---
res:
  bibo_abstract:
  - 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. @eng
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Benjamin
      foaf_name: Harrop-Griffiths, Benjamin
      foaf_surname: Harrop-Griffiths
  - foaf_Person:
      foaf_givenName: Rowan
      foaf_name: Killip, Rowan
      foaf_surname: Killip
  - foaf_Person:
      foaf_givenName: Monica
      foaf_name: Visan, Monica
      foaf_surname: Visan
      foaf_workInfoHomepage: http://www.librecat.org/personId=056daca0-b8d1-11f0-964f-f91054abf8ca
  bibo_doi: 10.1017/fmp.2024.4
  bibo_volume: 12
  dct_date: 2024^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2050-5086
  dct_language: eng
  dct_publisher: Cambridge University Press@
  dct_title: Sharp well-posedness for the cubic NLS and mKdV in H^s(R)@
...