The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier–Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier–Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε 1 as the viscosity ε tends to zero.