American Institute of Mathematical Sciences

Advanced
December  2007, 6(4): 1023-1041. doi: 10.3934/cpaa.2007.6.1023

Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

Daniela De Silva 1, , Nataša Pavlović 2, , Gigliola Staffilani 3, and  Nikolaos Tzirakis 4,

1. 

Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027, United States
2. 

Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712, United States
3. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
4. 

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, United States

Received  October 2006 Revised  June 2007 Published  September 2007

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^s(\mathbb R^n )$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the "$I$-method" combined with a local in time Morawetz estimate.
Keywords: global well-posedness, Schrödinger equations, $I$-method, Morawetz estimates..
Mathematics Subject Classification: 35Q5.
Citation: Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
[1]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[2]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[3]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[4]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
[5]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[6]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[7]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37
[8]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010
[9]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
[10]

Chao Yang. Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4631-4642. doi: 10.3934/dcdss.2021136
[11]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[12]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[13]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[14]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[15]

Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57
[16]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[17]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
[18]

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144
[19]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[20]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy