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

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

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.
Citation: Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

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