Stratified Fluids and Completely Integrable PDEs

分层流体和完全可积偏微分方程

• 批准号: 2006212
• 负责人: Yilun Wu
• 金额: $ 16.4万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006212&HistoricalAwards=false
• 关键词: Stratified; Fluids; Completely; Integrable; PDEs

In the study of water waves in the ocean and atmospheric waves in the air, it is customary to use asymptotic models to replace general equations of fluid mechanics. These models provide efficient descriptions of the underlying dynamics by suppressing unimportant details of the fluid motion. This is achieved by imposing certain scaling regimes of the wave amplitudes, wavelengths, and fluid depths. This project is concerned with the development of methods critical in the understanding of long-time behavior of some major asymptotic models of the two-layer fluid problem. Such scenarios arise in internal ocean waves layered by water of different salinity, such as are important for medium-term climate modeling including the El Nino effect, as well as long-term climate prediction. Sub-surface internal waves can also interact with surface waves and produce phenomena like rogue waves that can damage or destroy ocean vessels or fixed structures. To study the underlying equations mathematically, special solution methods were proposed in the 1980s. These methods attempt to transform the complicated nonlinear dynamics in the original equations into well predictable linear dynamics. However, rigorous mathematical analysis of the feasibility of these methods is missing. By carrying out a combination of research and educational activities, the PI will develop the mathematical theory for these methods, and make connections to partner disciplines, while attracting prospective students into the related fields. In the current project, the PI will develop complete integrability theories for the Benjamin-Ono (BO) equation and the intermediate long wave (ILW) equation. They are both 1D asymptotic models for the two-layer fluid problem. BO and ILW are important nonlinear dispersive equations with Hamiltonian structures. Furthermore, formal action-angle diffeomorphisms known as the direct scattering transforms (DST) were conjectured to map the dynamics on phase space to angular translations on infinite dimensional tori. If the DST can be inverted (IST), one will have a powerful method to study long-time asymptotics of these equations. However, currently there is no large data IST theory for BO, nor is there even a small data DST theory for ILW. Like those for many other completely integrable equations, the IST theories for BO and ILW can be formulated as Riemann-Hilbert (RH) problems. Nevertheless, the RH problems for BO and ILW involve certain nonlocal jump conditions. Lack of theory for nonlocal RH problems hinders the solution of BO and ILW. For BO, the PI will use a newly discovered relation between the scattering functions to reduce the RH problem to inverting a Fredholm operator of zero index. The PI will then attempt to use a new identity to show that this Fredholm operator has trivial kernel. For ILW, the PI will recast the DST problem as inverting a convolution type operator and prove certain uniform estimates on the convolution kernel. The PI will construct the IST theory for ILW by regularization and solution in certain weighted spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在研究海洋中的水波和空气中的大气波时，通常用渐近模型来代替流体力学的一般方程。这些模型通过抑制流体运动的不重要细节来提供对潜在动力学的有效描述。这是通过对波的振幅、波长和流体深度施加一定的缩放机制来实现的。这个项目是关于发展的方法，在理解的长期行为的一些主要的渐近模型的两层流体问题的关键。这种情景出现在由不同盐度的水分层的内海浪中，例如对于包括厄尔尼诺效应在内的中期气候建模以及长期气候预测都很重要。次表面内波也可以与表面波相互作用，并产生像流氓波一样的现象，可以损坏或摧毁海洋船只或固定结构。为了从数学上研究基本方程，在20世纪80年代提出了特殊的求解方法。这些方法试图将原始方程中复杂的非线性动力学转化为可预测的线性动力学。然而，这些方法的可行性缺乏严格的数学分析。通过开展研究和教育活动的结合，PI将开发这些方法的数学理论，并与合作伙伴学科建立联系，同时吸引潜在的学生进入相关领域。在目前的项目中，PI将为Benjamin-Ono（BO）方程和中长波（ILW）方程开发完整的可积性理论。它们都是两层流体问题的一维渐近模型。BO和ILW是一类重要的具有Hamilton结构的非线性色散方程。此外，被称为直接散射变换（DST）的形式作用角同态映射相空间上的动力学到无限维环面上的角平移。如果DST可以反演（IST），人们将有一个强大的方法来研究这些方程的长时间渐近性。然而，目前还没有大数据IST理论的BO，也没有甚至是一个小数据DST理论的ILW。像许多其他完全可积方程一样，BO和ILW的IST理论可以表述为Riemann-Hilbert（RH）问题。然而，BO和ILW的RH问题涉及某些非局部跳跃条件。非局部RH问题理论的缺乏阻碍了BO和ILW的求解。对于BO，PI将使用散射函数之间的新发现的关系来将RH问题减少到反转零指标的Fredholm算子。PI将尝试使用一个新的恒等式来证明这个Fredholm算子具有平凡核。对于ILW，PI将把DST问题改写为卷积型算子的反演，并证明卷积核上的某些一致估计。PI将通过在特定加权空间中的正则化和求解来构建ILW的IST理论。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。