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Global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems

Published online by Cambridge University Press:  14 November 2011

Lee Da-tsin(Li Ta-tsien)
Lee Da-tsin(Li Ta-tsien)
Affiliation:
Department of Mathematics, Fudan University, Shanghai, China and Collège de France, Paris, France
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Synopsis

For the Cauchy problem for strictly hyperbolic systems with general eigenvalues, we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data; on the other hand, we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications. The present paper includes the corresponding results in earlier papers by several authors as special cases.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 3-4 , 1981 , pp. 255 - 261
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Courant, R. and Friedrichs, K. O.. Supersonic flow and shock waves (New York: Interscience, 1948).Google Scholar
2Jeffrey, A.. Quasilinear hyperbolic systems and waves. Research Notes in Mathematics (Bath: Pitman, 1976).Google Scholar
3Chao-hao, Gu. Shock-free global solutions for quasilinear hyperbolic systems. Selected works of the Mathematics Department of Fudan University, 1960, 36–39.Google Scholar
4Lax, P. D.. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Mathematical Phys. 5 (1964), 611–613.CrossRefGoogle Scholar
5Keller, J. B. and Ting, L.. Periodic vibrations of systems governed by nonlinear partial differential equations. Comm. Pure Appl. Math. 19 (1966), 371–420.CrossRefGoogle Scholar
6Johnson, J. L.. Global continuous solutions of hyperbolic systems of quasilinear equations. Bull. Amer. Math. Soc. 73 (1967), 639–641.CrossRefGoogle Scholar
7Yamaguti, M. and Nishida, T.. On some global solutions for quasilinear hyperbolic equations. Funkcial. Ekvac. 11 (1968), 51–57.Google Scholar
8Jeffrey, A. and Donato, A.. The occurrence of singularities in solutions of homogeneous systems of two first order quasilinear hyperbolic equations with smooth initial data. Wave Motion 1 (1979), 177–185.CrossRefGoogle Scholar
9Hoff, D.. A constructive theory for shock-free, isentropic flow, preprint.Google Scholar
10Yanenko, N. N.. On discontinuities of solutions of quasilinear equations. Uspehi Mat. Nauk 10 (1955), 195–202.Google Scholar
11Rozdestvenskii, B. L.. On discontinuities of solutions of quasilinear equations. Mat. Sb. 47(89) (1959), 485–494.Google Scholar
12Rozdestvenskii, B. L.. Discontinuous solutions of hyperbolic systems of quasilinear equations. Uspehi Mat. Nauk 15 (1960), §8.Google Scholar
13Klainerman, S. and Majda, A.. Formation of singularities for wave equations including the nonlinear vibrating string, preprint.Google Scholar
14Ta-tsien, Li and Wen-tzu, Yu. Solutions régulières globales des systèmes hyperboliques quasi linéaires d'ordre 1, to appear.Google Scholar
15Douglis, A.. Some existence theorems for hyperbolic systems of partial differential equations in two independent variables. Comm. Pure Appl. Math. 2 (1952), 119–154.CrossRefGoogle Scholar
16Da-tsin, Lee and Wen-tzu, Yu. Cauchy's problems for quasilinear hyperbolic systems of first order partial differential equations. Math. Progress Sinica 7 (1964), 152–171.Google Scholar