Reduced fluid models for self-propelled particles interacting through alignment
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Published version
Author(s)
Bostan, Mihai
Carrillo de la Plata, J
Type
Journal Article
Abstract
The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self-propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.
Date Issued
2017-06-30
Date Acceptance
2017-02-28
Citation
Mathematical Models and Methods in Applied Sciences, 2017, 27 (7), pp.1255-1299
ISSN
0218-2025
Publisher
World Scientific Publishing
Start Page
1255
End Page
1299
Journal / Book Title
Mathematical Models and Methods in Applied Sciences
Volume
27
Issue
7
Copyright Statement
© 2017 The Author(s). 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 - https://creativecommons.org/licenses/by/4.0/) License. Further distribution of this work is permitted, provided the original work is properly cited.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/P031587/1
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Vlasov-like equations
measure solutions
swarming
Cucker-Smale model
Vicsek model
Laplace-Beltrami operator
generalized collision invariants
MEAN-FIELD LIMIT
LARMOR RADIUS APPROXIMATION
PLANCK-LANDAU EQUATION
PHASE-TRANSITION
ASYMPTOTIC FLOCKING
BOLTZMANN-EQUATION
DRIVEN PARTICLES
MAGNETIC-FIELDS
CONTINUUM-LIMIT
ANIMAL GROUPS
0102 Applied Mathematics
Applied Mathematics
Publication Status
Published
Date Publish Online
2017-05-18