A new resolution space for nonlinear Schrödinger equations and applications

Abstract

Resolution spaces play a central role in constructing solutions for nonlinear partial differential equations. One of the main goals in the area of nonlinear dispersive PDEs has been to construct effective resolution spaces which capture the known bilinear restrictions estimates for free solutions. In this paper we propose a new structure for the Schrödinger equation which effectively replicates the classical bilinear $L^2_{t,x}$ estimate. In addition, the new structure has the property that its "dual" is an effective candidate for a space for the forcing in the linear inhomogeneous Schrödinger equation, a feature that has been elusive so far in the literature. As an application, we show how these structures can recover the known global well-posedness results for derivative NLS with null structure, with Schrödinger Maps being one such model.