The work of Cédric Villani

Proceedings of the International Congress of Mathematicians 2010, ICM 2010

Volumn , Issue , 2010, Pages 86-98

  Yau, Horng Tzer ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84877904575     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (52)

1. Alexandre, R., Desvillettes, L., Villani, C. and Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152, no.4, 327-355 (2000).
2. Alexandre, R. and Villani, C.: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55, no.1, 30-70, (2002).
3. Alexandre, R. and Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincar Anal. Non Linaire 21, no. 1, 61-95, (2004).
4. Backus, G: Linearized plasma oscillations in arbitrary electron velocity distributions. J. Mathematical Phys. 1 (1960), 178-191; erratum, 559.
5. Bardos, C.; Golse, F.; Levermore, C. D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46 (1993), no. 5, 667-753.
6. Baranger, C.; Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana 21, no. 3, 819-841 (2005).
7. Bobylev, A. V. and Cercignani, C.: On the rate of entropy production for the Boltzmann equation. J. Stat. Phys. 94, 603-618 (1999).
8. Bonciocat, A. I. and Sturm, K. S.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256, no. 9, 2944-2966 (2009).
9. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math., 44, no. 4, 375-417 (1991).
10. Caglioti, E., and Maffei, C.: Time asymptotics for solutions of Vlasov-Poisson equation in a circle. J. Statist. Phys. 92, 1-2, 301-323 (1998).
11. Cercignani, C.: H-theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech. 34, 231-241 (1982).
12. Carlen, E. and Carvalho, M.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67, 575-608 (1992)
13. Cordero-Erausquin, D.; R. McCann and Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146, no. 2, 219-257 (2001).
14. Cordero-Erausquin, D.; Nazaret, B. and Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182 (2004), no. 2, 307-332.
15. Desvillettes, L. and Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159, 245-316 (2005).
16. DiPerna, R. J., and Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2) 130, 2 (1989), 321-366.
17. Golse, F. and Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004), no. 1, 81-161.
18. Chung, F. R. K. and Yau, S.T.: Logarithmic Harnack inequalities Math. Res. Lett. 3 (1996), no. 6, 793-812.
19. Gualdani M.; Mischler S., Mouhot C.:Factorization for non-symmetric operators and exponential H-theorem, arXiv:1006.5523
20. Figalli, A., Rifford, L. and Villani, C.: Nearly round spheres look convex. Preprint (2009).
21. Figalli, A.; Maggi, F.; Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities, to appear in Invent. Math.
22. Gressman, P. and Strain, R.: Global strong solutions of the Boltzmann equation without angular cut-off, arXiv:0912.0888v1
23. Gressman, P. and Strain, R.: Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials, arXiv:1002.3639.
24. Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305-353.
25. Guo, Y.: The Landau equation in a periodic box. Comm. Math. Phys. 231 (2002), no. 3, 391-434.
26. Hwang, J.-H. and Velazquez, J.: On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem. Indiana Univ. Math. J. 58, no. 6, 2623-2660 (2009).
27. Landau, L.: On the vibration of the electronic plasma. J. Phys. USSR 10 (1946), 25.
28. Lin, Y. and Yau, S.T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17 (2010), no. 2, 343-356.
29. Lin, Z. W., Zeng, C. C.: Small BGK waves and nonlinear Landau damping, arXiv:1003.3005
30. Loeper, G. and Villani, C.: Regularity of optimal transport in curved geometry: the nonfocal case. To appear in Duke Math. J.
31. McCann, R. J.: A convexity principle for interacting gases. Adv. Math. 128 (1997), no. 1, 153-179
32. McCann, R. J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11, no. 3, 589-608 (2001).
33. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Comm. Math. Phys. 261, no. 3, 629-672 (2006).
34. Mouhot, C.; Strain, R.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. 87, no. 5, 515-535 (2007).
35. Mouhot, C., and Villani, C.: On Landau damping. arXiv:0904.2760, (2009).
36. Lott, J. and Villani, C.: Ricci curvature via optimal transport. Ann. Math. 169 (2009), 903-991
37. Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 (2009), no. 3, 810-864.
38. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174.
39. Otto, F. and Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), no. 2, 361-400.
40. Penrose, O.: Electrostatic instability of a non-Maxwellian plasma. Phys. Fluids 3, 258-265 (1960).
41. von Renesse, M.-K. and Sturm, K.-T.: Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58 (2005), no. 7, 923-940.
42. Saenz, A. W.: Long-time behavior of the electric potential and stability in the linearized Vlasov theory. J. Mathematical Phys. 6 (1965), 859-875.
43. Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65-131.
44. Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133-177.
45. Toscani, G. and Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203, 667-706 (1999)
46. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143, no. 3, 273-307 (1998).
47. Villani, C.: Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234, 3, 455-490 (2003).
48. Villani, C.: A review of mathematical topics in collisional kinetic theory. In Handbook of mathematical fluid dynamics, Vol. I. North-Holland, Amsterdam, 2002, pp. 71-305.
49. Villani, C.: Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.
50. Villani, C.: Hypocoercivity. Mem. Amer. Math. Soc. 202, no. 950 (2009).
51. Villani, C.: Optimal transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2009.
52. Yu, S. H.: Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., (2010).