The kinetic Fokker-Planck equation with mean field interaction

Arnaud Guillin; Wei Liu; Liming Wu; Chaoen Zhang

doi:10.1016/j.matpur.2021.04.001

The kinetic Fokker-Planck equation with mean field interaction

Arnaud Guillin^*
, Wei Liu
, Liming Wu
, Chaoen Zhang

^*Corresponding author for this work

Université Clermont Auvergne
Wuhan University

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H¹(μ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

Original language	English
Pages (from-to)	1-23
Number of pages	23
Journal	Journal des Mathematiques Pures et Appliquees
Volume	150
DOIs	https://doi.org/10.1016/j.matpur.2021.04.001
State	Published - Jun 2021
Externally published	Yes

Keywords

Hypocoercivity
Logarithmic Sobolev inequality
Lyapunov conditions
Mean field interaction
Poincaré inequalities

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10.1016/j.matpur.2021.04.001

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title = "The kinetic Fokker-Planck equation with mean field interaction",

abstract = "We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H1(μ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.",

keywords = "Hypocoercivity, Logarithmic Sobolev inequality, Lyapunov conditions, Mean field interaction, Poincar{\'e} inequalities",

author = "Arnaud Guillin and Wei Liu and Liming Wu and Chaoen Zhang",

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year = "2021",

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T1 - The kinetic Fokker-Planck equation with mean field interaction

AU - Guillin, Arnaud

AU - Liu, Wei

AU - Wu, Liming

AU - Zhang, Chaoen

PY - 2021/6

Y1 - 2021/6

N2 - We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H1(μ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

AB - We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H1(μ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

KW - Hypocoercivity

KW - Logarithmic Sobolev inequality

KW - Lyapunov conditions

KW - Mean field interaction

KW - Poincaré inequalities

UR - https://www.scopus.com/pages/publications/85104073239

U2 - 10.1016/j.matpur.2021.04.001

DO - 10.1016/j.matpur.2021.04.001

M3 - 文章

AN - SCOPUS:85104073239

SN - 0021-7824

VL - 150

SP - 1

EP - 23

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -

The kinetic Fokker-Planck equation with mean field interaction

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Keywords

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