A diffusive limit for nonlinear discrete velocity models

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 1, 2003, Pages 35-58

  Bellouquid, A ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

Broadwell model; Carleman model; Diffusive limit; Discrete Boltzmann equation; Incompressible Stokes equations

Indexed keywords

EID: 0037288629     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002374     Document Type: Article

References (39)

1. M. Babovsky, A kinetic multiscale model, Math. Models Methods Appl. Sci. 12 (2002) 309-332.
2. C. Bardos, F. Golse and D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris 309 (1989) 727-732.
3. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I, Formal derivations, Transp. Theory Statist. Phys. 21 (1992) 531-555.
4. C. Bardos, F. Golse and D. Levermore, Fluid dynamical limits of discrete velocity kinetic equations, in Advances in Kinetic Theory and Continuum Mechanics, eds. R. Gatignol and Soubbaramayer (Springer-Verlag, 1991), pp. 57-71.
5. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations: Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (1993) 667-753.
6. C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equations, Math. Models Methods Appl. Sci. 2 (1991) 235-257.
7. N. Bellomo and S. Kawashima, The discrete Boltzmann equation with multiple collisions, global existence and stability for the initial value problem, J. Maths. Phys. 31 (1990) 245-253.
8. N. Bellomo and T. Gustafsson, The discrete Boltzmann equation: A review of the mathematical aspects of the initial and initial-boundary value problems, Rev. Math. Phys. 3 (1991) 137-162.
9. A. Bellouquid, Limite asymptotique pour le Modèle de Carleman, C. R. Acad. Sci. Paris 321 (1995) 655-658.
10. A. Bellouquid, The hydrodynamical limit of the nonlinear Boltzmann equation, J. Transp. Theor Statist. Phys. 28 (1999) 25-57.
11. A. Bellouquid, The hydrodynamical limit of discrete kinetic equations, C. R. Acad. Sci. Paris 330 (2000) 951-956.
12. A. Bellouquid, The nonlinear incompressible Navier-Stokes for the nonlinear discrete velocity models, submitted.
13. A. Bellouquid, Thèse de Doctorat (1995), Université Paris 7.
14. L. Bonilla and J. S. Soler, High-field limit of the Vlasov-Poisson-Fokker-Planck system: A comparison of different perturbation methods, Math. Models Methods Appl. Sci. 11 (2001) 1457-1468.
15. J. M. Bony, Existence globale à données de Cauchy petites pour les modules discrets de l'équation de Boltzmann, preprint centre de mathématiques, Ecole Polytechnique 91128 Palaiseau.
16. J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. Fluids 7 (1964) 1243-1247.
17. H. Cabannes, The Discrete Boltzmann Equation (Theory and Applications), Lecture Notes, Univ. of California, Berkeley (1980).
18. R. Caflish, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980) 651-666.
19. R. Caflish and G. Papanicolaou, The fluid-dynamical limit of a nonlinear model Boltzmann equation, Comm. Pure Appl. Math. 32 (1979) 589-616.
20. T. Carlemann, Problèmes mathématiques dans la théorie cinétique des gaz, Appsala. Publications scientifiques de l'institut Mittag-Leffler (1957).
21. A. De Masi, R. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Maths. 42 (1989) 321-366.
22. R. J. Diperna and P. L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results, Ann. Math. 130 (1989) 321-366.
23. R. Gatignol, Théorie Cinétique des Gaz à Répartition Discrète de Vitesses, Lecture Notes in Phys. 36 (Springer-Verlag, 1975).
24. F. Golse and D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation: Convergence proofs, preprint, Ecole Normale, DMA, 01-12 (2001).
25. D. Gorsch, Generalized discrete velocity models, Math. Models Methods Appl. Sci. 12 (2002) 49-76.
26. H. Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Amer. Math. Soc. Symp. On Applications of Partial Differential Equations (1964), pp. 154-183.
27. H. Grad, Asymptotic therory of the Boltzmann equation II, in Rarefied Gas Dynamics 1, ed. J. Laurmann (New York Acad. Press, 1963), pp. 26-59.
28. H. Grad, Singular and non uniform limits of solutions of the Boltzmann equation, Transport Theory, 1, SIAM-AMS Proceedings (1969), pp. 269-308.
29. S. Harris, Approach to equilibrium in a moderately dense discrete velocity gas, Phys. Fluids 9 (1966) 1328.
30. K. Inoue and T. Nishida, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Appl. Math. Optim. 3 (1976) 259-272.
31. S. Kawashima, 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. S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation, Comm. Math. Phys. 109 (1987) 563-589.
33. T. G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973) 259-272.
34. M. Lachowicz, Asymptotic analysis of nonlinear kinetic equations - The hydrodynamic limit, in Lecture Notes on the Mathematical Theory of the Boltzmann Equation, ed. N. Bellomo (World Scientific, 1995).
35. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934) 193-248.
36. P. Lions and G. Toscani, Diffusive limits for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamer. 13 (1997) 473-513.
37. P. Lions and N. Masmoudi, From Boltzmann equation to the equation of incompressible fluid mechanics, I, Arch. Rational. Mech. Anal. 158 (2001) 173-193.
38. T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the incompressible Euler equation, Comm. Math. Phys. 61 (1978) 119-148.
39. S. Ukai and K. Asano, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J. 12 (1983) 311-332.