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Well-posedness for nonlinear Dirac equations in one dimension. (English) Zbl 1248.35170
Summary: 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 (). 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.

35Q41Time-dependent Schrödinger equations, Dirac equations
35L70Nonlinear second-order hyperbolic equations
35L71Semilinear second-order hyperbolic equations
35B44Blow-up (PDE)
35B65Smoothness and regularity of solutions of PDE