Bounded-domain Stokes residual estimate — worked interpretation

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. In SI units, let $R>0$ metres, ${\rho }_0>0$ kilograms per cubic metre, $\mu >0$ pascal-seconds, ${\mathrm{\Omega }}_r>0$ inverse seconds, and $p_0$ pascals. On the ball $B_R=\{(x,y,z)\in {\mathbb{R}}^3:x^2+y^2+z^2<R^2\}$, let the spherical wall rotate rigidly about the $z$-axis with angular velocity $(0,0,{\mathrm{\Omega }}_r)$ and impose no slip. Define

$u_{\mathrm{N}\mathrm{S}}=u_{\mathrm{S}}=(-{\mathrm{\Omega }}_{r}y,{\mathrm{\Omega }}_{r}x,0),p_{\mathrm{S}}=p_0,p_{\mathrm{N}\mathrm{S}}=p_0+\frac{{\rho }_0{\mathrm{\Omega }}_r^2}{2}(x^2+y^2)\mathrm{.}$

The two velocity fields are smooth and divergence-free, have the same wall velocity, and satisfy respectively the stationary constant-density incompressible Navier–Stokes equation and the restricted steady Stokes comparison equation with zero body acceleration. Let $C_{B_R}>0$ be an explicitly assumed constant such that $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(B_R)}\le C_{B_R}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(B_R)}$ for every $C^1$ zero-trace vector field; this example does not establish a value for it. Their velocity difference is zero, while the convective residual is $f={\rho }_0{\mathrm{\Omega }}_r^2(x,y,0)=\mathrm{\nabla }(p_{\mathrm{N}\mathrm{S}}-p_{\mathrm{S}})$. The residual estimates therefore hold for this supplied Poincare input. This exact example has nonzero pressure residual but zero velocity error; it does not demonstrate a nonzero velocity discrepancy.