# Two-phase flows through porous media
described by a Cahn–Hilliard–Brinkman model
with dynamic boundary conditions

Pierluigi Colli 111 Dipartimento di Matematica “F. Casorati”, Università di
Pavia, and Research Associate at the IMATI – C.N.R. Pavia, 27100 Pavia, Italy
<EMAIL_ADDRESS>giusch04@unipv.it). Patrik Knopf 222Department of
Mathematics, University of Regensburg, 93040 Germany (patrik.knopf@ur.de).
Giulio Schimperna 11footnotemark: 1 Andrea Signori 333Dipartimento di
Matematica, Politecnico di Milano, 20133 Milano, Italy
(andrea.signori@polimi.it).

Abstract

We investigate a new diffuse-interface model that describes creeping two-phase
flows (i.e., flows exhibiting a low Reynolds number), especially flows that
permeate a porous medium. The system of equations consists of a Brinkman
equation for the volume averaged velocity field as well as a convective
Cahn–Hilliard equation with dynamic boundary conditions for the phase-field,
which describes the location of the two fluids within the domain. The dynamic
boundary conditions are incorporated to model the interaction of the fluids
with the wall of the container more precisely. In particular, they allow for a
dynamic evolution of the contact angle between the interface separating the
fluids and the boundary, and also for a convection-induced motion of the
corresponding contact line. For our model, we first prove the existence of
global-in-time weak solutions in the case where regular potentials are used in
the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness
of the weak solution under suitable additional assumptions. Moreover, we
further prove the existence of weak solutions in the case of singular
potentials. Therefore, we regularize such singular potentials by a Yosida
approximation, such that the results for regular potentials can be applied,
and eventually pass to the limit in this approximation scheme.

Keywords: two-phase flows, porous media, Cahn–Hilliard equation, Brinkman
equation, dynamic boundary conditions, bulk-surface interaction.
Mathematics Subject Classification: 35D30, 35K35, 35K86, 35B65, 76D03, 76T06

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and main results
    1. 2.1 Notation
    2. 2.2 General assumptions
    3. 2.3 Preliminaries
    4. 2.4 The Cahn–Hilliard–Brinkman system with regular potentials
    5. 2.5 The Cahn–Hilliard–Brinkman system with singular potentials
  3. 3 Analysis of the Cahn–Hilliard–Brinkman system with regular potentials
    1. 3.1 Existence of weak solutions in the case $K>0$
    2. 3.2 The limit $K\to 0$ and existence of a weak solution in the case $K=0$
    3. 3.3 Uniqueness of the weak solution for regular potentials
  4. 4 Analysis of the Cahn–Hilliard–Brinkman system with singular potentials
    1. 4.1 Yosida regularization
    2. 4.2 Uniform estimates
    3. 4.3 Passage to the limit and conclusion of the proof
  5. Appendix: Some calculus for bulk-surface function spaces

## 1 Introduction

The mathematical study of two-phase flows is an important topic in many areas
of applied science such as engineering, chemistry and biology. To predict the
motion of two fluids, it is crucial to understand how the interface between
these fluids evolves. To provide a mathematical description of this interface,
two fundamental methods have been developed: the sharp-interface approach and
the diffuse-interface approach. In the former, the interface is represented as
a hypersurface evolving in the surrounding domain. The occurring quantities
(e.g., the velocity fields) are then described by a free boundary problem. In
the latter, the fluids are represented by a phase-field function which is
expected to attain values close to $1$ in the region where the first fluid is
present, and close to $-1$ in the region where the second fluid is located.
However, unlike in sharp-interface models, this phase-field does not jump
between the values $1$ and $-1$ but exhibits a continuous transition between
these values in a thin tubular neighborhood around the boundary between the
fluids. This tubular neighborhood is referred to as the diffuse interface and
its thickness is proportional to a small parameter $\epsilon>0$. For a
comparison of sharp-interface methods and diffuse-interface methods, we refer
to [28, 4, 35, 55]. We point out that, even though the sharp-interface and the
diffuse-interface approach are conceptually different, they can, in general,
be related by the sharp-interface limit in which a parameter related to the
thickness of the diffuse interface is sent to zero.

In the context of diffuse-interface models, such models in which the phase-
field is described by a Cahn–Hilliard type equation have become particularly
popular. One of the most widely used models for describing the motion of two
viscous, incompressible fluids with matched (constant) densities is the Model
H. It was first proposed in [45] and was later rigorously derived in [43]. The
PDE system consists of an incompressible Navier–Stokes equation coupled with a
convective Cahn–Hilliard equation. In terms of mathematical analysis, the
Model H was investigated quite extensively, see, e.g., in [1, 10, 30, 41].
Further generalizations of this model can be found in [51, 11, 27, 57, 58, 35,
44, 29].

One drawback of the Model H is that it can merely be used to describe the
situation in which the fluids have the same individual density. To overcome
this issue, a thermodynamically consistent diffuse-interface model for
incompressible two-phase flows with possibly unmatched densities was derived
in the seminal work [6]. This model is usually referred to as the AGG model.
Concerning mathematical analysis of this model, we refer the reader to [2, 3,
7, 39, 38, 5]. The connection between the AGG model and the two-phase
Navier–Stokes free boundary problem is explained in [6, 4].

Even though the AGG model and the Model H subject to the classical boundary
conditions (i.e., a no-slip boundary condition for the velocity field and
homogeneous Neumann boundary conditions for the convective Cahn–Hilliard
equation) are well suited to describe the motion of the fluids in the the
interior of the considered domain, they still inherit some limitations from
the underlying (convective) Cahn–Hilliard system with homogeneous Neumann
boundary conditions. The main limitations are:

  1. (L1)

The homogeneous Neumann condition on the phase-field enforces that the diffuse
interface always intersects the boundary at a perfect ninety degree contact
angle. This will not be fulfilled in many applications. In general, the
contact angle might even change dynamically over the course of time.

  2. (L2)

The no-slip boundary condition on the velocity field makes the model not very
suitable to describe general moving contact line phenomena. As the trace of
the velocity field at the boundary is fixed to be identically zero, any motion
of the contact line of the diffuse interface can be caused only by diffusive
but not by convective effects.

  3. (L3)

The mass of the fluids in the bulk is conserved. Therefore, a transfer of
material between the bulk and the boundary (caused, e.g., by absorption
processes or chemical reactions) cannot be described.

A more detailed discussion can be found in [40].

To overcome the aforementioned restrictions (L1) and (L2), a class of dynamic
boundary conditions was derived in [56]. It involves an Allen–Cahn type
dynamic boundary condition for the phase-field coupled to a generalized
Navier-slip boundary condition for the velocity field. The Model H subject to
this boundary condition was analyzed in [31] whereas the AGG model subject to
this boundary condition was investigated in [32].

Recently, a thermodynamically consistent generalization of the AGG model
subject to another class of dynamic boundary conditions was derived in [40].
Here, the boundary condition consists of a convective surface Cahn–Hilliard
equation and a generalized Navier-slip boundary condition. Compared to the
models studied in [56, 32, 31], the Navier–Stokes–Cahn–Hilliard system
introduced in [40] provides more regularity for the boundary quantities and
therefore, the uniqueness of weak solutions can be established in two space
dimensions. Moreover, due to the fourth-order dynamic boundary condition of
Cahn–Hilliard type, the model in [40] is not only capable of overcoming the
limitations (L1) and (L2) but also (L3).

In the present paper, we particularly want to consider the situation of
creeping flows meaning that the Reynolds number

$\displaystyle\mathrm{Re}=\frac{uL}{\nu}$

associated with the fluids is very small ($\mathrm{Re}\ll 1$). This occurs if
the flow speed $u$ and/or the characteristic length $L$ of the flow are small
compared to the kinematic viscosity $\nu$. In this situation, it is not
necessary to describe the time evolution of the velocity field by the full
Navier–Stokes equation. Since advective inertial forces are small compared to
viscous forces, the material derivative can be neglected. This leads to the
Stokes equation. If a creeping flow through a porous medium is to be
considered, an additional term accounting for the permeability needs to be
included. The velocity field is then determined by the Brinkman equation.

Therefore, in this paper, we study the following Cahn–Hilliard–Brinkman system
with dynamic boundary conditions:

$\displaystyle-\mathop{\rm
div}\nolimits(2\nu(\varphi)\mathbf{D}{\boldsymbol{v}})+\lambda(\varphi){\boldsymbol{v}}+\nabla
p=\mu\nabla\varphi\quad$ in $Q$, (1.1a) $\displaystyle\mathop{\rm
div}\nolimits({\boldsymbol{v}})=0\quad$ in $Q$, (1.1b)
$\displaystyle\partial_{t}\varphi+\mathop{\rm
div}\nolimits(\varphi{\boldsymbol{v}})-\mathop{\rm
div}\nolimits(M_{\Omega}(\varphi)\nabla\mu)=0\quad$ in $Q$, (1.1c)
$\displaystyle\mu=-\epsilon\Delta\varphi+\tfrac{1}{\epsilon}F^{\prime}(\varphi)\quad$
in $Q$, (1.1d) $\displaystyle\partial_{t}\psi+\mathop{\rm
div}\nolimits_{\Gamma}(\psi{\boldsymbol{v}})-\mathop{\rm
div}\nolimits_{\Gamma}(M_{\Gamma}(\psi)\nabla_{\Gamma}\theta)=0\quad$ on
$\Sigma$, (1.1e)
$\displaystyle\theta=-\epsilon_{\Gamma}\Delta_{\Gamma}\psi+\tfrac{1}{\epsilon_{\Gamma}}G^{\prime}(\psi)+\partial_{\bf
n}\varphi\quad$ on $\Sigma$, (1.1f) $\displaystyle K\partial_{\bf
n}\varphi=\psi-\varphi\quad$ on $\Sigma$, (1.1g) $\displaystyle
M_{\Omega}(\varphi)\partial_{\bf n}\mu={\boldsymbol{v}}\cdot{\bf n}=0\quad$ on
$\Sigma$, (1.1h)
$\displaystyle[2\nu(\varphi)\,\mathbf{D}{\boldsymbol{v}}\,{\bf
n}+\gamma(\psi){\boldsymbol{v}}]_{\tau}=-[\psi\nabla_{\Gamma}\theta]_{\tau}\quad$
on $\Sigma$, (1.1i) $\displaystyle\varphi(0)=\varphi_{0}\quad$ in $\Omega$,
(1.1j) $\displaystyle\psi(0)=\psi_{0}\quad$ on $\Gamma$. (1.1k)

It can be regarded as a variant of the Navier–Stokes–Cahn–Hilliard model
derived in [40], where the incompressible Navier–Stokes equation is replaced
by the incompressible Brinkman/Stokes equation $\big{(}$(1.1a),(1.1b)$\big{)}$
to describe the situation of a creeping two-phase flow.

In system (1.1), $\Omega\subset{\mathbb{R}}^{d}$ with $d\in\\{2,3\\}$ is a
bounded domain with boundary $\Gamma:=\partial\Omega$, $T>0$ is a given final
time, and for brevity, the notation $Q:=\Omega\times(0,T)$ and
$\Sigma=\Gamma\times(0,T)$ is used. The vector-valued function
${\boldsymbol{v}}:Q\to{\mathbb{R}}^{d}$ stands for the volume averaged
velocity field associated with the fluid mixture and

$\displaystyle\mathbf{D}{\boldsymbol{v}}=\frac{1}{2}\big{(}\nabla{\boldsymbol{v}}+(\nabla{\boldsymbol{v}})^{\top}\big{)}$

denotes the associated symmetric gradient. For the sake of simplicity, we will
usually refrain from writing the trace operator. For instance, we will often
simply write ${\boldsymbol{v}}$ instead of ${\boldsymbol{v}}|_{\Gamma}$.
Nevertheless, in some instances, where confusion may arise, we will employ the
explicit notation. For any vector field ${\boldsymbol{w}}$ on the boundary, we
will write
${\boldsymbol{w}}_{\tau}:={\boldsymbol{w}}-({\boldsymbol{w}}\cdot{\bf n}){\bf
n}$ to denote its tangential component. The symbols $\nabla_{\Gamma}$ and
$\mathop{\rm div}\nolimits_{\Gamma}$ denote the surface gradient and the
surface divergence, respectively, and $\Delta_{\Gamma}$ stands for the
Laplace–Beltrami operator.

The functions $\varphi:Q\to{\mathbb{R}}$ and $\mu:Q\to{\mathbb{R}}$ denote the
phase-field and the chemical potential in the bulk, respectively, whereas
$\psi:\Sigma\to{\mathbb{R}}$ and $\theta:\Sigma\to{\mathbb{R}}$ represent the
phase-field and the chemical potential on the boundary, respectively.
Furthermore, the parameters $\epsilon$ and $\epsilon_{\Gamma}$ are positive
real numbers which are related to the thickness of the diffuse interface in
the bulk and on the surface, respectively. Therefore, these constants are
usually chosen to be quite small. However, as their values do not have any
impact on the mathematical analysis, we will simply fix
$\epsilon=\epsilon_{\Gamma}=1$ in the subsequent sections. The phase-fields
$\varphi$ and $\psi$ are directly related by the coupling condition (1.1g),
where $K$ is a given nonnegative constant.

From a physical point of view, the kinematic viscosity $\nu(\varphi)$ and the
permeability coefficient $\lambda(\varphi)$ in the Brinkman/Stokes equation
(1.1a) can be expressed as

$\displaystyle\nu(\varphi)=\frac{\eta(\varphi)}{\varrho}\quad\text{and}\quad\lambda(\varphi)=\frac{\eta(\varphi)}{\varkappa},$

where $\eta(\varphi)>0$ denotes the dynamic viscosity, and the constants
$\varrho>0$ and $\varkappa>0$ stand for the porosity and the intrinsic
permeability of the porous medium, respectively. If both $\nu(\varphi)$ and
$\lambda(\varphi)$ are positive, (1.1a) is the (quasi-stationary) Brinkman
equation which describes the flow through a porous medium. However, if the
porosity $\varkappa$ is large compared to the viscosity $\eta(\varphi)$, the
function $\lambda(\varphi)$ is very small and can be neglected. In this case
we enter the Stokes regime, where no porous media is considered (or the
effects of the porous medium are at least negligible). In the formal limit
$\varkappa\to\infty$ or $\lambda(\varphi)\to 0$, (1.1a) degenerates to the
Stokes equation. In our analysis, we will be able to handle the Brinkman case
($\nu(\varphi)>0$ and $\lambda(\varphi)>0$) and the Stokes case
($\nu(\varphi)>0$ and $\lambda(\varphi)\equiv 0$) simultaneously. On the other
hand, if $\lambda(\varphi)$ remains positive and the porosity $\varrho$ is
large compared to the dynamic viscosity $\eta(\varphi)$ such that
$\nu(\varphi)$ can be neglected, (1.1a) degenerates to Darcy’s law. However,
we are not able to handle this case in terms of mathematical analysis as due
to the absence of spatial derivatives of the velocity field in (1.1a), we
would not obtain enough regularity to define the trace of ${\boldsymbol{v}}$
on the boundary in a reasonable manner.

The functions $F^{\prime}$ and $G^{\prime}$ are the derivatives of double-well
potentials $F$ and $G$, respectively. Especially in applications related to
materials science, a physically relevant choice for $F$ and/or $G$ is the
logarithmic potential, which is also referred to as the Flory–Huggins
potential. It is given by

$\displaystyle
W_{\mathrm{log}}(s)=\frac{\Theta}{2}\big{[}(1+s)\,\ln(1+s)+(1-s)\,\ln(1-s)\big{]}+\frac{\Theta_{c}}{2}(1-s^{2}),$
(1.2)

for all $s\in(-1,1)$. Here, ${\Theta>0}$ is the absolute temperature of the
mixture, and $\Theta_{c}$ is a critical temperature such that phase separation
will occur in case $0<\Theta<\Theta_{c}$. The logarithmic potential is
classified as a singular potential since its derivative $F^{\prime}$ diverges
to $\pm\infty$ when its argument approaches $\pm 1$. It is often approximated
by a polynomial double-well potential

$\displaystyle W_{\mathrm{pol}}(s)=\frac{\alpha}{4}(s^{2}-1)^{2}\quad\text{for
all $s\in(-1,1)$},$ (1.3)

where $\alpha>0$ is a suitable constant. Another very commonly used singular
potential is the double-obstacle potential, which is given by

$\displaystyle W_{\mathrm{obst}}(s)=\begin{cases}\frac{1}{2}(1-s^{2})&\text{if
$|s|\leq 1$,}\\\ +\infty&\text{else}.\end{cases}$ (1.4)

In the case $K=0$, the convective bulk-surface Cahn–Hilliard subsystem
(1.1c)–(1.1h) is a special case of the one introduced in [40] since for the
chemical potential $\mu$, a homogeneous Neumann type boundary condition is
imposed in (1.1h). This corresponds to the choice $L=\infty$ in [40].
Therefore, by system (1.1), we describe a situation where no transfer of
material between bulk and boundary occurs. However, its is important that due
to the boundary conditions (1.1e)–(1.1i), the model (1.1) allows for dynamic
changes of the contact angle as well as for a convection-induced motion of the
contact line. This means that the limitations (L1) and (L2) explained above
can be overcome. It is worth mentioning that this setup of dynamic boundary
conditions for the Cahn–Hilliard equation (without coupling to a velocity
equation) was originally derived in [50] by the Energetic Variational
Approach. This system was further investigated in [19, 33, 53, 46]. For
similar works on the Cahn–Hilliard equation with Cahn–Hilliard type dynamic
boundary conditions, we refer to [49, 47, 34, 42, 14, 17, 18, 64].

In contrast to the model introduced in [40], the phase-fields $\varphi$ and
$\psi$ are not just coupled by the trace relation $\varphi|_{\Sigma}=\psi$ on
$\Sigma$, but by the more general Robin type coupling condition
$K\partial_{\bf n}\varphi=\psi-\varphi$ with $K\geq 0$ (see (1.1g)). This also
includes the trace relation via the choice $K=0$. The coupling condition
(1.1g) was first used in [16] for an Allen–Cahn type dynamic boundary
condition, and later in [46] for a Cahn–Hilliard type dynamic boundary
condition. In particular, it was rigorously shown in [46] that the Dirichlet
type coupling condition $\varphi|_{\Sigma}=\psi$ on $\Sigma$ can be recovered
in the asymptotic limit $K\to 0$. From a physical point of view, the boundary
condition (1.1g) with $K>0$ makes sense if the materials on the boundary may
be different from those in the bulk. For instance, this might be the case if
the materials on the boundary are transformed by chemical reactions. Apart
from this, the boundary condition (1.1g) with $K>0$ has a crucial advantage
for the mathematical analysis concerning the construction of weak solutions.
The reason is that in the case $K=0$, the Dirichlet type coupling condition
for the phase-fields already fixes one degree of freedom and is therefore a
bad match for the no-mass-flux boundary condition $\eqref{eq:8}_{1}$. This
seems to make the direct construction of weak solutions by a Faedo–Galerkin
scheme impossible. However, employing (1.1g) with $K>0$, such problems do not
arise. Hence, our strategy is to first prove the existence of weak solutions
for $K>0$ by a Faedo–Galerkin approach. Afterwards, we construct a weak
solution associated with $K=0$ by passing to the limit $K\to 0$ in a suitable
sense.

An important property of the system (1.1) (for any $K\geq 0$) is its
thermodynamic consistency with respect to the free energy functional

$\displaystyle E_{K}(\varphi,\psi)$
$\displaystyle:=\int_{\Omega}\Big{(}\frac{\epsilon}{2}|\nabla\varphi|^{2}+\frac{1}{\epsilon}F(\varphi)\Big{)}+\int_{\Gamma}\Big{(}\frac{\epsilon_{\Gamma}}{2}|\nabla_{\Gamma}\psi|^{2}+\frac{1}{\epsilon_{\Gamma}}G(\psi)\Big{)}$
$\displaystyle\quad+\frac{\sigma(K)}{2}\int_{\Gamma}(\psi-\varphi)^{2},$ (1.5)

where $\sigma(K)=K^{-1}$ if $K>0$ and $\sigma(K)=0$ if $K=0$. This means that
sufficiently regular solutions of (1.1) satisfy the energy dissipation law

$\displaystyle\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}E_{K}(\varphi,\psi)&=-\int_{\Omega}\lambda(\varphi)|{\boldsymbol{v}}|^{2}-\int_{\Omega}M_{\Omega}(\varphi)|\nabla\mu|^{2}-\int_{\Gamma}M_{\Gamma}(\psi)|\nabla_{\Gamma}\theta|^{2}\\\
&\quad-2\int_{\Omega}\nu(\varphi)|\mathbf{D}{\boldsymbol{v}}|^{2}-\int_{\Gamma}\gamma(\psi)|{\boldsymbol{v}}|^{2},\end{aligned}$

on $[0,T]$, where all the terms on the right-hand side are non-positive and
can be interpreted as the dissipation rate. Compared to the model in [40], the
additional term
$\int_{\Omega}\lambda(\varphi)\left|{\boldsymbol{v}}\right|^{2}$ arises due to
dissipative effects caused by the porous medium.

As mentioned above, due to the usage of the no-mass-flux condition
$M_{\Omega}(\varphi)\partial_{\bf n}\mu=0$ on $\Sigma$ (see (1.1h)), we do not
describe any transfer of material between bulk and surface. This entails that
the bulk mass and the surface mass are conserved separately, i.e.,
sufficiently regular solutions satisfy the mass conservation laws

$\displaystyle\frac{1}{|\Omega|}\int_{\Omega}\varphi(t)$
$\displaystyle=\frac{1}{|\Omega|}\int_{\Omega}\varphi_{0}=:m_{0}$ (1.6)
$\displaystyle\frac{1}{|\Gamma|}\int_{\Gamma}\psi(t)$
$\displaystyle=\frac{1}{|\Gamma|}\int_{\Gamma}\psi_{0}=:m_{\Gamma 0}$ (1.7)

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

The structure of the paper is as follows. In Section 2 we collect some
notation, assumptions, preliminaries and important tools as well as the main
results of our work. In the case of regular potentials, the
Cahn–Hilliard–Brinkman system is analyzed in Section 3. There, we first
establish the existence of a weak solution in the case $K>0$, where phase-
fields are coupled by a Robin type boundary condition. By sending $K\to 0$, we
then obtain the existence of a weak solution also in the case $K=0$, where the
phase-fields are coupled by a trace relation. Afterwards, we prove uniqueness
in all cases $K\geq 0$ under suitable additional assumptions. Eventually, in
Section 4, we show the existence of a weak solution to the
Cahn–Hilliard–Brinkman system with singular potentials. This is done based on
the results established in Section 3 by approximating the singular potentials
by means of Yosida regularizations and finally passing to the limit.

## 2 Preliminaries and main results

### 2.1 Notation

Throughout the manuscript, $\Omega$ is a bounded domain in ${\mathbb{R}}^{d}$,
$d\in\\{2,3\\}$, with Lipschitz boundary $\Gamma:=\partial\Omega$ and ${\bf
n}$ is the associated outward unit normal vector field. We write $|\Omega|$
and $|\Gamma|$ to denote the Lebesgue measure of $\Omega$ and the Hausdorff
measure of $\Gamma$, respectively. For any given Banach space $X$, we denote
its norm by $\left\|\,\cdot\,\right\|_{X}$, its dual space by $X^{*}$ and the
duality pairing between $X^{*}$ and $X$ by
$\mathopen{\langle}\cdot,\cdot\mathclose{\rangle}_{X}$. Besides, if $X$ is a
Hilbert space, we write $(\cdot,\cdot)_{X}$ to denote the corresponding inner
product. For every $1\leq p\leq\infty$, $k\geq 0$ and $s>0$, the standard
Lebesgue spaces, Sobolev–Slobodeckij spaces and Sobolev spaces defined on
$\Omega$ are denoted by $L^{p}(\Omega)$, $W^{k,p}(\Omega)$ and
$H^{s}(\Omega)$, and their standard norms are denoted by
$\left\|\,\cdot\,\right\|_{L^{p}(\Omega)}$
$\left\|\,\cdot\,\right\|_{W^{k,p}(\Omega)}$ and
$\left\|\,\cdot\,\right\|_{H^{s}(\Omega)}$, respectively. It is well known
that the spaces $H^{0}(\Omega)=L^{2}(\Omega)$ and
$H^{k}(\Omega)=W^{k,2}(\Omega)$ for all $k\in{\mathbb{N}}$ can be identified,
and these spaces are Hilbert spaces. The Lebesgue spaces, Sobolev–Slobodeckij
spaces and Sobolev spaces on the boundary $\Gamma$ are defined analogously.
For more details, we refer to [62, 63]. Moreover, for any Banach spaces $X$
and $Y$, their intersection $X\cap Y$ is also a Banach space subject to the
norm

$\displaystyle\left\|v\right\|_{X\cap
Y}:=\left\|v\right\|_{X}+\left\|v\right\|_{Y},\quad v\in X\cap Y.$

As some spaces will appear very frequently, we introduce the shortcuts:

$\displaystyle H$ $\displaystyle:=L^{2}(\Omega),\quad$ $\displaystyle
H_{\Gamma}$ $\displaystyle:=L^{2}(\Gamma),\quad$ $\displaystyle V$
$\displaystyle:=H^{1}(\Omega),\quad$ $\displaystyle{V}_{\Gamma}$
$\displaystyle:=H^{1}(\Gamma),$ $\displaystyle{\bf{H}}$
$\displaystyle:=L^{2}(\Omega;{\mathbb{R}}^{d}),\quad$
$\displaystyle\mathbf{H}_{\Gamma}$
$\displaystyle:=L^{2}(\Gamma;{\mathbb{R}}^{d}),\quad$ $\displaystyle{\bf{V}}$
$\displaystyle:=H^{1}(\Omega;{\mathbb{R}}^{d}),$ $\displaystyle{\mathbb{H}}$
$\displaystyle:=L^{2}(\Omega;{\mathbb{R}}^{d\times d}).$

We further introduce the spaces of solenoidal (divergence-free) velocity
fields:

$\displaystyle{\bf H}_{\sigma,{\bf n}}$
$\displaystyle:=\big{\\{}{\boldsymbol{w}}\in{\bf{H}}:\mathop{\rm
div}\nolimits({\boldsymbol{w}})=0\;\text{in
$\Omega$},\;{\boldsymbol{w}}|_{\Gamma}\cdot{\bf n}=0\;\text{on
$\Gamma$}\big{\\}},$ $\displaystyle{\bf V}_{\sigma,{\bf n}}$
$\displaystyle:={\bf{V}}\cap{\bf H}_{\sigma,{\bf n}}.$

We point out that in the definition of ${\bf H}_{\sigma,{\bf n}}$, the
relation $\mathop{\rm div}\nolimits({\boldsymbol{w}})=0$ in $\Omega$ is to be
understood in the sense of distributions. This already implies
${\boldsymbol{w}}|_{\Gamma}\cdot{\bf n}\in H^{-1/2}(\Gamma)$, and therefore,
the relation ${\boldsymbol{w}}|_{\Gamma}\cdot{\bf n}=0$ on $\Gamma$ is well-
defined. As ${\bf H}_{\sigma,{\bf n}}$ and ${\bf V}_{\sigma,{\bf n}}$ are
closed linear subspaces of the Hilbert spaces ${\bf{H}}$ and ${\bf{V}}$,
respectively, they are also Hilbert spaces. We further introduce the bulk-
surface product spaces

$\displaystyle{\cal H}$ $\displaystyle:=H\times H_{\Gamma},\quad{\cal
V}:=V\times{V}_{\Gamma},$ $\displaystyle{\cal V}_{K}$
$\displaystyle:=\begin{cases}{\cal V}&\text{if $K>0$},\\\
\\{(w,w_{\Gamma})\in{\cal V}:w_{\Gamma}={w}|_{\Gamma}\,\,\text{on
$\Gamma$}\\}&\text{if $K=0$},\end{cases}$

and endow them with the corresponding inner products

$\displaystyle\big{(}(v,v_{\Gamma}),(w,w_{\Gamma})\big{)}_{\cal H}$
$\displaystyle:=(v,w)_{H}+{(v_{\Gamma},w_{\Gamma})}_{H_{\Gamma}}$ for all
$(v,v_{\Gamma}),(w,w_{\Gamma})\in{{\cal H}}$,
$\displaystyle\big{(}(v,v_{\Gamma}),(w,w_{\Gamma})\big{)}_{\cal V}$
$\displaystyle:=(v,w)_{V}+{(v_{\Gamma},w_{\Gamma})}_{{V}_{\Gamma}}$ for all
$(v,v_{\Gamma}),(w,w_{\Gamma})\in{\cal V}$,

so that ${\cal H}$, ${\cal V}$ and ${\cal V}_{K}$ are Hilbert spaces. For any
$v\in{V^{*}}$ and $v_{\Gamma}\in{V}_{\Gamma}^{*}$, we define the generalized
mean values by

$\displaystyle\mathopen{\langle}v\mathclose{\rangle}_{\Omega}:=\frac{1}{|\Omega|}\,\mathopen{\langle}v,1\mathclose{\rangle}_{V}\,,\quad\mathopen{\langle}v_{\Gamma}\mathclose{\rangle}_{\Gamma}:=\frac{1}{|\Gamma|}\,\mathopen{\langle}v_{\Gamma},1\mathclose{\rangle}_{{V}_{\Gamma}}\,,$
(2.1)

where $1$ represents the constant function assuming value $1$ in $\Omega$ and
on $\Gamma$, respectively. To introduce a weak formulation of (1.1), it will
be useful to define the function

$\displaystyle\sigma:[0,\infty)\to[0,\infty),\quad\sigma(r)=\begin{cases}\frac{1}{r}&\text{if
$r>0$},\\\ 0&\text{if $r=0$}\end{cases}$ (2.2)

to handle the cases $K>0$ and $K=0$ simultaneously.

### 2.2 General assumptions

  1. (A1)

The set $\Omega\subset{\mathbb{R}}^{d}$ with $d\in\\{2,3\\}$ is a bounded
Lipschitz domain.

  2. (A2)

The mobility functions $M_{\Omega}:{\mathbb{R}}\to{\mathbb{R}}$ and
$M_{\Gamma}:{\mathbb{R}}\to{\mathbb{R}}$ are continuous, bounded and uniformly
positive. This means that there exist positive constants $M_{1}$, $M_{2}$,
$M_{\Gamma,1}$ and $M_{\Gamma,2}$ such that

$\displaystyle 0<M_{1}\leq M_{\Omega}(r)\leq M_{2},\quad 0<M_{\Gamma,1}\leq
M_{\Gamma}(r)\leq M_{\Gamma,2}\quad\text{for all $r\in{\mathbb{R}}.$}$

  3. (A3)

The viscosity function $\nu:{\mathbb{R}}\to{\mathbb{R}}$ and the friction
parameter $\gamma:{\mathbb{R}}\to{\mathbb{R}}$ are continuous, bounded and
uniformly positive. Namely, there exist positive constants $\nu_{1}$,
$\nu_{2}$, $\gamma_{1}$ and $\gamma_{2}$ such that

$\displaystyle 0<\nu_{1}\leq\nu(r)\leq\nu_{2},\quad
0<\gamma_{1}\leq\gamma(r)\leq\gamma_{2}\quad\text{for all
$r\in{\mathbb{R}}.$}$

Furthermore, the permeability function $\lambda:{\mathbb{R}}\to{\mathbb{R}}$
is continuous, bounded and nonnegative, that is, there exists a nonnegative
constant $\lambda_{1}$ such that

$\displaystyle 0\leq\lambda(r)\leq\lambda_{1}\quad\text{for all
$r\in{\mathbb{R}}.$}$

### 2.3 Preliminaries

Throughout the paper, we will frequently use the following bulk-surface
Poincaré inequality:

######  Lemma 2.1.

There exists a constant $C_{P}^{\prime}$ depending only on $\Omega$ such that

$\|u\|_{H}\leq C_{P}^{\prime}\left(\|\nabla u\|_{{\bf
H}}+\|u\|_{{H}_{\Gamma}}\right)\quad\text{for all $u\in V$.}$ (2.3)

A proof of this inequality can be found, for instance, in [61, Chapter II,
Section 1.4].

We further recall the following result on interpolation between Sobolev
spaces:

######  Lemma 2.2.

Let $U\subset{\mathbb{R}}^{m}$ with $m\in\mathbb{N}$ be a bounded Lipschitz
domain, and suppose that $\theta\in(0,1)$ and $r,s_{0},s_{1}\in{\mathbb{R}}$
satisfy

$r=(1-\theta)s_{0}+\theta s_{1}.$

We further assume that $U$ is of class $C^{\ell}$ with an integer
$\ell\geq\max\\{s_{0},s_{1}\\}$. Then, there exist positive constants $C_{U}$
and $C_{\partial U}$ depending only on $U$, $r$, $s_{0}$, $s_{1}$ and $\theta$
such that the following interpolation inequalities hold:

$\displaystyle\|f\|_{H^{r}(U)}$ $\displaystyle\leq
C_{U}\|f\|_{H^{s_{0}}(U)}^{1-\theta}\|f\|_{H^{s_{1}}(U)}^{\theta},$ (2.4)
$\displaystyle\|f\|_{H^{r}(\partial U)}$ $\displaystyle\leq C_{\partial
U}\|f\|_{H^{s_{0}}(\partial U)}^{1-\theta}\|f\|_{H^{s_{1}}(\partial
U)}^{\theta}.$ (2.5)

Inequality (2.4) follows from an interpolation result shown in [62, Sec 4.3.1,
Theorem 1 and Remark 1], whereas (2.5) follows from an interpolation result
presented in [63, Sec 7.4.5, Remark 2].

### 2.4 The Cahn–Hilliard–Brinkman system with regular potentials

First, we present our mathematical results for system (1.1) in the case of
regular double-well potentials $F$ and $G$. As mentioned above, we simply set
$\epsilon=\epsilon_{\Gamma}=1$, as the exact values of these interface
parameters do not have any impact on the mathematical analysis as long as they
are positive.

#### 2.4.1 Assumptions for regular potentials

  1. (R1)

The potentials $F:{\mathbb{R}}\to[0,\infty)$ and $G:{\mathbb{R}}\to[0,\infty)$
are continuously differentiable, and there exist exponents
$p,q\in{\mathbb{R}}$ with

$\displaystyle p\in\begin{cases}[2,\infty)&\text{if $d=2$,}\\\ [2,6]&\text{if
$d=3$,}\end{cases}\quad\text{and}\quad q\in[2,\infty)$ (2.6)

as well as constants $c_{F^{\prime}},c_{G^{\prime}}\geq 0$ such that

$\displaystyle\left|F^{\prime}(r)\right|$ $\displaystyle\leq
c_{F^{\prime}}(1+\left|r\right|^{p-1}),$ (2.7)
$\displaystyle\left|G^{\prime}(r)\right|$ $\displaystyle\leq
c_{G^{\prime}}(1+\left|r\right|^{q-1})$ (2.8)

for all $r\in{\mathbb{R}}$. This implies that there exist constants
$c_{F},c_{G}\geq 0$ such that $F$ and $G$ fulfill the growth conditions

$\displaystyle F(r)$ $\displaystyle\leq c_{F}(1+\left|r\right|^{p}),$ (2.9)
$\displaystyle\quad G(r)$ $\displaystyle\leq c_{G}(1+\left|r\right|^{q})$
(2.10)

for all $r\in{\mathbb{R}}$.

  2. (R2)

In addition to (R1), $F$ and $G$ are twice continuously differentiable and
there exist constants $c_{F^{\prime\prime}},c_{G^{\prime\prime}}\geq 0$ such
that

$\displaystyle\left|F^{\prime\prime}(r)\right|$ $\displaystyle\leq
c_{F^{\prime\prime}}(1+\left|r\right|^{p-2}),$ (2.11)
$\displaystyle\left|G^{\prime\prime}(r)\right|$ $\displaystyle\leq
c_{G^{\prime\prime}}(1+\left|r\right|^{q-2})$ (2.12)

for all $r\in{\mathbb{R}}$, where $p$ and $q$ are the exponents introduced in
(2.6).

#### 2.4.2 Definition of weak solutions for regular potentials

######  Definition 2.3.

Let $K\geq 0$ be arbitrary. Suppose that (A1)–(A3) and (R1) are fulfilled and
let $(\varphi_{0},\psi_{0})\in{\cal V}_{K}$ be any initial data. A quintuplet
$({\boldsymbol{v}},\varphi,\mu,\psi,\theta)$ is called a weak solution of the
Cahn–Hilliard–Brinkman system (1.1) if the following conditions are fulfilled:

  1. (i)

The functions ${\boldsymbol{v}}$, $\varphi$, $\mu$, $\psi$ and $\theta$ have
the regularity

$\displaystyle{\boldsymbol{v}}$ $\displaystyle\in L^{2}(0,T;{\bf
V}_{\sigma,{\bf n}}),\quad{\boldsymbol{v}}|_{\Gamma}\in L^{2}(0,T;{\bf
H}_{\Gamma}),$ $\displaystyle(\varphi,\psi)$ $\displaystyle\in
H^{1}(0,T;{{\cal V}_{K}^{*}})\cap C^{0}([0,T];{\cal H})\cap
L^{\infty}(0,T;{\cal V}_{K}),$ $\displaystyle(\mu,\theta)$ $\displaystyle\in
L^{2}(0,T;{\cal V}).$

  2. (ii)

The variational formulation

$\displaystyle\begin{aligned}
&2\int_{\Omega}\nu(\varphi)\mathbf{D}{\boldsymbol{v}}:\mathbf{D}{\boldsymbol{w}}+\int_{\Omega}\lambda(\varphi){\boldsymbol{v}}\cdot{\boldsymbol{w}}+\int_{\Gamma}\gamma(\psi){\boldsymbol{v}}\cdot{\boldsymbol{w}}\\\
&\quad=-\int_{\Omega}\varphi\nabla\mu\cdot{\boldsymbol{w}}-\int_{\Gamma}\psi\nabla_{\Gamma}\theta\cdot{\boldsymbol{w}},\end{aligned}$
(2.13a)
$\displaystyle\mathopen{\langle}\partial_{t}\varphi,\zeta\mathclose{\rangle}_{V}-\int_{\Omega}\varphi{\boldsymbol{v}}\cdot\nabla\zeta+\int_{\Omega}M_{\Omega}(\varphi)\nabla\mu\cdot\nabla\zeta=0,$
(2.13b)
$\displaystyle\mathopen{\langle}\partial_{t}\psi,{\zeta_{\Gamma}}\mathclose{\rangle}_{{V}_{\Gamma}}-\int_{\Gamma}\psi{\boldsymbol{v}}\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}+\int_{\Gamma}M_{\Gamma}(\psi)\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}=0,$
(2.13c) $\displaystyle\begin{aligned}
\int_{\Omega}\mu\eta+\int_{\Gamma}\theta{\eta_{\Gamma}}&=\int_{\Omega}\nabla\varphi\cdot\nabla\eta+\int_{\Omega}F^{\prime}(\varphi)\eta+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}{\eta_{\Gamma}}\\\
&\quad+\int_{\Gamma}G^{\prime}(\psi){\eta_{\Gamma}}+\sigma(K)\int_{\Gamma}(\psi-\varphi)({\eta_{\Gamma}}-\eta)\end{aligned}$
(2.13d)

holds a.e. in $[0,T]$ for all ${\boldsymbol{w}}\in{\bf V}_{\sigma,{\bf n}}$,
$\zeta\in V$, ${\zeta_{\Gamma}}\in V_{\Gamma}$ and
$(\eta,{\eta_{\Gamma}})\in{\cal V}_{K}$.

  3. (iii)

The initial conditions are satisfied in the follwing sense:

$\displaystyle\varphi(0)=\varphi_{0}\quad\hbox{a.e.\
in~{}$\Omega$},\quad\psi(0)=\psi_{0}\quad\hbox{a.e.\ on~{}$\Gamma$}.$

  4. (iv)

The weak energy dissipation law

$\displaystyle
E_{K}\big{(}\varphi(t),\psi(t)\big{)}+2\int_{0}^{t}\int_{\Omega}\nu(\varphi)|\mathbf{D}{\boldsymbol{v}}|^{2}+\int_{0}^{t}\int_{\Omega}\lambda(\varphi)|{\boldsymbol{v}}|^{2}+\int_{0}^{t}\int_{\Gamma}\gamma(\psi)|{\boldsymbol{v}}|^{2}$
$\displaystyle\qquad+\int_{0}^{t}\int_{\Omega}M_{\Omega}(\varphi)|\nabla\mu|^{2}+\int_{0}^{t}\int_{\Gamma}M_{\Gamma}(\psi)|\nabla_{\Gamma}\theta|^{2}$
$\displaystyle\quad\leq E_{K}(\varphi_{0},\psi_{0})$ (2.14)

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

#### 2.4.3 Existence of a weak solution in the case $K>0$

We first show the existence of a weak solution to the Cahn–Hilliard–Brinkman
system (1.1) in the case $K>0$.

######  Theorem 2.4.

Let $K>0$ be arbitrary. Suppose that (A1)–(A3) and (R1) are fulfilled and let
$(\varphi_{0},\psi_{0})\in{\cal V}_{K}$ be any initial data. Then, the
Cahn–Hilliard–Brinkman system (1.1) possesses at least one weak solution
$({\boldsymbol{v}},\varphi,\mu,\psi,\theta)$ in the sense of Definition 2.3,
which further satisfies $(\mu,\theta)\in L^{4}(0,T;{\cal H})$.

Let us now assume that the domain $\Omega$ is of class $C^{\ell}$ with
$\ell\in\\{2,3\\}$. If $d=3$, we further assume $p<6$, and if $\ell=3$, we
further assume that (R2) holds. Then, we have the additional regularities

$\displaystyle(\varphi,\psi)$ $\displaystyle\in
L^{4}\big{(}0,T;H^{2}(\Omega)\times H^{2}(\Gamma)\big{)}\qquad\text{in case
$\ell\in\\{2,3\\}$},$ (2.15a) $\displaystyle(\varphi,\psi)$ $\displaystyle\in
L^{2}\big{(}0,T;H^{3}(\Omega)\times H^{3}(\Gamma)\big{)}\qquad\text{in case
$\ell=3$},$ (2.15b)

and the equations (1.1d), (1.1f) and (1.1g) are fulfilled in the strong sense,
that is, almost everywhere in $Q$ and on $\Sigma$, respectively. In the case
$\ell=3$, we further have

$\displaystyle(\varphi,\psi)\in C^{0}\big{(}[0,T];{\cal V}\big{)}.$ (2.16)

#### 2.4.4 The limit $K\to 0$ and existence of a weak solution in the case
$K=0$

We now investigate the limit $K\to 0$ in which the boundary condition (1.1g)
formally tends to the Dirichlet condition $\psi=\varphi|_{\Gamma}$ almost
everywhere on $\Sigma$. In the following theorem, we send $K\to 0$ in system
(1.1) to prove the existence of a weak solution to (1.1) in the case $K=0$,
and we further specify the convergence properties of this asymptotic limit.

######  Theorem 2.5.

Suppose that (A1)–(A3) and (R1) are fulfilled and let
$(\varphi_{0},\psi_{0})\in{\cal V}_{0}$ be any initial data. Let
$({K_{n}})_{n\in{\mathbb{N}}}$ be a sequence of positive real numbers such
that ${K_{n}}\to 0$ as $n\to\infty$. For any $n\in{\mathbb{N}}$, let
$({\boldsymbol{v}}^{K_{n}},\varphi^{K_{n}},\mu^{K_{n}},\psi^{K_{n}},\theta^{K_{n}})$
denote any weak solution corresponding to ${K_{n}}>0$ in the sense of
Definition 2.3. Then, there exists a quintuplet of functions
$({\boldsymbol{v}}^{0},\varphi^{0},\mu^{0},\psi^{0},\theta^{0})$ with
$\varphi^{0}|_{\Gamma}=\psi^{0}$ a.e. on $\Sigma$ such that for any
$s\in[0,1)$,

$\displaystyle{\boldsymbol{v}}^{K_{n}}$ $\displaystyle\to{\boldsymbol{v}}^{0}$
$\displaystyle\quad\text{weakly in $L^{2}(0,T;{\bf V}_{\sigma,{\bf n}})$},$
(2.17a) $\displaystyle{\boldsymbol{v}}^{K_{n}}|_{\Gamma}$
$\displaystyle\to{\boldsymbol{v}}^{0}|_{\Gamma}$
$\displaystyle\quad\text{weakly in $L^{2}(0,T;\mathbf{H}_{\Gamma})$},$ (2.17b)
$\displaystyle\varphi^{K_{n}}$ $\displaystyle\to\varphi^{0}$
$\displaystyle\quad\text{weakly-${}^{*}$ in $L^{\infty}(0,T;V)$, weakly in
$H^{1}(0,T;V^{*})$},$ $\displaystyle\qquad\text{strongly in
$C^{0}([0,T];H^{s}(\Omega))$, and {a.e.\ in~{}$Q$}},$ (2.17c)
$\displaystyle\psi^{K_{n}}$ $\displaystyle\to\psi^{0}$
$\displaystyle\quad\text{weakly-${}^{*}$ in $L^{\infty}(0,T;{V}_{\Gamma})$,
weakly in $H^{1}(0,T;{V}_{\Gamma}^{*})$},$ $\displaystyle\qquad\text{strongly
in $C^{0}([0,T];H^{s}(\Gamma))$, and {a.e.\ on~{}$\Sigma$}},$ (2.17d)
$\displaystyle\mu^{K_{n}}$ $\displaystyle\to\mu^{0}$
$\displaystyle\quad\text{weakly in $L^{2}(0,T;V)$},$ (2.17e)
$\displaystyle\theta^{K_{n}}$ $\displaystyle\to\theta^{0}$
$\displaystyle\quad\text{weakly in $L^{2}(0,T;{V}_{\Gamma})$},$ (2.17f)
$\displaystyle\varphi^{K_{n}}|_{\Gamma}-\psi^{K_{n}}$ $\displaystyle\to 0$
$\displaystyle\quad\text{strongly in $L^{\infty}(0,T;{H}_{\Gamma})$, and
{a.e.\ on~{}$\Sigma$}},$ (2.17g)

as $n\to\infty$ along a non-relabeled subsequence. Moreover, the limit
$({\boldsymbol{v}}^{0},\varphi^{0},\mu^{0},\psi^{0},\theta^{0})$ is a weak
solution of the Cahn–Hilliard–Brinkman model (1.1) in the sense of Definition
2.3 with $K=0$.

Let us now assume that the domain $\Omega$ is of class $C^{\ell}$ with
$\ell\in\\{2,3\\}$. If $d=3$, we further assume $p\leq 4$, and if $\ell=3$, we
further assume that (R2) holds. Then, we have the additional regularities

$\displaystyle(\varphi^{0},\psi^{0})$ $\displaystyle\in
L^{2}\big{(}0,T;H^{2}(\Omega)\times H^{2}(\Gamma)\big{)}\qquad\text{in case
$\ell\in\\{2,3\\}$},$ (2.18a) $\displaystyle(\varphi^{0},\psi^{0})$
$\displaystyle\in L^{2}\big{(}0,T;H^{3}(\Omega)\times
H^{3}(\Gamma)\big{)}\qquad\text{in case $\ell=3$},$ (2.18b)

and the equations (1.1d) and (1.1f) are fulfilled in the strong sense.
Moreover, in the case $\ell=3$, we further have

$\displaystyle(\varphi^{0},\psi^{0})$ $\displaystyle\in
C^{0}\big{(}[0,T];{\cal V}_{0}\big{)}\cap L^{4}\big{(}0,T;H^{2}(\Omega)\times
H^{2}(\Gamma)\big{)},$ (2.19) $\displaystyle(\mu^{0},\theta^{0})$
$\displaystyle\in L^{4}\big{(}0,T;{\cal H}\big{)}.$ (2.20)

######  Remark 2.6.

The additional assumption $p\leq 4$ in the second paragraph of the above
theorem was merely imposed to avoid unnecessary technicalities in the proof
for additional regularity of the phase-fields. However, it should be possible
to obtain the same regularities also for $p\in(4,6)$. This could be
established by approximating the potential $F$ by a sequence
$(F_{k})_{k\in{\mathbb{N}}}$ of potentials satisfying (R1) with $p\leq 4$.
Then, proceeding similarly as in the proof of Theorem 2.4 (see Subsection
3.1.5) additional uniform bounds on the corresponding weak solutions
$({\boldsymbol{v}}_{k},\varphi_{k},\mu_{k},\psi_{k},\theta_{k})$ would have to
be derived to obtain the weak convergences
$(\varphi_{k},\psi_{k})\to(\varphi,\psi)$ in
$L^{2}\big{(}0,T;H^{2}(\Omega)\times H^{2}(\Gamma)\big{)}$ in case
$\ell\in\\{2,3\\}$, and $(\varphi_{k},\psi_{k})\to(\varphi,\psi)$ in
$L^{2}\big{(}0,T;H^{3}(\Omega)\times H^{3}(\Gamma)\big{)}$ in case $\ell=3$,
which imply the desired regularities.

#### 2.4.5 Stability and uniqueness of the weak solution in the general case
$K\geq 0$

In the case of regular potentials, constant mobilities, and a constant
viscosity, we are able to prove the uniqueness of the weak solutions
established in Theorem 2.4 provided that the following assumption on the
potentials $F$ and $G$ holds.

  1. (Renumi)

The potentials $F:{\mathbb{R}}\to[0,\infty)$ and $G:{\mathbb{R}}\to[0,\infty)$
are three times continuously differentiable, and there exist exponents
$p,q\in{\mathbb{R}}$ with

$\displaystyle p\in\begin{cases}[3,\infty)&\text{if $d=2$,}\\\ [3,6)&\text{if
$d=3$ and $K>0$,}\\\ [3,4]&\text{if $d=3$ and
$K=0$,}\end{cases}\qquad\text{and}\qquad q\in[3,\infty)$ (2.21)

as well as constants $c_{F^{(3)}},c_{G^{(3)}}\geq 0$ such that the third-order
derivatives satisfy

$\displaystyle\left|F^{(3)}(r)\right|$ $\displaystyle\leq
c_{F^{(3)}}(1+\left|r\right|^{p-3}),$ (2.22)
$\displaystyle\left|G^{(3)}(r)\right|$ $\displaystyle\leq
c_{G^{(3)}}(1+\left|r\right|^{q-3})$ (2.23)

for all $r\in{\mathbb{R}}$.

We point out that (Renumi) implies (R1) and (R2) with $p$ and $q$ being chosen
as in (2.21). The restriction $p\in[3,4]$ in the case $d=3$ and $K=0$ is
imposed since in Theorem 2.5, higher regularity properties for the phase-
fields were established only for $p\leq 4$. However, as pointed out in Remark
2.6, it should be possible to relieve this restriction such that also in the
case $d=3$ and $K=0$, merely $p\in[3,6)$ would have to be imposed in (Renumi).

######  Theorem 2.7.

Suppose that (A1) and (Renumi) are fulfilled and let $K\geq 0$ be arbitrary.
In addition to (A2) and (A3), we further assume that $\gamma$ and $\lambda$
are Lipschitz continuous functions and that the functions $\nu$, $M_{\Omega}$
and $M_{\Gamma}$ reduce to positive constants denoted by the same symbols. For
any $i\in\\{1,2\\}$, let $(\varphi_{0,i},\psi_{0,i})\in{\cal V}_{K}$ be any
pair of initial data, and let
$({\boldsymbol{v}}_{i},\varphi_{i},\mu_{i},\psi_{i},\theta_{i})$ be a
corresponding weak solution in the sense of Definition 2.3. Then, the
stability estimate

$\displaystyle\left\|{\boldsymbol{v}}_{1}-{\boldsymbol{v}}_{2}\right\|_{L^{2}(0,T;{\bf{V}})}+\left\|\varphi_{1}-\varphi_{2}\right\|_{L^{\infty}(0,T;V)}+\left\|\mu_{1}-\mu_{2}\right\|_{L^{2}(0,T;V)}+\left\|\psi_{1}-\psi_{2}\right\|_{L^{\infty}(0,T;{V}_{\Gamma})}$
$\displaystyle\quad+\left\|\theta_{1}-\theta_{2}\right\|_{L^{2}(0,T;{V}_{\Gamma})}\leq
C_{S}\big{(}\left\|\varphi_{0,1}-\varphi_{0,2}\right\|_{V}+\left\|\psi_{0,1}-\psi_{0,2}\right\|_{{V}_{\Gamma}}\big{)}$
(2.24)

holds for a constant $C_{S}\geq 0$ depending only on $K$, $\Omega$, $T$, the
initial data and the constants introduced in (A1)–(A3) and (Renumi). In
particular, choosing $(\varphi_{0,1},\psi_{0,1})=(\varphi_{0,2},\psi_{0,2})$,
this entails the uniqueness of the corresponding weak solution.

### 2.5 The Cahn–Hilliard–Brinkman system with singular potentials

We now consider the system (1.1) for a general class of singular potentials.
For those, we manage to establish just existence of weak solutions due to the
lower regularity at disposal. Recall that $\epsilon=\epsilon_{\Gamma}=1$ as
mentioned above.

#### 2.5.1 Assumptions for singular potentials

For the potentials $F$ and $G$, we now make the following assumptions.

  1. (S1)

The potentials $F$ and $G$ can be decomposed as
$F=\widehat{\beta}+\widehat{\pi}$ and
$G={\widehat{\beta}}_{\Gamma}+{\widehat{\pi}}_{\Gamma}$.

Here, $\widehat{\beta},{\widehat{\beta}}_{\Gamma}:{\mathbb{R}}\to[0,\infty]$
are lower semicontinuous and convex functions with $\widehat{\beta}(0)=0$ and
${\widehat{\beta}}_{\Gamma}(0)=0$. For brevity, we define

$\displaystyle\beta:=\partial\widehat{\beta}\quad\text{and}\quad\beta_{\Gamma}:=\partial{\widehat{\beta}_{\Gamma}}.$

Moreover, we suppose that $\widehat{\pi},{\widehat{\pi}}_{\Gamma}\in
C^{1}({\mathbb{R}})$ with Lipschitz continuous derivatives
$\pi:=\widehat{\pi}^{\prime}$ and
$\pi_{\Gamma}:={\widehat{\pi}}_{\Gamma}^{\prime}$.

We point out that $\beta$ and ${\beta_{\Gamma}}$ are maximal monotone graphs
in ${\mathbb{R}}\times{\mathbb{R}}$ whose effective domains are denoted by
$D(\beta)$ and $D(\beta_{\Gamma})$, respectively. In particular, as $0$ is a
minimum point of both $\widehat{\beta}$ and ${\widehat{\beta}}_{\Gamma}$, it
turns out that $0\in\beta(0)$ and $0\in{\beta_{\Gamma}}(0)$. Finally, we
denote by $\beta^{\circ}$ the minimal section of the graph $\beta$, which is
defined as

$\displaystyle\beta^{\circ}(r)\coloneqq\Big{\\{}r^{\ast}\in
D(\beta):\left|r^{\ast}\right|=\min_{s\in\beta(r)}\left|s\right|\Big{\\}}\qquad\text{for
all $r\in D(\beta)$}$

(see, e.g., [12]). The same definition applies to $\beta^{\circ}_{\Gamma}$ for
${\beta_{\Gamma}}$.

  2. (S2)

We also assume the growth condition

$\displaystyle\lim_{r\to+\infty}\frac{\widehat{\beta}(r)}{|r|^{2}}=+\infty\,.$
(2.25)

Moreover, we demand $D(\beta_{\Gamma})\subseteq D(\beta)$, and postulate that
the boundary graph dominates the bulk graph in the following sense:

$\displaystyle\exists\,\kappa_{1},\kappa_{2}>0:\quad|\beta^{\circ}(r)|\leq\kappa_{1}|\beta_{\Gamma}^{\circ}(r)|+\kappa_{2}\quad\text{for
every $r\in D(\beta_{\Gamma})$.}$ (2.26)

Here, $\beta^{\circ}$ and $\beta_{\Gamma}^{\circ}$ are the minimal sections
introduced in (S1).

Note that all the examples of potentials given in (1.2)–(1.4) fulfill the
assumptions (S1) and (S2), provided that the boundary potential dominates the
one in the bulk as demanded in (2.26). In particular, the only scenario where
a singular and a regular potential may coexist is the case in which the
boundary potential is the singular one. This assumption has first been made in
[13] and was used afterwards in several contributions in the literature, see,
e.g., [15, 21, 24, 23, 25, 22, 17, 18, 19, 14]. However, in some other works
such as [36, 37] different compatibility conditions were assumed.

#### 2.5.2 Definition of weak solutions for singular potentials

######  Definition 2.8.

Let $K\geq 0$ be arbitrary. Suppose that (A1)–(A3), (S1) and (S2) are
fulfilled and let $(\varphi_{0},\psi_{0})\in{\cal V}_{K}$ be any initial data
satisfying

$\displaystyle\widehat{\beta}(\varphi_{0})$ $\displaystyle\in
L^{1}(\Omega),\quad$ $\displaystyle
m_{0}:=\mathopen{\langle}\varphi_{0}\mathclose{\rangle}_{\Omega}\in{\rm
int}(D(\beta)),$ (2.27a) $\displaystyle\widehat{\beta}_{\Gamma}(\psi_{0})$
$\displaystyle\in L^{1}(\Gamma),\quad$ $\displaystyle m_{\Gamma
0}:=\mathopen{\langle}\psi_{0}\mathclose{\rangle}_{\Gamma}\in{\rm
int}(D(\beta_{\Gamma})).$ (2.27b)

Then, $({\boldsymbol{v}},\varphi,\xi,\mu,\psi,\xi_{\Gamma},\theta)$ is called
a weak solution of the Cahn–Hilliard–Brinkman system (1.1) if the following
conditions are fulfilled:

  1. (i)

The functions ${\boldsymbol{v}}$, $\varphi$, $\xi$, $\mu$, $\psi$,
$\xi_{\Gamma}$ and $\theta$ have the regularity

$\displaystyle{\boldsymbol{v}}$ $\displaystyle\in L^{2}(0,T;{\bf
V}_{\sigma,{\bf n}}),\quad{\boldsymbol{v}}|_{\Gamma}\in L^{2}(0,T;{\bf
H}_{\Gamma}),$ $\displaystyle(\varphi,\psi)$ $\displaystyle\in H^{1}(0,T;{\cal
V}_{K}^{*})\cap C^{0}([0,T];{{\cal H}})\cap L^{\infty}(0,T;{\cal
V}_{K}),\quad$ $\displaystyle(\xi,\xi_{\Gamma})$ $\displaystyle\in
L^{2}(0,T;{\cal H}),$ $\displaystyle(\mu,\theta)$ $\displaystyle\in
L^{2}(0,T;{\cal V}).$

  2. (ii)

The variational formulation

$\displaystyle\begin{aligned}
&2\int_{\Omega}\nu(\varphi)\mathbf{D}{\boldsymbol{v}}:\mathbf{D}{\boldsymbol{w}}+\int_{\Omega}\lambda(\varphi){\boldsymbol{v}}\cdot{\boldsymbol{w}}+\int_{\Gamma}\gamma(\psi){\boldsymbol{v}}\cdot{\boldsymbol{w}}\\\
&\quad=-\int_{\Omega}\varphi\nabla\mu\cdot{\boldsymbol{w}}-\int_{\Gamma}\psi\nabla_{\Gamma}\theta\cdot{\boldsymbol{w}},\end{aligned}$
(2.28a)
$\displaystyle\mathopen{\langle}\partial_{t}\varphi,\zeta\mathclose{\rangle}_{V}-\int_{\Omega}\varphi{\boldsymbol{v}}\cdot\nabla\zeta+\int_{\Omega}M_{\Omega}(\varphi)\nabla\mu\cdot\nabla\zeta=0,$
(2.28b)
$\displaystyle\mathopen{\langle}\partial_{t}\psi,{\zeta_{\Gamma}}\mathclose{\rangle}_{{V}_{\Gamma}}-\int_{\Gamma}\psi{\boldsymbol{v}}\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}+\int_{\Gamma}M_{\Gamma}(\psi)\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}=0,$
(2.28c) $\displaystyle\begin{aligned}
\int_{\Omega}\mu\eta+\int_{\Gamma}\theta{\eta_{\Gamma}}&=\int_{\Omega}\nabla\varphi\cdot\nabla\eta+\int_{\Omega}\xi\eta+\int_{\Omega}\pi(\varphi)\eta+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}{\eta_{\Gamma}}\\\
&\quad+\int_{\Gamma}\xi_{\Gamma}{\eta_{\Gamma}}+\int_{\Gamma}\pi_{\Gamma}(\psi){\eta_{\Gamma}}+\sigma(K)\int_{\Gamma}(\psi-\varphi)({\eta_{\Gamma}}-\eta)\end{aligned}$
(2.28d)

holds a.e. in $[0,T]$ for all ${\boldsymbol{w}}\in{\bf V}_{\sigma,{\bf n}}$,
$\zeta\in V$, ${\zeta_{\Gamma}}\in V_{\Gamma}$ and
$(\eta,{\eta_{\Gamma}})\in{\cal V}_{K}$, where

$\displaystyle\xi\in\beta(\varphi)\quad\hbox{a.e.\
in~{}$Q$},\quad\xi_{\Gamma}\in\beta_{\Gamma}(\psi)\quad\hbox{a.e.\
on~{}$\Sigma$}.$

  3. (iii)

The initial conditions are satisfied in the follwing sense:

$\displaystyle\varphi(0)=\varphi_{0}\quad\hbox{a.e.\
in~{}$\Omega$},\quad\psi(0)=\psi_{0}\quad\hbox{a.e.\ on~{}$\Gamma$}.$

  4. (iv)

The weak energy dissipation law

$\displaystyle
E_{K}\big{(}\varphi(t),\psi(t)\big{)}+2\int_{0}^{t}\int_{\Omega}\nu(\varphi)|\mathbf{D}{\boldsymbol{v}}|^{2}+\int_{0}^{t}\int_{\Omega}\lambda(\varphi)|{\boldsymbol{v}}|^{2}+\int_{0}^{t}\int_{\Gamma}\gamma(\psi)|{\boldsymbol{v}}|^{2}$
$\displaystyle\qquad+\int_{0}^{t}\int_{\Omega}M_{\Omega}(\varphi)|\nabla\mu|^{2}+\int_{0}^{t}\int_{\Gamma}M_{\Gamma}(\psi)|\nabla_{\Gamma}\theta|^{2}$
$\displaystyle\quad\leq E_{K}(\varphi_{0},\psi_{0})$

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

#### 2.5.3 Existence of a weak solution

######  Theorem 2.9.

Let $K\geq 0$ be arbitrary. Suppose that (A1)–(A3) and (S1)–(S2) are
fulfilled. Let $(\varphi_{0},\psi_{0})\in{\cal V}_{K}$ denote any initial data
satisfying (2.27). In the case $K=0$, let the domain $\Omega$ be of class
$C^{2}$. Then, the Cahn–Hilliard–Brinkman system (1.1) admits at least one
weak solution $({\boldsymbol{v}},\varphi,\xi,\mu,\psi,\xi_{\Gamma},\theta)$ in
the sense of Definition 2.8. In all cases, if the domain $\Omega$ is at least
of class $C^{2}$, it holds that

$\displaystyle\varphi\in L^{2}(0,T;H^{2}(\Omega)),\quad\psi\in
L^{2}(0,T;H^{2}(\Gamma))$ (2.29)

and the equations

$\displaystyle\mu$ $\displaystyle=-\Delta\varphi+\xi+\pi(\varphi)\quad$ in
$Q$, (2.30) $\displaystyle\theta$
$\displaystyle=-\Delta_{\Gamma}\psi+\xi_{\Gamma}+\pi_{\Gamma}(\psi)+\partial_{\bf
n}\varphi\quad$ $\displaystyle\text{on $\Sigma$},$ (2.31) $\displaystyle
K\partial_{\bf n}\varphi$ $\displaystyle=\varphi-\psi\quad$ on $\Sigma$ (2.32)

are fulfilled in the strong sense.

## 3 Analysis of the Cahn–Hilliard–Brinkman system with regular potentials

### 3.1 Existence of weak solutions in the case $K>0$

###### Proof of Theorem 2.4.

We intend to construct a weak solution to system (1.1) by discretizing the
weak formulation (2.13) by means of a Faedo–Galerkin scheme. In this proof,
the letter $C$ will denote generic positive constants that may depend on $K$,
$\Omega$, $T$, the initial data and the constants introduced in (A1)–(A3), and
may change their value from line to line. Recall that, as $K$ is assumed to be
positive here, we have $\sigma(K)=\frac{1}{K}$.

#### 3.1.1 Construction of local-in-time approximate solutions

It is well known that the Poisson–Neumann eigenvalue problem

$\displaystyle-\Delta u=\lambda_{\Omega}u\;\;\text{in
$\Omega$},\quad\partial_{\bf n}u=0\;\;\text{on $\Gamma$}$ (3.1)

possesses countably many eigenvalues and a corresponding sequence
$\\{u_{i}\\}_{i\in\mathbb{N}}\subset V$ of $H$-normalized eigenfunctions which
form an orthonormal basis of $H$ and an orthogonal Schauder basis of $V$.
Similarly, invoking the spectral theorem for compact self-adjoint operators,
it follows that the eigenvalue problem

$\displaystyle-\Delta_{\Gamma}v=\lambda_{\Gamma}v\;\;\text{on $\Gamma$}$ (3.2)

for the Laplace–Beltrami operator possesses countably many eigenvalues and a
corresponding sequence $\\{v_{i}\\}_{i\in\mathbb{N}}\subset V_{\Gamma}$ of
$H_{\Gamma}$-normalized eigenfunctions which form an orthonormal basis of
$H_{\Gamma}$ and an orthogonal Schauder basis of $V_{\Gamma}$. For any
$k\in\mathbb{N}$, we now define the finite-dimensional subspaces

$\displaystyle V_{k}$ $\displaystyle:={\rm span}\\{u_{i}:1\leq i\leq
k\\}\subset V,$ $\displaystyle V_{\Gamma,k}$ $\displaystyle:={\rm
span}\\{v_{j}:1\leq j\leq k\\}\subset V_{\Gamma},$ $\displaystyle{\cal V}_{k}$
$\displaystyle:={\rm span}\\{(u_{i},v_{j}):1\leq i,j\leq k\\}\subset{\cal V}.$

We point out that, due to the above considerations, the inclusions

$\displaystyle\bigcup_{k\in\mathbb{N}}V_{k}\subseteq
V,\qquad\bigcup_{k\in\mathbb{N}}V_{\Gamma,k}\subseteq
V_{\Gamma},\qquad\bigcup_{k\in\mathbb{N}}{\cal V}_{k}\subseteq{\cal V}$

are dense. To construct a sequence of approximate solutions, we now make the
ansatz

$\displaystyle\begin{aligned}
\varphi_{k}(x,t)&=\sum_{i=1}^{k}a_{i}^{k}(t)u_{i}(x),\quad&&\psi_{k}(x,t)=\sum_{i=1}^{k}b_{i}^{k}(t)v_{i}(x),\\\
\mu_{k}(x,t)&=\sum_{i=1}^{k}c_{i}^{k}(t)u_{i}(x),\quad&&\theta_{k}(x,t)=\sum_{i=1}^{k}d_{i}^{k}(t)v_{i}(x)\end{aligned}$
(3.3)

for every $k\in\mathbb{N}$, where the coefficients
$\mathbf{a}^{k}:=(a_{1}^{k},...,a_{k}^{k})^{\top}$,
$\mathbf{b}^{k}:=(b_{1}^{k},...,b_{k}^{k})^{\top}$,
$\mathbf{c}^{k}:=(c_{1}^{k},...,c_{k}^{k})^{\top}$,
$\mathbf{d}^{k}:=(d_{1}^{k},...,d_{k}^{k})^{\top}$ are still to be determined.

Let now $k\in{\mathbb{N}}$ and $t\in[0,T]$ be arbitrary. We consider the
bilinear form

$\displaystyle\mathcal{B}_{k,t}:{\bf V}_{\sigma,{\bf n}}\times{\bf
V}_{\sigma,{\bf n}}\to{\mathbb{R}},$
$\displaystyle({\boldsymbol{v}},{\boldsymbol{w}})\mapsto
2\int_{\Omega}\nu\big{(}\varphi_{k}(t)\big{)}\,\mathbf{D}{\boldsymbol{v}}:\mathbf{D}{\boldsymbol{w}}+\int_{\Omega}\lambda\big{(}\varphi_{k}(t)\big{)}\,{\boldsymbol{v}}\cdot{\boldsymbol{w}}+\int_{\Gamma}\gamma\big{(}\psi_{k}(t)\big{)}\,{\boldsymbol{v}}\cdot{\boldsymbol{w}},$

which is related to the weak formulation of the Brinkman equation with Navier-
slip boundary condition. It is obvious that $\mathcal{B}_{k,t}$ is symmetric,
and, in view of (A3), it is easy to see that $\mathcal{B}_{k,t}$ is
continuous. We further recall that every ${\boldsymbol{v}}\in{\bf
V}_{\sigma,{\bf n}}$ satisfies $\mathop{\rm div}\nolimits({\boldsymbol{v}})=0$
a.e. in $\Omega$, and ${\boldsymbol{v}}\cdot{\bf n}=0$ a.e. on $\Gamma$. By
means of (A3), integration by parts and the Poincaré inequality presented in
Lemma 2.1, we deduce

$\displaystyle\mathcal{B}_{k,t}({\boldsymbol{v}},{\boldsymbol{v}})$
$\displaystyle\geq
2\nu_{1}\int_{\Omega}\,\mathbf{D}{\boldsymbol{v}}:\mathbf{D}{\boldsymbol{v}}+\gamma_{1}\int_{\Gamma}|{\boldsymbol{v}}|^{2}$
$\displaystyle=\nu_{1}\int_{\Omega}|\nabla{\boldsymbol{v}}|^{2}+\gamma_{1}\int_{\Gamma}|{\boldsymbol{v}}|^{2}\geq
C\|{\boldsymbol{v}}\|_{{\bf{V}}}^{2}$

for all ${\boldsymbol{v}}\in{\bf V}_{\sigma,{\bf n}}$. This means that the
bilinear form $\mathcal{B}_{k,t}$ is coercive in ${\bf V}_{\sigma,{\bf n}}$.
Hence, the Lax–Milgram lemma implies that there exists a unique function
${\boldsymbol{v}}_{k}(t)\in{\bf V}_{\sigma,{\bf n}}$ solving

$\displaystyle\mathcal{B}_{k,t}\big{(}{\boldsymbol{v}}_{k}(t),{\boldsymbol{w}}\big{)}=-\int_{\Omega}\varphi_{k}(t)\nabla\mu_{k}(t)\cdot{\boldsymbol{w}}-\int_{\Gamma}\psi_{k}(t)\nabla_{\Gamma}\theta_{k}(t)\cdot{\boldsymbol{w}}$

for all ${\boldsymbol{w}}\in{\bf V}_{\sigma,{\bf n}}$. As $t\in[0,T]$ was
arbitrary, this defines a function ${\boldsymbol{v}}_{k}:[0,T]\to{\bf
V}_{\sigma,{\bf n}}$. We point out that by construction,
${\boldsymbol{v}}_{k}$ depends continuously on the coefficients
$\mathbf{a}^{k}$, $\mathbf{b}^{k}$, $\mathbf{c}^{k}$ and $\mathbf{d}^{k}$.

We now want to adjust the coefficient vectors $\mathbf{a}^{k}$,
$\mathbf{b}^{k}$, $\mathbf{c}^{k}$ and $\mathbf{d}^{k}$ such that the
discretized weak formulation

$\displaystyle
2\int_{\Omega}\nu(\varphi_{k})\mathbf{D}{\boldsymbol{v}}_{k}:\mathbf{D}{\boldsymbol{w}}+\int_{\Omega}\lambda(\varphi_{k}){\boldsymbol{v}}_{k}\cdot{\boldsymbol{w}}+\int_{\Gamma}\gamma(\psi_{k}){\boldsymbol{v}}_{k}\cdot{\boldsymbol{w}}$
$\displaystyle\quad=-\int_{\Omega}\varphi_{k}\nabla\mu_{k}\cdot{\boldsymbol{w}}-\int_{\Gamma}\psi_{k}\nabla_{\Gamma}\theta_{k}\cdot{\boldsymbol{w}},$
(3.4a)
$\displaystyle\mathopen{\langle}\partial_{t}\varphi_{k},\zeta\mathclose{\rangle}_{V}-\int_{\Omega}\varphi_{k}{\boldsymbol{v}}_{k}\cdot\nabla\zeta+\int_{\Omega}M_{\Omega}(\varphi_{k})\nabla\mu_{k}\cdot\nabla\zeta=0,$
(3.4b)
$\displaystyle\mathopen{\langle}\partial_{t}\psi_{k},{\zeta_{\Gamma}}\mathclose{\rangle}_{{V}_{\Gamma}}-\int_{\Gamma}\psi_{k}{\boldsymbol{v}}_{k}\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}+\int_{\Gamma}M_{\Gamma}(\psi_{k})\nabla_{\Gamma}\theta_{k}\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}=0,$
(3.4c)
$\displaystyle\int_{\Omega}\mu_{k}\eta+\int_{\Gamma}\theta_{k}{\eta_{\Gamma}}=\int_{\Omega}\nabla\varphi_{k}\cdot\nabla\eta+\int_{\Omega}F^{\prime}(\varphi_{k})\eta+\int_{\Gamma}\nabla_{\Gamma}\psi_{k}\cdot\nabla_{\Gamma}{\eta_{\Gamma}}$
$\displaystyle\hskip
86.11084pt+\int_{\Gamma}G^{\prime}(\psi_{k}){\eta_{\Gamma}}+\frac{1}{K}\int_{\Gamma}(\psi_{k}-\varphi_{k})({\eta_{\Gamma}}-\eta)$
(3.4d)

for all test functions ${\boldsymbol{w}}\in{\bf V}_{\sigma,{\bf n}}$,
$\zeta\in V_{k}$, ${\zeta_{\Gamma}}\in V_{\Gamma,k}$ and
$(\eta,{\eta_{\Gamma}})\in{\cal V}_{k}$, and the initial conditions

$\displaystyle\varphi_{k}(0)=\varphi_{0,k}:=\mathbb{P}_{V_{k}}(\varphi_{0})\quad\text{and}\quad\psi_{k}(0)=\psi_{0,k}:=\mathbb{P}_{V_{\Gamma,k}}(\psi_{0})$
(3.5)

are fulfilled. With the symbol $\mathbb{P}_{V_{k}}$ we denote the
$H$-orthogonal projection of $V$ onto $V_{k}$ whereas
$\mathbb{P}_{V_{\Gamma,k}}$ denotes the $H_{\Gamma}$-orthogonal projection of
$V_{\Gamma}$ onto $V_{\Gamma,k}$.

Choosing $\zeta=u_{j}$ in (3.4b) and $\zeta_{\Gamma}=v_{j}$ in (3.4c) for
$j=1,...,k$, we infer that $(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$ is
determined by a system of $2k$ nonlinear ordinary differential equations
subject to the initial conditions

$\displaystyle[\mathbf{a}^{k}]_{i}(0)=a_{i}^{k}(0)=\left(\varphi_{0},u_{i}\right)_{H}\quad\text{and}\quad[\mathbf{b}^{k}]_{i}(0)=b_{i}^{k}(0)=\left(\psi_{0},v_{i}\right)_{H_{\Gamma}}$

for all $i\in\\{1,...,k\\}$. In particular, since the functions $M_{\Omega}$
and $M_{\Gamma}$ are continuous and ${\boldsymbol{v}}_{k}$ depends
continuously on the coefficients $\mathbf{a}^{k}$, $\mathbf{b}^{k}$,
$\mathbf{c}^{k}$ and $\mathbf{d}^{k}$, the same holds for the right-hand side
of this ODE system. Moreover, choosing $(\eta,\eta_{\Gamma})=(u_{j},0)$ and
$(\eta,\eta_{\Gamma})=(0,v_{j})$ for $j=1,...,k$ in (3.4), respectively, we
find that the coefficients $\mathbf{c}^{k}$ and $\mathbf{d}^{k}$ are
explicitly given by $2k$ algebraic equations whose right-hand side depends
continuously on $\mathbf{a}^{k}$ and $\mathbf{b}^{k}$. This allows us to
replace $\mathbf{c}^{k}$ and $\mathbf{d}^{k}$ in the right-hand side of the
aforementioned ODE system to obtain a closed ODE system for the vector-valued
function $(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$ whose right-hand side
depends continuously on $(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$.
Consequently, the Cauchy–Peano theorem implies the existence of at least one
local-in-time solution

$(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}:[0,T_{k}^{*})\cap[0,T]\to{\mathbb{R}}^{2k}$

to the corresponding initial value problem. Here, we take $T_{k}^{*}>0$ as
large as possible meaning that $[0,T_{k}^{*})\cap[0,T]$ is the right-maximal
time interval of this solution. We can now reconstruct

$(\mathbf{c}^{k},\mathbf{d}^{k})^{\top}:[0,T_{k}^{*})\cap[0,T]\to{\mathbb{R}}^{2k}$

by the aforementioned system of $2k$ algebraic equations. Without loss of
generality, we now assume $T_{k}^{*}\leq T$ to simplify the notation.
Recalling the ansatz (3.3) as well as the construction of
${\boldsymbol{v}}_{k}$, we obtain an approximate solution
$({\boldsymbol{v}}_{k},\varphi_{k},\mu_{k},\psi_{k},\theta_{k})$ with

$\displaystyle{\boldsymbol{v}}_{k}\in C^{1}([0,T_{k}^{*});{\bf V}_{\sigma,{\bf
n}})\quad\text{and}\quad(\varphi_{k},\psi_{k}),(\mu_{k},\theta_{k})\in
C^{1}([0,T_{k}^{*});{\cal V}),$

which fulfills the discretized weak formulation (3.4) on the time interval
$[0,T_{k}^{*})$.

#### 3.1.2 Uniform estimates

Let now $T_{k}\in(0,T_{k}^{*})$ be arbitrary. We derive suitable estimates for
the approximate solutions
$({\boldsymbol{v}}_{k},\varphi_{k},\mu_{k},\psi_{k},\theta_{k})$ that are
uniform in $k$ and $T_{k}$. These estimates will allow us to extend the
approximate solutions onto the whole interval $[0,T]$ and to extract suitable
convergent subsequences.

First estimate. We first test (3.4a) by ${\boldsymbol{v}}_{k}$, (3.4b) by
$\mu_{k}$, (3.4c) by $\theta_{k}$, and (3.4) by
$-(\partial_{t}\varphi_{k},\partial_{t}\psi_{k})$. We then add these equations
and integrate the resulting equation with respect to time from $0$ to an
arbitrary $t\in[0,T_{k}]$. We obtain

$\displaystyle
E_{K}\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}+2\int_{0}^{t}\int_{\Omega}\nu(\varphi_{k})\left|\mathbf{D}{\boldsymbol{v}}_{k}\right|^{2}+\int_{0}^{t}\int_{\Omega}\lambda(\varphi_{k})\left|{\boldsymbol{v}}_{k}\right|^{2}+\int_{0}^{t}\int_{\Gamma}\gamma(\psi_{k})\left|{\boldsymbol{v}}_{k}\right|^{2}$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}M_{\Omega}(\varphi_{k})\left|\nabla\mu_{k}\right|^{2}+\int_{0}^{t}\int_{\Gamma}M_{\Gamma}(\psi_{k})\left|\nabla_{\Gamma}\theta_{k}\right|^{2}\leq
E_{K}(\varphi_{0,k},\psi_{0,k})$ (3.6)

for every $t\in[0,T_{k}]$. Due to (3.5) and the assumptions on the initial
data, we have

$\displaystyle\left\|\varphi_{0,k}\right\|_{V}\leq
C\left\|\varphi_{0}\right\|_{V}\leq
C\quad\text{and}\quad\left\|\psi_{0,k}\right\|_{{V}_{\Gamma}}\leq
C\left\|\psi_{0}\right\|_{{V}_{\Gamma}}\leq C.$ (3.7)

In view of the growth conditions from (R1), this directly implies

$\displaystyle\left\|F\big{(}\varphi_{0,k}\big{)}\right\|_{L^{1}(\Omega)}\leq
C,\quad\left\|G\big{(}\psi_{0,k}\big{)}\right\|_{L^{1}(\Gamma)}\leq
C\quad\text{and thus},\quad E_{K}(\varphi_{0,k},\psi_{0,k})\leq C.$ (3.8)

Hence, using the conditions in (A2) and (A3), a straightforward computation
yields

$\displaystyle\left\|\mathbf{D}{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};{\mathbb{H}})}+\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};\mathbf{H}_{\Gamma})}+\left\|\nabla\varphi_{k}\right\|_{L^{\infty}(0,T_{k};{\bf{H}})}+\left\|\nabla_{\Gamma}\psi_{k}\right\|_{L^{\infty}(0,T_{k};{\bf{H}}_{\Gamma})}$
$\displaystyle\quad+\left\|\nabla\mu_{k}\right\|_{L^{2}(0,T_{k};{\bf{H}})}+\left\|\nabla_{\Gamma}\theta_{k}\right\|_{L^{2}(0,T_{k};{\bf{H}}_{\Gamma})}\leq
C.$ (3.9)

Invoking the Poincaré inequality presented in Lemma 2.1, we directly infer

$\displaystyle\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};{\bf
V}_{\sigma,{\bf n}})}\leq C.$ (3.10)

Next, taking $\zeta=\frac{1}{|\Omega|}$ in (3.4b), and
$\zeta_{\Gamma}=\frac{1}{|\Gamma|}$ in (3.4c), we infer

$\displaystyle\mathopen{\langle}\varphi_{k}(t)\mathclose{\rangle}_{\Omega}=\mathopen{\langle}\varphi_{k,0}\mathclose{\rangle}_{\Omega},\quad\mathopen{\langle}\psi_{k}(t)\mathclose{\rangle}_{\Gamma}=\mathopen{\langle}\psi_{{k,0}}\mathclose{\rangle}_{\Gamma}\quad\text{for
all $t\in[0,T_{k}]$}.$

Hence, in view of (3.7) and (3.1.2), we use the Poincaré–Wirtinger inequality
to conclude

$\displaystyle\left\|\varphi_{k}\right\|_{L^{\infty}(0,T_{k};V)}+\left\|\psi_{k}\right\|_{L^{\infty}(0,T_{k};{V}_{\Gamma})}\leq
C.$ (3.11)

Second estimate. Let now $\zeta\in L^{2}(0,T_{k};V)$ and ${\zeta_{\Gamma}}\in
L^{2}(0,T_{k};{V}_{\Gamma})$ be arbitrary test functions. Testing (3.4b) with
${\overline{\zeta}}:=\mathbb{P}_{V_{k}}(\zeta)$ and exploiting (3.1.2)–(3.11)
along with Sobolev’s embeddings, we obtain

$\displaystyle\left|\int_{0}^{T_{k}}\mathopen{\langle}\partial_{t}\varphi_{k},\zeta\mathclose{\rangle}_{V}\;\right|=\left|\int_{0}^{T_{k}}\mathopen{\langle}\partial_{t}\varphi_{k},{\overline{\zeta}}\mathclose{\rangle}_{V}\;\right|$
$\displaystyle\quad=\left|\int_{0}^{T_{k}}\\!\\!\int_{\Omega}\varphi_{k}{\boldsymbol{v}}_{k}\cdot\nabla{\overline{\zeta}}-\int_{0}^{T_{k}}\\!\\!\int_{\Omega}M_{\Omega}(\varphi_{k})\nabla\mu_{k}\cdot\nabla{\overline{\zeta}}\;\right|$
$\displaystyle\quad\leq\left(\left\|\varphi_{k}\right\|_{L^{\infty}(0,T_{k};L^{4}(\Omega))}\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};(L^{4}(\Omega))^{d})}+M_{2}\left\|\nabla\mu_{k}\right\|_{L^{2}(0,T_{k};{\bf
H})}\right)\left\|{\overline{\zeta}}\right\|_{L^{2}(0,T_{k};V)}$
$\displaystyle\quad\leq C\left\|\zeta\right\|_{L^{2}(0,T_{k};V)}.$ (3.12)

Hence, taking the supremum over all $\zeta\in L^{2}(0,T_{k};V)$ with
$\left\|\zeta\right\|_{L^{2}(0,T_{k};V)}\leq 1$, we deduce

$\displaystyle\left\|\partial_{t}\varphi_{k}\right\|_{L^{2}(0,T_{k};V^{*})}\leq
C.$ (3.13)

Proceeding similarly and testing (3.4c) with
${{\overline{\zeta}}_{\Gamma}}:=\mathbb{P}_{V_{\Gamma,k}}({\zeta_{\Gamma}})$,
we obtain the estimate

$\displaystyle\left|\int_{0}^{T_{k}}\mathopen{\langle}\partial_{t}\psi_{k},{\zeta_{\Gamma}}\mathclose{\rangle}_{{V}_{\Gamma}}\;\right|$
$\displaystyle\leq\Bigl{(}C\left\|\psi_{k}\right\|_{L^{\infty}(0,T_{k};{V}_{\Gamma})}\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};{\bf
V}_{\sigma,{\bf
n}})}+M_{\Gamma,2}\left\|\nabla_{\Gamma}\theta_{k}\right\|_{L^{2}(0,T_{k};{\bf
H}_{\Gamma})}\Bigr{)}\left\|{{\overline{\zeta}}}_{\Gamma}\right\|_{L^{2}(0,T_{k};{V}_{\Gamma})}$
$\displaystyle\leq
C\left\|{\zeta_{\Gamma}}\right\|_{L^{2}(0,T_{k};{V}_{\Gamma})}.$ (3.14)

Taking the supremum over all ${\zeta_{\Gamma}}\in L^{2}(0,T_{k};{V}_{\Gamma})$
with $\left\|{\zeta_{\Gamma}}\right\|_{L^{2}(0,T_{k};{V}_{\Gamma})}\leq 1$, we
conclude

$\displaystyle\left\|\partial_{t}\psi_{k}\right\|_{L^{2}(0,T_{k};V_{\Gamma}^{*})}\leq
C.$ (3.15)

Third estimate. Next, we want to derive uniform bounds on $\mu_{k}$ in
$L^{4}(0,T_{k};H)\cap L^{2}(0,T_{k};V)$ and on $\theta_{k}$ in
$L^{4}(0,T_{k};{H}_{\Gamma})\cap L^{2}(0,T_{k};{V}_{\Gamma})$. Therefore, we
choose arbitrary functions $\eta\in L^{1}(0,T_{k};V)$ and $\eta_{\Gamma}\in
L^{1}(0,T_{k};{V}_{\Gamma})$ and we set
${\overline{\eta}}:=\mathbb{P}_{V_{k}}(\eta)$ and
${\overline{\eta_{\Gamma}}}:=\mathbb{P}_{V_{\Gamma,k}}(\eta_{\Gamma})$.
Testing (3.4) by $({\overline{\eta}},{\overline{{\eta_{\Gamma}}}})$, recalling
the growth conditions from (R1) as well as the uniform bounds (3.1.2) and
(3.10), we use Hölder’s inequality and Sobolev’s embedding theorem to derive
the estimate

$\displaystyle\int_{0}^{T_{k}}\big{|}\mathopen{\langle}(\mu_{k},\theta_{k}),(\eta,{\eta_{\Gamma}})\mathclose{\rangle}_{{\cal
V}}\big{|}=\int_{0}^{T_{k}}\big{|}\mathopen{\langle}(\mu_{k},\theta_{k}),({\overline{\eta}},{\overline{{\eta_{\Gamma}}}})\mathclose{\rangle}_{{\cal
V}}\big{|}$
$\displaystyle\quad\leq\int_{0}^{T_{k}}\Big{[}\left\|\nabla\varphi_{k}\right\|_{{\bf
H}}\left\|\nabla{\overline{\eta}}\right\|_{{\bf
H}}+\left\|F^{\prime}(\varphi_{k})\right\|_{L^{\frac{6}{5}}}(\Omega)\left\|{\overline{\eta}}\right\|_{L^{6}(\Omega)}+\left\|\nabla_{\Gamma}\psi_{k}\right\|_{{\bf
H}_{\Gamma}}\left\|\nabla_{\Gamma}{\overline{{\eta_{\Gamma}}}}\right\|_{{\bf
H}_{\Gamma}}$
$\displaystyle\qquad\qquad\quad+\left\|G^{\prime}(\psi_{k})\right\|_{{H}_{\Gamma}}\left\|{\overline{{\eta_{\Gamma}}}}\right\|_{{H}_{\Gamma}}+\tfrac{1}{K}\left\|\psi_{k}-\varphi_{k}\right\|_{{H}_{\Gamma}}\left\|{\overline{{\eta_{\Gamma}}}}-{\overline{\eta}}\right\|_{{H}_{\Gamma}}\Big{]}$
$\displaystyle\quad\leq
C(1+\left\|\varphi_{k}\right\|_{L^{\infty}(0,T_{k};V)}^{p-1}+\left\|\psi_{k}\right\|_{L^{\infty}(0,T_{k};{V}_{\Gamma})}^{q-1})\left\|(\eta,{\eta_{\Gamma}})\right\|_{L^{1}(0,T_{k};{\cal
V})}$

in $[0,T_{k}]$. Taking the supremum over all $(\eta,{\eta_{\Gamma}})\in
L^{1}(0,T_{k};{\cal V})$ with
$\left\|(\eta,{\eta_{\Gamma}})\right\|_{L^{1}(0,T_{k};{\cal V})}\leq 1$, and
using (3.11), we infer

$\displaystyle\left\|(\mu_{k},\theta_{k})\right\|_{L^{\infty}(0,T_{k};{\cal
V}^{*})}\leq C.$ (3.16)

We further have

$\displaystyle\left\|(\mu_{k},\theta_{k})\right\|^{2}_{\cal H}$
$\displaystyle=\mathopen{\langle}(\mu_{k},\theta_{k}),(\mu_{k},\theta_{k})\mathclose{\rangle}_{{\cal
V}}$ $\displaystyle\leq C\left\|(\mu_{k},\theta_{k})\right\|_{{{\cal
V}^{*}}}\big{(}\left\|(\mu_{k},\theta_{k})\right\|_{\cal
H}+\left\|(\nabla\mu_{k},\nabla_{\Gamma}\theta_{k})\right\|_{{\bf H}\times{\bf
H}_{\Gamma}}\big{)}$
$\displaystyle\leq\frac{1}{2}\left\|(\mu_{k},\theta_{k})\right\|^{2}_{\cal
H}+C\left\|(\mu_{k},\theta_{k})\right\|_{{{\cal
V}^{*}}}\left\|(\nabla\mu_{k},\nabla_{\Gamma}\theta_{k})\right\|_{{\bf
H}\times{\bf H}_{\Gamma}}+C\left\|(\mu_{k},\theta_{k})\right\|_{{{\cal
V}^{*}}}^{2}.$

Hence, squaring and integrating this estimate with respect to time, we use
(3.1.2) and (3.16) to conclude

$\displaystyle\left\|(\mu_{k},\theta_{k})\right\|_{L^{4}(0,T_{k};{\cal
H})}\leq C.$ (3.17)

In particular, we thus have

$\displaystyle\left\|\mu_{k}\right\|_{L^{2}(0,T_{k};V)}+\left\|\theta_{k}\right\|_{L^{2}(0,T_{k};{V}_{\Gamma})}\leq
C.$ (3.18)

Overall estimate. Combining (3.1.2)–(3.11), (3.13), (3.15), (3.17) and (3.18),
we obtain the overall uniform estimate

$\displaystyle\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T_{k};{\bf
V}_{\sigma,{\bf n}})\cap
L^{2}(0,T_{k};\mathbf{H}_{\Gamma})}+\left\|\varphi_{k}\right\|_{H^{1}(0,T_{k};{V^{*}})\cap
L^{\infty}(0,T_{k};V)}+\left\|\psi_{k}\right\|_{H^{1}(0,T_{k};{{V}_{\Gamma}^{*}})\cap
L^{\infty}(0,T_{k};{V}_{\Gamma})}$
$\displaystyle\quad+\left\|\mu_{k}\right\|_{L^{4}(0,T_{k};H)\cap
L^{2}(0,T_{k};V)}+\left\|\theta_{k}\right\|_{L^{4}(0,T_{k};{H}_{\Gamma})\cap
L^{2}(0,T_{k};{V}_{\Gamma})}\leq C.$ (3.19)

#### 3.1.3 Extension of the approximate solution onto the whole time interval
$[0,T]$

In Step 1, we constructed the coefficients
$(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$ as a solution of a nonlinear system
of ODEs existing on its right-maximal time interval $[0,T_{k}^{*})\cap[0,T]$.
We now assume that $T_{k}^{*}\leq T$. By the definition of the approximate
solutions given in (3.3) and the uniform bound (3.1.2), we infer that for any
$T_{k}\in[0,T_{k}^{*})$, all $t\in[0,T_{k}]$, and all $i\in\\{1,...,k\\}$,

$\displaystyle|a_{i}^{k}(t)|+|b_{i}^{k}(t)|$
$\displaystyle=\big{|}\left(\varphi_{k}(t),u_{i}\right)_{H}\big{|}+\big{|}\left(\psi_{k}(t),v_{i}\right)_{{H}_{\Gamma}}\big{|}$
$\displaystyle\leq\|\varphi_{k}\|_{L^{\infty}(0,T_{k};H)}+\|\psi_{k}\|_{L^{\infty}(0,T_{k};{H}_{\Gamma})}\leq
C.$

This means that the solution $(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$ is
bounded on the time interval $[0,T_{k}^{*})$ by a constant that is independent
of $T_{k}$ and $k$. Hence, according to classical ODE theory, the solution can
thus be extended beyond the time $T_{k}^{*}$. However, as the solution was
assumed to be right-maximal, this is a contradiction. We thus have
$T_{k}^{*}>T$, which directly implies $[0,T_{k}^{*})\cap[0,T]=[0,T]$. This
means that the solution $(\mathbf{a}^{k},\mathbf{b}^{k})^{\top}$ of the ODE
system actually exists on the whole time interval $[0,T]$. As the coefficients
$\mathbf{c}^{k}$ and $\mathbf{d}^{k}$ can be reconstructed from
$\mathbf{a}^{k}$ and $\mathbf{b}^{k}$ by the corresponding system of algebraic
equations, they also exist on the whole time interval $[0,T]$. Recalling (3.3)
and the construction of ${\boldsymbol{v}}_{k}$ this directly entails that the
approximate solution
$({\boldsymbol{v}}_{k},\varphi_{k},\mu_{k},\psi_{k},\theta_{k})$ actually
exists in $[0,T]$. Hence, choosing $T_{k}=T$ in (3.1.2), we eventually
conclude

$\displaystyle\left\|{\boldsymbol{v}}_{k}\right\|_{L^{2}(0,T;{\bf
V}_{\sigma,{\bf n}})\cap
L^{2}(0,T;\mathbf{H}_{\Gamma})}+\left\|\varphi_{k}\right\|_{H^{1}(0,T;V^{*})\cap
L^{\infty}(0,T;V)}+\left\|\psi_{k}\right\|_{H^{1}(0,T;{V}_{\Gamma}^{*})\cap
L^{\infty}(0,T;{V}_{\Gamma})}$
$\displaystyle\quad+\left\|\mu_{k}\right\|_{L^{4}(0,T;H)\cap
L^{2}(0,T;V)}+\left\|\theta_{k}\right\|_{L^{4}(0,T;{H}_{\Gamma})\cap
L^{2}(0,T;{V}_{\Gamma})}\leq C.$ (3.20)

#### 3.1.4 Convergence to a weak solution as $k\to\infty$

Considering the uniform estimate (3.1.3), we use the Banach–Alaoglu theorem
and the Aubin–Lions–Simon lemma to infer that there exist functions
${\boldsymbol{v}}$, $\varphi$, $\mu$, $\psi$ and $\theta$ such that for any
$s\in[0,1)$,

$\displaystyle{\boldsymbol{v}}_{k}$ $\displaystyle\to{\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(0,T;{\bf V}_{\sigma,{\bf n}})$},$
(3.21a) $\displaystyle{\boldsymbol{v}}_{k}|_{\Gamma}$
$\displaystyle\to{\boldsymbol{v}}|_{\Gamma}$ $\displaystyle\quad\text{weakly
in $L^{2}(0,T;\mathbf{H}_{\Gamma})$},$ (3.21b) $\displaystyle\varphi_{k}$
$\displaystyle\to\varphi$ $\displaystyle\quad\text{weakly-${}^{*}$ in
$L^{\infty}(0,T;V)$, weakly in $H^{1}(0,T;V^{*})$},$
$\displaystyle\qquad\text{strongly in $C^{0}([0,T];H^{s}(\Omega))$, and {a.e.\
in~{}$Q$}},$ (3.21c) $\displaystyle\psi_{k}$ $\displaystyle\to\psi$
$\displaystyle\quad\text{weakly-${}^{*}$ in $L^{\infty}(0,T;{V}_{\Gamma})$,
weakly in $H^{1}(0,T;{V}_{\Gamma}^{*})$},$ $\displaystyle\qquad\text{strongly
in $C^{0}([0,T];H^{s}(\Gamma))$, and {a.e.\ on~{}$\Sigma$}},$ (3.21d)
$\displaystyle\mu_{k}$ $\displaystyle\to\mu$ $\displaystyle\quad\text{weakly
in $L^{2}(0,T;V)\cap L^{4}(0,T;H)$},$ (3.21e) $\displaystyle\theta_{k}$
$\displaystyle\to\theta$ $\displaystyle\quad\text{weakly in
$L^{2}(0,T;{V}_{\Gamma})\cap L^{4}(0,T;{H}_{\Gamma})$},$ (3.21f)

as $k\to\infty$ along a non-relabeled subsequence. In particular, this shows
that the functions ${\boldsymbol{v}}$, $\varphi$, $\psi$, $\mu$ and $\theta$
have the regularity demanded in Definition 2.3(i).

Due to the trace theorem, the strong convergence from (3.21) (with
$s>\frac{1}{2}$) directly yields

$\displaystyle\varphi_{k}|_{\Gamma}\to\varphi|_{\Gamma}\quad\text{strongly in
$C^{0}([0,T];{H}_{\Gamma})$}.$ (3.22)

Recalling the growth conditions from (R1), we further deduce from the uniform
bound (3.1.3) that

$\displaystyle\left\|F^{\prime}(\varphi_{k})\right\|_{L^{\frac{6}{5}}(Q)}\leq
C\quad\text{and}\quad\left\|G^{\prime}(\varphi_{k})\right\|_{L^{2}(\Sigma)}\leq
C.$

Hence, there exist weakly convergent subsequences in the respective spaces. As
$F^{\prime}$ and $G^{\prime}$ are continuous, we use the pointwise
convergences from (3.21) and (3.21) to conclude

$\displaystyle F^{\prime}(\varphi_{k})$ $\displaystyle\to F^{\prime}(\varphi)$
$\displaystyle\quad\text{weakly in $L^{\frac{6}{5}}(Q)$ and {a.e.\
in~{}$Q$}},$ (3.23) $\displaystyle G^{\prime}(\psi_{k})$ $\displaystyle\to
G^{\prime}(\psi)$ weakly in $L^{2}(\Sigma)$ and a.e. on $\Sigma$ (3.24)

since the weak limit and the pointwise limit must coincide (see, e.g., [26,
Proposition 9.2c]). Furthermore, it follows from the pointwise convergences in
(3.21) and (3.21) that, as $k\to\infty$,

$\displaystyle M_{\Omega}(\varphi_{k})$ $\displaystyle\to
M_{\Omega}(\varphi),$ $\displaystyle\quad\nu(\varphi_{k})$
$\displaystyle\to\nu(\varphi),\quad\lambda(\varphi_{k})\to\lambda(\varphi)$
$\displaystyle\quad\text{{a.e.\ in~{}$Q$}},$ (3.25) $\displaystyle
M_{\Gamma}(\psi_{k})$ $\displaystyle\to M_{\Gamma}(\psi),$
$\displaystyle\quad\gamma(\psi_{k})$ $\displaystyle\to\gamma(\psi)$ a.e. on
$\Sigma$ (3.26)

as the functions $M_{\Omega}$, $M_{\Gamma}$, $\nu$, $\lambda$ and $\gamma$ are
continuous. Since, due to (A2) and (A3), these functions are also bounded, we
use Lebesgue’s dominated convergence theorem along with the weak convergences
in (3.21) to infer that

$\displaystyle\nu(\varphi_{k})\mathbf{D}{\boldsymbol{v}}_{k}$
$\displaystyle\to\nu(\varphi)\mathbf{D}{\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(Q;{\mathbb{R}}^{d\times d})$},$
(3.27) $\displaystyle\lambda(\varphi_{k}){\boldsymbol{v}}_{k}$
$\displaystyle\to\lambda(\varphi){\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(Q;{\mathbb{R}}^{d})$},$ (3.28)
$\displaystyle M_{\Omega}(\varphi_{k})\nabla\mu_{k}$ $\displaystyle\to
M_{\Omega}(\varphi)\nabla\mu$ $\displaystyle\quad\text{weakly in
$L^{2}(Q;{\mathbb{R}}^{d})$},$ (3.29)
$\displaystyle\gamma(\psi_{k}){\boldsymbol{v}}_{k}$
$\displaystyle\to\gamma(\psi){\boldsymbol{v}}$ $\displaystyle\quad\text{weakly
in $L^{2}(\Sigma;{\mathbb{R}}^{d})$},$ (3.30) $\displaystyle
M_{\Gamma}(\psi_{k})\nabla_{\Gamma}\theta_{k}$ $\displaystyle\to
M_{\Gamma}(\psi)\nabla_{\Gamma}\theta$ $\displaystyle\quad\text{weakly in
$L^{2}(\Sigma;{\mathbb{R}}^{d})$}.$ (3.31)

Combining the convergences (3.21a)–(3.21f), (3.22)–(3.24) and (3.27)–(3.31),
it is straightforward to pass to the limit as $k\to\infty$ in the discretized
weak formulation (3.4) to conclude that the quintuplet
$({\boldsymbol{v}},\varphi,\mu,\psi,\theta)$ fulfills the variational
formulation (2.13) for all test functions ${\boldsymbol{w}}\in{\bf
V}_{\sigma,{\bf n}}$,

$\displaystyle\zeta\in\bigcup_{k\in\mathbb{N}}V_{k}\subseteq
V,\qquad{\zeta_{\Gamma}}\in\bigcup_{k\in\mathbb{N}}V_{\Gamma,k}\subseteq
V_{\Gamma},\qquad(\eta,{\eta_{\Gamma}})\in\bigcup_{k\in\mathbb{N}}{\cal
V}_{k}\subseteq{\cal V}.$

Hence, because of density, (2.13) holds true for all test functions
${\boldsymbol{w}}\in{\bf V}_{\sigma,{\bf n}}$, $\zeta\in V$,
${\zeta_{\Gamma}}\in V_{\Gamma}$ and $(\eta,{\eta_{\Gamma}})\in{\cal V}={\cal
V}_{K}$. This verifies Definition 2.3(ii).

Moreover, we deduce from (3.5) that

$\displaystyle\big{(}\varphi_{k}(0),\psi_{k}(0)\big{)}\to\big{(}\varphi_{0},\psi_{0}\big{)}\quad\text{strongly
in ${\cal H}$}$

as the orthogonal projections converge strongly in $H$ and in ${H}_{\Gamma}$,
respectively. On the other hand, it follows from the strong convergences in
(3.21) and (3.21) that

$\displaystyle\big{(}\varphi_{k}(0),\psi_{k}(0)\big{)}\to\big{(}\varphi(0),\psi(0)\big{)}\quad\text{strongly
in ${\cal H}$.}$

Hence, due to the uniqueness of the limit, this verifies Definition 2.3(iii).

We still need to establish the weak energy dissipation law. Therefore, let
$\rho\in C^{\infty}([0,T])$ be an arbitrary nonnegative test function.
Employing the convergences (3.21) and (3.21), the weak lower semicontinuity of
the mappings

$\displaystyle L^{2}(0,T;V)$
$\displaystyle\ni\zeta\mapsto\int_{0}^{T}\left\|\nabla\zeta(t)\right\|_{{\bf
H}}^{2}\rho(t),$ $\displaystyle L^{2}(0,T;{V}_{\Gamma})$
$\displaystyle\ni\xi\mapsto\int_{0}^{T}\left\|\nabla_{\Gamma}\xi(t)\right\|_{{\bf
H}_{\Gamma}}^{2}\rho(t),$

as well as Fatou’s lemma, we deduce

$\displaystyle\int_{0}^{T}E_{K}\big{(}\varphi(t),\psi(t)\big{)}\,\rho(t)\leq\underset{k\to\infty}{\lim\inf}\;\int_{0}^{T}E_{K}\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}\,\rho(t).$
(3.32)

Proceeding similarly as above, we derive the convergences

$\displaystyle\sqrt{\nu(\varphi_{k})}\,\mathbf{D}{\boldsymbol{v}}_{k}$
$\displaystyle\to\sqrt{\nu(\varphi)}\,\mathbf{D}{\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(Q;{\mathbb{R}}^{d\times d})$},$
(3.33) $\displaystyle\sqrt{\lambda(\varphi_{k})}\,{\boldsymbol{v}}_{k}$
$\displaystyle\to\sqrt{\lambda(\varphi)}\,{\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(Q;{\mathbb{R}}^{d})$},$ (3.34)
$\displaystyle\sqrt{M_{\Omega}(\varphi_{k})}\,\nabla\mu_{k}$
$\displaystyle\to\sqrt{M_{\Omega}(\varphi)}\,\nabla\mu$
$\displaystyle\quad\text{weakly in $L^{2}(Q;{\mathbb{R}}^{d})$},$ (3.35)
$\displaystyle\sqrt{\gamma(\psi_{k})}\,{\boldsymbol{v}}_{k}$
$\displaystyle\to\sqrt{\gamma(\psi)}\,{\boldsymbol{v}}$
$\displaystyle\quad\text{weakly in $L^{2}(\Sigma;{\mathbb{R}}^{d})$},$ (3.36)
$\displaystyle\sqrt{M_{\Gamma}(\psi_{k})}\,\nabla_{\Gamma}\theta_{k}$
$\displaystyle\to\sqrt{M_{\Gamma}(\psi)}\,\nabla_{\Gamma}\theta$
$\displaystyle\quad\text{weakly in $L^{2}(\Sigma;{\mathbb{R}}^{d})$}.$ (3.37)

Hence, employing (3.1.2), (3.32) and weak lower semicontinuity, we eventually
obtain

$\displaystyle\int_{0}^{T}E_{K}\big{(}\varphi(t),\psi(t)\big{)}\rho(t)$
$\displaystyle\qquad+\int_{0}^{T}\int_{\Omega}\Big{[}2\nu(\varphi)\left|\mathbf{D}{\boldsymbol{v}}\right|^{2}\rho(t)+\lambda(\varphi)\left|{\boldsymbol{v}}\right|^{2}\rho(t)+M_{\Omega}(\varphi)\left|\nabla\mu\right|^{2}\rho(t)\Big{]}$
$\displaystyle\qquad+\int_{0}^{T}\int_{\Gamma}\Big{[}\gamma(\psi)\left|{\boldsymbol{v}}\right|^{2}\rho(t)+M_{\Gamma}(\psi)\left|\nabla_{\Gamma}\theta\right|^{2}\rho(t)\Big{]}$
$\displaystyle\quad\leq\underset{k\to\infty}{\liminf}\Bigg{\\{}\int_{0}^{T}E_{K}\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}\rho(t)$
$\displaystyle\qquad+\int_{0}^{T}\int_{\Omega}\Big{[}2\nu(\varphi_{k})\left|\mathbf{D}{\boldsymbol{v}}_{k}\right|^{2}\rho(t)+\lambda(\varphi_{k})\left|{\boldsymbol{v}}_{k}\right|^{2}\rho(t)+M_{\Omega}(\varphi_{k})\left|\nabla\mu_{k}\right|^{2}\rho(t)\Big{]}$
$\displaystyle\qquad+\int_{0}^{T}\int_{\Gamma}\Big{[}\gamma(\psi_{k})\left|{\boldsymbol{v}}_{k}\right|^{2}\rho(t)+M_{\Gamma}(\psi_{k})\left|\nabla_{\Gamma}\theta_{k}\right|^{2}\rho(t)\Big{]}\Bigg{\\}}$
$\displaystyle\quad\leq\underset{k\to\infty}{\lim}\int_{0}^{T}E_{K}\big{(}\varphi_{0,k},\psi_{0,k}\big{)}\rho(t)=\int_{0}^{T}E_{K}\big{(}\varphi_{0},\psi_{0}\big{)}\rho(t).$

Here, invoking the growth conditions from (R1), the equality in the last line
follows by means of Lebesgue’s general convergence theorem (see [8, Section
3.25]) since the orthogonal projections in (3.5) converge strongly in $V$ and
in ${V}_{\Gamma}$, respectively. As the nonnegative test function $\rho$ was
arbitrary, this proves that the weak energy dissipation law stated in ((iv))
holds for almost all $t\in[0,T]$. As the time integral in this inequality is
continuous with respect to $t$ and since the mapping $t\mapsto
E_{K}\big{(}\varphi(t),\psi(t)\big{)}$ is lower semicontinuous, we conclude
that ((iv)) actually holds true for all $t\in[0,T]$. This means that
Definition 2.3(iv) is verified.

We have thus shown that the quintuplet
$({\boldsymbol{v}},\varphi,\mu,\psi,\theta)$ is a weak solution in the sense
of Definition 2.3.

#### 3.1.5 Additional regularity for the phase-fields

To prove additional regularity for the phase-fields, we assume $\Omega$ to be
of class $C^{\ell}$ with $\ell\in\\{2,3\\}$, and we return to the
Faedo–Galerkin scheme. First of all, applying regularity theory for Poisson’s
equation with an inhomogeneous Neumann boundary condition (see, e.g., [52,
Theorem 4.18] or [59, s. 5, Proposition 7.7]), we infer that the
eigenfunctions $u_{i}$ exhibit the regularity $u_{i}\in H^{\ell}(\Omega)$ for
all $i\in{\mathbb{N}}$. Moreover, applying regularity theory for elliptic
equations on submanifolds (see, e.g., [59, s. 5, Theorem 1.3]), we conclude
that the eigenfunctions $v_{i}$ have the regularity $v_{i}\in
H^{\ell}(\Gamma)$ for all $i\in{\mathbb{N}}$. This directly entails

$\displaystyle\varphi_{k}(t)\in V_{k}\cap
H^{\ell}(\Omega)\quad\text{and}\quad\psi_{k}(t)\in V_{\Gamma,k}\cap
H^{\ell}(\Gamma)$

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

Let now $t\in[0,T]$ be arbitrary. We infer from (3.4) that there exists a null
set $\mathcal{N}\subset[0,T]$ such that

$\displaystyle\int_{\Omega}\nabla\varphi_{k}(t)\cdot\nabla\eta+\int_{\Gamma}\nabla_{\Gamma}\psi_{k}(t)\cdot\nabla_{\Gamma}{\eta_{\Gamma}}+\frac{1}{K}\int_{\Gamma}\big{(}\psi_{k}(t)-\varphi_{k}(t)\big{)}({\eta_{\Gamma}}-\eta)$
$\displaystyle\quad=\int_{\Omega}\Big{(}\mu_{k}(t)-\mathbb{P}_{V_{k}}\big{[}F^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{]}\Big{)}\,\eta+\int_{\Gamma}\Big{(}\theta_{k}(t)-\mathbb{P}_{V_{\Gamma,k}}\big{[}G^{\prime}\big{(}\psi_{k}(t)\big{)}\big{]}\Big{)}\,{\eta_{\Gamma}}$
(3.38)

for all test functions $(\eta,{\eta_{\Gamma}})\in{\cal V}$ and all
$t\in[0,T]\setminus\mathcal{N}$.

Let now $t\in[0,T]\setminus\mathcal{N}$ be arbitrary. Then, (3.1.5) entails
that the pair $\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}$ is a weak solution of
the bulk-surface elliptic problem

$\displaystyle-\Delta\varphi_{k}(t)$ $\displaystyle=f_{k}(t)$
$\displaystyle\quad\text{in $\Omega$},$ (3.39a)
$\displaystyle-\Delta_{\Gamma}\psi_{k}(t)+\partial_{\bf n}\varphi_{k}(t)$
$\displaystyle=g_{k}(t)$ $\displaystyle\quad\text{on $\Gamma$},$ (3.39b)
$\displaystyle K\partial_{\bf n}\varphi_{k}(t)$
$\displaystyle=\psi_{k}(t)-\varphi_{k}(t)$ $\displaystyle\quad\text{on
$\Gamma$},$ (3.39c)

where

$\displaystyle f_{k}(t)$
$\displaystyle:=\mu_{k}(t)-\mathbb{P}_{V_{k}}\big{[}F^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{]}$
$\displaystyle\in V_{k}\subset V,$ $\displaystyle g_{k}(t)$
$\displaystyle:=\theta_{k}(t)-\mathbb{P}_{V_{\Gamma,k}}\big{[}G^{\prime}\big{(}\psi_{k}(t)\big{)}\big{]}$
$\displaystyle\in V_{\Gamma,k}\subset{V}_{\Gamma}.$

In the following, without loss of generality, we assume that the growth
conditions from (R1) hold with $p\in[5,6)$. Most part of the computations to
follow resemble those performed in [40]. Therefore, we just highlight the key
steps for the reader’s convenience. Using the uniform estimate (3.1.3), the
Gagliardo–Nirenberg inequality and Young’s inequality, we derive the estimate

$\displaystyle\big{\|}F^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{\|}_{H}$
$\displaystyle\leq C+C\|\varphi_{k}(t)\|_{L^{2(p-1)}(\Omega)}^{p-1}$
$\displaystyle\leq
C+C\|\varphi_{k}(t)\|_{L^{6}(\Omega)}^{\frac{p+2}{2}}\|\varphi_{k}(t)\|_{H^{2}(\Omega)}^{\frac{p-4}{2}}$
$\displaystyle\leq
C\varepsilon^{-1}+\varepsilon\|\varphi_{k}(t)\|_{H^{2}(\Omega)},$ (3.40)

for any $\varepsilon\in(0,1)$. Moreover, due to (R1) and (3.1.3), we have

$\displaystyle\big{\|}G^{\prime}\big{(}\psi_{k}(t)\big{)}\big{\|}_{{H}_{\Gamma}}\leq
C+C\|\psi_{k}(t)\|_{L^{2(q-1)}(\Gamma)}^{q-1}\leq C.$ (3.41)

We first consider the case $\ell=2$. Applying regularity theory for elliptic
problems with bulk-surface coupling (see [48, Theorem 3.3]), we deduce
$\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}\in H^{2}(\Omega)\times
H^{2}(\Gamma)$ with

$\displaystyle\left\|\varphi_{k}(t)\right\|_{H^{2}(\Omega)}^{2}+\left\|\psi_{k}(t)\right\|_{H^{2}(\Gamma)}^{2}\leq
C\left\|f_{k}(t)\right\|_{H}^{2}+C\left\|g_{k}(t)\right\|_{{H}_{\Gamma}}^{2}$
$\displaystyle\quad\leq
C\left\|\mu_{k}(t)\right\|_{H}^{2}+C\left\|\theta_{k}(t)\right\|_{{H}_{\Gamma}}^{2}+C(1+\varepsilon^{-2})+C\varepsilon^{2}\|\varphi_{k}(t)\|_{H^{2}(\Omega)}^{2}.$

After choosing $\varepsilon\in(0,1)$ sufficiently small, we absorb the term
$C\varepsilon^{2}\|\varphi_{k}(t)\|_{H^{2}(\Omega)}^{2}$ by the left-hand
side. Squaring and integrating the resulting inequality with respect to $t$
from $0$ to $T$, we use (3.1.3) to arrive at the uniform estimate

$\displaystyle\left\|\varphi_{k}\right\|_{L^{4}(0,T;H^{2}(\Omega))}+\left\|\psi_{k}\right\|_{L^{4}(0,T;H^{2}(\Gamma))}\leq
C.$ (3.42)

Consequently, we have, as $k\to\infty$,

$\displaystyle\varphi_{k}$ $\displaystyle\to\varphi\quad\text{weakly in
$L^{4}(0,T;H^{2}(\Omega))$},$ $\displaystyle\psi_{k}$
$\displaystyle\to\psi\quad\text{weakly in $L^{4}(0,T;H^{2}(\Gamma))$},$

after another subsequence extraction. This proves the regularity assertion
(2.15a). As a direct consequence, it follows via integration by parts and the
fundamental lemma of calculus that the equations (1.1d), (1.1f) and (1.1g) are
now even satisfied in the strong sense (i.e., a.e. in $Q$ and on $\Sigma$,
respectively).

Let us now consider the case $\ell=3$. Employing the uniform estimate (3.1.3),
the Gagliardo–Nirenberg inequality, Agmon’s inequality and Young’s inequality,
we derive the estimate

$\displaystyle\big{\|}F^{\prime\prime}\big{(}\varphi_{k}(t)\big{)}\nabla\varphi_{k}(t)\big{\|}_{{\bf
H}}$ $\displaystyle\leq C\|\nabla\varphi_{k}(t)\|_{{\bf
H}}+C\||\varphi_{k}(t)|^{p-2}\nabla\varphi_{k}(t)\|_{{\bf H}}$
$\displaystyle\leq
C+C\|\varphi_{k}(t)\|_{L^{2(p-2)}(\Omega)}^{p-2}\|\nabla\varphi_{k}(t)\|_{{\bf{L}}^{\infty}(\Omega)}$
$\displaystyle\leq
C+C\|\varphi_{k}(t)\|_{L^{6}(\Omega)}^{\frac{p+1}{2}}\|\varphi_{k}(t)\|_{H^{2}(\Omega)}^{\frac{p-5}{2}+\frac{1}{2}}\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{\frac{1}{2}}$
$\displaystyle\leq
C+C\|\varphi_{k}(t)\|_{H^{2}(\Omega)}^{\frac{p-4}{2}}\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{\frac{1}{2}}$
$\displaystyle\leq
C+C\|\varphi_{k}(t)\|_{H^{1}(\Omega)}^{\frac{p-4}{4}}\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{\frac{p-4}{4}+\frac{1}{2}}$
$\displaystyle\leq C+C\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{\frac{p-2}{4}}$
$\displaystyle\leq
C\varepsilon^{-1}+\varepsilon\|\varphi_{k}(t)\|_{H^{3}(\Omega)}$ (3.43)

for any $\varepsilon\in(0,1)$. In combination with (3.1.5), this proves that

$\displaystyle\big{\|}F^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{\|}_{V}\leq
C\varepsilon^{-1}+\varepsilon\|\varphi_{k}(t)\|_{H^{3}(\Omega)}.$ (3.44)

In a similar fashion, we derive the estimate

$\displaystyle\big{\|}G^{\prime}\big{(}\psi_{k}(t)\big{)}\big{\|}_{{V}_{\Gamma}}\leq
C\varepsilon^{-1}+\varepsilon\|\psi_{k}(t)\|_{H^{3}(\Gamma)}$ (3.45)

for any $\varepsilon\in(0,1)$. Applying regularity theory for elliptic
problems with bulk-surface coupling (see [48, Theorem 3.3]), we infer
$\big{(}\varphi_{k}(t),\psi_{k}(t)\big{)}\in H^{3}(\Omega)\times
H^{3}(\Gamma)$ with

$\displaystyle\left\|\varphi_{k}(t)\right\|_{H^{3}(\Omega)}^{2}+\left\|\psi_{k}(t)\right\|_{H^{3}(\Gamma)}^{2}\leq
C\left\|f_{k}(t)\right\|_{V}^{2}+C\left\|g_{k}(t)\right\|_{{V}_{\Gamma}}^{2}$
$\displaystyle\quad\leq
C\left\|\mu_{k}(t)\right\|_{V}^{2}+C\left\|\theta_{k}(t)\right\|_{{V}_{\Gamma}}^{2}+C\varepsilon^{-2}+C\varepsilon^{2}\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{2}+C\varepsilon^{2}\|\psi_{k}(t)\|_{H^{3}(\Gamma)}^{2}.$

After choosing $\varepsilon\in(0,1)$ sufficiently small, we absorb the terms
$C\varepsilon^{2}\|\varphi_{k}(t)\|_{H^{3}(\Omega)}^{2}$ and
$C\varepsilon^{2}\|\psi_{k}(t)\|_{H^{3}(\Gamma)}^{2}$ by the left-hand side.
Integrating the resulting estimate with respect to $t$ from $0$ to $T$, we use
(3.1.3) to eventually obtain the uniform estimate

$\displaystyle\left\|\varphi_{k}\right\|_{L^{2}(0,T;H^{3}(\Omega))}+\left\|\psi_{k}\right\|_{L^{2}(0,T;H^{3}(\Gamma))}\leq
C.$ (3.46)

Therefore, after another subsequence extraction, we have, as $k\to\infty$,

$\displaystyle\varphi_{k}$ $\displaystyle\to\varphi\quad\text{weakly in
$L^{2}(0,T;H^{3}(\Omega))$},$ $\displaystyle\psi_{k}$
$\displaystyle\to\psi\quad\text{weakly in $L^{2}(0,T;H^{3}(\Gamma))$}.$

Hence, the regularity assertion (2.15b) is verified, and the regularity (2.16)
follows directly from Proposition A.1.

This means that all claims are established and thus, the proof is complete. ∎

### 3.2 The limit $K\to 0$ and existence of a weak solution in the case $K=0$

###### Proof of Theorem 2.5.

In this proof, the letter $C$ will denote generic positive constants that may
depend on $\Omega$, $T$, the initial data and the constants introduced in
(A1)–(A3), but not on ${K_{n}}$ or $n$. Such constants may also change their
value from line to line.

First of all, as the initial data were prescribed as
$(\varphi_{0},\psi_{0})\in{\cal V}_{0}$, they satisfy the Dirichlet type
coupling condition $\varphi_{0}|_{\Gamma}=\psi_{0}$ a.e. on $\Gamma$. In view
of the definition of the energy functional in (1), this means that the
${K_{n}}$-depending term in the energy $E_{K_{n}}(\varphi_{0},\psi_{0})$
vanishes. It thus holds

$\displaystyle E_{K_{n}}(\varphi_{0},\psi_{0})=E_{0}(\varphi_{0},\psi_{0})\leq
C\quad\text{for all $n\in{\mathbb{N}}$}.$ (3.47)

According to Definition 2.3(iv), the solutions
$({\boldsymbol{v}}^{K_{n}},\varphi^{K_{n}},\mu^{K_{n}},\psi^{K_{n}},\theta^{K_{n}})$
satisfy the weak energy dissipation law. By the definition of $E_{K_{n}}$, we
have

$\displaystyle
E_{K_{n}}\big{(}\varphi^{K_{n}}(t),\psi^{K_{n}}(t)\big{)}+2\int_{0}^{t}\int_{\Omega}\nu(\varphi^{K_{n}})|\mathbf{D}{\boldsymbol{v}}^{K_{n}}|^{2}+\int_{0}^{t}\int_{\Omega}\lambda(\varphi^{K_{n}})|{\boldsymbol{v}}^{K_{n}}|^{2}$
$\displaystyle\qquad+\int_{0}^{t}\int_{\Gamma}\gamma(\psi^{K_{n}})|{\boldsymbol{v}}^{K_{n}}|^{2}+\int_{0}^{t}\int_{\Omega}M_{\Omega}(\varphi^{K_{n}})|\nabla\mu^{K_{n}}|^{2}+\int_{0}^{t}\int_{\Gamma}M_{\Gamma}(\psi^{K_{n}})|\nabla_{\Gamma}\theta^{K_{n}}|^{2}$
$\displaystyle\quad\leq E_{K_{n}}(\varphi_{0},\psi_{0})\leq C$

for all $t\in[0,T]$ and all $n\in{{\mathbb{N}}}$. In particular, recalling
that the potentials $F$ and $G$ are nonnegative, this directly yields

$\displaystyle\left\|\varphi^{K_{n}}-\psi^{K_{n}}\right\|_{{H}_{\Gamma}}^{2}\leq
C{K_{n}}\quad\text{for all $t\in[0,T]$ and all $n\in{\mathbb{N}}$.}$ (3.48)

Testing (2.13b) and (2.13c) written for
$({\boldsymbol{v}}^{K_{n}},\varphi^{K_{n}},\mu^{K_{n}},\psi^{K_{n}},\theta^{K_{n}})$
by the constant functions $\frac{1}{|\Omega|}$ and $\frac{1}{|\Gamma|}$,
respectively, we infer

$\displaystyle{\langle\varphi^{K_{n}}(t)\rangle}_{\Omega}={\langle\varphi_{0}\rangle}_{\Omega}\quad\text{and}\quad{\langle\psi^{K_{n}}(t)\rangle}_{\Gamma}={\langle\psi_{0}\rangle}_{\Gamma}$

for all $t\in[0,T]$ and all $n\in{\mathbb{N}}$. Hence, proceeding analogously
as in the proof of Theorem 2.4 (Subsection 3.1.2, First and Second estimates),
we derive the uniform bound

$\displaystyle\left\|{\boldsymbol{v}}^{K_{n}}\right\|_{L^{2}(0,T;{\bf
V}_{\sigma,{\bf n}})\cap
L^{2}(0,T;\mathbf{H}_{\Gamma})}+\left\|\nabla\mu^{K_{n}}\right\|_{L^{2}(0,T;{\bf{H}})}+\left\|\nabla_{\Gamma}\theta^{K_{n}}\right\|_{L^{2}(0,T;\mathbf{H}_{\Gamma})}$
$\displaystyle\quad+\left\|\varphi^{K_{n}}\right\|_{H^{1}(0,T;{V^{*}})\cap
L^{\infty}(0,T;V)}+\left\|\psi^{K_{n}}\right\|_{H^{1}(0,T;{{V}_{\Gamma}^{*}})\cap
L^{\infty}(0,T;{V}_{\Gamma})}\leq C.$ (3.49)

We now test (2.13d) written for
$({\boldsymbol{v}}^{K_{n}},\varphi^{K_{n}},\mu^{K_{n}},\psi^{K_{n}},\theta^{K_{n}})$
by $(\eta,0)$, where $\eta\in C^{\infty}_{c}(\Omega)$ is an arbitrary test
function. Using (3.2) along with Hölder’s inequality, we infer that

$\displaystyle\left|\int_{\Omega}\mu^{K_{n}}\eta\right|\leq\left\|\varphi^{K_{n}}\right\|_{V}\left\|\eta\right\|_{V}+\left\|F^{\prime}(\varphi^{K_{n}})\right\|_{L^{\frac{6}{5}}(\Omega)}\left\|\eta\right\|_{L^{6}(\Omega)}\leq
C\left\|\eta\right\|_{V}$ (3.50)

entailing that $\mu^{K_{n}}$ is bounded in $L^{\infty}(0,T;V^{*})$. Then, by
means of the generalized Poincaré inequality (see [8, Section 8.16] or [33,
Lemma A.1]), we infer that

$\left\|\mu^{K_{n}}\right\|_{H}\leq
C\big{(}1+\left\|\nabla\mu^{K_{n}}\right\|_{\bf{H}}\big{)}.$

For more details see, e.g., [33, Proof of Lemma 4.5]. Hence, in combination
with (3.50), we conclude

$\displaystyle\left\|\mu^{K_{n}}\right\|_{L^{2}(0,T;V)}\leq C.$ (3.51)

In order to derive an analogous estimate for $\theta^{K_{n}}$, we first choose
$\eta\equiv 1$ and ${\eta_{\Gamma}}\equiv 0$ in (2.13d). Employing (3.2), we
obtain

$\displaystyle\left|\frac{1}{{K_{n}}}\int_{\Gamma}(\psi^{K_{n}}-\varphi^{K_{n}})\right|\leq\left\|\mu^{K_{n}}\right\|_{L^{1}(\Omega)}+\left\|F^{\prime}(\varphi^{K_{n}})\right\|_{L^{1}(\Omega)}\leq
C\left\|\mu^{K_{n}}\right\|_{H}+C.$ (3.52)

Let us now take $\eta\equiv 0$ and ${\eta_{\Gamma}}\equiv 1$ in (2.13d). Using
(3.2) and (3.52), we deduce

$\displaystyle\left|\int_{\Gamma}\theta^{K_{n}}\right|$
$\displaystyle\leq\left\|G^{\prime}(\psi^{K_{n}})\right\|_{L^{1}(\Gamma)}+\left|\frac{1}{{K_{n}}}\int_{\Gamma}(\psi^{K_{n}}-\varphi^{K_{n}})\right|\leq
C\left\|\mu^{K_{n}}\right\|_{H}+C.$

Employing Poincaré’s inequality, we thus infer

$\displaystyle\left\|\theta^{K_{n}}\right\|_{{H}_{\Gamma}}\leq
C\left\|\nabla_{\Gamma}\theta^{K_{n}}\right\|_{\mathbf{H}_{\Gamma}}+C\left\|\mu^{K_{n}}\right\|_{H}+C.$

Squaring and integrating this estimate with respect to time over $[0,T]$, we
eventually conclude

$\displaystyle\left\|\theta^{K_{n}}\right\|_{L^{2}(0,T;{H}_{\Gamma})}\leq C.$
(3.53)

In summary, combining (3.2), (3.51) and (3.53), we have thus shown

$\displaystyle\left\|{\boldsymbol{v}}^{K_{n}}\right\|_{L^{2}(0,T;{\bf
V}_{\sigma,{\bf n}})\cap
L^{2}(0,T;\mathbf{H}_{\Gamma})}+\left\|\mu^{K_{n}}\right\|_{L^{2}(0,T;V)}+\left\|\theta^{K_{n}}\right\|_{L^{2}(0,T;{V}_{\Gamma})}$
$\displaystyle\quad+\left\|\varphi^{K_{n}}\right\|_{H^{1}(0,T;{V^{*}})\cap
L^{\infty}(0,T;V)}+\left\|\psi^{K_{n}}\right\|_{H^{1}(0,T;{{V}_{\Gamma}^{*}})\cap
L^{\infty}(0,T;{V}_{\Gamma})}\leq C.$ (3.54)

As in Subsection 3.1.4, we deduce the existence of functions
$({\boldsymbol{v}}^{0},\varphi^{0},\mu^{0},\psi^{0},\theta^{0})$ such that the
convergences (2.17a)–(2.17f) hold along a non-relabeled subsequence. Moreover,
the estimate (3.48) directly implies (2.17g) and thus, all convergences in
(2.17) are established. In particular, due to the trace theorem, we also have

$\displaystyle\varphi^{K_{n}}|_{\Gamma}-\psi^{K_{n}}\;\to\;\varphi^{0}|_{\Gamma}-\psi^{0}\quad\text{strongly
in $C^{0}([0,T];{H}_{\Gamma})$}.$ (3.55)

In combination with (3.48), this proves that $\varphi^{0}|_{\Gamma}=\psi^{0}$
a.e. on $\Sigma$ due to uniqueness of the limit. Proceeding further as in
Subsection 3.1.4, we eventually show that the quintuplet
$({\boldsymbol{v}}^{0},\varphi^{0},\mu^{0},\psi^{0},\theta^{0})$ is a weak
solution of the Cahn–Hilliard–Brinkman system (1.1) in the sense of Definition
2.3.

It remains to verify the additional regularity assertions. Without loss of
generality, we merely consider the case $d=3$. The case $d=2$ can be handled
analogously but is even easier as the Sobolev embeddings in two dimensions are
better. We infer from (2.13d) written for the solution
$({\boldsymbol{v}}^{0},\varphi^{0},\mu^{0},\psi^{0},\theta^{0})$ and $K=0$
that there exists a null set $\mathcal{N}\subset[0,T]$ such that

$\displaystyle\int_{\Omega}\nabla\varphi^{0}(t)\cdot\nabla\eta+\int_{\Gamma}\nabla_{\Gamma}\psi^{0}(t)\cdot\nabla_{\Gamma}{\eta_{\Gamma}}$
$\displaystyle\quad=\int_{\Omega}\Big{(}\mu^{0}(t)-F^{\prime}\big{(}\varphi^{0}(t)\big{)}\Big{)}\,\eta+\int_{\Gamma}\Big{(}\theta^{0}(t)-G^{\prime}\big{(}\psi^{0}(t)\big{)}\Big{)}\,{\eta_{\Gamma}}$
(3.56)

for all $t\in[0,T]\setminus\mathcal{N}$ and all test functions
$(\eta,{\eta_{\Gamma}})\in{\cal V}_{0}$.

Let now $t\in[0,T]\setminus\mathcal{N}$ be arbitrary. We infer from (3.2) that
the pair $\big{(}\varphi^{0}(t),\psi^{0}(t)\big{)}$ is a weak solution of the
bulk-surface elliptic problem

$\displaystyle-\Delta\varphi^{0}(t)$ $\displaystyle=f(t)$
$\displaystyle\quad\text{in $\Omega$},$ (3.57a)
$\displaystyle-\Delta_{\Gamma}\psi^{0}(t)+\partial_{\bf n}\varphi^{0}(t)$
$\displaystyle=g(t)$ $\displaystyle\quad\text{on $\Gamma$},$ (3.57b)
$\displaystyle\varphi^{0}(t)|_{\Gamma}$ $\displaystyle=\psi^{0}(t)$
$\displaystyle\quad\text{on $\Gamma$},$ (3.57c)

where

$\displaystyle
f(t):=\mu^{0}(t)-F^{\prime}\big{(}\varphi^{0}(t)\big{)}\quad\text{and}\quad
g(t):=\theta^{0}(t)-G^{\prime}\big{(}\psi^{0}(t)\big{)}.$

Let us first consider the case $\ell=2$. As we assumed that the growth
conditions in (R1) are fulfilled with $p\leq 4$, we have

$\displaystyle\left\|F^{\prime}\big{(}\varphi^{0}(t)\big{)}\right\|_{H}$
$\displaystyle\leq C+C\left\|\varphi^{0}\right\|_{L^{6}(\Omega)}^{3}\leq C,$
(3.58)
$\displaystyle\left\|G^{\prime}\big{(}\psi^{0}(t)\big{)}\right\|_{{H}_{\Gamma}}$
$\displaystyle\leq C+C\left\|\psi^{0}\right\|_{L^{2(q-1)}(\Gamma)}^{q-1}\leq
C.$ (3.59)

Hence, applying regularity theory for elliptic problems with bulk-surface
coupling (see [48, Theorem 3.3]), we find that
$\big{(}\varphi^{0}(t),\psi^{0}(t)\big{)}\in H^{2}(\Omega)\times
H^{2}(\Gamma)$ with

$\displaystyle\left\|\varphi^{0}(t)\right\|_{H^{2}(\Omega)}^{2}+\left\|\psi^{0}(t)\right\|_{H^{2}(\Gamma)}^{2}\leq
C\left\|f^{0}(t)\right\|_{H}^{2}+C\left\|g^{0}(t)\right\|_{{H}_{\Gamma}}^{2}$
$\displaystyle\quad\leq
C+C\left\|\mu^{0}(t)\right\|_{H}^{2}+C\left\|\theta^{0}(t)\right\|_{{H}_{\Gamma}}^{2}.$

Since $\mu^{0}\in L^{2}(0,T;H)$ and $\theta^{0}\in L^{2}(0,T;{H}_{\Gamma})$,
this proves (2.18a).

We now consider the case $\ell=3$. Recalling that the growth conditions in
(R1) are fulfilled with $p\leq 4$, we use (3.2) to derive the estimates

$\displaystyle\left\|F^{\prime\prime}\big{(}\varphi^{0}(t)\big{)}\nabla\varphi^{0}(t)\right\|_{\bf
H}$ $\displaystyle\leq C\left\|\nabla\varphi^{0}(t)\right\|_{{\bf
H}}+C\left\||\varphi^{0}(t)|^{2}\,\nabla\varphi^{0}(t)\right\|_{{\bf H}}$
$\displaystyle\leq
C+C\left\|\varphi^{0}(t)\right\|_{L^{6}(\Omega)}^{2}\,\left\|\nabla\varphi^{0}(t)\right\|_{L^{6}(\Omega)}$
$\displaystyle\leq C+C\left\|\varphi^{0}(t)\right\|_{H^{2}(\Omega)}$

and

$\displaystyle\left\|G^{\prime\prime}\big{(}\psi^{0}(t)\big{)}\nabla_{\Gamma}\psi^{0}(t)\right\|_{{\bf
H}_{\Gamma}}$ $\displaystyle\leq C\left\|\nabla\psi^{0}(t)\right\|_{{\bf
H}_{\Gamma}}+C\left\||\psi^{0}(t)|^{q-2}\,\nabla\psi^{0}(t)\right\|_{{\bf
H}_{\Gamma}}$ $\displaystyle\leq
C+C\left\|\psi^{0}(t)\right\|_{L^{{2(q-2)}}}^{q-2}\left\|\nabla\psi^{0}(t)\right\|_{L^{4}(\Gamma)}$
$\displaystyle\leq C+C\left\|\psi^{0}(t)\right\|_{H^{2}(\Gamma)}.$

In combination with (3.58) and (3.59), these estimates directly imply

$\displaystyle\left\|F^{\prime}\big{(}\varphi^{0}(t)\big{)}\right\|_{V}$
$\displaystyle\leq C+C\left\|\varphi^{0}\right\|_{H^{2}(\Omega)},$
$\displaystyle\left\|G^{\prime}\big{(}\psi^{0}(t)\big{)}\right\|_{{V}_{\Gamma}}$
$\displaystyle\leq C+C\left\|\psi^{0}(t)\right\|_{H^{2}(\Gamma)}.$

Now, applying regularity theory for elliptic problems with bulk-surface
coupling (see [48, Theorem 3.3]), we infer

$\displaystyle\left\|\varphi^{0}(t)\right\|_{H^{3}(\Omega)}^{2}+\left\|\psi^{0}(t)\right\|_{H^{3}(\Gamma)}^{2}\leq
C\left\|f^{0}(t)\right\|_{V}^{2}+C\left\|g^{0}(t)\right\|_{{V}_{\Gamma}}^{2}$
$\displaystyle\quad\leq
C+C\left\|\mu^{0}(t)\right\|_{V}^{2}+C\left\|\theta^{0}(t)\right\|_{{V}_{\Gamma}}^{2}+C\left\|\varphi^{0}(t)\right\|_{H^{2}(\Omega)}^{2}+C\left\|\psi^{0}(t)\right\|_{H^{2}(\Gamma)}^{2}.$

Recalling $\mu^{0}\in L^{2}(0,T;V)$, $\theta^{0}\in L^{2}(0,T;{V}_{\Gamma})$
and that (2.18a) with $\ell=2$ is already verified, this proves (2.18b) in the
case $\ell=3$. By means of Proposition A.1(b), we directly infer
$(\varphi^{0},\psi^{0})\in C^{0}([0,T];{\cal V}_{0})$. Moreover, via
interpolation between $L^{\infty}(0,T;{\cal V}_{0})$ and
$L^{2}(0,T;H^{3}(\Omega)\times H^{3}(\Gamma))$ (cf. Lemma 2.2), we further get

$(\varphi^{0},\psi^{0})\in L^{4}(0,T;H^{2}(\Omega)\times H^{2}(\Gamma)).$

This means that (2.19) is established. Eventually, a simple comparison
argument based on (2.13d) yields (2.20).

This means that all assertions are verified and thus, the proof is complete. ∎

### 3.3 Uniqueness of the weak solution for regular potentials

In this subsection, we are going to prove Theorem 2.7 for regular potentials
and $K\geq 0$. To prove the theorem, we use some ideas devised in [40].

###### Proof of Theorem 2.7.

In this proof, the the letter $C$ will denote generic positive constants that
may depend on $\Omega$, $T$, the initial data and the constants introduced in
(A1)–(A3). Such constants may also change their value from line to line. We
first introduce the following notation for the differences of the solution
components:

$\displaystyle{\boldsymbol{v}}:={\boldsymbol{v}}_{1}-{\boldsymbol{v}}_{2},\quad\varphi:=\varphi_{1}-\varphi_{2},\quad\mu:=\mu_{1}-\mu_{2},\quad\psi:=\psi_{1}-\psi_{2},\quad\theta:=\theta_{1}-\theta_{2}.$

Recall that $M,M_{\Gamma}$ and $\nu$ are assumed to be constant. We thus infer
that the quintuplet $({\boldsymbol{v}},\varphi,\mu,\psi,\theta)$ satisfies the
variational formulation

$\displaystyle
2\nu\int_{\Omega}\mathbf{D}{\boldsymbol{v}}:\mathbf{D}{\boldsymbol{w}}+\int_{\Omega}(\lambda(\varphi_{1})-\lambda(\varphi_{2})){\boldsymbol{v}}_{1}\cdot{\boldsymbol{w}}$
$\displaystyle\qquad+\int_{\Omega}\lambda(\varphi_{2}){\boldsymbol{v}}\cdot{\boldsymbol{w}}+\int_{\Gamma}(\gamma(\psi_{1})-\gamma(\psi_{2})){\boldsymbol{v}}_{1}\cdot{\boldsymbol{w}}+\int_{\Gamma}\gamma(\psi_{2}){\boldsymbol{v}}\cdot{\boldsymbol{w}}$
$\displaystyle\quad=-\int_{\Omega}(\varphi\nabla\mu_{1}+\varphi_{2}\nabla\mu)\cdot{\boldsymbol{w}}-\int_{\Gamma}(\psi\nabla_{\Gamma}\theta_{1}+\psi_{2}\nabla_{\Gamma}\theta)\cdot{\boldsymbol{w}},$
(3.60a)
$\displaystyle\mathopen{\langle}\partial_{t}\varphi,\zeta\mathclose{\rangle}_{V}-\int_{\Omega}(\varphi{\boldsymbol{v}}_{1}+\varphi_{2}{\boldsymbol{v}})\cdot\nabla\zeta+{M}\int_{\Omega}\nabla\mu\cdot\nabla\zeta=0,$
(3.60b)
$\displaystyle\mathopen{\langle}\partial_{t}\psi,{\zeta_{\Gamma}}\mathclose{\rangle}_{{V}_{\Gamma}}-\int_{\Gamma}(\psi{\boldsymbol{v}}_{1}+\psi_{2}{\boldsymbol{v}})\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}+{M_{\Gamma}}\int_{\Gamma}\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}{\zeta_{\Gamma}}=0$
(3.60c)

almost everywhere in $(0,T)$ for all test functions ${\boldsymbol{w}}\in{\bf
V}_{\sigma,{\bf n}}$, $\zeta\in V$, and ${\zeta_{\Gamma}}\in{V}_{\Gamma}$, and
the equations

$\displaystyle\mu$
$\displaystyle=-\Delta\varphi+F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\quad$
in $Q$, (3.61) $\displaystyle\theta$
$\displaystyle=-\Delta_{\Gamma}\psi+G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})+\partial_{\bf
n}\varphi\quad$ on $\Sigma$ (3.62)

are fulfilled in the strong sense due the higher regularities established in
Theorem 2.4 and Theorem 2.5.

We now test (3.60a) by ${\boldsymbol{v}}$, (3.60b) by $\varphi+\mu$, (3.60c)
by $\psi+\theta$, and add the resulting equations. After some cancellations
and rearrangements, we obtain

$\displaystyle
2\nu\left\|\mathbf{D}{\boldsymbol{v}}\right\|^{2}_{\mathbb{H}}+\int_{\Omega}\lambda(\varphi_{2})|{{\boldsymbol{v}}}|^{2}+\int_{\Gamma}\gamma(\psi_{2})|{{\boldsymbol{v}}}|^{2}+\mathopen{\langle}\partial_{t}\varphi,\varphi+\mu\mathclose{\rangle}_{V}$
$\displaystyle\qquad+{M}\left\|\nabla\mu\right\|^{2}_{{\bf
H}}+\mathopen{\langle}\partial_{t}\psi,\psi+\theta\mathclose{\rangle}_{{V}_{\Gamma}}+{M_{\Gamma}}\left\|\nabla_{\Gamma}\theta\right\|^{2}_{{\bf
H}_{\Gamma}}$
$\displaystyle\quad=-\int_{\Omega}(\lambda(\varphi_{1})-\lambda(\varphi_{2})){\boldsymbol{v}}_{1}\cdot{\boldsymbol{v}}-\int_{\Gamma}(\gamma(\psi_{1})-\gamma(\psi_{2})){\boldsymbol{v}}_{1}\cdot{\boldsymbol{v}}$
$\displaystyle\qquad-\int_{\Omega}\varphi\nabla\mu_{1}\cdot{\boldsymbol{v}}-\int_{\Gamma}\psi\nabla_{\Gamma}\theta_{1}\cdot{\boldsymbol{v}}$
$\displaystyle\qquad+\int_{\Omega}(\varphi{\boldsymbol{v}}_{1}+\varphi_{2}{\boldsymbol{v}})\cdot\nabla\varphi+\int_{\Omega}\varphi{\boldsymbol{v}}_{1}\cdot\nabla\mu$
$\displaystyle\qquad+\int_{\Gamma}(\psi{\boldsymbol{v}}_{1}+\psi_{2}{\boldsymbol{v}})\cdot\nabla_{\Gamma}\psi+\int_{\Gamma}\psi{\boldsymbol{v}}_{1}\cdot\nabla_{\Gamma}\theta$
$\displaystyle\qquad-{M}\int_{\Omega}\nabla\mu\cdot\nabla\varphi-{M_{\Gamma}}\int_{\Gamma}\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}\psi=:\sum_{i=1}^{10}I_{i}.$
(3.63)

We point out that, as a consequence of Theorem 2.4 and Theorem 2.5, it holds
that $(\varphi_{i},\psi_{i})\in L^{2}(0,T;(H^{3}(\Omega)\times
H^{3}(\Gamma))\cap{\cal V}_{K})$, $i=1,2$. Next, by using (3.61) and (3.62),
along with the chain rule formula in Proposition A.1, we observe that the
duality terms on the left-hand side can be reformulated as

$\displaystyle\mathopen{\langle}\partial_{t}\varphi,\varphi+\mu\mathclose{\rangle}_{V}+\mathopen{\langle}\partial_{t}\psi,\psi+\theta\mathclose{\rangle}_{{V}_{\Gamma}}$
$\displaystyle\quad=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}{\left\|\varphi\right\|^{2}_{H}+\left\|\psi\right\|^{2}_{{H}_{\Gamma}}}\Big{)}+\mathopen{\langle}(\partial_{t}\varphi,\partial_{t}\psi),\big{(}-\Delta\varphi,-\Delta_{\Gamma}\psi+\partial_{\bf
n}\varphi\big{)}\mathclose{\rangle}_{{\cal V}}$
$\displaystyle\qquad+\mathopen{\langle}\partial_{t}\varphi,F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\mathclose{\rangle}_{V}+\mathopen{\langle}\partial_{t}\psi,G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\mathclose{\rangle}_{{V}_{\Gamma}}$
$\displaystyle\quad=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}{\left\|\varphi\right\|^{2}_{V}+\left\|\psi\right\|^{2}_{{V}_{\Gamma}}}+\sigma(K)\left\|\psi-\varphi\right\|^{2}_{{H}_{\Gamma}}\Big{)}$
$\displaystyle\qquad+\mathopen{\langle}\partial_{t}\varphi,F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\mathclose{\rangle}_{V}+\mathopen{\langle}\partial_{t}\psi,G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\mathclose{\rangle}_{{V}_{\Gamma}}.$
(3.64)

Using (A2) and (A3) as well as (3.64), we deduce from (3.63) that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\left\|\varphi\right\|_{V}^{2}+\left\|\psi\right\|_{{V}_{\Gamma}}^{2}+\sigma(K)\left\|\psi-\varphi\right\|^{2}_{{H}_{\Gamma}}\Big{)}$
$\displaystyle\quad+2\nu\left\|\mathbf{D}{\boldsymbol{v}}\right\|^{2}_{\mathbb{H}}+\lambda_{1}\left\|{\boldsymbol{v}}\right\|_{{\bf
H}}^{2}+\gamma_{1}\left\|{\boldsymbol{v}}\right\|_{{\bf
H}_{\Gamma}}^{2}+{M}\left\|\nabla\mu\right\|^{2}_{{\bf
H}}+{M_{\Gamma}}\left\|\nabla_{\Gamma}\theta\right\|^{2}_{{\bf H}_{\Gamma}}$
$\displaystyle\qquad\leq-\mathopen{\langle}\partial_{t}\varphi,F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\mathclose{\rangle}_{V}-\mathopen{\langle}\partial_{t}\psi,G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\mathclose{\rangle}_{{V}_{\Gamma}}+\sum_{i=1}^{10}I_{i}.$

We now intend to control the terms $I_{i}$, $i=1,...,10$, by means of Hölder’s
inequality, Young’s inequality, the Lipschitz continuity of $\lambda$ and
$\gamma$, and integration by parts along with Sobolev’s embeddings and the
trace theorem. For a positive $\delta$ yet to be chosen, we derive the
following estimates:

$\displaystyle{I_{1}}$ $\displaystyle\leq
C\left\|\varphi\right\|_{{L^{4}(\Omega)}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Omega)}}\left\|{\boldsymbol{v}}\right\|_{{{\bf
H}}}\leq\delta\left\|{\boldsymbol{v}}\right\|^{2}_{{\bf
H}}+C_{\delta}\left\|{\boldsymbol{v}}_{1}\right\|^{2}_{{\bf
V}}\left\|\varphi\right\|_{V}^{2},$ $\displaystyle{I_{2}}$ $\displaystyle\leq
C\left\|\psi\right\|_{{L^{4}(\Gamma)}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Gamma)}}\left\|{\boldsymbol{v}}\right\|_{{\mathbf{H}_{\Gamma}}}\leq\delta\left\|{\boldsymbol{v}}\right\|^{2}_{{\bf{V}}}+C_{\delta}\left\|{\boldsymbol{v}}_{1}\right\|^{2}_{{\bf{V}}}\left\|\psi\right\|^{2}_{{V}_{\Gamma}},$
$\displaystyle{I_{3}+I_{4}}$
$\displaystyle=\int_{\Omega}\mu_{1}\nabla\varphi\cdot{\boldsymbol{v}}-\int_{\Gamma}\psi\nabla_{\Gamma}\theta_{1}\cdot{\boldsymbol{v}}$
$\displaystyle{\leq\left\|\mu_{1}\right\|_{{L^{4}(\Omega)}}\left\|\nabla\varphi\right\|_{{{\bf{H}}}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{L}}^{4}(\Omega)}}+\left\|\psi\right\|_{{L^{4}(\Gamma)}}\left\|\nabla_{\Gamma}\theta_{1}\right\|_{{\mathbf{H}_{\Gamma}}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{L}}^{4}(\Gamma)}}}$
$\displaystyle{\leq{C}\left\|\mu_{1}\right\|_{V}\left\|\varphi\right\|_{V}\left\|{\boldsymbol{v}}\right\|_{{\bf
V}}+C\left\|\theta_{1}\right\|_{{V}_{\Gamma}}\left\|\psi\right\|_{{V}_{\Gamma}}\left\|{\boldsymbol{v}}\right\|_{\bf
V}}$ $\displaystyle{\leq 2\delta\left\|{\boldsymbol{v}}\right\|^{2}_{\bf
V}+C_{\delta}\left\|\mu_{1}\right\|^{2}_{V}\left\|\varphi\right\|^{2}_{V}+C_{\delta}\left\|\theta_{1}\right\|_{{V}_{\Gamma}}^{2}\left\|\psi\right\|_{{V}_{\Gamma}}^{2},}$
$\displaystyle{I_{5}+I_{6}}$
$\displaystyle\leq(\left\|\varphi\right\|_{{L^{4}(\Omega)}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Omega)}}+\left\|\varphi_{2}\right\|_{{L^{4}(\Omega)}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{L}}^{4}(\Omega)}})\left\|\nabla\varphi\right\|_{{{\bf{H}}}}$
$\displaystyle\quad+\left\|\varphi\right\|_{L^{4}(\Omega)}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Omega)}}\left\|\nabla\mu\right\|_{{\bf
H}}$ $\displaystyle\leq\frac{M}{4}\left\|\nabla\mu\right\|_{{\bf
H}}^{2}+\delta\left\|{\boldsymbol{v}}\right\|_{{\bf
V}}^{2}+(C\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf{V}}}+C\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf{V}}}^{2}+C_{\delta}\left\|\varphi_{2}\right\|_{V}^{2})\left\|\varphi\right\|_{V}^{2},$
$\displaystyle{I_{7}+I_{8}}$
$\displaystyle\leq{(\left\|\psi\right\|_{{L^{4}(\Gamma)}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Gamma)}}+\left\|\psi_{2}\right\|_{{L^{4}(\Gamma)}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{L}}^{4}(\Gamma)}})\left\|\nabla_{\Gamma}\psi\right\|_{{\mathbf{H}_{\Gamma}}}}$
$\displaystyle\quad+\left\|\psi\right\|_{L^{4}(\Gamma)}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Gamma)}}\left\|\nabla_{\Gamma}\theta\right\|_{\mathbf{H}_{\Gamma}}$
$\displaystyle{\leq\frac{M_{\Gamma}}{4}\left\|\nabla_{\Gamma}\theta\right\|_{\mathbf{H}_{\Gamma}}^{2}+\delta\left\|{\boldsymbol{v}}\right\|_{{\bf
V}}^{2}+(C\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf{V}}}+C\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf{V}}}^{2}+C_{\delta}\left\|\psi_{2}\right\|_{{V}_{\Gamma}}^{2})\left\|\psi\right\|_{{V}_{\Gamma}}^{2},}$
$\displaystyle{I_{9}+I_{10}}$
$\displaystyle\leq{\frac{{M}}{4}\left\|\nabla\mu\right\|^{2}_{{\bf
H}}+\frac{{M_{\Gamma}}}{4}\left\|\nabla_{\Gamma}\theta\right\|^{2}_{{\bf
H}_{\Gamma}}+C\left\|\nabla\varphi\right\|^{2}_{{\bf
H}}+C\left\|\nabla_{\Gamma}\psi\right\|^{2}_{{\bf H}_{\Gamma}}.}$

Furthermore, the terms in the last line of (3.64) can be estimated by

$\displaystyle\big{|}\mathopen{\langle}\partial_{t}\varphi,F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\mathclose{\rangle}_{V}\big{|}+\big{|}\mathopen{\langle}\partial_{t}\psi,G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\mathclose{\rangle}_{{V}_{\Gamma}}\big{|}$
$\displaystyle\quad\leq\left\|\partial_{t}\varphi\right\|_{V^{*}}\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{V}+\left\|\partial_{t}\psi\right\|_{{V}_{\Gamma}^{*}}\left\|G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\right\|_{{V}_{\Gamma}}.$
(3.65)

By means of a comparison argument in (3.60b), we obtain

$\displaystyle\left\|\partial_{t}\varphi\right\|_{V^{*}}$
$\displaystyle=\sup_{\left\|\zeta\right\|_{{V}}\leq
1}|\mathopen{\langle}\partial_{t}\varphi,\zeta\mathclose{\rangle}_{V}|$
$\displaystyle\leq
C(\left\|\varphi\right\|_{{L^{4}(\Omega)}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{L}}^{4}(\Omega)}}+\left\|\varphi_{2}\right\|_{{L^{4}(\Omega)}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{L}}^{4}(\Omega)}}+\left\|\nabla\mu\right\|_{{\bf
H}})$ $\displaystyle\leq
C(\left\|\varphi\right\|_{V}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{V}}}}+\left\|\varphi_{2}\right\|_{{V}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{V}}}}+\left\|\nabla\mu\right\|_{{\bf
H}}).$

Similarly, using (3.60c), we derive the estimate

$\displaystyle\left\|\partial_{t}\psi\right\|_{{V}_{\Gamma}^{*}}=\sup_{\left\|\zeta_{\Gamma}\right\|_{{{V}_{\Gamma}}}\leq
1}|\mathopen{\langle}\partial_{t}\psi,\zeta_{\Gamma}\mathclose{\rangle}_{{V}_{\Gamma}}|\leq
C(\left\|\psi\right\|_{{V}_{\Gamma}}\left\|{\boldsymbol{v}}_{1}\right\|_{{{\bf{V}}}}+\left\|\psi_{2}\right\|_{{{V}_{\Gamma}}}\left\|{\boldsymbol{v}}\right\|_{{{\bf{V}}}}+\left\|\nabla_{\Gamma}\theta\right\|_{{\bf
H}_{\Gamma}}).$

By employing equation (3.60b) as well as (Renumi), we can bound the norm on
the right-hand side of (3.65) as follows:

$\displaystyle\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{V}^{2}=\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{H}^{2}+\left\|F^{\prime\prime}(\varphi_{1})\nabla\varphi_{1}-F^{\prime\prime}(\varphi_{2})\nabla\varphi_{2}\right\|_{{\bf{H}}}^{2}$
$\displaystyle\quad=\int_{\Omega}|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})|^{2}+\int_{\Omega}|F^{\prime\prime}(\varphi_{1})\nabla\varphi|^{2}+\int_{\Omega}|F^{\prime\prime}(\varphi_{1})-F^{\prime\prime}(\varphi_{2})|^{2}|\nabla\varphi_{2}|^{2}$
$\displaystyle\quad\leq\int_{\Omega}\Big{|}\int_{0}^{1}F^{\prime\prime}\big{(}s\varphi_{1}+(1-s)\varphi_{2}\big{)}{\rm
ds}\Big{|}^{2}\varphi^{2}\;+\;C(\left\|\varphi_{1}\right\|_{{L^{\infty}(\Omega)}}^{2(p-2)}+1)\left\|\nabla\varphi\right\|^{2}_{{\bf
H}}$
$\displaystyle\qquad+\int_{\Omega}\Big{|}\int_{0}^{1}F^{(3)}\big{(}s\varphi_{1}+(1-s)\varphi_{2}\big{)}{\rm
ds}\Big{|}^{2}\varphi^{2}\,|\nabla\varphi_{2}|^{2}$ $\displaystyle\quad\leq
C(\left\|\varphi_{1}\right\|^{2(p-2)}_{{L^{3(p-2)}(\Omega)}}+\left\|\varphi_{2}\right\|^{2(p-2)}_{{L^{3(p-2)}(\Omega)}}+1)\left\|\varphi\right\|_{V}^{2}+C(\left\|\varphi_{1}\right\|_{{L^{\infty}(\Omega)}}^{2(p-2)}+1)\left\|\varphi\right\|^{2}_{V}$
$\displaystyle\qquad+C(\left\|\varphi_{1}\right\|^{2(p-3)}_{{L^{12(p-3)}(\Omega)}}+\left\|\varphi_{2}\right\|^{2(p-3)}_{{L^{12(p-3)}(\Omega)}}+1)\left\|\varphi_{2}\right\|_{W^{1,4}(\Omega)}^{2}\left\|\varphi\right\|_{V}^{2}.$
(3.66)

We now recall the restrictions on $p$ and $q$ demanded in (2.21). In
particular, we have $p\leq 6$ if $d=3$. In the case $d=2$ we assume, without
loss of generality, that $p\geq 5$. Using Agmon’s inequality as well as
interpolation between Sobolev spaces (see Lemma 2.2), we derive the estimates

$\displaystyle\left\|\varphi_{1}\right\|_{{L^{\infty}(\Omega)}}^{2(p-2)}\leq
C\left\|\varphi_{1}\right\|_{H^{s}(\Omega)}^{2(p-2)}\leq
C\left\|\varphi_{1}\right\|_{H^{1}(\Omega)}^{2p-8}\left\|\varphi_{1}\right\|_{H^{2}(\Omega)}^{4}\leq
C\left\|\varphi_{1}\right\|_{H^{2}(\Omega)}^{4}$ $\displaystyle\text{for
$d=2$},$
$\displaystyle\left\|\varphi_{1}\right\|_{{L^{\infty}(\Omega)}}^{2(p-2)}\leq
C\left\|\varphi_{1}\right\|_{H^{1}(\Omega)}^{(p-2)}\left\|\varphi_{1}\right\|_{H^{2}(\Omega)}^{(p-2)}\leq
C\left\|\varphi_{1}\right\|_{H^{2}(\Omega)}^{4}$ $\displaystyle\text{for
$d=3$},$ $\displaystyle\left\|\varphi_{2}\right\|_{W^{1,4}(\Omega)}^{2}\leq
C\left\|\varphi_{2}\right\|_{H^{1}(\Omega)}^{\frac{1}{2}}\left\|\varphi_{2}\right\|_{H^{2}(\Omega)}^{\frac{3}{2}}\leq
C\left\|\varphi_{2}\right\|_{H^{2}(\Omega)}^{\frac{3}{2}}$
$\displaystyle\text{for $d=2,3$},$

where, in the first inequality, $s=\frac{2p}{2(p-2)}\in(1,2)$. We thus infer
from (3.3) that

$\displaystyle\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{V}^{2}\leq
C\Lambda\left\|\varphi\right\|_{V}^{2}$

with a the time-dependent function $\Lambda$ that is given by

$\displaystyle\Lambda$
$\displaystyle:=(1+\left\|\varphi_{1}\right\|^{4}_{H^{2}{(\Omega)}})+\big{(}1+\left\|\varphi_{2}\right\|^{\frac{3}{2}}_{H^{2}{(\Omega)}}\big{)}\sum_{i=1,2}\big{(}1+\left\|\varphi_{i}\right\|^{2(p-2)}_{{L^{3(p-2)}(\Omega)}}+\left\|\varphi_{i}\right\|^{2(p-3)}_{{L^{12(p-3)}(\Omega)}}\big{)}.$

From (2.15a) and (2.19), we know that $\varphi_{2}\in
L^{4}(0,T;H^{2}(\Omega))$.

In the case $d=2$, we simply have

$\displaystyle\varphi_{i}\in L^{\infty}(0,T;L^{3(p-2)}(\Omega))\cap
L^{\infty}(0,T;L^{12(p-3)}(\Omega)),\quad i=1,2,$

due to the Sobolev embedding $H^{1}(\Omega){\hookrightarrow}L^{r}(\Omega)$ for
all $r\in(1,\infty)$.

In the case $d=3$, we use interpolation between Sobolev spaces (Lemma 2.2) to
derive the embedding

$\displaystyle L^{\infty}(0,T;H^{1}(\Omega))\cap
L^{2}(0,T;H^{3}(\Omega)){\hookrightarrow}L^{\rho}(0,T;L^{\frac{6\rho}{\rho-8}}(\Omega))\quad\text{for
any $\rho>8$.}$ (3.67)

Since $\varphi_{i}\in L^{\infty}(0,T;H^{1}(\Omega))\cap
L^{2}(0,T;H^{3}(\Omega))$, $i=1,2$, we infer

$\displaystyle\varphi_{i}\in L^{16}(0,T;L^{12}(\Omega))\cap
L^{\frac{48}{5}}(0,T;L^{36}(\Omega)),\quad i=1,2,$

by choosing $\rho=16$ and $\rho=\frac{48}{5}$ in (3.67), respectively.

In summary, by means of Hölder’s inequality, we conclude

$\displaystyle t\mapsto{\Lambda(t)}\in L^{1}(0,T)\quad\text{for $d=2,3$.}$

Arguing in a similar fashion, and recalling (2.23) as well as the regularity
in Theorem 2.4, we find that

$\displaystyle\left\|G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\right\|_{{V}_{\Gamma}}^{2}$
$\displaystyle\quad\leq(\left\|\psi_{1}\right\|^{2(q-2)}_{{V}_{\Gamma}}+\left\|\psi_{2}\right\|^{2(q-2)}_{{V}_{\Gamma}}+1)\left\|\psi\right\|_{{V}_{\Gamma}}^{2}+C(\left\|\psi_{1}\right\|_{L^{\infty}(\Gamma)}^{2(q-2)}+1)\left\|\psi\right\|_{{V}_{\Gamma}}^{2}$
$\displaystyle\qquad+C\big{(}1+\left\|\psi_{1}\right\|_{{V}_{\Gamma}}^{{2(q-3)}}+\left\|\psi_{2}\right\|_{{V}_{\Gamma}}^{2(q-3)}\big{)}\left\|\psi_{2}\right\|_{H^{2}(\Gamma)}^{2}\left\|\psi\right\|^{2}_{{V}_{\Gamma}}$
$\displaystyle\quad\leq
C(\left\|\psi_{1}\right\|_{L^{\infty}(\Gamma)}^{2(q-2)}+\left\|\psi_{2}\right\|_{H^{2}(\Gamma)}^{2}+1)\left\|\psi\right\|_{{V}_{\Gamma}}^{2}.$

In view of (2.21), we assume, without loss of generality, that $q\geq 5$.
Recalling that the boundary $\Gamma$ is a $(d-1)$-dimensional submanifold of
${\mathbb{R}}^{d}$ with $d\in\\{2,3\\}$, we have
$H^{s}(\Gamma){\hookrightarrow}L^{\infty}(\Gamma)$ for every $s>1$. Hence, via
interpolation between Sobolev spaces (see Lemma 2.2) we obtain the estimate

$\displaystyle\left\|\psi_{1}\right\|_{L^{\infty}(\Gamma)}^{2(q-2)}\leq
C\left\|\psi_{1}\right\|_{H^{s}(\Gamma)}^{2(q-2)}\leq
C\left\|\psi_{1}\right\|_{{V}_{\Gamma}}^{2q-8}\left\|\psi_{1}\right\|_{H^{2}(\Gamma)}^{4}\leq
C\left\|\psi_{1}\right\|_{H^{2}(\Gamma)}^{4},$

where $s=\frac{2q}{2(q-2)}\in(1,2)$. We thus conclude that

$\displaystyle\left\|G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\right\|_{{V}_{\Gamma}}^{2}\leq
C\Theta\left\|\psi\right\|_{{V}_{\Gamma}}^{2}$

with a time-dependent function $\Theta$ that is given by

$\displaystyle
t\mapsto\Theta(t):=C(1+\left\|\psi_{1}(t)\right\|^{4}_{H^{2}(\Gamma)}+\left\|\psi_{2}(t)\right\|^{2}_{H^{2}(\Gamma)})\in
L^{1}(0,T).$

Therefore, upon collecting the above computations, the integral in (3.65) can
be estimated with the help of Young’ inequality as

$\displaystyle-\mathopen{\langle}\partial_{t}\varphi,F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\mathclose{\rangle}_{V}-\mathopen{\langle}\partial_{t}\psi,G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\mathclose{\rangle}_{{V}_{\Gamma}}$
$\displaystyle\quad\leq
C\left\|\partial_{t}\varphi\right\|_{V^{*}}\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{V}+C\left\|\partial_{t}\psi\right\|_{{V}_{\Gamma}^{*}}\left\|G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\right\|_{{V}_{\Gamma}}$
$\displaystyle\quad\leq\delta(\left\|\partial_{t}\varphi\right\|_{V^{*}}^{2}+\left\|\partial_{t}\psi\right\|_{{V}_{\Gamma}^{*}}^{2})+C_{\delta}(\left\|F^{\prime}(\varphi_{1})-F^{\prime}(\varphi_{2})\right\|_{V}^{2}+\left\|G^{\prime}(\psi_{1})-G^{\prime}(\psi_{2})\right\|_{{V}_{\Gamma}}^{2})$
$\displaystyle\quad\leq\delta C(\left\|\nabla\mu\right\|^{2}_{{\bf
H}}+\left\|\nabla_{\Gamma}\theta\right\|^{2}_{\mathbf{H}_{\Gamma}})+\delta
C\left\|\varphi_{2}\right\|_{L^{\infty}(0,T;V)}^{2}\left\|{\boldsymbol{v}}\right\|_{{\bf
V}}^{2}+C_{\delta}(\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf
V}}^{2}+\Lambda)\left\|\varphi\right\|_{V}^{2}$ $\displaystyle\qquad+\delta
C\left\|\psi_{2}\right\|_{L^{\infty}(0,T;{V}_{\Gamma})}^{2}\left\|{\boldsymbol{v}}\right\|_{{\bf
V}}^{2}+C_{\delta}(\left\|{\boldsymbol{v}}_{1}\right\|_{{\bf
V}}^{2}+\Theta)\left\|\psi\right\|_{{V}_{\Gamma}}^{2}$

for a constant $\delta>0$ yet to be chosen. Finally, we adjust
$\delta\in(0,1)$ in such a way that

$\displaystyle\delta\max\Big{\\{}4,C,C\left\|\varphi_{2}\right\|_{L^{\infty}(0,T;V)}^{2},C\left\|\psi_{2}\right\|_{L^{\infty}(0,T;{V}_{\Gamma})}^{2}\Big{\\}}\leq\frac{1}{4}\min\Big{\\{}{{M}},{{M_{\Gamma}}},{C_{P}(\nu,\gamma_{1})}\Big{\\}},$

where $C_{P}(\nu,\gamma_{1})$ is a Poincaré constant such that
$2\nu\left\|\mathbf{D}{\boldsymbol{v}}\right\|^{2}_{\mathbb{H}}+\gamma_{1}\left\|{\boldsymbol{v}}\right\|^{2}_{{\bf
H}_{\Gamma}}\geq
C_{P}(\nu,\gamma_{1})\left\|{\boldsymbol{v}}\right\|^{2}_{{\bf{V}}}$. Thus, we
integrate over time and employ Gronwall’s lemma to deduce that

$\displaystyle\left\|{\boldsymbol{v}}_{1}-{\boldsymbol{v}}_{2}\right\|_{L^{2}(0,T;{\bf{V}})}+\left\|\varphi_{1}-\varphi_{2}\right\|_{L^{\infty}(0,T;V)}+\left\|\nabla\mu_{1}-\nabla\mu_{2}\right\|_{L^{2}(0,T;{\bf{H}})}$
$\displaystyle\qquad+\left\|\psi_{1}-\psi_{2}\right\|_{L^{\infty}(0,T;{V}_{\Gamma})}+\left\|\nabla_{\Gamma}\theta_{1}-\nabla_{\Gamma}\theta_{2}\right\|_{L^{2}(0,T;\mathbf{H}_{\Gamma})}$
$\displaystyle\quad\leq
C(\left\|\varphi_{0,1}-\varphi_{0,2}\right\|_{V}+\left\|\psi_{0,1}-\psi_{0,2}\right\|_{{V}_{\Gamma}}).$

Finally, by a comparison argument in (3.61) and (3.62), we infer that
$(\mu,\theta)$ is bounded in $L^{2}(0,T;{\cal H})$ by the same right-hand side
as the above inequality. This leads to (2.24) and thus, the proof is complete.
∎

## 4 Analysis of the Cahn–Hilliard–Brinkman system with singular potentials

We are now dealing with the proof of the existence of weak solutions for
singular potentials. Our strategy is to approximate the convex parts of the
singular potentials $F$ and $G$ satisfying (S1) and (S2) by means of a Yosida
regularization. In this way, the approximate potentials are regular and
exhibit quadratic growth and we can thus use Theorem 2.4 and Theorem 2.5 to
obtain suitable approximate solutions. We then derive uniform estimates with
respect to the approximation parameter, and eventually pass to the limit. In
the forthcoming analysis, the splitting $F^{\prime}=\beta+\pi$ and
$G^{\prime}={\beta_{\Gamma}}+\pi_{\Gamma}$ from (S1) will be adopted.

### 4.1 Yosida regularization

As mentioned, we rely on a Yosida regularization to smooth the singular parts
of the potentials $F$ and $G$. For any $\varepsilon\in(0,1)$, we approximate
the maximal monotone graphs $\beta$ and ${\beta_{\Gamma}}$ by

$\displaystyle{\beta_{\varepsilon}}(r)$
$\displaystyle:=\frac{1}{\varepsilon}\Big{(}r-\big{(}I+\varepsilon\beta\big{)}^{-1}(r)\Big{)},\quad\beta_{\Gamma,\varepsilon}(r):=\frac{1}{\varepsilon}\Big{(}r-\big{(}I+\varepsilon{\beta_{\Gamma}}\big{)}^{-1}(r)\Big{)},\quad
r\in{\mathbb{R}}.$

Then, the condition in (2.26) implies that

$\bigl{|}\beta_{\varepsilon}(r)\bigr{|}\leq\kappa_{1}\bigl{|}\beta_{\Gamma,\varepsilon}(r)\bigr{|}+\kappa_{2}\quad\text{for
all $r\in\mathbb{R}$ and all $\varepsilon\in(0,1)$}$ (4.1)

(see, e.g., [15, Appendix]), where $\kappa_{1}$ and $\kappa_{2}$ are the
constants introduced in (2.26). Next, we define
$F_{\varepsilon}:=\widehat{\beta_{\varepsilon}}+\widehat{\pi},$
$G_{\varepsilon}:=\widehat{\beta}_{\Gamma,\varepsilon}+\widehat{\pi}_{\Gamma},$
where

$\displaystyle\widehat{\beta_{\varepsilon}}(r):=\int_{0}^{r}{\beta_{\varepsilon}}(s)\,{\rm
ds},\quad\widehat{\beta}_{\Gamma,\varepsilon}(r):=\int_{0}^{r}\beta_{\Gamma,\varepsilon}(s)\,{\rm
ds},\quad r\in{\mathbb{R}}.$

It is well-known that for every $r\in{\mathbb{R}}$,

$\displaystyle 0\leq\widehat{\beta_{\varepsilon}}(r)\leq\widehat{\beta}(r)$
$\displaystyle\quad\forall\varepsilon\in(0,1),\quad$
$\displaystyle\widehat{\beta_{\varepsilon}}(r)$
$\displaystyle\nearrow\widehat{\beta}(r)$ monotonically as $\varepsilon\to 0,$
(4.2a) $\displaystyle|{\beta_{\varepsilon}}(r)|\leq|\beta^{\circ}(r)|$
$\displaystyle\quad\forall\varepsilon\in(0,1),\quad$
$\displaystyle{\beta_{\varepsilon}}(r)$ $\displaystyle\to\beta^{\circ}(r)$ as
$\varepsilon\to 0$. (4.2b)

Analogous properties hold for $\beta_{\Gamma,\varepsilon}$. Moreover, owing to
the growth condition (2.25), $\widehat{\beta_{\varepsilon}}$ fulfills the
following growth condition:

$\displaystyle\begin{aligned} &\hbox{For every $M>0$ there exist $C_{M}>0$ and
$\varepsilon_{M}\in(0,1)$ such that}\\\ &\widehat{\beta_{\varepsilon}}(r)\geq
M\,r^{2}-C_{M}\quad\hbox{for every $r\in{\mathbb{R}}$ and every
$\varepsilon\in(0,\varepsilon_{M})$}\,.\end{aligned}$ (4.3)

This property is checked in detail in the paper [20, beginning of Section 3].
Obviously, as a consequence, a similar condition holds for
$\widehat{\beta}_{\Gamma,\varepsilon}$ since (4.1) entails that

$\displaystyle\widehat{\beta_{\varepsilon}}(r)\leq\kappa_{1}\widehat{\beta}_{\Gamma,\varepsilon}(r)+\kappa_{2}|r|\quad\text{for
every $r\in{\mathbb{R}},\ \varepsilon\in(0,1)$},$ (4.4)

thanks to ${\beta_{\varepsilon}}(0)=\beta_{\Gamma,\varepsilon}(0)=0$ and since
$\widehat{\beta_{\varepsilon}}$ and $\widehat{\beta}_{\Gamma,\varepsilon}$
have the same sign. Due to their construction by the Yosida approximation,
$\beta_{\varepsilon}$ and $\beta_{\Gamma,\varepsilon}$ are Lipschitz
continuous and have at most linear growth. Hence,
$\widehat{\beta}_{\varepsilon}$ and $\widehat{\beta}_{\Gamma,\varepsilon}$
have at most quadratic growth. Moreover, (4.3) and (4.4) along with the (at
most) quadratic growth of $\widehat{\pi}$ and ${\widehat{\pi}}_{\Gamma}$ (cf.
(S1)), imply that both $F_{\varepsilon}$ and $G_{\varepsilon}$ are bounded
from below by negative constants independent of $\varepsilon$. We can thus
assume, without loss of generality, that $F_{\varepsilon}$ and
$G_{\varepsilon}$ are nonnegative (otherwise, we add the modulus of their