# On the Euler-Alignment system with weakly singular communication weights

## TL;DR

This paper investigates the pressureless Euler equations with weakly singular nonlocal alignment interactions, revealing a critical threshold phenomenon and diverse global behaviors depending on initial conditions, thus extending understanding of biological flocking models.

## Contribution

It introduces analysis of the Euler-Alignment system with weakly singular interactions, identifying a critical threshold and highlighting unique behaviors for critical initial data.

## Key findings

- Critical threshold phenomenon similar to bounded interactions
- Existence of diverse global behaviors for critical initial data
- Insights into the structure of weakly singular alignment operators

## Abstract

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of complex biological systems modeling animal flocks. For such Euler-Alignment system with bounded interactions, a critical threshold phenomenon is proved by Tadmor-Tan in 2014, where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained by Do-Kiselev-Ryzhik-Tan in 2018, for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.