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Erschienen in:
Kinetic and Moment Models for Cell Motion in Fiber Structures Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

Singular Cucker–Smale Dynamics

verfasst von : Piotr Minakowski, Piotr B. Mucha, Jan Peszek, Ewelina Zatorska

Erschienen in: Active Particles, Volume 2

Verlag: 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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Vorheriges Kapitel Kinetic Equations and Self-organized Band Formations
Nächstes Kapitel A Stochastic-Statistical Residential Burglary Model with Finite Size Effects
Fußnoten
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 zi.
 
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Metadaten
Titel
Singular Cucker–Smale Dynamics
verfasst von
Piotr Minakowski
Piotr B. Mucha
Jan Peszek
Ewelina Zatorska
Copyright-Jahr
2019
Buch
Active Particles, Volume 2

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

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

Copyright-Jahr: 2019

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

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