Sharp conditions to avoid collisions in singular Cucker–Smale interactions

• Published in: Nonlinear analysis: real world applications Vol. 37; pp. 317 - 328
• Main Authors: Carrillo, José A., Choi, Young-Pil, Mucha, Piotr B., Peszek, Jan
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.10.2017; Elsevier BV
• Subjects: Collision avoidance; Cucker–Smale system; Flocking models; Global in time existence; Non-collisioness gases; Singular communication weight; Weight; Non-collisioness gases; Global in time existence; Cucker–Smale system; Flocking models; Singular communication weight
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2017.02.017

We consider the Cucker–Smale flocking model with a singular communication weight ψ(s)=s−α with α>0. We provide a critical value of the exponent α in the communication weight leading to global regularity of solutions or finite-time collision between particles. For α≥1, we show that there is no collision between particles in finite time if they are placed in different positions initially. For α≥2 we investigate a version of the Cucker–Smale model with expanded singularity i.e. with weight ψδ(s)=(s−δ)−α, δ≥0. For such model we provide a uniform with respect to the number of particles estimate that controls the δ-distance between particles. In case of δ=0 it reduces to the estimate of collision avoidance.