New York Journal of Mathematics
Volume 11 (2005) 57-80


Terence Tao

Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data

Published: February 28, 2005
Keywords: Nonlinear Schrödinger equation, Strichartz estimates, Morawetz inequalities, spherical symmetry, energy bounds
Subject: 35Q55

In any dimension n ≧ 3, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical nonlinear Schrödinger equation
i ut + Δ u = |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, 1999a and 1999b, and Grillakis, 2000. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, improving 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 nonlinearity.


The author is a Clay Prize Fellow and is supported by the Packard Foundation.

Author information

Department of Mathematics, UCLA, Los Angeles CA 90095-1555