We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman-Machedon type. It has been known for some time that the Klein-Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
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Piero D'Ancona, Damiano Foschi, Sigmund Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. J. Eur. Math. Soc. 9 (2007), no. 4, pp. 877–899DOI 10.4171/JEMS/100