Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

Guès, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin

doi:10.1090/S0894-0347-04-00470-9

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability
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by Olivier Guès, Guy Métivier, Mark Williams and Kevin Zumbrun

J. Amer. Math. Soc. 18 (2005), 61-120

DOI: https://doi.org/10.1090/S0894-0347-04-00470-9

Published electronically: October 14, 2004

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Abstract:

We use energy estimates to study the long time stability of multidimensional planar viscous shocks $\psi (x_1)$ for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength are allowed, subject to an appropriate Evans function condition. The main tools are a conjugation argument that allows us to replace the eigenvalue equation by a problem in which the $x_1$ dependence of the coefficients is removed and degenerate Kreiss-type symmetrizers designed to cope with the vanishing of the Evans function for zero frequency.

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