Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

Abstract

In this paper we investigate the asymptotic stability of viscous shock wave for a onedimensional isentropic model of viscous gas with density dependent viscosity by a weighted energy method developed in the papers of Matsumura-Mei (1997) and Hashimoto-Matsumura (2007). Under the condition that the viscosity coefficient is given as a function of the absolute temperature which is determined by the Chapman-Enskog expansion theory in rarefied gas dynamics, any viscous shock wave is shown to be asymptotically stable for small initial perturbations with integral zero. This generalizes the previous result of Matsumua-Nishihara (1985) where the viscosity coefficient is given by a constant and a restriction on the strength of the viscous shock wave is assumed. This also analytically assures the spectral stability in the Zumbrun’s theory for any viscous shock wave in our specific case.