May  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, 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), 267-297. doi: 10.1007/BF02788703.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 987-1014 doi: 10.1002/cpa.20029.  Google Scholar

[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), 767-865 doi: 10.4007/annals.2008.167.767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d = 2$,, preprint, ().   Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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  Google Scholar

[27]

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

[28]

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

[29]

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

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 987-1014 doi: 10.1002/cpa.20029.  Google Scholar

[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), 767-865 doi: 10.4007/annals.2008.167.767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d = 2$,, preprint, ().   Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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  Google Scholar

[27]

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

[28]

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

[29]

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

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