Mass-threshold global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation

Let $d\geq 1$, $\mu\in\{-1,+1\}$, and $u_0\in L^2(\mathbb{R}^d)$. Consider the mass-critical nonlinear Schrödinger equation

$$i\partial_tu+\Delta u=\mu|u|^{\frac{4}{d}}u.$$

In the focusing case, assume additionally that $M(u_0)<M(\widetilde{Q})$, where $\widetilde{Q}$ is the unique positive radial solution of

$$\Delta\widetilde{Q}+\widetilde{Q}^{1+\frac{4}{d}}=\widetilde{Q}.$$

Global existence and scattering conjecture. All maximal-lifespan solutions are globally well-posed and scatter.

This asserts that the ground-state mass is the minimal obstruction to global well-posedness and scattering in the focusing equation, while no such obstruction occurs in the defocusing equation. The source presents this as a widely believed conjecture; its status is not specified here.