Finite- and Infinite-Time Cluster Formation for Alignment Dynamics on the Real Line

TL;DR

This paper characterizes the formation and structure of clusters in 1D Euler-alignment systems, showing that initial data determines where and when clustering occurs, including finite-time singularities and asymptotic clusters.

Contribution

It provides the first analysis of the structure of finite-time singularities and asymptotic clusters in 1D Euler-alignment systems based solely on initial data.

Findings

Locations of clustering can be determined from initial data.

Finite-time singularity set structure is characterized.

Cluster sizes relate to flux in scalar balance law formulation.

Abstract

We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated to a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated to a scalar balance law formulation of the system.