5 problems

Consider a vortex patch whose boundary has a single corner of acute angle, and define the order of a cusp at $x=x_0$ by … where $f_1$ and $f_2$ represent the graphs of the two side…

Let $S_{\mu,1,1}$ denote the set of maximizers associated with impulse $\mu$, and let $\mu$ be the impulse parameter. Small-impulse uniqueness conjecture. There exists a universal…

Let $\Omega$ be the bounded open set defining a Sadovskii vortex patch as described in Theorem 1. Detailed-shape conjecture. The set $\Omega$ is connected, convex, and makes a righ…

Persistence-of-Cat's-eyes conjecture. The conclusion of Theorem holds for all $s>0$.

Limiting-solutions conjecture. The singular solutions with $90$-degree corners seen in numerical computations exist as the weak limits of patches along $\mathscr C$ as $s\to\infty$…