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December  2020, 19(12): 5567-5580. doi: 10.3934/cpaa.2020251

Mathematical analysis of bump to bucket problem

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

2. 

LAMFA UMR 7352 CNRS, Université de Picardie Jules Verne, 80039 Amiens CEDEX 1, France

3. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China

* Corresponding author

Received  April 2020 Revised  July 2020 Published  December 2020 Early access  October 2020

Fund Project: O. Goubet acknowledges the financial support of both SODDA research project funded by Region Hauts-de-France and FEDER from E.C., and S2R 2018 - Action 4.3 of UPJV. S. Li is supported by the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264)

In this article, several systems of equations which model surface water waves generated by a sudden bottom deformation (bump) are studied. Because the effect of such deformation are often approximated by assuming the initial water surface has a deformation (bucket), this procedure is investigated and we prove rigorously that by using the correct bucket, the solutions of the regularized bump problems converge to the solution of the bucket problem.

Citation: Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

J. L. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. derivation and linear theory., Journal of Nonlinear Science. 12 (2002) 283–318. doi: 10.1007/s00332-002-0466-4.

[3]

J. BonaM. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Part Ⅱ: the nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[4]

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.

[5]

D. Dutykh and F. Dias, Energy of tsunami waves generated by bottom motion, Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 465 (2009) 725–744. doi: 10.1098/rspa.2008.0332.

[6]

D. Dutykh and F. Dias, Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting, Mathematics and Computers in Simulation, 80 (2009), 837-848.  doi: 10.1016/j.matcom.2009.08.036.

[7]

D. Dutykh and F. Dias, Influence of sedimentary layering on tsunami generation, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1268-1275.  doi: 10.1016/j.cma.2009.07.011.

[8]

D. DutykhF. Dias and Y. Kervella, Linear theory of wave generation by a moving bottom, Comptes Rendus Mathematique, 343 (2006), 499-504.  doi: 10.1016/j.crma.2006.09.016.

[9]

D. Dutykh and H. Kalisch, Boussinesq modeling of surface waves due to underwater landslides, arXiv: 1112.5083.

[10]

D. DutykhD. MitsotakisL. B. Chubarov and Y. I. Shokin, On the contribution of the horizontal sea-bed displacements into the tsunami generation process, Ocean Model., 56 (2012), 43-56. 

[11]

H. Fujiwara and T. Iguchi, A shallow water approximation for water waves over a moving bottom. Nonlinear dynamics in partial differential equations, Adv. Stud. Pure Math., 64, Math. Soc. Japan, Tokyo, 2015, 77–88. doi: 10.2969/aspm/06410077.

[12]

T. Iguchi, A mathematical analysis of tsunami generation in shallow water due to seabed deformation, P. Roy. Soc. Edinb. A, 141 (2011), 551-608.  doi: 10.1017/S0308210509001279.

[13]

T. Jamin, L. Gordillo, G. Ruiz-Chavarría, M. Berhanu and E. Falcon, Generation of surface waves by an underwater moving bottom: Experiments and application to tsunami modelling, arXiv: 1404.0312.

[14]

D. Lannes, The water waves problem: mathematical analysis and asymptotics, Am. Math. Soc., 188 (2013) doi: 10.1090/surv/188.

[15]

S. J. Lee, Generation Of Long Water Waves By Moving Disturbances., PhD thesis, California Institute of Technology, 1985.

[16]

D. Mitsotakis, Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves, Math. Comput. Simulat., 80 (2009), 860-873.  doi: 10.1016/j.matcom.2009.08.029.

[17]

H. NersisyanD. Dutykh and E. Zuazua, Generation of two-dimensional water waves by moving bottom disturbances, IMA J. Appl. Math., 80 (2015), 1235-1253.  doi: 10.1093/imamat/hxu051.

[18]

L. Nirenberg, A strong maximum principle for parabolic equations, Commun. Pure Appl. Math., 6 (1953), 167-177.  doi: 10.1002/cpa.3160060202.

[19]

G. Sadaka, Propagation of a tsunami wave., arXiv: 1210.4260.

[20]

T. Saito, Dynamic tsunami generation due to sea-bottom deformation: Analytical representation based on linear potential theory, Earth Planets Space, 65 (2013), 1411-1423. 

[21]

M. Schonbek, Existence of solutions for the Boussinesq system of equations, J. Differ. Equ., 42 (1981), 325-352.  doi: 10.1016/0022-0396(81)90108-X.

[22]

S. Y. Sekerzh-Zenkovich, Analytical study of a potential model of tsunami with a simple source of piston type. 1. exact solution. creation of tsunami, Russ. J. Math. Phys., 19 (2012), 385-393.  doi: 10.1134/S1061920812030107.

[23]

S. Y. Sekerzh-Zenkovich, Analytic study of a potential model of tsunami with a simple source of piston type. 2. asymptotic formula for the height of tsunami in the far field, Russ. J Math. Phys., 20 (2013), 542-546.  doi: 10.1134/S1061920813040134.

[24]

S. Y. Sekerzh-Zenkovich, Analytical study of the tsunami potential model with a simple piston-like source. 3. application of the model in the inverse problem related to the japanese tsunami 2011, Russ. J. Math. Phys., 21 (2014), 504-508.  doi: 10.1134/S1061920814040086.

[25]

M. Tulin, C. Yih, D. Wu, T. Wu and V. Shanmuganathan, Three-dimensional nonlinear long waves due to moving surface pressure. Proceedings of 14th Symposium on Naval Hydrodynamics, National Academy Press, Washington, DC, 1982.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

J. L. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. derivation and linear theory., Journal of Nonlinear Science. 12 (2002) 283–318. doi: 10.1007/s00332-002-0466-4.

[3]

J. BonaM. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Part Ⅱ: the nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[4]

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.

[5]

D. Dutykh and F. Dias, Energy of tsunami waves generated by bottom motion, Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 465 (2009) 725–744. doi: 10.1098/rspa.2008.0332.

[6]

D. Dutykh and F. Dias, Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting, Mathematics and Computers in Simulation, 80 (2009), 837-848.  doi: 10.1016/j.matcom.2009.08.036.

[7]

D. Dutykh and F. Dias, Influence of sedimentary layering on tsunami generation, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1268-1275.  doi: 10.1016/j.cma.2009.07.011.

[8]

D. DutykhF. Dias and Y. Kervella, Linear theory of wave generation by a moving bottom, Comptes Rendus Mathematique, 343 (2006), 499-504.  doi: 10.1016/j.crma.2006.09.016.

[9]

D. Dutykh and H. Kalisch, Boussinesq modeling of surface waves due to underwater landslides, arXiv: 1112.5083.

[10]

D. DutykhD. MitsotakisL. B. Chubarov and Y. I. Shokin, On the contribution of the horizontal sea-bed displacements into the tsunami generation process, Ocean Model., 56 (2012), 43-56. 

[11]

H. Fujiwara and T. Iguchi, A shallow water approximation for water waves over a moving bottom. Nonlinear dynamics in partial differential equations, Adv. Stud. Pure Math., 64, Math. Soc. Japan, Tokyo, 2015, 77–88. doi: 10.2969/aspm/06410077.

[12]

T. Iguchi, A mathematical analysis of tsunami generation in shallow water due to seabed deformation, P. Roy. Soc. Edinb. A, 141 (2011), 551-608.  doi: 10.1017/S0308210509001279.

[13]

T. Jamin, L. Gordillo, G. Ruiz-Chavarría, M. Berhanu and E. Falcon, Generation of surface waves by an underwater moving bottom: Experiments and application to tsunami modelling, arXiv: 1404.0312.

[14]

D. Lannes, The water waves problem: mathematical analysis and asymptotics, Am. Math. Soc., 188 (2013) doi: 10.1090/surv/188.

[15]

S. J. Lee, Generation Of Long Water Waves By Moving Disturbances., PhD thesis, California Institute of Technology, 1985.

[16]

D. Mitsotakis, Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves, Math. Comput. Simulat., 80 (2009), 860-873.  doi: 10.1016/j.matcom.2009.08.029.

[17]

H. NersisyanD. Dutykh and E. Zuazua, Generation of two-dimensional water waves by moving bottom disturbances, IMA J. Appl. Math., 80 (2015), 1235-1253.  doi: 10.1093/imamat/hxu051.

[18]

L. Nirenberg, A strong maximum principle for parabolic equations, Commun. Pure Appl. Math., 6 (1953), 167-177.  doi: 10.1002/cpa.3160060202.

[19]

G. Sadaka, Propagation of a tsunami wave., arXiv: 1210.4260.

[20]

T. Saito, Dynamic tsunami generation due to sea-bottom deformation: Analytical representation based on linear potential theory, Earth Planets Space, 65 (2013), 1411-1423. 

[21]

M. Schonbek, Existence of solutions for the Boussinesq system of equations, J. Differ. Equ., 42 (1981), 325-352.  doi: 10.1016/0022-0396(81)90108-X.

[22]

S. Y. Sekerzh-Zenkovich, Analytical study of a potential model of tsunami with a simple source of piston type. 1. exact solution. creation of tsunami, Russ. J. Math. Phys., 19 (2012), 385-393.  doi: 10.1134/S1061920812030107.

[23]

S. Y. Sekerzh-Zenkovich, Analytic study of a potential model of tsunami with a simple source of piston type. 2. asymptotic formula for the height of tsunami in the far field, Russ. J Math. Phys., 20 (2013), 542-546.  doi: 10.1134/S1061920813040134.

[24]

S. Y. Sekerzh-Zenkovich, Analytical study of the tsunami potential model with a simple piston-like source. 3. application of the model in the inverse problem related to the japanese tsunami 2011, Russ. J. Math. Phys., 21 (2014), 504-508.  doi: 10.1134/S1061920814040086.

[25]

M. Tulin, C. Yih, D. Wu, T. Wu and V. Shanmuganathan, Three-dimensional nonlinear long waves due to moving surface pressure. Proceedings of 14th Symposium on Naval Hydrodynamics, National Academy Press, Washington, DC, 1982.

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