Bounded-domain Stokes residual estimate — formal worked field

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $L_x,L_z,h,\mu >0$, define ${\mathbb{T}}_L=\mathbb{R}\mathrm{/}(L\mathbb{Z})$ for $L>0$, and set $Q={\mathbb{T}}_{L_x}\times (0,h)\times {\mathbb{T}}_{L_z}$. Regard fields on $Q$ as periodic lifts to $\mathbb{R}\times (0,h)\times \mathbb{R}$ with periods $L_x,L_z$ in $x,z$, and integrate with Lebesgue measure over one fundamental cell in each periodic coordinate. Set $w(x,y,z)=(\sin (\pi y\mathrm{/}h),0,0)$, $q=0$, and $f=\mu (\pi \mathrm{/}h{)}^{2}w$. Then $w$ is a smooth divergence-free periodic field with zero trace on both $y$ walls and satisfies $-\mu \mathrm{\Delta }w+\mathrm{\nabla }q=f$. Its norms satisfy $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(Q)}^2=L_x{L}_{z}h\mathrm{/}2$ and $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(Q)}^2=(\pi \mathrm{/}h{)}^2\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(Q)}^2$, so $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(Q)}=(h\mathrm{/}\pi )\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(Q)}\le h\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(Q)}$. This is a formal worked field for the periodic-channel Poincare clause; the general bounded $C^1$ Euclidean residual theorem is not applied to this periodic cell.