Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system
Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system
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DOI：
10.1080/03605302.2023.2202720
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发表时间：
2021-08
影响因子：
1.9
通讯作者：
T. Leslie;Changhui Tan
中科院分区：
文献类型：
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作者：
T. Leslie;Changhui Tan
摘要本文建立了具有测量值密度、有界速度和局部可积通信协议的一维欧拉对准系统弱解的全局适定性理论。对低正则性理论的令人满意的理解是一个迫切感兴趣的问题，因为光滑解可能在有限时间内失去正则性。然而，除了一类非常特殊的对齐相互作用外，目前还不存在这样的理论。我们证明了一维欧拉对准系统的动力学可以用一个非局部标量平衡定律有效地描述，该定律的熵条件作为一个熵选择原则，决定了欧拉对准系统的唯一弱解。此外，粘性粒子cucker - small动力学可以近似描述系统的弱解。我们的方法受到了Brenier和Grenier关于无压欧拉方程的工作的启发。
Abstract We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.