10 problems

Consider smooth solutions of the axisymmetric Euler equations with swirl on a periodic cylinder. Hou–Luo conjecture. Such solutions could develop finite-time singularities at the b…

There exist a time $T>0$ and smooth functions … with smooth initial data … for smooth $f_1,f_2,f_3$ and a smooth even angular profile $\phi(\theta)$ adapted to…

Consider axisymmetric Euler flow without swirl, with initial vorticity consisting of two anti-parallel vortex rings … where $f\geq 0$ is localized near $(r_0,z_0)$ with $r_0>0$ and…

Let $N>1$ and $0<\varepsilon\ll 1$. Suppose that the initial vorticity $\omega_0^\varepsilon$ is decomposed into $N$ blobs, each with circulation $\gamma>0$, sharply concentrated a…

For an axisymmetric Euler flow without swirl satisfying the symmetry and sign conditions described above, let $R(t)$ denote the relevant vorticity-growth scale. Childress's conject…

For the axisymmetric swirl-free Euler equations, let $v$ denote the velocity field and let $C^{1,\alpha}$ be the space of velocity fields with one derivative that is Hölder continu…

Expanding-ring conjecture. The singular set of the solutions of the cited self-similar Euler construction propagates within the natural symmetry class as the expanding ring…

Consider smooth axisymmetric Euler solutions in the axisymmetric coordinate formulation, with vorticity non-vanishing on the axis of symmetry. Artificial-boundary conjecture. There…

Radial smoothness conjecture. The initial vorticity $\omega_0$ can be $C^\infty$ as a function of $\rho$. This sharpens the preceding conjecture by requiring the lack of regularity…

Let $\alpha<1/3$. Subcritical Hölder blow-up conjecture. There exists a compactly supported, axisymmetric, no-swirl local Euler solution with initial regularity $C^\alpha_c$ that b…