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中出现的那些应用，理解流体和等离子体的动力学，其中耗散力（例如，摩擦力弱是至关重要的。 例如，机翼上方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，例如大幅改变燃油效率。 该项目的重点是更好地了解耗散和稳定性的平衡配置和相关问题，在这一制度使用数学分析。 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，它也提供了一个很好的机会，培养新的，多才多艺的研究人员谁是流利的科学应用和复杂的数学分析。 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、直接级联和流体中小尺度拟能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由于内在利益而产生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。 研究了三个一般领域：（A）用于理解增强耗散和瞬态不混合的分析工具;（B）线性和非线性随机强迫问题，以理解在数学上可访问的设置中的统计上稳定的直接叶栅;（C）估计亚临界转变阈值和理解层流的不稳定性，如管流，在无限和有限的规则性。 最后，自然延伸到亚椭圆问题，如碰撞限制在动力学理论，也可以考虑。 这项工作主要是数学分析;然而，计算机实验可能会进行，以提供初步的见解，并为应用数学本科生提供无障碍的培训机会。 研究生也参加了研究活动。

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.