Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder. Assuming that the derivatives of density, velocity, external force with respect to the third co-ordinate are sufficiently small in some norms we prove large time regular solutions without any restriction on the existence time. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.