[Une note sur l’hypocoercivité pour les équations cinétiques avec équilibres à queue lourde]
Dans cet article, on s’intéresse au comportement en temps long d’équations cinétiques linéaires dont les équilibres locaux sont à queue lourde. Notre contribution principale concerne l’équation de Lévy–Fokker–Planck cinétique, pour laquelle nous adaptons des techniques d’hypocoercivité afin de démontrer la convergence exponentielle des solutions vers un équilibre global. En comparant au cas de l’équation de Fokker–Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l’opérateur non-local de Lévy–Fokker–Planck et à la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitons également le cas de l’équation de BGK à queue lourde.
In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy–Fokker–Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker–Planck equation, the issues here concern the lack of symmetry of the non-local Lévy–Fokker–Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.
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@article{CRMATH_2020__358_3_333_0,
author = {Ayi, Nathalie and Herda, Maxime and Hivert, H\'el\`ene and Tristani, Isabelle},
title = {A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium},
journal = {Comptes Rendus. Math\'ematique},
pages = {333--340},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {3},
year = {2020},
doi = {10.5802/crmath.46},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.46/}
}
TY - JOUR AU - Ayi, Nathalie AU - Herda, Maxime AU - Hivert, Hélène AU - Tristani, Isabelle TI - A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium JO - Comptes Rendus. Mathématique PY - 2020 SP - 333 EP - 340 VL - 358 IS - 3 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.46/ DO - 10.5802/crmath.46 LA - en ID - CRMATH_2020__358_3_333_0 ER -
%0 Journal Article %A Ayi, Nathalie %A Herda, Maxime %A Hivert, Hélène %A Tristani, Isabelle %T A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium %J Comptes Rendus. Mathématique %D 2020 %P 333-340 %V 358 %N 3 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.46/ %R 10.5802/crmath.46 %G en %F CRMATH_2020__358_3_333_0
Ayi, Nathalie; Herda, Maxime; Hivert, Hélène; Tristani, Isabelle. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium. Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 333-340. doi : 10.5802/crmath.46. http://www.numdam.org/articles/10.5802/crmath.46/
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