Analysis of Incompressible Flows with Rigid and Free Boundaries

流体与边界的相互作用在自然界、科学和工程中普遍存在，在生物学、航空航天、海洋和地学等领域的实际应用中具有重要意义。飞机与周围空气之间、水与海床之间的边界可以是刚性的，也可以是油层中水与空气或水与油之间的自由边界。尽管在应用中得到了广泛的应用，但在对这些相互作用进行严格的数学分析时，许多基本问题仍然具有挑战性。边界的存在，特别是自由边界的存在，使得模拟这一现象的基本偏微分方程具有高度的非线性和非局部性。这个项目将进一步加深对方程的数学理解，在方程中，流体与刚性边界、自由边界或两者都相互作用。在这个项目中研究的问题是由物理学(液滴形成和流体喷射)、经济学(最优输送)和工程学(油藏工程)驱动的。在这项工作中获得的分析性理解将增加我们对实践中使用的基本物理现象和数学模型的理解。这项研究将涉及研究生和博士后学者。这个项目将研究几个当前感兴趣的数学问题，涉及具有刚性或自由边界的流体的相互作用：1.具有低正则性和大变化边界的Muskat问题的适定性和全局正则性。2.在最优输运和对流流体中出现的主动矢量输运方程的长时间行为。3.浅水小振幅周期重力行波的不稳定性。4.两种液滴形成模型的有限时间夹断奇点和无限时间夹断奇点。该项目将结合微局部分析、调和分析、分叉理论和频谱理论的工具。一些开发的工具和模型将应用于偏微分方程组的理论和计算方面的其他问题。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。

Interactions of fluids with boundaries are ubiquitous in nature, science and engineering, and are of great importance in practical applications ranging from biology to aerospace, oceanography and geoscience. The boundaries can be rigid as between an aircraft and the surrounding air or between water and the seabed or can be free as between water and the air or between water and oil in an oil reservoir. Despite being widely used in applications, many fundamental issues in the rigorous mathematical analysis of these interactions remain challenging. The presence of boundaries, especially free boundaries, makes the underlying partial differential equations modeling the phenomena highly nonlinear and nonlocal. This project will further the mathematical understanding of equations in which fluids interact with either rigid boundaries, free boundaries, or both. The problems studied in this project are motivated by physics (drop formation and fluid jets), economics (optimal transport), and engineering (reservoir engineering). The analytical understanding gained in this work will increase our understanding of the basic physical phenomena and mathematical models used in practice. This research will involve graduate students and postdoctoral scholars. This project will study several mathematical problems of current interest involving the interaction of fluids with rigid or free boundaries: 1. Well-posedness and global regularity of the Muskat problem with boundaries of low regularity and large variation. 2. Long-time behavior of an active vectorial transport equation arising in optimal transport and convective fluids. 3. Instabilities of small amplitude periodic traveling gravity waves in shallow water. 4. Finite-time and infinite-time pinch-off singularities for two models of drop formation. The project will combine tools from microlocal analysis, harmonic analysis, bifurcation theory, and spectral theory. Some of the tools and models developed will have applications to other problems in the theory and computational aspects of partial differential equations.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.