Geodesic
Some remarks on multidimensional systems of conservation laws
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 225-233.

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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. 
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Bressan, Alberto. Some remarks on multidimensional systems of conservation laws. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 225-233. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a5/

[1] L. Ambrosio, Transport equation and the Cauchy problem for $BV$ vector fields. Preprint 2003. | DOI | MR | Zbl

[2] P. Brenner, The Cauchy problem for the symmetric hyperbolic systems in $L_{p}$. Math. Scand., 19, 1966, 27-37. | fulltext EuDML | MR | Zbl

[3] A. Bressan, Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. Oxford University Press, 2000. | Zbl

[4] A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova, 110, 2003, 103-117. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[5] A. Bressan, A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova, 110, 2003, 97-102. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[6] A. Bressan - T.P. Liu - T. Yang, $L^{1}$ stability estimates for $n \times n$ conservation laws. Arch. Rational Mech. Anal., 149, 1999, 1-22. | DOI | MR | Zbl

[7] C. Dafermos, Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin 1999. | Zbl

[8] R. Diperna - P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 1989, 511-517. | fulltext EuDML | DOI | MR | Zbl

[9] L. Gärding, Problèmes de Cauchy pour les systèmes quasi-linéaires d’ordre un strictement hyperboliques. In: Les E.D.P’s, vol. 117, Paris 1963, Colloques Internationaux du CNRS, 33-40. | MR | Zbl

[10] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18, 1965, 697-715. | MR | Zbl

[11] S. Kruzhkov, First-order quasilinear equations with several space variables. Math. USSR Sbornik, 10, 1970, 217-273. | Zbl

[12] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Springer-Verlag, New York 1984. | DOI | MR | Zbl

[13] E.Y. Panov, 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. | DOI | MR | Zbl

[14] J. Rauch, $BV$ estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Comm. Math. Phys., 106, 1986, 481-484. | fulltext mini-dml | MR | Zbl

[15] D. Serre, Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, 47, 1997, 139-153. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[16] D. Serre, Systems of Conservation Laws I. Cambridge University Press, 2000. | MR | Zbl

[17] J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York 1983. | MR | Zbl

[18] Y. Zheng, Systems of conservation laws: two-dimensional Riemann problems. Birkhäuser, 2001. | DOI | MR | Zbl

[19] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York 1989. | DOI | MR | Zbl