An error estimate uniform in time for spectral semi-galerkin approximations of the nonhomogeneous navier-stokes equations

We consider the spectral semi-Galerkin method applied to the nonhomogeneous Navier-Stokes equations, which describes the motion of miscibles fluids. Under certain conditions it is known that the aproximate solutions constructed by using this method converge to a global strong solution of these equations. In this paper we prove that these solutions satisfy an optimal uniform in time error estimate in the H¹—norm for the velocity. We also derive an uniform error estimate in the L ^∞—norm for the density and an improved error estimate in the L²—norm for the velocity.