Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

TL;DR

This paper establishes a comprehensive theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity, using entropy conditions and sticky particle dynamics to ensure uniqueness and approximation.

Contribution

It introduces a novel low-regularity framework for the 1D Euler-alignment system, linking entropy conditions with sticky particle dynamics for solution selection.

Findings

Unique weak solutions characterized by entropy conditions.

Approximation of solutions via sticky particle Cucker-Smale dynamics.

Extension of low-regularity theory to measure-valued densities.

Abstract

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker--Smale dynamics. Our approach is inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.