Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

TL;DR

This paper studies the global existence and long-term behavior of smooth and weak solutions to a fractional Euler alignment system with singular kernels, revealing conditions for flock formation and energy conservation.

Contribution

It establishes new global existence results for smooth and weak solutions in the critical case and extends the framework for analyzing long-term flocking behavior.

Findings

Global smooth solutions exist under null initial entropy and small data conditions.

Weak solutions satisfy an Onsager-type energy criterion.

Density convergence to a flock depends on initial entropy size.

Abstract

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $\phi(x):=|x|^{-(n+\alpha)}$ for $\alpha\in(0,2)$. Here, we consider the critical case $\alpha=1$ and establish a couple of global existence results of smooth solutions, together with a full description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. In fact, using Duhamel's approach we get that any solution is almost Lipschitz-continuous in space. We extend the notion of weak solution for $\alpha\in[1,2)$ and prove the existence of global Leray-Hopf solutions. Furthermore, we give an anisotropic Onsager-type criteria for the validity of the natural energy law for weak solutions of the system. Finally, we provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution depending solely on the size of the initial entropy.