From particle to kinetic and hydrodynamic descriptions of flocking

Pages: 415 - 435, Volume 1, Issue 3, September 2008

doi:10.3934/krm.2008.1.415       Abstract        Full Text (271.2K)       Related Articles

Seung-Yeal Ha - Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea (email)
Eitan Tadmor - Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, United States (email)

Abstract: We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.

Keywords:  flocking, particles, kinetic formulation, moments, hydrodynamic formulation.
Mathematics Subject Classification:  Primary: 92D25,74A25; Secondary: 76N10.

Received: June 2008;      Revised: June 2008;      Published: August 2008.