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.
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
@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},
year = {2023},
publisher = {R\'eseau th\'ematique AEDP du CNRS},
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
[1] Fractional hypocoercivity, Commun. Math. Phys., Volume 390 (2022) no. 3, pp. 1369-1411 | DOI | Zbl | MR
[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 | Zbl | MR
[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 | Zbl | DOI | MR
[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 | MR | Numdam | Zbl
[11] Dispersion Laws for plane wave propagation, The Boltzmann Equation (A. Grünbaum, ed.), Courant Institute, 1971
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