We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena.

中文翻译：

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同步现象的流体动力学模型

我们提出了一种新的同步现象流体动力学模型，它是一种具有非局部相互作用力的无压欧拉系统。该系统可以从具有惯性​​的 Kuramoto 模型正式推导出来，该模型是相互作用相位振荡器的经典模型，广泛用于研究同步现象，通过单动力学闭合假设下的动力学描述。对于所提出的系统，我们首先建立了经典解决方案的本地时间存在性和唯一性。对于相同固有频率的情况，我们在初始配置的适当假设下提供同步估计。我们还分析了导致经典解决方案的有限时间爆炸或全局时间存在的关键阈值。特别是，我们提出的模型表现出有限时间的爆炸现象，这在经典 Kuramoto 模型中没有观察到，即使对于自然频率具有平滑分布函数。最后，我们对同步、有限时间爆炸、相变和滞后现象进行了数值研究。