Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, p. 947-967

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.

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.

Received : 2015-03-31
Revised : 2016-07-15
Accepted : 2016-09-15
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3101
Classification:  76P05,  35Q20
Keywords: Relativistic Boltzmann equation, Robertson–Walker, inhomogeneous, mild solution
@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--Walker space-time},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {947-967},
     doi = {10.5802/aif.3101},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 67 (2017) no. 3, pp. 947-967. doi : 10.5802/aif.3101. http://www.numdam.org/item/AIF_2017__67_3_947_0/

[1] Bancel, Daniel Problème Cauchy pour l’équation de Boltzmann en relativité générale, Ann. Inst. Henri Poincaré, Sect. A, Tome 18 (1973), pp. 263-284

[2] Choquet-Bruhat, Yvonne Problème de Cauchy pour le système intégro-differentiel d’Einstein–Liouville, Ann. Inst. Fourier, Tome 21 (1971) no. 3, pp. 181-201 | Article

[3] Csernai, L. P. Introduction to Relativistic heavy ion collision, John Wiley and Sons (1994), 322 pages

[4] Dudyński, Marek; Ekiel-Jeżewska, Maria L. On the linearized relativistic Boltzmann equation. I: Existence of solutions, Commun. Math. Phys., Tome 115 (1988) no. 4, pp. 607-629 | Article

[5] Dudyński, Marek; Ekiel-Jeżewska, Maria L. Relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., Tome 48 (2007) no. 1, pp. 39-47

[6] Glassey, Robert T. The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Other Titles in Applied Mathematics (1996), xii+241 pages

[7] Glassey, Robert T. Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Commun. Math. Phys., Tome 264 (2006) no. 3, pp. 705-724 | Article

[8] Glassey, Robert T.; Strauss, Walter A. On the derivatives of the collision map of relativistic particles, Transp. Theory Stat. Phys., Tome 20 (1991) no. 1, pp. 55-68 | Article

[9] Lee, Ho Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson–Walker space-time (2013) (https://arxiv.org/abs/1307.5688v1 )

[10] Lee, Ho; Rendall, Alan D. The Einstein–Boltzmann system and positivity, J. Hyperbolic Differ. Equ., Tome 10 (2013) no. 1, pp. 77-104 | Article

[11] Lee, Ho; Rendall, Alan D. The spatially homogeneous relativistic Boltzmann equation with hard potential (2013) (https://arxiv.org/abs/1301.0106v1 )

[12] Lichnerowicz, André Théorie Rélativiste de la Gravitation et de l’Electromagnétisme, Masson et Cie, Editeurs, Relativité générale et théories unitaires (1955), 300 pages

[13] Noutchegueme, Norbert; Dongo, David Global existence of solutions for the Einstein–Boltzmann system in a Bianchi type I spacetime for arbitrarily large initial data, Class. Quantum Grav., Tome 23 (2006) no. 9, pp. 2979-3003 | Article

[14] Noutchegueme, Norbert; Dongo, David; Takou, Étienne Global existence of solutions for the relativistic Boltzmann equation with arbitrarily large initial data on a Bianchi type I space-time., Gen. Relativ. Gravit., Tome 37 (2005) no. 12, pp. 2047-2062 | Article

[15] Noutchegueme, Norbert; Takou, Étienne Global existence of solutions for the Einstein–Boltzmann system with cosmological constant in the Robertson–Walker space-time., Commum. Math. Sci., Tome 4 (2006) no. 2, pp. 291-314 | Article

[16] Strain, Robert M. Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys., Tome 3000 (2010) no. 2, pp. 529-597 | Article

[17] Strain, Robert M. Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., Tome 42 (2010) no. 4, pp. 1568-1601 | Article