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

TL;DR

This paper develops new correlation estimates for nonlinear Schrödinger equations in one and two dimensions, leading to simplified proofs of scattering results and extending understanding of nonlinear wave behavior.

Contribution

It introduces novel interaction Morawetz estimates in 1D and 2D, with two different proofs in 2D, and applies these to prove scattering for supercritical nonlinear Schrödinger equations.

Findings

Established new correlation estimates in 1D and 2D.

Provided a direct proof of Nakanishi's $H^1$ scattering result.

Proved scattering below the energy space for certain supercritical 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.