On the Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models

Abstract

The goal of this note is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy $\mathcal{H} = \int ρ\log ρ\,\mbox{d}x$, and the measure of deviation from uniformity is given by a well-known conserved quantity $e=u^{\prime }+{\mathcal{L}}_{\psi }\rho$, where $u$ is velocity and ${\mathcal{L}}_{\psi }$ is the communication operator with kernel $\psi$. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study.

accepted version. 10 pages, 0 figures