7 problems

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…

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…

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…

Let $(r,z)$ belong to the cylindrical domain $\{(r,z):0\leq r\leq1\}$. Hou–Luo conjecture. There exists a smooth solution to the axisymmetric Euler equation on this cylindrical dom…