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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 |
$\begin{equation} u_t+u_x+u_{xxx}+uu_x = 0, ~~~~~ u(x, 0) = φ(x), ~~~~~ 0 < x < L, \ t>0~~~~~(0.1)\end{equation}$ |
$\begin{equation} B_1u = h_1(t), ~~~~~B_2 u = h_2 (t), ~~~~~ B_3 u = h_3 (t) ~~~~~t>0~~~~~(0.2) \end{equation} $ |
$ 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, $ |
$a_{ij}, \ b_{ij}$ |
$j = 0, 1, 2$ |
$ i = 1, 2, 3$ |
$a_{ij}, \ b_{ij}$ |
$H^s (0, L)$ |
$s \ge 0$ |
$φ ∈ H^s (0, L)$ |
$h_j$ |
$j = 1, 2, 3$ |
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.
|
show all references
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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