Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering
  We consider two classes of defocusing energy-supercritical nonlinear Schr\"odinger equations in dimensions $d\geq 5$. 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. 
Rowan Killip and Monica Visan