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Random attractors for dissipative systems with rough noises
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 |
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.
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. |
[2] |
T. Bodineau, I. 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. |
[3] |
L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. |
[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.
|
[5] |
N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. |
[6] |
C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge
University Press, Cambridge, 2000, From basic concepts to actual calculations. |
[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. |
[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. |
[9] |
T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. |
[10] |
T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. |
[11] |
R. Esposito, Y. Guo, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
H. Grad,
On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[18] |
J.-P. Guiraud,
Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490.
|
[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.
|
[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. |
[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. |
[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. |
[23] |
C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. |
[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. |
[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.
|
[26] |
H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991.
doi: 10.1007/978-3-642-84371-6. |
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. |
[2] |
T. Bodineau, I. 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. |
[3] |
L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. |
[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.
|
[5] |
N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. |
[6] |
C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge
University Press, Cambridge, 2000, From basic concepts to actual calculations. |
[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. |
[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. |
[9] |
T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. |
[10] |
T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. |
[11] |
R. Esposito, Y. Guo, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
H. Grad,
On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[18] |
J.-P. Guiraud,
Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490.
|
[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.
|
[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. |
[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. |
[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. |
[23] |
C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. |
[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. |
[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.
|
[26] |
H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991.
doi: 10.1007/978-3-642-84371-6. |


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