11 problems

Finn's conjecture. If $a$ is sufficiently small, the starting problem admits a solution that tends to a stationary solution as $t\to\infty$.

Let $u$ be a small steady solution of the three-dimensional Navier–Stokes equation in an exterior domain, and suppose its leading asymptotic term at infinity is a Landau solution.…

Let $n=2$, let $B_R$ be a disk of radius $R$, and let $\beta$ be a fixed perimeter or area constraint. For a bounded simply connected open set $\Omega\subset\mathbb{R}^2$, write…

Global existence conjecture. If

Second Robin eigenvalue conjecture. If $\mathcal B$ is a disk, then

Payne's conjecture. For every smooth exterior domain $U$,

Let $\Omega\subset\mathbb{R}^n$ be a smooth exterior domain, and let $\Delta_D$ and $\Delta_N$ denote the Dirichlet and Neumann Laplacians on $\Omega$, respectively. The Hassell–Si…

Extension conjecture. The preceding blow-up theorems should remain valid under the weaker assumption

Consider the nonlocal dispersal equation in the exterior of a compact obstacle $K\subset\mathbb{R}^N$, without assuming that $K$ is convex, and let $u$ be the entire solution const…

Geometric and radiality conjecture. Theorem cannot be generalised to other geometries, and the radiality assumptions in Theorems and are necessary.

Let $u$ be a steady Navier–Stokes solution in a three-dimensional exterior domain, and let $\tilde U^b$ be a regularization of the Landau solution $U^b$, where $b$ is the total for…