Global large solutions to 3-D inhomogeneous Navier–Stokes system with one slow variable

Abstract

Under a nonlinear smallness condition on the isotropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the horizontal components of the initial velocity, which have to be exponentially small compared with the critical anisotropic Besov norm to the third component of the initial velocity, we investigate the global wellposedness of 3-D inhomogeneous incompressible Navier–Stokes equations (1.2) in the critical Besov spaces. The novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field. In the second part, we apply the same idea to prove the global wellposedness of (1.2) with some large data which are slowly varying in one direction.