2021/05/25 by Daniel Lear, Lear, Daniel
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2105.12064
openalex publication_date 2021/05/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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). 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.