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

TL;DR

This paper establishes the global well-posedness of the 3-D incompressible anisotropic Navier-Stokes equations for small initial data in specific Besov-Sobolev spaces, including high oscillatory data.

Contribution

It introduces new scaling invariant Besov-Sobolev spaces and proves global well-posedness under smallness conditions stronger than existing norms.

Findings

Global well-posedness for small initial data in anisotropic spaces

Applicable to high oscillatory initial data

Conditions depend on horizontal viscosity strength

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, $B^{-1+\frac{2}{p},{1/2}}_{p}$ and $B^{-1+\frac{2}{p},{1/2}}_{p}(T)$, $p\geq2$. 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 $B^{-1+\frac{2}{p},{1/2}}_{p}$ norm. In particular, our results imply the global wellposedness of (\textit{ANS}) with high oscillatory initial data.