ L k L \mathcal{L} L R 0 \mathcal{R}_0 R 0 C K 0 CK_0 C K 0 ζ μ ( ⋅ − z ) \zeta_\mu(\cdot-z) ζ μ ( ⋅ − z ) B ( z , 1 / ( 16 μ ) ) B(z,1/(16\mu)) B ( z , 1/ ( 16 μ )) C μ 2 C\mu^2 C μ 2 dist ( x , z ) ≤ 1 / ( 8 μ ) \operatorname{dist}(x,z)\le1/(8\mu) dist ( x , z ) ≤ 1/ ( 8 μ ) C ( 1 + dist ( x , z ) − 2 ) C(1+\operatorname{dist}(x,z)^{-2}) C ( 1 + dist ( x , z ) − 2 )
∣ L g ( x ) ∣ + ∣ R 0 g ( x ) ∣ ≤ C ( μ − 2 + dist ( x , z ) 2 ) − 1 ≤ C μ γ / N Ψ μ ( x ) , \begin{aligned}
|\mathcal{L}g(x)|+|\mathcal{R}_0g(x)|\le C(\mu^{-2}+\operatorname{dist}(x,z)^2)^{-1} \\
&\le C\mu^{\gamma/N}\Psi_\mu(x),
\end{aligned} ∣ L g ( x ) ∣ + ∣ R 0 g ( x ) ∣ ≤ C ( μ − 2 + dist ( x , z ) 2 ) − 1 ≤ C μ γ / N Ψ μ ( x ) , where the last inequality uses z ∈ L k ⊂ Z k z\in L_k\subset Z_k z ∈ L k ⊂ Z k
∣ L [ g ( λ ⋅ ) ] ( x ) ∣ ≤ C λ − 1 ∥ f ∥ ∞ μ γ / N Ψ μ ( λ x ) . |\mathcal{L}[g(\lambda\cdot)](x)|\le C\lambda^{-1}\|f\|_\infty\mu^{\gamma/N}\Psi_\mu(\lambda x). ∣ L [ g ( λ ⋅ )] ( x ) ∣ ≤ C λ − 1 ∥ f ∥ ∞ μ γ / N Ψ μ ( λ x ) . By (3.7), the commutator is bounded by
C λ − 1 ∥ f ∥ C 1 μ γ / N K 0 ∗ Ψ μ ( λ ⋅ ) . C\lambda^{-1}\|f\|_{C^1}\mu^{\gamma/N}K_0*\Psi_\mu(\lambda\cdot). C λ − 1 ∥ f ∥ C 1 μ γ / N K 0 ∗ Ψ μ ( λ ⋅ ) . Together with (3.13), this proves (3.15).
For g = Ψ μ − f Ψ μ g=\Psi_\mu-f\Psi_\mu g = Ψ μ − f Ψ μ
∣ R 0 g ∣ ≤ C K 0 ∗ Ψ μ ≤ C Ψ μ . |\mathcal{R}_0g|\le CK_0*\Psi_\mu\le C\Psi_\mu. ∣ R 0 g ∣ ≤ C K 0 ∗ Ψ μ ≤ C Ψ μ . Now (3.3), (3.7), and (3.13) give (3.16) in the same way.
One step of the iteration: the construction
We construct ( ρ + , u + ) = ( ρ + θ , u + w ) (\rho^+,u^+)=(\rho+\theta,u+w) ( ρ + , u + ) = ( ρ + θ , u + w ) θ d \theta_d θ d ∇ w \nabla w ∇ w
Setting and statement
Fix n ≥ 0 n \ge0 n ≥ 0 λ n \lambda_n λ n δ n \delta_n δ n
λ = λ n + 1 , μ = λ N , ω = δ n − 4 . \lambda= \lambda_{n+1}, \qquad\mu= \lambda^N, \qquad\omega= \delta_n^{-4}. λ = λ n + 1 , μ = λ N , ω = δ n − 4 . The integers λ \lambda λ μ \mu μ ω > 0 \omega> 0 ω > 0 I ′ I^\prime I ′ J ′ J^\prime J ′ I ′ ⋐ J ′ ⊂ J I^\prime\Subset J^\prime\subset J I ′ ⋐ J ′ ⊂ J τ = dist ( I ′ , ∂ J ′ ) \tau= \operatorname{dist}(I^\prime,\partial J^\prime) τ = dist ( I ′ , ∂ J ′ ) r 0 ≤ r − < r + ≤ r B r_0 \le r_- < r_+ \le r_B r 0 ≤ r − < r + ≤ r B d = ( r + − r − ) / ( 64 N + 40 ) d = (r_+ - r_-)/(64N + 40) d = ( r + − r − ) / ( 64 N + 40 ) Z = ( ρ , u , Π , R , T ) Z = (\rho,u,\Pi,R,T) Z = ( ρ , u , Π , R , T ) ( R , T ) (R,T) ( R , T ) I ′ × B ( x 0 , r − ) I^\prime\times B(x_0,r_-) I ′ × B ( x 0 , r − ) n = 0 n = 0 n = 0 Z = Z ∗ Z = Z_\ast Z = Z ∗
Proposition 4.1 (One step). Assume that
∥ ( ρ , u , R , T ) ∥ C t , x 20 ≤ λ n N 4 , ∥ u ∥ ∞ ≤ 2 δ n − 1 , τ − 1 + d − 1 ≤ λ n , \|(\rho,u,R,T)\|_{C^ {20}_{t,x}} \le\lambda_n^{N^4}, \qquad\|u\|_\infty\le2\delta_n^{-1}, \qquad\tau^{-1} + d^{-1} \le\lambda_n, ∥ ( ρ , u , R , T ) ∥ C t , x 20 ≤ λ n N 4 , ∥ u ∥ ∞ ≤ 2 δ n − 1 , τ − 1 + d − 1 ≤ λ n , ∥ ( R , T ) ∥ C t L p ≤ 2 δ n 8 if n ≥ 1. (26) \|(R,T)\|_{C_tL^p} \le2\delta_n^8 \quad\text{if } n \ge1. \tag*{(26)} ∥ ( R , T ) ∥ C t L p ≤ 2 δ n 8 if n ≥ 1. (26) There is a 0 a_0 a 0 r 0 r_0 r 0 r B r_B r B Z ∗ Z_\ast Z ∗ a > a 0 a > a_0 a > a 0 Z Z Z Z + = ( ρ + , u + , Π + , R + , T + ) Z^+ = (\rho^+,u^+,\Pi^+,R^+,T^+) Z + = ( ρ + , u + , Π + , R + , T + ) Z + − Z Z^+ - Z Z + − Z ( R + , T + ) (R^+,T^+) ( R + , T + )
K ′ × B ( x 0 , 1 2 ( r − + r + ) ) K^\prime\times B\left(x_0,\tfrac{1}{2}(r_- + r_+)\right) K ′ × B ( x 0 , 2 1 ( r − + r + ) ) for a compact interval K ′ ⊂ J ′ K^\prime\subset J^\prime K ′ ⊂ J ′
The new defects satisfy
∥ ( R + , T + ) ∥ C t L p ≤ δ n + 1 8 , \|(R^+,T^+)\|_{C_tL^p} \le\delta_{n+1}^8, ∥ ( R + , T + ) ∥ C t L p ≤ δ n + 1 8 , and the total density perturbation θ = ρ + − ρ \theta= \rho^+ - \rho θ = ρ + − ρ
θ ≥ − δ n 2 ∣ R ∣ − δ n 4 ∣ T ∣ − δ n 2 + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ . (27) \theta\ge-\delta_n^2|R|-\delta_n^4|T|-\delta_n^2
+\delta_{n+1}^2|R^+|+\delta_{n+1}^4|T^+|. \tag*{(27)} θ ≥ − δ n 2 ∣ R ∣ − δ n 4 ∣ T ∣ − δ n 2 + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣. (27) The density and velocity increments satisfy
∥ ρ + − ρ ∥ C t L p ≤ δ n 2 , \|\rho^+ - \rho\|_{C_tL^p} \le\delta_n^2, ∥ ρ + − ρ ∥ C t L p ≤ δ n 2 , ∥ u + − u ∥ C t H s ≤ δ n + 1 16 , ∥ u + ∥ ∞ ≤ δ n + 1 − 1 . \|u^+ - u\|_{C_tH^s} \le\delta_{n+1}^{16}, \qquad\|u^+\|_\infty\le\delta_{n+1}^{-1}. ∥ u + − u ∥ C t H s ≤ δ n + 1 16 , ∥ u + ∥ ∞ ≤ δ n + 1 − 1 . If n ≥ 1 n \ge1 n ≥ 1
∥ ρ + u + − ρ u ∥ C t L 1 + ∥ ρ + u + ⊗ u + − ρ u ⊗ u ∥ C t L 1 + ∥ Π + − Π ∥ L t , x 1 ≤ δ n 2 . \|\rho^+u^+ - \rho u\|_{C_tL^1}
+ \|\rho^+u^+ \otimes u^+ - \rho u \otimes u\|_{C_tL^1}
+ \|\Pi^+ - \Pi\|_{L^1_{t,x}}
\le\delta_n^2. ∥ ρ + u + − ρ u ∥ C t L 1 + ∥ ρ + u + ⊗ u + − ρ u ⊗ u ∥ C t L 1 + ∥ Π + − Π ∥ L t , x 1 ≤ δ n 2 . Finally,
∥ ( ρ + , u + , R + , T + ) ∥ C t , x 3 + ∥ Π + − Π ∥ C t , x 3 ≤ 1 2 λ N 2 . \|(\rho^+,u^+,R^+,T^+)\|_{C^3_{t,x}} + \|\Pi^+ - \Pi\|_{C^3_{t,x}} \le\tfrac{1}{2}\lambda^{N^2}. ∥ ( ρ + , u + , R + , T + ) ∥ C t , x 3 + ∥ Π + − Π ∥ C t , x 3 ≤ 2 1 λ N 2 . The positive terms in (4.2) allow for the density removed when correcting R + R^+ R + T + T^+ T + Z + Z^+ Z + C C C r 0 r_0 r 0 r B r_B r B C ∗ C_\ast C ∗ Z ∗ Z_\ast Z ∗ n n n a a a a a a n n n
Cutoffs for the principal perturbations
We first localize the coefficients and the velocity perturbation. For j ≥ 0 j \ge0 j ≥ 0
B j = B ( x 0 , r − + 2 j d ) . B_j = B(x_0,r_- + 2jd). B j = B ( x 0 , r − + 2 j d ) . All spatial cutoffs below take values in [ 0 , 1 ] [0,1] [ 0 , 1 ] ∥ ⋅ ∥ C l ≤ C l d − l \|\cdot\|_{C^l} \le C_l d^{-l} ∥ ⋅ ∥ C l ≤ C l d − l l ≥ 0 l \ge0 l ≥ 0
χ R ∈ C c ∞ ( B 1 ) , χ R = 1 on B ( x 0 , r − + d ) , \chi_R \in C^\infty_c(B_1), \qquad\chi_R = 1 \text{ on } B(x_0,r_- + d), χ R ∈ C c ∞ ( B 1 ) , χ R = 1 on B ( x 0 , r − + d ) , χ u ∈ C c ∞ ( B 3 ) , χ u = 1 on B 2 . (28) \chi_u \in C^\infty_c(B_3), \qquad\chi_u = 1 \text{ on } B_2. \tag*{(28)} χ u ∈ C c ∞ ( B 3 ) , χ u = 1 on B 2 . (28) Let η , η u \eta,\eta_u η , η u I ′ ⋐ J ′ I^\prime\Subset J^\prime I ′ ⋐ J ′ χ u = 1 \chi_u = 1 χ u = 1 supp χ R \operatorname{supp}\chi_R supp χ R η u = 1 \eta_u = 1 η u = 1 supp η \operatorname{supp}\eta supp η
Cancellation of the defects
We seek nonnegative scalar weights a k a_k a k [ 0 , 1 ] × T 3 [0,1]\times\mathbb{T}^3 [ 0 , 1 ] × T 3 R R R T T T V k = ω k V_k=\omega k V k = ω k k ∈ Λ k\in\Lambda k ∈ Λ Λ T \Lambda_T Λ T
To keep the weights nonnegative and smooth at zeros of the defects, define the nonnegative scalar function
α ( t , x ) = ( 4 δ n 16 χ R ( x ) 4 + ∣ R ( t , x ) ∣ 2 + ω − 2 ∣ T ( t , x ) ∣ 2 ) 1 / 2 . (29) \alpha(t,x)=\left(4\delta_n^{16}\chi_R(x)^4+|R(t,x)|^2+\omega^{-2}|T(t,x)|^2\right)^{1/2}. \tag*{(29)} α ( t , x ) = ( 4 δ n 16 χ R ( x ) 4 + ∣ R ( t , x ) ∣ 2 + ω − 2 ∣ T ( t , x ) ∣ 2 ) 1/2 . (29) Define the symmetric matrix field of trace three
S ( t , x ) = Id + 4 ω 2 ( u ⊗ u − ∣ u ∣ 2 3 Id ) − T − 1 3 Tr ( T ) Id 6 ω α . (30) S(t,x)=\operatorname{Id}+\frac{4}{\omega^2}\left(u\otimes u-\frac{|u|^2}{3}\operatorname{Id}\right)-\frac{T-\frac{1}{3}\operatorname{Tr}(T)\operatorname{Id}}{6\omega\alpha}. \tag*{(30)} S ( t , x ) = Id + ω 2 4 ( u ⊗ u − 3 ∣ u ∣ 2 Id ) − 6 ω α T − 3 1 Tr ( T ) Id . (30) The quotients R / α R/\alpha R / α T / α T/\alpha T / α R = T = 0 R=T=0 R = T = 0
a k ( t , x ) = η ( t ) 2 α ω [ 1 + ( 12 u ω − R 2 α ) ⋅ k ] , k ∈ Λ R , a_k(t,x)=\frac{\eta(t)^2\alpha}{\omega}\left[1+\left(\frac{12u}{\omega}-\frac{R}{2\alpha}\right)\cdot k\right],\qquad k\in\Lambda_R, a k ( t , x ) = ω η ( t ) 2 α [ 1 + ( ω 12 u − 2 α R ) ⋅ k ] , k ∈ Λ R , a k ( t , x ) = 3 η ( t ) 2 α ω Γ k ( S ) , k ∈ Λ T . (31) a_k(t,x)=\frac{3\eta(t)^2\alpha}{\omega}\Gamma_k(S),\qquad k\in\Lambda_T. \tag*{(31)} a k ( t , x ) = ω 3 η ( t ) 2 α Γ k ( S ) , k ∈ Λ T . (31) The scalar part of the second moment is absorbed into the pressure through the scalar function
π ( t , x ) = η ( t ) 2 [ 8 ω α ( 1 − ∣ u ∣ 2 ω 2 ) + 1 3 Tr T ] . (32) \pi(t,x)=\eta(t)^2\left[8\omega\alpha\left(1-\frac{|u|^2}{\omega^2}\right)+\frac{1}{3}\operatorname{Tr}T\right]. \tag*{(32)} π ( t , x ) = η ( t ) 2 [ 8 ω α ( 1 − ω 2 ∣ u ∣ 2 ) + 3 1 Tr T ] . (32) Here and below all direction indices range over Λ \Lambda Λ
Define the smooth scalar fields
f k ( t , x ) = ( ∂ t + V k ⋅ ∇ ) a k ( t , x ) . f_k(t,x)=(\partial_t+V_k\cdot\nabla)a_k(t,x). f k ( t , x ) = ( ∂ t + V k ⋅ ∇ ) a k ( t , x ) . They are supported in supp η × B 1 \operatorname{supp}\eta\times B_1 supp η × B 1
Proposition 4.2 (Coefficients). For sufficiently large a > 0 a>0 a > 0 a k a_k a k π \pi π supp η × B 1 \operatorname{supp}\eta\times B_1 supp η × B 1 a k ≥ 0 a_k\geq0 a k ≥ 0
∑ k a k ( V k − u ) = − R , \sum_k a_k(V_k-u)=-R, k ∑ a k ( V k − u ) = − R , ∑ k a k ( V k ⊗ V k − u ⊗ u ) = π Id − T . (33) \sum_k a_k(V_k\otimes V_k-u\otimes u)=\pi\operatorname{Id}-T. \tag*{(33)} k ∑ a k ( V k ⊗ V k − u ⊗ u ) = π Id − T . (33) and
∑ k a k = 24 η 2 α ω . (34) \sum_k a_k=\frac{24\eta^2\alpha}{\omega}. \tag*{(34)} k ∑ a k = ω 24 η 2 α . (34) Their pointwise sizes obey
∑ k a k ≤ 24 ω ( 2 δ n 8 + ∣ R ∣ + ∣ T ∣ ω ) , \sum_k a_k\leq\frac{24}{\omega}\left(2\delta_n^8+|R|+\frac{|T|}{\omega}\right), k ∑ a k ≤ ω 24 ( 2 δ n 8 + ∣ R ∣ + ω ∣ T ∣ ) , ∣ π ∣ ≤ C ( ω δ n 8 + ω ∣ R ∣ + ∣ T ∣ ) . (35) |\pi|\leq C(\omega\delta_n^8+\omega|R|+|T|). \tag*{(35)} ∣ π ∣ ≤ C ( ω δ n 8 + ω ∣ R ∣ + ∣ T ∣ ) . (35) For 0 ≤ j ≤ 10 0\leq j\leq10 0 ≤ j ≤ 10
∥ a k ∥ C t , x j + 1 ≤ δ n + 1 − 1 ω − 1 , ∥ π ∥ C t , x j + 1 ≤ δ n + 1 − 1 ω . (36) \|a_k\|_{C^{j+1}_{t,x}}\leq\delta_{n+1}^{-1}\omega^{-1},\qquad\|\pi\|_{C^{j+1}_{t,x}}\leq\delta_{n+1}^{-1}\omega. \tag*{(36)} ∥ a k ∥ C t , x j + 1 ≤ δ n + 1 − 1 ω − 1 , ∥ π ∥ C t , x j + 1 ≤ δ n + 1 − 1 ω . (36) The material derivatives f k f_k f k
∥ f k ∥ C j ≤ C ω ∥ a k ∥ C j + 1 ≤ C δ n + 1 − 1 , 0 ≤ j ≤ 9. (37) \|f_k\|_{C^j}\leq C\omega\|a_k\|_{C^{j+1}}\leq C\delta_{n+1}^{-1},\qquad0\leq j\leq9. \tag*{(37)} ∥ f k ∥ C j ≤ C ω ∥ a k ∥ C j + 1 ≤ C δ n + 1 − 1 , 0 ≤ j ≤ 9. (37) Proof. We verify the algebraic identities and positivity first, then estimate the derivatives of the coefficients.
Step 1: Positivity and smoothness of the weights.
The open sets
O 1 = B ( x 0 , r − + d ) , O 2 = T 3 ∖ B ( x 0 , r − ) ‾ O_1=B(x_0,r_-+d),\qquad O_2=\mathbb{T}^3\setminus\overline{B(x_0,r_-)} O 1 = B ( x 0 , r − + d ) , O 2 = T 3 ∖ B ( x 0 , r − ) cover the torus. By (4.3) and (4.4), on O 1 O_1 O 1 χ R \chi_R χ R α ≥ 2 δ n 8 \alpha\geq2\delta_n^8 α ≥ 2 δ n 8 O 2 O_2 O 2 α = 2 δ n 8 χ R \alpha=2\delta_n^8\chi_R α = 2 δ n 8 χ R α \alpha α R / α R/\alpha R / α T / ( ω α ) T/(\omega\alpha) T / ( ω α )
∣ R ∣ ≤ α , ∣ T ∣ ≤ ω α . |R|\leq\alpha,\qquad|T|\leq\omega\alpha. ∣ R ∣ ≤ α , ∣ T ∣ ≤ ω α . By (4.1) and ω = δ n − 4 \omega=\delta_n^{-4} ω = δ n − 4 a > 0 a>0 a > 0
12 ∣ u ∣ ω ≤ 24 δ n 3 ≤ 1 4 , 4 ∣ u ∣ 2 ω 2 ≤ 16 δ n 6 ≤ 10 − 3 . \frac{12|u|}{\omega}\le24\delta_n^3\le\frac{1}{4},\qquad\frac{4|u|^2}{\omega^2}\le16\delta_n^6\le10^{-3}. ω 12∣ u ∣ ≤ 24 δ n 3 ≤ 4 1 , ω 2 4∣ u ∣ 2 ≤ 16 δ n 6 ≤ 1 0 − 3 . Consequently,
∣ 12 u ω − R 2 α ∣ ≤ 3 4 , ∣ S − Id ∣ ≤ 10 − 3 + 1 6 < 0.17. \left|\frac{12u}{\omega}-\frac{R}{2\alpha}\right|\le\frac{3}{4},\qquad|S-\operatorname{Id}|\le10^{-3}+\frac{1}{6}<0.17. ω 12 u − 2 α R ≤ 4 3 , ∣ S − Id ∣ ≤ 1 0 − 3 + 6 1 < 0.17. The coordinate weights are therefore nonnegative, as are the geometric weights by Lemma 3.9. Their spatial support is contained in B 1 B_1 B 1 χ R \chi_R χ R η 2 \eta^2 η 2
Step 2: The continuity defect.
For i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3
a e i + a − e i = 2 η 2 α ω a_{e_i}+a_{-e_i}=\frac{2\eta^2\alpha}{\omega} a e i + a − e i = ω 2 η 2 α and
ω ( a e i − a − e i ) = η 2 ( 24 α ω u − R ) ⋅ e i . \omega(a_{e_i}-a_{-e_i})=\eta^2\left(\frac{24\alpha}{\omega}u-R\right)\cdot e_i. ω ( a e i − a − e i ) = η 2 ( ω 24 α u − R ) ⋅ e i . For k ∈ Λ T k\in\Lambda_T k ∈ Λ T Tr S = 3 \operatorname{Tr}S=3 Tr S = 3
∑ k ∈ Λ T a k = 6 η 2 α ω Tr S = 18 η 2 α ω . \sum_{k\in\Lambda_T}a_k=\frac{6\eta^2\alpha}{\omega}\operatorname{Tr}S=\frac{18\eta^2\alpha}{\omega}. k ∈ Λ T ∑ a k = ω 6 η 2 α Tr S = ω 18 η 2 α . Adding the two groups yields
∑ k a k = 24 η 2 α ω , ∑ k a k V k = ( ∑ k a k ) u − η 2 R . \sum_k a_k=\frac{24\eta^2\alpha}{\omega},\qquad\sum_k a_kV_k=\left(\sum_k a_k\right)u-\eta^2R. k ∑ a k = ω 24 η 2 α , k ∑ a k V k = ( k ∑ a k ) u − η 2 R . As η = 1 \eta=1 η = 1 R ≠ 0 R\ne0 R = 0
Step 3: The stress.
By (4.6) and Lemma 3.9, the two groups contribute
∑ k ∈ Λ R a k V k ⊗ V k = 2 η 2 ω α Id , ∑ k ∈ Λ T a k V k ⊗ V k = 6 η 2 ω α S . \sum_{k\in\Lambda_R}a_kV_k\otimes V_k=2\eta^2\omega\alpha\operatorname{Id},\qquad\sum_{k\in\Lambda_T}a_kV_k\otimes V_k=6\eta^2\omega\alpha S. k ∈ Λ R ∑ a k V k ⊗ V k = 2 η 2 ω α Id , k ∈ Λ T ∑ a k V k ⊗ V k = 6 η 2 ω α S . Insert (4.5) and use η 2 T = T \eta^2T=T η 2 T = T
∑ k a k V k ⊗ V k = 8 η 2 ω α Id + 24 η 2 α ω ( u ⊗ u − ∣ u ∣ 2 3 Id ) − ( T − Tr T 3 Id ) = ( ∑ k a k ) u ⊗ u + π Id − T . \begin{aligned}
\sum_k a_kV_k\otimes V_k
=8\eta^2\omega\alpha\operatorname{Id}
+\frac{24\eta^2\alpha}{\omega}\left(u\otimes u-\frac{|u|^2}{3}\operatorname{Id}\right)
-\left(T-\frac{\operatorname{Tr}T}{3}\operatorname{Id}\right) \\
&=\left(\sum_k a_k\right)u\otimes u+\pi\operatorname{Id}-T.
\end{aligned} k ∑ a k V k ⊗ V k = 8 η 2 ω α Id + ω 24 η 2 α ( u ⊗ u − 3 ∣ u ∣ 2 Id ) − ( T − 3 Tr T Id ) = ( k ∑ a k ) u ⊗ u + π Id − T . The last equality uses (4.9) and (4.7), and proves the second identity in (4.8). Finally, (4.4) gives
α ≤ 2 δ n 8 χ R 2 + ∣ R ∣ + ∣ T ∣ ω , \alpha\le2\delta_n^8\chi_R^2+|R|+\frac{|T|}{\omega}, α ≤ 2 δ n 8 χ R 2 + ∣ R ∣ + ω ∣ T ∣ , which gives (4.10), using ∣ u ∣ ≤ ω |u|\le\omega ∣ u ∣ ≤ ω
Step 4: Derivative estimates.
All norms in this step are C t , x 1 C^1_{t,x} C t , x 1 δ n − 8 \delta_n^{-8} δ n − 8 λ n N 4 \lambda_n^{N^4} λ n N 4
∥ η 2 ∥ + ∥ χ R i ∥ ≤ C λ n 10 N 4 ( i = 2 , 4 ) . \|\eta^2\|+\|\chi_R^i\|\le C\lambda_n^{10N^4}\qquad(i=2,4). ∥ η 2 ∥ + ∥ χ R i ∥ ≤ C λ n 10 N 4 ( i = 2 , 4 ) . On O 1 O_1 O 1 ( R , T / ω ) / ( 2 δ n 8 ) (R,T/\omega)/(2\delta_n^8) ( R , T / ω ) / ( 2 δ n 8 ) C λ n 2 N 4 C\lambda_n^{2N^4} C λ n 2 N 4 z ↦ z / ( 1 + ∣ z ∣ 2 ) 1 / 2 z\mapsto z/(1+|z|^2)^{1/2} z ↦ z / ( 1 + ∣ z ∣ 2 ) 1/2 z ↦ ( 1 + ∣ z ∣ 2 ) 1 / 2 z\mapsto(1+|z|^2)^{1/2} z ↦ ( 1 + ∣ z ∣ 2 ) 1/2
∥ α ∥ + ∥ R / α ∥ + ∥ T / ( ω α ) ∥ ≤ C λ n 30 N 4 . \|\alpha\|+\|R/\alpha\|+\|T/(\omega\alpha)\|\le C\lambda_n^{30N^4}. ∥ α ∥ + ∥ R / α ∥ + ∥ T / ( ω α ) ∥ ≤ C λ n 30 N 4 . On O 2 O_2 O 2 α = 2 δ 8 χ R 2 \alpha= 2\delta^8\chi_R^2 α = 2 δ 8 χ R 2 R = T = 0 R = T = 0 R = T = 0 ∥ S ∥ + ∥ Γ k ( S ) ∥ ≤ C λ n 30 N 4 \|S\| + \|\Gamma_k(S)\| \le C\lambda_n^{30}N^4 ∥ S ∥ + ∥ Γ k ( S ) ∥ ≤ C λ n 30 N 4
ω ∥ a k ∥ + ω − 1 ∥ π ∥ ≤ C λ n 100 N 4 ≤ λ γ / 2 ≤ δ n + 1 − 1 . \omega\|a_k\| + \omega^{-1}\|\pi\| \le C\lambda_n^{100}N^4 \le\lambda^{\gamma/2} \le\delta_{n+1}^{-1}. ω ∥ a k ∥ + ω − 1 ∥ π ∥ ≤ C λ n 100 N 4 ≤ λ γ /2 ≤ δ n + 1 − 1 . Here λ = λ n b \lambda= \lambda_n^b λ = λ n b 100 N 4 < γ b / 2 100N^4 < \gamma b/2 100 N 4 < γ b /2 a > 0 a > 0 a > 0 V k V_k V k
Moving profiles and the velocity perturbation
The velocity perturbation must equal V k − u V_k-u V k − u k k k V k − u V_k-u V k − u
Let y k y_k y k L k L_k L k s ∗ s_* s ∗ ζ \zeta ζ χ ∗ \chi_* χ ∗ ζ μ \zeta_\mu ζ μ
Θ k ( t , x ) = ζ μ ( λ x − λ ω t k − y k ) , V k = ω k . \Theta_k(t,x)=\zeta_\mu(\lambda x-\lambda\omega t k-y_k), \qquad V_k=\omega k. Θ k ( t , x ) = ζ μ ( λ x − λω t k − y k ) , V k = ω k . Choose representatives y k ∈ R 3 y_k\in\mathbb{R}^3 y k ∈ R 3 j ∈ Z 3 j\in\mathbb{Z}^3 j ∈ Z 3 X k , j ( t ) = λ − 1 ( j + y k ) + ω t k X_{k,j}(t)=\lambda^{-1}(j+y_k)+\omega tk X k , j ( t ) = λ − 1 ( j + y k ) + ω t k λ X k , j ( t ) ∈ L k \lambda X_{k,j}(t)\in L_k λ X k , j ( t ) ∈ L k t t t L k L_k L k k k k u u u
W k ( t , x ) = η u ( t ) ∑ j ∈ Z 3 curl x [ χ u ( x ) χ ∗ ( λ μ ( x − X k , j ) ) A k , j ( t , x ) ] , (38) W_k(t,x)=\eta_u(t)\sum_{j\in\mathbb{Z}^3}\operatorname{curl}_x\left[\chi_u(x)\chi_*\bigl(\lambda\mu(x-X_{k,j})\bigr)A_{k,j}(t,x)\right], \tag*{(38)} W k ( t , x ) = η u ( t ) j ∈ Z 3 ∑ curl x [ χ u ( x ) χ ∗ ( λ μ ( x − X k , j ) ) A k , j ( t , x ) ] , (38) A k , j ( t , x ) = ∫ 0 1 s v k ( t , X k , j + s ( x − X k , j ) ) × ( x − X k , j ) d s , v k ( t , x ) = ω k − u ( t , x ) . A_{k,j}(t,x)=\int_0^1 s\,v_k\bigl(t,X_{k,j}+s(x-X_{k,j})\bigr)\times(x-X_{k,j})\,ds,\qquad v_k(t,x)=\omega k-u(t,x). A k , j ( t , x ) = ∫ 0 1 s v k ( t , X k , j + s ( x − X k , j ) ) × ( x − X k , j ) d s , v k ( t , x ) = ω k − u ( t , x ) . The vector field v k v_k v k A k , j A_{k,j} A k , j w ( t , x ) = ∑ k W k ( t , x ) w(t,x)=\sum_k W_k(t,x) w ( t , x ) = ∑ k W k ( t , x ) u + = u + w u^+=u+w u + = u + w
Lemma 4.1 (Moving profiles and velocity perturbation). The profiles Θ k \Theta_k Θ k
Θ k ≥ 0 , ∫ Θ k = 1 , ∥ Θ k ∥ L q ≤ C μ 3 − 3 / q ( 1 ≤ q ≤ ∞ ) . (39) \Theta_k\ge0,\qquad\int\Theta_k=1,\qquad\|\Theta_k\|_{L^q}\le C\mu^{3-3/q}\qquad(1\le q\le\infty). \tag*{(39)} Θ k ≥ 0 , ∫ Θ k = 1 , ∥ Θ k ∥ L q ≤ C μ 3 − 3/ q ( 1 ≤ q ≤ ∞ ) . (39) ∂ t Θ k + V k ⋅ ∇ Θ k = 0. \partial_t\Theta_k+V_k\cdot\nabla\Theta_k=0. ∂ t Θ k + V k ⋅ ∇ Θ k = 0. For sufficiently large a > 0 a>0 a > 0 W k W_k W k
⋃ j B ( X k , j ( t ) , ( 8 λ μ ) − 1 ) ⊂ C k , \bigcup_j B\bigl(X_{k,j}(t),(8\lambda\mu)^{-1}\bigr)\subset C_k, j ⋃ B ( X k , j ( t ) , ( 8 λ μ ) − 1 ) ⊂ C k , where the time-independent tubes are
C k = { x ∈ T 3 : dist ( λ x , L k ) ≤ 1 / ( 8 μ ) } . C_k=\{x\in\mathbb{T}^3:\operatorname{dist}(\lambda x,L_k)\le1/(8\mu)\}. C k = { x ∈ T 3 : dist ( λ x , L k ) ≤ 1/ ( 8 μ )} . The tubes C k C_k C k
∣ supp W k ( t ) ∣ ≤ C μ − 3 . (40) |\operatorname{supp}W_k(t)|\le C\mu^{-3}. \tag*{(40)} ∣ supp W k ( t ) ∣ ≤ C μ − 3 . (40) On the density profiles the total velocity is exact:
a k Θ k ( u + − V k ) = 0. (41) a_k\Theta_k(u^+-V_k)=0. \tag*{(41)} a k Θ k ( u + − V k ) = 0. (41) For 0 ≤ j ≤ 10 0\le j\le10 0 ≤ j ≤ 10
∣ ∇ j W k ∣ ≤ C ω ( λ μ ) j 1 supp W k , |\nabla^jW_k|\le C\omega(\lambda\mu)^j\mathbf{1}_{\operatorname{supp}W_k}, ∣ ∇ j W k ∣ ≤ C ω ( λ μ ) j 1 supp W k , ∥ W k ∥ C t , x j ≤ C ω ( λ μ ω ) j . (42) \|W_k\|_{C^j_{t,x}}\le C\omega(\lambda\mu\omega)^j. \tag*{(42)} ∥ W k ∥ C t , x j ≤ C ω ( λ μ ω ) j . (42) Finally, the full perturbation satisfies
∣ w ∣ ≤ C ω , ∥ w ∥ C t H s ≤ δ n + 1 16 . |w|\le C\omega,\qquad\|w\|_{C_tH^s}\le\delta_{n+1}^{16}. ∣ w ∣ ≤ C ω , ∥ w ∥ C t H s ≤ δ n + 1 16 . Proof. The main point is the identity (4.16). The curl construction gives zero divergence automatically; we show that it agrees with V k − u V_k-u V k − u a k Θ k a_k\Theta_k a k Θ k Step 1: Support and incompressibility.
The properties of Θ k \Theta_k Θ k (38) , because supp χ ∗ ⊂ B 1 / 8 \operatorname{supp}\chi_* \subset B_{1/8} supp χ ∗ ⊂ B 1/8 λ X k , j ∈ L k \lambda X_{k,j} \in L_k λ X k , j ∈ L k 1 / ( 4 μ ) < s ∗ 1/(4\mu)<s_* 1/ ( 4 μ ) < s ∗ X k , j ( t ) X_{k,j}(t) X k , j ( t ) j ∈ { 0 , … , λ 3 − 1 } j\in\{0,\ldots,\lambda^3-1\} j ∈ { 0 , … , λ 3 − 1 } λ 3 \lambda^3 λ 3 ( 8 λ μ ) − 1 (8\lambda\mu)^{-1} ( 8 λ μ ) − 1 T 3 \mathbb{T}^3 T 3 (40) . Each W k W_k W k
Step 2: Exact transport on each profile.
For a divergence-free field v v v X X X curl ( a × b ) = a div b − b div a + ( b ⋅ ∇ ) a − ( a ⋅ ∇ ) b \operatorname{curl}(a\times b)=a\operatorname{div}b-b\operatorname{div}a+(b\cdot\nabla)a-(a\cdot\nabla)b curl ( a × b ) = a div b − b div a + ( b ⋅ ∇ ) a − ( a ⋅ ∇ ) b
curl x [ s v ( X + s ( x − X ) ) × ( x − X ) ] = 2 s v ( X + s ( x − X ) ) + s 2 ( ( x − X ) ⋅ ∇ ) v ( X + s ( x − X ) ) = d d s [ s 2 v ( X + s ( x − X ) ) ] . \begin{aligned}
\operatorname{curl}_x\left[s\,v(X+s(x-X))\times(x-X)\right]
=2s\,v(X+s(x-X))+s^2((x-X)\cdot\nabla)v(X+s(x-X)) \\
&=\frac{\mathrm{d}}{\mathrm{d}s}\left[s^2v(X+s(x-X))\right].
\end{aligned} curl x [ s v ( X + s ( x − X )) × ( x − X ) ] = 2 s v ( X + s ( x − X )) + s 2 (( x − X ) ⋅ ∇ ) v ( X + s ( x − X )) = d s d [ s 2 v ( X + s ( x − X )) ] . Integrating in s s s curl x A k , j = v k \operatorname{curl}_x A_{k,j}=v_k curl x A k , j = v k supp Θ k \operatorname{supp}\Theta_k supp Θ k x x x B ( X k , j , ( 16 λ μ ) − 1 ) B(X_{k,j},(16\lambda\mu)^{-1}) B ( X k , j , ( 16 λ μ ) − 1 ) χ ∗ ( λ μ ( x − X k , j ) ) = 1 \chi_*(\lambda\mu(x-X_{k,j}))=1 χ ∗ ( λ μ ( x − X k , j )) = 1 supp a k ⊂ supp η × B 1 \operatorname{supp}a_k\subset\operatorname{supp}\eta\times B_1 supp a k ⊂ supp η × B 1 (28) and Lemma 3.1 give χ u = η u = 1 \chi_u=\eta_u=1 χ u = η u = 1 supp a k \operatorname{supp}a_k supp a k W k = v k = ω k − u W_k=v_k=\omega_k-u W k = v k = ω k − u supp ( a k Θ k ) \operatorname{supp}(a_k\Theta_k) supp ( a k Θ k ) W k ′ = 0 W_{k'}=0 W k ′ = 0 k ′ ≠ k k'\ne k k ′ = k u + = V k u^+=V_k u + = V k
Step 3: Derivative estimates.
Fix X = X k , j ( t ) X=X_{k,j}(t) X = X k , j ( t ) ∣ x − X ∣ ≤ ( 8 λ μ ) − 1 |x-X|\le(8\lambda\mu)^{-1} ∣ x − X ∣ ≤ ( 8 λ μ ) − 1 (26) , ∥ u ∥ C x h ≤ λ N 4 ≤ ω ( λ μ ) h \|u\|_{C^h_x}\le\lambda^{N^4}\le\omega(\lambda\mu)^h ∥ u ∥ C x h ≤ λ N 4 ≤ ω ( λ μ ) h 1 ≤ h ≤ 20 1\le h\le20 1 ≤ h ≤ 20 ∣ v k ∣ ≤ 2 ω |v_k|\le2\omega ∣ v k ∣ ≤ 2 ω ∥ ∇ h v k ∥ ∞ ≤ 2 ω ( λ μ ) h \|\nabla^h v_k\|_\infty\le2\omega(\lambda\mu)^h ∥ ∇ h v k ∥ ∞ ≤ 2 ω ( λ μ ) h A k , j A_{k,j} A k , j
∣ ∇ h A k , j ∣ ≤ C ( ∣ x − X ∣ ∥ ∇ h v k ∥ ∞ + ∥ ∇ h − 1 v k ∥ ∞ ) ≤ C ω ( λ μ ) h − 1 . \begin{aligned}
|\nabla^h A_{k,j}|\le C\left(|x-X|\|\nabla^h v_k\|_\infty+\|\nabla^{h-1}v_k\|_\infty\right) \\
&\le C\omega(\lambda\mu)^{h-1}.
\end{aligned} ∣ ∇ h A k , j ∣ ≤ C ( ∣ x − X ∣∥ ∇ h v k ∥ ∞ + ∥ ∇ h − 1 v k ∥ ∞ ) ≤ C ω ( λ μ ) h − 1 . For h = 0 h=0 h = 0 χ ∗ , j = χ ∗ ( λ μ ( x − X ) ) \chi_{*,j}=\chi_*(\lambda\mu(x-X)) χ ∗ , j = χ ∗ ( λ μ ( x − X ))
curl x [ χ u χ ∗ , j A k , j ] = χ u χ ∗ , j v k + χ ∗ , j ∇ χ u × A k , j + χ u λ μ ( ∇ χ ∗ ) ( λ μ ( x − X ) ) × A k , j , \begin{aligned}
\operatorname{curl}_x[\chi_u\chi_{*,j}A_{k,j}]
=\chi_u\chi_{*,j}v_k+\chi_{*,j}\nabla\chi_u\times A_{k,j} \\
&+\chi_u\lambda\mu(\nabla\chi_*)(\lambda\mu(x-X))\times A_{k,j},
\end{aligned} curl x [ χ u χ ∗ , j A k , j ] = χ u χ ∗ , j v k + χ ∗ , j ∇ χ u × A k , j + χ u λ μ ( ∇ χ ∗ ) ( λ μ ( x − X )) × A k , j , and ∥ χ u ∥ C i ≤ C d − i ≤ C ( λ μ ) i \|\chi_u\|_{C^i}\le Cd^{-i}\le C(\lambda\mu)^i ∥ χ u ∥ C i ≤ C d − i ≤ C ( λ μ ) i (42) , since the balls for different j j j X k , j ′ = ω k X'_{k,j}=\omega_k X k , j ′ = ω k χ ∗ , j \chi_{*,j} χ ∗ , j x − X x-X x − X C λ μ ω C\lambda\mu\omega C λ μ ω v k v_k v k u u u X X X C ω λ μ C\omega\lambda\mu C ω λ μ η u \eta_u η u C τ − 1 ≤ C λ n C\tau^{-1}\le C\lambda^n C τ − 1 ≤ C λ n
Step 4: The Sobolev estimate.
The spatial bound in (42) gives ∣ W k ∣ ≤ C ω |W_k|\le C\omega ∣ W k ∣ ≤ C ω (40) , we obtain
∥ W k ∥ L 2 ≤ C ω μ − 3 / 2 , ∥ ∇ 2 W k ∥ L 2 ≤ C ω ( λ μ ) 2 μ − 3 / 2 . \|W_k\|_{L^2}\le C\omega\mu^{-3/2},\qquad
\|\nabla^2W_k\|_{L^2}\le C\omega(\lambda\mu)^2\mu^{-3/2}. ∥ W k ∥ L 2 ≤ C ω μ − 3/2 , ∥ ∇ 2 W k ∥ L 2 ≤ C ω ( λ μ ) 2 μ − 3/2 . Interpolation gives
∥ w ∥ H s ≤ C ω μ − 3 / 2 ( λ μ ) s = C λ 4 γ / b + s + N ( s − 3 / 2 ) ≤ C λ − 1 ≤ δ n + 1 16 \|w\|_{H^s}\le C\omega\mu^{-3/2}(\lambda\mu)^s
=C\lambda^{4\gamma/b+s+N(s-3/2)}
\le C\lambda^{-1}\le\delta_{n+1}^{16} ∥ w ∥ H s ≤ C ω μ − 3/2 ( λ μ ) s = C λ 4 γ / b + s + N ( s − 3/2 ) ≤ C λ − 1 ≤ δ n + 1 16 for sufficiently large a > 0 a>0 a > 0 N ( 3 / 2 − s ) > s + 2 N(3/2-s)>s+2 N ( 3/2 − s ) > s + 2 16 γ < 1 16\gamma<1 16 γ < 1
The density perturbation
The principal density increment is
θ p ( t , x ) = ∑ k a k ( t , x ) ( Θ k ( t , x ) − 1 ) . (43) \theta_p(t,x)=\sum_k a_k(t,x)(\Theta_k(t,x)-1). \tag*{(43)} θ p ( t , x ) = k ∑ a k ( t , x ) ( Θ k ( t , x ) − 1 ) . (43) Since a k , Θ k ≥ 0 a_k,\Theta_k\ge0 a k , Θ k ≥ 0 ∑ k a k \sum_k a_k ∑ k a k
The stationary density correction
Applying Proposition 4.2 at the following stage shows that the density removed by its principal perturbation is bounded by
C ( δ n + 1 4 ∣ R + ∣ + δ n + 1 8 ∣ T + ∣ + δ n + 1 12 ) . C(\delta_{n+1}^{4}|R^+|+\delta_{n+1}^{8}|T^+|+\delta_{n+1}^{12}). C ( δ n + 1 4 ∣ R + ∣ + δ n + 1 8 ∣ T + ∣ + δ n + 1 12 ) . For sufficiently large a > 0 a>0 a > 0 δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ \delta_{n+1}^{2}|R^+|+\delta_{n+1}^{4}|T^+| δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ − ( ∇ w + ∇ w T ) -(\nabla w+\nabla w^{\mathsf T}) − ( ∇ w + ∇ w T ) λ μ \lambda\mu λ μ
The stationary profile of Lemma 3.11 covers the whole periodic line traversed by each moving bump. Its lower bound on the tubes and Lemma 4.1 give
∣ ∇ w ∣ ≤ C λ μ ω μ − 2 + γ / N Ψ μ ( λ x ) = C λ 1 − N + γ ω Ψ μ ( λ x ) . |\nabla w| \le C\lambda\mu\omega\,\mu^{-2+\gamma/N}\Psi_\mu(\lambda x)=C\lambda^{1-N+\gamma}\omega\Psi_\mu(\lambda x). ∣∇ w ∣ ≤ C λ μ ω μ − 2 + γ / N Ψ μ ( λ x ) = C λ 1 − N + γ ω Ψ μ ( λ x ) . Thus the profile can supply enough density on the cutoff regions at small L p L^p L p Lemma 3.11 .
To see how adding density helps, first disregard spatial localization and work on a time interval where the cutoffs equal one. Consider
θ d = δ n + 1 16 ( Ψ μ ( λ x ) − ∫ Ψ μ ) . \theta_d=\delta_{n+1}^{16}\left(\Psi_\mu(\lambda x)-\int\Psi_\mu\right). θ d = δ n + 1 16 ( Ψ μ ( λ x ) − ∫ Ψ μ ) . This function has zero mass, ∂ t θ d = 0 \partial_t\theta_d=0 ∂ t θ d = 0 ∥ θ d ∥ L p ≤ C δ n + 1 16 \|\theta_d\|_{L^p}\le C\delta_{n+1}^{16} ∥ θ d ∥ L p ≤ C δ n + 1 16 u + u^+ u + θ d \theta_d θ d θ d u + \theta_d u^+ θ d u + θ d u + ⊗ u + \theta_d u^+\otimes u^+ θ d u + ⊗ u + ∣ u + ∣ ≤ C ω |u^+|\le C\omega ∣ u + ∣ ≤ C ω
δ n + 1 2 ∣ θ d u + ∣ + δ n + 1 4 ∣ θ d u + ⊗ u + ∣ ≤ C δ n + 1 16 ( δ n + 1 2 ω + δ n + 1 4 ω 2 ) Ψ μ ( λ x ) ≤ 1 4 δ n + 1 16 Ψ μ ( λ x ) . \delta_{n+1}^2|\theta_d u^+|+\delta_{n+1}^4|\theta_d u^+\otimes u^+|
\le C\delta_{n+1}^{16}(\delta_{n+1}^2\omega+\delta_{n+1}^4\omega^2)\Psi_\mu(\lambda x)\le\frac14\delta_{n+1}^{16}\Psi_\mu(\lambda x). δ n + 1 2 ∣ θ d u + ∣ + δ n + 1 4 ∣ θ d u + ⊗ u + ∣ ≤ C δ n + 1 16 ( δ n + 1 2 ω + δ n + 1 4 ω 2 ) Ψ μ ( λ x ) ≤ 4 1 δ n + 1 16 Ψ μ ( λ x ) . Here δ n + 1 2 ω = λ − 2 γ + 4 γ / b → 0 \delta_{n+1}^2\omega=\lambda^{-2\gamma+4\gamma/b}\to0 δ n + 1 2 ω = λ − 2 γ + 4 γ / b → 0 θ d \theta_d θ d
We now localize this correction. Choose
χ m ∈ C c ∞ ( B 5 + 2 m ) , χ m = 1 on B 4 + 2 m , 0 ≤ m ≤ 8 N . \chi_m\in C^\infty_c(B_{5+2m}),\qquad\chi_m=1\ \text{on }B_{4+2m},\qquad0\le m\le8N. χ m ∈ C c ∞ ( B 5 + 2 m ) , χ m = 1 on B 4 + 2 m , 0 ≤ m ≤ 8 N . Thus χ m + 1 = 1 \chi_{m+1}=1 χ m + 1 = 1 d d d supp χ m \operatorname{supp}\chi_m supp χ m η d \eta_d η d Lemma 3.1 ; it is one near supp η u \operatorname{supp}\eta_u supp η u
c m = ∫ χ m ( x ) Ψ μ ( λ x ) d x ∫ χ m ( x ) d x , (44) c_m=\frac{\int\chi_m(x)\Psi_\mu(\lambda x)\,dx}{\int\chi_m(x)\,dx}, \tag*{(44)} c m = ∫ χ m ( x ) d x ∫ χ m ( x ) Ψ μ ( λ x ) d x , (44) and set
θ d ( t , x ) = δ n + 1 16 η d ( t ) ∑ m = 0 8 N λ − m / 2 χ m ( x ) ( Ψ μ ( λ x ) − c m ) . (45) \theta_d(t,x)=\delta_{n+1}^{16}\eta_d(t)\sum_{m=0}^{8N}\lambda^{-m/2}\chi_m(x)\bigl(\Psi_\mu(\lambda x)-c_m\bigr). \tag*{(45)} θ d ( t , x ) = δ n + 1 16 η d ( t ) m = 0 ∑ 8 N λ − m /2 χ m ( x ) ( Ψ μ ( λ x ) − c m ) . (45) Each summand has zero spatial integral. The positive contribution is δ n + 1 16 η d E \delta_{n+1}^{16}\eta_d E δ n + 1 16 η d E
E ( x ) = ∑ m = 0 8 N λ − m / 2 χ m ( x ) Ψ μ ( λ x ) . (46) E(x)=\sum_{m=0}^{8N}\lambda^{-m/2}\chi_m(x)\Psi_\mu(\lambda x). \tag*{(46)} E ( x ) = m = 0 ∑ 8 N λ − m /2 χ m ( x ) Ψ μ ( λ x ) . (46) It is supported in B 16 N + 5 B_{16N+5} B 16 N + 5 E ≥ Ψ μ ( λ ⋅ ) E\ge\Psi_\mu(\lambda\cdot) E ≥ Ψ μ ( λ ⋅ ) B 4 B_4 B 4
The successive cutoffs are needed because η d ′ \eta_d' η d ′ R 0 [ χ 0 ( Ψ μ ( λ ⋅ ) − c 0 ) ] \mathcal{R}_0[\chi_0(\Psi_\mu(\lambda\cdot)-c_0)] R 0 [ χ 0 ( Ψ μ ( λ ⋅ ) − c 0 )] supp χ 0 \operatorname{supp}\chi_0 supp χ 0 Lemmas 3.11 and 3.5, together with the mean estimate in Lemma 4.2 below, give
∣ R 0 [ χ 0 ( Ψ μ ( λ ⋅ ) − c 0 ) ] ( x ) ∣ ≤ C λ − 1 d − 1 χ 1 ( x ) Ψ μ ( λ x ) + C λ − 1 d − 1 . |\mathcal{R}_0[\chi_0(\Psi_\mu(\lambda\cdot)-c_0)](x)|\le C\lambda^{-1}d^{-1}\chi_1(x)\Psi_\mu(\lambda x)+C\lambda^{-1}d^{-1}. ∣ R 0 [ χ 0 ( Ψ μ ( λ ⋅ ) − c 0 )] ( x ) ∣ ≤ C λ − 1 d − 1 χ 1 ( x ) Ψ μ ( λ x ) + C λ − 1 d − 1 . The next positive density term has coefficient λ − 1 / 2 \lambda^{-1/2} λ − 1/2 λ − 1 \lambda^{-1} λ − 1 θ d \theta_d θ d λ − 4 N \lambda^{-4N} λ − 4 N λ − 4 N ∥ Ψ μ ∥ ∞ ≤ C λ − 2 N \lambda^{-4N}\|\Psi_\mu\|_\infty\le C\lambda^{-2N} λ − 4 N ∥ Ψ μ ∥ ∞ ≤ C λ − 2 N Lemma 5.4 includes the time cutoff in this estimate.
The spatial moment correction
The stationary correction has zero mass. We now correct the mass of θ p \theta_p θ p θ p + θ d \theta_p+\theta_d θ p + θ d ϕ 0 ∈ C c ∞ ( B ( x 0 , r 0 / 2 ) ) \phi_0\in C^\infty_c(B(x_0,r_0/2)) ϕ 0 ∈ C c ∞ ( B ( x 0 , r 0 /2 )) ∫ ϕ 0 = 1 \int\phi_0=1 ∫ ϕ 0 = 1 ϕ i = − ∂ i ϕ 0 \phi_i=-\partial_i\phi_0 ϕ i = − ∂ i ϕ 0 i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3
∫ ϕ 0 = 1 , ∫ ( x − x 0 ) ϕ 0 = 0 , \int\phi_0=1,\qquad\int(x-x_0)\phi_0=0, ∫ ϕ 0 = 1 , ∫ ( x − x 0 ) ϕ 0 = 0 , ∫ ϕ i = 0 , ∫ ( x − x 0 ) j ϕ i = δ i j . (47) \int\phi_i=0,\qquad\int(x-x_0)_j\phi_i=\delta_{ij}. \tag*{(47)} ∫ ϕ i = 0 , ∫ ( x − x 0 ) j ϕ i = δ ij . (47) Set
θ c ( t , x ) = − ϕ 0 ( x ) ∫ θ p ( t , y ) d y − ∑ i = 1 3 ϕ i ( x ) ∫ ( y − x 0 ) i ( θ p + θ d ) ( t , y ) d y . (48) \theta_c(t,x)=-\phi_0(x)\int\theta_p(t,y)\,dy-\sum_{i=1}^{3}\phi_i(x)\int(y-x_0)_i(\theta_p+\theta_d)(t,y)\,dy. \tag*{(48)} θ c ( t , x ) = − ϕ 0 ( x ) ∫ θ p ( t , y ) d y − i = 1 ∑ 3 ϕ i ( x ) ∫ ( y − x 0 ) i ( θ p + θ d ) ( t , y ) d y . (48) We finally define
θ ( t , x ) = ( θ p + θ c + θ d ) ( t , x ) , ρ + = ρ + θ . (49) \theta(t,x)=(\theta_p+\theta_c+\theta_d)(t,x),\qquad\rho^+=\rho+\theta. \tag*{(49)} θ ( t , x ) = ( θ p + θ c + θ d ) ( t , x ) , ρ + = ρ + θ . (49) Products of x − x 0 x-x_0 x − x 0 B B B
Lemma 4.2 (Density and spatial moments). The functions θ p , θ c , θ d \theta_p,\theta_c,\theta_d θ p , θ c , θ d θ \theta θ supp η d × B 16 N + 5 \operatorname{supp}\eta_d\times B_{16N+5} supp η d × B 16 N + 5
∫ θ d = 0 , ∫ θ = 0 , ∫ ( x − x 0 ) θ = 0. (50) \int\theta_d=0,\qquad\int\theta=0,\qquad\int(x-x_0)\theta=0. \tag*{(50)} ∫ θ d = 0 , ∫ θ = 0 , ∫ ( x − x 0 ) θ = 0. (50) Moreover,
c m ≤ C , ∥ E ∥ L p ≤ C , E ≥ Ψ μ ( λ ⋅ ) on B 4 , c_m\le C,\qquad\|E\|_{L^p}\le C,\qquad E\ge\Psi_\mu(\lambda\cdot)\quad\text{on }B_4, c m ≤ C , ∥ E ∥ L p ≤ C , E ≥ Ψ μ ( λ ⋅ ) on B 4 , and
θ p ≥ − ∑ k a k , θ d ≥ δ n + 1 16 η d E − C δ n + 1 16 . (51) \theta_p\ge-\sum_k a_k,\qquad\theta_d\ge\delta^{16}_{n+1}\eta_dE-C\delta^{16}_{n+1}. \tag*{(51)} θ p ≥ − k ∑ a k , θ d ≥ δ n + 1 16 η d E − C δ n + 1 16 . (51) With g = Ψ μ − ∫ Ψ μ g=\Psi_\mu-\int\Psi_\mu g = Ψ μ − ∫ Ψ μ
∣ c m − ∫ Ψ μ ∣ = ∣ ∫ χ m g ( λ ⋅ ) ∣ ∫ χ m ≤ C λ − 1 d − 1 . (52) \left|c_m-\int\Psi_\mu\right|=\frac{\left|\int\chi_mg(\lambda\cdot)\right|}{\int\chi_m}\le C\lambda^{-1}d^{-1}. \tag*{(52)} c m − ∫ Ψ μ = ∫ χ m ∫ χ m g ( λ ⋅ ) ≤ C λ − 1 d − 1 . (52) θ d = δ n + 1 16 η d ∑ m λ − m / 2 [ χ m g ( λ x ) − ( c m − ∫ Ψ μ ) χ m ] . (53) \theta_d=\delta^{16}_{n+1}\eta_d\sum_m\lambda^{-m/2}\left[\chi_mg(\lambda x)-\left(c_m-\int\Psi_\mu\right)\chi_m\right]. \tag*{(53)} θ d = δ n + 1 16 η d m ∑ λ − m /2 [ χ m g ( λ x ) − ( c m − ∫ Ψ μ ) χ m ] . (53) The stationary corrector also obeys
∣ θ d ∣ ≤ C δ n + 1 16 η d E . |\theta_d|\le C\delta^{16}_{n+1}\eta_dE. ∣ θ d ∣ ≤ C δ n + 1 16 η d E . Proof. The support and smoothness follow from (4.18)–(4.24), Proposition 4.2, and the chosen cutoffs. Equation (4.19) gives ∫ θ d = 0 \int\theta_d=0 ∫ θ d = 0
As χ m = 1 \chi_m=1 χ m = 1 B ( x 0 , r 0 ) B(x_0,r_0) B ( x 0 , r 0 ) c m ≤ ∥ Ψ μ ∥ L 1 / ∣ B ( x 0 , r 0 ) ∣ ≤ C c_m\le\|\Psi_\mu\|_{L^1}/|B(x_0,r_0)|\le C c m ≤ ∥ Ψ μ ∥ L 1 /∣ B ( x 0 , r 0 ) ∣ ≤ C Ψ μ ≥ 1 \Psi_\mu\ge1 Ψ μ ≥ 1 ∣ θ d ∣ ≤ C δ n + 1 16 η d E |\theta_d|\le C\delta^{16}_{n+1}\eta_dE ∣ θ d ∣ ≤ C δ n + 1 16 η d E ∥ E ∥ L p ≤ ∑ m λ − m / 2 ∥ Ψ μ ∥ L p ≤ C \|E\|_{L^p}\le\sum_m\lambda^{-m/2}\|\Psi_\mu\|_{L^p}\le C ∥ E ∥ L p ≤ ∑ m λ − m /2 ∥ Ψ μ ∥ L p ≤ C B 4 B_4 B 4 χ 0 = 1 \chi_0=1 χ 0 = 1 a k , Θ k ≥ 0 a_k,\Theta_k\ge0 a k , Θ k ≥ 0 c m ≤ C c_m\le C c m ≤ C
For the mean estimate, apply Lemma 3.4 to χ m g ( λ ⋅ ) \chi_mg(\lambda\cdot) χ m g ( λ ⋅ ) ∫ χ m ≥ ∣ B ( x 0 , r 0 ) ∣ \int\chi_m\ge|B(x_0,r_0)| ∫ χ m ≥ ∣ B ( x 0 , r 0 ) ∣ ∫ Ψ μ \int\Psi_\mu ∫ Ψ μ
The continuity defect
We first compute the error left by the transported profiles. By Lemma 4.1 and the definition of f k f_k f k
∂ t [ a k ( Θ k − 1 ) ] + div [ a k ( Θ k − 1 ) V k ] = ( Θ k − 1 ) f k . (54) \partial_t[a_k(\Theta_k-1)]+\operatorname{div}[a_k(\Theta_k-1)V_k]=(\Theta_k-1)f_k. \tag*{(54)} ∂ t [ a k ( Θ k − 1 )] + div [ a k ( Θ k − 1 ) V k ] = ( Θ k − 1 ) f k . (54) The sum of these errors, together with the two density corrections, determines a single scalar potential P ( t , x ) P(t,x) P ( t , x )
Δ P = ∑ k ( Θ k − 1 ) f k + ∂ t θ c + ∂ t θ d , ∫ P = 0. \Delta P=\sum_k(\Theta_k-1)f_k+\partial_t\theta_c+\partial_t\theta_d,\qquad\int P=0. Δ P = k ∑ ( Θ k − 1 ) f k + ∂ t θ c + ∂ t θ d , ∫ P = 0. The right-hand side has integral ∂ t ∫ θ = 0 \partial_t\int\theta=0 ∂ t ∫ θ = 0 P P P ∇ P \nabla P ∇ P
For compact support, put U = B 16 N + 8 U=B_{16N+8} U = B 16 N + 8 x 0 x_0 x 0
χ ∈ C c ∞ ( B 16 N + 7 ) , χ = 1 on B 16 N + 6 . \chi\in C^\infty_c(B_{16N+7}),\qquad\chi=1\quad\text{on }B_{16N+6}. χ ∈ C c ∞ ( B 16 N + 7 ) , χ = 1 on B 16 N + 6 . Its gradient is separated from all density increments by 2 d 2d 2 d ( 32 N + 16 ) d < ( r + − r − ) / 2 (32N+16)d<(r_+-r_-)/2 ( 32 N + 16 ) d < ( r + − r − ) /2
U ⊂ B ( x 0 , 1 2 ( r − + r + ) ) . U\subset B\left(x_0,\tfrac12(r_-+r_+)\right). U ⊂ B ( x 0 , 2 1 ( r − + r + ) ) . Multiplying ∇ P \nabla P ∇ P χ \chi χ ∇ χ ⋅ ∇ P \nabla\chi\cdot\nabla P ∇ χ ⋅ ∇ P
R c ( t , x ) = − B U [ ∇ χ ⋅ ∇ P ] ( t , x ) + 1 2 curl x [ φ 0 ( x ) ∑ k ∫ a k ( t , y ) ( Θ k ( t , y ) − 1 ) ( y − x 0 ) × V k d y ] . R_c(t,x)=-\mathcal{B}_U[\nabla\chi\cdot\nabla P](t,x)
+\frac{1}{2}\operatorname{curl}_x\left[\varphi_0(x)\sum_k\int a_k(t,y)(\Theta_k(t,y)-1)(y-x_0)\times V_k\,\mathrm{d}y\right]. R c ( t , x ) = − B U [ ∇ χ ⋅ ∇ P ] ( t , x ) + 2 1 curl x [ φ 0 ( x ) k ∑ ∫ a k ( t , y ) ( Θ k ( t , y ) − 1 ) ( y − x 0 ) × V k d y ] . Radial symmetry of χ \chi χ Lemma 3.7 ensure that the two localization terms χ ∇ P \chi\nabla P χ ∇ P B U [ ∇ χ ⋅ ∇ P ] \mathcal{B}_U[\nabla\chi\cdot\nabla P] B U [ ∇ χ ⋅ ∇ P ]
Lemma 4.3 (The continuity equation). For sufficiently large a > 0 a>0 a > 0
R + = ( ρ − ∑ k a k ) w + ( θ c + θ d ) u + + χ ∇ P + R c R^+=\left(\rho-\sum_k a_k\right)w+(\theta_c+\theta_d)u^++\chi\nabla P+R_c R + = ( ρ − k ∑ a k ) w + ( θ c + θ d ) u + + χ ∇ P + R c is smooth and supported in supp η d × U \operatorname{supp}\eta_d\times U supp η d × U
ρ + u + − R + − ( ρ u − R ) = ∑ k a k ( Θ k − 1 ) V k − χ ∇ P − R c , (55) \rho^+u^+-R^+-(\rho u-R)=\sum_k a_k(\Theta_k-1)V_k-\chi\nabla P-R_c, \tag*{(55)} ρ + u + − R + − ( ρ u − R ) = k ∑ a k ( Θ k − 1 ) V k − χ ∇ P − R c , (55) and
∂ t ρ + + div ( ρ + u + − R + ) = 0. \partial_t\rho^++\operatorname{div}(\rho^+u^+-R^+)=0. ∂ t ρ + + div ( ρ + u + − R + ) = 0. The modified momentum increment has zero spatial integral and zero angular momentum:
∫ [ ρ + u + − R + − ( ρ u − R ) ] = 0 , \int\left[\rho^+u^+-R^+-(\rho u-R)\right]=0, ∫ [ ρ + u + − R + − ( ρ u − R ) ] = 0 , ∫ ( x − x 0 ) × [ ρ + u + − R + − ( ρ u − R ) ] = 0. (56) \int(x-x_0)\times\left[\rho^+u^+-R^+-(\rho u-R)\right]=0. \tag*{(56)} ∫ ( x − x 0 ) × [ ρ + u + − R + − ( ρ u − R ) ] = 0. (56) Proof. Since χ = 1 \chi=1 χ = 1 Δ P \Delta P Δ P
∫ ∇ χ ⋅ ∇ P = − ∫ χ Δ P = 0. \int\nabla\chi\cdot\nabla P=-\int\chi\Delta P=0. ∫ ∇ χ ⋅ ∇ P = − ∫ χ Δ P = 0. Thus Lemma 3.7 defines the first term of R c R_c R c
div ( χ ∇ P + R c ) = Δ P . \operatorname{div}(\chi\nabla P+R_c)=\Delta P. div ( χ ∇ P + R c ) = Δ P . By Lemma 4.1, u + = V k u^+=V_k u + = V k supp ( a k Θ k ) \operatorname{supp}(a_k\Theta_k) supp ( a k Θ k ) ρ + u + \rho^+u^+ ρ + u +
ρ + u + − ρ u = ∑ k a k ( Θ k − 1 ) V k − R + ( ρ − ∑ k a k ) w + ( θ c + θ d ) u + . \rho^+u^+-\rho u=\sum_k a_k(\Theta_k-1)V_k-R+\left(\rho-\sum_k a_k\right)w+(\theta_c+\theta_d)u^+. ρ + u + − ρ u = k ∑ a k ( Θ k − 1 ) V k − R + ( ρ − k ∑ a k ) w + ( θ c + θ d ) u + . This proves (4.30). Its divergence, the transport identity (4.29), and the defining equation for P P P
The first spatial moment of θ \theta θ Lemma 4.2 . Integrating the continuity equation against x − x 0 x-x_0 x − x 0
χ ∇ P = ∇ ( χ P ) − P ∇ χ . \chi\nabla P=\nabla(\chi P)-P\nabla\chi. χ ∇ P = ∇ ( χ P ) − P ∇ χ . The gradient has zero angular momentum by Lemma 3.6 , and ( x − x 0 ) × ∇ χ = 0 (x-x_0)\times\nabla\chi=0 ( x − x 0 ) × ∇ χ = 0 Lemma 3.7 . Finally, Lemma 3.6 and ∫ φ 0 = 1 \int\varphi_0=1 ∫ φ 0 = 1 R c R_c R c
The momentum defect and pressure
The same transport identity, multiplied by V k V_k V k π Id \pi\operatorname{Id} π Id − π -\pi − π ∇ P \nabla P ∇ P
Π + = Π − π + χ ∂ t P . (57) \Pi^+=\Pi-\pi+\chi\partial_tP. \tag*{(57)} Π + = Π − π + χ ∂ t P . (57) Only derivatives of the outer cutoff and the compact correction R c R_c R c T c T_c T c
Lemma 4.4 (The momentum equation). For sufficiently large a > 0 a>0 a > 0
T c = S U [ χ ∑ k R ( ( Θ k − 1 ) f k V k ) − ( ∑ k R ( ( Θ k − 1 ) f k V k ) ) ∇ χ + ( ∂ t P ) ∇ χ − ∂ t R c ] (58) T_c=S_U\left[\chi\sum_k\mathcal{R}\left((\Theta_k-1)f_kV_k\right)-\left(\sum_k\mathcal{R}\left((\Theta_k-1)f_kV_k\right)\right)\nabla\chi+(\partial_tP)\nabla\chi-\partial_tR_c\right] \tag*{(58)} T c = S U [ χ k ∑ R ( ( Θ k − 1 ) f k V k ) − ( k ∑ R ( ( Θ k − 1 ) f k V k ) ) ∇ χ + ( ∂ t P ) ∇ χ − ∂ t R c ] (58) is well defined. The symmetric stress
T + = χ ∑ k R [ ( Θ k − 1 ) f k V k ] − ( ∇ w + ∇ w T ) T^+=\chi\sum_k\mathcal{R}\left[(\Theta_k-1)f_kV_k\right]-(\nabla w+\nabla w^{\mathsf T}) T + = χ k ∑ R [ ( Θ k − 1 ) f k V k ] − ( ∇ w + ∇ w T ) + ( ρ − ∑ k a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) +\left(\rho-\sum_k a_k\right)(u\otimes w+w\otimes u+w\otimes w) + ( ρ − k ∑ a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) + ( θ c + θ d ) u + ⊗ u + + T c +(\theta_c+\theta_d)u^+\otimes u^++T_c + ( θ c + θ d ) u + ⊗ u + + T c completes Z + = ( ρ + , u + , Π + , R + , T + ) Z^+=(\rho^+,u^+,\Pi^+,R^+,T^+) Z + = ( ρ + , u + , Π + , R + , T + ) Z + − Z Z^+-Z Z + − Z ( R + , T + ) (R^+,T^+) ( R + , T + ) supp η d × U \operatorname{supp}\eta_d\times U supp η d × U
ρ + u + ⊗ u + − ρ u ⊗ u = ∑ k a k ( Θ k − 1 ) V k ⊗ V k + π Id − T \rho^+u^+\otimes u^+-\rho u\otimes u
=\sum_k a_k(\Theta_k-1)V_k\otimes V_k+\pi\operatorname{Id}-T ρ + u + ⊗ u + − ρ u ⊗ u = k ∑ a k ( Θ k − 1 ) V k ⊗ V k + π Id − T + ( ρ − ∑ k a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) + ( θ c + θ d ) u + ⊗ u + . (59) +\left(\rho-\sum_k a_k\right)(u\otimes w+w\otimes u+w\otimes w)
+(\theta_c+\theta_d)u^+\otimes u^+. \tag*{(59)} + ( ρ − k ∑ a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) + ( θ c + θ d ) u + ⊗ u + . (59) Proof. We verify the cancellation, then the two compatibility conditions for the compactly supported symmetric inverse divergence.
Step 1: The transported profiles and pressure.
The density decomposition and Lemma 4.1 give
ρ + u + ⊗ u + = ∑ k a k Θ k V k ⊗ V k + ( ρ − ∑ k a k + θ c + θ d ) u + ⊗ u + . \rho^+u^+\otimes u^+=\sum_k a_k\Theta_kV_k\otimes V_k+\left(\rho-\sum_k a_k+\theta_c+\theta_d\right)u^+\otimes u^+. ρ + u + ⊗ u + = k ∑ a k Θ k V k ⊗ V k + ( ρ − k ∑ a k + θ c + θ d ) u + ⊗ u + . Subtracting ρ u ⊗ u \rho u\otimes u ρ u ⊗ u (59) . Moreover,
∂ t ∑ k a k ( Θ k − 1 ) V k + div ∑ k a k ( Θ k − 1 ) V k ⊗ V k = ∑ k ( Θ k − 1 ) f k V k . \partial_t\sum_k a_k(\Theta_k-1)V_k+\operatorname{div}\sum_k a_k(\Theta_k-1)V_k\otimes V_k=\sum_k(\Theta_k-1)f_kV_k. ∂ t k ∑ a k ( Θ k − 1 ) V k + div k ∑ a k ( Θ k − 1 ) V k ⊗ V k = k ∑ ( Θ k − 1 ) f k V k . Insert this identity, the modified momentum increment of Lemma 4.3, and (4.32) into the momentum equation. Since div ( ∇ w + ∇ w T ) = Δ w \operatorname{div}(\nabla w+\nabla w^{\mathsf T})=\Delta w div ( ∇ w + ∇ w T ) = Δ w
∑ k ( Θ k − 1 ) f k V k − div ( χ ∑ k R [ ( Θ k − 1 ) f k V k ] ) − ∂ t ( χ ∇ P + R c ) + ∇ ( χ ∂ t P ) . \sum_k(\Theta_k-1)f_kV_k-\operatorname{div}\left(\chi\sum_k\mathcal{R}\left[(\Theta_k-1)f_kV_k\right]\right)-\partial_t(\chi\nabla P+R_c)+\nabla(\chi\partial_tP). k ∑ ( Θ k − 1 ) f k V k − div ( χ k ∑ R [ ( Θ k − 1 ) f k V k ] ) − ∂ t ( χ ∇ P + R c ) + ∇ ( χ ∂ t P ) . By Lemma 3.3 and the product rule, this is exactly the vector field in brackets in (4.33).
Step 2: The compact correction.
The residual just computed is compactly supported in U U U T c T_c T c T c T_c T c div u + = 0 \operatorname{div}u^+=0 div u + = 0
The terms θ d u + \theta_du^+ θ d u + θ d u + ⊗ u + \theta_du^+\otimes u^+ θ d u + ⊗ u + θ d \theta_d θ d ∇ P \nabla P ∇ P R 0 ∂ t θ d R_0\partial_t\theta_d R 0 ∂ t θ d
One step of the iteration: the estimates
We prove a lower bound for the total density perturbation θ \theta θ δ n + 1 16 η d E \delta_{n+1}^{16}\eta_d E δ n + 1 16 η d E
We keep the hypotheses of Proposition 4.1 and the fields of Lemmas 4.3 and 4.4 . The formulas in these lemmas express the defects in terms of w w w P P P R c , T c R_c,T_c R c , T c θ \theta θ δ n + 1 = λ − γ \delta_{n+1}=\lambda^{-\gamma} δ n + 1 = λ − γ ω = λ 4 γ / b \omega=\lambda^{4\gamma/b} ω = λ 4 γ / b μ = λ N \mu=\lambda^N μ = λ N λ n = λ 1 / b \lambda_n=\lambda^{1/b} λ n = λ 1/ b 1000 ( N 4 + 20 ) < γ b 1000(N^4+20)<\gamma b 1000 ( N 4 + 20 ) < γ b a > 0 a>0 a > 0
∥ ( ρ , u , R , T ) ∥ C t , x 20 + τ − 20 + d − 20 ≤ λ γ / 1000 . (60) \|(\rho,u,R,T)\|_{C^{20}_{t,x}}+\tau^{-20}+d^{-20}\leq\lambda^{\gamma/1000}. \tag*{(60)} ∥ ( ρ , u , R , T ) ∥ C t , x 20 + τ − 20 + d − 20 ≤ λ γ /1000 . (60) Together with Lemma 3.1 and the defining spatial cutoff bounds, this also controls derivatives of the cutoffs up to order 20 by C λ γ / 1000 C\lambda^{\gamma/1000} C λ γ /1000
Localization and moment errors
Since 24 γ < 1 / 4 24\gamma<1/4 24 γ < 1/4 C λ − 1 / 2 C\lambda^{-1/2} C λ − 1/2 δ n + 1 24 / 100 \delta_{n+1}^{24}/100 δ n + 1 24 /100 a > 0 a>0 a > 0 θ c \theta_c θ c R c , T c R_c,T_c R c , T c supp ∇ χ \operatorname{supp}\nabla\chi supp ∇ χ 2 d 2d 2 d
Lemma 5.1 (Localization and spatial moment corrections). For sufficiently large a > 0 a>0 a > 0
∥ θ c ∥ C t , x 6 ≤ λ − 3 / 4 . (61) \|\theta_c\|_{C^6_{t,x}}\leq\lambda^{-3/4}. \tag*{(61)} ∥ θ c ∥ C t , x 6 ≤ λ − 3/4 . (61) The compact corrections of Lemmas 4.3 and 4.4 obey
∥ R c ∥ C t , x 2 + ∥ T c ∥ C 0 ≤ λ − 1 / 2 . (62) \|R_c\|_{C^2_{t,x}}+\|T_c\|_{C^0}\leq\lambda^{-1/2}. \tag*{(62)} ∥ R c ∥ C t , x 2 + ∥ T c ∥ C 0 ≤ λ − 1/2 . (62) Outside B 4 B_4 B 4
∣ χ ∑ k R 0 [ ( Θ k − 1 ) f k ] ∣ + ∣ χ ∑ k R [ ( Θ k − 1 ) f k V k ] ∣ ≤ λ − 3 / 4 . (63) \left|\chi\sum_k\mathcal{R}_0[(\Theta_k-1)f_k]\right|+\left|\chi\sum_k\mathcal{R}[(\Theta_k-1)f_kV_k]\right|\leq\lambda^{-3/4}. \tag*{(63)} χ k ∑ R 0 [( Θ k − 1 ) f k ] + χ k ∑ R [( Θ k − 1 ) f k V k ] ≤ λ − 3/4 . (63) Proof. We first estimate the spatial averages and the potentials away from their sources, then apply the compact inverse divergence operators.
Step 1: The spatial moment correction.
By Lemma 4.1, time derivatives of an average against Θ k − 1 \Theta_k-1 Θ k − 1 h h h [ 0 , 1 ] × T 3 [0,1]\times\mathbb{T}^3 [ 0 , 1 ] × T 3
∂ t j ∫ h ( Θ k − 1 ) = ∫ [ ( ∂ t + V k ⋅ ∇ ) j h ] ( Θ k − 1 ) . (64) \partial_t^j\int h(\Theta_k-1)=\int[(\partial_t+V_k\cdot\nabla)^j h](\Theta_k-1). \tag*{(64)} ∂ t j ∫ h ( Θ k − 1 ) = ∫ [( ∂ t + V k ⋅ ∇ ) j h ] ( Θ k − 1 ) . (64) Apply Lemma 3.4 to g = ζ μ ( ⋅ − y k ) − 1 g=\zeta_\mu(\cdot-y_k)-1 g = ζ μ ( ⋅ − y k ) − 1 L 1 L^1 L 1 h = a k h=a_k h = a k h = ( x − x 0 ) i a k h=(x-x_0)_i a_k h = ( x − x 0 ) i a k 0 ≤ j ≤ 6 0\leq j\leq6 0 ≤ j ≤ 6
∣ ∂ t j ∫ h ( Θ k − 1 ) ∣ ≤ C λ − 1 ω j ∥ a k ∥ C t , x j + 1 ≤ C λ − 1 ω j δ n + 1 . \begin{aligned}
\left|\partial_t^j\int h(\Theta_k-1)\right|\leq C\lambda^{-1}\omega^j\|a_k\|_{C^{j+1}_{t,x}} \\
&\leq C\lambda^{-1}\omega^j\delta_{n+1}.
\end{aligned} ∂ t j ∫ h ( Θ k − 1 ) ≤ C λ − 1 ω j ∥ a k ∥ C t , x j + 1 ≤ C λ − 1 ω j δ n + 1 . For the stationary corrector, put g = Ψ μ − ∫ Ψ μ g=\Psi_\mu-\int\Psi_\mu g = Ψ μ − ∫ Ψ μ L 1 L^1 L 1 Lemma 3.11 . We use the mean estimate and the centered representation of θ d \theta_d θ d Lemma 4.2 . Apply Lemma 3.4 to ( x − x 0 ) i χ m g ( λ ) (x-x_0)_i\chi_m g(\lambda) ( x − x 0 ) i χ m g ( λ ) Lemma 3.1 to η d \eta_d η d η d \eta_d η d 0 ≤ j ≤ 6 0\leq j\leq6 0 ≤ j ≤ 6
∣ ∂ t j ∫ ( x − x 0 ) i θ d ∣ ≤ C δ n + 1 16 τ − j λ − 1 d − 1 . \left|\partial_t^j\int(x-x_0)_i\theta_d\right|\leq C\delta_{n+1}^{16}\tau^{-j}\lambda^{-1}d^{-1}. ∂ t j ∫ ( x − x 0 ) i θ d ≤ C δ n + 1 16 τ − j λ − 1 d − 1 . It follows that the mass and three first-moment coefficients of θ c \theta_c θ c Lemma 4.2 , together with their first six time derivatives, are bounded by
C λ − 1 ( ω 6 δ n + 1 − 1 + τ − 6 d − 1 ) ≤ C λ − 1 + 2 γ , C\lambda^{-1}\left(\omega^6\delta_{n+1}^{-1}+\tau^{-6}d^{-1}\right)\leq C\lambda^{-1+2\gamma}, C λ − 1 ( ω 6 δ n + 1 − 1 + τ − 6 d − 1 ) ≤ C λ − 1 + 2 γ , where we used (5.1) and ω = λ 4 γ / b \omega=\lambda^{4\gamma/b} ω = λ 4 γ / b ϕ i \phi_i ϕ i
∥ θ c ∥ C t , x 6 ≤ C λ − 1 + 2 γ . (65) \|\theta_c\|_{C^6_{t,x}}\leq C\lambda^{-1+2\gamma}. \tag*{(65)} ∥ θ c ∥ C t , x 6 ≤ C λ − 1 + 2 γ . (65) For sufficiently large a > 0 a>0 a > 0
Step 2: The potentials away from their sources.
Fix 0 ≤ i , j ≤ 2 0 \le i,j \le2 0 ≤ i , j ≤ 2 L ∈ { R 0 , Δ − 1 P ≠ 0 } L \in\{R_0,\Delta^{-1}\mathbb{P}_{\ne0}\} L ∈ { R 0 , Δ − 1 P = 0 }
Θ k − 1 = g k ( λ x − λ ω t k ) , g k = ζ μ ( ⋅ − y k ) − 1. \Theta_k-1=g_k(\lambda x-\lambda\omega t_k),\qquad g_k=\zeta_\mu(\cdot-y_k)-1. Θ k − 1 = g k ( λ x − λω t k ) , g k = ζ μ ( ⋅ − y k ) − 1. The profile g k g_k g k L 1 L^1 L 1 supp f k ⊂ supp η × B 1 \operatorname{supp} f_k\subset\operatorname{supp}\eta\times B_1 supp f k ⊂ supp η × B 1 d d d B 1 B_1 B 1
∣ ∂ t j ∇ i L [ ( Θ k − 1 ) f k ] ∣ ≤ C d − 8 λ − 4 ( λ ω ) 2 ∥ f k ∥ C t , x 6 ≤ C d − 8 λ − 2 ω 3 ∥ a k ∥ C t , x 7 ≤ C λ − 2 + 2 γ . (66) \begin{aligned}
\left|\partial_t^j\nabla^i\mathcal{L}[(\Theta_k-1)f_k]\right|
\le Cd^{-8}\lambda^{-4}(\lambda\omega)^2\|f_k\|_{C^6_{t,x}} \\
&\le Cd^{-8}\lambda^{-2}\omega^3\|a_k\|_{C^7_{t,x}} \\
&\le C\lambda^{-2+2\gamma}. \tag*{(66)}
\end{aligned} ∂ t j ∇ i L [( Θ k − 1 ) f k ] ≤ C d − 8 λ − 4 ( λω ) 2 ∥ f k ∥ C t , x 6 ≤ C d − 8 λ − 2 ω 3 ∥ a k ∥ C t , x 7 ≤ C λ − 2 + 2 γ . (66) The last bound uses Proposition 4.2 and (5.1). The same argument applies to R [ ( Θ k − 1 ) f k V k ] \mathcal{R}[(\Theta_k-1)f_kV_k] R [( Θ k − 1 ) f k V k ] ω \omega ω
For the other two terms defining P P P
∣ ∂ t j ∇ i L ∂ t θ c ∣ ≤ C λ − 1 + 2 γ . \left|\partial_t^j\nabla^i\mathcal{L}\partial_t\theta_c\right|
\le C\lambda^{-1+2\gamma}. ∂ t j ∇ i L ∂ t θ c ≤ C λ − 1 + 2 γ . For θ d \theta_d θ d g = Ψ μ − ∫ Ψ μ g=\Psi_\mu-\int\Psi_\mu g = Ψ μ − ∫ Ψ μ dist ( x , B 16 N + 5 ) ≥ d \operatorname{dist}(x,B_{16N+5})\ge d dist ( x , B 16 N + 5 ) ≥ d
∣ ∇ i L [ χ m g ( λ ⋅ ) ] ( x ) ∣ ≤ C d − 12 λ − 4 . \left|\nabla^i\mathcal{L}[\chi_m g(\lambda\cdot)](x)\right|
\le Cd^{-12}\lambda^{-4}. ∇ i L [ χ m g ( λ ⋅ )] ( x ) ≤ C d − 12 λ − 4 . The mean correction is bounded by C λ − 1 d − 3 C\lambda^{-1}d^{-3} C λ − 1 d − 3 η d \eta_d η d C λ − 1 + 2 γ C\lambda^{-1+2\gamma} C λ − 1 + 2 γ d d d B 16 N + 5 B_{16N+5} B 16 N + 5
∣ ∂ t j ∇ i P ∣ + ∣ ∂ t j ∇ i ∇ P ∣ + ∣ ∂ t j ∇ i ∑ k R [ ( Θ k − 1 ) f k V k ] ∣ ≤ λ − 3 / 4 ( 0 ≤ i , j ≤ 2 ) . (67) \left|\partial_t^j\nabla^iP\right|+\left|\partial_t^j\nabla^i\nabla P\right|
+\left|\partial_t^j\nabla^i\sum_k\mathcal{R}[(\Theta_k-1)f_kV_k]\right|
\le\lambda^{-3/4}
\qquad(0\le i,j\le2). \tag*{(67)} ∂ t j ∇ i P + ∂ t j ∇ i ∇ P + ∂ t j ∇ i k ∑ R [( Θ k − 1 ) f k V k ] ≤ λ − 3/4 ( 0 ≤ i , j ≤ 2 ) . (67) Step 3: Averages of the transported profiles.
By Lemma 3.4, (5.5), and Proposition 4.2,
∣ ∂ t j ∑ k ∫ f ( Θ k − 1 ) f k V k ∣ ≤ C λ − 1 ω j + 1 ∑ k ∥ f k ∥ C j + 1 ≤ C λ − 1 ω 3 δ n + 1 − 1 ( j = 0 , 1 ) . \begin{aligned}
\left|\partial_t^j\sum_k\int f(\Theta_k-1)f_kV_k\right|
\le C\lambda^{-1}\omega^{j+1}\sum_k\|f_k\|_{C^{j+1}} \\
&\le C\lambda^{-1}\omega^3\delta_{n+1}^{-1}
\qquad(j=0,1).
\end{aligned} ∂ t j k ∑ ∫ f ( Θ k − 1 ) f k V k ≤ C λ − 1 ω j + 1 k ∑ ∥ f k ∥ C j + 1 ≤ C λ − 1 ω 3 δ n + 1 − 1 ( j = 0 , 1 ) . Applying the same averaging identity to ( x − x 0 ) × a k V k (x-x_0)\times a_kV_k ( x − x 0 ) × a k V k
∣ ∂ t j ∫ ( x − x 0 ) × ∑ k a k ( Θ k − 1 ) V k d x ∣ ≤ C λ − 1 ω 3 δ n + 1 − 1 ≤ λ − 3 / 4 , 0 ≤ j ≤ 2. (68) \left|\partial_t^j\int(x-x_0)\times\sum_k a_k(\Theta_k-1)V_k\,dx\right|
\le C\lambda^{-1}\omega^3\delta_{n+1}^{-1}
\le\lambda^{-3/4},
\qquad0\le j\le2. \tag*{(68)} ∂ t j ∫ ( x − x 0 ) × k ∑ a k ( Θ k − 1 ) V k d x ≤ C λ − 1 ω 3 δ n + 1 − 1 ≤ λ − 3/4 , 0 ≤ j ≤ 2. (68) Thus the coefficients of the curl in R c R_c R c
Step 4: The compact corrections.
The formula for R c R_c R c supp ∇ χ \operatorname{supp}\nabla\chi supp ∇ χ
∥ R c ∥ C t , x 2 ≤ C d − 3 λ − 3 / 4 . \|R_c\|_{C^2_{t,x}}\le Cd^{-3}\lambda^{-3/4}. ∥ R c ∥ C t , x 2 ≤ C d − 3 λ − 3/4 . The formula for T c T_c T c supp ∇ χ \operatorname{supp}\nabla\chi supp ∇ χ ∂ t R c \partial_tR_c ∂ t R c
∥ R c ∥ C t , x 2 + ∥ T c ∥ C 0 ≤ C d − 3 λ − 3 / 4 ≤ λ − 1 / 2 , \|R_c\|_{C^2_{t,x}}+\|T_c\|_{C^0}
\le Cd^{-3}\lambda^{-3/4}\le\lambda^{-1/2}, ∥ R c ∥ C t , x 2 + ∥ T c ∥ C 0 ≤ C d − 3 λ − 3/4 ≤ λ − 1/2 , where d − 3 ≤ λ 3 / b d^{-3}\le\lambda^{3/b} d − 3 ≤ λ 3/ b B 4 B_4 B 4 B 1 ⊃ supp f k B_1\supset\operatorname{supp}f_k B 1 ⊃ supp f k 6 d 6d 6 d
The density lower bound
We prove the lower bound for the full density perturbation required in Proposition 4.1.
Lemma 5.2 (Lower bound for the density perturbation). For sufficiently large a > 0 a>0 a > 0 θ = θ p + θ c + θ d \theta=\theta_p+\theta_c+\theta_d θ = θ p + θ c + θ d
θ ≥ − ∑ k a k + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ − C δ n + 1 16 . (69) \theta\geq-\sum_k a_k+\delta_{n+1}^2|R^+|+\delta_{n+1}^4|T^+|-C\delta_{n+1}^{16}. \tag*{(69)} θ ≥ − k ∑ a k + δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ − C δ n + 1 16 . (69) In particular, θ \theta θ (27) from Proposition 4.1.
We first compare the defects from w w w δ n + 1 16 η d E \delta_{n+1}^{16}\eta_dE δ n + 1 16 η d E χ R 0 ∂ t θ d \chi R_0\partial_t\theta_d χ R 0 ∂ t θ d θ d u + \theta_du^+ θ d u + θ d u + ⊗ u + \theta_du^+\otimes u^+ θ d u + ⊗ u +
Lemma 5.3 (Principal defect terms). For sufficiently large a > 0 a>0 a > 0
∣ ρ − ∑ k a k ∣ ≤ δ n + 1 − 1 . (70) \left|\rho-\sum_k a_k\right|\leq\delta_{n+1}^{-1}. \tag*{(70)} ρ − k ∑ a k ≤ δ n + 1 − 1 . (70) The new defects of Lemmas 4.3 and 4.4 satisfy
δ n + 1 2 ∣ R + − θ d u + − χ R 0 ∂ t θ d ∣ + δ n + 1 4 ∣ T + − θ d u + ⊗ u + ∣ ≤ 1 4 δ n + 1 16 η d E + λ − 1 / 2 . \delta_{n+1}^2\left|R^+-\theta_du^+-\chi R_0\partial_t\theta_d\right|+\delta_{n+1}^4\left|T^+-\theta_du^+\otimes u^+\right|
\leq\frac14\delta_{n+1}^{16}\eta_dE+\lambda^{-1/2}. δ n + 1 2 R + − θ d u + − χ R 0 ∂ t θ d + δ n + 1 4 T + − θ d u + ⊗ u + ≤ 4 1 δ n + 1 16 η d E + λ − 1/2 . Proof. We compare the products containing w w w Ψ μ ( λ x ) \Psi_\mu(\lambda x) Ψ μ ( λ x )
Step 1: Comparison with the stationary profile.
The old-scale bound (60) and Proposition 4.2 give (70) . Lemmas 4.1 and 3.11 imply
∣ w ∣ ≤ C ω 1 supp w , ∣ ∇ w ∣ ≤ C λ μ ω 1 supp w , |w|\leq C\omega\mathbf{1}_{\operatorname{supp}w},\qquad|\nabla w|\leq C\lambda\mu\omega\mathbf{1}_{\operatorname{supp}w}, ∣ w ∣ ≤ C ω 1 supp w , ∣∇ w ∣ ≤ C λ μ ω 1 supp w , 1 supp w ≤ C μ − 2 + γ / N Ψ μ ( λ x ) . \mathbf{1}_{\operatorname{supp}w}\leq C\mu^{-2+\gamma/N}\Psi_\mu(\lambda x). 1 supp w ≤ C μ − 2 + γ / N Ψ μ ( λ x ) . Since ∣ u ∣ ≤ ω |u|\leq\omega ∣ u ∣ ≤ ω w w w
∣ ( ρ − ∑ k a k ) w ∣ + ∣ ( ρ − ∑ k a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) ∣ ≤ C δ n + 1 − 1 ω 2 μ − 2 + γ / N Ψ μ ( λ x ) . \left|\left(\rho-\sum_k a_k\right)w\right|+\left|\left(\rho-\sum_k a_k\right)(u\otimes w+w\otimes u+w\otimes w)\right|
\leq C\delta_{n+1}^{-1}\omega^2\mu^{-2+\gamma/N}\Psi_\mu(\lambda x). ( ρ − k ∑ a k ) w + ( ρ − k ∑ a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) ≤ C δ n + 1 − 1 ω 2 μ − 2 + γ / N Ψ μ ( λ x ) . The viscous term is bounded by
∣ ∇ w + ∇ w T ∣ ≤ C λ μ ω μ − 2 + γ / N Ψ μ ( λ x ) . |\nabla w+\nabla w^{T}|\leq C\lambda\mu\omega\mu^{-2+\gamma/N}\Psi_\mu(\lambda x). ∣∇ w + ∇ w T ∣ ≤ C λ μ ω μ − 2 + γ / N Ψ μ ( λ x ) . For the oscillatory potentials, write Θ k − 1 = g ( λ x ) \Theta_k-1=g(\lambda x) Θ k − 1 = g ( λ x ) g = ζ μ ( ⋅ − z ) − 1 g=\zeta_\mu(\mathord\cdot-z)-1 g = ζ μ ( ⋅ − z ) − 1 z = y k + λ ω t k ∈ L k z=y_k+\lambda\omega t k\in L_k z = y k + λω t k ∈ L k ∥ f k ∥ C 1 ≤ C δ n + 1 − 1 \|f_k\|_{C^1}\leq C\delta_{n+1}^{-1} ∥ f k ∥ C 1 ≤ C δ n + 1 − 1 Lemma 3.11 gives
∣ R 0 [ ( Θ k − 1 ) f k ] ∣ + ∣ R [ ( Θ k − 1 ) f k V k ] ∣ ≤ C λ − 1 ω δ n + 1 − 1 μ γ / N Ψ μ ( λ x ) . |R_0[(\Theta_k-1)f_k]|+|R[(\Theta_k-1)f_kV_k]|
\leq C\lambda^{-1}\omega\delta_{n+1}^{-1}\mu^{\gamma/N}\Psi_\mu(\lambda x). ∣ R 0 [( Θ k − 1 ) f k ] ∣ + ∣ R [( Θ k − 1 ) f k V k ] ∣ ≤ C λ − 1 ω δ n + 1 − 1 μ γ / N Ψ μ ( λ x ) . The sum of these bounds is a multiple of Ψ μ ( λ x ) \Psi_\mu(\lambda x) Ψ μ ( λ x )
C δ n + 1 − 1 ( λ − 1 ω μ γ / N + ω 2 μ − 2 + γ / N + λ μ ω μ − 2 + γ / N ) ≤ C λ − 1 + 3 γ ≤ 1 8 δ n + 1 24 . C\delta_{n+1}^{-1}\left(\lambda^{-1}\omega\mu^{\gamma/N}+\omega^2\mu^{-2+\gamma/N}+\lambda\mu\omega\mu^{-2+\gamma/N}\right)
\leq C\lambda^{-1+3\gamma}\leq\frac18\delta_{n+1}^{24}. C δ n + 1 − 1 ( λ − 1 ω μ γ / N + ω 2 μ − 2 + γ / N + λ μ ω μ − 2 + γ / N ) ≤ C λ − 1 + 3 γ ≤ 8 1 δ n + 1 24 . Here μ γ / N = λ γ \mu^{\gamma/N}=\lambda^\gamma μ γ / N = λ γ μ − 2 + γ / N = λ − 2 N + γ \mu^{-2+\gamma/N}=\lambda^{-2N+\gamma} μ − 2 + γ / N = λ − 2 N + γ 27 γ < 1 27\gamma<1 27 γ < 1
Step 2: Localization and the compact corrections.
All terms estimated in Step 1 are supported in supp η u \operatorname{supp}\eta_u supp η u η d = 1 \eta_d=1 η d = 1 Lemma 3.1 . On B 4 B_4 B 4 Ψ μ ( λ x ) ≤ E ( x ) \Psi_\mu(\lambda x)\leq E(x) Ψ μ ( λ x ) ≤ E ( x ) Lemma 4.2 . Outside B 4 B_4 B 4 λ − 3 / 4 \lambda^{-3/4} λ − 3/4
It remains to estimate θ c u + \theta_cu^+ θ c u + θ c u + ⊗ u + \theta_cu^+\otimes u^+ θ c u + ⊗ u + χ R 0 ∂ t θ c \chi R_0\partial_t\theta_c χ R 0 ∂ t θ c R c , T c R_c,T_c R c , T c 3.3 , together with ∣ u + ∣ ≤ C ω |u^+|\leq C\omega ∣ u + ∣ ≤ C ω C δ n + 1 2 λ − 1 / 2 + C λ − 3 / 4 ω 2 C\delta_{n+1}^2\lambda^{-1/2}+C\lambda^{-3/4}\omega^2 C δ n + 1 2 λ − 1/2 + C λ − 3/4 ω 2 R + , T + R^+,T^+ R + , T + Δ P \Delta P Δ P Lemma 4.3 proves the claim for sufficiently large a > 0 a>0 a > 0
Lemma 5.4 (Time derivative of the density corrector). *For sufficiently large a > 0 a>0 a > 0
δ n + 1 2 ∣ χ R 0 ∂ t θ d ∣ ≤ 1 8 δ n + 1 16 η d E + 1 20 δ n + 1 24 . \delta_{n+1}^{2}|\chi\mathcal{R}_{0}\partial_t\theta_d|\leq\frac{1}{8}\delta_{n+1}^{16}\eta_d E+\frac{1}{20}\delta_{n+1}^{24}. δ n + 1 2 ∣ χ R 0 ∂ t θ d ∣ ≤ 8 1 δ n + 1 16 η d E + 20 1 δ n + 1 24 . Proof. This is the estimate for which the nested cutoffs in θ d \theta_d θ d m m m λ − 4 N \lambda^{-4N} λ − 4 N η d \eta_d η d
Let g = Ψ μ − f Ψ μ g=\Psi_\mu-f\Psi_\mu g = Ψ μ − f Ψ μ Lemma 4.2 replaces η d \eta_d η d η d ′ \eta'_d η d ′ Lemma 3.11 gives
∣ R 0 [ χ m g ( λ ⋅ ) ] ∣ ≤ C λ − 1 d − 1 Ψ μ ( λ x ) . |\mathcal{R}_0[\chi_m g(\lambda\cdot)]|\leq C\lambda^{-1}d^{-1}\Psi_\mu(\lambda x). ∣ R 0 [ χ m g ( λ ⋅ )] ∣ ≤ C λ − 1 d − 1 Ψ μ ( λ x ) . At distance at least d d d supp χ m \operatorname{supp}\chi_m supp χ m C d − 10 λ − 4 ≤ λ − 3 Cd^{-10}\lambda^{-4}\leq\lambda^{-3} C d − 10 λ − 4 ≤ λ − 3 m m m m < 8 N m<8N m < 8 N
λ − m / 2 ∣ R 0 [ χ m g ( λ ⋅ ) ] ∣ ≤ C λ − 1 / 2 d − 1 λ − ( m + 1 ) / 2 χ m + 1 Ψ μ ( λ x ) + λ − 3 − m / 2 . \lambda^{-m/2}|\mathcal{R}_0[\chi_mg(\lambda\cdot)]|\leq C\lambda^{-1/2}d^{-1}\lambda^{-(m+1)/2}\chi_{m+1}\Psi_\mu(\lambda x)
+\lambda^{-3-m/2}. λ − m /2 ∣ R 0 [ χ m g ( λ ⋅ )] ∣ ≤ C λ − 1/2 d − 1 λ − ( m + 1 ) /2 χ m + 1 Ψ μ ( λ x ) + λ − 3 − m /2 . The first line is a small multiple of the next positive summand of E E E m = 8 N m=8N m = 8 N ∥ Ψ μ ∥ ∞ ≤ C μ 2 \|\Psi_\mu\|_\infty\leq C\mu^2 ∥ Ψ μ ∥ ∞ ≤ C μ 2 Lemma 3.11 give
λ − 4 N ∣ R 0 [ χ 8 N g ( λ ⋅ ) ] ∣ ≤ C λ − 4 N − 1 d − 1 μ 2 . \lambda^{-4N}|\mathcal{R}_0[\chi_{8N}g(\lambda\cdot)]|\leq C\lambda^{-4N-1}d^{-1}\mu^2. λ − 4 N ∣ R 0 [ χ 8 N g ( λ ⋅ )] ∣ ≤ C λ − 4 N − 1 d − 1 μ 2 . The mean corrections are bounded by Lemma 4.2 , while ∣ R 0 χ m ∣ ≤ C |\mathcal{R}_0\chi_m|\leq C ∣ R 0 χ m ∣ ≤ C Lemma 3.3 . The old-scale bound (60) gives C δ n + 1 2 τ − 1 d − 1 λ − 1 / 2 ≤ λ − 1 / 4 C\delta_{n+1}^{2}\tau^{-1}d^{-1}\lambda^{-1/2}\leq\lambda^{-1/4} C δ n + 1 2 τ − 1 d − 1 λ − 1/2 ≤ λ − 1/4 Lemma 3.1 and sum the preceding bounds:
δ n + 1 2 ∣ χ R 0 ∂ t θ d ∣ ≤ δ n + 1 16 η d 1 − 1 / ( 16 N ) λ − 1 / 4 ∑ m < 8 N λ − ( m + 1 ) / 2 χ m + 1 Ψ μ ( λ x ) + C δ n + 1 16 ( λ − 3 / 4 + λ − 4 N − 3 / 4 μ 2 ) . \delta_{n+1}^{2}|\chi\mathcal{R}_0\partial_t\theta_d|
\leq\delta_{n+1}^{16}\eta_d^{1-1/(16N)}\lambda^{-1/4}
\sum_{m<8N}\lambda^{-(m+1)/2}\chi_{m+1}\Psi_\mu(\lambda x)
+C\delta_{n+1}^{16}\left(\lambda^{-3/4}+\lambda^{-4N-3/4}\mu^2\right). δ n + 1 2 ∣ χ R 0 ∂ t θ d ∣ ≤ δ n + 1 16 η d 1 − 1/ ( 16 N ) λ − 1/4 m < 8 N ∑ λ − ( m + 1 ) /2 χ m + 1 Ψ μ ( λ x ) + C δ n + 1 16 ( λ − 3/4 + λ − 4 N − 3/4 μ 2 ) . The second line is at most δ n + 1 24 / 40 \delta_{n+1}^{24}/40 δ n + 1 24 /40 a > 0 a>0 a > 0 λ − 4 N μ 2 = λ − 2 N \lambda^{-4N}\mu^2=\lambda^{-2N} λ − 4 N μ 2 = λ − 2 N 24 γ < 1 / 4 24\gamma<1/4 24 γ < 1/4
η d 1 − 1 / ( 16 N ) λ − 1 / 4 ≤ η d 8 + C N λ − 4 N . \eta_d^{1-1/(16N)}\lambda^{-1/4}\leq\frac{\eta_d}{8}+C_N\lambda^{-4N}. η d 1 − 1/ ( 16 N ) λ − 1/4 ≤ 8 η d + C N λ − 4 N . The weighted sum is bounded by E E E ∥ E ∥ ∞ ≤ C μ 2 \|E\|_\infty\leq C\mu^2 ∥ E ∥ ∞ ≤ C μ 2 Lemma 3.11 . Thus its contribution is at most
1 8 δ n + 1 16 η d E + C δ n + 1 16 λ − 2 N ≤ 1 8 δ n + 1 16 η d E + 1 40 δ n + 1 24 . \frac{1}{8}\delta_{n+1}^{16}\eta_dE+C\delta_{n+1}^{16}\lambda^{-2N}
\leq\frac{1}{8}\delta_{n+1}^{16}\eta_dE+\frac{1}{40}\delta_{n+1}^{24}. 8 1 δ n + 1 16 η d E + C δ n + 1 16 λ − 2 N ≤ 8 1 δ n + 1 16 η d E + 40 1 δ n + 1 24 . This proves the stated bound.
Proof of Lemma 5.2. By the input bounds in Proposition 4.1 and Lemma 4.1, ∣ u + ∣ ≤ ∣ u ∣ + ∣ w ∣ ≤ C ω |u^+|\leq|u|+|w|\leq C\omega ∣ u + ∣ ≤ ∣ u ∣ + ∣ w ∣ ≤ C ω Lemma 4.2 and δ n + 1 2 ω = λ − 2 γ + 4 γ / b \delta_{n+1}^{2}\omega=\lambda^{-2\gamma+4\gamma/b} δ n + 1 2 ω = λ − 2 γ + 4 γ / b
δ n + 1 2 ∣ θ d u + ∣ + δ n + 1 4 ∣ θ d u + ⊗ u + ∣ ≤ C δ n + 1 16 η d E ( δ n + 1 2 ω + δ n + 1 4 ω 2 ) ≤ 1 8 δ n + 1 16 η d E . \delta_{n+1}^{2}|\theta_du^+|+\delta_{n+1}^{4}|\theta_du^+\otimes u^+|
\leq C\delta_{n+1}^{16}\eta_dE(\delta_{n+1}^{2}\omega+\delta_{n+1}^{4}\omega^2)
\leq\frac{1}{8}\delta_{n+1}^{16}\eta_dE. δ n + 1 2 ∣ θ d u + ∣ + δ n + 1 4 ∣ θ d u + ⊗ u + ∣ ≤ C δ n + 1 16 η d E ( δ n + 1 2 ω + δ n + 1 4 ω 2 ) ≤ 8 1 δ n + 1 16 η d E . Combining this bound with Lemmas 5.3 and 5.4 gives
δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ ≤ 1 2 δ n + 1 16 η d E + 1 10 δ n + 1 24 , (71) \delta_{n+1}^{2}|R^+|+\delta_{n+1}^{4}|T^+|
\leq\frac{1}{2}\delta_{n+1}^{16}\eta_dE+\frac{1}{10}\delta_{n+1}^{24}, \tag*{(71)} δ n + 1 2 ∣ R + ∣ + δ n + 1 4 ∣ T + ∣ ≤ 2 1 δ n + 1 16 η d E + 10 1 δ n + 1 24 , (71) because C λ − 1 / 2 ≤ δ n + 1 24 / 20 C\lambda^{-1/2}\leq\delta_{n+1}^{24}/20 C λ − 1/2 ≤ δ n + 1 24 /20
By Lemmas 4.2 and 5.1, the full density perturbation satisfies
θ ≥ − ∑ k a k + δ n + 1 16 η d E − C δ n + 1 16 − λ − 3 / 4 . \theta\geq-\sum_k a_k+\delta_{n+1}^{16}\eta_dE-C\delta_{n+1}^{16}-\lambda^{-3/4}. θ ≥ − k ∑ a k + δ n + 1 16 η d E − C δ n + 1 16 − λ − 3/4 . Comparing this with (71) proves (69) , after absorbing the uniformly small terms into C δ n + 1 16 C\delta_{n+1}^{16} C δ n + 1 16 ω = δ n − 4 \omega=\delta_n^{-4} ω = δ n − 4
∑ k a k ≤ 48 δ n 12 + 24 δ n 4 ∣ R ∣ + 24 δ n 8 ∣ T ∣ ≤ 48 δ n 12 + δ n 2 ∣ R ∣ + δ n 4 ∣ T ∣ \sum_k a_k\leq48\delta_n^{12}+24\delta_n^{4}|R|+24\delta_n^{8}|T|
\leq48\delta_n^{12}+\delta_n^{2}|R|+\delta_n^{4}|T| k ∑ a k ≤ 48 δ n 12 + 24 δ n 4 ∣ R ∣ + 24 δ n 8 ∣ T ∣ ≤ 48 δ n 12 + δ n 2 ∣ R ∣ + δ n 4 ∣ T ∣ for sufficiently large a > 0 a>0 a > 0 48 δ n 12 + C δ n + 1 16 ≤ δ n 2 48\delta_n^{12}+C\delta_{n+1}^{16}\leq\delta_n^2 48 δ n 12 + C δ n + 1 16 ≤ δ n 2
Integral estimates
The pointwise bound already controls the defects in L p L^p L p
Lemma 5.5 (Integral estimates). For sufficiently large a > 0 a>0 a > 0
∥ ( R + , T + ) ∥ C t L p ≤ δ n + 1 8 (72) \|(R^+,T^+)\|_{C_tL^p}\leq\delta_{n+1}^8 \tag*{(72)} ∥ ( R + , T + ) ∥ C t L p ≤ δ n + 1 8 (72) and
∥ ρ + − ρ ∥ C t L p ≤ δ n 2 . (73) \|\rho^+-\rho\|_{C_tL^p}\leq\delta_n^2. \tag*{(73)} ∥ ρ + − ρ ∥ C t L p ≤ δ n 2 . (73) If n ≥ 1 n\geq1 n ≥ 1
∥ ρ + u + − ρ u ∥ C t L 1 + ∥ ρ + u + ⊗ u + − ρ u ⊗ u ∥ C t L 1 + ∥ Π + − Π ∥ L t 1 L x 1 ≤ δ n 2 . (74) \|\rho^+u^+-\rho u\|_{C_tL^1}+\|\rho^+u^+\otimes u^+-\rho u\otimes u\|_{C_tL^1}
+\|\Pi^+-\Pi\|_{L_t^1L_x^1}\leq\delta_n^2. \tag*{(74)} ∥ ρ + u + − ρ u ∥ C t L 1 + ∥ ρ + u + ⊗ u + − ρ u ⊗ u ∥ C t L 1 + ∥ Π + − Π ∥ L t 1 L x 1 ≤ δ n 2 . (74) Proof. The defect estimates of Lemmas 5.3 and 5.4, together with Lemma 4.2 and ∣ u + ∣ ≤ C ω |u^+|\leq C\omega ∣ u + ∣ ≤ C ω
∥ R + ∥ L p ≤ C δ n + 1 14 , ∥ T + ∥ L p ≤ C δ n + 1 12 . \|R^+\|_{L^p}\leq C\delta_{n+1}^{14},\qquad\|T^+\|_{L^p}\leq C\delta_{n+1}^{12}. ∥ R + ∥ L p ≤ C δ n + 1 14 , ∥ T + ∥ L p ≤ C δ n + 1 12 . For sufficiently large a > 0 a>0 a > 0
Step 1: The density increment.
For n ≥ 1 n\geq1 n ≥ 1 ∥ a k ∥ L p ≤ ∥ ∑ k a k ∥ L p ≤ C δ n 12 \|a_k\|_{L^p}\leq\|\sum_k a_k\|_{L^p}\leq C\delta_n^{12} ∥ a k ∥ L p ≤ ∥ ∑ k a k ∥ L p ≤ C δ n 12 n = 0 n=0 n = 0 Z ∗ Z_* Z ∗ ∥ a k ∥ ∞ ≤ C ∗ ω − 1 = C ∗ δ 0 4 \|a_k\|_\infty\leq C_*\omega^{-1}=C_*\delta_0^4 ∥ a k ∥ ∞ ≤ C ∗ ω − 1 = C ∗ δ 0 4 ∥ a k ∥ L p ≤ C ∗ δ n 4 \|a_k\|_{L^p}\leq C_*\delta_n^4 ∥ a k ∥ L p ≤ C ∗ δ n 4
Since 3 N ( 1 − 1 / p ) < 3 N ( p − 1 ) < γ / b 3N(1-1/p)<3N(p-1)<\gamma/b 3 N ( 1 − 1/ p ) < 3 N ( p − 1 ) < γ / b ∥ ζ μ ∥ L p ≤ C μ 3 − 3 / p ≤ C δ n − 1 \|\zeta_\mu\|_{L^p}\leq C\mu^{3-3/p}\leq C\delta_n^{-1} ∥ ζ μ ∥ L p ≤ C μ 3 − 3/ p ≤ C δ n − 1
∥ θ p ∥ L p ≤ C ∗ δ n − 1 ( δ n 4 + λ − 1 δ n + 1 2 ω − 1 ) + C ∗ δ n 4 ≤ C ∗ δ n 3 . \|\theta_p\|_{L^p}\leq C_*\delta_n^{-1}\left(\delta_n^4+\lambda^{-1}\delta_{n+1}^2\omega^{-1}\right)+C_*\delta_n^4
\leq C_*\delta_n^3. ∥ θ p ∥ L p ≤ C ∗ δ n − 1 ( δ n 4 + λ − 1 δ n + 1 2 ω − 1 ) + C ∗ δ n 4 ≤ C ∗ δ n 3 . Lemmas 5.1 and 4.2 give
∥ θ c ∥ ∞ ≤ λ − 3 / 4 , ∥ θ d ∥ L p ≤ C δ n + 1 16 . \|\theta_c\|_\infty\leq\lambda^{-3/4},\qquad\|\theta_d\|_{L^p}\leq C\delta_{n+1}^{16}. ∥ θ c ∥ ∞ ≤ λ − 3/4 , ∥ θ d ∥ L p ≤ C δ n + 1 16 . For sufficiently large a > 0 a>0 a > 0 θ p \theta_p θ p δ n 2 \delta_n^2 δ n 2 n ≥ 1 n\geq1 n ≥ 1
Step 2: The momentum increment.
Use the modified momentum identity in Lemma 4.3. By Lemma 3.4 with q = 1 q=1 q = 1
∥ ∑ k a k ( Θ k − 1 ) V k ∥ L 1 ≤ ω ∑ k ( 2 ∥ a k ∥ L 1 + C λ − 1 ∥ a k ∥ C 1 ) ≤ C δ n 8 + C λ − 1 δ n + 1 − 1 . \left\|\sum_k a_k(\Theta_k-1)V_k\right\|_{L^1}
\leq\omega\sum_k\left(2\|a_k\|_{L^1}+C\lambda^{-1}\|a_k\|_{C^1}\right)
\leq C\delta_n^8+C\lambda^{-1}\delta_{n+1}^{-1}. k ∑ a k ( Θ k − 1 ) V k L 1 ≤ ω k ∑ ( 2∥ a k ∥ L 1 + C λ − 1 ∥ a k ∥ C 1 ) ≤ C δ n 8 + C λ − 1 δ n + 1 − 1 . By Lemma 3.4 and Proposition 4.2,
∥ R 0 [ ( Θ k − 1 ) f k ] ∥ L 1 ≤ C λ − 1 ∥ f k ∥ C 1 ≤ C λ − 1 δ n + 1 − 1 . \|\mathcal{R}_0[(\Theta_k-1)f_k]\|_{L^1}
\leq C\lambda^{-1}\|f_k\|_{C^1}
\leq C\lambda^{-1}\delta_{n+1}^{-1}. ∥ R 0 [( Θ k − 1 ) f k ] ∥ L 1 ≤ C λ − 1 ∥ f k ∥ C 1 ≤ C λ − 1 δ n + 1 − 1 . For ∂ t θ d \partial_t\theta_d ∂ t θ d
∥ R 0 ∂ t θ d ∥ L 1 ≤ C τ − 1 λ − 1 d − 1 . \|\mathcal{R}_0\partial_t\theta_d\|_{L^1}\leq C\tau^{-1}\lambda^{-1}d^{-1}. ∥ R 0 ∂ t θ d ∥ L 1 ≤ C τ − 1 λ − 1 d − 1 . The decomposition of Δ P \Delta P Δ P ∥ χ ∇ P ∥ L 1 ≤ λ − 1 / 2 \|\chi\nabla P\|_{L^1}\leq\lambda^{-1/2} ∥ χ ∇ P ∥ L 1 ≤ λ − 1/2 a > 0 a>0 a > 0 R c R_c R c
∥ ρ + u + − ρ u ∥ L 1 ≤ ∥ ∑ k a k ( Θ k − 1 ) V k ∥ L 1 + ∥ χ ∇ P + R c ∥ L 1 + ∥ R + ∥ L p + ∥ R ∥ L p ≤ C δ n 8 . (75) \|\rho^+u^+-\rho u\|_{L^1}
\leq
\left\|\sum_k a_k(\Theta_k-1)V_k\right\|_{L^1}
+\|\chi\nabla P+R_c\|_{L^1}
+\|R^+\|_{L^p}+\|R\|_{L^p}
\leq C\delta_n^8. \tag*{(75)} ∥ ρ + u + − ρ u ∥ L 1 ≤ k ∑ a k ( Θ k − 1 ) V k L 1 + ∥ χ ∇ P + R c ∥ L 1 + ∥ R + ∥ L p + ∥ R ∥ L p ≤ C δ n 8 . (75) Step 3: The momentum flux.
We use the flux identity in Lemma 4.4. The principal oscillatory term is bounded by Lemma 3.4 and the coefficient bound in Step 1:
C ω 2 ( δ n 12 + λ − 1 δ n + 1 − 1 ω − 1 ) ≤ C δ n 4 . C\omega^2\left(\delta_n^{12}+\lambda^{-1}\delta_{n+1}^{-1}\omega^{-1}\right)\le C\delta_n^4. C ω 2 ( δ n 12 + λ − 1 δ n + 1 − 1 ω − 1 ) ≤ C δ n 4 . The scalar pressure correction has the same size. Indeed, Proposition 4.2 gives
∥ π ∥ L 1 ≤ C ( ω δ n 8 + ω ∥ R ∥ L 1 + ∥ T ∥ L 1 ) ≤ C δ n 4 , \begin{aligned}
\|\pi\|_{L^1}\le C\left(\omega\delta_n^8+\omega\|R\|_{L^1}+\|T\|_{L^1}\right) \\
&\le C\delta_n^4,
\end{aligned} ∥ π ∥ L 1 ≤ C ( ω δ n 8 + ω ∥ R ∥ L 1 + ∥ T ∥ L 1 ) ≤ C δ n 4 , and ∥ T ∥ L 1 ≤ 2 δ n 8 \|T\|_{L^1}\le2\delta_n^8 ∥ T ∥ L 1 ≤ 2 δ n 8
For the remaining products, combine (5.11) with the support and velocity bounds of Lemma 4.1:
∥ ( ρ − ∑ k a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) ∥ L 1 ≤ C δ n + 1 − 1 ω 2 μ − 3 . \left\|\left(\rho-\sum_k a_k\right)(u\otimes w+w\otimes u+w\otimes w)\right\|_{L^1}
\le C\delta_{n+1}^{-1}\omega^2\mu^{-3}. ( ρ − k ∑ a k ) ( u ⊗ w + w ⊗ u + w ⊗ w ) L 1 ≤ C δ n + 1 − 1 ω 2 μ − 3 . By Lemma 5.1 and ∣ u + ∣ ≤ C ω |u^+|\le C\omega ∣ u + ∣ ≤ C ω ∥ θ c u + ⊗ u + ∥ L 1 ≤ λ − 1 / 2 \|\theta_cu^+\otimes u^+\|_{L^1}\le\lambda^{-1/2} ∥ θ c u + ⊗ u + ∥ L 1 ≤ λ − 1/2 ∣ u + ∣ ≤ C ω |u^+|\le C\omega ∣ u + ∣ ≤ C ω
∥ θ d u + ⊗ u + ∥ L 1 ≤ C δ n + 1 16 ω 2 ∥ E ∥ L 1 ≤ C δ n + 1 12 . \begin{aligned}
\|\theta_du^+\otimes u^+\|_{L^1}\le C\delta_{n+1}^{16}\omega^2\|E\|_{L^1} \\
&\le C\delta_{n+1}^{12}.
\end{aligned} ∥ θ d u + ⊗ u + ∥ L 1 ≤ C δ n + 1 16 ω 2 ∥ E ∥ L 1 ≤ C δ n + 1 12 . Adding these estimates bounds the full flux increment by C δ n 4 C\delta_n^4 C δ n 4
Step 4: The pressure increment.
For the pressure in Lemma 4.4, recall Π + − Π = − π + χ ∂ t P \Pi^+-\Pi=-\pi+\chi\partial_tP Π + − Π = − π + χ ∂ t P ∥ π ∥ L 1 ≤ C δ n 4 \|\pi\|_{L^1}\le C\delta_n^4 ∥ π ∥ L 1 ≤ C δ n 4
∂ t [ ( Θ k − 1 ) f k ] = − div [ ( Θ k − 1 ) f k V k ] + ( Θ k − 1 ) ( ∂ t + V k ⋅ ∇ ) f k . \partial_t\left[(\Theta_k-1)f_k\right]
=-\operatorname{div}\left[(\Theta_k-1)f_kV_k\right]
+(\Theta_k-1)(\partial_t+V_k\cdot\nabla)f_k. ∂ t [ ( Θ k − 1 ) f k ] = − div [ ( Θ k − 1 ) f k V k ] + ( Θ k − 1 ) ( ∂ t + V k ⋅ ∇ ) f k . Apply Δ − 1 P ≠ 0 \Delta^{-1}\mathbb{P}_{\ne0} Δ − 1 P = 0 div \operatorname{div} div
∥ ∂ t Δ − 1 P ≠ 0 [ ( Θ k − 1 ) f k ] ∥ L 1 ≤ C λ − 1 ( ω ∥ f k ∥ C 1 + ∥ ( ∂ t + V k ⋅ ∇ ) f k ∥ C 1 ) ≤ C λ − 1 ω δ n + 1 − 1 . \begin{aligned}
\left\|\partial_t\Delta^{-1}\mathbb{P}_{\ne0}\left[(\Theta_k-1)f_k\right]\right\|_{L^1}
\le C\lambda^{-1}\left(\omega\|f_k\|_{C^1}+\|(\partial_t+V_k\cdot\nabla)f_k\|_{C^1}\right) \\
&\le C\lambda^{-1}\omega\delta_{n+1}^{-1}.
\end{aligned} ∂ t Δ − 1 P = 0 [ ( Θ k − 1 ) f k ] L 1 ≤ C λ − 1 ( ω ∥ f k ∥ C 1 + ∥ ( ∂ t + V k ⋅ ∇ ) f k ∥ C 1 ) ≤ C λ − 1 ω δ n + 1 − 1 . Further, ∥ Δ − 1 P ≠ 0 ∂ t 2 θ c ∥ L 1 ≤ C λ − 3 / 4 \|\Delta^{-1}\mathbb{P}_{\ne0}\partial_t^2\theta_c\|_{L^1}\le C\lambda^{-3/4} ∥ Δ − 1 P = 0 ∂ t 2 θ c ∥ L 1 ≤ C λ − 3/4
∥ Δ − 1 ∂ t 2 θ d ∥ L 1 ≤ C τ − 2 λ − 1 d − 1 . \|\Delta^{-1}\partial_t^2\theta_d\|_{L^1}\le C\tau^{-2}\lambda^{-1}d^{-1}. ∥ Δ − 1 ∂ t 2 θ d ∥ L 1 ≤ C τ − 2 λ − 1 d − 1 . Hence ∥ Π + − Π ∥ L t , x 1 ≤ C δ n 4 + λ − 1 / 2 \|\Pi^+-\Pi\|_{L^1_{t,x}}\le C\delta_n^4+\lambda^{-1/2} ∥ Π + − Π ∥ L t , x 1 ≤ C δ n 4 + λ − 1/2 a > 0 a>0 a > 0 1 3 δ n 2 \frac{1}{3}\delta_n^2 3 1 δ n 2 □ \square □
Derivative bounds
It remains to control the growth of the smooth norms. We use a coarse polynomial bound: the construction costs at most a power of λ \lambda λ N N N λ N 2 \lambda^{N^2} λ N 2
Lemma 5.6 (Derivative bounds). For sufficiently large a > 0 a>0 a > 0
∥ ( ρ + , u + , R + , T + ) ∥ C t , x 3 + ∥ Π + − Π ∥ C t , x 3 ≤ 1 2 λ N 2 . \|(\rho^+,u^+,R^+,T^+)\|_{C^3_{t,x}}+\|\Pi^+-\Pi\|_{C^3_{t,x}}\le\frac{1}{2}\lambda^{N^2}. ∥ ( ρ + , u + , R + , T + ) ∥ C t , x 3 + ∥ Π + − Π ∥ C t , x 3 ≤ 2 1 λ N 2 . Proof. We first bound the density and velocity perturbations, then the new errors. The losses from cutoff derivatives are controlled by (5.1) and absorbed in the extra powers of δ n + 1 − 1 \delta_{n+1}^{-1} δ n + 1 − 1
Step 1: The perturbations and transport errors.
For 0 ≤ j ≤ 6 0\le j\le6 0 ≤ j ≤ 6 C t , x j C^j_{t,x} C t , x j
∥ Θ k ∥ ≤ C μ 3 ( λ μ ω ) j , \|\Theta_k\|\le C\mu^3(\lambda\mu\omega)^j, ∥ Θ k ∥ ≤ C μ 3 ( λ μ ω ) j , ∥ W k ∥ ≤ C ω ( λ μ ω ) j , \|W_k\|\le C\omega(\lambda\mu\omega)^j, ∥ W k ∥ ≤ C ω ( λ μ ω ) j , ∥ Ψ μ ( λ ⋅ ) ∥ ≤ C μ 2 ( λ μ ω ) j . \|\Psi_\mu(\lambda\cdot)\|\le C\mu^2(\lambda\mu\omega)^j. ∥ Ψ μ ( λ ⋅ ) ∥ ≤ C μ 2 ( λ μ ω ) j . By Proposition 4.2,
∥ a k ∥ ≤ δ n + 1 − 1 , ∥ π ∥ + ∥ f k ∥ ≤ C δ n + 1 − 1 ω . \|a_k\| \le\delta_{n+1}^{-1}, \qquad\|\pi\|+\|f_k\| \le C\delta_{n+1}^{-1}\omega. ∥ a k ∥ ≤ δ n + 1 − 1 , ∥ π ∥ + ∥ f k ∥ ≤ C δ n + 1 − 1 ω . Apply Lemma 3.2 to the density perturbations of Lemma 4.2. Its bound c m ≤ C c_m \le C c m ≤ C
∥ θ p ∥ ≤ δ n + 1 − 2 μ 3 ( λ μ ω ) j , \|\theta_p\| \le\delta_{n+1}^{-2}\mu^3(\lambda\mu\omega)^j, ∥ θ p ∥ ≤ δ n + 1 − 2 μ 3 ( λ μ ω ) j , ∥ θ d ∥ ≤ δ n + 1 4 μ 2 ( λ μ ω ) j , \|\theta_d\| \le\delta_{n+1}^{4}\mu^2(\lambda\mu\omega)^j, ∥ θ d ∥ ≤ δ n + 1 4 μ 2 ( λ μ ω ) j , while ∥ θ c ∥ ≤ 1 \|\theta_c\| \le1 ∥ θ c ∥ ≤ 1 P P P 1 ≤ j ≤ 6 1 \le j \le6 1 ≤ j ≤ 6
∥ Δ P ∥ C t , x j − 1 + ∥ ∑ k ( Θ k − 1 ) f k V k ∥ C t , x j − 1 ≤ δ n + 1 − 2 μ 3 ( λ μ ω ) j . \|\Delta P\|_{C_{t,x}^{j-1}}+\left\|\sum_k(\Theta_k-1)f_kV_k\right\|_{C_{t,x}^{j-1}} \le\delta_{n+1}^{-2}\mu^3(\lambda\mu\omega)^j. ∥Δ P ∥ C t , x j − 1 + k ∑ ( Θ k − 1 ) f k V k C t , x j − 1 ≤ δ n + 1 − 2 μ 3 ( λ μ ω ) j . Here each derivative of a cutoff costs at most a power of τ − 1 + d − 1 ≤ λ n \tau^{-1}+d^{-1}\le\lambda_n τ − 1 + d − 1 ≤ λ n θ c \theta_c θ c
Step 2: The new defects and pressure. Apply Lemmas 3.3 and 3.7 to P P P R c R_c R c 0 ≤ j ≤ 5 0\le j\le5 0 ≤ j ≤ 5
∥ P ∥ C t , x j + ∥ ∇ P ∥ C t , x j + ∥ R c ∥ C t , x j ≤ δ n + 1 − 3 μ 3 ( λ μ ω ) j + 1 . \|P\|_{C_{t,x}^j}+\|\nabla P\|_{C_{t,x}^j}+\|R_c\|_{C_{t,x}^j}\le\delta_{n+1}^{-3}\mu^3(\lambda\mu\omega)^{j+1}. ∥ P ∥ C t , x j + ∥∇ P ∥ C t , x j + ∥ R c ∥ C t , x j ≤ δ n + 1 − 3 μ 3 ( λ μ ω ) j + 1 . The formula for T c T_c T c
∥ T c ∥ C t , x 3 ≤ δ n + 1 − 4 μ 3 ( λ μ ω ) 6 ω . \|T_c\|_{C_{t,x}^3}\le\delta_{n+1}^{-4}\mu^3(\lambda\mu\omega)^6\omega. ∥ T c ∥ C t , x 3 ≤ δ n + 1 − 4 μ 3 ( λ μ ω ) 6 ω . These estimates apply to all fields of Lemma 4.4. The products are controlled by Lemma 3.2, and the final symmetric antidivergence by Lemma 3.8. Together with the velocity bound, all the required norms are dominated by
C δ n + 1 − 5 μ 3 ω 2 ( λ μ ω ) 9 ≤ λ 50 N ≤ λ N 2 / 2 . C\delta_{n+1}^{-5}\mu^3\omega^2(\lambda\mu\omega)^9\le\lambda^{50N}\le\lambda^{N^2/2}. C δ n + 1 − 5 μ 3 ω 2 ( λ μ ω ) 9 ≤ λ 50 N ≤ λ N 2 /2 . Here 50 N < N 2 / 2 50N<N^2/2 50 N < N 2 /2 1 2 λ N 2 \frac12\lambda^{N^2} 2 1 λ N 2 a > 0 a>0 a > 0
Proof of Proposition 4.1
Proof of Proposition 4.1. Take the fields Z + Z^+ Z + K ′ = supp η d K'=\operatorname{supp}\eta_d K ′ = supp η d U ⊂ B ( x 0 , 1 2 ( r − + r + ) ) U\subset B(x_0,\frac12(r_-+r_+)) U ⊂ B ( x 0 , 2 1 ( r − + r + ))
The increment bounds follow from (5.14), Lemma 4.1 and
∥ u + ∥ ∞ ≤ 2 δ n − 1 + C ω ≤ δ n + 1 − 1 . \|u^+\|_\infty\le2\delta_n^{-1}+C\omega\le\delta_{n+1}^{-1}. ∥ u + ∥ ∞ ≤ 2 δ n − 1 + C ω ≤ δ n + 1 − 1 . The nonlinear estimates are (5.15), and the derivative bound is Lemma 5.6.
Regularization and proof of Proposition 2.1
Proposition 4.1 controls three derivatives, whereas its application at the next stage requires twenty. We recover these derivatives by regularization. The linear form of the approximate equations lets us mollify without introducing a new equation error; we then choose the support gaps and close the induction.
Regularization
We mollify differences from the fixed approximate solution in the linear variables of Definition 2.1.
Let Z ∗ = ( ρ ∗ , u ∗ , Π ‾ , R ∗ , T ∗ ) Z_*=(\rho_*,u_*,\overline{\Pi},R_*,T_*) Z ∗ = ( ρ ∗ , u ∗ , Π , R ∗ , T ∗ ) m ∗ m_* m ∗ Φ ∗ \Phi_* Φ ∗ κ ∈ C ∞ ( R 4 ) \kappa\in C^\infty(\mathbb{R}^4) κ ∈ C ∞ ( R 4 ) κ h ( t , x ) = h − 4 κ ( t / h , x / h ) \kappa_h(t,x)=h^{-4}\kappa(t/h,x/h) κ h ( t , x ) = h − 4 κ ( t / h , x / h ) F F F [ 0 , 1 ] × T 3 [0,1]\times\mathbb{T}^3 [ 0 , 1 ] × T 3 K ⊂ ( 0 , 1 ) × T 3 K\subset(0,1)\times\mathbb{T}^3 K ⊂ ( 0 , 1 ) × T 3 0 < h < 1 2 dist ( K , { 0 , 1 } × T 3 ) 0<h<\frac12\operatorname{dist}(K,\{0,1\}\times\mathbb{T}^3) 0 < h < 2 1 dist ( K , { 0 , 1 } × T 3 ) F h = κ h ∗ F F_h=\kappa_h*F F h = κ h ∗ F F F F F h ( t , x ) = 0 F_h(t,x)=0 F h ( t , x ) = 0 dist ( ( t , x ) , K ) > h \operatorname{dist}((t,x),K)>h dist (( t , x ) , K ) > h k ≥ 0 k\ge0 k ≥ 0
∥ F h − F ∥ C t , x k ≤ 2 h ∥ F ∥ C t , x k + 1 , ∥ F h ∥ C t , x k ≤ C k h − max { k − 3 , 0 } ∥ F ∥ C t , x 3 . (76) \|F_h-F\|_{C_{t,x}^k}\le2h\|F\|_{C_{t,x}^{k+1}},\qquad\|F_h\|_{C_{t,x}^k}\le C_kh^{-\max\{k-3,0\}}\|F\|_{C_{t,x}^3}. \tag*{(76)} ∥ F h − F ∥ C t , x k ≤ 2 h ∥ F ∥ C t , x k + 1 , ∥ F h ∥ C t , x k ≤ C k h − m a x { k − 3 , 0 } ∥ F ∥ C t , x 3 . (76) Lemma 6.1 (Regularization). Let Z = ( ρ , u , Π , R , T ) Z=(\rho,u,\Pi,R,T) Z = ( ρ , u , Π , R , T ) Z − Z ∗ Z-Z_\ast Z − Z ∗ K ⊂ ( 0 , 1 ) × T 3 K\subset(0,1)\times\mathbb{T}^3 K ⊂ ( 0 , 1 ) × T 3 h h h
ρ ′ = ρ ∗ + ( ρ − ρ ∗ ) h , u ′ = u ∗ + ( u − u ∗ ) h , Π ′ = Π ‾ + ( Π − Π ‾ ) h . (77) \begin{aligned}
\rho'&=\rho_\ast+(\rho-\rho_\ast)h,\\
u'&=u_\ast+(u-u_\ast)h,\\
\Pi'&=\overline{\Pi}+(\Pi-\overline{\Pi})h. \tag*{(77)}
\end{aligned} ρ ′ u ′ Π ′ = ρ ∗ + ( ρ − ρ ∗ ) h , = u ∗ + ( u − u ∗ ) h , = Π + ( Π − Π ) h . (77) Recover the defects from the regularized modified momentum and flux:
R ′ = ρ ′ u ′ − m ∗ − ( m − m ∗ ) h , T ′ = ρ ′ u ′ ⊗ u ′ − Φ ∗ − ( Φ − Φ ∗ ) h . (78) \begin{aligned}
R'&=\rho'u'-m_\ast-(m-m_\ast)h,\\
T'&=\rho'u'\otimes u'-\Phi_\ast-(\Phi-\Phi_\ast)h. \tag*{(78)}
\end{aligned} R ′ T ′ = ρ ′ u ′ − m ∗ − ( m − m ∗ ) h , = ρ ′ u ′ ⊗ u ′ − Φ ∗ − ( Φ − Φ ∗ ) h . (78) Then Z ′ = ( ρ ′ , u ′ , Π ′ , R ′ , T ′ ) Z'=(\rho',u',\Pi',R',T') Z ′ = ( ρ ′ , u ′ , Π ′ , R ′ , T ′ ) Z ′ ( t , x ) = Z ∗ ( t , x ) Z'(t,x)=Z_\ast(t,x) Z ′ ( t , x ) = Z ∗ ( t , x ) dist ( ( t , x ) , K ) > h \operatorname{dist}((t,x),K)>h dist (( t , x ) , K ) > h
Set the positive constant
H = 1 + ∥ ( ρ , u , R , T ) ∥ C t , x 3 + ∥ Π − Π ‾ ∥ C t , x 3 + ∥ ( ρ ∗ , u ∗ , R ∗ , T ∗ ) ∥ C t , x 3 . \mathcal{H}=1+\|(\rho,u,R,T)\|_{C^3_{t,x}}+\|\Pi-\overline{\Pi}\|_{C^3_{t,x}}+\|(\rho_\ast,u_\ast,R_\ast,T_\ast)\|_{C^3_{t,x}}. H = 1 + ∥ ( ρ , u , R , T ) ∥ C t , x 3 + ∥Π − Π ∥ C t , x 3 + ∥ ( ρ ∗ , u ∗ , R ∗ , T ∗ ) ∥ C t , x 3 . The approximation error satisfies
∥ Z ′ − Z ∥ C t , x 0 + ∥ u ′ − u ∥ C t H s + ∥ ρ ′ u ′ − ρ u ∥ C t L 1 + ∥ ρ ′ u ′ ⊗ u ′ − ρ u ⊗ u ∥ C t L 1 ≤ C 0 h H 3 . (79) \|Z'-Z\|_{C^0_{t,x}}+\|u'-u\|_{C_tH^s}+\|\rho'u'-\rho u\|_{C_tL^1}+\|\rho'u'\otimes u'-\rho u\otimes u\|_{C_tL^1}\le C_0h\mathcal{H}^3. \tag*{(79)} ∥ Z ′ − Z ∥ C t , x 0 + ∥ u ′ − u ∥ C t H s + ∥ ρ ′ u ′ − ρ u ∥ C t L 1 + ∥ ρ ′ u ′ ⊗ u ′ − ρ u ⊗ u ∥ C t L 1 ≤ C 0 h H 3 . (79) The higher derivatives obey
∥ ( ρ ′ , u ′ , R ′ , T ′ ) ∥ C t , x 20 ≤ C ∗ h − 17 H 3 , (80) \|(\rho',u',R',T')\|_{C^{20}_{t,x}}\le C_\ast h^{-17}\mathcal{H}^3, \tag*{(80)} ∥ ( ρ ′ , u ′ , R ′ , T ′ ) ∥ C t , x 20 ≤ C ∗ h − 17 H 3 , (80) while the pressure satisfies
∥ Π ′ − Π ‾ ∥ C t , x 3 ≤ ∥ Π − Π ‾ ∥ C t , x 3 . (81) \|\Pi'-\overline{\Pi}\|_{C^3_{t,x}}\le\|\Pi-\overline{\Pi}\|_{C^3_{t,x}}. \tag*{(81)} ∥ Π ′ − Π ∥ C t , x 3 ≤ ∥Π − Π ∥ C t , x 3 . (81) Here C 0 C_0 C 0 κ \kappa κ C ∗ C_\ast C ∗ ∥ ( ρ ∗ , u ∗ , R ∗ , T ∗ ) ∥ C t , x 20 \|(\rho_\ast,u_\ast,R_\ast,T_\ast)\|_{C^{20}_{t,x}} ∥ ( ρ ∗ , u ∗ , R ∗ , T ∗ ) ∥ C t , x 20
Proof. By the linear formulation in Definition 2.1, the differences of Z Z Z Z ∗ Z_\ast Z ∗
∂ t ( ρ − ρ ∗ ) + div ( m − m ∗ ) = 0 , \partial_t(\rho-\rho_\ast)+\operatorname{div}(m-m_\ast)=0, ∂ t ( ρ − ρ ∗ ) + div ( m − m ∗ ) = 0 , ∂ t ( m − m ∗ ) + div ( Φ − Φ ∗ ) + ∇ ( Π − Π ‾ ) − Δ ( u − u ∗ ) = 0. \partial_t(m-m_\ast)+\operatorname{div}(\Phi-\Phi_\ast)+\nabla(\Pi-\overline{\Pi})-\Delta(u-u_\ast)=0. ∂ t ( m − m ∗ ) + div ( Φ − Φ ∗ ) + ∇ ( Π − Π ) − Δ ( u − u ∗ ) = 0. and these differences are supported in K K K ρ ′ u ′ − R ′ = m ∗ + ( m − m ∗ ) h \rho'u'-R'=m_\ast+(m-m_\ast)h ρ ′ u ′ − R ′ = m ∗ + ( m − m ∗ ) h ρ ′ u ′ ⊗ u ′ − T ′ = Φ ∗ + ( Φ − Φ ∗ ) h \rho'u'\otimes u'-T'=\Phi_\ast+(\Phi-\Phi_\ast)h ρ ′ u ′ ⊗ u ′ − T ′ = Φ ∗ + ( Φ − Φ ∗ ) h Z ∗ Z_\ast Z ∗ Z ′ Z' Z ′ div u ′ = 0 \operatorname{div}u'=0 div u ′ = 0 T ′ T' T ′ dist ( ( t , x ) , K ) > h \operatorname{dist}((t,x),K)>h dist (( t , x ) , K ) > h ρ ′ = ρ ∗ \rho'=\rho_\ast ρ ′ = ρ ∗ u ′ = u ∗ u'=u_\ast u ′ = u ∗ Π ′ = Π ‾ \Pi'=\overline{\Pi} Π ′ = Π R ′ = R ∗ R'=R_\ast R ′ = R ∗ T ′ = T ∗ T'=T_\ast T ′ = T ∗
Step 1: Approximation in low norms.
In this step C C C κ \kappa κ ∥ ρ − ρ ∗ ∥ C 3 + ∥ u − u ∗ ∥ C 3 + ∥ Π − Π ‾ ∥ C 3 ≤ 3 H \|\rho-\rho_\ast\|_{C^3}+\|u-u_\ast\|_{C^3}+\|\Pi-\overline{\Pi}\|_{C^3}\le3\mathcal{H} ∥ ρ − ρ ∗ ∥ C 3 + ∥ u − u ∗ ∥ C 3 + ∥Π − Π ∥ C 3 ≤ 3 H ∥ m − m ∗ ∥ C 1 ≤ C H 2 \|m-m_\ast\|_{C^1}\le C\mathcal{H}^2 ∥ m − m ∗ ∥ C 1 ≤ C H 2 ∥ Φ − Φ ∗ ∥ C 1 ≤ C H 3 \|\Phi-\Phi_\ast\|_{C^1}\le C\mathcal{H}^3 ∥Φ − Φ ∗ ∥ C 1 ≤ C H 3
∥ ρ ′ − ρ ∥ C 0 + ∥ u ′ − u ∥ C 2 + ∥ Π ′ − Π ∥ C 0 ≤ C h H , \|\rho'-\rho\|_{C^0}+\|u'-u\|_{C^2}+\|\Pi'-\Pi\|_{C^0}\le Ch\mathcal{H}, ∥ ρ ′ − ρ ∥ C 0 + ∥ u ′ − u ∥ C 2 + ∥ Π ′ − Π ∥ C 0 ≤ C h H , since ρ ′ − ρ = ( ρ − ρ ∗ ) h − ( ρ − ρ ∗ ) \rho'-\rho=(\rho-\rho_\ast)h-(\rho-\rho_\ast) ρ ′ − ρ = ( ρ − ρ ∗ ) h − ( ρ − ρ ∗ ) u u u Π \Pi Π ∥ u ′ − u ∥ C t H s ≤ C ∥ u ′ − u ∥ C 2 ≤ C h H \|u'-u\|_{C_tH^s}\le C\|u'-u\|_{C^2}\le Ch\mathcal{H} ∥ u ′ − u ∥ C t H s ≤ C ∥ u ′ − u ∥ C 2 ≤ C h H ∥ u ′ ∥ ∞ + ∥ ρ ′ ∥ ∞ ≤ C H \|u'\|_\infty+\|\rho'\|_\infty\le C\mathcal{H} ∥ u ′ ∥ ∞ + ∥ ρ ′ ∥ ∞ ≤ C H
R ′ − R = ( ρ ′ u ′ − ρ u ) − [ ( m − m ∗ ) h − ( m − m ∗ ) ] , R'-R=(\rho'u'-\rho u)-\big[(m-m_\ast)h-(m-m_\ast)\big], R ′ − R = ( ρ ′ u ′ − ρ u ) − [ ( m − m ∗ ) h − ( m − m ∗ ) ] , because R = ρ u − m ∗ − ( m − m ∗ ) R=\rho u-m_\ast-(m-m_\ast) R = ρ u − m ∗ − ( m − m ∗ ) C h H 2 Ch\mathcal{H}^2 C h H 2 ρ ′ u ′ − ρ u = ( ρ ′ − ρ ) u ′ + ρ ( u ′ − u ) \rho'u'-\rho u=(\rho'-\rho)u'+\rho(u'-u) ρ ′ u ′ − ρ u = ( ρ ′ − ρ ) u ′ + ρ ( u ′ − u ) ∥ T ′ − T ∥ C 0 + ∥ ρ ′ u ′ ⊗ u ′ − ρ u ⊗ u ∥ C 0 ≤ C h H 3 \|T'-T\|_{C^0}+\|\rho'u'\otimes u'-\rho u\otimes u\|_{C^0}\le Ch\mathcal{H}^3 ∥ T ′ − T ∥ C 0 + ∥ ρ ′ u ′ ⊗ u ′ − ρ u ⊗ u ∥ C 0 ≤ C h H 3 ∣ T 3 ∣ = 1 |T^3|=1 ∣ T 3 ∣ = 1 L 1 L^1 L 1 C 0 C^0 C 0
Step 2: Higher derivatives.
By (6.1), for 0 ≤ i ≤ 20 0\le i\le20 0 ≤ i ≤ 20
∥ ρ ′ ∥ C i + ∥ u ′ ∥ C i ≤ C ∗ + C h − max { i − 3 , 0 } H . \|\rho'\|_{C^i}+\|u'\|_{C^i}\le C_\ast+Ch^{-\max\{i-3,0\}}\mathcal{H}. ∥ ρ ′ ∥ C i + ∥ u ′ ∥ C i ≤ C ∗ + C h − m a x { i − 3 , 0 } H . The regularized modified momentum and flux satisfy
∥ ( m − m ∗ ) h ∥ C 20 + ∥ ( Φ − Φ ∗ ) h ∥ C 20 ≤ C h − 17 H 3 . \|(m-m_\ast)h\|_{C^{20}}+\|(\Phi-\Phi_\ast)h\|_{C^{20}}\le Ch^{-17}\mathcal{H}^3. ∥ ( m − m ∗ ) h ∥ C 20 + ∥ ( Φ − Φ ∗ ) h ∥ C 20 ≤ C h − 17 H 3 . In differentiating ρ ′ u ′ \rho'u' ρ ′ u ′ ρ ′ u ′ ⊗ u ′ \rho'u' \otimes u' ρ ′ u ′ ⊗ u ′ h − 1 h^{-1} h − 1
∑ l max { i l − 3 , 0 } ≤ 17 whenever ∑ l i l = 20. \sum_l \max\{i_l-3,0\} \le17 \qquad\text{whenever} \qquad\sum_l i_l=20. l ∑ max { i l − 3 , 0 } ≤ 17 whenever l ∑ i l = 20. Leibniz’ rule therefore gives the bound for R ′ R' R ′ T ′ T' T ′ (80) . Finally, Π ′ − Π ˉ = ( Π − Π ˉ ) h \Pi'-\bar{\Pi}=(\Pi-\bar{\Pi})_h Π ′ − Π ˉ = ( Π − Π ˉ ) h C 3 C^3 C 3 (81) .
Proof of Proposition 2.1
Proof. At each stage we first regularize the difference from Z ∗ Z_* Z ∗
Step 1: Choice of the support margins.
Write J = ( t − , t + ) J=(t_-,t_+) J = ( t − , t + ) ℓ > 0 \ell>0 ℓ > 0 K ∗ ⊂ ( t − + 2 ℓ , t + − 2 ℓ ) × B K_* \subset(t_-+2\ell,t_+-2\ell)\times B K ∗ ⊂ ( t − + 2 ℓ , t + − 2 ℓ ) × B
I n = [ t − + ℓ ( 1 + 2 − n ) , t + − ℓ ( 1 + 2 − n ) ] . I_n=\left[t_-+\ell(1+2^{-n}),t_+-\ell(1+2^{-n})\right]. I n = [ t − + ℓ ( 1 + 2 − n ) , t + − ℓ ( 1 + 2 − n ) ] . and
I n ′ = [ t − + ℓ ( 1 + 3 4 2 − n ) , t + − ℓ ( 1 + 3 4 2 − n ) ] . I'_n=\left[t_-+\ell\left(1+\frac{3}{4}2^{-n}\right),t_+-\ell\left(1+\frac{3}{4}2^{-n}\right)\right]. I n ′ = [ t − + ℓ ( 1 + 4 3 2 − n ) , t + − ℓ ( 1 + 4 3 2 − n ) ] . Then I n ⊂ I n ′ ⊂ int I n + 1 I_n\subset I'_n\subset\operatorname{int} I_{n+1} I n ⊂ I n ′ ⊂ int I n + 1 K ∗ ⊂ I 0 × B K_*\subset I_0\times B K ∗ ⊂ I 0 × B τ n = dist ( I n ′ , ∂ I n + 1 ) \tau_n=\operatorname{dist}(I'_n,\partial I_{n+1}) τ n = dist ( I n ′ , ∂ I n + 1 ) ℓ 2 − n / 4 \ell2^{-n}/4 ℓ 2 − n /4
r ∞ = 1 2 ( r 0 + r B ) , r n = r ∞ − ( r ∞ − r 0 ) 2 − n , r_\infty=\frac{1}{2}(r_0+r_B), \qquad r_n=r_\infty-(r_\infty-r_0)2^{-n}, r ∞ = 2 1 ( r 0 + r B ) , r n = r ∞ − ( r ∞ − r 0 ) 2 − n , and set
r 0 − = r 0 , r n − = 1 2 ( r n − 1 + r n ) ( n ≥ 1 ) . r_0^-=r_0, \qquad r_n^-=\frac{1}{2}(r_{n-1}+r_n)\quad(n\ge1). r 0 − = r 0 , r n − = 2 1 ( r n − 1 + r n ) ( n ≥ 1 ) . All supports will lie in the compact subset
K ∞ = [ t − + ℓ , t + − ℓ ] × B ‾ ( x 0 , r ∞ ) ⊂ Q . K_\infty=[t_-+\ell,t_+-\ell]\times\overline{B}(x_0,r_\infty)\subset Q. K ∞ = [ t − + ℓ , t + − ℓ ] × B ( x 0 , r ∞ ) ⊂ Q . We use the spatial gap and regularization scale
d n = r n + 1 − r n 64 N + 40 , h n = λ n − N 3 . d_n=\frac{r_{n+1}-r_n}{64N+40}, \qquad h_n=\lambda_n^{-N^3}. d n = 64 N + 40 r n + 1 − r n , h n = λ n − N 3 . Since inverse support gaps grow at most like C 2 n C2^n C 2 n λ n = a b n \lambda_n=a^{b^n} λ n = a b n a > 0 a>0 a > 0 n n n
τ n − 1 + d n − 1 ≤ λ n , \tau_n^{-1}+d_n^{-1}\le\lambda_n, τ n − 1 + d n − 1 ≤ λ n , h n < min { τ n , ℓ / 2 } ( n ≥ 0 ) , h_n<\min\{\tau_n,\ell/2\}\qquad(n\ge0), h n < min { τ n , ℓ /2 } ( n ≥ 0 ) , h n < 1 4 ( r n − r n − 1 ) ( n ≥ 1 ) . (82) h_n<\frac{1}{4}(r_n-r_{n-1})\qquad(n\ge1). \tag*{(82)} h n < 4 1 ( r n − r n − 1 ) ( n ≥ 1 ) . (82) Step 2: The bounds to be propagated.
We construct Z n Z_n Z n Z 0 = Z ∗ Z_0=Z_* Z 0 = Z ∗ Z n − Z ∗ Z_n-Z_* Z n − Z ∗ ( R n , T n ) (R_n,T_n) ( R n , T n ) I n × B ‾ ( x 0 , r n − ) I_n\times\overline{B}(x_0,r_n^-) I n × B ( x 0 , r n − )
∥ ( ρ n , u n , R n , T n ) ∥ C t , x 3 + ∥ Π n − Π ˉ ∥ C t , x 3 ≤ λ n N 2 , \|(\rho_n,u_n,R_n,T_n)\|_{C^3_{t,x}}+\|\Pi_n-\bar{\Pi}\|_{C^3_{t,x}}\le\lambda_n^{N^2}, ∥ ( ρ n , u n , R n , T n ) ∥ C t , x 3 + ∥ Π n − Π ˉ ∥ C t , x 3 ≤ λ n N 2 , ∥ u n ∥ ∞ ≤ δ n − 1 , \|u_n\|_\infty\le\delta_n^{-1}, ∥ u n ∥ ∞ ≤ δ n − 1 , ∥ ( R n , T n ) ∥ C t L p ≤ δ n 8 ( n ≥ 1 ) . (83) \|(R_n,T_n)\|_{C_tL^p}\le\delta_n^8\qquad(n\ge1). \tag*{(83)} ∥ ( R n , T n ) ∥ C t L p ≤ δ n 8 ( n ≥ 1 ) . (83) For the density, we require the direct pointwise lower bound
ρ n ≥ ρ ∗ − δ 0 + δ n 2 ∣ R n ∣ + δ n 4 ∣ T n ∣ + 2 ∑ j ≥ n δ j 2 . (84) \rho_n\ge\rho_*-\delta_0+\delta_n^2|R_n|+\delta_n^4|T_n|+2\sum_{j\ge n}\delta_j^2. \tag*{(84)} ρ n ≥ ρ ∗ − δ 0 + δ n 2 ∣ R n ∣ + δ n 4 ∣ T n ∣ + 2 j ≥ n ∑ δ j 2 . (84) The defect terms cover the density removed by the next principal perturbation; the tail covers the remaining uniform losses. The initial conditions hold for sufficiently large a > 0 a>0 a > 0 Z ∗ Z_* Z ∗
δ 0 2 ∥ R ∗ ∥ ∞ + δ 0 4 ∥ T ∗ ∥ ∞ + 2 ∑ j ≥ 0 δ j 2 ≤ δ 0 . \delta_0^2\|R_*\|_\infty+\delta_0^4\|T_*\|_\infty+2\sum_{j\ge0}\delta_j^2\le\delta_0. δ 0 2 ∥ R ∗ ∥ ∞ + δ 0 4 ∥ T ∗ ∥ ∞ + 2 j ≥ 0 ∑ δ j 2 ≤ δ 0 . Step 3: Regularization before the perturbation. Assume the preceding bounds at level n n n n = 0 n=0 n = 0 Z 0 ′ = Z 0 Z'_0=Z_0 Z 0 ′ = Z 0 n ≥ 1 n\ge1 n ≥ 1
K = I n × B ‾ ( x 0 , r n − ) , h = h n , K=I_n\times\overline{B}(x_0,r_n^-), \qquad h=h_n, K = I n × B ( x 0 , r n − ) , h = h n , and denote the result by Z n ′ Z'_n Z n ′ H \mathcal{H} H 3 λ n N 2 3\lambda_n^{N^2} 3 λ n N 2
∥ Z n ′ − Z n ∥ C t , x 0 + ∥ u n ′ − u n ∥ C t H s + ∥ ρ n ′ u n ′ − ρ n u n ∥ C t L 1 + ∥ ρ n ′ u n ′ ⊗ u n ′ − ρ n u n ⊗ u n ∥ C t L 1 ≤ 27 C 0 λ n − N 3 + 3 N 2 ≤ λ n − N 3 / 2 ≤ δ n 16 . (85) \begin{aligned}
\lVert Z'_n-Z_n\rVert_{C^0_{t,x}}+\lVert u'_n-u_n\rVert_{C_tH^s}
&+\lVert\rho'_n u'_n-\rho_nu_n\rVert_{C_tL^1}\\
&+\lVert\rho'_n u'_n\otimes u'_n-\rho_nu_n\otimes u_n\rVert_{C_tL^1}\\
&\le27C_0\lambda_n^{-N^3+3N^2}\le\lambda_n^{-N^3/2}\le\delta_n^{16}. \tag*{(85)}
\end{aligned} ∥ Z n ′ − Z n ∥ C t , x 0 + ∥ u n ′ − u n ∥ C t H s + ∥ ρ n ′ u n ′ − ρ n u n ∥ C t L 1 + ∥ ρ n ′ u n ′ ⊗ u n ′ − ρ n u n ⊗ u n ∥ C t L 1 ≤ 27 C 0 λ n − N 3 + 3 N 2 ≤ λ n − N 3 /2 ≤ δ n 16 . (85) By (6.5),
∥ ( ρ n ′ , u n ′ , R n ′ , T n ′ ) ∥ C t , x 2 ≤ 27 C ∗ λ n 17 N 3 + 3 N 2 ≤ λ n N 4 / 2 ≤ λ n N 4 . \lVert(\rho'_n,u'_n,R'_n,T'_n)\rVert_{C^2_{t,x}}
\le27C_*\lambda_n^{17N^3+3N^2}
\le\lambda_n^{N^4/2}\le\lambda_n^{N^4}. ∥( ρ n ′ , u n ′ , R n ′ , T n ′ ) ∥ C t , x 2 ≤ 27 C ∗ λ n 17 N 3 + 3 N 2 ≤ λ n N 4 /2 ≤ λ n N 4 . By (6.7), the regularized support is contained in I n ′ × B ‾ ( x 0 , r n ) I'_n\times\overline{B}(x_0,r_n) I n ′ × B ( x 0 , r n ) K ∗ K_* K ∗ R n ′ , T n ′ R'_n,T'_n R n ′ , T n ′
∥ u n ′ ∥ ∞ ≤ 2 δ n − 1 , ∥ ( R n ′ , T n ′ ) ∥ C t L p ≤ 2 δ n 8 ( n ≥ 1 ) , \lVert u'_n\rVert_\infty\le2\delta_n^{-1},\qquad
\lVert(R'_n,T'_n)\rVert_{C_tL^p}\le2\delta_n^8\qquad(n\ge1), ∥ u n ′ ∥ ∞ ≤ 2 δ n − 1 , ∥( R n ′ , T n ′ ) ∥ C t L p ≤ 2 δ n 8 ( n ≥ 1 ) , and
ρ n ′ ≥ ρ ∗ − δ 0 + δ n 2 ∣ R n ′ ∣ + δ n 4 ∣ T n ′ ∣ + 2 ∑ j ≥ n δ j 2 − 3 δ n 16 . (86) \rho'_n\ge\rho_*-\delta_0+\delta_n^2|R'_n|+\delta_n^4|T'_n|+2\sum_{j\ge n}\delta_j^2-3\delta_n^{16}. \tag*{(86)} ρ n ′ ≥ ρ ∗ − δ 0 + δ n 2 ∣ R n ′ ∣ + δ n 4 ∣ T n ′ ∣ + 2 j ≥ n ∑ δ j 2 − 3 δ n 16 . (86) Consequently, Z n ′ Z'_n Z n ′
I ′ = I n ′ , J ′ = int I n + 1 , I'=I'_n,\qquad J'=\operatorname{int} I_{n+1}, I ′ = I n ′ , J ′ = int I n + 1 , r − = r n , r + = r n + 1 , r_-=r_n,\qquad r_+=r_{n+1}, r − = r n , r + = r n + 1 , τ = τ n , d = d n . \tau=\tau_n,\qquad d=d_n. τ = τ n , d = d n . Step 4: The next perturbation. Let Z n + 1 = ( Z n ′ ) + Z_{n+1}=(Z'_n)^+ Z n + 1 = ( Z n ′ ) + Z n + 1 − Z n ′ Z_{n+1}-Z'_n Z n + 1 − Z n ′ I n + 1 × B ‾ ( x 0 , r n + 1 − ) I_{n+1}\times\overline{B}(x_0,r_{n+1}^-) I n + 1 × B ( x 0 , r n + 1 − ) Z n ′ − Z ∗ Z'_n-Z_* Z n ′ − Z ∗ n + 1 n+1 n + 1
By Proposition 4.1 and (6.6),
∥ ( ρ n + 1 , u n + 1 , R n + 1 , T n + 1 ) ∥ C t , x 3 + ∥ Π n + 1 − Π ‾ ∥ C t , x 3 ≤ 1 2 λ n + 1 N 2 + ∥ Π n ′ − Π ‾ ∥ C t , x 3 ≤ 1 2 λ n + 1 N 2 + λ n N 2 ≤ λ n + 1 N 2 . \begin{aligned}
\lVert(\rho_{n+1},u_{n+1},R_{n+1},T_{n+1})\rVert_{C^3_{t,x}}
+\lVert\Pi_{n+1}-\overline{\Pi}\rVert_{C^3_{t,x}}
&\le\frac12\lambda_{n+1}^{N^2}+\lVert\Pi'_n-\overline{\Pi}\rVert_{C^3_{t,x}}\\
&\le\frac12\lambda_{n+1}^{N^2}+\lambda_n^{N^2}
\le\lambda_{n+1}^{N^2}.
\end{aligned} ∥( ρ n + 1 , u n + 1 , R n + 1 , T n + 1 ) ∥ C t , x 3 + ∥ Π n + 1 − Π ∥ C t , x 3 ≤ 2 1 λ n + 1 N 2 + ∥ Π n ′ − Π ∥ C t , x 3 ≤ 2 1 λ n + 1 N 2 + λ n N 2 ≤ λ n + 1 N 2 . The remaining norm bounds in (6.8) follow from Proposition 4.1. For the density, apply its lower bound for the full perturbation to the regularized fields and use (6.11):
ρ n + 1 ≥ ρ ∗ − δ 0 + δ n + 1 2 ∣ R n + 1 ∣ + δ n + 1 4 ∣ T n + 1 ∣ + 2 ∑ j ≥ n δ j 2 − δ n 2 − 3 δ n 16 . \rho_{n+1}\ge\rho_*-\delta_0+\delta_{n+1}^2|R_{n+1}|+\delta_{n+1}^4|T_{n+1}|
+2\sum_{j\ge n}\delta_j^2-\delta_n^2-3\delta_n^{16}. ρ n + 1 ≥ ρ ∗ − δ 0 + δ n + 1 2 ∣ R n + 1 ∣ + δ n + 1 4 ∣ T n + 1 ∣ + 2 j ≥ n ∑ δ j 2 − δ n 2 − 3 δ n 16 . Since 3 δ n 16 ≤ δ n 2 3\delta_n^{16}\le\delta_n^2 3 δ n 16 ≤ δ n 2 2 ∑ j ≥ n + 1 δ j 2 2\sum_{j\ge n+1}\delta_j^2 2 ∑ j ≥ n + 1 δ j 2 n + 1 n+1 n + 1
Step 5: Convergence bounds and the density lower bound. The common support property follows from the construction of K ∞ K_\infty K ∞
∥ ρ n + 1 − ρ n ∥ C t L p + ∥ u n + 1 − u n ∥ C t H s ≤ 2 δ n 16 + δ n 2 + δ n + 1 16 ≤ δ n . \lVert\rho_{n+1}-\rho_n\rVert_{C_tL^p}+\lVert u_{n+1}-u_n\rVert_{C_tH^s}
\le2\delta_n^{16}+\delta_n^2+\delta_{n+1}^{16}
\le\delta_n. ∥ ρ n + 1 − ρ n ∥ C t L p + ∥ u n + 1 − u n ∥ C t H s ≤ 2 δ n 16 + δ n 2 + δ n + 1 16 ≤ δ n . This proves the density and velocity increment bound. The defect bound (2.3) is the last estimate in (6.8).
For n ≥ 1 n\ge1 n ≥ 1 δ n 2 + 3 δ n 16 ≤ δ n \delta_n^2+3\delta_n^{16}\le\delta_n δ n 2 + 3 δ n 16 ≤ δ n ∥ Π n ′ − Π n ∥ L 1 ≤ ∥ Z n ′ − Z n ∥ C 0 \lVert\Pi'_n-\Pi_n\rVert_{L^1}\le\lVert Z'_n-Z_n\rVert_{C^0} ∥ Π n ′ − Π n ∥ L 1 ≤ ∥ Z n ′ − Z n ∥ C 0
Finally, dropping the nonnegative defect and tail terms in (6.9) gives ρ n ≥ ρ ∗ − δ 0 \rho_n\ge\rho_*-\delta_0 ρ n ≥ ρ ∗ − δ 0
Elementary auxiliary estimates
We prove Lemmas 3.1, 3.2, and 3.3. The cutoff construction gives the nested supports and the derivative bound used for positivity; the kernel calculation gives both the periodic scaling and the endpoint integral estimates.
Time cutoffs
Proof of Lemma 3.1. We first construct a smooth increasing cutoff on the real line, then rescale it at the two endpoints of I I I
Step 1: A cutoff with a controlled derivative.
Define η 0 ∈ C ∞ ( R ; [ 0 , 1 ] ) \eta_0 \in C^\infty(\mathbb{R};[0,1]) η 0 ∈ C ∞ ( R ; [ 0 , 1 ])
η 0 ( t ) = { 0 , t ≤ 0 , e − 1 / t e − 1 / t + e − 1 / ( 1 − t ) , 0 < t < 1 , 1 , t ≥ 1. \eta_0(t)=
\begin{cases}
0, & t \le0,\\
\dfrac{e^{-1/t}}{e^{-1/t}+e^{-1/(1-t)}}, & 0<t<1,\\
1, & t \ge1.
\end{cases} η 0 ( t ) = ⎩ ⎨ ⎧ 0 , e − 1/ t + e − 1/ ( 1 − t ) e − 1/ t , 1 , t ≤ 0 , 0 < t < 1 , t ≥ 1. On ( 0 , 1 ) (0,1) ( 0 , 1 )
η 0 ′ = η 0 ( 1 − η 0 ) ( t − 2 + ( 1 − t ) − 2 ) ≥ 0. \eta_0'=\eta_0(1-\eta_0)\left(t^{-2}+(1-t)^{-2}\right)\ge0. η 0 ′ = η 0 ( 1 − η 0 ) ( t − 2 + ( 1 − t ) − 2 ) ≥ 0. For 0 < t ≤ 1 / 2 0<t\le1/2 0 < t ≤ 1/2 η 0 ( t ) ≤ e 2 e − 1 / t \eta_0(t)\le e^2e^{-1/t} η 0 ( t ) ≤ e 2 e − 1/ t
∣ η 0 ′ ∣ η 0 − 1 + 1 / ( 16 N ) ≤ 2 t − 2 η 0 1 / ( 16 N ) ≤ 2 e 1 / ( 8 N ) t − 2 e − 1 / ( 16 N t ) ≤ C N . |\eta_0'|\eta_0^{-1+1/(16N)}
\le2t^{-2}\eta_0^{1/(16N)}
\le2e^{1/(8N)}t^{-2}e^{-1/(16Nt)}
\le C_N. ∣ η 0 ′ ∣ η 0 − 1 + 1/ ( 16 N ) ≤ 2 t − 2 η 0 1/ ( 16 N ) ≤ 2 e 1/ ( 8 N ) t − 2 e − 1/ ( 16 N t ) ≤ C N . For t ≥ 1 / 2 t\ge1/2 t ≥ 1/2 η 0 ≥ 1 / 2 \eta_0\ge1/2 η 0 ≥ 1/2 η 0 ′ \eta_0' η 0 ′
∣ η 0 ′ ∣ ≤ C N η 0 1 − 1 / ( 16 N ) on R . (87) |\eta_0'|\le C_N\eta_0^{1-1/(16N)}
\quad\text{on }\mathbb{R}. \tag*{(87)} ∣ η 0 ′ ∣ ≤ C N η 0 1 − 1/ ( 16 N ) on R . (87) Step 2: The three nested cutoffs.
Set
η ( t ) = η 0 ( 10 ( t − t − ) τ + 2 ) η 0 ( 10 ( t + − t ) τ + 2 ) , \eta(t)=\eta_0\left(\frac{10(t-t_-)}{\tau}+2\right)\eta_0\left(\frac{10(t_+-t)}{\tau}+2\right), η ( t ) = η 0 ( τ 10 ( t − t − ) + 2 ) η 0 ( τ 10 ( t + − t ) + 2 ) , and define η u \eta_u η u η d \eta_d η d 2 2 2 4 4 4 6 6 6 η = 1 \eta=1 η = 1 [ t − − τ / 10 , t + + τ / 10 ] [t_- - \tau/10,t_+ + \tau/10] [ t − − τ /10 , t + + τ /10 ]
supp η ⊂ [ t − − τ / 5 , t + + τ / 5 ] ⊂ { η u = 1 } , \operatorname{supp}\eta\subset[t_- - \tau/5,t_+ + \tau/5]\subset\{\eta_u=1\}, supp η ⊂ [ t − − τ /5 , t + + τ /5 ] ⊂ { η u = 1 } , supp η u ⊂ [ t − − 2 τ / 5 , t + + 2 τ / 5 ] ⊂ { η d = 1 } , \operatorname{supp}\eta_u\subset[t_- - 2\tau/5,t_+ + 2\tau/5]\subset\{\eta_d=1\}, supp η u ⊂ [ t − − 2 τ /5 , t + + 2 τ /5 ] ⊂ { η d = 1 } , supp η d ⊂ [ t − − 3 τ / 5 , t + + 3 τ / 5 ] ⋐ J ′ . \operatorname{supp}\eta_d\subset[t_- - 3\tau/5,t_+ + 3\tau/5]\Subset J'. supp η d ⊂ [ t − − 3 τ /5 , t + + 3 τ /5 ] ⋐ J ′ . The gaps in the definitions make each cutoff one near the preceding support. The desired C j C^j C j τ < 1 \tau<1 τ < 1 η d \eta_d η d [ 0 , 1 ] [0,1] [ 0 , 1 ] C N τ − 1 η d 1 − 1 / ( 16 N ) C_N\tau^{-1}\eta_d^{1-1/(16N)} C N τ − 1 η d 1 − 1/ ( 16 N ) η d ′ \eta_d' η d ′
Calculus inequalities
Proof of Lemma 3.2. Leibniz’ rule gives the product estimate, since the binomial coefficients for a derivative of order at most j j j 2 j 2^j 2 j
For the composition estimate, ∥ Γ ∘ v ∥ ∞ ≤ ∥ Γ ∥ C 0 ( K ) \|\Gamma\circ v\|_\infty\le\|\Gamma\|_{C^0(\mathcal{K})} ∥Γ ∘ v ∥ ∞ ≤ ∥Γ ∥ C 0 ( K )
For 1 ≤ i ≤ j 1\le i\le j 1 ≤ i ≤ j i i i C ( j , m ) C(j,m) C ( j , m )
( ∇ l Γ ) ( v ) [ ∂ α 1 v , … , ∂ α l v ] , (\nabla^l\Gamma)(v)[\partial^{\alpha_1}v,\ldots,\partial^{\alpha_l}v], ( ∇ l Γ ) ( v ) [ ∂ α 1 v , … , ∂ α l v ] , where the multi-indices satisfy
∣ α a ∣ ≥ 1 , ∑ a = 1 l ∣ α a ∣ = i . |\alpha_a|\ge1,\qquad\sum_{a=1}^{l}|\alpha_a|=i. ∣ α a ∣ ≥ 1 , a = 1 ∑ l ∣ α a ∣ = i . Here 1 ≤ l ≤ i 1\le l\le i 1 ≤ l ≤ i ( t , x ) (t,x) ( t , x )
∥ ∇ l Γ ∥ C 0 ( K ) ∥ v ∥ C t , x j l ≤ ∥ Γ ∥ C j ( K ) ( 1 + ∥ v ∥ C t , x j ) j . \|\nabla^l\Gamma\|_{C^0(\mathcal{K})}\|v\|_{C^j_{t,x}}^l
\le\|\Gamma\|_{C^j(\mathcal{K})}(1+\|v\|_{C^j_{t,x}})^j. ∥ ∇ l Γ ∥ C 0 ( K ) ∥ v ∥ C t , x j l ≤ ∥Γ ∥ C j ( K ) ( 1 + ∥ v ∥ C t , x j ) j . This proves the composition bound. For positive-order derivatives, only derivatives of Γ \Gamma Γ 1 , … , j 1,\ldots,j 1 , … , j K \mathcal{K} K
Periodic kernels
Proof of Lemma 3.3. We identify the singularity of each inverse Laplacian, then deduce the operator bounds and the scaling identities.
Step 1: The periodic Green functions.
The mean-zero periodic Green function has Fourier coefficients
G ^ ( 0 ) = 0 , G ^ ( k ) = − 1 4 π 2 ∣ k ∣ 2 ( k ∈ Z 3 ∖ { 0 } ) . \widehat{G}(0)=0,\qquad\widehat{G}(k)=-\frac{1}{4\pi^{2}|k|^{2}}\quad(k\in\mathbb{Z}^{3}\setminus\{0\}). G ( 0 ) = 0 , G ( k ) = − 4 π 2 ∣ k ∣ 2 1 ( k ∈ Z 3 ∖ { 0 }) . Thus G 2 = G ∗ G G_{2}=G*G G 2 = G ∗ G Δ G 2 = G \Delta G_{2}=G Δ G 2 = G Δ − 2 P ≠ 0 f = G 2 ∗ f \Delta^{-2}\mathbb{P}_{\ne0}f=G_{2}*f Δ − 2 P = 0 f = G 2 ∗ f
Choose a scalar χ ∈ C ∞ ( B ( 0 , 1 / 2 ) ) \chi\in C^{\infty}(B(0,1/2)) χ ∈ C ∞ ( B ( 0 , 1/2 )) B ( 0 , 1 / 4 ) B(0,1/4) B ( 0 , 1/4 ) Δ ( − 1 / ( 4 π ∣ z ∣ ) ) = δ 0 \Delta(-1/(4\pi|z|))=\delta_{0} Δ ( − 1/ ( 4 π ∣ z ∣ )) = δ 0
G ( z ) + χ ( z ) 4 π ∣ z ∣ G(z)+\frac{\chi(z)}{4\pi|z|} G ( z ) + 4 π ∣ z ∣ χ ( z ) has a smooth Laplacian and hence is smooth by elliptic regularity. Similarly, Δ ∣ z ∣ = 2 / ∣ z ∣ \Delta|z|=2/|z| Δ∣ z ∣ = 2/∣ z ∣
G 2 ( z ) + χ ( z ) ∣ z ∣ 8 π ∈ C ∞ ( T 3 ) . G_{2}(z)+\frac{\chi(z)|z|}{8\pi}\in C^{\infty}(\mathbb{T}^{3}). G 2 ( z ) + 8 π χ ( z ) ∣ z ∣ ∈ C ∞ ( T 3 ) . Both Green functions are smooth away from the origin. Differentiating these expressions proves the asserted kernel bounds.
Step 2: Kernel bounds and boundedness.
By definition, the kernel of R 0 \mathcal{R}_{0} R 0 ∇ G \nabla G ∇ G R \mathcal{R} R ∇ G \nabla G ∇ G ∇ 3 G 2 \nabla^{3}G_{2} ∇ 3 G 2 Δ − 1 P ≠ 0 \Delta^{-1}\mathbb{P}_{\ne0} Δ − 1 P = 0 G G G
All these kernels are integrable and have zero mean. Young’s inequality gives boundedness on L q L^{q} L q 1 ≤ q ≤ ∞ 1\le q\le\infty 1 ≤ q ≤ ∞ C t , x j C^{j}_{t,x} C t , x j
Step 3: Scaling and integrable derivatives.
The nonzero Fourier multipliers of R 0 \mathcal{R}_{0} R 0 R \mathcal{R} R − 1 -1 − 1 Δ − 1 P ≠ 0 \Delta^{-1}\mathbb{P}_{\ne0} Δ − 1 P = 0 − 2 -2 − 2 λ \lambda λ
g ( λ ⋅ ) ^ ( λ k ) = g ^ ( k ) , g ( λ ⋅ ) ^ ( k ) = 0 if k ∉ λ Z 3 . \widehat{g(\lambda\cdot)}(\lambda k)=\widehat{g}(k),\qquad\widehat{g(\lambda\cdot)}(k)=0\quad\text{if }k\notin\lambda\mathbb{Z}^{3}. g ( λ ⋅ ) ( λk ) = g ( k ) , g ( λ ⋅ ) ( k ) = 0 if k ∈ / λ Z 3 . Applying the multipliers at these frequencies proves the scaling identities.
Finally, the kernel of Δ − 2 ∂ α P ≠ 0 \Delta^{-2}\partial^{\alpha}\mathbb{P}_{\ne0} Δ − 2 ∂ α P = 0 ∂ α G 2 \partial^{\alpha}G_{2} ∂ α G 2 1 − ∣ α ∣ 1-|\alpha| 1 − ∣ α ∣ ∣ α ∣ ≤ 3 |\alpha|\le3 ∣ α ∣ ≤ 3
Xingyu Li
Department of Mathematics and Geosciences, University of Trieste, via Valerio 12/1, Trieste, 34127 Italy
Email address: xingyuli92@gmail.com
