We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the N-vortex system in the half-plane is nonintegrable for \(N>2\), which was suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in \({\mathbb {R}}^n,\;n\geqslant 3\) with certain symmetry can be regarded as point vortex motion of these 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper.

我们从调查整个平面，半平面和象限中的Euler方程的平面点涡旋运动开始。然后，我们通过使用半平面中2-涡旋系统轨道的显式形式来证明2-涡旋系统的非碰撞性质。我们还证明了\（N> 2 \）在半平面上的N涡旋系统是不可积的，这是先前在没有严格证明的情况下通过数值实验提出的。\（{\ mathbb {R}} ^ n，\; n \ geqslant 3 \）中的斜均值曲率（或双正态）流具有一定对称性的可以看作是这些二维湖方程的点涡运动。我们比较了欧拉方程和湖方程的点涡运动。解决了半平面象限中的点涡旋运动与同轴涡旋环的双向运动，球积膜之间有趣的相似之处。我们还在本文中提出了一些未解决的问题。