Validity of the Boltzmann equation with an external force

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June 2011, 4(2): 499-515. doi: 10.3934/krm.2011.4.499

Validity of the Boltzmann equation with an external force

Raffaele Esposito ^1, , Yan Guo ^2, and Rossana Marra ^3,

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

Received December 2010 Published April 2011

Abstract
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We establish local-in-time validity of the Boltzmann equation in the presence of an external force deriving from a $C^2$ potential.

Keywords: Rarified gases, derivation, external force., Boltzmann equation.

Mathematics Subject Classification: Primary: 76P05, 82C40; Secondary: 76A2.

Citation: Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic and Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499

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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