Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)
1. Statement of the problem and a mathematical model
Carbon dioxide, which is present under normal atmospheric conditions and also emitted as industrial output, attacks reinforced concrete structures by destroying their protection against corrosion. The process is called carbonation. The core reaction can be described as
CO2(g→aq) +Ca(OH)2(aq)H−→2OCaCO3(aq) +H2O. (1.1) Its impact on concrete microstructure is significant and possible repairs are often expensive. Therefore there is need of models capable to predict the depth ofCO2 penetration in concrete structures accurately. We consider the carbonation pene- tration in a wall whose chemistry, humidity level, and microstructure are known [6]. Experiments show that the zone of reaction is narrowly confined to the in- terface between the unreacted solid and the product layer, i.e., the region where calcium carbonate precipitates to the solid matrix. See [5, 6, 8] for more details on this motivating practical problem.
This work was completed with the partial support of DFG-SPP1122Prediction of the Course of Physicochemical Damage Processes Involving Mineral Materials.
318 A. Muntean and M. B¨ohm
We idealize the reaction front by a surface Γ(t). Let the positive x-axis be directed normally to Γ(t) and into the uncarbonated part. The basic geometry is sketched in Figure 1 (a). Att = 0, we assume that the origin located at x= 0 is behind the reaction interface Γ(t). Assuming that the reactants, which depend only on the real variablesx and t, are available to reaction, we expect that the reaction interface moves as x = s(t) for t ∈ ST :=]0, T[ such that s(0) = s0, where T ∈]0,+∞], s0 ∈]0, L[, and L ∈]0,+∞[ are given, see Figure 1 (b). We denote the mass concentration of the reactants and products as follows: ¯u1 :=
[CO2(aq)], ¯u2:= [CO2(g)], ¯u4:= [CaCO3(aq)], and ¯u5:= [H2O] are the chemical species present in the region Ω1(t) := [0, s(t)[; ¯u3 := [Ca(OH)2(aq)] and ¯u6 :=
[H2O] are species present in Ω2(t) :=]s(t), L]. For ease of notation, we use the
(a) (b)
Figure 1. (a) Basic geometry for thePΓmodel. The box A is the region which our model refers to. (b) Schematic 1D geometry. The reactants of (1.1) are spatially segregated at any timet.
set of indices I := I1∪ {4} ∪ I2, where I1 := {1,2,5} points out the active concentrations in Ω1(t), andI2:={3,6}refers to the active concentrations living in Ω2(t). Specifically, we take into account thatCaCO3(aq) is not transported in Ω := Ω1(t)∪Γ(t)∪Ω2(t), therefore the only partly dissipative character of the model. Then, we are led to discuss themoving-boundary problem of determining the concentrations ¯ui(x, t), i∈ I and the interface position s(t) which satisfy for allt∈ST the equations
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
(φφwu¯i),t+ (−Diνi2φφwu¯i,x)x = +fi,Henry, x∈Ω1(t), i∈ {1,2}, (φφwu¯3),t+ (−D3φφwu¯3,x)x = +fDiss, x∈Ω2(t),
(φφwu¯4),t = +fPrec+fReac Γ, x=s(t)∈Γ(t), (φ¯u5),t+ (−D5φ¯u5,x)x = +fReac Γ, x∈Ω1(t),
(φ¯u6),t+ (−D6φ¯u6,x)x = 0, x∈Ω2(t).
(1.2)
The initial and boundary conditions areφφwνi2u¯i(x,0) = ˆui0, i ∈ I, x ∈ Ω(0), φφwνi2u¯i(0, t) =λi,i∈ I1, ¯ui,x(L, t) = 0,i∈ I2,x∈Ω2(t), wheret∈ST. Specific to our problem, we impose the following interface conditions
⎧⎨
⎩
[j1ãn]Γ(t) = −η˜Γ(s(t), t) +s(t)[φφwu¯1]Γ(t), [jiãn]Γ(t) = s(t)[φφwνi2u¯i]Γ(t), i∈ {2,5,6}, [j3ãn]Γ(t) = −η˜Γ(s(t), t) +s(t)[φφwu¯3]Γ(t),
(1.3)
s(t) =α η˜Γ(s(t), t)
φφwu¯3(s(t), t), s(0) =s0, (1.4) whereν12 =ν32 := 1,ν22 := φφa
w, ν52 =ν62 := φ1
w, νi := 1 (i∈ I, ∈ I − {2}), ji := −Diνiφφwu¯i (i, ∈ I1 ∪ I2) are the corresponding diffusive fluxes, and α > 0. Here Di, L, and s0 are strictly positive constants, λi are prescribed in agreement with the environmental conditions to which Ω – a part of the wall (cf. Figure 1 (b)) – is exposed [6]. The initial conditions ˆui0 >0 are determined by the chemistry of the cement. We assume ˆui0 = λi. The hardened mixture of aggregate, cement, and water (i.e., the concrete) imposes ranges for the porosity φ >0 and also for the water and air fractions,φw>0 andφa >0, see [6]. Since the active concentrations are small, the constant porosity assumption is valid. The productions termsfi,Henry,fDiss,fPrec, andfReac Γ are sources or sinks by Henry- like interfacial transfer mechanisms, dissolution, precipitation, and carbonation reactions. Typical examples are (cf. [4, 8]):
fi,Henry:= (−1)iPi(φφwu¯1−Qiφφau¯2)(Pi>0, Qi>0), i∈ {1,2},
fDiss:=−S3,diss(φφwu¯3−u3,eq), S3,diss>0, fPrec:= 0, fReac Γ:= ˜ηΓ. (1.5) Let ¯u denote the vector of concentrations (¯u1, . . . ,u¯6)t and MΛ be the set of parameters Λ := (Λ1, . . . ,Λm)tthat are needed to describe the reaction rate. For our purposes, it suffices to assume thatMΛis a non-empty compact subset ofRm+. We introduce the function
¯
ηΓ :R6×MΛ→R+ by ¯ηΓ(¯u(x, t),Λ) :=kφφwu¯p1¯(x, t))¯uq3¯(x, t), x=s(t). (1.6) In (1.6)m:= 3 and Λ :={p,¯ q, kφφ¯ w} ∈R3+. We define the rate of reaction (1.1)
˜
ηΓ(s(t), t), which arises in (1.3)–(1.5), by
˜
ηΓ(s(t), t) := ¯ηΓ(¯u(s(t), t),Λ). (1.7) (1.6) is aspecialchoice of ¯ηΓ and represents the classical power-law ansatz. Other choices of ¯ηΓ may be employed, too. See [4, 8], e.g.
Note that some equations are valid in Ω1(t), while others act in Ω2(t) or at Γ(t)! All of the three space domains are varying in time and they area prioriun- known. The interface conditions require some explanation. The term ˜ηΓ(s(t), t)≈ αs(t) denotes the number of moles per volume that are transported by diffusion to the reaction interface. For convenience, we assume here ˜ηΓ(s(t), t) =s(t). The expression±φφwu(s(t), t)s¯ (t) accounts for the mass flux induced by the motion of the boundary Γ(t) in order to preserve conservation of mass. The conditions (1.3) express jumps in the gradients of concentrations across Γ(t). They are typical interface relations for a surface reaction mechanism, i.e., the classical Rankine- Hugoniot jump relations cf. [1], Section 1.2.E, e.g. The non-local law (1.4) governs the dynamics of the reaction interface. The latter interface condition is derived via first principles in the 1D case and for simple 2D geometries in [4, 8]. This kind of relationship is missing in the approach by Brieger and Wittmann [5]. It is however needed to complete the model formulation and allows the determination of the interface location once the reactants at Γ(t) are known. The setting is applicable
320 A. Muntean and M. B¨ohm
when the reaction rate is very rapid and the diffusion of the gaseousCO2 is suf- ficiently slow, or in other terms, when the characteristic time of the carbonation reaction is much smaller than the characteristic time of diffusion of the fastest species. This difference in the characteristic times causes the concentrations of the active chemical species and their gradient to have a jump at Γ(t). The magnitude of the jump typically depends on the concentration itself. The system (1.2)–(1.7) forms the sharp-interface carbonation model PΓ, or shortly (PΓ). The model PΓ consists of a coupled semi-linear system of parabolic equations that has a moving a prioriunknown internal boundary Γ(t), where the reaction (1.1) is assumed to take place. The coupling between the equations and the nonlinearities comes from the influence of the chemical reaction on the transport part and also from the dependence of the moving regions Ω1(t) and Ω2(t) on s(t). Other nonlinearities might be introduced by different assumptions on the production terms.
We present results on the well-posedness of the sharp-interface model PΓ and prove some useful estimates. To do this, we firstly fix the moving boundary by means of Landau-like transformations. Then we define the weak solutions to the transformed model and state its well-posedness locally in time. Since upper and lower bounds on the weak solution and on the shut-down time of the process are available, the maximum estimates become uniformly in time and the weak solution can be extended up to a global one, see the results and remarks in Sections 2 and 4. In Section 3, a simulation example shows the typical behavior of active concentrations and interface penetration into a real concrete wall. The model shows qualitatively good results when the numerical solution is compared with measured penetration depth profiles.
2. Main results
For eachi∈ I1∪I2, we denoteHi:=L2(a, b), where we set [a, b] := [0,1] fori∈ I1
and [a, b] := [1,2] for i ∈ I2. Moreover, H:=2
i∈I1∪I2Hi, Vi ={u∈ H1(a, b) : ui(a) = 0}, i∈ I1,Vi:=H1(a, b), i∈ I2, andV=2
i∈I1∪I2Vi, see [8]. In addition,
| ã |:=|| ã ||L2(a,b)and|| ã ||:=|| ã ||H1(a,b). If (Xi:i∈ I) is a sequence of given sets Xi, thenX|I1∪I2| denotes the product2
i∈I1∪I2Xi :=X1×X2×X3×X5×X6. Note that sometimesu(1) andu,y(1) replaceu(1, t) andu,y(1, t), respectively.
We re-formulate the modelPΓin terms of macroscopic quantities by perform- ing the transformation of all concentrations into volume-based concentrations via ˆ
ui := φφwu¯i, i∈ {1,3,4},uˆ2 :=φφau¯2,uˆi :=φ¯ui, i∈ {5,6}. We map (PΓ) onto a domain with fixed boundaries. To this effect, we employ the Landau transfor- mations (x, t) ∈ [0, s(t)]×S¯T −→ (y, τ) ∈ [a, b]×S¯T, y = s(t)x and τ = t, for i∈ I1, (x, t)∈[s(t), L]×S¯T −→(y, τ)∈[a, b]×S¯T,y=a+Lx−−s(t)s(t) andτ=t, for i∈ I2.We re-labelτ bytand introduce the new concentrations, which act in the auxiliaryy-tplane byui(y, t) := ˆui(x, t)−λi(t) for ally∈[a, b] andt∈ST. Thus,
the model equations are reduced to (ui+λi),t− 1
s2(t)(Diui,y),y=fi(u+λ) +ys(t)
s(t)ui,y, i∈ I1, (2.1) (ui+λi),t− 1
(L−s(t))2(Diui,y),y=fi(u+λ) + (2−y) s(t)
L−s(t)ui,y, i∈ I2, where uis the vector of concentrations (u1, u2, u3, u5, u6)t and λ := (λ1, λ2, λ3, λ5, λ6)t represents the boundary data. The transformed initial, boundary, and interface conditions are
ui(y,0) = 0, i∈ I1∪ I2, ui(a, t) = 0, i∈ I1, ui,y(b, t) = 0, i∈ I2, (2.2)
−D1
s(t)u1,y(1) =ηΓ(1, t) +s(t)(u1(1) +λ1),−D2
s(t)u2,y(1) =s(t)(u2(1) +λ2), (2.3)
−D3
L−s(t)u3,y(1) =ηΓ(1, t)−s(t)(u3(1) +λ3), (2.4) D5
s(t)u5,y(1)− D6
L−s(t)u6,y(1) =s(t)(u5(1) +λ5−u6(1)−λ6), (2.5) whereηΓ(1, t) denotes the reaction rate that acts in they-tplane. This is defined by ηΓ(1, t) := ¯ηΓ(¯u(ys(t), t) +λ(t),Λ), y∈[0,1], t∈ST (2.6) for a given Λ∈MΛ. Finally, two ode’s
s(t) =ηΓ(1, t) and ˆu4(t) =f4(ˆu(s(t), t)) a.e.t∈ST, (2.7) complete the model formulation, where ˆu:= (ˆu1,uˆ2,ˆu3,ˆu4,ˆu5,uˆ6)t. We also assume s(0) =s0>0,uˆ4(s(0),0) = ˆu40≥0. (2.8) The transformed model equations are collected in (2.1)–(2.8). Letϕ:= (ϕ1,ϕ2,ϕ3, ϕ5,ϕ6)t ∈ V be an arbitrary test function, and take t ∈ ST. To write the weak formulation of (2.1)–(2.8) in a compact form, we introduce the notation:
⎧⎪
⎪⎨
⎪⎪
⎩
a(s, u, ϕ) := s12
i∈I1(Diui,y, ϕi,y) +(L−1s)2
i∈I2(Diui,y, ϕi,y), bf(u, ϕ) :=
i∈I1∪I2(fi(u), ϕi), e(s, s, u, ϕ) := 1s
i∈I1gi(s, s, u(1))ϕi(1) + L1−s
i∈I2gi(s, s, u(1))ϕi(1), h(s, s, u,y, ϕ) := ss
i∈I1(yui,y, ϕi) +Ls−s
i∈I2((2−y)ui,y, ϕi),
(2.9) for any u ∈ V and λ ∈ W1,2(ST)|I1∪I2|. Furthermore, v4(t) := ˆu4(s(t), t) for t∈ST. The terma(ã) incorporates the diffusive part of the model,bf(ã) comprises volume productions,e(ã) sums up reaction terms acting on Γ(t), andh(ã) is a non- local term due to fixing of the domain. For our application (see (1.5) and (1.6)), the interface termsgi(i∈ I1∪ I2) are given by
g1(s, s, u) :=ηΓ(1, t) +s(t)u1(1), g2(s, s, u) :=s(t)u2(1),
g3(s, s, u) :=−ηΓ(1, t) +s(t)u3(1), g5(s, s, u) :=g6(s, s, u) = 0, (2.10)
322 A. Muntean and M. B¨ohm whereas the volume termsfi(i∈ I) are defined as
⎧⎨
⎩
f1(u) := P1(Q1u2−u1), f4(ˆu) := +˜ηΓ(s(t), t), f2(u) := −P2(Q2u2−u1), f5(u+λ) := +ηΓ(1, t), f3(u) := S3,diss(u3,eq−u3), f6(u) := 0.
(2.11)
SetMηΓ := supu(y,t)∈K{ηΓ(1, t) :y∈[a, b], t∈ST} K:=2
i∈I[0, ki] ,where
⎧⎨
⎩
ki := max{ui0(y) +λi(t), λi(t) :y∈[a, b], t∈S¯T}, i= 1,2,3, k4 := max{uˆ40(x) +MηΓT :x∈[0, s(t)], t∈S¯T},
kj := max{ui0(y) +λi(t) +MηΓT :y∈[a, b], t∈S¯T}, j= 5,6.
(2.12)
Definition 2.1. (Local Weak Solution) We call the triple (u, v4, s) a local weak solutionto (2.1)–(2.8) if there is aSδ:=]0, δ[ withδ∈]0, T] such that
v4∈W1,4(Sδ), s∈W1,4(Sδ), (2.13) u∈W21(Sδ;V,H)∩[ ¯Sδ→L∞(a, b)]|I1∪I2|∩L∞(Sδ;C0,12−([a, b]|I1∪I2|)), (2.14)
⎧⎪
⎪⎨
⎪⎪
⎩
(u(t), ϕ) +a(s, u, ϕ) +e(s, s, u, ϕ) =bf(u(t) +λ(t), ϕ) +h(s, s, u,y, ϕ)−(λ(t), ϕ) for allϕ∈V, a.e.t∈Sδ,
s(t) =ηΓ(1, t), v4(t) =f4(ˆu(s(t), t)) a.e.t∈Sδ, u(0) =u0∈H, s(0) =s0, v4(0) = ˆu40.
(2.15)
There is some freedom in choosing the exact structure of the reaction rate ηΓ. The only assumptions that are needed are the following:
(A) There exists a positive constantCη=Cη(Λ, u, λ, T) such that
¯
ηΓ(¯u(s(t), t),Λ)≤Cηu(s(t), t) for all¯ t∈ST.
(B) The reaction rateηΓ (defined cf. (2.6)) is locally Lipschitz with respect to all variables. More precisely, let (u(i), v4(i), s(i)) be two solutions corresponding to the sets of data Di := (u(i)0 , λ(i), . . . ,Λ(i))t, wherei ∈ {1,2}. Set ∆u :=u(2)−u(1),
∆v4:=v(2)4 −v4(1), ∆λ:=λ(2)2 −λ(1), ∆u0:=u(2)0 −u(1)0 , and ∆Λ := Λ(2)−Λ(1). The Lipschitz condition on ∆ηΓ := ∆¯ηΓ = ¯ηΓ(¯u(2),Λ(2))−η¯Γ(¯u(1),Λ(1)) reads:
There exists a positive constantcL=cL(D1,D2) such that the inequality|∆ηΓ| ≤ cL(|∆u|+|∆Λ|) holds locally pointwise. For a particular choice of ¯ηΓ, Λ, and hence cL, see (1.6).
(C1) k3≥max{|u3,eq(t)|: t∈S¯T}; (C2) P1Q1k2≤P1k1; P2k1≤P2Q2k2; (C3) Q2> Q1.
Theorem 2.2 (Local Existence and Uniqueness,[8]). Assume the hypotheses (A)–
(C2)and let the following conditions (2.16)–(2.20)be satisfied:
u3,eq∈L2(ST), λ∈W1,2(ST)|I1∪I2|, λ(t)≥0 a.e.t∈S¯T, (2.16) u0∈L∞(a, b)|I1∪I2|, u0(y) +λ(0)≥0a.e.y∈[a, b], (2.17)
ˆ
u40∈L∞(0, s0),ˆu4(x,0)≥0 a.e.x∈[0, s0], (2.18) min{min
S¯T
{u3,eq(t)}, S3,diss, P1, Q1, P2, Q2}>0, (2.19) 0< s0≤s(t)≤L0< Lfor allt∈S¯T. (2.20) Then the following assertions hold:
(a) There exists a δ ∈]0, T[ such that the problem (2.1)–(2.8) admits a unique local solution onSδ in the sense of Definition2.1;
(b) 0≤ui(y, t) +λi(t)≤ki a.e.y∈[a, b] (i∈ I1∪ I2) for allt∈Sδ. Moreover, 0≤uˆ4(x, t)≤k4 a.e.x∈[0, s(t)]for allt∈Sδ;
(c) v4, s∈W1,∞(Sδ).
Remark 2.3. Let (u(i), v4(i), s(i))(i ∈ {1,2}) be two local weak solutions on Sδ. Correspondingly, let (u(i)0 , λ(i),Λ(i)) be the initial, boundary and reaction data.
Then the functionH×W1,2(Sδ)|I1∪I2|×MΛ →W21(Sδ,V,H)×W1,4(Sδ)2 that maps (u0, λ,Λ)t into (u, v4, s)t is Lipschitz in the following sense: There exists a constantc=c(δ, s0,uˆ40, L, ki)>0 such that
||∆u||2W21(Sδ,V,H)∩L∞(Sδ,H)+||∆v4||2W1,4(Sδ)∩L∞(Sδ)+||∆s||2W1,4(Sδ)∩L∞(Sδ) (2.21)
≤c
||∆u0||2H∩L∞([a,b]|I1∪I2|)+||∆λ||2W1,2(Sδ)∩L∞(Sδ)+ max
MΛ |∆Λ|2
. Furthermore, we were able to show that small changes in the concentration fields induce small displacements of the position of the reaction front, for details see [8].
A straightforward consequence of this aspect is that the main output of the model (the penetration curveversustime, see Figure 2 (c), e.g.) is stable with respect to small perturbations in the reaction rate structure.
Proposition 2.4 (Strict Lower Bounds,[8]). Assume that the hypotheses of Theo- rem2.2 are satisfied. In addition, if 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 2.2(b) are strict for all times.
Theorem 2.2 and Remark 2.3 report on the well-posedness of (2.1)–(2.8) with respect to the time interval Sδ. In the sequel, we extend this local well- posed solution up to a global solution. Firstly, we assume that the hypotheses of Proposition 2.4 hold. In this case, for anarbitraryL0∈]s0, L[ there is aTfin<+∞ such that s(Tfin) = L0. Thus, Tfin is the time when Γ(t) has penetrated all of ]s0, L0[. We refer to it as the final carbonation time or shut-down time of the (carbonation) process. Physically reasonable restrictions on the life span of the weak solution (hence, onTfin) are given in Proposition 2.6 (iii). See also [10, 12, 13]
324 A. Muntean and M. B¨ohm
for some closely related scenarios. The next results are direct consequences of Theorem 2.2 and Proposition 2.4.
Proposition 2.5 (Strict Monotonicity of the Reaction Interface). If the hypotheses of Proposition 2.4 are satisfied, then the position s ∈ W1,∞(Sδ) of the interface Γ(t)is strictly monotonic increasing on Sδ.
Proposition 2.6 (Basic Estimates). Let (u, v4, s) be the unique local solution to (2.1)–(2.8) that fulfills the hypotheses of Proposition 2.4. Then the following esti- mates hold:
(i) ηmin< s(t)< ηmax for all t∈Sδ; (ii) s0≤s(t)≤s0+ηmaxtfor allt∈Sδ; (iii) Lη0−s0
max < Tfin< Lη0−s0
min ,
whereηmin andηmax denote uniform lower and upper bounds onη˜Γ.
Proof. By Theorem 2.2 (b) and Proposition 2.4, (i) and (ii) are straightforward.The equation fors in (2.7) leads to Tfin−t0=s(Tfin)
s(t0) 1
˜
ηΓ(s)ds(t0 ∈[0, Tfin[), see [2].
We apply the mean-value theorem and estimate the reaction rate ˜ηΓ from below by using the non-trivial uniform lower bounds on the reactants (i.e., onu1andu3), and afterwards from above, by means of the corresponding maximum estimates.
In this way, it yields (iii).
Theorem 2.7 (Global Solvability). Assume that the hypotheses of Proposition 2.4 are satisfied. Then the time intervalSTfin:=]0, Tfin[of (global) solvability of (2.1)–
(2.8) is finite and is characterized by
Tfin=s−1(L0). (2.22)
Proof. The finiteness of the length ofSTfinis a consequence of Proposition 2.6 (iii).
The uniform maximum estimates of concentrations together with the nonnegativ- ity of concentrations imply that the region2
i∈I[0, ki]×[s0, s0+TfinMηΓ], which confines the graph of (u, v4, s), remains invariant along the physically relevantin- terval ]0, Tfin[ of existence of the weak solution. The strictly positive constantMηΓ is defined by (2.12), while the value ofTfinobeys thea prioriestimate pointed out in Proposition 2.6 (iii). Note that the invariant region is independent ofu,s,x, or tand thatTfincan bea posterioricalculated via (2.22). By the strict monotonicity ofs(cf. Proposition 2.5) andW1,∞(STfin)→C( ¯STfin), we obtain (2.22).