Characterization of collision kernels
Desvillettes, Laurent ; Salvarani, Francesco
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003), p. 345-355 / Harvested from Numdam

In this paper we show how abstract physical requirements are enough to characterize the classical collision kernels appearing in kinetic equations. In particular Boltzmann and Landau kernels are derived.

Publié le : 2003-01-01
DOI : https://doi.org/10.1051/m2an:2003030
Classification:  76P05
@article{M2AN_2003__37_2_345_0,
     author = {Desvillettes, Laurent and Salvarani, Francesco},
     title = {Characterization of collision kernels},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {37},
     year = {2003},
     pages = {345-355},
     doi = {10.1051/m2an:2003030},
     mrnumber = {1991205},
     zbl = {1047.76114},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2003__37_2_345_0}
}
Desvillettes, Laurent; Salvarani, Francesco. Characterization of collision kernels. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) pp. 345-355. doi : 10.1051/m2an:2003030. http://gdmltest.u-ga.fr/item/M2AN_2003__37_2_345_0/

[1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 (2000) 327-355. | Zbl 0968.76076

[2] R. Alexandre and C. Villani, On the Landau approximation in plasma physics. To appear in Ann. I.H.P. An. non linéaire. | Numdam | MR 2037247 | Zbl 1044.83007

[3] A.V. Bobylev, The Boltzmann equation and the group transformations. Math. Models Methods Appl. Sci. 3 (1993) 443-476. | Zbl 0782.35078

[4] C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases. Springer Verlag, New York (1994). | MR 1307620 | Zbl 0813.76001

[5] L. Desvillettes, Boltzmann's kernel and the spatially homogeneous Boltzmann equation. Riv. Mat. Univ. Parma 6 (2001) 1-22. | Zbl 1078.76059

[6] L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions. J. Statist. Phys. 104 (2001) 1173-1189. | Zbl 1051.82022

[7] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 (2000) 179-259. | Zbl 0946.35109

[8] D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model. Comm. Math. Phys. 113 (1987) 209-230. | Zbl 0642.60057

[9] G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota interna No. 358, Istituto di Fisica, Università di Roma (1973).

[10] I.M. Guelfand and N.Y. Vilenkin, Les distributions, Tome IV, Applications de l'analyse harmonique. Dunod, Paris (1967). | Zbl 0219.46032

[11] L. Hörmander, The analysis of linear partial differential operators I. Springer Verlag, Berlin (1983). | Zbl 0521.35001

[12] R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Comm. Math. Phys. 105 (1986) 189-203. | Zbl 0609.76083

[13] R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in the vacuum: erratum and improved result. Comm. Math. Phys. 121 (1989) 143-146. | Zbl 0850.76600

[14] O. Lanford, Time evolution of large classical systems. Springer Verlag, Lecture Notes in Phys. 38 (1975) 1-111. | Zbl 0329.70011

[15] R.W. Preisendorfer, A mathematical foundation for radiative transfer. J. Math. Mech. 6 (1957) 685-730. | Zbl 0078.42503