人群运动建模中的拥堵处理
Handling congestion in crowd motion modeling
AI总结：
本文在非光滑框架下处理人群运动中的拥堵问题，通过将实际速度定义为所有允许速度中与期望速度最接近的（最小二乘意义），并利用Wasserstein距离揭示宏观层面的扫描过程性质，从而在非光滑演化下获得存在性结果。
摘要：
我们在此处理人群运动建模中的拥堵问题，采用非光滑框架：人与人之间的接触不是预先避免的，而是实际发生的，并且在模型中明确考虑。我们将方法限制在非常基本的行为原则上，以专注于模型非光滑特性带来的特定问题。我们假设个体倾向于根据期望或自发的速度移动。我们通过假设演化在每一时刻实现个体倾向与全局约束（禁止重叠）之间的瞬时平衡来处理拥堵：实际速度被定义为在所有允许速度中与期望速度最接近的（最小二乘意义）。我们在微观和宏观设定下发展这些原则，并展示度量之间的Wasserstein距离框架如何允许在宏观层面恢复问题的扫描过程性质，从而使得尽管演化过程具有非光滑特性，仍能获得存在性结果。比较了微观和宏观方法，并研究了这两个描述层次的相似性和深刻差异。
英文摘要：
We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontanous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.
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