Well-posedness of a moving two-reaction-strips problem modeling chemical corrosion of porous media Adrian Muntean CASA – Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Technical University of Eindhoven, The Netherlands

Abstract We deal with a one-dimensional coupled system of semi-linear reaction-diffusion equations in two a priori unknown moving phases driven by a non-local kinetic condition. The PDEs system models the penetration of gaseous CO2 in unsaturated cement-based porous materials (concrete). The main issue is that CO2 diffusion and fast reaction with Ca(OH)2 in concrete lead to a sudden drop of alkalinity near the steel reinforcement. This effect initiates the chemical corrosion of the material. We address some aspects concerning the existence, uniqueness and stability of local weak solutions. We also present our concept of global solvability of the movingboundary system in question. We illustrate the application of our model to the prediction of carbonation penetration into ordinary Portland concrete samples. Key words: Moving boundary, strips-concentrated reaction, Stefan problem, kinetic condition, a priori estimates, weak solutions, well-posedness MSC: primary 35R35; secondary 74F25, 35D05

1

Introduction

We present and analyse a two-reaction-strips moving-boundary model for concrete carbonation – a reaction-diffusion process by which carbon dioxide from the ambient air penetrates the concrete, dissolves in pore water and reacts with Email address: [email protected] (Adrian Muntean). Funding by the German Science Foundation (DFG) via the grant SPP 1122 Prediction of the course of physicochemical damage processes involving mineral materials is acknowledged. 1

calcium hydroxide, which is available by dissolution from the solid matrix, to form calcium carbonate and water. The reduced chemistry of the process is H O

2 CO2 (g → aq) + Ca(OH)2 (s → aq) −→ CaCO3 (aq → s) + H2 O.

(1)

We refer the reader to [7,8,36,28] and references therein for more details on the carbonation reaction and its negative as well as positive effects on concrete’s microstructure. We only mention that (1) typically facilitates corrosion, and hence, it drastically limits the service-life of concrete-based materials. In this paper, we exploit an experimental observation - called the phenolphthalein test - to offer a detailed modeling of the process when one assumes that the carbonation reaction is concentrated in two distinct strips. The phe-

Fig. 1. Slices of partly carbonated concrete samples sprayed with phenolphthalein (courtesy of Prof. Dr. M. Setzer and Dr. U. Dahme, University of Essen-Duisburg, Germany). Two distinct type of regions can be noticed; (1) colorless (low alkaline) parts, and (2) red (high alkaline) parts. Here we assume that one tiny reaction strip (say Ω (t)) is positioned at the sharp interface separating the two colors, and behind it, a second much wider reaction strip (say Ω1 (t)) is present. We assume that the latter strip fills the colorless region. Ω2 (t) points out the highly alkaline region.

nolphthalein measurement is limited to providing an indication of the position of the pH value (about 9) and does not show the changes which may occur in the colorless (partly carbonated) part [8,38]. This qualitative information may be turned into a quantitative one as soon as one examines pH levels inside pores and then at the macro-scale as in [31,15] involving also homogenizationbased arguments like in [27]. However, we find difficult to account for the whole ionic soup existing in concrete. We rather prefer to use the macroscopic information pointed out in Fig. 1: As the carbonation reaction proceeds, several reaction strips form and move inwards. In the inner parts of the strips, the alkaline species are depleted with possibly different rates. Obviously, the central question is now: How fast do these strips move into the material? The main goal of this paper is to understand the motion of these internal reaction strips, and hence, to predict in an accurate way the penetration of CO2 in ordinary (Portland) concrete. 2

The results presented here are parts from the author’s PhD thesis [22] and have been announced in [20]. Related aspects were addressed for a moving sharp-interface scenario in [21,23]. Our reaction-diffusion scenario (section 2) remotely resembles to the classical Stefan problem modeling the melting of ice cubes; see [1] for a nice introduction to modeling aspects around this theme and [19] for a collection of known facts on the mathematical analysis of Stefan-like problems. A special feature of our moving-boundary model is the presence of a kinetic condition for driving the main reaction strip. Such moving-boundary condition has been also used to model further (conceptually related) nonequilibrium processes like the swelling of glassy polymers [11], supercooling and superheating effects in solidification [3,39], sulfate corrosion in sewer pipes [5] and fast dissolution/precipitation reactions [30], e.g. The paper is organized in the following manner. The two-reaction strips movingboundary model is introduced in section 2. Our concept of weak solution and the main results are included in section 3. The bulk of the paper (i.e. section 4) consists of the proofs for the local existence, uniqueness and stability with respect to data and parameters of the weak solutions (theorem 3 and theorem 4). The reader is referred to [22] to see proofs for the rest of the statements. In section 5, we illustrate numerically a real situation relying on experimental data extracted from [7]. We also briefly evaluate the model and numerical approach.

2

The moving-two-reaction-strips model

Let > 0 be sufficiently small. Denote by uˆ` (` ∈ {1, 2}) the CO2 concentration in air and water in Ω1 (t) ∪ Ω (t), uˆr (r ∈ {3, 6}) the Ca(OH)2 and H2 O concentrations in Ω2 (t), and uˆm (m ∈ {5, 7}) and uˆ4 the H2 O, Ca(OH)2 and CaCO3 concentrations in Ω1 (t) ∪ Ω (t). Here uˆ := (ˆ u1 , uˆ2 , uˆ3 , uˆ4 , uˆ5 , uˆ6 , uˆ7 )t is the vector of concentrations, while the moving domains are defined by Ω1 (t) := ]0, s(t)−/2[, Ω (t) :=]s(t)−/2, s(t)+/2[ and Ω2 := Ω−Ω1 (t) ∪ Ω (t), where Ω := [0, L[ (L > 0). Here s(t) stands for the position of the a priori unknown center of Ω (t). Set Γr (t) := {x = s(t) + /2}. The geometrical setting is depicted in Fig. 2. We also use the following sets of indices I1 := {1, 2, 5, 7}, I2 := {3, 6} and I := I1 ∪ {4} ∪ I2 . The moving-boundary problem consists of finding the active concentrations uˆi and the position s(t) which satisfy for all t ∈ ST :=]0, T [ (T > 0) the 3

Fig. 2. Two reaction zones. Carbonation is assumed to take place simultaneously in Ω1 (t) and Ω (t) with different reaction rates.

mass-balance equations

uˆ`,t + (−D` uˆ`,x )x = f` (ˆ u), x ∈ Ω1 (t) ∪ Ω (t), ` = 1, 2,

uˆr,t + (−Dr uˆr,x )x = fr (ˆ u), x ∈ Ω2 (t), r ∈ {3, 6}, u ˆm,t + (−Dm uˆm,x )x = fm (ˆ u), x ∈ Ω1 (t) ∪ Ω (t), m ∈ {5, 7}.

(2)

The precise definition of the productions fi is given in (7). The initial and boundary conditions are uˆi (x, 0) = uˆi0 for i ∈ I, x ∈ Ω, uˆi (0, t) = λi for i ∈ I1 − {7}, u7,x (0, t) = 0, t ∈ ST and uˆi,x (L, t) = 0 for i ∈ I2 , t ∈ ST . The interface and transmission conditions ([37,14]) imposed across Γr (t) are defined via [−Di uˆi,x · n]Γr (t) = s0 (t)[ˆ ui ]Γr (t) (i ∈ I1 ∪ I2 ).

(3)

Set v4 (t) := uˆ4 (x, t) for x ∈ Ω1 (t) ∪ Ω (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

R 0

s (t) =

η (ˆ u, Λ )(x, t)dx , ˆ3 (x, t)dx Ω (t) u

Ω (t)

R

(4)

vˆ40 (t) = f4 (v4 (t)) a.e. t ∈ ST with s(0) = s0 > 0, vˆ4 (0) = uˆ40 > 0,

(5)

where η is the reaction rate in Ω (t). The model equations are collected in (2)-(5). 4

2.1

Comments on the expression (4) of the velocity s0 (t)

Due to the explicit representation of the velocity law s0 of the strip Ω (t) the moving-boundary system (2)-(5) is said to be with kinetic condition very much in the spirit of [39]. As mentioned in [11] (see section 2.2 loc. cit.), the advantage of this sort of model is the finite speed of the reaction strip propagation. Of course, the structure of the reaction rates have to be carefully chosen, otherwise blow up situations near moving thin strips may occur [24]. Expression (4) shows that the speed of the strip Ω (t) is proportional to the strength of the production by reaction in Ω (t). It also incorporates the following effect: If a large amount of Ca(OH)2 is present in Ω (t), then the speed s0 will correspondingly decrease. (4) represents a volumetric dilation of a similar expression derived via first principles in [22] (section 2.3.1) for the case of a moving interface-concentrated reaction.

2.2

Freezing moving reaction strips. Production rates. Model parameters

To deal with (2)-(5), we firstly freeze the moving strips by employing suitable Landau transformations; see details in section 4.3 or in [17,22], e.g. 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 ∈ V be an arbitrary test function 2 and take t ∈ ST . To write the weak formulation of (2)-(5) in fixed-domains, we introduce the notation: a(s, u, ϕ)

:=

1 s+ 2

P

i∈I1 (Di ui,y , ϕi,y )

bf (u, s, ϕ) := s +

e(s0 , u, ϕ) := s+1 2 h(s0 , u , ϕ) := P ,y

2

P

+

1 L−s− 2

P

i∈I2 (Di ui,y , ϕi,y ),

i∈I1 (fi (u), ϕi ) + L − s −

2

P

i∈I2 (fi (u), ϕi ),

P2

0 i=1 s (t)(ui (1, t) + λi (t))ϕi (1),

i∈I1 (s

0

yui,y , ϕi ) +

P

i∈I2 (s

0

(6)

(2 − y)ui,y , ϕi )

for any u ∈ V and λ ∈ W 1,2 (ST )|I1 ∪I2 | . The intervening function spaces are introduced in section 4.2. The term a(·) incorporates the diffusive part of the model, bf (·) comprises various productions, e(·) sums up some of the reaction terms acting on y = 1 and h(·) is a non-local term due to fixing of the domain. 2

The selection of the evolution triple (V, H, V∗ ) is standard.

5

For our application, the production terms fi (i ∈ I1 ∪ I2 ) are given by f1 (u) := P1 (Q1 u2 − u1 ) − η1 (u, Λ1 ) − η (u, Λ ), f2 (u) := −P2 (Q2 u2 − u1 ),

f3 (u) := S3,diss (u3,eq − u3 ), f4 (ˆ u) := +η1 (ˆ u, Λ1 ) + η (ˆ u, Λ ), f5 (u) := +η1 (u, Λ1 ) + η (u, Λ ), f6 (u) := 0, f7 (u) := S7,diss (u7,eq − u7 ) − η1 (u, Λ1 ) − η (u, Λ ).

(7)

In this framework, η1 and η denote carbonation reaction rates in Ω1 (t) and Ω (t), while each of the vectors Λ1 , Λ ∈ MΛ (MΛ is a compact set in Rm + away from zero) contain m ∈ N reaction parameters. Select ki := max{ui0 + λi (t), λi (t) : t ∈ S¯T }, i = 1, 2, 3, 6, 7,

k := max{ˆ u (x) + M T : x ∈ [0, s(t) + /2], t ∈ S¯T },

4 40 k5 := max{u60 + λ6 (t) + M T,

M1 D5 −M L

(8)

, k6 , λ5 (t) : t ∈ S¯T },

where we set Y

K :=

i∈I1 ∪I2

[0, ki ], M :=

1 max{f4 (ˆ u)}, M1 := max{f4 (ˆ u)}, u∈K k3∗ u∈K

and k3∗ > 0 is a given constant lower bound of the content of Ca(OH)2 in Ω (t) valid for all t ∈ ST . Additionally, let k¯ denote the sum k¯ = 2

7 X

k` .

(9)

`=1

Note that y ∈ [a, b] (i ∈ I1 ∪ I2 ) means here the following: If j ∈ {1, 2} and i ∈ Ij , then y ∈ [−1 + j, j]. The model parameters as well as the initial and boundary data need to satisfy the following assumptions: λ ∈ W 1,2 (ST )|I1 ∪I2 | , λ(t) ≥ 0 a.e. t ∈ S¯T , ∈ L∞ (ST ), u`,eq (t) ≥ 0 a.e. t ∈ S¯T , ` ∈ {3, 7},

(10) (11)

u0 ∈ L∞ (a, b)|I1 ∪I2 | , u0 (y) + λ(0) ≥ 0 a.e. y ∈ [a, b], uˆ40 ∈ L∞ (0, s0 + /2), uˆ4 (x, 0) ≥ 0 a.e. x ∈ [0, s0 + /2], min{S3,diss , S7,diss , P1 , Q1 , P2 , Q2 , D` (` ∈ I1 ∪ I2 )} > 0.

(12) (13) (14) (15)

u`,eq

6

2.3

How to define the production by reaction within a strip?

Knowing for a moment t ∈ ST the location of the reaction strip Ω (t) (i.e. the position s(t) and the value of ), we can localize the strips Ω1 (t) and Ω2 (t) in a straightforward manner. Interestingly, in the chemical engineering literature (see [13,26], e.g.) one defines what volume- and surface- concentrated reaction are, but it appears to be no consensus on how to define reaction concentrated within sub-domains (like strips). The standard procedure is to avoid employing strip-concentrated reactions via approximating the production by reaction in that strip either by a surface production (if the strip is sufficiently thin, see Gurtin’s pillbox lemma [14]) or by a volume production (if the width of the strip is comparable to the width of the reference volume). Herein, we model the reaction production inside the strip and mimic the strip motion by means of the moving-boundary system (2)-(5). If the two reactants completely segregate, then we expect the strip Ω (t) to degenerate into the interface Γ(t) positioned at x = s(t); see [11,9,22] for closely related aspects. In that case, the structure of the moving-boundary system changes dramatically (as it is described in [23]) and the solution vector typically looses regularity. We assume for a moment that no secondary carbonation effects can be noticed behind the (main) reaction strip Ω (t). In this case, Ω1 (t) can be defined via Ω1 (t) := {x ∈ Ω : φφw u¯3 (x, t) = 0}, where φ, φa and φw denote the total porosity and the air and water fractions, respectively. u¯ represents a mass concentration at the pore scale that is connected with the averaged concentration uˆ via (26); see [4,22], e.g. The strip Ω (t) may be defined in several ways. We are aware of the next six (non-equivalent) definitions: (i) We can rely on the typical behavior of reactants profiles vanishing in a fast reaction mechanism (see, for instance, Fig. 2 in [11] or Fig. 5-2 in [26], p. 94) to account for Ω (t) := {x ∈ Ω : φφw u¯1 (x, t) ⊂]0, k1 ] and φφw u¯3 (x, t) ⊂]0, k3 ]}.

(16)

(ii) Denote by c∗ > 0 the mass concentration of CO2 (aq) necessary to consume one mole of Ca(OH)2 (aq). We can define Ω (t) to be the region in which either u¯1 ≥ c∗ and 0 < u¯3 ≤ c∗ , or 0 ≤ u¯1 ≤ c∗ and u¯3 ≥ c∗ . (iii) Another possibility is to define the strip Ω (t) (as in [29], e.g.) to be the place where the solubility product c∗∗ (of the carbonation reaction) is above its equilibrium value and the concentrations φφw u¯i (i ∈ {1, 3}) are below their maximum values ki , i.e. k1 k3 ≥ u¯1 u¯3 ≥ c∗∗ 2 , provided k1 ≥ u¯1 > 0 7

and k3 ≥ u¯3 > 0. A slightly different concept employs the notion of extent of reaction [13] to define the reaction strip. (iv) In [2], the authors define asymptotically the strip via Ω (t) := {x ∈ Ω : x − s(t) = O()}. (v) Usually, people from the phase-field community (see chapter 1 in [12] or [10], e.g.) consider the moving internal strip to be an interface of width O() that is located on a plane curve Γ(t; ). The movement of Γ is typically tracked via the evolution of a function r(x, t; ) that represents the distance from x to Γ. Thus, Γ(t; ) := {r(x, t; ) = 0} defines our Ω (t). (vi) We can also define Ω (t) in an a posteriori way as the location where the reaction rate reaches its maximum. This procedure employs the concept of degree of reaction and is often used to define the strip location in case of isolines models, see [18,33] and references therein. In this paper, we adopt definition (16). Hence, Ω (t) can be seen as a sort of mushy region where both reactants (i.e. CO2 (aq) and Ca(OH)2 (aq)) are living together.

3

Main Results

Definition 1 (Weak formulation) We call the triple (u, v4 , s) a local weak solution to problem (2)-(5) if there is a Sδ :=]0, δ[ with δ ∈]0, T [ such that s0 +

< s(δ) + ≤ L0 , 2 2

v4 ∈ W 1,2 (Sδ ),

(17)

s ∈ W 1,2 (Sδ ),

(18)

u ∈ W21 (Sδ ; V, H) ∩ L∞ (Sδ , H),

(19)

s+

2

P

i∈I1 ((ui (t) + λi (t))t , ϕ) + L − s −

2

P

i∈I2 ((ui (t)

+ λi (t))t , ϕ)

+a(s, u, ϕ) + e(s0 , u, ϕ) = bf (u + λ(t), s, ϕ) + h(s0 , u,y , ϕ) for all ϕ ∈ V, a.e. t ∈ Sδ , s0 , v40 cf. (35) a.e. in Sδ , u(0) = u0 ∈ H.

(20)

The assumptions on the reaction rates and material parameters, which are needed in the sequel, are the following: 8

(A) Fix Λ ∈ MΛ and take arbitrarily ` ∈ {1, }. Let η` (ˆ u, Λ) > 0, if uˆ1 > 0 and uˆ7 > 0, and η` (ˆ u, Λ) = 0, otherwise. Assume η` to be bounded for any fixed uˆ1 ∈ R. (B) The reaction rate η` : R7 × MΛ → R+ (` ∈ {1, }) is locally Lipschitz. (C1) k` ≥ maxS¯T {|u`,eq (t)| : t ∈ S¯T }, ` ∈ {3, 7}; D5 > M L; (C2) P1 Q1 k2 ≤ P1 k1 ; P2 k1 ≤ P2 Q2 k2 ; (C3) Q2 > Q1 . Theorem 2 Assume the hypotheses (A)-(C2) and let the conditions (10)(15) be satisfied. If s ∈ W 1,2 (Sδ ) with s0 ≥ 0 a.e. in Sδ and s(0) = s0 is given, then the problem (2)-(5) admits a unique local solution on Sδ in the sense of Definition 1 (formulated for given s); Theorem 3 (Local existence and uniqueness) Assume the hypotheses (A)(C2) and let the conditions (10)-(15) be satisfied. Then the following assertions hold: (a) There exists a δ ∈]0, T [ such that the problem (2)-(5) admits a unique local solution on Sδ in the sense of Definition 1; (b) 0 ≤ ui (y, t) + λi (t) ≤ ki a.e. y ∈ [a, b] (i ∈ I1 ∪ I2 ) for all t ∈ Sδ . Moreover, 0 ≤ uˆ4 (x, t) ≤ k4 a.e. x ∈ [0, s(t) + /2] for all t ∈ Sδ , and v4 , s ∈ W 1,∞ (Sδ ). By (A) and (B), we deduce that η` (0, Λ) = 0 for all Λ ∈ MΛ and ` ∈ {1, }. Therefore, there exists a positive constant C`,η = C`,η (Λ, λ, u¯1 , Tfin ) such that the inequality η¯` (¯ u(y, t)), Λ) ≤ C`,η u¯(y, t)

(21)

holds locally for all t ∈ ST . (i)

Let ` ∈ {1, } be fixed arbitrarily. Select i ∈ {1, 2} and let (u(i) , v4 , si ) be two weak solutions on Sδ in the sense of Definition 1. They correspond to the sets of (i) (i) data Di := (u0 , λ(i) , Ξ(i) , Υ(i) , Λ(i) )t , where u0 , λ(i) , Ξ(i) , Υ(i) , and Λ(i) denote the respective initial data, boundary data, and the model parameters describing diffusion, dissolution mechanisms, and carbonation reaction, respectively. (i) (i) (i) In this context, we have Ξ(i) := (D` (` ∈ I1 ∪ I2 ), Pk (k ∈ {1, 2}), Qk (k ∈ (i) (i) {1, 2}), S3,diss )t ⊂ MΞ and Υ(i) = (u3,eq ) ⊂ MΥ , i ∈ {1, 2}. Here MΞ and MΥ 2 are compact subsets of R10 + and L (Sδ ). (2)

(1)

Set ∆u := u(2) − u(1) , ∆v4 := v4 − v4 , ∆s := s2 − s1 , ∆λ := λ(2) − λ(1) , (2) (1) ∆u0 := u0 −u0 , ∆Ξ := Ξ(2) −Ξ(1) , ∆Υ := Υ(2) −Υ(1) , ∆Λ := Λ(2) −Λ(1) , and (2) (1) (2) (2) (1) (1) ∆η` := η˜` − η˜` := η¯` (¯ u , Λ(2) )− η¯` (¯ u , Λ(1) ). The Lipschitz condition on η` reads: There exists a constant cL = cL (D1 , D2 ) > 0 such that the inequality |∆η` | ≤ cL (|∆u| + |∆Λ|) holds locally pointwise, see (B). Having these notations available, we can state the following two results.

9

(i)

Theorem 4 (Continuity w.r.t data and parameters) Let (u(i) , v4 , si ) (i ∈ {1, 2}) be two weak solutions on Sδ and assume the hypotheses of Theorem 3 to (i) be satisfied. Let (u0 , λ(i) , Λ(i) ) be the initial, boundary and reaction data. Then the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W21 (Sδ , V, H) × W 1,2 (Sδ )2 that maps (u0 , λ, Ξ, Υ, Λ)t into (u, v4 , s)t is Lipschitz in the following sense: There exists a constant c = c(δ, , s0 , uˆ40 , L, ki , θ, cL ) > 0 (i ∈ I1 ∪ I2 ) such that ||∆u||2W21 (Sδ ,V,H)∩L∞ (Sδ ,H) + ||∆v4 ||2W 1,2 (Sδ )∩L∞ (Sδ ) + ||∆s||2W 1,2 (Sδ )∩L∞ (Sδ )

≤ c ||∆u0 ||2H∩L∞ ([a,b]|I1 ∪I2 | ) + ||∆λ||2(W 1,2 (S

2

+c max |∆Ξ| + MΞ

δ

||∆Υ||2MΥ ∩L∞ (Sδ )

)∩L∞ (S

δ

))|I1 ∪I2 | 2

+ max |∆Λ|

MΛ

. (22)

A direct corollary of Theorem 4 is the following result. Proposition 5 (Stability of the moving strip Ω (t)) Assume that the hypotheses of Theorem 4 are satisfied. Then the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W 1,2 (Sδ ) that maps the data (u0 , λ, Ξ, Υ, Λ)t into the position of the strip s is Lipschitz in the following sense: There exists a constant c = c(δ, , s0 , uˆ40 , L, ki ,θ, cL ) > 0 (i ∈ I1 ∪ I2 ) such that

||∆s||2W 1,2 (Sδ )∩L∞ (Sδ ) ≤ c ||∆u0 ||2H∩L∞ ([a,b]|I1 ∪I2 | ) + ||∆λ||2(W 1,2 (S

2

+c max |∆Ξ| + MΞ

δ

||∆Υ||2L2 (Sδ )∩L∞ (Sδ )

)∩L∞ (S

δ

))|I1 ∪I2 | 2

+ max |∆Λ| MΛ

.

(23) Proposition 6 (Strict lower bounds) Assume that the hypotheses of Theorem 3 are satisfied. If, additionally, the restriction (C3) holds and the initial and boundary data are strictly positive, then there exists a range of parameters such that the positivity estimates stated in Theorem 3 (b) are strict for all times. Theorem 3 reports on the existence of locally in time weak solutions of problem (2)-(5) with respect to the time interval Sδ . Assume that the hypotheses of Proposition 6 hold. Therefore, for an arbitrarily given L0 ∈]s0 , L[ there exists Tfin ≤ +∞ such that s(Tfin ) +

= L0 . 2

(24)

Here, Tfin represents the time when Ω (t) has penetrated all of ]s0 , L0 [, or in 10

other words, when Ω (t) has touched x = L0 . We call Tfin the final time or shut-down time of the reaction-diffusion process. Owing to Theorem 3 and −s0 −s0 < Tfin < Lη0min . This helps us to conclude Proposition 6, we obtain that Lη0max with the next two results. Proposition 7 (Practical estimates) Let (u, v4 , s) be the unique local solution to (2)-(5) that fulfills the hypotheses of Proposition 6. Then the following estimates hold: (i) ηmin < s0 (t) < ηmax for all t ∈ Sδ ; (ii) (Localization of the strip Ω (t)) s0 + 2 ≤ s(t) ≤ s0 + 2 + δM for all t ∈ Sδ ; −s0 −s0 (iii) Lη0max < Tfin < Lη0min , where Tfin satisfies (24). (iv) If Ω1 (t)∪Ω (t) = {x ∈]0, L[: uˆ4 (x, t) > 0} for t ∈ S¯Tfin , then Ω1 (t1 )∪Ω (t1 ) ⊂ Ω1 (t2 ) ∪ Ω (t2 ) for t1 < t2 , t1 , t2 ∈ S¯Tfin . R

Here ηmin and ηmax denote uniform lower and upper bounds of

Ω (t)

ηΩ (t) (u(x,t))dx

R Ω (t)

u3 (x,t)dx

.

Note that ηmax can be chosen to be ηmax := M + 1, e.g. The proof of (i)-(iv) is straightforward. The statements of Proposition 7 have an obvious practical interpretation: (i) bounds from below and above the penetration speed, (iii) delimitates the time needed to finish the chemical reaction (1), while (iv) simply points out that the region Ω1 (t)∪Ω (t), where carbonates are detected, increases with the time. Theorem 8 (Global solvability) Assume that the hypotheses of Proposition 6 are satisfied. Then the time interval STfin :=]0, Tfin [ of (global) solvability −1 of problem (2)-(5) is finite and is characterized by Tfin = s L0 − 2 . Proposition 9 (Change of coordinates) Assume that the hypotheses of Theorem 3 are satisfied. Additionally, if u3,eq ∈ W 1,2 (Sδ ), uˆ0 ∈ H 1 (0, s0 )|I1 | × H 1 (s0 , L)|I2 | , uˆ0 and λ satisfy the compatibility conditions, then the solution (ˆ u, v4 , s) is mapped via inverse Landau transformations into the physical x-t plane and

(ˆ u, v4 , s) ∈

Y

ˆ i (t)) × W 1,2 (ST )2 , H 1 (STfin , H fin

(25)

i∈I1 ∪I2

ˆ i (t) := L2 (0, s(t) + /2) for i ∈ I1 and H ˆ i (t) := L2 (s(t) + /2, L) for where H i ∈ I2 . The proof from appendix B of [22] can be adapted to this situation. 11

4

4.1

Proofs of Theorem 3 and Theorem 4

Working strategy

We proceed in the following fashion: Firstly, we fix the moving boundaries and employ the weak formulation for the PDEs system defined in fixed domains. Next, we prove the positivity of the concentrations as well as a series of practical estimates like L∞ -bounds on concentrations and speed s0 of the strip Ω (t), energy estimates, non-trivial lower bounds on the final time of the process, etc. Having this wealth of estimates available, we cast the the existence and uniqueness problem in the framework of Banach’s contraction principle. The stability of the weak solution with respect to data and the model parameters is obtained in a straightforward manner by estimating via Gronwall’s inequality the difference of weak formulations written for two distinct solutions and parameters sets.

4.2

Function spaces

In this paper, we use the following spaces of functions and norms: The space Hi = L2 (a, b) is equipped with the norm |u|Hi := R

R

b a

u2 (y)dy

1 2

1

and with the scalar product (u, v)Hi := ab u(y)v(y)dy 2 for all u, v ∈ Hi Q (i ∈ I1 ∪ I2 ). The intermediate space H = i∈I1 ∪I2 Hi is normed by |u|H = P

1

2 2 for all u ∈ H and equipped with the standard scalar prodi∈I1 ∪I2 |ui |Hi uct. The spaces Vi := {u ∈ H 1 (0, 1) : u(0) = 0} (i ∈ I1 ), Vi := H 1 (1, 2) (i ∈ I2 ) and Vw := {(u5 , u6 ) ∈ H 1 (0, 1) × H 1 (1, 2) : u5 (0) = 0, u5 (1) = u6 (1)} = {(u5 , u6 ) ∈ V5 × V6 : u5 (1) = u6 (1)} are endowed with the norms ||u||Vi = |u,y |Hi , i ∈ {1, 2}, ||(u5 , u6 )||Vw = |u5,y |H5 + |u6,y |H6 , and with the corresponding scalar products. We use the space Vh := {(u3 , u7 ) ∈ H 1 (1, 2) × H 1 (0, 1) : u3 (1) = u7 (1)} with the norm ||(u3 , u7 )||Vh = |u3 |V3 + |u7 |V7 . We use V = V1 × V2 × Vw × Vh equipped with the corresponding norm.

For p ∈ [1, +∞] and k a positive integer, W k,p (Ω) are standard Sobolev spaces. [S → B] denotes the set of all functions defined on S with the range in ¯ B), m ∈ N, is the set of continuous functions of the form B, while C m (S, S¯ 3 t 7−→ u(t) ∈ B, that have continuous derivatives up to order m. The space Lp (S, B) (1 ≤ p < ∞) is the set of equivalence classes of Bochner-measurable 1 R functions u ∈ [S → B] for which ||u||Lp (S,B) = ( S ||u(t)||pB dt) p < +∞. We denote by L∞ (S, B) the vector space of all equivalence classes of Bochner measurable functions u ∈ [S → B] for which esssupt∈S ||u(t)||B < ∞. The set 12

W21 (S; V, H) := {v ∈ L2 (S, V) : v 0 ∈ L2 (S, V∗ )} forms a Banach space with the norm ||u||W21 = ||u||L2 (S,V) + ||u0 ||L2 (S,V∗ ) . More details on the mentioned function spaces can be found in [40], e.g.

4.3

Immobilizing moving strips

Let > 0 be sufficiently small and fixed. We reformulate the model in terms of macroscopic quantities by performing the transformation of all concentrations into volume-based concentrations via uˆi := φφw u¯i , i ∈ {1, 3, 4, 7}, uˆ2 := φφa u¯2 , uˆi := φ¯ ui , i ∈ {5, 6}

(26)

in a domain with fixed boundaries. To immobilize the moving reaction strips Ω1 (t) and Ω (t), we employ the Landau transformations (x, t) ∈ [0, s(t) + x and τ = t, for i ∈ I1 , (x, t) ∈ /2] × S¯T 7−→ (y, τ ) ∈ [a, b] × S¯T , y = s(t)+/2 x−s(t)−/2 [s(t) + /2, L] × S¯T 7−→ (y, τ ) ∈ [a, b] × S¯T , y = a + L−s(t)−/2 and τ = t, for i ∈ I2 . We relabel τ by t and introduce the new concentrations, which act in the auxiliary y-t plane by ui (y, t) := uˆi (x, t) − λi (t) for all y ∈ [a, b] and t ∈ ST . The model equations are reduced to

Di 0 s(t) + (ui + λi )i,t − ui,yy = s (t)yui,y 2 s(t) + 2 fi (u + λ), i ∈ I1 , + s(t) + 2

(27)

Di (ui + λi )i,t − ui,yy = s0 (t)(2 − y)ui,y L − s(t) − 2 L − s(t) − 2 fi (u + λ), i ∈ I2 + L − s(t) − 2

(28)

where u is the vector of concentrations (u1 , u2 , u3 , u5 , u6 , u7 )t and λ represents the boundary data (λ1 ,λ2 ,λ3 ,λ5 ,λ6 , λ7 )t . Formally, we employ λ3 = λ7 = λ6 = 0. The transformed initial 3 , boundary and transmission conditions (at Γr (t) for t ∈ ST ) are 3

We have ui0 (y) = u ˆi0 (x) − λi (0), where x = y s0 + s0 + 2 + (y − 1)(L − s0 − 2 ) for i ∈ I2 .

13

2

for i ∈ I1 and x =

ui (y, 0) = ui0 (y), y ∈ [a, b], i ∈ I1 ∪ I2 , ui (a, t) = 0, i ∈ I1 − {7}, u7,y (0, t) = ui,y (b, t) = 0, i ∈ I2 , u3 (1, t) = u7 (1, t), u5 (1, t) = u6 (1, t), −Di ui,y (1, t) = s0 (t)(ui (1, t) + λi (t)), i ∈ {1, 2}, s(t) + 2 −D3 −D7 u7,y (1, t), u3,y (1, t) = L − s(t) − 2 s(t) + 2 −D5 −D6 u5,y (1, t) = u6,y (1, t). s(t) + /2 L − s(t) − /2

(29) (30) (31) (32) (33) (34)

We keep unchanged the notation v4 (t) := uˆ4 (x, t) for x ∈ Ω1 (t) ∪ Ω (t) and t ∈ ST . Two ode’s R

η˜ (ˆ u)(x, t)dx and vˆ40 (t) = f4 (v4 (t)) a.e. t ∈ ST , u ˆ (x, t)dx 3 Ω (t)

Ω (t)

s0 (t) = cs () R

(35)

complete the model formulation, where uˆ := (ˆ u1 , uˆ2 , uˆ3 ,ˆ u4 ,ˆ u5 , uˆ6 , uˆ7 )t and cs () > 0. For this application, we expect cs () = O(). η1 and η denote now reaction rates that acts in the y-t plane and are defined by η` (y, t) := η¯` (ˆ u(ys(t) + y/2, t) + λ(t), Λ), y ∈ [0, 1],

(36)

for given 4 Λ ∈ MΛ . Finally, we assume s(0) = s0 > 0, vˆ4 (0) = uˆ40 > 0.

(37)

The transformed model equations are collected in (27)-(37).

4.4

Basic Estimates

Lemma 10 (Elementary inequalities) Let cξ > 0, ξ > 0, θ ∈ [ 12 , 1[ and s ∈ W 1,1 (Sδ ). (i) There exists the constant cˆ = cˆ(θ) > 0 such that |ui |∞ ≤ cˆ|ui |1−θ ||ui ||θ

(38)

4

Generally, two distinct sets of reaction parameters Λ1 and Λ describe the structure of η1 and η . For simplicity, we denote both by Λ.

14

for all ui ∈ Vi , where i ∈ I1 ∪ I2 . (ii) It holds |ui |1−θ ||ui ||θ ≤ ξ||ui || + cξ |ui |

(39)

for all ui ∈ Vi , where i ∈ I1 ∪ I2 . (iii) Let ϕ ∈ V with ϕ = (ϕ1 , . . . , ϕ6 )t , t ∈ Sδ , cˆ as in (i), and ξ, cξ as in (ii). Then we have for i ∈ I1 and j ∈ I2 the following inequalities: 1 |s0 (t)| |s0 (t)| (yϕi,y , ϕi ) = {ϕi (1)2 − |ϕi |2 } s(t) 2 s(t) 1 |s0 (t)| 2 ≤ {ˆ c |ϕi |2(1−θ) ||ϕi ||2θ − |ϕi |2 }; 2 s(t) |s0 (t)| |s0 (t)| |ϕi (1)|2 ≤ |ϕi |2∞ s(t) s(t) 2θ−1 2 1 ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ × s(t) 1−θ |s0 (t)| 1−θ |ϕi |2 ; s (t) 2 2θ |ϕi (1)| 1 2 2 2θ−2 2(1−θ) −1 ≤ |ϕ | ≤ c ˆ s(t) |ϕ | s(t) ||ϕ || i ∞ i i s2 (t) s2 (t) 2(θ−1) 2 ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ |s(t)| 1−θ |ϕi |2 ; s (t) 2 2θ−1 2 |ϕi (1)| ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ |s(t)| 1−θ |ϕi |2 ; s(t) s (t) 0 1 |s0 (t)| 1 |s0 (t)| |s (t)| ((2 − y)ϕj,y , ϕj ) = |ϕj (1)|2 + |ϕj |2 . L − s(t) 2 L − s(t) 2 L − s(t) (iv) The inequality |a + b|p ≤

p−1 (1 + ξ)p−1 |a|p + 1 + 1 |b|p for p ∈ [1, ∞[ ξ |a|p + |b|p

holds for arbitrary a, b ∈ R and ξ > 0. (v) For all θ ∈ [0, 1], a, b, c ∈ R+ , ξ > 0, ξ¯ > 0, cξ¯ := where

1 p

+

1 q

(40)

for p ∈]0, 1[

1 2ξ¯2

and cξ :=

1 q

1 √ p

(ξp)q

,

= 1 with p ∈]1, ∞[, the inequality

ξ¯ abθ c1−θ ≤ a2 + ξcξ¯b2 + cξ¯cξ c2 2

(41)

holds. 15

We refer to (38) as interpolation inequality. In applications, Lemma 10 (iii) will be used with s(t) replaced by s(t) + 2 . We introduce the positive constant K1 by

2

K1 := 1 + max{P1 Q1 , P2 } + cξ cˆ1−θ +

cξ cˆ4 + cξ cˆ4 (D32 + D72 ). 4 h

(42)

h

Notice that K1 depends on cˆ, cf ∈ R+ , θ ∈ 21 , 1 and ξ > 0. The strict positivity of ξ within a compact subset of ]0, ∞[ implies that cξ < ∞. Therefore, the constant K1 is finite. Furthermore, we denote by η1 the term η1 := η1 + η .

(43)

Theorem 11 (Positivity and L∞ -estimates) Let the triple (u, v4 , s) as in Definition 1 satisfy the assumptions (A)-(C2). Then the following statements hold: (i) (Positivity) u(t) + λ(t) ≥ 0 in V a.e. t ∈ Sδ . (ii) (L∞ -estimates) There exists a constant c = c(k` ) > 0(` ∈ I1 ∪ I2 ), where k` is cf. (8), such that u(t) + λ(t) ≤ c in V a.e. t ∈ Sδ . (iii) (Positivity and boundedness of CaCO3 within Ω (t)) 0 < uˆ40 ≤ uˆ4 (x, t) ≤ uˆ40 + δM a.e. x ∈ Ω (t), a.e. t ∈ Sδ . Proof. The statement (iii) is obvious as soon as the estimates ensuring (i) and (ii) hold. We show simultaneously the non-negativity and the maximum bounds of concentrations like we did in the proof of Theorem 3.3 in [23]. Since some of the arguments are sometimes the same, we do not always repeat them. In such cases, we prefer to give only the proof idea. We start with getting upper bounds for u1 and u2 . Once this step is done, the local Lipschitz condition on η` (` ∈ {1, }) becomes global. Select in the weak formulation (1) the test function ϕ := ((u1 + λ1 − k1 )+ , (u2 + λ2 − k2 )+ , 0, 0, 0, 0)t ∈ V, where we denote by ϕi (i ∈ {1, 2}) the expression ϕi := (ui + λi − ki )+ . We obtain 2 q X 1 1 1 d|ϕ|2 + || Di ϕi ||2 = − 2 2 dt s+ s + 2 i=1

2 X

s i=1 2

0

(ϕi (1)2 + ki ϕi (1))

+ P1 (Q1 (u2 + λ2 − k2 ), ϕ1 ) − P1 (u1 + λ1 − k1 , ϕ1 ) + P1 (Q1 k2 − k1 , ϕ1 ) − (η1 , ϕ1 ) + P2 (Q2 (u2 + λ2 − k2 ), ϕ1 ) + P2 (u1 + λ1 − k1 , ϕ2 ) 2 X s0 2 2 − P2 (Q2 k2 − k1 , ϕ2 ) + |ϕ (1)| − |ϕ | . i i 2 s + 2 i=1

16

(44)

By (C2), −η1 ϕ1 (1) ≤ 0 and −

2 X

s0 |ϕi (1)|2 + ki ϕi (1) +

i=1

s0 s0 |ϕi (1)|2 − |ϕi |2 ≤ 0, 2 2

it yields that 2 q X 1 1 d|ϕ|2 + Di ϕi ||2 ≤ K1 |ϕ|2 . || 2 2 dt s + 2 i=1

(45)

Since ϕ(0) = 0, the Gronwall’s inequality shows that u1 (t) + λ1 (t) ≤ k1 and u2 (t) + λ2 (t) ≤ k2 a.e. t ∈ Sδ . We establish now maximum bounds for u3 and u7 by setting in the weak formulation the test function ϕ := (0, 0, (u3 +λ3 −k3 )+ , 0, 0, (u7 +λ7 −k7 )+ )t ∈ V, where we denote ϕi := (ui + λi − ki )+ for i ∈ {3, 7}. For a convenient choice of k3 , k7 , we have that (ϕ3 , ϕ7 ) ∈ Vh . We obtain q 1d 1 1d |ϕ7 |2 + L − s − |ϕ3 |2 + D7 ϕ7 ||2 || 2 2 dt 2 2 dt s + 2 q 1 D7 D3 2 2 2 + || D3 ϕ3 || = |ϕ7 (1)| + |ϕ3 (1)| L−s− 2 s+ 2 L−s− 2 + L−s− S3,diss (u3,eq − k3 , ϕ3 ) − S3,diss |ϕ3 |2 2 −(η1 , ϕ7 ) + S7,diss (u7,eq − k7 , ϕ7 ) − S7,diss |ϕ7 |2 + s+ 2 0 h i s |ϕ7 (1)|2 + |ϕ3 (1)|2 − |ϕ7 |2 + |ϕ3 |2 . + (46) 2

s+

By (C1) and the interpolation inequality, we margin above the r.h.s of (46) by means of the following expression:

ξ

||ϕ7 ||2 s+

2

"

+K1

2 + ξ

||ϕ3 ||2 L−s−

s0 (s + ) + D7 2

2

2

2

+ s0 (L − s − ) + D3 2

2 #

|ϕ|2 .

(47)

We take ξ ∈]0, min{D3 , D7 }] and apply Gronwall inequality to get u3 (t) + λ3 (t) ≤ k3 and u7 (t) + λ7 (t) ≤ k7 a.e. t ∈ Sδ . However, in order to be able to effect this step, we need to ensure that s0 ∈ L∞ (Sδ ). The reaction rate η has to stay bounded with respect to the variable u3 for fixed u1 , therefore we use the boundedness assumption from (A). 17

Let us show the maximum estimates for u5 and u6 . Set ϕ5 = (u5 + λ5 − k5 y)+ and ϕ6 = (u6 + λ6 − k6 )+ , where (ϕ5 , ϕ6 ) ∈ Vw , and obtain

q 1d 1d 1 D5 ϕ5 ||2 |ϕ5 |2 + L − s − |ϕ6 |2 + || 2 2 dt 2 2 dt s + 2 q 1 k5 D5 ||ϕ5 || 2 + = (η1 , ϕ5 ) || D6 ϕ6 || + L−s− 2 s + 2 i s0 h + |ϕ5 (1)|2 + |ϕ6 (1)|2 − |ϕ5 |2 + |ϕ6 |2 2 + s0 k5 ||ϕ5 ||.

s+

(48)

By Cauchy-Schwarz’s and Poincar´e’s inequalities, there exists a constant M1 = M1 (k1 , k3 , Λ) > 0 such that |(η1 , ϕ5 )| ≤ M1 ||ϕ5 ||. Rearranging the terms in (48) and applying Poincar´e’s and interpolation inequalities, we obtain:

q 1d 1 1d |ϕ5 |2 + L − s − |ϕ6 |2 + D5 ϕ5 ||2 || s+ 2 2 dt 2 2 dt s + 2

!

q 1 k5 D5 2 + + M1 k5 ||ϕ5 || || D6 ϕ6 || ≤ M1 − L−s− 2 s + 2 ||ϕ6 ||2 ||ϕ5 ||2 +ξ 2 + ξ 2 s + 2 L − s − 2 "

+ K1 If 0

0 take place. Although k5 t − λ5 if the restrictions k5 ≥ 2M s+ 2 both restrictions admit obvious physical motivations, the upper bound k5 t is a sort of unconfortable because of its dependence on time. The choice (b2) does not bring more insight than the use of (b1). (c) The L∞ -estimates ki (i ∈ I) can be made independent on the choice of . We introduce the constant K2 by h iL P1 Q1 2 2 2 (1 + L) + S7,diss + S3,diss + S7,diss 2 2 1 4 2 2 + 2cξ cˆ (D3 + D7 + ). 2

K2 := P2 +

(52)

Clearly, it yields that K2 ∈ R∗+ . Lemma 12 (Energy estimates) Let the triple (u, v4 , s) satisfy Definition 1. Then the following statements hold:

(i)

|u(t) + λ(t)|2 ≤ α(t) exp

Zt

β(τ )dτ a.e. t ∈ Sδ ,

(53)

0

(ii)

|u(t) + λ(t)|2 ≤ α(t) +

Zt

t Z β(s)α(s) exp β(τ )dτ ds a.e. t ∈ Sδ ;(54) s

0

19

t Z β(τ )dτ a.e. t ∈ Sδ , ||u(τ ) + λ(τ )||2 dτ ≤ d−1 0 α(t) exp

Zt

(iii)

(55)

0

0

where )

(

Di Di d0 := min min , min i∈I1 L + i∈I2 L − s0 − 2

.

2

(56)

The factors α(t) and β(t) are defined by L 2 2 S3,diss |u3,eq |2∞ + S7,diss |u7,eq |2∞ 2 t 2 Z 2 a(τ )dτ α(t) := |u(0) + λ(0)| + m0

a(t) :=

(57) (58)

0

0

s β(t) := 2 + + K2 2

s+ 2

2

+ L−s− 2

2 !

,

(59)

where K2 satisfies (52). u,t ∈ L2 (Sδ , V∗ ), u ∈ C(S¯δ , H).

(iii)

(60)

Proof. We set in the variational formulation (20) the test function ϕ := (u + λ)t ∈ V and obtain the expression

s+

2

1 s+

X 2 i∈I1

d |ϕi |2 + dt i∈I1 X

2 +

q

|| Di ϕi ||2 +

L−s−

2

2

1 L−s−

X 2 i∈I2

d X |ϕi |2 dt i∈I2

q

|| Di ϕi ||2 =

3 X

Ij ,

(61)

j=1

where 2 X

s0 (t) D7 D3 2 2 2 |ϕi (1)| + |ϕ7 (1)| + |ϕ7 (1)| , s+ 2 L−s− 2 i=1 s(t) + 2 X X I2 := s + (fi (ϕi ) + L − s − (fi (ϕi ), ϕi ), 2 i∈I1 2 i∈I2

I1 := −

I3 := s0 (t)

X i∈I1

(yϕi,y , ϕi ) + s0 (t)

X

((2 − y)ϕi,y , ϕi ).

(62)

i∈I2

By the positivity of s0 and the interpolation inequality with θ = 21 , we have 20

|I1 | ≤ ξ

||ϕ7 ||2 s+

2

2 + ξ

||ϕ3 ||2 L−s−

2

2 2 + K2 |ϕ| .

(63)

Cauchy-Schwarz’s inequality and the means inequality show

h | P1 Q1 (ϕ2 , ϕ1 ) − P1 |ϕ1 |2 |I2 | ≤ s(t) + 2 − (η1 , ϕ1 + ϕ7 ) − P2 Q2 |ϕ2 |2 + P2 (ϕ1 , ϕ2 ) i i h 2 + S7,diss (u7,eq , ϕ7 ) − |ϕ7 | + L − s − S3,diss (u3,eq , ϕ3 ) − |ϕ3 |2 | 2 " 2 |u7,eq |2∞ S7,diss P1 Q1 2 2 2 2 (|ϕ1 | + |ϕ2 | ) + P2 (|ϕ1 | + |ϕ2 | ) + ≤L 2 2 # 2 2 2 2 2 2 S |ϕ7 | LS3,diss |u3,eq |∞ LS3,diss |ϕ3 | + 7,diss + + 2 2 2 L 2 2 ≤ S |u3,eq |2∞ + S7,diss |u7,eq |2∞ + K2 |ϕ|2 (64) 2 3,diss

Integrating by parts in the two sums of I3 , we have

s0 s0 |ϕ(1)|2 + |ϕ|2 2 2 X ||ϕi ||2 X ||ϕi ||2 ≤ξ + ξ 2 2 i∈I1 s + i∈I2 L − s − 2 2

|I3 | ≤

2 s0 + cξ cˆ1−θ + 2

"

s+ 2

2θ 1−θ

+ L−s− 2

2θ 1−θ

!#

|ϕ|2 .

(65)

For θ = 21 , we acquire

|I3 | ≤ ξ

||ϕ7 ||2

X i∈I1

s+

s0 + + K2 2 "

2

2 + ξ

Select ξ ∈]0, mini∈I1 ∪I2 the proof of (i).

s+ 2 Di ]. 2

||ϕi ||2

X i∈I1

2

L−s−

2

2

+ L−s− 2

2 !#

|ϕ|2 .

(66)

The application of Gronwall’s inequality completes

The proof of (ii) - (iv) follow similarly as in [22] (Lemma 3.4.23). Note that the energy estimates (53) - (55) do not depend on the choice of . 21

4.5

Proof of Theorem 3

The proof of Theorem 3 relies on the use of the Banach fixed-point principle. We consider the time interval Sδ =]0, δ[, t ∈ Sδ , with δ set as in Definition 1. Let σ = supSδ s0 (:= M + 1). Let Yδ be the set of functions r ∈ W 1,2 (Sδ ) such that r(0) = 0. (Yδ , ρ) is a complete metric space, where the metric ρ is defined by ρ : Yδ × Yδ → R+ such that

(67)

ρ(r1 , r2 ) = |r20 − r10 |L2 (Sδ ) for all r1 , r2 ∈ Yδ . Let M (Sδ ) := {r ∈ Yδ : r ∈ [s0 , σδ + s0 ], r0 ≥ 0, |r0 |L2 (Sδ ) ≤ σδ}.

(68)

Clearly, M (Sδ ) is a non-empty closed subset of Yδ . Define a mapping T : M (Sδ ) → W 1,p (Sδ ), p ∈ [1, ∞] in the following way: For any s ∈ W 1,2 (Sδ ) satisfying (35) and s − s0 ∈ M (Sδ ), we define T : M (Sδ ) 3 s˜ := s − s0 7−→ u cf. (20) , v4 cf. (35) 7−→ r − s0 := r˜ ∈ W 1,p (Sδ ).

(69)

We introduce the metric space (M (Sδ ), ρ), where M (Sδ ) := {r ∈ W 1,2 (Sδ ) : r(0) = s0 , r(t) − s0 ∈ [0, δσ],

(70)

r0 (t) ≥ 0 a.e. t ∈ Sδ , |r0 |L2 (Sδ ) < σ} = 6 ∅ and ρ : M (Sδ ) × M (Sδ ) → R+ defined by

(71)

ρ(r1 , r2 ) = |r20 − r10 |L2 (Sδ ) for all r1 , r2 ∈ M (Sδ ). M (Sδ ) is a complete space with respect to the metric ρ. Lemma 13 (The fixed-point operator) Let the assumptions of Theorem 3 be fulfilled. Then we have: (i) T : M (Sδ ) → M (Sδ ) for sufficiently small δ. (ii) There is a strictly positive constant χ = χ(s0 , L0 , Cη , θ, cˆ, δ, ki , D` ), where i ∈ I and ` ∈ I1 ∪ I2 , such that ρ(T s2 , T s1 ) ≤ δχρ(s2 , s1 )

for all s1 , s2 ∈ M (Sδ ).

22

(72)

Proof. The proof of (i) follows the lines of Lemma 3.3.25 from [22]. We concentrate on the proof of (ii), i.e. the Lipschitz continuity of T . Let w(j) := (wj1 , wj2 , . . ., wj7 ) (j ∈ {1, 2}) be two distinct concentrations vectors and let sj be the corresponding strips positions. Set wt := (w1 , w2 , . . . , w7 ) := w(2) − w(1) ∈ V, where wi := w2i − w1i := u2i + λ2i − u1i − λ1i for all i ∈ I1 ∪ I2 . Let us denote ∆s(t) := s2 (t) − s1 (t), ∆s0 (t) := s02 (t) − s01 (t), etc. We consider the fixed-point operator T : si ∈ M (Sδ ) 7−→ solution wi of the variational problem (20) 7−→ ri ∈ W 1,2 (Sδ ),

(73)

where ri0 (t) satisfies (35) a.e. t ∈ Sδ and ri (0) = s0 , i = 1, 2. Claim 14 Let α be an arbitrary positive real number. There is a strictly positive constant c = c(cs (), , k3∗ , k3 , α) such that the inequality Z

|∆r0 (τ )|α dτ ≤ c(cs (), , k3∗ , k3 , α)

Z

Z

|w|α dxdτ

(74)

Sδ Ω (τ )

Sδ

holds. By Claim 14, it yields that Z

|∆r0 (τ )|2 dτ ≤ δK2 sup |w(t)|2 ,

(75)

t∈Sδ

Sδ

where K2 is defined to be the constant c(cs (), , k3∗ , k3 , α) in which we put α = 2. We aim to estimate supt∈Sδ |w(t)|2 in order to obtain the contractivity of the operator T . We proceed as follows: We subtract the variational formulation written for the solution w(1) from that one written for w(2) and employ in both formulations the same test function wt := w(2) − w(1) ∈ V. The proof relies on convenient manipulations of the positivity of the concentrations, and of the maximum and energy estimates (see Lemma 11 and Lemma 12) to support the application of Banach’s fixed-point principle. For each j ∈ {1, 2}, (20) yields

sj (t) + 2

X i∈I1

(j) (wi,t (t), wi )

+ L − sj (t) − 2

X

(j)

(wi,t (t), wi )

i∈I2

+a(sj , w(j) , w) + e(s0j , w(j) , w) = bf (w(j) , sj , w) + h(sj , s0j , wy(j) , w). We rearrange the terms in (76) and obtain 23

(76)

s2 + 2

X 1d i∈I1

X1d |wi (t)| + L − s2 − |wi (t)| 2 dt 2 i∈I2 2 dt 1 X q + || Di wi ||2 s2 + 2 i∈I1

+

1 L − s2 −

X 2 i∈I2

q

|| Di wi ||2 ≤

5 X

|Jj |,

(77)

j=1

where the right-hand side of (77) is defined by means of the following terms:

J1 := −∆s

X

(w1i,t , wi ) + ∆s

i∈I1

J2 := −

2

(w1i,t , wi )

i∈I2

∆s s1 +

X

X 2

s2 +

Di (w1i,y , wi,y )

i∈I1

∆s L − s1 −

2

X

L − s2 −

2

Di (w1i,y , wi,y ),

i∈I2

J3 := bf (w2 , s2 , w) − bf (w1 , s1 , w), J4 := e(s01 , w1 , w) − e(s02 , w2 , w) + +

1 L − s2 −

1 s2 +

X q 2 i∈I1

| Di wi (1)|2

X q 2 i∈I2

| Di wi (1)|2 ,

J5 := h(s02 , w2,y , w) − h(s01 , w1,y , w). In order to simplify the writing of the estimates, we introduce the constant K3 cˆ4 := 1 + cξ + max Di + c + (|I1 | + |I2 |)(cˆ c)2 2 2 i∈I1 ∪I2 !

2 X 1 c2 ¯ as in (9)) + k¯2 + 1 + cξ k¯2 + cξ cξ¯ + + cξ + k( ki2 + cξ cˆ1−θ . 4 2 i=1

¯ ki , Di Note that the K3 is strictly positive and depends mainly on θ, ξ, ξ, (i ∈ I1 ∪ I2 ), s0 , etc. It does neither depend on the solution, nor on . The bound of |J1 | reads |J1 | ≤

|∆s|2 |w|2 |w1,t |2 + . 2 2

(78)

By Cauchy-Schwarz’s and the arithmetic-geometric means inequalities, we obtain 24

|J2 | ≤ +

|∆s| s1 +

2

X 2

s2 +

Di ||w1i ||||wi ||

i∈I1

|∆s|

2 2

L − s1 − X

≤ξ

i∈I1

X

2

L − s2 −

Di ||w1i ||||wi ||

i∈I2

X ||wi || ||wi ||2 +ξ 2 (s2 + )2 i∈I2 L − s2 − 2

2 2 X X Di ||w1i || Di ||w1i || 2 + cξ 2 + 2 |∆s|

s1 +

i∈I1

X

≤ξ

i∈I1

2

||wi ||2

X

2

i∈I2

s2 +

L − s1 −

i∈I2

2 + ξ

||wi ||2

L − s2 −

+ K3

2

2

2

1

s1 +

2

2 +

1 L − s1 −

2

2 2 2 ||w1 || |∆s|

(79)

Finally, for |J5 | we have

2

s01

s01

1 + 2K3 |J5 | ≤ 3 + 3ξ¯ + 4K3 L s1 + L − s1 −

2

2

2 2 |∆s|

+ 3 + 3ξ¯ + 6K3 |∆s0 |2

2

||wi ||

X

+ ξ(3 + 2cξ¯)

i∈I1

( s2 +

2

2

2 +

||wi ||

X i∈I2

L − s2 −

2

2

+ K3 χ1 (t)|w|2 ,

(80)

where the expression of χ1 (t) is given by

χ1 (t) :=

1 s2 +

+ + +

2

s01 s1 +

+

1 L − s2 −

!2

2 s01

L − s1 −

2

s2 + 2

!2 2

+ 2

1 s2 +

1 + s2 +

L − s2 − 2

(s01 )2 L − s1 −

2

2

L − s2 −

2

25

2 +

1 L − s2 −

2

! 2

2

+ 1 + L − s2 −

! 2

2

for a.e. t ∈ Sδ . To estimate the remaining terms, namely |J3 | and |J4 |, we need to take into account the exact structure of bf (·) and e(·). We obtain the bound on |J3 | as follows: (2) J3 := s2 + [P1 (Q1 w22 − w21 , w1 ) − (η1 , w1 )] 2 (1) − s1 + [P1 (Q1 w12 − w11 , w1 ) − (η1 , w1 )] 2 P2 (Q2 w22 − w21 , w2 ) + s1 + P2 (Q2 w12 − w11 , w2 ) − s2 + 2 2 + (L − s2 − )S3,diss (u3,eq − w23 , w3 ) 2 − (L − s1 − )S3,diss (u3,eq − w13 , w3 ) 2 (1) (2) + (η1 , w5 ) − (η1 , w5 ) + (s2 + )S7,diss (u7,eq − w27 , w7 ) − (s1 + )S7,diss (u7,eq − w17 , w7 ) 2 2 (1) (2) − (η1 , w7 ) + (η1 , w7 ),

and hence, [(−∆η1 , w1 ) + P1 Q1 (w2 , w1 ) − P1 |w1 |2 ] 2 + s2 + [−P2 Q2 |w2 |2 + P2 (w1 , w2 )] − L − s2 − S3,diss |w3 |2 ] 2 2 + s2 + (∆η1 , w5 ) 2 + s2 + S7,diss |w7 |2 + s2 + (−∆η1 , w7 ) 2 2 (1) + ∆s[P1 (Q1 w12 − w11 , w1 ) − (η1 , w1 )] − ∆s[P2 (Q2 w12 − w11 , w2 )] − ∆sS3,diss (u3,eq − w13 , w3 )

J3 ≤ s2 +

(1)

(1)

+ ∆s(η1 , w5 ) + ∆sS7,diss (u7,eq − w27 , ϕ7 ) + ∆s(η1 , w7 ) ≤ ρ1 |∆s|2 + ρ2 K3 |w|2 ,

(81)

where the constants ρ1 and ρ2 depend on P1 , P2 , Q1 , Q2 , S3,diss , S7,diss and cL . After some manipulations, we obtain

|J4 | ≤

X i∈I1 ∪I2

−

2 X i=1

Di

1 s2 +

s02 w2i wi (1)

+

2

1 + L − s2 −

2 X

s01 w1i wi (1)

i=1

26

! 2

|w(1)|2

L i∈I1 ∪I2 Di |∆s0 |2 ¯2 |w(1)|2 . + k |w(1)|2 + ≤ 2 s2 + 2 L − s2 − 2 P

Using twice the interpolation inequality in (82) with θ =

|J4 | ≤

1 2

(82)

and obtain

X ||wi ||2 X |∆s0 |2 ||wi ||2 +ξ 2 + ξ 2 2 i∈I1 s2 + i∈I2 L − s2 − 2 2

2 + s2 + 2

+ K3

2

+ L − s2 − 2

2 !

|w|2

(83)

We introduce the time-dependent factors

2

γ1 :=

1 |w1,t | 1 2 + K3 2 + 2 ||w1 || 2 s1 + 2 L − s1 − 2

+ 3 + 3ξ¯ + 4K3

s01 s1 +

2

+ 2K3

s02 L − s2 −

!2 2

+ ρ1 ,

γ2 := 4 + 3ξ¯ + 6K3 , χ1 + ρ2 + 3 + s2 + 2

γ3 := K3

2

+ L − s2 − 2

2 !

,

and reformulate the right-hand side of (77) as X q X q 1d X 1 1 2 2 |wi | + || Di wi || + || Di wi ||2 (84) 2 2 2 dt i∈I s2 + 2 i∈I1 L − s2 − 2 i∈I2

≤

X X ||wi ||2 7 ||wi ||2 13 ξ + 2ξc + ξ + 2ξcξ¯ 2 2 ξ¯ 2 2 i∈I2 L − s2 − i∈I1 s2 + 2 2

+γ1 |∆s|2 + γ2 |∆s0 |2 + γ3 |w|2 . Select ξ and ξ¯ in such a way that we may neglect the first two terms of √ P L|| Di wi ||2 the right-hand side of (84) when comparing them to i∈I1 and 2 s2 + 2 ) ( √ P L|| Di wi ||2 2 , respectively. The only problem is that γ1 depends on ||w1 || i∈I2 (L−s2 − 2 ) and |w1,t |. Employing conveniently the geometrical restrictions on s1 , s2 , L and R , and the energy estimate (53), we can show that Sδ γ1 (τ )dτ is independent of ||w1 || and |w1,t |. After a redefinition of γ1 , we obtain the inequality 5 Z Sδ 5

Z

γ1 (τ )|∆s(τ )|2 dτ ≤ δ 2 max |γ1 (t)| t∈S¯δ

|∆s0 (τ )|2 dτ,

(85)

Sδ

Note that (85) follows directly by means of the mean-value theorem for integrals.

27

where its right-hand side does not depend on ||w1 || and |w1,t | anymore. By (85) and integration along Sδ , we express (84) as

2

2

Z

2

|w| ≤ |w(0)| + 2 max δ |γ1 (t)| + |γ2 (t)| t∈S¯δ

0

2

|∆s (τ )| dτ +

Zt

2γ3 (τ )|w(τ )|2 dτ.

0

Sδ

By Gronwall’s inequality (with w(0) = 0), we gain the estimate 0 2

2

R2

|w| ≤ δ max |∆s | γ4 (t)e

Sδ

γ3 (τ )dτ

t∈S¯δ

,

(86)

where the factor γ4 is given by γ4 (t) := 2 (δ 2 |γ1 (t)| + |γ2 (t)|) for all t ∈ Sδ . Compare now (86) to (72). For each δ > 0 sufficiently small, we choose χ > 0 such that the operator T becomes contractive on Sδ . For instance, a possible choice of χ is R

2γ3 (τ )dτ

maxt∈S¯δ γ4 (t)e Sδ . χ := (1 + c(cs (), , k3∗ , k3 , α)) δ The local existence and uniqueness result via Banach’s Contraction Principle. If χT ≥ 1, additional steps are needed. Then δ > 0 has to be taken such that χδ < 1, and in this case, we consider problem (2)-(5) with respect to the time interval [0, δ]. By replacing T with δ in the above argument we obtain only a local solution. Now consider the problem on [δ, 2δ]. By applying the technique used at the outset to this problem, we obtain a unique solution that extends the previous one from the interval [0, δ] to [δ, 2δ]. It is clear that this procedure may be repeated on the interval [2δ, 3δ] and so on. After a number of steps, we gain that the solution of (2)-(5) is valid on [0, Tfin ]. Note that δ and χ (1 + c(cs (), , k3∗ , k3 , α)) do not depend on the selection of . Finally, it only remains to prove (74). Proof of Claim 14. Employing the definition of r0 , a simple calculation shows that

1 Z |∆r0 (τ )|α dτ = cs () Sδ

=

Z Sδ

R

R

Ω (τ ) ∆η (y, τ )dy − R R w (y, τ )dy Ω (τ )

23

(2) Ω (τ ) η (y, τ )dy

Ω (τ )

w3 (y, τ )dy

α Z

w3 (y, τ )dy dτ.

2 Ω (τ )

(87) 28

By (40), there exists a constant cα > 0 such that 6 for any a, b ∈ R+ we have (a + b)α ≤ cα (aα + bα ) .

(88)

Therefore, we may conclude that there exits a constant c¯ = c¯(cs (), , k3∗ , k3 , α) > 0 such that Z Sδ

ccs ()Cηα Z Z |∆η |α (y, τ )dydτ |∆r (τ )| dτ ≤ k3∗ α α 0

α

Sδ Ω (τ )

ccs () + 2α 2α k3 ≤ c¯

Z

Z

Z

Z

|w|α dydτ

Sδ Ω (τ )

|w|α dydτ.

(89)

Sδ Ω (τ )

For example, we take c¯ := proof of the claim.

4.6

ccs ()Cηα

Ω

k3∗ α α

+

ccs () k3 2α 2α

with c ≥ cα . This concludes the

Proof of Theorem 4 (i)

Let (u(i) , v4 , s(i) ) (i ∈ {1, 2}) be two weak solutions on Sδ (in the sense of Definition 1), which satisfy the assumptions of Theorem 3. We want to show that the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W21 (Sδ , V, H) × W 1,2 (Sδ )2 that maps (u0 , λ, Ξ, Υ, Λ)t into (u, v4 , s)t is Lipschitz continuous in the sense of (22). Let Γi (t) be the centerlines of two arbitrarily chosen reaction strips Ω(i) (t) (i ∈ {1, 2}). By (15), the positions si (t) of the interfaces Γi (t) satisfy the geometrical restriction 0 < si0 := si (0) ≤ si (t) −

< si (t) + ≤ Li0 < L for all t ∈ Sδ . 2 2

Denoting s0 := max{s10 , s20 } and L0 := min{L10 , L20 }, the common space domain travelled by Γi (t) is the interval ]s0 , L0 [. Within this frame we only discuss the non-trivial case, namely s0 < L0 . Set L∗ () := min{min{si (t) + , L − si (t) − : i = 1, 2}} > 0, 2 2 t∈S¯δ

(90)

Take cα ≥ max{(1 + ς)α−1 , (1 + 1ς )α−1 } for ς > 0 and α ∈ [1, ∞[. If α ∈]0, 1[, then cα ≥ 1.

6

29

(i)

D0 := min{Dj : j ∈ I1 ∪ I2 , i ∈ {1, 2}} > 0.

(91) (1)

We subtract the weak formulation (20) of the solution (u(1) , v4 , s1 ) from that (2) written in terms of (u(2) , v4 , s2 ). Choosing as test function w = (u(2) −u(1) )t + (2) (1) (2) (1) (λ(2) − λ(1) )t ∈ V, i.e. wi = ui − ui + λi − λi ∈ Vi for all i ∈ I1 ∪ I2 , we obtain: s2 + 2

2

L − s2 − |wi | + dt 2

X d i∈I1

2

1 + L − s2 −

2

X d

1 |wi | + dt s2 +

i∈I2

X 2 i∈I2

2

q

(1)

|| Di w||2 ≤

X 2 i∈I1 11 X

1 s2 +

J2 := −∆s

X 2 i∈I1

X

(2)

Di |wi (1)|2 +

|Jj |,

j=1

(w1i,t , wi ) + ∆s

i∈I1

J3 := −

1 s2 +

X 2 i∈I1

2

X

X 2 i∈I2

(2)

Di |wi (1)|2

(w1i,t , wi )

i∈I2

|∆Di |(w1i,y , wi,y )

∆s

+

1 L − s2 −

X

2

(1)

Di (w1i,y , wi,y ))

s2 + s1 + i∈I1 X 1 − |∆Di |(w1i,y , wi,y ) L − s2 − 2 i∈I2 +

∆s L − s1 −

J4 := −s02

2 X

2

L − s2 −

wi (1)2 − ∆s0

i=1

X

2 X

2

(1)

Di (w1i,y , wi,y ))

i∈I2

w1i (1)wi (1)

i=1

i h (2) (2) (2) P1 Q1 w22 − w21 , w1 − η1 , w1 J5 := s2 + 2 i h (1) (1) (1) − s1 + P1 Q1 w12 − w11 , w1 − η1 , w1 2

(2) (2) J6 := − s2 + P2 Q2 w22 − w21 , w2 2 (1) + s1 + Q2 w12 − w11 , w1 2 (2) (2) S3,diss (u3,eq − w3 , w3 ) J7 := L − s2 − 2

30

(1)

|| Di w||2

where the right-hand side of (92) is given by:

J1 :=

q

(92)

(1) (1) S3,diss (u3,eq − w3 , w3 ) 2 (2) (2) (1) (1) J8 := s2 + S7,diss (u7,eq − w7 , w7 ) − s1 + S7,diss (u7,eq − w7 , w7 ) 2 2 X X (2) (1) J9 := η1 , wj − η1 , wj s2 + s1 + 2 2 j∈{5,7} j∈{5,7}

− L − s1 −

J10 := s02

X

(yw2i,y , wi ) − s01

X

(yw1i,y , wi )

i∈I1

i∈I1

J11 := s02

X

((2 − y)w2i,y , wi ) − s01

i∈I2

X

((2 − y)w1i,y , wi ).

(93)

i∈I2

Let K4 be the following positive constant (1)

K4 := 1 + 2ˆ c4 max Di + i∈I1 ∪I2

(1) (1) 2 Q1 P 1

+

2

(1)

k 1 + k2 + M1 + cˆ4 cξ 2

2 k 2 (1) (2) 2 L2 (2) + 2 P1 Q1 + 1 + Q1 k2 2 2

(2)

+ S3,diss + S3,diss |u3,eq |∞

+ +

(1) 2

P2

2 X h `∈{3,7}

+

(2)

LQ2 k2

2

2

+ L|u3,eq |2∞

(2)

(2)

P2 Q2 k2

2

+ 2 2 i cξ (1) (2) 2 2 + cˆ4 . S`,diss + L |u`,eq | + S`,diss + M1 2

(94)

Clearly, K4 depends , ξ and all model parameters. It stays stays bounded for ξ > 0. Let us now proceed to estimate the terms |Ji | (i ∈ {1, . . . , 11}). The main tools used in this context are the positivity, L∞ - and energy estimates combined with the interpolation inequality, Cauchy-Schwarz’s and Young’s inequalities as well as some further elementary algebraic inequalities. We obtain 1 X ||wi ||2 1 X ||wi ||2 2 |J1 | ≤ ξ 2 + ξ 2 + K4 |w| . 2 i∈I1 s2 + 2 i∈I2 L − s2 − 2 2 |J2 | ≤ |∆s|2

|J3 | ≤ 2ξ

|w1,t |2 |w1 |2 + . 2 2

X i∈I1

(96)

||wi ||2

X

2

i∈I2

s2 +

2 + 2ξ

(95)

||wi ||2

L − s2 −

31

2

2 +

+ K4

1

s1 +

2

L − s1 −

2

||w1i ||2 |∆s|2

X i∈I1 ∪I2

||w1i ||2 |∆Di |2 ,

X

+ K4

2

2 +

1

(97)

i∈I1 ∪I2

|∆s0 |2 2

|J4 | ≤ K4 |w(1)|2 + ≤ξ

||wi ||2

X i∈I1

s2 +

2

|∆s0 |2 + + K4 2

2 + ξ

"

||wi ||2

X i∈I2

s1 + 2

2

L − s2 −

2

2 +

+ L − s1 − 2

2 #

|w|2 .

(98)

|∆s|2 |∆P |2 |∆Q|2 + + + K4 |w|2 . |J5 | ≤ 2 2 2

(99)

Note that |J6 | ≤ |J5 |. |∆s|2 |w3 |2 + K4 + L|∆u3,eq |2∞ + 2 2 |∆s|2 |w7 |2 |J8 | ≤ + K4 + L|∆u7,eq |2∞ + 2 2 |J7 | ≤

L |∆S3,diss |2 , 2 L |∆S7,diss |2 , 2

(100) (101)

and

|J9 | ≤ |∆s|2 + K4

|w5 |2 + |w7 |2 1 + |∆η1 |2 . 2 2

(102)

Integrating by parts the terms (ywi,y , wi ) and ((2−y)wj,y , wj ) ((i, j) ∈ I1 ×I2 ) and applying conveniently the interpolation inequality with θ = 21 , we obtain bounds on |J10 | and |J11 | as follows:

|J10 | ≤ s02

X i∈I1

|wi (1)|2 |wi |2 − 2 2

+

i∈I1

2

≤

X ||wi ||2

!

2

ξ X ||wi || 2 + 1 + K4 s 2 + 2 i∈I1 s2 + 2 2

|J11 | ≤ s02

X i∈I2

|wi (1)|2 |wi |2 + 2 2

+

i∈I2

32

2

X

|wi |2

i∈I1

2 !

X ||wi ||2

!

|∆s0 |2 + (s02 )2

X

|wi |2 ,

i∈I1

|∆s0 |2 +

X i∈I2

|wi |2

≤

||wi ||2 ξ X 2 i∈I2 L − s2 − 2 2

+ 1 + K4

L − s2 − 2

2 !

(s02 )2

X

|wi |2 .

(103)

i∈I2

By (B) and (88) (with α = 2 and c2 = 2), it results that the term 21 |∆η1 |2 can be bounded from above by c2L |w|2 +c2L |∆Λ|2 . By (95)-(103), we conclude that the right-hand side of (92) can be estimated from above by

4ξ

||wi ||2

X i∈I1

s2 +

2

2 + 4ξ

||wi ||2

X i∈I2

L − s2 −

2

2 0 2 2 + γ1 |∆s| + γ2 |∆s |

+γ3 |∆P |2 + |∆Q|2 +

|∆Di |2

X i∈I1 ∪I2

X

+

|∆S`,diss |2 + |∆u`,eq |2∞ + |∆Λ|2 + γ4 |w|2 .

(104)

`∈{3,7}

In (104), the factors γi (i ∈ {1, . . . , 4}) are defined by

γ1 := 3 + cξ

X

(1) 2

Di

||w1i ||2 +

i∈I1 ∪I2

γ2 :=

|w1,t |2 , 2

X ||w2i ||2 1 + , 2 i∈I1 ∪I2 2

γ3 := 1 + L + K4

X

||w1i ||2 + c2L ,

i∈I1 ∪I2

γ4 := 3 +

c2L

+ 7K4 + K4 1 +

2|s02 |2

"

s2 + 2

2

+ L − s2 − 2

2 #

. (105)

Select D0 ξ ∈ 0, 4

with D0 cf. (91)

(106)

and neglect the first two sums of (104). By (85), (104), (106) and Gronwall’s inequality, the variational formulation (92) leads to the estimate α |w| ≤ e L∗ () 2

R Sδ

2γ4 (τ )dτ

,

(107)

where L∗ () is cf. (90) and α is defined by 33

α := L∗ ()|w(0)|2 + 2δ max δ 2 |γ1 (t)| + |γ2 (t)| max |∆s0 |2 t∈S¯δ

t∈S¯δ

+ 2δ max |γ3 (t)| t∈S¯δ

2

× max |∆Ξ| + MΞ

||∆Υ||2MΥ ∩L∞ (Sδ )

2

+ max |∆Λ| MΛ

.

Bounding above the term |∆s0 |2 by c|∆η |2 and using (B), we obtain the conclusion of the Theorem. Although the value of the positive constant c can be exactly calculated, for our purpose is sufficient to ensure that c is finite. Note that c depends on the choice of , cs () and k3∗ .

5

Numerical illustration

In this section, we illustrate the expected behavior of the reaction-front position and concentration profiles in the framework of the moving two-reactionstrips model. The basic numerical approach is to immobilize the moving strips, to discretize the PDEs in space, and then to integrate (via MATLAB) the resulting semi-discrete system in time [5,34,22]. The numerical examples are obtained by choosing n = 80. The initial value ¯up10 uˆq70 . A typical k¯ is about 500. To of CaCO3 (aq) is calculated by uˆ40 = kˆ calculate λ1 and λ2 , we split the ambient concentration c¯ of CO2 at the exposed c¯Q c¯ boundary via Henry’s law, i.e. (λ1 , λ2 ) = 1+Q , 1+Q . λ3 incorporates the chemistry of the Portland cement from [7]. Compare [18] and chapter 4 of [22] for more details on the used reference parameters.

(a)

(b)

(c)

Fig. 3. (a)+(b) CO2 (aq) and CO2 (g) concentration profiles vs. space. Each curve refers to time t = i years, i ∈ {1, . . . , 18}. (c): Reaction strip location vs. the experimental points “ ◦ ” (from [7]) after Tfin = 18 years of exposure.

We illustrate the effect of the secondary carbonation in Fig. 5–6, where we point out the formation of internal reaction zones. More precisely, we note that two distinct zones of Ca(OH)2 (aq) depletion are formed and progress into the material. It is worth mentioning that, in order to obtain alike behavior (see Fig. 6, e.g.), rather strong prerequisites on the model parameters have to be 34

(a)

(b)

Fig. 4. (a) Concentration profiles of CO2 (aq) vs. time and space position. (b) CaCO3 (aq) concentration profiles vs. space. Each curve refers to time t = i years, i ∈ {1, . . . , 18}.

(c) Fig. 5. Effect of secondary carbonation: Two internal zones of depletion of Ca(OH)2 (aq) are formed and progress into the material. Each curve refers to time t = i years, i ∈ {1, . . . , 18}.

fulfilled: (1) A strong production by dissolution within a short time interval has to apply behind Γr (t), see the basic geometry in Fig. 2; (2) The newly produced Ca(OH)2 is (almost) not transported, thus a possible leaching out of the alkaline species is hindered; The new content of Ca(OH)2 basically waits to react with CO2 (g) arriving from the exterior boundary. (3) CO2 (g) has to arrive quickly to the location of Ω (t). Consequently, the porous matrix is assumed to be highly permeable and the transfer of mass from the air parts into the wet parts of the pores is considered to be rapid. (4) On the other hand, the strip Ω (t) should advance in such a way that the primary carbonation finishes. (5) The precipitation of CaCO3 (aq) is taken to be instantaneous (and hence its transport may be neglected). In Fig. 6 (a), we distinguish between two segregated internal strips where CaCO3 (aq) is produced. That one which is more visible corresponds to Ω1 (t). We see in Fig. 6 (b) that the secondary carbonation acts as a retarder with 35

(a)

(b)

Fig. 6. Effect of secondary carbonation: (a) Two internal zones of production of CaCO3 develop into the material. (b) The parameter R rises (R = 0, 1, 5, 10), while the penetration depth gradually decreases.

respect to the overall carbonation process in the sense that: If the secondary carbonation phenomenon occurs, then the speed of the strip Ω (t) decreases. To visualize such behaviors, we take about the half of the speed s0 used to obtain the previous plots and alter progressively a reaction parameter R ([22], appendix A). Here R compares the characteristic time of the carbonation reaction taking place in Ω1 (t) with the characteristic time of the carbonation reaction within Ω (t). Consequently, if R = 0, then no secondary carbonation acts behind the strip Ω (t). If R 6= 0 and if the dissolution overcomes the carbonation in the region Ω1 (t) (in the sense that Ca(OH)2 is produced in such quantity that CO2 (g) is practically stopped somewhere within Ω1 (t)), then Ω1 epsilon(t) may stop its motion. Nevertheless, since in carbonation tests an advanced degree of the hydration reaction is usually ensured [16,8], alike behavior is not really expected to happen. Note that by adopting the assumptions (C1)-(C3) from section 3, we practically impose that the velocity of the layer Ω (t) is strictly positive in the time interval STfin . In this example, it appears that the agreement between the experimentally observed motion trajectory of the reaction strip and concentration profiles and that computed via the proposed model is reasonable.

Acknowledgements

It is the author’s pleasure to express his deep gratitude to his advisor and mentor, Michael B¨ohm (University of Bremen, Germany), as well as to many others who contributed with suggestions and support. 36

References

[1] V. Alexiades, A. Solomon, Mathematical Modeling of Melting and Freezing Processes. (Hemisphere, Washington, 1993). [2] M. Z. Bazant, H. A. Stone, Asymptotics of reaction-diffusion fronts with one static and one diffusing reactant. Physica D, 147 (2000), 95–121. [3] J. M. Ball, D. Knderlehrer, P. Podio-Guidugli, M. Slemrod (eds) Evolving Phase Interfaces in Solids. (Springer, Berlin, 1999). [4] J. Bear, Dynamics of Fluids in Porous Media. (Dover Publications Inc., NY, 1972) [5] M. B¨ohm, J. Devinny, F. Jahani, G. Rosen, On a moving-boundary system modeling corrosion of sewer pipes. Appl. Math. Comput. 92(1998), 247-269. [6] M. B¨ohm, R. Showalter, Diffusion in fissured media. SIAM J. Appl. Math. 16(1985), 500-509. [7] D. Bunte, Zum Karbonatisierungsbedingten Verlust der Dauerhaftigkeit von Aussenbauteuilen aus Stahlbeton, Dissertation, TU Braunschweig, 1994. ´ [8] T. Chaussadent, Etats des lieux et r´eflexions sur la carbonatation du beton arm´e. LCPC, Paris, 1999. [9] E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura, H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Analysis RWA, 5(2004), 645–665. [10] H. Emmerich, The Diffuse Interface Approach in Materials. ScienceThermodynamic Concepts and Applications of Phase-Field Models. (Springer, Berlin, 2002). [11] A. Fasano, M. Primicerio, R. Ricci, Limiting behaviour of some problems in diffusive penetration. Rendiconti di Matematica, Serie VII, 10(1990), 39–57. [12] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces. (SIAM, Philadelphia, 1988). [13] G. Froment, K. B. Bischoff, Chemical Reactor Analysis and Design. (Wiley, New York, 1991) [14] M. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane. (Clarendon, Oxford, 2005). [15] Ishida, T., K. Maekawa, Modeling of pH profile in pore water baesd on mass transport and chemical equilibrium theory. Concrete Library of JSCE 37(2001), 131–146. [16] B. Lagerblad, Carbon dioxide uptake during concrete life cycle-State of the art. NI-project 03018, Swedish Cement and Concrete Research Institute, Stockholm, 2005.

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[17] H. G. Landau, Heat conduction in a melting solid, Quart. Appl. Mech. Math., 8 (1950), 81–94. [18] S. A. Meier, M. A. Peter, A. Muntean, M. B¨ohm, Dynamics of the internal reaction layer arising during carbonation of concrete. Chem. Engng. Sci. 62(2007), 1125-1137. [19] A. Meirmanov, The Stefan Problem (Walter de Gruyter, Berlin, 1991). [20] A. Muntean, M. B¨ohm, A moving two-reaction zones model: global existence of solutions. PAMM - Proc. Appl. Math. Mech. 6(2006), 649-650. [21] A. Muntean, M. B¨ohm, Dynamics of a moving reaction interface in a concrete wall. In Free and Moving Boundary Problems. Theory and Applications (Proceedings of the conference FBP 2005), J. F. Rodrigues, I. N. Figuieredo, L. Santos (eds.) Int. Series of Numerical Mathematics, 154:317-326, Birkh¨auser, Basel, 2006. [22] A. Muntean, A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation, (Cuvillier, G¨ottingen, 2006). [23] A. Muntean, M. B¨ohm, A moving-boundary problem for concrete carbonation: global existence and uniqueness of weak solutions. CASA-Report 07-25. Department of Mathematics and Computer Science, TU Eindhoven, 2007. [24] A. Muntean, Concentration blow up in a two-phase non-equilibrium model with source term. Meccanica 42(2007), 409–411. [25] A. Narimanyan, A. Muntean, Mathematical modeling driven by two industrial applications: a moving-boundary approach. Math. Mod. Anal. 11(2006), 295314. [26] P. Ortoleva, Geochemical Self-Organization. (OUP, Oxford, 1994). [27] M. A. Peter, Homogenization in domains with evolving microstructure. C. R. M´ecanique 335(2007), 7, 357-362. [28] V. G. Papadakis, C. G. Vayenas, M. N. Fardis, A reaction engineering approach to the problem of complex carbonation. AIChE Journal 35(1989), 1639. [29] S. J. Preece, J. Billingham, A. C. King, On the initial stages of cement hydration. J. Engng. Math. 40(2001), 43–58. [30] T. L. van Noorden, I. S. Pop, M. R¨oger, Crystal dissolution and precipitation in porous media. L1 -contraction and uniqueness. Discr. Cont. Dyn. Syst. (suppl. 2007), 1013–1020. [31] I. Rubinstein, Electro-Diffusion of Ions. (SIAM, Philadelphia, 1990). [32] A. V. Saetta, B. A. Schrefler, R. V. Vitaliani, 2D model for carbonation, moisture/heat flow in porous materials. Cement and Concrete Research 32(1995) 939-941

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[33] A. Schmidt, A. Muntean, M. B¨ohm, Moving carbonation fronts around corners: a self-adaptive finite element approach. In Transport in concrete: Nano- to Macrostructure. TRANSCON 07 (ed. by M. Setzer), pp. 467-476. (Aedificatio Publishers, Freiburg, 2007). [34] A. M. Soane, M. K. Gobbert, Th. Seidman, Numerical exploration of a system of reaction diffusion equations with internal and transient layers. Nonlinear Analysis RWA 6(2005), (5), 914–934. [35] A. Steffens, D. Dinkler, H. Ahrens, Modeling carbonation for corrosion risk prediction of concrete structures. Cement Concrete Research 32(2002), 935– 941. [36] H. F. W. Taylor, Cement Chemistry. (Thomas Telford, London, 1997). [37] R. Temam, A. Miranville, Mathematical Modeling in Continuum Mechanics. (CUP, Cambridge, 2005). [38] K. Tuutti, Corrosion of steel in concrete. Swedish Cement and Concrete Research Institute (CBI, Stockholm, 1982). [39] A. Visintin, A new model for supercooling and superheating effects. IMA J. Appl. Math. 36(1986), 141–157. [40] E. Zeidler, Nonlinear Functional Analysis and Its Applications. Linear Monotone Operators. vol. II/A, (Springer Verlag, Berlin, 1990).

39

Abstract We deal with a one-dimensional coupled system of semi-linear reaction-diffusion equations in two a priori unknown moving phases driven by a non-local kinetic condition. The PDEs system models the penetration of gaseous CO2 in unsaturated cement-based porous materials (concrete). The main issue is that CO2 diffusion and fast reaction with Ca(OH)2 in concrete lead to a sudden drop of alkalinity near the steel reinforcement. This effect initiates the chemical corrosion of the material. We address some aspects concerning the existence, uniqueness and stability of local weak solutions. We also present our concept of global solvability of the movingboundary system in question. We illustrate the application of our model to the prediction of carbonation penetration into ordinary Portland concrete samples. Key words: Moving boundary, strips-concentrated reaction, Stefan problem, kinetic condition, a priori estimates, weak solutions, well-posedness MSC: primary 35R35; secondary 74F25, 35D05

1

Introduction

We present and analyse a two-reaction-strips moving-boundary model for concrete carbonation – a reaction-diffusion process by which carbon dioxide from the ambient air penetrates the concrete, dissolves in pore water and reacts with Email address: [email protected] (Adrian Muntean). Funding by the German Science Foundation (DFG) via the grant SPP 1122 Prediction of the course of physicochemical damage processes involving mineral materials is acknowledged. 1

calcium hydroxide, which is available by dissolution from the solid matrix, to form calcium carbonate and water. The reduced chemistry of the process is H O

2 CO2 (g → aq) + Ca(OH)2 (s → aq) −→ CaCO3 (aq → s) + H2 O.

(1)

We refer the reader to [7,8,36,28] and references therein for more details on the carbonation reaction and its negative as well as positive effects on concrete’s microstructure. We only mention that (1) typically facilitates corrosion, and hence, it drastically limits the service-life of concrete-based materials. In this paper, we exploit an experimental observation - called the phenolphthalein test - to offer a detailed modeling of the process when one assumes that the carbonation reaction is concentrated in two distinct strips. The phe-

Fig. 1. Slices of partly carbonated concrete samples sprayed with phenolphthalein (courtesy of Prof. Dr. M. Setzer and Dr. U. Dahme, University of Essen-Duisburg, Germany). Two distinct type of regions can be noticed; (1) colorless (low alkaline) parts, and (2) red (high alkaline) parts. Here we assume that one tiny reaction strip (say Ω (t)) is positioned at the sharp interface separating the two colors, and behind it, a second much wider reaction strip (say Ω1 (t)) is present. We assume that the latter strip fills the colorless region. Ω2 (t) points out the highly alkaline region.

nolphthalein measurement is limited to providing an indication of the position of the pH value (about 9) and does not show the changes which may occur in the colorless (partly carbonated) part [8,38]. This qualitative information may be turned into a quantitative one as soon as one examines pH levels inside pores and then at the macro-scale as in [31,15] involving also homogenizationbased arguments like in [27]. However, we find difficult to account for the whole ionic soup existing in concrete. We rather prefer to use the macroscopic information pointed out in Fig. 1: As the carbonation reaction proceeds, several reaction strips form and move inwards. In the inner parts of the strips, the alkaline species are depleted with possibly different rates. Obviously, the central question is now: How fast do these strips move into the material? The main goal of this paper is to understand the motion of these internal reaction strips, and hence, to predict in an accurate way the penetration of CO2 in ordinary (Portland) concrete. 2

The results presented here are parts from the author’s PhD thesis [22] and have been announced in [20]. Related aspects were addressed for a moving sharp-interface scenario in [21,23]. Our reaction-diffusion scenario (section 2) remotely resembles to the classical Stefan problem modeling the melting of ice cubes; see [1] for a nice introduction to modeling aspects around this theme and [19] for a collection of known facts on the mathematical analysis of Stefan-like problems. A special feature of our moving-boundary model is the presence of a kinetic condition for driving the main reaction strip. Such moving-boundary condition has been also used to model further (conceptually related) nonequilibrium processes like the swelling of glassy polymers [11], supercooling and superheating effects in solidification [3,39], sulfate corrosion in sewer pipes [5] and fast dissolution/precipitation reactions [30], e.g. The paper is organized in the following manner. The two-reaction strips movingboundary model is introduced in section 2. Our concept of weak solution and the main results are included in section 3. The bulk of the paper (i.e. section 4) consists of the proofs for the local existence, uniqueness and stability with respect to data and parameters of the weak solutions (theorem 3 and theorem 4). The reader is referred to [22] to see proofs for the rest of the statements. In section 5, we illustrate numerically a real situation relying on experimental data extracted from [7]. We also briefly evaluate the model and numerical approach.

2

The moving-two-reaction-strips model

Let > 0 be sufficiently small. Denote by uˆ` (` ∈ {1, 2}) the CO2 concentration in air and water in Ω1 (t) ∪ Ω (t), uˆr (r ∈ {3, 6}) the Ca(OH)2 and H2 O concentrations in Ω2 (t), and uˆm (m ∈ {5, 7}) and uˆ4 the H2 O, Ca(OH)2 and CaCO3 concentrations in Ω1 (t) ∪ Ω (t). Here uˆ := (ˆ u1 , uˆ2 , uˆ3 , uˆ4 , uˆ5 , uˆ6 , uˆ7 )t is the vector of concentrations, while the moving domains are defined by Ω1 (t) := ]0, s(t)−/2[, Ω (t) :=]s(t)−/2, s(t)+/2[ and Ω2 := Ω−Ω1 (t) ∪ Ω (t), where Ω := [0, L[ (L > 0). Here s(t) stands for the position of the a priori unknown center of Ω (t). Set Γr (t) := {x = s(t) + /2}. The geometrical setting is depicted in Fig. 2. We also use the following sets of indices I1 := {1, 2, 5, 7}, I2 := {3, 6} and I := I1 ∪ {4} ∪ I2 . The moving-boundary problem consists of finding the active concentrations uˆi and the position s(t) which satisfy for all t ∈ ST :=]0, T [ (T > 0) the 3

Fig. 2. Two reaction zones. Carbonation is assumed to take place simultaneously in Ω1 (t) and Ω (t) with different reaction rates.

mass-balance equations

uˆ`,t + (−D` uˆ`,x )x = f` (ˆ u), x ∈ Ω1 (t) ∪ Ω (t), ` = 1, 2,

uˆr,t + (−Dr uˆr,x )x = fr (ˆ u), x ∈ Ω2 (t), r ∈ {3, 6}, u ˆm,t + (−Dm uˆm,x )x = fm (ˆ u), x ∈ Ω1 (t) ∪ Ω (t), m ∈ {5, 7}.

(2)

The precise definition of the productions fi is given in (7). The initial and boundary conditions are uˆi (x, 0) = uˆi0 for i ∈ I, x ∈ Ω, uˆi (0, t) = λi for i ∈ I1 − {7}, u7,x (0, t) = 0, t ∈ ST and uˆi,x (L, t) = 0 for i ∈ I2 , t ∈ ST . The interface and transmission conditions ([37,14]) imposed across Γr (t) are defined via [−Di uˆi,x · n]Γr (t) = s0 (t)[ˆ ui ]Γr (t) (i ∈ I1 ∪ I2 ).

(3)

Set v4 (t) := uˆ4 (x, t) for x ∈ Ω1 (t) ∪ Ω (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

R 0

s (t) =

η (ˆ u, Λ )(x, t)dx , ˆ3 (x, t)dx Ω (t) u

Ω (t)

R

(4)

vˆ40 (t) = f4 (v4 (t)) a.e. t ∈ ST with s(0) = s0 > 0, vˆ4 (0) = uˆ40 > 0,

(5)

where η is the reaction rate in Ω (t). The model equations are collected in (2)-(5). 4

2.1

Comments on the expression (4) of the velocity s0 (t)

Due to the explicit representation of the velocity law s0 of the strip Ω (t) the moving-boundary system (2)-(5) is said to be with kinetic condition very much in the spirit of [39]. As mentioned in [11] (see section 2.2 loc. cit.), the advantage of this sort of model is the finite speed of the reaction strip propagation. Of course, the structure of the reaction rates have to be carefully chosen, otherwise blow up situations near moving thin strips may occur [24]. Expression (4) shows that the speed of the strip Ω (t) is proportional to the strength of the production by reaction in Ω (t). It also incorporates the following effect: If a large amount of Ca(OH)2 is present in Ω (t), then the speed s0 will correspondingly decrease. (4) represents a volumetric dilation of a similar expression derived via first principles in [22] (section 2.3.1) for the case of a moving interface-concentrated reaction.

2.2

Freezing moving reaction strips. Production rates. Model parameters

To deal with (2)-(5), we firstly freeze the moving strips by employing suitable Landau transformations; see details in section 4.3 or in [17,22], e.g. 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 ∈ V be an arbitrary test function 2 and take t ∈ ST . To write the weak formulation of (2)-(5) in fixed-domains, we introduce the notation: a(s, u, ϕ)

:=

1 s+ 2

P

i∈I1 (Di ui,y , ϕi,y )

bf (u, s, ϕ) := s +

e(s0 , u, ϕ) := s+1 2 h(s0 , u , ϕ) := P ,y

2

P

+

1 L−s− 2

P

i∈I2 (Di ui,y , ϕi,y ),

i∈I1 (fi (u), ϕi ) + L − s −

2

P

i∈I2 (fi (u), ϕi ),

P2

0 i=1 s (t)(ui (1, t) + λi (t))ϕi (1),

i∈I1 (s

0

yui,y , ϕi ) +

P

i∈I2 (s

0

(6)

(2 − y)ui,y , ϕi )

for any u ∈ V and λ ∈ W 1,2 (ST )|I1 ∪I2 | . The intervening function spaces are introduced in section 4.2. The term a(·) incorporates the diffusive part of the model, bf (·) comprises various productions, e(·) sums up some of the reaction terms acting on y = 1 and h(·) is a non-local term due to fixing of the domain. 2

The selection of the evolution triple (V, H, V∗ ) is standard.

5

For our application, the production terms fi (i ∈ I1 ∪ I2 ) are given by f1 (u) := P1 (Q1 u2 − u1 ) − η1 (u, Λ1 ) − η (u, Λ ), f2 (u) := −P2 (Q2 u2 − u1 ),

f3 (u) := S3,diss (u3,eq − u3 ), f4 (ˆ u) := +η1 (ˆ u, Λ1 ) + η (ˆ u, Λ ), f5 (u) := +η1 (u, Λ1 ) + η (u, Λ ), f6 (u) := 0, f7 (u) := S7,diss (u7,eq − u7 ) − η1 (u, Λ1 ) − η (u, Λ ).

(7)

In this framework, η1 and η denote carbonation reaction rates in Ω1 (t) and Ω (t), while each of the vectors Λ1 , Λ ∈ MΛ (MΛ is a compact set in Rm + away from zero) contain m ∈ N reaction parameters. Select ki := max{ui0 + λi (t), λi (t) : t ∈ S¯T }, i = 1, 2, 3, 6, 7,

k := max{ˆ u (x) + M T : x ∈ [0, s(t) + /2], t ∈ S¯T },

4 40 k5 := max{u60 + λ6 (t) + M T,

M1 D5 −M L

(8)

, k6 , λ5 (t) : t ∈ S¯T },

where we set Y

K :=

i∈I1 ∪I2

[0, ki ], M :=

1 max{f4 (ˆ u)}, M1 := max{f4 (ˆ u)}, u∈K k3∗ u∈K

and k3∗ > 0 is a given constant lower bound of the content of Ca(OH)2 in Ω (t) valid for all t ∈ ST . Additionally, let k¯ denote the sum k¯ = 2

7 X

k` .

(9)

`=1

Note that y ∈ [a, b] (i ∈ I1 ∪ I2 ) means here the following: If j ∈ {1, 2} and i ∈ Ij , then y ∈ [−1 + j, j]. The model parameters as well as the initial and boundary data need to satisfy the following assumptions: λ ∈ W 1,2 (ST )|I1 ∪I2 | , λ(t) ≥ 0 a.e. t ∈ S¯T , ∈ L∞ (ST ), u`,eq (t) ≥ 0 a.e. t ∈ S¯T , ` ∈ {3, 7},

(10) (11)

u0 ∈ L∞ (a, b)|I1 ∪I2 | , u0 (y) + λ(0) ≥ 0 a.e. y ∈ [a, b], uˆ40 ∈ L∞ (0, s0 + /2), uˆ4 (x, 0) ≥ 0 a.e. x ∈ [0, s0 + /2], min{S3,diss , S7,diss , P1 , Q1 , P2 , Q2 , D` (` ∈ I1 ∪ I2 )} > 0.

(12) (13) (14) (15)

u`,eq

6

2.3

How to define the production by reaction within a strip?

Knowing for a moment t ∈ ST the location of the reaction strip Ω (t) (i.e. the position s(t) and the value of ), we can localize the strips Ω1 (t) and Ω2 (t) in a straightforward manner. Interestingly, in the chemical engineering literature (see [13,26], e.g.) one defines what volume- and surface- concentrated reaction are, but it appears to be no consensus on how to define reaction concentrated within sub-domains (like strips). The standard procedure is to avoid employing strip-concentrated reactions via approximating the production by reaction in that strip either by a surface production (if the strip is sufficiently thin, see Gurtin’s pillbox lemma [14]) or by a volume production (if the width of the strip is comparable to the width of the reference volume). Herein, we model the reaction production inside the strip and mimic the strip motion by means of the moving-boundary system (2)-(5). If the two reactants completely segregate, then we expect the strip Ω (t) to degenerate into the interface Γ(t) positioned at x = s(t); see [11,9,22] for closely related aspects. In that case, the structure of the moving-boundary system changes dramatically (as it is described in [23]) and the solution vector typically looses regularity. We assume for a moment that no secondary carbonation effects can be noticed behind the (main) reaction strip Ω (t). In this case, Ω1 (t) can be defined via Ω1 (t) := {x ∈ Ω : φφw u¯3 (x, t) = 0}, where φ, φa and φw denote the total porosity and the air and water fractions, respectively. u¯ represents a mass concentration at the pore scale that is connected with the averaged concentration uˆ via (26); see [4,22], e.g. The strip Ω (t) may be defined in several ways. We are aware of the next six (non-equivalent) definitions: (i) We can rely on the typical behavior of reactants profiles vanishing in a fast reaction mechanism (see, for instance, Fig. 2 in [11] or Fig. 5-2 in [26], p. 94) to account for Ω (t) := {x ∈ Ω : φφw u¯1 (x, t) ⊂]0, k1 ] and φφw u¯3 (x, t) ⊂]0, k3 ]}.

(16)

(ii) Denote by c∗ > 0 the mass concentration of CO2 (aq) necessary to consume one mole of Ca(OH)2 (aq). We can define Ω (t) to be the region in which either u¯1 ≥ c∗ and 0 < u¯3 ≤ c∗ , or 0 ≤ u¯1 ≤ c∗ and u¯3 ≥ c∗ . (iii) Another possibility is to define the strip Ω (t) (as in [29], e.g.) to be the place where the solubility product c∗∗ (of the carbonation reaction) is above its equilibrium value and the concentrations φφw u¯i (i ∈ {1, 3}) are below their maximum values ki , i.e. k1 k3 ≥ u¯1 u¯3 ≥ c∗∗ 2 , provided k1 ≥ u¯1 > 0 7

and k3 ≥ u¯3 > 0. A slightly different concept employs the notion of extent of reaction [13] to define the reaction strip. (iv) In [2], the authors define asymptotically the strip via Ω (t) := {x ∈ Ω : x − s(t) = O()}. (v) Usually, people from the phase-field community (see chapter 1 in [12] or [10], e.g.) consider the moving internal strip to be an interface of width O() that is located on a plane curve Γ(t; ). The movement of Γ is typically tracked via the evolution of a function r(x, t; ) that represents the distance from x to Γ. Thus, Γ(t; ) := {r(x, t; ) = 0} defines our Ω (t). (vi) We can also define Ω (t) in an a posteriori way as the location where the reaction rate reaches its maximum. This procedure employs the concept of degree of reaction and is often used to define the strip location in case of isolines models, see [18,33] and references therein. In this paper, we adopt definition (16). Hence, Ω (t) can be seen as a sort of mushy region where both reactants (i.e. CO2 (aq) and Ca(OH)2 (aq)) are living together.

3

Main Results

Definition 1 (Weak formulation) We call the triple (u, v4 , s) a local weak solution to problem (2)-(5) if there is a Sδ :=]0, δ[ with δ ∈]0, T [ such that s0 +

< s(δ) + ≤ L0 , 2 2

v4 ∈ W 1,2 (Sδ ),

(17)

s ∈ W 1,2 (Sδ ),

(18)

u ∈ W21 (Sδ ; V, H) ∩ L∞ (Sδ , H),

(19)

s+

2

P

i∈I1 ((ui (t) + λi (t))t , ϕ) + L − s −

2

P

i∈I2 ((ui (t)

+ λi (t))t , ϕ)

+a(s, u, ϕ) + e(s0 , u, ϕ) = bf (u + λ(t), s, ϕ) + h(s0 , u,y , ϕ) for all ϕ ∈ V, a.e. t ∈ Sδ , s0 , v40 cf. (35) a.e. in Sδ , u(0) = u0 ∈ H.

(20)

The assumptions on the reaction rates and material parameters, which are needed in the sequel, are the following: 8

(A) Fix Λ ∈ MΛ and take arbitrarily ` ∈ {1, }. Let η` (ˆ u, Λ) > 0, if uˆ1 > 0 and uˆ7 > 0, and η` (ˆ u, Λ) = 0, otherwise. Assume η` to be bounded for any fixed uˆ1 ∈ R. (B) The reaction rate η` : R7 × MΛ → R+ (` ∈ {1, }) is locally Lipschitz. (C1) k` ≥ maxS¯T {|u`,eq (t)| : t ∈ S¯T }, ` ∈ {3, 7}; D5 > M L; (C2) P1 Q1 k2 ≤ P1 k1 ; P2 k1 ≤ P2 Q2 k2 ; (C3) Q2 > Q1 . Theorem 2 Assume the hypotheses (A)-(C2) and let the conditions (10)(15) be satisfied. If s ∈ W 1,2 (Sδ ) with s0 ≥ 0 a.e. in Sδ and s(0) = s0 is given, then the problem (2)-(5) admits a unique local solution on Sδ in the sense of Definition 1 (formulated for given s); Theorem 3 (Local existence and uniqueness) Assume the hypotheses (A)(C2) and let the conditions (10)-(15) be satisfied. Then the following assertions hold: (a) There exists a δ ∈]0, T [ such that the problem (2)-(5) admits a unique local solution on Sδ in the sense of Definition 1; (b) 0 ≤ ui (y, t) + λi (t) ≤ ki a.e. y ∈ [a, b] (i ∈ I1 ∪ I2 ) for all t ∈ Sδ . Moreover, 0 ≤ uˆ4 (x, t) ≤ k4 a.e. x ∈ [0, s(t) + /2] for all t ∈ Sδ , and v4 , s ∈ W 1,∞ (Sδ ). By (A) and (B), we deduce that η` (0, Λ) = 0 for all Λ ∈ MΛ and ` ∈ {1, }. Therefore, there exists a positive constant C`,η = C`,η (Λ, λ, u¯1 , Tfin ) such that the inequality η¯` (¯ u(y, t)), Λ) ≤ C`,η u¯(y, t)

(21)

holds locally for all t ∈ ST . (i)

Let ` ∈ {1, } be fixed arbitrarily. Select i ∈ {1, 2} and let (u(i) , v4 , si ) be two weak solutions on Sδ in the sense of Definition 1. They correspond to the sets of (i) (i) data Di := (u0 , λ(i) , Ξ(i) , Υ(i) , Λ(i) )t , where u0 , λ(i) , Ξ(i) , Υ(i) , and Λ(i) denote the respective initial data, boundary data, and the model parameters describing diffusion, dissolution mechanisms, and carbonation reaction, respectively. (i) (i) (i) In this context, we have Ξ(i) := (D` (` ∈ I1 ∪ I2 ), Pk (k ∈ {1, 2}), Qk (k ∈ (i) (i) {1, 2}), S3,diss )t ⊂ MΞ and Υ(i) = (u3,eq ) ⊂ MΥ , i ∈ {1, 2}. Here MΞ and MΥ 2 are compact subsets of R10 + and L (Sδ ). (2)

(1)

Set ∆u := u(2) − u(1) , ∆v4 := v4 − v4 , ∆s := s2 − s1 , ∆λ := λ(2) − λ(1) , (2) (1) ∆u0 := u0 −u0 , ∆Ξ := Ξ(2) −Ξ(1) , ∆Υ := Υ(2) −Υ(1) , ∆Λ := Λ(2) −Λ(1) , and (2) (1) (2) (2) (1) (1) ∆η` := η˜` − η˜` := η¯` (¯ u , Λ(2) )− η¯` (¯ u , Λ(1) ). The Lipschitz condition on η` reads: There exists a constant cL = cL (D1 , D2 ) > 0 such that the inequality |∆η` | ≤ cL (|∆u| + |∆Λ|) holds locally pointwise, see (B). Having these notations available, we can state the following two results.

9

(i)

Theorem 4 (Continuity w.r.t data and parameters) Let (u(i) , v4 , si ) (i ∈ {1, 2}) be two weak solutions on Sδ and assume the hypotheses of Theorem 3 to (i) be satisfied. Let (u0 , λ(i) , Λ(i) ) be the initial, boundary and reaction data. Then the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W21 (Sδ , V, H) × W 1,2 (Sδ )2 that maps (u0 , λ, Ξ, Υ, Λ)t into (u, v4 , s)t is Lipschitz in the following sense: There exists a constant c = c(δ, , s0 , uˆ40 , L, ki , θ, cL ) > 0 (i ∈ I1 ∪ I2 ) such that ||∆u||2W21 (Sδ ,V,H)∩L∞ (Sδ ,H) + ||∆v4 ||2W 1,2 (Sδ )∩L∞ (Sδ ) + ||∆s||2W 1,2 (Sδ )∩L∞ (Sδ )

≤ c ||∆u0 ||2H∩L∞ ([a,b]|I1 ∪I2 | ) + ||∆λ||2(W 1,2 (S

2

+c max |∆Ξ| + MΞ

δ

||∆Υ||2MΥ ∩L∞ (Sδ )

)∩L∞ (S

δ

))|I1 ∪I2 | 2

+ max |∆Λ|

MΛ

. (22)

A direct corollary of Theorem 4 is the following result. Proposition 5 (Stability of the moving strip Ω (t)) Assume that the hypotheses of Theorem 4 are satisfied. Then the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W 1,2 (Sδ ) that maps the data (u0 , λ, Ξ, Υ, Λ)t into the position of the strip s is Lipschitz in the following sense: There exists a constant c = c(δ, , s0 , uˆ40 , L, ki ,θ, cL ) > 0 (i ∈ I1 ∪ I2 ) such that

||∆s||2W 1,2 (Sδ )∩L∞ (Sδ ) ≤ c ||∆u0 ||2H∩L∞ ([a,b]|I1 ∪I2 | ) + ||∆λ||2(W 1,2 (S

2

+c max |∆Ξ| + MΞ

δ

||∆Υ||2L2 (Sδ )∩L∞ (Sδ )

)∩L∞ (S

δ

))|I1 ∪I2 | 2

+ max |∆Λ| MΛ

.

(23) Proposition 6 (Strict lower bounds) Assume that the hypotheses of Theorem 3 are satisfied. If, additionally, the restriction (C3) holds and the initial and boundary data are strictly positive, then there exists a range of parameters such that the positivity estimates stated in Theorem 3 (b) are strict for all times. Theorem 3 reports on the existence of locally in time weak solutions of problem (2)-(5) with respect to the time interval Sδ . Assume that the hypotheses of Proposition 6 hold. Therefore, for an arbitrarily given L0 ∈]s0 , L[ there exists Tfin ≤ +∞ such that s(Tfin ) +

= L0 . 2

(24)

Here, Tfin represents the time when Ω (t) has penetrated all of ]s0 , L0 [, or in 10

other words, when Ω (t) has touched x = L0 . We call Tfin the final time or shut-down time of the reaction-diffusion process. Owing to Theorem 3 and −s0 −s0 < Tfin < Lη0min . This helps us to conclude Proposition 6, we obtain that Lη0max with the next two results. Proposition 7 (Practical estimates) Let (u, v4 , s) be the unique local solution to (2)-(5) that fulfills the hypotheses of Proposition 6. Then the following estimates hold: (i) ηmin < s0 (t) < ηmax for all t ∈ Sδ ; (ii) (Localization of the strip Ω (t)) s0 + 2 ≤ s(t) ≤ s0 + 2 + δM for all t ∈ Sδ ; −s0 −s0 (iii) Lη0max < Tfin < Lη0min , where Tfin satisfies (24). (iv) If Ω1 (t)∪Ω (t) = {x ∈]0, L[: uˆ4 (x, t) > 0} for t ∈ S¯Tfin , then Ω1 (t1 )∪Ω (t1 ) ⊂ Ω1 (t2 ) ∪ Ω (t2 ) for t1 < t2 , t1 , t2 ∈ S¯Tfin . R

Here ηmin and ηmax denote uniform lower and upper bounds of

Ω (t)

ηΩ (t) (u(x,t))dx

R Ω (t)

u3 (x,t)dx

.

Note that ηmax can be chosen to be ηmax := M + 1, e.g. The proof of (i)-(iv) is straightforward. The statements of Proposition 7 have an obvious practical interpretation: (i) bounds from below and above the penetration speed, (iii) delimitates the time needed to finish the chemical reaction (1), while (iv) simply points out that the region Ω1 (t)∪Ω (t), where carbonates are detected, increases with the time. Theorem 8 (Global solvability) Assume that the hypotheses of Proposition 6 are satisfied. Then the time interval STfin :=]0, Tfin [ of (global) solvability −1 of problem (2)-(5) is finite and is characterized by Tfin = s L0 − 2 . Proposition 9 (Change of coordinates) Assume that the hypotheses of Theorem 3 are satisfied. Additionally, if u3,eq ∈ W 1,2 (Sδ ), uˆ0 ∈ H 1 (0, s0 )|I1 | × H 1 (s0 , L)|I2 | , uˆ0 and λ satisfy the compatibility conditions, then the solution (ˆ u, v4 , s) is mapped via inverse Landau transformations into the physical x-t plane and

(ˆ u, v4 , s) ∈

Y

ˆ i (t)) × W 1,2 (ST )2 , H 1 (STfin , H fin

(25)

i∈I1 ∪I2

ˆ i (t) := L2 (0, s(t) + /2) for i ∈ I1 and H ˆ i (t) := L2 (s(t) + /2, L) for where H i ∈ I2 . The proof from appendix B of [22] can be adapted to this situation. 11

4

4.1

Proofs of Theorem 3 and Theorem 4

Working strategy

We proceed in the following fashion: Firstly, we fix the moving boundaries and employ the weak formulation for the PDEs system defined in fixed domains. Next, we prove the positivity of the concentrations as well as a series of practical estimates like L∞ -bounds on concentrations and speed s0 of the strip Ω (t), energy estimates, non-trivial lower bounds on the final time of the process, etc. Having this wealth of estimates available, we cast the the existence and uniqueness problem in the framework of Banach’s contraction principle. The stability of the weak solution with respect to data and the model parameters is obtained in a straightforward manner by estimating via Gronwall’s inequality the difference of weak formulations written for two distinct solutions and parameters sets.

4.2

Function spaces

In this paper, we use the following spaces of functions and norms: The space Hi = L2 (a, b) is equipped with the norm |u|Hi := R

R

b a

u2 (y)dy

1 2

1

and with the scalar product (u, v)Hi := ab u(y)v(y)dy 2 for all u, v ∈ Hi Q (i ∈ I1 ∪ I2 ). The intermediate space H = i∈I1 ∪I2 Hi is normed by |u|H = P

1

2 2 for all u ∈ H and equipped with the standard scalar prodi∈I1 ∪I2 |ui |Hi uct. The spaces Vi := {u ∈ H 1 (0, 1) : u(0) = 0} (i ∈ I1 ), Vi := H 1 (1, 2) (i ∈ I2 ) and Vw := {(u5 , u6 ) ∈ H 1 (0, 1) × H 1 (1, 2) : u5 (0) = 0, u5 (1) = u6 (1)} = {(u5 , u6 ) ∈ V5 × V6 : u5 (1) = u6 (1)} are endowed with the norms ||u||Vi = |u,y |Hi , i ∈ {1, 2}, ||(u5 , u6 )||Vw = |u5,y |H5 + |u6,y |H6 , and with the corresponding scalar products. We use the space Vh := {(u3 , u7 ) ∈ H 1 (1, 2) × H 1 (0, 1) : u3 (1) = u7 (1)} with the norm ||(u3 , u7 )||Vh = |u3 |V3 + |u7 |V7 . We use V = V1 × V2 × Vw × Vh equipped with the corresponding norm.

For p ∈ [1, +∞] and k a positive integer, W k,p (Ω) are standard Sobolev spaces. [S → B] denotes the set of all functions defined on S with the range in ¯ B), m ∈ N, is the set of continuous functions of the form B, while C m (S, S¯ 3 t 7−→ u(t) ∈ B, that have continuous derivatives up to order m. The space Lp (S, B) (1 ≤ p < ∞) is the set of equivalence classes of Bochner-measurable 1 R functions u ∈ [S → B] for which ||u||Lp (S,B) = ( S ||u(t)||pB dt) p < +∞. We denote by L∞ (S, B) the vector space of all equivalence classes of Bochner measurable functions u ∈ [S → B] for which esssupt∈S ||u(t)||B < ∞. The set 12

W21 (S; V, H) := {v ∈ L2 (S, V) : v 0 ∈ L2 (S, V∗ )} forms a Banach space with the norm ||u||W21 = ||u||L2 (S,V) + ||u0 ||L2 (S,V∗ ) . More details on the mentioned function spaces can be found in [40], e.g.

4.3

Immobilizing moving strips

Let > 0 be sufficiently small and fixed. We reformulate the model in terms of macroscopic quantities by performing the transformation of all concentrations into volume-based concentrations via uˆi := φφw u¯i , i ∈ {1, 3, 4, 7}, uˆ2 := φφa u¯2 , uˆi := φ¯ ui , i ∈ {5, 6}

(26)

in a domain with fixed boundaries. To immobilize the moving reaction strips Ω1 (t) and Ω (t), we employ the Landau transformations (x, t) ∈ [0, s(t) + x and τ = t, for i ∈ I1 , (x, t) ∈ /2] × S¯T 7−→ (y, τ ) ∈ [a, b] × S¯T , y = s(t)+/2 x−s(t)−/2 [s(t) + /2, L] × S¯T 7−→ (y, τ ) ∈ [a, b] × S¯T , y = a + L−s(t)−/2 and τ = t, for i ∈ I2 . We relabel τ by t and introduce the new concentrations, which act in the auxiliary y-t plane by ui (y, t) := uˆi (x, t) − λi (t) for all y ∈ [a, b] and t ∈ ST . The model equations are reduced to

Di 0 s(t) + (ui + λi )i,t − ui,yy = s (t)yui,y 2 s(t) + 2 fi (u + λ), i ∈ I1 , + s(t) + 2

(27)

Di (ui + λi )i,t − ui,yy = s0 (t)(2 − y)ui,y L − s(t) − 2 L − s(t) − 2 fi (u + λ), i ∈ I2 + L − s(t) − 2

(28)

where u is the vector of concentrations (u1 , u2 , u3 , u5 , u6 , u7 )t and λ represents the boundary data (λ1 ,λ2 ,λ3 ,λ5 ,λ6 , λ7 )t . Formally, we employ λ3 = λ7 = λ6 = 0. The transformed initial 3 , boundary and transmission conditions (at Γr (t) for t ∈ ST ) are 3

We have ui0 (y) = u ˆi0 (x) − λi (0), where x = y s0 + s0 + 2 + (y − 1)(L − s0 − 2 ) for i ∈ I2 .

13

2

for i ∈ I1 and x =

ui (y, 0) = ui0 (y), y ∈ [a, b], i ∈ I1 ∪ I2 , ui (a, t) = 0, i ∈ I1 − {7}, u7,y (0, t) = ui,y (b, t) = 0, i ∈ I2 , u3 (1, t) = u7 (1, t), u5 (1, t) = u6 (1, t), −Di ui,y (1, t) = s0 (t)(ui (1, t) + λi (t)), i ∈ {1, 2}, s(t) + 2 −D3 −D7 u7,y (1, t), u3,y (1, t) = L − s(t) − 2 s(t) + 2 −D5 −D6 u5,y (1, t) = u6,y (1, t). s(t) + /2 L − s(t) − /2

(29) (30) (31) (32) (33) (34)

We keep unchanged the notation v4 (t) := uˆ4 (x, t) for x ∈ Ω1 (t) ∪ Ω (t) and t ∈ ST . Two ode’s R

η˜ (ˆ u)(x, t)dx and vˆ40 (t) = f4 (v4 (t)) a.e. t ∈ ST , u ˆ (x, t)dx 3 Ω (t)

Ω (t)

s0 (t) = cs () R

(35)

complete the model formulation, where uˆ := (ˆ u1 , uˆ2 , uˆ3 ,ˆ u4 ,ˆ u5 , uˆ6 , uˆ7 )t and cs () > 0. For this application, we expect cs () = O(). η1 and η denote now reaction rates that acts in the y-t plane and are defined by η` (y, t) := η¯` (ˆ u(ys(t) + y/2, t) + λ(t), Λ), y ∈ [0, 1],

(36)

for given 4 Λ ∈ MΛ . Finally, we assume s(0) = s0 > 0, vˆ4 (0) = uˆ40 > 0.

(37)

The transformed model equations are collected in (27)-(37).

4.4

Basic Estimates

Lemma 10 (Elementary inequalities) Let cξ > 0, ξ > 0, θ ∈ [ 12 , 1[ and s ∈ W 1,1 (Sδ ). (i) There exists the constant cˆ = cˆ(θ) > 0 such that |ui |∞ ≤ cˆ|ui |1−θ ||ui ||θ

(38)

4

Generally, two distinct sets of reaction parameters Λ1 and Λ describe the structure of η1 and η . For simplicity, we denote both by Λ.

14

for all ui ∈ Vi , where i ∈ I1 ∪ I2 . (ii) It holds |ui |1−θ ||ui ||θ ≤ ξ||ui || + cξ |ui |

(39)

for all ui ∈ Vi , where i ∈ I1 ∪ I2 . (iii) Let ϕ ∈ V with ϕ = (ϕ1 , . . . , ϕ6 )t , t ∈ Sδ , cˆ as in (i), and ξ, cξ as in (ii). Then we have for i ∈ I1 and j ∈ I2 the following inequalities: 1 |s0 (t)| |s0 (t)| (yϕi,y , ϕi ) = {ϕi (1)2 − |ϕi |2 } s(t) 2 s(t) 1 |s0 (t)| 2 ≤ {ˆ c |ϕi |2(1−θ) ||ϕi ||2θ − |ϕi |2 }; 2 s(t) |s0 (t)| |s0 (t)| |ϕi (1)|2 ≤ |ϕi |2∞ s(t) s(t) 2θ−1 2 1 ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ × s(t) 1−θ |s0 (t)| 1−θ |ϕi |2 ; s (t) 2 2θ |ϕi (1)| 1 2 2 2θ−2 2(1−θ) −1 ≤ |ϕ | ≤ c ˆ s(t) |ϕ | s(t) ||ϕ || i ∞ i i s2 (t) s2 (t) 2(θ−1) 2 ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ |s(t)| 1−θ |ϕi |2 ; s (t) 2 2θ−1 2 |ϕi (1)| ξ ≤ 2 ||ϕi ||2 + cξ cˆ1−θ |s(t)| 1−θ |ϕi |2 ; s(t) s (t) 0 1 |s0 (t)| 1 |s0 (t)| |s (t)| ((2 − y)ϕj,y , ϕj ) = |ϕj (1)|2 + |ϕj |2 . L − s(t) 2 L − s(t) 2 L − s(t) (iv) The inequality |a + b|p ≤

p−1 (1 + ξ)p−1 |a|p + 1 + 1 |b|p for p ∈ [1, ∞[ ξ |a|p + |b|p

holds for arbitrary a, b ∈ R and ξ > 0. (v) For all θ ∈ [0, 1], a, b, c ∈ R+ , ξ > 0, ξ¯ > 0, cξ¯ := where

1 p

+

1 q

(40)

for p ∈]0, 1[

1 2ξ¯2

and cξ :=

1 q

1 √ p

(ξp)q

,

= 1 with p ∈]1, ∞[, the inequality

ξ¯ abθ c1−θ ≤ a2 + ξcξ¯b2 + cξ¯cξ c2 2

(41)

holds. 15

We refer to (38) as interpolation inequality. In applications, Lemma 10 (iii) will be used with s(t) replaced by s(t) + 2 . We introduce the positive constant K1 by

2

K1 := 1 + max{P1 Q1 , P2 } + cξ cˆ1−θ +

cξ cˆ4 + cξ cˆ4 (D32 + D72 ). 4 h

(42)

h

Notice that K1 depends on cˆ, cf ∈ R+ , θ ∈ 21 , 1 and ξ > 0. The strict positivity of ξ within a compact subset of ]0, ∞[ implies that cξ < ∞. Therefore, the constant K1 is finite. Furthermore, we denote by η1 the term η1 := η1 + η .

(43)

Theorem 11 (Positivity and L∞ -estimates) Let the triple (u, v4 , s) as in Definition 1 satisfy the assumptions (A)-(C2). Then the following statements hold: (i) (Positivity) u(t) + λ(t) ≥ 0 in V a.e. t ∈ Sδ . (ii) (L∞ -estimates) There exists a constant c = c(k` ) > 0(` ∈ I1 ∪ I2 ), where k` is cf. (8), such that u(t) + λ(t) ≤ c in V a.e. t ∈ Sδ . (iii) (Positivity and boundedness of CaCO3 within Ω (t)) 0 < uˆ40 ≤ uˆ4 (x, t) ≤ uˆ40 + δM a.e. x ∈ Ω (t), a.e. t ∈ Sδ . Proof. The statement (iii) is obvious as soon as the estimates ensuring (i) and (ii) hold. We show simultaneously the non-negativity and the maximum bounds of concentrations like we did in the proof of Theorem 3.3 in [23]. Since some of the arguments are sometimes the same, we do not always repeat them. In such cases, we prefer to give only the proof idea. We start with getting upper bounds for u1 and u2 . Once this step is done, the local Lipschitz condition on η` (` ∈ {1, }) becomes global. Select in the weak formulation (1) the test function ϕ := ((u1 + λ1 − k1 )+ , (u2 + λ2 − k2 )+ , 0, 0, 0, 0)t ∈ V, where we denote by ϕi (i ∈ {1, 2}) the expression ϕi := (ui + λi − ki )+ . We obtain 2 q X 1 1 1 d|ϕ|2 + || Di ϕi ||2 = − 2 2 dt s+ s + 2 i=1

2 X

s i=1 2

0

(ϕi (1)2 + ki ϕi (1))

+ P1 (Q1 (u2 + λ2 − k2 ), ϕ1 ) − P1 (u1 + λ1 − k1 , ϕ1 ) + P1 (Q1 k2 − k1 , ϕ1 ) − (η1 , ϕ1 ) + P2 (Q2 (u2 + λ2 − k2 ), ϕ1 ) + P2 (u1 + λ1 − k1 , ϕ2 ) 2 X s0 2 2 − P2 (Q2 k2 − k1 , ϕ2 ) + |ϕ (1)| − |ϕ | . i i 2 s + 2 i=1

16

(44)

By (C2), −η1 ϕ1 (1) ≤ 0 and −

2 X

s0 |ϕi (1)|2 + ki ϕi (1) +

i=1

s0 s0 |ϕi (1)|2 − |ϕi |2 ≤ 0, 2 2

it yields that 2 q X 1 1 d|ϕ|2 + Di ϕi ||2 ≤ K1 |ϕ|2 . || 2 2 dt s + 2 i=1

(45)

Since ϕ(0) = 0, the Gronwall’s inequality shows that u1 (t) + λ1 (t) ≤ k1 and u2 (t) + λ2 (t) ≤ k2 a.e. t ∈ Sδ . We establish now maximum bounds for u3 and u7 by setting in the weak formulation the test function ϕ := (0, 0, (u3 +λ3 −k3 )+ , 0, 0, (u7 +λ7 −k7 )+ )t ∈ V, where we denote ϕi := (ui + λi − ki )+ for i ∈ {3, 7}. For a convenient choice of k3 , k7 , we have that (ϕ3 , ϕ7 ) ∈ Vh . We obtain q 1d 1 1d |ϕ7 |2 + L − s − |ϕ3 |2 + D7 ϕ7 ||2 || 2 2 dt 2 2 dt s + 2 q 1 D7 D3 2 2 2 + || D3 ϕ3 || = |ϕ7 (1)| + |ϕ3 (1)| L−s− 2 s+ 2 L−s− 2 + L−s− S3,diss (u3,eq − k3 , ϕ3 ) − S3,diss |ϕ3 |2 2 −(η1 , ϕ7 ) + S7,diss (u7,eq − k7 , ϕ7 ) − S7,diss |ϕ7 |2 + s+ 2 0 h i s |ϕ7 (1)|2 + |ϕ3 (1)|2 − |ϕ7 |2 + |ϕ3 |2 . + (46) 2

s+

By (C1) and the interpolation inequality, we margin above the r.h.s of (46) by means of the following expression:

ξ

||ϕ7 ||2 s+

2

"

+K1

2 + ξ

||ϕ3 ||2 L−s−

s0 (s + ) + D7 2

2

2

2

+ s0 (L − s − ) + D3 2

2 #

|ϕ|2 .

(47)

We take ξ ∈]0, min{D3 , D7 }] and apply Gronwall inequality to get u3 (t) + λ3 (t) ≤ k3 and u7 (t) + λ7 (t) ≤ k7 a.e. t ∈ Sδ . However, in order to be able to effect this step, we need to ensure that s0 ∈ L∞ (Sδ ). The reaction rate η has to stay bounded with respect to the variable u3 for fixed u1 , therefore we use the boundedness assumption from (A). 17

Let us show the maximum estimates for u5 and u6 . Set ϕ5 = (u5 + λ5 − k5 y)+ and ϕ6 = (u6 + λ6 − k6 )+ , where (ϕ5 , ϕ6 ) ∈ Vw , and obtain

q 1d 1d 1 D5 ϕ5 ||2 |ϕ5 |2 + L − s − |ϕ6 |2 + || 2 2 dt 2 2 dt s + 2 q 1 k5 D5 ||ϕ5 || 2 + = (η1 , ϕ5 ) || D6 ϕ6 || + L−s− 2 s + 2 i s0 h + |ϕ5 (1)|2 + |ϕ6 (1)|2 − |ϕ5 |2 + |ϕ6 |2 2 + s0 k5 ||ϕ5 ||.

s+

(48)

By Cauchy-Schwarz’s and Poincar´e’s inequalities, there exists a constant M1 = M1 (k1 , k3 , Λ) > 0 such that |(η1 , ϕ5 )| ≤ M1 ||ϕ5 ||. Rearranging the terms in (48) and applying Poincar´e’s and interpolation inequalities, we obtain:

q 1d 1 1d |ϕ5 |2 + L − s − |ϕ6 |2 + D5 ϕ5 ||2 || s+ 2 2 dt 2 2 dt s + 2

!

q 1 k5 D5 2 + + M1 k5 ||ϕ5 || || D6 ϕ6 || ≤ M1 − L−s− 2 s + 2 ||ϕ6 ||2 ||ϕ5 ||2 +ξ 2 + ξ 2 s + 2 L − s − 2 "

+ K1 If 0

0 take place. Although k5 t − λ5 if the restrictions k5 ≥ 2M s+ 2 both restrictions admit obvious physical motivations, the upper bound k5 t is a sort of unconfortable because of its dependence on time. The choice (b2) does not bring more insight than the use of (b1). (c) The L∞ -estimates ki (i ∈ I) can be made independent on the choice of . We introduce the constant K2 by h iL P1 Q1 2 2 2 (1 + L) + S7,diss + S3,diss + S7,diss 2 2 1 4 2 2 + 2cξ cˆ (D3 + D7 + ). 2

K2 := P2 +

(52)

Clearly, it yields that K2 ∈ R∗+ . Lemma 12 (Energy estimates) Let the triple (u, v4 , s) satisfy Definition 1. Then the following statements hold:

(i)

|u(t) + λ(t)|2 ≤ α(t) exp

Zt

β(τ )dτ a.e. t ∈ Sδ ,

(53)

0

(ii)

|u(t) + λ(t)|2 ≤ α(t) +

Zt

t Z β(s)α(s) exp β(τ )dτ ds a.e. t ∈ Sδ ;(54) s

0

19

t Z β(τ )dτ a.e. t ∈ Sδ , ||u(τ ) + λ(τ )||2 dτ ≤ d−1 0 α(t) exp

Zt

(iii)

(55)

0

0

where )

(

Di Di d0 := min min , min i∈I1 L + i∈I2 L − s0 − 2

.

2

(56)

The factors α(t) and β(t) are defined by L 2 2 S3,diss |u3,eq |2∞ + S7,diss |u7,eq |2∞ 2 t 2 Z 2 a(τ )dτ α(t) := |u(0) + λ(0)| + m0

a(t) :=

(57) (58)

0

0

s β(t) := 2 + + K2 2

s+ 2

2

+ L−s− 2

2 !

,

(59)

where K2 satisfies (52). u,t ∈ L2 (Sδ , V∗ ), u ∈ C(S¯δ , H).

(iii)

(60)

Proof. We set in the variational formulation (20) the test function ϕ := (u + λ)t ∈ V and obtain the expression

s+

2

1 s+

X 2 i∈I1

d |ϕi |2 + dt i∈I1 X

2 +

q

|| Di ϕi ||2 +

L−s−

2

2

1 L−s−

X 2 i∈I2

d X |ϕi |2 dt i∈I2

q

|| Di ϕi ||2 =

3 X

Ij ,

(61)

j=1

where 2 X

s0 (t) D7 D3 2 2 2 |ϕi (1)| + |ϕ7 (1)| + |ϕ7 (1)| , s+ 2 L−s− 2 i=1 s(t) + 2 X X I2 := s + (fi (ϕi ) + L − s − (fi (ϕi ), ϕi ), 2 i∈I1 2 i∈I2

I1 := −

I3 := s0 (t)

X i∈I1

(yϕi,y , ϕi ) + s0 (t)

X

((2 − y)ϕi,y , ϕi ).

(62)

i∈I2

By the positivity of s0 and the interpolation inequality with θ = 21 , we have 20

|I1 | ≤ ξ

||ϕ7 ||2 s+

2

2 + ξ

||ϕ3 ||2 L−s−

2

2 2 + K2 |ϕ| .

(63)

Cauchy-Schwarz’s inequality and the means inequality show

h | P1 Q1 (ϕ2 , ϕ1 ) − P1 |ϕ1 |2 |I2 | ≤ s(t) + 2 − (η1 , ϕ1 + ϕ7 ) − P2 Q2 |ϕ2 |2 + P2 (ϕ1 , ϕ2 ) i i h 2 + S7,diss (u7,eq , ϕ7 ) − |ϕ7 | + L − s − S3,diss (u3,eq , ϕ3 ) − |ϕ3 |2 | 2 " 2 |u7,eq |2∞ S7,diss P1 Q1 2 2 2 2 (|ϕ1 | + |ϕ2 | ) + P2 (|ϕ1 | + |ϕ2 | ) + ≤L 2 2 # 2 2 2 2 2 2 S |ϕ7 | LS3,diss |u3,eq |∞ LS3,diss |ϕ3 | + 7,diss + + 2 2 2 L 2 2 ≤ S |u3,eq |2∞ + S7,diss |u7,eq |2∞ + K2 |ϕ|2 (64) 2 3,diss

Integrating by parts in the two sums of I3 , we have

s0 s0 |ϕ(1)|2 + |ϕ|2 2 2 X ||ϕi ||2 X ||ϕi ||2 ≤ξ + ξ 2 2 i∈I1 s + i∈I2 L − s − 2 2

|I3 | ≤

2 s0 + cξ cˆ1−θ + 2

"

s+ 2

2θ 1−θ

+ L−s− 2

2θ 1−θ

!#

|ϕ|2 .

(65)

For θ = 21 , we acquire

|I3 | ≤ ξ

||ϕ7 ||2

X i∈I1

s+

s0 + + K2 2 "

2

2 + ξ

Select ξ ∈]0, mini∈I1 ∪I2 the proof of (i).

s+ 2 Di ]. 2

||ϕi ||2

X i∈I1

2

L−s−

2

2

+ L−s− 2

2 !#

|ϕ|2 .

(66)

The application of Gronwall’s inequality completes

The proof of (ii) - (iv) follow similarly as in [22] (Lemma 3.4.23). Note that the energy estimates (53) - (55) do not depend on the choice of . 21

4.5

Proof of Theorem 3

The proof of Theorem 3 relies on the use of the Banach fixed-point principle. We consider the time interval Sδ =]0, δ[, t ∈ Sδ , with δ set as in Definition 1. Let σ = supSδ s0 (:= M + 1). Let Yδ be the set of functions r ∈ W 1,2 (Sδ ) such that r(0) = 0. (Yδ , ρ) is a complete metric space, where the metric ρ is defined by ρ : Yδ × Yδ → R+ such that

(67)

ρ(r1 , r2 ) = |r20 − r10 |L2 (Sδ ) for all r1 , r2 ∈ Yδ . Let M (Sδ ) := {r ∈ Yδ : r ∈ [s0 , σδ + s0 ], r0 ≥ 0, |r0 |L2 (Sδ ) ≤ σδ}.

(68)

Clearly, M (Sδ ) is a non-empty closed subset of Yδ . Define a mapping T : M (Sδ ) → W 1,p (Sδ ), p ∈ [1, ∞] in the following way: For any s ∈ W 1,2 (Sδ ) satisfying (35) and s − s0 ∈ M (Sδ ), we define T : M (Sδ ) 3 s˜ := s − s0 7−→ u cf. (20) , v4 cf. (35) 7−→ r − s0 := r˜ ∈ W 1,p (Sδ ).

(69)

We introduce the metric space (M (Sδ ), ρ), where M (Sδ ) := {r ∈ W 1,2 (Sδ ) : r(0) = s0 , r(t) − s0 ∈ [0, δσ],

(70)

r0 (t) ≥ 0 a.e. t ∈ Sδ , |r0 |L2 (Sδ ) < σ} = 6 ∅ and ρ : M (Sδ ) × M (Sδ ) → R+ defined by

(71)

ρ(r1 , r2 ) = |r20 − r10 |L2 (Sδ ) for all r1 , r2 ∈ M (Sδ ). M (Sδ ) is a complete space with respect to the metric ρ. Lemma 13 (The fixed-point operator) Let the assumptions of Theorem 3 be fulfilled. Then we have: (i) T : M (Sδ ) → M (Sδ ) for sufficiently small δ. (ii) There is a strictly positive constant χ = χ(s0 , L0 , Cη , θ, cˆ, δ, ki , D` ), where i ∈ I and ` ∈ I1 ∪ I2 , such that ρ(T s2 , T s1 ) ≤ δχρ(s2 , s1 )

for all s1 , s2 ∈ M (Sδ ).

22

(72)

Proof. The proof of (i) follows the lines of Lemma 3.3.25 from [22]. We concentrate on the proof of (ii), i.e. the Lipschitz continuity of T . Let w(j) := (wj1 , wj2 , . . ., wj7 ) (j ∈ {1, 2}) be two distinct concentrations vectors and let sj be the corresponding strips positions. Set wt := (w1 , w2 , . . . , w7 ) := w(2) − w(1) ∈ V, where wi := w2i − w1i := u2i + λ2i − u1i − λ1i for all i ∈ I1 ∪ I2 . Let us denote ∆s(t) := s2 (t) − s1 (t), ∆s0 (t) := s02 (t) − s01 (t), etc. We consider the fixed-point operator T : si ∈ M (Sδ ) 7−→ solution wi of the variational problem (20) 7−→ ri ∈ W 1,2 (Sδ ),

(73)

where ri0 (t) satisfies (35) a.e. t ∈ Sδ and ri (0) = s0 , i = 1, 2. Claim 14 Let α be an arbitrary positive real number. There is a strictly positive constant c = c(cs (), , k3∗ , k3 , α) such that the inequality Z

|∆r0 (τ )|α dτ ≤ c(cs (), , k3∗ , k3 , α)

Z

Z

|w|α dxdτ

(74)

Sδ Ω (τ )

Sδ

holds. By Claim 14, it yields that Z

|∆r0 (τ )|2 dτ ≤ δK2 sup |w(t)|2 ,

(75)

t∈Sδ

Sδ

where K2 is defined to be the constant c(cs (), , k3∗ , k3 , α) in which we put α = 2. We aim to estimate supt∈Sδ |w(t)|2 in order to obtain the contractivity of the operator T . We proceed as follows: We subtract the variational formulation written for the solution w(1) from that one written for w(2) and employ in both formulations the same test function wt := w(2) − w(1) ∈ V. The proof relies on convenient manipulations of the positivity of the concentrations, and of the maximum and energy estimates (see Lemma 11 and Lemma 12) to support the application of Banach’s fixed-point principle. For each j ∈ {1, 2}, (20) yields

sj (t) + 2

X i∈I1

(j) (wi,t (t), wi )

+ L − sj (t) − 2

X

(j)

(wi,t (t), wi )

i∈I2

+a(sj , w(j) , w) + e(s0j , w(j) , w) = bf (w(j) , sj , w) + h(sj , s0j , wy(j) , w). We rearrange the terms in (76) and obtain 23

(76)

s2 + 2

X 1d i∈I1

X1d |wi (t)| + L − s2 − |wi (t)| 2 dt 2 i∈I2 2 dt 1 X q + || Di wi ||2 s2 + 2 i∈I1

+

1 L − s2 −

X 2 i∈I2

q

|| Di wi ||2 ≤

5 X

|Jj |,

(77)

j=1

where the right-hand side of (77) is defined by means of the following terms:

J1 := −∆s

X

(w1i,t , wi ) + ∆s

i∈I1

J2 := −

2

(w1i,t , wi )

i∈I2

∆s s1 +

X

X 2

s2 +

Di (w1i,y , wi,y )

i∈I1

∆s L − s1 −

2

X

L − s2 −

2

Di (w1i,y , wi,y ),

i∈I2

J3 := bf (w2 , s2 , w) − bf (w1 , s1 , w), J4 := e(s01 , w1 , w) − e(s02 , w2 , w) + +

1 L − s2 −

1 s2 +

X q 2 i∈I1

| Di wi (1)|2

X q 2 i∈I2

| Di wi (1)|2 ,

J5 := h(s02 , w2,y , w) − h(s01 , w1,y , w). In order to simplify the writing of the estimates, we introduce the constant K3 cˆ4 := 1 + cξ + max Di + c + (|I1 | + |I2 |)(cˆ c)2 2 2 i∈I1 ∪I2 !

2 X 1 c2 ¯ as in (9)) + k¯2 + 1 + cξ k¯2 + cξ cξ¯ + + cξ + k( ki2 + cξ cˆ1−θ . 4 2 i=1

¯ ki , Di Note that the K3 is strictly positive and depends mainly on θ, ξ, ξ, (i ∈ I1 ∪ I2 ), s0 , etc. It does neither depend on the solution, nor on . The bound of |J1 | reads |J1 | ≤

|∆s|2 |w|2 |w1,t |2 + . 2 2

(78)

By Cauchy-Schwarz’s and the arithmetic-geometric means inequalities, we obtain 24

|J2 | ≤ +

|∆s| s1 +

2

X 2

s2 +

Di ||w1i ||||wi ||

i∈I1

|∆s|

2 2

L − s1 − X

≤ξ

i∈I1

X

2

L − s2 −

Di ||w1i ||||wi ||

i∈I2

X ||wi || ||wi ||2 +ξ 2 (s2 + )2 i∈I2 L − s2 − 2

2 2 X X Di ||w1i || Di ||w1i || 2 + cξ 2 + 2 |∆s|

s1 +

i∈I1

X

≤ξ

i∈I1

2

||wi ||2

X

2

i∈I2

s2 +

L − s1 −

i∈I2

2 + ξ

||wi ||2

L − s2 −

+ K3

2

2

2

1

s1 +

2

2 +

1 L − s1 −

2

2 2 2 ||w1 || |∆s|

(79)

Finally, for |J5 | we have

2

s01

s01

1 + 2K3 |J5 | ≤ 3 + 3ξ¯ + 4K3 L s1 + L − s1 −

2

2

2 2 |∆s|

+ 3 + 3ξ¯ + 6K3 |∆s0 |2

2

||wi ||

X

+ ξ(3 + 2cξ¯)

i∈I1

( s2 +

2

2

2 +

||wi ||

X i∈I2

L − s2 −

2

2

+ K3 χ1 (t)|w|2 ,

(80)

where the expression of χ1 (t) is given by

χ1 (t) :=

1 s2 +

+ + +

2

s01 s1 +

+

1 L − s2 −

!2

2 s01

L − s1 −

2

s2 + 2

!2 2

+ 2

1 s2 +

1 + s2 +

L − s2 − 2

(s01 )2 L − s1 −

2

2

L − s2 −

2

25

2 +

1 L − s2 −

2

! 2

2

+ 1 + L − s2 −

! 2

2

for a.e. t ∈ Sδ . To estimate the remaining terms, namely |J3 | and |J4 |, we need to take into account the exact structure of bf (·) and e(·). We obtain the bound on |J3 | as follows: (2) J3 := s2 + [P1 (Q1 w22 − w21 , w1 ) − (η1 , w1 )] 2 (1) − s1 + [P1 (Q1 w12 − w11 , w1 ) − (η1 , w1 )] 2 P2 (Q2 w22 − w21 , w2 ) + s1 + P2 (Q2 w12 − w11 , w2 ) − s2 + 2 2 + (L − s2 − )S3,diss (u3,eq − w23 , w3 ) 2 − (L − s1 − )S3,diss (u3,eq − w13 , w3 ) 2 (1) (2) + (η1 , w5 ) − (η1 , w5 ) + (s2 + )S7,diss (u7,eq − w27 , w7 ) − (s1 + )S7,diss (u7,eq − w17 , w7 ) 2 2 (1) (2) − (η1 , w7 ) + (η1 , w7 ),

and hence, [(−∆η1 , w1 ) + P1 Q1 (w2 , w1 ) − P1 |w1 |2 ] 2 + s2 + [−P2 Q2 |w2 |2 + P2 (w1 , w2 )] − L − s2 − S3,diss |w3 |2 ] 2 2 + s2 + (∆η1 , w5 ) 2 + s2 + S7,diss |w7 |2 + s2 + (−∆η1 , w7 ) 2 2 (1) + ∆s[P1 (Q1 w12 − w11 , w1 ) − (η1 , w1 )] − ∆s[P2 (Q2 w12 − w11 , w2 )] − ∆sS3,diss (u3,eq − w13 , w3 )

J3 ≤ s2 +

(1)

(1)

+ ∆s(η1 , w5 ) + ∆sS7,diss (u7,eq − w27 , ϕ7 ) + ∆s(η1 , w7 ) ≤ ρ1 |∆s|2 + ρ2 K3 |w|2 ,

(81)

where the constants ρ1 and ρ2 depend on P1 , P2 , Q1 , Q2 , S3,diss , S7,diss and cL . After some manipulations, we obtain

|J4 | ≤

X i∈I1 ∪I2

−

2 X i=1

Di

1 s2 +

s02 w2i wi (1)

+

2

1 + L − s2 −

2 X

s01 w1i wi (1)

i=1

26

! 2

|w(1)|2

L i∈I1 ∪I2 Di |∆s0 |2 ¯2 |w(1)|2 . + k |w(1)|2 + ≤ 2 s2 + 2 L − s2 − 2 P

Using twice the interpolation inequality in (82) with θ =

|J4 | ≤

1 2

(82)

and obtain

X ||wi ||2 X |∆s0 |2 ||wi ||2 +ξ 2 + ξ 2 2 i∈I1 s2 + i∈I2 L − s2 − 2 2

2 + s2 + 2

+ K3

2

+ L − s2 − 2

2 !

|w|2

(83)

We introduce the time-dependent factors

2

γ1 :=

1 |w1,t | 1 2 + K3 2 + 2 ||w1 || 2 s1 + 2 L − s1 − 2

+ 3 + 3ξ¯ + 4K3

s01 s1 +

2

+ 2K3

s02 L − s2 −

!2 2

+ ρ1 ,

γ2 := 4 + 3ξ¯ + 6K3 , χ1 + ρ2 + 3 + s2 + 2

γ3 := K3

2

+ L − s2 − 2

2 !

,

and reformulate the right-hand side of (77) as X q X q 1d X 1 1 2 2 |wi | + || Di wi || + || Di wi ||2 (84) 2 2 2 dt i∈I s2 + 2 i∈I1 L − s2 − 2 i∈I2

≤

X X ||wi ||2 7 ||wi ||2 13 ξ + 2ξc + ξ + 2ξcξ¯ 2 2 ξ¯ 2 2 i∈I2 L − s2 − i∈I1 s2 + 2 2

+γ1 |∆s|2 + γ2 |∆s0 |2 + γ3 |w|2 . Select ξ and ξ¯ in such a way that we may neglect the first two terms of √ P L|| Di wi ||2 the right-hand side of (84) when comparing them to i∈I1 and 2 s2 + 2 ) ( √ P L|| Di wi ||2 2 , respectively. The only problem is that γ1 depends on ||w1 || i∈I2 (L−s2 − 2 ) and |w1,t |. Employing conveniently the geometrical restrictions on s1 , s2 , L and R , and the energy estimate (53), we can show that Sδ γ1 (τ )dτ is independent of ||w1 || and |w1,t |. After a redefinition of γ1 , we obtain the inequality 5 Z Sδ 5

Z

γ1 (τ )|∆s(τ )|2 dτ ≤ δ 2 max |γ1 (t)| t∈S¯δ

|∆s0 (τ )|2 dτ,

(85)

Sδ

Note that (85) follows directly by means of the mean-value theorem for integrals.

27

where its right-hand side does not depend on ||w1 || and |w1,t | anymore. By (85) and integration along Sδ , we express (84) as

2

2

Z

2

|w| ≤ |w(0)| + 2 max δ |γ1 (t)| + |γ2 (t)| t∈S¯δ

0

2

|∆s (τ )| dτ +

Zt

2γ3 (τ )|w(τ )|2 dτ.

0

Sδ

By Gronwall’s inequality (with w(0) = 0), we gain the estimate 0 2

2

R2

|w| ≤ δ max |∆s | γ4 (t)e

Sδ

γ3 (τ )dτ

t∈S¯δ

,

(86)

where the factor γ4 is given by γ4 (t) := 2 (δ 2 |γ1 (t)| + |γ2 (t)|) for all t ∈ Sδ . Compare now (86) to (72). For each δ > 0 sufficiently small, we choose χ > 0 such that the operator T becomes contractive on Sδ . For instance, a possible choice of χ is R

2γ3 (τ )dτ

maxt∈S¯δ γ4 (t)e Sδ . χ := (1 + c(cs (), , k3∗ , k3 , α)) δ The local existence and uniqueness result via Banach’s Contraction Principle. If χT ≥ 1, additional steps are needed. Then δ > 0 has to be taken such that χδ < 1, and in this case, we consider problem (2)-(5) with respect to the time interval [0, δ]. By replacing T with δ in the above argument we obtain only a local solution. Now consider the problem on [δ, 2δ]. By applying the technique used at the outset to this problem, we obtain a unique solution that extends the previous one from the interval [0, δ] to [δ, 2δ]. It is clear that this procedure may be repeated on the interval [2δ, 3δ] and so on. After a number of steps, we gain that the solution of (2)-(5) is valid on [0, Tfin ]. Note that δ and χ (1 + c(cs (), , k3∗ , k3 , α)) do not depend on the selection of . Finally, it only remains to prove (74). Proof of Claim 14. Employing the definition of r0 , a simple calculation shows that

1 Z |∆r0 (τ )|α dτ = cs () Sδ

=

Z Sδ

R

R

Ω (τ ) ∆η (y, τ )dy − R R w (y, τ )dy Ω (τ )

23

(2) Ω (τ ) η (y, τ )dy

Ω (τ )

w3 (y, τ )dy

α Z

w3 (y, τ )dy dτ.

2 Ω (τ )

(87) 28

By (40), there exists a constant cα > 0 such that 6 for any a, b ∈ R+ we have (a + b)α ≤ cα (aα + bα ) .

(88)

Therefore, we may conclude that there exits a constant c¯ = c¯(cs (), , k3∗ , k3 , α) > 0 such that Z Sδ

ccs ()Cηα Z Z |∆η |α (y, τ )dydτ |∆r (τ )| dτ ≤ k3∗ α α 0

α

Sδ Ω (τ )

ccs () + 2α 2α k3 ≤ c¯

Z

Z

Z

Z

|w|α dydτ

Sδ Ω (τ )

|w|α dydτ.

(89)

Sδ Ω (τ )

For example, we take c¯ := proof of the claim.

4.6

ccs ()Cηα

Ω

k3∗ α α

+

ccs () k3 2α 2α

with c ≥ cα . This concludes the

Proof of Theorem 4 (i)

Let (u(i) , v4 , s(i) ) (i ∈ {1, 2}) be two weak solutions on Sδ (in the sense of Definition 1), which satisfy the assumptions of Theorem 3. We want to show that the function H × W 1,2 (Sδ )|I1 ∪I2 | × MΞ × MΥ × MΛ → W21 (Sδ , V, H) × W 1,2 (Sδ )2 that maps (u0 , λ, Ξ, Υ, Λ)t into (u, v4 , s)t is Lipschitz continuous in the sense of (22). Let Γi (t) be the centerlines of two arbitrarily chosen reaction strips Ω(i) (t) (i ∈ {1, 2}). By (15), the positions si (t) of the interfaces Γi (t) satisfy the geometrical restriction 0 < si0 := si (0) ≤ si (t) −

< si (t) + ≤ Li0 < L for all t ∈ Sδ . 2 2

Denoting s0 := max{s10 , s20 } and L0 := min{L10 , L20 }, the common space domain travelled by Γi (t) is the interval ]s0 , L0 [. Within this frame we only discuss the non-trivial case, namely s0 < L0 . Set L∗ () := min{min{si (t) + , L − si (t) − : i = 1, 2}} > 0, 2 2 t∈S¯δ

(90)

Take cα ≥ max{(1 + ς)α−1 , (1 + 1ς )α−1 } for ς > 0 and α ∈ [1, ∞[. If α ∈]0, 1[, then cα ≥ 1.

6

29

(i)

D0 := min{Dj : j ∈ I1 ∪ I2 , i ∈ {1, 2}} > 0.

(91) (1)

We subtract the weak formulation (20) of the solution (u(1) , v4 , s1 ) from that (2) written in terms of (u(2) , v4 , s2 ). Choosing as test function w = (u(2) −u(1) )t + (2) (1) (2) (1) (λ(2) − λ(1) )t ∈ V, i.e. wi = ui − ui + λi − λi ∈ Vi for all i ∈ I1 ∪ I2 , we obtain: s2 + 2

2

L − s2 − |wi | + dt 2

X d i∈I1

2

1 + L − s2 −

2

X d

1 |wi | + dt s2 +

i∈I2

X 2 i∈I2

2

q

(1)

|| Di w||2 ≤

X 2 i∈I1 11 X

1 s2 +

J2 := −∆s

X 2 i∈I1

X

(2)

Di |wi (1)|2 +

|Jj |,

j=1

(w1i,t , wi ) + ∆s

i∈I1

J3 := −

1 s2 +

X 2 i∈I1

2

X

X 2 i∈I2

(2)

Di |wi (1)|2

(w1i,t , wi )

i∈I2

|∆Di |(w1i,y , wi,y )

∆s

+

1 L − s2 −

X

2

(1)

Di (w1i,y , wi,y ))

s2 + s1 + i∈I1 X 1 − |∆Di |(w1i,y , wi,y ) L − s2 − 2 i∈I2 +

∆s L − s1 −

J4 := −s02

2 X

2

L − s2 −

wi (1)2 − ∆s0

i=1

X

2 X

2

(1)

Di (w1i,y , wi,y ))

i∈I2

w1i (1)wi (1)

i=1

i h (2) (2) (2) P1 Q1 w22 − w21 , w1 − η1 , w1 J5 := s2 + 2 i h (1) (1) (1) − s1 + P1 Q1 w12 − w11 , w1 − η1 , w1 2

(2) (2) J6 := − s2 + P2 Q2 w22 − w21 , w2 2 (1) + s1 + Q2 w12 − w11 , w1 2 (2) (2) S3,diss (u3,eq − w3 , w3 ) J7 := L − s2 − 2

30

(1)

|| Di w||2

where the right-hand side of (92) is given by:

J1 :=

q

(92)

(1) (1) S3,diss (u3,eq − w3 , w3 ) 2 (2) (2) (1) (1) J8 := s2 + S7,diss (u7,eq − w7 , w7 ) − s1 + S7,diss (u7,eq − w7 , w7 ) 2 2 X X (2) (1) J9 := η1 , wj − η1 , wj s2 + s1 + 2 2 j∈{5,7} j∈{5,7}

− L − s1 −

J10 := s02

X

(yw2i,y , wi ) − s01

X

(yw1i,y , wi )

i∈I1

i∈I1

J11 := s02

X

((2 − y)w2i,y , wi ) − s01

i∈I2

X

((2 − y)w1i,y , wi ).

(93)

i∈I2

Let K4 be the following positive constant (1)

K4 := 1 + 2ˆ c4 max Di + i∈I1 ∪I2

(1) (1) 2 Q1 P 1

+

2

(1)

k 1 + k2 + M1 + cˆ4 cξ 2

2 k 2 (1) (2) 2 L2 (2) + 2 P1 Q1 + 1 + Q1 k2 2 2

(2)

+ S3,diss + S3,diss |u3,eq |∞

+ +

(1) 2

P2

2 X h `∈{3,7}

+

(2)

LQ2 k2

2

2

+ L|u3,eq |2∞

(2)

(2)

P2 Q2 k2

2

+ 2 2 i cξ (1) (2) 2 2 + cˆ4 . S`,diss + L |u`,eq | + S`,diss + M1 2

(94)

Clearly, K4 depends , ξ and all model parameters. It stays stays bounded for ξ > 0. Let us now proceed to estimate the terms |Ji | (i ∈ {1, . . . , 11}). The main tools used in this context are the positivity, L∞ - and energy estimates combined with the interpolation inequality, Cauchy-Schwarz’s and Young’s inequalities as well as some further elementary algebraic inequalities. We obtain 1 X ||wi ||2 1 X ||wi ||2 2 |J1 | ≤ ξ 2 + ξ 2 + K4 |w| . 2 i∈I1 s2 + 2 i∈I2 L − s2 − 2 2 |J2 | ≤ |∆s|2

|J3 | ≤ 2ξ

|w1,t |2 |w1 |2 + . 2 2

X i∈I1

(96)

||wi ||2

X

2

i∈I2

s2 +

2 + 2ξ

(95)

||wi ||2

L − s2 −

31

2

2 +

+ K4

1

s1 +

2

L − s1 −

2

||w1i ||2 |∆s|2

X i∈I1 ∪I2

||w1i ||2 |∆Di |2 ,

X

+ K4

2

2 +

1

(97)

i∈I1 ∪I2

|∆s0 |2 2

|J4 | ≤ K4 |w(1)|2 + ≤ξ

||wi ||2

X i∈I1

s2 +

2

|∆s0 |2 + + K4 2

2 + ξ

"

||wi ||2

X i∈I2

s1 + 2

2

L − s2 −

2

2 +

+ L − s1 − 2

2 #

|w|2 .

(98)

|∆s|2 |∆P |2 |∆Q|2 + + + K4 |w|2 . |J5 | ≤ 2 2 2

(99)

Note that |J6 | ≤ |J5 |. |∆s|2 |w3 |2 + K4 + L|∆u3,eq |2∞ + 2 2 |∆s|2 |w7 |2 |J8 | ≤ + K4 + L|∆u7,eq |2∞ + 2 2 |J7 | ≤

L |∆S3,diss |2 , 2 L |∆S7,diss |2 , 2

(100) (101)

and

|J9 | ≤ |∆s|2 + K4

|w5 |2 + |w7 |2 1 + |∆η1 |2 . 2 2

(102)

Integrating by parts the terms (ywi,y , wi ) and ((2−y)wj,y , wj ) ((i, j) ∈ I1 ×I2 ) and applying conveniently the interpolation inequality with θ = 21 , we obtain bounds on |J10 | and |J11 | as follows:

|J10 | ≤ s02

X i∈I1

|wi (1)|2 |wi |2 − 2 2

+

i∈I1

2

≤

X ||wi ||2

!

2

ξ X ||wi || 2 + 1 + K4 s 2 + 2 i∈I1 s2 + 2 2

|J11 | ≤ s02

X i∈I2

|wi (1)|2 |wi |2 + 2 2

+

i∈I2

32

2

X

|wi |2

i∈I1

2 !

X ||wi ||2

!

|∆s0 |2 + (s02 )2

X

|wi |2 ,

i∈I1

|∆s0 |2 +

X i∈I2

|wi |2

≤

||wi ||2 ξ X 2 i∈I2 L − s2 − 2 2

+ 1 + K4

L − s2 − 2

2 !

(s02 )2

X

|wi |2 .

(103)

i∈I2

By (B) and (88) (with α = 2 and c2 = 2), it results that the term 21 |∆η1 |2 can be bounded from above by c2L |w|2 +c2L |∆Λ|2 . By (95)-(103), we conclude that the right-hand side of (92) can be estimated from above by

4ξ

||wi ||2

X i∈I1

s2 +

2

2 + 4ξ

||wi ||2

X i∈I2

L − s2 −

2

2 0 2 2 + γ1 |∆s| + γ2 |∆s |

+γ3 |∆P |2 + |∆Q|2 +

|∆Di |2

X i∈I1 ∪I2

X

+

|∆S`,diss |2 + |∆u`,eq |2∞ + |∆Λ|2 + γ4 |w|2 .

(104)

`∈{3,7}

In (104), the factors γi (i ∈ {1, . . . , 4}) are defined by

γ1 := 3 + cξ

X

(1) 2

Di

||w1i ||2 +

i∈I1 ∪I2

γ2 :=

|w1,t |2 , 2

X ||w2i ||2 1 + , 2 i∈I1 ∪I2 2

γ3 := 1 + L + K4

X

||w1i ||2 + c2L ,

i∈I1 ∪I2

γ4 := 3 +

c2L

+ 7K4 + K4 1 +

2|s02 |2

"

s2 + 2

2

+ L − s2 − 2

2 #

. (105)

Select D0 ξ ∈ 0, 4

with D0 cf. (91)

(106)

and neglect the first two sums of (104). By (85), (104), (106) and Gronwall’s inequality, the variational formulation (92) leads to the estimate α |w| ≤ e L∗ () 2

R Sδ

2γ4 (τ )dτ

,

(107)

where L∗ () is cf. (90) and α is defined by 33

α := L∗ ()|w(0)|2 + 2δ max δ 2 |γ1 (t)| + |γ2 (t)| max |∆s0 |2 t∈S¯δ

t∈S¯δ

+ 2δ max |γ3 (t)| t∈S¯δ

2

× max |∆Ξ| + MΞ

||∆Υ||2MΥ ∩L∞ (Sδ )

2

+ max |∆Λ| MΛ

.

Bounding above the term |∆s0 |2 by c|∆η |2 and using (B), we obtain the conclusion of the Theorem. Although the value of the positive constant c can be exactly calculated, for our purpose is sufficient to ensure that c is finite. Note that c depends on the choice of , cs () and k3∗ .

5

Numerical illustration

In this section, we illustrate the expected behavior of the reaction-front position and concentration profiles in the framework of the moving two-reactionstrips model. The basic numerical approach is to immobilize the moving strips, to discretize the PDEs in space, and then to integrate (via MATLAB) the resulting semi-discrete system in time [5,34,22]. The numerical examples are obtained by choosing n = 80. The initial value ¯up10 uˆq70 . A typical k¯ is about 500. To of CaCO3 (aq) is calculated by uˆ40 = kˆ calculate λ1 and λ2 , we split the ambient concentration c¯ of CO2 at the exposed c¯Q c¯ boundary via Henry’s law, i.e. (λ1 , λ2 ) = 1+Q , 1+Q . λ3 incorporates the chemistry of the Portland cement from [7]. Compare [18] and chapter 4 of [22] for more details on the used reference parameters.

(a)

(b)

(c)

Fig. 3. (a)+(b) CO2 (aq) and CO2 (g) concentration profiles vs. space. Each curve refers to time t = i years, i ∈ {1, . . . , 18}. (c): Reaction strip location vs. the experimental points “ ◦ ” (from [7]) after Tfin = 18 years of exposure.

We illustrate the effect of the secondary carbonation in Fig. 5–6, where we point out the formation of internal reaction zones. More precisely, we note that two distinct zones of Ca(OH)2 (aq) depletion are formed and progress into the material. It is worth mentioning that, in order to obtain alike behavior (see Fig. 6, e.g.), rather strong prerequisites on the model parameters have to be 34

(a)

(b)

Fig. 4. (a) Concentration profiles of CO2 (aq) vs. time and space position. (b) CaCO3 (aq) concentration profiles vs. space. Each curve refers to time t = i years, i ∈ {1, . . . , 18}.

(c) Fig. 5. Effect of secondary carbonation: Two internal zones of depletion of Ca(OH)2 (aq) are formed and progress into the material. Each curve refers to time t = i years, i ∈ {1, . . . , 18}.

fulfilled: (1) A strong production by dissolution within a short time interval has to apply behind Γr (t), see the basic geometry in Fig. 2; (2) The newly produced Ca(OH)2 is (almost) not transported, thus a possible leaching out of the alkaline species is hindered; The new content of Ca(OH)2 basically waits to react with CO2 (g) arriving from the exterior boundary. (3) CO2 (g) has to arrive quickly to the location of Ω (t). Consequently, the porous matrix is assumed to be highly permeable and the transfer of mass from the air parts into the wet parts of the pores is considered to be rapid. (4) On the other hand, the strip Ω (t) should advance in such a way that the primary carbonation finishes. (5) The precipitation of CaCO3 (aq) is taken to be instantaneous (and hence its transport may be neglected). In Fig. 6 (a), we distinguish between two segregated internal strips where CaCO3 (aq) is produced. That one which is more visible corresponds to Ω1 (t). We see in Fig. 6 (b) that the secondary carbonation acts as a retarder with 35

(a)

(b)

Fig. 6. Effect of secondary carbonation: (a) Two internal zones of production of CaCO3 develop into the material. (b) The parameter R rises (R = 0, 1, 5, 10), while the penetration depth gradually decreases.

respect to the overall carbonation process in the sense that: If the secondary carbonation phenomenon occurs, then the speed of the strip Ω (t) decreases. To visualize such behaviors, we take about the half of the speed s0 used to obtain the previous plots and alter progressively a reaction parameter R ([22], appendix A). Here R compares the characteristic time of the carbonation reaction taking place in Ω1 (t) with the characteristic time of the carbonation reaction within Ω (t). Consequently, if R = 0, then no secondary carbonation acts behind the strip Ω (t). If R 6= 0 and if the dissolution overcomes the carbonation in the region Ω1 (t) (in the sense that Ca(OH)2 is produced in such quantity that CO2 (g) is practically stopped somewhere within Ω1 (t)), then Ω1 epsilon(t) may stop its motion. Nevertheless, since in carbonation tests an advanced degree of the hydration reaction is usually ensured [16,8], alike behavior is not really expected to happen. Note that by adopting the assumptions (C1)-(C3) from section 3, we practically impose that the velocity of the layer Ω (t) is strictly positive in the time interval STfin . In this example, it appears that the agreement between the experimentally observed motion trajectory of the reaction strip and concentration profiles and that computed via the proposed model is reasonable.

Acknowledgements

It is the author’s pleasure to express his deep gratitude to his advisor and mentor, Michael B¨ohm (University of Bremen, Germany), as well as to many others who contributed with suggestions and support. 36

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