On the exponential weak flocking for the kinetic Cucker–Smale model with non-compact support

In this paper, we study the propagation of the second spatial-velocity moments for the kinetic Cucker–Smale model with non-compact spatial support. In contrast to compact support, non-compact support leads to a lower bound of zero for the communication weight, which makes the previous approach break down. To address this challenge, we consider two types of initial distributions: exponential decay distributions and polynomial decay distributions. Moreover, our approach uses the infinite-particle mean-field approximation as an intermediary step to analyze the kinetic Cucker–Smale model, with conservation laws of mass and momentum. When initial distributions belong to the aforementioned types of decaying classes and coupling strength exceeds a certain threshold, we show the weak flocking behavior of the kinetic Cucker–Smale model. Specifically, the second velocity moment of the solution centered around the initial average velocity converges to zero, and the second spatial moment around the position of the center of mass remains uniformly bounded in time. The emergence of weak flocking behavior illustrates that even for non-compact support, a certain degree of aggregation can be maintained for the kinetic Cucker–Smale model, as long as the initial distribution exhibits relative concentration.