Global existence and stability of nearly aligned flocks

TL;DR

This paper proves the global existence, stability, and exponential flocking behavior of solutions in a hydrodynamic model of collective behavior, under small initial velocity variations, in multi-dimensional settings.

Contribution

It establishes global well-posedness and stability results for a hydrodynamic flocking model with small initial velocity differences, extending previous work to multi-dimensional cases.

Findings

Global regular solutions exist for small initial velocity variations.

Solutions align and flock exponentially fast.

Limiting flock configurations are stable.

Abstract

We study regularity of a hydrodynamic singular model of collective behavior introduced in \cite{ST1}. In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data $(u,\rho)$ with small velocity variations $|u(x) - u(y)| < \e$ relative to its higher order norms, gives rise to a unique global regular solution which aligns and flocks exponentially fast. Moreover, we prove that the limiting flocks are stable.