Structure stability of steady supersonic shear flow with inflow boundary conditions

We study the existence and zero viscous limit of smooth solutions to steady compressible Navier-Stokes equations near plane shear flow between two moving parallel walls. Under the assumption $0<L\ll 1$, we prove that for any plane supersonic shear flow ${\mathbf{U}}^0=(\mu (x_2),0)$, there exist smooth solutions near ${\mathbf{U}}^0$ to steady compressible Navier-Stokes equations in a 2-dimension domain $\mathrm{\Omega }=(0,L)\times (0,2)$. Moreover, based on the uniform-in-$\epsilon$ estimates, we establish the zero viscosity limit of the solutions obtained above to the solutions of the steady Euler equations.