论文标题
在随机奇异的cucker-smale型号上：适应性，避免碰撞和羊群
On the Stochastic Singular Cucker--Smale Model: Well-Posedness, Collision-Avoidance and Flocking
论文作者
论文摘要
我们研究了涉及奇异性和噪声的Cucker--Smale（C-S）植入系统。我们首先在第一次碰撞时间之前展示了随机奇异的C-S系统的局部强度良好，这是一个明确的停止时间。然后，为了与原点上的高阶奇异性进行通信（在$ψ（r）= r^{ - α} $的情况下，我们可以通过在有限时间内显示碰撞避免碰撞来确定全球良好的态度，前提是没有初始碰撞，并且初始碰撞量没有任何积极的顺序。最后，当$ψ$的下限为零时，我们研究了解决方案的较大时间行为，并分别为恒定和正方形的集成强度提供条件植入或无条件羊群的出现。
We study the Cucker--Smale (C-S) flocking systems involving both singularity and noise. We first show the local strong well-posedness for the stochastic singular C-S systems before the first collision time, which is a well defined stopping time. Then, for communication with higher order singularity at origin (corresponding to $α\ge1$ in the case of $ψ(r)=r^{-α}$), we establish the global well-posedness by showing the collision-avoidance in finite time, provided that there is no initial collisions and the initial velocities have finite moment of any positive order. Finally, we study the large time behavior of the solution when $ψ$ is of zero lower bound, and provide the emergence of conditional flocking or unconditional flocking in the mean sense, for constant and square integrable intensity respectively.
