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

Okamoto, Mamoru

Adv. Differ. Equ. 18, No. 1-2, 179-199 (2013).

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