论文标题
点涡流动力学通过符号减少的整合性：调查
Integrability of point-vortex dynamics via symplectic reduction: a survey
论文作者
论文摘要
点 - 涡流动力学描述了对二维流形上不可压缩的欧拉方程的理想化的，非平滑的解决方案。在各个领域上几个点旋转的几点涡流的可集成性结果是一个生动的主题，文献中散布了许多结果和技术。在这里，我们为$ n = 2 $，$ 3 $或$ 4 $ point-vortices（以及更通用的汉密尔顿系统）提供了一个统一的框架，以$ n = 2 $，$ 3 $或$ 4 $的框架为基础，基于符号减少理论。该方法适用于任何二维流形；我们在球体，平面，双曲机和平坦的圆环上进行说明。二维湍流的进步促使对整合性的系统研究，这使2D Euler方程的长期行为与点 - 涡流的可集成性问题桥接。附录中给出了解决方案画廊。
Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on 2-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 2-dimensional manifold; 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 2-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.
