KdV on an incoming tide. (30th November 2021)

• Record Type: Journal Article
• Authors: Laurens, Thierry
• Is Part Of: Nonlinearity. Volume 35:Number 1(2022)
• Page Start: 343
• Page End: 387
• Publication Date: 2021-11-30
• Subjects: KdV equation -- dispersive PDEs -- integrable systems; 35Q53; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac37f5
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Given smooth step-like initial data V (0, x ) on the real line, we show that the Korteweg–de Vries equation is globally well-posed for initial data u ( 0, x ) ∈ V ( 0, x ) + H − 1 ( R ) . The proof uses our general well-posedness result (2021 arXiv:2104.11346 ). As a prerequisite, we show that KdV is globally well-posed for H 3 ( R ) perturbations of step-like initial data. In the case V ≡ 0, we obtain a new proof of the Bona–Smith theorem (Bona and Smith 1975 Trans. R. Soc. A 278 555–601) using the low-regularity methods that established the sharp well-posedness of KdV in H −1 (Killip and Vişan 2019 Ann. Math. 190 249–305).