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In this paper, we first prove the global well-posedness of 3-D anisotropicNavier-Stokes system provided that the vertical viscous coefficient of thesystem is sufficiently large compared to some critical norm of the initialdata. Then we shall construct a family of initial data, $u_{0,\nu},$ which varyfast 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 classical3-D Navier-Stokes system provided that $\nu$ is sufficiently large.

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