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

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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.