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

Abstract

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega$ in $R^d$ under the assumptions that $\Omega$ 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 $\Omega$. The smoothness conditions on $\Omega$ 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.