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

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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

Abstract
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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 and 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, Savoirs Actuels. InterEditions et Editions du CNRS, Paris, 1991.
[2]	C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175. doi: 10.1142/S0218202509004078.
[3]	R. Barros and W. Choi, On regularizing the strongly nonlinear model for two-dimensional internal waves, Physica D, 264 (2013), 27-34. doi: 10.1016/j.physd.2013.08.010.
[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, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[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, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010.
[6]	J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. (9),89 (2008), 538-566. doi: 10.1016/j.matpur.2008.02.003.
[7]	W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103. doi: 10.1017/S0022112096002133.
[8]	W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36. doi: 10.1017/S0022112099005820.
[9]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[10]	V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260. doi: 10.1137/090761100.
[11]	V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS:Math. Models Methods Appl. Sci., 24 (2014), 1-65. doi: 10.1142/S0218202513500462.
[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, SIAM J. Math. Anal., 47 (2015), 240-290. doi: 10.1137/130947064.
[13]	V. Duchêne, S. Israwi and R. Talhouk, Shallow water asymptotic models for the propagation of internal waves, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 239-269. doi: 10.3934/dcdss.2014.7.239.
[14]	A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.
[15]	P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.
[16]	D. Lannes, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567. doi: 10.1007/s00205-012-0604-6.
[17]	D. Lannes, Water Waves: Mathematical Analysis and Asymptotics, volume 188, Mathematical Surveys and Monographs, AMS, 2013. doi: 10.1090/surv/188.
[18]	Z. L. Mal'tseva, Unsteady long waves in a two-layer fluid, Dinamika Sploshn. Sredy, 93-94 (1989), 96-110.
[19]	Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems, J. Phys. Soc. Japan, 62 (1993), 1902-1916.
[20]	M. Miyata, An internal solitary wave of large amplitude, La mer, 23 (1985), 43-48.
[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, Stud. Appl. Math., 122 (2009), 275-294. doi: 10.1111/j.1467-9590.2009.00433.x.
[22]	V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.

show all references

References:

[1]	S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels. InterEditions et Editions du CNRS, Paris, 1991.
[2]	C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175. doi: 10.1142/S0218202509004078.
[3]	R. Barros and W. Choi, On regularizing the strongly nonlinear model for two-dimensional internal waves, Physica D, 264 (2013), 27-34. doi: 10.1016/j.physd.2013.08.010.
[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, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[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, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010.
[6]	J. L. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. (9),89 (2008), 538-566. doi: 10.1016/j.matpur.2008.02.003.
[7]	W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103. doi: 10.1017/S0022112096002133.
[8]	W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36. doi: 10.1017/S0022112099005820.
[9]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[10]	V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260. doi: 10.1137/090761100.
[11]	V. Duchêne, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS:Math. Models Methods Appl. Sci., 24 (2014), 1-65. doi: 10.1142/S0218202513500462.
[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, SIAM J. Math. Anal., 47 (2015), 240-290. doi: 10.1137/130947064.
[13]	V. Duchêne, S. Israwi and R. Talhouk, Shallow water asymptotic models for the propagation of internal waves, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 239-269. doi: 10.3934/dcdss.2014.7.239.
[14]	A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.
[15]	P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.
[16]	D. Lannes, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567. doi: 10.1007/s00205-012-0604-6.
[17]	D. Lannes, Water Waves: Mathematical Analysis and Asymptotics, volume 188, Mathematical Surveys and Monographs, AMS, 2013. doi: 10.1090/surv/188.
[18]	Z. L. Mal'tseva, Unsteady long waves in a two-layer fluid, Dinamika Sploshn. Sredy, 93-94 (1989), 96-110.
[19]	Y. Matsuno, A unified theory of nonlinear wave propagation in two-layer fluid systems, J. Phys. Soc. Japan, 62 (1993), 1902-1916.
[20]	M. Miyata, An internal solitary wave of large amplitude, La mer, 23 (1985), 43-48.
[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, Stud. Appl. Math., 122 (2009), 275-294. doi: 10.1111/j.1467-9590.2009.00433.x.
[22]	V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.

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