Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global existence and limiting behavior of unidirectional flocks for the
  fractional Euler Alignment system   In this note we continue our study of unidirectional solutions to
hydrodynamic Euler alignment systems with strongly singular communication
kernels $\phi(x):=|x|^{-(n+\alpha)}$ for $\alpha\in(0,2)$. Here, we consider
the critical case $\alpha=1$ 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 $\alpha\in[1,2)$ 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.

17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg