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 $\mathrm{\Omega }=(0,L)\times (0,2)$. More precisely, under the assumption that the Mach number $\eta <{\text{va}}^{\text{f}12+}$ and $L\ll 1$, 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 $\text{va}$ tends to zero.