Almost global existence for Hamiltonian PDEs on compact manifolds

TL;DR

This paper establishes almost global existence of small solutions for Hamiltonian PDEs on compact manifolds, extending previous results to more general settings and specific equations like Klein-Gordon and Schrödinger.

Contribution

It provides an abstract framework for almost global existence under weak non-resonance conditions, applicable to a wide class of Hamiltonian PDEs on compact manifolds.

Findings

Proves almost global existence for nonlinear Klein-Gordon equations on compact manifolds.

Extends results to nonlinear Schrödinger equations near ground states.

First to achieve almost global existence without specific manifold assumptions.

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 $\varepsilon \ll 1$ in the Sobolev space $H^s \times H^{s-1}$ exist and remain in $H^s \times H^{s-1}$ for polynomial times $|t| \leq \varepsilon^{-r}$. 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{\"o}dinger equations close to ground states and nonlinear Klein--Gordon equations on $\mathbb{R}^d$ with positive quadratic potentials.