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Active Particles, Volume 2Singular Cucker–Smale Dynamics

Active Particles, Volume 2: Singular Cucker–Smale Dynamics [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Active Particles, Volume 2Singular Cucker–Smale Dynamics

Part of the Modeling and Simulation in Science, Engineering and Technology Book Series
Editors: Bellomo, Nicola; Degond, Pierre; Tadmor, Eitan
Minakowski, Piotr; Mucha, Piotr B.; Peszek, Jan; Zatorska, Ewelina
Active Particles, Volume 2 — Aug 23, 2019
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42 pages

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Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2019
ISBN
978-3-030-20296-5
Pages
201 –243
DOI
10.1007/978-3-030-20297-2_7
Publisher site
See Chapter on Publisher Site

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.]

Published: Aug 23, 2019

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