Instability of singular equilibria of a wave kinetic equation

2024/06/07 by Miguel Escobedo, Escobedo, Miguel, Angeliki Menegaki +1 · 3 citations
Computer Science · Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation

paper · pdf · doi:10.48550/arxiv.2406.05280

openalex publication_date 2024/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider the singular Rayleigh-Jeans equilibrium of the 4-waves kinetic turbulence equation for the three dimensional Schrödinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the solution of the wave kinetic equation when the initial data has the form of Rayleigh-Jeans, truncated at large values of the energy. The initial value problem for the linearization around the singular Rayleigh-Jeans equilibria is then solved in several functional spaces. Then, long time convergence to a Dirac measure at the origin is described in detail for some of the solutions. This determines a basin of attraction of the Dirac measure.

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• On the optimal local well-posedness of the wave kinetic equation in L^r
• 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
• Stability of stationary solutions in Acoustic wave turbulence

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