Bounded Stokes pressure sensors

Assume Countable Choice. Let $d\in \{2,3\}$ and let $D\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz open set. Put $Q:=L_0^2(D)=\{q\in L^2(D;\mathbb{R}):{\int }_{D}qdx=0\}$ with the $L^2$ norm, and write $Q^{\mathrm{\prime }}$ for the continuous real dual of $Q$. A bounded Stokes pressure sensor on $D$ is a bounded real-linear functional $S:Q\to \mathbb{R}$, where bounded means that its operator norm $\mathrm{\parallel }S{\mathrm{\parallel }}_{Q^{\mathrm{\prime }}}:={\sup }_{\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(D)}\le 1}\mathrm{\mid }S(q)\mathrm{\mid }$ is finite. The reading of $S$ at a normalized pressure $q\in Q$ is $S(q)$.

Guermond, MATH 661, Chapter 53, §§53.1 and 53.2.1, weak Stokes formulation and zero-mean pressure gauge.

Boffi–Brezzi–Fortin, Mixed Finite Element Methods and Applications, §8.2.1, pp. 462–463, mixed formulation and pressure modulo constants.

The normalized pressure-space convention $Q=L_0^2(D)$ is supplied by Stokes velocity and pressure spaces and weak forms, under Countable Choice.