The Filippov characteristic flow for the aggregation equation with mildly singular potentials

TL;DR

This paper introduces a new PDE-based approach to establish existence and uniqueness of measure solutions for the aggregation equation with mildly singular potentials, using Filippov flows and contraction arguments.

Contribution

It presents an alternative to gradient flow methods by constructing a Filippov characteristic flow and proves solution equivalence and numerical convergence.

Findings

Existence of a Filippov flow for the aggregation equation.

Uniqueness of measure solutions via contraction in transport distances.

Convergence of a numerical scheme for measure solutions.

Abstract

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.