BV solutions for a hydrodynamic model of flocking--type with all-to-all interaction kernel

TL;DR

This paper proves the global existence of entropy weak solutions with concentration for a 1D hydrodynamic flocking model with all-to-all interactions, and demonstrates conditions for asymptotic flocking behavior.

Contribution

It establishes the existence of solutions with concentration for the model and identifies conditions leading to flocking, advancing understanding of hydrodynamic flocking models.

Findings

Global existence of entropy weak solutions with concentration

Conditions for time-asymptotic flocking behavior

Solutions applicable for BV initial data with finite mass

Abstract

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration 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. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking.