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

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.
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

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