Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions

DOI10.1215/S0012-7094-07-14015-8zbMath1187.35246arXivmath/0609692OpenAlexW2133328036WikidataQ106312491 ScholiaQ106312491MaRDI QIDQ2458992

Xiaoyi Zhang, Terence C. Tao, Monica Visan

Publication date: 5 November 2007

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0609692

zbMATH Keywords

scatteringglobal well-posednessmass-critical nonlinear Schrödinger equation

Mathematics Subject Classification ID

Scattering theory for PDEs (35P25) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) NLS equations (nonlinear Schrödinger equations) (35Q55) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Blow-up in context of PDEs (35B44)

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Cites Work

• Unnamed Item
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• Unnamed Item
• L\({}^ 2\) concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity
• Scattering theory in the energy space for a class of nonlinear Schrödinger equations
• Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrö\-dinger equation in the radial case
• The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension.
• The nonlinear Schrödinger equation. Self-focusing and wave collapse
• A sharp bilinear restriction estimate for paraboloids
• On nonlinear Schrödinger equations. II: \(H^ S\)-solutions and unconditional well-posedness
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• On the blow up phenomenon of the critical nonlinear Schrödinger equation
• Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrö\-dinger equation for radial data
• Existence of blow-up solutions in the energy space for the critical generalized KdV equation
• On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case
• The cauchy problem for the critical nonlinear Schrödinger equation in Hs
• Minimal-mass blowup solutions of the mass-critical NLS
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4
• Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation
• On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations
• Endpoint Strichartz estimates
• Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case
• The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities
• Time decay for the nonlinear Klein-Gordon equation
• Regularity of solutions to the free Schrödinger equation with radial initial data

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