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

Ha, Seung-Yeal; Liu, Jian-Guo

Commun. Math. Sci. 7, No. 2, 297-325 (2009).

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.

MSC:

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

Keywords:

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

Citations:

Zbl 1166.92323