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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 |
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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