Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Propagation of the mono-kinetic solution in the Cucker–Smale-type kinetic equations

Pages: 1221 – 1231



Moon-Jin Kang (Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, South Korea)

Jeongho Kim (Institute of New Media and Communications, Seoul National University, Seoul, South Korea)


In this paper, we study the propagation of the mono-kinetic distribution in the Cucker–Smale-type kinetic equations. More precisely, if the initial distribution is a Dirac mass for the variables other than the spatial variable, then we prove that this “mono-kinetic” structure propagates in time. For that, we first obtain the stability estimate of measure-valued solutions to the kinetic equation, by which we ensure the uniqueness of the mono-kinetic solution in the class of measure-valued solutions with compact supports. We then show that the mono-kinetic distribution is a special measure-valued solution. The uniqueness of the measure-valued solution implies the desired propagation of mono-kinetic structure.


hydrodynamic equations, kinetic equation, mono-kinetic solution, the Cucker–Smale model, the thermomechanical Cucker–Smale model

2010 Mathematics Subject Classification

35Q35, 35Q70

The work of M.-J. Kang is partially supported by the NRF-2019R1C1C1009355.

The work of J. Kim was supported by the Basic Research Lab Program through the National Research Foundation of Korea (NRF) funded by the MSIT(2018R1A4A1059976).

Received 7 September 2019

Accepted 11 February 2020

Published 23 September 2020