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

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- by Jerry L. Bona, S. M. Sun and Bing-Yu Zhang
- Trans. Amer. Math. Soc.
**354**(2002), 427-490 - DOI: https://doi.org/10.1090/S0002-9947-01-02885-9
- Published electronically: September 26, 2001
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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 \[ (0.1)\qquad \qquad \quad \left \{ \begin {array}{l} \eta _t+\eta _x+\eta \eta _x +\eta _{xxx} =0 , \quad \mbox {for} \ x, t \geq 0, \cr \ \cr \eta (x,0) = \phi (x),\qquad \quad \eta (0,t) =h(t),\end {array}\right . \qquad \qquad \qquad \quad \] 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 $s>\frac 34$ 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.## References

- T. B. Benjamin, J. L. Bona, and J. J. Mahony,
*Model equations for long waves in nonlinear dispersive systems*, Philos. Trans. Roy. Soc. London Ser. A**272**(1972), no. 1220, 47–78. MR**427868**, DOI 10.1098/rsta.1972.0032 - J. L. Bona and P. J. Bryant,
*A mathematical model for long waves generated by wavemakers in non-linear dispersive systems*, Proc. Cambridge Philos. Soc.**73**(1973), 391–405. MR**339651**, DOI 10.1017/s0305004100076945 - Jerry L. Bona and Min Chen,
*A Boussinesq system for two-way propagation of nonlinear dispersive waves*, Phys. D**116**(1998), no. 1-2, 191–224. MR**1621920**, DOI 10.1016/S0167-2789(97)00249-2 - J. L. Bona, M. Chen and J.-C. Saut,
*Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media*,*Part*I.*Derivation and the linear theory*. To appear. - W. H. Ruan,
*One-parameter family of invariant sets for nonweakly coupled nonlinear parabolic systems*, J. Math. Anal. Appl.**189**(1995), no. 3, 763–780. MR**1312552**, DOI 10.1006/jmaa.1995.1050 - Jerry L. Bona and Laihan Luo,
*A generalized Korteweg-de Vries equation in a quarter plane*, Applied analysis (Baton Rouge, LA, 1996) Contemp. Math., vol. 221, Amer. Math. Soc., Providence, RI, 1999, pp. 59–125. MR**1647197**, DOI 10.1090/conm/221/03118 - J. L. Bona, W. G. Pritchard, and L. R. Scott,
*An evaluation of a model equation for water waves*, Philos. Trans. Roy. Soc. London Ser. A**302**(1981), no. 1471, 457–510. MR**633485**, DOI 10.1098/rsta.1981.0178 - Jerry Bona and Ridgway Scott,
*Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces*, Duke Math. J.**43**(1976), no. 1, 87–99. MR**393887** - J. L. Bona and R. Smith,
*The initial-value problem for the Korteweg-de Vries equation*, Philos. Trans. Roy. Soc. London Ser. A**278**(1975), no. 1287, 555–601. MR**385355**, DOI 10.1098/rsta.1975.0035 - Jerry Bona and Ragnar Winther,
*The Korteweg-de Vries equation, posed in a quarter-plane*, SIAM J. Math. Anal.**14**(1983), no. 6, 1056–1106. MR**718811**, DOI 10.1137/0514085 - Jerry L. Bona and Ragnar Winther,
*The Korteweg-de Vries equation in a quarter plane, continuous dependence results*, Differential Integral Equations**2**(1989), no. 2, 228–250. MR**984190** - Jerry L. Bona and Bing-Yu Zhang,
*The initial-value problem for the forced Korteweg-de Vries equation*, Proc. Roy. Soc. Edinburgh Sect. A**126**(1996), no. 3, 571–598. MR**1396279**, DOI 10.1017/S0308210500022915 - J. Bourgain,
*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal.**3**(1993), no. 2, 107–156. MR**1209299**, DOI 10.1007/BF01896020 - J. Bourgain,
*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal.**3**(1993), no. 2, 107–156. MR**1209299**, DOI 10.1007/BF01896020 - B. A. Bubnov,
*Solvability in the large of nonlinear boundary value problems for the Korteweg-de Vries equation in a bounded domain*, Differentsial′nye Uravneniya**16**(1980), no. 1, 34–41, 186 (Russian). MR**559805** - Amy Cohen Murray,
*Solutions of the Korteweg-de Vries equation from irregular data*, Duke Math. J.**45**(1978), no. 1, 149–181. MR**470533** - Amy Cohen,
*Existence and regularity for solutions of the Korteweg-de Vries equation*, Arch. Rational Mech. Anal.**71**(1979), no. 2, 143–175. MR**525222**, DOI 10.1007/BF00248725 - T. Colin and J.-M. Ghidaglia,
*An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval*, to appear in Adv. Diff. Eq.. - P. Constantin and J.-C. Saut,
*Local smoothing properties of dispersive equations*, J. Amer. Math. Soc.**1**(1988), no. 2, 413–439. MR**928265**, DOI 10.1090/S0894-0347-1988-0928265-0 - W. Craig, T. Kappeler, and W. Strauss,
*Gain of regularity for equations of KdV type*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**9**(1992), no. 2, 147–186 (English, with French summary). MR**1160847**, DOI 10.1016/S0294-1449(16)30243-8 - Theodore E. Dushane,
*On existence and uniqueness for a new class of nonlinear partial differential equations using compactness methods and differential difference schemes*, Trans. Amer. Math. Soc.**188**(1974), 77–96. MR**338585**, DOI 10.1090/S0002-9947-1974-0338585-5 - A. V. Faminskiĭ,
*The Cauchy problem and the mixed problem in the half strip for equations of Korteweg-de Vries type*, Dinamika Sploshn. Sredy**63**(1983), 152–158, 162 (Russian). MR**809894** - A. V. Faminskii,
*A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations*, (Russian) Dinamika Sploshn. Sredy**258**(1988), 54–94. - A. S. Fokas and B. Pelloni,
*The solution of certain initial boundary-value problems for the linearized Korteweg-de Vries equation*, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.**454**(1998), no. 1970, 645–657. MR**1638297**, DOI 10.1098/rspa.1998.0178 - A. S. Fokas and A. R. Its,
*Integrable equations on the half-infinite line*, Chaos Solitons Fractals**5**(1995), no. 12, 2367–2376. Solitons in science and engineering: theory and applications. MR**1368225**, DOI 10.1016/0960-0779(94)E0105-X - A. S. Fokas and A. R. It⋅s,
*An initial-boundary value problem for the Korteweg-de Vries equation*, Math. Comput. Simulation**37**(1994), no. 4-5, 293–321. Solitons, nonlinear wave equations and computation (New Brunswick, NJ, 1992). MR**1308105**, DOI 10.1016/0378-4754(94)00021-2 - A. S. Fokas and A. R. It⋅s,
*Soliton generation for initial-boundary value problems*, Phys. Rev. Lett.**68**(1992), no. 21, 3117–3120. MR**1163545**, DOI 10.1103/PhysRevLett.68.3117 - J. Ginibre and Y. Tsutsumi,
*Uniqueness of solutions for the generalized Korteweg-de Vries equation*, SIAM J. Math. Anal.**20**(1989), no. 6, 1388–1425. MR**1019307**, DOI 10.1137/0520091 - J. Ginibre, Y. Tsutsumi, and G. Velo,
*Existence and uniqueness of solutions for the generalized Korteweg de Vries equation*, Math. Z.**203**(1990), no. 1, 9–36. MR**1030705**, DOI 10.1007/BF02570720 - J. Ginibre and G. Velo,
*Smoothing properties and retarded estimates for some dispersive evolution equations*, Comm. Math. Phys.**144**(1992), no. 1, 163–188. MR**1151250** - C. J. Everett Jr.,
*Annihilator ideals and representation iteration for abstract rings*, Duke Math. J.**5**(1939), 623–627. MR**13** - J. L. Hammack,
*A note on tsunamis: their generation and propagation in an ocean of uniform depth*, J. Fluid Mech.**60**(1973), 769–799. - Joseph L. Hammack and Harvey Segur,
*The Korteweg-de Vries equation and water waves. II. Comparison with experiments*, J. Fluid Mech.**65**(1974), 289–313. MR**366198**, DOI 10.1017/S002211207400139X - Josef Mall,
*Ein Satz über die Konvergenz von Kettenbrüchen*, Math. Z.**45**(1939), 368–376 (German). MR**40**, DOI 10.1007/BF01580290 - T. Kato,
*Quasilinear equations of evolution, with applications to partial differential equations*, Springer Lecture Notes in Math.**448**(1975), 27–50. - Tosio Kato,
*On the Korteweg-de Vries equation*, Manuscripta Math.**28**(1979), no. 1-3, 89–99. MR**535697**, DOI 10.1007/BF01647967 - Tosio Kato,
*The Cauchy problem for the Korteweg-de Vries equation*, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979) Res. Notes in Math., vol. 53, Pitman, Boston, Mass.-London, 1981, pp. 293–307. MR**631399** - Tosio Kato,
*On the Cauchy problem for the (generalized) Korteweg-de Vries equation*, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR**759907** - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*On the (generalized) Korteweg-de Vries equation*, Duke Math. J.**59**(1989), no. 3, 585–610. MR**1046740**, DOI 10.1215/S0012-7094-89-05927-9 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Oscillatory integrals and regularity of dispersive equations*, Indiana Univ. Math. J.**40**(1991), no. 1, 33–69. MR**1101221**, DOI 10.1512/iumj.1991.40.40003 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Well-posedness of the initial value problem for the Korteweg-de Vries equation*, J. Amer. Math. Soc.**4**(1991), no. 2, 323–347. MR**1086966**, DOI 10.1090/S0894-0347-1991-1086966-0 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle*, Comm. Pure Appl. Math.**46**(1993), no. 4, 527–620. MR**1211741**, DOI 10.1002/cpa.3160460405 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices*, Duke Math. J.**71**(1993), no. 1, 1–21. MR**1230283**, DOI 10.1215/S0012-7094-93-07101-3 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*A bilinear estimate with applications to the KdV equation*, J. Amer. Math. Soc.**9**(1996), no. 2, 573–603. MR**1329387**, DOI 10.1090/S0894-0347-96-00200-7 - S. N. Kruzhkov and A. V. Faminskiĭ,
*Generalized solutions of the Korteweg-de Vries equation*, Dokl. Akad. Nauk SSSR**261**(1981), no. 6, 1296–1298 (Russian). MR**640840** - Alfred Rosenblatt,
*Sur les points singuliers des équations différentielles*, C. R. Acad. Sci. Paris**209**(1939), 10–11 (French). MR**85** - J.-L. Lions and E. Magenes,
*Non-homogeneous boundary value problems and applications. Vol. I*, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR**0350177** - Robert M. Miura,
*The Korteweg-de Vries equation: a survey of results*, SIAM Rev.**18**(1976), no. 3, 412–459. MR**404890**, DOI 10.1137/1018076 - David L. Russell and Bing Yu Zhang,
*Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation*, J. Math. Anal. Appl.**190**(1995), no. 2, 449–488. MR**1318405**, DOI 10.1006/jmaa.1995.1087 - Robert L. Sachs,
*Classical solutions of the Korteweg-de Vries equation for nonsmooth initial data via inverse scattering*, Comm. Partial Differential Equations**10**(1985), no. 1, 29–98. MR**773211**, DOI 10.1080/03605308508820371 - J.-C. Saut,
*Applications de l’interpolation non linéaire à des problèmes d’évolution non linéaires*, J. Math. Pures Appl. (9)**54**(1975), 27–52 (French). MR**454374** - J. C. Saut and R. Temam,
*Remarks on the Korteweg-de Vries equation*, Israel J. Math.**24**(1976), no. 1, 78–87. MR**454425**, DOI 10.1007/BF02761431 - Anders Sjöberg,
*On the Korteweg-de Vries equation: existence and uniqueness*, J. Math. Anal. Appl.**29**(1970), 569–579. MR**410135**, DOI 10.1016/0022-247X(70)90068-5 - Per Sjölin,
*Regularity of solutions to the Schrödinger equation*, Duke Math. J.**55**(1987), no. 3, 699–715. MR**904948**, DOI 10.1215/S0012-7094-87-05535-9 - Elias M. Stein and Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR**0304972** - S. M. Sun,
*The Korteweg-de Vries equation on a periodic domain with singular-point dissipation*, SIAM J. Control Optim.**34**(1996), no. 3, 892–912. MR**1384958**, DOI 10.1137/S0363012994269491 - L. Tartar,
*Interpolation non linéaire et régularité*, J. Functional Analysis**9**(1972), 469–489 (French). MR**0310619**, DOI 10.1016/0022-1236(72)90022-5 - R. Temam,
*Sur un problème non linéaire*, J. Math. Pures Appl. (9)**48**(1969), 159–172 (French). MR**261183** - Michael Mudi Tom,
*Smoothing properties of some weak solutions of the Benjamin-Ono equation*, Differential Integral Equations**3**(1990), no. 4, 683–694. MR**1044213** - Yoshio Tsutsumi,
*The Cauchy problem for the Korteweg-de Vries equation with measures as initial data*, SIAM J. Math. Anal.**20**(1989), no. 3, 582–588. MR**990865**, DOI 10.1137/0520041 - Luis Vega,
*Schrödinger equations: pointwise convergence to the initial data*, Proc. Amer. Math. Soc.**102**(1988), no. 4, 874–878. MR**934859**, DOI 10.1090/S0002-9939-1988-0934859-0 - N. J. Zabusky and C. J. Galvin,
*Shallow-water waves, the Korteweg-de Vries equation and solitons*, J. Fluid Mech.**47**(1971), 811–824. - Bing Yu Zhang,
*Boundary stabilization of the Korteweg-de Vries equation*, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993) Internat. Ser. Numer. Math., vol. 118, Birkhäuser, Basel, 1994, pp. 371–389. MR**1313527** - Bing Yu Zhang,
*Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values*, J. Funct. Anal.**129**(1995), no. 2, 293–324. MR**1327180**, DOI 10.1006/jfan.1995.1052 - Bing Yu Zhang,
*Analyticity of solutions of the generalized Kortweg-de Vries equation with respect to their initial values*, SIAM J. Math. Anal.**26**(1995), no. 6, 1488–1513. MR**1356456**, DOI 10.1137/S0036141093242600 - Bing Yu Zhang,
*A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain*, Differential Integral Equations**8**(1995), no. 5, 1191–1204. MR**1325553**

## Bibliographic 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
- MR Author ID: 310235
- Email: bzhang@math.uc.edu
- 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. - © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1862556