An energy method for averaging lemmas

TL;DR

This paper presents a novel energy-based duality approach to velocity averaging lemmas in kinetic theory, extending classical results to dual spaces and offering a more robust, inequality-based method with potential applications in kinetic equations.

Contribution

Introduces a new energy method for velocity averaging lemmas that extends classical results to dual spaces and simplifies proofs through inequalities.

Findings

Develops a duality principle for kinetic transport equations.

Extends averaging lemmas to cases with dual space densities and sources.

Provides a functional analytic framework with potential applications in kinetic theory.

Abstract

This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach -- based upon the classical energy method -- provides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view. They are motivated by the abstract regularity theory of kinetic transport equations. However, we may recall that velocity averaging lemmas have profound implications in kinetic theory and its related physical models. In particular, the precise formulation of such results has the potential to lead to important applications to the regularity of renormalizations of Boltzmann-type equations, as well as kinetic formulations of gas dynamics, for instance.