On scalar hyperbolic conservation laws with a discontinuous flux. (English) Zbl 1217.35117
Summary: We study the Cauchy problem for scalar hyperbolic conservation laws with a flux that can have jump discontinuities. We introduce new concepts of entropy weak and measure-valued solutions that are consistent with the standard ones if the flux is continuous. Having various definitions of solutions to the problem, we then answer the question what kind of properties the flux should possess in order to establish the existence and/or uniqueness of solutions of a particular type. In any space dimension we establish the existence of a measure-valued entropy solution for a flux having countable jump discontinuities. Under the additional assumption of the Hölder continuity of the flux at zero, we prove the uniqueness of an entropy measure-valued solution, and as a consequence, we establish the existence and uniqueness of a weak entropy solution. If we restrict ourselves to one spatial dimension, we prove the existence of weak solutions to the problem where the flux has merely monotone jumps; in such a setting we do not require any continuity of the flux at zero.
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 |
References:
| [1] | Amann, H.; Escher, J., Analysis II, 2008, Birkhäuser Verlag · Zbl 1146.26001 |
| [2] | Audusse, E.Perthame, B., Proc. R. Soc. Edinb., Sec. A Math.135, 253 (2005). · Zbl 1071.35079 |
| [3] | Bulíček, M.et al., Adv. Calc. Var.2, 109 (2009). |
| [4] | Crandall, M. G., Israel J. Math.12, 108 (1972). · Zbl 0246.35018 |
| [5] | Dias, J.-P.Figueira, M.Rodrigues, J.-F., J. Math. Fluid Mech.7, 153 (2005). · Zbl 1065.35184 |
| [6] | DiPerna, R. J., Arch. Rational Mech. Anal.88, 223 (1985). · Zbl 0616.35055 |
| [7] | Evans, L. C., Weak Convergence Methods for Nonlinear Partial Differential Equations, 74, 1990, Conference Board of the Mathematical Sciences: Conference Board of the Mathematical Sciences, Washington |
| [8] | Francfort, G.Murat, F.Tartar, L., Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.7, 23 (2004). · Zbl 1115.35047 |
| [9] | Gimse, T., SIAM J. Math. Anal.24, 279 (1993). · Zbl 0801.35081 |
| [10] | Kružkov, S. N., Mat. Sb. (N.S.)81, 228 (1970). |
| [11] | Perthame, B., Kinetic Formulation of Conservation Laws, 21, 2002, Oxford Univ. Press · Zbl 1030.35002 |
| [12] | Panov, E. Yu., J. Hyperbolic Diff. Eqns.6, 525 (2009). |
| [13] | Rajagopal, K. R., Appl. Math.48, 279 (2003). · Zbl 1099.74009 |
| [14] | Rajagopal, K. R., J. Fluid Mech.550, 243 (2006). · Zbl 1097.76009 |
| [15] | Rajagopal, K. R., Z. Angew. Math. Phys.58, 309 (2007). · Zbl 1113.74006 |
| [16] | Szepessy, A., Comm. Partial Diff. Eqns.14, 1329 (1989). · Zbl 0704.35022 |
| [17] | Szepessy, A., Arch. Rational Mech. Anal.107, 181 (1989). · Zbl 0702.35155 |
| [18] | Tartar, L., Systems of Nonlinear Partial Differential Equations, 111 (Reidel, 1983) pp. 263-285. · Zbl 0514.00014 |
| [19] | Tartar, L., Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symp.39, (Pitman, 1979) pp. 136-212. · Zbl 0437.35004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
