---
OA_place: repository
OA_type: green
_id: '22028'
abstract:
- lang: eng
  text: "We prove global well-posedness of the Korteweg–de Vries equation for\r\ninitial
    data in the space H^−1(R). This is sharp in the class of H^s(R) spaces.\r\nEven
    local well-posedness was previously unknown for s < −3/4. The proof\r\nis based
    on the introduction of a new method of general applicability for the\r\nstudy
    of low-regularity well-posedness for integrable PDE, informed by the\r\nexistence
    of commuting flows. In particular, as we will show, completely\r\nparallel arguments
    give a new proof of global well-posedness for KdV with\r\nperiodic H−1 data, shown
    previously by Kappeler and Topalov, as well as\r\nglobal well-posedness for the
    fifth order KdV equation in L^2(R).\r\nAdditionally, we give a new proof of the
    a priori local smoothing bound\r\nof Buckmaster and Koch for KdV on the line.
    Moreover, we upgrade this\r\nestimate to show that convergence of initial data
    in H^−1(R) guarantees\r\nconvergence of the resulting solutions in L^2loc(R ×
    R). Thus, solutions with\r\nH^−1(R) initial data are distributional solutions."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- 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: Killip R, Vişan M. KdV is well-posed in H^-1. <i>Annals of Mathematics</i>.
    2019;190(1):249-305. doi:<a href="https://doi.org/10.4007/annals.2019.190.1.4">10.4007/annals.2019.190.1.4</a>
  apa: Killip, R., &#38; Vişan, M. (2019). KdV is well-posed in H^-1. <i>Annals of
    Mathematics</i>. Annals of Mathematics. <a href="https://doi.org/10.4007/annals.2019.190.1.4">https://doi.org/10.4007/annals.2019.190.1.4</a>
  chicago: Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” <i>Annals
    of Mathematics</i>. Annals of Mathematics, 2019. <a href="https://doi.org/10.4007/annals.2019.190.1.4">https://doi.org/10.4007/annals.2019.190.1.4</a>.
  ieee: R. Killip and M. Vişan, “KdV is well-posed in H^-1,” <i>Annals of Mathematics</i>,
    vol. 190, no. 1. Annals of Mathematics, pp. 249–305, 2019.
  ista: Killip R, Vişan M. 2019. KdV is well-posed in H^-1. Annals of Mathematics.
    190(1), 249–305.
  mla: Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” <i>Annals of
    Mathematics</i>, vol. 190, no. 1, Annals of Mathematics, 2019, pp. 249–305, doi:<a
    href="https://doi.org/10.4007/annals.2019.190.1.4">10.4007/annals.2019.190.1.4</a>.
  short: R. Killip, M. Vişan, Annals of Mathematics 190 (2019) 249–305.
das_tickbox: '1'
date_created: 2026-06-19T07:37:37Z
date_published: 2019-07-05T00:00:00Z
date_updated: 2026-06-22T11:12:40Z
day: '05'
doi: 10.4007/annals.2019.190.1.4
extern: '1'
external_id:
  arxiv:
  - '1802.04851'
intvolume: '       190'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1802.04851
month: '07'
oa: 1
oa_version: Preprint
page: 249-305
publication: Annals of Mathematics
publication_identifier:
  issn:
  - 0003-486X
publication_status: published
publisher: Annals of Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: KdV is well-posed in H^-1
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 190
year: '2019'
...