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.
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.
@article{CRMATH_2015__353_7_623_0, 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}, publisher = {Elsevier}, volume = {353}, number = {7}, year = {2015}, doi = {10.1016/j.crma.2015.04.013}, language = {en}, }
TY - JOUR AU - Thierry Bodineau AU - Isabelle Gallagher AU - Laure Saint-Raymond TI - From hard spheres dynamics to the Stokes–Fourier equations: An $ {L}^{2}$ analysis of the Boltzmann–Grad limit JO - Comptes Rendus. Mathématique PY - 2015 SP - 623 EP - 627 VL - 353 IS - 7 PB - Elsevier DO - 10.1016/j.crma.2015.04.013 LA - en ID - CRMATH_2015__353_7_623_0 ER -
%0 Journal Article %A Thierry Bodineau %A Isabelle Gallagher %A Laure Saint-Raymond %T From hard spheres dynamics to the Stokes–Fourier equations: An $ {L}^{2}$ analysis of the Boltzmann–Grad limit %J Comptes Rendus. Mathématique %D 2015 %P 623-627 %V 353 %N 7 %I Elsevier %R 10.1016/j.crma.2015.04.013 %G en %F CRMATH_2015__353_7_623_0
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/
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[2] Equilibrium time correlation functions in the low density limit, J. Stat. Phys., Volume 22 (1980), pp. 237-257
[3] 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 analysis of the Boltzmann–Grad limit, in preparation.
[5] 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] 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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