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

Abstract

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, $BM{O}^{-1}.$ Yet $u_{0,\nu }$ generates a unique global solution to the classical 3-D Navier-Stokes system provided that $\nu$ is sufficiently large.

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