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.

流体动力学的应用在科学和工程中无处不在，从生物学到地质，海洋学和航空航天。该项目着重于在实用流体动力学应用中常见的一类数学模型：自由结合问题。在这样的系统中，流体流是通过在边界是动态的域中配制的部分微分方程的解决方案来建模的，并且根据与流体场的耦合而发展。自由边缘问题是流体动力学上最具挑战性的问题之一，并且由于其臭名昭著的复杂性，严重的非局部性和隐性非线性而更普遍地分析部分微分方程。该项目的广泛目标是开发新方法来提高对这类模型的了解。该项目旨在研究通用解决方案的长期存在，规律性和行为，以及特殊解决方案的稳定性。该项目还将为研究生参与研究提供机会。该项目将研究三种不同性质的模型：Muskat问题，水波和自由界限不可压缩的多孔中等方程。关于多孔培养基中恒定密度的流体的Muskat问题，目的是建立全球存在和独特性，并研究大型解决方案的一相问题的长期行为。特殊解决方案及其稳定性的建设将是另一个重点。对于水波，该项目打算严格证明经典的二维Stokes波的不稳定性，用于具有和没有表面张力效应的二维和三维扰动。将研究自由性不可压缩的多孔培养基方程，以了解稳态的大量密度曲线和稳定性分析的局部良好性。所有三个方程都是准线性的，是退化的抛物线或双曲线或混合的。将从谐波分析，微局部分析，潜在理论，光谱理论和分叉理论提出的一组工具结合起来，以解决这些具有挑战性的问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来获得支持的。