# Global regularity of two-dimensional flocking hydrodynamics

## TL;DR

This paper analyzes the conditions under which smooth solutions to two-dimensional flocking hydrodynamics remain globally regular and exhibit flocking behavior, based on initial configuration thresholds.

## Contribution

It derives sharp critical thresholds in initial conditions that guarantee global regularity and flocking in 2D Euler systems with velocity alignment.

## Key findings

- Global regularity persists under sub-critical initial conditions.
- Initial divergence and spectral gap determine long-term behavior.
- Flocking behavior is linked to initial phase space thresholds.

## Abstract

We study the systems of Euler equations which arise from agent-based dynamics driven by velocity \emph{alignment}. It is known that smooth solutions of such systems must flock, namely -- the large time behavior of the velocity field approaches a limiting "flocking" velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration which characterize the global regularity and hence flocking behavior of such \emph{two-dimensional} systems. Specifically, we prove for that a large class of \emph{sub-critical} initial conditions such that the initial divergence is "not too negative" and the initial spectral gap is "not too large", global regularity persists for all time.