Characterization of collision kernels
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 345-355.

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.

DOI : 10.1051/m2an:2003030
Classification : 76P05
Mots clés : Boltzmann, Landau, collision kernels
@article{M2AN_2003__37_2_345_0,
     author = {Desvillettes, Laurent and Salvarani, Francesco},
     title = {Characterization of collision kernels},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {345--355},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
     doi = {10.1051/m2an:2003030},
     mrnumber = {1991205},
     zbl = {1047.76114},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2003030/}
}
TY  - JOUR
AU  - Desvillettes, Laurent
AU  - Salvarani, Francesco
TI  - Characterization of collision kernels
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2003
SP  - 345
EP  - 355
VL  - 37
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2003030/
DO  - 10.1051/m2an:2003030
LA  - en
ID  - M2AN_2003__37_2_345_0
ER  - 
%0 Journal Article
%A Desvillettes, Laurent
%A Salvarani, Francesco
%T Characterization of collision kernels
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2003
%P 345-355
%V 37
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2003030/
%R 10.1051/m2an:2003030
%G en
%F M2AN_2003__37_2_345_0
Desvillettes, Laurent; Salvarani, Francesco. Characterization of collision kernels. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 345-355. doi : 10.1051/m2an:2003030. http://archive.numdam.org/articles/10.1051/m2an:2003030/

[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

[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 | Zbl

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

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

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

[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

[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

[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

[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

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

[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

[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

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

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

Cité par Sources :