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

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arxiv 2210.11189 v2 pith:U2RULZL2 submitted 2022-10-20 math.AP math-phmath.MP

Angeliki Menegaki This is my paper · ORCID
classification math.APmath-phmath.MP
keywords aroundrayleigh-jeansequationequilibriumkineticoperatorsolutionsstability
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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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  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.

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