From hard spheres dynamics to the Stokes–Fourier equations: An $ {L}^{2}$ analysis of the Boltzmann–Grad limit

Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure

doi:10.1016/j.crma.2015.04.013

Partial differential equations/Mathematical physics

From hard spheres dynamics to the Stokes–Fourier equations: An

L^{2}

analysis of the Boltzmann–Grad limit
[De la dynamique des sphères dures aux équations de Stokes–Fourier : Une analyse

L^{2}

de la limite de Boltzmann–Grad]

Présenté par : the Editorial Board

Bodineau, Thierry ¹ ; Gallagher, Isabelle ² ; Saint-Raymond, Laure ³

¹ CNRS & École polytechnique, Centre de mathématiques appliquées, route de Saclay, 91128 Palaiseau, France
² Université Paris-Diderot, Institut de mathématiques de Jussieu, Paris Rive Gauche, 75205 Paris cedex 13, France
³ Université Pierre-et-Marie-Curie & École normale supérieure, Département de mathématiques et applications, 11, rue Pierre-et-Marie-Curie, 75231 Paris cedex 05, France

Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 623-627.

Résumé
Abstract

Les équations de Stokes–Fourier sont obtenues, en dimension 2, comme dynamique limite d'un système de N sphères dures de diamètre ε quand $N \to \infty$ , $ε \to 0$ , $N ε = α \to \infty$ , en utilisant l'équation de Boltzmann linéarisée comme étape intermédiaire. Notre preuve est basée sur la stratégie de Lanford [6] et sur la procédure de troncature développée dans [3] pour améliorer le temps de convergence. La principale nouveauté ici est que les estimations a priori uniformes viennent d'une borne $L^{2}$ sur la donnée initiale, dont la propagation en temps repose sur un argument fin de symétrie et une étude systématique des recollisions.

We derive the Stokes–Fourier equations in dimension 2 as the limiting dynamics of a system of N hard spheres of diameter ε when $N \to \infty$ , $ε \to 0$ , $N ε = α \to \infty$ , using the linearized Boltzmann equation as an intermediate step. Our proof is based on the strategy of Lanford [6], and on the pruning procedure developed in [3] to improve the convergence time. The main novelty here is that uniform a priori estimates come from a $L^{2}$ bound on the initial data, the time propagation of which involves a fine symmetry argument and a systematic study of recollisions.

Reçu le : 2015-04-16
Accepté le : 2015-04-17
Publié le : 2015-05-07

DOI : 10.1016/j.crma.2015.04.013

Affiliations des auteurs :

Bodineau, Thierry ¹ ; Gallagher, Isabelle ² ; Saint-Raymond, Laure ³

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     title = {From hard spheres dynamics to the {Stokes{\textendash}Fourier} equations: {An} $ {L}^{2}$ analysis of the {Boltzmann{\textendash}Grad} limit},
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Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure. From hard spheres dynamics to the Stokes–Fourier equations: An $ {L}^{2}$ analysis of the Boltzmann–Grad limit. Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 623-627. doi : 10.1016/j.crma.2015.04.013. https://www.numdam.org/articles/10.1016/j.crma.2015.04.013/

Bibliographie
Cité par

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[3] Bodineau, T.; Gallagher, I.; Saint-Raymond, L. The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math. (2015), pp. 1-61 (in press) | DOI

[4] T. Bodineau, I. Gallagher, L. Saint-Raymond, From hard spheres dynamics to the Stokes–Fourier equations: an $L^{2}$ analysis of the Boltzmann–Grad limit, in preparation.

[5] Gallagher, I.; Saint-Raymond, L.; Texier, B. From Newton to Boltzmann: The Case of Hard-Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, Switzerland, 2014

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Jabin, Pierre-Emmanuel; Wang, Zhenfu Mean Field Limit for Stochastic Particle Systems, Active Particles, Volume 1 (2017), p. 379 | DOI:10.1007/978-3-319-49996-3_10
Ayi, Nathalie From Newton’s Law to the Linear Boltzmann Equation Without Cut-Off, Communications in Mathematical Physics, Volume 350 (2017) no. 3, p. 1219 | DOI:10.1007/s00220-016-2821-6

Cité par 3 documents. Sources : Crossref