Vol. 12, No. 3, 2019

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A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment

Alessio Figalli and Moon-Jin Kang

Vol. 12 (2019), No. 3, 843–866
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.

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Keywords
hydrodynamic limit, kinetic Cucker–Smale model, local alignment, pressureless Euler system, relative entropy, Wasserstein distance
Mathematical Subject Classification 2010
Primary: 35Q70
Secondary: 35B25
Milestones
Received: 22 January 2018
Revised: 23 April 2018
Accepted: 29 June 2018
Published: 7 October 2018
Authors
Alessio Figalli
Department of Mathematics
ETH Zürich
Zürich
Switzerland
Moon-Jin Kang
Department of Mathematics & Research Institute of Natural Sciences
Sookmyung Women’s University
Seoul
South Korea