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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 |
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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