The Vlasov alignment model with high order power-law potentials

Abstract

Consensus behavior is a notable emergence phenomenon in nature. It is only recently that consensus behavior has been demonstrated in the Vlasov alignment and Euler alignment models with low-order power-law potentials, i.e. \(U(r)=r^{\alpha }, \ \alpha \in [1,4)\). Note that the attraction between particles weakens as \(\alpha \) grows, so it is interesting to consider the high-order power-law potential case. By some macroscopic and microscopic Lyapunov functionals, for any \(\alpha \in (2,\infty )\) and any long range communication weight, we establish both the weak and strong consensus and their precise convergence rates for the Vlasov alignment and Euler alignment models.