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

Serre, Denis

doi:10.1090/S0002-9947-01-02831-8

La transition vers l’instabilité pour les ondes de choc multi-dimensionnelles
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by Denis Serre

Trans. Amer. Math. Soc. 353 (2001), 5071-5093

DOI: https://doi.org/10.1090/S0002-9947-01-02831-8

Published electronically: July 17, 2001

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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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