Steady compressible Nevier-Stokes flow in a square

Abstract

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 $(v,\rho )\in W_p^2(Q)\times W_p^1(Q)$ 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.

29 pages