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A semigroup approach to the convergence rate of a collisionless gas
Sorbonne Université, CNRS, Laboratoire de Probabilité, Statistique et Modélisation, F-75005 Paris, France |
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 [
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. |
[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. |
[3] |
L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatshefte für Mathematik, 123 (1997), 285–298.
doi: 10.1007/BF01326764. |
[4] |
J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Comprehensive studies in mathematics. Springer Berlin Heidelberg, 1976. |
[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. |
[6] |
L. Boltzmann, Lectures on Gas Theory, University of California Press, Berkeley-Los Angeles, Calif. 1964. |
[7] |
J. A. Cañizo and S. Mischler, Doeblin-Harris theory for stochastic operators and semigroups, In preparation, 2019. |
[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. |
[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. |
[10] |
R. Esposito, Y. Guo, C. 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. |
[11] |
S. N. Evans,
Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.
doi: 10.1214/aoap/1015345298. |
[12] |
C. Goulaouic,
Prolongements de foncteurs d'interpolation et applications, Annales de l'Institut Fourier, 18 (1968), 1-98.
doi: 10.5802/aif.277. |
[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. |
[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. |
[15] |
M. Hairer, Convergence of Markov processes, Lecture Notes available at http://www.hairer.org/notes/Convergence.pdf, 2016. |
[16] |
S. Janson,
Interpolation of subcouples and quotient couples, Ark. Mat., 31 (1993), 307-338.
doi: 10.1007/BF02559489. |
[17] |
H-W. Kuo, T-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. |
[18] |
H-W. Kuo, T-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. |
[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. |
[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. |
[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. |
[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. |
[23] |
J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de matemática. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, 1968. |
[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. |
[25] |
T. Tsuji, K. 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. |
[26] |
C. Villani., Hypocoercivity., American Mathematical Society, 2009.
doi: 10.1090/S0065-9266-09-00567-5. |
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. |
[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. |
[3] |
L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatshefte für Mathematik, 123 (1997), 285–298.
doi: 10.1007/BF01326764. |
[4] |
J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Comprehensive studies in mathematics. Springer Berlin Heidelberg, 1976. |
[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. |
[6] |
L. Boltzmann, Lectures on Gas Theory, University of California Press, Berkeley-Los Angeles, Calif. 1964. |
[7] |
J. A. Cañizo and S. Mischler, Doeblin-Harris theory for stochastic operators and semigroups, In preparation, 2019. |
[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. |
[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. |
[10] |
R. Esposito, Y. Guo, C. 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. |
[11] |
S. N. Evans,
Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.
doi: 10.1214/aoap/1015345298. |
[12] |
C. Goulaouic,
Prolongements de foncteurs d'interpolation et applications, Annales de l'Institut Fourier, 18 (1968), 1-98.
doi: 10.5802/aif.277. |
[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. |
[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. |
[15] |
M. Hairer, Convergence of Markov processes, Lecture Notes available at http://www.hairer.org/notes/Convergence.pdf, 2016. |
[16] |
S. Janson,
Interpolation of subcouples and quotient couples, Ark. Mat., 31 (1993), 307-338.
doi: 10.1007/BF02559489. |
[17] |
H-W. Kuo, T-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. |
[18] |
H-W. Kuo, T-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. |
[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. |
[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. |
[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. |
[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. |
[23] |
J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de matemática. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, 1968. |
[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. |
[25] |
T. Tsuji, K. 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. |
[26] |
C. Villani., Hypocoercivity., American Mathematical Society, 2009.
doi: 10.1090/S0065-9266-09-00567-5. |
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