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

该项目涉及对物理边界附近的流体流动的数学分析，这是流体动力学中的一个基本问题。认识到物体在空气(或水)中运动时所受的阻力是在物体周围的薄层上产生的，这一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力学中的实际重要性是巨大的：计算物体上的阻力和附近的能量耗散率，这对燃料效率有影响；理解机翼失速问题，升力与攻角的关系；甚至加强固体壁面附近的换热。从数学上讲，这些问题涉及到在粘性极限消失时Navier-Stokes方程的解的行为，以及随后的边界层的稳定性。这个项目将开发新的分析工具来研究消失粘性极限的有效性，边界层中奇点的形成，并探索导致动态边界层分离的非线性稳定机制。这些理论信息将导致对真实流体流动的更准确的简化模型和更精细的预测。该项目将培训本科生、研究生和研究生研究应用数学中的现代研究问题和研究它们的工具。该项目旨在开发新的数学工具，促进我们对消失的粘性极限的理解：在存在特征物理边界的情况下，当粘性接近零时，不可压缩的Navier-Stokes方程的解是否收敛到不可压缩的Euler方程的解。对于固定的外部参数，消失的粘性极限等价于无限大的雷诺数极限，因此这一问题对于流体流动中湍流开始的研究是至关重要的。PI和合作者将通过建立更稳健的有限时间爆破情景来解决Prandtl边界层方程中出现的奇点问题。近层流边界层的稳定性将通过对Prandtl系统和高阶模型的亚胁性分析来研究。最后，PI将开发新的能量方法，将欧拉方法和拉格朗日方法结合在一起，以证明在高正则性区域中粘度极限的消失。其目标是开发非线性的、适用于解决方案的方法。将指导研究生和博士后研究员，并将其纳入研究活动。

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.