Mathematical Analysis of Water Waves and Other Fluid Models

尽管水波在物理和工程中无处不在并具有重要意义，但它仍然很难分析或预测，这在很大程度上是因为存在未知的、时变的流体域，以及时空图案影响能量和流体粒子传输的非线性方式。虽然目前有许多分析工具用于研究小幅度的水波，但该项目的重点是开发新的机械来构造大幅度的波，并调查它们在小扰动下持续存在的能力。这个项目的另一个目标是了解能量如何随着流体和固体之间的相互作用而演变。该项目的结果将促进水波的数学理论，并有助于理解流体力学中的其他数学模型和其他相关的应用科学和工程分支。这项研究还包括与研究生和博士后研究人员的培训和合作。本项目将对由水波和其他流体模型引起的一些非线性偏微分方程组进行严格的研究，将一些分析技术扩展到相关背景下，并将研究整合到本科生和研究生的教育中。具体地说，关于水波问题和其他具有物理意义的系统解的存在性和定性性质，我们提出了三个不同的方向：(1)使用一种新的全局分叉理论机制来构造通道中两相流体系统的大振幅前型解和孤立波解，并将这一整体理论扩展到处理按更一般的对称群演化的行波；(2)证明了孤立波对某些长波水波模型的谱稳定性；(3)建立了流体相互作用模型的精确能量方程，得到了有边界区域内Navier-Stokes解的弱无粘极限的充分条件。这项研究的主要内容和技术包括椭圆型偏微分方程的方法、分叉和度理论和稳定性分析，以及能量估计和傅立叶分析。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Despite their ubiquity and importance in physics and engineering, water waves remain very difficult to analyze or to predict, largely because of the presence of the unknown, time-varying fluid domain and the nonlinear way the spatiotemporal patterns affect the transport of energy and fluid particles. While many analytical tools currently exist for studying water waves of small amplitude, this project focuses on developing new machinery to construct large-amplitude waves and to investigate their ability to persist under small perturbations. Another objective of this project is to understand how energy evolves in response to the interaction between a fluid and a solid. Results from this project will advance the mathematical theory of water waves and contribute to understanding of other mathematical models in fluid mechanics and other related branches of applied science and engineering. This research also involves training and collaboration with graduate students and postdoctoral researchers.This project will develop a rigorous study of some nonlinear partial differential equations arising from water waves and other fluid models, extending some of the analytic techniques to related contexts, and integrating the research to the education of undergraduate and graduate students. Specifically, the proposed research addresses three separate directions regarding the existence and qualitative properties of the solutions to the water wave problem as well as other physically significant systems: (1) using a novel global bifurcation theoretic machinery to construct large-amplitude front-type solutions and solitary wave solutions to a two-phase fluid system in a channel, and extend this global theory to handle traveling waves that evolve according to more general symmetry groups; (2) proving the spectral stability of solitary waves to some long-wave water wave model which are indefinite energy saddles; (3) establishing exact energy equality for a fluid-interaction model and deriving sufficient conditions for the weak inviscid limit of Navier-Stokes solutions in domains with boundary. The main ingredients and techniques involved in the study include methods from elliptic partial differential equations, bifurcation and degree theory and stability analysis, together with energy estimates and Fourier analysis.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.