Long time justification of wave turbulence theory

TL;DR

This paper rigorously extends the derivation of the wave kinetic equation to arbitrarily long times, covering the full lifespan of the equation, marking a significant advancement in the mathematical understanding of wave turbulence.

Contribution

It provides the first long-time, large data derivation of the wave kinetic equation in a nonlinear wave kinetic limit.

Findings

Extended the justification to cover the full lifespan of the WKE

Achieved the first large data, long-time derivation in wave turbulence

Validated the wave kinetic equation for arbitrarily long times

Abstract

In a series of previous works (arXiv:2104.11204, arXiv:2110.04565, arXiv:2301.07063), we gave a rigorous derivation of the homogeneous wave kinetic equation (WKE) up to small multiples of the kinetic timescale, which corresponds to short time solutions to the wave kinetic equation. In this work, we extend this justification to arbitrarily long times that cover the full lifespan of the WKE. This is the first large data, long-time derivation ever obtained in any nonlinear (particle or wave) collisional kinetic limit.