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Asymptotic stability of viscous shock wave for a one-dimensional isentropic model of viscous gas with density dependent viscosity. (English) Zbl 1242.76059

The authors investigate the asymptotic stability of viscous shock waves for a one-dimensional isentropic model of a viscous gas with density dependent viscosity by a weighted energy method developed in the papers of A. Matsumura and M. Mei [Osaka J. Math. 34, No. 3, 589–603 (1997; Zbl 0945.74539)] and I. Hashimoto and A. Matsumura [Methods Appl. Anal. 14, No. 1, 45–59 (2007; Zbl 1149.35057)]. 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 A. Matsumura and K. Nishihara [Japan J. Appl. Math. 2, 17–25 (1985; Zbl 0602.76080)] where the viscosity coefficient was given by a constant, and a restriction on strength of viscous shock wave was assumed. This also analytically assures the spectral stability in Zumbrun’s theory for any viscous shock wave in the present specific case.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76L05 Shock waves and blast waves in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
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