Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

Integrability of point-vortex dynamics via symplectic reduction: a survey

Viviani, Milo;Modin, Klas
2021

Abstract

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.
2021
Settore MATH-05/A - Analisi numerica
Point-vortex dynamics, Integrable systems; Euler equations; Symplectic reduction
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/148743
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