Vol. 3, No. 2, 2021

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An energy method for averaging lemmas

Diogo Arsénio and Nicolas Lerner

Vol. 3 (2021), No. 2, 319–362
Abstract

We introduce 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, 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.

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Keywords
kinetic transport equation, averaging lemmas, energy method, duality principle, maximal regularity, hypoellipticity, dispersion
Mathematical Subject Classification
Primary: 35B65, 35Q20, 35Q49, 35Q83, 82C40
Milestones
Received: 29 June 2020
Revised: 16 January 2021
Accepted: 23 February 2021
Published: 31 July 2021
Authors
Diogo Arsénio
New York University Abu Dhabi
Abu Dhabi
United Arab Emirates
Nicolas Lerner
Institut de Mathématiques de Jussieu
Sorbonne Université
Campus Pierre et Marie Curie
Paris
France