Limites hydrodynamiques de l'équation de Boltzmann
Séminaire Bourbaki : volume 2000/2001, exposés 880-893, Astérisque, no. 282 (2002), Talk no. 893, 41 p.
@incollection{SB_2000-2001__43__365_0,
     author = {Villani, C\'edric},
     title = {Limites hydrodynamiques de l'\'equation de {Boltzmann}},
     booktitle = {S\'eminaire Bourbaki : volume 2000/2001, expos\'es 880-893},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:893},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {282},
     year = {2002},
     zbl = {01969562},
     mrnumber = {1975186},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2000-2001__43__365_0/}
}
TY  - CHAP
AU  - Villani, Cédric
TI  - Limites hydrodynamiques de l'équation de Boltzmann
BT  - Séminaire Bourbaki : volume 2000/2001, exposés 880-893
AU  - Collectif
T3  - Astérisque
N1  - talk:893
PY  - 2002
DA  - 2002///
IS  - 282
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/SB_2000-2001__43__365_0/
UR  - https://zbmath.org/?q=an%3A01969562
UR  - https://www.ams.org/mathscinet-getitem?mr=1975186
LA  - fr
ID  - SB_2000-2001__43__365_0
ER  - 
%0 Book Section
%A Villani, Cédric
%T Limites hydrodynamiques de l'équation de Boltzmann
%B Séminaire Bourbaki : volume 2000/2001, exposés 880-893
%A Collectif
%S Astérisque
%Z talk:893
%D 2002
%N 282
%I Société mathématique de France
%G fr
%F SB_2000-2001__43__365_0
Villani, Cédric. Limites hydrodynamiques de l'équation de Boltzmann, in Séminaire Bourbaki : volume 2000/2001, exposés 880-893, Astérisque, no. 282 (2002), Talk no. 893, 41 p. http://www.numdam.org/item/SB_2000-2001__43__365_0/

[Al] S. Alinhac - Blowup for nonlinear hyperbolic equations, Progress in nonlinear differential equations and their applications, 17, Birkhäuser, Boston 1995. | MR | Zbl

[AK] V. Arnold, B. Khesin - Topological methods in hydrodynamics, Springer-Verlag, New York 1998. | MR | Zbl

[As] K. Asano - Fluid dynamical limit of the Boltzmann equation, I, in "Proceedings of the 13th International Conference on Transport Theory (Riccione 1993), vol. 24 (1995), pp. 329-345. | MR | Zbl

[AU] K. Asano, S. Ukai - On the fluid dynamical limit of the Boltzmann equation, in "Recent topics in nonlinear PDE" (Hiroshima 1983), North-Holland, Amsterdam 1984, pp. 1-19. | MR | Zbl

[BGL1] C. Bardos, F. Golse, C. D. Levermore - Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris, Série I Math. 309 (1989), 727-732. | MR | Zbl

[BGL2] C. Bardos, F. Golse, C. D. Levermore - Fluid dynamical limits of kinetic equations, I : Formal derivation. J. Statist. Phys. 63 (1991), 323- 344. | MR

[BGL3] C. Bardos, F. Golse, C. D. Levermore - Fluid dynamical limits of kinetic equations, II : Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46, 5 (1993), 667-753. | MR | Zbl

[BGL4] C. Bardos, F. Golse, C. D. Levermore - Acoustic and Stokes limits for the Boltzmann equation, C. R. Acad. Sci. Paris, Série I Math. 327, 3 (1998), 323-328. | MR | Zbl

[BGL5] C. Bardos, F. Golse, C. D. Levermore - The acoustic limit for the Boltzmann equation, Arch. Rational Mech. Anal. 153, 3 (2000), 177-204. | MR | Zbl

[BU] C. Bardos, S. Ukai - The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci. 1, 2 (1991), 235-257. | MR | Zbl

[Ba] G. K. Batchelor - An introduction to fluid dynamics, 2nd paperback ed., Cambridge University Press, Cambridge 1999. | MR | Zbl

[Bo] F. Bouchut - Introduction à la théorie mathématique des équations cinétiques, Session « L'Etat de la Recherche » de la SMF (1998), in « Kinetic equations and asymptotic theory », F. Bouchut, F. Golse, M. Pulvirenti, coll. « Series in Appl. Math. », Gauthier-Villars, 2000.

[Bn] Y. Brenier - Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25, 3-4 (2000), 737-754. | MR | Zbl

[Br] H. Brezis - Analyse fonctionnelle, Théorie et applications, Masson, Paris 1983. | MR | Zbl

[Ca1] R. E. Caflisch - The Boltzmann equation with a soft potential, Comm. Math. Phys. 74 (1980), 71-109. | MR | Zbl

[Ca2] R. E. Caflisch - The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33, 5 (1980), 651-666. | MR | Zbl

[Ca3] R. E. Caflisch - Asymptotics of the Boltzmann equation and fluid dynamics, in "Kinetic theory and gas dynamics", Springer, Vienna 1988, pp. 95-133. | MR

[Cn] M. Cannone - Ondelettes, paraproduits et Navier-Stokes, Diderot Ed., Paris 1995. | MR | Zbl

[Cr] T. Carleman - Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1932), 369-424. | JFM | Zbl

[C1] C. Cercignani - The Boltzmann equation and its applications, 2nd ed., Springer, New York 1988. | MR | Zbl

[C2] C. Cercignani - Rarefied gas dynamics, From basic concepts to actual calculations, Cambridge University Press, Cambridge 2000. | MR | Zbl

[CIP] C. Cercignani, R. Illner, M. Pulvirenti - The Mathematical theory of dilute gases, Springer, New York 1994. | MR | Zbl

[CKNS] L. Caffarelli, R. Kohn, L. Nirenberg - Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. | MR | Zbl

[CM] A. J. Chorin, J. E. Marsden - A mathematical introduction to fluid mechanics, 3rd ed., Springer-Verlag, New York 1993. | MR | Zbl

[Co] F. Comets - Limites hydrodynamiques, Séminaire Bourbaki, Vol. 1990-91, Exp. no. 735, Astérisque 201-203 (1992), 167-192. | Numdam | MR | Zbl

[D] C. Dafermos - The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202-212. | MR | Zbl

[DMEL] A. De Masi, R. Esposito, J. L. Lebowitz - Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42, 8 (1989), 1189-1214. | MR | Zbl

[DG] B. Desjardins, E. Grenier - Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 1986 (1999), 2271-2279. | MR | Zbl

[DGLM] B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi - Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9) 78, 5 (1999), 461-471. | MR | Zbl

[DPL1] R. J. Dipierna, P.-L. Lions - On the Cauchy problem for the Boltzmann equation : Global existence and weak stability, Ann. of Math. (2) 130 (1989), 312-366. | MR | Zbl

[DPL2] R. J. Dipierna, P.-L. Lions - Global solutions of Boltzmann's equation and the entropy inequality, Arch. Rational Mech. Anal. 114 (1991), 47-55. | MR | Zbl

[EM] D. Ebin, J. E. Marsden - Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 40 (1970), 102-163. | MR | Zbl

[EP] R. S. Ellis, M. A. Pinsky - The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125- 156. | MR | Zbl

[E] D. Enskog - Kinetische Theorie des Vorgänge in mässig verdünnten Gasen, Uppsala, Almqvist & Wiksell, Trad. Kinetic Theory of Processes in Dilute Gases, in "Kinetic Theory" 3, S.G. Brush, Ed., Pergamon Press, Oxford, 1972, 125-225.

[ELM1] R. Esposito, J. L. Lebowitz, R. Marra - Hydrodynamic limit of the stationary Boltzmann equation in a slab, Comm. Math. Phys. 160, 1 (1994), 49-80. | MR | Zbl

[ELM2] R. Esposito, J. L. Lebowitz, R. Marra - The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, J. Statist. Phys. 78, 1-2 (1995), 389-412. | MR | Zbl

[ELM3] R. Esposito, J. L. Lebowitz, R. Marra - On the derivation of hydrodynamics from the Boltzmann equation, Phys. Fluids 11, 8 (1999), 2354-2366. | MR | Zbl

[Ge1] P. Gérard - Solutions globales du problème de Cauchy pour l'équation de Boltzmann (d'après R.J. DiPerna et P.-L. Lions), Séminaire Bourbaki, Vol. 1988-89, Astérisque 161-162, Exp. no. 699, 5 (1989), 257-281. | Numdam | MR | Zbl

[Ge2] P. Gérard - Résultats récents sur les fluides parfaits incompressibles bidimensionnels (d'après J.-Y. Chemin et J.-M. Delort), Séminaire Bourbaki, Vol. 1991-92, Astérisque 206, Exp. no. 757, 5 (1992), 411-444. | Numdam | MR | Zbl

[GM] Y. Giga, T. Miyakawa - A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J. 50, 2 (1983), 505-515. | MR | Zbl

[Gl] J. Glimm - Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. | MR | Zbl

[Go] F. Golse - From kinetic to macroscopic models, Session « L'Etat de la Recherche » de la SMF (1998). In « Kinetic equations and asymptotic theory », F. Bouchut, F. Golse, M. Pulvirenti, coll. « Series in Appl. Math. », Gauthier-Villars, 2000.

[GL] F. Golse, C. D. Levermore - Stokes-Fourier and acoustic limits for Boltzmann equations : Convergence proofs, Prépublication, 2001. | MR

[GLSR] F. Golse, C. D. Levermore, L. Saint-Raymond - La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques, Sém. EDP École polytechnique, exposé no. XIX, avril 2000. | Numdam | MR | Zbl

[GLPS] F. Golse, P.-L. Lions, B. Perthame, R. Sentis - Regularity of the moments of the solution of the transport equation, J. Funct. Anal. 76 (1988), 110-125. | MR | Zbl

[GPS] F. Golse, B. Perthame, R. Sentis - Un résultat de compacité pour les équations de transport et applications au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris, Série I Math. 301 (1985), 341-344. | MR | Zbl

[GSR] F. Golse, L. Saint-Raymond - The Navier-Stokes limit of the Boltzmann equation : Convergence proof, Manuscrit, 2001. | MR

[Gr1] H. Grad - On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331-407. | MR | Zbl

[Gr2] H. Grad - Principles of the kinetic theory of gases, in "Flügge's Handbuch des Physik", vol. XII. Springer-Verlag, 1958, pp. 205-294. | MR

[Gr3] H. Grad - Asymptotic theory of the Boltzmann equation, II, in "Rarefied Gas Dynamics", 3rd Symposium (1962), pp. 26-59. | MR

[Gre] E. Grenier - Fluides en rotation et ondes d'inertie, C. R. Acad. Sci. Paris Série I Math. 321, 6 (1995), 711-714. | MR | Zbl

[GPV] M. Z. Guo, G. C. Papanicolaou, S. R. S. Varadhan - Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys. 118, 1 (1988), 31-59. | MR | Zbl

[H] D. Hilbert - Begründung des kinetischen Gastheorie, Math. Annalen 72 (1912), 562-577. Trad. Foundations of the Kinetic Theory of Gases, in "Kinetic Theory" 3, S.G. Brush, Ed., Pergamon Press, Oxford 1972, 89-101. | JFM | MR

[IP] R. Illner, M. Pulvirenti - Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result for "Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum", Comm. Math. Phys. 105, 2 (1986), 189-203; Comm. Math. Phys. 121, 1 (1989), 143-146. | Zbl

[K] M. Kac - Probability and related topics in physical sciences, Interscience Publishers, London-New York 1959. | MR | Zbl

[KL] C. Kipnis, C. Landim - Scaling limits of interacting particle systems, Springer-Verlag, Berlin 1999. | MR | Zbl

[KM] S. Klainerman, A. Majda - Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34, 4 (1981), 481-524. | MR | Zbl

[Ld] O. A. Ladyzhenskaya - The mathematical theory of viscous incompressible flow, Gordon and Breach, New York 1969. | MR | Zbl

[LL] L. D. Landau, E. M. Lifshitz - Fluid mechanics, Pergamon Press, London 1959. | MR | Zbl

[La] O. E. Lanford Iii - Time evolution of large classical systems, In "Dynamical systems, theory and applications", Springer, Berlin 1975, pp. 1-111. Lecture Notes in Phys., Vol. 38. | MR | Zbl

[Le1] J. Leray - Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82. | Zbl

[Le2] J. Leray - Essai sur les mouvements plans d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. | JFM | MR

[Le3] J. Leray - Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl. 13 (1934), 331-418. | JFM

[LeMa] C. D. Levermore, N. Masmoudi - Travail en préparation.

[Li1] P.-L. Lions - Compactness in Boltzmann's equation via Fourier integral operators and applications, I, J. Math. Kyoto Univ. 34, 2 (1994), 391-427. | MR | Zbl

[Li2] P.-L. Lions - Compactness in Boltzmann's equation via Fourier integral operators and applications, II, J. Math. Kyoto Univ. 34, 2 (1994), 429-461. | MR | Zbl

[Li3] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 1 : Incompressible models, The Clarendon Press Oxford University Press, New York 1996. | MR | Zbl

[Li4] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 2 : Compressible models. The Clarendon Press Oxford University Press, New York 1998. | MR | Zbl

[LM1] P.-L. Lions, N. Masmoudi - Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9) 77, 6 (1998), 585-627. | MR | Zbl

[LM2] P.-L. Lions, N. Masmoudi - Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Série I Math. 329, 5 (1999), 387-392. | MR | Zbl

[LM3] P.-L. Lions, N. Masmoudi - From Boltzmann equations to incompressible fluid mechanics equations, I, Arch. Rational Mech. Anal. 158, 3 (2001), 173- 193. | MR | Zbl

[LM4] P.-L. Lions, N. Masmoudi - From Boltzmann equations to incompressible fluid mechanics equations, II, Arch. Rational Mech. Anal. 158, 3 (2001) 195-211. | MR | Zbl

[LPS] P.-L. Lions, B. Perthame, P. Souganidis - Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49, 6 (1996), 599-638. | MR | Zbl

[LPT1] P.-L. Lions, B. Perthame, E. Tadmor - A kinetic formulation of multi-dimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7, 1 (1994), 169-191. | MR | Zbl

[LPT2] P.-L. Lions, B. Perthame, E. Tadmor - Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys. 163, 2 (1994), 415-431. | MR | Zbl

[MR] N. B. Maslova, Y. R. Romanovski - Justification of the Hilbert method in the theory of kinetic equations, Zh. Vychisl. Mat. i Mat. Fiz. 27, 11 (1987), 1680-1689,1759. | MR | Zbl

[M] C. Morrey - On the derivation of the equation of hydrodynamics from statistical mechanics, Comm. Pure Appl. Math. 8 (1955), 279-326. | MR | Zbl

[N] T. Nishida - Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys. 61, 2 (1978), 119-148. | MR | Zbl

[OVY] S. Olla, S. R. S. Varadhan, H.-T. Yau - Hydrodynamical limit for a Hamiltonian system with weak noise, Comm. Math. Phys. 155, 3 (1993), 523-560. | MR | Zbl

[QY] J. Quastel, H.-T. Yau - Lattice gases, large deviations, and the incompressible Navier-Stokes equations, Ann. of Math. (2) 148, 1 (1998), 51-108. | MR | Zbl

[SR] L. Saint-Raymond - From the Boltzmann-BGK to the Navier-Stokes equations, Prépublication, 2001.

[Se] S. Schochet - Fast singular limits of hyperbolic PDEs, J. Differential Equations 114, 2 (1994), 476-512. | MR | Zbl

[Si] T. Sideris - Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101, 4 (1985), 475-485. | MR | Zbl

[SATSB] Y. Sone, K. Aoki, S. Takata, H. Sugimoto, A. V. Bobylev - Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit : examination by asymptotic analysis and numerical computation of the Boltzmann equation, Phys. Fluids 8, 2 (1996), 628-638. Erratum dans Phys. Fluids 8, 3 (1996), 841. | MR | Zbl

[S] H. Spohn - Large scale dynamics of interacting particles, Texts and Monographs in Physics, Springer-Verlag, Berlin 1991. | Zbl

[T] R. Temam - Navier-Stokes equations. Theory and numerical analysis, 3rd ed., North-Holland, Amsterdam 1984. | MR | Zbl

[TM] C. Truesdell, R. Muncaster - Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas, Academic Press, New York 1980. | MR

[UA] S. Ukai, K. Asano - The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J. 12, 3, part 1 (1983), 311-332. | MR | Zbl

[Va] S. R. S. Varadhan - Entropy methods in hydrodynamic scaling, In "Nonequilibrium problems in many-particle systems" (Montecatini 1992), Springer, Berlin, 1993, pp. 112-145. | MR | Zbl

[Vi] C. Villani - A review of mathematical problems in collisional kinetic theory, à paraître dans "Handbook of fluid mechanics", D. Serre et S. Friedlander, Ed. | Zbl

[YI] H.-T. Yau - Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22, 1 (1991), 63-80. | MR | Zbl

[Y2] H.-T. Yau - Asymptotic solutions to dynamics of many-body systems and classical continuum equations, Lecture at Current developments in mathematics, 1998. | MR | Zbl