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Transactions of the American Mathematical Society

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A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

Authors: Jerry L. Bona, S. M. Sun and Bing-Yu Zhang
Journal: Trans. Amer. Math. Soc. 354 (2002), 427-490
MSC (2000): Primary 35Q53; Secondary 76B03, 76B15
Published electronically: September 26, 2001
MathSciNet review: 1862556
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Abstract: The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

\begin{displaymath}(0.1)\qquad\qquad\quad \left\{ \begin{array}{l} \eta _t+\eta ... ...uad \eta(0,t) =h(t),\end{array}\right. \qquad\qquad\qquad\quad \end{displaymath}

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data $\phi$ in the class $H^s(R^+)$ for $s>\frac34$ and boundary data $h$ in $H^{(1+s)/3}_{loc} (R^+)$, whereas global well-posedness is shown to hold for $\phi \in H^s (R^+) , h\in H^{\frac{7+3s}{12}}_{loc} (R^+) $ when $1\leq s\leq 3$, and for $\phi \in H^s(R^+) , h\in H^{(s+1)/3}_{loc} (R^+) $ when $s\geq 3$. In addition, it is shown that the correspondence that associates to initial data $\phi $and boundary data $h$ the unique solution $u$ of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

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Additional Information

Jerry L. Bona
Affiliation: Department of Mathematics, Texas Institute for Computational and Applied Mathematics, University of Texas, Austin, Texas 78712
Address at time of publication: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607

S. M. Sun
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Bing-Yu Zhang
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221

Keywords: Korteweg-de Vries equation, KdV equation in a quarter plane, non-homogeneous problems, well-posedness
Received by editor(s): April 19, 2000
Received by editor(s) in revised form: January 8, 2001
Published electronically: September 26, 2001
Additional Notes: JLB was partially supported by the National Science Foundation and by the W. M. Keck Foundation.
SMS was partially supported by National Science Foundation grant DMS-9971764.
BYZ was partially supported by a Taft Competitive Faculty Fellowship. Part of the work was done while BYZ was a Research Fellow of the Texas Institute for Computational and Applied Mathematics at the University of Texas at Austin.
The line of argument in Section 3 reflects a very helpful suggestion by a referee, for which the authors are grateful.
Article copyright: © Copyright 2001 American Mathematical Society