Abstract
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.
Highlights
Abstract. — The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation
(1)The dimensionless Green-Naghdi equations traditionally involve two other dimensionless parameters ε and β defined as amplitude of surface variations ε=
This section is devoted to a reformulation of the shoreline problem for the GreenNaghdi equations (1.2)
Summary
This section is devoted to a reformulation of the shoreline problem for the GreenNaghdi equations (1.2). The first step is to fix the free-boundary — As usual with free boundary problems, we first use a diffeomorphism mapping the moving domain (X(t), +∞) into a fixed domain (X0, +∞) for some time independent X0. — Composing the first equation of (1.2) with the Lagrangian mapping (2.1), and with η defined in (2.2), we obtain h ∂th + η ∂xu = 0; when combined with the relation. We recover the classical fact that in Lagrangian variables, the water depth is given in terms of η and of the water depth h0 at t = 0,. In Lagrangian variables, the Green-Naghdi equations reduce to the above equation on η complemented by the equation on u obtained by composing the second equation of (1.2) with φ,.
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