Tensor products and Correlation Estimates with applications to Nonlinear Schr\"odinger equations

Abstract

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schr\"odinger equations and derive a direct proof of Nakanishi's $H^1$ scattering result for every $L^{2}$-supercritical nonlinearity. We also prove scattering below the energy space for a certain class of $L^{2}$-supercritical equations.