9 problems

Let $u$ solve the incompressible Navier–Stokes equations on $[0,1]\times\mathbb{R}^3$, … For any $\alpha\in[0,1/2)$, consider initial data $u_0\in H^\alpha\cap L^{3,\infty}$ and so…

Elgindi's blow-up conjecture. For every $\alpha\in(0,1/3)$, there exists a compactly supported $C_c^{1,\alpha}$ axisymmetric swirl-free local solution to the Euler equations that d…

The self-similar axisymmetric Euler equations admit no solutions with velocity discontinuities at finitely many points satisfying the Rankine–Hugoniot conditions … Here $\theta$ is…

Longer-time approximation conjecture. An approximation result of the form $$

Let $u$ be a classical axisymmetric solution of the axisymmetric Navier–Stokes equations, and let a critical Besov norm replace the weak-$L^3$ norm appearing in Theorem. Critical B…

Let $u^0\not\equiv 0$ belong to $H^s_{as\&df}(\mathbb{R}^3)$ with $s>\frac{5}{2}$, and let $\omega^0$ be its associated vorticity. Assume that $\omega^0$ is odd in $z$ and satisfie…

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 the original three-dimensional incompressible axisymmetric Navier–Stokes equations and the possibility of focusing blowup at the symmetry axis $r=0$. Two-scale blowup excl…

Let the three-dimensional incompressible Euler equations be equipped with Kida's high-symmetry initial data. Boratav–Pelz conjecture. The Euler equations with these initial data sh…