Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inviscid models generalizing the 2D Euler and the surface
  quasi-geostrophic equations   Any classical solution of the 2D incompressible Euler equation is global in
time. However, it remains an outstanding open problem whether classical
solutions of the surface quasi-geostrophic (SQG) equation preserve their
regularity for all time. This paper studies solutions of a family of active
scalar equations in which each component $u_j$ of the velocity field $u$ is
determined by the scalar $\theta$ through $u_j =\mathcal{R} \Lambda^{-1}
P(\Lambda) \theta$ where $\mathcal{R}$ is a Riesz transform and
$\Lambda=(-\Delta)^{1/2}$. The 2D Euler vorticity equation corresponds to the
special case $P(\Lambda)=I$ while the SQG equation to the case $P(\Lambda)
=\Lambda$. We develop tools to bound $\|\nabla u||_{L^\infty}$ for a general
class of operators $P$ and establish the global regularity for the Loglog-Euler
equation for which $P(\Lambda)= (\log(I+\log(I-\Delta)))^\gamma$ with $0\le
\gamma\le 1$. In addition, a regularity criterion for the model corresponding
to $P(\Lambda)=\Lambda^\beta$ with $0\le \beta\le 1$ is also obtained.

3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int