On an $L^2$ critical Boltzmann equation

TL;DR

This paper establishes the existence and uniqueness of large global scattering solutions to a two-dimensional $L^2$ critical Boltzmann equation with constant collision kernel, for initial data close to Gaussian distributions.

Contribution

It proves the existence of global solutions in an $L^2$ critical setting for the Boltzmann equation with specific initial data constraints.

Findings

Existence of large global scattering solutions in 2D

Solutions are unique if initial data is Schwartz

Solutions maintain Schwartz regularity over time

Abstract

We prove the existence of a class of large global scattering solutions of Boltzmann's equation with constant collision kernel in two dimensions. These solutions are found for $L^2$ perturbations of an underlying initial data which is Gaussian jointly in space and velocity. Additionally, the perturbation is required to satisfy natural physical constraints for the total mass and second moments, corresponding to conserved or controlled quantities. The space $L^2$ is a scaling critical space for the equation under consideration. If the initial data is Schwartz then the solution is unique and again Schwartz on any bounded time interval.