Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence and temporal asymptotics of solutions for the two and
  half dimensional Hall MHD   In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking
the velocity field $u$ and the magnetic field $B$ of the form
$u(t,x,y)=\left(\nabla^{\perp}\phi(t,x,y), W(t,x,y)\right)$ and
$B(t,x,y)=\left(\nabla^{\perp}\psi(t,x,y), Z(t,x,y)\right)$.
  We begin with the Hall equations (without the effect of the fluid part). We
first show the long time behavior of weak solutions and weak-strong uniqueness.
We then proceed to prove the existence of unique strong solutions locally in
time and to derive a blow-up criterion. We also demonstrate that the strong
solution exists globally in time and decay algebraically if some smallness
conditions are imposed. We further improve the decay rates of $\psi$ using the
structure of the equation of $\psi$. As a consequence of the decay rates of
$(\psi,Z)$, we find the asymptotic profiles of $(\psi,Z)$. We finally show that
a small perturbation of initial data near zero can be extended to a small
perturbations near harmonic functions.
  In the presence of the fluid filed, the results, by comparison, fall short of
the previous ones in the absence of the fluid part. We prove two results: the
existence of unique strong solutions locally in time and a blow-up criterion,
and the existence of unique strong solutions globally in time with some
smallness condition on initial data.

15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg