# Regularisation and separation for evolving surface Cahn-Hilliard equations

Diogo Caetano Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
UK<EMAIL_ADDRESS><EMAIL_ADDRESS>Charles M. Elliott
11footnotemark: 1
Andrea Poiatti22footnotemark: 2 Diogo Caetano Mathematics Institute,
University of Warwick, Coventry CV4 7AL, United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS>Charles M.
Elliott11footnotemark: 1 Maurizio Grasselli Dipartimento di Matematica,
Politecnico di Milano, Milano, 20133, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Andrea Poiatti22footnotemark: 2

###### Abstract

We consider the Cahn-Hilliard equation with constant mobility and logarithmic
potential on a two-dimensional evolving closed surface embedded in
$\mathbb{R}^{3}$, as well as a related weighted model. The well-posedness of
weak solutions for the corresponding initial value problems on a given time
interval $[0,T]$ have already been established by the first two authors. Here
we first prove some regularisation properties of weak solutions in finite
time. Then, we show the validity of the strict separation property for both
the problems. This means that the solutions stay uniformly away from the pure
phases $\pm 1$ from any positive time on. This property plays an essential
role to achieve higher-order regularity for the solutions. Also, it is a
rigorous validation of the standard double-well approximation. The present
results are a twofold extension of the well-known ones for the classical
equation in planar domains.

##  1 Introduction

The Cahn-Hilliard equation in an evolving surface setting has recently been
studied in [13]. More precisely, having fixed $T>0$, considering a family of
closed, connected, oriented surfaces $\Gamma(t)\subset\mathbb{R}^{3}$ such
that its evolution is given a priori as a flow determined by the (sufficiently
smooth) velocity field $\mathbf{V}$, the evolving surface Cahn-Hilliard
equation reads

$\begin{cases}\dot{u}+u\nabla_{\Gamma}\cdot\mathbf{V}-\nabla_{\Gamma}\cdot(u\mathbf{V}_{a}^{\tau})+\Delta_{\Gamma}^{2}u-\Delta_{\Gamma}F^{\prime}(u)=0,\quad\text{
in }\,\,\mathcal{G}_{T},\\\ u(0)=u_{0},\quad\text{ in
}\,\,\Gamma(0),\end{cases}$ (1.1)

where $\mathcal{G}_{T}:=\bigcup_{t\in[0,T]}\\{t\\}\times\Gamma(t)$. Here
$\mathbf{V}_{a}^{\tau}$ corresponds to the difference between the tangential
component of $\mathbf{V}$ and an advective velocity $\textbf{V}_{a}$ on the
surface. The quantity $u$ can be interpreted as the difference between the
concentration of two immiscible substances which are present on the surface.
In the same contribution, the following related weighted Cahn-Hilliard system
has also been analysed

$\displaystyle\begin{cases}\dot{\rho}+\rho\nabla_{\Gamma}\cdot\mathbf{V}=0,\quad\text{
in }\,\,\mathcal{G}_{T},\\\
\rho\dot{c}-\nabla_{\Gamma}\cdot\left(\rho\nabla_{\Gamma}\left(-\dfrac{1}{\rho}\Delta_{\Gamma}c+F^{\prime}(c)\right)\right)&=0,\quad\text{
in }\,\,\mathcal{G}_{T},\\\ c(0)=c_{0},\,\,\rho(0)\equiv 1,\quad\text{ in
}\,\,\Gamma(0),\end{cases}$ (1.2)

where $\rho$ is a suitable weight function transported by the surface
evolution (in particular, it can be interpreted as the total mass density) and
$c$ can be seen as the relative (i.e., dimensionless) concentration difference
between the two substances on the surface. In the equations above, equipped
with suitable initial conditions, $\nabla_{\Gamma}$ denotes the tangential
gradient on the surface $\Gamma(\cdot)$, $\Delta_{\Gamma}$ is the Laplace-
Beltrami operator and $\dot{u}$ denotes the material time derivative of $u$
(see Section 2 for more details). The functional framework in this work is the
same as in [13]. Model (1.2) is a simplified version of the one presented in
[49, 56], which also includes a coupling with an equation for the surface
deformation, i.e., the Kirchhoff-Love thin shell equation. We decided to
concentrate on the single Cahn-Hilliard equation in the same formulation
arising from the model in [56], so that this analysis, although being
interesting per se, could also be exploited (and extended) to take the
evolution of the surface into account as in [49, 56]; in our case, this
evolution is given a priori. In both models, the main physical property is the
conservation of total mass. In particular (see also Remark 4.1 for more
details), in the case of model (1.1) this implies, $u$ denoting the difference
density, that we have

$\displaystyle\int_{\Gamma(t)}u\equiv\int_{\Gamma_{0}}u_{0},$ (1.3)

whereas, for model (1.2), it holds

$\displaystyle\int_{\Gamma(t)}\rho c\equiv\int_{\Gamma_{0}}c_{0},$ (1.4)

$\rho c$ being this time the difference density between the two substances and
having assumed $\rho(0)\equiv 1$.

This work aims at proving that weak solutions to equations (1.1) and (1.2)
regularise in finite time and enjoy the instantaneous strict separation
property from the pure phases, i.e., for each $\tau>0$, there exists
$\delta>0$ depending on $\tau,\ T$ and the data such that

$\displaystyle\|u(t)\|_{L^{\infty}(\Gamma(t))}\leq 1-\delta,\quad\text{ for
a.a. }t\in[\tau,T].$ (1.5)

The present results answer to some regularity issues left open (see [13,
5.1.3]).

The classical Cahn-Hilliard equation was introduced in [15] (see also [14])
for the study of spinodal decomposition in binary alloys. More precisely, it
models the phenomenon of phase separation of an alloy of two components when
the temperature of the system is quenched to a critical temperature, resulting
on a spatially separated two-phase structure. On a smooth bounded domain
$\Omega\subset\mathbb{R}^{d}$, where $d=1,2,3$, it reads as

$\displaystyle\begin{cases}\dot{u}=\Delta w,\quad\text{ in
}\Omega\times(0,T),\\\ w=-\Delta u+F^{\prime}(u),\quad\text{ in
}\Omega\times(0,T),\end{cases}$ (1.6)

where $w$ is the so called chemical potential and is complemented with the
(no-flux) boundary and initial conditions

$\displaystyle\partial_{n}u=\partial_{n}w=0\,\,\text{ on
}\partial\Omega\times(0,T)\quad\text{ and }\quad u(0)=u_{0}\,\,\text{ in
}\Omega.$

Note that we have assumed constant mobility and set it equal to $1$. It has
since then inspired numerous works in other areas of science; to name a few,
versions of the Cahn-Hilliard equation have been used to study population
dynamics [18], tumour growth models [38] and they have been exploited in image
processing analysis [11, 12] (see also the recent book [42] for other
examples). It is also worth recalling that phase separation has recently
become a paradigm in cell biology (see, for instance, [21, 22]). For example,
Cahn-Hilliard equations have been used to model solid tumour growth (see,
e.g., [17, 52, 53] and the book [20, Part I, Chap. 5]), dynamics of plasma
membranes and multicomponent vesicles [8, 9, 10, 16, 39, 41, 50, 54]. In some
of these cases, such as sorting in biological membranes, phase separation and
coarsening take place in a thin, evolving layer of self-organising molecules,
which in continuum-based approach can be modelled as a material surface. This
justifies the recent interest in the Cahn-Hilliard equation posed on evolving
surfaces. In particular, one of the most interesting phenomenon which has been
modelled by evolving-surface Cahn-Hilliard equations (see, e.g., [31, 56]) is
the lipid rafts formation on cell membranes, which are composed of lipids,
proteins, and cholesterol. Whereas proteins mediate traffic and serve as
signalling devices, lipids provide a fluid matrix within which transmembrane
proteins are free to move. The separation of lipids into two immiscible liquid
phases is often linked to the formation of rafts in cell membranes. These
rafts are heterogeneous, highly dynamic, sterol, and sphingolipid-enriched
domains that compartmentalise cellular processes and they are thought to be in
the liquid-ordered phase. Rafts are believed to play an important role in
regulating protein activity ([37]) that may in turn affect biological
processes such as trafficking and signalling and are known to be central to
the replication of viruses ([48]). All of the above phenomena involve elastic
bending of cell membranes being fully coupled with the irreversible processes
of lipid flow, the diffusion of lipids and proteins, and the surface binding
of proteins. Comprehensive membrane models which include these effects are
needed to fully understand the complex physical behavior of biological
membranes. In our work we make a first stride in the direction of the analysis
of these highly complex models. In particular, as already noticed, taking
inspiration from such models (see, e.g., [49, 56]), we do not consider the
elasticity of the surface, that would determine an evolution equation for the
surface itself, which we assume to be given, but we only study the Cahn-
Hilliard equation arising from such models. The natural direction for future
work is then to consider the fully coupled system, where the evolution of the
surface is itself part of problem, see for instance [1, 2, 49, 56].  In
equation (1.6), $u$ is again to be thought of as the difference between the
concentrations of the two components in the mixture. The function $F$ is the
homogeneous free energy (potential) of the system, and is defined as follows

$\displaystyle
F(r)=\dfrac{\theta}{2}\left((1+r)\ln(1+r)+(1-r)\ln(1-r)\right)-\dfrac{\theta_{0}}{2}r^{2},\quad
r\in[-1,1],$ (1.7)

where $\theta$, $\theta_{0}$ are absolute temperatures and satisfy
$0<\theta<\theta_{0}$. This ensures that $F$ has a double-well shape. From the
modelling point of view, the logarithmic terms are related to the entropy of
the system, while the quadratic term accounts for demixing effects. It is the
competition between the two terms that gives rise to the spatially distributed
phase separation. The potential defined in this way is called singular,
whereas many authors considered a proper approximation, which avoids the fact
that $F^{\prime}$ is unbounded at the pure phases $\pm 1$. The most common
choice is a polynomial of fourth degree, typically of the form
$F(r)=\frac{1}{4}(r^{2}-u_{\theta}^{2})^{2}$, where $u_{\theta}>0$ and
$-u_{\theta}<0$ are the minima of (1.7); this guarantees that $F$ has a
symmetric double-well shape also with minima at $\pm u_{\theta}$. This is
usually referred as a regular potential. This case was also taken into account
in [13]. However, the polynomial approximation does not ensure the existence
of physical solutions, that is, solutions whose values are in $[-1,1]$, due to
the lack of comparison principles for the Cahn-Hilliard equation. We refer to
the original proof of well-posedness with (1.7) in [27], as well as the survey
articles [26, 45] and the recent book [42] for an overview of the mathematical
results for (1.6).

As far as the evolving surface version is concerned, both systems (1.1) and
(1.2) are treated in [13], where the authors establish existence, uniqueness
and stability of solutions for the different cases where $F$ is a smooth,
logarithmic or double obstacle potential, respectively. Some regularity
results are also proved. We refer also to [28, 47, 55] for different
derivations of the equation and some numerical results, and to [56] for a
rigorous modelling source for the weighted system (1.2).  Nevertheless, for
the sake of completeness we will give a short derivation of each model in
Sections 3.1 and 4.1, respectively. Interest in these equations is part of the
more general problem of considering partial differential equations on domains
or surfaces that evolve in time which is presently being vastly studied, since
these have been seen to provide more realistic models for physical and
biological phenomena. Some examples are [7, 25, 30, 32, 35, 51]. Not only are
these relevant for applications, but they also raise interesting modelling,
numerical and computational questions, as well as challenging problems from
the point of view of mathematical analysis. We refer in particular to [29] for
a detailed exposition of the numerical analysis of such problems and to [4, 5,
6] for an abstract functional framework well-suited for the treatment of such
problems.

The importance of establishing the strict separation property is twofold.

  * •

First it is essential when one considers higher-order regularity of weak
solutions, due to the behaviour of $F$ and its derivatives close to $\pm 1$.
Indeed, note that

$\displaystyle
F^{\prime}(r)=\dfrac{\theta}{2}\ln\dfrac{1+r}{1-r}-\theta_{0}r\quad\text{ and
}\quad
F^{\prime\prime}(r)=\dfrac{\theta}{2}\left(\dfrac{1}{1+r}+\dfrac{1}{1-r}\right)-\theta_{0}$

are both singular when $r\to\pm 1$, and even though the structure of
$F^{\prime}$ can be exploited in order to obtain some estimates, it is more
challenging to do so for $F^{\prime\prime}$. As noted in, e.g., [36], we have

$\displaystyle F^{\prime\prime}(r)\leq Ce^{C|F^{\prime}(r)|},$ (1.8)

which precludes us from controlling $F^{\prime\prime}$ in $L^{p}$-spaces in
terms of the $L^{p}$-norms of $F^{\prime}$. It is then important to study
conditions that ensure the integrability of $F^{\prime\prime}$. It is clear
that establishing strict separation from the pure phases as in (1.5) is
crucial to achieve this. Furthermore, the strict separation property is a
fundamental ingredient in the study of longtime behavior of solutions (see
[36, 43]).

  * •

Secondly, if the strict separation property holds then, being the solution
away from $\pm 1$, the logarithmic potential $F$ is smooth and can be
dominated by a polynomial. Therefore, this result can be viewed as a rigorous
justification of the usual aforementioned polynomial double-well
approximation.

Separation from the pure phases for dimensions $d\geq 3$ is unknown even in
the fixed domain setting, in the case of constant mobility. This apparently
technical restriction is related to the growth condition (1.8) (see [34] for a
detailed analysis). As a consequence, our results are restricted to two-
dimensional surfaces. The strict separation for $d=2$ was first established
for (1.1) in [43]. Then, a more general argument was introduced in [36]. For
an up-to-date picture of the state-of-the-art, the reader is referred to [34],
where new proofs and further generalizations are given.

Here, not only we extend the result for two-dimensional planar domains for
equation (1.1), but we also prove the first separation result for the weighted
model (1.2).

This article on one hand complements [13] by proving instantaneous
regularisation of the weak solutions and on the other hand extends the
validity of the strict separation property for the local Cahn-Hilliard
equation with constant mobility to the setting of evolving surfaces in
$\mathbb{R}^{3}$ in two cases. It is also worth observing that our approach to
the estimates for solutions to the approximate problems allows us to forgo
some of the assumptions made in [13] (see, e.g., [13, Assumption $A_{P}$]),
generalising the results therein. For the sake of completeness, here we also
include the proof of continuous dependence on the initial data, entailing
uniqueness for the general system (1.2), which was omitted in [13].

The paper is structured as follows. In Section 2 we briefly recall the setting
from [13] on the evolution of the surfaces and state some additional
regularity assumptions. Section 3 is devoted to the analysis of the first
system (1.1) and we establish higher-order regularity for the solution $u$ and
prove that it satisfies the strict separation property. Finally, in Section 4
we introduce a better suited Galerkin approximation for the alternative
weighted model (1.2) and prove analogous regularity results as well as the
strict separation property. We also include three appendices for the sake of
completeness: in Appendix A we present the proofs of some propositions which
are only stated in the main body of the paper. Appendix B collects some
preliminary results essential to obtain the control of higher-order
derivatives of the logarithmic potential (in particular, we report a Moser-
Trudinger type inequality valid on compact Riemannian manifolds and a
generalized Young’s inequality), whereas in Appendix C we show the validity of
a well-known embedding result for Bochner spaces also in the evolving space
setting.

##  2 Surface motion: assumptions

We refer to the setting and the notation of [23]. To be more precise, we
consider $T>0$ and a $C^{2}$-evolving surface $\\{\Gamma(t)\\}_{t\in[0,T]}$ in
$\mathbb{R}^{3}$, i.e. a closed, connected, orientable $C^{2}$-surface
$\Gamma_{0}$ in $\mathbb{R}^{3}$ together with a smooth flow map

$\displaystyle\Phi\colon[0,T]\times\Gamma_{0}\to\mathbb{R}^{3}$

such that

  * (i)

denoting $\Gamma(t):=\Phi_{t}^{0}(\Gamma_{0})$, the map

$\Phi_{t}^{0}:=\Phi(t,\cdot)\colon\Gamma_{0}\to\Gamma(t)$

is a $C^{2}$-diffeomorphism, with inverse map

$\Phi_{0}^{t}\colon\Gamma(t)\to\Gamma_{0};$

  * (ii)

$\Phi_{0}^{0}=\text{id}_{\Gamma_{0}}$.

It follows from the definition above that, for each $t\in[0,T]$, $\Gamma(t)$
is also a closed, connected, orientable $C^{2}$-surface. In addition to the
assumptions on the surface motion, we  first recall the same hypothesis as
[13, $\mathbf{A}_{\Phi}$]:

Assumption $\mathbf{A}_{\Phi}$: There exists a velocity field
$\textbf{V}:[0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ with regularity

$\textbf{V}\in C^{0}([0,T];C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})),$

such that, for any $t\in[0,T]$ and every $x\in\Gamma_{0}$,

$\displaystyle\dfrac{d}{dt}\,\Phi_{t}^{0}(x)$
$\displaystyle=\mathbf{V}\left(t,\Phi_{t}^{0}(x)\right),\quad\text{ in }[0,T]$
(2.1) $\displaystyle\Phi_{0}^{0}(x)$ $\displaystyle=x.$ (2.2)

In addition to $\mathbf{A}_{\Phi}$ we will also suppose, where necessary, that
the velocity field
$\mathbf{V}\colon[0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfies

Assumptions $\textbf{B}_{\Phi}$:

  1. 1.

$\mathbf{V}$ is such that

$\displaystyle\mathbf{V}\in
C^{1}([0,T];C^{2}(\mathbb{R}^{3},\mathbb{R}^{3}));$ (2.3)

  2. 2.

The advective tangential velocity $\mathbf{V}_{a}$ is such that

$\displaystyle\mathbf{V}_{a}\in
C^{1}([0,T];C^{1}(\mathbb{R}^{3},\mathbb{R}^{3})).$ (2.4)

Note that the above assumptions $\textbf{B}_{\Phi}$ require more regularity in
time for the map $\Phi_{0}^{(\cdot)}$ than AΦ. Denoting by $\mathbf{V}_{\tau}$
and $\mathbf{V}_{\nu}$ the tangential and normal components of $\mathbf{V}$,
respectively,  Assumptions $\textbf{B}_{\Phi}$ imply, in particular, for
$t\in[0,T],$

$\displaystyle\|\mathbf{V}_{\tau}(t)\|_{C^{2}(\Gamma(t))},\|\mathbf{V}_{\nu}(t)\|_{C^{2}(\Gamma(t))}\leq\|\mathbf{V}(t)\|_{C^{2}(\Gamma(t))}\leq
C_{\mathbf{V}},$
$\displaystyle\|\mathbf{V}_{a}(t)\|_{C^{0}(\Gamma(t))},\|\partial^{\bullet}\mathbf{V}_{a}^{\tau}(t)\|_{C^{0}(\Gamma(t))}\leq
C_{\mathbf{V}},$ (2.5)

for $t\in[0,T]$ and for some $C_{\mathbf{V}}>0$ independent of time, where
(see [13]) $\mathbf{V}_{a}^{\tau}:=\mathbf{V}_{\tau}-\mathbf{V}_{a}$. whereas
the single extra Assumption $\textbf{B}_{\Phi}.2.$ implies

$\displaystyle\|\partial^{\bullet}\mathbf{V}_{a}(t)\|_{C^{0}(\Gamma(t))}\leq
C_{\mathbf{V}},$ (2.6)

for $t\in[0,T]$ and for some $C_{\mathbf{V}}>0$ independent of time.

In this setting we can define the normal material time derivative of a scalar
quantity $u$ on $\Gamma(t)$ by

$\displaystyle\partial^{\circ}u:={\color[rgb]{0,0,0}\hat{u}}_{t}+\nabla{\color[rgb]{0,0,0}\hat{u}}\cdot\mathbf{V}_{\nu},$

where ${\color[rgb]{0,0,0}\hat{u}}$ denotes any extension of $u$ to a (space-
time) neighbourhood of $\Gamma(t)$, and its full material time derivative, or
strong time derivative, as

$\displaystyle\partial^{\bullet}u:=\partial^{\circ}u+\nabla_{\Gamma}u\cdot\mathbf{V}_{\tau}={\color[rgb]{0,0,0}\hat{u}}_{t}+\nabla{\color[rgb]{0,0,0}\hat{u}}\cdot\mathbf{V},$
(2.7)

These definitions take into account not only the evolution of the quantity $u$
but also the movement of points in the surface. The definition in (2.7) can be
abstracted to a weaker sense as follows. Let $u\in L^{2}_{H^{1}}$. A
functional $v\in L^{2}_{H^{-1}}$ is said to be the weak time derivative of
$u$, and we write $v=\partial^{\bullet}u$, if, for any
$\eta\in\mathcal{D}_{H^{1}}(0,T)$, we have

$\displaystyle\int_{0}^{T}\langle v(t),\eta(t)\rangle_{H^{-1}\times
H^{1}}=-\int_{0}^{T}(u(t),\partial^{\bullet}\eta(t))_{L^{2}}-\int_{0}^{T}\int_{\Gamma(t)}u(t)\eta(t)\nabla_{\Gamma(t)}\cdot\mathbf{V}(t).$
(2.8)

Observe that $\partial^{\bullet}\eta$ is the strong material derivative of
$\eta$. We will use $\partial^{\bullet}u$ for both the strong and weak
material derivatives.

Thanks to [24, Lemma 2.6], we observe that, for a sufficiently regular vector
field $\mathbf{f}$, there holds

$\partial^{\bullet}(\nabla_{\Gamma}\cdot\mathbf{f})=\partial^{\bullet}\underline{D}^{\Gamma(t)}_{l}\mathbf{f}_{l}=\underline{D}^{\Gamma(t)}_{l}\partial^{\bullet}\mathbf{f}_{l}-A_{lr}(\mathbf{V})\underline{D}^{\Gamma(t)}_{r}\mathbf{f}_{l},\quad
l,r=1,2,3.$

Here $\underline{D}^{\Gamma(t)}_{i}$, $i=1,2,3$, is the $i^{th}$ component of
the tangential gradient and

$A_{lr}(\mathbf{V})=\underline{D}^{\Gamma(t)}_{l}\mathbf{V}_{r}-\nu_{s}\nu_{l}\underline{D}^{\Gamma(t)}_{r}\mathbf{V}_{s},\quad
s=1,2,3,$

with $\nu$ standing for the normal vector field to the surface. Therefore, by
(2.3) we can set $\mathbf{f}=\mathbf{V}$ in the previous equality and deduce
that, due to the regularity of $\Gamma(t)$, we also have

$\displaystyle\|\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V}(t)\|_{C^{0}(\Gamma(t))}\leq
C_{\mathbf{V}},\quad\text{ for any }t\in[0,T].$ (2.9)

Moreover, denoting by $J_{t}^{0}$ and $J_{0}^{t}$, the change of area element
from $\Gamma_{0}$ to $\Gamma(t)$ and the one from $\Gamma(t)$ to $\Gamma_{0}$,
respectively, we have, for any $\eta:\Gamma(t)\to\mathbb{R}$,

$\int_{\Gamma(t)}\eta
d\Gamma=\int_{\Gamma_{0}}\widetilde{\eta}J_{t}^{0}d\Gamma_{0},$

with $\widetilde{\eta}(p)=\eta(\Phi_{t}^{0}(p)),\text{ for any
}p\in\Gamma_{0}$. Note that
$J_{t}^{0}(\cdot)=\left|\text{det}D_{\Gamma_{0}}\Phi_{t}^{0}(\cdot)\right|$.
Then, due to the previous assumptions, we deduce

$\displaystyle\frac{1}{C_{J}}\leq\|J_{t}^{0}\|_{C^{0}(\Gamma(t))}\leq
C_{J},\quad\text{ for any }t\in[0,T],$ (2.10)

where $C_{J}$ can be chosen to be independent of $t$. This also implies that
there exists a constant $C_{\Gamma}>0$, independent of $t$, such that

$\displaystyle\frac{|\Gamma_{0}|}{C_{\Gamma}}\leq|\Gamma(t)|\leq
C_{\Gamma}|\Gamma_{0}|,$ (2.11)

for any $t\in[0,T]$. Here $|\Gamma_{0}|$ stands for the Lebesgue surface
measure. For any integrable function $v$ over a surface $\Gamma$ of positive
measure, we set

$(v)_{\Gamma}=\dfrac{1}{|\Gamma|}\int_{\Gamma}v.$

General agreement. The symbol $C>0$ will denote a generic constant, depending
only on the structural parameters of the problem and $T$, but independent of
time $t$ and of the approximating indices $\delta,M$ (unless otherwise
specified).

##  3 The first model

In this section we consider system (1.1), which is a model proposed, e.g., in
[46] and [13, Problem 4.1]. We start by briefly recalling its derivation.

###  3.1 Derivation

Fix $t\in[0,T]$ and consider a scalar quantity
$u=u(t)\colon\Gamma(t)\to\mathbb{R}$, to be thought of, in our context, as the
concentration difference between two immiscible substances present in a
mixture on the surface. The main property for $u$ is that its total mass is
conserved, so that (1.3) holds. Starting from the balance law

$\displaystyle\dfrac{d}{dt}\int_{P(t)}u=-\int_{\partial P(t)}q\cdot\mu,$

on every portion $P(t)\subset\Gamma(t)$ evolving under the purely normal
velocity $\mathbf{V}_{\nu}$, where $q$ is a flux to be defined later on, we
are led to

$\displaystyle\int_{P(t)}\partial^{\circ}u+u\,\text{div}_{\Gamma}\mathbf{V}_{\nu}+\text{div}_{\Gamma}q=0.$

Since this holds for every region we obtain the equation

$\displaystyle\partial^{\circ}u+u\,\text{div}_{\Gamma}\mathbf{V}_{\nu}+\text{div}_{\Gamma}q=0\quad\text{
on }\Gamma(t).$

Now, the flux $q$ consists of an advective term $q_{a}=u\mathbf{V}_{a}$, where
$\mathbf{V}_{a}$ is tangential and represents the particle velocity in the
fluid, and a diffusive part
$q_{d}=-\nabla_{\Gamma}(-\Delta_{\Gamma}u+F^{\prime}(u))$. Substituting in the
equation above leads to

$\displaystyle\partial^{\circ}u+u\,\text{div}_{\Gamma}\mathbf{V}_{\nu}+\text{div}_{\Gamma}(u\mathbf{V}_{a})=\Delta_{\Gamma}\left(-\Delta_{\Gamma}u+F^{\prime}(u)\right).$
(3.1)

This is the physical equation that we wish to consider. In order to apply the
functional framework developed in [4], we work under Assumption
$\mathbf{A_{\Phi}}$. To incorporate the tangential component
$\mathbf{V}_{\tau}$, which is to be interpreted as an arbitrary
parametrisation of the surface $\Gamma(t)$, we add and subtract a term
$\text{div}_{\Gamma}(u\mathbf{V}_{\tau})$ to (3.1), resulting in

$\displaystyle\partial^{\bullet}u+u\,\text{div}_{\Gamma}\mathbf{V}+\text{div}_{\Gamma}(u(\mathbf{V}_{a}-\mathbf{V}_{\tau}))=\Delta_{\Gamma}\left(-\Delta_{\Gamma}u+F^{\prime}(u)\right),$
(3.2)

or equivalently

$\displaystyle\partial^{\bullet}u+u\,\text{div}_{\Gamma}\mathbf{V}+\text{div}_{\Gamma}(u(\mathbf{V}_{a}-\mathbf{V}_{\tau}))$
$\displaystyle=\Delta_{\Gamma}w,$
$\displaystyle-\Delta_{\Gamma}u+F^{\prime}(u)$ $\displaystyle=w.$

###  3.2 Weak formulation

In order to introduce the variational formulation of the Cahn-Hilliard systems
we need to introduce the abstract framework of [4, 5] for the definition of
the time-dependent function spaces. We do it in a summarised way and refer the
reader to [4, 5] for a more rigorous and detailed explanation of how these are
constructed and how they can be abstracted to a more generalised setting.

We use the flow map $\Phi_{t}^{0}$ to define pullback and pushforward
operators

$\displaystyle\phi_{-t}u=u\circ\Phi_{t}^{0}\colon\Gamma_{0}\to\mathbb{R}\,\,$
$\displaystyle\text{ for }u\colon\Gamma(t)\to\mathbb{R},$
$\displaystyle\phi_{t}v=v\circ\Phi_{0}^{t}\colon\Gamma(t)\to\mathbb{R}\,\,$
$\displaystyle\text{ for }v\colon\Gamma_{0}\to\mathbb{R}$

and suppose, for $t\in[0,T]$, $X(t)$ to be a Banach space of functions over
$\Gamma(t)$; typically $L^{2}(\Gamma(t))$ or some higher-order Sobolev space
$H^{k}(\Gamma(t))$, but also other $L^{p}(\Gamma(t))$ or even the dual space
$H^{-1}(\Gamma(t)):=(H^{1}(\Gamma(t))^{*}$. We consider functions of the form

$\displaystyle u\colon[0,T]$
$\displaystyle\to\bigcup_{t\in[0,T]}X(t)\times\\{t\\},\quad
t\mapsto(\overline{u}(t),t)$

and identify $u(t)\equiv\overline{u}(t)$; in practice, we want to see $u(t)$
as an element of the space $X(t)$. The function spaces are defined as follows.

  * (i)

For $p\in[1,\infty]$, $u\in L^{p}_{X}$ if
$t\mapsto\phi_{-(\cdot)}\overline{u}(\cdot)\in L^{p}(0,T;X_{0})$ with the norm

$\displaystyle\|u\|_{L^{p}_{X}}:=\begin{cases}\left(\int_{0}^{T}\|u(t)\|_{X(t)}^{p}\right)^{1/p}&\text{
if }p<\infty\\\ \operatorname*{ess\,sup}_{t\in[0,T]}\|u(t)\|_{X(t)}&\text{ if
}p=\infty\end{cases};$

if $X(t)=H(t)$ are Hilbert spaces, then so is $L^{2}_{H}$ with the inner
product

$\displaystyle(u,v)_{L^{2}_{H}}$
$\displaystyle:=\int_{0}^{T}(u(t),v(t))_{H(t)}.$

  * (ii)

For $k\in\mathbb{N}\cup\\{0\\}$, $u\in C^{k}_{X}$ if $t\mapsto\phi_{-t}u(t)\in
C^{k}([0,T];X_{0})$ and we define its time derivatives as

$\displaystyle\partial^{\bullet,s}u(t)=\phi_{t}\dfrac{d^{s}}{dt^{s}}\phi_{-t}u(t),\quad
s=1,\dots,k.$

We will denote $\partial^{\bullet,1}=\partial^{\bullet}$; it is easy to see
that the latter produces the same definition as in (2.7). These spaces are
endowed with the norm

$\displaystyle\|u\|_{C^{k}_{X}}:=\sup_{t\in[0,T]}\|u(t)\|_{X(t)}+\sum_{s=1}^{k}\sup_{t\in[0,T]}\|\partial^{\bullet,s}u(t)\|_{X(t)}.$

  * (iii)

A test function $u\in\mathcal{D}_{X}(0,T)$ if $t\mapsto\phi_{-t}u(t)\in
C_{c}^{\infty}((0,T);X_{0}).$

  * (iv)

We define the Banach space

$\displaystyle H^{1}_{H^{-1}}:=\left\\{u\in
L^{2}_{H^{1}}\colon\partial^{\bullet}u\in L^{2}_{H^{-1}}\right\\}\,\text{ with
}\,\,\|u\|_{H^{1}_{H^{-1}}}:=\|u\|_{L^{2}_{H^{1}}}+\|\partial^{\bullet}u\|_{L^{2}_{H^{-1}}}.$

  * (v)

Similarly, $H^{1}_{L^{2}}$ denotes the space of those $u\in L^{2}_{H^{1}}$
which have a more regular weak time derivative $\partial^{\bullet}u\in
L^{2}_{L^{2}}$.

######  Remark 3.1.

We point out that in order to define the spaces $L^{p}_{X}$ we are implicitly
assuming that the family $(\phi_{t},X(t))_{t\in[0,T]}$ is compatible in the
sense of [13, Assumption 3.1]. Moreover, in the cases where $X(t)$ is a space
of functions with some regularity, it is also assumed that the differentiation
makes sense on $\Gamma(t)$. In our setting, these assumptions require
appropriate regularity of the flow map $\Phi$ and the velocity field
$\mathbf{V}$. More precisely, if for some $k\in\mathbb{N}$ we have

$\displaystyle\Phi_{(\cdot)}^{0},\Phi_{0}^{(\cdot)}\in
C^{1}([0,T];C^{k}(\mathbb{R}^{3},\mathbb{R}^{3}))\quad\text{ and
}\quad\mathbf{V}\in C^{0}([0,T];C^{k}(\mathbb{R}^{3},\mathbb{R}^{3})),$

then:

  * (i)

$\Gamma(t)$ is a $C^{k}$-surface, and it makes sense to define strong and weak
derivatives up to order $k$ for functions on $\Gamma(t)$ (in our case we have
$k=2$);

  * (ii)

the pair $(\phi_{t},X(t))_{t}$ is compatible where
$X=L^{p},\,C^{0},\,H^{-1},\,W^{r,p},\,C^{r}$ for $p\in[1,\infty]$ and $1\leq
r\leq k$ (see e.g. [13, Section 7] for examples).

Notice that the assumptions in Sec.2 are enough to guarantee that all the
function spaces involved are well defined, as well as to ensure compatibility
with the evolution.

Let us now explicitly recall some notation introduced in [13, Section 3.2]. In
particular, for $t\in[0,T]$, we define the following bilinear forms:

  1. 1.

for $\eta,\phi\in L^{2}(\Gamma(t))$, the zero order terms

$m(t;\eta,\phi):=\int_{\Gamma(t)}\eta\phi,\quad
g(t;\eta,\phi):=\int_{\Gamma(t)}\eta\phi\nabla_{\Gamma}\cdot\mathbf{V}(t);$

  2. 2.

for $\eta\in L^{2}(\Gamma(t))$, $\phi\in H^{1}(\Gamma(t))$, the first order
term

$a_{N}(t;\eta,\phi):=\int_{\Gamma(t)}\eta\mathbf{V}_{a}^{\tau}(t)\cdot\nabla_{\Gamma}\phi;$

  3. 3.

for $\eta,\phi\in H^{1}(\Gamma(t))$, the second order terms

$a_{S}(t;\eta,\phi):=\int_{\Gamma(t)}\nabla_{\Gamma}\eta\cdot\nabla_{\Gamma}\phi,\quad
b(t;\eta,\phi):=\int_{\Gamma(t)}B(\mathbf{V}(t))\nabla_{\Gamma}\eta\cdot\nabla_{\Gamma}\phi,$

with
${B}(\mathbf{V})=(\nabla_{\Gamma}\cdot\mathbf{V})\mathbf{I}-2\mathbf{D}(\mathbf{V})$,
where $\mathbf{D}(\mathbf{V})$ stands for the symmetrized rate tensor;

  4. 4.

for $\eta\in H^{-1}(\Gamma(t))$, $\phi\in H^{1}(\Gamma(t))$, the duality
pairing

$m_{\star}(t;\eta,\phi):=\langle\eta,\phi\rangle_{H^{-1}(\Gamma(t)),H^{1}(\Gamma(t))}.$

For the sake of simplicity, from now on we will omit the explicit dependence
on time and we will denote $\|\cdot\|_{L^{2}(\Gamma(t))}$ only by $\|\cdot\|$.
In conclusion, we recall [13, Prop.2.8], see also [23, Sec.8.2]) in

######  Proposition 3.2.

The following transport formulas hold:

  * (a)

For $\eta,\phi\in H^{1}_{H^{-1}}$,

$\displaystyle\dfrac{d}{dt}m(\eta,\phi)=m_{\ast}(\partial^{\bullet}\eta,\phi)+m_{\ast}(\partial^{\bullet}\phi,\eta)+g(\eta,\phi).$

  * (b)

If additionally
$\nabla_{\Gamma}\partial^{\bullet}\eta,\nabla_{\Gamma}\partial^{\bullet}\phi\in
L^{2}_{L^{2}}$, then

$\displaystyle\dfrac{d}{dt}a_{S}(\eta,\phi)=a_{S}(\partial^{\bullet}\eta,\phi)+a_{S}(\eta,\partial^{\bullet}\phi)+b(\eta,\phi).$

  * (c)

For $\eta,\phi\in H^{1}_{L^{2}}$, with
$\nabla_{\Gamma}\partial^{\bullet}\phi\in L^{2}_{L^{2}}$, the following
identity holds

$\displaystyle\frac{d}{dt}a_{N}(\eta,\phi)$
$\displaystyle=a_{N}(\partial^{\bullet}\eta,\phi)+a_{N}(\eta,\partial^{\bullet}\phi)$
$\displaystyle+\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))\eta\cdot\nabla_{\Gamma}\phi,$
(3.3)

for almost any $t\in[0,T]$, where $B_{adv}$ is a vector field given by

$B_{adv}(\mathbf{V}_{a}^{\tau},\mathbf{V})_{i}:=(\partial^{\bullet}\mathbf{V}_{a}^{\tau})_{i}+(\nabla_{\Gamma}\cdot\mathbf{V})(\mathbf{V}_{a}^{\tau})_{i}-\sum_{j=1}^{3}(\mathbf{V}_{a}^{\tau})_{j}\underline{D}^{\Gamma(t)}_{j}\mathbf{V}_{i},\quad
i=1,2,3.$

In the above we use the same notation as in [29]. We postpone the proof of
this result to Appendix A.

Let us now recall the compatibility condition on the initial datum $u_{0}$ in
order to obtain the existence of a (weak) solution to the Cahn-Hilliard
equation with logarithmic potential. As noted in [13], there is an interplay
between the evolution of the surfaces $\Gamma(t)$ and the admissible initial
conditions. In effect, setting

$m_{\eta}(t):=\frac{1}{|\Gamma(t)|}\left|\int_{\Gamma_{0}}\eta\right|,$

we require that the initial datum $u_{0}$ satisfies

$\displaystyle\max_{t\in[0,T]}m_{u_{0}}(t)<1.$ (3.4)

Although it might seem unnatural to prescribe a condition involving the
initial data and the area of the surfaces at all future times, this assumption
has an interesting physical meaning. Let us introduce the maximum shrinkage
ratio $S_{R}$ on $[0,T]$, which is a priori prescribed by the evolution of the
surfaces $\Gamma(t)$, as:

$S_{R}:=\max_{t\in[0,T]}\frac{|\Gamma_{0}|}{|\Gamma(t)|}.$

Condition (3.4) then means that

$\displaystyle|(u_{0})_{\Gamma_{0}}|S_{R}<1.$ (3.5)

Since at time $t=0$ we already assume $|(u_{0})_{\Gamma_{0}}|<1$ for the
initial mass (meaning that we do not start with a one-phase mixture),
condition (3.5) shows that the absolute value of the initial mass,
$|(u_{0})_{\Gamma_{0}}|$, compensates for the maximum shrinkage ratio $S_{R}$
of the evolving surface $\Gamma(t)$ on $[0,T]$. Therefore, the higher the
value of $S_{R}$ (i.e., the smaller the surface $\Gamma(t)$ becomes over
$[0,T]$), the further $u_{0}$ must be from the pure phases $\pm 1$ in
$\Gamma_{0}$.

Moreover, recalling that the Cahn-Hilliard dynamics implies that the total
mass of the solution is preserved, namely,

$\int_{\Gamma(t)}u(t)\equiv\int_{\Gamma_{0}}u_{0},\quad\text{for all
}t\in[0,T],$

we can write

$|(u(t))_{\Gamma(t)}|=m_{u_{0}}(t)$

and thus the condition $m_{u_{0}}(t)<1$ for every $t\in[0,T]$ is the dynamic
counterpart of

$|(u(t))_{\Omega}|=\left|\dfrac{1}{|\Omega|}\int_{\Omega}u(t)\,dx\right|=\left|\dfrac{1}{|\Omega|}\int_{\Omega}u_{0}\,dx\right|=|(u_{0})_{\Omega}|<1$

which holds in the case of a static bounded domain $\Omega$.

For future use, let us then define the set $\mathcal{I}_{0}$ of admissible
initial conditions as

$\mathcal{I}_{0}:=\left\\{\eta\in H^{1}(\Gamma_{0}):|\eta|\leq 1\text{ a.e. on
}\Gamma_{0},\ \text{and }\left|(\eta)_{\Gamma_{0}}\right|S_{R}<1\right\\}.$

We recall that the free energy is given by

$E^{CH}[u]=\int_{\Gamma(t)}\left(\frac{|\nabla u|^{2}}{2}+F(u)\right),$

with $F$ logarithmic potential. Note that, if $u_{0}\in\mathcal{I}_{0}$, then
$u_{0}$ has finite energy. Indeed $\|u_{0}\|_{L^{\infty}(\Gamma_{0})}\leq 1$
implies $F(u_{0})\in L^{1}(\Gamma_{0})$. We can now recall the result obtained
in [13, Theorems 5.14, 5.15] for

$\displaystyle
F(s)=\frac{\theta}{2}((1+r)\ln(1+r)+(1-r)\ln(1-r))+\frac{1-r^{2}}{2},\quad
r\in(-1,1),$ (3.6)

where $0<\theta<1$ ($F$ is extended by continuity at $r=\pm 1$). We set
$F_{ln}(r):=(1+r)\ln(1+r)+(1-r)\ln(1-r)$ for the sake of simplicity and denote
$\varphi(s)=F^{\prime}_{ln}(s)$. Note that this choice corresponds to the
singular potential (1.7) with $\theta_{0}=1$, up to a translation to get
directly $F\geq 0$ in $[-1,1]$.

######  Remark 3.3.

Concerning the logarithmic potential, there exists a constant $C>0$ such that

$\displaystyle\frac{\theta}{2}\varphi^{\prime}(s)\leq
e^{C\left|\frac{\theta}{2}\varphi(s)\right|+C},\quad\forall s\in(-1,1).$ (3.7)

We now let Assumption $\mathbf{A}_{\Phi}$ hold: we have (see [13])

######  Theorem 3.4.

Let $u_{0}\in\mathcal{I}_{0}$ and $F:[-1,1]\to\mathbb{R}$ be given by (3.6).
Then there exists a unique pair $(u,w)$ with

$u\in L^{\infty}_{H^{1}}\cap H^{1}_{H^{-1}}\quad\text{ and }\quad w\in
L^{2}_{H^{1}},$

such that, for almost any $t\in(0,T]$, $|u(t)|<1$ almost everywhere on
$\Gamma(t)$ and $u$ satisfies, for all $\eta\in L^{2}_{H^{1}}$ and almost any
$t\in[0,T]$,

$\displaystyle
m_{\star}(\partial^{\bullet}u,\eta)+g(u,\eta)+a_{N}(u,\eta)+a_{S}(w,\eta)=0,$
(3.8) $\displaystyle
a_{S}(u,\eta)+\frac{\theta}{2}m(\varphi(u),\eta)-m(u,\eta)-m(w,\eta)=0,$ (3.9)

where $u(0)=u_{0}$ almost everywhere in $\Gamma_{0}$. The solution $u$ also
satisfies the additional regularity

$u\in C^{0}_{L^{2}}\cap L^{\infty}_{L^{p}}\cap L^{2}_{H^{2}},$

for all $p\in[1,+\infty)$. Furthermore, if $u_{0},v_{0}\in\mathcal{I}_{0}$ are
such that $(u_{0})_{\Gamma_{0}}=(v_{0})_{\Gamma_{0}}$, and $u,v$ are the
solutions of the system with $u(0)=u_{0}$ and $v(0)=v_{0}$, then there exists
a constant $C>0$ independent of $t$, such that, for almost any $t\in[0,T]$,

$\|u(t)-v(t)\|_{H^{-1}(\Gamma(t))}\leq
e^{Ct}\|u_{0}-v_{0}\|_{H^{-1}(\Gamma(t))}.$

######  Remark 3.5.

We notice that the regularity stated in Theorem 3.4 can be slightly improved.
In particular, since $u\in L^{2}_{H^{2}}$ solves the problem, for almost all
$t\in[0,T]$,

$-\Delta_{\Gamma}u(t)=w(t)-F^{\prime}(u(t))\in L^{2}({\Gamma(t)}),$

we are allowed to multiply by $-\Delta_{\Gamma}u\in L^{2}(\Gamma(t))$ for
almost any $t\in[0,T]$. Recalling that $\varphi^{\prime}>0$, after an
integration by parts, being $\Gamma(t)$ closed and $u\in L^{\infty}_{H^{1}}$,
we obtain

$\displaystyle{\left\|\Delta_{\Gamma}u\right\|}^{2}$
$\displaystyle\leq{\left\|\Delta_{\Gamma}u\right\|}^{2}+\frac{\theta}{2}m(\varphi^{\prime}(u),|\nabla_{\Gamma}u|^{2})\leq{\left\|\nabla_{\Gamma}w\right\|}{\left\|\nabla_{\Gamma}u\right\|}+\|\nabla_{\Gamma}u\|^{2}\leq
C(1+{\left\|\nabla_{\Gamma}w\right\|}),$

and knowing that $w\in L^{2}_{H^{1}}$, we infer $u\in L^{4}_{H^{2}}$.

###  3.3 Regularisation and strict separation property

In this subsection we need Assumptions $\textbf{B}_{\Phi}$. We first show that
the approximating solution given by the Galerkin scheme is consequently more
regular. This allows us to perform higher-order regularisation estimates and
to establish the strict separation property.

####  3.3.1 Galerkin approximation

Motivated by the approach in [13], we start by considering the Galerkin
approximation with the approximated potential $F^{\delta}\in
C^{2}(\mathbb{R})$ defined by

$F^{\delta}(r):=F_{ln}^{\delta}(r)+\frac{1-r^{2}}{2},\quad\forall
r\in\mathbb{R},$

where

$\displaystyle
F^{\delta}_{ln}(r)=\dfrac{\theta}{2}\begin{cases}(1-r)\ln(\delta)+(1+r)\ln(2-\delta)+\frac{(1-r)^{2}}{2\delta}+\frac{(1+r)^{2}}{2(2-\delta)}-1,&\,r\geq
1-\delta\\\ (1+r)\ln(1+r)+(1-r)\ln(1-r),&\,|r|\leq 1-\delta\\\
(1+r)\ln(\delta)+(1-r)\ln(2-\delta)+\frac{(1+r)^{2}}{2\delta}+\frac{(1-r)^{2}}{2(2-\delta)}-1,&\,r\leq-1+\delta\end{cases}$

and denote $\varphi^{\delta}=(F_{ln}^{\delta})^{\prime}$. Let us recall [13,
Sec.4.1] and consider a basis $\\{\chi_{j}^{0}:j\in\mathbb{N}\\}$ orthonormal
in $L^{2}(\Gamma_{0})$ and orthogonal in $H^{1}(\Gamma_{0})$ consisting of
smooth functions such that $\chi_{1}^{0}$ is constant (for example consider
the eigenfunctions of the Laplace-Beltrami operator). We then transport this
basis using the flow map. This gives
$\\{\chi_{j}^{t}:=\phi_{t}(\chi_{j}^{0}):j\in\mathbb{N}\\}\subset
H^{1}(\Gamma(t))$ and we define the finite-dimensional spaces

$\displaystyle V_{M}(t):=\\{\chi_{j}^{t}\colon j=1,\dots,M\\}.$ (3.10)

The goal in this section is to find an approximating solution pair
$(u^{M},w^{M})$ in these spaces; more precisely, for each $M\in\mathbb{N}$, we
aim to find functions $u^{M},w^{M}\in L^{2}_{V_{M}}$ with
$\partial^{\bullet}u^{M}\in L^{2}_{V_{M}}$ such that, for any $\eta\in
L^{2}_{V_{M}}$ and all $t\in[0,T]$,

$\displaystyle
m(\partial^{\bullet}u^{M},\eta)+g({\color[rgb]{0,0,0}u^{M}},\eta)+a_{N}(u^{M},\eta)+a_{S}(w^{M},\eta)=0,$
(3.11) $\displaystyle
a_{S}(u^{M},\eta)+m((F^{\delta})^{\prime}(u^{M}),\eta)-m(w^{M},\eta)=0,$
(3.12)

and $u^{M}(0)=P_{M}^{0}u_{0}$ almost everywhere in $\Gamma_{0}$ ($P_{M}^{0}$
is the $L^{2}$ orthogonal projector operator at $t=0$ (see [13, Sec.4.1]). We
first prove a refinement of [13, Prop.4.4], namely,

######  Proposition 3.6.

Let Assumptions $\textbf{B}_{\Phi}$ hold. Then there exists a unique local
solution pair to (3.11)-(3.12). In particular there exist functions
$(u^{M},w^{M})$ satisfying (3.11)-(3.12) on an interval $[0,t^{\star})$,
$0\leq t^{\star}\leq T$, together with the initial condition
$u^{M}(0)=P_{M}^{0}u_{0}$. The functions are of the form

$u^{M}(t)=\sum_{i=1}^{M}u_{i}^{M}(t)\chi_{i}^{t},\qquad
w^{M}(t)=\sum_{i=1}^{M}w_{i}^{M}(t)\chi_{i}^{t},\qquad t\in[0,t^{\star}),$

with $u_{i}^{M}\in C^{2}([0,t^{\star}))$ and $w_{i}^{M}\in
C^{2}([0,t^{\star}))$, for every $i\in\\{1,\ldots,M\\}$.

###### Proof.

We consider the matrix form of the equations, where we denote
$\mathbf{u}^{M}(t)=(u_{1}^{M}(t),\ldots,u_{M}^{M}(t))$ and
$\mathbf{w}^{M}(t)=(w_{1}^{M}(t),\ldots,w_{M}^{M}(t))$,

$\displaystyle
M(t)\dot{\mathbf{u}}^{M}(t)+{G}(t)\mathbf{{u}}^{M}(t)+A_{N}(t)\mathbf{u}^{M}(t)+A_{S}(t)\mathbf{w}^{M}(t)=0,$
$\displaystyle
A_{S}(t)\mathbf{u}^{M}(t)+(\mathbf{F}^{\delta})^{\prime}(\mathbf{{u}}^{M}(t))-M(t)\mathbf{w}^{M}(t)=0.$

Here

$\displaystyle(M(t))_{ij}=m(t;\chi_{i}^{t},\chi_{j}^{t}),\qquad({G}(t))_{ij}=g(t;{\color[rgb]{0,0,0}\chi_{j}^{t}},\chi_{i}^{t}),$
$\displaystyle(A_{S}(t))_{ij}=a_{S}(t;\chi^{t}_{i},\chi^{t}_{j}),\qquad(A_{N}(t))_{ij}=a_{N}(t;\chi_{j}^{t},\chi_{i}^{t}),$

and

$(\mathbf{F}^{\delta})^{\prime}(\mathbf{u}^{M}(t))_{j}=m(t;(F^{\delta})^{\prime}(u^{M}(t)),\chi_{j}^{t}).$

We now observe that actually these matrices enjoy more regularity than noted
in [13]. Indeed, let us consider for example $M_{ij}$. By [13, Prop. 2.8],
recalling that $\partial^{\bullet}\chi_{i}^{t}\equiv 0$ for any
$i=1,\ldots,M$, we have

$\frac{d}{dt}M_{ij}=g(\chi^{t}_{i},\chi^{t}_{j})\in C^{0}([0,T]),$

due to the regularity assumptions on $\Gamma(t)$, on the corresponding flow
map and on the velocity field. We also have

$\frac{d}{dt}(M^{-1})_{ij}=-\left(M^{-1}\left(\frac{d}{dt}M\right)M^{-1}\right)_{ij}\in
C^{0}([0,T]).$

Similarly, we get

$\frac{d}{dt}(A_{S}(t))_{ij}=b(\chi^{t}_{i},\chi^{t}_{j})\in C^{0}([0,T]),$

and, by (3.3),

$\frac{d}{dt}(A_{N}(t))_{ij}=\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))\chi^{t}_{j}\cdot\nabla_{\Gamma}\chi_{i}^{t}\in
C^{0}([0,T]).$

Moreover, by Proposition 3.2 point (a),

$\displaystyle\frac{d}{dt}({G}(t))_{ij}=m(\chi_{i}^{t}$
$\displaystyle\partial^{\bullet}(\nabla_{\Gamma}\cdot\mathbf{V}),\chi_{j}^{t})+g\left(\chi_{i}^{t}\nabla_{\Gamma}\cdot\mathbf{V},\chi_{j}^{t}\right)\in
C^{0}([0,T]),$

thanks to (2.9). Furthermore, we get

$\frac{d}{dt}(\mathbf{F}^{\delta})^{\prime}(\mathbf{y})_{j}=g\left((F^{\delta})^{\prime}\left(\sum_{i=1}^{M}y_{i}\chi_{i}^{t}\right),\chi_{j}^{t}\right)\in
C^{0}([0,T]).$

Recalling then that $(F^{\delta})^{\prime}$ is $C^{1,1}(\mathbb{R})$, ensuring
the spatial $C^{1,1}$-continuity of the $\mathbf{y}$-Jacobian of
$\mathbf{y}\mapsto(\textbf{F}^{\delta})^{\prime}(\mathbf{y})$, the result
follows from the general ODE theory. ∎

We now obtain the energy estimate for the Galerkin approximation, without
letting $M\rightarrow\infty$ (cf. [13]). This is essential in order to
maintain the necessary regularity to perform higher-order estimates.

######  Proposition 3.7.

Denoting by $(u_{\delta}^{M},w_{\delta}^{M})$ the solution pair to
(3.11)-(3.12), we have the energy estimate

$\displaystyle\sup_{t\in[0,T]}\tilde{E}^{CH}[t;u_{\delta}^{M}(t)]+\int_{0}^{T}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}dt\leq
C_{\delta}(T),$ (3.13)

for some $C_{\delta}(T)$ depending on $\delta$ and $T$, but not on $M$, where
$\tilde{E}^{CH}:=E^{CH}+\tilde{C}$ for some $\tilde{C}>0$ chosen so that
$\tilde{E}^{CH}\geq 0$. Moreover, it holds

$\displaystyle\sup_{t\in[0,T]}\tilde{E}^{CH}[u_{\delta}^{M}(t)]$
$\displaystyle+\frac{1}{4}\int_{0}^{T}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right),$ (3.14)

for some $C(T)>0$ independent of $M,\delta$ but possibly depending on $T$.

######  Remark 3.8.

We observe that, even though the energy $E^{CH}$ is not necessarily non-
negative, it is bounded from below. Indeed, there exists $C=C(\theta)$ such
that $F\geq C(\theta)$, and therefore

$\displaystyle E^{CH}[u]\geq
C(\theta)|\Gamma(t)|\geq\dfrac{C(\theta)}{C_{\Gamma}}|\Gamma_{0}|,$ (3.15)

where $C_{\Gamma}$ is defined in (2.11). Note that $\tilde{C}$ (see the
previous statement) depends on $T$, $\Gamma_{0}$ and $\theta$, as well as on
an upper bound for the $C^{1}$-norm of the flow map $\Phi$.

###### Proof.

Keep $M,\delta$ fixed and denote by $C$ a generic positive constant
independent of $M$ and $\delta$. We consider the Galerkin approximation and
the results given by Proposition 3.6. Here we denote the solution by
$(u_{\delta}^{M},w_{\delta}^{M})$ to emphasize the dependence on $\delta$. We
then note that the total mass of $u_{\delta}^{M}$ is preserved in time, since
$\eta\equiv 1\in V_{M}$ for any $M\in\mathbb{N}$. We then come back to the
energy estimate as in the proof of [13, Prop.5.7, a)]:

$\displaystyle\frac{d}{dt}\tilde{E}^{CH}[u_{\delta}^{M}]+\frac{1}{2}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq-g({\color[rgb]{0,0,0}u_{\delta}^{M}},w_{\delta}^{M})+C_{0}+C_{1}\tilde{E}^{CH}[u_{\delta}^{M}],$
(3.16)

where $\tilde{E}_{CH}:={E}_{CH}+\tilde{C}$, with $\tilde{C}>0$ a suitable
constant so that $\tilde{E}_{CH}\geq 0$. Thanks to the bound on $\mathbf{V}$,
by Young and Poincaré’s inequalities we have

$\displaystyle-g({\color[rgb]{0,0,0}u_{\delta}^{M}},w_{\delta}^{M})$
$\displaystyle\leq
C\|{\color[rgb]{0,0,0}u_{\delta}^{M}}\|\|w_{\delta}^{M}\|\leq
C\|{\color[rgb]{0,0,0}u_{\delta}^{M}}\|(|m(w_{\delta}^{M},1)|+\|\nabla_{\Gamma}w_{\delta}^{M}\|)$
$\displaystyle\leq\frac{1}{4}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+C(\|{\color[rgb]{0,0,0}u_{\delta}^{M}}\|^{2}+|m(w_{\delta}^{M},1)|^{2})$
$\displaystyle\leq\frac{1}{4}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+C(1+\|{\color[rgb]{0,0,0}\nabla_{\Gamma}u_{\delta}^{M}}\|^{2}+|m(w_{\delta}^{M},1)|^{2}),$

where we exploited the fact that, by mass conservation, Poincaré’s inequality
and (2.11),

$\displaystyle\|u_{\delta}^{M}\|$
$\displaystyle\leq\frac{1}{|\Gamma(t)|}|m(u_{\delta}^{M},1)|+\|\nabla_{\Gamma}u_{\delta}^{M}\|\leq
C(|m(P_{M}^{0}u_{0},1)|+\|\nabla_{\Gamma}u_{\delta}^{M}\|)\leq
C(1+\|\nabla_{\Gamma}u_{\delta}^{M}\|),$ (3.17)

recalling that $\|P_{M}^{0}u_{0}\|\leq\|u_{0}\|\leq C$. For further use, this
gives

$\displaystyle\frac{d}{dt}\tilde{E}^{CH}[u_{\delta}^{M}]+\frac{1}{4}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq
C_{0}+C_{1}\tilde{E}^{CH}[u_{\delta}^{M}]+C_{2}|m(w_{\delta}^{M},1)|^{2},$
(3.18)

with $C_{0},C_{1},C_{2}>0$ independent of $M,\delta$. Since estimate (3.46)
cannot be performed in the Galerkin context (note that, differently from the
static domain case, here it is not ensured that
$\Delta_{\Gamma}u_{\delta}^{M}\in L^{2}_{V_{M}}$), we now first obtain an
estimate depending on $\delta$ but not on $M$, and in a second time, after
passing to limit as $M\to\infty$, we obtain estimates uniform in $\delta$,
thanks to (3.46). Therefore, we now exploit
$|F^{\prime}_{\delta}(u_{\delta}^{M})|\leq C_{\delta}|u_{\delta}^{M}|$, being
${\color[rgb]{0,0,0}|F^{\prime\prime}_{\delta}|}\leq C_{\delta}$ and
$F_{\delta}^{\prime}(0)=F^{\prime}(0)=0$, to deduce

$\displaystyle|m(w_{\delta}^{M},1)|\leq|m(F_{\delta}^{\prime}(u_{\delta}^{M}),1)|\leq
C_{\delta}\|u_{\delta}^{M}\|_{L^{1}(\Gamma(t))}\leq
C_{\delta}\|u_{\delta}^{M}\|$ (3.19)

with $C_{\delta}>0$ depending on $\delta$, so that, by Young and Poincaré’s
inequalities, we find

$-g({\color[rgb]{0,0,0}u_{\delta}^{M}},w_{\delta}^{M})\leq
C\|u_{\delta}^{M}\|\|w_{\delta}^{M}\|\leq
C_{\delta}\left(1+\|u_{\delta}^{M}\|^{2}\right)+\frac{3}{8}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2},$

and thus, from (3.16) and (3.17),

$\displaystyle\frac{d}{dt}\tilde{E}^{CH}[u_{\delta}^{M}]+\frac{1}{8}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq{\color[rgb]{0,0,0}C_{\delta}}\tilde{E}^{CH}[u_{\delta}^{M}]+C_{\delta}.$

Therefore Gronwall’s Lemma, together with (3.17) and [13, Lemma 5.6] give

$\displaystyle\sup_{t\in[0,T]}\tilde{E}^{CH}[t;u_{\delta}^{M}(t)]+\sup_{t\in[0,T]}\|u_{\delta}^{M}(t)\|^{2}+\int_{0}^{T}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}dt\leq
C_{\delta},$ (3.20)

independently of $M$. Moreover, by Gronwall’s Lemma and [13, Lemma 5.6],
applied to (3.18), we also get (3.14). ∎

####  3.3.2 Regularisation and strict separation property

The main result of this section is the following, letting Assumptions
$\textbf{B}_{\Phi}$ hold.

######  Theorem 3.9.

Let the assumptions of Theorem 3.4 and Assumptions $\textbf{B}_{\Phi}$ hold.
Denote by $(u,w)$ the (unique) weak solution to (3.8)-(3.9) with $u(0)=u_{0}$.

  * (i)

There exists a constant $C=C(T,{E}^{CH}(u_{0}))>0$ such that, for almost any
$t\in[0,T]$,

$\displaystyle
t\|w\|_{H^{1}(\Gamma(t))}^{2}+\int_{0}^{t}s\|\partial^{\bullet}u\|^{2}_{H^{1}(\Gamma(s))}ds\leq
C(T,{E}^{CH}(u_{0})).$ (3.21)

  * (ii)

For any $0<\tau\leq T$, there exist constants $C=C(T,\tau,{E}^{CH}(u_{0}))>0$
and $C_{p}=C_{p}(T,\tau,p,{E}^{CH}(u_{0}))>0$ such that, for almost any
$t\in[\tau,T]$,

$\displaystyle\|w\|_{H^{1}(\Gamma(t))}\leq C(T,\tau,{E}^{CH}(u_{0})),$ (3.22)
$\|\varphi(u)\|_{L^{p}(\Gamma(t))}+\|\varphi^{\prime}(u)\|_{L^{p}(\Gamma(t))}\leq
C_{p}(T,\tau,p,{E}^{CH}(u_{0})),\quad\forall p\in[2,\infty),$
$\|u\|_{H^{2}(\Gamma(t))}\leq C(T,\tau,{E}^{CH}(u_{0})).$

  * (iii)

There exists $\xi=\xi(T,\tau,{E}^{CH}(u_{0}))>0$ such that

$\displaystyle\|u\|_{L^{\infty}(\Gamma(t))}\leq 1-{\xi},\quad\text{ for a.a.
}t\in[\tau,T].$

######  Remark 3.10.

Notice that actually we could say more about the separation property, if we
assume the regularity on the flow map and the velocity field as in Lemma C.1.
Indeed, for any $\tau>0$, we have $u\in L^{\infty}_{H^{2}}(\tau,T)$ and
$\partial^{\bullet}u\in L^{2}_{H^{1}}(\tau,T)$ (we use the symbol
$L^{q}_{X}(\tau,T)$ to mean that the set of times is $[\tau,T]$ instead of
$[0,T]$, typical of $L^{q}_{X}$). By the embedding result shown in Appendix C,
we infer that $u\in C^{0}_{H^{3/2}}(\tau,T)$, thus by the embedding
$H^{3/2}(\Gamma(t))\hookrightarrow C^{0}(\Gamma(t))$, $u\in
C^{0}_{C^{0}}(\tau,T)$, implying

$\displaystyle\sup_{t\in[\tau,T]}\|u\|_{C^{0}(\Gamma(t))}\leq 1-{\xi}.$

###### Proof.

From now on, we will omit the dependence on ${E}^{CH}(u_{0})$, since it is
considered understood due to Prop. 3.7.

Part (i). Step 1. Limit as $M\to\infty$. We test (3.11) with
$\eta=\partial^{\bullet}w^{M}_{\delta}\in L^{2}_{V_{M}}$, obtaining

$\displaystyle
m(\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})+g({\color[rgb]{0,0,0}u^{M}_{\delta}},\partial^{\bullet}w^{M}_{\delta})+a_{N}(u^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})+a_{S}(w^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})=0.$
(3.23)

Observe now that, by Proposition 3.2, for $\eta\in L^{2}_{V_{M}}$ such that
$\partial^{\bullet}\eta\in L^{2}_{V_{M}}$, we have

$\displaystyle\frac{d}{dt}m(w^{M}_{\delta},\eta)=m(\partial^{\bullet}w^{M}_{\delta},\eta)+m(w^{M}_{\delta},\partial^{\bullet}\eta)+g(w^{M}_{\delta},\eta),$
(3.24)

but also (see (3.12))

$\displaystyle\frac{d}{dt}m(w^{M}_{\delta},\eta)=\frac{d}{dt}\left(a_{S}(u^{M}_{\delta},\eta)+m((F^{\delta})^{\prime}(u^{M}_{\delta}),\eta)\right)$
(3.25)

Moreover, again by Proposition 3.2, we infer

$\frac{d}{dt}a_{S}(u^{M}_{\delta},\eta)=a_{S}(\partial^{\bullet}u^{M}_{\delta},\eta)+a_{S}(u^{M}_{\delta},\partial^{\bullet}\eta)+b(u^{M}_{\delta},\eta).$

On the other hand, exploiting the chain rule, we get

$\displaystyle\frac{d}{dt}m((F^{\delta})^{\prime}(u^{M}_{\delta}),\eta)=m((F^{\delta})^{\prime\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\eta)$
$\displaystyle+m((F^{\delta})^{\prime}(u^{M}_{\delta}),\partial^{\bullet}\eta)+g((F^{\delta})^{\prime}(u^{M}_{\delta}),\eta).$

Therefore, comparing (3.24) with (3.25), we obtain

$\displaystyle m($
$\displaystyle\partial^{\bullet}w^{M}_{\delta},\eta)+m(w^{M}_{\delta},\partial^{\bullet}\eta)+g(w^{M}_{\delta},\eta)=m((F^{\delta})^{\prime\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\eta)$
$\displaystyle+m((F^{\delta})^{\prime}(u^{M}_{\delta}),\partial^{\bullet}\eta)+g((F^{\delta})^{\prime}(u^{M}_{\delta}),\eta)+a_{S}(\partial^{\bullet}u^{M}_{\delta},\eta)+a_{S}(u^{M}_{\delta},\partial^{\bullet}\eta)+b(u^{M}_{\delta},\eta),$

but since $\partial^{\bullet}\eta$ is still an admissible function in (3.12),
we infer

$\displaystyle m(\partial^{\bullet}w^{M}_{\delta},\eta)$
$\displaystyle=-g(w^{M}_{\delta},\eta)+m((F^{\delta})^{\prime\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\eta)+g((F^{\delta})^{\prime}(u^{M}_{\delta}),\eta)+a_{S}(\partial^{\bullet}u^{M}_{\delta},\eta)+b(u^{M}_{\delta},\eta).$

Thus, setting $\eta=\partial^{\bullet}u_{\delta}^{M}\in L^{2}_{V_{M}}$ (note
that $\partial^{\bullet,2}u_{\delta}^{M}\in L^{2}_{V_{M}}$ by Proposition
3.6), we get

$\displaystyle
m(\partial^{\bullet}w^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle=-g(w^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+m((F^{\delta})^{\prime\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle+g((F^{\delta})^{\prime}(u^{M}_{\delta}),\partial^{\bullet}u^{M}_{\delta})+a_{S}(\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+b(u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta}).$
(3.26)

Using once more Proposition 3.2, on account of the regularity given by
Proposition 3.6, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}=a_{S}(w^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})+\frac{1}{2}b(w^{M}_{\delta},w^{M}_{\delta}).$
(3.27)

Furthermore, consider the term $g$ in (3.23). By Proposition 3.2, we have

$\displaystyle\frac{d}{dt}g({\color[rgb]{0,0,0}u^{M}_{\delta}},w^{M}_{\delta})$
$\displaystyle=g({\color[rgb]{0,0,0}u^{M}_{\delta}},\partial^{\bullet}w^{M}_{\delta})+m(\partial^{\bullet}({\color[rgb]{0,0,0}u^{M}_{\delta}}\nabla_{\Gamma}\cdot\mathbf{V}),w^{M}_{\delta})+g({\color[rgb]{0,0,0}u^{M}_{\delta}}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})$
$\displaystyle=g({\color[rgb]{0,0,0}u^{M}_{\delta}},\partial^{\bullet}w^{M}_{\delta})+m(\partial^{\bullet}u^{M}_{\delta}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+m({\color[rgb]{0,0,0}u^{M}_{\delta}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+g({\color[rgb]{0,0,0}u^{M}_{\delta}}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta}).$

Therefore we get

$\displaystyle
g({\color[rgb]{0,0,0}u^{M}_{\delta}},\partial^{\bullet}w^{M}_{\delta})=\frac{d}{dt}$
$\displaystyle
g({\color[rgb]{0,0,0}u^{M}_{\delta}},w^{M}_{\delta})-m(\partial^{\bullet}u_{\delta}^{M}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})-m({\color[rgb]{0,0,0}u^{M}_{\delta}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})-g({\color[rgb]{0,0,0}u^{M}_{\delta}}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta}).$
(3.28)

Recalling (3.3), we have

$\displaystyle\frac{d}{dt}a_{N}(u^{M}_{\delta},w^{M}_{\delta})=a_{N}(\partial^{\bullet}u^{M}_{\delta},$
$\displaystyle
w^{M}_{\delta})+a_{N}(u^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})+\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta},$

that is,

$\displaystyle
a_{N}(u^{M}_{\delta},\partial^{\bullet}w^{M}_{\delta})=\frac{d}{dt}a_{N}(u^{M}_{\delta},$
$\displaystyle
w^{M}_{\delta})-a_{N}(\partial^{\bullet}u^{M}_{\delta},w^{M}_{\delta})-\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta}.$
(3.29)

Thanks to Proposition 3.2, we deduce

$\displaystyle\frac{d}{dt}\frac{\theta}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),1)=\frac{\theta}{2}g(\varphi_{\delta}(u^{M}_{\delta}),\partial^{\bullet}u^{M}_{\delta})+\frac{\theta}{2}\int_{\Gamma(t)}$
$\displaystyle
F_{ln}^{\delta}(u^{M}_{\delta})\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V}+\frac{\theta}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),\nabla_{\Gamma}\cdot\mathbf{V}).$
(3.30)

This last equality is necessary, since in the Galerkin setting we are not able
to retrieve a uniform estimate for
$\|\varphi_{\delta}(u_{\delta}^{M})\|_{L^{2}_{L^{2}}}$ but only for its
$L^{2}$-orthogonal projection over $V_{M}$(t), whereas, thanks to (3.20), we
are able to uniformly estimate
$\sup_{t\in[0,T]}\|F^{\delta}_{ln}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))}$.
Collecting (3.23)-(3.29), we obtain

$\displaystyle\frac{d}{dt}\Bigg{(}\frac{1}{2}$
$\displaystyle\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+a_{N}(u^{M}_{\delta},w^{M}_{\delta})+g({\color[rgb]{0,0,0}u_{\delta}^{M}},w^{M}_{\delta})\Bigg{)}+\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle\hskip
17.07164pt+m((F^{\delta})^{\prime\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle=g(w^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})-g((F^{\delta})^{\prime}(u^{M}_{\delta}),\partial^{\bullet}u^{M}_{\delta})-b(u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+\frac{1}{2}b(w^{M}_{\delta},w^{M}_{\delta})$
$\displaystyle\hskip
17.07164pt+m(\partial^{\bullet}u^{M}_{\delta}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+m({\color[rgb]{0,0,0}u_{\delta}^{M}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})$
$\displaystyle\hskip
17.07164pt+g({\color[rgb]{0,0,0}u_{\delta}^{M}}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+a_{N}(\partial^{\bullet}u^{M}_{\delta},w^{M}_{\delta})$
$\displaystyle\hskip
17.07164pt+\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta}.$

In particular, substituting the explicit value of $F^{\delta}$, setting
$\varphi_{\delta}:=(F^{\delta}_{ln})^{\prime}$, and exploiting (3.30), we find

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+a_{N}(u^{M}_{\delta},w^{M}_{\delta})+g(u^{M}_{\delta},w^{M}_{\delta})+\frac{\theta}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),1)\right)$
$\displaystyle+\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+\frac{\theta}{2}m(\varphi_{\delta}^{\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle=g(w^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+m(\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+g(u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})-b(u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle+\frac{1}{2}b(w^{M}_{\delta},w^{M}_{\delta})+m(\partial^{\bullet}u^{M}_{\delta}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+m({\color[rgb]{0,0,0}u_{\delta}^{M}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})$
$\displaystyle+g({\color[rgb]{0,0,0}u_{\delta}^{M}}\nabla_{\Gamma}\cdot\mathbf{V},w^{M}_{\delta})+a_{N}(\partial^{\bullet}u^{M}_{\delta},w^{M}_{\delta})+\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta}$
$\displaystyle+\frac{\theta}{2}\int_{\Gamma(t)}F_{ln}^{\delta}(u^{M}_{\delta})\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V}+\frac{\theta}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),\nabla_{\Gamma}\cdot\mathbf{V}).$

Recalling that $\varphi_{\delta}^{\prime}=(F^{\delta}_{ln})^{\prime\prime}\geq
0$, we have
$\frac{\theta}{2}m(\varphi_{\delta}^{\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})\geq
0$. Therefore, setting

$\displaystyle\mathcal{Q}^{M}_{\delta}:=\frac{1}{2}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+a_{N}(u^{M}_{\delta},w^{M}_{\delta})+g({\color[rgb]{0,0,0}u_{\delta}^{M}},w^{M}_{\delta})+\frac{\theta}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),1),$
(3.31)

and multiplying by $t\in(0,T]$ the above inequality, we get

$\displaystyle{\color[rgb]{0,0,0}\frac{d}{dt}\left(t\mathcal{Q}^{M}_{\delta}(t)\right)}+t\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+t\frac{\theta}{2}m(\varphi_{\delta}^{\prime}(u^{M}_{\delta})\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle=\mathcal{Q}^{M}_{\delta}(t)+g(w^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})+tm(\partial^{\bullet}u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle+g(u^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})-b(u^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})+\frac{t}{2}b(w^{M}_{\delta},w^{M}_{\delta})$
$\displaystyle+m(\partial^{\bullet}u^{M}_{\delta}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})+m({\color[rgb]{0,0,0}u_{\delta}^{M}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})$
$\displaystyle+g({\color[rgb]{0,0,0}u_{\delta}^{M}}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})+a_{N}(\partial^{\bullet}u^{M}_{\delta},tw^{M}_{\delta})+t\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta}$
$\displaystyle+\frac{\theta
t}{2}\int_{\Gamma(t)}F_{ln}^{\delta}(u^{M}_{\delta})\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V}+\frac{\theta
t}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),\nabla_{\Gamma}\cdot\mathbf{V}).$

We now recall that, by Poincaré’s inequality,

$\displaystyle\|w^{M}_{\delta}\|\leq
C(\|\nabla_{\Gamma}w^{M}_{\delta}\|+|m(w_{\delta}^{M},1)|).$ (3.32)

Then, on account of (2.3) and using Hölder’s and Young’s inequalities, we
deduce

$\displaystyle
g(w^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})+m(\partial^{\bullet}u^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})$
$\displaystyle+g(u^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})-b(u^{M}_{\delta},t\partial^{\bullet}u^{M}_{\delta})+\frac{t}{2}b(w^{M}_{\delta},w^{M}_{\delta})$
$\displaystyle{\color[rgb]{0,0,0}+}\frac{\theta
t}{2}\int_{\Gamma(t)}F_{ln}^{\delta}(u^{M}_{\delta})\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V}+\frac{\theta
t}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),\nabla_{\Gamma}\cdot\mathbf{V})$
$\displaystyle\leq
Ct\|w^{M}_{\delta}\|\|\partial^{\bullet}u^{M}_{\delta}\|+t\|\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle+t{\color[rgb]{0,0,0}\|u^{M}_{\delta}\|}\|\partial^{\bullet}u^{M}_{\delta}\|+Ct\|\nabla_{\Gamma}u_{\delta}^{M}\|\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|+Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}$
$\displaystyle+Ct\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))}+Ct\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))}$
$\displaystyle\leq
Ct\|\partial^{\bullet}u^{M}_{\delta}\|^{2}+Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+\frac{t}{2}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle+Ct(\|u^{M}_{\delta}\|^{2}+|m(w_{\delta}^{M},1)|^{2}+\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))}),$

where we exploited (2.9) and (3.32). Analogously, recalling (2.5), (2.9) and
(3.32), thanks to Cauchy-Schwarz and Young’s inequalities, we find

$\displaystyle
m(\partial^{\bullet}u^{M}_{\delta}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})+m({\color[rgb]{0,0,0}u_{\delta}^{M}}\partial^{\bullet}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})$
$\displaystyle+g({\color[rgb]{0,0,0}u_{\delta}^{M}}\nabla_{\Gamma}\cdot\mathbf{V},tw^{M}_{\delta})+a_{N}(\partial^{\bullet}u^{M}_{\delta},tw^{M}_{\delta})+t\int_{\Gamma(t)}B_{adv}(\mathbf{V}_{a}^{\tau}(t),\mathbf{V}(t))u^{M}_{\delta}\cdot\nabla_{\Gamma}w^{M}_{\delta}$
$\displaystyle\leq
Ct\|\partial^{\bullet}u^{M}_{\delta}\|({\color[rgb]{0,0,0}|m(w_{\delta}^{M},1)|}+\|\nabla_{\Gamma}w^{M}_{\delta}\|)+Ct{\color[rgb]{0,0,0}\|u^{M}_{\delta}\|}({\color[rgb]{0,0,0}|m(w_{\delta}^{M},1)|}+\|\nabla_{\Gamma}w^{M}_{\delta}\|)$
$\displaystyle+Ct{\color[rgb]{0,0,0}\|u^{M}_{\delta}\|}({\color[rgb]{0,0,0}|m(w_{\delta}^{M},1)|}+\|\nabla_{\Gamma}w^{M}_{\delta}\|)+Ct\|\partial^{\bullet}u^{M}_{\delta}\|\|\nabla_{\Gamma}w^{M}_{\delta}\|+Ct\|u^{M}_{\delta}\|\|\nabla_{\Gamma}w^{M}_{\delta}\|$
$\displaystyle\leq
Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+Ct\|\partial^{\bullet}u^{M}_{\delta}\|^{2}+Ct(1+{\color[rgb]{0,0,0}|m(w_{\delta}^{M},1)|^{2}}+\|u^{M}_{\delta}\|^{2})$

Collecting the above inequalities, we eventually obtain

$\displaystyle{\color[rgb]{0,0,0}\frac{d}{dt}\left(t\mathcal{Q}^{M}_{\delta}(t)\right)}+\frac{t}{2}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle\leq\mathcal{Q}^{M}_{\delta}(t)+Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+Kt\|\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle{\color[rgb]{0,0,0}+Ct(\|u^{M}_{\delta}\|^{2}+|m(w_{\delta}^{M},1)|^{2}+\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))})},$
(3.33)

where we explicitly denoted by $K>0$, for further use, the constant in front
of $\|\partial^{\bullet}u^{M}_{\delta}\|$, which has to be controlled in a
uniform way. Since in the Galerkin setting we are not able to find a uniform
estimate for $\|\partial^{\bullet}u^{M}_{\delta}\|_{L^{2}_{H^{-1}}}$, we
cannot exploit the classical interpolation inequality
$\|\partial^{\bullet}u_{\delta}^{M}\|\leq
C\|\partial^{\bullet}u^{M}_{\delta}\|_{H^{-1}(\Gamma(t))}^{1/2}\|\partial^{\bullet}u^{M}_{\delta}\|_{H^{1}(\Gamma(t))}^{1/2}$
as in the case of nonevolving surfaces. Therefore, we test (3.11) with
$\eta=t\partial^{\bullet}u_{\delta}^{M}\in L^{2}_{V_{M}}$, obtaining

$t\|\partial^{\bullet}u^{M}_{\delta}\|^{2}+tg({\color[rgb]{0,0,0}u^{M}_{\delta}},\partial^{\bullet}u^{M}_{\delta})+ta_{N}(u^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})+ta_{S}(w^{M}_{\delta},\partial^{\bullet}u^{M}_{\delta})=0,$

from which, by standard estimates, for some $\kappa$ sufficiently small to be
chosen later on, we obtain

$t\|\partial^{\bullet}u^{M}_{\delta}\|^{2}\leq\frac{t}{2}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}+t\kappa\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+Ct\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+Ct{\color[rgb]{0,0,0}\|u^{M}_{\delta}\|^{2}},$

that is,

$\frac{t}{2}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}\leq
t\kappa\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+Ct\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+Ct{\color[rgb]{0,0,0}\|u^{M}_{\delta}\|^{2}}.$

We can now add this inequality multiplied by $\omega=4K$ to (3.33), choosing
$\kappa=\frac{1}{16K}$. This gives, for all $t\in[0,T]$,

$\displaystyle{\frac{d}{dt}\left(t\mathcal{Q}^{M}_{\delta}(t)\right)}+\frac{t}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+{Kt}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle\leq\mathcal{Q}^{M}_{\delta}(t)+Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+Ct(\|u^{M}_{\delta}\|^{2}$
$\displaystyle+|m(w_{\delta}^{M},1)|^{2}+\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))}).$
(3.34)

Recalling Proposition 3.7 and (3.19), from (3.34) we get, for all $t\in[0,T]$

$\displaystyle{\frac{d}{dt}\left(t\mathcal{Q}^{M}_{\delta}(t)\right)}+\frac{t}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}+{Kt}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}$
$\displaystyle\leq\mathcal{Q}^{M}_{\delta}(t)+Ct\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}+C_{\delta}t,$
(3.35)

where $C_{\delta}>0$ depends on $\delta$.  Observe now that

$\displaystyle|a_{N}(u^{M}_{\delta},w^{M}_{\delta})|\leq
C\|u_{\delta}^{M}\|\|\nabla_{\Gamma}w^{M}_{\delta}\|\leq
C\|u_{\delta}^{M}\|^{2}+\frac{1}{8}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2},$
(3.36)

and, by Poincaré’s inequality,

$\displaystyle|g(u^{M}_{\delta},w^{M}_{\delta})|\leq
C\|u_{\delta}^{M}\|\|w_{\delta}^{M}\|\leq
C(\|u_{\delta}^{M}\|^{2}+|m(w_{\delta}^{M},1)|^{2})+\frac{1}{8}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2},$
(3.37)

so that (see Proposition 3.7, (3.19) and (3.32))

$|g(u^{M}_{\delta},w^{M}_{\delta})|\leq
C_{\delta}+\frac{1}{8}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2},$

Moreover, owing to (3.20), we have

$\frac{\theta}{2}|g(F_{ln}^{\delta}(u^{M}_{\delta}),1)|\leq
C\|F_{ln}^{\delta}(u^{M}_{\delta})\|_{L^{1}(\Gamma(t))}\leq C_{\delta}.$

implying that there exists $\widehat{C}{\color[rgb]{0,0,0}{}_{\delta}}>0$,
depending on $\delta$, such that

$\frac{1}{4}\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}-\widehat{C}{\color[rgb]{0,0,0}{}_{\delta}}\leq\mathcal{Q}^{M}_{\delta}(t)\leq
C{\color[rgb]{0,0,0}{}_{\delta}}(1+\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}),\quad\forall\,t\in[0,T].$

Note now that, due to the energy estimate (3.20), we have that
$\mathcal{Q}^{M}_{\delta}$ is uniformly bounded in $L^{1}(0,T)$. Thus,
defining
$\widetilde{\mathcal{Q}}^{M}_{\delta}:=\mathcal{Q}_{\delta}^{M}+\widehat{C}{\color[rgb]{0,0,0}{}_{\delta}}\geq
0$, we obtain from (3.35)

$\displaystyle{\color[rgb]{0,0,0}\frac{d}{dt}\left(t\widetilde{\mathcal{Q}}^{M}_{\delta}(t)\right)}+{Kt}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}+\frac{t}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}\leq\widetilde{\mathcal{Q}}^{M}_{\delta}(t)+C{\color[rgb]{0,0,0}{}_{\delta}}t\widetilde{\mathcal{Q}}^{M}_{\delta}(t)+{C}{\color[rgb]{0,0,0}{}_{\delta}}t.$
(3.38)

Therefore, by Gronwall’s Lemma applied to
$y(t)=t\widetilde{\mathcal{Q}}^{M}_{\delta}(t)$, recalling that
$\widetilde{\mathcal{Q}}^{M}_{\delta}$ is uniformly (in $M$) bounded in
$L^{1}(0,T)$, we infer

$t\widetilde{\mathcal{Q}}^{M}_{\delta}(t)\leq
C{\color[rgb]{0,0,0}{}_{\delta}}(T),\quad\forall\,t\in[0,T],$

entailing, recalling (3.32),

$\displaystyle
t\|w^{M}_{\delta}\|_{H^{1}(\Gamma(t))}^{2}+\int_{0}^{t}s\|\partial^{\bullet}u^{M}_{\delta}\|^{2}_{H^{1}(\Gamma(s))}ds\leq
C{\color[rgb]{0,0,0}{}_{\delta}}(T),\quad\forall\,t\in[0,T],$ (3.39)

where $C_{\delta}(T)>0$ depends on $\delta$ but is independent of $M$. For
further use we also integrate in time (3.34), to get, for all $t\in[0,T]$,

$\displaystyle{t\mathcal{Q}^{M}_{\delta}(t)}+\int_{0}^{t}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}ds+\int_{0}^{t}{Ks}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}ds$
$\displaystyle\leq\int_{0}^{t}\mathcal{Q}^{M}_{\delta}(s)ds+C\int_{0}^{t}s\|\nabla_{\Gamma}w^{M}_{\delta}\|^{2}ds$
$\displaystyle+C\int_{0}^{t}s(\|u^{M}_{\delta}\|^{2}+|m(w_{\delta}^{M},1)|^{2}+\|F_{ln}^{\delta}(u_{\delta}^{M})\|_{L^{1}(\Gamma(t))})ds.$
(3.40)

Then by the definition (3.31) of $\mathcal{Q}_{\delta}^{M}$, (3.36) and
(3.37), we also deduce that, for any $t\in[0,T]$,

$\displaystyle\int_{0}^{t}\mathcal{Q}_{\delta}^{M}(s)ds$ $\displaystyle\leq
C(T)\int_{0}^{T}\left(\tilde{E}^{CH}[t;u_{\delta}^{M}(t)]+\|u_{\delta}^{M}\|^{2}\right)dt+C(T)\int_{0}^{T}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}dt$
$\displaystyle\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right),$

having applied (3.14), (3.17) and the conservation of mass. This, together
with (3.40), gives

$\displaystyle{t\mathcal{Q}^{M}_{\delta}(t)}+\int_{0}^{t}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}ds+\int_{0}^{t}{Ks}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}ds\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right),$ (3.41)

where again we applied (3.14) and (3.17) for the last summands in the right-
hand side of (3.40). Note also that, by the definition (3.31) of
$\mathcal{Q}_{\delta}^{M}$, from (3.41) we infer

$\displaystyle\frac{t}{2}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+\int_{0}^{t}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}ds+\int_{0}^{t}{Ks}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}ds$
$\displaystyle\leq C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right)$
$\displaystyle-
ta_{N}(u^{M}_{\delta},w^{M}_{\delta})-tg(u_{\delta}^{M},w^{M}_{\delta})-\frac{\theta
t}{2}g(F_{ln}^{\delta}(u^{M}_{\delta}),1)$ $\displaystyle\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right)+\frac{t}{4}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+C(T)(1+\tilde{E}^{CH}[t;u_{\delta}^{M}]),$

due to (3.17), (3.36) and (3.37). This implies, by (3.14),

$\displaystyle\frac{t}{4}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}+\int_{0}^{t}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u^{M}_{\delta}\|^{2}ds+\int_{0}^{t}{Ks}\|\partial^{\bullet}u^{M}_{\delta}\|^{2}ds\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta}^{M},1)|^{2}dt\right),$ (3.42)

for any $t\in[0,T]$, where $C(T)$ is independent of $M,\delta$. By standard
compactness arguments, exploiting the regularity we have just obtained,
arguing as in [13, Sec.4.2], we can pass to the limit as $M\to\infty$ and
obtain the existence of a couple $(u_{\delta},w_{\delta})$ satisfying for any
$\eta\in L^{2}_{H^{1}}$ and all $t\in[0,T]$,

$\displaystyle
m_{\star}(\partial^{\bullet}u_{\delta},\eta)+g(u_{\delta},\eta)+a_{N}(u_{\delta},\eta)+a_{S}(w_{\delta},\eta)=0,$
(3.43) $\displaystyle
a_{S}(u_{\delta},\eta)+m((F^{\delta})^{\prime}(u_{\delta}),\eta)-m(w_{\delta},\eta)=0,$
(3.44)

and $u_{\delta}(0)=u_{0}$ almost everywhere in $\Gamma_{0}$. Moreover, it
holds

$\displaystyle\sup_{t\in[0,T]}\tilde{E}^{CH}[t;u_{\delta}(t)]+{\color[rgb]{0,0,0}\sup_{t\in[0,T]}\|u_{\delta}(t)\|^{2}}+\int_{0}^{T}\|w_{\delta}\|^{2}_{H^{1}(\Gamma(t))}dt$
$\displaystyle\leq C_{\delta}(T),$ $\displaystyle
t\|w_{\delta}\|_{H^{1}(\Gamma(t))}^{2}+\int_{0}^{t}s\|\partial^{\bullet}u_{\delta}\|^{2}_{H^{1}(\Gamma(s))}ds$
$\displaystyle\leq
C{\color[rgb]{0,0,0}{}_{\delta}}(T),\quad\forall\,t\in[0,T],$

for some $C_{\delta}(T)>0$ possibly depending on $\delta$. Now note that,
since clearly (see [13, Sec.4.2]) we have $u_{\delta}^{M}\to u_{\delta}$ in
$L^{2}_{L^{2}}$ as $M\to\infty$, and being
$|F^{\prime}_{\delta}(c_{\delta}^{M})|\leq C_{\delta}|c_{\delta}^{M}|$ and
${|F^{\prime\prime}_{\delta}|}\leq C_{\delta}$, we have, since
$m(w_{\delta}^{M},1)=m(F^{\prime}(u_{\delta}^{M}),1)$ and
$m(w_{\delta},1)=m(F^{\prime}(u_{\delta}),1)$,

$\displaystyle\left|\int_{0}^{T}\left(|m(w_{\delta}^{M},1)|^{2}-|m(w_{\delta},1)|^{2}\right)dt\right|$
$\displaystyle\leq\int_{0}^{T}\left(|m(F^{\prime}(u_{M}^{\delta}),1)|+|m(F^{\prime}(u^{\delta}),1)|\right)\left|m(F^{\prime}(u_{M}^{\delta})-F^{\prime}(u^{\delta}),1)\right|dt$
$\displaystyle\leq
C_{\delta}(T)\|u_{\delta}^{M}-u_{\delta}\|_{L^{2}_{L^{2}}}\to 0\quad\text{as
}M\to\infty,$

having also exploited the uniform controls on $u_{\delta}^{M}$ and
$u_{\delta}$ in $L^{\infty}_{L^{2}}$. Therefore, these result, together with
the bound (3.39) (which gives weak lower sequential semicontinuity of the
norms on the left-hand side of (3.42)), is enough to pass to the limit as
$M\to\infty$ in (3.42), so that it holds

$\displaystyle\operatorname*{ess\,sup}_{t\in[0,T]}\left(\frac{t}{4}{\|\nabla_{\Gamma}w_{\delta}\|^{2}}\right)+\int_{0}^{T}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u_{\delta}\|^{2}ds+\int_{0}^{T}{Ks}\|\partial^{\bullet}u_{\delta}\|^{2}ds\leq
C(T)\left(1+\int_{0}^{T}|m(w_{\delta},1)|^{2}ds\right),$ (3.45)

where this time $C(T)>0$ is a constant independent of $\delta$.

Step 2. Limit as $\delta\to 0$. In order to pass to the limit in $\delta$ we
need to find a more refined energy estimate. In particular, let us first
observe that again, from (3.43), it holds the conservation of total mass, by
choosing $\eta\equiv 1$. Then we test (3.43) by $\eta=u_{\delta}$, inferring

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u_{\delta}\|^{2}=m_{\star}(\partial_{t}u_{\delta},u_{\delta})+\frac{1}{2}g(u_{\delta},u_{\delta})=-\frac{1}{2}g(u_{\delta},u_{\delta})-a_{N}(u_{\delta},u_{\delta})-a_{S}(w_{\delta},u_{\delta}).$

Integrating by parts, exploiting also (3.44), we get, by the definition of
$F^{\prime\prime}_{\delta}$,

$\displaystyle
a_{S}(w_{\delta},u_{\delta})=-\int_{\Gamma(t)}w_{\delta}\Delta_{\Gamma}u_{\delta}$
$\displaystyle=\|\Delta_{\Gamma}u_{\delta}\|^{2}+\int_{\Gamma(t)}F_{\delta}^{\prime\prime}(u_{\delta})|\nabla_{\Gamma}u_{\delta}|^{2}$
$\displaystyle=\|\Delta_{\Gamma}u_{\delta}\|^{2}+\int_{\Gamma(t)}\varphi^{\prime}_{\delta}(u_{\delta})|\nabla_{\Gamma}u_{\delta}|^{2}-\|\nabla_{\Gamma}u_{\delta}\|^{2}.$
(3.46)

Notice that by [13, Prop. 4.12] the solution to (3.43)-(3.44) is unique and
thus it enjoys the regularity $L^{2}_{H^{2}}$ as in [13, Thm. 4.14]. As a
consequence, the second equation (i.e. the one for $w_{\delta}$) holds
pointwise almost everywhere, and therefore the regularity of $u_{\delta}$ is
sufficient to perform rigorously the calculation above. Recalling now that
$\varphi_{\delta}^{\prime}\geq 0$ and by the regularity of V, we end up with

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u_{\delta}\|^{2}+\|\Delta_{\Gamma}u_{\delta}\|^{2}$
$\displaystyle\leq-\frac{1}{2}g(u_{\delta},u_{\delta})-a_{N}(u_{\delta},u_{\delta})+\|\nabla_{\Gamma}u_{\delta}\|^{2}\leq
C(\|u_{\delta}\|^{2}+\|\nabla_{\Gamma}u_{\delta}\|^{2}).$

We can now apply the following basic interpolation inequality, which can be
obtained after an integration by parts and an application of Cauchy-Schwarz
and Young’s inequalities:

$\|\nabla_{\Gamma}u_{\delta}\|^{2}=-\int_{\Gamma(t)}u_{\delta}\Delta_{\Gamma}u_{\delta}\leq\frac{1}{2}\|u_{\delta}\|^{2}+\frac{1}{2}\|\Delta_{\Gamma}u_{\delta}\|^{2},$

allowing to infer, by Young’s inequality,

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u_{\delta}\|^{2}+\frac{1}{2}\|\Delta_{\Gamma}u_{\delta}\|^{2}\leq
C\|u_{\delta}\|^{2},$

so that, in the end, by Gronwall’s Lemma, we deduce

$\displaystyle\sup_{t\in[0,T]}\|u_{\delta}\|^{2}+\int_{0}^{T}\|\Delta_{\Gamma}u_{\delta}\|^{2}ds\leq
C(T),$ (3.47)

uniformly in $\delta$. Concerning the energy estimate, by the same arguments
as for (3.16) we get:

$\displaystyle\frac{d}{dt}\tilde{E}^{CH}[u_{\delta}]+\frac{1}{2}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq-g(u_{\delta},w_{\delta})+C_{0}+C_{1}\tilde{E}^{CH}[u_{\delta}].$
(3.48)

Thanks to (3.47) and the bound on $\mathbf{V}$, by Poincaré’s inequality we
have

$\displaystyle-g(u_{\delta},w_{\delta})\leq C\|u_{\delta}\|\|w_{\delta}\|\leq
C\|w_{\delta}\|\leq\widetilde{C}(|m(w_{\delta},1)|+\|\nabla_{\Gamma}w_{\delta}\|),$
(3.49)

with $\widetilde{C}>0$ a suitable constant independent of $\delta$. Then,
using known arguments together with the conservation of total mass (see also
the proof of [13, Prop.5.7, a)]) we deduce, from [13, (5.1.7)],

$\displaystyle|m(w_{\delta},u_{\delta})|\leq\frac{1-\alpha}{8\widetilde{C}}\|\nabla_{\Gamma}w_{\delta}\|^{2}+C\|\nabla_{\Gamma}u_{\delta}\|^{2}+\alpha|m(w_{\delta},1)|,$
(3.50)

where $\alpha\in[0,1)$ is a constant such that

$\displaystyle 0\leq m_{u_{0}}(t)\leq\alpha<1,$ (3.51)

for any $t\in[0,T]$, whose existence is guaranteed being
$\sup_{[0,T]}m_{u_{0}}(t)<1$ by assumption. Now, from [13, (5.1.5)],

$\displaystyle|m(w_{\delta},1)|\leq
C\left(1+\|\nabla_{\Gamma}u_{\delta}\|^{2}\right)+|m(w_{\delta},u_{\delta})|,$
(3.52)

thus by (3.50) we infer

$\displaystyle|m(w_{\delta},1)|\leq
C\left(1+\|\nabla_{\Gamma}u_{\delta}\|^{2}\right)+\frac{1}{8\widetilde{C}}\|\nabla_{\Gamma}w_{\delta}\|^{2},$
(3.53)

so that, together with (3.48) and (3.49), we deduce

$\displaystyle\frac{d}{dt}\tilde{E}^{CH}[u_{\delta}]+\frac{3}{8}\|\nabla_{\Gamma}w_{\delta}^{M}\|^{2}\leq
C(1+\tilde{E}^{CH}[u_{\delta}]),$

entailing, by Gronwall’s Lemma,

$\displaystyle\operatorname*{ess\,sup}_{t\in[0,T]}\tilde{E}^{CH}[u_{\delta}]+\int_{0}^{T}\|\nabla_{\Gamma}w_{\delta}\|^{2}dt\leq
C(T),$ (3.54)

uniformly in $\delta$. Observe now that by (3.54) we can deduce a better
estimate of $|m(w_{\delta},u_{\delta})|$. In particular, from [13, (5.1.6)],
(3.54) and the conservation of mass, by Poincaré’s inequality we infer

$\displaystyle|m(w_{\delta},u_{\delta})|$ $\displaystyle\leq
C\|\nabla_{\Gamma}w_{\delta}\|\|u_{\delta}-(u_{\delta})_{\Gamma(t)}\|+m_{u_{0}}(t)|m(w_{\delta},1)|$
$\displaystyle\leq C\|\nabla_{\Gamma}w_{\delta}\|+\alpha|m(w_{\delta},1)|,$

for $\alpha$ already defined in (3.51). On account of (3.47), (3.54), we have
$\|u_{\delta}\|_{H^{1}(\Gamma(t))}\leq C(T)$ for almost any $t\in[0,T]$, so
that, since $|\varphi_{\delta}(r)|\leq\varphi_{\delta}(r)r+1$ and
$m(w_{\delta},1)=-m(u_{\delta},1)+m(\varphi_{\delta}(u_{\delta}),1)$,

$\displaystyle|m(w_{\delta},1)|$
$\displaystyle\leq|m(u_{\delta},1)|+|m(\varphi_{\delta}(u_{\delta}),1)|$
$\displaystyle\leq|m(u_{\delta},1)|+|m(\varphi_{\delta}(u_{\delta}),u_{\delta})|+|\Gamma(t)|$
$\displaystyle\leq
C+\|u_{\delta}\|^{2}+\|\nabla_{\Gamma}u_{\delta}\|^{2}+|m(w_{\delta},u_{\delta})|$
$\displaystyle\leq
C(1+\|\nabla_{\Gamma}w_{\delta}\|)+\alpha|m(w_{\delta},1)|,$

recalling
$m(\varphi_{\delta}(u_{\delta}),u_{\delta})=m(w_{\delta},u_{\delta})-\|\nabla
u_{\delta}\|^{2}$. This implies, being $\alpha<1$, by Poincaré’s inequality,

$\displaystyle|m(w_{\delta},1)|\leq C(1+\|\nabla_{\Gamma}w_{\delta}\|),$
(3.55)

from which we infer, together with (3.54),

$\int_{0}^{T}|m(w_{\delta},1)|^{2}\leq C(T),$

entailing, by (3.45),

$\displaystyle\operatorname*{ess\,sup}_{t\in[0,T]}\left({\frac{t}{4}\|\nabla_{\Gamma}w_{\delta}\|^{2}}\right)+\int_{0}^{T}\frac{s}{4}\|\nabla_{\Gamma}\partial^{\bullet}u_{\delta}\|^{2}ds+\int_{0}^{T}{Ks}\|\partial^{\bullet}u_{\delta}\|^{2}ds\leq
C(T).$ (3.56)

Following [13, Sec.4.2], [13, Propositions 5.8-5.10] and [13, Lemma 5.11], the
uniform estimates (3.47), (3.54) and (3.56) are then enough, by standard
compactness arguments, to pass to the limit as $\delta\rightarrow 0$,
obtaining a weak solution with the same properties as the one of Theorem 3.4.
Therefore, this solution coincides with the one of Theorem 3.4 by uniqueness,
and it also enjoys (3.21) and (3.22). This concludes the proof of Part (i).

Part (ii). Let us fix $\tau>0$. We clearly have that
$\|w\|_{H^{1}(\Gamma(t))}\leq C$ for almost any $t\geq\tau$. Let then
introduce the cutoff function

$\displaystyle h_{k}(r)=\begin{cases}1-\frac{1}{k},\qquad r>1-\frac{1}{k},\\\
r,\qquad-1+\frac{1}{k}\leq r\leq 1-\frac{1}{k},\\\ -1+\frac{1}{k},\qquad
r<-1+\frac{1}{k},\end{cases}$ (3.57)

which is Lipschitz continuous. Then we set $u_{k}=h_{k}(u)$. Being $u$ in
$L^{\infty}_{H^{1}}$, we have that the chain rule holds giving

$\nabla_{\Gamma}u_{k}=\chi_{[-1+\frac{1}{k},1-\frac{1}{k}]}(u)\nabla_{\Gamma}u.$

Accordingly, for any $k>1$ and $p\geq 2$,
$f_{k}=\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k})$
is well defined and belongs to $L^{\infty}_{H^{1}}$ and satisfies

$\nabla_{\Gamma}\left(\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k})\right)=(p-1)\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi^{\prime}(u_{k})\nabla_{\Gamma}u_{k}.$

If we now set $\eta=f_{k}$ in (3.9), we infer that

$\displaystyle(p-1)\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi^{\prime}(u_{k})\nabla_{\Gamma}u\cdot\nabla_{\Gamma}u_{k}+\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k})\frac{\theta}{2}\varphi(u)=\int_{\Gamma(t)}\widehat{w}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k}),$

where $\widehat{w}=w+u$. Being $F_{ln}$ strictly convex, the first term in the
left-hand side is nonnegative. Moreover, since $\varphi$ is increasing, by the
definition of $u_{k}$, we immediately infer

$\displaystyle\varphi(u_{k})^{2}\leq\varphi(u)\varphi(u_{k}),\quad\forall k>1$
(3.58)

and thus for the second term we have

$\displaystyle\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k})\frac{\theta}{2}\varphi(u)\geq\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p}$

Regarding the right-hand side, we easily get, by the Sobolev embedding
$H^{1}(\Gamma(t))\hookrightarrow L^{p}(\Gamma(t))$,

$\displaystyle\int_{\Gamma(t)}\widehat{w}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{p-2}\frac{\theta}{2}\varphi(u_{k})$
$\displaystyle\leq\frac{1}{2}\left\|\frac{\theta}{2}\varphi(u_{k})\right\|^{p}_{L^{p}(\Gamma(t))}+C\|\widehat{w}\|^{p}_{L^{p}(\Gamma(t))}$
$\displaystyle\leq\frac{1}{2}\left\|\frac{\theta}{2}\varphi(u_{k})\right\|^{p}_{L^{p}(\Gamma(t))}+C_{p}\|\widehat{w}\|^{p}_{H^{1}(\Gamma(t))},$

with $C_{p}>0$ depending on $p$. Collecting the above estimates, being ${u}\in
L^{\infty}_{H^{1}}$ and $\|w\|_{H^{1}(\Gamma(t))}\leq C$ for almost any
$t\geq\tau$, we immediately deduce

$\displaystyle\|\varphi(u_{k})\|_{L^{p}(\Gamma(t))}\leq
C_{p}(T,\tau,p),\quad\forall p\in[2,\infty),$ (3.59)

for almost any $t\in[\tau,T]$. Therefore, there exists $\zeta\in
L^{\infty}_{L^{p}}(\tau,T)$ such that, up to subsequences
$\varphi(u_{k})\overset{\ast}{\rightharpoonup}\zeta$ in
$L^{\infty}_{L^{p}}(\tau,T)$ as $k\rightarrow\infty$. Now observe that, being
$|u|<1$ almost everywhere on $\Gamma(t)$ (for almost any $t\in(0,T)$), and
since, for almost any $t\in(0,T)$, $u_{k}\to u$ as $k\to\infty$ almost
everywhere on $\Gamma(t)$, we get $\varphi(u_{k})\to\varphi(u)$ almost
everywhere. It is now immediate to deduce by (3.59) that, for $p=2$,
$\|\varphi(u_{k})\|_{L^{2}_{L^{2}}}\leq C(\tau,T)$, therefore we can apply
[13, Thm.B.2] on $[\tau,T]$ to infer $\varphi(u_{k})\rightharpoonup\varphi(u)$
in $L^{2}_{L^{2}}(\tau,T)$. By uniqueness of the weak limit we therefore infer
$\zeta=\varphi(u)$. Then, by weak⋆ sequential lower semicontinuity we get

$\displaystyle\operatorname*{ess\,sup}_{t\in[\tau,T]}\|\varphi(u)\|_{L^{p}(\Gamma(t))}\leq
C_{p}(T,\tau,p),\quad\forall p\in[2,\infty).$ (3.60)

Concerning $\varphi^{\prime}$, we consider
$g_{k}=\frac{\theta}{2}\varphi(u_{k})e^{L\frac{\theta}{2}|\varphi(u_{k})|}\in
L^{\infty}_{H^{1}}$, for some arbitrary $L>0$, and we observe that

$\displaystyle\nabla_{\Gamma}\left(\frac{\theta}{2}\varphi(u_{k})e^{L\frac{\theta}{2}|\varphi(u_{k})|}\right)=\frac{\theta}{2}\varphi^{\prime}(u_{k})\left(1+L\frac{\theta}{2}\left|\varphi(u_{k})\right|\right)e^{L\frac{\theta}{2}|\varphi(u_{k})|}\nabla_{\Gamma}u_{k}.$

Therefore, considering again (3.9) with $\eta=g_{k}$, we get

$\displaystyle\int_{\Gamma(t)}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}u_{k}\frac{\theta}{2}\varphi^{\prime}(u_{k})\left(1+L\frac{\theta}{2}\left|\varphi(u_{k})\right|\right)e^{L\frac{\theta}{2}|\varphi(u_{k})|}+\int_{\Gamma(t)}\frac{\theta}{2}\varphi(u)\frac{\theta}{2}$
$\displaystyle\varphi(u_{k})e^{L\frac{\theta}{2}|\varphi(u_{k})|}$
$\displaystyle=\int_{\Gamma(t)}\widehat{w}\frac{\theta}{2}\varphi(u_{k})e^{L\frac{\theta}{2}|\varphi(u_{k})|}.$

Again the first term in the left-hand side is nonnegative, whereas, exploiting
again (3.58), we obtain in the end

$\displaystyle\int_{\Gamma(t)}\left(\frac{\theta}{2}\varphi(u_{k})\right)^{2}e^{L\frac{\theta}{2}|\varphi(u_{k})|}\leq\int_{\Gamma(t)}\widehat{w}\frac{\theta}{2}\varphi(u_{k})e^{L\frac{\theta}{2}|\varphi(u_{k})|}$

By Lemma B.4, we get

$\displaystyle\int_{\Gamma(t)}|\widehat{w}|\left|\frac{\theta}{2}\varphi(u_{k})\right|e^{L\left|\frac{\theta}{2}\varphi(u_{k})\right|}\leq\frac{1}{2}\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{2}e^{L\left|\frac{\theta}{2}\varphi(u_{k})\right|}+\int_{\Gamma(t)}e^{N|\widehat{w}|},$
(3.61)

implying

$\displaystyle\frac{1}{2}\int_{\Gamma(t)}\left|\frac{\theta}{2}\varphi(u_{k})\right|^{2}e^{L\left|\frac{\theta}{2}\varphi(u_{k})\right|}\leq\int_{\Gamma(t)}e^{N|\widehat{w}|}$
(3.62)

for any $L>0$ and some $N=N(L)$. Now we exploit (2.10), and then apply Lemma
B.3 with $u=N\widetilde{\widehat{w}}$ and the manifold
$\mathcal{M}=\Gamma_{0}$ (with the corresponding metric) to infer

$\displaystyle\int_{\Gamma(t)}e^{N|\widehat{w}|}=\int_{\Gamma_{0}}e^{N|\widetilde{\widehat{w}}|}J_{t}^{0}d\Gamma_{0}$
$\displaystyle\leq
C\int_{\Gamma_{0}}e^{N|\widetilde{\widehat{w}}|}d\Gamma_{0}\leq
Ce^{CN^{2}\|\widetilde{\widehat{w}}\|^{2}_{H^{1}(\Gamma_{0})}}\leq
Ce^{CN^{2}\|{\widehat{w}}\|^{2}_{H^{1}(\Gamma(t))}},$

where in the last estimate we exploited the property that
$(H^{1}(\Gamma(t)),\phi_{t})_{t\in[0,T]}$ is a compatible space, where
$\phi_{-t}v=\widetilde{v}$. Recalling now property (3.7), we infer that, for
some $\tilde{C}>0$ sufficiently large,

$\displaystyle\left(\frac{\theta}{2}\varphi^{\prime}(s)\right)^{p}\leq
e^{pC}\left(\tilde{C}+\left|\frac{\theta}{2}\varphi(s)\right|^{2}e^{pC\left|\frac{\theta}{2}\varphi(s)\right|}\right),\quad\forall
s\in(-1,1),$ (3.63)

thus, taking $L=pC$ in (3.62) and recalling that
$\|\widehat{w}\|_{H^{1}(\Gamma(t))}\leq C$ for almost any $t\geq\tau$, we end
up with

$\|\varphi^{\prime}(u_{k})\|_{L^{p}(\Gamma(t))}\leq C_{p}(T,\tau,p),$

implying, by the same arguments used for $\varphi(u_{k})$, applied in this
case to $\varphi^{\prime}(u_{k})$, that

$\displaystyle\operatorname*{ess\,sup}_{t\in[\tau,T]}\|\varphi^{\prime}(u)\|_{L^{p}(\Gamma(t))}\leq
C_{p}(T,\tau,p),\quad\forall p\in[2,\infty).$ (3.64)

Therefore, exploiting elliptic regularity and recalling that $u\in
L^{\infty}_{H^{1}}$, we obtain

$\|u\|_{H^{2}(\Gamma(t))}\leq\left(C+\|\Delta_{\Gamma(t)}u\|\right)\leq
C\left(1+\|w\|+\|\varphi(u)\|\right)\leq C(T,\tau),$

for almost any $t\in[\tau,T]$.

Part (iii). If we now apply the chain rule to $\varphi(u)$ (which is possible,
e.g., by approximation with the truncated functions $u_{k}$ and then passing
to the limit as $k\rightarrow\infty$) we obtain

$\displaystyle\nabla_{\Gamma}\varphi(u)=\varphi^{\prime}(u)\nabla_{\Gamma}u,\quad\text{
for a.a. }\ t\in[\tau,T].$

Then, for almost any $t\in[\tau,T]$, we have that

$\|\nabla_{\Gamma}\varphi(u)\|_{L^{p}(\Gamma(t))}\leq\|\varphi^{\prime}(u)\|_{L^{2p}(\Gamma(t))}\|\nabla_{\Gamma}u\|_{L^{2p}(\Gamma(t))}\leq
C_{p}(T,\tau,p),$

by the Sobolev embedding $H^{2}(\Gamma(t))\hookrightarrow W^{1,q}(\Gamma(t))$
for every $q\geq 2$. Therefore we get

$\|\varphi(u)\|_{W^{1,p}(\Gamma(t))}\leq C_{p}(T,\tau,p),$

for every $p\geq 2$. This implies, choosing, e.g., $p=3$, that

$\displaystyle\|\varphi(u)\|_{L^{\infty}(\Gamma(t))}\leq C(T,\tau),\quad\text{
for a.a. }t\in[\tau,T],$ (3.65)

by the embedding $W^{1,3}(\Gamma(t))\hookrightarrow L^{\infty}(\Gamma(t))$.
Therefore, being $u(t)\in H^{2}(\Gamma(t))\hookrightarrow C^{0}(\Gamma(t))$
for almost any $t\in[0,T]$, it follows from the singularities of $\varphi$ at
$\pm 1$ and the estimate (3.65) that we can find $\xi=\xi(T,\tau)>0$ such that

$\displaystyle\|u\|_{L^{\infty}(\Gamma(t))}\leq 1-{\xi},\quad\text{ for a.a.
}t\in[\tau,T],$

that is, the strict separation property holds. This concludes the proof. ∎

##  4 The second model

We now consider the alternative weighted model (1.2), which is analysed in
[13, Sec.6] (and references therein).  In particular, as noticed in the
Introduction, this is a simplified version of the model presented in [49, 56],
in which the problem is governed by two coupled fourth-order nonlinear PDEs
that live on an evolving two-dimensional manifold. For the phase transitions,
the PDE is the Cahn-Hilliard equation for curved surfaces, which can be
derived from surface mass balance in the framework of irreversible
thermodynamics. For the surface deformation, the PDE is the (vector-valued)
Kirchhoff–Love thin shell equation. In our work, we study only the Cahn-
Hilliard equation in the same formulation arising from the model in [56], so
that this analysis could, in a future work, be extended to consider the
complete model as [49, 56], in which the evolution of the surface is part of
the model itself. We now show a sketch of the derivation of our model (1.2),
taken directly from [56].

###  4.1 Derivation

For consistency with the literature, see e.g. [49, 56], to derive this system
we start with a description of the surfaces $\\{\Gamma(t)\\}_{t}$ given by the
flow $\Phi_{t}^{0}\colon\Gamma_{0}\to\Gamma(t)$ as in $\mathbf{A}_{\Phi}$.
Assume that the surface $\Gamma(t)$ consists of two species with the mass
densities per unit area $\rho_{1}$ and $\rho_{2}$. The total mass of each
species is assumed to be conserved. This entails, for the total density
$\rho:=\rho_{1}+\rho_{2}$, the balance law

$\displaystyle\dfrac{d}{dt}\int_{P(t)}\rho\equiv 0,$

for any surface patch $P(t)\subset\Gamma(t)$ evolving under the full velocity
$\mathbf{V}$. By the Reynolds transport theorem, see also Proposition 3.2, and
the arbitrariness of $P(t)$ we easily deduce

$\displaystyle\partial^{\bullet}\rho+\rho\nabla_{\Gamma}\cdot\textbf{V}=0,\quad\rho(0)=\widehat{\rho},$

where $\widehat{\rho}$ is the initial density. This equation shows in
particular that we have the identity

$\displaystyle\rho(t,\Phi_{t}^{0}(p))=\frac{\widehat{\rho}}{J_{t}^{0}(p)},\quad\forall
p\in\Gamma_{0},$ (4.1)

where $J_{t}^{0}$ is the area change defined in (2.10).

For the process on the surface we now introduce the dimensionless
concentrations $c_{i}=\rho_{i}/\rho$, for $i=1,2$. Since $c_{1}+c_{2}=1$, it
is sufficient to consider $c_{1}$ to model the local density fractions of the
two species. The mass of species 1 in the membrane patch $P(t)$, evolving
under $\mathbf{V}$, may only change due to a diffusive mass flux $q_{d}$ at
the boundary, so that

$\displaystyle\int_{P(t)}\rho\partial^{\bullet}{c}_{1}=\dfrac{d}{dt}\int_{P(t)}\rho
c_{1}=-\int_{\partial
P(t)}q_{d}\cdot\boldsymbol{\mu}=-\int_{P(t)}\nabla_{\Gamma}\cdot q_{d},$ (4.2)

where $\boldsymbol{\mu}$ denotes the outer unit conormal of $\partial P(t)$.
The last identity comes from the fact that we can directly consider, without
loss of generality, $q_{d}$ to be purely tangential to $\Gamma(t)$. The
diffusive flux $q_{d}$ is related to the chemical potential in the following
sense. In analogy to 3D problems ([15]), the Cahn-Hillard energy per reference
area takes the form

$\Psi_{CH}=\Psi_{mix}(c_{1},T)+\Psi_{i}(J_{t}^{0},\nabla_{\Gamma}c_{1}),$

where $T$ is the absolute temperature and, up to setting some constants to $1$
for simplicity,

$\displaystyle\Psi_{mix}(c_{1},T)=T\left(c_{1}\ln(c_{1})+(1-c_{1})\ln(1-c_{1})\right)+c_{1}(1-c_{1}),$
$\displaystyle\Psi_{i}(J_{t}^{0},\nabla_{\Gamma}c_{1})=\frac{\lambda}{2}J_{t}^{0}|\nabla_{\Gamma}c_{1}|^{2},$

with $\lambda>0$ a coefficient related to the width of the phase interface. As
noted in [56, (35)], the area change $J^{0}_{t}$ is included in $\Psi_{i}$,
since $\Psi_{CH}$ is postulated to be an energy w.r.t. the reference
configuration $\Gamma_{0}$, while $\nabla_{\Gamma}c_{1}$ refers to the current
configuration (it can be viewed as having units of 1/(current length)). If we
then want a Cahn-Hilliard energy per current area instead of one per reference
area we simply need to multiply $\Psi_{CH}$ by $J^{t}_{0}=(J_{t}^{0})^{-1}$;
in effect, we have (current area)/(reference area)$=J^{0}_{t}$, thus,
heuristically,

$\frac{\text{energy}}{\text{current area}}=\frac{\text{reference
area}}{\text{current area}}\frac{\text{energy}}{\text{reference
area}}=\frac{1}{J_{t}^{0}}\frac{\text{energy}}{\text{reference area}},$

so that the total energy on the current configuration $\Gamma(t)$ can be
defined as

$\displaystyle
E_{CH}^{\rho}(c_{1}):=\int_{\Gamma(t)}J^{t}_{0}\Psi_{CH}=\int_{\Gamma(t)}J^{t}_{0}\Psi_{mix}+\frac{\lambda}{2}\int_{\Gamma(t)}|\nabla_{\Gamma}c_{1}|^{2}.$
(4.3)

Then, following the thermodynamical derivation in [56, Appendix A] one can
obtain that the chemical potential $w$ is defined by 111As already noticed,
this is performed in [56, Appendix A] in the case of an elastic surface, but
we keep the same derivation as a first step in the analysis of the more
complex model.

$\displaystyle w=J^{0}_{t}\frac{\delta E_{CH}^{\rho}}{\delta
c_{1}}=-J^{0}_{t}\lambda\Delta_{\Gamma}c_{1}+\Psi^{\prime}_{mix}(c_{1}),$
(4.4)

and the diffusive flux is (see [56, (121)])

$q_{d}=-\frac{M}{J^{0}_{t}}\nabla_{\Gamma}w,$

where $M>0$ is a mobility coefficient. The presence of the term $1/J^{0}_{t}$
in $q_{d}$ can again be heuristically explained by the fact that $w$ is
defined per reference area, whereas $q_{d}$ is defined per current area. All
in all, by setting for simplicity $M=1$, $\lambda=1$ and $\widehat{\rho}\equiv
1$, so that $J^{0}_{t}=1/\rho$, from (4.2) we are led to

$\displaystyle\rho\dot{c}_{1}-\nabla_{\Gamma}\cdot\left(\rho\nabla_{\Gamma}\left(-\dfrac{1}{\rho}\Delta_{\Gamma}c_{1}+\Psi^{\prime}_{mix}(c_{1})\right)\right)=0.$

Problem (1.2) can be retrieved by rewriting the equation for the dimensionless
concentration difference $c:=\frac{\rho_{1}-\rho_{2}}{\rho}={2c_{1}-1}$ and
making a proper rescaling on $\Psi_{mix}$ so that it coincides with the
potential $F$ defined in (1.7).

######  Remark 4.1.

Two observations are timely regarding the relation between models (1.1) and
(1.2).

  * 1.

As observed in [55], model (1.2) is similar to model (1.1), but neither is a
particular case of the other. In model (1.1), the equation is written for the
conserved variable $u=\rho c$ and under the assumption that the free energy
functional depends on the concentration difference $u$ rather than on the
relative concentration difference $c$ as in model (1.2). Formally, systems
(1.1) and (1.2) are the same problem for inextensible membranes (i.e.,
$\nabla_{\Gamma}\cdot\textbf{V}\equiv 0$) and if one assumes $\rho=const$.
Here the situation partially resembles the coupling of a two-component
compressible fluid flow with dissipative Ginzburg–Landau interface dynamics
discussed in [40], where, depending on the choice of the variables to define
the energy functional, the effects of compressibility have to be considered in
the definition of the chemical potential. These differences between the two
models explain why the conserved quantities (1.3)-(1.4) are in principle
different.

  * 2.

In the derivation of (1.2), we think of the tangential component
$\mathbf{V}_{\tau}$ as fixing a parametrisation of $\Gamma(t)$, as well as
describing advection on the surface; note that the balance laws are considered
on portions evolving under the full velocity $\mathbf{V}$. This is consistent
with the framework of the physics literature, namely [49, 56], in which this
tangential component $\textbf{V}_{\tau}$ is part of the unknowns. We can also
see this model in the light of (1.1) by heuristically taking
$\mathbf{V}_{\tau}=\mathbf{V}_{a}$, so that the term involving both velocities
vanishes and the notion of a material time derivative coincides with both the
one in our analytical framework, i.e. involving the parametrisation, and the
usual physical notion of including the advection of material points on the
surface.

###  4.2 Weak formulation

Let Assumption $\mathbf{A_{\Phi}}$ hold. Then the problem reads: find a pair
$(c,w)$ such that, for all $\eta\in L^{2}_{H^{1}}$,

$\displaystyle
m_{\star}(\rho\partial^{\bullet}c,\eta)+\int_{\Gamma(t)}\rho\nabla_{\Gamma}w\cdot\nabla_{\Gamma}\eta=0,$
(4.5)
$\displaystyle\int_{\Gamma(t)}\nabla_{\Gamma}c\cdot\nabla_{\Gamma}\eta+\int_{\Gamma(t)}\rho
F^{\prime}(c)\eta=\int_{\Gamma(t)}\rho w\eta,$ (4.6) $\displaystyle
c(0)=c_{0},\qquad\text{ a.e. on }\Gamma_{0}.$ (4.7)

Here the weight function $\rho$, i.e., the total density, is determined by the
transport equation

$\displaystyle\partial^{\bullet}\rho+\rho\nabla_{\Gamma}\cdot\mathbf{V}=0,$
(4.8)

with $\rho(0)\equiv 1$ on $\Gamma_{0}$. As noticed in Section 4.1, we have an
interesting characterisation of $\rho$:

$\displaystyle\rho(t,\Phi(t,p))=(J_{t}^{0}(p))^{-1},\quad\forall
p\in\Gamma_{0}.$ (4.9)

In particular, we also have the bounds

$\displaystyle 0<\frac{1}{C_{\rho}}\leq\rho\leq
C_{\rho},\quad|\nabla_{\Gamma}\rho|\leq C_{\rho},$ (4.10)

with $C_{\rho}>1$. Note that the total energy of the system reads (see also
(4.3), but with respect to the relative concentration difference $c$)

$E^{\rho}_{CH}(c):=\int_{\Gamma(t)}\left(\frac{|\nabla_{\Gamma}c|^{2}}{2}+\rho
F(c)\right).$

We recall the following result (see [13, Thm.6.2]). The statement below and
its proof involve the weighted inverse Laplacian operator, which we denote as
$A_{\rho}^{-1}$, define to be, for any

$\displaystyle f\in H^{-1}(\Gamma(t))\quad\text{ such that
}\quad{\color[rgb]{0,0,0}m_{\star}(\rho f,1)}=0,$

the unique solution $\zeta:=A_{\rho}^{-1}f$ to the problem

$\displaystyle\int_{\Gamma(t)}\rho\nabla_{\Gamma}\zeta\cdot\nabla_{\Gamma}\eta$
$\displaystyle={\color[rgb]{0,0,0}m_{\star}(\rho f,\eta)}\quad\text{ with
}\quad\int_{\Gamma(t)}\zeta=0.$ (4.11)

We define also, for such elements $f$, the weighted norm

$\displaystyle\|f\|_{\rho,-1}:=\|\sqrt{\rho}\,\nabla_{\Gamma}\zeta\|=\langle\rho
f,\zeta\rangle.$ (4.12)

Note that there exist $C_{1},C_{2}>0$ such that
$C_{1}\|\nabla_{\Gamma}\zeta\|\leq\|f\|_{\rho,-1}\leq
C_{2}\|\nabla_{\Gamma}\zeta\|.$

######  Theorem 4.2.

Let $c_{0}\in H^{1}(\Gamma_{0})$, $|c_{0}|\leq 1$, $|(c_{0})_{\Gamma_{0}}|<1$
and $F:[-1,1]\to\mathbb{R}$ be given by (3.6). Then there exists a unique pair
$(c,w)$ with

$c\in L^{\infty}_{H^{1}}\cap H^{1}_{H^{-1}},\quad w\in L^{2}_{H^{1}},$

such that, for almost any $t\in[0,T]$, $|c(t)|<1$ almost everywhere in
$\Gamma(t)$ and $(c,w)$ satisfies, for almost any $t\in[0,T]$, (4.5)-(4.6),
with $c(0)=c_{0}$ almost everywhere in $\Gamma_{0}$. The solution $c$ also
satisfies the additional regularity

$c\in C^{0}_{L^{2}}\cap L^{\infty}_{L^{p}}\cap L^{2}_{H^{2}},$

for all $p\in[1,+\infty)$. Furthermore, if $c_{0,1},c_{0,2}\in
H^{1}(\Gamma_{0})$, satisfying the existence assumptions, are such that
$(c_{0,1})_{\Gamma_{0}}=(c_{0,2})_{\Gamma_{0}}$, and $c_{1},c_{2}$ are the
solutions of the system with $c_{1}(0)=c_{0,1}$ and $c_{2}(0)=c_{0,2}$, then
there exists a constant $C>0$ independent of $t$, such that, for almost any
$t\in[0,T]$,

$\displaystyle\|c_{1}(t)-c_{2}(t)\|_{\rho,-1}\leq
e^{Ct}\|c_{0,1}-c_{0,2}\|_{\rho,-1}.$ (4.13)

######  Remark 4.3.

The assumption $|(c_{0})_{\Gamma_{0}}|<1$ is necessary and standard when
dealing with Cahn-Hilliard equations, since it simply excludes that the
initial datum is a pure phase, i.e, $c_{0}\equiv 1$ or $c_{0}\equiv-1$.
Indeed, in these cases, the phase separation phenomenon would not take place,
due to the presence of one single substance.

The proof of existence follows from the uniform estimates obtained in the
proof of Theorem 4.6 below. For completeness, we now include a proof of
continuous dependence on the initial data, entailing uniqueness, which was
omitted in [13].

###### Proof.

(Stability of weak solutions to (4.5)-(4.7)) The density $\rho$ is determined
by a single ordinary differential equation which can be solved explicitly, and
it is thus unique. Now suppose $(c_{1},w_{1})$ and $(c_{2},w_{2})$ both solve
(4.5), (4.6), with initial conditions $c_{1}(0)=c_{0,1}$ and
$c_{2}(0)=c_{0,2}$, such that $(c_{0,1})_{\Gamma_{0}}=(c_{0,2})_{\Gamma_{0}}$.
Denote $\xi^{c}=c_{1}-c_{2}$, $\xi^{w}=w_{1}-w_{2}$. Subtracting the
corresponding equations leads to

$\displaystyle{\color[rgb]{0,0,0}m_{\star}(\rho\partial^{\bullet}\xi^{c},\eta)}+\int_{\Gamma(t)}\rho\nabla_{\Gamma}\xi^{w}\cdot\nabla_{\Gamma}\eta$
$\displaystyle=0$ (4.14)
$\displaystyle\int_{\Gamma(t)}\nabla_{\Gamma}\xi^{c}\cdot\nabla_{\Gamma}\eta+\int_{\Gamma(t)}\rho(F^{\prime}(c_{1})-F^{\prime}(c_{2}))\eta$
$\displaystyle=\int_{\Gamma(t)}\rho\xi^{w}\eta.$ (4.15)

Using the ODE (4.8) for $\rho$ and (4.5), noting that
$\xi^{c}(0)=c_{0,1}-c_{0,2}$, we obtain

$\displaystyle\dfrac{d}{dt}\int_{\Gamma(t)}\rho\xi^{c}={\color[rgb]{0,0,0}m_{\star}(\rho\partial^{\bullet}\xi^{c},1)}=0\quad\Longrightarrow\quad\int_{\Gamma(t)}\rho\xi^{c}=\int_{\Gamma_{0}}c_{0,1}-c_{0,2}\
\equiv 0.$

Therefore the weighted inverse Laplacian $(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$
is well defined, and taking $\eta=(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$ in
(4.14) we get

$\displaystyle{\color[rgb]{0,0,0}m_{\star}(\rho\partial^{\bullet}\xi^{c},(\Delta_{\Gamma,\rho}^{-1})\xi^{c})}+\int_{\Gamma(t)}\rho\nabla_{\Gamma}\xi^{w}\cdot\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}=0.$
(4.16)

Now note:

$\displaystyle{\color[rgb]{0,0,0}m_{\star}(\rho\partial^{\bullet}\xi^{c},(\Delta_{\Gamma,\rho}^{-1})\xi^{c})}$
$\displaystyle=\dfrac{d}{dt}\left[\int_{\Gamma(t)}\rho\xi^{c}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\right]-\int_{\Gamma(t)}\rho\xi^{c}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$
$\displaystyle=\dfrac{d}{dt}\left[\int_{\Gamma(t)}\rho|\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}|^{2}\right]-\int_{\Gamma(t)}\rho\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\cdot\nabla_{\Gamma}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$
$\displaystyle=\dfrac{d}{dt}\|\xi^{c}\|_{\rho,-1}^{2}-\int_{\Gamma(t)}\rho\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\cdot\nabla_{\Gamma}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$

and

$\displaystyle\int_{\Gamma(t)}\rho\nabla_{\Gamma}\xi^{w}\cdot\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}=\int_{\Gamma(t)}\rho\xi^{w}\xi^{c}$

so that (4.16) becomes

$\displaystyle\dfrac{d}{dt}\|\xi^{c}\|_{\rho,-1}^{2}+\int_{\Gamma(t)}\rho\xi^{w}\xi^{c}$
$\displaystyle=\int_{\Gamma(t)}\rho\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\cdot\nabla_{\Gamma}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}.$
(4.17)

We now also test (4.15) with $\eta=\xi^{c}$ to obtain

$\displaystyle\|\nabla_{\Gamma}\xi^{c}\|^{2}+\int_{\Gamma(t)}\rho(F^{\prime}(c_{1})-F^{\prime}(c_{2}))\xi^{c}=\int_{\Gamma(t)}\rho\xi^{w}\xi^{c}.$
(4.18)

Due to the structure of the logarithmic potential $F$, being $F_{ln}$ strictly
convex, we have the estimate

$\displaystyle\int_{\Gamma(t)}\rho(F^{\prime}(c_{1})-F^{\prime}(c_{2}))\xi^{c}\geq-C\|{\color[rgb]{0,0,0}\sqrt{\rho}}\xi^{c}\|{\color[rgb]{0,0,0}{}^{2}}$

from where (4.18) becomes

$\displaystyle\|\nabla_{\Gamma}\xi^{c}\|^{2}\leq\int_{\Gamma(t)}\rho\xi^{w}\xi^{c}+C\|{\color[rgb]{0,0,0}\sqrt{\rho}}\xi^{c}\|^{2}.$
(4.19)

Adding (4.17) and (4.19) the terms involving the product $\xi^{w}\xi^{c}$
cancel out and we are led to

$\displaystyle\dfrac{d}{dt}\|\xi^{c}\|_{\rho,-1}^{2}+\|\nabla_{\Gamma}\xi^{c}\|^{2}$
$\displaystyle\leq
C\|{\color[rgb]{0,0,0}\sqrt{\rho}}\xi^{c}\|^{2}+\int_{\Gamma(t)}\rho\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\cdot\nabla_{\Gamma}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}.$
(4.20)

We now estimate the first term on the right-hand side as

$\displaystyle
C\|{\color[rgb]{0,0,0}\sqrt{\rho}}\xi^{c}\|^{2}=C\int_{\Gamma(t)}\rho|\xi^{c}|^{2}$
$\displaystyle=C\int_{\Gamma(t)}\rho\nabla_{\Gamma}\xi^{c}\cdot\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\leq\dfrac{1}{4}\|\nabla_{\Gamma}\xi^{c}\|^{2}+C\|\xi^{c}\|_{\rho,-1}^{2}.$

For the second term, we note that

$\displaystyle\int_{\Gamma(t)}\rho\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}\cdot\nabla_{\Gamma}\partial^{\bullet}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}$
$\displaystyle\leq\dfrac{1}{2}\dfrac{d}{dt}\int_{\Gamma(t)}\rho|\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}|^{2}+C\int_{\Gamma(t)}\rho|\nabla_{\Gamma}(\Delta_{\Gamma,\rho}^{-1})\xi^{c}|^{2}$
$\displaystyle\leq\dfrac{1}{2}\dfrac{d}{dt}\|\xi^{c}\|_{\rho,-1}^{2}+C\|\xi^{c}\|_{\rho,-1}^{2}.$

All in all, we obtain from (4.20) the estimate

$\displaystyle\dfrac{d}{dt}\|\xi^{c}\|^{2}_{\rho,-1}+\|\nabla_{\Gamma}\xi^{c}\|^{2}\leq
C\|\xi^{c}\|^{2}_{\rho,-1},$

and an application of Gronwall’s Lemma implies the continuous dependence
estimate stated in (4.13). Then uniqueness follows by setting $c_{0,1}\equiv
c_{0,2}$. ∎

######  Remark 4.4.

Notice that the regularity stated in Theorem 4.2 can be slightly improved as
in the case of the first model (see Remark 3.5). In particular, since $c\in
L^{2}_{H^{2}}$ solves the problem, for almost all $t\in[0,T]$,

$-\frac{1}{\rho}\Delta_{\Gamma}c(t)=w(t)-F^{\prime}(c(t))\in
L^{2}({\Gamma(t)}),$

we are allowed to multiply by $-\Delta_{\Gamma}c\in L^{2}(\Gamma(t))$ for
almost any $t\in[0,T]$. Recalling that $\varphi^{\prime}>0$, after an
integration by parts, being $\Gamma(t)$ closed and $c\in L^{\infty}_{H^{1}}$,
we obtain, by (4.10),

$\displaystyle C{\left\|\Delta_{\Gamma}c\right\|}^{2}\leq
m(\frac{1}{\rho}\Delta_{\Gamma}c,\Delta_{\Gamma}c)+\frac{\theta}{2}m(\varphi^{\prime}(c),|\nabla_{\Gamma}c|^{2})\leq{\left\|\nabla_{\Gamma}w\right\|}{\left\|\nabla_{\Gamma}c\right\|}+\|\nabla_{\Gamma}c\|^{2}\leq
C(1+{\left\|\nabla_{\Gamma}w\right\|}),$

and knowing that $w\in L^{2}_{H^{1}}$, we infer $c\in L^{4}_{H^{2}}$.

We make use of the following weighted $L^{2}$ and $H^{1}$ products, whose
induced norms are equivalent norms on $L^{2}(\Gamma(t))$ and
$H^{1}(\Gamma(t))$, respectively:

$\displaystyle(f,g)_{\rho}:=m(\rho
f,g),\qquad(f,g)_{1,\rho}:=(f,g)_{\rho}+\int_{\Gamma(t)}\rho\nabla_{\Gamma}f\cdot\nabla_{\Gamma}g.$
(4.21)

Notice that for $t=0$ these definitions coincide with the natural norms on
these spaces, being $\rho(0,x)\equiv 1$ for any $x\in\Gamma_{0}$. We now give
an extension of Proposition 3.2, whose proof is shown in Appendix A:

######  Proposition 4.5.

For $\eta,\phi\in H^{1}_{L^{2}}$, with
$\nabla_{\Gamma}\partial^{\bullet}\phi\in L^{2}_{L^{2}}$ and
$\nabla_{\Gamma}\partial^{\bullet}\eta\in L^{2}_{L^{2}}$, the following
identity holds

$\displaystyle\frac{d}{dt}(\nabla_{\Gamma}\eta,\nabla_{\Gamma}\phi)_{\rho}$
$\displaystyle=m(\partial^{\bullet}\rho\nabla_{\Gamma}\eta,\nabla_{\Gamma}\phi)+(\nabla_{\Gamma}\partial^{\bullet}\eta,\nabla_{\Gamma}\phi)_{\rho}+(\nabla_{\Gamma}\eta,\nabla_{\Gamma}\partial^{\bullet}\phi)_{\rho}+\int_{\Gamma(t)}\rho
B(\mathbf{V})\nabla_{\Gamma}\eta\cdot\nabla_{\Gamma}\phi$
$\displaystyle=(\nabla_{\Gamma}\partial^{\bullet}\eta,\nabla_{\Gamma}\phi)_{\rho}+(\nabla_{\Gamma}\eta,\nabla_{\Gamma}\partial^{\bullet}\phi)_{\rho}+\int_{\Gamma(t)}\rho\widetilde{B}(\mathbf{V})\nabla_{\Gamma}\eta\cdot\nabla_{\Gamma}\phi,$
(4.22)

for almost any $t\in[0,T]$, where $B$ is the vector field given in Section 3.2
and $\widetilde{B}(\mathbf{V}):=-2\mathbf{D}(\mathbf{V})$.

In conclusion, being
$\partial^{\bullet}\rho=-\rho\nabla_{\Gamma}\cdot\mathbf{V}$, we infer by
Proposition 3.2

$\displaystyle\frac{d}{dt}(\eta,\phi)_{\rho}=(\partial^{\bullet}\eta,\phi)_{\rho}+(\eta,\partial^{\bullet}\phi)_{\rho},\quad\forall\,\eta,\phi\in
H^{1}_{L^{2}},$ (4.23)

retrieving the classical integration by parts formula as in the case of a
fixed manifold. This is not surprising, in the sense that the term $\rho$
accounts for local stretching or compressing of the surfaces and somehow
annihilates the effects of the evolving surface.

###  4.3 Regularisation and strict separation property

For this model we only need to assume $\mathbf{A_{\Phi}}$, without any extra
regularity hypothesis. Our main result is the following

######  Theorem 4.6.

Let the assumptions of Theorem 4.2 hold. Denote by $(c,w)$ the (unique) weak
solution to (4.5)-(4.7).

  * (i)

There exists a constant $C=C(T,E_{CH}^{\rho}(u_{0}))>0$ such that, for almost
any $t\in[0,T]$,

$\displaystyle
t\|w\|_{H^{1}(\Gamma(t))}^{2}+\int_{0}^{t}s\|\partial^{\bullet}c\|^{2}_{H^{1}(\Gamma(s))}ds\leq
C(T,E_{CH}^{\rho}(u_{0})).$ (4.24)

  * (ii)

For any $0<\tau\leq T$, there exist constants
$C=C(T,\tau,E_{CH}^{\rho}(u_{0}))>0$ and
$C_{p}=C_{p}(T,\tau,p,E_{CH}^{\rho}(u_{0}))>0$ such that, for almost any
$t\in[\tau,T]$

$\displaystyle\|w\|_{H^{1}(\Gamma(t))}\leq C(T,\tau,E_{CH}^{\rho}(u_{0})),$
(4.25)
$\|\varphi(c)\|_{L^{p}(\Gamma(t))}+\|\varphi^{\prime}(c)\|_{L^{p}(\Gamma(t))}\leq
C_{p}(T,\tau,p,E_{CH}^{\rho}(u_{0})),\quad\forall p\in[2,\infty),$
$\|c\|_{H^{2}(\Gamma(t))}\leq C(T,\tau,E_{CH}^{\rho}(u_{0})).$

  * (iii)

There exists $\xi=\xi(T,\tau,E_{CH}^{\rho}(u_{0}))>0$, such that

$\displaystyle\|c\|_{L^{\infty}(\Gamma(t))}\leq 1-{\xi},\quad\text{ for almost
any }t\in[\tau,T].$

######  Remark 4.7.

For this second model the simple assumption AΦ is enough. Indeed, due to the
presence of $\rho$ we do not need, e.g., (2.9). For this reason, it is our
belief that the second model seems more natural in the context of evolving
surfaces.

######  Remark 4.8.

As in Remark 3.10, assuming the regularity stated in Lemma C.1, also in this
case we actually obtain that, for any $\tau>0$, $c\in C^{0}_{C^{0}}(\tau,T)$
and thus

$\displaystyle\sup_{t\in[\tau,T]}\|c\|_{C^{0}(\Gamma(t))}\leq 1-{\xi}.$

####  4.3.1 Galerkin approximation

We need a slight revision of the Galerkin approximation scheme. Let us recall
[13, Sec.4.1] and consider a basis $\\{\chi_{j}^{0}:j\in\mathbb{N}\\}$
orthonormal in $L^{2}(\Gamma_{0})$ and orthogonal in $H^{1}(\Gamma_{0})$
consisting of smooth functions such that $\chi_{1}^{0}$ is constant (for
example consider the eigenfunctions of the Laplace-Beltrami operator). We then
transport this basis using the flow map. This gives
$\\{\chi_{j}^{t}:=\phi_{t}(\chi_{j}^{0}):j\in\mathbb{N}\\}\subset
H^{1}(\Gamma(t))$. Observe that this basis is still an orthonormal basis in
$L^{2}(\Gamma(t))$ endowed with the norm induced by (4.21). Indeed, thanks to
(4.9), we have

$(\chi_{i}^{t},\chi_{j}^{t})_{\rho}=m(\rho\chi_{i}^{t},\chi_{j}^{t})=\int_{\Gamma_{0}}\chi_{i}^{0}\chi_{j}^{0}=\delta_{ij},\quad\forall\,i,j\in\mathbb{N}.$

Notice that the same basis is such that
$\overline{\text{span}\\{\chi_{j}^{t}:j\in\mathbb{N}\\}}^{H^{1}(\Gamma(t))}=H^{1}(\Gamma(t))$,
but we are not able to show any orthogonality property in this space, even
endowing the space with the norm induced by (4.21). We can then introduce the
finite dimensional spaces

$V_{M}(t)=\text{span}\\{\chi_{j}^{t}:1\leq j\leq M\\}\subset
H^{1}(\Gamma(t)),$

and, exploiting the orthogonality with respect to the equivalent $L^{2}$ inner
product (4.21)1, we can give an explicit expression to the $L^{2}$-orthogonal
projector $P_{M}^{\rho}(t):L^{2}(\Gamma(t))\to V_{M}(t)$ as

$P_{M}^{\rho}(t)f:=\sum_{j=1}^{M}(f,\chi_{j}^{t})_{\rho}\chi_{j}^{t}=\sum_{j=1}^{M}\left(\int_{\Gamma_{0}}\widetilde{f}\chi_{j}^{0}\right)\chi_{j}^{t}=\phi_{t}(P_{M}^{0}\widetilde{f}),$

where $P_{M}^{0}=P^{\rho}_{M}(0)$ is the orthogonal projector over $V_{M}(0)$
(endowed with the canonical $L^{2}$-norm). Clearly this implies
$\|P_{M}^{\rho}(t)f\|_{\rho}\leq\|f\|_{\rho}$ uniformly in $M$, for any $t>0$.
Following [4, Sec.7], we now introduce the matrix

$\mathbf{A}_{t}^{0}:=(D\Phi_{t}^{0})^{T}D\Phi_{t}^{0}+\nu_{0}\otimes\nu_{0},$

where $\nu_{0}$ is the normal to $\Gamma_{0}$. This matrix is invertible in
$\mathbb{R}^{3}$ thanks to the extension in the normal direction. We then have
that (see [4, (7.5)])

$\displaystyle\phi_{-t}(\nabla_{\Gamma}h)=D\Phi_{t}^{0}(\mathbf{A}_{t}^{0})^{-1}\nabla_{\Gamma_{0}}(\phi_{-t}(h))=D\Phi_{t}^{0}(\mathbf{A}_{t}^{0})^{-1}\nabla_{\Gamma_{0}}\widetilde{h}.$
(4.26)

Note also that, for any $t\in[0,T]$,

$\displaystyle\|D\Phi_{t}^{0}(\mathbf{A}_{t}^{0})^{-1}\|_{C^{0}(\Gamma(t))}\leq
C,$ (4.27)

being, thanks to assumption AΦ, $\Phi_{(\cdot)}^{0}\in
C^{1}([0,T];C^{2}(\mathbb{R}^{3};\mathbb{R}^{3}))$. Therefore, again by (4.9),
the properties of the orthogonal basis $\\{\chi_{j}^{0}:j\in\mathbb{N}\\}$ in
$H^{1}(\Gamma_{0})$, and the compatibility of the space
$(H^{1}(\Gamma(t)),\phi_{t})_{t\in[0,T]}$,

$\displaystyle\int_{\Gamma(t)}\rho\nabla_{\Gamma}w\cdot\nabla_{\Gamma}P_{M}^{\rho}(t)f$
$\displaystyle=\int_{\Gamma_{0}}\nabla_{\Gamma_{0}}\widetilde{w}(D\Phi_{t}^{0})^{T}(\mathbf{A}_{t}^{0})^{-T}D\Phi_{t}^{0}(\mathbf{A}_{t}^{0})^{-1}\nabla_{\Gamma_{0}}P_{M}^{0}\widetilde{f}$
$\displaystyle\leq
C\|\nabla_{\Gamma_{0}}\widetilde{w}\|_{L^{2}(\Gamma_{0})}\|\nabla_{\Gamma_{0}}P_{M}^{0}\widetilde{f}\|_{L^{2}(\Gamma_{0})}$
$\displaystyle\leq
C\|\nabla_{\Gamma_{0}}\widetilde{w}\|_{L^{2}(\Gamma_{0})}\|\nabla_{\Gamma_{0}}\widetilde{f}\|_{L^{2}(\Gamma_{0})}$
$\displaystyle\leq C\|\nabla_{\Gamma}{w}\|\|\nabla_{\Gamma}{f}\|,$ (4.28)

for any $w,f\in H^{1}(\Gamma(t))$. We can also prove that the time derivative
$\partial^{\bullet}$ commutes with the projector $P_{M}^{\rho}(t)$. Indeed,
since $\partial^{\bullet}\chi_{j}^{t}\equiv 0$ for any $j\in\mathbb{N}$, we
have

$\displaystyle\begin{split}\partial^{\bullet}(P_{M}^{\rho}(t)f)&=\sum_{i=1}^{M}\partial^{\bullet}(f,\chi_{i}^{t})_{\rho}\chi_{i}^{t}\\\
&=\sum_{i=1}^{M}\left((\partial^{\bullet}f,\chi_{i}^{t})_{\rho}+m(f\partial^{\bullet}\rho,\chi_{i}^{t})+g(\rho
f,\chi_{i}^{t})\right)\chi_{i}^{t}\\\
&=\sum_{i=1}^{M}(\partial^{\bullet}f,\chi_{i}^{t})_{\rho}\chi_{i}^{t}\\\
&=P_{M}^{\rho}(t)\partial^{\bullet}f,\quad\forall f\in
H^{1}_{L^{2}},\end{split}$ (4.29)

where we exploited the fact that
$\partial^{\bullet}\rho=-\rho\nabla_{\Gamma}\cdot\mathbf{V}$.

We then consider the Galerkin approximation with the approximated potential
$F^{\delta}$ ($\varphi^{\delta}=(F_{ln}^{\delta})^{\prime}$) of the original
problem. More precisely, given the spaces $V_{M}$ as in (3.10), for each
$M\in\mathbb{N}$, find functions $c_{\delta}^{M},w_{\delta}^{M}\in
L^{2}_{V_{M}}$ with $\partial^{\bullet}c_{\delta}^{M}\in L^{2}_{V_{M}}$ such
that, for any $\eta\in L^{2}_{V_{M}}$ and all $t\in[0,T]$,

$\displaystyle(\partial^{\bullet}c_{\delta}^{M},\eta)_{\rho}+(\nabla_{\Gamma}w_{\delta}^{M},\nabla_{\Gamma}\eta)_{\rho}=0,$
(4.30) $\displaystyle
w_{\delta}^{M}=P_{M}^{\rho}(t)(-\frac{1}{\rho}\Delta_{\Gamma(t)}c_{\delta}^{M}+(F^{\delta})^{\prime}(c_{\delta}^{M})),$
(4.31) $\displaystyle c_{\delta}^{M}(0)=P_{M}^{0}c_{0},\quad\text{ a.e. on
}\Gamma_{0}.$ (4.32)

Notice that this formulation, written with respect to the $\rho$ inner
product, is very similar the classical weak formulation of CH equation in
bounded domains of $\mathbb{R}^{2}$ (or $\mathbb{R}^{3}$). For this problem we
then have

######  Proposition 4.9.

Let Assumption $\textbf{A}_{\Phi}$ hold. Then there exists a unique local
solution $(c_{\delta}^{M},w_{\delta}^{M})$ to (4.30)-(4.32). In particular
there exist functions $(c^{M},w^{M})$ satisfying (4.30)-(4.31) on an interval
$[0,t^{\star})$, $0\leq t^{\star}\leq T$, together with (4.32). The functions
are of the form (omitting for simplicity the dependence on $\delta$)

$c_{\delta}^{M}(t)=\sum_{i=1}^{M}c_{i}^{M}(t)\chi_{i}^{t},\qquad
w_{\delta}^{M}(t)=\sum_{i=1}^{M}w_{i}^{M}(t)\chi_{i}^{t},\qquad
t\in[0,t^{\star}),$

with $c_{i}^{M}\in C^{2}([0,t^{\star}))$ and $w_{i}^{M}\in
C^{2}([0,t^{\star}))$, for every $i\in\\{1,\ldots,M\\}$.

###### Proof.

We consider the matrix form of the equations, where, as before, we set
$\mathbf{c}^{M}(t)=(c_{1}^{M}(t),\ldots,c_{M}^{M}(t))$ and
$\mathbf{w}^{M}(t)=(w_{1}^{M}(t),\ldots,w_{M}^{M}(t))$,

$\displaystyle
M^{\rho}(t)\dot{\mathbf{c}}^{M}(t)+A_{S}^{\rho}(t)\mathbf{w}^{M}(t)=0,$
$\displaystyle
A_{S}(t)\mathbf{c}^{M}(t)+(\mathbf{F}^{\delta}_{\rho})^{\prime}(\mathbf{{c}}^{M}(t))-M^{\rho}(t)\mathbf{w}^{M}(t)=0.$

Here

$\displaystyle(M^{\rho}(t))_{ij}=(\chi_{i}^{t},\chi_{j}^{t})_{\rho}=\delta_{ij},$
$\displaystyle(A_{S}(t))_{ij}=a_{S}(t;\chi^{t}_{i},\chi^{t}_{j}),$
$\displaystyle(A_{S}^{\rho}(t))_{ij}=(\nabla_{\Gamma}\chi^{t}_{i},\nabla_{\Gamma}\chi^{t}_{j})_{\rho},$
$\displaystyle(\mathbf{F}^{\delta}_{\rho})^{\prime}(\mathbf{c}^{M}(t))_{j}=((F^{\delta})^{\prime}(c^{M}(t)),\chi_{j}^{t})_{\rho}.$

We now observe that again these matrices are more regular. Indeed,
$M^{\rho}_{ij}$ is actually the identity matrix. Similarly, we get

$\frac{d}{dt}(A_{S}(t))_{ij}=b(\chi^{t}_{i},\chi^{t}_{j})\in C^{0}([0,T]),$

and the same goes for $A_{S}^{\rho}$, by (4.22). Recalling then that
$(F^{\delta})^{\prime}$ is $C^{1,1}(\mathbb{R})$, the result follows from the
general theory of ODEs. ∎

We establish some a priori estimates for the solutions of the Galerkin
approximation.

######  Proposition 4.10.

For the approximating solution pair $(u_{\delta}^{M},w_{\delta}^{M})$ we have:

$\displaystyle\sup_{t\in[0,T]}\tilde{E}_{CH}^{\rho}(c_{\delta}^{M})+\frac{1}{2}\int_{0}^{T}\int_{\Gamma(t)}|\nabla_{\Gamma}w_{\delta}^{M}|^{2}dt\leq
C(T)$ (4.33)
$\displaystyle\|\partial^{\bullet}c_{\delta}^{M}\|_{L^{2}_{H^{-1}}}\leq
C(T),\quad\text{ and }\quad\|w_{\delta}^{M}\|_{L^{2}_{H^{1}}}\leq C_{\delta},$
(4.34)

where as before $\tilde{E}_{CH}^{\rho}:=E^{\rho}_{CH}+\tilde{C}$ for some
$\tilde{C}>0$ chosen so that $\tilde{E}_{CH}^{\rho}\geq 0$.

###### Proof.

We consider the Galerkin approximation of Proposition 4.9. First notice that
we have the conservation of the total mass, by choosing $\eta\equiv 1$:

$\displaystyle\frac{d}{dt}\int_{\Gamma(t)}\rho
c_{\delta}^{M}=\int_{\Gamma(t)}\rho\partial^{\bullet}c_{\delta}^{M}\equiv 0.$
(4.35)

Arguing as in [13, (6.2.1)] we can obtain

$\displaystyle\frac{d}{dt}E_{CH}^{\rho}(c_{\delta}^{M})+\int_{\Gamma(t)}\rho|\nabla_{\Gamma}w_{\delta}^{M}|^{2}dt=b(c_{\delta}^{M},c_{\delta}^{M}).$
(4.36)

Observe also that, by the conservation of total mass,  see (1.4) and (4.35),

$(\widetilde{c}_{\delta}^{M})_{\Gamma_{0}}\equiv\dfrac{\int_{\Gamma(t)}\rho
c_{\delta}^{M}}{|\Gamma_{0}|}\equiv(c_{0})_{\Gamma_{0}}.$

In the end we easily get

$\displaystyle\frac{d}{dt}\tilde{E}^{\rho}_{CH}(c_{\delta}^{M})+\frac{1}{2}\int_{\Gamma(t)}\rho|\nabla_{\Gamma}w_{\delta}^{M}|^{2}dt\leq
C\tilde{E}^{\rho}_{CH}(c_{\delta}^{M}),$ (4.37)

where
$\tilde{E}^{\rho}_{CH}(c_{\delta}^{M}):={E}^{\rho}_{CH}(c_{\delta}^{M})+C$, so
that $\tilde{E}^{\rho}_{CH}(c_{\delta}^{M})\geq 0$. Therefore, an application
of Gronwall’s Lemma gives

$\displaystyle\sup_{t\in[0,T]}\tilde{E}_{CH}^{\rho}(c_{\delta}^{M})+\frac{1}{2}\int_{0}^{T}\int_{\Gamma(t)}|\nabla_{\Gamma}w_{\delta}^{M}|^{2}dt\leq
C(T)\tilde{E}^{\rho}_{CH}(P_{M}^{0}c_{0}),$ (4.38)

and, recalling the properties of $P_{M}^{0}$ (see, e.g., [13, Lemma 5.6]), we
have

$\displaystyle\tilde{E}^{\rho}_{CH}(P_{M}^{0}c_{0})\leq CT.$ (4.39)

This clearly allows us to extend the maximal time from $t^{\star}$ to $T$.
Concerning the estimate of the mean value of $c$, since in this model this
quantity is not conserved, being conserved the product $\rho c$, we can
observe, as in [13], that

$\displaystyle\frac{d}{dt}\int_{\Gamma(t)}c_{\delta}^{M}=\int_{\Gamma(t)}\partial^{\bullet}c_{\delta}^{M}+\int_{\Gamma(t)}c_{\delta}^{M}\nabla_{\Gamma}\cdot\mathbf{V}.$

If we now take $\eta=P_{M}^{\rho}(t)\frac{1}{\rho}$ in (4.30), we have, being
$\partial^{\bullet}c^{M}_{\delta}\in L^{2}_{V_{M}}$ and $P_{M}^{\rho}(t)$
self-adjoint,

$\displaystyle\left(\partial^{\bullet}c_{\delta}^{M},P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}=m(\partial^{\bullet}c_{\delta}^{M},1)=-\left(\nabla_{\Gamma}w_{\delta}^{M},\nabla_{\Gamma}P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}.$

Therefore, integrating by parts
$\int_{\Gamma(t)}c_{\delta}^{M}\nabla_{\Gamma}\cdot\mathbf{V}$, recalling that
$\Gamma(t)$ is closed, we get

$\displaystyle\frac{d}{dt}\int_{\Gamma(t)}c_{\delta}^{M}=-\left(\nabla_{\Gamma}w_{\delta}^{M},\nabla_{\Gamma}P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}-\int_{\Gamma(t)}\nabla_{\Gamma}c_{\delta}^{M}\cdot\mathbf{V},$

so that

$\displaystyle\int_{\Gamma(t)}c_{\delta}^{M}=\int_{\Gamma_{0}}P^{0}_{M}c_{0}-\int_{0}^{t}\left(\nabla_{\Gamma(s)}w_{\delta}^{M},\nabla_{\Gamma(s)}P_{M}^{\rho}(s)\frac{1}{\rho}\right)_{\rho}ds-\int_{0}^{t}\int_{\Gamma(s)}\nabla_{\Gamma}c_{\delta}^{M}\cdot\mathbf{V}ds.$

Thus, by (4.10), (4.39), the properties of $P_{M}^{0}$, and (4.28), we deduce

$\displaystyle\left|\int_{\Gamma(t)}c_{\delta}^{M}\right|\leq
C+C\int_{0}^{t}\left\|\nabla_{\Gamma(s)}w^{M}_{\delta}\right\|\left\|\nabla_{\Gamma(s)}\frac{1}{\rho}\right\|ds+C(T)$
$\displaystyle\leq
C\left(1+\int_{0}^{t}\|\nabla_{\Gamma}w_{\delta}^{M}\|ds\right)+C(T)\leq
C(T).$

Combining this result with (4.38) and Poincaré’s inequality, we infer that

$\displaystyle\|c^{M}_{\delta}\|_{L^{\infty}_{H^{1}}}\leq C(T),$ (4.40)

independently of $M$ and $\delta$. Observe now that, by (4.28), we can find a
uniform estimate for $\partial^{\bullet}c^{M}_{\delta}$. Indeed we can write,
for any $\eta\in L^{2}_{H^{1}}$, being $P^{\rho}_{M}(t)$ self-adjoint (with
respect to (4.21)1) and by (4.28)  and (4.40),

$\displaystyle{\color[rgb]{0,0,0}m_{\star}(\partial^{\bullet}c^{M}_{\delta},\eta)}$
$\displaystyle=(P_{M}^{\rho}(t)\partial^{\bullet}c^{M}_{\delta},\eta)=\left(P_{M}^{\rho}(t)\partial^{\bullet}c^{M}_{\delta},\frac{1}{\rho}\eta\right)_{\rho}$
$\displaystyle=\left(\partial^{\bullet}c^{M}_{\delta},P_{M}^{\rho}(t){\color[rgb]{0,0,0}\left(\frac{1}{\rho}\eta\right)}\right)_{\rho}$
$\displaystyle=-\left(\nabla_{\Gamma}w_{\delta}^{M},\nabla_{\Gamma}P_{M}^{\rho}(t){\color[rgb]{0,0,0}\left(\frac{1}{\rho}\eta\right)}\right)_{\rho}$
$\displaystyle\leq
C\|\nabla_{\Gamma}w_{\delta}^{M}\|\left\|\nabla_{\Gamma}\frac{\eta}{\rho}\right\|+C\|c_{\delta}^{M}\|\left\|\nabla_{\Gamma}\frac{\eta}{\rho}\right\|$
$\displaystyle\leq
C{\color[rgb]{0,0,0}\left(\|\nabla_{\Gamma}w_{\delta}^{M}\|+1\right)\|\eta\|_{H^{1}(\Gamma(t))}},$

which implies, by (4.38),

$\displaystyle\|\partial^{\bullet}c^{M}_{\delta}\|_{L^{2}_{H^{-1}}}\leq C(T),$
(4.41)

independently of $M$ and $\delta$. We now need a control over the mean value
of $w_{\delta}^{M}$. This can be obtained by testing (4.31) with
$\eta=P_{M}^{\rho}(t)\frac{1}{\rho}$ being $w\in L^{2}_{V_{M}}$. On account of
(4.28), we have

$\displaystyle\left(w_{\delta}^{M},\frac{1}{\rho}\right)_{\rho}=\left(w_{\delta}^{M},P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}$
$\displaystyle=\left(-\frac{1}{\rho}\Delta
c_{\delta}^{M},P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}+\left(F^{\prime}_{\delta}(c_{\delta}^{M}),P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}$
$\displaystyle=m\left(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}P_{M}^{\rho}(t)\frac{1}{\rho}\right)+\left(F^{\prime}_{\delta}(c_{\delta}^{M}),P_{M}^{\rho}(t)\frac{1}{\rho}\right)_{\rho}$
$\displaystyle\leq
C\|\nabla_{\Gamma}c_{\delta}^{M}\|\left\|\nabla_{\Gamma}\left(P^{\rho}_{M}(t)\frac{1}{\rho}\right)\right\|+C\|F^{\prime}_{\delta}(c_{\delta}^{M})\|\left\|P_{M}^{\rho}(t)\frac{1}{\rho}\right\|$
$\displaystyle\leq
C\|\nabla_{\Gamma}c_{\delta}^{M}\|\left\|\nabla_{\Gamma}\frac{1}{\rho}\right\|+C_{\delta}\|c_{\delta}^{M}\|\leq
C_{\delta},$

by (4.10) and (4.38). Here we have also applied the fact that
$|F^{\prime}_{\delta}(c_{\delta}^{M})|\leq C_{\delta}|c_{\delta}^{M}|$, being
${\color[rgb]{0,0,0}|F^{\prime\prime}_{\delta}|}\leq C_{\delta}$ and
$F_{\delta}^{\prime}(0)=F^{\prime}(0)=0$. In the Galerkin scheme we are not
able to retrieve a uniform-in-$\delta$ estimate for the mean value of
$w_{\delta}^{M}$. Indeed, we should be able to control the
$L^{\infty}(\Gamma(t))$ norm of $\rho P_{M}^{\rho}(t)\frac{1}{\rho}$ and then
control $\int_{\Gamma(t)}|F^{\prime}_{\delta}(c_{\delta}^{M})|$, but this does
not seem feasible. Therefore, we will need to pass to the limit in $M$ first.
Then this control will be obtained independently of $\delta$. The above bound
entails, thanks to Poincaré’s inequality combined with (4.33),

$\displaystyle\|w_{\delta}^{M}\|_{L^{2}_{H^{1}}}\leq C_{\delta}.$ (4.42)

∎

####  4.3.2 Proof of Theorem 4.6

Part (i). We now need to find higher-order estimates. In particular, we set
$\eta=\partial^{\bullet}w_{\delta}^{M}\in L^{2}_{V_{M}}$ in (4.30), to get

$\displaystyle(\partial^{\bullet}c_{M}^{\delta},\partial^{\bullet}w^{M}_{\delta})_{\rho}+(\nabla_{\Gamma}w_{M}^{\delta},\nabla_{\Gamma}\partial^{\bullet}w_{M}^{\delta})_{\rho}=0.$
(4.43)

Observe that, by (4.22),

$\displaystyle\frac{1}{2}\frac{d}{dt}(\nabla_{\Gamma}w_{\delta}^{M},\nabla_{\Gamma}w_{\delta}^{M})_{\rho}=(\nabla_{\Gamma}w_{M}^{\delta},\nabla_{\Gamma}\partial^{\bullet}w_{M}^{\delta})_{\rho}+\int_{\Gamma(t)}\rho\widetilde{B}(\mathbf{V})\nabla_{\Gamma}w_{\delta}^{M}\cdot\nabla_{\Gamma}w_{\delta}^{M}.$
(4.44)

On the other hand, by Proposition 3.2, we have, for any $\eta\in
L^{2}_{V_{M}}$ such that $\partial^{\bullet}\eta\in L^{2}_{V_{M}}$,

$\displaystyle\frac{d}{dt}m(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}\eta)$
$\displaystyle=m(\nabla_{\Gamma}\partial^{\bullet}c_{\delta}^{M},\nabla_{\Gamma}\eta)+m(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}\partial^{\bullet}\eta)+\int_{\Gamma(t)}{B}(\mathbf{V})\nabla_{\Gamma}c_{\delta}^{M}\cdot\nabla_{\Gamma}\eta$

and, by (4.23),

$\displaystyle\frac{d}{dt}(w_{\delta}^{M},\eta)_{\rho}=(\partial^{\bullet}w_{\delta}^{M},\eta)_{\rho}+(w,\partial^{\bullet}\eta)_{\rho}.$

Furthermore, we have, using again (4.23),

$\displaystyle\frac{d}{dt}(F^{\prime}_{\delta}(c_{\delta}^{M}),\eta)_{\rho}=(F^{\prime\prime}_{\delta}(c_{\delta}^{M}),\partial^{\bullet}c_{\delta}^{M}\eta)_{\rho}+(F^{\prime}_{\delta}(c_{\delta}^{M}),\partial^{\bullet}\eta)_{\rho}.$

We now recall that

$\displaystyle\frac{d}{dt}(w_{\delta}^{M},\eta)_{\rho}=\frac{d}{dt}m(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}\eta)+\frac{d}{dt}(F^{\prime}_{\delta}(c_{\delta}^{M}),\eta)_{\rho},$

thus

$\displaystyle(\partial^{\bullet}w_{\delta}^{M},\eta)_{\rho}+(w,\partial^{\bullet}\eta)_{\rho}$
$\displaystyle=m(\nabla_{\Gamma}\partial^{\bullet}c_{\delta}^{M},\nabla_{\Gamma}\eta)+m(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}\partial^{\bullet}\eta)$
$\displaystyle+\int_{\Gamma(t)}{B}(\mathbf{V})\nabla_{\Gamma}c_{\delta}^{M}\cdot\nabla_{\Gamma}\eta$
$\displaystyle+(F^{\prime\prime}_{\delta}(c_{\delta}^{M}),\partial^{\bullet}c_{\delta}^{M}\eta)_{\rho}+(F^{\prime}_{\delta}(c_{\delta}^{M}),\partial^{\bullet}\eta)_{\rho}.$

By noticing that, being $\partial^{\bullet}\eta\in L^{2}_{V_{M}}$,

$(w,\partial^{\bullet}\eta)_{\rho}=m(\nabla_{\Gamma}c_{\delta}^{M},\nabla_{\Gamma}\partial^{\bullet}\eta)+(F^{\prime}_{\delta}(c_{\delta}^{M}),\partial^{\bullet}\eta)_{\rho},$

we infer, choosing $\eta=\partial^{\bullet}c_{\delta}^{M}$ (which is
sufficiently regular thanks to Proposition 4.9),