一般区域上N-涡问题周期解的稳定性
Stability of periodic solutions of the N-vortex problem in general domains
摘要：
研究了一般区域$Ω\subset \mathbb{R}^2$上$N$ -涡问题的一类周期解的稳定性.所讨论的解从全平面涡系的刚性旋转构型和与$Ω$的Dirichlet Laplacian相关联的Robin函数的临界点$a_0\inΩ$分叉。在初始旋转配置的线性稳定性条件下，可以验证的例子组成的多达4个涡，我们表明，诱导的解决方案的线性稳定性是完全由临界点$a_0$的类型。如果$a_0$是鞍形，则它们不稳定。否则，它们在某种线性意义上是稳定的。证明使用的多重特征值，这是适用于合适的庞加莱部分的分歧的标准。除了线性稳定性之外，赫尔曼的最后几何定理使我们能够证明在$N=2$涡旋的情况下等能轨道稳定解的存在性。
英文摘要：
We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $Ω\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point $a_0\inΩ$ of the Robin function associated to the Dirichlet Laplacian of $Ω$. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. Otherwise they are stable in a certain linear sense. The proof uses a criterion for the bifurcation of multiple eigenvalues, which is applied to suitable Poincaré sections. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.
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