Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space

Abstract

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (ANS). We prove the global wellposedness for ANS provided the initial horizontal data are sufficient small in the scaling invariant Besov-Sobolev type space \({B^{0,\frac{1}{2}}}\) . In particular, the result implies the global wellposedness of ANS with large initial vertical velocity.