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Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability


Authors: Olivier Guès, Guy Métivier, Mark Williams and Kevin Zumbrun
Journal: J. Amer. Math. Soc. 18 (2005), 61-120
MSC (2000): Primary 35L60; Secondary 35B35, 35B45, 35K55, 35L65, 76L05
DOI: https://doi.org/10.1090/S0894-0347-04-00470-9
Published electronically: October 14, 2004
MathSciNet review: 2114817
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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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Additional Information

Olivier Guès
Affiliation: LATP, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille, France
Email: gues@cmi.univ-mrs.fr

Guy Métivier
Affiliation: MAB, Université de Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
Email: Guy.Metivier@math.u-bordeaux.fr

Mark Williams
Affiliation: University of North Carolina, Department of Mathematics, CB 3250, Phillips Hall, Chapel Hill, NC 27599
Email: williams@email.unc.edu

Kevin Zumbrun
Affiliation: Indiana University, Department of Mathematics, Rawles Hall, Bloomington, IN 47405
Email: kzumbrun@indiana.edu

DOI: https://doi.org/10.1090/S0894-0347-04-00470-9
Keywords: Viscous shock stability, Evans-Lopatinski condition, Kreiss symmetrizers
Received by editor(s): September 18, 2002
Published electronically: October 14, 2004
Additional Notes: Research was supported in part by European network HYKE grants HPRN-CT-2002-00282 (O.G.) and HPRN-CT-2002-00282 (G.M.) and by NSF grants DMS-0070684 (M.W.) and DMS-0070765 (K.Z.).
Article copyright: © Copyright 2004 American Mathematical Society
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