KdV is well-posed in $H^{-1}$

Abstract

We prove global well-posedness of the Korteweg--de Vries equation for initial data in the space $H^{-1}(\mathbb{R})$. This is sharp in the class of $H^s(\mathbb{R})$spaces. Even local well-posedness was previously unknown for $s\lt -3/4$. 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 $L^2(\mathbb{R})$.

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 $H^{-1}(\mathbb{R})$ guarantees convergence of the resulting solutions in $L^2_{\mathrm{loc}}(\mathbb{R}\times \mathbb{R})$. Thus, solutions with $H^{-1}(\mathbb{R})$ initial data are distributional solutions.