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A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior

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Abstract

This survey summarizes and illustrates the main qualitative properties of hydrodynamics models for collective behavior. These models include a velocity consensus term together with attractive–repulsive potentials leading to non-trivial flock profiles. The connection between the underlying particle systems and the swarming hydrodynamic equations is performed through kinetic theory modeling arguments. We focus on Lagrangian schemes for the hydrodynamic systems showing the different qualitative behaviors of the systems and its capability of keeping properties of the original particle models. We illustrate the known results concerning large-time profiles and blowup in finite time of the hydrodynamic systems to validate the numerical scheme. We finally explore the unknown situations making use of the numerical scheme showcasing a number of conjectures based on the numerical results.

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Acknowledgements

J. A. C. was partially supported by the Royal Society via a Wolfson Research Merit Award. J. A. C. and Y. -P. C. were partially supported by EPSRC grant EP/K008404/1. Y. -P. C. was supported by the ERC-Stating grant HDSPCONTR “High-Dimensional Sparse Optimal Control”. S. P. P. was partially supported by a Erasmus+ scholarship.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington, SW7 2AZ, UK

    José A. Carrillo

  2. Fakultät für Mathematik, Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Young-Pil Choi

  3. ETSIAE, Technical University of Madrid, Pza. de Cardenal Cisneros, 3, 28040, Madrid, Spain

    Sergio P. Perez

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Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

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Carrillo, J.A., Choi, YP., Perez, S.P. (2017). A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_7

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