Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remarks on well-posedness of the generalized surface quasi-geostrophic
  equation   In this paper, we are concerned with the Cauchy problem of the generalized
surface quasi-geostrophic (SQG) equation in which the velocity field is
expressed as $u=K\ast\omega$, where $\omega=\omega(x,t)$ is an unknown function
and $K(x)=\frac{x^\perp}{|x|^{2+2\alpha}}, 0\le\alpha\le \frac12.$ When
$\alpha=0$, it is the two-dimensional Euler equations. When $\alpha=\frac 12$,
it corresponds to the inviscid SQG. We will prove that if the existence
interval of the smooth solution to the generalized SQG for some
$0<\alpha_0\le\frac12$ is $[0,T]$, then under the same initial data, the
existence interval of the generalized SQG with $\alpha$ which is close to
$\alpha_0$ will keep on $[0,T]$. As a byproduct, our result implies that the
construction of the possible singularity of the smooth solution of the Cauchy
problem to the generalized SQG with $\alpha>0$ will be subtle, in comparison
with the singularity presented in [Kiselev et al 2016]. To prove our main
results, the difference between the two solutions and meanwhile the
approximation of the singular integrals will be dealt with. Some new uniform
estimates with respect to $\alpha$ on the singular integrals and commutator
estimates will be shown in this paper.

14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci