Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model

In this paper, we analyze the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852–862], which describes the collective behavior of an ensemble of organisms, animals, or devices. This kinetic version introduced in [S.-Y. Ha and E. Tadmor, Kinet. Relat. Models, 1 (2008), pp. 415–435] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [F. Cucker and S. Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast in velocity to the mean velocity of the initial condition, while in space they will converge towards a translational flocking solution.

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