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Commutator method for averaging lemmas

Pierre-Emmanuel Jabin, Hsin-Yi Lin and Eitan Tadmor

Vol. 15 (2022), No. 6, 1561–1584
Abstract

We introduce a commutator method with multiplier to prove averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. Our method requires only elementary techniques in Fourier analysis and highlights a new range of assumptions that are sufficient for the velocity average to be in L2([0,T],Hx12). Our result provides a direct proof (without interpolation) and improves the regularizing result for the measure-valued solutions to scalar conservation laws in space dimension 1.

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Keywords
velocity-averaging lemma, kinetic transport equation, dispersion, singular integral, multiplier, scalar conservation law, measure-valued solution
Mathematical Subject Classification
Primary: 35B65
Secondary: 35L65, 35Q83, 42B37
Milestones
Received: 12 May 2020
Revised: 12 November 2020
Accepted: 16 February 2021
Published: 10 November 2022
Authors
Pierre-Emmanuel Jabin
Department of Mathematics and Huck Institutes of the Life Sciences
Pennsylvania State University
State College, PA
United States
Hsin-Yi Lin
CIRES
University of Colorado
Boulder, CO
United States
Eitan Tadmor
Department of Mathematics and Institute for Physical Sciences and Technology
University of Maryland
College Park, MD
United States