On the ill-posedness of kinetic wave equations

TL;DR

This paper establishes a precise threshold distinguishing well-posedness from ill-posedness in kinetic wave equations derived from quasilinear Schrödinger models, highlighting the role of initial data smoothness and validating a gain-only approach.

Contribution

It identifies a sharp ill-posedness/well-posedness threshold for kinetic wave equations and confirms the equivalence of gain-only and full equations in this context.

Findings

Well-posedness proven using a collisional averaging estimate

Ill-posedness causes instantaneous loss of smoothness

Gain-only and full equations share the same well-posedness threshold

Abstract

In this article we identify a sharp ill-posedness/well-posedness threshold for kinetic wave equations (KWE) derived from quasilinear Schr\"{o}dinger models. We show well-posedness using a collisional averaging estimate proved in our earlier work \cite{AmLe}. Ill-posedness manifests as instantaneous loss of smoothness for well-chosen initial data. We also prove that both the gain-only and full equation share the same well-posedness threhold, thus legitimizing a gain-only approach to solving 4-wave kinetic equations.