Low-regularity scattering conjecture for defocusing nonlinear wave equations

Let $s_c\geq0$, $mu=-1$, and let $(u_0,u_1)\in\dot{H}^{s_c}\times\dot{H}^{s_c-1}$. Low regularity conjecture. The corresponding solution $u$ is global and scatters: there exist unique $(u_0^{\pm},u_1^{\pm})\in\dot{H}^{s_c}_x(\mathbb{R}^d)\times\dot{H}^{s_c-1}$ such that

$$\lim_{t\to\pm\infty}\|u(t)-S(t)(u_0^{\pm},u_1^{\pm})\|_{\dot{H}_x^{s_c}(\mathbb{R}^d)}=0.$$

This asserts global scattering for arbitrary data at the critical regularity, without assuming an a priori critical-norm bound. It is presented as an open extension of the critical-norm results.