## Well-posedness and ill-posedness of the Cauchy problem for the Maxwell-Dirac system in $$1+1$$ space time dimensions.(English)Zbl 1261.35126

Summary: We completely determine the range of Sobolev regularity for the Maxwell-Dirac system in $$1+1$$ space time dimensions to be well-posed locally in the case that the initial data of the Dirac part regularity is of $$L^2$$. The well-posedness follows from the standard energy estimates. Outside the range for the well-posedness, we show either the flow map is not continuous or not twice differentiable at zero.

### MSC:

 35Q40 PDEs in connection with quantum mechanics 35Q61 Maxwell equations 35L70 Second-order nonlinear hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs
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