Global wellposed problem for the 3-D incompressible anisotropic Navier–Stokes equations

Abstract

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier–Stokes equations (ANS). In order to do so, we first introduce the scaling invariant Besov–Sobolev type spaces, B p − 1 + 2 p , 1 2 and B p − 1 + 2 p , 1 2 ( T ) , p ⩾ 2 . Then, we prove the global wellposedness for (ANS) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than B p − 1 + 2 p , 1 2 norm. In particular, our results imply the global wellposedness of (ANS) with high oscillatory initial data.