Vortex Motion of the Euler and Lake Equations

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.