On the optimal local well-posedness of the wave kinetic equation in $L^r$

TL;DR

This paper establishes the local well-posedness of the wave kinetic equation in almost critical weighted L^r spaces, using purely kinetic methods without Fourier analysis, extending previous results.

Contribution

It provides a unified kinetic approach to well-posedness in weighted L^r spaces for the wave kinetic equation, covering a broad range of r values.

Findings

Proves local well-posedness in weighted L^r spaces for 2 ≤ r ≤ ∞

Develops a kinetic-only method avoiding Fourier analysis

Extends previous results to a wider class of function spaces

Abstract

In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted $L^r$ spaces with $2 \leq r \leq \infty.$ The proof builds on ideas from our earlier works \cite{AmLe24, AmLemain25}. Our approach is based solely on kinetic tools, with no appeal to Fourier theory.