Local well-posedness below the charge norm for the Dirac-Klein-Gordon   system in two space dimensions
  We prove that the Cauchy problem for the Dirac-Klein-Gordon equations in two space dimensions is locally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor, and an associated range of spaces of positive index for the meson field. In particular, we can go below the charge norm, that is, the $L^2$ norm of the spinor. We hope that this can have implications for the global existence problem, since the charge is conserved. Our result relies on the null structure of the system, and bilinear space-time estimates for the homogeneous wave equation. 
Piero D'Ancona, Damiano Foschi, Sigmund Selberg