二维空间拟周期重力水波的局部适定性
Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions
AI总结：
本文首次证明了二维重力水波方程在空间拟周期初始条件下的局部适定性，通过全纯坐标和希尔伯特变换计算狄利克雷-涅曼算子，并利用Littlewood-Paley分解建立乘积和交换估计。
摘要：
我们首次证明了具有空间拟周期初始条件的二维重力水波方程的局部适定性。我们用全纯坐标表示解，这相当于运动方程的共形映射形式。这允许我们通过希尔伯特变换计算狄利克雷-涅曼算子，其在空间拟周期设定下具有简单形式。我们使用适应于拟周期设定的Littlewood-Paley分解，并在此框架内建立乘积和交换估计。证明的关键步骤是推导具有拟周期初始数据的线性化水波方程的拟线性能量估计。
英文摘要：
We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data.
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