Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long-term regularity of 3D gravity water waves   We study a fundamental model in fluid mechanics--the 3D gravity water wave
equation, in which an incompressible fluid occupying half the 3D space flows
under its own gravity. In this paper we show long-term regularity of solutions
whose initial data is small but not localized. Our results include: almost
global wellposedness for unweighted Sobolev initial data and global
wellposedness for weighted Sobolev initial data with weight $|x|^\alpha$, for
any $\alpha > 0$. In the periodic case, if the initial data lives on an $R$ by
$R$ torus, and $\epsilon$ close to the constant solution, then the life span of
the solution is at least $R/(\epsilon^2(\log R)^2)$.

33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga