-
Previous Article
Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound
- DCDS-S Home
- This Issue
-
Next Article
Global attractor for damped forced nonlinear logarithmic Schrödinger equations
Large-time existence for one-dimensional Green-Naghdi equations with vorticity
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 |
This essay is concerned with the one-dimensional Green-Naghdi equations in the presence of a non-zero vorticity according to the derivation in [
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. |
[1] |
Darryl D. Holm, Ruiao Hu. Nonlinear dispersion in wave-current interactions. Journal of Geometric Mechanics, 2022 doi: 10.3934/jgm.2022004 |
[2] |
Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 |
[3] |
Lijun Zhang, Yixia Shi, Maoan Han. Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2917-2926. doi: 10.3934/dcdss.2020217 |
[4] |
Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133 |
[5] |
Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153 |
[6] |
Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217 |
[7] |
Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065 |
[8] |
Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109 |
[9] |
Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 |
[10] |
Vincent Duchêne, Christian Klein. Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021300 |
[11] |
Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186 |
[12] |
Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773 |
[13] |
Luigi Roberti. The surface current of Ekman flows with time-dependent eddy viscosity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2463-2477. doi: 10.3934/cpaa.2022064 |
[14] |
Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089 |
[15] |
Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen, Alexandre M. Bayen. Comparison of two data assimilation algorithms for shallow water flows. Networks and Heterogeneous Media, 2009, 4 (2) : 409-430. doi: 10.3934/nhm.2009.4.409 |
[16] |
E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure and Applied Analysis, 2016, 15 (2) : 457-475. doi: 10.3934/cpaa.2016.15.457 |
[17] |
Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393 |
[18] |
Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2415-2431. doi: 10.3934/cpaa.2022053 |
[19] |
Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016 |
[20] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]