Well-posedness for nonlinear Dirac equations in one dimension

Abstract

We completely determine the range of Sobolev regularity for the Dirac- Klein-Gordon system, the quadratic nonlinear Dirac equations, and the wave-map equation to be well posed locally in time on the real line. For the Dirac-Klein-Gordon system, we can continue those local solutions in nonnegative Sobolev spaces by the charge conservation. In particular, we obtain global well-posedness in the space where both the spinor and scalar fields are only in $L^2\left(\mathbb{R}\right)$. Outside the range for well-posedness, we show either that some solutions exit the Sobolev space instantly or that the solution map is not twice differentiable at zero.