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 2 . We show existence if a solution ( v , ρ ) ∈ W p 2 ( Q ) × W p 1 ( Q ) that is a small perturbation of a constant flow ( v ¯ ≡ [ 1 , 0 ] , ρ ¯ ≡ 1 ). We also show that this solution is unique in a class of small perturbations of the constant flow ( v ¯ , ρ ¯ ) . 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.