On the energy transfer towards large values of wavenumbers for solutions of 4-wave kinetic equations

TL;DR

This paper demonstrates that solutions to 4-wave kinetic equations transfer energy to large wavenumbers over time and establishes a global well-posedness theory for these solutions under broad dispersion relations.

Contribution

It proves energy transfer to high wavenumbers and develops a global well-posedness framework for solutions with general dispersion relations.

Findings

Energy transfer towards large wavenumbers is proven.

Global well-posedness for mild solutions is established.

Results hold under very general dispersion relations.

Abstract

Inspired by the fundamental work of Escobedo and Velazquez \cite{EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}, we prove that solutions of 4-wave kinetic equations, under very general forms of the dispersion relations, exhibit the transfer of energy towards large values of wavenumbers as time evolves. We also provide a global well-posedness theory for mild solutions of the equation under general forms of the dispersion relations.