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

DOI:
https://doi.org/10.1090/S0002-9939-00-05845-7

Published electronically:
October 4, 2000

MathSciNet review:
1814146

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Abstract | References | Similar Articles | Additional Information

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 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 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:
https://doi.org/10.1090/S0002-9939-00-05845-7

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