The Inviscid Limit and Boundary Layer Theory for Stationary Navier-Stokes Flows

众所周知，粘性不可压缩流流过固体时会形成边界层，即，靠近物体边界的非常薄的区域，在该区域中，流动快速地调整到周围的自由流流动。边界层无处不在：它们在空气或水中运动的物体周围形成，如飞机、轮船和汽车，以及当风吹过建筑物和陆地时靠近地面。该项目旨在建立关于边界层稳定性的严格数学结果，建立在近年来主要研究者和合作者取得的最新数学进展的基础上。这些数学结果将为各种科学应用提供信息，例如，通过提供更简单的简化模型，该简化模型捕获边界层流的定性行为，然后可以用于生成精确的流体动力学模拟。这个项目致力于二维定常不可压缩纳维尔-斯托克斯流的边界层和无粘极限的严格数学分析。三种制度被确定为研究，对应于三种不同的定性行为的流。第一个制度将研究全球层流，自相似边界层的稳定性。这些边界层已在实验中得到了非常精确的证实。第二种制度是反向流动制度。这些流动是在边界层分离后观察到的。在数学上，这一制度将需要引进新的技术，从混合型问题，自由边界问题，和许多其他偏微分方程学科。第三个领域包括研究边界层分离的项目，这是自然界中观察到的一种奇异现象。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。

It is well known that viscous incompressible flows moving past solid bodies develop boundary layers, i.e., very thin regions near the body's boundary where the flow rapidly adjusts to the surrounding free stream flow. Boundary layers are ubiquitous: they develop around bodies moving in air or water, such as airplanes, ships, and cars, and near the ground as wind blows over buildings and land. This project aims to establish rigorous mathematical results regarding the stability of boundary layers, building upon recent mathematical advances made by the principal investigator and collaborators in recent years. These mathematical results will inform a diverse range of scientific applications, e.g., by providing simpler reduced models which capture the qualitative behavior of boundary layer flows that can then be used to generate accurate fluid dynamics simulations. This project is devoted to the rigorous mathematical analysis of boundary layers and the inviscid limit for two dimensional, stationary, incompressible Navier-Stokes flows. Three regimes are identified for study which correspond to three different qualitative behaviors of the flow. The first regime will study the stability of globally laminar, self-similar boundary layers. These boundary layers have been confirmed to remarkable accuracy in experiments. The second regime is that of reversed flows. These flows are observed after boundary layer separation. Mathematically, this regime will require the introduction of new techniques from mixed-type problems, free boundary problems, and many other PDE disciplines. The third regime consists of projects studying boundary layer separation, which is a singularity phenomenon observed in nature. Here, large tangential gradients require higher-order boundary layer models to be studied.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.