Global Solutions For Systems of Quadratic Nonlinear Schr\"odinger Equations in 3D

Abstract

In this work, we prove global well-posedness 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.