Pollution Transfer as Optimal Mass Transport Problem

In this paper, we use mass transportation theory to study pollution transfer in porous media. We show the existence of a L2−regular vector field defined by a W1,1− optimal transport map. A sufficient condition for solvability of our model, is given by a (non homogeneous) transport equation with a source defined by a measure. The mathematical framework used, allows us to show in some specifical cases, existence of solution for a nonlinear PDE deriving from the modelling. And we end by numerical simulations.


Introduction
Pollution problems are interesting and important topic in physics, in mathematical physics, in chemistry, in biology and even in complex sciences.And it is a great challenge to understand the problems in all its dimensions.But there are numerous mathematical obstacles to tackle this challenge even if an acceptable mathematical model is considered.In fact, in general, there are more mathematical unknown than relations (equations inequations, inclusions..).And then, in front of these obstacles, one used to introduce in some cases, physical experimental law in order to reduce the number of variables.Thus, the mathematical study becomes possible with suitable assumptions.We invite the reader to see papers [I.Faye, A. Sy and D. Seck (2008)], [L.Ndiaye, A. Sy and D. Seck (2012)] and the references therein for more details.One of our aim is to propose ways to weaken some hypotheses in such a study.In fact, we shall avoid to use always experimental laws to reduce unknown variables.In this paper, after the modelling of the physical problem, we endeavour to use mass transportation and PDE theories to study pollution in porous media.
Given two distributions µ and ν on R d with equal total mass, the classical generic Monge transportation problem consists in finding maps T : R d → R d verifying T # µ = ν, i.e ( maps transport µ to ν) and minimizing : where c is the cost when moving µ to ν.These maps will be called optimal transport maps.For the existence of solutions, we recommend to see [L.Ambrosio (2000), (2003)], [L.Caffarelli and al. (2002)], [L.C.Evans and W. Gangbo (1999)], [A.Pratelli (2003)].In particularly Sudakov have studied in [V.N.Sudakov (1979)] the existence of optimal map transportation when c(x, y) = ∥x − y∥ and µ is absolutely continuous with respect to the Lebesgue measure L d .
It is noted that there are many cases where the Monge transportation problem doesn't give a solution, Kantorovich considered the relaxed version of the Monge problem.In this framework, the transportation problem consists in finding among all admissible measures γ : R d × R d → R + having µ and ν as marginals, those which solve the minimization problem The Monge-Kantorovich problem obtained, depends only on the two distributions µ and ν, and the cost c which may be a function of the path connecting x to y.
When the unknowns of the problem are the distributions µ and ν, the Monge-Kantorovich mass transportation problem can modelise an optimal urban design problem.When the unknown is the transportation network, we can modelise an irrigation problem as well as design public transportation networks.
• ε(x, t), the effective porosity given by ε(x, t) = dV l dV total , where dV l is an element of the volume of the fluid and dV total an element of the total volume ; • σ(x, t), the porosity given by σ(x, t) = dV v dV total where dV v is an element of the volume of the vacuum in the medium.
If we consider Ω ∈ R n any elementary domain of the porous domain D, then the mass of the considered fluid is given by M(Ω, t) = ∫ Ω dm; where dm is a mass element of the fluid.In fact it is given by the following expression: dm = ρ(x, t)ε(x, t); where ρ(x, t) is the density. of the fluid For our model we shall use these notations: ρ s [kg/m 3 ] stands for the density of the solution given by ρ s = dm solution dv solution W(x, t) the fraction of the mass (concentration): dm solution is a mass element of the solution and dm solute is a mass element of the pollutant.

Mass Conservation Principle for the Solution
Let dm solution = ρ s dv solution = ρ s dv solution dv total dv toal = ρ s ε(x, t)dv total then the mass transport equation is given by The balance law claims that the variation of the mass with respect to the time in Ω is equal to the flux exchanged through the boundary of Ω with velocity V.This means Hence, By the Green formula we obtain

Mass Conservation Principle for the Pollutant
Here we consider for example that our pollutant liquid is: water mixed to chemical concentration.From the definition of W(x, t) and the expression of dm solution , we deduce the following relation And then the total mass is given by Using the balance law as in the previous subsection for this mass, we have: where J is the flux of dispersion diffusion.And finally we have ∂ ∂t

Momentum Conservation Principle
This section aims to complete the part devoted to the modeling by giving some basic formulas on the conservation of the momentum.And we sum up them in the following remark.
1.If the porous medium is homogeneous then the Darcy's law is given by where e 3 is third vector of the canonical basis of R 3 ; p is the pressure, ⃗ g = ge 3 is the gravity field, K the intrinsic permeability tensor, λ the dynamic viscosity and K/λ the hydraulic conductivity.For mathematical reasons, we are going to assume, if necessary the following ellipticity condition: 2. If we have some weak concentration the flux of dispersion diffusion J is determined by the Fick law where D be the tensor of dispersion diffusion.In many works it is assumed that the following ellipticity condition: is satisfied: D = (d i j ) 1≤i, j≤n ; d i j ξ i ξ j ≥ α 2 ∥ξ∥ 2 , where α 2 a positive constant.
3. In some interesting experimental cases, it is supposed that the density ρ s satisfies the following expression: ρ 0 , T 0 , p 0 are respectively the density, temperature, pressure at the initial time, they are data , T and p stand for the temperature and the pressure and they are unknown.And γ, β T , β p are given constants.
Finally the proposed model is summarized in a system of equations as follows: It is important to underline that the above system of equations is very complicated to solve by direct methods.It is easy to list that ρ s , p, T, W, ε are unknown for the two first equations, because from the last three equations some unknowns can be substituted in the two first ones.
We are then going, to make some realistic hypotheses.

Transport Equation and Mass Conservation Formulas
Before coming to our question let us recall some basic but important results.For additional details about these results and even there refinements see for example [L.Ambrosio (2003)], [L.Ambrosio (2000)] and [C.Villani (2008)].
At first for the mass conservation formula, lets consider a C 1 open set , noted Ω (or more generally a C 1 manifold), and T ∈ (0, +∞].Let ξ(t, x) be a measurable vector field defined on [0, T ) × Ω.Let (µ t ) 0≤t≤T be a time dependent family of probability measures on Ω, continuous in time for the weak topology, such that then the following two statements are equivalent: 1. µ = µ t (dx) is a weak solution of the linear transport partial differential equation 2. µ t is the law at time t of a random solution T t (x) of the following equation If moreover ξ is locally Lipschitz then (T t ) 0≤t<T defines a deterministic flow and the above second item can be rewritten as follows where γ 0 ∈ C 1 (M), for additional details see [B.Dacorogna and J. Moser (1990)], [J.Moser (1965)] and [C.Villani (2008)].In our case, since we work mainly in R N , we can use γ 0 (x) = ∥x∥ 2 , the square of the Euclidian norm.In some specific cases we shall use other functions γ 0 .For the sake of simplifying the computations we can omit the constant C without loss of generality, it is quite possible to do with.
Let µ 0 = ρ 0 ν, µ 1 = ρ 1 ν be two probability measures on an open set Ω, assume that ρ 0 , ρ 1 are bounded from below by a constant.Further we assume that ρ 0 and ρ 1 are locally Lipschitz and that equation (1−t)ρ 0 (x)+tρ 1 (x) with the associated flow (T t (x) 0≤t≤1 ) and a family (µ t ) 0<t<1 of probability measures by µ . So µ t satisfies the formula of conservation of mass, therefore µ t = (T t ) # µ 0 .In particular T 1 pushes µ 0 forward to µ 1 .Now, we are in situation to discuss on how to transform the general problem of pollution into a transport equation.This will be a guidance for our study of the subject as an optimal mass transport problem.Let us consider the general system describing the evolution of the pollution in a given porous medium.
Expanding the second equation of ( 6), we have And, thanks to the first equation, the above second equation becomes The main discussion is to see if it is possible to transform this latter equation into another one expressed by means of a measure µ t and a vector field ξ to be found.Let us suppose that J is given and µ t 0. For the vector field ξ(t, x) equal to J µ t (x) , it is easy to see that: To get a positive answer, it suffices that the following equation makes sense in a framework of functional spaces to be precised later.
But let us point out that it will be quite possible to give a meaning to µ t (x) as a distribution whenever W, V ρ and ε, are in the distribution space.The time derivative of the measure µ t is to be understood in the weak sense.
If we are in the situation where µ t exists, it will be interesting too, to look for the existence of ερ s and W.
Now, let us give two non null concentration distributions: x) and ν = e −γ 0 (x) vol(dx), we set µ 0 = νW 0 , µ 1 = νW 1 .In this case, for µ Replacing div J by its expression, we have: Finally we can claim that: whenever it is possible to solve the following system of PDE { And from that situation, we have: In the next section, we are going to be more precised by giving details on all what we have just done as discussions in this section.

Mass Transport Problem
In this section, we suppose that • the space of measures acting is a time-space domain Q =]0, T [×Ω where Ω is a bounded Lipschitz open subset of R d with outward normal vector n Ω .In the sequel we are going to consider T ≡ 1.
• the mass density ρ(t, x) at the position x and time t is a Borel measure supported on • the velocity field v(t, x) of a particle at (t, x) is a Borel vectorial measure supported on Q; • the velocity field ξ(t, x) of the flow at (t, x) is a Borel vectorial measure supported on The Monge-Kantorovich mass transportation problem consists in solving the following optimization problem: with the constraints: where Ψ is an integral functional on the R 1+d -valued measures defined on Q.Note that (10) is the continuity equation of our mass transportation model.
Then the following problem where Ψ is an integral functional on the R 1+d -valued measures defined on Q by with the constraints: (12) admits a solution.
The minimum value called W(µ 0 , µ 1 ) of the minimization problem is the square of the L 2 Kantorovich distance i.e.
Proof.Under the hypotheses of the theorem, it suffices to apply the proofs of the main results in [Y.Brenier (2003)] and [G.Buttazzo and al. (2007)].
Remark 3.2.As mentioned in [G.Buttazzo and al. (2007)], mass should move along • straight lines when Ω is a convex set; • geodesic curves when Ω is not convex.
• It is important to point out that the above theorem with if we consider : then we can say that the problem where Ψ is an integral functional on the R 1+d -valued measures defined on Q by with the constraints: (14) admits a solution.This means that (µ(t, x), V(t, x)) does exist.
• For additional details about the existence of the optimal transport and the candidate map, we refer the reader to some references such as [L.Ambrosio and al. (2005)], [F. Santambrosio (2015)] (Theorem 8.1 and the previous needed results), [J.Benamou and Y. Bernier (2005)].But let us give some hints that we shall connect to system of PDE modelling the pollution problem which is our subject of studies.
And we can state the following theorem.
Theorem 3.3.Let us consider the following functional environment defined by: V(µ 0 , µ 1 ) be the set of all (µ, ξ) = (µ(t, .),ξ(t, .))such that where P ac (R d )) is the space of absolutely continuous (with respect of Legesgue measure) probability measures on R d ; in fact it can be identified with a subspace of L 1 (R d ) and w * − P ac (R d )) stands for the set P ac (R d )) endowed with the weak − * topology.Then, under the above functional framework, to solve the following system of PDE standing for a model of pollution it suffices to be able to solve the following transport equation where the existence of V, ερ s , and µ t is ensured by the existence of the optimal transport maps.
Before proving this theorem by optimal mass transportation theory, let's give some foundamental and important notions and results ( see for instance [W.Gangbo (2004)]).
Let's suppose that in a region (place) X ⊂ R d , we have sand piles and in another region Y in R d , we have holes to be filled by the sand.
In optimal mass tranport the aim is to minimize the cost to move the sand from X to Y.
For this, we shall introduce a model expressed as follows: Let µ o be the mass density of the sand and µ 1 the mass density of the holes.
Suppose that: for any (x, y) ∈ X × Y we associate c(x, y) > 0 the transport cost from the point x to the point y.Let T be the strategy of transport used for which x is moved to T x.
T satisfies the conservation of the mass expressed as follows: Whenever this above equality is satisfied, we say that T is a strategy to move µ o onto µ 1 and the notation used, is And then the total cost to transport the sand with density equal to µ o onto the holes with capacity (density ) µ 1 is given by: The Monge's problem consists in finding a solution of the following problem inf

Relaxation of the Monge's Formulation: The Kantorovich's Formulation
Let X and Y be the supports respectively of the measures µ o et µ 1 , and let T : X −→ 2 Y be a plan (relation) where 2 Y = {A, A ⊂ Y}.
For any x ∈ X, we associate the measure γ x supported by the set T x which explains us how the mass located in x is distributed through T x.Then the transport cost from x to T x is given by: ∫ {y∈Y /∃x∈X ; y=T x} c(x, y) dγ x (y).
The total transport cost from µ o to µ 1 is equal to It is convenient to define a measure γ on X × Y which contains the information coded in [T, {γ x } x∈X ] by : ∫ and the measure γ has to satisfy the following conservation mass conditions: ∀A ⊂ X and B ⊂ Y borelian sets.
The Kantorovich's problem, in terms of γ is a relaxation of the Monge's problem.
And it is quite possible to extend the set T (µ o , µ 1 ) of plans defined by T : X −→ Y such that T # µ o = µ 1 to a bigger set Γ(µ o , µ 1 ) and the function defined by I : Thus if T (µ o , µ 1 ) ∅ , then we have inf Ī.
Definition 3.4.Let µ o and µ 1 be two measures defined respectively on X and Y.
i) The map T : X −→ Ycarries µ o to µ 1 and one notes ii) Let γ be a measure defined on X × Y, then the projection pro j Xγ (resp.proj Yγ ) is a measure defined on X (resp.onY) as follows: iii) µ o and µ 1 are to be said marginals of a measure γ on X × Y if µ o = pro j X γ and µ 1 = pro Y γ and we say that γ ∈ Γ(µ o , µ 1 ) and γ is the transport scheme from µ o in µ 1 .
Let 's review the Monge's problem: The aim is to find a minimum for the following functional ∫ where T ∈ T (µ o , µ 1 ) i.e T # µ o = µ 1 .And the Kantorovich's problem consists in finding minimum for the following functional ∫ where γ ∈ Γ(µ o , µ 1 ).
Remark 3.5.It is not easy to show the existence of the minimum for the Monge's problem.But for the Kantorovich's, it is possible most of the time to show easier the existence of the minimum.
We can sum up one of the first main results in optimal mass transportation as follows: Theorem 3.6.Given µ and ν probability measures on a compact domain Ω ⊂ R d there exists an optimal transport plan γ for the cost c(x, y) = h(x − y) with h strictly convex.It is unique and of the form (id, T ) # µ, provided µ is absolutely continuous and ∂Ω is negligible.Moreover, there exists a Kantorovich potential ϕ, and T and the potentials ϕ are linked by T (x) = x − (∇h) −1 (∇ϕ(x)).
For the proof see for instance [F.Santambrosio 2015].
Proof. of the theorem 3.3 Existence of the two optimal transport maps T t and S t : From this step, we shall deduce the existence of ερ s ; V.
It suffices to apply the Benamou-Brenier theorem and techniques in the proof see for instance [J.Benamou and Y. Bernier (2000)], and [F.Santambrosio (2015)] to each of the following compatibility equations: As a consequence of the above theorem 3.6, we can particularize the above theorem to the quadratic case c(x, y) = 1 2 |x−y| 2 , thus getting the existence of an optimal transport map for a convex function u.
, and we define ξ 1 (t, x) to be 0 whenever µ t ≡ 0. And in the same way we have t if m t 0, and we define ξ 2 (t, x) to be 0 whenever m t ≡ 0. Let us note that ξ 1 , ξ 2 ∈ L 2 (dµ(t, x)).In fact: where |µ ′ |(t) denotes the metric derivative at time t of the curve t → µ(t, .)with respect to the Wasserstein distance d = W 2 .
The above equality holds also for ξ 2 .
The general definition for the metric derivative, can be expressed as follows: Let P 2 (Ω) = {µ ∈ P(Ω) : And finally, having at hands µ t , ερ s and V.

Existence of the Two Optimal Transport Maps W :
To find W, we have to look at the following PDE , ∂(ερ s W) After expansion, it is equivalent to : Let us set now µ t ξ 1 = J, then ∂µ t ∂t + divJ = 0. Combining the two last equalities, we get From the first part of the proof, µ t is known and then ∂µ t ∂t is a datum.By the hypothesis ( * * ) we get W.
This theorem makes sense if it is possible to solve the above transport equation which is not an easy task.

Discussion
Let set m t (x) := m(t, x) = ε(t, x)ρ s (t, x).The Cauchy problem becomes It is important to point out that the transport equation si still a great of interest for the mathematician community working on PDE topic and and physicists.
Let's suppose at first that m t 0, and set f t = 1 m t ∂µ t ∂t and V = b t .The question is to see if under BV or Sobolev regularity of f t and b t , the above Cauchy problem gets solution.And if yes is it possible to get information about the uniqueness, to define a flow?
The authors are not aware that the answer is always positive in very weak hypotheses or if the question is closed.But they know that relevant works due to [R.Diperma and P.L. Lions (1989)], [L.Ambrosio (2004)], [L.Ambrosio (2005)] and [C. De Lellis (2007)] on these questions have been done.
In [R.Diperma and P.L. Lions (1989)], DiPerna and Lions showed existence, uniqueness and stability of regular Lagrangian flows for Sobolev vector fields with bounded divergence.
Let's now give some particular cases for the solvability of our transport equation: Diperma and P.L. Lions (1989)], pp 4 and • When m t := m(x) does not depend on t and m(x) > 0, then if V satisfies the same hypotheses as in the above first item, formally we should have W(t, X(t)) = µ t (X(t)) m(x) + W 0 (x) in the distribution sense; where Ẋ = V, X(0) = x.
After these two particular cases, let us point out that the regularity of V is a priori L 2 (dµ(t, x)) and f t is a measure and may be singular.
Another important fact is the regularity of the optimal transport map.In fact, since in our case the cost function c is in quadratic form as explained above, for We end this discussion by claiming that the questions seem to be not easy according and it should be interesting to come back to this problem in next works.
In the sequel of this paper we shall be particularily interested by the case where ξ(t, x) = J (1−t)W 0 (x)+tW 1 (x) and the linear interpolation for the measure: µ t = (1 − t)µ 0 + tµ 1 .And we shall study the following system of PDE combined with initial and boundary condition if necessary A particular non linear boundary value problem is studied too.

Partial Differential Equations
In this section we are going to solve one type of non linear partial equations deriving from basic model (5).

Combination with an Experimental Law
This subsection is devoted to deal with a particular case considering an experimental law with suitable hypotheses which are expressed as follows: • H-1 the fluid density of the solution ρ s > 0 and is a constant; • H-2 the hydraulic conductivity tensor K λ and D are positive constants; • H-3 the evolution is isotherm i.e. the temperature T in the medium is a constant.
Because of the hypothesis H-1, the equations ( 1) and (2) become Using hypothesis H-3, we can establish a relation between the pressure p and the concentration W: Replacing J and q by its values, we have: The equation ( 27) implies that Wdivq = −W ∂ε ∂t and therefore Without loosing of generality in the reasoning , we give up the term −ρ s g K µ ∂W ∂z = 0 in the above equation.Finally we are going to consider in the sequel, the below Cauchy problem translating the evolution of the pollution: Let us set a = γK β p µD and b = γ Dβ p , then the transport equation becomes: Remark 3.8.Let's suppose that the space is equipped with a probability measure ν(dx) = exp(−a γ 0 )vol(dx).
If we define a vector field that assumed locally Lipschitz by: with associated flow (X t (x)) 0≤t≤1 and a family (µ t ) 0≤t≤1 of probability measures given by: Then we remark that And finally we have the following relation translating the continuity equation: Next we are going to focus the following type of PDE where f is a bounded function with respect to the time t and the variable x.It is easy to remark that these types of equations translate a family of equations translating the evolution of the pollution.

Let us consider
Replacing the expression ( 35) in (34), we obtain in the system Theorem 3.9.Let W(t, x) be the solution of ( 36).Setting u(t, x) := ψ(W(t, x)), then u(t, x) is the solution of the following partial differential equation, with ψ(x) solution of the following ordinary differential equation.
In order to compute ψ, let us suppose that the integral Conversely, if ψ is in the form (40) and u(x, t) the solution of (37), the function W(x, t) = ψ −1 (u(x, t)) is solution of (36), see [I? Faye, A. Sy and D. Seck ( 2008)] for details.

Numerical Simulations
In the numerical results, we suppose that the porosity ε = 1 so that, (x,t) .
Our process to realize numerical simulations is organized as follows: 1 And finally we do some numerical representation in order to illustrate the theoretical study of our subject.

Example 1
In this example, the following data are used:

Extensions
This section is devoted to set issues as perspectives for this paper.
1. Instead of limiting numerical simulations in the case of linear interpolation we think that it should be interesting to try numerical analysis in the case of optimal mass transportation by considering:  (45) and minimal additional hypotheses needed, like in [J.Benamou and Y. Bernier (2000)], [L.Ambrosio and al. (2005)] and [F. Santambrosio (2015)].
2. In the case where γ 0 = 0, ε is the only constant: by taking into account mass conservation principle the following equation is reduced to: For µ 0 = W 0 ν and µ 1 = W 1 ν where ν is taken as above, J = F (∇ x W, W) where F is a given function and ξ(t, x) = F (∇ x W) (1−t)W 0 (x)+tW 1 (x) , we have: It would be interesting to investigate the two following cases: • if F (∇ x W, W) = |∇ x W| p−2 ∇ x W, 1 ≤ p < +∞ we have in the last equation of the above system, the p-Laplacian equation ; • if F (∇ x W, W) = ρ s ∇ x W m , m ≥ 1 we have at hands, the familiar term of the porous media equation.

W
1 1+t and a Neumann non-homogeneous boundary condition on Γ N an a Neumann homogeneous boundary condition on Γ ∂ερ s ∂t + div(ρ s εV) = 0 ερ(0, x) = m 0 a convenient initial data −(ερ s ∂W ∂t + ερ s V • ∇W) − ∇γ 0 .J = 1 ν ∂µ t ∂t , is Lipschitz, continuous in x, uniformly in t and uniformly bounded, then every solution of the same continuity equation with µ t << L d for every time is necessarily obtained as µ t = (Φ t ) # µ 0 , where Φ t is the flow associated to ξ Another important reminder for our work is the Moser's technique for coupling smooth positive probability measures.And this technique works in the case of Riemannian manifolds M equipped with a reference probability measure . We solve the below boundary value problem by Matlab software;