The Analysis and PDE Seminar will take place on Monday, November 7th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Global well-posedness for the energy critical Massive Maxwell-Klein-Gordon equation with small data

Abstract: We discuss the global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $R^{1+d}$ $(d\ge 4)$ for data with small critical Sobolev norm. This extends to $m^2>0$ the results of Krieger-Sterbenz-Tataru ($d=4,5$) and Rodnianski-Tao ($d\ge 6$).

The proof is based on 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 are proved 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 ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}^{-1}$ in conjunction with endpoint Strichartz estimates in Lorentz spaces.