In the first part of this thesis we established a maximal regularity result to the Stokes equations in exterior domains with moving boundary. This leads to existence of solutions to the Navier-Stokes equations globally in time for small data. Secondly, we consider Leray's problem on the decay of weak solutions to the Navier-Stokes equations in an exterior domain with non-homogeneous Dirichlet boundary data. It is shown that the solution decays polynomially.
