The Interctitical Defocusing Nonlinear Schr?dinger Equations with Radial Initial Data in Dimensions Four and Higher

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英文篇名：The Interctitical Defocusing Nonlinear Schr?dinger Equations with Radial Initial Data in Dimensions Four and Higher 作者：Chuanwei ; Gao ; Changxing ; Miao ; Jianwei ; Yang-Urbain 英文作者：Chuanwei Gao;Changxing Miao;Jianwei Yang-Urbain;The Graduate School of China Academy of Engineering Physics;Institute of Applied Physics and Computational Mathematics;Department of Mathematics, Beijing Institute of Technology;LAGA (UMR CNRS 7539), Université; 英文关键词：Nonlinear Schr?dinger equation;;scattering;;frequency-localized Morawetz estimae;;weighted Strichartz space 中文刊名：BJYY 英文刊名：分析,理论与应用(英文版) 机构：The Graduate School of China Academy of Engineering Physics;Institute of Applied Physics and Computational Mathematics;Department of Mathematics, Beijing Institute of Technology;LAGA (UMR CNRS 7539), Université Paris 13, Sorbonne Paris Cité,Villetaneuse, France; 出版日期：2019-06-15 出版单位：Analysis in Theory and Applications 年：2019 期：v.35 基金：supported in part by the National Natural Science Foundation of China under grant No.11671047 and No.11726005;; supported by the LabEx MME-DII 语种：英文; 页：BJYY201902005 页数：30 CN：02 ISSN：32-1631/O1 分类号：91-120

摘要

In this paper, we consider the defocusing nonlinear Schr?dinger equation in space dimensions d≥4. We prove that if u is a radial solution which is priori bounded in the critical Sobolev space, that is,■, then u is global and scatters. In practise,we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases d ≥ 4 and 0 < s_c<1/2. The results in this paper extend the work of [27, Commun. PDEs, 40(2015), 265–308] to higher dimensions.
        In this paper, we consider the defocusing nonlinear Schr?dinger equation in space dimensions d≥4. We prove that if u is a radial solution which is priori bounded in the critical Sobolev space, that is,■, then u is global and scatters. In practise,we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases d ≥ 4 and 0 < s_c<1/2. The results in this paper extend the work of [27, Commun. PDEs, 40(2015), 265–308] to higher dimensions.

引文

[1]J.Bourgain,Global wellposedness of defocusing critical nonlinear Schr?dinger equation in the radial case,J.Amer.Math.Soc.,12(1999),145-171.
    [2]T.Cazenave,Semilinear schr?dinger equations,American Mathematical Soc.,2003.
    [3]T.Cazenave and F.B.Weissler,The Cauchy problem for the nonlinear Schrodinger equations in Hs,Non-Linear Anal.,14(10)(1990),807-836.
    [4]J.Colliander,M.Keel,G.Staffilani,H.Takaoka,and T.Tao,Global existence and scattering for rough solutions of a nonlinear Schr?dinger equation on R3,Commun.Pure Appl.Math.,57(8)(2003),987-1014.
    [5]J.Colliander,M.Keel,G.Staffilani,H.Takaoka,and T.Tao,Global well-posedness and scattering for the energy-critical nonlinear Schr?dinger equation in R3,Ann.Math.,167(2008),767-865.
    [6]B.Dodson,Global well-posedness and scattering for the defocusing,L2-critical,nonlinear Schr?dinger equation when d≥3,J.Amer.Math.Soc.,25(2012),429-463.
    [7]B.Dodson,Global well-posedness and scattering for the defocusing,L2-critical,nonlinear Schr?dinger equation when d=2,Duke Math.J.,165(2016),3435-3516.
    [8]B.Dodson,Global well-posedness and scattering for the defocusing,L2-critical,nonlinear Schr?dinger equation when d=1,Amer.J.Math.,138(2016),531-569.
    [9]B.Dodson,Global well-posedness and scattering for the mass critical nonlinear Schr?dinger equation with mass below the mass of the ground state,Adv.Math.,285(2015),1589-1618.
    [10]B.Dodson,Global well-posedness and scattering for the focusing,energy-critical nonlinear Schr?dinger problem in dimension d=4 for initial data below a ground state threshold,arXiv preprint arXiv:1409.1950,(2014).
    [11]B.Dodson,C.Miao,J.Murphy,and J.Zheng,The defocusing quintic NLS in four space dimensions,Annales de l’Institut Henri Poincare/Analyse non lineaire,34(2017),759-787.
    [12]J.Holmer and S.Roudenko,A sharp condition for scattering of the radial 3D cubic nonlinear Schr?dinger equation,Commun.Math.Phys.,282(2)(2008),435-467.
    [13]M.Grillakis,On nonlinear Schr?dinger equations,Commun.PDE,25(2000),1827-1844.
    [14]C.E.Kenig and F.Merle,Scattering for˙H1/2 bounded solutions to the cubic,defocusing NLS in 3 dimensions,Trans.Amer.Math.Soc.,362(2010),1937-1962.
    [15]C.E.Kenig and F.Merle,Global well-posedness,scattering and blow-up for the energycritical,focusing,non-linear Schr?dinger equation in the radial case,Invent.Math.,166(2006),645-675.
    [16]C.E.Kenig and F.Merle,Scattering for˙H1/2 bounded solutions to the cubic,defocusing NLS in 3 dimensions,Trans.Amer.Math.Soc.,362(2010),1937-1962.
    [17]R.Killip and M.Visan,The focusing energy-critical nonlinear Schr?dinger equation in dimensions five and higher,Amer.J.Math.,132(2010),361-424.
    [18]R.Killip and M.Visan,Energy-supercritical NLS:critical˙Hs-bounds imply scattering,Commun.PDE,35(2010),945-987.
    [19]R.Killip and M.Visan,Nonlinear Schr?dinger equations at critical regularity,Evolution Equations,325-437,Clay Math.Proc.,17,Amer.Math.Soc.,Providence,RI,2013.
    [20]R.Killip and M.Visan,Global well-posedness and scattering for the defocusing quintic NLSin three dimensions,Anal.Partial Differential Equations,5(4)(2011),855-885.
    [21]R.Killip,T.Tao,and M.Visan,The cubic nonlinear Schr?dinger equation in two dimensions with radial data,J.Euro.Math.Soc.,(JEMS)11(2009),1203-1258.
    [22]R.Killip,M.Visan and X.Zhang,The mass-critical nonlinear Schr?dinger equation with radial data in dimensions three and higher,Anal.PDE,1(2008),229-266.
    [23]C.Lu and J.Zheng,The radial defocusing energy-supercritical NLS in dimension four,J.Differential Equations,262(8)(2017),4390-4414.
    [24]C.Miao,J.Murphy,and J.Zheng,The defocusing energy-supercritical NLS in four space dimensions,J.Funct.Anal.,267(6)(2014),1662-1724.
    [25]J.Murphy,Inter-critical NLS:critical˙Hs-bounds imply scattering,SIAM J.Math.Anal.,46(2014),939-997.
    [26]J.Murphy,The defocusing˙H1/2-critical NLS in high dimensions,Discrete Contin.Dyn.Syst.Series A,34(2014),733-748.
    [27]J.Murphy,The radial defocusing nonlinear Schr?dinger equation in three space dimensions,Commun.Partial Differential Equations,40(2015),265-308.
    [28]E.Ryckman and M.Visan,Global well-posedness and scattering for the defocusing energycritical nonlinear Schr?dinger equation in R1+4,Amer.J.Math.,129(2007),1-60.
    [29]E.M.Stein,Harmonic Analysis:Real-Variable Methods,Orthogonality,and Oscillatory Integrals,Princeton University Press,2006.
    [30]T.Tao,Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schr?dinger equation for radial data,New York J.Math.,11(2005),57-80.
    [31]T.Tao,M.Visan,and X.Zhang,Global well-posedness and scattering for the defocusing mass-critical nonlinear Schr?dinger equation for radial data in high dimensions,Duke Math.J.,140(2007),165-202.
    [32]T.Tao,M.Visan,and X.Zhang,Minimal-mass blowup solutions of the mass-critical NLS,Forum Math.,20(2008),881-919.
    [33]M.C.Vilela,Regularity of solutions to the free Schr?dinger equation with radial initial data,Illinois J.Math.,45(2001),361-370.
    [34]M.Visan,The defocusing energy-critical nonlinear Schr?dinger equation in higher dimensions,Duke Math.J.,138(2007),281-374.
    [35]M.Visan,The Defocusing Energy-Critical Nonlinear Schr?dinger Equation in Dimensions Five and Higher,Ph.D.Thesis,UCLA,2006.
    [36]M.Visan,Global well-posedness and scattering for the defocusing cubic nonlinear Schr?dinger equation in four dimensions,Int.Math.Research Notices,5(2012),1037-1067.