American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[4]

C. Cercignani, H-Theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.  Google Scholar

[5]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Commun. in Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Feller, On the integral equation of renewal theory, Ann. Math. Stat., 12 (1941), 243-267. doi: 10.1214/aoms/1177731708.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. Villani, "Hypocoercivity,'' Memoirs of the Amer. Math. Soc., 202 (2009), no. 950.  Google Scholar

[14]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066. doi: 10.1007/BF02183613.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[4]

C. Cercignani, H-Theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.  Google Scholar

[5]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Commun. in Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Feller, On the integral equation of renewal theory, Ann. Math. Stat., 12 (1941), 243-267. doi: 10.1214/aoms/1177731708.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. Villani, "Hypocoercivity,'' Memoirs of the Amer. Math. Soc., 202 (2009), no. 950.  Google Scholar

[14]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066. doi: 10.1007/BF02183613.  Google Scholar

[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.  Google Scholar

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