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.
Mots clés : Well-posedness theory, shallow water models, quasilinear dispersive systems
Vincent Duchêne 1 ; Samer Israwi 2
CC-BY 4.0
@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 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/ 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://ambp.centre-mersenne.org/articles/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, Tome 25 (2018) no. 1, pp. 21-74. doi : 10.5802/ambp.372. https://ambp.centre-mersenne.org/articles/10.5802/ambp.372/
[1] A minicourse on the low Mach number limit, Discrete Contin. Dyn. Syst., Ser. S, Volume 1 (2008) no. 3, pp. 365-404 | Zbl
[2] 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] 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] Large time existence for 3D water waves and asymptotics, Invent. Math., Volume 171 (2008) no. 3, pp. 485-541 | Zbl
[5] 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] Nonlinear shallow water theories for coastal waves., Surveys in Geophysics, Volume 25 (2004) no. 3-4, pp. 315-337 | DOI
[7] Discrete asymptotic equations for long wave propagation, SIAM J. Numer. Anal., Volume 54 (2016), pp. 3280-3299 | Zbl
[8] Multi-dimensional hyperbolic partial differential equations. First-order systems and applications., Oxford Mathematical Monographs, Oxford University Press, 2007, xxv+508 pages | Zbl
[9] 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] 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] Anelastic limits for Euler-type systems, AMRX, Appl. Math. Res. Express, Volume 2010 (2010), pp. 119-141 | Zbl
[12] Long-time effects of bottom topography in shallow water, Physica D, Volume 98 (1996) no. 2-4, pp. 258-286 | Zbl
[13] Long-time shallow-water equations with a varying bottom, J. Fluid Mech., Volume 349 (1997), pp. 173-189 | Zbl
[14] Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., Volume 759 (2014), pp. 642-675 | Zbl
[15] 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] Practical use of variational principles for modeling water waves, Phys. D, Volume 241 (2012) no. 1, pp. 25-36 | DOI | Zbl
[17] Hamiltonian long-wave approximations to the water waves problem, Wave Motion, Volume 19 (1994) no. 4, pp. 367-389 | Zbl
[18] Numerical simulation of gravity waves, J. Comput. Phys., Volume 108 (1993) no. 1, pp. 73-83 | Zbl
[19] Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, Volume 5 (1992) no. 2, pp. 497-522 | Zbl
[20] 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] 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] Résultats récents sur la limite incompressible, Bourbaki seminar. Volume 2003/2004 (Astérisque), Volume 299, Société Mathématique de France, 2005, pp. 29-57 | Zbl
[23] A kinematic conservation law in free surface flow, Nonlinearity, Volume 28 (2015) no. 6, pp. 1805-1821 | Zbl
[24] Generalized vorticity for bubbly liquid and dispersive shallow water equations, Contin. Mech. Thermodyn., Volume 13 (2001) no. 6, pp. 365-382 | Zbl
[25] A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., Volume 78 (1976), pp. 237-246 | Zbl
[26] Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion, Phys. Fluids, Volume 31 (1988) no. 8, pp. 2371-2372 | Zbl
[27] A shallow water approximation for water waves, J. Math. Kyoto Univ., Volume 49 (2009) no. 1, 2, pp. 13-55 | Zbl
[28] Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., Theory Methods Appl., Volume 74 (2011) no. 1, pp. 81-93 | Zbl
[29] The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., Volume 58 (1975), pp. 181-205 | Zbl
[30] A derivation of the Green-Naghdi equations for irrotational flows, J. Eng. Math., Volume 40 (2001) no. 1, pp. 17-42 | Zbl
[31] The water waves problem. Mathematical analysis and asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, 2013, xx+321 pages | Zbl
[32] 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] 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] A numerical scheme for the Green-Naghdi model, J. Comput. Phys., Volume 229 (2010) no. 6, pp. 2034-2045 | Zbl
[35] 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] 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] A shallow-water approximation to the full water waves problem., Commun. Pure Appl. Math., Volume 59 (2006) no. 9, pp. 1225-1285 | Zbl
[38] 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] The singular limit of the Water-Waves equations in the rigid lid regime (2015) (https://arxiv.org/abs/1512.02424)
[40] 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] 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] Weakly dispersive nonlinear gravity waves, J. Fluid Mech., Volume 157 (1985), pp. 519-531 | Zbl
[43] Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl., Volume 215 (1997) no. 2, pp. 471-484 | Zbl
[44] Long waves on a beach, J. Fluid Mech., Volume 27 (1967), pp. 815-827 | Zbl
[45] Hamiltonian fluid mechanics, Annual Review of Fluid Mechanics, Volume 20 (1988), pp. 225-256 | DOI
[46] 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] Contribution à l’étude des écoulements permanents et variables dans les canaux, La Houille Blanche, Volume 6 (1953), pp. 830-872
[48] Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics, Advances in Geophysics, Volume 32 (1990), pp. 287-338 | DOI
[49] 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] Korteweg-de Vries equation and generalizations. III, J. Math. Phys., Volume 10 (1969) no. 3, pp. 536-539 | Zbl
[51] Partial differential equations. III Nonlinear equations, Applied Mathematical Sciences, 117, Springer, 1997, xxii+715 pages | Zbl
[52] Variational methods and applications to water waves, Proc. R. Soc. Lond., Ser. A, Volume 299 (1967), pp. 6-25 | Zbl
[53] 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
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