2013, 6(4): 789-800. doi: 10.3934/krm.2013.6.789

Transport coefficients in the $2$-dimensional Boltzmann equation

1. 

Department of Mathematics, Karlstad University, SE-651 88 Karlstad

2. 

International Research Center M&MOCS, Università di L'Aquila, Cisterna di Latina, 04012

Received  July 2013 Revised  August 2013 Published  November 2013

We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the $2$-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.
Citation: Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789
References:
[1]

G. Basile, C. Bernardin and S. Olla, Thermal conductivity for a momentum conserving model,, Commun. in Mathematical Physics, 287 (2009), 67. doi: 10.1007/s00220-008-0662-7.

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).

[3]

R. Esposito and M. Pulvirenti, Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas,, Kinetic and Related Models, 3 (2010), 281. doi: 10.3934/krm.2010.3.281.

[4]

D.T. Morelli, J. Heremans, M. Sakamoto and C. Uher, Anisotropic heat conduction in diacetylenes,, Phys. Rev. Lett., 57 (1986), 869. doi: 10.1103/PhysRevLett.57.869.

[5]

O. E. Lanford III, The evolution of Large Classical systems,, in Dynamical Systems, 38 (1975), 1.

[6]

J. C. Maxwell, On the dynamical theory of gases,, Phil. Trans. Royal Soc. London, 157 (1867), 49.

[7]

A. Smontara, J. C. Lasjaunas and R. Maynard, Phonon poiseuille flow in quasi-one-dimensional single crystals,, Phys. Rev. Lett., 77 (1996), 5397. doi: 10.1103/PhysRevLett.77.5397.

[8]

A. V. Sologubenko, K. Giann H. R. Ott, A. Vietkine and A. Revcolevschi, Heat transport by lattice and spin excitations in the spin-chain compounds $SrCuO_2$ and $Sr_2CuO_3$,, Phys. Rev. B, 64 (2001).

[9]

S.Ukai, On the spectrum of the space-independent Boltzmann operator,, J. Nuclear Energy, 19 (1965), 833.

[10]

S.Ukai, On the existence of global solution of mixed problem for non-linear Boltzmann equation,, Proc. Japan. Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.

[11]

V. S. Vladimirov, Equations of Mathematical Physics,, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, (1971).

show all references

References:
[1]

G. Basile, C. Bernardin and S. Olla, Thermal conductivity for a momentum conserving model,, Commun. in Mathematical Physics, 287 (2009), 67. doi: 10.1007/s00220-008-0662-7.

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).

[3]

R. Esposito and M. Pulvirenti, Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas,, Kinetic and Related Models, 3 (2010), 281. doi: 10.3934/krm.2010.3.281.

[4]

D.T. Morelli, J. Heremans, M. Sakamoto and C. Uher, Anisotropic heat conduction in diacetylenes,, Phys. Rev. Lett., 57 (1986), 869. doi: 10.1103/PhysRevLett.57.869.

[5]

O. E. Lanford III, The evolution of Large Classical systems,, in Dynamical Systems, 38 (1975), 1.

[6]

J. C. Maxwell, On the dynamical theory of gases,, Phil. Trans. Royal Soc. London, 157 (1867), 49.

[7]

A. Smontara, J. C. Lasjaunas and R. Maynard, Phonon poiseuille flow in quasi-one-dimensional single crystals,, Phys. Rev. Lett., 77 (1996), 5397. doi: 10.1103/PhysRevLett.77.5397.

[8]

A. V. Sologubenko, K. Giann H. R. Ott, A. Vietkine and A. Revcolevschi, Heat transport by lattice and spin excitations in the spin-chain compounds $SrCuO_2$ and $Sr_2CuO_3$,, Phys. Rev. B, 64 (2001).

[9]

S.Ukai, On the spectrum of the space-independent Boltzmann operator,, J. Nuclear Energy, 19 (1965), 833.

[10]

S.Ukai, On the existence of global solution of mixed problem for non-linear Boltzmann equation,, Proc. Japan. Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.

[11]

V. S. Vladimirov, Equations of Mathematical Physics,, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, (1971).

[1]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[2]

Taposh Kumar Das, Óscar López Pouso. New insights into the numerical solution of the Boltzmann transport equation for photons. Kinetic & Related Models, 2014, 7 (3) : 433-461. doi: 10.3934/krm.2014.7.433

[3]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[4]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145

[5]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[6]

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

[7]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

[8]

Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237

[9]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551

[10]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

[11]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17

[12]

Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083

[13]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[14]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[15]

Carlo Cercignani. The Boltzmann equation in the 20th century. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 83-94. doi: 10.3934/dcds.2009.24.83

[16]

Valeriano Comincioli, Lucia Della Croce, Giuseppe Toscani. A Boltzmann-like equation for choice formation. Kinetic & Related Models, 2009, 2 (1) : 135-149. doi: 10.3934/krm.2009.2.135

[17]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[18]

Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435

[19]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[20]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]