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 N=2, 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.
Highlights
The governing equations for an incompressible inviscid fluid were formulated by Euler in 1757 [11] and have since been cardinal in the ever growing field of hydrodynamics
101 years after Euler’s influential paper on incompressible fluids, Helmholtz [14] showed that the 2D Euler equations exhibit special solutions consisting of a finite number of point-vortices
Our motivation for questions of integrability of point-vortex dynamics originates from recent numerical results for 2D Euler equations indicating that integrability of point-vortex dynamics, rather than prevailing statistical mechanics based theories, is central for predicting the long-time behaviour of solutions [34]
Summary
The governing equations for an incompressible inviscid fluid were formulated by Euler in 1757 [11] and have since been cardinal in the ever growing field of hydrodynamics. 101 years after Euler’s influential paper on incompressible fluids, Helmholtz [14] showed that the 2D Euler equations exhibit special solutions consisting of a finite number of point-vortices. Concerning the planar case they say: “For N = 3 one can check that the motion is (completely) integrable in the sense that the (non-abelian) reduced phase spaces are points” They go on to say: “ one can see that the dynamics of 3 point-vortices is (completely) integrable by exhibiting 3 independent integrals in involution...”. Our motivation for questions of integrability of point-vortex dynamics originates from recent numerical results for 2D Euler equations indicating that integrability of point-vortex dynamics, rather than prevailing statistical mechanics based theories, is central for predicting the long-time behaviour of solutions [34]. In Appendix A, we provide a gallery of point-vortex solutions—a sort of ‘visual summary’ of the paper
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