A gradient load observed through Stokes pressure

Since $D$ is nonempty and open it contains a ball, so $\mathrm{\mid }D\mathrm{\mid }>0$; boundedness gives $\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$. Hence $x_1\in L^2(D)$ and ${\bar{x}}_1$ is finite. Thus $\varphi \in L^2(D)$, ${\int }_D\varphi dx=0$, and its weak gradient is $\mathrm{\nabla }\varphi =e_1$.

For $v\in C_c^{\mathrm{\infty }}(D;{\mathbb{R}}^d)$, weak integration by parts gives ${\int }_D\mathrm{\nabla }\varphi \cdot vdx=-{\int }_D\varphi \mathrm{div}vdx$. By the $H^1$-closure in [F1], approximate any $v\in V$ by such fields; both integrals converge by Cauchy–Schwarz, so the identity holds for all $v\in V$. Pointwise $\mathrm{\mid }\mathrm{div}v\mathrm{\mid }=\mathrm{\mid }\mathrm{tr}\mathrm{\nabla }v\mathrm{\mid }\le \sqrt{d}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}_F$, whence $\mathrm{\mid }{F}_{\varphi }(v)\mathrm{\mid }\le \sqrt{d}\mathrm{\parallel }\varphi {\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$ and $F_{\varphi }\in V^{\mathrm{\prime }}$.

For every $v\in V$, $a_{\nu }(0,v)+b(v,\varphi )=-{\int }_D\varphi \mathrm{div}vdx=F_{\varphi }(v)$ by step 2.1, while $b(0,q)=0$ for every $q\in Q$. Thus $(0,\varphi )$ satisfies both weak Stokes equations. Uniqueness in [F2] identifies it with the solution, so the pressure notation in [F3] gives $p_{F_{\varphi }}=\varphi$.

By [F6], $S_{\varphi }(q)={\int }_D\varphi qdx$ is bounded on $Q$; by [F4], [F5], and $\varphi \in Q$, it is a bounded gauge-invariant sensor. Since a nonempty open $D$ contains a ball and no ball is contained almost everywhere in a hyperplane $x_1={\bar{x}}_1$, $\varphi$ is not zero almost everywhere; therefore ${\int }_D{\varphi }^{2}dx>0$.