We rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.

中文翻译：

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平均场极限：从粒子描述到宏观方程

我们从具有对齐相互作用的集群模型的牛顿型粒子描述中严格推导出具有速度和聚集方程中的非局部耗散项的无压欧拉式方程和具有非局部速度场的聚集方程。至关重要的是，我们利用调制动能的离散版本以及有界 Lipschitz 距离进行测量，以控制由于非局部相互作用而导致的时间导数中的项。