Abstract
A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field limit of the particle system is obtained. Furthermore, the time-asymptotic flocking behavior of the solution to the kinetic equation is also derived. The technics used for local wellposedness and stability follow from similar ideas to those have been used in [3,14,22]. While in order to extend the local result globally, the main contribution here is to generate a series of new estimates for this Vlasov type equation, which imply that the growing of the characteristics can be controlled globally. Further estimates also show the long time flocking phenomena.
Highlights
In the last few years, two strategies have been used to describe these phenomena in applied mathematics literature: particle dynamics [9, 10, 15, 21] and continuum models for mesoscopic or macroscopic quantities [6, 17, 19, 26, 27]
In the works by Ha and Tadmor [19], Carrillo et al [4], a statistical description of the interacting agent system based on mesoscopic models by means of kinetic equations were studied
An in depth study of kinetic models is very important in understanding the phenomena since this class of models play a role of bridge between the particle models and the macroscopic models
Summary
We will focus on a general collision-avoiding flocking system proposed by Felipe Cucker and Jiu-Gang Dong [8] It is a new simple dynamical system which describes the emergency of flocking, especially the behavior of collision-avoidance. We start with presenting a well-known approach to the wellposedness (existence, uniqueness and stability) of the kinetic model, where some basic knowledge of optimal transport theory [29] will be used This method was used in [3] aiming to give a global wellposedness result for a general system. The second goal of this paper is to prove that the kinetic model (2) exhibits time-asymptotic flocking behavior when the interaction rate has a uniformly non-negative lower bound. The kinetic velocity fluctuation decay in time and the kinetic velocity fluctuation is bounded in time
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