10 problems

A vortex patch is a solution of the three-dimensional incompressible Euler equations whose initial vorticity has patch-like, conormally smooth structure; its boundary is initially…

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…

Consider a vortex patch whose boundary has a single corner of acute angle, without any imposed symmetry. Cohen–Danchin's conjecture. Any such acute single corner would cusp immedia…

Let $\Omega(t)\subset\mathbb R^2$ be the evolving region of a vortex patch, with boundary $\partial\Omega(t)$, and suppose that the initial boundary $\partial\Omega(0)$ is a smooth…

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…

Let $\mathscr{S}_m$ be the global $m$-fold bifurcation branch of uniformly rotating vortex patches emanating from the Rankine vortex, with $m\geq 3$, and let $s$ denote its paramet…

A vortex patch is a vorticity distribution of the form $\omega(x,t)=\lambda I_{A_t}$, where $\lambda\in\mathbb R$ is the vorticity strength and $A_t\subset\mathbb R^2$ is the patch…

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 $s\to\infty$…