A Schauder theory for the Stokes equations in rough domains

Abstract

We consider the steady Stokes equations in a bounded domain with forcing in divergence form supplemented with no-slip boundary conditions. We provide a maximal regularity theory in Campanato spaces (inlcuding $\mathrm{BMO}$ and $C^{0,\alpha }$ for $0<\alpha <1$ as special cases) under minimal assumptions on the regularity of the underlying domain. Our approach is based on pointwise multipliers in Campanto spaces.

To appear in Indiana Univ. Math. J