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Variations on conservation laws for the wave equation

Author: Cathleen Synge Morawetz
Journal: Bull. Amer. Math. Soc. 37 (2000), 141-154
MSC (2000): Primary 35Lxx, 35Mxx, 35Qxx, 83Cxx
Published electronically: January 21, 2000
MathSciNet review: 1751947
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Abstract: The first part of this paper, presented as an Emmy Noether lecture in connection with the ICM in Berlin in August 1998, gives some examples of using Noether's theorem for conservation laws for Tricomi-like equations and for the wave equation. It is also shown that equations which are semilinear variations of the wave equation can very often be handled similarly. The type of estimate obtained can even be used to get otherwise unobtainable local estimates for regularity.

The fourth part is an introduction to the relation of black holes to the wave equation mainly showing the results of D. Christodoulou. His results use much more difficult estimates not corresponding at all to those in the first part of the paper.

References [Enhancements On Off] (What's this?)

  • 1. J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145–171. MR 1626257, 10.1090/S0894-0347-99-00283-0
  • 2. Choquet-Bruhat, Y. Theorème d'existence pour certain systemes d'equations aux derivées partielles nonlineaires. Acta Mathematica, 88 (1952), 141-225.
  • 3. Demetrios Christodoulou, The formation of black holes and singularities in spherically symmetric gravitational collapse, Comm. Pure Appl. Math. 44 (1991), no. 3, 339–373. MR 1090436, 10.1002/cpa.3160440305
    Demetrios Christodoulou, Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Comm. Pure Appl. Math. 46 (1993), no. 8, 1131–1220. MR 1225895, 10.1002/cpa.3160460803
  • 4. Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR 1316662
  • 5. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), no. 4, 487–505. MR 786279, 10.1007/BF01168155
  • 6. Heisenberg, W. Mesonerzeugung als Stosswellen problem. Zeit. Physik 33 (1952), 65-79.
  • 7. M. D. Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. (2) 119 (1960), 1743–1745. MR 0115757
  • 8. Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
  • 9. Lemaître, G. L'univers en expansion, Ann. Soc. Sci. Bruxelles A 53 (1933), 51-85.
  • 10. C. S. Morawetz and D. Ludwig, An inequality for the reduced wave operator and the justification of geometrical optics, Comm. Pure Appl. Math. 21 (1968), 187–203. MR 0223136
  • 11. Michell, J. On the Means of Discovering the Distance, Magnitude, Etc. of the Fixed Stars, in Consequence of the Diminution of their Light, in Case such a Dimunution Should be Found to Take Place in Any of Them and Such Other Data Should be Procured from Observations as Would be Further Necessary for That Purpose, Phil. Trans. of RS of London 74 (1784), 55, presented 1783.
  • 12. Cathleen S. Morawetz, Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation, Proc. Roy. Soc. London. Ser. A. 236 (1956), 141–144. MR 0079712
  • 13. Cathleen S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561–568. MR 0132908
    Cathleen S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962), 349–361. MR 0151712
  • 14. Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291–296. MR 0234136
  • 15. Cathleen S. Morawetz and Walter A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1–31. MR 0303097
  • 16. Noether, E. Invarianter beliebiger Differentialausdrücke. Nachr.Ges.d.Wiss.zn Göttingen (Math.Phys. Kl.), 37-44, (1918). Collected papers pp. 240-257.
  • 17. Oppenheimer, J.R. and Snyder, H. On continued gravitational contraction. Phys. Rev. 56 (1939), 455.
  • 18. Payne, C.H. Stellar Atmospheres: A Contribution to the Observational Study of High Temperatures in the Reversing Layers of Stars. Harvard Observatory, 1925.
  • 19. Schwarzschild, K. Sitzber. Preuss. Akad. Wiss. Physik-Math. Kl. 189 (1916).
  • 20. Jalal Shatah and Michael Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503–518. MR 1247991, 10.2307/2946554
  • 21. Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 0512086
  • 22. J. L. Synge, The gravitational field of a particle, Proc. Roy. Irish Acad. Sect. A. 53 (1950), 83–114. MR 0039426
  • 23. Kip S. Thorne, Black holes and time warps, Commonwealth Fund Book Program, W. W. Norton & Co. Inc., New York, 1994. Einstein’s outrageous legacy; With a foreword by Stephen Hawking and an introduction by Frederick Seitz. MR 1267060

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Additional Information

Cathleen Synge Morawetz
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012

Received by editor(s): July 1, 1999
Received by editor(s) in revised form: October 6, 1999
Published electronically: January 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society