Granular media equation with double-well external landscape: limiting steady state

TL;DR

This paper establishes a simple initial condition criterion for the granular media equation with a double-well landscape, ensuring convergence to a unique steady state with positive center of mass, using functional inequalities and probabilistic methods.

Contribution

It provides a novel simple condition on initial states that guarantees convergence to the steady state in the granular media equation with a double-well landscape.

Findings

Identifies a specific initial condition ensuring convergence to the steady state.

Uses functional inequalities and Laplace method for analysis.

Connects the granular media equation to McKean-Vlasov diffusion.

Abstract

In this paper, we give a simple condition on the initial state of the granular media equation which ensures that the limit as the time goes to infinity is the unique steady state with positive center of mass. To do so, we use functional inequalities, Laplace method and McKean-Vlasov diffusion (which corresponds to the probabilistic interpretation of the granular media equation).