On the Stochastic Singular Cucker--Smale Model: Well-Posedness, Collision-Avoidance and Flocking

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 $\alpha\ge1$ in the case of $\psi(r)=r^{-\alpha}$), 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 $\psi$ 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.