STOKES RESOLVENT ESTIMATES IN SPACES OF BOUNDED FUNCTIONS

The Stokes equation on a domain Omega subset of 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 < infinity. The situation is very different for the case p = infinity 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-infinity-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,C-sigma (Omega), or a non-C-0-analytic semigroup on L-sigma(infinity)(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 types of boundary conditions as, e.g., Robin boundary conditions.