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A mathematical PDE perspective on the Chapman-Enskog expansion


Author: Laure Saint-Raymond
Journal: Bull. Amer. Math. Soc. 51 (2014), 247-275
MSC (2010): Primary 76P05, 35Q20, 35C20
DOI: https://doi.org/10.1090/S0273-0979-2013-01440-X
Published electronically: December 3, 2013
MathSciNet review: 3166041
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents in a synthetic way some recent advances on hydrodynamic limits of the Boltzmann equation. It aims at bringing a new light to these results by placing them in the more general framework of asymptotic expansions of Chapman-Enskog type, and by discussing especially the issues of regularity and truncation.


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  • [1] Diogo Arsénio and Laure Saint-Raymond, Compactness in kinetic transport equations and hypoellipticity, J. Funct. Anal. 261 (2011), no. 10, 3044-3098. MR 2832590 (2012h:35029), https://doi.org/10.1016/j.jfa.2011.07.020
  • [2] D. Arsénio and L. Saint-Raymond. From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. In preparation (2013).
  • [3] Claude Bardos, François Golse, and C. David Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (1993), no. 5, 667-753. MR 1213991 (94g:82039), https://doi.org/10.1002/cpa.3160460503
  • [4] Claude Bardos and Seiji Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 235-257. MR 1115292 (93a:76023), https://doi.org/10.1142/S0218202591000137
  • [5] A. V. Bobylëv, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR 262 (1982), no. 1, 71-75 (Russian). MR 647994 (83g:82034)
  • [6] A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys. 124 (2006), no. 2-4, 371-399. MR 2264613 (2007i:82066), https://doi.org/10.1007/s10955-005-8087-6
  • [7] T. Bodineau, I. Gallagher, and L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard spheres. Submitted (2013).
  • [8] T. Bodineau, I. Gallagher, and L. Saint-Raymond. Linear fluid models as scaling limits of systems of particles. Work in progress (2013).
  • [9] Stefano Bianchini and Alberto Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223-342. MR 2150387 (2007i:35160), https://doi.org/10.4007/annals.2005.161.223
  • [10] Russel E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980), no. 5, 651-666. MR 586416 (81j:76072), https://doi.org/10.1002/cpa.3160330506
  • [11] Russel E. Caflisch and Basil Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys. 86 (1982), no. 2, 161-194. MR 676183 (84d:82022)
  • [12] Sydney Chapman and T. G. Cowling, The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, New York, 1960. MR 0116537 (22 #7324)
  • [13] Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028 (95i:82082)
  • [14] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245-316. MR 2116276 (2005j:82070), https://doi.org/10.1007/s00222-004-0389-9
  • [15] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321-366. MR 1014927 (90k:82045), https://doi.org/10.2307/1971423
  • [16] R. J. DiPerna, P.-L. Lions, and Y. Meyer, $ L^p$ regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 3-4, 271-287 (English, with French summary). MR 1127927 (92g:35036)
  • [17] François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110-125. MR 923047 (89a:35179), https://doi.org/10.1016/0022-1236(88)90051-1
  • [18] I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: the case of hard-spheres and short-range potentials, to appear in Zürich Lectures Adv. Math. (2013).
  • [19] F. Golse, D. Levermore, and L. Saint-Raymond. La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques, Séminaire Equations aux dérivées partielles (Polytechnique) (1999-2000).
  • [20] 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 (2005f:76003), https://doi.org/10.1007/s00222-003-0316-5
  • [21] François Golse and Laure Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9) 91 (2009), no. 5, 508-552 (English, with English and French summaries). MR 2517786 (2010j:35378), https://doi.org/10.1016/j.matpur.2009.01.013
  • [22] François Golse and Laure Saint-Raymond, Velocity averaging in $ L^1$ for the transport equation, C. R. Math. Acad. Sci. Paris 334 (2002), no. 7, 557-562 (English, with English and French summaries). MR 1903763 (2003f:35044), https://doi.org/10.1016/S1631-073X(02)02302-6
  • [23] Alexander N. Gorban and Iliya V. Karlin, Structure and approximations of the Chapman-Enskog expansion for the linearized Grad equations, Transport Theory Statist. Phys. 21 (1992), no. 1-2, 101-117. MR 1149364 (92m:82117), https://doi.org/10.1080/00411459208203524
  • [24] Iliya V. Karlin and Alexander N. Gorban, Hydrodynamics from Grad's equations: what can we learn from exact solutions?, Ann. Phys. 11 (2002), no. 10-11, 783-833. MR 1957348 (2004e:82050), https://doi.org/10.1002/1521-3889(200211)11:10/11$ \langle $783::AID-ANDP783$ \rangle $3.0.CO;2-V
  • [25] A. K. Gorban and I. Karlin, Hilbert's $ 6$th Problem: exact and approximate hydrodynamic manifolds for kinetic equations Bull. Amer. Math. Soc. 51 (2014), no. 2, 186-246.
  • [26] Harold Grad, Asymptotic theory of the Boltzmann equation. II, Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, Academic Press, New York, 1963, pp. 26-59. MR 0156656 (27 #6577)
  • [27] E. Grenier, Quelques limites singulières oscillantes, Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, École Polytech., Palaiseau, 1995, pp. Exp. No. XXI, 13 (French). MR 1362569 (96m:82076)
  • [28] E. H. Hauge, Exact and Chapman-Enskog solutions of the Boltzmann equation for the Lorentz model, Phys. Fluids 13 (1970), 1201-1208. MR 0278672 (43 #4402)
  • [29] David Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912), no. 4, 562-577 (German). MR 1511713, https://doi.org/10.1007/BF01456676
  • [30] Oscar E. Lanford III, Time evolution of large classical systems, Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, 1975, pp. 1-111. Lecture Notes in Phys., Vol. 38. MR 0479206 (57 #18653)
  • [31] J. Leray. Etude de diverses équations intégrales non linéaires et quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 9 (1933), 1-82.
  • [32] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications J. Math. Kyoto Univ. 34 (1994), 391-427, 429-461, 539-584.
  • [33] Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251 (98b:76001)
  • [34] Pierre-Louis Lions and Nader Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 5, 387-392 (French, with English and French summaries). MR 1710123 (2000e:76103), https://doi.org/10.1016/S0764-4442(00)88611-5
  • [35] Tai Ping Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), no. 1, 147-177. MR 0435618 (55 #8576)
  • [36] Tai-Ping Liu and Shih-Hsien Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004), no. 1, 133-179. MR 2044894 (2005f:82101), https://doi.org/10.1007/s00220-003-1030-2
  • [37] Laure Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003), no. 1, 47-80. MR 1952079 (2004d:35038), https://doi.org/10.1007/s00205-002-0228-3
  • [38] Laure Saint-Raymond, Hydrodynamic limits: some improvements of the relative entropy method, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 705-744 (English, with English and French summaries). MR 2526399 (2010g:76028), https://doi.org/10.1016/j.anihpc.2008.01.001
  • [39] Laure Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, vol. 1971, Springer-Verlag, Berlin, 2009. MR 2683475 (2012f:82079)
  • [40] Steven Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), no. 2, 476-512. MR 1303036 (95k:35131), https://doi.org/10.1006/jdeq.1994.1157
  • [41] Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475-485. MR 815196 (87d:35127)
  • [42] M. Slemrod. Admissibility of weak solutions for the compressible Euler equations, $ n\geq 2$. To appear in Philosophical Transactions of the Royal Society A (2013).
  • [43] Horng-Tzer Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22 (1991), no. 1, 63-80. MR 1121850 (93e:82035), https://doi.org/10.1007/BF00400379

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Additional Information

Laure Saint-Raymond
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, Paris, France
Address at time of publication: 45 rue d’Ulm, 75230 Paris Cedex 05, France
Email: Laure.Saint-Raymond@ens.fr

DOI: https://doi.org/10.1090/S0273-0979-2013-01440-X
Received by editor(s): July 20, 2013
Published electronically: December 3, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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