Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Journal of Statistical Physics

Volumn 111, Issue 3-4, 2003, Pages 931-966

  Guérin, Hélène ^a   Méléard, Sylvie ^a,b  

^a UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^b University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

Interacting particle systems; Landau equation with soft potential; Monte Carlo algorithm; Nonlinear stochastic differential equations; Soft potential Boltzmann equations without cutoff

Indexed keywords

EID: 0347123342     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022858517569     Document Type: Article

References (28)

1. A. A. Arsenev and O. E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik 69:465-478 (1991).
2. C. Buet, S. Cordier, P. Degond, and M. Lemou, Fast algorithms for numerical, conservative and entropic approximations of the Fokker-Planck-Landau equation, J. Comput. Phys. 133:310-322 (1997).
3. C. Buet, S. Cordier, and B. Lucquin-Desreux, The grazing collision limit for the Boltzmann-Lorentz model, Asymptot. Anal. 25:93-107 (2001).
4. P. Degon and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Mod. Meth. Appl. Sci. 2:167-182 (1992).
5. L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys. 21:259-276 (1992).
6. L. Desvillettes, About the regularizing properties of the non-Cut-off Kac equation, Comm. Math. Phys. 168:416-440 (1995).
7. L. Desvillettes, C. Graham, and S. Méléard, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff, Stochastic Process. Appl. 84:115-135 (1999).
8. N. El Karoui and J. P. Lepeltier, Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson, Z. Wahr. Verw. Geb. 39:111-133 (1977).
9. N. Fournier, Existence and regularity study for a 2-dimensional Kac equation without cutoff by a probabilistic approach, Ann. Appl. Probab. 10:434-462 (2000).
10. N. Fournier and S. Méléard, Monte-Carlo approximations and fluctuations for 2D Boltzmann equations without cutoff, Inhomogeneous random systems (Cergy-Pontoise, 2000), Markov Process. Related Fields 7:159-191 (2001).
11. N. Fournier and S. Méléard, A Markov process associated with a Boltzmann equation without cutoff and for non Maxwell molecules, J. Statist. Phys. 104:359-385 (2001).
12. N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff, Math. Comp. 71:583-604 (2002).
13. T. Goudon, Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau: influence des collisions rasantes, CRAS Paris 324:265-270 (1997).
14. C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab. 25:115-132 (1997).
15. C. Graham and S. Méléard, Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations, Comm. Math. Phys. 205:551-569 (1999).
16. H. Guérin, Solving Landau equation for some soft potentials through a probabilistic approach, to appear in Ann. Appl. Probab. (2002).
17. H. Guérin, Existence and regularity of a weak function-solution for some Landau equation with a stochastic approach, Stochastic Process. Appl. 101:303-325 (2002).
18. J. Horowitz and R. L. Karandikar, Martingale problem associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989, E. Cinlar, K. L. Chung, and R. K. Getoor, eds. (Birkhäuser, Boston, 1990).
19. J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes (Springer-Verlag, 1987).
20. E. M. Lifchitz and L. P. Pitaevskii, Physical Kinetics - Course in Theorical Physics (Pergamon, Oxford, 10, 1981).
21. S. Méléard, A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation, Ann. Appl. Probab. 10:1197-1211 (2000).
22. L. Pareschi, G. Toscani, and C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, to appear in Numer. Math. (2002).
23. A. S. Sznitman, Topics in Propagation of Chaos, Ecole d'été de Saint-Flour XIX, 1989, L.N. in Math. 1464 (Springer, 1991).
24. H. Tanaka, On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules, in Proc. Intern. Symp. SDE, Kyoto (1976), pp. 409-425.
25. H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46:67-105 (1978).
26. T. Takizuka and H. Abe, A binary collision model for plasma simulation with a particle code, J. Comput. Phys. 25:205-219 (1977).
27. C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, ARMA 143:273-307 (1998).
28. W. X. Wang, M. Okamoto, N. Nakajima, and S. Murakami, Vector implementation of nonlinear Monte-Carlo Coulomb algorithms, J. Comput. Phys. 128:209-229 (1996).