Global Solutions for Systems of Quadratic Nonlinear Schrödinger Equations in 3D

Abstract

In this thesis, we prove existence of global solutions and scattering for systems of quadratic nonlinear Schr\"odinger equations in the critical three-dimensional case, for small, localized data. For the terms corresponding to the nonlinearity $u\bar{u}$, we need to do an $\epsilon$ regularization of this part of the nonlinearity. In order to tackle quadratic space-time resonances, after performing a Littlewood--Paley decomposition, we use integration by parts in the Duhamel term, to take advantage of the oscillations when space-time resonances are absent