Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori

TL;DR

This paper establishes almost global existence results for certain Hamiltonian PDEs on flat tori, using a novel nonresonance condition that extends to various equations and stability analyses.

Contribution

It introduces a weaker nonresonance condition for Hamiltonian PDEs on tori, enabling almost global existence proofs for multiple equations and stability results.

Findings

Proves almost global existence for nonlinear Schrödinger, beam, and quantum hydrodynamical equations.

Applies the abstract result to stability of plane waves in NLS.

Develops a nonresonance condition weaker than Bourgain's Lemma.

Abstract

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schr\"odinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.