On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

Journal of Statistical Physics

Volumn 131, Issue 4, 2008, Pages 749-781

  Fournier, Nicolas ^a   Guérin, Hélène ^b  

^a UFR des Sciences et Technologie   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Boltzmann equation without cutoff; Long range interaction; Quadratic cost; Uniqueness; Wasserstein distance

Indexed keywords

EID: 41849151052     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-008-9511-5     Document Type: Article

References (23)

1. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327-355 (2000)
2. Bhatt, A., Karandikar, R.: Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21(4), 2246-2268 (1993)
3. Cercignani, C.: The Boltzmann Equation and its Applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988)
4. Desvillettes, L.: Boltzmann's Kernel and the Spatially Homogeneous Boltzmann Equation. Riv. Mat. dell'Univ. Parma 6(4), 1-22 (2001) (special issue)
5. Desvillettes, L., Graham, C., Méléard, S.: Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stoch. Process. Appl. 84(1), 115-135 (1999)
6. Desvillettes, L., Mouhot, C.: Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. arXiv eprint math.AP/0606307 (2006)
7. Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)
8. Fontbona, J., Guérin, H., Méléard, S.: Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Preprint (2007)
9. Fournier, N.: Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff. J. Stat. Phys. 125(4), 927-946 (2006)
10. Fournier, N., Méléard, S.: A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput. 71(238), 583-604 (2002)
11. Fournier, N., Méléard, S.: A weak criterion of absolute continuity for jump processes: application to the Boltzmann equation. Bernoulli 8(4), 537-558 (2002)
12. Fournier, N., Mouhot, C.: On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Preprint (2007)
13. Horowitz, J., Karandikar, R.L.: Martingale problems associated with the Boltzmann equation. In: Seminar on Stochastic Processes, San Diego, CA, 1989. Progr. Probab., vol. 18, pp. 75-122. Birkhäuser, Boston (1990)
14. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam (1981)
15. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften, vol. 288. Springer, Berlin (1987)
16. Lu, X., Mouhot, C.: About measures solutions of the spatially homogeneous Boltzmann equation. Work in progress
17. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 467-501 (1999)
18. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheor. Verwandte. Geb. 46(1), 67-105 (1978)
19. Tanaka, H.: On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In: Proceedings of the International Symposium on Stochastic Differential Equations. Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976, pp. 409-425. Wiley, New York (1978)
20. Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94(3-4), 619-637 (1999)
21. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273-307 (1998)
22. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, vol. I, pp. 71-305. North-Holland, Amsterdam (2002)
23. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003)