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

TL;DR

This paper proves resolvent estimates for the Stokes operator in bounded nonsmooth domains, establishing the generation of bounded analytic semigroups in spaces of bounded solenoidal functions, with sharp smoothness conditions.

Contribution

It introduces new estimates linking pressure and velocity in $L^q$ averages, extending resolvent analysis to nonsmooth domains and exterior domains.

Findings

Resolvent estimates hold in $L^ abla$ spaces for Lipschitz and $C^1$ domains.

The Stokes operator generates a bounded analytic semigroup in these spaces.

Smoothness conditions on the domain are proven to be sharp.

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.