Abstract: In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels ? (x) := | x| (n+? ) for ? in (0, 2). Here, we consider the critical case ? = 1 and establish a couple of global existence results of smooth solutions, together with a description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. We extend the notion of weak solution and prove the existence of global Leray-Hopf solutions for ? in [1, 2). Furthermore, we give anisotropic Onsager-type criteria for the validity of the natural energy law of the system. Finally, we provide quantitative estimates that show how far the density of the limiting flock is from a uniform distribution, depending solely on the size of the initial entropy.
