Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

2004/02/09 by Terence Tao, Tao, Terence · 3 citations
Mathematics · Medicine · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Soft tissue tumor case studies #math.AP #msc:35Q55

paper · pdf · doi:10.48550/arxiv.math/0402130

23 pages, no figures, to appear, New York J. Math. This is the final version

openalex publication_date 2004/02/09 · arxiv created 2005/02/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In any dimension n ≥ 3, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schrödinger equation i u_t + Δu = |u|^(4)/(n-2) u in \R × \Rⁿ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions n ≥ 6 some new technical difficulties arise because of the very low power of the non-linearity.

Cited by

• Finite-mass soliton-type rigidity and four-channel reduction for the three-dimensional nonradial focusing energy-critical nonlinear Schrödinger equation
• Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes
• Energy critical fourth-order Schrödinger equation system with power-type nonlinearities in the radial case

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