L²-stability near equilibrium for the 4 waves kinetic equation

2022/10/20 by Angeliki Menegaki, Menegaki, Angeliki · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2210.11189

openalex publication_date 2022/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L² - stability for initial data close to Rayleigh-Jeans.

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• An energy cascade finite volume scheme for a mixed 3- and 4-wave kinetic equation arising from the theory of finite-temperature trapped Bose gases

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