Uniqueness of the stagnation point in each cat's-eye region

Prove that each cat's-eye-shaped region enclosed by the inner and outer streamlines of an m-fold symmetric V-state with 90-degree corners contains exactly one stagnation point of the co-rotating flow.

Background

The paper proves that, in the region between the patch boundary and a second outer streamline, stagnation points are confined to specified symmetry line segments and that at least one occurs on each such segment.

It does not establish that each cat's-eye-shaped region contains only one stagnation point. The authors connect this unresolved question with a previously conjectured cat's-eye structure for m-fold symmetric V-states.

References

In fact, there is at least one along each of these line segments, which corresponds to the global minima of $\phi_\dag$; see Theorem \ref{thm: heteroclinic trajectory}. Nevertheless, it is non-trivial to confirm that there is only one stagnation point in ${(r\cos\th,r\sin\th)\in \hat{D}_0\setminus \overline{D_0}:\;\th\in (0,\f{2\pi}{m})}$, which is one of the cat's-eye-shaped regions enclosed by $\pa D_0$ and $\pa\hat{D}_0$ shown in Figure \ref{fig:12-fold_intro}(B). This issue is related to the conjectured cat's-eye-type structure in the flow field of an $m$-fold symmetric V-state Conjecture 1.4, but unfortunately, such desired feature seems to crucially rely on more detailed geometric properties of $D_0$ as well as $\hat{D}_0$.

— Uniformly Rotating Vortex Patches with 90-Degree Corners  (2608.13134 - Huang et al., 13 Aug 2026) in Remark following Theorem “Characterizations of the flow field in the co-rotating frame”