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

Publication date: 20 October 2022

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.

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