\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 50, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/50\hfil Existence and uniqueness]
{Existence and uniqueness of the solution to a
3D thermoviscoelastic system}

\author[Elena Bonetti \&  Giovanna Bonfanti\hfil EJDE--2003/50\hfilneg]
{Elena Bonetti \&  Giovanna Bonfanti}

\address{Elena Bonetti \newline
Dipartimento di Matematica
``F. Casorati", Universit\`a di Pavia\\
Via Ferrata 1, 27100 Pavia, Italy}
\email{bonetti@dimat.unipv.it}

\address{Giovanna Bonfanti \newline
Dipartimento di Matematica, Universit\`a di Brescia\\
Via Branze 38, 25123 Brescia, Italy}
\email{bonfanti@ing.unibs.it}

\date{}
\thanks{Submitted December 9, 2002. Published April 29, 2003.}
\subjclass[2000]{74D10, 35K60, 74A15}
\keywords{3D thermoviscoelastic system, thermomechanical modelling, \newline\indent
nonlinear PDE's system, existence and uniqueness results}

\begin{abstract}
 This paper presents results on existence and uniqueness of solutions
 to a three-dimensional thermoviscoelastic system.  The constitutive
 relations of the model are recovered by a free energy functional
 and a pseudo-potential of dissipation. Using a fixed point
 argument, combined with an a priori estimates-passage to the limit
 technique, we prove a local existence result for a related initial
 and boundary values problem. Hence, uniqueness of the solution is
 proved on the whole time interval, as well as positivity of the
 absolute temperature.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\modulo}[1]{|#1|}

\section{Introduction to the model}

In this paper we deal with a three-dimensional model for
thermoviscoelastic systems, which is derived by  two energy
functionals: the free energy and the pseudo-potential of
dissipation. In particular, we refer to a  modelling approach by
Fr\'emond which is fully detailed and justified in \cite{Frlib},
where it is applied to different thermomechanics phenomena. Thus,
we first introduce the state variables of the model, which are the
absolute temperature $\theta$ and the linearized symmetric strain
tensor $\varepsilon(\mathbf{u})$ ($\mathbf{u}$ is the vector of
small displacements), specified by
\begin{equation}
\varepsilon_{ij}(\mathbf{u})=\frac12
(\partial_{x_i}u_j+\partial_{x_j}u_i),\quad
i,j=1,2,3,\quad\mathbf{u}=(u_1,u_2,u_3).\label{strain}
\end{equation}
Hence, the thermomechanical equilibrium of the system is described by
a free energy functional $\Psi$, which depends on the   non-dissipative
variables. We make precise this functional  as follows
\begin{equation}
\Psi(\theta,\varepsilon(\mathbf{u}))=-c_s\theta\log\theta+\frac12
\varepsilon(\mathbf{u})
K \varepsilon(\mathbf{u})+\alpha(\theta)\mathop{\rm tr}\varepsilon(\mathbf{u}),
\label{enfree}
\end{equation}
where $K$ denotes the elastic tensor, $c_s>0$ the heat capacity of the
system, and $\alpha(\theta)$ a thermal expansion coefficient acting
only on the trace of the strain tensor. More precisely,
we let
\begin{equation}
\alpha(\theta)=\alpha\theta,\quad\alpha\in\mathbb{R}. \label{alfa}
\end{equation}
Note that here and in the sequel, in regard of simplicity, we do
not use a specific notation for the tensorial product, while in
general  by $\cdot$ we denote the scalar product in $\mathbb{R}^n$. As
usual in elasticity theory, we assume the material to be
homogeneous and isotropic and we let
\begin{equation}
K\varepsilon(\mathbf{u})=\lambda\mathop{\rm
tr}\varepsilon(\mathbf{u})\mathbf{1}+2\mu\varepsilon(\mathbf{u}),\label{elasticita}
\end{equation}
where $\lambda$ and $\mu$ stand for the Lam\'e constants and
$\mathbf{1}$  for the identity matrix. Hence, we include
dissipation in the model by following the approach proposed by
Moreau (cf. \cite{Frlib} and references therein)  and introduce a
pseudo-potential of dissipation $\Phi$ depending on the
dissipative variables $\nabla\theta$ and $\varepsilon(\mathbf{u}_t)$,
related to the heat flux and the evolution of deformations,
respectively. In particular, we set
\begin{equation}
\Phi(\nabla\theta,\varepsilon(\mathbf{u}_t))={{k_0}\over{2\theta}}
\modulo{\nabla\theta}^2
+\frac12 \varepsilon(\mathbf{u}_t)B\varepsilon(\mathbf{u}_t),
\label{pseudodiss}
\end{equation}
where $k_0>0$ and $B$ is a positive definite  (symmetric) matrix. Now,
letting the system be located in a smooth bounded domain
$\Omega\subset\mathbb{R}^3$, during a finite time interval $[0,T]$, $T>0$, we
introduce the universal  balance laws of thermomechanics, that are the energy balance
\begin{equation}
e_t+\mathop{\rm div}\mathbf{q}=r+\sigma\varepsilon(\mathbf{u}_t)\quad\hbox{in
}\Omega\times(0,T), \label{intener}
\end{equation}
and the momentum balance (in which we account for macroscopic
accelerations)
\begin{equation}
\mathbf{u}_{tt}-\mathop{\rm div}\sigma=\mathbf{G}\quad\hbox{in
}\Omega\times(0,T), \label{mom}
\end{equation}
whose ingredients will be specified in a moment. In advance, let
us point out that by the subscript $t$ we denote the partial
derivative operator ${\partial\over{\partial t}}$. In \eqref{intener}
$e$ is the internal energy of the system, $\mathbf{q}$ the heat
flux, $r$ stands for  an external heat source, and the term
$\sigma\varepsilon(\mathbf{u}_t)$, $\sigma$ being the stress tensor,
accounts for mechanically induced heat sources; $\mathbf{G}$ in
\eqref{mom} denotes a volume force applied to the structure  from the
exterior. Next, \eqref{intener} and \eqref{mom} are complemented by suitable
boundary conditions. Denoting by $\mathbf{n}$ the outward unit
normal vector to the boundary $\Gamma:=\partial\Omega$, we let
\begin{equation}
-\mathbf{q}\cdot\mathbf{n}=h\quad\hbox{on
}\Gamma\times(0,T),\label{flussob}
\end{equation}
namely we assume that the heat flux through the boundary  is
known. Then, we prescribe homogeneous Dirichlet boundary
conditions on the displacement $\mathbf{u}$ and the velocity $\mathbf{u}_t$.
\begin{equation}
\mathbf{u}=\mathbf{u}_t=\mathbf{0}\quad\hbox{on }\Gamma\times(0,T). \label{spostb}
\end{equation}

Now, we  make precise the constitutive relations for dissipative and
non dissipative quantities in the model, in terms of the functionals
$\Psi$ and $\Phi$ (cf. \eqref{enfree} and \eqref{pseudodiss}). We have, as usual, the internal energy $e$ related
to the entropy
\begin{equation}
s=-{{\partial\Psi}\over{\partial\theta}},\label{entropia}
\end{equation}
by the Helmholtz relation
$
e=\Psi+s\theta. %\label{energia}
$
Hence, concerning the stress tensor $\sigma$, we distinguish a
non-dissipative contribution $\sigma^{nd}$,
\begin{equation}
\sigma^{nd}={{\partial\Psi}\over{\partial\varepsilon(\mathbf{u})}},\label{stressnd}
\end{equation}
and a dissipative one
\begin{equation}
\sigma^d={{\partial\Phi}\over{\partial\varepsilon(\mathbf{u}_t)}},\label{stressd}
\end{equation}
and let
$\sigma=\sigma^{nd}+\sigma^d$.
% \label{stresstot}
Finally, by following the approach by Fr\'emond, we derive the
usual Fourier heat flux law by the pseudo-potential of dissipation. To
this aim, let us introduce the dissipative vector
\begin{equation}
\mathbf{Q}^d=-{{\partial\Phi}\over{\partial\nabla\theta}},\label{Qd}
\end{equation}
related to $\mathbf{q}$ by
\begin{equation}
\mathbf{q}=\theta\mathbf{Q}^d. \label{Qdq}
\end{equation}
Note that, due to \eqref{pseudodiss}, from \eqref{Qd} and \eqref{Qdq} it follows the
Fourier law governing the heat flux
\begin{equation}
\mathbf{q}=-k_0\nabla\theta.\label{fourier}
\end{equation}
Henceforth, if we substitute \eqref{entropia}--\eqref{Qdq} in \eqref{intener}, applying the
chain rule and after cancelling some terms, we can equivalently write
the balance of the energy as follows
\begin{equation}
\theta(s_t+\mathop{\rm div}\mathbf{Q}^d-R)=\sigma^d\varepsilon(\mathbf{u}_t)-\mathbf{Q}^d\cdot\nabla\theta=\partial\Phi(\varepsilon(\mathbf{u}_t),\nabla\theta)\cdot(\varepsilon(\mathbf{u}_t),\nabla\theta),
\label{CDI}
\end{equation}
where $R=r/\theta$ and $\partial\Phi$ denotes the subdifferential
of $\Phi$ w.r.t. (with respect to)  the dissipative variables
$(\varepsilon(\mathbf{u}_t),\nabla\theta)$. Note that, since
$\Phi$ in \eqref{pseudodiss} is such that
$$
\Phi(\mathbf{0})=0,\; \Phi\geq0, \Phi \mbox{ is  convex w.r.t. the
dissipative variables,}
$$
the right-hand side of \eqref{CDI} turns out to be  non-negative.
This fact as well as the positivity of the absolute temperature
$\theta$  are sufficient to guarantee the thermodynamical
consistence of the model. Now, we are  in the position of
recovering the PDE's system describing
 the model we have introduced above, written in terms of the unknowns $\theta$
and $\mathbf{u}$. After specifying \eqref{entropia}--\eqref{Qdq} by  \eqref{enfree} and
\eqref{pseudodiss} (cf. also \eqref{alfa}),  we substitute in \eqref{intener} and
\eqref{mom} the constitutive relations. We obtain in
$\Omega\times(0,T)$
\begin{gather}
c_s\theta_t-k_0\Delta\theta-\alpha\theta\mathop{\rm div}\mathbf{u}_t
=B\varepsilon(\mathbf{u}_t)\varepsilon(\mathbf{u}_t)+r, \label{enI}\\
\mathbf{u}_{tt}-\mathop{\rm
div}(B\varepsilon(\mathbf{u}_t)+K\varepsilon(\mathbf{u})
+\alpha\theta\mathbf{1})=\mathbf{G}, \label{sposI}
\end{gather}
and fix the Cauchy conditions in $\Omega$
\begin{gather}
\theta(0)=\theta_0, \label{cauchyt}\\
\mathbf{u}(0)=\mathbf{u}_0,\quad\mathbf{u}_t(0)={\mathbf{u}}_1. \label{cauchyu}
\end{gather}
Finally, due to \eqref{flussob} and \eqref{spostb}, we complete  the above
system by the following  boundary conditions
\begin{gather}
k_0\partial_n\theta=h\quad\hbox{on }\Gamma\times(0,T),\label{boundt}\\
\mathbf{u}=\mathbf{u}_t=\mathbf{0}\quad\hbox{on }\Gamma\times(0,T).
\label{bounduII}
\end{gather}

Before stating and proving our main existence and uniqueness
theorems, let us recall some related results in the literature. In
the one-dimensional setting, we recall the papers \cite{Daf,DafHs}
where the authors prove a global existence result for classical
solutions to a thermoviscoelastic system for solidlike materials,
in the case when  one of the endpoints is  stress-free. In
\cite{ChHof}, an existence result is given for pinned and
thermally insulated endpoints in  the special case of  a
one-dimensional model for shape-memory alloys. Other related
existence and uniqueness results for this kind of models, and
assuming a Landau-Devonshire or Landau-Ginzburg free-energy, can
be found, e.g., in \cite{HZ,NSZ,HofZoc,SZ}. Hence, in \cite{RacZh}
the solid-solid phase transition problem in nonlinear
thermoviscoelasticity is investigated in terms of the asymptotic
behaviour of the solutions. Finally, in a recent contribution
\cite{Wat} the theory for solidlike and gaseous materials is
treated in an unified manner. In particular, the main result of
that paper is the global existence and uniqueness of the solution
to an initial and boundary value problem corresponding to pinned
endpoints held at a constant temperature. Notice that all the
above results are given in the one-dimensional setting. Concerning
the derivation of the thermoviscoelastic model, one can refer,
e.g., to \cite{Ger}. In the three-dimensional case, we recall the
homogenization approach by \cite{FrSu} for a fairly simplified
quasi-static model, but retaining the nonlinear dissipative
contributions, and a related global existence result given in
\cite{BlGu}. More precisely, exploiting the techniques of
renormalized solutions for parabolic equations with $L^1$ data,
and under suitable assumptions on the thermal stress with respect
to the temperature, in \cite{BlGu} existence of small solutions is
established. Then, this result is extended for arbitrary data
under stronger assumptions on the thermal stress. Nonetheless, we
point out that our setting is different not only for the
techniques we use (mainly exploiting $L^2$ arguments). Indeed, in
\cite{FrSu,BlGu} the state quantities are derived by a free energy
functional obtained approximating the thermal energy contribution
$-\theta\log\theta$ in \eqref{enfree} by a first order
approximation of $\log\theta$. In particular, the energy equation
\eqref{enI} we deal with is not included in the framework of
\cite{BlGu}. Other results concerning a thermoviscoelastic system
in the multi-dimensional case (with the same approximation of the
free energy), can be found in \cite{Shi} where the author proves a
global existence and uniqueness result of small solutions,
assuming smooth and sufficiently small data.

The outline of the present work is as follows. In Section 2 we
introduce a variational formulation of the problem and we state
the main results (Theorems \ref{thm2.1} and \ref{thm2.2}). Section 3 is
devoted to the proof of the local existence result (Theorem \ref{thm2.1})
performed by a fixed point procedure. Section 4 is concerned with
the proof of the uniqueness result (Theorem \ref{thm2.2}) established by
some (global) contracting estimates. Finally, in Section 5 we
derive some global estimates on the solution (Proposition 5.3)
exploiting the positivity of the temperature (Theorem \ref{thm5.1}), which
is proved by  a maximum principle argument. Throughout the paper
some comments and remarks are given.

\section{Variational formulation and main results}

The aim of this section is to introduce   an abstract  formulation
of the initial and boundary values problem \eqref{enI}--\eqref{bounduII} and
state related existence and uniqueness results. We set the problem
in the domain $\Omega$, which is a bounded smooth domain in
$\mathbb{R}^3$, and investigate its evolution during a finite time
interval $[0,T]$. We first introduce the Hilbert triplet
$(V,H,V')$, where
\[
H:=L^2(\Omega),\quad V:= H^1(\Omega),
\]
and identify, as usual, $H $ with its dual space $H'$, so that
$V \hookrightarrow H \hookrightarrow V'$,
with dense and continuous embeddings. Besides, let the symbol
$\norm{\cdot}_X$ denote the norm either of some space  $X$ or $X^3$
and denote by
$\langle\cdot,\cdot\rangle$
the duality pairing between $V'$ and $V$.
Then, the associated Riesz isomorphism $J:V\rightarrow V'$ and the
scalar product in $V$  and in $V'$ can be specified by
\begin{equation}
\langle Jv_1,v_2\rangle:=((v_1,v_2)),\quad((u_1,u_2))_*:=\langle
u_1,J^{-1}u_2\rangle,\label{riesz}
\end{equation}
for $v_i\in V$, $u_i\in V'$, $i=1,2$.
Hence, we set the further Hilbert space
\[
\mathbf{W}:=H^1_0(\Omega)^3,
\]
endowed with the usual norm. In addition, let  introduce on $\mathbf{W}\times\mathbf{W}$
two bilinear continuous
symmetric forms: $a(\cdot,\cdot)$ defined by
\[
a(\mathbf{u},\mathbf{v})=\lambda\int_\Omega\mathop{\rm
div}\mathbf{u}\mathop{\rm
div}\mathbf{v}+2\mu\sum_{i,j=1}^3\int_\Omega\varepsilon_{ij}(\mathbf{u})\varepsilon_{ij}(\mathbf{v}),
\]
and $b(\cdot,\cdot)$ defined by
\[
b(\mathbf{u},\mathbf{v})=\sum_{i,j=1}^3\int_\Omega
b_{ij}\varepsilon_{ij}(\mathbf{u})\varepsilon_{ij}(\mathbf{v}),\quad B=(b_{ij}).
\]
In the sequel, just for the sake of simplicity but without loss of
generality, we let
$b_{ij}=1$,  $i,j=1,2,3$, so that
\begin{equation}
b(\mathbf{u},\mathbf{u})=\norm{\varepsilon(\mathbf{u})}^2_H,\label{bsimple}
\end{equation}
and fix $c_s=\alpha=k_0=1$.
Hence, recalling  Korn's inequality,
we are allowed to infer that there exists a positive constant
$\widetilde c$, depending only on $\lambda$, $\mu$, and $\Omega$,
such that
\begin{equation}
a(\mathbf{v},\mathbf{v})\geq\widetilde c\
\norm{\mathbf{v}}^2_\mathbf{W},\label{korn}
\end{equation}
for any $\mathbf{v}$ in $\mathbf{W}$.
Note also that
\begin{equation}
b(\mathbf{v},\mathbf{v})\geq\widehat c\
\norm{\mathbf{v}}^2_\mathbf{W},\quad\widehat c>0.\label{poin}
\end{equation}
 Next, to set our problem in the abstract
framework of the dual spaces $V'$ and $\mathbf{W}'$, we introduce the
operators
\begin{gather}
A:V\rightarrow V',\quad \langle Au,v\rangle=\int_\Omega\nabla
u\cdot\nabla v,\quad u,v\in V, \label{operA}\\
\mathcal{A}:\mathbf{W}\rightarrow\mathbf{W}',\quad_{\mathbf{W}'}\langle\mathcal{A}\mathbf{u},\mathbf{v}\rangle_\mathbf{W}=a(\mathbf{u},\mathbf{v}),
\quad\mathbf{u},\mathbf{v}\in\mathbf{W}, \label{opercalA}\\
\mathcal{B}:\mathbf{W}\rightarrow\mathbf{W}',\quad_{\mathbf{W}'}\langle\mathcal{B}\mathbf{u},\mathbf{v}\rangle_\mathbf{W}
=b(\mathbf{u},\mathbf{v}),\quad\mathbf{u},\mathbf{v}\in\mathbf{W}, \label{opercalB}\\
\mathcal{H}:H\rightarrow\mathbf{W}',\quad_{\mathbf{W}'}\langle\mathcal{H} u,\mathbf{v}\rangle_\mathbf{W}
=\int_\Omega u\mathop{\rm div}\mathbf{v},\quad u\in H,\mathbf{v}\in\mathbf{W}. \label{opercalH}
\end{gather}
In particular, let us point out that  $\mathcal{H}$ turns out to be  a
continuous and linear operator $V\subset H\rightarrow H^3\subset
\mathbf{W}'$, i.e. there exists a positive constant $c$ such that
\begin{equation}
\norm{\mathcal{H} v}_H\leq c\norm{v}_V,\quad\forall v\in V.\label{stimaHH}
\end{equation}
Now we set hypotheses on the data. Concerning the Cauchy
conditions \eqref{cauchyt}--\eqref{cauchyu}, we assume
\begin{equation} \label{regcauchyt}
\theta_0\in H,\quad \mathbf{u}_0\in\mathbf{W}\cap H^2(\Omega)^3,\quad
{\mathbf{u}}_1\in\mathbf{W}. %\label{regcauchyu}
\end{equation}
Then, dealing with the other functions, we prescribe that
\begin{equation} \label{regrhG} % regh  regG
r\in L^2(0,T;H), \quad h\in L^2(0,T;L^2(\Gamma)), \quad
\mathbf{G}\in L^2(0,T;H^3)
\end{equation}
and introduce the functions $\mathcal{R}$ and $\mathcal{G}$ specified by
\begin{gather}
\langle\mathcal{R} (t),v\rangle=\int_\Omega r (t)
v+\int_\Gamma h (t)v_{|_\Gamma},\quad v\in V, \quad\hbox {for a.a. }
t\in (0,T) \label{Rcal}\\
{}_{\mathbf{W}'}\langle\mathcal{G} (t),\mathbf{v}\rangle_\mathbf{W}=\int_\Omega\mathbf{G} (t)\cdot\mathbf{v},
 \quad \mathbf{v}\in \mathbf{W} , \quad \hbox {for a.a. } t\in (0,T) \label{Gcal}
\end{gather}
so that, by \eqref{regrhG}, it is natural to postulate
\begin{equation} \label{regRR} % regGG
\mathcal{R}\in L^2(0,T;V'),\quad \mathcal{G}\in L^2(0,T;H^3).
\end{equation}
Here is a precise formulation of the  abstract  problem. \smallskip

\noindent\textbf{Problem $P_a$.}\quad Find $(\theta,\mathbf{u})$ satisfying
\eqref{cauchyt}--\eqref{cauchyu} and such that, a.e. in $(0,T)$, (cf. also
\eqref{bsimple})
\begin{gather}
\theta_t+A\theta=\theta\mathop{\rm div}\mathbf{u}_t+
\mathcal{R}+\modulo{\varepsilon(\mathbf{u}_t)}^2\quad\hbox{in }V',\label{enas}\\
\mathbf{u}_{tt}+\mathcal{B}\mathbf{u}_t+\mathcal{A}\mathbf{u}
+\mathcal{H}\theta=\mathcal{G}\quad\hbox{in }\mathbf{W}'. \label{sposas}
\end{gather}
Using a fixed point theorem, combined with contracting estimates, we prove
the following local in time existence  result.

\begin{theorem} \label{thm2.1}
Let \eqref{regcauchyt}--\eqref{regrhG} and \eqref{regRR} hold.
Then, there exist $T_0\in(0,T]$ and a pair of functions
$(\theta,\mathbf{u})$ solving Problem $P_a$ in $(0,T_0)$ and
fulfilling
\begin{gather}
\theta\in H^1(0,T_0;V')\cap C^0([0,T_0];H)\cap
L^2(0,T_0;V),\label{regtheta}\\
\mathbf{u}\in H^2(0,T_0;H^3)\cap W^{1,\infty}(0,T_0;\mathbf{W})\cap H^1(0,T_0;H^2(\Omega)^3).
\label{regu}
\end{gather}
\end{theorem}

The proof of this theorem is given in the following section, using
the Schauder theorem. Next, in Section 4, performing  local in time
contracting estimates, which can be iterated on the whole time
interval $[0,T]$, we show that this solution is unique. Indeed, the
following theorem holds.

\begin{theorem} \label{thm2.2}
Let $(\theta,\mathbf{u})$ fulfilling \eqref{regtheta}--\eqref{regu} be a solution to
Problem $P_a$ during a time interval $[0,T_0]$. Then, this solution
is unique on the whole time interval $[0,T_0]$.
\end{theorem}

\begin{remark} \label{rmk2.3} \rm
Note that, in spite of the
fact that we are  able to prove only a local in time existence
result, uniqueness of a solution to Problem $P_a$ with regularity
\eqref{regtheta}--\eqref{regu} can be stated on the whole interval $[0,T]$.
In particular, if we were able to extend Theorem \ref{thm2.1} to a global
existence result, Theorem \ref{thm2.2} guarantees the (global) uniqueness
of this solution.
\end{remark}

\section{Proof of the existence result} % sec. 3

In this section, we detail the proof of the local existence result
stated in Theorem \ref{thm2.1}. To this aim, we apply the Schauder
theorem to a suitable operator $\mathcal{T}$ constructed as it
will be specified in a moment. Now, for $D>0$, consider the set
\[
X:=\big\{\mathbf{v}\in H^1(0,T_0;(W^{1,4}_0(\Omega))^3):
\norm{\mathbf{v}}_{H^1(0,T_0;(W^{1,4}_0(\Omega))^3)}\leq D \},
\]
where $T_0\in(0,T]$  will be fixed later, in such a way that
\begin{itemize}
\item $\mathcal{T}$ maps X into itself;
\item $\mathcal{T}$ is compact;
\item $\mathcal{T}$ is continuous.
\end{itemize}
\textbf{First step}.\quad Take an arbitrary $\widetilde{\mathbf{u}}\in X$
and substitute $\mathbf{u}$ in \eqref{enas}.  In particular, notice that  the
right hand side of \eqref{enas} turns out to be in
$L^1(0,T;H)+L^2(0,T;V')$, as by construction of $X$
$\modulo{\varepsilon(\widetilde{\mathbf{u}}_t)}^2\in L^1(0,T;H)$ via the
H\"older inequality (cf. also \eqref{regRR}). Thus, we can apply the
result of \cite[Theorem 3.2, p. 256]{Baiocchi} to infer that
there exists a unique
\begin{equation}
\theta:=\mathcal{T}_1(\widetilde{\mathbf{u}})\in [W^{1,1}(0,T;H)+H^1(0,T;V')]\cap
C^0([0,T];H)\cap L^2(0,T;V),\label{thetaSc}
\end{equation}
solving the corresponding equation with associated Cauchy condition
\eqref{cauchyt}.
Moreover, if we test the equation by $\theta$ and integrate over
$(0,t)$, with $t$ arbitrary in $(0,T)$, after some integrations by parts, applying the H\"older
inequality, owing to the continuous embedding $H^1(\Omega)\hookrightarrow
L^4(\Omega)$ and trace theorems, we have (cf. \eqref{cauchyt} and \eqref{regcauchyt})
\begin{equation} \label{stI}
\begin{aligned}
&\frac12 \norm{\theta(t)}^2_H-\frac12
 \norm{\theta_0}^2_H+\norm{\nabla\theta}^2_{L^2(0,t;H)} \\
&\leq c\Big(\int_0^t\norm{\theta}_V
 \norm{\mathop{\rm div}\widetilde{\mathbf{u}}_t}_{L^4(\Omega)}\norm{\theta}_H+
 \int_0^t\big(\norm{\widetilde{\mathbf{u}}_t}^2_{W^{1,4}_0(\Omega)}+
 \norm{r}_H\big)\norm{\theta}_H\\
&\quad +\int_0^t\norm{h}_{L^2(\Gamma)}\norm{\theta}_V\Big).
\end{aligned}
\end{equation}
For the rest of this article, we will denote by $c$ possibly
different positive constants depending only on the data of the problem.
The Young inequality
\begin{equation}
pq\leq{1\over{2\delta}}p^2+{\delta\over2}q^2,\quad\delta>0,\quad
p,q\in\mathbb{R},\label{young}
\end{equation}
leads to
\begin{equation} \label{stII}
\begin{aligned}
&\frac12 \norm{\theta(t)}^2_H+\frac12 \norm{\theta}^2_{L^2(0,t;V)}\\
&\leq \frac12 \norm{\theta_0}^2_H+c\Big(\norm{\theta}^2_{L^2(0,t;H)}
+\norm{h}^2_{L^2(0,t;L^2(\Gamma))}\\
&+\int_0^t(\norm{\widetilde{\mathbf{u}}_t}^2_{W^{1,4}_0(\Omega)}
+\norm{r}_H)\norm{\theta}_H +\int_0^t\norm{\mathop{\rm
div}\widetilde{\mathbf{u}}_t}^2_{L^4(\Omega)}
\norm{\theta}^2_H\Big).
\end{aligned}
\end{equation}
By the  generalized version of the Gronwall lemma
introduced in \cite{Baiocchi} and recalling the definition of $X$, due to \eqref{stII} we deduce that there exists a constant $C_1$
depending on $T$, $\Omega$, $h$, $r$, $\theta_0$, and $D$,
such that
\begin{equation}
\norm{\theta}_{L^\infty(0,T;H)\cap L^2(0,T;V)}\leq C_1.\label{stIII}
\end{equation}
\textbf{Second step.}\quad
After fixing $\theta=\mathcal{T}_1(\widetilde{\mathbf{u}})$
in \eqref{sposas}, standard results on parabolic equations (see e.g.
\cite{DuLi}) ensure that there exists a unique corresponding
solution $\mathbf{u}:=\mathcal{T}_2(\theta)$ satisfying
 \eqref{cauchyu} (cf. \eqref{bounduII}). Concerning the regularity
and the boundedness of $\mathbf{u}$, let us first test equation \eqref{sposas}
by $\mathbf{u}_t$ and integrate over $(0,t)$ (cf. \eqref{korn} and
\eqref{regcauchyt})
\begin{equation} \label{stIV}
\begin{aligned}
&\norm{\mathbf{u}_t(t)}^2_H+\norm{\mathbf{u}_t}^2_{L^2(0,t;\mathbf{W})}+\norm{\mathbf{u}(t)}^2_\mathbf{W} \\
&\leq c\Big(\norm{\mathbf{u}_1}^2_H+\norm{\mathbf{u}_0}^2_\mathbf{W}+\int_0^t\norm{\theta}_H
\norm{\mathop{\rm div}\mathbf{u}_t}_H+\int_0^t\norm{\mathcal{G}}_{\mathbf{W}'}\norm{\mathbf{u}_t}_\mathbf{W}\Big)\\
&\leq \frac12 \norm{\mathbf{u}_t}^2_{L^2(0,t;\mathbf{W})}+c\left(1
+\norm{\theta}^2_{L^2(0,t;H)}+\norm{\mathcal{G}}^2_{L^2(0,T;\mathbf{W}')}\right).
\end{aligned}
\end{equation}
Then, \eqref{stIII} yields
\begin{equation}
\norm{\mathbf{u}}_{W^{1,\infty}(0,T;H^3)\cap H^1(0,T;\mathbf{W})}\leq
C_2,\label{stV}
\end{equation}
for a suitable constant $C_2$, with the same dependence of   $C_1$
and depending in addition on $\mathbf{u}_0$, $\mathbf{u}_1$, and $\mathcal{G}$ (cf.
\eqref{stIII}). Next, we can formally test by $\mathbf{u}_{tt}$ and integrate
over $(0,t)$. We have, after integrating by parts,
\begin{equation}
\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+\frac12
\norm{\varepsilon(\mathbf{u}_t)(t)}^2_{H}-\frac12
\norm{\varepsilon(\mathbf{u}_t)(0)}^2_H\leq\sum_{i=1}^3
\modulo{I_i(t)},\label{stVI}
\end{equation}
where (some notation is only formal)
\begin{gather}
I_1(t)=-a(\mathbf{u}(t),\mathbf{u}_t(t))+a(\mathbf{u}_0,\mathbf{u}_1)+\int_0^ta(\mathbf{u}_t,\mathbf{u}_t),\label{stVII}\\
I_2(t)=-\int_0^t\int_\Omega\mathcal{H}\theta\mathbf{u}_{tt}, \label{stVIII} \\
I_3(t)=\int_0^t\int_\Omega\mathbf{G}\cdot\mathbf{u}_{tt}. \label{stIX}
\end{gather}
Now, we can handle the above integrals as follows.
First, by the definition of $a$ and due to \eqref{regcauchyt} and \eqref{stV}, we
can infer
\begin{equation} \label{stXI}
\begin{aligned}
\modulo{I_1(t)}&\leq c\left(\norm{\mathbf{u}_{0}}^2_\mathbf{W}
+\norm{\mathbf{u}_1}^2_\mathbf{W}\right)+\delta\norm{\mathbf{u}_t(t)}^2_\mathbf{W}
+C_\delta\norm{\mathbf{u}}^2_{L^\infty(0,T;\mathbf{W})}+c\int_0^t\norm{\mathbf{u}_t}^2_\mathbf{W}\\
&\leq c+\delta\norm{\mathbf{u}_t(t)}^2_\mathbf{W},
\end{aligned}
\end{equation}
where $\delta>0$ will be chosen later. Analogously, by definition of
$\mathcal{H}$, \eqref{stimaHH}, and  \eqref{young}, we get
\begin{equation}
\modulo{I_2(t)}\leq
\delta\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+C_\delta\norm{\theta}^2_{L^2(0,t;V)}\leq
\delta\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+c.\label{stXII}
\end{equation}
Finally, we can infer
\begin{equation}
\modulo{I_3(t)}\leq\delta\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+C_\delta\norm{\mathbf{G}}^2_{L^2(0,T;H^3)}\leq\delta\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+c.\label{stXIII}
\end{equation}
Thus, for a suitable choice of $\delta$ and combining \eqref{stXI}--\eqref{stXIII}
with \eqref{stVI} one obtains
\begin{equation}
\norm{\mathbf{u}_{tt}}^2_{L^2(0,t;H^3)}+\norm{\mathbf{u}_t(t)}^2_\mathbf{W}\leq c+\frac12
\norm{\mathbf{u}_t}_{L^\infty(0,t;\mathbf{W})},\label{stXV}
\end{equation}
and consequently
\begin{equation}
\norm{\mathbf{u}}_{H^2(0,T;H^3)\cap W^{1,\infty}(0,T;\mathbf{W})}\leq
C_3,\label{stXVbis}
\end{equation}
where the dependence of $C_3$ easily follows by the previous estimates.
Now, a comparison in \eqref{sposas}   leads to
\begin{equation}
\norm{\mathcal{B}\mathbf{u}_t+\mathcal{A}\mathbf{u}}_{L^2(0,T;H^3)}\leq
c,\label{stXVIprima}
\end{equation}
so that, recalling \eqref{bounduII}, \eqref{regcauchyt}, and the definition of
the operators $\mathcal{B}$, $\mathcal{A}$, we deduce
\begin{equation}
\norm{\mathbf{u}}_{H^1(0,T;H^2(\Omega)^3)}\leq C_4,\label{stXVI}
\end{equation}
for a  constant $C_4$ with the same dependence  of the previous
constants.

\noindent
\textbf{Third step.}\quad Now, our aim is to find $T_0$
such that the operator
\begin{equation}
\mathcal{T}:X\rightarrow
X,\quad\mathcal{T}(\widetilde{\mathbf{u}}):=\mathcal{T}_2(\mathcal{T}_1(\widetilde{\mathbf{u}})),
\label{operaTT}
\end{equation}
turns out to be well-defined. Note that thanks to a
Gagliardo-Nirenberg estimate (cf. \cite{GaglNir}), due to \eqref{stXVbis} and \eqref{stXVI},
there exists $C_5$ such that
\begin{equation}
\norm{\mathcal{T}(\widetilde{\mathbf{u}})}_{W^{1,{8/3}}(0,T;W^{1,4}_0(\Omega)^3)}\leq
C_5. \label{TbendefI}
\end{equation}
By the H\"older inequality  we have
\begin{equation}
\norm{\mathcal{T}(\widetilde{\mathbf{u}})}_{H^1(0,T;W^{1,4}_0(\Omega)^3)}\leq c
T^{1/8}\norm{\mathcal{T}(\widetilde{\mathbf{u}})}_{W^{1,{8/3}}(0,T;W^{1,4}_0(\Omega)^3)}.
\label{TbendefII}
\end{equation}
Thus, to ensure that $\mathcal{T}(\widetilde{\mathbf{u}})\in X$, i.e.
\begin{equation}
\norm{\mathcal{T}(\widetilde{\mathbf{u}})}_{H^1(0,T_0;W^{1,4}_0(\Omega)^3)}\leq
T_0^{1/8} C_6\leq D,\label{TbendefIIbis}
\end{equation}
we can find
$T_0\in(0,T]$ such that, e.g., $T_0\leq D^8/ C_6^8$. %\label{Tbendef}
Next, we observe that the above argument leads to (cf. \eqref{stXVbis}--\eqref{stXVI})
\begin{equation}
\norm{\mathcal{T}(\widetilde{\mathbf{u}})}_{H^2(0,T_0;H^3)\cap
W^{1,\infty}(0,T_0;\mathbf{W})\cap H^1(0,T_0;H^2(\Omega)^3)}\leq
c,\label{stXVII}
\end{equation}
for a constant $c$ independent of the choice of  $\widetilde{\mathbf{u}}\in
X$, which ensures that $\mathcal{T}$ is a  compact operator. From now on,
for the sake of simplicity, we directly refer to $T$ instead of
$T_0$. Hence, to achieve the proof of the Schauder theorem, it
remains to show that $\mathcal{T}$ is continuous with respect to the
natural strong topology induced in $X$ by
$H^1(0,T;W^{1,4}_0(\Omega)^3)$. Towards this goal, we consider a
sequence in $X$ such that
\begin{equation}
\widetilde{\mathbf{u}}_n\rightarrow\widetilde{\mathbf{u}}\quad\hbox{in
}H^1(0,T;W^{1,4}_0(\Omega)^3),\label{stXVIII}
\end{equation}
and let
\begin{equation}
\theta_n=\mathcal{T}_1(\widetilde{\mathbf{u}}_n),\quad\mathbf{u}_n=\mathcal{T}_2(\theta_n).\label{stXIX}
\end{equation}
Proceeding as for the previous estimates, we can find a positive
constant $c$ not depending on $n$ such that
\begin{gather}
\norm{\theta_n}_{[W^{1,1}(0,T;H)+H^1(0,T;V')]\cap
L^\infty(0,T;H)\cap L^2(0,T;V)}\leq c, \label{stXX} \\
\norm{\mathbf{u}_n}_{H^2(0,T;H^3)\cap W^{1,\infty}(0,T;\mathbf{W})\cap
H^1(0,T;H^2(\Omega)^3)}\leq c. \label{stXXI}
\end{gather}
By well-known weak and weak star compactness results the following
convergences are satisfied, at least for some suitable subsequences,
\begin{gather}
\theta_n\rightharpoonup ^* \theta\quad\hbox{in }
L^\infty(0,T;H)\cap L^2(0,T;V), \label{stXXII}\\
\mathbf{u}_n\rightharpoonup ^* \mathbf{u}\quad\hbox{in }H^2(0,T;H^3)\cap
W^{1,\infty}(0,T;\mathbf{W})\cap H^1(0,T;H^2(\Omega)^3). \label{stXXIII}
\end{gather}
In particular, by compactness we have (cf. \cite{Lions,Simon})
\begin{equation}
\mathbf{u}_n\rightarrow\mathbf{u}\quad\hbox{in
}H^1(0,T;W^{1,4}_0(\Omega)^3),\label{stXXIV}
\end{equation}
so that to conclude the proof it remains to verify that (cf. \eqref{stXVIII})
\begin{equation}
\mathbf{u}=\mathcal{T}(\widetilde{\mathbf{u}}).\label{stXXV}
\end{equation}
We first show that $\theta=\mathcal{T}_1(\widetilde{\mathbf{u}})$ in
\eqref{stXXII}. To
this aim, we write \eqref{enas} for $\widetilde{\mathbf{u}}_n$ and
$\widetilde{\mathbf{u}}$, take the difference and test by
$\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}})$. After an integration over $(0,t)$
and performing an analogous estimate as \eqref{stII}, we can infer
\begin{equation} \label{stXXVI}
\begin{aligned}
&\frac12 \norm{(\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}}))(t)}^2_H
+\norm{\nabla(\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}}))}^2_{L^2(0,t;H)} \\
&\leq c\int_0^t\norm{\theta_n}_H\norm{\mathop{\rm div}(\widetilde{\mathbf{u}}_n
-\widetilde{\mathbf{u}})_t}_{L^4(\Omega)}\norm{\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}})}_V\\
&\quad +c\int_0^t\norm{\mathop{\rm div}\widetilde{\mathbf{u}}_t}_{L^4(\Omega)}\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}_V\norm{\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}})}_H\\
&\quad
+\int_0^t\big(\norm{\varepsilon(\widetilde{\mathbf{u}}_{n_t})}^2_{L^4(\Omega)}
-\norm{\varepsilon(\widetilde{\mathbf{u}}_t)}^2_{L^4(\Omega)}\big)
\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}_H\leq\delta\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}^2_{L^2(0,t;V)} \\
&\quad +C_\delta\Big(\int_0^t\norm{\theta_n}^2_H
\norm{\mathop{\rm div}(\widetilde{\mathbf{u}}_n-\widetilde{\mathbf{u}})_t}^2_{L^4(\Omega)}
+\int_0^t\norm{\mathop{\rm div}\widetilde{\mathbf{u}}_t}^2_{L^4(\Omega)}\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}^2_H\Big)\\
&\quad +\int_0^t\big( \norm{\varepsilon(\widetilde{\mathbf{u}}_{n_t})}^2_{L^4(\Omega)}
-\norm{\varepsilon(\widetilde{\mathbf{u}}_t)}^2_{L^4(\Omega)}\big)
\norm{\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}})}_H.
\end{aligned}
\end{equation}
Thus, for a suitable choice of $\delta$, we can get the following
estimate
\begin{equation} \label{stXXVIbis}
\begin{aligned}
&\norm{(\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}}))(t)}^2_H+\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}^2_{L^2(0,t;V)}\\
&\leq c\Big(\norm{\theta_n}^2_{L^\infty(0,t;H)}
\norm{\mathop{\rm div}(\widetilde{\mathbf{u}}_n-\widetilde{\mathbf{u}})_t}^2_{L^2(0,t;L^4(\Omega))}\\
&\quad +\int_0^t\big(1+\norm{\mathop{\rm div}\widetilde{\mathbf{u}}_t}^2_{L^4(\Omega)}\big)
\norm{\theta_n-\mathcal{T}_1(\widetilde{\mathbf{u}})}^2_H \\
&\quad +\int_0^t\big(\norm{\varepsilon(\widetilde{\mathbf{u}}_{n_t})}^2_{L^4(\Omega)}
-\norm{\varepsilon(\widetilde{\mathbf{u}}_t)}^2_{L^4(\Omega)}\big)\norm{\theta_n
-\mathcal{T}_1(\widetilde{\mathbf{u}})}_H\Big).
\end{aligned}
\end{equation}
We can apply  the Gronwall lemma to \eqref{stXXVIbis} and, thanks to
\eqref{stXXIV} and  \eqref{stXX}, we get
\begin{equation}
\lim_{n\rightarrow+\infty}\norm{\theta_n-\mathcal{T}_1(\widetilde
u)}_{L^\infty(0,T;H)\cap L^2(0,T;V)}=0,\label{stXXVItris}
\end{equation}
and eventually $\theta$ in \eqref{stXXII} can be  identified with
$\mathcal{T}_1(\widetilde{\mathbf{u}})$. Notice in particular that  the following
strong convergence holds
\begin{equation}
\theta_n\rightarrow\theta\quad\hbox{in }C^0([0,T];H)\cap
L^2(0,T;V).\label{convI}
\end{equation}
Now, it is a standard matter to observe that \eqref{stXXIV} and
\eqref{convI} allow us to pass to the limit as $n\rightarrow+\infty$ in
\eqref{sposas} and, thanks to the uniqueness result for the limit
equation, once $\theta$ is fixed, eventually  identify $\mathbf{u}$ with
$\mathcal{T}_2(\theta)$, from which \eqref{stXXV} easily follows. Finally, we
have to complete the proof of
 \eqref{regtheta}. To this aim, after adding to both sides of \eqref{enas}
$\theta$, we test  by $J^{-1}\theta_t$
(cf. \eqref{riesz}) and integrate over $(0,t)$. By definition of $J$ and
using the H\"older inequality, we get (cf. \eqref{regu} and \eqref{convI})
\begin{equation} \label{regfinaleth}
\begin{aligned}
&\norm{\theta_t}^2_{L^2(0,t;V')}+\norm{\theta(t)}^2_H\\
&\leq c\Big(\norm{\theta_0}^2_H+\int_0^t\norm{\theta}_H
\norm{\mathop{\rm div}\mathbf{u}_t}_{L^4(\Omega)}\norm{J^{-1}\theta_t}_{V}\\
&\quad +\int_0^t\norm{\varepsilon(\mathbf{u}_t)}_{L^4(\Omega)}
\norm{\varepsilon(\mathbf{u}_t)}_H\norm{J^{-1}\theta_t}_{V}\\
&\quad +\int_0^t\norm{\mathcal{R}}_{V'}\norm{J^{-1}\theta_t}_V
+\int_0^t\norm{\theta}_H\norm{J^{-1}\theta_t}_H\Big)\\
&\leq\frac12 \norm{\theta_t}^2_{L^2(0,t;V')}
+c\Big(1+\norm{\mathop{\rm div}\mathbf{u}_t}^2_{L^2(0,T;L^4(\Omega))}
\norm{\theta}^2_{L^\infty(0,t;H)}\\
&\quad +\norm{\varepsilon(\mathbf{u}_t)}^2_{L^2(0,T;L^4(\Omega)^3)}
\norm{\varepsilon(\mathbf{u}_t)}^2_{L^\infty(0,T;H^3)}+
\norm{\theta}^2_{L^2(0,T;H)}\Big),
\end{aligned}
\end{equation}
from which \eqref{regtheta} is easily deduced.
\medskip

\begin{remark} \label{rmk3.1} \rm
Due to the regularity of $\theta$ and $\mathbf{u}$ and
\eqref{regcauchyt}, we could iterate the above argument, starting
from $T_0$ and so on. Nonetheless, even if we can extend the
existence result to some interval $[0,T_0^*]$, with $T_0<T_0^*\leq
T$, we are not allowed to infer that a solution exists on the
whole interval $[0,T]$.
\end{remark}

\section{Proof of the uniqueness result}

This section is devoted to the proof of the uniqueness result in
Theorem \ref{thm2.2}, i.e. we show uniqueness of the solution to
the system \eqref{enas}--\eqref{sposas},
\eqref{cauchyt}--\eqref{cauchyu}, during $[0,T_0]$, with the
regularity specified by  \eqref{regtheta}--\eqref{regu}. Let us
point out that this uniqueness result is stated  in any interval
$[0,T_0]$, with $T_0\in[0,T]$ (cf. Remark 2.3).

First, we make some remarks about the notation to be used.
 We assume that \eqref{enas}--\eqref{sposas},
\eqref{cauchyt}--\eqref{cauchyu} admit two solutions
\begin{equation}
\mathcal{S}_1=\{\theta_1,\mathbf{u}_1\},\quad
\mathcal{S}_2=\{\theta_2,\mathbf{u}_2\},\label{duesol}
\end{equation}
whose regularity is given by \eqref{regtheta}--\eqref{regu}. We denote by
$\widehat{f}$ the difference of two functions $f_1$ and $f_2$, i.e.
\begin{equation}
\widehat{f}=f_1-f_2,\label{diffeffe}
\end{equation}
and make use of two trivial identities
\begin{equation}
p_1q_1-p_2q_2=\widehat{pq}=\hat{p}q_2+p_1\hat{q}=\hat{p}q_1 +p_2\hat{q},
\label{diffab}
\end{equation}
so that, without loss of generality, in the sequel let us rewrite
\eqref{diffab} and subsequent computations omitting subscripts, i.e.
\begin{equation}
\widehat{pq}=\hat{p}q+p\hat{q}.\label{diffabgen}
\end{equation}
Now, we take the difference of \eqref{enas}, written for $\mathcal{S}_1$
and $\mathcal{S}_2$, and integrate it in time
\begin{equation}
\widehat{\theta}+1*A\widehat{\theta}=1*\widehat{\theta\mathop{\rm div}\mathbf{u}_t}
+1*\widehat{\modulo{\varepsilon(\mathbf{u}_t)}^2},
\label{endiff}
\end{equation}
where by $*$ we denote the usual convolution product over $(0,t)$,
namely
\begin{equation}
(p*q)(t)=\int_0^tp(t-s)q(s)\,ds.\label{conv}
\end{equation}
Then, we can test \eqref{endiff} by $\widehat{\theta}$ and integrate over
$(0,t)$. After an integration by parts in time and eploiting
\eqref{diffabgen}, we can infer
\begin{equation}
\|\widehat{\theta}\|^2_{L^2(0,t;H)}+\frac12 \|(1*\nabla\widehat{\theta})(t)\|^2_{H}
\leq\sum_{i=4}^6I_i(t),
\label{difI}
\end{equation}
where the integrals $I_i(t)$, $i=4,5,6$, are specified as follows
\begin{gather}
I_4(t)=\int_0^t\int_\Omega(1*\theta\mathop{\rm div}\widehat{\mathbf{u}}_t)\widehat{\theta},
\label{Ii}\\
I_5(t)=\int_0^t\int_\Omega(1*\widehat{\theta}\mathop{\rm div}\mathbf{u}_t)\widehat{\theta},
\label{Iii}\\
I_6(t)=2\int_0^t\int_\Omega(1*\varepsilon(\mathbf{u}_t)\widehat{\varepsilon(\mathbf{u}_t)})
\widehat{\theta}\,. \label{Iiii}
\end{gather}

We first treat the integral $I_4$ recalling \eqref{regtheta}, integrating
by parts in time,  and applying
the Young inequality. We have
\begin{equation} \label{difII}
\begin{aligned}
\modulo{I_4(t)}
&\leq\int_\Omega\modulo{(1*\widehat{\theta})(t)}
\int_0^t\modulo{\theta\mathop{\rm div}\widehat{\mathbf{u}}_t}
+\int_0^t\int_\Omega\modulo{1*\widehat{\theta}}\modulo{\theta\mathop{\rm div}\widehat{\mathbf{u}}_t}\\
&\leq\|(1*\widehat{\theta})(t)\|_{L^4(\Omega)}\int_0^t
\norm{\mathop{\rm div}\widehat{\mathbf{u}}_t}_H\norm{\theta}_{L^4(\Omega)}\\
&\quad +\int_0^t\|1*\widehat{\theta}\|_{L^4(\Omega)}
 \|\mathop{\rm div}\widehat{\mathbf{u}}_t\|_H
 \norm{\theta}_{L^4(\Omega)}\\
&\leq\delta\|1*\widehat{\theta}\|^2_{L^\infty(0,t;V)}
+c_5\int_0^t\norm{\theta}^2_V\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})},
\end{aligned}
\end{equation}
for a suitable positive constant $c_5$ depending  only on the data of
the problem and $\delta>0$.
Analogously, by \eqref{regu} we have
\begin{equation} \label{difIII}
\begin{aligned}
\modulo{I_5(t)}&\leq\int_\Omega |(1*\widehat{\theta})(t)|\int_0^t
|\mathop{\rm div}\mathbf{u}_t\widehat{\theta}|+\int_0^t\int_\Omega |1*\widehat{\theta}|\,|\mathop{\rm div}\mathbf{u}_t\widehat{\theta}|\\
&\leq\|(1*\widehat{\theta})(t)\|_V\int_0^t\norm{\mathop{\rm div}\mathbf{u}_t}_{L^4(\Omega)}
\|\widehat{\theta}\|_H+\int_0^t\|1*\widehat{\theta}\|_V\norm{\mathop{\rm div}\mathbf{u}_t}_{L^4(\Omega)}
\|\widehat{\theta}\|_H \\
&\leq\delta\|1*\widehat{\theta}\|^2_{L^\infty(0,t;V)}
+c_6\int_0^t\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}\|\widehat{\theta}\|^2_{L^2(0,t;H)}.
\end{aligned}
\end{equation}
Finally, by analogous arguments we obtain
\begin{equation} \label{difIV}
\begin{aligned}
\modulo{I_6(t)}&\leq2\int_\Omega |(1*\widehat{\theta})(t)|
\int_0^t|\varepsilon(\mathbf{u}_t)\varepsilon(\widehat{\mathbf{u}}_t)|
+2\int_0^t|1*\widehat{\theta}|\,|\varepsilon(\mathbf{u}_t)\varepsilon(\widehat{\mathbf{u}}_t)|\\
&\leq\delta\|1*\widehat{\theta}\|^2_{L^\infty(0,t;V)}
+c_7\int_0^t\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})}.
\end{aligned}
\end{equation}
Now, we test the difference of \eqref{sposas}  by $\widehat{\mathbf{u}}_t$,
integrate over $(0,t)$, and infer, using fairly standard
arguments,
\begin{equation}
\norm{\widehat{\mathbf{u}}_t(t)}^2_H+\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})}+\norm{\widehat{\mathbf{u}}(t)}^2_\mathbf{W}\leq
c_9\|\widehat{\theta}\|^2_{L^2(0,t;H)}+\frac12
\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})}, \label{difV}
\end{equation}
for a suitable positive constant $c_9$.
Hence, we combine \eqref{difII}--\eqref{difIV} with \eqref{difI} and add the resulting
inequality to \eqref{difV} multiplied by ${1/({2c_9})}$. In particular, after recalling that
\[
\|(1*\widehat{\theta})(t)\|_H\leq T^{1/2}\|\widehat{\theta}\|_{L^2(0,t;H)},
\]
we can take e.g. $c_8=\min\big\{\frac12 ,{1\over{4T}}\big\}$, fix
$\delta=c_8/6$, and write
\begin{equation}
\begin{aligned}
&\frac14
\|\widehat{\theta}\|^2_{L^2(0,t;H)}+c_8\|(1*\widehat{\theta})(t)\|^2_{V}
+\frac 1{2c_9} \norm{\widehat{\mathbf{u}}_t(t)}^2_H+{1\over{4c_9}}
\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})}\\
&\leq {{c_8}\over2} \|1*\widehat{\theta}\|^2_{L^\infty(0,t;V)}
+\Big(c_5\int_0^t\norm{\theta}^2_V+c_7\int_0^t
\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}\Big)
\norm{\widehat{\mathbf{u}}_t}^2_{L^2(0,t;\mathbf{W})}\\
&+c_6\int_0^t\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}
\|\widehat{\theta}\|^2_{L^2(0,t;H)}.
\end{aligned} \label{difVI}
\end{equation}
Note that because of \eqref{regtheta}--\eqref{regu},
$\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}$ and $\norm{\theta}^2_V$
belong to $L^1(0,T)$. Thus, we can find $\tau$ sufficiently small
such that, e.g.
\begin{equation}
\begin{aligned}
&\max\Big\{c_5\int_{\widehat t}^{\widehat t+\tau}
\norm{\theta}^2_V,c_6\int_{\widehat t}^{\widehat t+\tau}
\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3},c_7\int_{\widehat
t}^{\widehat t+\tau}\norm{\mathbf{u}_t}^2_{H^2(\Omega)^3}\Big\}\\
&\leq\min\big\{{1\over8},{1\over{8c_9}}\big\},
\end{aligned} \label{difVII}
\end{equation}
for any $\widehat t\in[0,T-\tau]$. Thus, at the end, by combining
\eqref{difVI} with \eqref{difVII}, we eventually get
\begin{equation}
\frac18 \|\widehat{\theta}\|^2_{L^2(0,t;H)}+c_8
\|(1*\widehat{\theta})(t)\|^2_V+ \frac1{2c_9}
\norm{\widehat{\mathbf{u}}_t(t)}^2_H
\leq{{c_8}\over2} \|1*\widehat{\theta}\|^2_{L^\infty(0,t;V)},\label{difVIII}
\end{equation}
for any $t\in(0,\tau)$. Now, from \eqref{difVIII} it easily
follows
\begin{equation}
\widehat{\theta}=\widehat{\mathbf{u}}=0\quad\hbox{a.e. in }\Omega\times(0,\tau).\label{difIX}
\end{equation}
Finally, as we can repeat the same estimates for any interval
$(\widehat t,\tau+\widehat t)$, by iterating the above procedure
we are allowed to extend \eqref{difIX} to the whole interval $(0,T)$,
which concludes the proof of the uniqueness result.

\section{Positivity of the temperature and global estimates}

In this section, we aim to prove positivity of the temperature,  ensuring that the second principle
of thermodynamics is satisfied, as we have discussed  in the introduction.
Hence,
in the second part of the section, mainly exploiting the positivity of the temperature, we are able to prove some
global estimates on the local solution of Problem $P_a$, which do not depend on the
choice of $T_0$. In particular, these estimates could be  extended to the whole initial
interval $(0,T)$. We first detail the proof of the positivity of
$\theta$.

\begin{theorem} \label{thm5.1}
Let the hypotheses of Theorem \ref{thm2.1} hold and suppose in addition
\begin{gather}
r\geq0\quad\hbox{a.e. in }\Omega\times(0,T), \label{rpos}\\
h\geq0\quad\hbox{a.e. in }\Gamma\times(0,T), \label{hpos}\\
\theta_0\geq0\quad\hbox{a.e. in }\Omega. \label{thetaopos}
\end{gather}
Then, the solution $(\theta,\mathbf{u})$ to Problem $P_a$ is such that
\begin{equation}
\theta(x,t)\geq0\quad\hbox{for
a.a. }(x,t)\in\Omega\times(0,T_0).\label{tempos}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk5.2} \rm
By construction \eqref{rpos}--\eqref{hpos}  imply that $\mathcal{R}$ defined in
\eqref{Rcal} is non-negative in the sense of distributions, i.e. for
any positive test function $\phi$ one has
$\langle\mathcal{R}(t),\phi\rangle\geq0$, %\label{RRpos}
for a.a. $t$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm5.1}]
We use a maximum principle argument.
Thus, we  test \eqref{enas} by $-\theta^-$,  $f^-$ denoting the so-called
negative part of a function $f$, i.e. $f^-:=\max\{0,-f\}$, and integrate over
$(0,t)$. After an integration by parts in time,  exploiting
\eqref{rpos}--\eqref{thetaopos}, and owing to the H\"older inequality, we can infer
that
\begin{equation} \label{stposIV}
\begin{aligned}
&\frac12 \norm{\theta^-(t)}^2_H+\norm{\nabla\theta^-}^2_{L^2(0,t;H)} \\
&\leq \int_0^t\norm{\theta^-}_H\norm{\theta^-}_{L^4(\Omega)}
\norm{\mathop{\rm div}\mathbf{u}_t}_{L^4(\Omega)}
-\int_0^t\!\int_\Omega r\theta^--\int_0^t\!\int_\Gamma
h\theta^-_{|_\Gamma}-\int_0^t\!\int_\Omega\modulo{\varepsilon(\mathbf{u}_t)}^2\theta^-\\
&\leq c\int_0^t\norm{\theta^-}_H\norm{\theta^-}_{V}
\norm{\mathop{\rm div}\mathbf{u}_t}_{L^4(\Omega)}.
\end{aligned}
\end{equation}
Hence, to handle the right hand side
of \eqref{stposIV} we  apply the Young inequality  and get, by definition of the $V$-norm,
\begin{equation} \label{stposIVbis}
 \frac12 \norm{\theta^-(t)}^2_H+\frac12
\norm{\theta^-}^2_{L^2(0,t;V)}
\leq
c\Big(\norm{\theta^-}^2_{L^2(0,t;H)}
+\int_0^t\norm{\mathop{\rm div}\mathbf{u}_t}^2_{L^4(\Omega)}\norm{\theta^-}^2_H\Big).
\end{equation}
Then, since $\norm{\mathop{\rm div}\mathbf{u}_t}^2_{L^4(\Omega)}$ belongs to
$L^1(0,T)$ (cf. \eqref{regu}), the generalized version of the Gronwall
lemma introduced in  \cite{Baiocchi} allows us to deduce
\begin{equation}
\norm{\theta^-}_{L^\infty(0,T;H)\cap L^2(0,T;V)}\leq0,\label{stposV}
\end{equation}
which eventually gives \eqref{tempos}.
\end{proof}

In the last part of this section, we show some uniform estimates
on the local solution $(\theta,\mathbf{u})$ of Problem $P_a$, which
actually  could be extended on the whole interval $(0,T)$. Indeed,
the following Proposition holds.

\begin{proposition} \label{prop5.3}
Under the assumptions of Theorem \ref{thm5.1}, there exists a positive constant
$c$ depending only on $\Omega$, $T$, and the data of the problem, such
that the following bound is fulfilled by the solution $(\theta,\mathbf{u})$ to
Problem $P_a$
\begin{equation}
\norm{\theta}_{L^\infty(0,T;L^1(\Omega))}+\norm{\mathbf{u}}_{W^{1,\infty}(0,T;H^3)\cap
L^\infty(0,T;\mathbf{W})}\leq c.\label{stglobI}
\end{equation}
\end{proposition}

\begin{proof}
The estimating process, used for proving \eqref{stglobI}, mainly
exploits the positivity of the temperature stated by Theorem
\ref{thm5.1}. We formally proceed by testing \eqref{enas} by the
constant function 1 and  \eqref{sposas} by $\mathbf{u}_t$. Then,
we add the resulting equations and finally integrate over $(0,t)$.
As some terms cancel, after an integration by parts in time, we
eventually write
\begin{equation} \label{stglobII}
\begin{aligned}
&\int_\Omega\theta(t)+\norm{\mathbf{u}_t(t)}^2_H+\norm{\mathbf{u}(t)}^2_\mathbf{W} \\
&\leq c\Big(\int_\Omega\theta_0+\norm{\mathbf{u}_1}^2_H
+\norm{\mathbf{u}_0}^2_\mathbf{W}+\norm{\mathcal{R}}_{L^2(0,T;V')}+\int_0^t{_{\mathbf{W}'}
\langle \mathcal{G},\mathbf{u}_t\rangle_\mathbf{W}}\Big) \\
&\leq c\Big(1+\norm{\mathcal{R}}_{L^2(0,T;V')}+\int_0^t
\norm{\mathbf{G}}_H\norm{\mathbf{u}_t}_H\Big).
\end{aligned}
\end{equation}
Thus, using the Gronwall lemma, one can easily obtain \eqref{stglobI},
after observing that by \eqref{tempos} there holds
\[
\int_\Omega\theta(t)=\norm{\theta(t)}_{L^1(\Omega)}.
\]
\end{proof}

\subsection*{Acknowledgement}
 We would like to thank Professor Michel Fr\'emond for having proposed us
this interesting problem and for several  suggestions especially concerning
the modelling aspects.


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