Ill-posedness for the Maxwell-Dirac system below charge in space dimension three and lower

2020/02/22 by Selberg, Sigmund, Tesfahun, Achenef · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.09717

Abstract

The Maxwell-Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension d=3 the system is charge-critical, that is, L²-critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions d=1,2, global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if d=3), by demonstrating ill-posedness below the charge regularity. In fact, for d ≤ 3 we exhibit a spinor datum belonging to H^s(\mathbb R^d) for s<0, and to L^p(\mathbb R^d) for 1 ≤ p < 2, but not to L²(\mathbb R^d), which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.

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