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, $B^{-1+frac{2}{p},{1/2}}_{p}$ a
nd $B^{-1+frac{2}{p},{1/2}}_{p}(T)$, $pgeq2$. 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.
In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $mathbb{R}^N$, $Ngeq2$ (or exte
rior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).
