Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in 1+4 dimensions

Abstract

We prove that the Maxwell-Klein-Gordon equations on ${\mathbb{R}}^{1+4}$ relative to the Coulomb gauge are locally well-posed for initial data in $H^{1+\epsilon }$ for all $\epsilon >0$. This builds on previous work by Klainerman and Machedon who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as well as cubic terms.