×
Haidar, Mohammad; El Arwadi, Toufic; Israwi, Samer

On the Green-Naghdi equations with surface tension in the Camassa-Holm scale. (English) Zbl 1469.76021

Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 67, No. 1, 83-110 (2021).
Summary: The aim of this paper is to analyze the water waves problem for uneven bottom under the influence of surface tension. For that, we consider an asymptotic model of the 1D Green-Naghdi equations in Camassa-Holm scale and derive in a formal way by using the Whitham technique the Camassa-Holm equation under the influence of surface tension. After that, the well-posdeness of the obtained Camassa-Holm equation is proved by using the Picard iterative scheme which proves that there is no loss of regularity of the solution relative to the initial condition. Also, the \(H^s\)-consistency, the stability and the convergency of the solution with the 1D Green-Naghdi model in Camassa-Holm scale are showed. Finally, the aspect of breaking wave for the Camassa-Holm equation is discussed in the presence of surface tension.

Cited in 3 Documents

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45 Capillarity (surface tension) for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics

Keywords:

water wave; asymptotic model; Green-Naghdi equations; Camassa-Holm scale; surface tension; well-posedness; Picard iterative scheme; convergence
Cite Review PDF
Full Text: DOI

References:

[1] Alvarez-Samaniego, B.; Lannes, D., A Nash-Moser theorem for singular evolution equations. application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57, 97-131 (2008) · Zbl 1144.35007 · doi:10.1512/iumj.2008.57.3200
[2] Alinhac, S., Gérard, P.: Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. InterEditions, Paris; Editions du Centre National de la Recherche Scientifique (CNRS), Meudon, 190 pp (1991) · Zbl 0791.47044
[3] Abd-Elaziz, E.; Marin, M.; Othman, M., On the effect of Thomson and initial stress in a Thermo-Porous elastic solid under G-N electromagnetic theory, Symmetry, 11, 3, 413 (2019) · Zbl 1425.74170 · doi:10.3390/sym11030413
[4] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[5] Coutand, D.; Shkoller, S., Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20, 829-930 (2007) · Zbl 1123.35038 · doi:10.1090/S0894-0347-07-00556-5
[6] Craig, W., An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits, Commun. Partial Differ. Equ., 10, 787-1003 (1985) · Zbl 0577.76030 · doi:10.1080/03605308508820396
[7] Green, AE; Naghdi, PM, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78, 237-246 (1976) · Zbl 0351.76014 · doi:10.1017/S0022112076002425
[8] Green, AE; Laws, N.; Naghdi, PM, On the theory of water waves, Proc. Roy. Soc. (London) Ser. A, 338, 43-55 (1974) · Zbl 0289.76010
[9] Haidar, M.; El Arwadi, T.; Israwi, S., Existence of a regular solution for 1D Green-Naghdi equations with surface tension at a large time instant, Bound. Value Probl., 136, 1-20 (2018) · Zbl 1499.76024
[10] Haidar, M.; El Arwadi, T.; Israwi, S., Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension, J. Appl. Math. Comput., 63, 655-681 (2020) · Zbl 1481.76045 · doi:10.1007/s12190-020-01333-8
[11] Israwi, S.; Mourad, A., An explicit solution with correctors for the Green-Naghdi equations, Mediterr. J. Math, 11, 519-532 (2014) · Zbl 1292.35264 · doi:10.1007/s00009-013-0356-z
[12] Israwi, S., Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74, 83-93 (2011) · Zbl 1381.86012 · doi:10.1016/j.na.2010.08.019
[13] Israwi, S., Variable depth KDV equations and generalizations to more nonlinear regimes, Math. Model. Numer. Anal., 44, 2, 347-370 (2009) · Zbl 1258.76040 · doi:10.1051/m2an/2010005
[14] Lannes, D., The Water Waves Problem (2013), Providence: American Mathematical Society, Providence · Zbl 1410.35003 · doi:10.1090/surv/188
[15] Lannes, D., Well-posedness of the water waves equations, J. Amer. Math. Soc., 18, 605-654 (2005) · Zbl 1069.35056 · doi:10.1090/S0894-0347-05-00484-4
[16] Lannes, D.; Bonneton, P., Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids, 21, 016601 (2009) · Zbl 1183.76294 · doi:10.1063/1.3053183
[17] Marin, M.; Craciun, E.; Pop, N., Considerations on mixed initial- boundary value problems for micopolar porous bodies, Dynam. Syst. Appl., 25, 175-196 (2016) · Zbl 1430.74010
[18] Nalimov, V. I.: The Cauchy-Poison problem. (Russian) Dinamika Splosn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, 254, 104-210 (1974)
[19] Ogawa, M.; Tani, A., It free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl Sci., 12, 12, 1725-1740 (2002) · Zbl 1023.76007 · doi:10.1142/S0218202502002306
[20] Shatah, J.; Zeng, C., Geometry and a priori estimates for free boundary problems of the Euler’s equation, Commun. Pure Appl Math., 61, 698-744 (2008) · Zbl 1174.76001 · doi:10.1002/cpa.20213
[21] Wu, S., Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130, 39-72 (1997) · Zbl 0892.76009 · doi:10.1007/s002220050177
[22] Wu, S., Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12, 2, 445-495 (1999) · Zbl 0921.76017 · doi:10.1090/S0894-0347-99-00290-8
[23] Yosihara, H., Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18, 1, 49-96 (1982) · Zbl 0493.76018 · doi:10.2977/prims/1195184016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.