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Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle


Author: J.-M. Delort
Journal: Trans. Amer. Math. Soc. 361 (2009), 4299-4365
MSC (2000): Primary 35L70, 35S50
DOI: https://doi.org/10.1090/S0002-9947-09-04747-3
Published electronically: March 13, 2009
MathSciNet review: 2500890
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Abstract: We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on $ \mathbb{S}^1$ remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.


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Additional Information

J.-M. Delort
Affiliation: Université Paris 13, Institut Galilée, CNRS, UMR 7539, Laboratoire Analyse, Géométrie et Applications, 99, Avenue J.-B. Clément, F-93430 Villetaneuse, France
Email: delort@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04747-3
Keywords: Quasi-linear Klein-Gordon equation, long-time stability, paradifferential calculus
Received by editor(s): September 19, 2007
Published electronically: March 13, 2009
Additional Notes: This work was partially supported by the ANR project Equa-disp.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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