December  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), 67-98. doi: 10.1007/s00220-008-0662-7.  Google Scholar

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994.  Google Scholar

[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), 281-297. doi: 10.3934/krm.2010.3.281.  Google Scholar

[4]

D.T. Morelli, J. Heremans, M. Sakamoto and C. Uher, Anisotropic heat conduction in diacetylenes, Phys. Rev. Lett., 57 (1986), 869-872. doi: 10.1103/PhysRevLett.57.869.  Google Scholar

[5]

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, 38 (1975), 1-111.  Google Scholar

[6]

J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Royal Soc. London, 157 (1867), 49-88. Google Scholar

[7]

A. Smontara, J. C. Lasjaunas and R. Maynard, Phonon poiseuille flow in quasi-one-dimensional single crystals, Phys. Rev. Lett., 77 (1996), 5397-5400. doi: 10.1103/PhysRevLett.77.5397.  Google Scholar

[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), 054412. Google Scholar

[9]

S.Ukai, On the spectrum of the space-independent Boltzmann operator, J. Nuclear Energy, Parts A/B, 19 (1965), 833-848. Google Scholar

[10]

S.Ukai, On the existence of global solution of mixed problem for non-linear Boltzmann equation, Proc. Japan. Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.  Google Scholar

[11]

V. S. Vladimirov, Equations of Mathematical Physics, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, 3 Marcel Dekker, Inc., New York 1971.  Google Scholar

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), 67-98. doi: 10.1007/s00220-008-0662-7.  Google Scholar

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994.  Google Scholar

[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), 281-297. doi: 10.3934/krm.2010.3.281.  Google Scholar

[4]

D.T. Morelli, J. Heremans, M. Sakamoto and C. Uher, Anisotropic heat conduction in diacetylenes, Phys. Rev. Lett., 57 (1986), 869-872. doi: 10.1103/PhysRevLett.57.869.  Google Scholar

[5]

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, 38 (1975), 1-111.  Google Scholar

[6]

J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Royal Soc. London, 157 (1867), 49-88. Google Scholar

[7]

A. Smontara, J. C. Lasjaunas and R. Maynard, Phonon poiseuille flow in quasi-one-dimensional single crystals, Phys. Rev. Lett., 77 (1996), 5397-5400. doi: 10.1103/PhysRevLett.77.5397.  Google Scholar

[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), 054412. Google Scholar

[9]

S.Ukai, On the spectrum of the space-independent Boltzmann operator, J. Nuclear Energy, Parts A/B, 19 (1965), 833-848. Google Scholar

[10]

S.Ukai, On the existence of global solution of mixed problem for non-linear Boltzmann equation, Proc. Japan. Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.  Google Scholar

[11]

V. S. Vladimirov, Equations of Mathematical Physics, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, 3 Marcel Dekker, Inc., New York 1971.  Google Scholar

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