A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. (English) Zbl 1177.92003

Summary: We present a simple proof on the formation of flocking in the F. Cucker and S. Smale system [Jpn. J. Math. (3) 2, No. 1, 197–227 (2007; Zbl 1166.92323)] based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to the flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.


92C17 Cell movement (chemotaxis, etc.)
82C22 Interacting particle systems in time-dependent statistical mechanics
92D50 Animal behavior
37N25 Dynamical systems in biology
82D99 Applications of statistical mechanics to specific types of physical systems
34A99 General theory for ordinary differential equations


Zbl 1166.92323
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