VANISHING VISCOSITY LIMIT TO RAREFACTION WAVE WITH VACUUM FOR 1-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

The vanishing viscosity limit of the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity epsilon(rho) = epsilon rho(alpha) (alpha > 0) is considered in the present paper. It is proven that given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to the compressible Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the convergence rate depending on a is obtained for all alpha > 0. The main difficulty in our proof lies in the degeneracies of the density and the density-dependent viscosity at the vacuum region in the vanishing viscosity limit.