## Variations on conservation laws for the wave equation

HTML articles powered by AMS MathViewer

- by Cathleen Synge Morawetz PDF
- Bull. Amer. Math. Soc.
**37**(2000), 141-154 Request permission

## 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

- 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**, DOI 10.1090/S0894-0347-99-00283-0 - Choquet-Bruhat, Y. Theorème d’existence pour certain systemes d’equations aux derivées partielles nonlineaires.
*Acta Mathematica*,**88**(1952), 141–225. - 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**, DOI 10.1002/cpa.3160440305 - 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** - 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**, DOI 10.1007/BF01168155 - Heisenberg, W. Mesonerzeugung als Stosswellen problem.
*Zeit. Physik***33**(1952), 65–79. - M. D. Kruskal,
*Maximal extension of Schwarzschild metric*, Phys. Rev. (2)**119**(1960), 1743–1745. MR**115757**, DOI 10.1103/PhysRev.119.1743 - Peter D. Lax and Ralph S. Phillips,
*Scattering theory*, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR**0217440** - Lemaître, G. L’univers en expansion,
*Ann. Soc. Sci. Bruxelles A***53**(1933), 51–85. - 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**223136**, DOI 10.1002/cpa.3160210206 - 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. - Cahit Arf,
*Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper*, J. Reine Angew. Math.**181**(1939), 1–44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - 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**132908**, DOI 10.1002/cpa.3160140327 - Cathleen S. Morawetz,
*Time decay for the nonlinear Klein-Gordon equations*, Proc. Roy. Soc. London Ser. A**306**(1968), 291–296. MR**234136**, DOI 10.1098/rspa.1968.0151 - 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**303097**, DOI 10.1002/cpa.3160250103 - Noether, E. Invarianter beliebiger Differentialausdrücke.
*Nachr.Ges.d.Wiss.zn Göttingen (Math.Phys. Kl.)*, 37–44, (1918). Collected papers pp. 240–257. - Oppenheimer, J.R. and Snyder, H. On continued gravitational contraction.
*Phys. Rev.***56**(1939), 455. - Payne, C.H.
*Stellar Atmospheres: A Contribution to the Observational Study of High Temperatures in the Reversing Layers of Stars*. Harvard Observatory, 1925. - Schwarzschild, K. Sitzber. Preuss. Akad. Wiss. Physik-Math. Kl. 189 (1916).
- Jalal Shatah and Michael Struwe,
*Regularity results for nonlinear wave equations*, Ann. of Math. (2)**138**(1993), no. 3, 503–518. MR**1247991**, DOI 10.2307/2946554 - 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**512086** - Charles Hopkins,
*Rings with minimal condition for left ideals*, Ann. of Math. (2)**40**(1939), 712–730. MR**12**, DOI 10.2307/1968951 - 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**

## Additional Information

**Cathleen Synge Morawetz**- Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
- Email: morawetz@cims.nyu.edu
- Received by editor(s): July 1, 1999
- Received by editor(s) in revised form: October 6, 1999
- Published electronically: January 21, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**37**(2000), 141-154 - MSC (2000): Primary 35Lxx, 35Mxx, 35Qxx, 83Cxx
- DOI: https://doi.org/10.1090/S0273-0979-00-00857-0
- MathSciNet review: 1751947