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 $\epsilon (\rho) = \epsilon \rho^{\alpha} (\alpha \gt 0)$ 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 $\alpha$ is obtained for all $\alpha \gt 0$. 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.

Keywords

compressible Navier-Stokes equations, vanishing viscosity limit, density-dependent viscosity, rarefaction wave, vacuum

2010 Mathematics Subject Classification

35L60, 35L65, 76N15

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Published 3 December 2014