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
HTML articles powered by AMS MathViewer

by Denis Serre PDF

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

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.

References

A. I. Akhiezer, G. Ia. Liubarskii, and R. V. Polovin, The stability of shock waves in magnetohydrodynamics, Soviet Physics. JETP 35 (8) (1959), 507–511 (731–737 Ž. Eksper. Teoret. Fiz.). MR 0102324
S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. 31 (1998), no. 1-2, 243–263. MR 1487544, DOI 10.1016/S0362-546X(96)00309-4
S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 150 (1999), no. 1, 23–55. MR 1738168, DOI 10.1007/s002050050179
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.
C. Carasso, M. Rascle, and D. Serre, Étude d’un modèle hyperbolique en dynamique des câbles, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 573–599 (French, with English summary). MR 826225, DOI 10.1051/m2an/1985190405731
Jerome J. Erpenbeck, Stability of step shocks, Phys. Fluids 5 (1962), 1181–1187. MR 155515, DOI 10.1063/1.1706503
G. Richard 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., vol. 52, Springer, New York, 1993, pp. 93–110. MR 1240334, DOI 10.1007/978-1-4613-8348-2_{5}
H. Freistühler. Hyperbolic systems of conservation laws with rotationally equivariant flux function. Mat. Aplic. Comp., 11 (1992), pp 167-188.
Heinrich Freistühler, A short note on the persistence of ideal shock waves, Arch. Math. (Basel) 64 (1995), no. 4, 344–352. MR 1319006, DOI 10.1007/BF01198091
H. Freistühler, Some results on the stability of non-classical shock waves, J. Partial Differential Equations 11 (1998), no. 1, 25–38. MR 1618406
H. Freistühler, The persistence of ideal shock waves, Appl. Math. Lett. 7 (1994), no. 6, 7–11. MR 1340723, DOI 10.1016/0893-9659(94)90085-X
H. Freistühler, K. Zumbrun. Examples of unstable overcompressive shock waves, Rédaction non publiée, RWTH Aachen, Février 1998.
E. Isaacson and B. Temple, The Riemann problem near a hyperbolic singularity. II, III, SIAM J. Appl. Math. 48 (1988), no. 6, 1287–1301, 1302–1318. MR 968831, DOI 10.1137/0148079
E. Isaacson, D. Marchesin, B. Plohr, and B. Temple, The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems. I, SIAM J. Appl. Math. 48 (1988), no. 5, 1009–1032. MR 960467, DOI 10.1137/0148059
Robert A. Gardner and Kevin Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251, DOI 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1
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.
A. Jeffrey and T. Taniuti, Non-linear wave propagation. With applications to physics and magnetohydrodynamics, Academic Press, New York-London, 1964. MR 0167137
Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI 10.1002/cpa.3160230304
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI 10.1090/memo/0328
Tai Ping Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218–234. MR 369939, DOI 10.1016/0022-0396(75)90091-1
Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422, DOI 10.1090/memo/0275
A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
Denis Serre, Systems of conservation laws. 2, Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems; Translated from the 1996 French original by I. N. Sneddon. MR 1775057
David G. Schaeffer and Michael Shearer, The classification of $2\times 2$ systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40 (1987), no. 2, 141–178. MR 872382, DOI 10.1002/cpa.3160400202
K. Zumbrun, D. Serre. Viscous and inviscid stability of multidimensional shock fronts. Indiana Univ. Math. J., 48 (1999), pp 937-992.