# The sharp interface limit of a Navier–Stokes/Allen–Cahn system with constant
mobility: Convergence rates by a relative energy approach

Sebastian Hensel  Hausdorff Center for Mathematics, Universität Bonn,
Endenicher Allee 62, 53115 Bonn, Germany<EMAIL_ADDRESS>and
Yuning Liu  NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China, and
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan
Road North, Shanghai, 200062, China<EMAIL_ADDRESS>
###### Abstract.

We investigate the sharp interface limit of a diffuse interface system that
couples the Allen–Cahn equation with the instationary Navier–Stokes system in
a bounded domain in $\mathbb{R}^{d}$ with $d\in\\{2,3\\}$. This model is used
to describe a propagating front in a viscous incompressible flow with the
width of the transition layer being characterized by a small parameter
$\varepsilon>0$. We show that the solutions converge to a limit two-phase
fluid system with surface tension that couples the mean curvature flow and the
Navier–Stokes system. The main assumptions are that the evolution of the limit
system is sufficiently regular and that the associated evolving interface does
not intersect the boundary of the container. For quantitatively well-prepared
initial data, we even establish an optimal convergence rate. This is the first
rigorous result of this kind which is valid in all physically relevant ambient
dimensions.

Keywords: Allen–Cahn equation, complex fluid, relative entropy, mean curvature
flow, sharp interface limit.

Mathematical Subject Classification: Primary: 76T99; Secondary: 76D45, 76D05,
35R35, 53E10, 35K57.

###### Contents

  1. 1 Introduction
    1. 1.1 Context
    2. 1.2 The phase-field model and its sharp interface limit
  2. 2 Main result
  3. 3 The relative energy functional
    1. 3.1 Definition of the relative energy functional
    2. 3.2 Coercivity properties of the relative energy functional
    3. 3.3 Error in the phase indicators
  4. 4 Estimate on the time evolution of the relative energy
  5. 5 Time evolution of the error in the phase indicators
  6. 6 Proof of main results
  7. A Well-posedness of the sharp interface limit model
    1. A.1 Preliminaries
    2. A.2 The work of Abels and Moser [zbMATH06951007]
    3. A.3 Proof of Proposition 8.

##  1\. Introduction

###  1.1. Context

Curvature driven interface evolution is a challenging topic in PDE theory due
to the inherent emergence of topology changes. Indeed, classical descriptions
(i.e., via parametrization of the evolving interface) cease to work once the
solution approaches the first time of such a topology change. One popular tool
to describe the dynamics even after these consists of phase-field models,
where instead of a sharp interface one considers a diffuse interface layer of
finite width (which is typically small related to a given scaling parameter
$\varepsilon$). In the specific example of the evolution of two
(macroscopically) immiscible fluids in the presence of surface tension, an
associated fundamental phase-field model is the so-called model H coupling
Navier–Stokes dynamics with the fourth-order Cahn–Hilliard dynamics. This
model was in fact introduced in [GURTIN1996, HohenbergHalperin].

Next to obvious PDE questions concerning existence and uniqueness of solutions
to such a phase-field model, another highly relevant and natural question is
the consistency with a sharp interface model. Indeed, by formally taking the
limit $\varepsilon\to 0$ one hopes to identify the underlying dynamics in
terms of the sharp interface model under consideration, and therefore to
justify the transition to the phase-field model. Needless to say, for
numerical applications apart from qualitative convergence results also
quantitative results in the form of convergence rates in certain norms are of
interest.

In the context of model H (and even for more general models allowing, e.g., a
non-constant density), the formal derivation of the associated sharp interface
limit model by means of asymptotic expansion techniques is performed in
[AbelsGarckeGruen]. Rigorous justifications of these arguments are scarce
though. For global-in-time convergence of solutions to model H (or a related
model) to a rather weak solution to the sharp interface limit model (i.e., a
concept of varifold solutions), one may consult [AbelsLengeler] (or
[AbelsRoeger], respectively). Short-time (quantitative) convergence to strong
solutions of the sharp interface limit model is in turn shown in the recent
works [Abels2021a] and [Abels2021b], where, however, only the setting of the
stationary Stokes operator is treated. At the time of this writing, the
rigorous derivation of convergence rates incorporating the full Navier–Stokes
dynamics in model H remains an important open problem in the field.

In the present work, we study instead of model H (or related models) a phase-
field model with fluid mechanical coupling in ambient dimension
$d\in\\{2,3\\}$ which is based on the second-order Allen–Cahn operator. We
further consider the scaling regime of constant mobility, and with respect to
the fluid mechanical modeling we restrict ourselves to the case of
Navier–Stokes dynamics with constant density and constant viscosity (we refer
to Subsection 1.2 for a mathematical formulation of the model). Phase-field
models of such Navier–Stokes/Allen–Cahn type were first introduced in
[LiuShen, Jiang2017]. Formal asymptotic expansions (cf. [Abels2022a]) suggest
that the associated sharp-interface limit model is given by a Navier–Stokes
two-phase flow with surface tension, where, as a consequence of the constant
mobility assumption, the interface separating the two fluid phases is not
merely transported by the fluids but is also subject to mean curvature flow
(we again refer to Subsection 1.2 for a mathematical formulation of the
corresponding model). In particular, the mass of each individual fluid phase
is not preserved and the resulting model may be interpreted as a simplified
model for two-phase fluid flow incorporating phase transitions and surface
tension. For global-in-time existence of weak solutions to the phase-field
model, we refer to [MR1329830] (see also [Gal2010] for longtime behavior of
solutions).

In this setting, the main result of the present work rigorously justifies the
formally derived sharp interface limit model in ambient dimension
$d\in\\{2,3\\}$. Our convergence result holds true on the time horizon of
existence of a (sufficiently regular) strong solution of the sharp interface
limit. We even derive sharp convergence rates in strong norms under the
assumption of correspondingly well-prepared initial data. For precise
mathematical statements of these two main results, we refer to Theorem 1 and
Corollary 2 below. We mention that a similar convergence result was recently
established in [Abels2022] (cf. also [Jiang2022]) by a completely different
approach (we provide further comments on the methods later). The results of
[Abels2022] are, however, restricted to ambient dimension $d=2$, so that our
work, to the best of our knowledge, is the first to rigorously justify the
above sharp interface limit in all physically relevant dimensions. It has to
be said, though, that the authors of [Abels2022] are in addition able to treat
the regime of different viscosities. Even though we expect this to be
manageable also in the framework of our strategy (without restrictions on the
ambient dimension), this may very well lead to a doubling of the length of the
present paper (cf. the corresponding challenges in [Fischer2020c]). For this
reason, we restrict ourselves to the most basic setting. We finally mention
that [AbelsLiu] contains a preliminary convergence result preceding ours and
the one of [Abels2022], where the authors replace the full Navier–Stokes
dynamics by the stationary Stokes operator (and again only in ambient
dimension $d=2$).

Without coupling to a fluid mechanical system, the rigorous convergence of
phase-field models to sharp interface evolutions is of course an already
extensively studied subject in the literature. For (qualitative and/or
quantitative) convergence of solutions of the Allen–Cahn equation towards
various weak or strong notions of solutions for mean curvature flow, one may
consult, e.g., the works [Evans1992, DeMottoni1995, ilmanen, mugnai-roeger,
serfaty, Fischer2020b, Hensel2021l]. For the inclusion of constant contact
angles, corresponding results can be found in [Owen1992, Katsoulakis1995,
Barles1998, Lio2003, Mizuno2015, Kagaya2018a, Abels2019, Abels2021,
Hensel2021d, Hensel2021c], whereas (qualitative and/or quantitative)
convergence of solutions of the vectorial Allen–Cahn equation towards
evolution by multiphase mean curvature flow is the subject of [LauxSimon18,
Fischer2022]. The results on the connection of the Allen–Cahn equation with
mean curvature flow have a fourth-order analogue, namely, the convergence of
solutions of the Cahn–Hilliard equation towards evolution by Mullins–Sekerka
flow as, e.g., shown in [Chen, Alikakos1994, Le2008] (cf. also the discussion
in [serfaty]). For a corresponding result with disparate mobilities, we refer
to the recent work [Kroemer2021]. Finally, we mention [Laux2021] for a scaling
limit result modeling nematic-isotropic phase transitions in the context of
Landau–De Gennes theory of liquid crystals.

In order to establish the above convergence results, a variety of techniques
and frameworks is used depending on the precise goals (i.e., qualitative long-
time convergence towards weak solutions of the sharp interface limit model vs.
quantitative convergence towards sufficiently smooth solutions of the sharp
interface limit model until the latter run into their first topology change).
In fact, most of the above works are either based on notions from geometric
measure theory (e.g., varifolds and their first variations), gradient flow
techniques in the spirit of [Sandier2004] (cf. also [serfaty]), or combining
rigorous asymptotic expansions with a stability analysis of the linearized
operator associated with the phase-field model. Especially in the context of
the previously mentioned quantitative convergence results incorporating a
fluid mechanical coupling, the latter strategy seems to be the only one used
so far to the best of our knowledge.

However, in the recent inspiring work [Fischer2020b], a new approach for the
derivation of convergence rates was introduced and implemented for the
simplest setting of the Allen–Cahn equation posed on the full space
$\mathbb{R}^{d}$. This strategy is closest to the gradient flow perspective in
the sense that it first generates a distance measure for the difference of the
phase-field and sharp interface solutions based on the phase-field energy, and
then estimates its time evolution by a Gronwall-type argument based, amongst
other things, on the dissipation structure of the phase-field model. The
method developed in [Fischer2020b] is therefore reminiscent of a well-
established technique to establish (weak-strong) uniqueness of solutions in
the context of a variety of classical continuum mechanics models (e.g.,
incompressible and compressible Navier–Stokes flow, or conservation laws): the
so-called relative entropy method. In fact, the work [Fischer2020b] draws
motivation from recent results extending the relative entropy method (and thus
weak-strong uniqueness) to problems incorporating geometric evolution. This
was first implemented for binormal curvature flow of curves in
$\mathbb{R}^{3}$ in [JerrardSmets], or for Navier–Stokes two-phase flow with
surface tension in [Fischer2020c]. Subsequent extensions are able to deal with
multiphase mean curvature flow [Fischer2020a] (cf. also [Hensel2021]), contact
angle problems [Hensel2021m, Hensel2021d], or a novel notion of varifold
solutions for mean curvature flow in the spirit of ideas from De Giorgi
[Hensel2021l]. In view of these developments, it is not surprising that also
[Fischer2020b] already led to several follow-up works in the context of
scaling limits for phase-field models, see [Laux2021, Hensel2021c, Liu2021,
Fischer2022].

In the present work, we continue this story and extend the approach from
[Fischer2020b] to the case of a Navier–Stokes/Allen–Cahn model with constant
mobility. We refer the reader to Section 2 and especially Section 3 for a
mathematical account on our strategy.

###  1.2. The phase-field model and its sharp interface limit

Let $\Omega\subset{\mathbb{R}^{d}}$, $d\in\\{2,3\\}$, be a bounded domain with
orientable and $C^{2}$-boundary. We then consider the following most basic
Navier–Stokes/Allen–Cahn problem in $\Omega$

$\displaystyle\partial_{t}v_{\varepsilon}+(v_{\varepsilon}\cdot\nabla)v_{\varepsilon}$
$\displaystyle=\Delta
v_{\varepsilon}-\nabla\pi_{\varepsilon}-\nabla\cdot\big{(}\varepsilon\nabla\varphi_{\varepsilon}\otimes\nabla\varphi_{\varepsilon}\big{)}$
$\displaystyle\text{in }\Omega\times(0,T_{0}),$ (1a) $\displaystyle\nabla\cdot
v_{\varepsilon}$ $\displaystyle=0$ $\displaystyle\text{in
}\Omega\times(0,T_{0}),$ (1b)
$\displaystyle\partial_{t}\varphi_{\varepsilon}+(v_{\varepsilon}\cdot\nabla)\varphi_{\varepsilon}$
$\displaystyle=\Delta\varphi_{\varepsilon}-\smash{\frac{1}{\varepsilon^{2}}W^{\prime}(\varphi_{\varepsilon})}$
$\displaystyle\text{in }\Omega\times(0,T_{0}),$ (1c) $\displaystyle
v_{\varepsilon}(\cdot,0)$ $\displaystyle=v_{\varepsilon,0}$
$\displaystyle\text{in }\Omega,$ (1d)
$\displaystyle\varphi_{\varepsilon}(\cdot,0)$
$\displaystyle=\varphi_{\varepsilon,0}\in[-1,1]$ $\displaystyle\text{in
}\Omega,$ (1e) $\displaystyle(\varphi_{\varepsilon},v_{\varepsilon})$
$\displaystyle=(-1,0)$ $\displaystyle\text{on }\partial\Omega\times(0,T_{0}).$
(1f)

Here, $W$ denotes a double-well potential satisfying standard assumptions
(see, e.g., [AbelsLiu]). We fix the associated surface tension constant by

$\displaystyle c_{0}:=\int_{-1}^{1}\sqrt{2W(r)}\,dr.$

In the sharp interface limit $\varepsilon\to 0$ one expects to obtain a
simplified model for a two-phase fluid flow with phase transitions, which in
our case reads

$\displaystyle\partial_{t}v+(v\cdot\nabla)v$ $\displaystyle=\Delta
v-\nabla\pi$ $\displaystyle\text{in
}\big{(}\Omega{\times}(0,T_{*})\big{)}\setminus\mathcal{I},$ (2a)
$\displaystyle\nabla\cdot v$ $\displaystyle=0$ $\displaystyle\text{in
}\Omega{\times}(0,T_{*}),$ (2b)
$\displaystyle\partial_{t}\chi+(v\cdot\nabla)\chi$
$\displaystyle=-H_{\mathcal{I}}|\nabla\chi|$ $\displaystyle\text{in
}\Omega{\times}(0,T_{*}),$ (2c) $\displaystyle[\\![v]\\!]$ $\displaystyle=0,$
$\displaystyle\text{on }\mathcal{I},$ (2d)
$\displaystyle[\\![2\nabla^{\mathrm{sym}}v-p\mathrm{Id}]\\!]\frac{\nabla\chi}{|\nabla\chi|}$
$\displaystyle=c_{0}H_{\mathcal{I}}\frac{\nabla\chi}{|\nabla\chi|}$
$\displaystyle\text{on }\mathcal{I},$ (2e) $\displaystyle v(\cdot,0)$
$\displaystyle=v_{0}$ $\displaystyle\text{in }\Omega,$ (2f)
$\displaystyle\chi(\cdot,0)$ $\displaystyle=\chi_{0}$ $\displaystyle\text{in
}\Omega,$ (2g) $\displaystyle(\chi,v)$ $\displaystyle=(0,0)$
$\displaystyle\text{on }\partial\Omega{\times}(0,T_{*}).$ (2h)

In the above system, $\chi$ denotes a time-dependent characteristic function.
From now on, we assume that we are provided with a sufficiently regular
solution $(\chi,v)$ for the sharp interface limit model (for precise
assumptions, we refer to Definition 4 below). In particular,
$\operatorname{supp}|\nabla\chi|\cap(\Omega{\times}[0,T_{*}))$ shall model a
sufficiently regular interface

$\mathcal{I}={\bigcup_{t\in[0,T_{*})}\mathcal{I}(t){\times}\\{t\\}}$ (3)

in space-time associated with an open and sufficiently regular set

$\Omega_{+}={\bigcup_{t\in[0,T_{*})}\Omega_{+}(t){\times}\\{t\\}}\subset\Omega{\times}(0,T_{*}).$

The map $\chi$ is assumed to be $\equiv 1$ in $\Omega_{+}$ as well as $\equiv
0$ in $(\Omega{\times}[0,T_{*}))\setminus\overline{\Omega_{+}}$. Denoting for
all $t\in(0,T_{*})$ by $\mathbf{n}_{\mathcal{I}}(\cdot,t)$ the associated unit
normal along $\mathcal{I}(t)$ pointing inside $\Omega_{+}(t)$, the evolution
equation for the interface then translates into (all scalar geometric
quantities are oriented with respect to the normal vector field
$\mathbf{n}_{\mathcal{I}}$)

$\displaystyle V_{\mathcal{I}}$ $\displaystyle=\mathbf{n}_{\mathcal{I}}\cdot
v+H_{\mathcal{I}}$ $\displaystyle\text{on }\mathcal{I}.$ (4)

Here, $V_{\mathcal{I}}(\cdot,t)$ and $H_{\mathcal{I}}(\cdot,t)$ denote the
normal speed and the scalar mean curvature of the interface $\mathcal{I}(t)$,
$t\in[0,T_{*})$. Finally, in order to avoid issues originating from contact
point dynamics, we assume throughout the rest of this work that

$\displaystyle\overline{\Omega_{+}(t)}$ $\displaystyle\subset\Omega$
$\displaystyle\text{for all }t\in[0,T_{*}).$ (5)

##  2\. Main result

The main result of the present work is concerned with a rigorous convergence
result for solutions of the phase-field model (1a)–(1f) (even quantitatively
with sharp convergence rates when starting from well-prepared initial data)
towards strong solutions of the sharp interface limit model (2a)–(2h). As
already mentioned in the introduction, our approach is not based on the
strategy of combining rigorous asymptotic expansions with stability estimates
for the underlying linearized operator. In contrast, our results are
facilitated by the introduction of two error functionals which aim to encode
the difference between a solution of the phase-field model and a solution for
the expected sharp interface limit. More precisely, directly inspired by the
recent work of Fischer, Laux and Simon [Fischer2020b] on the (quantitative)
convergence of solutions of the Allen–Cahn equation to classical solutions of
mean curvature flow, we will work on $[0,T_{*})$ with a relative energy

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$
$\displaystyle:=\int_{\Omega}\frac{1}{2}\big{|}v_{\varepsilon}{-}v\big{|}^{2}\,dx$
(6) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}\big{|}^{2}+\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon}\big{)}-\xi\cdot\nabla\Big{(}\int_{-1}^{\varphi_{\varepsilon}}\sqrt{2W(r)}\,dr\Big{)}\,dx$

as well as with a suitable measure for the difference in the phase indicators

$\displaystyle
E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]:=\int_{\Omega}\bigg{|}c_{0}\chi-\int_{-1}^{\varphi_{\varepsilon}}\sqrt{2W(r)}\,dr\bigg{|}|\vartheta|\,dx.$
(7)

In the above definitions, the vector field $\xi$ will be a suitable extension
of the unit normal vector field of the smoothly evolving interface
$\mathcal{I}$ whereas $\vartheta$ will be a suitable truncation of the
associated signed distance function (for precise assumptions on $\xi$ and
$\vartheta$, we refer to Subsection 3.1 and Subsection 3.3, respectively).
Both qualitative and quantitative convergence then follows from a Gronwall-
type stability estimate for the two error functionals (6) and (7). The precise
statement reads as follows.

######  Theorem 1 (Error estimates for general initial data).

Let $d\in\\{2,3\\}$, let $\Omega\subset{\mathbb{R}^{d}}$ be a bounded domain
with orientable and $C^{2}$-boundary $\partial\Omega$, and consider two finite
time horizons $0<T_{*}\leq T_{0}<\infty$. Let $\varepsilon\in(0,1)$ and let
$(\varphi_{\varepsilon},v_{\varepsilon})$ be a weak solution of the
Navier–Stokes/Allen–Cahn system  _( 1a)–(1f)_ with time horizon $T_{0}$ and
initial data $(\varphi_{\varepsilon,0},v_{\varepsilon,0})$ in the sense of
Definition 3. Finally, let $(\chi,v)$ be a strong solution for the sharp
interface limit model  _( 2a)–(2h)_ with time horizon $T_{*}$ and initial data
$(\chi_{0},v_{0})$ in the sense of Definition 4.

Then, for a.e. $T\in(0,T_{*})$ there exists a constant
$C=C(\chi,v,T)\in(0,\infty)$, a continuous vector field
$\xi{=}\xi(\chi,v,T)\colon\Omega{\times}[0,T]\to{\mathbb{R}^{d}}$, and a
continuous weight
$\vartheta{=}\vartheta(\chi,v,T)\colon\Omega{\times}[0,T]\to\mathbb{R}$ such
that the relative energy $E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$ and
the error in the phase indicators
$E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]$ defined by (6) and (7),
respectively, satisfy the estimates

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)$
$\displaystyle\leq
e^{Ct}\big{(}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)+E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)\big{)},$
$\displaystyle\text{for a.e.\ }t\in[0,T],$ (8) $\displaystyle
E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](t)$ $\displaystyle\leq
e^{Ct}\big{(}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)+E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)\big{)},$
$\displaystyle\text{for a.e.\ }t\in[0,T].$ (9)

Based on the straightforward construction of well-prepared initial data for
the phase indicator (see, e.g., [Fischer2020b]), the above theorem leads to a
sharp $L^{1}$-convergence rate for the phase-field and to a sharp
$L^{2}$-convergence rate for the fluid velocity.

######  Corollary 2 (Sharp convergence rates for well-prepared initial data).

Let the assumptions and notation of Theorem 1 be in place. For well-prepared
initial data in the sense of

$\displaystyle
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)+E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)\leq
C(\chi_{0})\varepsilon^{2},$ (10)

where $C(\chi_{0})\in(0,\infty)$ denotes a constant depending only on the
initial geometry of the strong sharp interface limit solution, it follows that
for a.e. $T\in(0,T_{*})$ there exists a constant $C=C(\chi,v,T)\in(0,\infty)$
such that

$\displaystyle\big{\|}(v_{\varepsilon}{-}v)(\cdot,t)\big{\|}_{L^{2}(\Omega)}+\bigg{\|}\bigg{(}c_{0}\chi(\cdot,t){-}\int_{-1}^{\varphi_{\varepsilon}(\cdot,t)}\sqrt{2W(r)}\,dr\bigg{)}\bigg{\|}_{L^{1}(\Omega)}\leq
Ce^{Ct}\varepsilon$ (11)

for a.e. $t\in[0,T]$.

The proof of Theorem 1 is given in Section 6 and is based on the following key
ingredients. As a preliminary, we provide throughout the remainder of this
section precise definitions for the solution concepts associated with the
system of PDEs (1a)–(1f) and (2a)–(2h), respectively. Section 3 introduces the
definition of the two error functionals for which Theorem 1 claims the
stability estimates (8) and (9), respectively. We also collect several
important coercivity properties of these error functionals. These in turn are
needed to appropriately estimate preliminary bounds for the time evolution of
the two error functionals as derived in Section 4 and Section 5, respectively.

Starting point of our journey is the following precise concept of weak
solutions for the phase-field model.

######  Definition 3 (Weak solutions of the Navier–Stokes/Allen–Cahn system
(1a)–(1f)).

Let $d\in\\{2,3\\}$, let $\Omega\subset{\mathbb{R}^{d}}$ be a bounded domain
with orientable and $C^{2}$-boundary, and consider a finite time horizon
$0<T_{0}<\infty$. Let $\varepsilon\in(0,1)$, and consider an initial velocity
field $v_{\varepsilon,0}\in H^{1}(\Omega)$ in combination with an initial
order parameter $\varphi_{\varepsilon,0}\in H^{1}(\Omega)$ such that
$\varphi_{\varepsilon,0}\in[-1,1]$ holds true $\mathcal{L}^{d}$-a.e. in
$\Omega$, $\nabla\cdot v_{\varepsilon,0}=0$ in the distributional sense in
$\Omega$, as well as $(\varphi_{\varepsilon,0}{-}(-1),v_{\varepsilon,0})\in
H^{1}_{0}(\Omega,\mathbb{R}{\times}{\mathbb{R}^{d}})$.

A pair of measurable maps
$(\varphi_{\varepsilon},v_{\varepsilon})\colon\Omega{\times}(0,T_{0})\to\mathbb{R}{\times}{\mathbb{R}^{d}}$
is called a _weak solution of the Navier–Stokes/Allen–Cahn system _( 1a)–(1f)_
with time horizon $T_{0}$ and initial data
$(\varphi_{\varepsilon,0},v_{\varepsilon,0})$_ if the following conditions are
satisfied:

  * i)

In terms of regularity, it has to hold

$\displaystyle v_{\varepsilon}$ $\displaystyle\in
L^{2}\big{(}0,T_{0};H^{1}(\Omega;{\mathbb{R}^{d}})\big{)}\cap
L^{\infty}\big{(}0,T_{0};L^{2}(\Omega;{\mathbb{R}^{d}})\big{)},$ (12a)
$\displaystyle\varphi_{\varepsilon}$ $\displaystyle\in
H^{1}\big{(}0,T_{0};L^{2}(\Omega;[-1,1])\big{)}\cap
L^{2}\big{(}0,T_{0};H^{2}(\Omega)\big{)},$ (12b)

such that in addition (1f) holds in form of

$\displaystyle\big{(}\varphi_{\varepsilon}(\cdot,t){-}(-1),v_{\varepsilon}(\cdot,t)\big{)}\in
H^{1}_{0}(\Omega,\mathbb{R}{\times}{\mathbb{R}^{d}})\quad\text{for a.e.\
}t\in(0,T_{0}).$ (12c)

  * ii)

The velocity field $v_{\varepsilon}$ is a solution of (1a)–(1b) and (1d) in
the sense that

$\displaystyle\int_{\Omega}v_{\varepsilon}(\cdot,T)\cdot\eta(\cdot,T)\,dx-\int_{\Omega}v_{\varepsilon,0}\cdot\eta(\cdot,0)\,dx$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v_{\varepsilon}\cdot\partial_{t}\eta\,dxdt+\int_{0}^{T}\int_{\Omega}v_{\varepsilon}\otimes
v_{\varepsilon}:\nabla\eta\,dxdt$ (12d) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ -\int_{0}^{T}\int_{\Omega}\nabla
v_{\varepsilon}:\nabla\eta\,dxdt+\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\otimes\nabla\varphi_{\varepsilon}:\nabla\eta\,dxdt$

holds true for a.e. $T\in(0,T_{0})$ and all $\eta\in
C^{\infty}_{c}([0,T_{0});C^{\infty}_{c}(\Omega;{\mathbb{R}^{d}}))$ with
$\nabla\cdot\eta=0$, as well as

$\displaystyle(\nabla\cdot v_{\varepsilon})(\cdot,t)=0$ (12e)

for a.e. $t\in(0,T_{0})$ in the distributional sense in $\Omega$.

  * iii)

The order parameter $\varphi_{\varepsilon}$ satisfies (1c) and (1e) in form of

$\displaystyle\partial_{t}\varphi_{\varepsilon}+(v_{\varepsilon}\cdot\nabla)\varphi_{\varepsilon}$
$\displaystyle=\Delta\varphi_{\varepsilon}-\smash{\frac{1}{\varepsilon^{2}}W^{\prime}(\varphi_{\varepsilon})}$
$\displaystyle\text{a.e.\ in }\Omega\times(0,T_{0}),$ (12f)
$\displaystyle\varphi_{\varepsilon}(\cdot,t)$
$\displaystyle\to\varphi_{\varepsilon,0}$ $\displaystyle\text{strongly in
}L^{2}(\Omega)\text{ as }t\downarrow 0.$ (12g)

  * iv)

Defining for every admissible pair $(\varphi,v)$ the total energy functional
by

$\displaystyle
E[\varphi,v]:=\int_{\Omega}\frac{1}{2}|v|^{2}\,dx+\int_{\Omega}\frac{\varepsilon}{2}|\nabla\varphi|^{2}+\frac{1}{\varepsilon}W\big{(}\varphi\big{)}\,dx,$
(12h)

sharp energy dissipation is required in form of

$\displaystyle
E[\varphi_{\varepsilon}(\cdot,T),v_{\varepsilon}(\cdot,T)]+\int_{0}^{T}\int_{\Omega}\big{|}\nabla
v_{\varepsilon}\big{|}^{2}+\varepsilon\big{|}\big{(}\partial_{t}{+}(v_{\varepsilon}\cdot\nabla)\big{)}\varphi_{\varepsilon}\big{|}^{2}\,dxdt$
(12i) $\displaystyle\leq E[\varphi_{\varepsilon,0},v_{\varepsilon,0}]$

for a.e. $T\in(0,T_{0})$.

We conclude this section with a precise concept of strong solutions for the
sharp interface limit model.

######  Definition 4 (Strong solutions of the sharp interface limit system
(2a)–(2h)).

Let $d\in\\{2,3\\}$, let $\Omega\subset{\mathbb{R}^{d}}$ be a bounded domain
with orientable and $C^{2}$-boundary, and consider a finite time horizon
$0<T_{*}<\infty$. Consider an initial velocity field $v_{0}\in(H^{1}_{0}\cap
C^{1})(\Omega;{\mathbb{R}^{d}})$ in combination with an initial phase
indicator $\chi_{0}\in BV(\Omega;\\{0,1\\})$ such that $\nabla\cdot v_{0}=0$
pointwise in $\Omega$, the set $\Omega_{+}(0):=\\{\chi_{0}{=}1\\}$ yields an
open subset of $\Omega$ consisting of finitely many connected components with
$\overline{\Omega_{+}(0)}\subset\Omega$, the associated boundary
$\partial\Omega_{+}(0)$ is orientable and uniformly of class $C^{3}$, and
finally the following compatibility conditions hold true:
$\displaystyle[\\![v_{0}]\\!]=0\leavevmode\nobreak\ $
$\displaystyle\text{on}\leavevmode\nobreak\ \mathcal{I}(0),$
$\displaystyle(\mathrm{Id}{-}\mathbf{n}_{\mathcal{I}(0)}\otimes\mathbf{n}_{\mathcal{I}(0)})[\\![\nabla^{\mathrm{sym}}v_{0}\cdot\mathbf{n}_{\mathcal{I}(0)}]\\!]=0\leavevmode\nobreak\
$ $\displaystyle\text{on}\leavevmode\nobreak\ \mathcal{I}(0).$

A pair of measurable maps
$(\chi,v)\colon\Omega{\times}(0,T_{*})\to\\{0,1\\}{\times}{\mathbb{R}^{d}}$ is
called a _strong solution of the sharp interface limit system _( 2a)–(2h)_
with time horizon $T_{*}$ and initial data $(\chi_{0},v_{0})$_ if the
following requirements hold true:

  * i)

The fluid velocity $v$ is subject to the regularity

$\displaystyle v\in
H^{1}\big{(}0,T_{*};L^{2}(\Omega;{\mathbb{R}^{d}})\big{)}\cap
L^{2}\big{(}0,T_{*};H^{1}(\Omega;{\mathbb{R}^{d}})\big{)}$ (13a)

such that the corresponding boundary condition (2h) as well as the
corresponding initial condition (2f) are satisfied in form of

$\displaystyle v(\cdot,t)$ $\displaystyle\in
H^{1}_{0}(\Omega;{\mathbb{R}^{d}})$ $\displaystyle\text{for a.e.\
}t\in(0,T_{*}),$ (13b) $\displaystyle v(\cdot,t)$ $\displaystyle\to v_{0}$
$\displaystyle\text{strongly in }L^{2}(\Omega)\text{ as }t\downarrow 0.$ (13c)

Furthermore, for each $T\in[0,T_{*})$ there exists $C=C(T)\in(0,\infty)$ such
that

$\displaystyle v$ $\displaystyle\in
L^{\infty}\big{(}0,T;C^{1}(\Omega;{\mathbb{R}^{d}})\big{)},$ (13d)

with a corresponding bound on the highest-order derivative of the form

$\displaystyle\|\nabla v\|_{L^{\infty}(\Omega{\times}(0,T))}\leq C.$ (13e)

  * ii)

The velocity field $v$ solves (2a)–(2b) and (2d)–(2e) in the sense that

$\displaystyle\int_{\Omega}v(\cdot,T)\cdot\eta(\cdot,T)\,dx-\int_{\Omega}v_{0}\cdot\eta(\cdot,0)\,dx$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v\cdot\partial_{t}\eta\,dxdt-\int_{0}^{T}\int_{\Omega}\eta\cdot(v\cdot\nabla)v\,dxdt$
(13f) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\int_{0}^{T}\int_{\Omega}\nabla
v:\nabla\eta\,dxdt+c_{0}\int_{0}^{T}\int_{\Omega}\frac{\nabla\chi}{|\nabla\chi|}\otimes\frac{\nabla\chi}{|\nabla\chi|}:\nabla\eta\,d|\nabla\chi|dt$

holds true for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}([0,T_{*});C^{\infty}_{c}(\Omega;{\mathbb{R}^{d}}))$ with
$\nabla\cdot\eta=0$, as well as

$\displaystyle(\nabla\cdot v)(\cdot,t)=0$ (13g)

for a.e. $t\in(0,T_{*})$ in the distributional sense in $\Omega$.

  * iii)

Define for every $t\in(0,T_{*})$ the set
$\Omega_{+}(t):=\\{\chi(\cdot,t){=}1\\}$. There then exists a map
$\Psi\colon\overline{\Omega}{\times}[0,T_{*})\to\overline{\Omega}$ such that
$\Psi(\cdot,0)=\mathrm{Id}$, $\Psi(\cdot,t)\in
C^{3}\text{-Diffeo}(\overline{\Omega},\overline{\Omega})$ for all
$t\in(0,T_{*})$, and $\Omega_{+}(t)=\Psi(\Omega_{+}(0),t)$ for all
$t\in(0,T_{*})$. With respect to regularity in time, we require that for all
$T\in(0,T_{*})$ it holds

$\displaystyle\Psi$ $\displaystyle\in
C^{1}\big{(}[0,T];C^{1}(\overline{\Omega};{\mathbb{R}^{d}})\big{)}\cap
C\big{(}[0,T];C^{3}(\overline{\Omega};{\mathbb{R}^{d}})\big{)}.$ (13h)

Note that the corresponding initial condition (2g) is thus a consequence of
(13h). Finally, the corresponding boundary condition (2h) is satisfied in form
of

$\displaystyle\overline{\Omega_{+}(t)}$ $\displaystyle\subset\Omega$
$\displaystyle\text{for all }t\in(0,T_{*}).$ (13i)

  * iv)

The geometric evolution equation (2c) is required to be satisfied in its
strong form given by (4), where the space-time interface
$\mathcal{I}={\bigcup_{t\in[0,T_{*})}\mathcal{I}(t){\times}\\{t\\}}$ is
defined by $\mathcal{I}(t):=\partial\Omega_{+}(t)$, $t\in[0,T_{*})$.

In the appendix, we provide some detailed comments on how to obtain such
strong solutions for the sharp interface limit model starting from the work of
Abels and Moser [zbMATH06951007].

##  3\. The relative energy functional

###  3.1. Definition of the relative energy functional

In this subsection, we follow ideas first developed in [Fischer2020b]. We
start by defining the appropriate $BV$-pendant of $c_{0}\chi$ at the level of
the phase-field model via

$\displaystyle\psi_{\varepsilon}(x,t):=\int_{-1}^{\varphi_{\varepsilon}(x,t)}\sqrt{2W}(r)\,dr,\quad(x,t)\in\Omega\times(0,T_{0}).$
(14)

We then introduce a “unit-normal vector field”
$\mathbf{n}_{\varepsilon}\in\mathbb{S}^{d-1}$ by means of

$\displaystyle\mathbf{n}_{\varepsilon}(\cdot,t):=\begin{cases}\frac{\nabla\varphi_{\varepsilon}(\cdot,t)}{|\nabla\varphi_{\varepsilon}(\cdot,t)|}&\text{if
}\nabla\varphi_{\varepsilon}(\cdot,t)\neq 0,\\\
\mathrm{s}&\text{else},\end{cases}\qquad t\in(0,T_{0}),$ (15)

with $\mathrm{s}\in\mathbb{S}^{d-1}$ being a fixed but otherwise arbitrary
unit vector. Note that because of the previous two definitions we always have
the relations

$\displaystyle\mathbf{n}_{\varepsilon}|\nabla\varphi_{\varepsilon}|=\nabla\varphi_{\varepsilon}\quad\text{and}\quad\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|=\nabla\psi_{\varepsilon}\quad\text{throughout}\quad\Omega\times(0,T_{0}).$
(16)

We now define a measure for the difference between the phase-field
approximation $(\varphi_{\varepsilon},v_{\varepsilon})$ and the classical
solution $(\chi,v)$ for the sharp interface limit model as follows: for
$t\in(0,T_{*})\subset(0,T_{0})$ let

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)$ (17)
$\displaystyle:=\int_{\Omega}\frac{1}{2}\big{|}(v_{\varepsilon}{-}v)(\cdot,t)\big{|}^{2}\,dx+\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}(\cdot,t)\big{|}^{2}+\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon}(\cdot,t)\big{)}-(\xi\cdot\nabla\psi_{\varepsilon})(\cdot,t)\,dx.$

The vector field $\xi$ associated with the strong sharp interface limit
solution $(\chi,v)$ is yet to be determined. Note already that the above
functional has a form reminiscent of that of a relative entropy. Indeed,

$\displaystyle
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]=E[\varphi_{\varepsilon},v_{\varepsilon}]-\int_{\Omega}v_{\varepsilon}\cdot
v+\int_{\Omega}\frac{1}{2}|v|^{2}\,dx-\int_{\Omega}\xi\cdot\nabla\psi_{\varepsilon}\,dx,$
(18)

where we dropped for notational convenience the dependence on the time
variable. Continuing in this fashion, a straightforward computation moreover
shows

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$ (19)
$\displaystyle=\int_{\Omega}\frac{1}{2}|v_{\varepsilon}-v|^{2}+\int_{\Omega}\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}+\int_{\Omega}(1-\xi\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|.$

Ensuring non-negativity of $E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$
thus motivates $|\xi|\leq 1$ as an absolute minimal assumption on the vector
field $\xi$.

The main goal of our approach is to derive a stability estimate on
$E[\varphi_{\varepsilon},v_{\varepsilon}\,|\,\chi,v]$ in terms of the initial
distance $E_{0}:=E[\varphi_{\varepsilon,0},v_{\varepsilon,0}|\chi_{0},v_{0}]$
by a Gronwall argument, cf. the stability estimate (8) in Theorem 1. This in
turn necessitates control on the time evolution of the relative energy. In
particular, we will need an appropriate control on the time evolution of the
vector field $\xi$. To this end, it turns out to be beneficial (mostly for
clarity of exposition and efficient organization of terms in the relative
energy inequality) to introduce a second vector field $B$. One should keep in
mind the following interpretation of the pair $(\xi,B)$: the vector field
$\xi$ will represent a suitable extension of the unit normal vector field
$\mathbf{n}_{\mathcal{I}}$ for the smoothly evolving sharp interface
$\mathcal{I}$, whereas $B$ shall denote a suitable extension of the associated
normal velocity vector of $\mathcal{I}$. More precisely, we impose the
following conditions on the pair $(\xi,B)$ which will turn out to be
sufficient.

For every $T\in[0,T_{*})$ there are $c=c(\chi,T)\in(0,1)$ and
$C=C(\chi,v,T)\in(0,\infty)$ such that it holds (employing the convenient
notation $\operatorname{dist}(\cdot,\mathcal{I})$ representing the space-time
map
$\overline{\Omega}{\times}[0,T_{*})\ni(x,t)\mapsto\operatorname{dist}(x,\mathcal{I}(t))$):

  * •

(Regularity estimates) In terms of qualitative regularity, it has to hold

$\displaystyle\xi$ $\displaystyle\in
C^{0,1}\big{(}[0,T];L^{\infty}(\Omega;{\mathbb{R}^{d}})\big{)}\cap
L^{\infty}\big{(}[0,T];C^{1,1}_{c}(\Omega;{\mathbb{R}^{d}})\big{)},$ (20a)
$\displaystyle B$ $\displaystyle\in
L^{\infty}\big{(}[0,T];C^{0,1}_{c}(\overline{\Omega};{\mathbb{R}^{d}})\big{)},$
(20b)

with a bound on the corresponding highest-order derivatives in form of

$\displaystyle\|(\partial_{t}\xi,\nabla^{2}\xi,\nabla
B)\|_{L^{\infty}(\Omega{\times}(0,T))}$ $\displaystyle\leq C.$ (20c)

  * •

(Coercivity and consistency) It holds

$\displaystyle|\xi|$ $\displaystyle\leq
1{-}c\min\\{\operatorname{dist}^{2}(\cdot,\mathcal{I}),1\\}$
$\displaystyle\text{a.e.\ on }\Omega{\times}[0,T]$ (20d) $\displaystyle\xi$
$\displaystyle=\mathbf{n}_{\mathcal{I}}\text{ and
}\nabla\cdot\xi=-H_{\mathcal{I}}$ $\displaystyle\text{on }\mathcal{I}.$ (20e)

  * •

(Approximate transport of $\xi$ by $B$) We have

$\displaystyle\big{|}\partial_{t}\xi+(B\cdot\nabla)\xi+(\nabla
B)^{\mathsf{T}}\xi\big{|}$ $\displaystyle\leq
C\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}$ $\displaystyle\text{a.e.\
on }\Omega{\times}[0,T],$ (20f)
$\displaystyle\big{|}\xi\cdot(\partial_{t}+B\cdot\nabla)\xi\big{|}$
$\displaystyle\leq C\min\\{\operatorname{dist}^{2}(\cdot,\mathcal{I}),1\\}$
$\displaystyle\text{a.e.\ on }\Omega{\times}[0,T].$ (20g)

  * •

(Interpretation of $B$ as a normal velocity) It holds

$\displaystyle\big{|}(B{-}v)\cdot\xi+\nabla\cdot\xi\big{|}$ $\displaystyle\leq
C\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}$ $\displaystyle\text{a.e.\
on }\Omega{\times}[0,T],$ (20h)
$\displaystyle\big{|}\xi\cdot(\xi\cdot\nabla)B\big{|}$ $\displaystyle\leq
C\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}$ $\displaystyle\text{a.e.\
on }\Omega{\times}[0,T].$ (20i)

###  3.2. Coercivity properties of the relative energy functional

We collect several useful coercivity estimates for
$E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$, dropping in the process
again for notational convenience the dependence on the time variable. The
proof of these estimates is provided afterwards.

######  Lemma 5.

Let the assumptions and notation of Subsection 3.1 be in place. First, the
relative energy $E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$ provides
control on the error in the fluid velocity by means of

$\displaystyle\int_{\Omega}\frac{1}{2}|v_{\varepsilon}-v|^{2}\,dx\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v].$ (21)

Second, it entails a “tilt-excess” type control on the geometry in the form of

$\displaystyle\int_{\Omega}(1-\mathbf{n}_{\varepsilon}\cdot\xi)|\nabla\psi_{\varepsilon}|\,dx\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v].$ (22)

Third, one obtains control on the lack of equipartition of energy through

$\displaystyle\int_{\Omega}\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}\,dx\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v].$ (23)

Fourth, for every $T\in[0,T_{*})$ there exists a constant
$C=C(\chi,v,T)\in(0,\infty)$ such that for all times in $[0,T]$ it holds

$\displaystyle\int_{\Omega}|\mathbf{n}_{\varepsilon}-\xi|^{2}|\nabla\psi_{\varepsilon}|\,dx+\int_{\Omega}\min\\{\operatorname{dist}^{2}(\cdot,\mathcal{I}),1\\}|\nabla\psi_{\varepsilon}|\,dx$
$\displaystyle\leq CE[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v],$ (24)
$\displaystyle\int_{\Omega}|\mathbf{n}_{\varepsilon}-\xi|^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dx+\int_{\Omega}\min\\{\operatorname{dist}^{2}(\cdot,\mathcal{I}),1\\}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dx$
$\displaystyle\leq CE[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v],$ (25)

as well as finally

$\displaystyle\int_{\Omega}\big{(}\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}+\sqrt{1{-}\mathbf{n}_{\varepsilon}\cdot\xi}\big{)}\big{|}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{|}\,dx\leq
CE[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v].$ (26)

###### Proof.

The assertions (22)–(26) are essentially already contained and proved in
[Fischer2020b, Subsection 2.3]. For the sake of completeness, we re-produce
the straightforward argument here.

By (14) and (15), we can write (17) as the sum of three integrals with each
featuring a non-negative integrand:

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)$
$\displaystyle=\int_{\Omega}\frac{1}{2}\big{|}(v_{\varepsilon}{-}v)(\cdot,t)\big{|}^{2}\,dx$
$\displaystyle+\int_{\Omega}|\nabla\psi_{\varepsilon}(\cdot,t)|-(\xi\cdot\nabla\psi_{\varepsilon})(\cdot,t)\,dx$
$\displaystyle+\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}(\cdot,t)\big{|}^{2}+\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon}(\cdot,t)\big{)}-|\nabla\psi_{\varepsilon}(\cdot,t)|\,dx.$

Using (14) and the chain rule, we can complete the square in the last integral
above, and this proves (21), (22) and (23). The estimate (24) follows from
(20d) and (22).

Next, adding zero and employing Young’s inequality in the form of

$\displaystyle\ \varepsilon\left|\nabla\varphi_{\varepsilon}\right|^{2}$
$\displaystyle=\left|\nabla\psi_{\varepsilon}\right|+\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|\left(\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W\left(\varphi_{\varepsilon}\right)}\right)$
$\displaystyle\leq\left|\nabla\psi_{\varepsilon}\right|+\frac{1}{2}\varepsilon\left|\nabla\varphi_{\varepsilon}\right|^{2}+\frac{1}{2}\left(\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W\left(\varphi_{\varepsilon}\right)}\right)^{2}$

we infer

$\displaystyle\int_{\Omega}(1-\mathbf{n}_{\varepsilon}\cdot\xi)\varepsilon\left|\nabla\varphi_{\varepsilon}\right|^{2}dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leq
2\int_{\Omega}(1-\mathbf{n}_{\varepsilon}\cdot\xi)\left|\nabla\psi_{\varepsilon}\right|\,dx+4\int_{\Omega}\left(\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W\left(\varphi_{\varepsilon}\right)}\right)^{2}\,dx$
$\displaystyle\overset{\eqref{eq:coercivityRelEntropy2},\eqref{eq:coercivityRelEntropy3}}{\leq}CE[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v],$
(27)

which then together with
$2(1-\mathbf{n}_{\varepsilon}\cdot\xi)\geq|\mathbf{n}_{\varepsilon}-\xi|^{2}$
and (20d) leads to (25). Finally

$\displaystyle\int_{\Omega}\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}\big{|}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{|}\,dx$
$\displaystyle\overset{\eqref{eq:conditionsXiB4}}{\leq}C\int_{\Omega}\sqrt{1-\mathbf{n}_{\varepsilon}\cdot\xi}\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|\left|\sqrt{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W\left(\varphi_{\varepsilon}\right)}\right|\,dx.$

So (26) is a consequence of (3.2) and the Cauchy–Schwarz inequality. ∎

###  3.3. Error in the phase indicators

The second step in the proof consists of deriving an estimate for the error in
the phase indicators $\varphi_{\varepsilon}$ and $\chi$ based on the stability
estimate for the relative energy
$E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$. This in turn is done by
means of a stability estimate à la Gronwall (cf. the estimate (9) in Theorem
1) for an appropriate error functional defined for all $t\in[0,T_{*})$ as
follows:

$\displaystyle
E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](t):=\int_{\Omega}\big{|}\psi_{\varepsilon}(\cdot,t)-c_{0}\chi(\cdot,t)\big{|}\big{|}\vartheta(\cdot,t)\big{|}\,dx.$
(28)

Here, $\vartheta\colon\overline{\Omega}{\times}[0,T_{*})\to[-1,1]$ denotes a
(by the velocity field $B$ transported) time-dependent weight function
satisfying the following conditions, which in turn will be sufficient to
derive a stability estimate for
$E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]$.

For every $T\in[0,T_{*})$ there are $c=c(\chi,T)\in(0,1)$ and
$C=C(\chi,v,T)\in(1,\infty)$ such that it holds (employing again the
convenient notation $\operatorname{dist}(\cdot,\mathcal{I})$ representing the
space-time map
$\overline{\Omega}{\times}[0,T_{*})\ni(x,t)\mapsto\operatorname{dist}(x,\mathcal{I}(t))$):

  * •

(Regularity estimates) We have

$\displaystyle\vartheta\in
C^{0,1}\big{(}[0,T];L^{\infty}(\overline{\Omega})\big{)}\cap
L^{\infty}\big{(}[0,T];C^{0,1}(\overline{\Omega})\big{)}$ (29a)

with a corresponding estimate for the highest-order derivatives in form of

$\displaystyle\|(\partial_{t}\vartheta,\nabla\vartheta)\|_{L^{\infty}(\Omega{\times}(0,T))}$
$\displaystyle\leq C.$ (29b)

  * •

(Coercivity and consistency) It holds

$\displaystyle
c\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}\leq|\vartheta|$
$\displaystyle\leq C\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}$
$\displaystyle\text{a.e.\ on }\Omega{\times}[0,T],$ (29c)

as well as for all $t\in[0,T_{*})$

$\displaystyle\vartheta(\cdot,t)$ $\displaystyle<0$ $\displaystyle\text{a.e.\
in the interior of }\\{\chi(\cdot,t){=}1\\}\cap\Omega,$ (29d)
$\displaystyle\vartheta(\cdot,t)$ $\displaystyle>0$ $\displaystyle\text{a.e.\
in the interior of }\\{\chi(\cdot,t){=}0\\}\cap\Omega.$ (29e)

  * •

(Transport equation) Finally, it is required that

$\displaystyle\big{|}\partial_{t}\vartheta+(B\cdot\nabla)\vartheta\big{|}$
$\displaystyle\leq C\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}$
$\displaystyle\text{a.e.\ on }\Omega{\times}[0,T].$ (29f)

As in the case of the relative energy functional, we conclude the discussion
of this subsection by recording useful coercivity properties of the error
functional $E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]$ (dropping yet again
the dependence on the time variable). More precisely, for every
$T\in[0,T_{*})$ there exists a constant $C=C(\chi,v,T)\in(0,\infty)$ such that
for all times in $[0,T]$ it obviously holds that

$\displaystyle\int_{\Omega}\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}|\psi_{\varepsilon}-c_{0}\chi|\,dx\leq
CE_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi].$ (30)

More importantly, it holds for all $\lambda\in(0,1)$

$\displaystyle\int_{\Omega}|c_{0}\chi{-}\psi_{\varepsilon}||v_{\varepsilon}{-}v|\,dx$
$\displaystyle\leq\frac{C}{\lambda}\big{(}E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]{+}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]\big{)}+\lambda\int_{\Omega}|\nabla
v_{\varepsilon}{-}\nabla v|^{2}\,dx.$ (31)

###### Proof of the coercivity estimate (31).

Let $\delta=\delta(\chi,T)\in(0,\smash{\frac{1}{2}}]$ be the thickness of the
tubular neighborhood
$B_{\delta}(\mathcal{I}(t)):=\\{x\in{\mathbb{R}^{d}}\colon\operatorname{dist}(x,\mathcal{I}(t))<\delta\\}$
of $\mathcal{I}(t)$. For fixed $t\in[0,T]$, we have

$\displaystyle\int_{\Omega\setminus
B_{\delta}(\mathcal{I}(t))}|c_{0}\chi{-}\psi_{\varepsilon}||v_{\varepsilon}{-}v|\,dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\overset{\eqref{eq:coercivityBulkError1}}{\leq}\frac{1}{\delta}\int_{\Omega}\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}|c_{0}\chi{-}\psi_{\varepsilon}||v_{\varepsilon}{-}v|\,dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\overset{\eqref{eq:conditionsWeakSolPhaseField2}}{\leq}\frac{C}{\delta}\int_{\Omega}\sqrt{\min\\{\operatorname{dist}(\cdot,\mathcal{I}),1\\}}|c_{0}\chi{-}\psi_{\varepsilon}|^{1/2}|v_{\varepsilon}{-}v|\,dx$
$\displaystyle\overset{\eqref{eq:coercivityBulkError1},\,\eqref{eq:coercivityRelEntropy1}}{\leq}\frac{C}{\delta}\big{(}E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi]{+}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]\big{)}.$

It remains to estimate the integral over $B_{\delta}(\mathcal{I}(t))$, and we
shall use the following elementary inequality

$\displaystyle\left(\int_{0}^{\tau}g(r)dr\right)^{2}\leq
2\|g\|_{L^{\infty}(0,\tau)}\int_{0}^{\tau}rg(r)dr,\quad\forall g(r)\geq
0,\tau\geq 0.$ (32)

Recall that $\mathbf{n}_{\mathcal{I}}(y,t)$ is the inward normal vector at
$y\in\mathcal{I}(t)$. By the area formula and the one-dimensional
Gagliardo–Nirenberg–Sobolev inequality,

$\displaystyle\int_{B_{\delta}(\mathcal{I}(t))}|c_{0}\chi{-}\psi_{\varepsilon}||v_{\varepsilon}{-}v|\,dx$
$\displaystyle\leavevmode\nobreak\ \leq
C\int_{\mathcal{I}(t)}\int_{-\delta}^{\delta}|c_{0}\chi{-}\psi_{\varepsilon}||v_{\varepsilon}{-}v|(y{+}\mathbf{n}_{\mathcal{I}}(y,t)r)\,drd\mathcal{H}^{d-1}(y)$
$\displaystyle\leavevmode\nobreak\ \leq
C\int_{\mathcal{I}(t)}\sup_{|r|\leq\delta}|v_{\varepsilon}{-}v|(y{+}\mathbf{n}_{\mathcal{I}}(y,t)r)\left(\int_{-\delta}^{\delta}\left|c_{0}\chi{-}\psi_{\varepsilon}\right|(y{+}\mathbf{n}_{\mathcal{I}}(y,t)r)dr\right)d\mathcal{H}^{d-1}(y)$
$\displaystyle\stackrel{{\scriptstyle\eqref{def: sobolev
1d}}}{{\leq}}C\int_{\mathcal{I}(t)}\left\|(v_{\varepsilon}{-}v)(y{+}\mathbf{n}_{\mathcal{I}}r)\right\|_{L^{2}((-\delta,\delta);dr)}^{\frac{1}{2}}\left\|(v_{\varepsilon}{-}v)(y{+}\mathbf{n}_{\mathcal{I}}r)\right\|_{H^{1}((-\delta,\delta);dr)}^{\frac{1}{2}}$
$\displaystyle\qquad\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\cdot\sqrt{\int_{-\delta}^{\delta}\left|r\|c_{0}\chi{-}\psi_{\varepsilon}\right|(y{+}\mathbf{n}_{\mathcal{I}}r)\,dr}\,d\mathcal{H}^{d-1}(y).$

By Hölder’s and Young’s inequalities, the terms in the last step can be
controlled by the right hand side of (31). ∎

##  4\. Estimate on the time evolution of the relative energy

The aim of this section is to derive an estimate representing the key
ingredient for the derivation of the asserted stability estimate (8) for the
relative energy.

######  Proposition 6.

Let the assumptions and notation of Theorem 1 be in place, let $(\xi,B)$ be a
pair of vector fields subject to  _( 20a)–(20c)_ as well as (20e), and let the
relative energy $E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v]$ be given by
(17). Define the quantity

$\displaystyle
H_{\varepsilon}:=-\varepsilon\Delta\varphi_{\varepsilon}+\frac{1}{\varepsilon}W^{\prime}(\varphi_{\varepsilon})\quad\text{in
}\Omega\times(0,T_{0}).$ (33)

(This notation serves as a reminder that $H_{\varepsilon}$ plays the role of a
scalar curvature.) Then, the following relative energy inequality holds true

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](T)$
$\displaystyle\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)-\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}-\nabla v|^{2}\,dxdt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}-\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(v_{\varepsilon}-v)\cdot\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}v\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(c_{0}\chi-\psi_{\varepsilon})\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}(\nabla\cdot\xi)\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{|}(B-v)\cdot\xi+\nabla\cdot\xi\big{|}^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}|\nabla\cdot\xi|^{2}\bigg{(}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}-\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\bigg{)}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{\sqrt{\varepsilon}}\big{(}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\big{)}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{(}\partial_{t}\xi+(B\cdot\nabla)\xi+(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot(\mathbf{n}_{\varepsilon}-\xi)|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\cdot\big{(}\partial_{t}\xi+(B\cdot\nabla)\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\nabla
B:(\xi-\mathbf{n}_{\varepsilon})\otimes(\xi-\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)(1-\xi\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}-\xi\otimes\xi\big{)}:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\otimes\xi:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$

for a.e. $T\in(0,T_{*})$.

###### Proof.

We split the proof into six steps.

Step 1: Time evolution of kinetic energy terms. We claim that

$\displaystyle\int_{\Omega}\frac{1}{2}\big{|}(v_{\varepsilon}{-}v)(\cdot,T)\big{|}^{2}\,dx-\int_{\Omega}\frac{1}{2}\big{|}v_{\varepsilon,0}{-}v_{0}\big{|}^{2}\,dx$
$\displaystyle=\int_{\Omega}\frac{1}{2}|v_{\varepsilon}(\cdot,T)|^{2}\,dx-\int_{\Omega}\frac{1}{2}|v_{\varepsilon,0}|^{2}\,dx+\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\,dxdt$ (34) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{|}\nabla(v_{\varepsilon}{-}v)\big{|}^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}(v_{\varepsilon}{-}v)\cdot\big{(}(v_{\varepsilon}{-}v)\cdot\nabla\big{)}v\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)v\,dxdt-\int_{0}^{T}\int_{\Omega}c_{0}\chi((v_{\varepsilon}{-}v)\cdot\nabla)(\nabla\cdot\xi)\,dxdt$

for a.e. $T\in(0,T_{*})$. For a proof of (34), we first observe that by (ii))
and the regularity of the sharp interface limit solution it holds

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\eta\cdot\big{(}\partial_{t}+(v\cdot\nabla)\big{)}v\,dxdt+\int_{0}^{T}\int_{\Omega}\nabla
v:\nabla\eta\,dxdt$ (35) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-c_{0}\int_{0}^{T}\int_{\mathcal{I}}H_{\mathcal{I}}\mathbf{n}_{\mathcal{I}}\cdot\eta\,d\mathcal{H}^{d-1}dt$

for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}(\Omega{\times}[0,T];{\mathbb{R}^{d}})$ with
$\nabla\cdot\eta=0$. Instead of directly exploiting the information provided
by (35), it is convenient to post-process the curvature term before. More
precisely, for any admissible solenoidal test vector field in (35), it follows
from plugging in the second identity of (20e) as well as integrating by parts
that

$\displaystyle-
c_{0}\int_{0}^{T}\int_{\mathcal{I}}H_{\mathcal{I}}\mathbf{n}_{\mathcal{I}}\cdot\eta\,d\mathcal{H}^{d-1}dt$
$\displaystyle=c_{0}\int_{0}^{T}\int_{\mathcal{I}}(\nabla\cdot\xi)\mathbf{n}_{\mathcal{I}}\cdot\eta\,d\mathcal{H}^{d-1}dt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}c_{0}\chi(\eta\cdot\nabla)(\nabla\cdot\xi)\,dxdt.$

Hence, (35) updates to

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\eta\cdot\big{(}\partial_{t}+(v\cdot\nabla)\big{)}v\,dxdt+\int_{0}^{T}\int_{\Omega}\nabla
v:\nabla\eta\,dxdt$ (36) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}c_{0}\chi(\eta\cdot\nabla)(\nabla\cdot\xi)\,dxdt$

for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}(\Omega{\times}[0,T];{\mathbb{R}^{d}})$ with
$\nabla\cdot\eta=0$. By means of standard mollification arguments, one may
plug in the choice $\eta=v_{\varepsilon}-v$ into (36) resulting in the
identity

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}\int_{\Omega}(v_{\varepsilon}{-}v)\cdot\big{(}\partial_{t}+(v\cdot\nabla)\big{)}v\,dxdt+\int_{0}^{T}\int_{\Omega}\nabla
v:\nabla(v_{\varepsilon}{-}v)\,dxdt$ (37) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}c_{0}\chi((v_{\varepsilon}{-}v)\cdot\nabla)(\nabla\cdot\xi)\,dxdt$

for a.e. $T\in(0,T_{*})$. Next to the previous display, we also trivially have
by the regularity of the sharp interface limit solution as well as the
solenoidality of the velocity fields $v_{\varepsilon}$ and $v$, respectively,
that

$\displaystyle\int_{\Omega}\frac{1}{2}|v(\cdot,T)|^{2}\,dx-\int_{\Omega}\frac{1}{2}|v_{0}|^{2}\,dx$
(38)
$\displaystyle=\int_{0}^{T}\int_{\Omega}v\cdot\partial_{t}v\,dxdt=\int_{0}^{T}\int_{\Omega}v\cdot\partial_{t}v\,dxdt+\int_{0}^{T}\int_{\Omega}v\cdot(v_{\varepsilon}\cdot\nabla)v\,dxdt$

for a.e. $T\in(0,T_{*})$. Finally, up to a standard mollification argument one
may insert $\eta=v$ as a test vector field in (ii)) entailing

$\displaystyle\int_{\Omega}v_{\varepsilon}(\cdot,T)\cdot
v(\cdot,T)\,dx-\int_{\Omega}v_{\varepsilon,0}\cdot v_{0}\,dx$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v_{\varepsilon}\cdot\partial_{t}v\,dxdt+\int_{0}^{T}\int_{\Omega}v_{\varepsilon}\cdot(v_{\varepsilon}\cdot\nabla)v\,dxdt$
(39) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\int_{0}^{T}\int_{\Omega}\nabla v_{\varepsilon}:\nabla
v\,dxdt+\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\otimes\nabla\varphi_{\varepsilon}:\nabla
v\,dxdt$

for a.e. $T\in(0,T_{*})$. Hence, adding first (37) to (38) and then
subtracting (39) from the resulting identity produces the claim (34).

Step 2: Time evolution of interfacial energy terms. We next claim that

$\displaystyle\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}(\cdot,T)\big{|}^{2}+\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon}(\cdot,T)\big{)}-(\xi\cdot\nabla\psi_{\varepsilon})(\cdot,T)\,dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon,0}\big{|}^{2}+\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon,0}\big{)}-(\xi(\cdot,0)\cdot\nabla\psi_{\varepsilon,0})\,dx$
$\displaystyle=\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}(\cdot,T)\big{|}^{2}{+}\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon}(\cdot,T)\big{)}\,dx-\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon,0}\big{|}^{2}{+}\frac{1}{\varepsilon}W\big{(}\varphi_{\varepsilon,0}\big{)}\,dx$
(40) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}|H_{\varepsilon}|^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}|H_{\varepsilon}|^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)(v_{\varepsilon}\cdot\nabla)\psi_{\varepsilon}\,dxdt-\int_{0}^{T}\int_{\Omega}\partial_{t}\xi\cdot\nabla\psi_{\varepsilon}\,dxdt$

for a.e. $T\in(0,T_{*})$. For a proof of (40), we start with an integration by
parts which combined with the fundamental theorem of calculus, the regularity
of the phase-field $\varphi_{\varepsilon}$, a standard mollifier approximation
(w.r.t. the spatial variable) $\xi_{k}:=\theta_{k}\ast\xi$ for the vector
field $\xi$ (recall for this the regularity requirements (20a) and (20c)), the
condition (12g), and finally another integration by parts ensures

$\displaystyle\int_{\Omega}(\xi\cdot\nabla\psi_{\varepsilon})(\cdot,T)\,dx-\int_{\Omega}\xi(\cdot,0)\cdot\nabla\psi_{\varepsilon,0}\,dx$
(41)
$\displaystyle=-\bigg{(}\int_{\Omega}(\nabla\cdot\xi)(\cdot,T)\psi_{\varepsilon}(\cdot,T)\,dx-\int_{\Omega}(\nabla\cdot\xi)(\cdot,0)\psi_{\varepsilon,0}\,dx\bigg{)}$
$\displaystyle=-\lim_{k\to\infty}\int_{0}^{T}\int_{\Omega}(\nabla\cdot\partial_{t}\xi_{k})\psi_{\varepsilon}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\partial_{t}\psi_{\varepsilon}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\partial_{t}\xi\cdot\nabla\psi_{\varepsilon}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\partial_{t}\psi_{\varepsilon}\,dxdt.$

By the chain rule, the condition (12f), and the definition (33), it also
follows

$\displaystyle\partial_{t}\psi_{\varepsilon}=-\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}-(v_{\varepsilon}\cdot\nabla)\psi_{\varepsilon}.$
(42)

Hence, one obtains the claim (40) from plugging in (42) into (41) and adding
(in a self-evident way) several zeros.

Step 3: Exploiting sharp energy dissipation. The combination of the two
identities (34) and (40) from the previous two steps together with the sharp
energy dissipation estimate (12i), the condition (12f), and finally the
definition (33) entail

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](T)$
$\displaystyle\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)-\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}-\nabla v|^{2}\,dxdt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(v_{\varepsilon}{-}v)\cdot\big{(}(v_{\varepsilon}{-}v)\cdot\nabla\big{)}v\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}|H_{\varepsilon}|^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}c_{0}\chi((v_{\varepsilon}{-}v)\cdot\nabla)(\nabla\cdot\xi)\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)(v_{\varepsilon}\cdot\nabla)\psi_{\varepsilon}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)v\,dxdt-\int_{0}^{T}\int_{\Omega}\partial_{t}\xi\cdot\nabla\psi_{\varepsilon}\,dxdt$
$\displaystyle=:E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)-\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}-\nabla v|^{2}\,dxdt$ (43) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(v_{\varepsilon}{-}v)\cdot\big{(}(v_{\varepsilon}{-}v)\cdot\nabla\big{)}v\,dxdt+\mathrm{Res}^{(1)}(T).$

The remainder of the proof takes care of suitably rewriting the residual term
$\mathrm{Res}^{(1)}$. To this end, we start with an auxiliary result.

Step 4: A well-known identity for the phase-field curvature operator. We claim
that it holds

$\displaystyle-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)\eta\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}(\eta\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\eta)|\nabla\psi_{\varepsilon}|\,dxdt$
(44) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\eta)\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}\,dxdt$

for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}(\Omega{\times}[0,T];{\mathbb{R}^{d}})$. In particular, we
record that

$\displaystyle-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)\eta\,dxdt=-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}(\eta\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt$
(45)

for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}(\Omega{\times}[0,T];{\mathbb{R}^{d}})$ with
$\nabla\cdot\eta=0$.

Indeed, as we may compute by an integration by parts and (16)

$\displaystyle-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)\eta\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\varepsilon\Delta\varphi_{\varepsilon}(\eta\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt+\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\otimes\eta:\nabla^{2}\varphi_{\varepsilon}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\varepsilon\Delta\varphi_{\varepsilon}(\eta\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}\frac{1}{2}(\nabla\cdot\eta)\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$

it follows in combination with completing a square that

$\displaystyle-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)\eta\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\varepsilon\Delta\varphi_{\varepsilon}(\eta\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt+\int_{0}^{T}\int_{\Omega}(\nabla\cdot\eta)\frac{1}{\varepsilon}W(\varphi_{\varepsilon})\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\eta)|\nabla\psi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\eta)\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}\,dxdt$

for a.e. $T\in(0,T_{*})$ and all $\eta\in
C^{\infty}_{c}(\Omega{\times}[0,T];{\mathbb{R}^{d}})$. Further integrating by
parts in the second term on the right hand side of the last display, we obtain
the claim (44) from recalling the definition (33) of $H_{\varepsilon}$.

Step 5: Exploiting control on the time evolution of $\xi$. We next claim that

$\displaystyle\mathrm{Res}^{(1)}(T)$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\big{(}\partial_{t}\xi+(B\cdot\nabla)\xi+(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot(\mathbf{n}_{\varepsilon}-\xi)|\nabla\psi_{\varepsilon}|\,dxdt$
(46) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\cdot\big{(}\partial_{t}{+}(B\cdot\nabla)\big{)}\xi|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\nabla
B:(\xi-\mathbf{n}_{\varepsilon})\otimes(\xi-\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)(1-\xi\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}-\xi\otimes\xi\big{)}:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\otimes\xi:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\mathrm{Res}^{(2)}(T)$

for a.e. $T\in(0,T_{*})$, where

$\displaystyle\mathrm{Res}^{(2)}(T)$
$\displaystyle:=-\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}|H_{\varepsilon}|^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
(47) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}c_{0}\chi((v_{\varepsilon}{-}v)\cdot\nabla)(\nabla\cdot\xi)\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)((v_{\varepsilon}{-}B)\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}\big{(}(v{-}B)\cdot\mathbf{n}_{\varepsilon}\big{)}|\nabla\varphi_{\varepsilon}|\,dxdt.$

For a proof of (46) and (47), we first record that it follows from plugging in
$\eta=v$ as a test function in (45), which is indeed an admissible choice
after a standard mollification argument, that

$\displaystyle-\int_{0}^{T}\int_{\Omega}\varepsilon\nabla\varphi_{\varepsilon}\cdot(\nabla\varphi_{\varepsilon}\cdot\nabla)v\,dxdt=-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}(v\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|\,dxdt$
(48)

for a.e. $T\in(0,T_{*})$. The next step in the computation takes care of the
term involving the time derivative of the vector field $\xi$. Adding and
subtracting the anticipated PDE for the time evolution of the vector field
$\xi$, cf. the left hand side of (20f), and analogously for the anticipated
equation for the time evolution of its length, cf. the left hand side of
(20g), we get making also use of (16)

$\displaystyle-\int_{0}^{T}\int_{\Omega}\nabla\psi_{\varepsilon}\cdot\partial_{t}\xi\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\big{(}\partial_{t}\xi{+}(B\cdot\nabla)\xi{+}(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{(}(B\cdot\nabla)\xi+(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\big{(}\partial_{t}\xi{+}(B\cdot\nabla)\xi{+}(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot(\mathbf{n}_{\varepsilon}{-}\xi)|\nabla\psi_{\varepsilon}|\,dxdt$
(49) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\cdot\big{(}\partial_{t}{+}(B\cdot\nabla)\big{)}\xi|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\nabla
B:\xi\otimes(\xi-\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\cdot(B\cdot\nabla)\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$

for a.e. $T\in(0,T_{*})$. We further rewrite the last term in the preceding
identity by repeatedly adding zero and also using that
$(B\cdot\nabla)\xi=\nabla\cdot(\xi\otimes B)-(\nabla\cdot B)\xi$ as follows

$\displaystyle\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\cdot(B\cdot\nabla)\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)(1-\xi\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}(\mathrm{Id}-\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}):\nabla
B\,|\nabla\psi_{\varepsilon}|\,dxdt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{(}\nabla\cdot(\xi\otimes
B)\big{)}\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}:\nabla
B\,|\nabla\psi_{\varepsilon}|\,dxdt.$

In order to combine the last two terms, we compute based on a standard
mollifier approximation (w.r.t. the spatial variable)
$\xi_{k}:=\theta_{k}\ast\xi$ and $B_{k}:=\theta_{k}\ast B$ for the vector
fields $\xi$ and $B$, respectively (recall the regularity assumptions
(20a)–(20c)), another integration by parts as well as (16) that

$\displaystyle\int_{0}^{T}\int_{\Omega}\big{(}\nabla\cdot(\xi\otimes
B)\big{)}\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=-\lim_{k\to\infty}\int_{0}^{T}\int_{\Omega}\psi_{\varepsilon}\nabla\cdot\big{(}\nabla\cdot(\xi_{k}\otimes
B_{k})\big{)}\,dxdt$
$\displaystyle=-\lim_{k\to\infty}\int_{0}^{T}\int_{\Omega}\psi_{\varepsilon}\nabla\cdot\big{(}\nabla\cdot(B_{k}\otimes\xi_{k})\big{)}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)B\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt+\int_{0}^{T}\int_{\Omega}\mathbf{n}_{\varepsilon}\otimes\xi:\nabla
B\,|\nabla\psi_{\varepsilon}|\,dxdt.$

All in all, feeding back into (49) the previous two displays we consequently
obtain

$\displaystyle-\int_{0}^{T}\int_{\Omega}\nabla\psi_{\varepsilon}\cdot\partial_{t}\xi\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\big{(}\partial_{t}\xi{+}(B\cdot\nabla)\xi{+}(\nabla
B)^{\mathsf{T}}\xi\big{)}\cdot(\mathbf{n}_{\varepsilon}{-}\xi)|\nabla\psi_{\varepsilon}|\,dxdt$
(50) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\cdot\big{(}\partial_{t}{+}(B\cdot\nabla)\big{)}\xi|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\nabla
B:(\xi-\mathbf{n}_{\varepsilon})\otimes(\xi-\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)(1-\xi\cdot\mathbf{n}_{\varepsilon})|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)B\cdot\mathbf{n}_{\varepsilon}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\mathrm{Id}-\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}):\nabla
B\,|\nabla\psi_{\varepsilon}|\,dxdt$

for a.e. $T\in(0,T_{*})$. Finally appealing to (44) with the admissible choice
of the test vector field $\eta=B$ (again after a standard mollification
argument) in form of

$\displaystyle-\int_{0}^{T}\int_{\Omega}(\mathrm{Id}-\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}):\nabla
B\,|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}H_{\varepsilon}(B\cdot\mathbf{n}_{\varepsilon})|\nabla\varphi_{\varepsilon}|$
(51) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\int_{0}^{T}\int_{\Omega}(\nabla\cdot
B)\frac{1}{2}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt-\frac{1}{\sqrt{\varepsilon}}\sqrt{2W(\varphi_{\varepsilon})}\Big{)}^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}-\xi\otimes\xi\big{)}:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\xi\otimes\xi:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$

for a.e. $T\in(0,T_{*})$, the claims (46) and (47) therefore follow from the
combination of the identities (48), (50) and (51).

Step 6: Generating dissipation squares. For the final step, we claim that

$\displaystyle\mathrm{Res}^{(2)}$
$\displaystyle\leq-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
(52) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}-\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(c_{0}\chi-\psi_{\varepsilon})\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}(\nabla\cdot\xi)\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{|}(B-v)\cdot\xi+\nabla\cdot\xi\big{|}^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}|\nabla\cdot\xi|^{2}\bigg{(}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}-\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\bigg{)}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{\sqrt{\varepsilon}}\big{(}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\big{)}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt$

for a.e. $T\in(0,T_{*})$. Indeed, once we established (52) the asserted
relative energy inequality from Proposition 6 is a consequence of collecting
the estimates and identities from (43), (46) and (47), respectively.

For a proof of (52), we start by spending half of the available interfacial
energy dissipation in form of

$\displaystyle-\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}|H_{\varepsilon}|^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
(53) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}|H_{\varepsilon}|^{2}\,dxdt+\int_{0}^{T}\int_{\Omega}\frac{1}{2}|\nabla\cdot\xi|^{2}\Big{(}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{)}^{2}\,dxdt.$

We proceed by combining the remaining terms. Adding zero, recalling (16), and
integrating by parts (using in the process the solenoidality of the velocity
fields) results in

$\displaystyle-\int_{0}^{T}\int_{\Omega}c_{0}\chi\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}(\nabla\cdot\xi)\,dxdt-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\big{(}(v_{\varepsilon}-B)\cdot\mathbf{n}_{\varepsilon}\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}(c_{0}\chi-\psi_{\varepsilon})\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}(\nabla\cdot\xi)\,dxdt$
(54) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\big{(}(v-B)\cdot\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt.$

Moreover, adding zero as well as spending the remaining half of the available
interfacial energy dissipation leads to

$\displaystyle-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}|H_{\varepsilon}|^{2}\,dxdt-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}\big{(}(v-B)\cdot\mathbf{n}_{\varepsilon}\big{)}|\nabla\varphi_{\varepsilon}|\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}-\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\Big{|}^{2}\,dxdt$
(55) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{2}|(B-v)\cdot\xi|^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\varphi_{\varepsilon}|\,dxdt.$

Two simple steps remain until we reach the desired final form (52). The first
makes use of
$|\nabla\psi_{\varepsilon}|=\sqrt{2W(\varphi_{\varepsilon})}|\nabla\varphi_{\varepsilon}|$
which in turn allows to combine

$\displaystyle-\int_{0}^{T}\int_{\Omega}H_{\varepsilon}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\varphi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\frac{1}{\sqrt{\varepsilon}}\big{(}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\big{)}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt.$
(56)

The second consists of collecting the square, adding zero and exploiting the
simple estimate $(a+b)^{2}\leq 2(a^{2}+b^{2})$ in form of

$\displaystyle\int_{0}^{T}\int_{\Omega}\frac{1}{2}|(B-v)\cdot\xi|^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt+\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\big{|}\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\nabla\cdot\xi)\big{(}(B-v)\cdot\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}\big{(}(B{-}v)\cdot\xi{+}\nabla\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\,dxdt+\big{(}\sqrt{2W(\varphi_{\varepsilon})}{-}\varepsilon|\nabla\varphi_{\varepsilon}|\big{)}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
$\displaystyle\leq\int_{0}^{T}\int_{\Omega}\big{|}(B-v)\cdot\xi+\nabla\cdot\xi\big{|}^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$
(57) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}|\nabla\cdot\xi|^{2}\bigg{(}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}-\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\bigg{)}^{2}\,dxdt.$

Hence, the combination of (53)–(57) implies the claim (52) and thus concludes
the derivation of the relative energy inequality. ∎

##  5\. Time evolution of the error in the phase indicators

######  Lemma 7.

Let the assumptions and notation of Theorem 1 be in place, let $(\xi,B)$ be a
pair of vector fields subject to (20a) and (20b), respectively, let
$\vartheta$ be a time-dependent weight satisfying (29a) as well as
(29d)–(29e), and recall finally the notation (15) and (33). The time evolution
of the error in the phase indicators (28) may then be represented as follows:
it holds

$\displaystyle E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](T)$
$\displaystyle=E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}-c_{0}\chi)\big{(}\partial_{t}\vartheta+(B\cdot\nabla)\vartheta\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}-c_{0}\chi)\vartheta\nabla\cdot
B\,dxdt$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}-c_{0}\chi)\big{(}(v-v_{\varepsilon})\cdot\nabla)\vartheta\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot\xi\big{)}\big{(}|\nabla\psi_{\varepsilon}|-\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\Big{(}\big{(}(B-v)\cdot\xi\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\Big{)}\vartheta\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\Big{(}\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}+\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}(\nabla\cdot\xi)\Big{)}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta(\nabla\cdot\xi)\Big{|}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\vartheta\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{)}\,dxdt$

for a.e. $T\in(0,T_{*})$.

###### Proof.

First, since for a.e. $t\in(0,T_{0})$ we have
$|\varphi_{\varepsilon}(\cdot,t)|\leq 1$ $\mathcal{L}^{d}$-a.e. in $\Omega$,
which in turn is guaranteed by a maximum principle argument and the analogous
condition for the initial phase-field $\varphi_{\varepsilon,0}$, it follows
that $\psi_{\varepsilon}\in[0,c_{0}]$ and thus from the conditions (29d) and
(29e) that

$\displaystyle
E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](T)=\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)(\cdot,T)\,\vartheta(\cdot,T)\,dx$
(58)

for a.e. $T\in(0,T_{*})$. Second, as a consequence of $\vartheta\equiv 0$
along $\mathcal{I}$, cf. again to this end (29d) and (29e), we also observe
that it holds
$c_{0}\int_{0}^{T}\int_{\Omega}\vartheta\partial_{t}\chi\,dxdt=0$ for a.e.
$T\in(0,T_{*})$. Hence, based on these two properties we may compute by an
application of the fundamental theorem of calculus, the regularity of the
phase-field $\varphi_{\varepsilon}$, the condition (12g), adding zero, as well
as the product rule that

$\displaystyle E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](T)$
$\displaystyle=E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)\partial_{t}\vartheta\,dxdt+\int_{0}^{T}\int_{\Omega}\vartheta\partial_{t}\psi_{\varepsilon}\,dxdt$
$\displaystyle=E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)(\partial_{t}\vartheta{+}(B\cdot\nabla)\vartheta)\,dxdt$
(59) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)\nabla\cdot\big{(}B\vartheta\big{)}\,dxdt+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)\vartheta\nabla\cdot
B\,dxdt$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\partial_{t}\psi_{\varepsilon}\,dxdt$

for a.e. $T\in(0,T_{*})$. Integrating by parts and exploiting again that
$\vartheta\equiv 0$ along $\mathcal{I}$, we may upgrade (59) to

$\displaystyle E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](T)$
$\displaystyle=E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)(\partial_{t}\vartheta{+}(B\cdot\nabla)\vartheta)\,dxdt$
(60) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)\vartheta\nabla\cdot
B\,dxdt$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta(\partial_{t}\psi_{\varepsilon}{+}(B\cdot\nabla)\psi_{\varepsilon})\,dxdt$

for a.e. $T\in(0,T_{*})$. The remainder of the proof takes care of suitably
post-processing the last right hand side term from the previous display
incorporating an advective derivative for the $BV$-approximation
$\psi_{\varepsilon}$ of $c_{0}\chi$.

To this end, inserting in a first step the identity (42), making use of (16)
in a second step, and finally adding zero yields

$\displaystyle\int_{0}^{T}\int_{\Omega}\vartheta(\partial_{t}\psi_{\varepsilon}{+}(B\cdot\nabla)\psi_{\varepsilon})\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\vartheta(B\cdot\nabla)\psi_{\varepsilon}\,dxdt-\int_{0}^{T}\int_{\Omega}\vartheta\Big{(}\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}+(v_{\varepsilon}\cdot\nabla)\psi_{\varepsilon}\Big{)}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}|\nabla\psi_{\varepsilon}|\,dxdt$
(61) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}\vartheta\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(v-v_{\varepsilon})\cdot\nabla\big{)}\psi_{\varepsilon}\,dxdt.$

Note next that we may rewrite

$\displaystyle\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(v-v_{\varepsilon})\cdot\nabla\big{)}\psi_{\varepsilon}\,dxdt$
(62)
$\displaystyle=-\int_{0}^{T}\int_{\Omega}\psi_{\varepsilon}\big{(}(v{-}v_{\varepsilon})\cdot\nabla)\vartheta\,dxdt=-\int_{0}^{T}\int_{\Omega}(\psi_{\varepsilon}{-}c_{0}\chi)\big{(}(v{-}v_{\varepsilon})\cdot\nabla)\vartheta\,dxdt.$

Indeed, the first equality simply relies on the solenoidality of the velocity
fields $v_{\varepsilon}$ and $v$, respectively, in combination with an
integration by parts. The second equality in turn exploits $\vartheta\equiv 0$
along $\mathcal{I}$ which—by the solenoidality of the velocity fields—allows
to smuggle in the same term but with $\psi_{\varepsilon}$ being replaced by
$c_{0}\chi$. Furthermore, adding zero several times entails

$\displaystyle\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot\xi\big{)}|\nabla\psi_{\varepsilon}|\,dxdt-\int_{0}^{T}\int_{\Omega}\vartheta\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\vartheta\big{(}(B-v)\cdot\xi\big{)}\big{(}|\nabla\psi_{\varepsilon}|-\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\big{)}\,dxdt$
(63) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\Big{(}\big{(}(B-v)\cdot\xi\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\Big{)}\vartheta\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta\Big{(}\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}+\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}(\nabla\cdot\xi)\Big{)}\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{)}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\vartheta(\nabla\cdot\xi)\Big{|}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\vartheta\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\Big{(}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}\Big{)}\,dxdt.$

Feeding back (61)–(63) into (60) thus yields the desired result. ∎

##  6\. Proof of main results

###### Proof of Theorem 1.

The proof is split into six steps.

Step 1: Construction of the triple $(\xi,B,\vartheta)$. Fix $T\in[0,T_{*})$.
Due to the assumed regularity of the evolving geometry underlying the strong
solutions $(\chi,v)$, cf. item iii) of Definition 4, there exists a small
scale $\delta=\delta(\chi,T)\in(0,\smash{\frac{1}{2}}]$ such that the signed
distance $s_{\mathcal{I}}(\cdot,t)$ to $\mathcal{I}(t)$ as well as the nearest
point projection $P_{\mathcal{I}}(\cdot,t)$ onto $\mathcal{I}(t)$,
$t\in[0,T]$, are regular in the sense that

$\displaystyle s_{\mathcal{I}}$ $\displaystyle\in(C^{1}_{t}C^{1}_{x}\cap
C_{t}C^{3}_{x})\big{(}\overline{B_{2\delta}(\mathcal{I})}\big{)},$ (64)
$\displaystyle\big{(}(x,t)\mapsto
P_{\mathcal{I}}(x,t):=(x{-}(s_{\mathcal{I}}\nabla
s_{\mathcal{I}})(x,t),t)\big{)}$
$\displaystyle\in(C^{1}_{t}C_{x}{\cap}C_{t}C^{2}_{x})\big{(}\overline{B_{2\delta}(\mathcal{I})}\big{)}.$
(65)

We remark in this context that the signed distance is oriented by the
requirement $\nabla s_{\mathcal{I}}=\mathbf{n}_{\mathcal{I}}$ on
$\mathcal{I}$, and that for every $r\in(0,1]$ the associated space-time
tubular neighborhood $B_{r}(\mathcal{I})$ of $\mathcal{I}$ is defined by

$\displaystyle
B_{r}(\mathcal{I}):=\bigcup_{t\in[0,T]}B_{r}(\mathcal{I}(t))\times\\{t\\}.$
(66)

Due to (13h) and (13i), we may further choose $\delta$ small enough such that

$\displaystyle\overline{B_{2\delta}(\mathcal{I}(t))}\subset\Omega\quad\text{for
all }t\in[0,T].$ (67)

Next, let $\bar{\eta}\colon\mathbb{R}\to[0,1]$ be a smooth and even profile
with $\operatorname{supp}\bar{\eta}\subset[-1,1]$ and with quadratic decay at
the origin in the sense of

$\displaystyle c_{\bar{\eta}}r^{2}\leq 1-\bar{\eta}(r)\leq
C_{\bar{\eta}}r^{2}\quad\text{for all }r\in[-1,1],$ (68)

for some constants $0<c_{\bar{\eta}}<C_{\bar{\eta}}<\infty$. We further choose
a smooth and even profile $\widetilde{\eta}\colon\mathbb{R}\to[0,1]$ such that
$\operatorname{supp}\widetilde{\eta}\subset[-2,2]$ and

$\displaystyle\widetilde{\eta}\equiv 1\quad\text{on }[-1,1].$ (69)

We then define two quadratic cutoffs

$\displaystyle\eta_{\mathcal{I}}(x,t)$
$\displaystyle:=\bar{\eta}\big{(}s_{\mathcal{I}}(x,t)/\delta\big{)},$
$\displaystyle(x,t)\in{\mathbb{R}^{d}}{\times}[0,T],$ (70)
$\displaystyle\widetilde{\eta}_{\mathcal{I}}(x,t)$
$\displaystyle:=\widetilde{\eta}\big{(}s_{\mathcal{I}}(x,t)/\delta\big{)},$
$\displaystyle(x,t)\in{\mathbb{R}^{d}}{\times}[0,T].$ (71)

Note that

$\displaystyle\operatorname{supp}\eta_{\mathcal{I}}(\cdot,t)$
$\displaystyle\subset\overline{B_{\delta}(\mathcal{I}(t))}$
$\displaystyle\text{for all }t\in[0,T],$ (72)
$\displaystyle\operatorname{supp}\widetilde{\eta}_{\mathcal{I}}(\cdot,t)$
$\displaystyle\subset\overline{B_{2\delta}(\mathcal{I}(t))}$
$\displaystyle\text{for all }t\in[0,T],$ (73)
$\displaystyle\widetilde{\eta}_{\mathcal{I}}(\cdot,t)$ $\displaystyle\equiv 1$
$\displaystyle\text{on }\overline{B_{\delta}(\mathcal{I}(t))}\text{ for all
}t\in[0,T].$ (74)

With these ingredients in place, we define the vector fields $\xi$ and $B$ by
means of

$\displaystyle\xi$ $\displaystyle:=\eta_{\mathcal{I}}\nabla s_{\mathcal{I}}$
$\displaystyle\text{on }\Omega\times[0,T],$ (75) $\displaystyle B$
$\displaystyle:=\big{(}\big{(}(v\circ P_{\mathcal{I}})\cdot\nabla
s_{\mathcal{I}}\big{)}-(\Delta s_{\mathcal{I}})\circ
P_{\mathcal{I}}\big{)}\,\widetilde{\eta}_{\mathcal{I}}\nabla s_{\mathcal{I}}$
$\displaystyle\text{on }\Omega\times[0,T],$ (76)

where $v$ is the velocity field of the strong solution.

For a construction of a suitable weight $\vartheta$, we first fix a smooth and
odd map $\bar{\vartheta}\colon\mathbb{R}\to[-1,1]$ representing a suitable
truncation of the (negative of the) identity in the sense that

$\displaystyle\bar{\vartheta}\equiv 1\text{ on
}(-\infty,-1]\quad\text{and}\quad\bar{\vartheta}\equiv-1\text{ on
}[1,\infty),$ (77) $\displaystyle\bar{\vartheta}>0\text{ in
}(-1,0)\quad\text{and}\quad\bar{\vartheta}<0\text{ in }(0,1),$ (78)
$\displaystyle c_{\bar{\vartheta}}r\leq|\bar{\vartheta}(r)|\leq
C_{\bar{\vartheta}}r\quad\text{for all }r\in[-1,1],$ (79)

for some constants $0<c_{\bar{\vartheta}}<C_{\bar{\vartheta}}<\infty$. We may
then define

$\displaystyle\vartheta$
$\displaystyle:=\bar{\vartheta}\big{(}s_{\mathcal{I}}/\delta\big{)}$
$\displaystyle\text{on }\Omega\times[0,T].$ (80)

Step 2: Proof of the conditions  _( 20a)–(20i)_. The required regularity
(20a)–(20c) is immediate from the definitions (75) and (76) as well as the
regularity (13d), (64) and (65) of the associated building blocks.
Furthermore, because of (72), (73) and (67) it follows that the vector fields
$\xi$ and $B$ are indeed compactly supported within $\Omega$.

We next note that the coercivity estimate (20d) directly follows from the
definition (75) and the lower bound from (68). For a proof of the consistency
conditions (20e), $\xi=\mathbf{n}_{\mathcal{I}}$ along $\mathcal{I}$ simply
follows from the definition (75) as well as $\bar{\eta}(0)=0$ whereas
$\nabla\cdot\xi=-H_{\mathcal{I}}$ along $\mathcal{I}$ is a consequence of the
definition (75), $\bar{\eta}^{\prime}(0)=0$ as well as the well-known identity
$\Delta s_{\mathcal{I}}=-H_{\mathcal{I}}$ along $\mathcal{I}$.

We proceed with a proof of the (approximate) evolution equations (20f) and
(20g). The argument is based on the claim

$\displaystyle\big{(}\partial_{t}s_{\mathcal{I}}+(B\cdot\nabla)s_{\mathcal{I}}\big{)}(\cdot,t)\equiv
0\quad\text{on }\overline{B_{\delta}(\mathcal{I}(t))},\,t\in[0,T].$ (81)

Since $\partial_{t}s_{\mathcal{I}}=-V_{\mathcal{I}}$ along $\mathcal{I}$, the
identity (81) is indeed valid as a consequence of the definition (76), the
property (74), item iv) of Definition 4 in form of
$V_{\mathcal{I}}=(B\cdot\nabla)s_{\mathcal{I}}$ along $\mathcal{I}$, as well
as $\partial_{t}s_{\mathcal{I}}=(\partial_{t}s_{\mathcal{I}})\circ
P_{\mathcal{I}}$ (the latter following from a straightforward computation
based on differentiating $s_{\mathcal{I}}\circ P_{\mathcal{I}}\equiv 0$). Note
then that (81) together with the chain rule immediately implies

$\displaystyle\big{(}\partial_{t}\eta_{\mathcal{I}}+(B\cdot\nabla)\eta_{\mathcal{I}}\big{)}(\cdot,t)\equiv
0\quad\text{on }\overline{B_{\delta}(\mathcal{I}(t))},\,t\in[0,T].$ (82)

This in turn directly entails (20g) due to the simple observation
$|\xi|^{2}=\eta_{\mathcal{I}}^{2}$, which itself follows from the definition
(75). Furthermore, (82) reduces the proof of (20f) to a proof of

$\displaystyle\big{(}\partial_{t}\nabla s_{\mathcal{I}}{+}(B\cdot\nabla)\nabla
s_{\mathcal{I}}{+}(\nabla B)^{\mathsf{T}}\nabla
s_{\mathcal{I}}\big{)}(\cdot,t)\equiv 0\quad\text{on
}\overline{B_{\delta}(\mathcal{I}(t))},\,t\in[0,T].$ (83)

However, (83) simply follows from taking the spatial gradient of (81).

We finally note that (20h) and (20i) are straightforward consequences of the
definitions (75) and (76), the regularity of the associated building blocks,
as well as the already established property (20e).

Step 3: Proof of the conditions  _( 29a)–(29f)_. The regularity requirements
(29a) and (29b) are immediate from the definition (80) and the regularity
(64). The coercivity estimate (29c) as well as the sign conditions (29d)–(29e)
in turn follow directly from the definition (80) and the properties (77)–(79).
Finally, the approximate transport equation (29f) simply results from a
combination of (81) and the chain rule.

Step 4: Derivation of the stability estimates (8) and (9). Applying (21)–(24)
and (20f)–(20g) to the inequality in Proposition 6, we have for a.e.
$T\in(0,T_{*})$ that

$\displaystyle E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](T)$
$\displaystyle\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)+C\int_{0}^{T}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)dt-\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}-\nabla v|^{2}\,dxdt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
(84) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{2\varepsilon}\Big{|}H_{\varepsilon}-\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\Big{|}^{2}\,dxdt$
(85) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}(c_{0}\chi-\psi_{\varepsilon})\big{(}(v_{\varepsilon}-v)\cdot\nabla\big{)}(\nabla\cdot\xi)\,dxdt$
(86) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\big{|}(B-v)\cdot\xi+\nabla\cdot\xi\big{|}^{2}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}\,dxdt$
(87) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\frac{1}{\sqrt{\varepsilon}}\big{(}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\big{)}\big{(}(v-B)\cdot(\mathbf{n}_{\varepsilon}-\xi)\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|\,dxdt$
(88) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\int_{0}^{T}\int_{\Omega}\big{(}\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}-\xi\otimes\xi\big{)}:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt$
(89) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\int_{0}^{T}\int_{\Omega}\xi\otimes\xi:\nabla
B\,\big{(}\varepsilon|\nabla\varphi_{\varepsilon}|^{2}-|\nabla\psi_{\varepsilon}|\big{)}\,dxdt.$
(90)

The integral (86) is estimated by (31) with a sufficiently small $\lambda>0$.
The integral (87) can be estimated using (20h) and (25). To estimate (88), we
first employ Cauchy–Schwarz’s inequality, followed by Young’s inequality with
a sufficiently small prefactor in order to exploit the sign of (84), and then
conclude with (25). To estimate (89), we write

$\displaystyle\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}:\nabla
B=\mathbf{n}_{\varepsilon}\otimes(\mathbf{n}_{\varepsilon}{-}\xi):\nabla
B+(\xi\cdot\nabla)B\cdot(\mathbf{n}_{\varepsilon}{-}\xi)+\xi\otimes\xi:\nabla
B$

so that we can estimate

$\displaystyle|\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}:\nabla
B-\xi\otimes\xi:\nabla B|\leq C\sqrt{1-\mathbf{n}_{\varepsilon}\cdot\xi}.$

Because of this, (89) can be estimated by using (26). Finally, (90) can be
bounded by means of (20i) and (26). All in all,

$\displaystyle
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](T)+\frac{1}{2}\int_{0}^{T}\int_{\Omega}|\nabla
v_{\varepsilon}-\nabla v|^{2}\,dxdt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{4\varepsilon}\Big{|}H_{\varepsilon}+\sqrt{2W(\varphi_{\varepsilon})}(\nabla\cdot\xi)\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{4\varepsilon}\Big{|}H_{\varepsilon}-\big{(}(B-v)\cdot\xi\big{)}\varepsilon|\nabla\varphi_{\varepsilon}|\Big{|}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leq
E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](0)+C\int_{0}^{T}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)+E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](t)dt.$

Regarding the evolution of the bulk error in Lemma 7, similar considerations
based in addition on (25), (26), (29b), (29c), (29d) and (29f) also lead to

$\displaystyle E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](T)$
$\displaystyle\leq
E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](0)+C\int_{0}^{T}E[\varphi_{\varepsilon},v_{\varepsilon}|\chi,v](t)+E_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](t)dt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{4}|\nabla v_{\varepsilon}-\nabla
v|^{2}\,dxdt$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{8}\Big{(}\big{(}(B-v)\cdot\xi\big{)}\sqrt{\varepsilon}|\nabla\varphi_{\varepsilon}|-\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}\Big{)}^{2}\,dxdt$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\int_{\Omega}\frac{1}{8}\Big{(}\frac{H_{\varepsilon}}{\sqrt{\varepsilon}}+\frac{\sqrt{2W(\varphi_{\varepsilon})}}{\sqrt{\varepsilon}}(\nabla\cdot\xi)\Big{)}^{2}\,dxdt.$

The above two inequalities together with Grönwall’s inequality therefore allow
to conclude the proof. ∎

###### Proof of Corollary 2.

We proceed in two steps.

Step 1: From $E_{\mathrm{vol}}$-control to $L^{1}(\Omega)$-control of the
error in the phase indicators. We claim that for all $T\in[0,T_{*})$ there
exists a constant $C=C(\chi,v,T)\in(0,\infty)$ such that

$\displaystyle\big{\|}c_{0}\chi(\cdot,t)-\psi_{\varepsilon}(\cdot,t)\big{\|}_{L^{1}(\Omega)}^{2}\leq
CE_{\mathrm{vol}}[\varphi_{\varepsilon}|\chi](t)$ (91)

holds true for all $t\in[0,T]$. For the simple proof based on a slicing
argument and an application of Fubini’s theorem, we refer, e.g., to
[Fischer2020b, Proof of Theorem 1, Step 2].

Step 2: Derivation of the sharp convergence rate (11). This now immediately
follows from post-processing the stability estimates (8) and (9) by means of
the bound (91) and the assumption (10). ∎

##  Appendix A Well-posedness of the sharp interface limit model

The goal of this part is to construct a strong solution in the sense of
Definition 4. We consider the following system which is equivalent to
(2a)–(2h) (with $c_{0}=1$):

$\displaystyle\partial_{t}v+(v\cdot\nabla)v$ $\displaystyle=\Delta
v+\nabla\pi$ $\displaystyle\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\mathring{\Omega}(t),$ (92a) $\displaystyle\nabla\cdot v$ $\displaystyle=0$
$\displaystyle\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\mathring{\Omega}(t),$ (92b) $\displaystyle[\\![2Dv-\pi\leavevmode\nobreak\
\mathrm{Id}]\\!]\cdot\mathbf{n}_{\mathcal{I}}$
$\displaystyle=H_{\mathcal{I}}\mathbf{n}_{\mathcal{I}}$
$\displaystyle\leavevmode\nobreak\ \text{on}\leavevmode\nobreak\
\mathcal{I}(t),$ (92c) $\displaystyle[\\![v]\\!]$ $\displaystyle=0$
$\displaystyle\leavevmode\nobreak\ \text{on}\leavevmode\nobreak\
\mathcal{I}(t),$ (92d) $\displaystyle V_{\mathcal{I}}$
$\displaystyle=\mathbf{n}_{\mathcal{I}}\cdot v+H_{\mathcal{I}}$
$\displaystyle\leavevmode\nobreak\ \text{on}\leavevmode\nobreak\
\mathcal{I}(t).$ (92e)

Here $\mathring{\Omega}(t):=\Omega_{+}(t)\cup\Omega_{-}(t)$ is the bulk region
and $Dv:=\nabla^{\mathrm{sym}}v$ is the symmetric part of the flow gradient.
The major difference of (92) to the system studied in [KohnePruessWilke] is
that in the latter the interface is purely transported by the fluid velocity,
i.e., $V_{\mathcal{I}}=\mathbf{n}_{\mathcal{I}}\cdot v$.

######  Proposition 8.

Let $p>d+2$ and $\mathcal{I}(0)\subset\Omega$ be a $C^{3}$ closed surface.
Assume that the initial velocity field $v_{0}$ satisfies $v_{0}\in
W_{p}^{2-2/p}(\mathring{\Omega}(0))\cap W^{1}_{p}(\Omega)$ together with the
following compatibility conditions:

$\displaystyle\operatorname{div}v_{0}=0\leavevmode\nobreak\ $
$\displaystyle\text{in}\leavevmode\nobreak\ \mathring{\Omega}(0),$ (93)
$\displaystyle v_{0}=0\leavevmode\nobreak\ $
$\displaystyle\text{on}\leavevmode\nobreak\ \partial\Omega,$ (94)
$\displaystyle[\\![v_{0}]\\!]=0\leavevmode\nobreak\ $
$\displaystyle\text{on}\leavevmode\nobreak\ \mathcal{I}(0),$ (95)
$\displaystyle\Pi_{\mathcal{I}(0)}[\\![\nabla^{\mathrm{sym}}v_{0}\cdot\mathbf{n}_{\mathcal{I}}]\\!]=0\leavevmode\nobreak\
$ $\displaystyle\text{on}\leavevmode\nobreak\ \mathcal{I}(0),$ (96)

where
$\Pi_{\mathcal{I}(t)}=\mathrm{Id}-\mathbf{n}_{\mathcal{I}(t)}\otimes\mathbf{n}_{\mathcal{I}(t)}$
is the tangential projection. Then there exists $T>0$ such that (92) has a
unique solution $(v,\pi,\mathcal{I})$ with

$\displaystyle v\in W_{p}^{1}\left(0,T;L^{p}(\Omega)\right)\cap
L^{p}\left(0,T;W_{p}^{2}(\mathring{\Omega}(t))\cap W_{p}^{1}(\Omega)\right),$
(97a) $\displaystyle\pi\in
L^{p}\left(0,T;\mathring{W}_{p}^{1}(\mathring{\Omega}(t))\right)\leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ \int_{\Omega}\pi(\cdot,t)=0,$ (97b)
$\displaystyle[\\![\pi]\\!]\in
W_{p}^{1-\frac{1}{p},\frac{1}{2}(1-\frac{1}{p})}(\mathcal{I}(t)).$ (97c)

Moreover, the free boundary
$\mathcal{I}=\cup_{t\in(0,T)}\mathcal{I}(t){\times}\\{t\\}$ is parametrized
through the diffeomorphism
$\Theta_{h}(x,t)\colon\mathring{\Omega}(0)\mapsto\mathring{\Omega}(t)$ defined
by

$\displaystyle\Theta_{h}(x,t):=x+\zeta\left(\frac{\operatorname{dist}(x,\mathcal{I}(0))}{\delta}\right)h(P_{\mathcal{I}(0)}(x),t)\,\mathbf{n}_{\mathcal{I}(0)}(x)\quad\text{
for all }x\in{\mathbb{R}^{d}},$ (98)

where $P_{\mathcal{I}(t)}$ is the nearest point projection to
$\mathcal{I}(t)$, $\zeta\in C^{\infty}({\mathbb{R}^{d}})$ satisfies

$\left|\nabla\zeta(s)\right|\leq 4\leavevmode\nobreak\ \text{in
${\mathbb{R}^{d}}$ and }\leavevmode\nobreak\ \zeta(s)=1\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ |s|\leq 1/2,\leavevmode\nobreak\
\zeta(s)=0\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ |s|\geq 1,$ (99)

and $h$ is the height function

$\displaystyle h\in
W_{p}^{2-\frac{1}{2p}}\big{(}J;L^{p}(\mathcal{I}(0))\big{)}\cap
L^{p}\big{(}J;W_{p}^{4-\frac{1}{p}}(\mathcal{I}(0))\big{)}\leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ h|_{t=0}=0.$ (100)

Throughout the next three subsection, we sketch the main ingredients for a
proof of Proposition 8 and thus conclude with the regularity stated in
Definition 4.

###  A.1. Preliminaries

We will employ the following notation:

$\displaystyle\mathring{W}^{k}_{p}(\Omega)$
$\displaystyle=\\{\partial^{\alpha}f\in L^{p}(\Omega),|\alpha|=k\\},$ (101a)
$\displaystyle W^{k}_{p}(\Omega)$ $\displaystyle=\\{\partial^{\alpha}f\in
L^{p}(\Omega),|\alpha|\leq k\\},$ (101b) $\displaystyle L^{p}_{(0)}(\Omega)$
$\displaystyle=\Big{\\{}f\in L^{p}(\Omega),\int_{\Omega}f\,dx=0\Big{\\}}.$
(101c)

We need elementary results from interpolation theory and maximal regularity
theory of parabolic system, see to this end, for instance, [MR1345385, Section
4.10] and [MR1783238]. For a Banach space $(X,\|\cdot\|_{X})$ and
$s\in(0,1),p\geq 1$, we also recall the Sobolev–Slobodeckij space
$W_{p}^{s}(J;X)$ normed by

$\displaystyle\|f\|_{W_{p}^{s}(J;X)}:=\|f\|_{L^{p}(J;X)}+[f]_{s,p}<\infty,$
(102)

where $J=[0,T]$ and

$\displaystyle[f]_{s,p,X}:=\left(\int_{J}\int_{J}\frac{\|f(t)-f(\tau)\|_{X}^{p}}{|t-\tau|^{ps+1}}dtd\tau\right)^{\frac{1}{p}}.$
(103)

For $s\in(m,m+1)$ with $m$ being a positive integer, we finally define

$\displaystyle\|f\|_{W_{p}^{s}(J;X)}:=\|f\|_{W_{p}^{m}(J;X)}+\max_{|\alpha|=m}[\partial^{\alpha}f]_{s-m,p,X}.$
(104)

Now we state the required interpolation inequalities involving these spaces.
We denote by $(E_{0},E_{1})_{\theta,p}$ the real interpolation of Banach
spaces $E_{0}$ and $E_{1}$. For $\theta\in[0,1]$ and Banach spaces
$X_{0},X_{1}$, we then have

$\displaystyle W_{p}^{1}\left(J;X_{0}\right)\cap L_{p}\left(J;X_{1}\right)$
$\displaystyle\hookrightarrow
W^{1-\theta}_{p}\left(J;(X_{0},X_{1})_{\theta,p}\right),$ (105) $\displaystyle
W_{p}^{1}\left(J;X_{0}\right)\cap L_{p}\left(J;X_{1}\right)$
$\displaystyle\hookrightarrow
C^{1-\theta-\frac{1}{p}}\left(J;(X_{0},X_{1})_{\theta,p}\right),$ (106)

where the second embedding holds provided $1-\theta\geq\frac{1}{p}$. In
particular,

$\displaystyle W_{p}^{1}\left(J;X_{0}\right)\cap
L_{p}\left(J;X_{1}\right)\hookrightarrow
C^{0}\left(J;(X_{0},X_{1})_{1-1/p,p}\right).$ (107)

We will also rely on the following inequality

$\displaystyle\|f\|_{W^{s}_{p}(J;X)}\leqslant
C_{s,s^{\prime},p}T^{\left(s^{\prime}-s\right)+\frac{1}{p}}\|f\|_{C^{s^{\prime}}(J;X)}$
(108)

provided that $0<s<s^{\prime}\leqslant 1,0<T\leqslant 1$. Indeed,

$\displaystyle[f]_{s,p,X}^{p}$ $\displaystyle\overset{\eqref{slobo
norm}}{=}\int_{J}\int_{J}\frac{\|f(t)-f(\tau)\|_{X}^{p}}{|t-\tau|^{ps+1}}dtd\tau$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leqslant\int_{J}\int_{J}|t-\tau|^{p\left(s^{\prime}-s\right)-1}dtd\tau\|f\|_{C^{s^{\prime}}(J;X)}^{p}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leqslant
C_{s^{\prime},s,p}T^{p\left(s^{\prime}-s\right)+1}\|f\|_{C^{s^{\prime}}(J;X)}^{p}$

for all $0<s<s^{\prime}\leqslant 1$, and this implies (108).

###  A.2. The work of Abels and Moser [zbMATH06951007]

We shall first reduce (92) to a system with fixed domains. Assume
$\operatorname{dist}(\mathcal{I},\partial\Omega)>4\delta$. For $h\in
C^{2}(\mathcal{I}(0)\times J)$ with $\|h\|_{L^{\infty}_{x,t}}<2\delta$, the
Hanzawa transformation (98) is a family of diffeomorphisms

$\Theta_{h}(\cdot,t)\colon\mathcal{I}(0)\mapsto\mathcal{I}(t)\leavevmode\nobreak\
\text{ and }\leavevmode\nobreak\
\mathring{\Omega}(0)\mapsto\mathring{\Omega}(t).$

Recalling (99), we see that for a fixed $t$, $\Theta_{h}(\cdot,t)$ is the
identity map on ${\mathbb{R}^{d}}\backslash B_{\delta}(\mathcal{I}(t))$; in
particular near $\partial\Omega.$ Moreover
$\operatorname{det}\nabla\Theta_{h}\geq c>0$ and

$\quad\left\|\nabla\Theta_{h}\right\|_{L^{\infty}_{x,t}}+\left\|\nabla\Theta_{h}^{-1}\right\|_{L^{\infty}_{x,t}}\leq
C\left(1+\|h\|_{C^{1}(\mathcal{I}(0)\times J)}\right)$

with $c,C>0$ independent of $h$. For technical reasons, it is simpler to
replace $h\circ P_{\mathcal{I}(0)}$ in (98) by an extension $Eh$ where

$\displaystyle E\colon X_{0}$
$\displaystyle:=W_{p}^{1-1/p}(\mathcal{I}(0))\mapsto
W_{p}^{1}\left(B_{\delta}(\mathcal{I}(0))\right)$ (109)

with $E$ being a bounded and linear extension operator of class

$\displaystyle
E\in\mathscr{L}\left(W_{p}^{k-1/p}(\mathcal{I}(0)),W_{p}^{k}\left(B_{\delta}(\mathcal{I}(0))\right)\right),\quad\forall
k\in\\{1,\ldots,4\\}.$ (110)

Based on this extension operator, we define a modified Hanzawa transform by

$\displaystyle
y=\tilde{\Theta}_{h}(x,t):=x+\zeta\left(\frac{\operatorname{dist}(x,\mathcal{I}(0))}{\delta}\right)(Eh)(x,t)\mathbf{n}_{\mathcal{I}(0)}(x),\quad\forall
x\in{\mathbb{R}^{d}}.$ (111)

Let $(v,\pi)$ be a solution of (92). We then define the transformed solution
by

$\displaystyle\tilde{v}(x,t):=v(\tilde{\Theta}_{h}(x,t),t),\quad\tilde{\pi}(x,t):=\pi(\tilde{\Theta}_{h}(x,t),t),$
(112)

for $(x,t)\in\Omega\times(0,T)$. Equivalently for $y\in\mathring{\Omega}(t)$,
we have

$\displaystyle
v(y,t):=\tilde{v}(\tilde{\Theta}_{h}^{-1}(y,t),t),\quad\pi(y,t):=\tilde{\pi}(\tilde{\Theta}_{h}^{-1}(y,t),t).$
(113)

To identify the system of PDEs satisfied by $(\tilde{v},\tilde{\pi})$, we
introduce the following notation:

$\displaystyle\nabla^{h}$
$\displaystyle:=((\nabla\tilde{\Theta}_{h})^{-1})^{\mathsf{T}}\nabla,$ (114a)
$\displaystyle D^{h}\mathbf{v}$
$\displaystyle:=\frac{1}{2}\left(\nabla^{h}\mathbf{v}+(\nabla^{h}\mathbf{v})^{\mathsf{T}}\right),$
(114b) $\displaystyle\operatorname{div}^{h}\tilde{v}$
$\displaystyle:=\operatorname{tr}\nabla^{h}\tilde{v},$ (114c) $\displaystyle
T^{h}(\tilde{v},\tilde{\pi})$
$\displaystyle:=2D^{h}\tilde{v}-\tilde{\pi}\leavevmode\nobreak\ \mathrm{Id},$
(114d) $\displaystyle H^{h}$
$\displaystyle:=H_{\mathcal{I}}\circ\tilde{\Theta}_{h}(x,t)\colon\mathcal{I}(0){\times}J\rightarrow{\mathbb{R}^{d}},$
(114e) $\displaystyle\mathbf{n}^{h}$
$\displaystyle:=\mathbf{n}_{\mathcal{I}}\circ\tilde{\Theta}_{h}(x,t)\colon\mathcal{I}(0){\times}J\rightarrow{\mathbb{R}^{d}}.$
(114f)

Then it follows from (114a) and (113) that

$\displaystyle\nabla_{y}v|_{y=\tilde{\Theta}_{h}(x,t)}=\nabla^{h}\tilde{v}(x).$
(115)

Similarly, by (112) we obtain
$\partial_{t}\tilde{v}(x)=(\partial_{t}v+\partial_{t}\tilde{\Theta}_{h}\cdot\nabla
v)|_{y=\tilde{\Theta}_{h}(x,t)}.$ With these definitions and formulas, the
system (92) can be rewritten as one over a fixed domain. For simplicity we
split it into two parts, one describing the hydrodynamics, and one for the
evolution of the interface:

$\displaystyle\partial_{t}\tilde{v}+\tilde{v}\cdot\nabla^{h}\tilde{v}$
$\displaystyle=\operatorname{div}^{h}\nabla^{h}\tilde{v}+\nabla^{h}\tilde{\pi}+\nabla^{h}\tilde{v}\,\partial_{t}\tilde{\Theta}_{h}$
$\displaystyle\text{ in }\mathring{\Omega}(0)\times(0,T),$ (116a)
$\displaystyle\operatorname{div}^{h}\tilde{v}$ $\displaystyle=0$
$\displaystyle\text{ in }\mathring{\Omega}(0)\times(0,T),$ (116b)
$\displaystyle[\\![2D^{h}\tilde{v}-\tilde{\pi}\mathrm{Id}]\\!]\cdot\mathbf{n}^{h}$
$\displaystyle=H^{h}\mathbf{n}^{h}$ $\displaystyle\text{ on
}\mathcal{I}(0)\times(0,T),$ (116c) $\displaystyle[\\![\tilde{v}]\\!]$
$\displaystyle=0$ $\displaystyle\text{ on }\mathcal{I}(0)\times(0,T),$ (116d)
$\displaystyle\left.\tilde{v}\right|_{\partial\Omega}$ $\displaystyle=0$
$\displaystyle\text{ on }\partial\Omega\times(0,T),$ (116e)
$\displaystyle\left.\tilde{v}\right|_{t=0}$ $\displaystyle=v_{0}$
$\displaystyle\text{ in }\Omega,$ (116f)

as well as

$\displaystyle\partial_{t}h$
$\displaystyle=H^{h}+\tilde{v}\cdot\mathbf{n}^{h}$ $\displaystyle\text{ on
}\mathcal{I}(0)\times(0,T),$ (117a) $\displaystyle h|_{t=0}$ $\displaystyle=0$
$\displaystyle\text{ on }\mathcal{I}(0).$ (117b)

To state the regularity of solutions to the above system, we finally introduce
the following function spaces:

$\displaystyle\mathbb{V}_{p}:=W_{p}^{1}\left(J;L^{p}(\Omega)\right)\cap
L^{p}\left(J;W_{p}^{2}(\mathring{\Omega}(0))\cap W_{p}^{1}(\Omega)\right),$
(118a)
$\displaystyle\mathbb{P}_{p}:=\Big{\\{}L^{p}\left(J;\mathring{W}_{p}^{1}(\mathring{\Omega}(0))\right)\Big{|}\int_{\Omega}\tilde{\pi}(\cdot,t)=0,$
$\displaystyle\qquad\qquad\qquad\qquad\leavevmode\nobreak\
[\\![\tilde{\pi}]\\!]\in
W_{p}^{1-1/p,\frac{1}{2}\left(1-1/p\right)}(\mathcal{I}(0){\times}(0,T))\Big{\\}},$
(118b)
$\displaystyle\mathbb{H}_{p}:=W_{p}^{1}\left(J;W_{p}^{1-1/p}(\mathcal{I}(0))\right)\cap
L^{p}\left(J;W_{p}^{3-1/p}(\mathcal{I}(0))\right).$ (118c)

######  Lemma 9.

The following embeddings hold true:

$\displaystyle\mathbb{V}_{p}\hookrightarrow
C(J;W_{p}^{2-2/p}(\mathring{\Omega}(0))),$ (119a)
$\displaystyle\mathbb{H}_{p}\hookrightarrow
C\left(J;W_{p}^{3-3/p}(\mathcal{I}(0))\right)\hookrightarrow
C(J;C^{2}(\mathcal{I}(0))).$ (119b)

###### Proof.

To compute the interpolation of Sobolev–Slobodeckij spaces we recall

$\displaystyle\left(W_{p}^{m_{1}}(U),W_{p}^{m_{2}}(U)\right)_{\theta,p}=W_{p}^{s}(U),\qquad
s=(1-\theta)m_{1}+\theta m_{2},$ (120)

where $\theta\in(0,1)$ and $U$ is an open set or a closed compact manifold. We
deduce from (120) that

$\displaystyle\left(W_{p}^{1-1/p}(U),W_{p}^{3-1/p}(U)\right)_{1-1/p,p}$
$\displaystyle=W_{p}^{3-\frac{3}{p}}(U)\hookrightarrow C^{2}(U),$ (121)
$\displaystyle\left(L_{p}(U),W_{p}^{2}(U)\right)_{1-1/p,p}$
$\displaystyle=W_{p}^{2-\frac{2}{p}}(U).$ (122)

These combined with the interpolation inequality (107) leads to (119a) and the
first embedding in (119b). Concerning the second embedding in (119b), as
$p>d+2$ and $\mathcal{I}(0)$ is $(d{-}1)$-dimensional, we have
$3-\frac{3}{p}-\frac{d-1}{p}>2$. Then the result follows from Morrey’s
embedding. ∎

The following result is a simpler version of Abels and Moser [zbMATH06951007,
Theorem 4.1].

######  Proposition 10.

Let the assumptions of Proposition 8 be in place. Let $p>d+2$ and $2<q<3$ with
$1+\frac{d+2}{p}>\frac{d+2}{q}$. Then there exists $T>0$ such that the system
consisting of (116) and (117) has a unique solution with

$\displaystyle(\tilde{v},\tilde{\pi},h)\in\mathbb{V}_{q}\times\mathbb{P}_{q}\times\mathbb{H}_{p}.$
(123)

We make a comment on the regularity of $(\tilde{v},\tilde{\pi})$.

######  Remark 11.

According to the regularity of $\tilde{v}$ by (118a) and the embedding (119a),
we must have $v_{0}(\cdot)=\tilde{v}(\cdot,0)\in
W_{q}^{2-2/q}(\mathring{\Omega}(0))$. If $q>3$, then the trace estimate
implies $\nabla v_{0}\in W_{q}^{1-2/q}(\mathring{\Omega}(0))\hookrightarrow
W_{q}^{1-3/q}(\mathcal{I}(0))$. In order to guarantee that the restrictions
from outer and inner domains $\Omega^{\pm}(0)$ give the same trace, an
additional compatibility condition must be added. Such a condition is caused
by the possible jump of $\nabla v_{0}$ across $\mathcal{I}(0)$. To avoid such
a compatibility condition, Abels and Moser [zbMATH06951007] assume $q<3$, and
this causes unbalanced regularities (123). However, in the current system
(116), we work in the regime of coinciding shear viscosities of the two
phases. It turns out that this assumption necessitates continuity of the flow
gradient across the interface, see Lemma 12 below. In particular, we can
simply assume $v_{0}\in W_{q}^{2-2/q}(\Omega)$.

######  Lemma 12.

Under the assumption of coinciding shear viscosities $\mu_{+}=\mu_{-}$ of the
two fluid phases, it follows that the flow gradient $\nabla v$ is continuous
across $\mathcal{I}$, i.e., $[\\![\nabla v]\\!]=0$. As a result, we can
improve (97a) to

$\displaystyle v\in W_{p}^{1}\left(J;L^{p}(\Omega)\right)\cap
L^{p}\left(J;W_{p}^{2}\left(\Omega\right)\right).$ (124)

###### Proof.

In the proof, we shall abbreviate $\mathbf{n}_{\mathcal{I}}$ by $\mathbf{n}$.
It follows from (97a) that

$\nabla v\in W^{1}_{p}(\mathring{\Omega}(t))\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for a.e.\ }t\in[0,T],$ (125)

and thus $[\\![\nabla v]\\!]$ makes sense. By (92d), we deduce that the
tangential derivative of $v$ do not jump across $\mathcal{I}$, i.e.,

$\displaystyle[\\![\nabla^{\tan}v_{i}]\\!]=0\leavevmode\nobreak\ \text{ on
}\mathcal{I}\leavevmode\nobreak\ \text{ for }\leavevmode\nobreak\ 1\leq i\leq
d,$ (126)

where $\nabla^{\tan}v_{i}:=(\mathrm{Id}-\mathbf{n}\otimes\mathbf{n})\nabla
v_{i}$ is the tangential gradient of $v_{i}$. Thus, one can verify that
$\nabla\cdot v=0$ in $\Omega$ in the sense of distribution (see [Alberti2003,
Lemma 2.5] for an even more general situation), i.e., for any $\varphi\in
C_{c}^{1}(\Omega)$ it holds $\int_{\Omega}v\cdot\nabla\varphi\,dx=0$. This in
turn implies by the regularity of $v$ that $\nabla\cdot v=0$ a.e. in $\Omega$,
and therefore by (126)

$\displaystyle[\\![\mathbf{n}\otimes\mathbf{n}\colon\nabla
v]\\!]=[\\![(\mathbf{n}\otimes\mathbf{n}-\mathrm{Id})\colon\nabla
v]\\!]+[\\![\nabla\cdot v]\\!]=0.$ (127)

Next, multiplying (92c) by any vector field $\mathbf{t}$ orthogonal to
$\mathbf{n}$, we deduce from (126)

$\displaystyle
0=[\\![2\nabla^{\mathrm{sym}}v:\mathbf{n}\otimes\mathbf{t}]\\!]=[\\![\nabla
v:\mathbf{t}\otimes\mathbf{n}]\\!].$ (128)

Hence, the claim of Lemma 12 follows from (126), (127) and (128). ∎

###  A.3. Proof of Proposition 8.

The regularities of $(\tilde{v},\tilde{\pi})$ and $\tilde{h}$ in (123) are not
balanced. It remains to show further integrability of
$(\tilde{v},\tilde{\pi})$ by solving (116) separately under some additional
compatibility conditions. A complete proof of this will be quite lengthy and
technical, and we thus only give a sketch of the proof. To this end, we write
the system (116) in an abstract form

$\displaystyle\mathscr{L}\begin{pmatrix}\tilde{v}\\\
\tilde{\pi}\end{pmatrix}=\mathscr{N}\begin{pmatrix}\tilde{v}\\\
\tilde{\pi}\end{pmatrix}$ (129)

where

$\displaystyle\mathscr{L}\begin{pmatrix}\tilde{v}\\\
\tilde{\pi}\end{pmatrix}:=\left(\begin{array}[]{c}\partial_{t}\tilde{v}-\Delta\tilde{v}-\nabla\tilde{\pi}\\\
\operatorname{div}\tilde{v}\\\
[\\![2D\tilde{v}-\tilde{\pi}\mathrm{Id}]\\!]\mathbf{n}_{\mathcal{I}}\\\
\tilde{v}|_{t=0}\end{array}\right)$ (134)

is the linearized operator and $\mathscr{N}$ is the nonlinear one:

$\displaystyle\mathscr{N}\begin{pmatrix}\tilde{v}\\\
\tilde{\pi}\end{pmatrix}:=\left(\begin{array}[]{c}\mathbf{a}\left(\tilde{v},\tilde{\pi},h\right)-\tilde{v}\cdot\nabla^{h}\tilde{v}+\nabla^{h}\tilde{v}\,\partial_{t}\tilde{\Theta}_{h}\\\
(\operatorname{div}\tilde{v}-\operatorname{div}^{h}\tilde{v})-\fint_{\Omega}(\operatorname{div}\tilde{v}-\operatorname{div}^{h}\tilde{v})dx\\\
\mathbf{b}\left(\tilde{v},\tilde{\pi},h\right)+H^{h}\mathbf{n}^{h}\\\
v_{0}\end{array}\right).$ (139)

In (139), we used the definitions from (114) and

$\displaystyle\mathbf{a}(\tilde{v},\tilde{\pi},h)$
$\displaystyle:=(\operatorname{div}^{h}\nabla^{h}\tilde{v}-\Delta\tilde{v})+(\nabla^{h}-\nabla)\tilde{\pi},$
(140a) $\displaystyle\mathbf{b}(\tilde{v},\tilde{\pi},h)$
$\displaystyle:=[\\![2D\tilde{v}-\tilde{\pi}\leavevmode\nobreak\
\mathrm{Id}]\\!](\mathbf{n}_{\mathcal{I}}-\mathbf{n}^{h})+2[\\![D\tilde{v}-D^{h}\tilde{v})]\\!]\mathbf{n}^{h}.$
(140b)

######  Remark 13.

It is easy to verify that the operator equation (129) is equivalent to (116)
except that the second equation in the latter, i.e.,
$\operatorname{div}^{h}\tilde{v}=0$, is replaced by

$\displaystyle\operatorname{div}^{h}\tilde{v}=\fint_{\Omega}\operatorname{div}^{h}\tilde{v}\,dx$
(141)

after simplification. It is obvious that the former equation implies the
latter one. The opposite direction is proved in [MR3062573, p. 51]:

$\displaystyle\fint_{\Omega}\operatorname{div}^{h}\tilde{v}\,dx\int_{\Omega}|\det\nabla\tilde{\Theta}(x,t)|\,dx$
$\displaystyle\overset{\eqref{div
integral}}{=}\int_{\Omega}\operatorname{div}^{h}\tilde{v}|\det\nabla\tilde{\Theta}(x,t)|\,dx$
$\displaystyle\overset{\eqref{gradient
equ}}{=}\int_{\Omega}(\operatorname{div}_{y}v)|_{y=\tilde{\Theta}_{h}(x,t)}|\det\nabla\tilde{\Theta}(x,t)|\,dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
=\int_{\Omega}\operatorname{div}_{y}v(y)\,dy=0.$

Recall from (117) that $h$ starts from $0$. By the regularity
$h\in\mathbb{H}_{p}$, cf. (118c), it can be shown that, within a short time
period $[0,T]$, the nonlinear operator $\mathscr{N}$ defined by (139) is
locally Lipschitz continuous between a pair of Banach spaces. Moreover, we
shall show that $\mathscr{L}$ is an isomorphism between this pair. These two
results together with Banach’s fixed point theorem then lead to the proof of
Proposition 8.

The invertibility of the linear operator $\mathscr{L}$ from (134) corresponds
to the solvability of the following linear system:

$\displaystyle\partial_{t}\tilde{v}-\Delta\tilde{v}+\nabla\tilde{\pi}$
$\displaystyle=f$ $\displaystyle\text{ in }\mathring{\Omega}(0)\times(0,T),$
(142a) $\displaystyle\operatorname{div}\tilde{v}$ $\displaystyle=g$
$\displaystyle\text{ in }\mathring{\Omega}(0)\times(0,T),$ (142b)
$\displaystyle[\\![2D\tilde{v}-\tilde{\pi}\leavevmode\nobreak\
\mathrm{Id}]\\!]\leavevmode\nobreak\ \mathbf{n}_{\mathcal{I}}$
$\displaystyle=w$ $\displaystyle\text{ on }\mathcal{I}(0)\times(0,T),$ (142c)
$\displaystyle[\\![\tilde{v}]\\!]$ $\displaystyle=0$ $\displaystyle\text{ on
}\mathcal{I}(0)\times(0,T),$ (142d) $\displaystyle\tilde{v}|_{\partial\Omega}$
$\displaystyle=0$ $\displaystyle\text{ on }\partial\Omega\times(0,T),$ (142e)
$\displaystyle\tilde{v}|_{t=0}$ $\displaystyle=\tilde{v}_{0}$
$\displaystyle\text{ in }\Omega.$ (142f)

Here, we assume $(f,g,w)\in
L^{p}_{x,t}\times\mathbb{G}_{p}\times\mathbb{W}_{p}$ where

$\displaystyle\mathbb{G}_{p}:=\left\\{g\in
L^{p}\left(0,T;W^{1}_{p}(\mathring{\Omega}(0))\right)\Big{|}\int_{\Omega}g(t,\cdot)=0\text{
for a.e.\ }t\in(0,T)\right\\},$ (143a)
$\displaystyle\mathbb{W}_{p}:=W^{\frac{1}{2}(1-\frac{1}{p})}_{p}\Big{(}0,T;L^{p}(\mathcal{I}(0))\Big{)}\cap
L^{p}\Big{(}0,T;W^{1-\frac{1}{p}}_{p}(\mathcal{I}(0))\Big{)},$ (143b)

and we assume that $(g,w,\tilde{v}_{0})$ satisfy the following compatibility
conditions:

$\displaystyle\tilde{v}_{0}$ $\displaystyle\in
W_{p}^{2-2/p}(\mathring{\Omega}(0)),$ (144a)
$\displaystyle\operatorname{div}\tilde{v}_{0}$ $\displaystyle=g|_{t=0}$
$\displaystyle\text{in}\leavevmode\nobreak\ \mathring{\Omega}(0)$ (144b)
$\displaystyle\tilde{v}_{0}|_{\partial\Omega}$ $\displaystyle=0,$ (144c)
$\displaystyle[\\![\tilde{v}_{0}]\\!]$ $\displaystyle=0$
$\displaystyle\text{on}\leavevmode\nobreak\ \mathcal{I}(0),$ (144d)
$\displaystyle\Pi_{\mathcal{I}}\left([\\![2D\tilde{v}_{0}]\\!]\cdot\mathbf{n}_{\mathcal{I}}\right)\Big{|}_{t=0}$
$\displaystyle=\Pi_{\mathcal{I}}w\Big{|}_{t=0}.$ (144e)

######  Remark 14.

We note that the integral constraint in (143a) follows by integrating (142b)
and using (142d) and (142e). The compatibility condition (144b) follows by
taking $t=0$ in (142b). Finally, (144e) is a consequence of taking the
tangential projection of (142c) and then restricting it to $t=0$.

The solvability of the system (142) is due to the following maximal regularity
result in [MR2548875].

######  Lemma 15.

Assuming the compatibility conditions (144), the system (142) has a unique
solution with the following estimate under notations in (118):