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
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show all references

References:
[1]

Journal d'Analyse Mathematique, 4 (1998), 267-297. doi: 10.1007/BF02788703.  Google Scholar

[2]

International Mathematical Research Notices, 5 (1998), 253-283. doi: 10.1155/S1073792898000191.  Google Scholar

[3]

American Mathematical Society, American Mathematical Society Colloquium Publications, Providence, RI. 1999.  Google Scholar

[4]

Instituto de Matematica - UFRJ - Rio de Janeiro, 1996. Google Scholar

[5]

Manuscripta Math., 61 (1988), 477-494. doi: 10.1007/BF01258601.  Google Scholar

[6]

Nonlinear Analysis, 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[7]

Mathematical Research Letters, 9 (2002), 659-682.  Google Scholar

[8]

Communications on Pure and Applied Mathematics, 21 (2004), 987-1014 doi: 10.1002/cpa.20029.  Google Scholar

[9]

Annals of Mathematics. Second Series, 167 (2008), 767-865 doi: 10.4007/annals.2008.167.767.  Google Scholar

[10]

Discrete and Continuous Dynamical Systems A, 21 (2007), 665-686. doi: 10.3934/dcds.2008.21.665.  Google Scholar

[11]

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]

Communications in Mathematical Physics, 144 (1992), 163-188.  Google Scholar

[14]

Journal de Mathmatiques Pures et Appliques, 9 (1985), 363-401.  Google Scholar

[15]

International Mathematics Research Notices, 21 (1998), 1117-1156. doi: 10.1155/S107379289800066X.  Google Scholar

[16]

American Journal of Mathematics, 120 (1998), 945-957.  Google Scholar

[17]

Transactions of the American Mathematical Society, 362 (2010), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9.  Google Scholar

[18]

Clay Lecture Notes 2009. Available from: http://www.math.ucla.edu/~ visan/lecturenotes.html. Google Scholar

[19]

Journal of Functional Analysis, 30 (1978), 245-263. doi: 10.1016/0022-1236(78)90073-3.  Google Scholar

[20]

Discrete and Continuous Dynamical Systems. Series A., 24 (2009), 1307-1323. doi: 10.3934/dcds.2009.24.1307.  Google Scholar

[21]

Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[22]

Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[23]

CBMS Regional Conference Series in Mathematics, 73. American Mathematical Society, Providence, RI, 1989.  Google Scholar

[24]

Duke Mathematical Journal, 44 (1977), 705-714.  Google Scholar

[25]

Mathematical Research Letters, 19 (2012), 431-451. Google Scholar

[26]

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]

Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[28]

Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[29]

American Mathematical Society, Mathematical Surveys and Monographs, 31 Providence, RI, 2000.  Google Scholar

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