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LOG-TRANSFORM AND THE WEAK HARNACK INEQUALITY FOR KINETIC FOKKER-PLANCK EQUATIONS

Part of: Partial differential equations Close-to-elliptic equations and systems

Published online by Cambridge University Press:  16 May 2022

Jessica Guerand  and
Cyril Imbert Open the ORCID record for Cyril Imbert [Opens in a new window]
Jessica Guerand
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB Cambridge, United Kingdom (jg900@cam.ac.uk)
Cyril Imbert*
Affiliation:
Département de Mathématiques et applications, École Normale Supérieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, France
*
cyril.imbert@ens.fr
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Abstract

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

Keywords

kinetic theoryFokker–Planck equationHölder continuityPoincaré inequalityHarnack inequality

MSC classification

Primary: 35H10: Hypoelliptic equations
Secondary: 35B65: Smoothness and regularity of solutions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 6 , November 2023 , pp. 2749 - 2774
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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