12 problems

Let $D$ be a worm domain whose $\overline\partial$-Neumann problem is not globally hypoelliptic. Let $A$ be its boundary annulus, let $V\subset\mathbb C^2$ be an arbitrarily small…

Kraichnan's assertion. For every given $bcp,bcn\fteq 0$, sufficiently small $b5$ should ensure the existence of global-in-time strong solutions for large perturbations of the equil…

Let $U_1({\bf r})$ be an initial condition with a spatial symmetry, and let … be another initial condition with a different spatial symmetry or without spatial symmetry, where…

Small-stress global-estimate conjecture. Global estimates should be obtainable when $\tau$ is sufficiently small.

Beck–Flandoli–Gubinelli–Maurelli's conjecture. If the LPS condition does not hold, namely if

St. Venant's conjecture. Among simply connected domains of a given volume $v$, the torsional rigidity $T(\Omega)$ is maximized by the round ball.

For a one-dimensional Boussinesq system, let $T_\epsilon$ denote the lifespan of its solution, and impose the natural no-cavitation condition on the initial data. The known possibi…

Classification conjecture. All minimizers for $S$ are of the form

Navier–Stokes global regularity conjecture. For every $\nu>0$ and every such initial data $u_0$, such a global smooth finite-energy solution exists.

Boundary conjecture. Type II solutions are the boundary of the set of all initial data that lead to blow-up for the nonlinear wave equation.

Let $B$ be a ball in $R^N$, and let $lambda^D_1(Omega)$ denote the first eigenvalue of the Dirichlet Laplacian on $Omega$. For domains $Omega\tsubset R^N$ with $|Omega|=|B|$, Rayle…

Generic domain controllability conjecture. The generic approximate controllability property proved when either $(\Omega,V)$ or $(\Omega,W)$ is fixed should also hold with respect t…