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Validity of the Boltzmann equation with an external force
1. | Dipartimento di Matematica pura ed Applicata, Università dell’Aquila, Via Vetoio - Coppito, L’Aquila, 67100 |
2. | Division of Applied Mathematics, Brown University, Providence, RI 02812, United States |
3. | Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma |
References:
[1] |
L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Rat. Mech. Anal., 198 (2010), 125-187.
doi: 10.1007/s00205-010-0292-z. |
[2] |
S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction I: Derivation of kinetic and hydrodynamic equation, Jour. Statist. Phys., 101 (2000), 1087-1136.
doi: 10.1023/A:1026481706240. |
[3] |
A. V. Bobylev, A. Hansen, J. Piasecki and E. H. Hauge, From the Liouville equation to the generalized Boltzmann equation for magnetotransport in the 2D Lorentz model, Jour. Statist. Phys., 102 (2001), 1133-1150.
doi: 10.1023/A:1004880010020. |
[4] |
E. A. Carlen, M. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity, 16 (2003), 1075-1105.
doi: 10.1088/0951-7715/16/3/316. |
[5] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, 1994. |
[6] |
L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field, Bull. Sci. Math., 128 (2004), 39-46.
doi: 10.1016/j.bulsci.2003.09.003. |
[7] |
R. Duan, T. Yang and C. Zhu, Global existence to Boltzmann equation with external force in infinite vacuum, Journal of Mathematical Physics, 46 (2005), 253-277.
doi: 10.1063/1.1899985. |
[8] |
R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Communications in Mathematical Physics, 277 (2008), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[9] |
R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture, Commun. Math. Phys., 296 (2010), 1-33.
doi: 10.1007/s00220-010-1009-8. |
[10] |
R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq regime, Jour. Stat. Phys., 90 (1998), 1129-1178.
doi: 10.1023/A:1023223226585. |
[11] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.
doi: 10.1007/s002200100391. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., LV (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo, The Vlasov-Boltzmann-Maxwell system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[14] |
O. E. Lanford III, The evolution of large classical systems, in "Dynamical Systems, Theory and Applications,'' J. Moser ed., Lecture Notes in Physics, Springer Berlin, 35 (1975), 1-111. |
[15] |
S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Analysis and Applications (Singapore), 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
show all references
References:
[1] |
L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Rat. Mech. Anal., 198 (2010), 125-187.
doi: 10.1007/s00205-010-0292-z. |
[2] |
S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction I: Derivation of kinetic and hydrodynamic equation, Jour. Statist. Phys., 101 (2000), 1087-1136.
doi: 10.1023/A:1026481706240. |
[3] |
A. V. Bobylev, A. Hansen, J. Piasecki and E. H. Hauge, From the Liouville equation to the generalized Boltzmann equation for magnetotransport in the 2D Lorentz model, Jour. Statist. Phys., 102 (2001), 1133-1150.
doi: 10.1023/A:1004880010020. |
[4] |
E. A. Carlen, M. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity, 16 (2003), 1075-1105.
doi: 10.1088/0951-7715/16/3/316. |
[5] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, 1994. |
[6] |
L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field, Bull. Sci. Math., 128 (2004), 39-46.
doi: 10.1016/j.bulsci.2003.09.003. |
[7] |
R. Duan, T. Yang and C. Zhu, Global existence to Boltzmann equation with external force in infinite vacuum, Journal of Mathematical Physics, 46 (2005), 253-277.
doi: 10.1063/1.1899985. |
[8] |
R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Communications in Mathematical Physics, 277 (2008), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[9] |
R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture, Commun. Math. Phys., 296 (2010), 1-33.
doi: 10.1007/s00220-010-1009-8. |
[10] |
R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq regime, Jour. Stat. Phys., 90 (1998), 1129-1178.
doi: 10.1023/A:1023223226585. |
[11] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.
doi: 10.1007/s002200100391. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., LV (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo, The Vlasov-Boltzmann-Maxwell system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[14] |
O. E. Lanford III, The evolution of large classical systems, in "Dynamical Systems, Theory and Applications,'' J. Moser ed., Lecture Notes in Physics, Springer Berlin, 35 (1975), 1-111. |
[15] |
S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Analysis and Applications (Singapore), 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
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