Stokes Resolvent Estimates in Spaces of Bounded Functions

The Stokes equation on a domain $\mathrm{\Omega }\subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\mathrm{\infty }$. The situation is very different for the case $p=\mathrm{\infty }$ 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 $L^{\mathrm{\infty }}$-type estimates to the Stokes equation. They imply in particular that the Stokes operator generates a $C_0$-analytic semigroup of angle $\pi \mathrm{/}2$ on $C_{0,\sigma }(\mathrm{\Omega })$, or a non-$C_0$-analytic semigroup on $L_{\sigma }^{\mathrm{\infty }}(\mathrm{\Omega })$ 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.