Convergence of regularized nonlocal interaction energies

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 $Γ$-converge to the unregularized energy and minimizers converge to minimizers. We then apply our results to prove $Γ$-convergence of the gradient flows, when restricted to the space of measures with bounded density.

28 pages, 1 figure. This updated version includes a new section (section 4) discussing the natural extensions of the results in sections 2 and 3 to more general interaction kernels.This version is to appear in SIAM J. Math. Anal