On the Existence of Global Weak Solutions to 1 D Pollutant Transport Model

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.


Introduction
The purpose of this paper is devoted to the existence of global weak solutions to 1D pollutant transport model.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.
This work follows the work done in (Roamba, Zabsonré & Zongo, 2017).In (Roamba, Zabsonré & Zongo, 2017) as in this present work, we use a model of transport of pollutant in 1D formally derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013).In (Roamba, Zabsonré & Zongo, 2017), the authors showed the existence of global weak solutions of similar model derived in (Fernandez-Nieto, Narbona-Reina & Zabsonré 2013).To lead well this result, the authors considered the condition according to which h 2 ≤ h 1 (the water layer is more important than the layer of the pollutant).We suppose in this paper the existence of molecular interactions between molecules and this leads us to use the force of Van Der Waals which is given by V(h 1 ) = (Kitavtsev, Laurenc ¸ot & 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, Zabsonré& Zongo, 2017).The model studied is read as follow: (2) where (t, x) ∈ (0, T ) × [0, 1].
These equations represent a system composed of two layers of immiscible fluids.
We denote h 1 , h 2 respectively, the water and the pollutant heights, u 1 is the water velocity.We introduce the ratio of densities r = ρ 2 ρ 1 where ρ 1 and ρ 2 denoted respectively the densities of the water and the pollutant.ν 1 is the kinematic viscosity; g is the constant gravity.
The coefficients σ, r 1 and β are respectively the coefficients of the interfaz tension, quadratic friction and positive slip length parameters.a and b are respectively depending on the friction at the interfaz and coefficient of the viscosity of the pollutant.α and ε are positive constants.
From a theoretical point of view, several studies have been carried out on the existence of global weak solutoins of shallow-water equations model.We cite among others some results concerning the existence of global weak solutions of the transport models.
The existence of weak solutions for a viscous sedimentation model is obtained by assuming smallness of the data in (Toumbou, Le Roux & Sene, 2007) in which the viscosity term is the form −ν∆u.The authors in (Zabsonré Lucas & Fern ndez-Nieto, 2009) studied the stability of global weak solutions for a sediment transport model in two-dimensional with the term νdiv(h∇u).
The stability result is obtained without any restriction on the data and by using a mathematical entropy introduced first in (Bresch & Desjardins, 2003) namely BD entropy.The BD entropy allows us to get regularity on ∂ x √ h, see lemma 2. The authors in (Bresch, Desjardins & Grard-Varet, 2007;Bresch, Desjardins & Lin, 2003;Marche, 2007;Mellet & Vasseur, 2007) use the BD entropy inequality to get the existence results of global weak solutions for Shallow-Water and viscous compressible Navier-Stokes equations.In (Kitavtsev, Laurenc ¸ot & Niethammer, 2011) the authors proved the existence of global weak solutions for one-dimensional lubrication models that describe the dewetting process of nanoscopic thin polymer films on hy-drophobyzed substrates and take into account a large slippage at the polymer-substrate interface.In their work, the authors have used an intermolecular force that is very crucial in ours.
In the analysis, our contribution is to show the existence of global weak solutions for a similar model studied in (Roamba, Zabsonré & 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 complete the system studied with the initial conditions h 1 (0, x) = h 1 0 (x), h 2 (0, x) = h 2 0 (x), (h 1 u 1 )(0, x) = m 0 (x) in (0, 1). (5) where φ will be defined later on (see ( 13)).
Our paper is organized as follows.First of all, we will give in the Section 2, the definition of global weak solutions, secondly, we will establish a classical energy equality and the "mathematical BD entropy", which entails some regularities on the unknown.Thirdly, we will give a proposition allowing us to limit inferiorly the height of water which is very fundamental for the continuation since this limit study gives us additional regularities on the unknowns.Fourthly, we will give an existence theorem of global weak solutions.And finally, we will give the proof of existence Theorem including the limits passage.

Main Results
Definition 1.We say that (h 1 , h 2 , u 1 ) is a weak solution of ( 1)-( 3), with the initial condition verifying the entropy inequality (12) for all smooth test functions ϕ = ϕ(t, x) with ϕ(T, ) = 0, we have: Lemma 1. (Energy inequality) For classical solutions of the system (1)-( 3), the following inequality holds d dt where the potential function U is the indefinite integral of V defined by U(h Proof: First, we multiply the momentum equation by u 1 and we integrate from 0 to 1.We use the mass conservation equation of the first layer for simplification.Then, we obtain Now, let us simplify each term: The equation for the thin film flow gives us : ) and we have: Substituting all these terms in (11), we get (10) by integrating under 0 to T .

The term
Corollary 1.Let (h 1 , h 2 , u 1 ) be a solution of model (1) − (2).Then, thanks to Lemma 1 we have: We will need furthermore, some additional regularities on h 1 and this will be achieved through an additional BD entropy inequality presented in the next lemma.
Lemma 2. For smooth solutions (h 1 , h 2 , u 1 ) of model (1) − (3) satisfying the classical energy equality of the Lemma 1, we have the following mathematical BD entropy inequality: ] We need these results to prove the above lemma.
Proposition 1.If h 1 has the regularities established in corollary 2.1, then there exists constants c 1 and c 2 such as 0 < c 1 < h 1 < c 2 .
Proof: (Lemma 2) Let us multiply the equation ( 2) by ∂ x φ(h 1 ), integrate with respect to x and use an integration by parts, and using (1), we have: On the one hand, a further integration by parts of the first integral of ( 14), equation ( 1), and the energy equality (10) give On the other hand,we can write the third and the fourth integrals of ( 14) as (See (Kitavtsev, Laurenc ¸ot & Niethammer, 2011)).
In Lemma 2 all the terms, excepted are controlled since they have the good sign.The control of the term ba, Zabsonr?& Traor? 2016).The term be absorbed thanks to the work done in (Kitavtsev, Laurençot & Niethammer, 2011).It remains for us to control the term Proposition 2. There exists a constant C such as ∥ (ε Proof: We have: we can write: We will now look at the two terms to the right of the above inequality separately.For the first one, we have: . For the second one, since h 2 ∈ L ∞ (0, T ; H 1 (0, 1)) and (Marche, 2007) for instance.Which completes the proof.
Lemma 3.For classical solutions of the system (1) − (3) with a first component h 1 , we have ] .
Proof: Using the elementary inequality for h 1 > 0, and the definition of E, we obtain: that the statement of the lemma follows.

In the Corollary 1, the estimate
2. We have the additional regularities thanks to Corollary 1: Remark 4. We have the following additional regularities: 1. h 1 and h 2 are bounded in L ∞ (0, T ; L 2 (0, 1)).

Convergences
The objective of this section is to prove the Theorem 1.Let (h k 1 , h k 2 , u k 1 ) be a sequence of weak solutions with initial data Such approximate solutions can be built by a regularization of capillary effect.
We want to prove the strong convergence for this term.We know that 2 ); consequently, Fatou lemma reads: In particular, m is equal to zero for almost every x where h 1 (t, x) vanishes.Then, we can define the limit velocity taking u k 1 (t, x) = m(t,x) h 1 (t,x) since h 1 (t, x) 0 .So we have a link between the limits m(t, x) = h 1 (t, x)u 1 (t, x) and: Moreover, we can use Fatou lemma again to write As m k and h k 1 converge almost everywhere, the sequence of Moreover, for all M positive √ h 1 u 1 1 |u|≤M ( still assuming that h k 1 does not vanish).Then, almost everywhere, we obtain the convergence of ( ) k Finally, let us consider the following norm: is bounded in this space.So, as we have seen previously, the first integral tends to zero.Let us study the other two terms: for all M > 0. When M tends to the infinity, our two integrals tend to zero.Then

Conclusion
In this paper, we shown the existence of global weak solutions to 1D pollutant transport model.We have considered the existence of molecular interactions between the molecules and we added the term ε∂ 2 x h 2 to the level of the transport equation.This has led us to the use of a molecular interaction force in this work.We obtained a result of existence of