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.

Published 1 January 2009