AI 中文总结
本研究证明单连通有界域中均匀旋转涡斑仅存在于圆盘域，并在单位圆盘小扰动域中首次构造了Euler方程的拟周期涡斑解，揭示了几何效应对涡结构持久性的关键影响。
AI 中文摘要
我们首先证明了在有界单连通平面域中Euler方程均匀旋转涡斑解的一个刚性结果。即，如果一个具有实解析边界的单连通有界域包含一个非平凡涡斑，该涡斑在经历均匀刚体旋转时严格保持在域内，那么外部域必须是以旋转中心为圆心的圆盘。证明结合了旋转坐标系公式与调和唯一延拓，以及跨解析边界弧的广义Schwarz反射原理。这些工具表明，除非外部域是旋转不变的，否则速度必须在包围斑块的开区域中消失，这与斑块产生的非零环量相矛盾。同理，我们证明只有圆形域才具有圆形点涡轨道。作为第二个结果，我们在单位圆盘$\mathbb{D}$的小扰动域中构造了Euler方程的拟周期涡斑解。解的振幅与域变形的大小相关联，使得经典Rankine涡$b\cdot\mathbb{D}$（$b\in(0,1)$）可被视为平衡态。我们的分析相对于未变形情形中的构造是微扰的，所得的拟周期解对大多数内半径$b$的值存在。这提供了在非径向域中Euler方程拟周期涡斑解的首次构造。此外，结合上述刚性结果，这表明在周期情形下，这些解不可能源于纯粹的刚体旋转。
英文摘要
We first prove a rigidity result for uniformly rotating vortex patch solutions to Euler equations in bounded simply-connected planar domains. Namely, if a bounded simply-connected domain with real-analytic boundary contains a nontrivial vortex patch that remains strictly inside the domain while undergoing a uniform rigid rotation, then the ambient domain must be a disc centered at the rotation center. The proof combines the rotating-frame formulation with harmonic unique continuation and the generalized Schwarz's reflection principle across analytic boundary arcs. These tools imply that, unless the outer domain is rotationally invariant, the velocity must vanish in an open region surrounding the patch, contradicting the nonzero circulation generated by the patch. In the same spirit, we show that only circular domains possess a circular point vortex orbit. As a second result, we construct quasi-periodic vortex patch solutions to the Euler equations in domains that are small perturbations of the unit disc $\mathbb{D}$. The amplitudes of the solutions are linked to the size of the domain deformation, allowing the classical Rankine vortices $b\cdot\mathbb{D}$, $b\in(0,1)$, to be regarded as equilibrium states. Our analysis is perturbative with respect to the construction developed in the undeformed case, and the resulting quasi-periodic solutions exist for most values of the inner radius $b$. This provides the first construction of quasi-periodic vortex patch solutions to the Euler equations in non-radial domains. Moreover, combined with the rigidity result established above, this shows that, in the periodic case, these solutions cannot arise from a purely rigid rotation.
Comments50 pages
