A MOVING-BOUNDARY PROBLEM FOR CONCRETE
CARBONATION: GLOBAL EXISTENCE AND UNIQUENESS OF
WEAK SOLUTIONS∗
ADRIAN MUNTEAN† AND MICHAEL BÖHM‡
Abstract. This paper deals with a one-dimensional coupled system of semi-linear parabolic
equations with a kinetic condition on the moving boundary. The latter furnishes the driving force
for the moving boundary. The main results are a (global) existence- and uniqueness theorem, and
non-trivial lower and upper estimates for the velocity of the moving boundary.
The system under consideration is modelled on the so-called carbonation of concrete - a prototypical chemical-corrosion process in a porous solid – concrete – which incorporates slow diffusive
transport, interfacial exchange between wet and dry parts of the pores and, in particular, a fast reaction in thin layers, here idealized as as a moving-boundary surface in the solid. We include simulation
results showing that the model captures the qualitative behaviour of the carbonation process.
Key words. moving-boundary problem, reaction-diffusion equations, Stefan-like problem with
kinetic condition, a priori estimates, lower and upper bounds, concrete carbonation
AMS subject classifications. Primary, 35 R 35; Secondary, 74 F 25, 35 D 05
1. Introduction. We study a two phase moving-boundary system with kinetic
condition arising in the modeling of the concrete corrosion. Starting from the problem formulated in [30], we show existence, uniqueness and practical upper and lower
bounds for the solution to a coupled PDEs-ODE system. The moving boundary
represents in this framework the locus where a fast but non-instantaneous agressive
chemical reaction (called carbonation, see (1.1)) is localized. Due to the presence
of the moving boundary and of the various production terms by dissolution, precipitation and mass transfer at the water/air interfaces in the pores, the system is
strongly coupled, and hence, the derivation of a priori bounds of the solution becomes
non-trivial.
The physical process can be summarized as follows: 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 loss of protection is basically induced by the transformation of dissolved calcium
hydroxide (forming the protective pH ambient) into calcite. The loss of protection
means in this context that once the calcium hydroxide is reacted away, the concretebased material can be easily attacked by sulfate or chloride ions [9, 3, 13, 26], e.g.
On this way the reinforcement becomes subject of corrosion, and hence, spalling or
other unwanted effects can occur. The overall reaction-diffusion scenario is called
carbonation. The core reaction can be described, in a first approximation [40, 34, 26],
as
(1.1)
H O
2
CO2 (g → aq) + Ca(OH)2 (aq) −→
CaCO3 (aq) + H2 O.
∗ This work was partially supported by the German Science Foundation (DFG) via the grant
SPP 1122 entitled Prediction of the course of physicochemical damage processes involving mineral
materials. Financial support from the ESF grant SV 1577 is also acknowledged.
† Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics
and Computer Science, Technical University of Eindhoven, The Netherlands ([email protected]).
‡ Centre for Industrial Mathematics (ZeTeM), Department of Mathematics and Computer Science,
University of Bremen, Germany ([email protected]).
1
2
A. MUNTEAN AND M. BÖHM
The phenomenology of the process is apparently simple: Molecules of gaseous CO2
from the atmosphere penetrate the concrete via the unsaturated porous matrix. After
entering the air part of the pores, CO2 is transported through the gaseous phase and
is dissolved in the aqueous phase, where it is further transported towards the place
where reaction (1.1) takes place. The second reactant, i.e. Ca(OH)2 , is initially in
the solid matrix. It arrives in the aqueous phase of the pores through a relatively
strong dissolution process. Water and CaCO3 are the reaction products. CaCO3
precipitates instantaneously to the concrete fabrics and the water produced by (1.1)
steadily distributes within the pores.
Due to the density change produced when transforming Ca(OH)2 into CaCO3 ,
the impact of the carbonation process on concrete micro-structure is significant and
possible repairs are often expensive. Therefore there is need of models capable to
predict the depth of CO2 penetration in concrete structures accurately. More details
on this important durability issue can be found in [12, 13, 30] and references cited
therein. Experiments show that the zone of reaction is narrowly confined to the
interface between the unreacted solid and the product layer, i.e. the region where
calcium carbonate precipitates to the solid matrix. In Fig. 1.1, such a macroscopic
sharp reaction interface separating the carbonated region from the uncarbonated one
is pointed out. Our aim is to understand the way this type of reaction interface
Fig. 1.1. Slice of partly carbonated concrete piece sprayed with phenolphthalein. Two distinct
zones can be distinguished where the two reactants live. Courtesy of Prof. M. Setzer and Dr. U.
Dahme, University of Essen-Duisburg, Germany.
penetrate the material. A few relevant questions, which need to be addressed, are:
• Why is the moving-boundary modeling strategy applicable to carbonation?
• How can one define the interface position?
• How fast does the interface move into the material?
The reader can find in [28, 32, 30] some of our answers. In this paper, we focus on
the unidimensional motion of the interface. Therefore, transport and reaction near
corners or around macroscopic fissures, which are typically occurring in porous media
(see [10, 11], e.g.), can not be described here. Despite this geometrical restriction,
the problem is much more general than we state it in the context of carbonation. A
wealth of other reaction diffusion scenarios arising in geochemistry ([33], e.g.), polymer
industry ([1, 41], e.g.) or life sciences ([16], e.g.) may be tackled by conceptually close
moving-boundary modeling strategies. The modeling, analysis and simulation of alike
non-equilibrium scenarios in two and three space dimensions are sources rich in open
problems.
The paper is organized as follows. In section 2 we describe the moving-boundary
model that we propose to model the penetration of the carbonation interface in concrete. In section 3, we introduce some notation and function spaces in order to prepare
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
3
a functional framework where the problem can be tackled. This is also the place where
we present our weak formulation and state the main results. The bulk of the proofs is
given in section 4. The aim of section 5 is to illustrate numerically a simple carbonation scenario, which is modeled as mentioned in section 2. We conclude the paper with
section 6, where we shortly evaluate the moving-boundary model from both analysis
and modeling points of view.
The results of this paper have been announced in [32]. They constitute a part of
the results from the PhD thesis [30] of the first author.
2. Moving-sharp interface carbonation model. We consider the carbonation penetration in a wall made of concrete whose chemistry, humidity level, and
micro-structure are well known [40]. Let the positive x-axis be directed normally to
the reaction interface, say Γ(t), pointing into the uncarbonated part. The basic geometry is sketched in Fig. 2.1. At initial time t = 0, we assume that the origin located at
x = 0 is behind the reaction interface Γ(t). Assuming that the reactants, whose mass
concentration only depends on the real variables x and t, are separated but available
for reaction, we expect that the reaction interface moves as x = s(t) for t ∈ ST :=]0, T [
such that s(0) = s0 . Here T ∈]0, +∞], s0 ∈]0, L[, and L ∈]0, +∞[ are given, see Fig.
2.1 (center). Note that the case s0 = 0 is excluded for two reasons: First, it describes a
different process, namely the surface initialisation of carbonation coupled with carbonation in the interior, which leads to different models [15]. Secondly, it complicates the
mathematical analysis. Among others one would have to take care of the degeneracies
induced by the Landau transformation (3.1). We refer to [17, 18] to somehow related
subjects involving conditions like s0 = 0. We denote the mass concentration of the reactants and products as follows: ū1 := [CO2 (aq)], ū2 := [CO2 (g)], ū4 := [CaCO3 (aq)]
and ū5 := [H2 O] are the chemical species present in the region Ω1 (t) := [0, s(t)[;
ū3 := [Ca(OH)2 (aq)] and ū6 := [H2 O] are species present in Ω2 (t) :=]s(t), L]. For
Fig. 2.1. Left: Basic geometry for the moving sharp-interface model. The box A is the region
which our model refers to. Center: Schematic 1D geometry. The reactants are spatially segregated
at any time t. Right: Definition of the interface position.
ease of notation, we use the set of indices I := I1 ∪ {4} ∪ I2 , where I1 := {1, 2, 5}
points out the active concentrations in Ω1 (t) and I2 := {3, 6} refers to the active
concentrations living in Ω2 (t). Specifically, we take into account that CaCO3 (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 the moving-boundary problem of
determining the concentrations ūi (x, t), i ∈ I and the interface position s(t) which
4
A. MUNTEAN AND M. BÖHM
satisfy for all t ∈ ST the equations

(φφw ūi ),t + (−Di νi2 φφw ūi,x )x




 (φφw ū3 ),t + (−D3 φφw ū3,x )x
(φφw ū4 ),t
(2.1)


(φū5 ),t + (−D5 φū5,x )x



(φū6 ),t + (−D6 φū6,x )x
= fi,Henry , x ∈ Ω1 (t), i ∈ {1, 2},
= fDiss , x ∈ Ω2 (t),
= fP rec + fReacΓ , x = s(t) ∈ Γ(t),
= 0, x ∈ Ω1 (t),
= 0, x ∈ Ω2 (t).
The initial and boundary conditions are φφw νi2 ūi (x, 0) = ûi0 (x), i ∈ I, x ∈ Ω(0),
φφw νi2 ūi (0, t) = λi (t), i ∈ I1 , ūi,x (L, t) = 0, i ∈ I2 , where t ∈ ST . Specific to our
problem, we impose the following interface conditions

−η̃Γ (s(t), t) + s0 (t)[φφw ū1 ]Γ(t) ,
 [j1 · n]Γ(t) =
[ji · n]Γ(t) = η̃Γ (s(t), t)δ5i + s0 (t)[φφw νi2 ūi ]Γ(t) , i ∈ {2, 5, 6},
(2.2)

[j3 · n]Γ(t) =
−η̃Γ (s(t), t) + s0 (t)[φφw ū3 ]Γ(t) ,
(2.3)
s0 (t) = α
η̃Γ (s(t), t)
=: ψ̃Γ (s(t), t), s(0) = s0 ,
φφw ū3 (s(t), t)
where ν12 = ν32 := 1, ν22 := φφwa , ν52 = ν62 := φ1w , νi` := 1 (i ∈ I, ` ∈ I − {2}),
δij (i, j ∈ I) is Kronecker’s symbol, ji := −Di νi` φφw ūi (i, ` ∈ I1 ∪ I2 ) are the
corresponding effective 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 a concrete sample (cf. Fig. 2.1 (b)) - is exposed, see [37, 13]. An
argument for the boundary conditions (2.2) is based on the so-called pillbox lemma (see
[24]). The initial conditions ûi0 > 0 are determined by the chemistry of the cement.
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 and
φa > 0. Since the active concentrations are small, the constant-porosity assumption
([7, 40]) is valid. The productions terms fi,Henry , fDiss , fP rec and fReacΓ are sources
or sinks by Henry-like interfacial transfer mechanisms, dissolution, precipitation, and
carbonation reactions. Typical examples are:
fi,Henry := (−1)i Pi (φφw ū1 − Qi φφa ū2 )(Pi > 0, Qi > 0), i ∈ {1, 2},
(2.4)
fDiss := −S3,diss (φφw ū3 − u3,eq ), S3,diss > 0, fP rec := 0, fReacΓ := η̃Γ .
In (2.4), η̃Γ (s(t), t) denotes the carbonation reaction rate. It is defined in the following
fashion: Let ū = (ū1 , . . . , ū6 )t be the vector of concentrations and MΛ the set of
parameters Λ := (Λ1 , . . . , Λm )t chosen to describe the reaction rate. For our purposes,
it suffices at this moment to assume that MΛ is a non-empty compact subset of Rm
+.
We introduce the function
(2.5) η̄Γ : R6 × MΛ → R+ by η̄Γ (ū(x, t), Λ) := kφφw ūp1 (x, t))ūq3 (x, t), x = s(t).
In (2.5), m := 3 and Λ := {p, q, kφφw } ∈ R3+ . We define the reaction rate η̃Γ (s(t), t)
and the term ψ̃Γ (s(t), t) in (2.3) by
(2.6)
η̃Γ (s(t), t) := η̄Γ (ū(s(t), t), Λ), ψ̃Γ (s(t), t) := ψ̄Γ (ū(s(t), t), Λ),
where η̄Γ is given by (2.5) and represents the classical power-law ansatz [22]. In the
engineering literature, there is a whole variety of reaction rates used in the context
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
5
of carbonation ([15, 34, 36]); (2.5), with p > 0 and q > 0, is the ansatz most widely
used. Note that some mass-balance equations act in Ω1 (t), while other act in Ω2 (t)
or at Γ(t). All of the three space regions are varying in time and they are a priori
unknown.
The system (2.1)-(2.6) forms the sharp-interface carbonation model. We abbreviate it as (PΓ ). The model consists of a coupled semi-linear system of parabolic
equations that has a moving a priori unknown internal boundary Γ(t), where the carbonation reaction is assumed to take place. The coupling between the equations and
the non-linearities 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).
2.1. Remarks around (2.2) and (2.3). The interface conditions require further explanation. The term η̃Γ (s(t), t) ≈ α1 s0 (t) denotes the number of grams per
volume and time that is transported by diffusion to the interface Γ(t). In (2.2),
±φφw ū(s(t), t)s0 (t) accounts for the mass flux induced by the motion of Γ(t) in order to preserve the conservation of mass. The conditions (2.2) 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.
[5], section 1.2.E, e.g.
The law (2.3), which we call kinetic or non-equilibrium condition, governs the
dynamics of the reaction interface. (2.3) is exact for the 1D case and has been derived
via first principles in [30]. We need the kinetic condition to complete the model formulation. We rely on (2.3) to determine the position of the interface once the reactants
concentration at Γ(t) is known. Kinetic laws show in many situations a regularizing
effect by ensuring the global (in time) existence of the solutions. Nevertheless, if they
are posed inappropriately, then they can bring about a blow up in concentration (see
[23, 31], e.g.) or in the speed s0 (t) of the interface, and hence, all regularizing effects
are lost. Further examples of moving-boundary problems with kinetic conditions are
treated, for instance, in [16, 41, 43].
The present setting is only applicable when the reaction rate is very rapid and
the diffusion of the gaseous CO2 is sufficiently 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. The quotient of the characteristic times may
cause 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. Notice that when dealing with reaction-diffusion scenarios one typically imposes
the continuity of concentrations accross interfaces. A special feature brought in by
(2.1)-(2.6) is that concentration fields are not obliged to be continuous everywhere.
They may have finite jumps at Γ(t). At the macroscopic level it is not a priori clear
what actually happens at Γ(t) with the reactants. For our case, complete reaction at
Γ(t) would immediately imply the case of infinitely fast chemical reaction in which
the reactants practically vanish at Γ(t) (see [14], e.g.) We refer the reader to [35, 38]
for concrete reaction-diffusion scenarios, where discontinuities in concentrations at
moving boundaries arise.
3. Main results.
3.1. Preliminaries. For each i ∈ I1 ∪ I2 , we denote Hi := L2 (a,
Q b) and set
[a, b] := [0, 1] for i ∈ I1 and [a, b] := [1, 2] for i ∈ I2 . Moreover, H := i∈I1 ∪I2 Hi ,
Q
Vi = {u ∈ H 1 (a, b) : ui (a) = 0}, i ∈ I1 , Vi := H 1 (a, b), i ∈ I2 , and V = i∈I1 ∪I2 Vi .
In addition, |·| := ||·||L2 (a,b) and ||·|| := ||·||H 1 (a,b) . If (Xi : i ∈ I) is a sequence of given
6
A. MUNTEAN AND M. BÖHM
Q
sets Xi , then X |I1 ∪I2 | denotes the product i∈I1 ∪I2 Xi := X1 × X2 × X3 × X5 × X6 .
Details on the Sobolev and Lp -spaces, which appear in the paper but are not listed
here, can be found in [44].
Note that sometimes u(1) and u,y (1) replace u(1, t) and u,y (1, t), respectively.
We reformulate the system (2.1)-(2.6): Let ûi := φφw ūi , i ∈ {1, 3, 4}, û2 :=
φφa ū2 , ûi := φūi , i ∈ {5, 6} and write down (PΓ ) on fixed domains. As result of
this procedure, we obtain the transformed model (3.3)-(3.13).
Let t ∈ ST be arbitrarily fixed. In our setting, the fixed-domain transformations
[27] read:
x
for i ∈ I1 ,
s(t)
x − s(t)
(3.2) (x, t) ∈ [s(t), L] × S̄T −
7 → (y, t) ∈ [a, b] × S̄T , y = a +
for i ∈ I2 .
L − s(t)
(3.1) (x, t) ∈ [0, s(t)] × S̄T 7−→ (y, t) ∈ [a, b] × S̄T , y =
We introduce the notation ui (y, t) := ûi (x, t) − λi (t) for all y ∈ [a, b] and t ∈ ST . The
model equations become
s0 (t)
ui,y , i ∈ I1 ,
s(t)
1
s0 (t)
(3.4)(ui + λi ),t −
(Di ui,y ),y = fi (u + λ) + (2 − y)
ui,y , i ∈ I2 ,
2
(L − s(t))
L − s(t)
(3.3)
(ui + λi ),t −
1
s2 (t)
(Di ui,y ),y = fi (u + λ) + y
where u is the vector of concentrations (u1 , u2 , u3 , u5 , u6 )t and λ represents the boundary data (λ1 ,λ2 ,λ3 ,λ5 ,λ6 )t . We make use of λ3 and λ6 only for notational simplicity
(λ3 := λ6 := 0). The vectors of concentrations u0 and λ are assumed to be compatible,
i.e.
(3.5)
u0i (0) = λi (0), and hence ûi (0) = 0 for i ∈ I1
The transformed initial, boundary and interface conditions are
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
ui (y, 0) = ui0 (y), i ∈ I1 ∪ I2 , ui (a, t) = 0, i ∈ I1 , ui,y (b, t) = 0, i ∈ I2 ,
−D1
u1,y (1) = ηΓ (1, t) + s0 (t)(u1 (1) + λ1 ),
s(t)
−D2
u2,y (1) = s0 (t)(u2 (1) + λ2 ),
s(t)
−D3
u3,y (1) = −ηΓ (1, t) + s0 (t)(u3 (1) + λ3 ),
L − s(t)
−D5
D6
u5,y (1) +
u6,y (1) = ηΓ (1, t), u5 (1) + λ5 = u6 (1) + λ6 ,
s(t)
L − s(t)
where ηΓ (1, t) denotes the reaction rate that acts in the y-t plane. This is defined by
(3.11)
ηΓ (1, t) := η̄Γ (ū(ys(t), t) + λ(t), Λ), y ∈ [0, 1],
for given Λ ∈ MΛ and η̄Γ as in (2.5). ψΓ (1, t) is defined analogously. We also mention
that ui0 (y) = ûi0 (x) − λi (0), where x = ys0 , y ∈ [0, 1] for i ∈ I1 , and x = s0 + (y −
1)(L − s0 ), y ∈ [1, 2] for i ∈ I2 . Finally, two ordinary differential equations
(3.12)
s0 (t) = ψΓ (1, t) and v40 (t) = f4 (v4 (t)) a.e. t ∈ ST ,
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
7
where v4 (t) := û4 (s(t), t) for t ∈ ST , complete the model formulation. Furthermore,
we take
(3.13)
s(0) = s0 , v4 (0) = û40 .
Let ϕ := (ϕ1 ,ϕ2 ,ϕ3 ,ϕ5 ,ϕ6 )t ∈ V be an arbitrary test function and take t ∈ ST .
To write down the weak formulation of (3.3)-(3.13) in a compact form, we introduce
the notation:
P
P

1
a(s, u, ϕ)
:= 1s i∈I1 (Di ui,y , ϕi,y ) + L−s

i∈I2 (Di ui,y , ϕi,y ),

P
P

bf (u, s, ϕ)
:= sP i∈I1 (fi (u), ϕi ) + (L − s) i∈I2 (fi (u), ϕi ),
(3.14)
0
e(s0 , u, ϕ)
:=

i (1),
i∈I1 ∪I2 gi (s, s , u(1))ϕ

P
P

0
0
0
h(s , u,y , ϕ) := s
(yu
,
ϕ
)
+
s
i,y
i
i∈I1
i∈I2 ((2 − y)ui,y , ϕi ),
for any u ∈ V and λ ∈ W 1,2 (ST )|I1 ∪I2 | . The term a(·) incorporates the diffusive part
of the model, bf (·) comprises volume productions, e(·) sums up reaction terms acting
on Γ(t) and h(·) is a non-local term due to fixing of the domain. For our application,
the interface terms gi (i ∈ I1 ∪ I2 ) are given by

 g1 (s, s0 , u) := ηΓ (1, t) + s0 (t)u1 (1), g2 (s, s0 , u) := s0 (t)u2 (1),
g3 (s, s0 , u) := ηΓ (1, t) − s0 (t)u3 (1), g5 (s, s0 , u) := ηΓ (1, t),
(3.15)

g6 (s, s0 , u) := 0,
whereas the volume terms

 f1 (u) :=
f2 (u) :=
(3.16)

f3 (u) :=
fi (i ∈ I) are defined as
P1 (Q1 u2 − u1 ),
f4 (û) := +η̃Γ (s(t), t),
−P2 (Q2 u2 − u1 ),
f5 (u) := 0,
S3,diss (u3,eq − u3 ), f6 (u) := 0.
We assume that the initial and boundary data as well as the model parameters satisfy
the restrictions:
(3.17)
(3.18)
λ ∈ W 1,2 (ST )|I1 ∪I2 | , λ(t) ≥ 0 a.e. t ∈ S̄T ,
u3,eq ∈ L∞ (ST ), u3,eq (t) ≥ 0 a.e. t ∈ S̄T ,
(3.19)
(3.20)
(3.21)
(3.22)
u0 ∈ L∞ (a, b)|I1 ∪I2 | , u0 (y) + λ(0) ≥ 0 a.e. y ∈ [a, b],
û40 ∈ L∞ (0, s0 ), û4 (x, 0) > 0 a.e. x ∈ [0, s0 ],
s0 > 0, L0 < L < +∞, s0 < L0 ,
min{S3,diss , P1 , Q1 , P2 , Q2 , D` (` ∈ I1 ∪ I2 )} > 0.
We denote
(3.23)
m0 := min{s0 , L − L0 },
M0 := max{L0 , L − s0 }.
Set
K :=
(3.24)
Y
[0, ki ],
i∈I1 ∪I2
and, for fixed Λ ∈ MΛ , we take
(3.25)
MηΓ := max{η̄Γ (ū, Λ)}.
ū∈K
8
A. MUNTEAN AND M. BÖHM
In (3.24) we set

ki := max{ui0 (y) + λi (t), λi (t) : y ∈ [a, b], t ∈ S̄T }, i = 1, 2, 3, 6,



k4 := max{û40 (x) + MηΓ T : x ∈ [0, s(t)], t ∈ S̄T },
(3.26)
k5 := max{u50 (y) + λ5 (t), λ6 (t), κ : y ∈ [a, b], t ∈ S̄T },



k6 := k5 ,
where
(3.27)
κ :=
L0
D5 − MηΓ LL0
MηΓ +
L
|λ5,t |∞ + 1 .
2
Definition 3.1. (Local Weak Solution) We call the triple (u, v4 , s) a local weak
solution to problem (3.3)-(3.13) if there is a δ ∈]0, T ] with Sδ :=]0, δ[ such that
(3.28)
(3.29)
(3.30)
s0 < s(δ) ≤ L0 ,
v4 ∈ W 1,4 (Sδ ),
s ∈ W 1,4 (Sδ ),
u ∈ W21 (Sδ ; V, H) ∩ [S̄δ 7→ L∞ (a, b)]|I1 ∪I2 | ,
For all ϕ ∈ V and a.e. t ∈ Sδ we have
 P
P
s i∈I1 (ui,t (t), ϕi ) + (L − s) i∈I2 (ui,t (t), ϕi ) + a(s, u, ϕ)



0

+e(s
ϕ) + h(s0 , u,y , ϕ)

P , u + λ, ϕ) = bf (u + λ, s, P
−s i∈I1 (λi,t (t), ϕi ) − (L − s) i∈I2 (λi,t (t), ϕi ),
(3.31)


s0 (t) = ηΓ (1, t), v40 (t) = f4 (v4 (t)) a.e. t ∈ Sδ ,



u(0) = u0 ∈ H, s(0) = s0 , v4 (0) = û40 .
3.2. Hypotheses on the model parameters. The only assumptions that are
needed are the following:
(A) Fix Λ ∈ MΛ . Let η̄Γ (ū, Λ) > 0, if ū1 > 0 and ū3 > 0, and η̄Γ (ū, Λ) = 0, otherwise.
For any fixed ū1 ∈ R, η̄Γ is bounded.
(B) The reaction rate η̄Γ : R6 × MΛ → R+ is locally Lipschitz. This restricts the
choice of p and q in (2.5).
(C1) 1 > k3 ≥ maxS̄T {|u3,eq (t)| : t ∈ S̄T }; D5 − MηΓ L > 0;
(C2) P1 Q1 k2 ≤ P1 k1 ; P2 k1 ≤ P2 Q2 k2 ;
(C3) Q2 > Q1 .
(A)-(C) can be interpreted in the following way: (A) means that the reaction takes
place if both CO2 (aq) and Ca(OH)2 (aq) are present. The last part of (A) prevents
the unreacted region to vanish completely. (B) is mainly needed from mathematical
reasons (it simply helps proving the local existence of weak solutions). On the other
hand, (B) represents a quite natural assumption if one finally wants a PDE model
whose solution depends continuously on data and parameters. (C1) is needed to
establish the L∞ -estimates on u3 and u5 . It says that the equilibrium concentration
of Ca(OH)2 (aq) is uniformly bounded by 1 and the diffusion of moisture should be
sufficiently strong to spread away the water produced by reaction (1.1). (C2) and
(C3) are rather technical. They both suggests that the transfer of CO2 from the air
phase into the pore water is fast. (C2) is used to get the L∞ -estimates on u1 and
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
9
u2 , while (C3) prevents the vanishing of these two concentrations at the interface
position.
By (A) and (B), we deduce that ηΓ (0, Λ) = 0 for all Λ ∈ MΛ . For all ū ∈ R6
there is an -neighbourhood U (ū) and a positive constant Cη = Cη (Λ, λ, , Tfin ) such
that the inequality
(3.32)
η̄Γ (ū(s(t), t), Λ) ≤ Cη |ū(s(t), t)|
holds for all t ∈ ST . (3.32) can be reformulated as
(3.33)
ηΓ (1, t) ≤ Cη |u(1, t)| for all t ∈ ST .
Note also that there exists a function cg = cg (Cη ) such that
(3.34)
|e(s0 , u(1), ϕ(1))| ≤ cg |u(1)||ϕ(1)|
for all ϕ ∈ V
and a constant cf = cf (Cη , K1 ) > 0 such that
(3.35)
|bf (u, s, ϕ)| ≤ cf |u3,eq |2∞ + |u|2 + |ϕ|2
for all ϕ ∈ V,
where K1 > 0 is a constant depending on the material parameters entering fi (i ∈ I),
i.e. P1 , P2 , Q1 , Q2 , and S3,diss . The exact structure of cg , cf and K1 is dictated by
the definition of the production terms fi and gi (i ∈ I), see (3.16) and (3.15). Since
ψΓ (1, t) has essentially the same structure as ηΓ (1, t), it also satisfies (A) and (B).
3.3. Local solvability. We have the following results.
Theorem 3.2. Assume the hypotheses (A)-(C2) and let the conditions (3.17)(3.22) be satisfied. If s ∈ W 1,4 (Sδ ) with s0 ≥ 0 a.e. in Sδ and s(0) = s0 is given, then
the problem (3.3)-(3.13) admits a unique weak solution in the sense of Definition 3.1
(formulated for given s).
Proof. Although this problem is non-linear, it is however standard. The existence and uniqueness of the weak solutions can be shown as for the model problems
presented in [42, 44], and therefore we omit the proof.
Theorem 3.3 (Local Existence and Uniqueness). Assume the hypotheses (A)(C2) and let the conditions (3.17)-(3.22) be satisfied. Then the following assertions
hold:
(a) There exists a δ ∈]0, T [ such that the problem (3.3)-(3.13) admits a unique local
solution on Sδ in the sense of Definition 3.1;
(b) 0 ≤ ui (y, t) + λi (t) ≤ ki a.e. y ∈ [a, b] (i ∈ I1 ∪ I2 ) for all t ∈ Sδ . Moreover,
0 ≤ û4 (x, t) ≤ k4 a.e. x ∈ [0, s(t)] for all t ∈ Sδ ;
(c) v4 , s ∈ W 1,∞ (Sδ ).
It is worth mentioning that if the assumptions of Theorem 3.3 hold, then we can
additionally prove the estimate u5 (y, t) + λ5 (t) ≤ k5 y for a.e. y ∈ [0, 1] and all t ∈ Sδ ;
see Remark 3.4.7 of [30].
Proposition 3.4 (Strict Lower Bounds). Assume that the hypotheses of Theorem 3.3 are satisfied. If, additionally, (C3) holds and the initial and boundary data
are strictly positive, then there exists a range of reasonable parameters1 such that the
positivity estimates stated in Theorem 3.3 (b) are strict for all times.
The main physical motivation why such range of parameters ensuring the existence
of strict lower bounds (see Proposition 3.4) may be found is that the carbonation
1 See Lemma 4.5 and Lemma 4.4 in section 4. A list of concrete reasonable parameters for which
Proposition 3.4 holds, is given in [30] (Remark 3.4.31).
10
A. MUNTEAN AND M. BÖHM
process can be viewed as a one-stage non-catalytic gas-solid reaction. By Theorem
3.3 (b) and Proposition 3.4, there exist constants ψmin , ψmax ∈ R+ such that
(3.36)
0 < ψmin ≤ ψΓ (1, t) ≤ ψmax < ∞
for all t ∈ Sδ . By (2.5), ψmax is independent of δ and we may take ψmax := MηΓ (see
(3.25)). ψmin typically depends on δ.
3.4. Global solvability. We say that problem (3.3)-(3.13) is globally solvable,
if for each L0 ∈]s0 , L[ there is a solution on ]s0 , L0 [ in the sense of Definition 3.1. The
case L0 = L is excluded for similar reasons as s0 = 0; see the corresponding note in
section 2.
In order to obtain the global solvability of our problem, we start with assuming
that the hypotheses of Proposition 3.4 hold. In this case, for an arbitrarily fixed
L0 ∈]s0 , L[ there is a moment Tfin = Tfin (L0 ) ∈]0, ∞[ such that
(3.37)
s(Tfin ) = L0 .
Thus Tfin denotes 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 process. Physically reasonable
restrictions on the life span of the weak solution (hence, on Tfin ) are given in Proposition 3.6 (iii). The next three results are direct consequences of Theorem 3.3 and
Proposition 3.4.
Proposition 3.5 (Strict Monotonicity of the Reaction Interface). If the assumptions of Proposition 3.4 are satisfied, then the position s ∈ W 1,∞ (Sδ ) of the interface
Γ(t) is strictly monotone increasing on Sδ .
Proof. The conclusion is straightforward if one combines the definition of s0 and
the strict positivity of concentrations.
Proposition 3.6 (Practical Estimates). Let (u, v4 , s) be the unique local solution
to (3.3)-(3.13) that fulfills the hypotheses of Proposition 3.4. Then the following
estimates hold:
(i) ψmin < s0 (t) < ψmax for all t ∈ Sδ ;
(ii) s0 ≤ s(t) ≤ s0 + ψmax t for all t ∈ Sδ ;
−s0
−s0
(iii) Lψ0max
< Tfin < Lψ0min
, where Tfin satisfies (3.37);
(iv) If Ω1 (t) = {x ∈]0, L[: û4 (x, t) > 0}, t ∈ S̄Tfin , then Ω1 (t1 ) ⊂ Ω1 (t2 ) for
t1 < t2 , t1 , t2 ∈ S̄Tfin .
Here ψmin and ψmax are as in (3.36).
Proof. By Theorem 3.3 (b) and Proposition 3.4, (i) and (ii) are straightforward.
R s(T ) 1
The equation for s0 in (3.12) leads to Tfin − t0 = s(t0fin
ds with t0 ∈ [0, Tfin [,
)
ψ̃ (s)
Γ
see [8]. We apply the mean-value theorem and estimate ψ̃Γ from below by using the
non-trivial uniform lower bounds on the reactants (i.e. on u1 and u3 ), and afterwards
from above, by means of the corresponding L∞ -estimates and obtain (iii). By (3.12)
and (i), one gets (iv).
The statements of Proposition 3.6 have a clear practical meaning: the finite propagation speed is established in (i); (ii) points out a linear asymptotic behavior of the
speed of the reaction interface with respect to time. This is only a rough estimate.
Considering the investigations reported in [4, 6, 19, 39], we expect that better asymptotic estimates could be obtained. Inequalities in (iii) estimate Tfin from above and
below. This is the time that reaction (1.1) needs to fully carbonate [s0 , L0 ]. The
upper bound is the most helpful indicator from the practical viewpoint. (iv) shows
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
11
that the carbonated zone progresses into the material; for somewhat similar situations
see [25] (Corollary 2.5, pp. 284-285), [20] (Theorem 9.1, pp. 84-87) or [21] (Corollary
2.4), e.g.
Theorem 3.7 (Global Solvability). Let L0 ∈]s0 , L[. Assume that the hypotheses of Proposition 3.4 are satisfied. Then the time interval STfin :=]0, Tfin [ of global
solvability of problem (3.3)-(3.13) is finite and is characterized by
(3.38)
Tfin = s−1 (L0 ).
Proof. The finiteness of the length of STfin is a consequence of Proposition 3.6 (iii).
The L∞ -estimates of the concentrations together with their non-negativity imply that
Y
(u(y, t), v4 (t), s(t)) ∈
[0, ki ] × [s0 , s0 + Tfin MηΓ ],
i∈I
for all y ∈ [a, b] and t ≥ 0. The strictly positive constant MηΓ is given by (3.26),
while the value of Tfin obeys the a priori estimate pointed out in Proposition 3.6 (iii).
Note that the invariant region is independent of u, s, x or t and that Tfin can be a
posteriori calculated via (3.38). By the strict monotonicity of s (cf. Proposition 3.5)
and W 1,∞ (STfin ) ,→ C(S̄Tfin ), we obtain (3.38).
3.5. Return to the physical domain. Using the inverse Landau transformation corresponding to (3.1) and (3.2), we map the solution back to the physical
domain. We have
Proposition 3.8 (Change of Coordinates). Assume that the hypotheses of
Theorem 3.3 are satisfied. Additionally, if u3,eq ∈ W 1,2 (Sδ ), û0 ∈ H 1 (0, s0 )|I1 | ×
H 1 (s0 , L)|I2 | , û0 and λ satisfy the compatibility conditions (3.5), then the solution
(û, v4 , s) acts in the physical x-t plane and
"
#
Y
1
(3.39)
(û, v4 , s) ∈
H (STfin , Ĥi (t)) × W 1,4 (STfin )2 ,
i∈I1 ∪I2
where Ĥi (t) := L2 (0, s(t)) for i ∈ I1 and Ĥi (t) := L2 (s(t), L) for i ∈ I2 .
Proof. We employ the inverse Landau transformations. The rest of the proof
relies on a lifting regularity argument and change of variables in the Bochner integral;
see the proof of Proposition 3.4.17 in [30] for details.
4. Proofs of the main results. We prove the results stated in 3.3 and 3.4.
This section is organized in the following manner: We obtain a series of positivity,
L∞ - and energy estimates in 4.1. Section 4.2 contains the proof of the local existence
of solutions, while in section 4.3 we indicate a way to ensure the strict positivity of
the active concentrations.
4.1. Basic Estimates. The first lemma contains several inequalities which are
needed for following the estimates. It may be proved by essentially standard methods.
Lemma 4.1 (Some Basic Estimates). Let cξ > 0, ξ > 0, θ ∈ [ 21 , 1[ and s ∈
W 1,1 (Sδ ).
(i) There exists the constant ĉ = ĉ(θ) > 0 such that
(4.1)
|ui |∞ ≤ ĉ|ui |1−θ ||ui ||θ
12
A. MUNTEAN AND M. BÖHM
for all ui ∈ Vi , where i ∈ I1 ∪ I2 .
(ii) It holds
|ui |1−θ ||ui ||θ ≤ ξ||ui || + cξ |ui |
(4.2)
for all ui ∈ Vi , where i ∈ I1 ∪ I2 .
(iii) Let ϕ ∈ V with ϕ = (ϕ1 , . . . , ϕ6 )t , t ∈ Sδ , ĉ as in (i), and ξ, cξ as in (ii). Then
we have for i ∈ I1 and j ∈ I2 the following inequalities:
1 |s0 (t)|
1 |s0 (t)| 2
|s0 (t)|
(yϕi,y , ϕi ) =
{ϕi (1)2 − |ϕi |2 } ≤
{ĉ |ϕi |2(1−θ) ||ϕi ||2θ − |ϕi |2 };
s(t)
2 s(t)
2 s(t)
2θ−1
2
1
|s0 (t)|
|s0 (t)|
ξ
|ϕi (1)|2 ≤
|ϕi |2∞ ≤ 2 ||ϕi ||2 + cξ ĉ 1−θ × s(t) 1−θ |s0 (t)| 1−θ |ϕi |2 ;
s(t)
s(t)
s (t)
2θ
|ϕi (1)|2
1
≤ 2 |ϕi |2∞ ≤ ĉ2 s(t)2θ−2 |ϕi |2(1−θ) s(t)−1 ||ϕi ||
2
s (t)
s (t)
2(θ−1)
2
ξ
≤ 2 ||ϕi ||2 + cξ ĉ 1−θ |s(t)| 1−θ |ϕi |2 ;
s (t)
2θ−1
2
ξ
|ϕi (1)|2
≤ 2 ||ϕi ||2 + cξ ĉ 1−θ |s(t)| 1−θ |ϕi |2 ;
s(t)
s (t)
|s0 (t)|
1 |s0 (t)|
1 |s0 (t)|
((2 − y)ϕj,y , ϕj ) =
|ϕj (1)|2 +
|ϕj |2 .
L − s(t)
2 L − s(t)
2 L − s(t)
Proof. (i) The case θ ∈] 21 , 1[ follows from H θ (a, b) ,→ C([a, b]) and from an
interpolation inequality (see Theorem 5.9 in [2], e.g.). The case θ = 12 is discussed
in [44] (Example 21.62, p. 285), e.g. To get (ii) we use Young’s inequality. Young’s
inequality and the integration by parts are the necessary tools to prove (iii). For
instance, the fact that for each i ∈ I2 we have ((2 − y)ϕi,y , ϕi ) = ϕi (1)2 − (ϕi , (2 −
y)ϕi,y ) + |ϕi |2 shows the last statement in (iii).
The special choice θ = 21 can be further used to simplify the estimates. By this
2θ−1
2θ−1
choice, the sum s(t) 1−θ + (L − s(t)) 1−θ becomes 2. Note also that for any ξ > 0,
there exists a constant cξ > 0 such that
(4.3)
|ϕi (z)|2 ≤ ξ||ϕi ||2 + cξ |ϕi |2
for any ϕi ∈ Vi (i ∈ I1 ∪ I2 ) and z ∈ [a, b]. (4.3) is a straightforward consequence
of Lemma 4.1 (i). (4.3) can also be proved without using the interpolation inequality
(as in (i)) by adapting Lemma 1 of [41] or Lemma 1 of [16] to our setting.
Let K1 be the following positive constant
2
2
3ĉ MηΓ
K1 := 1 + P2 Q2 + max{P1 Q1 , P2 Q2 } + cξ
(4.4)
+ cξ ĉ4 D32 .
2
The constant K1 is dependent on the material parameters explicitly shown in (4.4),
but is independent of the solution and of the length of the time interval.
Theorem 4.2 (Positivity and L∞ -Estimates). Let the triple (u, v4 , s) as in
Definition 3.1 satisfy the assumptions (A)-(C2). Then the following statements hold:
(i) (Positivity) u(t) + λ(t) ≥ 0 in V for all t ∈ Sδ .
(ii) (L∞ -estimates) Let ` ∈ I1 ∪ I2 be arbitrarily fixed. There exists a constant
k` > 0 (see (3.26)) such that u` (t) + λ` (t) ≤ k` in V` (` ∈ I − {4, 5}) for all
t ∈ Sδ . In addition, there exists a constant k5 > 0 such that u5 (t) ≤ k5 y a.e.
y ∈ [0, 1] and all t ∈ Sδ .
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
13
(iii) (Localization of the interface)
s0 ≤ s(t) ≤ s0 + δMηΓ for all t ∈ Sδ , where MηΓ is given in (3.26).
(iv) (Positivity and boundedness of v4 at Γ(t))
0 < û40 ≤ v4 (t) ≤ û40 + δMηΓ for all t ∈ Sδ .
Proof. The key idea of dealing with (i) and (ii) is to choose appropriate test
functions ϕ ∈ V in the weak formulation (3.31). We prove (i) and (ii) simultaneously
by following the next steps:
(Step 1) Get L∞ -estimates on u1 , u2 and u3 (thus Cη becomes independent of
u1 );
(Step 2) Show the positivity of u1 , u2 and u3 ;
(Step 3) Show the positivity of u5 and u6 ;
(Step 4) Get L∞ -estimates on u5 and u6 .
We adopt this strategy because of the presence of the term −ηΓ (1)ϕ1 (1)−ηΓ (1)ϕ3 (1)+
ηΓ (1)ϕ5 (1) at the r.h.s. of the weak formulation, while ηΓ has the properties stated
in (2.5), (A) and (B). Let ki (i ∈ I) be as in (3.26). We show that these values are
L∞ -estimates that we are looking for.
Step 1: We start by looking for L∞ -estimates of u1 and u2 . We put in (3.31) the
test function ϕ ∈ ((u1 + λ1 − k1 )+ , (u2 + λ2 − k2 )+ , 0, 0, 0)t ∈ V. We therefore have
2
1X
s d
|ϕ|2 +
(Di ϕi,y , ϕi,y ) = − [ηΓ + s0 (u1 (1) + λ1 )] ϕ1 (1)
2 dt
s i=1
− s0 (u2 (1) + λ2 )ϕ2 (1) + sP1 (Q1 ϕ2 − ϕ1 + Q1 k2 − k1 , ϕ1 )
(4.5)
− sP2 (Q2 ϕ2 − ϕ1 + Q2 k2 − k1 , ϕ2 ) + s0
2
X
(yϕi,y , ϕi ).
i=1
Divide (4.5) by s and integrate by parts
s0
s
P2
i=1 (yϕi,y , ϕi ).
By (C2), we obtain:
2
1 X
1 d
s0
(Di ϕi,y , ϕi,y ) + |ϕ(1)|2
|ϕ|2 + 2
2 dt
s i=1
s
−
ηΓ
s0
s0
s0
ϕ1 (1) − (k1 ϕ1 (1) + k2 ϕ2 (1)) + max{P1 Q1 , P2 Q2 }|ϕ|2 + |ϕ(1)|2 − |ϕ|2 .
s
s
s
s
Cancelling the term
the inequality
(4.6)
s0
2
s |ϕ(1)|
and using the positivity of ϕ1 , ϕ2 and ηΓ , we are led to
2
2
X
1 d|ϕ|2
1 X p
+ 2
|| Di ϕi ||2 ≤ K1
|ϕi |2 .
2 dt
s i=1
i=1
Applying Gronwall’s inequality we obtain ui (t) + λi (t) ≤ ki (i ∈ {1, 2}) for all t ∈ Sδ .
To complete Step 1, we still need to estimate u3 from above. For this purpose, let
us choose ϕ = (0, 0, (u3 + λ3 − k3 )+ , 0, 0)t ∈ V, i.e. ϕ3 := (u3 + λ3 − k3 )+ ∈ V3 . By
means of this test function, we obtain
1
D3
D3 (u3,y , ϕ3,y ) +
|ϕ3 (1)|
L−s
L−s
D3
=
|ϕ3 (1)| − [ηΓ − s0 (u3 (1) + λ3 )] ϕ3 (t)
L−s
+ (L − s)S3,diss (u3,eq − (u3 + λ3 ), ϕ3 ) + s0 ((2 − y)u3,y , ϕ3 ).
(L − s)((u3 + λ3 ),t , ϕ3 ) +
(4.7)
14
A. MUNTEAN AND M. BÖHM
Dividing by L − s and integrating afterwards by parts the expression
y)ϕ3,y , ϕ3 ), we have
(4.8)
s0
2(L−s) ((2
−
1 d|ϕ3 |2
1
D3
D3 ||ϕ3 ||2 =
|ϕ3 (1)|2
+
2
2 dt
(L − s)
(L − s)2
ηΓ
s0
−
ϕ3 (1) +
(ϕ3 (1) + k3 )ϕ3 (1)
L−s
L−s
s0 |ϕ3 (1)|2 + |ϕ3 |2
.
+ S3,diss (u3,eq − k3 − ϕ3 , ϕ3 ) +
2(L − s)
By (C1), we obtain
D3
3s0
1 d|ϕ3 |2
1
s0
2
2
D
||ϕ
||
≤
+
+
|ϕ
(1)|
+
|ϕ3 |2
3
3
3
2 dt
(L − s)2
(L − s)2
2(L − s)
2
" #
1
1−θ
||ϕ3 ||2
D3
3s0
s0
2
2θ
≤ξ
+ cξ ĉ
+
(L − s)
|ϕ3 |2 .
+
(L − s)2
(L − s)2
2(L − s)
2(L − s)
Via Gronwall’s inequality we have that u3 + λ3 ≤ k3 for all t ∈ Sδ , provided that
s0 ∈ L∞ (Sδ ), i.e. for given u1 the reaction rate η̃Γ stays bounded.
Until now we have shown the boundedness of u1 and u2 without requiring the
boundedness of ηΓ . The boundedness of u3 needs that of s0 , and hence, ηΓ has to be
bounded for fixed u1 . The positivity of ηΓ is necessary in each Step.
Step 2: We continue with proving the positivity property of u1 and u2 . Setting
ϕ := (−(u1 + λ1 )− , −(u2 + λ2 )− , 0, 0, 0)t ∈ V, i.e. ϕi := −(ui + λi )− ∈ Vi (i ∈ {1, 2}),
we get:
2
s d|ϕ|2
1X
+
||Di ϕi ||2 = −ηΓ ϕ1 (1) + s0 |ϕ(1)|2
2 dt
s i=1
+ sP1 (Q1 (u1 + λ2 ), ϕ1 ) − sP1 |ϕ1 |2 − sP2 Q2 (u2 + λ2 , ϕ2 )
(4.9)
+ sP2 (u1 + λ1 , ϕ2 ) + s0
2
X
(yϕi,y , ϕi ).
i=1
Noting that ±η̃Γ (u1 (1) + λ1 )− = 0 and dividing the expression (4.9) by s, we obtain
2
1 d|ϕ|2
1 X
s0
+ 2
||Di ϕi ||2 ≤ |ϕ(1)|2 − P1 |ϕ|2 + P2 Q2 |ϕ2 |2
2 dt
s i=1
s
2
+ min{P1 Q1 , P2 Q2 }|ϕ|2 +
(4.10)
≤
s0 X
|ϕi (1)|2 − |ϕi |2
2s i=1
3s0
||ϕ||2
|ϕ(1)|2 + K1 |ϕ|2 ≤ ξ 2 + 2K1 |ϕ|2 .
2s
s
Let ξ ∈]0, min{D1 , D2 }]. Gronwall’s inequality shows the positivity property. Finally,
we focus on showing the positivity of u3 . We obtain this property as follows: With
ϕ := (0, 0, −(u3 + λ3 )− , 0, 0)t ) ∈ V or ϕ3 := −(u3 + λ3 )− ∈ V3 , we are led to
p
L−s d
1
|ϕ3 |2 +
|| D3 ϕ3 ||2 = −ηΓ ϕ3 (1) − s0 |ϕ3 (1)|2
2 dt
L−s
s0
D3
(4.11)
|ϕ(1)|2 + (L − s)S3,diss (u3,eq , ϕ3 ) + ϕ23 + (|ϕ3 (1)|2 + |ϕ3 |2 ).
+
L−s
2
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
15
0
Note that −ηΓ ϕ3 (1)− s2 |ϕ3 (1)|2 ≤ 0 and (L−s)S3,diss (u3,eq , ϕ3 ≤ 0. Now, we employ
again the interpolation and Young inequalities in (4.11) to obtain
||ϕ3 ||2
s0
1 d
D3
2
2
||ϕ3 || ≤ ξ
+ K1 +
(4.12)
|ϕ3 | +
|ϕ3 |2 .
2 dt
(L − s)2
(L − s)2
2
By ξ ∈]0, D3 ( and the use of Gronwall’s inequality, we complete this step.
Step 3: We recall here the arguments from Step 2. Since the proof is similar, we do
not repeat it. Our choice of test function concerning this case is ϕ = (0, 0, 0, −(u5 +
λ5 )− , −(u6 − λ6 )− )t ∈ V. No additional restrictions on the model parameters are
needed.
Step 4: As test function we put ϕ := (0, 0, 0, (u5 − k5 y)+ , (u6 − k6 )+ )t ∈ V. By
ϕ5 (1) = ϕ6 (1), Cauchy-Schwarz’ inequality, and the embeddings H 1 (Sδ ) ,→ C(S̄δ ),
1
H 1 (Sδ ) ,→ H 2 (Sδ ) and H 1 (Sδ ) ,→ L∞ (Sδ ) imply the following estimates:
(4.13)
s d|ϕ5 |2
L − s d|ϕ6 |2
D5
+
+
||ϕ5 ||2
2 dt
2
dt
s
k5 D5
D6
s
+
||ϕ5 || +
||ϕ6 ||2 ≤ MηΓ |ϕ5 |∞ − (λ5,t , ϕ5 )
s
L−s
2
s0
s0
s0
s0
2
2
2
2
+ |ϕ5 (1)| − |ϕ5 | + |ϕ6 (1)| + |ϕ6 |
2
2
2
2
s
0
+ k5 s L|ϕ5 |∞ + |λ5,t ||ϕ5 |∞
2
||ϕ5 ||2
L
≤ (MηΓ + |λ5,t | + k5 MηΓ L)||ϕ5 || + ξ
2
s2
0
1
s
+ K1 (s0 s2θ ) 1−θ (|ϕ5 |2 + |ϕ6 |2 ).
+
2
Choose ξ ∈]0, s0 k5 D5 ] and recall the definition of k5 (cf. (3.26)), then
(4.14)
MηΓ +
L
k5 D5
|λ5,t |∞ + k5 MηΓ L ≤
2
L0
is fulfilled. Application of Gronwall’s inequality shows that u5 (t) ≤ k5 y and u6 (t) ≤ k6
for all t ∈ Sδ a.e. y ∈ [a, b].
Steps 1-4 complete the proof of (i) and (ii). The definitions of s0 and v40 together
with the statement (ii) prove (iii) and (iv).
Condition (4.14) is sufficient to prevent a possible blow up in the concentration
u5 . It basically says that, in order to avoid a blow up situation, the water produced
at Γ(t) should diffuse sufficiently quickly away from the interface.
Lemma 4.3 (Energy Estimates). Assume that (A)-(C2) hold and let the triple
(u, v4 , s) be as in Definition 3.1. The following statements hold a.e. in Sδ :
Z t
(4.15)
|u(t) + λ(t)|2 ≤ α(t) exp
β(τ )dτ ;
0
(4.16)
|u(t) + λ(t)|2 ≤ α(t) +
Z
t
||u(τ ) + λ(τ )|| dτ ≤
(4.17)
0
ds;
s
t
2
β(τ )dτ
0
Z
t
Z
β(s)α(s) exp
d−1
0 α(t) exp
Z
t
β(τ )dτ
t0
,
16
A. MUNTEAN AND M. BÖHM
where
(4.18)
s0 Di
(L − L0 )Di
d0 := min min 2 , min
, m0 as in (3.23).
i∈I1 L m0 i∈I2 (L − s0 )2 m0
The factors a(t), α(t) and β(t) are given by
(4.19)
(4.20)
(4.21)
(L − s(t))2 K2
(s0 (t))2
+
,
2
2
Z t
2
α(t) := |ϕ(0)|2 +
a(τ )dτ,
m0 0
"
2 #
s0 (t)
D3
s0 (t)
1
β(t) :=
+ K2 2 +
+
,
2
L − s(t)
2
m0
a(t) :=
whereas
2
K2 := 1 + (S3,diss |u3,eq |∞ ) +
(4.22)
LP1 Q1
+ cξ ĉ4 .
2
Furthermore, we have
u ∈ L2 (Sδ , V), u,t ∈ L2 (Sδ , V∗ ), u ∈ C(S̄δ , H).
(4.23)
Proof. Inserting the test function ϕ := u + λ ∈ V in the variational formulation
(3.31), we find the energy estimates (4.16) and (4.17) in the following way: By
s X d
L−s X d
|ϕi (t)|2 +
|ϕi (t)|2 + a(s, ϕ, ϕ)
2
dt
2
dt
i∈I1
i∈I2
+e(s0 , ϕ, ϕ) = bf (ϕ, s, ϕ) + h(s0 , ϕ,y , ϕ) for t ∈ Sδ ,
(4.24)
it yields
m0 d
s0 X
L − L0 X
|ϕ(t)|2 + 2
Di ||ϕi ||2 +
Di ||ϕi ||2
2 dt
s
(L − s)2
i∈I1
i∈I2
≤ −ηΓ ϕ1 (1) − s0 (|ϕ1 (1)|2 + |ϕ2 (1)|2 ) − ηΓ ϕ3 (1) + s0 |ϕ3 (1)|2
D3
+ ηΓ |ϕ5 (1)| +
|ϕ3 (1)|2
L−s
+ sP1 Q1 (ϕ2 , ϕ1 ) − P1 |ϕ1 |2 − sP2 Q2 |ϕ2 |2 + sP2 (ϕ1 , ϕ2 )
+ (L − s)S3,diss |u3,eq |∞ |ϕ3 | − (L − s)S3,diss |ϕ3 |2
2 s0 X
2
s0 X
+
|ϕi (1)|2 − |ϕi |2 +
|ϕi (1)|2 + |ϕi |2
2
2
i∈I1
i∈I2
≤ s0 |ϕ3 (1)|2 + s0 |ϕ5 (1)| +
D3
LP1 Q1
|ϕ3 (1)|2 +
|ϕ1 |2 + |ϕ2 |2
L−s
2
LP2
|ϕ1 |2 + |ϕ2 |2 + (L − s)S3,diss |u3,eq |∞ |ϕ3 |
2
s0 X
s0 X
+
|ϕi (1)|2 +
|ϕi |2
2
2
+
i∈I1 ∪I2
i∈I2
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
17
2
(s0 )2
[(L − s)S3,diss |u3,eq |∞ ]
D3
≤
+
+ 1+
2
2
L−s
X
X
LP1 Q1
s0
s0
2
|ϕi (1)| +
|ϕi |2 .
+
+
2
2
2
(4.25)
i∈I1 ∪I2
i∈I1 ∪I2
We employ the interpolation inequality to bound the term
X
D3
s0
1+
|ϕi (1)|2
+
L−s
2
i∈I1 ∪I2
from above by
X ||ϕi || 2θ
D3
s0
ĉ2
|ϕi |2(1−θ)
+
s2θ
L−s
2
s
i∈I1
2θ
0
X
D3
s
||ϕi ||
2θ
2
+ 1+
+
(L − s)
|ϕi |2(1−θ)
ĉ
L−s
2
L−s
1+
i∈I2
(4.26)
+cξ
X ||ϕi ||2
X ||ϕi ||2
+
ξ
≤ξ
s2
(L − s)2
i∈I2
i∈I1
1
i
X 2θ
2θ
D3
s0 1−θ h 1−θ
1+
+
s
+ (L − s) 1−θ |ϕi |2 .
L−s
2
i∈I1 ∪I2
Set θ = 12 . We then obtain
m0 d
s0 X
L − L0 X
|ϕ(t)|2 + 2
Di ||ϕi ||2 +
Di ||ϕi ||2
2 dt
s
(L − s)2
i∈I1
(4.27)
i∈I2
X ||ϕi ||2
X ||ϕi ||2
≤ξ
+ξ
+ a(t) + β(t)|ϕ(t)|2 .
2
s
(L − s)2
i∈I1
i∈I2
Set ξ ∈ ]0, mini∈I1 ∪I2 {s0 Di , (L − L0 )Di }[. We obtain via Gronwall’s inequality the
estimate (4.15), where the factors a(t), α(t) and β(t) are given as in (4.19)-(4.21).
Gronwall’s inequality yields (4.16). Choose d0 as in (4.18) and note also that d0 > 0.
The proof of (4.17) relies on (4.15) and integration by parts. To get (4.23), we have
to discuss the Bochner measurability and integrability of u and u,t in the same way as
in Lemma 26.1 in [42], pp.395-396 and Remark 27.1 in [42]), p.405. If u ∈ L2 (Sδ , V)
and u,t ∈ L2 (Sδ , V∗ ), then u ∈ W12 (Sδ ; V, H) (see [44] for this space). The embedding
W12 (Sδ ; V, H) ,→ C(S̄δ ; H) shows the second part of (4.23).
4.2. Proof of Theorem 3.3. The proof relies on the use of Banach’s fixed-point
principle.
Proof. The set
(4.28)
M (Sδ ) := {r ∈ W 1,4 (Sδ ) : r(t) ∈ [s0 , MηΓ δ + s0 ]
for all t ∈ Sδ , r0 (t) ≥ 0 for a.e. t ∈ Sδ , |r0 |L4 (Sδ ) ≤ δMηΓ }
equipped with the metric
ρ(r1 , r2 ) = |r20 − r10 |L4 (Sδ ) for all r1 , r2 ∈ M (Sδ )
18
A. MUNTEAN AND M. BÖHM
is a non-empty closed subset of W 1,4 (Sδ ). (M (Sδ ), ρ) forms a complete metric space.
We show that the map
(4.29)
T : M (Sδ ) → W 1,4 (Sδ ),
(4.30)
T : s 7−→ (u, v4 ) cf. ((3.31), (3.12)) (see also Theorem 3.2)
7−→ r with r̃0 (t) = ψΓ (1, t), r(0) = r0
is a strictly contractive self-map, provided
0 < δ ≤ δ0 ( see (4.58) for the choice of δ0 ).
(4.31)
By Theorem 3.2, T does indeed map M (Sδ ) into W 1,4 (Sδ ). Because of r(t) = s0 +
Rt
ψ (1, τ )dτ , yields r0 ∈ L4 (Sδ ). By r ∈ W 1,∞ (Sδ ) ,→ W 1,p (Sδ ) for all 1 ≤ p ≤ ∞,
0 Γ
it results that r ∈ W 1,4 (Sδ ). By (3.12) and the positivity of concentrations, we get
that r0 ≥ 0 a.e. in Sδ . Furthermore, the L∞ -estimates on concentrations ensure that
|r0 |L4 (Sδ ) ≤ MηΓ δ.
We show: T is strictly contractive. To this end, let si ∈ M (Sδ ), i = 1, 2, and
ri = T (si ), where
T : si 7−→ (ui , v4i ) 7−→ ri , i = 1, 2.
Set w := w2 − w1 , where wi := (wi1 , wi2 , wi3 , wi5 , wi6 )t , λi := (λi1 , λi2 , λi3 , λi5 , λi6 )t ,
wij := uij − λij (i ∈ {1, 2}, j ∈ I1 ∪ I2 ). Furthermore, we set ∆ψΓ (t) := ψΓ (s2 (t), t) −
ψΓ (s1 (t), t), ∆η̃Γ (t) := η̃Γ (s2 (t), t) − η̃Γ (s1 (t), t), ∆s(t) := s2 (t) − s1 (t), ∆s0 (t) :=
s02 (t) − s01 (t), ∆r(t) := r2 (t) − r1 (t), ∆r0 (t) := r20 (t) − r10 (t) and ∆v4 = v41 − v42 .
The key idea of the proof relies on the fact that T improves integrability, i.e.
T : W 1,2 (Sδ ) → W 1,4 (Sδ ).
By (A) and (3.12), we note that
Z
Z
Z
0
4
4
|∆r (τ )| dτ =
|∆ψ̃Γ (τ )| dτ ≤ Cη
|w(1, τ )|4 dτ.
Sδ
Sδ
Sδ
Applying the interpolation inequality2 (4.1) with θ = 21 , we get
Z
Z
(4.32)
|∆r0 (τ )|4 dτ ≤ Cη ĉ4 sup |w(t)|2
||w(τ )||2 dτ.
Sδ
t∈S̄δ
Sδ
R
We want to bound the quantities supt∈Sδ |w(t)|2 and Sδ ||w(τ )||2 dτ from above. To
this end, we subtract the variational formulation (3.31) written for w2 from that one
written for w1 , where in both expressions we use the test function w = w2 − w1 ∈ V.
It yields
s2
X1 d
X1 d
1 X p
|wi (t)|2 + (L − s2 ))
|wi (t)|2 +
|| Di wi ||2
2 dt
2 dt
s2
i∈I2
i∈I1
i∈I1
X p
1
+
|| Di wi ||2
(L − s2 )
i∈I2
2 The
same argument shows that ∆v4 belongs to W 1,4 (Sδ ).
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
≤ −∆s
X
(w1i,t , wi ) + ∆s
X
19
(w1i,t , wi )
i∈I2
i∈I1
1 X p
| Di wi (1)|2
s2
+
i∈I1
X p
1
∆s X
+
| Di wi (1)|2 +
Di (w1i,y , wi,y )
L − s2
s1 s2
i∈I2
i∈I1
X
∆s
−
Di (w1i,y , wi,y )
(L − s1 )(L − s2 )
i∈I2
+ e(s01 , w1 , w) − e(s02 , w2 , w) + bf (w2 , s2 , w) − bf (w1 , s1 , w)
+ h(s02 , w2,y , w) − h(s01 , w1,y , w) ≤
(4.33)
5
X
J` ,
`=1
where we are given
J1 := −∆s
X
(w1i,t , wi ) + ∆s
i∈I1
J2 :=
X
(w1i,t , wi ),
i∈I2
X
∆s
∆s X
(Di w1i,y , wi,y ) −
(Di w1i,y , wi,y ),
s1 s2
(L − s1 )(L − s2 )
i∈I1
i∈I2
J3 := bf (w2 , s2 , w) − bf (w1 , s1 , w),
J4 := e(s01 , w1 , w) − e(s02 , w2 , w) +
1 X p
| Di wi (1)|2
s2
i∈I1
X p
1
+
| Di wi (1)|2 ,
L − s2
i∈I2
J5 := h(s02 , w2,y , w) − h(s01 , w1,y , w).
To estimate J` (` ∈ {1, . . . , 5}), we repeatedly use Lemma 4.1 and Theorem 4.2 together with the application of Cauchy-Schwarz’s and Young’s inequalities. To shorten
further the notation, we employ the positive constant
K3 := 1 + cξ
X
Di +
i∈I1 ∪I2
(4.34)
P1 Q1 + P2
+ k32 + S3,diss |u3,eq |2∞
2
+ (P2 k1 )2 + (P1 Q1 k2 )2 + 2cξ c̄2 L2 ĉ4 + cξ (1 + k̄ 2 ) + cξ cξ̄ c2 ĉ2 ,
P
0 +s0
where c̄ = c̄(Cη , k1 , k2 , k3 ) ≥ 3Cη + 3MηΓ + k1 + k12 + k32 + sL−L
i∈I1 ∪I2 Di . It
0 (L−L0 )
is worth noticing that K3 is bounded above, invariant with respect to t and depends
on the choice of L, L0 , s0 , cξ , cξ̄ , ĉ, θ, ki and Di (i ∈ I1 ∪ I2 ).
P5
We proceed with finding out an upper margin of `=1 |J` |. The bound of |J1 | is
|J1 | ≤
(4.35)
|∆s|2
|w|2
|w1,t |2 +
.
2
2
By Cauchy-Schwarz and the arithmetic-geometric means inequalities, we obtain
|J2 | ≤
X
|∆s| X
|∆s|
Di ||w1i ||||wi || +
Di ||w1i ||||wi ||
s1 s2
(L − s1 )(L − s2 )
i∈I1
i∈I2
20
A. MUNTEAN AND M. BÖHM
X Di ||w1i || 2
X ||wi ||2
X ||wi ||2
≤ξ
+ cξ
+
ξ
s22
(L − s2 )2
i∈I1
i∈I2
i∈I1
s21
+
X Di ||w1i || 2
i∈I2
!
(L − s1 )2
X ||wi ||2
X ||wi ||2
1
1
≤ξ
+
ξ
+
K
+
||w1 ||2 |∆s|2
3
s22
(L − s2 )2
s21
(L − s1 )2
i∈I1
i∈I2
|∆s|2
P1 Q1 + P2
2
2
|J3 | ≤
+ (P1 Q1 k2 ) + (P2 k1 ) +
(|w1 |2 + |w2 |2 )
2
2
|∆s|2
|∆s|2
3
2
+ S3,diss
+ k32 |w3 |2 ≤ |∆s|2 + K3 |w|2 .
(4.36)
+
|u3,eq |2∞ |w3 |2 +
2
2
2
To bound |J4 |, we use (A) and the special structure of the boundary term e(s0 , wi , w).
We obtain
|J4 | ≤ | ∆η̃Γ w1 (1) + s02 w1 (1)2 + ∆s0 k1 w1 (1) + s02 w1 (1)2 + ∆s0 k1 w2 (1)
+ ∆η̃Γ w3 (1) − s02 w3 (1)2 + ∆s0 k3 w3 (1) − ∆ηΓ w5 (1) (−1)|
X
1
1
+
+
Di |wi (1)|2
s0
L − L0
i∈I1 ∪I2
(4.37)
X ||wi ||2
X ||wi ||2
3
+ K3 |w|2 .
+ξ
≤ |∆s0 |2 + ξ
2
2
s2
(L − s2 )2
i∈I1
i∈I2
We split the term J5 into two components J51 and J52 such that J5 = J51 +J52 , where
J51 := s2
s02 X
s0 X
(yw2i,y , wi ) − s1 1
(yw1i,y , wi )
s2
s1
i∈I1
J52
i∈I1
s02 X
s01 X
:= (L − s2 )
((2 − y)w2i,y , wi ) − (L − s1 )
((2 − y)w1i,y , wi ).
L − s2
L − s1
i∈I2
i∈I2
Rearranging the last two expressions, we obtain
X
1
|∆s0 |
s01
s0 X
|J51 | ≤
+
|∆s|
|(yw2i,y , wi )| + 1
|(ywi,y , wi )|,
L
s2
s1 s2
s1
i∈I1
i∈I1
X
1
|∆s0 |
s01
|J52 | ≤
+
|∆s|
|((2 − y)w2i,y , wi )|
L
L − s2
(L − s1 )(L − s2 )
i∈I2
(4.38)
s01 X
+
|((2 − y)wi,y , wi )|.
L − s1
i∈I2
Fix i ∈ I1 arbitrarily. Let us now estimate each of the terms J5k (k ∈ {1, 2}). We
have
Z 1
Z 1
(yw2i,y , wi ) = w2i (1)wi (1) −
w2i wi dy −
yw2i wi,y dy.
0
0
There exists a constant c = c(k̄) > 0 such that
(4.39)
|(yw2i,y , wi )| ≤ c|wi (1)| + |w2i ||wi | + |w2i |||wi ||.
|∆s|2
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
21
(4.39), (4.1) together with the embedding L∞ (a, b) ,→ L2 (a, b) and the arithmeticgeometric means inequality provide upper bounds for terms like
|∆s0 |
s0
|(yw2i,y , wi )| and 1 |∆s||(yw2i,y , wi )|
s2
s1 s2
as follows:
|∆s0 |
|wi |2
||wi ||θ
|∆s0 |2
|(yw2i,y , wi )| ≤ cĉ|∆s0 | θ |wi |1−θ sθ−1
+
+ k̄ 2 2
2
s2
2
s2
s2
+ cξ k̄ 2
|∆s0 |2
||wi ||2
+ξ 2
2
s2
2
s01
|∆s|2
s01
||wi ||θ
s01
1−θ
+
|∆s||(yw2i,y , wi )| ≤ cĉ
|∆s| θ |wi |
+ k̄
|wi |2
s1 s2
2
s1 s2
s2
s1 s1−θ
2
+ξ
||wi ||2
+ cξ
s22
s01
s1
2
k̄ 2 |∆s|2 .
By Lemma 4.1 (in which we set θ = 12 ), we gain the estimate
0 2
s01
||wi ||2
s1
|(ywi,y , wi )| ≤ ξ 2 + K3
s22 |wi |2 .
s1
s2
s1
Collecting the last three inequalities for all i ∈ I1 , we obtain
"
#
0 2 !
X
1
1
s1
1
||wi ||2
2
0 2
|J51 | ≤
|∆s|
+ K3
+ K3 + 3ξ 2
+ |∆s |
L
2
s1
2
s2
i∈I1
0 2 ! X
s1
1
+
s22
+ K3
|wi |2
2
s2
s1
i∈I1
θ
θ
X cĉs0
||wi ||
1
1−θ
0 ||wi ||
1−θ θ−1
|∆s|
(4.40)
|w
|
+
cĉ|∆s
|
|w
|
s
|wi |2 .
+
i
i
2
1−θ
θ
θ
s
s
s
s
1 2
2
2
i∈I
1
To deal with the last two terms in (4.40), we use the following strategy:
s01
||wi ||θ
2
¯
|∆s|
|wi |1−θ ≤ ξ|∆s|
+
1−θ
θ
s
s
s
1
2
2
i∈I1
1
1−θ
X ||wi ||2
X
1
s01
1−θ
+
c
c
(cĉ)
|wi |2 .
+ξcξ̄
ξ
ξ̄
1−θ
s22
s
s
1
2
i∈I1
i∈I1
X
(4.41)
cĉ
The last term is estimated in the same manner. We obtain
0 2 !
s1
1
2
|J51 | ≤ |∆s| 2 + 2ξ¯ + 4K3
+ |∆s0 |2 2 + 2ξ¯ + 4K3 +
L
s1
X ||wi ||2
+
s22
i∈I1
" 0 2 # X
1
1
s1
1
2
+ K3
1+
+
s2 +
|wi |2 .
s2
s2
s1
s2
+ ξ(3 + 2cξ̄ )
i∈I1
22
A. MUNTEAN AND M. BÖHM
The bound on |J52 | follows similarly. This reads
1
|J52 | ≤ (1 + ξ¯ + 2K3 )|∆s0 |2 +
L
+ ξ(3 + 2cξ̄ )
X
i∈I2
1 + ξ¯ + 2K3
s01
L − s1
2 !
|∆s|2 +
||wi ||2
+
(L − s2 )2
"
1
(s01 )2
1
+ K3
+
+
+
L − s2
(L − s2 )2
(L − s1 )(L − s2 )
X
1
2
×
+ (L − s2 )
|wi |2 .
L − s2
s01
L − s1
2
×
i∈I2
Finally, it yields
(4.42)
"
0 2
2 !
0
s
s
1
1
1
|J5 | ≤ 3 + 3ξ¯ + 4K3
+ 2K3
|∆s|2
L
s1
L − s1
+ 3 + 3ξ¯ + 6K3 |∆s0 |2
!
X ||wi ||2
X ||wi ||2
+ ξ(3 + 2cξ̄ )
+
+ K3 χ1 (t)|w|2 ,
s22
(L − s2 )2
i∈I1
i∈I2
where the expression of χ1 (t) is given by
0 2 s1 (t)
1
1
1
1
1
2
+
+ 2
+
+
s
(t)
+
2
s2 (t) L − s2 (t) s2 (t) (L − s2 (t))2
s1 (t)
s2 (t)
2
0
0
2
s1 (t)
1
(s1 (t))
+
(L − s2 (t))2 +
+
,
L − s1 (t)
L − s2 (t)
(L − s1 (t))(L − s2 (t))
χ1 (t) :=
for a.e. t ∈ Sδ . Summing up the bounds on |Ji |, it yields
|J| ≤ a(t)|∆s|2 + b(t)|∆s0 |2 + c(t)|w|2
(4.43)
¯
+d(ξ, ξ)
X ||wi ||2
X ||wi ||2
+
2
s2
(L − s2 )2
i∈I1
!
,
i∈I2
where the factors a(t), b(t) and c(t) (with t ∈ Sδ ) are defined by
|w1,t |2
3
1
1
a(t) :=
+ + K3
+
||w1 (t)||2
2
2
s21 (t) (L − s1 (t))2
"
0 2
2 #
0
s
(t)
s
(t)
1
1
+ L 3 + 3ξ¯ + 4K3
+ 2K3
,
s1 (t)
L − s1 (t)
3
+ L(3 + 3ξ¯ + 6K3 ),
2
1
c(t) := + 2K3 + LK3 χ1 (t),
2
b(t) :=
¯ := 2ξ + Lξ(3 + 2c ), where c = c (ξ).
¯ Clearly, for any choice of ξ > 0 and
and d(ξ, ξ)
ξ̄
ξ̄
ξ̄
¯
¯ <β <
ξ > 0 there exist two real constants αξξ̄ and βξξ̄ such that 0 < αξξ̄ < d(ξ, ξ)
ξ ξ̄
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
23
∞. On the other hand, there exists a constant C = C(L, L0 , s0 , K3 , MηΓ ) ∈ R∗+ such
that
c(t) ≤ C a.e. in Sδ .
(4.44)
Insert (4.43) in (4.33). Choosing conveniently sufficiently small ξ > 0 and ξ¯ > 0, we
are led to
Z t
Z t
|w(t)|2 + d0
||w(τ )||2 dτ ≤ |w(0)|2 +
a(τ )|∆s(τ )|2 + b(τ )|∆s0 (τ )|2 dτ
0
0
Z t
2
(4.45)
+
c(τ )|w(τ )| dτ,
0
where the strictly positive constant d0 is given by
(4.46)
d0 := min{min
i∈I1
¯
¯
s0 Di − d(ξ, ξ)
(L − L0 )Di − d(ξ, ξ)
, min
}
2
2
i∈I2
L
(L − s0 )
We denote
N2 (t)
(with C as in (4.44)),
C
N2 (t) := c(t)N3 (t),
Z t
N3 (t) := |w(0)|2 +
a(τ )|∆s(τ )|2 + b(τ )|∆s0 (τ )|2 dτ.
N1 (t) :=
0
The application of Gronwall’s inequality in (4.45) leads to
Z t
2
(4.47)
|w(t)| ≤ N3 (t) exp
c(τ )dτ a.e. t ∈ Sδ .
0
It should be noted that the presence of the factors ||w1 (t)||2 and |w1,t |2 in the expresRt
sion of a(t) is not disturbing. Our only concern is to ensure, e.g., that 0 ||w1 (τ )||2 |∆s(τ )|dτ
stays bounded for a.e. t ∈ Sδ . This follows due to the the energy estimate (4.17) and
the inequality
Z τ
Z t
Z t
d
||w1 (s)||2 ds |∆s(τ )|2 dτ ≤ |∆s(t)|2
||w1 (τ )||2 dτ.
dτ
0
0
0
Also, it results that
Z t
1
||w(τ )||2 dτ ≤ (N1 (t) + N3 (t))eCt a.e. t ∈ Sδ .
(4.48)
d0
0
Moreover, the inequality
2
t
Z
|∆s0 (τ )|2 dτ
|∆s(t)| ≤ t
(4.49)
0
provides the estimate
Z
2
|∆s(τ )| dτ ≤ δ
(4.50)
Sδ
2
Z
Sδ
|∆s0 (τ )|2 dτ.
24
A. MUNTEAN AND M. BÖHM
We show that
M1 := exp
(4.51)
!
δ
Z
max{δ 2 a(t) + b(t)}
c(τ )dτ
t∈S̄δ
0
and
(4.52)
M2 :=
2
exp(Cδ) max{δ 2 a(t) + b(t)} ,
d0
t∈S̄δ
satisfy
√
2
0
|w(t)| ≤ M1 δ
(4.53)
12
t
Z
4
|∆s (τ )| dτ
0
and
√
t
Z
2
Z
||w(τ )|| dτ ≤ M2 δ
(4.54)
0
21
t
0
4
|∆s (τ )| dτ
.
0
The constants M1 and M2 depend only on δ, θ, s0 , L, L0 , C, cξ , cξ̄ and ki (i ∈ I).
The statements (4.53) and (4.54) follow in a straightforward way. On one hand,
by (4.44), (4.49) and (4.50), we have
Z t
Z t
|w(t)|2 ≤ N3 (t) exp
c(τ )dτ = exp
c(τ ) ×
0
0
Z t
Z t
2
2
0
2
× |w(0)| +
a(τ )|∆s(τ )| + b(τ )|∆s (τ )| dτ ≤ exp
c(τ )dτ ×
0
Z
×
t
0
Z t
12
√
2
0
2
0
4
δ a(τ ) + b(τ ) |∆s (τ )| dτ ≤ M1 δ
|∆s (τ )| dτ ,
0
0
where M1 satisfies (4.51). On the other hand, we use (4.47), (4.48), and the positivity
of N1 (0) and of N10 (t) (for a.e. t ∈ Sδ ), to establish
Z t
Z t
1
2
Cτ
||w(τ )|| dτ ≤
N3 (t) +
N2 (τ )e dτ
d0
0
0
≤
1
(N3 (t) + N1 (t)) eCt
d0
2
2
≤
exp (Ct) max{δ a(t) + b(t)} ×
d0
t∈S̄δ
Z
×
t
0
2
Z
|∆s (τ )| dτ ≤ M2
0
t
0
2
√
Z
|∆s (τ )| dτ ≤ M2 δ
0
t
0
4
|∆s (τ )| dτ
0
12
,
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
25
where M2 is given by (4.52). Combining (4.53), (4.54) in (4.32), it results that
Z t
Z t
|∆r0 (τ )|4 dτ ≤ M3
(4.55)
|∆s0 (τ )|4 dτ,
t0
t0
where the constant M3 is given by
M3 := Cη ĉ4 M1 M2 δ.
(4.56)
We set
χ := min{Cη ĉ4 M1 M2 ,
(4.57)
1
},
δ+1
where we put
0<δ≤
(4.58)
L0
:= δ0 .
MηΓ
By (4.58), we ensure s(δ) ≤ L0 . Finally, using the Lipschitz constant δχ < 1, the
strict contractivity of the fixed-point operator T is proven.
We now have the existence of a unique weak solution in the time interval Sδ =]0, δ[.
The same argument can be repeated to gain the existence and uniqueness of the weak
solution with respect to time intervals like ]kδ, (k + 1)δ[, where the free factor k ∈ N
satisfies the property k + 1 < 1δ Tfin , where Tfin is given by (3.37).This proves Theorem
3.3. Moreover, if Tfin < +∞, then the last time interval on which the solution is proved
to exist is ]km δ, Tfin [, where km := sup{k ∈ N : k + 1 < 1δ Tfin }. Since the uniqueness
is granted on each of the intervals ]0, δ[, . . ., ]kδ, (k + 1)δ[, . . ., ](km − 1)δ, km δ[, it
actually holds on the whole ]0, Tfin [.
4.3. Sketch of the proof of Proposition 3.4. The idea of the proof is a refined
version of the arguments that we have used to show the positivity of concentrations.
We focus on getting lower bounds for Ca(OH)2 (aq), CO2 (aq) and CO2 (g) concentrations and only tersely suggest how the lower bounds for the other concentrations are
obtained.
We choose in the weak formulation (3.31) the test function
−
− [u3 − u∗3 γ3 (t)] , for i = 3
(4.59)
ϕi :=
∈ Vi for all i ∈ I1 ∪ I2 .
0,
otherwise
In (4.59), the function γ3 ∈ C 1 (S̄δ ) has to be determined such that γ30 (t) ≤ 0 for all
t ∈ Sδ and γ3 (0) = 1. On this way, we obtain the following identity
D3
1
||ϕ3 ||2 =
(−ηΓ (u(1) + λ), ϕ3 (1)) +
(L − s)2
L−s
+ s0 (u3 (1) + λ3 , ϕ3 (1)) + S3,diss (u3 + λ3 − u3,eq , ϕ3 ) +
s0
+
((2 − y)ϕ3,y , ϕ3 ).
L−s
((u3 + λ3 ),t , ϕ3 ) +
(4.60)
We collect some of the terms in (4.60), add the positive term −(ζu∗3 γ3 (t) − θu∗3 , ϕ3 )
to its right-hand side, and conveniently select u∗3 within the interval ]0, minS̄δ λ3 (t)[.
The constants ζ > 0 and θ > 0 are chosen such that
(4.61)
ζγ3 (t) − θ > 0 for all t ∈ [0, ∞[ and lim (ζγ3 (t) − θ) > 0.
t→∞
26
A. MUNTEAN AND M. BÖHM
Thus, we obtain
1 d
1
D3
||ϕ3 ||2 ≤
|ϕ3 |2 +
(−ηΓ (u(1) + λ)+
2 dt
(L − s)2
L−s
+ s0 (u3 (1) + λ3 , ϕ3 (1)) + S3,diss |ϕ3 |2 + S3,diss (λ3 − u3,eq + u∗3 γ3 (t), ϕ3 ) −
s0
((2 − y)ϕ3,y , ϕ3 ).
−(u∗3 γ30 (t) + λ03 , ϕ3 ) − (ζu∗3 γ3 (t) − θu∗3 , ϕ3 ) +
L−s
(4.62)
We state the following auxiliary result:
Lemma 4.4. Assume that the hypotheses of Theorem 3.3 are satisfied. Additionally, let λ3 ∈ W 1,2 (Sδ ) and set σ := −S3,diss + ζ, ρ := u1∗ (λ03 − S3,diss λ3
3
+S3,diss u3,eq ), χ := ρ − θ, where ζ and θ are positive constants satisfying (4.61). The
following statements hold:
(i) If λ3 = const., and if
σθ
(4.63)
}, +∞ and θ ∈]ρ, σ + ρ[,
ζ ∈ max{S3,diss ,
θ−ρ
then the function
γ3 (t) = −
(4.64)
χ σ + χ −σt
+
e
for all t ∈ [0, ∞[
σ
σ
is the unique positive solution of the problem
(4.65)
γ30 (t) + σγ3 (t) + χ(t) = 0, γ30 (t) < 0 for all t ∈]0, ∞[ with γ3 (0) = 1.
(ii) If λ3 = λ3 (t) 6= const., ρ ∈ L2+ (0, ∞), and
Z t
σ−σ
(4.66)
χ(τ )eστ dτ + χ(t)eσt > 0 for all t ∈ [0, ∞[,
0
then
(4.67)
γ3 (t) =
Z
1−
t
στ
χ(τ )e dτ
e−σt for all t ∈ Sδ
0
is the unique positive solution of (4.65).
Proof. [of Lemma 4.4] The choice of the test function (4.64) (or (4.67)) relies
on the sign restrictions ρ > 0, σ > 0, χ < 0 as well as on (4.61), (4.63), and (4.66).
These estimates support the existence and uniqueness of a strictly positive and bounded
function γ3 . The statements (i) and (ii) follow by straightforward verification.
Now, with γ3 (t) as in Lemma 4.4 we force the fourth, fifth and sixth term from
the right-hand side of (4.62) to vanish. Combining Young’s inequality and the interD3
d
s0
2
polation inequality (4.1), we obtain: 12 dt
|ϕ3 |2 + (L−s)
2 ||ϕ3 || ≤ L−s ((2−y)ϕ3,y , ϕ3 ) ≤
ξ ||ϕ3 ||2
2 (L−s)2
c
2
1
2θ−1
+ 2ξ ĉ 1−θ |s0 | 1−θ (L − s) 1−θ |ϕ3 |2 , where we select ξ ∈]0, 2D3 ]. Since ϕ3 (0) = 0,
we can use Gronwall’s inequality to conclude that u3 ≥ u∗3 γ3 (t) > 0 for all t ∈ Sδ .
Note that the choice of γ3 does not depend on δ.
We choose in the weak formulation (3.31) the test functions
−
− [ui − u∗i γi (t)] , for i ∈ {1, 2}
(4.68) ϕi :=
∈ Vi for all i ∈ I1 ∪ I2 .
0,
otherwise
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
27
Select i ∈ {1, 2}. In (4.68) u∗i > 0 are given constants and the functions γi ∈ C 1 (S̄δ )
have to be determined such that γi0 (t) ≤ 0 for all t ∈ Sδ and γi (0) = 1. It makes sense
to look for u∗i in the interval ]0, minS̄δ λi (t)[. We want to determine the function γi
such that u∗i γi margins the concentration ui from below. Moreover, for sufficiently
large time t we want that u∗i γi (t) decreases to some strictly positive value. We have
that
1
D1
s0
(u
,
ϕ
)
+
(η
(u
+
λ),
ϕ
(1))
+
(u1 (1) + λ1 , ϕ1 (1))
1,y
1,y
Γ
1
s2
s
s
s0
= P1 (Q1 (u2 + λ2 ) − (u1 + λ1 ), ϕ1 ) + y (u1,y , ϕ1 ),
s
s0
D2
((u2 + λ2 ),t , ϕ2 ) + 2 (u2,y , ϕ2,y ) + (u2 (1) + λ2 , ϕ2 (1)) = −P2 (Q2 (u2 + λ2 )
s
s
s0
− (u1 + λ1 ), ϕ2 ) + y (u2,y , ϕ2 ).
s
((u1 + λ1 ),t , ϕ1 ) +
Taking ϕi as in (4.68), we obtain
1 d
D1
1
s0
|ϕ1 |2 + 2 ||ϕ1 ||2 = − (ηΓ (u + λ), ϕ1 (1)) − |ϕ1 (1)|2
2 dt
s
s
s
s0 ∗
s0
+ y (ϕ1,y , ϕ1 ) − (u1 γ1 (t) + λ1 , ϕ1 (1)) − (λ01 + P1 λ1 − P1 Q1 λ2 , ϕ1 )
s
s
(4.69)
− (u∗1 γ10 , ϕ1 ) + P1 Q1 (u∗2 γ2 (t), ϕ1 ) − P1 (u∗1 γ1 (t), ϕ1 ) + P1 (Q1 ϕ2 − ϕ1 , ϕ1 )
and respectively,
1 d
D2
s0
s0
s0
|ϕ2 |2 + 2 ||ϕ1 ||2 = − |ϕ2 (1)|2 + y (ϕ2,y , ϕ2 ) − (u∗2 γ2 (t) + λ2 , ϕ2 (1))
2 dt
s
s
s
s
− (λ02 + P2 Q2 λ2 − P2 λ1 , ϕ2 ) − (u∗2 γ20 , ϕ2 ) + P2 Q2 (−u∗2 γ2 (t), ϕ2 )
(4.70)
+ P2 (u∗1 γ1 (t), ϕ2 ) − P2 (Q2 ϕ2 − ϕ1 , ϕ2 ).
We make use of the next auxiliary result:
Lemma 4.5. Assume that (C3) and the hypotheses of Theorem 3.3 are satisfied.
Additionally, let λ1 ∈ C 2 (S̄δ ), λ2 ∈ C 1 (S̄δ ), u∗1 > 0 and u∗2 > 0. The problem
(4.71)-(4.73)
(4.71)
(4.72)
(4.73)
u∗2
λ01 + P1 λ1 − P1 Q1 λ2
γ
+
P
γ
=
−
2
1
1
u∗1
u∗1
∗
0
u
λ + P2 Q2 λ2 − P2 λ1
γ20 + P2 Q2 γ2 − P2 2∗ γ1 = − 2
u1
u∗2
γ1 (0) = γ2 (0) = 1, γi0 ≤ 0 in Sδ .
γ10 − P1 Q1
has a unique positive solution in C 1 (S̄δ ) × C 1 (S̄δ ).
Proof. [of Lemma 4.5] The idea of the proof is rather simple: We formulate
(4.71)-(4.73) as a second-order non-homogeneous differential equation. We solve this
explicitly. Explicit ranges of model parameters are needed in order to ensure (4.73).
The conclusion of the lemma follows by straightforward verification.
By Lemma 4.5, several terms from (4.69) and (4.70) cancel out. More precisely,
the first four terms as well as the last one on the right-hand side of (4.69) remain,
but the other vanish. On the right-hand side of (4.70), the first three terms and the
last one stay, but the rest of them vanish. In this way, we obtain once more the
28
A. MUNTEAN AND M. BÖHM
inequalities employed for the positivity of the concentrations u1 and u2 . We conclude
via the same Gronwall-type argument that
u1 > u∗1 γ1 (t) and u2 > u∗2 γ2 (t) for a.e. t ∈ Sδ .
Since reaction (1.1) produces water, there is no difficulty to show that the initial
conditions û50 and û60 are the strict lower bounds of u5 and u6 .
5. Illustration of CO2 penetration in a concrete wall. We consider an 18
years old concrete wall made of Portland cement (CEM 1), whose chemistry and
outdoor exposure conditions are described in Table 3.1 of [13]. The indicator test
emphasizes a thin macroscopic front penetrating the material and separating carbonated from non-carbonated phases; see Fig. 1.1. We employ a FEM Galerkin scheme
to approximate the weak solution to (PΓ ). We proceed as follows: We immobilize the
moving boundary and discretize the PDE system in space. Afterwards, we integrate
the obtained stiff ODE system in time using MATLAB. The numerical procedure is
explained in detail in chapter 4 of [30], while a priori and a posteriori error estimates
for the semi-discrete approximation are derived in [29]. The plots in Fig. 5.1–5.2
(a)
(b)
(c)
Fig. 5.1. (a)+(b) CO2 (aq) and Ca(OH)2 (aq) profiles vs. space. Each curve refers to time
t = i years, i ∈ {1, . . . , 18}. (c): Interface position vs. the experimental points “ ◦ ” after Tfin = 18
years of exposure.
(a)
(b)
(c)
Fig. 5.2. (a) CaCO3 (aq) profiles vs. space. Each curve refers to time t = i years, i ∈
{1, . . . , 18}. (b)+(c) Concentration of CO2 (aq) and Ca(OH)2 (aq) vs. time and space.
show the solution of (PΓ ). Observe that steep concentration gradients arise near Γ(t)
(cf. Fig. 5.1 (b), Fig. 5.2 (c), e.g.) and the calculated interface position is in the
experimental range, see Fig. 5.1 (c) and Fig. 5.3. Furthermore, Fig. 5.2 (a) shows
a gradual increase in the concentration of CaCO3 (aq) within Ω1 (t). It visualizes the
expansion of Ω1 (t) and also points out the shrinking of Ω2 (t). The results in Fig. 5.3
indicate a strong dependence of the penetration speed on the structure of the reaction
A MOVING-BOUNDARY PROBLEM FOR CONCRETE CARBONATION
(a)
29
(b)
Fig. 5.3. (a) Interface position vs. the experimental points “ ◦ ” when varying the reaction
order p = 1.5, 1.3, 1, 0.9 while q = 1. (b) Interface position vs. the experimental points “ ◦ ” when
varying the effective dimensional diffusion coefficient of CO2 (g) as follows: D22 , D2 , 2D2 and 4D2 .
rate η̃Γ and on the range of the effective diffusion coefficient of CO2 (g). Changing the
partial reaction order p from 0.9 to 1.5 produces a significant increase of the reaction
rate, which finally results in a higher penetration depth. The penetration depth obtained with p = 1.5 is at least twice bigger than that obtained for p = 1 (compare
the curve 1 with the curve 4 in Fig. 5.3 (a)). Alterations of the exponent q may
lead to drastic changes in the penetration depth as well. An increase in the effective
diffusivity of CO2 (g) produces a significant increase in the penetration depth. In Fig.
5.3 (b), we observe that if CO2 (g) encounters difficulties to travel to the reaction
zone, then the speed of this zone is correspondingly smaller. On the other hand, if
the matrix has large pores, then a fast advancement of CO2 (g) molecules is to be
expected. Another issue is illustrated in Fig. 5.4 and Table 5.1. Namely, we use the
standard set of parameters (see [30] (appendix D)) to compare the numerical lower
bounds with the theoretical lower bounds in the case of Ca(OH)2 (aq) concentration.
Fig. 5.4. Theoretical vs. numerical lower bounds of Ca(OH)2 (aq) concentration. The broken
line is the computed profile of Ca(OH)2 (aq), while the continuous line is the proposed theoretical
lower bound.
For the chosen parameter set, where we additionally select u∗3 /5 = 10−3 and take γ3 (t)
as in (4.64), the theoretical lower bounds underestimate the numerical ones along the
whole computation time. This underestimation of the lower bounds yields an overestimation of the final time of the process Tfin . Also, we observe a decrease in time of
the numerical lower bound for Ca(OH)2 (aq), see the broken curve in Fig. 5.4. This
effect is mainly due to the continuous depletion of alkaline species by carbonation.
On the other hand, a slight increase in time of the theoretical lower bound on the
30
A. MUNTEAN AND M. BÖHM
Time (years)
2
5
11
16
18
u3 (theor.)
0.02119
0.02267
0.02392
0.02437
0.02449
u3 (numer.)
0.07115
0.06751
0.06268
0.05959
0.05847
Table 5.1
Selection of lower bounds of Ca(OH)2 (aq) concentration at different front positions.
same species can be noticed in Fig. 5.4 (the continuous curve). The latter effect is a
direct consequence of the influence of the production term by dissolution.
6. Conclusions. We formulate a moving-boundary model to describe the penetration of a sharp-reaction interface in concrete. Results concerning the global existence and uniqueness of positive weak solutions to the proposed model are presented.
A simulation example illustrates the typical behavior of active concentrations and interface penetration into a concrete wall. The model shows qualitatively good results
when the numerical solution is compared with measured penetration depth profiles.
The setting can be extended (with minor modifications) to account for more reaction
interfaces simultaneously penetrating an unsaturated reactive mineral material.
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