Finite-time blowup conjecture for axisymmetric swirl-free Euler flow in dimensions at least four

Let $u^0 ot\equiv 0$ belong to $H^s_{as\&df}(\mathbb{R}^d)$, where $d\geq 4$ and $s>1+\frac{d}{2}$, and let $\omega^0$ be its associated vorticity. Assume that $\omega^0$ is odd in $z$ and satisfies $\omega^0(r,z)\geq 0$ for all $z>0$. Let

$$u\in C\left([0,T_{\max}); H^s_{df}(\mathbb{R}^d) \right) \cap C^1\left([0,T_{\max}),H^{s-1}_{df}(\mathbb{R}^d)\right)$$

be the smooth Euler solution with initial data $u^0$. Finite-time blowup conjecture. The maximal existence time satisfies $T_{\max}<+\infty$, so the solution blows up in finite time. The preceding sign and monotonicity results motivate this conjecture, but the finite-time singularity formation claim in dimensions at least four remains open.