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).

show all references

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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