American Institute of Mathematical Sciences

Advanced
November  2015, 14(6): 2203-2230. doi: 10.3934/cpaa.2015.14.2203

An improved result for the full justification of asymptotic models for the propagation of internal waves

Ralph Lteif 1, , Samer Israwi 2, and  Raafat Talhouk 2,

1. 

LAMA, UMR 5127 CNRS, Universit\'e de Savoie Mont Blanc, 73376 Le Bourget du lac cedex, France
2. 

Laboratory of Mathematics-EDST and Faculty of Sciences I, Lebanese University, Beirut, Lebanon

Received  November 2014 Revised  July 2015 Published  September 2015

We consider here asymptotic models that describe the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with uneven bottoms. The aim of this paper is to show that the full justification result of the model obtained by Duchêne, Israwi and Talhouk [SIAM J. Math. Anal., 47(1), 240–-290], in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data, can be improved in two directions. The first direction is taking into account medium amplitude topography variations and the second direction is allowing strong nonlinearity using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of the Camassa-Holm regime for the well-posedness and stability results.
Keywords: full justification, asymptotic model, medium amplitude., variable topography, Internal waves.
Mathematics Subject Classification: 76B55, 35C20, 35Q35, 35L6.
Citation: Ralph Lteif, Samer Israwi, Raafat Talhouk. An improved result for the full justification of asymptotic models for the propagation of internal waves. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2203-2230. doi: 10.3934/cpaa.2015.14.2203
References:
[1]

S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser,, \emph{Savoirs Actuels. InterEditions et Editions du CNRS, (1991).   Google Scholar

[2]

C. T. Anh, Influence of surface tension and bottom topography on internal waves,, \emph{Math. Models Methods Appl. Sci.}, 19 (2009), 2145.  doi: 10.1142/S0218202509004078.  Google Scholar

[3]

R. Barros and W. Choi, On regularizing the strongly nonlinear model for two-dimensional internal waves,, \emph{Physica D}, 264 (2013), 27.  doi: 10.1016/j.physd.2013.08.010.  Google Scholar

[4]

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,, \emph{J. Nonlinear Sci.}, 12 (2002), 283.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[5]

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,, \emph{Nonlinearity}, 17 (2004), 925.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[6]

J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves,, \emph{J. Math. Pures Appl. (9)}, 89 (2008), 538.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[7]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system,, \emph{J. Fluid Mech.}, 313 (1996), 83.  doi: 10.1017/S0022112096002133.  Google Scholar

[8]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system,, \emph{J. Fluid Mech.}, 396 (1999), 1.  doi: 10.1017/S0022112099005820.  Google Scholar

[9]

W. Craig and C. Sulem, Numerical simulation of gravity waves,, \emph{J. Comput. Phys.}, 108 (1993), 73.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[10]

V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2229.  doi: 10.1137/090761100.  Google Scholar

[11]

V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves,, \emph{M3AS:Math. Models Methods Appl. Sci.}, 24 (2014), 1.  doi: 10.1142/S0218202513500462.  Google Scholar

[12]

V. Duchêne, S. Israwi and R. Talhouk, A new fully justified asymptotic model for the propagation of internal waves in the Camassa-Holm regime,, \emph{SIAM J. Math. Anal.}, 47 (2015), 240.  doi: 10.1137/130947064.  Google Scholar

[13]

V. Duchêne, S. Israwi and R. Talhouk, Shallow water asymptotic models for the propagation of internal waves,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 7 (2014), 239.  doi: 10.3934/dcdss.2014.7.239.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth,, \emph{J. Fluid Mech.}, 78 (1976), 237.   Google Scholar

[15]

P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, \emph{Nonlinearity}, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[16]

D. Lannes, A stability criterion for two-fluid interfaces and applications,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 481.  doi: 10.1007/s00205-012-0604-6.  Google Scholar

[17]

D. Lannes, Water Waves: Mathematical Analysis and Asymptotics,, volume 188, (2013).  doi: 10.1090/surv/188.  Google Scholar

[18]

Z. L. Mal'tseva, Unsteady long waves in a two-layer fluid,, \emph{Dinamika Sploshn. Sredy}, 93-94 (1989), 93.   Google Scholar

[19]

Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems,, \emph{J. Phys. Soc. Japan}, 62 (1993), 1902.   Google Scholar

[20]

M. Miyata, An internal solitary wave of large amplitude,, \emph{La mer}, 23 (1985), 43.   Google Scholar

[21]

A. Ruiz de Zárate, D. G. A. Vigo, A. Nachbin and W. Choi, A higher-order internal wave model accounting for large bathymetric variations,, \emph{Stud. Appl. Math.}, 122 (2009), 275.  doi: 10.1111/j.1467-9590.2009.00433.x.  Google Scholar

[22]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, \emph{J. Appl. Mech. Tech. Phys.}, 9 (1968), 190.   Google Scholar

show all references

References:
[1]

S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser,, \emph{Savoirs Actuels. InterEditions et Editions du CNRS, (1991).   Google Scholar

[2]

C. T. Anh, Influence of surface tension and bottom topography on internal waves,, \emph{Math. Models Methods Appl. Sci.}, 19 (2009), 2145.  doi: 10.1142/S0218202509004078.  Google Scholar

[3]

R. Barros and W. Choi, On regularizing the strongly nonlinear model for two-dimensional internal waves,, \emph{Physica D}, 264 (2013), 27.  doi: 10.1016/j.physd.2013.08.010.  Google Scholar

[4]

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,, \emph{J. Nonlinear Sci.}, 12 (2002), 283.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[5]

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,, \emph{Nonlinearity}, 17 (2004), 925.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[6]

J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves,, \emph{J. Math. Pures Appl. (9)}, 89 (2008), 538.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[7]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system,, \emph{J. Fluid Mech.}, 313 (1996), 83.  doi: 10.1017/S0022112096002133.  Google Scholar

[8]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system,, \emph{J. Fluid Mech.}, 396 (1999), 1.  doi: 10.1017/S0022112099005820.  Google Scholar

[9]

W. Craig and C. Sulem, Numerical simulation of gravity waves,, \emph{J. Comput. Phys.}, 108 (1993), 73.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[10]

V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2229.  doi: 10.1137/090761100.  Google Scholar

[11]

V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves,, \emph{M3AS:Math. Models Methods Appl. Sci.}, 24 (2014), 1.  doi: 10.1142/S0218202513500462.  Google Scholar

[12]

V. Duchêne, S. Israwi and R. Talhouk, A new fully justified asymptotic model for the propagation of internal waves in the Camassa-Holm regime,, \emph{SIAM J. Math. Anal.}, 47 (2015), 240.  doi: 10.1137/130947064.  Google Scholar

[13]

V. Duchêne, S. Israwi and R. Talhouk, Shallow water asymptotic models for the propagation of internal waves,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 7 (2014), 239.  doi: 10.3934/dcdss.2014.7.239.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth,, \emph{J. Fluid Mech.}, 78 (1976), 237.   Google Scholar

[15]

P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, \emph{Nonlinearity}, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[16]

D. Lannes, A stability criterion for two-fluid interfaces and applications,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 481.  doi: 10.1007/s00205-012-0604-6.  Google Scholar

[17]

D. Lannes, Water Waves: Mathematical Analysis and Asymptotics,, volume 188, (2013).  doi: 10.1090/surv/188.  Google Scholar

[18]

Z. L. Mal'tseva, Unsteady long waves in a two-layer fluid,, \emph{Dinamika Sploshn. Sredy}, 93-94 (1989), 93.   Google Scholar

[19]

Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems,, \emph{J. Phys. Soc. Japan}, 62 (1993), 1902.   Google Scholar

[20]

M. Miyata, An internal solitary wave of large amplitude,, \emph{La mer}, 23 (1985), 43.   Google Scholar

[21]

A. Ruiz de Zárate, D. G. A. Vigo, A. Nachbin and W. Choi, A higher-order internal wave model accounting for large bathymetric variations,, \emph{Stud. Appl. Math.}, 122 (2009), 275.  doi: 10.1111/j.1467-9590.2009.00433.x.  Google Scholar

[22]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, \emph{J. Appl. Mech. Tech. Phys.}, 9 (1968), 190.   Google Scholar

[1]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051
[2]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[3]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927
[4]

Elena Rossi. A justification of a LWR model based on a follow the leader description. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 579-591. doi: 10.3934/dcdss.2014.7.579
[5]

Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118
[6]

Raphael Stuhlmeier. Internal Gerstner waves on a sloping bed. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3183-3192. doi: 10.3934/dcds.2014.34.3183
[7]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1
[8]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[9]

Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167
[10]

Ryszard Rudnicki, Radoslaw Wieczorek. Does assortative mating lead to a polymorphic population? A toy model justification. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 459-472. doi: 10.3934/dcdsb.2018031
[11]

Zengji Du, Zhaosheng Feng. Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1899-1920. doi: 10.3934/cpaa.2018090
[12]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[13]

Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183
[14]

Mateusz Kluczek. Nonhydrostatic Pollard-like internal geophysical waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5171-5183. doi: 10.3934/dcds.2019210
[15]

Hung-Chu Hsu. Exact azimuthal internal waves with an underlying current. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4391-4398. doi: 10.3934/dcds.2017188
[16]

Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066
[17]

Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175
[18]

Sebastian Aniţa, Bedreddine Ainseba. Internal eradicability for an epidemiological model with diffusion. Mathematical Biosciences & Engineering, 2005, 2 (3) : 437-443. doi: 10.3934/mbe.2005.2.437
[19]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[20]

Peter E. Kloeden, Jacson Simsen, Petra Wittbold. Asymptotic behavior of coupled inclusions with variable exponents. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1001-1016. doi: 10.3934/cpaa.2020046

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences