Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2

TL;DR

This paper investigates the long-term behavior of small, non-smooth solutions to nonlinear Klein-Gordon equations on the sphere S^2, showing near preservation of low harmonic energies over long times using advanced normal form and probabilistic techniques.

Contribution

It introduces new multilinear estimates and probabilistic bounds to analyze the dynamics of nonlinear Klein-Gordon equations in low regularity settings on S^2.

Findings

Low harmonic energies are almost preserved for long times.

Modes exchange energy only when oscillating at similar frequencies.

New probabilistic bounds on eigenfunction products are established.

Abstract

We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times of order $\epsilon$^--r , where r >> 1 is an arbitrarily large number and $\epsilon$ << 1 is the norm of the initial datum in the energy space H^1 x L^2. Roughly speaking, it means that, in order to exchange energy, modes have to oscillate at the same frequency. The proof relies on new multilinear estimates on Hamiltonian vector fields to put the system in Birkhoff normal form. They are derived from new probabilistic bounds on products of Laplace eigenfunctions that we obtain using Levy's concentration inequality.