On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω = Ω0 × (0, L) ∈ R. We show existence of a solution (v, ρ) ∈ W 2 p (Ω) × W 1 p (Ω), where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (v̄ ≡ [1, 0, 0], ρ̄ ≡ 1). We also show that this solution is unique in a class of small perturbations of (v̄, ρ̄). The term u · ∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (v, ρ) that is bounded in W 2 p (Ω) × W 1 p (Ω) and satisfies the Cauchy condition in a larger space L∞(0, L;L2(Ω0)) what enables us to deduce that the weak limit of a subsequence of (v, ρ) is in fact a strong solution to our problem.