Condensation and non-condensation times for 4-wave kinetic equations

TL;DR

This paper proves that solutions to 4-wave kinetic equations with broad dispersion relations undergo finite-time condensation at the origin, and it offers estimates on when solutions do not condense, advancing understanding of wave kinetic dynamics.

Contribution

It establishes finite-time condensation for 4-wave kinetic equations under more general conditions than previous studies.

Findings

Solutions develop condensation at the origin in finite time.

Provides estimates on non-condensation times.

Extends results to broader dispersion relations.

Abstract

Inspired by the pioneering 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 dispersion relations, develop condensation at the origin in finite time, under weaker conditions on the initial data than the ones considered in \cite{EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}. We also provide some estimates on the non-condensation times of the solutions.