16:10-16:30 Tuesday, 6 May 2025 Auditorium, Rectorate
Abstract
The goal of this talk is to present the multi-scale theory for a new model of alignment, called the s-model, alongside results on the long-time behavior of a Fokker-Planck-Navier-Stokes system describing interacting particles in a fluid. The s-model generalizes the Cucker-Smale model by allowing for a broader class of velocity averaging protocols, while preserving the 1D conservation law, $\partial_t e + \partial_x(ue) = 0$, $e = \partial_x u + s$, which plays a critical role in the regularity theory as well as the long-time behavior. We will present our results on the mean field and hydrodynamic passage between all scales of description as well as the long-time behavior. In the second part of the talk, we examine the long-time behavior of Cucker-Smale particles in a fluid, coupled via a drag force. In particular, we present the unconditional alignment of the particle and fluid velocities for the Fokker-Planck-Navier-Stokes system. Previous results of Carillo and Choi were conditioned on the sufficient integrability of the macroscopic particle density, which is not known a priori. We provide an unconditional alignment of the particle and fluid velocity by utilizing the Fisher information.