Time quasi-periodic gravity water waves in finite depth

• 成果类型: Article
• 署名作者: Baldi, Pietro; Berti, Massimiliano; Haus, Emanuele; Montalto, Riccardo
• 署名单位: University of Naples Federico II; International School for Advanced Studies (SISSA); University of Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0812-2
• 发表日期: 2018
• 页码: 739-911
• 关键词: nonlinear schrodinger-equation; partial-differential-equations; implicit function theorems; small divisor problems; global-solutions; kam; SYSTEM; perturbations; REGULARITY; fluid

摘要

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutionsnamely periodic and even in the space variable xof a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equationsthe highest order x-derivative appears in the nonlinear term but not in the linearization at the originand the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash-Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.