zbMATH — the first resource for mathematics

Lipschitz continuous solutions to the Cauchy problem for quasi-linear hyperbolic systems. (English) Zbl 1257.35119
Summary: Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasilinear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.

35L45 Initial value problems for first-order hyperbolic systems
35D30 Weak solutions to PDEs
35L60 First-order nonlinear hyperbolic equations
Full Text: DOI
[1] Cesari, L., A boundary-value problem for quasilinear hyperbolic systems in the Schauder canonic form, Ann. Squola Norm. Sup. Pisa, 4(1), 1974, 311–358. · Zbl 0307.35063
[2] CirinĂ , M., Nonlinear hyperbolic problems with solutions on preassigned sets, Michigan Math. J., 17, 1970, 193–209. · Zbl 0201.42702 · doi:10.1307/mmj/1029000466
[3] Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1955. · Zbl 0064.33002
[4] Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. II, Wiley, New York, 1962. · Zbl 0099.29504
[5] Courant, R. and Lax, P., Cauchy’s problem for nonlinear hyperbolic differential equations in two independent variables, Annali di matmatica, 40, 1955, 161–166. · Zbl 0065.32901 · doi:10.1007/BF02416530
[6] Douglis, A., Some existence theorems for hyperbolic systems of partial differential equations in two independent variables, Comm. Pure Appl. Math., 5, 1952, 119–154. · Zbl 0047.09101 · doi:10.1002/cpa.3160050202
[7] Douglis, A., The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data, Comm. Pure Appl. Math., 14, 1961, 267–284. · Zbl 0117.31102 · doi:10.1002/cpa.3160140307
[8] Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[9] Hartman, P. and Wintner, A., On hyperbolic partial differential equations, Amer. J. Math., 74, 1952, 834–864. · Zbl 0048.33302 · doi:10.2307/2372229
[10] Hoff, D., Locally Lipschitz solutions of a single conservation law in several space variables, J. Diff. Eq., 42(2), 1981, 215–233. · Zbl 0465.34003 · doi:10.1016/0022-0396(81)90027-9
[11] Hoff, D., A characterization of the blow-up time for the solution of a conservation law in several space variables, Comm. Part. Diff. Eq., 7(2), 1982, 141–151. · Zbl 0501.35054 · doi:10.1080/03605308208820220
[12] Kruzkov, S. N., First order quasilinear equations with several independent variables, USSR Sbornik, 10(2), 1970, 217–243. · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[13] Li, T. T., Global Classical Solutions for Quasilinear Hyperbolic Systems, Wiley Chichester, New York, Paris, 1994. · Zbl 0841.35064
[14] Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Appl. Math., Vol. 3, American Institute of Mathematical Sciences and Higher Education Press, Springfield, 2010.
[15] Li, T. T. and Yu, W. C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Math. Ser. V, Durham, NC, 1985.
[16] Myshkis, A. D. and Filimonov, A. M., Continuous solutions of quasilinear hyperbolic systems with two independent variables (in Russian), Differ. Uravn., 17(3), 1981, 488–500; Translation in Differ. Eq., 17(3), 1981, 336–345. · Zbl 0459.35052
[17] Myshkis, A. D. and Filimonov, A. M., On the global continuous solvability of a mixed problem for onedimensional hyperbolic systems of quasilinear equations (in Russian), Differ. Uravn., 44(3), 2008, 413–427; Translation in Differ. Eq., 44(3), 2008, 394–407. · Zbl 1152.35071 · doi:10.1134/S0012266108030129
[18] Peng, Y. J. and Yang, Y. F., Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, J. Math. Phy., 52(5), 2011, 053702, 23 pages. · Zbl 1317.49050 · doi:10.1063/1.3591133
[19] Wang, R. H. and Wu, Z. Q., On mixed initial boundary value problem for quasilinear hyperbolic system of partial differential equations in two independent variables (in Chinese), Acta Scientiarum Naturalium of Jilin University, 2, 1963, 459–502.
[20] Zhou, M. Q., Function Theory of Real Variables (in Chinese), Peking University Press, Beijing, 2008.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.