On estimates for the Stokes flow in a space of bounded functions

In this paper, we study regularizing effects of the composition operator S(t)P∂ for the Stokes semigroup S(t) and the Helmholtz projection P in a space of bounded functions. We establish new a priori L∞-estimates of the operator S(t)P∂ for a certain class of domains including bounded and exterior domains. They imply unique existence of mild solutions of the Navier-Stokes equations in a space of bounded functions.