Abstract
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.
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Ambrosio, L., Gigli, N., Savarhé ,G.: Gradient Flows in Metric Spaces and in The Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2008
Bakry D., Barthe F., Cattiaux P., Guillin A.: A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case. Electron. Commun. Probab., 13, 60–66 (2008)
Benedetto D., Caglioti E., Carrillo J.A., Pulvirenti M.: A non Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91(5/6), 979–990 (1998)
Bolley F., Gentil I., Guillin A.: Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations. J. Funct. Anal. 263(8), 2430–2457 (2012)
Bolley F., Guillin A., Malrieu F.: Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. Math. Mod. Num. Anal. 44(5), 867–884 (2010)
Calvez V., Carrillo J.A.: Refined asymptotics for the subcritical Keller–Segel system and related functional inequalities. Proc. AMS 140, 3515–3530 (2012)
Carrillo J.A., McCann R.J., Villani C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19(3), 971–1018 (2003)
Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263 (2006)
Carrillo, J.A., Toscani, G.: Wasserstein metric and large-time asymptotics of nonlinear diffusion equations. In: New Trends in Mathematical Physics. World Scientific, Singapore, 2005
Carrillo J.A., Toscani G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 7(6), 75–198 (2007)
Cattiaux P., Guillin A., Malrieu F.: Probabilistic approach for granular media equations in the non uniformly convex case. Prob. Theor. Rel. Fields 140(1–2), 19–40 (2008)
Cordero-Erausquin, D., Gangbo, W., Houdré, C.: Inequalities for generalized entropy and optimal transportation. In: Recent Advances in the Theory and Applications of Mass Transport, Contemporary Mathematics, Vol. 353. AMS, Providence, 2004
Daneri, S., Savaré, G.: Lecture notes on gradient flows and optimal transport. Séminaires et Congrès, SMF (2012, to appear)
Lisini S.: Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM Contr. Opt. Calc. Var. 15, 712–740 (2009)
Malrieu F.: Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13(2), 540–560 (2003)
McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)
Tugaut, J.: Convergence to the equilibria for self-stabilizing processes in double well landscape. Ann. Prob. (2012, to appear)
Tugaut, J.: Self-stabilizing processes in multi-wells landscape in R d—Invariant probabilities. J. Theor. Prob. (2012, to appear)
Villani, C.: Optimal Transport, Old and New. Grund. Math. Wiss., Vol. 338. Springer, Berlin, 2009
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Communicated by P.-L. Lions
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Bolley, F., Gentil, I. & Guillin, A. Uniform Convergence to Equilibrium for Granular Media. Arch Rational Mech Anal 208, 429–445 (2013). https://doi.org/10.1007/s00205-012-0599-z
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DOI: https://doi.org/10.1007/s00205-012-0599-z