Diffusion and kinetic transport with very weak confinement

Emeric Bouin; Jean Dolbeault; Christian Schmeiser

doi:10.3934/krm.2020012

Article Contents

2020, Volume 13, Issue 2: 345-371. Doi: 10.3934/krm.2020012

This issue Previous Article Trend to the equilibrium for the Fokker-Planck system with an external magnetic field Next Article Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases

Diffusion and kinetic transport with very weak confinement

1.
CEREMADE (CNRS UMR n° 7534), PSL university, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
2.
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

^* Corresponding author: Emeric Bouin
^* Corresponding author: Emeric Bouin

Received: December 31, 2018

Revised: August 31, 2019

Published: January 2020

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Abstract

This paper is devoted to Fokker-Planck and linear kinetic equations with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diffusion wins over the confinement although the potential diverges at infinity. When there is no confinement potential, it is possible to rely on Fourier analysis and mode-by-mode estimates for the kinetic equations. Here we develop an alternative approach based on moment estimates and Caffarelli-Kohn-Nirenberg inequalities of Nash type for diffusion and kinetic equations.

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Mathematics Subject Classification: Primary: 35B40, 35Q84; Secondary: 82C40, 76P05, 26D10.

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