CAREER: Inviscid Limits and Stability at High Reynolds Numbers

在流体力学的许多应用中，例如在大气和海洋科学、航空航天工程和高能或聚变等离子体物理中，了解耗散力(例如摩擦)较弱的流体和等离子体的动力学是至关重要的。例如，机翼上方空气层的稳定性以及附近的能量消耗可能会对飞机产生重大影响，例如大幅改变燃油效率。该项目的重点是利用数学分析更好地理解平衡构型的耗散和稳定性以及在该制度下的相关问题。该项目有助于为流体力学中关于混合、耗散和稳定性的更广泛的数学理论奠定基础。由于它既需要纯粹的数学创新，也需要应用数学创新，它也提供了一个极好的机会来培养在科学应用和复杂的数学分析方面都很流利的新的、多才多艺的研究人员。该项目的工作包括研究生。该项目旨在进一步阐明非线性稳定性、直接叶栅和流体中小尺度的拟能或能量耗散等基本问题，并扩展严格的数学理论来理解这些现象。研究人员和他的合作者将重点放在由于内在兴趣而产生广泛影响的基本问题上，以及为解决这些问题而开发的新工具。研究了三个一般领域：(A)用于理解增强的耗散和瞬变混合的分析工具；(B)线性和非线性随机强迫问题，以了解在数学上可达的设置下的统计平稳的直接叶栅；(C)估计亚临界转变阈值和理解无限和有限规则的层流流动，如管流的不稳定性。最后，也可以考虑亚椭圆型问题的自然扩展，例如动力学理论中的无碰撞极限。这项工作主要是数学分析；然而，可以进行计算机实验，以便提供初步的见解，并为本科生提供应用数学方面的可获得的培训机会。研究生被包括在研究活动中。

In many applications of fluid mechanics, such as those arising in atmosphere and ocean sciences, aerospace engineering, and high-energy or fusion plasma physics, understanding the dynamics of fluids and plasmas where dissipative forces (e.g., friction) are weak is crucial. For example, the stability of the layer of air over a wing, and the dissipation of energy nearby, can have major implications for the aircraft, such as drastically changing the fuel efficiency. The focus of the project is to better understand dissipation and stability of equilibrium configurations and related problems in this regime using mathematical analysis. The project helps lay the foundation for a wider mathematical theory on mixing, dissipation, and stability in fluid mechanics. As it requires both pure and applied mathematical innovations, it also presents an excellent opportunity to train new, versatile researchers who are fluent in both scientific applications and sophisticated mathematical analysis. Graduate students are included in the work of the project.The project aims to further elucidate basic questions of nonlinear stability, direct cascades, and the dissipation of enstrophy or energy at small scales in fluids and to expand the rigorous mathematical theory for understanding these phenomena. The investigator and his collaborators focus on fundamental questions that will have broad impact due to intrinsic interest, as will the new tools developed to solve them. Three general areas are studied: (A) analysis tools for understanding enhanced dissipation and transient unmixing; (B) linear and nonlinear stochastically forced problems in order to understand statistically stationary direct cascades in mathematically accessible settings; (C) estimating the subcritical transition thresholds and understanding instabilities for laminar flows, such as pipe flow, in infinite and finite regularity. Finally, natural extensions to hypoelliptic problems, such as collisionless limits in kinetic theory, may also be considered. The work is primarily mathematical analysis; however, computer experiments may be performed in order to provide preliminary insights and to provide accessible training opportunities for undergraduates in applied mathematics. Graduate students are included in the research activities.