Inevitable monokineticity of strongly singular alignment

TL;DR

This paper proves that measure-valued solutions to the strongly singular Cucker-Smale model are monokinetic, enabling a rigorous derivation of the macroscopic fractional Euler-alignment system and establishing existence of solutions from particle systems.

Contribution

It demonstrates monokineticity of solutions for the strongly singular Cucker-Smale model and derives the macroscopic fractional Euler-alignment system directly from particle dynamics.

Findings

Weak measure-valued solutions are monokinetic under mild conditions.

Rigorous derivation of fractional Euler-alignment system from kinetic Cucker-Smale equation.

Existence of weak solutions for the fractional Euler-alignment system in 1D and 2D.

Abstract

We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker-Smale model with singularity of order $\alpha$ greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker-Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker-Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from kinetic Cucker-Smale equation without the need to perform any hydrodynamical limit. This suggests superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker-Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker-Smale particle system to the fractional Euler-alignment model. This leads to the final result -- existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in $\mathbb{R}^1$, including the possibility of vacuum. Existence can be extended to $\mathbb{R}^2$ under the a priori assumption that the density of the mean-field limit has no atoms.