Archive for Rational Mechanics and Analysis

, Volume 179, Issue 2, pp 217–263

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

  • José A. Carrillo
  • Robert J. McCann
  • Cédric Villani
Article

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • José A. Carrillo
    • 1
  • Robert J. McCann
    • 2
  • Cédric Villani
    • 3
  1. 1.ICREA-Dept. de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSPAIN
  2. 2.Department of MathematicsUniversity of TorontoTorontoCANADA
  3. 3.U.M.P.AEcole Normale Supérieure de LyonFRANCE

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