Sharp conditions to avoid collisions in singular Cucker-Smale interactions

TL;DR

This paper analyzes the Cucker-Smale flocking model with singular communication weights, identifying conditions on the singularity exponent that prevent collisions and ensure global regularity of solutions.

Contribution

It establishes critical thresholds for the singularity exponent in the communication weight that guarantee collision avoidance and regularity in the model.

Findings

For $oldsymbol{ ext{α} ext{ ≥ } 1}$, particles do not collide if initially separated.

For $oldsymbol{ ext{α} ext{ ≥ } 2}$, a uniform estimate controls the particle distances in the expanded singularity model.

The results provide conditions for global regularity and collision avoidance in singular Cucker-Smale interactions.

Abstract

We consider the Cucker-Smale flocking model with a singular communication weight $\psi(s) = s^{-\alpha}$ with $\alpha > 0$. We provide a critical value of the exponent $\alpha$ in the communication weight leading to global regularity of solutions or finite-time collision between particles. For $\alpha \geq 1$, we show that there is no collision between particles in finite time if they are placed in different positions initially. For $\alpha\geq 2$ we investigate a version of the Cucker-Smale model with expanded singularity i.e. with weight $\psi_\delta(s) = (s-\delta)^{-\alpha}$, $\delta\geq 0$. For such model we provide a uniform with respect to the number of particles estimate that controls the $\delta$-distance between particles. In case of $\delta = 0$ it reduces to the estimate of non-collisioness.