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Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK
2 Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, Château Gombert 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France

Special Issues: Mathematical Methods in the Biosciences

We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.
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Keywords Vlasov-like equations; swarming; Cucker-Smale model; Vicsek model

Citation: Pedro Aceves-Sánchez, Mihai Bostan, Jose-Antonio Carrillo, Pierre Degond. Hydrodynamic limits for kinetic flocking models of Cucker-Smale type. Mathematical Biosciences and Engineering, 2019, 16(6): 7883-7910. doi: 10.3934/mbe.2019396

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