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

众所周知，流经固体的粘性不可压缩流动会形成边界层，即在物体边界附近的非常薄的区域，在那里流动迅速地调整到周围的自由流动。边界层无处不在：它们围绕在空中或水中运动的物体，如飞机、轮船和汽车，并随着风吹过建筑物和陆地而在地面附近发展。这个项目旨在建立关于边界层稳定性的严格的数学结果，建立在主要研究人员和合作者近年来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来捕捉边界层流动的定性行为，然后可以用于生成准确的流体动力学模拟。本项目致力于对二维、定常、不可压缩的N-S流动的边界层和无粘边界进行严格的数学分析。确定了三个区域用于研究，它们对应于三种不同的流动质量行为。第一个方案将研究全球层流、自相似边界层的稳定性。这些边界层在实验中已被证实具有显著的准确性。第二个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。从数学上讲，这一制度将需要从混合类型问题、自由边界问题和许多其他PDE学科中引入新的技术。第三个机制包括研究边界层分离的项目，这是在自然界中观察到的一种奇特现象。在这里，大的切向梯度需要研究更高阶的边界层模式。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。

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.