On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions

Journal of Statistical Physics

Volumn 89, Issue 3-4, 1997, Pages 751-776

  Goudon, T ^a  

^a UNIVERSITÉ BORDEAUX 1   (France)

Author keywords

Grazing collisions; Homogeneous Boltzmann equation; Kinetic equation; Landau Fokker Planck equation

Indexed keywords

EID: 0037823935     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02765543     Document Type: Article

References (34)

1. L. Arkeryd, On the Boltzmann equation, Arch. Rat. Mech. Anal. 45:1-34 (1972).
2. L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rat. Mech. Anal. 77:11-21 (1981).
3. A. Arsenev and O. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation. Math. Sbornyk 181:4, 465 477 (1991).
4. A. Bers and J. L. Delcroix, Physique des plasmas (InterEdition/CNRS, 1994).
5. H. Brezis, Analyse fonctionnelle (Masson, 1993).
6. C. Cercignani, The Boltzmann equation and its applications (Springer-Verlag, 1988).
7. C. Cercignani, R. Illner, and M. Pulvirenti, The mathematical theory of dilute gases (Springer-Verlag, 1994).
8. F. Chvala and R. Pettersson, Weak solutions of the linear Boltzmann equation with very soft interactions, J. Math. Anal. and Appl. 191:360-379 (1995).
9. F. Chvala, T. Gustafsson, and R. Pettersson, On solutions to the linear Boltzmann equation with external electromagnetic force, SIAM J. Math. Anal. 24:3, 583-602 (1993).
10. P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann operator in the Coulomb case, Math. Models and Meth. in the Appl. Sci. 2:2, 167-182 (1992).
11. L. Desvillettes, On the asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory in Slat. Phys. 21:259-276 (1992).
12. L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rat. Mech. Anal. 123:387-404 (1993).
13. R. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equation: Global existence and weak stability, Ann. Math. 130:321-366 (1989).
14. R. Di Perna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys. 120:1-23 (1988).
15. T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rat. Mech. Anal. 82:1-12 (1983).
16. T. Goudon, Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau Fokker-Planck: Influence des collisions rasantes, CRAS 324:265-270 (1997).
17. T. Goudon, Sur quelques questions relatives à la théorie cinétique des gaz et à l'équation de Boltzmann, Thèse Université Bordeaux 1 (1997).
18. H. Grad, Asymptotic theory of the Boltzmann equation in Rarefied Gas Dynamic, Third Symposium, Paris, Laurmann Ed., pp. 26-59 (Ac. Press 1963).
19. K. Hamdache, Sur l'existence globale et le comportement asymptotique de quelques solutions de l'équation de Boltzmann, Thèse Université Paris 6 (1986).
20. K. Hamdache, Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck, CRAS 302:187-190 (1986).
21. L. Hormander, The analysis of linear pde, Vol. 1 (Springer, 1983).
22. N. A. Krall and A. W. Trivelpiece, Principles of plasma physics (Mc Graw-Hill, 1964).
23. E. Lifshitz and L. Pitaevski, Cinétique Physique. Coll. "Physique Théorique," L. Landau E. Lifshitz (Mir, 1990).
24. P. L. Lions. Compactness in Boltzmann equation via Fourier integral operators and applications. Part I, II and III, J. Math. Kyoto Univ. 34:2, (39) 391-461 (1994), and J. Math. Kyoto Univ. 34:3, 539-584 (1994).
25. P. L. Lions. Compactness in Boltzmann equation via Fourier integral operators and applications. Part I, II and III, J. Math. Kyoto Univ. 34:2, (39) 391-461 (1994), and J. Math. Kyoto Univ. 34:3, 539-584 (1994).
26. R. Pettersson, Existence theorems for the linear space homogeneous transport equation, IMA J. Appl. Math. 30:81-105 (1983).
27. R. Pettersson, On solutions and higher moments for the linear Boltzmann equation with infinite-range forces, IMA J. Appl. Math. 38:151-166 (1987).
28. R. Pettersson, On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces, J. Stat. Phys. 59:1/2, 403-440 (1990).
29. M. Reed and B. Siimon, Methods of modern mathematical physics., Vol. 2 (Ac. Press, 1970).
30. S. Semmes, A primer on Hardy spaces, Comm. Part. Diff. Eqt. 19:277-319 (1994).
31. p(0, T; B), Ann. Mat. Pura Appl. IV:146, 65-96 (1987).
32. S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math. 1:141-156 (1984).
33. C. Villani, On a newr class of weak solutions to the Boltzmannn and Landau equations. Personal communication (1997).
34. B. Wennberg, On moments and uniqueness for solutions to the space homogeneous Boltzmann equation. Transp. Th. Stat. Phys. 24:4, 533-539 (1994).