Emergence of multi-cluster configurations from attractive and repulsive interactions

We discuss a first-order CuckerSmale-type consensus model with attractive and repulsive interactions and present upper and lower bound estimates on the number of asymptotic point-clusters depending on the relative ranges of interactions and coupling strength. When the number of agents approaches infinity, we introduce a scalar conservation law with a non-local flux for a macroscopic description. We show that the corresponding conservation law admits a classical solution for sufficiently smooth initial data, which illustrates the shock avoidance effect due to the non-locality of the interactions. We also study the dynamics of special Dirac-Comb-type solutions consisting of two and three point-clusters.