Well-posedness of a nonlinear boundary value problem for the Korteweg–de Vries equation on a bounded domain

• Published in: Journal of mathematical analysis and applications Vol. 448; no. 2; pp. 797 - 814
• Main Authors: Caicedo, Miguel Andres, Zhang, Bing-Yu
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2017
• Subjects: Korteweg–de Vries equation; Nonlinear boundary-value problem; Well-posedness; Nonlinear boundary-value problem; Korteweg–de Vries equation; Well-posedness
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2016.11.032

Considered in this article is an initial-boundary-value problem (IBVP) for the Korteweg–de Vries equationut+ux+uux+uxxx=0 posed on a finite interval I=(0,L) subject to the initial conditionu(x,0)=ϕ(x)forx∈(0,L) and the nonhomogeneous nonlinear boundary conditionsuxx(0,t)+u(0,t)−16u2(0,t)=h(t),u(L,t)=0,ux(L,t)=0fort≥0, which was derived by Rosier [37] as a model to investigate the motion of water waves in a long canal with a moving boundary at the left of the canal (wavemaker) and a fixed boundary at the right by using Lagrangian coordinates. It is shown here, aided by the hidden regularities (or sharp Kato smoothing properties) of the associated linear problem, that the IBVP is well-posed in the space Hs(0,L) for any s≥0 and thus addresses a question left open by Rosier in [37].