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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

La transition vers l'instabilité pour les ondes de choc multi-dimensionnelles


Author: Denis Serre
Journal: Trans. Amer. Math. Soc. 353 (2001), 5071-5093
MSC (1991): Primary 35L50; Secondary 35L65, 35L67
Published electronically: July 17, 2001
MathSciNet review: 1852095
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Abstract:

We consider multi-dimensional shock waves. We study their stability in Hadamard's sense, following Erpenbeck and Majda's strategy. When the unperturbed shock is close to a Lax shock which is already $1$-d unstable, we show, under a generic hypothesis, that it cannot be strongly stable. We also characterize strong instability in terms of a sign of an explicit quadratic form. In most cases, the instability under 1-d perturbations, which occurs for exceptional shock waves, characterizes a transition between weak stability and strong instability in the multi-dimensional setting.


RÉSUMÉ. Nous considérons la stabilité des ondes de choc multi-dimensionnelles, en suivant la stratégie d'Erpenbeck et Majda. Lorsque le choc non perturbé est proche d'un choc de Lax longitudinalement instable, nous montrons, moyennant une hypothèse générique, que des ondes de surface sont présentes, empêchant ainsi la stabilité forte. Nous donnons aussi un critère d'instabilité forte en termes de signe d'une certaine forme quadratique. L'instabilité $1$-d d'un choc est en général facile à établir, car elle revêt un caractère exceptionnel. Elle apparaît comme une transition entre la stabilité faible et l'instabilité dans le contexte multi-d.


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Additional Information

Denis Serre
Affiliation: Unité de Mathématiques Pures et Appliquées, (CNRS UMR #5669), ENS Lyon, 46, Allée d’Italie, 69364 Lyon Cedex 07, France
Email: serre@umpa.ens-lyon.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02831-8
PII: S 0002-9947(01)02831-8
Keywords: Shock waves, Kreiss-Lopatinski condition, evolutionary condition
Received by editor(s): September 8, 1999
Received by editor(s) in revised form: December 21, 2000
Published electronically: July 17, 2001
Additional Notes: Travail effectué en accomplissement du projet TMR “Hyperbolic conservation laws", contract #ERB FMRX-CT96-0033
Article copyright: © Copyright 2001 American Mathematical Society