论文标题
一维单数cucker-smale模型：均匀的平均场限制和合同性
One dimensional singular Cucker-Smale model: uniform-in-time mean-field limit and contractivity
论文作者
论文摘要
我们分析了一个尺寸的cucker-smale（在简短的CS中）模型，具有弱的奇异通信权重$ψ（x）= | x |^{ - β} $，$β\ in（0,1）$。我们首先建立了针对动力学CS方程的度量值解决方案的全球存在。为此，我们使用变量的适当更改将粒子CS模型重新调整为一阶粒子系统，并为该粒子系统提供均匀的时间稳定性。然后，我们扩展了单个CS粒子系统的稳定性估计。通过使用该稳定性估计，我们会及时构建针对动力学CS方程的度量值解决方案。此外，作为均匀稳定性估计的直接应用，我们显示了从粒子系统到$ p $ -P $ -WASSERSTEIN距离的定量均匀时均值限制，并在[1，\ infty] $中使用$ p $ -P \。我们的结果在平均场限制的意义（即，由与粒子系统相关的经验测量值近似，均具有均值的测量解决方案，从而赋予了测量值溶液的唯一性。一阶模型的相似结果也遵循副产品。我们还通过采用累积粒子分布的伪式侵入来重新重新重新制定源自一阶模型的连续性型方程作为整数分化方程。通过使用修改后的$ p $ - WASSERSTEIN距离，我们为Continuum方程的绝对连续解决方案提供了合同性估计。
We analyze the one dimensional Cucker-Smale (in short CS) model with a weak singular communication weight $ψ(x) = |x|^{-β}$ with $β\in (0,1)$. 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.
