Communications in Mathematical Sciences
Volume 18 (2020)
Propagation of the mono-kinetic solution in the Cucker–Smale-type kinetic equations
Pages: 1221 – 1231
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
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