# Cahn–Hilliard Approach to Some Degenerate Parabolic Equations with Dynamic Boundary Conditions

Takeshi Fukao

► To cite this version:

Takeshi Fukao. Cahn–Hilliard Approach to Some Degenerate Parabolic Equations with Dynamic Boundary Conditions. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.282-291, 10.1007/978-3-319-55795-3_26 . hal-01626916

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Submitted on 31 Oct 2017

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# Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions

Takeshi Fukao

Department of Mathematics, Faculty of Education

Kyoto University of Education

1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto 612-8522 Japan

E-mail: fukao@kyokyo-u.ac.jp

No Institute Given

**Abstract.** In this paper the well-posedness of some degenerate parabolic equations with a dynamic boundary condition is considered. To characterize the target degenerate parabolic equation from the Cahn-Hilliard system, the nonlinear term coming from the convex part of the double-well potential is chosen using a suitable maximal monotone graph. The main topic of this paper is the existence problem under an assumption for this maximal monotone graph for treating a wider class. The existence of a weak solution is proved.

**Key words:** degenerate parabolic equation, dynamic boundary condition, weak solution, Cahn-Hilliard system.

AMS (MOS) subject classification: 35K65, 35K30, 47J35

## 1 Introduction

The relationship between the Allen-Cahn equation [2] and the motion by mean curvature is interesting as the singular limit of the following form:

$$ \frac{\partial u}{\partial t} - \Delta u + \frac{1}{\varepsilon^2}(u^3 - u) = 0 \quad \text{in } Q := (0, T) \times \Omega, $$

as $\varepsilon \searrow 0$, where $0 < T < +\infty$ and $\Omega \subset \mathbb{R}^d$ for $d = 2, 3$, which is a bounded domain with smooth boundary $\Gamma$. For example, Bronsard and Kohn presented a pioneering result in [5], and subsequently many related results have been obtained. A similar concept in this framework, the Cahn-Hilliard system [7], is connected to motion by the Mullins-Sekerka law [18] in the limit of

$$ \begin{gathered} \frac{\partial u}{\partial t} - \Delta \mu = 0 \quad \text{in } Q, \\ \mu = -\varepsilon \Delta u + \frac{1}{\varepsilon}(u^3 - u) \quad \text{in } Q \end{gathered} \tag{1.1} $$
as $\varepsilon \searrow 0$. For both of these, the target problems are sharp interface models in a classical sense and a powerful analysis tool seems to be the method of matched asymptotic expansions (see [1, ?, ?] and the references in these papers).

In this paper, we discuss this relation from a different view point. To do so, we begin with the following degenerate parabolic equation:

$$ \frac{\partial u}{\partial t} - \Delta\beta(u) = g \quad \text{in } Q, \qquad (1.2) $$

where $g: \Omega \to \mathbb{R}$ is a given source. This equation is characterized by the choice of $\beta: \mathbb{R} \to \mathbb{R}$. For example, if we choose $\beta$ to be a piecewise linear function of the form

$$ \beta(r) := \begin{cases} k_s r & r < 0, \\ 0 & 0 \le r \le L, \\ k_\ell(r - L) & r > L; \end{cases} \qquad (1.3) $$

where $k_s$ and $k_\ell > 0$ represent the heat conductivities of the solid and liquid regions, respectively, and $L > 0$ is the latent heat constant, then (1.2) is the weak formulation of the Stefan problem, or the “enthalpy formulation,” where the unknown $u$ denotes the enthalpy and $\beta(u)$ denotes the temperature. The informant of the sharp interface, in other words the Stefan condition, is hidden in the weak formulation. Another example is the weak formulation of the Hele-Shaw problem. If we choose $\beta$ to be the inverse of the Heaviside function

$$ \mathcal{H}(r) := \begin{cases} 0 & \text{if } r < 0, \\ [0, 1] & \text{if } r = 0, \quad \text{for all } r \in \mathbb{R}, \\ 1 & \text{if } r > 0 \end{cases} $$

so that $\beta$ is the multivalued function $\beta(r) := \mathcal{H}^{-1}(r) = \partial I_{[0,1]}(r)$ for all $r \in [0, 1]$, then (1.2) can be stated as

$$ \xi \in \beta(u), \quad \frac{\partial u}{\partial t} - \Delta\xi = g \quad \text{in } Q, $$

where $\partial I_{[0,1]}$ is the subdifferential of the indicator function $I_{[0,1]}$ on the interval $[0, 1]$, the unknown $u$ denotes the order parameter. Details about weak formulations may be found in Visintin [21]. Weak formulations for this kind of sharp interface model are the focus of this paper. Therefore, we use the terms “Stefan problem” and “Hele-Shaw problem” in the sense of weak formulations throughout this paper.

Recently, the author considered the approach to the following Cahn-Hilliard system for the Stefan problem in [13]:

$$ \frac{\partial u}{\partial t} - \Delta\mu = 0 \quad \text{in } Q, \qquad (1.4) $$

$$ \mu = -\varepsilon\Delta u + \beta(u) + \varepsilon\pi(u) - f \quad \text{in } Q, \qquad (1.5) $$
with a dynamic boundary condition of the form

$$ \frac{\partial u}{\partial t} + \partial_{\nu} \mu - \Delta_{\Gamma} \mu = 0 \quad \text{on } \Sigma := (0, T) \times \Gamma, \qquad (1.6) $$

$$ \mu = \varepsilon \partial_{\nu} u - \varepsilon \Delta_{\Gamma} u + \beta(u) + \varepsilon \pi(u) - f_{\Gamma} \quad \text{on } \Sigma, \qquad (1.7) $$

where the symbol $\partial_\nu$ denotes the normal derivative on the boundary $\Gamma$ outward from $\Omega$, the symbol $\Delta_\Gamma$ stands for the Laplace-Beltrami operator on $\Gamma$ (see, e.g., [15, Chapter 3]), $\beta$ is defined by (1.3), and $\pi: \mathbb{R} \to \mathbb{R}$ is a piecewise linear function defined by $\pi(r) := L/2$ if $r < 0$, $\pi(r) := L/2 - r$ if $0 \le r \le L$ and $\pi(r) := -L/2$ if $r > L$. Thanks to this choice, system (1.4)-(1.7) has the structure of a Cahn-Hilliard system. This problem originally comes from [14]. Formally, if we let $\varepsilon \searrow 0$ in (1.4)-(1.7), then we can see that the Cahn-Hilliard system (1.4)-(1.7) converges in a suitable sense to the following Stefan problem with a dynamic boundary condition:

$$ \frac{\partial u}{\partial t} - \Delta\beta(u) = -\Delta f \quad \text{in } Q, \\ \frac{\partial u}{\partial t} + \partial_{\nu}\beta(u) - \Delta_{\Gamma}\beta(u) = \partial_{\nu}f_{\Gamma} - \Delta_{\Gamma}f_{\Gamma} \quad \text{on } \Sigma. $$

Here, we should take care of the difference between the order and position of $\varepsilon$ in (1.1) and (1.5) even when $\beta(u) = u^3$ and $\pi(u) = -u$. In [13], $\beta$ is assumed to satisfy the following condition:

$\beta$ is a maximal monotone graph in $\mathbb{R} \times \mathbb{R}$, and is a subdifferential $\beta = \partial \hat{\beta}$ of some proper, lower semicontinuous, and convex function $\hat{\beta}: \mathbb{R} \to [0, +\infty]$ satisfying $\hat{\beta}(0) = 0$ with some effective domain $D(\beta)$. This implies $\beta(0) = 0$. Moreover, there exist two constants $c, \tilde{c} > 0$ such that

$$ \hat{\beta}(r) \geq c|r|^2 - \tilde{c} \quad \text{for all } r \in \mathbb{R}. \qquad (1.8) $$

It is easy to see that (1.2) represents a large number of problems, including the porous media equation, the nonlinear diffusion equation of Penrose-Fife type, the fast diffusion equation, and so on. However, to apply this approach from the Cahn-Hilliard system to these wider classes of the degenerate parabolic equation, the growth condition (1.8) is too strong (see, e.g. [12]). Therefore, in this paper based on the essential idea from [10], we relax the assumption in (1.8). This is the different point from the previous work [13]. See also [3, ?] for related problems of interest.

**Notation.** Let $H := L^2(\Omega)$, $V := H^1(\Omega)$, $H_\Gamma := L^2(\Gamma)$ and $V_\Gamma := H^1(\Gamma)$ with the usual norms $|\cdot|_H$, $|\cdot|_V$, $|\cdot|_{H_\Gamma}$, $|\cdot|_{V_\Gamma}$ and inner products $(\cdot, \cdot)_H$, $(\cdot, \cdot)_V$, $(\cdot, \cdot)_{H_\Gamma}$, $(\cdot, \cdot)_{V_\Gamma}$, respectively, and let $\mathbf{H} := H \times H_\Gamma$, $\mathbf{V} := \{(z, z_\Gamma) \in V \times V_\Gamma : z_\Gamma = z|_\Gamma \text{ a.e. on } \Gamma\}$ and $\mathbf{W} := H^2(\Omega) \times H^2(\Gamma)$. Then $\mathbf{H}$, $\mathbf{V}$ and $\mathbf{W}$ are Hilbert spaces with the inner product

$$ (u, z)_H := (u, z)_H + (u_\Gamma, z_\Gamma)_{H_\Gamma} \quad \text{for all } u, z \in \mathbf{H}, $$
and the related norm is analogously defined as one of $\mathbf{V}$ or $\mathbf{W}$. Define $m: \mathbf{H} \to \mathbb{R}$ by

$$m(\boldsymbol{z}) := \frac{1}{|\boldsymbol{\Omega}| + |\boldsymbol{\Gamma}|} \left\{ \int_{\boldsymbol{\Omega}} z dx + \int_{\boldsymbol{\Gamma}} z_{\Gamma} d\Gamma \right\} \quad \text{for all } \boldsymbol{z} \in \mathbf{H},$$

where $|\boldsymbol{\Omega}| := \int_{\boldsymbol{\Omega}} 1 dx$ and $|\boldsymbol{\Gamma}| := \int_{\boldsymbol{\Gamma}} 1 d\Gamma$. The symbol $\mathbf{V}^*$ denotes the dual space of $\mathbf{V}$, and the pair $\langle \cdot, \cdot \rangle_{\mathbf{V}^*, \mathbf{V}}$ denotes the duality pairing between $\mathbf{V}^*$ and $\mathbf{V}$. Moreover, define the bilinear form $a(\cdot, \cdot) : \mathbf{V} \times \mathbf{V} \to \mathbb{R}$ by

$$a(\mathbf{u}, \boldsymbol{z}) := \int_{\boldsymbol{\Omega}} \nabla \mathbf{u} \cdot \nabla z dx + \int_{\boldsymbol{\Gamma}} \nabla_{\Gamma} u_{\Gamma} \cdot \nabla_{\Gamma} z_{\Gamma} d\Gamma \quad \text{for all } \mathbf{u}, \boldsymbol{z} \in \mathbf{V},$$

where $\nabla_{\Gamma}$ denotes the surface gradient on $\Gamma$ (see, e.g., [15, Chapter 3]). We introduce the subspace $\mathbf{H}_0 := \{\boldsymbol{z} \in \mathbf{H} : m(\boldsymbol{z}) = 0\}$ of $\mathbf{H}$ and $\mathbf{V}_0 := \mathbf{V} \cap \mathbf{H}_0$, with their norms $|\boldsymbol{z}|_{\mathbf{H}_0} := |\boldsymbol{z}|_{\mathbf{H}}$ for all $\boldsymbol{z} \in \mathbf{H}_0$ and $|\boldsymbol{z}|_{\mathbf{V}_0} := a(\boldsymbol{z}, \boldsymbol{z})^{1/2}$ for all $\boldsymbol{z} \in \mathbf{V}_0$. Then the duality mapping $\mathbf{F}: \mathbf{V}_0 \to \mathbf{V}_0^*$ is defined by $\langle \mathbf{F}\boldsymbol{z}, \tilde{\boldsymbol{z}} \rangle_{\mathbf{V}_0^*, \mathbf{V}_0} := a(\boldsymbol{z}, \tilde{\boldsymbol{z}})$ for all $\boldsymbol{z}, \tilde{\boldsymbol{z}} \in \mathbf{V}_0$ and the inner product in $\mathbf{V}_0^*$ is defined by $(\boldsymbol{z}_1^*, \boldsymbol{z}_2^*)_{\mathbf{V}_0^*, \mathbf{V}_0} := \langle \boldsymbol{z}_1^*, \mathbf{F}^{-1}\boldsymbol{z}_2^* \rangle_{\mathbf{V}_0^*, \mathbf{V}_0}$ for all $\boldsymbol{z}_1^*, \boldsymbol{z}_2^* \in \mathbf{V}_0^*$. Moreover, define $\mathbf{P}: \mathbf{H} \to \mathbf{H}_0$ by $\mathbf{P}\boldsymbol{z} := \boldsymbol{z} - m(\boldsymbol{z})\mathbf{1}$ for all $\boldsymbol{z} \in \mathbf{H}$, where $\mathbf{1} := (1, 1)$. Thus we obtain the dense and compact embeddings $\mathbf{V}_0 \hookrightarrow \mathbf{H}_0 \hookrightarrow \mathbf{V}_0^*$. See [8, ?] for further details.

## 2 Existence of the weak solution

In this section, we state an existence theorem for the weak solution of a degenerate parabolic equation with a dynamic boundary condition of the following form:

$$\xi \in \beta(u), \quad \frac{\partial u}{\partial t} - \Delta \xi = g \quad \text{a.e. in } Q,$$

$$\xi_{\Gamma} \in \beta(u_{\Gamma}), \quad \xi_{\Gamma} = \xi|_{\Gamma}, \quad \frac{\partial u_{\Gamma}}{\partial t} + \partial_{\nu} \xi - \Delta_{\Gamma} \xi_{\Gamma} = g_{\Gamma} \quad \text{a.e. on } \Sigma,$$

$$u(0) = u_0 \quad \text{a.e. in } \Omega, \quad u_{\Gamma}(0) = u_{0\Gamma} \quad \text{a.e. on } \Gamma,$$

where $\beta, g, g_{\Gamma}, u_0$ and $u_{0\Gamma}$ satisfy the following assumptions:

(A1) $\beta$ is a maximal monotone graph in $\mathbb{R} \times \mathbb{R}$, and is a subdifferential $\beta = \partial \hat{\beta}$ of some proper, lower semicontinuous, and convex function $\hat{\beta}: \mathbb{R} \to [0, +\infty]$ satisfying $\hat{\beta}(0) = 0$ in some effective domain $D(\beta)$. This implies that $\beta(0) = 0$;

(A2) $g \in L^2(0, T; \mathbf{H}_0)$;

(A3) $\mathbf{u}_0 := (u_0, u_{0\Gamma}) \in \mathbf{H}$ with $m_0 \in \text{int}D(\beta)$, and the compatibility conditions $\hat{\beta}(u_0) \in L^1(\Omega), \hat{\beta}(u_{0\Gamma}) \in L^1(\Gamma)$ hold.

We remark that the growth condition of $\hat{\beta}$ in (A1) and the regularity of $\mathbf{u}_0$ in (A3) are relaxations from a previous related result [13] (cf. (1.8)).
**Theorem 2.1.** Under assumptions (A1)-(A3), there exists at least one pair $(\mathbf{u}, \boldsymbol{\xi})$ of functions $\mathbf{u} \in H^1(0, T; \mathbf{V}^*) \cap L^2(0, T; \mathbf{H})$ and $\boldsymbol{\xi} \in L^2(0, T; \mathbf{V})$ such that $\boldsymbol{\xi} \in \beta(u)$ a.e. in $Q$, $\xi_{\Gamma} \in \beta(u_{\Gamma})$ and $\xi_{\Gamma} = \xi|_{\Gamma}$ a.e. on $\Sigma$, and that satisfy

$$ 
\begin{aligned}
& \langle u'(t), z \rangle_{V^*, V} + \langle u'_\Gamma(t), z_\Gamma \rangle_{V_\Gamma^*, V_\Gamma} + \int_\Omega \nabla \xi(t) \cdot \nabla z dx + \int_\Gamma \nabla_\Gamma \xi_\Gamma(t) \cdot \nabla_\Gamma z_\Gamma d\Gamma \\
& = \int_\Omega g(t) z dx + \int_\Gamma g_\Gamma(t) z_\Gamma d\Gamma \quad \text{for all } \mathbf{z} := (z, z_\Gamma) \in \mathbf{V} \quad (2.1)
\end{aligned}
$$

for a.a. $t \in (0, T)$ with $u(0) = u_0$ a.e. in $\Omega$ and $u_\Gamma(0) = u_{0\Gamma}$ a.e. on $\Gamma$.

The continuous dependence is completely the same as in a previous result [13, Theorem 2.2]. Therefore, we devolve the uniqueness problem on [13].

### 3 Proof of the main theorem

In this section, we prove the main theorem. The strategy of the proof is similar to that of [13, Theorem 2.1]. However, to relax the assumption we use a different uniform estimate. Let us start with an approximate problem. Recall the Yosida approximation $\beta_\lambda : \mathbb{R} \to \mathbb{R}$ and the related Moreau-Yosida regularization $\hat{\beta}_\lambda$ of $\hat{\beta} : \mathbb{R} \to \mathbb{R}$ (see, e.g., [4]). We see that $0 \le \hat{\beta}_\lambda(r) \le \hat{\beta}(r)$ for all $r \in \mathbb{R}$. Moreover, we define the following proper, lower semicontinuous, and convex functional $\varphi : \mathbf{H}_0 \to [0, +\infty]$:

$$ \varphi(\mathbf{z}) := \begin{cases} \frac{1}{2} \int_{\Omega} |\nabla z|^2 dx + \frac{1}{2} \int_{\Gamma} |\nabla_{\Gamma} z_{\Gamma}|^2 d\Gamma & \text{if } \mathbf{z} \in \mathbf{V}_0, \\ +\infty & \text{otherwise.} \end{cases} $$

The subdifferential $\partial\varphi$ on $\mathbf{H}_0$ is characterized by $\partial\varphi(\mathbf{z}) = (-\Delta z, \partial_\nu z - \Delta_\Gamma z_\Gamma)$ with $\mathbf{z} \in D(\partial\varphi) = \mathbf{W} \cap \mathbf{V}_0$ (see, e.g., [9, Lemma C]). By virtue of the well-known theory of evolution equations (see, e.g., [8, ?, ?, ?]), for each $\varepsilon \in (0, 1]$ and $\lambda \in (0, 1]$, there exist $\mathbf{v}_{\varepsilon, \lambda} \in H^1(0, T; \mathbf{H}_0) \cap L^\infty(0, T; \mathbf{V}_0) \cap L^2(0, T; \mathbf{W})$ and $\boldsymbol{\mu}_{\varepsilon, \lambda} \in L^2(0, T; \mathbf{V})$ such that

$$ 
\begin{aligned}
& \lambda \mathbf{v}'_{\varepsilon, \lambda}(t) + \mathbf{F}^{-1}(\mathbf{v}'_{\varepsilon, \lambda}(t)) + \varepsilon \partial\varphi(\mathbf{v}_{\varepsilon, \lambda}(t)) \\
& = \mathbf{P}(-\hat{\beta}_\lambda(\mathbf{u}_{\varepsilon, \lambda}(t)) - \varepsilon \boldsymbol{\pi}(\mathbf{u}_{\varepsilon, \lambda}(t)) + \mathbf{f}(t)) \quad \text{in } \mathbf{H}_0 \quad (3.1)
\end{aligned}
$$

for a.a. $s \in (0, T)$ with $\mathbf{v}_{\varepsilon, \lambda}(0) = \mathbf{v}_{0\varepsilon}$ in $\mathbf{H}_0$, where $\mathbf{v}_{0\varepsilon} \in \mathbf{V}_0$ solves the auxiliary problem $\mathbf{v}_{0\varepsilon} + \varepsilon \partial\varphi(\mathbf{v}_{0\varepsilon}) = \mathbf{v}_0$ in $\mathbf{H}_0$ so that there exists a constant $C > 0$ such that

$$ |\mathbf{v}_{0\varepsilon}|_{\mathbf{H}_0}^2 \le C, \quad \varepsilon |\mathbf{v}_{0\varepsilon}|_{\mathbf{V}_0}^2 \le C, \quad (3.2) $$

$$ \int_{\Omega} \hat{\beta}(v_{0\varepsilon} + m_0) dx \le C, \quad \int_{\Gamma} \hat{\beta}(v_{0\varepsilon} + m_0) d\Gamma \le C. $$
Moreover, $\mathbf{u}_{\varepsilon,\lambda} := \mathbf{v}_{\varepsilon,\lambda} + m_0 \mathbf{1}$, $m_0 := m(\mathbf{u}_0)$ and $\mathbf{1} := (1,1)$, and $\beta_{\lambda}(\mathbf{z}) := (\beta_{\lambda}(\mathbf{z}), \beta_{\lambda}(z_{\Gamma}))$ and $\pi(\mathbf{z}) := (\pi(\mathbf{z}), \pi(z_{\Gamma}))$ for all $\mathbf{z} \in \mathbf{H}$, where $\pi: D(\pi) = \mathbb{R} \to \mathbb{R}$ is a Lipschitz continuous function with a Lipschitz constant $L_{\pi}$ that breaks the monotonicity in $\beta + \varepsilon\pi$; $\mathbf{f} \in L^2(0,T; D(\partial\varphi))$ is the solution of $\mathbf{g}(t) = \partial\varphi(\mathbf{f}(t))$ in $\mathbf{H}_0$ for a.a. $t \in (0,T)$. Namely, from [9, Lemma C], we can choose $\mathbf{f}(t) := (f(t), f_{\Gamma}(t))$ to satisfy

$$ \begin{cases} -\Delta f(t) = g(t) & \text{a.e. in } \Omega, \\ \partial_{\nu} f(t) - \Delta_{\Gamma} f_{\Gamma}(t) = g_{\Gamma}(t) & \text{a.e. on } \Gamma, \end{cases} \quad \text{for a.a. } t \in (0, T). \quad (3.3) $$

### 3.1 Uniform estimates for approximate solutions

The key strategy in the proof is to obtain uniform estimates independent of $\varepsilon > 0$ and $\lambda > 0$, after which we consider the limiting procedures $\lambda \searrow 0$ and $\varepsilon \searrow 0$. Recall (3.1) in the equivalent form

$$ \mathbf{v}'_{\varepsilon,\lambda}(s) + \mathbf{F}(\mathbf{P}\mathbf{\mu}_{\varepsilon,\lambda}(s)) = \mathbf{0} \quad \text{in } \mathbf{V}^*_{0}, \quad (3.4) $$

$$ \begin{gathered} \mu_{\varepsilon,\lambda}(s) = \lambda \mathbf{v}'_{\varepsilon,\lambda}(s) + \varepsilon \partial\varphi(\mathbf{v}_{\varepsilon,\lambda}(s)) + \beta_{\lambda}(\mathbf{u}_{\varepsilon,\lambda}(s)) + \varepsilon \pi(\mathbf{u}_{\varepsilon,\lambda}(s)) - \mathbf{f}(s) \quad \text{in } \mathbf{H} \quad (3.5) \\ \mathbf{v}_{\varepsilon,\lambda}(t) \in \mathbf{V}^*_{0} \quad \text{for a.a. } t \in (0,T]. \end{gathered} $$

for a.a. $s \in (0,T)$. Moreover, if we put $\varepsilon_0 := \min\{1, 1/(4L_{\pi}^2)\}$, then we have:

**Lemma 3.1.** There exist positive constants $M_1, M_2$ independent of $\varepsilon \in (0, \varepsilon_0]$ and $\lambda \in (0, 1]$ such that

$$ \lambda |\mathbf{v}_{\varepsilon,\lambda}(t)|_{H_0}^2 + |\mathbf{v}_{\varepsilon,\lambda}(t)|_{V_0^*}^2 \le M_1, $$

$$ \frac{\varepsilon}{2} \int_0^t |\mathbf{v}_{\varepsilon,\lambda}(s)|_{V_0}^2 ds + 2 \int_0^t |\hat{\beta}_{\lambda}(u_{\varepsilon,\lambda}(s))|_{L^1(\Omega)} ds + 2 \int_0^t |\hat{\beta}_{\lambda}(u_{\Gamma,\varepsilon,\lambda}(s))|_{L^1(\Gamma)} ds \le M_2 $$

for all $t \in [0, T]$.

*Proof.* Multiplying (3.1) by $\mathbf{v}_{\varepsilon,\lambda}(s) \in V_0$, we have

$$ \begin{aligned} \lambda(\mathbf{v}'_{\varepsilon,\lambda}(s), \mathbf{v}_{\varepsilon,\lambda}(s))_{H_0} + (\mathbf{v}'_{\varepsilon,\lambda}(s), \mathbf{v}_{\varepsilon,\lambda}(s))_{V_0^*} + \varepsilon(\partial\varphi(\mathbf{v}_{\varepsilon,\lambda}(s)), \mathbf{v}_{\varepsilon,\lambda}(s))_{H_0} \\ + (\mathbf{P}\beta_{\lambda}(\mathbf{v}_{\varepsilon,\lambda}(s) + m_0 \mathbf{1}), \mathbf{v}_{\varepsilon,\lambda}(s))_{H_0} = (\mathbf{f}(s) - \varepsilon \mathbf{P}\pi(\mathbf{v}_{\varepsilon,\lambda}(s) + m_0 \mathbf{1}), \mathbf{v}_{\varepsilon,\lambda}(s))_{H_0} \end{aligned} $$

for a.a. $s \in (0, T)$. Using the definition of the subdifferential, we see that

$$ \begin{aligned} & \frac{\lambda}{2} \frac{d}{ds} |\mathbf{v}_{\varepsilon,\lambda}(s)|_{H_0}^2 + \frac{1}{2} \frac{d}{ds} |\mathbf{v}_{\varepsilon,\lambda}(s)|_{V_0^*}^2 + \frac{\varepsilon}{2} |\mathbf{v}_{\varepsilon,\lambda}(s)|_{V_0}^2 \\ & + |\hat{\beta}_{\lambda}(u_{\varepsilon,\lambda}(s))|_{L^1(\Omega)} + |\hat{\beta}_{\lambda}(u_{\Gamma,\varepsilon,\lambda}(s))|_{L^1(\Gamma)} \\ & \le (|\Omega| + |\Gamma|) \hat{\beta}(m_0) + \frac{1}{2} |\mathbf{v}_{\varepsilon,\lambda}(s)|_{V_0^*}^2 + |\mathbf{f}(s)|_{V_0}^2 + L_{\pi}^2 \varepsilon^2 |\mathbf{v}_{\varepsilon,\lambda}(s)|_{V_0}^2 \quad \text{for a.a. } s \in (0, T). \end{aligned} $$

Taking $\varepsilon \in (0, \varepsilon_0]$ and using the Gronwall inequality, we obtain the existence of $M_1$ and $M_2$ independent of $\varepsilon \in (0, \varepsilon_0]$ and $\lambda \in (0, 1]$ satisfying the conclusion. $\square$
### Lemma 3.2. There exists a positive constant $M_3$, independent of $\varepsilon \in (0, \varepsilon_0]$ and $\lambda \in (0, 1]$, such that

$$ 
\begin{aligned}
& 2\lambda \int_0^t |\boldsymbol{v}'_{\varepsilon,\lambda}(s)|_{H_0}^2 ds + \int_0^t |\boldsymbol{v}'_{\varepsilon,\lambda}(s)|_{V_0^*}^2 ds + \varepsilon |\boldsymbol{v}_{\varepsilon,\lambda}(t)|_{V_0}^2 \\
& + 2|\hat{\beta}_{\lambda}(u_{\varepsilon,\lambda}(t))|_{L^1(\Omega)} + 2|\hat{\beta}_{\lambda}(u_{\Gamma,\varepsilon,\lambda}(t))|_{L^1(\Gamma)} \le M_3, \\
& \int_0^t |\boldsymbol{P}\boldsymbol{\mu}_{\varepsilon,\lambda}(s)|_{V_0}^2 ds \le M_3 \quad \text{for all } t \in [0, T].
\end{aligned}
$$

*Proof.* Multiplying (3.1) by $\boldsymbol{v}'_{\varepsilon,\lambda}(s) \in H_0$, we have

$$ 
\begin{aligned}
& \lambda |\boldsymbol{v}'_{\varepsilon,\lambda}(s)|_{H_0}^2 + \frac{1}{2} |\boldsymbol{v}'_{\varepsilon,\lambda}(s)|_{V_0^*}^2 + \varepsilon \frac{d}{ds} \varphi(\boldsymbol{v}_{\varepsilon,\lambda}(s)) + \frac{d}{ds} \int_{\Omega} \hat{\beta}_{\lambda}(u_{\varepsilon,\lambda}(s)) dx \\
& + \frac{d}{ds} \int_{\Gamma} \hat{\beta}_{\lambda}(u_{\Gamma,\varepsilon,\lambda}(s)) d\Gamma \le L_{\pi}^2 \varepsilon^2 |\boldsymbol{v}_{\varepsilon,\lambda}(s)|_{V_0}^2 + |\boldsymbol{f}(s)|_{V_0}^2 \quad \text{for a.a. } s \in (0, T).
\end{aligned}
$$

Integrating this over $(0, t)$ with respect to $s$, we see that there exists a positive constant $M_3$, independent of $\varepsilon \in (0, \varepsilon_0]$ and $\lambda \in (0, 1]$, such that the first estimate holds. Next, multiplying (3.4) by $\boldsymbol{P}\boldsymbol{\mu}_{\varepsilon,\lambda}(s) \in V_0$ and integrating the resultant over $(0, t)$ with respect to $s$, we obtain the second estimate. $\square$

The previous two lemmas are essentially the same as [13, Lemmas 3.1 and 3.2]. The next uniform estimate is the point of emphasis in this paper.

### Lemma 3.3. There exists positive constant $M_4$, independent of $\varepsilon \in (0, 1]$ and $\lambda \in (0, 1]$, such that

$$ 
\begin{aligned}
|\boldsymbol{u}_{\varepsilon,\lambda}(t)|_{H}^2 &\le M_4 \left(1 + \frac{\lambda}{\varepsilon}\right), \quad |\boldsymbol{v}_{\varepsilon,\lambda}(t)|_{H_0}^2 \le M_4 \left(1 + \frac{\lambda}{\varepsilon}\right), \\
\lambda |\boldsymbol{v}_{\varepsilon,\lambda}(t)|_{V_0}^2 + \varepsilon \int_0^t |\partial\varphi(\boldsymbol{v}_{\varepsilon,\lambda}(s))|_{H_0}^2 ds &\le M_4 \left(1 + \frac{\lambda}{\varepsilon}\right) \quad \text{for all } t \in [0, T].
\end{aligned}
$$

*Proof.* Multiplying (3.4) by $\boldsymbol{v}_{\varepsilon,\lambda}(s) \in V_0$ and using the fact $(d/ds)m(\boldsymbol{u}_{\varepsilon,\lambda}(s)) = 0$, we have

$$ (\boldsymbol{u}'_{\varepsilon,\lambda}(s), \boldsymbol{u}_{\varepsilon,\lambda}(s))_{H} + a(\boldsymbol{\mu}_{\varepsilon,\lambda}(s), \boldsymbol{u}_{\varepsilon,\lambda}(s)) = 0 $$

for a.a. $s \in (0, T)$ (see [13, Remark 3]). On the other hand, multiplying (3.5) by $\partial\varphi(\boldsymbol{v}_{\varepsilon,\lambda}(s)) \in H_0$ and integrating by parts, we have

$$ 
\begin{aligned}
a(\boldsymbol{\mu}_{\varepsilon,\lambda}(s), \boldsymbol{u}_{\varepsilon,\lambda}(s)) ={}& \frac{\lambda}{2} \frac{d}{ds} a(\boldsymbol{u}_{\varepsilon,\lambda}(s), \boldsymbol{u}_{\varepsilon,\lambda}(s)) + \varepsilon |\partial\varphi(\boldsymbol{v}_{\varepsilon,\lambda}(s))|_{H_0}^2 \\
& + \int_{\Omega} \beta'_{\lambda}(u_{\varepsilon,\lambda}(s)) |\nabla u_{\varepsilon,\lambda}(s)|^2 dx + \int_{\Gamma} \beta'_{\lambda}(u_{\Gamma,\varepsilon,\lambda}(s)) |\nabla_{\Gamma} u_{\Gamma,\varepsilon,\lambda}(s)|^2 d\Gamma \\
& + \varepsilon (\boldsymbol{\pi}(\boldsymbol{u}_{\varepsilon,\lambda}(s)), \partial\varphi(\boldsymbol{u}_{\varepsilon,\lambda}(s)))_{H} + (\partial\varphi(\boldsymbol{f}(s)), \boldsymbol{u}_{\varepsilon,\lambda}(s))_{H}
\end{aligned}
$$
for a.a. $s \in (0, T)$. Using the Lipschitz continuity of $\pi$ and (3.3), we see that
there exists a positive constant $C_\pi$ such that

$$ \frac{d}{ds} |\mathbf{u}_{\varepsilon, \lambda}(s)|_{\mathbf{H}}^{2} + \lambda \frac{d}{ds} |\mathbf{v}_{\varepsilon, \lambda}(s)|_{\mathbf{V}_{0}}^{2} + \varepsilon |\partial \varphi(\mathbf{v}_{\varepsilon, \lambda}(s))|_{\mathbf{H}_{0}}^{2} \leq C_{\pi} (|\mathbf{u}_{\varepsilon, \lambda}(s)|_{\mathbf{H}}^{2} + 1) + |\mathbf{g}(s)|_{\mathbf{H}_{0}}^{2} $$

for a.a. $s \in (0, T)$. Then, using (3.2) and the Gronwall inequality, we deduce that

$$ \begin{aligned} |\mathbf{u}_{\varepsilon, \lambda}(t)|_{\mathbf{H}}^{2} + \lambda |\mathbf{v}_{\varepsilon, \lambda}(t)|_{\mathbf{V}_{0}}^{2} &\leq \left\{ |v_{0\varepsilon} + m_{0} \mathbf{1}_{\mathbf{H}}|_{\mathbf{H}}^{2} + \lambda |v_{0\varepsilon}|_{\mathbf{V}_{0}}^{2} + C_{\pi} T + |\mathbf{g}|_{L^{2}(0,T; \mathbf{H}_{0})}^{2} \right\} e^{C_{\pi} T} \\ &\leq \left\{ 2C + 2|m_{0}|^{2}(|\Omega| + |\Gamma|) + \frac{\lambda}{\varepsilon} C + C_{\pi} T + |\mathbf{g}|_{L^{2}(0,T; \mathbf{H}_{0})}^{2} \right\} e^{C_{\pi} T} \end{aligned} $$

for all $t \in [0, T]$. That is, there exists a positive constant $M_4$ independent of $\varepsilon \in (0, 1]$ and $\lambda \in (0, 1]$ such that the uniform estimates hold. □

Lemma 3.4. There exists positive constant $M_5$, independent of $\varepsilon \in (0, 1]$ and $\lambda \in (0, 1]$, such that

$$ \begin{aligned} \int_0^t |\boldsymbol{\mu}_{\varepsilon, \lambda}(s)|_{\mathbf{V}_0}^2 ds &\le M_5 \left(1 + \frac{\lambda}{\varepsilon}\right), \\ \int_0^t |\boldsymbol{\beta}_{\lambda}(\mathbf{u}_{\varepsilon, \lambda}(s))|_{\mathbf{H}}^2 ds &\le M_5 \left(1 + \frac{\lambda}{\varepsilon}\right) \quad \text{for all } t \in [0, T]. \end{aligned} $$

Using Lemmas 3.1 to 3.3, the proofs of these uniform estimates are completely
the same as those for [9, Lemmas 4.3 and 4.4]. Therefore, we omit the proof.

## 3.2 Limiting procedure

From the previous uniform estimates, we can consider the limit as $\lambda \searrow 0$. More precisely, for each $\varepsilon \in (0, \varepsilon_0]$, there exists a subsequence $\{\lambda_k\}_{k \in \mathbb{N}}$ with $\lambda_k \searrow 0$ as $k \to +\infty$ and a quadruplet $(\mathbf{v}_\varepsilon, \mathbf{v}_\varepsilon^*, \boldsymbol{\mu}_\varepsilon, \boldsymbol{\xi}_\varepsilon)$ of $\mathbf{v}_\varepsilon \in H^1(0, T; \mathbf{V}_0^*) \cap L^\infty(0, T; \mathbf{V}_0) \cap L^2(0, T; \mathbf{W})$, $\mathbf{v}_\varepsilon^* \in L^2(0, T; \mathbf{H}_0)$, $\boldsymbol{\mu}_\varepsilon \in L^2(0, T; \mathbf{V})$, $\boldsymbol{\xi}_\varepsilon \in L^2(0, T; \mathbf{H})$, such that

$$ \begin{aligned} \mathbf{v}_{\varepsilon, \lambda_k} &\to \mathbf{v}_\varepsilon \quad \text{weakly star in } L^\infty(0, T; \mathbf{H}_0), & \mathbf{v}_{\varepsilon, \lambda_k} &\to \mathbf{v}_\varepsilon \quad \text{weakly in } L^2(0, T; \mathbf{V}_0), \\ \lambda_k \mathbf{v}_{\varepsilon, \lambda_k}' &\to \mathbf{0} \quad \text{in } L^2(0, T; \mathbf{H}_0), & \mathbf{v}_{\varepsilon, \lambda_k}' &\to \mathbf{v}_\varepsilon' \quad \text{weakly in } L^2(0, T; \mathbf{V}_0^*), \\ \mathbf{u}_{\varepsilon, \lambda_k} &\to \mathbf{u}_\varepsilon := \mathbf{v}_\varepsilon + m_0 \mathbf{1} \quad \text{weakly star in } L^\infty(0, T; \mathbf{V}), \\ \partial \varphi(\mathbf{v}_{\varepsilon, \lambda_k}) &\to \mathbf{v}_\varepsilon^* \quad \text{weakly in } L^2(0, T; \mathbf{H}_0), & \boldsymbol{\mu}_{\varepsilon, \lambda_k} &\to \boldsymbol{\mu}_\varepsilon \quad \text{weakly in } L^2(0, T; \mathbf{V}), \\ \boldsymbol{\beta}_{\lambda_k}(\mathbf{u}_{\varepsilon, \lambda_k}) &\to \boldsymbol{\xi}_\varepsilon \quad \text{weakly in } L^2(0, T; \mathbf{H}) \quad \text{as } k \to +\infty. \end{aligned} $$

From the compactness theorem (see, e.g., [20, Section 8, Corollary 4]), this gives

$$ \begin{aligned} \mathbf{v}_{\varepsilon, \lambda_k} &\to \mathbf{v}_\varepsilon \quad \text{in } C([0, T]; \mathbf{H}_0), & \mathbf{u}_{\varepsilon, \lambda_k} &\to \mathbf{u}_\varepsilon \quad \text{in } C([0, T]; \mathbf{H}), \\ \boldsymbol{\pi}(\mathbf{u}_{\varepsilon, \lambda_k}) &\to \boldsymbol{\pi}(\mathbf{u}_\varepsilon) \quad \text{in } C([0, T]; \mathbf{H}) \quad \text{as } k \to +\infty. \end{aligned} $$
Moreover, from the demi-closedness of $\partial\varphi$ and [4, Proposition 2.2], we see that $v_{\varepsilon}^* = \partial\varphi(v_{\varepsilon})$ in $L^2(0, T; H_0)$ and $\xi_{\varepsilon} \in \beta(\mathbf{u}_{\varepsilon})$ in $L^2(0, T; H)$. From these facts, we deduce from (3.4) and (3.5) that

$$v'_{\varepsilon}(t) + F(P\mu_{\varepsilon}(t)) = 0 \quad \text{in } V_0^*, \qquad (3.6)$$

$$\xi_{\varepsilon}(t) \in \beta(\mathbf{u}_{\varepsilon}(t)), \quad \mu_{\varepsilon}(t) = \varepsilon\partial\varphi(v_{\varepsilon}(t)) + \xi_{\varepsilon}(t) + \varepsilon\pi(\mathbf{u}_{\varepsilon}(t)) - f(t) \quad \text{in } H \qquad (3.7)$$

for a.a. $t \in (0, T)$, with $v_{\varepsilon}(0) = v_{0\varepsilon}$ in $H$. We also have the regularity $\mathbf{u}_{\varepsilon} \in H^1(0, T; V^*) \cap L^{\infty}(0, T; V) \cap L^2(0, T; W)$. Now, taking the limit inferior as $\lambda \searrow 0$ on the uniform estimates, $\lambda/\varepsilon \searrow 0$ for all $\varepsilon \in (0, \varepsilon_0]$, and we therefore obtain the same kind of uniform estimates as in the previous lemmas independent of $\varepsilon \in (0, \varepsilon_0]$.

**Proof of Theorem 2.1.** By using the estimates for $v_{\varepsilon}, \mathbf{u}_{\varepsilon}, \mu_{\varepsilon}$ and $\xi_{\varepsilon}$, there exist a subsequence $\{\varepsilon_k\}_{k \in \mathbb{N}}$ with $\varepsilon_k \searrow 0$ as $k \to +\infty$ and functions $v \in H^1(0, T; V_0^*) \cap L^{\infty}(0, T; H_0)$, $\mathbf{u} \in H^1(0, T; V^*) \cap L^{\infty}(0, T; H)$, $\mu \in L^2(0, T; V)$ and $\xi \in L^2(0, T; H)$ such that

$$v_{\varepsilon_k} \to v \quad \text{weakly star in } L^{\infty}(0, T; H_0),$$

$$\mathbf{u}_{\varepsilon_k} \to \mathbf{u} = v + m_0 \mathbf{1} \quad \text{weakly star in } L^{\infty}(0, T; H), \quad \varepsilon_k v_{\varepsilon_k} \to 0 \quad \text{in } L^{\infty}(0, T; V_0),$$

$$v'_{\varepsilon_k} \to v' \quad \text{weakly in } L^2(0, T; V_0^*), \quad \mathbf{u}'_{\varepsilon_k} \to \mathbf{u}' \quad \text{weakly in } L^2(0, T; V^*),$$

$$\mu_{\varepsilon_k} \to \mu \quad \text{weakly in } L^2(0, T; V), \quad \xi_{\varepsilon_k} \to \xi \quad \text{weakly in } L^2(0, T; H),$$

$$\varepsilon_k \pi(\mathbf{u}_{\varepsilon_k}) \to 0 \quad \text{in } L^{\infty}(0, T; H) \quad \text{as } k \to +\infty.$$

From the Ascoli–Arzelà theorem, we also have

$$v_{\varepsilon_k} \to v \quad \text{in } C([0, T]; V_0^*), \quad \mathbf{u}_{\varepsilon_k} \to \mathbf{u} \quad \text{in } C([0, T]; V^*) \quad \text{as } k \to +\infty.$$

Now, multiplying (3.7) by $\eta \in L^2(0, T; V)$ and integrating over $(0, T)$, we obtain

$$\int_0^T (\mu_{\varepsilon_k}(t), \eta(t))_H dt = \varepsilon_k \int_0^T a(v_{\varepsilon_k}(t), \eta(t)) dt + \int_0^T (\xi_{\varepsilon_k}(t), \eta(t))_H dt \\ + \varepsilon_k \int_0^T (\pi(\mathbf{u}_{\varepsilon_k}(t)), \eta(t))_H dt - \int_0^T (f(t), \eta(t))_H dt. \qquad (3.8)$$

Letting $k \to \infty$, we obtain

$$\int_0^T (\mu(t), \eta(t))_H dt = \int_0^T (\xi(t) - f(t), \eta(t))_H dt \quad \text{for all } \eta \in L^2(0, T; V),$$

namely, $\mu = \xi - f$ in $L^2(0, T; H)$. This implies the regularity of $\xi \in L^2(0, T; V)$, that is, $\xi_\Gamma = \xi|_\Gamma$ a.e. on $\Sigma$. Next, we take $\eta := \mathbf{u}_{\varepsilon_k} \in L^2(0, T; V)$ in (3.8), so
that

$$ \limsup_{k \to +\infty} \int_0^T (\xi_{\varepsilon_k}(t), u_{\varepsilon_k}(t))_H dt \le \int_0^T \langle u(t), \mu(t) \rangle_{V^*, V} dt + \int_0^T \langle f(t), u(t) \rangle_H dt \\ = \int_0^T (\xi(t), u(t))_H dt. $$

Thus, applying [4, Proposition 2.2] we have $\xi \in \beta(\mathbf{u})$ in $L^2(0, T; H)$, and so we obtain $\xi \in \beta(\mathbf{u})$ a.e. in $\Omega$. $\xi_{\Gamma} \in \beta(u_{\Gamma})$ a.e. on $\Gamma$. Finally, letting $k \to +\infty$ and applying Hahn-Banach extension theorem of bounded linear functional on $V$ to $V \times V_{\Gamma}$, then we see that (3.6) gives (2.1) for a.a. $t \in (0, T)$, with $u(0) = u_0$ a.e. in $\Omega$ and $u_{\Gamma}(0) = u_{0\Gamma}$ a.e. on $\Gamma$. $\square$

## Acknowledgments

The author is indebted to professor Pierluigi Colli, who kindly gave him the opportunity for fruitful discussions. The author is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Number 26400164. Last but not least, the author is also grateful to the referee for the careful reading of the manuscript.

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