Discrete Cucker--Smale Flocking Model with a Weakly Singular Weight
Jan Peszek · SIAM Journal on Mathematical Analysis · 2015
The Cucker--Smale flocking model is one of the many aggregation and flocking models describing a collective, self-driven motion of self-propelled particles with some predefined tendency (e.g., to form a flock, to move in a specific direction). We are interested in the discrete Cucker--Smale flocking model with a singular kernel $\psi(s)=s^{-\alpha}$ with $0<\alpha$. The Cucker--Smale model with a singular kernel possesses particularly interesting and difficult-to-analyze dynamics. Even the most basic question of well-posedness for the particle system remains open. In this paper we prove that in case of $\alpha<\frac{1}{2}$ the velocity component of a certain type of weak solutions is absolutely continuous. This result enables us to obtain existence and uniqueness of global solutions.