Abstract:
The aim of this talk is to introduce a first order system of macroscopic equations describing the dynamics of an infinitely large ensemble of Cucker–Smale particles with singular interactions. It is reminiscent of the pressureless Eulerian dynamics with alignment effects, that is the natural macroscopic counterpart of the classical Cucker–Smale model once inertia is neglected in the momentum equation. We will focus on singular influence functions, that have been recently proposed as a way to avoid the odd behavior exhibited by regular weights when we have isolated groups separated by large distances.
Since there is just a handful of contributions to that singular model, we first introduce a singular hyperbolic scaling of the kinetic Cucker–Smale model with regular weights, friction and noise depending on a parameter ?. Next, we derive the rigorous hydrodynamic singular limit, giving rise to measure-valued solutions of the macroscopic system as ? vanishes. Finally, we prove that for small enough initial data, the singular macroscopic system enjoys a unique global-in-time solution with Sobolev-type regularity. Most of this analysis relies on deriving appropriate regularity estimates for the alignment term, that can be regarded as a commutator of singular integrals. |