Stokes resolvent estimates in spaces of bounded functions

The Stokes equation on a domain Ω Rⁿ is well understood in the Lp-setting for a large class of domains including bounded and exterior domains with smooth boundaries pro- vided 1 <p <∞. The situation is very different for the case p = ∞ 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^∞-type estimates to the Stokes equation. They imply in par- ticular that the Stokes operator generates a C0-analytic semigroup of angle π/2 on C_0,α(Ω), or a non- C0-analytic semigroup on L^∞_α (Ω) 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.