On the global classical solution to compressible Euler system with singular velocity alignment

TL;DR

This paper establishes local and global well-posedness results for a compressible Euler system with singular velocity alignment, demonstrating exponential convergence to flocking behavior in large animal group models.

Contribution

It provides the first rigorous analysis of well-posedness and flocking dynamics for the Euler-alignment system with singular velocity interactions.

Findings

Local well-posedness of the system

Global well-posedness for small initial data

Exponential convergence to flocking state

Abstract

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.