Uniform Convergence to Equilibrium for Granular Media

We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media. We prove a uniform exponential convergence to equilibrium for degenerately convex and nonconvex interaction or confinement potentials, improving in particular results by J. A. Carrillo, R. J. McCann and C. Villani. The method is based on studying the dissipation of the Wasserstein distance between a solution and the steady state.

[1]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .

[2]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[3]  S. Glotzer,et al.  Time-course gait analysis of hemiparkinsonian rats following 6-hydroxydopamine lesion , 2004, Behavioural Brain Research.

[4]  A. Guillin,et al.  Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation , 2009, 0906.1417.

[5]  J. Tugaut,et al.  Self-stabilizing Processes in Multi-wells Landscape in ℝd-Invariant Probabilities , 2014 .

[6]  F. Malrieu.  Convergence to equilibrium for granular media equations and their Euler schemes , 2003 .

[7]  Julian Tugaut,et al.  Convergence to the equilibria for self-stabilizing processes in double-well landscape , 2013, 1305.5725.

[8]  Giuseppe Toscani,et al.  WASSERSTEIN METRIC AND LARGE-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS , 2005 .

[9]  Vincent Calvez,et al.  Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities , 2010, 1007.2837.

[10]  G. Toscani,et al.  Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations , 2007 .

[11]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .

[12]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[13]  Christian Houdré,et al.  Inequalities for generalized entropy and optimal transportation , 2003 .

[14]  Emanuele Caglioti,et al.  A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media , 1998 .

[15]  S. Lisini.  Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces , 2009 .

[16]  C. Villani.  Optimal Transport: Old and New , 2008 .

[17]  D. Bakry,et al.  A simple proof of the Poincaré inequality for a large class of probability measures , 2008 .

[18]  Sara Daneri,et al.  Lecture notes on gradient flows and optimal transport , 2010, Optimal Transport.

[19]  V. Sidoravicius,et al.  New Trends in Mathematical Physics , 2009 .

[20]  P. Cattiaux,et al.  Probabilistic approach for granular media equations in the non-uniformly convex case , 2006, math/0603541.