Revista Matemática Iberoamericana


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Volume 19, Issue 3, 2003, pp. 971–1018
DOI: 10.4171/RMI/376

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates

José A. Carrillo[1], Robert J. McCann[2] and Cédric Villani[3]

(1) Department of Mathematics, Imperial College London, Huxley Building, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM
(2) Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., M5S 2E4, TORONTO, ONTARIO, CANADA
(3) UMPA, CNRS, École Normale Supérieure de Lyon, 46 allée d'Italie, 69364, LYON CEDEX 07, FRANCE

The long-time asymptotics of certain nonlinear, nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [BCCP98] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogeneous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities, via either the Bakry-Emery method or the abstract approach of Otto and Villani [OV00].

Keywords: Rates of convergence, generalized log-Sobolev inequalities, Wasserstein distance, inelastic collision models

Carrillo José, McCann Robert, Villani Cédric: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003), 971-1018. doi: 10.4171/RMI/376