Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine Holes

Abstract

For every prescribed integer $N\ge 2$, we construct uniformly rotating unit-vorticity vortex patches $\omega =1_D$ 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 ${\epsilon }^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size ${\epsilon }^{2N}$. 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.

arXiv comment39 pages, 1 figure