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

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.