# Small data global well-posedness for a Boltzmann equation via bilinear spacetime estimates

## TL;DR

This paper establishes global well-posedness and scattering for a 2D Boltzmann equation with constant collision kernel in the critical Lebesgue space, using new bilinear spacetime estimates and iteration techniques.

## Contribution

It introduces a novel bilinear spacetime estimate for the collision gain term and applies Kaniel-Shinbrot iteration to prove well-posedness in the critical space.

## Key findings

- Global well-posedness for small initial data in L^2_{x,v}
- New bilinear spacetime estimate for collision gain term
- Application of Kaniel-Shinbrot iteration in this context

## Abstract

We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision "gain" term in Boltzmann's equation, combined with a novel application of the Kaniel-Shinbrot iteration.