On the speed of approach to equilibrium for a collisionless gas

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March 2011, 4(1): 87-107. doi: 10.3934/krm.2011.4.87

On the speed of approach to equilibrium for a collisionless gas

Kazuo Aoki ^1, and François Golse ^2,

1.	Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Kyoto 606-8501
2.	Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz, 91128 Palaiseau Cedex, France

Received September 2010 Revised November 2010 Published January 2011

Abstract
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We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform $L^p$ bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall.

Keywords: diffuse reflection; kinetic theory of gases, Free-molecular gas, renewal equation., approach to equilibrium.

Mathematics Subject Classification: Primary: 82C40; Secondary: 35F10, 35B4.

Citation: Kazuo Aoki, François Golse. On the speed of approach to equilibrium for a collisionless gas. Kinetic and Related Models, 2011, 4 (1) : 87-107. doi: 10.3934/krm.2011.4.87

References:

[1]	L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764.
[2]	A. V. Bobylev, The theory of the nonlinear, spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Rev. C. Math. Phys., 7 (1988), 111-233
[3]	A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618. doi: 10.1023/A:1004537522686.
[4]	C. Cercignani, H-Theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.
[5]	L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Commun. in Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.
[6]	L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Math. Sci., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.
[7]	L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.
[8]	W. Feller, On the integral equation of renewal theory, Ann. Math. Stat., 12 (1941), 243-267. doi: 10.1214/aoms/1177731708.
[9]	Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. for Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.
[10]	T. Tsuji, K. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, J. Stat. Phys., 140 (2010), 518-543. doi: 10.1007/s10955-010-9997-5.
[11]	S. Ukai, N. Point and H. Ghidouche, Sur la solution globale du problème mixte de l'équation de Boltzmann non linéaire, (French)[On the global solution of the mixed problem for the nonlinear Boltzmann equation], J. Math. Pures Appl. (9), 57 (1978), 203-229.
[12]	C. Villani, Cercignani's conjecture is sometimes true and almost always true, Commun. Math. Phys., 234 (2003), 455-490. doi: 10.1007/s00220-002-0777-1.
[13]	C. Villani, "Hypocoercivity,'' Memoirs of the Amer. Math. Soc., 202 (2009), no. 950.
[14]	B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066. doi: 10.1007/BF02183613.
[15]	S.-H. Yu, Stochastic formulation for the initial boundary value problems of the Boltzmann equation, Arch. for Ration. Mech. Anal., 192 (2009), 217-274. doi: 10.1007/s00205-008-0139-z.

show all references

References:

[1]	L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764.
[2]	A. V. Bobylev, The theory of the nonlinear, spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Rev. C. Math. Phys., 7 (1988), 111-233
[3]	A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618. doi: 10.1023/A:1004537522686.
[4]	C. Cercignani, H-Theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.
[5]	L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Commun. in Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.
[6]	L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Math. Sci., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.
[7]	L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.
[8]	W. Feller, On the integral equation of renewal theory, Ann. Math. Stat., 12 (1941), 243-267. doi: 10.1214/aoms/1177731708.
[9]	Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. for Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.
[10]	T. Tsuji, K. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, J. Stat. Phys., 140 (2010), 518-543. doi: 10.1007/s10955-010-9997-5.
[11]	S. Ukai, N. Point and H. Ghidouche, Sur la solution globale du problème mixte de l'équation de Boltzmann non linéaire, (French)[On the global solution of the mixed problem for the nonlinear Boltzmann equation], J. Math. Pures Appl. (9), 57 (1978), 203-229.
[12]	C. Villani, Cercignani's conjecture is sometimes true and almost always true, Commun. Math. Phys., 234 (2003), 455-490. doi: 10.1007/s00220-002-0777-1.
[13]	C. Villani, "Hypocoercivity,'' Memoirs of the Amer. Math. Soc., 202 (2009), no. 950.
[14]	B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066. doi: 10.1007/BF02183613.
[15]	S.-H. Yu, Stochastic formulation for the initial boundary value problems of the Boltzmann equation, Arch. for Ration. Mech. Anal., 192 (2009), 217-274. doi: 10.1007/s00205-008-0139-z.

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