Some remarks on multidimensional systems of conservation laws

Alberto Bressan

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 225-233
  • ISSN: 1120-6330

Abstract

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This note is concerned with the Cauchy problem for hyperbolic systems of conservation laws in several space dimensions. We first discuss an example of ill-posedness, for a special system having a radial symmetry property. Some conjectures are formulated, on the compactness of the set of flow maps generated by vector fields with bounded variation.

How to cite

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Bressan, Alberto. "Some remarks on multidimensional systems of conservation laws." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 225-233. <http://eudml.org/doc/252274>.

@article{Bressan2004,
abstract = {This note is concerned with the Cauchy problem for hyperbolic systems of conservation laws in several space dimensions. We first discuss an example of ill-posedness, for a special system having a radial symmetry property. Some conjectures are formulated, on the compactness of the set of flow maps generated by vector fields with bounded variation.},
author = {Bressan, Alberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Hyperbolic system; Conservation laws; Several space dimensions},
language = {eng},
month = {12},
number = {3-4},
pages = {225-233},
publisher = {Accademia Nazionale dei Lincei},
title = {Some remarks on multidimensional systems of conservation laws},
url = {http://eudml.org/doc/252274},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Bressan, Alberto
TI - Some remarks on multidimensional systems of conservation laws
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 225
EP - 233
AB - This note is concerned with the Cauchy problem for hyperbolic systems of conservation laws in several space dimensions. We first discuss an example of ill-posedness, for a special system having a radial symmetry property. Some conjectures are formulated, on the compactness of the set of flow maps generated by vector fields with bounded variation.
LA - eng
KW - Hyperbolic system; Conservation laws; Several space dimensions
UR - http://eudml.org/doc/252274
ER -

References

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  1. AMBROSIO, L., Transport equation and the Cauchy problem for B ⁢ V vector fields. Preprint 2003. Zbl1075.35087MR2096794DOI10.1007/s00222-004-0367-2
  2. BRENNER, P., The Cauchy problem for the symmetric hyperbolic systems in L p . Math. Scand., 19, 1966, 27-37. Zbl0154.11304MR212427
  3. BRESSAN, A., Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. Oxford University Press, 2000. Zbl0997.35002
  4. BRESSAN, A., An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova, 110, 2003, 103-117. Zbl1114.35123MR2033003
  5. BRESSAN, A., A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova, 110, 2003, 97-102. Zbl1114.05002MR2033002
  6. BRESSAN, A. - LIU, T.P. - YANG, T., L 1 stability estimates for n × n conservation laws. Arch. Rational Mech. Anal., 149, 1999, 1-22. Zbl0938.35093MR1723032DOI10.1007/s002050050165
  7. DAFERMOS, C., Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin1999. Zbl1196.35001
  8. DIPERNA, R. - LIONS, P.L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 1989, 511-517. Zbl0696.34049MR1022305DOI10.1007/BF01393835
  9. GÄRDING, L., Problèmes de Cauchy pour les systèmes quasi-linéaires d’ordre un strictement hyperboliques. In: Les E.D.P’s, vol. 117, Paris1963, Colloques Internationaux du CNRS, 33-40. Zbl0239.35013MR171079
  10. GLIMM, J., Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18, 1965, 697-715. Zbl0141.28902MR194770
  11. KRUZHKOV, S., First-order quasilinear equations with several space variables. Math. USSR Sbornik, 10, 1970, 217-273. Zbl0215.16203
  12. MAJDA, A., Compressible fluid flow and systems of conservation laws in several space variables. Springer-Verlag, New York1984. Zbl0537.76001MR748308DOI10.1007/978-1-4612-1116-7
  13. PANOV, E.Y., On the theory of generalized entropy solutions of the Cauchy problem for a class of non-strictly hyperbolic systems of conservation laws. Sbornik: Mathematics, 191, 2000, 121-150. Zbl0954.35107MR1753495DOI10.1070/SM2000v191n01ABEH000450
  14. RAUCH, J., B ⁢ V estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Comm. Math. Phys., 106, 1986, 481-484. Zbl0619.35073MR859822
  15. SERRE, D., Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, 47, 1997, 139-153. Zbl0864.35069MR1437182
  16. SERRE, D., Systems of Conservation Laws I. Cambridge University Press, 2000. Zbl0936.35001MR1775057
  17. SMOLLER, J., Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York1983. Zbl0807.35002MR688146
  18. ZHENG, Y., Systems of conservation laws: two-dimensional Riemann problems. Birkhäuser, 2001. Zbl0971.35002MR1839813DOI10.1007/978-1-4612-0141-0
  19. ZIEMER, W.P., Weakly Differentiable Functions. Springer-Verlag, New York1989. Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3

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