Weak and strong solutions for the stokes approximation of non-homogeneous incompressible Navier-Stokes equations

In this paper, the Dirichlet problem of Stokes approximate of non-homogeneous incompressible Navier-Stokes equations is studied. It is shown that there exist global weak solutions as well as global and unique strong solution for this problem, under the assumption that initial density rho(0)(x) is bounded away from 0 and other appropriate assumptions (see Theorem 1 and Theorem 2). The semi-Galerkin method is applied to construct the approximate solutions and a prior estimates are made to elaborate upon the compactness of the approximate solutions.