Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient

In this paper, we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. Then we shall construct a family of initial data, $u_{0,\nu},$ which vary fast enough in the vertical variable and which are not small in the space, $BMO^{-1}.$ Yet $u_{0,\nu}$ generates a unique global solution to the classical 3-D Navier-Stokes system provided that $\nu$ is sufficiently large.