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Fluid models with phase transition for kinetic equations in swarming

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

E-mail Address: mihai.bostan@univ-amu.fr

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and

José Antonio Carrillo

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

E-mail Address: carrillo@maths.ox.ac.uk

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https://doi.org/10.1142/S0218202520400163Cited by:5 (Source: Crossref)

This article is part of the issue:

Special Issue on “Recent Results and Challenges” in Behavioral Systems
Guest Editors: N. Bellomo, F. Brezzi and J. Soler

Abstract

We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.

Communicated by N. Bellomo, F. Brezzi and J. Soler

Keywords:

AMSC: 92D50, 82C40, 92C10

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Vol. 30, No. 10

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History

Received 30 November 2019

Revised 6 April 2020

Accepted 8 May 2020

Published: 7 August 2020

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Fluid models with phase transition for kinetic equations in swarming

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