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

Angeliki Menegaki

doi:10.3934/krm.2023031

Article Contents

2024, Volume 17, Issue 4: 514-532. Doi: 10.3934/krm.2023031

This issue Previous Article Next Article Growth-fragmentation model for a population presenting heterogeneity in growth rate: Malthus parameter and long-time behavior

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

Angeliki Menegaki

Institut des hautes études scientifiques, France

Received: November 1, 2022

Revised: July 1, 2023

Early access: September 1, 2023

Published online: August 1, 2024

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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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Mathematics Subject Classification: Primary: 35BXX, 45G05, 35B40; Secondary: 35Q20, 35Q55.

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Figure 1. In $ 2 $-dim, as long as $ |k|, |k_i|\leq k_c $ for all $ i = 1, 2, 3 $ we allow collisions. The dashed rectangular corresponding to a collision when the resulting wavenumbers are outside of the domain is not allowed in our set-up

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