Rectangular evolving distributional pressure

Extend the real-linear supplier $R$ to complex mean-zero tests by $R_{\mathbb{C}}(h_1+i{h}_2)=R{h}_1+iR{h}_2$, where $h_1,h_2$ are real. Their integrals both vanish. Real linearity makes this extension complex-linear; applying the supplier bounds to the two parts preserves fixed-support continuity, the same output-support set, and $\mathrm{div}{R}_{\mathbb{C}}h=h$. In the rest of the proof $R$ denotes this extension. On $U=(0,\mathrm{\infty })\times \mathrm{\Omega }$, $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ and $u\otimes u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. For a compact $K\subset U$, let $p_{m,K}$ be the maximum of the suprema on $K$ of all component derivatives of order at most $m$. The same local estimate $\mathrm{\mid }{\int }_{K}f\psi \mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(K)}{p}_{0,K}(\psi )$ shows that the regular pairings of $u$ and $u\otimes u$ are continuous on every fixed-support test space, hence are distributions. The residual functional $⟨T,\psi ⟩={\int }_U(u\cdot {\mathrm{\partial }}_t\psi +(u\otimes u):{\mathrm{\nabla }}_x\psi +\nu u\cdot {\mathrm{\Delta }}_x\psi )dtdx$ is well-defined and satisfies $\mathrm{\mid }⟨T,\psi ⟩\mathrm{\mid }\le C_d(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(K)}+\mathrm{\parallel }u\otimes u{\mathrm{\parallel }}_{L^1(K)})p_{1,K}(\psi )+C_d\nu \mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(K)}{p}_{2,K}(\psi )$ for every test supported in $K$. Thus its absolute value is a seminorm whose restriction to each fixed-support test space is continuous; by the test-function topology in [F4], $T$ is a spacetime distribution. By [F3], this formula is exactly the residual $-{\mathrm{\partial }}_{t}u-\mathrm{div}(u\otimes u)+\nu \mathrm{\Delta }u$ of those regular distributions. For a scalar test $\varphi (t,x)$ put ${\varphi }_0=\varphi -\theta (x){\int }_{\mathrm{\Omega }}\varphi (t,y)dy$ and define $⟨\pi ,\varphi ⟩=-⟨T,R_x{\varphi }_0⟩\mathrm{.}$ The map $R$ is applied at each time. The spatial supports of ${\varphi }_0(t,\cdot )$ lie in one compact subset of $\mathrm{\Omega }$, since they are contained in the union of the spatial support of $\varphi$ and $\mathrm{supp}\theta$. For each time derivative order, $t\mapsto {\mathrm{\partial }}_t^j{\varphi }_0(t,\cdot )$ is continuous in every fixed-support test seminorm. Linearity and continuity of $R$ on that fixed-support space imply that $t\mapsto R{\varphi }_0(t,\cdot )$ is smooth with derivatives $R{\mathrm{\partial }}_t^j{\varphi }_0$; the support condition in [F2] puts all outputs in one compact spatial set. Thus $R_x{\varphi }_0$ is a smooth spacetime test with compact temporal support, and the displayed functional is continuous by the same fixed-support estimate for $T$. It is a distribution.

For a compact smooth vector test $\psi$, $h={\mathrm{div}}_x\psi$ has spatial integral zero by coordinate integration, so $h\in D_0(\mathrm{\Omega })$ and $\psi -Rh$ is compactly supported, smooth, and divergence free. By [F1], $⟨T,\psi -Rh⟩=0$. Moreover, $⟨{\mathrm{\nabla }}_x\pi ,\psi ⟩=-⟨\pi ,h⟩=⟨T,Rh⟩=⟨T,\psi ⟩\mathrm{.}$ Substituting the definition of $T$ proves the full distributional momentum equation. The normalization holds because $(a\theta {)}_0=0$.

If ${\pi }_1,{\pi }_2$ have the same spatial gradient, their difference $q$ has ${\mathrm{\nabla }}_{x}q=0$. For an arbitrary scalar spacetime test $\varphi$, put $a(t)={\int }_{\mathrm{\Omega }}\varphi (t,x)dx$ and ${\varphi }_0=\varphi -a\theta$. Step 1.1 gives a spacetime vector test $R_x{\varphi }_0$ with spatial divergence ${\varphi }_0$. Hence $⟨q,{\varphi }_0⟩=-⟨{\mathrm{\nabla }}_{x}q,R_x{\varphi }_0⟩=0$, and $⟨q,\varphi ⟩=⟨q,a\theta ⟩=0$ by the shared normalization. Thus $q=0$, proving uniqueness conditional on [F2].