Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system

Abstract

We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in $1+3$ 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.