Contractions in the 2-Wasserstein length space and thermalization of granular media

Archive for Rational Mechanics and Analysis

Volumn 179, Issue 2, 2006, Pages 217-263

  Carrillo, José A ^a   McCann, Robert J ^b   Villani, Cédric ^c  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITY OF TORONTO   (Canada)

^c UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 32544445429     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-005-0386-1     Document Type: Article

References (51)

1. AGUEH, M.: Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. PhD Thesis, Georgia Institute of Technology, 2002
2. AGUEH, M., GHOUSSOUB, N., KANG, X.: Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14, 215-244 (2004)
3. AGUEH, M., GHOUSSOUB, N., KANG, X.: The optimal evolution of the free energy of interacting gases and its applications. C. R. Math. Acad. Sci. Paris, Ser. 1337, 173-178 (2003)
4. AGUEH, M.: Asymptotic behavior for doubly degenerate parabolic equations. C. R. Math. Acad. Sci. Paris, Ser. I 337, 331-336 (2003)
5. ALBEVERIO, S., RÖCKNER, M.: Classical Dirichlet froms on topological vector spacesclosability and a Cameron-Martin formula. J. Funct. Anal. 88, 395-436 (1990)
6. ALT, H.W., LUCKHAUS, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311-341 (1983)
7. AMBROSIO, L.A., GIGLI, N., SAVARÉ, G.: Gradient flows with metric and differentiable structures, and applications to the Wasserstein space. To appear in the proceedings of the meeting "Nonlinear Evolution Equations" held in the Academy of Lincei in Rome.
8. AMBROSIO, L.A., GIGLI, N., SAVARÉ, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics, Birkhäuser, 2005
9. BALL, K., CARLEN, E.A., LIEB, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463-482 (1994)
10. BAKRY, E., EMERY, M.: Diffusions hypercontractives. In: Sem. Pmbab. XIX LNM 1123. Springer, New York, 1985, pp. 177-206
11. BENEDETTO, D., CAGLIOTI, E., PULVIRENTI, M.: A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31, 615-641 (1997)
12. BENEDETTO, D., CAGLIOTI, E., CARRILLO, J.A., PULVIRENTI, M.: A non-maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91, 979-990 (1998)
13. BENEDETTO, D., CAGLIOTI, E., GOLSE, F., PULVIRENTI, M.: A hydrodynamic model arising in the context of granular media. Comput. Math. Appl. 38, 121-131 (1999)
14. BERTSCH, M., HILHORST, D.: A density dependent diffusion equation in population dynamics: stabilization to equilibrium. SIAM J. Math. Anal. 17, 863-883 (1986)
15. BIANE, P., SPEICHER, R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37, 581-606 (2001)
16. BLOWER, G.: Displacement convexity for the generalized orthogonal ensemble. J. Statist. Phys. 116, 1359-1387 (2004)
17. BOLLEY, F., BRENIER, Y., LOEPER, G.: Contractive metrics for scalar conservation laws. To appear in Journal of Hyperbolic Differential Equations.
18. BRENIER, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375-417 (1991)
19. CARLEN, E., GANGBO, W.: Constrained steepest descent in the 2-Wasserstein metric. Annals Math. 157, 807-846 (2003)
20. CARRILLO, J.A., FELLNER, K.: Long time asymptotics via entropy methods for diffusion dominated equations. Asymptotic Analysis 42, 29-54 (2005)
21. CARRILLO, J.A., GUALDANI, M.P., TOSCANI, G.: Finite speed of propagation for the porous medium equation by mass transportation methods. C. R. Math. Acad. Sci. Paris, Ser. I 338, 815-818 (2004)
22. CARRILLO, J.A., JUNGEL, A., MARKOWICH, P.A., TOSCANI, G., UNTERREITER, A.: Entropy dissipation methods for degenerate parabolic systems and generalized Sobolev inequalities. Monatsh. Math. 133, 1-82 (2001)
23. CARRILLO, J.A., MCCANN, R.J., VILLANI, C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Matemática Iberoamericana 19, 1-48 (2003)
24. 1 -decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113-141 (2000)
25. CARRILLO, J.A., TOSCANI, G.: Wasserstein metric and large-time asymptotics of nonlinear diffusion equations. In: New Trends in Mathematical Physics, (In Honour of the Salvatore Rionero 70th Birthday). World Scientific, 2005
26. CORDERO-ERAUSQUIN, D.: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257-269 (2002)
27. CORDERO-ERAUSQUIN, D., GANGBO, W., HOUDRE, C.: Inequalities for generalized entropy and optimal transportation. In: Recent advances in the theory and applications of mass transport, 73-94, Contemp. Math. 353, Amer. Math. Soc., Providence, RI, 2004
28. DUDLEY, R.M.: Probabilities and metrics - Convergence of laws on metric spaces, with a view to statistical testing. Universitet Matematisk Institut, Aarhus, Denmark, 1976
29. GANGBO, W., MCCANN, R.J.: Shape recognition via Wasserstein distance. Quart. J. Appl. Math. 4, 705-737 (2000)
30. GIVENS, C.R., SHORTT, R.M.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31, 231-240 (1984)
31. GROMOV, M.: Structures métriques pour les variétés riemanniennes. Lafontaine, J., and Pansu, P. (eds.) Cedic/Fernand Nathan, Paris, 1981
32. GROMOV, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Lafontaine, J., Pansu, P. (eds.) With appendices by S. Semmes. Birkhauser, Boston, 1999
33. GROSS, L.: Logarithmic Sobolev inequalities. Amer. J. of Math. 97, 10610-1083 (1975)
34. KANTOROVICH, L.V., RUBINSTEIN, G.S.: On a space of completely additive functions. Vestnik Leningrad. Univ. 13, 52-59 (1958)
35. Li, H., TOSCANI, G.: Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172, 407-428 (2004)
36. LOTT, J., VILLANI, C.: Ricci curvature for metric-measure spaces via optimal transport. Preprint a thttp://www.math.lsa.umich.edu/~lott/
37. MCCANN, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309-323 (1995)
38. MCCANN, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153-179 (1997)
39. MCCANN, R.J.: Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys. 195, 699-723 (1998)
40. MCCANN, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589-608 (2001)
41. OTTO, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101-174 (2001)
42. OTTO, F., VILLANI, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal 173, 361-400 (2001)
43. RACHEV, S.T., RÜSCHENDORF, L.: Mass transportation problems. In: Probability and its Applications. Springer-Verlag, New York, 1998
44. STURM, K.-T.: Convex functionals of probability measures and nonlinear diffusions on manifolds. To appear in J. Math. Pures Appl.
45. STURM, K.-T.: Generalized Ricci bounds and convergence of metric measure spaces. To appear in C. R. Acad. Sci. Paris Sér. I Math.
46. STURM, K.-T.: On the geometry of metric measure spaces. SFB Preprint #203, Bonn. http://www-wt.iam.uni-borin.de/~sturm/en/index.html
47. STURM, K.-T., VON RENESSE, M.-K.: Transport inequalities, gradient estimates, entropy and Ricci curvature. To appear in Comm. Pure Appl. Math.
48. TALAGRAND, M. : Transportation cost for Gaussian and other transport measures. Geom. Func. Anal. 6, 587-600 (1996)
49. TOSCANI, G.: One-dimensional kinetic models of granular flows. RAIRO Model. Math. Anal. Numér. 34, 1277-1291 (2000)
50. VILLANI, C.: Topics in optimal transportation. Graduate Studies in Mathematics Vol. 58. Amer. Math. Soc, Providence, 2003
51. WASSERSTEIN, L.N.: Markov processes over denumerable products of spaces describing large systems of automata. Problems of Information Transmission 5, 47-52 (1969)