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One-dimensional kinetic models of granular flows

Published online by Cambridge University Press:  15 April 2002

Giuseppe Toscani
Giuseppe Toscani*
Affiliation:
Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy. ([email protected])
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Abstract

We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance for the finite time extinction of the solution.

Keywords

Granular gasesBoltzmann equationlong-time behavior of solutions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1277 - 1291
Copyright
© EDP Sciences, SMAI, 2000

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References

R. Alexandre and C. Villani, On the Boltzmann equation for long range interactions and the Landau approximation in plasma physics. Preprint DMA, École Normale Supérieure (1999).
Arkeryd, L., Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rational Mech. Anal. 77 (1981) 11-21. CrossRef
G.I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge Univ. Press, New York (1996).
Benedetto, D., Caglioti, E. and Pulvirenti, M., A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31 (1997) 615-641. CrossRef
Benedetto, D., Caglioti, E. and Pulvirenti, M., Erratum: A kinetic equation for granular media [RAIRO Modél. Math. Anal. Numér. 31 (1997) 615-641]. ESAIM: M2AN 33 (1999) 439-441. CrossRef
Bizon, C., Shattuck, J.B., Swift, M.D., McCormick, W.D. and Swinney, H.L., Pattern in 2D vertically oscillated granular layers: simulation and experiments. Phys. Rev. Lett. 80 (1998) 57-60. CrossRef
Bobylev, A.V., J.A-Carrillo and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Statist. Phys. 98 (2000) 743-773. CrossRef
C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases. Springer Ser. Appl. Math. Sci. 106, Springer-Verlag, New York (1994).
Desvillettes, L., About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys. 168 (1995) 417-440. CrossRef
Du, Y., Li, H. and Kadanoff, L.P., Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74 (1995) 1268-1271. CrossRef
Goldman, D., Shattuck, M.D., Bizon, C., McCormick, W.D., Swift, J.B. and Swinney, H.L., Absence of inelastic collapse in a realistic three ball model. Phys. Rev. E 57 (1998) 4831-4833. CrossRef
Goldhirsch, I., Scales and kinetics of granular flows. Chaos 9 (1999) 659-672. CrossRef
M. Kac, Probability and related topics in the physical sciences. New York (1959).
Kantorovich, L., On translation of mass (in Russian). Dokl. AN SSSR 37 (1942) 227-229.
Landau, L., Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung. Phys. Z. Sowjet. 10 (1936) 154. Trad.: The transport equation in the case of Coulomb interactions, in Collected papers of L.D. Landau, D. ter Haar Ed., Pergamon Press, Oxford (1981) 163-170.
McNamara, S. and Young, W.R., Inelastic collapse and clumping in a one-dimensional granular medium. Phys. Fluids A 4 (1992) 496-504. CrossRef
McNamara, S. and Young, W.R., Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A 5 (1993) 34-45. CrossRef
G. Naldi, L. Pareschi and G. Toscani, Spectral methods for a singular Boltzmann equation for granular flows and numerical quasi elastic limit. Preprint (2000).
Toscani, G., The grazing collision asymptotic of the non cut-off Kac equation. RAIRO Modél. Math. Anal. Numér. 32 (1998) 763-772. CrossRef
I. Vaida, Theory of statistical Inference and Information. Kluwer Academic Publishers, Dordrecht (1989).
Vasershtein, L.N., Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii 5 (1969) 64-73.
C. Villani, Contribution à l'étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas. Ph.D. thesis, Univ. Paris-Dauphine (1998).
Villani, C., On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 (1998) 273-307. CrossRef
C. Villani, Contribution à l'étude mathématique des collisions en théorie cinétique. Ceremade, Paris IX-Dauphine, January 24 (2000).
Zolotarev, V.M., Probability Metrics. Theory Probab. Appl. 28 (1983) 278-302. CrossRef