The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 455-518.

The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove this assumption. The problem then becomes a free-boundary problem since the position of the shoreline is unknown and driven by the solution itself. For the (hyperbolic) nonlinear shallow water equation, this problem is very related to the vacuum problem for a compressible gas. The Green-Naghdi equation include additional nonlinear, dispersive and topography terms with a complex degenerate structure at the boundary. In particular, the degeneracy of the topography terms makes the problem loose its quasilinear structure and become fully nonlinear. Dispersive smoothing also degenerates and its behavior at the boundary can be described by an ODE with regular singularity. These issues require the development of new tools, some of which of independent interest such as the study of the mixed initial boundary value problem for dispersive perturbations of characteristic hyperbolic systems, elliptic regularization with respect to conormal derivatives, or general Hardy-type inequalities.

Les équations de Green-Naghdi sont une perturbation non linéaire et dispersive des équations de Saint-Venant, dont la précision est meilleure d’un ordre de grandeur. Elles sont utilisées pour la simulation des écoulements côtiers, en particulier dans des zones où la profondeur s’annule (rivage). Le caractère bien posé des équations de Green-Naghdi (et leur justification en tant que modèle asymptotique pour les équations des vagues) a été étudié en détail, mais toujours sous l’hypothèse que la hauteur d’eau ne s’annule pas. L’objectif de cet article est s’affranchir de cette hypothèse. Le problème devient donc un problème à frontière libre car la position du rivage est inconnue. Pour les équations (hyperboliques) de Saint-Venant, ce problème est relié au problème du vide pour les gaz compressibles. Les équations de Green-Naghdi introduisent des termes supplémentaires non linéaires et dispersifs, ainsi que des termes de topographie complexes et dont la structure dégénère près du rivage. En particulier, la dégénérescence des termes de topographie fait perdre aux équations leur caractère quasi-linéaire. La régularisation dispersive propre aux équations de Green-Naghdi dégénère également au rivage ; elle peut néanmoins être décrite à l’aide d’EDO à singularité régulière. Ces problèmes requièrent le développement de nouveaux outils, dont certains sont d’un intérêt qui dépasse le cadre de cette étude. Par exemple : étude du problème aux limites mixte pour des perturbations dispersives de systèmes hyperboliques, régularisation elliptique par rapport aux dérivées conormales et inégalités de type Hardy générales.

Accepted:
Published online:
DOI: 10.5802/jep.76
Classification: 35F61, 35J70, 35L80, 35Q53, 76B15
Keywords: Green-Naghdi equations, shoreline, degenerate nonlinear dispersive equations, degenerate elliptic equation
Mot clés : Équations de Green-Naghdi, rivage, équations non linéaires dispersives dégénérées, équations elliptiques dégénérées
David Lannes 1; Guy Métivier 2

1 Institut de Mathématiques de Bordeaux, Université de Bordeaux et CNRS UMR 5251 351 Cours de la Libération, 33405 Talence Cedex, France
2 Institut de Mathématiques de Bordeaux, Université de Bordeaux et CNRS UMR 5251 351 Cours de la Libération, 33405 Talence Cedex, France
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David Lannes; Guy Métivier. The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 455-518. doi : 10.5802/jep.76. https://jep.centre-mersenne.org/articles/10.5802/jep.76/

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