Optimal local well-posedness theory for the kinetic wave equation

TL;DR

This paper establishes local existence and uniqueness results for the space-homogeneous 4-wave kinetic equation in wave turbulence, accommodating general dispersion relations and nearly critical weighted spaces.

Contribution

It introduces new local well-posedness theorems for the kinetic wave equation with broad dispersion relations in nearly critical weighted spaces.

Findings

Proved local existence and uniqueness for the kinetic wave equation.

Extended well-posedness results to general radial dispersion relations.

Operated within nearly critical weighted function spaces.

Abstract

We prove local existence and uniqueness results for the (space-homogeneous) 4-wave kinetic equation in wave turbulence theory. We consider collision operators defined by radial, but general dispersion relations satisfying suitable bounds, and we prove two local well-posedness theorems in nearly critical weighted spaces.