On the Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models

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 $\mathcal{H} = \int \rho \log \rho \,\mbox{d}x$, and the measure of deviation from uniformity is given by a well-known conserved quantity $e = u' + \mathcal{L}_\psi \rho$, 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.