doi: 10.3934/dcds.2021177
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

UMPA (UMR CNRS 5669), École Normale Superieur de Lyon, 46 allée d'Italie, 69364 LYON, France

Received  April 2021 Revised  September 2021 Early access November 2021

Fund Project: The author thanks Sergio Simonella, Raphael Winter and the anonymous referees for their suggestions which helped to improve the paper. We also thank Laure Saint Raymond for suggesting the problem

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of $ N $ hard spheres of diameter $ \epsilon $ in a box $ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling $ N\epsilon^2 = 1 $, $ \epsilon\rightarrow 0 $ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

Citation: Corentin Le Bihan. Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021177
References:
[1]

R. K. Alexander, The Infinite Hard-Sphere System, ProQuest LLC, Ann Arbor, MI, 1975, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7615092, Thesis (Ph.D.)–University of California, Berkeley.  Google Scholar

[2]

T. BodineauI. Gallagher and L. Saint-Raymond, A microscopic view of the Fourier law, Comptes Rendus Physique, 20 (2019), 402-418.  doi: 10.1016/j.crhy.2019.08.002.  Google Scholar

[3]

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

[4]

S. Caprino and M. Pulvirenti, The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177 (1996), 63-81.   Google Scholar

[5]

N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. Google Scholar

[6]

C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000, From basic concepts to actual calculations. Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[8]

R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1.  Google Scholar

[9]

T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. Google Scholar

[10]

T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. Google Scholar

[11]

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

[12]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Paper No. 1,119 pp. doi: 10.1007/s40818-017-0037-5.  Google Scholar

[13]

R. Esposito and R. Marra, Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180 (2020), 773-809.  doi: 10.1007/s10955-020-02528-w.  Google Scholar

[14]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[15]

V. Gerasimenko and I. Gapyak, Low-density asymptotic behavior of observables of hard sphere fluids, Adv. Math. Phys., 2018 (2018), No 6252919, 11 pp. doi: 10.1155/2018/6252919.  Google Scholar

[16]

S. Goldstein, J. L. Lebowitz and E. Presutti, Mechanical system with stochastic boundaries, Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory, Esztergom 1979, Colloq. Math. Soc. Janos Bolyai, 27 (1981), 403–419. Google Scholar

[17]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

[18]

J.-P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490.   Google Scholar

[19]

J.-P. Guiraud, Problème aux limites interieur pour l'équation de Boltzmann en regime stationnaire, faiblement non linéaire, J. Méc., Paris, 11 (1972), 183-231.   Google Scholar

[20]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: "Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum", [Comm. Math. Phys., 105 (1986), 189-203; MR0849204 (88d: 82061)] and "Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum" [ibid. 113 (1987), 79–85; MR0918406 (89b: 82052)] by Pulvirenti, Comm. Math. Phys., 121 (1989), 143–146, URL http://projecteuclid.org/euclid.cmp/1104178007.  Google Scholar

[21]

O. E. Lanford, III Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Phys., Vol. 38, 1975, 1–111.  Google Scholar

[22]

C. Le Bihan, Convergence D'un Système de Sphères Dures Vers une Solution de L'équation de Stokes-Fourier Avec Bord, Master's thesis, ENS de Lyon, 2019. Google Scholar

[23]

C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. Google Scholar

[24]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[25]

M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Acad. Sin. (N.S.), 10 (2015), 171-204.   Google Scholar

[26]

H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991. doi: 10.1007/978-3-642-84371-6.  Google Scholar

show all references

References:
[1]

R. K. Alexander, The Infinite Hard-Sphere System, ProQuest LLC, Ann Arbor, MI, 1975, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7615092, Thesis (Ph.D.)–University of California, Berkeley.  Google Scholar

[2]

T. BodineauI. Gallagher and L. Saint-Raymond, A microscopic view of the Fourier law, Comptes Rendus Physique, 20 (2019), 402-418.  doi: 10.1016/j.crhy.2019.08.002.  Google Scholar

[3]

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

[4]

S. Caprino and M. Pulvirenti, The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177 (1996), 63-81.   Google Scholar

[5]

N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. Google Scholar

[6]

C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000, From basic concepts to actual calculations. Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[8]

R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1.  Google Scholar

[9]

T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. Google Scholar

[10]

T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. Google Scholar

[11]

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

[12]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Paper No. 1,119 pp. doi: 10.1007/s40818-017-0037-5.  Google Scholar

[13]

R. Esposito and R. Marra, Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180 (2020), 773-809.  doi: 10.1007/s10955-020-02528-w.  Google Scholar

[14]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[15]

V. Gerasimenko and I. Gapyak, Low-density asymptotic behavior of observables of hard sphere fluids, Adv. Math. Phys., 2018 (2018), No 6252919, 11 pp. doi: 10.1155/2018/6252919.  Google Scholar

[16]

S. Goldstein, J. L. Lebowitz and E. Presutti, Mechanical system with stochastic boundaries, Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory, Esztergom 1979, Colloq. Math. Soc. Janos Bolyai, 27 (1981), 403–419. Google Scholar

[17]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

[18]

J.-P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490.   Google Scholar

[19]

J.-P. Guiraud, Problème aux limites interieur pour l'équation de Boltzmann en regime stationnaire, faiblement non linéaire, J. Méc., Paris, 11 (1972), 183-231.   Google Scholar

[20]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: "Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum", [Comm. Math. Phys., 105 (1986), 189-203; MR0849204 (88d: 82061)] and "Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum" [ibid. 113 (1987), 79–85; MR0918406 (89b: 82052)] by Pulvirenti, Comm. Math. Phys., 121 (1989), 143–146, URL http://projecteuclid.org/euclid.cmp/1104178007.  Google Scholar

[21]

O. E. Lanford, III Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Phys., Vol. 38, 1975, 1–111.  Google Scholar

[22]

C. Le Bihan, Convergence D'un Système de Sphères Dures Vers une Solution de L'équation de Stokes-Fourier Avec Bord, Master's thesis, ENS de Lyon, 2019. Google Scholar

[23]

C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. Google Scholar

[24]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[25]

M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Acad. Sin. (N.S.), 10 (2015), 171-204.   Google Scholar

[26]

H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991. doi: 10.1007/978-3-642-84371-6.  Google Scholar

Figure 1.  Reflection of particle $ i $ at time $ \tau $
Figure 2.  Evolution of $ \bar{\omega}_i $ when particle $ i $ has a reflection at time $ \tau $
Figure 3.  Pseudotrajectory associated with collision parameters $ ((1, -)_3, (2, +)_4, (1, +)_5, (4, -)_6) $
Figure 4.  A shift of particle $ j $ in $ \mathcal{P}^{i, j, i'}_1 $
Figure 5.  Virtual particle
[1]

Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji Zhang. Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064

[2]

Mario Pulvirenti, Sergio Simonella. On the cardinality of collisional clusters for hard spheres at low density. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3903-3914. doi: 10.3934/dcds.2021021

[3]

Yusheng Jia, Weishi Liu, Mingji Zhang. Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022

[4]

Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068

[5]

Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305

[6]

Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459

[7]

Hongxu Chen. Cercignani-Lampis boundary in the Boltzmann theory. Kinetic & Related Models, 2020, 13 (3) : 549-597. doi: 10.3934/krm.2020019

[8]

Dohyun Kim. Asymptotic behavior of a second-order swarm sphere model and its kinetic limit. Kinetic & Related Models, 2020, 13 (2) : 401-434. doi: 10.3934/krm.2020014

[9]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

[10]

Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014

[11]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[12]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343

[13]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[14]

A. V. Bobylev, E. Mossberg. On some properties of linear and linearized Boltzmann collision operators for hard spheres. Kinetic & Related Models, 2008, 1 (4) : 521-555. doi: 10.3934/krm.2008.1.521

[15]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[16]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, Johannes Lankeit. The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6529-6546. doi: 10.3934/dcds.2020289

[17]

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

[18]

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

[19]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741

[20]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (31)
  • HTML views (28)
  • Cited by (0)

Other articles
by authors

[Back to Top]