Communications in Mathematical Sciences

Volume 14 (2016)

Number 8

Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier–Stokes equations with density dependent viscous coefficient

Pages: 2215 – 2228

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n8.a5

Authors

Alexis F. Vasseur (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Lei Yao (School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an, China)

Abstract

We prove the nonlinear stability of viscous shock waves of arbitrary amplitudes to one-dimensional compressible isentropic Navier–Stokes equations with density dependent viscosity. Under the assumption that the viscous coefficient is given as a power function of density, any viscous shock wave is shown to be nonlinear stable for small initial perturbations with integral zero. In contrast to previous related results [A. Matsumura, K. Nishihara, Japan J. Appl. Math., 2, 17–25, 1985; A. Matsumura, Y. Wang, Methods Appl. Anal., 17, 279–290, 2010], there is no restriction on the power index of the viscous coefficient and the amplitudes of the viscous shock wave in our result.

Keywords

nonlinear stability, compressible isentropic Navier–Stokes equations, viscous shock wave, energy estimate

2010 Mathematics Subject Classification

35B40, 35Q30, 76N10

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Published 26 October 2016