Existence of 90-degree-corner V-states for every symmetry order

Prove that, for every integer m≥3, m-fold symmetric uniformly rotating vortex patches with 90-degree corners exist as weak limits of patches along each bifurcation branch.

Background

The paper constructs m-fold symmetric uniformly rotating vortex patches with 90-degree corners for all m≥12. It places this construction in the context of the global bifurcation branches of smooth m-fold symmetric V-states developed in earlier work.

The authors report a conjecture that the limiting patches along every such branch should possess 90-degree corners for every m≥3. Since the present construction covers only sufficiently large m, the conjecture remains unresolved for the full range of symmetry orders.