Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds

TL;DR

This paper studies a collective behavior model with intrinsic interactions on spheres and other manifolds, establishing well-posedness, mean-field approximation, and conditions for consensus formation over time.

Contribution

It introduces a measure solution framework for the model on Riemannian manifolds and analyzes long-term behavior, including consensus conditions, on spheres and other manifolds.

Findings

Well-posedness of measure solutions on sphere and other manifolds.

Conditions for asymptotic consensus formation.

Analysis of long-time behavior of solutions.

Abstract

We investigate a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions (defined via mass transport) on sphere, as well as investigate the mean-field particle approximation. We study the long-time behaviour of solutions to the model on sphere, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. Well-posedness of solutions and the formation of consensus are also investigated for other manifolds (e.g., a hypercylinder).