The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher

Abstract

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iu t Cu D ˙juj 4=d u for large spherically symmetric L 2x ‫ޒ.‬ d / initial data in dimensions d 3.In the focusing case we require that the mass is strictly less than that of the ground state.As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.4 d u (1-1) where u is a complex-valued function of spacetime ‫ޒ‬ ‫ޒ‬ d .Here D ˙1, with D 1 known as the defocusing equation and D 1 as the focusing equation.The name "mass-critical" refers to the fact that the scaling symmetry u.t; x/ 7 !u .t;x/ WD d 2 u. 2 t; 1 x/ leaves both the equation and the mass invariant.The mass of a solution isand is conserved under the flow.In this paper, we investigate the Cauchy problem for (1-1) for spherically symmetric L 2 x ‫ޒ.‬ d / initial data in dimensions d 3 by adapting the recent argument from [Killip et al. 2007], which treated the case d D 2. Before describing our results, we need to review some background material.We begin by making the notion of a solution more precise: MSC2000: 35Q55.

Locations

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• arXiv (Cornell University) - PDF
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