### Well-posedness and ill-posedness of the Cauchy problem for the Maxwell-Dirac system in $1+1$ space time dimensions

Mamoru Okamoto

#### Abstract

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.

#### Article information

Source
Adv. Differential Equations, Volume 18, Number 1/2 (2013), 179-199.

Dates
First available in Project Euclid: 18 December 2012

Okamoto, Mamoru. Well-posedness and ill-posedness of the Cauchy problem for the Maxwell-Dirac system in $1+1$ space time dimensions. Adv. Differential Equations 18 (2013), no. 1/2, 179--199. https://projecteuclid.org/euclid.ade/1355867485