Translator Disclaimer
2019 A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment
Alessio Figalli, Moon-Jin Kang
Anal. PDE 12(3): 843-866 (2019). DOI: 10.2140/apde.2019.12.843

Abstract

We consider the kinetic Cucker–Smale model with local alignment as a mesoscopic description for the flocking dynamics. The local alignment was first proposed by Karper, Mellet and Trivisa (2014), as a singular limit of a normalized nonsymmetric alignment introduced by Motsch and Tadmor (2011). The existence of weak solutions to this model was obtained by Karper, Mellet and Trivisa (2014), and in the same paper they showed the time-asymptotic flocking behavior. Our main contribution is to provide a rigorous derivation from a mesoscopic to a macroscopic description for the Cucker–Smale flocking models. More precisely, we prove the hydrodynamic limit of the kinetic Cucker–Smale model with local alignment towards the pressureless Euler system with nonlocal alignment, under a regime of strong local alignment. Based on the relative entropy method, a main difficulty in our analysis comes from the fact that the entropy of the limit system has no strict convexity in terms of density variable. To overcome this, we combine relative entropy quantities with the 2-Wasserstein distance.

Citation

Download Citation

Alessio Figalli. Moon-Jin Kang. "A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment." Anal. PDE 12 (3) 843 - 866, 2019. https://doi.org/10.2140/apde.2019.12.843

Information

Received: 22 January 2018; Revised: 23 April 2018; Accepted: 29 June 2018; Published: 2019
First available in Project Euclid: 25 October 2018

zbMATH: 06986455
MathSciNet: MR3864212
Digital Object Identifier: 10.2140/apde.2019.12.843

Subjects:
Primary: 35Q70
Secondary: 35B25

Rights: Copyright © 2019 Mathematical Sciences Publishers

JOURNAL ARTICLE
24 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.12 • No. 3 • 2019
MSP
Back to Top