Global well-posedness of the three dimensional incompressible anisotropic Navier–Stokes system

In this paper, we study the global well-posed problem for the three dimensional incompressible anisotropic Navier–Stokes system (ANS) with initial data in the scaling invariant Besov–Sobolev type spaces. We prove that (ANS) has a unique global solution provided that the initial vertical velocity is large while initial horizontal data are sufficiently small compared with the horizontal viscosity. In particular, our result implies the global well-posedness of (ANS) with highly oscillating initial data.