Estimates of anisotropic Sobolev spaces with mixed norms for the Stokes system in a half-space

Abstract

We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in $${\mathbb {R}}^n_+ \times (0,T)$$ . We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier–Stokes equations in a half-space in dimension three.