We show that the locations where finite- and infinite-time clustering occur for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated with a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated with a scalar balance law formulation of the system.

中文翻译：

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用于实线上对准动态的有限和无限时间簇形成

我们证明，仅使用初始数据就可以确定一维欧拉对准系统发生有限时间和无限时间聚类的位置。我们目前的工作提供了有限时间奇点集结构和与弱解相关的渐近簇的第一个结果。在许多情况下，簇的最终大小可以直接从与系统的标量平衡定律公式相关的通量中读出。