Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres (Memoirs of the American Mathematical Society, 1103)

By J.-M. Delort

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.

• Format: Paperback
• Language: English
• ISBN: 1470409836
• ISBN13: 9781470409838
• Release Date: March 2015
• Publisher: Amer Mathematical Society
• Length: 80 Pages
• Weight: 0.32 lbs.
• Dimensions: 9.8" x 0.3" x 7.0"