Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

Kramer, Eugene; Rivas, Ivonne; Zhang, Bing-Yu

doi:10.1051/cocv/2012012

Kramer, Eugene ; Rivas, Ivonne ; Zhang, Bing-Yu

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 358-384.

Résumé

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0,L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev space H^s(0,L) for s > -3/4, which provides a positive answer to one of the open questions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001) 1463-1492].

MR Zbl

DOI : 10.1051/cocv/2012012

Classification : 35Q53
Mots clés : The kortweg-de Vries equation, well-posedness, non-homogeneous boundary value problem

@article{COCV_2013__19_2_358_0,
     author = {Kramer, Eugene and Rivas, Ivonne and Zhang, Bing-Yu},
     title = {Well-posedness of a class of non-homogeneous boundary value problems of the {Korteweg-de} {Vries} equation on a finite domain},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {358--384},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012012},
     mrnumber = {3049715},
     zbl = {1273.35238},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012012/}
}

TY  - JOUR
AU  - Kramer, Eugene
AU  - Rivas, Ivonne
AU  - Zhang, Bing-Yu
TI  - Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 358
EP  - 384
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012012/
DO  - 10.1051/cocv/2012012
LA  - en
ID  - COCV_2013__19_2_358_0
ER  -

%0 Journal Article
%A Kramer, Eugene
%A Rivas, Ivonne
%A Zhang, Bing-Yu
%T Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 358-384
%V 19
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2012012/
%R 10.1051/cocv/2012012
%G en
%F COCV_2013__19_2_358_0

Kramer, Eugene; Rivas, Ivonne; Zhang, Bing-Yu. Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 358-384. doi : 10.1051/cocv/2012012. http://www.numdam.org/articles/10.1051/cocv/2012012/

Bibliographie
Cité par

[1] T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Proc. R. Soc. London A 272 (1972) 47-78. | MR | Zbl

[2] 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) 457-510. | MR | Zbl

[3] J.L. Bona, S.M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane. Trans. Amer. Math. Soc. 354 (2002) 427-490. | MR | Zbl

[4] J.L. Bona, S.M. Sun and B.-Y. Zhang, Forced oscillations of a damped korteweg-de Vries equation in a quarter plane. Commun. Partial Differ. Equ. 5 (2003) 369-400. | MR | Zbl

[5] J.L. Bona, S.M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation on a finite domain. Commun. Partial Differ. Equ. 28 (2003) 1391-1436. | MR | Zbl

[6] J.L. Bona, S.M. Sun and B.-Y. Zhang, Conditional and unconditional well posedness of nonlinear evolution equations. Adv. Differ. Equ. 9 (2004) 241-265. | MR | Zbl

[7] J.L. Bona, S.M. Sun and B.-Y. Zhang, Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications. Dyn. Partial Differ. Equ. 3 (2006) 1-69. | MR | Zbl

[8] J.L. Bona, S.M. Sun and B.-Y. Zhang, Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane. Ann. Henri Poincaré 25 (2008) 1145-1185. | Numdam | MR | Zbl

[9] J.L. Bona, S.M. Sun and B.-Y. Zhang, Nonhomogeneous problem for the Korteweg-de Vries equation in a bounded domain II. J. Differ. Equ. 247 (2009) 2558-2596. | MR | Zbl

[10] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I : Shrödinger equations. Geom. Funct. Anal. 3 (1993) 107-156. | MR | Zbl

[11] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II : the KdV-equation. Geom. Funct. Anal. 3 (1993) 209-262. | MR | Zbl

[12] B.A. Bubnov, Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain. Differ. Equ. 15 (1979) 17-21. | MR | Zbl

[13] B.A. Bubnov, Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations. Differ. Equ. 16 (1980) 24-30. | MR | Zbl

[14] E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877-899. | MR | Zbl

[15] E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Henri Poincaré 26 (2009) 457-475. | Numdam | MR | Zbl

[16] T. Colin and J.-M. Ghidaglia, Un problème aux limites pour l'équation de Korteweg-de Vries sur un intervalle borné (French) [A boundary value problem for the Korteweg-de Vries equation on a bounded interval] Journées équations aux Drives Partielles, Saint-Jean-de-Monts, Exp. No. III, École Polytech., Palaiseau (1997), p. 10. | Numdam | MR | Zbl

[17] T. Colin and J.-M. Ghidaglia, Un problème mixte pour l'équation de Korteweg-de Vries sur un intervalle borné (French) [A mixed initial-boundary value problem for the Korteweg-de Vries equation on a bounded interval]. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 599-603. | MR | Zbl

[18] T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem fo the Korteweg-de Vries equation posed on a finite interval. Adv. Differ. Equ. 6 (2001) 1463-1492. | MR | Zbl

[19] T. Colin and M. Gisclon, An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation. Nonlinear Anal. 46 (2001) 869-892. | MR | Zbl

[20] J.E. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line. Commun. Partial Differ. Equ. 27 (2002) 2187-2266. | MR | Zbl

[21] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length. J. Eur. Math. Soc. 6 (2004) 367-398. | MR | Zbl

[22] A.V. Faminskii, On an initial boundary value problem in a bounded domain for the generalized Korteweg-de Vries equation, International Conference on Differential and Functional Differential Equations (Moscow, 1999). Funct. Differ. Equ. 8 (2001) 183-194. | MR | Zbl

[23] A.V. Faminskii, Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation. Differ. Integral Equ. 20 (2007) 601-642. | MR | Zbl

[24] J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time. J. Differ. Equ. 74 (1988) 369-390. | Zbl

[25] J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations. J. Differ. Equ. 110 (1994) 356-359. | MR | Zbl

[26] J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation. Commun. Partial Differ. Equ. 31 (2006) 1151-1190. | MR | Zbl

[27] T. Kappeler and P. Topalov, Global well-posedness of KdV in H-1(T,R). Duke Math. J. 135 (2006) 327-360. | MR | Zbl

[28] T. Kato, On the Korteweg-de Vries equation. Manuscr. Math. 28 (1979) 89-99. | Zbl

[29] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Advances in Mathematics Supplementary Studies, Stud. Appl. Math. 8 (1983) 93-128. | MR | Zbl

[30] C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. 4 (1991) 323-347. | Zbl

[31] C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equations via the contraction principle. Commun. Pure Appl. Math. 46 (1993) 527-620 | MR | Zbl

[32] C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applicatios to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573-603. | MR | Zbl

[33] V. Komornik, D.L. Russell and B.-Y. Zhang, Stabilization de l'equation de Korteweg-de Vries. C. R. Acad. Sci. Paris 312 (1991) 841-843. | MR | Zbl

[34] E.F. Kramer and B.-Y. Zhang, Nonhomogeneous boundary value problems for the Korteweg-de Vries equation on a bounded domain. J. Syst. Sci. Complex 23 (2010) 499-526. | MR | Zbl

[35] L. Monilet, A note on ill-posedness for the KdV equation. Differ. Integral Equ. 24 (2011) 759-765. | MR | Zbl

[36] L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation. Int. Math. Res. Not. (2002) 1979-2005. | MR | Zbl

[37] L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation : the periodic case. arXiv:10054805V1[Math AP] (2010). | MR | Zbl

[38] A.F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM : COCV 11 (2005) 473-486. | Numdam | MR | Zbl

[39] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983). | MR | Zbl

[40] G. Perla-Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping. Q. Appl. Math. 60 (2002) 111-129. | MR | Zbl

[41] I. Rivas, M. Usman and B.-Y. Zhang, Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-de Vries equation on a finite domain. Math. Control Rel. Fields 1 (2011) 61-81. | MR | Zbl

[42] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM : COCV 2 (1997) 33-55. | Numdam | MR | Zbl

[43] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation. SIAM J. Control Optim. 45 (2006) 927-956. | MR | Zbl

[44] L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation : recent progresses. J. Syst. Sci. Complex. 22 (2009) 647-682. | MR | Zbl

[45] D.L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993) 659-676. | MR | Zbl

[46] D.L. Russell and B.-Y. 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) 449-488. | MR | Zbl

[47] L. Tartar, Interpolation non linéaire et régularité. J. Funct. Anal. 9 (1972) 469-489. | MR | Zbl

[48] B.-Y. Zhang, Boundary stabilization of the Korteweg-de Vries equations, Proc. of International Conference on Control and Estimation of Distributed Parameter Systems : Nonlinear Phenomena. Vorau, Styria, Austria (1993). International Series of Numer. Math. 118 (1994) 371-389. | MR | Zbl

[49] B.-Y. Zhang, A remark on the Cauchy problem for the Korteweg de-Vries equation on a periodic domain. Differ. Integral Equ. 8 (1995) 1191-1204. | MR | Zbl

[50] B.-Y. Zhang, Analyticity of solutions for the generalized Korteweg de-Vries equation with respect to their initial datum. SIAM J. Math. Anal. 26 (1995) 1488-1513. | MR | Zbl

[51] B.-Y. Zhang, Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values. J. Func. Anal. 129 (1995) 293-324. | MR | Zbl

[52] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 37 (1999) 543-565. | MR | Zbl

Cité par Sources :