December  2020, 13(6): 1071-1106. doi: 10.3934/krm.2020038

A semigroup approach to the convergence rate of a collisionless gas

Sorbonne Université, CNRS, Laboratoire de Probabilité, Statistique et Modélisation, F-75005 Paris, France

Received  November 2019 Revised  June 2020 Published  December 2020 Early access  September 2020

We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension $ n \in \{2,3\} $. By semigroup arguments, we prove that in the $ L^1 $ norm, the polynomial rate of convergence $ \frac{1}{(t+1)^{n-}} $ given in [25], [17] and [18] can be extended to any $ C^2 $ domain, with standard assumptions on the initial data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependency of the rate with respect to the initial distribution is detailed. Our study includes the case where the temperature at the boundary varies. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [7]. We also compare our model with a free-transport equation with absorbing boundary.

Citation: Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic & Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038
References:
[1]

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

[2]

L. Arkeryd and C. Cercignani, A global existence theorem for the initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal, Archive for Rational Mechanics and Analysis, 125 (1993), 271-287.  doi: 10.1007/BF00383222.  Google Scholar

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L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatshefte für Mathematik, 123 (1997), 285–298. doi: 10.1007/BF01326764.  Google Scholar

[4]

J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Comprehensive studies in mathematics. Springer Berlin Heidelberg, 1976.  Google Scholar

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A. Bernou and N. Fournier, A coupling approach for the convergence to equilibrium for a collisionless gas, arXiv e-prints, page arXiv: 1910.02739, Oct. 2019. Google Scholar

[6]

L. Boltzmann, Lectures on Gas Theory, University of California Press, Berkeley-Los Angeles, Calif. 1964.  Google Scholar

[7]

J. A. Cañizo and S. Mischler, Doeblin-Harris theory for stochastic operators and semigroups, In preparation, 2019. Google Scholar

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, New York : Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[9]

R. Douc, G. Fort and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Processes and their Applications, 119 (2009), 897 – 923. doi: 10.1016/j.spa.2008.03.007.  Google Scholar

[10]

R. EspositoY. GuoC. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Communications in Mathematical Physics, 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2.  Google Scholar

[11]

S. N. Evans, Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.  doi: 10.1214/aoap/1015345298.  Google Scholar

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C. Goulaouic, Prolongements de foncteurs d'interpolation et applications, Annales de l'Institut Fourier, 18 (1968), 1-98.  doi: 10.5802/aif.277.  Google Scholar

[13]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana University Mathematics Journal, 43 (1994), 255-320.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[14]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Archive for Rational Mechanics and Analysis, 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[15]

M. Hairer, Convergence of Markov processes, Lecture Notes available at http://www.hairer.org/notes/Convergence.pdf, 2016. Google Scholar

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S. Janson, Interpolation of subcouples and quotient couples, Ark. Mat., 31 (1993), 307-338.  doi: 10.1007/BF02559489.  Google Scholar

[17]

H-W. KuoT-P. Liu and L-C. Tsai, Free molecular flow with boundary effect, Communications in Mathematical Physics, 318 (2013), 375-409.  doi: 10.1007/s00220-013-1662-9.  Google Scholar

[18]

H-W. KuoT-P. Liu and L-C. Tsai, Equilibrating Effects of Boundary and Collision in Rarefied Gases, Communications in Mathematical Physics, 328 (2014), 421-480.  doi: 10.1007/s00220-014-2042-9.  Google Scholar

[19]

H-W. Kuo, Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas, Journal of Statistical Physics, 161 (2015), 743-800.  doi: 10.1007/s10955-015-1355-1.  Google Scholar

[20]

J. C. Maxwell, Ⅳ. On the dynamical theory of gases, Philosophical Transactions of the Royal Society of London, 157 (1867), 49-88.  doi: 10.1098/rstl.1867.0004.  Google Scholar

[21]

S. Mischler, On the trace problem for solutions of the Vlasov equation, Communications in Partial Differential Equations, 25 (1999), 1415-1443.  doi: 10.1080/03605300008821554.  Google Scholar

[22]

M. Mokhtar-Kharroubi and D. Seifert, Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry, Journal of Functional Analysis, 275 (2018), 2404-2452.  doi: 10.1016/j.jfa.2018.08.005.  Google Scholar

[23]

J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de matemática. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, 1968.  Google Scholar

[24]

Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, 1st ed edition, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[25]

T. TsujiK. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, Journal of Statistical Physics, 140 (2010), 518-543.  doi: 10.1007/s10955-010-9997-5.  Google Scholar

[26]

C. Villani., Hypocoercivity., American Mathematical Society, 2009. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

show all references

References:
[1]

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

[2]

L. Arkeryd and C. Cercignani, A global existence theorem for the initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal, Archive for Rational Mechanics and Analysis, 125 (1993), 271-287.  doi: 10.1007/BF00383222.  Google Scholar

[3]

L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatshefte für Mathematik, 123 (1997), 285–298. doi: 10.1007/BF01326764.  Google Scholar

[4]

J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Comprehensive studies in mathematics. Springer Berlin Heidelberg, 1976.  Google Scholar

[5]

A. Bernou and N. Fournier, A coupling approach for the convergence to equilibrium for a collisionless gas, arXiv e-prints, page arXiv: 1910.02739, Oct. 2019. Google Scholar

[6]

L. Boltzmann, Lectures on Gas Theory, University of California Press, Berkeley-Los Angeles, Calif. 1964.  Google Scholar

[7]

J. A. Cañizo and S. Mischler, Doeblin-Harris theory for stochastic operators and semigroups, In preparation, 2019. Google Scholar

[8]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, New York : Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[9]

R. Douc, G. Fort and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Processes and their Applications, 119 (2009), 897 – 923. doi: 10.1016/j.spa.2008.03.007.  Google Scholar

[10]

R. EspositoY. GuoC. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Communications in Mathematical Physics, 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2.  Google Scholar

[11]

S. N. Evans, Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.  doi: 10.1214/aoap/1015345298.  Google Scholar

[12]

C. Goulaouic, Prolongements de foncteurs d'interpolation et applications, Annales de l'Institut Fourier, 18 (1968), 1-98.  doi: 10.5802/aif.277.  Google Scholar

[13]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana University Mathematics Journal, 43 (1994), 255-320.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[14]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Archive for Rational Mechanics and Analysis, 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[15]

M. Hairer, Convergence of Markov processes, Lecture Notes available at http://www.hairer.org/notes/Convergence.pdf, 2016. Google Scholar

[16]

S. Janson, Interpolation of subcouples and quotient couples, Ark. Mat., 31 (1993), 307-338.  doi: 10.1007/BF02559489.  Google Scholar

[17]

H-W. KuoT-P. Liu and L-C. Tsai, Free molecular flow with boundary effect, Communications in Mathematical Physics, 318 (2013), 375-409.  doi: 10.1007/s00220-013-1662-9.  Google Scholar

[18]

H-W. KuoT-P. Liu and L-C. Tsai, Equilibrating Effects of Boundary and Collision in Rarefied Gases, Communications in Mathematical Physics, 328 (2014), 421-480.  doi: 10.1007/s00220-014-2042-9.  Google Scholar

[19]

H-W. Kuo, Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas, Journal of Statistical Physics, 161 (2015), 743-800.  doi: 10.1007/s10955-015-1355-1.  Google Scholar

[20]

J. C. Maxwell, Ⅳ. On the dynamical theory of gases, Philosophical Transactions of the Royal Society of London, 157 (1867), 49-88.  doi: 10.1098/rstl.1867.0004.  Google Scholar

[21]

S. Mischler, On the trace problem for solutions of the Vlasov equation, Communications in Partial Differential Equations, 25 (1999), 1415-1443.  doi: 10.1080/03605300008821554.  Google Scholar

[22]

M. Mokhtar-Kharroubi and D. Seifert, Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry, Journal of Functional Analysis, 275 (2018), 2404-2452.  doi: 10.1016/j.jfa.2018.08.005.  Google Scholar

[23]

J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de matemática. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, 1968.  Google Scholar

[24]

Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, 1st ed edition, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[25]

T. TsujiK. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, Journal of Statistical Physics, 140 (2010), 518-543.  doi: 10.1007/s10955-010-9997-5.  Google Scholar

[26]

C. Villani., Hypocoercivity., American Mathematical Society, 2009. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

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