The Korteweg–de Vries equation on the half-line with Robin and Neumann data in low regularity spaces. (September 2022)

• Record Type: Journal Article
• Authors: Himonas, A. Alexandrou; Yan, Fangchi
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: primary 35Q55 35G31 35G16 37K10; Korteweg–de Vries equation -- Integrability -- Initial–boundary value problems -- Robin and Neumann conditions -- Fokas unified transform method -- Well-posedness in Sobolev spaces -- Bourgain spaces -- Linear and bilinear estimates; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113008
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The well-posedness of the initial–boundary value problem (ibvp) for the Korteweg–de Vries equation on the half-line is studied for initial data u 0 ( x ) in spatial Sobolev spaces H s ( 0, ∞ ), s > − 3 / 4, and Robin and Neumann boundary data φ ( t ) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.