@article{22028,
  abstract     = {We prove global well-posedness of the Korteweg–de Vries equation for
initial data in the space H^−1(R). This is sharp in the class of H^s(R) spaces.
Even local well-posedness was previously unknown for s < −3/4. The proof
is based on the introduction of a new method of general applicability for the
study of low-regularity well-posedness for integrable PDE, informed by the
existence of commuting flows. In particular, as we will show, completely
parallel arguments give a new proof of global well-posedness for KdV with
periodic H−1 data, shown previously by Kappeler and Topalov, as well as
global well-posedness for the fifth order KdV equation in L^2(R).
Additionally, we give a new proof of the a priori local smoothing bound
of Buckmaster and Koch for KdV on the line. Moreover, we upgrade this
estimate to show that convergence of initial data in H^−1(R) guarantees
convergence of the resulting solutions in L^2loc(R × R). Thus, solutions with
H^−1(R) initial data are distributional solutions.},
  author       = {Killip, Rowan and Visan, Monica},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {1},
  pages        = {249--305},
  publisher    = {Annals of Mathematics},
  title        = {{KdV is well-posed in H^-1}},
  doi          = {10.4007/annals.2019.190.1.4},
  volume       = {190},
  year         = {2019},
}