Finite time energy cascade for mixed $3-$ and $4-$wave kinetic equations

TL;DR

This paper investigates a wave kinetic equation with mixed 3- and 4-wave interactions, showing that solutions exhibit rapid energy transfer to high frequencies, including finite-time blow-up scenarios.

Contribution

It introduces a novel kinetic model combining 3- and 4-wave processes and analyzes energy cascade phenomena in finite temperature Bose gases.

Findings

Solutions show immediate energy cascade to high frequencies.

Energy transfer to infinity occurs in finite time for certain initial data.

The model captures complex wave interaction dynamics in Bose gases.

Abstract

In this work we study a kinetic equation whose collision operator comprises three distinct wave interaction mechanisms: one representing a 3-wave process, and two corresponding to 4-wave processes. This wave kinetic equation describes the temporal evolution of the density function of the thermal cloud of a finite temperature trapped Bose gas. We establish that, for a broad class of initial data, solutions exhibit an immediate cascade of energy towards arbitrarily large frequencies. Furthermore, for other classes of initial conditions, we demonstrate that the energy is transferred to infinity in finite time.