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December  2014, 7(6): 1149-1163. doi: 10.3934/dcdss.2014.7.1149

Well-posedness for the BBM-equation in a quarter plane

Jerry L. Bona 1, , Hongqiu Chen 2, and  Chun-Hsiung Hsia 3,

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

Received  July 2013 Revised  December 2013 Published  June 2014

The so-called wave-maker problem for the $BBM$-equation is studied on the half-line. Improving on earlier results, global well-posedness is established for square-integrable initial data and boundary data that is only assumed to be locally bounded.
Keywords: BBM equation, wave-maker problem, initial-boundary-value problems, nonlinear dispersive wave equations., Long wave propagation.
Mathematics Subject Classification: Primary: 35Q35, 35A53, 76B03; Secondary: 35Q51, 35Q86, 76B1.
Citation: Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149
References:
[1]

L. Abdelohaub, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Physica D, 40 (1989), 360-392. doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[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, Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[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), 391-405. doi: 10.1017/S0305004100076945.  Google Scholar

[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, Series A, 23 (2009), 1253-1275. doi: 10.3934/dcds.2009.23.1253.  Google Scholar

[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), 921-940. doi: 10.3934/dcds.2005.13.921.  Google Scholar

[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), 503-522. doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[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, G. Goldstein and F. Neubrander), Lecture Notes in Pure and Appl. Math., 168, Marcel Dekker, New York, 1995, 65-94.  Google Scholar

[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, Series A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.  Google Scholar

[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), 427-490. doi: 10.1090/S0002-9947-01-02885-9.  Google Scholar

[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), 1391-1436. doi: 10.1081/PDE-120024373.  Google Scholar

[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, Series A, 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[15]

J. L. Bona and V. Varlamov, Wave generation by a moving boundary, Contemp. Math., 371 (2005), 41-71. doi: 10.1090/conm/371/06847.  Google Scholar

[16]

J. L. Bona and R. Winther, The Korteweg-de Vries equation posed in a quarter plane, SIAM J. Math. Anal., 14 (1983), 1056-1106. doi: 10.1137/0514085.  Google Scholar

[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), 228-250.  Google Scholar

[18]

M. Chen, Equations for bi-directional waves over an uneven bottom, Mathematics and Computers in Simulation, 62 (2003), 3-9. doi: 10.1016/S0378-4754(02)00193-3.  Google Scholar

[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), 1463-1492.  Google Scholar

[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, Method & Application, 46 (2001), 869-892. doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[21]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Commun. Partial Differential Eqns., 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157.  Google Scholar

[22]

B. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), 635-654. doi: 10.1007/PL00005413.  Google Scholar

[23]

J. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60 (1973), 769-799. doi: 10.1017/S0022112073000479.  Google Scholar

[24]

J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Eqns., 31 (2006), 1151-1190. doi: 10.1080/03605300600718503.  Google Scholar

[25]

H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discrete Cont. Dynamical Systems, 6 (2000), 1-20. doi: 10.3934/dcds.2000.6.1.  Google Scholar

[26]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. 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), 360-392. doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[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, Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[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), 391-405. doi: 10.1017/S0305004100076945.  Google Scholar

[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, Series A, 23 (2009), 1253-1275. doi: 10.3934/dcds.2009.23.1253.  Google Scholar

[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), 921-940. doi: 10.3934/dcds.2005.13.921.  Google Scholar

[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), 503-522. doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[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, G. Goldstein and F. Neubrander), Lecture Notes in Pure and Appl. Math., 168, Marcel Dekker, New York, 1995, 65-94.  Google Scholar

[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, Series A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.  Google Scholar

[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), 427-490. doi: 10.1090/S0002-9947-01-02885-9.  Google Scholar

[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), 1391-1436. doi: 10.1081/PDE-120024373.  Google Scholar

[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, Series A, 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[15]

J. L. Bona and V. Varlamov, Wave generation by a moving boundary, Contemp. Math., 371 (2005), 41-71. doi: 10.1090/conm/371/06847.  Google Scholar

[16]

J. L. Bona and R. Winther, The Korteweg-de Vries equation posed in a quarter plane, SIAM J. Math. Anal., 14 (1983), 1056-1106. doi: 10.1137/0514085.  Google Scholar

[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), 228-250.  Google Scholar

[18]

M. Chen, Equations for bi-directional waves over an uneven bottom, Mathematics and Computers in Simulation, 62 (2003), 3-9. doi: 10.1016/S0378-4754(02)00193-3.  Google Scholar

[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), 1463-1492.  Google Scholar

[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, Method & Application, 46 (2001), 869-892. doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[21]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Commun. Partial Differential Eqns., 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157.  Google Scholar

[22]

B. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), 635-654. doi: 10.1007/PL00005413.  Google Scholar

[23]

J. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60 (1973), 769-799. doi: 10.1017/S0022112073000479.  Google Scholar

[24]

J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Eqns., 31 (2006), 1151-1190. doi: 10.1080/03605300600718503.  Google Scholar

[25]

H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discrete Cont. Dynamical Systems, 6 (2000), 1-20. doi: 10.3934/dcds.2000.6.1.  Google Scholar

[26]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. Google Scholar

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