Asymptotic Behavior and Collision Avoidance in the Cucker–Smale Model

Abstract

For the long-range-communicated Cucker-Smale (C-S) model, asymptotic flocking can be obtained by taking full use of a key equality about the kinetic energy. However, such behavior does not exist for the short-range-communicated C-S model with general initial data. In this note, we first establish a new equality about the second-order velocity-position moments for general C-S models. Then, we use this equality to deduce a new type of asymptotic behavior for the short-range-communicated C-S model. Such behavior not only reflects an essential feature of the short-range communication weight, but also has some potential applications in the complex multicluster problem. Besides, collision avoidance is also obtained for the C-S model with some singular short-range communication weights.