Energy-supercritical NLS: critical ${\dot{H}}^s$-bounds imply scattering

We consider two classes of defocusing energy-supercritical nonlinear Schr"odinger equations in dimensions $d\ge 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$,then$u$ is global and scatters.