# On a nonlocal two-phase flow with convective heat transfer

Šárka Nečasová<EMAIL_ADDRESS>John Sebastian H. Simon<EMAIL_ADDRESS>
###### Abstract

We study a system describing the dynamics of a two-phase flow of
incompressible viscous fluids influenced by the convective heat transfer of
Caginalp-type. The separation of the fluids is expressed by the order
parameter which is of diffuse interface and is known as the Cahn-Hilliard
model. We shall consider a nonlocal version of the Cahn-Hilliard model which
replaces the gradient term in the free energy functional into a spatial
convolution operator acting on the order parameter and incorporate with it a
potential that is assumed to satisfy an arbitrary polynomial growth. The order
parameter is influenced by the fluid velocity by means of convection, the
temperature affects the interface via a modification of the Landau-Ginzburg
free energy. The fluid is governed by the Navier–Stokes equations which is
affected by the order parameter and the temperature by virtue of the
capillarity between the two fluids. The temperature on the other hand
satisfies a parabolic equation that considers latent heat due to phase
transition and is influenced by the fluid via convection. The goal of this
paper is to prove the global existence of weak solutions and show that, for an
appropriate choice of sequence of convolutional kernels, the solutions of the
nonlocal system converges to its local version.

###### keywords:

nonlocal Cahn–Hilliard equations , Boussinesq equations , well-posedness ,
nonlocal-to-local convergence

###### MSC:

[2020] 45K05 , 76D03 , 76T06 , 35B40

[]organization=Institute of Mathematics, Czech Academy of
Sciences,city=Prague, country=Czech Republic

##  1 Introduction

The study of multi-phase flows has been of interest among experts due to their
complexities and robust properties. One of the well-known models is the so-
called Cahn-Hilliard equation which models a binary immiscible fluid that
assumes a diffuse interface between the two fluids [1, 2]. Since then several
modifications have been introduced to take into account the physical and
biological implications of the model, see eg. [3, 4] for application to tumor
growth, [5, 6] for binary fluids with unmatched densities, [7, 8] for binary
magnetic fluids, [9] for separation of diblock copolymers, and [10] for image
reconstruction.

Aside from modifying the original system or coupling it with other physical
models to describe more physical (or biological) phenomena, the Cahn-Hilliard
equations have been recasted to consider long-range particle interactions on
the interface of the two fluids. This version of the Cahn-Hilliard equations,
known to many as the nonlocal Cahn-Hilliard equations, was introduced by G.
Giacomin and J. Lebowitz [11, 12]. Such model has been the subject of many
expositions, among them are the following works [13, 14, 15, 16, 17, 18, 19].
These modifications serve as motivations for this article. In particular, we
will consider an incompressible two-phase flow undergoing phase separation due
to diffusion and is influenced by temperature by means of a Caginalp-type
equation. The dynamics of the binary fluids described by the Cahn-Hilliard
equations will make use of the aforementioned nonlocal system.

Let $\Omega\subset\mathbb{R}^{d}$ be a fixed domain – with sufficiently smooth
boundary $\partial\Omega$ – which a two-phase fluid occupies, and $T>0$ be a
fixed final time. We denote by
$({\operatorname{{\mathbf{u}}}},p):\Omega\times(0,T)\to\mathbb{R}^{d}\times\mathbb{R}$
the velocity-pressure pairing of the fluid,
$\theta:\Omega\times(0,T)\to\mathbb{R}$ the relative temperature around the
critical temperature $\theta_{c}=0$, and
$\varphi:\Omega\times(0,T)\to\mathbb{R}$ the (relative) concentration of the
binary fluid.

The concentration $\varphi$ is described by the Cahn–Hilliard equation with a
transport term influenced by the average velocity
${\operatorname{{\mathbf{u}}}}$, which takes values $\varphi\in[-1,1]$, where
$\varphi=1$ or $\varphi=-1$ corresponds to the two different phases of the
fluids. To be specific, the dynamics of $\varphi$ are governed by

$\displaystyle D_{t}\varphi=m\Delta\mu,$

where $D_{t}=\partial_{t}+{\operatorname{{\mathbf{u}}}}\cdot\nabla$ is the
material derivative due to the fluid velocity, $\mu$ is the chemical
potential, and $m$ corresponds to the mobility of the interface. Such model –
if one, for the meantime, ignores the contribution of the fluid velocity – is
based on the so-called H-model which considers a diffuse interface between two
interacting fluids [1, 20].

The average velocity and pressure of the fluids is governed by the
Navier–Stokes equations and is influenced by the order parameter $\varphi$ and
the relative temperature $\theta$ by a Korteweg-type force proportional to
$(\mu-\ell_{c}\theta)\nabla\varphi$ where $\ell_{c}$ is a parameter concerning
latent heat [21, Appendix]. A Boussinesq approximation which incorporates a
constant gravitational force with the linearized equation of state expressed
as
$\ell(\varphi,\theta){\operatorname{\mathbf{g}}}=(\alpha_{0}+\alpha_{1}\varphi+\alpha_{2}\theta){\operatorname{\mathbf{g}}}$
is also assumed to affect the fluid. To be specific, the fluid is governed by
the equation

$\displaystyle
D_{t}{\operatorname{{\mathbf{u}}}}-\operatorname{\mathrm{div}}(\nu(\varphi)2\mathrm{D}{\operatorname{{\mathbf{u}}}})+\nabla
p=\mathcal{K}(\mu-\ell_{c}\theta)\nabla\varphi+\ell(\varphi,\theta){\operatorname{\mathbf{g}}}+{\operatorname{{\mathbf{q}}}}.$

Here $\mathcal{K}$ is a capillarity constant, $\nu(\varphi)>0$ denotes the
fluid viscosity, ${\operatorname{{\mathbf{q}}}}$ denotes an external force and
the operator $\mathrm{D}$ is defined as
$\mathrm{D}{\operatorname{{\mathbf{u}}}}=(\nabla{\operatorname{{\mathbf{u}}}}+\nabla{\operatorname{{\mathbf{u}}}}^{\top})/2$.
Ignoring for the meantime the contribution of the temperature $\theta$, the
combination of the two equations previously introduced is the well-known
Navier–Stokes/Cahn–Hilliard system which has been considered in numerous
expositions, see for example the papers [5, 7, 22, 23, 24, 25, 26, 27] and the
references therein.

The equation concerning the temperature is based on Caginalp’s interpretation
of free boundary problems derived from phase trasition [28]. By introducing
the quantity $H(\theta,\varphi):=\theta-\ell_{h}\varphi$ we consider the
dynamics governed by the transport equation

$\displaystyle
D_{t}H(\theta,\varphi)-\kappa\Delta\theta={\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}+z,$

where $\ell_{h}$ is a constant related to the latent heat,
${\operatorname{\mathbf{g}}}$ is the gravitational force, and $z$ is an
external heat source. The Laplacian above comes from the assumption that the
heat flux follows Fourier’s thermal conduction law.

Before we go any further, let us talk about the chemical potential $\mu$.
Here, we note that the usual Landau–Ginzburg free energy functional is written
as

$\displaystyle\tilde{\mathcal{E}}(\varphi,\theta):=\int_{\Omega}\left(\frac{\xi}{2}|\nabla\varphi|^{2}+\eta
F(\varphi)+\ell_{c}\theta\varphi\right)\operatorname{d\\!}x,$

where the chemical potential $\mu$ is obtained by the first variation of
$\mathcal{E}(\cdot,\theta)$, i.e., $\mu=-\xi\Delta\varphi+\eta
F^{\prime}(\varphi)+\ell_{c}\theta$. Such functional is used in [29, 30] to
model a nonisothermal diffuse interface two-phase flow where the author showed
the existence of solutions and its application to optimal control problems.

Recent developments, however, saw the applicability of nonlocal energy
functionals, for example in tumor growth models [4] and in exhibiting sharp
interface without having to taking asymptotic limit of the diffusive constant
[15]. In this paper, we shall modify the temperate Landau free energy
functional into a nonlocal one by changing the gradient into a nonlocal
spatial operator. To be specific, we shall use the non-local Landau–Ginzburg
free energy functional

$\displaystyle\mathcal{E}(\varphi,\theta):=$
$\displaystyle\frac{1}{4}\int_{\Omega}\int_{\Omega}J(x-y)(\varphi(x)-\varphi(y))^{2}\operatorname{d\\!}x\operatorname{d\\!}y+\int_{\Omega}\left(\eta
F(\varphi(x))+\ell_{c}\theta(x)\varphi(x)\right)\operatorname{d\\!}x,$

where $J:\mathbb{R}^{2}\to\mathbb{R}$ is a sufficiently smooth function that
satisfies $J(x)=J(-x)$. The first variation of $\mathcal{E}(\cdot,\theta)$
will then give us the chemical potential $\mu=a\varphi-J\ast\varphi+\eta
F^{\prime}(\varphi)+\ell_{c}\theta$, where

$\displaystyle a(x):=\int_{\Omega}J(x-y)\operatorname{d\\!}y\ \text{ and }\
(J\ast\varphi)(x)=\int_{\Omega}J(x-y)\varphi(y)\operatorname{d\\!}y.$ (1.1)

Such energy functional has been proven to take into account microscopic
interactions that describe phase segregation [11, 12]. We also mention [17,
18] where the authors proved the convergence of the nonlocal system to the
local one which also serve as a justification of the contention that
$\tilde{\mathcal{E}}$ is the macroscopic limit of the energy functional
$\mathcal{E}$.

Combining all these, we get the non-local Cahn-Hilliard-Boussinesq equations

$\displaystyle\partial_{t}\varphi+{\operatorname{{\mathbf{u}}}}\cdot\nabla\varphi=m\Delta\mu,\quad\mu=a\varphi-J\ast\varphi+\eta
F^{\prime}(\varphi)+\ell_{c}\theta,$ (1.2a) $\displaystyle\begin{aligned}
\partial_{t}{\operatorname{{\mathbf{u}}}}+({\operatorname{{\mathbf{u}}}}\cdot\nabla){\operatorname{{\mathbf{u}}}}{-}\operatorname{\mathrm{div}}(\nu(\varphi)2\mathrm{D}{\operatorname{{\mathbf{u}}}})+\nabla
p=\mathcal{K}(\mu-\ell_{c}\theta)\nabla\varphi+\ell(\varphi,\theta){\operatorname{\mathbf{g}}}+{\operatorname{{\mathbf{q}}}},\end{aligned}$
(1.2b)
$\displaystyle\partial_{t}\theta-\ell_{h}\partial_{t}\varphi+{\operatorname{{\mathbf{u}}}}\cdot\nabla(\theta-\ell_{h}\varphi)-\kappa\Delta\theta={\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}+z,$
(1.2c)

all of which are satisfied in the space-time domain $Q:=\Omega\times I$,
where $I=(0,T)$, with the divergence-free assumption
$\operatorname{\mathrm{div}}{\operatorname{{\mathbf{u}}}}=0$ in $Q$, the
boundary boundary conditions
$\frac{\partial\mu}{\partial{\operatorname{\mathbf{n}}}}=0$,
${\operatorname{{\mathbf{u}}}}=0$, and
$\frac{\partial\theta}{\partial{\operatorname{\mathbf{n}}}}=0$ on
$\Gamma:=\partial\Omega\times I$, and initial conditions
$\varphi(0)=\varphi_{0}$,
${\operatorname{{\mathbf{u}}}}(0)={\operatorname{{\mathbf{u}}}}_{0}$, and
$\theta(0)=\theta_{0}$ in $\Omega$.

The purpose of this article is to show the existence of weak solutions through
spectral Galerkin method and to provide proof that, under appropriate
conditions and choice of the convolutional kernel, solutions of the nonlocal
system converge to its local version. We shall apply the techniques used in
[31] for the Cahn-Hilliard system with periodic boundary conditions and
sigular free energy density, by [17] for the Cahn-Hilliard equations with
singular potentials and $W^{1,1}$ kernels, and [6] for a
Cahn–Hilliard–Navier–Stokes equations with unmatched densities.

The novelty in this article are as follows:

(i) compared to the works of P. Colli et al. [16] and S. Frigeri et al. [19]
we considered a nonisothermal system, where the influence of the temperature
is based on G. Caginalp [28]; (ii) aside from modifying the local free energy
functional into the nonlocal version in [29, 30], we considered a non-constant
viscosity dependent on the order parameter, a generalized potential of
polynomial growth, and considered not only the two-dimensional case but also
delved into the system in three dimensions.

The subsequent parts of this paper is as follows: the next section is devoted
to the mathematical setting such as the functional spaces and the necessary
embeddings. Section 3 deals with the existence of weak solutions for (1.2)
where we use the spectral Galerkin method as a discretization which gets to
converge to the original continuous system. The final section establishes
convergence of the nonlocal system to its local version by using the framework
introduced in [32, 33] and the $\Gamma$-convergence in [34, 35].

##  2 Preliminaries

Let us introduce the function spaces upon which the analysis will anchor on.
Given a Banach space $X$ with its dual $X^{*}$, the duality pairing is simply
denoted as $\langle x^{*},x\rangle_{X}$ for $x^{*}\in X^{*}$ and $x\in X$. We
denote by $L^{p}(\Omega)$ the space of Lebesgue integrable function of order
$p\geq 1$ and the usual Sobolev-Slobodeckij spaces are denoted as
$W^{s,p}(\Omega)$ with $H^{s}(\Omega)=W^{s,2}(\Omega)$. For simplicity, we use
$\|\cdot\|$ and $(\cdot,\cdot)_{\Omega}$ to denote the norm and inner product
in $L^{2}(\Omega)$ or $L^{2}(\Omega)^{d}$, for $d=2,3$, $\|\cdot\|_{L^{p}}$
for the norm in $L^{p}(\Omega)$ or $L^{p}(\Omega)^{d}$, and
$\|\cdot\|_{W^{s,p}}$ the norm in $W^{s,p}(\Omega)$ or $W^{s,p}(\Omega)^{d}$.
To take into account the incompressibility of the fluid we utilize the
following solenoidal spaces $V_{\sigma}=\\{{\operatorname{{\mathbf{u}}}}\in
H^{1}_{0}(\Omega)^{d}:\operatorname{\mathrm{div}}{\operatorname{{\mathbf{u}}}}=0\text{
in }\Omega\\}$, and $H_{\sigma}=\\{{\operatorname{{\mathbf{u}}}}\in
L^{2}(\Omega)^{d}:\operatorname{\mathrm{div}}{\operatorname{{\mathbf{u}}}}=0\text{
in
}\Omega,\,{\operatorname{{\mathbf{u}}}}\cdot{\operatorname{\mathbf{n}}}=0\text{
on }\partial\Omega\\}$.

Let us define $V_{s}:=D(B^{s/2})$, where $B=-\Delta$ with homogeneous Neumann
boundary conditions. We hence have

$V_{2}=D(B)=\left\\{\varphi\in
H^{2}(\Omega):\frac{\partial\varphi}{\partial{\operatorname{\mathbf{n}}}}=0\text{
on }\partial\Omega\right\\}.$

We also use the notation $H:=V_{0}=L^{2}(\Omega)$ and
$V:=V_{1}=H^{1}(\Omega)$. We note that $B:D(B)\subset H\to H$ is an unbounded
linear operator in $H$, and $B^{-1}:H\to H$ is a self-adjoint compact operator
on $H$, which is also maximal monotone. We obtain a sequence of eigenvalues
$\nu_{j}$ with $0<\nu_{1}\leq\nu_{2}\leq\ldots\leq\nu_{j}\to\infty$, with the
associated eigenfunctions $\psi_{j}\in D(B)$ such that
$B\psi_{j}=\nu_{j}\psi_{j}$ for all $j$. The set $\\{\psi_{j}\\}$ is an
orthonormal eigenbasis for $H$ while orthogonal in $V$ and $D(B)$.

Let us introduce the Stokes operator $A:D(A)\subset H_{\sigma}\to H_{\sigma}$
defined as $A=-P\Delta$, where $P:L^{2}(\Omega)^{d}\to H_{\sigma}$ is the
Leray projector. We note that $D(A)=H^{2}(\Omega)^{d}\cap V_{\sigma}$ whenever
$\Omega$ is of either of class $\mathcal{C}^{2}$ or a convex polygonal domain.
A sequence of eigenvalues $\lambda_{j}$ with
$0\leq\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{j}\to\infty$ can be
obtained with the associated eigenfunctions $v_{j}\in D(A)$ so that
$Av_{j}=\lambda_{j}v_{j}$. The set $\\{v_{j}\\}$ is an orthonormal eigenbasis
for $H_{\sigma}$ which is orthogonal in $D(A)$.

Let ${\operatorname{{\mathbf{u}}}}\in V_{\sigma}$ and $\varphi\in H$, the
operator $\mathcal{A}:V_{\sigma}\times H\to V_{\sigma}^{*}$ is defined as

$\langle\mathcal{A}({\operatorname{{\mathbf{u}}}},\varphi),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}=2(\nu(\varphi)\mathrm{D}{\operatorname{{\mathbf{u}}}},\mathrm{D}{\operatorname{\mathbf{v}}})_{\Omega}.$

The trilinear form $b:[H^{1}(\Omega)^{d}]^{3}\to\mathbb{R}$ coming out of the
weak formulation of the Navier–Stokes equations is defined as
$b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})=(({\operatorname{{\mathbf{u}}}}\cdot\nabla){\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})_{\Omega}$.
Such trilinear form is continuous and known to satisfy
$b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})=-b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{w}}},{\operatorname{\mathbf{v}}})$
for all
${\operatorname{{\mathbf{u}}}},{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}}\in
V_{\sigma}$. From $b$, we also define $\mathcal{B}:V_{\sigma}\times
V_{\sigma}\to V_{\sigma}^{*}$ as
$\langle\mathcal{B}({\operatorname{{\mathbf{u}}}},{\operatorname{\mathbf{v}}}),{\operatorname{\mathbf{w}}}\rangle_{V_{\sigma}}=b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})$
for
${\operatorname{{\mathbf{u}}}},{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}}\in
V_{\sigma}$. From which we shall use the notation
$\mathcal{B}({\operatorname{{\mathbf{u}}}},{\operatorname{{\mathbf{u}}}}):=\mathcal{B}({\operatorname{{\mathbf{u}}}})$
for simplicity.

To take into account the transport terms involving the scalar functions, we
introduce the operators $\mathcal{C}_{1}:V_{\sigma}\times V\to V^{*}$ and
$\mathcal{C}_{2}:V\times V\to V_{\sigma}^{*}$ respectively defined as

$\displaystyle\langle\mathcal{C}_{1}({\operatorname{\mathbf{v}}},\varphi),\psi\rangle_{V}=({\operatorname{\mathbf{v}}}\cdot\nabla\varphi,\psi)_{\Omega},$
$\displaystyle\langle\mathcal{C}_{2}(\varphi,\psi),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}=({\operatorname{\mathbf{v}}}\cdot\nabla\varphi,\psi)_{\Omega}=(\psi\nabla\varphi,{\operatorname{\mathbf{v}}})_{\Omega}.$

The following inequalities are well-known but we reiterate them here for
completeness.

######  Lemma 2.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open and bounded domain that is at
least of Lipschitzian class.

  1. I.

Rellich–Kondrachov: If $1\leq mp<n$ by setting $p^{*}=np/(n-mp)$ then the
embedding $W^{m,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$ is continuous,
while the embedding $W^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact
whenever $1\leq q<p^{*}$. If $mp\geq n$ then we have the compact embedding
$W^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ whenever $1\leq q<\infty$.

  2. II.

Gagliardo–Nirenberg: Let $1\leq p_{1},p_{2},p_{3}\leq+\infty$,
$j,m\in\mathbb{N}$ with $j<m$ and $\theta\in[0,1]$ satisfy the relation

$\displaystyle\frac{1}{p_{1}}=\theta\left(\frac{1}{p_{3}}-\frac{m}{n}\right)+\frac{1-\theta}{p_{2}},\quad\frac{j}{m}\leq\theta\leq
1.$

Then if $u\in L^{p_{2}}(\Omega)\cap W^{m,p_{3}}(\Omega)$ we get, for some
$c>0$ independent of $u$,

$\displaystyle\|u\|_{L^{p_{1}}}\leq
c\|u\|_{W^{m,p_{3}}}^{\theta}\|u\|_{L^{p_{2}}}^{1-\theta}.$

  3. III.

Interpolation in $L^{p}$: Let $u\in L^{p_{1}}(\Omega)$, $p_{2}<p_{1}$ and
$p_{\theta}=p_{1}p_{2}/[(1-\theta)p_{1}+\theta p_{2}]$. There exists $c>0$
independent of $u$ such that

$\displaystyle\|u\|_{L^{p_{\theta}}}\leq
c\|u\|_{L^{p_{1}}}^{\theta}\|u\|_{L^{p_{2}}}^{1-\theta}.$

By virtue of Gagliardo–Nirenberg inequality the following properties hold true
for any
${\operatorname{{\mathbf{u}}}},{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}}\in
V_{\sigma}$:

$\displaystyle|b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})|\leq
c\|{\operatorname{{\mathbf{u}}}}\|_{H_{\sigma}}^{1/2}\|{\operatorname{{\mathbf{u}}}}\|_{V_{\sigma}}^{1/2}\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}\|{\operatorname{\mathbf{w}}}\|_{V_{\sigma}},$
$\displaystyle\text{for }d=3$ (2.1)
$\displaystyle|b({\operatorname{{\mathbf{u}}}};{\operatorname{\mathbf{v}}},{\operatorname{\mathbf{w}}})|\leq
c\|{\operatorname{{\mathbf{u}}}}\|_{H_{\sigma}}^{1/2}\|{\operatorname{{\mathbf{u}}}}\|_{V_{\sigma}}^{1/2}\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}|{\operatorname{\mathbf{w}}}\|_{H_{\sigma}}^{1/2}\|{\operatorname{\mathbf{w}}}\|_{V_{\sigma}}^{1/2},$
$\displaystyle\text{for }d=2.$ (2.2)

Functions which are defined on the interval mapped to a real Banach space $X$
shall also be utilized. We denote by $C(I;X)$ the functions which are
continuous from $I$ to $X$ upon which we use the norm
$\|u\|_{C(I;X)}:=\displaystyle\sup_{t\in\overline{I}}\|u(t)\|_{X}$. The
Bochner spaces $L^{p}(I;X)$ with the norm

$\displaystyle\|u\|_{L^{p}(I;X)}=\left\\{\begin{aligned}
&\left(\int_{I}\|u(t)\|_{X}\operatorname{d\\!}t\right)^{1/p}&&\text{if
}p<\infty,\\\ &\mathrm{ess}\sup_{t\in I}\|u(t)\|_{X}&&\text{if
}p=\infty,\end{aligned}\right.$

shall also be utilized, as well as the space $W^{m,p}(I;X):=\\{u\in
L^{p}(I;X):\partial^{j}_{t}u\in L^{p}(I;X),0\leq j\leq m\\}$.

We recall Aubin–Lions–Simon embedding. Let $X,Y,Z$ be Banach spaces such that
the embeddings $X\hookrightarrow Y$ and $Y\hookrightarrow Z$ are compact and
continuous, respectively. Given $1\leq p,q\leq\infty$, the space
$W^{p,q}(X,Z):=\\{u\in L^{p}(I;X):\partial_{t}u\in L^{q}(I;Z)\\}$ is compactly
embedded to $L^{p}(I;Y)$ if $p<+\infty$. On the other hand, if $p=+\infty$ and
$q>1$ the embedding $W^{p,q}(X,Z)\hookrightarrow C(\overline{I};Y)$ is
compact.

##  3 Existence of weak solution

In this section we provide the existence of weak solutions to system (1.2). By
weak solution we use the following notion:

######  Definition 3.1.

Suppose that $\varphi_{0}\in H$ with $F(\varphi_{0})\in L^{1}(\Omega)$,
${\operatorname{{\mathbf{u}}}}_{0}\in H_{\sigma}$, and $\theta_{0}\in H$. We
say that $[\varphi,{\operatorname{{\mathbf{u}}}},\theta]$ is a weak solution
to (1.2) (with the boundary condition incorporated) on $[0,T]$ corresponding
to the initial conditions if

  * $\bullet$

$[\varphi,{\operatorname{{\mathbf{u}}}},\theta]$ including $\mu$ satisfies

$\displaystyle\left\\{\begin{aligned} &\varphi\in L^{\infty}(I;H)\cap
L^{2}(I;V),\\\ &\mu\in L^{2}(I;V),\\\ &\partial_{t}\varphi\in
L^{4/d}(I;V^{*}),\end{aligned}\right.$ $\displaystyle\left\\{\begin{aligned}
&{\operatorname{{\mathbf{u}}}}\in L^{\infty}(I;H_{\sigma})\cap
L^{2}(I;V_{\sigma}),\\\ &\partial_{t}{\operatorname{{\mathbf{u}}}}\in
L^{4/d}(I;V_{\sigma}^{*}),\end{aligned}\right.$
$\displaystyle\left\\{\begin{aligned} &\theta\in L^{\infty}(I;H)\cap
L^{2}(I;V),\\\ &\partial_{t}\theta\in L^{4/d}(I;V^{*}),\end{aligned}\right.$

  * $\bullet$

letting $\rho(x,\varphi):=a(x)\varphi+F^{\prime}(\varphi)$ the following
system of variational problems hold:

$\displaystyle\begin{aligned}
&\langle\partial_{t}\varphi(t),\psi\rangle_{V}+({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla\varphi(t),\psi)_{\Omega}+(m(\varphi)\nabla\rho(t),\nabla\psi)_{\Omega}\\\
&=(m(\varphi)\nabla(J\ast\varphi(t)),\nabla\psi)_{\Omega}-\ell_{c}(m(\varphi)\nabla\theta(t),\nabla\psi)_{\Omega},\end{aligned}$
(3.1a) $\displaystyle\begin{aligned}
&\langle\partial_{t}{\operatorname{{\mathbf{u}}}}(t),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+(({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla){\operatorname{{\mathbf{u}}}}(t),{\operatorname{\mathbf{v}}})_{\Omega}+2(\nu(\varphi)\mathrm{D}{\operatorname{{\mathbf{u}}}}(t),\mathrm{D}{\operatorname{\mathbf{v}}})\\\
&=\mathcal{K}(\operatorname{\mathbf{v}}\cdot\nabla\varphi(t),\mu(t)-\ell_{c}\theta(t))_{\Omega}+(\ell(\varphi(t),\theta(t)){\operatorname{\mathbf{g}}},{\operatorname{\mathbf{v}}})_{\Omega}+\langle{\operatorname{{\mathbf{q}}}},{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}\end{aligned}$
(3.1b) $\displaystyle\begin{aligned}
&\langle\partial_{t}\theta(t),\vartheta\rangle_{V}-\ell_{h}\langle\partial_{t}\varphi(t),\vartheta\rangle_{V}++\kappa(\nabla\theta(t),\nabla\vartheta)\\\
&=({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla(\ell_{h}\varphi(t)-\theta(t)),\vartheta)_{\Omega}+({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}(t),\vartheta)_{\Omega}+\langle
z,\vartheta\rangle_{V}\end{aligned}$ (3.1c)

for all $\psi\in V$, ${\operatorname{\mathbf{v}}}\in V_{\sigma}$,
$\vartheta\in V$ and almost every $t\in(0,T)$

  * $\bullet$

the initial conditions hold in weak sense, i.e.

$\displaystyle({\varphi}(t),{\psi})_{\Omega}\to({\varphi}_{0},{\psi})_{\Omega}\text{
as }t\to 0\quad\forall{\psi}\in V,$ (3.2)
$\displaystyle({\operatorname{{\mathbf{u}}}}(t),{\operatorname{\mathbf{v}}})_{\Omega}\to({\operatorname{{\mathbf{u}}}}_{0},{\operatorname{\mathbf{v}}})_{\Omega}\text{
as }t\to 0\quad\forall{\operatorname{\mathbf{v}}}\in V_{\sigma},$ (3.3)
$\displaystyle({\theta}(t),{\vartheta})_{\Omega}\to({\theta}_{0},{\vartheta})_{\Omega}\text{
as }t\to 0\quad\forall{\vartheta}\in V.$ (3.4)

Note that
$({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla\varphi(t),\psi)_{\Omega}=-({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla\psi,\varphi(t))_{\Omega}$.
This implies that by taking $\psi=1$, we get
$\langle\partial_{t}\varphi(t),1\rangle_{V}=0$. Whence, we get a total mass
conservation for the order parameter.

To be able to provide such existence, we have to impose the following
assumptions on the kernel $J$, the viscosity $\nu$, the potential $F$, and the
external forces ${\operatorname{{\mathbf{q}}}}$ and $z$:

  * (A1)

$J\in W^{1,1}(\mathbb{R}^{d})$, $J(x)=J(-x)$, and $a(x)\geq 0$ a.e.
$x\in\Omega$.

  * (A2)

We assume that the mobility is constant upon which we assume $m(\varphi)=1$,
while the viscosity function $\nu$ is locally Lipschitz on $\mathbb{R}$ and
that there exists $\underline{\nu},\overline{\nu}>0$ such that
$\underline{\nu}\leq\nu(s)\leq\overline{\nu}$ for all $s\in\mathbb{R}$.

  * (A3)

$F\in C^{2,1}_{loc}(\mathbb{R})$ and that there exists $c_{0}>0$ such that
$F^{\prime\prime}(s)+a(x)\geq c_{0}$, for all $s\in\mathbb{R}$ and a.e.
$x\in\Omega$.

  * (A4)

There exists $c_{1}>\frac{1}{2}\|J\|_{L^{1}(\mathbb{R}^{d})}$ and
$c_{2}\in\mathbb{R}$ such that $F(s)\geq c_{1}s^{2}-c_{2},$ for all
$s\in\mathbb{R}$.

  * (A5)

There exists $c_{3}>0$, $c_{4}\geq 0$, and $p\in(1,2]$ such that
$|F^{\prime}(s)|^{p}\leq c_{3}|F(s)|+c_{4}$ for all $s\in\mathbb{R}$.

  * (A6)

$\operatorname{{\mathbf{q}}}\in L^{2}(I;V_{\sigma}^{*})$, and $z\in
L^{2}(I;V^{*})$.

We mention that (A1) is a standard assumption for the nonlocal Cahn-Hilliard
equations which properly serves our purpose of establishing existence of weak
solutions and is satisfied by the sequence of kernels for establising
nonlocal-to-local convergence, nevertheless we mention [14] for a slightly
stricter assumption and [19, Remark 1] for a weaker assumption.

Note that (A3) implies that $F$ can be written as
$F(s)=G(s)-\frac{a^{*}}{2}s^{2}$, where $a^{*}=\|a\|_{L^{\infty}(\Omega)}$ and
$G\in C^{2,1}(\mathbb{R})$ is a strictly convex function. While (A5) accounts
for an arbitrary polynomial growth for $F$.

We also mention Korn’s inequality which will be useful in the subsequent
analyses

######  Lemma 3.2.

There exists a constant $c_{\rm K}>0$ such that the inequalities,
$\|D(\operatorname{\mathbf{v}})\|\leq\|\operatorname{\mathbf{v}}\|_{V_{\sigma}}\leq
c_{\rm K}\|D(\operatorname{\mathbf{v}})\|$, hold for any
$\operatorname{\mathbf{v}}\in V$.

Lastly, let us introduce the total energy for the system which is given by:

$\displaystyle\begin{aligned}
2\mathbb{E}(\varphi,{\operatorname{{\mathbf{u}}}},\theta):=&\,\frac{1}{\ell_{c}}\left\\{\frac{1}{2}\int_{\Omega}\int_{\Omega}J(x-y)(\varphi(x)-\varphi(y))^{2}\operatorname{d\\!}y\operatorname{d\\!}x+2\int_{\Omega}F(\varphi(x))\operatorname{d\\!}x\right\\}\\\
&+\frac{1}{\mathcal{K}\ell_{c}}\int_{\Omega}|{\operatorname{{\mathbf{u}}}}(x)|^{2}\operatorname{d\\!}x+\frac{1}{\ell_{h}}\int_{\Omega}|\theta(x)|^{2}\operatorname{d\\!}x,\end{aligned}$
(3.5)

and the dissipation functional denoted and written as:

$\displaystyle\mathbb{D}(\varphi,\mu,{\operatorname{{\mathbf{u}}}},\theta):=\frac{1}{\ell_{c}}\|\nabla\mu\|^{2}+\frac{2}{\mathcal{K}\ell_{c}}\|\sqrt{\nu(\varphi)}\mathrm{D}{\operatorname{{\mathbf{u}}}}\|^{2}+\frac{\kappa}{\ell_{h}}\|\nabla\theta\|^{2}$
(3.6)

We also use the following notation for the nonlocal energy

$\displaystyle\mathbb{E}_{nl}(\varphi):=\frac{1}{2}\int_{\Omega}\int_{\Omega}J(x-y)(\varphi(x)-\varphi(y))^{2}\operatorname{d\\!}y\operatorname{d\\!}x.$

The proof of existence of weak solutions that we shall provide will be based
to that of [16, Theorem 1]. We provide a comprehensive proof for completeness
as well as to simplify the computations we go through for the asymptotic
analysis in the next section. We also point out that instead of beginning with
a higher regularity for the initial data of the order parameter, as done in
the aforementioned reference, we go directly into using Yosida approximation
for the initial data. The promised existence of weak solution is summarized
and proven below.

######  Theorem 3.3.

Suppose that the assumptions on the initial data is as in Definition 3.1.
Additionally, assume that (A1)-(A6) hold. Then, for any $T>0$ there exists a
weak solution $[\varphi,{\operatorname{{\mathbf{u}}}},\theta]$ to the system
(1.2) on $[0,T]$. Furthermore, the following energy inequality holds for
almost every $t>0$

$\displaystyle\begin{aligned}
&\mathbb{E}(\varphi(t),{\operatorname{{\mathbf{u}}}}(t),\theta(t))+\int_{0}^{t}\mathbb{D}(\varphi(s),\mu(s),{\operatorname{{\mathbf{u}}}}(s),\theta(s))\operatorname{d\\!}s\\\
&\leq\mathbb{E}(\varphi_{0},{\operatorname{{\mathbf{u}}}}_{0},\theta_{0})+\int_{0}^{t}\big{\\{}\langle{\operatorname{{\mathbf{q}}}}(s),{\operatorname{{\mathbf{u}}}}(s)\rangle_{V_{\sigma}}+(\ell(\varphi(s),\theta(s)){\operatorname{\mathbf{g}}},{\operatorname{{\mathbf{u}}}}(s))_{\Omega}\\\
&\ \
+\langle{z}(s),{\theta}(s)\rangle_{V}+({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}(s),\theta(s))_{\Omega}\big{\\}}\operatorname{d\\!}s.\end{aligned}$
(3.7)

###### Proof.

The existence of solutions is established using spectral Galerkin method. We
project the system (3.1a)-(3.1c) onto the finite dimensional spaces spanned by
the orthonormal eigenfunctions of $H$ and $H_{\sigma}$. To be precise, we use
the finite dimensional spaces $H^{n}:=\textrm{span}\\{\psi_{j}\\}_{j=1}^{n}$
and
$H_{\sigma}^{n}:=\textrm{span}\\{{\operatorname{\mathbf{v}}}_{j}\\}_{j=1}^{n}$.
We also utilize the orthogonal projections $P_{\sigma}^{n}:H_{\sigma}\to
H_{\sigma}^{n}$ and $P^{n}:H\to H^{n}$ respectively defined as
$P_{\sigma}^{n}{\operatorname{{\mathbf{u}}}}=\sum_{j=1}^{n}({\operatorname{{\mathbf{u}}}},{\operatorname{\mathbf{v}}}_{j})_{\Omega}{\operatorname{\mathbf{v}}}_{j}$
and $P^{n}\varphi=\sum_{j=1}^{n}(\varphi,\psi_{j})_{\Omega}\psi_{j}$ for any
${\operatorname{{\mathbf{u}}}}\in H_{\sigma}$ and $\varphi\in H$.

The projected variational problem now reads

$\displaystyle\begin{aligned}
&\langle\partial_{t}\varphi^{n}(t),\psi\rangle_{V}+({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla\varphi^{n}(t),\psi)_{\Omega}+(\nabla\rho^{n}(t),\nabla\psi)_{\Omega}\\\
&=(\nabla(J\ast\varphi^{n}(t)),\nabla\psi)_{\Omega}-\ell_{c}(\nabla\theta^{n}(t),\nabla\psi)_{\Omega},\end{aligned}$
(3.8a) $\displaystyle\begin{aligned}
&\langle\partial_{t}{\operatorname{{\mathbf{u}}}}^{n}(t),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+(({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla){\operatorname{{\mathbf{u}}}}^{n}(t),{\operatorname{\mathbf{v}}})_{\Omega}+2(\nu(\varphi^{n})\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}(t),\mathrm{D}{\operatorname{\mathbf{v}}})\\\
&=\mathcal{K}(\operatorname{\mathbf{v}}\cdot\nabla\varphi^{n}(t),\mu^{n}(t)-\ell_{c}\theta^{n}(t))_{\Omega}+(\ell(\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}},{\operatorname{\mathbf{v}}})_{\Omega}+\langle{\operatorname{{\mathbf{q}}}}^{n}(t),{\operatorname{\mathbf{v}}})\rangle_{V_{\sigma}},\end{aligned}$
(3.8b) $\displaystyle\begin{aligned}
&\langle\partial_{t}\theta^{n}(t),\vartheta\rangle_{V}-\ell_{h}\langle\partial_{t}\varphi^{n}(t),\vartheta\rangle_{V}+\kappa(\nabla\theta^{n}(t),\nabla\vartheta)\\\
&=({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla(\ell_{h}\varphi^{n}(t)-\theta^{n}(t)),\vartheta)_{\Omega}+({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}^{n}(t),\vartheta)_{\Omega}+\langle
z^{n}(t),\vartheta\rangle_{V}\end{aligned}$ (3.8c)

with $\rho(\cdot,\varphi^{n}):=a(\cdot)\varphi^{n}+F^{\prime}(\varphi^{n})$,
$\mu^{n}=P^{n}(\rho(\cdot,\varphi^{n})-J\ast\varphi^{n}+\ell_{c}\theta^{n})$,
$\varphi^{n}(0)=\varphi^{n}_{0}$,
${\operatorname{{\mathbf{u}}}}^{n}(0)={\operatorname{{\mathbf{u}}}}_{0}^{n}$,
and $\theta^{n}(0)=\theta^{n}_{0}$. Here the discrete initial conditions for
the fluid velocity and temperature are obtained using the projections, i.e.,
${\operatorname{{\mathbf{u}}}}_{0}^{n}=P_{\sigma}^{n}{\operatorname{{\mathbf{u}}}}_{0}$
and $\theta^{n}_{0}=P^{n}\theta_{0}$, while the initial condition for order
parameter is obtained as $\varphi^{n}_{0}=(I+\frac{1}{n}B)^{-1}\varphi_{0}\in
D(B)$.

Using the ansatz

$\displaystyle\varphi^{n}(t)=\sum_{j=1}^{n}a^{j}(t)\psi_{j},\quad{\operatorname{{\mathbf{u}}}}^{n}(t)=\sum_{j=1}^{n}b^{j}(t){\operatorname{\mathbf{v}}}_{j},\quad\theta^{n}(t)=\sum_{j=1}^{n}c^{j}(t)\psi_{j}$

system (3.8a)-(3.8c) can be written as an ode involving the variables
$\boldsymbol{a}(t):=[a^{1}(t),a^{2}(t),\ldots,a^{n}(t)]^{\top}$,
$\boldsymbol{b}(t):=[b^{1}(t),b^{2}(t),\ldots,b^{n}(t)]^{\top}$, and
$\boldsymbol{c}(t):=[c^{1}(t),c^{2}(t),\ldots,c^{n}(t)]^{\top}$ whose
existence on the space $C^{1}([0,t^{n}];\mathbb{R}^{n})$, for some
$t^{n}\in(0,+\infty]$, can be established quite easily using, for example,
Cauchy–Lipschitz theorem.

We now derive a priori estimates that establish $t^{n}=T$, and that the
elements $\varphi^{n}$, ${\operatorname{{\mathbf{u}}}}^{n}$ and $\theta^{n}$
are uniformly bounded under appropriate norms. For the meantime, we shall drop
the time variable $t$ for simplicity. Let us begin with taking
$\psi=\mu^{n}(t)$ in (3.8a) to get

$\displaystyle\begin{aligned}
(\partial_{t}\varphi^{n},\mu^{n})_{\Omega}+&\,({\operatorname{{\mathbf{u}}}}^{n}\cdot\nabla\varphi^{n},\mu^{n})_{\Omega}+(\nabla\rho^{n},\nabla\mu^{n})_{\Omega}=(\nabla(J\ast\varphi^{n}),\nabla\mu^{n})_{\Omega}-\ell_{c}(\nabla\theta^{n},\nabla\mu^{n})_{\Omega}.\end{aligned}$
(3.9)

Here, we used the notation
$\rho^{n}(t):=P^{n}\rho(\cdot,\varphi^{n}(t))=\mu^{n}(t)+P^{n}(J\ast\varphi^{n}(t))-\ell_{c}P^{n}(\theta^{n})$.

Furthermore, the first term on the left-hand side of (3.9) can be written by
using the explicit expression for $\mu^{n}$ and using the fact that
$(\phi_{1},J\ast\phi_{2})_{\Omega}=(\phi_{2},J\ast\phi_{1})_{\Omega}$ from the
property $J(x)=J(-x)$. Indeed, we have

$\displaystyle\begin{aligned}
&(\partial_{t}\varphi^{n},\mu^{n})_{\Omega}=(\partial_{t}\varphi^{n},a\varphi^{n}+F^{\prime}(\varphi^{n})-J\ast\varphi^{n}+\ell_{c}\theta^{n})_{\Omega}\\\
&=\frac{1}{2}\frac{d}{dt}\left(\int_{\Omega}a|\varphi^{n}(x)|^{2}\operatorname{d\\!}x+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x-\int_{\Omega}\varphi^{n}(x)(J\ast\varphi^{n}(x))\operatorname{d\\!}x\right)+\ell_{c}(\partial_{t}\varphi^{n},\theta^{n})_{\Omega}\\\
&=\frac{1}{2}\frac{d}{dt}\left(\mathbb{E}_{nl}(\varphi^{n})+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\right)+\ell_{c}(\partial_{t}\varphi^{n},\theta^{n})_{\Omega}.\end{aligned}$
(3.10)

Using the definition of $\rho^{n}$, and (3.10) we rewrite (3.9) as

$\displaystyle\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\left(\mathbb{E}_{nl}(\varphi^{n})+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\right)+\|\nabla\mu^{n}\|^{2}\\\
&=({\operatorname{{\mathbf{u}}}}^{n}\cdot\nabla\mu^{n},\varphi^{n})_{\Omega}+(\nabla
J\ast\varphi^{n},\nabla\mu^{n})_{\Omega}-(\nabla
P^{n}(J\ast\varphi^{n}),\nabla\mu^{n})_{\Omega}-\ell_{c}(\partial_{t}\varphi^{n},\theta^{n})_{\Omega}.\end{aligned}$
(3.11)

Let us now test (3.8b) with
${\operatorname{\mathbf{v}}}={\operatorname{{\mathbf{u}}}}^{n}(t)$ to get

$\displaystyle\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|{\operatorname{{\mathbf{u}}}}^{n}\|^{2}+2\|\sqrt{\nu(\varphi)}\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}\|^{2}=\mathcal{K}({\operatorname{{\mathbf{u}}}}^{n}\cdot\nabla\varphi^{n},\mu^{n}-\ell_{c}\theta^{n})_{\Omega}+(\ell(\varphi^{n},\theta^{n}){\operatorname{\mathbf{g}}},{\operatorname{{\mathbf{u}}}}^{n})_{\Omega}+\langle{\operatorname{{\mathbf{q}}}}^{n},{\operatorname{{\mathbf{u}}}}^{n}\rangle_{V_{\sigma}}\end{aligned}$
(3.12)

Similarly, substituting $\vartheta=\theta^{n}(t)$ into (3.8c) gives us

$\displaystyle\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\theta^{n}\|^{2}+\kappa\|\nabla\theta^{n}\|^{2}=\ell_{h}(\partial_{t}\varphi^{n},\theta^{n})_{\Omega}+\ell_{h}({\operatorname{{\mathbf{u}}}}^{n}\cdot\nabla\varphi^{n},\theta^{n})_{\Omega}+({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}^{n},\theta^{n})_{\Omega}+\langle
z^{n},\theta^{n}\rangle_{V}\end{aligned}$ (3.13)

To reconcile the computations above we divide the equations (3.11), (3.12) and
(3.13) by the constants $\ell_{c}$, $\mathcal{K}\ell_{c}$ and $\ell_{h}$,
respectively, and add the results together. Such process yields

$\displaystyle\begin{aligned}
&\frac{d}{dt}\mathbb{E}(\varphi^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))+\mathbb{D}(\varphi^{n}(t),\mu^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))\\\
&=\frac{1}{\ell_{c}}\left\\{(\nabla
J\ast\varphi^{n}(t),\nabla\mu^{n}(t))_{\Omega}-(\nabla
P^{n}(J\ast\varphi^{n}(t)),\nabla\mu^{n}(t))_{\Omega}\right\\}\\\ &\ \ \
+\frac{1}{\mathcal{K}\ell_{c}}\left\\{(\ell(\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}},{\operatorname{{\mathbf{u}}}}^{n}(t))_{\Omega}+\langle{\operatorname{{\mathbf{q}}}}^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t)\rangle_{V_{\sigma}}\right\\}\\\
&\ \ \
+\frac{1}{\ell_{h}}\left\\{({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))_{\Omega}+\langle
z^{n}(t),\theta^{n}(t)\rangle_{V}\right\\}\end{aligned}$ (3.14)

The next step is to gain some estimates for the right-hand side of (3.14). Let
us begin with the terms multiplied with $1/\ell_{c}$:

$\displaystyle\begin{aligned} |(&\nabla
J\ast\varphi^{n}(t),\nabla\mu^{n}(t))_{\Omega}-(\nabla
P^{n}(J\ast\varphi^{n}(t)),\nabla\mu^{n}(t))_{\Omega}|\\\ &\leq\|\nabla
J\|_{L^{1}}\|\varphi^{n}(t)\|\|\nabla\mu^{n}(t)\|+\|\nabla
P^{n}(J\ast\varphi^{n}(t))\|\|\nabla\mu^{n}(t)\|\\\ &\leq\|\nabla
J\|_{L^{1}}\|\varphi^{n}(t)\|\|\nabla\mu^{n}(t)\|+\|B^{1/2}P^{n}(J\ast\varphi^{n}(t))\|\|\nabla\mu^{n}(t)\|\\\
&\leq\|\nabla J\|_{L^{1}}\|\varphi^{n}(t)\|\|\nabla\mu^{n}(t)\|+\|\nabla
J\ast\varphi^{n}(t)\|\|\nabla\mu^{n}(t)\|\\\ &\leq
c\|J\|_{W^{1,1}}\|\varphi^{n}(t)\|\|\nabla\mu^{n}(t)\|\leq\frac{1}{2}\|\nabla\mu^{n}(t)\|^{2}+\frac{c}{2}\|J\|_{W^{1,1}}^{2}\|\varphi^{n}(t)\|^{2}\end{aligned}$
(3.15)

In the computations above, we utilized Hölder’s inequality as well as Young’s
convolution inequality. The last estimate, on the other hand is achieved using
Young’s inequality. Similarly, the terms with $1/\mathcal{K}\ell_{c}$ are
estimated with the Hölder’s and Young’s inequalities, and the constancy of the
gravity $\operatorname{\mathbf{g}}$. The computation is shown below:

$\displaystyle\begin{aligned}
|(\ell(&\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}},{\operatorname{{\mathbf{u}}}}^{n}(t))_{\Omega}+\langle{\operatorname{{\mathbf{q}}}}^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t)\rangle_{V_{\sigma}}|\\\
&\leq\|\ell(\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}}\|\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|+\|{\operatorname{{\mathbf{q}}}}^{n}(t)\|_{V_{\sigma}^{*}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|_{V_{\sigma}}\\\
&\leq
c\left(1+\frac{\alpha\mathcal{K}}{2}\|\varphi^{n}(t)\|^{2}+\frac{\mathcal{K}\ell_{c}}{4\ell_{h}}\|\theta^{n}(t)\|^{2}+\frac{1}{4}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|^{2}\right)+\frac{c_{\rm
K}}{4\underline{\nu}}\|{\operatorname{{\mathbf{q}}}}^{n}(t)\|_{V_{\sigma}^{*}}^{2}+\frac{\underline{\nu}}{c_{\rm
K}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|_{V_{\sigma}}^{2},\end{aligned}$
(3.16)

where $c>0$ depends on $\mathcal{K}$, $\ell_{h}$, $\ell_{c}$, $\alpha$ (which
is defined as $\alpha:=2c_{1}-\|J\|_{L^{1}}>0$ based on assumption (A4)) and
$\operatorname{\mathbf{g}}$, the constant $c_{\rm K}$ on the other hand
corresponds to that in Korn’s inequality. Lastly, we employ the same
techniques we previously utilized to the terms multiplied with $1/\ell_{h}$

$\displaystyle\begin{aligned}
|({\operatorname{\mathbf{g}}}&\cdot{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))_{\Omega}+\langle
z^{n}(t),\theta^{n}(t)\rangle_{V}|\\\ &\leq
c\left(\frac{1}{4}\|\theta^{n}(t)\|^{2}+\frac{\ell_{h}}{4\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|^{2}\right)+\frac{1}{2\kappa}\|z^{n}(t)\|_{V^{*}}^{2}+\frac{\kappa}{2}\|\theta^{n}(t)\|_{V}^{2}.\end{aligned}$
(3.17)

Plugging estimates (3.15), (3.16) and (3.17) into (3.14), and using the lower
bound for the viscosity and Korn’s inequality will yield

$\displaystyle\begin{aligned}
&\frac{d}{dt}\mathbb{E}(\varphi^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))+\widehat{\mathbb{D}}(\mu^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))\\\
&\leq
c\left(1+\frac{\alpha}{2\ell_{c}}\|\varphi^{n}(t)\|^{2}+\frac{1}{2\ell_{h}}\|\theta^{n}(t)\|^{2}+\frac{1}{2\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|^{2}\right)\\\
&\ \ \ +\frac{c}{2}\|J\|_{W^{1,1}}^{2}\|\varphi^{n}(t)\|^{2}+\frac{c_{\rm
K}}{4\underline{\nu}\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{q}}}}^{n}(t)\|_{V_{\sigma}^{*}}^{2}+\frac{1}{2\kappa\ell_{h}}\|z^{n}(t)\|_{V^{*}}^{2},\end{aligned}$
(3.18)

where
$\widehat{\mathbb{D}}(\mu,{\operatorname{{\mathbf{u}}}},\theta):=\frac{1}{2\ell_{c}}\|\nabla\mu\|^{2}+\frac{\underline{\nu}}{c_{\rm
K}\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{u}}}}\|_{V_{\sigma}}^{2}+\frac{\kappa}{2\ell_{h}}\|\nabla\theta\|^{2}$.

To this end, let us note that the contribution of $\varphi^{n}$ in the total
energy functional $\mathbb{E}$ can be estimated from below. First, we can
rewrite such expression using (1.1) and bound it below using Young’s
convolution inequality:

$\displaystyle\begin{aligned}
\frac{1}{2}&\int_{\Omega}\int_{\Omega}J(x-y)(\varphi^{n}(x)-\varphi^{n}(y))^{2}\operatorname{d\\!}x\operatorname{d\\!}y+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\\\
&=\int_{\Omega}a(x)\varphi^{n}(x)^{2}\operatorname{d\\!}x-\int_{\Omega}(J\ast\varphi^{n})(x)\varphi^{n}(x)\operatorname{d\\!}x+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\\\
&\geq\int_{\Omega}a(x)\varphi^{n}(x)^{2}\operatorname{d\\!}x-\|J\|_{L^{1}}\int_{\Omega}\varphi^{n}(x)^{2}\operatorname{d\\!}x+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x.\end{aligned}$

From assumption (A4) and the nonegativity of $a(\cdot)$ we further minorize
the left-hand side of the expression above:

$\displaystyle\begin{aligned}
\frac{1}{2}&\int_{\Omega}\int_{\Omega}J(x-y)(\varphi^{n}(x)-\varphi^{n}(y))^{2}\operatorname{d\\!}x\operatorname{d\\!}y+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\\\
&\geq\int_{\Omega}a(x)\varphi^{n}(x)^{2}\operatorname{d\\!}x-\|J\|_{L^{1}}\int_{\Omega}\varphi^{n}(x)^{2}\operatorname{d\\!}x+2\int_{\Omega}F(\varphi^{n}(x))\operatorname{d\\!}x\\\
&\geq(2c_{1}-\|J\|_{L^{1}})\int_{\Omega}\varphi^{n}(x)^{2}\operatorname{d\\!}x-2c_{2}|\Omega|=:\alpha\|\varphi^{n}\|^{2}-c.\end{aligned}$
(3.19)

Integrating (3.18) over $(0,t)$ for $t\in(0,t^{n}]$ and utilizing (3.19) –
after some rearrangement – gives us

$\displaystyle\begin{aligned}
&\widehat{\mathbb{E}}(\varphi^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))+\int_{0}^{t}\widehat{\mathbb{D}}(\mu^{n}(s),{\operatorname{{\mathbf{u}}}}^{n}(s),\theta^{n}(s))\operatorname{d\\!}s\\\
&\leq\mathbb{E}(\varphi^{n}(0),{\operatorname{{\mathbf{u}}}}^{n}(0),\theta^{n}(0))+\frac{c_{\rm
K}}{4\underline{\nu}\mathcal{K}\ell_{c}}\int_{0}^{t}\|{\operatorname{{\mathbf{q}}}}^{n}(s\|_{V_{\sigma}^{*}}^{2}\operatorname{d\\!}s+\frac{1}{2\kappa\ell_{h}}\int_{0}^{t}\|z^{n}(s)\|_{V^{*}}^{2}\operatorname{d\\!}s\\\
&\ \ \
+c\left(1+\int_{0}^{t}\left\\{\frac{c}{2}\|J\|_{W^{1,1}}^{2}\|\varphi^{n}(s)\|^{2}+\widehat{\mathbb{E}}(\varphi^{n}(s),\theta^{n}(s),{\operatorname{{\mathbf{u}}}}^{n}(s))\right\\}\operatorname{d\\!}s\right),\end{aligned}$

where
$\widehat{\mathbb{E}}(\varphi,{\operatorname{{\mathbf{u}}}},\theta):=\frac{\alpha}{2\ell_{c}}\|\varphi\|^{2}+\frac{1}{2\ell_{h}}\|\theta\|^{2}+\frac{1}{2\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{u}}}}\|^{2}$.
Grönwall’s inequality assures us of the existence of a constant $c>0$ such
that

$\displaystyle\begin{aligned}
&\sup_{t\in(0,t^{n}]}\widehat{\mathbb{E}}(\varphi^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))+\int_{0}^{t^{n}}\widehat{\mathbb{D}}(\mu^{n}(s),{\operatorname{{\mathbf{u}}}}^{n}(s),\theta^{n}(s))\operatorname{d\\!}s\\\
&\leq
ce^{cT}\left(1+\mathbb{E}(\varphi^{n}(0),{\operatorname{{\mathbf{u}}}}^{n}(0),\theta^{n}(0))+\|{\operatorname{{\mathbf{q}}}}\|_{L^{2}(I;V_{\sigma}^{*})}+\|z\|_{L^{2}(I;V^{*})}\right).\end{aligned}$
(3.20)

Note that the constant $c>0$ above is independent of $n$, so that further
estimate on
$\mathbb{E}(\varphi^{n}(0),{\operatorname{{\mathbf{u}}}}^{n}(0),\theta^{n}(0))$
with independence on $n$ will yield uniform boundedness of the left-hand side
of (3.20) which implies

$\displaystyle\|\varphi^{n}\|_{L^{\infty}(I;H)}\leq M$ (3.21)
$\displaystyle\|\nabla\mu^{n}\|_{L^{2}(I;L^{2}(\Omega)^{2})}\leq M$ (3.22)
$\displaystyle\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})\cap
L^{2}(I;V_{\sigma})}\leq M$ (3.23)
$\displaystyle\|{\theta}^{n}\|_{L^{\infty}(I;H)}\leq M.$ (3.24)
$\displaystyle\|\nabla{\theta}^{n}\|_{L^{2}(I;L^{2}(\Omega)^{2})}\leq M.$
(3.25)

Indeed, from the definition of the orthogonal projections $P^{n}$ and
$P^{n}_{\sigma}$ whose norms are bounded by 1 in the linear spaces
$\mathcal{L}(H,H)$ and $\mathcal{L}(H_{\sigma},H_{\sigma})$, respectively, we
see that
$\|{\operatorname{{\mathbf{u}}}}^{n}(0)\|_{H_{\sigma}}\leq\|{\operatorname{{\mathbf{u}}}}_{0}\|_{H_{\sigma}}$
and $\|\theta^{n}(0)\|_{H}\leq\|\theta_{0}\|_{H}$. For the contribution of the
order parameter, we first note that the Neumann operator $B$ is a maximal
monotone operator hence $\varphi^{n}_{0}\to\varphi_{0}$ in $H$. Hence, there
exists $c>0$ independent of $n$ such that

$\displaystyle\mathbb{E}_{nl}(\varphi^{n}_{0})+2\int_{\Omega}F(\varphi^{n}_{0}(x))\operatorname{d\\!}x\leq
c\|J\|_{L^{1}}\|\varphi^{n}_{0}\|^{2}+2\int_{\Omega}F(\varphi^{n}_{0}(x))\operatorname{d\\!}x$
$\displaystyle\leq
c\|J\|_{L^{1}}\|\varphi_{0}\|^{2}+2\int_{\Omega}F(\varphi^{n}_{0}(x))\operatorname{d\\!}x.$

The next challenge lies on controlling the nonlinear part by the initial data.
As previously remarked, we can write $F$ – by virtue of (A3) – as
$F(s)=G(s)-\frac{a^{*}}{2}s^{2}$, hence

$\displaystyle\int_{\Omega}F(\varphi^{n}_{0}(x))\operatorname{d\\!}x=\int_{\Omega}G(\varphi^{n}_{0}(x))\operatorname{d\\!}x-\frac{a^{*}}{2}\|\varphi^{n}_{0}\|^{2}.$

Notably the function $h=G^{\prime}$ is monotonically increasing which we can
suppose to satisfy $h(0)=0$. If we multiply $h(\varphi^{n}_{0})$ by
$-(\varphi_{0}-\varphi_{0}^{n})=-\frac{1}{n}B\varphi_{0}^{n}$ and integrate
over $\Omega$ one gets

$\displaystyle\begin{aligned}
\int_{\Omega}h(\varphi^{n}_{0})(\varphi_{0}^{n}-\varphi_{0})\operatorname{d\\!}x&=-\frac{1}{n}\int_{\Omega}h(\varphi^{n}_{0})B\varphi_{0}^{n}\operatorname{d\\!}x=-\frac{1}{n}\int_{\Omega}\left\\{\nabla
h(\varphi^{n}_{0})\cdot\nabla\varphi_{0}^{n}\right\\}\operatorname{d\\!}x\\\
&=-\frac{1}{n}\int_{\Omega}\left\\{h^{\prime}(\varphi^{n}_{0})|\nabla\varphi_{0}^{n}|^{2}\right\\}\operatorname{d\\!}x\leq
0.\end{aligned}$

Expanding $G$ about $\varphi^{n}_{0}$, the convexity of $G$ then implies

$\displaystyle\begin{aligned}
\int_{\Omega}G(\varphi^{n}_{0})\operatorname{d\\!}x\leq\int_{\Omega}G(\varphi_{0})+h(\varphi^{n}_{0})(\varphi_{0}^{n}-\varphi_{0})\operatorname{d\\!}x\leq\int_{\Omega}G(\varphi_{0})\operatorname{d\\!}x.\end{aligned}$

Using Fatou’s lemma, we get

$\displaystyle\limsup_{n\to\infty}\int_{\Omega}F(\varphi^{n}_{0})\operatorname{d\\!}x\leq\int_{\Omega}G(\varphi_{0})\operatorname{d\\!}x-\frac{a^{*}}{2}\|\varphi_{0}\|^{2}=\int_{\Omega}F(\varphi_{0})\operatorname{d\\!}x.$

In summary, we get the energy estimate

$\displaystyle\begin{aligned}
&\sup_{t\in[0,T]}\widehat{\mathbb{E}}(\varphi^{n}(t),{\operatorname{{\mathbf{u}}}}^{n}(t),\theta^{n}(t))+\int_{0}^{T}\widehat{\mathbb{D}}(\mu^{n}(s),{\operatorname{{\mathbf{u}}}}^{n}(s),\theta^{n}(s))\operatorname{d\\!}s\\\
&\leq
ce^{cT}\left(1+\mathbb{E}(\varphi_{0},{\operatorname{{\mathbf{u}}}}_{0},\theta_{0})+\frac{c_{\rm
K}}{4\underline{\nu}\mathcal{K}\ell_{c}}\|{\operatorname{{\mathbf{q}}}}\|_{L^{2}(I;V_{\sigma}^{*})}+\frac{1}{2\kappa\ell_{h}}\|z\|_{L^{2}(I;V^{*})}\right)\end{aligned}$

which validates (3.21)–(3.25).

On the other hand, if we test (3.8c) with $\vartheta=1$ we get

$\displaystyle\left|\int_{\Omega}\theta^{n}(x)\operatorname{d\\!}x\right|\leq
c\left(\|\theta_{0}\|_{H}+\|{\operatorname{{\mathbf{u}}}^{n}}\|_{L^{\infty}(I;H_{\sigma})}+\|z\|_{L^{2}(I;V^{*})}\right).$

The inequality above, together with (3.23) and (3.25), imply

$\displaystyle\|{\theta}^{n}\|_{L^{\infty}(I;H)\cap L^{2}(I;V)}\leq M.$ (3.26)

At this moment, we shall derive some estimates for $\varphi^{n}$ in
$L^{2}(I;V)$. First, we use the definition of $\mu^{n}$, assumption (A3), and
Young’s convolution and Young’s inequalities to get

$\displaystyle\begin{aligned} &(\nabla\varphi^{n},\nabla\mu^{n})_{\Omega}\\\
&=(\nabla\varphi^{n},(F^{\prime\prime}(\varphi^{n})+a)\nabla\varphi^{n}+\varphi^{n}\nabla
a-\nabla J\ast\varphi^{n}+\ell_{c}\nabla\theta^{n})_{\Omega}\\\ &\geq
c_{0}\|\nabla\varphi^{n}\|^{2}-2\|J\|_{W^{1,1}}\|\nabla\varphi^{n}\|\|\varphi^{n}\|+\ell_{c}(\nabla\varphi^{n},\nabla\theta^{n})\\\
&\geq\frac{3c_{0}}{4}\|\nabla\varphi^{n}\|^{2}-\frac{4}{c_{0}}\|J\|_{W^{1,1}}\|\varphi^{n}\|^{2}+\ell_{c}(\nabla\varphi^{n},\nabla\theta^{n})\end{aligned}$
(3.27)

Using Hölder’s and Young’s inequalities on
$(\nabla\varphi^{n},\nabla\mu^{n})_{\Omega}$, on the other hand, one gets
$(\nabla\varphi^{n},\nabla\mu^{n})_{\Omega}\leq(c_{0}/4)\|\nabla\varphi^{n}\|^{2}+(1/c_{0})\|\nabla\mu^{n}\|^{2}$.
Combining this with (3.27) we have

$\displaystyle\begin{aligned}
&\frac{c_{0}}{2}\|\nabla\varphi^{n}\|^{2}\leq\left|\frac{1}{c_{0}}\|\nabla\mu^{n}\|^{2}+\frac{4}{c_{0}}\|J\|_{W^{1,1}}\|\varphi^{n}\|^{2}-\ell_{c}(\nabla\varphi^{n},\nabla\theta^{n})\right|\\\
&\leq\frac{1}{c_{0}}\|\nabla\mu^{n}\|^{2}+\frac{4}{c_{0}}\|J\|_{W^{1,1}}\|\varphi^{n}\|^{2}+\frac{\ell_{c}^{2}}{c_{0}}\|\nabla\theta^{n}\|+\frac{c_{0}}{4}\|\nabla\varphi^{n}\|^{2}.\end{aligned}$
(3.28)

Moving the last term on the right-hand side of (3.28) to the other side,
utilizing the total mass conservation of the order parameter, and the
inequalities (3.21), (3.22) and (3.26) we get the uniform boundedness of
$\varphi^{n}$ in ${L^{2}(I;V)}$, i.e.,

$\displaystyle\|\varphi^{n}\|_{L^{2}(I;V)}\leq M.$ (3.29)

To strengthen (3.22) let us prove that the average of $\mu^{n}$ – which we
denote as
$\widehat{\mu^{n}}:=\frac{1}{|\Omega|}\int_{\Omega}\mu^{n}\operatorname{d\\!}x$–
over $\Omega$ is uniformly bounded as well. Indeed, by the definition of
$\mu^{n}$ and by assumption (A5) we have

$\displaystyle\begin{aligned}
\left|\int_{\Omega}\mu^{n}\operatorname{d\\!}x\right|&=\left|\int_{\Omega}P^{n}(a\varphi^{n}-J\ast\varphi^{n}+F^{\prime}(\varphi^{n})+\ell_{c}\theta^{n})\operatorname{d\\!}x\right|\\\
&\leq\int_{\Omega}|F^{\prime}(\varphi^{n})|\operatorname{d\\!}x+\ell_{c}\sqrt{|\Omega|}\|\theta^{n}\|\\\
&\leq
c_{3}\int_{\Omega}|F(\varphi^{n})|\operatorname{d\\!}x+c_{4}|\Omega|+\ell_{c}\sqrt{|\Omega|}\|\theta^{n}\|.\end{aligned}$

Here we used the fact that
$\int_{\Omega}P^{n}(a\varphi^{n}-J\ast\varphi^{n})\operatorname{d\\!}x=0$. To
close the estimate above we note that $\|F(\varphi^{n})\|_{L^{1}}$ can be
shown to be uniformly bounded by integrating (3.18) over $(0,T)$, using
Gronwall’s inequality and using the upper bound for
$\mathbb{E}(\varphi^{n}(0),{\operatorname{{\mathbf{u}}}}^{n}(0),\theta^{n}(0))$.
The boundedness above, together with (3.22) and Poincaré-Wirtinger inequality
implies

$\displaystyle\|\mu^{n}\|_{L^{2}(I;V)}\leq M.$ (3.30)

Before we move on, we mention – by virtue of assumption (A5) – that
$\|\rho(\cdot,\varphi^{n})\|_{L^{\infty}(I;L^{p}(\Omega))}$ is uniformly
bounded where $p\in(1,2]$ is as in the aforementioned assumption. Indeed, we
have

$\displaystyle\begin{aligned}
\|\rho(\cdot,\varphi^{n})\|_{L^{p}}&\leq\|a\|_{L^{\infty}}\|\varphi^{n}\|_{L^{p}}+\left(\int_{\Omega}|F^{\prime}(\varphi^{n}(x))|^{p}\operatorname{d\\!}x\right)^{1/p}\\\
&\leq\|a\|_{L^{\infty}}\|\varphi^{n}\|+\left(\int_{\Omega}c_{3}|F(\varphi^{n}(x))|+c_{4}\operatorname{d\\!}x\right)^{1/p}\\\
&\leq\|a\|_{L^{\infty}}\|\varphi^{n}\|+c_{3}\|F(\varphi^{n})\|_{L^{1}}^{1/p}+c_{4}{|\Omega|}^{1/p}.\end{aligned}$

The uniform boundedness follows from the boundedness of
$\|F(\varphi^{n})\|_{L^{1}}$ and estimate (3.21), i.e.

$\displaystyle\|\rho(\cdot,\varphi^{n})\|_{L^{\infty}(I;L^{p}(\Omega))}\leq
M.$ (3.31)

Our next aim is to establish some boundedness for the time differentiated
variables. Let us begin with the time derivative of the fluid velocity
${\operatorname{{\mathbf{u}}}}^{n}$. We rewrite (3.8b) as

$\displaystyle\begin{aligned}
\langle&\partial_{t}{\operatorname{{\mathbf{u}}}}^{n}(t),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+\langle
P^{n}_{\sigma}\mathcal{B}(\operatorname{{\mathbf{u}}}^{n}(t)),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+\langle
P^{n}_{\sigma}\mathcal{A}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\operatorname{\mathbf{v}}\rangle_{V_{\sigma}}\\\
&=\mathcal{K}\langle
P^{n}_{\sigma}\mathcal{C}_{2}(\varphi^{n}(t),\mu^{n}(t)-\ell_{c}\theta^{n}(t)),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+(P^{n}_{\sigma}\ell(\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}},{\operatorname{\mathbf{v}}})_{\Omega}+\langle
P^{n}_{\sigma}{\operatorname{{\mathbf{q}}}}(t),{\operatorname{\mathbf{v}}})\rangle_{V_{\sigma}}.\end{aligned}$
(3.32)

Using the antisymmetry of $b({\operatorname{{\mathbf{u}}}};\cdot,\cdot)$ for
${\operatorname{{\mathbf{u}}}}\in V_{\sigma}$, and the estimates (2.1) and
(2.2) we get

$\displaystyle\begin{aligned}
\|P^{n}_{\sigma}\mathcal{B}(\operatorname{{\mathbf{u}}}^{n}(t))\|_{V_{\sigma}^{*}}&\leq\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in
V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}|b(\operatorname{{\mathbf{u}}}^{n}(t);\operatorname{{\mathbf{u}}}^{n}(t),{\operatorname{\mathbf{v}}})|\\\
&\leq\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}c\left\\{\begin{matrix}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{H_{\sigma}}^{1/2}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{V_{\sigma}}^{3/2}\|\operatorname{\mathbf{v}}\|_{V_{\sigma}}&\text{for
}d=3\\\
\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{H_{\sigma}}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{V_{\sigma}}\|\operatorname{\mathbf{v}}\|_{V_{\sigma}}&\text{for
}d=2\end{matrix}\right.\\\ &\leq
c\left\\{\begin{matrix}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{H_{\sigma}}^{1/2}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{V_{\sigma}}^{3/2}&\text{for
}d=3\\\
\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{H_{\sigma}}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{V_{\sigma}}&\text{for
}d=2.\end{matrix}\right.\end{aligned}$

For $d=3$, we thus see that

$\displaystyle\|P^{n}_{\sigma}\mathcal{B}({\operatorname{{\mathbf{u}}}}^{n})\|_{L^{4/3}(I;V_{\sigma}^{*})}\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}^{1/2}\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{2}(I;V_{\sigma})}^{3/2}\leq
M^{2},$ (3.33)

while when $d=2$ we have

$\displaystyle\|P^{n}_{\sigma}\mathcal{B}({\operatorname{{\mathbf{u}}}}^{n})\|_{L^{2}(I;V_{\sigma}^{*})}\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{2}(I;V_{\sigma})}\leq
M^{2}.$ (3.34)

For the viscosity term we use Hölder’s inequality and assumption (A2) for both
$d=2,3$ to get

$\displaystyle\begin{aligned}
\|P^{n}_{\sigma}\mathcal{A}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t))\|_{V_{\sigma}^{*}}=\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in
V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}2(\nu(\varphi^{n}(t))\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}(t),\mathrm{D}{\operatorname{\mathbf{v}}})_{\Omega}&\\\
\leq 2\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}\|\nu(\varphi^{n}(t))\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\mathrm{D}{\operatorname{\mathbf{v}}}\|\leq
2\overline{\nu}\|\operatorname{{\mathbf{u}}}^{n}(t)\|_{V_{\sigma}}.&\end{aligned}$
(3.35)

This implies that
$\|P^{n}_{\sigma}\mathcal{A}({\operatorname{{\mathbf{u}}}}^{n},\varphi^{n})\|_{L^{2}(I;V_{\sigma}^{*})}\leq
M$.

For the term that takes into account the capillarity we first deal with the
case $d=3$ as follows:

$\displaystyle\begin{aligned}
\|P^{n}_{\sigma}\mathcal{C}_{2}(\varphi^{n}(t),\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|_{V_{\sigma}^{*}}&=\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in
V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}|\langle
P^{n}_{\sigma}\mathcal{C}_{2}(\ell_{c}\theta^{n}(t)-\mu^{n}(t),\varphi^{n}(t)),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}|\\\
&=\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}|(\varphi^{n}(t)\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t)),{\operatorname{\mathbf{v}}})_{\Omega}|\\\
&\leq\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}\|\varphi^{n}(t)\|_{L^{3}}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|\|{\operatorname{\mathbf{v}}}\|_{L^{6}}\\\
&\leq\inf_{\begin{subarray}{c}{\operatorname{\mathbf{v}}}\in V_{\sigma}\\\
\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}=1\end{subarray}}\|\varphi^{n}(t)\|_{L^{3}}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|\|{\operatorname{\mathbf{v}}}\|_{V_{\sigma}}.\end{aligned}$
(3.36)

We note that to be able to reach the third line, we used Hölder’s inequality,
while Rellich-Kondrachov embedding theorem — which ensures us of the
continuous embeddings $W^{1,2}(\Omega)\hookrightarrow L^{6}(\Omega)$ — to get
the last inequality. Now, the interpolation of $L^{p}$ spaces, with
$\theta=\frac{1}{2}$, $p_{1}=6$ and $p_{2}=2$, gives us

$\displaystyle\|\varphi\|_{L^{3}}\leq
c\|\varphi\|^{1/2}_{L^{6}}\|\varphi\|^{1/2}.$ (3.37)

Using (3.37) to (3.36), and the continuous embedding
$W^{1,2}(\Omega)\hookrightarrow L^{6}(\Omega)$ lead to

$\displaystyle\begin{aligned}
\|P^{n}_{\sigma}\mathcal{C}_{2}(\varphi^{n}(t),\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|_{V_{\sigma}^{*}}&\leq
c\|\varphi^{n}(t)\|^{1/2}\|\varphi^{n}(t)\|^{1/2}_{L^{6}}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|\\\
&\leq
c\|\varphi^{n}(t)\|^{1/2}\|\varphi^{n}(t)\|_{V}^{1/2}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|.\end{aligned}$

For the next computations, we employ Young’s inequality
$\alpha\beta\leq\frac{\varepsilon}{p}\alpha^{p}_{1}+\frac{1}{p_{2}\varepsilon^{p^{\prime}/p}}\beta^{p^{\prime}}$,
which holds for any $\varepsilon,\alpha,\beta>0$ and $1/p_{1}+1/p_{2}=1$.
Taking $p_{1}=3$, $p_{2}=3/2$, $\alpha=\|\varphi^{n}(t)\|_{V}^{2/3}$, and
$\beta=\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|^{4/3}$ and from the
estimates (3.21), (3.22), (3.26) and (3.29) we get the following estimates for
the three dimensional case:

$\displaystyle\begin{aligned}
&\|P^{n}_{\sigma}\mathcal{C}_{2}(\varphi^{n},\mu^{n}-\ell_{c}\theta^{n})\|_{L^{4/3}(I;V_{\sigma}^{*})}\\\
&=c\left(\int_{0}^{T}\left(\|\varphi^{n}(t)\|^{\frac{1}{2}}\|\varphi^{n}(t)\|_{V}^{\frac{1}{2}}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|\right)^{\frac{4}{3}}\operatorname{d\\!}t\right)^{\frac{3}{4}}\\\
&\leq
c\|\varphi^{n}\|_{L^{\infty}(I;H)}^{\frac{1}{2}}\left(\int_{0}^{T}\|\varphi^{n}(t)\|_{V}^{\frac{2}{3}}\|\nabla(\mu^{n}(t)-\ell_{c}\theta^{n}(t))\|^{\frac{4}{3}}\operatorname{d\\!}t\right)^{\frac{3}{4}}\\\
&\leq
c\|\varphi^{n}\|_{L^{\infty}(I;H)}^{\frac{1}{2}}\left\\{\|\varphi^{n}\|_{L^{2}(I;V)}^{\frac{3}{2}}+\|\nabla(\mu^{n}-\ell_{c}\theta^{n}(t))\|_{L^{2}(I;L^{2}(\Omega)^{2})}^{\frac{3}{2}}\right\\}\\\
&\leq M^{2}.\end{aligned}$ (3.38)

For the two dimensional case, we rewrite
$(\mu^{n}-\ell_{c}\theta^{n})\nabla\varphi^{n}$ as

$\displaystyle(\mu^{n}-\ell_{c}\theta^{n})\nabla\varphi^{n}=\nabla\left(F(\varphi^{n})+a(\cdot)\frac{(\varphi^{n})^{2}}{2}\right)-\nabla
a(\cdot)\frac{(\varphi^{n})^{2}}{2}-(J\ast\varphi^{n})\nabla\varphi^{n}.$
(3.39)

This allows us to have

$\displaystyle\begin{aligned}
&\int_{0}^{T}((\mu^{n}(t)-\ell_{c}\theta^{n}(t))\nabla\varphi^{n}(t),\operatorname{\mathbf{v}}(t))_{\Omega}\operatorname{d\\!}t\\\
&\leq\int_{0}^{T}|(\nabla
a\frac{(\varphi(t)^{n})^{2}}{2},\operatorname{\mathbf{v}}(t))_{\Omega}|\operatorname{d\\!}t+\int_{0}^{T}|((\nabla
J\ast\varphi^{n}(t))\varphi^{n}(t),\operatorname{\mathbf{v}}(t))_{\Omega}|\operatorname{d\\!}t\\\
&\leq c(\|\nabla a\|_{L^{\infty}}\|+\|\nabla
J\|_{L^{1}})\int_{0}^{T}\|\varphi^{n}(t)\|^{2}_{L^{4}}\|\operatorname{\mathbf{v}}(t)\|_{H_{\sigma}}\operatorname{d\\!}t\\\
&\leq c(\|\nabla a\|_{L^{\infty}}\|+\|\nabla
J\|_{L^{1}})\|\varphi^{n}\|^{2}_{L^{4}(I;L^{4}(\Omega))}\|\operatorname{\mathbf{v}}\|_{L^{2}(I;V_{\sigma})}.\end{aligned}$
(3.40)

From Gagliardo-Nirenberg inequality, (3.21) and (3.29), we infer that

$\displaystyle\begin{aligned}
\|\varphi^{n}\|_{L^{4}(I;L^{4}(\Omega))}^{4}&\leq\int_{0}^{T}\|\varphi^{n}(t)\|_{L^{2}}^{2}\|\nabla\varphi^{n}(t)\|_{L^{2}}^{2}\operatorname{d\\!}t\leq\|\varphi^{n}\|_{L^{\infty}(I;H)}^{2}\|\varphi^{n}\|_{L^{2}(I;V)}^{2}\leq
M^{4}.\end{aligned}$ (3.41)

Therefore,

$\displaystyle\|P^{n}_{\sigma}\mathcal{C}_{2}(\varphi^{n},\mu^{n}-\ell_{c}\theta^{n})\|_{L^{2}(I;V_{\sigma}^{*})}\leq
M^{2}.$ (3.42)

The remaining terms in (3.32) are trivially handled and it can be easily shown
that

$\displaystyle\|P^{n}_{\sigma}\ell(\varphi^{n}(t),\theta^{n}(t)){\operatorname{\mathbf{g}}}\|_{L^{2}(I;V_{\sigma}^{*})}\leq
1+2M,$ (3.43)
$\displaystyle\|P^{n}_{\sigma}{\operatorname{{\mathbf{q}}}}(t)\|_{L^{2}(I;V_{\sigma}^{*})}\leq
1+\|{\operatorname{{\mathbf{q}}}}(t)\|_{L^{2}(I;V_{\sigma}^{*})},$ (3.44)

where (3.43) is achieved from (3.21) and (3.26), while (3.44) is owed from the
fact that $P^{n}_{\sigma}\in\mathcal{L}(V_{\sigma}^{*},V_{\sigma}^{*})$.
Combining the estimates (3.33), (3.34), (3.35), (3.38), (3.42), (3.43), and
(3.44), we get the estimate

$\displaystyle\|\partial_{t}{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{q}(I;V_{\sigma}^{*})}\leq
c(1+(1+M)M+\|{\operatorname{{\mathbf{q}}}}(t)\|_{L^{2}(I;V_{\sigma}^{*})})$
(3.45)

for some constant $c>0$, and where $q=4/3$ if $d=3$, and $q=2$ if $d=2$.

Now let us derive estimates for the time derivative of the order parameter. To
do this we first rewrite (3.8a)

$\displaystyle\langle\partial_{t}\varphi^{n}(t),\psi\rangle_{V_{s}}+\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}+\langle
P^{n}B(\mu^{n}(t)),\psi\rangle_{V_{s}}=0,$

with $\mu^{n}=\rho(\cdot,\varphi^{n})-J\ast\varphi^{n}+\ell_{c}\theta^{n}$.
Before going further, let us mention that the estimates will highly depend on
$p\in(1,2]$ from assumption (A5) due to the appearance of
$\rho(\cdot,\varphi^{n})$ in the definition of $\mu^{n}$ which we have proven
to be bounded in $L^{\infty}(I;L^{p}(\Omega))$. Due to the limited of
regularity of $\rho(\cdot,\varphi^{n})$ we are compelled to consider test
functions $\psi\in V_{s}$ for some $s\geq 2$ that will allow us to utilize the
Sobolev embedding $H^{s-2}(\Omega)\hookrightarrow L^{p^{\prime}}(\Omega)$
where $p^{\prime}$ is the Hölder conjugate of $p$. If such embedding holds
then Hölder’s inequality, (A1), and (3.31) will imply

$\displaystyle\begin{aligned} |\langle
P^{n}B(\mu^{n}(t)),\psi\rangle_{V_{s}}|&\leq|(\nabla\rho(\cdot,\varphi^{n}),\nabla\psi)_{\Omega}|+|(\nabla
J\ast\varphi^{n},\nabla\psi)_{\Omega}|+\ell_{c}|(\nabla\theta^{n},\nabla\psi)_{\Omega}|\\\
&\leq|(\rho(\cdot,\varphi^{n}),\Delta\psi)_{\Omega}|+\|\nabla
J\|_{L^{1}}\|\varphi^{n}\|\|\nabla\psi\|+\ell_{c}|(\theta^{n},\Delta\psi)_{\Omega}|\\\
&\leq\|\rho(\cdot,\varphi^{n})\|_{L^{p}}\|\Delta\psi\|_{L^{p^{\prime}}}+\|\nabla
J\|_{L^{1}}\|\varphi^{n}\|\|\psi\|_{V_{s}}+\ell_{c}\|\theta^{n}\|\|\Delta\psi\|_{L^{p^{\prime}}}\\\
&\leq(\|\rho(\cdot,\varphi^{n})\|_{L^{p}}+\|\nabla
J\|_{L^{1}}\|\varphi^{n}\|+\ell_{c}\|\theta^{n}\|)\|\psi\|_{V_{s}}.\end{aligned}$

Using (3.21), (3.26) and (3.31), we thus get

$\displaystyle\|P^{n}B(\mu^{n}(t))\|_{L^{\infty}(I;V_{s}^{*})}\leq M.$

Let us now take our gaze onto the term $\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}$.
We know that the embedding $H^{s-2}(\Omega)\hookrightarrow
L^{p^{\prime}}(\Omega)$ — to maintain the validity of the previous computation
— holds if either of the following cases is true:

  * 1.

for $2\leq p^{\prime}\leq+\infty$ if $(s-2)2>d$,

  * 2.

for $2\leq p^{\prime}<+\infty$ if $(s-2)2=d$,

  * 3.

for $2\leq p^{\prime}\leq 2d/(d-(s-2)2)$ if $(s-2)2<d$.

The first two cases instantly yields a good estimate for $\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V}$.
Indeed, from Hölder’s inequality, and from the fact that $(s-2)2\geq d$
implies $(s-1)2>d$ — which validates the embedding
$H^{s-1}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ — we get

$\displaystyle\begin{aligned} |\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}|&=|({\operatorname{{\mathbf{u}}}}^{n}\cdot\nabla\psi,\varphi^{n})_{\Omega}|\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{L^{\infty}}\|\varphi^{n}\|\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{H^{s-1}}\|\varphi^{n}\|\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\varphi^{n}\|\|\psi\|_{V_{s}}\end{aligned}$
(3.46)

The third case and the fact that $p^{\prime}$ is the Hölder conjugate of $p$
implies that it would suffice to just consider $\frac{(4-d)p+2d}{2p}\leq
s<\frac{d+4}{2}$. The arbitrariness of $p\in(1,2]$ also calls for us to
consider $p\in(1,d/(d-1))$, $p=d/(d-1)$, and $p\in(3/2,2]$ exclusively for
$d=3$. Let us first consider the following cases:

  * 1.

$\frac{(4-d)p+2d}{2p}\leq s$ and $p\in(1,d/(d-1))$,

  * 2.

$\frac{(4-d)p+2d}{2p}<s$ and $p=d/(d-1)$.

We see that the scenarios above imply $(s-1)2>d$, hence the embedding
$H^{s-1}(\Omega)\hookrightarrow L^{\infty}(\Omega)$. Following the same
computations as in (3.46), we get

$\displaystyle\begin{aligned} |\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}|&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\varphi^{n}\|\|\psi\|_{V_{s}}\end{aligned}$

Using (3.21) and (3.23) we see that

$\displaystyle\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t))\|_{L^{\infty}(I;V_{s}^{*})}\leq
M^{2},$

if either $(s-2)2\geq d$, $\frac{(4-d)p+2d}{2p}\leq s$ and $p\in(1,d/(d-1))$,
or $\frac{(4-d)p+2d}{2p}<s$ and $p=d/(d-1)$.

The case $\frac{(4-d)p+2d}{2p}=s$ and $p=d/(d-1)$ implies $(s-1)2=d$ which
gives us the embedding $H^{s-1}(\Omega)\hookrightarrow L^{r}(\Omega)$ for any
$2<r<+\infty$. Using Hölder’s inequality, the embedding previously mentioned,
and the interpolation in $L^{p}$ with $\theta=(r-4)/r$, $p=2$, and $q=4$, we
arrive at the following computation:

$\displaystyle\begin{aligned} |\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}|&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{L^{r}}\|\varphi^{n}\|_{L^{\frac{2r}{r-2}}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{H^{s-1}}\|\varphi^{n}\|^{\frac{r-4}{r}}\|\varphi^{n}\|_{L^{4}}^{\frac{4}{r}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\psi\|_{V_{s}}\|\varphi^{n}\|^{\frac{r-4}{r}}\|\varphi^{n}\|_{V}^{\frac{4}{r}}.\end{aligned}$

Using (3.21), (3.23) and (3.29), we get that the case currently being
considered yields

$\displaystyle\begin{aligned}
\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n},\varphi^{n})\|_{L^{\frac{r}{2}}(I;V_{s}^{*})}&\leq\left(\int_{0}^{T}(\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\varphi^{n}(t)\|^{\frac{r-4}{r}}\|\varphi^{n}(t)\|_{V}^{\frac{4}{r}})^{\frac{r}{2}}\operatorname{d\\!}t\right)^{2/r}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\|\varphi^{n}\|_{L^{\infty}(I;H)}^{\frac{r-4}{r}}\|\varphi^{n}\|_{L^{2}(I;V)}^{\frac{4}{r}}\end{aligned}$

Lastly, $\frac{3}{2}<p\leq 2$ and
$s=\frac{(4-d)p+2d}{2p}=\frac{1}{2}+\frac{3}{p}$ we get that $(s-1)2<3=d$.
This implies the embedding $H^{s-1}(\Omega)\hookrightarrow L^{r}(\Omega)$ with
$r=\frac{3p}{2p-3}$. By letting $r^{\prime}>1$ be such that
$\frac{1}{r}+\frac{1}{r^{\prime}}=\frac{1}{2}$, together with $L^{p}$
interpolation (with $\theta=\frac{20p-30}{9p}$, $p_{1}=5$, and $p_{2}=2$), and
the embedding $H^{1}(\Omega)\hookrightarrow L^{5}(\Omega)$ we get the
following estimate

$\displaystyle\begin{aligned} |\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),\varphi^{n}(t)),\psi\rangle_{V_{s}}|&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{L^{r}}\|\varphi^{n}\|_{L^{r^{\prime}}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\nabla\psi\|_{H^{s-1}}\|\varphi^{n}\|^{\frac{30-11p}{9p}}\|\varphi^{n}\|_{L^{5}}^{\frac{10}{9}\frac{2p-3}{p}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|\|\psi\|_{V_{s}}\|\varphi^{n}\|^{\frac{30-11p}{9p}}\|\varphi^{n}\|_{V}^{\frac{10}{9}\frac{2p-3}{p}}.\end{aligned}$

From the $L^{\infty}(I;H_{\sigma})$ and $L^{\infty}(I;H)$ estimates of
${\operatorname{{\mathbf{u}}}}^{n}$ and $\varphi^{n}$, respectively, and the
$L^{2}(I;V)$ estimate of $\varphi^{n}$ from (3.29) we infer that

$\displaystyle\begin{aligned}
\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n},\varphi^{n})\|_{L^{\frac{9p}{5(2p-3)}}(I;V_{s}^{*})}&\leq\left(\int_{0}^{T}\left(\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\varphi^{n}\|^{\frac{30-11p}{9p}}\|\varphi^{n}(t)\|_{V}^{\frac{10}{9}\frac{2p-3}{p}}\right)^{\frac{9p}{5(2p-3)}}\operatorname{d\\!}t\right)^{\frac{5(2p-3)}{9p}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\|\varphi^{n}\|^{\frac{30-11p}{9p}}_{L^{\infty}(I;H)}\|\varphi^{n}\|_{L^{2}(I;V)}^{\frac{10}{9}\frac{2p-3}{p}}.\end{aligned}$

From all these we finally get the following estimates for
$\partial_{t}\varphi^{n}$.

$\displaystyle\begin{aligned}
\|\partial_{t}\varphi^{n}\|_{L^{\infty}(I;V_{s}^{*})\cap
L^{\frac{r}{2}}(I;V_{\frac{d+2}{2}}^{*})}\leq M+M^{2},\end{aligned}$ (3.47)

where $r>2$ and if any of the following cases hold:

  * 1.

$(s-2)2\geq d$,

  * 2.

$1<p<\frac{d}{d-1}$ with $\frac{d+4}{2}>s\geq\frac{(4-d)p+2d}{2p}$, or

  * 3.

$p=\frac{d}{d-1}$ with $\frac{d+4}{2}>s>\frac{(4-d)p+2d}{2p}$.

Lastly, if $d=3$ and $\frac{3}{2}<p\leq 2$ we have

$\displaystyle\begin{aligned}
\|\partial_{t}\varphi^{n}\|_{L^{\frac{9p}{5(2p-3)}}(I;V_{s}^{*})}\leq
M+M^{2}.\end{aligned}$ (3.48)

For the time derivative of the temperature, we rewrite (3.8c) as

$\displaystyle\begin{aligned}
&\langle\partial_{t}H(\theta^{n}(t),\varphi^{n}(t)),\vartheta\rangle_{V_{s}}+\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t))),\vartheta\rangle_{V_{s}}\\\
&+\kappa\langle P^{n}\mathcal{B}\theta^{n}(t),\vartheta\rangle_{V_{s}}=\langle
P^{n}({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}^{n}(t)),\vartheta\rangle_{V_{s}}+\langle
z^{n}(t),\vartheta\rangle_{V_{s}},\end{aligned}$ (3.49)

where $H(\theta^{n}(t),\varphi^{n}(t))=\theta^{n}(t)-\ell_{h}\varphi^{n}(t)$.
Apparently, we also have to work on the space $V_{s}$ due to the appearance of
the time derivative of the order parameter. Of course, the linear parts of
(3.49) can be handled conveniently, and in fact — because of the embedding
$H^{s-2}(\Omega)\hookrightarrow L^{2}(\Omega)$ — we have

$\displaystyle\|P^{n}\mathcal{B}\theta^{n}(t)\|_{V_{s}^{*}}$
$\displaystyle=\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}|\langle
P^{n}\mathcal{B}\theta^{n}(t),\vartheta\rangle_{V_{s}}|=\inf_{\begin{subarray}{c}{\vartheta}\in
V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}|(\theta^{n}(t),\Delta\vartheta)_{\Omega}|$
$\displaystyle=\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}\|\theta^{n}(t)\|\|\Delta\vartheta\|\leq\|\theta^{n}(t)\|.$

Hence, we obtain from (3.26)

$\displaystyle\|P^{n}\mathcal{B}\theta^{n}\|_{L^{\infty}(I;V_{s}^{*})}\leq\|\theta^{n}\|_{L^{\infty}(I;H)}\leq
M.$

Hölder’s inequality, the constancy of the gravitational parameter
$\operatorname{\mathbf{g}}$ and (3.23), on the other hand, give us

$\displaystyle\|P^{n}({\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}^{n})\|_{L^{\infty}(I;V_{s}^{*})}\leq
c\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\leq M.$

For the transport term, we shall take advantage of the embedding
$H^{s-1}(\Omega)\hookrightarrow L^{r}(\Omega)$ for some $2<r\leq+\infty$ as
illustrated in the previous computations for the time derivative of
$\varphi^{n}$. For this purpose, we shall skip some parts which we have
covered previously, but mention the values of $r$ that we used.

Case 1: $r=+\infty$. Before we begin, let us mention that due to (3.21),
(3.26) and (3.29) we get that $H(\theta^{n},\varphi^{n})$ belongs to
$L^{\infty}(I;H)\cap L^{2}(I;V)$, and in fact we have

$\displaystyle\begin{aligned}
\|H(\theta^{n},\varphi^{n})\|_{L^{\infty}(I;H)\cap L^{2}(I;V)}\leq
M.\end{aligned}$ (3.50)

Such property is global and does not only cover the current case.

Going back, we see that by utilizing Hölder’s inequality, we get

$\displaystyle\begin{aligned}
\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t)))\|_{V_{s}^{*}}&=\inf_{\begin{subarray}{c}{\vartheta}\in
V_{s}\\\ \|\vartheta\|_{V_{s}}=1\end{subarray}}|\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t))),\vartheta\rangle_{V_{s}}|\\\
&=\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}|({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla\vartheta,H(\theta^{n}(t),\varphi^{n}(t)))_{\Omega}|\\\
&\leq\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\nabla\vartheta\|_{L^{\infty}}\|H(\theta^{n}(t),\varphi^{n}(t))\|\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|H(\theta^{n}(t),\varphi^{n}(t))\|.\end{aligned}$

From (3.21) and (3.23) we hence achieve

$\displaystyle\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t)))\|_{L^{\infty}(I;V_{s}^{*})}\leq
M^{2}.$

Case 2: $2<r<+\infty$. This particular case is divided in two subcases. The
first one is when $\frac{(4-d)p+2d}{2p}=s$ and $p=d/(d-1)$, which implies
$2<r<+\infty$ can be chosen arbitrarily. In this instance, we see that

$\displaystyle\begin{aligned}
\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t)))\|_{V_{s}^{*}}&=\inf_{\begin{subarray}{c}{\vartheta}\in
V_{s}\\\ \|\vartheta\|_{V_{s}}=1\end{subarray}}|\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t))),\vartheta\rangle_{V_{s}}|\\\
&=\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}|({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla\vartheta,H(\theta^{n}(t),\varphi^{n}(t)))_{\Omega}|\\\
&\leq\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\nabla\vartheta\|_{L^{r}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{L^{\frac{2r}{r-2}}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|H(\theta^{n}(t),\varphi^{n}(t))\|^{\frac{r-4}{r}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{V}^{\frac{4}{r}}.\end{aligned}$

Here, the last inequality is achieved by additionally employing
$L^{p}$-interpolation with $\theta=(r-4)/r$, $p_{1}=4$ and $p_{2}=2$, and the
embedding $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$. We thus get

$\displaystyle\begin{aligned}
&\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n},H(\theta^{n},\varphi^{n}))\|_{L^{\frac{r}{2}}(I;V_{s}^{*})}\\\
&\leq\left(\int_{0}^{T}\left(\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|H(\theta^{n}(t),\varphi^{n}(t))\|^{\frac{r-4}{r}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{V}^{\frac{4}{r}}\right)^{\frac{r}{2}}\operatorname{d\\!}t\right)^{2/r}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\|H(\theta^{n},\varphi^{n})\|_{L^{\infty}(I;H)}^{\frac{r-4}{r}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{L^{2}(I;V)}^{\frac{4}{r}}\leq
M^{2},\end{aligned}$

where the last inequality follows from (3.23) and (3.50).

The second subcase is when $d=3$ and $\frac{3}{2}<p\leq 2$, which gives us a
particular value $r=\frac{3p}{2p-3}$. By additionally utilizing $L^{p}$
interpolation with $\theta=\frac{30-11p}{9p}$, $p_{1}=5$, and $p_{2}=2$ and
the embedding $H^{1}(\Omega)\hookrightarrow L^{5}(\Omega)$ we get

$\displaystyle\begin{aligned}
\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t)))\|_{V_{s}^{*}}&=\inf_{\begin{subarray}{c}{\vartheta}\in
V_{s}\\\ \|\vartheta\|_{V_{s}}=1\end{subarray}}|\langle
P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n}(t),H(\theta^{n}(t),\varphi^{n}(t))),\vartheta\rangle_{V_{s}}|\\\
&=\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}|({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla\vartheta,H(\theta^{n}(t),\varphi^{n}(t)))_{\Omega}|\\\
&\leq\inf_{\begin{subarray}{c}{\vartheta}\in V_{s}\\\
\|\vartheta\|_{V_{s}}=1\end{subarray}}\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|\nabla\vartheta\|_{L^{r}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{L^{\frac{2r}{r-2}}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|H(\theta^{n}(t),\varphi^{n}(t))\|^{\frac{30-11p}{9p}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{V}^{\frac{10}{9}\frac{2p-3}{p}}\end{aligned}$

We further estimate with respect to the time variable as follows:

$\displaystyle\begin{aligned}
&\|P^{n}\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}^{n},H(\theta^{n},\varphi^{n}))\|_{L^{\tilde{p}}(I;V_{s}^{*})}\\\
&\leq\left(\int_{0}^{T}\left(\|{\operatorname{{\mathbf{u}}}}^{n}(t)\|\|H(\theta^{n}(t),\varphi^{n}(t))\|^{\frac{30-11p}{9p}}\|H(\theta^{n}(t),\varphi^{n}(t))\|_{V}^{\frac{2}{\tilde{p}}}\right)^{{\tilde{p}}}\operatorname{d\\!}t\right)^{\frac{1}{\tilde{p}}}\\\
&\leq\|{\operatorname{{\mathbf{u}}}}^{n}\|_{L^{\infty}(I;H_{\sigma})}\|H(\theta^{n},\varphi^{n})\|_{L^{\infty}(I;H_{\sigma})}^{\frac{30-11p}{9p}}\|H(\theta^{n},\varphi^{n})\|_{L^{2}(I;V)}^{\frac{10}{9}\frac{2p-3}{p}}\leq
M^{2},\end{aligned}$

where $\tilde{p}=\frac{9p}{5(2p-3)}$.

Finally, the computations above give us the estimate for the temperature
variable $\partial_{t}\theta^{n}$

$\displaystyle\begin{aligned}
\|\partial_{t}\theta^{n}\|_{L^{\infty}(I;V_{s}^{*})\cap
L^{\frac{r}{2}}(I;V_{\frac{d+2}{2}}^{*})}\leq M+M^{2}\end{aligned}$ (3.51)

where $r>2$ and if any of the following cases hold:

  * 1.

$(s-2)2\geq d$,

  * 2.

$1<p<\frac{d}{d-1}$ with $\frac{d+4}{2}>s\geq\frac{(4-d)p+2d}{2p}$, or

  * 3.

$p=\frac{d}{d-1}$ with $\frac{d+4}{2}>s>\frac{(4-d)p+2d}{2p}$.

And if $d=3$ and $\frac{3}{2}<p\leq 2$ we have

$\displaystyle\begin{aligned}
\|\partial_{t}\theta^{n}\|_{L^{\frac{9p}{5(2p-3)}}(I;V_{s}^{*})}\leq
M+M^{2}.\end{aligned}$ (3.52)

The estimates (3.21)–(3.26), (3.29), (3.30) and (3.31) imply the existence of
$\varphi\in L^{\infty}(I;H)\cap L^{2}(I;V)$, $\mu\in L^{2}(I;V)$, $\rho\in
L^{\infty}(I;L^{p}(\Omega))$, $\operatorname{{\mathbf{u}}}\in
L^{\infty}(I;H_{\sigma})\cap L^{2}(I;V_{\sigma})$ and $\theta\in
L^{\infty}(I;H)\cap L^{2}(I;V)$ for which — up to a sebsequence — the
following properties hold

$\displaystyle\varphi^{n}\rightharpoonup\varphi$ $\displaystyle\text{in
}L^{2}(I;V)$ (3.53)
$\displaystyle\varphi^{n}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}$
$\displaystyle\text{in }L^{\infty}(I;H)$ (3.54)
$\displaystyle\mu^{n}\rightharpoonup\mu$ $\displaystyle\text{in }L^{2}(I;V)$
(3.55)
$\displaystyle\rho(\cdot,\varphi^{n})\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\rho$
$\displaystyle\text{in }L^{\infty}(I;L^{p}(\Omega))$ (3.56)
$\displaystyle\operatorname{{\mathbf{u}}}^{n}\rightharpoonup\operatorname{{\mathbf{u}}}$
$\displaystyle\text{in }L^{2}(I;V_{\sigma})$ (3.57)
$\displaystyle\operatorname{{\mathbf{u}}}^{n}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\operatorname{{\mathbf{u}}}$
$\displaystyle\text{in }L^{\infty}(I;H_{\sigma})$ (3.58)
$\displaystyle\theta^{n}\rightharpoonup\theta$ $\displaystyle\text{in
}L^{2}(I;V)$ (3.59)
$\displaystyle\theta^{n}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\theta$
$\displaystyle\text{in }L^{\infty}(I;H).$ (3.60)

Estimates (3.45), (3.47), (3.48), (3.51) and (3.52), further gives us the
convergences of the time derivatives:

$\displaystyle\partial_{t}{\operatorname{{\mathbf{u}}}}^{n}\rightharpoonup\partial_{t}{\operatorname{{\mathbf{u}}}}\text{
in
}L^{4/d}(I;V_{\sigma}^{*}),\quad\partial_{t}\varphi^{n}\rightharpoonup\partial_{t}\varphi\text{
and }\partial_{t}\theta^{n}\rightharpoonup\partial_{t}\theta\text{ in
}L^{q}(I;V^{*}_{s})$ (3.61)

for $q=\frac{9p}{5(2p-3)}$ and $s=\frac{p+6}{2p}$ if $d=3$ and
$\frac{3}{2}<p\leq 2$, and for $q=\frac{r}{2}$ and $s=\frac{d+2}{2}$ — with
$r>2$ taken arbitrarily — if $p=\frac{d}{d-1}$ and $d=2,3$,

$\displaystyle\partial_{t}\varphi^{n}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\partial_{t}\varphi,\qquad\partial_{t}\theta^{n}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\partial_{t}\theta$
$\displaystyle\text{in }L^{\infty}(I;V^{*}_{s})$ (3.62)

for either $(s-2)2\geq d$, $1<p<\frac{d}{d-1}$ with
$\frac{d+4}{2}>s\geq\frac{(4-d)p+2d}{2p}$, or $p=\frac{d}{d-1}$ with
$\frac{d+4}{2}>s>\frac{(4-d)p+2d}{2p}$.

Taking advantage of Aubin–Lions–Simon embedding theorems would give us the
compact embedding $W^{2,4/d}(V_{\sigma},V_{\sigma}^{*})\hookrightarrow
L^{2}(I;H_{\sigma})$. Similarly, the embedding
$W^{2,q}(V,V_{s}^{*})\hookrightarrow L^{2}(I;H)$ holds for the following cases

  * 1.

$q=\frac{9p}{5(2p-3)}$ and $s=\frac{p+6}{2p}$ if $d=3$ and $\frac{3}{2}<p\leq
2$;

  * 2.

$q=\frac{r}{2}$, $r>2$ taken arbitrarily, and $s=\frac{d+2}{2}$ if
$p=\frac{d}{d-1}$ $d=2,3$; and

  * 3.

$q=+\infty$ if either $(s-2)2\geq d$; $1<p<\frac{d}{d-1}$ with
$\frac{d+4}{2}>s\geq\frac{(4-d)p+2d}{2p}$; or $p=\frac{d}{d-1}$ with
$\frac{d+4}{2}>s>\frac{(4-d)p+2d}{2p}$.

From these, we get the following strong convergences

$\displaystyle\operatorname{{\mathbf{u}}}^{n}\to\operatorname{{\mathbf{u}}}\text{
in }L^{2}(I;H_{\sigma}),\quad\varphi^{n}\to\varphi\text{ in
}L^{2}(I;H),\quad\theta^{n}\to\theta\text{ in }L^{2}(I;H)$ (3.63)

all of which converge a.e. in $\Omega\times(0,T)$.

The passage of the limit for the order parameter and the fluid velocity
follows that of in [16]. Although such steps would be straightforward to
establish, we highlight the crucial points for which we prove that the triple
$[\varphi,{\operatorname{{\mathbf{u}}}},\theta]$ is a weak solution of the
system in the sense of Definition 3.1. Firstly, we point out some direct
consequences of (3.63).

  * 1.

By definition of $\rho(\cdot,\varphi^{n})$, the regularity of $F$, and the
pointwise convergence of the order parameter, $\rho(\cdot,\varphi^{n})\to
a(\cdot)\varphi+F^{\prime}(\varphi)$ a.e. in $\Omega\times(0,T)$, from which
we also infer $\rho=a(\cdot)\varphi+F^{\prime}(\varphi)$.

  * 2.

The strong convergence of the order parameter in $L^{2}(I;H)$ implies
$J\ast\varphi^{n}\to J\ast\varphi$ in $L^{2}(I;V)$. Such convergence and the
definition of $\mu^{n}$ imply $\mu=\rho-J\ast\varphi+\ell_{c}\theta$. Indeed,
for any $\psi\in H^{n}$ and $\chi\in C_{0}^{\infty}(0,T)$, we get

$\displaystyle\int_{0}^{T}(\mu^{n},\psi)_{\Omega}\chi\operatorname{d\\!}t$
$\displaystyle=\int_{0}^{T}(\rho(\cdot,\varphi^{n})-J\ast\varphi^{n}+\ell_{c}\theta^{n},\psi)_{\Omega}\chi\operatorname{d\\!}t$
$\displaystyle\to\int_{0}^{T}(\rho-J\ast\varphi+\ell_{c}\theta,\psi)_{\Omega}\chi\operatorname{d\\!}t,$

where we used the pointwise convergence of $\rho$, the strong convergence of
convolution, the strong convergence of the temperature variable in
$L^{2}(I;H)$, and the density of $\rm{span}\\{\psi_{j}\\}$ in $V$. We thus
infer that $\mu=\rho-J\ast\varphi+\ell_{c}\theta$ from (3.55), and that
$\rho\in L^{2}(I;V)$.

Moving forward, we multiply (3.8a), (3.8b), (3.8c) by
$\chi,\mathfrak{w},\omega\in C_{0}^{\infty}(0,T)$, and integrate over the
interval $(0,T)$ and take advantage of (3.53)–(3.63). Since some of the terms
in the resulting integral equation can be handled quite easily, we highlight
only some of the parts which we deemed to be crucial.

  * 1.

The term $(\nabla\rho(\cdot,\varphi^{n}),\nabla\psi)_{\Omega}$ is handled by
passing the derivative to the test function, i.e.
$-(\rho(\cdot,\varphi^{n}),\Delta\psi)$, and utilizing (3.56) with the
knowledge that $\psi\in V_{s}$ and $H^{s-2}\hookrightarrow L^{p^{\prime}}$.

  * 2.

Due to the convergence $J\ast\varphi^{n}\to J\ast\varphi$ in $L^{2}(I;V)$ we
also achieve convergence for the term $(\nabla
J\ast\varphi^{n},\nabla\psi)_{\Omega}$.

  * 3.

The expression involving the transport term
$({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla\varphi^{n}(t),\psi)_{\Omega}$
can be established to converge to
$({\operatorname{{\mathbf{u}}}}(t)\cdot\nabla\varphi(t),\psi)_{\Omega}$ by
utilizing the convergences (3.57), (3.53) and (3.63). The same argument is
used to show the convergence of the transport term occurring in the equation
for the temperature, i.e., the term
$({\operatorname{{\mathbf{u}}}}^{n}(t)\cdot\nabla(\theta^{n}(t)-\ell_{h}\varphi^{n}(t)),\vartheta)_{\Omega}$.

  * 4.

The strong convergence of the order parameter in (3.63), the imposed
properties of $\nu(\cdot)$ in (A2), and the dominated convergence theorem
imply $\nu(\varphi^{n})\to\nu(\varphi)$ in $L^{p}(I;L^{p}(\Omega))$ for any
$1\leq p<+\infty$. From the convergence of the viscosity term above and (3.57)
we implore that
$\nu(\varphi^{n})\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}\rightharpoonup\nu(\varphi)\mathrm{D}{\operatorname{{\mathbf{u}}}}$
in $L^{2}(I;L^{2}(\Omega)^{d\times d})$. The weak and strong convergences of
the velocity vector in $L^{2}(I;V_{\sigma})$ and $L^{2}(I;H_{\sigma})$,
respectively, give us the convergence for the trilinear form in (3.8b) [36,
Lemma III.3.2].

The linear terms can be handled quite easily using the convergences we have in
hand. For the terms concerning the time derivative, we pass the time
derivatives to the functions $\chi,\mathfrak{w},\omega\in C_{0}^{\infty}(0,T)$
and use (3.63). From these we see that the triple
$[\varphi,{\operatorname{{\mathbf{u}}}},\theta]$ satisfies (3.1a), (3.1b) and
(3.1c) by virtue of the density of
$\rm{span}\\{{\operatorname{\mathbf{v}}}_{j}\\}$ and $\rm{span}\\{\psi_{j}\\}$
in $V_{\sigma}$ and $V$.

Now we suppose $\chi,\mathfrak{w},\omega\in C_{\infty}(0,T)$ with values equal
to one at the initial time $t=0$ and equal to zero at the terminal time $t=T$
and apply the same passage of the limit. Multiplying the same functions to
(3.1a), (3.1b) and (3.1c) and apply integration by parts eventually lead to

$\displaystyle({\varphi}(0)-{\varphi}_{0},{\psi})_{\Omega}=0\quad\forall{\psi}\in
V,$
$\displaystyle({\operatorname{{\mathbf{u}}}}(0)-{\operatorname{{\mathbf{u}}}}_{0},{\operatorname{\mathbf{v}}})_{\Omega}=0\quad\forall{\operatorname{\mathbf{v}}}\in
V_{\sigma},$
$\displaystyle({\theta}(0)-{\theta}_{0},{\vartheta})_{\Omega}=0\quad\forall{\vartheta}\in
V.$

We also mention that due to the Aubin–Lions lemma we have the embeddings
$W^{+\infty,4/d}(H_{\sigma},V_{\sigma}^{*})\hookrightarrow
C(\overline{I};H_{\sigma})$ and $W^{+\infty,q}(H,V_{s}^{*})\hookrightarrow
C(\overline{I};H)$. These validate the evaluation of the weak solutions at the
initial time $t=0$, and shows the satisfaction of (3.2), (3.3) and (3.4).

The improved regularity of $\rho\in L^{2}(I;V)$ lets us write (3.1a) as

$\displaystyle\langle\partial_{t}\varphi(t),\psi\rangle_{V}+\langle\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}}(t),\varphi(t)),\psi\rangle_{V}+\langle
B\mu(t),\psi\rangle_{V}$ $\displaystyle=0$ $\displaystyle\forall\psi\in V.$

The transport term can thus be estimated — following analogous arguments as in
(3.38) and (3.42) — as

$\displaystyle\|\mathcal{C}_{1}({\operatorname{{\mathbf{u}}}},\varphi)\|_{L^{4/d}(I;V^{*})}$
$\displaystyle\leq M^{2}.$

This improves the regularity of the time derivative as well, i.e.
$\partial_{t}\varphi\in L^{4/d}(I,V^{*})$. The space upon which the time
derivative of the temperature inherits the improvement gained by the order
parameter as well, i.e. $\partial_{t}\theta\in L^{4/d}(I,V^{*})$.

Another consequence of the embeddings above is that up to a subsequence we
have $\varphi^{n}(t)\to\varphi(t)$ in $H$ and almost everywhere in $\Omega$.
Consequently, by virtue of Fatou’s lemma

$\displaystyle\int_{\Omega}F(\varphi(t))\operatorname{d\\!}x\leq\liminf_{n\to\infty}\int_{\Omega}F(\varphi^{n}(t))\operatorname{d\\!}x.$
(3.64)

We also have, from the weak lower semi-continuity of the nonlocal energy
$\mathbb{E}_{nl}$ and the weak convergence of $\varphi^{n}$ to $\varphi$,

$\displaystyle\int_{\Omega}\mathbb{E}_{nl}(\varphi(t))\operatorname{d\\!}x\leq\liminf_{n\to\infty}\int_{\Omega}\mathbb{E}_{nl}(\varphi^{n}(t))\operatorname{d\\!}x.$
(3.65)

We recall that $J\ast\varphi^{n}\to J\ast\varphi$ in $L^{2}(I;V)$ which
implies $P^{n}(J\ast\varphi^{n})\to J\ast\varphi$ in $L^{2}(I;V)$, and
$\sqrt{\nu(\varphi^{n})}\mathrm{D}{\operatorname{{\mathbf{u}}}}^{n}\rightharpoonup\sqrt{\nu(\varphi)}\mathrm{D}{\operatorname{{\mathbf{u}}}}$
in $L^{2}(I;L^{2}(\Omega)^{d\times d})$. Integrating (3.14) over $[0,t]$,
utilizing (3.64), (3.65) and (3.53)–(3.60), and taking advantage of the weak
lower semi-continuity of norms, we achieve the energy inequality (3.7).

∎

##  4 Nonlocal-to-Local Convergence

The purpose of the section is to show, under an appropriate choice of the
kernel $J$, that solutions of the nonlocal system converges to its local
version. Notably, we should be able to determine by which convergence we shall
anchor on, we also recall that the only assumption we imposed for the kernel
is (A1), so it is imperative that such assumption is not violated by the
particular kinds of kernel we shall consider.

Suppose $\gamma\in(0,d-1)$, $\eta\in C^{1}([0,+\infty);[0,+\infty))$ such that
the map $s\mapsto|\eta^{\prime}(s)|s^{d-1-\gamma}$ is in
$L^{1}(\mathbb{R}^{+})$, and satisfies the renormalization

$\displaystyle\int_{0}^{+\infty}\eta(s)s^{d+1-\gamma}\operatorname{d\\!}s=\frac{2}{C_{d}},\text{
where }C_{d}:=\int_{S^{d-1}}|\sigma\cdot
e_{1}|^{2}\operatorname{d\\!}\mathcal{H}^{d-1}(\sigma).$

We define the family of mollifiers $(\eta_{\varepsilon})_{\varepsilon>0}$ as
$\eta_{\varepsilon}(s)=\frac{1}{\varepsilon^{d}}\eta(s/\varepsilon)$ for
$s\geq 0$, from which we define the kernel
$J_{\varepsilon}:\mathbb{R}^{d}\to\mathbb{R}$ by
$J_{\varepsilon}(x)=\frac{1}{\varepsilon^{2-\gamma}}\frac{\eta_{\varepsilon}(|x|)}{|x|^{\gamma}}$.
Evidently, for any $\varepsilon>0$ $J_{\varepsilon}(x)=J_{\varepsilon}(-x)$
for all $x\in\Omega$ and
$a_{\varepsilon}(x)=\int_{\Omega}J_{\varepsilon}(x-y)\operatorname{d\\!}y\geq
0$ for almost every $x\in\Omega$. Meanwhile, from [17, Lemma 3.1] we infer
that $J_{\varepsilon}\in W^{1,1}(\mathbb{R}^{d})$.

For a given $\varepsilon>0$ we now define the corresponding nonlocal energy
functional to each kernel $J_{\varepsilon}$ as

$\displaystyle\mathbb{E}_{nl}^{\varepsilon}(\varphi):=\frac{1}{2}\int_{\Omega}\int_{\Omega}J_{\varepsilon}(x-y)(\varphi(x)-\varphi(y))^{2}\operatorname{d\\!}y\operatorname{d\\!}x.$

And for each $\epsilon>0$ and each kernel $J_{\varepsilon}$, Theorem 3.3
assures us of the existence of solutions of (1.2) in the sense of Definition
3.1 which we shall denote as
$[\varphi_{\varepsilon},{\operatorname{{\mathbf{u}}}}_{\varepsilon},\theta_{\varepsilon}]$
together with the chemical potential
$\mu_{\varepsilon}:=a_{\varepsilon}\varphi_{\varepsilon}-J_{\varepsilon}\ast\varphi_{\varepsilon}+F^{\prime}(\varphi_{\varepsilon})+\ell_{c}\theta_{\varepsilon}$.
We also denote by $\mathbb{E}^{\varepsilon}$ the total energy of the system
(1.2) which we recall to be written as in (3.5) but now with the kernel
$J_{\varepsilon}$.

Our goal is to establish that the solutions
$[\varphi_{\varepsilon},{\operatorname{{\mathbf{u}}}}_{\varepsilon},\theta_{\varepsilon}]$
converge to a triple
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
that solves the local version of system (1.2) as $\varepsilon\to 0$. In
particular, we shall show that
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
satisfies — in weak sense — the Cahn–Hilliard–Bousinessq system similar to
what was proposed in [29]:

$\displaystyle\partial_{t}\varphi+{\operatorname{{\mathbf{u}}}}\cdot\nabla\varphi=\Delta\mu,\quad\mu=-\Delta\varphi+\eta
F^{\prime}(\varphi)+\ell_{c}\theta,$ (4.1a) $\displaystyle\begin{aligned}
\partial_{t}{\operatorname{{\mathbf{u}}}}+({\operatorname{{\mathbf{u}}}}\cdot\nabla){\operatorname{{\mathbf{u}}}}{-}\operatorname{\mathrm{div}}(\nu(\varphi)2\mathrm{D}{\operatorname{{\mathbf{u}}}})+\nabla
p=\mathcal{K}(\mu-\ell_{c}\theta)\nabla\varphi+\ell(\varphi,\theta){\operatorname{\mathbf{g}}}+{\operatorname{{\mathbf{q}}}},\end{aligned}$
(4.1b)
$\displaystyle\partial_{t}\theta-\ell_{h}\partial_{t}\varphi+{\operatorname{{\mathbf{u}}}}\cdot\nabla(\theta-\ell_{h}\varphi)-\kappa\Delta\theta={\operatorname{\mathbf{g}}}\cdot{\operatorname{{\mathbf{u}}}}+z.$
(4.1c)

with the incompressibility condition
$\operatorname{\mathrm{div}}{\operatorname{{\mathbf{u}}}}=0$ in $Q$, the
incorporated initial conditions $\varphi(0)=\varphi_{0}$,
${\operatorname{{\mathbf{u}}}}(0)={\operatorname{{\mathbf{u}}}}_{0}$, and
$\theta(0)=\theta_{0}$ in $\Omega$ and closed with the boundary conditions
$\frac{\partial\mu}{\partial{\operatorname{\mathbf{n}}}}=0$,
${\operatorname{{\mathbf{u}}}}=0$, and
$\frac{\partial\theta}{\partial{\operatorname{\mathbf{n}}}}=0$ on $\Gamma$. We
mention that the total energy for the local system can be written as

$\displaystyle\begin{aligned}
&2\widetilde{\mathbb{E}}(\varphi,{\operatorname{{\mathbf{u}}}},\theta):=\frac{1}{\ell_{c}}\int_{\Omega}\frac{1}{2}|\nabla\varphi(x)|^{2}+F(\varphi(x))\operatorname{d\\!}x+\frac{1}{\mathcal{K}\ell_{c}}\int_{\Omega}|{\operatorname{{\mathbf{u}}}}(x)|^{2}\operatorname{d\\!}x+\frac{1}{\ell_{h}}\int_{\Omega}|\theta(x)|^{2}\operatorname{d\\!}x,\end{aligned}$
(4.2)

while the local energy is defined for any $\varphi\in V$ as

$\displaystyle\mathbb{E}_{l}(\varphi):=\frac{1}{2}\int_{\Omega}|\nabla\varphi(x)|^{2}\operatorname{d\\!}x.$

Finally, the purpose of this section is to establish the following result.

######  Theorem 4.1.

Given $\varepsilon>0$ let
$[\varphi_{\varepsilon,0},{\operatorname{{\mathbf{u}}}}_{\varepsilon,0},\theta_{\varepsilon,0}]\in
H\times H_{\sigma}\times H$, and assume that there exists
$m_{\Omega}\in(-1,1)$ such that $\widehat{\varphi_{\varepsilon,0}}=m_{\Omega}$
for any $\varepsilon>0$. Suppose that there exists
$[\widetilde{\varphi}_{0},\widetilde{\operatorname{{\mathbf{u}}}}_{0},\widetilde{\theta}_{0}]\in
V\times H_{\sigma}\times H$ such that the following convergences hold:

$\displaystyle\varphi_{\varepsilon,0}\to\widetilde{\varphi}_{0}\text{ in
}H,\quad{\operatorname{{\mathbf{u}}}}_{\varepsilon,0}\to\widetilde{\operatorname{{\mathbf{u}}}}_{0}\text{
in }H_{\sigma},\quad\theta_{\varepsilon,0}\to\widetilde{\theta}_{0}\text{ in
}H$
$\displaystyle\mathbb{E}^{\varepsilon}(\varphi_{\varepsilon,0},{\operatorname{{\mathbf{u}}}}_{\varepsilon,0},\theta_{\varepsilon,0})\to\widetilde{\mathbb{E}}(\widetilde{\varphi}_{0},\widetilde{\operatorname{{\mathbf{u}}}}_{0},\widetilde{\theta}_{0}).$

If for any $\varepsilon>0$,
$[\varphi_{\varepsilon},{\operatorname{{\mathbf{u}}}}_{\varepsilon},\theta_{\varepsilon}]$
solves (1.2) in weak sense with initial conditions
$[\varphi_{\varepsilon,0},{\operatorname{{\mathbf{u}}}}_{\varepsilon,0},\theta_{\varepsilon,0}]$,
then as $\varepsilon\to 0$

$\displaystyle\varphi_{\varepsilon}\rightharpoonup\widetilde{\varphi}\text{ in
}L^{2}(I;V)$ (4.3)
$\displaystyle\varphi_{\varepsilon}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\widetilde{\varphi}\text{
in }L^{\infty}(I;H)$ (4.4)
$\displaystyle\varphi_{\varepsilon}\to\widetilde{\varphi}\text{ in
}L^{2}(I;H)\text{ and a.e. in }\Omega\times(0,T)$ (4.5)
$\displaystyle{\operatorname{{\mathbf{u}}}}_{\varepsilon}\rightharpoonup\widetilde{\operatorname{{\mathbf{u}}}}\text{
in }L^{2}(I;V_{\sigma})$ (4.6)
$\displaystyle{\operatorname{{\mathbf{u}}}}_{\varepsilon}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\widetilde{\operatorname{{\mathbf{u}}}}\text{
in }L^{\infty}(I;H_{\sigma})$ (4.7)
$\displaystyle{\operatorname{{\mathbf{u}}}}_{\varepsilon}\to\widetilde{\operatorname{{\mathbf{u}}}}\text{
in }L^{2}(I;H_{\sigma})\text{ and a.e. in }\Omega\times(0,T)$ (4.8)
$\displaystyle\theta_{\varepsilon}\rightharpoonup\widetilde{\theta}\text{ in
}L^{2}(I;V)$ (4.9)
$\displaystyle\theta_{\varepsilon}\operatorname{\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}}\widetilde{\theta}\text{
in }L^{\infty}(I;H)$ (4.10)
$\displaystyle\theta_{\varepsilon}\to\widetilde{\theta}\text{ in
}L^{2}(I;H)\text{ and a.e. in }\Omega\times(0,T)$ (4.11)

where
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
solves (4.1) in weak sense with
$[\widetilde{\varphi}(0),\widetilde{\operatorname{{\mathbf{u}}}}(0),\widetilde{\theta}(0)]=[\widetilde{\varphi}_{0},\widetilde{\operatorname{{\mathbf{u}}}}_{0},\widetilde{\theta}_{0}]$
and

$\displaystyle\widetilde{\varphi}\in W^{+\infty,4/d}(H;V^{*})\cap
L^{2}(I;V)\cap L^{2}(I;V_{2})$ (4.12)
$\displaystyle\widetilde{\operatorname{{\mathbf{u}}}}\in
W^{+\infty,4/d}(H_{\sigma};V_{\sigma}^{*})\cap L^{2}(I;V_{\sigma})$ (4.13)
$\displaystyle\widetilde{\theta}\in W^{+\infty,4/d}(H;V^{*})\cap L^{2}(I;V)$
(4.14)

where
$\widetilde{\mu}=-\Delta\widetilde{\varphi}+F^{\prime}(\widetilde{\varphi})+\ell_{c}\widetilde{\theta}\in
L^{2}(I;H)$. Furthermore, for any $t\in[0,T]$ the following energy inequality
holds

$\displaystyle\begin{aligned}
&\widetilde{\mathbb{E}}(\widetilde{\varphi}(t),\widetilde{\operatorname{{\mathbf{u}}}}(t),\widetilde{\theta}(t))+\int_{0}^{t}\mathbb{D}(\widetilde{\varphi}(s),\widetilde{\mu}(s),\widetilde{\operatorname{{\mathbf{u}}}}(s),\widetilde{\theta}(s))\operatorname{d\\!}s\\\
&\leq\widetilde{\mathbb{E}}(\widetilde{\varphi}_{0},\widetilde{\operatorname{{\mathbf{u}}}}_{0},\widetilde{\theta}_{0})+\int_{0}^{t}\big{\\{}\langle{\operatorname{{\mathbf{q}}}}(s),\widetilde{\operatorname{{\mathbf{u}}}}(s)\rangle_{V_{\sigma}}+(\ell(\widetilde{\varphi}(s),\widetilde{\theta}(s)){\operatorname{\mathbf{g}}},\widetilde{\operatorname{{\mathbf{u}}}}(s))_{\Omega}\\\
&\ \
+\langle{z}(s),\widetilde{\theta}(s)\rangle_{V}+({\operatorname{\mathbf{g}}}\cdot\widetilde{\operatorname{{\mathbf{u}}}}(s),\widetilde{\theta}(s))_{\Omega}\big{\\}}\operatorname{d\\!}s.\end{aligned}$
(4.15)

By saying
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
solves (4.1) in weak sense, we mean it to satisfy the variational problem

$\displaystyle\begin{aligned}
\langle\partial_{t}\widetilde{\varphi}(t),\psi\rangle_{V}+&\,(\widetilde{\operatorname{{\mathbf{u}}}}(t)\cdot\nabla\widetilde{\varphi}(t),\psi)_{\Omega}+(\nabla\widetilde{\mu}(t),\nabla\psi)_{\Omega}=0\end{aligned}$
(4.16a) $\displaystyle\begin{aligned}
&\langle\partial_{t}\widetilde{\operatorname{{\mathbf{u}}}}(t),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}+((\widetilde{\operatorname{{\mathbf{u}}}}(t)\cdot\nabla)\widetilde{\operatorname{{\mathbf{u}}}}(t),{\operatorname{\mathbf{v}}})_{\Omega}+2(\nu(\varphi)\mathrm{D}\widetilde{\operatorname{{\mathbf{u}}}}(t),\mathrm{D}{\operatorname{\mathbf{v}}})\\\
&=\mathcal{K}(\operatorname{\mathbf{v}}\cdot\nabla\widetilde{\varphi}(t),(\widetilde{\mu}(t)-\ell_{c}\widetilde{\theta}(t)))_{\Omega}+(\ell(\widetilde{\varphi}(t),\widetilde{\theta}(t)){\operatorname{\mathbf{g}}},{\operatorname{\mathbf{v}}})_{\Omega}+\langle{\operatorname{{\mathbf{q}}}}(t),{\operatorname{\mathbf{v}}}\rangle_{V_{\sigma}}\end{aligned}$
(4.16b) $\displaystyle\begin{aligned}
&\langle\partial_{t}\widetilde{\theta}(t),\vartheta\rangle_{V}-\ell_{h}\langle\partial_{t}\widetilde{\varphi}(t),\vartheta\rangle_{V}+\kappa(\nabla\widetilde{\theta}(t),\nabla\vartheta)\\\
&=(\widetilde{\operatorname{{\mathbf{u}}}}(t)\cdot\nabla(\ell_{h}\widetilde{\varphi}(t)-\widetilde{\theta}(t)),\vartheta)_{\Omega}+({\operatorname{\mathbf{g}}}\cdot\widetilde{\operatorname{{\mathbf{u}}}}(t),\vartheta)_{\Omega}+\langle
z(t),\vartheta\rangle_{V}\end{aligned}$ (4.16c)

for all $\psi\in V$, ${\operatorname{\mathbf{v}}}\in V_{\sigma}$,
$\vartheta\in V$ and almost every $t\in(0,T)$.

To proceed with the proof of the theorem above we shall need the following
Lemma which was proven in [17, Lemma 3.3].

######  Lemma 4.2.

If $\varphi_{1},\varphi_{2}\in V$ then

$\displaystyle\lim_{\varepsilon\to
0}\mathbb{E}^{\varepsilon}_{nl}(\varphi_{1})=\mathbb{E}_{l}(\varphi_{1}),$
(4.17) $\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}(a_{\varepsilon}(x)\varphi_{1}(x)-J_{\varepsilon}\ast\varphi_{1}(x))\varphi_{2}(x)\operatorname{d\\!}x=\int_{\Omega}\nabla\varphi_{1}(x)\cdot\nabla\varphi_{2}(x)\operatorname{d\\!}x.$
(4.18)

Furthermore, if $\\{\varphi_{\varepsilon}\\}\subset H$ is a sequence that
converges strongly to $\varphi\in H$ in $H$ then

$\displaystyle\mathbb{E}_{l}(\varphi)\leq\liminf_{\varepsilon\to
0}\mathbb{E}^{\varepsilon}_{nl}(\varphi_{\varepsilon}).$ (4.19)

We mention that due to the computations in the previous section, the proof of
the main theorem will be simplified. Mainly because the arguments for
establishing the uniform boundedness of the discretized solutions can be done
analogously for the solutions of the nonlocal system with the newly specified
convolution kernels. We are now in the position to prove the main result of
this section.

###### Proof of Theorem 4.1.

Since every solution
$[\varphi_{\varepsilon},{\operatorname{{\mathbf{u}}}}_{\varepsilon},\theta_{\varepsilon}]$
satisfies the energy inequality (3.7), following the same arguments to be able
to reach (3.20) we get

$\displaystyle\begin{aligned}
&\sup_{t\in[0,T]}\widehat{\mathbb{E}}(\varphi_{\varepsilon}(t),{\operatorname{{\mathbf{u}}}}_{\varepsilon}(t),\theta_{\varepsilon}(t))+\int_{0}^{T}\widehat{\mathbb{D}}(\mu_{\varepsilon}(s),{\operatorname{{\mathbf{u}}}}_{\varepsilon}(s),\theta_{\varepsilon}(s))\operatorname{d\\!}s\leq
M,\end{aligned}$

for some constant $M>0$ independent of $\varepsilon>0$. This provides uniform
boundedness of $\\{\varphi_{\varepsilon}\\}$ in $L^{\infty}(I;H)$ and
$L^{\infty}(I;H)$, $\\{{\operatorname{{\mathbf{u}}}}_{\varepsilon}\\}$ in
$L^{\infty}(I;H_{\sigma})\cap L^{2}(I;V_{\sigma})$ and
$\\{\nabla\mu_{\varepsilon}\\}$ in $L^{2}(I;L^{2}(\Omega)^{d})$. Testing
(3.1c) with $\vartheta=1$ and integrating over $(0,T)$ we get

$\displaystyle\left|\int_{\Omega}\theta_{\varepsilon}\operatorname{d\\!}x\right|\leq
c\left(\|{\operatorname{{\mathbf{u}}}_{\varepsilon}}\|_{L^{\infty}(I;H_{\sigma})}+\|z\|_{L^{2}(I;V^{*})}\right),$

and thus the uniform boundedness of $\\{\theta_{\varepsilon}\\}$ in
$L^{2}(I;V)$.

Following the same steps as in the proof of Theorem 3.3, we can show that
(3.18) holds but with the superscript $n$ replaced with the subscript
$\varepsilon$ and the superscripts on the external force
${\operatorname{{\mathbf{q}}}}^{n}$ and external heat source $z^{n}$ removed.
Taking the integral over $(0,T)$ on such estimate and using the assumption on
convergence of the nonlocal total energy to the local total energy evaluated
at the initial data shows that $\\{F(\varphi_{\varepsilon})\\}$ is uniformly
bounded in $L^{\infty}(I;L^{1}(\Omega))$. From assumption (A5), we thus get

$\displaystyle\begin{aligned}
\left|\int_{\Omega}\mu_{\varepsilon}\operatorname{d\\!}x\right|&=\left|\int_{\Omega}a_{\varepsilon}\varphi_{\varepsilon}-J\ast\varphi_{\varepsilon}+F^{\prime}(\varphi_{\varepsilon})+\ell_{c}\theta_{\varepsilon}\operatorname{d\\!}x\right|\\\
&\leq
c_{3}\int_{\Omega}|F(\varphi_{\varepsilon})|\operatorname{d\\!}x+c_{4}|\Omega|+\ell_{c}\sqrt{|\Omega|}\|\theta_{\varepsilon}\|.\end{aligned}$

which implies uniform boundedness of $\\{\mu_{\varepsilon}\\}$ in
$L^{2}(I;V)$.

From these, we establish the existence of
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
and $\widetilde{\mu}$ such that (4.3), (4.4), (4.6), (4.7), (4.9), (4.10) and
$\mu_{\varepsilon}\rightharpoonup\widetilde{\mu}$ in $L^{2}(I;V)$. Following
analogous arguments as in the well-posedness of the nonlocal system, we derive
uniform boundedness for the time derivatives of the variables
$[\varphi_{\varepsilon},{\operatorname{{\mathbf{u}}}}_{\varepsilon},\theta_{\varepsilon}]$,
i.e.

$\displaystyle\|\partial_{t}\varphi_{\varepsilon}\|_{L^{4/d}(I;V^{*})}+\|\partial_{t}{\operatorname{{\mathbf{u}}}}_{\varepsilon}\|_{L^{4/d}(I;V_{\sigma}^{*})}+\|\partial_{t}\theta_{\varepsilon}\|_{L^{4/d}(I;V^{*})}\leq
M.$ (4.20)

Due to Aubin-Lions-Simon we infer that
$W^{+\infty,4/d}(H_{\sigma},V^{*}_{\sigma})\hookrightarrow
C(\overline{I};H_{\sigma})$, $W^{+\infty,4/d}(H,V^{*})\hookrightarrow
C(\overline{I};H)$, $W^{2,4/d}(V_{\sigma},V^{*}_{\sigma})\hookrightarrow
L^{2}({I};H_{\sigma})$ and $W^{2,4/d}(V,V^{*})\hookrightarrow L^{2}({I};H)$
are compact which give us the strong and a.e. convergences (4.5), (4.8) and
(4.11). The convergences we have in hand, by passing through the limits, are
enough to establish that
$[\widetilde{\varphi},\widetilde{\operatorname{{\mathbf{u}}}},\widetilde{\theta}]$
satisfies (4.16a), (4.16b) and (4.16c).

Multiplying $G^{\prime}(\varphi_{\varepsilon})$ and $\mu_{\varepsilon}$, using
the second equation in (1.2a), integrating over $\Omega$, and rearranging the
resulting integral, one gets by virtue of Hölder and Young inequalities

$\displaystyle\begin{aligned}
&\int_{\Omega}|G^{\prime}(\varphi_{\varepsilon}(x))|^{2}\operatorname{d\\!}x+\frac{1}{2}\int_{\Omega}\int_{\Omega}J_{\varepsilon}(x-y)\delta
G^{\prime}_{\varphi}(x,y)(\varphi_{\varepsilon}(x)-\varphi_{\varepsilon}(y))\operatorname{d\\!}x\operatorname{d\\!}y\\\
&=\int_{\Omega}(\mu_{\varepsilon}(x)+a^{*}\varphi_{\varepsilon}(x)-\ell_{c}\theta_{\varepsilon}(x))G^{\prime}(\varphi_{\varepsilon}(x))\operatorname{d\\!}x\leq
M+\frac{1}{2}\int_{\Omega}|G^{\prime}(\varphi_{\varepsilon}(x))|^{2}\operatorname{d\\!}x,\end{aligned}$

where $M>0$ is a constant independent of $\varepsilon>0$ and $\delta
G^{\prime}_{\varphi}(x,y):=(G^{\prime}(\varphi_{\varepsilon}(x))-G^{\prime}(\varphi_{\varepsilon}(y)))$.
Since $G$ is a strictly convex function, we see the the second term on the
left-hand side above is nonnegative, and thus the uniform boundedness of
$\\{F^{\prime}(\varphi_{\varepsilon})\\}$ in $L^{2}(I;H)$. This implies the
existence of ${\operatorname{\mathbf{f}}}\in L^{2}(I;H)$ such the
$F^{\prime}(\varphi_{\varepsilon})\rightharpoonup{\operatorname{\mathbf{f}}}$
in $L^{2}(I;H)$. Furthermore, by virtue of Egorov’s theorem (see footnote in
[13, p. 1093]), the strong convergence (4.5) implies
$F^{\prime}(\varphi_{\varepsilon})\to F^{\prime}(\widetilde{\varphi})$ a.e.,
and hence in $L^{2}(I;L^{2}(\Omega))$. On the other hand, since
$\mu_{\varepsilon}$ and $\theta_{\varepsilon}$ are bounded in $L^{2}(I;H)$,
with independence on $\varepsilon>0$, we get uniform boundedness of
$a_{\varepsilon}(x)\varphi_{\varepsilon}-J_{\varepsilon}\ast\varphi_{\varepsilon}$
in $L^{2}(I;H)$ which implies the existence of ${\operatorname{\mathbf{g}}}\in
L^{2}(I;H)$ such that
$a_{\varepsilon}(x)\varphi_{\varepsilon}-J_{\varepsilon}\ast\varphi_{\varepsilon}\rightharpoonup{\operatorname{\mathbf{g}}}$
in $L^{2}(I;H)$.

Referring to [17, Lemma 3.2] $\mathbb{E}^{\varepsilon}_{nl}$ is Gâteaux
differentiable which can be computed for any $\varphi,\psi$ as

$\displaystyle\begin{aligned} \langle
D\mathbb{E}_{nl}^{\varepsilon}(\varphi),\psi\rangle&=\frac{1}{2}\int_{\Omega}\int_{\Omega}J_{\varepsilon}(x-y)(\varphi(x)-\varphi(y))(\psi(x)-\psi(y))\operatorname{d\\!}x\operatorname{d\\!}y\\\
&=\int_{\Omega}(a_{\varepsilon}(x)\varphi(x)-J_{\varepsilon}\ast\varphi(x))\psi(x)\operatorname{d\\!}x.\end{aligned}$

The second order Gâteaux derivative as a bilinear form in $H$ can also be
computed for any $\varphi,\psi_{1},\psi_{2}$ as

$\displaystyle\langle
D^{2}\mathbb{E}_{nl}^{\varepsilon}(\varphi),[\psi_{1},\psi_{2}]\rangle=\frac{1}{2}\int_{\Omega}\int_{\Omega}J_{\varepsilon}(x-y)(\psi_{1}(x)-\psi_{1}(y))(\psi_{2}(x)-\psi_{2}(y))\operatorname{d\\!}x\operatorname{d\\!}y.$

The nonegativity of $J_{\varepsilon}$ thus imply that $\langle
D^{2}\mathbb{E}_{nl}^{\varepsilon}(\varphi),[\psi,\psi]\rangle\geq 0$ for any
$\varphi,\psi\in H$. Letting $\psi\in L^{2}(I;V)$ such that
$\widehat{\psi}=m_{\Omega}$, we take the Taylor’s expansion of
$\mathbb{E}_{nl}^{\varepsilon}$ about $\varphi_{\varepsilon}$ so that

$\displaystyle\begin{aligned}
&\int_{0}^{T}\mathbb{E}_{nl}^{\varepsilon}(\psi(t))\operatorname{d\\!}t=\int_{0}^{T}\mathbb{E}_{nl}^{\varepsilon}(\varphi_{\varepsilon}(t))+\langle
D\mathbb{E}_{nl}^{\varepsilon}(\varphi_{\varepsilon}(t)),(\psi(t)-\varphi_{\varepsilon}(t))\rangle\\\
&\ \ +\frac{1}{2}\langle
D^{2}\mathbb{E}_{nl}^{\varepsilon}(\widetilde{\varphi}),(\psi(t)-\varphi_{\varepsilon}(t))^{2}\rangle\operatorname{d\\!}t\\\
&\geq\int_{0}^{T}\mathbb{E}_{nl}^{\varepsilon}(\varphi_{\varepsilon}(t))+(a_{\varepsilon}(x)\varphi_{\varepsilon}(t)-J_{\varepsilon}\ast\varphi_{\varepsilon}(t),\psi(t)-\varphi_{\varepsilon}(t))_{\Omega}\operatorname{d\\!}t.\end{aligned}$
(4.21)

Taking advantage of the convergences (4.17) and (4.19) in Lemma 4.2, Fatou’s
lemma, and the convergence $\varphi_{\varepsilon}\to\widetilde{\varphi}$ in
$C(\overline{I};H)$ we infer that

$\displaystyle\begin{aligned}
\int_{0}^{T}\mathbb{E}_{l}(\widetilde{\varphi}(t))+({\operatorname{\mathbf{g}}}(t),\psi(t)-\widetilde{\varphi}(t))_{\Omega}\operatorname{d\\!}t\leq\int_{0}^{T}\mathbb{E}_{l}(\psi(t))\operatorname{d\\!}t\end{aligned}$
(4.22)

This implies that
${\operatorname{\mathbf{g}}}\in\partial\mathbb{E}_{l}(\widetilde{\varphi})$,
from which we further infer — by additionally considering the
differentiability and convexity of $\mathbb{E}_{l}$ — that
$\langle{\operatorname{\mathbf{g}}},\psi\rangle=\langle
D\mathbb{E}_{l}(\widetilde{\varphi}),\psi\rangle$ for any $\psi\in
L^{2}(I;V)$, i.e.

$\displaystyle\int_{0}^{T}\int_{\Omega}{\operatorname{\mathbf{g}}}\psi\operatorname{d\\!}x\operatorname{d\\!}t=\int_{0}^{T}\int_{\Omega}\nabla\widetilde{\varphi}\cdot\nabla\psi\operatorname{d\\!}x\operatorname{d\\!}t,$
(4.23)

and we further conclude that
${\operatorname{\mathbf{g}}}=-\Delta\widetilde{\varphi}$ in $L^{2}(I;H)$ and
$\frac{\partial\widetilde{\varphi}}{\partial{\operatorname{\mathbf{n}}}}=0$
almost everywhere in $\Gamma$, for more details see [18, pages 142–143] for
example.

Summarizing the computations above, we were able to show that
$\widetilde{\mu}=-\Delta\widetilde{\varphi}+F^{\prime}(\widetilde{\varphi})+\ell_{c}\widetilde{\theta}$
in $L^{2}(I;H)$ and that $\widetilde{\varphi}\in L^{2}(I;V_{2})$.

Finally, passing to the limits would show that the equations (4.16a)–(4.16c)
hold. While, Lemma 4.2, the weak lower semicontinuity of norms, the assumption
on the energy convergence of the initial data, and the convergences we have in
hand show that (4.15) holds.

∎

## Acknowledgement

Š. N. and J.S.H.S have been supported by Præmium Academiæ of Š. Nečasová.
Moreover, Š. N. has been supported by by the Czech Science Foundation (GAČR)
through project GA22-01591S. The Institute of Mathematics CAS is supported by
RVO:67985840.

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