Well-posedness and Long Time Behavior of the Euler Alignment System with Adaptive Communication Strength

TL;DR

This paper introduces a new flocking model with adaptive communication strength that preserves entropy and ensures well-posedness, capturing realistic flocking behavior and extending classical results to more general settings.

Contribution

The paper develops a well-posedness theory for a novel flocking model with evolving communication strength, unifying features of Cucker-Smale and Motsch-Tadmor models.

Findings

The model exhibits alignment and flocking behavior similar to classical models.

Entropy law preservation leads to better long-term behavior analysis.

Numerical simulations confirm qualitative similarities with established flocking models.

Abstract

We study a new flocking model which has the versatility to capture the physically realistic qualitative behavior of the Motsch-Tadmor model, while also retaining the entropy law, which lends to a similar 1D global well-posedness analysis to the Cucker-Smale model. This is an improvement to the situation in the Cucker-Smale case, which may display the physically unrealistic behavior that large flocks overpower the dynamics of small, far away flocks; and it is an improvement in the situation in the Motsch-Tadmor case, where 1D global well-posedness is not known. The new model was proposed in arXiv:2211.00117v3 and has a similar structure to the Cucker-Smale and Motsch-Tadmor hydrodynamic systems, but with a new feature: the communication strength is not fixed, but evolves in time according to its own transport equation along the Favre-filtered velocity field. This transport of the communication strength is precisely what preserves the entropy law. A variety of phenomenological behavior can be obtained from various choices of the initial communication strength, including the aforementioned Motsch-Tadmor-like behavior. We develop the general well-posedness theory for the new model and study the long time behavior -- including alignment, strong flocking in 1D, and entropy estimates to estimate the distribution of the limiting flock, all of which extend the classical results of the Cucker-Smale case. In addition, we provide numerical evidence to show the similar qualitative behavior