The wave kinetic theory of three wave and four wave models

Abstract

In this thesis, we introduce new techniques for studying the random series expansion of dispersive PDEs. We take a quadratic KdV type and a cubic Klein-Gordon type equation as examples to demonstrate the different techniques in three wave and four wave models. For three wave models, we introduce a counting argument to handle the degeneracy problems of the resonance surface and the loss of derivative problem. For four wave models, we introduce a novel renormalization argument and prove a renormalized Wick theorem. We provide a heuristic argument that this renormalization is able to remove all bad terms from the $L^2$ mass term, combining with an almost cancellation identity of the regular pairing and the Deng-Hani's Feynman diagram analysis \cite{deng2021full}, \cite{deng2023derivation}.