A semigroup approach to the convergence rate of a collisionless gas

Previous Article
Averaging of highly-oscillatory transport equations
KRM Home
This Issue
Next Article

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.

Keywords: Transport equations, Maxwellian diffusion boundary conditions, subgeometric Harris' theorem, explicit, collisionless gas.

Mathematics Subject Classification: Primary:35B40, 82C40;Secondary:35F16.

Citation: Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic and 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.
[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.

[1]	Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102
[2]	José A. Cañizo, Chuqi Cao, Josephine Evans, Havva Yoldaş. Hypocoercivity of linear kinetic equations via Harris's Theorem. Kinetic and Related Models, 2020, 13 (1) : 97-128. doi: 10.3934/krm.2020004
[3]	Marc Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinetic and Related Models, 2015, 8 (2) : 281-308. doi: 10.3934/krm.2015.8.281
[4]	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
[5]	Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252
[6]	Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001
[7]	Vandana Sharma, Jyotshana V. Prajapat. Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 109-135. doi: 10.3934/dcds.2021109
[8]	Guillaume Bal, Alexandre Jollivet. Boundary control for transport equations. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022014
[9]	Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054
[10]	Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations and Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61
[11]	Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions. Networks and Heterogeneous Media, 2013, 8 (3) : 727-744. doi: 10.3934/nhm.2013.8.727
[12]	Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
[13]	Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279
[14]	Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295
[15]	Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations and Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062
[16]	Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic and Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001
[17]	Shijin Deng, Weike Wang, Shih-Hsien Yu. Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Networks and Heterogeneous Media, 2007, 2 (3) : 383-395. doi: 10.3934/nhm.2007.2.383
[18]	Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
[19]	Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
[20]	Mohan Mallick, Sarath Sasi, R. Shivaji, S. Sundar. Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2022, 21 (2) : 705-726. doi: 10.3934/cpaa.2021195

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Armand Bernou
on Google Scholar
Armand Bernou

2020 Impact Factor: 1.432

Tools

Article outline

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Armand Bernou
on Google Scholar
Armand Bernou

[Back to Top]