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

Ting Zhang; Daoyuan Fang

doi:10.1007/s00220-008-0631-1

paper

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

arXiv:0712.2652 · doi:10.1007/s00220-008-0631-1

Abstract

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

39 pages