SU\G(𝔸)/K·U

Global well-posedness and scattering for the defocusing, L 2 -critical, nonlinear Schrödinger equation when d = 1

Benjamin Dodson

Type: Article

Publication Date: 2016-04-01

Citations: 81

DOI: https://doi.org/10.1353/ajm.2016.0016

Abstract

In this paper we prove global well-posedness and scattering for the defocusing, one dimensional mass-critical nonlinear Schr{\"o}dinger equation. We make use of a long-time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in earlier papers for dimensions $d\geq 3$ and $d = 2$.

Locations

  • American Journal of Mathematics - View
  • arXiv (Cornell University) - View - PDF

Similar Works

Action Title Year Authors
+ PDF Chat Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d ≥3$ 2011 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{ö}dinger equation when $d \geq 3$ 2009 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{\ 2009 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 1$ 2010 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 1$ 2010 Benjamin Dodson
+ Global Well-posedness and Scattering for the Defocusing Cubic nonlinear Schrödinger equation in Four Dimensions 2011 Monica Vişan
+ The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions 2005 Monica Vişan
+ Global well-posedness and scattering for the mass critical nonlinear Schr{ö}dinger equation with mass below the mass of the ground state 2011 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\R^{1+4}$ 2005 E. Ryckman
Monica Vişan
+ Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state 2015 Benjamin Dodson
+ Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state 2011 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>d</mml:mi><mml:mo>⩾</mml:mo><mml:mn>9</mml:mn></mml:math> 2011 Changxing Miao
Guixiang Xu
Lifeng Zhao
+ Global well - posedness and scattering for the focusing, energy - critical nonlinear Schrödinger problem in dimension $d = 4$ for initial data below a ground state threshold 2014 Benjamin Dodson
+ PDF Chat Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d=2 2016 Benjamin Dodson
+ Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{ö}dinger equation in higher dimensions 2009 Benjamin Dodson
+ PDF Chat Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4 2007 E. Ryckman
Monica Vişan
+ PDF Chat Scattering theory for the defocusing fourth-order Schrödinger equation 2016 Changxing Miao
Jiqiang Zheng
+ Stability of energy-critical nonlinear Schr\"odinger equations in high dimensions 2005 Terence Tao
Monica Vişan
+ Stability of energy-critical nonlinear Schrödinger equations in high dimensions 2005 Terence Tao
Monica Vişan
+ PDF Chat Remark on energy‐critical NLS in 5D 2015 Xing Cheng
Yanfang Gao
Jiqiang Zheng

Works That Cite This (82)

Action Title Year Authors
+ Long time behavior of stochastic NLS with a small multiplicative noise 2021 Chenjie Fan
Weijun Xu
Zehua Zhao
+ PDF Chat On scattering asymptotics for the 2D cubic resonant system 2022 Kailong Yang
Zehua Zhao
+ PDF Chat Symplectic Non-Squeezing for the Cubic NLS on the Line 2017 Rowan Killip
Monica Vişan
Xiaoyi Zhang
+ A new proof of scattering for the 5D radial focusing Hartree equation 2020 Fanfei Meng
+ PDF Chat Numerical simulations for the energy-supercritical nonlinear wave equation 2020 Jason Murphy
Yanzhi Zhang
+ Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation 2017 Luiz Gustavo Farah
Carlos M. Guzmán
+ Almost sure scattering for the energy-critical nonlinear wave equation 2019 Bjoern Bringmann
+ Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases 2023 Deng Zhang
+ The $L^{2}$ weak sequential convergence of radial mass critical NLS solutions with mass above the ground state 2016 Chenjie Fan
+ Subcritical approximations to stochastic defocusing mass-critical nonlinear Schrödinger equation on $\mathbb{R}$ 2018 Chenjie Fan
Weijun Xu

Works Cited by This (91)

Action Title Year Authors
+ Global well - posedness and scattering for the focusing, energy - critical nonlinear Schrödinger problem in dimension $d = 4$ for initial data below a ground state threshold 2014 Benjamin Dodson
+ Singular Integrals and Differentiability Properties of Functions. 1971 Elias M. Stein
+ PDF Chat Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case 1999 Jean Bourgain
+ Nonlinear Dispersive Equations 2006 Terence Tao
+ Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power 1993 Franck Merle
+ Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals 2002 Elias M. Stein
Timothy S Murphy
+ PDF Chat Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d ≥3$ 2011 Benjamin Dodson
+ Global well-posedness and scattering for the defocusing, mass - critical generalized KdV equation 2013 Benjamin Dodson
+ Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ<sup>3</sup> 2004 J. Colliander
M. Keel
Gigliola Staffilani
Hideo Takaoka
Terence Tao
+ PDF Chat Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation 2007 Pascal Bégout
Ana Vargas