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-way 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.

中文翻译：

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多维周期性水波的长期存在

我们证明了用2维周期性接口全重力毛细水波系统的寿命延长的结果：对于足够小的尺寸的初始数据\（{\ varepsilon} \） ，光滑解存在最多的顺序的时间\（ {\ varepsilon ^ {-5/3 +}} \），几乎适用于重力和表面张力参数的所有值。除方程的拟线性性质外，主要困难是要处理与二次和三次相互作用有关的弱小除数边界，并随最大频率的大小而增长。为了克服这个困难，我们使用（1）方程的（Hamiltonian）结构，该结构在共振超表面附近提供额外的平滑度；（2）与时间可逆性相关的另一种结构特性，使我们能够处理“平凡”的三次共振。 ，（3）基于计数自变量的三向和四向调制函数的尖锐小除数下界，以及（4）傅立叶空间中的部分正态变换和对称化自变量。