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

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 Hs(Rd) for s<0, and to Lp(Rd) for 1≤p<2, but not to L2(Rd), which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.