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).

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