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Comptes Rendus Mathématique
Volume 353, n° 7
pages 623-627 (juillet 2015)
Doi : 10.1016/j.crma.2015.04.013
Received : 16 April 2015 ;  accepted : 17 April 2015
From hard spheres dynamics to the Stokes–Fourier equations: An   analysis of the Boltzmann–Grad limit
De la dynamique des sphères dures aux équations de Stokes–Fourier : Une analyse   de la limite de Boltzmann–Grad
 

Thierry Bodineau a , Isabelle Gallagher b , Laure Saint-Raymond c
a CNRS & École polytechnique, Centre de mathématiques appliquées, route de Saclay, 91128 Palaiseau, France 
b Université Paris-Diderot, Institut de mathématiques de Jussieu, Paris Rive Gauche, 75205 Paris cedex 13, France 
c Université Pierre-et-Marie-Curie & École normale supérieure, Département de mathématiques et applications, 11, rue Pierre-et-Marie-Curie, 75231 Paris cedex 05, France 

Abstract

We derive the Stokes–Fourier equations in dimension 2 as the limiting dynamics of a system of N hard spheres of diameter ε when  ,  ,  , 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   bound on the initial data, the time propagation of which involves a fine symmetry argument and a systematic study of recollisions.

The full text of this article is available in PDF format.
Résumé

Les équations de Stokes–Fourier sont obtenues, en dimension 2, comme dynamique limite d'un système de N sphères dures de diamètre ε quand  ,  ,  , 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   sur la donnée initiale, dont la propagation en temps repose sur un argument fin de symétrie et une étude systématique des recollisions.

The full text of this article is available in PDF format.


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