Global well-posedness below the charge norm for the Dirac-Klein-Gordon   system in one space dimension
  We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high frequency part. To prove the nonlinear smoothing we rely on the null structure of the DKG system, and bilinear estimates in Bourgain-Klainerman-Machedon spaces. 
Sigmund Selberg