Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Interior and boundary regularity for the Navier-Stokes equations in the
  critical Lebesgue spaces   We study regularity criteria for the $d$-dimensional incompressible
Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times
\mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and
the pressure $p$ satisfies a local condition $\|p\|_{L_{2-1/d}(Q(z_0,1)\cap
(0,T)\times \mathbb{R}^d_+)}\leq K$ for some constant $K>0$ uniformly in $z_0$,
then $u$ is regular up to the boundary in $(0,T)\times \mathbb{R}^d_+$.
Furthermore, when $T=\infty$, $u$ tends to zero as $t\rightarrow \infty$. We
also study the local problem in half unit cylinder $Q^+$ and prove that if
$u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is
H\"{o}lder continuous in the closure of the set $Q^+(1/4)$. This generalizes a
result by Escauriaza, Seregin, and \v{S}ver\'{a}k to higher dimensions and
domains with boundary.

12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics