Blow up dynamics for smooth equivariant solutions to the energy critical   Schr\"odinger map
  We consider the energy critical Schr\"odinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. 
Frank Merle and Pierre Rapha\"el and Igor Rodnianski
math.AP