TAILIEUCHUNG - Đề tài " Global well-posedness and scattering for the energy-critical nonlinear Schr¨odinger equation in R3 "

We obtain global well-posedness, scattering, and global L10 spacetime t,x bounds for energy-class solutions to the quintic defocusing Schr¨dinger equao tion in R1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. | Annals of Mathematics Global well-posedness and scattering for the energy-critical nonlinear Schr odinger equation in R3 By J. Colliander M. Keel G. Staffilani H. Takaoka and T. Tao Annals of Mathematics 167 2008 767 865 Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R3 By J. Colliander M. Keel G. Staffilani H. Takaoka and T. Tao Abstract We obtain global well-posedness scattering and global L1X spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equation in R1 3 which is energy-critical. In particular this establishes global existence of classical solutions. Our work extends the results of Bourgain 4 and Grillakis 20 which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the Morawetz inequality by an interaction variant first used in 12 13 . The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular this interaction estimate together with an almost-conservation argument controlling the movement of L2 mass in frequency space rules out the possibility of energy concentration. Contents 1. Introduction . Critical NLS and main result . Notation 2. Local conservation laws 3. Review of Strichartz theory in R1 3 . Linear Strichartz estimates . Bilinear Strichartz estimates . Quintilinear Strichartz estimates . Local well-posedness and perturbation theory . is supported in part by . Grant DMS-0100595 . Grant . 250233-03 and the Sloan Foundation. . was supported in part by . Grant DMS-0303704 and by the McKnight and Sloan Foundations. . is supported in part by . Grant DMS-0100375 . Grant DMS-0111298 through the IAS and the Sloan Foundation. . is .

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