Mathematical Analysis of Water Waves and Other Fluid Models

尽管水波在物理和工程中无处不在，也很重要，但它仍然很难分析或预测，这主要是因为存在未知的、时变的流体域，以及时空模式影响能量和流体粒子传输的非线性方式。虽然目前已有许多分析工具用于研究小振幅水波，但本项目侧重于开发新的机器来构建大振幅水波并研究它们在小扰动下持续存在的能力。该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。该项目的研究成果将促进水波的数学理论，并有助于理解流体力学和其他应用科学与工程相关分支的其他数学模型。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。本项目将对水波及其他流体模型的非线性偏微分方程进行严谨的研究，将一些分析技术扩展到相关的环境中，并将研究整合到本科生和研究生的教育中。具体而言，拟议的研究涉及三个不同的方向，涉及水波问题以及其他物理上重要的系统的解决方案的存在性和定性：(1)利用一种新的全局分岔理论机制，构造了通道内两相流体系统的大振幅前型解和孤波解，并将这一全局理论推广到处理更一般对称群演化的行波；(2)对具有无限能量鞍的长波水波模型证明了孤波的谱稳定性；(3)建立了流体-相互作用模型的精确能量等式，给出了有边界区域内Navier-Stokes解弱无粘极限的充分条件。该研究涉及的主要成分和技术包括椭圆偏微分方程、分岔和度理论和稳定性分析方法，以及能量估计和傅里叶分析。该奖项反映了美国国家科学基金会的法定使命，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，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.