On an inhomogeneous slip-inflow boundary value problem for a steady viscous compressible channel flow

Abstract

We prove the existence and uniqueness of a strong solution to the steady isentropic compressible Navier–Stokes equations with inflow boundary condition for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary conditions on the inflow and outflow part of the boundary and the full Navier boundary conditions on the wall ${\mathrm{\Gamma }}_0$ for the velocity field are considered. For our result, there are no restrictions on the amplitude of friction coefficients α, and only the assumption that the viscosity coefficient μ is appropriately large is required. 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 the continuity equation, establish the a priori estimates for a linearized system and apply the fixed point argument.

Keywords:

• Inhomogeneous boundary conditions
• compressible Navier–Stokes system
• strong solution

Maths:

• 35-11

Disclosure statement

No potential conflict of interest was reported by the author(s).