The resolution of the Navier-Stokes equations in anisotropic spaces

Abstract

In this paper we prove global existence and uniqueness for solutions of the dimensional Navier-Stokes equations with small initial data in spaces which are $H^{\delta_i}$ in the i-th direction, $\delta_1 + \delta_2 + \delta_3 = 1/2, –1/2 < \delta_i < 1/2$ and in a space which is L^2 in the first two directions and B^{1/2}_{2,1} in the third direction where $H$ and $B$ denote the usual homogeneous Sobolev and Besov spaces.