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Instability of singular equilibria of a wave kinetic equation

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arxiv 2406.05280 v1 pith:YIDBNS6O submitted 2024-06-07 math.AP math-phmath.MP

Miguel Escobedo , Angeliki Menegaki This is my paper · ORCID
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keywords equationdirackineticmeasurerayleigh-jeanssingularequilibriainitial
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abstract

We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schr\"{o}dinger 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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Entropy Structures and Long-Time Relaxation for 3-Wave Kinetic Equations

    math.AP 2026-05 unverdicted novelty 8.0 of 10

    New entropy structures from one-sided balance conditions on interaction weights yield global weak L1_loc solutions to 3-wave kinetic equations and prove their local relaxation to zero equilibrium.

  2. On the optimal local well-posedness of the wave kinetic equation in $L^r$

    math.AP 2025-11 conditional novelty 6.0 of 10

    Local well-posedness of the 3D wave kinetic equation holds in almost critical weighted L^r spaces for all 2≤r≤∞, extending the prior L^2/L^∞ cases.

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