Sharp Global Well-posedness and Scattering of the Boltzmann Equation

TL;DR

This paper establishes the first sharp global well-posedness and scattering results for the 3D Boltzmann equation with Maxwellian particles and soft potential, using advanced bilinear spacetime estimates.

Contribution

It introduces novel bilinear spacetime estimates for the gain term, enabling sharp global results at low regularity in the critical space.

Findings

Proves global well-posedness for small initial data in critical space.

Maintains $L^{1}$ regularity if initial data is in $L^{1}$.

First 3D sharp global result for the Boltzmann equation.

Abstract

We consider the 3D Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff. We prove sharp global well-posedness with initial data small in the scaling-critical space. The solution also remains in $L^{1}$ if the initial datum is in $L^{1}$, even at such low regularity. The key to existence, uniqueness and regularity criteria is the new bilinear spacetime estimates for the gain term, the proof of which is based on novel techniques from nonlinear dispersive PDEs including the atomic $U$-$V$ spaces, multi-linear frequency analysis, dispersive estimates, etc. To our knowledge, this is the first 3D sharp global result for the Boltzmann equation.