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Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

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Abstract

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 — will quantify contractivity (limit expansivity) of the flow.

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Authors and Affiliations

  1. ICREA-Dept. de Matemàtiques, Universitat Autònoma de Barcelona, E-08193, Bellaterra, SPAIN

    José A. Carrillo

  2. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3, CANADA

    Robert J. McCann

  3. U.M.P.A, Ecole Normale Supérieure de Lyon, F-69364, Lyon Cedex 07, FRANCE

    Cédric Villani

Authors
  1. José A. Carrillo
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  2. Robert J. McCann
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  3. Cédric Villani
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Correspondence to José A. Carrillo.

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Communicated by L. Ambrosio

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Carrillo, J., McCann, R. & Villani, C. Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media. Arch. Rational Mech. Anal. 179, 217–263 (2006). https://doi.org/10.1007/s00205-005-0386-1

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