Kenig–Merle critical norm conjecture for nonlinear Schrödinger equations

Let $d\geq 1$ and $\alpha>1$ be such that the critical regularity $s_c\geq 0$. Let $u:I\times\mathbb{R}^d\to\mathbb{C}$ be a maximal-lifespan solution to the nonlinear Schrödinger equation such that

$$u\in L_t^\infty\dot{H}_x^{s_c}(I\times\mathbb{R}^d).$$

Kenig–Merle critical norm conjecture. If $u$ has bounded critical Sobolev norm in this sense, then $u$ is global and scatters to a free solution.

This is a conditional global well-posedness and scattering statement for critical nonlinear Schrödinger equations. The surrounding discussion says that unconditional global well-posedness and scattering for large data at non-conserved critical regularity remains open; the precise status of this conditional formulation is not established by the supplied material.