Inhomogeneous incompressible viscous flows with slowly varying initial data

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3-D inhomogeneous incompressible Navier-Stokes system in the whole space~$\R^3$. This class of data is based on functions which vary slowly in one direction. The idea is that 2-D inhomogeneous Navier-Stokes system with large data is globally well-posedness and we construct the 3-D approximate solutions by the 2-D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2-D inhomogeneous Navier-Stokes system. We obtained the same optimal decay estimates as the solutions of 2-D homogeneous Navier-Stokes system.