Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Break-down criterion for the water-wave equation   We study the break-down mechanism of smooth solution for the gravity
water-wave equation of infinite depth. It is proved that if the mean curvature
$\kappa$ of the free surface $\Sigma_t$, the trace $(V,B)$ of the velocity at
the free surface, and the outer normal derivative $\frac {\pa P} {\pa
\textbf{n}}$ of the pressure $P$ satisfy \beno &&\displaystyle\sup_{t\in
[0,T]}\|\kappa(t)\|_{L^p\cap L^2}+\int_0^T\|(\na V, \na
B)(t)\|_{L^\infty}^6dt<+\infty, &\displaystyle\inf_{(t,x,y)\in [0,T]\times
\Sigma_t}-\frac {\pa P} {\pa \textbf{n}}(t,x,y)\ge c_0, \eeno for some $p>2d$
and $c_0>0$, then the solution can be extended after $t=T$.

32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
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
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga