Large-time existence for one-dimensional Green-Naghdi equations with vorticity. (English) Zbl 07451802

Summary: This essay is concerned with the one-dimensional Green-Naghdi equations in the presence of a non-zero vorticity according to the derivation in [5], and with the addition of a small surface tension. The Green-Naghdi system is first rewritten as an equivalent system by using an adequate change of unknowns. We show that solutions to this model may be obtained by a standard Picard iterative scheme. No loss of regularity is involved with respect to the initial data. Moreover solutions exist at the same level of regularity as for first order hyperbolic symmetric systems, i.e. with a regularity in Sobolev spaces of order \(s>3/2\).


76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
35Q31 Euler equations
Full Text: DOI


[1] S. Alinhac and P. Gérard, Opérateurs Pseudo-différentiels et Théorème de Nash-Moser, Savoirs Actuels, InterEditions, Paris; Éditions du Centre national de la recherche scientifique, Meudon, 1991.
[2] B. Alvarez-Samaniego; D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171, 485-541 (2008) · Zbl 1131.76012
[3] S. V. Basenkova, N. N. Morozov and O. P. Pogutse, Dispersive effects in two-dimensional hydrodynamics, Dokl. Akad. Nauk, 293 (1985), 818-822 (transl. Sov. Phys. Dokl., 32 (1987), 262-264). · Zbl 0632.76018
[4] P. Bonneton; F. Chazel; D. Lannes; F. Marche; M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230, 1479-1498 (2011) · Zbl 1391.76066
[5] A. Castro; D. Lannes, Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., 759, 642-675 (2014) · Zbl 1446.76077
[6] A. Castro; D. Lannes, Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity, Indiana Univ. Math. J., 64, 1169-1270 (2015) · Zbl 1329.35242
[7] Q. Chen; J. T. Kirby; R. A. Dalrymple; A. B. Kennedy; A. Chawla, Boussinesq modeling of wave transformation, breaking, and runup, Part II: Two horizontal dimensions, J. Waterway Port Coastal Ocean Engrg., 126, 48-56 (2000)
[8] Q. Chen; J. T. Kirby; R. A. Dalrymple; F. Shi; E. B. Thornton, Boussinesq modeling of longshore currents, J. Geophys. Res., 108, 3362-3379 (2003)
[9] R. Cienfuegos; E. Barthélemy; P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations, Part I: Model development and analysis, Int. J. Numer. Meth. Fluids, 51, 1217-1253 (2006) · Zbl 1158.76361
[10] V. Duchêne; S. Israwi, Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems, Ann. Math. Blaise Pascal, 25, 21-74 (2018) · Zbl 1405.35159
[11] V. Duchêne; S. Israwi; R. Talhouk, A new fully justified asymptotic model for the propagation of internal waves in the Camassa-Holm regime, SIAM J. Math. Anal., 47, 240-290 (2015) · Zbl 1317.76027
[12] V. Duchêne; S. Israwi; R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137, 356-415 (2016) · Zbl 1356.35175
[13] D. Dutykh; D. Clamond; P. Milewski; D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, European J. Appl. Math., 24, 761-787 (2013) · Zbl 1400.65053
[14] A. E. Green; N. Laws; P. M. Naghdi, On the theory of water waves, Proc. Royal Soc. London Ser. A, 338, 43-55 (1974) · Zbl 0289.76010
[15] A. E. Green; P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78, 237-246 (1976) · Zbl 0351.76014
[16] T. Iguchi, A shallow water approximation for water waves, J. Math. Kyoto Univ., 49, 13-55 (2009) · Zbl 1421.76020
[17] S. Israwi, Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74, 81-93 (2011) · Zbl 1381.86012
[18] S. Israwi; H. Kalisch, Approximate conservation laws in the KdV equation, Phys. Lett. A, 383, 854-858 (2019) · Zbl 1480.76016
[19] T. Kano; T. Nishida, Sur les ondes de surface de l’eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ., 19, 335-370 (1979) · Zbl 0419.76013
[20] T. Kato; G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 891-907 (1988) · Zbl 0671.35066
[21] M. Kazolea; A. I. Delis; I. K. Nikolos; C. E. Synolakis, An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations, Coastal Eng., 69, 42-66 (2012)
[22] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232, 495-539 (2006) · Zbl 1099.35191
[23] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids, 21 (2009), 016601. · Zbl 1183.76294
[24] D. Lannes; F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Physics, 282, 238-268 (2015) · Zbl 1351.76114
[25] O. Le Métayer; S. Gavrilyuk; S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys., 229, 2034-2045 (2010) · Zbl 1303.76105
[26] Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59, 1225-1285 (2006) · Zbl 1169.76012
[27] N. Makarenko, The second long-wave approximation in the Cauchy-Poisson problem, Dyn. Contin. Media, 77, 56-72 (1986) · Zbl 0628.35023
[28] G. Métivier, Para-differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, Vol. 5, Scuola Norm. Sup. Pisa, 2008.
[29] L. V. Ovsjannikov, Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification, In: Appl. Meth. Funct. Anal. Probl. Mech. (IUTAM/IMU-Symp., Marseille, 1975), Lect. Notes Math. 503, Springer, 1976,426-437.
[30] M. Ricchiuto; A. G. Filippini, Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries, J. Comput. Physics, 271, 306-341 (2014) · Zbl 1349.76257
[31] M. E. Taylor, Partial Differential Equations III, Applied Mathematical Sciences, 117, Springer, 2011.
[32] G. Wei; J. T. Kirby; S. T. Grilli; R. Subramanya, A fully nonlinear Boussinesq model for surface waves, Part I. Highly nonlinear unsteady waves, J. Fluid Mech., 294, 71-92 (1995) · Zbl 0859.76009
[33] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Applied Mech. and Techn. Phys., 9, 190-194 (1968)
[34] Y. Zhang; A. B. Kennedy; N. Panda; C. Dawson; J. J. Westerink, Boussinesq-Green-Naghdi rotational water wave theory, Coastal Engrg., 73, 13-27 (2013)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.