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、航空航天工程以及高能或聚变等离子体物理学中出现的应用中，了解耗散力（例如摩擦）较弱的流体和等离子体的动力学至关重要。例如，机翼上的空气层的稳定性，以及附近能量的耗散，可能对飞机产生重大影响，比如彻底改变燃油效率。项目的重点是利用数学分析来更好地理解平衡构型的耗散和稳定性以及在该状态下的相关问题。该项目有助于为流体力学中混合、耗散和稳定性的更广泛的数学理论奠定基础。由于它需要纯数学和应用数学的创新，它也提供了一个极好的机会来培养新的，多才多艺的研究人员，他们在科学应用和复杂的数学分析方面都很流利。研究生也包括在该项目的工作中。该项目旨在进一步阐明流体中非线性稳定性、直接级联和小尺度熵或能量耗散的基本问题，并扩展理解这些现象的严格数学理论。研究者和他的合作者专注于那些由于内在兴趣而具有广泛影响的基本问题，以及为解决这些问题而开发的新工具。研究了三个主要领域：(A)理解增强耗散和瞬态解混的分析工具；(B)线性和非线性随机强迫问题，以便在数学上可访问的设置中理解统计平稳的直接级联；(C)估计亚临界过渡阈值并理解层流（如管道流）在无限和有限规则下的不稳定性。最后，还可以考虑对准椭圆问题的自然扩展，如运动论中的无碰撞极限。这项工作主要是数学分析；然而，为了提供初步的见解，并为应用数学的本科生提供可访问的训练机会，可以进行计算机实验。研究生也包括在研究活动中。

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.