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March  2011, 4(1): 333-344. doi: 10.3934/krm.2011.4.333

On the Kac model for the Landau equation

Evelyne Miot 1, , Mario Pulvirenti 2, and  Chiara Saffirio 2,

1. 

Laboratoire de Mathématiques, Université Paris-Sud 11, bât. 425, 91405 Orsay, France
2. 

Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza - Roma, P.le A. Moro, 5 00185 Roma, Italy, Italy

Received  October 2010 Revised  October 2010 Published  January 2011

We introduce a $N$-particle system which approximates, in the mean-field limit, the solutions of the Landau equation with Coulomb singularity. This model plays the same role as the Kac's model for the homogeneous Boltzmann equation. We use compactness arguments following [11].
Keywords: propagation of chaos., Kac's model for the Landau equation, N-particle system, N-particle hierarchy.
Mathematics Subject Classification: 35D30, 35Q20, 35Q70, 35Q8.
Citation: Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic & Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333
References:
[1]

A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, (Russian), Mat. Sb., 181 (1992), 435-446; translation in Math. USSR-Sb., 69 (1991), 465-478.  Google Scholar

[2]

R. Balescu, "Equilibrium and Nonequilibrium Statistical Mechanics,'' John Wiley & Sons, New-York, 1975.  Google Scholar

[3]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298. doi: 10.1080/03605300008821513.  Google Scholar

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259. doi: 10.1080/03605300008821512.  Google Scholar

[5]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543.  Google Scholar

[6]

M. Kac, Foundations of kinetic theory, in "Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability," University of California Press, Berkeley and Los Angeles (1956).  Google Scholar

[7]

A. I. Khinchin, "Mathematical Foundations of Information Theory," New York: Dover, 1957.  Google Scholar

[8]

L. P. Pitaevskii and E. M. Lifshitz, "Course of Theoretical Physics. Vol. 10," Pergamon Press, Oxford-Elmsford, N.Y. 1981.  Google Scholar

[9]

R. Peyre, Some ideas about quantitative convergence of collision models to their mean field limit, J. Stat. Phys., 136 (2009), 1105-1130. doi: 10.1007/s10955-009-9820-3.  Google Scholar

[10]

M. Pulvirenti, The weak-coupling limit of large classical and quantum systems, in "International Congress of Mathematicians," Eur. Math. Soc., Zürich (2006).  Google Scholar

[11]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

show all references

References:
[1]

A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, (Russian), Mat. Sb., 181 (1992), 435-446; translation in Math. USSR-Sb., 69 (1991), 465-478.  Google Scholar

[2]

R. Balescu, "Equilibrium and Nonequilibrium Statistical Mechanics,'' John Wiley & Sons, New-York, 1975.  Google Scholar

[3]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298. doi: 10.1080/03605300008821513.  Google Scholar

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259. doi: 10.1080/03605300008821512.  Google Scholar

[5]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543.  Google Scholar

[6]

M. Kac, Foundations of kinetic theory, in "Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability," University of California Press, Berkeley and Los Angeles (1956).  Google Scholar

[7]

A. I. Khinchin, "Mathematical Foundations of Information Theory," New York: Dover, 1957.  Google Scholar

[8]

L. P. Pitaevskii and E. M. Lifshitz, "Course of Theoretical Physics. Vol. 10," Pergamon Press, Oxford-Elmsford, N.Y. 1981.  Google Scholar

[9]

R. Peyre, Some ideas about quantitative convergence of collision models to their mean field limit, J. Stat. Phys., 136 (2009), 1105-1130. doi: 10.1007/s10955-009-9820-3.  Google Scholar

[10]

M. Pulvirenti, The weak-coupling limit of large classical and quantum systems, in "International Congress of Mathematicians," Eur. Math. Soc., Zürich (2006).  Google Scholar

[11]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

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