We analyze the one dimensional Cucker–Smale (in short CS) model with a weak singular communication weight $\psi (x)=|x{|}^{-\beta }$ with $\beta \in (0,1)$. 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 $p\in [1,\infty ]$. Our result gives the uniqueness of measure-valued solution in the sense of mean-field limits, i.e., 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.

我们分析了具有奇异通信权重的一维Cucker-Smale（简称CS）模型 $\psi （X）=|X{|}^{-\beta }$ 和 $\beta \in （0，\mathrm{1个}）$。我们首先建立动力学CS方程的量值解的全局时间存在性。为此，我们使用变量的适当更改来将粒子CS模型重新构造为一阶粒子系统，并为该粒子系统提供时间均匀的稳定性。然后，我们将奇异CS粒子系统的稳定性估计值扩展。通过使用该稳定性估计，我们可以及时构造全局动力学CS方程的量值解决方案。此外，作为时间均匀性稳定估计的直接应用，我们展示了从粒子系统到动力学CS方程在p -Wasserstein距离下的定量时间平均均场极限，其中$p\in [\mathrm{1个}，\infty ]$。我们的结果给出了在均值场极限意义上度量值解的唯一性，即唯一存在于与粒子系统相关的经验度量近似的度量值解。一阶模型的类似结果也作为副产品出现。通过使用累积粒子分布的拟逆，我们还将源自一阶模型的连续型方程重新构造为积分微分方程。通过使用修正的p -Wasserstein距离，我们为连续方程的绝对连续解提供了收缩率估计。