On the Construction of Approximate Solutions for the 1D Pollutant Transport Model

Yacouba Zongo,Boulaye Yira,Brahima Roamba

doi:10.5539/jmr.v12n3p1

Abstract

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, Zabsonr&eacute;, 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.

Highlights

We consider a bilayer model of immiscible fluids where the upper layer can be represented by a Reynolds lubrifications model and the lower layer by a shallow water model
The stability arguments in this paper apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model
This article was the subject of the construction of global weak solutions of a model of pollutant transport in dimension 1

Summary

Introduction

We consider a bilayer model of immiscible fluids where the upper layer can be represented by a Reynolds lubrifications model and the lower layer by a shallow water 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). (α > 0), see (Kitavtsev, Laurencot & Niethammer, 2011; Seemann, Herminghaus & Jacobs, 2001) This force of Van Der Waals allows us to lower the height of water which allows us to get around hypothesis made in (Roamba, Zabsonre & Zongo, 2017). We give the proof of existence Theorem including the limits passage in the section

Construction of Approximate Solutions

We have the additional regularities thanks to Corollary 1:

Conclusion