\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 41, pp. 1--29.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/41\hfil Thermoelastic Solid and Viscous Thermofluid]
{Generalized solutions to linearized equations of thermoelastic
solid and viscous thermofluid}

\author[A. M. Meirmanov, S. A. Sazhenkov\hfil EJDE-2007/41\hfilneg]
{Anvarbek M. Meirmanov, Sergey A. Sazhenkov}  % in alphabetical order

\address{Anvarbek M. Meirmanov \newline
Center for Advanced Mathematics and Physics \\
National University of Sciences and Technology \\
Peshawar Road, Rawalpindi 46000, Pakistan.\newline
Mathematical Department, Belgorod State University\\
 Ul. Pobedy 85, Belgorod 308015, Russia}
\email{anvarbek@list.ru}

\address{Sergey A. Sazhenkov \newline
Center for Advanced Mathematics and Physics \\
National University of Sciences and Technology \\
Peshawar Road, Rawalpindi 46000, Pakistan.\newline
Institute of Hydrodynamics \\
Siberian Division of Russian Academy of Sciences \\
Prosp. Lavrentieva 15, Novosibirsk 630090, Russia}
\email{sazhenkovs@yahoo.com}
\urladdr{http://sazhenkovs.narod.ru/Serezha.html}

\thanks{Submitted July 20, 2006. Published March 9, 2007.}
\subjclass[2000]{35D05, 35Q30, 74A10, 80A17, 93B18}
\keywords{Thermoelastic solid; viscous thermofluid;
compressibility; \hfill\break\indent
linearization; existence and uniqueness theory;
weak generalized solutions}

\begin{abstract}
Within the framework of continuum mechanics, the full description
of joint motion of elastic bodies and compressible viscous fluids
with taking into account thermal effects is given by the system
consisting of the mass, momentum, and energy balance equations,
the first and the second laws of thermodynamics, and an additional
set of thermomechanical state laws. The present paper is devoted
to the investigation of this system. Assuming that variations of
the physical characteristics of the thermomechanical system of the
fluid and the solid are small about some rest state, we derive the
linearized non-stationary dynamical model, prove its
well-posedness, establish additional refined global integral
bounds for solutions, and further deduce the linearized
incompressible models and models incorporating absolutely rigid
skeleton, as asymptotic limits.
\end{abstract}


\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction} \label{Introduction}

We are interested in proposing a mathematical description for
small perturbations in the thermomechanical system consisting of
interacting elastic solid and viscous compressible fluid. In our
study we aim to integrate purely mechanical, thermodynamical, and
heat transfer effects altogether under one umbrella. Within such
unified approach, a thermoconductive elastic body may be named a
\emph{thermoelastic solid}. Also we use the term
\emph{thermofluids}, which has been introduced quite recently
for a subject that analyzes systems and processes involved in
energy, various forms of energy, and transfer of energy in fluids
\cite[Sec. I.1]{MAS}.

The basic mathematical concept of reciprocal motion of
thermoelastic solids and viscous thermofluids incorporates the
classical conservation laws of continuum mechanics, the first and
the second laws of thermodynamics, and a set of state laws
specifying individual thermomechanical behavior of the components
of media. In this article this concept is called \emph{Model
O}$_1$ and is stated in the beginning of Sec. \ref{Basic}. It is
quite universal and spans a large variety of different phenomena
in nature and technology. (A rather general relevant observation
may be found, for example, in \cite[Secs. I.5--I.7, VI]{MAS}.) At
the same time this model is very complex and highly nonlinear.
Therefore, some physically reasonable simplifications are
necessary in view of further applications to natural problems and
in engineering.

In investigation of small perturbations it is suitable to simplify
\emph{Model O}$_1$ by implementing the classical formalism of
linearization about a rest state \cite[Sec. V.7]{GER}. As the
result, the linearized model arises, whose core consists of the
heat equation coupled with the non-stationary compressible Stokes
system in the fluid phase and the system of the wave equations (in
the elasticity theory also called Lam\'{e}'s equations) in the
solid phase. The linearization procedure and the precise
formulations of the resulting \emph{Model O}$_2$ and its
dimensionless version \emph{Model A} are outlined in Secs.
\ref{Basic} and \ref{Problem}.

Also in Sec. \ref{Problem} we introduce the notion of generalized
solutions of initial-boundary value problem for \emph{Model A}
and formulate the first main result of this article -- Theorem
\ref{theorem1} on existence and uniqueness of solutions to
\emph{Model A}. The proof of this theorem relies on classical
methods in the theory of generalized solutions of equations of
mathematical physics and is fulfilled in Secs. \ref{Energy} and
\ref{Galerkin}.

After this, we are interested in studying of the limiting regimes
in \emph{Model A}, arising as some coefficients grow infinitely.
To this end, in Secs. \ref{Pressures} and \ref{Deformation}
additional global integral bounds are established for the pressure
distributions (see Theorem \ref{proposition1e}) and for the
deformation tensor in the solid phase (see Theorem
\ref{proposition1f}). These bounds constitute the second main
result of this article. With their help, in Secs.
\ref{Incompressibility} and \ref{Solidification} \emph{Models
B1, B2}, and \emph{B3} of incompressible media and
\emph{Models C1} and \emph{C2} of viscous thermofluid
contained in an absolutely rigid heat-conducting skeleton are
established as respective incompressibility and solidification
asymptotic limits of \emph{Model A} (see Theorems
\ref{theorem4}, \ref{theorem5}, and \ref{theorem6}). These models
are the third main result of this article.

\section{Basic Nonlinear Formulation and Linearization} \label{Basic}

Let $\Omega$ be an open bounded set in $\mathbb{R}^3$ with a
smooth boundary. From now on we assume that $\Omega$ is the cube
with the side of a size $L_0$, i.e., $\Omega=(0,L_0)^3$. Next,
assume that at time $t=0$ $\Omega$ is occupied by a solid
component $\Omega_s$ and by a fluid component
$\Omega_f=\Omega\setminus \overline{\Omega}_s$ such that the interface
between the components $\Gamma:=\partial \Omega_f \cap
\partial \Omega_s$ is a rather smooth surface or a finite union
of such surfaces. The both solid and fluid phases obey the
fundamental conservation laws, which have the following forms in
Lagrangian variables $\mathbf{x}$ and $t$ \cite[Sec. V.5]{GER},
\cite[Sec. 10]{OVS}. The balance of mass equation is
\begin{equation} \label{(1a)}
\rho_0 =\rho J, \quad \mathbf{x} \mbox{ in } \Omega_f \mbox{ or }
\Omega_s,\quad t>0,
\end{equation}
the balance of momentum equation is
\begin{equation} \label{(2a)}
\rho_0 \frac{\partial^2 \mathbf{w}}{\partial t^2} = J\mathop{\rm div}_x (\mathbf{T} \mathbf{P}) - J
\mathbf{T} \mathbf{P} \mathbf{T}^{-1} \mathop{\rm div}_x (\mathbf{T}^t)^{-1} +\rho_0 \mathbf{F},\quad \mathbf{x} \mbox{ in
} \Omega_f \mbox{ or } \Omega_s,\quad t>0,
\end{equation}
and the balance of energy equation is
\begin{equation} \label{(3a)}
\begin{gathered}
\rho_0 \frac{\partial U}{\partial t} = -J\mathop{\rm div}_x (\mathbf{T}^{-1} \mathbf{
q})+ J\mathbf{q}\cdot \mathop{\rm div}_x (\mathbf{T}^t)^{-1} + J \mathbf{P} :
\frac{\partial \mathbb{E}}{\partial t} +\Psi,\\
\mathbf{q}= -\varkappa (\mathbf{T}^t)^{-1} \nabla_x \vartheta, \quad \mathbf{x} \mbox{ in }
\Omega_f \mbox{ or } \Omega_s,\quad t>0,
\end{gathered}
\end{equation}
where
\begin{equation} \label{(4a)}
\mathbf{T}=\nabla_x \mathbf{r}, \quad \mathbf{r}-\mathbf{x}=\mathbf{w}, \quad J=\det
\mathbf{T}, \quad 2\mathbb{E}= \mathbf{T}^t \mathbf{T}-\mathbf{I}.
\end{equation}

The thermodynamical state of the both phases is governed by the
first law of thermodynamics
\begin{equation} \label{(5a)}
\vartheta ds= dU +pd\frac{1}{\rho}
\end{equation}
and a thermodynamical state equation
\begin{equation} \label{(6a)}
U-\vartheta s=\mathcal{F}(\rho,\vartheta).
\end{equation}

We postulate that thermomechanical behavior in the solid phase is
described by the Duhamel--Neumann law of linear thermoelasticity,
which is consistent with thermodynamical relations \eqref{(5a)}
and \eqref{(6a)} \cite[Sec. 1.6]{LL7}, \cite[Sec. 10]{OVS}:
\begin{equation}  \label{(7a)}
\begin{gathered}
\mathbf{P}:= \mathbf{P}_s =\Big(-p_0 -\eta \gamma_s (\vartheta
-\vartheta_*) +(\eta -\frac{2}{3}\lambda) \mathop{\rm div}_x
\mathbf{w}\Big) \mathbf{I} +2\lambda \mathbf{D}(x,\mathbf{w}), \\
\mathbf{x}\in \Omega_s,\quad t>0,
\end{gathered}
\end{equation}
and that thermomechanical behavior in the fluid phase is described
by the Stokes state equation, which has the following form in
Lagrangian variables \cite[Sec. V.5]{GER}, \cite[Sec. 6]{OVS}:
\begin{equation} \label{(8a)}
\begin{gathered}
\mathbf{P}:= \mathbf{P}_f =\Bigl(-p +\Bigl(\nu
-\frac{2}{3}\mu\Bigr) \mbox{tr\,} \Bigl((\mathbf{T}^t)^{-1}
\frac{\partial \mathbb{E}}{\partial t} \mathbf{T}^{-1}\Bigr)\Bigr) \mathbf{I} +
2\mu \Bigl((\mathbf{T}^t)^{-1} \frac{\partial \mathbb{E}}{\partial t}
\mathbf{T}^{-1}\Bigr), \\ \mathbf{x}\in \Omega_f, \quad t>0.
\end{gathered}
\end{equation}
In  \eqref{(1a)}--\eqref{(8a)}, $\mathbf{w}$, $\vartheta$, $\rho$, $p$,
$U$, and $s$ are unknown displacement, temperature, density,
pressure, specific intrinsic energy, and entropy, respectively.
The medium under consideration is a two-parameter thermomechanical
system. Further we refer $\vartheta$ and $\rho$ to as the
independent thermodynamical parameters. The triple $\mathbf{
r}=(r_1,r_2,r_3)$ is the set of spatial Eulerian coordinates of
particles of solid or fluid. The vector $\mathbf{q}$ is the heat
flux. The second equation in \eqref{(3a)} is the Fourier law for
the heat flux.

$\mathbf{T}$ is the distortion tensor. $\mathbf{T}^t$, $\mathbf{T}^{-1}$, and
$(\mathbf{T}^t)^{-1}$ are its conjugate, inverse and inverse conjugate,
respectively. $\mathbb{E}$ is the deformation tensor. It is
connected with the displacement vector by the formula
$$
2\mathbb{E} =(\nabla_x \mathbf{w})^t +\nabla_x \mathbf{w} +(\nabla_x \mathbf{w})^t
\nabla_x \mathbf{w}.
$$
$\mathbf{D}(x,\mathbf{w})$ is the symmetric part of the gradient $\nabla_x \mathbf{w}$,
i.e.,
$$
2\mathbf{D}(x,\mathbf{w}) =(\nabla_x \mathbf{w})^t +\nabla_x \mathbf{w}.
$$
$\mathbf{P}$ is the stress tensor. Note that, due to the identity
\cite[Sec. V.4]{GER}, \cite[Sec. 6]{OVS}
$$
\frac{\partial \mathbb{E}}{\partial t} =\mathbf{T}^t \mathbf{D}(r,\mathbf{v}) \mathbf{T},
$$
where $\mathbf{D}(r,\mathbf{v})=(1/2)((\nabla_r \mathbf{v})^t +\nabla_r \mathbf{v})$ is the
deformation rate tensor in Eulerian variables $\mathbf r$ and $t$,
and $\mathbf{v}$ is the velocity of particles of fluid or solid,
\eqref{(8a)} reduces in Eulerian variables to the well-known form
\begin{equation*}
\mathbf{P}_f =\Bigl(-p +\Bigl(\nu -\frac{2}{3}\mu\Bigr) \mathop{\rm div}_r \mathbf{v}\Bigr)
\mathbf{I} +2\mu \mathbf{D}(r,\mathbf{v}), \quad \mathbf{r}\in \Omega_f(t), \quad t>0.
\end{equation*}

Coefficients $\varkappa$, $\nu$, $\mu$, $\eta$, $\lambda$, and
$\gamma_s$ are given. They are a thermal conductivity, bulk and
shear viscosity coefficients of the fluid, bulk and shear elastic
modules of the solid, and a thermal extension of the solid,
respectively. In general, they may depend on the thermodynamical
parameters $\vartheta$ and $\rho$. In line with the thermodynamics
fundamentals, we have \cite[Chap. 5]{LL6}, \cite[Chap. 1]{LL7}
\begin{equation} \label{(9a)}
\varkappa, \nu, \mu, \eta, \lambda, \gamma_s >0, \quad
\nu>\frac{2}{3}\mu,\quad \eta>\frac{2}{3}\lambda.
\end{equation}

Constant positive coefficients $\vartheta_*$ and $p_0$ are given
temperature of some rest state and atmosphere pressure,
respectively.

Functions $\rho_0$, $\mathbf{F}$, $\Psi$, and $\mathcal{F}$ are given.
They are an initial distribution of density, a density of mass
distributed forces, a volumetric density of exterior heat
application, and a specific free energy, respectively.
Thermodynamical behavior in the solid and the fluid is different,
in general. Therefore we are given, in fact, two distinct
functions $\mathcal{F}_s$ and $\mathcal{F}_f$ such that
\begin{equation*}
\mathcal{F}(\rho,\vartheta)=\begin{cases} \mathcal{
F}_f(\rho,\vartheta) &\mbox{if }  \mathbf{x}\in \Omega_f,\quad t\geq 0,\\
\mathcal{F}_s(\rho,\vartheta) &\mbox{if }  \mathbf{x}\in \Omega_s,\quad
t\geq 0
\end{cases}
\end{equation*}
and two distinct functions $\Psi_s$ and $\Psi_f$ such that
\begin{equation*}
\Psi(\mathbf{x},t)=\begin{cases} \Psi_f(\mathbf{x},t) &\mbox{if }
\mathbf{x}\in \Omega_f,\quad t\geq 0,\\
\Psi_s(\mathbf{x},t) &\mbox{if }  \mathbf{x}\in \Omega_s,\quad t\geq 0.
\end{cases}
\end{equation*}

Similarly, we have two distinct coefficients of
thermoconductivity, $\varkappa_s$ in the solid phase and
$\varkappa_f$ in the liquid phase.

In terms of the indicator function of the fluid phase
\begin{equation} \label{(10a)}
\bar{\chi}(\mathbf{x})=\begin{cases} 1 &\mbox{if } \mathbf{x}\in \Omega_f,\\
0 &\mbox{if } \mathbf{x} \in \Omega_s, \end{cases}
\end{equation}
we may write
\begin{gather} \label{(11a)}
 \mathcal{F}(\rho,\vartheta)=\bar{\chi} \mathcal{F}_f
(\rho,\vartheta) +(1-\bar{\chi}) \mathcal{F}_s(\rho,\vartheta)
\quad
\forall\, \mathbf{x}\in \Omega,\quad t\geq 0,\\
\label{(11a-2)}  \Psi(\mathbf{x},t)=\bar{\chi} \Psi_f
(\mathbf{x},t)+(1-\bar{\chi}) \Psi_s (\mathbf{x},t) \quad
\forall\, \mathbf{x}\in \Omega,\quad t\geq 0,\\
 \label{(11a-3)}
\bar{\varkappa}(\rho,\vartheta)=\bar{\chi} \varkappa_f
(\rho,\vartheta)+(1-\bar{\chi}) \varkappa_s (\rho,\vartheta) \quad
\forall\, \mathbf{x}\in \Omega,\quad t\geq 0.
\end{gather}
Analogously,
\begin{equation} \label{(12a)}
\mathbf{P}=\bar{\chi}\mathbf{P}_f +(1-\bar{\chi})\mathbf{P}_s \quad \forall\, \mathbf{x}\in
\Omega,\quad t\geq 0.
\end{equation}

Interactions between the fluid and the solid are governed by the
classical conditions on discontinuity surfaces
\cite[Sec. II.3]{GER}, \cite[Sec. 12]{OVS}, \cite[Sec. 2]{VVP}. In order to
state these conditions, we introduce the following notation. For
any $\mathbf{x}_0\in \Gamma$ and for any function $\varphi(\mathbf{x})$,
continuous in the interior of $\Omega_s$ and in the interior of
$\Omega_f$, denote
\begin{equation} \label{(jump)}
\begin{gathered}
 \varphi_{(s)}(\mathbf{x}_0) =\lim_{\mathbf{x}\to \mathbf{x}_0,\;
\mathbf{x}\in \Omega_s} \varphi(\mathbf{x}),\quad
\varphi_{(f)}(\mathbf{x}_0) =\lim_{\mathbf{x}\to \mathbf{x}_0,\;
\mathbf{x}\in \Omega_f} \varphi(\mathbf{x}),\\
  [\varphi](\mathbf{x}_0)=\varphi_{(s)}(\mathbf{x}_0)
-\varphi_{(f)}(\mathbf{x}_0).
\end{gathered}
\end{equation}

\begin{remark} \label{remark1a} \rm
Clearly, $\Gamma$ is immovable in Lagrangian coordinates in the
sense that its any parametrization in Lagrangian variables does
not depend on $t$.
\end{remark}

Thus, in terms of notation \eqref{(jump)} we write down the
conditions on $\Gamma$ as follows: the continuity of temperature
\begin{equation} \label{(13a)}
[\vartheta]=0, \quad \mathbf{x}_0\in \Gamma,\; t\geq 0,
\end{equation}
the continuity of displacement
\begin{equation} \label{(14a)}
[\mathbf{w}]=0,\quad \mathbf{x}_0\in \Gamma,\; t\geq 0,
\end{equation}
the continuity of normal stress
\begin{equation} \label{(15a)}
[\mathbf{P}\mathbf{n}]=0,\quad \mathbf{x}_0\in \Gamma,\; t\geq 0,
\end{equation}
and the continuity of the normal heat flux
\begin{equation} \label{(16a)}
[\varkappa (\mathbf{T}^t)^{-1} \nabla_x \vartheta \cdot \mathbf{n}]=0,
\quad \mathbf{x}_0\in \Gamma,\quad t\geq 0.
\end{equation}

In  \eqref{(15a)}--\eqref{(16a)} $\mathbf{n}(\mathbf{x}_0)$ is the
unit normal to $\Gamma$ at a point $\mathbf{x}_0 \in \Gamma$. We suppose
that $\mathbf{n}$ is pointing into $\Omega_f$.

The conditions of continuity of temperature and displacement on
$\Gamma$ reflect the local thermodynamical equilibrium and no-slip
effect on the interface, respectively. Besides, the no-slip
condition \eqref{(14a)} includes the mass conservation law,
applied to $\Gamma$, and manifests that $\Gamma$ is a contact
discontinuity, which means that the solid and the fluid do not
exchange particles. The conditions of continuity of normal stress
and normal heat flux are the respective momentum and energy
conservation laws on $\Gamma$.

Equations \eqref{(1a)}--\eqref{(8a)} and \eqref{(13a)}--\eqref{(16a)}
constitute the closed nonlinear model of joint motion of
thermoelastic solid and viscous thermofluid. By the standard
procedure \cite[Sec 8.1]{GER}, thermodynamical parameters $s$,
$p$, and $U$ may be expressed in terms of $\rho$ and $\vartheta$
from \eqref{(5a)} and \eqref{(6a)} by virtue of the free
energy $\mathcal{F}$:
\begin{equation} \label{(17a)}
s=-\mathcal{F}'_\vartheta (\rho,\vartheta), \quad p=\rho^2
\mathcal{F}'_\rho (\rho,\vartheta), \quad U=-\vartheta \mathcal{
F}'_\vartheta(\rho, \vartheta) +\mathcal{F}(\rho,\vartheta).
\end{equation}

Insert \eqref{(17a)} into \eqref{(3a)} and \eqref{(8a)},
and discard \eqref{(5a)} and \eqref{(6a)} to get the
equivalent system of equations and interface conditions, which we
name \emph{Model O}$_1$.

Now simplify \emph{Model O}$_1$, applying the classical formal
procedure of linearization \cite[Sec. V.7]{GER},
 \cite[Sec. 2.2--2.3]{TEM} to it. Assume that $\rho_s$ and $\rho_f$ are mean
constant densities of the solid and the liquid at rest. Suppose
that the temperature in this rest state is equal to the constant
temperature $\vartheta_*$, which was introduced in
\eqref{(7a)}. Denote
\begin{equation}
\begin{gathered}
c_s^*= \mathcal{F}_s (\rho_s, \vartheta_*),  \quad
c_{s\rho}= \mathcal{F}'_{s\rho}(\rho_s, \vartheta_*), \quad
c_{s\vartheta}= \mathcal{F}'_{s\vartheta} (\rho_s,\vartheta_*),\\
c_{s\rho \rho} =\mathcal{F}''_{s\rho\rho}(\rho_s,\vartheta_*), \quad
c_{s\rho\vartheta}= \mathcal{F}''_{s\rho\vartheta}(\rho_s,\vartheta_*), \quad
c_{s\vartheta\vartheta}= \mathcal{F}''_{s\vartheta\vartheta}(\rho_s,\vartheta_*),
\\
c_f^*= \mathcal{F}_f (\rho_f, \vartheta_*), \quad
c_{f\rho}= \mathcal{F}'_{f\rho}(\rho_f, \vartheta_*), \quad
c_{f\vartheta}=\mathcal{F}'_{f\vartheta} (\rho_f,\vartheta_*),\\
c_{f\rho \rho} =\mathcal{F}''_{f\rho\rho} (\rho_f,\vartheta_*), \quad
c_{f\rho\vartheta}= \mathcal{F}''_{f\rho\vartheta}(\rho_f,\vartheta_*), \quad
c_{f\vartheta\vartheta}= \mathcal{F}''_{f\vartheta\vartheta}(\rho_f,\vartheta_*).
\end{gathered}\label{(18a)}
\end{equation}
Expand \eqref{(17a)} for $p$ and $U$ in Taylor's series and
skip terms of orders higher than one to get the linearized
expressions
\begin{gather} \label{(19a)}
 p(\rho,\vartheta) =c_{f\rho} \rho_f^2 +(2 c_{f\rho} \rho_f +
c_{f\rho\rho} \rho_f^2)(\rho-\rho_f) + c_{f\rho \vartheta}
\rho_f^2 (\vartheta
-\vartheta_*),\\
 U_s(\rho,\vartheta) =-c_{s\vartheta} \vartheta_* + c_s^* +
(c_{s\rho} - c_{s\rho \vartheta} \vartheta_*)(\rho-\rho_s)
-c_{s\vartheta \vartheta} \vartheta_* (\vartheta -\vartheta_*),
\label{(20a)}\\
\label{(21a)}  U_f(\rho,\vartheta) =-c_{f\vartheta} \vartheta_* +
c_f^* +(c_{f\rho} - c_{f\rho \vartheta} \vartheta_*) (\rho-\rho_f)
-c_{f\vartheta \vartheta} \vartheta_* (\vartheta -\vartheta_*).
\end{gather}

\begin{remark} \label{remark2a} \rm
In \eqref{(19a)} and
\eqref{(21a)} notice that $p(\rho_f, \vartheta_*)=p_0$ due to
interface condition \eqref{(15a)}. Also notice that from the
physical point of view it is observed for the most of continuous
media that $p(\rho,\vartheta)$ is increasing in $\rho$ and
$\vartheta$ and that $U_f(\rho,\vartheta)$ and
$U_s(\rho,\vartheta)$ are increasing in $\vartheta$ and decreasing
in $\rho$. Thus $c_{f\rho\vartheta}>0$, $2 c_{f\rho} +
c_{f\rho\rho}\rho_f >0$, $c_{s\vartheta\vartheta}<0$, $c_{s\rho}
-c_{s\rho \vartheta}\vartheta_*<0$, $c_{f\vartheta\vartheta}<0$,
and $c_{f\rho} -c_{f\rho\vartheta}\vartheta<0$.
\end{remark}

Substituting \eqref{(19a)}--\eqref{(21a)} into \emph{Model
O}$_1$, expanding the terms of thus obtained equations in Taylor's
series in $\rho$, $\vartheta$, $\mathbf{w}$, and derivatives of $\mathbf{w}$ with
respect to $\mathbf{x}$ and $t$ about the rest state, and discarding the
terms of orders higher than one, we arrive at the following
linearized dynamical model of the system \emph{thermoelastic
body -- compressible viscous thermofluid}, which we call further
\emph{Model O}$_2$.

\noindent {\bf \emph{Statement of Model O}$_2$.} The motion in
the solid phase is governed by the classical equations of the
linear thermoelasticity theory:
\begin{gather} \label{(22a)}
  \rho_s \frac{\partial^2 \mathbf{w}}{\partial t^2} =\mathop{\rm div}_x
\mathbf{P}_s +\rho_s \mathbf{F},\quad \mathbf{x}\in \Omega_s,\; t>0,\\
 \label{(23a)}
  -\rho_s c_{s \vartheta \vartheta} \vartheta_*
\frac{\partial \vartheta}{\partial t} =\varkappa_s \Delta_x
\vartheta -\vartheta_* \gamma_s \eta \frac{\partial}{\partial t}
\mathop{\rm div}_x \mathbf{w} +\Psi_s,\quad \mathbf{x}\in \Omega_s,\; t>0,\\
   \mathbf{P}_s= \Bigl(-p_0 -\gamma_s \eta
(\vartheta-\vartheta_*) +\Bigl(\eta-\frac{2}{3}\lambda\Bigr)
\mathop{\rm div}_x \mathbf{w}\Bigr) \mathbf{I} +2\lambda \mathbf{D}(x,\mathbf{w}),\quad \mathbf{x}\in \Omega_s,\;
t>0,\\
\label{(24a)}\\
\label{(25a)}   \rho=\rho_s (1-\mathop{\rm div}_x \mathbf{w}), \quad
\mathbf{x}\in \Omega_s,\; t>0.
\end{gather}

The motion of the liquid phase is described by the linearized
classical model of liquid and gases:
\begin{gather} \label{(26a)}
  \rho_f \frac{\partial^2 \mathbf{w}}{\partial t^2}= \mathop{\rm div}_x
\mathbf{P}_f +\rho_f \mathbf{F}, \quad \mathbf{x}\in \Omega_f,\; t>0,
\\
 \label{(27a)}
  -\rho_f c_{f \vartheta\vartheta} \vartheta_*
\frac{\partial \vartheta}{\partial t} = \varkappa_f \Delta_x
\vartheta -\vartheta_* c_{f\rho\vartheta} \rho_f^2
\frac{\partial}{\partial t} \mathop{\rm div}_x \mathbf{w} +\Psi_f,\quad \mathbf{x}\in
\Omega_f,\; t>0,
\\
 \label{(28a)} \begin{aligned}
\mathbf{P}_f &=\Bigl(-p_0 -c_{f\rho\vartheta}\rho_f^2
(\vartheta-\vartheta_*) +
(2 c_{f\rho} +c_{f\rho\rho} \rho_f)\rho_f^2 \mathop{\rm div}_x \mathbf{w}\\
&\quad +\bigl(\nu- \frac{2}{3}\mu\bigr) \mathop{\rm div}_x \frac{\partial
\mathbf{w}}{\partial t}\Bigr) \mathbf{I} + 2\mu \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}\Bigr), \quad \mathbf{x}\in \Omega_f,\; t>0,
\end{aligned}
\\
 \label{(29a)}
\rho=\rho_f (1-\mathop{\rm div}_x \mathbf{w}), \quad \mathbf{x}\in \Omega_f,\; t>0.
\end{gather}

The linearized conditions on the interface between the solid and
the fluid are
\begin{gather} \label{(30a)}
[\vartheta]=0, \quad \mathbf{x}_0\in \Gamma,\; t\geq 0,\\
 \label{(31a)}
 [\mathbf{w}]=0,\quad \mathbf{x}_0\in \Gamma,\; t\geq 0,\\
 \label{(32a)}
 \bigl(\mathbf{P}_s(\vartheta_{(s)},\mathbf{w}_{(s)})-
\mathbf{P}_f(\vartheta_{(f)},\mathbf{w}_{(f)})\bigr) \mathbf{n} =0, \quad
\mathbf{x}_0\in \Gamma,\; t\geq 0,\\
 \label{(33a)}
 \bigl(\varkappa_s \nabla_x \vartheta_{(s)} -\varkappa_f \nabla_x
\vartheta_{(f)}\bigr) \cdot \mathbf{n}=0, \quad \mathbf{x}_0\in \Gamma,\;
t\geq 0.
\end{gather}

\begin{remark} \label{remark3a} \rm
Transforming \emph{Model O}$_1$ to
Euler variables $(\mathbf{r},t)$ and applying the same
linearization formalism as above, one arrives exactly at
\emph{Model O}$_2$. Therefore in linearized setting the Lagrange
and Euler descriptions of the thermomechanical system under
consideration coincide.
\end{remark}

\section{Problem Formulation and Statement of the Existence and
 Uniqueness Theorem} \label{Problem}

In line with later asymptotic analysis in the present and
forthcoming papers, we bring \emph{Model O}$_2$ to a
dimensionless form and absorb the interface conditions on $\Gamma$
in the equations by introducing a uniform description of the both
phases.

More precisely, choose the diameter $L_0$ of the domain $\Omega$,
a characteristic duration of physical processes $\tau_0$,
acceleration of gravity $g$, atmosphere pressure $p_0$, mean
density of air $\rho_0$  at the temperature 273 $K$ under
atmosphere pressure, and the temperature difference $\vartheta_0$
between the thawing and freezing points of water under atmosphere
pressure as characteristic scales of length, time, density of mass
distributed forces, pressure, density of matter, and temperature,
respectively, and denote $\theta=\vartheta-\vartheta_*$.

Next, introduce the dimensionless variables (with primes) by the
formulas
\begin{equation}  \label{(34a)}
\begin{gathered}
 \mathbf{x}=L_0 \mathbf{x}',\quad t=\tau_0 t',\quad \mathbf{w}=L_0 \mathbf{w}',\\
 p=p_0 p',\quad \rho=\rho_0 \rho',\quad
\theta=\vartheta_0 \theta',
\end{gathered}
\end{equation}
the dimensionless vector of distributed mass forces, volumetric
densities of exterior heat application, thermal conductivity
coefficients, and mean densities of the solid and the fluid at
rest (all with primes), respectively, by the formulas
\begin{gather}
\label{(35a)}  \mathbf{F}=g\mathbf{F}',\quad \Psi_s = \frac{p_0
\vartheta_*}{\tau_0 \vartheta_0} \Psi_s',\quad \Psi_f = \frac{p_0
\vartheta_*}{\tau_0 \vartheta_0} \Psi_f',\\
\label{(35a-2)}  \varkappa_s =\frac{L_0^2 p_0
\vartheta_*}{\tau_0 \vartheta_0^2} \varkappa_s',\quad \varkappa_f
= \frac{L_0^2 p_0 \vartheta_*}{\tau_0 \vartheta_0^2}
\varkappa_f',\quad \rho_s= \rho_0 \rho_s',\quad \rho_f =\rho_0
\rho_f',
\end{gather}
and the dimensionless ratios by the formulas
\begin{equation} \label{(36a)}
 \begin{gathered}   \alpha_\tau
=\frac{\gamma_0 L_0^2}{c_0^2 \tau_0^2}, \quad
\alpha_F=\frac{\gamma_0 g L_0}{c_0^2}, \quad
\alpha_\nu=\frac{1}{\tau_0 p_0}
\bigl(\nu -\frac{2}{3}\mu\bigr),\\
\alpha_\eta =\frac{1}{p_0}\Bigl(\eta-\frac{2}{3}\lambda\Bigr), \quad
 \alpha_\lambda =\frac{2\lambda}{p_0}, \quad
 \alpha_p
=\frac{\gamma_0 c^2}{c_0^2}\rho_f',\\
 c_0^2=\frac{\gamma_0 p_0}{\rho_0}, \quad
\alpha_{\theta s}=\frac{\gamma_s \eta \vartheta_0}{p_0}, \quad
 \alpha_{\theta f}= \frac{c_{f\rho \vartheta}
\rho_f^2
\vartheta_0}{p_0},\\
 \alpha_\mu =\frac{2\mu}{\tau_0 p_0}, \quad
c_{pf}=-\frac{c_{f\vartheta \vartheta} \rho_f \vartheta_0^2}{p_0}, \quad
 c_{ps}=-\frac{c_{s\vartheta\vartheta} \rho_s
\vartheta_0^2}{p_0},
\end{gathered}
\end{equation}
where $c=\sqrt{2 c_{f\rho} \rho_f +c_{f\rho\rho} \rho_f^2}$ and
$\gamma_0=7/5$. Quantity $c_0$ is the speed of sound in air at the
temperature 273 $K$ under atmosphere pressure. Quantity $c$ is the
speed of sound in the considered fluid at the temperature
$\vartheta_*$ under atmosphere pressure. Dimensionless constant
$\gamma_0$ is the ratio of specific heats (in other terms, the
polytropic exponent) for air at the temperature 273 $K$ under
atmosphere pressure \cite[Appendix 4]{MAS}, \cite[Sec. 2.4]{TEM}.

\begin{remark} \label{remark4a} \rm
On the strength of \eqref{(9a)} and Remark \ref{remark2a},
all dimensionless constants on the left-hand sides of relations
\eqref{(36a)} are positive.
\end{remark}

Now, first shift the pressure scale on the constant value $p_0$ so
that the stress tensors \eqref{(24a)} and \eqref{(28a)} become
vanishing at the rest state. Next, multiply \eqref{(22a)} and
\eqref{(26a)} by $L/p_0$, \eqref{(23a)} and \eqref{(27a)} by
$\tau_0 \vartheta_0/(p_0\vartheta_*)$, and divide
\eqref{(24a)} and \eqref{(28a)} by $p_0$. After this, substitute
expressions \eqref{(34a)}--\eqref{(36a)} into the resulting
equations and then omit primes. Thus, \emph{Model O}$_2$ is
brought to a dimensionless form. Finally, use the notation
\eqref{(10a)}--\eqref{(12a)} and additionally set
\begin{equation} \label{(37a)}
\begin{gathered}
 \bar{\rho} =\bar{\chi} \rho_f +(1-\bar{\chi}) \rho_s,\quad
\bar{\alpha}_\theta
=\bar{\chi} \alpha_{\theta f} +(1-\bar{\chi})\alpha_{\theta s},\\
  \bar{c}_p=\bar{\chi} c_{pf} +(1-\bar{\chi})
c_{ps}, \quad \bar{\varkappa} =\bar{\chi} \varkappa_f
+(1-\bar{\chi}) \varkappa_s
\end{gathered}
\end{equation}
and introduce the dimensionless pressures $p$, $q$, and $\pi$ in
order to wrap the dimensionless \emph{Model O}$_2$ into the
following form.

\noindent{\bf \emph{Statement of Model A.}} In the space-time cylinder
$Q=\Omega\times (0,T)$, where $\Omega=(0,1)^3$ and
$T=\mbox{const}>0$, it is necessary to find a displacement vector
$\mathbf{w}$, a temperature distribution $\theta$, and distributions of
pressures $p$, $q$, and $\pi$, which satisfy the equations
\begin{gather} \label{(38a)}
  \alpha_\tau \bar{\rho} \frac{\partial^2
\mathbf{w}}{\partial t^2}=\mathop{\rm div}_x \mathbf{P} + \alpha_F \bar{\rho} \mathbf{F},\\
 \label{(39a)}
  \mathbf{P} =  \bar{\chi}\Bigl(-q\mathbf{I} + \alpha_\mu
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr)\Bigr) +(1-\bar{\chi})\bigl(-\pi \mathbf{I}
+\alpha_\lambda \mathbf{D}(x,\mathbf{w})\bigr)
-\bar{\alpha}_\theta \theta \mathbf{I},\\
 \label{(40a)}
  \bar{c}_p \frac{\partial \theta}{\partial t}
=\mathop{\rm div}_x (\bar{\varkappa} \nabla_x \theta) -\bar{\alpha}_\theta
\frac{\partial}{\partial t} \mathop{\rm div}_x \mathbf{w} +\Psi,\\
 \label{(41a)}
  p+\bar{\chi} \alpha_p \mathop{\rm div}_x \mathbf{w}=0,\\
 \label{(42a)}
  q=p+\frac{\alpha_\nu}{\alpha_p}\frac{\partial
p}{\partial t},\\
 \label{(43a)}
  \pi +(1-\bar{\chi}) \alpha_\eta \mathop{\rm div}_x \mathbf{w}=0.
\end{gather}
We endow \emph{Model A} with initial data
\begin{equation} \label{(44a)}
\mathbf{w}|_{t=0}=\mathbf{w}_0,\quad \mathbf{w}_t|_{t=0}=\mathbf{v}_0,\quad \theta|_{t=0}
=\theta_0,\quad \mathbf{x}\in \Omega
\end{equation}
and homogeneous boundary conditions
\begin{equation} \label{(45a)}
\mathbf{w}=0,\quad \theta=0,\quad \mathbf{x} \in \partial \Omega, \quad t\geq 0.
\end{equation}

\begin{remark} \label{remark5a} \rm
On the strength of classical theory of conservation laws of
continuum mechanics \cite[Sec. II.3]{GER},
\eqref{(38a)}--\eqref{(43a)} yield interface conditions
\eqref{(30a)}--\eqref{(33a)} (in the dimensionless form). Thus
\eqref{(38a)}--\eqref{(43a)} are equivalent to \emph{Model
O}$_2$.
\end{remark}

Generalized solutions of \emph{Model A} are understood in the
following sense.

\begin{definition} \label{definition1a}
Five functions $(\mathbf{w},\theta,p,q,\pi)$ are called a generalized
solution of \emph{Model A} if they satisfy the regularity
conditions
\begin{equation} \label{(46a)}
\mathbf{w},\,\frac{\partial \mathbf{w}}{\partial t},\, \mathbf{D}(x,\mathbf{w}),\, \bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr),\,\mathop{\rm div}_x\mathbf{w},\,
\theta,\, \nabla_x \theta \in L^2(Q),
\end{equation}
boundary conditions \eqref{(45a)} in the trace sense, equations
\eqref{(41a)}--\eqref{(43a)} a.e. in $Q$, and the integral
equalities
\begin{equation} \label{(47a)}
\int_Q \Bigl(\alpha_\tau \bar{\rho} \frac{\partial \mathbf{w}}{\partial
t}\cdot \frac{\partial \mathbf{\varphi}}{\partial t}
-\mathbf{P}:\nabla_x \mathbf{\varphi} + \alpha_F \bar{\rho} \mathbf{F}\cdot
\mathbf{\varphi}\Bigr) d\mathbf{x} dt +\int_\Omega \alpha_\tau \bar{\rho}
\mathbf{v}_0 \cdot \mathbf{\varphi}|_{t=0} d\mathbf{x} =0
\end{equation}
for all smooth $\mathbf{\varphi}=\mathbf{\varphi}(\mathbf{x},t)$ such
that $\mathbf{\varphi}|_{\partial \Omega} =\mathbf{
\varphi}|_{t=T}=0$ and
\begin{equation} \label{(48a)}
\begin{aligned}
&\int_Q \Bigl(\bar{c}_p \theta \frac{\partial \psi}{\partial t} -
\bar{\varkappa} \nabla_x \theta \cdot \nabla_x \psi
+\bar{\alpha}_\theta (\mathop{\rm div}_x \mathbf{w}) \frac{\partial
\psi}{\partial t} +\Psi \psi\Bigr) d\mathbf{x} dt\\
&+\int_\Omega \bigl(\bar{c}_p \theta_0 +\bar{\alpha}_\theta \mathop{\rm div}_x
\mathbf{w}_0\bigr) \psi|_{t=0} d\mathbf{x} =0
\end{aligned}
\end{equation}
for all smooth $\psi= \psi(\mathbf{x},t)$ such that
$\psi|_{\partial \Omega} = \psi|_{t=T}=0$.
\end{definition}

The first main result of the paper is the following theorem on
existence and uniqueness of solutions to \emph{Model A}.

\begin{theorem} \label{theorem1}
Whenever $\mathbf{w}_0 \in \mathaccent"7017 {W^1_2}(\Omega)$,
$\mathbf{v}_0,\theta_0 \in L^2(\Omega)$, and $\mathbf{F},\Psi\in L^2(Q)$,
\emph{Model A} has a unique generalized solution
$(\mathbf{w},\,\theta,\,p,\,q,\,\pi)$ in the sense of Definition
\ref{definition1a}.
\end{theorem}

\section{The Energy Estimate and Uniqueness of Solution of Model
A} \label{Energy}

Construction of the energy estimate is based on introducing the
alternative equivalent definition of generalized solutions of
\emph{Model A} and on a special choice of test functions in the
integral equalities in this definition.
Namely, we state:

\begin{definition} \label{definition1c} \rm
Five functions $(\mathbf{w},\theta,p,q,\pi)$ are called a generalized
solution of \emph{Model A} if they satisfy the regularity
conditions
\begin{gather} \label{(1c)}
\mathbf{w},\,\frac{\partial \mathbf{w}}{\partial t},\, \mathbf{D}(x,\mathbf{w}),\, \bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr),\,\mathop{\rm div}_x\mathbf{w},\,
\theta,\, \nabla_x \theta \in L^2(Q),
\\
\label{(2c)} \begin{aligned}
& \mbox{functions $t\mapsto \frac{\partial
\mathbf{w}}{\partial t}(t), \; t\mapsto \bar{c}_p \theta(t)
+\bar{\alpha}_\theta \mathop{\rm div}_x \mathbf{w}(t)$ are weakly }\\
 &\mbox{continuous mappings of the interval $[0,T]$
 into $L^2(\Omega)$,}
\end{aligned}
\end{gather}
boundary conditions \eqref{(45a)} in the trace sense, equations
\eqref{(41a)}--\eqref{(43a)} a.e. in $Q$, and the integral
equalities
\begin{equation} \label{(3c)}
\begin{aligned}
&\int_0^\tau \int_\Omega \Bigl(\alpha_\tau \bar{\rho}
\frac{\partial \mathbf{w}}{\partial t}\cdot \frac{\partial \mathbf{
\varphi}}{\partial t} -\mathbf{P}:\nabla_x \mathbf{\varphi} + \alpha_F
\bar{\rho} \mathbf{F}\cdot \mathbf{\varphi}\Bigr) d\mathbf{x} dt\\
& = \int_\Omega
\alpha_\tau \bar{\rho} \frac{\partial \mathbf{w}}{\partial
t}(\mathbf{x},\tau)\cdot \mathbf{\varphi}(\mathbf{x},\tau) d\mathbf{x} -\int_\Omega
\alpha_\tau \bar{\rho} \mathbf{v}_0 \cdot \mathbf{\varphi}|_{t=0} d\mathbf{x},
\quad \forall\, \tau\in [0,T]
\end{aligned}
\end{equation}
for all smooth $\mathbf{\varphi}=\mathbf{\varphi}(\mathbf{x},t)$ such
that $\mathbf{\varphi}|_{\partial \Omega}=0$ and
\begin{equation} \label{(4c)}
\begin{aligned}
&\int_0^\tau \int_\Omega \Bigl[\Bigl(\bar{c}_p \theta +
\bar{\alpha}_\theta \mathop{\rm div}_x \mathbf{w}\Bigr) \frac{\partial \psi}{\partial
t} - \bar{\varkappa}
\nabla_x \theta \cdot \nabla_x \psi + \Psi \psi \Bigr] d\mathbf{x} dt\\
&=\int_\Omega \bigl(\bar{c}_p \theta(\mathbf{x},\tau) +\bar{\alpha}_\theta
\mathop{\rm div}_x \mathbf{w}(\mathbf{x},\tau)\bigr) \psi(\mathbf{x},\tau) d\mathbf{x} -\int_\Omega
\bigl(\bar{c}_p \theta_0 +\bar{\alpha}_\theta \mathop{\rm div}_x \mathbf{w}_0\bigr)
\psi|_{t=0} d\mathbf{x},
\end{aligned}
\end{equation}
for all $ \tau\in [0,T]$ and all smooth $\psi= \psi(\mathbf{x},t)$ such that
$\psi|_{\partial \Omega} =0$.
\end{definition}

\begin{remark} \label{remark1c} \rm
Definitions \ref{definition1a} and \ref{definition1c} are equivalent.
The fact that a generalized
solution in the sense of Definition \ref{definition1c} is a
generalized solution in the sense of Definition \ref{definition1a}
is quite obvious. The inverse proposition is true thanks to the
simple standard considerations. Its justification can be fulfilled
similarly to, for example, \cite[Sec. III.1]{AKM}.
\end{remark}

Now let us follow the track of considerations of
\cite[Chap. 2, Sec. 5.2]{LIO}.

Fix arbitrary $\tau_*,\tau_{**}\in (0,\tau)$, $\tau_*<\tau_{**}$.
Take a continuous piece-wise linear function on $[0,\tau]$ such
that $\phi_m(t)=1$ if $\tau_*+(2/m)<t<\tau_{**} -(2/m)$ and
$\phi_m(t)=0$ if $t>\tau_{**} -(1/m)$ and $t<\tau_* +(1/m)$. Take
a regularizing sequence $\omega_n\in C_0^\infty (\mathbf{R})$ such that
$$
\omega_n(t)=\omega_n(-t),\quad \omega_n(t)\geq 0,\quad
\int_{-\infty}^\infty \omega_n(t)dt=1, \quad \mbox{supp\,}
\omega_n \subset \Bigl[-\frac{1}{n},\frac{1}{n}\Bigr].
$$
For $n>2m$, set
\begin{equation} \label{(5c)}
\mathbf{\varphi}_{mn} =\Bigl(\Bigl(\phi_m \frac{\partial
\mathbf{w}}{\partial t}\Bigr) *\omega_n *\omega_n\Bigr)\phi_m, \quad
\psi_{mn}=((\phi_m\theta)*\omega_n *\omega_n) \phi_m,
\end{equation}
where the asterisk$*$ means the integral convolution in $\mathbf{R}$, and
substitute for $\mathbf{\varphi}$ and $\psi$ into
\eqref{(3c)} and \eqref{(4c)}, respectively. Clearly, this choice
of test functions is valid due to regularity properties
\eqref{(1c)} and \eqref{(2c)}.

Insert \eqref{(41a)}--\eqref{(43a)} into  \eqref{(3c)}
(with $\mathbf{\varphi}=\mathbf{\varphi}_{mn}$) and then sum the
result with \eqref{(4c)} (with $\psi=\psi_{mn}$). In thus
obtained equality represent
\begin{equation} \label{(5c-2)}
\begin{aligned}
&\int_0^\tau \int_\Omega \bar{\alpha}_\theta \theta \mathop{\rm div}_x \mathbf{
\varphi}_{mn} d\mathbf{x} dt \\
&=\int_0^\tau \int_\Omega \bar{\alpha}_\theta
\bigl((\phi_m \theta)*\omega_n\bigr) \frac{\partial}{\partial t}
\mathop{\rm div}_x\bigl((\phi_m \mathbf{w})*\omega_n\bigr) d\mathbf{x} dt\\
&\quad -\int_0^\tau \int_\Omega \bar{\alpha}_\theta \bigl((\phi_m
\theta)*\omega_n\bigr) \mathop{\rm div}_x\Bigl(\Bigl(\frac{\partial
\phi_m}{\partial t}\mathbf{w}\Bigr)*\omega_n\Bigr) d\mathbf{x} dt
\end{aligned}
\end{equation}
and
%\begin{equation} \label{(5c-3)}
\begin{align*}
\int_0^\tau \int_\Omega \bar{\alpha}_\theta \mathop{\rm div}_x \mathbf{w}
\frac{\partial \psi_{mn}}{\partial t} d\mathbf{x} dt
& =\int_0^\tau
\int_\Omega \bar{\alpha}_\theta \mathop{\rm div}_x \mathbf{w} \bigl((\phi_m
\theta)*\omega_n *\omega_n\bigr) \frac{\partial \phi_m}{\partial
t} d\mathbf{x} dt\\
&\quad +\int_0^\tau \int_\Omega \bar{\alpha}_\theta \mathop{\rm div}_x \bigl((\phi_m
\mathbf{w})*\omega_n \bigr) \frac{\partial}{\partial t} \bigl((\phi_m
\theta)*\omega_n\bigr) d\mathbf{x} dt.
\end{align*}
Then we integrate by parts with respect to $t$ in the last summand of
the above expression and combine similar terms.

Applying the arguments of \cite[Chap. 2, Sec. 5.2]{LIO}, after
some technical transformations and passage to the limit as
$n\nearrow \infty$, $m\nearrow \infty$, $\tau_*\searrow 0$, and
$\tau_{**}\nearrow \tau$, successively, we finally arrive at the
\emph{energy identity}, as follows:
\begin{equation}
\begin{aligned}
&   \frac{1}{2}\alpha_\tau\Bigl\|\sqrt{\bar{\rho}}
\frac{\partial \mathbf{w}}{\partial t}(\tau)\Bigr\|_{2,\Omega}^2
+\frac{1}{2} \alpha_\eta \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}(\tau)\|_{2,\Omega}^2 +\frac{1}{2} \alpha_p
\|\bar{\chi} \mathop{\rm div}_x \mathbf{w}(\tau)\|_{2,\Omega}^2\\
 &  + \frac{1}{2} \alpha_\lambda
\|(1-\bar{\chi}) \mathbf{D}(x,\mathbf{w}(\tau))\|_{2,\Omega}^2
+\frac{1}{2} \|\sqrt{\bar{c}_p} \theta (\tau)\|_{2,\Omega}^2\\
 &   +\alpha_\nu \Bigl\|\bar{\chi} \mathop{\rm div}_x
\frac{\partial \mathbf{w}}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2
+\alpha_\mu \Bigl\|\bar{\chi} \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2
+\|\sqrt{\bar{\varkappa}} \nabla_x \theta\|_{2,\Omega\times (0,\tau)}^2 \\
 &=   \frac{1}{2}\alpha_\tau
\|\sqrt{\bar{\rho}} \mathbf{v}_0 \|_{2,\Omega}^2 +\frac{1}{2} \alpha_\eta
\|(1-\bar{\chi}) \mathop{\rm div}_x \mathbf{w}_0\|_{2,\Omega}^2 +\frac{1}{2} \alpha_p
\|\bar{\chi} \mathop{\rm div}_x \mathbf{w}_0\|_{2,\Omega}^2\\
 &\quad  + \frac{1}{2} \alpha_\lambda
\|(1-\bar{\chi}) \mathbf{D}(x,\mathbf{w}_0)\|_{2,\Omega}^2 +\frac{1}{2}
\|\sqrt{\bar{c}_p} \theta_0\|_{2,\Omega}^2\\
 &\quad  -\int_0^\tau \int_\Omega
\Bigl(\alpha_F \bar{\rho} \mathbf{F}\cdot \frac{\partial \mathbf{w}}{\partial t} +
\Psi \theta \Bigr) d\mathbf{x} dt, \quad \forall\, \tau\in [0,T].
\end{aligned}\label{(6c)}
\end{equation}
Discarding all the terms on the left-hand side except for the
first and the fifth ones and applying the Cauchy--Schwartz
inequality on the right-hand side, we get
\begin{equation} \label{(7c)}
\begin{aligned}
&\frac{1}{2}\alpha_\tau\Bigl\|\sqrt{\bar{\rho}} \frac{\partial
\mathbf{w}}{\partial t}(\tau)\Bigr\|_{2,\Omega}^2 +\frac{1}{2}
\|\sqrt{\bar{c}_p} \theta (\tau)\|_{2,\Omega}^2\\
&\leq \frac{1}{2}\alpha_\tau \|\sqrt{\bar{\rho}} \mathbf{v}_0
\|_{2,\Omega}^2 +\frac{1}{2} \|\sqrt{\bar{c}_p}
\theta_0\|_{2,\Omega}^2 +\int_0^\tau \frac{1}{2} \alpha_\tau
\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{w}}{\partial
t}(t)\Bigr\|_{2,\Omega}^2 dt\\
&\quad  +\int_0^\tau \frac{1}{2}
\|\sqrt{\bar{c}_p} \theta(t)\|_{2,\Omega}^2 dt
+\frac{\alpha_F^2}{2\alpha_\tau} \|\sqrt{\bar{\rho}}
\mathbf{F}\|_{2,\Omega \times (0,\tau)}^2 +
\frac{1}{2\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega\times
(0,\tau)}^2\\
&\quad +\frac{1}{2}\alpha_\eta \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}_0\|_{2,\Omega}^2 +\frac{1}{2} \alpha_p \|\bar{\chi} \mathop{\rm div}_x
\mathbf{w}_0\|_{2,\Omega}^2\\
&\quad +\frac{1}{2} \alpha_\lambda \|(1-\bar{\chi})
\mathbf{D}(x,\mathbf{w}_0)\|_{2,\Omega}^2, \quad \forall\, \tau\in [0,T].
\end{aligned}
\end{equation}
Applying Grownwall's lemma to this inequality, we conclude that
%\begin{equation} \label{(8c)}
\begin{align*}
&\frac{1}{2} \alpha_\tau \Bigl\|\sqrt{\bar{\rho}} \frac{\partial
\mathbf{w}}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2 +\frac{1}{2}
\|\sqrt{\bar{c}_p} \theta \|_{2,\Omega\times (0,\tau)}^2\\
&\leq \int_0^\tau \Bigl[\frac{\alpha_F^2}{2\alpha_\tau}
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega\times (0,t)}^2
+\frac{1}{2\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega\times
(0,t)}^2\Bigr] e^{\tau-t} dt\\
&\quad +e^\tau\Bigl[\frac{1}{2} \alpha_\tau \|\sqrt{\bar{\rho}}
\mathbf{v}_0\|_{2,\Omega}^2 +\frac{1}{2} \|\sqrt{\bar{c}_p}
\theta_0\|_{2,\Omega}^2\Bigr]\\
&\quad +(e^\tau-1)\Bigl[\frac{1}{2} \alpha_\eta \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}_0\|_{2,\Omega}^2
+\frac{1}{2} \alpha_p \|\bar{\chi} \mathop{\rm div}_x \mathbf{w}_0\|_{2,\Omega}^2\\
&\quad +\frac{1}{2} \alpha_\lambda \|(1-\bar{\chi})
\mathbf{D}(x,\mathbf{w}_0)\|_{2,\Omega}^2\Bigr],\quad \forall\, \tau\in [0,T].
\end{align*}
%\end{equation}
Combining energy identity \eqref{(6c)} with the above estimate and
\eqref{(7c)}, we finally establish the following result.

\begin{proposition} \label{proposition1c}
Let $\mathbf{F},\Psi\in L^2(Q)$, $\mathbf{w}_0 \in \mathaccent"7017 {W_2^1}(\Omega)$,
 and $\mathbf{v}_0,\theta_0\in L^2(\Omega)$.
Assume that five functions $(\mathbf{w},\theta,p,q,\pi)$ are a generalized
solution of Model A. Then $\mathbf{w}$ and $\theta$ satisfy the energy
estimate
\begin{equation} \label{(12c)}
\begin{aligned}
&\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\tau\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{w}}{\partial
t}(t)\Bigr\|_{2,\Omega}^2
+\frac{1}{2} \mathop{\rm ess\,sup}{t\in [0,\tau]}
\|\sqrt{\bar{c}_p} \theta (t)\|_{2,\Omega}^2
\\
&\quad +\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]} \alpha_\eta \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}(t)\|_{2,\Omega}^2 +\frac{1}{2}
\mathop{\rm ess\,sup}_{t\in [0,\tau]}
 \alpha_p \|\bar{\chi} \mathop{\rm div}_x \mathbf{w}(t)\|_{2,\Omega}^2
\\
&\quad  + \frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\lambda \|(1-\bar{\chi}) \mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega}^2
+\alpha_\nu \Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial
t}\Bigr\|_{2,\Omega\times (0,\tau)}^2
\\
&\quad +\alpha_\mu \Bigl\|\bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2
+\|\sqrt{\bar{\varkappa}} \nabla_x \theta\|_{2,\Omega\times (0,\tau)}^2\\
&\leq \frac{\alpha_F^2}{2 \alpha_\tau}  \|\sqrt{\bar{\rho}}
\mathbf{F}\|_{2,\Omega\times (0,\tau)}^2 +
\frac{1}{2\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega \times (0,\tau)}^2
\\
&\quad +\int_0^\tau \Bigl[\frac{\alpha_F^2}{2 \alpha_\tau}
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega\times (0,t)}^2 +
\frac{1}{2\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega \times
(0,t)}^2\Bigr] e^{\tau-t} dt
\\
&\quad +\frac{e^\tau +1}{2} \Bigl[ \alpha_\tau \|\sqrt{\bar{\rho}} \mathbf{v}_0
\|_{2,\Omega}^2 + \|\sqrt{\bar{c}_p}
\theta_0\|_{2,\Omega}^2\Bigr]
\\
&\quad + \frac{e^\tau}{2} \Bigl[ \alpha_\eta
\|(1-\bar{\chi}) \mathop{\rm div}_x \mathbf{w}_0\|_{2,\Omega}^2 + \alpha_p
\|\bar{\chi} \mathop{\rm div}_x
\mathbf{w}_0\|_{2,\Omega}^2
\\
&\quad + \alpha_\lambda \|(1-\bar{\chi}) \mathbf{D}(x,\mathbf{w}_0)\|_{2,\Omega}^2\Bigr]
 := C_{\rm en}(\tau),
 \quad \forall\, \tau\in [0,T].
\end{aligned}
\end{equation}
\end{proposition}

\begin{corollary} \label{corollary1c}
The displacement $\mathbf{w}$ belongs to the space
$L^\infty(0,T;\mathaccent"7017 {W_2^1}(\Omega))$ and admits the
bound
\begin{equation} \label{(13c)}
\mathop{\rm ess\,sup}_{t\in [0,T]}
\|\mathbf{w}(t)\|_{W_2^1(\Omega)}
 \leq C\bigl(C_{\rm en}(T),C_k(\Omega)\bigr),
\end{equation}
where $C_k(\Omega)$ depends only on geometry of $\partial \Omega$.
\end{corollary}

\begin{proof} On the strength of the Newton-Leibnitz formula,
\begin{equation} \label{(14c)}
\begin{aligned}
&\int_\Omega \bar{\chi}(\mathbf{x}) \mathbf{D}(x,\mathbf{w}(\tau)): \mathbb{G}(\mathbf{x})d\mathbf{x} \\
& =\int_0^\tau \int_\Omega \bar{\chi}(\mathbf{x}) \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}(t)\Bigr):\mathbb{G}(\mathbf{x}) d\mathbf{x} dt\\
&\quad + \int_\Omega \bar{\chi}(\mathbf{x}) \mathbf{D}(x,\mathbf{w}_0):\mathbb{G}(\mathbf{x}) d\mathbf{x} \quad
 \forall\, \mathbb{G}\in (L^2(\Omega))^{3\times 3}, \quad
 \forall\, \tau\in [0,T].
\end{aligned}
\end{equation}
Substitute $\mathbb{G}(\mathbf{x})=\mathbf{D}(x,\mathbf{w}(\tau))$ and use inequality
\eqref{(7c)} and the Cauchy-Schwartz inequality with some
positive $\varepsilon_0$ and $\varepsilon_1$ to get
\begin{align*}
&\mathop{\rm ess\,sup}_{t\in [0,\tau]} \int_\Omega
\bar{\chi} |\mathbf{D}(x,\mathbf{w}(t))\|^2 d\mathbf{x}\\
& \leq \int_0^\tau \int_\Omega
\bar{\chi} \Bigl[\frac{1}{2\varepsilon_0}
\Bigl|\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}(t)\Bigr)\Bigr|^2
+\frac{\varepsilon_0}{2} |\mathbf{D}(x,\mathbf{w}(\tau))|^2\Bigr]d\mathbf{x} dt\\
&\quad \times \int_\Omega \bar{\chi} \Bigl[\frac{1}{2\varepsilon_1}
|\mathbf{D}(x,\mathbf{w}_0)|^2
+\frac{\varepsilon_1}{2}|\mathbf{D}(x,\mathbf{w}(\tau))|^2\Bigr]d\mathbf{x}\\
&\leq \int_0^\tau \int_\Omega \bar{\chi} \frac{1}{2\varepsilon_0}
\Bigl|\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}(t)\Bigr)\Bigr|^2d\mathbf{x}
dt +\Bigl(\frac{\tau
\varepsilon_0}{2}+\frac{\varepsilon_1}{2}\Bigr)
\mathop{\rm ess\,sup}_{t\in [0,\tau]} \int_\Omega \bar{\chi}
|\mathbf{D}(x,\mathbf{w}(t))|^2 d\mathbf{x}\\
&\quad +\int_\Omega \bar{\chi} \frac{1}{2\varepsilon_1} |\mathbf{D}(x,\mathbf{w}_0)|^2
d\mathbf{x}.
\end{align*}
Choosing here $\varepsilon_0=1/(2\tau)$ and $\varepsilon_1=1/2$ we
derive
\begin{equation} \label{(15c)}
\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|\bar{\chi}\mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega}^2
 \leq \tau \bigl\|\bar{\chi}
\mathbf{D}\bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\bigr)\bigr\|_{2,\Omega\times (0,\tau)}^2 +\|\bar{\chi}
\mathbf{D}(x,\mathbf{w}_0)\|_{2,\Omega}^2,
\end{equation}
for all $\tau\in [0,T]$. We end up with estimate \eqref{(13c)},
by combining inequality \eqref{(15c)} with energy estimate
\eqref{(12c)} and Korn's inequality
\begin{equation} \label{Korn}
\|\mathbf{\phi}\|_{\mathaccent"7017 {W_2^1}(\Omega)} \leq
C_k(\Omega) \|\mathbf{D}(x,\mathbf{\phi})\|_{2,\Omega},
\end{equation}
which is valid for all functions $\mathbf{\phi}\in L^2(\Omega)$
such that $\mathbf{D}(x,\mathbf{\phi})\in L^2(\Omega)$ and $\mathbf{
\phi}$ vanishes in the trace sense on some open subset of
$\partial \Omega$ \cite[Chap. III, Sec. 3.2]{RESH}.
\end{proof}

\begin{corollary} \label{corollary2c}
Let $\mathbf{F},\Psi \in L^2(Q)$, $\mathbf{v}_0,\theta_0 \in L^2(\Omega)$, and
$\mathbf{w}_0 \in \mathaccent"7017 {W^1_2}(\Omega)$. If Model A is
solvable, then there exists exactly one generalized solution.
\end{corollary}

\begin{proof} Since \emph{Model A} is linear, the uniqueness
assertion amounts to the proposition that, if $\mathbf{F},\mathbf{v}_0,\mathbf{w}_0=0$
and $\Psi,\theta_0 =0$, then there is only trivial solution. The
latter proposition is obvious due to energy estimate
\eqref{(12c)}.
\end{proof}

\section{Galerkin's Approximations and Existence of Solutions
to Model A} \label{Galerkin}

Let $\{\mathbf{\phi}_l\} \subset C^\infty_0(\Omega)^3$ and
$\{\psi_l\}\subset C^\infty_0(\Omega)$ be total systems in
$\mathaccent"7017 {W^1_2}(\Omega)^3$ and
$\mathaccent"7017 {W^1_2}(\Omega)$, respectively.

We construct Galerkin's approximations of the displacement vector
and of the temperature distribution in the forms
\begin{equation} \label{(1d)}
\mathbf{w}_n(\mathbf{x},t)=\sum_{l=1}^n a_l(t) \mathbf{\phi}_l(\mathbf{x}), \quad
\theta_n(\mathbf{x},t) =\sum_{l=1}^n b_l(t) \psi_l(\mathbf{x}),
\end{equation}
where unknown functions $a_l(t)$ and $b_l(t)$ $(l=1,\dots,n)$ are
found from Galerkin's system
\begin{equation}  \label{(2d)}
\begin{aligned}
&\sum_{l=1}^n \frac{d^2 a_l(t)}{dt^2} \int_\Omega \alpha_\tau
\bar{\rho}(\mathbf{x}) \mathbf{\phi}_l(\mathbf{x})\cdot \mathbf{\phi}_j(\mathbf{x})
d\mathbf{x}\\
&=-\sum_{l=1}^n \frac{da_l(t)}{dt} \int_\Omega \bar{\chi}(\mathbf{x})
\bigl( \alpha_\nu \mathop{\rm div}_x \mathbf{\phi}_l(\mathbf{x}) \cdot \mathop{\rm div}_x\mathbf{
\phi}_j(\mathbf{x})
 +\alpha_\mu \mathbf{D}(x,\mathbf{\phi}_l(\mathbf{x})):
\mathbf{D}(x,\mathbf{\phi}_j(\mathbf{x}))\bigr) d\mathbf{x}\\
&\quad -\sum_{l=1}^n a_l(t) \int_\Omega \bigl[\bar{\chi}(\mathbf{x}) \alpha_p
\mathop{\rm div}_x \mathbf{\phi}_l(\mathbf{x})\cdot \mathop{\rm div}_x\mathbf{\phi}_j(\mathbf{x})
+(1-\bar{\chi}(\mathbf{x})) \alpha_\eta \mathop{\rm div}_x \mathbf{\phi}_l(\mathbf{x})\cdot
\mathop{\rm div}_x \mathbf{\phi}_j(\mathbf{x})\\
&\quad +(1-\bar{\chi}(\mathbf{x})) \alpha_\lambda \mathbf{D}(x,\mathbf{\phi}_l(\mathbf{x})):
\mathbf{D}(x,\mathbf{\phi}_j(\mathbf{x}))\bigr]d\mathbf{x}\\
&\quad  +\sum_{l=1}^n b_l(t)
\int_\Omega \bar{\alpha}_\theta(\mathbf{x}) \psi_l(\mathbf{x}) \mathop{\rm div}_x
\mathbf{\phi}_j(\mathbf{x})d\mathbf{x}
 +\int_\Omega \alpha_F \bar{\rho}(\mathbf{x})
\mathbf{F}_n(\mathbf{x},t)\cdot \mathbf{\phi}_j(\mathbf{x})d\mathbf{x},
\end{aligned}
\end{equation}
$j=1,\dots,n$, where $\mathbf{F}_n$ is a given approximation of $\mathbf{F}$ such
that
\begin{equation} \label{(2d-2)}
\mathbf{F}_n \in C^\infty(Q), \quad \mathbf{F}_n \rightarrow \mathbf{F} \mbox{ in } L^2(Q)
\quad \mbox{as } n\nearrow \infty,
\end{equation}
\begin{equation} \label{(3d)}
\begin{aligned}
&\sum_{l=1}^n \frac{db_l(t)}{dt} \int_\Omega \bar{c}_p(\mathbf{x})
\psi_l(\mathbf{x}) \psi_j(\mathbf{x})d\mathbf{x} \\
&=-\sum_{l=1}^n b_l(t) \int_\Omega
\bar{\varkappa}(\mathbf{x}) \nabla_x
\psi_l(\mathbf{x})\cdot \nabla_x \psi_j(\mathbf{x}) d\mathbf{x}\\
&\quad +\sum_{l=1}^n \frac{da_l(t)}{dt} \int_\Omega \bar{\alpha}_\theta
(\mathbf{x}) (\mathop{\rm div}_x\mathbf{\phi}_l(\mathbf{x})) \psi_j(\mathbf{x}) d\mathbf{x} +\int_\Omega
\Psi_n (\mathbf{x},t) \psi_j (\mathbf{x})d\mathbf{x},
\end{aligned}
\end{equation}
$j=1,\dots,n$, where $\Psi_n$ is a given approximation of $\Psi$
such that
\begin{equation} \label{(3d-2)}
\Psi_n \in C^\infty(Q), \quad \Psi_n \rightarrow \Psi \mbox{ in }
L^2(Q) \quad \mbox{as } n\nearrow \infty.
\end{equation}
Since $\alpha_\tau \bar{\rho}(\mathbf{x})>0$, $\bar{c}_p(\mathbf{x})>0$ and the
sets $(\mathbf{\phi}_1,\dots,\mathbf{\phi}_n)$ and
$(\psi_1,\dots,\psi_n)$ are linearly independent in
$L^2(\Omega)^3$ and $L^2(\Omega)$, respectively, then the matrices
$$
\mathbb{A}_n =\Bigl(\int_\Omega \alpha_\tau \bar{\rho}(\mathbf{x})
\mathbf{\phi}_l(\mathbf{x})\cdot \mathbf{\phi}_j(\mathbf{x}) d\mathbf{x}
\Bigr)_{l,j=1}^n,\quad \mathbb{B}_n =\Bigl(\int_\Omega
\bar{c}_p(\mathbf{x}) \psi_l(\mathbf{x}) \psi_j(\mathbf{x})d\mathbf{x}\Bigr)_{l,j=1}^n
$$
are invertible, due to the classical theory of Hilbert spaces.
Hence, setting
\begin{gather*}
  c_l(t)=\frac{da_l(t)}{dt},\quad \mathbf{a}_n (t)
=(a_1(t),\dots,a_n(t)),\\
 \mathbf{b}_n (t) =(b_1(t),\dots,b_n(t)),\quad \mathbf{c}_n
(t) =(c_1(t),\dots,c_n(t)),
\\
\mathbb{A}_n^{(1)} =\Bigl(\int_\Omega \bar{\chi}(\mathbf{x}) \bigl(
\alpha_\nu \mathop{\rm div}_x \mathbf{\phi}_l(\mathbf{x}) \cdot  \mathop{\rm div}_x\mathbf{
\phi}_j(\mathbf{x})
 +\alpha_\mu \mathbf{D}(x,\mathbf{\phi}_l(\mathbf{x})):
\mathbf{D}(x,\mathbf{\phi}_j(\mathbf{x}))\bigr) d\mathbf{x}\Bigr)_{l,j=1}^n,
\\
\begin{aligned}
\mathbb{A}_n^{(2)} &=\Bigl(\int_\Omega \bigl[\bar{\chi}(\mathbf{x})
\alpha_p \mathop{\rm div}_x \mathbf{\phi}_l(\mathbf{x})\cdot \mathop{\rm div}_x\mathbf{
\phi}_j(\mathbf{x}) +(1-\bar{\chi}(\mathbf{x})) \alpha_\eta \mathop{\rm div}_x \mathbf{
\phi}_l(\mathbf{x})\cdot \mathop{\rm div}_x \mathbf{
\phi}_j(\mathbf{x})\\
&\quad +(1-\bar{\chi}(\mathbf{x})) \alpha_\lambda \mathbf{D}(x,\mathbf{\phi}_l(\mathbf{x})):
\mathbf{D}(x,\mathbf{\phi}_j(\mathbf{x}))\bigr]d\mathbf{x}\Bigr)_{l,j=1}^n,
\end{aligned}
\\
\mathbb{A}_n^{(3)} =\Bigl(\int_\Omega \bar{\alpha}_\theta(\mathbf{x})
\psi_l(\mathbf{x}) \mathop{\rm div}_x \mathbf{\phi}_j(\mathbf{x})d\mathbf{x}\Bigr)_{l,j=1}^n,
\\
\mathbb{B}_n^{(1)} =\Bigl(\int_\Omega \bar{\varkappa}(\mathbf{x})
\nabla_x \psi_l(\mathbf{x})\cdot \nabla_x \psi_j(\mathbf{x}) d\mathbf{x}\Bigr)_{l,j=1}^n,
\\
\mathbb{B}_n^{(2)} =\Bigl(\int_\Omega \bar{\alpha}_\theta (\mathbf{x})
(\mathop{\rm div}_x\mathbf{\phi}_l(\mathbf{x})) \psi_j(\mathbf{x}) d\mathbf{x}\Bigr)_{l,j=1}^n,
\\
\tilde{\mathbf{F}}_n(t) =\Bigl(\int_\Omega \alpha_F \bar{\rho}(\mathbf{x})
\mathbf{F}_n(\mathbf{x},t)\cdot \mathbf{\phi}_j(\mathbf{x})d\mathbf{x}\Bigr)_{j=1}^n,
\\
\tilde{\Psi}_n(t) =\Bigl(\int_\Omega \Psi_n(\mathbf{x},t)
\psi_j(\mathbf{x})d\mathbf{x}\Bigr)_{j=1}^n,
\end{gather*}
we see that system \eqref{(2d)}--\eqref{(3d)} is equivalent to the
system of the first-order linear differential equations with
constant coefficients in the normal form:
\begin{gather} \label{(4d)}
 \frac{d\mathbf{c}_n(t)}{dt} = -\mathbb{A}_n^{-1}
\mathbb{A}_n^{(1)} \mathbf{c}_n(t) -\mathbb{A}_n^{-1} \mathbb{
A}_n^{(2)} \mathbf{a}_n(t)
  +\mathbb{A}_n^{-1} (\mathbb{A}_n^{(3)})^t \mathbf{b}_n(t)
+\mathbb{A}_n^{-1}\tilde{\mathbf{F}}_n(t),\\
\label{(5d)}  \frac{d\mathbf{b}_n(t)}{dt} =
-\mathbb{B}_n^{-1} \mathbb{B}_n^{(1)} \mathbf{b}_n(t) +\mathbb{
B}_n^{-1} (\mathbb{B}_n^{(2)})^t \mathbf{c}_n(t)
+\mathbb{B}_n^{-1} \tilde{\Psi}_n(t),\\
\label{(6d)}  \frac{d\mathbf{a}_n(t)}{dt} = \mathbf{c}_n(t).
\end{gather}
System \eqref{(2d)}--\eqref{(3d)} or, equivalently, system
\eqref{(4d)}--\eqref{(6d)}, is supplemented with initial data
\begin{gather} \label{(7d)}
 a_l(t)|_{t=0} =a_l^0 :=
\int_\Omega \mathbf{w}_0(\mathbf{x})\cdot \mathbf{\phi}_l(\mathbf{x}) d\mathbf{x},\\
\label{(8d)}  \quad b_l(t)|_{t=0} =b_l^0
:= \int_\Omega \theta_0(\mathbf{x}) \psi_l(\mathbf{x}) d\mathbf{x},\\
\label{(9d)}  \quad c_l(t)|_{t=0} =c_l^0
:= \int_\Omega \mathbf{v}_0(\mathbf{x})\cdot \mathbf{\phi}_l(\mathbf{x})
d\mathbf{x}.
\end{gather}

On the strength of the classical theory of systems of first-order
ordinary linear differential equations, the Cauchy problem
\eqref{(4d)}--\eqref{(9d)} has a unique infinitely smooth solution
$(\mathbf{a}_n(t),\mathbf{b}_n(t),\mathbf{c}_n(t))$ for any
$n\in \mathbb{N}$. This amounts to the following result.

\begin{proposition} \label{proposition1d}
Galerkin's system \eqref{(2d)}--\eqref{(3d)}, supplemented with
initial data \eqref{(7d)}--\eqref{(9d)}, has a unique smooth
solution $(a_1(t),\dots,a_n(t),b_1(t),\dots,b_n(t))$ on the interval
$[0,T]$ for any $n\in \mathbb{N}$.
\end{proposition}

The approximate distributions of pressures now can be found from
the equations
\begin{gather} \label{(10d)}
 p_n(\mathbf{x},t)=-\bar{\chi}(\mathbf{x}) \alpha_p \mathop{\rm div}_x \mathbf{w}_n(\mathbf{x},t),\\
\label{(11d)}
q_n(\mathbf{x},t)=p_n(\mathbf{x},t)+\frac{\alpha_\nu}{\alpha_p}\frac{\partial
p_n(\mathbf{x},t)}{\partial t},\\
 \label{(12d)}
\pi_n(\mathbf{x},t)=-(1-\bar{\chi}(\mathbf{x})) \alpha_\eta \mathop{\rm div}_x \mathbf{w}_n(\mathbf{x},t).
\end{gather}

\begin{remark} \label{remark1d} \rm
Using the standard technics \cite[Sec. III.3]{LAD} we easily
conclude that five approximate functions
$(\mathbf{w}_n,\,\theta_n,\,p_n,\,q_n,\,\pi_n)$, which are obtained by
virtue of \eqref{(1d)}--\eqref{(3d)} and
\eqref{(7d)}--\eqref{(12d)}, are a generalized solution of
\emph{Model A}, provided with the given approximate functions
$\mathbf{F}_n(\mathbf{x},t)$ and $\Psi_n (\mathbf{x},t)$, and with the initial data
\begin{gather} \label{(13d-1)}
  \mathbf{w}_n(\mathbf{x},t)|_{t=0} =\sum_{l=1}^n a_l^0 \mathbf{
\phi}_l (\mathbf{x}),\\
\label{(13d-2)}   \frac{\partial
\mathbf{w}_n(\mathbf{x},t)}{\partial t}|_{t=0} =\sum_{l=1}^n c_l^0 \mathbf{
\phi}_l(\mathbf{x}),\\
\label{(13d-3)}   \theta_n(\mathbf{x},t)|_{t=0}
=\sum_{l=1}^n b_l^0 \psi_l(\mathbf{x}),
\end{gather}
where $a_l^0$, $b_l^0$, and $c_l^0$ are given by
\eqref{(7d)}--\eqref{(9d)}, i.e., they are Fourier coefficients of
initial data $\mathbf{w}_0$, $\mathbf{v}_0$, and $\theta_0$, and hence initial
data \eqref{(13d-1)}--\eqref{(13d-3)} are the partial Fourier sums of $\mathbf{w}_0$,
$\mathbf{v}_0$, and $\theta_0$.
\end{remark}

Due to this remark and energy estimate \eqref{(12c)}, the sequence
$(\mathbf{w}_n,\,\theta_n,\,p_n,\,q_n,\,\pi_n)$ has a weak limiting point
$(\mathbf{w},\,\theta,\,p,\,q,\,\pi)$ as $n\nearrow \infty$, and, due to
linearity of \emph{Model A}, the functions $\mathbf{w}$, $\theta$, $p$,
$q$, and $\pi$ are a generalized solution of \emph{Model A},
provided with initial data $\mathbf{w}_0$, $\mathbf{v}_0$, and $\theta_0$, which
completes the proof of Theorem \ref{theorem1}.

\section{Additional Estimates for the Pressures}
\label{Pressures}

Energy inequality \eqref{(12c)} includes some estimates for the
pressures $p$, $q$, and $\pi$, due to
\eqref{(41a)}--\eqref{(43a)}. However, these estimates are not
applicable for analysis of incompressible limiting regimes,
because they are not uniform in $\alpha_p$ and $\alpha_\eta$, as
one or both of these coefficients grow infinitely. Hence, in order
to do such analysis, it is necessary to obtain additional bounds
on the pressures.
We start with justification of the following technical result.

\begin{lemma} \label{lemma1e}
Let $\mathbf{F},\,\partial \mathbf{F}/\partial t,\, \Psi,\, \partial \Psi/\partial
 t\in L^2(Q)$, and initial data be homogeneous, i.e.,
 \begin{equation} \label{(5e)}
 \mathbf{w}_0=0,\quad \mathbf{v}_0=0,\quad \theta_0 =0.
 \end{equation}
Then the following bound is valid:
\begin{equation} \label{(12c-2)}
\begin{aligned}
&\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\tau\Bigl\|\sqrt{\bar{\rho}} \frac{\partial^2 \mathbf{w}}{\partial
t^2}(t)\Bigr\|_{2,\Omega}^2+\frac{1}{2}
\mathop{\rm ess\,sup}_{t\in [0,\tau]} \Bigl\|\sqrt{\bar{c}_p}
\frac{\partial
\theta}{\partial t} (t)\Bigr\|_{2,\Omega}^2
\\
&+\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\eta \Bigl\|(1-\bar{\chi}) \mathop{\rm div}_x \frac{\partial
\mathbf{w}}{\partial t}(t)\Bigr\|_{2,\Omega}^2 +\frac{1}{2}
\mathop{\rm ess\,sup}_{t\in [0,\tau]} \alpha_p
\Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial t}(t)\Bigr\|_{2,\Omega}^2
\\
& + \frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\lambda \Bigl\|(1-\bar{\chi}) \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}(t)\Bigr)\Bigr\|_{2,\Omega}^2 +\alpha_\nu
\Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial^2 \mathbf{w}}{\partial
t^2}\Bigr\|_{2,\Omega\times (0,\tau)}^2
\\
& +\alpha_\mu \Bigl\|\bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial^2 \mathbf{w}}{\partial
t^2}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2
+\Bigl\|\sqrt{\bar{\varkappa}} \frac{\partial}{\partial t}
\nabla_x \theta\Bigr\|_{2,\Omega\times (0,\tau)}^2
\\
&\leq \frac{\alpha_F^2}{2 \alpha_\tau}  \Bigl\|\sqrt{\bar{\rho}}
\frac{\partial \mathbf{F}}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2
+\frac{1}{2\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega \times (0,\tau)}^2
\\
&\quad +\int_0^\tau \Bigl[\frac{\alpha_F^2}{2 \alpha_\tau}
\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{F}}{\partial
t}\Bigr\|_{2,\Omega\times (0,t)}^2 +
\frac{1}{2\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega \times
(0,t)}^2\Bigr] e^{\tau-t} dt
\\
&\quad +\frac{e^\tau +1}{2} \Bigl[\frac{\alpha_F^2}{\alpha_\tau}
\|\sqrt{\bar{\rho}} \mathbf{F}|_{t=0} \|_{2,\Omega}^2 +
\frac{1}{\min\{c_{pf},c_{ps}\}} \|\Psi|_{t=0}\|_{2,\Omega}^2\Bigr]
 := C^{(2)}_{\rm en}(\tau),
\end{aligned}
\end{equation}
for all $\tau\in [0,T]$.
\end{lemma}

\begin{proof} We take advantage of the fact that the generalized
solution is unique and may be constructed, using Galerkin's
method.

First of all, we notice that the values of all the constants
$a_l^0$, $b_l^0$, and $c_l^0$, defined in
\eqref{(7d)}--\eqref{(9d)}, are equal to zero. Passing to the
limit in the right hand sides of \eqref{(2d)}--\eqref{(3d)}
(or, equivalently, of \eqref{(4d)}--\eqref{(6d)}), we
conclude that
\begin{equation} \label{(10e)}
\alpha_\tau \bar{\rho} \frac{\partial^2 \mathbf{w}_n}{\partial
t^2}\Bigl|_{t=0} = \alpha_F \bar{\rho} \mathbf{F}_n|_{t=0}, \quad
\bar{c}_p \frac{\partial \theta_n}{\partial t}\Bigl|_{t=0} =
\Psi_n|_{t=0}.
\end{equation}
Next, differentiating Galerkin's system \eqref{(2d)}--\eqref{(3d)}
and equations \eqref{(10d)}--\eqref{(12d)} with respect to $t$, on
the strength of Remark \ref{remark1d} and \eqref{(10e)} we
conclude that the derivatives
$$
\Bigl(\frac{\partial \mathbf{w}_n}{\partial t},\,\frac{\partial
\theta_n}{\partial t},\, \frac{\partial p_n}{\partial t},\,
\frac{\partial q_n}{\partial t},\, \frac{\partial \pi_n}{\partial
t}\Bigr)
$$
are the generalized solutions of \emph{Model A}, provided with
the dimensionless approximate density of distributed mass forces
$\partial \mathbf{F}_n/\partial t$ and volumetric density of heat
application $\partial \Psi_n /\partial t$, and with the initial
data \eqref{(10e)} and \eqref{(13d-2)}. Now passing to the limit
as $n\nearrow\infty$ in energy estimate \eqref{(12c)}, we
immediately
derive bound \eqref{(12c-2)}.
\end{proof}

Next, we introduce the normalized pressures $\tilde{p}$,
$\tilde{q}$, and $\tilde{\pi}$ as follows:
\begin{gather} \label{(1e)}
\tilde{p}(\mathbf{x},t) =
 -\bar{\chi}(\mathbf{x}) \alpha_p \mathop{\rm div}_x \mathbf{w}(\mathbf{x},t)
 +\frac{\bar{\chi}(\mathbf{x})}{\mathop{\rm meas}\Omega_f}
 \int_\Omega \bar{\chi}(\mathbf{x}) \alpha_p \mathop{\rm div}_x \mathbf{w}(\mathbf{x},t)d\mathbf{x},
\\    \label{(2e)}
\begin{aligned}
 \tilde{\pi}(\mathbf{x},t) & =  -(1-\bar{\chi}(\mathbf{x})) \alpha_\eta \mathop{\rm div}_x \mathbf{w}(\mathbf{x},t)\\
& \quad  +\frac{1-\bar{\chi}(\mathbf{x})}{\mathop{\rm meas}\Omega_s}
\int_\Omega (1-\bar{\chi}(\mathbf{x}))
 \alpha_\eta \mathop{\rm div}_x \mathbf{w}(\mathbf{x},t)d\mathbf{x},
\end{aligned}\\
  \label{(3e)}
 \tilde{q}(\mathbf{x},t)  =   \tilde{p}(\mathbf{x},t) +\frac{\alpha_\nu}{\alpha_p}
 \frac{\partial \tilde{p}(\mathbf{x},t)}{\partial t}.
 \end{gather}

 \begin{remark} \label{remark1e} \rm
By the straightforward  calculation, using Green's formula, we conclude
that integral equalities \eqref{(47a)}
 and \eqref{(3c)} with  \eqref{(39a)} being inserted into them,
 are valid with $\tilde{q}$ and $\tilde{\pi}$ on the places of $q$ and
 $\pi$, respectively, and that
 \begin{equation} \label{(4e)}
 \int_\Omega \tilde{p}d\mathbf{x} =\int_\Omega \tilde{\pi} d\mathbf{x}
 =\int_\Omega \tilde{q} d\mathbf{x} =0.
 \end{equation}
 This means that all the results previously obtained in this article,
 in particular, Theorem \ref{theorem1} and energy estimate \eqref{(12c)},
  remain true for
 the modified model, which appears if we substitute
equations \eqref{(41a)}--\eqref{(43a)} in the statement of
\emph{Model A} by equations \eqref{(1e)}--\eqref{(3e)}.
 \end{remark}

 For the normalized pressures $\tilde{p}$, $\tilde{q}$, and
 $\tilde{\pi}$ we prove the following result.

 \begin{theorem} \label{proposition1e}
 Let $\mathbf{F},\,\partial \mathbf{F}/\partial t,\, \Psi,\, \partial \Psi/\partial
 t\in L^2(Q)$ and initial data be homogeneous, i.e., satisfy
 \eqref{(5e)}.
 Then the normalized pressures $\tilde{p}$, $\tilde{q}$, and
 $\tilde{\pi}$ satisfy the estimates
\begin{gather} \label{(7e)}
\begin{aligned}
 \|\tilde{q}\|_{2,Q}^2 + \|\tilde{\pi}\|_{2,Q}^2
& \leq  C_{\rm inc} \cdot \Bigl(\|\mathbf{F}\|_{2,Q}^2  + \|\Psi\|_{2,Q}^2
+\Bigl\| \frac{\partial \mathbf{F}}{\partial  t}\Bigr\|_{2,Q}^2 \\
 +\Bigl\|\frac{\partial \Psi}{\partial t}\Bigr\|_{2,Q}^2
 &\quad  +\| \mathbf{F}|_{t=0}\|_{2,\Omega}^2  +\|\Psi|_{t=0}\|_{2,\Omega}^2\Bigr)
 := C^*_{\rm inc},
 \end{aligned}
\\
 \label{(6e)}
\|\tilde{p}\|_{2,Q}^2 \leq C^*_{\rm inc} +C_{\rm en}(T),
 \end{gather}
 where $C_{\rm inc}=C_{\rm inc} (T,\Omega, \alpha_\tau, \alpha_F,
\alpha_\mu, \alpha_\lambda,
\alpha_{\theta f}, \alpha_{\theta s}, \rho_f, \rho_s, c_{pf},
c_{ps})$.
\end{theorem}

\begin{remark} \rm
We emphasize that the constants $C_{\rm en}(T)$ and $C_{\rm inc}$ do not
depend on $\alpha_p$ and $\alpha_\eta$, which implies that the
obtained bounds are applicable for studying asymptotic as
$\alpha_p$ and $\alpha_\eta$ grow infinitely. Moreover, notice
that the constants $C_{\rm en}(T)$ and $C_{\rm inc}$ also do not depend on
geometry of $\Omega_f$ and $\Omega_s$. This fact seems to be very
useful in view of possible studies of homogenization topics in
periodic structures (like in \cite{BP,SP}) for \emph{Model A}.
\end{remark}

\begin{proof} From Lemma \ref{lemma1e}, we have the bound
\begin{equation} \label{(9e)}
\begin{aligned}
&\frac{1}{2} \alpha_\tau \Bigl\|\sqrt{\bar{\rho}}\frac{\partial^2
\mathbf{w}}{\partial t^2}\Bigr\|_{2,\Omega\times (0,\tau)}^2 +\frac{1}{2}
\Bigl\|\sqrt{\bar{c}_p} \frac{\partial \theta}{\partial
t}\Bigr\|_{2,\Omega \times (0,\tau)}^2\\
&\leq \int_0^\tau \Bigl[\frac{\alpha_F^2}{2\alpha_\tau}
\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{F}}{\partial
t}\Bigr\|_{2,\Omega \times (0,t)}^2 +
\frac{1}{2\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega \times (0,t)}^2\Bigr] e^{\tau-t} dt\\
&\quad +\frac{\alpha_F^2}{2\alpha_\tau} \Bigl\|\sqrt{\bar{\rho}}
\frac{\partial \mathbf{F}}{\partial t}\Bigr\|_{2,\Omega \times (0,\tau)}^2
+ \frac{1}{2\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega \times (0,\tau)}^2\\
&\quad + \frac{e^\tau +1}{2} \Bigl[\frac{\alpha_F^2}{\alpha_\tau}
\bigl\|\sqrt{\bar{\rho}} \mathbf{F}|_{t=0}\bigr\|_{2,\Omega}^2
+\frac{1}{\min\{c_{pf},c_{ps}\}} \bigl\|\Psi|_{t=0}\bigr\|_{2,\Omega}^2\Bigr]\\
&\leq \frac{e^\tau +1}{2} \Bigl[\frac{\alpha_F^2}{\alpha_\tau}
\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{F}}{\partial
t}\Bigr\|_{2,\Omega \times (0,\tau)}^2
+\frac{1}{\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega \times (0,\tau)}^2\\
&\quad +\frac{\alpha_F^2}{\alpha_\tau} \bigl\|\sqrt{\bar{\rho}}
\mathbf{F}|_{t=0}\bigr\|_{2,\Omega}^2 +\frac{1}{\min\{c_{pf},c_{ps}\}}
\bigl\|\Psi|_{t=0}\bigr\|_{2,\Omega}^2 \Bigr],
 \quad \forall\,
\tau\in [0,T]\,.
\end{aligned}
\end{equation}
Justification of estimates \eqref{(6e)}--\eqref{(7e)} is based on
use of this bound, on a special choice of test functions in
integral equality \eqref{(3c)}, and on application of energy
inequality \eqref{(12c)} and Remark \ref{remark1e}.

We integrate the first term in  \eqref{(3c)} by parts,
substitute \eqref{(39a)} and \eqref{(42a)}, and replace $p$
and $\pi$ by $\tilde{p}$ and $\tilde{\pi}$, respectively, which is
legal due to Remark \ref{remark1e}. Thus we get
\begin{equation} \label{(12e)}
\begin{aligned}
&\int_{\Omega \times (0,\tau)} \Bigl(\alpha_\tau \bar{\rho}
\frac{\partial^2 \mathbf{w}}{\partial t^2}\cdot \mathbf{\varphi}
-\tilde{z} \mathop{\rm div}_x\mathbf{\varphi} +\bar{\chi} \alpha_\mu
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr): \mathbf{D}(x,\mathbf{
\varphi})\\
& +(1-\bar{\chi}) \alpha_\lambda
\mathbf{D}(x,\mathbf{w}):\mathbf{D}(x,\mathbf{\varphi}) -\bar{\alpha}_\theta \theta
\mathop{\rm div}_x \mathbf{\varphi} -\alpha_F \bar{\rho} \mathbf{F} \cdot \mathbf{
\varphi}\Bigr) d\mathbf{x} dt =0,
\end{aligned}
\end{equation}
for all $\tau \in [0,T]$. Here we denote $\tilde{z}:= \tilde{q} +\tilde{\pi}$
for briefness.

Now our aim is to choose a test function $\mathbf{\varphi}$ in
this equality such that $\mathop{\rm div}_x \mathbf{\varphi} =\tilde{z}$ and
all integrals make perfect sense and admit estimates in terms of
$\|\tilde{z}\|_{2,\Omega\times (0,\tau)}$ independently of
$\alpha_p$ and $\alpha_\eta$. Evidently, if this choice of
$\mathbf{\varphi}$ is possible then it leads directly to a bound
on $\|\tilde{z}\|_{2,\Omega\times (0,\tau)}$, uniform in
$\alpha_p$ and $\alpha_\eta$.
 We succeed to pick up such a test
function as follows. Introduce successively $\varphi_1
=\varphi_1(\mathbf{x},t)$ as the solution of the Dirichlet problem for
Poisson's equation on $\Omega$ for $t\in [0,T]$:
\begin{equation} \label{(13e)}
\Delta_x \varphi_1 =\tilde{z},\quad \varphi_1\big|_{\partial \Omega}
=0
\end{equation}
and $\mathbf{\varphi}_2 =\mathbf{\varphi}_2(\mathbf{x},t)$ such that
\begin{equation} \label{(14e)}
\mathop{\rm div}_x \mathbf{\varphi}_2 =0, \quad \mathbf{
\varphi}_2|_{\partial \Omega} =\nabla_x \varphi_1|_{\partial
\Omega}.
\end{equation}

Note that due to the classical theory of elliptic equations, a
solution of problem \eqref{(13e)} exists, is uniquely defined by
$\tilde{z}$, and admits the bound \cite[Sec. II.7]{LAD}
\begin{equation} \label{(15e)}
\|\varphi_1\|_{\mathaccent"7017 {W_2^2}(\Omega)} \leq
C_1(\Omega) \|\tilde{z}\|_{2,\Omega} \quad \forall\, t\in [0,T],
\end{equation}
and that, on the strength of the construction in \cite[Chap. I,
Sec. 2.1]{LAD2} and property \eqref{(4e)}, a vector-function
$\mathbf{\varphi}_2$ satisfying \eqref{(14e)} may be found
such that
\begin{equation} \label{(16e)}
\|\mathbf{\varphi}_2\|_{W_2^1(\Omega)} \leq C_0(\Omega)
\|\nabla_x \varphi_1\|_{2,\partial \Omega} \quad \forall\, t\in
[0,T],
\end{equation}
which immediately implies the bound
\begin{equation} \label{(17e)}
\|\mathbf{\varphi}_2\|_{W_2^1(\Omega)} \leq C_2(\Omega)
\|\varphi_1\|_{\mathaccent"7017 {W_2^2}(\Omega)} \quad
\forall\, t\in [0,T],
\end{equation}
thanks to the well-known trace theorem \cite[Sec. I.6]{LAD}. In
inequalities \eqref{(15e)}--\eqref{(17e)} the constants
$C_0(\Omega)$, $C_1(\Omega)$, and $C_2(\Omega)$ depend merely on
regularity of $\partial \Omega$ and make sense, for example, for
$C^2$-piecewise smooth surfaces $\partial \Omega$.

The sum $\mathbf{\varphi}=\nabla_x \varphi_1 -\mathbf{
\varphi}_2$ is a valid test function for integral equality
\eqref{(12e)}. Inserting it and rearranging terms, we obtain the
equality
\begin{equation} \label{(18e)}
\begin{aligned}
&\int_{\Omega\times (0,\tau)} \tilde{z}^2 d\mathbf{x} dt\\
&=\int_{\Omega\times (0,\tau)} \Bigl(\alpha_\tau \bar{\rho}
\frac{\partial^2 \mathbf{w}}{\partial t^2} \cdot (\nabla_x \varphi_1
-\mathbf{\varphi}_2) +\bar{\chi} \alpha_\mu
\mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}\Bigr): \mathbf{D}(x,\nabla_x \varphi_1)\\
&\quad -\bar{\chi} \alpha_\mu \mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr): \mathbf{D}(x,\mathbf{\varphi}_2) + (1-\bar{\chi})
\alpha_\lambda
\mathbf{D}(x,\mathbf{w}):\mathbf{D}(x,\nabla_x \varphi_1)\\
&\quad -(1-\bar{\chi}) \alpha_\lambda \mathbf{D}(x,\mathbf{w}):\mathbf{D}(x,\mathbf{\varphi}_2)
-\bar{\alpha}_\theta \theta \tilde{z}
 -\alpha_F \bar{\rho} \mathbf{F} \cdot (\nabla_x
\varphi_1 -\mathbf{\varphi}_2)\Bigr) d\mathbf{x} dt,
\end{aligned}
\end{equation}
for all $\tau \in [0,T]$, due to \eqref{(13e)} and
\eqref{(14e)}.

Applying H\"{o}lder's and the Cauchy-Schwartz inequalities,
bounds \eqref{(9e)} and \eqref{(15e)}--\eqref{(17e)}, and energy
inequality \eqref{(12c)} to the right hand side of equality
\eqref{(18e)}, we estimate
\begin{equation} \label{(19e)}
\begin{aligned}
&\|\tilde{z}\|_{2,\Omega \times (0,\tau)}^2 \\
& \leq \frac{1}{2}\Bigl(\frac{1}{\varepsilon_1}
+\frac{1}{\varepsilon_2}\Bigr) \Bigl[\alpha_\tau
\Bigl\|\sqrt{\bar{\rho}}\frac{\partial^2 \mathbf{w}}{\partial
t^2}\Bigr\|_{2,\Omega \times (0,\tau)}^2 +\alpha_F
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega \times (0,\tau)}^2\Bigr]\\
&\quad +\frac{1}{2}\Bigl(\frac{1}{\varepsilon_3}
+\frac{1}{\varepsilon_4}\Bigr)\Bigl[\alpha_\mu \Bigl\|\bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2 +\alpha_\lambda
\|(1-\bar{\chi})\mathbf{D}(x,\mathbf{w})\|_{2,\Omega\times (0,\tau)}^2\Bigr]\\
&\quad +\frac{1}{2\varepsilon_5} \|\sqrt{\bar{\alpha}_\theta}
\theta\|_{2,\Omega \times (0,\tau)}^2 + \frac{(\alpha_\tau
+\alpha_F) \max\{\rho_f,\rho_s\} \varepsilon_1}{2} \|\nabla_x
\varphi_1\|_{2,\Omega\times (0,\tau)}^2\\
&\quad +\frac{\max\{\alpha_\mu,\alpha_\lambda\} \varepsilon_3}{2}
\|\mathbf{D}(x,\nabla_x \varphi_1)\|_{2,\Omega \times (0,\tau)}^2
+\frac{\max\{\alpha_\mu,\alpha_\lambda\} \varepsilon_4}{2}
\|\mathbf{D}(x,\mathbf{\varphi}_2)\|_{2,\Omega \times (0,\tau)}^2\\
&\quad +\frac{(\alpha_\tau +\alpha_F) \max\{\rho_f,\rho_s\}
\varepsilon_2}{2} \|\mathbf{\varphi}_2\|_{2,\Omega\times
(0,\tau)}^2 + \frac{\max \{\alpha_{\theta f}, \alpha_{\theta s}\}
\varepsilon_5}{2} \|\tilde{z}\|_{2,\Omega \times (0,\tau)}^2
\\
&\leq \frac{e^\tau+1}{2}\Bigl(\frac{1}{\varepsilon_1}
+\frac{1}{\varepsilon_2}\Bigr)
\Bigl[\frac{\alpha_F^2}{\alpha_\tau}
\Bigl\|\sqrt{\bar{\rho}}\frac{\partial \mathbf{F}}{\partial
t}\Bigr\|_{2,\Omega \times (0,\tau)}^2
+\frac{1}{\min\{c_{pf},c_{ps}\}} \Bigl\|\frac{\partial
\Psi}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2 \\
&\quad +\frac{\alpha_F^2}{\alpha_\tau} \bigl\|\sqrt{\bar{\rho}}
\mathbf{F}|_{t=0}\bigr\|_{2,\Omega}^2 + \frac{1}{\min\{c_{pf},c_{ps}\}}
\bigl\| \Psi|_{t=0}\bigr\|_{2,\Omega}^2 + \alpha_F
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega \times (0,\tau)}^2\Bigr]\\
&\quad +\frac{1}{2} \Bigl( \frac{1}{\varepsilon_3}
+\frac{1}{\varepsilon_4}\Bigr)
\Biggl[\frac{\alpha_F^2}{\alpha_\tau} \|\sqrt{\bar{\rho}}
\mathbf{F}\|_{2,\Omega \times (0,\tau)}^2 +\frac{1}{\min\{c_{pf},c_{ps}\}}
\|\Psi\|_{2,\Omega \times (0,\tau)}^2\\
&\quad + \int_0^\tau \Bigl(\frac{\alpha_F^2}{\alpha_\tau}
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega \times (0,t)}^2 +
\frac{1}{\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega \times
(0,t)}^2\Bigr) e^{\tau-t} dt \Biggr]\\
&\quad +\frac{1}{2\varepsilon_5} \frac{\max\{\alpha_{\theta f},
\alpha_{\theta s}\}}{\min\{c_{pf},c_{ps}\}}
\Biggl[\frac{\alpha_F^2}{\alpha_\tau}  \|\sqrt{\bar{\rho}}
\mathbf{F}\|_{2,\Omega \times (0,\tau)}^2 +
\frac{1}{\min\{c_{pf},c_{ps}\}}
\|\Psi\|_{2,\Omega \times (0,\tau)}^2\\
&\quad +\int_0^\tau \Bigl(\frac{\alpha_F^2}{\alpha_\tau}
\|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega \times (0,t)}^2 +
\frac{1}{\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega \times
(0,t)}^2\Bigr) e^{\tau-t} dt\Biggr]\\
&\quad  +\frac{C_1^2(\Omega)}{2} \Bigl[(\alpha_\tau +\alpha_F)
\max\{\rho_f,\rho_s\} (\varepsilon_1 +C_2^2(\Omega)
\varepsilon_2)\\
&\quad +\max\{\alpha_\mu,\alpha_\lambda\} (\varepsilon_3 +C_2^2(\Omega)
\varepsilon_4) +\max\{\alpha_{\theta f},\alpha_{\theta s}\}
\varepsilon_5\Bigr] \|\tilde{z}\|_{2,\Omega\times (0,\tau)}^2.
\end{aligned}
\end{equation}
Choosing
\begin{gather*}
\varepsilon_1 =2/(5 C_1^2(\Omega) (\alpha_\tau +\alpha_F)
\max\{\rho_f,\rho_s\}),\\
  \varepsilon_2 =2/(5 C_1^2(\Omega) C_2^2(\Omega)
(\alpha_\tau +\alpha_F) \max\{\rho_f,\rho_s\}),\\
 \varepsilon_3 =2/(5 C_1^2(\Omega)
\max\{\alpha_\mu,\alpha_\lambda\}),\\
 \varepsilon_4 =2/(5 C_1^2(\Omega) C_2^2(\Omega)
\max\{\alpha_\mu,\alpha_\lambda\}),\\
 \varepsilon_5 =2/(5 C_1^2(\Omega)
\max\{\alpha_{\theta f},\alpha_{\theta s}\}),
\end{gather*}
we deduce the estimate
\begin{equation} \label{(6e-2)}
\begin{aligned}
& \frac{1}{2}\|\tilde{z}\|_{2,\Omega\times (0,\tau)}^2 \\
& \leq  C_*^{(1)} \Bigl[\frac{\alpha_F^2}{\alpha_\tau}
 \|\sqrt{\bar{\rho}} \mathbf{F}\|_{2,\Omega\times (0,\tau)}^2
 +\frac{1}{\min\{c_{pf},c_{ps}\}} \|\Psi\|_{2,\Omega\times
 (0,\tau)}^2\Bigr]\\
&\quad  +C_*^{(2)} \Bigl[\frac{\alpha_F^2}{\alpha_\tau}
 \Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{F}}{\partial
 t}\Bigr\|_{2,\Omega\times (0,\tau)}^2
 +\frac{1}{\min\{c_{pf},c_{ps}\}}\Bigl\|\frac{\partial
 \Psi}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
&\quad  +\frac{\alpha_F^2}{\alpha_\tau} \|\sqrt{\bar{\rho}}
 \mathbf{F}|_{t=0}\|_{2,\Omega}^2 + \frac{1}{\min\{c_{pf},c_{ps}\}}
 \| \Psi|_{t=0}\|_{2,\Omega}^2 +\alpha_F \|\sqrt{\bar{\rho}}
 \mathbf{F}\|_{2,\Omega\times (0,\tau)}^2\Bigr],
 \end{aligned}
\end{equation}
where
\begin{gather*}
\begin{aligned}
C_*^{(1)} &=(5/4) (e^\tau +1) C_1^2(\Omega) (1+C_2^2(\Omega))
\max\{\alpha_\mu,\alpha_\lambda\} \\
&\quad +(5/4) (e^\tau +1) C_1^2(\Omega) (\max\{\alpha_{\theta
f},\alpha_{\theta s}\})^2/\min\{c_{pf},c_{ps}\},
\end{aligned}
\\
C_*^{(2)}= (5/4) (e^\tau +1) C_1^2(\Omega)(1+C_2^2(\Omega))
(\alpha_\tau + \alpha_F) \max\{\rho_f,\rho_s\}.
\end{gather*}
Since the supports of $\tilde{q}$ and $\tilde{\pi}$ do not
intersect, we have
$$
\|\tilde{z}\|_{2,\Omega\times (0,\tau)}^2
=\|\tilde{q}\|_{2,\Omega\times (0,\tau)}^2 +
\|\tilde{\pi}\|_{2,\Omega\times (0,\tau)}^2.
$$
Thus bound \eqref{(7e)} immediately follows from inequality \eqref{(6e-2)}.
Finally, estimate \eqref{(6e)} follows from bound \eqref{(7e)},
energy estimate \eqref{(12c)}, and equation \eqref{(3e)}.
\end{proof}

\section{Additional Estimates for the Deformation Tensor}
 \label{Deformation}

Also we are interested in investigating limiting regimes arising
as $\alpha_\lambda$ grows infinitely. In order to fulfill this
study, it is necessary to establish additional bounds on solution
of \emph{Model A}, which should be in certain sense uniform in
$\alpha_\lambda$.

In this line, the fundamental estimate immediately follows from
energy inequality \eqref{(12c)}:
\begin{equation} \label{(1f)}
\mathop{\rm ess\,sup}_{ t\in [0,\tau]} \|(1-\bar{\chi})
\mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega}^2 \leq C_{\rm en}(\tau)/\alpha_\lambda, \quad
\forall\, \tau\in [0,T].
\end{equation}

\begin{remark} \label{remark1f} \rm
Assume that $(1-\bar{\chi})\mathbf{D}(x,\mathbf{w}_0)=0$,
which implies that $C_{\rm en}(\tau)$ is independent of
$\alpha_\lambda$. Then from bound \eqref{(1f)} it obviously
follows that
$$
\|(1-\bar{\chi})\mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega} \longrightarrow 0\quad
\mbox{ as } \alpha_\lambda \nearrow \infty.
$$
Since the kernel of
the operator $\mathbf{\phi} \mapsto \mathbf{D}(x,\mathbf{\phi})$ is the
set of absolutely rigid body motions, i.e., translations and
rotations $\mathbf{\phi}(\mathbf{x}) =\mathbf{x}_0 +\mathbf{\omega}\times \mathbf{x}$
($\mathbf{x}_0,\mathbf{\omega}=const$) \cite[Chap. III, \S 2.1]{RESH}, we
see that the infinite growth of $\alpha_\lambda$ leads to
\emph{solidification} limiting regimes.
\end{remark}

The following assertion provides one more useful bound in the case
of potential (e.g., gravitational) mass forces under additional
assumption on geometry of $\Omega_s$:

\begin{assumption} \label{assumption1f} \rm
Let the Lebesgue measure of $\partial \Omega \cap \partial
\Omega_s$ be strictly positive and $\Omega_s$ be connected.
\end{assumption}

\begin{theorem} \label{proposition1f}
Let Assumption \ref{assumption1f} hold. Suppose that initial data satisfy
\eqref{(5e)}, the derivative $\partial \Psi/\partial t$ belongs to
$L^2(Q)$, and the given density of distributed mass forces has the
form
\begin{equation} \label{(2f)}
\mathbf{F}(\mathbf{x},t) =\nabla_x \Phi(\mathbf{x},t)
\end{equation}
with some potential $\Phi \in W_2^1(Q)$ such that
$(\partial/\partial t) \nabla_x \Phi \in L^2(Q)$.
Then the following bound holds:
\begin{equation} \label{(3f)}
\begin{aligned}
&\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,T]}
\alpha_\tau \alpha_\lambda \Bigl\|\sqrt{\bar{\rho}} \frac{\partial
\mathbf{w}}{\partial t}(t)\Bigr\|_{2,\Omega}^2 + \alpha_\nu \alpha_\lambda
\Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial
t}\Bigr\|_{2,Q}^2\\
&+ \alpha_\mu \alpha_\lambda \Bigl\|\bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr)\Bigr\|_{2,Q}^2 +
\frac{1}{4} \frac{\alpha_\eta}{\alpha_\lambda} \mathop{\rm ess\,sup}_{t\in [0,T]}  \Bigl\|(1-\bar{\chi})\alpha_\lambda
\mathop{\rm div}_x \mathbf{w}(t)\Bigr\|_{2,\Omega}^2\\
&+\frac{1}{4}\frac{\alpha_p}{\alpha_\lambda} \mathop{\rm ess\,sup}_{t\in [0,T]}  \Bigl\|\bar{\chi} \alpha_\lambda \mathop{\rm div}_x
\mathbf{w}(t)\Bigr\|_{2,\Omega}^2 + \frac{1}{4} \mathop{\rm ess\,sup}_{t\in [0,T]}  \|(1-\bar{\chi}) \alpha_\lambda
\mathbf{D}(x, \mathbf{w}(t))\|_{2,\Omega}^2\\
&\leq C_{\rm sol}  \Bigl(1+\frac{\alpha_\lambda}{\alpha_\eta}
+\frac{\alpha_\lambda}{\alpha_p}\Bigr) \Bigl(\|\Phi\|_{W_2^1(Q)}^2
+\Bigl\|\frac{\partial}{\partial t} \nabla_x \Phi\Bigr\|_{2,Q}^2
+\|\Psi\|_{2,Q}^2 +\Bigl\|\frac{\partial \Psi}{\partial
t}\Bigr\|_{2,Q}^2\Bigr),
\end{aligned}
\end{equation}
where $C_{\rm sol}=C_{\rm sol} (T,\Omega,\bar{\chi},\alpha_{\theta},
\alpha_F, \bar{\rho}, \bar{c}_p)$.
\end{theorem}

\begin{remark} \label{remark2e} \rm
We emphasize that the constant
$C_{\rm sol}$ does not depend on $\alpha_p$, $\alpha_\eta$, and
$\alpha_\lambda$. Thus estimate \eqref{(3f)} becomes uniform in
$\alpha_\lambda \nearrow \infty$, whenever it is assumed that
$\alpha_\eta$ and $\alpha_p$ also grow infinitely such that
\begin{equation} \label{(23e)}
\alpha_\eta =O(\alpha_\lambda),\quad \alpha_p =O(\alpha_\lambda)
\quad \mbox{as} \quad \alpha_\lambda\nearrow \infty.
\end{equation}
\end{remark}

\begin{remark} \label{remark3f} \rm
On the strength of \eqref{(41a)}--\eqref{(43a)} and energy
estimate \eqref{(12c)}, we observe that
\begin{gather*}
\frac{\alpha_p}{\alpha_\lambda} \mathop{\rm ess\,sup}_{t\in [0,T]}  \Bigl\|\bar{\chi} \alpha_\lambda \mathop{\rm div}_x
\mathbf{w}(t)\Bigr\|_{2,\Omega}^2  =  \frac{\alpha_\lambda}{\alpha_p}
\mathop{\rm ess\,sup}_{t\in [0,T]}
\|p(t)\|_{2,\Omega}^2,\\
 \frac{\alpha_\eta}{\alpha_\lambda}
\mathop{\rm ess,sup}_{t\in [0,T]} \Bigl\|(1-\bar{\chi})\alpha_\lambda
\mathop{\rm div}_x \mathbf{w}(t)\Bigr\|_{2,\Omega}^2  =
\frac{\alpha_\lambda}{\alpha_\eta} \mathop{\rm ess\,sup}_{t\in [0,T]}
 \|\pi(t)\|_{2,\Omega}^2,\\
 \|q\|_{2,Q} \leq \|p\|_{2,Q} \leq \|p\|_{2,Q} + \alpha_\nu
\Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial
t}\Bigr\|_{2,Q}  \leq  \|p\|_{2,Q} + \sqrt{\alpha_\nu
C_{\rm en}(T)}.
\end{gather*}
These formulas and estimate \eqref{(3f)} yield that the pressures
$p$, $q$, and $\pi$ stay bounded in $L^2(Q)$ as $\alpha_\lambda$,
$\alpha_\eta$, and $\alpha_p$ grow infinitely such that limiting
relations \eqref{(23e)} hold.
\end{remark}

\begin{proof}[Proof of Theorem \ref{proposition1f}]
 Substitute
\eqref{(39a)}, \eqref{(41a)}--\eqref{(43a)}, \eqref{(5e)} into
\eqref{(3c)} to get
\begin{equation} \label{(4f)}
\begin{aligned}
&\int_0^\tau \int_\Omega \Bigl(\alpha_\tau \bar{\rho}
\frac{\partial \mathbf{w}}{\partial t}\cdot \frac{\partial \mathbf{
\varphi}}{\partial t} -\bar{\chi} \alpha_p \mathop{\rm div}_x\mathbf{w} \cdot \mathop{\rm div}_x
\mathbf{\varphi} -\bar{\chi} \alpha_\nu \mathop{\rm div}_x \frac{\partial
\mathbf{w}}{\partial t} \cdot \mathop{\rm div}_x \mathbf{\varphi}\\
&-\bar{\chi}\alpha_\mu \mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr)
:\mathbf{D}(x,\mathbf{\varphi}) -(1-\bar{\chi}) \alpha_\eta \mathop{\rm div}_x \mathbf{w}
\cdot \mathop{\rm div}_x \mathbf{\varphi}\\
&-(1-\bar{\chi}) \alpha_\lambda \mathbf{D}(x,\mathbf{w}):\mathbf{D}(x,\mathbf{\varphi})
+\bar{\alpha}_\theta \theta \mathop{\rm div}_x \mathbf{\varphi} -\alpha_F
\bar{\rho} \mathbf{F}\cdot \mathbf{\varphi}\Bigr) d\mathbf{x} dt\\
&=\int_\Omega \alpha_\tau \bar{\rho} \frac{\partial \mathbf{w}}{\partial t}
(\mathbf{x},\tau) \cdot \mathbf{\varphi}(\mathbf{x},\tau) d\mathbf{x}, \quad \forall\,
\tau\in [0,T].
\end{aligned}
\end{equation}
On the strength of Lemma \ref{lemma1e}, $\mathbf{\varphi}
=\alpha_\lambda \partial \mathbf{w}/\partial t$ is a valid test function
for  \eqref{(4f)}. Inserting it and  \eqref{(2f)} into
\eqref{(4f)}, we arrive at the equality
\begin{equation} \label{(5f)}
\begin{aligned}
&\frac{1}{2} \alpha_\tau \alpha_\lambda \Bigl\|\sqrt{\bar{\rho}}
\frac{\partial \mathbf{w}}{\partial t}(\tau)\Bigr\|_{2,\Omega}^2 +
\frac{1}{2} \alpha_\lambda \alpha_p \|\bar{\chi} \mathop{\rm div}_x
\mathbf{w}(\tau)\|_{2,\Omega}^2\\
& +\alpha_\nu \alpha_\lambda \Bigl\|\bar{\chi}
\mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial t}\Bigr\|_{2,\Omega\times
(0,\tau)}^2 + \alpha_\mu \alpha_\lambda \Bigl\|\bar{\chi}
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
& +\frac{1}{2} \alpha_\eta \alpha_\lambda \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}(\tau)\|_{2,\Omega}^2 +\frac{1}{2} \|(1-\bar{\chi})
\alpha_\lambda
\mathbf{D}(x,\mathbf{w}(\tau))\|_{2,\Omega}^2\\
&=-\int_0^\tau \int_\Omega \Bigl(\alpha_F \alpha_\lambda \bar{\rho}
\nabla_x \Phi \cdot \frac{\partial \mathbf{w}}{\partial t} -\alpha_\lambda
\bar{\alpha}_\theta \theta \mathop{\rm div}_x \frac{\partial \mathbf{w}}{\partial
t}\Bigr) d\mathbf{x} dt,\quad \forall\, \tau\in [0,T].
\end{aligned}
\end{equation}
Using integration by parts on the right-hand side of this
equality, we have
\begin{equation} \label{(8f)}
\begin{aligned}
&\frac{1}{2} \alpha_\tau \alpha_\lambda \Bigl\|\sqrt{\bar{\rho}}
\frac{\partial \mathbf{w}}{\partial t}(\tau)\Bigr\|_{2,\Omega}^2
+\alpha_\nu \alpha_\lambda \Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial
\mathbf{w}}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
& + \alpha_\mu
\alpha_\lambda \Bigl\|\bar{\chi} \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2 +
\frac{1}{2} \alpha_\lambda \alpha_p \|\bar{\chi} \mathop{\rm div}_x
\mathbf{w}(\tau)\|_{2,\Omega}^2\\
& +\frac{1}{2} \alpha_\lambda \alpha_\eta
\|(1-\bar{\chi}) \mathop{\rm div}_x \mathbf{w}(\tau)\|_{2,\Omega}^2
+\frac{1}{2}\|(1-\bar{\chi}) \alpha_\lambda
\mathbf{D}(x,\mathbf{w}(\tau))\|_{2,\Omega}^2
\\
&= \int_\Omega \alpha_F
\alpha_\lambda \rho_f \Phi(\mathbf{x},\tau) \mathop{\rm div}_x \mathbf{w}(\mathbf{x},\tau) d\mathbf{x}
-\int_0^\tau \int_\Omega \alpha_F \alpha_\lambda \rho_f
\frac{\partial \Phi}{\partial t} \mathop{\rm div}_x \mathbf{w}
d\mathbf{x} dt\\
&\quad-\int_\Omega \alpha_F \alpha_\lambda (1-\bar{\chi}(\mathbf{x})) (\rho_s
-\rho_f)
\nabla_x \Phi(\mathbf{x},\tau)\cdot \mathbf{w}(\mathbf{x},\tau) d\mathbf{x}\\
&\quad +\int_0^\tau \int_\Omega \alpha_F \alpha_\lambda (1-\bar{\chi})
(\rho_s -\rho_f) \Bigl(\frac{\partial}{\partial t} \nabla_x
\Phi\Bigr)
\cdot \mathbf{w} d\mathbf{x} dt\\
&\quad +\int_\Omega \alpha_\lambda \bar{\alpha}_\theta(\mathbf{x})
\theta(\mathbf{x},\tau)
\mathop{\rm div}_x \mathbf{w}(\mathbf{x},\tau) d\mathbf{x}\\
&\quad -\int_0^\tau \int_\Omega \alpha_\lambda \bar{\alpha}_\theta
\frac{\partial \theta}{\partial t} \mathop{\rm div}_x \mathbf{w} d\mathbf{x} dt, \quad
\forall\, \tau\in [0,T].
\end{aligned}
\end{equation}
Now, on the right-hand side apply the Cauchy-Schwartz and Korn's
inequality \eqref{Korn} (with $\Omega_s$ substituted for $\Omega$)
along with a simple estimate
\begin{equation} \label{(16f)}
\|\phi\|_{2,\Omega\times (0,\tau)}^2 \leq \tau
 \mathop{\rm ess\,sup}_{t\in [0,\tau]} \|\phi(t)\|_{2,\Omega}^2,\quad
\forall\, \phi\in L^\infty(0,T;L^2(\Omega))
\end{equation}
to obtain
%\begin{equation} \label{(17f)}
\begin{align*}
& \frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}\alpha_\tau \alpha_\lambda
\Bigl\|\sqrt{\bar{\rho}} \frac{\partial \mathbf{w}}{\partial
t}(t)\Bigr\|_{2,\Omega}^2 +\alpha_\nu \alpha_\lambda
\Bigl\|\bar{\chi} \mathop{\rm div}_x \frac{\partial
\mathbf{w}}{\partial t}\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
& + \alpha_\mu
\alpha_\lambda \Bigl\|\bar{\chi} \mathbf{D}\Bigl(x,\frac{\partial
\mathbf{w}}{\partial t}\Bigr)\Bigr\|_{2,\Omega\times (0,\tau)}^2 +
\frac{1}{2}\mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\lambda \alpha_\eta \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}(t)\|_{2,\Omega}^2\\
&+\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\alpha_\lambda \alpha_p \|\bar{\chi} \mathop{\rm div}_x \mathbf{w}(t)\|_{2,\Omega}^2 +
\frac{1}{2} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|(1-\bar{\chi})\alpha_\lambda\mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega}^2\\
& \leq \frac{\alpha^2_F \rho_f^2}{2\varepsilon_0}
\mathop{\rm ess\,sup}_{t\in [0,\tau]}\|(1-\bar{\chi})
\Phi(t)\|_{2,\Omega}^2 + \frac{\alpha^2_F
\rho_f^2}{2\varepsilon_1} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|\bar{\chi} \Phi(t)\|_{2,\Omega}^2\\
&\quad +\frac{\alpha^2_F \rho_f^2}{2\varepsilon_2} \Bigl\|(1-\bar{\chi})
\frac{\partial \Phi}{\partial t}\Bigr\|_{2,\Omega\times(0,\tau)}^2
+\frac{\alpha^2_F \rho_f^2}{2\varepsilon_3} \Bigl\|\bar{\chi}
\frac{\partial \Phi}{\partial
t}\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
&\quad +\frac{\alpha_F^2(\rho_s-\rho_f)^2}{2\varepsilon_4}
\mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|(1-\bar{\chi})\nabla_x\Phi(t)\|_{2,\Omega}^2\\
&\quad + \frac{\alpha_F^2(\rho_s-\rho_f)^2}{2\varepsilon_5}
\Bigl\|(1-\bar{\chi})\frac{\partial}{\partial t}
\nabla_x\Phi\Bigr\|_{2,\Omega\times (0,\tau)}^2\\
&\quad  +\frac{\alpha_{\theta s}^2}{2\varepsilon_6}
\mathop{\rm ess\,sup}_{t\in [0,\tau]} \|(1-\bar{\chi})
\theta(t)\|_{2,\Omega}^2 +\frac{\alpha_{\theta
f}^2}{2\varepsilon_7} \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|\bar{\chi} \theta(t)\|_{2,\Omega}^2\\
&\quad +\frac{\alpha_{\theta s}^2}{2\varepsilon_8} \Bigl\|(1-\bar{\chi})
\frac{\partial \theta}{\partial
t}\Bigr\|_{2,\Omega\times(0,\tau)}^2 +\frac{\alpha_{\theta
f}^2}{2\varepsilon_9} \Bigl\|\bar{\chi}\frac{\partial
\theta}{\partial
t}\Bigr\|_{2,\Omega\times(0,\tau)}^2\\
&\quad +\frac{\alpha_\lambda^2}{2} (\varepsilon_0 +\tau \varepsilon_2
+\varepsilon_6 +\tau \varepsilon_8) \mathop{\rm ess\,sup}_{t\in [0,\tau]} \|(1-\bar{\chi}) \mathop{\rm div}_x
\mathbf{w}(t)\|_{2,\Omega}^2\\
&\quad +\frac{\alpha_\lambda^2}{2} (\varepsilon_1 + \tau \varepsilon_3
+\varepsilon_7 +\tau \varepsilon_9) \mathop{\rm ess\,sup}_{t\in [0,\tau]}
\|\bar{\chi} \mathop{\rm div}_x \mathbf{w}(t)\|_{2,\Omega}^2\\
&\quad +\frac{(C_k(\Omega_s))^2}{2} (\varepsilon_4 +\tau \varepsilon_5)
\mathop{\rm ess\,sup}_{t\in [0,\tau]} \|(1-\bar{\chi})
\alpha_\lambda^2 \mathbf{D}(x,\mathbf{w}(t))\|_{2,\Omega}^2, \quad
 \forall\, \tau \in
[0,T].
\end{align*}
%\end{equation}
Choose
\begin{gather*}
\varepsilon_0 = \varepsilon_6 =
\frac{\alpha_\eta}{8\alpha_\lambda}, \quad \varepsilon_1 =
\varepsilon_7 = \frac{\alpha_p}{8\alpha_\lambda}, \quad
\varepsilon_2 =\varepsilon_8 = \frac{\alpha_\eta}{8 \tau
\alpha_\lambda},
\\
\varepsilon_3 = \varepsilon_9= \frac{\alpha_p}{8\tau
\alpha_\lambda}, \quad \varepsilon_4=\frac{1}{4(C_k(\Omega_s))^2},
\quad \varepsilon_5 =\frac{1}{4\tau (C_k(\Omega_s))^2},
\end{gather*}
and set $\tau=T$. Applying energy inequality \eqref{(12c)} and
Lemma \ref{lemma1e} in order to estimate the norms of $\theta$ and
$\partial \theta/\partial t$ in the above estimate,%\eqref{(17f)}
 we arrive finally
at inequality \eqref{(3f)}.
\end{proof}

\section{Incompressibility Limits} \label{Incompressibility}

In this section we prove the theorem, which explains the limiting
behavior of solutions of \emph{Model A}, provided with
homogeneous initial data, as the coefficients $\alpha_p$ and
$\alpha_\eta$ grow infinitely:

\begin{theorem} \label{theorem4}
Let $\mathbf{F},\, \partial \mathbf{F}/\partial t,\, \Psi,\, \partial
\Psi/\partial t \in L^2(Q)$, and initial data be homogeneous,
i.e., satisfy \eqref{(5e)}. Suppose that in \emph{Model A}
the positive constants $\rho_f$, $\rho_s$, $\alpha_\tau$,
$\alpha_\nu$, $\alpha_\mu$, $\alpha_\lambda$, $\alpha_{\theta f}$,
$\alpha_{\theta s}$, $\alpha_F$, $c_{pf}$, $c_{ps}$,
$\varkappa_f$, and $\varkappa_s$ are fixed and the coefficients
$\alpha_p$ and $\alpha_\eta$ depend on a small parameter
$\varepsilon>0$: $\alpha_p=\alpha_p^\varepsilon$,
$\alpha_\eta=\alpha_\eta^\varepsilon$. Let
\begin{equation} \label{(1g)}
\lim_{\varepsilon\searrow 0} \alpha_p^\varepsilon
=\alpha_p^0, \quad \lim_{\varepsilon\searrow 0}
\alpha_\eta^\varepsilon =\alpha_\eta^0.
\end{equation}
For a fixed $\varepsilon>0$ by
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ denote the generalized solution of \emph{Model
A} such that the pressures $p^\varepsilon$, $q^\varepsilon$, and
$\pi^\varepsilon$ are normalized, i.e., satisfy equations
\eqref{(1e)}--\eqref{(3e)} instead of
\eqref{(41a)}--\eqref{(43a)}.

Then the following assertions hold:

1) If $\alpha_p^0 \in (0,\infty)$ and $\alpha_\eta^0=\infty$, then
the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ is convergent such that
\begin{gather} \label{(2g-1)}
\mathbf{w}^\varepsilon \underset{\varepsilon\searrow 0}{\longrightarrow}
\mathbf{w},\; \frac{\partial \mathbf{w}^\varepsilon}{\partial t}
\underset{\varepsilon\searrow 0}{\longrightarrow} \frac{\partial
\mathbf{w}}{\partial t} \quad \mbox{weakly* in } L^\infty(0,T;
\mathaccent"7017 {W_2^1}(\Omega)),
\\
 \theta^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} \theta \quad \mbox{ weakly in }
L^2(0,T; \mathaccent"7017 {W_2^1}(\Omega)),\quad
 \mbox{weakly* in } L^\infty(0,T; L^2(\Omega)), \label{(2g-2)}
\\
\label{(3g)} p^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} p, \; q^\varepsilon
\underset{\varepsilon\searrow 0}{\longrightarrow} q, \;
\pi^\varepsilon \underset{\varepsilon\searrow 0}{\longrightarrow}
\pi \quad \mbox{ weakly in } L^2(Q),
\end{gather}
and the five limiting functions $(\mathbf{w},\theta,p,q,\pi)$ are the
generalized solution of \emph{Model B1} with
$\alpha_p=\alpha_p^0$. (Statement of \emph{Model B1} and the
notion of its generalized solution are given immediately after
formulation of the theorem.)

2) If $\alpha_p^0 =\infty$ and $\alpha_\eta^0\in (0,\infty)$, then
the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ is convergent such that the limiting relations
\eqref{(2g-1)}--\eqref{(3g)} hold true and, moreover, $p=q$ and
the four limiting functions $(\mathbf{w},\theta,q,\pi)$ are the
generalized solution of \emph{Model B2} with
$\alpha_\eta=\alpha_\eta^0$. (Statement of \emph{Model B2} and
the notion of its generalized solution are given after statement
of the definition of generalized solution of \emph{Model B1}.)

3) If $\alpha_p^0 =\alpha_\eta^0= \infty$, then the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ is convergent such that the limiting relations
\eqref{(2g-1)}--\eqref{(3g)} hold true and, moreover, $p=q$ and
the triple of limiting functions $(\mathbf{w},\theta,{\mathfrak H}
:= q+\pi)$ is the generalized solution of
\emph{Model B3}. (Statement of \emph{Model B3} and the notion
of its generalized solution are given after statement of the
definition of generalized solution of \emph{Model B2}.)
\end{theorem}

\noindent {\bf \emph{Statement of Model B1.}} In the space-time
cylinder $Q$ it is necessary to find a displacement vector $\mathbf{w}$, a
temperature distribution $\theta$, and distributions of pressures
$p$, $q$, and $\pi$, which satisfy equations
\eqref{(38a)}--\eqref{(40a)}, \eqref{(1e)}, and \eqref{(3e)},
incompressibility condition in the solid phase
\begin{equation} \label{(4g)}
(1-\bar{\chi}) \mathop{\rm div}_x \mathbf{w} =0, \quad (\mathbf{x},t)\in Q,
\end{equation}
homogeneous initial data \eqref{(5e)}, and homogeneous boundary
conditions \eqref{(45a)}.

\begin{definition} \label{definition1g} \rm
The set of functions $(\mathbf{w},\theta,p,q,\pi)$ is called a generalized
solution of \emph{Model B1} if they satisfy regularity
conditions \eqref{(46a)}, condition $\pi \in L^2(Q)$,
 initial data \eqref{(5e)}, integral equalities \eqref{(47a)} and
\eqref{(48a)}, and equations \eqref{(39a)}, \eqref{(1e)},
\eqref{(3e)}, and \eqref{(4g)} a.e. in $Q$.
\end{definition}


\noindent {\bf \emph{Statement of Model B2.}} In the space-time
cylinder $Q$ it is necessary to find a displacement vector $\mathbf{w}$, a
temperature distribution $\theta$, and distributions of pressures
$q$, and $\pi$, which satisfy equations
\eqref{(38a)}--\eqref{(40a)} and \eqref{(2e)}, incompressibility
condition in the liquid phase
\begin{equation} \label{(5g)}
\bar{\chi} \mathop{\rm div}_x \mathbf{w} =0, \quad (\mathbf{x},t)\in Q,
\end{equation}
homogeneous initial data \eqref{(5e)}, and homogeneous boundary
conditions \eqref{(45a)}.

\begin{definition} \label{definition2g} \rm
The set of functions $(\mathbf{w},\theta,q,\pi)$ is called a generalized
solution of \emph{Model B2} if they satisfy regularity
conditions \eqref{(46a)}, condition $q\in L^2(Q)$, initial data
\eqref{(5e)}, integral equalities \eqref{(47a)} and \eqref{(48a)},
and equations \eqref{(39a)}, \eqref{(2e)}, and \eqref{(5g)} a.e.
in $Q$.
\end{definition}

\noindent {\bf \emph{Statement of Model B3.}} In the space-time
cylinder $Q$ it is necessary to find a displacement vector $\mathbf{w}$, a
temperature distribution $\theta$, and a distribution of pressure
${\mathfrak H}$, which satisfy equations \eqref{(38a)},
\eqref{(40a)}, and
\begin{equation} \label{(6g)}
\mathbf{P}=-({\mathfrak H}+ \bar{\alpha}_\theta \theta) \mathbf{I} +\bar{\chi}
\alpha_\mu \mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial t}\Bigr) +
(1-\bar{\chi}) \alpha_\lambda \mathbf{D}(x,\mathbf{w}),\quad (\mathbf{x},t)\in Q,
\end{equation}
incompressibility condition in the both phases
\begin{equation} \label{(7g)}
\mathop{\rm div}_x \mathbf{w} =0, \quad (\mathbf{x},t)\in Q,
\end{equation}
homogeneous initial data \eqref{(5e)}, and homogeneous boundary
conditions \eqref{(45a)}.

\begin{definition} \label{definition3g} \rm
The triple $(\mathbf{w},\theta,{\mathfrak H})$ is called a generalized
solution of \emph{Model B3} if it satisfies regularity
conditions \eqref{(46a)}, condition ${\mathfrak H}\in L^2(Q)$,
initial data \eqref{(5e)}, integral equalities \eqref{(47a)} and
\eqref{(48a)}, and equations \eqref{(6g)} and \eqref{(7g)} a.e. in
$Q$.
\end{definition}

\begin{remark} \label{remark1g} \rm
Clearly Theorem \ref{theorem4} provides the existence results for
generalized solutions to \emph{Models B1--B3}, as a byproduct.
At the same time, existence and uniqueness of generalized
solutions of \emph{Models B1--B3} may be justified without
considering \emph{Model A} firstly, but starting from
Definitions \ref{definition1g}--\ref{definition3g}, introducing
the proper Galerkin's approximations, and keeping track of the
proof of Theorem \ref{theorem1}. Thus, \emph{Models B1--B3} are
well-posed.
\end{remark}

\begin{remark} \label{remark2g} \rm
Equations \eqref{(4g)}, \eqref{(5g)},
and \eqref{(7g)} are indeed the conditions of incompressibility,
since, on the strength of continuity equations \eqref{(25a)} and
\eqref{(29a)}, they imply that $\rho\equiv \rho_s$ in
$\Omega_s\times (0,T)$, $\rho\equiv \rho_f$ in $\Omega_f\times
(0,T)$, and $\rho\equiv \bar{\rho}$ in $Q$, respectively.
Consequently, \emph{Models B1, B2}, and \emph{B3} are
respective solid, liquid, and total incompressibility limits of
\emph{Model A}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{theorem4}]
 We verify assertion 3 of the theorem only. Justification of
assertions 1 and 2 is quite similar.

If $\alpha_p^\varepsilon,\alpha_\eta^\varepsilon
\underset{\varepsilon\searrow 0}{\longrightarrow} \infty$ then
there exist a subsequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ and a limiting set of functions $(\mathbf{w},\theta,p,q,
\pi)$ such that limiting relations \eqref{(2g-1)}--\eqref{(3g)}
hold true, due to energy estimate \eqref{(12c)}, Lemma
\ref{lemma1e}, Corollary \ref{corollary1c}, Theorem
\ref{proposition1e}, and homogeneous initial conditions
\eqref{(5e)}. Energy estimate \eqref{(12c)} yields that
\begin{gather*}
 \mathop{\rm ess\,sup}{t\in [0,T]} \|\bar{\chi} \mathop{\rm div}_x
\mathbf{w}^\varepsilon\|_{2,\Omega}^2 \leq
2(C_{\rm en}(T)/\alpha_p^\varepsilon) \underset{\varepsilon\searrow
0}{\longrightarrow} 0,\\
 \mathop{\rm ess\,sup}{t\in [0,T]} \|(1-\bar{\chi})
\mathop{\rm div}_x \mathbf{w}^\varepsilon\|_{2,\Omega}^2 \leq
2(C_{\rm en}(T)/\alpha_\eta^\varepsilon) \underset{\varepsilon\searrow
0}{\longrightarrow} 0,
\end{gather*}
since initial data are homogeneous and therefore $C_{\rm en}(T)$ does
not depend on $\varepsilon$. Hence  \eqref{(7g)} holds true for
the limiting function $\mathbf{w}$.

Next, substitute  \eqref{(1e)} into  \eqref{(3e)} and pass
to the limit as $\varepsilon \searrow 0$, using limiting relations
\eqref{(2g-1)} and \eqref{(3g)}, to get
$$
q=p-\bar{\chi} \alpha_\nu (\partial/\partial t) \mathop{\rm div}_x \mathbf{w}.
$$
By  \eqref{(7g)} this equality yields
$q=p$ on $Q$.

Now, from the limiting relations
\eqref{(2g-1)}--\eqref{(3g)}, passing to the limit in
\eqref{(47a)} and \eqref{(48a)} as $\varepsilon \searrow 0$, we
conclude that the triple $(\mathbf{w},\theta,q+\pi)$ is a generalized
solution of \emph{Model B3}.

To complete the proof of assertion 3, it remains to
notice that due to Remark \ref{remark1g} the solution of
\emph{Model B3} is unique. Hence the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ has exactly one (weak) limiting point and
therefore converges entirely, and there is no need to shift to a
subsequence.
\end{proof}

\section{Solidification Limits} \label{Solidification}

In this section we observe limiting behavior of solutions of
\emph{Model A}, as the coefficient $\alpha_\lambda$ grows
infinitely.
First, we prove the following result.

\begin{theorem} \label{theorem5}
Let $\mathbf{F},\, \partial \mathbf{F}/\partial t,\, \Psi,\, \partial
\Psi/\partial t \in L^2(Q)$, and initial data be homogeneous,
i.e., satisfy \eqref{(5e)}. Let Assumption \ref{assumption1f}
hold.

 Suppose that in \emph{Model A}
the positive constants $\rho_f$, $\rho_s$, $\alpha_\tau$,
$\alpha_p$, $\alpha_\eta$, $\alpha_\nu$, $\alpha_\mu$,
$\alpha_{\theta f}$, $\alpha_{\theta s}$, $\alpha_F$, $c_{pf}$,
$c_{ps}$, $\varkappa_f$, and $\varkappa_s$ are fixed and the
coefficient $\alpha_\lambda$ depends on a small parameter
$\varepsilon>0$, $\alpha_\lambda=\alpha_\lambda^\varepsilon$. Let
\begin{equation} \label{(1h)}
\lim_{\varepsilon\searrow 0} \alpha_\lambda^\varepsilon
=\infty.
\end{equation}
By
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ denote the generalized solution of \emph{Model
A} corresponding to a fixed $\varepsilon>0$.

Then the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,\pi^\varepsilon)$
is convergent such that
\begin{gather} \label{(2h)}
\mathbf{w}^\varepsilon \underset{\varepsilon\searrow 0}{\longrightarrow}
\mathbf{w},\; \frac{\partial \mathbf{w}^\varepsilon}{\partial t}
\underset{\varepsilon\searrow 0}{\longrightarrow} \frac{\partial
\mathbf{w}}{\partial t} \quad \mbox{weakly* in }
L^\infty(0,T;\mathaccent"7017  {W_2^1}(\Omega)),
\\
\label{(3h)} (1-\bar{\chi}) \nabla_x \mathbf{w}^\varepsilon
\underset{\varepsilon\searrow 0}{\longrightarrow} 0,\;
\pi^\varepsilon \underset{\varepsilon\searrow 0}{\longrightarrow}
0 \quad \mbox{strongly in } L^\infty(0,T;L^2(\Omega)),
\\
 \label{(4h)} \theta^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} \theta \quad \mbox{ weakly in }
L^2(0,T;\mathaccent"7017 {W_2^1}(\Omega)),
 \quad \mbox{weakly* in } L^\infty(0,T; L^2(\Omega)),
\\
\label{(5h)} p^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} p, \; q^\varepsilon
\underset{\varepsilon\searrow 0}{\longrightarrow} q \quad
  \mbox{weakly in } L^2(Q),
\end{gather}
and the four limiting functions $(\mathbf{w},\theta,p,q)$ are the
generalized solution of \emph{Model C1}.
\end{theorem}

\noindent {\bf \emph{Statement of Model C1.}} In the space-time
cylinder $Q$ it is necessary to find a displacement vector $\mathbf{w}$, a
temperature distribution $\theta$, and distributions of pressures
$p$ and $q$, which satisfy the equations
\begin{gather} \label{(6h)}
 \alpha_\tau \rho_f \frac{\partial^2 \mathbf{w}}{\partial
t^2}=\mathop{\rm div}_x \mathbf{P}_f + \alpha_F \rho_f \mathbf{F}, \quad \mbox{in }
\Omega_f\times (0,T),\\
 \label{(7h)}
 \mathbf{P}_f = -(q+\alpha_{\theta f} \theta)\mathbf{I}  +
\alpha_\mu \mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}}{\partial
t}\Bigr)\Bigr),
\quad \mbox{in } \Omega_f\times (0,T),\\
\label{(8h)}   (1-\bar{\chi})\mathbf{w}=0, \quad \mbox{in }
Q,\\
 \label{(9h)} \bar{c}_p \frac{\partial
\theta}{\partial t} =\mathop{\rm div}_x (\bar{\varkappa} \nabla_x \theta)
-\bar{\chi} \alpha_{\theta f} \frac{\partial}{\partial t} \mathop{\rm div}_x
\mathbf{w} +\Psi, \quad \mbox{in } Q,\\
\label{(10h)}   p+\bar{\chi} \alpha_p \mathop{\rm div}_x \mathbf{w}=0,
\quad \mbox{in } Q,\\
\label{(11h)}
q=p+\frac{\alpha_\nu}{\alpha_p}\frac{\partial p}{\partial t},
\quad \mbox{in } Q,
\end{gather}
homogeneous initial data \eqref{(5e)}, and homogeneous boundary
conditions
\begin{equation} \label{(12h)}
\mathbf{w}=0, \quad \mbox{for } \mathbf{x} \in \partial \Omega_f, \; t\geq 0;\quad
\theta=0,\quad \mbox{for } \mathbf{x} \in \partial \Omega, \; t\geq 0.
\end{equation}

\begin{definition} \label{definition1h} \rm
A set of functions $(\mathbf{w},\theta,p,q)$ is called a generalized solution
of \emph{Model C1} if they satisfy the regularity conditions
\begin{equation} \label{(13h)}
\mathbf{w},\,\frac{\partial \mathbf{w}}{\partial t},\, \nabla_x \mathbf{w},\, \theta,\,
\nabla_x \theta \in L^2(Q),
\end{equation}
equations \eqref{(7h)}, \eqref{(8h)}, \eqref{(10h)}, and
\eqref{(11h)} a.e. in $Q$, and the integral equalities
\begin{equation} \label{(14h)}
\int_0^T \int_{\Omega_f} \Bigl(\alpha_\tau \rho_f \frac{\partial
\mathbf{w}}{\partial t}\cdot \frac{\partial \mathbf{\varphi}}{\partial t}
-\mathbf{P}_f:\nabla_x \mathbf{\varphi} + \alpha_F \rho_f \mathbf{F}\cdot
\mathbf{\varphi}\Bigr) d\mathbf{x} dt  =0
\end{equation}
for all smooth $\mathbf{\varphi}=\mathbf{\varphi}(\mathbf{x},t)$ such
that $\mathbf{\varphi}|_{\partial \Omega_f} =\mathbf{
\varphi}|_{t=T}=0$ and
\begin{equation} \label{(15h)}
\int_Q \Bigl(\bar{c}_p \theta \frac{\partial \psi}{\partial t} -
\bar{\varkappa} \nabla_x \theta \cdot \nabla_x \psi + \bar{\chi}
\alpha_{\theta f} (\mathop{\rm div}_x \mathbf{w}) \frac{\partial \psi}{\partial t}
+\Psi \psi\Bigr) d\mathbf{x} dt =0
\end{equation}
for all smooth $\psi= \psi(\mathbf{x},t)$ such that $\psi|_{\partial
\Omega} = \psi|_{t=T}=0$.
\end{definition}

\begin{remark} \label{remark1h} \rm
As in Remark \ref{remark1g}, we notice that Theorem
\ref{theorem5} provides the existence result for \emph{Model C1}
as a byproduct, and that existence and uniqueness of generalized
solutions of \emph{Model C1} may be justified independently of
Theorem \ref{theorem5} by considerations similar to the proof of
Theorem \ref{theorem1}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{theorem5}]
Firstly,  we have that
limiting relations \eqref{(2h)}, \eqref{(4h)}, and \eqref{(5h)}
hold true for some subsequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon)$
due to energy estimate \eqref{(12c)} and Lemma \ref{lemma1e}.

Secondly, bound \eqref{(1f)} immediately implies that the limiting
displacement vector $\mathbf{w}$ satisfies the equality
$(1-\bar{\chi})\mathbf{D}(x,\mathbf{w})=0$ in $Q$. On the strength of Remark
\ref{remark1f} and Assumption \ref{assumption1f}, this equality
yields that  \eqref{(8h)} holds. From bound \eqref{(1f)},
Assumption \ref{assumption1f}, homogeneity of initial data (see
 \eqref{(5e)}), and Korn's inequality \eqref{Korn} (with
$\Omega_s$ substituted for $\Omega$) it follows that
\begin{align*}
&\mathop{\rm ess\,sup}_{t\in[0,\tau]}
\bigl(\|(1-\bar{\chi}) \mathbf{w}^\varepsilon(t)\|_{2,\Omega}^2 +
\|(1-\bar{\chi}) \nabla_x
\mathbf{w}^\varepsilon(t)\|_{2,\Omega}^2\bigr)\\
&\leq (C_k(\Omega_s))^2 \mathop{\rm ess\,sup}_{t\in[0,\tau]}
 \|(1-\bar{\chi}) \mathbf{D}(x,\mathbf{w}^\varepsilon(t))\|_{2,\Omega}^2
\underset{\varepsilon\searrow 0}{\longrightarrow} 0,
\end{align*}
which proves limiting relations \eqref{(3h)}. (By $C_k(\Omega_s)$
the constant in Korn's inequality is denoted.)

Next, from the limiting relations
\eqref{(2h)}--\eqref{(5h)}, passing to the limit in
\eqref{(47a)}, \eqref{(48a)}, \eqref{(41a)}, and \eqref{(42a)} as
$\varepsilon \searrow 0$, we conclude that the four functions
$(\mathbf{w},\theta,p,q)$ are a generalized solution of \emph{Model C1}.

To complete the proof, it remains to notice that due to
Remark \ref{remark1h} the solution of \emph{Model C1} is unique.
Hence the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon)$
has exactly one (weak) limiting point and therefore converges
entirely and there is no need to shift to a subsequence.
\end{proof}

To complete this article, we derive a rather peculiar asymptotic
behavior, which is observed thanks to additional bound \eqref{(3f)}.
Namely, we prove the following result.

\begin{theorem} \label{theorem6}
Let Assumption \ref{assumption1f} hold, initial data be
homogeneous, i.e., satisfy \eqref{(5e)}, the given density of
distributed mass forces have the form
\begin{equation} \label{(16h)}
\mathbf{F}(\mathbf{x},t) =\nabla_x \Phi(\mathbf{x},t)
\end{equation}
with some potential $\Phi \in W^1_2(Q)$ such that
$(\partial/\partial t) \nabla_x \Phi \in L^2(Q)$, and the given
volumetric density of exterior heat application be such that
$\Psi,\partial \Psi/\partial t\in L^2(Q)$.

Suppose that in \emph{Model A} the positive constants $\rho_f$,
$\rho_s$, $\alpha_\tau$, $\alpha_\nu$, $\alpha_\mu$,
$\alpha_{\theta f}$, $\alpha_{\theta s}$, $\alpha_F$, $c_{pf}$,
$c_{ps}$, $\varkappa_f$, and $\varkappa_s$ are fixed and the
coefficients $\alpha_\lambda$, $\alpha_p$, and $\alpha_\eta$
depend on a small parameter $\varepsilon>0$: $\alpha_\lambda
=\alpha_\lambda^\varepsilon$, $\alpha_p =\alpha_p^\varepsilon$,
and $\alpha_\eta =\alpha_\eta^\varepsilon$. Let limiting relations
\eqref{(23e)} hold, or, equivalently,
\begin{equation} \label{(17h)}
\lim_{\varepsilon\searrow 0} \alpha_\lambda^\varepsilon
=\infty,\quad \lim_{\varepsilon\searrow 0}
\frac{\alpha_p^\varepsilon}{\alpha_\lambda^\varepsilon}
=\alpha_p^0 \in (0,\infty),\quad \lim_{\varepsilon\searrow
0} \frac{\alpha_\eta^\varepsilon}{\alpha_\lambda^\varepsilon}
=\alpha_\eta^0 \in (0,\infty).
\end{equation}
By
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ denote the generalized solution of \emph{Model
A} corresponding to a fixed $\varepsilon>0$.

Then the sequence
$(\mathbf{w}^\varepsilon,\theta^\varepsilon,p^\varepsilon,q^\varepsilon,
\pi^\varepsilon)$ is convergent and there exist functions
$(\mathbf{u},\theta,p)$ such that
\begin{gather} \label{(18h)}
 \mathbf{w}^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} 0,\; \frac{\partial \mathbf{w}^\varepsilon}{\partial
t} \underset{\varepsilon\searrow 0}{\longrightarrow} 0 \quad
\mbox{strongly in } L^\infty(0,T;L^2(\Omega)),
\\
\label{(19h)} \theta^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} \theta \quad \mbox{weakly in }
L^2(0,T;\mathaccent"7017 {W^1_2}(\Omega)),
\quad \mbox{weakly* in } L^\infty (0,T;L^2(\Omega)),
\\
\label{(20h)} p^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} p,\; q^\varepsilon
\underset{\varepsilon\searrow 0}{\longrightarrow} p,
\\
\label{(21h)} \pi^\varepsilon \underset{\varepsilon\searrow
0}{\longrightarrow} -(1-\bar{\chi}) \alpha_\eta^0 \mathop{\rm div}_x \mathbf{u},
\\
\label{(21h-2)} (1-\bar{\chi}) \alpha_\lambda^\varepsilon
\mathbf{w}^\varepsilon \underset{\varepsilon\searrow 0}{\longrightarrow}
\mathbf{u} \quad \mbox{weakly* in }L^\infty (0,T;L^2(\Omega)),
\\
 \label{(22h)} (1-\bar{\chi}) \mathbf{D}(x,\alpha_\lambda^\varepsilon
\mathbf{w}^\varepsilon) \underset{\varepsilon\searrow 0}{\longrightarrow}
(1-\bar{\chi}) \mathbf{D}(x,\mathbf{u}) \quad \mbox{weakly* in } L^\infty
(0,T;L^2(\Omega)),
\end{gather}
and the triple $(\mathbf{u},\theta,p)$ is the generalized solution of
\emph{Model C2}, formulated below.
\end{theorem}

\noindent {\bf \emph{Statement of Model C2.}}
Find successively the following:\\
{\bf (C2a)} a temperature distribution $\theta$, solving
the initial-boundary value problem for the heat equation
\begin{eqnarray} \label{(23h)}
& & \bar{c}_p \frac{\partial \theta}{\partial t} =\mathop{\rm div}_x
(\bar{\varkappa}\nabla_x \theta) +\Psi, \mbox{in } Q,\\
\label{(24h)}  & &  \theta|_{t=0}=0, \mbox{ on }
\Omega,\quad \theta|_{\partial \Omega} =0, \mbox{ for } t>0;
\end{eqnarray}
{\bf (C2b)} a hydrostatic pressure  $p$ in the liquid
phase from the equilibrium equation
\begin{equation} \label{(25h)}
p=-\alpha_{\theta f} \theta +\alpha_F \rho_f \Phi, \mbox{ in }
\Omega_f\times (0,T);
\end{equation}
{\bf (C2c)} an up-scaled displacement vector field $\mathbf{u}$,
solving the mixed problem for the 3D system of stationary wave
equations
\begin{gather} \label{(26h)}
 \mathop{\rm div}_x \mathbf{D}(x,\mathbf{u}) +\alpha_\eta^0 \nabla_x \mathop{\rm div}_x \mathbf{u} =
\alpha_{\theta s} \nabla_x \theta -\alpha_F \rho_s \nabla_x \Phi,
 \mbox{ in } \Omega_s,\\
\label{(27h)}  \mathbf{D}(x,\mathbf{u})\cdot \mathbf{n} +\alpha_\eta^0
(\mathop{\rm div}_x \mathbf{u})\mathbf{n} =(\alpha_{\theta s} \theta -\alpha_F
\rho_f \Phi)\mathbf{n},
\mbox{ on } \Gamma,\\
\label{(28h)}  \mathbf{u}=0, \mbox{ on } (\partial \Omega_s\setminus
\Gamma).
\end{gather}
(Here $\mathbf{n}$ is the outward unit normal to $\partial
\Omega_s$. Variable $t$ appears in the system as a parameter.
Also, recall that $\Gamma:= \partial\Omega_s \cap
\partial \Omega_f$ and that the Lebesgue measure of $\partial
\Omega_s\setminus \Gamma$ is positive due to Assumption
\ref{assumption1f}.)

\begin{definition} \label{definition2h} \rm
The triple $(\mathbf{u},\theta,p)$ is called a generalized solution of
\emph{Model C2} if it satisfies the regularity conditions
\begin{equation} \label{(29h)}
\theta,\,\nabla_x\theta,\,p,\,(1-\bar{\chi})\nabla_x\mathbf{u} \in
L^\infty(0,T;L^2(\Omega)),
\end{equation}
and the integral equalities
\begin{equation} \label{(30h)}
\begin{aligned}
&\int_Q \Bigl(\bigl(\bar{\chi} p -(1-\bar{\chi}) \alpha_\eta^0
\mathop{\rm div}_x \mathbf{u} +\bar{\alpha}_\theta \theta\bigr)\mathop{\rm div}_x \mathbf{
\varphi}\\
& -(1-\bar{\chi}) \mathbf{D}(x,\mathbf{u}):\mathbf{D}(x,\mathbf{\varphi})
+\alpha_F \bar{\rho} \nabla_x\Phi \cdot \mathbf{\varphi}\Bigr)
d\mathbf{x} dt =0
\end{aligned}
\end{equation}
for all smooth $\mathbf{\varphi}=\mathbf{\varphi}(\mathbf{x},t)$ such
that $\mathbf{\varphi}|_{\partial \Omega} =\mathbf{
\varphi}|_{t=T} =0$ and
\begin{equation} \label{(31h)}
\int_Q \Bigl(\bar{c}_p \theta \frac{\partial \psi}{\partial t}
-\bar{\varkappa} \nabla_x \theta \cdot \nabla_x \psi +\Psi
\psi\Bigr) d\mathbf{x} dt=0
\end{equation}
for all smooth $\psi=\psi(\mathbf{x},t)$ such that $\psi|_{\partial
\Omega} =\psi|_{t=T}=0$.
\end{definition}

\begin{remark} \label{remark2h} \rm
By the standard arguments it is easy
to verify that integral equalities \eqref{(30h)} and \eqref{(31h)}
are equivalent to equations \eqref{(23h)}--\eqref{(28h)} in the
distributions sense. Due to the well-known theory of second-order
elliptic and parabolic equations \cite{LU,LSU} there exists
exactly one generalized solution of \emph{Model C2}, whenever
$\Phi$ and $\Psi$ satisfy conditions of Theorem \ref{theorem6} and
whenever we prescribe some certain values to $(1-\bar{\chi})p$ and
$\bar{\chi}\mathbf{u}$. (For example, we may simply set
$(1-\bar{\chi})p=\bar{\chi}\mathbf{u}=0$.)
\end{remark}

\begin{proof}[Proof of Theorem \ref{theorem6}]
 From energy estimate \eqref{(12c)}, bound \eqref{(3f)}, limiting
relations \eqref{(17h)}, and Remark \ref{remark3f}, we conclude
that limiting relations \eqref{(19h)}--\eqref{(22h)} hold true, at
least for some subsequence, and that limiting relation
\eqref{(18h)} and relations
\begin{equation} \label{(32h)}
\bar{\chi} \alpha_\nu \mathop{\rm div}_x \frac{\partial
\mathbf{w}^\varepsilon}{\partial t} \underset{\varepsilon\searrow
0}{\longrightarrow} 0,\quad \bar{\chi} \alpha_\mu
\mathbf{D}\Bigl(x,\frac{\partial \mathbf{w}^\varepsilon}{\partial t}\Bigr)
\underset{\varepsilon\searrow 0}{\longrightarrow} 0\quad
\mbox{strongly in } L^2(Q)
\end{equation}
are fulfilled for the entire sequence $\varepsilon\searrow 0$.

The limiting functions $\mathbf{u}$, $p$, and $\theta$ satisfy regularity
conditions \eqref{(29h)} due to relations
\eqref{(19h)}--\eqref{(22h)} and Korn's inequality
$$
\|\mathbf{u}(t)\|_{W^1_2(\Omega_s)} \leq
C_k(\Omega_s)\|\mathbf{D}(x,\mathbf{u}(t))\|_{2,\Omega_s},\quad \forall\, t\in
[0,T].
$$
Substituting \eqref{(41a)}--\eqref{(43a)}, \eqref{(5e)}, and
\eqref{(16h)} into integral equality \eqref{(47a)}, and passing to
the limit as $\varepsilon\searrow 0$, we arrive at integral
equality \eqref{(30h)} due to relations
\eqref{(18h)}--\eqref{(22h)} and \eqref{(32h)}. Analogously we
justify that $\theta$ satisfies integral equality \eqref{(31h)}.
Finally, it remains to notice that, since solution of
\emph{Model C2} is unique, the sequence
$((1-\bar{\chi})\alpha_\lambda^\varepsilon \mathbf{w}^\varepsilon,\,
\theta^\varepsilon,\, p^\varepsilon,\, q^\varepsilon,\,
\pi^\varepsilon)$ has exactly one (weak) limiting point and
therefore converges entirely. \end{proof}

\subsection*{Acknowledgments}
The authors were partially supported by the grants of Higher
Education Commission of Pakistan under National Research Program
for Universities. The titles of the grants are:
\emph{Homogenization of Underground Flows} (A. M. Meirmanov) and
\emph{Modern Mathematical Analysis for Phenomenon of Anisotropic
Diffusion and Acoustic Wave Propagation in Porous Media} (S. A.
Sazhenkov). Also, S. A. Sazhenkov was partially supported by
Russian Foundation for Basic Research (grants 03-01-00829 and
07-01-00309).


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\end{document}