# A hydrodynamic model for synchronization phenomena

## TL;DR

This paper introduces a new hydrodynamic model derived from the Kuramoto model with inertia to study synchronization, revealing finite-time blow-up phenomena and providing analytical and numerical insights.

## Contribution

The paper develops a novel pressureless Euler-based hydrodynamic model for synchronization, establishing well-posedness, synchronization conditions, and blow-up analysis, which extends classical Kuramoto models.

## Key findings

- Finite-time blow-up phenomena observed in the model.
- Synchronization estimates for identical natural frequencies.
- Numerical simulations showing phase transitions and hysteresis.

## Abstract

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.