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
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.
Anal. PDE, Volume 12, Number 3 (2019), 843-866.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q70: PDEs in connection with mechanics of particles and systems
Secondary: 35B25: Singular perturbations
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