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 $\varepsilon<sup>{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon<sup>{2N}$. 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.

• The paper effectively constructs uniformly rotating vortex patches with an arbitrarily large number of genuine holes for the 2D incompressible Euler equation over various structures.
• Uniformly, the paper finds results in which hole sizes have a multiplicative size correction of $N$ and angular velocities within $(0, 1/2)$ interval. The outer boundary's correction includes $igcirc = rac{M_{}N(P) A{} }}_{ {N}{2}}\quadextu{O}( ho)$
• Vortex patches converge to collision-free limiting domains and their shapes become less radical in the long run.
• follow_up_questions_displayed_in_order

This paper by Zhilong Xue and Weicheng Zhan constructs, for every integer $N\ge2$, uniformly rotating vortex patches $\omega=\mathbf{1}_D$ for the two-dimensional incompressible Euler equation such that $D$ is connected and $\mathbb{R}^2\setminus D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes: the vorticity is a single characteristic function, not a multilevel or sign-changing configuration. The angular velocities lie in $(0,1/2)$, consistent with known rigidity constraints, and the domains converge in measure to the Rankine disk as the holes collapse. To the authors' knowledge this is the first rigorous existence result of its kind for characteristic-function patches with an arbitrary prescribed number of holes.

Context and problem

A vortex patch is a Yudovich solution with vorticity $\omega_0=\mathbf{1}_{D_0}$; boundary regularity persists globally at the $C^{k,\gamma}$ level (2608.19627), while Kiselev and Luo showed ill-posedness at the $C^2$ level. A patch is a $V$-state with angular velocity $\omega=\mathbf{1}_D$0 if it rotates rigidly, equivalently if the relative stream function $\omega=\mathbf{1}_D$1 is constant on each connected component of $\omega=\mathbf{1}_D$2. The literature contains extensive bifurcation theory from the Rankine disk and Kirchhoff ellipses, doubly connected (annular) branches, desingularization of point-vortex equilibria into multipatch configurations, and rigidity results forcing nonradial $\omega=\mathbf{1}_D$3-states to satisfy $\omega=\mathbf{1}_D$4.

The paper distinguishes three structurally different problems. A union of disjoint positive patches has a disconnected vortical region. A multilevel state $\omega=\mathbf{1}_D$5 carries vorticity $\omega=\mathbf{1}_D$6 inside the inner components; the recent construction of Baroncini–Cantero–García–Hassainia–Mateu produces such states, where the inner value equals $\omega=\mathbf{1}_D$7 locally and continuation to vanishing inner vorticity is left open. The present work instead enforces the coefficient exactly one from the outset, so every hole has identically zero vorticity. A further obstruction is that no radial seed exists: a connected radially symmetric patch domain is a disk or annulus, hence admits at most one hole, so there is nothing to bifurcate from.

Main theorem

For each $\omega=\mathbf{1}_D$8, fixing $\omega=\mathbf{1}_D$9 and $D$0, the authors produce a family $D$1 indexed by a small parameter $D$2 such that:

• all sets are simply connected $D$3 domains, the holes are pairwise disjoint and compactly contained in $D$4;
• $D$5 is connected, $D$6-invariant under the order-$D$7 dihedral group, and $D$8 has exactly $D$9 bounded components;
• $\mathbb{R}^2\setminus D$0 is a global Yudovich solution;
• as $\mathbb{R}^2\setminus D$1, the outer radius profile tends to $\mathbb{R}^2\setminus D$2, the hole charts tend to a fixed template chart $\mathbb{R}^2\setminus D$3, and $\mathbb{R}^2\setminus D$4.

Quantitative asymptotics are also established: the outer boundary satisfies

$\mathbb{R}^2\setminus D$5

the hole boundaries deform at scale $\mathbb{R}^2\setminus D$6 via a fixed shape function $\mathbb{R}^2\setminus D$7, the angular velocity correction is $\mathbb{R}^2\setminus D$8, and the symmetric-difference area obeys $\mathbb{R}^2\setminus D$9. Here $N$0 is the first nonvanishing complex moment of the seed. If the microscopic parameter $N$1 is sent to zero along nonresonant values, then $N$2 and $N$3.

Proof architecture

Fixed nonresonant template. The construction starts from García's regular $N$4-polygon branch of co-rotating unit-vorticity patches (2608.19627). After amplitude homogeneity normalization, the branch has angular velocity $N$5. A key technical step restores the full target space $N$6: García's reduced formulation eliminates the first sine mode, and the authors reconstruct the missing mode through an explicit selector $N$7 built from his divided-difference regularization, obtaining a $N$8 extension to $N$9 and proving that the augmented linearization $(0,1/2)$0 is a bounded isomorphism. Since $(0,1/2)$1 is strictly increasing, resonant values $(0,1/2)$2 form a discrete set accumulating only at zero, so a nonresonant $(0,1/2)$3 can be fixed once and for all; thereafter only $(0,1/2)$4 varies.

Exact complement identity. Inside the Rankine disk, $(0,1/2)$5. Removing a small copy $(0,1/2)$6 of the co-rotating template yields the identity

$(0,1/2)$7

whose right-hand side is tangent to every component of $(0,1/2)$8. Consequently all $(0,1/2)$9 inner boundary equations are solved exactly before the outer circle is perturbed — the entire defect resides on the unit circle.

Two scales. The $\omega_0=\mathbf{1}_{D_0}$0 symmetry annihilates moments of orders $\omega_0=\mathbf{1}_{D_0}$1, so the first nonradial exterior multipole of the holes is the $\omega_0=\mathbf{1}_{D_0}$2-th one, producing an outer residual of size $\omega_0=\mathbf{1}_{D_0}$3. The resulting outer correction is harmonic near the origin; its leading invariant term has degree $\omega_0=\mathbf{1}_{D_0}$4, which after evaluation at $\omega_0=\mathbf{1}_{D_0}$5 and the $\omega_0=\mathbf{1}_{D_0}$6 renormalization feeds back into the inner equation at scale $\omega_0=\mathbf{1}_{D_0}$7. These weights coincide when $\omega_0=\mathbf{1}_{D_0}$8 (where the feedback reduces to the Hessian/strain block) and differ for $\omega_0=\mathbf{1}_{D_0}$9 (where symmetry forces the Hessian contribution to vanish). Keeping them separate yields the sharp exponents above.

Renormalized system and gluing. With $C^{k,\gamma}$0, $C^{k,\gamma}$1, $C^{k,\gamma}$2, both residuals admit continuous extensions to $C^{k,\gamma}$3 with jointly operator-norm-continuous derivatives. The limiting affine system has lower-triangular derivative

$C^{k,\gamma}$4

where $C^{k,\gamma}$5 is the nonresonant Rankine operator with multiplier $C^{k,\gamma}$6 on the $C^{k,\gamma}$7-mode subspace (invertible precisely because of the nonresonance condition), and $C^{k,\gamma}$8 encodes the degree-$C^{k,\gamma}$9 harmonic jet of the outer deformation sampled at the hole scale. Both diagonal blocks are isomorphisms, so a continuous-parameter implicit-function theorem (proved in an appendix for one-sided parameters) yields a unique solution branch near the explicit limiting root $C^2$0, with rate $C^2$1. The passage from the boundary criterion to a weak Euler solution follows by Reynolds' transport formula plus Yudovich uniqueness.

Limitations and open questions

The construction is local in $C^2$2 for a fixed seed: no estimates uniform as $C^2$3 are claimed, and the constants depend on the chosen nonresonant $C^2$4. The result establishes existence but not stability or global continuation of the branches. In bounded fluid domains the ansatz transfers only partially: in a disk the Green kernel's regular part contributes at order one to the outer-scale equation, so the argument would require invertibility of the corresponding disk $C^2$5-state linearization in the $C^2$6 class; in an annulus the ansatz fails outright since the polygon collapses to the origin, and a new seed problem with prescribed harmonic circulation would be needed. Whether such analogues exist remains open, as does the global continuation question identified in the related multilevel work.

Conclusion

The paper provides a rigorous two-scale desingularization that converts a co-rotating polygonal patch configuration into genuine zero-vorticity holes inside a perturbed Rankine vortex, for arbitrary hole number $C^2$7. The exact complement identity, the separation of the exterior-multipole scale $C^2$8 from the harmonic-feedback scale $C^2$9, and the triangular implicit-function argument together yield not only existence but explicit leading-order corrections to the geometry and angular velocity, all within the rigidity-compatible interval $V$0.