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Tensor products and correlation estimates with applications to nonlinear Schrödinger equations
Author(s) -
Colliander James,
Tzirakis Nikolaos,
Grillakis Manoussos G.
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20278
Subject(s) - mathematics , nonlinear system , dimension (graph theory) , bilinear interpolation , tensor (intrinsic definition) , conservation law , diagonal , sobolev space , bilinear form , commutator , class (philosophy) , pure mathematics , space (punctuation) , mathematical analysis , algebra over a field , quantum mechanics , geometry , physics , artificial intelligence , computer science , linguistics , statistics , lie conformal algebra , philosophy
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 two‐dimensional 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 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. © 2009 Wiley Periodicals, Inc.