Local weak solvability of a moving boundary problem describing swelling along a halfline

We obtain the local well-posedness of a moving boundary problem that describes the swelling of a pocket of water within an infinitely thin elongated pore. Our result involves fine a priori estimates of the moving boundary evolution, Banach fixed point arguments as well as an application of the general theory of evolution equations governed by subdifferentials.


Introduction
We wish to understand which effect the water-triggered micro-swelling of pores can have at observable scales of concrete-based materials. Such topic is especially relevant in cold regions, where buildings exposed to extremely low temperatures undergo freezing and build microscopic ice lenses that ultimately lead to the mechanical damage of the material; see, for instance, [18]. One way to tackle this issue from a theoretical point of view is to get a better picture of the transport of moisture. Our long-term goal is to build a macro-micro model for moisture transport suitable for cementitious mixtures, where at the macroscopic scale the transport of moisture follows a porous-media-like equation, while at the microscopic scale the moisture is involved in an adsorption-desorption process leading to a strong local swelling of the pores. Such a perspective would lead to a system of partial differential equations with distributed microstructures, see [8,9] for related settings. In this paper, we propose a one-dimensional microscopic problem posed on a halfline with a moving boundary at one of the ends. The moving boundary conditions encode the swelling mechanism, while a diffusion equation is responsible to providing water content for the swelling to take place.
Since we are interested in how far the water content can actually push the a priori unknown moving boundary of swelling, we assume that pore depth is infinite although the actual physical length is finite. Our target here is to show the wellposedness of the model.
Let us now describe briefly the setting of our equations. The timespan is [0, T ] while the pore is [a, +∞), with a, T ∈ (0, +∞). The variables are t ∈ [0, T ] and z ∈ [a, +∞). The boundary z = a denotes the edge of the pore in contact with wetness. The interval [a, s(t)] indicates the region of diffusion of the water content u(t), where s(t) is the moving interface of the water region. The function u(t) acts in the non-cylindrical region Q s (T ) defined by Q s (T ) := {(t, z)|0 < t < T, a < z < s(t)}.
Our free boundary problem, which we denote by (P) u0,s0,h , reads: Find the pair (u(t, z), s(t)) satisfying u t − ku zz = 0 for (t, z) ∈ Q s (T ), (1.1) − ku z (t, a) = β(h(t) − Hu(t, a)) for t ∈ (0, T ), (1.2) − ku z (t, s(t)) = u(t, s(t))s t (t) for t ∈ (0, T ), (1.3) s t (t) = a 0 (u(t, s(t)) − ϕ(s(t))) for t ∈ (0, T ), (1.4) s(0) = s 0 , u(0, z) = u 0 (z) for z ∈ [a, s 0 ]. (1.5) Here k is a diffusion constant, β is a given adsorption function on R that is equal to 0 for negative input and takes a positive value for positive input, h is a given moisture threshold function on [0, T ], H and a 0 are further given (positive) constants, ϕ is our swelling function defined on R, while s 0 and u 0 are the initial data. From the physical perspective, (1.1) is the diffusion equation displacing u in the unknown region [a, s]; the boundary condition (1.2), imposed at z = a, implies that the moisture content h inflows if h is present at z = a in a larger amount than u. The boundary condition (1.3) at z = s(t) describes the mass conservation at the moving boundary. Indeed, if the flux u z (t, a) at z = a is active on the time interval [t, t + ∆t] for t > 0, namely, s t (t) > 0, then, it holds that s(t) a u(t, z)dz − ku z (t, a)∆t = s(t+∆t) a u(t + ∆t, z)dz.
Hence, by dividing ∆t in both side and letting ∆t → 0 we formally obtain that − ku z (t, a) = s(t) a u t (t, z)dz + s t u(t, s(t)).
This formal argument motivates the structure of the moving boundary condition (1.3). The ordinary differential equation (1.4) describes the growth rate of the free boundary s and it is determined by the balance between the water content u(t, s(t)) at z = s(t) and the swelling expression ϕ(s(t)). It is worth mentioning at this stage that the function ϕ(s(t)) limits the growth of the moving boundary.
From the mathematical point of view, our free boundary problem resembles remotely the classical one phase Stefan problem and its variations for handling superheating, phase transitions, evaporation; compare [15,16,17,19] and references cited therein. Our work contributes to the existing mathematical modeling work of swelling by Fasano and collaborators (see [6,7], e.g.) as well as other authors cf. e.g. [20]. The main difference between these papers and our formulation lies in the choice of the boundary conditions (1.2) and (1.3). Most of the cited settings impose an homogeneous Dirichlet boundary condition at one of the boundaries, while we impose flux boundary conditions at both boundaries. Relation (1.2) will be used in a forthcoming work to connect the microscopic moving boundary discussed here to a macroscopic transport equation.
It is worth mentioning that the literature contains already a number of free boundary problems posed for the corrosion of porous materials. We review here the closest contributions to our setting. For instance, we refer to Muntean and Böhm [12] who proposed a well-posed free boundary problem as mathematical model for the concrete carbonation process in one space dimension; Aiki and Muntean [3,4,5] proved the existence and uniqueness of a solution for a simplified Muntean-Böhmmodel and obtained the large-time behavior of the free boundary as t → ∞. Also, in [1,14], Sato et al. proposed a free boundary problem as a mathematical model of single pore adsorption, a setting very close to ours, and showed the existence of a solution locally in time; Aiki and Murase guaranteed in [3] the existence of a solution globally in time and established the large time behaviour of this solution. Recently, based on the results of Sato et al. [14] and Aiki and Murase [2], Kumazaki et al. proposed in [11] a multiscale model of moisture transport with adsorption, coupling in a particular fashion a macroscopic diffusion equation with the microscopic picture of the model proposed by Sato et al. in [14] and ensured the local existence of a solution of this two-scale problem. We refer the reader to [8,9,13] and references cited therein for comprehensive descriptions of modeling, mathematical analysis and numerical approximation of reaction-diffusion systems posed on multiple space scales in the absence of free or moving boundaries.
The paper is organized as follows: In Section 2, we state the used notation and assumptions as well as our main theorem concerning the existence and uniqueness of a solution for the moving boundary problem. In Section 3, we consider an auxiliary problem focused on finding u for given s and prove the existence of a solution of this problem by relying on the abstract theory of evolution equations governed by timedependent subdifferentials. By using the result of Section 4, we finally prove our main theorem by suitably applying Banach's fixed point theorem and the maximum principle.

Notation and assumptions
In this framework, we use the following basic notations. We denote by | · | X the norm for a Banach space X. The norm and the inner product of a Hilbert space H are denoted by | · | H and (·, ·) H , respectively. Particularly, for Ω ⊂ R, we use the standard notation of the usual Hilbert spaces L 2 (Ω), H 1 (Ω) and H 2 (Ω).
Throughout this paper, we assume the following restrictions on the model parameters and functions: (A1) a, a 0 , H, k and T are positive constants.
For T > 0, let s be a function on [0, T ] and u be a function on Q s (T ) := {(t, z)|0 ≤ t ≤ T, a < s(t)}.
Next, we define our concept of solution to (P) u0,s0,h on [0, T ] in the following way: Definition 2.1. We call that pair (s, u) a solution to (P) u0,s0,h on [0, T ] if the following conditions (S1)-(S6) hold: The main result of this paper is concerned with the existence and uniqueness of a locally in time solution in the sense of Definition 2.1 to the problem (P) u0,s0,h . This result is stated in the next Theorem.
By using the functionũ, we consider now the following problem (P)ũ 0 ,s0,h : In the rest of the paper, we focus on proving Theorem 2.4, which finally will turn to provide candidate solutions for Theorem 2.2.
, then the following statements hold: (1) There exists positive constant C 0 and C 1 such that the following inequalities hold: (2) For t ∈ [0, T ], the functional ψ t is proper, lower semi-continuous, and convex on L 2 (0, 1).
where η is arbtrary positive constant. By taking η suitably in (3.5) and putting In the case u(1) ≤ ϕ(a), then σ(u(1)) = ϕ(a) so that we have the similarly inequality (3.6). Also, we have that where c β is the same constant as in (A3). By adding (3.6) and (3.7), it yields Also, it holds that Therefore, by (3.8) and the estimate of u(0) we see that the statement (1) of Lemma 3.1 holds.
By Lemma 3.1 we obtain the following existence result concerning the solutions to problem (AP)ũ 0 ,f,s,h .
Using the structure of the boundary conditions, we obtain Combining this inequality with (3.11), it follows that Here, we use the Sobolev's embedding therem in one dimensional case: where C e is a positive constant in Sobolev's embedding. By using (3.13), we have (3.14) Taking C 2 = C e (a 0 c ϕ + c β H) and using Young's inequality leads to Now, we put δ s such that s(t) − a ≥ δ s for t ∈ [0, T ]. By (3.14), we obtain Now, by setting Denote by C 3 the coefficient of I(t) arising in the right-hand side. Using Gronwall's inequality to (3.16) gives This implies that that there exists a small T 1 ≤ T such that Γ T1 is a contraction mapping on W 1,2 (Q(T )) ∩ L ∞ (0, T ; H 1 (0, 1)). Therefore, by Banach's fixed point theorem we can findũ ∈ W 1,2 (Q(T )) ∩ L ∞ (0, T ; H 1 (0, 1)) such that Γ T1 (ũ) =ũ. In other words, we can find a solutionũ of (AP)ũ 0,s,h on [0, T 1 ]. Since T 1 is indepedent of the choice of initial value, by repeating the argument of the local existence result, we can extend the solutionũ beyond T 1 . This argument completes the proof of the Lemma.
As next step, for given s ∈ W 1,2 (0, T ) with a < s < L on [0, T ], we construct a solution to problem (AP)ũ 0 ,s,h . Proof. We choose a sequence {s n } ⊂ W 1,∞ (0, T ) and a < δ < L satisfying s n (t) − a ≥ δ on [0, T ] for each n ∈ N, s n → s in W 1,2 (0, T ) as n → ∞. By Lemma 3.3 we can take a sequence {ũ n } of solutions to (AP)ũ 0 ,sn,h on [0, T ]. Then, we see that For the second term in the left hand side, it holds that a 0 σ(ũ n (t, 1))(σ(ũ n (t, 1)) − ϕ(s n (t)))ũ n (t, 1) Accordingly, by a 0 (σ(ũ n (t, 1))) 2ũ n (t, 1) ≥ 0 we obtain that a 0 ϕ(s n (t))σ(ũ n (t, 1))ũ n (t, 1) Using (3.13) it follows that As a consequence, we see from the above two estimates and (3.17) that We denote now the coefficient of |ũ n | 2 L 2 (0,1) in the above inequality by F (t). Then, F ∈ L 1 (0, T ) and Gronwall's inequality yields that Next, for each n ∈ N and h > 0, we can write For the second term of (3.19), we obtain We name as I 1 , I 2 and I 3 the three terms in the last identity. We proceed with estimating them from bellow. For the first term I 1 , using the same notation g 1 and g 2 cf. the proof of Lemma 3.1, it holds that Next, for the term I 2 we have The term I 3 can be dealt with as follows  Combining all these lower bounds and using the fact that t → k/(s n (t)−a) 2 |ũ ny (t)| 2 is continuous on [0, T ], we obtain lim inf Applying this result to (3.19) and letting h → 0, we observe Using Lemma 3.1, we estimate now from above each of the terms J i for 1 ≤ i ≤ 6 that pinpoint each term from the the right-hand side of the above inequality. By using σ(r) ≤ |r| + ϕ(a) for r ∈ R the following upper bounds hold true: Finally, by combining all these estimates, we obtain that Therefore, by setting and using Gronwall's lemma, we have that Therefore, by l ∈ L 2 (0, T ) and combining the latter inequality with (A2) we see that the right hand side of (3.20) is bounded. From this result, we infer that the sequence {ũ n } is bounded in W 1,2 (0, T ; L 2 (0, 1)) and the sequence {ψ (·) (ũ n (·))} is bounded in L ∞ (0, T ). Finally, this result in combination with Lemma 3.1, (3.18) and (3.20) means that the sequence {ũ n } is bounded in W 1,2 (0, T ; L 2 (0, 1)) ∩ L ∞ (0, T ; H 1 (0, 1)). Therefore, we can take a sequence {n k } ⊂ {n} such that for someũ ∈ W 1,2 (0, T ; L 2 (0, 1))∩L ∞ (0, T ; H 1 (0, 1)),ũ n →ũ weakly in W 1,2 (0, T ; L 2 (0, 1)), weakly -* in L ∞ (0, T ; H 1 (0, 1)) and in C(Q(T )) as k → ∞. By letting k → ∞, we get thatũ is a solution of (AP)ũ 0 ,s,h on [0, T ].

Local existence
In this section, using the results obtained in Section 3 we first show that (P)ũ 0,s0 ,h has a solution locally in time. Throughout the rest of this section, we assume (A1)-(A5). Let T > 0 and set L > s 0 . We define the set This setting is constructed such that, relying on (3.18) and (3.20) in Lemma 3.4, the inequality in the next Lemma holds true. where C = C(T,ũ 0 , K, L, h) depending on T ,ũ 0 , K, L and h.