On a non-homogeneous bi-layer shallow-water problem: smoothness and uniqueness results (original) (raw)

On a nonhomogeneous bi-layer shallow-water problem: an existence theorem

Differential and Integral Equations, 2004

In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.

On a bi-layer shallow-water problem

Nonlinear Analysis-real World Applications, 2003

In this paper, we prove an existence and uniqueness result for a bi-layer shallow water model in depth-mean velocity formulation. Some smoothness results for the solution are also obtained. In a previous work we proved the same results for a one-layer problem. Now the di culty arises from the terms coupling the two layers. In order to obtain the energy estimate, we use a special basis which allows us to bound these terms.

A bidimensional bi-layer shallow-water model

DOAJ (DOAJ: Directory of Open Access Journals), 2017

The existence of global weak solutions in a periodic domain for a non-linear viscous bi-layer shallow-water model with capillarity effects and extra friction terms in a two-dimensional space has been proved in [21]. The main contribution of this article is to show the existence of global weak solutions without friction term or capillary effect following the ideas of [20] for the two dimensional case.

On an one-dimensional bi-layer shallow-water problem* 1

Nonlinear analysis, 2003

This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial di erential equations governing the one-dimensional ow of two superposed shallow layers of immiscible viscous uid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a ÿrst-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange ows through straits with a sill and a contraction. ?

Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model

Nonlinear Analysis-real World Applications, 2009

We consider a non-linear viscous bi-layer shallow water model with capillarity effects and extra friction terms in a two-dimensional space. This system is issued from a derivation of a three-dimensional Navier-Stokes equations with water-depth depending on friction coefficients. We prove an existence result for global weak solution in a periodic domain Ω = T 2 .

Uniqueness of strong solution for a 1D viscous bi-layer Shallow Water model

Annals of the University of Craiova - Mathematics and Computer Science Series, 2012

The aim of this paper is to prove the uniqueness of strong solutionof a one dimensional viscous bilayer shallow water model. Our analy-sis is based on some new useful estimate namely BD entropy and on amethod developed by Mellet and Vasseur in [14] to prove the existenceand uniqueness on some compressible one dimensional Navier-Stokessystem. Under suitable assumptions on the solutions and using Gron-wall Lemma, we obtain the uniqueness of strong solution. We performour analysis in periodic domain with periodic boundaries conditions.

Models and methods for two-layer shallow water flows

Journal of Computational Physics

Two-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method.

Three-Layer Approximation of Two-Layer Shallow Water Equations

Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of di↵erent (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.

Existence of global weak solutions for a viscous 2D bilayer Shallow Water model

Comptes Rendus Mathematique, 2011

We consider a system composed by two immiscible fluids in two-dimensional space that can be modelized by a bilayer Shallow Water equations with extra friction terms and capillary effects. We give an existence theorem of global weak solutions in a periodic domain. Résumé Nous considérons un système composé par deux fluides immiscibles dans un domaine bi-dimensionnel pouvantêtre représenté par un modèle bicouche visqueux de Saint-Venant avec des termes de friction additionnels et des effets de capillarité. Nous donnons un théorème d'existence de solutions faibles globales dans un domaine périodique. Version française abrégée Dans cette note, nous nous intéressonsà l'étude de l'existence de solutions faibles globales en temps d'un modèle bicouche visqueux de Saint-Venant dérivé dans [6]. Notons que dans le cas d'une couche, dans [1] et [4] les auteurs ont obtenu l'existence de solutions faibles grâceà une nouvelle entropie (BD) introduite premièrement par Bresch et Desjardins dans [1]. On peut trouver d'autres résultats sur l'existence de solutions pour des modèles bicouche de Saint-Venant dans [3] et [5]. Dans ces modèles, les termes couplant les deux fluides compliquent le passageà la limite. Dans [7] uneétude du modèle bicouche mais où les termes de friction ontété simplifiés aété faite. Les termes de friction couplant les deux couches dans le