Convolution estimates for the Boltzmann gain operator with hard spheres

TL;DR

This paper establishes new convolution estimates for the Boltzmann gain operator with hard potentials, including hard spheres, using a novel cancellation mechanism to handle energy-absorbing collisions.

Contribution

It introduces a new method for convolution estimates in the Boltzmann equation that accounts for the rare but problematic energy-absorbing collisions in hard potentials.

Findings

Proves moment-preserving polynomially weighted convolution estimates.

Handles the critical case of hard-spheres.

Uses geometric identities and angular averaging techniques.

Abstract

We prove new moment-preserving polynomially weighted convolution estimates for the gain operator of the Boltzmann equation with hard potentials, including the critical case of hard-spheres. Our approach relies crucially on a novel cancellation mechanism dealing with the pathological case of energy-absorbing collisions (that is, collisions that accumulate energy to only one of the outgoing particles). This difficulty is specific to hard potentials, and is not present for Maxwell molecules. Our method quantifies the heuristic that, while energy-absorbing collisions occur with non-trivial probability, they are statistically rare, and therefore do not affect the overall averaging behavior of the gain operator. At the technical level, our proof relies solely on tools from kinetic theory, such as geometric identities and angular averaging.