Global weak velocity in the no-slip vessel model

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be any nonempty bounded open set in Cartesian coordinates measured in metres. Define $D_{\sigma }(\mathrm{\Omega })=\{a\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3):\mathrm{div}a=0\}$, let $H$ be its $L^2(\mathrm{\Omega })$ closure, and let $V$ be its closure in $H_0^1(\mathrm{\Omega };{\mathbb{R}}^3)$, where $H_0^1$ is the $H^1$ closure of compactly supported smooth fields and $H^1$ has the norm formed from the $L^2$ norms of a field and its first weak derivatives. Let ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mu >0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, $\nu =\mu \mathrm{/}{\rho }_0$, and $u_0\in H$ be an initial velocity in $\mathrm{m}{\mathrm{s}}^{-1}$. Assume the Axiom of Choice and Countable Choice. Then the adopted unforced no-slip model has

$$u\in C_w([0,\mathrm{\infty });H)\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V),$$

where $C_w$ means continuity of every fixed $H$-pairing and the time-dependent spaces are Bochner spaces of strongly measurable maps, with $u(0)=u_0$, strong $H$ trace at zero, and, for every compactly time-supported $C^1$ curve $\varphi :[0,\mathrm{\infty })\to V$,

$${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}[(u,{\mathrm{\partial }}_t\varphi {)}_{L^2}+(u\otimes u,\mathrm{\nabla }\varphi {)}_{L^2}-\nu (\mathrm{\nabla }u,\mathrm{\nabla }\varphi {)}_{L^2}]dt+(u_0,\varphi (0){)}_{L^2}=0.$$

It satisfies the initial energy inequality

$$\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_2^{2}ds\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2(t\ge 0),$$

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

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

Here spatial norms and pairings use unnormalized Lebesgue measure, $(u\otimes u):\mathrm{\nabla }\varphi ={\sum }_{i,j}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_i$, and $\mathrm{\nabla }u:\mathrm{\nabla }\varphi ={\sum }_{i,j}{\mathrm{\partial }}_j{u}_i{\mathrm{\partial }}_j{\varphi }_i$. Thus the SI kinetic energy $E(t)=\frac{{\rho }_0}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }u(t,x){\mathrm{\mid }}^{2}dx$ obeys

$$E(t)+\mu {\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u(s,x){\mathrm{\mid }}^{2}dxds\le E(0)\mathrm{.}$$

The no-slip condition is only the stated $H_0^1$ closure condition; the result supplies no pressure or pointwise wall trace.

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

Lenya Ryzhik, Lecture Notes for Math 256B, Version 2024, §§5.2–5.5, printed pp. 39–51