An intrinsic aggregation model on the special orthogonal group $\mathrm{SO}(3)$: well-posedness and collective behaviours

TL;DR

This paper introduces an aggregation model on the special orthogonal group $SO(3)$, establishing well-posedness and analyzing long-term collective behaviors, supported by numerical experiments illustrating various asymptotic patterns.

Contribution

It presents the first well-posedness results for an intrinsic aggregation model on $SO(3)$ and provides conditions for consensus formation, advancing understanding of collective dynamics on manifolds.

Findings

Existence of measure-valued solutions established

Conditions for asymptotic consensus derived

Numerical experiments demonstrate diverse collective patterns

Abstract

We investigate an aggregation model with intrinsic interactions on the special orthogonal group $SO(3)$. We consider a smooth interaction potential that depends on the squared intrinsic distance, and establish local and global existence of measure-valued solutions to the model via optimal mass transport techniques. We also study the long-time behaviours of such solutions, where we present sufficient conditions for the formation of asymptotic consensus. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.