Gradient flows for non-smooth interaction potentials

TL;DR

This paper studies the evolution of particle densities under non-smooth, convex interaction potentials, extending previous results to more general cases and characterizing when the dynamics follow a continuity equation with a nonlocal velocity.

Contribution

It generalizes the analysis of nonlocal interaction equations to include non-smooth, convex potentials and identifies conditions for the dynamics to be governed by a continuity equation.

Findings

Extended the class of potentials for which solutions are well-understood.

Characterized the cases where the evolution follows a continuity equation.

Provided conditions under which the velocity field is well-defined.

Abstract

We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or \lambda-convex) potential, possibly not smooth at several points, generalizing the results of [CDFLS]. We also identify the cases in which the dynamic is still governed by the continuity equation with well-characterized nonlocal velocity field. Reference: [CDFLS] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 (2011), 229--271.