# American Institute of Mathematical Sciences

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

## 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

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.
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. 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., (). [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. [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., (). [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.

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##### 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., (). [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. [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., (). [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.
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