KdV is well-posed in H–1

By Rowan Killip, Monica Visan. Annals of Mathematics, 2018

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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 $s<-3/4$. 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. 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 $L^2_\text{loc}(\mathbb{R}\times\mathbb{R})$. Thus, solutions with $H^{-1}(\mathbb{R})$ initial data are distributional solutions.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/743d86f9479578e3cab4021b076811e3d1cbf8b8