Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces

TL;DR

This paper advances the well-posedness theory of the inhomogeneous 4-wave kinetic equation and constructs solutions for its associated hierarchy, employing novel combinatorial methods and the Hewitt-Savage theorem.

Contribution

It extends the analysis of the inhomogeneous 4-wave kinetic equation to polynomially decaying data and develops a framework for the associated hierarchy solutions.

Findings

Global well-posedness for initial data with polynomial decay.

Construction of solutions for the hierarchy using wave kinetic solutions.

Control of factorial growth via a combinatorial board game argument.

Abstract

Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series.