Sharp regularizing estimates for the gain term of the Boltzmann collision operator

TL;DR

This paper establishes sharp regularizing estimates for the gain term of the Boltzmann collision operator across various models, precisely characterizing its smoothing and convolution properties in Sobolev and Lebesgue spaces.

Contribution

It provides the first sharp regularizing estimates for the gain term, matching the kinetic kernel's exponent and extending to broader Lebesgue space settings without weight loss.

Findings

Sharp regularization exponents in Sobolev spaces

Broader Lebesgue space applicability of estimates

No weight loss in homogeneous Sobolev space estimates

Abstract

We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator including hard sphere, hard potential and Maxwell molecule models. Our new estimates characterize both regularization and convolution properties of the gain term and have the following features. The regularizing exponent is sharp both in the $L^2$ based inhomogeneous and homogeneous Sobolev spaces which is exact the exponent of the kinetic part of collision kernel. The functions in these estimates belong to a wider scope of (weighted) Lebesgue spaces than the previous regularizing estimates. Furthermore, for the estimates in homogeneous Sobolev spaces, we only need functions lying in Lebesgue spaces instead of weighted Lebesgue spaces, i.e., no loss of weight occurs in this case.