Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems

TL;DR

This paper investigates the geometric structure of mass concentration sets in pressureless Euler alignment systems, revealing a deep link between entropy functions and the support of singular measures, with implications for flocking behavior and concentration set dimensions.

Contribution

It establishes a novel correspondence between entropy functions and the support of limiting measures, providing new insights into the structure and properties of flocking dynamics in Euler alignment systems.

Findings

Concentration sets form unions of $C^1$ hypersurfaces.

A sharp upper bound on the dimension of concentration sets.

A detailed analysis of well-posedness, flocking, and stability.

Abstract

We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $\phi$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + \phi*\rho_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of $e_0$. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.