Stokes Resolvent Estimates in Spaces of Bounded Functions

TL;DR

This paper introduces a new approach to analyze the Stokes equation in spaces of bounded functions, establishing the generation of analytic semigroups and providing new a priori estimates, extending understanding beyond traditional $L^p$-spaces.

Contribution

It presents a novel method inspired by Masuda-Stewart techniques to obtain $L^$-type estimates for the Stokes operator, applicable to various boundary conditions.

Findings

Stokes operator generates a $C_0$-analytic semigroup of angle $$ on $C_{0,}(0)$.

The method applies to different boundary conditions, including Robin boundary conditions.

New a priori $L^$-type estimates are established for the Stokes equation.

Abstract

The Stokes equation on a domain $\Omega \subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\infty$. The situation is very different for the case $p=\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. Nevertheless it was recently proved by the first and the second author of this paper by a contradiction argument that the Stokes operator generates an analytic semigroup on spaces of bounded functions for a large class of domains. This paper presents a new approach as well as new a priori $L^\infty$-type estimates to the Stokes equation. They imply in particular that the Stokes operator generates a $C_0$-analytic semigroup of angle $\pi/2$ on $C_{0,\sigma}(\Omega)$, or a non-$C_0$-analytic semigroup on $L^\infty_\sigma(\Omega)$ for a large class of domains. The approach presented is inspired by the so called Masuda-Stewart technique for elliptic operators. It is shown furthermore that the method presented applies also to different type of boundary conditions as, e.g., Robin boundary conditions.