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Explicit flock solutions for Quasi-Morse potentials

Published online by Cambridge University Press:  15 April 2014

J. A. CARRILLO ,
Y. HUANG  and
S. MARTIN
J. A. CARRILLO
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: carrillo@imperial.ac.uk, yanghong.huang@imperial.ac.uk, stephan.martin@imperial.ac.uk
Y. HUANG
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: carrillo@imperial.ac.uk, yanghong.huang@imperial.ac.uk, stephan.martin@imperial.ac.uk
S. MARTIN
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: carrillo@imperial.ac.uk, yanghong.huang@imperial.ac.uk, stephan.martin@imperial.ac.uk
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Abstract

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We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, which are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness are proven for three space dimensions, while existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.

Keywords

Integral equationsSwarming patternsNon-local problemsBessel functions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 5 , October 2014 , pp. 553 - 578
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence http://creativecommons.org/licenses/by/3.0/
Copyright
Copyright © Cambridge University Press 2014

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