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Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres
About this Title
J.-M. Delort, Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, Avenue J.-B. Clément, F-93430 Villetaneuse
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 234, Number 1103
ISBNs: 978-1-4704-0983-8 (print); 978-1-4704-2030-7 (online)
DOI: https://doi.org/10.1090/memo/1103
Published electronically: July 28, 2014
Keywords: Hamiltonian quasi-linear Klein-Gordon equations,
Almost global existence,
Birkhoff normal forms
MSC: Primary 35L72, 35S50, 37K45
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of the main theorem
- 3. Symbolic calculus
- 4. Quasi-linear Birkhoff normal forms method
- 5. Proof of the main theorem
- A. Appendix
Abstract
The Hamiltonian $\int _X(\lvert {\partial _t u}\rvert ^2 + \lvert {\nabla u}\rvert ^2 + \mathbf {m}^2\lvert {u}\rvert ^2)\,dx$, defined on functions on $\mathbb {R}\times X$, where $X$ is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. We consider perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of $u$. The associated PDE is then a quasi-linear Klein-Gordon equation. We show that, when $X$ is the sphere, and when the mass parameter $\mathbf {m}$ is outside an exceptional subset of zero measure, smooth Cauchy data of small size $\epsilon$ give rise to almost global solutions, i.e. solutions defined on a time interval of length $c_N\epsilon ^{-N}$ for any $N$. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on $u$) or to the one dimensional problem.
The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.
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