$L^2$-stability near equilibrium for the $4$ waves kinetic equation

TL;DR

This paper investigates the long-term stability of solutions near equilibrium for the four waves kinetic equation in wave turbulence, demonstrating $L^2$-stability for initial data close to Rayleigh-Jeans equilibrium under certain conditions.

Contribution

It establishes $L^2$-stability near equilibrium for the four waves kinetic equation with weakly perturbed dispersion relations, extending understanding of solution behavior in wave turbulence.

Findings

Linearized operator is coercive around equilibrium.

Proves $L^2$-stability for nonlinear solutions near equilibrium.

Results hold for cut-off frequencies and weak dispersion perturbations.

Abstract

We consider the four waves spatial homogeneous kinetic equation arising in wave turbulence theory. We study the long-time behaviour and existence of solutions around the Rayleigh-Jeans equilibrium solutions. For cut-off'd frequencies, we show that for dispersion relations weakly perturbed around the quadratic case, the linearized operator around the Rayleigh-Jeans equilibria is coercive. We then pass to the fully nonlinear operator, showing an $L^2$ - stability for initial data close to Rayleigh-Jeans.