The Analysis and PDE Seminar will take place on Monday, November 21th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Scattering below the ground state for the radial focusing NLS

Abstract: We consider scattering below the ground state for the radial cubic focusing NLS in three dimensions. Holmer and Roudenko originally proved this via concentration compactness and a localized virial estimate. We present a simplified proof that avoids the use of concentration compactness, relying instead on the radial Sobolev embedding and a virial/Morawetz hybrid. This is joint work with Ben Dodson.

The Analysis and PDE Seminar will take place on Monday, November 14th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Finite depth gravity water waves In holomorphic coordinates

Abstract: In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth paper.

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}$ $(d \geq 4)$ 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 \geq 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 $ \Box^{-1} $ in conjunction with endpoint Strichartz estimates in Lorentz spaces.