Energy Critical NLS in two space dimensions
  We investigate the initial value problem for a defocusing nonlinear Schr\"odingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case. 
Jim Colliander, Slim Ibrahim, Mohamed Majdoub and Nader Masmoudi