---
OA_place: repository
OA_type: green
_id: '22049'
abstract:
- lang: eng
  text: '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.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
  full_name: Tao, Terence
  last_name: Tao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Xiaoyi
  full_name: Zhang, Xiaoyi
  last_name: Zhang
citation:
  ama: Tao T, Vişan M, Zhang X. Minimal-mass blowup solutions of the mass-critical
    NLS. <i>Forum Mathematicum</i>. 2008;20(5):881-919. doi:<a href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>
  apa: Tao, T., Vişan, M., &#38; Zhang, X. (2008). Minimal-mass blowup solutions of
    the mass-critical NLS. <i>Forum Mathematicum</i>. De Gruyter. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>
  chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “Minimal-Mass Blowup Solutions
    of the Mass-Critical NLS.” <i>Forum Mathematicum</i>. De Gruyter, 2008. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>.
  ieee: T. Tao, M. Vişan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical
    NLS,” <i>Forum Mathematicum</i>, vol. 20, no. 5. De Gruyter, pp. 881–919, 2008.
  ista: Tao T, Vişan M, Zhang X. 2008. Minimal-mass blowup solutions of the mass-critical
    NLS. Forum Mathematicum. 20(5), 881–919.
  mla: Tao, Terence, et al. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.”
    <i>Forum Mathematicum</i>, vol. 20, no. 5, De Gruyter, 2008, pp. 881–919, doi:<a
    href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>.
  short: T. Tao, M. Vişan, X. Zhang, Forum Mathematicum 20 (2008) 881–919.
das_tickbox: '1'
date_created: 2026-06-19T07:53:12Z
date_published: 2008-11-03T00:00:00Z
date_updated: 2026-06-25T08:15:22Z
day: '03'
doi: 10.1515/forum.2008.042
extern: '1'
external_id:
  arxiv:
  - math/0609690
intvolume: '        20'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0609690
mathsc:
- 35Q55
month: '11'
oa: 1
oa_version: Preprint
page: 881-919
publication: Forum Mathematicum
publication_identifier:
  eissn:
  - 1435-5337
  issn:
  - 0933-7741
publication_status: published
publisher: De Gruyter
quality_controlled: '1'
scopus_import: '1'
status: public
title: Minimal-mass blowup solutions of the mass-critical NLS
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2008'
...