In these notes we report on a work in collaboration with Thierry Bodineau and Laure Saint-Raymond, where we show how the heat equation can be obtained from a deterministic system of hard spheres when the number of particles goes to infinity while their radius simultaneously goes to zero. As suggested by Hilbert in his sixth problem, the kinetic theory of Boltzmann is used as an intermediate level of description.
@article{JEDP_2014____A2_0,
author = {Gallagher, Isabelle},
title = {From classical mechanics to kinetic theory and fluid dynamics},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {2},
pages = {1--14},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2014},
doi = {10.5802/jedp.105},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.105/}
}
TY - JOUR AU - Gallagher, Isabelle TI - From classical mechanics to kinetic theory and fluid dynamics JO - Journées équations aux dérivées partielles PY - 2014 SP - 1 EP - 14 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.105/ DO - 10.5802/jedp.105 LA - en ID - JEDP_2014____A2_0 ER -
%0 Journal Article %A Gallagher, Isabelle %T From classical mechanics to kinetic theory and fluid dynamics %J Journées équations aux dérivées partielles %D 2014 %P 1-14 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.105/ %R 10.5802/jedp.105 %G en %F JEDP_2014____A2_0
Gallagher, Isabelle. From classical mechanics to kinetic theory and fluid dynamics. Journées équations aux dérivées partielles (2014), article no. 2, 14 p. doi : 10.5802/jedp.105. http://www.numdam.org/articles/10.5802/jedp.105/
[1] R. Alexander, The Infinite Hard Sphere System, Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley, 1975. | MR
[2] H. van Beijeren, O. E. Lanford, J. L. Lebowitz, H. Spohn, Equilibrium Time Correlation Functions in the Low Density Limit. Jour. Stat. Phys. 22, (1980), 237-257. | MR | Zbl
[3] Th. Bodineau, I. Gallagher, L. Saint-Raymond. Limite de diffusion linéaire pour un système déterministe de sphères dures, C. R. Math. Acad. Sci. Paris 352 (2014), no. 5, 411-419. | MR | Zbl
[4] T. Bodineau, I. Gallagher and L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, in revision at Inventiones Mathematicae.
[5] C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Verlag, New York NY, 1994. | MR | Zbl
[6] C. Cercignani, V. I. Gerasimenko, D. I. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishers, Netherlands, 1997. | MR | Zbl
[7] I. Gallagher, L. Saint-Raymond, B. Texier. From Newton to Boltzmann : the case of hard-spheres and short-range potentials, Zürich Lectures in Advanced Mathematics 18 2014. | MR
[8] D. Hilbert, Sur les problèmes futurs des mathématiques, in Compte-Rendu du 2ème Congrès International de Mathématiques, Gauthier-Villars, Paris (1902), 58-114.
[9] O. E. Lanford, Time evolution of large classical systems, Lect. Notes in Physics 38, J. Moser ed., 1-111, Springer Verlag (1975). | MR | Zbl
[10] J. Lebowitz, H. Spohn, Steady state self-diffusion at low density. J. Statist. Phys. 29 (1982), 39-55. | MR | Zbl
[11] M. Pulvirenti, C. Saffirio, S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys. 64 (2014), no. 2, 64 pp. | MR | Zbl
[12] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, Springer-Verlag 1971, 2009. | MR | Zbl
[13] H. Spohn, Large scale dynamics of interacting particles, Springer-Verlag 174 (1991). | Zbl
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