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

Thierry Bodineau; Isabelle Gallagher; Laure Saint-Raymond

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

Presented by: the Editorial Board

Thierry Bodineau ¹; Isabelle Gallagher ²; Laure Saint-Raymond 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, Volume 353 (2015) no. 7, pp. 623-627.

Abstract
Résumé

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.

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.

Received: 2015-04-16
Accepted: 2015-04-17
Published online: 2015-05-07

DOI: 10.1016/j.crma.2015.04.013

Author's affiliations:

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

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     author = {Thierry Bodineau and Isabelle Gallagher and Laure Saint-Raymond},
     title = {From hard spheres dynamics to the {Stokes{\textendash}Fourier} equations: {An} $ {L}^{2}$ analysis of the {Boltzmann{\textendash}Grad} limit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {623--627},
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     year = {2015},
     doi = {10.1016/j.crma.2015.04.013},
     language = {en},
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Thierry Bodineau; Isabelle Gallagher; Laure Saint-Raymond. From hard spheres dynamics to the Stokes–Fourier equations: An $ {L}^{2}$ analysis of the Boltzmann–Grad limit. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 623-627. doi : 10.1016/j.crma.2015.04.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.013/

References
Cited by

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[2] H. van Beijeren; O.E. Lanford; J.L. Lebowitz; H. Spohn Equilibrium time correlation functions in the low density limit, J. Stat. Phys., Volume 22 (1980), pp. 237-257

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

[6] O.E. Lanford Time evolution of large classical systems (J. Moser, ed.), Lecture Notes in Physics, vol. 38, Springer Verlag, 1975, pp. 1-111

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