Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

TL;DR

This paper investigates local minimizers of one-dimensional interaction energies with repulsive-attractive power-law potentials, establishing uniqueness under certain Wasserstein metrics and characterizing stability of steady states.

Contribution

It proves the uniqueness of the sum of two Dirac masses as local minimizers under the $ ext{W}_ ext{lambda}$ metric and characterizes stability for the $ ext{W}_ ext{infinity}$ metric.

Findings

Sum of two Dirac masses is the unique local minimizer under $ ext{W}_ ext{lambda}$ for $1 \\le \\lambda<\\infty$.

Stability of steady states depends on the powers of the interaction potentials.

Characterization of stability in the $ ext{W}_ ext{infinity}$ metric.

Abstract

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $\lambda-$Wasserstein metric topology with $1\le \lambda<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.