Lie groupoids in fluid dynamics
Anton Izosimov (University of Arizona)
Abstract: In 1966, V. Arnold showed that the Euler equation describing the motion of an ideal fluid on a Riemannian manifold can be regarded as the geodesic flow of a right-invariant metric on the Lie group of volume-preserving diffeomorphisms. 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. However, the scope of application of Arnold’s approach is limited to problems whose symmetries form a group. At the same time, there are many problems in fluid dynamics, such as free boundary problems, fluid-structure interactions, as well as discontinuous fluid flows, whose symmetries should instead be regarded as a groupoid. In the talk, I will discuss an extension of Arnold's theory from Lie groups to Lie groupoids. The example of vortex sheet motion (i.e. fluids with discontinuities) will be addressed in detail. The talk is based on ongoing work with B. Khesin.
mathematical physicsalgebraic geometrydifferential geometryrepresentation theorysymplectic geometry
Audience: researchers in the topic
( video )
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