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

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