Double milling in self-propelled swarms from kinetic theory

José A. Carrillo; M. R. D’Orsogna; V. Panferov

doi:10.3934/krm.2009.2.363

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2009, Volume 2, Issue 2: 363-378. Doi: 10.3934/krm.2009.2.363

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Double milling in self-propelled swarms from kinetic theory

1.
ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra
2.
Department of Mathematics, California State University, Northridge, CA 91330-8313

Received: November 30, 2008

Revised: February 28, 2009

Published: May 2009

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Abstract

We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.

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Mathematics Subject Classification: 92D50, 82C40, 82C22, 92C15.

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