11 problems

Consider the 3D incompressible Euler equations on $[0,T]\times \mathbb{T}^3$. A weak solution is understood in the usual distributional sense, and $C^{\beta}$ denotes the Hölder sp…

Let $d\in\{2,3\}$. Let $u\in C^0([0,1];C^\alpha(\mathbb T^d))\cap C^\alpha([0,1];C^0(\mathbb T^d))$ be a weak solution of the incompressible Euler equations, for some Hölder expone…

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

Exceptional-support conjecture. Part (i) of the theorem on invariant measures already contains essentially all counterexamples to non-invariance: for…

Hamel–Nadirashvili's weaker conjecture. If $z\in D$ is the only stagnation point of $\mathbf{v}$ in $\overline{D}$, meaning that $|\mathbf{v}(z)|=0$ and $|\mathbf{v}|>0$ in…

Hamel–Nadirashvili's conjecture. If $|\mathbf{v}|>0$ in $\overline{D}\backslash\{z\}$, then $z$ is the origin and $\mathbf{v}$ is a circular flow.

Let solutions of the two-dimensional Euler equation evolve as $t\to\pm\infty$. A solution is called generic in the informal sense intended by the conjecture, and its weak limit set…

Let $(\omega,u)$ be a solution of the two-dimensional incompressible Euler equation on $\mathbb{T}^2$, and consider its weak limit set as $t\to\pm\infty$. A solution lies on a comp…

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…

Let $D$ be an open non-empty disk and let $z\in D$. Let $v$ be a $C^2(\overline{D}\!\setminus\!\{z\})$ and bounded flow solving the Euler equations and satisfying $v\cdot n=0$ on…

Generic energy-profile regularity conjecture. A generic solution with regularity at most $1/3$ fails to conserve energy and has an energy profile of minimal regularity. More precis…