Zero Viscosity Limit of Steady Compressible Shear Flow with Navier-Slip Boundary

We investigate the existence and the zero viscosity limit of steady compressible shear flow with Navier-slip boundary condition in the absence of any external force in a two-dimension domain Ω=(0,L)×(0,2). More precisely, under the assumption that the Mach number η<¹²⁺ and L1, we prove the existence of smooth solutions to steady compressible Naiver-Stokes equations near plane Poiseuille-Couette flow as well as the convergence of the solutions obtained above to the solutions of steady incompressible Euler equations when the viscous tends to zero.