Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and Uniqueness of the Solution to the Anisotropic
  Quasi-Geostrophic Equations in the Sobolev Space   In this paper, we focus on the two-dimensional surface quasi-geostrophic
equation with fractional horizontal dissipation and vertical thermal diffusion
which represents a general case of the classical surface quasi-geostrophic
equation. On the one hand, we will show the local existence and uniqueness of
the solution in Sobolev space $H^{2-2\alpha}(\mathbb{R}^2)\cap
H^{2-2\beta}(\mathbb{R}^2)$, which is the critical space in the classical case.
Furthermore, we will demonstrate that the solution is global even when the
initial data is very small. Finally, we will study the asymptotic
representation of our global solution in infinity.

18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th