7 problems

Let $u$ be a solution of the focusing energy-critical wave equation in five spatial dimensions, with lifespan endpoint that may be finite or infinite. A soliton is a rescaled and t…

Rigidity conjecture. Any solution to the energy-critical wave equation with the compactness property is a travelling wave.

Ground state conjecture. The solution extends uniquely to a global solution $u\in C(\mathbb{R}:\dot{H}^1(\mathbb{R}^d))$ of

Let $d\geq 3$, let $I$ be the lifespan interval, and let $u:I\times\mathbf{R}^{d}\to\mathbf{C}$ solve the focusing energy-critical nonlinear Schrödinger equation with…

Consider the focusing energy-critical wave equation in space dimension $N\in\{3,4,5\}$. A solution has the compactness property if its trajectory stays in a compact subset of the e…

Minimal kinetic-energy conjecture. If $E_*<\|\nabla W\|_2$, then

Non-$L^2$ threshold classification conjecture. Without assuming $u_0\in L^2$, the only solution whose interval of definition is not finite is $W^+$ up to the symmetries of the equa…