Instability of singular equilibria of a wave kinetic equation
Abstract
We consider the singular Rayleigh-Jeans equilibrium of the 4-waves kinetic turbulence equation for the three dimensional Schr\"odinger 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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