AI 中文总结
本文针对平面欧拉方程，构造了带有任意N≥2个真实零涡量孔洞的均匀旋转单位涡量涡块，通过移除收缩共旋转多边形构型实现，确定了边界与角速度的一阶修正。
AI 中文摘要
对于每个给定的整数N≥2，我们为平面欧拉方程构造均匀旋转、单位涡量的涡块ω=1_D，其中D是连通的，且ℝ²\backslash D恰好有N个有界连通分支。这些分支是真实的零涡量孔洞，而非异号涡块或携带第二个非零涡量水平的区域。角速度位于与刚性兼容的区间(0,1/2)内，当孔洞坍缩时，这些区域依测度收敛到Rankine圆盘。构造过程从N个单位涡量涡块的固定共旋转多边形构型出发，从Rankine圆盘中移除该构型的一个收缩空间副本。一个精确补恒等式在外部圆被扰动前解决了所有内边界方程。剩余缺陷由第N阶外多极子产生，大小为ε^(N+2)。所得的外修正以ε^(2N)的规模反馈到归一化内问题中。对外、内方程分别进行重正化，得到一个极限仿射系统，其导数为下三角形式。其对角块为非共振Rankine算子和固定种子构型的角速度增广线性化。我们还确定了外边界、孔洞边界和角速度的一阶修正。
英文摘要
For every prescribed integer $N\ge2$, we construct uniformly rotating unit-vorticity vortex patches $ω=\mathbf{1}_D$ for the planar Euler equation such that $D$ is connected and $\mathbb{R}^2\backslash D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes, rather than opposite-sign vortex patches or regions carrying a second nonzero vorticity level. The angular velocities lie in the rigidity-compatible interval $(0,1/2)$, and the domains converge in measure to the Rankine disk as the holes collapse. The construction starts from a fixed co-rotating polygonal configuration of $N$ unit-vorticity vortex patches and removes a shrinking spatial copy of that configuration from the Rankine disk. An exact complement identity solves all inner-boundary equations before the outer circle is perturbed. The remaining defect is generated by the $N$-th exterior multipole and has size $\varepsilon^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon^{2N}$. Separate renormalization of the outer and inner equations produces a limiting affine system with a lower-triangular derivative. Its diagonal blocks are the nonresonant Rankine operator and the angular-velocity-augmented linearization of the fixed seed configuration. We also determine the first corrections to the outer boundary, the hole boundaries, and the angular velocity.
Comments39 pages, 1 figure
