Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus of size . Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain ‐type norm is small. This energy inequality is of “quintic” type: if the norm is , then the increment of the high‐order energies is controlled for times of the order , consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order , in a suitable scaling regime relating and . For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable ‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size are regular for times of the order . In particular, on the real line, solutions exist for times of order . This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data.

中文翻译：

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关于二维重力波的波湍流理论，I：确定性能量估计


我们本文的目标是启动对波湍流的严格研究以及水波模型波动力学方程 (WKE) 的推导。近年来，这个问题在半线性模型（例如薛定谔方程或多维 KdV 型方程）中受到了强烈关注。然而，我们这里的情况有所不同，因为水波方程是拟线性的，并且由于不可避免的导数损失而无法通过杜哈梅尔公式的迭代来构造解。这是我们设计解决此问题的新策略的两篇论文中的第一篇。我们研究二维重力水波系统的解决方案。在无旋情况下，该系统可以简化为一维界面上的演化方程，该界面是一个尺寸为 的大环面。我们的第一个主要结果是确定性能量不等式，它在某种类型的范数很小的情况下，可以长时间控制（可能很大的）解的索博列夫范数。这种能量不等式是“五次”类型：如果范数为 ，则高阶能量的增量被控制为 次次，与系统的近似四次可积性一致。在本系列的第二篇论文中，我们将展示如何使用这种能量估计和随机性参数的传播来证明在与 和 相关的适当缩放机制中，达到 次数级的概率规律性结果。对于我们的第二个主要结果，我们使用合适的 Strichartz 型范数将五次能量不等式与自举论证相结合，以证明具有 Sobolev 数据大小的确定性解对于 次数级是正则的。特别是，在实线上，存在有序 次数的解。 这大大改善了所有早期使用小 Sobolev 数据的二维重力水波的寿命延长结果。