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 ∈ R. We show existence if a solution (v, ρ) ∈ W 2 p (Q) ×W 1 p (Q) that is a small perturbation of a constant flow (v̄ ≡ [1, 0], ρ̄ ≡ 1). We also show that this solution is unique in a class of small perturbations of the constant flow (v̄, ρ̄). In order show the existence of the solution we adapt the techniques know from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument. MSC: 35Q30; 76N10