Communications in Mathematical Sciences

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

Seung-Yeal Ha and Jian-Guo Liu

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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.

Article information

Commun. Math. Sci., Volume 7, Number 2 (2009), 297-325.

First available in Project Euclid: 27 May 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92C17: Cell movement (chemotaxis, etc.)
Secondary: 82C22: Interacting particle systems [See also 60K35] 82C40: Kinetic theory of gases

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


Ha, Seung-Yeal; Liu, Jian-Guo. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7 (2009), no. 2, 297--325.

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