In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels φ(x):=∣x∣−(n+α) for α∈(0,2)
Nearby in the stack
. Here, we consider the critical case
α=1
and establish a couple of global existence results of smooth solutions, together with a full 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. In fact, using Duhamel's approach we get that any solution is almost Lipschitz-continuous in space. We extend the notion of weak solution for
α∈[1,2)
and prove the existence of global Leray-Hopf solutions. Furthermore, we give an anisotropic Onsager-type criteria for the validity of the natural energy law for weak solutions of the system. Finally, we provide a series of 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.
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