KdV is well-posed in H^-1

article KdV is well-posed in H^-1 published yes Rowan Killip author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca 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. Annals of Mathematics2019 eng Annals of Mathematics 0003-486X 1802.0485110.4007/annals.2019.190.1.4 1901249-305 yes R. Killip and M. Vişan, “KdV is well-posed in H^-1,” Annals of Mathematics, vol. 190, no. 1. Annals of Mathematics, pp. 249–305, 2019. Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” Annals of Mathematics, 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>. Killip, R., & Vişan, M. (2019). KdV is well-posed in H^-1. Annals of Mathematics. 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> Killip R, Vişan M. KdV is well-posed in H^-1. Annals of Mathematics. 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> Killip R, Vişan M. 2019. KdV is well-posed in H^-1. Annals of Mathematics. 190(1), 249–305. Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” Annals of Mathematics. 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>. R. Killip, M. Vişan, Annals of Mathematics 190 (2019) 249–305. 220282026-06-19T07:37:37Z2026-06-22T11:12:40Z