Calculus of Variations and Geometric Measure Theory
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A. Figalli - M. J. Kang

A rigorous derivation from the kinetic cucker-smale model to the pressureless euler system with nonlocal alignment

created by figalli on 03 Jun 2019

[BibTeX]

Accepted Paper

Inserted: 3 jun 2019
Last Updated: 3 jun 2019

Journal: Anal. PDE
Year: 2019

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, as a singular limit of a normalized non-symmetric alignment introduced by Motsch and Tadmor. The existence of weak solutions to this model is obtained in the paper of Karper, Mellet and Trivis. The time-asymptotic flocking behavior is shown in this article. Our main contribution is to provide a rigorous derivation from mesoscopic to mascroscopic 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 strictly convexity in terms of density variable. To overcome this, we combine relative entropy quantities with the 2-Wasserstein distance.


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