Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 843-866.

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

Alessio Figalli and Moon-Jin Kang

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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.

Article information

Source
Anal. PDE, Volume 12, Number 3 (2019), 843-866.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1540432872

Digital Object Identifier
doi:10.2140/apde.2019.12.843

Mathematical Reviews number (MathSciNet)
MR3864212

Zentralblatt MATH identifier
06986455

Subjects
Primary: 35Q70: PDEs in connection with mechanics of particles and systems
Secondary: 35B25: Singular perturbations

Keywords
hydrodynamic limit kinetic Cucker–Smale model local alignment pressureless Euler system relative entropy Wasserstein distance

Citation

Figalli, Alessio; Kang, Moon-Jin. A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 12 (2019), no. 3, 843--866. doi:10.2140/apde.2019.12.843. https://projecteuclid.org/euclid.apde/1540432872


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