Vanishing Viscosity Limit of the Compressible Navier-Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier-Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier-Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier-Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.