[ beforeskip=.2em plus 1pt,]toclinesection

# Well-posedness of a bulk-surface
convective Cahn–Hilliard system
with dynamic boundary conditions

Patrik Knopf 111 Faculty for Mathematics, University of Regensburg, 93053
Regensburg, Germany
<EMAIL_ADDRESS>jonas.stange@ur.de).  Jonas Stange 111 Faculty for
Mathematics, University of Regensburg, 93053 Regensburg, Germany
<EMAIL_ADDRESS>jonas.stange@ur.de).

This is a preprint version of the paper. Please cite as:
P. Knopf, J. Stange, [Journal] xx:xx 000-000 (2023),
https://doi.org/...

Abstract

We consider a general class of bulk-surface convective Cahn–Hilliard systems
with dynamic boundary conditions. In contrast to classical Neumann boundary
conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for
dynamic changes of the contact angle between the diffuse interface and the
boundary, a convection-induced motion of the contact line as well as
absorption of material by the boundary. The coupling conditions for bulk and
surface quantities involve parameters $K,L\in[0,\infty]$, whose choice
declares whether these conditions are of Dirichlet, Robin or Neumann type. We
first prove the existence of a weak solution to our model in the case
$K,L\in(0,\infty)$ by means of a Faedo–Galerkin approach. For all other cases,
the existence of a weak solution is then shown by means of the asymptotic
limits, where $K$ and $L$ are sent to zero or to infinity, respectively.
Eventually, we establish higher regularity for the phase-fields, and we prove
the uniqueness of weak solutions given that the mobility functions are
constant.
Keywords: convective Cahn–Hilliard equation, bulk-surface interaction, dynamic
boundary conditions, dynamic contact angle, moving contact line.
Mathematics Subject Classification: 35K35, 35D30, 35A01, 35A02, 35Q92,

###### Contents

  1. 1 Introduction
  2. 2 Notation, assumptions and preliminaries
    1. 2.1 Notation
    2. 2.2 Assumptions
    3. 2.3 Preliminaries
  3. 3 Main results
  4. 4 Existence of weak solutions in the case $K,L\in(0,\infty)$
  5. 5 Asymptotic limits and existence of weak solutions to the limit models
    1. 5.1 The limit $K\rightarrow 0$ and the existence of a weak solution if $(K,L)\in\\{0\\}\times(0,\infty)$
    2. 5.2 The limit $K\rightarrow\infty$ and the existence of a weak solution if $(K,L)\in\\{\infty\\}\times(0,\infty)$
    3. 5.3 The limit $L\rightarrow 0$ and the existence of a weak solution if $(K,L)\in[0,\infty]\times\\{0\\}$
    4. 5.4 The limit $L\rightarrow\infty$ and the existence of a weak solution if $(K,L)\in[0,\infty]\times\\{\infty\\}$
  6. 6 Higher regularity, continuous dependence and uniqueness
    1. 6.1 Higher spatial regularity for the phase-fields
    2. 6.2 Continuous dependence and uniqueness
  7. Appendix: Some calculus for bulk-surface function spaces

## 1 Introduction

In recent times, the description of immiscible two-phase flows in a bounded
domain by diffuse-interface models has become very popular. This is especially
because those systems can usually be handled more easily in terms of
mathematical analysis than their sharp-interface counterparts. In such models,
the location of the two fluids inside the container is represented by an order
parameter, the so-called phase-field. The time evolution of this phase-field
function is often described by a convective Cahn–Hilliard equation. There, the
velocity field of the mixture enters via the material derivative of the phase-
field. The time evolution of the velocity field is usually described by a
fluid equation, e.g., the Navier–Stokes equation. Very frequently used
Navier–Stokes–Cahn–Hilliard models for two-phase flows are the Model H (see
[25, 27]), which covers the case where both fluids have the same individual
density, and the Abels–Garcke–Grün model (AGG model) (see [1]), which is even
capable of describing the situation where both fluids have different (i.e.,
unmatched) densities.

For the Model H and the AGG model, the standard choice are homogeneous Neumann
boundary conditions for the Cahn–Hilliard quantities (i.e., the phase-field
and the chemical potential) as well as a no-slip boundary condition for the
velocity field. However, these choices lead to some crucial limitations of
these models:

  * •

The contact angle between the diffuse interface and the boundary of the domain
is fixed at ninety degrees at all times, which is unrealistic for many real-
world applications.

  * •

The motion of the contact line, where the diffuse interface intersects the
boundary of the domain, is driven only by diffusion. This means that a motion
of the contact line caused by convection is not taken into account.

  * •

No transfer of material between the bulk and the boundary of the domain is
allowed. Therefore, any absorption of material by the boundary cannot be
described.

For a more detailed discussion of these issues, we refer the reader to [23].

To overcome these limitations, a new Navier–Stokes–Cahn–Hilliard model with
dynamic boundary conditions was derived in [23]. It can be regarded as an
extension of the AGG model and is capable of describing two-phase flows with
unmatched densities. In the case of matched densities, some first analytic
results (namely the existence of weak solution and their uniqueness under
certain additional assumptions) were presented in [23]. The case of unmatched
densities, however, is much more involved. This is mainly because in the
Navier–Stokes equation, the density function then depends on the phase-field,
and a further flux term related to the interfacial motion will appear. In the
case of unmatched densities, at least for certain parameter choices in the
dynamic boundary conditions, the existence of a weak solution to the model
introduced in [23] was shown in [17] by an implicit time discretization
scheme. A different strategy to prove the existence of weak solutions to the
model proposed in [23], in a unified framework for all parameter choices in
the dynamic boundary conditions, is to first analyze the underlying bulk-
surface convective Cahn–Hilliard subsystem separately. Using this information,
a weak solution to the full Navier–Stokes–Cahn–Hilliard system is then to be
constructed by means of a suitable fixed point argument.

This motivates us to investigate the following bulk-surface convective
Cahn–Hilliard system with dynamic boundary conditions:

$\displaystyle\partial_{t}\phi+\textnormal{div}(\phi\boldsymbol{v})=\textnormal{div}(m_{\Omega}(\phi)\nabla\mu)$
$\displaystyle\text{in}\ Q,$ (1.1a)
$\displaystyle\mu=-\epsilon\Delta\phi+\epsilon^{-1}F^{\prime}(\phi)$
$\displaystyle\text{in}\ Q,$ (1.1b)
$\displaystyle\partial_{t}\psi+\textnormal{div}_{\Gamma}(\psi\boldsymbol{v}_{\tau})=\textnormal{div}_{\Gamma}(m_{\Gamma}(\psi)\nabla_{\Gamma}\theta)-\beta
m_{\Omega}(\phi)\partial_{\mathbf{n}}\mu$ $\displaystyle\text{on}\ \Sigma,$
(1.1c)
$\displaystyle\theta=-\epsilon_{\Gamma}\kappa\Delta_{\Gamma}\psi+\epsilon_{\Gamma}^{-1}G^{\prime}(\psi)+\alpha\epsilon\partial_{\mathbf{n}}\phi$
$\displaystyle\text{on}\ \Sigma,$ (1.1d) $\displaystyle\begin{cases}\epsilon
K\partial_{\mathbf{n}}\phi=\alpha\psi-\phi&\text{if}\ K\in[0,\infty),\\\
\partial_{\mathbf{n}}\phi=0&\text{if}\ K=\infty\end{cases}$
$\displaystyle\text{on}\ \Sigma,$ (1.1e)
$\displaystyle\begin{cases}Lm_{\Omega}(\phi)\partial_{\mathbf{n}}\mu=\beta\theta-\mu&\text{if}\
L\in[0,\infty),\\\ m_{\Omega}(\phi)\partial_{\mathbf{n}}\mu=0&\text{if}\
L=\infty\end{cases}$ $\displaystyle\text{on}\ \Sigma,$ (1.1f)
$\displaystyle\phi|_{t=0}=\phi_{0}$ $\displaystyle\text{in}\ \Omega,$ (1.1g)
$\displaystyle\psi|_{t=0}=\psi_{0}$ $\displaystyle\text{on}\ \Gamma.$ (1.1h)

Here, $\Omega\subset\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ is a bounded domain
with boundary $\Gamma\coloneqq\partial\Omega$, $T>0$ is a prescribed final
time, and we set $Q\coloneqq\Omega\times(0,T)$ and
$\Sigma\coloneqq\Gamma\times(0,T)$. The outward unit normal vector field on
$\Gamma$ is denoted by $\mathbf{n}$. For any vector field $\boldsymbol{w}$,
the subscript $\tau$ denotes its tangential component, i.e.
$\boldsymbol{w}_{\tau}=\boldsymbol{w}-(\boldsymbol{w}\cdot\mathbf{n})\mathbf{n}$.
Moreover, $\nabla_{\Gamma}$, $\textnormal{div}_{\Gamma}$ and $\Delta_{\Gamma}$
denote the surface gradient, the surface divergence, and the Laplace-Beltrami
operator on $\Gamma$, respectively.

The function $\phi:Q\rightarrow\mathbb{R}$ is the bulk phase-field. It is an
order parameter, which represents the distribution of the two immiscible
materials within the domain $\Omega$. Similarly, the surface phase-field
$\psi:\Sigma\rightarrow\mathbb{R}$ represents the distribution of two
materials on the surface. The functions $F$ and $G$ are double-well
potentials, which lead to the effect that $\phi(t)$ and $\psi(t)$ will attain
values close to $\pm 1$ in most parts of $\Omega$ and $\Gamma$ (which
correspond to the pure phases of the materials). In the remaining intermediate
regions, which are called diffuse interfaces, $\phi$ and $\psi$ are expected
to exhibit a continuous transition between the values $-1$ and $1$. The
functions $\mu:Q\rightarrow\mathbb{R}$ and
$\theta:\Sigma\rightarrow\mathbb{R}$ represent the bulk chemical potential and
the surface chemical potential, respectively. The non-negative, scalar
functions $m_{\Omega}(\phi)$ and $m_{\Gamma}(\psi)$ are called mobilities.
They depend on the phase-fields and describe where diffusion processes occur.
Moreover, the function $\boldsymbol{v}:Q\rightarrow\mathbb{R}^{d}$ is a
prescribed velocity field that corresponds to the flow of the two materials in
the bulk. Furthermore, $\epsilon>0$ is a parameter related to the thickness of
the diffuse interface in the bulk, whereas $\epsilon_{\Gamma}>0$ corresponds
to the width of the diffuse interface on the boundary. The parameter
$\kappa>0$ acts as a weight for the surface Dirichlet energy
$\psi\mapsto\int_{\Gamma}\left|\nabla_{\Gamma}\psi\right|^{2}\;\mathrm{d}\Gamma$,
which has a smoothing effect on the phase separation at the boundary.

The time evolution of $(\phi,\mu)$ is determined by the bulk convective
Cahn–Hilliard subsystem $\big{(}\eqref{CH:1},\eqref{CH:2}\big{)}$, whereas the
evolution of $(\psi,\theta)$ is described by the surface convective
Cahn–Hilliard subsystem $\big{(}\eqref{CH:3},\eqref{CH:4}\big{)}$, which is
coupled to the bulk by expressions involving the normal derivatives
$\partial_{\mathbf{n}}\phi$ and $\partial_{\mathbf{n}}\mu$. The bulk and
surface quantities are further coupled by the boundary conditions (1.1e) and
(1.1f), which depend on parameters $K,L\in[0,\infty]$ and
$\alpha,\beta\in\mathbb{R}$.

In (1.1e) and (1.1f), the parameters $K,L\in[0,\infty]$ are used to
distinguish different cases, each corresponding to a certain solution
behaviour related to a physical phenomenon. The case that has been studied
most extensively in the literature is $K=0$. In this case, (1.1e) is to be
interpreted as the Dirichlet type boundary condition $\phi=\alpha\psi$ on
$\Sigma$. This choice makes particular sense along with $\alpha=1$, if the
materials on the boundary are simply considered as an extension of those in
the bulk. Originally, the boundary condition (1.1f) was introduced in the non-
convective case (i.e., $\boldsymbol{v}\equiv 0$) in the following literature:

  * •

The choice $L=0$ was proposed in [14] and [24]. Then, (1.1f) can be restated
as the Dirichlet condition $\mu=\beta\theta$ on $\Sigma$, which means that the
chemical potentials $\mu$ and $\theta$ are always in a chemical equilibrium.
In this case, a rapid transfer of material between bulk and boundary can be
expected (see, e.g., [29]).

  * •

The choice $L=\infty$ was introduced in [32]. In this case, (1.1f) is a
homogeneous Neumann boundary condition on $\mu$, which means that the mass
flux between bulk and surface is zero. Consequently, no transfer of material
between bulk and surface will occur.

  * •

The choice $L\in(0,\infty)$ was first used in [29] to interpolate between the
extreme cases $L=0$ and $L=\infty$. In this case, a transfer of material
between bulk and surface will occur, and the number $L^{-1}$ is related to the
rate of absorption of bulk material by the boundary (cf. [29]).

In the convective case (i.e., with a nontrivial velocity field
$\boldsymbol{v}$), the boundary condition (1.1f) was also used in the
Navier–Stokes–Cahn–Hilliard model derived in [23] and in the
Cahn–Hilliard–Brinkman model studied in [12].

However, we also want to cover the case, where the materials on the boundary
are not the same as those in the bulk. For instance, this is the case if the
bulk materials are transformed at the boundary by a chemical reaction. In this
case, the surface phase-field might not be proportional to the trace of the
bulk phase-field, so the choice $K=0$ would not be appropriate. In our model,
this is taken into account by the choice $K\in(0,\infty]$. In the case
$K=\infty$, (1.1f) degenerates to a homogeneous Neumann boundary condition for
the phase-field. For most applications, this might not be the preferred choice
as it is known that such a Neumann boundary condition enforces the diffuse
interface to always intersect the boundary at a perfect angle of ninety
degrees. In fact, this is one of the aforementioned limitations we usually
want to overcome by the usage of dynamic boundary conditions. However, we
include the case $K=\infty$ for the sake of completeness. In the case
$K\in(0,\infty)$, (1.1f) can be regarded as a Robin type boundary condition.
It is suitable to describe a scenario, where $\psi$ and the trace of $\phi$
are not proportional, but it still allows for dynamical changes of the contact
angle between the diffuse interface and the boundary. In the context of bulk-
surface Cahn–Hilliard equations, condition (1.1f) with $K\in(0,\infty)$ was
used in [28] (in the non-convective case) and in [12] (in the convective
case). In [28] and [12], it was further shown that the limit $K\to 0$ can be
used to recover the boundary condition (1.1f) with $K=0$.

We further want to point out that including the case $K\in(0,\infty)$ also
helps our mathematical analysis. This is because the existence of a weak
solution to (1.1) can be shown by a suitable Faedo–Galerkin scheme only in the
cases, where the boundary conditions are of the same type (i.e., $K=L=0$,
$K=L=\infty$ or $0<K,L<\infty$), as otherwise, the spaces of admissible test
functions in the weak formulation do not match. In this paper, we will first
construct a weak solution of (1.1) in the case $0<K,L<\infty$. Then, we prove
the existence of a weak solution in the remaining cases by sending the
parameters $K$ and $L$ to zero or to infinity, respectively.

Let us now discuss some important properties of our model. The system (1.1) is
associated with the total energy

$\displaystyle\begin{split}E_{K}(\phi,\psi)&=\int_{\Omega}\frac{\epsilon}{2}\left|\nabla\phi\right|^{2}+\epsilon^{-1}F(\phi)\;\mathrm{d}x+\int_{\Gamma}\frac{\epsilon_{\Gamma}\kappa}{2}\left|\nabla_{\Gamma}\psi\right|^{2}+\epsilon_{\Gamma}^{-1}G(\psi)\;\mathrm{d}\Gamma\\\
&\quad+\mathds{h}(K)\int_{\Gamma}\frac{1}{2}\left|\alpha\psi-\phi\right|^{2}\;\mathrm{d}\Gamma,\end{split}$
(1.2)

where the function

$\displaystyle\mathds{h}(K)\coloneqq\begin{cases}K^{-1},&\text{if
}K\in(0,\infty),\\\ 0,&\text{if }K\in\\{0,\infty\\}\end{cases}$

is used to distinguish the different cases corresponding to the choice of $K$.
Sufficiently regular solutions of the system (1.1) satisfy the mass
conversation law

$\displaystyle\begin{dcases}\beta\int_{\Omega}\phi(t)\;\mathrm{d}x+\int_{\Gamma}\psi(t)\;\mathrm{d}\Gamma=\beta\int_{\Omega}\phi_{0}\;\mathrm{d}x+\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma,&\textnormal{if
}L\in[0,\infty),\\\
\int_{\Omega}\phi(t)\;\mathrm{d}x=\int_{\Omega}\phi_{0}\;\mathrm{d}x\quad\textnormal{and}\quad\int_{\Gamma}\psi(t)\;\mathrm{d}\Gamma=\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma,&\textnormal{if
}L=\infty\end{dcases}$ (1.3)

for all $t\in[0,T]$. As mentioned above, this means that a transfer of
material between bulk and surface is allowed only in the cases
$L\in[0,\infty)$. Moreover, sufficiently regular solutions satisfy the energy
identity

$\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}E_{K}(\phi,\psi)&=\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\mu\;\mathrm{d}x+\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta\;\mathrm{d}\Gamma-\int_{\Omega}m_{\Omega}(\phi)\left|\nabla\mu\right|^{2}\;\mathrm{d}x\\\
&\qquad-\int_{\Gamma}m_{\Gamma}(\psi)\left|\nabla_{\Gamma}\theta\right|^{2}\;\mathrm{d}\Gamma-\mathds{h}(L)\int_{\Gamma}(\beta\theta-\mu)^{2}\;\mathrm{d}\Gamma\\\
\end{split}$ (1.4)

for all $t\in[0,T]$. We point out that in the non-convective case (i.e.,
$\boldsymbol{v}=0$), the right-hand side of (1.4) is clearly non-positive.
This means that the energy dissipates over the course of time and the terms on
the right-hand side of (1.4) can be interpreted as the dissipation rate. In
the convective case (i.e., $\boldsymbol{v}$ is nontrivial), the energy
identity (1.4) does, in general, not imply dissipation of the energy. However,
if the velocity field is not just prescribed but determined by a Navier–Stokes
equation (cf. [23]) or a Brinkman/Stokes equation (cf. [12]), an energy
dissipation law for the corresponding total energy can be obtained.

#### A brief overview of related literature.

In the non-convective case, we refer to [19, 20, 8, 7, 6, 13, 37] for
analytical results and to [34, 35, 26, 33, 2, 3] for numerical results on the
Cahn–Hilliard equation with a dynamic boundary condition of Cahn–Hilliard
type. The nonlocal Cahn–Hilliard equation with a nonlocal dynamic boundary
condition of Cahn–Hilliard type was proposed and analyzed in [31]. Results on
the Cahn–Hilliard equation with dynamic boundary conditions of second-order
(e.g., of Allen–Cahn or Wentzell type) can be found, for instance, in [5, 9,
38, 39, 42, 18, 21, 4].

The convective Cahn–Hilliard equation with dynamic boundary conditions was
analyzed, for instance, in [10, 11, 22], and also in [15, 16, 23, 17] as part
of a system in which the velocity field is described by an additional fluid
equation.

For more information on the Cahn–Hilliard equation with classical homogeneous
Neumann boundary conditions or dynamic boundary conditions, we refer to the
recent review paper [41] as well as the book [36].

#### Structure of this paper.

In Section 2 we collect some notation, assumptions, preliminaries and
important tools. After introducing the notion of weak solutions of (1.1), our
main results are stated in Section 3. In Section 4, we construct a weak
solution in the case $(K,L)\in(0,\infty)^{2}$ via a Faedo–Galerkin approach.
Afterwards, in Section 5, we investigate the asymptotic limits $K\rightarrow
0$, $K\rightarrow\infty$, $L\rightarrow 0$ and $L\rightarrow\infty$, which
prove the existence of weak solutions of (1.1) in the limit cases. In Section
6, under suitable additional assumptions, we establish higher spatial
regularity for the phase-fields in the context of weak solutions. In the case
of constant mobilities, we eventually prove the continuous dependence of weak
solutions on the velocity field and the initial data. In particular, this
continuous dependence result directly entails the uniqueness of the weak
solution.

## 2 Notation, assumptions and preliminaries

In this section, we introduce some notation, assumptions and preliminaries
that are supposed to hold throughout the remainder of this paper.

### 2.1 Notation

Let us first introduce some basic notation.

  1. (N1)

$\mathbb{N}$ denotes the set of natural numbers excluding zero, whereas
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$.

  2. (N2)

Let $\Omega\subset\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ be a bounded Lipschitz
domain in $\mathbb{R}^{d}$, and let $\Gamma\coloneqq\partial\Omega$ denote its
boundary. For any $s\geq 0$ and $p\in[1,\infty]$, the Lebesgue and Sobolev
spaces for functions mapping from $\Omega$ to $\mathbb{R}$ are denoted as
$L^{p}(\Omega)$ and $W^{s,p}(\Omega)$. We write $\|\cdot\|_{L^{p}(\Omega)}$
and $\|\cdot\|_{W^{s,p}(\Omega)}$ to denote the standard norms on these
spaces. In the case $p=2$, we use the notation
$H^{s}(\Omega)=W^{s,2}(\Omega)$. In particular, $H^{0}(\Omega)$ can be
identified with $L^{2}(\Omega)$. The Lebesgue and Sobolev spaces on $\Gamma$
are denoted by $L^{p}(\Gamma)$ and $W^{s,p}(\Gamma)$ along with the
corresponding norms $\|\cdot\|_{L^{p}(\Gamma)}$ and
$\|\cdot\|_{W^{s,p}(\Gamma)}$, respectively. For vector-valued functions
mapping from $\Omega$ into $\mathbb{R}^{d}$, we use the notation
$\mathbf{L}^{p}(\Omega)$, $\mathbf{W}^{s,p}(\Omega)$ and
$\mathbf{H}^{s}(\Omega)$. For any real numbers $s\geq 0$ and $p\in[1,\infty]$
and any Banach space $X$, the Bochner spaces of functions mapping from an
interval $I$ into $X$ are denoted by $L^{p}(I;X)$ and $W^{s,p}(I;X)$.
Furthermore, for any interval $I$ and any Banach space $X$, the space $C(I;X)$
denotes the set of continuous functions mapping from $I$ to $X$.

  3. (N3)

For any Banach space $X$, its dual space is denoted by $X^{\prime}$. The
corresponding duality pairing of elements $\phi\in X^{\prime}$ and $\zeta\in
X$ is denoted by $\langle\phi,\zeta\rangle_{X}$. If $X$ is a Hilbert space, we
write $\left(\cdot,\cdot\right)_{X}$ to denote its inner product.

  4. (N4)

For any bounded domain $\Omega\subset\mathbb{R}^{d}$ ($d\in\mathbb{N}$) with
Lipschitz boundary $\Gamma$, $u\in H^{1}(\Omega)^{\prime}$ and $v\in
H^{1}(\Gamma)^{\prime}$, we write

$\displaystyle{\langle
u\rangle}_{\Omega}\coloneqq\frac{1}{\left|\Omega\right|}\langle
u,1\rangle_{H^{1}(\Omega)},\qquad{\langle
v\rangle}_{\Omega}\coloneqq\frac{1}{\left|\Gamma\right|}\langle
v,1\rangle_{H^{1}(\Gamma)}$

to denote the generalized means of $u$ and $v$, respectively. Here,
$\left|\Omega\right|$ denotes the $d$-dimensional Lebesgue measure of
$\Omega$, whereas $\left|\Gamma\right|$ denotes the $(d-1)$-dimensional
Hausdorff measure of $\Gamma$. If $u\in L^{1}(\Omega)$ or $v\in
L^{1}(\Gamma)$, the generalized mean can be expressed as

$\displaystyle{\langle
u\rangle}_{\Omega}=\frac{1}{\left|\Omega\right|}\int_{\Omega}u\;\mathrm{d}x,\qquad{\langle
v\rangle}_{\Gamma}=\frac{1}{\left|\Gamma\right|}\int_{\Gamma}v\;\mathrm{d}\Gamma,$

respectively.

  5. (N5)

For any bounded domain $\Omega\subset\mathbb{R}^{d}$ ($d\in\mathbb{N}$) with
Lipschitz boundary $\Gamma:=\partial\Omega$, we introduce the space

$\displaystyle\mathbf{H}^{1}_{\mathbf{n}}(\Omega)\coloneqq\big{\\{}\boldsymbol{v}\in{H}^{1}(\Omega;\mathbb{R}^{d}):\boldsymbol{v}\cdot\mathbf{n}=0\text{
a.e.\leavevmode\nobreak\ on }\Gamma\big{\\}}.$

### 2.2 Assumptions

We make the following general assumptions.

  1. (A1)

$\Omega$ is a non-empty, bounded Lipschitz domain in $\mathbb{R}^{d}$ with
$d\in\\{2,3\\}$, whose boundary is denoted by $\Gamma\coloneqq\partial\Omega$.
Moreover, $T>0$ denotes an arbitrary final time and for brevity, we use the
notation

$\displaystyle
Q\coloneqq\Omega\times(0,T),\quad\Sigma\coloneqq\Gamma\times(0,T).$

  2. (A2)

The constants in system (1.1) satisfy $\epsilon,\epsilon_{\Gamma},\kappa>0$,
and $\alpha,\beta\in\mathbb{R}$ with
$\alpha\beta\left|\Omega\right|+\left|\Gamma\right|\neq 0$. (The latter
condition is required to apply a certain bulk-surface Poincaré inequality, see
(P6).) Since the choice of $\epsilon_{\Gamma},\epsilon$ and $\kappa$ has no
impact on the mathematical analysis, we will simply set (without loss of
generality) $\epsilon=\epsilon_{\Gamma}=\kappa=1$ in the remainder of this
paper.

  3. (A3)

The mobility functions $m_{\Omega}:\mathbb{R}\rightarrow\mathbb{R}$ and
$m_{\Gamma}:\mathbb{R}\rightarrow\mathbb{R}$ are bounded, continuous and
uniformly positive. This means that there exist constants
$m_{\Omega}^{\ast},M_{\Omega}^{\ast},m_{\Gamma}^{\ast},M_{\Gamma}^{\ast}>0$
such that

$\displaystyle 0<m_{\Omega}^{\ast}\leq m_{\Omega}(s)\leq
M_{\Omega}^{\ast}\quad\text{and}\quad 0<m_{\Gamma}^{\ast}\leq
m_{\Gamma}(s)\leq M_{\Gamma}^{\ast}\quad\text{for all $s\in\mathbb{R}$}.$

  4. (A4)

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

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

as well as constants $c_{F^{\prime}},c_{G^{\prime}}\geq 0$ such that the
first-order derivatives satisfy the growth conditions

$\displaystyle\left|F^{\prime}(s)\right|$ $\displaystyle\leq
c_{F^{\prime}}(1+\left|s\right|^{p-1}),$ (2.1)
$\displaystyle\left|G^{\prime}(s)\right|$ $\displaystyle\leq
c_{G^{\prime}}(1+\left|s\right|^{q-1})$ (2.2)

for all $s\in\mathbb{R}$. These assumptions already imply the existence of
constants $c_{F},c_{G}\geq 0$ such that $F$ and $G$ satisfy the growth
conditions

$\displaystyle F(s)$ $\displaystyle\leq c_{F}(1+\left|s\right|^{p}),$ (2.3)
$\displaystyle G(s)$ $\displaystyle\leq c_{G}(1+\left|s\right|^{q})$ (2.4)

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

  5. (A5)

There exist constants $a_{F},a_{G}>0$ and $b_{F},b_{G}\geq 0$ such that the
potentials $F$ and $G$ introduced in (A4) satisfy

$\displaystyle F(s)$ $\displaystyle\geq a_{F}\left|s\right|^{2}-b_{F},$ (2.5)
$\displaystyle G(s)$ $\displaystyle\geq a_{G}\left|s\right|^{2}-b_{G}$ (2.6)

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

  6. (A6)

The potentials $F$ and $G$ introduced in (A4) are twice continuously
differentiable, and there exist constants
$c_{F^{\prime\prime}},c_{G^{\prime\prime}}\geq 0$ such that the second-order
derivatives of $F$ and $G$ satisfy the growth conditions

$\displaystyle\left|F^{\prime\prime}(s)\right|$ $\displaystyle\leq
c_{F^{\prime\prime}}(1+\left|s\right|^{p-2}),$ (2.7)
$\displaystyle\left|G^{\prime\prime}(s)\right|$ $\displaystyle\leq
c_{G^{\prime\prime}}(1+\left|s\right|^{q-2})$ (2.8)

for all $s\in\mathbb{R}$. We point out that these assumptions already imply
the growth conditions (2.1)–(2.4) stated in (A4).

######  Remark 2.1.

A standard choice for $F$ and $G$ is the polynomial double-well potential

$W:\mathbb{R}\to\mathbb{R},\quad W(s)=\tfrac{1}{4}(s^{2}-1)^{2}.$

It satisfies the assumptions (A4)–(A6) with $p=q=4$. However, singular
potentials such as the logarithmic Flory–Huggins potential or the double-
obstacle potential are not admissible as they do not satisfy any polynomial
growth condition. Nevertheless, the investigation of system (1.1) with
singular potentials is an interesting topic for future research.

### 2.3 Preliminaries

  1. (P1)

For any real numbers $s\geq 0$ and $p\in[1,\infty]$, we set

$\displaystyle\mathcal{L}^{p}\coloneqq L^{p}(\Omega)\times
L^{p}(\Gamma),\quad\text{and}\quad\mathcal{H}^{s}\coloneqq H^{s}(\Omega)\times
H^{s}(\Gamma).$

As usual, we identify $\mathcal{L}^{2}$ with $\mathcal{H}^{0}$. Note that
$\mathcal{H}^{s}$ is a Hilbert space with respect to the inner product

$\displaystyle\big{(}\left(\phi,\psi\right),\left(\zeta,\xi\right)\big{)}_{\mathcal{H}^{s}}\coloneqq\left(\phi,\zeta\right)_{H^{s}(\Omega)}+\left(\psi,\xi\right)_{H^{s}(\Gamma)}\quad\text{for
all }\left(\phi,\psi\right),\left(\zeta,\xi\right)\in\mathcal{H}^{s}$

and its induced norm
$\|\cdot\|_{\mathcal{H}^{s}}\coloneqq\left(\cdot,\cdot\right)_{\mathcal{H}^{s}}^{1/2}$.
We recall that the duality pairing can be expressed as

$\displaystyle\langle\left(\phi,\psi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}}=\left(\phi,\zeta\right)_{L^{2}(\Omega)}+\left(\psi,\xi\right)_{L^{2}(\Gamma)}$

if $\left(\phi,\psi\right)\in\mathcal{L}^{2}$ and
$\left(\zeta,\xi\right)\in\mathcal{H}^{1}$.

  2. (P2)

Let $L\in[0,\infty]$ and $\beta\in\mathbb{R}$. We introduce the closed linear
subspace

$\displaystyle\mathcal{D}_{\beta}\coloneqq\\{(\phi,\psi)\in\mathcal{H}^{1}:\phi=\beta\psi\text{
a.e.\leavevmode\nobreak\ on }\Gamma\\}\subset\mathcal{H}^{1}$

and define

$\displaystyle\mathcal{H}_{L,\beta}^{1}\coloneqq\begin{cases}\mathcal{H}^{1},&\text{if
}L\in(0,\infty],\\\ \mathcal{D}_{\beta},&\text{if }L=0.\end{cases}$

Endowed with the inner product
$\left(\cdot,\cdot\right)_{\mathcal{H}_{L,\beta}^{1}}\coloneqq\left(\cdot,\cdot\right)_{\mathcal{H}^{1}}$
and its induced norm, the space $\mathcal{H}_{L,\beta}^{1}$ is a Hilbert
space. Moreover, we define the product

$\displaystyle\langle\left(\phi,\psi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}_{L,\beta}^{1}}\coloneqq\left(\phi,\zeta\right)_{L^{2}(\Omega)}+\left(\psi,\xi\right)_{L^{2}(\Gamma)}$

for all $\left(\phi,\psi\right),\left(\zeta,\xi\right)\in\mathcal{L}^{2}$. By
means of the Riesz representation theorem, this product can be extended to a
duality pairing on
$(\mathcal{H}_{L,\beta}^{1})^{\prime}\times\mathcal{H}_{L,\beta}^{1}$, which
will also be denoted as
$\langle\cdot,\cdot\rangle_{\mathcal{H}_{L,\beta}^{1}}$.

  3. (P3)

Let $L\in[0,\infty]$ and $\beta\in\mathbb{R}$. We define the closed linear
subspace

$\displaystyle\mathcal{V}_{L,\beta}^{1}$
$\displaystyle\coloneqq\begin{cases}\\{\left(\phi,\psi\right)\in\mathcal{H}^{1}_{L,\beta}:\beta\left|\Omega\right|{\langle\phi\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi\rangle}_{\Gamma}=0\\},&\text{if\leavevmode\nobreak\
}L\in[0,\infty),\\\
\\{\left(\phi,\psi\right)\in\mathcal{H}^{1}:{\langle\phi\rangle}_{\Omega}={\langle\psi\rangle}_{\Gamma}=0\\},&\text{if\leavevmode\nobreak\
}L=\infty.\end{cases}$

Note that $\mathcal{V}_{L,\beta}^{1}$ is a Hilbert space with respect to the
inner product $\left(\cdot,\cdot\right)_{\mathcal{H}^{1}}$ and its induced
norm.

  4. (P4)

Let $L\in[0,\infty]$ and $\beta\in\mathbb{R}$. We set

$\displaystyle\mathds{h}(L)\coloneqq\begin{cases}L^{-1},&\text{if
}L\in(0,\infty),\\\ 0,&\text{if }L\in\\{0,\infty\\},\end{cases}$

and we define a bilinear form on $\mathcal{H}^{1}\times\mathcal{H}^{1}$ by

$\displaystyle\big{(}\left(\phi,\psi\right),\left(\zeta,\xi\right)\big{)}_{L,\beta}\coloneqq$
$\displaystyle\int_{\Omega}\nabla\phi\cdot\nabla\zeta\;\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
$\displaystyle\quad+\mathds{h}(L)\int_{\Gamma}(\beta\psi-\phi)(\beta\xi-\zeta)\;\mathrm{d}\Gamma$

for all $\left(\phi,\psi\right),\left(\zeta,\xi\right)\in\mathcal{H}^{1}$.
Moreover, we set

$\displaystyle\|\left(\psi,\phi\right)\|_{L,\beta}\coloneqq\big{(}\left(\phi,\psi\right),\left(\phi,\psi\right)\big{)}_{L,\beta}^{1/2}$

for all $\left(\phi,\psi\right)\in\mathcal{H}^{1}$. The bilinear form
$\left(\cdot,\cdot\right)_{L,\beta}$ defines an inner product on
$\mathcal{V}^{1}_{L,\beta}$, and $\|\cdot\|_{L,\beta}$ defines a norm on
$\mathcal{V}^{1}_{L,\beta}$, that is equivalent to the norm
$\|\cdot\|_{\mathcal{H}^{1}}$ (see [30, Corollary A.2]). Moreover, the space
$\big{(}\mathcal{V}^{1}_{L,\beta},\left(\cdot,\cdot\right)_{L,\beta},\|\cdot\|_{L,\beta}\big{)}$
is a Hilbert space.

  5. (P5)

For any $L\in[0,\infty]$ and $\beta\in\mathbb{R}$, we define the space

$\displaystyle\mathcal{V}_{L,\beta}^{-1}\coloneqq\begin{cases}\\{\left(\phi,\psi\right)\in(\mathcal{H}^{1}_{L,\beta})^{\prime}:\beta\left|\Omega\right|{\langle\phi\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi\rangle}_{\Gamma}=0\\},&\text{if\leavevmode\nobreak\
}L\in[0,\infty),\\\
\\{\left(\phi,\psi\right)\in(\mathcal{H}^{1})^{\prime}:{\langle\phi\rangle}_{\Omega}={\langle\psi\rangle}_{\Gamma}=0\\},&\text{if\leavevmode\nobreak\
}L=\infty.\end{cases}$

Using the Lax-Milgram theorem, one can show that for any
$\left(\phi,\psi\right)\in\mathcal{V}_{L,\beta}^{-1}$, there exists a unique
weak solution
$\mathcal{S}_{L,\beta}(\phi,\psi)=\big{(}\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\big{)}\in\mathcal{V}^{1}_{L,\beta}$
to the following elliptic problem with bulk-surface coupling:

$\displaystyle-\Delta\mathcal{S}_{L,\beta}^{\Omega}$
$\displaystyle=-\phi\qquad$ $\displaystyle\text{in }\Omega,$ (2.9a)
$\displaystyle-\Delta_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}+\beta\partial_{\mathbf{n}}\mathcal{S}_{L,\beta}^{\Omega}$
$\displaystyle=-\psi$ $\displaystyle\text{on }\Gamma,$ (2.9b) $\displaystyle
L\partial_{\mathbf{n}}\mathcal{S}_{L,\beta}^{\Omega}$
$\displaystyle=\beta\mathcal{S}_{L,\beta}^{\Gamma}-\mathcal{S}_{L,\beta}^{\Omega}\qquad$
$\displaystyle\text{on }\Gamma.$ (2.9c)

This means that $\mathcal{S}_{L,\beta}(\phi,\psi)$ satisfies the weak
formulation

$\displaystyle\big{(}S_{L,\beta}(\phi,\psi),\left(\zeta,\xi\right)\big{)}_{L,\beta}=-\langle\left(\phi,\psi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}_{L,\beta}}$

for all test functions $\left(\xi,\zeta\right)\in\mathcal{H}^{1}_{L,\beta}$.
Consequently, we have

$\displaystyle\|\mathcal{S}_{L,\beta}(\phi,\psi)\|_{\mathcal{H}^{1}}\leq
C\|\left(\phi,\psi\right)\|_{(\mathcal{H}^{1}_{L,\beta})^{\prime}}$ (2.10)

for all $\left(\phi,\psi\right)\in\mathcal{V}^{-1}_{L,\beta}$, for a constant
$C\geq 0$ depending only on $\Omega$, $L$ and $\beta$. We can thus define the
solution operator

$\displaystyle\mathcal{S}_{L,\beta}:\mathcal{V}_{L,\beta}^{-1}\rightarrow\mathcal{V}^{1}_{L,\beta},\quad\left(\phi,\psi\right)\mapsto\mathcal{S}_{L,\beta}(\phi,\psi)=\big{(}\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\big{)},$

as well as an inner product and its induced norm on
$\mathcal{V}^{-1}_{L,\beta}$ by

$\displaystyle\big{(}\left(\phi,\psi\right),\left(\zeta,\xi\right)\big{)}_{L,\beta,\ast}$
$\displaystyle\coloneqq\big{(}\mathcal{S}_{L,\beta}(\phi,\psi),\mathcal{S}_{L,\beta}(\zeta,\xi)\big{)}_{L,\beta},$
$\displaystyle\|\left(\phi,\psi\right)\|_{L,\beta,\ast}$
$\displaystyle\coloneqq\big{(}\left(\phi,\psi\right),\left(\phi,\psi\right)\big{)}_{L,\beta,\ast}^{1/2},$

for
$\left(\phi,\psi\right),\left(\zeta,\xi\right)\in\mathcal{V}_{L,\beta}^{-1}$.
This norm is equivalent to the norm
$\|\cdot\|_{(\mathcal{H}^{1}_{L,\beta})^{\prime}}$ on
$\mathcal{V}_{L,\beta}^{-1}$. For the case $L\in(0,\infty)$, we refer the
reader to [30, Theorem 3.3 and Corollary 3.5] for a proof of these statements.
In the other cases, the results can be proven analogously.

  6. (P6)

We further recall the following bulk-surface Poincaré inequalitiy, which has
been established in [30, Lemma A.1]:
Let $K\in[0,\infty)$ and $\alpha,\beta\in\mathbb{R}$ with
$\alpha\beta\left|\Omega\right|+\left|\Gamma\right|\neq 0$ be arbitrary. Then
there exists a constant $C_{P}>0$ depending only on $K,\alpha,\beta$ and
$\Omega$ such that

$\displaystyle\|\left(\phi,\psi\right)\|_{\mathcal{L}^{2}}\leq
C_{P}\|\left(\phi,\psi\right)\|_{K,\alpha}$ (2.11)

for all $\left(\phi,\psi\right)\in\mathcal{H}^{1}_{K,\alpha}$ satisfying
$\beta\left|\Omega\right|{\langle\phi\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi\rangle}_{\Gamma}=0$.

We conclude this section by presenting a simple inequality that will be
frequently used in our mathematical analysis.

######  Lemma 2.2.

Let $K\in[0,\infty]$ and $\alpha\in\mathbb{R}$. Then, it holds

$\displaystyle\|\left(\zeta,\xi\right)\|^{2}_{\mathcal{L}^{2}}\leq
2\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}\|\left(\nabla\zeta,\nabla_{\Gamma}\xi\right)\|_{\mathcal{L}^{2}}+\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}^{2}.$

for all $\left(\zeta,\xi\right)\in\mathcal{H}^{1}_{K,\alpha}$.

  * Proof.

Let $(\zeta,\xi)\in\mathcal{H}^{1}_{K,\alpha}$ be arbitrary. Recalling the
definition of $\mathcal{H}^{1}_{K,\alpha}$ (see (P2)), we have

$\|(\zeta,\xi)\|_{\mathcal{H}^{1}_{K,\alpha}}=\|(\zeta,\xi)\|_{\mathcal{H}^{1}}\leq\|(\zeta,\xi)\|_{\mathcal{L}^{2}}+\|(\nabla\zeta,\nabla_{\Gamma}\xi)\|_{\mathcal{L}^{2}}.$

Using Young’s inequality, we deduce

$\displaystyle\|\left(\zeta,\xi\right)\|_{\mathcal{L}^{2}}^{2}$
$\displaystyle=\langle\left(\zeta,\xi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}_{K,\alpha}}$
$\displaystyle\leq\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}\|\left(\zeta,\xi\right)\|_{\mathcal{H}^{1}_{K,\alpha}}$
$\displaystyle\leq\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}\|\left(\zeta,\xi\right)\|_{\mathcal{L}^{2}}+\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}\|\left(\nabla\zeta,\nabla_{\Gamma}\xi\right)\|_{\mathcal{L}^{2}}$
$\displaystyle\leq\tfrac{1}{2}\|\left(\zeta,\xi\right)\|_{\mathcal{L}^{2}}^{2}+\tfrac{1}{2}\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}^{2}+\|\left(\zeta,\xi\right)\|_{(\mathcal{H}^{1}_{K,\alpha})^{\prime}}\|\left(\nabla\zeta,\nabla_{\Gamma}\xi\right)\|_{\mathcal{L}^{2}}.$

Hence, the claim directly follows. ∎

## 3 Main results

As mentioned in (A2), we set $\epsilon=\epsilon_{\Gamma}=\kappa=1$. This does
not mean any loss of generality as the exact values of $\epsilon$,
$\epsilon_{\Gamma}$ and $\kappa$ do not have any impact on the mathematical
analysis (as long as they are positive). By this choice, the system (1.1) can
be restated as follows:

$\displaystyle\partial_{t}\phi+\textnormal{div}(\phi\boldsymbol{v})=\textnormal{div}(m_{\Omega}(\phi)\nabla\mu)$
$\displaystyle\text{in}\ Q,$ (3.1a)
$\displaystyle\mu=-\Delta\phi+F^{\prime}(\phi)$ $\displaystyle\text{in}\ Q,$
(3.1b)
$\displaystyle\partial_{t}\psi+\textnormal{div}_{\Gamma}(\psi\boldsymbol{v}_{\tau})=\textnormal{div}_{\Gamma}(m_{\Gamma}(\psi)\nabla_{\Gamma}\theta)-\beta
m_{\Omega}(\phi)\partial_{\mathbf{n}}\mu$ $\displaystyle\text{on}\ \Sigma,$
(3.1c)
$\displaystyle\theta=-\Delta_{\Gamma}\psi+G^{\prime}(\psi)+\alpha\partial_{\mathbf{n}}\phi$
$\displaystyle\text{on}\ \Sigma,$ (3.1d)
$\displaystyle\begin{cases}K\partial_{\mathbf{n}}\phi=\alpha\psi-\phi&\text{if}\
K\in[0,\infty),\\\ \partial_{\mathbf{n}}\phi=0&\text{if}\ K=\infty\end{cases}$
$\displaystyle\text{on}\ \Sigma,$ (3.1e)
$\displaystyle\begin{cases}Lm_{\Omega}(\phi)\partial_{\mathbf{n}}\mu=\beta\theta-\mu&\text{if}\
L\in[0,\infty),\\\ m_{\Omega}(\phi)\partial_{\mathbf{n}}\mu=0&\text{if}\
L=\infty\end{cases}$ $\displaystyle\text{on}\ \Sigma,$ (3.1f)
$\displaystyle\phi|_{t=0}=\phi_{0}$ $\displaystyle\text{in}\ \Omega,$ (3.1g)
$\displaystyle\psi|_{t=0}=\psi_{0}$ $\displaystyle\text{on}\ \Gamma.$ (3.1h)

The total energy associated with this system reads as

$\displaystyle\begin{split}E_{K}(\phi,\psi)&=\int_{\Omega}\frac{1}{2}\left|\nabla\phi\right|^{2}+F(\phi)\;\mathrm{d}x+\int_{\Gamma}\frac{1}{2}\left|\nabla_{\Gamma}\psi\right|^{2}+G(\psi)\;\mathrm{d}\Gamma+\mathds{h}(K)\int_{\Gamma}\frac{1}{2}\left|\alpha\psi-\phi\right|^{2}\;\mathrm{d}\Gamma.\end{split}$

We now introduce the notion of a weak solution to system (3.1).

######  Definition 3.1 (Weak solutions of system (3.1) for
$K,L\in[0,\infty]$).

Suppose that assumptions (A1)–(A4) hold. Let $K,L\in[0,\infty]$, let
$(\phi_{0},\psi_{0})\in\mathcal{H}^{1}_{K,\alpha}$ be arbitrary initial data
and let $\boldsymbol{v}\in L^{\infty}(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}_{\mathbf{n}}^{1})$ be a given velocity field. The
quadruplet $(\phi,\psi,\mu,\theta)$ is called a weak solution of the system
(3.1) on $[0,T]$ if the following properties hold:

  1. (i)

The functions $\phi$, $\psi$, $\mu$ and $\theta$ have the following
regularity:

$\displaystyle\left(\phi,\psi\right)\in C([0,T];\mathcal{L}^{2})\cap
H^{1}(0,T;(\mathcal{H}_{L,\beta}^{1})^{\prime})\cap
L^{\infty}(0,T;\mathcal{H}_{K,\alpha}^{1}),$ (3.2a)
$\displaystyle\left(\mu,\theta\right)\in
L^{2}(0,T,\mathcal{H}_{L,\beta}^{1}).$ (3.2b)

  2. (ii)

The functions $\phi$ and $\psi$ satisfy the initial conditions

$\displaystyle\phi|_{t=0}=\phi_{0}\quad\text{a.e.\leavevmode\nobreak\ in
}\Omega,\quad\text{and}\quad\psi|_{t=0}=\psi_{0}\quad\text{a.e.\leavevmode\nobreak\
on }\Gamma.$ (3.3)

  3. (iii)

The functions $\phi$, $\psi$, $\mu$ and $\theta$ satisfy the weak formulation

$\displaystyle\langle\left(\partial_{t}\phi,\partial_{t}\psi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}_{L,\beta}^{1}}-\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
$\displaystyle\quad=-\int_{\Omega}m_{\Omega}(\phi)\nabla\mu\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Gamma}m_{\Gamma}(\psi)\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
(3.4a)
$\displaystyle\qquad-\mathds{h}(L)\int_{\Gamma}(\beta\theta-\mu)(\beta\xi-\zeta)\;\mathrm{d}\Gamma,$
$\displaystyle\int_{\Omega}\mu\,\eta\;\mathrm{d}x+\int_{\Gamma}\theta\,\vartheta\;\mathrm{d}\Gamma$
$\displaystyle\quad=\int_{\Omega}\nabla\phi\cdot\nabla\eta+F^{\prime}(\phi)\eta\;\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}\vartheta+G^{\prime}(\psi)\vartheta\;\mathrm{d}\Gamma$
(3.4b)
$\displaystyle\qquad+\mathds{h}(K)\int_{\Gamma}(\alpha\psi-\phi)(\alpha\vartheta-\eta)\;\mathrm{d}\Gamma,$

a.e. on $[0,T]$ for all
$\left(\zeta,\xi\right)\in\mathcal{H}_{L,\beta}^{1},\left(\eta,\vartheta\right)\in\mathcal{H}_{K,\alpha}^{1}$.

  4. (iv)

The functions $\phi$ and $\psi$ satisfy the mass conservation law

$\displaystyle\begin{dcases}\beta\int_{\Omega}\phi(t)\;\mathrm{d}x+\int_{\Gamma}\psi(t)\;\mathrm{d}\Gamma=\beta\int_{\Omega}\phi_{0}\;\mathrm{d}x+\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma,&\textnormal{if
}L\in[0,\infty),\\\
\int_{\Omega}\phi(t)\;\mathrm{d}x=\int_{\Omega}\phi_{0}\;\mathrm{d}x\quad\textnormal{and}\quad\int_{\Gamma}\psi(t)\;\mathrm{d}\Gamma=\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma,&\textnormal{if
}L=\infty\end{dcases}$ (3.5)

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

  5. (v)

The functions $\phi,\psi,\mu$ and $\theta$ satisfy the energy inequality

$\displaystyle\begin{split}&E_{K}(\phi(t),\psi(t))+\int_{0}^{t}\int_{\Omega}m_{\Omega}(\phi)\left|\nabla\mu\right|^{2}\;\mathrm{d}x\;\mathrm{d}s+\int_{0}^{t}\int_{\Gamma}m_{\Gamma}(\psi)\left|\nabla_{\Gamma}\theta\right|^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad+\mathds{h}(L)\int_{0}^{t}\int_{\Gamma}(\beta\theta-\mu)^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad-\int_{0}^{t}\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\mu\;\mathrm{d}x\;\mathrm{d}s-\int_{0}^{t}\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\leq E_{K}(\phi_{0},\psi_{0})\end{split}$ (3.6)

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

Now, we are ready to formulate the main results of this paper. The first main
result provides the existence of a weak solution to system (3.1) in all cases
$(K,L)\in[0,\infty]^{2}$.

######  Theorem 3.2.

(Existence of weak solutions of (3.1)) Suppose that the assumptions (A1)–(A4)
hold. Let $K,L\in[0,\infty]$, let
$\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}_{K,\alpha}$ be arbitrary
initial data, and let $\boldsymbol{v}\in L^{\infty}(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}_{\mathbf{n}}^{1})$ be a given velocity field. In the case
$K\in\\{0,\infty\\}$, we further assume that (A5) holds. Then, there exists a
weak solution $(\phi,\psi,\mu,\theta)$ of the system (3.1) in the sense of
Definition 3.1, which has the additional regularity

$\displaystyle(\mu,\theta)\in L^{4}(0,T;\mathcal{L}^{2})\quad\text{if
}K\in(0,\infty].$ (3.7)

Let us now assume that (A6) holds, that the domain $\Omega$ is of class
$C^{k}$ for $k\in\\{2,3\\}$, and in the case $d=3$, we further assume that
(A4) holds with $p\leq 4$. Then, we additionally have

$\displaystyle\left(\phi,\psi\right)\in L^{4}(0,T;\mathcal{H}^{2})\quad$
$\displaystyle\text{if $K\in(0,\infty]$},$ (3.8)
$\displaystyle\left(\phi,\psi\right)\in L^{2}(0,T;\mathcal{H}^{k})$
$\displaystyle\text{for $k=2,3$},$ (3.9)
$\displaystyle\left(\phi,\psi\right)\in C([0,T];\mathcal{H}^{1})\quad$ if
$(K,L)\in[0,\infty]\times(0,\infty]$ and $k=3$ (3.10)

and the equations

$\displaystyle\mu=-\Delta\phi+F^{\prime}(\phi)$
$\displaystyle\text{a.e.\leavevmode\nobreak\ in }Q,$
$\displaystyle\theta=-\Delta_{\Gamma}\psi+G^{\prime}(\psi)+\alpha\partial_{\mathbf{n}}\phi$
$\displaystyle\text{a.e.\leavevmode\nobreak\ on }\Sigma,$
$\displaystyle\begin{cases}K\partial_{\mathbf{n}}\phi=\alpha\psi-\phi&\text{if}\
K\in[0,\infty),\\\ \partial_{\mathbf{n}}\phi=0&\text{if}\ K=\infty\end{cases}$
$\displaystyle\text{a.e.\leavevmode\nobreak\ on }\Sigma$

are fulfilled in the strong sense.

######  Remark 3.3.

The continuity property (3.10) also holds in the case $K=L=0$ provided that
the potentials $F$ and $G$ to satisfy the compatibility condition

$\displaystyle F(s)=\beta G(\alpha s)\quad\text{for all\leavevmode\nobreak\
}s\in\mathbb{R}.$

In that case, we have
$\left(F^{\prime}(\phi),G^{\prime}(\psi)\right)\in\mathcal{H}^{1}_{0,\beta}=\mathcal{D}_{\beta}$,
which allows us to use Proposition A.1 similarly as in Section 6.1.

  * Proof of Theorem 3.2.

The existence of a weak solution in the sense of Definition 3.1 is established
in

    * –

Theorem 4.1 if $(K,L)\in(0,\infty)\times(0,\infty)$,

    * –

Theorem 5.2 if $(K,L)\in\\{0\\}\times(0,\infty)$,

    * –

Theorem 5.4 if $(K,L)\in\\{\infty\\}\times(0,\infty)$,

    * –

Theorem 5.6 if $(K,L)\in[0,\infty]\times\\{0\\}$,

    * –

Theorem 5.8 if $(K,L)\in[0,\infty]\times\\{\infty\\}$.

Moreover, in the case $K\in(0,\infty]$, the additional regularity (3.7)
follows from the corresponding aforementioned theorems. All remaining results
are shown in Theorem 6.1. ∎

######  Remark 3.4.

We point out that Theorem 5.2, Theorem 5.4, Theorem 5.6 and Theorem 5.8 are
not only useful to prove Theorem 3.2, but also provide further valuable
insights about the asymptotic limits $K\to 0$, $K\to\infty$, $L\to 0$ and
$L\to\infty$, respectively, on the level of weak solutions.

Our second main result shows continuous dependence of weak solutions on the
velocity field and the initial data in the case of constant mobilities. As a
direct consequence, this entails uniqueness of the weak solution.

######  Theorem 3.5.

(Continuous dependence and uniqueness) Suppose that the assumptions (A1)–(A4)
hold, and that the mobility functions $m_{\Omega}$ and $m_{\Gamma}$ are
constant. Let $K,L\in[0,\infty]$, let $\left(\phi_{0}^{1},\psi_{0}^{1}\right)$
and $\left(\phi_{0}^{2},\psi_{0}^{2}\right)\in\mathcal{H}^{1}_{K,\alpha}$ be
two pairs of initial data, which satisfy

$\left(\phi_{0}^{1}-\phi_{0}^{2},\psi_{0}^{1}-\psi_{0}^{2}\right)\in\mathcal{V}^{-1}_{L,\beta}$
(3.11)

and let $\boldsymbol{v}_{1},\boldsymbol{v}_{2}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}_{\mathbf{n}}^{1})$ be
two given velocity fields. Suppose that
$(\phi_{1},\psi_{1},\mu_{1},\theta_{1})$ and
$(\phi_{2},\psi_{2},\mu_{2},\theta_{2})$ are weak solutions in the sense of
Definition 3.1 corresponding to
$(\phi_{0}^{1},\psi_{0}^{1},\boldsymbol{v}_{1})$ and
$(\phi_{0}^{2},\psi_{0}^{2},\boldsymbol{v}_{2})$, respectively. Then, the
continuous dependence estimate

$\displaystyle\begin{split}&\big{\|}\big{(}\phi_{1}(t)-\phi_{2}(t),\psi_{1}(t)-\psi_{2}(t)\big{)}\big{\|}^{2}_{L,\beta,\ast}\\\\[3.99994pt]
&\leq\big{\|}\big{(}\phi_{0}^{1}-\phi_{0}^{2},\psi_{0}^{1}-\psi_{0}^{2}\big{)}\big{\|}^{2}_{L,\beta,\ast}\exp\left(C\int_{0}^{t}\mathcal{F}(\tau)\;\mathrm{d}\tau\right)\\\
&\quad+\int_{0}^{t}\big{\|}\big{(}\boldsymbol{v}_{1}(s)-\boldsymbol{v}_{2}(s),\boldsymbol{v}_{1,\tau}(s)-\boldsymbol{v}_{2,\tau}(s)\big{)}\big{\|}^{2}_{\mathcal{L}^{3}}\exp\left(C\int_{s}^{t}\mathcal{F}(\tau)\;\mathrm{d}\tau\right)\;\mathrm{d}s\end{split}$
(3.12)

holds for almost all $t\in[0,T]$, where $\mathcal{F}\coloneqq
1+\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}^{2}$
and the constant $C>0$ depends only on $\Omega$, the parameters of the system
and the initial data.

In particular, if $(\phi_{0}^{1},\psi_{0}^{1})=(\phi_{0}^{2},\psi_{0}^{2})$
a.e. in $\Omega\times\Gamma$ and $\boldsymbol{v}_{1}=\boldsymbol{v}_{2}$ a.e.
in $Q$, estimate (3.12) ensures uniqueness of the corresponding weak solution.

The proof of Theorem 3.5 will be presented in Subsection 6.2.

## 4 Existence of weak solutions in the case $K,L\in(0,\infty)$

######  Theorem 4.1.

Suppose that the assumptions (A1)–(A4) hold. Let
$\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}$ be arbitrary initial data,
$\boldsymbol{v}\in L^{\infty}(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}_{\mathbf{n}}^{1})$ and $K,L\in(0,\infty)$. Then there
exists a weak solution of the system (3.1) in the sense of Definition 3.1
which further satisfies $\left(\mu,\theta\right)\in
L^{4}(0,T;\mathcal{L}^{2})$.

  * Proof.

Step 1: Discretization via a Faedo–Galerkin scheme. It is well known that the
problems

$\displaystyle\begin{cases}\begin{aligned}
-\Delta\zeta&=\lambda_{\Omega}\zeta,\quad&&\text{in }\Omega,\\\
\partial_{\mathbf{n}}\zeta&=0,\quad&&\text{on
}\Gamma,\end{aligned}\end{cases}\quad-\Delta_{\Gamma}\xi=\lambda_{\Gamma}\xi\quad\text{on
}\Gamma,$

have countable many eigenvalues, which can be written as increasing sequences
$\\{\lambda_{\Omega}^{j}\\}_{j\in\mathbb{N}}$ and
$\\{\lambda_{\Gamma}^{j}\\}_{j\in\mathbb{N}}$, respectively. The associated
eigenfunctions $\\{\zeta_{j}\\}_{j\in\mathbb{N}}\subset H^{1}(\Omega)$ and
$\\{\xi_{j}\\}_{j\in\mathbb{N}}\subset H^{1}(\Gamma)$ form an orthonormal
Schauder basis of $L^{2}(\Omega)$ and $L^{2}(\Gamma)$, respectively. In
particular, we fix the eigenfunctions associated to the first eigenvalues
$\lambda_{\Omega}^{1}=\lambda_{\Gamma}^{1}=0$ as
$\zeta_{1}\equiv\left|\Omega\right|^{-1/2}$ and
$\xi_{1}\equiv\left|\Gamma\right|^{-1/2}$. Moreover, the eigenfunctions
$\\{\zeta_{j}\\}_{j\in\mathbb{N}}$ and $\\{\xi_{j}\\}_{j\in\mathbb{N}}$ also
form an orthogonal Schauder basis of $H^{1}(\Omega)$ and $H^{1}(\Gamma)$,
respectively.

For any $m\in\mathbb{N}$, we introduce the finite-dimensional subspaces

$\displaystyle\mathcal{A}_{m}=\textnormal{span}\\{\zeta_{1},\ldots,\zeta_{m}\\}\subset
H^{1}(\Omega),\quad\mathcal{B}_{m}=\textnormal{span}\\{\xi_{1},\ldots,\xi_{m}\\}\subset
H^{1}(\Gamma),$

along with the orthogonal $L^{2}(\Omega)$-projection
$\mathbb{P}_{\mathcal{A}_{m}}$, and the orthogonal $L^{2}(\Gamma)$-projection
$\mathbb{P}_{\mathcal{B}_{m}}$. In particular, there exist constants
$C_{\Omega},C_{\Gamma}>0$ depending only on $\Omega$ and $\Gamma$,
respectively, such that for all $\zeta\in H^{1}(\Omega)$ and $\xi\in
H^{1}(\Gamma)$,

$\displaystyle\|\mathbb{P}_{\mathcal{A}_{m}}\zeta\|_{H^{1}(\Omega)}\leq
C_{\Omega}\|\zeta\|_{H^{1}(\Omega)}\quad\text{and}\quad\|\mathbb{P}_{\mathcal{B}_{m}}\xi\|_{H^{1}(\Gamma)}\leq
C_{\Gamma}\|\xi\|_{H^{1}(\Gamma)}.$

For any $m\in\mathbb{N}$, and $t\in[0,T]$, we make the ansatz

$\displaystyle\phi_{m}(t)$
$\displaystyle\coloneqq\sum_{j=1}^{m}a_{j}^{m}(t)\,\zeta_{j}\quad\text{a.e.\leavevmode\nobreak\
in }\Omega,$ $\displaystyle\qquad\psi_{m}(t)$
$\displaystyle\coloneqq\sum_{j=1}^{m}b_{j}^{m}(t)\,\xi_{j}\quad\text{a.e.\leavevmode\nobreak\
on }\Gamma,$ $\displaystyle\mu_{m}(t)$
$\displaystyle\coloneqq\sum_{j=1}^{m}c_{j}^{m}(t)\,\zeta_{j}\quad\text{a.e.\leavevmode\nobreak\
in }\Omega,$ $\displaystyle\qquad\theta_{m}(t)$
$\displaystyle\coloneqq\sum_{j=1}^{m}d_{j}^{m}(t)\,\xi_{j}\quad\text{a.e.\leavevmode\nobreak\
on }\Gamma.$

Here, the scalar, time-dependent coefficients
$a_{j}^{m},b_{j}^{m},c_{j}^{m},d_{j}^{m}$, $j=1,\ldots,m$ are assumed to be
continuously differentiable functions that are not yet determined. They need
to be designed in a way such that the discretized weak formulation

$\displaystyle\langle\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}}-\int_{\Omega}\phi_{m}\boldsymbol{v}\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Gamma}\psi_{m}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
$\displaystyle\quad=-\int_{\Omega}m_{\Omega}(\phi_{m})\nabla\mu_{m}\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Gamma}m_{\Gamma}(\psi_{m})\nabla_{\Gamma}\theta_{m}\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
(4.2a)
$\displaystyle\qquad-L^{-1}\int_{\Gamma}(\beta\theta_{m}-\mu_{m})(\beta\xi-\zeta)\;\mathrm{d}\Gamma,$
$\displaystyle\int_{\Omega}\mu_{m}\,\zeta\;\mathrm{d}x+\int_{\Gamma}\theta_{m}\,\xi\;\mathrm{d}\Gamma$
$\displaystyle\quad=\int_{\Omega}\nabla\phi_{m}\cdot\nabla\zeta+F^{\prime}(\phi_{m})\zeta\;\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi_{m}\cdot\nabla_{\Gamma}\xi+G^{\prime}(\psi_{m})\xi\;\mathrm{d}\Gamma$
(4.2b)
$\displaystyle\qquad+K^{-1}\int_{\Gamma}(\alpha\psi_{m}-\phi_{m})(\alpha\xi-\zeta)\;\mathrm{d}\Gamma,$

holds on $[0,T]$ for all
$\left(\zeta,\xi\right)\in\mathcal{A}_{m}\times\mathcal{B}_{m}$, supplemented
with the initial conditions

$\displaystyle\phi_{m}(0)=\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0})\quad\text{a.e.\leavevmode\nobreak\
in
}\Omega,\qquad\psi_{m}(0)=\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\quad\text{a.e.\leavevmode\nobreak\
on }\Gamma.$ (4.3)

Now, consider the corresponding coefficient vectors
$\mathbf{a}^{m}\coloneqq(a^{m}_{1},\ldots,a^{m}_{m}),\mathbf{b}^{m}\coloneqq(b^{m}_{1},\ldots,b^{m}_{m})$,
$\mathbf{c}^{m}\coloneqq(c^{m}_{1},\ldots,c^{m}_{m})$ and
$\mathbf{d}^{m}\coloneqq(d^{m}_{1},\ldots,d^{m}_{m})$. Testing (4.2a) with
$\left(\zeta_{1},\xi_{1}\right),\ldots,\left(\zeta_{m},\xi_{m}\right)$, we
conclude that $(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$ is determined by a
system of $2m$ ordinary differential equations, whose right-hand side depends
continuously on $\mathbf{a}^{m},\mathbf{b}^{m},\mathbf{c}^{m}$ and
$\mathbf{d}^{m}$. In view of (4.3), this system is subject to the initial
conditions

$\displaystyle[\mathbf{a}^{m}]_{j}(0)$
$\displaystyle=a^{m}_{j}(0)=\left(\phi_{0},\zeta_{j}\right),\quad\quad$
$\displaystyle\text{for all }j\in\\{1,\ldots,m\\},$
$\displaystyle[\mathbf{b}^{m}]_{j}(0)$
$\displaystyle=b^{m}_{j}(0)=\left(\psi_{0},\xi_{j}\right),$
$\displaystyle\text{for all }j\in\\{1,\ldots,m\\}.$

Moreover, testing (4.2b) with
$(\zeta_{1},\xi_{1}),\ldots,(\zeta_{m},\xi_{m})$, we infer that
$(\mathbf{c}^{m},\mathbf{d}^{m})^{\top}$ is explicitly given by a system of
$2m$ algebraic equations, whose right-hand side depends continuously on
$\mathbf{a}^{m}$ and $\mathbf{b}^{m}$. Thus, replacing
$(\mathbf{c}^{m},\mathbf{d}^{m})^{\top}$ in the right-hand side of the
aforementioned ODE system by this algebraic description, we obtain a closed
$2m$-dimensional ODE system for $(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$,
whose right-hand side depends continuously on
$(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$. We can thus apply the Cauchy–Peano
theorem to obtain a local solution
$(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}:[0,T_{m}^{\ast})\cap[0,T]\rightarrow\mathbb{R}^{2m}$
with $T_{m}^{\ast}>0$ to the corresponding initial value problem. Without loss
of generality, we assume that $T_{m}^{\ast}\leq T$ and that
$(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$ is non-extendable, i.e.,
$T_{m}^{\ast}$ is chosen as large as possible. Now, we can reconstruct
$(\mathbf{c}^{m},\mathbf{d}^{m})^{\top}:[0,T_{m}^{\ast})\rightarrow\mathbb{R}^{2m}$
by the aforementioned $2m$-dimensional system of algebraic equations. In view
of (4.1), we thus have shown the existence of functions

$\displaystyle\left(\phi_{m},\psi_{m}\right)\in
C^{1}([0,T_{m}^{\ast});\mathcal{H}^{1}),\quad\quad\left(\mu_{m},\theta_{m}\right)\in
C^{1}([0,T_{m}^{\ast});\mathcal{H}^{1})$

solving (4.2) on the time interval $[0,T_{m}^{\ast})$ subject to the initial
conditions (4.3).
Step 2: Uniform estimates. We establish suitable estimates for each
approximate solution $(\phi_{m},\psi_{m},\mu_{m},\theta_{m})$, which are
uniform with respect to the index $m$. In particular, let $T_{m}<T_{m}^{\ast}$
be arbitrary. In the following, let $C$ denote a generic non-negative constant
depending only on the initial data and the constants introduced in (A3)–(A4)
including the final time $T$, but not on $m$ or $T_{m}$.

Testing (4.2a) with $\left(\mu_{m},\theta_{m}\right)$, (4.2b) with
$-\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right)$, adding the
resulting equations, and integrating with respect to time from $0$ to $t$, we
derive the discrete energy identity

$\displaystyle\begin{split}&E_{K}(\phi_{m}(t),\psi_{m}(t))+\int_{0}^{t}\int_{\Omega}m_{\Omega}(\phi_{m})\left|\nabla\mu_{m}\right|^{2}\;\mathrm{d}x\;\mathrm{d}s+\int_{0}^{t}\int_{\Gamma}m_{\Gamma}(\psi_{m})\left|\nabla_{\Gamma}\theta_{m}\right|^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad\quad+L^{-1}\int_{0}^{t}\int_{\Gamma}(\beta\theta_{m}-\mu_{m})^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad=E_{K}(\phi_{m}(0),\psi_{m}(0))+\int_{0}^{t}\int_{\Omega}\phi_{m}\boldsymbol{v}\cdot\nabla\mu_{m}\;\mathrm{d}x\;\mathrm{d}s+\int_{0}^{t}\int_{\Gamma}\psi_{m}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta_{m}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
\end{split}$ (4.4)

for all $t\in[0,T_{m}]$. Now, due to the growth conditions on $F,G$ (see
(A4)), the Sobolev embeddings $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and
$H^{1}(\Gamma)\hookrightarrow L^{q}(\Gamma)$, the trace embedding
$H^{1}(\Omega)\hookrightarrow L^{2}(\Gamma)$, and the properties of the
projections $\mathbb{P}_{\mathcal{A}_{m}}$ and $\mathbb{P}_{\mathcal{B}_{m}}$,
we use the initial conditions to infer

$\displaystyle\begin{split}&E_{K}(\phi_{m}(0),\psi_{m}(0))\\\
&\leq\frac{1}{2}\|\nabla\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0})\|^{2}_{L^{2}(\Omega)}+\frac{1}{2}\|\nabla_{\Gamma}\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\|^{2}_{L^{2}(\Gamma)}+c_{F}\left(\left|\Omega\right|+\|\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0})\|^{p}_{L^{p}(\Omega)}\right)\\\
&\quad+c_{G}\left(\left|\Gamma\right|+\|\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\|^{q}_{L^{q}(\Gamma)}\right)+\frac{1}{2K}\Big{(}\left|\alpha\right|\|\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\|_{L^{2}(\Gamma)}+\|\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0})\|_{L^{2}(\Gamma)}\Big{)}^{2}\\\
&\leq
C\|\left(\phi_{0},\psi_{0}\right)\|_{\mathcal{H}^{1}}^{2}+C\left(1+\|\phi_{0}\|^{p}_{H^{1}(\Omega)}\right)+C\left(1+\|\psi_{0}\|^{q}_{H^{1}(\Gamma)}\right).\end{split}$
(4.5)

To obtain a suitable bound on the energy and the integral terms on the left-
hand side of (4.4), we have to deal with the convective terms appearing on the
right-hand side. To this end, we start by deriving for all $t\in[0,T_{m}]$ the
estimate

$\displaystyle\int_{0}^{t}\int_{\Omega}\phi_{m}\boldsymbol{v}\cdot\nabla\mu_{m}\;\mathrm{d}x\;\mathrm{d}s+\int_{0}^{t}\int_{\Gamma}\psi_{m}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\mu_{m}\;\mathrm{d}\Gamma\;\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|\phi_{m}\|_{L^{6}(\Omega)}\|\boldsymbol{v}\|_{L^{3}(\Omega)}\|\nabla\mu_{m}\|_{L^{2}(\Omega)}+\|\psi_{m}\|_{L^{6}(\Gamma)}\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}\|\nabla_{\Gamma}\theta_{m}\|_{L^{2}(\Gamma)}\;\mathrm{d}s$
$\displaystyle\begin{split}&\leq\frac{m_{\Omega}^{\ast}}{2}\int_{0}^{t}\|\nabla\mu_{m}\|^{2}_{L^{2}(\Omega)}\;\mathrm{d}s+\frac{m_{\Gamma}^{\ast}}{2}\int_{0}^{t}\|\nabla_{\Gamma}\theta_{m}\|^{2}_{L^{2}(\Gamma)}\;\mathrm{d}s\\\
&\quad+C\int_{0}^{t}\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}\|\phi_{m}\|_{H^{1}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\|\psi_{m}\|_{H^{1}(\Gamma)}^{2}\;\mathrm{d}s\end{split}$
(4.6)
$\displaystyle\begin{split}&\leq\frac{m_{\Omega}^{\ast}}{2}\int_{0}^{t}\|\nabla\mu_{m}\|^{2}_{L^{2}(\Omega)}\;\mathrm{d}s+\frac{m_{\Gamma}^{\ast}}{2}\int_{0}^{t}\|\nabla_{\Gamma}\theta_{m}\|^{2}_{L^{2}(\Gamma)}\;\mathrm{d}s\\\
&\quad+C\int_{0}^{t}\left(\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\|\left(\phi_{m},\psi_{m}\right)\|^{2}_{\mathcal{H}^{1}}\;\mathrm{d}s,\end{split}$

using again the embeddings $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and
$H^{1}(\Gamma)\hookrightarrow L^{6}(\Gamma)$. Recalling that $F,G\geq 0$, and
that the mobility functions $m_{\Omega}$ and $m_{\Gamma}$ are uniformly
positive (see (A3)), we infer from (4.5) and (Proof.) that

$\displaystyle\begin{split}&\frac{1}{2}\|\left(\phi_{m},\psi_{m}\right)\|_{K,\alpha}^{2}+\frac{m_{\Omega}^{\ast}}{2}\int_{0}^{t}\int_{\Omega}\left|\nabla\mu_{m}\right|^{2}\;\mathrm{d}x\;\mathrm{d}s+\frac{m_{\Gamma}^{\ast}}{2}\int_{0}^{t}\int_{\Gamma}\left|\nabla_{\Gamma}\theta_{m}\right|^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad\quad+L^{-1}\int_{0}^{t}\int_{\Gamma}(\beta\theta_{m}-\mu_{m})^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad\leq
C+C\int_{0}^{t}\left(\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\|\left(\phi_{m},\psi_{m}\right)\|^{2}_{\mathcal{H}^{1}}\;\mathrm{d}s\\\
&\quad\leq
C+C\int_{0}^{t}\left(\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\|\left(\phi_{m},\psi_{m}\right)\|^{2}_{K,\alpha}\;\mathrm{d}s\end{split}$
(4.7)

for all $t\in[0,T_{m}]$. Here, the last inequality follows from the bulk-
surface Poincaré inequality (P6), which usage is justified by the following
reasoning. First, testing (4.2a) with $(\beta,1)$ yields the discrete mass
conservation law

$\displaystyle\beta\int_{\Omega}\phi_{m}(t)\;\mathrm{d}x+\int_{\Gamma}\psi_{m}(t)\;\mathrm{d}\Gamma=\beta\int_{\Omega}\phi_{0}\;\mathrm{d}x+\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma$
(4.8)

for all $t\in[0,T_{m}]$, where we used that

$\displaystyle\beta\left|\Omega\right|{\langle\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0})\rangle}_{\Omega}+\left|\Gamma\right|{\langle\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\rangle}_{\Gamma}=\big{(}\left(\mathbb{P}_{\mathcal{A}_{m}}(\phi_{0}),\mathbb{P}_{\mathcal{B}_{m}}(\psi_{0})\right),\left(\beta,1\right)\big{)}_{\mathcal{L}^{2}}$
$\displaystyle\quad=\big{(}\left(\phi_{0},\psi_{0}\right),\left(\beta,1\right)\big{)}_{\mathcal{L}^{2}}=\beta\left|\Omega\right|{\langle\phi_{0}\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi_{0}\rangle}_{\Gamma}.$

Hence, considering

$\displaystyle
c\coloneqq\frac{\beta\left|\Omega\right|{\langle\phi_{0}\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi_{0}\rangle}_{\Gamma}}{\beta^{2}\left|\Omega\right|+\left|\Gamma\right|},$

and invoking (4.8), we infer

$\displaystyle\beta\left|\Omega\right|{\langle\phi_{m}(t)-\beta
c\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi_{m}(t)-c\rangle}_{\Gamma}=0\quad\text{for
all }t\in[0,T_{m}].$

This allows us to use the bulk-surface Poincaré inequality (P6), which yields

$\displaystyle\begin{split}\|\left(\phi_{m},\psi_{m}\right)\|_{\mathcal{L}^{2}}&\leq\|\left(\phi_{m}-\beta
c,\psi_{m}-c\right)\|_{\mathcal{L}^{2}}+\|\left(\beta
c,c\right)\|_{\mathcal{L}^{2}}\\\ &\leq
C_{P}\|\left(\phi_{m},\psi_{m}\right)\|_{K,\alpha}+C.\end{split}$ (4.9)

Thus, using Gronwall’s lemma, we conclude from (4.7) that

$\displaystyle\|\left(\phi_{m}(t),\psi_{m}(t)\right)\|_{K,\alpha}^{2}$
$\displaystyle\leq
C\exp\left(C\int_{0}^{t}\left(\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\;\mathrm{d}s\right)$
$\displaystyle\leq
C\exp\left(C\int_{0}^{T}\left(\|\boldsymbol{v}\|_{L^{3}(\Omega)}^{2}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\;\mathrm{d}s\right)$

for all $t\in[0,T_{m}]$. We thus readily deduce that

$\displaystyle\|\left(\phi_{m},\psi_{m}\right)\|_{L^{\infty}(0,T_{m};\mathcal{H}^{1})}^{2}+K^{-1}\|\alpha\psi_{m}-\phi_{m}\|^{2}_{L^{\infty}(0,T_{m};L^{2}(\Gamma))}\leq
C.$ (4.10)

In view of (4.7), this additionally entails

$\displaystyle\|\left(\nabla\mu_{m},\nabla_{\Gamma}\theta_{m}\right)\|^{2}_{L^{2}(0,T_{m};\mathcal{L}^{2})}+L^{-1}\|\beta\theta_{m}-\mu_{m}\|^{2}_{L^{2}(0,T_{m};L^{2}(\Gamma))}$
$\displaystyle\leq C.$ (4.11)

Next, we derive a uniform estimate for $\left(\mu_{m},\theta_{m}\right)$ in
the full $\mathcal{H}^{1}$ norm. To this end, let
$\left(\zeta^{\ast},\xi^{\ast}\right)\in\mathcal{H}^{1}$ and consider
$\left(\zeta,\xi\right)\coloneqq\left(\mathbb{P}_{\mathcal{A}_{m}}(\zeta^{\ast}),\mathbb{P}_{\mathcal{B}_{m}}(\xi^{\ast})\right)$.
By the growth conditions (2.1) and (2.2) from (A4), the Sobolev embeddings
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and $H^{1}(\Gamma)\hookrightarrow
L^{2q}(\Gamma)$, and the trace theorem $H^{1}(\Omega)\hookrightarrow
L^{2}(\Gamma)$, we have

$\displaystyle\begin{split}\left|\langle\left(\mu_{m},\theta_{m}\right),\left(\zeta^{\ast},\xi^{\ast}\right)\rangle_{\mathcal{H}^{1}}\right|&=\left|\langle\left(\mu_{m},\theta_{m}\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}}\right|\\\
&\leq\|\nabla\phi_{m}\|_{L^{2}(\Omega)}\|\nabla\zeta\|_{L^{2}(\Omega)}+\|F^{\prime}(\phi_{m})\|_{L^{6/5}(\Omega)}\|\zeta\|_{L^{6}(\Omega)}\\\
&\quad+\|\nabla_{\Gamma}\psi_{m}\|_{L^{2}(\Gamma)}\|\nabla_{\Gamma}\xi\|_{L^{2}(\Gamma)}+\|G^{\prime}(\psi_{m})\|_{L^{2}(\Gamma)}\|\xi\|_{L^{2}(\Gamma)}\\\
&\quad+K^{-1}\|\alpha\psi_{m}-\phi_{m}\|_{L^{2}(\Gamma)}\|\alpha\zeta-\xi\|_{L^{2}(\Gamma)}\\\
&\leq
C\Big{(}1+\|\phi_{m}\|_{H^{1}(\Omega)}+\|\phi_{m}\|_{L^{6}(\Omega)}^{5}\Big{)}\|\zeta\|_{H^{1}(\Omega)}\\\
&\quad+C\Big{(}1+\|\psi_{m}\|_{H^{1}(\Gamma)}+\|\psi_{m}\|_{L^{2q}(\Gamma)}^{q}\Big{)}\|\xi\|_{H^{1}(\Gamma)}\\\
&\leq
C\left(1+\|\phi_{m}\|^{5}_{H^{1}(\Omega)}+\|\psi_{m}\|^{q}_{H^{1}(\Gamma)}\right)\|\left(\zeta^{\ast},\xi^{\ast}\right)\|_{\mathcal{H}^{1}}\end{split}$
(4.12)

on $[0,T_{m}]$. Taking the supremum over all
$\left(\zeta^{\ast},\xi^{\ast}\right)\in\mathcal{H}^{1}$ with
$\|\left(\zeta^{\ast},\xi^{\ast}\right)\|_{\mathcal{H}^{1}}\leq 1$, and
exploiting (4.10), we conclude that

$\displaystyle\|\left(\mu_{m},\theta_{m}\right)\|_{L^{\infty}(0,T_{m};(\mathcal{H}^{1})^{\prime})}\leq
C.$ (4.13)

Now, due to Lemma 2.2, we have

$\displaystyle\|\left(\mu_{m},\theta_{m}\right)\|_{\mathcal{L}^{2}}^{2}\leq
2\|\left(\mu_{m},\theta_{m}\right)\|_{(\mathcal{H}^{1})^{\prime}}\|\left(\nabla\mu_{m},\nabla_{\Gamma}\theta_{m}\right)\|_{\mathcal{L}^{2}}+\|\left(\mu_{m},\theta_{m}\right)\|_{(\mathcal{H}^{1})^{\prime}}^{2}$
(4.14)

on $[0,T_{m}]$. Utilizing (4.11) and (4.13), we deduce from (4.14) that

$\displaystyle\|\left(\mu_{m},\theta_{m}\right)\|_{L^{4}(0,T_{m};\mathcal{L}^{2})}\leq
C.$ (4.15)

In view of (4.11) we additionally infer

$\displaystyle\|\left(\mu_{m},\theta_{m}\right)\|_{L^{2}(0,T_{m};\mathcal{H}^{1})}\leq
C.$ (4.16)

Lastly, we derive a uniform estimate of the time derivatives. Therefore, we
consider again
$\left(\zeta,\xi\right)\coloneqq\left(\mathbb{P}_{\mathcal{A}_{m}}(\zeta^{\ast}),\mathbb{P}_{\mathcal{B}_{m}}(\xi^{\ast})\right)$
for an arbitrary $\left(\zeta^{\ast},\xi^{\ast}\right)\in\mathcal{H}^{1}$.
Recalling that the mobility functions $m_{\Omega}$ and $m_{\Gamma}$ are
bounded (see (A3)), we use Hölder’s inequality as well as the continuous
embeddings $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$,
$H^{1}(\Gamma)\hookrightarrow L^{4}(\Gamma)$ and $H^{1}(\Omega)\hookrightarrow
L^{4}(\Gamma)$ to infer that

$\displaystyle\begin{split}\left|\langle\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right),\left(\zeta^{\ast},\xi^{\ast}\right)\rangle_{\mathcal{H}^{1}}\right|&=\left|\langle\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}}\right|\\\
&\leq\|\phi_{m}\|_{H^{1}(\Omega)}\|\boldsymbol{v}\|_{H^{1}(\Omega)}\|\zeta\|_{H^{1}(\Omega)}+C\|\mu_{m}\|_{H^{1}(\Omega)}\|\zeta\|_{H^{1}(\Omega)}\\\
&\quad+\|\psi_{m}\|_{H^{1}(\Gamma)}\|\boldsymbol{v}\|_{H^{1}(\Omega)}\|\xi\|_{H^{1}(\Gamma)}+C\|\theta_{m}\|_{H^{1}(\Gamma)}\|\xi\|_{H^{1}(\Gamma)}\\\
&\quad+L^{-1}\|\beta\theta_{m}-\mu_{m}\|_{L^{2}(\Gamma)}\|\beta\xi-\zeta\|_{L^{2}(\Gamma)}\\\
&\leq
C\Big{(}\|\left(\phi_{m},\psi_{m}\right)\|_{\mathcal{H}^{1}}+\|\left(\mu_{m},\theta_{m}\right)\|_{\mathcal{H}^{1}}\Big{)}\|\left(\zeta^{\ast},\xi^{\ast}\right)\|_{\mathcal{H}^{1}}\end{split}$

on $[0,T_{m}]$. Hence, after taking the supremum over all
$\left(\zeta^{\ast},\xi^{\ast}\right)\in\mathcal{H}^{1}$ satisfying
$\|\left(\zeta^{\ast},\xi^{\ast}\right)\|_{\mathcal{H}^{1}}\leq 1$, taking the
square of the resulting inequality and integrating in time over $[0,T_{m}]$,
we deduce that

$\displaystyle\|\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right)\|_{L^{2}(0,T_{m};(\mathcal{H}^{1})^{\prime})}^{2}\leq
C\Big{(}\|\left(\phi_{m},\psi_{m}\right)\|_{L^{\infty}(0,T_{m};\mathcal{H}^{1})}^{2}+\|\left(\mu_{m},\theta_{m}\right)\|_{L^{2}(0,T_{m};\mathcal{H}^{1})}^{2}\Big{)}.$

Exploiting the uniform estimates (4.10) and (4.16) we conclude

$\displaystyle\|\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right)\|_{L^{2}(0,T_{m};(\mathcal{H}^{1})^{\prime})}\leq
C.$ (4.17)

Step 3: Extension of the approximate solution onto $[0,T]$. Using the
definition of the approximate solution (4.1) and the uniform estimate (4.10),
we obtain for any $T_{m}\in[0,T_{m}^{\ast})$, all $t\in[0,T_{m}]$ and
$j\in\\{1,\ldots,m\\}$

$\displaystyle\left|a_{j}^{m}(t)\right|+\left|b_{j}^{m}(t)\right|$
$\displaystyle=\left|\left(\phi_{m}(t),\zeta_{j}\right)_{L^{2}(\Omega)}\right|+\left|\left(\psi_{m}(t),\xi_{j}\right)_{L^{2}(\Gamma)}\right|$
$\displaystyle\leq\|\phi_{m}\|_{L^{\infty}(0,T_{m};L^{2}(\Omega))}+\|\psi_{m}\|_{L^{\infty}(0,T_{m};L^{2}(\Gamma))}\leq
C.$

This shows that the solution $(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$ is
bounded on the time interval $[0,T_{m}^{\ast})$ by a constant that is
independent of $m$ and $T_{m}^{\ast}$. Hence, by classical ODE theory, this
allows us to extend the solution beyond the time $T_{m}^{\ast}$. However, as
$\left(\mathbf{a}^{m},\mathbf{b}^{m}\right)^{\top}$ was assumed to be non-
extendable, this is a contradiction. We thus infer that the solution
$\left(\mathbf{a}^{m},\mathbf{b}^{m}\right)^{\top}$ actually exists on
$[0,T]$. Since the coefficients
$\left(\mathbf{c}^{m},\mathbf{d}^{m}\right)^{\top}$ can be reconstructed from
$(\mathbf{a}^{m},\mathbf{b}^{m})^{\top}$, we further conclude that the
coefficients $(\mathbf{c}^{m},\mathbf{d}^{m})^{\top}$ also exist on $[0,T]$.
This automatically entails that the approximate solution
$(\phi_{m},\psi_{m},\mu_{m},\theta_{m})$ exists on $[0,T]$ and satisfies the
discretized weak formulation (4.2) on $[0,T]$ for all $m\in\mathbb{N}$.
Additionally, as the particular choice of $T_{m}$ did not play any role in the
derivation of the uniform estimates established in Step 2, we infer that the
estimates (4.10), (4.11), (4.15), (4.16) and (4.17) remain true after
replacing $T_{m}$ with $T$. In summary, we conclude that for each
$m\in\mathbb{N}$, the approximate solution
$(\phi_{m},\psi_{m},\mu_{m},\theta_{m})$ satisfies the uniform estimate

$\displaystyle\begin{split}&\|\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}+\|\left(\phi_{m},\psi_{m}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}+\|\alpha\psi_{m}-\phi_{m}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\\\
&\;\;+\|\left(\mu_{m},\theta_{m}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}+\|\left(\mu_{m},\theta_{m}\right)\|_{L^{4}(0,T;\mathcal{L}^{2})}+\|\beta\theta_{m}-\mu_{m}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (4.18)

Step 4: Convergence to a weak solution. In view of the uniform estimate
(4.18), the Banach–Alaoglu theorem and the Aubin–Lions–Simon lemma imply the
existence of functions $\phi,\psi,\mu$ and $\theta$ such that

$\displaystyle\left(\partial_{t}\phi_{m},\partial_{t}\psi_{m}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi,\partial_{t}\psi\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$ (4.19)
$\displaystyle\left(\phi_{m},\psi_{m}\right)$
$\displaystyle\rightarrow\left(\phi,\psi\right)$ $\displaystyle\text{weakly-
star in }L^{\infty}(0,T;\mathcal{H}^{1}),$ $\displaystyle\text{strongly in
}C([0,T];\mathcal{H}^{s})\text{ for all }s\in[0,1),$ (4.20)
$\displaystyle\left(\mu_{m},\theta_{m}\right)$
$\displaystyle\rightarrow\left(\mu,\theta\right)$ $\displaystyle\text{weakly
in }L^{4}(0,T;\mathcal{L}^{2})\cap L^{2}(0,T;\mathcal{H}^{1}),$ (4.21)
$\displaystyle\beta\theta_{m}-\mu_{m}$
$\displaystyle\rightarrow\beta\theta-\mu$ $\displaystyle\text{weakly in
}L^{2}(0,T;L^{2}(\Gamma)),$ (4.22) $\displaystyle\alpha\psi_{m}-\phi_{m}$
$\displaystyle\rightarrow\alpha\psi-\phi$ $\displaystyle\text{weakly-star in
}L^{\infty}(0,T;L^{2}(\Gamma)),$ (4.23)

as $m\rightarrow\infty$, along a non-relabeled subsequence. From these
convergences, we readily obtain the desired regularities (3.2a)–(3.2b) of the
functions $\phi,\psi,\mu$ and $\theta$. Hence, Definition 3.1(i) is fulfilled.

Using Sobolev’s embedding theorem, we infer from (4.20) that

$\displaystyle\phi_{m}$ $\displaystyle\rightarrow\phi\quad\quad$
$\displaystyle\text{strongly in }C([0,T];L^{r}(\Omega))\text{ for all
}r\in[2,6),\text{ and a.e.\leavevmode\nobreak\ in }Q,$ (4.24)
$\displaystyle\psi_{m}$ $\displaystyle\rightarrow\psi$
$\displaystyle\text{strongly in }C([0,T];L^{r}(\Gamma))\text{ for all
}r\in[2,\infty),\text{ and a.e.\leavevmode\nobreak\ on }\Sigma,$ (4.25)

as $m\rightarrow\infty$, after another subsequence extraction. Further, by the
trace theorem, (4.20) additionally implies that

$\displaystyle\alpha\psi_{m}-\phi_{m}\rightarrow\alpha\psi-\phi\quad\quad\text{strongly
in }C([0,T];L^{2}(\Gamma))$ (4.26)

as $m\rightarrow\infty$, after another subsequence extraction. Moreover, due
to the growth conditions on $F^{\prime}$ from (A4), the uniform estimate
(4.10) and the Sobolev embedding $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$,
we deduce that $F^{\prime}(\phi_{m})$ is bounded in $L^{6/5}(Q)$ uniformly in
$m\in\mathbb{N}$. Hence, there exists a function $f\in L^{6/5}(Q)$ such that
$F^{\prime}(\phi_{m})\rightarrow f$ weakly in $L^{6/5}(Q)$ as
$m\rightarrow\infty$. Considering that $F^{\prime}(\phi_{m})\rightarrow
F^{\prime}(\phi)$ a.e. in $Q$ as $m\rightarrow\infty$ due to (4.24), we infer
from a convergence principle based on Egorov’s theorem that
$F^{\prime}(\phi)=f$. We thus conclude that

$\displaystyle F^{\prime}(\phi_{m})\rightarrow
F^{\prime}(\phi)\quad\quad\text{weakly in }L^{6/5}(Q)\text{ and
a.e.\leavevmode\nobreak\ in }Q$ (4.27)

as $m\rightarrow\infty$. Recalling the growth conditions on $G$ and
$G^{\prime}$ (see (A4)), the convergence in (4.25) in combination with
Lebesgue’s general convergence theorem yields

$\displaystyle G(\psi_{m})$ $\displaystyle\rightarrow G(\psi)\quad\quad$
$\displaystyle\text{strongly in }L^{1}(\Sigma),\text{ and
a.e.\leavevmode\nobreak\ on }\Sigma,$ (4.28) $\displaystyle
G^{\prime}(\psi_{m})$ $\displaystyle\rightarrow G^{\prime}(\psi)$
$\displaystyle\text{strongly in }L^{2}(\Sigma),\text{ and
a.e.\leavevmode\nobreak\ on }\Sigma,$ (4.29)

as $m\rightarrow\infty$. Furthermore, by means of Lebesgue’s dominated
convergence theorem, it follows from (4.24), (4.25) and (A3) that

$\displaystyle m_{\Omega}(\phi_{m})$ $\displaystyle\rightarrow
m_{\Omega}(\phi)\quad$ $\displaystyle\text{strongly in }L^{r}(Q),\text{ and
a.e.\leavevmode\nobreak\ in Q},$ (4.30) $\displaystyle m_{\Gamma}(\psi_{m})$
$\displaystyle\rightarrow m_{\Gamma}(\psi)$ $\displaystyle\text{strongly in
}L^{r}(\Sigma),\text{ and a.e.\leavevmode\nobreak\ on }\Sigma,$ (4.31)

for all $r\in[2,\infty)$, as $m\rightarrow\infty$. Moreover, as the mobility
functions $m_{\Omega}$ and $m_{\Gamma}$ are bounded (see (A3)), we use
Lebesgue’s general convergence theorem along with the weak-strong convergence
principle to find that

$\displaystyle\sqrt{m_{\Omega}(\phi_{m})}\nabla\mu_{m}$
$\displaystyle\rightarrow\sqrt{m_{\Omega}(\phi)}\nabla\mu\quad\quad$
$\displaystyle\text{weakly in }L^{2}(Q),$ (4.32)
$\displaystyle\sqrt{m_{\Gamma}(\psi_{m})}\nabla_{\Gamma}\theta_{m}$
$\displaystyle\rightarrow\sqrt{m_{\Gamma}(\psi)}\nabla_{\Gamma}\theta$
$\displaystyle\text{weakly in }L^{2}(\Sigma),$ (4.33) $\displaystyle
m_{\Omega}(\phi_{m})\nabla\mu_{m}$ $\displaystyle\rightarrow
m_{\Omega}(\phi)\nabla\mu$ $\displaystyle\text{weakly in }L^{2}(Q),$ (4.34)
$\displaystyle m_{\Gamma}(\psi_{m})\nabla_{\Gamma}\theta_{m}$
$\displaystyle\rightarrow m_{\Gamma}(\psi)\nabla_{\Gamma}\theta$
$\displaystyle\text{weakly in }L^{2}(\Sigma),$ (4.35)

as $m\rightarrow\infty.$ Lastly, exploiting the convergences (4.21), (4.24)
and (4.25), and using again the weak-strong convergence principle, we infer

$\displaystyle\phi_{m}\,\boldsymbol{v}\cdot\nabla\mu_{m}$
$\displaystyle\rightarrow\phi\,\boldsymbol{v}\cdot\nabla\mu\quad\quad\quad$
$\displaystyle\text{strongly in }L^{1}(Q),\text{ and a.e.\leavevmode\nobreak\
in }Q,$ (4.36)
$\displaystyle\psi_{m}\,\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta_{m}$
$\displaystyle\rightarrow\psi\,\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta$
$\displaystyle\text{strongly in }L^{1}(\Sigma),\text{ and
a.e.\leavevmode\nobreak\ on }\Sigma,$ (4.37)

as $m\rightarrow\infty$.

We now multiply both equations of the discretized weak formulation (4.2) with
an arbitrary test function $\sigma\in C_{c}^{\infty}([0,T])$ and integrate in
time from $0$ to $T$. Using the convergences (4.19)–(4.37) established above,
we may pass to the limit $m\rightarrow\infty$ to infer that

$\displaystyle\int_{0}^{T}\langle\left(\partial_{t}\phi,\partial_{t}\psi\right),\left(\zeta_{j},\xi_{j}\right)\rangle_{\mathcal{H}^{1}}\,\sigma\;\mathrm{d}t-\int_{0}^{T}\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\zeta_{j}\,\sigma\;\mathrm{d}x\;\mathrm{d}t-\int_{0}^{T}\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\xi_{j}\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}m_{\Omega}(\phi)\nabla\mu\cdot\nabla\zeta_{j}\,\sigma\;\mathrm{d}x\;\mathrm{d}t-\int_{0}^{T}\int_{\Gamma}m_{\Gamma}(\psi)\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}\xi_{j}\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t$
(4.38a)
$\displaystyle\quad-L^{-1}\int_{0}^{T}\int_{\Gamma}(\beta\theta-\mu)(\beta\xi_{j}-\zeta_{j})\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t,$
$\displaystyle\int_{0}^{T}\int_{\Omega}\mu\,\zeta_{j}\,\sigma\;\mathrm{d}x\;\mathrm{d}t+\int_{0}^{T}\int_{\Gamma}\theta\,\xi_{j}\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t$
(4.38b)
$\displaystyle=\int_{0}^{T}\int_{\Omega}\nabla\phi\cdot\nabla\zeta_{j}\,\sigma+F^{\prime}(\phi)\zeta_{j}\,\sigma\;\mathrm{d}x\;\mathrm{d}t+\int_{0}^{T}\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}\xi_{j}\,\sigma+G^{\prime}(\psi)\xi_{j}\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t$
$\displaystyle\quad+K^{-1}\int_{0}^{T}\int_{\Gamma}(\alpha\psi-\phi)(\alpha\xi_{j}-\zeta_{j})\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t,$

hold for all $j\in\mathbb{N}$ and all test functions $\sigma\in
C_{c}^{\infty}([0,T])$. Since
$\text{span}\\{\left(\zeta_{j},\xi_{j}\right):j\in\mathbb{N}\\}$ is dense in
$\mathcal{H}^{1}$, this proves that the quadruplet $(\phi,\psi,\mu,\theta)$
satisfies the weak formulation (3.4) for all test functions
$\left(\zeta,\xi\right)\in\mathcal{H}^{1}$. We thus conclude that Definition
3.1(iii) is fulfilled.

Since the approximate solutions $\left(\phi_{m},\psi_{m}\right)$ satisfy the
discrete mass conservation law (see (4.8)) for all $m\in\mathbb{N}$, we can
use the convergence (4.20) to deduce that $\left(\phi,\psi\right)$ satisfies
the mass conservation law (3.5) as well. Thus, Definition 3.1(iv) is
fulfilled.
Concerning the initial conditions, we infer from (4.3), due to the convergence
properties of the projections, that

$\displaystyle\left(\phi_{m}(0),\psi_{m}(0)\right)\rightarrow\left(\phi_{0},\psi_{0}\right)\quad\text{strongly
in }\mathcal{L}^{2}$

as $m\rightarrow\infty$. Additionally, the convergences (4.24) and (4.25)
yield

$\displaystyle\left(\phi_{m}(0),\psi_{m}(0)\right)\rightarrow\left(\phi(0),\psi(0)\right)\quad\text{strongly
in }\mathcal{L}^{2}$

as $m\rightarrow\infty$. We deduce that $\phi(0)=\phi_{0}$ a.e. in $\Omega$
and $\psi(0)=\psi_{0}$ a.e. on $\Gamma$. This proves that Definition 3.1(ii)
is fulfilled.

To verify the energy inequality (3.6), we consider an arbitrary non-negative
test function $\sigma\in C_{c}^{\infty}(0,T)$. Multiplying (4.4) with $\sigma$
and integrating in time from $0$ to $T$ yields

$\displaystyle\begin{split}&\int_{0}^{T}E_{K}(\phi_{m}(t),\psi_{m}(t))\,\sigma(t)\;\mathrm{d}t\\\
&\quad\quad+\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\left(m_{\Omega}(\phi_{m})\left|\nabla\mu_{m}\right|^{2}-\phi_{m}\boldsymbol{v}\cdot\nabla\mu_{m}\right)\sigma(t)\;\mathrm{d}x\;\mathrm{d}s\;\mathrm{d}t\\\
&\quad\quad+\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}\left(m_{\Gamma}(\psi_{m})\left|\nabla_{\Gamma}\theta_{m}\right|^{2}-\psi_{m}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta_{m}\right)\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\\\
&\quad\quad+L^{-1}\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}(\beta\theta_{m}-\mu_{m})^{2}\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\\\
&\
=\int_{0}^{T}E_{K}(\phi_{m}(0),\psi_{m}(0))\,\sigma(t)\;\mathrm{d}t.\end{split}$
(4.39)

As $m\rightarrow\infty$, we infer from (4.28) that

$\displaystyle\int_{0}^{T}\left(\int_{\Gamma}G(\psi_{m})\;\mathrm{d}\Gamma\right)\sigma(t)\;\mathrm{d}t\rightarrow\int_{0}^{T}\left(\int_{\Gamma}G(\psi)\;\mathrm{d}\Gamma\right)\sigma(t)\;\mathrm{d}t.$
(4.40)

Additionally, using (4.24) and (A4), Fatou’s lemma implies

$\displaystyle\int_{0}^{T}\left(\int_{\Omega}F(\phi)\;\mathrm{d}x\right)\sigma(t)\;\mathrm{d}t\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}\left(\int_{\Omega}F(\phi_{m})\;\mathrm{d}x\right)\sigma(t)\;\mathrm{d}t.$
(4.41)

Moreover, from the strong convergence in (4.26) we obtain

$\displaystyle\int_{0}^{T}\left(\int_{\Gamma}\left|\alpha\psi_{m}-\phi_{m}\right|^{2}\;\mathrm{d}\Gamma\right)\sigma(t)\;\mathrm{d}t\rightarrow\int_{0}^{T}\left(\int_{\Gamma}\left|\alpha\psi-\phi\right|^{2}\;\mathrm{d}\Gamma\right)\sigma(t)\;\mathrm{d}t,$
(4.42)

as $m\rightarrow\infty$. Lastly, in view of (4.20) and the weak lower
semicontinuity of norms, we have

$\displaystyle\begin{split}&\int_{0}^{T}\left(\frac{1}{2}\|\nabla\phi\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|\nabla_{\Gamma}\psi\|_{L^{2}(\Gamma)}^{2}\right)\sigma(t)\;\mathrm{d}t\\\
&\quad\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}\left(\frac{1}{2}\|\nabla\phi_{m}\|^{2}_{L^{2}(\Omega)}+\frac{1}{2}\|\nabla_{\Gamma}\psi_{m}\|^{2}_{L^{2}(\Gamma)}\right)\sigma(t)\;\mathrm{d}t.\end{split}$
(4.43)

Combining (4.40)–(4.43) yields

$\displaystyle\int_{0}^{T}E_{K}(\phi(t),\psi(t))\,\sigma(t)\;\mathrm{d}t\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}E_{K}(\phi_{m}(t),\psi_{m}(t))\,\sigma(t)\;\mathrm{d}t.$
(4.44)

As a consequence of (4.32), (4.33) and the weak lower semicontinuity of norms
with respect to the weak convergence, another application of Fatou’s lemma
entails that

$\displaystyle\begin{split}&\int_{0}^{T}\int_{0}^{t}\int_{\Omega}m_{\Omega}(\phi)\left|\nabla\mu\right|^{2}\,\sigma(t)\;\mathrm{d}x\;\mathrm{d}s\;\mathrm{d}t+\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}m_{\Gamma}(\psi)\left|\nabla_{\Gamma}\theta\right|^{2}\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\\\
&\leq\liminf_{m\rightarrow\infty}\left[\int_{0}^{T}\int_{0}^{t}\int_{\Omega}m_{\Omega}(\phi_{m})\left|\nabla\mu_{m}\right|^{2}\,\sigma(t)\;\mathrm{d}x\;\mathrm{d}s\;\mathrm{d}t\right.\\\
&\qquad\qquad\qquad\left.+\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}m_{\Gamma}(\psi_{m})\left|\nabla_{\Gamma}\theta_{m}\right|^{2}\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\right].\end{split}$
(4.45)

Further, due to the convergences (4.36) and (4.37), we find that

$\displaystyle\begin{split}&\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\phi_{m}\boldsymbol{v}\cdot\nabla\mu_{m}\,\sigma(t)\;\mathrm{d}x\;\mathrm{d}s\;\mathrm{d}t+\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}\psi_{m}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta_{m}\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\\\
&\quad\longrightarrow\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\mu\,\sigma(t)\;\mathrm{d}x\;\mathrm{d}s\;\mathrm{d}t+\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\end{split}$
(4.46)

as $m\rightarrow\infty$. Additionally, as convex and continuous functionals
are weakly lower semicontinuous, we conclude from (4.22) that

$\displaystyle\liminf_{m\rightarrow\infty}\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}(\beta\theta_{m}-\mu_{m})^{2}\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t\leq\int_{0}^{T}\int_{0}^{t}\int_{\Gamma}(\beta\theta-\mu)^{2}\,\sigma(t)\;\mathrm{d}\Gamma\;\mathrm{d}s\;\mathrm{d}t.$
(4.47)

Lastly, recalling the growth conditions on $F$ and $G$ (see (A4)), we use the
initial conditions (4.3) as well as the convergence properties of the
projections $\mathbb{P}_{\mathcal{A}_{m}}$ and $\mathbb{P}_{\mathcal{B}_{m}}$
together with Lebesgue’s general convergence theorem to infer that

$\displaystyle\lim_{m\rightarrow\infty}E_{K}(\phi_{m}(0),\psi_{m}(0))=E_{K}(\phi_{0},\psi_{0}).$
(4.48)

Collecting (4.45)–(4.48) finally yields

$\displaystyle\begin{split}&E_{K}(\phi(t),\psi(t))+\int_{0}^{t}\int_{\Omega}m_{\Omega}(\phi)\left|\nabla\mu\right|^{2}\;\mathrm{d}x\;\mathrm{d}s+\int_{0}^{t}\int_{\Gamma}m_{\Gamma}(\psi)\left|\nabla_{\Gamma}\theta\right|^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad-\int_{0}^{t}\int_{\Omega}\phi\boldsymbol{v}\cdot\nabla\mu\;\mathrm{d}x\;\mathrm{d}s-\int_{0}^{t}\int_{\Gamma}\psi\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\theta\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\quad+L^{-1}\int_{0}^{t}\int_{\Gamma}(\beta\theta-\mu)^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\leq E_{K}(\phi_{0},\psi_{0})\end{split}$ (4.49)

for almost every $t\in[0,T]$. To prove that (4.49) holds for every
$t\in[0,T]$, first note that all integral terms in (4.49) depend continuously
on time. Furthermore, due to $(\phi,\psi)\in C([0,T];\mathcal{L}^{2})$, we
deduce that the following functions

$\displaystyle t\mapsto\|\nabla\phi(t)\|_{L^{2}(\Omega)}^{2},\quad
t\mapsto\|\nabla_{\Gamma}\psi(t)\|_{L^{2}(\Gamma)}^{2},\quad
t\mapsto\int_{\Omega}F(\phi(t))\;\mathrm{d}x,\quad
t\mapsto\int_{\Gamma}G(\psi(t))\;\mathrm{d}\Gamma$

are lower semicontinuous on $[0,T]$. For the first two functions, this is a
consequence of the lower semicontinuity of the respective norms, while for the
last two functions, it follows from Fatou’s lemma. This already entails that
the weak solution $(\phi,\psi,\mu,\theta)$ satisfies the energy inequality for
all times $t\in[0,T]$. Thus, Definition 3.1(v) is fulfilled.
We have therefore shown that the quadruplet $(\phi,\psi,\mu,\theta)$ is a weak
solution of system (3.1) in the sense of Definition 3.1. ∎

######  Remark 4.2.

This proof should work similarly in the cases $K=L=0$ and $K=L=\infty$. In the
first case, one uses a Faedo–Galerkin scheme as in [23] based on
eigenfunctions of a suitable bulk-surface elliptic problem with Dirichlet type
coupling condition (cf. [30, Theorem 3.3]). In the second case, the bulk-
surface Cahn–Hilliard equation (3.1) reduces to two, uncoupled Cahn–Hilliard
equations, one in $\Omega$ and one on $\Gamma$. Therefore, a Faedo–Galerkin
basis as in the above proof can be used. Also note that, while we do not have
the bulk-surface Poincaré inequality at our disposal anymore, we can use the
standard Poincaré inequality for functions with vanishing mean, since the
approximate solutions satisfy the mass conservation law (3.5) for $L=\infty$.

## 5 Asymptotic limits and existence of weak solutions to the limit models

In this section, we investigate the asymptotic limits $K\rightarrow 0$ and
$K\rightarrow\infty$, and $L\rightarrow 0$ and $L\rightarrow\infty$ of the
system (3.1).

######  Remark 5.1.

In this section, we will need the additional assumption (A5) when approaching
the limit cases $K\in\\{0,\infty\\}$. This is because the constant in the
bulk-surface Poincaré inequality (P6) depends on $K$ in some way, but we do
not know this dependence explicitly. In particular, it is unclear how this
constant behaves if we send $K$ to zero or infinity, respectively. Therefore,
we cannot rely on this Poincaré inequality to obtain suitable uniform bounds,
but instead, we use (A5) to directly obtain uniform bounds from the energy
functional.

### 5.1 The limit $K\rightarrow 0$ and the existence of a weak solution if
$(K,L)\in\\{0\\}\times(0,\infty)$

######  Theorem 5.2.

(Asymptotic limit $K\rightarrow 0$) Suppose that the assumptions (A1)–(A5)
hold, let $\left(\phi_{0},\psi_{0}\right)\in\mathcal{D}_{\alpha}$ be arbitrary
initial data, $L\in(0,\infty)$ and $\boldsymbol{v}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}^{1}_{\mathbf{n}})$.
For any $K\in(0,\infty)$, let $(\phi_{K},\psi_{K},\mu_{K},\theta_{K})$ denote
a weak solution of the system (3.1) in the sense of Definition 3.1 with
initial data $\left(\phi_{0},\psi_{0}\right)$. Then, there exists a quadruplet
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ with
$\phi_{\ast}=\alpha\psi_{\ast}$ a.e. on $\Sigma$ such that

$\displaystyle\left(\partial_{t}\phi_{K},\partial_{t}\psi_{K}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi_{\ast},\partial_{t}\psi_{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$
$\displaystyle\left(\phi_{K},\psi_{K}\right)$
$\displaystyle\rightarrow\left(\phi_{\ast},\psi_{\ast}\right)$
$\displaystyle\text{weakly-star in }L^{\infty}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{strongly in }C([0,T];\mathcal{H}^{s})\text{ for all
}s\in[0,1),$ $\displaystyle\left(\mu_{K},\theta_{K}\right)$
$\displaystyle\rightarrow\left(\mu_{\ast},\theta_{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\alpha\psi_{K}-\phi_{K}$ $\displaystyle\rightarrow 0$
$\displaystyle\text{strongly in }L^{\infty}(0,T,L^{2}(\Gamma)),$

as $K\rightarrow 0$, up to subsequence extraction, with

$\displaystyle\|\alpha\psi_{K}-\phi_{K}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\leq
C\sqrt{K},$ (5.1)

and the limit $(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ is a weak
solution to the system (3.1) in the sense of Definition 3.1 with $K=0$.

######  Remark 5.3.

  1. (a)

As the right-hand side in (5.1) tends to zero as $K\to 0$, this explains why
the Dirichlet type boundary condition $\phi_{*}=\alpha\psi_{*}$ a.e. on
$\Sigma$ appears in the limit model corresponding to $K=0$.

  2. (b)

The result of Theorem 5.2 remains valid if we replace the initial data
$\left(\phi_{0},\psi_{0}\right)\in\mathcal{D}_{\alpha}$ by a sequence
$\left(\phi_{0,K},\psi_{0,K}\right)\in\mathcal{H}^{1}$ satisfying

$\displaystyle E_{K}(\phi_{0,K},\psi_{0,K})\rightarrow
E_{0}(\phi_{0},\psi_{0})\quad\text{as $K\rightarrow 0$}$

for some pair $\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}$ (see also
[28, Theorem 2.3]). In this case, one can show that $\phi_{0}=\alpha\psi_{0}$
a.e. on $\Gamma$, and that $\left(\phi_{\ast},\psi_{\ast}\right)$ is a weak
solution corresponding to the initial data $\left(\phi_{0},\psi_{0}\right)$.

  * Proof of Theorem 5.2.

We consider an arbitrary sequence $(K_{m})_{m\in\mathbb{N}}\subset(0,\infty)$
such that $K_{m}\rightarrow 0$ as $m\rightarrow\infty$ and a corresponding
sequence of weak solutions
$(\phi_{K_{m}},\psi_{K_{m}},\mu_{K_{m}},\theta_{K_{m}})$ to the initial data
$\left(\phi_{0},\psi_{0}\right)$. In this proof, we use the letter $C$ to
denote generic positive constants independent of $K_{m}$ and $m$. In order to
prove suitable uniform estimates, we make use of the energy inequality and the
additional growth assumptions made on $F$ and $G$ (see (A5)). Let now
$m\in\mathbb{N}$ be arbitrary. First, note that $\phi_{0}=\alpha\psi_{0}$ a.e.
on $\Gamma$ directly implies

$\displaystyle E_{K_{m}}(\phi_{0},\psi_{0})=E_{0}(\phi_{0},\psi_{0})\leq C.$
(5.2)

Next, due to the growth condition (A5), we find that

$\displaystyle\begin{split}&\frac{1}{2}\int_{\Omega}\left|\nabla\phi_{K_{m}}\right|^{2}\;\mathrm{d}x+\int_{\Omega}F(\phi_{K_{m}})\;\mathrm{d}x+\frac{1}{2}\int_{\Gamma}\left|\nabla_{\Gamma}\psi_{K_{m}}\right|^{2}\;\mathrm{d}\Gamma+\int_{\Gamma}G(\psi_{K_{m}})\;\mathrm{d}\Gamma\\\
&\geq\frac{1}{2}\int_{\Omega}\left|\nabla\phi_{K_{m}}\right|^{2}\;\mathrm{d}x+a_{F}\int_{\Omega}\left|\phi_{K_{m}}\right|^{2}\;\mathrm{d}x-b_{F}\left|\Omega\right|\\\
&\qquad+\frac{1}{2}\int_{\Gamma}\left|\nabla_{\Gamma}\psi_{K_{m}}\right|^{2}\;\mathrm{d}\Gamma+a_{G}\int_{\Gamma}\left|\psi_{K_{m}}\right|^{2}\;\mathrm{d}\Gamma-
b_{G}\left|\Gamma\right|\\\ &\geq
c\|\left(\phi_{K_{m}},\psi_{K_{m}}\right)\|^{2}_{\mathcal{H}^{1}}-b_{F}\left|\Omega\right|-b_{G}\left|\Gamma\right|,\end{split}$

where $c=\min\\{\frac{1}{2},a_{F},a_{G}\\}$. Then, using the energy inequality
(3.6), and bounding the convective terms as in (Proof.), we deduce that

$\displaystyle\begin{split}&c\|\left(\phi_{K_{m}},\psi_{K_{m}}\right)\|_{\mathcal{H}^{1}}^{2}+\frac{1}{2}K_{m}^{-1}\int_{\Gamma}\left|\alpha\psi_{K_{m}}-\phi_{K_{m}}\right|^{2}\;\mathrm{d}\Gamma+\frac{m_{\Omega}^{*}}{2}\int_{0}^{t}\int_{\Omega}\left|\nabla\mu_{K_{m}}\right|^{2}\;\mathrm{d}x\;\mathrm{d}s\\\
&\quad+\frac{m_{\Gamma}^{*}}{2}\int_{0}^{t}\int_{\Gamma}\left|\nabla_{\Gamma}\theta_{K_{m}}\right|^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s+L^{-1}\int_{0}^{t}\int_{\Gamma}(\beta\theta_{K_{m}}-\mu_{K_{m}})^{2}\;\mathrm{d}\Gamma\;\mathrm{d}s\\\
&\leq
C+C\int_{0}^{t}\left(\|\boldsymbol{v}\|^{2}_{L^{3}(\Omega)}+\|\boldsymbol{v}_{\tau}\|_{L^{3}(\Gamma)}^{2}\right)\|\left(\phi_{K_{m}},\psi_{K_{m}}\right)\|_{\mathcal{H}^{1}}^{2}\;\mathrm{d}s\end{split}$

a.e. on $[0,T]$. Thus, using Gronwall’s lemma, we readily infer that

$\displaystyle\begin{split}&\|\left(\phi_{K_{m}},\psi_{K_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}+K_{m}^{-1/2}\|\alpha\psi_{K_{m}}-\phi_{K_{m}}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\\\
&\quad+\|\left(\nabla\mu_{K_{m}},\nabla_{\Gamma}\theta_{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{L}^{2})}+\|\beta\theta_{K_{m}}-\mu_{K_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (5.3)

This already entails (5.1). We now test the weak formulation (3.4b) for
$(\mu_{K_{m}},\theta_{K_{m}})$ with $(\eta,\vartheta)\in\mathcal{D}_{\alpha}$.
As $\eta=\alpha\vartheta$ a.e. on $\Gamma$, the corresponding boundary term
involving $K_{m}$ vanishes. Therefore, arguing as in the proof of Theorem 3.2,
we deduce

$\displaystyle\|\left(\mu_{K_{m}},\theta_{K_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{D}_{\alpha}^{\prime})}\leq
C.$ (5.4)

Now, using Ehrling’s lemma, we find that

$\displaystyle\|\left(\mu_{K_{m}},\theta_{K_{m}}\right)\|_{\mathcal{L}^{2}}\leq\|\left(\nabla\mu_{K_{m}},\nabla_{\Gamma}\theta_{K_{m}}\right)\|_{\mathcal{L}^{2}}+C\|\left(\mu_{K_{m}},\theta_{K_{m}}\right)\|_{\mathcal{D}_{\alpha}^{\prime}}.$

In combination with (5.3) and (5.4), this yields

$\displaystyle\|\left(\mu_{K_{m}},\theta_{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}\leq
C.$ (5.5)

Lastly, proceeding similarly as in the derivation of (4.17) in the proof of
Theorem 3.2, it follows from the weak formulation (3.4a) and estimate (5.3)
that

$\displaystyle\|\left(\partial_{t}\phi_{K_{m}},\partial_{t}\psi_{K_{m}}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}\leq
C.$ (5.6)

In summary, combining (5.3), (5.5) and (5.6), we have thus shown that

$\displaystyle\begin{split}&\|\left(\partial_{t}\phi_{K_{m}},\partial_{t}\psi_{K_{m}}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}+\|\left(\phi_{K_{m}},\psi_{K_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}\\\
&\quad+\|\left(\mu_{K_{m}},\theta_{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}+\|\beta\theta_{K_{m}}-\mu_{K_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (5.7)

Using the Banach–Alaoglu theorem and the Aubin–Lions–Simon lemma, we deduce
the existence of functions
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ such that

$\displaystyle\left(\partial_{t}\phi_{K_{m}},\partial_{t}\psi_{K_{m}}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi_{\ast},\partial_{t}\psi_{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$ (5.8)
$\displaystyle\left(\phi_{K_{m}},\psi_{K_{m}}\right)$
$\displaystyle\rightarrow\left(\phi_{\ast},\psi_{\ast}\right)$
$\displaystyle\text{weakly-star in }L^{\infty}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{strongly in }C([0,T];\mathcal{H}^{s})\text{ for all
}s\in[0,1),$ (5.9) $\displaystyle\left(\mu_{K_{m}},\theta_{K_{m}}\right)$
$\displaystyle\rightarrow\left(\mu_{\ast},\theta_{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$ (5.10)

as $m\rightarrow\infty$, along a non-relabeled subsequence. Additionally, due
to (5.9) and the trace theorem, we infer

$\displaystyle\alpha\psi_{K_{m}}-\phi_{K_{m}}\rightarrow\alpha\psi_{\ast}-\phi_{\ast}\quad\quad\text{strongly
in }C([0,T];L^{2}(\Gamma))$ (5.11)

as $m\rightarrow\infty$, which, in combination with (5.3), already entails
$\phi_{\ast}=\alpha\psi_{\ast}$ a.e. on $\Sigma$, i.e.,
$\left(\phi_{\ast}(t),\psi_{\ast}(t)\right)\in\mathcal{D}_{\alpha}$ for almost
all $t\in[0,T]$. It is then clear that from the convergence properties
(5.8)–(5.10) the quadruplet
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ has the desired
regularity as stated in Definition 3.1(i).
Further, due to (5.9), we infer that

$\displaystyle\left(\phi_{K_{m}}(0),\psi_{K_{m}}(0)\right)\rightarrow\left(\phi_{\ast}(0),\psi_{\ast}(0)\right)\quad\text{strongly
in }\mathcal{L}^{2}$ (5.12)

as $m\rightarrow\infty$, and therefore $\phi_{\ast}(0)=\phi_{0}$ a.e. in
$\Omega$ and $\psi_{\ast}(0)=\psi_{0}$ a.e. on $\Gamma$. Thus, Definition
3.1(ii) is fulfilled.
Regarding the weak formulation, we consider
$\left(\zeta,\xi\right)\in\mathcal{H}^{1}$ and
$\left(\eta,\vartheta\right)\in\mathcal{D}_{\alpha}$ as test functions,
multiply the resulting equations with $\sigma\in C_{c}^{\infty}([0,T])$ and
integrate in time from $0$ to $T$. We obtain

$\displaystyle\begin{split}&\int_{0}^{T}\langle\left(\partial_{t}\phi_{K_{m}},\partial_{t}\psi_{K_{m}}\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}^{1}}\,\sigma\;\mathrm{d}t-\int_{0}^{T}\int_{\Omega}\phi_{K_{m}}\boldsymbol{v}\cdot\nabla\zeta\,\sigma\;\mathrm{d}x\;\mathrm{d}t\\\
&\quad-\int_{0}^{T}\int_{\Gamma}\psi_{K_{m}}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\xi\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t\\\
&=-\int_{0}^{T}\int_{\Omega}m_{\Omega}(\phi_{K_{m}})\nabla\mu_{K_{m}}\cdot\nabla\zeta\,\sigma\;\mathrm{d}x\;\mathrm{d}t\\\
&\quad-\int_{0}^{T}\int_{\Gamma}m_{\Gamma}(\psi_{K_{m}})\nabla_{\Gamma}\theta_{K_{m}}\cdot\nabla_{\Gamma}\xi\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t\\\
&\quad-L^{-1}\int_{0}^{T}\int_{\Gamma}(\beta\theta_{K_{m}}-\mu_{K_{m}})(\beta\xi-\zeta)\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t,\end{split}$
(5.13a)
$\displaystyle\begin{split}&\int_{0}^{T}\int_{\Omega}\mu_{K_{m}}\,\eta\,\sigma\;\mathrm{d}x\;\mathrm{d}t+\int_{0}^{T}\int_{\Gamma}\theta_{K_{m}}\,\vartheta\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t\\\
&=\int_{0}^{T}\int_{\Omega}\nabla\phi_{K_{m}}\cdot\nabla\eta\,\sigma+F^{\prime}(\phi_{K_{m}})\eta\,\sigma\;\mathrm{d}x\;\mathrm{d}t\\\
&\quad+\int_{0}^{T}\int_{\Gamma}\nabla_{\Gamma}\psi_{K_{m}}\cdot\nabla_{\Gamma}\vartheta\,\sigma+G^{\prime}(\psi_{K_{m}})\vartheta\,\sigma\;\mathrm{d}\Gamma\;\mathrm{d}t.\end{split}$
(5.13b)

Proceeding similarly as in the proof of Theorem 3.2, the convergences
(5.8)–(5.10) allow us to pass to the limit $m\rightarrow\infty$, and deduce
that $(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ satisfies the
desired weak formulation (3.4). Thus, Definition 3.1(iii) is fulfilled.
The mass conservation law follows by passing to the limit $m\rightarrow\infty$
in the mass conservation law for $\left(\phi_{K_{m}},\psi_{K_{m}}\right)$.
Alternatively, one can test the weak formulation (3.4a) with
$\left(\beta,1\right)\in\mathcal{H}^{1}$, integrate with respect to time from
$0$ to $t$, and employ the fundamental theorem of calculus to infer (3.5). We
conclude that Definition 3.1(iv) is satisfied.
As the boundary term involving $K_{m}$ in the energy is non-negative, we note
that

$\displaystyle E_{0}(\phi_{K_{m}}(t),\psi_{K_{m}}(t))\leq
E_{K_{m}}(\phi_{K_{m}}(t),\psi_{K_{m}}(t))$ (5.14)

for all $t\in[0,T]$, and thus

$\displaystyle\begin{split}\int_{0}^{T}E_{0}(\phi_{\ast}(t),\psi_{\ast}(t))\,\sigma(t)\;\mathrm{d}t&\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}E_{0}(\phi_{K_{m}}(t),\psi_{K_{m}}(t))\,\sigma(t)\;\mathrm{d}t\\\
&\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}E_{K_{m}}(\phi_{K_{m}}(t),\psi_{K_{m}}(t))\,\sigma(t)\;\mathrm{d}t\end{split}$
(5.15)

for all non-negative test functions $\sigma\in C_{c}^{\infty}(0,T)$. Here, the
first inequality follows by proceeding similarly as in the proof of Theorem
3.2. We now use the energy inequality (3.6) written for
$(\phi_{K_{m}},\psi_{K_{m}},\mu_{K_{m}},\theta_{K_{m}})$ to further bound the
right-hand side of (5.15). Using lower semicontinuity arguments (similar to
those in the proof of Theorem 3.2), and recalling (5.2), passing to the limit
$m\to\infty$ leads to the corresponding energy inequality for
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$.
This proves that the quadruplet
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ is a weak solution of
(3.1) in the sense of Definition 3.1 with $K=0$. ∎

### 5.2 The limit $K\rightarrow\infty$ and the existence of a weak solution if
$(K,L)\in\\{\infty\\}\times(0,\infty)$

######  Theorem 5.4.

(Asymptotic limit $K\rightarrow\infty$) Suppose that the assumptions (A1)–(A5)
hold, let $\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}$ be arbitrary
initial data, $L\in(0,\infty)$ and $\boldsymbol{v}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}^{1}_{\mathbf{n}})$.
For any $K\in(0,\infty)$, let $(\phi_{K},\psi_{K},\mu_{K},\theta_{K})$ denote
a weak solution of the system (3.1) in the sense of Definition 3.1 with
initial data $\left(\phi_{0},\psi_{0}\right)$. Then, there exists a quadruplet
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ such that

$\displaystyle\left(\partial_{t}\phi^{K},\partial_{t}\psi^{K}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi^{\ast},\partial_{t}\psi^{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$
$\displaystyle\left(\phi^{K},\psi^{K}\right)$
$\displaystyle\rightarrow\left(\phi^{\ast},\psi^{\ast}\right)$
$\displaystyle\text{weakly-star in }L^{\infty}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{strongly in }C([0,T];\mathcal{H}^{s})\text{ for all
}s\in[0,1),$ $\displaystyle\left(\mu^{K},\theta^{K}\right)$
$\displaystyle\rightarrow\left(\mu^{\ast},\theta^{\ast}\right)$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\cap
L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\frac{1}{K}\left(\alpha\psi^{K}-\phi^{K}\right)$
$\displaystyle\rightarrow 0$ $\displaystyle\text{strongly in
}L^{\infty}(0,T,L^{2}(\Gamma)),$

as $K\rightarrow\infty$, up to subsequence extraction, with

$\displaystyle\frac{1}{K}\|\alpha\psi^{K}-\phi^{K}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\leq\frac{C}{\sqrt{K}},$
(5.16)

and the limit $(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ is a weak
solution to the system (3.1) in the sense of Definition 3.1 with $K=\infty$,
which additionally satisfies $\left(\mu^{\ast},\theta^{\ast}\right)\in
L^{4}(0,T;\mathcal{L}^{2})$.

######  Remark 5.5.

Suppose that for any $K\in(0,\infty)$, the phase-field $\phi^{K}$ is
sufficiently regular such that the boundary condition (3.1e) holds in the
strong sense, that is

$\displaystyle
K\partial_{\mathbf{n}}\phi^{K}=\alpha\psi^{K}-\phi^{K}\quad\text{a.e.\leavevmode\nobreak\
on $\Sigma$}.$

This is actually fulfilled under additional assumptions on the regularity of
$\Gamma$ and the parameter $p$ (if $d=3$), see Theorem 6.1. Then, estimate
(5.16) can be reformulated as

$\displaystyle\|\partial_{\mathbf{n}}\phi^{K}\|_{L^{2}(\Sigma)}\leq\frac{C}{\sqrt{K}}.$

As the right-hand side of this inequality tends to zero as $K\to\infty$, this
explains why the homogeneous Neumann boundary condition
$\partial_{\mathbf{n}}\phi^{*}=0$ a.e. on $\Sigma$ appears in the limit model
corresponding to $K=\infty$.

  * Proof of Theorem 5.4.

We consider an arbitrary sequence $(K_{m})_{m\in\mathbb{N}}\subset(0,\infty)$
such that $K_{m}\rightarrow\infty$ as $m\rightarrow\infty$. Without loss of
generality, we assume that $K_{m}\in[1,\infty)$ for all $m\in\mathbb{N}$. For
any $m\in\mathbb{N}$, let
$(\phi^{K_{m}},\psi^{K_{m}},\mu^{K_{m}},\theta^{K_{m}})$ denote a weak
solution of the system (3.1) in the sense of Definition 3.1 with initial data
$\left(\phi_{,}\psi_{0}\right)$ corresponding to the parameter $K_{m}$. In
this proof, we use the letter $C$ to denote generic positive constants
independent of $K_{m}$ and $m$. Let now $m\in\mathbb{N}$ be arbitrary.

As we have seen in the proof of Theorem 3.2 (see 4.5), the initial energy
satisfies

$\displaystyle E_{K_{m}}(\phi^{K_{m}}(0),\psi^{K_{m}}(0))\leq
C(1+K_{m}^{-1})\leq C$ (5.17)

since $K_{m}\geq 1$ for all $m\in\mathbb{N}$. This allows us to use the same
argumentation as in the proof of Theorem 5.2, and we infer

$\displaystyle\begin{split}&\|\left(\phi^{K_{m}},\psi^{K_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}+K_{m}^{-1/2}\|\alpha\psi^{K_{m}}-\phi^{K_{m}}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\\\
&\quad+\|\left(\nabla\mu^{K_{m}},\nabla_{\Gamma}\theta^{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{L}^{2})}+\|\beta\theta^{K_{m}}-\mu^{K_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (5.18)

This already implies (5.16). We now test the weak formulation (3.4b) with
$\left(\zeta,\xi\right)\in\mathcal{H}^{1}$. Using again that $K_{m}\geq 1$, we
find that

$\displaystyle\|\left(\mu^{K_{m}},\theta^{K_{m}}\right)\|_{L^{\infty}(0,T;(\mathcal{H}^{1})^{\prime})}\leq
C.$ (5.19)

This estimate allows us to apply Lemma 2.2, and using (5.18) and (5.19), we
conclude

$\displaystyle\|\left(\mu^{K_{m}},\theta^{K_{m}}\right)\|_{L^{4}(0,T;\mathcal{L}^{2})}\leq
C.$ (5.20)

In combination with (5.18) we thus deduce

$\displaystyle\|\left(\mu^{K_{m}},\theta^{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}\leq
C.$ (5.21)

Lastly, exploiting (5.18), we conclude from the weak formulation (3.4a) by a
comparison argument that

$\displaystyle\|\left(\partial_{t}\phi^{K_{m}},\partial_{t}\psi^{K_{m}}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}\leq
C.$ (5.22)

In summary, combining (5.18) and (5.20)–(5.22), we thus have shown that

$\displaystyle\begin{split}&\|\left(\partial_{t}\phi^{K_{m}},\partial_{t}\psi^{K_{m}}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}+\|\left(\phi^{K_{m}},\psi^{K_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}\\\
&\quad+\|\left(\mu^{K_{m}},\theta^{K_{m}}\right)\|_{L^{4}(0,T;\mathcal{L}^{2})}+\|\left(\mu^{K_{m}},\theta^{K_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}\leq
C.\end{split}$ (5.23)

In view of the uniform estimate (5.23), the Banach–Alaoglu theorem and the
Aubin–Lions–Simon lemma imply the existence of functions
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ such that

$\displaystyle\left(\partial_{t}\phi^{K_{m}},\partial_{t}\psi^{K_{m}}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi^{\ast},\partial_{t}\psi^{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$ (5.24)
$\displaystyle\left(\phi^{K_{m}},\psi^{K_{m}}\right)$
$\displaystyle\rightarrow\left(\phi^{\ast},\psi^{\ast}\right)$
$\displaystyle\text{weakly-star in }L^{\infty}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{strongly in }C([0,T];\mathcal{H}^{s})\text{ for all
}s\in[0,1),$ (5.25) $\displaystyle\left(\mu^{K_{m}},\theta^{K_{m}}\right)$
$\displaystyle\rightarrow\left(\mu^{\ast},\theta^{\ast}\right)$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\cap
L^{2}(0,T;\mathcal{H}^{1}),$ (5.26)

as $m\rightarrow\infty$, along a non-relabeled subsequence. Additionally,
thanks to (5.18) and the trace theorem, we infer

$\displaystyle\frac{1}{K_{m}}(\alpha\psi^{K_{m}}-\phi^{K_{m}})\rightarrow
0\quad\text{strongly in }L^{\infty}(0,T;L^{2}(\Gamma))$ (5.27)

as $m\rightarrow\infty$. We readily deduce that Definition 3.1(i) is
satisfied.

Using the Sobolev embedding $\mathcal{H}^{s}\hookrightarrow\mathcal{L}^{2}$
for $s\in(0,1)$ along with (5.25), we find that

$\displaystyle\left(\phi^{K_{m}}(0),\psi^{K_{m}}(0)\right)\rightarrow\left(\phi^{\ast}(0),\psi^{\ast}(0)\right)\quad\text{strongly
in }\mathcal{L}^{2},$ (5.28)

as $m\rightarrow\infty$. In view of the initial conditions satisfied by
$(\phi^{K_{m}},\psi^{K_{m}})$, we deduce that Definition 3.1(ii) is fulfilled.

As in the proof of Theorem 5.2, the convergences (5.24)–(5.27) are sufficient
to pass to the limit in the weak formulations (3.4) associated with $K_{m}$ to
conclude that the limit $(\phi^{*},\psi^{*},\mu^{*},\theta^{*})$ fulfils the
weak formulation (3.4) associated with $K=\infty$. Thus, we infer that
Definition 3.1(iii) is satisfied.

The mass conservation law as stated in Definition 3.1(iv) also follows with
the same reasoning as in the proof of Theorem 5.2.

For the energy inequality, note that for all non-negative $\sigma\in
C_{c}^{\infty}(0,T)$, we have

$\displaystyle\begin{split}\int_{0}^{T}E_{\infty}(\phi^{\ast}(t),\psi^{\ast}(t))\,\sigma(t)\;\mathrm{d}t&\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}E_{\infty}(\phi^{K_{m}}(t),\psi^{K_{m}}(t))\,\sigma(t)\;\mathrm{d}t\\\
&\leq\liminf_{m\rightarrow\infty}\int_{0}^{T}E_{K_{m}}(\phi^{K_{m}}(t),\psi^{K_{m}}(t))\,\sigma(t)\;\mathrm{d}t.\end{split}$
(5.29)

The first inequality follows the same argumentation that we already have seen
before, whereas the second inequality is due to (5.27). We finish the proof by
mimicking the proof of the energy inequality in Theorem 3.2, which is based on
the convergence results (5.24)–(5.26) and the assumptions (A3)–(A4).

We thus have shown that the quadruplet
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ is a weak solution to the
system (3.1) in the sense of Definition 3.1 for $K=\infty$. ∎

### 5.3 The limit $L\rightarrow 0$ and the existence of a weak solution if
$(K,L)\in[0,\infty]\times\\{0\\}$

In this section, we investigate the asymptotic limits $L\rightarrow 0$ and
$L\rightarrow\infty$ of the system (3.1) for fixed $K\in[0,\infty]$.

######  Theorem 5.6.

(Asymptotic limit $L\rightarrow 0$) Suppose that the assumptions (A1)–(A4)
hold, let $\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}_{K,\alpha}$ be
arbitrary initial data for $K\in[0,\infty]$, and $\boldsymbol{v}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}^{1}_{\mathbf{n}})$. In
the case $K\in\\{0,\infty\\}$, we further assume that (A5) holds. For any
$L\in(0,\infty)$, let $(\phi_{L},\psi_{L},\mu_{L},\theta_{L})$ denote a weak
solution of the system (3.1) in the sense of Definition 3.1 with initial data
$\left(\phi_{0},\psi_{0}\right)$. Then, there exists a quadruplet
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ with
$\mu_{\ast}=\beta\theta_{\ast}$ a.e. on $\Sigma$ such that

$\displaystyle\left(\partial_{t}\phi_{L},\partial_{t}\psi_{L}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi_{\ast},\partial_{t}\psi_{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{D}_{\beta}^{\prime}),$
$\displaystyle\left(\phi_{L},\psi_{L}\right)$
$\displaystyle\rightarrow\left(\phi_{\ast},\psi_{\ast}\right)$
$\displaystyle\text{weakly-star in
}L^{\infty}(0,T;\mathcal{H}^{1}_{K,\alpha}),$ $\displaystyle\text{strongly in
}C([0,T];\mathcal{H}^{s})\text{ for all }s\in[0,1),$
$\displaystyle\left(\mu_{L},\theta_{L}\right)$
$\displaystyle\rightarrow\left(\mu_{\ast},\theta_{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\ \text{if
}K\in(0,\infty],$ $\displaystyle\beta\theta_{L}-\mu_{L}$
$\displaystyle\rightarrow 0$ $\displaystyle\text{strongly in
}L^{2}(0,T;L^{2}(\Gamma)),$

as $L\rightarrow 0$, up to subsequence extraction, with

$\displaystyle\|\beta\theta_{L}-\mu_{L}\|_{L^{2}(\Sigma)}\leq C\sqrt{L},$
(5.30)

and the quadruplet $(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ is a
weak solution to the system (3.1) in the sense of Definition 3.1 with $L=0$,
which additionally satisfies $\left(\mu_{\ast},\theta_{\ast}\right)\in
L^{4}(0,T;\mathcal{L}^{2})$ in the case $K\in(0,\infty]$.

######  Remark 5.7.

As the right-hand side in (5.30) tends to zero as $L\to 0$, this explains why
the Dirichlet type boundary condition $\mu_{*}=\beta\theta_{*}$ a.e. on
$\Sigma$ appears in the limit model corresponding to $L=0$.

  * Proof of Theorem 5.6.

We consider an arbitrary sequence $(L_{m})_{m\in\mathbb{N}}\subset(0,\infty)$
such that $L_{m}\rightarrow\infty$ as $m\rightarrow\infty$ and a corresponding
weak solution $(\phi_{L_{m}},\psi_{L_{m}},\theta_{L_{m}},\mu_{L_{m}})$ to the
system (3.1) in the sense of Definition 3.1 to the initial data
$\left(\phi_{0},\psi_{0}\right)$. In this proof, we denote by $C$ arbitrary
positive constants independent of $L_{m}$ and $m$, which may change their
value from line to line. Let now $m\in\mathbb{N}$ be arbitrary.

If $K\in[0,\infty)$, we conclude from the energy inequality (3.6) that

$\displaystyle\begin{split}&\|\left(\phi_{L_{m}},\psi_{L_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}+\mathds{h}(K)^{1/2}\|\alpha\psi_{L_{m}}-\phi_{L_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\\\
&\quad+\|\left(\nabla\mu_{L_{m}},\nabla_{\Gamma}\theta_{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{L}^{2})}+L_{m}^{-1/2}\|\beta\theta_{L_{m}}-\mu_{L_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (5.31)

If $K=\infty$, we do not have the bulk-surface Poincaré inequality at our
disposal. Instead, we have to argue as in the proof of Theorem 5.2 and make
use of the additional assumption (A5) to obtain the estimate (5.31). In
particular, (5.31) yields (5.30). Arguing as in the proof of Theorem 3.2, we
additionally infer that

$\displaystyle\|\left(\mu_{L_{m}},\theta_{L_{m}}\right)\|_{L^{\infty}(0,T;(\mathcal{H}^{1}_{K,\alpha})^{\prime})}\leq
C.$ (5.32)

If $K\in(0,\infty]$, we may use Lemma 2.2 and further obtain

$\displaystyle\|\left(\mu_{L_{m}},\theta_{L_{m}}\right)\|_{L^{4}(0,T;\mathcal{L}^{2})}\leq
C,$ (5.33)

which in combination with (5.31) yields

$\displaystyle\|\left(\mu_{L_{m}},\theta_{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}\leq
C.$ (5.34)

In the case $K=0$, we argue analogously as in the proof of Theorem 5.2 to
deduce that

$\displaystyle\|\left(\mu_{L_{m}},\theta_{L_{m}}\right)\|_{L^{2}(0;T;\mathcal{L}^{2})}\leq
C,$ (5.35)

from which we obtain (5.34) as well.

For the time derivatives, we again proceed similarly as in the proof of
Theorem 3.2 but choose the test function space $\mathcal{D}_{\beta}$ instead
of $\mathcal{H}^{1}$. We then obtain due to the weak formulation (3.4a) and
the uniform bound (5.31) that

$\displaystyle\|\left(\phi_{L_{m}},\psi_{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{D}_{\beta}^{\prime})}\leq
C.$ (5.36)

In view of the uniform estimates (5.31), (5.33), (5.34) and (5.36), the
Banach–Alaoglu theorem and the Aubin–Lions–Simon lemma imply the existence of
functions $\phi_{\ast},\psi_{\ast},\mu_{\ast}$ and $\theta_{\ast}$ such that

$\displaystyle\left(\partial_{t}\phi_{L_{m}},\partial_{t}\psi_{L_{m}}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi_{\ast},\partial_{t}\psi_{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{D}_{\beta}^{\prime}),$
(5.37) $\displaystyle\left(\phi_{L_{m}},\psi_{L_{m}}\right)$
$\displaystyle\rightarrow\left(\phi_{\ast},\psi_{\ast}\right)$
$\displaystyle\text{weakly-star in
}L^{\infty}(0,T;\mathcal{H}^{1}_{K,\alpha}),$ $\displaystyle\text{strongly in
}C([0,T];\mathcal{H}^{s})\text{ for all }s\in[0,1),$ (5.38)
$\displaystyle\left(\mu_{L_{m}},\theta_{L_{m}}\right)$
$\displaystyle\rightarrow\left(\mu_{\ast},\theta_{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\ \text{if
}K\in(0,\infty],$ (5.39) $\displaystyle\beta\theta_{L_{m}}-\mu_{L_{m}}$
$\displaystyle\rightarrow\beta\theta_{\ast}-\mu_{\ast}$
$\displaystyle\text{weakly in }L^{2}(0,T;L^{2}(\Gamma)),$ (5.40)

as $m\rightarrow\infty$, along a non-relabeled subsequence. Furthermore, we
conclude from (5.31) that

$\displaystyle\|\beta\theta_{L_{m}}-\mu_{L_{m}}\|_{L^{2}(\Sigma)}\leq
C\sqrt{L_{m}}\rightarrow 0,$ (5.41)

as $m\rightarrow\infty$. In combination with (5.40) we infer that
$\mu_{\ast}=\beta\theta_{\ast}$ a.e. on $\Sigma$ due to the uniqueness of the
limit. Proceeding similarly as in the case $K\rightarrow 0$ (see the proof of
Theorem 5.2), we eventually show that the quadruplet
$(\phi_{\ast},\psi_{\ast},\mu_{\ast},\theta_{\ast})$ is a weak solution of the
system (3.1) in the sense of Definition 3.1 for $L=0$. ∎

### 5.4 The limit $L\rightarrow\infty$ and the existence of a weak solution if
$(K,L)\in[0,\infty]\times\\{\infty\\}$

######  Theorem 5.8.

(Asymptotic limit $L\rightarrow\infty$) Suppose that the assumptions (A1)–(A4)
hold, let $\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}_{K,\alpha}$ be
arbitrary initial data for $K\in[0,\infty]$, and $\boldsymbol{v}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}^{1}_{\mathbf{n}})$. In
the case $K\in\\{0,\infty\\}$, we further assume that (A5) holds. For any
$L\in(0,\infty)$, let $(\phi^{L},\psi^{L},\mu^{L},\theta^{L})$ denote a weak
solution to the system (3.1) in the sense of Definition 3.1 with initial data
$\left(\phi_{0},\psi_{0}\right)$. Then there exists a quadruplet
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ such that

$\displaystyle\left(\partial_{t}\phi^{L},\partial_{t}\psi^{L}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi^{\ast},\partial_{t}\psi^{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$
$\displaystyle\left(\phi^{L},\psi^{L}\right)$
$\displaystyle\rightarrow\left(\phi^{\ast},\psi^{\ast}\right)$
$\displaystyle\text{weakly-star in
}L^{\infty}(0,T;\mathcal{H}^{1}_{K,\alpha}),$ $\displaystyle\text{strongly in
}C([0,T];\mathcal{H}^{s})\text{ for all }s\in[0,1),$
$\displaystyle\left(\mu^{L},\theta^{L}\right)$
$\displaystyle\rightarrow\left(\mu^{\ast},\theta^{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\ \text{if
}K\in(0,\infty],$
$\displaystyle\frac{1}{L}\left(\beta\theta^{L}-\mu^{L}\right)$
$\displaystyle\rightarrow 0$ $\displaystyle\text{strongly in
}L^{2}(0,T;L^{2}(\Gamma)),$

as $L\rightarrow\infty$, up to subsequence extraction, with

$\displaystyle\frac{1}{L}\|\beta\theta^{L}-\mu^{L}\|_{L^{2}(\Sigma)}\leq\frac{C}{\sqrt{L}},$
(5.42)

and the quadruplet $(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ is a
weak solution to the system (3.1) in the sense of Definition 3.1 for
$L=\infty$, which additionally satisfies
$\left(\mu^{\ast},\theta^{\ast}\right)\in L^{4}(0,T;\mathcal{L}^{2})$ in the
case $K\in(0,\infty]$.

######  Remark 5.9.

Assuming that for any $L\in(0,\infty)$, the chemical potential $\mu^{L}$ is
sufficiently regular such that the boundary condition (3.1f) holds in the
strong sense, that is

$\displaystyle
Lm_{\Omega}(\phi^{L})\partial_{\mathbf{n}}\mu^{L}=\beta\theta^{L}-\mu^{L}\quad\text{a.e.\leavevmode\nobreak\
on $\Sigma$},$

estimate (5.42) can be reformulated as

$\displaystyle\|\partial_{\mathbf{n}}\mu^{L}\|_{L^{2}(\Sigma)}\leq\frac{C}{\sqrt{L}}.$

As the right-hand side of this inequality tends to zero as $L\to\infty$, this
explains why the homogeneous Neumann boundary condition
$\partial_{\mathbf{n}}\mu^{*}=0$ a.e. on $\Sigma$ appears in the limit model
corresponding to $L=\infty$.

  * Proof of Theorem 5.8.

We consider an arbitrary sequence $(L_{m})\subset(0,\infty)$ such that
$L_{m}\rightarrow\infty$ as $m\rightarrow\infty$ and a corresponding weak
solution $(\phi^{L_{m}},\psi^{L_{m}},\theta^{L_{m}},\mu^{Lm})$ to (3.1) in the
sense of Definition 3.1 to the initial data $\left(\phi_{0},\psi_{0}\right)$.
Since $L_{m}\rightarrow\infty$ as $m\rightarrow\infty$, we can assume without
loss of generality that $L_{m}\in[1,\infty)$ for all $m\in\mathbb{N}$. In this
proof, we use the letter $C$ to denote generic positive constants independent
of $L_{m}$ and $m$, which may change their value from line to line. Let now
$m\in\mathbb{N}$ be arbitrary.

Using the same argumentation as in the proof of Theorem 5.6, we infer

$\displaystyle\begin{split}&\|\left(\phi^{L_{m}},\psi^{L_{m}}\right)\|_{L^{\infty}(0,T;\mathcal{H}^{1})}+\mathds{h}(K)^{1/2}\|\alpha\psi^{L_{m}}-\phi^{L_{m}}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\\\
&\quad+\|\left(\nabla\mu^{L_{m}},\nabla_{\Gamma}\theta^{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{L}^{2})}+L_{m}^{-1/2}\|\beta\theta^{L_{m}}-\mu^{L_{m}}\|_{L^{2}(0,T;L^{2}(\Gamma))}\leq
C.\end{split}$ (5.43)

In particular, (5.43) already entails (5.42). If $K\in(0,\infty]$, we again
derive the estimate

$\displaystyle\|\left(\mu^{L_{m}},\theta^{L_{m}}\right)\|_{L^{4}(0,T;\mathcal{L}^{2})}\leq
C,$ (5.44)

while in the case $K=0$, we merely obtain

$\displaystyle\|\left(\mu^{L_{m}},\theta^{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{L}^{2})}\leq
C.$ (5.45)

In both cases, we thus get

$\displaystyle\|\left(\mu^{L_{m}},\theta^{L_{m}}\right)\|_{L^{2}(0,T;\mathcal{H}^{1})}\leq
C.$ (5.46)

For the time derivatives, we proceed similarly as in the proof of Theorem 3.2
and obtain due to the weak formulation (3.4a) and the uniform bounds (5.43)
that

$\displaystyle\|\left(\phi^{L_{m}},\psi^{L_{m}}\right)\|_{L^{2}(0,T;(\mathcal{H}^{1})^{\prime})}\leq
C\left(1+\frac{1}{\sqrt{L_{m}}}\right)\leq C.$ (5.47)

Here we additionally used $L_{m}\geq 1$ for all $m\in\mathbb{N}$. In view of
the uniform estimates (5.43), (5.44)–(5.46) and (5.47), the Banach–Alaoglu
theorem and the Aubin–Lions–Simon lemma imply the existence of functions
$\phi^{\ast},\psi^{\ast},\mu^{\ast}$ and $\theta^{\ast}$ such that

$\displaystyle\left(\partial_{t}\phi^{L_{m}},\partial_{t}\psi^{L_{m}}\right)$
$\displaystyle\rightarrow\left(\partial_{t}\phi^{\ast},\partial_{t}\psi^{\ast}\right)\quad\quad$
$\displaystyle\text{weakly in }L^{2}(0,T;(\mathcal{H}^{1})^{\prime}),$ (5.48)
$\displaystyle\left(\phi^{L_{m}},\psi^{L_{m}}\right)$
$\displaystyle\rightarrow\left(\phi^{\ast},\psi^{\ast}\right)$
$\displaystyle\text{weakly-star in
}L^{\infty}(0,T;\mathcal{H}^{1}_{K,\alpha}),$ $\displaystyle\text{strongly in
}C([0,T];\mathcal{H}^{s})\text{ for all }s\in[0,1),$ (5.49)
$\displaystyle\left(\mu^{L_{m}},\theta^{L_{m}}\right)$
$\displaystyle\rightarrow\left(\mu^{\ast},\theta^{\ast}\right)$
$\displaystyle\text{weakly in }L^{2}(0,T;\mathcal{H}^{1}),$
$\displaystyle\text{weakly in }L^{4}(0,T;\mathcal{L}^{2})\ \text{if
}K\in(0,\infty],$ (5.50)

as $m\rightarrow\infty$, along a non-relabeled subsequence. Due to (5.43), we
additionally have

$\displaystyle\frac{1}{L_{m}}\|\beta\theta^{L_{m}}-\mu^{L_{m}}\|_{L^{2}(\Sigma)}\leq\frac{C}{\sqrt{L_{m}}}\rightarrow
0$ (5.51)

as $m\rightarrow\infty$. Arguing further as in the case $K\rightarrow\infty$
(see the proof of Theorem 5.4), we eventually show that the quadruplet
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ satisfies Definition
3.1(i)–3.1(iii) and 3.1(v). For the mass conservation law, simply note that
the weak formulations for $\phi^{\ast}$ and $\psi^{\ast}$, as well as the test
functions, are not coupled, which allows us to use $\zeta\equiv 1$ and
$\xi\equiv 1$ as test functions, respectively. Integrating the resulting
equations in time from $0$ to $t$, and employing the fundamental theorem of
calculus, we infer

$\displaystyle\int_{\Omega}\phi^{\ast}(t)\;\mathrm{d}x=\int_{\Omega}\phi_{0}\;\mathrm{d}x\quad\text{and}\quad\int_{\Gamma}\psi^{\ast}(t)\;\mathrm{d}\Gamma=\int_{\Gamma}\psi_{0}\;\mathrm{d}\Gamma$

for all $t\in[0,T]$, which shows that
$(\phi^{\ast},\psi^{\ast},\mu^{\ast},\theta^{\ast})$ is a solution of the
system (3.1) in the sense of Definition 3.1 for $L=\infty$. ∎

We conclude this section with the following remark.

######  Remark 5.10.

If the uniqueness of the respective limiting models is known (e.g. if the
mobility functions are constant, cf. Theorem 3.5), we even obtain convergence
of the full sequence (not only a subsequence) in Theorem 5.2, Theorem 5.4,
Theorem 5.6 and Theorem 5.8, respectively, by means of a standard subsequence
argument.

## 6 Higher regularity, continuous dependence and uniqueness

In this section, we present the results on higher regularity for the phase-
fields as well as the continuous dependence and uniqueness of weak solutions
to the system (3.1) as stated in Theorem 3.2 and Theorem 3.5, respectively.

### 6.1 Higher spatial regularity for the phase-fields

######  Theorem 6.1.

(Higher regularity) Suppose that the assumptions (A1)–(A6) hold, and let
$K,L\in[0,\infty]$,
$\left(\phi_{0},\psi_{0}\right)\in\mathcal{H}^{1}_{K,\alpha}$ and
$\boldsymbol{v}\in L^{\infty}(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}^{1}_{\mathbf{n}})$. Suppose that the domain $\Omega$ is
of class $C^{k}$ for $k\in\\{2,3\\}$, and in the case $d=3$, we further assume
that (A4) holds with $p\leq 4$. Then, if a weak solution
$(\phi,\psi,\mu,\theta)$ of the system (3.1) in the sense of Definition 3.1
exists, it additionally satisfies

$\displaystyle\left(\phi,\psi\right)$ $\displaystyle\in
L^{4}(0,T;\mathcal{H}^{2})\quad$ $\displaystyle\text{if\leavevmode\nobreak\
}K\in(0,\infty],$ (6.1) $\displaystyle\left(\phi,\psi\right)$
$\displaystyle\in L^{2}(0,T;\mathcal{H}^{k})$ $\displaystyle\text{for
$k\in\\{2,3\\}$},$ (6.2) $\displaystyle\left(\phi,\psi\right)$
$\displaystyle\in C([0,T];\mathcal{H}^{1})\quad$ if
$(K,L)\in[0,\infty]\times(0,\infty]$ and $k=3$ (6.3)

as well as

$\displaystyle\mu=-\Delta\phi+F^{\prime}(\phi)$
$\displaystyle\text{a.e.\leavevmode\nobreak\ in }Q,$ (6.4)
$\displaystyle\theta=-\Delta_{\Gamma}\psi+G^{\prime}(\psi)+\alpha\partial_{\mathbf{n}}\phi$
$\displaystyle\text{a.e.\leavevmode\nobreak\ on }\Sigma,$ (6.5)
$\displaystyle\begin{cases}K\partial_{\mathbf{n}}\phi=\alpha\psi-\phi&\text{if}\
K\in[0,\infty),\\\ \partial_{\mathbf{n}}\phi=0&\text{if}\ K=\infty\end{cases}$
$\displaystyle\text{a.e.\leavevmode\nobreak\ on }\Sigma.$ (6.6)

  * Proof.

Let $(\phi,\psi,\mu,\theta)$ be a weak solution to the system (3.1) in the
sense of Definition 3.1. We infer from the weak formulation (3.4b) that

$\displaystyle\begin{split}&\int_{\Omega}\nabla\phi(t)\cdot\nabla\eta\;\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi(t)\cdot\nabla_{\Gamma}\vartheta\;\mathrm{d}\Gamma+\mathds{h}(K)\int_{\Gamma}(\alpha\psi(t)-\phi(t))(\alpha\vartheta-\eta)\;\mathrm{d}\Gamma\\\
&\quad=\int_{\Omega}\big{(}\mu(t)-F^{\prime}(\phi(t))\big{)}\eta\;\mathrm{d}x+\int_{\Gamma}\big{(}\theta(t)-G^{\prime}(\psi(t))\big{)}\vartheta\;\mathrm{d}\Gamma\end{split}$
(6.7)

for almost all $t\in[0,T]$ and all
$\left(\eta,\vartheta\right)\in\mathcal{H}^{1}_{K,\alpha}$. If
$K\in[0,\infty)$, this means that the pair $\left(\phi(t),\psi(t)\right)$ is a
weak solution of the bulk-surface elliptic problem

$\displaystyle-\Delta\phi(t)$ $\displaystyle=f(t)\qquad$
$\displaystyle\text{in }\Omega,$
$\displaystyle-\Delta_{\Gamma}\psi(t)+\alpha\partial_{\mathbf{n}}\phi(t)$
$\displaystyle=g(t)$ $\displaystyle\text{on }\Gamma,$ $\displaystyle
K\partial_{\mathbf{n}}\phi(t)$ $\displaystyle=\alpha\psi(t)-\phi(t)\quad$
$\displaystyle\text{on }\Gamma,$

for almost all $t\in[0,T]$, in the sense of [30, Definition 3.1], where

$\displaystyle f(t)=\mu(t)-F^{\prime}(\phi(t)),\quad\text{and}\quad
g(t)=\theta(t)-G^{\prime}(\psi(t)).$

If $K=\infty$, we may choose $\left(\eta,0\right)\in\mathcal{H}^{1}$ and
$\left(0,\vartheta\right)\in\mathcal{H}^{1}$ as test functions in (6.7),
respectively. This yields that $\phi(t)$ is a weak solution to the Poisson-
Neumann problem

$\displaystyle-\Delta\phi(t)$ $\displaystyle=f(t)\quad$
$\displaystyle\text{in\leavevmode\nobreak\ }\Omega,$ (6.9a)
$\displaystyle\partial_{\mathbf{n}}\phi(t)$ $\displaystyle=0$
$\displaystyle\text{on\leavevmode\nobreak\ }\Gamma$ (6.9b)

for almost all $t\in[0,T]$, whereas $\psi(t)$ is a weak solution to the
elliptic problem

$\displaystyle-\Delta_{\Gamma}\psi(t)=g(t)\quad\text{on\leavevmode\nobreak\
}\Gamma$ (6.10)

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

Recalling that $p\leq 4$, we use the growth conditions on $F^{\prime}$ and
$G^{\prime}$ (see (A4)), the Sobolev embeddings $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)$ and $H^{1}(\Gamma)\hookrightarrow L^{2(q-1)}(\Gamma)$ and the
fact that $\left(\phi,\psi\right)\in
L^{\infty}(0,T;\mathcal{H}^{1}_{K,\alpha})$ to derive the estimate

$\displaystyle\|F^{\prime}(\phi(t))\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C+C\|\phi(t)\|^{p-1}_{L^{2(p-1)}(\Omega)}\leq C,$ (6.11)

as well as

$\displaystyle\|G^{\prime}(\psi(t))\|_{L^{2}(\Gamma)}\leq
C+C\|\psi(t)\|^{q-1}_{L^{2(q-1)}(\Gamma)}\leq C,$ (6.12)

for almost all $t\in[0,T]$. In particular, the estimates (6.11) and (6.12)
imply $\left(f(t),g(t)\right)\in\mathcal{L}^{2}$ with

$\displaystyle\|f(t)\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C+\|\mu(t)\|_{L^{2}(\Omega)},$ (6.13) $\displaystyle\|g(t)\|_{L^{2}(\Gamma)}$
$\displaystyle\leq C+\|\theta(t)\|_{L^{2}(\Gamma)},$ (6.14)

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

We first consider the case $k=2$. In the case $K\in[0,\infty)$, we deduce from
regularity theory for elliptic problems with bulk-surface coupling (see [30,
Theorem 3.3]) that $\left(\phi(t),\psi(t)\right)\in\mathcal{H}^{2}$ with

$\displaystyle\begin{split}\|\left(\phi(t),\psi(t)\right)\|_{\mathcal{H}^{2}}^{2}&\leq