The critical-norm scattering conjecture for defocusing NLS outside a convex obstacle

Let $\Omega\subset\mathbb{R}^3$ be the exterior domain of a smooth, compact, strictly convex obstacle, and consider the defocusing nonlinear Schrödinger equation with Dirichlet boundary conditions. For $\alpha>4/3$, set $s_c=3/2-2/\alpha$, and let $u:I\times\Omega\to\mathbb{C}$ be a maximal-lifespan solution. Exterior-domain critical-norm scattering conjecture. If

$$u\in L_t^\infty \dot{H}_D^{s_c}(I\times\Omega),$$

then $u$ is global and scatters as $t\to\pm\infty$. This is the exterior-domain analogue of the Euclidean critical-norm conjecture. The paper studies the three-dimensional strictly convex setting, while the asserted result is not established in the full range $\alpha>4/3$.