eBook

Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres

Delort, J.-M.

2014

Overview

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. 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 \\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.

• Publisher: American Mathematical Society
• Subject: Hamiltonian systems / Klein-Gordon equation / Sphere / Wave equation
• ISBN: 9781470409838, 1470409836