Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves
Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves
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DOI：
10.1080/03605300903296736
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发表时间：
2009-01
影响因子：
1.9
通讯作者：
T. Alazard;Guy M'etivier
中科院分区：
文献类型：
--
作者：
T. Alazard;Guy M'etivier
本文研究基本域为对称菱形的三维双周期重力波的先验C ∞ 正则性。这种波的存在是一个长期悬而未决的问题，最近由 Iooss 和 Plotnikov 解决。主要困难在于，与传统的自由边界问题不同，对于三维纯重力波，简化的边界系统不是椭圆形的，这会导致除数较小的问题。我们的主要结果断言，满足丢番图条件的足够平滑的金刚石波自动为 C ∞。特别是，我们证明了 Iooss 和 Plotnikov 定义的解是 C ∞。两个值得注意的技术方面是：(i) 不需要微小条件；(ii) 我们获得了 Dirichlet 到 Neumann 算子的精确平行线性化公式。
This paper is concerned with a priori C ∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ∞. In particular, we prove that the solutions defined by Iooss and Plotnikov are C ∞. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.