Emergent dynamics of Cucker–Smale flocking particles in a random environment
Emergent dynamics of Cucker–Smale flocking particles in a random environment
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DOI：
10.1016/j.jde.2016.11.017
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发表时间：
2017-02
影响因子：
2.4
通讯作者：
Seung‐Yeal Ha;J. Jeong;S. E. Noh;Qinghua Xiao;Xiongtao Zhang
中科院分区：
文献类型：
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作者：
Seung‐Yeal Ha;J. Jeong;S. E. Noh;Qinghua Xiao;Xiongtao Zhang
提出了一个新的动力学Cucker-Smale-Fokker-Planck（CS-FP）型方程，该方程描述了随机环境中无限多Cucker-Smale粒子系综的动力学行为. CS-FP方程的渐近动力学表现出一种阈值现象，依赖于耦合强度和噪声强度之间的相对强度。在小耦合区域，噪声效应变得占主导地位，这导致速度方差以指数方式快速增加到无穷大。与此相反，速度对齐效应是强的，在大耦合区，速度方差趋于零指数快速。对于一个充分光滑的初始数据，我们证明了CS-FP方程古典解的整体存在性。对于具有度量依赖的通信权的动力学CS-FP方程，我们给出了从随机CS模型到动力学CS-FP方程的无收敛速度的时间上一致的平均场极限.
We present a new kinetic Cucker–Smale–Fokker–Planck (CS-FP) type equation with a degenerate diffusion, which describes the dynamics for an ensemble of infinitely many Cucker–Smale particles in a random environment. The asymptotic dynamics of the CS-FP equation exhibits a threshold-like phenomenon depending on the relative strength between the coupling strength and the noise strength. In the small coupling regime, the noise effect becomes dominant, which induces the velocity variance to increase to infinity exponentially fast. In contrast, the velocity alignment effect is strong in the large coupling regime, and the velocity variance tends to zero exponentially fast. We present the global existence of classical solutions to the CS-FP equation for a sufficiently smooth initial datum without smallness in its size. For the kinetic CS-FP equation with a metric dependent communication weight, we provide a uniform-in-time mean-field limit from the stochastic CS-model to the kinetic CS-FP equation without convergence rate.