Abstract
The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
Highlights
The determination of suitable boundary conditions has been recognized as an important problem in geophysical fluid dynamics, in particular for the problem of Limited Areas Models (LAMs); see e.g. [41]
We have addressed the question of the boundary conditions for the one-dimensional shallow water equation for one layer, rigorously in [32] and computationally in [38]
In this article we want to address the question of the boundary conditions which are suitable for two superposed layers of fluid governed by the shallow water equations
Summary
The determination of suitable boundary conditions has been recognized as an important problem in geophysical fluid dynamics, in particular for the problem of Limited Areas Models (LAMs); see e.g. [41]. The two-layer system that we consider essentially reduces to a one-layer shallow water equation for the so-called barotropic mode, and another similar system for the so-called baroclinic mode Both systems relate to isentropic gas dynamics which has been extensively studied following the program of Di Perna [9, 10]; see e.g. In the long range, we aim to study initial boundary value problems for nonlinear primitive equations, following the linear case studied in [35, 36], and we believe that the present work is a useful step in that direction since two-layer shallow water equations essentially correspond to two modes of the vertical expansion of the primitive equations ([29], [39], [35, 36]).
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