AI 中文总结
研究相空间扩展的动力学Cucker-Smale模型的弱稳定性，通过有界Lipschitz通信权重函数得出有限时间Osgood型弱稳定性，因速度尾部无界解算子非Lipschitz连续，还得到独立同分布采样结果，即经验测度期望收敛到测度值解。
AI 中文摘要
我们研究了相空间扩展设置下动力学Cucker-Smale（简称KCS）模型的弱稳定性，它可从无限Cucker-Smale模型在平均场极限中形式推导得出。对于有界Lipschitz通信权重函数，我们得到了具有指数速度尾部和有限空间二阶矩的测度值解的有限时间Osgood型弱稳定性。与相空间受限设置不同，KCS模型的解算子关于初始数据不是Lipschitz连续的，这是由于速度尾部无界以及对齐力缺乏统一的Lipschitz界。作为弱稳定性的应用，我们得到了一个独立同分布采样结果：独立初始样本产生的经验测度在任何有限时间区间内期望收敛到相应动力学模型的测度值解。
英文摘要
We study the weak stability of the kinetic Cucker-Smale (in short, KCS) model in a phase-spatially extended setting, which can be formally derived from the infinite Cucker-Smale model in the mean-field limit. For a bounded Lipschitz communication weight function, we derive finite-time Osgood-type weak stability for measure-valued solutions with exponential velocity tails and finite spatial second moments. Unlike the phase-spatially confined setting, the solution operator to the KCS model is not Lipschitz continuous with respect to initial data. This is due to the unbounded velocity tail and the corresponding absence of a uniform Lipschitz bound for the alignment force. As an application of weak stability, we obtain an i.i.d. sampling consequence: empirical measures generated from independent initial samples converge to the measure-value solution for the corresponding kinetic model in any finite time interval, in expectation.
Comments35 pages
