Isotropic Wave Turbulence with simplified kernels: existence, uniqueness and mean-field limit for a class of instantaneous coagulation-fragmentation processes

TL;DR

This paper proves existence and uniqueness of solutions for a simplified isotropic 4-wave kinetic equation with linear growth kernels and establishes conditions under which stochastic coagulation-fragmentation particle systems approximate these solutions.

Contribution

It introduces a simplified kernel model for isotropic wave turbulence and demonstrates the mean-field limit linking stochastic particle systems to the kinetic equation.

Findings

Existence and uniqueness of solutions for linear growth kernels.

Conditions for stochastic particle systems to approximate the kinetic equation.

Analysis of coagulation-fragmentation phenomena in wave turbulence.

Abstract

The isotropic 4-wave kinetic equation is considered in its weak formulation using model (simplified) homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting where the kernels have a rate of growth at most linear. We also consider finite stochastic particle systems undergoing instantaneous coagulation- fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).