Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems
Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 1, pp. 21-74.

In this paper we address the Cauchy problem for two systems modeling the propagation of long gravity waves in a layer of homogeneous, incompressible and inviscid fluid delimited above by a free surface, and below by a non-necessarily flat rigid bottom. Concerning the Green–Naghdi system, we improve the result of Alvarez–Samaniego and Lannes [5] in the sense that much less regular data are allowed, and no loss of derivatives is involved. Concerning the Boussinesq–Peregrine system, we improve the lower bound on the time of existence provided by Mésognon-Gireau [40]. The main ingredient is a physically motivated change of unknowns revealing the quasilinear structure of the systems, from which energy methods are implemented.

Published online:
DOI: 10.5802/ambp.372
Classification: 35L45,  35Q35,  76B15
Keywords: Well-posedness theory, shallow water models, quasilinear dispersive systems
Vincent Duchêne 1; Samer Israwi 2

1 Univ. Rennes 1, CNRS, IRMAR - UMR 6625, 35000 Rennes, France
2 Mathématiques, Faculté des sciences I et École doctorale des sciences et technologie Université Libanaise Beyrouth, Liban
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AMBP_2018__25_1_21_0,
     author = {Vincent Duch\^ene and Samer Israwi},
     title = {Well-posedness of the {Green{\textendash}Naghdi} and {Boussinesq{\textendash}Peregrine} systems},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {21--74},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {25},
     number = {1},
     year = {2018},
     doi = {10.5802/ambp.372},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.372/}
}
TY  - JOUR
AU  - Vincent Duchêne
AU  - Samer Israwi
TI  - Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems
JO  - Annales mathématiques Blaise Pascal
PY  - 2018
DA  - 2018///
SP  - 21
EP  - 74
VL  - 25
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.372/
UR  - https://doi.org/10.5802/ambp.372
DO  - 10.5802/ambp.372
LA  - en
ID  - AMBP_2018__25_1_21_0
ER  - 
%0 Journal Article
%A Vincent Duchêne
%A Samer Israwi
%T Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems
%J Annales mathématiques Blaise Pascal
%D 2018
%P 21-74
%V 25
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.372
%R 10.5802/ambp.372
%G en
%F AMBP_2018__25_1_21_0
Vincent Duchêne; Samer Israwi. Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems. Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 1, pp. 21-74. doi : 10.5802/ambp.372. https://ambp.centre-mersenne.org/articles/10.5802/ambp.372/

[1] Thomas Alazard A minicourse on the low Mach number limit, Discrete Contin. Dyn. Syst., Ser. S, Volume 1 (2008) no. 3, pp. 365-404 | Zbl

[2] Thomas Alazard; Guy Métivier Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves, Commun. Partial Differ. Equations, Volume 34 (2009) no. 12, pp. 1632-1704 | Zbl

[3] Serge Alinhac Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Commun. Partial Differ. Equations, Volume 14 (1989) no. 2, pp. 173-230 | Zbl

[4] Borys Alvarez-Samaniego; David Lannes Large time existence for 3D water waves and asymptotics, Invent. Math., Volume 171 (2008) no. 3, pp. 485-541 | Zbl

[5] Borys Alvarez-Samaniego; David Lannes A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations., Indiana Univ. Math. J., Volume 57 (2008) no. 1, pp. 97-131 | Zbl

[6] Éric Barthélemy Nonlinear shallow water theories for coastal waves., Surveys in Geophysics, Volume 25 (2004) no. 3-4, pp. 315-337 | DOI

[7] Stevan Bellec; Mathieu Colin; Mario Ricchiuto Discrete asymptotic equations for long wave propagation, SIAM J. Numer. Anal., Volume 54 (2016), pp. 3280-3299 | Zbl

[8] Sylvie Benzoni-Gavage; Denis Serre Multi-dimensional hyperbolic partial differential equations. First-order systems and applications., Oxford Mathematical Monographs, Oxford University Press, 2007, xxv+508 pages | Zbl

[9] Jerry L. Bona; Ronald Smith The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. roy. Soc. London, Ser. A, Volume 278 (1975), pp. 555-601 | Zbl

[10] Philippe Bonneton; Florent Chazel; David Lannes; Fabien Marche; Marion Tissier A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., Volume 230 (2011) no. 4, pp. 1479-1498 | Zbl

[11] Didier Bresch; Guy Métivier Anelastic limits for Euler-type systems, AMRX, Appl. Math. Res. Express, Volume 2010 (2010), pp. 119-141 | Zbl

[12] Roberto Camassa; Darryl D. Holm; C. David Levermore Long-time effects of bottom topography in shallow water, Physica D, Volume 98 (1996) no. 2-4, pp. 258-286 | Zbl

[13] Roberto Camassa; Darryl D. Holm; C. David Levermore Long-time shallow-water equations with a varying bottom, J. Fluid Mech., Volume 349 (1997), pp. 173-189 | Zbl

[14] Angel Castro; David Lannes Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., Volume 759 (2014), pp. 642-675 | Zbl

[15] Marx Chhay; Denys Dutykh; Didier Clamond On the multi-symplectic structure of the Serre-Green-Naghdi, J. Phys. A, Math. Theor., Volume 49 (2016) no. 3, 03LT01, 7 pages | Zbl

[16] Didier Clamond; Denys Dutykh Practical use of variational principles for modeling water waves, Phys. D, Volume 241 (2012) no. 1, pp. 25-36 | DOI | Zbl

[17] Walter Craig; Mark D. Groves Hamiltonian long-wave approximations to the water waves problem, Wave Motion, Volume 19 (1994) no. 4, pp. 367-389 | Zbl

[18] Walter Craig; Catherine Sulem Numerical simulation of gravity waves, J. Comput. Phys., Volume 108 (1993) no. 1, pp. 73-83 | Zbl

[19] Walter Craig; Catherine Sulem; Pierre-Louis Sulem Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, Volume 5 (1992) no. 2, pp. 497-522 | Zbl

[20] Vincent Duchêne; Samer Israwi; Raafat Talhouk A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., Volume 137 (2016) no. 3, pp. 356-415 | Zbl

[21] Hiroyasu Fujiwara; Tatsuo Iguchi A shallow water approximation for water waves over a moving bottom, Nonlinear dynamics in partial differential equations (Advanced Studies in Pure Mathematics), Volume 64, Mathematical Society of Japan, 2015, pp. 77-88 | Zbl

[22] Isabelle Gallagher Résultats récents sur la limite incompressible, Bourbaki seminar. Volume 2003/2004 (Astérisque), Volume 299, Société Mathématique de France, 2005, pp. 29-57 | Zbl

[23] Sergey L. Gavrilyuk; Henrik Kalisch; Zahra Khorsand A kinematic conservation law in free surface flow, Nonlinearity, Volume 28 (2015) no. 6, pp. 1805-1821 | Zbl

[24] Sergey L. Gavrilyuk; Vladimir M. Teshukov Generalized vorticity for bubbly liquid and dispersive shallow water equations, Contin. Mech. Thermodyn., Volume 13 (2001) no. 6, pp. 365-382 | Zbl

[25] Albert E. Green; Paul Mansour Naghdi A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., Volume 78 (1976), pp. 237-246 | Zbl

[26] Darryl D. Holm Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion, Phys. Fluids, Volume 31 (1988) no. 8, pp. 2371-2372 | Zbl

[27] Tatsuo Iguchi A shallow water approximation for water waves, J. Math. Kyoto Univ., Volume 49 (2009) no. 1, 2, pp. 13-55 | Zbl

[28] Samer Israwi Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., Theory Methods Appl., Volume 74 (2011) no. 1, pp. 81-93 | Zbl

[29] Tosio Kato The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., Volume 58 (1975), pp. 181-205 | Zbl

[30] Jang Whan Kim; Kwang June Bai; R. Cengiz Ertekin; William C. Webster A derivation of the Green-Naghdi equations for irrotational flows, J. Eng. Math., Volume 40 (2001) no. 1, pp. 17-42 | Zbl

[31] David Lannes The water waves problem. Mathematical analysis and asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, 2013, xx+321 pages | Zbl

[32] David Lannes; Philippe Bonneton Derivation of asymptotic two-dimensional time-dependent equations for surface water waves propagation, Phys. Fluids, Volume 21 (2009) no. 1, 016601, 9 pages | Zbl

[33] David Lannes; Fabien Marche A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Phys., Volume 282 (2015), pp. 238-268 | Zbl

[34] Olivier Le Métayer; Sergey L. Gavrilyuk; Sarah Hank A numerical scheme for the Green-Naghdi model, J. Comput. Phys., Volume 229 (2010) no. 6, pp. 2034-2045 | Zbl

[35] C. David Levermore; Marcel Oliver; Edriss S. Titi Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J., Volume 45 (1996) no. 2, pp. 479-510 | Zbl

[36] Yi A. Li Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations, J. Nonlinear Math. Phys., Volume 9 (2002), pp. 99-105 | Zbl

[37] Yi A. Li A shallow-water approximation to the full water waves problem., Commun. Pure Appl. Math., Volume 59 (2006) no. 9, pp. 1225-1285 | Zbl

[38] Yoshimasa Matsuno Hamiltonian structure for two-dimensional extended Green-Naghdi equations, Proc. R. Soc. Lond., Ser. A, Volume 472 (2016) no. 2190, 20160127, 24 pages | Zbl

[39] Benoît Mésognon-Gireau The singular limit of the Water-Waves equations in the rigid lid regime (2015) (https://arxiv.org/abs/1512.02424)

[40] Benoît Mésognon-Gireau The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations, Adv. Differ. Equ., Volume 22 (2017) no. 7-8, pp. 457-504 | Zbl

[41] Benoît Mésognon-Gireau The Cauchy problem on large time for the Water Waves equations with large topography variations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 1, pp. 89-118 | Zbl

[42] John Miles; Rick Salmon Weakly dispersive nonlinear gravity waves, J. Fluid Mech., Volume 157 (1985), pp. 519-531 | Zbl

[43] Marcel Oliver Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl., Volume 215 (1997) no. 2, pp. 471-484 | Zbl

[44] D. Howell Peregrine Long waves on a beach, J. Fluid Mech., Volume 27 (1967), pp. 815-827 | Zbl

[45] Rick Salmon Hamiltonian fluid mechanics, Annual Review of Fluid Mechanics, Volume 20 (1988), pp. 225-256 | DOI

[46] Fernando J. Seabra-Santos; Dominique P. Renouard; André Temperville Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle, J. Fluid Mech., Volume 176 (1987), pp. 117-134 | DOI

[47] Fraçois Serre Contribution à l’étude des écoulements permanents et variables dans les canaux, La Houille Blanche, Volume 6 (1953), pp. 830-872

[48] Theodore G. Shepherd Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics, Advances in Geophysics, Volume 32 (1990), pp. 287-338 | DOI

[49] Piyanuch Siriwat; Chompit Kaewmanee; Sergey V. Meleshko Symmetries of the hyperbolic shallow water equations and the Green-Naghdi model in lagrangian coordinates, International Journal of Non-Linear Mechanics, Volume 86 (2016), pp. 185-195 | DOI

[50] C. H. Su; Clifford S. Gardner Korteweg-de Vries equation and generalizations. III, J. Math. Phys., Volume 10 (1969) no. 3, pp. 536-539 | Zbl

[51] Michael E. Taylor Partial differential equations. III Nonlinear equations, Applied Mathematical Sciences, 117, Springer, 1997, xxii+715 pages | Zbl

[52] Gerald B. Whitham Variational methods and applications to water waves, Proc. R. Soc. Lond., Ser. A, Volume 299 (1967), pp. 6-25 | Zbl

[53] Vladimir E. Zakharov Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, Volume 9 (1968) no. 2, pp. 190-194 | DOI

Cited by Sources: