Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long-time existence for multi-dimensional periodic water waves   We prove an extended lifespan result for the full gravity-capillary water
waves system with a $2$ dimensional periodic interface: for initial data of
sufficiently small size $\varepsilon$, smooth solutions exist up to times of
the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and
surface tension parameters. Besides the quasilinear nature of the equations,
the main difficulty is to handle the weak small divisors bounds for quadratic
and cubic interactions, growing with the size of the largest frequency. To
overcome this difficulty we use (1) the (Hamiltonian) structure of the
equations which gives additional smoothing close to the resonant hypersurfaces,
(2) another structural property, connected to time-reversibility, that allows
us to handle "trivial" cubic resonances, (3) sharp small divisors lower bounds
on three and four-waves modulation functions based on counting arguments, and
(4) partial normal form transformations and symmetrization arguments in the
Fourier space. Our theorem appears to be the first extended lifespan result for
quasilinear equations with non-trivial resonances on a multi-dimensional torus.

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The categories of this paper are: supr-con
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph