Minimal kinetic-energy conjecture for blowup solutions to the focusing energy-critical nonlinear Schrödinger equation

Let $d\geq 3$ and let $u:I\times\mathbb{R}^d\to\mathbb{C}$ be a solution to the focusing energy-critical nonlinear Schrödinger equation. Let

$$W(x)=\frac{1}{\left(1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$

be the stationary ground-state solution, and define

$$E_*:=\sup_{t\in I}\|\nabla u(t)\|_2.$$

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

$$\int_I\int_{\mathbb{R}^d}|u(t,x)|^{\frac{2(d+2)}{d-2}}\,dx\,dt\leq C(E_*)<\infty.$$

This asserts that every solution whose kinetic energy remains strictly below that of the ground state has finite scattering size, and hence cannot exhibit blowup in the sense associated with infinite scattering size. The statement is the threshold conjecture for global behavior in the focusing energy-critical equation; the supplied text presents it as a belief and gives no resolution status.