Delort J.-M. Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

American Mathematical Society, 2015. — 92 p. — (Memoirs of the American Mathematical Society v.234, №1103) — ISBN: 1470409836.The Hamiltonian 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.
The author considers 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. The author shows that, when X is the sphere, and when the mass parameter is outside an exceptional subset of zero measure, smooth Cauchy data of small size give rise to almost global solutions, i.e. solutions defined on a time interval of length. 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.- Introduction
- Statement of the main theorem
- Symbolic calculus
- Quasi-linear Birkhoff normal forms method
- Proof of the main theorem
Appendix