Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-25T00:42:23.623Z Has data issue: false hasContentIssue false
  • English
  • Français

The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type

Published online by Cambridge University Press:  14 February 2012

A. Jeffrey
A. Jeffrey
Affiliation:
Department of Engineering Mathematics, University of Newcastle upon Tyne
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper is concerned with quasilinear hyperbolic systems of conservation type and establishes two main results. The first is that when a general system is considered in one space dimension and time, then the exceptional nature of a characteristic field implies the coincidence of a shock with one of the characteristics of that field. The second result involves the demonstration by example that quasilinear hyperbolic systems of conservation type may possess solutions that become unbounded after only a finite time, even though they are exceptional.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 1-2 , 1977 , pp. 1 - 8
Copyright
Copyright © Royal Society of Edinburgh 1977

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

1Jeffrey, A.Quasilinear hyperbolic systems and waves (London: Pitman, 1976).Google Scholar
2Courant, R. and Hilbert, D.Methods of mathematical physics, II (New York: Interscience, 1962).Google Scholar
3Lax, P. D.Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537–566.CrossRefGoogle Scholar
4John, F.Formation of singularities of solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math. 27 (1974), 377–405.CrossRefGoogle Scholar
5Glimm, J. and Lax, P. D.Decay of solutions of systems of nonlinear hyperbolic conservation laws. Amer. Math. Soc. Memoir 101 (1970), 1–111.Google Scholar
6Jeffrey, A.Breakdown of the solution to a completely exceptional system of hyperbolic equations. J. Math. Anal. Appl. 45 (1974), 375–381.CrossRefGoogle Scholar
7Levine, H. A.Some non-existence and instability theorems for solutions of formally parabolic equations of the form Put = −Au+F(u). Arch. Rational Mech. Anal. 51 (1973), 371–386.CrossRefGoogle Scholar
8Knops, R. J., Levine, H. A. and Payne, L. E.Non-existence, instability and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics. Arch. Rational Mech. Anal. 55 (1974), 52–72.CrossRefGoogle Scholar
9Keller, J.On solutions of nonlinear wave equations. Comm. Pure Appl. Math. 10 (1957), 523–532.CrossRefGoogle Scholar
10Glassey, R.Blow-up theorems for nonlinear wave equations. Math. Z. 132 (1973), 183–203.CrossRefGoogle Scholar