The incompressible navier–stokes for the nonlinear discrete velocity models

Journal of Nonlinear Mathematical Physics

Volumn 9, Issue 4, 2002, Pages 426-445

  Bellouquid, A ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036435154     PISSN: 14029251     EISSN: 17760852     Source Type: Journal    
DOI: 10.2991/jnmp.2002.9.4.4     Document Type: Article

References (39)

1. Babovsky, M. 2002. A Kinetic Multiscale Model. Math. Models Meth. Appl. Sci., 12:209–234.
2. Bardos, C, Golse, F, and Levermore, D. 1989. Sur les limites asymptotiques de la théorie cinétique conduisant a la dynamique des fluides incompressibles. C. R. Acad. Sci. Paris, 309:727–732.
3. Bardos, C, Golse, F, and Levermore, D. 1992. Fluid Dynamic Limits of Kinetic Equations I, Formal Derivations. Transp. Theory Statist. Phys., 21:531–555.
4. Bardos, C, Golse, F, and Levermore, D. 1991. “Fluid Dynamical Limits of Discrete Velocity Kinetic Equations”. In Advances in Kinetic Theory and Continuum Mechanics, Edited by:Gatignol, et Soubbaramayer R., 57–71. Berlin, New York:Springer-Verlag.
5. Bardos, C, Golse, F, and Levermore, D. 1993. Fluid Dynamic Limits of Kinetic Equations:Convergence Proofs for the Boltzmann Equation. Comm. Pure Appl. Math., 46:667–753.
6. Bardos, C, and Ukai, S. 1991. The Classical Incompressible Navier–Stokes Limit of the Boltzmann Equations. Math. Models Meth. Appl. Sci., 2:235–257.
7. Bellomo, N, and Kawashima, S. 1990. The Discrete Boltzmann Equation with Multiple Collisions, Global Existence and Stability for the Initial Value Problem. J. Math. Phys., 31:245–253.
8. Bellomo, N, and Gustafsson, T. 1991. The Discrete Boltzmann Equation:A Review of the Mathematical Aspects of The Initial and Initial-Boundary Value Problems. Reviews in Math. Phys., 3:137–162.
9. Bellouquid, A. 1995. Limite asymptotique pour le Modèle de Carleman. C. R. Acad. Sci. Paris, 321:655–658.
10. Bellouquid, A. 1999. The Hydrodynamical Limit of the Nonlinear Boltzmann Equation. J. Transp. Theor. Stati. Phys., 28:25–57.
11. Bellouquid, A. 2000. The Hydrodynamical Limit of Discrete Kinetic Equations. C. R. Acad. Sci. Paris, 330:951–956.
12. Bellouquid, A. 2003. Diffusive Limit for Nonlinear Discrete Velocity Models. To appear in Math. Models Meth. Appl. Sci., 13
13. Bellouquid A, Thèse de Doctorat, Université paris (7)1996
14. Bonilla, L, and Soler, J S. 2001. High-Field Limit of the Vlasov Poisson Fokker Plank System-A Comparison of Different Perturbation Methods. Math. Models Meth. Appl. Sci., 11:1457–1468.
15. Bony J M, Existence globale à données de Cauchy Petites pour les modèles Discrets de l’Equation de Boltzmann, Preprint centre de Mathématiques Ecole Polytechnique 91128 palaiseau
16. Boltzmann, L. 1872. Weitere Studien uber das Warmegleichgewicht unter Gas-molekulen. Sitzungs. Akad. Wiss. Wein, 66:275–370. in English:Further Studies on the Thermal Equilibrium of Gas Molecules, in Kinetic Theory, Vol. 2, Editor:Brush S G, Pergamon Press, London, 1966, 88–174
17. Broadwell, J E. 1964. Shock Structure in a Simple Discrete Velocity Gas. The Physics of Fluids, 7:1243–1247.
18. Cabannes H, The Discrete Boltzmann Equation (Theory and Applications), Lecture Notes, Univ. of California, Berkely, 1980
19. Caflish, R, and Papanicolaou, G. 1979. The Fluid-Dynamical Limit of a Nonlinear Model Boltzmann Equation. Comm. Pure Appl. Math., 32:589–616.
20. Carlemann T, Problémes mathématiques dans la théorie cinétique des gaz. Appsala. Publications scientifiques de l’institut Mittag-Leffler, 1957
21. De, Masi A, Esposito, R, and Lebowitz, J L. 1989. Incompressible Navier–Stokes and Euler Limits of the Boltzmann Equation. Comm. Pure Appl. Maths., 42:321–366.
22. Diperna, R J, and Lions, P L. 1989. On the Cauchy Problem for the Boltzmann Equation:Global Existence and Weak Stability Results. Ann. Math., 130:321–366.
23. Gatignol, R. 1975. Théorie cinétique des gaz à répartition discrète de vitesses, Lecture Notes in Phys, Vol. 36, New York:Springer-Verlag.
24. Golse, F, Lions, P, Perthame, B, and Sentis, R. 1988. Regularity of the Moments of of the Solution of a Transport Equation. J. of Funct. Anal., 76:110–125.
25. Golse F, Levermore D, Stokes–Fourier and Acoustic Limits for the Boltzmann Equation:Convergence Proofs, Preprint, Ecole normale supérieure, DMA-01–12, 2001
26. Golse, F, and Saint-Raymond, L. 2001. The Navier–Stokes Limit for the Boltzmann Equation. C. R. Acad. Sci. Paris, 333:897–902.
27. Gorsch, D. 2002. Generalized Discrete Velocity Models. Math. Models Meth. Appl. Sci., 12:49–75.
28. Hammouch J, These de l‘Universite Paris (7)1996
29. Harris, S. 1966. Approach to Equilibrium in a Moderately Dense Discrete Velocity Gas. The Physics of Fluids, 9:1328–1334.
30. Inoue, K, and Nishida, T. 1976. On the Broadwell Model of the Boltzmann Equation for a Simple Discrete Velocity Gas. Appl. Math. Optim., 3:259–272.
31. Kawashima S, Global Existence and Stability of Solutions for Discrete Velocity Models of the Boltzmann Equation, Lecture Notes in Num. Appl. Anal., 1983, 59–85. Recent Topics in Nolinear PDE, Hiroshima, 1983
32. Kawashima, S. 1987. Large-Time Behavior of Solutions of the Discrete Boltzmann Equation. Commun. Math. Phys., 109:563–589.
33. Kurtz, T G. 1973. Convergence of Sequences of Semigroups of Nonlinear Operators with an Application to Gas Kinetics. Trans. Amer. Math. Soc., 186:259–272.
34. Lachowicz M, Asymptotic Analysis of Nonlinear Kinetic Equations The Hydrodynamic Limit, in Lecture Notes on the Mathematical Theory of the Boltzmann Equation, Editor:Bellomo N, World Sci., London, Singapore, 1995
35. Leray, J. 1934. Sur le mouvement d‘un liquide visqueux emplissant l’espace. Acta Math., 63:193–248.
36. Lions, P, and Toscani, G. 1997. Diffusive Limits for Finite Velocity Boltzmann Kinetic Models. Revista Mat. Iberoamer., 13:473–513.
37. Lions, P, and Masmoudi, N. 2001. From Boltzmann Equation to the Equation of Incompressible Fluid Mechanics I. Arch. Rat. Mech. Anal., 158:173–193.
38. Maxwell, J C. 1866. On the Dynamical Theory of Gases. Phil. Trans. Roy. Soc. London, 157:49–88. also in The Scientific Papers of James Clerk Maxwell, Vol. 2, Dover, New York, 1965, 26–78
39. Tartar, L. 1975. Existence globale pour un systeme semi lineaire de la theorie cinetique des gas, seminaire Goulaouic Schwartz. Ecole Polytechnique Palaiseau, France,:475–485.