Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces

In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces B and B-4(-1/2, 1/2) (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that (ANS(nu)) has a unique global solution with initial data u(0) = (u(0)(h), u(0)(3)) which satisfies parallel to u(0)(h)parallel to B exp(C/nu(4) parallel to u(0)(3)parallel to(4)(B)) <= c(0)nu or parallel to u(0)(h)parallel to(B4-1/2, 1/2) exp(C/nu(4)parallel to u(0)(3)parallel to(4)(B4-1/2, 1/2)) <= c(0)nu for some c(0) sufficiently small. To overcome the difficulty that Gronwall's inequality can not be applied in the framework of Chemin-Lerner type spaces, <(L-t(p))over tilde>(B), we introduced here sort of weighted Chemin-Lerner type spaces, <(L-t, f(2))over tilde>(B) for some apropriate L-1 function f(t).