On an inhomogeneous slip-inflow boundary value problems for a steady viscous compressible channel flows

We prove the existence and uniqueness of strong solutions to the steady isentropic compressible Navier-Stokes equations with inflow boundary conditions for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary condition on inflow and outflow part of the boundary while the full Navier boundary conditions on the wall $\Gamma_{0}$ for the velocity filed are considered. For our result, there are no restrictions on the amplitude of friction coefficients $\alpha$, and the appropriately large hypothesis for the viscous coefficients $\mu$ is enough. One of the substantial ingredients of our proof is an elegant transformation induced by the flow field. With the help of this transformation, we can overcome the difficulties caused by the hyperbolicity of continuity equation, establish the a priori estimates for a linearized system and apply the fixed point arguments.