[Équation de Boltzmann relativiste inhomogène dans l’espace-temps de Robertson–Walker avec des donnés initiales petites]
Dans cet article, nous considérons le problème de Cauchy pour l’équation de Boltzmann relativiste avec des données initiales petites. Nous supposons que la fonction de distribution dépend du temps, de la position et de l’impulsion. Le noyau de collision considéré ici est pour le cas des potentiels durs et l’espace-temps dans lequel l’étude est faite est celui de Robertson–Walker. Nous prouvons un théorème d’existence et d’unicité globale (dans le temps) d’une solution généralisée dans un espace à poids approprié.
In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson–Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space.
Révisé le :
Accepté le :
Publié le :
Keywords: Relativistic Boltzmann equation, Robertson–Walker, inhomogeneous, mild solution
Mot clés : équation de Boltzmann relativiste, Robertson–Walker, inhomogène, solution généralisée
Takou, Étienne 1 ; Ciake Ciake, Fidèle L. 2
CC-BY-ND 4.0
@article{AIF_2017__67_3_947_0,
author = {Takou, \'Etienne and Ciake Ciake, Fid\`ele L.},
title = {Inhomogeneous relativistic {Boltzmann} equation near vacuum in the {Robertson{\textendash}Walker} space-time},
journal = {Annales de l'Institut Fourier},
pages = {947--967},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {67},
number = {3},
year = {2017},
doi = {10.5802/aif.3101},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3101/}
}
TY - JOUR AU - Takou, Étienne AU - Ciake Ciake, Fidèle L. TI - Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time JO - Annales de l'Institut Fourier PY - 2017 SP - 947 EP - 967 VL - 67 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3101/ DO - 10.5802/aif.3101 LA - en ID - AIF_2017__67_3_947_0 ER -
%0 Journal Article %A Takou, Étienne %A Ciake Ciake, Fidèle L. %T Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time %J Annales de l'Institut Fourier %D 2017 %P 947-967 %V 67 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3101/ %R 10.5802/aif.3101 %G en %F AIF_2017__67_3_947_0
Takou, Étienne; Ciake Ciake, Fidèle L. Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time. Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 947-967. doi : 10.5802/aif.3101. https://aif.centre-mersenne.org/articles/10.5802/aif.3101/
[1] Problème Cauchy pour l’équation de Boltzmann en relativité générale, Ann. Inst. Henri Poincaré, Sect. A, Volume 18 (1973), pp. 263-284
[2] Problème de Cauchy pour le système intégro-differentiel d’Einstein–Liouville, Ann. Inst. Fourier, Volume 21 (1971) no. 3, pp. 181-201 | DOI
[3] Introduction to Relativistic heavy ion collision, John Wiley and Sons, 1994, 322 pages
[4] On the linearized relativistic Boltzmann equation. I: Existence of solutions, Commun. Math. Phys., Volume 115 (1988) no. 4, pp. 607-629 | DOI
[5] Relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., Volume 48 (2007) no. 1, pp. 39-47
[6] The Cauchy problem in kinetic theory, Other Titles in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), 1996, xii+241 pages
[7] Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Commun. Math. Phys., Volume 264 (2006) no. 3, pp. 705-724 | DOI
[8] On the derivatives of the collision map of relativistic particles, Transp. Theory Stat. Phys., Volume 20 (1991) no. 1, pp. 55-68 | DOI
[9] Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson–Walker space-time (2013) (https://arxiv.org/abs/1307.5688v1)
[10] The Einstein–Boltzmann system and positivity, J. Hyperbolic Differ. Equ., Volume 10 (2013) no. 1, pp. 77-104 | DOI
[11] The spatially homogeneous relativistic Boltzmann equation with hard potential (2013) (https://arxiv.org/abs/1301.0106v1)
[12] Théorie Rélativiste de la Gravitation et de l’Electromagnétisme, Relativité générale et théories unitaires, Masson et Cie, Editeurs, 1955, 300 pages
[13] Global existence of solutions for the Einstein–Boltzmann system in a Bianchi type I spacetime for arbitrarily large initial data, Class. Quantum Grav., Volume 23 (2006) no. 9, pp. 2979-3003 | DOI
[14] Global existence of solutions for the relativistic Boltzmann equation with arbitrarily large initial data on a Bianchi type I space-time., Gen. Relativ. Gravit., Volume 37 (2005) no. 12, pp. 2047-2062 | DOI
[15] Global existence of solutions for the Einstein–Boltzmann system with cosmological constant in the Robertson–Walker space-time., Commum. Math. Sci., Volume 4 (2006) no. 2, pp. 291-314 | DOI
[16] Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys., Volume 3000 (2010) no. 2, pp. 529-597 | DOI
[17] Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., Volume 42 (2010) no. 4, pp. 1568-1601 | DOI
Cité par Sources :
