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

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

2007-12-17

arxiv.org/abs/0712.2652v2

Mathematics

Analysis of PDEs

39 pages

Scientific paper

10.1007/s00220-008-0631-1

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.

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