Existence of suitable weak solutions to the Navier–Stokes equations in time varying domains

Abstract

We consider suitable weak solutions to the Navier–Stokes equations in time varying domains. We develop Schauder theory for the approximate Stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary. Existence of suitable weak solutions in time varying domains follows from compactness in Lebesgue and Sobolev spaces.