Global Stability for Charged Scalar Fields on Minkowski Space
  We prove that the charge-scalar field (also known as the massless Maxwell-Klein-Gordon) equations are globally stable on (3+1) dimensional Minkowski space for small initial data in certain gauge covariant weighted Sobolev spaces. These spaces can be chosen as to be almost scale invariant with respect to the homogeneity of the equations. This result is valid for initial data with non-zero charge that is also non-stationary at space-like infinity. The method of proof is a tensor-geometric approach which is based on a certain family of weighted bilinear L2 space-time estimates. 
Hans Lindblad, Jacob Sterbenz