From classical mechanics to kinetic theory and fluid dynamics

Isabelle Gallagher[1]

  • [1] Université Paris-Diderot Institut de Mathématiques de Jussieu Paris Rive Gauche 75013 Paris, France

Journées Équations aux dérivées partielles (2014)

  • page 1-14
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Gallagher, Isabelle. "From classical mechanics to kinetic theory and fluid dynamics." Journées Équations aux dérivées partielles (2014): 1-14. <http://eudml.org/doc/275483>.

@article{Gallagher2014,
abstract = {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.},
affiliation = {Université Paris-Diderot Institut de Mathématiques de Jussieu Paris Rive Gauche 75013 Paris, France},
author = {Gallagher, Isabelle},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
pages = {1-14},
publisher = {Groupement de recherche 2434 du CNRS},
title = {From classical mechanics to kinetic theory and fluid dynamics},
url = {http://eudml.org/doc/275483},
year = {2014},
}

TY - JOUR
AU - Gallagher, Isabelle
TI - From classical mechanics to kinetic theory and fluid dynamics
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 14
AB - 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.
LA - eng
UR - http://eudml.org/doc/275483
ER -

References

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  1. R. Alexander, The Infinite Hard Sphere System, Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley, 1975. MR2625918
  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. Zbl0508.60089MR560556
  3. Th. Bodineau, I. Gallagher, L. Saint-Raymond. Limite de diffusion linéaire pour un système déterministe de sphères dures, C. R. Math. Acad. Sci. Paris352 (2014), no. 5, 411-419. Zbl1291.35210MR3194248
  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. Zbl1337.35107
  5. C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Verlag, New York NY, 1994. Zbl0813.76001MR1307620
  6. C. Cercignani, V. I. Gerasimenko, D. I. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishers, Netherlands, 1997. Zbl0933.82001MR1472233
  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. Zbl1315.82001MR3157048
  8. D. Hilbert, Sur les problèmes futurs des mathématiques, in Compte-Rendu du 2ème Congrès International de Mathématiques, Gauthier-Villars, Paris (1902), 58-114. Zbl32.0084.06
  9. O. E. Lanford, Time evolution of large classical systems, Lect. Notes in Physics 38, J. Moser ed., 1-111, Springer Verlag (1975). Zbl0329.70011MR479206
  10. J. Lebowitz, H. Spohn, Steady state self-diffusion at low density. J. Statist. Phys. 29 (1982), 39-55. Zbl0511.60098MR676928
  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. Zbl1296.82051MR3190204
  12. L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, Springer-Verlag1971, 2009. Zbl1171.82002MR2683475
  13. H. Spohn, Large scale dynamics of interacting particles, Springer-Verlag174 (1991). Zbl0742.76002

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