Long-time behaviour of interaction models on Riemannian manifolds with bounded curvature

TL;DR

This paper studies the long-term dynamics of a nonlocal PDE on Riemannian manifolds with bounded curvature, focusing on conditions leading to consensus and quantifying the convergence rate.

Contribution

It provides new analytical conditions for consensus formation and convergence rates on curved manifolds, supported by numerical simulations on the rotation group.

Findings

Consensus states form under specific conditions for attractive potentials.

The convergence rate of the solution's support diameter is quantified.

Numerical simulations confirm the analytical results on the rotation group.

Abstract

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution's support. The analytical results are supported by numerical simulations for the equation set up on the rotation group.