Stability of energy-critical nonlinear Schr\"odinger equations in high   dimensions
  We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite, Strichartz norms. For dimensions $n \leq 6$, this theory is a standard extension of the small data well-posedness theory based on iteration in Strichartz spaces. However, in dimensions $n > 6$ there is an obstruction to this approach because of the subquadratic nature of the nonlinearity (which makes the derivative of the nonlinearity non-Lipschitz). We resolve this by iterating in exotic Strichartz spaces instead. The theory developed here will be applied in a subsequent paper of the second author, to establish global well-posedness and scattering for the defocusing energy-critical equation for large energy data. 
Terence Tao, Monica Visan