Well-posedness of an interaction model on Riemannian manifolds

TL;DR

This paper studies a mathematical model of collective behavior on Riemannian manifolds, proving local and global well-posedness of solutions using optimal transport and Lipschitz continuity concepts.

Contribution

It establishes the well-posedness of measure-valued solutions for interaction models on Riemannian manifolds, extending to global solutions under certain curvature conditions.

Findings

Local well-posedness for regular potentials

Global well-posedness on manifolds with nonpositive curvature

Use of Lipschitz vector fields via parallel transport

Abstract

We investigate a model for collective behaviour with intrinsic interactions on smooth Riemannian manifolds. For regular interaction potentials, we establish the local well-posedness of measure-valued solutions defined via optimal mass transport. We also extend our result to the global well-posedness of solutions for manifolds with nonpositive bounded sectional curvature. The core concept underlying the proofs is that of Lipschitz continuous vector fields in the sense of parallel transport.