# American Institute of Mathematical Sciences

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] 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

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
 [1] Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021 [2] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 [3] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [4] Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019 [5] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376 [6] Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024 [7] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [8] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [9] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [10] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [11] Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 [12] Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013 [13] Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100 [14] Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030 [15] Yosra Soussi. Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021022 [16] Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047 [17] Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014 [18] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [19] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [20] Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

2019 Impact Factor: 1.338

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS