Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T23:26:43.082Z Has data issue: false hasContentIssue false

Existence and uniqueness of discontinuous solutions for a hyperbolic system

Published online by Cambridge University Press:  14 November 2011

Feimin Huang
Affiliation:
Institute of Mathematics, Shantou University, Shantou 515063, P.R. China Institute of Applied Mathematics, Academia Sinica, Beijing 100080, P.R. China

Extract

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ding, X.. On a non-strictly hyperbolic system (Preprint 167, Department of Mathematics, University of Jyvaskyla, 1993).Google Scholar
2Ding, X. and Wang, Z.. Existence and uniqueness of discontinuous solutions defined by Lebesgue–Stieltjes integral. Science in China (Series A) 39 (1996), 807–19.Google Scholar
3Forestier, A. and LeFloch, P.. Multivalued solutions to some non-linear and non-strictly hyperbolic systems. Japan. J. Indust. Appl. Math. 9 (1992), 123.CrossRefGoogle Scholar
4Hopf, E.. The partial differential equation u t, + uux = μuxx. Comm. Pure Appl. Math. 3 (1950), 201–30.CrossRefGoogle Scholar
5Huang, F.. Existence and uniqueness of discontinuous solutions for a class of nonstrictly hyperbolic systems (in prep.).Google Scholar
6Huang, F., Li, C. and Wang, Z.. Solutions containing delta-waves of Cauchy problems for a nonstrictly hyperbolic system. Acta Mathematicae Applicatae Sinica 11 (1995), 429–46.CrossRefGoogle Scholar
7Lax, P.. Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10 (1957), 537–66.CrossRefGoogle Scholar
8Li, C. and Yang, X.. Riemann problems of a class of non-strictly hyperbolic equations (Preprint 19, Institute of Mathematics, Shantou University, 1994).Google Scholar
9Oleinik, O.. Discontinuous solutions of nonlinear differential equations. Usp. Math. Nauk (N.S.) 12 (1957), 373; English translation: Amer. Math. Soc. Transl. Ser. 2 26 (1963), 95–172.Google Scholar
10Tan, D., Zheng, T. and Zhang, Y.. Delta-shock wave as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differential Equations 112 (1994), 132.CrossRefGoogle Scholar