Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine Holes

Abstract

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.

Disclosure

“of CPSF (Grant No. GZB20260747) and the China Postdoctoral Science Foundation (Grant No. 2026M793420). W. Zhan is supported by the National Natural Science Foundation of China (Grant No. 12571126). Declaration on the Use of Generative AI The authors used OpenAI’s ChatGPT (GPT-5.6 Sol) to assist with the writing, language polishing, and checking of the manuscript. All mathemat- ical arguments, calculations, and proofs were independently reviewed and verified by the auth”

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2026-08-22T03:06:06.384743Z

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