American Institute of Mathematical Sciences

Advanced
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. 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. 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. 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. Google Scholar

[5]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations,, Commun. in Math. Phys., 123 (1989), 687. 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. 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. 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. 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. 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. 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], 57 (1978), 203. Google Scholar

[12]

C. Villani, Cercignani's conjecture is sometimes true and almost always true,, Commun. Math. Phys., 234 (2003), 455. doi: 10.1007/s00220-002-0777-1. Google Scholar

[13]

C. Villani, "Hypocoercivity,'', Memoirs of the Amer. Math. Soc., 202 (2009). Google Scholar

[14]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation,, J. Stat. Phys., 86 (1997), 1053. 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. 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. 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. 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. 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. Google Scholar

[5]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations,, Commun. in Math. Phys., 123 (1989), 687. 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. 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. 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. 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. 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. 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], 57 (1978), 203. Google Scholar

[12]

C. Villani, Cercignani's conjecture is sometimes true and almost always true,, Commun. Math. Phys., 234 (2003), 455. doi: 10.1007/s00220-002-0777-1. Google Scholar

[13]

C. Villani, "Hypocoercivity,'', Memoirs of the Amer. Math. Soc., 202 (2009). Google Scholar

[14]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation,, J. Stat. Phys., 86 (1997), 1053. 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. doi: 10.1007/s00205-008-0139-z. Google Scholar

[1]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004
[2]

Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies, Dirk Roose. Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 855-874. doi: 10.3934/dcdsb.2009.11.855
[3]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317
[4]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[5]

Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049
[6]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
[7]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
[8]

Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020
[9]

Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic & Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333
[10]

Katherine A. Bold, Karthikeyan Rajendran, Balázs Ráth, Ioannis G. Kevrekidis. An equation-free approach to coarse-graining the dynamics of networks. Journal of Computational Dynamics, 2014, 1 (1) : 111-134. doi: 10.3934/jcd.2014.1.111
[11]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[12]

Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065
[13]

Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
[14]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[15]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[16]

P.K. Newton. N-vortex equilibrium theory. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 411-418. doi: 10.3934/dcds.2007.19.411
[17]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[18]

Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007
[19]

Pierre Monmarché. Hypocoercive relaxation to equilibrium for some kinetic models. Kinetic & Related Models, 2014, 7 (2) : 341-360. doi: 10.3934/krm.2014.7.341
[20]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences