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December  2011, 4(4): 873-900. doi: 10.3934/krm.2011.4.873

Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach

Naoufel Ben Abdallah 1, , Antoine Mellet 2, and  Marjolaine Puel 3,

1. 

Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9, France
2. 

Department of Mathematics, University of Maryland, College Park, MD 20742
3. 

Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France

Received  July 2010 Published  November 2011

We develop a Hilbert expansion approach for the derivation of fractional diffusion equations from the linear Boltzmann equation with heavy tail equilibria.
Keywords: fractional diffusion, Linear Boltzmann equation, anomalous diffusion limit..
Mathematics Subject Classification: 76P05, 26A33, 22E46, 53C35, 57S2.
Citation: Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic & Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873
References:
[1]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. A. M. S., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[2]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567.  Google Scholar

[3]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency,, Math. Models Methods Appl. Sci., ().   Google Scholar

[4]

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: 10.2977/prims/1195188427.  Google Scholar

[5]

A. V. Bobylev, J. A. Carrillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys., 98 (2000), 743-773. doi: 10.1023/A:1018627625800.  Google Scholar

[6]

A. V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: Exact solutions and power like tails, J. Stat. Phys., 124 (2006), 497-516. doi: 10.1007/s10955-006-9044-8.  Google Scholar

[7]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversibles processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.  Google Scholar

[8]

M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys., 109 (2002), 407-432. doi: 10.1023/A:1020437925931.  Google Scholar

[9]

T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation, Journal Diff. Equations, 189 (2003), 17-45. doi: 10.1016/S0022-0396(02)00096-7.  Google Scholar

[10]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81. doi: 10.1063/1.1666510.  Google Scholar

[11]

N. Masmoudi and M.-L. Tayeb, On the diffusion limit of a semiconductor Boltzmann-Poisson system without micro-reversible process, Comm. Partial Differential Equations, 35 (2010), 1163-1175.  Google Scholar

[12]

A. Mellet, Diffusion limit of a nonlinear kinetic model without the detailed balance principle, Monatshefte f. Mathematik, 134 (2002), 305-329. doi: 10.1007/s605-002-8265-1.  Google Scholar

[13]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[14]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.  Google Scholar

[15]

D. A. Mendis and M. Rosenberg, Cosmic dusty plasma, Annu. Rev. Astron. Astrophys., 32 (1994), 419-463. doi: 10.1146/annurev.aa.32.090194.002223.  Google Scholar

[16]

A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model, J. Statist. Phys., 114 (2004), 1453-1480. doi: 10.1023/B:JOSS.0000013964.98706.00.  Google Scholar

[17]

D. Summers and R. M. Thorne, The modified plasma dispersion function, Phys. Fluids, 83 (1991), 1835-1847. Google Scholar

show all references

References:
[1]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. A. M. S., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[2]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567.  Google Scholar

[3]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency,, Math. Models Methods Appl. Sci., ().   Google Scholar

[4]

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: 10.2977/prims/1195188427.  Google Scholar

[5]

A. V. Bobylev, J. A. Carrillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys., 98 (2000), 743-773. doi: 10.1023/A:1018627625800.  Google Scholar

[6]

A. V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: Exact solutions and power like tails, J. Stat. Phys., 124 (2006), 497-516. doi: 10.1007/s10955-006-9044-8.  Google Scholar

[7]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversibles processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.  Google Scholar

[8]

M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys., 109 (2002), 407-432. doi: 10.1023/A:1020437925931.  Google Scholar

[9]

T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation, Journal Diff. Equations, 189 (2003), 17-45. doi: 10.1016/S0022-0396(02)00096-7.  Google Scholar

[10]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81. doi: 10.1063/1.1666510.  Google Scholar

[11]

N. Masmoudi and M.-L. Tayeb, On the diffusion limit of a semiconductor Boltzmann-Poisson system without micro-reversible process, Comm. Partial Differential Equations, 35 (2010), 1163-1175.  Google Scholar

[12]

A. Mellet, Diffusion limit of a nonlinear kinetic model without the detailed balance principle, Monatshefte f. Mathematik, 134 (2002), 305-329. doi: 10.1007/s605-002-8265-1.  Google Scholar

[13]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[14]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.  Google Scholar

[15]

D. A. Mendis and M. Rosenberg, Cosmic dusty plasma, Annu. Rev. Astron. Astrophys., 32 (1994), 419-463. doi: 10.1146/annurev.aa.32.090194.002223.  Google Scholar

[16]

A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model, J. Statist. Phys., 114 (2004), 1453-1480. doi: 10.1023/B:JOSS.0000013964.98706.00.  Google Scholar

[17]

D. Summers and R. M. Thorne, The modified plasma dispersion function, Phys. Fluids, 83 (1991), 1835-1847. Google Scholar

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