# On the Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models

## TL;DR

This paper analyzes the limiting structure of self-organized flocks in 1D hydrodynamic Euler Alignment models, providing estimates on how the density deviates from uniformity based on entropy and conserved quantities.

## Contribution

It introduces quantitative estimates linking density distortion to entropy and conserved quantities across various kernel types in Euler Alignment models.

## Key findings

- Density approaches uniformity as entropy decreases.
- Conserved quantity $e$ measures deviation from uniform density.
- Results apply to Lipschitz, singular, and topological kernels.

## Abstract

The goal of this note is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy $\mathcal{H} = \int \rho \log \rho \,\mbox{d}x$, and the measure of deviation from uniformity is given by a well-known conserved quantity $e = u' + \mathcal{L}_\psi \rho$, where $u$ is velocity and $\mathcal{L}_\psi $ is the communication operator with kernel $\psi$. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study.