[Modèles cinétiques des superfluides : des résultats mathématiques]
Les contributions mathématiques de X.G. Lu (J. Statist. Phys. 98 (5/6) (2000) 1335–1394) et de M. Escobedo et al. (Electronic J. Differential Equations, Monograph 4 (2003)) qui sont présentées dans cette Note constituent la première avancée dans la compréhension de la dynamique superfluide et notamment de la condensation de Bose–Einstein grâce aux modèles cinétiques. L'équation de Boltzmann–Nordheim, qui permet de décrire l'évolution d'un gaz quantique dilué constitué de bosons, pose de nombreux problèmes mathématiques. Néanmoins, sous une hypothèse non physique de troncature des collisions à basse énergie, on peut montrer que le problème de Cauchy homogène en espace est bien posé. De plus, le système relaxe vers l'équilibre (en un sens faible), avec apparition d'une singularité en temps infini si la masse initiale est supercritique : cela correspond à la formation d'un condensat de Bose–Einstein.
The mathematical contributions by X.G. Lu (J. Statist. Phys. 98 (5/6) (2000) 1335–1394) and by M. Escobedo et al. (Electronic J. Differential Equations, Monograph 4 (2003)) presented in this Note constitute the first stage in the understanding of the superfluid dynamics, especially of the Bose–Einstein condensation, by means of kinetic models. The Boltzmann–Nordheim equation, which is physically relevant to describe dilute quantum Bose gases, sets important mathematical problems. Nevertheless, under an unphysical truncation of the collision cross-section at low energies, it has been proved that the spatially homogeneous Cauchy problem is well-posed. Furthermore, relaxation towards equilibrium holds in a weak sense, with the appearance of a singularity in infinite time if the initial mass is supercritical, which corresponds to the formation of a Bose–Einstein condensate.
Mots-clés : Condensation de Bose–Einstein, Équation cinétique, Problème de Cauchy, Relaxation vers l'équilibre
Laure Saint-Raymond 1
@article{CRPHYS_2004__5_1_65_0,
author = {Laure Saint-Raymond},
title = {Kinetic models for superfluids: a review of mathematical results},
journal = {Comptes Rendus. Physique},
pages = {65--75},
publisher = {Elsevier},
volume = {5},
number = {1},
year = {2004},
doi = {10.1016/j.crhy.2004.01.005},
language = {en},
}
Laure Saint-Raymond. Kinetic models for superfluids: a review of mathematical results. Comptes Rendus. Physique, Bose-Einstein condensates: recent advances in collective effects, Volume 5 (2004) no. 1, pp. 65-75. doi : 10.1016/j.crhy.2004.01.005. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2004.01.005/
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