@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}, series = {Ast\'erisque}, note = {talk:893}, pages = {365--405}, publisher = {Soci\'et\'e math\'ematique de France}, number = {282}, year = {2002}, mrnumber = {1975186}, zbl = {01969562}, 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 SP - 365 EP - 405 IS - 282 PB - Société mathématique de France UR - http://www.numdam.org/item/SB_2000-2001__43__365_0/ 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 %P 365-405 %N 282 %I Société mathématique de France %U http://www.numdam.org/item/SB_2000-2001__43__365_0/ %G fr %F SB_2000-2001__43__365_0
Villani, Cédric. Limites hydrodynamiques de l'équation de Boltzmann, dans Séminaire Bourbaki : volume 2000/2001, exposés 880-893, Astérisque, no. 282 (2002), Exposé no. 893, 41 p. http://www.numdam.org/item/SB_2000-2001__43__365_0/
[Al] Blowup for nonlinear hyperbolic equations, Progress in nonlinear differential equations and their applications, 17, Birkhäuser, Boston 1995. | MR | Zbl
-[AK] Topological methods in hydrodynamics, Springer-Verlag, New York 1998. | MR | Zbl
, -[As] 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] 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] 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] Fluid dynamical limits of kinetic equations, I : Formal derivation. J. Statist. Phys. 63 (1991), 323- 344. | MR
, , -[BGL3] 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] Acoustic and Stokes limits for the Boltzmann equation, C. R. Acad. Sci. Paris, Série I Math. 327, 3 (1998), 323-328. | MR | Zbl
, , -[BGL5] The acoustic limit for the Boltzmann equation, Arch. Rational Mech. Anal. 153, 3 (2000), 177-204. | MR | Zbl
, , -[BU] The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci. 1, 2 (1991), 235-257. | MR | Zbl
, -[Ba] An introduction to fluid dynamics, 2nd paperback ed., Cambridge University Press, Cambridge 1999. | MR | Zbl
-[Bo] 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] Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25, 3-4 (2000), 737-754. | MR | Zbl
-[Br] Analyse fonctionnelle, Théorie et applications, Masson, Paris 1983. | MR | Zbl
-[Ca1] The Boltzmann equation with a soft potential, Comm. Math. Phys. 74 (1980), 71-109. | MR | Zbl
-[Ca2] The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33, 5 (1980), 651-666. | MR | Zbl
-[Ca3] Asymptotics of the Boltzmann equation and fluid dynamics, in "Kinetic theory and gas dynamics", Springer, Vienna 1988, pp. 95-133. | MR
-[Cn] Ondelettes, paraproduits et Navier-Stokes, Diderot Ed., Paris 1995. | MR | Zbl
-[Cr] Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1932), 369-424. | JFM | Zbl
-[C1] The Boltzmann equation and its applications, 2nd ed., Springer, New York 1988. | MR | Zbl
-[C2] Rarefied gas dynamics, From basic concepts to actual calculations, Cambridge University Press, Cambridge 2000. | MR | Zbl
-[CIP] The Mathematical theory of dilute gases, Springer, New York 1994. | MR | Zbl
, , -[CKNS] Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. | MR | Zbl
, , -[CM] A mathematical introduction to fluid mechanics, 3rd ed., Springer-Verlag, New York 1993. | MR | Zbl
, -[Co] Limites hydrodynamiques, Séminaire Bourbaki, Vol. 1990-91, Exp. no. 735, Astérisque 201-203 (1992), 167-192. | Numdam | MR | Zbl
-[D] The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202-212. | MR | Zbl
-[DMEL] Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42, 8 (1989), 1189-1214. | MR | Zbl
, , -[DG] 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] 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] On the Cauchy problem for the Boltzmann equation : Global existence and weak stability, Ann. of Math. (2) 130 (1989), 312-366. | MR | Zbl
, -[DPL2] Global solutions of Boltzmann's equation and the entropy inequality, Arch. Rational Mech. Anal. 114 (1991), 47-55. | MR | Zbl
, -[EM] Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 40 (1970), 102-163. | MR | Zbl
, -[EP] The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125- 156. | MR | Zbl
, -[E] 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] Hydrodynamic limit of the stationary Boltzmann equation in a slab, Comm. Math. Phys. 160, 1 (1994), 49-80. | MR | Zbl
, , -[ELM2] The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, J. Statist. Phys. 78, 1-2 (1995), 389-412. | MR | Zbl
, , -[ELM3] On the derivation of hydrodynamics from the Boltzmann equation, Phys. Fluids 11, 8 (1999), 2354-2366. | MR | Zbl
, , -[Ge1] 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] 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] A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J. 50, 2 (1983), 505-515. | MR | Zbl
, -[Gl] Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. | MR | Zbl
-[Go] 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] Stokes-Fourier and acoustic limits for Boltzmann equations : Convergence proofs, Prépublication, 2001. | MR
, -[GLSR] 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] Regularity of the moments of the solution of the transport equation, J. Funct. Anal. 76 (1988), 110-125. | MR | Zbl
, , , -[GPS] 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] The Navier-Stokes limit of the Boltzmann equation : Convergence proof, Manuscrit, 2001. | MR
, -[Gr1] On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331-407. | MR | Zbl
-[Gr2] Principles of the kinetic theory of gases, in "Flügge's Handbuch des Physik", vol. XII. Springer-Verlag, 1958, pp. 205-294. | MR
-[Gr3] Asymptotic theory of the Boltzmann equation, II, in "Rarefied Gas Dynamics", 3rd Symposium (1962), pp. 26-59. | MR
-[Gre] Fluides en rotation et ondes d'inertie, C. R. Acad. Sci. Paris Série I Math. 321, 6 (1995), 711-714. | MR | Zbl
-[GPV] Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys. 118, 1 (1988), 31-59. | MR | Zbl
, , -[H] 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] 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] Probability and related topics in physical sciences, Interscience Publishers, London-New York 1959. | MR | Zbl
-[KL] Scaling limits of interacting particle systems, Springer-Verlag, Berlin 1999. | MR | Zbl
, -[KM] 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] The mathematical theory of viscous incompressible flow, Gordon and Breach, New York 1969. | MR | Zbl
-[LL] Fluid mechanics, Pergamon Press, London 1959. | MR | Zbl
, -[La] 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] Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82. | Numdam | Zbl
-[Le2] Essai sur les mouvements plans d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. | JFM | MR
-[Le3] Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl. 13 (1934), 331-418. | JFM | Numdam
-[LeMa]
, - Travail en préparation.[Li1] Compactness in Boltzmann's equation via Fourier integral operators and applications, I, J. Math. Kyoto Univ. 34, 2 (1994), 391-427. | MR | Zbl
-[Li2] Compactness in Boltzmann's equation via Fourier integral operators and applications, II, J. Math. Kyoto Univ. 34, 2 (1994), 429-461. | MR | Zbl
-[Li3] Mathematical topics in fluid mechanics. Vol. 1 : Incompressible models, The Clarendon Press Oxford University Press, New York 1996. | MR | Zbl
-[Li4] Mathematical topics in fluid mechanics. Vol. 2 : Compressible models. The Clarendon Press Oxford University Press, New York 1998. | MR | Zbl
-[LM1] Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9) 77, 6 (1998), 585-627. | MR | Zbl
, -[LM2] Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Série I Math. 329, 5 (1999), 387-392. | MR | Zbl
, -[LM3] From Boltzmann equations to incompressible fluid mechanics equations, I, Arch. Rational Mech. Anal. 158, 3 (2001), 173- 193. | MR | Zbl
, -[LM4] From Boltzmann equations to incompressible fluid mechanics equations, II, Arch. Rational Mech. Anal. 158, 3 (2001) 195-211. | MR | Zbl
, -[LPS] 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] A kinetic formulation of multi-dimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7, 1 (1994), 169-191. | MR | Zbl
, , -[LPT2] Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys. 163, 2 (1994), 415-431. | MR | Zbl
, , -[MR] 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] On the derivation of the equation of hydrodynamics from statistical mechanics, Comm. Pure Appl. Math. 8 (1955), 279-326. | MR | Zbl
-[N] 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] Hydrodynamical limit for a Hamiltonian system with weak noise, Comm. Math. Phys. 155, 3 (1993), 523-560. | MR | Zbl
, , -[QY] Lattice gases, large deviations, and the incompressible Navier-Stokes equations, Ann. of Math. (2) 148, 1 (1998), 51-108. | MR | Zbl
, -[SR] From the Boltzmann-BGK to the Navier-Stokes equations, Prépublication, 2001.
-[Se] Fast singular limits of hyperbolic PDEs, J. Differential Equations 114, 2 (1994), 476-512. | MR | Zbl
-[Si] Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101, 4 (1985), 475-485. | MR | Zbl
-[SATSB] 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] Large scale dynamics of interacting particles, Texts and Monographs in Physics, Springer-Verlag, Berlin 1991. | Zbl
-[T] Navier-Stokes equations. Theory and numerical analysis, 3rd ed., North-Holland, Amsterdam 1984. | MR | Zbl
-[TM] Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas, Academic Press, New York 1980. | MR
, -[UA] The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J. 12, 3, part 1 (1983), 311-332. | MR | Zbl
, -[Va] Entropy methods in hydrodynamic scaling, In "Nonequilibrium problems in many-particle systems" (Montecatini 1992), Springer, Berlin, 1993, pp. 112-145. | MR | Zbl
-[Vi] A review of mathematical problems in collisional kinetic theory, à paraître dans "Handbook of fluid mechanics", D. Serre et S. Friedlander, Ed. | Zbl
-[YI] Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22, 1 (1991), 63-80. | MR | Zbl
-[Y2] Asymptotic solutions to dynamics of many-body systems and classical continuum equations, Lecture at Current developments in mathematics, 1998. | MR | Zbl
-