Uniqueness of m-fold V-states with 90-degree corners

Establish uniqueness, within a suitable class of patches, of the m-fold symmetric uniformly rotating vortex patches with 90-degree corners constructed in the paper.

Background

The existence theorem produces m-fold symmetric V-states with 90-degree corners for all m≥12, while numerical computations yield apparently robust solutions and no competing solutions in the simulations.

The authors explicitly state that rigorous uniqueness has not been proved. The unresolved issue is whether the numerically observed solution is the only solution in an appropriately specified class of patches.

References

It is noteworthy that, although the uniqueness of the $m$-fold symmetric V-states with 90-degree corners has not been rigorously established, we can robustly obtain the presented numerical solutions in the simulations with no other solutions found, which seems to suggest the uniqueness in a suitable class of patches.

— Uniformly Rotating Vortex Patches with 90-Degree Corners  (2608.13134 - Huang et al., 13 Aug 2026) in Remark following Theorem “Existence of V-states with 90-degree corners and their characterizations”