Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Strong ill-posedness of logarithmically regularized 2D Euler equations
  in the borderline Sobolev Space   Logarithmically regularized 2D Euler equations are active scalar equations
with the non-local velocity $u = \nabla^\perp \Delta^{-1}T_\gamma \omega$ for
the scalar $\omega$. Two types of the regularizing operator $T_\gamma$ with a
parameter $\gamma> 0$ are considered: $T_\gamma = \ln^{-\gamma} (e+|\nabla|)$
and $T_\gamma = \ln^{-\gamma} (e-\Delta)$. These models regularize the 2D Euler
equation for the vorticity (conventionally corresponding to the $\gamma=0$
case), which results in their local well-posedness in the borderline Sobolev
space $H^1(\mathrm{R}^2)\cap\dot{H}^{-1}(\mathrm{R}^2)$ when $\gamma>\frac 12$.
In this paper, we examine the regularized models in the remaining regime
$\gamma\leq \frac 12$ and establish the strong ill-posedness in the borderline
space. This completely solves the well-posedness problem of the regularized
models in the borderline space by closing the gap between the local
well-posedness result for $\gamma>\frac 12$ and the strong ill-posedness for
$\gamma = 0$.

19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci