# Fixed-Point and STILS Method to Solve a Coupled System of Transport Equations

Authors: Daouda Sangare
Journal: Journal of Applied Mathematics
 (2022)

---

## Abstract

In this paper, a coupled system of two transport equations is studied. The techniques are a fixed-point and Space-Time Integrated Least Square (STILS) method. The nonstationary advective transport equation is transformed to a “stationary” one by integrating space and time. Using a variational formulation and an adequate Poincare inequality, we prove the existence and the uniqueness of the solution. The transport equation with a nonlinear feedback is solved using a fixed-point method.


---

## Body

## 1. Introduction

This work is motivated by the crystal dissolution and precipitation model in saturated porous medium [1]. In [1], the authors present a macroscopic model describing ions transport by fluid flow in a porous medium undergoing dissolution and precipitation reactions. Such models received much attention during the past years (e.g., [2–4]).All these papers deal with the upscaled formulation of the phenomena. A rigorous justification, starting from a well-posed microscopic (pore-scale) model and applying a suitable upscaling, has been given for important classes of problems. For instance, [5] presents homogenization as a method for upscaling and contains an overview with particular emphasis on porous media flow including chemical reactions. In this respect, we also mention [6], where the reaction rates and isotherms are linear (see also [7] or [8]), where nonlinear cases as well as multivalued interface conditions are analyzed.In [9], the authors study the pore-scale analogue of the model proposed in [1], which is built on Stokes flow in the pores, transport of dissolved ions by convection and diffusion, and dissolution-precipitation reactions on the surface of the porous skeleton. They use regularisation techniques and a fixed-point argument to obtain existence of a weak solution in general domains. The results obtained are a rigorous justification of the macroscopic model in [1].In this paper, we give a mathematical analysis of the macroscopic model in [1] using the fixed-point theorem and the STILS method to solve the transport equations.The least squares method is widely used to solve partial differential equations. We can refer to [10, 11] for application on elasticity and fluid mechanics problems. Some general mathematical results have been obtained for this method in the case of first-order time-dependent conservations laws. With the space-time objects below, the STILS method transforms a nonstationary problem into a “stationary” problem by integrating space and time. This “stationary” problem is of advective form. For instance, in [12], an equivalence between the advective formulation and that of anisotropic diffusion is established.The STILS method is originated to [13, 14]. In [13, 14], a least squares method is used to solve a 2D stationary first-order conservation equation with regularity assumptions on the advection velocity. A comparison between the least squares solution and the renormalized solution in the sense of [15] for some equations is given in [16]. The STILS method leads to some numerical schemes which are much simpler than the usual ones (like the streamline diffusion method, the characteristic method, and the discontinuous finite element method with flux limiter). Some numerical examples are presented in [17–19].In this paper, we make use of both the fixed-point theorem and the so called STILS method to solve nonlinear transport equations in the model described below.This paper is organized as follows. Section2 provides a brief presentation of the model proposed in [1]. Assumptions and preliminary results useful to the resolution are also presented. Section 3 is devoted to an existence and uniqueness theorem. The fixed-point theorem and the STILS method are both used to deal with the above mentioned model.


## 2. Model Equations and Problem Description

### 2.1. Model Equations

It is useful to recall the model equations suggested in [1] without going into the details. We assume having two species M1 and M2, for example, ions, say M1 being a cation and M2 an anion. In addition, there may be a crystalline solid M12 present at the porous skeleton. M1 and M2 may precipitate at the surface of the porous skeleton to form M12, and conversely, the crystalline solid may dissolve. The stoichiometry of the reaction is supposed to be as follows:
(1)M12⇌nM1+mM2.n and m denote positive numbers. Let ci,i=1,2, be the molar concentration of Mi in the solution relative to the water volume, and let c12 be the molar concentration of M12 relative to the mass of the porous skeleton. The particle M12 is attached to the surface of the porous skeleton and thus is immobile. The conservation of the corresponding total masses leads to the partial differential equations
(2)∂∂tθc1+nρ∂∂tc12−divθD∇c1−q∗c1=0,(3)∂∂tθc2+mρ∂∂tc12−divθD∇c2−q∗c2=0,where the water content θ is supposed to be constant and not affected by the reaction (1), ρ is the bulk density, D is the diffusion/dispersion tensor, and q∗ is the specific discharge vector. If we define
(4)c=mc1−nc2,then equations (2) and (3) imply that the quantity c verifies
(5)∂∂tc−divD∇c−q∗θc=0.Another equation forc12 results from a description of the precipitation and dissolution processes. Following the detailed discussion in [1], we have
(6)ρθ∂∂tc12−k∗rp−rd=0,where rd and rP=kprc1,c2 are, respectively, the dissolution and precipitation rates and k∗ the reaction velocity. r is a nonlinear smooth nonnegative function depending on c1 and c2. A typical example is leading to
(7)rx,y=xnym.Summarizing the discussion done in about precipitation-dissolution reaction, we have for the crystalline solid the equation(8)ρθ∂∂tc12∈k∗kprc1,c2−kdHc12or equivalently
(9)ρθ∂∂tc12=k∗kprc1,c2−kdw,where
(10)w∈Hc12,which means0≤w≤1 for c12=0 and w=1 for c12>0.H is the set-valued Heaviside function defined by
(11)Hu=1foru>0,0,1foru=0,0foru<0.We make now some assumptions which lead to the model we study in this paper. Equation (4) gives
(12)c2=1nmc1−c.Then, in this model, we consider equations (2), (5), and (9).If the dispersive transport is negligible compared to the advective transport, it is reasonable to tendD to zero. This assumption cancels the corresponding terms in (2) and (5). We assume also that c12>0 anywhere; it means w=1 in (9). The velocity of the solute transport q=q∗/θ in (2) is the gradient of the hydraulic potential. Setting
(13)c12≔nρθc12,α≔k∗kp,β≔kdkp,the model equations have the following form:
(14)∂∂tc1+∂∂tc12+q·∇c1=0,(15)∂∂tc+q·∇c=0,(16)∂∂tc12=αrc1,1nmc1−c−β.In this paper, we study the coupled system (14)–(16) in addition to the appropriate boundary conditions.


### 2.2. Problem Description

#### 2.2.1. Notations

LetΩ⊂ℝd,d=1,2,3, be a domain with a Lipschitz boundary ∂Ω satisfying the cone property.IfT>0 is given, set ΩT=Ω×0,T. Let
(17)Γ−=x∈∂Ω,qx·nx<0,Γ+=x∈∂Ω,qx·nx>0,where nx is the outer normal to ∂Ω at x. Let
(18)ΓT−=Γ−×0,T∪Ω×0,a=q,1t,∇t=∇,∂∂tt.LetHmΩ be the classical Hilbert space of order m, and
(19)H01Ω=φ∈H1Ω;φ=0on∂Ω.We also define the following Hilbert spaceV:
(20)V=ϕ∈L2ΩT;a·∇tϕ∈L2ΩT,equipped with the graph norm:
(21)vV2=vL2ΩT2+a·∇tvL2ΩT2,∀v∈V.Let(22)Ha,ΩT;Γ−=ϕ∈Vsuchthatϕ=0onΓ−×0,T.The problem consists in finding(23)c,c1:ΩT⟶ℝ,satisfying the following partial differential equation system
(24)P1∂c∂t+q·∇c=0inΩT,c=gonΓT−,cx,0=c0xinΩ,P2∂c1∂t+q·∇c1=rc1,1nmc1−c−βinΩT,c1=g1onΓT−,c1x,0=c10xinΩ.Equations (14) and (16) give system P2. In systems P1 and P2, initial conditions and inflow boundary conditions are given for c and c1. The technique is to solve first P1 and after P2. With the solutions c and c1 and equation (16), we obtain c12.


#### 2.2.2. Preliminary Results

The STILS method will be used to solve the problemP1 like in [12, 20]. The method leads to a variational formulation problem, and we use the classical Lax-Milgram theorem to find the solution. So we give the following lemma to prove the bilinear map coercivity.Lemma 1 (curved Poincaré inequality).
There exists a constantk=2T such that
(25)uL2ΩT≤ka·∇tuL2ΩT,∀u∈Ha,ΩT;Γ−.The proof is in [20].LetHX,x,t be a polynomial with respect to X with bounded coefficients hi defined in ΩT.
(26)HX,x,t=∑i=0nhix,tXi.Letai,i=0,⋯,n be defined by
(27)ai=supΩThix,t.P2 is also a transport problem with a nonlinear feedback. We use both STILS method and fixed-point theorem to solve the problem. So we need c1kk∈ℕ to be bounded in L∞ΩT; hence, we give the following lemma.Lemma 2.
LetT be a positive real number and u0∈DΩT a positive function such that
(28)T<supΩTu0∑i=0naisupΩTu0i.
Letukk∈ℕ be the sequence of functions defined by
(29)a·∇tuk=Huk−1,x,tinΩT,uk=0onΓT−.
Then, we have(30)∀k∈ℕ,supΩTuk≤supΩTu0.Proof.
Assume that there existsk∈ℕ∗ such that
(31)supΩTuj≤supΩTu0,∀j=0,⋯,k−1.
We know that the velocity fieldq is bounded in its time component and has a constant dot product (equal to 1, too) with the vector 0,⋯,0,1. This ensures (see Proposition 7 in [12]) that ΩT is filled by the characteristics in the following ways:
There exists anS>0 such that for almost each point x,t of ΩT, there exists an integral curve
(32)ζ:0,S⟶ΩTsuchthat∂ζ∂s=aζthat connect x,t to the space-time inflow boundary:
(33)ζ0=x0,t0∈ΓT−,ζt=x,t,∀t∈0,S.
On this integral curve (characteristic), system (29) gives
(34)ukxt,t=∫t0tHuk−1,xs,sds.
Hence,(35)ukxt,t≤T∑i=0naisupΩTu0i≤supΩTu0,thanks to (28). Moreover, since ΩT is filled by characteristics, we have
(36)supΩTuk≤supΩTu0.
Finally, we conclude that for anyn∈ℕ, supΩTun≤supΩTu0. This ends the proof of the lemma.


## 2.1. Model Equations

It is useful to recall the model equations suggested in [1] without going into the details. We assume having two species M1 and M2, for example, ions, say M1 being a cation and M2 an anion. In addition, there may be a crystalline solid M12 present at the porous skeleton. M1 and M2 may precipitate at the surface of the porous skeleton to form M12, and conversely, the crystalline solid may dissolve. The stoichiometry of the reaction is supposed to be as follows:
(1)M12⇌nM1+mM2.n and m denote positive numbers. Let ci,i=1,2, be the molar concentration of Mi in the solution relative to the water volume, and let c12 be the molar concentration of M12 relative to the mass of the porous skeleton. The particle M12 is attached to the surface of the porous skeleton and thus is immobile. The conservation of the corresponding total masses leads to the partial differential equations
(2)∂∂tθc1+nρ∂∂tc12−divθD∇c1−q∗c1=0,(3)∂∂tθc2+mρ∂∂tc12−divθD∇c2−q∗c2=0,where the water content θ is supposed to be constant and not affected by the reaction (1), ρ is the bulk density, D is the diffusion/dispersion tensor, and q∗ is the specific discharge vector. If we define
(4)c=mc1−nc2,then equations (2) and (3) imply that the quantity c verifies
(5)∂∂tc−divD∇c−q∗θc=0.Another equation forc12 results from a description of the precipitation and dissolution processes. Following the detailed discussion in [1], we have
(6)ρθ∂∂tc12−k∗rp−rd=0,where rd and rP=kprc1,c2 are, respectively, the dissolution and precipitation rates and k∗ the reaction velocity. r is a nonlinear smooth nonnegative function depending on c1 and c2. A typical example is leading to
(7)rx,y=xnym.Summarizing the discussion done in about precipitation-dissolution reaction, we have for the crystalline solid the equation(8)ρθ∂∂tc12∈k∗kprc1,c2−kdHc12or equivalently
(9)ρθ∂∂tc12=k∗kprc1,c2−kdw,where
(10)w∈Hc12,which means0≤w≤1 for c12=0 and w=1 for c12>0.H is the set-valued Heaviside function defined by
(11)Hu=1foru>0,0,1foru=0,0foru<0.We make now some assumptions which lead to the model we study in this paper. Equation (4) gives
(12)c2=1nmc1−c.Then, in this model, we consider equations (2), (5), and (9).If the dispersive transport is negligible compared to the advective transport, it is reasonable to tendD to zero. This assumption cancels the corresponding terms in (2) and (5). We assume also that c12>0 anywhere; it means w=1 in (9). The velocity of the solute transport q=q∗/θ in (2) is the gradient of the hydraulic potential. Setting
(13)c12≔nρθc12,α≔k∗kp,β≔kdkp,the model equations have the following form:
(14)∂∂tc1+∂∂tc12+q·∇c1=0,(15)∂∂tc+q·∇c=0,(16)∂∂tc12=αrc1,1nmc1−c−β.In this paper, we study the coupled system (14)–(16) in addition to the appropriate boundary conditions.


## 2.2. Problem Description

### 2.2.1. Notations

LetΩ⊂ℝd,d=1,2,3, be a domain with a Lipschitz boundary ∂Ω satisfying the cone property.IfT>0 is given, set ΩT=Ω×0,T. Let
(17)Γ−=x∈∂Ω,qx·nx<0,Γ+=x∈∂Ω,qx·nx>0,where nx is the outer normal to ∂Ω at x. Let
(18)ΓT−=Γ−×0,T∪Ω×0,a=q,1t,∇t=∇,∂∂tt.LetHmΩ be the classical Hilbert space of order m, and
(19)H01Ω=φ∈H1Ω;φ=0on∂Ω.We also define the following Hilbert spaceV:
(20)V=ϕ∈L2ΩT;a·∇tϕ∈L2ΩT,equipped with the graph norm:
(21)vV2=vL2ΩT2+a·∇tvL2ΩT2,∀v∈V.Let(22)Ha,ΩT;Γ−=ϕ∈Vsuchthatϕ=0onΓ−×0,T.The problem consists in finding(23)c,c1:ΩT⟶ℝ,satisfying the following partial differential equation system
(24)P1∂c∂t+q·∇c=0inΩT,c=gonΓT−,cx,0=c0xinΩ,P2∂c1∂t+q·∇c1=rc1,1nmc1−c−βinΩT,c1=g1onΓT−,c1x,0=c10xinΩ.Equations (14) and (16) give system P2. In systems P1 and P2, initial conditions and inflow boundary conditions are given for c and c1. The technique is to solve first P1 and after P2. With the solutions c and c1 and equation (16), we obtain c12.


### 2.2.2. Preliminary Results

The STILS method will be used to solve the problemP1 like in [12, 20]. The method leads to a variational formulation problem, and we use the classical Lax-Milgram theorem to find the solution. So we give the following lemma to prove the bilinear map coercivity.Lemma 1 (curved Poincaré inequality).
There exists a constantk=2T such that
(25)uL2ΩT≤ka·∇tuL2ΩT,∀u∈Ha,ΩT;Γ−.The proof is in [20].LetHX,x,t be a polynomial with respect to X with bounded coefficients hi defined in ΩT.
(26)HX,x,t=∑i=0nhix,tXi.Letai,i=0,⋯,n be defined by
(27)ai=supΩThix,t.P2 is also a transport problem with a nonlinear feedback. We use both STILS method and fixed-point theorem to solve the problem. So we need c1kk∈ℕ to be bounded in L∞ΩT; hence, we give the following lemma.Lemma 2.
LetT be a positive real number and u0∈DΩT a positive function such that
(28)T<supΩTu0∑i=0naisupΩTu0i.
Letukk∈ℕ be the sequence of functions defined by
(29)a·∇tuk=Huk−1,x,tinΩT,uk=0onΓT−.
Then, we have(30)∀k∈ℕ,supΩTuk≤supΩTu0.Proof.
Assume that there existsk∈ℕ∗ such that
(31)supΩTuj≤supΩTu0,∀j=0,⋯,k−1.
We know that the velocity fieldq is bounded in its time component and has a constant dot product (equal to 1, too) with the vector 0,⋯,0,1. This ensures (see Proposition 7 in [12]) that ΩT is filled by the characteristics in the following ways:
There exists anS>0 such that for almost each point x,t of ΩT, there exists an integral curve
(32)ζ:0,S⟶ΩTsuchthat∂ζ∂s=aζthat connect x,t to the space-time inflow boundary:
(33)ζ0=x0,t0∈ΓT−,ζt=x,t,∀t∈0,S.
On this integral curve (characteristic), system (29) gives
(34)ukxt,t=∫t0tHuk−1,xs,sds.
Hence,(35)ukxt,t≤T∑i=0naisupΩTu0i≤supΩTu0,thanks to (28). Moreover, since ΩT is filled by characteristics, we have
(36)supΩTuk≤supΩTu0.
Finally, we conclude that for anyn∈ℕ, supΩTun≤supΩTu0. This ends the proof of the lemma.


## 2.2.1. Notations

LetΩ⊂ℝd,d=1,2,3, be a domain with a Lipschitz boundary ∂Ω satisfying the cone property.IfT>0 is given, set ΩT=Ω×0,T. Let
(17)Γ−=x∈∂Ω,qx·nx<0,Γ+=x∈∂Ω,qx·nx>0,where nx is the outer normal to ∂Ω at x. Let
(18)ΓT−=Γ−×0,T∪Ω×0,a=q,1t,∇t=∇,∂∂tt.LetHmΩ be the classical Hilbert space of order m, and
(19)H01Ω=φ∈H1Ω;φ=0on∂Ω.We also define the following Hilbert spaceV:
(20)V=ϕ∈L2ΩT;a·∇tϕ∈L2ΩT,equipped with the graph norm:
(21)vV2=vL2ΩT2+a·∇tvL2ΩT2,∀v∈V.Let(22)Ha,ΩT;Γ−=ϕ∈Vsuchthatϕ=0onΓ−×0,T.The problem consists in finding(23)c,c1:ΩT⟶ℝ,satisfying the following partial differential equation system
(24)P1∂c∂t+q·∇c=0inΩT,c=gonΓT−,cx,0=c0xinΩ,P2∂c1∂t+q·∇c1=rc1,1nmc1−c−βinΩT,c1=g1onΓT−,c1x,0=c10xinΩ.Equations (14) and (16) give system P2. In systems P1 and P2, initial conditions and inflow boundary conditions are given for c and c1. The technique is to solve first P1 and after P2. With the solutions c and c1 and equation (16), we obtain c12.


## 2.2.2. Preliminary Results

The STILS method will be used to solve the problemP1 like in [12, 20]. The method leads to a variational formulation problem, and we use the classical Lax-Milgram theorem to find the solution. So we give the following lemma to prove the bilinear map coercivity.Lemma 1 (curved Poincaré inequality).
There exists a constantk=2T such that
(25)uL2ΩT≤ka·∇tuL2ΩT,∀u∈Ha,ΩT;Γ−.The proof is in [20].LetHX,x,t be a polynomial with respect to X with bounded coefficients hi defined in ΩT.
(26)HX,x,t=∑i=0nhix,tXi.Letai,i=0,⋯,n be defined by
(27)ai=supΩThix,t.P2 is also a transport problem with a nonlinear feedback. We use both STILS method and fixed-point theorem to solve the problem. So we need c1kk∈ℕ to be bounded in L∞ΩT; hence, we give the following lemma.Lemma 2.
LetT be a positive real number and u0∈DΩT a positive function such that
(28)T<supΩTu0∑i=0naisupΩTu0i.
Letukk∈ℕ be the sequence of functions defined by
(29)a·∇tuk=Huk−1,x,tinΩT,uk=0onΓT−.
Then, we have(30)∀k∈ℕ,supΩTuk≤supΩTu0.Proof.
Assume that there existsk∈ℕ∗ such that
(31)supΩTuj≤supΩTu0,∀j=0,⋯,k−1.
We know that the velocity fieldq is bounded in its time component and has a constant dot product (equal to 1, too) with the vector 0,⋯,0,1. This ensures (see Proposition 7 in [12]) that ΩT is filled by the characteristics in the following ways:
There exists anS>0 such that for almost each point x,t of ΩT, there exists an integral curve
(32)ζ:0,S⟶ΩTsuchthat∂ζ∂s=aζthat connect x,t to the space-time inflow boundary:
(33)ζ0=x0,t0∈ΓT−,ζt=x,t,∀t∈0,S.
On this integral curve (characteristic), system (29) gives
(34)ukxt,t=∫t0tHuk−1,xs,sds.
Hence,(35)ukxt,t≤T∑i=0naisupΩTu0i≤supΩTu0,thanks to (28). Moreover, since ΩT is filled by characteristics, we have
(36)supΩTuk≤supΩTu0.
Finally, we conclude that for anyn∈ℕ, supΩTun≤supΩTu0. This ends the proof of the lemma.


## 3. Existence and Uniqueness Theorem

Theorem 3.
Letg,g1∈L∞ΓT−×L∞ΓT− and c0,c10∈L∞Ω×L∞Ω. System (24) has an unique solution c,c1∈V2.Proof.
The proof will be done in two steps.
Step one. In the systemP1, we assume that the divergence of the velocity q is zero; then, we can use the result in [12] or in [21, 22] cited in [12] to prove the existence of the boundary trace on ΓT of a function in V.
So, using the extensiond∈V of g in system P1, this later is equivalent to searching w in Ha,ΩT;Γ− such that
(37)PMCa·∇tw−F=0inΩT,w=0onΓT−,wx,0=c0x−dx,00inΩ,with F=−∂d/∂t−1/θq·∇d. Now, like in [12], we define the following convex quadratic form J:
(38)Jw=∫ΩTa·∇tw−F2dxdt.
The Gateau derivative ofJ is
(39)DJwφ=∫ΩTa·∇tw−Fa·∇tφdxdt.
Hence,w is the solution of (37) if and only if w satisfies
(40)∫ΩTa·∇twa·∇tφdxdt=∫ΩTFa·∇tφdxdt,∀φ∈Ha,ΩT;Γ−.
Using the Lax-Milgram theorem, we have the solution of the variational problem (40). Lemma 1 is used to prove the bilinear application coercivity.
Step two. This last step is devoted to the problemP2.
We set(41)Gc1=−αrc1,1nmc1−c−β,which is a polynomial function with respect to c1 with coefficients depending on c. Then, P3 becomes
(42)a·∇tc1==Gc1inΩT,c1=g1inΓT−,c1x,0=c10xinΩ.
This is a transport equation with a nonlinear feedbackGc1 due to the chemical reactions (dissolution-precipitation). To solve it, we first split (42) into the following systems:
(43)PT1a·∇tc¯==Gc¯+c~inΩT,c¯=0onΓT−,c¯x,0=0inΩ.(44)PT2a·∇tc~==0inΩT,c~=g1onΓT−,c~x,0=c10xinΩ.
In order to solve the nonlinear system (43), we use the fixed-point theory. For T>0 and c0∈DΩT satisfying (28), let ckk∈ℕ be the sequence of functions defined by
(45)PT1ka·∇tck=Gck−1+c~inΩT,ck=0onΓT−,ckx,0=0inΩ.
Using Lemma2, we prove that ckk∈ℕ is bounded in L∞ΩT.
Putc1k=ck+c~ solution of
(46)a·∇tc1k=Gc1k−1inΩT,c1k=g1onΓT−,c1kx,0=c10xinΩ.
Let us show thatc1kk∈ℕ is a Cauchy sequence in V and that its limit c1 is the solution of (42).
Letp and q be integers such that
(47)a·∇tc1p=Gc1p−1,(48)a·∇tc1q=Gc1q−1.
Equations (47) and (48) give
(49)a·∇tc1p−c1q=Gc1p−1−Gc1q−1.
By a variational formulation, (49) give
(50)∫ΩTa·∇tc1p−c1qa·∇twdxdt=∫ΩTGc1p−1−Gc1q−1a·∇twdxdt,∀w∈V.
We choosew=c1p−c1q then
(51)∫ΩTa·∇tc1p−c1qa·∇tc1p−c1qdxdt=∫ΩTGc1p−1−Gc1q−1a·∇tc1p−c1qdxdt.
Hence,(52)a·∇tc1p−c1qL2ΩT2=∫ΩTGc1p−1−Gc1q−1a·∇tc1p−c1qdxdt.
Since the functionG is regular enough, we have
(53)Gc1p−1−Gc1q−1=G′sc1p−1c1q−1.
So using the Cauchy-Schwarz inequality in the right hand side of (52), we get
(54)a·∇tc1p−c1qL2ΩT2≤max0≤s≤γG′sa·∇tc1p−c1qL2ΩTc1p−1−c1q−1L2ΩT.
Hence,(55)a·∇tc1p−c1qL2ΩT≤max0≤s≤γG′sc1p−1−c1q−1L2ΩT.
Letτ=max0≤s≤γG′s; we have
(56)a·∇tc1p−c1qL2ΩT≤τc1p−1−c1q−1L2ΩT.
Using the curved Poincaré inequality of Lemma1, we get
(57)c1p−c1qV≤2τTc1p−1−c1q−1V.
So, ifT<1/2τ, the sequence c1kk∈ℕ is a Cauchy sequence in V which is a Hilbert space then converges to c1∈V.
Let us now prove that the limitc1 of c1kk∈ℕ is the solution of (42). For this purpose, we write
(58)a·∇tc1−Gc1=a·∇tc1−a·∇tc1k+a·∇tc1k−Gc1k−1+Gc1k−1−Gc1,∀k∈ℕ.
Sincea·∇tc1−a·∇tc1k tends to zero in L2ΩT and a·∇tc1k−Gc1k−1=0, it remains to be proved that Gc1k−1−Gc1 tends to zero. We have
(59)Gc1k−1−Gc1=G′sc1k−1−c1k.
We know thatG′ is bounded and c1k−1 tends to c1 in V so Gc1k−1−Gc1 tends to zero. Then, c1 is the solution of the first equation of (42). The boundary and initial conditions are obtained by the continuity of the trace mapping.


## 4. Conclusion

The existence and uniqueness solutions of a coupled transport equations have been proved by a fixed-point and STILS method. Free divergence of the transport velocity is considered in this work, but the method used can be extended to no free divergence case. The techniques used in this paper can be applied in other coupled nonstationary systems. The spatial operator inN dimension, for example, can be replaced by an operator in N+1 dimension, where the N+1 component is the time derivative. The obtained result can be improved adding a concrete example like solving a salt-wedge intrusion model or a coupled system arising in epidemiology. All those things and the numerical simulation will be done in the future work.



---

*Source: 2705591-2022-08-22.xml*