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Sharp ill-posedness for the Maxwell-Dirac equations in one space dimension

Research paper by Sigmund Selberg, Achenef Tesfahun

Indexed on: 24 Jan '19Published on: 24 Jan '19Published in: arXiv - Mathematics - Analysis of PDEs



Abstract

The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with $L^2$ data for the spinor. We also prove that this result is sharp, in the sense that well-posedness fails for spinor data in $H^s$ with $s<0$, as well as in $L^p$ with $1 \le p < 2$. More precisely, we give an explicit example of such data for which no local solution can exist. Our proof of well-posedness applies to a class of systems which includes also the Dirac-Klein-Gordon system, but it does not require any null structure in the system.