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 $B_{2,1}^{1/2}$ in the third direction where $H$ and $B$ denote the usual homogeneous Sobolev and Besov spaces.