Diffusion limit of Fokker−Planck equation with heavy tail equilibria
Nasreddine, Elissar1 ; Puel, Marjolaine2
1 Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse cedex 9, France.
2 Laboratoire Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice cedex 2, France.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 1-17.

This paper is devoted to the diffusion limit of the Fokker−Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We prove that for an appropriate time scale, in a suitable weighted Sobolev space, the small mean free path limit gives rise to a diffusion equation.

DOI : 10.1051/m2an/2014020
Classification : 60J60, 60J70, 35Q84
Mots-clés : Fokker Plank, diffusion limit, heavy tail, Cauchy distribution
Nasreddine, Elissar 1 ; Puel, Marjolaine 2

1 Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse cedex 9, France.
2 Laboratoire Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice cedex 2, France. 
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Nasreddine, Elissar; Puel, Marjolaine. Diffusion limit of Fokker−Planck equation with heavy tail equilibria. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 1-17. doi : 10.1051/m2an/2014020. http://archive.numdam.org/articles/10.1051/m2an/2014020/

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size. Numerical Solutions of Nonlinear Problems. INRIA, Rocquencourt (1984). | MR | Zbl

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency. M3AS 21 (2011) 2249–2262. | MR | Zbl

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach. KRM 4 (2011) 873–900. | DOI | MR | Zbl

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci. 15 (1979) 53–157. | DOI | MR | Zbl

P. Cattiaux, E. Nasreddine and M. Puel, Diffusion limit of Fokker−Planck equation with heavy tails equilibria: a probabilistic approach including anomalous rate (preprint). | MR

L. Cesbron, A. Mellet and K. Trivisa, Anomalous transport of particles in plasma physics. Appl. Math. Lett. 25 (2012) 2344–2348. | DOI | MR | Zbl

P. Degond, Global existence of smooth solutions for the Vlasov−Fokker−Planck equation in one and two spaces dimensions, vol. 19 of Ann. Sci. E.N.S 4e Ser. (1986) 519–542. | Numdam | MR | Zbl

P. Degond. Macroscopic limits of the Boltzmann equation: a review. Modeling and computational methods for kinetic equations, vol. 357 of Model. Simul. Sci. Eng. Technol. Birkhauser Boston, Boston, MA (2004). | MR | Zbl

P. Degond and P. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker−Planck equation. Proc. of the conference on mathematical methods applied to kinetic equations, Paris, 1985. Transp. Theory Stat. Phys. 16 (1987) 589–636. | MR | Zbl

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49 (2000) 1175–1198. | MR | Zbl

P. Garbaczewski, V. Stephanovich and D. Kedzierski, Heavy-tailed targets and (ab)normal asymptotics in diffusive motion. Physica A 390 (2011) 990–1008. | DOI

E. Larsen and J. Keller. Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15 (1974) 75–81. | DOI | MR

J.L. Lions, Equations différentielles opérationnelles et problèmes aux limites. Vol. 9. Springer, Berlin 1961 (2003) 371–398. | MR | Zbl

E. Lutz, Anomalous diffusion and Tsallis statistics in an optical lattice. Phys. Rev. A 67 (2003) 051402. | DOI

E. Lutz, Power-law tail distributions and nonergodicity. Phys. Rev. Lett. 93 (2004) 190602. | DOI

A. Mellet, Fractional diffusion limit for collisional kinetic equations: a moments method. Indiana Univ. Math. J. 59 (2010) 1333–1360. | DOI | MR | Zbl

A. Mellet, S. Mishler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199 (2011) 493–525. | DOI | MR | Zbl

P.L. Lions and G. Toscani. Diffusive limits for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoamer. 13 (1997) 473–513. | DOI | MR | Zbl

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