Uniform convergence to equilibrium for granular media

TL;DR

This paper proves uniform exponential convergence to equilibrium for a nonlinear, nonlocal equation modeling granular media, extending previous results by analyzing Wasserstein distance dissipation.

Contribution

It introduces a new method based on Wasserstein distance dissipation to establish convergence for both convex and non-convex potentials, improving prior results.

Findings

Established uniform exponential convergence to equilibrium.

Applied the method to both convex and non-convex potentials.

Enhanced understanding of long-term behavior in granular media models.

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 non convex 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.