无限图上Cucker-Smale模型的集群行为

The collective behavior of the Cucker-Smale model on the in nite graphs

本文研究两类无限图上Cucker-Smale模型的集群行为.第一类图为全连通的无限图,本文得到在初值关于位置无界的情形下,仍然会出现对应的集群行为(速度的一致性),称之为编队行为.第二类图为局部连接有限的无限图.首先研究Laplace算子谱的下确界大于0的局部有限无限图,得到Cucker-Smale模型发生群集行为的充分条件.其次研究具有Poincaré不等式的局部有限无限图,借助图的几何性质,得到时变图的衰减性,并将其应用在Cucker-Smale模型上,得到Cucker-Smale模型的非线性稳定性(小初值的群集行为).

In this paper,we study the collective behavior of the Cucker-Smale model on two types of infinite graphs.The first type of graph is a fully connected infinite graph.We obtain that when the initial value is unbounded with respect to the position,the corresponding collective behavior(the consistency of velocity)still emerges.We call it the formation behavior.The second type of graph is an infinite graph with locally finite connections.We first study the locally finite infinite graphs with the infimum spectrum of the Laplace operator strictly positive and obtain sufficient conditions to guarantee the emergence of the flocking behavior.Then,we study the locally finite infinite graphs with the Poincar´e inequality.With the help of the geometric properties of the graph,we obtain the ultra-contractive properties of the time-varying graph,and apply them to the Cucker-Smale model to obtain the nonlinear stability(the flocking behavior of small initial values).