Stokes Resolvent Estimates in Spaces of Bounded Functions

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 $L_{\sigma }^{\infty }(\Omega )$ 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.

22 pages, to appear in Ann. Sci. Éc. Norm. Supér. (4)