Hypocoercivity of linear kinetic equations via Harris's Theorem

José A. Cañizo; Chuqi Cao; Josephine Evans; Havva Yoldaş

doi:10.3934/krm.2020004

Article Contents

2020, Volume 13, Issue 1: 97-128. Doi: 10.3934/krm.2020004

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Hypocoercivity of linear kinetic equations via Harris's Theorem

1.
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain
2.
CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris, France
3.
BCAM Basque Center for Applied Mathematics, Alameda Mazarredo, 14, 48009 Bilbao, Spain

^*Corresponding author: Chuqi Cao
^*Corresponding author: Chuqi Cao

The authors would like to thank S. Mischler, C. Mouhot, J. Féjoz and R. Ortega for some useful discussion and ideas for some parts in the paper

Received: January 31, 2019

Revised: July 31, 2019

Published: January 2020

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Abstract

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $ (x,v) \in \mathbb{T}^d \times \mathbb{R}^d $ or on the whole space $ (x,v) \in \mathbb{R}^d \times \mathbb{R}^d $ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $ L^1 $ or weighted $ L^1 $ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.

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Mathematics Subject Classification: Primary: 35B40, 82B40, 47A35; Secondary: 34G10.

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