Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces

TL;DR

This paper develops Lyapunov functionals to analyze the convergence to equilibrium of damped Euler equations with interaction forces in Wasserstein distance, including the overdamped limit relevant for granular media modeling.

Contribution

It introduces new tools for constructing Lyapunov functionals on probability measures to study convergence in Wasserstein distance for damped Euler systems with interaction forces.

Findings

Proves convergence to equilibrium in Wasserstein distance for the damped Euler system.

Analyzes the overdamped limit to a nonlocal equation in granular media modeling.

Provides rigorous proofs for specific one-dimensional cases.

Abstract

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension.