# American Institute of Mathematical Sciences

2013, 33(5): 1905-1926. doi: 10.3934/dcds.2013.33.1905

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

 1 970 Evans Hall, number 3840, UC Berkeley mathematics, Berkeley, CA 94720-3840, United States

Received  October 2011 Revised  September 2012 Published  December 2012

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.
Citation: Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
##### References:
 [1] J. Bourgain, Scattering in the energy space and below in 3D NLS,, Journal d'Analyse Mathematique, 4 (1998), 267. 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. doi: 10.1155/S1073792898000191. [3] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (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. 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. 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. [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. doi: 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. doi: 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. 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. doi: 10.3934/cpaa.2011.10.397. [12] B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d = 2$,, preprint, (). [13] J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations,, Communications in Mathematical Physics, 144 (1992), 163. [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. [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. doi: 10.1155/S107379289800066X. [16] M. Keel and T. Tao, Endpoint strichartz estimates,, American Journal of Mathematics, 120 (1998), 945. [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. 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: , (2009). [19] J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation,, Journal of Functional Analysis, 30 (1978), 245. 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. doi: 10.3934/dcds.2009.24.1307. [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). [22] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton University Press, (1993). [23] W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (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. [25] Q. Su, Global well - posedness and scattering for the defocusing, cubic NLS in $\mathbbR^3$,, Mathematical Research Letters, 19 (2012), 431. [26] T. Tao, "Nonlinear Dispersive Equations,", CBMS Regional Conference Series in Mathematics, 106 (2006). [27] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2. [28] M. E. Taylor, "Partial Differential Equations I - III,", Springer-Verlag, (1996). doi: 10.1007/978-1-4684-9320-7. [29] M. E. Taylor, "Tools for PDE,", American Mathematical Society, 31 (2000).

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##### References:
 [1] J. Bourgain, Scattering in the energy space and below in 3D NLS,, Journal d'Analyse Mathematique, 4 (1998), 267. 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. doi: 10.1155/S1073792898000191. [3] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (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. 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. 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. [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. doi: 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. doi: 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. 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. doi: 10.3934/cpaa.2011.10.397. [12] B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d = 2$,, preprint, (). [13] J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations,, Communications in Mathematical Physics, 144 (1992), 163. [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. [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. doi: 10.1155/S107379289800066X. [16] M. Keel and T. Tao, Endpoint strichartz estimates,, American Journal of Mathematics, 120 (1998), 945. [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. 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: , (2009). [19] J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation,, Journal of Functional Analysis, 30 (1978), 245. 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. doi: 10.3934/dcds.2009.24.1307. [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). [22] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton University Press, (1993). [23] W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (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. [25] Q. Su, Global well - posedness and scattering for the defocusing, cubic NLS in $\mathbbR^3$,, Mathematical Research Letters, 19 (2012), 431. [26] T. Tao, "Nonlinear Dispersive Equations,", CBMS Regional Conference Series in Mathematics, 106 (2006). [27] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2. [28] M. E. Taylor, "Partial Differential Equations I - III,", Springer-Verlag, (1996). doi: 10.1007/978-1-4684-9320-7. [29] M. E. Taylor, "Tools for PDE,", American Mathematical Society, 31 (2000).
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