Weak continuation of the adopted periodic viscous model

Let $L>0$ be the numerical period in the declared SI metre unit, with physical side length $L\mathrm{m}$; let ${\rho }_0>0$ be constant in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ and $\mu >0$ constant in $\mathrm{P}\mathrm{a}\mathrm{s}$. Set $\nu =\mu \mathrm{/}{\rho }_0>0$. On ${\mathbb{T}}_L^3$, use normalized Lebesgue measure $L^{-3}dx$, and let $H$ be the real $L^2$ vector fields whose divergence vanishes distributionally, with unrestricted mean mode, and $V=H\cap H^1$, where $H^1$ consists of fields and first weak derivatives in $L^2$. Here $H_{\sigma }^s$ means the real Fourier-weighted Sobolev space with weights $(1+L^2\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^s$, Fourier divergence condition $k\cdot \widehat{u}(k)=0$ for $k\mathrm{\ne }0$, and unrestricted constant mode. Let $u_0\in H$ be an initial velocity measured in $\mathrm{m}{\mathrm{s}}^{-1}$. Assume Countable Choice. Then the adopted unforced periodic model has a global weak velocity $u\in C_w([0,\mathrm{\infty });H)\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$, where $C_w$ means continuity of each fixed $H$-pairing and the time-dependent spaces are strongly measurable Bochner spaces, with $u(0)=u_0$ and strong $H$ trace at zero. For every smooth real periodic divergence-free test field $\varphi$ compactly supported in time it satisfies

$${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}[(u,{\mathrm{\partial }}_t\varphi )+(u\otimes u,\mathrm{\nabla }\varphi )-\nu (\mathrm{\nabla }u,\mathrm{\nabla }\varphi )]dt+(u_0,\varphi (0))=0,$$

where the spatial pairings use the normalized measure and $(u\otimes u):\mathrm{\nabla }\varphi ={\sum }_{i,j}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_i$. It also satisfies, for every $t\ge 0$,

$$\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_2^{2}ds\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2,$$

and there is a full-measure set $S\subset (0,\mathrm{\infty })$ such that for every $s\in S\cup \{0\}$ and every $t\ge s$,

$$\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_s^t\mathrm{\parallel }\mathrm{\nabla }u(\tau ){\mathrm{\parallel }}_2^{2}d\tau \le \mathrm{\parallel }u(s){\mathrm{\parallel }}_2^2\mathrm{.}$$

In SI units the kinetic energy is $K(t)=\frac{{\rho }_0}{2}{\int }_{[0,L{]}^3}\mathrm{\mid }u(t,x){\mathrm{\mid }}^{2}dx$, and the inequality implies

$$K(t)+\mu {\int }_0^t{\int }_{[0,L{]}^3}\mathrm{\mid }\mathrm{\nabla }u(s,x){\mathrm{\mid }}^{2}dxds\le K(0)\mathrm{.}$$

There is also a full-measure set $S\subset (0,\mathrm{\infty })$ such that for every $s\in S\cup \{0\}$ and every $t\ge s$,

$$K(t)+\mu {\int }_s^t{\int }_{[0,L{]}^3}\mathrm{\mid }\mathrm{\nabla }u(\tau ,x){\mathrm{\mid }}^{2}dxd\tau \le K(s)\mathrm{.}$$

This is a conditional model deduction about velocity; it asserts no pressure regularity, weak uniqueness, global smoothness, or empirical validity.

Leonid Ryzhik, Lecture Notes for Math 256B, Version 2024, §§5.1–5.5, printed pp. 37–51

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, Theorem 3, §5.2, printed pp. 14–16