Wave turbulence refers to the statistical theory of weakly nonlinear dispersive waves. In the weakly turbulent regime of a system of dispersive waves, its statistics can be described via a coarse-grained dynamics, governed by the kinetic wave equation. Remarkably, kinetic wave equations admit exact power-law solutions, called Kolmogorov-Zakharov spectra, which resemble Kolmogorov spectrum of hydrodynamic turbulence, and is often interpreted as a transient equilibrium between excitation and dissipation. In this talk, we will outline a local well-posedness result for kinetic wave equation for a toy model for wave turbulence. The result includes well-posedness near K-Z spectra, and demonstrates a surprising smoothing effect of the kinetic wave equation. The talk is based on the joint work with Pierre Germain (ICL) and Katherine Zhiyuan Zhang (Northeastern).
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