Paper detail

Convergence of regularized nonlocal interaction energies

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.