The Navier-Stokes Limit of the Boltzmann equation for bounded collision kernels

Inventiones Mathematicae

Volumn 155, Issue 1, 2004, Pages 81-161

  Golse, François ^a   Saint Raymond, Laure ^b  

^a INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0742323595     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-003-0316-5     Document Type: Article

References (66)

1. Agoshkov, V.: Space of functions with differential difference characteristics and smoothness of solutions of the transport equation. Dokl. Akad. Nauk SSSR 276, 1289-1293 (1984)
2. Arkeryd, L., Nouri, A.: A compactness result related to the stationary Boltzmann equation in a slab with applications to the existence theory. Indiana Univ. Math. J. 44, 815-839 (1995)
3. Asano, K.: Conference at the 4th International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics. Kyoto 1991
4. Bardos, C., Degond, P.: Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst. H. Poincaré, Anal. Non Linéaire 2, 101-118 (1985)
5. Bardos, C., Golse, F., Levermore, D.: Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C. R. Acad. Sci., Paris 309, 727-732 (1989)
6. Bardos, C., Golse, F., Levermore, D.: Fluid Dynamic Limits of Kinetic Equations I: Formal Derivations. J. Stat. Phys. 63, 323-344 (1991)
7. Bardos, C., Golse, F., Levermore, C.D.: Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Commun. Pure Appl. Math. 46, 667-753 (1993)
8. Bardos, C., Golse, F., Levermore, C.D.: Acoustic and Stokes Limits for the Boltzmann Equation. C. R. Acad. Sci., Paris 327, 323-328 (1999)
9. Bardos, C., Golse, F., Levermore, C.D.: The Acoustic Limit for the Boltzmann Equation. Arch. Ration. Mech. Anal. 153, 177-204 (2000)
10. Bardos, C., Levermore, C.D.: Kinetic Equations and an Incompressible Fluid Dynamical Limit that Recovers Viscous Heating. In preparation (2003)
11. Bardos, C., Ukai, S.: The Classical Incompressible Navier-Stokes Limit of the Boltzmann Equation. Math. Models Methods Appl. Sci. 1, 235-257 (1991)
12. Lagha-Benabdallah, A.: Limites des équations d'un fluide compressible lorsque la compressibilité tend vers zéro. (French) Fluid dynamics (Varenna, 1982), 139-165. Lecture Notes Math. 1047. Berlin: Springer 1984
13. Lagha-Benabdallah, A.: Limites des équations d'un fluide compressible lorsque la compressibilité tend vers zéro. (French) Fluid dynamics (Varenna, 1982), 139-165. Lecture Notes Math. 1047. Berlin: Springer 1984
14. Bouchut, F.: Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. 9, 313-327 (2002)
15. Bouchut, F., Desvillettes, L.: A proof of the smoothing properties of the positive part of Boltzmann's kernel. Rev. Mat. Iberoam. 14, 47-61 (1998)
16. Bouchut, F., Golse, F., Pulvirenti, M.: Kinetic Equations and Asymptotic Theory, ed. by B. Perthame, L. Desvillettes, Series in Applied Mathematics 4. Paris: Gauthier-Villars 2000
17. Caflisch, R.: The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Commun. Math. Phys. 74, 71-95 (1980)
18. Castella, F., Perthame, B.: Estimations de Strichartz pour les équations de transport cinétique. C. R. Acad. Sci., Paris, Sér. I, Math. 322, 535-540 (1996)
19. Cercignani, C.: Mathematical Methods in Kinetic Theory. New York: Plenum Press 1990
20. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. New York: Springer 1994
21. Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics. Chicago: The University of Chicago Press 1988
22. DeMasi, A., Esposito, R., Lebowitz, J.: Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation. Commun. Pure Appl. Math. 42, 1189-1214 (1990)
23. Desvillettes, L., Golse, F.: On a model Boltzmann equation without angular cutoff. Differ. Integral Equ. 13, 567-594 (2000)
24. DiPerna, R.J., Lions, P.-L.: On the Cauchy Problem for the Boltzmann Equation: Global Existence and Weak Stability Results. Ann. Math. 130, 321-366 (1990)
25. Dunford, N., Pettis, B.J.: Linear Operations on Summable Functions. Trans. Am. Math. Soc. 47, 323-392 (1940)
26. Golse, F., Levermore, C.D.: The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation. Commun. Pure Appl. Math. 55, 336-393 (2002)
27. Golse, F., Levermore, C.D., Saint-Raymond, L.: La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques. Séminaire: Equations aux Dérivées Partielles, 1999-2000, Exp. No. XIX, 23 pp., Ecole Polytech., Palaiseau 2000
28. Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the Moments of the Solution of a Transport Equation. J. Funct. Anal. 76, 110-125 (1988)
29. Golse, F., Perthame, B., Sentis, R.: Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale de l'opérateur de transport. C. R. Acad. Sci., Paris 301, 341-344 (1985)
30. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit for the Boltzmann equation. C. R. Acad. Sci., Paris 333, 897-902 (2001)
31. 1 for the transport equation. C. R. Acad. Sci., Paris 334, 557-562 (2002)
32. Golse, F., Saint-Raymond, L.: Work in preparation
33. Grad, H.: Asymptotic theory of the Boltzmann equation. II. 1963 Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, pp. 26-59
34. Grenier, E.: Fluides en rotation et ondes d'inertie. C. R. Acad. Sci., Paris 321, 711-714 (1995)
35. Grothendieck, A.: Topological vector spaces. New York: Gordon Breach 1973
36. Hilbert, D.: Mathematical Problems. International Congress of Mathematicians, Paris 1900. Translated and reprinted in Bull. Am. Math. Soc. 37, 407-436 (2000)
37. Hilbert, D.: Mathematical Problems. International Congress of Mathematicians, Paris 1900. Translated and reprinted in Bull. Am. Math. Soc. 37, 407-436 (2000)
38. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481-524 (1981)
39. Lachowicz, M.: On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9, 342-366 (1987)
40. Landau, L., Lifshitz, E.: Course of theoretical physics. Vol. 6. Fluid mechanics. Oxford: Pergamon Press 1987
41. Lanford, O.: Time evolution of large classical systems. In: "Dynamical systems, theory and applications" (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1-111. Lecture Notes Phys. 38. Berlin: Springer 1975
42. Lanford, O.: Time evolution of large classical systems. In: "Dynamical systems, theory and applications" (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1-111. Lecture Notes Phys. 38. Berlin: Springer 1975
43. Leray, J.: Sur le mouvement d'un fluide visqueux emplissant l'espace. Acta Math. 63, 193-248 (1934)
44. Levermore, C.D., Masmoudi, N.: From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System. Work in preparation
45. Lions, J.-L.: Théorèmes de trace et d'interpolation I, II. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 389-403 (1959); 14, 317-331 (1960)
46. Lions, J.-L.: Théorèmes de trace et d'interpolation I, II. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 389-403 (1959); 14, 317-331 (1960)
47. Lions, P.-L.: Conditions at infinity for Boltzmann's equation. Commun. Partial. Differ. Equations 19, 335-367 (1994)
48. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press 1996
49. Lions, P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications I. J. Math. Kyoto Univ. 34, 391-427 (1994)
50. Lions, P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications II. J. Math. Kyoto Univ. 34, 429-461 (1994)
51. Lions, P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications III. J. Math. Kyoto Univ. 34, 539-584 (1994)
52. Lions, P.-L.: On Boltzmann and Landau equations. Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 346, 191-204 (1994)
53. Lions, P.-L., Masmoudi, N.: Une approche locale de la limite incompressible. (French) C. R. Acad. Sci., Paris, Sér. I, Math. 329, 387-392 (1999)
54. Lions, P.-L., Masmoudi, N.: From Boltzmann Equations to Navier-Stokes Equations I. Arch. Ration. Mech. Anal. 158, 173-193 (2001)
55. Lions, P.-L., Masmoudi, N.: From Boltzmann Equations to the Stokes and Euler Equations II. Arch. Ration. Mech. Anal. 158, 195-211 (2001)
56. Masmoudi, N., Saint-Raymond, L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. To appear in Commun. Pure Appl. Math.
57. Perthame, B.: Global Existence to the BGK Model of the Boltzmann Equation. J. Differ. Equations 82, 191-205 (1989)
58. Quastel, J., Yau, H.-T.: Lattice gases, large deviations, and the incompressible Navier-Stokes equations. Ann. Math. (2) 148, 51-108 (1998)
59. Saint-Raymond, L.: Discrete time Navier-Stokes limit for the BGK Boltzmann equation. Commun. Partial Differ. Equations 27, 149-184 (2002)
60. Saint-Raymond, L.: From the Boltzmann BGK Equation to the Navier-Stokes System. Ann. Sci. Éc. Norm. Supér., IV. Sér. 36, 271-317 (2003)
61. Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equations 114, 476-512 (1994)
62. p(0, T; B). Ann. Mat. Pura Appl., IV. Ser. 146, 65-96 (1987)
63. Sone, Y.: Asymptotic Theory of Flow of a Rarefied Gas over a Smooth Boundary II. In: Rarefied Gas Dynamics. Vol. II, pp. 737-749, ed. by D. Dini. Pisa: Editrice Tecnico Scientifica 1971
64. Sone, Y.: Kinetic Theory and Fluid Mechanics. Boston: Birkhäuser 2002
65. Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203, 667-706 (1999)
66. Villani, C.: Limites hydrodynamiques de l'équation de Boltzmann [d'après C. Bardos, F. Golse, D. Levermore, Lions, P.-L., N. Masmoudi, L. Saint-Raymond]. Séminaire Bourbaki, vol. 2000-2001, Exp. 893