Difference energy for given smooth incompressible and symmetric solutions

Assume Countable Choice. Let $d\in \{1,2,3\}$, $T>0$, and $Q_d={\mathbb{R}}^d\mathrm{/}(2\pi \mathbb{Z}{)}^d$ with normalized measure $d\mu =(2\pi {)}^{-d}dx$. For $z:Q_d\to {\mathbb{R}}^M$, use componentwise Fourier coefficients $\widehat{z}(k)={\int }_{Q_d}z(x)e^{-ik\cdot x}d\mu (x)$, weights $w_k=(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, and the weighted coefficient completion norm $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^s}^2={\sum }_{k\in {\mathbb{Z}}^d}{w}_k^{2s}\mathrm{\mid }\widehat{z}(k){\mathrm{\mid }}^2$; the normalized Fourier isometry gives $\mathrm{\parallel }z{\mathrm{\parallel }}_2^2={\sum }_k\mathrm{\mid }\widehat{z}(k){\mathrm{\mid }}^2$, where $\mathrm{\parallel }z{\mathrm{\parallel }}_2=({\int }_{Q_d}\mathrm{\mid }z{\mathrm{\mid }}^{2}d\mu {)}^{1\mathrm{/}2}$ and the pointwise component norm is Euclidean. In particular, $H^0=L^2$. For a map $z:[0,T]\times Q_d\to {\mathbb{R}}^M$, smooth on the closed cylinder means that for every $m\in {\mathbb{N}}_0$ and spatial multi-index $\alpha$, ${\mathrm{\partial }}_t^m{\mathrm{\partial }}_x^{\alpha }z$ exists on $(0,T)\times Q_d$ and extends continuously to $[0,T]\times Q_d$; spatial derivatives are those of its periodic lift. For a spatial vector field define $\mathrm{div}z={\sum }_{j=1}^d{\mathrm{\partial }}_j{z}_j$ and $\mathrm{\Delta }z={\sum }_{j=1}^d{\mathrm{\partial }}_j^{2}z$. Write $\mathrm{\nabla }z$ for the spatial Jacobian and $\mathrm{\parallel }\mathrm{\nabla }z(t){\mathrm{\parallel }}_{\mathrm{\infty }}={\sup }_{x\in Q_d}\mathrm{\mid }\mathrm{\nabla }z(t,x){\mathrm{\mid }}_F$, where $\mathrm{\mid }\cdot {\mathrm{\mid }}_F$ is the Euclidean Frobenius matrix norm. Let $P$ be the coefficientwise vector multiplier with $P_0=I$ and $P_k=I-k{k}^{\mathsf{T}}\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}^2$ for $k\mathrm{\ne }0$. The equations below hold pointwise on $(0,T)\times Q_d$ and at the time endpoints by continuous extension.

(i) If $\nu \ge 0$ and $u,v\in C^{\mathrm{\infty }}([0,T]\times Q_d;{\mathbb{R}}^d)$ satisfy, for each $z\in \{u,v\}$, $z_t+P((z\cdot \mathrm{\nabla })z)=\nu \mathrm{\Delta }z,\mathrm{div}z=0,$ then for every $t\in [0,T]$, with $w=u-v$, $\mathrm{\parallel }w(t){\mathrm{\parallel }}_2^2\le \mathrm{\parallel }w(0){\mathrm{\parallel }}_2^2\exp \text{\hspace{0.17em}⁣}(2{\int }_0^t\mathrm{\parallel }\mathrm{\nabla }v(\tau ){\mathrm{\parallel }}_{\mathrm{\infty }}d\tau )\mathrm{.}$

(ii) Let $N\ge 1$ and, for $1\le j\le d$, let $A_j:{\mathbb{R}}^N\to {\mathbb{R}}^{N\times N}$ be a fixed real-linear map whose image consists of symmetric matrices. If $q,r\in C^{\mathrm{\infty }}([0,T]\times Q_d;{\mathbb{R}}^N)$ satisfy, for each $z\in \{q,r\}$, $z_t+{\sum }_{j=1}^d{A}_j(z){\mathrm{\partial }}_{j}z=0,$ then for every $t\in [0,T]$, $\mathrm{\parallel }q(t)-r(t){\mathrm{\parallel }}_2^2\le \mathrm{\parallel }q(0)-r(0){\mathrm{\parallel }}_2^2\exp \text{\hspace{0.17em}⁣}(C_{d,N,A}{\int }_0^t(\mathrm{\parallel }\mathrm{\nabla }q(\tau ){\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }\mathrm{\nabla }r(\tau ){\mathrm{\parallel }}_{\mathrm{\infty }})d\tau ),$ where $C_{d,N,A}$ depends only on $d,N$ and the operator norms of the finitely many maps $A_j$. For either pair $(z_1,z_2)=(u,v)$ or $(q,r)$, put $w=z_1-z_2$. Let $\mathrm{\Gamma }\ge 0$ be any uniform rate for which the corresponding squared $L^2$ estimate above gives $\mathrm{\parallel }w(t){\mathrm{\parallel }}_2^2\le \mathrm{\parallel }w(0){\mathrm{\parallel }}_2^2{e}^{\mathrm{\Gamma }t}$. If $s\ge 1$ and ${\max }_{t\in [0,T]}\max (\mathrm{\parallel }{z}_1(t){\mathrm{\parallel }}_{H^s},\mathrm{\parallel }{z}_2(t){\mathrm{\parallel }}_{H^s})\le M$ for some $M\ge 0$, define $B_s(M)=\{\begin{array}{ll}1,& s=1,\\ (2M{)}^{(s-1)\mathrm{/}s},& s>1.\end{array}$ Then for every $t\in [0,T]$, $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^{s-1}}\le B_s(M)\mathrm{\parallel }w(0){\mathrm{\parallel }}_2^{1\mathrm{/}s}{e}^{\mathrm{\Gamma }t\mathrm{/}(2s)}\mathrm{.}$ These are conditional estimates for already given smooth maps on the stated cylinder and assert no existence theorem. All quantities are mathematical and no physical units are assigned to $\nu$.

Erick Schulz, Introduction to the Navier–Stokes Equations, McGill MATH 595 notes (2014).

Terence Tao, 254A Notes 1: Local well-posedness of the Navier–Stokes equations (2018).