Cazenave–Kenig–Merle scattering conjecture for the energy-critical nonlinear Schrödinger equation

Let $d\geq 3$, let $\mu\in\{1,-1\}$, and consider the energy-critical nonlinear Schrödinger equation

$$i\partial_t u=-\frac{1}{2}\Delta u+\mu|u|^{\frac{4}{d-2}}u,\qquad u(0)=u_0\in\dot{H}^1(\mathbb{R}^d).$$

Its Hamiltonian is

$$E_{\Delta}(u)=\int \frac{1}{2}|\nabla u|^2+\mu\frac{d-2}{d}|u|^{\frac{2d}{d-2}}\,dx.$$

Cazenave–Kenig–Merle conjecture. When $\mu=1$, solutions exist globally and scatter: for every $u_0\in\dot{H}^1(\mathbb{R}^d)$, there is a unique global solution $u:\mathbb{R}\times\mathbb{R}^d\to\mathbb{C}$ satisfying

$$S_{\mathbb{R}}(u):=\int_{\mathbb{R}}\int_{\mathbb{R}^d}|u(t,x)|^{\frac{2(d+2)}{d-2}}\,dx\,dt\leq C(E_{\Delta}(u_0))<\infty,$$

and there are $u_{\pm}\in\dot{H}^1(\mathbb{R}^d)$ such that

$$\lim_{t\to\pm\infty}\left\|u(t)-e^{\pm\frac{it\Delta}{2}}u_{\pm}\right\|_{\dot{H}^1}=0,$$

with $u_0\mapsto u_{\pm}(u_0)$ homeomorphisms of $\dot{H}^1$. When $\mu=-1$, the same conclusion holds provided

$$E_{\Delta}(u_0)<E_{\Delta}(W),\qquad \|\nabla u_0\|_{L^2}<\|\nabla W\|_{L^2},$$

where

$$W(x)=\frac{1}{\left(1+\frac{2|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}\in\dot{H}^1(\mathbb{R}^d)$$

is the ground state solving $\frac{1}{2}\Delta W+|W|^{\frac{4}{d-2}}W=0$. The conjecture has been proved for the defocusing equation, for radial initial data in the focusing case when $d\geq3$, and for arbitrary initial data in the focusing case when $d\geq5$; the remaining cases are open.