# Global weak solutions in nonlinear 3D thermoelasticity

Tomasz Cieślak  Institute of Mathematics, Polish Academy of Sciences
<EMAIL_ADDRESS>,  Boris Muha  Department of Mathematics, Faculty of Science,
University of Zagreb<EMAIL_ADDRESS>and  Srđan Trifunović  Department for
Mathematics and Informatics, University of Novi Sad
<EMAIL_ADDRESS>
###### Abstract.

Here we study a nonlinear thermoelasticity hyperbolic-parabolic system
describing the balance of momentum and internal energy of a heat-conducting
elastic body, preserving the positivity of temperature. So far, no global
existence results in such a natural case were available. Our result is
obtained by using thermodynamically justified variables which allow us to
obtain an equivalent system in which the internal energy balance is replaced
with entropy balance. For this system, a concept of weak solution with defect
measure is introduced, which satisfies entropy inequality instead of balance
and has a positive temperature almost everywhere. Then, the global existence,
consistency and weak-strong uniqueness are shown in the cases where heat
capacity and heat conductivity are both either constant or non-constant. Let
us point out that this is the first result concerning global existence for
large initial data in nonlinear thermoelasticity where the model is in full
accordance with the laws of thermodynamics.

###### 2020 Mathematics Subject Classification:

35M13, 35D30, 74A15, 80A17

##  1\. Introduction and main result

Let $\Omega\in\mathbb{R}^{3}$ be a bounded Lipschitz domain. We study the
physical model of a heat-conducting elastic body described by two state
variables – the displacement
$\textbf{u}:[0,T)\times\Omega\rightarrow\mathbb{R}^{3}$ and temperature
$\theta:[0,T)\times\Omega\rightarrow\mathbb{R}$. The two governing equations
are the balance of momentum and internal energy. In this paper, we will focus
on the following two models, physically viable for high and low temperatures,
respectively. Their derivation is given in section 3.

###  1.1. The model with constant heat capacity and heat conductivity

We deal with the following problem:

(1) $\begin{cases}\mathbf{u}_{tt}=\nabla\cdot(\nabla\mathbf{u}-\mu\theta
I),&\quad\text{ in }(0,T)\times\Omega,\\\
\theta_{t}-\Delta\theta=-\mu\theta\nabla\cdot\mathbf{u}_{t},&\quad\text{ in
}(0,T)\times\Omega,\\\ \mathbf{u}=0,\partial_{n}\theta=0,&\quad\mbox{on
}(0,T)\times\partial\Omega,\\\
\mathbf{u}(0,\cdot)=\mathbf{u}_{0},~{}\mathbf{u}_{t}(0,\cdot)=\mathbf{v}_{0},~{}\theta(0,\cdot)=\theta_{0}>0,&\end{cases}$

where constant $\mu>0$ is a given quantity.

Such a system arises as a simplest nonlinear thermoelasticity model. So far,
the problem of global existence of solutions has not been solved for this
system.

First of all, let us notice that the equation $\eqref{system0}_{2}$ can
formally be divided by $\theta>0$ to obtain the entropy balance

(2)
$\left(\ln\theta+\mu\nabla\cdot\mathbf{u}\right)_{t}-\Delta\ln\theta=|\nabla\ln\theta|^{2}\;\;\mbox{on}\;\;(0,T)\times\Omega.$

Next, multiplying $\eqref{system0}_{1}$ with $\mathbf{u}_{t}$, summing up with
$\eqref{system0}_{2}$ and integrating over $(0,t)\times\Omega$ for $t\in(0,T]$
gives the total energy balance

$\displaystyle\int_{\Omega}\theta(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
$\displaystyle=\int_{\Omega}\theta_{0}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

while subtracting $\eqref{ent:eq1}$ integrated over $(0,t)\times\Omega$ from
the total energy balance gives rise to the total dissipation balance

$\displaystyle\int_{\Omega}(\theta-\ln\theta)(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)+\int_{0}^{t}\int_{\Omega}|\nabla{\ln\theta}|^{2}$
(3)
$\displaystyle=\int_{\Omega}(\theta_{0}-\ln\theta_{0})+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}.$

Hence, denoting $\tau:=\ln\theta$, we introduce the following definition of
weak solution:

######  Definition 1.1.

We say that $(\mathbf{u},\tau)$ is a weak solution to the problem
$(\ref{system0})$ with defect measure if:

  * •

The initial data is of regularity

$\displaystyle\mathbf{u}_{0}\in H_{0}^{1}(\Omega),\quad\mathbf{v}_{0}\in
L^{2}(\Omega),\quad\tau_{0},e^{\tau_{0}}\in L^{1}(\Omega).$

Moreover111In Appendix, it is proved that $\mathbf{u}_{t}\in
C_{w}(0,T;L^{2}(\Omega))$ for any weak solution., $\mathbf{v}_{0}$ is attained
in $C_{w}(0,T;L^{2}(\Omega))$ and $\lim\limits_{t\to
0}\int_{\Omega}{\tau}(t)\geq\int_{\Omega}{\tau}_{0}$;

  * •

$\mathbf{u}\in L^{\infty}(0,T;H_{0}^{1}(\Omega))\cap
W^{1,\infty}(0,T;L^{2}(\Omega))$,
$\tau\in L^{\infty}(0,T;L^{1}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$,
$e^{\tau}\in L^{\infty}(0,T;L^{1}(\Omega))$;

  * •

The momentum equation

$\displaystyle\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\boldsymbol{\varphi}_{t}-\int_{0}^{T}\int_{\Omega}\nabla\mathbf{u}:\nabla\boldsymbol{\varphi}+\mu\int_{0}^{T}\left\langle\theta,\nabla\cdot\boldsymbol{\varphi}\right\rangle_{[\mathcal{M}^{+};C](\overline{\Omega})}$
(4) $\displaystyle=-\int_{\Omega}\mathbf{v}_{0}\cdot\boldsymbol{\varphi},$

holds for all test function $\boldsymbol{\varphi}\in
C_{0}^{\infty}([0,T)\times\Omega)$, where $\theta\in
L^{\infty}(0,T;\mathcal{M}^{+}(\overline{\Omega}))$ satisfies

$\displaystyle d\theta=e^{\tau}dx+f,$

and $f\geq 0$ is a singular part of $\theta$ supported on a set of measure
zero;

  * •

The entropy equation

(5)
$\displaystyle\int_{0}^{T}\int_{\Omega}\tau\phi_{t}-\int_{0}^{T}\int_{\Omega}\nabla\tau\cdot\nabla\phi+\mu\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\nabla\phi+\left\langle\boldsymbol{\sigma},\phi\right\rangle_{[\mathcal{M}^{+};C]([0,T]\times\overline{\Omega})}$
$\displaystyle=-\int_{\Omega}\tau_{0}\phi_{0},$

holds for all $\phi\in C_{0}^{\infty}([0,T)\times\Omega)$, where
$\boldsymbol{\sigma}\in\mathcal{M}^{+}([0,T]\times\overline{\Omega})$
satisfies

$\displaystyle\boldsymbol{\sigma}\geq|\nabla\tau|^{2};$

  * •

The energy inequality

(6)
$\displaystyle\int_{\Omega}d\theta(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
$\displaystyle\leq\int_{\Omega}e^{\tau_{0}}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

holds for all $t\in(0,T]$.

######  Remark 1.1.

(1) In this definition, the internal energy equation $\eqref{system0}_{2}$ is
replaced with the entropy equation $\eqref{ent:eq1}$. Let us point out the key
ideas behind this. First, the nonlinearity
$-\mu\theta\nabla\cdot\mathbf{u}_{t}$ in $\eqref{system0}_{2}$ is not expected
to be even integrable for weak solutions, and the compactness for such a term
seems unreachable even in 1D (see [9]). The entropy equation provides us with
some additional information, in particular the total dissipation identity
(inequality) $\eqref{total:dissipation:balance:lin}$, which gives us the
estimates of $\ln\theta$ and $\nabla\ln\theta$ and thus ensures the positivity
of $\theta$. It also plays a crucial role in the proof of weak-strong
uniqueness. Finally, the measure $\boldsymbol{\sigma}$ appears since the
nonlinear term $|\nabla\ln\theta|^{2}$ is only bounded $L_{t,x}^{1}$ and
therefore we cannot rule out concentrations which would give rise to a defect
measure.
(2) In the above definition, the unknown functions are the displacement
$\mathbf{u}$ and the thermal entropy $\tau=\ln\theta$. However, this identity,
due to lack of uniform integrability of $\theta$, might not hold on a set of
measure zero, which is an issue since $\theta$ is only a measure. Whether or
not this integrability can be obtained is an interesting and important
problem.
(3) Such a concept of a weak solution was inspired by the work of Feireisl and
Novotny [15], where heat-conducting compressible fluids governed by the full
Navier-Stokes-Fourier system are studied. However, the idea of weak solutions
with defect measures dates back to work of DiPerna and Lions [13] and
Alexandre and Villani [1], where the concept of weak solutions with defect
measure was studied in the context of the Boltzmann equation.

The main result for this model concerns global-in-time existence of solutions
to (1) in the sense of the above definition. Moreover, we show that solutions
satisfying the above definition are consistent, i.e. if a weak solution
satisfies certain additional regularity property, then it solves (1)
pointwise. Next, we show that in case a classical solution starting from
initial data $(\theta_{0},\mathbf{u}_{0},\mathbf{v}_{0})$ exists, then our
weak solution and classical solution coincide. To be more precise, the
following theorem holds.

######  Theorem 1.1 (Main result I).

Let $\mathbf{u}_{0}\in H_{0}^{1}(\Omega)$, $\mathbf{v}_{0}\in L^{2}(\Omega)$,
$\tau_{0}\in L^{1}(\Omega)$ with $e^{\tau_{0}}\in L^{1}(\Omega)$. Then, one
has the following:

  1. Global existence. There exists a global weak solution $(\mathbf{u},\tau)$ in the sense of Definition 1.1;

  2. Consistency. If a weak solution in the sense of Definition 1.1 is smooth, then the solution solves (1) pointwise;

  3. Weak-strong uniqueness. Let $(\mathbf{u},\tau)$ be a weak solution in the sense of Definition 1.1. Then, it satisfies the relative entropy inequality $\eqref{relent}$. Moreover, if $(\tilde{\mathbf{u}},\tilde{\tau})$ is a classical solution to the problem $(\ref{system0})$ and

$\displaystyle\tau_{0}=\tilde{\tau}_{0},\qquad\mathbf{u}_{0}=\tilde{\mathbf{u}}_{0},\qquad\mathbf{v}_{0}=\tilde{\mathbf{v}}_{0},$

then

$\displaystyle\mathbf{u}\equiv\tilde{\mathbf{u}},\qquad\tau\equiv\tilde{\tau}.$

The approach developed here was inspired by the theory of Feireisl and Novotny
from [15], where the authors deal with viscous compressible heat-conducting
fluids.

###  1.2. The model with non-constant heat capacity and heat conductivity

The problem we choose to study is the following:

(7)
$\displaystyle\begin{cases}\mathbf{u}_{tt}=\nabla\cdot(\nabla\mathbf{u}-\mu\theta
I),&\mbox{in }(0,T)\times\Omega,\\\
(\theta+\theta^{\alpha})_{t}-\nabla\cdot\big{(}(1+\theta^{\beta})\nabla\theta\big{)}=-\mu\theta\nabla\cdot\mathbf{u}_{t},&\mbox{in
}(0,T)\times\Omega,\\\ \mathbf{u}=0,~{}\partial_{n}\theta=0,&\mbox{on
}(0,T)\times\partial\Omega,\\\
\mathbf{u}(0,\cdot)=\mathbf{u}_{0},~{}\mathbf{u}_{t}(0,\cdot)=\mathbf{v}_{0},~{}\theta(0,\cdot)=\theta_{0},&\end{cases}$

where $\alpha>1$ and $\beta,\mu>0$. The entropy equation reads

$\displaystyle\left(\ln\theta+\frac{\alpha}{\alpha-1}\theta^{\alpha-1}+\mu\nabla\cdot\mathbf{u}\right)_{t}-\nabla\cdot\left(\frac{(1+\theta^{\beta})\nabla\theta}{\theta}\right)$
(8) $\displaystyle=\frac{(1+\theta^{\beta})|\nabla\theta|^{2}}{\theta^{2}},$

on $(0,T)\times\Omega$. The total energy balance for this system is now

(9)
$\displaystyle\int_{\Omega}(\theta+\theta^{\alpha})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
$\displaystyle=\int_{\Omega}(\theta_{0}+\theta_{0}^{\alpha})+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

while the total dissipation balance is

$\displaystyle\int_{\Omega}\left(\theta+\theta^{\alpha}-\ln\theta-\frac{\alpha}{\alpha-1}\theta^{\alpha-1}\right)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{(1+\theta^{\beta})|\nabla\theta|^{2}}{\theta^{2}}$
(10)
$\displaystyle=\int_{\Omega}\left(\theta_{0}+\theta_{0}^{\alpha}-\ln\theta_{0}-\frac{\alpha}{\alpha-1}\theta_{0}^{\alpha-1}\right)+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2}.$

This leads to the following:

######  Definition 1.2.

We say that $(\mathbf{u},\theta)$ is a weak solution to the problem (7) with
defect measure if:

  * •

The initial data is of regularity

$\displaystyle\mathbf{u}_{0}\in H_{0}^{1}(\Omega),\quad\mathbf{v}_{0}\in
L^{2}(\Omega),\quad\theta_{0}\in L^{\alpha}(\Omega),\quad{\ln\theta}_{0}\in
L^{1}(\Omega).$

Moreover, $\mathbf{v}_{0}$ is attained in $C_{w}(0,T;L^{2}(\Omega))$ and

$\displaystyle\lim\limits_{t\to
0}\int_{\Omega}\left(\ln\theta+\frac{\alpha}{\alpha-1}\theta^{\alpha-1}\right)(t)\geq\int_{\Omega}\left(\ln\theta_{0}+\frac{\alpha}{\alpha-1}\theta_{0}^{\alpha-1}\right);$

  * •

$\mathbf{u}\in L^{\infty}(0,T;H_{0}^{1}(\Omega))\cap
W^{1,\infty}(0,T;L^{2}(\Omega))$,
$\theta\in L^{\infty}(0,T;L^{\alpha}(\Omega))$,
$\theta^{\beta}\in L^{1}(0,T;L^{3}(\Omega))$,
$\theta^{\frac{\alpha}{2}+\frac{\beta}{2}}\in
L^{2}(0,T;L^{\frac{3}{2}}(\Omega))$,
$\nabla\theta^{\frac{s}{2}}\in L^{2}((0,T)\times\Omega)$, for all
$s\in(0,\beta]$,
$\ln\theta\in L^{\infty}(0,T;L^{1}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$;

  * •

The momentum equation

(11)
$\displaystyle\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\boldsymbol{\varphi}_{t}-\int_{0}^{T}\int_{\Omega}\nabla\mathbf{u}:\nabla\boldsymbol{\varphi}+\mu\int_{0}^{T}\int_{\Omega}\theta\nabla\cdot\boldsymbol{\varphi}$
$\displaystyle=-\int_{\Omega}v_{0}\cdot\boldsymbol{\varphi},$

holds for all test function $\boldsymbol{\varphi}\in
C_{0}^{\infty}([0,T)\times\Omega)$;

  * •

The entropy balance

$\displaystyle\int_{0}^{T}\int_{\Omega}(\ln\theta+\frac{\alpha}{\alpha-1}\theta^{\alpha-1})\phi_{t}-\int_{0}^{T}\int_{\Omega}\left(\frac{(1+\theta^{\beta})\nabla\theta\cdot\nabla\phi}{\theta}\right)$
$\displaystyle\quad+\mu\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\nabla\phi+\left\langle\boldsymbol{\sigma},\phi\right\rangle_{[\mathcal{M}^{+};C]([0,T]\times\overline{\Omega})}\
$ (12)
$\displaystyle=-\int_{\Omega}(\ln\theta_{0}+\frac{\alpha}{\alpha-1}\theta_{0}^{\alpha-1})\phi_{0},$

holds for all non-negative $\phi\in C_{0}^{\infty}([0,T)\times\Omega)$, where
$\boldsymbol{\sigma}\in\mathcal{M}^{+}([0,T]\times\overline{\Omega})$
satisfies

$\displaystyle\boldsymbol{\sigma}\geq\frac{(1+\theta^{\beta})|\nabla\theta|^{2}}{\theta^{2}};$

  * •

The energy inequality

$\displaystyle\int(\theta+\theta^{\alpha})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
(13)
$\displaystyle\leq\int_{\Omega}(\theta_{0}+\theta_{0}^{\alpha})+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

holds for all $t\in(0,T]$.

We shall show the following result concerning the problem (7):

######  Theorem 1.2 (Main result II).

Let $\mathbf{u}_{0}\in H_{0}^{1}(\Omega)$, $\mathbf{v}_{0}\in L^{2}(\Omega)$,
$\theta_{0}\in L^{\alpha}(\Omega)$ with $\ln\theta_{0}\in L^{1}(\Omega)$, and
let $\alpha>1,\beta>0$. Then, one has the following:

  1. Global existence. There exists a weak solution $(\mathbf{u},\theta)$ in the sense of Definition 1.2;

  2. Consistency. If a weak solution in the sense of Definition 1.1 is smooth, then the solution solves both (7) and (8) pointwise;

  3. Weak-strong uniqueness. Let $(\mathbf{u},\theta)$ be a weak solution in the sense of Definition 1.2. Then, it satisfies the relative entropy inequality $\eqref{rel:ent:nonlin}$. Moreover, if $(\tilde{\mathbf{u}},\tilde{\theta})$ is a classical solution to the problem $(\ref{system:nonlin})$ and $\alpha\geq 2,\beta=2$, then

$\displaystyle\theta_{0}=\tilde{\theta}_{0}>0,\qquad\mathbf{u}_{0}=\tilde{\mathbf{u}}_{0},\qquad\mathbf{v}_{0}=\tilde{\mathbf{v}}_{0},$

implies

$\displaystyle\mathbf{u}\equiv\tilde{\mathbf{u}},\qquad\theta\equiv\tilde{\theta}.$

######  Remark 1.2.

The above result also holds when heat conductivity is constant, i.e. when
$\alpha>1$ and $\beta=0$.

##  2\. Comparison with known results and discussion

In this section, we compare our result with the known literature. The study of
existence of solutions to systems of thermoelasticity dates back to 1960s,
however the initial work was related to the linear simplification (linear
thermoelasticity is addressed in detail shortly after). Nonlinear systems of
elasticity require more sophisticated methods. Under some particular choices
of nonlinear couplings $b(\theta,\mathbf{u}_{tx})$ replacing the term
$-\mu\theta\nabla\cdot\mathbf{u}_{t}$ in the second equation of (1), smooth,
classical solutions were known in the literature in the spatially one-
dimensional setting locally in time for any data and globally for small data.
This was investigated by various authors and one of the pioneer works was
Slemrod’s paper [30]. Still, assumptions required to guarantee existence of
smooth solutions with nonnegative temperature were so specific, that the
simplest system (1) was not covered. See the discussion in [9] and results in
[4, 17, 27]. The classical smooth local-in-time solutions with nonnegative
temperature were obtained recently in [9]. Let us also recall earlier similar
results in [19, 28]. Still, even in 1D case, the problem of existence of
global-in-time solutions is widely opened. In 1D, one could mention the global
existence of measure-valued solutions in [9, Theorem 2]. The regularity of
such a solution is very weak and there seems no hope in that case for any sort
of weak-strong uniqueness.

Recently, a notable attention is being given to problems in adiabatic
thermoelasticity – the system in which the heat flux is neglected, a valid
approximation in thermodynamical systems in which processes happen in a small
time interval and thus not allow for heat flux to take effect. From a
mathematical point of view, this is a purely hyperbolic system. Adiabatic
thermoviscoelastic system was studied by C. Christoforou and A. Tzavaras in
[6]. They use the method of relative entropy in order to prove the zero
viscosity limit, thus going from thermoviscoelasticity with heat flux to
adiabatic thermoelastic system. Note that the existence of viscous solutions
is not shown. Their next results obtained together with Galanopoulou in [7, 8]
concern some adiabatic thermoelastic problems. In the first paper, a measure-
valued solution to the thermoelastic problem is obtained. It requires the use
of Young measures as well as an embedding of the considered system into the
hyperbolic conservation laws framework. It requires some structural
assumptions (see [7, Remark 1, p. 6182]). It seems that the solution obtained
in [8], again a measure-valued one, requires similar strong structural
assumptions. The studies of weak-strong uniqueness of the weak solutions
obtained in the above papers were performed in [16].

Let us comment that the question concerning regularity of the obtained
solution is not clear. One cannot a priori exclude the finite-time blowup of
first derivatives, see [12].

Finally, let us mention that related system of thermoviscoelasticity was also
extensively studied, see e.g. [11, 21], [29, Chapter 12] and references
within. However, in this case the system is of parabolic - parabolic type and
therefore analysis of such system is much different form the analysis
presented here.

###  2.1. Linear thermoelasticity

Under the assumption that $\theta$ is close to a state $\theta_{0}=const$, the
system of equations $\eqref{system0}_{1,2}$ can be reduced to a linear one
(see for instance [18, Chapter 4]):

(14)
$\displaystyle\begin{cases}\mathbf{u}_{tt}=\nabla\cdot(\nabla\mathbf{u}-\mu\theta
I),\\\
\theta_{t}-\Delta\theta=-\mu\theta_{0}\nabla\cdot\mathbf{u}_{t}.\end{cases}$

This is a well-known system which has been studied extensively by many
authors. To the best of our knowledge the first result was due to Dafermos
[10], followed by plethora of other works, see e.g. [18], [23, Chapter 7],
[24, Chapter 3], [25], and references within.

Let us point out some important properties of $\eqref{linearTh}$. First of
all, the equation $\eqref{linearTh}_{2}$ is actually an entropy equation, the
entropy is $\theta+\mu\theta_{0}\nabla\cdot\mathbf{u}$, the entropy flux is
$-\nabla\theta$, while the entropy production is zero. To obtain the
corresponding heat equation, one multiplies $\eqref{linearTh}_{2}$ with
$\theta$ which gives

$\displaystyle\frac{1}{2}(\theta^{2})_{t}-\frac{1}{2}\Delta(\theta^{2})+|\nabla\theta|^{2}=-\mu\theta_{0}\theta\nabla\cdot\mathbf{u}_{t}.$

From a physical point of view, there are three major issues here - the entropy
production is zero, there is a dissipation coming from temperature
$|\nabla\theta|^{2}$ and the system allows temperature to be negative (the
maximum principle isn’t satisfied). Moreover, this model predicts that
temperature will oscillate around (or simply stay close to) a stationary
state, which is not physical, since mechanical energy tends to irreversibly
transfer to heat, thus increasing it in the process.

While this linear system has nice mathematical properties and serves as a good
model for the study of stability of thermoelastic bodies, it is physical
relevant only in the certain regimes under quite restrictive assumptions.
There have been efforts throughout last four decades to study more nonlinear
models, but most of them are restricted to special 1D cases or the developed
theory requires special forms of nonlinearities. In this paper, we are finally
able to bridge this gap and introduce a meaningful notion of a weak solution
for a model in nonlinear thermoelasticity, which is in full agreement with the
laws of thermodynamics. Moreover, these solutions are global-in-time for large
data, coincide with the regular ones (provided such exists) and are also
consistent in a sense that if they are smooth, they satisfy the equations
pointwise. Finally, let us point out that the results also hold in lower
dimensions 1 and 2.

##  3\. Physical derivation

The present section is devoted to the physical derivation of our model as well
as some related models. When deriving the model, we obtain candidates for
natural variables in which a model should be examined. This way we are led to
the formal version of the important estimates on the one hand, and to the
natural framework in which the weak solution is defined on the other hand.

###  3.1. Derivation of the model

We start with a system of equations modeling the balance of momentum and
internal energy

(15)
$\displaystyle\begin{cases}\rho_{S}\mathbf{u}_{tt}=\nabla\cdot\boldsymbol{\sigma},\\\
e_{t}+\nabla\cdot\mathbf{q}=\boldsymbol{\sigma}\cdot\nabla\mathbf{u}_{t},\end{cases}$

where the state variables are the displacement vector and the temperature
$\mathbf{u},\theta$, respectively, and given functions are the density of the
elastic body $\rho_{S}$, the stress tensor $\boldsymbol{\sigma}$, the internal
energy $e$ and the internal energy flux $\mathbf{q}$, while
$\boldsymbol{\sigma}\cdot\nabla\mathbf{u}_{t}$ represents the work done by the
elastic body. We next assume

$\displaystyle\rho_{S}\equiv
1,\quad\boldsymbol{\sigma}=\boldsymbol{\sigma}(\theta,\nabla\mathbf{u}),\quad
e=e(\theta,\nabla\mathbf{u}),\quad\mathbf{q}=\mathbf{q}(\theta,\nabla\theta).$

The free energy function $\psi=\psi(\theta,\mathbf{F})$, where
$\psi:\mathbb{R}\times\mathbb{R}^{3\times 3}\to\mathbb{R}$, is defined as
$\psi:=e-\theta s$, where $s$ is the entropy, while its Frechét derivative
w.r.t. $\mathbf{F}$ is denoted as $\psi_{\mathbf{F}}$. As a consequence of the
Clausius-Duhem inequality (see e.g. [18, Lemma 1.1]), one must impose

$\displaystyle\boldsymbol{\sigma}(\theta,\nabla\mathbf{u})=\psi_{\mathbf{F}}(\theta,\nabla\mathbf{u}),\quad
s(\theta,\nabla\mathbf{u})=-\psi_{\theta}(\theta,\nabla\mathbf{u}).$

It is useful to write these relations in the following form of Gibbs’
relations:

(16) $\displaystyle\theta
s(\theta,\nabla\mathbf{u})_{\theta}=e(\theta,\nabla\mathbf{u})_{\theta},$ (17)
$\displaystyle\theta
s(\theta,\nabla\mathbf{u})_{\mathbf{F}}=e(\theta,\nabla\mathbf{u})_{\mathbf{F}}-\boldsymbol{\sigma}(\theta,\nabla\mathbf{u}).$

Dividing the internal energy equation $\eqref{original:system}_{2}$ by
$\theta$ (provided that $\theta>0$), we arrive at the entropy equation

(18) $\displaystyle
s_{t}+\nabla\cdot\left(\frac{\mathbf{q}}{\theta}\right)=-\frac{\mathbf{q}\cdot\nabla\theta}{\theta^{2}},$

where $\frac{\mathbf{q}}{\theta}$ is the entropy flux and
$-\frac{\mathbf{q}\cdot\nabla\theta}{\theta^{2}}$ is the entropy production.
In accordance with the second law of thermodynamics, we have to impose a non-
negative entropy production

(19) $\displaystyle\mathbf{q}\cdot\nabla\theta\leq 0.$

The total energy balance can be obtained by multiplying
$\eqref{original:system}_{1}$ with $\mathbf{u}_{t}$ and summing up with
$\eqref{original:system}_{2}$ to obtain

(20)
$\displaystyle\frac{d}{dt}\mathcal{E}+\nabla\cdot(\boldsymbol{\sigma}\mathbf{u}_{t}+\mathbf{q})=0,$

where the total energy is

$\displaystyle\mathcal{E}=\frac{1}{2}|\mathbf{u}_{t}|^{2}+e(\theta,\nabla\mathbf{u}).$

Now, assuming the following boundary conditions on $\partial\Omega$

(21) $\displaystyle(\boldsymbol{\sigma}\mathbf{u}_{t})\cdot\mathbf{n}$
$\displaystyle=$ $\displaystyle 0,$ (22)
$\displaystyle\mathbf{q}\cdot\mathbf{n}$ $\displaystyle=$ $\displaystyle 0,$

we integrate (20) over $(0,t)\times\Omega$ to obtain

(23) $\displaystyle\int_{\Omega}\mathcal{E}(t)=\int_{\Omega}\mathcal{E}(0),$

for any $t\in(0,T]$, so the total energy is conserved.

In order to simplify the model, we first assume that energy can be decomposed
as

(24) $\displaystyle
e(\theta,\nabla\mathbf{u})=e_{1}(\theta)+e_{2}(\nabla\mathbf{u}),$

so $(\ref{gibbs1})$ implies

$\displaystyle
s(\theta,\nabla\mathbf{u})_{\theta}=\frac{e_{1}(\theta)}{\theta},$

which by integrating w.r.t. $\theta$ one concludes that entropy can also be
decomposed

(25) $\displaystyle
s(\theta,\nabla\mathbf{u})=s_{1}(\theta)+s_{2}(\nabla\mathbf{u}),$

and the relations $(\ref{gibbs1})$ and $(\ref{gibbs2})$ then simplify to

(26) $\displaystyle\theta s_{1}(\theta)_{\theta}=e_{1}(\theta)_{\theta},$ (27)
$\displaystyle\theta
s_{2}(\nabla\mathbf{u})_{\mathbf{F}}=e_{2}(\nabla\mathbf{u})_{\mathbf{F}}-\boldsymbol{\sigma}(\theta,\nabla\mathbf{u}).$

The original system (15) now simplifies to

(28)
$\displaystyle\begin{cases}\mathbf{u}_{tt}=\nabla\cdot(e_{2}(\nabla\mathbf{u})_{\mathbf{F}}-\theta
s_{2}(\nabla\mathbf{u})_{\mathbf{F}}),\\\
e_{1}(\theta)_{t}+\nabla\cdot\mathbf{q}(\theta,\nabla\theta)=-\theta
s_{2}(\nabla\mathbf{u})_{t},\end{cases}$

while the entropy equation stays the same. Note that the internal energy
equation $(\ref{original:system})_{2}$ now becomes the heat equation
$(\ref{original:system:s})_{2}$, since the elastic energy
$e_{2}(\nabla\mathbf{u})$ is canceled out.

######  Remark 3.1.

Let us point out an important observation. In the above system, due to
decoupling of the energy $\eqref{en:dec}$, the stress tensor is of the form

$\displaystyle\boldsymbol{\sigma}(\theta,\nabla\mathbf{u})=e_{2}(\nabla\mathbf{u})_{\mathbf{F}}-\theta
s_{2}(\nabla\mathbf{u})_{\mathbf{F}},$

while the work transfered to heat becomes $-\theta
s_{2}(\nabla\mathbf{u})_{t}$. This is very restrictive in $\theta$ and
completely determines the coupling.

Finally, the following assumptions of thermodynamical stability are assumed on
the energy and entropy:

(29) $\displaystyle e_{1}(\theta),~{}e_{1}(\theta)_{\theta}>0,\quad\mbox{for
all }\theta>0,$ (30) $\displaystyle\lim\limits_{\theta\to
0^{+}}s_{1}(\theta)=-\infty,$

and there exist constants $c_{1}>0$ and $c_{2}$ such that222This assumption is
automatically satisfied if, for example, $e_{2}$ is convex and $s_{2}$ is
concave.

(31) $\displaystyle e_{2}(\mathbf{F})-c_{1}s_{2}(\mathbf{F})\geq c_{2},$

for all $\mathbf{F}\in\mathbb{R}^{3\times 3}$. Now, for any constant
$\overline{\theta}>0$, we introduce the Helmholtz function

$\displaystyle
H_{\overline{\theta}}(\theta,\nabla\mathbf{u}):=e(\theta,\nabla\mathbf{u})-\overline{\theta}s(\theta,\nabla\mathbf{u}).$

Now, we multiply $(\ref{original:entropy})$ with $\overline{\theta}$,
integrate over $(0,T)\times\Omega$ and then subtract from
$(\ref{original:energy})$, which then gives us the total dissipation balance

(32)
$\displaystyle\int_{\Omega}H_{\overline{\theta}}(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\overline{\theta}\int_{0}^{t}\int_{\Omega}-\frac{\mathbf{q}(\theta,\nabla\theta)\cdot\nabla\theta}{\theta^{2}}$
$\displaystyle=\int_{\Omega}H_{\overline{\theta}}(0)+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}.$

First, by choosing $\overline{\theta}=c_{1}$, from (29), (30) and (31), one
directly has that $\theta>0$, at least a.e. Next, noticing that the condition
(29) implies $s_{1}(\theta)_{\theta}>0$, one has by (26)

$\displaystyle(e_{1}-c_{1}s_{1})_{\theta}=(e_{1})_{\theta}-c_{1}(s_{1})_{\theta}=\underbrace{(e_{1})_{\theta}-\theta(s_{1})_{\theta}}_{=0}+(\theta-
c_{1})(s_{1})_{\theta}.$

Therefore the function $e_{1}-c_{1}s_{1}$ attains its global minimum on
$\mathbb{R}^{+}$ at $\theta=c_{1}$. This combined with $(\ref{stability2})$
implies the coercivity of the Helmholtz function

(33) $\displaystyle
H_{c_{1}}(\theta,\nabla\mathbf{u})=e(\theta,\nabla\mathbf{u})-c_{1}s(\theta,\nabla\mathbf{u})\geq
C,$

where $C$ only depends on $c_{1}$ and $c_{2}$ from $(\ref{stability2})$. Thus,
(32) for $\overline{\theta}=c_{1}$ implies the boundedness of the entropy
production.

###  3.2. Prescribing the constitutive functions

First, we will assume that the deformation is small, so that the following
assumptions are physical:

$\displaystyle e_{2}(\nabla\mathbf{u})=\frac{1}{2}|\nabla\mathbf{u}|^{2},\quad
s_{2}(\nabla\mathbf{u})=\mu\nabla\cdot\mathbf{u},$

where $\mu>0$ is the heat expansion coefficient and $\nabla\cdot\mathbf{u}$
represents the linearization of Jacobian, and thus approximates the volume.
Next, the internal energy flux is taken as

$\displaystyle\mathbf{q}(\theta,\nabla\theta)=-\kappa(\theta)\nabla\theta,$

where $\kappa(\theta)>0$ is the heat conductivity. Finally, we assume that the
body is fixed at the boundary and the internal energy flux through the
boundary is zero:

$\displaystyle\mathbf{u}=0,\quad\partial_{n}\theta=0,\quad\mbox{on
}(0,T)\times\partial\Omega.$

Supplementing with initial data, this leads to the following problem:

$\displaystyle\begin{cases}\mathbf{u}_{tt}=\nabla\cdot(\nabla\mathbf{u}-\mu\theta
I),&\quad\text{ in }(0,T)\times\Omega,\\\
c_{V}(\theta)\theta_{t}-\nabla\cdot(\kappa(\theta)\nabla\theta)=-\mu\theta\nabla\cdot\mathbf{u}_{t},&\quad\text{
in }(0,T)\times\Omega,\\\ \mathbf{u}=0,~{}\partial_{n}\theta=0,&\quad\text{on
}(0,T)\times\partial\Omega,\\\
\mathbf{u}(0,\cdot)=\mathbf{u}_{0},~{}\mathbf{u}_{t}(0,\cdot)=\mathbf{v}_{0},~{}\theta(0,\cdot)=\theta_{0},&\end{cases}$

where $c_{V}(\theta):=e_{1}(\theta)_{\theta}$ is the heat capacity. It remains
to prescribe $c_{V}(\theta)$ and $\kappa(\theta)$.

####  3.2.1. High temperature model

For many homogeneous materials, the heat capacity is known to be almost
constant at high temperatures due to the Dulong-Petit law. On the other hand,
heat conductivity tends to decrease as $1/\theta$ due to high concentration of
phonons which then start to scatter. Nevertheless, $1/\theta$ can be
approximated with a constant for large $\theta$. This leads to the first
model:

$\displaystyle c_{V}(\theta)=1,\quad\kappa(\theta)=1.$

####  3.2.2. Low temperature model

At low temperatures, the heat capacity increases fast, usually as $\theta^{3}$
due to the Debye law. Moreover, the heat conductivity at this temperature is
mostly only affected by heat capacity and thus also behaves as $\theta^{3}$.
To keep the result general, the following is chosen:

(34) $\displaystyle
c_{V}(\theta)=1+\alpha\theta^{\alpha-1},\quad\kappa(\theta)=1+\theta^{\beta},$

where $\alpha>1$ and $\beta>0$. Here, the constant part was added due to
mathematical reasons. In particular, the constant in $c_{V}(\theta)$ ensures
that $\ln\theta$ will be bounded, so the temperature will not vanish a.e,
while the constant part in $\kappa(\theta)$ is added to provide enough control
over $\ln\theta$. The conditions $\alpha>1$ and $\beta>0$ are chosen so that
$c_{V}(\theta)$ and $\kappa(\theta)$ are growing functions.

##  4\. Proof of Theorem 1.1

In our approach, it is crucial to work with thermodynamical variables, i.e. to
use equation (2). However, this equation is formally equivalent with (1)2
under assumption $\theta>0$, which needs to be proved. Therefore, we split our
proof in two parts. In the first part, we construct approximate solutions
$({\bf u}^{n},\theta^{n})$ to equation (1) such that $\theta^{n}>0$. In the
second step, we introduce thermodynamical variables and pass to the limit in
approximations parameters to obtain a weak solution in sense of Definition
1.1.

###  4.1. Step 1: Approximate problem

We introduce a smooth basis
$\\{{\boldsymbol{\varphi}}_{i}\\}_{i\in\mathbb{N}}$ of $H_{0}^{1}(\Omega)^{3}$
and denote $V_{n}:={\rm span}\\{{\boldsymbol{\varphi}}_{i}\\}_{1\leq i\leq
n}$. The approximate problem is defined as follows.

######  Definition 4.1 (Approximate problem).

We say that $({\bf u}^{n},\theta^{n})\in W^{1,\infty}(0,T;V_{n})\times
L^{2}(0,T;H^{1}(\Omega))$ is a solution to the approximate problem if for all
${\boldsymbol{\varphi}}\in V_{n}$ and $\phi\in H^{1}(\Omega)$ the following
equations are satisfied in $\mathcal{D}^{\prime}(0,T)$

(35)
$\displaystyle\frac{d^{2}}{dt^{2}}\int_{\Omega}\mathbf{u}^{n}\cdot{\boldsymbol{\varphi}}+\int_{\Omega}\nabla\mathbf{u}^{n}:\nabla{\boldsymbol{\varphi}}=\mu\int_{\Omega}\theta^{n}\nabla\cdot{\boldsymbol{\varphi}},$
(36)
$\displaystyle\frac{d}{dt}\int_{\Omega}\theta^{n}\phi+\int_{\Omega}\nabla\theta^{n}\cdot\nabla\phi=-\mu\int_{\Omega}\theta^{n}(\nabla\cdot\mathbf{u}^{n}_{t})\phi.$

Moreover, for the initial data, we choose
$0<c_{n}\leq\theta^{n}(0)=\theta_{0}^{n}\in H^{1}(\Omega)$ as a regularization
of $\theta_{0}$ such that $\partial_{n}\theta_{0}^{n}=0$ on $\partial\Omega$,
$\theta_{0}^{n}\to\theta_{0}$ in $L^{1}(\Omega)$ as $n\to\infty$ and
$\int_{\Omega}\theta_{0}^{n}\leq\int_{\Omega}\theta_{0}$ for all
$n\in\mathbb{N}$. For the displacement, we choose
$\mathbf{u}^{n}(0)=P_{V_{n}}{\bf u}_{0}$,
$\mathbf{u}_{t}^{n}(0)=P_{V_{n}}\mathbf{v}_{0}$, where $P_{V_{n}}$ is an
orthogonal projection onto $V_{n}$.

######  Remark 4.1.

Note that we are actually using the heat equation in approximate problem,
while later we will switch to the entropy equation in order to pass to the
limit $n\to\infty$.

######  Lemma 4.1.

Let $({\bf u}^{n},\theta^{n})\in W^{1,\infty}(0,T;V_{n})\times
L^{2}(0,T;H^{1}(\Omega))$ be a solution to the approximate problem given by
Definition 4.1. Then the following energy equality is satisfied:

(37) $\displaystyle\int_{\Omega}\theta^{n}(t)+\frac{1}{2}\int_{\Omega}|{\bf
u}^{n}_{t}(t)|^{2}+\frac{1}{2}\int_{\Omega}|\nabla{\bf
u}^{n}(t)|^{2}=E^{n}_{0},\quad\text{ for a.a. }t\in[0,T],$

where
$E^{n}_{0}=\int_{\Omega}\theta_{0}^{n}+\frac{1}{2}\int_{\Omega}|P_{V_{n}}\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla
P_{V_{n}}{\bf u}_{0}|^{2}$ is the energy of the initial data projected onto
finite-dimensional subspace $V_{n}$.

###### Proof.

We take ${\boldsymbol{\varphi}}={\bf u}_{t}^{n}$ in (35), $\phi=1$ in (36),
add the resulting equations and integrate over $(0,t)$. Notice that the terms
on the right hand sides cancel. ∎

First we prove a version of the standard maximum principle result for
parabolic equations which we will use to prove strict positivity of the
temperature.

######  Lemma 4.2.

Let $a\in L^{1}(0,T;L^{\infty}(\Omega))$ and $\theta\in
L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ be the solution to the
following parabolic problem:

$\displaystyle\begin{cases}\theta_{t}-\Delta\theta=-a\theta,&\quad\text{ in
}(0,T)\times\Omega,\\\ \partial_{n}\theta=0,&\quad\text{ on
}(0,T)\times\partial\Omega,\\\ \theta(0)=\theta_{0}^{n}>0.&\end{cases}$

Then

$\displaystyle\theta(t,{\bf x})\geq\left(\min_{{\bf
x}\in\Omega}\theta_{0}^{n}({\bf
x})\right)\exp\left(-\int_{0}^{t}\|a(s)\|_{L^{\infty}(\Omega)}ds\right).$

###### Proof.

We define $C_{0}=\min_{{\bf x}\in\Omega}\theta_{0}^{n}({\bf x})$ and

$\displaystyle d(t,{\bf x})=\theta(t,{\bf
x})-C_{0}\exp\left(-\int_{0}^{t}\|a(s)\|_{L^{\infty}(\Omega)}ds\right).$

By straightforward calculation $d$ satisfies the following equation:

$\displaystyle d_{t}-\Delta
d+ad=C_{0}\exp\left(-\int_{0}^{t}\|a(s)\|_{L^{\infty}(\Omega)}ds\right)\underbrace{(\|a\|_{L^{\infty}}-a)}_{\geq
0}.$

Therefore

(38) $\displaystyle d_{t}-\Delta d+ad\geq 0,\quad d(0)\geq 0.$

We multiply (38) with $d^{-}:=\max\\{-d,0\\}$ to get:

$\displaystyle\frac{1}{2}\frac{d}{dt}\|d^{-}\|^{2}_{L^{2}(\Omega)}+\|\nabla
d^{-}\|^{2}_{L^{2}(\Omega)}\leq-a\|d^{-}\|_{L^{2}(\Omega)}^{2}.$

Since $d(0)\geq 0$, we have $d^{-}(0)=0$ and therefore by the Gronwall
inequality we get $d^{-}=0$ which finishes the proof. ∎

######  Proposition 1.

For every $n\in\mathbb{N}$ there exists a solution $(\mathbf{u},\theta)$ to
the approximate problem in the sense of Definition 4.1. Moreover, one has
$\theta\in L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$.

###### Proof.

We prove existence by using Schaefer’s fixed point theorem. We define the
fixed point set:

$X_{n}=C^{1}([0,T];V_{n}).$

We equip $X_{n}$ with the following norm: $\|{\bf
u}\|_{X_{n}}=\max_{t\in[0,T]}\\{\|{\bf u}(t)\|_{L^{2}(\Omega)},\|{\bf
u}_{t}(t)\|_{L^{2}(\Omega)}\\}$. Fixed point mapping $\mathcal{A}$ is defined
in the following way. Let ${\bf u}\in X_{n}$. We define $\theta({\bf u})$ by
solving (36) with ${\bf u}^{n}$ on the right hand side. More precisely,
$\theta({\bf u}^{n})$ satisfies the following equation for every $\phi\in
H^{1}(\Omega)$:

(39) $\displaystyle\frac{d}{dt}\int_{\Omega}\theta({\bf
u})\phi+\int_{\Omega}\nabla\theta({\bf
u})\cdot\nabla\phi=-\mu\int_{\Omega}\theta({\bf
u})(\nabla\cdot\mathbf{u}_{t})\phi.$

Existence of a weak solution $\theta({\bf u})\in L^{2}(0,T;H^{1}(\Omega))$ is
a consequence of the standard parabolic theory. Moreover, by taking
$\phi=\theta({\bf u})$, we obtain the following estimate:

(40) $\displaystyle\frac{1}{2}\frac{d}{dt}\|\theta({\bf
u})\|^{2}_{L^{2}(\Omega)}+\|\nabla\theta({\bf
u})\|^{2}_{L^{2}(\Omega)}\leq\mu\|\nabla\cdot{\bf
u}_{t}\|_{L^{\infty}(\Omega)}\|\theta({\bf u})\|^{2}_{L^{2}(\Omega)}.$

However, since $V_{n}$ is a finite dimensional space, all norms are equivalent
so we also have

(41) $\displaystyle\|\nabla\cdot{\bf u}_{t}\|_{L^{\infty}(\Omega)}\leq
C_{n}\|{\bf u}_{t}\|_{L^{2}(\Omega)}.$

Note that the constant in (41) depends on $n$, but in the proof of this Lemma
$n$ is fixed so this dependence does not influence the proof. By combining
(40) and (41), and using Gronwall’s inequality we get:

(42) $\displaystyle\|\theta({\bf u})\|_{L^{2}(0,T;H^{1}(\Omega))}\leq
C_{n}(\|{\bf u}\|_{X_{n}}),$

where $C_{n}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a monotone function. Now,
$\mathcal{A}{\bf u}$ is obtained by solving (35) with $\theta({\bf u})$
instead of $\theta$ on the right hand side. More precisely, $\mathcal{A}{\bf
u}$ satisfies the following equations:

(43) $\displaystyle\frac{d^{2}}{dt^{2}}\int_{\Omega}(\mathcal{A}{\bf
u})\cdot{\boldsymbol{\varphi}}+\int_{\Omega}\nabla(\mathcal{A}{\bf
u}):\nabla{\boldsymbol{\varphi}}=\mu\int_{\Omega}\theta({\bf
u})\nabla\cdot{\boldsymbol{\varphi}},\quad{\boldsymbol{\varphi}}\in V_{n}.$

Existence of $\mathcal{A}{\bf u}\in C^{1}([0,T];V_{n})$ is obtained by solving
a linear ODE system (43). Therefore operator $\mathcal{A}:X_{n}\to X_{n}$ is
well defined. Let us prove that is satisfies the assumptions of Schaefer’s
fixed point theorem.

Continuity of $\mathcal{A}$. Let ${\bf u}^{m}$ be a convergent sequence in
$X_{n}$ and ${\bf u}=\lim\limits_{m\to\infty}{\bf u}^{m}$. Let
$\psi^{m}:=\theta({\bf u})-\theta^{m}({\bf u})$. By (39), $\psi^{m}$ satisfies
the following equality with zero initial data:

$\displaystyle\frac{d}{dt}\int_{\Omega}\psi^{m}\phi+\int_{\Omega}\nabla\psi^{m}\cdot\nabla\phi$
$\displaystyle=-\mu\int_{\Omega}\theta({\bf
u})(\nabla\cdot\mathbf{u}_{t}-\nabla\cdot{\bf
u}_{t}^{m})\phi-\mu\int_{\Omega}(\nabla\cdot{\bf u}_{t}^{m})\psi^{m}\phi.$

By taking $\phi=\psi^{m}$, we arrive at

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\psi^{m}\|^{2}_{L^{2}(\Omega)}+\|\nabla\psi^{m}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq\mu\|\nabla\cdot(\mathbf{u}_{t}-{\bf
u}_{t}^{m})\|_{L^{\infty}(\Omega)}\|\theta({\bf
u})\|_{L^{2}(\Omega)}\|\psi^{m}\|_{L^{2}(\Omega)}$
$\displaystyle\quad+\mu\|\psi^{m}\|^{2}_{L^{2}(\Omega)}\|\nabla\cdot{\bf
u}_{t}^{m}\|_{L^{\infty}(\Omega)}.$

Next, (42) and boundedness of $\|\nabla\cdot{\bf
u}_{t}^{m}\|_{L^{\infty}(\Omega)}$, together with Gronwall’s inequality, yield
a bound

$\|\nabla\psi^{m}\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}\leq
C_{n}\|\nabla\cdot(\mathbf{u}_{t}-{\bf
u}_{t}^{m})\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}.$

This, in turn, allows us to obtain:

(44) $\displaystyle\|\psi^{m}\|_{L^{2}(0,T;H^{1}(\Omega))}\leq
C_{n}\|\nabla\cdot(\mathbf{u}_{t}-{\bf
u}_{t}^{m})\|_{L^{\infty}(0,T;L^{\infty}(\Omega))},$

where $C_{n}$ is a constant independent of $m$. Now, let us denote
$\mathbf{a}^{m}=\mathcal{A}{\bf u}-\mathcal{A}{\bf u}^{m}$. By definition of
operator $\mathcal{A}$ in (43), $\mathbf{a}^{m}$ satisfy the following
equations with zero initial data:

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{\Omega}\mathbf{a}^{m}\cdot{\boldsymbol{\varphi}}+\int_{\Omega}\nabla\mathbf{a}^{m}:\nabla{\boldsymbol{\varphi}}=\mu\int_{\Omega}\psi^{m}\nabla\cdot{\boldsymbol{\varphi}},\;{\boldsymbol{\varphi}}\in
V_{n}.$

By taking ${\boldsymbol{\varphi}}=\mathbf{a}_{t}^{m}$, we get

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\|\mathbf{a}_{t}^{m}\|^{2}_{L^{2}(\Omega)}+\|\nabla\mathbf{a}^{m}\|^{2}_{L^{2}(\Omega)}\right)$
$\displaystyle\leq\|\nabla\psi^{m}\|_{L^{2}(\Omega)}\|\mathbf{a}_{t}^{m}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C_{n}\|\nabla\cdot(\mathbf{u}_{t}-{\bf
u}_{t}^{m})\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\|\mathbf{a}_{t}^{m}\|_{L^{2}(\Omega)}.$

Since all norms on $V_{n}$ are equivalent we have
$\|\nabla\cdot(\mathbf{u}_{t}-{\bf
u}_{t}^{m})\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\to 0$ and therefore, by
Gronwall’s and Young’s inequalities, $\|\mathbf{a}^{m}\|_{X_{n}}\to 0$. Thus
we proved that $\mathcal{A}$ is a continuous function on $X_{n}$.

Compactness of $\mathcal{A}$. Let ${\bf u}\in X_{n}$, $\|{\bf u}\|_{X_{n}}\leq
R$. By (43), we have

$\displaystyle\|(\mathcal{A}{\bf u})_{tt}\|_{L^{2}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq\|\Delta(\mathcal{A}{\bf
u})\|_{L^{2}(0,T;L^{2}(\Omega))}+\mu\|\nabla(\theta({\bf
u}))\|_{L^{2}(0,T;L^{2}(\Omega))}\leq C_{n}(R).$

Here we have used continuity of $\mathcal{A}$ and equivalence of norms on
$X_{n}$ to bound the first term, and (42) to bound the second. Since $X_{n}$
is finite dimensional, these estimates immediately give compactness of
operator $\mathcal{A}$ due to the compactness of the embedding
$H^{1}(0,T)\hookrightarrow C[0,T]$.

Boundedness of $Y=\\{{\bf u}\in X_{n}:(\exists\lambda\in(0,1])\;{\bf
u}=\lambda\mathcal{A}{\bf u}\\}$.

Let ${\bf u}\in Y$ and $\lambda\in(0,1]$ such that ${\bf
u}=\lambda\mathcal{A}{\bf u}$. Then $(\frac{1}{\lambda}{\bf u},\theta({\bf
u}))$ is a solution to the approximate problem given by Defintion 4.1.
Therefore, the following energy equality is satisfied by Lemma 4.1:

$\displaystyle\int_{\Omega}\theta({\bf
u})(t)+\frac{1}{2}\frac{1}{\lambda}\int_{\Omega}|{\bf
u}_{t}(t)|^{2}+\frac{1}{2}\frac{1}{\lambda}\int_{\Omega}|\nabla{\bf
u}(t)|^{2}$
$\displaystyle=\int_{\Omega}\theta_{0}^{n}+\frac{1}{2}\frac{1}{\lambda}\int_{\Omega}|P_{V_{n}}\mathbf{v}_{0}|^{2}+\frac{1}{2}\frac{1}{\lambda}\int_{\Omega}|\nabla
P_{V_{n}}\mathbf{u}_{0}|^{2},\quad t\in[0,T].$

Therefore,

$\displaystyle\|{\bf u}\|_{C^{1}([0,T;V_{n})}\leq CE_{0}^{n},\quad{\bf u}\in
Y.$

Thus, by Schaefer’s fixed point theorem there exists a fixed point ${\bf u}$
of the mapping $\mathcal{A}$. Moreover, by Lemma 4.2, $\theta({\bf u})$ is a
strictly positive function. Therefore, $({\bf u},\theta({\bf u}))$ is a
solution to the approximate problem in the sense of Definition 4.1.

Finally, since $\theta\nabla\cdot\mathbf{u}_{t}\in L^{2}((0,T)\times\Omega)$,
there is a unique solution $\tilde{\theta}\in L^{2}(0,T;H^{2}(\Omega))\cap
H^{1}(0,T;L^{2}(\Omega))$ to the equation

$\displaystyle\tilde{\theta}_{t}-\Delta\tilde{\theta}=-\mu\theta\nabla\cdot\mathbf{u}_{t},$

so $\theta\equiv\tilde{\theta}$ and the improved regularity is thus obtained.
∎

####  4.1.1. Passing to the limit

######  Lemma 4.3.

There exists $\theta\in L^{\infty}(0,T;\mathcal{M}^{+}(\Omega))$ and ${\bf
u}\in L^{\infty}(0,T;H^{1}(\Omega))\cap W^{1,\infty}(0,T;L^{2}(\Omega))$ such
that the following convergence properties hold (on a subsequence)

$\displaystyle\theta^{n}\rightharpoonup\theta,$ $\displaystyle\quad\text{
weakly* in }L^{\infty}(0,T;\mathcal{M}^{+}(\Omega)),$ $\displaystyle{\bf
u}^{n}\rightharpoonup{\bf u},$ $\displaystyle\quad\text{ weakly* in
}L^{\infty}(0,T;H^{1}(\Omega)),$ $\displaystyle{\bf
u}_{t}^{n}\rightharpoonup{\bf u}_{t},$ $\displaystyle\quad\text{ weakly* in
}L^{\infty}(0,T;L^{2}(\Omega)).$

###### Proof.

As a direct consequence of energy equality given in Lemma 4.1, the following
uniform estimate holds:

(45) $\displaystyle\|\theta^{n}\|_{L^{\infty}(0,T;L^{1}(\Omega))}+\|{\bf
u}\|_{{L^{\infty}(0,T;H^{1}(\Omega))}}+\|{\bf
u}_{t}\|_{{L^{\infty}(0,T;L^{2}(\Omega))}}\leq E_{0},$

where $n\in\mathbb{N}$ and $E_{0}=\|\theta_{0}\|_{L^{1}(\Omega)}+\|\nabla{\bf
u}_{0}\|_{L^{2}(\Omega)}+\|\mathbf{v}_{0}\|_{L^{2}(\Omega)}$. Therefore, the
existence of a weakly* convergence subsequence follows from standard results
about weak* compactness of a ball in the corresponding function spaces. ∎

By using Lemma 4.3 we can pass to the limit in equation (35) and prove that
$({\bf u},\theta)$ satisfy the following variational equation:

$\displaystyle-\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot{\boldsymbol{\varphi}}_{t}+\int_{0}^{T}\int_{\Omega}\nabla\mathbf{u}:\nabla{\boldsymbol{\varphi}}=\mu\int_{0}^{T}\int_{\Omega}\nabla\cdot{\boldsymbol{\varphi}}d\theta,$

for all ${\boldsymbol{\varphi}}\in C^{\infty}_{0}([0,T)\times\Omega)$.
However, weak* convergences from Lemma 4.3 are not enough for passing to the
limit in (36) because it contains nonlinear terms. Since approximate solutions
are smooth and $\theta^{n}\geq C_{n}>0$, $n\in\mathbb{N}$ (see Lemma 4.2), we
can divide the heat equation $\eqref{APP2}$ by $\theta^{n}$, so by using a new
unknown $\tau^{n}=\ln\theta^{n}$, we can rewrite (36) in the form of entropy
equation:

$\displaystyle\int_{0}^{T}\int_{\Omega}\tau^{n}\phi_{t}-\int_{0}^{T}\int_{\Omega}\nabla\tau^{n}\cdot\nabla\phi+\mu\int_{0}^{T}\int_{\Omega}\mathbf{u}^{n}_{t}\cdot\nabla\phi+\int_{0}^{T}\int_{\Omega}|\nabla\tau^{n}|^{2}\phi$
(46) $\displaystyle=-\int_{\Omega}\tau_{0}\phi(0,x),\qquad\phi\in
C_{0}^{\infty}([0,T)\times\Omega),$

where $\tau_{0}^{n}=\ln\theta_{0}^{n}$. Now, one can also show that the total
dissipation balance $\eqref{total:dissipation:balance:lin}$ holds for the
approximate solutions, so we get the following additional estimate:

(47)
$\displaystyle\|\nabla\tau^{n}\|_{L^{2}(0,T;L^{2}(\Omega))}+\|\tau^{n}\|_{L^{\infty}(0,T;L^{1}(\Omega))}\leq
C.$

Therefore there exists $\tau\in L^{2}(0,T;H^{1}(\Omega))$ and measure
$\boldsymbol{\sigma}\in\mathcal{M}^{+}([0,T]\times\overline{\Omega})$ such
that

$\displaystyle\tau_{n}\rightharpoonup\tau$ $\displaystyle\quad{\rm
weakly}\;{\rm in}\;L^{2}(0,T;H^{1}(\Omega)),$
$\displaystyle|\nabla\tau^{n}|^{2}\rightharpoonup\boldsymbol{\sigma}$
$\displaystyle\quad{\rm weakly*}\;{\rm
in}\;\mathcal{M}^{+}([0,T]\times\overline{\Omega}).$

Using these convergence properties we can pass to the limit in (36) to obtain
the entropy equation (5). It remains to identify relations between $\tau$,
$\theta$ and $\boldsymbol{\sigma}.$ First, we prove
$\boldsymbol{\sigma}\geq|\nabla\tau|^{2}$. For $\phi\in
C([0,T]\times\overline{\Omega})$, we define mapping
$I_{\phi}:L^{2}(\Omega)\to\mathbb{R}$ with

$I_{\phi}(f)=\int_{\Omega}f^{2}\phi.$

Notice that this is a continuous mapping and convex for $\phi\geq 0$. In
particular, it is lower semicontinuous in strong $L^{2}$ topology. Therefore,
by Mazur’s lemma (see e.g. [3, Thm. 9.1]), $I_{\phi}$ is weakly lower
semicontiunous. Thus we have:

$\int_{\Omega}|\nabla\tau|^{2}\phi=I_{\phi}(\nabla\tau)=I_{\phi}(\lim_{n}\nabla\tau^{n})\leq\lim_{n}I_{\phi}(\nabla\tau^{n})=\langle\boldsymbol{\sigma},\phi\rangle,$

which gives $\boldsymbol{\sigma}\geq|\nabla\tau|^{2}$ .

From equation (4.1.1), we immediately get that time derivatives of $\tau^{n}$
are uniformly bounded in $L^{1}(0,T;W^{-1,p^{*}}(\Omega))$, $p>3$, i.e. we
have the following estimate:

$\|\tau_{t}^{n}\|_{L^{1}(0,T;W^{-1,p^{*}}(\Omega))}\leq C.$

Combining this estimate with (47) and Aubin-Lions lemma [2] we get that
sequence $\tau^{n}$ strongly converges (after passing to a subsequence) in
$L^{1}((0,T)\times\Omega)$.

Therefore, $\theta^{n}$ converges almost everywhere (on a subseqence), i.e. we
have

$\ln\theta^{n}=\tau^{n}\to\tau\quad{\rm a.e.}\;{\rm in}\;(0,T)\times\Omega,$

which also implies

$\theta^{n}=e^{\tau_{n}}\to e^{\tau}\quad{\rm a.e.}\;{\rm
in}\;(0,T)\times\Omega.$

We can now use the theorem of Egorov, which gives us that for every
$\varepsilon>0$ there is a set $A_{\varepsilon}\subset(0,T)\times\Omega$ such
that $|((0,T)\times\Omega)\setminus A_{\varepsilon}|<\varepsilon$ and
$\theta^{n}\to e^{\tau}$ uniformly on $A_{\varepsilon}$, and in light of
uniqueness of uniform and distributional limits, the continuous part of
$d\theta$ equals $e^{\tau}$ on $A_{\varepsilon}$. By letting $\varepsilon\to
0$, one obtains that the continuous part of $\theta$ is $e^{\tau}$ a.e. on
$(0,T)\times\Omega$. Note that we do not rule out that $\theta$ also includes
a singular part supported on sets of Lebesgue measure $0$ and therefore cannot
conclude $d\theta=e^{\tau}$. Nevertheless, we will prove that inequality
$e^{\tau}\leq d\theta$ holds in the sense of measures. Let $E$ be a set of all
points where $\theta^{n}$ does not converge to $\theta$. It is a set of
measure zero, and singular part of measure $\theta$ (we name it $f$) is
supported in $E$. Now by Fatou’s lemma, for any non-negative $\phi\in
C([0,T]\times\overline{\Omega})$ we have:

$\int_{\Omega}e^{\tau}\phi=\int_{\Omega\setminus
E}e^{\tau}\phi=\int_{\Omega\setminus
E}\lim_{n}\theta^{n}\phi\leq\lim_{n}\int_{\Omega\setminus
E}\theta^{n}\phi=\int_{\Omega}\phi d\theta.$

Therefore, for any non-negative $\phi\in C([0,T]\times\overline{\Omega})$, we
have

$\int_{\Omega}e^{\tau}\phi\leq\int_{\Omega}\phi d\theta.$

This concludes the proof of the existence part of Theorem 1.1.

###  4.2. Consistency

Assume that $(\mathbf{u},\tau)$ is a weak solution in the sense of Definition
1.1 which is smooth. The goal is to prove that then $(\mathbf{u},e^{\tau})$ is
a classical solution to the system $\eqref{system0}$. First, it is obvious
that $\eqref{system0}_{1}$ is satisfied and $\mathbf{u}=0$ on
$(0,T)\times\partial\Omega$. Moreover, since the momentum equation is
satisfied pointwise, this implies that $\theta$ is a function so
$e^{\tau}=\theta$. Now, for any $t\in(0,T]$, let $\psi_{n}\in C(\mathbb{R})$
be a sequence of functions defined as

$\displaystyle\psi_{n}(\tau):=\begin{cases}1,&\text{for }\tau\leq t,\\\
1-n(\tau-t),&\text{for }t<\tau\leq t+\frac{1}{n},\\\ 0,&\text{for
}\tau>t+\frac{1}{n},\end{cases}$

so by choosing $\boldsymbol{\varphi}=\mathbf{u}\psi_{n}$ in $\eqref{momweak}$
and $\phi=\theta\psi_{n}$ in (5) and then summing them up and letting
$n\to\infty$, one has

$\displaystyle\int_{\Omega}\theta(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)\geq\int_{\Omega}\theta_{0}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

which compared to $\eqref{enweak}$ implies that $\eqref{enweak}$ has to hold
as equality. Let us now prove that $\eqref{entweak}$ holds as identity. We
assume the opposite - there exists a non-negative test function $\tilde{\phi}$
such that

$\displaystyle\int_{0}^{T}\int_{\Omega}{\ln\theta}\tilde{\phi}_{t}-\int_{0}^{T}\int_{\Omega}\nabla{\ln\theta}\cdot\nabla\tilde{\phi}$
$\displaystyle\quad+\mu\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\nabla\tilde{\phi}+\int_{0}^{T}\int_{\Omega}|\nabla{\ln\theta}|^{2}\tilde{\phi}$
(48) $\displaystyle<-\int_{\Omega}{\ln\theta}_{0}\tilde{\phi}_{0}.$

and w.l.o.g. $\tilde{\phi}\leq\theta$. Then, by choosing
$\phi=\theta-\tilde{\phi}$ in (5) and summing it up with (48) gives us

$\displaystyle\int_{\Omega}\theta(T)+\mu\int_{0}^{T}\int_{\Omega}\theta\nabla\cdot\mathbf{u}_{t}>\int_{\Omega}\theta_{0},$

which combined with $\eqref{momweak}$ for $\boldsymbol{\varphi}=\mathbf{u}$
implies

$\displaystyle\int_{\Omega}\theta(T)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(T)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(T)$
$\displaystyle>\int_{\Omega}\theta_{0}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}|^{2},$

so we have a contradiction. Thus (5) holds as an identity for all test
functions, and therefore pointwise. Now, due to the sufficient regularity of
the functions, one has

$\displaystyle\int_{0}^{T}\int_{\Omega}\big{(}{\ln\theta}_{t}-\Delta{\ln\theta}+\mu\nabla\cdot\mathbf{u}_{t}-|\nabla{\ln\theta}|^{2}\big{)}\phi=-\int_{0}^{T}\int_{\partial\Omega}\partial_{n}\ln\theta\phi.$

For any test function $\varphi\in C^{\infty}([0,T]\times\partial\Omega)$, we
can now choose a sequence of test functions $\phi_{n}$ converging to
$\phi=\varphi\chi_{\partial\Omega}$, so
$\int_{0}^{T}\int_{\partial\Omega}\partial_{n}\ln\theta\varphi=0$. Since the
test function $\varphi$ was arbitrary, one has $\partial_{n}{\ln\theta}=0$, so
$\partial_{n}\theta=0$. Finally, by multiplying $\eqref{ent:eq1}$ with
$\theta$, one obtains that the energy equation $\eqref{system0}_{2}$ holds
pointwise.

###  4.3. Weak-strong uniqueness

In this section we prove the weak-strong uniqueness part of Theorem 1.1. To
this end, we use a relative entropy method. Our proof mostly follows the ideas
from [14], where weak-strong uniqueness of Navier-Stokes-Fourier system
describing heat-conducting compressible fluids is studied. However, comparing
to [14] (and Section 5.4), how we deal with some terms towards the end of the
proof is different. We make use of the specific simple form of the system to
avoid dealing with quadratic term $|\theta-\Theta|^{2}$, since this would
require us to have at least quadratic heat energy (see
$\eqref{quadtratic:control}$). Last but not least, due to defect measure
$\theta$ appearing in the momentum equation $\eqref{momweak}$, the relative
entropy method is extended to its measure-valued version.

For a weak solution $(\mathbf{u},\tau)$ in sense of Definition 1.1 with
measures $\theta,\boldsymbol{\sigma}$, and smooth functions $\mathbf{U}\in
C_{0}^{\infty}([0,T]\times\Omega)$ and $\Theta\in
C^{\infty}([0,T]\times\Omega)$ with $\Theta>0$, $\partial_{n}\Theta=0$ on
$(0,T)\times\partial\Omega$ and $\mathcal{T}:=\ln\Theta$, we define the
relative entropy inequality

$\displaystyle\int_{\Omega}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}|\nabla\tau-\nabla\mathcal{T}|^{2}\Theta$
$\displaystyle\leq\int_{\Omega}\left(e^{\tau_{0}}-\Theta(0)-\Theta(0)(\tau_{0}-\mathcal{T}(0))\right)$
$\displaystyle+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(0)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(0)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\nabla\cdot\nabla\mathbf{u}-\mu\nabla\Theta\big{)}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\tau-\mathcal{T})(\Theta_{t}-\Delta\Theta+\mu\Theta\nabla\cdot\mathbf{U}_{t})$
(49)
$\displaystyle\quad-\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}~{}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right),$

where the measure-valued relative entropy is defined as

$\displaystyle
d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right):=d\theta-\left(\Theta-\Theta(\tau-\mathcal{T})\right)dx.$

Moreover, we notice the following:

######  Lemma 4.4.

For $\theta=e^{\tau}dx+f$, where $f\geq 0$ is a singular part of $\theta$, for
any $\mathcal{T}=\Theta$, we have

(50) $\displaystyle 0\leq
e^{\tau}-\Theta-\Theta(\tau-\mathcal{T})\leq\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right),$

so $\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)$ is non-
negative and $\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)=0$
if and only if $\tau=\mathcal{T}$, $e^{\tau}=\Theta$ and $f=0$.

####  4.3.1. Any weak solution satisfies the relative entropy inequality

We start with testing $\eqref{momweak}$ with $\chi_{(0,t]}\mathbf{U}_{t}$ to
obtain

$\displaystyle\int_{\Omega}(\mathbf{u}_{t}\cdot\mathbf{U}_{t})(t)-\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\mathbf{U}_{tt}+\int_{0}^{t}\int_{\Omega}\nabla\mathbf{u}:\nabla\mathbf{U}_{t}-\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}d\theta$
$\displaystyle=\int_{\Omega}\mathbf{v}_{0}\cdot\mathbf{V}_{0}$

and expressing

$\displaystyle\int_{0}^{t}\int_{\Omega}\nabla\mathbf{u}:\nabla\mathbf{U}_{t}=-\int_{0}^{t}\int_{\Omega}\mathbf{u}\cdot\Delta\mathbf{U}_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\Delta\mathbf{U}-\int_{\Omega}\mathbf{u}\cdot\Delta\mathbf{U}~{}\Big{|}_{0}^{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\Delta\mathbf{U}+\int_{\Omega}\nabla\mathbf{u}:\nabla\mathbf{U}~{}\Big{|}_{0}^{t},$

we obtain

$\displaystyle\int_{\Omega}(\mathbf{u}_{t}\cdot\mathbf{U}_{t})(t)+\int_{\Omega}\big{(}\nabla\mathbf{u}:\nabla\mathbf{U}\big{)}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}\big{)}-\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}d\theta$
(51)
$\displaystyle=\int_{\Omega}\mathbf{v}_{0}\cdot\mathbf{V}_{0}+\int_{\Omega}\nabla\mathbf{u}_{0}:\nabla\mathbf{U}_{0}.$

Since

$\displaystyle\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}\big{)}+\int_{0}^{t}\int_{\Omega}\mathbf{U}_{t}\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}\big{)}-\Big{(}\frac{1}{2}\int_{\Omega}|\mathbf{U}_{t}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{U}|^{2}\Big{)}\Big{|}_{0}^{t},$

we subtract $\eqref{id1}$ from the energy inequality $\eqref{enweak}$ to
obtain

$\displaystyle\int_{\Omega}\theta(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}-\mu\nabla\Theta\big{)}$
$\displaystyle\quad-\mu\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\nabla\Theta+\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}d\theta$
(52)
$\displaystyle\leq\int_{\Omega}e^{\tau_{0}}+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}-\mathbf{V}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}-\nabla\mathbf{U}_{0}|^{2},$

where we have also added and subtracted the term
$\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\mu\nabla\Theta$.
Next, in $\eqref{entweak}$ we choose $\phi=\chi_{(0,t]}\Theta$ to obtain

$\displaystyle\int_{0}^{t}\int_{\Omega}{{\tau}}\Theta_{t}+\int_{0}^{t}\int_{\Omega}{{\tau}}\Delta\Theta+\mu\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\nabla\Theta+\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}|^{2}\Theta$
(53)
$\displaystyle\leq\int_{\Omega}({{\tau}}\Theta)(t)-\int_{\Omega}{{\tau}}_{0}\Theta_{0},$

so by adding and subtracting the terms
$\int_{0}^{t}\int_{\Omega}{\mathcal{T}}\Theta_{t}$,
$\int_{0}^{t}\int_{\Omega}{\mathcal{T}}\Delta\Theta$ and
$\mu\int_{0}^{t}\int_{\Omega}\Theta\nabla\cdot\mathbf{U}_{t}$, one has

$\displaystyle\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Theta_{t}+\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Delta\Theta+\int_{0}^{t}\int_{\Omega}{\mathcal{T}}\Delta\Theta+\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}|^{2}\Theta$
$\displaystyle\quad+\mu\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\nabla\Theta-\mu\int_{0}^{t}\int_{\Omega}\Theta\nabla\cdot\mathbf{U}_{t}$
(54)
$\displaystyle\leq\int_{\Omega}({{\tau}}\Theta)(t)-\int_{\Omega}{{\tau}}_{0}\Theta_{0}-\int_{0}^{t}\int_{\Omega}{\mathcal{T}}\Theta_{t}.$

Now, since

$\displaystyle{\mathcal{T}}\Theta_{t}=-\Theta_{t}+({\mathcal{T}}\Theta)_{t},$
$\displaystyle\int_{0}^{t}\int_{\Omega}{\mathcal{T}}\Delta\Theta=\int_{0}^{t}\int_{\Omega}\Delta{\mathcal{T}}\Theta=\int_{0}^{t}\underbrace{\int_{\Omega}\Delta\Theta}_{=0}-\int_{0}^{t}\int_{\Omega}|\nabla{\mathcal{T}}|^{2}\Theta,$

and

$\displaystyle\int_{0}^{t}\int_{\Omega}\big{(}|\nabla{{\tau}}|^{2}-|\nabla{\mathcal{T}}|^{2}\big{)}\Theta$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}-\nabla{\mathcal{T}}|^{2}\Theta+2\int_{0}^{t}\int_{\Omega}\nabla{\mathcal{T}}\cdot(\nabla{{\tau}}-\nabla{\mathcal{T}})\Theta$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}-\nabla{\mathcal{T}}|^{2}\Theta-2\int_{0}^{t}\int_{\Omega}\nabla\cdot(\underbrace{\nabla{\mathcal{T}}\Theta}_{=\nabla\Theta})({{\tau}}-{\mathcal{T}})$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}-\nabla{\mathcal{T}}|^{2}\Theta-2\int_{0}^{t}\int_{\Omega}\Delta\Theta({{\tau}}-{\mathcal{T}}),$

where we have used the identity $a^{2}-b^{2}=(a-b)^{2}+2b(a-b)$, one arrives
at

$\displaystyle\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Theta_{t}+\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Delta\Theta+\mu\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\nabla\Theta$
$\displaystyle-\mu\int_{0}^{t}\int_{\Omega}\Theta\nabla\cdot\mathbf{U}_{t}+\int_{0}^{t}\int_{\Omega}(|\nabla{{\tau}}|^{2}-|\nabla{\mathcal{T}}|^{2})\Theta$
(55)
$\displaystyle\leq\int_{\Omega}({{\tau}}\Theta-{\mathcal{T}}\Theta+\Theta)(t)-\int_{\Omega}({{\tau}}_{0}\Theta_{0}-{\mathcal{T}}_{0}\Theta_{0}+\Theta_{0}).$

We now sum up $\eqref{id3}$ and $\eqref{id4}$ to obtain

$\displaystyle\int_{\Omega}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}|\nabla{{\tau}}-\nabla{\mathcal{T}}|^{2}\Theta$
$\displaystyle\leq\int_{\Omega}e^{\tau_{0}}-\Theta(0)-\Theta(0)(\tau_{0}-\mathcal{T}(0))$
$\displaystyle+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(0)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{u}|^{2}(0)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\nabla\cdot\nabla\mathbf{u}-\mu\nabla\Theta\big{)}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Theta_{t}+\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Delta\Theta-\mu\int_{0}^{t}\int_{\Omega}(d\theta-\Theta)\nabla\cdot\mathbf{U}_{t}.$

Finally, noticing that the last three terms can be transformed as follows,

$\displaystyle\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Theta_{t}-\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Delta\Theta+\mu\int_{0}^{t}\int_{\Omega}(d\theta-\Theta)\nabla\cdot\mathbf{U}_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})(\Theta_{t}-\Delta\Theta)\pm\mu\int_{0}^{t}\int_{\Omega}({{\tau}}-{\mathcal{T}})\Theta\nabla\cdot\mathbf{U}_{t}$
$\displaystyle\quad+\mu\int_{0}^{t}\int_{\Omega}(d\theta-\Theta)\nabla\cdot\mathbf{U}_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}(\tau-\mathcal{T})(\Theta_{t}-\Delta\Theta+\mu\Theta\nabla\cdot\mathbf{U}_{t})$
$\displaystyle\quad+\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right),$

we arrive at $\eqref{relent}$.

####  4.3.2. The proof of weak-strong uniqueness

Provided that $(\mathbf{U},e^{\mathcal{T}}=\Theta)$ is a classical solution of
$\eqref{system0}$ such that $\Theta_{0}=\theta_{0},\mathbf{U}_{0}={\bf
u}_{0},$ and $\mathbf{V}_{0}=\mathbf{v}_{0}$, the inequality $\eqref{relent}$
gives us

$\displaystyle\int_{\Omega}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}|\nabla{\tau}-\nabla\mathcal{T}|^{2}\Theta$
$\displaystyle\leq-\mu\int_{0}^{t}\int_{\Omega}\nabla\cdot\mathbf{U}_{t}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)$
$\displaystyle\leq\mu\int_{0}^{t}\left(||\nabla\cdot\mathbf{U}_{t}||_{L^{\infty}(\Omega)}\int_{\Omega}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)\right)$
$\displaystyle\leq
C\int_{0}^{t}\int_{\Omega}d\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)(s)~{}ds,$

and since $\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)$ is
non-negative and $C>0$ doesn’t depend on $t$, we conclude by Gronwall’s
inequality that
$\mathcal{E}\left(\theta,\tau\big{|}\Theta,\mathcal{T}\right)=0$ for all
$t\in[0,T]$. In turn, on the one hand, in view of the above inequality, we
have $\mathbf{u}\equiv\mathbf{U}$, on the other hand, Lemma 4.4 yields
$\tau\equiv\mathcal{T},\theta\equiv\Theta$. The proof is finished.

##  5\. Proof of Theorem 1.2

###  5.1. Global existence – the construction of approximate solutions

####  5.1.1. The approximate problem

Before we start, let us point out that in order to solve
$\eqref{APP:nonlin}_{2}$ given below, one needs at least $C^{2,\alpha}$
boundary, which is higher than Lipschitz. However, this issue can be dealt
with by constructing a sequence of smooth domains which converge to $\Omega$,
and one can easily show that the weak solutions in the sense of Definition 1.2
are sequentially stable with respect to this domain convergence. Such
construction of domains is originally due to Nečas [26] (in Russian), and it
can also be found in [5, Chapter 1, Theorem 8.3.1]. Thus, for simplicity, let
$\Omega$ be of $C^{2,\alpha}$ regularity.

Now, similarly as in previous section, we introduce a basis
$\\{\boldsymbol{\varphi}_{i}\\}_{i\in\mathbb{N}}$ of $H_{0}^{1}(\Omega)$ and
denote $V_{n}:={\rm span}\\{\boldsymbol{\varphi}_{i}\\}_{1\leq i\leq n}$. The
problem of interest is the following:

######  Definition 5.1 (Approximate problem).

We say that $\mathbf{u}\in C^{2}(0,T;V_{n})$ and $\theta\in
L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ is a solution to the
approximate problem if the following equations are satisfied

(56)
$\displaystyle\begin{cases}\int_{\Omega}\mathbf{u}_{tt}\cdot\boldsymbol{\varphi}+\int_{\Omega}\nabla\mathbf{u}:\nabla\boldsymbol{\varphi}-\mu\int_{\Omega}\theta\nabla\cdot\boldsymbol{\varphi}=0,~{}\text{
for all }t\in(0,T),~{}\boldsymbol{\varphi}\in V_{n},\\\\[5.69054pt]
e_{1}(\theta)_{t}-\nabla\cdot(\kappa(\theta)\nabla\theta)+\mu\theta\nabla\cdot\mathbf{u}_{t}+\delta\theta^{2}-\delta\frac{1}{\theta^{2}}=0,\end{cases}$

where $e_{1}(\theta)=\theta+\theta^{\alpha}$ and
$\kappa(\theta)=1+\theta^{\beta}$ and $\delta>0$ is an approximation
parameter. The initial data $\mathbf{u}(0,\cdot)=\mathbf{u}_{0}^{n}$,
$\mathbf{u}_{t}(0,\cdot)=\mathbf{v}_{0}^{n}$ and
$\theta(0,\cdot)=\theta_{0}^{n}$ are chosen in the same way as in Definition
4.1.

######  Remark 5.1.

(1) The role of the additional terms $\delta\theta^{2}$ and
$-\delta\theta^{-2}$ is to ensure that the approximate temperature is
uniformly bounded from below and above by a constant. This is in contrast with
the case studied in previous section, since here we are relying on the
comparison principle instead of the maximum principle.
(2) In this section, the constructed approximate solutions depend on
$n,\delta$, so using the notational convention from previous section, we
should write $(\mathbf{u}^{n,\delta},\theta^{n,\delta})$. However, in this
section we will omit this superscript for simplicity of notation. We will
emphasize these indices when passing to the limit.
(3) The proof of existence and convergence of approximate solutions presented
in this section is inspired by ideas from [15, Chapter 3].

####  5.1.2. Estimates of approximate solutions

We start with the following:

######  Lemma 5.1.

Let $({\bf u},\theta)$ be a solution in the sense of Definiton 5.1 such that
$\theta\geq c>0$. Then, $(\mathbf{u},\theta)$ satisfy the entropy balance
equation

$\displaystyle\left(\ln\theta+\frac{\alpha}{\alpha-1}\theta^{\alpha-1}+\mu\nabla\cdot\mathbf{u}\right)_{t}-\nabla\cdot\left(\frac{\kappa(\theta)\nabla\theta}{\theta}\right)+\delta\theta-\delta\theta^{-3}=\frac{\kappa(\theta)|\nabla\theta|^{2}}{\theta^{2}}.$
(57)

Moreover, for all $t\in(0,T]$, the following total energy balance

$\displaystyle\int_{\Omega}(\theta+\theta^{\alpha})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)+\delta\int_{0}^{t}\int_{\Omega}\theta^{2}$
(58)
$\displaystyle=\delta\int_{0}^{t}\int_{\Omega}\theta^{-2}+\int_{\Omega}(\theta_{0}^{n}+(\theta_{0}^{n})^{\alpha})+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}^{n}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}^{n}|^{2},$

and the total dissipation balance

$\displaystyle\int_{\Omega}\left(\theta+\theta^{\alpha}-\ln\theta-\frac{\alpha}{\alpha-1}\theta^{\alpha-1}\right)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{(1+\theta^{\beta})|\nabla\theta|^{2}}{\theta^{2}}+\delta\int_{0}^{t}\int_{\Omega}(\theta^{2}+\theta^{-3})$
$\displaystyle=\delta\int_{0}^{t}\int_{\Omega}(\theta+\theta^{-2})+\int_{\Omega}\left(\theta_{0}^{n}+(\theta_{0}^{n})^{\alpha}-\ln\theta_{0}^{n}-\frac{\alpha}{\alpha-1}(\theta_{0}^{n})^{\alpha-1}\right)$
(59)
$\displaystyle\quad+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}^{n}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}^{n}|^{2},$

are satisfied.

###### Proof.

The entropy balance $\eqref{app:ent:bal}$ is obtained by dividing
$\eqref{APP:nonlin}_{2}$ by $\theta$. The identity $\eqref{app:en:id:nonlin}$
follows by taking ${\boldsymbol{\varphi}}={\bf u}_{t}\chi_{[0,t]}$ in
$\eqref{APP:nonlin}_{1}$ and summing it with $\eqref{APP:nonlin}_{2}$
integrated over $(0,t)\times\Omega$, while $\eqref{diss:nonlin}$ is obtained
by subtracting $\eqref{app:ent:bal}$ integrated over $(0,t)\times\Omega$ from
$\eqref{app:en:id:nonlin}$. ∎

######  Lemma 5.2 (Comparison principle).

Let $\mathbf{u}\in C^{2}(0,T;V_{n})$ and
$\underline{\theta},\overline{\theta}$ be a sub and super solutions to the
equation $\eqref{APP:nonlin}_{2}$, that is they satisfy
$\eqref{APP:nonlin}_{2}$ where $"="$ is replaced with $"\leq"$ and $"\geq"$,
respectively. Denote
$K(\theta):=\int_{0}^{\theta}\kappa(s)~{}ds=\theta+\frac{\theta^{\beta+1}}{\beta+1}$.
If

  1. (1)

$\underline{\theta},\overline{\theta}\in L^{2}(0,T;H^{1}(\Omega))$,
$\underline{\theta}_{t},\overline{\theta}_{t}\in L^{2}((0,T)\times\Omega)$,
$\Delta K(\underline{\theta}),\Delta K(\overline{\theta})\in
L^{2}((0,T)\times\Omega)$;

  2. (2)

$0<{\rm ess~{}inf}_{(0,T)\times\Omega}\underline{\theta}\leq{\rm
ess~{}sup}_{(0,T)\times\Omega}\underline{\theta}<\infty$,
$0<{\rm ess~{}inf}_{(0,T)\times\Omega}\overline{\theta}\leq{\rm
ess~{}sup}_{(0,T)\times\Omega}\overline{\theta}<\infty$;

  3. (3)

$\underline{\theta}(0,\cdot)\leq\overline{\theta}(0,\cdot)$,

then, one has

(60) $\displaystyle\underline{\theta}\leq\overline{\theta},\quad\text{a.e. on
}(0,T)\times\Omega.$

###### Proof.

First, it is easy to see that the following inequality holds

$\displaystyle(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))_{t}-\Delta(K(\underline{\theta})-K(\overline{\theta}))$
(61)
$\displaystyle\leq-\mu(\underline{\theta}-\overline{\theta})\nabla\cdot\mathbf{u}_{t}-\delta(\underline{\theta}^{2}-\overline{\theta}^{2})+\delta(\underline{\theta}^{-2}-\overline{\theta}^{-2}).$

Now, introducing a function

$\displaystyle{\rm sgn}^{+}(a):=\begin{cases}0,&\quad a\leq 0,\\\ 1,&\quad
a>0,\end{cases}$

we multiply $\eqref{comp:ineq}$ with ${\rm
sgn}^{+}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))$ to obtain

$\displaystyle{\rm
sgn}^{+}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))\Big{[}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))_{t}-\Delta(K(\underline{\theta})-K(\overline{\theta}))\Big{]}$
$\displaystyle\leq C{\rm
sgn}^{+}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))|\underline{\theta}-\overline{\theta}|,$

where $C$ depends on $\delta$, $\mathbf{u}$ and upper and lower bounds for
$\underline{\theta}$ and $\overline{\theta}$. Now, since $e_{1},K$ and
$x\mapsto x$ are all increasing, one has

$\displaystyle{\rm
sgn}^{+}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))={\rm
sgn}^{+}(K(\underline{\theta})-K(\overline{\theta}))={\rm
sgn}^{+}(\underline{\theta}-\overline{\theta}),$

so the above inequality can be written as

$\displaystyle{\rm
sgn}^{+}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))\big{[}e_{1}(\underline{\theta})-e_{1}(\overline{\theta})\big{]}_{t}$
$\displaystyle\quad-{\rm
sgn}^{+}(K(\underline{\theta})-K(\overline{\theta}))\Delta\big{[}K(\underline{\theta})-K(\overline{\theta})\big{]}$
(62) $\displaystyle\leq C{\rm
sgn}^{+}(\underline{\theta}-\overline{\theta})|\underline{\theta}-\overline{\theta}|.$

Denoting $a^{+}:=\max\\{a,0\\}$, and noticing that for any function $f$ that
$\partial_{t}f^{+}={\rm sgn}^{+}(f)f_{t}$ and

$\displaystyle\int_{\Omega}\Delta f~{}{\rm sgn}^{+}(f)\leq 0,\quad\text{if
}\partial_{n}f=0\text{ on }\partial\Omega,$

provided the mentioned derivatives of $f$ are integrable, we integrate
$\eqref{comp:ineq2}$ over $(0,t)\times\Omega$ for any $t\in(0,T)$ to obtain

$\displaystyle\int_{\Omega}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))^{+}(t)\leq
C\int_{0}^{t}\int_{\Omega}{\rm
sgn}^{+}(\underline{\theta}-\overline{\theta})|\underline{\theta}-\overline{\theta}|=C\int_{0}^{t}\int_{\Omega}(\underline{\theta}-\overline{\theta})^{+}.$

Now, since $x\mapsto e_{1}(x)$ is strictly increasing and differentiable with
derivatives strictly bounded from bellow for all $x>0$, one has that
$|\underline{\theta}-\overline{\theta}|\leq
c|e_{1}(\underline{\theta})-e_{1}(\overline{\theta})|$, where $c$ depends on
upper and lower bounds for $\underline{\theta}$ and $\overline{\theta}$ and
$e_{1}$, so it leads to

$\displaystyle\int_{\Omega}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))^{+}(t)\leq
C\int_{0}^{t}\int_{\Omega}(e_{1}(\underline{\theta})-e_{1}(\overline{\theta}))^{+},$

and the conclusion follows by Gronwall’s lemma.

∎

######  Lemma 5.3.

Assume that $\mathbf{u}\in C^{2}(0,T;V_{n})$ and $\theta\in
L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ solve the equation
$\eqref{APP:nonlin}_{2}$. Then, there are constants
$0<\underline{\theta}\leq\overline{\theta}<\infty$ such that

(63)
$\displaystyle\underline{\theta}\leq\theta\leq\overline{\theta},\quad\text{a.e.
on }(0,T)\times\Omega,$

and the following estimate holds

$\displaystyle{\rm
ess~{}sup}_{(0,T)}||\theta||_{H^{1}(\Omega)}^{2}+\int_{0}^{T}\left(||\theta_{t}||_{L^{2}(\Omega)}^{2}+\left|\left|\Delta
K(\theta)\right|\right|_{L^{2}(\Omega)}^{2}\right)$ (64) $\displaystyle\leq
C\left(\max\limits_{x\in\Omega}e_{1}(\theta_{0}^{n}),~{}||K(\theta_{0}^{n})||_{L^{2}(\Omega)},~{}||\mathbf{u}||_{C^{1}(0,T;L^{2}(\Omega))},\underline{\theta},~{}\overline{\theta}\right).$

###### Proof.

First, it is easy to see that there exists a small enough constant
$\underline{\theta}>0$ which satisfies the following inequality

$\displaystyle\mu\underline{\theta}\nabla\cdot\mathbf{u}_{t}+\delta\underline{\theta}^{2}\leq\delta\frac{1}{\underline{\theta}^{2}}$

so $\underline{\theta}$ is a sub solution to $\eqref{APP:nonlin}_{2}$.
Similarly, we can choose a large enough constant $\overline{\theta}$ so that

$\displaystyle\delta\overline{\theta}^{2}\geq\delta\frac{1}{\overline{\theta}^{2}}-\mu\overline{\theta}\nabla\cdot\mathbf{u}_{t},$

so $\overline{\theta}$ is a super solution to $\eqref{APP:nonlin}_{2}$. Thus,
by previous lemma, the inequality $\eqref{up:low:bound}$ follows.

Next, in order to prove $\eqref{overall:bound}$, first we multiply
$\eqref{APP:nonlin}_{2}$ with $K(\theta)_{t}$ and integrate over
$(0,t)\times\Omega$ which gives us

$\displaystyle\int_{0}^{t}\int_{\Omega}\left(1+\alpha\theta^{\alpha-1}\right)\kappa(\theta)|\theta_{t}|^{2}+\int_{\Omega}|\nabla
K(\theta)|^{2}(t)$ $\displaystyle=\int_{\Omega}|\nabla
K(\theta_{0}^{n})|^{2}-\mu\int_{0}^{t}\int_{\Omega}K(\theta)_{t}\theta\nabla\cdot\mathbf{u}_{t}-\delta\int_{0}^{t}\int_{\Omega}(\theta^{2}-\theta^{-2})K(\theta)_{t}$
$\displaystyle\leq
C\left(\max\limits_{x\in\Omega}e_{1}(\theta_{0}^{n}),||K(\theta_{0}^{n})||_{L^{2}(\Omega)},||\mathbf{u}||_{C^{1}(0,T;L^{2}(\Omega))}\right)+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\theta_{t}|^{2},$

since $\theta$ is bounded from above by $\eqref{up:low:bound}$, which yields

$\displaystyle\int_{0}^{T}\int_{\Omega}|\theta_{t}|^{2}+{\rm
ess~{}sup}_{(0,T)}||\nabla\theta||_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\left(\max\limits_{x\in\Omega}e_{1}(\theta_{0}^{n}),~{}||K(\theta_{0}^{n})||_{L^{2}(\Omega)},~{}||\mathbf{u}||_{C^{1}(0,T;L^{2}(\Omega))}\right).$

Similarly, multiplying with $\Delta K(\theta)$ and using the obtained estimate
yields

$\displaystyle\int_{0}^{T}\int_{\Omega}|\Delta K(\theta)|^{2}+{\rm
ess~{}sup}_{(0,T)}||\nabla\theta||_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\left(\max\limits_{x\in\Omega}e_{1}(\theta_{0}^{n}),~{}||K(\theta_{0}^{n})||_{L^{2}(\Omega)},~{}||\mathbf{u}||_{C^{1}(0,T;L^{2}(\Omega))}\right),$

so the proof is finished. ∎

####  5.1.3. Solving the approximate problem

######  Lemma 5.4.

Let $\mathbf{u}\in C^{2}(0,T;V_{n})$. Then, there exists a solution $\theta\in
L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ to the equation
$\eqref{APP:nonlin}_{2}$. Moreover, this solution is unique and satisfies
$0<\underline{\theta}\leq\theta\leq\overline{\theta}<\infty$ a.e. on
$(0,T)\times\Omega$, where $\underline{\theta},\overline{\theta}$ are
constants.

###### Proof.

Before we start, in order to write $\eqref{APP:nonlin}_{2}$ as a quasilinear
parabolic PDE, we divide $\eqref{APP:nonlin}_{2}$ by
$1+\alpha\theta^{\alpha+1}$ to obtain

$\displaystyle\theta_{t}-\frac{(1+\theta^{\beta})}{1+\alpha\theta^{\alpha-1}}\Delta\theta+\frac{\mu\theta\nabla\cdot\mathbf{u}_{t}-\beta|\nabla\theta|^{2}\theta^{\beta-1}+\delta\theta^{2}-\delta\theta^{-2}}{1+\alpha\theta^{\alpha-1}}=0.$

Now, for any $\omega>0$, in order to ensure uniform parabolicity (at least for
a fixed $\omega$), we introduce

$\displaystyle
f_{\omega}(x):=\frac{\sqrt{x^{2}+\omega^{2}}}{1+\omega\sqrt{x^{2}+\omega^{2}}},$

and use it to approximate some functions in the equation. This will be denoted
with the notation $[\cdot]_{\omega}:=f_{\omega}(\cdot)$ (for example
$[\theta]_{\omega}=f_{\omega}(\theta)$). Note that
$\frac{\omega}{1+\omega^{2}}\leq f_{\omega}(x)<\frac{1}{\omega}$ and

$\displaystyle-1<f_{\omega}^{\prime}(x)=\frac{x}{\sqrt{x^{2}+\omega^{2}}(1+\omega\sqrt{x^{2}+\omega^{2}})}<1,$

for all $x\in\mathbb{R}$. We aim to solve the following system in the
classical sense:

(65)
$\displaystyle\begin{cases}&\theta_{t}-a(\theta)\Delta\theta+b(t,x,\theta,\nabla\theta)=0,\\\
&a(\theta)\partial_{n}\theta=0,\\\
&\theta(0,\cdot)=\theta_{0,\omega}^{n},\end{cases}$

where $\theta_{0,\omega}^{n}$ is a regularization of $\theta_{0}^{n}$ and

$\displaystyle
a(\theta):=\frac{[1+\langle\theta\rangle_{\omega}^{\beta}]_{\omega}}{[1+\alpha\langle\theta\rangle_{\omega}^{\alpha-1}]_{\omega}},$
$\displaystyle b(t,x,\theta,\nabla\theta):=$
$\displaystyle\quad\frac{\mu\theta\nabla\cdot\mathbf{u}_{t}(t,x)-\beta|\nabla\theta|^{2}f_{\omega}^{\prime}(1+\langle\theta\rangle_{\omega}^{\beta})(\langle\theta\rangle_{\omega}^{\beta})^{\prime}+\delta\theta^{2}-\delta\frac{1}{\theta^{2}+\omega^{2}}}{[1+\alpha\langle\theta\rangle_{\omega}^{\alpha-1}]_{\omega}},$

with $\langle x\rangle_{\omega}^{s}$ being the regularization of
$x\mapsto|x|^{s}$, for $s>0$. Since the functions $a$ and $b$ satisfy all the
necessary properties, the classical solution to this problem follows by the
classical quasilinear parabolic theory (see [22, Chapter VI, Theorem 7.1] or
[15, Theorem 10.24]). Now, by multiplying $\eqref{app:system}_{1}$ with
$[1+\alpha\langle\theta\rangle_{\omega}^{\alpha-1}]_{\omega}$, one has

$\displaystyle e_{\omega}(\theta)_{t}-\Delta
K_{\omega}(\theta)+\mu\theta\nabla\cdot\mathbf{u}_{t}+\delta\theta^{2}-\delta\frac{1}{\theta^{2}+\omega^{2}}=0,$

where

$\displaystyle e_{\omega}(x)=\int_{0}^{x}[1+\alpha\langle
x\rangle_{\omega}^{\alpha-1}]_{\omega}\ dx,\quad
K_{\omega}(x)=\int_{0}^{x}[1+\langle x\rangle_{\omega}^{\beta}]_{\omega}\ dx.$

This equation has the same structure as the original one
$\eqref{APP:nonlin}_{2}$ and allows us to prove the comparison principle and
the estimates in the same way as in lemmas 5.2 and 5.3. Passing to the limit
in $\omega\to 0$, we obtain a solution in $L^{2}(0,T;H^{2}(\Omega))\cap
H^{1}(0,T;L^{2}(\Omega))$. The uniqueness and the lower and upper bounds now
follow by the comparison principle given in Lemma 5.2. ∎

######  Lemma 5.5.

There exists a solution $\mathbf{u}\in C^{2}(0,T;V_{n})$ and $\theta\in
L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ to the approximate
problem $\eqref{APP:nonlin}$.

###### Proof.

The proof is based on a fixed-point argument and it doesn’t differ from the
one given in Proposition 1, so we omit it. ∎

###  5.2. Global existence – the convergence of approximate solutions

Denote the approximate solution obtained in previous subsection as
$(\mathbf{u}^{n},\theta^{n})$. The goal is to pass to the limit $n\to\infty$
and $\delta\to 0$ in approximate momentum equation $\eqref{APP:nonlin}_{1}$,
approximate entropy balance equation $\eqref{app:ent:bal}$ and the total
energy balance $\eqref{app:en:id:nonlin}$, and show that the limiting
functions are a weak solution in the sense of Definition 1.2. This will be
done in one limit by fixing $\delta=1/n$, say.

Before we proceed, let us summarize the uniform estimates for
$(\mathbf{u}^{n},\theta^{n})$, which come from Lemma 5.1 (note that the
$\delta$ terms appearing on right-hand sides are absorbed):

(66)
$\displaystyle||\mathbf{u}^{n}||_{L^{\infty}(0,T;H^{1}(\Omega))}+||\mathbf{u}_{t}^{n}||_{L^{\infty}(0,T;L^{2}(\Omega))}\leq
C,$ (67) $\displaystyle{\rm
ess~{}sup}_{(0,T)}\int_{\Omega}(e_{1}(\theta^{n})+|s_{1}(\theta^{n})|)+\int_{0}^{T}\int_{\Omega}\frac{\kappa(\theta^{n})|\nabla\theta^{n}|^{2}}{(\theta^{n})^{2}}\leq
C,$ (68)
$\displaystyle\delta\int_{0}^{T}\int_{\Omega}\left(\frac{1}{(\theta^{n})^{3}}+(\theta^{n})^{2}\right)\leq
C,$

where $e_{1}(\theta^{n})=\theta^{n}+(\theta^{n})^{\alpha}$,
$s_{1}(\theta^{n})=\ln\theta^{n}+\frac{\alpha}{\alpha-1}(\theta^{n})^{\alpha-1}$
and $\kappa(\theta^{n})=1+(\theta^{n})^{\beta}$. Note that this implies
$(\theta^{n})^{\beta-2}|\nabla\theta^{n}|^{2}\in L^{1}((0,T)\times\Omega)$, so
$\nabla((\theta^{n})^{\frac{\beta}{2}})\in L^{2}((0,T)\times\Omega)$ and
consequently

(69) $\displaystyle||(\theta^{n})^{\beta}||_{L^{1}(0,T;L^{3}(\Omega))}\leq C.$

As a direct consequence of the above inequalities, one has:

######  Lemma 5.6.

There exists $\theta\in L^{\infty}(0,T;L^{\alpha}(\Omega))$, ${\bf u}\in
L^{\infty}(0,T;H^{1}(\Omega))\cap W^{1,\infty}(0,T;L^{2}(\Omega))$ and
$\boldsymbol{\sigma}\in\mathcal{M}^{+}([0,T]\times\overline{\Omega})$ such
that the following convergence properties hold (on a subsequence)

$\displaystyle\theta^{n}\rightharpoonup\theta,$ $\displaystyle\quad\text{
weakly* in }L^{\infty}(0,T;L^{\alpha}(\Omega)),$ $\displaystyle{\bf
u}^{n}\rightharpoonup{\bf u},$ $\displaystyle\quad\text{ weakly* in
}L^{\infty}(0,T;H^{1}(\Omega)),$ $\displaystyle{\bf
u}_{t}^{n}\rightharpoonup{\bf u}_{t},$ $\displaystyle\quad\text{ weakly* in
}L^{\infty}(0,T;L^{2}(\Omega)),$
$\displaystyle\frac{k(\theta^{n})|\nabla\theta^{n}|^{2}}{(\theta^{n})^{2}}+\delta\frac{1}{(\theta^{n})^{3}}\rightharpoonup\boldsymbol{\sigma},$
$\displaystyle\quad{\rm weakly*}\;{\rm
in}\;\mathcal{M}^{+}([0,T]\times\overline{\Omega}),$

where $\frac{k(\theta)|\nabla\theta|^{2}}{\theta^{2}}\leq\boldsymbol{\sigma}$,
in the sense of measures.

Note that these weak convergences are enough to pass to the limit in the
approximate momentum equation $\eqref{APP:nonlin}_{1}$, so it remains to pass
to the limit in the approximate entropy balance equation $\eqref{app:ent:bal}$
and the total energy balance $\eqref{app:en:id:nonlin}$. This is the focus of
the remainder of this section.

####  5.2.1. Pointwise convergence of $\theta$ – case $\alpha\neq 2$

First, note that when $\alpha>3\beta$, then $(\theta^{n})^{\alpha}$ is only in
$L^{1}(\Omega)$ for a.a. $t\in(0,T)$. Therefore, to keep the proof general, we
choose

$\displaystyle\mathbf{U}_{n}:=\left[{\ln\theta}^{n}+\frac{\alpha}{\alpha-1}(\theta^{n})^{\alpha-1},~{}\mu\mathbf{u}_{t}^{n}-\frac{\kappa(\theta^{n})\nabla\theta^{n}}{\theta^{n}}\right],\quad\mathbf{V}_{n}:=[T_{k}(\theta^{n}),0],$

where $T_{k}$ is the cut-off function

$\displaystyle T_{k}(x)=\begin{cases}x,&\text{for }x<k,\\\ k,&\text{for }x\geq
k.\end{cases}$

We will rely on the following well-known div-curl lemma due to Murat [20] (in
French), stated here for the convenience of the reader (see also [15, Theorem
10.21] for proof in English):

######  Lemma 5.7.

Assume that

$\displaystyle\mathbf{U}_{n}\rightharpoonup\mathbf{U},\quad\text{weakly in
}L^{p}((0,T)\times\Omega),$
$\displaystyle\mathbf{V}_{n}\rightharpoonup\mathbf{V},\quad\text{weakly in
}L^{q}((0,T)\times\Omega),$

where $1/p+1/q=1/r<1$. In addition333Here, ${\rm
div}_{t,x}\mathbf{U}_{n}:=\partial_{t}\mathbf{U}_{n}^{1}+\sum_{i=1}^{3}\partial_{x_{i}}\mathbf{U}_{n}^{i+1}$
is the time-space divergence operator, while ${\rm
curl}_{t,x}\mathbf{V}_{n}:=\nabla_{t,x}\mathbf{V}_{n}-\nabla_{t,x}^{T}\mathbf{V}_{n}$
is the time-space curl operator., let ${\rm div}_{t,x}\mathbf{U}_{n}$ and
${\rm curl}_{t,x}\mathbf{V}_{n}$ be precompact in
$W^{-1,s}((0,T)\times\Omega)$, for some $s>1$. Then,

$\displaystyle\mathbf{U}_{n}\cdot\mathbf{V}_{n}\rightharpoonup\mathbf{U}\cdot\mathbf{V},\quad\text{weakly
in }L^{r}((0,T)\times\Omega).$

By using this lemma for $p=\min\\{\frac{\alpha}{\alpha-1},2\\}$ and any
$q>\max\\{\alpha,2\\}$ and $s\in[1,\frac{4}{3})$, one has

(70)
$\displaystyle\overline{\left(\ln\theta+\frac{\alpha-1}{\alpha}\theta^{\alpha-1}\right)T_{k}(\theta)}=\left(\overline{\ln\theta}+\overline{\frac{\alpha-1}{\alpha}\theta^{\alpha-1}}\right)\overline{T_{k}(\theta)},$

where the notation $\overline{f}$ is used to represent the weak limit of
$f^{n}$. Now, since $\ln x$, $x^{\alpha-1}$ and $T_{k}(x)$ are non-decreasing
functions, one has by [15, Theorem 10.19(i)]

$\displaystyle\overline{\ln\theta}~{}\overline{T_{k}(\theta)}\leq\overline{\ln\theta
T_{k}(\theta)},\qquad\overline{\theta^{\alpha-1}}~{}\overline{T_{k}(\theta)}\leq\overline{\theta^{\alpha-1}T_{k}(\theta)},$

which together with $\eqref{div:curl:id}$ imply

$\displaystyle\overline{\ln\theta}~{}\overline{T_{k}(\theta)}=\overline{\ln\theta
T_{k}(\theta)},\qquad\overline{\theta^{\alpha-1}}~{}\overline{T_{k}(\theta)}=\overline{\theta^{\alpha-1}T_{k}(\theta)}.$

Now, by the second identity and [15, Theorem 10.19(ii)], one concludes444For
the purpose of this theorem, one can extend $x\mapsto x^{\alpha-1}$ with
$-|x|^{\alpha-1}$ to $\mathbb{R}^{-}$, but since $\theta>0$, this makes no
difference.

(71) $\displaystyle
T_{k}((\overline{\theta^{\alpha-1}})^{\frac{1}{\alpha-1}})=\overline{T_{k}(\theta)}.$

We want to pass to the limit $k\to\infty$. First, note that by [15, Corollary
10.2]

$\displaystyle\int_{0}^{T}\int_{\Omega}|T_{k}(\theta^{n})-\theta^{n}|^{s}=\int_{\\{\theta^{n}\geq
k\\}}|k-\theta^{n}|^{s}\leq\int_{\\{\theta^{n}\geq k\\}}|\theta^{n}|^{s}$
$\displaystyle\leq\frac{1}{k^{\alpha-s}}\sup_{n>0}||\theta^{n}||_{L^{\alpha}((0,T)\times\Omega)}^{\alpha}\leq\frac{C}{k^{\alpha-s}},$

for any $s\in[0,\alpha]$, so

$\displaystyle\left|\int_{0}^{T}\int_{\Omega}(\overline{T_{k}(\theta)}-\theta)\varphi\right|=\lim\limits_{n\to\infty}\left|\int_{0}^{T}\int_{\Omega}(T_{k}(\theta^{n})-\theta^{n})\varphi\right|$
$\displaystyle\leq\lim\limits_{n\to\infty}||T_{k}(\theta^{n})-\theta^{n}||_{L^{s}(0,T;L^{s}(\Omega))}||\varphi||_{L^{s^{*}}(0,T;L^{s^{*}}(\Omega))}$
$\displaystyle\leq\frac{C}{k^{\alpha-s}}||\varphi||_{L^{s^{*}}(0,T;L^{s^{*}}(\Omega))}.$

As $k\to\infty$, one concludes that
$\overline{T_{k}(\theta^{n})}\rightharpoonup\theta$ in
$L^{s}(0,T;L^{s}(\Omega))$ for any $s\in[1,\alpha)$, so by passing to the
limit $k\to\infty$ in $\eqref{weak:lim}$ implies

$\displaystyle(\overline{\theta^{\alpha-1}})^{\frac{1}{\alpha-1}}=\theta\implies\overline{\theta^{\alpha-1}}=\theta^{\alpha-1}\text{
a.e. on }(0,T)\times\Omega.$

Now, if $\alpha>2$, function $x\mapsto x^{\alpha-1}$ for $x\geq 0$ is strictly
convex so one can use [15, Theorem 10.20] to conclude that
$\theta^{n}\to\theta$ a.e. on $(0,T)\times\Omega$. If $\alpha\in(1,2)$, the
same conclusion follows by convexity of $x\mapsto-x^{\alpha-1}$ for $x\geq 0$.

####  5.2.2. Pointwise convergence of $\theta$ – case $\alpha=2$

In this case, function $x\mapsto x^{\alpha-1}$ is a linear function which is
not strictly convex or concave. Thus, we choose

$\displaystyle\mathbf{U}_{n}:=\left[{\ln\theta}_{n}+2\theta^{n},\mu\mathbf{u}_{t}^{n}-\frac{\kappa(\theta^{n})\nabla\theta^{n}}{\theta^{n}}\right],\quad\mathbf{V}_{n}:=[(\theta^{n})^{2},0],$

so we can conclude in the same way that

$\displaystyle\overline{\ln\theta}~{}\overline{\theta^{2}}=\overline{\ln\theta\theta^{2}},\qquad\theta\overline{\theta^{2}}=\overline{\theta^{3}}.$

Now, due to [15, Theorem 10.19(ii)], one has
$\overline{\theta^{2}}=\theta^{2}$ so the convexity of $x\mapsto x^{2}$ gives
the a.e. convergence of $\theta$ due to [15, Theorem 10.20].

####  5.2.3. Passing to the limit

Here, we make use of a.e. convergence of $\theta^{n}$ and the uniform bounds
to prove the convergence of all the nonlinear terms. First, due to uniform
boundedness of $\theta^{n}$ in $L^{\infty}(0,T;L^{\alpha}(\Omega))$ in
$\eqref{est2:nonlin}$, one has

$\displaystyle(\theta^{n})^{\alpha-1}\to\theta^{\alpha-1},\quad\text{in
}L^{1}((0,T)\times\Omega).$

Moreover, from $\eqref{est2:nonlin}$ and $\eqref{est3:nonlin}$, one has that
$\ln\theta^{n}\in L^{\infty}(0,T;L^{1}(\Omega))$ and $\nabla\ln\theta\in
L^{2}((0,T)\times\Omega)$, so by imbedding $\ln\theta^{n}\in
L^{2}(0,T;L^{6}(\Omega))$ which gives us

$\displaystyle\ln\theta^{n}\to\ln\theta,\quad\text{in
}L^{1}((0,T)\times\Omega).$

The above two convergences give us

$\displaystyle s_{1}(\theta^{n})\to s_{1}(\theta),\quad\text{in
}L^{1}((0,T)\times\Omega).$

Next, since $\nabla\ln\theta^{n}$ and $\nabla((\theta^{n})^{\frac{\beta}{2}})$
are uniformly bounded in $L^{2}((0,T)\times\Omega)$, one has

$\displaystyle\nabla\ln\theta^{n}\rightharpoonup\nabla\ln\theta\text{ and
}\nabla((\theta^{n})^{\frac{\beta}{2}})\rightharpoonup\nabla(\theta^{\frac{\beta}{2}}),\quad\text{in
}L^{2}((0,T)\times\Omega).$

On the other hand, due to $\eqref{est2:nonlin}$ and $\eqref{est4:nonlin}$, by
interpolation, one can conclude that $(\theta^{n})^{\frac{\beta}{2}}\in
L^{p}((0,T)\times\Omega)$, for some $p>2$, so

$\displaystyle(\theta^{n})^{\frac{\beta}{2}}\to\theta^{\frac{\beta}{2}},\quad\text{in
}L^{2}((0,T)\times\Omega).$

Combining the previous two convergences, one has

$\displaystyle\nabla\ln\theta^{n}+(\theta^{n})^{\frac{\beta}{2}}\nabla((\theta^{n})^{\frac{\beta}{2}})=\frac{\kappa(\theta^{n})\nabla\theta^{n}}{\theta^{n}}\rightharpoonup\frac{\kappa(\theta)\nabla\theta}{\theta},\quad\text{in
}L^{1}((0,T)\times\Omega).$

Finally, due to $\eqref{est3:nonlin}$,

$\displaystyle 0\leq\int_{0}^{T}\int_{\Omega}\delta\theta\leq
C\delta^{\frac{1}{2}}\left(\int_{0}^{T}\int_{\Omega}\left|\delta^{\frac{1}{2}}\theta^{n}\right|^{2}\right)^{\frac{1}{2}}\leq
C\delta^{\frac{1}{2}},$ $\displaystyle
0\leq\int_{0}^{T}\int_{\Omega}\delta\frac{1}{(\theta^{n})^{2}}\leq
C\delta^{\frac{1}{3}}\left(\int_{0}^{T}\int_{\Omega}\left|\delta^{\frac{2}{3}}\frac{1}{(\theta^{n})^{3}}\right|^{\frac{3}{2}}\right)^{\frac{2}{3}}\leq
C\delta^{\frac{1}{3}},$

so $\delta\theta^{n}$ and $\delta\frac{1}{(\theta^{n})^{2}}$ vanish as
$n\to\infty$. Thus, we can pass to the limit in the approximate entropy
balance equation $\eqref{app:ent:bal}$ and the total energy balance
$\eqref{app:en:id:nonlin}$ and conclude the desired result. Note that the
$\delta\int_{0}^{T}\int_{\Omega}(\theta^{n})^{2}$ is positive, so it doesn’t
affect the limiting energy inequality and therefore doesn’t need to vanish.

###  5.3. Consistency

This proof is the same as in the previous section and is therefore omitted.

###  5.4. Weak-strong uniqueness

For the sake of generality of the proof, we will restrain from writing the
explicit forms of functions $e_{1},s_{1}$ and $\kappa$ throughout the majority
of the proof, and only use their properties when they are needed. As in
Section 4.3, the following proof also follows the ideas from [14].

Now, let us introduce the relative entropy as

$\displaystyle\mathcal{E}(\theta\big{|}\Theta):=e_{1}(\theta)-e_{1}(\Theta)-\Theta(s_{1}(\theta)-s_{1}(\Theta))$
$\displaystyle=\theta+\theta^{\alpha}-\Theta-\Theta^{\alpha}-\Theta\Big{(}\ln\theta-\ln\Theta+\frac{\alpha}{\alpha-1}(\theta^{\alpha-1}-\Theta^{\alpha-1}\big{)}\Big{)},$

and for any smooth functions $\mathbf{U}\in C_{0}^{\infty}([0,T]\times\Omega)$
and $\Theta\in C^{\infty}([0,T]\times\Omega)$ with $\Theta>0$,
$\partial_{n}\Theta=0$ on $(0,T)\times\partial\Omega$, the relative entropy
inequality

$\displaystyle\int_{\Omega}\mathcal{E}(\theta\big{|}\Theta)(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)|\nabla\theta|^{2}}{\theta^{2}}\Theta$
$\displaystyle\leq\int_{\Omega}\mathcal{E}(\theta\big{|}\Theta)(0)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(0)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(0)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}-\mu\nabla\Theta\big{)}-\mu\int_{0}^{t}\int_{\Omega}(\theta-\Theta)\nabla\cdot\mathbf{U}_{t}$
(72)
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\Theta)\right)\Theta_{t}+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)}{\theta}\nabla\theta\cdot\nabla\Theta.$

####  5.4.1. Any weak solution satisfies the relative entropy inequality

First, in the same way as in $\eqref{id3}$, we have

$\displaystyle\int_{\Omega}(\theta+\theta^{\alpha})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\mathbf{U}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\mathbf{U}|^{2}(t)$
$\displaystyle-\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\big{(}-\mathbf{U}_{tt}+\Delta\mathbf{U}-\mu\nabla\Theta\big{)}$
$\displaystyle-\mu\int_{0}^{t}\int_{\Omega}(\mathbf{u}_{t}-\mathbf{U}_{t})\cdot\nabla\Theta+\mu\int_{0}^{t}\int_{\Omega}\theta\nabla\cdot\mathbf{U}_{t}$
(73)
$\displaystyle\leq\int_{\Omega}(\theta_{0}+\theta_{0}^{\alpha})+\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}-\mathbf{V}_{0}|^{2}+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}_{0}-\nabla\mathbf{U}_{0}|^{2},$

where $\mathbf{v}_{0}=\mathbf{u}_{t}(0,\cdot)$ and
$\mathbf{V}_{0}=\mathbf{U}_{t}(0,\cdot)$. Next, in $\eqref{entweak:nonlin}$ we
choose $\phi=\chi_{[0,t]}\Theta$ to obtain

$\displaystyle\int_{0}^{t}\int_{\Omega}s_{1}(\theta)\Theta_{t}-\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)}{\theta}\nabla\theta\cdot\nabla\Theta$
$\displaystyle\quad+\mu\int_{0}^{t}\int_{\Omega}\mathbf{u}_{t}\cdot\nabla\Theta+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)|\nabla\theta|^{2}}{\theta^{2}}\Theta$
(74)
$\displaystyle\leq\int_{\Omega}s_{1}(\theta(t))\Theta(t)-\int_{\Omega}s_{1}(\theta_{0})\Theta(0),$

and since

$\displaystyle\int_{0}^{t}\int_{\Omega}s_{1}(\theta)\Theta_{t}=\int_{0}^{t}\int_{\Omega}(s_{1}(\theta)-s_{1}(\Theta))\Theta_{t}+\int_{0}^{t}\int_{\Omega}s_{1}(\Theta)\Theta_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}(s_{1}(\theta)-s_{1}(\Theta))\Theta_{t}+\int_{0}^{t}\int_{\Omega}\big{(}(s_{1}(\Theta)\Theta)_{t}-e_{1}(\Theta)_{t}\big{)},$

we conclude that (72) holds.

####  5.4.2. Proof of weak-strong uniqueness

Now, if $(\tilde{\mathbf{u}},\tilde{\theta})$ is a classical solution to the
same problem with $\tilde{\mathbf{v}}_{0}=\mathbf{v}_{0}$,
$\tilde{\mathbf{u}}_{0}=\mathbf{u}_{0}$ and $\tilde{\theta}_{0}=\theta_{0}$,
one has

$\displaystyle\int_{\Omega}\mathcal{E}(\theta\big{|}\tilde{\theta})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\tilde{\mathbf{u}}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\tilde{\mathbf{u}}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)|\nabla\theta|^{2}}{\theta^{2}}\tilde{\theta}$
$\displaystyle\leq-\mu\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\nabla\cdot\tilde{\mathbf{u}}_{t}-\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)}{\theta}\nabla\theta\cdot\nabla\tilde{\theta}.$

First, we want to transform the RHS. We start with

$\displaystyle\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}+\mu\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\nabla\cdot\tilde{\mathbf{u}}_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}+\int_{0}^{t}\int_{\Omega}\underbrace{s_{1}(\tilde{\theta})_{\theta}\tilde{\theta}_{t}}_{=s_{1}(\tilde{\theta})_{t}}(\theta-\tilde{\theta})$
$\displaystyle\quad+\mu\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\nabla\cdot\tilde{\mathbf{u}}_{t}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})(s_{1}(\tilde{\theta})_{t}+\mu\nabla\cdot\tilde{\mathbf{u}}_{t}).$

Moreover, since

$\displaystyle\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\left(\nabla\cdot\left(\frac{\kappa(\tilde{\theta})\nabla\tilde{\theta}}{\tilde{\theta}}\right)+\frac{\kappa(\tilde{\theta})|\nabla\tilde{\theta}|^{2}}{\tilde{\theta}^{2}}\right)$
$\displaystyle=-\int_{0}^{t}\int_{\Omega}(\nabla\theta-\nabla\tilde{\theta})\cdot\nabla\tilde{\theta}\frac{\kappa(\tilde{\theta})}{\tilde{\theta}}+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\frac{\kappa(\tilde{\theta})\nabla\theta\cdot\nabla\tilde{\theta}}{\tilde{\theta}^{2}},$

we arrive at

$\displaystyle\int_{\Omega}\mathcal{E}(\theta\big{|}\tilde{\theta})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\tilde{\mathbf{u}}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\tilde{\mathbf{u}}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)|\nabla\theta|^{2}}{\theta^{2}}\tilde{\theta}-\int_{0}^{t}\int_{\Omega}\frac{\kappa(\theta)}{\theta}\nabla\theta\cdot\nabla\tilde{\theta}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\nabla\theta-\nabla\tilde{\theta})\cdot\nabla\tilde{\theta}\frac{\kappa(\tilde{\theta})}{\tilde{\theta}}+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})\frac{\kappa(\tilde{\theta})|\nabla\tilde{\theta}|^{2}}{\tilde{\theta}^{2}}$
$\displaystyle\leq-\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}.$

At this point, we need some specific form for $\kappa(\theta)$ in order to
obtain good estimates with the right signs. Therefore, we choose
$\kappa(\theta)=1+\theta^{2}$. Now, we calculate the terms corresponding to
the constant part of $\kappa$

$\displaystyle\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta|^{2}\tilde{\theta}-\int_{0}^{t}\int_{\Omega}\frac{\nabla\theta\cdot\nabla\tilde{\theta}}{\theta}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\nabla\theta-\nabla\tilde{\theta})\cdot\nabla\ln\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})|\nabla\ln\tilde{\theta}|^{2}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta|^{2}\tilde{\theta}-\int_{0}^{t}\int_{\Omega}\tilde{\theta}\nabla\ln\theta\cdot\nabla\ln\tilde{\theta}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\theta\nabla\ln\theta-\tilde{\theta}\nabla\ln\tilde{\theta})\cdot\nabla\ln\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})|\nabla\ln\tilde{\theta}|^{2}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta-\nabla\ln\tilde{\theta}|^{2}\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\tilde{\theta}-\theta)\nabla\ln\theta\cdot\nabla\ln\tilde{\theta}$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})|\nabla\ln\tilde{\theta}|^{2}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta-\nabla\ln\tilde{\theta}|^{2}\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\tilde{\theta}-\theta)\nabla\ln\tilde{\theta}\cdot(\nabla\ln\theta-\nabla\ln\tilde{\theta}).$

Similarly, corresponding to the second component $-\theta^{2}$

$\displaystyle\int_{0}^{t}\int_{\Omega}|\nabla\theta|^{2}\tilde{\theta}-\int_{0}^{t}\int_{\Omega}\theta\nabla\theta\cdot\nabla\tilde{\theta}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\nabla\theta-\nabla\tilde{\theta})\cdot\nabla\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\theta-\tilde{\theta})|\nabla\tilde{\theta}|^{2}$
$\displaystyle=\int_{0}^{t}\int_{\Omega}|\nabla\theta-\nabla\tilde{\theta}|^{2}\tilde{\theta}+\int_{0}^{t}\int_{\Omega}(\tilde{\theta}-\theta)\nabla\tilde{\theta}\cdot(\nabla\theta-\nabla\tilde{\theta}).$

Thus, we finally obtain

$\displaystyle\int_{\Omega}\mathcal{E}(\theta\big{|}\tilde{\theta})(t)+\frac{1}{2}\int_{\Omega}|\mathbf{u}_{t}-\tilde{\mathbf{u}}_{t}|^{2}(t)+\frac{1}{2}\int_{\Omega}|\nabla\mathbf{u}-\nabla\tilde{\mathbf{U}}|^{2}(t)$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}|\nabla\theta-\nabla\tilde{\theta}|^{2}\tilde{\theta}+\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta-\nabla\ln\tilde{\theta}|^{2}\tilde{\theta}$
$\displaystyle\leq-\int_{0}^{t}\int_{\Omega}\left(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})\right)\tilde{\theta}_{t}$
$\displaystyle\quad-\int_{0}^{t}\int_{\Omega}(\tilde{\theta}-\theta)\nabla\tilde{\theta}\cdot(\nabla\theta-\nabla\tilde{\theta})-\int_{0}^{t}\int_{\Omega}(\tilde{\theta}-\theta)\nabla\ln\tilde{\theta}\cdot(\nabla\ln\theta-\nabla\ln\tilde{\theta})$
$\displaystyle\leq\int_{0}^{t}||\tilde{\theta}_{t}||_{L^{\infty}(\Omega)}\int_{\Omega}|(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|$
$\displaystyle\quad+\int_{0}^{t}C(\underline{\theta})\big{(}||\nabla\tilde{\theta}||_{L^{\infty}(\Omega)}+||\nabla\ln\tilde{\theta}||_{L^{\infty}(\Omega)}\big{)}\int_{\Omega}|\theta-\tilde{\theta}|^{2}$
(75)
$\displaystyle\quad+\frac{\underline{\theta}}{2}\int_{0}^{t}\int_{\Omega}|\nabla\theta-\nabla\tilde{\theta}|^{2}+\frac{\underline{\theta}}{2}\int_{0}^{t}\int_{\Omega}|\nabla\ln\theta-\nabla\ln\tilde{\theta}|^{2}.$

Note that it is enough to prove

$\displaystyle|(s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta})\quad\text{ and
}\quad|\theta-\tilde{\theta}|^{2}\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),$

for $C>0$ that doesn’t depend on $t$, since this implies

$\displaystyle\mathcal{E}(\theta\big{|}\tilde{\theta})(t)\leq
C\int_{0}^{t}\mathcal{E}(\theta\big{|}\tilde{\theta})(s)~{}ds,$

and the conclusion follows as in section 4.3.2. The following result will be
useful:

######  Lemma 5.8.

Let $f,g:[a,b]\to\mathbb{R}$ be two $C^{2}$ functions and $s\in[a,b]$. If:

  1. (1)

$f(s)=g(s)=f^{\prime}(s)=g^{\prime}(s)=0$,

  2. (2)

$f^{\prime\prime}(s)>0$,

  3. (3)

$f(x)>0$, on $[a,b]\setminus\\{s\\}$,

then there is a constant $c>0$ such that

$\displaystyle f(x)\geq c|g(x)|,\quad\text{on }[a,b].$

###### Proof.

By Taylor’s theorem, we have

$\displaystyle
f(x)=\underbrace{f(s)+f^{\prime}(s)(x-s)}_{=0}+\frac{f^{\prime\prime}(s)}{2}(x-s)^{2}+r_{f}(x)(x-s)^{2}$
$\displaystyle=\left(\frac{f^{\prime\prime}(s)}{2}+r_{f}(x)\right)(x-s)^{2},\quad
x\in[a,b],$

where $r_{f}$ is a continuous function such that $r_{f}(s)=0$. Now, since
$f>0$ on $[a,b]\setminus\\{s\\}$, one has
$\frac{f^{\prime\prime}(s)}{2}+r_{f}(x)>0$ on $[a,b]\setminus\\{s\\}$, while
the continuity of $r_{f}$ and $r_{f}(s)=0$ implies that
$\frac{f^{\prime\prime}(s)}{2}+r_{f}(x)>0$ on entire $[a,b]$. Denote
$m=\min\limits_{x\in[a,b]}\frac{f^{\prime\prime}(s)}{2}+r_{f}(x)$. Applying
Taylor’s theorem to $g$, in the same way we have

$\displaystyle
g(x)=\left(\frac{g^{\prime\prime}(s)}{2}+r_{g}(x)\right)(x-s)^{2},\quad
x\in[a,b],$

so denoting
$M=\max\limits_{x\in[a,b]}\left|\frac{g^{\prime\prime}(s)}{2}+r_{g}(x)\right|$
and choosing $c=\frac{m}{M}$, we conclude

$\displaystyle c|g(x)|\leq
cM(x-s)^{2}=m(x-s)^{2}\leq\left(\frac{f^{\prime\prime}(s)}{2}+r_{f}(x)\right)(x-s)^{2}=f(x),$

and the proof is finished. ∎

Now, applying this lemma

$\displaystyle|s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),\quad\text{for
}\theta\in\left[\underline{\theta},\overline{\theta}\right].$

Note that here for each fixed $\tilde{\theta}$ there a constant
$c_{\tilde{\theta}}$ for which this inequality holds, so our $C>0$ is chosen
as
$\max\limits_{\tilde{\theta}\in[\underline{\theta},\overline{\theta}]}c_{\tilde{\theta}}$.
Next, we aim to control the function
$|s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|$
outside of $\left[\underline{\theta},\overline{\theta}\right]$. First,

$\displaystyle|s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|\leq
C(1+|\ln(\theta)|)\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),\quad\text{for
}\theta\in\left[0,\underline{\theta}\right],$

since the term $\ln(\theta)$ has a negative sign in
$\mathcal{E}(\theta\big{|}\tilde{\theta})$ and the other terms are bounded,
while for $\theta\geq\overline{\theta}$ the function $\theta^{\alpha}$
dominates $\theta^{\alpha-1}$ and $\ln\theta$ so

$\displaystyle|s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|\leq
C(1+e_{1}(\theta))\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),\quad\text{for
}\theta\geq\overline{\theta},$

which together imply

$\displaystyle|s_{1}(\theta)-s_{1}(\tilde{\theta})_{\theta}(\theta-\tilde{\theta})-s_{1}(\tilde{\theta})|\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}).$

Next, to control $|\theta-\tilde{\theta}|^{2}$, we start with

(76) $\displaystyle|\theta-\tilde{\theta}|^{2}\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),\quad\text{for
}\theta\in[0,\underline{\theta}]\cup[\underline{\theta},\overline{\theta}],$

by Lemma 5.8 and

$\displaystyle|\theta-\tilde{\theta}|^{2}\leq C(1+\theta^{2})\leq
C(1+e_{1}(\theta))\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}),\quad\text{for
}\theta\geq\overline{\theta},$

where we have used the growth condition $\alpha\geq 2$, so combining the above
inequalities implies

$\displaystyle|\theta-\tilde{\theta}|^{2}\leq
C\mathcal{E}(\theta\big{|}\tilde{\theta}).$

Thus, the proof is finished.

## Appendix

######  Lemma.

Let $\Omega$ be a Lipschitz domain and let $p\in(1,\infty]$. Assume that
$\theta\in L^{p}(0,T;\mathcal{M}(\Omega))$, and that $\mathbf{u}\in
L^{p}(0,T;H^{1}(\Omega))$ with $\mathbf{u}_{t}\in
L^{\infty}(0,T;L^{2}(\Omega))$ satisfies the following equation

$\displaystyle\int_{0}^{T}\int_{\Omega}\mathbf{u}_{t}\cdot\boldsymbol{\varphi}_{t}-\int_{0}^{T}\int_{\Omega}\mathbb{D}(\mathbf{u}):\nabla\boldsymbol{\varphi}+\mu\int_{0}^{T}{}_{\mathcal{M}(\Omega)}\langle\theta,\nabla\cdot\boldsymbol{\varphi}\rangle_{C_{0}(\Omega)}=-\int_{\Omega}\mathbf{v}_{0}\cdot\boldsymbol{\varphi},$

for all test function $\boldsymbol{\varphi}\in
H^{1}(0,T;C^{\infty}_{0}(\Omega))$, ${\boldsymbol{\varphi}}(T)=0$, where
$\mathbf{v}_{0}\in L^{2}(\Omega)$ is given. Then, $\mathbf{u}_{t}\in
C_{w}(0,T;L^{2}(\Omega))$.

###### Proof.

First, let us prove that . Since $\theta\in L^{p}(0,T;\mathcal{M}(\Omega))$,
one has from the equation for every $\boldsymbol{\varphi}\in
C_{0}^{\infty}((0,T)\times\Omega)$ that

$\displaystyle\left|\int_{0}^{T}\int_{\Omega}\mathbf{u}_{tt}\cdot\boldsymbol{\varphi}\right|\leq\big{(}||\mathbf{u}||_{L^{p}(0,T;H^{1}(\Omega))}+||\theta||_{L^{p}(0,T;\mathcal{M}^{+}(\Omega))}\big{)}||\boldsymbol{\varphi}||_{L^{p^{*}}(0,T;W^{2,\infty}(\Omega))},$

so $\mathbf{u}_{tt}\in L^{p^{*}}(0,T;[C_{0}^{\infty}(\Omega)]^{\prime})$, and
consequently $\mathbf{u}_{t}\in C(0,T;[C_{0}^{\infty}(\Omega)]^{\prime})$.

Now, for an arbitrary $\boldsymbol{\varphi}\in L^{2}(\Omega)$, let
$\boldsymbol{\varphi}_{n}\in C_{0}^{\infty}(\Omega)$ be such that
$||\boldsymbol{\varphi}_{n}-\boldsymbol{\varphi}||_{L^{2}(\Omega)}\to 0$, as
$n\to\infty$. Then, for almost all $t_{1},t_{2}$, one has

$\displaystyle\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot\boldsymbol{\varphi}$
$\displaystyle=\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot(\boldsymbol{\varphi}-\boldsymbol{\varphi}_{n})+\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot\boldsymbol{\varphi}_{n},$

so

$\displaystyle\lim\limits_{t_{2}\to
t_{1}}\left|\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot\boldsymbol{\varphi}\right|$
$\displaystyle\leq\lim\limits_{t_{2}\to
t_{1}}\left|\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot(\boldsymbol{\varphi}-\boldsymbol{\varphi}_{n})\right|+\underbrace{\lim\limits_{t_{2}\to
t_{1}}\left|\int_{\Omega}(\mathbf{u}_{t}(t_{1})-\mathbf{u}_{t}(t_{2}))\cdot\boldsymbol{\varphi}_{n}\right|}_{=0}$
$\displaystyle\leq
2||\mathbf{u}_{t}||_{L^{\infty}(0,T;L^{2}(\Omega))}||\boldsymbol{\varphi}-\boldsymbol{\varphi}_{n}||_{L^{2}(\Omega)}\to
0,\quad\text{as }n\to\infty.$

Now, we can give meaning to
$\int_{\Omega}\mathbf{u}_{t}(t)\cdot\boldsymbol{\varphi}$ for all $t$ by
constructing a sequence of $t_{n}$ such that
$\int_{\Omega}\mathbf{u}_{t}(t_{n})\cdot\boldsymbol{\varphi}$ is finite for
all $n$ and $t_{n}\to t$ as $n\to\infty$. From the above inequality,
$\int_{\Omega}\mathbf{u}_{t}(t_{n})\cdot\boldsymbol{\varphi}$ forms a Cauchy
sequence which converges to a unique limit
$a_{\boldsymbol{\varphi}}\in\mathbb{R}$ for any $\boldsymbol{\varphi}$,
independent of the choice of the sequence $\\{t_{n}\\}_{n\in\mathbb{N}}$. Now,
since
$\left|\int_{\Omega}\mathbf{u}_{t}(t_{n})\cdot\boldsymbol{\varphi}\right|\leq
C||\varphi||_{L^{2}(\Omega)}$, one has that $|a_{\boldsymbol{\varphi}}|\leq
C||\varphi||_{L^{2}(\Omega)}$. This combined with linearity of
$\boldsymbol{\varphi}\mapsto a_{\boldsymbol{\varphi}}$ allows us to identify
$a_{\boldsymbol{\varphi}}$ with a function $\mathbf{u}_{t}(t)\in
L^{2}(\Omega)$, by the Riesz theorem. The proof is now finished.
∎

Acknowledgements. T.C. was supported by the National Science Center of Poland
grant SONATA BIS 7 number UMO-2017/26/E/ST1/00989. B.M. was partially
supported by the Croatian Science Foundation (Hrvatska Zaklada za Znanost)
grant number IP-2018-01-3706. S.T. was supported by Provincial Secretariat for
Higher Education and Scientific Research of Vojvodina, Serbia, grant no
142-451-2593/2021-01/2.

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