Lie Groupoids and Infinite-Dimensional Dynamical Systems

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This project aims at developing new mathematical tools for studying the motion of fluids. Having an adequate mathematical language for their description is crucial for understanding such phenomena as formation of air turbulence in meteorology, as well as large- and small-scale structures in liquids and plasmas. Despite much effort, many aspects of fluid dynamics are still poorly understood and a breakthrough in this field only seems possible if a variety of different mathematical tools is used. One of the most promising directions is a geometric approach to fluids. This approach is known to work well for fluids confined to a fixed domain. The goal of the project is to extend the geometric language to more general settings, with applications including formation of waves, ocean currents, insight into vortex instabilities, and the study of motion of underwater vehicles. The investigator will actively involve graduate students in this project.Modern geometric fluid dynamics originated in the 1960s when V. Arnold proved that the Euler equation for an ideal fluid describes the geodesic flow of a right-invariant metric on the group of volume-preserving diffeomorphisms of the flow domain. This insight turned out to be indispensable for the study of Hamiltonian properties and conservation laws in hydrodynamics, fluid instabilities, topological properties of flows, as well as a powerful tool for obtaining sharper existence and uniqueness results for Euler-type equations. Furthermore, Arnold's group-theoretic description of incompressible fluids has also been shown to be applicable in many other fluid-related settings, including magnetohydrodynamics, compressible fluids, semi-geostrophic and the Korteweg-de Vries equations. However, the scope of applicability of Arnold's approach is limited to systems whose symmetries form a group. At the same time, there are many problems in fluid dynamics, such as free boundary problems, fluid-structure interactions, discontinuous fluid flows, as well as multiphase and stratified fluids, whose symmetries should instead be regarded as a groupoid. The aim of the project is to develop a paradigm of infinite-dimensional Lie groupoids in the context of various fluid-dynamical problems, as well as to apply this paradigm to approach a range of concrete questions that are of interest for applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.