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.

中文翻译：

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Cucker-Smale 型动力学方程中单动力学解的传播

在本文中，我们研究了 Cucker-Smale 型动力学方程中单动力学分布的传播。更准确地说，如果初始分布是空间变量以外的变量的狄拉克质量，那么我们证明这种“单动力”结构会及时传播。为此，我们首先获得动力学方程的测值解的稳定性估计，通过它我们确保单动力学解在具有紧支撑的测值解类中的唯一性。然后我们证明单动力学分布是一个特殊的测量值解。测量值解的唯一性意味着单动力结构的所需传播。