×

zbMATH — the first resource for mathematics

Contractions in the 2-Wasserstein length space and thermalization of granular media. (English) Zbl 1082.76105
Summary: An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) – if uniformly controlled – quantify contractivity (limit expansivity) of the flow.

MSC:
76T25 Granular flows
74E20 Granularity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. PhD Thesis, Georgia Institute of Technology, 2002 · Zbl 1103.35051
[2] Agueh, M., Ghoussoub, N., Kang, X.: Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14, 215–244 (2004) · Zbl 1122.82022
[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. I 337, 173–178 (2003) · Zbl 1095.49012
[4] Agueh, M.: Asymptotic behavior for doubly degenerate parabolic equations. C. R. Math. Acad. Sci. Paris, Ser. I 337, 331–336 (2003) · Zbl 1029.35144
[5] Albeverio, S., Röckner, M.: Classical Dirichlet froms on topological vector spaces–closability and a Cameron-Martin formula. J. Funct. Anal. 88, 395–436 (1990) · Zbl 0737.46036
[6] Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983) · Zbl 0497.35049
[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. · Zbl 1162.35349
[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 · Zbl 1090.35002
[9] Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463–482 (1994) · Zbl 0803.47037
[10] Bakry, E., Emery, M.: Diffusions hypercontractives. In: Sem. Probab. XIX LNM 1123. Springer, New York, 1985, pp. 177–206 · Zbl 0561.60080
[11] Benedetto, D., Caglioti, E., Pulvirenti, M.: A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31, 615–641 (1997) · Zbl 0888.73006
[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) · Zbl 0921.60057
[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) · Zbl 0946.76096
[14] Bertsch, M., Hilhorst, D.: A density dependent diffusion equation in population dynamics: stabilization to equilibrium. SIAM J. Math. Anal. 17, 863–883 (1986) · Zbl 0607.35052
[15] Biane, P., Speicher, R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37, 581–606 (2001) · Zbl 1020.46018
[16] Blower, G.: Displacement convexity for the generalized orthogonal ensemble. J. Statist. Phys. 116, 1359–1387 (2004) · Zbl 1097.82012
[17] Bolley, F., Brenier, Y., Loeper, G.: Contractive metrics for scalar conservation laws. To appear in Journal of Hyperbolic Differential Equations. · Zbl 1071.35081
[18] Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991) · Zbl 0738.46011
[19] Carlen, E., Gangbo, W.: Constrained steepest descent in the 2-Wasserstein metric. Annals Math. 157, 807–846 (2003) · Zbl 1038.49040
[20] Carrillo, J.A., Fellner, K.: Long time asymptotics via entropy methods for diffusion dominated equations. Asymptotic Analysis 42, 29–54 (2005) · Zbl 1211.35158
[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) · Zbl 1049.35110
[22] Carrillo, J.A., Jüngel, 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) · Zbl 0984.35027
[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) · Zbl 1073.35127
[24] Carrillo, J.A., Toscani, G.: Asymptotic L1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–141 (2000) · Zbl 0963.35098
[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 · Zbl 1089.76055
[26] Cordero-Erausquin, D.: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257–269 (2002) · Zbl 0998.60080
[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 · Zbl 1135.49026
[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 · Zbl 0355.60004
[29] Gangbo, W., McCann, R.J.: Shape recognition via Wasserstein distance. Quart. J. Appl. Math. 4, 705–737 (2000) · Zbl 1039.49038
[30] Givens, C.R., Shortt, R.M.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31, 231–240 (1984) · Zbl 0582.60002
[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 · Zbl 0953.53002
[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) · Zbl 0082.11001
[35] Li, H., Toscani, G.: Long–time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172, 407–428 (2004) · Zbl 1116.82025
[36] Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Preprint at http://www.math.lsa.umich.edu/ott/ · Zbl 1178.53038
[37] McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309–323 (1995) · Zbl 0873.28009
[38] McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997) · Zbl 0901.49012
[39] McCann, R.J.: Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys. 195, 699–723 (1998) · Zbl 0936.74029
[40] McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589–608 (2001) · Zbl 1011.58009
[41] Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101–174 (2001) · Zbl 0984.35089
[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) · Zbl 0985.58019
[43] Rachev, S.T., Rüschendorf, L.: Mass transportation problems. In: Probability and its Applications. Springer-Verlag, New York, 1998 · Zbl 0990.60500
[44] Sturm, K.-T.: Convex functionals of probability measures and nonlinear diffusions on manifolds. To appear in J. Math. Pures Appl. · Zbl 1259.49074
[45] Sturm, K.-T.: Generalized Ricci bounds and convergence of metric measure spaces. To appear in C. R. Acad. Sci. Paris Sér. I Math. · Zbl 1092.28010
[46] Sturm, K.-T.: On the geometry of metric measure spaces. SFB Preprint #203, Bonn. http://www-wt.iam.uni-bonn.de/turm/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. · Zbl 1078.53028
[48] Talagrand, M.: Transportation cost for Gaussian and other transport measures. Geom. Func. Anal. 6, 587–600 (1996) · Zbl 0859.46030
[49] Toscani, G.: One-dimensional kinetic models of granular flows. RAIRO Modél. Math. Anal. Numér. 34, 1277–1291 (2000) · Zbl 0981.76098
[50] Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics Vol. 58. Amer. Math. Soc, Providence, 2003 · Zbl 1106.90001
[51] Wasserstein, L.N.: Markov processes over denumerable products of spaces describing large systems of automata. Problems of Information Transmission 5, 47–52 (1969)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.