# Navier-Stokes-Cahn-Hilliard equations on evolving surfaces

Charles M. Elliott
Thomas Sales Mathematics Institute, Zeeman Building, University of Warwick,
Coventry CV4 7AL, UK<EMAIL_ADDRESS>Institute, Zeeman
Building, University of Warwick, Coventry CV4 7AL, UK<EMAIL_ADDRESS>
###### Abstract

We derive a system of equations which can be seen as an evolving surface
version of the diffuse interface “Model H” of Hohenberg and Halperin. We then
consider the well-posedness for the corresponding (tangential) system when one
prescribes the evolution of the surface. This well-posedness theory is
considered for smooth potentials with polynomial growth, and a
thermodynamically relevant singular potential - with some restrictions on
initial data in the latter case.

##  1 Introduction

The Navier-Stokes-Cahn-Hilliard system on a sufficiently smooth, closed,
oriented, evolving hypersurface,
$(\Gamma(t))_{t\in[0,T]}\subset\mathbb{R}^{3}$, is given by

$\displaystyle\rho\partial^{\bullet}\mathbf{u}=-\nabla_{\Gamma}p+pH\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\mathbf{u}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi)+\mathbf{F},$
(1.1) $\displaystyle\nabla_{\Gamma}\cdot\mathbf{u}=0,$ (1.2)
$\displaystyle\partial^{\bullet}\varphi=\nabla_{\Gamma}\cdot(M(\varphi)\nabla_{\Gamma}\mu),$
(1.3)
$\displaystyle\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi),$
(1.4)

where

$\mathbb{E}(\mathbf{u})=\frac{1}{2}\left(\nabla_{\Gamma}\mathbf{u}+(\nabla_{\Gamma}\mathbf{u})^{T}\right)$

is the rate of strain tensor. We assume a matched density and so we denote the
constant density by $\rho$, the velocity of the surface is denoted by
$\mathbf{u}$ which will often be decomposed as
$\mathbf{u}=\mathbf{u}_{\nu}+\mathbf{u}_{T}$, where
$\mathbf{u}_{\nu},\mathbf{u}_{T}$ are the normal and tangential components
respectively. The associated pressure is denoted as $p$, $\eta(\cdot)$ is the
variable viscosity which depends on $\varphi$, and $\mathbf{F}$ is some
external force. We have split the fourth order Cahn-Hilliard equation into two
second order equations (1.3), (1.4) for $\varphi,\mu$ where $\mu$ is the
associated chemical potential. In our derivations we consider a non-constant
mobility function $M(\cdot)$, but our analysis will consider a constant
mobility $M(\cdot)\equiv 1$. Lastly, $\boldsymbol{\nu}$ denotes the outer unit
normal to $\Gamma(t)$, and $H=\nabla_{\Gamma}\cdot\boldsymbol{\nu}$ the mean
curvature of $\Gamma(t)$ \- with the convention that a sphere has positive
mean curvature. The differential operators used will be discussed below.

As in [24, 34, 44], one may be interested in this system where the normal
component of the surface evolution is known a priori. That is,
$\mathbf{u}_{\nu}=V_{N}\boldsymbol{\nu}$ for some, sufficiently smooth, known
function $V_{N}$. For our theory, it is sufficient to assume that $V_{N}$ is a
$C^{3}$ function, and so we are working on a $C^{3}$ evolving surface,
$\Gamma(t)$. In this case one obtains the tangential Navier-Stokes-Cahn-
Hilliard system

$\displaystyle\rho\left(\mathbb{P}\partial^{\circ}\mathbf{u}_{T}+(\nabla_{\Gamma}\mathbf{u}_{T})\mathbf{u}_{T}+V_{N}\mathbb{H}\mathbf{u}_{T}-\frac{1}{2}\nabla_{\Gamma}V_{N}^{2}\right)=-\nabla_{\Gamma}\tilde{p}+\mathbb{P}\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\mathbf{u}_{T}))+\mu\nabla_{\Gamma}\varphi+\mathbf{F}_{T},$
(1.5) $\displaystyle\nabla_{\Gamma}\cdot\mathbf{u}_{T}=-HV_{N},$ (1.6)
$\displaystyle\partial^{\circ}\varphi+\nabla_{\Gamma}\varphi\cdot\mathbf{u}_{T}=\nabla_{\Gamma}\cdot(M(\varphi)\nabla_{\Gamma}\mu),$
(1.7)
$\displaystyle\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi).$
(1.8)

Here
$\tilde{p}=p+\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi)$
is a modified pressure. In both systems the pressure, and modified pressure,
are unknown due to the incompressibility constraint.

Variants of this model have been considered on (mainly stationary) surfaces in
[11, 12, 42, 46, 49, 51]. The main focus in the existing literature is on the
derivation and numerical simulation of such a system, but there has been
little consideration for the well-posedness thus far. As such, our work
considers the well-posedness of a somewhat simpler model (surface evolution
notwithstanding) which still captures the main features of the coupling of the
Navier-Stokes equations with the Cahn-Hilliard equation. Lastly we note that
the model in [11] considers the influence of a physically relevant bending
energy, and the model in [46] consider a variable density - in accordance with
the derivation in [2].

Some geometric differential notation The evolving two dimensional hypersurface
$\Gamma$ is assumed to be sufficiently smooth with a normal,
$\boldsymbol{\nu}$, and normal velocity, $V_{N}\boldsymbol{\nu}$. Geometric
quantities and differential operators are defined by

$\displaystyle\mathbb{P}=\mathbb{I}-\boldsymbol{\nu}\otimes\boldsymbol{\nu},$
$\displaystyle\nabla_{\Gamma}\phi=\mathbb{P}\nabla\phi^{e},\quad(\nabla_{\Gamma}\phi)_{i}=\underline{D}_{i}\phi,\quad\nabla_{\Gamma}\mathbf{v}=\mathbb{P}\nabla\mathbf{v}^{e}\mathbb{P},\quad(\nabla_{\Gamma}\mathbf{v})_{i,j}:=\underline{D}_{j}v_{i}$
$\displaystyle(\nabla_{\Gamma}\mathbf{v}\cdot\mathbf{w})_{i}=\underline{D}_{j}v_{i}w_{j},\quad\nabla_{\Gamma}\cdot\mathbf{v}=\text{tr}(\nabla_{\Gamma}\mathbf{v}),\quad\mathbb{H}=\nabla_{\Gamma}\boldsymbol{\nu},$

where we have used Einstein summation convention. Here $\phi$ and
$\mathbf{v},\mathbf{w}$ are scalar and vector fields respectively,
$\mathbb{I}$ denotes the identity, and $(\cdot)^{e}$ denotes an extension onto
a neighbourhood of $\Gamma(t)$. These expressions are independent of the
specific choice of extension. Likewise we use the following notation for the
normal time derivatives of scalar and vector fields,

$\displaystyle\partial^{\circ}\phi=\frac{\partial\phi^{e}}{\partial
t}+V_{N}\nabla\phi^{e}\cdot\boldsymbol{\nu},\quad\partial^{\circ}\mathbf{v}=\frac{\partial\mathbf{v}^{e}}{\partial
t}+V_{N}\nabla\mathbf{v}^{e}\cdot\boldsymbol{\nu}$

which are again independent of choice of extension. Note that this is the time
derivative along a trajectory evolving in the normal direction. We use
$\partial^{\bullet}$ to note the time derivative which follows also the
physical tangential flow $\mathbf{u}_{T}$,

$\displaystyle\partial^{\bullet}\phi=\partial^{\circ}\phi+\mathbf{u}_{T}\cdot\nabla_{\Gamma}\phi,\quad\partial^{\bullet}\mathbf{v}=\partial^{\circ}\mathbf{v}+\nabla_{\Gamma}\mathbf{v}\cdot\mathbf{u}_{T},$

On the other hand if we wish to use another tangential flow for a velocity
field $\mathbf{w}=V_{N}\boldsymbol{\nu}+\mathbf{w}_{T}$ with a tangential
vector field $\mathbf{w}_{T}$, we write

$\partial_{\mathbf{w}}^{\bullet}\phi=\partial^{\circ}\phi+\nabla_{\Gamma}\phi\cdot\mathbf{w}_{T},\quad\partial_{\mathbf{w}}^{\bullet}\mathbf{v}=\partial^{\circ}\mathbf{v}+\nabla_{\Gamma}\mathbf{v}\cdot\mathbf{w}_{T}.$

Throughout we will use the convention that a vector quantity will be denoted
in bold, e.g. $\mathbf{v}$, and a tensor quantity will be denoted in
blackboard bold, e.g. $\mathbb{P}$.

#### Applications

On a stationary Euclidean domain, the Navier-Stokes-Cahn-Hilliard equations
has found many applications, for example in studying thermocapillary flows
[9], and spinodal decomposition [31]. For further details and applications we
refer the reader to [8]. A more recent application to a modified version of
this system has been to the study chemotaxis, for example in modelling tumour
growth [37]. In this case there are suitable changes to allow for a transfer
of mass - adding further complications.

Another biological application which has been of interest in recent years is
in studying lipid bilayer membranes. It is known that the curvature of the
domain influences the dynamics of lipid membranes and enters through a bending
energy, for example the Jülicher-Lipowsky energy

$E^{\text{H}}[\varphi]=\int_{\Gamma}\frac{1}{2}\kappa(\varphi)(H-H_{0}(\varphi))^{2},$

as presented in [36]. Here $\kappa(\cdot)$ is the bending stiffness, and
$H_{0}(\cdot)$ is the spontaneous curvature which depends on the diffuse
interface - an example being $H_{0}(\varphi)=\Lambda\varphi$, for some
curvature coefficient $\Lambda\in\mathbb{R}$. We refer the reader to [23, 39]
for a discussion of these mechanisms, and to [32] for discussion and analysis
of diffuse interface models for phase separation on biological membranes. In
[28] the authors consider a model for the kinetics of a lipid bilayer
membrane, coupling the Cahn-Hilliard equation to the stationary Stokes
equations in a planar domain. This model has been extended to surfaces in the
recent works [11, 12] where the authors also include the contributions of the
relevant bending energy, as well as considering the full time-dependent
Navier-Stokes equations for the hydrodynamics.

Lastly we mention a different model of interest [13], in which the authors
consider the Cahn-Hilliard equations coupled with the Navier-Stokes equations
on a free surface determined by the Navier-Stokes equation in a bulk domain.
This work also proposes semi-discrete and fully-discrete numerical schemes,
and contains numerical examples.

##  2 Derivation of the surface Navier-Stokes-Cahn-Hilliard system

In this section we provide two derivations (1.5)-(1.8) - one by conservation
laws, and the other by considering a thin film limit.

###  2.1 Derivation by conservation laws

We follow a similar presentation to that of [31] and derive (1.1)-(1.4) by
using a balance of microstresses. Consider a binary mixture of a fluid, with
constituent densities $\rho_{1},\rho_{2}$. The total density

$\rho:=\rho_{1}+\rho_{2},$

is assumed to be constant. We define $c_{i}:=\frac{\rho_{i}}{\rho}$ to be the
corresponding concentration, so that $c_{1}+c_{2}=1$. Following the assumption
of [31], we assume that the momenta and kinetic energies of the constituent
components are negligible when computed relative to the gross motion of the
fluid. As such, we consider the gross velocity, $\mathbf{u}$, in our
derivation instead of the velocities of each component. By considering the
total momentum of the fluid in an arbitrary region it is clear to see that

$\mathbf{u}=c_{1}\mathbf{u}_{1}+c_{2}\mathbf{u}_{2},$

where $\mathbf{u}_{i}$ is the velocity of component with density $\rho_{i}$.
Throughout we consider an arbitrary material portion
$\Sigma(t)\subset\Gamma(t)$ whose boundary, $\partial\Sigma(t)$, moves with
conormal material velocity of the surface fluid
$\mathbf{u}_{T}\cdot\boldsymbol{\nu}_{\Sigma}$, where
$\boldsymbol{\nu}_{\Sigma}$ denotes the unit conormal vector for
$\partial\Sigma(t)$ (the outward unit normal vector which is tangential to
$\Sigma(t)$).

Since $\rho$ is constant, conservation of mass within the material region
$\Sigma(t)$ yields

$0=\frac{d}{dt}\int_{\Sigma(t)}\rho=\rho\frac{d}{dt}\int_{\Sigma(t)}1$

. Applying the transport theorem we obtain that

$0=\int_{\Sigma(t)}\nabla_{\Gamma}\cdot\mathbf{v},$

and since $\Sigma$ is arbitrary this yields

$\nabla_{\Gamma}\cdot\mathbf{v}=0,\ \mathrm{on}\ \Gamma(t).$

This shows that the material surface $\Gamma(t)$ has the property of local
inextensiblity, that is to say $\frac{d}{dt}|\Sigma(t)|=0$ for all
$\Sigma(t)\subset\Gamma(t)$ such that the boundary moves with conormal
material velocity $\mathbf{u}_{T}\cdot\boldsymbol{\nu}_{\Sigma}$. As a
consequence we have the property that the total area is preserved

$|\Gamma(t)|=|\Gamma_{0}|,$

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

For each component, $u_{i}$, we consider the mass balance

$\frac{d}{dt}\int_{\Sigma(t)}c_{i}=-\int_{\partial\Sigma(t)}\mathbf{q}_{i}\cdot\boldsymbol{\nu}_{\Sigma},$

where $\mathbf{q}_{i}$ is some flux vector to be determined. The normal
component of $\mathbf{q}_{i}$ does not contribute to the flux, and hence we
assume $\mathbf{q}_{i}$ is purely tangential. Using the transport theorem,
along with the incompressibility above we find that

$\frac{d}{dt}\int_{\Sigma(t)}c_{i}=\int_{\Sigma(t)}\partial^{\bullet}c_{i}.$

Using integration by parts on the boundary integral, one obtains

$\int_{\partial\Sigma(t)}\mathbf{q}_{i}\cdot\boldsymbol{\nu}_{\Sigma}=\int_{\Sigma(t)}\nabla_{\Gamma}\cdot\mathbf{q}_{i}-\int_{\Sigma(t)}\mathbf{q}_{i}\cdot\boldsymbol{\nu}H=\int_{\Sigma(t)}\nabla_{\Gamma}\cdot\mathbf{q}_{i},$

where $H$ denotes the mean curvature. Thus we obtain an equation for $u_{i}$,

$\partial^{\bullet}c_{i}=-\nabla_{\Gamma}\cdot\mathbf{q}_{i},\quad\text{on
}\Gamma(t).$

We define quantities
$\varphi=c_{1}-c_{2},\mathbf{q}=\mathbf{q}_{1}-\mathbf{q}_{2}$, and observe
that

$\displaystyle\partial^{\bullet}\varphi=-\nabla_{\Gamma}\cdot\mathbf{q},$
(2.1)

where we choose $\mathbf{q}$ later. Since $c_{1}+c_{2}=1$, we can revert back
to the individual concentrations by

$c_{1}=\frac{1+\varphi}{2},\quad c_{2}=\frac{1-\varphi}{2}.$

Lastly, one considers the linear momentum balance for the gross momentum, that
is

$\frac{d}{dt}\int_{\Sigma(t)}\rho\mathbf{u}=\int_{\partial\Sigma(t)}\mathbb{T}\boldsymbol{\nu}_{\Sigma}+\int_{\Sigma(t)}\mathbf{F},$

where $\mathbb{T}$ is the Cauchy stress tensor describing the stresses across
the surface. Integrating by parts together with $\Sigma(t)$ being arbitrary to

$\displaystyle\rho\partial^{\bullet}\mathbf{u}=\nabla_{\Gamma}\cdot\mathbb{T}+\mathbf{F},$
(2.2)

where again there is no term involving $\boldsymbol{\nu}$ coming from the
integration by parts111However there is, as we emphasise later, still a
component of this equation in the normal direction. as we assume $\mathbb{T}$
maps onto the tangent space of $\Gamma(t)$. Similarly, one can apply standard
arguments to show that the balance of angular momentum yields

$\displaystyle\mathbb{T}=\mathbb{T}^{T}.$ (2.3)

Next we consider a local dissipation inequality for the energy. For a region
$\Sigma(t)$, the energy is given by

$\int_{\Sigma(t)}E(\varphi,\nabla_{\Gamma}\varphi,\mathbf{u})=\int_{\Sigma(t)}E_{1}(\mathbf{u})+E_{2}(\varphi,\nabla_{\Gamma}\varphi)$

where

$E_{1}(\mathbf{z})=\frac{\rho}{2}|\mathbf{z}|^{2},\quad
E_{2}(x,\mathbf{z})=\frac{\varepsilon}{2}|\mathbf{z}|^{2}+\frac{1}{\varepsilon}F(x).$

We assume, as in [2, 31], that there is a local dissipation inequality given
by

$\displaystyle\frac{d}{dt}\int_{\Sigma(t)}E\leq\int_{\partial\Sigma(t)}\mathbb{T}\boldsymbol{\nu}_{\Sigma}\cdot\mathbf{u}+\int_{\partial\Sigma(t)}\partial^{\bullet}\varphi\,\boldsymbol{\xi}\cdot\boldsymbol{\nu}_{\Sigma}-\int_{\partial\Sigma(t)}\mu\,\mathbf{q}\cdot\boldsymbol{\nu}_{\Sigma}+\int_{\Gamma(t)}\mathbf{F}\cdot\mathbf{u},$
(2.4)

where $\mu$ is the difference of the chemical potentials, $\mu_{i}$, of each
component, and $\boldsymbol{\xi}$ is a stress (which we assume exists as in
[2, 31]) characterising the microforces across the boundary of a region - and
acts only in the tangential direction. This inequality is understood as being
an appropriate form of the second law of thermodynamics. The boundary terms
correspond to the work done by the macroscopic stresses in the fluid, the work
done by the microscopic stresses, and the change of potential energy
respectively.

As $\Sigma$ is arbitrary, we find that on $\Gamma(t)$ one has

$\partial^{\bullet}E-\nabla_{\Gamma}\cdot\left(\mathbb{T}\mathbf{u}\right)-\nabla_{\Gamma}\cdot\left(\partial^{\bullet}\varphi\boldsymbol{\xi}\right)+\nabla_{\Gamma}\cdot\left(\mu\mathbf{q}\right)-\mathbf{F}\cdot\mathbf{u}\leq
0,$

where we have used (2.3). We use (2.2) and the form of $E$ to see that this
implies

$\rho\partial^{\bullet}\mathbf{u}\cdot\mathbf{u}+\varepsilon\nabla_{\Gamma}\varphi\cdot\partial^{\bullet}\nabla_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi)\partial^{\bullet}\varphi-\rho\partial^{\bullet}\mathbf{u}\cdot\mathbf{u}-\mathbb{T}:\nabla_{\Gamma}\mathbf{u}-\nabla_{\Gamma}\cdot\left(\partial^{\bullet}\varphi\boldsymbol{\xi}\right)+\nabla_{\Gamma}\cdot\left(\mu\mathbf{q}\right)-\mathbf{F}\cdot\mathbf{u}\leq
0.$

The stress $\boldsymbol{\xi}$ is understood to have a microforce balance on an
arbitrary region $\Sigma(t)$, given by

$\int_{\partial\Sigma(t)}\boldsymbol{\xi}\cdot\boldsymbol{\nu}_{\Sigma}=\int_{\Sigma(t)}\sigma,$

from which we obtain

$\displaystyle\nabla_{\Gamma}\cdot\boldsymbol{\xi}+\sigma=0,\text{ on
}\Gamma(t),$ (2.5)

where $\sigma$ is a scalar function representing the internal forces in the
surface. Lastly we write $\mathbb{S}=\mathbb{T}+p\mathbb{P}$, where
$\mathbb{P}$ is the projection tensor, and
${p=-\frac{1}{2}\mathrm{tr}(\mathbb{T})}$ is the pressure. We note that
$\mathbb{S}$ maps onto tangent vectors, is trace-free, and we find

$\mathbb{P}:\nabla_{\Gamma}\mathbf{u}=\mathbb{I}:\nabla_{\Gamma}\mathbf{u}-\boldsymbol{\nu}\otimes\boldsymbol{\nu}:\nabla_{\Gamma}\mathbf{u}=\nabla_{\Gamma}\cdot\mathbf{u}=0.$

Next by using (2.1), (2.2), (2.5) in (2.4) it is straightforward to see that

$(\varepsilon\nabla_{\Gamma}\varphi-\boldsymbol{\xi})\cdot\partial^{\bullet}\nabla_{\Gamma}\varphi+\left(\sigma+\frac{1}{\varepsilon}F^{\prime}(\varphi)-\mu\right)\partial^{\bullet}\varphi-\left(\mathbb{S}+\nabla_{\Gamma}\varphi\otimes\boldsymbol{\xi}\right):\nabla_{\Gamma}\mathbf{u}+\mathbf{q}\cdot\nabla_{\Gamma}\mu-\mathbf{F}\cdot\mathbf{u}\leq
0,$

where we have also used

$\nabla_{\Gamma}(\partial^{\bullet}\varphi)=\partial^{\bullet}\nabla_{\Gamma}\varphi+(\nabla_{\Gamma}\mathbf{u})^{T}\nabla_{\Gamma}\varphi.$

As noted in [2, 31] the quantity

$-(\varepsilon\nabla_{\Gamma}\varphi-\boldsymbol{\xi})\cdot\partial^{\bullet}\nabla_{\Gamma}\varphi-\left(\sigma+\frac{1}{\varepsilon}F^{\prime}(\varphi)-\mu\right)\partial^{\bullet}\varphi+\left(\mathbb{S}+\nabla_{\Gamma}\varphi\otimes\boldsymbol{\xi}\right):\nabla_{\Gamma}\mathbf{u}-\mathbf{q}\cdot\nabla_{\Gamma}\mu-\mathbf{F}\cdot\mathbf{u}=:\mathcal{D},$

represents the dissipation, and hence the assumed inequality is equivalent to
$\mathcal{D}\geq 0$.

One then argues as in [2, 31] to show that if one allows
$\mathbb{S},\mathbf{q},\boldsymbol{\xi},\sigma$ to depend on
$\varphi,\nabla_{\Gamma}\varphi,\mu,\nabla_{\Gamma}\mu,\mathbb{E}(\mathbf{u})$
arbitrarily then the assumed dissipation inequality can fail to hold. The
argument requires one to assume the presence of general forces and external
mass supplies. In particular one finds that necessarily

$\displaystyle\varepsilon\nabla_{\Gamma}\varphi-\boldsymbol{\xi}=0,\quad\sigma+\frac{1}{\varepsilon}F^{\prime}(\varphi)-\mu=0.$
(2.6)

We then assume, as in [18], that the mass flux takes the form

$\mathbf{q}=-M(\varphi)\nabla_{\Gamma}\mu,$

for some mobility function $M(\cdot)$. Similarly, motivated by Newton’s
rheological law we assume

$\mathbb{S}+\varepsilon\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi=2\eta(\varphi)\mathbb{E}(\mathbf{u}),$

where

$\mathbb{E}(\mathbf{u})=\frac{1}{2}\left(\nabla_{\Gamma}\mathbf{u}+(\nabla_{\Gamma}\mathbf{u})^{T}\right)$

is the rate of strain tensor, and $\eta(\cdot)$ is a variable viscosity
(depending on the concentration). As noted in [2],
$\mathbb{S}+\varepsilon\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi$
represents the change in energy due to friction in the fluids, and is referred
to as the viscous strain tensor. In summary, this allows us to observe that

$\displaystyle\mathbb{T}=\underbrace{-p\mathbb{P}+2\eta(\varphi)\mathbb{E}(\mathbf{v})}_{\text{Newtonian
term}}-\underbrace{\varepsilon\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi}_{\text{Korteweg
term}},$ (2.7)

where the Newtonian term is the Boussinesq-Scriven ansatz seen in [16]. Then
by combining (2.5), (2.6) we find that

$\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi),$

and similarly combining (2.1), and the assumption on $\mathbf{q}$ we obtain

$\partial^{\bullet}\varphi=\nabla_{\Gamma}\cdot(M(\varphi)\nabla_{\Gamma}\mu),$

which are the relations for $\varphi,\mu$ as in [18]. Lastly we observe that

$\nabla_{\Gamma}\cdot(p\mathbb{P})=\nabla_{\Gamma}p-pH\boldsymbol{\nu},$

and hence by using (2.2), (2.7) we see

$\rho\partial^{\bullet}\mathbf{u}=-\nabla_{\Gamma}p+pH\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\mathbf{u}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi)+\mathbf{F}.$

Lastly by recalling that $\nabla_{\Gamma}\cdot\mathbf{u}=0$, we observe that
we have derived (1.1) - (1.4).

To obtain the tangential Navier-Stokes-Cahn-Hilliard system we assume that the
geometric motion of $\Gamma(t)$ is defined by the normal velocity field
$V_{N}\boldsymbol{\nu}$ so material continuity in the normal direction implies
the equation

$\mathbf{u}\cdot\boldsymbol{\nu}=V_{N}.$

To find the unknown $\mathbf{u}_{T}$ one considers the projection of this
momentum equation. For this we note that (as in [45])

$\displaystyle\partial^{\bullet}\varphi=\partial^{\circ}\varphi+\nabla_{\Gamma}\varphi\cdot\mathbf{u}_{T},$
$\displaystyle\mathbb{P}\partial^{\bullet}\mathbf{v}=\mathbb{P}\partial^{\circ}\mathbf{v}_{T}+(\nabla_{\Gamma}\mathbf{v}_{T})\mathbf{u}_{T}+V_{N}\mathbb{H}\mathbf{v}_{T}-\frac{1}{2}\nabla_{\Gamma}V_{N}^{2}.$

Similarly we compute

$\displaystyle\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi))=\Delta_{\Gamma}\varphi\nabla_{\Gamma}\varphi+\frac{1}{2}\nabla_{\Gamma}|\nabla_{\Gamma}\varphi|^{2}-\left(\nabla_{\Gamma}\varphi\cdot\mathbb{H}\nabla_{\Gamma}\varphi\right)\boldsymbol{\nu},$

and by using (1.4) we see

$-\varepsilon\Delta_{\Gamma}\varphi\nabla_{\Gamma}\varphi=\left(\mu-\frac{1}{\varepsilon}F^{\prime}(\varphi)\right)\nabla_{\Gamma}\varphi.$

Thus defining the modified pressure to be

$\tilde{p}=p+\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi),$

and recalling that $\nabla_{\Gamma}\cdot\mathbf{v}_{T}=-HV_{N}$ we obtain the
tangential Navier-Stokes-Cahn-Hilliard equations (1.5) - (1.8). The obvious
modification to the calculations in [45] also yields an equation for the
normal component

$\displaystyle\partial^{\bullet}V_{N}=-2\eta(\varphi)\left(\text{tr}(\mathbb{H}\nabla_{\Gamma}\mathbf{u}_{T})+V_{N}\text{tr}(\mathbb{H}^{2})\right)+\mathbf{u}_{T}\cdot\mathbb{H}\mathbf{u}_{T}-\mathbf{u}_{T}\cdot\nabla_{\Gamma}V_{N}+pH-\varepsilon\nabla_{\Gamma}\varphi\cdot\mathbb{H}\nabla_{\Gamma}\varphi+F_{\nu},$
(2.8)

where $F_{\nu}=\mathbf{F}\cdot\nu$, which must be satisfied and coupled with
(1.5)-(1.8) .

######  Remark 2.1.

(1.1)-(1.4) is a simplified form of the system derived in [11], where the
authors also consider the effects of bending/friction. It is useful to note
that these authors consider the modified pressure throughout. Hence neglecting
the effects of bending/friction terms and changing notation suitably one finds
the two systems are identical.

###  2.2 Derivation by a thin film limit

In this subsection we consider the thin film limit of relevant Navier-Stokes-
Cahn-Hilliard equations on an evolving Cartesian domain. This approach has
been considered for the heat equation [41], and the Navier-Stokes equations
[16, 40]. Besides use in derivation of a suitable system of surface Navier-
Stokes-Cahn-Hilliard equations there has been interest in using a thin film
approximation numerically [51] to study the limiting surface equations.

We assume throughout that $\Gamma(t)$ does not undergo a change in topology.
Indeed, in the presence of a change in topology the modelling of this
phenomenon is different and so one expects the systems (1.1)-(1.4) and
(1.5)-(1.8) won’t necessarily make sense. We discuss this more in Remark 2.5.
We also ignore the effect of the external force, $\mathbf{F}$, for brevity.

As before we still consider a closed oriented evolving surface, $\Gamma(t)$,
with a prescribed normal velocity $V_{N}$. We define $\Omega_{\gamma}(t)$ by

$\Omega_{\gamma}(t):=\\{x\in\mathbb{R}^{3}\mid|d(x,t)|<\gamma\\},$

where $d(x,t)$ is the signed distance function of $\Gamma(t)$, and $\gamma>0$
is sufficiently small. We then define the (noncylindrical) space-time domain

$Q_{\gamma,T}:=\bigcup_{t\in[0,T]}\Omega_{\gamma}(t)\times\\{t\\},$

where we pose our problem. We consider a Navier-Stokes-Cahn-Hilliard system on
$Q_{\gamma,T}$,

$\displaystyle\rho\left(\frac{\partial\mathbf{u}^{\gamma}}{\partial
t}+(\mathbf{u}^{\gamma}\cdot\nabla)\mathbf{u}^{\gamma}\right)=-\nabla
p^{\gamma}+\nabla\cdot(2\eta(\varphi^{\gamma})\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma}))-\varepsilon\nabla\cdot(\nabla\varphi^{\gamma}\otimes\nabla\varphi^{\gamma}),$
(2.9) $\displaystyle\nabla\cdot\mathbf{u}^{\gamma}=0,$ (2.10)
$\displaystyle\frac{\partial\varphi^{\gamma}}{\partial
t}-\nabla\cdot\left(M(\varphi^{\gamma})\nabla\mu^{\gamma}\right)+\mathbf{u}^{\gamma}\cdot\nabla\varphi^{\gamma}=0,$
(2.11)
$\displaystyle\mu^{\gamma}=-\varepsilon\Delta\varphi^{\gamma}+\frac{1}{\varepsilon}F^{\prime}(\varphi^{\gamma}),$
(2.12)

equipped with boundary conditions

$\displaystyle\mathbf{u}^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}=V_{N}^{\gamma},$
(2.13)
$\displaystyle\left[\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma})\boldsymbol{\nu}^{\gamma}\right]_{\text{tan}}=0,$
(2.14) $\displaystyle\nabla\varphi^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}=0,$
(2.15) $\displaystyle\nabla\mu^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}=0,$
(2.16)

on the lateral boundary $\partial_{\ell}Q_{\gamma,T}$ defined as

$\partial_{\ell}Q_{\gamma,T}:=\bigcup_{t\in[0,T]}\partial\Omega_{\gamma}(t)\times\\{t\\}.$

Here we are using the notation
$\mathbb{E}_{\Omega}:=\frac{1}{2}(\nabla\mathbf{u}^{\gamma}+\left(\nabla\mathbf{u}^{\gamma})^{T}\right)$
for the rate of strain tensor in $\Omega_{\gamma}$, and $[\cdot]_{\text{tan}}$
denoting the tangential component to $\partial\Omega_{\gamma}(t)$ of a vector
in $\mathbb{R}^{3}$. The condition (2.14) is sometimes referred to as the
perfect slip condition, and appears as a natural boundary condition. One may
expect different boundary conditions for $\varphi^{\gamma},\mu^{\gamma}$,
similar to the Robin-type condition seen for the heat equation in [41].
However the usual Neumann conditions are still sufficient our setting, and
retain the mass conservation property. To see this we use the Reynolds
transport theorem so that

$\displaystyle\frac{d}{dt}\int_{\Omega_{\gamma}(t)}\varphi^{\gamma}$
$\displaystyle=\int_{\Omega_{\gamma}(t)}\frac{\partial\varphi^{\gamma}}{\partial
t}+\int_{\partial\Omega_{\gamma}(t)}\varphi^{\gamma}V_{N}^{\gamma}$
$\displaystyle=\int_{\Omega_{\gamma}(t)}\left(\Delta\mu^{\gamma}-\mathbf{u}^{\gamma}\cdot\nabla\varphi^{\gamma}\right)+\int_{\partial\Omega_{\gamma}(t)}\varphi^{\gamma}V_{N}^{\gamma}$
$\displaystyle=\int_{\partial\Omega_{\gamma}(t)}\left(\nabla\mu^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}-\varphi^{\gamma}\mathbf{u}^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}+\varphi^{\gamma}V_{N}^{\gamma}\right)$
$\displaystyle=\int_{\partial\Omega_{\gamma}(t)}\nabla\mu^{\gamma}\cdot\boldsymbol{\nu}^{\gamma},$

where we have used (2.13) for the final equality. The Neumann condition for
$\varphi^{\gamma}$ follows in the usual way, without any need for Reynolds
transport theorem. We now expand
$\varphi^{\gamma},\mu^{\gamma},\mathbf{u}^{\gamma},p^{\gamma}$ in terms of the
signed distance function as:

$\displaystyle\varphi^{\gamma}(x,t)=\varphi^{0}(\pi(x,t),t)+d(x,t)\varphi^{1}(\pi(x,t),t)+d(x,t)^{2}\varphi^{2}(\pi(x,t),t)+\mathcal{O}(d(x,t)^{3}),$
$\displaystyle\mu^{\gamma}(x,t)=\mu^{0}(\pi(x,t),t)+d(x,t)\mu^{1}(\pi(x,t),t)+d(x,t)^{2}\mu^{2}(\pi(x,t),t)+\mathcal{O}(d(x,t)^{3}),$
$\displaystyle\mathbf{u}^{\gamma}(x,t)=\mathbf{u}^{0}(\pi(x,t),t)+d(x,t)\mathbf{u}^{1}(\pi(x,t),t)+d(x,t)^{2}\mathbf{u}^{2}(\pi(x,t),t)+\mathcal{O}(d(x,t)^{3}),$
$\displaystyle
p^{\gamma}(x,t)=p^{0}(\pi(x,t),t)+d(x,t)p^{1}(\pi(x,t),t)+d(x,t)^{2}p^{2}(\pi(x,t),t)+\mathcal{O}(d(x,t)^{3}),$

where here $\pi(x,t)$ is the closed point projection of $x$ onto $\Gamma(t)$.
This is uniquely defined on a small tubular neighbourhood,
$\mathcal{N}(\Gamma(t))$, of $\Gamma(t)$. Hence we have a requirement on
$\gamma$ being sufficiently small so that

$Q_{\gamma,T}\subseteq\bigcup_{t\in[0,T]}\mathcal{N}(\Gamma(t))\times\\{t\\}.$

Before considering the thin film limit we recall some preliminary results.
Firstly, we recall that

$\nabla d(x,t)=\boldsymbol{\nu}(\pi(x,t),t),\qquad\frac{\partial
d(x,t)}{\partial t}=-V_{N}(\pi(x,t),t),$

and from these one can show the following results.

######  Lemma 2.2 ([40], Lemma 2.7).

Let $f$ be a scalar of vector valued function on $\mathcal{G}_{T}$. Then the
spatial/temporal derivatives of the composite function $f(\pi(x,t),t)$ are
such that

$\displaystyle\nabla(f(\pi(x,t),t))=\nabla_{\Gamma}f(\pi(x,t),t)+d(x,t)\left(\mathbb{H}\nabla_{\Gamma}f\right)(\pi(x,t),t)+\mathcal{O}(d(x,t)^{2}),$
$\displaystyle\frac{\partial f(\pi(x,t),t)}{\partial
t}=\partial^{\circ}f(\pi(x,t),t)+d(x,t)((\nabla_{\Gamma}V_{N}\cdot\nabla_{\Gamma})f)(\pi(x,t),t)+\mathcal{O}(d(x,t)^{2}),$

for $(x,t)\in Q_{\gamma,T}$.

######  Lemma 2.3 ([40], Lemma 2.8).

Let $\mathbb{S}^{0},\mathbb{S}^{1}$ be $3\times 3$ matrix valued functions on
$\Gamma(t)$ for each $t\in(0,T)$. Then for $x\in\Omega_{\gamma}(t)$ set

$\mathbb{S}(x)=\mathbb{S}^{0}(\pi(x,t))+d(x,t)\mathbb{S}^{1}(\pi(x,t))+\mathcal{O}(d(x,t)^{2}).$

Then we have

$\nabla\cdot\mathbb{S}(x)=\nabla_{\Gamma}\cdot\mathbb{S}^{0}(\pi(x,t),t)+(\mathbb{S}^{1}(\pi(x,t),t))^{T}\boldsymbol{\nu}(\pi(x,t),t)+\mathcal{O}(d(x,t)),$

for $x\in\Omega_{\gamma}(t)$.

######  Theorem 2.4.

Let $(\varphi^{\gamma},\mu^{\gamma},\mathbf{u}^{\gamma},p^{\gamma})$ solve
(2.9)-(2.12) with boundary conditions (2.13)-(2.16). Then
$(\varphi^{0},\mu^{0},\mathbf{u}^{0},p^{0},p^{1})$ from the corresponding
expansion in terms of the signed distance functions solve

$\displaystyle\rho\partial^{\bullet}\mathbf{u}^{0}=-\nabla_{\Gamma}p^{0}+p^{1}\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{0}),$
(2.17) $\displaystyle\nabla_{\Gamma}\cdot\mathbf{v}^{0}=0,$ (2.18)
$\displaystyle\partial^{\bullet}\varphi^{0}=\nabla_{\Gamma}\cdot(M(\varphi^{0})\nabla_{\Gamma}\mu^{0}),$
(2.19)
$\displaystyle\mu^{0}=-\varepsilon\Delta_{\Gamma}\varphi^{0}+\frac{1}{\varepsilon}F^{\prime}(\varphi^{0}),$
(2.20)

on $\mathcal{G}_{T}$, such that $\mathbf{u}^{0}\cdot\boldsymbol{\nu}=V_{N}$.

###### Proof.

We abbreviate $\pi(x,t)$ to $\pi$, and $d(x,t)$ to $d$ throughout this proof.
Firstly by considering (2.13) we have222Here the $\pm\gamma$ corresponds to
the boundary of $\Omega_{\gamma}(t)$ consisting of two disjoint sets,
$\\{x\in\mathbb{R}^{3}\mid d(x,t)=\gamma\\},$ and $\\{x\in\mathbb{R}^{3}\mid
d(x,t)=-\gamma\\}$.

$\mathbf{u}^{0}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)\pm\gamma\mathbf{u}^{1}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)+\gamma^{2}\mathbf{u}^{2}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)+\mathcal{O}(\gamma^{3})=V_{N}(\pi,t),$

and so equating terms of order $\gamma^{k}$ for $k=0,1,2$, one finds that

$\displaystyle\mathbf{u}^{0}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)=V_{N}(\pi,t),$
$\displaystyle\mathbf{u}^{1}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)=0,$
$\displaystyle\mathbf{u}^{2}(\pi,t)\cdot\boldsymbol{\nu}(\pi,t)=0.$

Taking the gradient of $\mathbf{u}^{\gamma}$, we find

$\displaystyle\nabla\mathbf{u}^{\gamma}(x,t)=\nabla_{\Gamma}\mathbf{u}^{0}(\pi,t)+\boldsymbol{\nu}(\pi,t)\otimes\mathbf{u}^{1}(\pi,t)+d\left((\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0})(\pi,t)+\nabla_{\Gamma}\mathbf{u}^{1}(\pi,t)+2(\boldsymbol{\nu}\otimes\mathbf{u}^{2})(\pi,t)\right)+\mathcal{O}(d^{2}),$
(2.21)

and hence taking the trace of the above, using (2.10) and
$\mathbf{u}^{1}\cdot\boldsymbol{\nu}=0$, we obtain

$\nabla\cdot\mathbf{u}^{\gamma}=\nabla_{\Gamma}\cdot\mathbf{u}^{0}+\mathcal{O}(d),$

from which the zeroth order terms yield (2.18). Similar calculations let us
verify that

$\displaystyle\nabla\varphi^{\gamma}(x,t)=\nabla_{\Gamma}\varphi^{0}(\pi,t)+\varphi^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)+d\left((\mathbb{H}\nabla_{\Gamma}\varphi^{0})(\pi,t)+\nabla_{\Gamma}\varphi^{1}(\pi,t)+2(\varphi^{2}\boldsymbol{\nu})(\pi,t)\right)+\mathcal{O}(d^{2}),$
(2.22) $\displaystyle\nabla
p^{\gamma}(x,t)=\nabla_{\Gamma}p^{0}(\pi,t)+p^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)+\mathcal{O}(d).$
(2.23)

Similarly, by considering the transpose of (2.21) one finds

$\displaystyle\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma})(x,t)=\mathbb{S}^{0}(\pi,t)+d\mathbb{S}^{1}(\pi,t)+\mathcal{O}(d^{2}),$
(2.24)

where

$\displaystyle\mathbb{S}^{0}=\mathbb{E}(\mathbf{u}^{0})+\frac{\boldsymbol{\nu}\otimes\mathbf{u}^{1}+\mathbf{u}^{1}\otimes\boldsymbol{\nu}}{2},$
$\displaystyle\mathbb{S}^{1}=\frac{\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0}+(\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0})^{T}}{2}+\mathbb{E}(\mathbf{u}^{1})+\boldsymbol{\nu}\otimes\mathbf{u}^{2}+\mathbf{u}^{2}\otimes\boldsymbol{\nu}.$

Now, by our smoothness assumption on $\eta(\cdot)$ we may use Taylor’s theorem
to write

$\eta(\varphi^{\gamma}(x,t))=\eta(\varphi^{0}(\pi,t))+d\eta^{\prime}(\varphi^{*}(\pi,t))\varphi^{1}(\pi,t)+\mathcal{O}(d^{2}),$

where $\varphi^{*}(\pi,t)$ is some function valued between
$\varphi^{0}(\pi,t)$ and $\varphi^{\gamma}(x,t)$ which arises from the
remainder term in Taylor’s theorem. Using this, we find that

$\eta(\varphi^{\gamma}(x,t))\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma})(x,t)=\widetilde{\mathbb{S}}^{0}(\pi,t)+d\widetilde{S}^{1}(\pi,t)+\mathcal{O}(d^{2}),$

where

$\displaystyle\widetilde{\mathbb{S}}^{0}=\eta(\varphi^{0})\left(\mathbb{E}(\mathbf{u}^{0})+\frac{\boldsymbol{\nu}\otimes\mathbf{u}^{1}+\mathbf{u}^{1}\otimes\boldsymbol{\nu}}{2}\right),$
$\displaystyle\widetilde{\mathbb{S}}^{1}=\eta^{\prime}(\varphi^{*})\left(\mathbb{E}(\mathbf{u}^{0})+\frac{\boldsymbol{\nu}\otimes\mathbf{u}^{1}+\mathbf{u}^{1}\otimes\boldsymbol{\nu}}{2}\right)+\eta(\varphi^{0})\left(\frac{\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0}+(\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0})^{T}}{2}+\mathbb{E}(\mathbf{u}^{1})+\boldsymbol{\nu}\otimes\mathbf{u}^{2}+\mathbf{u}^{2}\otimes\boldsymbol{\nu}\right).$

Hence using Lemma 2.3 we see

$\nabla\cdot(2\eta(\varphi^{\gamma})\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma}))=2\nabla_{\Gamma}\cdot\widetilde{\mathbb{S}}^{0}+2(\widetilde{\mathbb{S}}^{1})^{T}\boldsymbol{\nu}+\mathcal{O}(d),$

and so we check which of these terms vanish. To do this we firstly note that
by rewriting (2.14) one has

$\mathbb{P}(\pi,t)\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma})\boldsymbol{\nu}(\pi,t)=0,\quad
x\in\partial\Omega_{\gamma}(t),$

and so by using (2.24) one finds

$\mathbb{P}(\pi,t){\mathbb{S}}^{0}(\pi,t)\boldsymbol{\nu}(\pi,t)\pm\gamma\mathbb{P}(\pi,t){\mathbb{S}}^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)+\mathcal{O}(\gamma^{2})=0,$

and hence

$\displaystyle\mathbb{P}(\pi,t){\mathbb{S}}^{0}(\pi,t)\boldsymbol{\nu}(\pi,t)=0,$
$\displaystyle\mathbb{P}(\pi,t){\mathbb{S}}^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)=0.$

Then by using the form of $\mathbb{S}^{0}$, and

$(\boldsymbol{\nu}\otimes\mathbf{u}^{1})\boldsymbol{\nu}=0,\quad(\mathbf{u}^{1}\otimes\boldsymbol{\nu}),\quad\mathbb{P}\mathbf{u}^{1}=\mathbf{u}^{1},$

in the above yields

$\mathbf{u}^{1}=-2\mathbb{P}(\pi,t)\mathbb{E}(\mathbf{u}^{0})\boldsymbol{\nu}(\pi,t).$

Thus we find

$\displaystyle\widetilde{\mathbb{S}}^{0}$
$\displaystyle=\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0})-\eta(\varphi^{0})(\boldsymbol{\nu}\otimes\boldsymbol{\nu})\mathbb{E}(\mathbf{u}^{0})\mathbb{P}-\eta(\varphi^{0})\mathbb{P}\mathbb{E}(\mathbf{u}^{0})(\boldsymbol{\nu}\otimes\boldsymbol{\nu})$
$\displaystyle=\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0})-\eta(\varphi^{0})(\boldsymbol{\nu}\otimes\boldsymbol{\nu})\mathbb{E}(\mathbf{u}^{0})(\boldsymbol{\nu}\otimes\boldsymbol{\nu})$
$\displaystyle=\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0}),$

where we have used
$\mathbb{P}=\mathbb{I}-(\boldsymbol{\nu}\otimes\boldsymbol{\nu})$ and
$\nabla_{\Gamma}\mathbf{u}^{0}=\mathbb{P}\nabla_{\Gamma}\mathbf{u}^{0}\mathbb{P}$.
Hence we find

$\nabla_{\Gamma}\cdot\widetilde{\mathbb{S}}^{0}=\nabla_{\Gamma}\cdot(\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0})).$

We now show that $(\widetilde{\mathbb{S}}^{1})^{T}\boldsymbol{\nu}=0.$
Firstly, we notice that from the above calculations it is clear that

$(\widetilde{\mathbb{S}}^{1})^{T}\boldsymbol{\nu}=\eta^{\prime}(\varphi^{*})\mathbb{E}(\mathbf{u}^{0})\boldsymbol{\nu}+\eta(\varphi^{0})\left(\frac{\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0}\boldsymbol{\nu}+(\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0})^{T}\boldsymbol{\nu}}{2}+\mathbb{E}(\mathbf{u}^{1})\boldsymbol{\nu}+(\boldsymbol{\nu}\otimes\mathbf{u}^{2})\boldsymbol{\nu}+(\mathbf{u}^{2}\otimes\boldsymbol{\nu})\boldsymbol{\nu}\right).$

Recalling that

$\mathbb{E}(\mathbf{u}^{0})\boldsymbol{\nu}=0=\mathbb{E}(\mathbf{u}^{1})\boldsymbol{\nu},\quad\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0}\boldsymbol{\nu}=0=(\mathbb{H}\nabla_{\Gamma}\mathbf{u}^{0})^{T}\boldsymbol{\nu},\quad(\boldsymbol{\nu}\otimes\mathbf{u}^{2})\boldsymbol{\nu}=0,\quad(\mathbf{u}^{2}\otimes\boldsymbol{\nu})\boldsymbol{\nu}=\mathbf{u}^{2},$

where we have used $\boldsymbol{\nu}\in\ker(\mathbb{H})$, and
$\mathbf{u}^{2}\cdot\boldsymbol{\nu}=0,$ we find

$(\widetilde{\mathbb{S}}^{1})^{T}\boldsymbol{\nu}=\eta(\varphi^{0})\mathbf{u}^{2}.$

Then using the form of $\mathbb{S}^{1}$ in
$\mathbb{P}(\pi,t){\mathbb{S}}^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)=0$ one finds
that $\mathbf{u}^{2}=0$, and so
$(\widetilde{\mathbb{S}}^{1})^{T}\boldsymbol{\nu}=0$. Hence

$\displaystyle\nabla\cdot(2\eta(\varphi^{\gamma})\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma}))=\nabla_{\Gamma}\cdot(2\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0}))+\mathcal{O}(d).$
(2.25)

The tensor product involving $\nabla\varphi^{\gamma}$ is dealt with similarly,
where it is straightforward to see that

$\nabla\varphi^{\gamma}\otimes\nabla\varphi^{\gamma}=\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{0}+d(\nabla_{\Gamma}\varphi^{1}\otimes\nabla_{\Gamma}\varphi^{0}+\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{1})+\mathcal{O}(d^{2}),$

and hence by using Lemma 2.3

$\nabla\cdot(\nabla\varphi^{\gamma}\otimes\nabla\varphi^{\gamma})=\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{0})+(\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{1}+\nabla_{\Gamma}\varphi^{1}\otimes\nabla_{\Gamma}\varphi^{0})\boldsymbol{\nu}+\mathcal{O}(d),$

where the latter term clearly vanishes as
$\nabla_{\Gamma}\varphi^{0},\nabla_{\Gamma}\varphi^{1}$ are tangential. Thus
one obtains

$\displaystyle\varepsilon\nabla\cdot(\nabla\varphi^{\gamma}\otimes\nabla\varphi^{\gamma})=\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{0})+\mathcal{O}(d).$
(2.26)

For the momentum equation, all that remains to discuss is the time derivative
and the advective term. The time derivative follows immediately from Lemma 2.2
as

$\displaystyle\frac{\partial\mathbf{u}^{\gamma}}{\partial
t}(x,t)=\partial^{\circ}\mathbf{u}^{0}(\pi,t)-V_{N}(\pi,t)\mathbf{u}^{1}(\pi,t)+\mathcal{O}(d),$
(2.27)

which we want to turn into a material time derivative by considering the
advective term. For the advective term we write
$(\mathbf{u}^{\gamma}\cdot\nabla)\mathbf{u}^{\gamma}=(\mathbf{u}^{\gamma})^{T}\nabla\mathbf{u}^{\gamma}.$
Hence from the expansion of $\mathbf{u}^{\gamma}$ and using Lemma 2.2 one
finds

$(\mathbf{u}^{\gamma})^{T}\nabla\mathbf{u}^{\gamma}=(\mathbf{u}^{0})^{T}\nabla_{\Gamma}\mathbf{u}^{0}+\mathbf{u}^{1}(\mathbf{u}^{0}\cdot\boldsymbol{\nu})+\mathcal{O}(d)=(\mathbf{u}^{0}\cdot\nabla_{\Gamma})\mathbf{u}^{0}+V_{N}\mathbf{u}^{1}+\mathcal{O}(d).$

Hence using this expression for the advection with (2.27) one finds

$\displaystyle\rho\left(\frac{\partial\mathbf{u}^{\gamma}}{\partial
t}+(\mathbf{u}^{\gamma}\cdot\nabla)\mathbf{u}^{\gamma}\right)=\rho\left(\partial^{\circ}\mathbf{u}^{0}+(\mathbf{u}^{0}\cdot\nabla_{\Gamma})\mathbf{u}^{0}\right)+\mathcal{O}(d)=\rho\partial^{\bullet}\mathbf{u}^{0}+\mathcal{O}(d),$
(2.28)

where here we understand $\partial^{\bullet}$ to mean the derivative along the
velocity field given by $V_{N}\boldsymbol{\nu}+\mathbf{u}^{0}$ \- that is the
tangential velocity is only considered up to the zeroth order term. This point
is made clearer by the notation of [40] where one would write this as
$\partial^{\bullet}_{\mathbf{u}^{0}}$. Now by combining (2.23), (2.25),
(2.26), (2.28) in (2.9) one obtains

$\rho\partial^{\bullet}\mathbf{u}^{0}=-\nabla_{\Gamma}p^{0}+p^{1}\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi^{0})\mathbb{E}(\mathbf{u}^{0}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi^{0}\otimes\nabla_{\Gamma}\varphi^{0})+\mathcal{O}(d),$

and as the functions $(\varphi^{0},\mu^{0},\mathbf{u}^{0},p^{0},p^{1})$ are
independent of $d$ one obtains (2.17) from the zeroth order terms.

It remains to show that (2.11),(2.12) give (2.19), (2.20) at zeroth order.
This is largely the same, so we skim the details. The advective term in (2.11)
is the main point of interest here. Using the expansions for
$\varphi^{\gamma},\mathbf{u}^{\gamma}$ and Lemma 2.2 one finds

$\displaystyle\mathbf{u}^{\gamma}\cdot\nabla\varphi^{\gamma}=\mathbf{u}^{0}\cdot\nabla_{\Gamma}\varphi^{0}+\mathbf{u}^{0}\cdot\boldsymbol{\nu}\varphi^{1}+\mathcal{O}(d).$
(2.29)

For the time derivative one uses Lemma 2.2 as before so that

$\frac{\partial\varphi^{\gamma}}{\partial
t}(x,t)=\partial^{\circ}\varphi^{0}(\pi,t)-V_{N}(\pi,t)\varphi^{1}(\pi,t)+\mathcal{O}(d).$

We combine this with (2.29), recalling that
$\mathbf{u}^{0}\cdot\boldsymbol{\nu}=V_{N}$, so that

$\displaystyle\frac{\partial\varphi^{\gamma}}{\partial
t}+\mathbf{u}^{\gamma}\cdot\nabla\varphi^{\gamma}=\partial^{\circ}\varphi^{0}+\mathbf{u}^{0}\cdot\nabla_{\Gamma}\varphi^{0}+\mathcal{O}(d)=\partial^{\bullet}\varphi^{0}+\mathcal{O}(d).$
(2.30)

It remains to consider the term $\nabla\cdot(M(\varphi^{\gamma})\nabla\mu)$,
which is dealt with almost identically to the term
$\nabla\cdot(2\eta(\varphi^{\gamma})\mathbb{E}_{\Omega}(\mathbf{u}^{\gamma}))$.
From Lemma 2.2 we see

$\nabla\mu^{\gamma}(x,t)=\nabla_{\Gamma}\mu^{0}(\pi,t)+\mu^{1}(\pi,t)\boldsymbol{\nu}(\pi,t)+d\left(\nabla_{\Gamma}\mu^{1}(\pi,t)+2\mu^{2}(\pi,t)\boldsymbol{\nu}(\pi,t)\right)+\mathcal{O}(d^{2}),$

and by Taylor’s theorem (assuming $M(\cdot)$ is sufficiently smooth)

$M(\varphi^{\gamma}(x,t))=M(\varphi^{0}(\pi,t))+dM^{\prime}(\varphi^{*}(\pi,t))\varphi^{1}(\pi,t)+\mathcal{O}(d^{2}),$

where we have abused notation and reused the $\varphi^{*}$ for the
intermediate point arising in the remainder - which is different from the
$\varphi^{*}$ before, but this does not matter. Combining these we find

$M(\varphi^{\gamma})\nabla\mu^{\gamma}=M(\varphi^{0})\nabla_{\Gamma}\mu^{0}+M(\varphi^{0})\mu^{1}\boldsymbol{\nu}+d\left(M^{\prime}(\varphi^{*})\varphi^{1}\nabla_{\Gamma}\mu^{0}+M^{\prime}(\varphi^{*})\varphi^{1}\mu^{1}\boldsymbol{\nu}+M(\varphi^{0})\nabla_{\Gamma}\mu^{1}+2M(\varphi^{0})\mu^{2}\boldsymbol{\nu}\right)\\\
+\mathcal{O}(d^{2}).$

Considering the gradient of this expression one finds

$\nabla(M(\varphi^{\gamma})\nabla\mu^{\gamma})=\nabla_{\Gamma}(M(\varphi^{0})\nabla_{\Gamma}\mu^{0})+\nabla_{\Gamma}(M(\varphi^{0})\mu^{1})\otimes\boldsymbol{\nu}+M(\varphi^{0})\mu^{1}\mathbb{H}+M^{\prime}(\varphi^{*})\varphi^{1}\boldsymbol{\nu}\otimes\nabla_{\Gamma}\mu^{0}\\\
+M^{\prime}(\varphi^{*})\varphi^{1}\mu^{1}\boldsymbol{\nu}\otimes\boldsymbol{\nu}+M(\varphi^{0})\boldsymbol{\nu}\otimes\nabla_{\Gamma}\mu^{1}+2M(\varphi^{0})\mu^{2}\boldsymbol{\nu}\otimes\boldsymbol{\nu}+\mathcal{O}(d),$

and taking the trace of the above yields

$\nabla\cdot(M(\varphi^{\gamma})\nabla\mu^{\gamma})=\nabla_{\Gamma}\cdot(M(\varphi^{0})\nabla_{\Gamma}\mu^{0})+\nabla_{\Gamma}(M(\varphi^{0})\mu^{1})\cdot\boldsymbol{\nu}+M(\varphi^{0})\mu^{1}H+M^{\prime}(\varphi^{*})\varphi^{1}\boldsymbol{\nu}\cdot\nabla_{\Gamma}\mu^{0}\\\
+M^{\prime}(\varphi^{*})\varphi^{1}\mu^{1}\boldsymbol{\nu}\cdot\boldsymbol{\nu}+M(\varphi^{0})\boldsymbol{\nu}\cdot\nabla_{\Gamma}\mu^{1}+2M(\varphi^{0})\mu^{2}\boldsymbol{\nu}\cdot\boldsymbol{\nu}+\mathcal{O}(d),$

which simplifies to

$\displaystyle\nabla\cdot(M(\varphi^{\gamma})\nabla\mu^{\gamma})=\nabla_{\Gamma}\cdot(M(\varphi^{0})\nabla_{\Gamma}\mu^{0})+M(\varphi^{0})\mu^{1}H+M^{\prime}(\varphi^{*})\varphi^{1}\mu^{1}+2M(\varphi^{0})\mu^{2}+\mathcal{O}(d).$
(2.31)

Now from the boundary condition (2.16) one finds

$0=\nabla\mu^{\gamma}\cdot\boldsymbol{\nu}^{\gamma}=\nabla_{\Gamma}\mu^{0}\cdot\boldsymbol{\nu}+\mu^{1}\boldsymbol{\nu}\cdot\boldsymbol{\nu}\pm\gamma\left(\nabla_{\Gamma}\mu^{1}\cdot\boldsymbol{\nu}+2\mu^{2}\boldsymbol{\nu}\cdot\boldsymbol{\nu}\right)+\mathcal{O}(\gamma^{2}).$

Thus equating terms of the same order in $\gamma$ one finds

$\displaystyle\nabla_{\Gamma}\mu^{0}\cdot\boldsymbol{\nu}+\mu^{1}=0,$
$\displaystyle\nabla_{\Gamma}\mu^{1}\cdot\boldsymbol{\nu}+2\mu^{2}=0,$

from which one concludes $\mu^{1}=0=\mu^{2}$ and hence (2.31) becomes

$\displaystyle\nabla\cdot(M(\varphi^{\gamma})\nabla\mu^{\gamma})=\nabla_{\Gamma}\cdot(M(\varphi^{0})\nabla_{\Gamma}\mu^{0})+\mathcal{O}(d).$
(2.32)

By combining (2.30), (2.32) in (2.11) one obtains (2.19) by considering the
zeroth order terms.

We skip the derivation of (2.20) from (2.12) and (2.15) as it follows the same
arguments as we have used so far. The only point worth mentioning is that we
assume the potential $F(\cdot)$ is $C^{2}$ so that one may indeed use Taylor’s
theorem as we have for previous terms. ∎

######  Remark 2.5.

  1. 1.

The key difference between the systems (1.1)-(1.4) and (2.17)-(2.20) is that
the latter system has two Lagrange multipliers to be determined, $p^{0}$ which
is understood as enforcing the divergence free constraint, and $p^{1}$ which
is understood as enforcing the normal velocity constraint. This difference is
the same as observed in a comparison of various derivations of the evolving
surface Navier-Stokes equations in [16]. Moreover, this only occurs in the
normal direction and has no bearing on our following analysis of the
tangential system (1.5)-(1.8).

  2. 2.

As remarked in [40] the equations (2.10) and (2.18) imply333Here we are using
$|\Omega_{\gamma}(t)|$ to denote the $\mathscr{L}^{3}$ Lebesgue measure of
$\Omega_{\gamma}(t)$, and $|\Gamma(t)|$ the $\mathcal{H}^{2}$ Hausdorff
measure of $\Gamma(t)$. that

$\displaystyle\frac{d}{dt}|\Omega_{\gamma}(t)|=0,$
$\displaystyle\frac{d}{dt}|\Gamma(t)|=0,$

respectively. However, one may also use a corollary of the coarea formula (see
[27] for example) proven in [40], Appendix A, to see that

$|\Omega_{\gamma}(t)|=\int_{\Omega_{\gamma}(t)}1=\int_{-\gamma}^{\gamma}\int_{\Gamma(t)}J(t;x,r),$

where $J(t;x,r)$ is a corresponding Jacobian of the form

$J(t;x,r)=1-rH(t;x)+r^{2}K(t;x),$

where $K(t;x)=\det(\mathbb{H}(t;x))$ is the Gaussian curvature of $\Gamma(t)$.
From this one finds

$|\Omega_{\gamma}(t)|=2\gamma|\Gamma(t)|+\frac{2\gamma^{3}}{3}\int_{\Gamma(t)}K(t;x),$

where one finds the mean curvature term vanishes by using the divergence
theorem and the fact that $\Gamma(t)$ is closed. Hence assuming that
$\frac{d}{dt}|\Gamma(t)|=0$ is not sufficient for the existence of a solution
to (2.17)-(2.20), one also requires

$\frac{d}{dt}\int_{\Gamma(t)}K(t;x)=0.$

As seen for the evolving surface Euler, and Navier-Stokes equations in [40]
this can be assured by imposing that $\Gamma(t)$ does not change its topology
as by the Gauss-Bonnet theorem, see [35],

$\int_{\Gamma(t)}K(t;x)=2\pi\chi(\Gamma(t)),$

where $\chi(\Gamma(t))$ is the Euler characteristic of $\Gamma(t)$.

###  2.3 Equivalence of the derived systems

In this subsection we discuss the equivalence of the systems (1.5)-(1.8) and
(2.17)-(2.20). Firstly, when considering (1.5)-(1.8), one notes that in
prescribing the normal velocity one does not have to solve (2.8) but this
equation must still be resolved for the normal component to be given by the
prescribed velocity. Hence there must be some normal force
$F_{\nu}\boldsymbol{\nu}$ such that the normal velocity one would obtain from
(2.8) is the prescribed normal velocity $V_{N}$. With this in mind the form of
(1.1)-(1.4) with a prescribed normal velocity becomes

$\displaystyle\rho\partial^{\bullet}\mathbf{u}=-\nabla_{\Gamma}p+pH\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\mathbf{u}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi)+F_{\nu}\boldsymbol{\nu},$
(2.33) $\displaystyle\nabla_{\Gamma}\cdot\mathbf{v}=0,$ (2.34)
$\displaystyle\partial^{\bullet}\varphi=\nabla_{\Gamma}\cdot(M(\varphi)\nabla_{\Gamma}\mu),$
(2.35)
$\displaystyle\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi).$
(2.36)

It is clear to see that one must have

$\displaystyle
F_{\nu}=\partial^{\bullet}V_{N}+2\eta(\varphi)\left(\text{tr}(\mathbb{H}\nabla_{\Gamma}\mathbf{u}_{T})-V_{N}\text{tr}(\mathbb{H}^{2})\right)-\mathbf{u}_{T}\cdot\mathbb{H}\mathbf{u}_{T}+\mathbf{u}_{T}\cdot\nabla_{\Gamma}V_{N}-pH+\varepsilon\nabla_{\Gamma}\varphi\cdot\mathbb{H}\nabla_{\Gamma}\varphi,$
(2.37)

which one can find directly from after solving (1.5)-(1.8).

Similarly, by considering the normal component of (2.17)-(2.20) one finds (up
to a change of notation)

$\displaystyle
p^{1}=\partial^{\bullet}V_{N}+2\eta(\varphi)\left(\text{tr}(\mathbb{H}\nabla_{\Gamma}\mathbf{u}_{T})-V_{N}\text{tr}(\mathbb{H}^{2})\right)-\mathbf{u}_{T}\cdot\mathbb{H}\mathbf{u}_{T}+\mathbf{u}_{T}\cdot\nabla_{\Gamma}V_{N}+\varepsilon\nabla_{\Gamma}\varphi\cdot\mathbb{H}\nabla_{\Gamma}\varphi=F_{\nu}+pH.$

Thus one finds that we may express (2.33)-(2.36) as

$\displaystyle\rho\partial^{\bullet}\mathbf{u}=-\nabla_{\Gamma}p+p^{1}\boldsymbol{\nu}+\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\mathbf{u}))-\varepsilon\nabla_{\Gamma}\cdot(\nabla_{\Gamma}\varphi\otimes\nabla_{\Gamma}\varphi),$
$\displaystyle\nabla_{\Gamma}\cdot\mathbf{v}=0,$
$\displaystyle\partial^{\bullet}\varphi=\nabla_{\Gamma}\cdot(M(\varphi)\nabla_{\Gamma}\mu),$
$\displaystyle\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi),$

which is precisely the form of (2.17)-(2.20) from the thin film limit (up to a
change of notation). This equivalence will also hold in the presence of some
tangential force, $\mathbf{F}_{T}$.

##  3 Notation, Function Spaces and Inequalities

###  3.1 Notation

Next we introduce some notation which will be used throughout. For a
$\mathcal{H}^{2}-$measurable set, $X\subset\mathbb{R}^{3}$ and a function
$f\in L^{1}(X)$, we denote the $\mathcal{H}^{2}$ measure of $X$ and the mean
value of $f$ on $X$ by

$|X|:=\mathcal{H}^{2}(X),\quad\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{X}f:=\frac{1}{|X|}\int_{X}f.$

The components of the tangential gradient are denoted by

$\nabla_{\Gamma}\phi=(\underline{D}_{1}\phi,\underline{D}_{2}\phi,\underline{D}_{3}\phi).$

The (scalar) Sobolev spaces on $\Gamma(t)$ are defined by

$\displaystyle H^{k,p}(\Gamma(t))=\left\\{\phi\in
L^{p}(\Gamma(t)\mid\underline{D}_{i}\phi\in H^{k-1,p}(\Gamma(t)),\
i=1,2,3\right\\},$

and $H^{0,p}(\Gamma(t)):=L^{p}(\Gamma(t)).$ We refer the reader to [22] for
further details. We also use the following notation for tangential vector-
valued Sobolev spaces,

$\displaystyle\mathbf{L}^{p}(\Gamma(t))=\left\\{\boldsymbol{\phi}\in
L^{p}(\Gamma(t))^{3}\mid\boldsymbol{\phi}\cdot\boldsymbol{\nu}=0\text{ almost
everywhere}\right\\},$
$\displaystyle\mathbf{H}^{k,p}(\Gamma(t))=\left\\{\boldsymbol{\phi}\in\mathbf{L}^{p}(\Gamma(t)\mid\underline{D}_{i}\boldsymbol{\phi}\in\mathbf{H}^{k-1,p}(\Gamma(t))\
i=1,2,3\right\\},$

where $\underline{D}_{i}\boldsymbol{\phi}$ denotes the ‘$i$’th column of
$\nabla_{\Gamma}\boldsymbol{\phi}$, and
$\mathbf{H}^{0,p}(\Gamma(t)):=\mathbf{L}^{p}(\Gamma(t))$. As is standard, in
the case $p=2$ we omit the $p$, and write $\mathbf{H}^{k}(\Gamma(t))$.
Similarly we write $\mathbf{H}^{-1}(\Gamma(t))$ for the dual space of
$\mathbf{H}^{1}(\Gamma(t))$.

###  3.2 Pushforward map and compatible time dependent spaces

From our assumptions we obtain the existence of a $C^{3}$ diffeomorphism

$\Phi_{t}^{n}:\Gamma_{0}\rightarrow\Gamma(t),$

which is defined as $\Phi_{t}^{n}(x_{0})=x(t)$, where $x(t)$ solves

$\frac{dx}{dt}=V_{N}(x(t),t)\boldsymbol{\nu}(x(t),t),\quad x(0)=x_{0}.$

We denote the corresponding inverse as
$\Phi_{-t}^{n}:\Gamma(t)\rightarrow\Gamma_{0}$. We then may use the framework
established in [6, 7], where we use the normal pushforward map defined by
$\Phi_{t}^{n}\phi=\phi\circ\Phi_{t}^{n}$, and the pullback
$\Phi_{-t}^{n}\psi=\psi\circ\Phi_{-t}^{n}$, for some functions $\phi,\psi$ on
$\Gamma_{0}$ and $\Gamma(t)$ respectively.

In can then be shown that we have compatibility of the pairs
$(H^{k,p}(\Gamma(t)),\Phi_{t}^{n})$,
$(\mathbf{H}^{k,p}(\Gamma(t)),\Phi_{t}^{n})$ for $k=0,1,2$, $p\in[1,\infty]$,
in the sense of [6]. However, a known issue with the associated pushforward
map is that is doesn’t necessarily preserve the divergence free properties of
solenoidal vector fields on $\Gamma_{0}$. We remedy this by using the Piola
transform as in [24, 45].

###  3.3 The Piola transform and time differentiation

It is clear that the differentials
$D\Phi_{t}^{n}(p):T_{p}\Gamma_{0}\rightarrow T_{\Phi_{t}^{n}(p)}\Gamma(t)$ are
invertible. We introduce the notation $J(p,t)=\det(D\Phi_{t}^{n}(p))$,
$J^{-1}(x,t)=\det(D\Phi_{-t}^{n}(x))=J(t,\Phi_{-t}^{n}x)^{-1}$,
$\mathbb{D}(p,t)=D\Phi_{t}^{n}(p))\mathbb{P}(p,0)$, and
$\mathbb{D}^{-1}(x,t)=D\Phi_{-t}^{n}(x))\mathbb{P}(x,t)$. These matrices are
such that $\mathbb{D}\mathbb{D}^{-1}=\mathbb{D}^{-1}\mathbb{D}=\mathbb{P}.$
One then defines the operator

$\mathbb{A}(p,t):=J^{-1}(\Phi_{t}^{n}(p),t)\mathbb{D}(p,t)+\boldsymbol{\nu}(\Phi_{t}^{n}(p),t)\otimes\boldsymbol{\nu}(p,0),$

for $p\in\Gamma_{0}$, $t\in[0,T]$. One can readily observes that

$\mathbb{A}(p,t)|_{T_{p}\Gamma_{0}}:T_{p}\Gamma_{0}\rightarrow
T_{\Phi_{t}^{n}(p)}\Gamma(t),\quad\mathbb{A}(p,t)|_{T_{p}\Gamma_{0}}:T_{p}\Gamma_{0}^{\perp}\rightarrow
T_{\Phi_{t}^{n}(p)}\Gamma(t)^{\perp}.$

We then define the Piola pushforward map, for a vector field
$\boldsymbol{\psi}$ on $\Gamma_{0}$ as

$\mathcal{P}_{t}\boldsymbol{\psi}(x)=\mathbb{A}(\Phi_{-t}^{n}(x),t)\boldsymbol{\psi}(\Phi_{-t}^{n}(x)),$

where it is known that for some sufficiently smooth, tangential vector field,
$\boldsymbol{\psi}$, on $\Gamma_{0}$, then
$\nabla_{\Gamma}\cdot\boldsymbol{\psi}=0$ almost everywhere $\Gamma_{0}$ if,
and only if, $\nabla_{\Gamma}\cdot\mathcal{P}_{t}\boldsymbol{\psi}=0$ almost
everywhere on $\Gamma(t)$.

One similarly defines an inverse operator $\mathbb{A}^{-1}$ by

$\mathbb{A}^{-1}(x,t):=J(\Phi_{-t}^{n}(x),t)\mathbb{D}^{-1}(x,t)+\boldsymbol{\nu}(\Phi_{-t}^{n}(x),0)\otimes\boldsymbol{\nu}(x,t),$

for $x\in\Gamma(t)$, $t\in[0,T]$. The Piola pullback is defined as one would
expect,

$\mathcal{P}_{-t}\boldsymbol{\psi}(p)=\mathbb{A}^{-1}(\Phi_{t}^{n}(p),t)\boldsymbol{\psi}(\Phi_{t}^{n}(p)),$

for a vector field $\boldsymbol{\psi}$ on $\Gamma(t)$. As one would expect
(and hope) this is such that $\nabla_{\Gamma}\cdot\boldsymbol{\psi}=0$ almost
everywhere on $\Gamma(t)$ if, and only if,
$\nabla_{\Gamma}\cdot\mathcal{P}_{-t}\boldsymbol{\psi}=0$ almost everywhere on
$\Gamma_{0}$. We now recall the following result.

######  Lemma 3.1 ([45], Lemma 3.1).

We have that $\mathbb{D},\mathbb{A}\in C^{2}(\Gamma_{0}\times[0,T])$, and
$\mathbb{D}^{-1},\mathbb{A}^{-1}\in C^{2}(\mathcal{G}_{T})$, and are hence
uniformly bounded in space and time.

We note that we have improved regularity compared to the result in [45], as we
assume $\Phi_{t}^{n}$ are $C^{3}$-diffeomorphisms instead of $C^{2}$.

This result is then used to show compatibility, in the sense of [7], of the
pairs $(\mathbf{H}^{k}(t),\mathcal{P}_{t})$ ($k=0,1,2$), and the divergence
free space $(\mathbf{V}_{\sigma}(t),\mathcal{P}_{t})$, which we discuss later.
We refer the reader to [45] for details. With this compatibility of spaces in
hand one can refer to a derivative associated to the Piola transform
pushforward/pullback maps, as in the sense of [7], defined by

$\partial^{*}\boldsymbol{\psi}=\mathcal{P}_{t}\left(\frac{d}{dt}\mathcal{P}_{-t}\boldsymbol{\psi}\right).$

We refer to this as the strong Piola derivative. The utility of this choice of
derivative is that for a sufficiently smooth vector field,
$\boldsymbol{\psi}$, on $\mathcal{G}_{T}$ we have

$\boldsymbol{\psi}(t)\cdot\boldsymbol{\nu}=0\Rightarrow\partial^{*}\boldsymbol{\psi}\cdot\boldsymbol{\nu}=0,\qquad\nabla_{\Gamma}\cdot\boldsymbol{\psi}(t)=0\Rightarrow\nabla_{\Gamma}\cdot\partial^{*}\boldsymbol{\psi}(t)=0.$

The corresponding weak Piola derivative is defined in the same way as the weak
material/normal derivative (see [7]). The normal derivative,
$\partial^{\circ}$, and the Piola derivative, $\partial^{*}$, are related
through the following.

######  Lemma 3.2 ([45], Lemma 3.6).

For $\boldsymbol{\psi}\in\mathcal{D}$ we have

$\displaystyle\partial^{\circ}\boldsymbol{\psi}=\partial^{*}\boldsymbol{\psi}-\mathbb{A}(\partial^{\circ}\mathbb{A}^{-1})\boldsymbol{\psi},$
(3.1)
$\displaystyle\mathbb{P}\partial^{\circ}\boldsymbol{\psi}=\partial^{*}\boldsymbol{\psi}-\mathbb{A}\mathbb{P}(\partial^{\circ}\mathbb{A}^{-1})\boldsymbol{\psi}.$
(3.2)

From here on we define
$\bar{\mathbb{A}}:=\mathbb{A}\mathbb{P}(\partial^{\circ}\mathbb{A}^{-1})\in
C^{1}(\mathcal{G}_{T})$.

###  3.4 Some bilinear and trilinear forms

Here we introduce some bilinear/trilinear forms to be used in our weak
formulation later.

$\displaystyle m(t;\phi,\psi)=\int_{\Gamma(t)}\phi\psi,$
$\displaystyle\mathbf{m}(t;\boldsymbol{\phi},\boldsymbol{\psi})=\int_{\Gamma(t)}\boldsymbol{\phi}\cdot\boldsymbol{\psi},$
$\displaystyle
m_{*}(t;\Lambda,\phi)=\langle\Lambda,\phi\rangle_{H^{-1}(\Gamma(t),H^{1}(\Gamma(t)))},$
$\displaystyle\mathbf{m}_{*}(t;\boldsymbol{\Lambda},\boldsymbol{\phi})=\langle\boldsymbol{\Lambda},\boldsymbol{\phi}\rangle_{\mathbf{V}_{\sigma}(t)^{\prime},\mathbf{V}_{\sigma}(t)},$
$\displaystyle
a(t;\phi,\psi)=\int_{\Gamma(t)}\nabla_{\Gamma}\phi\cdot\nabla_{\Gamma}\psi,$
$\displaystyle\mathbf{a}(t;\boldsymbol{\phi},\boldsymbol{\psi})=2\int_{\Gamma(t)}\mathbb{E}(\boldsymbol{\phi}):\mathbb{E}(\boldsymbol{\psi}),$
$\displaystyle\hat{\mathbf{a}}(t;\phi,\boldsymbol{\psi},\boldsymbol{\chi})=2\int_{\Gamma(t)}\phi\mathbb{E}(\boldsymbol{\psi}):\mathbb{E}(\boldsymbol{\chi}),$
$\displaystyle\mathbf{c}_{1}(t;\boldsymbol{\phi},\boldsymbol{\psi},\boldsymbol{\chi})=\int_{\Gamma(t)}(\nabla_{\Gamma}\boldsymbol{\phi})\boldsymbol{\psi}\cdot\boldsymbol{\chi},$
$\displaystyle\mathbf{c}_{2}(t;\phi,\psi,\boldsymbol{\chi})=\int_{\Gamma(t)}\phi\nabla_{\Gamma}\psi\cdot\boldsymbol{\chi},$
$\displaystyle\mathbf{l}(t;\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}(t;V_{N}\mathbb{H}\boldsymbol{\phi},\boldsymbol{\psi}),$
$\displaystyle\mathbf{d}_{1}(t;\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{c}_{1}(t;\boldsymbol{\phi},\widetilde{\mathbf{u}_{T}},\boldsymbol{\psi})+\mathbf{c}_{1}(t;\widetilde{\mathbf{u}_{T}},\boldsymbol{\phi},\boldsymbol{\psi}),$
$\displaystyle\mathbf{d}_{2}(t;\phi,\boldsymbol{\psi})=\hat{\mathbf{a}}(t;\phi,\boldsymbol{\widetilde{\mathbf{u}_{T}}},\boldsymbol{\psi}),$

for sufficiently smooth scalar functions $\phi,\psi$, vector functions
$\boldsymbol{\phi},\boldsymbol{\psi},\boldsymbol{\chi}$, and linear
functionals $\Lambda\in
H^{-1}(\Gamma(t)),\boldsymbol{\Lambda}\in\mathbf{V}_{\sigma}(t)^{\prime}$. We
will omit the $t$ argument throughout, as above. We note the following
antisymmetry properties of the trilinear forms
$\mathbf{c}_{1},\mathbf{c}_{2}$, which one can readily verify by using the
divergence theorem,

$\displaystyle\mathbf{c}_{1}(\boldsymbol{\phi},\boldsymbol{\phi},\boldsymbol{\chi})=-\mathbf{c}_{1}(\boldsymbol{\chi},\boldsymbol{\phi},\boldsymbol{\phi}),$
$\displaystyle\mathbf{c}_{1}(\boldsymbol{\phi},\boldsymbol{\phi},\boldsymbol{\phi})=0,$
$\displaystyle\mathbf{c}_{2}(\phi,\psi,\boldsymbol{\chi})=-\mathbf{c}_{2}(\psi,\phi,\boldsymbol{\chi}),$

for $\phi,\psi\in H^{1}(\Gamma(t))$ and
$\boldsymbol{\phi},\boldsymbol{\chi}\in\mathbf{V}_{\sigma}(t)$. We will use
these throughout.

We relate some of these bilinear forms to the normal derivatives using the
transport theorem.

######  Lemma 3.3 (Transport theorem).

  1. 1.

Let $\phi,\psi\in H^{1}_{H^{-1}}\cap L^{2}_{L^{2}}$, then

$\frac{d}{dt}m(\phi,\psi)=m_{*}(\partial^{\circ}\phi,\psi)+m_{*}(\partial^{\circ}\psi,\phi)+m(\phi,\psi
HV_{N}).$

Moreover, if we have that $\phi,\psi\in H^{1}_{H^{1}}$ then

$\frac{d}{dt}a(\phi,\psi)=a(\partial^{\circ}\phi,\psi)+a(\partial^{\circ}\phi,\psi)+b(\phi,\psi),$

where

$b(t;\phi,\psi):=\int_{\Gamma(t)}V_{N}(H\mathbb{I}-2\mathbb{H})\nabla_{\Gamma}\phi\cdot\nabla_{\Gamma}\psi.$

  2. 2.

Let $\boldsymbol{\phi},\boldsymbol{\psi}\in
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}\cap L^{2}_{\mathbf{V}_{\sigma}}$, then

$\frac{d}{dt}\mathbf{m}(\boldsymbol{\phi},\boldsymbol{\psi})=\mathbf{m}_{*}(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\psi})+m_{*}(\partial^{\circ}\boldsymbol{\psi},\boldsymbol{\phi})+m(\boldsymbol{\phi},\boldsymbol{\psi}HV_{N}).$

Moreover, if we have that $\boldsymbol{\phi},\boldsymbol{\psi}\in
H^{1}_{\mathbf{V}_{\sigma}}$ then

$\frac{d}{dt}\mathbf{a}(\boldsymbol{\phi},\boldsymbol{\phi})=a(\partial^{\circ}\boldsymbol{\phi},\boldsymbol{\phi})+a(\boldsymbol{\phi},\partial^{\circ}\boldsymbol{\psi})+\mathbf{b}(\boldsymbol{\phi},\boldsymbol{\psi}),$

where

$\mathbf{b}(t;\boldsymbol{\phi},\boldsymbol{\psi}):=\int_{\Gamma(t)}V_{N}(H\mathbb{I}-2\mathbb{H})\nabla_{\Gamma}\boldsymbol{\phi}:\nabla_{\Gamma}\boldsymbol{\psi}.$

The relevant form for the bilinear form $\mathbf{b}$ can be deduced from the
identity

$\partial^{\circ}\nabla_{\Gamma}\boldsymbol{\phi}=\nabla_{\Gamma}(\partial^{\circ}\boldsymbol{\phi})-(\nabla_{\Gamma}V_{N}\boldsymbol{\nu})^{T}\nabla_{\Gamma}\boldsymbol{\phi},$

and noting that
$\nabla_{\Gamma}(V_{N}\boldsymbol{\nu})=V_{N}\mathbb{H}+\nabla_{\Gamma}V_{N}\otimes\boldsymbol{\nu}$,
where it is clear that

$(\nabla_{\Gamma}V_{N}\otimes\boldsymbol{\nu})\nabla_{\Gamma}\boldsymbol{\phi}:\nabla_{\Gamma}\boldsymbol{\psi}=0=(\boldsymbol{\nu}\otimes\nabla_{\Gamma}V_{N})\nabla_{\Gamma}\boldsymbol{\phi}:\nabla_{\Gamma}\boldsymbol{\psi},$

due to the presence of the normal vector.

###  3.5 Preliminary rewriting of the system

In order to set up the weak formulation we rewrite the system (1.5)-(1.8) in
such a way that the unknown $\mathbf{u}_{T}$ is divergence free and we may
eliminate the pressure. To do this we consider the unique solution, $\Psi$, of
the elliptic PDE

$-\Delta_{\Gamma}\Psi(t)=H(t)V_{N}(t),$

on $\Gamma(t)$, subject to the constraint
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\Psi=0$, for all $t\in[0,T].$ We then
define $\widetilde{\mathbf{u}_{T}}=\nabla_{\Gamma}\Psi$, from which we see
that $\widetilde{\mathbf{u}_{T}}\in\mathbf{H}^{2,p}(\Gamma(t))$ for all
$p\in[1,\infty)$, and
$-\nabla_{\Gamma}\cdot\widetilde{\mathbf{u}_{T}}=HV_{N}$. Then defining
$\widehat{\mathbf{u}_{T}}:=\mathbf{u}_{T}-\widetilde{\mathbf{u}_{T}}$ we find
that

$\displaystyle\mathbb{P}\partial^{\circ}\widehat{\mathbf{u}_{T}}+(\nabla_{\Gamma}\widehat{\mathbf{u}_{T}})\widehat{\mathbf{u}_{T}}+V_{N}\mathbb{H}\widehat{\mathbf{u}_{T}}-\frac{1}{2}\nabla_{\Gamma}V_{N}^{2}=-\nabla_{\Gamma}\tilde{p}+\mathbb{P}\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\widehat{\mathbf{u}_{T}}))+\mu\nabla_{\Gamma}\varphi+\mathbf{F}_{T}-\mathfrak{D}_{1}(\varphi,\mathbf{u}_{T},\widetilde{\mathbf{u}_{T}}),$
(3.3) $\displaystyle\nabla_{\Gamma}\cdot\widehat{\mathbf{u}_{T}}=0,$ (3.4)
$\displaystyle\partial^{\circ}\varphi+\nabla_{\Gamma}\varphi\cdot\widehat{\mathbf{u}_{T}}=\Delta_{\Gamma}\mu-\mathfrak{D}_{2}(\varphi,\widetilde{\mathbf{u}_{T}}),$
(3.5)
$\displaystyle\mu=-\varepsilon\Delta_{\Gamma}\varphi+\frac{1}{\varepsilon}F^{\prime}(\varphi),$
(3.6)

where

$\displaystyle\mathfrak{D}_{1}(\varphi,\mathbf{u}_{T},\widetilde{\mathbf{u}_{T}})=\mathbb{P}\partial^{\circ}\widetilde{\mathbf{u}_{T}}+(\nabla_{\Gamma}\widetilde{\mathbf{u}_{T}})\widetilde{\mathbf{u}_{T}}+\nabla_{\Gamma}(\widetilde{\mathbf{u}_{T}})\widehat{\mathbf{u}_{T}}+\nabla_{\Gamma}(\widehat{\mathbf{u}_{T}})\widetilde{\mathbf{u}_{T}}+V_{N}\mathbb{H}\widetilde{\mathbf{u}_{T}}-\mathbb{P}\nabla_{\Gamma}\cdot(2\eta(\varphi)\mathbb{E}(\widetilde{\mathbf{u}_{T}}))$
$\displaystyle\mathfrak{D}_{2}(\varphi,\widetilde{\mathbf{u}_{T}})=\nabla_{\Gamma}\varphi\cdot\widetilde{\mathbf{u}_{T}}$

This suggest a new “body force”, $\mathbf{B}$, which is defined as

$\mathbf{B}=\mathbf{F}_{T}-(\nabla_{\Gamma}\widetilde{\mathbf{u}_{T}})\widetilde{\mathbf{u}_{T}}-\mathbb{P}\partial^{\circ}\widetilde{\mathbf{u}_{T}}-V_{N}\mathbb{H}\widetilde{\mathbf{u}_{T}}.$

From Appendix A we know $\widetilde{\mathbf{u}_{T}}\in
C^{0}_{\mathbf{H}^{2,p}}\cap C^{1}_{\mathbf{L}^{p}}$, for all
$p\in[1,\infty)$, and so for $\mathbf{F}_{T}\in L^{2}_{\mathbf{L}^{2}}$ one
can readily show that $\mathbf{B}\in L^{2}_{\mathbf{L}^{2}}$.

The above reformulation is formal, but for sufficiently smooth
$\widetilde{\mathbf{u}_{T}}$ we find that this holds in a weak setting. We
show the necessary regularity properties of $\Psi$ (and
$\widetilde{\mathbf{u}_{T}}$) in Appendix A, and will discuss this later. From
here on we will now denote $\widehat{\mathbf{u}_{T}}$ as $\mathbf{u}_{T}$ and
treat this as the unknown velocity. This formulation allows us to work in the
space of divergence free test functions, as is typical in the analysis of the
Navier-Stokes equations. As such we introduce some notation for a suitable
space of divergence free functions,

$\mathbf{V}_{\sigma}(t):=\left\\{\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))\mid\nabla_{\Gamma}\cdot\boldsymbol{\phi}=0\right\\}.$

Similarly we define $\mathbf{H}_{\sigma}(t)$ to be the following closure in
the $\|\cdot\|_{\mathbf{L}^{2}(\Gamma(t))}$ norm,

$\mathbf{H}_{\sigma}(t):=\left\\{\boldsymbol{\phi}\in
C^{1}(\Gamma(t))^{3}\mid\nabla_{\Gamma}\cdot\boldsymbol{\phi}=0,\boldsymbol{\phi}\cdot\boldsymbol{\nu}=0\right\\}^{\|\cdot\|_{\mathbf{L}^{2}(\Gamma(t))}}.$

Moreover, we have compact, dense embeddings

$\mathbf{V}_{\sigma}(t){\hookrightarrow}\mathbf{H}_{\sigma}(t){\hookrightarrow}\mathbf{V}_{\sigma}(t)^{\prime}.$

The appropriate weak formulation follows from multiplying by a sufficiently
smooth, solenoidal test function, $\boldsymbol{\phi}$, in (3.3) and a
sufficiently smooth test function, $\phi$, in (3.5), (3.6) and integrate over
$\Gamma(t)$. This yields (4.1)-(4.3) below. Notice now that, by using the
divergence theorem, one finds the pressure term vanishes as

$0=\int_{\Gamma(t)}\nabla_{\Gamma}\cdot(\tilde{p}\boldsymbol{\phi})=\int_{\Gamma(t)}\nabla_{\Gamma}\tilde{p}\cdot\boldsymbol{\phi}+\int_{\Gamma({t})}\tilde{p}\nabla_{\Gamma}\cdot\boldsymbol{\phi}.$

###  3.6 Inequalities

We end this section by recalling some useful inequalities. The following
results on Sobolev spaces are proven in [10, 33].

######  Theorem 3.4.

  1. 1.

[Poincaré inequality]
There exists a constant $C_{P}>0$, independent of $t\in[0,T]$, such that for
$f\in H^{1}(\Gamma(t))$ we have

$\left\|f-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}f\right\|_{L^{2}(\Gamma(t))}\leq
C_{P}\|\nabla_{\Gamma}f\|_{L^{2}(\Gamma(t))}.$

  2. 2.

[Sobolev Embeddings]

    1. (a)

Let $0\leq l\leq k$ be two integers, and $1\leq p<q$ be two real numbers such
that $\frac{1}{q}=\frac{1}{p}-\frac{k-l}{2}$. Then we have continuous
embedding $H^{k,p}(\Gamma(t))\hookrightarrow H^{l,q}(\Gamma(t))$.

    2. (b)

If $\frac{k-r-\alpha}{2}\geq\frac{1}{p}$, where $\alpha\in(0,1)$, then we have
continuous embedding $H^{k,p}(\Gamma(t))\hookrightarrow
C^{r+\alpha}(\Gamma(t))$.

Notice in particular that this implies

$H^{1}(\Gamma(t))\hookrightarrow L^{p}(\Gamma(t)),$

for all $p\in[1,\infty)$, and in fact these embeddings are compact. It can be
shown that the operator norm of the above continuous injections are
independent of time - which follows from the fact that we consider a compact
time interval and sufficiently smooth evolution of $\Gamma(t)$. We obtain
analogous Sobolev embeddings for the spaces $\mathbf{H}^{k,p}(\Gamma(t))$. We
also recall the following inequalities which are used throughout the analysis.

######  Lemma 3.5 (Ladyzhenskaya’s interpolation inequality, [45], Lemma 3.4).

For $\phi\in H^{1}(\Gamma(t))$ and
$\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$ we have

$\displaystyle\|\phi\|_{L^{4}(\Gamma(t))}\leq
C\|\phi\|_{L^{2}(\Gamma(t))}^{\frac{1}{2}}\|\phi\|_{H^{1}(\Gamma(t))}^{\frac{1}{2}},$
(3.7) $\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{L}^{4}(\Gamma(t))}\leq
C\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{1}{2}}\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}^{\frac{1}{2}},$
(3.8)

for a constant $C$ independent of $t$.

######  Lemma 3.6 (Korn’s inequality, [45], Lemma 3.2).

For $\boldsymbol{\phi}\in\mathbf{H}^{1}(\Gamma(t))$ we have

$\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\leq
C\left(\|\boldsymbol{\phi}\|_{\mathbf{L}^{2}(\Gamma(t))}+\|\mathbb{E}(\boldsymbol{\phi})\|_{\mathbf{L}^{2}(\Gamma(t))}\right),$
(3.9)

where the constant $C$ is independent of $t$.

######  Remark 3.7.

In fact, one can show an $L^{p}$ version of Korn’s inequality,

$\displaystyle\|\boldsymbol{\phi}\|_{\mathbf{H}^{1,p}(\Gamma(t))}\leq
C\left(\|\boldsymbol{\phi}\|_{\mathbf{L}^{p}(\Gamma(t))}+\|\mathbb{E}(\boldsymbol{\phi})\|_{\mathbf{L}^{p}(\Gamma(t))}\right),$
(3.10)

for $\boldsymbol{\phi}\in\mathbf{H}^{1,p}(\Gamma(t))$ and $p\in(1,\infty)$.
The argument for this on a stationary surface follows as in the proof of [34]
Lemma 4.1, where one uses an $L^{p}$ version of Korn’s inequality in Euclidean
space (see for example [21]) after considering a local parametrisation of
$\Gamma(t)$. The inequality then extends to an evolving surface with a
constant independent of $t$.

In order to establish energy estimates we use the following nonlinear
generalisation of the Grönwall inequality.

######  Lemma 3.8 (Bihari-LaSalle inequality, [14]).

Let $X,K:[0,T]\rightarrow\mathbb{R}$, be non-negative continuous444By density
this can be shown to extend to $K\in L^{1}([0,T])$. functions,
$\omega:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a decreasing continuous
function, and $k\geq 0$. Then the inequality

$X(t)\leq k+\int_{0}^{t}K(s)\omega(X(s))\,ds,$

implies the inequality

$\displaystyle
X(t)\leq\Omega^{-1}\left(\Omega(k)+\int_{0}^{t}K(s)\,ds\right),$ (3.11)

where

$\Omega(t)=\int_{t_{0}}^{t}\frac{1}{\omega(s)}\,ds,$

for some constant $t_{0}$ chosen so that

$\Omega(k)+\int_{0}^{t}K(s)\,ds\in\text{dom}(\Omega^{-1}).$

Finally we recall three results which will be used in proving uniqueness.

######  Lemma 3.9 ([26], Lemma 4.3).

Let $z\in H^{-1}(\Gamma(t))$ be such that $m_{*}(z,1)=0$. Define the inverse
Laplacian $\mathcal{G}z\in H^{1}(\Gamma(t))\in H^{1}(\Gamma(t))$ as the unique
solution of

$a(\mathcal{G}z,\phi)=m_{*}(z,\phi),$

for all $\phi\in H^{1}(\Gamma(t))$. If $z\in H^{1}_{H^{-1}}$ then

$\|\mathcal{G}z||_{H^{1}_{H^{1}}}\leq C\|z\|_{H^{1}_{H^{-1}}}.$

In fact in [26] it is assumed that $z\in H^{1}_{H^{1}}$ but examining the
proof it is sufficient to assume $z\in H^{1}_{H^{-1}}$.

######  Lemma 3.10 ([38], Lemma 2.2).

Let $m_{1},m_{2},S$ be non-negative functions on $(0,T)$ such that $m_{1},S\in
L^{1}(0,T)$ and $m_{2}\in L^{2}(0,T)$, with $S>0$ a.e. on $(0,T)$. Now suppose
$f,g$ are non-negative functions on $(0,T)$, $f$ is absolutely continuous on
$[0,T)$, such that555Here we are using the notation
$\log^{+}(x):=\max(0,\log(x))$.

$\displaystyle f^{\prime}(t)+g(t)\leq
m_{1}(t)f(t)+m_{2}(t)\left(f(t)g(t)\log^{+}\left(\frac{S(t)}{g(t)}\right)\right)^{\frac{1}{2}},$
(3.12)

holds a.e. on $(0,T)$, and $f(0)=0$. Then $f(t)\equiv 0$ on $[0,T)$.

The final result we mention is an evolving surface analogue of the Brezis-
Gallouët-Wainger inequality (which originates from work on the nonlinear
Schrödinger equation, see [17]).

######  Lemma 3.11.

For $\phi\in H^{2}(\Gamma(t))$, one has

$\displaystyle\|\phi\|_{L^{\infty}(\Gamma(t))}\leq
C\|\phi\|_{H^{1}(\Gamma(t))}\left(1+\log\left(1+\frac{C\|\phi\|_{H^{2}(\Gamma(t))}}{\|\phi\|_{H^{1}(\Gamma(t))}}\right)^{\frac{1}{2}}\right),$
(3.13)

for constants $C$ independent of $t$.

###### Proof.

From [30], Theorem 1.1 we see that for $\phi\in H^{2}(\Gamma_{0})$ one has

$\|\phi\|_{L^{\infty}(\Gamma_{0})}\leq
C\|\phi\|_{H^{1}(\Gamma_{0})}\left(1+\log\left(1+\frac{\|\phi\|_{H^{2}(\Gamma_{0})}}{\|\phi\|_{H^{1}(\Gamma_{0})}}\right)^{\frac{1}{2}}\right),$

where examining the proof one finds that our assumption that $\Gamma_{0}$ is
$C^{3}$ is sufficient. To see that one can choose the constant independent of
time we observe that
$\|\phi\|_{L^{\infty}(\Gamma(t))}=\|\Phi_{-t}^{n}\phi\|_{L^{\infty}(\Gamma_{0})}.$
Hence by pulling back to $\Gamma_{0}$ and using the above inequality, the
compatibility of the pairs $(H^{k}(\Gamma(t)),\Phi_{t}^{n})$ in the sense of
[7], and the monotonicity of $x\mapsto\log(x)^{\frac{1}{2}}$, it is clear that
the inequality holds on $\Gamma(t)$ with constants independent of $t$. We note
that this also introduces a constant into the logarithmic term. ∎

##  4 Weak formulation and well posedness theorems

We are now in a position to discuss the weak formulation. For simplicity we
assume constant mobility, $M(\cdot)\equiv 1$ and a scaling such that $\rho=1$.

###  4.1 Regular potential

We firstly consider a smooth potential, $F$, under the same assumptions as in
[18]. That is, we assume $F(r)=F_{1}(r)+F_{2}(r)$ for $F_{1},F_{2}\in
C^{2}(\mathbb{R})$ such that

  1. 1.

$F(r)\geq\beta$,

  2. 2.

$F_{1}\geq 0$ is convex,

  3. 3.

$\exists q\in[1,\infty)$ such that
$|F_{1}^{\prime}(r)|\leq\alpha|r|^{q}+\alpha$,

  4. 4.

$|F_{1}^{\prime}(r)|+|rF_{1}^{\prime}(r)|\leq\alpha F_{1}(r)+\beta$,

  5. 5.

$|F_{2}^{\prime}(r)|\leq\alpha|r|+\alpha$,

where $\alpha$ denote some non-negative constant, and $\beta$ some real
constant. We also assume the viscosity function, $\eta(\cdot)$ is Lipschitz
continuous. An typical example of a viscosity function is

$\eta(r)=\eta_{1}\frac{(1+r)}{2}+\eta_{2}\frac{(1-r)}{2},\quad r\in[-1,1],$

for two positive constants $\eta_{1},\eta_{2}$, which can then be suitably
extended to a $C^{2}$, Lipschitz continuous function on $\mathbb{R}$. The
previous section then allows the following weak formulation, using the
notation introduced in section 3.4.

Given initial data $\varphi_{0}\in
H^{1}(\Gamma_{0}),\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$, find $\varphi\in
H^{1}_{H^{-1}}\cap L^{2}_{H^{1}},\mu\in L^{2}_{H^{1}},\mathbf{u}_{T}\in
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}\cap L^{2}_{\mathbf{V}_{\sigma}}$ such
that

$\displaystyle\begin{split}\mathbf{m}_{*}(\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi})+\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{d}_{2}(\eta(\varphi),\boldsymbol{\phi})\\\
=\mathbf{m}(\mathbf{B},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})\end{split},$
(4.1) $\displaystyle
m_{*}(\partial^{\circ}\varphi,\phi)+a(\mu,\phi)+\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})+\mathbf{c}_{2}(\phi,\varphi,\widetilde{\mathbf{u}_{T}})=0,$
(4.2) $\displaystyle m(\mu,\phi)=\varepsilon
a(\varphi,\phi)+\frac{1}{\varepsilon}m(F^{\prime}(\varphi),\phi),$ (4.3)

for all $\phi\in H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$
for almost all $t\in[0,T]$, and such that $\varphi(0)=\varphi_{0}$,
$\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$ almost everywhere on $\Gamma_{0}$.

######  Theorem 4.1.

Let $\Gamma(t)$ be a $C^{3}$ evolving surface, $F$ a potential function
satisfying the assumptions at the beginning of the section, and
$\varphi_{0}\in H^{1}(\Gamma(0)),\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$ be
initial data. Then there exists a solution triple
$(\varphi,\mu,\mathbf{u}_{T})$ such that

$\displaystyle\varphi\in L^{\infty}_{H^{1}}\cap L^{2}_{H^{2}}\cap
H^{1}_{H^{-1}},$ $\displaystyle\mu\in L^{2}_{H^{1}},$
$\displaystyle\mathbf{u}_{T}\in L^{\infty}_{L^{2}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}},$

and solving (4.1)-(4.3) for all $\phi\in
H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$ for almost all
$t\in[0,T]$, and such that $\varphi(0)=\varphi_{0}$,
$\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$ almost everywhere on $\Gamma_{0}$.

###  4.2 Logarithmic potential

In this section we consider the well-posedness theory for the singular
logarithmic potential,

$F(r)=\frac{\theta}{2}\left((1+r)\log(1+r)-(1-r)\log(1-r)\right)+\frac{1-r^{2}}{2}=:\frac{\theta}{2}F_{\log}(r)+\frac{1-r^{2}}{2},$

for $\theta\in(0,1)$. Here $\theta$ can be understood as a temperature in the
system, which we have scaled for notational simplicity. As the equations
(1.5)-(1.8) consider the derivative of $F$ we introduce some shorthand
notation,

$f(r):=(F_{\log}(r))^{\prime}=\log\left(\frac{1+r}{1-r}\right).$

The corresponding version of (4.1)-(4.3) for the logarithmic potential is as
follows. Given initial data
$\varphi_{0}\in\mathcal{I}_{0},\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$,
find $\varphi\in H^{1}_{H^{-1}}\cap L^{2}_{H^{1}},\mu\in
L^{2}_{H^{1}},\mathbf{u}_{T}\in H^{1}_{\mathbf{V}_{\sigma}^{\prime}}\cap
L^{2}_{\mathbf{V}_{\sigma}}$ such that

$\displaystyle\begin{split}\mathbf{m}_{*}(\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi})+\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})+\mathbf{d}_{2}(\eta(\varphi),\boldsymbol{\phi})\\\
=\mathbf{m}(\mathbf{B},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})\end{split},$
(4.4) $\displaystyle
m_{*}(\partial^{\circ}\varphi,\phi)+a(\mu,\phi)+\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})+\mathbf{c}_{2}(\phi,\varphi,\widetilde{\mathbf{u}_{T}})=0,$
(4.5) $\displaystyle m(\mu,\phi)=\varepsilon
a(\varphi,\phi)+\frac{\theta}{2\varepsilon}m(f(\varphi),\phi)-\frac{1}{\varepsilon}m(\varphi,\phi),$
(4.6)

for all $\phi\in H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$
for almost all $t\in[0,T]$, and such that $\varphi(0)=\varphi_{0}$,
$\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$ almost everywhere on $\Gamma_{0}$.

An advantage of this singular potential is that it necessarily has “physical
solutions”, that is $|\varphi(t)|<1$ almost everywhere on $\Gamma(t)$ for
almost all $t\in[0,T]$, due to the logarithmic nonlinearity. As in [18, 19]
there is a constraint on the initial conditions which allow us to find such a
solution. For $t\in[0,T]$ and $\eta\in H^{1}(\Gamma_{0})$, we define

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

The following result is essentially unchanged from [18] Proposition 5.1 and
Corollary 5.2.

######  Proposition 4.2.

Let $\varphi_{0}\in H^{1}(\Gamma_{0})$, and
$\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$. If there exists $t^{*}\in[0,T]$
such that $m_{\varphi_{0}}(t)>1$ then there doesn’t exist a solution triple
$(\varphi,\mu,\mathbf{u}_{T})$ solving (4.4)-(4.6) such that $|\varphi|<1$
almost everywhere on $\Gamma(t)$ for almost all $t\in[0,T]$.

###### Proof.

Firstly, we notice that $t\mapsto m_{\varphi_{0}}(t)$ is continuous. Hence,
there exists some $\gamma>0$ such that

$t\in(t^{*}-\gamma,t^{*}+\gamma)\Rightarrow m_{\varphi_{0}}(t)>1.$

Now from testing (4.5) against $\phi\equiv 1$ we find the mass conservation
property

$\int_{\Gamma(t)}\varphi(t)=\int_{\Gamma_{0}}\varphi_{0},$

holds for almost all $t\in[0,T]$. In particular, as
$(t^{*}-\gamma,t^{*}+\gamma)$ has positive measure, the mass conservation
holds for almost all $t\in(t^{*}-\gamma,t^{*}+\gamma)$, and hence

$\frac{1}{|\Gamma(t)|}\left|\int_{\Gamma(t)}\varphi(t)\right|=\frac{1}{|\Gamma(t)|}\left|\int_{\Gamma_{0}}\varphi_{0}\right|=m_{\varphi_{0}}(t)>1.$

Then, for contradiction, suppose $\varphi(t)<1$ almost everywhere on
$\Gamma(t)$ for almost all $t\in[0,T]$. Then for almost all
$t\in(t^{*}-\gamma,t^{*}+\gamma)$ the above mass conservation holds, and
$|\varphi(t)|<1$. Hence

$\frac{1}{|\Gamma(t)|}\left|\int_{\Gamma(t)}\varphi(t)\right|\leq\frac{1}{|\Gamma(t)|}\int_{\Gamma(t)}\left|\varphi(t)\right|<1,$

and we have a contradiction. ∎

This motivates the choice of admissible initial conditions for the logarithmic
potential to be

$\mathcal{I}_{0}:=\left\\{\eta\in H^{1}(\Gamma_{0})\mid
E^{\text{CH}}[\eta;0]<\infty,\ m_{\eta}(t)<1\text{ for all
}t\in[0,T]\right\\}.$

We note there is no condition on the choice of initial velocity, so we still
choose $\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$.

######  Theorem 4.3.

Let $\Gamma(t)$ be a $C^{3}$ evolving surface, $F$ be the logarithmic
potential, and
$\varphi_{0}\in\mathcal{I}_{0},\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$ be
initial data. Then there exists a solution triple
$(\varphi,\mu,\mathbf{u}_{T})$ such that

$\displaystyle\varphi\in L^{\infty}_{H^{1}}\cap L^{2}_{H^{2}}\cap
H^{1}_{H^{-1}},$ $\displaystyle\mu\in L^{2}_{H^{1}},$
$\displaystyle\mathbf{u}_{T}\in L^{\infty}_{L^{2}}\cap
H^{1}_{\mathbf{V}_{\sigma}^{\prime}},$

solving (4.4)-(4.6) for all $\phi\in
H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$ for almost all
$t\in[0,T]$, and such that $\varphi(0)=\varphi_{0}$,
$\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$ almost everywhere on $\Gamma_{0}$.

##  5 Proof of existence

###  5.1 The regular potential

####  5.1.1 Galerkin approximation

We now show existence of a solution triple to (4.1)-(4.3) via the Faedo-
Galerkin method. For $M\in\mathbb{N}$ we define the Galerkin approximations

$\displaystyle\varphi^{M}(t)=\sum_{i=1}^{M}\alpha_{i}^{M}(t)\Phi_{t}^{n}\psi_{i},\qquad\mu^{M}(t)=\sum_{i=1}^{M}\beta_{i}^{M}(t)\Phi_{t}^{n}\psi_{i},$
$\displaystyle\mathbf{u}_{T}^{M}(t)=\sum_{i=1}^{M}\gamma_{i}^{M}(t)\mathcal{P}_{t}\boldsymbol{\chi}_{i},$

where $(\psi_{i})_{i=1,...\infty}$ form a countable basis of
$H^{1}(\Gamma_{0})$, and $(\boldsymbol{\chi}_{i})_{i=1,...\infty}$ form a
countable basis of $\mathbf{V}_{\sigma}(0)$, and hence pushforward onto
countable bases of $H^{1}(\Gamma(t))$ and $\mathbf{V}_{\sigma}(t)$
respectively. For example, by Hilbert-Schmidt theory, one can choose
$(\psi_{i})_{i=1,...\infty}$ to be the eigenfunctions of the Laplace-Beltrami
operator on $\Gamma_{0}$, and $(\boldsymbol{\chi}_{i})_{i=1,...\infty}$ to be
the eigenfunctions of the surface Stokes operator on $\Gamma_{0}$. We define
the following spaces:

$\displaystyle V^{M}(t):=\text{span}\\{\Phi_{t}^{n}\psi_{i}\mid
i=1,...,M\\}\leq H^{1}(\Gamma(t)),$
$\displaystyle\mathbf{V}_{\sigma}^{M}(t):=\text{span}\\{\mathcal{P}_{t}\boldsymbol{\chi}_{i}\mid
i=1,...,M\\}\leq\mathbf{V}_{\sigma}(t),$

and the corresponding projections as $P^{M}_{V}(t):H^{1}(\Gamma(t))\rightarrow
V^{M}(t),P^{M}_{\mathbf{V}}(t):\mathbf{H}_{\sigma}(t)\rightarrow\mathbf{V}_{\sigma}^{M}(t)$,
which are defined by

$\displaystyle(P^{M}_{V}(t)\phi,\psi)_{H^{1}(\Gamma(t))}=(\phi,\psi)_{H^{1}(\Gamma(t))},\
\forall\psi\in V^{M}(t),$
$\displaystyle(P^{M}_{\mathbf{V}}(t)\boldsymbol{\phi},\boldsymbol{\psi})_{\mathbf{L}^{2}(\Gamma(t))}=(\boldsymbol{\phi},\boldsymbol{\psi})_{\mathbf{L}^{2}(\Gamma(t))},\
\forall\boldsymbol{\psi}\in\mathbf{H}_{\sigma}(t).$

We emphasise that these are $H^{1}$ and $\mathbf{L}^{2}$ projections
respectively, as it is sufficient to choose initial data such that
$\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0)$, rather than
$\mathbf{V}_{\sigma}(0)$, but require $\varphi_{0}\in H^{1}(\Gamma_{0})$.

######  Lemma 5.1.

There exists a solution triple $(\varphi^{M},\mu^{M},\mathbf{u}_{T}^{M})$ on
$[0,t^{*})$ solving

$\displaystyle\begin{split}\mathbf{m}(\partial^{\circ}\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\hat{\mathbf{a}}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\mathbf{d}_{2}(\eta(\varphi^{M}),\boldsymbol{\phi})\\\
=\mathbf{m}(\mathbf{B},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu^{M},\varphi^{M},\boldsymbol{\phi}),\end{split}$
(5.1) $\displaystyle
m(\partial^{\circ}\varphi^{M},\phi)+a(\mu^{M},\phi)+\mathbf{c}_{2}(\phi,\varphi^{M},\mathbf{u}_{T}^{M})+\mathbf{c}_{2}(\phi,\varphi^{M},\widetilde{\mathbf{u}_{T}})=0,$
(5.2) $\displaystyle m(\mu^{M},\phi)=\varepsilon
a(\varphi^{M},\phi)+\frac{1}{\varepsilon}m(F^{\prime}(\varphi^{M}),\phi),$
(5.3)

for all $\phi\in V^{M}(t),\boldsymbol{\phi}\in V^{M}_{\sigma}(t)$ for almost
all $t\in[0,t^{*})$, and such that $\varphi^{M}(0)=P_{V}^{M}\varphi_{0}$,
$\mathbf{u}_{T}(0)=P_{\mathbf{V}}^{M}\mathbf{u}_{T,0}$ almost everywhere on
$\Gamma_{0}$.

###### Proof.

To begin we show the system (5.1)-(5.3) is equivalent to an ODE for the
coefficients $\alpha,\beta,\gamma$. Firstly, by (3.2) we note that we may
write

$\mathbf{m}(\partial^{\circ}\mathbf{u}_{T}^{M},\boldsymbol{\phi})=\mathbf{m}(\partial^{*}\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\mathbf{m}(\bar{\mathbb{A}}\mathbf{u}_{T}^{M},\boldsymbol{\phi}).$

Now we note that by definition of the strong derivative one finds

$\partial^{\circ}\Phi_{t}^{n}\psi_{i}=0,\qquad\partial^{*}\mathcal{P}_{t}\boldsymbol{\chi}_{i}=0,$

and hence

$\displaystyle\partial^{\circ}\varphi^{M}(t)=\sum_{i=1}^{M}(\alpha_{i}^{M})^{\prime}(t)\Phi_{t}^{n}\psi_{i},$
$\displaystyle\partial^{*}\mathbf{u}_{T}^{M}(t)=\sum_{i=1}^{M}(\gamma_{i}^{M})^{\prime}(t)\mathcal{P}_{t}\boldsymbol{\chi}_{i}.$

With this in hand, it is clear that testing (5.1) with
$\mathcal{P}_{t}\boldsymbol{\chi}_{k}$ yields

$\sum_{i=1}^{M}(\gamma_{i}^{M})^{\prime}\mathbf{m}(\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\sum_{i=1}^{M}\gamma_{i}^{M}\mathbf{m}(\bar{\mathbb{A}}\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\sum_{i=1}^{M}\gamma_{i}^{M}\hat{\mathbf{a}}\left(\eta\left(\sum_{j=1}^{M}\alpha_{j}^{M}\Phi_{t}^{n}\psi_{j}\right)\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k}\right)\\\
+\sum_{i=1}^{M}(\gamma_{i}^{M})^{2}\mathbf{c}_{1}(\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\sum_{i=1}^{M}\gamma_{i}^{M}\mathbf{l}(\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\sum_{i=1}^{M}\gamma_{i}^{M}\mathbf{d}_{1}(\mathcal{P}_{t}\boldsymbol{\chi}_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\mathbf{d}_{2}\left(\eta\left(\sum_{j=1}^{M}\alpha_{j}^{M}\Phi_{t}^{n}\psi_{j}\right),\mathcal{P}_{t}\boldsymbol{\chi}_{k}\right)\\\
=\mathbf{m}(\mathbf{B},\mathcal{P}_{t}\boldsymbol{\chi}_{k})+\sum_{i=1}^{M}\sum_{j=1}^{M}\alpha_{j}^{M}\beta_{i}^{M}\mathbf{c}_{2}(\Phi_{t}^{n}\psi_{i},\Phi_{t}^{n}\psi_{j},\mathcal{P}_{t}\boldsymbol{\chi}_{k}).$

Next testing (5.2) and (5.3) with $\Phi_{t}^{n}\psi_{k}$ we obtain

$\sum_{i=1}^{M}(\alpha_{i}^{M})^{\prime}m(\Phi_{t}^{n}\psi_{i},\Phi_{t}^{n}\psi_{k})+\sum_{i=1}^{M}\beta_{i}^{M}a(\Phi_{t}^{n}\psi_{i},\Phi_{t}^{n}\psi_{i})+\sum_{i=1}^{M}\sum_{j=1}^{M}\alpha_{i}^{M}\gamma_{j}^{M}\mathbf{c}_{2}(\Phi_{t}^{n}\psi_{k},\Phi_{t}^{n}\psi_{i},\mathcal{P}_{t}\boldsymbol{\chi}_{j})\\\
+\sum_{i=1}^{M}\alpha_{i}^{M}\mathbf{c}_{2}(\Phi_{t}^{n}\psi_{k},\Phi_{t}^{n}\psi_{i},\widetilde{\mathbf{u}_{T}})=0,$

and

$\displaystyle\sum_{i=1}^{M}\beta_{i}^{M}m(\Phi_{t}^{n}\psi_{i},\Phi_{t}^{n}\psi_{k})=\varepsilon\sum_{i=1}^{M}\alpha_{i}^{M}a(\Phi_{t}^{n}\psi_{i},\Phi_{t}^{n}\psi_{k})+\frac{1}{\varepsilon}m\left(F^{\prime}\left(\sum_{i=1}^{M}\alpha_{i}^{M}\Phi_{t}^{n}\psi_{i}\right),\Phi_{t}^{n}\psi_{k}\right).$

By considering the system this generates for $k=1,...,M$ one obtain an ODE
system for the vectors
$\alpha^{M}(t),\beta^{M}(t),\gamma^{M}(t)\in\mathbb{R}^{M}$, It is
straightforward to see that the nonlinearities are locally Lipschitz, but we
omit these details. Applying standard ODE theory one obtains the short time
existence of a solution triple $(\alpha^{M},\beta^{M},\gamma^{M})$. ∎

Next we establish existence on $[0,T]$ by use of energy estimates. To do this
we recall the Ginzburg-Landau functional,

$\displaystyle
E^{\text{CH}}[\varphi;t]:=\int_{\Gamma(t)}\frac{\varepsilon}{2}|\nabla_{\Gamma}\varphi|^{2}+\frac{1}{\varepsilon}F(\varphi)=\frac{\varepsilon}{2}a(\varphi,\varphi)+\frac{1}{\varepsilon}m(F(\varphi),1),$
(5.4)

which we use in the following result.

######  Lemma 5.2.

The solution triple $(\varphi^{M},\mu^{M},\mathbf{u}_{T}^{M})$ is such that

$\displaystyle\sup_{t\in[0,T]}\left(\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+E^{\text{CH}}[\varphi^{M};t]\right)+\frac{\eta_{*}}{2}\int_{0}^{T}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{1}{2}\int_{0}^{T}\|\nabla_{\Gamma}\mu^{M}\|_{L^{2}(\Gamma(t))}^{2}\leq
C,$ (5.5)

for a constant $C$ independent of $M$.

###### Proof.

To begin, we test (5.1) with $\mathbf{u}_{T}^{M}$ for

$\mathbf{m}(\partial^{\circ}\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})+\hat{\mathbf{a}}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})=\mathbf{m}(\mathbf{B},\mathbf{u}_{T}^{M})+\mathbf{c}_{2}(\mu^{M},\varphi^{M},\mathbf{u}_{T}^{M})-\mathbf{l}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})\\\
-\mathbf{d}_{2}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M}),$

where we have used
$\mathbf{c}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})=0$.
Next we notice that from (5.2) that

$\displaystyle\mathbf{c}_{2}(\mu^{M},\varphi^{M},\mathbf{u}_{T}^{M})=-m(\partial^{\circ}\varphi^{M},\mu^{M})-a(\mu^{M},\mu^{M})-\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}}),$

and from (5.3) we find

$m(\partial^{\circ}\varphi^{M},\mu^{M})=\varepsilon
a(\varphi^{M},\partial^{\circ}\varphi^{M})+\frac{1}{\varepsilon}m(F^{\prime}(\varphi^{M}),\partial^{\circ}\varphi^{M}).$

Hence combining these three equalities it is clear that

$\mathbf{m}(\partial^{\circ}\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})+\hat{\mathbf{a}}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})+\varepsilon
a(\varphi^{M},\partial^{\circ}\varphi^{M})+\frac{1}{\varepsilon}m(F^{\prime}(\varphi^{M}),\partial^{\circ}\varphi^{M})+a(\mu^{M},\mu^{M})=\mathbf{m}(\mathbf{B},\mathbf{u}_{T}^{M})\\\
-\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}})-\mathbf{l}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\mathbf{d}_{2}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M}).$
(5.6)

Recalling Lemma 3.3 we find that

$\displaystyle\mathbf{m}(\partial^{\circ}\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})=\frac{1}{2}\frac{d}{dt}\mathbf{m}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\frac{1}{2}\mathbf{m}(\mathbf{u}_{T}^{M},HV_{N}\mathbf{u}_{T}^{M}),$
$\displaystyle
a(\varphi^{M},\partial^{\circ}\varphi^{M})=\frac{1}{2}\frac{d}{dt}a(\varphi^{M},\varphi^{M})-\frac{1}{2}b(\varphi^{M},\varphi^{M}),$
$\displaystyle
m(F^{\prime}(\varphi^{M}),\partial^{\circ}\varphi^{M})=\frac{d}{dt}m(F(\varphi^{M}),1)-m(F(\varphi^{M}),HV_{N}).$

Next, by using the bound $\eta_{*}\leq\eta(\cdot)$ and the above we see

$\frac{1}{2}\frac{d}{dt}\mathbf{m}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})+\frac{d}{dt}E^{\text{CH}}[\varphi^{M};t]+\eta_{*}\mathbf{a}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})+a(\mu^{M},\mu^{M})=\mathbf{m}(\mathbf{B},\mathbf{u}_{T}^{M})+\frac{1}{2}\mathbf{m}(\mathbf{u}_{T}^{M},HV_{N}\mathbf{u}_{T}^{M})+\frac{\varepsilon}{2}b(\varphi^{M},\varphi^{M})\\\
+\frac{1}{\varepsilon}m(F(\varphi^{M}),HV_{N})-\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}})-\mathbf{l}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})-\mathbf{d}_{2}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M}).$
(5.7)

The focus now is bounding these terms on the right hand side. Firstly we find

$\mathbf{m}(\mathbf{B},\mathbf{u}_{T}^{M})+\frac{1}{2}\mathbf{m}(\mathbf{u}_{T}^{M},HV_{N}\mathbf{u}_{T}^{M})+\mathbf{l}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})\leq\left(\frac{1}{2}+\frac{1}{2}\|HV_{N}\|_{L^{\infty}(\Gamma(t))}+\|V_{N}\mathbb{H}\|_{\mathbf{L}^{\infty}(\Gamma(t))}\right)\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}\\\
+\frac{1}{2}\|\mathbf{B}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2},$ (5.8)

where we note that $H,V_{N}\in
C^{1}(\mathcal{G}_{T}),\mathbb{H}\in(C^{1}(\mathcal{G}_{T}))^{3\times 3}$ by
our assumptions. Similarly, this smoothness assumption on $HV_{N}$ allows us
to bound

$\displaystyle\frac{\varepsilon}{2}b(\varphi^{M},\varphi^{M})+\frac{1}{\varepsilon}m(F(\varphi^{M}),HV_{N})\leq
CE^{\text{CH}}[\varphi^{M};t],$ (5.9)

for a constant independent of $t$ and $M$. We now look at the terms introduced
by the influence of $\widetilde{\mathbf{u}_{T}}$. It is straightforward to see
that

$\displaystyle|\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})|\leq\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{H}^{1}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,\infty}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}$

where we have used the regularity result from Appendix A to bound
$\widetilde{\mathbf{u}_{T}}$. We then use (3.8) to see that

$\|\mathbf{u}_{T}^{M}\|_{\mathbf{H}^{1}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq
C\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{3}{2}}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{\frac{1}{2}},$

and hence from Young’s inequality we obtain

$\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{H}^{1}(\Gamma(t))}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}\leq\frac{\eta_{*}}{4}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{\frac{4}{3}}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$

All in all, this yields a bound on
$\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})$ given by

$\displaystyle|\mathbf{d}_{1}(\mathbf{u}_{T}^{M},\mathbf{u}_{T}^{M})|\leq\frac{\eta_{*}}{4}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\left(\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,\infty}(\Gamma(t))}+C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(t))}^{\frac{4}{3}}\right)\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(5.10)

Likewise it is straightforward to see that

$\displaystyle|\mathbf{d}_{2}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M})|\leq\frac{\eta_{*}}{4}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{2(\eta^{*})^{2}}{\eta_{*}}\|\mathbb{E}(\widetilde{\mathbf{u}_{T}})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}.$
(5.11)

We lastly consider the $\mathbf{c}_{2}$ term and see that

$\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}})=\int_{\Gamma(t)}\mu^{M}\nabla_{\Gamma}\varphi^{M}\cdot\widetilde{\mathbf{u}_{T}}=-\int_{\Gamma(t)}\varphi^{M}\nabla_{\Gamma}\mu^{M}\cdot\widetilde{\mathbf{u}_{T}}-\int_{\Gamma(t)}\varphi^{M}\mu^{M}\nabla_{\Gamma}\cdot\widetilde{\mathbf{u}_{T}},$

which follows from integration by parts. By construction of
$\widetilde{\mathbf{u}_{T}}$ we now find that

$-\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}})=\mathbf{c}_{2}(\varphi^{M},\mu^{M},\widetilde{\mathbf{u}_{T}})-m(\mu^{M},HV_{N}\varphi^{M}).$

The difficulty now is in establishing an $L^{2}$ bound on $\mu^{M}$. To do
this we test (5.3) with $\mu^{M}$ and use Young’s inequality to see that

$\|\mu^{M}\|_{L^{2}(\Gamma(t))}^{2}\leq\frac{1}{4}\|\nabla_{\Gamma}\mu^{M}\|^{2}+\varepsilon^{2}\|\nabla_{\Gamma}\varphi^{M}\|_{L^{2}(\Gamma(t))}^{2}+\frac{1}{2\varepsilon^{2}}\|F^{\prime}(\varphi^{M})\|_{L^{2}(\Gamma(t))}^{2}.$

One then deals with the nonlinearity here by using the growth conditions on
$F^{\prime}$, from which we readily see that

$\frac{1}{2\varepsilon^{2}}\|F^{\prime}(\varphi^{M})\|_{L^{2}(\Gamma(t))}^{2}\leq
C+C\|\varphi^{M}\|_{L^{2q}(\Gamma(t))}^{2q}+C\|\varphi^{M}\|_{L^{2}(\Gamma(t))}^{2},$

for some constants $C,q$ dependent on the choice of $F$. By using the Sobolev
embedding $H^{1}(\Gamma(t))\hookrightarrow L^{2q}(\Gamma(t))$ and the Poincaré
inequality (noting that $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{M}$ is constant and can be
bounded independent of $M$) it is straightforward to show that

$\|\varphi^{M}\|_{L^{2q}(\Gamma(t))}^{2q}+\|\varphi^{M}\|_{L^{2}(\Gamma(t))}^{2}\leq
C+C\|\nabla_{\Gamma}\varphi^{M}\|_{L^{2}(\Gamma(t))}^{2q}+C\|\nabla_{\Gamma}\varphi^{M}\|_{L^{2}(\Gamma(t))}^{2}.$

Combining these bounds one finds that

$\displaystyle|\mathbf{c}_{2}(\mu^{M},\varphi^{M},\widetilde{\mathbf{u}_{T}})|\leq
C+\frac{1}{2}\|\nabla_{\Gamma}\mu^{M}\|_{L^{2}(\Gamma(t))}^{2}+CE^{\text{CH}}[\varphi^{M};t]+CE^{\text{CH}}[\varphi^{M};t]^{q},$
(5.12)

for constants $C$ independent of $M$.

Finally, by using the estimates (5.8)-(5.12) in (5.7) and integrating over
$[0,t]$ we obtain an inequality of the form

$\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+E^{\text{CH}}[\varphi^{M};t]+\frac{\eta_{*}}{2}\int_{0}^{t}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{1}{2}\int_{0}^{t}\|\nabla_{\Gamma}\mu^{M}\|_{L^{2}(\Gamma(t))}^{2}\leq
k\\\
+\int_{0}^{t}K(s)\left(\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}+E^{\text{CH}}[\varphi^{M};s]+\left(\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}+E^{\text{CH}}[\varphi^{M};s]\right)^{q}\right)\,ds,$
(5.13)

where

$k=C+\frac{1}{2}\int_{0}^{T}\|\mathbf{B}\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}\,ds+\frac{2(\eta^{*})^{2}}{\eta_{*}}\int_{0}^{T}\|\mathbb{E}(\widetilde{\mathbf{u}_{T}})\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}\,ds+\frac{1}{2}\|P_{\mathbf{V}}^{M}\mathbf{u}_{T,0}\|_{\mathbf{L}^{2}(\Gamma(0))}^{2}+E^{\text{CH}}[P_{V}^{M}\varphi_{0};0],$

and

$\displaystyle
K(s)=C+\frac{1}{2}\|HV_{N}\|_{L^{\infty}(\Gamma(s))}+\|V_{N}\mathbb{H}\|_{\mathbf{L}^{\infty}(\Gamma(s))}+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,\infty}(\Gamma(s))}+C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(s))}^{\frac{4}{3}}.$

We can bound $k$ independently of $M$ by noting that

$\|P_{V}^{M}\varphi_{0}\|_{H^{1}(\Gamma(0))}\leq\|P_{V}^{M}\varphi_{0}\|_{H^{1}(\Gamma(0))},\qquad\|P^{M}_{\mathbf{V}}\mathbf{u}_{T,0}\|_{\mathbf{L}^{2}(\Gamma(0))}\leq\|\mathbf{u}_{T,0}\|_{\mathbf{L}^{2}(\Gamma(0))},$

and as above we can bound the potential term in $E^{\text{CH}}$ by using
Sobolev embeddings.

Lastly, we note that while $E^{\text{CH}}[\varphi^{M};t]$ is not necessarily
non-negative, it is bounded below. Hence we add some sufficiently large
constant to (5.13) so that the analogous inequality holds for the modified
energy,

$\widetilde{E^{\text{CH}}}[\varphi;t]:=E^{\text{CH}}[\varphi;t]+\tilde{\beta}\geq\frac{\varepsilon}{2}\|\nabla_{\Gamma}\varphi\|_{L^{2}(\Gamma(t))}^{2}.$

From our assumptions it is clear such a constant exists and depends only on
the evolution of $\Gamma(t)$ and choice of $F$. Thus one obtains (5.5) by
using Lemma 3.8 with $\omega(y)=y+y^{q}$, and choosing some sufficiently small
$t_{0}$, which is independent of $M$. ∎

By using the $L^{2}$ bound we established in the preceding proof, the Sobolev
embedding $H^{1}(\Gamma(t))\hookrightarrow L^{2q}(\Gamma(t))$, and the energy
estimate (5.5) it is clear that one obtains uniform $L^{2}_{H^{1}}$ bounds on
$\mu^{M}$. Likewise, using (3.9), one can establish uniform
$L^{2}_{\mathbf{H}^{1}}$ bounds for $\mathbf{u}_{T}^{M}$.

####  5.1.2 Passage to the limit

We have established uniform bounds for $\mathbf{u}_{T}^{M}$ in
$L^{\infty}_{\mathbf{H}_{\sigma}}$ and $L^{2}_{\mathbf{H^{1}}}$, for
$\varphi^{M}$ in $L^{\infty}_{H^{1}}$, and for $\mu^{M}$ in $L^{2}_{H^{1}}$.
Thus, there exist limiting functions $\mathbf{u}_{T},\varphi,\mu$ such that

$\displaystyle\mathbf{u}_{T}^{M}\rightharpoonup\mathbf{u}_{T},\text{ weakly in
}L^{2}_{\mathbf{V}_{\sigma}},$
$\displaystyle\mathbf{u}_{T}^{M}\overset{*}{\rightharpoonup}\mathbf{u}_{T},\text{
weak-}*\text{ in }L^{\infty}_{\mathbf{H}_{\sigma}},$
$\displaystyle\varphi^{M}\overset{*}{\rightharpoonup}\varphi,\text{
weak-}*\text{ in }L^{\infty}_{H^{1}},$
$\displaystyle\mu^{M}\rightharpoonup\mu,\text{ weakly in }L^{2}_{H^{1}}.$

Moreover, arguing as in [18], one can show that
$\varphi^{M}\rightarrow\varphi$ strongly in $L^{2}_{L^{2}}$. This is useful as
one cannot use (a variant of) the Aubin-Lions theorem to show strong
convergence of $\varphi,\mathbf{u}_{T}$ because we do not have uniform
estimates for $\varphi^{M}$ in $H^{1}_{H^{-1}}$ and $\mathbf{u}_{T}^{M}$ in
$H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$. Now by proceeding as in the proof of
[18], Proposition 4.10, one shows the existence of $\partial^{\circ}\varphi$,
but we omit the details. We compute a bound for $\partial^{\circ}\varphi$ in
$L^{2}_{H^{-1}}$ in Lemma 5.3 below.

Recalling the compact embeddings

${H^{1}}(\Gamma(t))\hookrightarrow L^{p}(\Gamma(t)),$

for all $p\in(1,\infty)$, by using [6], Theorem 6.2, one obtains strongly
convergent subsequences such that

$\displaystyle\varphi^{M}\rightarrow\varphi,\text{ strongly in
}L^{2}_{L^{p}}.$

The case $p=4$ will be useful when we discuss the passage to the limit of
$\mathbf{u}_{T}^{M}$. Similarly, as noted in [45], one has a compact embedding
$L^{2}_{\mathbf{V}_{\sigma}}\hookrightarrow L^{2}_{\mathbf{H}_{\sigma}}$, and
hence one can extract a subsequence such that

$\displaystyle\mathbf{u}_{T}^{M}\rightarrow\mathbf{u}_{T},\text{ strongly in
}L^{2}_{\mathbf{H}_{\sigma}}.$

We now discuss passage to the limit in the nonlinear terms. Firstly, it is
shown in [18] that

$F^{\prime}(\varphi^{M})\rightharpoonup F^{\prime}(\varphi),\text{ weakly in
}L^{2}_{L^{2}},$

where the authors use a generalisation of the dominated convergence theorem.
To see that we can pass to the limit in the $\mathbf{c}_{1}$ term one argues
as in [50] Lemma 3.2. Similarly, to see the convergence in the
$\mathbf{c}_{2}(\mu^{M},\varphi^{M},\boldsymbol{\phi})$ term in (5.1) we write

$\mathbf{c}_{2}(\mu^{M},\varphi^{M},\boldsymbol{\phi})=-\mathbf{c}_{2}(\varphi^{M},\mu^{M},\boldsymbol{\phi})=\mathbf{c}_{2}(\varphi-\varphi^{M},\mu^{M},\boldsymbol{\phi})-\mathbf{c}_{2}(\varphi,\mu^{M},\boldsymbol{\phi}).$

Firstly, we notice that
$\int_{0}^{T}\mathbf{c}_{2}(\varphi,\cdot,\boldsymbol{\phi})$ is an element of
$L^{2}_{H^{-1}}$ from the established bounds, and hence the weak convergence
of $\mu^{M}$ yields

$\int_{0}^{T}\mathbf{c}_{2}(\varphi,\mu^{M},\boldsymbol{\phi})\rightarrow\int_{0}^{T}\mathbf{c}_{2}(\varphi,\mu,\boldsymbol{\phi})$

as $M\rightarrow\infty$. For the other term, we use the strong convergence
$\varphi^{M}\rightarrow\varphi$ in $L^{2}_{L^{4}}$, and write

$\left|\int_{0}^{T}\mathbf{c}_{2}(\varphi-\varphi^{M},\mu^{M},\boldsymbol{\phi})\right|\leq\int_{0}^{T}\|\varphi-\varphi^{M}\|_{L^{4}(\Gamma(t))}\|\nabla_{\Gamma}\mu^{M}\|_{L^{2}(\Gamma(t))}\|\boldsymbol{\phi}\|_{\mathbf{L}^{4}(\Gamma(t))}\rightarrow
0.$

Thus we find

$\int_{0}^{T}\mathbf{c}_{2}(\mu^{M},\varphi^{M},\boldsymbol{\phi})\rightarrow-\int_{0}^{T}\mathbf{c}_{2}(\varphi,\mu,\boldsymbol{\phi})=\int_{0}^{T}\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi}),$

and by using standard localisation arguments one concludes that
$\mathbf{c}_{2}(\mu^{M},\varphi^{M},\boldsymbol{\phi})\rightarrow\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})$
for almost all $t\in[0,T]$ and all
$\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$.
$\mathbf{c}_{2}(\phi,\varphi^{M},\mathbf{u}_{T}^{M})$ converges by similar
logic, but with the added advantage that one can use strong convergence of
both $\varphi^{M}$ and $\mathbf{u}_{T}^{M}$.

The final nonlinear term to consider the convergence of is
$\hat{\mathbf{a}}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M},\boldsymbol{\phi})$.
This follows essentially the same calculations as above, owing to the
assumption that we have a Lipschitz continuous viscosity $\eta(\cdot)$. One
writes

$\hat{\mathbf{a}}(\eta(\varphi^{M}),\mathbf{u}_{T}^{M},\boldsymbol{\phi})=\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T}^{M},\boldsymbol{\phi})+\hat{\mathbf{a}}(\eta(\varphi^{M})-\eta(\varphi),\mathbf{u}_{T}^{M},\boldsymbol{\phi}),$

and as above we find
$\int_{0}^{T}\hat{\mathbf{a}}(\eta(\varphi),\cdot,\boldsymbol{\phi})$ is an
element of $L^{2}_{\mathbf{V}_{\sigma}^{\prime}}$. Likewise, the integral of
the second term over $[0,T]$ vanishes666Here one considers $\boldsymbol{\phi}$
sufficiently smooth so that $\mathbb{E}(\boldsymbol{\phi})\in
L^{\infty}_{\mathbf{L}^{4}}$ and extends to $\boldsymbol{\phi}\in
L^{2}_{\mathbf{V}_{\sigma}(t)}$ by density. as $M\rightarrow 0$ by using the
Lipschitz property of $\eta(\cdot)$ and the strong convergence of
$\varphi^{M}$ in $L^{2}_{L^{4}}$.

With these considerations, one argues as in [45] to see that
$\partial^{\circ}\mathbf{u}_{T}$ exists. We omit the details here as the main
point of interest is in the nonlinear terms. Now we know these derivatives
exist, we may show the following bounds.

######  Lemma 5.3.

We have $\partial^{\circ}\varphi\in L^{2}_{H^{-1}}$ and
$\partial^{\circ}\mathbf{u}_{T}\in L^{2}_{\mathbf{V}_{\sigma}^{\prime}}$.

###### Proof.

To see the bound for $\partial^{\circ}\varphi^{M}$ observe from (4.1) that

$|m(\partial^{\circ}\varphi,\phi)|\leq|a(\mu,\phi)|+|\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})|+|\mathbf{c}_{2}(\phi,\varphi,\widetilde{\mathbf{u}_{T}})|,$

and we recall

$\displaystyle\mathbf{c}_{2}(\phi,\varphi,\mathbf{u}_{T})=-\mathbf{c}_{2}(\varphi,\phi,\mathbf{u}_{T}),$
$\displaystyle\mathbf{c}_{2}(\phi,\varphi,\widetilde{\mathbf{u}_{T}})=m(\phi,HV_{N}\varphi)-\mathbf{c}_{2}(\varphi,\phi,\widetilde{\mathbf{u}_{T}}).$

From this it is clear to see that

$\frac{|m(\partial^{\circ}\varphi,\phi)|}{\|\phi\|_{H^{1}(\Gamma(t))}}\leq\|\nabla_{\Gamma}\mu\|_{L^{2}(\Gamma(t))}+\|\varphi\|_{L^{4}(\Gamma(t))}\|\mathbf{u}_{T}\|_{\mathbf{L}^{4}(\Gamma(t))}+\|HV_{N}\|_{L^{\infty}(\Gamma(t))}\|\varphi\|_{L^{2}(\Gamma(t))}\\\
+\|\varphi\|_{L^{4}(\Gamma(t))}\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{4}(\Gamma(t))},$

and hence using (3.7), (3.8), and the uniform bounds established by (5.5) we
obtain the $L^{2}_{H^{-1}}$ bound.

Similarly to bound $\partial^{\circ}\mathbf{u}_{T}$, we see from (5.1) that

$|\mathbf{m}(\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi})|\leq|\hat{\mathbf{a}}(\eta(\varphi),\mathbf{u}_{T},\boldsymbol{\phi})|+|\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})|+|\mathbf{l}(\mathbf{u}_{T},\boldsymbol{\phi})|+|\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})|+|\mathbf{d}_{2}(\eta(\varphi),\boldsymbol{\phi})|\\\
+|\mathbf{m}(\mathbf{B},\boldsymbol{\phi})|+|\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})|.$

Recall from properties of $\mathbf{c}_{1},\mathbf{c}_{2}$ that

$\displaystyle|\mathbf{c}_{1}(\mathbf{u}_{T},\mathbf{u}_{T},\boldsymbol{\phi})|=|\mathbf{c}_{1}(\boldsymbol{\phi},\mathbf{u}_{T},\mathbf{u}_{T})|\leq\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}\|\mathbf{u}_{T}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2},$
$\displaystyle|\mathbf{c}_{2}(\mu,\varphi,\boldsymbol{\phi})|=|\mathbf{c}_{2}(\varphi,\mu,\boldsymbol{\phi})|\leq
C\|\varphi\|_{L^{4}(\Gamma(t))}\|\nabla_{\Gamma}\mu\|_{L^{2}(\Gamma(t))}\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))},$

where we have used the Sobolev embedding
$\mathbf{H}^{1}(\Gamma(t))\hookrightarrow\mathbf{L}^{4}(\Gamma(t))$ in the
second inequality. The only other problematic term here is
$|\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})|$, which we bound as

$\displaystyle|\mathbf{d}_{1}(\mathbf{u}_{T},\boldsymbol{\phi})|$
$\displaystyle\leq|\mathbf{c}_{1}(\mathbf{u}_{T},\widetilde{\mathbf{u}_{T}},\boldsymbol{\phi})|+|\mathbf{c}_{1}(\widetilde{\mathbf{u}_{T}},\mathbf{u}_{T},\boldsymbol{\phi})|=|\mathbf{c}_{1}(\boldsymbol{\phi},\widetilde{\mathbf{u}_{T}},\mathbf{u}_{T})|+|\mathbf{c}_{1}(\boldsymbol{\phi},\mathbf{u}_{T},\widetilde{\mathbf{u}_{T}})|+2|\mathbf{m}(\mathbf{u}_{T}HV_{N},\boldsymbol{\phi})|$
$\displaystyle\leq
C\left(\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{4}(\Gamma(t))}+1\right)\|\mathbf{u}_{T}\|_{\mathbf{L}^{4}(\Gamma(t))}\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}.$

From these inequalities it is straightforward to see that

$\frac{|\mathbf{m}(\partial^{\circ}\mathbf{u}_{T},\boldsymbol{\phi})|}{\|\boldsymbol{\phi}\|_{\mathbf{H}^{1}(\Gamma(t))}}\leq\eta^{*}\|\mathbb{E}(\mathbf{u}_{T})\|_{\mathbf{L}^{2}(\Gamma(t))}+\|\mathbf{u}_{T}\|_{\mathbf{L}^{4}(\Gamma(t))}^{2}+C\|\varphi\|_{L^{4}(\Gamma(t))}\|\nabla_{\Gamma}\mu\|_{L^{2}(\Gamma(t))}+\|V_{N}\mathbb{H}\|_{\mathbf{L}^{\infty}(\Gamma(t))}\|\mathbf{u}_{T}\|_{\mathbf{L}^{2}(\Gamma(t))}\\\
+C\|\mathbf{u}_{T}\|_{\mathbf{L}^{2}(\Gamma(t))}+C\|\mathbf{u}_{T}\|_{\mathbf{L}^{4}(\Gamma(t))}\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{4}(\Gamma(t))}+\eta^{*}\|\mathbb{E}(\widetilde{\mathbf{u}_{T}})\|_{\mathbf{L}^{2}(\Gamma(t))}+\|\mathbf{B}\|_{\mathbf{L}^{2}(\Gamma(t))}.$

From this inequality it is straightforward to see how one obtains the
$L^{2}_{\mathbf{V}_{\sigma}^{\prime}}$ bound by using (3.8), (5.5), and
Sobolev embeddings where necessary. ∎

Lastly it remains to discuss the initial conditions. We note that
$\mathbf{u}_{T}^{M}(0)=P^{M}_{\mathbf{V}}\mathbf{u}_{T,0}$, and
$\varphi^{M}(0)=P^{M}_{V}\varphi_{0}$. These projections are such that
$P^{M}_{\mathbf{V}}\mathbf{u}_{T,0}\rightarrow\mathbf{u}_{T,0}$ strongly in
$\mathbf{H}_{\sigma}(0)$, and $P^{M}_{V}\varphi_{0}\rightarrow\varphi_{0}$
strongly in $H^{1}(\Gamma(0))$. By using standard arguments (again we refer to
[18, 45]) one can then verify that $\mathbf{u}_{T}(0)=\mathbf{u}_{T,0}$, and
$\varphi(0)=\varphi_{0}$. All in all we have shown Theorem 4.1.

###  5.2 The logarithmic potential

As is common in the literature, see for example [18, 25], we consider a
regularised version of this potential and use the preceding theory to show
existence. To this end, we choose $\delta\in(0,1)$ and define a regularised
function

$F_{\log}^{\delta}(r)=\begin{cases}(1-r)\log(\delta)+(1+r)\log(2-\delta)+\frac{(1-r)^{2}}{2\delta}+\frac{(1+r)^{2}}{2(2-\delta)},&r\geq
1-\delta,\\\ F_{\log}(r),&r\in(-1+\delta,1-\delta),\\\
(1+r)\log(\delta)+(1-r)\log(2-\delta)+\frac{(1+r)^{2}}{2\delta}+\frac{(1-r)^{2}}{2(2-\delta)},&r\leq-1+\delta.\end{cases}$

It is a straightforward calculation to see that $F_{\log}^{\delta}\in
C^{2}(\mathbb{R})$. As before, we also introduce shorthand notation
$f^{\delta}(r)=(F_{\log}^{\delta})^{\prime}(r)$.

We now focus on the following version of (4.4)-(4.6). We want to find
$\varphi^{\delta}\in H^{1}_{H^{-1}}\cap L^{\infty}_{H^{1}}$, $\mu^{\delta}\in
L^{2}_{H^{1}}$, and $\mathbf{u}_{T}^{\delta}\in
H^{1}_{\mathbf{V}_{\sigma}^{\prime}}\cap L^{2}_{\mathbf{V}_{\sigma}}$ such
that

$\displaystyle\begin{split}\mathbf{m}_{*}(\partial^{\circ}\mathbf{u}_{T}^{\delta},\boldsymbol{\phi})+\hat{\mathbf{a}}(\eta(\varphi^{\delta}),\mathbf{u}_{T}^{\delta},\boldsymbol{\phi})+\mathbf{c}_{1}(\mathbf{u}_{T}^{\delta},\mathbf{u}_{T}^{\delta},\boldsymbol{\phi})+\mathbf{l}(\mathbf{u}_{T}^{\delta},\boldsymbol{\phi})+\mathbf{d}_{1}(\mathbf{u}_{T}^{\delta},\boldsymbol{\phi})+\mathbf{d}_{2}(\eta(\varphi^{\delta}),\boldsymbol{\phi})\\\
=\mathbf{m}(\mathbf{B},\boldsymbol{\phi})+\mathbf{c}_{2}(\mu^{\delta},\varphi^{\delta},\boldsymbol{\phi})\end{split},$
(5.14) $\displaystyle
m_{*}(\partial^{\circ}\varphi^{\delta},\phi)+a(\mu^{\delta},\phi)+\mathbf{c}_{2}(\phi,\varphi^{\delta},\mathbf{u}_{T}^{\delta})+\mathbf{c}_{2}(\phi,\varphi^{\delta},\widetilde{\mathbf{u}_{T}}),$
(5.15) $\displaystyle m(\mu^{\delta},\phi)=\varepsilon
a(\varphi^{\delta},\phi)+\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),\phi)-\frac{1}{\varepsilon}m(\varphi^{\delta},\phi),$
(5.16)

for all $\phi\in H^{1}(\Gamma(t)),\boldsymbol{\phi}\in\mathbf{V}_{\sigma}(t)$
and almost all $t\in[0,T]$, such that
$\varphi^{\delta}(0)=\varphi_{0}\in\mathcal{I}_{0}$ and
$\mathbf{u}_{T}^{\delta}(0)=\mathbf{u}_{T,0}\in\mathbf{H}_{\sigma}(0).$

Existence then follows from Theorem 4.1. However, as our polynomial conditions
depend on our regularisation parameter, $\delta$, we now have to establish new
energy estimates before passing to the limit $\delta\rightarrow 0$.

######  Lemma 5.4.

The solution triple $(\varphi^{\delta},\mu^{\delta},\mathbf{u}_{T}^{\delta})$
is such that

$\displaystyle\sup_{t\in[0,T]}\left(\frac{1}{2}\|\mathbf{u}_{T}^{\delta}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+E^{\text{CH},\delta}[\varphi^{\delta};t]\right)+\frac{\eta_{*}}{2}\int_{0}^{T}\|\mathbb{E}(\mathbf{u}_{T}^{\delta})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{1}{2}\int_{0}^{T}\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}^{2}\leq
C,$ (5.17)

for a constant $C$ independent of $\delta$.

###### Proof.

For the interest of brevity we try to recycle as much of the proof of the
previous energy estimate. Examining the proof of Lemma 5.2 one finds that the
analogues of (5.6)-(5.11) still hold. Continuing in the same way, we find that

$-\mathbf{c}_{2}(\mu^{\delta},\varphi^{\delta},\widetilde{\mathbf{u}_{T}})=\mathbf{c}_{2}(\varphi^{\delta},\mu^{\delta},\widetilde{\mathbf{u}_{T}})-m(\mu^{\delta},HV_{N}\varphi^{\delta}).$

In particular, all we need to establish is an $L^{2}$ bound on $\mu^{\delta}$.
To do this we test (5.16) against $\varphi^{\delta}HV_{N}\in H^{1}(\Gamma(t))$
for

$\displaystyle m(\mu^{\delta},HV_{N}\varphi^{\delta})=\varepsilon
a(\varphi^{\delta},HV_{N}\varphi^{\delta})+\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),\varphi^{\delta}HV_{N})-\frac{1}{\varepsilon}m(\varphi^{\delta},\varphi^{\delta}HV_{N}).$
(5.18)

We then notice that $rf^{\delta}(r)\geq 0$, and so

$\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),\varphi^{\delta}HV_{N})\leq\frac{\theta}{2\varepsilon}\|HV_{N}\|_{L^{\infty}(\Gamma(t))}\|f^{\delta}(\varphi^{\delta})\varphi^{\delta}\|_{L^{1}(\Gamma(t))}=\frac{\theta}{2\varepsilon}\|HV_{N}\|_{L^{\infty}(\Gamma(t))}m(f^{\delta}(\varphi^{\delta}),\varphi^{\delta}),$

and using (5.16) we see

$\displaystyle\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),\varphi^{\delta})=m(\mu^{\delta},\varphi^{\delta})-\varepsilon
a(\varphi^{\delta},\varphi^{\delta})+\frac{1}{\varepsilon}m(\varphi^{\delta},\varphi^{\delta}).$
(5.19)

Now by using the inverse Laplacian, we see

$m(\mu^{\delta},\varphi^{\delta})=m\left(\mu^{\delta},\varphi^{\delta}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right)+m\left(\mu^{\delta},\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right)=a\left(\mu^{\delta},\mathcal{G}\left(\varphi^{\delta}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right)\right)+m\left(\mu^{\delta},\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right).$

Now by using the definition of the inverse Laplacian, and Poincaré’s
inequality one finds

$a\left(\mu^{\delta},\mathcal{G}\left(\varphi^{\delta}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right)\right)\leq
C\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))}.$

Also, as we chose $\varphi_{0}\in\mathcal{I}_{0}$ there exists some constant
$\alpha<1$ such that

$m\left(\mu^{\delta},\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\right)=m(\mu^{\delta},1)\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\Gamma(t)}\varphi^{\delta}\leq|m(\mu^{\delta},1)|m_{\varphi_{0}}(t)\leq\alpha|m(\mu^{\delta},1)|,$

and hence we find

$\displaystyle|m(\mu^{\delta},\varphi^{\delta})|\leq
C\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))}+\alpha|m(\mu^{\delta},1)|.$
(5.20)

Testing (5.15) against $\phi\equiv 1$ and noting that $f^{\delta}(r)\leq
rf^{\delta}(r)+1$,

$m(\mu^{\delta},1)=\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),1)-\frac{1}{\varepsilon}m(\varphi^{\delta},1)\leq\frac{\theta}{2\varepsilon}|\Gamma(t)|+\frac{\theta}{2\varepsilon}m(f^{\delta}(\varphi^{\delta}),\varphi^{\delta})-\frac{1}{\varepsilon}m(\varphi^{\delta},1),$

thus using (5.19) we see

$\displaystyle|m(\mu^{\delta},1)|\leq
C+C\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t)}^{2}+|m(\varphi^{\delta},\mu^{\delta})|,$
(5.21)

where we have used Young’s and Poincaré’s inequalities as appropriate. Now by
combining (5.20), (5.21) we see

$|m(\mu^{\delta},\varphi^{\delta})|\leq
C+C\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))}^{2}+C\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))},$

for constants which depend on $\alpha$. Now using this and (5.19) in (5.18)
one can show that

$|m(\mu^{\delta},HV_{N}\varphi^{\delta})|\leq
C+C\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))}^{2}+C\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))},$

for constants which depend on
$\sup_{t\in[0,T]}\|HV_{N}\|_{H^{1,\infty}(\Gamma(t))}$. Now by using Young’s
inequality it then follows that

$|\mathbf{c}_{2}(\mu^{\delta},\varphi^{\delta},\widetilde{\mathbf{u}_{T}})|\leq
C+C\|\nabla_{\Gamma}\varphi^{\delta}\|_{L^{2}(\Gamma(t))}^{2}+\frac{1}{2}\|\nabla_{\Gamma}\mu^{\delta}\|_{L^{2}(\Gamma(t))}^{2},$

which is the analogue of the bound (5.12). Combining this with the analogues
of (5.6)-(5.11) it is straightforward to see that

$\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+E^{\text{CH},\delta}[\varphi^{M};t]+\frac{\eta_{*}}{2}\int_{0}^{t}\|\mathbb{E}(\mathbf{u}_{T}^{M})\|_{\mathbf{L}^{2}(\Gamma(t))}^{2}+\frac{1}{2}\int_{0}^{t}\|\nabla_{\Gamma}\mu^{M}\|_{L^{2}(\Gamma(t))}^{2}\leq
k\\\
+\int_{0}^{t}K(s)\left(\frac{1}{2}\|\mathbf{u}_{T}^{M}\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}+E^{\text{CH},\delta}[\varphi^{M};s]\right)\,ds,$
(5.22)
$k=C+\frac{1}{2}\int_{0}^{T}\|\mathbf{B}\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}\,ds+\frac{2(\eta^{*})^{2}}{\eta_{*}}\int_{0}^{T}\|\mathbb{E}(\widetilde{\mathbf{u}_{T}})\|_{\mathbf{L}^{2}(\Gamma(s))}^{2}\,ds+\frac{1}{2}\|\mathbf{u}_{T,0}\|_{\mathbf{L}^{2}(\Gamma(0))}^{2}+E^{\text{CH},\delta}[\varphi_{0};0],$

and

$\displaystyle
K(s)=C+\frac{1}{2}\|HV_{N}\|_{L^{\infty}(\Gamma(s))}+\|V_{N}\mathbb{H}\|_{\mathbf{L}^{\infty}(\Gamma(s))}+\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{H}^{1,\infty}(\Gamma(s))}+C\|\widetilde{\mathbf{u}_{T}}\|_{\mathbf{L}^{\infty}(\Gamma(s))}^{\frac{4}{3}}.$

The claim then follows from Grönwall’s inequality once we establish that
$E^{\text{CH},\delta}[\varphi_{0};0]$ is independent of $\delta$. To see this
we find that

$E^{\text{CH},\delta}[\varphi_{0};0]=\int_{\Gamma(t)}\frac{\varepsilon|\nabla_{\Gamma}\varphi_{0}|^{2}}{2}+\frac{\theta}{2\varepsilon}F_{\log}^{\delta}(\varphi_{0})=\frac{\varepsilon}{2}\|\varphi_{0}\|_{H^{1}(\Gamma_{0})}^{2}+\frac{1}{2\varepsilon}\|1-\varphi_{0}\|_{L^{2}(\Gamma_{0})}^{2}+\frac{\theta}{2\varepsilon}\int_{\\{|\varphi_{0}|<1-\delta\\}}F_{\log}(\varphi_{0})\\\
+\frac{\theta}{2\varepsilon}\int_{\\{\varphi_{0}\geq
1-\delta\\}}F_{\log}^{\delta}(\varphi_{0})+\frac{\theta}{2\varepsilon}\int_{\\{\varphi_{0}\leq-1+\delta\\}}F_{\log}^{\delta}(\varphi_{0}),$

and so it is only necessary to bound the last three terms. Firstly, from the
definition of $F_{\log}(\cdot)$, it is straightforward to see that

$\frac{\theta}{2\varepsilon}\int_{\\{|\varphi_{0}|<1-\delta\\}}F_{\log}(\varphi_{0})\leq\frac{\theta\log(2)|\Gamma_{0}|}{\varepsilon}.$

For the terms involving $F_{\log}^{\delta}(\cdot)$ we use the set they’re
integrating over to bound them, for example

$\displaystyle\int_{\\{\varphi_{0}\geq
1-\delta\\}}(1-\varphi_{0})\log(\delta)+(1+\varphi_{0})\log(2-\delta)+\frac{(1-\varphi_{0})^{2}}{2\delta}+\frac{(1+\varphi_{0})^{2}}{2(2-\delta)}$
$\displaystyle\leq\int_{\Gamma_{0}}-\delta\log(\delta)+2\log(2)+\frac{\delta}{2}+2$
$\displaystyle\leq\left(\frac{1}{e}+2\log(2)+\frac{5}{2}\right)|\Gamma_{0}|,$

and the other term is bounded similarly. Thus we see
$E^{\text{CH},\delta}[\varphi_{0};0]$ is bounded independent of $\delta$ and
we obtain (5.17). ∎

Notice that this proof also gave us uniform bounds for $\mu^{\delta}$ in
$L^{2}_{L^{2}}$. We now use this energy estimate to obtain some further
$\delta$-independent bounds.

######  Lemma 5.5.

There exists a constant $C$ independent of $\delta$ such that:

  1. 1.

$\displaystyle\int_{0}^{T}\|\partial^{\circ}\mathbf{u}_{T}^{\delta}\|_{\mathbf{V}_{\sigma}(t)^{\prime}}^{2}+\int_{0}^{T}\|\partial^{\circ}\varphi^{\delta}\|_{H^{-1}(\Gamma(t))}^{2}\leq
C,$

  2. 2.

$\displaystyle\int_{0}^{T}\|f^{\delta}(\varphi^{\delta})\|_{L^{2}(\Gamma(t))}^{2}\leq
C.$

###### Proof.

The proof for the bounds on the time derivatives is identical to that of Lemma
5.3, and is hence omitted. The bound for the regularised potential term was
not required in the setting of a smooth potential as we used the polynomial
growth conditions and Sobolev embeddings to bound this term. Testing (5.16)
against $f^{\delta}(\varphi^{\delta})\in H^{1}(\Gamma(t))$ yields

$\|f^{\delta}(\varphi^{\delta})\|_{L^{2}(\Gamma(t)}^{2}=\frac{2\varepsilon}{\theta}m(\mu^{\delta},\varphi^{\delta})-\frac{2\varepsilon^{2}}{\theta}a(\varphi^{\delta},f^{\delta}(\varphi^{\delta}))+\frac{2}{\theta}m(\varphi^{\delta},f^{\delta}(\varphi^{\delta})).$

Now we notice that as $(f^{\delta})^{\prime}(\cdot)\geq 0$ one has

$-a(\varphi^{\delta},f^{\delta}(\varphi^{\delta}))=-\int_{\Gamma(t)}(f^{\delta})^{\prime}(\varphi^{\delta})\nabla_{\Gamma}\varphi^{\delta}\cdot\nabla_{\Gamma}\varphi^{\delta}\leq
0,$

hence by using Young’s inequality it is clear that

$\|f^{\delta}(\varphi^{\delta})\|_{L^{2}(\Gamma(t)}^{2}\leq\frac{8}{\theta^{2}}\|\varphi^{\delta}\|_{L^{2}(\Gamma(t))}^{2}+\varepsilon^{2}\|\mu^{\delta}\|_{L^{2}(\Gamma(t))}^{2}.$

The bound then follows from integrating in time and using the uniform
$L^{2}_{L^{2}}$ bounds. ∎

####  5.2.1 Passage to the limit

Now we want to pass to the limit as $\delta\rightarrow 0$. We have established
uniform bounds for $\mathbf{u}_{T}^{\delta}$ in
$L^{\infty}_{\mathbf{L}^{2}},L^{2}_{\mathbf{H^{1}}}$ and
$H^{1}_{\mathbf{V}_{\sigma}^{\prime}}$, hence there is a limiting function
$\mathbf{u}_{T}$ such that

$\displaystyle\mathbf{u}_{T}^{\delta}\rightharpoonup\mathbf{u}_{T},\text{
weakly in }L^{2}_{\mathbf{V}_{\sigma}},$
$\displaystyle\mathbf{u}_{T}^{\delta}\overset{*}{\rightharpoonup}\mathbf{u}_{T},\text{
weak-}*\text{ in }L^{\infty}_{\mathbf{L}^{2}},$
$\displaystyle\mathbf{u}_{T}^{\delta}\rightharpoonup\mathbf{u}_{T},\text{
weakly in }H^{1}_{\mathbf{V}_{\sigma}^{\prime}}.$

Likewise $\varphi^{\delta}$ is uniformly bounded in $L^{\infty}_{H^{1}}$ and
$H^{1}_{H^{-1}}$, and $\mu^{\delta}$ is uniformly bounded in $L^{2}_{H^{1}}$.
Thus, there exist limiting functions $\varphi,\mu$ such that

$\displaystyle\varphi^{\delta}\overset{*}{\rightharpoonup}\varphi,\text{
weak-}*\text{ in }L^{\infty}_{H^{1}},$
$\displaystyle\varphi^{\delta}\rightharpoonup\varphi,\text{ weakly in
}H^{1}_{H^{-1}},$ $\displaystyle\mu^{\delta}\rightharpoonup\mu,\text{ weakly
in }L^{2}_{H^{1}}.$

Lastly, the uniform bounds for $f^{\delta}(\varphi^{\delta})$ in
$L^{2}_{L^{2}}$ imply the existence of some $\tilde{\varphi}\in L^{2}_{L^{2}}$
such that

$\displaystyle f^{\delta}(\varphi^{\delta})\rightharpoonup\tilde{f},\text{
weakly in }L^{2}_{L^{2}}.$

One also has the same strong convergence properties as seen for smooth
potentials, which follow from the relevant compact embeddings.

It remains for us to show that the limiting functions are such that
$|\varphi(t)|<1$ almost everywhere on $\Gamma(t)$ for almost all $t\in[0,T]$,
and that $\tilde{f}=f(\varphi)$. From the piecewise definition of $f^{\delta}$
it is clear that these two issues are related. We firstly recall a result from
[18].

######  Lemma 5.6 ([18], Lemma 5.8).

There exist constants $C_{1},C_{2}$ independent of $\delta,t$ such that

$\int_{\Gamma(t)}[\varphi^{\delta}(t)-1]_{+}+\int_{\Gamma(t)}[-\varphi^{\delta}(t)-1]_{+}\leq
C_{1}\delta+\frac{C_{2}}{|\log(\delta)|},$

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

The proof of this result is unchanged with the coupling with the Navier-Stokes
equations, and hence follows without any adaptations. In particular, in the
limit $\delta\rightarrow 0$, we find that $|\varphi(t)|\leq 1$ almost
everywhere on $\Gamma(t)$ for almost all $t\in[0,T]$. There is still the issue
of the set of values such that $|\varphi(t)|=1$, which we discuss using the
same arguments as in [18].

######  Lemma 5.7.

For almost all $t\in[0,T]$ the set $\\{x\in\Gamma(t)\mid|\varphi(x,t)|=1\\}$
has $\mathcal{H}^{2}$ measure 0.

###### Proof.

To begin, we claim that up to a subsequence of $\delta\rightarrow 0$ we have
for almost all $t\in[0,T]$ that

$\lim_{\delta\rightarrow
0}f^{\delta}(\varphi^{\delta}(t))=\begin{cases}f(\varphi(t)),&\text{ if
}|\varphi(t)|<1\text{ almost everywhere on }\Gamma(t),\\\ \infty,&\text{
otherwise.}\end{cases}$

This is in fact exactly the content of [18], Lemma 5.10 - and so we do not
prove this. Then we have that

$\lim_{\delta\rightarrow
0}\varphi^{\delta}(t)f^{\delta}(\varphi^{\delta}(t))=\begin{cases}\varphi(t)f(\varphi(t)),&\text{
if }|\varphi(t)|<1\text{ almost everywhere on }\Gamma(t),\\\ \infty,&\text{
otherwise,}\end{cases}$

pointwise, almost everywhere on $\Gamma(t)$ for almost all $t\in[0,T]$. Now by
testing (5.16) against $\varphi^{\delta}$, integrating in time, and using the
established uniform bounds for $\varphi^{\delta}$ and $\mu^{\delta}$ one finds

$\int_{0}^{T}\int_{\Gamma(t)}\varphi^{\delta}f^{\delta}(\varphi^{\delta})\leq
C,$

for a constant $C$ independent of $\delta$. Now noting that
$rf^{\delta}(r)\geq 0$, by Fatou’s lemma one has

$\int_{0}^{T}\int_{\Gamma(t)}\liminf_{\delta\rightarrow
0}\varphi^{\delta}f^{\delta}(\varphi^{\delta})\leq\liminf_{\delta\rightarrow
0}\int_{0}^{T}\int_{\Gamma(t)}\varphi^{\delta}f^{\delta}(\varphi^{\delta})\leq
C.$

From the claim it is now evident that one necessarily has that for almost all
$t\in[0,T]$ the set $\\{x\in\Gamma(t)\mid|\varphi(x,t)|=1\\}$ has measure 0. ∎

In particular, we see that $f^{\delta}(\varphi^{\delta}(x,t))\rightarrow
f(\varphi(x,t))$ for almost all $x\in\Gamma(t),t\in[0,T]$. Hence, again by
using Fatou’s lemma, one finds

$\int_{0}^{T}\|f(\varphi)\|_{L^{2}(\Gamma(t))}^{2}=\int_{0}^{T}\int_{\Gamma(t)}\liminf_{\delta\rightarrow
0}|f^{\delta}(\varphi^{\delta})|^{2}\leq\liminf_{\delta\rightarrow