Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T07:29:07.386Z Has data issue: false hasContentIssue false
  • English
  • Français

Conservation laws and null divergences

Published online by Cambridge University Press:  24 October 2008

Peter J. Olver
Peter J. Olver
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Get access Rights & Permissions [Opens in a new window]

Extract

For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 3 , November 1983 , pp. 529 - 540
Copyright
Copyright © Cambridge Philosophical Society 1983

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

[1] Anderson, I. M. and Duchamp, T.. On the existence of global variational principles. Amer. J. Math. 102 (1980), 781–868.CrossRefGoogle Scholar
[2] Ball, J. M., Currie, J. C. and Olver, P. J.. Na Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981), 135–174:CrossRefGoogle Scholar
[3] Courant, R. and Hilbert, D.. Methods of Mathematical Physics, vol. II (Interscience, 1962).Google Scholar
[4] Gel'fand, I. M. and Dikii, L. A.. Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equation. Russian Math.Surveys 30 (1975), 77–113.CrossRefGoogle Scholar
[5] HöRmander, L.. Partial Differential Operators (Springer-Verlag, 1969).Google Scholar
[6] Jacobson, N.. Lectures in Abstract Algebra, vol. 3 (Van Nostrand, 1964).Google Scholar
[7] Knowles, J. K. and Sternberg, E.. On a class of conservation laws in linearized and finite elastostatics. Arch. Rational Mech. Anal. 44 (1972), 187–211.Google Scholar
[8] Lamb, G. L.Elements of Soliton Theory (Wiley-Interscience, 1980).Google Scholar
[9] Mount, K. R.. A remark on determinantal loci. J. London Math. Soc. 42 (1987), 595–598.Google Scholar
[10] Northcott, D. G.. Some remarks on the theory of ideals defined by matrices. Quart. J. Math. Oxford 14 (1963), 193–204.CrossRefGoogle Scholar
[11] Olver, P. J.. Euler operators and conservation laws of the BBM equation. Math. Proc. Cambridge Philos. Soc. 85 (1979), 143–160.Google Scholar
[12] Olver, P. J.. Applications of Lie Groups to Differential Equations (Lecture Notes, University of Oxford, 1980).Google Scholar
[13] Olver, P. J.. Hyperjacobians, determinantal ideals and weak solutions to variational problems. Proc. Roy. Soc. Edinburgh (to appear).Google Scholar
[14] Olver, P. J.. Conservation laws of free boundary problems and the classification of conservation laws for water waves. Trans. Amer. Math. Soc., 277, (1983), 353–380.Google Scholar
[15] Olver, P. J.. Conservation laws in elasticity. I. General results. Arch. Rational Mech. Anal. (to appear).Google Scholar
[16] Olver, P. J.. Conservation laws in elasticity. II. Linear isotropic homogeneous elastostatics. Arch. Rational Mech. Arch. (to appear).Google Scholar
[17] Ritt, J. F.. Differential Algebra (Dover, 1966).Google Scholar
[18] Shakiban, C.. A resolution of the Euler operator. II. Math. Proc. Cambridge Philos. Soc. 89 (1981), 501–510.CrossRefGoogle Scholar
[19] Strauss, W. A.. Nonlinear invariant wave equations. In Invariant Wave Equations, Lecture Notes in Physics, vol. 73 (Springer-Verlag, 1978), pp. 197–249.CrossRefGoogle Scholar
[20] Takens, F.. A global version of the inverse problem of the calculus of variations. J. Differential Geom. 14 (1979), 543–562.Google Scholar
[21] Tulczyjew, W. M.. The Lagrange complex. Bull. Soc. Math. France 105 (1977), 419–431.CrossRefGoogle Scholar
[22] Vasilenko, G. N.. Weak continuity of Jacobians. Siberian Math. J. 22 (1981), 355–360.Google Scholar
[23] Vinogradov, A. M.. A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints. Soviet Math. Dokl. 19 (1978), 144–148.Google Scholar