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Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition

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  • In this paper, we prove global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ and $u_{0} \in H^{s}(\mathbf{R}^{3})$, $s > 5/7$. To this end, we utilize a linear-nonlinear decomposition, similar to the decomposition used in [20] for the wave equation.
    Mathematics Subject Classification: Primary: 35Q55.

    Citation:

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  • [1]

    J. Bourgain, Scattering in the energy space and below in 3D NLS, Journal d'Analyse Mathematique, 4 (1998), 267-297.doi: 10.1007/BF02788703.

    [2]

    J. Bourgain, Refinements of Strichartz' inequality and applications to 2{D-NLS with critical nonlinearity, International Mathematical Research Notices, 5 (1998), 253-283.doi: 10.1155/S1073792898000191.

    [3]

    J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations," American Mathematical Society, American Mathematical Society Colloquium Publications, Providence, RI. 1999.

    [4]

    T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Instituto de Matematica - UFRJ - Rio de Janeiro, 1996.

    [5]

    T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494.doi: 10.1007/BF01258601.

    [6]

    T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Analysis, 14 (1990), 807-836.doi: 10.1016/0362-546X(90)90023-A.

    [7]

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Mathematical Research Letters, 9 (2002), 659-682.

    [8]

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbfR^{3}$, Communications on Pure and Applied Mathematics, 21 (2004), 987-1014doi: 10.1002/cpa.20029.

    [9]

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for the energy - critical nonlinear Schrödinger equation on $\mathbfR^3$, Annals of Mathematics. Second Series, 167 (2008), 767-865doi: 10.4007/annals.2008.167.767.

    [10]

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbfR^{2}$, Discrete and Continuous Dynamical Systems A, 21 (2007), 665-686.doi: 10.3934/dcds.2008.21.665.

    [11]

    J. Colliander and T. Roy, Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $\mathbbR^2$, Communications in Pure and Applied Analysis, 10 (2011), 397-414.doi: 10.3934/cpaa.2011.10.397.

    [12]

    B. DodsonGlobal well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d = 2$, preprint, arXiv:1006.1375,

    [13]

    J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Communications in Mathematical Physics, 144 (1992), 163-188.

    [14]

    J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, Journal de Mathmatiques Pures et Appliques, 9 (1985), 363-401.

    [15]

    M. Keel and T. Tao, Local and global well posedness of wave maps on $\mathbfR^{1 + 1}$ for rough data, International Mathematics Research Notices, 21 (1998), 1117-1156.doi: 10.1155/S107379289800066X.

    [16]

    M. Keel and T. Tao, Endpoint strichartz estimates, American Journal of Mathematics, 120 (1998), 945-957.

    [17]

    C. Kenig and F. Merle, Scattering for $\dot H^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Transactions of the American Mathematical Society, 362 (2010), 1937-1962.doi: 10.1090/S0002-9947-09-04722-9.

    [18]

    R. Killip and M. Visan, "Nonlinear Schrodinger Equations at Critical Regularity," Clay Lecture Notes 2009. Available from: http://www.math.ucla.edu/~ visan/lecturenotes.html.

    [19]

    J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, Journal of Functional Analysis, 30 (1978), 245-263.doi: 10.1016/0022-1236(78)90073-3.

    [20]

    T. Roy, Adapted linear - nonlinear decomposition and global well - posedness for solutions to the defocusing cubic wave equation on $\mathbbR^3$, Discrete and Continuous Dynamical Systems. Series A., 24 (2009), 1307-1323.doi: 10.3934/dcds.2009.24.1307.

    [21]

    E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, NJ, 1970.

    [22]

    E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, NJ, 1993.

    [23]

    W. Strauss, "Nonlinear Wave Equations," CBMS Regional Conference Series in Mathematics, 73. American Mathematical Society, Providence, RI, 1989.

    [24]

    R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Mathematical Journal, 44 (1977), 705-714.

    [25]

    Q. Su, Global well - posedness and scattering for the defocusing, cubic NLS in $\mathbbR^3$, Mathematical Research Letters, 19 (2012), 431-451.

    [26]

    T. Tao, "Nonlinear Dispersive Equations," CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006

    [27]

    M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhäuser, Boston, 1991.doi: 10.1007/978-1-4612-0431-2.

    [28]

    M. E. Taylor, "Partial Differential Equations I - III," Springer-Verlag, New York, 1996.doi: 10.1007/978-1-4684-9320-7.

    [29]

    M. E. Taylor, "Tools for PDE," American Mathematical Society, Mathematical Surveys and Monographs, 31 Providence, RI, 2000.

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