Stability of Rayleigh-Jeans equilibria in the kinetic FPU equation

TL;DR

This paper investigates the stability of Rayleigh-Jeans equilibria in the kinetic FPU equation, demonstrating nonlinear stability despite the absence of a spectral gap, with polynomial decay rates.

Contribution

It proves the nonlinear stability of Rayleigh-Jeans equilibria in the kinetic FPU equation, addressing challenges posed by the lack of spectral gap.

Findings

Nonlinear stability of Rayleigh-Jeans equilibria established.

Polynomial decay rates achieved despite spectral gap absence.

Analysis of the kinetic wave equation related to the FPU problem.

Abstract

We study the nonlinear dynamics of the kinetic wave equation associated to the FPU problem and prove stability of the non-singular Rayleigh-Jeans equilibria. The lack of a spectral gap for the linearized problem leads to polynomial decay, which we are able to leverage to obtain nonlinear stability.