Bounded no-slip Newtonian vessel model

Let $I\subseteq \mathbb{R}$ be an open time interval in seconds and let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a fixed nonempty bounded open vessel in Cartesian coordinates measured in metres. Assume Countable Choice for the stated Sobolev space conventions. Define $D_{\sigma }(\mathrm{\Omega })=\{a\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3):\mathrm{div}a=0\}$, where divergence is the distributional divergence (equal to the classical divergence for these smooth fields), $H$ as its closure in $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$, and $V$ as its closure in $H_0^1(\mathrm{\Omega };{\mathbb{R}}^3)$; here $H_0^1$ is the closure of compactly supported smooth vector fields in the $H^1$ norm, and $H^1$ uses square-integrable fields and first weak derivatives. The adopted no-slip condition means $u(t)\in V$, not a pointwise boundary value. This closure convention does not identify $V$ with every divergence-free member of $H_0^1$ on an arbitrary open set and does not assert a boundary trace theorem.

Take a constant mass density ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, a constant dynamic viscosity $\mu >0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, and zero body-force density in $\mathrm{N}{\mathrm{m}}^{-3}$. For classical fields $u\in C^1(I;C^2(\mathrm{\Omega };{\mathbb{R}}^3))$ and $p\in C^1(I;C^1(\mathrm{\Omega }))$, adopt $\mathrm{div}u=0$, the Cauchy stress $\sigma =-p{I}_3+2\mu D(u)$ with $D(u)=\frac{1}{2}(\mathrm{\nabla }u+(\mathrm{\nabla }u{)}^{\mathsf{T}})$, and the local balance ${\rho }_0({\mathrm{\partial }}_{t}u+\mathrm{div}(u\otimes u))=\mathrm{div}\sigma$ in the interior. The velocity has units $\mathrm{m}{\mathrm{s}}^{-1}$, pressure and stress have units $\mathrm{P}\mathrm{a}$, spatial derivatives have units ${\mathrm{m}}^{-1}$, and $\nu =\mu \mathrm{/}{\rho }_0$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. No temperature field is included.

For the weak regime, take $u\in C_w([0,\mathrm{\infty });H)\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$, with $u(0)=u_0\in H$ and strong $H$ trace at zero; $u_0$ is an initial velocity in $\mathrm{m}{\mathrm{s}}^{-1}$. Here $C_w$ means continuity of every fixed $H$-pairing, and each time-dependent Bochner space uses strongly measurable maps with the indicated norm, locally on every finite interval. Interpret this model through the solenoidal variational identity: for every compactly time-supported $C^1$ curve $\varphi :[0,\mathrm{\infty })\to V$,

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

Here all spatial pairings use unnormalized Lebesgue measure on $\mathrm{\Omega }$, $(u\otimes u):\mathrm{\nabla }\varphi ={\sum }_{i,j}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_i$, and $\mathrm{\nabla }u:\mathrm{\nabla }\varphi ={\sum }_{i,j}{\mathrm{\partial }}_j{u}_i{\mathrm{\partial }}_j{\varphi }_i$. In this weak formulation pressure is eliminated by the divergence-free tests; no distributional pressure or corner regularity is postulated. On a smoother boundary, a pointwise zero-trace reading would require a separate trace theorem. This is an adopted fixed-vessel model, not a deduction about every real wall.

Bounded no-slip solenoidal closure spaces: H, V, and the no-trace convention

Cartesian inertial-frame, SI, and Cauchy-stress conventions supplies only the formulation conventions used here: Cartesian inertial coordinates, velocity in metres per second, density in kilograms per cubic metre, stress and pressure in pascals, dynamic viscosity in pascal seconds, the row-gradient $L_{ij}={\mathrm{\partial }}_j{u}_i$, its symmetric part $D=(L+L^{\mathsf{T}})\mathrm{/}2$, and tension-positive traction $\sigma n$. The smooth local momentum law and the weak solution class in this Postulate are separately adopted model assumptions; no weak balance, energy inequality, boundary trace, or pressure regularity is inferred from a smooth material-balance theorem.

Lenya Ryzhik, Lecture Notes for Math 256B, Version 2024, §§1.2–1.3, printed pp. 6–12, and §§5.2–5.5, printed pp. 39–51

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, §1, printed pp. 1–3, and §§2.1–2.2, printed pp. 4–5