 

Terence Tao
Global wellposedness and scattering for the higherdimensional energycritical nonlinear Schrödinger equation for radial data


Published: 
February 28, 2005

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


Abstract
In any dimension n ≧ 3, we show that spherically symmetric bounded energy solutions
of the defocusing energycritical nonlinear Schrödinger
equation
i u_{t} + Δ u = u^{4/(n2)} u
in R × R^{n}
exist globally and scatter to free solutions;
this generalizes the three and fourdimensional 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 towertype bounds of Bourgain.
In higher dimensions n ≧ 6 some new technical difficulties arise because of the
very low power of the nonlinearity.


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


Author information
Department of Mathematics, UCLA, Los Angeles CA 900951555
tao@math.ucla.edu
http://www.math.ucla.edu/~tao/

