CC BY
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Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 28. С. 69-84
УДК 518.517
MSG 35R35, 35К49, 76S05
DOI https://doi.org/10.26516/1997-7670.2019.28.69
Global Existence of a Solution for a Multiscale Model Describing Moisture Transport in Concrete Materials *
K. Kumazaki
Nagasaki University, Nagasaki, Japan, Department of Education
Abstract. In the previous study [5] we proved the existence of a solution locally in time for a two-scale problem which is given as a mathematical model for moisture transport arising in a concrete carbonation process. The two-scale model consists of a diffusion equation of the relative humidity in a macro domain and the free boundary problems describing a wetting and drying process in infinite micro domains. In this paper, by improving the diffusion equation of the relative humidity based on the experimental result [3; 10], we construct a globally-in-time solution of the two scale model. For the global existence, we obtain uniform estimates and uniform boundedness of the solution with respect to time and use the method of extending local solutions.
Keywords: two-scale model, free boundary problem, quasilinear parabolic equation, moisture transport.
In our recent, work [5], we proposed a two-scale model describing moisture transport phenomena arising in a concrete carbonation process, and showed that our two-scale problem is solvable locally in time. In this paper, we improve an equation consisted of the two-scale model and prove the existence of our concept of solution globally in time.
Based on the setting of [5], let us describe our model. Our model consists of a macro domain Q, where Q is a bounded domain in [R3 which is occupied by concrete, and a micro domain for each In the macro domain Q
1. Introduction
* This work was supported by JSPS KAKENHI Grant Number JP16K17636
we consider the relative humidity h = h(t,x), where t is a time variable. On the other hand, we assume that the micro domain is the hole for each and consider the hole as an interval (0, L), where L is the depth of the hole. This interval (0, L) indicates the water drop region (0, s(t, x)) and the air region (s(t, x), L), and the boundary L denotes the edge of the hole in contact with h. Here, we regard the degree of saturation s = s(t, x) and the size of the water drop region, and assume that the relative humidity u in one hole acts in the following non-cylindrical domain QS(T) defined by
QS(T) := {{t,x,z)\{t,x) € (0,T) x tt,s(t,x) < z < L}.
Our two scale model, which we denote by (P), is as follows: Find a triplet (h,s,u) = (h(t,x),s(t,x),u(t,x,z)) satisfying the following equations, bo-
undary and initial conditions,
pvht - div(g(h)Vh) = s(l - f(h))v in Q(T) := (0, T) x Q, (1.1)
h = hb on S(T) := (0, T) x дП, (1.2)
pvut - kuzz = 0 on QS(T), (1.3)
u(t, x, L) = h{t, x) for (t, x) € Q(T), (1.4) kuz(t, x, s(t, x)) = (pw — pvu(t, x, s(t, x)))st(t, x) for (i, x) € Q(T), (1.5)
st(t, x) = a(u(t, x, s(t, x)) — <p(s(t, ж))) for (t, x) € Q(T), (1.6)
h{0, x) = ho(x) for x e Q, (1.7)
s(0,x) = so(x) for x £ Q, (1.8)
u(0, x, z) = Uo(x, z) for (x, z) € QSo, (1.9)
where = {(x,z) € Q x R|so(a;) < z < L}, pw and pv are the densities of H2O in water regions and in air regions, respectively, g is a continuous function on (0, 00), / and v are given functions on [R and on Q(T), respectively, k and a are positive constants, (p is a continuous function on [R, ho is an initial condition on Q, so is an initial position of the free boundary s and uo is an initial condition on (so,L).
From the physical point of view, (1.1) is the diffusion equation of h which is originally proposed by [7; 8], g is a diffusion coefficient. Also, v represents the concentration of CO2 and s(l — f(h))v represents the quantity of water generated by chemical reaction according to the level of the humidity, where / is a monotone function increasing on [0,1] such that /(0) > 0 and /(1) is almost 1. The forcing term is proposed by [3; 10] based on the experimental result that CO2 cannot be dissolved completely in low humidity, while the water in the pores hinders the diffusion of CO2 in high humidity, and 1 — f(h) is the reduction factor depending on h.
The system (1.3)-(1.6) with (1.8) and (1.9) is a free boundary problem describing a wetting and drying process in the hole for each and is
originally proposed by [9]. The equations (1.3) and (1.5) are derived from
the mass conservation for H2O in the air region and near the free boundary, respectively. Also, (1.4) means that each hole is exposed to air at the end of the hole, and (1.6) is the growth rate of the water drop region and ^ represents the rate of change from water to air in the hole.
The problem (P) is quite close to the two-scale model in [5]. Indeed, the two-scale setting and the system {(1.3)-(1.6), (1.8), (1.9)} is exactly same. The difference between (P) and the model in [5] is the diffusion equation of the relative humidity h: we consider (1.1) in (P), while we studied in [5] the following diffusion equation of the relative humidity:
pvht - div(g(h)Vh) = sv in Q(T), (1.10)
where pv, g, and v are the same constant and functions as in (P). As pointed out in [5], for the global existence it needs to satisfy that 0 < h < h* on [0, T], where h* is a positive constant with h* < 1. However, it is difficult for the solution h satisfying (1.10) to guarantee such boundedness on [0, T] for T > 0. To overcome this difficulty we improve (1.10) based on the experimental result [3; 10], and arrive to a more realistic equation (1.1). Under this improvement, we show the smallness of h satisfying (P) on [0, T] for T > 0 and establish the existence of a globally-in-time solution of (P).
2. Notations and Assumptions
In this paper we use the following notations. In general, for a Banach space X we denote by | ■ |X its norm. Particularly, we denote by H = L2(Q), and the norm and the inner product of H are simply denoted by | ■ | h and (■, -)h, respectively. Also, for Q c Rn for n = 1 or n = 3, H^q), H1 (Q) and H2(Q) are the usual Sobolev spaces.
Throughout this paper, we assume the following conditions:
(A1) Q is a bounded domain of R3 which has the boundary dQ in the class of C2.
(A2) k, a, k0 are positive constants satisfying k0 < 1.
(A3) G € C3((0, +rc>)) is such that g(r) := G'(r) > g0 for r > 0, where go is a positive constant and put
Cg = sup |g'(r)| + sup |g''(r)|.
(A4) f € C!(R) n W 1>~(R), 0 < f < 1 on R and f (r) = 1 for r > h*, where h* is a positive constant satisfying h* < 1, and put Cf = |f'|L~(R).
(A5) v € L°°(Q(T)) and vt € L2(0, T; H1^)) with v > 0 a.e. on Q(T).
(A6) hb € C2(Q(T)) and hbt € L2(0, T; H2(i})) with
Ko < hb < h* on Q(T),
(2.1)
where h* is the same constant as in (A4). (A7) ho € H2(Q) n with
Ko<h0< h* on Q, (2.2)
where h* is the same constant as in (A4), and Aho is bounded a.e. on Q. Also, ho satisfies that ho = hb{0) a.e. on <9Q.
(A8) <p € C^R) n W1'00^), <p = 0 on (—oo, 0], <p < 1 on [R, <p>' > 0 on (0, L] and for bo > 0 <p(L) — bo > 0. Also, we denote by (p the primitive function of cp with 0(0) = 0 and put C^ = |<£>'|l°o(R). (A9) Two positive constants pw and pv satisfy
Pw > pv(C<p + 2), 9aLpl < kpw.
(A10) so € H such that 0 < so < L — So for ¿o > 0 a.e. on Q, and uo € L°°(QS0) and the function x —> \uo(x)\Hi(SOtL^ is bounded a.e. on Q and uo(x, L) = h(x, 0) for x € Q, and 0 < uo < 1 a.e. on Qso.
Definition 1. Let h and s be functions on Q(T) and u be a function on QS(T), respectively, for T > 0. We say that a triplet (h,s,u) is a solution of (P) on [0,T] if the conditions (S1)-(S9) hold:
(51) h € W1'2^^-, H) n L°°(0,T-, H^Q)) n L2(0,T] H2(Q)) with h> 0 a.e. on Q(T).
(52) s, st € L°°(Q(T)), 0 < s < L on Q(T), u € L°°(QS(T)), ut, uzz € L2(Qs(T)) and (t,x) € Q(T) —> \uz(t,x, is bounded.
(53) pvht - AG(h) = s( 1 - f(h))v a.e. in Q(T).
(54) h = hb a.e. on S(T).
(55) pvut - kuzz = 0 on QS{T).
(56) u(t,x,L) = h(t,x) for a.e. (t,x) € Q(T).
(SI) kuz(t,x,s(t,x)) = (pw — pvu(t, x, s(t, x)))st(t, x) for a.e. (t,x) €
Q(T).
(58) st(t, x) = a(u(t, x, s(t, x)) — <p(s(t, x))) for a.e. (t, x) € Q(T).
(59) h{0, x) = ho{x), s(0, x) = so(a;) for u(0, x, z) = uo(x, z) for (x, z) € Qso.
Our main result of this paper is concerned with the existence of a globally-in-time solution for the problem (P).
Theorem 1. Let T > 0. If (Al)-(AIO) hold, then (P) has a unique solution (h,s,u) on [0,T] with no < h < h* a.e. on Q(T) and 0 < u < 1 a.e. onQs(T).
GLOBAL EXISTENCE OF A SOLUTION FOR A MULTISCALE MODEL 73 3. Mathematical model for moisture transport
For T > 0 and 5 € (0, L) we set
X(T,5) := {s € W 1>2(0,T; H)| 0 < s < L - 5,
|st| < 2a a.e. on Q(T),s(0) = so in Q}.
First, for given s € X(T, 5), we prove the existence of a solution of
(AP)(s) := (AP)(sA,ho)
on [0, T]. To do so, we put
Dk(T) := {z € C(0,T; H)| Ml2(o,t,h) + Ml-(o,t;hi(n)) < K},
where K is a positive constant determined later. For given h € DK (T), we consider (AP)(s, h) := (AP)(s, h, hb, ho):
Pvht - div(g(h)Vh) = s(1 - f (h))v in Q(T), h = hb on S (T), h(0,x) = h0(x) for x € Q.
By (A4) and (A5) we easily see that s(1 -f (h))v € L2(0,T; H)n(Q(T)). Hence, by using the result of [1;2] we have a solution h of (AP)(s, h) such that h € W 1'2(0, T; H) n (0, T; H 1(Q)) n L2(0, T; H2(Q)), h(t) - hb(t) € H0 (Q) for a.e. t € [0.T] and
ko < h < p(t) a.e. on Q for t € [0,T], (3.1)
where P(t) = L|v|L-(q(t))t + h* for t € [0, T]. Next, we note the useful property of a solution of (AP)(s, h).
Lemma 1. Let T > 0 and h be solutions of (AP)(s,h) on [0,T] for given s € X (T, 5) and h € DK (T). Then, there exists an increasing function Mi (T) > 0 (1 < i < 2) with respect to T which is independent of s and 5 such that
(i) |h|w1,2(o,t;H) + |h|l-(o,t;H!(Q)) < M1(T),
(ii) |Vh|l-(q(t)) < M2(T).
The estimate (i) is obtained by the standard calculation because |s(1 -f(h))v| < L|v|L-(q(t)) by s € X(T,5), (A4) and (A5). Also, from s(1 -f (h)) € L^(Q(T)) we can apply the same argument of Lemmas 3.1-3.4 in [1] and get M2(T) satisfying (ii). In particular, we note that these constants are independent of the choice of h.
Next, for fixed s € X(T, 5) we prove the continuous dependence of a solution h of (AP)(s, h) for given h € DK(T).
Lemma 2. Let h\ and h,2 be solutions of (AP)(s,hi) and (AP)(s,h2) for given hi, h,2 € Dk(T), respectively. Then, there exists C(T) depending on T such that
Mi) - h2{t)\% + f IV(fci(r) - h2(r))\2Hdr
Jo
<C{T) [ Ih^-h^lldT forte [0,T\. Jo
Proof By the subtraction of the equations for h\ and h2, it holds that
El±\hl(t) - h2(t)\2H + J V(G(hi(t)) - G(h2(t)))V(hi(t) ~ h2(t))dx
= L ^^-/(^(^"^-/(^(i))))^) (h^-h^dx. (3.2)
Here, by using (ii) of Lemma 1, we can estimate the second term of the left hand side of (3.2) as follows.
[ V(G(hi) - G(h2))V(h! - h2)dx J n
J n
>go\V{hi-h2)\2H-M2{T) [ \g(hi) - д^МЬ - h^dx
J n
>f |V(/n - h2)\l - Mf^hhl - h2\l, (3.3)
where go and Cg are the same positive constants as in (A3), and M2(T) is the same constant as in Lemma 1. Also, by (A4) and Holder inequality, we observe that
^ s(t) - f(hi(t)) - (1 - f(h2(t)))^v(t) {h^t) - h2{t))dx
^LCf^Loo^T^^t) -h2{t)\H\h{t) -h2{t)\H. (3.4)
Hence, by substituting (3.3) and (3.4) into (3.2) we obtain
f Jrlfci(i) - Ш\н + f |V(ft,(i) - Mt))|2H
+ (^"f«*™ + Unit) - мой for a-e. « 6
Therefore, by Gronwall's inequality, we see that there exists a positive constant C(T) depending on T such that Lemma 2 holds. □
From Lemma 2, for fixed s € X(T,S), we also see that the solution h of (AP)(s, h) for given h € Dk(T) is unique. Now, we prove the existence of a solution of (AP)(s) for given s € X(T, 5). By taking K = Mi(T), we define the mapping T : Dk(T) —> Dk(T) by T(h) = h, where h is a unique solution of (AP)(s, h) for given h. Then, by Lemma 2 we see that T is continuous with respect to C(0,T;H). Therefore, since Dk(T) is compact in C(0,T;H), by Schauder's fixed point theorem, we can find h € Dk(T) such that T(h) = h. This means that (AP)(s) has a unique solution on [0,T] for fixed s <E X(T,5).
Next, we prove the following boundedness of a solution h of (AP)(s) for given s eX(T, 5).
Lemma 3. Let T > 0, 5 € (0, L) and h be a solution of (AP)(s) for given s € X(T, 5). Then, n0 < h(t) < h* for t € [0, T] a.e. on Q, where h* is the same constant as in (A4)-
Proof First, we prove that h > kq for t € [0, T] a.e. on Q. By (2.1) it holds that [—hb + ko}+ = 0 a.e. on Q(T). Then, we have that
^jt\[~h{t) + K0]+\2H +go J IV[-h(t) + Ko]+\2dx < / -s(t)( 1 - f(h(t)))v(t)[-h(t) + K0]+dx for a.e. t € [0,T],
J n
Here, s(l — f(h))v > 0 a.e. on Q(T) so that it follows that
^i\l-h(t) + Ko}+\2H+go [ |V[—/¿(t) + Ko] + \2dx < 0 for a.e. t € [0,T]. * Ml J Q
Clearly, the second term is positive. Hence, we obtain
jt\[-h(t) + k0] + \2h < 0 for a.e. t € [0,T],
This result and (2.2) implies that h(t) > kq for t € [0, T] a.e. on Q. Next, we show that h < h* for t € [0, T] a.e. on Q. Since [hb - h*}+ = 0 a.e. on Q(T) by (2.1), it holds that
^jf\[h{t) - h*] + \2H+g0 ji | V[h(t) - h*] + \2dx
< [ s(t)( 1 - f(h(t)))v(t)[h(t) — h*]+dx for a.e. t € [0,T], J n
We note that the second term is positive and the right hand side is equal to 0 because f{h) = 1 by (A4) if h > h*. Hence, we obtain that
jt\[h(t) - h*]+\2H < 0 for a.e. t € [0,T],
Finally, by integrating over [0,t] for t € [0, T] and (2.2) we see that h < h* for t € [0,T] a.e. on Q. Thus, Lemma 3 is proved. □
At the end of this section, we give some properties of a solution h of (AP)(s) for given s <E X(T, 5).
Lemma 4. (i) Let T > 0, 5 £ (0,L) and h be a solution of (AP)(s) on [0, T] for given s € X{T,5). Then, there exists an increasing function Mi(T) > 0 (1 < i < 4) with respect to T which is independent of s such that
\h\w^2(0,T;H) + \h\L°°(0,T-,m(n)) < Ml(T), (3-5)
\Vh\L00mT)) < M2(T), (3.6)
\ht\L°°(0,T;H) + |V^|L2(0>r;H) < M3(T), (3.7)
Nl°°(Q(T)) < M4(T), (3.8)
where M\(T) and M2(T) are the same as in Lemma 1.
(ii) Let h\ and h2 be solutions of (^4P)(si) and (AP){s2) for s\, s2 € X(T,5), respectively. Then, there exists M5(T) > 0, M6(T) > 0 such that
IV(Mi) " h2{t))\2H + f |A(^i(r) - h2{r))\2Hdr Jo
< M5(T)\h - s2||2(o>r;H) fortG [0, T], (3.9)
and
\hlt - h2t\l2(0,t;H) < M6(T)\§1 - S2\l2(0,T;H) (3-10)
As mentioned in Lemma 1, since |s(l — f(h))v\ < L\v\lco(q(T^ (3.5) and (3.6) are obtained, and these estimates are independent of the choice of s. Also, by (A4), (A5) and (3.5), we see that (s( 1 - f(h))v)t € L2(0,T]H). On account of this, (3.7) can be obtained by using the same argument of [1]. Moreover, with the help of (A4), (A5), (A7) and (3.7) it holds that (s( 1 - f(h))v)t € L6(Q(T)) and s0( 1 - f(ho))v(0) € L°°(Q). From this result, we can have (3.8) by the technique of [6] or the same way of the proof of Lemma 3.2 in [5]. The continuous dependence estimate (3.9) and (3.10) are derived from the subtraction of the equations h\ and h2 by using (3.6) and the same proof of Lemma 3.3 of [5]. In this paper, we omit the precise proofs.
4. Free boundary problem
In this section, we note the obtained result for the free boundary problem with a given h on Q(T) for T > 0. First, we give a definition of a solution to (FBP)(^,s0,«o) satisfying (1.3)-(1.6), (1.8), and (1.9) for each x € Q. Throughout this section, we use the notation QS(X)(T) for the following domain: For each iGil,
Qs(x)(T) := {(t,z)\t € (0,T),s(t,x) < z < L}.
Definition 2. Let T > 0 and x G Q,, s = s(x) = s(-,x) be a function on [0,T] and u = u(x) = u(-,x, •) be a function on QS(X)(T). We say that a pair (s,u) is a solution of (FBP)(h,so,uo) := (FBP)(h(-,x),so(x),uo(x,-)) on [0, T] if the following conditions (D1)-(D6) hold:
(Dl) s(x) G M^1,oo(0,T),0 < s(x) < L on [0,T], u(x) G L°°(Q8ix)(T)), ut(x),uzz(x) G L2(Qs{x)(T)), t G [0,T] i y \uz(t, x, -)\L2(s(t,x),L) is bounded. (D2) pvut(x) - kuzz(x) = 0 on Qs(x)(T). (D3) u(t,x,L) = h(t) for a.e. t G [0,T],
(D4) kuz(t,x,s(t)) = (pw — pvu(t, x, s(t, x)))st(t, x) for a.e. t G [0, T], (D5) St{t, x) = a(u(t, x, s(t, x)) — <p(s(t, x))) for a.e. t G [0, T]. (D6) s(0, x) = so(x), u(0, x, z) = uo(x, z) for z G [so(œ), L].
To handle (FBP)(/&, so, «o)> by introducing the following function û
ù(t,x,y) = u(t,x,(l - y)s(t,x) +yL) for (t,y) G (0,T) x [0,1], (4.1)
we transform (FBP)(/&, so,uo), initially posed in a non-cylindrical domain, to the following problem (FBP)(/&, so,«o) in a cylindrical domain:
Pvût^ ~ (.L-s(t x))2Ûyy^ = PVL- s(tt)^'y^ in X ù(t,x, 1) = h(t, x) for t G [0,T], k
L - s(t x)™^'X' ^ = ^Pw ~ pv^ty x' for t G t0'
st(t, x) = a(u(t, x, 0) - ip(s(t, x))) for t G [0, T],
s(0,x)=So(x), ù(0,x,y)=uo(x,(l -y)s0(x) + yL)=\ u0(x,y) for yG[0,1]. Here, the condition (Dl) is equivalent to the following (S):
(s(x) G t71,oo(0,T), 0 < s(x) < L on [0,T], (S) < ù(x) G W1'2(0, T; L2(0,1)) n L°°(0, T; Hl{0,1)) n L°°((0, T) x (0,1)) [flL2(0, T; H2(0,1)).
The following result for (FBP)(h,so,uo) is already obtained in [4].
Theorem 2. Let h G W1,2(0, T; H)_n L2(0, T; H2(Q)), ht G L°°(Q(T)) with 0 < h < h* a.e. on Q(T), where h* is a positive constant with h* <1. Also, assume that sq G H is such that 0 < so < L — ô for ô > 0 a.e. on Q, and ùo G L°°(Q x (0,1)) and the function x —> |«o(®)|iîi(o,i) is bounded a.e. on Q and ûo(x, 1) = h(x, 0) for a.e. x G Q, and 0 < ûq < 1 a.e. on Q x (0,1). Then, (i) and (ii) hold:
(i) For any T > 0, (FBP)(h(-,x),so(x),ùo(x,-)) has a unique solution (s(-,x),ù(-, x, •)) on [0, T] such that ù G L°°(Q; 1^L2(0, T; L2(0,1))) n
L°°(Q; L°°(0, T; Hl{0,1))) nL°°(Q; L2(0, T; ii2(0,1))) nL°°(Q; L°°((0,1))), s € L°°(Q; M^1,oo(0,T)), 0 < û < 1 a.e. on Q x (0,1) for t € [0,T] and |st| < 2a a.e. on (0, T) x Q. for (s(-,x),u(-,x, •)) with u(-,x,-) =
Ù l~-s(Xx) ) on Qs(x)(T), there exists a positive constant C\ such that
7T T /L l^r(r)|2dzdr + ^ /L \uz(t)\2dz 1 Jo Js(t) 1 Js(t)
<\js \uoz\2dz + ^j^ \st{t)\\uz(t,s(r))|2dr
+ Ci [ (\sT(T)\2 + \hr(T,x)\2)dT + Ciforte[0,T] a.e. onQ. (4.2)
Jo
Moreover, let be a positive constant obtained by (4-2) which satisfies fs(t) \uz(t)\2dz < C2 for t € [0,T], ^en there exists a positive constant depending on k, a, hC^ and L and such thai
0 < s < L — for t € [0, T] a.e. on Q. (4.3)
(ii) For a.e. x € Q, let (si(-, x), ùi(-, x, •)) and (s2(-, x), Ù2(-, x, •)) be solutions of (FBP)(hi(-,x), so(x),ùo(x, •)) and (FBP)(h2(-, x), so(x),ûo(x, •)) on [0, T], respectively, then it holds that
\Ù1 (t) - W2(i)||2(nx(0)i)) + / \Ù1 y{r) - Ù2y(T)\2L2{n
+ |S1 -S2\loomH) < Cs\hl-h2\2wl,2(0tt.H) fort G [0,T], (4.4) where Cs is a positive constant depending on k, a, hC^, pw, pv, L and
5. Local and Global existence 5.1. Local existence
In this section, we prove the local existence of a solution of (P). First, for fixed T > 0 and 6' € (0, L) we define the mapping A : X(T, 5') ->■ X(T, 5') for 5' € (0, L) as follows: for each s € X(T, 6') we denote by h the unique solution of (AP)(s) on [0,T], Then, by Lemmas 3 and 4, h satisfies the assumption of Theorem 2 with ft* = h*. Next, we denote by s the unique solution (u,s) of (FBP)(ft, so,uo) on [0,T]. Here, by (i) of Lemma 4 and Theorem 2 it holds that 0 < s(t) < L - ¿* for t € [0, T] a.e. on Q and I St I <2a a.e. on Q(T), where ¿* is a positive constant obtained by Theorem
2. By Lemma 4 the estimate of \h\wi,2^0tT.Lco^ is independent of 6' so that we can choose € (0, L) independent of 6' in (i) of Theorem 2. Thus, we take 5' = 5* and infer that A : X(T', 6*) ->■ X(T', 6*) for any 0 <T' <T.
Now, we prove that there exists T' < T such that A is a contraction mapping on Wl'2(<d,T';H). Let hi be a solution of (AP)(sj) on [0,T"] and (■Ui, Si) = (Ui, A(§i)) be a solution of (FBP)(ftj, so,uo) on [0, T'\ for ¿ = 1,2. We note the result by Sobolev's embedding theorem in one dimension:
\ui(t,x,0) — v,2(t, x, 0)|2 <Ce\Ul(t,x) -U2(t,x)\L2{0A)\ui(t,x) U2{tj x) |iJl(o,l) j (5.1)
where Ce is a positive constant by Sobolev's embedding theorem. Then, by (4.4) and (5.1) it holds that
rT>
/ \A(s!)t - Mh)t\Hdt Jo
fT>
= / |o(ui(i, •, 0) - u2(t, ; 0) - (p(A(si)) - ^(A{~s2))?Hdt Jo
T' rTi
2Hdt
Jo
rT'
<2a2 ^ jf |ui(i, •, 0) - u2{t, 0)|2Hdt + C2 jf 1 |A(si) - A(s2)|
<2Cea2^J \ui - U2\L2(flx(0,l))\uiy - U2y\L2(Clx(0,l))dt
+ Jo \ui - u2\2L2(nxm)d?j +2a2C2^ |A(si)-A(s2)\2Hdt
<2Cea2VTCs\h1-h2\
in — 11-2\w1>2(0,T';H) ^ ^eC2rr'
rT' 10
By using (3.10) and
rT'
+ 2a2C2 I |A(si)-A{S2)\ldt. (5.2)
|A{hit)) - A(S2(t))& <T' [ |A(5x(r))T - A(52(r))T&dr
Jo
we can obtain from (5.2) that
2 (0,T';H)
Ci(rO|A(si)-A(^)|^1,2(0>r,;H)+C2(r,)|si-S2li2(0>T';H)
<(1+T/2)V
where C\{T') and ¿^(T') are positive constants depending on a, Ce, C3 as in (4.4), M6(T') as in (3.10) and T'. Hence, for a small 0 <T' <T such that 7^(1 + T/2)(Ci(T/) + C2(T')) < we see that A is a contraction
mapping on X(T',5*). Therefore, by Banach's fixed point theorem there exists one and only one s € X(T', such that A(s) = s in X(T', ¿*). This
is a unique solution of (P) on [0,T']. 5.2. Global existence
In this section, we establish a globally-in-time solution of (P). Let define T* := sup{T' > 0| (P) has a solution (h,s,u) on [0,T']}.
By the local existence result, it is clear that T* > 0. Now, we assume T* < T. Then, by Lemma 3, (i) of Lemma 4 and Theorem 2 it holds that for any t < T*,
ko < h(t) < h* a.e. on Q, 0 < u(t) < 1 a.e. on Q x [s(t, •), L], (5.3)
\h\wW(p,t;h) + \h\l°°(0,t;hhn)) < Ml(T), (5-4)
\st\ < 2a, \Vh\ < M2(T), < M4(T) a.e. on Q(T*), (5.5)
rt rL rL
/ / \ut{t)\2(1z(1t + / \uz{t)\2dz < M7{T) a.e. on Q, (5.6) Jo Js(t) Js(t)
where Mj(T) is a positive constant depending on uqz, pv, pw, k, a, M±(T) and C\. Then, by (i) of Theorem 2 and (5.6) we see that there exists a positive constant depending on Mj(T) such that
Next, by ht - AG(h) = s( 1 - f(h))v a.e. on Q(T*), |s(l - f(h))v| < L\v\l°°(q(t)) and (5.5) we infer that AG(h(t)) is bounded in L°°(Q) for a.e. t € [0, T*). Hence, from this result, (5.3), and go < g we have that
where C is a positive constant. Also, by (A6) and (5.4), h(t) — hb(t) is bounded in for a.e. t € [0,T*). Moreover, by (4.1), (5.6) and (5.7)
0 < s < L - ¿»for t € [0, T*) a.e. on Q.
(5.7)
\Ah(f)\boo{Çi) < С for a.e. t € [0,T*)
(5.8)
it follows that for t € [0,T*),
f [1\ùt(r)\2dydt= f [L ut{T)+uz{T){LT z)s.t{^ Jo Jo Jo J sir) L - S(T)
^T f fL \ut{r)\2dzdt + [* [L \uz{r)\2dzd
b* Jo Js(t) b* Jo Js(t)
2 2{ALa)2T
^ M7(T) a.e. on Q
(5.9)
and
rl t-L
uy(t)\2dy = J \uz{t){L - sm2L^s{t)dz
< L [ \uz(t)\2dz < LM7{T) a.e. on Q. (5.10)
Js(t)
Now, let N be a subset of [0, T*) with | JV| = 0 such that for t € [0, T*)\N, it holds that Vh(t) and Ah{t) are bounded in L°°(Q), h{t) — hb(t) is bounded in and u(t,x, 1) = h(t,x) for a.e. x € Q,. Since h(t), s(t) and
u(t) also satisfy (5.3), (5.4) and (5.7) for t € [0,T*) \ N, we can take a sequence {tn} C [0,T*)\ N such that tn —> T* as n —> oo and for some h(T*), s(T*) G H, rji G L°°(Q)( 1 <i< 3) and u(T*) e L2(Q x (0,1)) the following convergences hold:
h(tn) ->■ h(T*) in H, weakly in H1^), (5.11)
dih(tn) ->■ rn weakly - * in L°°(Q), (5.12)
h(tn) - hb(tn) ->■ h(T*) - hb(T*) weakly in (5.13)
s(tn) ->■ s(T*) weakly in H, (5.14)
«(in) ->■ u(T*) weakly in L2(Q x (0,1)) as n ->■ oo, (5.15)
where di is the weak derivative with respect to Xi.
Lemma 5. Let h(T*), s(T*) € H and u(T*) € L2(Q x (0,1)) be functions satisfying (5.11)-(5.15). Then, it holds that
(i) h(T*) e H2(Q) n Wl>°°(ii), no < h(T*) < h* a.e. on Q, Ah(T*) is bounded a.e. on Q and h{T*) = hb{T*) a.e. on dQ.
(ii) 0 < s(T*) < L — 6* a.e. on Q, where 6* is the same as in (5.7). (Hi) 0 < u(T*) < 1 a.e. on Qs(t*)> the function x —> \u(T*, ^)|_h"1(s(t*),l)
is bounded a.e. on Q and u(T*,x,L) = h(T*,x) for a.e.
Proof, (i) By (5.3) and (5.11) it is clear that k0 < h{T*) < h* a.e. on Q. Also, by (5.11) and (5.12), we have that ra = dih(T*) € L°°(Q). Next, we show that h(T*) e H2(Q) n l^1'00^) and Ah(T*) is bounded a.e. on Q. Let define —Ah(T*) in the following distribution sense:
(-Ah(T*),<p) = [ Vh(T*)Vipdx for <p e C0°°(n).
J n
Then, for <p e Cq°(Q) it holds that
(-Ah(T*),<p) = f -h(T*)Aipdx = lim f -h(tn)A<pdx J n J n
= lim / Vh(tn)Vipdx = lim / —Ah(tn)ipdx.
oo JQ n->oo JQ
Hence, by (5.8) it follows that (-Ah(T*),<p) < C\p\Li(n) for у € C0°°(Q). Accordingly, by Hahn-Banach's theorem, there exists I € L°°(Q) such that
{1,<р)ь°°(п),щп) = (-Ah(T*),<p) for <P e C0°°(Q). (5.16)
Consequently, by (5.16) and the regularity result of elliptic problems, we see that h(T*) € H2(Q), and therefore, I = -Ah(T*) e H. Then, by I € L°°(Q) we also obtain that Ah(T*) e L°°(Q). Thus, from these result (i) holds.
(ii) Since 0 < s(tn) < L — 5* a.e. on Q, [s(tra) — (L — ¿*)]+ = 0 and ~[s(tn) — (L — ¿*)]+ < ~(s(tn) - (L - 5*)), it follows that
|[e(T*)-(L-*„)]+& =([s(T*) - (L - 5,)]+, s(T*) - (L - 5,) - [s(tn) - (L - 5*)]+)H <([s(T*) - (L - ¿*)]+, s(T*) - (L - 6*) - (s(tn) — (L — St))H.
Hence, by (5.14), we derive that |[s(T*) - (L - 5*)}+\2H = 0 which implies that s(T*) < L — 5* a.e. on Q. Similarly, we have that 0 < s(T*) a.e. on Q.
(iii) By using (4.1) and (5.9), it holds that for the function u,
(2 2(4 La)2T\
— + ^—J M7(T) a.e. on Q.
This implies that u(tn) is a Cauchy sequence in L2(Q x (0,1)), namely, by (5.15) we see that
u(tn) ->■ u(T*) in L2(Q x (0,1)) as n ->■ oo. (5.17)
By (5.3) and (5.17) it is clear that 0 < й(Т*) < 1 a.e. on Q x (0,1). Also, from (5.11) and (5.17) there exists a subsequence {rik} С {n} and M\ с П with | Mi | =0 such that
h(tnk,x) -+h(T*,x) and u(tnk,x) -+й(Т*,х) in L2(0,1) (5.18)
as к —> oo for x € f2 \ Mi. Moreover, by (5.10), there exists M2 С П with |M2| = 0 such that uy(tnk) is bounded in L2(0,1) on Q \ M2. Then, for \ (Mi U M2), we can take a subsequence {tnk(x)} С {irafc} such that for some ¿¡(ж) € L2(0,1),
v,y(tnk(x), x) —)• {(ж) weakly in L2(0,1). (5.19)
Therefore, by (5.18) and (5.19), we see that {(ж) = йу(Т*,х) in L2(0,1) on n \ (Ml U M2), namely, й(Т*) € Hl(0,1) a.e. on Q.
Finally, we prove that й(Т*,х, 1) = h(T*,x) a.e. on Q. From (5.18) we see that u(tnk(x),x) —>• u(T*,x) in L2(0,1) and h(tnk(x),x) —>• h(T*,x) as
k ^ oo and u(T*,x) € Hl(<d, 1) for x € Q \ (Mi U M2). Here, by (5.1) it holds that
\u{tnk{x),x, 1) - u(T*,x, 1)|2 <Ce\u(tnk(x),x) -u(T*,x)\Hi{0A)\u(tnk(x),x) -u(T*,x)\L2(0,1).
Hence, by the convergence of u(tnk(x)) in L2(0,1) and the fact that u(tnk(x)) is bounded in #*((), 1) for a; € Q \ (Mi U M2) we have that u(tnk(x),x,l) —> u(T*, x, 1) as k —> oo, and therefore u(T*,x,l) = h(T*, x)
a.e. on Q. Thus, we see that u(t*,x,z) = u (t*,x, ¿Z^y*'^)) satisfies
(iii). ' □
By Lemma 5 we can consider h(T*), s(T*) and u(T*) as an initial data. Finally, by repeating the argument of the local existence, we see that the solution can be extended beyond T*. This is a contradiction of the definition of T*, and therefore, T* must coincide with T. Thus, we can show that (P) has a solution globally in time, and Theorem 1 is proved.
6. Conclusion
We studied a two-scale problem as a mathematical model describing moisture transport phenomena arising in concrete carbonation process. Under suitable assumptions, we established a globally-in-time solution of our two-scale model by the method of extending local solutions.
References
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Kota Kumazaki, Doctor of Sciences (Physics and Mathematics), Professor, Department of Education, Nagasaki University, 1-14, Bunkyo-cho, Nagasaki, 852-8521, Japan (e-mail: k.kumazaki@nagasaki-u.ac.jp)
Received 02.05.19
Существование глобального решения для многомерной модели влагопереноса в бетонных материалах
К. Кумазаки
Университет Нагасаки, Нагасаки, Япония
Аннотация. В предыдущем исследовании [5] мы доказали существование локального по времени решения для двумерной задачи, которая дается в качестве математической модели влагопереноса, возникающего в процессе карбонизации бетона. Двумерная модель состоит из уравнения диффузии относительной влажности в макро-области и задач со свободной границей, описывающих процесс смачивания и сушки в бесконечных микро-областях. В этой статье, улучшая уравнение диффузии относительной влажности на основе экспериментального результата [3;10], мы строим глобальное решение двумерной модели. Для доказательства существования глобального решения мы получили равномерные оценки и равномерную ограниченность решения по времени и использовали метод расширения локальных решений.
Ключевые слова: двумерная модель, задача со свободной границей, квазилинейное параболическое уравнение, уравнение влагопереноса.
Кота Кумазаки, доктор физико-математических наук, профессор, педагогический факультет, Университет Нагасаки, 1-14, Бункёмати, Нагасаки, 852-8521, Япония (e-mail: k.kumazaki@nagasaki-u.ac.jp)
Поступила в редакцию 02.05.19
