Hydrodynamic limit of granular gases to pressureless Euler in dimension 1
Authors:
Pierre-Emmanuel Jabin and Thomas Rey
Journal:
Quart. Appl. Math. 75 (2017), 155-179
MSC (2010):
Primary 35A23, 35B40, 35L67, 35L80
DOI:
https://doi.org/10.1090/qam/1442
Published electronically:
July 5, 2016
MathSciNet review:
3580099
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We investigate the behavior of granular gases in the limit of small Knudsen number, that is, very frequent collisions. We deal with the strongly inelastic case in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.
References
- Ricardo J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J. 58 (2009), no. 3, 999–1022. MR 2541357, DOI https://doi.org/10.1512/iumj.2009.58.3506
- Ricardo J. Alonso and Bertrand Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal. 42 (2010), no. 6, 2499–2538. MR 2733258, DOI https://doi.org/10.1137/100793979
- Ricardo J. Alonso and Bertrand Lods, Two proofs of Haff’s law for dissipative gases: the use of entropy and the weakly inelastic regime, J. Math. Anal. Appl. 397 (2013), no. 1, 260–275. MR 2968989, DOI https://doi.org/10.1016/j.jmaa.2012.07.045
- Dario Benedetto and Mario Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 899–905. MR 1866273, DOI https://doi.org/10.1051/m2an%3A2001141
- D. Benedetto, E. Caglioti, F. Golse, and M. Pulvirenti, A hydrodynamic model arising in the context of granular media, Comput. Math. Appl. 38 (1999), no. 7-8, 121–131. MR 1713167, DOI https://doi.org/10.1016/S0898-1221%2899%2900243-6
- D. Benedetto, E. Caglioti, and M. Pulvirenti, A one dimensional Boltzmann equation with inelastic collisions, Rend. Sem. Mat. Fis. Milano 67 (1997), 169–179 (2000). MR 1781039, DOI https://doi.org/10.1007/BF02930497
- Andrei Biryuk, Walter Craig, and Vladislav Panferov, Strong solutions of the Boltzmann equation in one spatial dimension, C. R. Math. Acad. Sci. Paris 342 (2006), no. 11, 843–848 (English, with English and French summaries). MR 2224633, DOI https://doi.org/10.1016/j.crma.2006.04.005
- J.-M. Bony, Solutions globales bornées pour les modèles discrets de l’équation de Boltzmann, en dimension $1$ d’espace, Journées “Équations aux derivées partielles” (Saint Jean de Monts, 1987) École Polytech., Palaiseau, 1987, pp. Exp. No. XVI, 10 (French). MR 920011
- François Bouchut and François James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations 24 (1999), no. 11-12, 2173–2189. MR 1720754, DOI https://doi.org/10.1080/03605309908821498
- Laurent Boudin, A solution with bounded expansion rate to the model of viscous pressureless gases, SIAM J. Math. Anal. 32 (2000), no. 1, 172–193. MR 1766512, DOI https://doi.org/10.1137/S0036141098346840
- Yann Brenier and Emmanuel Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal. 35 (1998), no. 6, 2317–2328. MR 1655848, DOI https://doi.org/10.1137/S0036142997317353
- Nikolai V. Brilliantov and Thorsten Pöschel, Kinetic theory of granular gases, Oxford Graduate Texts, Oxford University Press, Oxford, 2004. MR 2101911
- Carlo Cercignani, A remarkable estimate for the solutions of the Boltzmann equation, Appl. Math. Lett. 5 (1992), no. 5, 59–62. MR 1345903, DOI https://doi.org/10.1016/0893-9659%2892%2990065-H
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620
- Alina Chertock, Alexander Kurganov, and Yurii Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal. 45 (2007), no. 6, 2408—2441. MR 2361896, DOI https://doi.org/10.1137/050644124
- Weinan E, Yu. G. Rykov, and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177 (1996), no. 2, 349–380. MR 1384139
- François Golse and Laure Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (2004), no. 1, 81–161. MR 2025302, DOI https://doi.org/10.1007/s00222-003-0316-5
- François Golse and Laure Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma (7) 4** (2005), 1–144. MR 2197021
- P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134 (1983), 401–30.
- Feimin Huang and Zhen Wang, Well posedness for pressureless flow, Comm. Math. Phys. 222 (2001), no. 1, 117–146. MR 1853866, DOI https://doi.org/10.1007/s002200100506
- Moon-Jin Kang and Alexis F. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci. 25 (2015), no. 11, 2153–2173. MR 3368270, DOI https://doi.org/10.1142/S0218202515500542
- P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169–191. MR 1201239, DOI https://doi.org/10.1090/S0894-0347-1994-1201239-3
- P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
- S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior, J. Stat. Phys. 124 (2006), no. 2-4, 703–746. MR 2264623, DOI https://doi.org/10.1007/s10955-006-9097-8
- O. A. Oleĭnik, On Cauchy’s problem for nonlinear equations in a class of discontinuous functions, Doklady Akad. Nauk SSSR (N.S.) 95 (1954), 451–454 (Russian). MR 0064258
- Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
- Thomas Rey, Blow up analysis for anomalous granular gases, SIAM J. Math. Anal. 44 (2012), no. 3, 1544–1561. MR 2982722, DOI https://doi.org/10.1137/110835645
- Laure Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 271–317 (English, with English and French summaries). MR 1980313, DOI https://doi.org/10.1016/S0012-9593%2803%2900010-7
- J. Silk, A. Szalay, and Ya. B. Zeldovich, Large-scale structure of the universe, Scientific American 249 (1983), 72–80.
- Giuseppe Toscani, Hydrodynamics from the dissipative Boltzmann equation, Mathematical models of granular matter, Lecture Notes in Math., vol. 1937, Springer, Berlin, 2008, pp. 59–75. MR 2436466, DOI https://doi.org/10.1007/978-3-540-78277-3_3
- Isabelle Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal. 270 (2016), no. 5, 1922–1970. MR 3452720, DOI https://doi.org/10.1016/j.jfa.2015.09.025
- Cédric Villani, Mathematics of granular materials, J. Stat. Phys. 124 (2006), no. 2-4, 781–822. MR 2264625, DOI https://doi.org/10.1007/s10955-006-9038-6
- Zhigang Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum, Contin. Mech. Thermodyn. 22 (2010), no. 3, 239–249. MR 2602445, DOI https://doi.org/10.1007/s00161-009-0127-z
References
- Ricardo J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J. 58 (2009), no. 3, 999–1022. MR 2541357, DOI https://doi.org/10.1512/iumj.2009.58.3506
- Ricardo J. Alonso and Bertrand Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal. 42 (2010), no. 6, 2499–2538. MR 2733258, DOI https://doi.org/10.1137/100793979
- Ricardo J. Alonso and Bertrand Lods, Two proofs of Haff’s law for dissipative gases: the use of entropy and the weakly inelastic regime, J. Math. Anal. Appl. 397 (2013), no. 1, 260–275. MR 2968989, DOI https://doi.org/10.1016/j.jmaa.2012.07.045
- Dario Benedetto and Mario Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 899–905. MR 1866273, DOI https://doi.org/10.1051/m2an%3A2001141
- D. Benedetto, E. Caglioti, F. Golse, and M. Pulvirenti, A hydrodynamic model arising in the context of granular media, Comput. Math. Appl. 38 (1999), no. 7-8, 121–131. MR 1713167, DOI https://doi.org/10.1016/S0898-1221%2899%2900243-6
- D. Benedetto, E. Caglioti, and M. Pulvirenti, A one dimensional Boltzmann equation with inelastic collisions, Rend. Sem. Mat. Fis. Milano 67 (1997), 169–179 (2000). MR 1781039, DOI https://doi.org/10.1007/BF02930497
- Andrei Biryuk, Walter Craig, and Vladislav Panferov, Strong solutions of the Boltzmann equation in one spatial dimension, C. R. Math. Acad. Sci. Paris 342 (2006), no. 11, 843–848 (English, with English and French summaries). MR 2224633, DOI https://doi.org/10.1016/j.crma.2006.04.005
- J.-M. Bony, Solutions globales bornées pour les modèles discrets de l’équation de Boltzmann, en dimension $1$ d’espace, Journées “Équations aux derivées partielles” (Saint Jean de Monts, 1987) Exp. No. XVI, 10 pp., École Polytech., Palaiseau, 1987 (French). MR 920011
- François Bouchut and François James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations 24 (1999), no. 11-12, 2173–2189. MR 1720754, DOI https://doi.org/10.1080/03605309908821498
- Laurent Boudin, A solution with bounded expansion rate to the model of viscous pressureless gases, SIAM J. Math. Anal. 32 (2000), no. 1, 172–193 (electronic). MR 1766512, DOI https://doi.org/10.1137/S0036141098346840
- Yann Brenier and Emmanuel Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal. 35 (1998), no. 6, 2317–2328 (electronic). MR 1655848, DOI https://doi.org/10.1137/S0036142997317353
- Nikolai V. Brilliantov and Thorsten Pöschel, Kinetic theory of granular gases, Oxford Graduate Texts, Oxford University Press, Oxford, 2004. MR 2101911
- Carlo Cercignani, A remarkable estimate for the solutions of the Boltzmann equation, Appl. Math. Lett. 5 (1992), no. 5, 59–62. MR 1345903, DOI https://doi.org/10.1016/0893-9659%2892%2990065-H
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620
- Alina Chertock, Alexander Kurganov, and Yurii Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal. 45 (2007), no. 6, 2408—2441 (electronic). MR 2361896, DOI https://doi.org/10.1137/050644124
- Weinan E, Yu. G. Rykov, and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177 (1996), no. 2, 349–380. MR 1384139
- François Golse and Laure Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (2004), no. 1, 81–161. MR 2025302, DOI https://doi.org/10.1007/s00222-003-0316-5
- François Golse and Laure Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma (7) 4** (2005), 1–144. MR 2197021
- P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134 (1983), 401–30.
- Feimin Huang and Zhen Wang, Well posedness for pressureless flow, Comm. Math. Phys. 222 (2001), no. 1, 117–146. MR 1853866, DOI https://doi.org/10.1007/s002200100506
- Moon-Jin Kang and Alexis F. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci. 25 (2015), no. 11, 2153–2173. MR 3368270, DOI https://doi.org/10.1142/S0218202515500542
- P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169–191. MR 1201239, DOI https://doi.org/10.2307/2152725
- P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
- S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior, J. Stat. Phys. 124 (2006), no. 2-4, 703–746. MR 2264623, DOI https://doi.org/10.1007/s10955-006-9097-8
- O. A. Oleĭnik, On Cauchy’s problem for nonlinear equations in a class of discontinuous functions, Doklady Akad. Nauk SSSR (N.S.) 95 (1954), 451–454 (Russian). MR 0064258
- Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
- Thomas Rey, Blow up analysis for anomalous granular gases, SIAM J. Math. Anal. 44 (2012), no. 3, 1544–1561. MR 2982722, DOI https://doi.org/10.1137/110835645
- Laure Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 271–317 (English, with English and French summaries). MR 1980313, DOI https://doi.org/10.1016/S0012-9593%2803%2900010-7
- J. Silk, A. Szalay, and Ya. B. Zeldovich, Large-scale structure of the universe, Scientific American 249 (1983), 72–80.
- Giuseppe Toscani, Hydrodynamics from the dissipative Boltzmann equation, Mathematical models of granular matter, Lecture Notes in Math., vol. 1937, Springer, Berlin, 2008, pp. 59–75. MR 2436466, DOI https://doi.org/10.1007/978-3-540-78277-3_3
- Isabelle Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal. 270 (2016), no. 5, 1922–1970. MR 3452720, DOI https://doi.org/10.1016/j.jfa.2015.09.025
- Cédric Villani, Mathematics of granular materials, J. Stat. Phys. 124 (2006), no. 2-4, 781–822. MR 2264625, DOI https://doi.org/10.1007/s10955-006-9038-6
- Zhigang Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum, Contin. Mech. Thermodyn. 22 (2010), no. 3, 239–249. MR 2602445, DOI https://doi.org/10.1007/s00161-009-0127-z
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35A23,
35B40,
35L67,
35L80
Retrieve articles in all journals
with MSC (2010):
35A23,
35B40,
35L67,
35L80
Additional Information
Pierre-Emmanuel Jabin
Affiliation:
CSCAMM and Department of Mathematics, University of Maryland, College Park, Maryland 20742
MR Author ID:
660988
Email:
pjabin@umd.edu
Thomas Rey
Affiliation:
Laboratoire P. Painlevé, CNRS UMR 8524, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Email:
thomas.rey@math.univ-lille1.fr
Received by editor(s):
March 3, 2016
Published electronically:
July 5, 2016
Additional Notes:
The first author was partially supported by NSF Grant 1312142 and by NSF Grant RNMS (Ki-Net) 1107444.
The second author was partially supported by the team Inria/Rapsodi, Labex CEMPI (ANR-11-LABX-0007-01) and NSF Grant RNMS (Ki-Net) 1107444.
Article copyright:
© Copyright 2016
Brown University