Minimal-mass blowup solutions of the mass-critical NLS

We consider the minimal mass m 0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iu t + Δ u = μ| u | 4/ d u to blow up. If m 0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L 2 -critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in for the defocusing NLS in three and higher dimensions with spherically symmetric data.