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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
DOI: https://doi.org/10.1090/S0002-9947-01-02831-8
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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  • 1. A. I. Akhiezer, G. Ia. Liubarskii, R. V. Polovin. The stability of shock waves in magnetohydrodynamics. Soviet Physics JETP, 35 (1959), pp 507-511. MR 21:1117
  • 2. S. Benzoni-Gavage. Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal. TMA, 31 (1998), pp 243-263. MR 99b:76070
  • 3. S. Benzoni-Gavage. Stability of subsonic planar phase boundaries in a van der Waals fluid. Archive for Rational Mechanics and Analysis. 150 (1999), 23-55. MR 2001a:76109
  • 4. H. A. Bethe. On the theory of shock waves for an arbitrary equation of state. US Army report (1942). In: Classic papers in shock compression science, Springer, New York (1998), pp 421-492. CMP 98:14
  • 5. C. Carasso, M. Rascle, D. Serre. Etude d'un modèle hyperbolique en dynamique des câbles. RAIRO, Modél. Math. Anal. Numér. 19 (1985), pp 573-599. MR 87j:73068
  • 6. J. J. Erpenbeck. Stability of step shocks. Phys. Fluids, 5 (1962), pp 1181-1187. MR 27:5449
  • 7. G. R. Fowles. On the evolutionary condition for stationary plane waves in inert and reactive substances. Shock induced transitions and phase structures in general media. IMA vol. Math. Appl., 52, Springer, New York (1993), pp 93-110. MR 94g:76043
  • 8. H. Freistühler. Hyperbolic systems of conservation laws with rotationally equivariant flux function. Mat. Aplic. Comp., 11 (1992), pp 167-188.
  • 9. H. Freistühler. A short note on the persistence of ideal shock waves. Arch. Math., 64 (1995), pp 344-352. MR 96i:35084
  • 10. H. Freistühler. Some results on the stability of non-classical shock waves. J. PDEs, 11 (1998), pp 23-38. MR 99e:35142
  • 11. H. Freistühler. The persistence of ideal shock waves. Appl. Math. Lett., 7 (1994), pp 7-11. MR 96c:35114
  • 12. H. Freistühler, K. Zumbrun. Examples of unstable overcompressive shock waves, Rédaction non publiée, RWTH Aachen, Février 1998.
  • 13. E. Isaacson, B. Temple. The Riemann problem near a hyperbolic singularity, II, III. SIAM J. Appl. Math. 48 (1988), pp 1287-1301, 1302-1318. MR 89k:35140
  • 14. E. Isaacson, D. Marchesin, B. Plohr, B. Temple. The Riemann problem near a hyperbolic singularity, I. SIAM J. Appl. Math. 48 (1988), pp 1009-1032. MR 89k:35139
  • 15. R. Gardner, K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math, 51 (1998), pp 797-855. MR 99c:35152
  • 16. S. Godunov. Lois de conservation et intégrales d'énergie des équations hyperboliques. In Nonlinear hyperbolic problems, St-Etienne 1986. Lect. Notes Math. 1270 (1987), Springer-Verlag. CMP 20:02
  • 17. A. Jeffrey, T. Taniuti. Nonlinear wave propagation. With applications to physics and magnetohydrodynamics. Academic Press, New York (1964). MR 29:4410
  • 18. H.-O. Kreiss. Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math., 23 (1970), pp 277-298. MR 55:10862
  • 19. P. Lax. Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math, 10 (1957), pp 537-566. MR 20:176
  • 20. T.-P. Liu. Nonlinear stability of shock waves for viscous conservation laws, Memoirs of the Amer. Math. Soc., 328. Providence, 1985. MR 87a:35127
  • 21. T.-P. Liu. The Riemann problem for general systems of conservation laws. J. Diff. Eqn. 18 (1975), pp 218-234. MR 51:6168
  • 22. A. J. Majda. The stability of multi-dimensional shock fronts. Memoirs of the Amer. Math. Soc. 275. Providence, 1983. MR 84e:35100
  • 23. A. J. Majda. Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci, 53. Springer-Verlag, New York, 1984. MR 85e:35077
  • 24. D. Serre. Systems of conservation laws, II. Cambridge University Press. Cambridge (2000). MR 2001c:35146
  • 25. M. Shearer, D. Schaeffer. The classification of $2\times2$ systems of nonstrictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Math. 40 (1987), pp 141-178. MR 88a:35155
  • 26. K. Zumbrun, D. Serre. Viscous and inviscid stability of multidimensional shock fronts. Indiana Univ. Math. J., 48 (1999), pp 937-992. CMP 2000:07

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

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