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.
Categories: math.AP