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Null structure and almost optimal local well-posedness of the Maxwell-Dirac system
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 3, June 2010
- pp. 771-839
- 10.1353/ajm.0.0118
- Article
- Additional Information
We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized $L^2$ space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration.