Limites hydrodynamiques de l'équation de Boltzmann
Séminaire Bourbaki : volume 2000/2001, exposés 880-893, Astérisque, no. 282 (2000-2001), Exposé no. 893, p. 365-405
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     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},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {282},
     year = {2000-2001},
     note = {talk:893},
     pages = {365-405},
     zbl = {01969562},
     mrnumber = {1975186},
     language = {fr},
     url = {http://www.numdam.org/item/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 (2000-2001), Exposé no. 893, pp. 365-405. 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 1339762 | Zbl 0820.35001

[AK] V. Arnold, B. Khesin - Topological methods in hydrodynamics, Springer-Verlag, New York 1998. | MR 1612569 | Zbl 0902.76001

[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 1316925 | Zbl 0818.76075

[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 839266 | Zbl 0566.76063

[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 1054287 | Zbl 0697.35111

[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 1115587

[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 1213991 | Zbl 0817.76002

[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 1650310 | Zbl 0918.35109

[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 1771519 | Zbl 0973.76075

[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 1115292 | Zbl 0758.35060

[Ba] G.K. Batchelor - An introduction to fluid dynamics, 2nd paperback ed., Cambridge University Press, Cambridge 1999. | MR 1744638 | Zbl 0958.76001

[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 1748352 | Zbl 0970.35110

[Br] H. Brezis - Analyse fonctionnelle, Théorie et applications, Masson, Paris 1983. | MR 697382 | Zbl 0511.46001

[Ca1] R.E. Caflisch - The Boltzmann equation with a soft potential, Comm. Math. Phys. 74 (1980), 71-109. | MR 575897 | Zbl 0434.76066

[Ca2] R.E. Caflisch - The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33, 5 (1980), 651-666. | MR 586416 | Zbl 0424.76060

[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 1007995

[Cn] M. Cannone - Ondelettes, paraproduits et Navier-Stokes, Diderot Ed., Paris 1995. | MR 1688096 | Zbl 1049.35517

[Cr] T. Carleman - Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1932), 369-424. | JFM 59.0404.02 | Zbl 0006.40002

[C1] C. Cercignani - The Boltzmann equation and its applications, 2nd ed., Springer, New York 1988. | MR 1313028 | Zbl 0646.76001

[C2] C. Cercignani - Rarefied gas dynamics, From basic concepts to actual calculations, Cambridge University Press, Cambridge 2000. | MR 1744523 | Zbl 0961.76002

[CIP] C. Cercignani, R. Illner, M. Pulvirenti - The Mathematical theory of dilute gases, Springer, New York 1994. | MR 1307620 | Zbl 0813.76001

[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 673830 | Zbl 0509.35067

[CM] A.J. Chorin, J.E. Marsden - A mathematical introduction to fluid mechanics, 3rd ed., Springer-Verlag, New York 1993. | MR 1218879 | Zbl 0712.76008

[Co] F. Comets - Limites hydrodynamiques, Séminaire Bourbaki, Vol. 1990-91, Exp. no. 735, Astérisque 201-203 (1992), 167-192. | Numdam | MR 1157842 | Zbl 0758.76006

[D] C. Dafermos - The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202-212. | MR 328368 | Zbl 0262.35038

[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 1029125 | Zbl 0689.76024

[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 1702718 | Zbl 0934.76080

[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 1697038 | Zbl 0992.35067

[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 1014927 | Zbl 0698.45010

[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 1088276 | Zbl 0724.45011

[EM] D. Ebin, J.E. Marsden - Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 40 (1970), 102-163. | MR 271984 | Zbl 0211.57401

[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 609540 | Zbl 0286.35062

[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 1262191 | Zbl 0790.76072

[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 1317151 | Zbl 1078.82525

[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 1719963 | Zbl 1147.76383

[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 992213 | Zbl 0704.76041

[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 1206075 | Zbl 00434753

[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 705037 | Zbl 0519.35053

[Gl] J. Glimm - Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. | MR 194770 | Zbl 0141.28902

[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 1866367

[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 1813181 | Zbl 02124215

[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 923047 | Zbl 0652.47031

[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 808622 | Zbl 0591.45007

[GSR] F. Golse, L. Saint-Raymond - The Navier-Stokes limit of the Boltzmann equation : Convergence proof, Manuscrit, 2001. | MR 1873232

[Gr1] H. Grad - On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331-407. | MR 33674 | Zbl 0037.13104

[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 135535

[Gr3] H. Grad - Asymptotic theory of the Boltzmann equation, II, in "Rarefied Gas Dynamics", 3rd Symposium (1962), pp. 26-59. | MR 156656

[Gre] E. Grenier - Fluides en rotation et ondes d'inertie, C. R. Acad. Sci. Paris Série I Math. 321, 6 (1995), 711-714. | MR 1354711 | Zbl 0843.35073

[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 954674 | Zbl 0652.60107

[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 43.1055.03 | MR 1511713

[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 0850.76600

[K] M. Kac - Probability and related topics in physical sciences, Interscience Publishers, London-New York 1959. | MR 102849 | Zbl 0087.33003

[KL] C. Kipnis, C. Landim - Scaling limits of interacting particle systems, Springer-Verlag, Berlin 1999. | MR 1707314 | Zbl 0927.60002

[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 615627 | Zbl 0476.76068

[Ld] O.A. Ladyzhenskaya - The mathematical theory of viscous incompressible flow, Gordon and Breach, New York 1969. | MR 254401 | Zbl 0184.52603

[LL] L.D. Landau, E.M. Lifshitz - Fluid mechanics, Pergamon Press, London 1959. | MR 108121 | Zbl 0146.22405

[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 479206 | Zbl 0329.70011

[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 0006.16702

[Le2] J. Leray - Essai sur les mouvements plans d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. | JFM 60.0726.05 | MR 1555394

[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 60.0727.01

[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 1284432 | Zbl 0831.35139

[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 1284432 | Zbl 0831.35139

[Li3] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 1 : Incompressible models, The Clarendon Press Oxford University Press, New York 1996. | MR 1422251 | Zbl 0866.76002

[Li4] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 2 : Compressible models. The Clarendon Press Oxford University Press, New York 1998. | MR 1637634 | Zbl 0908.76004

[LM1] P.-L. Lions, N. Masmoudi - Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9) 77, 6 (1998), 585-627. | MR 1628173 | Zbl 0909.35101

[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 1710123 | Zbl 0937.35132

[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 1842343 | Zbl 0987.76088

[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 1842343 | Zbl 0987.76088

[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 1383202 | Zbl 0853.76077

[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 1201239 | Zbl 0820.35094

[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 1284790 | Zbl 0799.35151

[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 927914 | Zbl 0663.76090

[M] C. Morrey - On the derivation of the equation of hydrodynamics from statistical mechanics, Comm. Pure Appl. Math. 8 (1955), 279-326. | MR 70527 | Zbl 0065.19406

[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 503305 | Zbl 0381.76060

[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 1231642 | Zbl 0781.60101

[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 1652971 | Zbl 0919.35102

[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 1303036 | Zbl 0838.35071

[Si] T. Sideris - Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101, 4 (1985), 475-485. | MR 815196 | Zbl 0606.76088

[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 1371533 | Zbl 1027.76649

[S] H. Spohn - Large scale dynamics of interacting particles, Texts and Monographs in Physics, Springer-Verlag, Berlin 1991. | Zbl 0742.76002

[T] R. Temam - Navier-Stokes equations. Theory and numerical analysis, 3rd ed., North-Holland, Amsterdam 1984. | MR 769654 | Zbl 0568.35002

[TM] C. Truesdell, R. Muncaster - Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas, Academic Press, New York 1980. | MR 554086

[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 719971 | Zbl 0525.76062

[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 1296260 | Zbl 0791.60098

[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 01942873

[YI] H.-T. Yau - Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22, 1 (1991), 63-80. | MR 1121850 | Zbl 0725.60120

[Y2] H.-T. Yau - Asymptotic solutions to dynamics of many-body systems and classical continuum equations, Lecture at Current developments in mathematics, 1998. | MR 1772325 | Zbl 1036.82500