Global well-posedness and scattering for the defocusing, energy   -critical, nonlinear Schr{\"o}dinger equation in the exterior of a convex   obstacle when $d = 4$
  In this paper we prove that the energy - critical nonlinear Schr{\"o}dinger equation in the domain exterior to a convex obstacle is globally well - posed and scattering for initial data having finite energy. To prove this we utilize frequency localized Morawetz estimates adapted to an exterior domain. 
Benjamin Dodson
math.AP