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Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. (English) Zbl 1102.35064
Summary: The question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated. We show that no extra assumption of convexity or genuine nonlinearity with respect to the state variable of the flux function is required for the problem to be well-posed and prove it. The proof uses a kinetic formulation of the conservation law.

MSC:
35L65 Hyperbolic conservation laws
35R05 PDEs with low regular coefficients and/or low regular data
35L45 Initial value problems for first-order hyperbolic systems
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[1] DOI: 10.1017/S0308210500003863 · Zbl 1071.35079
[2] Bachmann F., Advances in Differential Equations 11 pp 1317– (2004)
[3] Bénilan , P. ( 1972 ). Equations D’évolution Dans un Espace de Banach Quelconque et Applications. Thèse d’Etat , Univerite Paris XI , Orsay .
[4] DOI: 10.1007/s00211-003-0503-8 · Zbl 1053.76047
[5] DOI: 10.1007/BF00752112 · Zbl 0616.35055
[6] Eymard R., Handbook of Numerical Analysis pp 713– (2000)
[7] Gagneux G., Analyse Mathématique de Modèles Non Linéaires de L’ingénierie Pétrolière (1996)
[8] DOI: 10.1023/A:1011574824970 · Zbl 0952.76085
[9] DOI: 10.1080/03605309508821159 · Zbl 0836.35090
[10] DOI: 10.1051/m2an:2001114 · Zbl 1032.76048
[11] DOI: 10.1093/imanum/22.4.623 · Zbl 1014.65073
[12] Karlsen K. H., Electron. J. Differential Equations 93 (2002)
[13] Karlsen K. H., Skr. K. Nor. Vidensk. Selsk. 3 pp 1– (2003)
[14] Kruzhkov S. N., Mat. Sb. (N.S.) 81 pp 228– (1970)
[15] Karlsen , K. H. , Towers , J. D. ( 2004 ). Convergence of the lax-friedrichs scheme and stability for conservation laws with a discontinuous space-time dependant flux . www.math.ntnu.no/conservation/2004/005.html . · Zbl 1112.65085
[16] Kuznetsov N. N., Ž. Vyčisl. Mat. i Mat. Fiz. 16 pp 1489– (1976)
[17] DOI: 10.1090/S0894-0347-1994-1201239-3
[18] DOI: 10.1016/S0021-7824(99)80003-8 · Zbl 0919.35088
[19] Perthane B., Oxford Lecture Series in Mathematics and its Applications 21, in: Kinetic Formulation of Conservation Laws (2002)
[20] DOI: 10.1017/S0308210500002870 · Zbl 1057.47056
[21] DOI: 10.1016/S0022-0396(03)00213-4 · Zbl 1036.35132
[22] DOI: 10.1007/s00205-003-0282-5 · Zbl 1059.35044
[23] DOI: 10.1142/S0218202503002477 · Zbl 1078.35011
[24] Towers J. D., Society for Industrial and Applied Mathematics J. Numer. Anal. 38 pp 681– (2000)
[25] Towers J. D., Society for Industrial and Applied Mathematics J. Numer. Anal. 39 pp 1197– (2001)
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