Communications in Mathematical Sciences

Volume 19 (2021)

Number 7

Kinetic and hydrodynamic models for the relativistic Cucker–Smale ensemble and emergent behaviors

Pages: 1945 – 1990



Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and Korea Institute for Advanced Study, Seoul, South Korea)

Jeongho Kim (Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, South Korea)

Tommaso Ruggeri (Department of Mathematics and Alma Mater Research Center on Applied Mathematics, University of Bologna, Italy)


We study emergent behaviors of relativistic kinetic and hydrodynamic models which can be derived from the relativistic Cucker–Smale (RCS) model. The RCS model was introduced to describe the flocking dynamics of relativistic particles in authors’ recent work. The proposed relativistic kinetic and hydrodynamic models correspond to relativistic counterparts of the Vlasov-type CS model and pressureless gas dynamics, respectively, and their classical limits can be made in any finite-time interval as the speed of light tends to infinity. For the proposed relativistic models, we provide a global well-posedness of classical solutions and asymptotic flocking estimates using Lyapunov functional approach.


emergent behavior, flocking, hydrodynamic model, kinetic model, relativistic Cucker–Smale model

2010 Mathematics Subject Classification

35Q70, 70H40, 83C10

Received 31 December 2020

Accepted 18 April 2021

Published 7 September 2021