# American Institute of Mathematical Sciences

October  2019, 39(10): 5543-5569. doi: 10.3934/dcds.2019244

## Propagation of long-crested water waves. Ⅱ. Bore propagation

 1 Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, MC 249, Chicago, IL 60607, USA 2 Université de Bordeaux, Institut de mathématiques de Bordeaux, UMR CNRS 5251, 351 cours de la Libération, 33405 Talence, France 3 Université Paris–Est Créteil, Laboratoire d'analyse et de mathématiques appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

Received  February 2017 Published  July 2019

This essay is concerned with long-crested waves such as those arising in bore propagation. Such motions obtain on rivers when a surge of water invades an otherwise constantly flowing stretch and in the run-up of waves in the near-shore zone of large bodies of water. The dominating feature of the motion is that, in a standard $xyz-$coordinate system in which $z$ increases in the direction opposite to which gravity acts and $x$ increases in the principal direction of propagation, the depth of the fluid approaches a constant value $h_0>0$ as $x \to +\infty$ and another value $h_1>h_0$ as $x \to -\infty$. In an earlier work, the authors developed theory for an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in that article applies to the sort of initial data arising in modeling bore propagation. However, well-posedness on the longer, Boussinesq time scale was not dealt with in the case of bore propagation, though such results were established for motions where $h_1 = h_0$.

We argue that without a well-posedness theory at least on the Boussinesq time scale, such models for bore-propagation may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.

The theory makes its way via the derivation of suitable approximations with which to compare the full solution. An interesting feature of the theory is the determination of dynamical boundary behavior that is not prescribed, but which the solution necessarily satisfies.

Citation: Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
##### References:
 [1] G. B. Airy, Tides and waves, Encyclopaedia Metropolitana, 5 (1845), 525–528, ed. E. Smedley, Hugh J. Rose, Henry J. Rose, London. [2] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM–equation and a Boussinesq system, Adv. Differential. Eq., 11 (2006), 121-166. [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.  doi: 10.1007/s00222-007-0088-4. [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Royal Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [5] T. B. Benjamin and M. J. Lighthill, On cnoidal waves and bores, Proc. Royal Soc. London Ser. A, 224 (1954), 448-460.  doi: 10.1098/rspa.1954.0172. [6] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Phys. D, 116 (1998), 191-224.  doi: 10.1016/S0167-2789(97)00249-2. [7] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4. [8] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ: The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010. [9] J. L. Bona, T. Colin and C. Guillopé, Propagation of long-crested water waves, Discrete Cont. Dynamical Systems Ser. A, 33 (2013), 599-628.  doi: 10.3934/dcds.2013.33.599. [10] J. L. Bona, T. Colin and D. Lannes, Long wave approximation for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1. [11] J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. London Ser. A, 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178. [12] J. L. Bona, S. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two dimensional theory, Differential Int. Eq., 7 (1994), 699-734. [13] J. L. Bona and R. Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Royal Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [14] P. Bonneton, J. Van de Loock, J.-P. Parisot, N. Bonneton, A. Sottolichio, G. Detandt, B. Castelle, V. Marieu and N. Pochon, On the occurrence of tidal bores. The Garonne River case, J. Coastal Res., Special Issue, 64 (2011), 1462-1466. [15] C. Burtea, Long time existence results for bore-type initial data for BBM-Boussinesq systems, J. Diff. Eq., 261 (2016), 4825-4860.  doi: 10.1016/j.jde.2016.07.014. [16] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On some Boussinesq systems in two space dimensions: Theory and numerical analysis, Math. Model. Numer. Anal., 41 (2007), 825-854.  doi: 10.1051/m2an:2007043. [17] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On initial-boundary value problems for a Boussinesq system of BBM–BBM type in a plane domain, Discrete Cont. Dynamical Systems Ser. A, 23 (2009), 1191-1204.  doi: 10.3934/dcds.2009.23.1191. [18] H. Favre, Étude théorique et expérimentale des ondes de translation dans les canaux découverts, Publications du laboratoire de recherche hydraulique annexé à l'École Polyechnique Fédérale de Zurich, 1935. [19] J. L. Hammack and H. Segur, The Kortweg-de Vries equation and water waves. Part 2. Comparison with experiments, J. Fluid Mech., 65 (1974), 289-313.  doi: 10.1017/S002211207400139X. [20] D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 19 (2006), 2853-2875.  doi: 10.1088/0951-7715/19/12/007. [21] M. Ming, J.-C. Saut and P. Zhang, Long time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214. [22] D. H. Peregrine, Calculation of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678. [23] S. V. Rajopadhye, Propagation of bores in incompressible fluids, Int. J. Modern Physics C, 4 (1993), 621-699. [24] S. V. Rajopadhye, Propagation of bores. Ⅱ. Three–dimensional theory, Nonlinear Anal., 27 (1996), 963-986.  doi: 10.1016/0362-546X(94)00358-O. [25] S. V. Rajopadhye, Some models for the propagation of bores, J. Diff. Eq., 217 (2005), 179-203.  doi: 10.1016/j.jde.2005.06.015. [26] J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012. [27] N. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons, J. Fluid Mech., 47 (1971), 811-824.

show all references

##### References:
 [1] G. B. Airy, Tides and waves, Encyclopaedia Metropolitana, 5 (1845), 525–528, ed. E. Smedley, Hugh J. Rose, Henry J. Rose, London. [2] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM–equation and a Boussinesq system, Adv. Differential. Eq., 11 (2006), 121-166. [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.  doi: 10.1007/s00222-007-0088-4. [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Royal Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [5] T. B. Benjamin and M. J. Lighthill, On cnoidal waves and bores, Proc. Royal Soc. London Ser. A, 224 (1954), 448-460.  doi: 10.1098/rspa.1954.0172. [6] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Phys. D, 116 (1998), 191-224.  doi: 10.1016/S0167-2789(97)00249-2. [7] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4. [8] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ: The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010. [9] J. L. Bona, T. Colin and C. Guillopé, Propagation of long-crested water waves, Discrete Cont. Dynamical Systems Ser. A, 33 (2013), 599-628.  doi: 10.3934/dcds.2013.33.599. [10] J. L. Bona, T. Colin and D. Lannes, Long wave approximation for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1. [11] J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. London Ser. A, 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178. [12] J. L. Bona, S. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two dimensional theory, Differential Int. Eq., 7 (1994), 699-734. [13] J. L. Bona and R. Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Royal Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [14] P. Bonneton, J. Van de Loock, J.-P. Parisot, N. Bonneton, A. Sottolichio, G. Detandt, B. Castelle, V. Marieu and N. Pochon, On the occurrence of tidal bores. The Garonne River case, J. Coastal Res., Special Issue, 64 (2011), 1462-1466. [15] C. Burtea, Long time existence results for bore-type initial data for BBM-Boussinesq systems, J. Diff. Eq., 261 (2016), 4825-4860.  doi: 10.1016/j.jde.2016.07.014. [16] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On some Boussinesq systems in two space dimensions: Theory and numerical analysis, Math. Model. Numer. Anal., 41 (2007), 825-854.  doi: 10.1051/m2an:2007043. [17] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On initial-boundary value problems for a Boussinesq system of BBM–BBM type in a plane domain, Discrete Cont. Dynamical Systems Ser. A, 23 (2009), 1191-1204.  doi: 10.3934/dcds.2009.23.1191. [18] H. Favre, Étude théorique et expérimentale des ondes de translation dans les canaux découverts, Publications du laboratoire de recherche hydraulique annexé à l'École Polyechnique Fédérale de Zurich, 1935. [19] J. L. Hammack and H. Segur, The Kortweg-de Vries equation and water waves. Part 2. Comparison with experiments, J. Fluid Mech., 65 (1974), 289-313.  doi: 10.1017/S002211207400139X. [20] D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 19 (2006), 2853-2875.  doi: 10.1088/0951-7715/19/12/007. [21] M. Ming, J.-C. Saut and P. Zhang, Long time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214. [22] D. H. Peregrine, Calculation of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678. [23] S. V. Rajopadhye, Propagation of bores in incompressible fluids, Int. J. Modern Physics C, 4 (1993), 621-699. [24] S. V. Rajopadhye, Propagation of bores. Ⅱ. Three–dimensional theory, Nonlinear Anal., 27 (1996), 963-986.  doi: 10.1016/0362-546X(94)00358-O. [25] S. V. Rajopadhye, Some models for the propagation of bores, J. Diff. Eq., 217 (2005), 179-203.  doi: 10.1016/j.jde.2005.06.015. [26] J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012. [27] N. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons, J. Fluid Mech., 47 (1971), 811-824.
 [1] Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 599-628. doi: 10.3934/dcds.2013.33.599 [2] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [3] Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 [4] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [5] Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065 [6] Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115 [7] Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59 [8] V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191 [9] Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239 [10] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212 [11] Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005 [12] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [13] Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161 [14] Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 635-648. doi: 10.3934/eect.2021019 [15] Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807 [16] Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843 [17] Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104 [18] Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917 [19] Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 [20] Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381

2021 Impact Factor: 1.588