Limites hydrodynamiques de l'équation de Boltzmann

 Access to full text

 Full (PDF)

1. [Al] S. Alinhac — Blowup for nonlinear hyperbolic equations, Progress in nonlinear differential equations and their applications, 17, Birkhäuser, Boston1995. Zbl0820.35001MR1339762
2. [AK] V. Arnold, B. Khesin — Topological methods in hydrodynamics, Springer-Verlag, New York1998. Zbl0902.76001MR1612569
3. [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. Zbl0818.76075MR1316925
4. [AU] K. Asano, S. Ukai — On the fluid dynamical limit of the Boltzmann equation, in "Recent topics in nonlinear PDE" (Hiroshima 1983), North-Holland, Amsterdam1984, pp. 1-19. Zbl0566.76063MR839266
5. [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. Zbl0697.35111MR1054287
6. [BGL2] C. Bardos, F. Golse, C.D. Levermore — Fluid dynamical limits of kinetic equations, I : Formal derivation. J. Statist. Phys.63 (1991), 323- 344. MR1115587
7. [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. Zbl0817.76002MR1213991
8. [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. Zbl0918.35109MR1650310
9. [BGL5] C. Bardos, F. Golse, C.D. Levermore — The acoustic limit for the Boltzmann equation, Arch. Rational Mech. Anal.153, 3 (2000), 177-204. Zbl0973.76075MR1771519
10. [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. Zbl0758.35060MR1115292
11. [Ba] G.K. Batchelor — An introduction to fluid dynamics, 2nd paperback ed., Cambridge University Press, Cambridge1999. Zbl0958.76001MR1744638
12. [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. 
13. [Bn] Y. Brenier — Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations25, 3-4 (2000), 737-754. Zbl0970.35110MR1748352
14. [Br] H. Brezis — Analyse fonctionnelle, Théorie et applications, Masson, Paris1983. Zbl0511.46001MR697382
15. [Ca1] R.E. Caflisch — The Boltzmann equation with a soft potential, Comm. Math. Phys.74 (1980), 71-109. Zbl0434.76066MR575897
16. [Ca2] R.E. Caflisch — The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math.33, 5 (1980), 651-666. Zbl0424.76060MR586416
17. [Ca3] R.E. Caflisch — Asymptotics of the Boltzmann equation and fluid dynamics, in "Kinetic theory and gas dynamics", Springer, Vienna1988, pp. 95-133. MR1007995
18. [Cn] M. Cannone — Ondelettes, paraproduits et Navier-Stokes, Diderot Ed., Paris1995. Zbl1049.35517MR1688096
19. [Cr] T. Carleman — Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math.60 (1932), 369-424. Zbl0006.40002JFM59.0404.02
20. [C1] C. Cercignani — The Boltzmann equation and its applications, 2nd ed., Springer, New York1988. Zbl0646.76001MR1313028
21. [C2] C. Cercignani — Rarefied gas dynamics, From basic concepts to actual calculations, Cambridge University Press, Cambridge2000. Zbl0961.76002MR1744523
22. [CIP] C. Cercignani, R. Illner, M. Pulvirenti - The Mathematical theory of dilute gases, Springer, New York1994. Zbl0813.76001MR1307620
23. [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. Zbl0509.35067MR673830
24. [CM] A.J. Chorin, J.E. Marsden — A mathematical introduction to fluid mechanics, 3rd ed., Springer-Verlag, New York1993. Zbl0712.76008MR1218879
25. [Co] F. Comets — Limites hydrodynamiques, Séminaire Bourbaki, Vol. 1990-91, Exp. no. 735, Astérisque201-203 (1992), 167-192. Zbl0758.76006MR1157842
26. [D] C. Dafermos — The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations14 (1973), 202-212. Zbl0262.35038MR328368
27. [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. Zbl0689.76024MR1029125
28. [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. Zbl0934.76080MR1702718
29. [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. Zbl0992.35067MR1697038
30. [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. Zbl0698.45010MR1014927
31. [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. Zbl0724.45011MR1088276
32. [EM] D. Ebin, J.E. Marsden — Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.40 (1970), 102-163. Zbl0211.57401MR271984
33. [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. Zbl0286.35062MR609540
34. [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. 
35. [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. Zbl0790.76072MR1262191
36. [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. Zbl1078.82525MR1317151
37. [ELM3] R. Esposito, J.L. Lebowitz, R. Marra - On the derivation of hydrodynamics from the Boltzmann equation, Phys. Fluids11, 8 (1999), 2354-2366. Zbl1147.76383MR1719963
38. [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, Astérisque161-162, Exp. no. 699, 5 (1989), 257-281. Zbl0704.76041MR992213
39. [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, Astérisque206, Exp. no. 757, 5 (1992), 411-444. Zbl1154.76321MR1206075
40. [GM] Y. Giga, T. Miyakawa — A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J.50, 2 (1983), 505-515. Zbl0519.35053MR705037
41. [Gl] J. Glimm — Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math.18 (1965), 697-715. Zbl0141.28902MR194770
42. [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. 
43. [GL] F. Golse, C.D. Levermore — Stokes-Fourier and acoustic limits for Boltzmann equations : Convergence proofs, Prépublication, 2001. MR1866367
44. [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. Zbl1255.82067MR1813181
45. [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. Zbl0652.47031MR923047
46. [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. Zbl0591.45007MR808622
47. [GSR] F. Golse, L. Saint-Raymond — The Navier-Stokes limit of the Boltzmann equation : Convergence proof, Manuscrit, 2001. MR1873232
48. [Gr1] H. Grad — On the kinetic theory of rarefied gases, Comm. Pure Appl. Math.2 (1949), 331-407. Zbl0037.13104MR33674
49. [Gr2] H. Grad — Principles of the kinetic theory of gases, in "Flügge's Handbuch des Physik", vol. XII. Springer-Verlag, 1958, pp. 205-294. MR135535
50. [Gr3] H. Grad — Asymptotic theory of the Boltzmann equation, II, in "Rarefied Gas Dynamics", 3rd Symposium (1962), pp. 26-59. MR156656
51. [Gre] E. Grenier - Fluides en rotation et ondes d'inertie, C. R. Acad. Sci. Paris Série I Math.321, 6 (1995), 711-714. Zbl0843.35073MR1354711
52. [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. Zbl0652.60107MR954674
53. [H] D. Hilbert — Begründung des kinetischen Gastheorie, Math. Annalen72 (1912), 562-577. Trad. Foundations of the Kinetic Theory of Gases, in "Kinetic Theory" 3, S.G. Brush, Ed., Pergamon Press, Oxford1972, 89-101. MR1511713JFM43.1055.03
54. [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. Zbl0850.76600
55. [K] M. Kac — Probability and related topics in physical sciences, Interscience Publishers, London-New York1959. Zbl0087.33003MR102849
56. [KL] C. Kipnis, C. Landim — Scaling limits of interacting particle systems, Springer-Verlag, Berlin1999. Zbl0927.60002MR1707314
57. [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. Zbl0476.76068MR615627
58. [Ld] O.A. Ladyzhenskaya — The mathematical theory of viscous incompressible flow, Gordon and Breach, New York1969. Zbl0184.52603MR254401
59. [LL] L.D. Landau, E.M. Lifshitz — Fluid mechanics, Pergamon Press, London1959. Zbl0146.22405MR108121
60. [La] O.E. Lanford III - Time evolution of large classical systems, In "Dynamical systems, theory and applications", Springer, Berlin1975, pp. 1-111. Lecture Notes in Phys., Vol. 38. Zbl0329.70011MR479206
61. [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. Zbl0006.16702
62. [Le2] J. Leray — Essai sur les mouvements plans d'un liquide visqueux emplissant l'espace, Acta Math.63 (1934), 193-248. MR1555394JFM60.0726.05
63. [Le3] J. Leray — Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl.13 (1934), 331-418. Zbl60.0727.01JFM60.0727.01
64. [LeMa] C.D. Levermore, N. Masmoudi — Travail en préparation. 
65. [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. Zbl0831.35139MR1284432
66. [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. Zbl0831.35139MR1284432
67. [Li3] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 1 : Incompressible models, The Clarendon Press Oxford University Press, New York1996. Zbl0866.76002MR1422251
68. [Li4] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 2 : Compressible models. The Clarendon Press Oxford University Press, New York1998. Zbl0908.76004MR1637634
69. [LM1] P.-L. Lions, N. Masmoudi — Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9) 77, 6 (1998), 585-627. Zbl0909.35101MR1628173
70. [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. Zbl0937.35132MR1710123
71. [LM3] P.-L. Lions, N. Masmoudi — From Boltzmann equations to incompressible fluid mechanics equations, I, Arch. Rational Mech. Anal.158, 3 (2001), 173- 193. Zbl0987.76088MR1842343
72. [LM4] P.-L. Lions, N. Masmoudi - From Boltzmann equations to incompressible fluid mechanics equations, II, Arch. Rational Mech. Anal.158, 3 (2001) 195-211. Zbl0987.76088MR1842343
73. [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. Zbl0853.76077MR1383202
74. [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. Zbl0820.35094MR1201239
75. [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. Zbl0799.35151MR1284790
76. [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. Zbl0663.76090MR927914
77. [M] C. Morrey — On the derivation of the equation of hydrodynamics from statistical mechanics, Comm. Pure Appl. Math.8 (1955), 279-326. Zbl0065.19406MR70527
78. [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. Zbl0381.76060MR503305
79. [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. Zbl0781.60101MR1231642
80. [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. Zbl0919.35102MR1652971
81. [SR] L. Saint-Raymond — From the Boltzmann-BGK to the Navier-Stokes equations, Prépublication, 2001. 
82. [Se] S. Schochet — Fast singular limits of hyperbolic PDEs, J. Differential Equations114, 2 (1994), 476-512. Zbl0838.35071MR1303036
83. [Si] T. Sideris — Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys.101, 4 (1985), 475-485. Zbl0606.76088MR815196
84. [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. Fluids8, 2 (1996), 628-638. Erratum dans Phys. Fluids8, 3 (1996), 841. Zbl1027.76649MR1371533
85. [S] H. Spohn — Large scale dynamics of interacting particles, Texts and Monographs in Physics, Springer-Verlag, Berlin1991. Zbl0742.76002
86. [T] R. Temam — Navier-Stokes equations. Theory and numerical analysis, 3rd ed., North-Holland, Amsterdam1984. Zbl0568.35002MR769654
87. [TM] C. Truesdell, R. Muncaster — Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas, Academic Press, New York1980. MR554086
88. [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. Zbl0525.76062MR719971
89. [Va] S.R.S. Varadhan — Entropy methods in hydrodynamic scaling, In "Nonequilibrium problems in many-particle systems" (Montecatini1992), Springer, Berlin, 1993, pp. 112-145. Zbl0791.60098MR1296260
90. [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. Zbl1170.82369
91. [YI] H.-T. Yau — Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys.22, 1 (1991), 63-80. Zbl0725.60120MR1121850
92. [Y2] H.-T. Yau — Asymptotic solutions to dynamics of many-body systems and classical continuum equations, Lecture at Current developments in mathematics, 1998. Zbl1036.82500MR1772325

1. François Golse, New Results in Velocity Averaging