Unconditional flocking for weak solutions to self-organized systems of Euler-type with all-to-all interaction kernel

TL;DR

This paper proves that in a one-dimensional hydrodynamic flocking model with all-to-all interactions, global weak solutions always tend to a flocking state exponentially fast, regardless of initial data specifics.

Contribution

It establishes unconditional flocking for all weak solutions with finite mass and positive density, without additional initial data restrictions.

Findings

Global entropy weak solutions exhibit unconditional flocking.

Convergence to flocking profile is exponential.

Results hold for initial data with finite mass and positive density.

Abstract

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in [2] to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.