A moving two-reaction zones model: global existence of solutions

Adrian Muntean∗ and Michael Bohm∗∗

Centre for Industrial Mathematics, FB3 (Mathematics), University of Bremen, Postfach 330440, 28334, Bremen, Germany

We address some aspects concerning the analysis of a moving-boundary system modeling concrete carbonation. The model

is based on the fact that carbonation might be considered as a reaction which is localized on two distinct a priori unknown

internal zones which progress into concrete. We report on the existence of local and global weak solutions. The main feature

of the problem is that the nonlinear coupling of the system occurs due to the moving boundary and nonlinearity of the involved

productions.

1 Introduction

In this note, we present a two-moving-reaction-zones model for concrete carbonation – a process by which carbon dioxide from

the ambient air penetrates the concrete, dissolves in pore water and reacts with calcium hydroxide (available by dissolution

from the solid matrix) to form calcium carbonate and water [2]. The reader is referred to [4] and references therein for

more details on the moving-interface modeling of this physico-chemical process. There is an interesting pattern which shows

localized reaction on two distinct a priori unknown internal zones that penetrate the material. We report in section 2 on the

existence of local and global weak solutions. The reader is referred to [4] for the proofs. Related results were addressed for a

moving sharp-interface scenario in [3].

Let ε > 0 be sufficiently small. Denote by u ( ∈ 1, 2) the CO2 concentration in air and water, ur (r ∈ 3, 6) the

Ca(OH)2 and H2O concentrations in Ω2(t), and um (m ∈ 5, 7) and u4 the H2O, Ca(OH)2 and CaCO3 concentrations

in Ω1(t) ∪ Ωε(t). Hence u := (u1, u2, u3, u4, u5, u6, u7)t forms the vector of concentrations defined in the moving domains

Ω1(t) :=]0, s(t)− ε/2[, Ωε(t) :=]s(t)− ε/2, s(t)+ ε/2[ and Ω2 := Ω−Ω1(t) ∪ Ωε(t), where Ω := [0, L[ (L > 0). Here s(t)stands for the position of the a priori unknown center of Ωε(t), see Fig. 1. Set Γr(t) := x = s(t) + ε/2. We also use the

following sets of indices I1 := 1, 2, 5, 7, I2 := 3, 6 and I := I1 ∪ 4 ∪ I2. Our moving-boundary problem consists of

Fig. 1 Left: Basic geometry of the region which our model refers to. Right: Illustration of the 1D geometry (magnification

of Box A). The domains Ωε(t) and Ω1(t) are the places where carbonation reaction is assumed to take place.

finding the vector u of active concentrations and the position s(t) which satisfy for all t ∈ ST :=]0, T [ (T > 0) the equations

⎧⎨⎩

u,t + (−Du,x)x = f(u), x ∈ Ω1(t) ∪ Ωε(t), ∈ 1, 2, (balance of CO2),ur,t + (−Drur,x)x = fr(u), x ∈ Ω2(t), r ∈ 3, 6, (balance of Ca(OH)2 and H2O),

um,t + (−Dmum,x)x = fm(u), x ∈ Ω1(t) ∪ Ωε(t), m ∈ 5, 7, (balance of Ca(OH)2 and H2O).(1)

In (1), Di are macroscopic diffusivities of species ui (i ∈ I1 ∪ I2). The precise definition of the productions fi is given in

(4). The initial and boundary conditions are ui(x, 0) = ui0 for i ∈ I, x ∈ Ω, ui(0, t) = λi for i ∈ I1 − 7, u7,x(0, t) = 0,

t ∈ ST and ui,x(L, t) = 0 for i ∈ I2, t ∈ ST . The transmission conditions imposed across the interface Γr(t) are defined via

[−Diui,x · n]Γr(t) = s′(t)[ui]Γr(t) (i ∈ I1 ∪ I2). (2)

Set v4(t) := u4(x, t) for x ∈ Ω1(t)∪Ωε(t) and t ∈ ST . Observe that x plays here the role of a parameter. The driving force of

the moving boundary are potential differences modeled via averaged reaction rates (see chapter 4 from [5] for an explanation

∗ Corresponding author: e-mail: muntean@math.uni-bremen.de, Phone: +49 421 218 9173, Fax: +49 421 218 9406∗∗ e-mail: mbohm@math.uni-bremen.de

PAMM · Proc. Appl. Math. Mech. 6, 649–650 (2006) / DOI 10.1002/pamm.200610305

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

of the physico-chemical motivation for treating reaction zones as free boundaries)

s′(t) = ε

∫Ωε(t)

ηε(u, Λε)(x, t)dx∫Ωε(t)

u3(x, t)dx, and v′4(t) = f4(v4(t)) a.e. t ∈ ST , provided s(0) = s0 > 0, v4(0) = u40 > 0, (3)

where ηε := ηε(u, Λε) is a power-law ansatz for the reaction rate acting in Ωε(t). The model equations are collected in (1)-(3).

To deal with (1)-(3) we firstly freeze the moving boundaries by employing suitable Landau transformations cf. [1, 4].

Denote by u the vector of concentrations (u1, u2, u3, u5, u6, u7)t in the fixed-domain formulation and by λ the boundary data

(λ1,λ2,λ3,λ5,λ6, λ7)t. Formally, we employ λ3 = λ7 = λ6 = 0. Let ϕ := (ϕ1,ϕ2,ϕ3,ϕ5,ϕ6, ϕ7)t ∈ be an arbitrary test

function1 and take t ∈ ST . For our problem, the production terms fi(i ∈ I1 ∪ I2) are given by⎧⎨⎩

f1(u) := P1(Q1u2 − u1) − η1(u, Λ1) − ηε(u, Λε), f2(u) := −P2(Q2u2 − u1),f3(u) := S3,diss(u3,eq − u3), f4(u) := +η1(u, Λ1) + ηε(u, Λε),f5(u) := +η1(u, Λ1) + ηε(u, Λε), f6(u) := 0, f7(u) := S7,diss(u7,eq − u7) − η1(u, Λ1) − ηε(u, Λε).

(4)

In this framework, η1 and ηε denote carbonation reaction rates in Ω1(t) and Ωε(t), while each of the vectors Λ1, Λε ∈ MΛ

(MΛ – a compact set in m+ away from zero) contain m reaction parameters. Set ki = ki(ui0, λi(t),Mε, T, D5, Λε) > 0

(i ∈ I), e.g k5 := maxu60 + λ6(t) + MεT, M1εD5L −Mε

, k6, λ5(t) : t ∈ ST , where we put K :=∏

i∈I1∪I2[0, ki], Mε :=

ε 1k∗3

maxu∈Kf4(u), M1ε := maxu∈Kf4(u), and k∗3 > 0 is a given constant lower bound of the content of Ca(OH)2 in

Ωε(t) valid for all t ∈ ST . The model parameters as well as the initial and boundary data need to satisfy2:

λ ∈ W 1,2(ST )|I1∪I2|, λ(t) ≥ 0 a.e. t ∈ ST , , u,eq ∈ L∞(ST ), u,eq(t) ≥ 0 a.e. t ∈ ST , ∈ 3, 7, (5)

u0 ∈ L∞(a, b)|I1∪I2|, u0(y) + λ(0) ≥ 0 a.e. y ∈ [a, b], u40 ∈ L∞(0, s0 + ε/2), (6)

u4(x, 0) ≥ 0 a.e. x ∈ [0, s0 + ε/2], minS3,diss, S7,diss, P1, Q1, P2, Q2, D( ∈ I1 ∪ I2) > 0. (7)

2 Main Results

The assumptions on the reaction rates and material parameters, which are needed in the sequel, are the following:

(A) Fix Λ ∈ MΛ and take arbitrarily ∈ 1, ε. Let η(u, Λ) > 0, if u1 > 0 and u7 > 0, and η(u, Λ) = 0, otherwise.

Assume η to be bounded for any fixed u1 ∈ . (B) The reaction rate η : 7 × MΛ → + ( ∈ 1, ε) is locally Lipschitz.

(C1) k ≥ maxST|u,eq(t)| : t ∈ ST , ∈ 3, 7; D5 > MεL; (C2) P1Q1k2 ≤ P1k1; P2k1 ≤ P2Q2k2; (C3) Q2 > Q1.

Theorem 2.1 (Local Existence and Uniqueness) Assume the hypotheses (A)-(C2) and let the conditions (5)-(7) be satisfied.Then the following assertions hold:(a) There exists a δ ∈]0, T [ such that the problem (1)-(3) admits a unique local weak solution on Sδ;(b) 0 ≤ ui(y, t)+λi(t) ≤ ki a.e. y ∈ [a, b] (i ∈ I1 ∪ I2) for all t ∈ Sδ . Moreover, 0 ≤ u4(x, t) ≤ k4 a.e. x ∈ [0, s(t)+ε/2]for all t ∈ Sδ , and v4, s ∈ W 1,∞(Sδ).

Proposition 2.2 (Strict Lower Bounds) Assume that the hypotheses of Theorem 2.1 are satisfied. If, additionally, therestriction (C3) holds and the initial and boundary data are strictly positive, then there exists a range of parameters such thatthe positivity estimates stated in Theorem 2.1 (b) are strict for all times.

Theorem 2.1 reports on the existence of locally in time weak solutions of problem (1)-(3) with respect to the time interval

Sδ . Assume that the hypotheses of Proposition 2.2 hold. Therefore, for an arbitrarily given L0 ∈]s0, L[ there exists Tfin ≤+∞ such that s(Tfin) + ε

2 = L0. Here, Tfin represents the time when Ωε(t) has penetrated all of ]s0, L0[, or in other words,

when Ωε(t) has touched x = L0. We call Tfin the final time or shut-down time of the reaction-diffusion process. Owing to

Theorem 2.1 and Proposition 2.2, we obtain that L0−s0ηmax

< Tfin < L0−s0ηmin

. This helps us to conclude with the next result.

Theorem 2.3 (Global Solvability) Assume that the hypotheses of Proposition 2.2 are satisfied. Then the time intervalSTfin :=]0, Tfin[ of (global) solvability of problem (1)-(3) is finite and is characterized by Tfin = s−1

(L0 − ε

2

).

Acknowledgements We thank the German Science Foundation (DFG) for financial support through the grant SPP1122.

References[1] H. G. Landau, Quart. Appl. Mech. Math. 8, 81 (1950).[2] K. Maekawa, T. Ishida and T. Kishii, J. Adv. Concr. Technology (Japan Concrete Institute) 1, (2), 91 (2003).[3] A. Muntean and M. Bohm, to appear in: Free and Moving Boundary Problems. Theory and Applications, edited by J. F. Rodrigues, L.

Santos, I. N. Figuiredo (Birkhauser, Basel, 2006).[4] A. Muntean, Ph.D. thesis, University of Bremen, (Cuvillier Verlag, Gottingen, 2006).[5] P. Ortoleva, Geochemical Self-Organization, (Oxford University Press, Oxford, 1994).

1 The selection of the evolution triple (,,∗) is standard. See [4] for its definition and use in the weak formulation.2 Note that y ∈ [a, b] (i ∈ I1 ∪ I2) means here the following: If j ∈ 1, 2 and i ∈ Ij , then y ∈ [−1 + j, j].

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Section 13 650

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