---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '22079'
abstract:
- lang: eng
  text: 'We prove that the cubic nonlinear Schrödinger equation (both focusing and
    defocusing) is globally well-posed in H^s(R) for any regularity s > −1/2. Well-posedness
    has long been known for s ≥ 0, see [55], but not previously for any s < 0. The
    scaling-critical value s = −1/2 is necessarily excluded here, since instantaneous
    norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion)
    well-posedness of the real- and complex-valued modified Korteweg–de Vries equations
    in H^s(R) for any s > −1/2. The best regularity achieved previously was s ≥ 1/4
    (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution
    map, we employ the method of commuting flows introduced in [37]. In stark contrast
    with our arguments in [37], an essential ingredient in this paper is the demonstration
    of a local smoothing effect for both equations. Despite the nonperturbative nature
    of the well-posedness, the gain of derivatives matches that of the underlying
    linear equation. To compensate for the local nature of the smoothing estimates,
    we also demonstrate tightness of orbits. The proofs of both local smoothing and
    tightness rely on our discovery of a new one-parameter family of coercive microscopic
    conservation laws that remain meaningful at this low regularity. '
article_number: e6
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Benjamin
  full_name: Harrop-Griffiths, Benjamin
  last_name: Harrop-Griffiths
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Harrop-Griffiths B, Killip R, Vişan M. Sharp well-posedness for the cubic NLS
    and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. 2024;12. doi:<a href="https://doi.org/10.1017/fmp.2024.4">10.1017/fmp.2024.4</a>
  apa: Harrop-Griffiths, B., Killip, R., &#38; Vişan, M. (2024). Sharp well-posedness
    for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/fmp.2024.4">https://doi.org/10.1017/fmp.2024.4</a>
  chicago: Harrop-Griffiths, Benjamin, Rowan Killip, and Monica Vişan. “Sharp Well-Posedness
    for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>. Cambridge
    University Press, 2024. <a href="https://doi.org/10.1017/fmp.2024.4">https://doi.org/10.1017/fmp.2024.4</a>.
  ieee: B. Harrop-Griffiths, R. Killip, and M. Vişan, “Sharp well-posedness for the
    cubic NLS and mKdV in H^s(R),” <i>Forum of Mathematics, Pi</i>, vol. 12. Cambridge
    University Press, 2024.
  ista: Harrop-Griffiths B, Killip R, Vişan M. 2024. Sharp well-posedness for the
    cubic NLS and mKdV in H^s(R). Forum of Mathematics, Pi. 12, e6.
  mla: Harrop-Griffiths, Benjamin, et al. “Sharp Well-Posedness for the Cubic NLS
    and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>, vol. 12, e6, Cambridge University
    Press, 2024, doi:<a href="https://doi.org/10.1017/fmp.2024.4">10.1017/fmp.2024.4</a>.
  short: B. Harrop-Griffiths, R. Killip, M. Vişan, Forum of Mathematics, Pi 12 (2024).
das_tickbox: '1'
date_created: 2026-06-19T08:26:10Z
date_published: 2024-04-02T00:00:00Z
date_updated: 2026-06-30T12:16:50Z
day: '02'
ddc:
- '500'
doi: 10.1017/fmp.2024.4
extern: '1'
external_id:
  arxiv:
  - '2003.05011'
has_accepted_license: '1'
intvolume: '        12'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1017/fmp.2024.4
mathsc:
- 35Q55
- 35Q53
- 37K10
month: '04'
oa: 1
oa_version: Published Version
publication: Forum of Mathematics, Pi
publication_identifier:
  eissn:
  - 2050-5086
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sharp well-posedness for the cubic NLS and mKdV in H^s(R)
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 12
year: '2024'
...