AI 中文总结
研究准周期通道中点涡的动力学，通过壳层公式构造输运轨迹，并利用遍历性建立临界点分布的零点计数定理，揭示其渐近密度与相位分布。
AI 中文摘要
我们研究了一个无界平面通道中单个点涡的动力学，该通道的界面在纵向上是准周期的。运动由Robin函数控制。我们引入了一个壳层（hull）公式，将准周期几何提升为有限维环面上的周期问题，并得到Robin函数的精确准周期表示。在自然的横截性条件下，我们构造了描述输运涡旋轨迹的全局准周期不变图，并证明了每个边界分量的完整邻域都被这样的图叶状化。在空间频率的Diophantine条件下，沿每个图的动力学可以被拉直为恒定漂移，因此涡旋运动在模平移意义下是准周期的。一个核心贡献涉及涡旋平衡态的分布。在没有基本空间胞腔的准周期设置中，我们将Robin函数的临界点与壳层环面上Kronecker流与超曲面的交叉点等同起来。利用唯一遍历性，我们建立了一个允许有限阶相切的通用零点计数定理，该定理给出了临界点渐近密度及其极限相位分布的显式几何通量公式。一旦临界壳层被构造，该结果是非微扰的。显式的周期和准周期模型揭示了临界点分布中的分岔和相变，而数值计算为解析结果提供了定量验证。
英文摘要
We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.
Comments89 pages
