@article{22049,
  abstract     = {We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 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 L2-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.},
  author       = {Tao, Terence and Visan, Monica and Zhang, Xiaoyi},
  issn         = {1435-5337},
  journal      = {Forum Mathematicum},
  number       = {5},
  pages        = {881--919},
  publisher    = {De Gruyter},
  title        = {{Minimal-mass blowup solutions of the mass-critical NLS}},
  doi          = {10.1515/forum.2008.042},
  volume       = {20},
  year         = {2008},
}