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arXiv · 1909.07525

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

Moon-Jin Kang·Jeongho Kim
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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.

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Moon-Jin Kang, Jeongho Kim. 2019-09-17. Propagation of the mono-kinetic solution in the Cucker-Smale-type kinetic equations. https://arxiv.org/abs/1909.07525

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