Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

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Sigmund Selberg, Norwegian University of Science and Technology
Fine Hall 110

In this talk I will present recent joint work with P. D'Ancona and D. Foschi on the classical Maxwell-Dirac system, which is the fundamental PDE in quantum electrodynamics. We show that the system has some special structural properties ("null" structure) which improve the regularity of solutions. To see this structure, however, one must consider the system as a whole: it cannot be seen in the individual component equations. For the multilinear forms that thus arise, we prove estimates in $X^{s,b}$ spaces at the scale invariant regularity up to some logarithmic losses, and as a consequence we obtain almost optimal local well-posedness by iteration.