Minimizing movement scheme for intrinsic aggregation on compact Riemannian manifolds

TL;DR

This paper develops a minimizing movement scheme to establish the existence and properties of measure-valued solutions for the aggregation equation on compact Riemannian manifolds, addressing technical challenges related to non-differentiability.

Contribution

It introduces a novel application of the minimizing movement scheme to intrinsic aggregation equations on Riemannian manifolds, overcoming non-differentiability issues at the cut locus.

Findings

Existence of measure-valued solutions for small time intervals

Finite speed of propagation of solutions

Handling non-differentiability at the cut locus

Abstract

Recently solutions to the aggregation equation on compact Riemannian Manifolds have been studied with different techniques. This work demonstrates the small time existence of measure-valued solutions for suitably regular intrinsic potentials. The main tool is the use of the minimizing movement scheme which together with the optimality conditions yield a finite speed of propagation. The main technical difficulty is non-differentiability of the potential in the cut locus which is resolved via the propagation properties of geodesic interpolations of the minimizing movement scheme and passes to the limit as the time step goes to zero.