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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM–equation and a Boussinesq system, Adv. Differential. Eq., 11 (2006), 121-166.   Google Scholar

[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.  Google Scholar

[4]

T. B. BenjaminJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. L. BonaM. 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.  Google Scholar

[8]

J. L. BonaM. 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.  Google Scholar

[9]

J. L. BonaT. 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.  Google Scholar

[10]

J. L. BonaT. 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.  Google Scholar

[11]

J. L. BonaW. 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.  Google Scholar

[12]

J. L. BonaS. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two dimensional theory, Differential Int. Eq., 7 (1994), 699-734.   Google Scholar

[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.  Google Scholar

[14]

P. BonnetonJ. Van de LoockJ.-P. ParisotN. BonnetonA. SottolichioG. DetandtB. CastelleV. Marieu and N. Pochon, On the occurrence of tidal bores. The Garonne River case, J. Coastal Res., Special Issue, 64 (2011), 1462-1466.   Google Scholar

[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.  Google Scholar

[16]

V. A. DougalisD. 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.  Google Scholar

[17]

V. A. DougalisD. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. MingJ.-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.  Google Scholar

[22]

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

[23]

S. V. Rajopadhye, Propagation of bores in incompressible fluids, Int. J. Modern Physics C, 4 (1993), 621-699.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

N. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons, J. Fluid Mech., 47 (1971), 811-824.   Google Scholar

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. Google Scholar

[2]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM–equation and a Boussinesq system, Adv. Differential. Eq., 11 (2006), 121-166.   Google Scholar

[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.  Google Scholar

[4]

T. B. BenjaminJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. L. BonaM. 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.  Google Scholar

[8]

J. L. BonaM. 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.  Google Scholar

[9]

J. L. BonaT. 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.  Google Scholar

[10]

J. L. BonaT. 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.  Google Scholar

[11]

J. L. BonaW. 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.  Google Scholar

[12]

J. L. BonaS. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two dimensional theory, Differential Int. Eq., 7 (1994), 699-734.   Google Scholar

[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.  Google Scholar

[14]

P. BonnetonJ. Van de LoockJ.-P. ParisotN. BonnetonA. SottolichioG. DetandtB. CastelleV. Marieu and N. Pochon, On the occurrence of tidal bores. The Garonne River case, J. Coastal Res., Special Issue, 64 (2011), 1462-1466.   Google Scholar

[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.  Google Scholar

[16]

V. A. DougalisD. 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.  Google Scholar

[17]

V. A. DougalisD. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. MingJ.-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.  Google Scholar

[22]

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

[23]

S. V. Rajopadhye, Propagation of bores in incompressible fluids, Int. J. Modern Physics C, 4 (1993), 621-699.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

N. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons, J. Fluid Mech., 47 (1971), 811-824.   Google Scholar

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