This expository article, written for the proceedings of the Journées EDP 2023, presents mainly joint works with Dolbeault and Laflèche [1] and Mouhot [3]. We will review some results about long time behaviour of linear kinetic equations for which the microscopic equilibrium (that is, the kernel of the reorientation operator) is typically a density with polynomial decay. There will be no space confinement and the reorientation operator could be of scattering, Fokker–Planck or Levy–Fokker–Planck types. We will first present a spectral approach a la Ellis and Pinsky that yields to a unified treatment of the macroscopic limits for this kind of equations and then focus on re-shaping the Dolbeault–Mouhot–Schmeiser -hypocoercivity method to get explicit rates of decay to zero in suitable weighted norms.
@incollection{JEDP_2023____A3_0,
author = {Emeric Bouin},
title = {Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
note = {talk:3},
pages = {1--7},
publisher = {R\'eseau th\'ematique AEDP du CNRS},
year = {2023},
doi = {10.5802/jedp.674},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.674/}
}
TY - JOUR AU - Emeric Bouin TI - Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails JO - Journées équations aux dérivées partielles N1 - talk:3 PY - 2023 SP - 1 EP - 7 PB - Réseau thématique AEDP du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.674/ DO - 10.5802/jedp.674 LA - en ID - JEDP_2023____A3_0 ER -
%0 Journal Article %A Emeric Bouin %T Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails %J Journées équations aux dérivées partielles %Z talk:3 %D 2023 %P 1-7 %I Réseau thématique AEDP du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.674/ %R 10.5802/jedp.674 %G en %F JEDP_2023____A3_0
Emeric Bouin. Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails. Journées équations aux dérivées partielles (2023), Talk no. 3, 7 p. doi : 10.5802/jedp.674. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.674/
[1] Fractional hypocoercivity, Commun. Math. Phys., Volume 390 (2022) no. 3, pp. 1369-1411 | DOI | MR | Zbl
[2] Quantitative fluid approximation with more invariants (2023) (in progress)
[3] Quantitative fluid approximation in transport theory: a unified approach, Probability and Mathematical Physics, Volume 3 (2022) no. 3, pp. 491-542 | DOI | MR | Zbl
[4] Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: the critical case, Kinet. Relat. Models, Volume 12 (2019) no. 4, pp. 727-748 | DOI | MR | Zbl
[5] Hypocoercivity for Linear Kinetic Equations Conserving Mass, Trans. Am. Math. Soc., Volume 367 (2015) no. 6, pp. 3807-3828 http://www.ams.org/tran/2015-367-06/s0002-9947-2015-06012-7/ (Accessed 2018-09-13) | DOI | MR | Zbl
[6] The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., Volume 54 (1975), pp. 125-156 | MR | Zbl
[7] Anomalous diffusion for multi-dimensional critical kinetic Fokker–Planck equations, Ann. Probab., Volume 48 (2020) no. 5, pp. 2359-2403 | DOI | MR | Zbl
[8] One dimensional critical kinetic Fokker-Planck equations, Bessel and stable processes, Commun. Math. Phys., Volume 381 (2021) no. 1, pp. 143-173 | DOI | MR | Zbl
[9] Diffusion approximation for Fokker Planck with heavy tail equilibria: a spectral method in dimension 1, Commun. Math. Phys., Volume 366 (2019) no. 2, pp. 709-735 | DOI | MR | Zbl
[10] Diffusion limit of Fokker-Planck equation with heavy tail equilibria, ESAIM, Math. Model. Numer. Anal., Volume 49 (2015) no. 1, pp. 1-17 | DOI | Numdam | MR | Zbl
[11] Dispersion Laws for plane wave propagation, The Boltzmann Equation (A. Grünbaum, ed.), Courant Institute, 1971
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