Liouville链：二维Euler方程的新混合涡平衡
Liouville chains: new hybrid vortex equilibria of the 2D Euler equation
摘要：
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。
英文摘要：
A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a "pure" point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to\infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a "Liouville link". In some cases, the emergent point vortex equilibrium as $A\to\infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a "Liouville chain". We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the 2D Euler equation.
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