Weak rectangular Newtonian velocity and pressure

Let $\mathrm{\Omega }={\prod }_{i=1}^3(a_i,b_i)\subset {\mathbb{R}}^3$ be a finite open rectangle in Cartesian coordinates measured in metres, with $a_i<b_i$, and write $U=(0,\mathrm{\infty })\times \mathrm{\Omega }\subset {\mathbb{R}}^4$ with coordinates $(t,x_1,x_2,x_3)$. For an open $O\subseteq {\mathbb{R}}^n$, set $\mathcal{D}(O)=C_c^{\mathrm{\infty }}(O;\mathbb{C})$ and, for each compact $K\subset O$, set ${\mathcal{D}}_K(O)=\{\psi \in \mathcal{D}(O):\mathrm{supp}\psi \subseteq K\}$ with Fréchet seminorms $p_m(\psi )={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_{x\in K}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\psi (x)\mathrm{\mid }$, $m\in {\mathbb{N}}_0$. Give $\mathcal{D}(O)$ its locally convex LF topology, the finest locally convex topology making every inclusion ${\mathcal{D}}_K(O)\to \mathcal{D}(O)$ continuous; real and finite-dimensional vector test-function spaces carry the corresponding real and componentwise topologies. Let ${\mathcal{D}}^{\mathrm{\prime }}(O)$ be the vector space of continuous complex-linear functionals on $\mathcal{D}(O)$, with bilinear pairing $⟨F,\psi ⟩=F(\psi )$; a spacetime distribution on $U$ is an element of ${\mathcal{D}}^{\mathrm{\prime }}(U)$. A locally integrable field $f$ is identified componentwise with the distribution $\psi \mapsto {\int }_{U}f\psi dtdx$. For $F\in {\mathcal{D}}^{\mathrm{\prime }}(O)$ and a multi-index $\alpha \in {\mathbb{N}}_0^n$, define $⟨{\mathrm{\partial }}^{\alpha }F,\psi ⟩=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}⟨F,{\mathrm{\partial }}^{\alpha }\psi ⟩$ for every $\psi \in \mathcal{D}(O)$. Derivatives of vector or matrix distributions are taken componentwise; for scalar $F\in {\mathcal{D}}^{\mathrm{\prime }}(U)$, ${\mathrm{\nabla }}_{x}F=({\mathrm{\partial }}_{x_i}F{)}_{i=1}^3$, for a matrix distribution $A$, $(\mathrm{div}A{)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{A}_{ij}$ for $i=1,2,3$, and $\mathrm{\Delta }$ acts componentwise as ${\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^2$.

Define $D_{\sigma }(\mathrm{\Omega })=\{a\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3):\mathrm{div}a=0\}$, where for smooth fields $\mathrm{div}a={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}{a}_i$, let $H$ be its closure in $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$, 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 vector fields and $H^1$ uses square-integrable fields and first weak derivatives. This convention does not identify $V$ with all divergence-free members of $H_0^1$ on an arbitrary open set and makes no pointwise trace assertion. Let ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mu >0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, $\nu =\mu \mathrm{/}{\rho }_0>0$, and $u_0\in H$ be an initial velocity in $\mathrm{m}{\mathrm{s}}^{-1}$. Assume AC and Countable Choice. In addition, assume the following rectangular test-divergence inverse: for $D_0(\mathrm{\Omega })=\{h\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R}):{\int }_{\mathrm{\Omega }}h=0\}$ with its subspace LF topology, a continuous linear map $R:D_0(\mathrm{\Omega })\to C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ with the componentwise LF topology satisfies $\mathrm{div}(Rh)=h$; for each compact input support there is one compact output support, and every fixed-support restriction is continuous for the test-function seminorms above.

Then there are a weak velocity $u\in C_w([0,\mathrm{\infty });H)\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$ and a real kinematic-pressure distribution $\pi \in {\mathcal{D}}^{\mathrm{\prime }}(U)$, meaning it takes real values on real-valued tests, with units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$. Here $C_w$ means continuity of every fixed $H$-pairing, and the time-dependent spaces use strongly measurable Bochner maps on finite intervals. The velocity has $u(0)=u_0$ and strong $H$ trace at zero, and for every compactly time-supported $C^1$ curve $\varphi :[0,\mathrm{\infty })\to V$ it satisfies

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

All spatial pairings use unnormalized Lebesgue measure, with $(u\otimes u{)}_{ij}=u_i{u}_j$, $(u\otimes u):\mathrm{\nabla }\varphi ={\sum }_{i,j}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_i$, and $\mathrm{\nabla }u:\mathrm{\nabla }\varphi ={\sum }_{i,j}{\mathrm{\partial }}_j{u}_i{\mathrm{\partial }}_j{\varphi }_i$. For every $t\ge 0$ it satisfies

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

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

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

Fix one real scalar test function $\theta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ with ${\int }_{\mathrm{\Omega }}\theta =1$. The following spacetime equation is an equality in ${\mathcal{D}}^{\mathrm{\prime }}(U{)}^3$ under the stated distributional-derivative convention, and this fixed normalization is

$${\mathrm{\partial }}_{t}u+\mathrm{div}(u\otimes u)+\mathrm{\nabla }\pi =\nu \mathrm{\Delta }u,$$ $$⟨\pi ,a(t)\theta (x)⟩=0$$

for every $a\in C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$. Such a $\theta$ exists because a product of normalized smooth compactly supported interval bumps belongs to $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and has integral one. The physical pressure distribution $p={\rho }_0\pi$ has units $\mathrm{P}\mathrm{a}$, is unique under this fixed scaled normalization, and satisfies

$${\rho }_0{\mathrm{\partial }}_{t}u+{\rho }_0\mathrm{div}(u\otimes u)+\mathrm{\nabla }p=\mu \mathrm{\Delta }u$$

as a spacetime distribution. The SI kinetic energy $E(t)=\frac{{\rho }_0}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }u(t,x){\mathrm{\mid }}^{2}dx$ obeys $E(t)+\mu {\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u(s,x){\mathrm{\mid }}^{2}dxds\le E(0)$. This conditional result asserts no pressure in $L^2$, corner regularity, or pointwise boundary traction.

Rectangular evolving distributional pressure under a continuous divergence inverse

Continuous right inverse of divergence on mean-zero rectangular tests

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, §3.5 and §5.2, printed pp. 6–8 and 14–16

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