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

2019/01/24 by Sigmund Selberg, Selberg, Sigmund, Achenef Tesfahun +1
Engineering · Mathematics · Physics and Astronomy · #35L60 #35L70 #35Q40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1901.08409

openalex publication_date 2019/01/24 · openalex created_date 2019/02/21 · openalex updated_date 2026/07/28

Abstract

The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with L² 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 ≤ 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.

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