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

流体动力学的应用在科学和工程中无处不在，从生物学到地质学，海洋学和航空航天。该项目侧重于在实际流体动力学应用中经常遇到的一类数学模型：自由边界问题。在这样的系统中，流体流动通过求解在其边界是动态的并且根据与流体场的耦合而演变的域中制定的偏微分方程来建模。自由边界问题是流体动力学中最具数学挑战性的问题之一，更广泛地说，由于其臭名昭著的复杂性，严重的非局部性和隐式非线性，在偏微分方程的分析中。该项目的主要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，规律性和行为，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：Muskat问题、水波和自由边界不可压缩多孔介质方程。关于多孔介质中常密度流体的Muskat问题，目的是建立整体存在性和唯一性，并研究单相问题大解的长时间行为。特别解决方案的构建及其稳定性将是另一个重点。对于水波，该项目旨在严格证明经典的二维斯托克斯波的不稳定性，无论是二维和三维扰动，表面张力的影响。自由边界的不可压缩多孔介质方程将被研究的局部适定性为一大类的密度分布和稳定性分析的稳定状态。这三个方程都是拟线性的，并且是退化抛物型或双曲型或混合型的。一套工具，从谐波分析，微局部分析，潜在的理论，光谱理论，和分叉理论将结合起来，并锐化，以解决这些具有挑战性的问题。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.