Singular Cucker-Smale Dynamics

TL;DR

This paper reviews singular flocking models from microscopic to macroscopic scales, discussing collision avoidance, solution existence, and analytical comparisons with related fluid models.

Contribution

It provides a comprehensive overview of singular Cucker-Smale models across different regimes, including new existence results and analytical comparisons.

Findings

Collision-avoidance in microscopic models

Existence of global measure-valued solutions

Comparison of macroscopic models with porous medium equations

Abstract

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation.