10 problems

Let $m\geq0$ and $N_0\geq2$, where $N_0$ is the rotation number of the initial immersed curve. Consider the area-preserving curvature flow of order $m$. Finite-time blowup conjectu…

Let $S$ be a mild solution of the Navier–Stokes strain equation without advection, with $S otag\in C\left([0,T_{\max});L^2_{st}\right)$, where $L^2_{st}$ denotes the space of square…

Let $u\in C\left([0,T_{\max}); \dot{H}^s_{\mathcal{M}}\right)$, with $s>\frac{5}{2}$, be an odd, permutation symmetric, $\sigma$-mirror symmetric solution of the Fourier-restricted…

Finite-time breakdown conjecture for the b-equations. If, for some operator $A$, the Euler–Arnold equation has $C^1$ solutions that break down in finite time, then for every $b>1$…

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…

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

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…

Parity conjecture. In general, $H^s$ solutions should develop singularities when $s<3/2$; if $m$ is odd, there exist analytic solutions that become singular in finite time; and if…

Let $T_*<\infty$ and let $\omega\in C^\infty_c(\mathbb{R}^3\times[0,T_*))$ be a smooth solution of the 3D Euler equation that becomes singular as $t\to T_*$. Minimal-intensity conj…

Finite-time blowup conjecture. There exists such a compact Riemannian manifold $(M,g)$, of some dimension $d>2$, and such a smooth solution that cannot be smoothly continued to the…