# American Institute of Mathematical Sciences

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

## On the speed of approach to equilibrium for a collisionless gas

 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.
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.
 [1] Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic and 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 and 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 and 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 and 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 and 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 and 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 and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003 [8] Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003 [9] Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic and Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020 [10] Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic and Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333 [11] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [12] 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 [13] Dong Li. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2453-2460. doi: 10.3934/dcds.2021198 [14] Vladimir Djordjić, Milana Pavić-Čolić, Nikola Spasojević. Polytropic gas modelling at kinetic and macroscopic levels. Kinetic and Related Models, 2021, 14 (3) : 483-522. doi: 10.3934/krm.2021013 [15] Chun Liu, Jan-Eric Sulzbach. The Brinkman-Fourier system with ideal gas equilibrium. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 425-462. doi: 10.3934/dcds.2021123 [16] Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic and Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020 [17] Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429 [18] Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic and Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621 [19] P.K. Newton. N-vortex equilibrium theory. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 411-418. doi: 10.3934/dcds.2007.19.411 [20] Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems and Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713

2020 Impact Factor: 1.432

## Metrics

• HTML views (0)
• Cited by (14)

• on AIMS