Convergence of Cahn–Hilliard systems to the Stefan problem with dynamic boundary conditions

Abstract

This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by the limit of an equation and a dynamic boundary condition of Cahn–Hilliard-type. Using this Cahn–Hilliard approach, the enthalpy formulation of the Stefan problem is characterized by the asymptotic limit of a fourth-order system that has a double-well structure. The main result obtained for the Stefan problem can also be applied to a wider class of degenerate parabolic equations by setting the nonlinearity to give the general maximal monotone graph.

Keywords

Stefan problem dynamic boundary condition weak solution Cahn–Hilliard system

1. Introduction

The Stefan problem is a well-known mathematical model that describes the solid–liquid phase transition. Among many results in the literature [17,21,26,29], the following enthalpy formulation of the Stefan problem with Dirichlet–Robin-type boundary conditions has been studied in the $L^{2}$ -framework [15]: $\begin{matrix} (1.1) & \frac{\partial u}{\partial t} - Δ β (u) = g in Q : = (0, T) \times Ω, \end{matrix}$ where $0 < T < + \infty$ is a finite time and $Ω \subset R^{d}$ , $d = 2 or 3$ , is a bounded domain with a smooth boundary Γ. The unknown $u : Q \to R$ denotes the enthalpy and $β (u)$ is the temperature; $g : Q \to R$ is a given heat source. In the model of the Stefan problem, $β : R \to R$ is a piecewise linear function of the form: $\begin{matrix} (1.2) & β (r) : = \{\begin{matrix} k_{s} r & r < 0, \\ 0 & 0 ⩽ r ⩽ L, \\ k_{ℓ} (r - L) & r > L; \end{matrix} \end{matrix}$ where $k_{s}$ , $k_{ℓ} > 0$ represent the heat conductivities of the solid and liquid regions, respectively, and $L > 0$ is the latent heat constant. Consider the initial-boundary value problem for this kind of partial differential equation. The dynamic boundary condition is a sort of differential equation that represents the dynamics on the boundary Γ. It is a boundary condition includes a time derivative, which is well treated by Dirichlet-, Neumann-, and Robin-type conditions for various problems. Under dynamic boundary conditions of the form: $\begin{matrix} (1.3) & \frac{\partial β (u)}{\partial t} + \partial_{ν} β (u) = g_{Γ} on Σ : = (0, T) \times Γ, \end{matrix}$ the existence and uniqueness of (1.1) has been studied using a subdifferential approach [1,2], where $\partial_{ν}$ denotes the normal derivative on Γ outward from Ω and $g_{Γ} : Σ \to R$ is a given heat source on the boundary. Results for a more general setting are given in [3].

In this paper, the well-posedness of the Stefan problem with the following dynamic boundary condition is studied: $\begin{matrix} (1.4) & \frac{\partial u}{\partial t} + \partial_{ν} β (u) - Δ_{Γ} β (u) = g_{Γ} on Σ, \end{matrix}$ where $Δ_{Γ}$ denotes the Laplace–Beltrami operator on Γ (see, e.g., [20, Chapter 3]). If we simultaneously consider (1.1) on the bulk Ω and (1.3) or (1.4) on the boundary Γ, then the setting of (1.4) is more natural than that of (1.3) from the viewpoint of total mass conservation [8,11,19].

The main idea of the existence result is to approximate the Stefan problem from the Cahn–Hilliard system. Consider the following system: For $ε > 0$ $\begin{array}{l} (1.5) & \frac{\partial u}{\partial t} - Δ μ = 0 in Q, \\ (1.6) & μ = - ε Δ u + β (u) + ε π (u) - f in Q, \end{array}$ with a dynamic boundary condition of the form: $\begin{array}{l} (1.7) & \frac{\partial u}{\partial t} + \partial_{ν} μ - Δ_{Γ} μ = 0 on Σ, \\ (1.8) & μ = ε \partial_{ν} u - ε Δ_{Γ} u + β_{Γ} (u) + ε π_{Γ} (u) - f_{Γ} on Σ . \end{array}$ The unknowns $u, μ : Q \to R$ represent the order parameter and chemical potential, respectively. Let us recall some basic concepts. The Cahn–Hilliard system is characterized by the nonlinear terms $β + π$ and $β_{Γ} + π_{Γ}$ , which are some derivatives of functions W and $W_{Γ}$ , respectively. These are usually referred to as double-well potentials; for example, $W (r) = W_{Γ} (r) = (1 / 4) {(r^{2} - 1)}^{2}$ . In this case, $β (r) = β_{Γ} (r) = r^{3}$ and $π (r) = π_{Γ} (r) = - r$ for all $r \in R$ . For the details, see [6,16]. Taking $β_{Γ} = β$ as (1.2) and letting $ε \to 0$ in (1.6), we obtain (1.1) as the limiting problem of (1.5) with $μ = β (u) - f$ and some modification of $- Δ f = g$ in Ω and $\partial_{ν} f - Δ_{Γ} f = g_{Γ}$ in Γ, with the setting that the trace $f_{|_{Γ}}$ of f coincides with $f_{Γ}$ . From (1.8), we can then characterize the dynamic boundary condition (1.4) by (1.7). The physical background of this approach can be found in [27] and the references therein.

The existence and uniqueness of the Cahn–Hilliard system (1.5)–(1.8) were first treated for the case $ε > 0$ [19]. Following this, many related problems were examined [8,9]. Recently, the author extended the pioneering work of [19] to the more general case in which β and $β_{Γ}$ are maximal monotone graphs. This included the singular case for the subdifferential $\partial I_{[- 1, 1]}$ of the indicator function $I_{[- 1, 1]}$ , and the case when π and $π_{Γ}$ are Lipschitz continuous functions. The well-posedness of strong and weak solutions for the initial-boundary problem under appropriate assumptions was discussed in [11], the essential idea of which came from [7,13].

The present paper proceeds as follows. In Section 2, the main theorem is stated. First, we prepare the notation used in this paper. Next, we recall a known result for some equations and dynamic boundary conditions of Cahn–Hilliard-type ${(P)}_{ε}$ in Proposition 2.1. The main theorem is related to the convergence of Cahn–Hilliard systems ${(P)}_{ε}$ to the Stefan problem (P) with dynamic boundary conditions. In Section 3, we obtain a uniform estimate that is useful in proving the main theorem. To guarantee sufficient regularity of the unknowns for ${(P)}_{ε}$ , we start from the approximate problem ${(P)}_{ε, λ}$ . After obtaining all necessary estimates, we correct similar uniform estimates for ${(P)}_{ε}$ . A proof of the main theorem is given in Section 4. The strategy of the proof proceeds in a standard manner. Based on the uniform estimates, we consider the limiting procedure $ε \to 0$ . The main theorem is applied under a more general assumption for β, namely that it is not only a non-decreasing piecewise linear function, as in (1.2), but also some maximal monotone graph. Therefore, we apply the monotonicity argument. Finally, in Section 5, the uniqueness of (P) is proved by showing its continuous dependence for the given data.

A detailed index of sections and subsections is as follows:

Introduction

Main results

Notation

Main theorem

Approximate solutions to the Cahn–Hilliard system ${(P)}_{ε}$

Uniform estimates

Uniform estimates for approximate solutions of ${(P)}_{ε, λ}$

Uniform estimates for approximate solutions of ${(P)}_{ε}$

Proof of the main theorem

Continuous dependence

2. Main results

The main theorem is stated in this section. Before doing so, we recall a previous result from [11] that plays an important role in this paper.

2.1. Notation

We use the spaces $H : = L^{2} (Ω)$ , $V : = H^{1} (Ω)$ , $H_{Γ} : = L^{2} (Γ)$ , and $V_{Γ} : = H^{1} (Γ)$ with the usual norms $| \cdot |_{H}$ , $| \cdot |_{V}$ , $| \cdot |_{H_{Γ}}$ , $| \cdot |_{V_{Γ}}$ and inner products ${(\cdot, \cdot)}_{H}$ , ${(\cdot, \cdot)}_{V}$ , ${(\cdot, \cdot)}_{H_{Γ}}$ , ${(\cdot, \cdot)}_{V_{Γ}}$ , respectively. Moreover, $H : = H \times H_{Γ}$ , $V : = {(z, z_{Γ}) \in V \times V_{Γ} : z_{Γ} = z_{|_{Γ}} a.e. on Γ}$ , and $W : = H^{2} (Ω) \times H^{2} (Γ)$ . Hereafter, we use a bold symbol $z$ to denote the pair $(z, z_{Γ})$ corresponding to the letter. Then, $H$ , $V$ , and $W$ are Hilbert spaces with the inner product $\begin{matrix} {(u, z)}_{H} : = {(u, z)}_{H} + {(u_{Γ}, z_{Γ})}_{H_{Γ}} for all u, z \in H, \end{matrix}$ and the related norm is analogously defined as one of $V$ or $W$ . Note that if $z \in V$ , then $z_{Γ}$ is exactly the trace $z_{|_{Γ}}$ of z on Γ, whereas if $z$ is just in $H$ , then $z \in H$ and $z_{Γ} \in H_{Γ}$ are independent. Define $m : H \to R$ by $\begin{matrix} (2.1) & m (z) : = \frac{1}{| Ω | + | Γ |} {\int_{Ω} z d x + \int_{Γ} z_{Γ} d Γ} for all z \in H, \end{matrix}$ where $| Ω | : = \int_{Ω} 1 d x$ and $| Γ | : = \int_{Γ} 1 d Γ$ . The symbol $V^{*}$ denotes the dual space of $V$ , and the pair ${⟨ \cdot, \cdot ⟩}_{V^{*}, V}$ denotes the duality pairing between $V^{*}$ and $V$ . Moreover, we define the bilinear form $a (\cdot, \cdot) : V \times V \to R$ by $\begin{matrix} a (u, z) : = \int_{Ω} \nabla u \cdot \nabla z d x + \int_{Γ} \nabla_{Γ} u_{Γ} \cdot \nabla_{Γ} z_{Γ} d Γ for all u, z \in V, \end{matrix}$ where $\nabla_{Γ}$ denotes the surface gradient on Γ (see, e.g., [20, Chapter 3]). We also introduce the subspace of $H$ as $H_{0} : = {z \in H : m (z) = 0}$ and $V_{0} : = V \cap H_{0}$ with their norms $| z |_{H_{0}} : = | z |_{H}$ for all $z \in H_{0}$ and $| z |_{V_{0}} : = a {(z, z)}^{1 / 2}$ for all $z \in V_{0}$ . Under the Poincaré–Wirtinger inequality (see, e.g., [11, Lemma A]), there exists a positive constant $c_{P}$ such that $\begin{matrix} (2.2) & c_{P} | z |_{V}^{2} ⩽ | z |_{V_{0}}^{2} for all z \in V_{0} . \end{matrix}$ Then, we can define the duality mapping $F : V_{0} \to V_{0}^{*}$ by $\begin{matrix} (2.3) & {⟨ F z, \tilde{z} ⟩}_{V_{0}^{*}, V_{0}} : = a (z, \tilde{z}) for all z, \tilde{z} \in V_{0} . \end{matrix}$ Using this, we can define the inner product in $V_{0}^{*}$ by $\begin{matrix} (2.4) & {(z_{1}^{*}, z_{2}^{*})}_{V_{0}^{*}} : = {⟨ z_{1}^{*}, F^{- 1} z_{2}^{*} ⟩}_{V_{0}^{*}, V_{0}} for all z_{1}^{*}, z_{2}^{*} \in V_{0}^{*} . \end{matrix}$ Then, we obtain the dense and compact embeddings $V_{0} ↪ ↪ H_{0} ↪ ↪ V_{0}^{*}$ ; see [11] for details of this setting. These are essentially the same as in previous studies [10,23–25].

2.2. Main theorem

In this subsection, we define the weak solution for the Stefan problem with a dynamic boundary condition. Then, we state the main theorem.

First, we present the target problem (P) of this paper, namely the Stefan problem with the dynamic boundary condition: $\begin{matrix} (2.5) & \begin{matrix} \frac{\partial u}{\partial t} - Δ β (u) = g a.e. in Q, \\ β (u_{Γ}) = β {(u)}_{|_{Γ}}, \frac{\partial u_{Γ}}{\partial t} + \partial_{ν} β (u) - Δ_{Γ} β (u_{Γ}) = g_{Γ} a.e. on Σ, \\ u (0) = u_{0} a.e. in Ω, u_{Γ} (0) = u_{0 Γ} a.e. on Γ, \end{matrix} \end{matrix}$ where the prototype Stefan problem is formulated by the setting (1.2). In this paper, the target problem (P) will be formulated in a more general setting; see Definition 2.1.

Remark 1.
The dynamic boundary condition comes from previous results [1,2] regarding not only the Laplace–Beltrami operator, but also the time derivative. Indeed, these previous studies treated $\partial β (u_{Γ}) / \partial t$ . In the case of (1.2), the first condition (2.5) implies that $u_{Γ}$ is not necessarily equal to the trace $u_{|_{Γ}}$ of u. More precisely, we will obtain $u_{Γ} = u_{|_{Γ}}$ except in the mushy region ${x \in Ω : 0 ⩽ u (x) ⩽ L}$ of the Stefan problem. This is because we can only expect $β (u (t)) \in V$ , whereas it may not hold that $u (t) \in V$ (weak solution).

Throughout this paper, we assume that

β is a maximal monotone graph in $R \times R$ , and is a subdifferential $β = \partial \hat{β}$ of some proper, lower semicontinuous, and convex function $\hat{β} : R \to [0, + \infty]$ satisfying $\hat{β} (0) = 0$ with some effective domain $D (β)$ . This implies $β (0) = 0$ . Moreover, there exist two constants $c_{1}$ , $c_{2} > 0$ such that $\begin{matrix} (2.6) & \hat{β} (r) ⩾ c_{1} | r |^{2} - c_{2} for all r \in R; \end{matrix}$

$g \in L^{2} (0, T; H_{0})$ ;

$u_{0} : = (u_{0}, u_{0 Γ}) \in V$ with $m_{0} : = m (u_{0}) \in int D (β)$ , and the compatibility conditions $\hat{β} (u_{0}) \in L^{1} (Ω)$ , $\hat{β} (u_{0 Γ}) \in L^{1} (Γ)$ hold.

We now define a weak solution.
Definition 2.1.
The pair $(u, ξ)$ of functions $u \in H^{1} (0, T; V^{}) \cap L^{\infty} (0, T; H)$ and $ξ \in L^{2} (0, T; V)$ is called the weak solution of (P) if $\begin{matrix} ξ \in β (u) a.e. in Q, ξ_{Γ} \in β (u_{Γ}), ξ_{Γ} = ξ_{|_{Γ}} a.e. on Σ, \end{matrix}$ and they satisfy $\begin{array}{l} {⟨ u^{'} (t), z ⟩}_{V^{}, V} + {⟨ u_{Γ}^{'} (t), z_{Γ} ⟩}_{V_{Γ}^{}, V_{Γ}} + \int_{Ω} \nabla ξ (t) \cdot \nabla z d x + \int_{Γ} \nabla_{Γ} ξ_{Γ} (t) \cdot \nabla_{Γ} z_{Γ} d Γ \\ (2.7) & = \int_{Ω} g (t) z d x + \int_{Γ} g_{Γ} (t) z_{Γ} d Γ for all z = (z, z_{Γ}) \in V \end{array}$ for a.a. $t \in (0, T)$ with $u (0) = u_{0}$ a.e. in Ω and $u_{Γ} (0) = u_{0 Γ}$ a.e. on Γ.

Our main theorem is now stated.
Theorem 2.1.
Assume* (A1)–(A3) hold. Then, there exists at least one weak solution $(u, ξ)$ of (P).

Moreover, we obtain the following continuous dependence for the given data:
Theorem 2.2.
Assume (A1). For $i = 1, 2$ , let $(u^{(i)}, ξ^{(i)})$ be a weak solution of (P) corresponding to the data $g^{(i)}$ and $u_{0}^{(i)}$ , which satisfy (A2) and (A3) with $m_{0} : = m (u_{0}^{(1)}) = m (u_{0}^{(2)})$ , respectively. Then, there exists a positive constant C depending only on T such that $\begin{matrix} (2.8) & {| u^{(1)} (t) - u^{(2)} (t) |}_{V_{0}^{}}^{2} ⩽ C {{| u_{0}^{(1)} - u_{0}^{(2)} |}_{V_{0}^{}}^{2} + \int_{0}^{T} {| g^{(1)} (s) - g^{(2)} (s) |}_{H_{0}}^{2} d s} \end{matrix}$ for all $t \in [0, T]$ . Moreover, if β is Lipschitz continuous, then $\begin{array}{l} \int_{0}^{t} {| ξ^{(1)} (s) - ξ^{(2)} (s) |}_{H}^{2} d s \\ (2.9) & ⩽ \frac{c_{β}}{2} C (1 + T) {{| u_{0}^{(1)} - u_{0}^{(2)} |}_{V_{0}^{}}^{2} + \int_{0}^{T} {| g^{(1)} (s) - g^{(2)} (s) |}_{H_{0}}^{2} d s} \end{array}$ for all* $t \in [0, T]$ , where $c_{β} > 0$ is the Lipschitz constant of β.

Note that the assumption $m (u_{0}^{(1)}) = m (u_{0}^{(2)})$ implies that $u^{(1)} (t) - u^{(2)} (t) \in H_{0}$ for a.a. $t \in (0, T)$ . This theorem implies the uniqueness of the weak solution of (P).
2.3. Approximate solutions to the Cahn–Hilliard system ${(P)}_{ε}$

In this subsection, we state the approximate problem for (P). For this purpose, we recall a previous result [11] for the equation and dynamic boundary condition of the Cahn–Hilliard-type ${(P)}_{ε}$ . This can be written as the following initial-boundary value problem (2.10)–(2.14): For $ε > 0$ , $\begin{array}{l} (2.10) & \frac{\partial u}{\partial t} - Δ μ = 0 a.e. in Q, \\ (2.11) & ξ \in β (u), μ = - ε Δ u + ξ + ε π (u) - f a.e. in Q, \\ (2.12) & u_{Γ} = u_{|_{Γ}}, μ_{Γ} = μ_{|_{Γ}}, \frac{\partial u_{Γ}}{\partial t} + \partial_{ν} μ - Δ_{Γ} μ_{Γ} = 0 a.e. on Σ, \\ (2.13) & ξ_{Γ} \in β (u_{Γ}), μ_{Γ} = ε \partial_{ν} u - ε Δ_{Γ} u_{Γ} + ξ_{Γ} + ε π (u_{Γ}) - f_{Γ} a.e. on Σ, \\ (2.14) & u (0) = u_{0} a.e. in Ω, u_{Γ} (0) = u_{0 Γ} a.e. on Γ, \end{array}$ where $f : Q \to R$ , $f_{Γ} : Σ \to R$ , $u_{0} : Ω \to R$ , and $u_{0 Γ} : Γ \to R$ are given data. In particular, f and $f_{Γ}$ are constructed by g and $g_{Γ}$ at a later point. Moreover, we assume that

$π : D (π) = R \to R$ is Lipschitz continuous.

In the main theorem, we treated the general setting (A1) of β for the Stefan problem with some suitable π. In the setting of (1.2), one example of

π : R \to R

is a piecewise linear function of the following form:

\begin{matrix} (2.15) & β (r) : = \{\begin{matrix} k_{s} r & if r < 0, \\ 0 & if 0 ⩽ r ⩽ L, \\ k_{ℓ} (r - L) & if r > L, \end{matrix} π (r) : = \{\begin{matrix} \frac{L}{2} & if r < 0, \\ \frac{L}{2} - r & if 0 ⩽ r ⩽ L, \\ - \frac{L}{2} & if r > L . \end{matrix} \end{matrix}

Of course, in this case,

ξ = β (u)

and

ξ_{Γ} = β (u_{Γ})

. Therefore, (A1) and (A4) hold; actually,

\hat{β}

is obtained by

\begin{matrix} \hat{β} (r) : = \{\begin{matrix} \frac{k_{s}}{2} r^{2} & if r < 0, \\ 0 & if 0 ⩽ r ⩽ L, \\ \frac{k_{ℓ}}{2} {(r - L)}^{2} & if r > L, \end{matrix} \end{matrix}

so that

β = \partial \hat{β}

, and this quadratic function in

\hat{β}

satisfies (2.6). This value of

ε π

enables us to realize the double-well structure of the potential

W = \hat{β} + ε \hat{π}

as the sum of primitives of β and

ε π

. Therefore,

{(P)}_{ε}

with the prototype setting (2.15) has the exact structure of the Cahn–Hilliard system. Viewed in terms of the Stefan problem, π is an artificial term. Therefore, hereafter, we simply assume that

\begin{matrix} (2.16) & | π^{'} (r) | ⩽ 1 for a.a. r \in R . \end{matrix}

Remark 2.
The treatment of the Cahn–Hilliard approach is completely independent of the choice of boundary condition. We will focus on the convergence from the Cahn–Hilliard system to the degenerate parabolic equation under the Neumann boundary condition in a forthcoming paper [12].

Here, we see that for each $g : = (g, g_{Γ}) \in L^{2} (0, T; H_{0})$ , there exists $f \in L^{2} (0, T; V_{0})$ such that $\begin{array}{l} (2.17) & \int_{Ω} \nabla f \cdot \nabla z d x + \int_{Γ} \nabla_{Γ} f_{Γ} \cdot \nabla_{Γ} z_{Γ} d Γ = \int_{Ω} g z d x + \int_{Γ} g_{Γ} z_{Γ} d Γ for all z \in V_{0}, \end{array}$ a.e. in $(0, T)$ (see [11, Lemma C], also (2.24) and (2.25)). To obtain the previous known results, we use the change of variable $v : = u - m_{0} 1$ , where $1 : = (1, 1)$ , the pair of constants. We also use $v_{0} : = u_{0} - m_{0} 1$ . We set $β (z) : = (β (z), β (z_{Γ}))$ , $π (z) : = (π (z), π (z_{Γ}))$ for all $z \in H$ . Under assumptions (A1)–(A3), [11, Theorem 2.2] leads to the following proposition for the existence and uniqueness of the equation and dynamic boundary condition of Cahn–Hilliard-type ${(P)}_{ε}$ with respect to our $β + ε π$ .
Proposition 2.1.
Under assumptions (A1)–(A4), there exists a triplet $(v_{ε}, μ_{ε}, ξ_{ε})$ of $\begin{array}{l} v_{ε} \in H^{1} (0, T; V_{0}^{}) \cap L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; W), \\ μ_{ε} \in L^{2} (0, T; V), ξ_{ε} \in L^{2} (0, T; H), \end{array}$ that satisfies* $\begin{array}{l} (2.18) & {⟨ v_{ε}^{'} (t), z ⟩}_{V_{0}^{}, V_{0}} + a (μ_{ε} (t), z) = 0 for all z \in V_{0}, \\ (2.19) & {(μ_{ε} (t), z)}_{H} = ε a (v_{ε} (t), z) + {(ξ_{ε} (t) + ε π (v_{ε} (t) + m_{0} 1) - f (t), z)}_{H} for all z \in V, \end{array}$ for a.a.* $t \in (0, T)$ , with $\begin{matrix} ξ_{ε} \in β (v_{ε} + m_{0}) a.e. in Q, ξ_{Γ, ε} \in β (v_{Γ, ε} + m_{0}) a.e. on Σ . \end{matrix}$ Moreover, $v_{ε} (0) = v_{0}$ holds in $H_{0}$ .

We call the solution obtained by this proposition a weak solution for the problem in (2.10)–(2.14). This proposition is a direct consequence of [11, Theorem 2.2]. Indeed, we assumed (A1)–(A4), and then, from the construction (2.17) of $f$ , obtained $f \in L^{2} (0, T; V_{0})$ . Thus, all of the conditions needed to apply [11, Theorem 2.2] were satisfied.

To obtain uniform estimates with respect to $ε \in (0, 1]$ , we must consider the approximate problem for ${(P)}_{ε}$ , which is the same strategy used to prove Proposition 2.1. Therefore, we give only a sketch of the proof here. For each $λ \in (0, 1]$ , consider the approximate problem ${(P)}_{ε, λ}$ $\begin{matrix} (2.20) & \begin{matrix} {⟨ v_{ε, λ}^{'} (t), z ⟩}_{V_{0}^{}, V_{0}} + a (μ_{ε, λ} (t), z) = 0 for all z \in V_{0}, \\ {(μ_{ε, λ} (t), z)}_{H} = λ {(v_{ε, λ}^{'} (t), z)}_{H} + ε a (v_{ε, λ} (t), z) \\ + {(β_{λ} (v_{ε, λ} (t) + m_{0} 1) + ε π (v_{ε, λ} (t) + m_{0} 1) - f (t), z)}_{H} for all z \in V, \end{matrix} \end{matrix}$ with $v_{ε, λ} (0) = v_{0}$ in $H_{0}$ , where $β_{λ} (z) : = (β_{λ} (z), β_{λ} (z_{Γ}))$ for all $z \in H$ and $β_{λ}$ is the Yosida approximation (see, e.g., [4,5,22]). Namely, $β_{λ} : R \to R$ along with the associated resolvent operator $J_{λ} : R \to R$ are defined by $\begin{matrix} β_{λ} (r) : = \frac{1}{λ} (r - J_{λ} (r)) : = \frac{1}{λ} (r - {(I + λ β)}^{- 1} (r)) for all r \in R . \end{matrix}$ Moreover, the related Moreau–Yosida regularization ${\hat{β}}_{λ}$ of $\hat{β} : R \to R$ fulfills $\begin{matrix} (2.21) & {\hat{β}}_{λ} (r) : = inf_{l \in R} {\frac{1}{2 λ} | r - l |^{2} + \hat{β} (l)} = \frac{1}{2 λ} {| r - J_{λ} (r) |}^{2} + \hat{β} (J_{λ} (r)) = \int_{0}^{r} β_{λ} (l) d l \end{matrix}$ for all $r \in R$ . We know the basic property $\begin{matrix} (2.22) & 0 ⩽ {\hat{β}}_{λ} (r) ⩽ \hat{β} (r) for all r \in R . \end{matrix}$

The problem ${(P)}_{ε, λ}$ can be solved (see, e.g., [10,11,14,23–25]); more precisely, there exist unique $v_{ε, λ} \in H^{1} (0, T; H_{0}) \cap L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; W)$ and $μ_{ε, λ} \in L^{2} (0, T; V)$ that satisfy $\begin{array}{l} λ v_{ε, λ}^{'} (t) + F^{- 1} v_{ε, λ}^{'} (t) + ε \partial φ (v_{ε, λ} (t)) \\ (2.23) & = P (- β_{λ} (v_{ε, λ} (t) + m_{0} 1) - ε π (v_{ε, λ} (t) + m_{0} 1) + f (t)) in H_{0} \end{array}$ for a.a. $t \in (0, T)$ with $v_{ε, λ} (0) = v_{0}$ in $H_{0}$ , where $φ : H_{0} \to [0, + \infty]$ is a proper, lower semicontinuous, and convex functional $\begin{matrix} (2.24) & φ (z) : = \{\begin{matrix} \frac{1}{2} \int_{Ω} | \nabla z |^{2} d x + \frac{1}{2} \int_{Γ} | \nabla_{Γ} z_{Γ} |^{2} d Γ & if z \in V_{0}, \\ + \infty & otherwise . \end{matrix} \end{matrix}$ Here, the subdifferential $\partial φ$ on $H_{0}$ is characterized by $\partial φ (z) = (- Δ z, \partial_{ν} z - Δ_{Γ} z_{Γ})$ with $z \in D (\partial φ) = W \cap V_{0}$ (see, e.g., [11, Lemma C]). We also note that $\begin{matrix} (2.25) & 2 φ (z) = a (z, z) = {⟨ F z, z ⟩}_{V_{0}^{}, V_{0}} = | z |_{V_{0}}^{2} for all z \in V_{0} . \end{matrix}$ Moreover, $P : H \to H_{0}$ is a projection defined by $P z : = z - m (z) 1$ for all $z \in H$ , and it satisfies $\begin{array}{l} {(z_{0}, P \tilde{z})}_{H_{0}} = {(z_{0}, \tilde{z})}_{H} for all z_{0} \in H_{0} and \tilde{z} \in H, \\ | P z |_{V_{0}} = | z |_{V_{0}} ⩽ | z |_{V} for all z \in V . \end{array}$ The standard strategy obtains a priori estimates with respect to $λ \in (0, 1]$ and considers the limiting procedure $λ \to 0$ . Remark 3.
Let $u_{ε} : = v_{ε} + m_{0} 1$ . Then, [11, Remark 2] means that (2.18) implies $\begin{matrix} {⟨ u_{ε}^{'} (t), z ⟩}_{V^{}, V} + {⟨ u_{Γ, ε}^{'} (t), z_{Γ} ⟩}_{V_{Γ}^{}, V_{Γ}} + \int_{Ω} \nabla μ_{ε} (t) \cdot \nabla z d x + \int_{Γ} \nabla_{Γ} μ_{Γ, ε} (t) \cdot \nabla_{Γ} z_{Γ} d Γ = 0 \end{matrix}$ for all $z \in V$ . Indeed, the fact that ${P z : z \in V, | z |_{V} ⩽ 1} \subset {z \in V_{0} : | z |_{V_{0}} ⩽ 1}$ implies $u_{ε}^{'} (t) \in V^{*}$ for all $t \in [0, T]$ . Therefore, the Hahn–Banach extension theorem works, because $V$ is a closed subspace of $V \times V_{Γ}$ . Moreover, thanks to the regularity $v_{ε} \in L^{2} (0, T; W)$ , we see that (2.19) implies $\begin{array}{l} μ_{ε} = - ε Δ u_{ε} + ξ_{ε} + ε π (u_{ε}) - f a.e. in Q, \\ μ_{Γ, ε} = ε \partial_{ν} u_{ε} - ε Δ_{Γ} u_{Γ, ε} + ξ_{Γ, ε} + ε π_{Γ} (u_{Γ, ε}) - f_{Γ} a.e. on Σ . \end{array}$

3. Uniform estimates

In this section, we obtain the uniform estimates needed to prove the main theorem.

3.1. Uniform estimates for approximate solutions of ${(P)}_{ε, λ}$

In this subsection, we obtain uniform estimates for the approximate solutions of ${(P)}_{ε, λ}$ . We recall the change of variable $u_{ε, λ} : = v_{ε, λ} + m_{0} 1$ .

Lemma 3.1.
There exist positive constants $M_{1}$ and $M_{2}$ , independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that $\begin{array}{l} λ {| v_{ε, λ} (t) |}_{H_{0}}^{2} + {| v_{ε, λ} (t) |}_{V_{0}^{}}^{2} ⩽ M_{1}, \\ \frac{ε}{2} \int_{0}^{t} {| v_{ε, λ} (s) |}_{V_{0}}^{2} d s + 2 \int_{0}^{t} {| {\hat{β}}_{λ} (u_{ε, λ} (s)) |}_{L^{1} (Ω)} d s + 2 \int_{0}^{t} {| {\hat{β}}_{λ} (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)} d s ⩽ M_{2} \end{array}$ for all* $t \in [0, T]$ .
Proof.
We test (2.23) at time $s \in (0, T)$ by $v_{ε, λ} (s) \in V_{0}$ , which is considered in the problem ${(P)}_{ε, λ}$ . Then, using (2.4), we have $\begin{array}{l} λ {(v_{ε, λ}^{'} (s), v_{ε, λ} (s))}_{H_{0}} + {(v_{ε, λ}^{'} (s), v_{ε, λ} (s))}_{V_{0}^{}} + ε {(\partial φ (v_{ε, λ} (s)), v_{ε, λ} (s))}_{H_{0}} \\ + {(P β_{λ} (v_{ε, λ} (s) + m_{0} 1), v_{ε, λ} (s))}_{H_{0}} \\ (3.1) & = {(f (s) - ε P π (v_{ε, λ} (s)), v_{ε, λ} (s))}_{H_{0}} \end{array}$ for a.a. $s \in (0, T)$ . Now, from the definition of the subdifferential, we have $\begin{array}{l} {(P β_{λ} (v_{ε, λ} (s) + m_{0} 1), v_{ε, λ} (s))}_{H_{0}} \\ = \int_{Ω} β_{λ} (u_{ε, λ} (s)) (u_{ε, λ} (s) - m_{0}) d x + \int_{Γ} β_{λ} (u_{Γ, ε, λ} (s)) (u_{Γ, ε, λ} (s) - m_{0}) d Γ \\ (3.2) & ⩾ \int_{Ω} {\hat{β}}_{λ} (u_{ε, λ} (s)) d x - {\hat{β}}_{λ} (m_{0}) | Ω | + \int_{Γ} {\hat{β}}_{λ} (u_{Γ, ε, λ} (s)) d Γ - {\hat{β}}_{λ} (m_{0}) | Γ | \end{array}$ for a.a. $s \in (0, T)$ . Next, recalling the fundamental property of the chain rule with (2.16) and using Young’s inequality, we have $\begin{array}{l} - ε {⟨ v_{ε, λ} (s), P π (v_{ε, λ} (s) + m_{0} 1) ⟩}_{V_{0}^{}, V_{0}} \\ ⩽ \frac{1}{4} {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} + ε^{2} \int_{Ω} {| \nabla π (v_{ε, λ} (s) + m_{0}) |}^{2} d x + ε^{2} \int_{Γ} {| \nabla_{Γ} π (v_{Γ, ε, λ} (s) + m_{0}) |}^{2} d Γ \\ (3.3) & ⩽ \frac{1}{4} {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} + ε^{2} {| v_{ε, λ} (s) |}_{V_{0}}^{2}, \end{array}$ and $\begin{matrix} (3.4) & {⟨ v_{ε, λ} (s), f (s) ⟩}_{V_{0}^{}, V_{0}} ⩽ \frac{1}{4} {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} + {| f (s) |}_{V_{0}}^{2} \end{matrix}$ for a.a. $s \in (0, T)$ . Thus, correcting (3.1)–(3.4), we recall the definition of the subdifferential and use (2.22), (2.25), and $φ (0) = 0$ to obtain $\begin{array}{l} λ \frac{d}{d s} {| v_{ε, λ} (s) |}_{H_{0}}^{2} + \frac{d}{d s} {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} + (ε - 2 ε^{2}) {| v_{ε, λ} (s) |}_{V_{0}}^{2} \\ + 2 {| {\hat{β}}_{λ} (u_{ε, λ} (s)) |}_{L^{1} (Ω)} + 2 {| {\hat{β}}_{λ} (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)} \\ (3.5) & ⩽ 2 (| Ω | + | Γ |) \hat{β} (m_{0}) + {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} + 2 {| f (s) |}_{V_{0}}^{2} \end{array}$ for a.a. $s \in (0, T)$ . If we take $ε \in (0, 1 / 4]$ , then $ε - 2 ε^{2} ⩾ ε / 2 > 0$ . By virtue of Gronwall’s inequality, we obtain $\begin{array}{l} λ {| v_{ε, λ} (t) |}_{H_{0}}^{2} + {| v_{ε, λ} (t) |}_{V_{0}^{}}^{2} \\ ⩽ (| v_{0} |_{H_{0}}^{2} + | v_{0} |_{V_{0}^{}}^{2} + 2 T (| Ω | + | Γ |) \hat{β} (m_{0}) + 2 | f |_{L^{2} (0, T; V_{0})}^{2}) e^{T} = : M_{1} \end{array}$ for all $t \in [0, T]$ . Next, integrating (3.5) over $(0, t)$ with respect to s, we obtain $\begin{matrix} \begin{matrix} \frac{ε}{2} \int_{0}^{t} {| v_{ε, λ} (s) |}_{V_{0}}^{2} d s + 2 \int_{0}^{t} {| {\hat{β}}_{λ} (u_{ε, λ} (s)) |}_{L^{1} (Ω)} d s + 2 \int_{0}^{t} {| {\hat{β}}_{λ} (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)} d s \\ ⩽ | v_{0} |_{H_{0}}^{2} + | v_{0} |_{V_{0}^{}}^{2} + 2 T (| Ω | + | Γ |) \hat{β} (m_{0}) + 2 | f |_{L^{2} (0, T; V_{0})}^{2} + \int_{0}^{t} {| v_{ε, λ} (s) |}_{V_{0}^{}}^{2} d s \\ ⩽ M_{1} (1 + T) : = M_{2} \end{matrix} \end{matrix}$ for all $t \in [0, T]$ . Thus, we have the conclusion. □
Lemma 3.2.
There exists a positive constant $M_{3}$ , independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that $\begin{array}{l} 2 λ \int_{0}^{t} {| v_{ε, λ}^{'} (s) |}_{H_{0}}^{2} d s + \int_{0}^{t} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} d s + ε {| v_{ε, λ} (t) |}_{V_{0}}^{2} \\ (3.6) & + 2 {| {\hat{β}}_{λ} (u_{ε, λ} (t)) |}_{L^{1} (Ω)} + 2 {| {\hat{β}}_{λ} (u_{Γ, ε, λ} (t)) |}_{L^{1} (Γ)} ⩽ M_{3}, \\ (3.7) & \int_{0}^{t} {| P μ_{ε, λ} (s) |}_{V_{0}}^{2} d s ⩽ M_{3} \end{array}$ for all* $t \in [0, T]$ .
Proof.
We test (2.23) at time $s \in (0, T)$ by $v_{ε, λ}^{'} (s) \in H_{0}$ . Then, using the same method as for (3.3), we have $\begin{matrix} - ε {⟨ v_{ε, λ}^{'} (s), P π (v_{ε, λ} (s) + m_{0} 1) ⟩}_{V_{0}^{}, V_{0}} ⩽ \frac{1}{4} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} + ε^{2} {| v_{ε, λ} (s) |}_{V_{0}}^{2}, \end{matrix}$ and $\begin{matrix} {⟨ v_{ε, λ}^{'} (s), f (s) ⟩}_{V_{0}^{}, V_{0}} ⩽ \frac{1}{4} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} + {| f (s) |}_{V_{0}}^{2} \end{matrix}$ for a.a. $s \in (0, T)$ . Then, using (2.4) and the chain rule, we deduce $\begin{array}{l} λ {| v_{ε, λ}^{'} (s) |}_{H_{0}}^{2} + \frac{1}{2} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} + ε \frac{d}{d s} φ (v_{ε, λ} (s)) \\ + \frac{d}{d s} \int_{Ω} {\hat{β}}_{λ} (v_{ε, λ} (s) + m_{0}) d x + \frac{d}{d s} \int_{Γ} {\hat{β}}_{λ} (v_{Γ, ε, λ} (s) + m_{0}) d Γ \\ (3.8) & ⩽ ε^{2} {| v_{ε, λ} (s) |}_{V_{0}}^{2} + {| f (s) |}_{V_{0}}^{2} \end{array}$ for a.a. $s \in (0, T)$ . Integrating (3.8) over $(0, t)$ with respect to s, we can find a positive constant $M_{3}$ , depending only on $| v_{0} |_{V_{0}}$ , $| \hat{β} (u_{0}) |_{L^{1} (Ω)}$ , $| \hat{β} (u_{0 Γ}) |_{L^{1} (Γ)}$ , $M_{2}$ , and $| f |_{L^{2} (0, T; V_{0})}$ , such that the aforementioned estimate (3.6) holds. Next, to obtain (3.7), we recall (2.20) (see also Remark 3). We have $\begin{matrix} (3.9) & μ_{ε, λ} (s) = λ v_{ε, λ}^{'} (s) + ε \partial φ (v_{ε, λ} (s)) + β_{λ} (u_{ε, λ} (s)) + ε π (u_{ε, λ} (s)) - f (s) in V, \end{matrix}$ for a.a. $s \in (0, T)$ . Therefore, the evolution equation (2.23) is equivalent to $\begin{matrix} (3.10) & v_{ε, λ}^{'} (s) + F (P μ_{ε, λ} (s)) = 0 in V_{0}^{}, for a.a. s \in (0, T) \end{matrix}$ with (3.9). We test (3.10) by $P μ_{ε, λ} (s) \in V_{0}$ , and integrate the resultant over $(0, t)$ with respect to s. Then, using (2.3) and Young’s inequality, we have $\begin{matrix} \int_{0}^{t} {| P μ_{ε, λ} (s) |}_{V_{0}}^{2} d s ⩽ \frac{1}{2} \int_{0}^{t} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} d s + \frac{1}{2} \int_{0}^{t} {| P μ_{ε, λ} (s) |}_{V_{0}}^{2} d s \end{matrix}$ for all $t \in [0, T]$ . Thus, using (3.6), we obtain (3.7). □
Lemma 3.3.
There exist two positive constants* $M_{4}$ and $M_{5}$ , independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that $\begin{array}{l} (3.11) & {| u_{ε, λ} (t) |}_{H}^{2} ⩽ M_{4}, {| v_{ε, λ} (t) |}_{H_{0}}^{2} ⩽ M_{4}, \\ (3.12) & \int_{0}^{t} {| β_{λ} (u_{ε, λ} (s)) |}_{L^{1} (Ω)}^{2} d s + \int_{0}^{t} {| β_{λ} (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)}^{2} d s ⩽ M_{5} \end{array}$ for all $t \in [0, T]$ .
Proof.
From (2.6) in (A1) and (2.21), $\begin{array}{l} {\hat{β}}_{λ} (r) & = \frac{1}{2 λ} {| r - J_{λ} (r) |}^{2} + \hat{β} (J_{λ} (r)) \\ ⩾ \frac{1}{2 λ} {| r - J_{λ} (r) |}^{2} + c_{1} {| J_{λ} (r) |}^{2} - c_{2} for all r \in R . \end{array}$ Therefore, from (3.6) in Lemma 3.2, we obtain $\begin{array}{l} \frac{1}{2 λ} {| u_{ε, λ} (t) - J_{λ} (u_{ε, λ} (t)) |}_{H}^{2} + c_{1} {| J_{λ} (u_{ε, λ} (t)) |}_{H}^{2} ⩽ \frac{M_{3}}{2} + c_{2} | Ω |, \\ \frac{1}{2 λ} {| u_{Γ, ε, λ} (t) - J_{λ} (u_{Γ, ε, λ} (t)) |}_{H_{Γ}}^{2} + c_{1} {| J_{λ} (u_{Γ, ε, λ} (t)) |}_{H_{Γ}}^{2} ⩽ \frac{M_{3}}{2} + c_{2} | Γ | \end{array}$ for all $t \in [0, T]$ . Thus, $λ \in (0, 1]$ implies that there exists some ${\tilde{M}}_{4} > 0$ , depending only on $c_{1}$ , $c_{2}$ , $M_{3}$ , $| Ω |$ , and $| Γ |$ , such that $\begin{array}{l} {| u_{ε, λ} (t) |}_{H}^{2} = & {| u_{ε, λ} (t) |}_{H}^{2} + {| u_{Γ, ε, λ} (t) |}_{H_{Γ}}^{2} \\ ⩽ & 2 {| u_{ε, λ} (t) - J_{λ} (u_{ε, λ} (t)) |}_{H}^{2} + 2 {| J_{λ} (u_{ε, λ} (t)) |}_{H}^{2} \\ + 2 {| u_{Γ, ε, λ} (t) - J_{λ} (u_{Γ, ε, λ} (t)) |}_{H_{Γ}}^{2} + 2 {| J_{λ} (u_{Γ, ε, λ} (t)) |}_{H_{Γ}}^{2} \\ ⩽ & {\tilde{M}}_{4} \end{array}$ for all $t \in [0, T]$ . Then, there exists some $M_{4} > 0$ , depending only on ${\tilde{M}}_{4}$ , $| m_{0} |$ , $| Ω |$ , and $| Γ |$ , such that $\begin{array}{l} {| v_{ε, λ} (t) |}_{H}^{2} & ⩽ 2 {| u_{ε, λ} (t) |}_{H}^{2} + 2 | m_{0} |^{2} (| Ω | + | Γ |) \\ ⩽ M_{4} \end{array}$ for all $t \in [0, T]$ . Thus, we have obtained (3.11). Next, recall the useful inequality in [18, Section 5]. Indeed, from assumption (A4), we have $m_{0} \in int D (β)$ (which is the criterion needed to apply the inequality), meaning that there exist two constants $c_{3}, c_{4} > 0$ such that $\begin{matrix} (3.13) & β_{λ} (r) (r - m_{0}) ⩾ c_{3} | β_{λ} (r) | - c_{4} \end{matrix}$ for all $r \in R$ and $λ \in (0, 1]$ . We can obtain $c_{3}$ and $c_{4}$ explicitly. Then, (3.2) can be improved by $\begin{array}{l} {(P β_{λ} (v_{ε, λ} (s) + m_{0} 1), v_{ε, λ} (s))}_{H_{0}} \\ = \int_{Ω} β_{λ} (u_{ε, λ} (s)) (u_{ε, λ} (s) - m_{0}) d x + \int_{Γ} β_{λ} (u_{Γ, ε, λ} (s)) (u_{Γ, ε, λ} (s) - m_{0}) d Γ \\ ⩾ c_{3} \int_{Ω} | β_{λ} (u_{ε, λ} (s)) | d x - c_{4} | Ω | + c_{3} \int_{Γ} | β_{λ} (u_{Γ, ε, λ} (s)) | d Γ - c_{4} | Γ | \end{array}$ for a.a. $s \in (0, T)$ . Therefore, using (2.4) and the monotonicity of $\partial φ$ , $\begin{array}{l} c_{3} \int_{Ω} | β_{λ} (u_{ε, λ} (s)) | d x + c_{3} \int_{Γ} | β_{λ} (u_{Γ, ε, λ} (s)) | d Γ \\ ⩽ c_{4} (| Ω | + | Γ |) + λ {| v_{ε, λ}^{'} (s) |}_{H_{0}} {| v_{ε, λ} (s) |}_{H_{0}} + {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}} {| v_{ε, λ} (s) |}_{V_{0}^{}} \\ + ε {| π (u_{ε, λ} (s)) |}_{H} {| v_{ε, λ} (s) |}_{H_{0}} + {| f (s) |}_{H_{0}} {| v_{ε, λ} (s) |}_{H_{0}} \end{array}$ for a.a. $s \in (0, T)$ . Now, squaring the above expression and using Lemma 3.1 and (3.11), we obtain $\begin{array}{l} {(c_{3} \int_{Ω} | β_{λ} (u_{ε, λ} (s)) | d x + c_{3} \int_{Γ} | β_{λ} (u_{Γ, ε, λ} (s)) | d Γ)}^{2} \\ ⩽ 5 c_{4}^{2} {(| Ω | + | Γ |)}^{2} + 5 λ^{2} M_{4} {| v_{ε, λ}^{'} (s) |}_{H_{0}}^{2} + 5 M_{1} {| v_{ε, λ}^{'} (s) |}_{V_{0}^{}}^{2} + 5 ε^{2} M_{4} {| π (u_{ε, λ} (s)) |}_{H}^{2} \\ (3.14) & + 5 M_{4} {| f (s) |}_{H_{0}}^{2} \end{array}$ for a.a. $s \in (0, T)$ . Thus, from (2.16) and (3.11), we have $\begin{array}{l} {| π (u_{ε, λ} (s)) |}_{H}^{2} & = {| π (u_{ε, λ} (s)) - π (m_{0} 1) + π (m_{0} 1) |}_{H}^{2} \\ ⩽ 2 {\int_{Ω} ({| v_{ε, λ} (s) |}^{2} + {| π (m_{0}) |}^{2}) d x + \int_{Γ} ({| v_{Γ, ε, λ} (s) |}^{2} + {| π (m_{0}) |}^{2}) d Γ} \\ ⩽ 2 {{| v_{ε, λ} (s) |}_{H_{0}}^{2} + {| π (m_{0}) |}^{2} (| Ω | + | Γ |)} \\ (3.15) & ⩽ 2 {M_{4} + {| π (m_{0}) |}^{2} (| Ω | + | Γ |)} = : {\tilde{M}}_{5} \end{array}$ for a.a. $s \in (0, T)$ . We integrate the resultant (3.14) over $(0, t)$ with respect to s. From Lemma 3.2, there exists a positive constant $M_{5}$ , depending only on $c_{3}$ , $c_{4}$ , T, $| Ω |$ , $| Γ |$ , $M_{1}$ , $M_{3}$ , $M_{4}$ , ${\tilde{M}}_{5}$ , and $| f |_{L^{2} (0, T; H_{0})}$ and independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that (3.12) holds. □

Now, recalling (3.9) and using the facts that $v_{ε, λ}^{'} (t)$ , $\partial φ (v_{ε, λ} (t)) \in H_{0}$ for a.a. $t \in (0, T)$ , we obtain $\begin{matrix} (3.16) & m (μ_{ε, λ} (t)) = m (β_{λ} (u_{ε, λ} (t)) + ε π (u_{ε, λ} (t)) - f (t)) for a.a. t \in (0, T) . \end{matrix}$ Lemma 3.4.
There exist two positive constants* $M_{6}$ and $M_{7}$ , independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that $\begin{array}{l} \int_{0}^{t} {| m (μ_{ε, λ} (s)) |}^{2} d s ⩽ M_{6}, \\ \int_{0}^{t} {| μ_{ε, λ} (s) |}_{V}^{2} d s ⩽ M_{7} \end{array}$ for all $t \in [0, T]$ .
Proof.
From (2.1) and (3.16), we have $\begin{array}{l} {| m (μ_{ε, λ} (s)) |}^{2} ⩽ & \frac{6}{{(| Ω | + | Γ |)}^{2}} {{| β_{λ} (u_{ε, λ} (s)) |}_{L^{1} (Ω)}^{2} + ε^{2} {| π (u_{ε, λ} (s)) |}_{L^{1} (Ω)}^{2} + | Ω | {| f (s) |}_{H}^{2} \\ + {| β_{λ} (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)}^{2} + {| π (u_{Γ, ε, λ} (s)) |}_{L^{1} (Γ)}^{2} + | Γ | {| f_{Γ} (s) |}_{H_{Γ}}^{2}} \end{array}$ for a.a. $s \in (0, T)$ . Then, by integrating over $(0, T)$ , it follows that there is a positive constant $M_{6}$ , depending only on T, $| Ω |$ , $| Γ |$ , $M_{4}$ , $M_{5}$ , $| π (m_{0}) |$ , and $| f |_{L^{2} (0, T; H_{0})}$ , such that the first estimate holds. Next, from the Poincaré–Wirtinger inequality (2.2), we obtain $\begin{array}{l} \int_{0}^{t} {| μ_{ε, λ} (s) |}_{V}^{2} d s & ⩽ 2 \int_{0}^{t} {| P μ_{ε, λ} (s) |}_{V}^{2} d s + 2 \int_{0}^{t} {| m (μ_{ε, λ} (s)) 1 |}_{V}^{2} d s \\ ⩽ \frac{2}{c_{P}} \int_{0}^{t} {| P μ_{ε, λ} (s) |}_{V_{0}}^{2} d s + 2 (| Ω | + | Γ |) \int_{0}^{t} {| m (μ_{ε, λ} (s)) |}^{2} d s \\ ⩽ \frac{2 M_{3}}{c_{P}} + 2 (| Ω | + | Γ |) M_{6} = : M_{7} \end{array}$ for all $t \in [0, T]$ . □
Lemma 3.5.
There exists a positive constant $M_{8}$ , independent of $ε \in (0, 1 / 4]$ and $λ \in (0, 1]$ , such that $\begin{matrix} (3.17) & \int_{0}^{t} {| β_{λ} (u_{ε, λ} (s)) |}_{H}^{2} d s ⩽ M_{8} \end{matrix}$ for all $t \in [0, T]$ .
Proof.
We test (3.9) by $β_{λ} (u_{ε, λ} (s)) \in V$ . Then, $\begin{array}{l} ε {(\partial φ (v_{ε, λ} (s)), β_{λ} (u_{ε, λ} (s)))}_{H} + {| β_{λ} (u_{ε, λ} (s)) |}_{H}^{2} \\ (3.18) & ⩽ {(μ_{ε, λ} (s) - λ v_{ε, λ}^{'} (s) - ε π (u_{ε, λ} (s)) + f (s), β_{λ} (u_{ε, λ} (s)))}_{H} \end{array}$ for a.a. $s \in (0, T)$ . Here, from the monotonicity of $β_{λ}$ , we have $\begin{array}{l} ε {(\partial φ (v_{ε, λ} (s)), β_{λ} (u_{ε, λ} (s)))}_{H} \\ = ε \int_{Ω} (- Δ v_{ε, λ} (s)) β_{λ} (u_{ε, λ} (s)) d x + ε \int_{Γ} \partial_{ν} v_{ε, λ} (s) β_{λ} (u_{Γ, ε, λ} (s)) d Γ \\ + ε \int_{Γ} (- Δ_{Γ} v_{Γ, ε, λ} (s)) β_{λ} (u_{Γ, ε, λ} (s)) d Γ \\ = ε \int_{Ω} β_{λ}^{'} (u_{ε, λ} (s)) {| \nabla v_{ε, λ} (s) |}^{2} d x + \int_{Γ} β_{λ}^{'} (u_{Γ, ε, λ} (s)) {| \nabla_{Γ} v_{Γ, ε, λ} (s) |}^{2} d Γ \\ (3.19) & ⩾ 0 \end{array}$ for a.a. $s \in (0, T)$ , where we have used the fact that $β_{λ} {(u_{ε, λ} (s))}_{|_{Γ}} = β_{λ} (u_{Γ, ε, λ} (s))$ a.e. on Γ. Moreover, by Young’s inequality, $\begin{array}{l} {(μ_{ε, λ} (s) - λ v_{ε, λ}^{'} (s) - ε π (u_{ε, λ} (s)) + f (s), β_{λ} (u_{ε, λ} (s)))}_{H} \\ (3.20) & ⩽ \frac{1}{2} {| β_{λ} (u_{ε, λ} (s)) |}_{H}^{2} + 2 {| μ_{ε, λ} (s) |}_{H}^{2} + 2 λ^{2} {| v_{ε, λ}^{'} (s) |}_{H_{0}}^{2} + 2 ε^{2} {| π (u_{ε, λ} (s)) |}_{H}^{2} + 2 {| f (s) |}_{H_{0}}^{2} \end{array}$ for a.a. $s \in (0, T)$ . Now, let us integrate (3.18)–(3.20) over $(0, T)$ with respect to s and combine the results. Recalling (2.22) and (3.15), and using Lemmas 3.2 and 3.4, we obtain $\begin{array}{l} \int_{0}^{t} {| β_{λ} (u_{ε, λ} (s)) |}_{H}^{2} d s ⩽ & 4 \int_{0}^{T} {| μ_{ε, λ} (s) |}_{H}^{2} d s + 4 λ^{2} \int_{0}^{T} {| v_{ε, λ}^{'} (s) |}_{H_{0}}^{2} d s \\ + 4 ε^{2} \int_{0}^{T} {| π (u_{ε, λ} (s)) |}_{H}^{2} d s + 4 \int_{0}^{T} {| f (s) |}_{H_{0}}^{2} d s \\ ⩽ & 4 M_{7} + 4 M_{3} + 4 T {\tilde{M}}_{5} + 4 | f |_{L^{2} (0, T; H_{0})}^{2} = : M_{8} \end{array}$ for all $t \in [0, T]$ . □

3.2. Uniform estimates for approximate solutions of ${(P)}_{ε}$

In this subsection, based on the result of Proposition 2.1, we obtain uniform estimates for the solutions of ${(P)}_{ε}$ . Actually, by obtaining additional uniform estimates (see [11]) and taking the limit in the approximate problem ${(P)}_{ε, λ}$ as $λ \to 0$ , Proposition 2.1 can be proved. Thus, for each $ε \in (0, 1 / 4]$ , there exists a triplet $(v_{ε}, μ_{ε}, ξ_{ε})$ of $\begin{array}{l} v_{ε} \in H^{1} (0, T; V_{0}^{*}) \cap L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; W), \\ μ_{ε} \in L^{2} (0, T; V), ξ_{ε} \in L^{2} (0, T; H), ξ_{ε} \in β (u_{ε}) in L^{2} (0, T; H) \end{array}$ that satisfies (2.18), (2.19), and $v_{ε} (0) = v_{0}$ in $H_{0}$ . Here, we also have $\begin{matrix} u_{ε} \in H^{1} (0, T; V^{*}) \cap L^{\infty} (0, T; V) \cap L^{2} (0, T; W), \end{matrix}$ with the relation $u_{ε} = v_{ε} + m_{0} 1$ . On taking the limit as $λ \to 0$ , we obtain the same kind of uniform estimates as in the previous lemmas, namely $\begin{array}{l} (3.21) & \int_{0}^{t} {| v_{ε}^{'} (s) |}_{V_{0}^{*}}^{2} d s ⩽ M_{3}, \int_{0}^{t} {| u_{ε}^{'} (s) |}_{V_{0}}^{2} d s ⩽ M_{3}, \\ (3.22) & ε {| v_{ε} (t) |}_{V_{0}}^{2} ⩽ M_{3}, \\ (3.23) & {| u_{ε} (t) |}_{H}^{2} ⩽ M_{4}, {| v_{ε} (t) |}_{H_{0}}^{2} ⩽ M_{4}, \\ (3.24) & \int_{0}^{t} {| μ_{ε} (s) |}_{V}^{2} d s ⩽ M_{7}, \\ (3.25) & \int_{0}^{t} {| ξ_{ε} (s) |}_{H}^{2} d s ⩽ M_{8} \end{array}$ for all $t \in [0, T]$ . Using these estimates, we can prove the main theorem.

4. Proof of the main theorem

In this subsection, we conclude the proof of the main theorem by taking the limit in the approximate problem ${(P)}_{ε}$ as $ε \to 0$ .

Proof of Theorem 2.1.
Using the estimates (3.21)–(3.25) stated in the previous section, there exist a subsequence ${ε_{k}}_{k \in N}$ with $ε_{k} \to 0$ as $k \to + \infty$ and some limit functions $u \in H^{1} (0, T; V^{}) \cap L^{\infty} (0, T; H)$ , $v \in H^{1} (0, T; V_{0}^{}) \cap L^{\infty} (0, T; H_{0})$ , $μ \in L^{2} (0, T; V)$ , and $ξ \in L^{2} (0, T; H)$ such that $\begin{array}{l} (4.1) & v_{ε_{k}} \to v weakly star in L^{\infty} (0, T; H_{0}), \\ (4.2) & u_{ε_{k}} \to u = v + m_{0} 1 weakly star in L^{\infty} (0, T; H), \\ (4.3) & ε_{k} v_{ε_{k}} \to 0 in L^{\infty} (0, T; V_{0}), \\ (4.4) & v_{ε_{k}}^{'} \to v^{'} weakly in L^{2} (0, T; V_{0}^{}), \\ (4.5) & u_{ε_{k}}^{'} \to u^{'} weakly in L^{2} (0, T; V^{}), \\ (4.6) & μ_{ε_{k}} \to μ weakly in L^{2} (0, T; V), \\ (4.7) & ξ_{ε_{k}} \to ξ weakly in L^{2} (0, T; H), \\ (4.8) & ε_{k} π (u_{ε_{k}}) \to 0 in L^{\infty} (0, T; H) \end{array}$ as $k \to + \infty$ . From (4.1), (4.2), (4.4), and (4.5), the well-known generalized Aubin–Lions lemma (see, e.g., [28, Section 8, Corollary 4]) gives $\begin{array}{l} (4.9) & v_{ε_{k}} \to v in C ([0, T]; V_{0}^{}), \\ (4.10) & u_{ε_{k}} \to u in C ([0, T]; V^{}) \end{array}$ as $k \to + \infty$ . Now, integrating (2.19) over $(0, T)$ , we find $\begin{array}{l} \int_{0}^{T} {(μ_{ε_{k}} (t), η (t))}_{H} d t = & ε_{k} \int_{0}^{T} a (v_{ε_{k}} (t), η (t)) d t + \int_{0}^{T} {(ξ_{ε_{k}} (t), η (t))}_{H} d t \\ + ε_{k} \int_{0}^{T} {(π (u_{ε_{k}} (t)), η (t))}_{H} d t - \int_{0}^{T} {(f (t), η (t))}_{H} d t \\ (4.11) & for all η \in L^{2} (0, T; V) . \end{array}$ Letting $k \to \infty$ , we can use (4.6), (4.3), (4.7), and (4.8) to obtain $\begin{matrix} \int_{0}^{T} {(μ (t), η (t))}_{H} d t = \int_{0}^{T} {(ξ (t) - f (t), η (t))}_{H} d t for all η \in L^{2} (0, T; V), \end{matrix}$ namely, $μ = ξ - f$ in $L^{2} (0, T; H)$ . Additionally, the information $μ + f \in L^{2} (0, T; V)$ gives the regularity of $ξ \in L^{2} (0, T; V)$ , namely $ξ_{Γ} = ξ_{|_{Γ}}$ a.e. on Σ. Next, we take $η : = u_{ε_{k}} \in L^{2} (0, T; V)$ in (4.11). Then, using the positivity $\begin{matrix} ε_{k} \int_{0}^{T} a (v_{ε_{k}} (t), u_{ε_{k}} (t)) d t = ε_{k} \int_{0}^{T} {| v_{ε_{k}} (t) |}_{V_{0}}^{2} d t ⩾ 0, \end{matrix}$ and recalling (4.2), (4.6), (4.8), and (4.10), we have $\begin{array}{l} \underset{k \to + \infty}{lim sup} \int_{0}^{T} {(ξ_{ε_{k}} (t), u_{ε_{k}} (t))}_{H} d t & ⩽ \int_{0}^{T} {⟨ u (t), μ (t) ⟩}_{V^{}, V} d t + \int_{0}^{T} {(f (t), u (t))}_{H} d t \\ = \int_{0}^{T} {(ξ (t), u (t))}_{H} d t . \end{array}$ Thus, applying [4, Proposition 2.2, p. 38] with (4.1) and (4.7), we can deduce that $\begin{matrix} ξ \in β (u) in L^{2} (0, T; H) \end{matrix}$ from the maximal monotonicity of β. Namely, we obtain $ξ \in β (u)$ a.e. in Q, $ξ_{Γ} \in β (u_{Γ})$ a.e. on Σ. Finally, integrating (2.18) over $(0, T)$ with respect to t and letting $k \to + \infty$ , we have $\begin{matrix} \int_{0}^{T} {⟨ v^{'} (t), η (t) ⟩}_{V_{0}^{}, V_{0}} d t + \int_{0}^{T} a (P μ (t), η (t)) d t = 0 for all η \in L^{2} (0, T; V_{0}) . \end{matrix}$ Next, using (2.3), we deduce $\begin{matrix} v^{'} (t) + F (P μ (t)) = 0 in V_{0}^{}, for a.a. t \in (0, T), \end{matrix}$ namely, $\begin{matrix} (4.12) & {⟨ v^{'} (t), z ⟩}_{V_{0}^{}, V_{0}} + a (ξ (t), z) = a (f (t), z) for all z \in V_{0} . \end{matrix}$ Therefore, recalling (2.17) and Remark 3, we finally obtain $\begin{array}{l} {⟨ u^{'} (t), z ⟩}_{V^{}, V} + {⟨ u_{Γ}^{'} (t), z_{Γ} ⟩}_{V_{Γ}^{}, V_{Γ}} + \int_{Ω} \nabla ξ (t) \cdot \nabla z d x + \int_{Γ} \nabla_{Γ} ξ_{Γ} (t) \cdot \nabla_{Γ} z_{Γ} d Γ \\ = \int_{Ω} g (t) z d x + \int_{Γ} g_{Γ} (t) z_{Γ} d Γ for all z = (z, z_{Γ}) \in V, \end{array}$ for a.a. $t \in (0, T)$ , with $u (0) = u_{0}$ a.e. in Ω and $u_{Γ} (0) = u_{0 Γ}$ a.e. on Γ. Thus, it transpires that the pair $(u, ξ)$ is a weak solution of (P). □

5. Continuous dependence

In this section, we prove the continuous dependence of the data. This theorem also guarantees the uniqueness of the component $u$ in the solution.

Proof of Theorem 2.2.

For $i = 1, 2$ , let $(u^{(i)}, ξ^{(i)})$ be a weak solution of (P) corresponding to the data ( $f^{(i)}$ , $u_{0}^{(i)}$ ). Set $m_{0} : = m (u_{0}^{(1)}) = m (u_{0}^{(2)})$ . From the weak formulation (4.12) of the Stefan problem (P) for $v^{(i)} = u^{(i)} - m_{0} 1$ , we obtain $\begin{matrix} (5.1) & {⟨ {(v^{(i)})}^{'} (s), z ⟩}_{V_{0}^{*}, V_{0}} + a (ξ^{(i)} (s), z) = a (f^{(i)} (s), z) for all z \in V_{0}, \end{matrix}$ for a.a. $s \in (0, T)$ . Now, $P ξ^{(i)} (s) = ξ^{(i)} (s) - m (ξ^{i} (s)) 1 \in V_{0}$ . Therefore, we see that $\begin{array}{l} a (ξ^{(i)} (s), F^{- 1} z_{0}) & = a (F^{- 1} z_{0}, P ξ^{(i)} (s)) \\ = {⟨ z_{0}, P ξ^{(i)} (s) ⟩}_{V_{0}^{*}, V_{0}} \\ = {(z_{0}, P ξ^{(i)} (s))}_{H_{0}} \\ (5.2) & = {(z_{0}, ξ^{(i)} (s))}_{H} for all z_{0} \in H_{0} \subset V_{0}^{*} . \end{array}$ Thus, we take the difference between (5.1) when $i = 1$ and when $i = 2$ . Setting $z : = F^{- 1} (v^{(1)} (s) - v^{(2)} (s)) \in V_{0}$ and using (2.3), we obtain $\begin{matrix} (5.3) & \begin{matrix} \frac{1}{2} \frac{d}{d s} {| v^{(1)} (s) - v^{(2)} (s) |}_{V_{0}^{*}}^{2} + {(v^{(1)} (s) - v^{(2)} (s), ξ^{(1)} (s) - ξ^{(2)} (s))}_{H} \\ ⩽ {⟨ v^{(1)} (s) - v^{(2)} (s), f^{(1)} (s) - f^{(2)} (s) ⟩}_{V_{0}^{*}, V_{0}} \\ ⩽ \frac{1}{2} {| v^{(1)} (s) - v^{(2)} (s) |}_{V_{0}^{*}}^{2} + \frac{1}{2} {| f^{(1)} (s) - f^{(2)} (s) |}_{V_{0}}^{2} \end{matrix} \end{matrix}$ for a.a. $s \in (0, T)$ . Now, in the variational formulation (2.17) used to construct $f$ , we can take $z : = f$ to obtain $\begin{matrix} | f |_{V_{0}}^{2} ⩽ \frac{1}{2 c_{P}} | g |_{H_{0}}^{2} + \frac{c_{P}}{2} | f |_{H_{0}}^{2} \end{matrix}$ a.e. in $(0, T)$ . Therefore, using the Poincaré–Wirtinger inequality (2.2), $\begin{matrix} (5.4) & {| f^{(1)} (s) - f^{(2)} (s) |}_{V_{0}}^{2} ⩽ \frac{1}{c_{P}} {| g^{(1)} (s) - g^{(2)} (s) |}_{H_{0}}^{2} \end{matrix}$ for a.a. $s \in (0, T)$ . Then, from the monotonicity of β, we can apply Gronwall’s inequality to deduce (2.8) for some positive constant C that depends only on $c_{P}$ and T. Here, we must take care to ensure that $v^{(1)} - v^{(2)} = u^{(1)} - u^{(2)}$ .

If β is a Lipschitz continuous function with the Lipschitz constant $c_{β} > 0$ , then $ξ = (β (u), β (u_{Γ}))$ and, with the help of the monotonicity of β, we have $\begin{matrix} (5.5) & {| ξ^{(1)} (s) - ξ^{(2)} (s) |}_{H}^{2} ⩽ c_{β} {(u^{(1)} (s) - u^{(2)} (s), ξ^{(1)} (s) - ξ^{(2)} (s))}_{H} . \end{matrix}$ Thus, combining (5.3), (5.4), (5.5), and (2.8) and integrating the resultant quantity over $(0, t)$ with respect to s, we obtain $\begin{array}{l} {| u^{(1)} (t) - u^{(2)} (t) |}_{V_{0}^{*}}^{2} + \frac{2}{c_{β}} \int_{0}^{t} {| ξ^{(1)} (s) - ξ^{(2)} (s) |}_{H}^{2} d s \\ ⩽ {| u_{0}^{(1)} - u_{0}^{(2)} |}_{V_{0}^{*}}^{2} + \int_{0}^{t} {| u^{(1)} (s) - u^{(2)} (s) |}_{V_{0}^{*}}^{2} d s + \frac{1}{c_{P}} \int_{0}^{t} {| g^{(1)} (s) - g^{(2)} (s) |}_{H_{0}}^{2} d s \\ ⩽ C (1 + T) {{| u_{0}^{(1)} - u_{0}^{(2)} |}_{V_{0}^{*}}^{2} + \int_{0}^{T} {| g^{(1)} (s) - g^{(2)} (s) |}_{H_{0}}^{2} d s} \end{array}$ for all $t \in [0, T]$ . That is, we have obtained (2.9). □

Footnotes

Acknowledgements

The author wishes to express his heartfelt gratitude to professors Goro Akagi and Ulisse Stefanelli, who kindly gave him the opportunity of exchange visits supported by the JSPS–CNR bilateral joint research project on Innovative Variational Methods for Evolution Equations. The author is also indebted to Professor Pierluigi Colli for fruitful discussions that helped to obtain the results in this paper. The author was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C), Grant Number 26400164. Last but not least, the author is grateful to the referee for their careful reading of the manuscript.

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Convergence of Cahn–Hilliard systems to the Stefan problem with dynamic boundary conditions

Abstract

Keywords

1. Introduction

2. Main results

2.1. Notation

2.2. Main theorem

3.1. Uniform estimates for approximate solutions of ( P ) ε , λ

4. Proof of the main theorem

Footnotes

Acknowledgements

References

3.1. Uniform estimates for approximate solutions of ${(P)}_{ε, λ}$