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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Validity of nonlinear geometric optics with times growing logarithmically


Authors: David Lannes and Jeffrey Rauch
Journal: Proc. Amer. Math. Soc. 129 (2001), 1087-1096
MSC (1991): Primary 35B25, 35B40, 35L60, 35Q60, 78A60, 35C20, 35F25
Published electronically: October 4, 2000
MathSciNet review: 1814146
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Abstract: The profiles (a.k.a. amplitudes) which enter in the approximate solutions of nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperbolic systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The approximations of nonlinear geometric optics typically have percentage error which tends to zero uniformly on bounded time intervals as the wavelength $\epsilon$ tends to zero. Under suitable hypotheses when the amplitude is uniformly bounded in space and time we show that the percentage error tends to zero uniformly on time intervals $[0,C\vert\ln \epsilon\vert]$ which grow logarithmically. The proof relies in an essential way on the fact that one has a corrector to the leading term of geometric optics.


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Additional Information

David Lannes
Affiliation: MAB, Université de Bordeaux I, 33405 Talence, France
Email: lannes@math.u-bordeaux.fr

Jeffrey Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: rauch@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05845-7
PII: S 0002-9939(00)05845-7
Keywords: Nonlinear geometric optics, large time asymptotics, corrector
Received by editor(s): June 30, 1999
Published electronically: October 4, 2000
Additional Notes: This research was partially supported by the U.S. National Science Foundation, and the NSF-CNRS cooperation program under grants number NSF-DMS-9500823 and NSF-INT-9314095 respectively, and the CNRS through the Groupe de Recherche G1180 POAN
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society