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.

This paper is devoted to the study of pollutant transport model by water in dimension one. The model studied extend the results obtained in ( Roamba, Zabsonré & Zongo, 2017) . However, our model does not take into account cold pressure term and the quadratic friction term as in (Roamba, Zabsonré & Zongo, 2017) which are considered regularizing terms to show the existence of global weak solutions of your model. Without these regularizing terms, we show the existence of global weak solutions in time with a periodic domain.

The purpose of this paper is to build sequences of suitably smooth approximate solutions to the 1D pollutant transport model that preserve the mathematical structure discovered in (Roamba, Zabsonré, Zongo, 2017). The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. We show that when the Reynold number goes to infinity, we have always an existence of global weak solutions result for the corresponding model.