KdV is wellposed in $H^{-1}$

Abstract

We prove global well-posedness of the Korteweg–de Vries equation for initial data in the space $H^{-1}(\Bbb{R})$. This is sharp in the class of $H^{s}(\Bbb{R})$ 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 5th order KdV equation in $L^2(\Bbb{R})$.

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

Authors

Rowan Killip

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA

Monica Vişan

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA