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

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.