CAREER: Nonlinear stability mechanisms and boundary layer singularities in fluid flows

该项目涉及物理边界附近流体流动的数学分析，这是流体动力学的一个基本问题。对物体在空气（或水）中运动时所施加的阻力发生在物体周围的薄层上的认识，导致了边界层理论的发展。边界层理论在空气动力学和流体动力学中的实际重要性是巨大的：计算物体的阻力和附近的能量耗散率，从而影响燃油效率；了解机翼失速问题，升力对迎角的依赖；甚至是固体壁面附近传热的增强。在数学上，这些问题涉及到Navier-Stokes方程在消失粘度极限下的解的行为，以及随之产生的边界层的稳定性。本项目将开发新的分析工具来研究粘性消失极限的有效性，边界层奇点的形成，并探索这些导致动态边界层分离的非线性稳定性机制。这些理论信息将导致更精确的简化模型和对实际流体流动的更精细的预测。该项目将在应用数学的现代研究问题和研究工具方面培养本科生、研究生和研究生的研究人员。该项目旨在开发新的数学工具，以进一步理解消失的粘度极限：不可压缩的Navier-Stokes方程的解是否收敛于不可压缩的欧拉方程的解，当粘度接近零时，在一个特征物理边界的存在。对于固定的外部参数，粘度消失极限等同于无限雷诺数极限，因此该问题对于研究流体流动中湍流的发生具有重要意义。PI和合作者将通过建立更强大的有限时间爆炸场景来解决普朗特边界层方程中奇点的出现。近层流边界层的稳定性将通过对普朗特系统和高阶模型的准矫顽力分析来研究。最后，PI将开发新的能量方法，将欧拉和拉格朗日方法结合起来，以证明高正则性体系中消失的粘度极限。目标是开发非线性的、解适应的方法。研究生和博士后将接受指导并参与研究活动。

The project concerns the mathematical analysis of fluid flows in the vicinity of physical boundaries, a fundamental problem in fluid dynamics. The realization that drag forces exerted on a body moving through air (or water) take place on a thin layer around the body lead to the development of boundary layer theory. The practical importance of boundary layer theory in aerodynamics and hydrodynamics is immense: computing the drag on an object and the energy dissipation rate nearby, with implications for fuel efficiency; understanding the problem of wing stall, the dependence of the lift on the angle of attack; and even the enhancement of heat transfer near solid walls. Mathematically, these questions concern the behavior of solutions to the Navier-Stokes equations in the vanishing viscosity limit, and the stability of the ensuing boundary layers. This project will develop new analytical tools to study the validity of the vanishing viscosity limit, the formation of singularities in boundary layers, and to explore the nonlinear stability mechanisms by which these lead to dynamic boundary layer separation. This theoretical information will lead to more accurate reduced models and finer predictions about real fluid flows. The project will train undergraduate, graduate, and post-graduate researchers, in modern research problems in applied mathematics and the tools to study them.This project aims to develop new mathematical tools that will further our understanding of the vanishing viscosity limit: the question whether solutions of the incompressible Navier-Stokes equations converge to solutions of the incompressible Euler equations as the viscosity approaches zero, in the presence of a characteristic physical boundary. For fixed external parameters, the vanishing viscosity limit is equivalent to the infinite Reynolds number limit, and thus this problem is of vital importance to the study of the onset of turbulence in fluid flows. The PI and collaborators will address the emergence of singularities in the Prandtl boundary layer equations by establishing more robust finite time blow-up scenarios. The stability of nearly laminar boundary layers will be studied via a hypocoercive analysis of the Prandtl system and of higher order models. Finally, the PI will develop new energy methods that intertwine Eulerian and Lagrangian approaches to prove the vanishing viscosity limit in high regularity regimes. The goal is to develop nonlinear, solution-adapted methods. Graduate students and post-doctoral fellows will be mentored and included in the research activities.