Communications in Mathematical Sciences

Volume 7 (2009)

Number 2

A simple proof of the Cucker-Smale flocking dynamics and mean-field limit

Pages: 297 – 325



Seung-Yeal Ha

Jian-Guo Liu


We present a simple proof on the formation of flocking to the Cucker-Smale system 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 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.


Flocking; swarming; emergence; self-driven particles system; autonomous agents; Vlasov equation; Lyapunov functional; measure valued solution; Kantorovich-Rubinstein distance

2010 Mathematics Subject Classification

Primary 92C17. Secondary 82C22, 82C40.

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