Convergence of regularized nonlocal interaction energies

TL;DR

This paper investigates how regularization affects nonlocal interaction energies, proving convergence of regularized energies and their minimizers to the original, and extending results to gradient flows in measure spaces.

Contribution

It establishes the $ ext{Γ}$-convergence of regularized nonlocal interaction energies and their gradient flows to the unregularized versions, with implications for numerical and analytical studies.

Findings

Regularized energies $ ext{Γ}$-converge to unregularized energies.

Minimizers of regularized energies converge to minimizers.

Gradient flows of regularized energies converge in the measure space.

Abstract

Inspired by numerical studies of the aggregation equation, we study the effect of regularization on nonlocal interaction energies. We consider energies defined via a repulsive-attractive interaction kernel, regularized by convolution with a mollifier. We prove that, with respect to the 2-Wasserstein metric, the regularized energies $\Gamma$-converge to the unregularized energy and minimizers converge to minimizers. We then apply our results to prove $\Gamma$-convergence of the gradient flows, when restricted to the space of measures with bounded density.