Topics in Vortex Dynamics: Extreme Events, Optimal Closures and New Equilibrium Solutions

我的研究计划的长期目标是为理论流体动力学领域的经典和新兴开放问题提供答案。尽管流体力学所依据的数学模型是在近两个世纪前引入的，但它们的许多基本特性仍有待充分理解。 作为第一个目标，我们将对受纳维-斯托克斯方程控制的不可压缩流体流动中的奇异行为进行系统搜索。有限时间奇点形成问题被克莱数学研究所认定为“千年难题”之一，是数学流体力学的核心问题之一。虽然这是数学分析中的一个问题，但我们将通过有条不紊地寻找表现出最极端行为的解决方案来对其进行新的阐述，以便了解这种行为是否可能导致奇点的形成。我们方法的一个关键新颖之处在于，将通过适当的变分优化问题的数值解来系统地寻找这种极端流。 第二类开放问题是由湍流简化模型的开发引发的。考虑到此类流动所表现出的多尺度复杂性，它们在数值计算中的完全解析始终是一个挑战，并且需要某种形式的近似来表示未解析的动力学。虽然迄今为止关于这个所谓的“封闭问题”的大部分工作都依赖于纯粹的经验方法，但我们将寻找数学上的最佳解决方案。除了提供设计此类关闭策略的新方法之外，我们的工作还将回答有关其基本性能限制的问题。 最后，我们将研究经典涡动力学中的问题，寻求发现新的平衡解族并探索它们的稳定性能。这项工作将填补理论流体动力学的重要空白，并将提供有关各种技术和生物应用中普遍存在的涡环行为的见解。 虽然数值优化广泛用于解决实际问题，但这里将采用它来阐明流体流动数学模型的内部工作原理。我们的许多研究问题都是专门设计的，旨在提供超出严格分析所能建立的新见解，以便这些见解将指出数学分析可以扩展和锐化的方向。鉴于该项目的跨学科性质，拟议的研究和培训将整合理论流体力学、应用和数值分析、优化和控制以及大规模科学计算的方法。从长远来看，我们的发现将对流体动力学发挥重要作用的各个领域产生影响，例如机械和化学工程、大气和环境科学以及数值天气预报。

The long-term objective of my research program is to provide answers to classical and emerging open questions in the area of theoretical fluid dynamics. Even though the mathematical models on which fluid mechanics is based were introduced nearly two centuries ago, many of their fundamental properties are still to be fully understood. As a first objective, we will carry out a systematic search for singular behavior in flows of incompressible fluids governed by the Navier-Stokes equation. Recognized by the Clay Mathematical Institute as one of its "Millennium Problems", the question of the finite-time singularity formation is one of the central problems in mathematical fluid mechanics. While this is a problem in mathematical analysis, we will cast a new light on it by methodically finding solutions exhibiting the most extreme behavior in order to understand whether such behavior may lead to singularity formation. A key novelty of our approach is that such extreme flows will be sought systematically via numerical solution of suitable variational optimization problems. The second class of open questions is motivated by the development of simplified models for turbulent flows. Given the multiscale complexity exhibited by such flows, their complete resolution in numerical computations will always be a challenge and some form of approximation is needed to represent unresolved dynamics. While most of the work on this so-called "closure problem" to date has relied on purely empirical approaches, we will look for mathematically optimal solutions. In addition to offering an new way to design such closure strategies, our work will answer questions about their fundamental performance limitations. Finally, we will study problems in classical vortex dynamics where we will seek to discover new families of equilibrium solutions and explore their stability properties. This effort will fill important gaps in theoretical hydrodynamics and will also provide insights about the behavior of vortex rings ubiquitous in various technological and biological applications. While numerical optimization is widely used to solve practical problems, here it will be employed to shed light on the inner workings of the mathematical models of fluid flow. Many of our research questions are specifically framed to provide new insights beyond what can be established with rigorous analysis, such that these insights will point to the direction in which the mathematical analysis can be extended and sharpened. Given the interdisciplinary nature of this program, the proposed research and training will integrate methods from theoretical fluid mechanics, applied and numerical analysis, optimization and control, as well as large-scale scientific computing. In the long term our findings will have impact on various fields where fluid dynamics plays an important role, such as mechanical and chemical engineering, atmospheric and environmental science as well as numerical weather prediction.