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Well-posedness for the BBM-equation in a quarter plane
1. | Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States |
2. | Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152 |
3. | Institute of Applied Mathematical Sciences, National Taiwan University, Taipei, Taiwan |
References:
[1] |
L. Abdelohaub, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves,, Physica D, 40 (1989), 360.
doi: 10.1016/0167-2789(89)90050-X. |
[2] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London, 272 (1972), 47.
doi: 10.1098/rsta.1972.0032. |
[3] |
B. Boczar-Karakiewicz, J. L. Bona, W. Romanczyk and E. B. Thornton, Seasonal and interseasonal variability of sand bars at Duck, NC, USA. Observations and model predictions,, submitted., (). Google Scholar |
[4] |
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.
doi: 10.1017/S0305004100076945. |
[5] |
J. L. Bona, On solitary waves and their role in the evolution of long waves,, in Applications of Nonlinear Analysis in the Physical Sciences (ed. H. Amann, (): 183. Google Scholar |
[6] |
J. L. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Discrete Continuous Dyn. Systems, 23 (2009), 1253.
doi: 10.3934/dcds.2009.23.1253. |
[7] |
J. L. Bona, H. Chen, S. Sun and B.-Y. Zhang, Comparison of quarter-plane and two-point boundary value problems: The BBM-equation,, Discrete Continuous Dyn. Systems, 13 (2005), 921.
doi: 10.3934/dcds.2005.13.921. |
[8] |
J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves,, J. Math. Anal. Appl., 75 (1980), 503.
doi: 10.1016/0022-247X(80)90098-0. |
[9] |
J. L. Bona and L. Luo, Initial-boundary-value problems for model equations for the propagation of long waves,, in Evolution Equations (ed. G. Gerreyra, (1995), 65.
|
[10] |
J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal. Soc. London, 302 (1981), 457.
doi: 10.1098/rsta.1981.0178. |
[11] |
J. L. Bona, S. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane,, Trans. American Math. Soc., 354 (2002), 427.
doi: 10.1090/S0002-9947-01-02885-9. |
[12] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain,, Comm. Partial Differential Eqns., 28 (2003), 1391.
doi: 10.1081/PDE-120024373. |
[13] |
J. L. Bona, S. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for the one-dimensional nonlinear Schrödinger equation,, to appear., (). Google Scholar |
[14] |
J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM-equation,, Discrete Continuous Dyn. Systems, 23 (2009), 1241.
doi: 10.3934/dcds.2009.23.1241. |
[15] |
J. L. Bona and V. Varlamov, Wave generation by a moving boundary,, Contemp. Math., 371 (2005), 41.
doi: 10.1090/conm/371/06847. |
[16] |
J. L. Bona and R. Winther, The Korteweg-de Vries equation posed in a quarter plane,, SIAM J. Math. Anal., 14 (1983), 1056.
doi: 10.1137/0514085. |
[17] |
J. L. Bona and R. Winther, The Korteweg-de Vries equation in a quarter plane, continuous dependence results,, Differential and Integral Eq., 2 (1989), 228.
|
[18] |
M. Chen, Equations for bi-directional waves over an uneven bottom,, Mathematics and Computers in Simulation, 62 (2003), 3.
doi: 10.1016/S0378-4754(02)00193-3. |
[19] |
T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval,, Adv. Differential Eq., 6 (2001), 1463.
|
[20] |
T. Colin and M. Gisclon, An initial-boundary-value problem that approximates the quarter-plane problem for the Korteweg-de Vries equation,, Nonlinear Analysis: Theory, 46 (2001), 869.
doi: 10.1016/S0362-546X(00)00155-3. |
[21] |
J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line,, Commun. Partial Differential Eqns., 27 (2002), 2187.
doi: 10.1081/PDE-120016157. |
[22] |
B. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval,, Numer. Math., 86 (2000), 635.
doi: 10.1007/PL00005413. |
[23] |
J. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth,, J. Fluid Mech., 60 (1973), 769.
doi: 10.1017/S0022112073000479. |
[24] |
J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation,, Comm. Partial Differential Eqns., 31 (2006), 1151.
doi: 10.1080/03605300600718503. |
[25] |
H. Kalisch and J. L. Bona, Models for internal waves in deep water,, Discrete Cont. Dynamical Systems, 6 (2000), 1.
doi: 10.3934/dcds.2000.6.1. |
[26] |
D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321. Google Scholar |
show all references
References:
[1] |
L. Abdelohaub, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves,, Physica D, 40 (1989), 360.
doi: 10.1016/0167-2789(89)90050-X. |
[2] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London, 272 (1972), 47.
doi: 10.1098/rsta.1972.0032. |
[3] |
B. Boczar-Karakiewicz, J. L. Bona, W. Romanczyk and E. B. Thornton, Seasonal and interseasonal variability of sand bars at Duck, NC, USA. Observations and model predictions,, submitted., (). Google Scholar |
[4] |
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.
doi: 10.1017/S0305004100076945. |
[5] |
J. L. Bona, On solitary waves and their role in the evolution of long waves,, in Applications of Nonlinear Analysis in the Physical Sciences (ed. H. Amann, (): 183. Google Scholar |
[6] |
J. L. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Discrete Continuous Dyn. Systems, 23 (2009), 1253.
doi: 10.3934/dcds.2009.23.1253. |
[7] |
J. L. Bona, H. Chen, S. Sun and B.-Y. Zhang, Comparison of quarter-plane and two-point boundary value problems: The BBM-equation,, Discrete Continuous Dyn. Systems, 13 (2005), 921.
doi: 10.3934/dcds.2005.13.921. |
[8] |
J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves,, J. Math. Anal. Appl., 75 (1980), 503.
doi: 10.1016/0022-247X(80)90098-0. |
[9] |
J. L. Bona and L. Luo, Initial-boundary-value problems for model equations for the propagation of long waves,, in Evolution Equations (ed. G. Gerreyra, (1995), 65.
|
[10] |
J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal. Soc. London, 302 (1981), 457.
doi: 10.1098/rsta.1981.0178. |
[11] |
J. L. Bona, S. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane,, Trans. American Math. Soc., 354 (2002), 427.
doi: 10.1090/S0002-9947-01-02885-9. |
[12] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain,, Comm. Partial Differential Eqns., 28 (2003), 1391.
doi: 10.1081/PDE-120024373. |
[13] |
J. L. Bona, S. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for the one-dimensional nonlinear Schrödinger equation,, to appear., (). Google Scholar |
[14] |
J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM-equation,, Discrete Continuous Dyn. Systems, 23 (2009), 1241.
doi: 10.3934/dcds.2009.23.1241. |
[15] |
J. L. Bona and V. Varlamov, Wave generation by a moving boundary,, Contemp. Math., 371 (2005), 41.
doi: 10.1090/conm/371/06847. |
[16] |
J. L. Bona and R. Winther, The Korteweg-de Vries equation posed in a quarter plane,, SIAM J. Math. Anal., 14 (1983), 1056.
doi: 10.1137/0514085. |
[17] |
J. L. Bona and R. Winther, The Korteweg-de Vries equation in a quarter plane, continuous dependence results,, Differential and Integral Eq., 2 (1989), 228.
|
[18] |
M. Chen, Equations for bi-directional waves over an uneven bottom,, Mathematics and Computers in Simulation, 62 (2003), 3.
doi: 10.1016/S0378-4754(02)00193-3. |
[19] |
T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval,, Adv. Differential Eq., 6 (2001), 1463.
|
[20] |
T. Colin and M. Gisclon, An initial-boundary-value problem that approximates the quarter-plane problem for the Korteweg-de Vries equation,, Nonlinear Analysis: Theory, 46 (2001), 869.
doi: 10.1016/S0362-546X(00)00155-3. |
[21] |
J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line,, Commun. Partial Differential Eqns., 27 (2002), 2187.
doi: 10.1081/PDE-120016157. |
[22] |
B. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval,, Numer. Math., 86 (2000), 635.
doi: 10.1007/PL00005413. |
[23] |
J. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth,, J. Fluid Mech., 60 (1973), 769.
doi: 10.1017/S0022112073000479. |
[24] |
J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation,, Comm. Partial Differential Eqns., 31 (2006), 1151.
doi: 10.1080/03605300600718503. |
[25] |
H. Kalisch and J. L. Bona, Models for internal waves in deep water,, Discrete Cont. Dynamical Systems, 6 (2000), 1.
doi: 10.3934/dcds.2000.6.1. |
[26] |
D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321. Google Scholar |
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