A non-homogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain II

• Published in: Journal of Differential Equations 2009-11, Vol.247 (9), p.2558-2596
• Main Authors: Bona, Jerry L., Sun, Shu Ming, Zhang, Bing-Yu
• Format: Article
• Language: English
• Subjects: Boundary smoothing; Initial–boundary-value problem; Korteweg–de Vries equation; Well-posedness
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2009.07.010

Studied here is an initial- and boundary-value problem for the Korteweg–de Vries equation ∂ u ∂ t + ∂ u ∂ x + u ∂ u ∂ x + ∂ 3 u ∂ x 3 = 0 , posed on a bounded interval I = { x : a ⩽ x ⩽ b } . This problem features non-homogeneous boundary conditions applied at x = a and x = b and is known to be well-posed in the L 2 -based Sobolev space H s ( I ) for any s > − 3 4 . It is shown here that this initial–boundary-value problem is in fact well-posed in H s ( I ) for any s > − 1 . Moreover, the solution map that associates the solution to the auxiliary data is not only continuous, but also analytic between the relevant function classes. The improvement on the previous theory comes about because of a more exacting appreciation of the damping that is inherent in the imposition of the boundary conditions.