Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

Armand Bernou; Kleber Carrapatoso; Stéphane Mischler; Isabelle Tristani

doi:10.4171/AIHPC/44

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

Armand Bernou
, Kleber Carrapatoso
, Stéphane Mischler
, Isabelle Tristani

Laboratoire de Probabilités
Sorbonne Université
Université Paris Dauphine
Département de Mathématiques et Applications 45
PSL research University & IPSL

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

17 Citations (Scopus)

Abstract

We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator; in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all types of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

Original language	English
Pages (from-to)	287-338
Number of pages	52
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	40
Issue number	2
DOIs	https://doi.org/10.4171/AIHPC/44
Publication status	Published - 1 Jan 2022

Keywords

Boltzmann equation
Kinetic equations
Korn inequality
Landau equation
Poincaré inequality
convergence to equilibrium
hypocoercivity
spectral gap

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title = "Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition",

abstract = "We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator; in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all types of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.",

keywords = "Boltzmann equation, Kinetic equations, Korn inequality, Landau equation, Poincar{\'e} inequality, convergence to equilibrium, hypocoercivity, spectral gap",

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T1 - Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

AU - Bernou, Armand

AU - Carrapatoso, Kleber

AU - Mischler, Stéphane

AU - Tristani, Isabelle

PY - 2022/1/1

Y1 - 2022/1/1

N2 - We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator; in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all types of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

AB - We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator; in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all types of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

KW - Boltzmann equation

KW - Kinetic equations

KW - Korn inequality

KW - Landau equation

KW - Poincaré inequality

KW - convergence to equilibrium

KW - hypocoercivity

KW - spectral gap

U2 - 10.4171/AIHPC/44

DO - 10.4171/AIHPC/44

M3 - Article

AN - SCOPUS:85130534318

SN - 0294-1449

VL - 40

SP - 287

EP - 338

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 2

ER -

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

Abstract

Keywords

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