Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon (and a possible vortex at the center) with more than three vertices (two if there is a vortex at the center), a relative equilibrium requires equal vorticities (on the polygon). We also determine all the relative equilibria with two concentric regular n-gons and the same vorticity on each n-gon. This result completes the classical studies for two regular n-gons when all the vortices have the same vorticity or when the total vorticity vanishes.
Liouville chains: new hybrid vortex equilibria of the 2D Euler equation
本文给出了平面上定常不可压缩欧拉方程的一大类新的精确解。这些混合解决方案包括一组固定点涡嵌入在一个背景海的刘维尔型涡是指数相关的流函数。输入的结构是一个“纯”点涡平衡的背景无旋流。纯点涡平衡也出现作为一个参数$A$的混合解决方案接近极限$A\to 0,\infty$。当$A\to 0$再现输入平衡时，$A\to\infty$产生新的纯点涡平衡。我们把由$A$连续参数化的混合平衡族称为“刘维尔链”。在某些情况下，作为$A\to\infty$的突现点涡平衡本身可以是第二类混合平衡的输入，在某种程度上，该混合平衡与另一个纯点涡平衡相联系。这样，刘维尔链就形成了一个“刘维尔链”。我们讨论几个例子的刘维尔链，并证明他们可以有一个有限或无限数量的链接。我们在这里证明了Crowdy（2003）和Krishnamurthy et al.（2019）发现的混合解类形成了一个这样的无限链中的前两个环节。我们还表明，Krishnamurthy等人（2020）最近研究的驻点涡平衡可以解释为Liouville链接的极限。我们的研究结果指出，这类平衡的2D欧拉方程的基础上丰富的理论结构。
