One dimensional singular Cucker?Smale model: Uniform-in-time mean-field limit and contractivity
One dimensional singular Cucker?Smale model: Uniform-in-time mean-field limit and contractivity
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一维奇异 Cucker?Smale 模型：均匀时间平均场极限和收缩性
DOI：
10.1016/j.jde.2021.04.002
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发表时间：
2021
影响因子：
2.4
通讯作者：
Zhang Xiongtao
中科院分区：
文献类型：
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作者：
Choi Young-Pil;Zhang Xiongtao
我们分析了具有弱奇异通信权的一维Cucker-Smale（简称CS）模型|X| − β，其中β∈（0，1）。首先，我们建立了动力学CS方程的测度值解的整体存在性。为此，我们使用一个适当的变量的变化，重新制定的粒子CS模型作为一个一阶粒子系统，并提供该粒子系统的时间一致稳定性。然后，我们扩展了奇异CS粒子系统的稳定性估计。通过使用该稳定性估计，我们构造了动力学CS方程在时间上的全局测度值解。此外，作为时间一致稳定性估计的直接应用，我们给出了从粒子系统到动力学CS方程在p-Wasserstein距离（p∈[1，∞]）上的时间一致平均场的定量极限.我们的结果给出了在平均场极限意义下的测度值解的唯一性，即由与粒子系统相关的经验测度所近似的测度值解是唯一存在的.一阶模型的类似结果也是一个副产品。我们还重新制定的连续型方程，这是来自一阶模型，作为一个积分微分方程，采用伪逆的累积颗粒分布。利用修正的p-Wasserstein距离，给出了连续方程绝对连续解的压缩性估计.
We analyze the one dimensional Cucker–Smale (in short CS) model with a weak singular communication weight ψ (x)=| x|− β with β∈(0, 1). We first establish a global-in-time existence of measure-valued solutions to the kinetic CS equation. For this, we use a proper change of variable to reformulate the particle CS model as a first-order particle system and provide the uniform-in-time stability for that particle system. We then extend this stability estimate for the singular CS particle system. By using that stability estimate, we construct the measure-valued solutions to the kinetic CS equation globally in time. Moreover, as a direct application of the uniform-in-time stability estimate, we show the quantitative uniform-in-time mean-field limit from the particle system to that kinetic CS equation in p-Wasserstein distance with p∈[1,∞]. Our result gives the uniqueness of measure-valued solution in the sense of mean-field limits, ie, the measure-valued solutions, approximated by the empirical measures associated to the particle system, uniquely exist. Similar results for the first-order model also follow as a by-product. We also reformulate the continuity-type equation, which is derived from the first-order model, as an integro-differential equation by employing the pseudo-inverse of the accumulative particle distribution. By making use of a modified p-Wasserstein distance, we provide the contractivity estimate for absolutely continuous solutions of the continuum equation.