具有非紧支集的Cucker-Smale型模型的时间一致稳定性和平均场极限
Uniform-in-time stability and mean-field limit of the Cucker-Smale-type model with noncompact support
AI总结：
研究完全非紧设置下无限Cucker-Smale型模型的时间一致稳定性和平均场极限，针对原始模型困难构造反例，引入新模型，在适当假设下建立其时间一致稳定性，推导其平均场极限及相应KCS型模型的时间一致稳定性。
AI中文摘要：
我们研究了在完全非紧设置下无限Cucker-Smale型（ICS型）模型 的时间一致稳定性和平均场极限。首先，我们阐明了当初始空间非紧时，原始ICS模型及其相应的动力学Cucker-Smale（KCS）模型中出现的主要困难。特别地，我们表明速度直径可能随时间保持不变，这使得基于位置和速度直径的经典方法无效。此外，我们构造了一个反例，表明原始KCS模型在非紧空间设置中不满足时间一致稳定性。为了克服这些障碍，我们引入了一种新的ICS型模型，其通信权重取决于成对空间矩。在适当的假设下，我们还建立了ICS型模型在非紧设置下关于初始数据的时间一致稳定性。最后，我们推导了ICS型模型的时间一致平均场极限，以及相应KCS型模型的时间一致稳定性。
英文摘要：
We study the uniform-in-time stability and mean-field limit of an infinite Cucker--Smale-type (ICS-type) model in a fully noncompact setting. First, we clarify the main difficulties arising from the original ICS model and its corresponding kinetic Cucker--Smale (KCS) model, when the spatial support is noncompact. In particular, we show that the velocity diameter may remain constant in time, which invalidates the classical approach based on position and velocity diameters. Moreover, we construct a counterexample showing that the original KCS model does not satisfy uniform-in-time stability in the noncompact spatial setting. To overcome these obstacles, we introduce a new ICS-type model whose communication weight depends on a pairwise spatial moment. Under suitable assumptions, we also establish the uniform-in-time stability of the ICS-type model with respect to initial data in the noncompact setting. Finally, we derive the uniform-in-time mean-field limit for the ICS-type model, together with uniform-in-time stability for the corresponding KCS-type model.
