Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation in 1+n

摘要

Using the same induction on energy argument in both the frequency space and the spatial space simultaneously as in [66. Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2008 ). Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 . Ann. Math. 167 : 767 - 865 . [CrossRef], [Web of Science ®]View all references, 3333. Ryckman , E. , Visan , M. ( 2007 ). Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in 1+4 . Amer. J. Math. 129 : 1 - 60 . [Web of Science ®]View all references, 3838. Visan , M. ( 2007 ). The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions . Duke Math. J. 138 : 281 - 374 . [CrossRef], [Web of Science ®]View all references], we obtain the global well-posedness and scattering of energy solutions of the defocusing energy-critical nonlinear Hartree equation in  × n (n ≥ 5), which removes the radial assumption on the data in [2525. Miao , C. , Xu , G. , Zhao , L. ( 2007 ). Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data . J. Funct. Anal. 253 : 605 - 627 . [CrossRef], [Web of Science ®]View all references]. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition 3.1 and Corollary 3.1) of the minimal energy blow up solutions, which cannot be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that -norm of minimal energy blow up solutions is bounded from below, the -norm is stronger than the potential energy.