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On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero

On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated with the zero eigenvalue as t→∞.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t \rightarrow \infty .$$\end{document} This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero

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References (39)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-022-00777-8
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated with the zero eigenvalue as t→∞.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t \rightarrow \infty .$$\end{document} This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned.

Journal

Journal of Evolution Equations – Springer Journals

Published: Mar 1, 2022

Keywords: Kinetic equation; Boundary operators; Nonzero velocities; Convergence to equilibrium

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