Topological and geometrical methods influid dynamics - Júnior, Guilherme Ferreira Vasconcelos (2024)
In this work we discuss an approach to describe the dynamics of fluid flows with vortex sheets using a framework of geodesics on a certain Lie groupoid of volume-preserving diffeomorphisms. This method draws inspiration from the group-theoretic technique proposed by V. Arnold in 1966, that deals with the ideal hydrodynamics problem of an inviscid incompressible fluid by modeling the system\'s equations of motions as geodesic flows of the right-invariant L^2-metric on a certain group of volume-preserving diffeomorphisms. One of the key aspects of dealing with a fluid flow with a vortex sheet is that it turns out that this falls in a category of problems whose symmetries do not form a Lie group, but a Lie groupoid. In order to completely describe the dynamics of such a system, we first introduce the main concepts of the theory of groupoids, algebroids, Fréchet manifolds and fluid Hamiltonian mechanics. Then, we present a technique proposed in a groundbreaking work of A. Izosimov and B. Khesin, that formalizes a way to deal with the hydrodynamics problem of fluid flows with vortex sheets and generalizes Arnolds approach for certain problems modeled via Euler-Arnold equations.