Almost global existence for Hamiltonian PDEs on compact manifolds

Abstract

We prove an abstract result of almost global existence of small solutions to semi-linear Hamiltonian partial differential equations satisfying very weak non resonance conditions and basic multilinear estimates. Thanks to works by Delort--Szeftel, these assumptions turn out to typically hold for Hamiltonian PDEs on any smooth compact boundaryless Riemannian manifold. As a main application, we prove the almost global existence of small solutions to nonlinear Klein--Gordon equations on such manifolds: for almost all mass, any arbitrarily large $r$ and sufficiently large $s$, solutions with initial data of sufficiently small size $\epsilon \ll 1$ in the Sobolev space $H^s\times H^{s-1}$ exist and remain in $H^s\times H^{s-1}$ for polynomial times $|t|\le {\epsilon }^{-r}$. This is the first result of almost global existence without specific assumptions on the compact manifold. We also apply this abstract result to nonlinear Schr{ö}dinger equations close to ground states and nonlinear Klein--Gordon equations on ${\mathbb{R}}^d$ with positive quadratic potentials.