Second-order swarming model: Large-time behavior and uniform-in-time mean-field limit

Abstract

We investigate the large-time behavior and uniform-in-time stability of solutions for a second-order swarming model characterized by quadratic confinement, power-law-type interactions in positions, and velocity alignment. We begin by analyzing the large-time behavior through the total energy and its modifications, establishing a framework for exponential consensus in both positions and velocities. We then demonstrate the uniform-in-time stability of solutions using these exponential consensus estimates. As a direct application, we construct global-in-time measure-valued solutions for the corresponding kinetic swarming model. Furthermore, we derive a quantitative uniform-in-time mean-field limit from the particle swarming model to the kinetic model in the second-order Wasserstein distance.