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August  2021, 14(8): 2947-2974. doi: 10.3934/dcdss.2021040

Large-time existence for one-dimensional Green-Naghdi equations with vorticity

Colette Guillopé 1, , Samer Israwi 2, and  Raafat Talhouk 2,,

1. 

LAMA, Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, F-94010, Créteil, France
2. 

Mathématiques, Faculté des sciences I et Laboratoire de mathématiques, École doctorale des sciences et technologie, Université Libanaise, Beyrouth, Liban

* Corresponding author: Raafat Talhouk

Received  June 2020 Revised  March 2021 Published  August 2021 Early access  April 2021

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 $.

Keywords: Green-Naghdi equations, shallow water, rotational flows, wave-current interactions, rip-currents, surface tension, existence of solutions.
Mathematics Subject Classification: 76B15, 35A01, 35Q35.
Citation: Colette Guillopé, Samer Israwi, Raafat Talhouk. Large-time existence for one-dimensional Green-Naghdi equations with vorticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2947-2974. doi: 10.3934/dcdss.2021040
References:
[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 and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.  doi: 10.1007/s00222-007-0088-4.

[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).

[4]

P. Bonneton, F. Chazel, D. Lannes, F. Marche and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230 (2011), 1479-1498.  doi: 10.1016/j.jcp.2010.11.015.

[5]

A. Castro and D. Lannes, Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., 759 (2014), 642-675.  doi: 10.1017/jfm.2014.593.

[6]

A. Castro and 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 (2015), 1169-1270.  doi: 10.1512/iumj.2015.64.5606.

[7]

Q. Chen, J. T. Kirby, R. A. Dalrymple, A. B. Kennedy and A. Chawla, Boussinesq modeling of wave transformation, breaking, and runup, Part II: Two horizontal dimensions, J. Waterway Port Coastal Ocean Engrg., 126 (2000), 48-56.  doi: 10.1061/(ASCE)0733-950X(2000)126:1(48).

[8]

Q. Chen, J. T. Kirby, R. A. Dalrymple, F. Shi and E. B. Thornton, Boussinesq modeling of longshore currents, J. Geophys. Res., 108 (2003), 3362-3379.  doi: 10.1029/2002JC001308.

[9]

R. Cienfuegos, E. Barthélemy and 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 (2006), 1217-1253.  doi: 10.1002/fld.1141.

[10]

V. Duchêne and S. Israwi, Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems, Ann. Math. Blaise Pascal, 25 (2018), 21-74.  doi: 10.5802/ambp.372.

[11]

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.

[12]

V. Duchêne, S. Israwi and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137 (2016), 356-415.  doi: 10.1111/sapm.12125.

[13]

D. Dutykh, D. Clamond, P. Milewski and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, European J. Appl. Math., 24 (2013), 761-787.  doi: 10.1017/S0956792513000168.

[14]

A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Royal Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072.

[15]

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.  doi: 10.1017/S0022112076002425.

[16]

T. Iguchi, A shallow water approximation for water waves, J. Math. Kyoto Univ., 49 (2009), 13-55.  doi: 10.1215/kjm/1248983028.

[17]

S. Israwi, Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74 (2011), 81-93.  doi: 10.1016/j.na.2010.08.019.

[18]

S. Israwi and H. Kalisch, Approximate conservation laws in the KdV equation, Phys. Lett. A, 383 (2019), 854-858.  doi: 10.1016/j.physleta.2018.12.009.

[19]

T. Kano and 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 (1979), 335-370.  doi: 10.1215/kjm/1250522437.

[20]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[21]

M. Kazolea, A. I. Delis, I. K. Nikolos and C. E. Synolakis, An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations, Coastal Eng., 69 (2012), 42-66.  doi: 10.1016/j.coastaleng.2012.05.008.

[22]

D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.  doi: 10.1016/j.jfa.2005.07.003.

[23]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183.

[24]

D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Physics, 282 (2015), 238-268.  doi: 10.1016/j.jcp.2014.11.016.

[25]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys., 229 (2010), 2034-2045.  doi: 10.1016/j.jcp.2009.11.021.

[26]

Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59 (2006), 1225-1285.  doi: 10.1002/cpa.20148.

[27]

N. Makarenko, The second long-wave approximation in the Cauchy-Poisson problem, Dyn. Contin. Media, 77 (1986), 56-72. 

[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. doi: 10.1007/BFb0088777.

[30]

M. Ricchiuto and A. G. Filippini, Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries, J. Comput. Physics, 271 (2014), 306-341.  doi: 10.1016/j.jcp.2013.12.048.

[31]

M. E. Taylor, Partial Differential Equations III, Applied Mathematical Sciences, 117, Springer, 2011.

[32]

G. Wei, J. T. Kirby, S. T. Grilli and R. Subramanya, A fully nonlinear Boussinesq model for surface waves, Part I. Highly nonlinear unsteady waves, J. Fluid Mech., 294 (1995), 71-92.  doi: 10.1017/S0022112095002813.

[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 (1968), 190-194.  doi: 10.1007/BF00913182.

[34]

Y. Zhang, A. B. Kennedy, N. Panda, C. Dawson and J. J. Westerink, Boussinesq-Green-Naghdi rotational water wave theory, Coastal Engrg., 73 (2013), 13-27.  doi: 10.1016/j.coastaleng.2012.09.005.

show all references

References:
[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 and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.  doi: 10.1007/s00222-007-0088-4.

[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).

[4]

P. Bonneton, F. Chazel, D. Lannes, F. Marche and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230 (2011), 1479-1498.  doi: 10.1016/j.jcp.2010.11.015.

[5]

A. Castro and D. Lannes, Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., 759 (2014), 642-675.  doi: 10.1017/jfm.2014.593.

[6]

A. Castro and 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 (2015), 1169-1270.  doi: 10.1512/iumj.2015.64.5606.

[7]

Q. Chen, J. T. Kirby, R. A. Dalrymple, A. B. Kennedy and A. Chawla, Boussinesq modeling of wave transformation, breaking, and runup, Part II: Two horizontal dimensions, J. Waterway Port Coastal Ocean Engrg., 126 (2000), 48-56.  doi: 10.1061/(ASCE)0733-950X(2000)126:1(48).

[8]

Q. Chen, J. T. Kirby, R. A. Dalrymple, F. Shi and E. B. Thornton, Boussinesq modeling of longshore currents, J. Geophys. Res., 108 (2003), 3362-3379.  doi: 10.1029/2002JC001308.

[9]

R. Cienfuegos, E. Barthélemy and 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 (2006), 1217-1253.  doi: 10.1002/fld.1141.

[10]

V. Duchêne and S. Israwi, Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems, Ann. Math. Blaise Pascal, 25 (2018), 21-74.  doi: 10.5802/ambp.372.

[11]

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.

[12]

V. Duchêne, S. Israwi and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137 (2016), 356-415.  doi: 10.1111/sapm.12125.

[13]

D. Dutykh, D. Clamond, P. Milewski and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, European J. Appl. Math., 24 (2013), 761-787.  doi: 10.1017/S0956792513000168.

[14]

A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Royal Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072.

[15]

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.  doi: 10.1017/S0022112076002425.

[16]

T. Iguchi, A shallow water approximation for water waves, J. Math. Kyoto Univ., 49 (2009), 13-55.  doi: 10.1215/kjm/1248983028.

[17]

S. Israwi, Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74 (2011), 81-93.  doi: 10.1016/j.na.2010.08.019.

[18]

S. Israwi and H. Kalisch, Approximate conservation laws in the KdV equation, Phys. Lett. A, 383 (2019), 854-858.  doi: 10.1016/j.physleta.2018.12.009.

[19]

T. Kano and 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 (1979), 335-370.  doi: 10.1215/kjm/1250522437.

[20]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[21]

M. Kazolea, A. I. Delis, I. K. Nikolos and C. E. Synolakis, An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations, Coastal Eng., 69 (2012), 42-66.  doi: 10.1016/j.coastaleng.2012.05.008.

[22]

D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.  doi: 10.1016/j.jfa.2005.07.003.

[23]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183.

[24]

D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Physics, 282 (2015), 238-268.  doi: 10.1016/j.jcp.2014.11.016.

[25]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys., 229 (2010), 2034-2045.  doi: 10.1016/j.jcp.2009.11.021.

[26]

Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59 (2006), 1225-1285.  doi: 10.1002/cpa.20148.

[27]

N. Makarenko, The second long-wave approximation in the Cauchy-Poisson problem, Dyn. Contin. Media, 77 (1986), 56-72. 

[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. doi: 10.1007/BFb0088777.

[30]

M. Ricchiuto and A. G. Filippini, Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries, J. Comput. Physics, 271 (2014), 306-341.  doi: 10.1016/j.jcp.2013.12.048.

[31]

M. E. Taylor, Partial Differential Equations III, Applied Mathematical Sciences, 117, Springer, 2011.

[32]

G. Wei, J. T. Kirby, S. T. Grilli and R. Subramanya, A fully nonlinear Boussinesq model for surface waves, Part I. Highly nonlinear unsteady waves, J. Fluid Mech., 294 (1995), 71-92.  doi: 10.1017/S0022112095002813.

[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 (1968), 190-194.  doi: 10.1007/BF00913182.

[34]

Y. Zhang, A. B. Kennedy, N. Panda, C. Dawson and J. J. Westerink, Boussinesq-Green-Naghdi rotational water wave theory, Coastal Engrg., 73 (2013), 13-27.  doi: 10.1016/j.coastaleng.2012.09.005.

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