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Kinetic and Moment Models for Cell Motion in Fiber Structures Read chapter Read first chapter

2019 | OriginalPaper | Chapter

Singular Cucker–Smale Dynamics

Authors : Piotr Minakowski, Piotr B. Mucha, Jan Peszek, Ewelina Zatorska

Published in: Active Particles, Volume 2

Publisher: Springer International Publishing

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Abstract

This chapter is dedicated to the singular models of flocking. We give an overview of the existing literature starting from microscopic Cucker–Smale (CS) model with singular communication weight, through its mesoscopic mean-field limit, up to the corresponding macroscopic regime. For the microscopic CS model and its selected variants, the collision-avoidance phenomenon is discussed. For the kinetic mean-field model, we sketch the existence of global-in-time measure-valued solutions, paying special attention to weak-atomic uniqueness of solutions. Ultimately, for the macroscopic singular model, we provide a summary of existence results for the Euler-type alignment system. This includes the existence of strong solutions on a one-dimensional torus, and the extension of this result to higher dimensions by restricting the size of the initial data. Additionally, we present the pressureless Navier–Stokes-type system corresponding to particular choice of alignment kernel. This system is then compared—analytically and numerically—to the porous medium equation.

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previous chapter Kinetic Equations and Self-organized Band Formations
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Footnotes
1
Which exists by Theorem 3.2.
 
2
See, for example, the Olympic Rings symbol simulated in the video available at https://​youtu.​be/​C7UDGRudsyA, produced within the work [20] through the manipulation of z i.
 
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Metadata
Title
Singular Cucker–Smale Dynamics
Authors
Piotr Minakowski
Piotr B. Mucha
Jan Peszek
Ewelina Zatorska
Copyright Year
2019
Book
Active Particles, Volume 2

Print ISBN: 978-3-030-20296-5

Electronic ISBN: 978-3-030-20297-2

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-20297-2_7

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