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

Abstract

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. RésuméNous 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 abrégéeDans 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]

Strong solutions for a 1D viscous bilayer shallow water model

“…The second proof of this section is that for the BD entropy estimate (18). Multiplying the mass equations by ' 0 " (h i," ) (for i = 1, 2), we get @ t ' " (h i," ) + @ x (' " (h i," ))v i," + ' 0 " (h i," )h i," @ x v i," = 0.…”

“…2 can be merged with the left hand side of Equation (18). Due to conditions (11) on the viscosities and (10) on ", the quantities (…”

“…In this paper, we are interested in another system of bilayer immiscible fluid obtained by derivation in [1]. These equations have been studied in [18,19]; the authors proved the existence of a global weak solution for viscous bilayer Shallow Water models with the BD inequality (but in two dimensions). This paper deals with strong solutions, in one dimension.…”

“…To obtain new bounds thanks to the BD entropy (in order to complete Corollary 3.3), it remains to control the last two terms of the left hand side of Equation (18). Since ' 0 " (h i," ) > 1 (for i = 1, 2), one can write:…”

Layer-averaged approximation of Navier–Stokes system with complex rheologies

“…Moreover, we can consider 𝑢 𝛼 ∈ P 0 as in [8] but 𝑤 𝛼 ∈ P 1 and the resulting model is also an extension of the model in that work to the case of non-layerwise constant shear 𝜏 𝑥𝑧,𝛼 . In particular, the difference between considering 𝑤 𝛼 layerwise linear or constant is the fact of including the term involving 𝜁 𝑥𝑧,𝛼 in 𝑄 𝑢 (42).…”