Global Well-Posedness near Rayleigh-Jeans Equilibria for the Cubic NLS Wave Kinetic Equation

Abstract

We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt ω\ddω)$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

Disclosure

“and Physical Sci- ences Research Council fellowship with reference UKRI2025. Miguel Escobedo is supported by MINECO through grant PID2023-146872OB-I00. We thank Pierre Germain and Clément Mouhot for useful discus- sions. AI use statement. AI tools were used for proofreading and for supplying the proof of Proposition 4.3, in particular for the idea of transforming the problem to the exponential variables. 2. Properties of the linear operator Lµ”

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2026-08-22T02:53:19.905137Z

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