KdV is well-posed in H-1

• 成果类型: Article
• 署名作者: Killip, Rowan; Visan, Monica
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.190.1.4
• 发表日期: 2019
• 页码: 249-305
• 关键词: de-vries equation; korteweg-devries equation; ill-posedness; cauchy-problem; SCATTERING; Operators; spectrum; mkdv

摘要

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-loc(2)(R x R). Thus, solutions with H-1 (R) initial data are distributional solutions.