Global existence and stability of nearly aligned flocks

Abstract

We study regularity of a hydrodynamic singular model of collective behavior introduced in ST1. In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data $(u,ρ)$ with small velocity variations |u(x) - u(y)| < 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.