# American Institute of Mathematical Sciences

September & December  2018, 8(3&4): 583-605. doi: 10.3934/mcrf.2018024

## General boundary value problems of the Korteweg-de Vries equation on a bounded domain

 1 Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-545, Brazil 2 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA 3 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA 4 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Bing-Yu Zhang

The paper is dedicated to Jiongmin Yong for his 60th birthday.
The authors thank anonymous referees for many helpful comments, corrections and suggestions

Received  March 2017 Revised  October 2017 Published  September 2018

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval
 $$$u_t+u_x+u_{xxx}+uu_x = 0, ~~~~~ u(x, 0) = φ(x), ~~~~~ 0 < x < L, \ t>0~~~~~(0.1)$$$
subject to the nonhomogeneous boundary conditions,
 $$$B_1u = h_1(t), ~~~~~B_2 u = h_2 (t), ~~~~~ B_3 u = h_3 (t) ~~~~~t>0~~~~~(0.2)$$$
where
 $B_i u = \sum\limits_{j = 0}^2 \left(a_{ij} \partial ^j_x u(0, t) + b_{ij}\partial ^j_x u(L, t)\right), ~~~~~i = 1, 2, 3,$
and
 $a_{ij}, \ b_{ij}$
(
 $j = 0, 1, 2$
and
 $i = 1, 2, 3$
) are real constants. Under some general assumptions imposed on the coefficients
 $a_{ij}, \ b_{ij}$
, the IBVP (0.1)-(0.2) is shown to be locally well-posed in the space
 $H^s (0, L)$
for any
 $s \ge 0$
with
 $φ ∈ H^s (0, L)$
and boundary values
 $h_j$
,
 $j = 1, 2, 3$
, belonging to some appropriate spaces with optimal regularity.
Citation: Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control and Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
##### References:
 [1] J. L. Bona and L. R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobelev Spaces, Duke Math. J., 43 (1976), 87-99.  doi: 10.1215/S0012-7094-76-04309-X. [2] J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [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.  doi: 10.1090/S0002-9947-01-02885-9. [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. Contemp. Math., 5 (2003), 369-400.  doi: 10.1142/S021919970300104X. [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, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373. [6] J. L. Bona, S. M. Sun and B.-Y. Zhang, Conditional and unconditional well posedness of nonlinear evolution equations, Adv. Differential Equations, 9 (2004), 241-265. [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.  doi: 10.4310/DPDE.2006.v3.n1.a1. [8] J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous problem for the Korteweg-de Vries equation in a bounded domain Ⅱ, J. Diff. Eq., 247 (2009), 2558-2596.  doi: 10.1016/j.jde.2009.07.010. [9] 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.  doi: 10.1007/BF01896020. [10] 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.  doi: 10.1007/BF01895688. [11] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math., 3 (1997), 115-159.  doi: 10.1007/s000290050008. [12] B. A. Bubnov, Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain, Differential Equations, 15 (1979), 17-21. [13] E. Cerpa, Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Field, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45. [14] T. Colin and J.-M. Ghidaglia, Un probléme aux limites pour l'équation de Korteweg-de Vries sur un intervalle boné, (French) Journes Equations aux Drives Partielles, No. Ⅲ, école Polytech., Palaiseau, (1997), 10 pp. [15] T. Colin and J.-M. Ghidaglia, Un probléme mixte pour l'équation de Korteweg-de Vries sur un intervalle boné. (French), C. R. Acad. Sci. Paris. Sér. I Math., 324 (1997), 599-603.  doi: 10.1016/S0764-4442(99)80397-8. [16] T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem fo the Korteweg-de Vries Equation posed on a finite interval, Adv. Differential Equations, 6 (2001), 1463-1492. [17] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1. [18] A. V. Faminskii, The Cauchy problem and the mixed problem in the half strip for equation of Korteweg-de Vries type, Dinamika Sploshn. Sredy, 63 (1983), 152-158. [19] 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; English transl. in Trans. Moscow Math. Soc., 51 (1989), 53-91. [20] A. V. Faminskii, Mixed problms fo the Korteweg-de Vries equation, Sbornik: Mathematics, 190 (1999), 903-935.  doi: 10.1070/SM1999v190n06ABEH000408. [21] J. Holmer, The Initial-Boundary Value Problem for the Korteweg-de Vries Equation, Comm. Partial Differential Equations, 31 (2006), 1151-1190.  doi: 10.1080/03605300600718503. [22] C. Jia, I. Rivas and B.-Y. Zhang, Lower regularity solutions for a class of non-homogeneous boundary values of the Kortweg-de Vries equation on a finite domain, Adv. Differential Equations, 19 (2014), 559-584. [23] T. Kato, On the Korteweg-de Vries Equation, Manuscripta mathematica, 28 (1979), 89-99.  doi: 10.1007/BF01647967. [24] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations, Advances in Mathematics Supplementary Studies, 8 (1983), 93-128. [25] C. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9. [26] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003. [27] 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.  doi: 10.1090/S0894-0347-1991-1086966-0. [28] C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3. [29] C. Kenig, G. Ponce and L. Vega, Well-Posedness and scattering results for teh generalized Korteweg-de Vries equations via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405. [30] C. Kenig, G. Ponce and L. Vega, A Bilinear Estimate with Applicatios to the KdV Equation, J. Amer. Math. Soc., 9 (1996), 573-603. [31] 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.  doi: 10.1007/s11424-010-0143-x. [32] E. F. Kramer, I. Rivas and B.-Y. Zhang, Well-posedness of a class of non-homogeneous boundary value problem of the Korteweg-de Vries equation on a finite domain, ESAIM Control Optim. Calc. Var., 19 (2013), 358-384.  doi: 10.1051/cocv/2012012. [33] 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 Relat. Fields, 1 (2011), 61-81.  doi: 10.3934/mcrf.2011.1.61. [34] L. Tartar, Interpolation non linèaire et régularité, J. Funct. Anal., 9 (1972), 469-489.

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##### References:
 [1] J. L. Bona and L. R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobelev Spaces, Duke Math. J., 43 (1976), 87-99.  doi: 10.1215/S0012-7094-76-04309-X. [2] J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [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.  doi: 10.1090/S0002-9947-01-02885-9. [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. Contemp. Math., 5 (2003), 369-400.  doi: 10.1142/S021919970300104X. [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, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373. [6] J. L. Bona, S. M. Sun and B.-Y. Zhang, Conditional and unconditional well posedness of nonlinear evolution equations, Adv. Differential Equations, 9 (2004), 241-265. [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.  doi: 10.4310/DPDE.2006.v3.n1.a1. [8] J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous problem for the Korteweg-de Vries equation in a bounded domain Ⅱ, J. Diff. Eq., 247 (2009), 2558-2596.  doi: 10.1016/j.jde.2009.07.010. [9] 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.  doi: 10.1007/BF01896020. [10] 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.  doi: 10.1007/BF01895688. [11] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math., 3 (1997), 115-159.  doi: 10.1007/s000290050008. [12] B. A. Bubnov, Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain, Differential Equations, 15 (1979), 17-21. [13] E. Cerpa, Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Field, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45. [14] T. Colin and J.-M. Ghidaglia, Un probléme aux limites pour l'équation de Korteweg-de Vries sur un intervalle boné, (French) Journes Equations aux Drives Partielles, No. Ⅲ, école Polytech., Palaiseau, (1997), 10 pp. [15] T. Colin and J.-M. Ghidaglia, Un probléme mixte pour l'équation de Korteweg-de Vries sur un intervalle boné. (French), C. R. Acad. Sci. Paris. Sér. I Math., 324 (1997), 599-603.  doi: 10.1016/S0764-4442(99)80397-8. [16] T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem fo the Korteweg-de Vries Equation posed on a finite interval, Adv. Differential Equations, 6 (2001), 1463-1492. [17] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1. [18] A. V. Faminskii, The Cauchy problem and the mixed problem in the half strip for equation of Korteweg-de Vries type, Dinamika Sploshn. Sredy, 63 (1983), 152-158. [19] 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; English transl. in Trans. Moscow Math. Soc., 51 (1989), 53-91. [20] A. V. Faminskii, Mixed problms fo the Korteweg-de Vries equation, Sbornik: Mathematics, 190 (1999), 903-935.  doi: 10.1070/SM1999v190n06ABEH000408. [21] J. Holmer, The Initial-Boundary Value Problem for the Korteweg-de Vries Equation, Comm. Partial Differential Equations, 31 (2006), 1151-1190.  doi: 10.1080/03605300600718503. [22] C. Jia, I. Rivas and B.-Y. Zhang, Lower regularity solutions for a class of non-homogeneous boundary values of the Kortweg-de Vries equation on a finite domain, Adv. Differential Equations, 19 (2014), 559-584. [23] T. Kato, On the Korteweg-de Vries Equation, Manuscripta mathematica, 28 (1979), 89-99.  doi: 10.1007/BF01647967. [24] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations, Advances in Mathematics Supplementary Studies, 8 (1983), 93-128. [25] C. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9. [26] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003. [27] 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.  doi: 10.1090/S0894-0347-1991-1086966-0. [28] C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3. [29] C. Kenig, G. Ponce and L. Vega, Well-Posedness and scattering results for teh generalized Korteweg-de Vries equations via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405. [30] C. Kenig, G. Ponce and L. Vega, A Bilinear Estimate with Applicatios to the KdV Equation, J. Amer. Math. Soc., 9 (1996), 573-603. [31] 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.  doi: 10.1007/s11424-010-0143-x. [32] E. F. Kramer, I. Rivas and B.-Y. Zhang, Well-posedness of a class of non-homogeneous boundary value problem of the Korteweg-de Vries equation on a finite domain, ESAIM Control Optim. Calc. Var., 19 (2013), 358-384.  doi: 10.1051/cocv/2012012. [33] 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 Relat. Fields, 1 (2011), 61-81.  doi: 10.3934/mcrf.2011.1.61. [34] L. Tartar, Interpolation non linèaire et régularité, J. Funct. Anal., 9 (1972), 469-489.
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