Gradient flows for non-smooth interaction potentials

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 λ-convex) potential, possibly not smooth at several points, generalizing the results of Carrillo et al. (2011). We also identify the cases in which the dynamic is still governed by the continuity equation with well-characterized nonlocal velocity field.