Abstract: The global well-posedness of 3-D incompressible Navier-Stokes equations is one of the most

important open problems in nonlinear PDE. In this talk, we first prove the global well-posedness of N-S

provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. 

Inspired by the above calculations, then we shall construct a family of initial data varying fast enough in

the vertical variable, which are not small in the $BMO^{-1}$ sense, but can still generate a unique global solution to N-S. 

Further, we can prove that as long as $\partial_3 u_0$ is small in some sense, then N-S is globally well-posedness. 

This result can cover the large solutions which are varying slowly in one direction, or strongly oscillating in one direction. 

This result is also true even if we replace the full Laplacian in N-S to the Laplacian in the horizontal variables, which is the so-called anisotropic N-S appearing in geophysical fluids.