January  2018, 38(1): 397-429. doi: 10.3934/dcds.2018019

Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received  September 2016 Revised  August 2017 Published  September 2017

The focus of the current paper is the higher order nonlinear dispersive equation which models unidirectional propagation of small amplitude long waves in dispersive media. The specific interest is in the initial-boundary value problem where spatial variable lies in $\mathbb R^+,$ namely, quarter plane problem. With proper requirement on initial and boundary condition, we show local and global well posedness.

Citation: Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019
References:
[1]

T. B. BenjaminJ. 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

[2]

B. Boczar-Karakiewicz, J. L. Bona, W. Romanczyk and E. B. Thornton, Seasonal and interseasonal vaiability of sand bars at Duck, NC, USA. Observations and model predictions}, submitted. Google Scholar

[3]

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

[4]

J. L. Bona, X. Carvajar, M. Panthee and M. Scialom, Higher-order Hamiltonian model for unidirectional water waves, to appear in Journal of Nonlinear Science and in https://arxiv.org/pdf/1509.08510.pdf. Google Scholar

[5]

J. L. BonaH. Chen and C.-H. Hsia, 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

[6]

J. L. BonaH. ChenS. 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

[7]

J. L. Bona and V. 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

[8]

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. , Marcel Dekker: New York, 168 (1995), 65-94. Google Scholar

[9]

J. L. BonaW. 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

[10]

J. L. BonaS. 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

[11]

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

[12]

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

[13]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.  Google Scholar

show all references

References:
[1]

T. B. BenjaminJ. 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

[2]

B. Boczar-Karakiewicz, J. L. Bona, W. Romanczyk and E. B. Thornton, Seasonal and interseasonal vaiability of sand bars at Duck, NC, USA. Observations and model predictions}, submitted. Google Scholar

[3]

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

[4]

J. L. Bona, X. Carvajar, M. Panthee and M. Scialom, Higher-order Hamiltonian model for unidirectional water waves, to appear in Journal of Nonlinear Science and in https://arxiv.org/pdf/1509.08510.pdf. Google Scholar

[5]

J. L. BonaH. Chen and C.-H. Hsia, 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

[6]

J. L. BonaH. ChenS. 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

[7]

J. L. Bona and V. 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

[8]

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. , Marcel Dekker: New York, 168 (1995), 65-94. Google Scholar

[9]

J. L. BonaW. 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

[10]

J. L. BonaS. 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

[11]

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

[12]

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

[13]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.  Google Scholar

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