Regularity and Asymptotic Behavior in Fluid Dynamics

偏微分方程是研究自然现象，如流体流动，传热和波传播的基本方程。它们有助于模拟大气、海洋、恒星和其他物理现象的动力学。该项目的目的是研究模拟流体动力学和其他涉及流体的复杂系统的偏微分方程，从而提高对流体特性和控制的理解。描述流体与固体和弹性体相互作用的数学模型也很有意义，可应用于工程和其他科学领域。该项目还为研究生提供培训和研究机会。这个项目解决了偏微分方程的解的存在性，规律性和定性性质，以及模拟流体与弹性体相互作用的系统。该项目考虑流体-结构相互作用模型，涉及流体与弹性固体相互作用，可压缩或不可压缩流体，以及粘性和非粘性设置。主要的重点是建立整体的存在性，唯一性和渐近行为的解决方案。此外，该项目解决了涉及板和弹性膜的流体-结构相互作用模型的解的存在性和定性行为。此外，该项目还研究了无粘极限问题和其他奇异极限，无论是在有界域中的Navier-Stokes方程还是在更复杂的物理环境中。该项目还应解决与不断变化的边界流体模型有关的存在性和规律性问题。一个重要的例子是具有自由界面的欧拉系统，有和没有表面张力。最后，该项目将考虑与Boussinesq系统有关的问题，包括与规律性的持久性和解决方案的长期行为有关的问题。该奖项反映了NSF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。

Partial differential equations are essential for studying natural phenomena such as fluid flow, heat transfer, and wave propagation. They help model the dynamics of the atmosphere, ocean, stars, and other physical phenomena. The aim of this project is to study partial differential equations that model fluid dynamics and other complex systems that involve fluids, allowing for improved understanding of fluid properties and control. Mathematical models describing fluid interactions with solid and elastic bodies are also of interest, with applications to engineering and other scientific fields. This project also provides training and research opportunities for graduate students. This project addresses the existence, regularity, and qualitative properties of solutions to partial differential equations and systems modeling fluids interacting with elastic bodies. The project considers fluid-structure interaction models that involve fluids interacting with elastic solids, for both compressible or incompressible fluids, and in viscous and inviscid settings. The primary focus is on establishing global existence, uniqueness, and asymptotic behavior of the solutions. Additionally, the project addresses the existence and qualitative behaviors of solutions to fluid-structure interaction models involving plates and elastic membranes. Furthermore, the project studies the inviscid limit problem and other singular limits, either for the Navier-Stokes equations in bounded domains or in more involved physical settings. The project shall also address existence and regularity issues related to fluid models with evolving boundaries. One important example is the Euler system with a free interface, both with and without surface tension. Finally, the project will consider questions concerning the Boussinesq system, including those relating to the persistence of regularity and the long-horizon behavior of solutions.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.