Almost conservation laws and global rough solutions to a Nonlinear   Schr\"odinger equation
  We prove an "almost conservation law" to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schr\"odinger equation in H^s(R^n) when n = 2, 3 and s > 4/7, 5/6, respectively. 
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao