Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stratified solutions for systems of conservation laws

Authors: Andrea Corli and Olivier Gues
Journal: Trans. Amer. Math. Soc. 353 (2001), 2459-2486
MSC (2000): Primary 35L65, 35L67; Secondary 35L45, 58G17
Published electronically: February 13, 2001
MathSciNet review: 1814078
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular'' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.

References [Enhancements On Off] (What's this?)

  • 1. G. Boillat: Chocs caractéristiques, C. R. Acad. Sci. Paris, Série A-B, 274 (1972), A1018-A1021. MR 46:4736
  • 2. A. Bressan: The semigroup approach to systems of conservation laws, Mat. Contemp. 10 (1996), 21-74. MR 97k:35158
  • 3. J. Chazarain and A. Piriou: Introduction to the theory of linear partial differential equations, North-Holland (1982). MR 83j:35001
  • 4. N. Dencker: On the propagation of polarization sets for systems of real principal type, J. Funct. Analysis 46 (1982), 351-372. MR 84c:58081
  • 5. W. E: Propagation of oscillations in the solutions of 1-D compressible fluid equations, Comm. Partial Differential Equations 17(3/4) (1992), 347-370. MR 93a:35120
  • 6. H. Freistühler: Linear degeneracy and shock waves, Math. Z. 207 (1991), 583-596. MR 92e:35108
  • 7. K. O. Friedrichs: Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345-392. MR 16:44c
  • 8. K. O. Friedrichs and P. D. Lax: Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686-1688. MR 44:3016
  • 9. J. Glimm: Solutions in the large for nonlinear systems of conservation laws, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 33:2976
  • 10. E. Godlewski and P. A. Raviart: Hyperbolic systems of conservation laws, Ellipses (1991). MR 95i:65146
  • 11. O. Guès: Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15 (1990), 595-645. MR 91i:35122
  • 12. A. Heibig: Error estimates for oscillatory solutions to hyperbolic systems of conservation laws, Comm. Partial Differential Equations 18 (1993), 281-304. MR 94c:35121
  • 13. P. D. Lax: Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 20:176
  • 14. G. Métivier: Ondes soniques, J. Math. Pures Appl., IX. Ser., 70(2) (1991), 197-268. MR 92g:35045
  • 15. J. Moser: A rapidly convergent iteration method and non-linear partial differential equations, Ann. Scuola Norm. Sup. Pisa 20 (1966), 265-315. MR 33:7667
  • 16. Y.-J. Peng: Solutions faibles globales pour l'équation d'Euler d'un fluide compressible avec de grandes données initiales, Comm. Partial Differential Equations, 17 (1992), 161-187. MR 93e:35083
  • 17. J. Rauch and M. Reed: Bounded, stratified and striated solutions of hyperbolic systems, in H. Brézis, J.-L. Lions (eds.): ``Nonlinear Partial Differential Equations and their applications'', Collège de France Seminar, vol. IX, Pitman (1988), 334-351. MR 90f:35122
  • 18. B. L. Rozhdenstventskyi and N. Yanenko: Systems of quasilinear equations and their applications to gas dynamics, A. M. S. Translations of Mathematical Monographs 55 (1983).
  • 19. D. Serre: Oscillations non-linéaires de haute fréquence; dim=1, in J.-L. Lions and H. Brézis (ed.): Séminaires Collège de France 12, Longman. MR 96e:35102
  • 20. D. Serre: Quelques méthodes d'étude de la propagation d'oscillations hyperboliques non linéaires, Sém. Equations aux Dérivées Partielles 1990-91, Ecole Polytechnique, exposé n. 20. MR 92k:35178
  • 21. D. Serre: Systèmes de lois de conservation II, Diderot éditeur, Arts et Sciences, 1996, Paris. MR 99e:35144
  • 22. B. Sévennec: Géométrie des systèmes hyperboliques de lois de conservation, Société Math. de France, Mémoire 56, Suppl. Bull. Soc. Math., France (1994). MR 95g:35123

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L65, 35L67, 35L45, 58G17

Retrieve articles in all journals with MSC (2000): 35L65, 35L67, 35L45, 58G17

Additional Information

Andrea Corli
Affiliation: Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy

Olivier Gues
Affiliation: Laboratoire J.-A. Dieudonné, UMR 6621 CNRS, Université de Nice - Sophia Antipolis, 06108 Nice, cedex 2, France

Keywords: Hyperbolic systems of conservation laws, linearly degenerate eigenvalue, weak solutions, stratified solutions
Received by editor(s): April 7, 1999
Received by editor(s) in revised form: January 7, 2000
Published electronically: February 13, 2001
Additional Notes: This research was performed at the “Laboratoire J. A. Dieudonné” of the University of Nice while the first author was a recipient of an Italian CNR grant, and at the University of Ferrara, which the second author thanks for its hospitality.
Article copyright: © Copyright 2001 American Mathematical Society