KdV is well-posed in H^-1

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.