We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

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具有松弛的非局部欧拉系统中的适定性和临界阈值

我们提出并研究了一个具有松弛的非局部欧拉系统，它在双曲标度限制下趋于严格的双曲系统。该系统在任何空间维度上的局部存在性和唯一性的独立证明。我们在一维设置中进一步推导出该系统的精确临界阈值。我们的结果表明，这种非局部系统承认对大量初始数据的全局平滑解决方案。因此，非局部速度正则化了无压欧拉系统中的一般有限时间分解。