American Institute of Mathematical Sciences

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September  2008, 1(3): 415-435. doi: 10.3934/krm.2008.1.415

From particle to kinetic and hydrodynamic descriptions of flocking

Seung-Yeal Ha 1, and  Eitan Tadmor 2,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, South Korea
2. 

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

Received  June 2008 Revised  June 2008 Published  August 2008

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: hydrodynamic formulation., moments, flocking, particles, kinetic formulation.
Mathematics Subject Classification: Primary: 92D25,74A25; Secondary: 76N1.
Citation: Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415
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