Lipschitz stability of bounded Stokes pressure observations

Assume Countable Choice. Let $d\in \{2,3\}$, let $D\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz open set, and let $\nu >0$. Set $V=H_0^1(D;{\mathbb{R}}^d)$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D)}$ and $Q=L_0^2(D)$ with the $L^2$ norm. For each $F\in V^{\mathrm{\prime }}$, let $(u_F,p_F)\in V\times Q$ be the unique pair satisfying $2\nu {\int }_D\epsilon (u_F):\epsilon (v)dx-{\int }_D{p}_F\mathrm{div}vdx=F(v)(v\in V),-{\int }_{D}q\mathrm{div}{u}_{F}dx=0(q\in Q),$ where $V^{\mathrm{\prime }}$ is the continuous real dual of $V$ and $\epsilon (u)=(\mathrm{\nabla }u+\mathrm{\nabla }{u}^T)\mathrm{/}2$. Write $C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D)$ for a finite constant in a bounded right inverse estimate $\mathrm{div}{B}_{D}q=q$ and $\mathrm{\parallel }\mathrm{\nabla }{B}_{D}q{\mathrm{\parallel }}_{L^2(D)}\le C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D)\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(D)}$ for $q\in Q$. For a positive integer $m$ and bounded pressure sensors $S_1,\dots ,S_m\in Q^{\mathrm{\prime }}$, define $\mathcal{O}(F)=(S_1(p_F),\dots ,S_m(p_F))\in {\mathbb{R}}^m$. Then $F\mapsto p_F$ and $\mathcal{O}$ are real-linear, and for all $F,G\in V^{\mathrm{\prime }}$ $\mathrm{\parallel }\mathcal{O}(F)-\mathcal{O}(G){\mathrm{\parallel }}_{{\mathbb{R}}^m}\le 3C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D){\left({\sum }_{j=1}^m\mathrm{\parallel }{S}_j{\mathrm{\parallel }}_{Q^{\mathrm{\prime }}}^2\right)}^{1\mathrm{/}2}\mathrm{\parallel }F-G{\mathrm{\parallel }}_{V^{\mathrm{\prime }}},$ where ${\mathbb{R}}^m$ has its Euclidean norm.

Guermond, MATH 661, Chapter 53, §§53.2.1–53.2.2, Lemma 53.9 and Theorem 53.11, weak formulation, divergence lifting, well-posedness and pressure stability.

Boffi–Brezzi–Fortin, Mixed Finite Element Methods and Applications, §§4.2.1–4.2.3, Example 4.2.2 and §8.2.1, pp. 225–229, 233–234, and 462–463, mixed Hilbert-space stability and Stokes pressure gauge.