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Journal of the European Mathematical Society


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Volume 11, Issue 6, 2009, pp. 1203–1258
DOI: 10.4171/JEMS/180

Published online: 2009-12-23

The cubic nonlinear Schrödinger equation in two dimensions with radial data

Rowan Killip[1], Terence Tao[2] and Monica Vișan[3]

(1) University of California Los Angeles, United States
(2) University of California Los Angeles, United States
(3) University of California Los Angeles, United States

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iut + u = ±|u|2 u for large spherically symmetric Lx2(ℝ2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.

Keywords: Nonlinear Schrödinger equation, scattering, mass-critical, virial identity

Killip Rowan, Tao Terence, Vișan Monica: The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc. 11 (2009), 1203-1258. doi: 10.4171/JEMS/180