Convolution inequalities for the Boltzmann collision operator

TL;DR

This paper establishes new integrability inequalities for the Boltzmann collision operator, including sharp Young's and Hardy-Littlewood-Sobolev type inequalities, using a convolution-based approach applicable to various potentials.

Contribution

It introduces a novel convolution reformulation and proves sharp inequalities for the Boltzmann collision operator, extending classical analysis techniques to kinetic theory.

Findings

Proved a sharp Young's inequality for hard potentials.

Derived a Hardy-Littlewood-Sobolev type inequality for soft potentials.

Established explicit constants depending on angular cross section integrability.

Abstract

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in $n$-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in $L^p$ we prove a Young's inequality for hard potentials, which is sharp for Maxwell molecules in the $L^2$ case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some $L^{s}_{weak}$ or $L^{s}$. The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.