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

DOI:
https://doi.org/10.1090/S0002-9947-01-02885-9

Published electronically:
September 26, 2001

MathSciNet review:
1862556

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

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 in the class for and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution 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

Email:
bona@math.utexas.edu

**S. M. Sun**

Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Email:
sun@math.vt.edu

**Bing-Yu Zhang**

Affiliation:
Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221

Email:
bzhang@math.uc.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02885-9

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