Steady compressible Nevier-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 in $\mathbf{R^2}$. We show existence of a strong solution $(v,\rho) \in W^2_p(Q) \times W^1_p(Q)$ that is a small perturbation of a constant flow $(\bar v \equiv [1,0],\bar \rho \equiv 1)$. We also show that this solution is unique in a class of small perturbations of the constant flow $(\bar v,\bar \rho)$. 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.