We consider a model for collective behaviour on Riemannian manifolds, with interactions modelled through a smooth interaction potential that depends on the squared intrinsic distance. We pay particular attention to the model set up on the special orthogonal group SO(3). We establish local and global existence of measure-valued solutions to the model on SO(3) 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 on SO(3). In addition, we investigate asymptotic consensus on more general manifolds. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

我们考虑了黎曼流形上的集体行为模型，其相互作用通过依赖于平方内在距离的平滑相互作用势建模。我们特别关注在特殊正交群SO (3)上建立的模型。我们通过最优质量传输技术为SO (3)模型建立局部和全局存在的测量值解。我们还研究了此类解决方案的长期行为，其中我们提出了在SO (3)上形成渐近共识的充分条件。此外，我们研究了更一般流形的渐近共识。分析结果用数值实验来说明，这些实验表现出各种渐近模式。