Regularity and Stability Analysis of Free-Boundary Problems in Fluid Dynamics

流体动力学自由边界问题的规律性和稳定性分析

• 批准号: 2205710
• 负责人: Huy Nguyen
• 金额: $ 34万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205710&HistoricalAwards=false
• 关键词: Regularity; Stability; Analysis; Free; Boundary

Applications of fluid dynamics are ubiquitous in science and engineering, ranging from biology to geology, oceanography, and aerospace. This project focuses on a class of mathematical models commonly encountered in practical fluid dynamics applications: free-boundary problems. In such systems, fluid flow is modeled by the solution to a partial differential equation formulated in a domain whose boundary is dynamic and evolves according to couplings with the fluid fields. Free-boundary problems are among the most mathematically challenging in fluid dynamics and more generally in the analysis of partial differential equations due to their notorious complexity, severe nonlocality, and implicit nonlinearity. The broad objective of this project is to develop new methods to advance understanding of this class of models. The project aims to study the long-time existence, regularity, and behavior of generic solutions, as well as the stability of special solutions. The project will also provide opportunities for involvement of graduate students in the research.The project will study three models of different nature: the Muskat problem, water waves, and the free-boundary incompressible porous medium equation. Regarding the Muskat problem for fluids with constant density in porous media, the aims are to establish global existence and uniqueness and investigate long-time behavior of large solutions for the one-phase problem. Construction of special solutions and their stability will be another focus. For water waves, the project intends to rigorously demonstrate the instability of the classic two-dimensional Stokes waves for both two-dimensional and three-dimensional perturbations with and without surface tension effects. The free-boundary incompressible porous medium equation will be investigated regarding local well-posedness for a large class of density profiles and stability analysis for steady states. All three equations are quasilinear and are either degenerate parabolic or hyperbolic or mixed. A set of tools from harmonic analysis, microlocal analysis, potential theory, spectral theory, and bifurcation theory will be combined and sharpened to tackle these challenging questions.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.

流体动力学在科学和工程中的应用无处不在，从生物学到地质学、海洋学和航空航天。本课题主要研究一类在实际流体力学应用中经常遇到的数学模型：自由边界问题。在这样的系统中，流体流动是由在一个区域中建立的偏微分方程解来模拟的，该区域的边界是动态的，并根据与流场的耦合而演变。自由边界问题由于其复杂性、严重的非局部性和隐含的非线性，在流体力学和偏微分方程组的分析中是最具数学挑战性的问题之一。这个项目的广泛目标是开发新的方法来促进对这类模型的理解。该项目旨在研究一般解的长期存在性、规律性和性态，以及特殊解的稳定性。该项目还将为研究生提供参与研究的机会。该项目将研究三种不同性质的模型：马斯卡特问题、水波和自由边界不可压缩多孔介质方程。对于多孔介质中具有恒定密度的流体的Muskat问题，目的是建立全局解的存在唯一性，并研究单相问题大解的长期行为。特殊解决方案的构建及其稳定性将是另一个重点。对于水波，该项目打算严格地证明经典的二维Stokes波在有和没有表面张力效应的情况下对二维和三维扰动的不稳定性。关于一大类密度分布的局部适定性和稳态的稳定性分析，将研究自由边界不可压缩多孔介质方程。这三个方程都是拟线性的，要么退化，要么是抛物线，要么是双曲型，要么是混合型。来自调和分析、微域分析、势能理论、频谱理论和分叉理论的一系列工具将被结合并磨砺以解决这些具有挑战性的问题。该奖项反映了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。