The large-time behavior of the Vlasov alignment model with power-law or Riesz potentials

We consider the Vlasov alignment model with pairwise attractive potentials. By using an artificial potential energy representing the velocity alignment, we construct a new Lyapunov functional, and then establish the large time behavior of the Vlasov alignment model with any regular power-law potential. Compared with previous results, which focus on the low order power-law potential case, we give a better estimate of convergence rates. More importantly, the consensus with polynomial rates is also established for the high order power-law potential case. In the meanwhile, for the Riesz and logarithmic potential case, the non-existence of flocking is proved. Besides, the above large time behavior also holds for the hydrodynamic model.(c) 2023 Elsevier B.V. All rights reserved.