13 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…

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 $S otagin Cleft([0,T{max});L^2{st}right)$, where $L^2{st}$ denotes the space of square-integ…

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 $u^0 otequiv 0$ belong to H^s{as&df}(mathbb{R}^d), where $dgeq 4$ and $s1+frac{d}{2}$, and let $omega^0$ be its associated vorticity. Assume that $omega^0$ is odd in $z$ and…

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

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…

Let $u_1$ be the nontrivial equilibrium of the complex Ginzburg–Landau equation, and let $\theta\in[-\pi/2,\pi/2]$. The strong unstable manifold is the $(1$-complex-dimensional$)$…

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…

Consider the one-dimensional system … with Biot–Savart law … and singular steady solution … where $c>0$ is suitable. Let $\omega_0$ be a smooth profile compactly supported inside…

Radial blowup profile conjecture. This class of radially symmetric decreasing solutions, including a Dirac mass at the origin, describes well the local behavior at the blowup time…