Abstract
We consider a one-dimensionnal bilayer model coupling shallow water and Reynolds lubrication equations with a molecular interactions between molecules. These molecular interactions give rise to intermolecular forces, namely the long-range van der Waals forces and short-range Born intermolecular forces. In this paper, an expression will be used to take into account all these intermolecular forces. Our model is a similar model studied in (Roamba, Zabsonré & Zongo, 2017). The model considered is represented by the two superposed immiscible fluids. A similar model was studied in (Zabsonré Lucas & Fernandez-Nieto, 2009) but the authors do not take into account the intermolecular forces. Without hypothesis about the unknowns as in (Roamba, Zabsonré & Zongo, 2017), we show the existence of global weak solution in time in a periodic domain.
Highlights
The purpose of this paper is devoted to the existence of global weak solutions to 1D pollutant transport model
A similar model was studied in (Zabsonre Lucas & Fernandez-Nieto, 2009) but the authors do not take into account the intermolecular forces
This model can be used to simulate for instance the evolution of a pollutant fluid over water. It is a bilayer model of two immiscible fluids where the upper layer can be represented by a Reynolds lubrifications model and the lower layer by a shallow water model
Summary
The purpose of this paper is devoted to the existence of global weak solutions to 1D pollutant transport model. In (Roamba, Zabsonre & Zongo, 2017), the authors showed the existence of global weak solutions of similar model derived in (Fernandez-Nieto, Narbona-Reina & Zabsonre 2013). To lead well this result, the authors considered the condition according to which h2 ≤ h1 (the water layer is more important than the layer of the pollutant). Our contribution is to show the existence of global weak solutions for a similar model studied in (Roamba, Zabsonre & Zongo, 2017) taking into acount the presence of intermolecular forces, namely long-range attractive van der Waals and short-range Born repulsive intermolecular forces without any hypothesis on unknown. We will give the proof of existence Theorem including the limits passage
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