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。尽管流体力学所基于的数学模型是在近两个世纪前提出的，但它们的许多基本性质仍有待完全理解。 作为第一个目标，我们将进行一个系统的搜索奇异行为的不可压缩流体的Navier-Stokes方程。有限时间奇点的形成问题是数学流体力学的核心问题之一，被美国克莱数学研究所列为“千年难题”之一。虽然这是数学分析中的一个问题，但我们将通过有系统地找到表现出最极端行为的解决方案，以了解这种行为是否会导致奇点的形成，从而对它进行新的阐述。我们的方法的一个关键的新奇是，这种极端的流量将寻求系统通过数值解合适的变分优化问题。 第二类开放性问题的动机是湍流简化模型的发展。鉴于这种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决将始终是一个挑战，需要某种形式的近似来表示未解决的动态。虽然大多数关于这个所谓的“封闭问题”的工作迄今为止都依赖于纯经验的方法，我们将寻找数学上的最佳解决方案。除了提供一种新的方法来设计这样的封闭策略，我们的工作将回答有关其基本性能限制的问题。 最后，我们将研究经典涡旋动力学中的问题，我们将寻求发现新的平衡解族并探索其稳定性。这项工作将填补理论流体力学的重要空白，也将提供有关在各种技术和生物应用中无处不在的涡环行为的见解。 虽然数值优化被广泛用于解决实际问题，但在这里，它将用于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，以提供超越严格分析所能建立的新见解，这些见解将指向数学分析可以扩展和锐化的方向。鉴于该计划的跨学科性质，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.