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

TL;DR

This paper proves that the mono-kinetic distribution in Cucker-Smale-type kinetic equations remains stable and propagates over time, ensuring the uniqueness of such solutions under certain initial conditions.

Contribution

It establishes the stability and propagation of mono-kinetic solutions in measure-valued solutions for Cucker-Smale-type kinetic equations, demonstrating their uniqueness.

Findings

Mono-kinetic distribution propagates over time.

Stability estimate ensures uniqueness of solutions.

Mono-kinetic solutions are a special class of measure-valued solutions.

Abstract

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.