$l^{2}$-decoupling and the unconditional uniqueness for the Boltzmann equation

TL;DR

This paper extends the $l^{2}$-decoupling theorem to the Boltzmann equation, establishing key estimates and proving unconditional uniqueness of solutions in both $ ext{R}^d$ and $ ext{T}^d$ settings.

Contribution

It introduces a unified hierarchy scheme to prove unconditional uniqueness for the Boltzmann equation with Maxwellian particles and soft potentials.

Findings

Proved Strichartz estimates for the linear Boltzmann problem on $ ext{T}^d$.

Established space-time bilinear estimates for the Boltzmann equation.

Achieved unconditional uniqueness of solutions for Maxwellian particles and soft potentials.

Abstract

We broaden the application of the $l^{2}$-decoupling theorem to the Boltzmann equation. We prove Strichartz estimates for the linear problem in the $\mathbb{T}^d$ setting. We establish space-time bilinear estimates, and hence the unconditional uniqueness of solutions to the $\mathbb{R}^d$ and $\mathbb{T}^d$ Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff, adopting a unified hierarchy scheme originally developed for the nonlinear Schr\"{o}dinger equation.