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Characterization of collision kernels

Published online by Cambridge University Press:  15 November 2003

Laurent Desvillettes  and
Francesco Salvarani
Laurent Desvillettes
Affiliation:
École Normale Supérieure de Cachan, Centre de Mathématiques et leurs Applications, 61 Avenue du Président Wilson, 94235 Cachan, France. [email protected].
Francesco Salvarani
Affiliation:
Università degli Studi di Pavia, Dipartimento di Matematica, Via Ferrata, 1, 27100 , Italy. [email protected].
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Abstract

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

Keywords

BoltzmannLandaucollision kernels.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 345 - 355
Copyright
© EDP Sciences, SMAI, 2003

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References

Alexandre, R., Desvillettes, L., Villani, C. and Wennberg, B., Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 (2000) 327-355. CrossRef
R. Alexandre and C. Villani, On the Landau approximation in plasma physics. To appear in Ann. I.H.P. An. non linéaire.
Bobylev, A.V., The Boltzmann equation and the group transformations. Math. Models Methods Appl. Sci. 3 (1993) 443-476. CrossRef
C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases. Springer Verlag, New York (1994).
Desvillettes, L., Boltzmann's kernel and the spatially homogeneous Boltzmann equation. Riv. Mat. Univ. Parma 6 (2001) 1-22.
Desvillettes, L. and Ricci, V., 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. CrossRef
Desvillettes, L. and Villani, C., On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 (2000) 179-259. CrossRef
Dürr, D., Goldstein, S. and Lebowitz, J., Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model. Comm. Math. Phys. 113 (1987) 209-230. CrossRef
G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota interna No. 358, Istituto di Fisica, Università di Roma (1973).
I.M. Guelfand and N.Y. Vilenkin, Les distributions, Tome IV, Applications de l'analyse harmonique. Dunod, Paris (1967).
L. Hörmander, The analysis of linear partial differential operators I. Springer Verlag, Berlin (1983).
Illner, R. and Pulvirenti, M., Global validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Comm. Math. Phys. 105 (1986) 189-203. CrossRef
Illner, R. and Pulvirenti, M., 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. CrossRef
O. Lanford, Time evolution of large classical systems. Springer Verlag, Lecture Notes in Phys. 38 (1975) 1-111. CrossRef
Preisendorfer, R.W., A mathematical foundation for radiative transfer. J. Math. Mech. 6 (1957) 685-730.