Steady compressible Navier–Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R2. We show existence if a solution that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of the constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.