Resolvent Estimates in $L^{\mathrm{\infty }}$ for the Stokes Operator in Nonsmooth Domains

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\mathrm{\Omega }$ in $R^d$ under the assumptions that $\mathrm{\Omega }$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $\mathrm{\Omega }$. The smoothness conditions on $\mathrm{\Omega }$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level.