We prove new interaction Morawetz-type (correlation) estimates in one and two dimensions. In dimension 2 the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the twodimensional 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 two-dimensional estimate to nonlinear Schrödinger equations and derive a direct proof of Nakanishi's H1 scattering result for every L2-supercritical nonlinearity. We also prove scattering below the energy space for a certain class of L2-supercritical equations.
ASJC Scopus subject areas
- Applied Mathematics