Global well-posedness and parametrices for critical Maxwell-Dirac and massive Maxwell-Klein-Gordon equations with small Sobolev data

In this thesis we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon (MKG) and for the massless Maxwell-Dirac (MD) equations, in the Coulomb gauge on $\\bbR^{1+d}$ $(d \\geq 4)$ for data with small critical Sobolev norm. For MKG, this work extends to the general case $ m^2 > 0 $ the results of Krieger-Sterbenz-Tataru ($d=4,5 $) and Rodnianski-Tao ($ d \\geq 6 $), who considered the case $ m=0$. We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp $ L^2 $ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr. To overcome logarithmic divergences we rely on an embedding property of $ \\Box^{-1} $ in conjunction with endpoint Strichartz estimates in Lorentz spaces. For MD, the main components of the proof consist of A) uncovering of the null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit and justify a deep analogy between MD and MKG, which says that the most difficult part of MD takes essentially the same form as parts of the Maxwell-Klein-Gordon structure. As a result, the aforementioned functional framework and parametrix construction become applicable.