Emergent dynamics of Cucker-Smale flocking particles in a random environment

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• 作者：Seung-Yeal Ha^a ; ^b ; ^{syha@snu.ac.kr} ; Jiin Jeong^c ; ^{jiin@snu.ac.kr} ; Se Eun Noh^d ; ^{senoh@mju.ac.kr} ; Qinghua Xiao^e ; ^{pdexqh@hotmail.com} ; Xiongtao Zhang^c ; ^{zhangxiongtao1987@gmail.com}
• 关键词：74A25 ; 76N10 ; 92D25
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2554-2591
• 全文大小：476 K
• 卷排序：262

文摘

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.