# The Rayleigh–Bénard problem for compressible fluid flows

Eduard Feireisl The work of E.F. was partially supported by the Czech Sciences
Foundation (GAČR), Grant Agreement 18–05974S. The Institute of Mathematics of
the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
This work is partially supported by the Simons Foundation Award No 663281
granted to the Institute of Mathematics of the Polish Academy of Sciences for
the years 2021-2023. Agnieszka Świerczewska-Gwiazda The work of A. Ś-G. was
partially supported by National Science Centre (Poland), agreement no
2017/27/B/ST1/01569.

###### Abstract

We consider the physically relevant fully compressible setting of the
Rayleigh–Bénard problem of a fluid confined between two parallel plates,
heated from the bottom, and subjected to the gravitational force. Under
suitable restrictions imposed on the constitutive relations we show that this
open system is dissipative in the sense of Levinson, meaning there exists a
bounded absorbing set for any global–in–time weak solution. In addition,
global–in–time trajectories are asymptotically compact in suitable topologies
and the system possesses a global compact trajectory attractor $\mathcal{A}$.
The standard technique of Krylov and Bogolyubov then yields the existence of
an invariant measure – a stationary statistical solution sitting on
$\mathcal{A}$. In addition, the Birkhoff–Khinchin ergodic theorem provides
convergence of ergodic averages of solutions belonging to $\mathcal{A}$ a.s.
with respect to the invariant measure.

Institute of Mathematics of the Academy of Sciences of the Czech Republic

Žitná 25, CZ-115 67 Praha 1, Czech Republic

Institute of Applied Mathematics and Mechanics, University of Warsaw

Banacha 2, 02-097 Warsaw, Poland

Keywords: Rayleigh–Bénard problem, Navier–Stokes–Fourier system, weak
solution, attractor, stationary statistical solution

###### Contents

  1. 1 Introduction
  2. 2 Principal hypotheses, weak solutions
    1. 2.1 Weak solution
    2. 2.2 Constitutive relations
  3. 3 Main results
  4. 4 Dissipativity
    1. 4.1 Mass conservation
    2. 4.2 Coercivity of the dissipative term
    3. 4.3 Energy estimates
    4. 4.4 Entropy estimates
    5. 4.5 Pressure estimates
    6. 4.6 Uniform bounds
      1. 4.6.1 Pressure estimates revisited
      2. 4.6.2 Proof of Lemma 4.1
    7. 4.7 Bounded absorbing sets
  5. 5 Applications, long–time behavior
    1. 5.1 Trajectory space
    2. 5.2 Attractor
    3. 5.3 Stationary statistical solutions
    4. 5.4 Convergence of ergodic means

##  1 Introduction

The Rayleigh–Bénard problem concerns the motion of a fluid confined between
two parallel planes, where the temperature of the bottom plane is maintained
at the level $\Theta_{\rm B}$, while the top plane has the ambient temperature
$\Theta_{\rm U}$, typically $\Theta_{\rm U}\leq\Theta_{\rm B}$. The only
volume force is due to gravitation acting in the downward vertical direction.
The fluid mass density $\varrho=\varrho(t,x)$, the temperature
$\vartheta(t,x)$, and the velocity ${\bf u}={\bf u}(t,x)$ obey the standard
system of equations of continuum fluid mechanics:

$\displaystyle\partial_{t}\varrho+{\rm div}_{x}(\varrho{\bf u})$
$\displaystyle=0,$ (1.1) $\displaystyle\partial_{t}(\varrho{\bf u})+{\rm
div}_{x}(\varrho{\bf u}\otimes{\bf u})+\nabla_{x}p(\varrho,\vartheta)$
$\displaystyle={\rm div}_{x}\mathbb{S}+\varrho\nabla_{x}G,$ (1.2)
$\displaystyle\partial_{t}(\varrho e(\varrho,\vartheta))+{\rm div}_{x}(\varrho
e(\varrho,\vartheta){\bf u})+\nabla_{x}{\bf q}$
$\displaystyle=\mathbb{S}:\mathbb{D}_{x}{\bf u}-p(\varrho,\vartheta){\rm
div}_{x}{\bf u},$ (1.3)

where we have denoted

the pressure $p=p(\varrho,\vartheta)$,

the (specific) internal energy $e=e(\varrho,\vartheta)$,

the viscous stress tensor $\mathbb{S}$,

the gravitational potential $G=-gx_{3}$,

the heat flux ${\bf q}$

the symmetric part of the velocity gradient $\mathbb{D}_{x}{\bf
u}=\frac{1}{2}\left(\nabla_{x}{\bf u}+\nabla_{x}^{t}{\bf u}\right)$.

For the sake of simplicity, we consider the periodic boundary conditions with
respect to the horizontal variables. Accordingly, the fluid domain $\Omega$
can be identified with

$\Omega=\mathbb{T}^{2}\times(0,1),$ (1.4)

where $\mathbb{T}^{2}$ is the two–dimensional flat torus. If the boundary
planes are impermeable and the viscous fluid sticks to them, the relevant
boundary conditions read  $\displaystyle{\bf u}|_{x_{3}=0}$
$\displaystyle={\bf u}|_{x_{3}=1}=0,$ (1.5)
$\displaystyle\vartheta|_{x_{3}=0}$ $\displaystyle=\Theta_{\rm B},\
\vartheta|_{x_{3}=1}=\Theta_{\rm U}.$ (1.6)

We suppose that the fluid is Newtonian, with the viscous stress

$\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u})=\mu(\vartheta)\left(\nabla_{x}{\bf u}+\nabla_{x}^{t}{\bf
u}-\frac{2}{3}{\rm div}_{x}{\bf u}\mathbb{I}\right)+\eta(\vartheta){\rm
div}_{x}{\bf u}\mathbb{I}.$ (1.7)

The heat flux is given by Fourier’s law

${\bf
q}(\vartheta,\nabla_{x}\vartheta)=-\kappa(\vartheta)\nabla_{x}\vartheta,$
(1.8)

where $\kappa$ is the conductivity. The field equations (1.1)–(1.3), endowed
with the constitutive relations (1.7), (1.8), will be referred to as
_Navier–Stokes–Fourier system_.

The behaviour of the fluid under the boundary conditions (1.5), (1.6) with a
sufficiently large difference between the top and bottom temperatures is a
prominent example of turbulence, see e.g. Davidson [13]. There is a large
amount of mathematical literature devoted to the asymptotic behaviour of
solutions to the Rayleigh–Bénard problem in the simplified incompressible
framework, where the system (1.1)–(1.3) is replaced by the Oberbeck–Boussinesq
approximation, see Constantin et al. [11], Foias, Manley and Temam [22], Cao
et al. [8] and the references therein. Recently, the problem motivated a
series of studies by Otto et al. [10], [30], [34] concerning the associated
scaling laws.

Much less seems to be known in the original and physically relevant framework
of compressible and heat conducting fluids. Here, the rigorous analysis is
hampered by the following notoriously known difficulties:

  * •

Navier–Stokes–Fourier system endowed with the boundary conditions (1.5), (1.6)
is an open system in the regime far from equilibrium. Unfortunately, the
existence of _global–in–time_ smooth solutions is known only in the case of
closed systems approaching the equilibrium solution in the long run, see
Matsumura and Nishida [28], [29], or Valli and Zajaczkowski [36].

  * •

The available theory of weak solutions developed in [16] (see also the
alternative approach by Bresch and Desjardins [6] and Bresch and Jabin [7])
applies to conservative or periodic boundary conditions pertinent to the
closed systems. Note that the dynamics of the Navier–Stokes–Fourier system
with _conservative_ boundary conditions is nowadays well understood, see [20].
Indeed, in accordance with the celebrated statement of Clausius:

“Die Energie der Welt ist konstant. Die Entropie der Welt strebt einem Maximum
zu”

Rudolf Clausius, Poggendorff’s Annals of Physics 1865 (125), 400;

any global–in–time weak solution of the Navier–Stokes–Fourier system with
conservative boundary conditions and driven by a conservative volume force
tends to an equilibrium, see e.g. [18], [20] or Novotný and Pokorný [31],
Novotný and Straškraba [32], [33]. Here, conservative means that the system is
thermally insulated, the boundary conditions (1.6) being replaced by

${\bf q}\cdot{\bf n}=q_{3}=0\ \mbox{on}\ \partial\Omega.$

  * •

The weak solutions are not (known to be) uniquely determined by the
initial/boundary data.

Recently, the theory of weak solutions has been extended to non–zero in/out
flux boundary conditions in [17], and, finally, to general Dirichlet boundary
conditions in [9]. In particular, the theory of weak solutions proposed in [9]
yields a suitable platform to attack the Rayleigh–Bénard problem (1.5), (1.6).
As pointed out above, the weak solutions are not known to be uniquely
determined by the initial/boundary data. Accordingly, we follow the approach
advocated by Sell [35] and Málek and Nečas [27] replacing the standard phase
space by the space of trajectories.

The principal objective of the paper is to establish the following two basic
results:

  * •

Levinson dissipativity or bounded absorbing set. Any global–in–time weak
solution to the Navier–Stokes–Fourier system endowed with the boundary
conditions (1.5), (1.6) enters eventually a bounded absorbing set. In
comparison with [15], we relax the hypothesis of the hard sphere pressure and
consider the physically relevant equation of state of a general monoatomic gas
with the effect of radiation proposed in [16].

  * •

Asymptotic compactness. Similarly to [21, Chapter 4, Theorem 4.2], we show
that any bounded family of global solutions is precompact in a suitable
topology of the trajectory space, whereas any of its accumulation points
represents a weak solution of the same problem.

Using the above results, we establish the existence of a compact trajectory
attractor, an invariant measure and the existence of stationary statistical
solutions generated by bounded trajectories. Finally, we also discuss the
existence of the ergodic averages in the spirit of [14].

As pointed out, the key point of the analysis is the Levinson dissipativity or
the existence of a universal bounded absorbing set for the “monoatomic”
equation of state introduced in [16, Chapters 1,2]. This is rather surprising
as this constitutive equation can be seen as a temperature dependent
counterpart of the isentropic pressure law
$p(\varrho)\approx\varrho^{\gamma}$, with $\gamma=\frac{5}{3}$. Note that for
the isentropic model, the existence of a bounded absorbing set is known only
if $\gamma>\frac{5}{3}$, see [19], whereas the limit case $\gamma=\frac{5}{3}$
requires smallness of the total mass of the fluid, see Wang and Wang [37].
Moreover, uniform boundedness of global trajectories for the
Navier–Stokes–Fourier system is a delicate issue. As is known, see [20], [21],
the energy of _all_ global–in–time solutions tends to infinity with growing
time as soon as the system is energetically closed and driven by a
non–conservative volume force.

Similarly to the incompressible Navier–Stokes system with conservative
boundary conditions studied by Málek and Nečas [27] and Sell [35], the large
time asymptotic behaviour of solutions to the Rayleigh–Bénard problem is
captured by the set of _entire_ trajectories $\mathcal{A}$ defined for all
$t\in R$ and with uniformly bounded total energy and mass. We show that the
set $\mathcal{A}$ is (i) non–empty, (ii) time shift invariant, and (iii)
compact if endowed by a suitable metrics. The standard Krylov–Bogolyubov
technique then yields the existence of an invariant measure supported in
$\mathcal{A}$ – a stationary statistical solution of the Rayleigh–Bénard
problem. Moreover, the standard Birkhoff–Khinchin ergodic theorems yields the
convergence of the ergodic averages a.s. with respect to the invariant
measure. Uniqueness of the invariant measure for solutions with the same total
mass remains an outstanding open problem.

The paper is organized as follows. In Section 2, we recall the principal
constitutive hypotheses and introduce the concept of weak solution. The main
results – the existence of a bounded absorbing set and asymptotic compactness
of global–in–time solutions – are stated in Section 3. In Section 4, we show
the existence of a bounded absorbing set in terms of the total energy. The
implications of the main results on the long–time behavior of the system are
discussed in Section 5.

##  2 Principal hypotheses, weak solutions

Following [9] we introduce the concept of weak solution to the
Navier–Stokes–Fourier system based on the combination of the balance equations
for the entropy and the _ballistic energy_

$E_{\tilde{\vartheta}}(\varrho,\vartheta,{\bf u})=E(\varrho,\vartheta,{\bf
u})-\tilde{\vartheta}\varrho s(\varrho,\vartheta),\ E(\varrho,\vartheta,{\bf
u})=\frac{1}{2}\varrho|{\bf u}|^{2}+\varrho e(\varrho,\vartheta),$

where $s$ is the entropy related to the other thermodynamic functions through
Gibbs’ equation

$\vartheta
Ds(\varrho,\vartheta)=De(\varrho,\vartheta)+pD\left(\frac{1}{\varrho}\right),$
(2.1)

and $\tilde{\vartheta}$ is an arbitrary continuously differentiable function
of $(t,x)$ satisfying

$\tilde{\vartheta}>0,\ \tilde{\vartheta}|_{x_{3}=0}=\Theta_{\rm B},\
\tilde{\vartheta}|_{x_{3}=1}=\Theta_{\rm U}.$ (2.2)

For the sake of simplicity, we suppose that $\Theta_{\rm B}$, $\Theta_{\rm U}$
are positive _constants_. A generalization of the results of the present paper
to the space or even time dependent boundary temperatures is possible with
obvious modifications in the proofs.

###  2.1 Weak solution

As we are interested in the long time behaviour of solutions, the specific
value of the initial data is irrelevant. We therefore consider solutions of
the Navier–Stokes–Fourier system defined on the open time interval
$(T,\infty)$, $T\in R$.

######  Definition 2.1.

We say that $(\varrho,\vartheta,{\bf u})$ is a _weak solution_ of the
Navier–Stokes–Fourier system (1.1)–(1.3), (1.7), (1.8), with the boundary
conditions (1.5), (1.6) defined on the time interval $(T,\infty)$ if the
following holds: • The solution belongs to the following regularity class:
$\displaystyle\varrho$ $\displaystyle\in L^{\infty}(0,T;L^{\gamma}(\Omega))\
\mbox{for some}\ \gamma>1,$ $\displaystyle{\bf u}$ $\displaystyle\in
L^{2}_{\rm loc}(T,\infty;W^{1,2}_{0}(\Omega;R^{3})),$
$\displaystyle\vartheta^{\beta/2},\ \log(\vartheta)$ $\displaystyle\in
L^{2}(0,T;W^{1,2}(\Omega)\ \mbox{for some}\ \beta\geq 2,$
$\displaystyle(\vartheta-\vartheta_{B})$ $\displaystyle\in
L^{2}(0,T;W^{1,2}_{0}(\Omega)).$ (2.3) • The equation of continuity (1.1)
along with its renormalization are satisfied in the sense of distributions,
specifically,
$\displaystyle\int_{T}^{\infty}\int_{\Omega}\left[\varrho\partial_{t}\varphi+\varrho{\bf
u}\cdot\nabla_{x}\varphi\right]\ \,{\rm d}{x}\,{\rm d}t$ $\displaystyle=0,$
(2.4)
$\displaystyle\int_{T}^{\infty}\int_{\Omega}\left[b(\varrho)\partial_{t}\varphi+b(\varrho){\bf
u}\cdot\nabla_{x}\varphi+\Big{(}b(\varrho)-b^{\prime}(\varrho)\varrho\Big{)}{\rm
div}_{x}{\bf u}\varphi\right]\ \,{\rm d}{x}\,{\rm d}t$ $\displaystyle=0$ (2.5)
for any $\varphi\in C^{1}_{c}((T,\infty)\times\overline{\Omega})$, and any
$b\in C^{1}(R)$, $b^{\prime}\in C_{c}(R)$. • The momentum equation (1.2) is
satisfied in the sense of distributions, $\displaystyle\int_{T}^{\infty}$
$\displaystyle\int_{\Omega}\left[\varrho{\bf
u}\cdot\partial_{t}\boldsymbol{\varphi}+\varrho{\bf u}\otimes{\bf
u}:\nabla_{x}\boldsymbol{\varphi}+p{\rm div}_{x}\boldsymbol{\varphi}\right]\
\,{\rm d}{x}\,{\rm d}t$
$\displaystyle=\int_{T}^{\infty}\int_{\Omega}\left[\mathbb{S}:\nabla_{x}\boldsymbol{\varphi}-\varrho\nabla_{x}G\cdot\boldsymbol{\varphi}\right]\
\,{\rm d}{x}\,{\rm d}t,$ (2.6) for any $\boldsymbol{\varphi}\in
C^{1}_{c}((T,\infty)\times\Omega;R^{3})$. • The internal energy equation (1.3)
is replaced by the entropy inequality
$\displaystyle-\int_{T}^{\infty}\int_{\Omega}\left[\varrho
s\partial_{t}\varphi+\varrho s{\bf u}\cdot\nabla_{x}\varphi+\frac{{\bf
q}}{\vartheta}\cdot\nabla_{x}\varphi\right]\ \,{\rm d}{x}\,{\rm
d}t\geq\int_{T}^{\infty}\int_{\Omega}\frac{\varphi}{\vartheta}\left[\mathbb{S}:\mathbb{D}_{x}{\bf
u}-\frac{{\bf q}\cdot\nabla_{x}\vartheta}{\vartheta}\right]\ \,{\rm
d}{x}\,{\rm d}t$ (2.7) for any $\varphi\in C^{1}_{c}((T,\infty)\times\Omega)$,
$\varphi\geq 0$; and the ballistic energy balance,
$\displaystyle-\int_{T}^{\infty}\partial_{t}\psi$
$\displaystyle\int_{\Omega}\left[\frac{1}{2}\varrho|{\bf u}|^{2}+\varrho
e-\tilde{\vartheta}\varrho s\right]\ \,{\rm d}{x}\,{\rm
d}t+\int_{T}^{\infty}\psi\int_{\Omega}\frac{\tilde{\vartheta}}{\vartheta}\left[\mathbb{S}:\mathbb{D}_{x}{\bf
u}-\frac{{\bf q}\cdot\nabla_{x}\vartheta}{\vartheta}\right]\ \,{\rm
d}{x}\,{\rm d}t$
$\displaystyle\leq\int_{T}^{\infty}\psi\int_{\Omega}\left[\varrho{\bf
u}\cdot\nabla_{x}G-\varrho s{\bf u}\cdot\nabla_{x}\tilde{\vartheta}-\frac{{\bf
q}}{\vartheta}\cdot\nabla_{x}\tilde{\vartheta}\right]\ \,{\rm d}{x}$ (2.8) for
any $\psi\in C^{1}_{c}(T;\infty)$, $\psi\geq 0$, and any $\tilde{\vartheta}\in
C^{1}([T;\infty)\times\overline{\Omega})$, $\tilde{\vartheta}>0,\
\tilde{\vartheta}|_{x_{3}=0}=\Theta_{\rm B},\
\tilde{\vartheta}|_{x_{3}=1}=\Theta_{\rm U}.$

The existence of global–in–time weak solutions under the constitutive
restrictions specified in the forthcoming section was proved in [9, Theorem
4.2]. In addition, the weak solutions comply with the weak–strong uniqueness
principle and coincide with strong solutions as soon as they are smooth.

###  2.2 Constitutive relations

Following [16, Chapters 1,2] we consider the equation of state

$p(\varrho,\vartheta)=p_{\rm m}(\varrho,\vartheta)+p_{\rm rad}(\vartheta),$

where $p_{\rm m}$ is the pressure of a general _monoatomic_ gas,

$p_{\rm m}(\varrho,\vartheta)=\frac{2}{3}\varrho e_{\rm
m}(\varrho,\vartheta),$ (2.9)

enhanced by the radiation pressure

$p_{\rm rad}(\vartheta)=\frac{a}{3}\vartheta^{4},\ a>0.$

Accordingly, the internal energy reads

$e(\varrho,\vartheta)=e_{\rm m}(\varrho,\vartheta)+e_{\rm
rad}(\varrho,\vartheta),\ e_{\rm
rad}(\varrho,\vartheta)=\frac{a}{\varrho}\vartheta^{4}.$

To identify the specific form of $p_{m}$ we successively employ several
physical principles, see [16, Chapter 1] for details.

  * •

Gibbs’ relation together with (2.9) yield

$p_{m}(\varrho,\vartheta)=\vartheta^{\frac{5}{2}}P\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)$

for a certain $P\in C^{1}[0,\infty)$. Consequently,

$p(\varrho,\vartheta)=\vartheta^{\frac{5}{2}}P\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)+\frac{a}{3}\vartheta^{4},\
e(\varrho,\vartheta)=\frac{3}{2}\frac{\vartheta^{\frac{5}{2}}}{\varrho}P\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)+\frac{a}{\varrho}\vartheta^{4},\
a>0.$ (2.10)

  * •

Hypothesis of thermodynamics stability, cf. Bechtel, Rooney, and Forrest [1],
expressed in terms of $P$, reads

$P(0)=0,\ P^{\prime}(Z)>0\ \mbox{for}\ Z\geq 0,\
0<\frac{\frac{5}{3}P(Z)-P^{\prime}(Z)Z}{Z}\leq c\ \mbox{for}\ Z>0.$ (2.11)

In particular, the function $Z\mapsto P(Z)/Z^{\frac{5}{3}}$ is decreasing, and
we suppose

$\lim_{Z\to\infty}\frac{P(Z)}{Z^{\frac{5}{3}}}=p_{\infty}>0.$ (2.12)

  * •

In accordance with Gibbs’ relation (2.1), the associated entropy takes the
form

$s(\varrho,\vartheta)=\mathcal{S}\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)+\frac{4a}{3}\frac{\vartheta^{3}}{\varrho},$
(2.13)

where

$\mathcal{S}^{\prime}(Z)=-\frac{3}{2}\frac{\frac{5}{3}P(Z)-P^{\prime}(Z)Z}{Z^{2}}<0.$
(2.14)

In addition, the Third law of thermodynamics, cf. Belgiorno [2], [3], requires
the entropy to vanish as soon as the abolute temperature approaches zero,

$\lim_{Z\to\infty}\mathcal{S}(Z)=0.$ (2.15)

Note that (2.11) – (2.15) imply

$0\leq\varrho\mathcal{S}\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)\leq
c\left(1+\varrho\log^{+}(\varrho)+\varrho\log^{+}(\vartheta)\right).$ (2.16)

As for the transport coefficients, we suppose that they are continuously
differentiable functions satisfying

$\displaystyle 0<\underline{\mu}(1+\vartheta)$
$\displaystyle\leq\mu(\vartheta),\
|\mu^{\prime}(\vartheta)|\leq\overline{\mu},$ $\displaystyle
0\leq\eta(\vartheta)$ $\displaystyle\leq\overline{\eta}(1+\vartheta),$
$\displaystyle 0<\underline{\kappa}(1+\vartheta^{\beta})$
$\displaystyle\leq\kappa(\vartheta)\leq\overline{\kappa}(1+\vartheta^{\beta}).$
(2.17)

The existence theory developed in [9] requires

$\beta>6.$ (2.18)

The state equation specified above, together with the fact that the transport
coefficients depend on the temperature, are pertinent to models of gaseous
stars discussed by Bormann [4], [5].

##  3 Main results

Our main result states that the Navier–Stokes–Fourier system in the
Rayleigh–Bénard regime (1.5), (1.6) admits a bounded absorbing set.

######  Theorem 3.1 (Bounded absorbing set).

Let $\Theta_{\rm B}$, $\Theta_{\rm U}$ be two strictly positive constants. Let
the pressure $p$, the internal energy $e$, and the entropy $s$ satisfy the
hypotheses (2.10)–(2.15). Let the transport coefficients $\mu$, $\eta$, and
$\kappa$ satisfy (2.17), (2.18). Then for any global–in–time weak solution
$(\varrho,\vartheta,{\bf u})$ defined on a time interval $(T,\infty)$, there
exists a constant $\mathcal{E}_{\infty}$ that depends only on $\Theta_{\rm
B}$, $\Theta_{\rm U}$, the total mass of the fluid $M=\int_{\Omega}\varrho\
\,{\rm d}{x},$ and the structural properties of $p=p(\varrho,\vartheta)$,
$e=a(\varrho,\vartheta)$, $s=s(\varrho,\vartheta)$ such that ${\rm
ess}\limsup_{t\to\infty}\int_{\Omega}E(\varrho,\vartheta,{\bf u})(t,\cdot)\
\,{\rm d}{x}\leq\mathcal{E}_{\infty}.$ (3.1) If, moreover, ${\rm
ess}\limsup_{t\to T+}\int_{\Omega}E(\varrho,\vartheta,{\bf u})(t,\cdot)\
\,{\rm d}{x}\leq\mathcal{E}_{0}<\infty,$ then the convergence is uniform in
$\mathcal{E}_{0}$. Specifically, for any $\varepsilon>0$, there exists a time
$T(\varepsilon,\mathcal{E}_{0})$ such that ${\rm
ess}\sup_{t>T(\varepsilon,\mathcal{E}_{0})}\int_{\Omega}E(\varrho,\vartheta,{\bf
u})(t,\cdot)\ \,{\rm d}{x}\leq\mathcal{E}_{\infty}+\varepsilon.$

######  Remark 3.2.

The same result can be shown for a general bounded domain with an arbitrary
(nonconstant) profile of the boundary temperature and a general potential
volume force ${\bf g}=\nabla_{x}G$, $G=G(x)$. In particular, the problem posed
in the inclined layer studied e.g. by Daniels et al. [12] can be included.

The existence of a bounded absorbing set for the isentropic
($p=a\varrho^{\gamma}$) Navier–Stokes system with the no–slip boundary
conditions was established in [19] under the condition $\gamma>\frac{5}{3}$,
see also Wang and Wang [37]. For similar results related to the conservative
boundary conditions see [18] and the monograph [21]. The existence of a
bounded absorbing set for the Navier–Stokes–Fourier system with general
Dirichlet boundary conditions was shown in [15] under rather restrictive
assumption postulating a hard sphere equation of state for the pressure. Note
that this considerably simplifies the analysis as _uniform_ bounds on the
fluid density are a priori available. It is exactly this missing piece of
information that makes the analysis of the present paper much more delicate.

The second result concerns the asymptotic compactness of bounded trajectories.

######  Theorem 3.3 (Asymptotic compactness).

Under the hypotheses of Theorem 3.1, let
$(\varrho_{n},\vartheta_{n},{\bf u}_{n})_{n=1}^{\infty}$ be a sequence of weak
solutions to the Navier–Stokes–Fourier system in the sense of Definition 2.1
on the time intervals $(T_{n},\infty),\ T_{n}\geq-\infty,\ T_{n}\to-\infty\
\mbox{as}\ n\to\infty,$ such that ${\rm ess}\sup_{t\to
T_{n}}\int_{\Omega}E(\varrho_{n},\vartheta_{n},{\bf u}_{n})(t,\cdot)\ \,{\rm
d}{x}\leq\mathcal{E}_{0},\ \int_{\Omega}\varrho\ \,{\rm d}{x}=M>0,$ uniformly
for $n\to\infty$. Then there is a subsequence (not relabelled) such that
$\displaystyle\varrho_{n}$ $\displaystyle\to\varrho\ \mbox{in}\ C_{\rm
weak}([-M,M];L^{\frac{5}{3}}(\Omega))\cap C([-M,M];L^{1}(\Omega)),$
$\displaystyle\vartheta_{n}$ $\displaystyle\to\vartheta\ \mbox{in}\
L^{q}((-M,M);L^{4}(\Omega))\ \mbox{for any}\ 1\leq q<\infty,$
$\displaystyle{\bf u}_{n}$ $\displaystyle\to{\bf u}\ \mbox{weakly in}\
L^{2}((-M,M);W^{1,2}(\Omega;R^{3}))$ (3.2) for any $M>0$, where the limit
$(\varrho,\vartheta,{\bf u})$ is an entire weak solution of the
Navier–Stokes–Fourier system defined for $t\in R$ and satisfying
$\int_{\Omega}E(\varrho,\vartheta,{\bf u})(t,\cdot)\ \,{\rm
d}{x}\leq\mathcal{E}_{\infty}\ \mbox{for a.a.}\ t\in R.$ (3.3)

The heart of the paper is the proof of Theorem 3.1. Once the uniform bounds on
the energy are established, the proof of Theorem 3.3 reduces to showing
compactness of a sequence of bounded solutions. To certain extent, this is
similar to the existence proof, where the only essential issue is the strong
(a.e. pointwise) convergence of the densities in (3.2). Unlike in the
existence proof, compactness of the densities at an appropriate “initial” time
is not available here. Fortunately, this problem is nowadays well understood
and we refer the reader to [14, Section 3, Theorem 3.1] for a detailed proof.

The next section is devoted to the proof of Theorem 3.1. In view of the
hypotheses (2.10), (2.12),

$p(\varrho,\vartheta)\approx\varrho^{\frac{5}{3}}+\vartheta^{4}.$

As already pointed out, the exponent $\gamma=\frac{5}{3}$ is critical in the
simplified isentropic case. To handle this problem we use the fact that (i)
the gravitational force acting on the fluid is of potential type, and (ii) the
entropy satisfies the Third law of thermodynamics, notably (2.15).

##  4 Dissipativity

Our goal is to prove Theorem 3.1. Suppose that we are given a global–in–time
solution $(\varrho,\vartheta,{\bf u})$ defined on a time interval
$(T,\infty)$. The proof of asymptotic boundedness leans on several estimates
that follow from the basic physical conservation laws. Here and hereafter, we
fix $\tilde{\vartheta}$ to be the unique solution of the Dirichlet problem

$\Delta_{x}\tilde{\vartheta}=0\ \mbox{in}\ \Omega,\
\tilde{\vartheta}|_{x_{3}=0}=\Theta_{\rm B},\
\tilde{\vartheta}|_{x_{3}=1}=\Theta_{\rm U}.$ (4.1)

As $\Theta_{\rm B}$, $\Theta_{\rm U}$ are constant, we easily compute

$\tilde{\vartheta}=\tilde{\vartheta}(x_{3})=\Theta_{\rm
B}+x_{3}\left({\Theta_{\rm U}}-\Theta_{\rm B}\right).$

Obviously, the same ansatz can be used in the case of general $x-$dependent
boundary data.

###  4.1 Mass conservation

It follows from the equation of continuity (2.4) that the total mass of the
fluid is a constant of motion,

$M=\int_{\Omega}\varrho(t,\cdot)\ \,{\rm d}{x}\ \mbox{for any}\ t>T.$ (4.2)

In addition, as the volume force is potential, we can write

$\int_{\Omega}\varrho{\bf u}\cdot\nabla_{x}G\ \,{\rm d}{x}=\frac{{\rm
d}}{\,{\rm d}t}\int_{\Omega}\varrho G\ \,{\rm d}{x},\ G=-x_{3}.$

Consequently, the ballistic energy balance (2.8) takes the form

$\displaystyle\frac{{\rm d}}{\,{\rm d}t}$
$\displaystyle\int_{\Omega}\left[\frac{1}{2}\varrho|{\bf u}|^{2}+\varrho
e-\tilde{\vartheta}\varrho s-\varrho G\right]\ \,{\rm
d}{x}+\int_{\Omega}\frac{\tilde{\vartheta}}{\vartheta}\left[\mathbb{S}:\mathbb{D}_{x}{\bf
u}-\frac{{\bf q}\cdot\nabla_{x}\vartheta}{\vartheta}\right]\ \,{\rm d}{x}$
$\displaystyle\leq-\int_{\Omega}\left[\varrho s{\bf
u}\cdot\nabla_{x}\tilde{\vartheta}+\frac{{\bf
q}}{\vartheta}\cdot\nabla_{x}\tilde{\vartheta}\right]\ \,{\rm d}{x}\
\mbox{in}\ \mathcal{D}^{\prime}(T,\infty).$ (4.3)

It is worth–noting that the same argument applies for a general Lipschitz
potential $G=G(x)$.

###  4.2 Coercivity of the dissipative term

It follows from the hypotheses (2.17) and Korn–Poincaré inequality that

$\displaystyle\int_{\Omega}\frac{\tilde{\vartheta}}{\vartheta}\left(\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\mathbb{D}_{x}{\bf u}-\frac{{\bf
q}(\vartheta,\nabla_{x}\vartheta)\cdot\nabla_{x}\vartheta}{\vartheta}\right)\
\,{\rm d}{x}$ $\displaystyle\quad\geq c\inf\\{\Theta_{\rm U},\Theta_{\rm
B}\\}\left(\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}+\|\nabla_{x}\vartheta^{\frac{\beta}{2}}\|^{2}_{L^{2}(\Omega;R^{3})}+\|\nabla_{x}\log(\vartheta)\|^{2}_{L^{2}(\Omega;R^{3})}\right).$

Consequently, adding the boundary integrals to the left–hand side and using
Poincaré inequality, we get

$\displaystyle\left(\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}+\|\vartheta^{\frac{\beta}{2}}\|^{2}_{W^{1,2}(\Omega)}+\|\log(\vartheta)\|^{2}_{W^{1,2}(\Omega)}\right)$
$\displaystyle\quad\leq c(\Theta_{\rm U},\Theta_{\rm
B})\left[1+\int_{\Omega}\frac{\tilde{\vartheta}}{\vartheta}\left(\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\mathbb{D}_{x}{\bf u}-\frac{{\bf
q}(\vartheta,\nabla_{x}\vartheta)\cdot\nabla_{x}\vartheta}{\vartheta}\right)\
\,{\rm d}{x}\right]$ (4.4)

###  4.3 Energy estimates

To simplify the ballistic energy inequality (4.3), we first realize, by virtue
of (4.1),

$\int_{\Omega}\frac{{\bf
q}(\vartheta,\nabla_{x}\vartheta)}{\vartheta}\cdot\nabla_{x}\tilde{\vartheta}\
\,{\rm
d}{x}=-\int_{\Omega}\frac{\kappa(\vartheta)}{\vartheta}\nabla_{x}\vartheta\cdot\nabla_{x}\tilde{\vartheta}\
\,{\rm
d}{x}=\int_{\partial\Omega}\mathcal{K}(\tilde{\vartheta})\nabla_{x}\tilde{\vartheta}\
{\rm d}\sigma_{x},$

where

$\mathcal{K}^{\prime}(\vartheta)=\kappa(\vartheta).$

Consequently, the ballistic energy inequality (4.3) reduces to

$\displaystyle\frac{{\rm d}}{\,{\rm d}t}$
$\displaystyle\int_{\Omega}\left(\frac{1}{2}\varrho|{\bf u}|^{2}+\varrho
e-\tilde{\vartheta}\varrho s-\varrho G\right)\ \,{\rm
d}{x}+\int_{\Omega}\frac{\tilde{\vartheta}}{\vartheta}\left(\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\mathbb{D}_{x}{\bf u}-\frac{{\bf
q}(\vartheta,\nabla_{x}\vartheta)\cdot\nabla_{x}\vartheta}{\vartheta}\right)\
\,{\rm d}{x}$ $\displaystyle\leq-\int_{\Omega}\left[\varrho s{\bf
u}\cdot\nabla_{x}\tilde{\vartheta}\right]\ \,{\rm d}{x}+c(\Theta_{\rm
U},\Theta_{\rm B}).$ (4.5)

###  4.4 Entropy estimates

In accordance with hypothesis (2.13),

$\int_{\Omega}\varrho s{\bf u}\cdot\nabla\tilde{\vartheta}\ \,{\rm
d}{x}=\int_{\Omega}\varrho\mathcal{S}\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right){\bf
u}\cdot\nabla_{x}\tilde{\vartheta}\ \,{\rm
d}{x}+\frac{4a}{3}\int_{\Omega}\vartheta^{3}{\bf
u}\cdot\nabla\tilde{\vartheta}\ \,{\rm d}{x},$

where, by virtue of (2.14), (2.15),

$\varrho\mathcal{S}\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)\leq\varrho\mathcal{S}(r)\
\mbox{provided}\ \frac{\varrho}{\vartheta^{\frac{3}{2}}}\geq r,\ \mbox{where}\
\mathcal{S}(r)\to 0\ \mbox{as}\ r\to\infty.$ (4.6)

If

$\frac{\varrho}{\vartheta^{\frac{3}{2}}}<r\ \mbox{or}\
\varrho<r\vartheta^{\frac{3}{2}},$

we get, by virtue of (2.16),

$0\leq\varrho\mathcal{S}\left(\frac{\varrho}{\vartheta^{\frac{3}{2}}}\right)\leq
c\left(1+r\vartheta^{\frac{3}{2}}\left[\log^{+}(r\vartheta^{\frac{3}{2}})+\log^{+}(\vartheta)\right]\right).$
(4.7)

As $\beta>6$ in hypothesis (2.17), we may combine (4.4) with (4.5)–(4.7) to
obtain

$\displaystyle\frac{{\rm d}}{\,{\rm d}t}$
$\displaystyle\int_{\Omega}\left(\frac{1}{2}\varrho|{\bf u}|^{2}+\varrho
e-\tilde{\vartheta}\varrho s-\varrho G\right)\ \,{\rm d}{x}$
$\displaystyle+c_{1}(\Theta_{\rm U},\Theta_{\rm B})\left(\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}+\|\vartheta^{\frac{\beta}{2}}\|^{2}_{W^{1,2}(\Omega)}+\|\log(\vartheta)\|^{2}_{W^{1,2}(\Omega)}\right)$
$\displaystyle\leq c_{2}(\Theta_{\rm U},\Theta_{\rm
B})\mathcal{S}(r)\int_{\Omega}\varrho|{\bf u}|\ \,{\rm
d}{x}+\Lambda(\Theta_{\rm U},\Theta_{\rm B},r),$
$\displaystyle\quad\quad\mbox{where}\ c_{1}>0,\ c_{2}>0,\ \mbox{and}\
\Lambda(r)\to\infty\ \mbox{if}\ r\to\infty.$ (4.8)

The main problem to conclude is the fact that forcing term
$\int_{\Omega}\varrho|{\bf u}|\ \,{\rm d}{x}$ on the right–hand side is not
directly controlled by the dissipation on the left–hand side. To this end, we
need the so–called pressure estimates which we recall in the next section.

###  4.5 Pressure estimates

To continue, we recall the inverse of the divergence known as Bogovskii
operator:

$\displaystyle\mathcal{B}:L^{q}_{0}(\Omega)$ $\displaystyle\equiv\left\\{f\in
L^{q}(\Omega)\ \Big{|}\ \int_{\Omega}f\ \,{\rm d}{x}=0\right\\}\to
W^{1,q}_{0}(\Omega,R^{d}),\ 1<q<\infty,$ $\displaystyle{\rm
div}_{x}\mathcal{B}[f]$ $\displaystyle=f,$ $\displaystyle\|\mathcal{B}[{\rm
div}_{x}{\bf g}]\|_{L^{r}(\Omega)}$ $\displaystyle\leq c\|{\bf
g}\|_{L^{r}(\Omega)},\ 1<r<\infty\ \mbox{whenever}\ {\bf g}\cdot{\bf
n}|_{\partial\Omega}=0.$ (4.9)

see e.g. Galdi [25, Chapter 3] or Geißert, Heck, and Hieber[26].

Now, the test function

$\varphi(t,x)=\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]$

in the momentum equation yields

$\displaystyle\int_{\tau}^{\tau+1}$
$\displaystyle\int_{\Omega}p(\varrho,\vartheta)b(\varrho)\ \,{\rm d}{x}\,{\rm
d}t=\int_{\tau}^{\tau+1}\frac{1}{|\Omega|}\left(\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right)\left(\int_{\Omega}p(\varrho,\vartheta)\ \,{\rm
d}{x}\right)\,{\rm d}t$
$\displaystyle-\int_{\tau}^{\tau+1}\int_{\Omega}\varrho({\bf u}\otimes{\bf
u}):\nabla_{x}\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\int_{\tau}^{\tau+1}\int_{\Omega}\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\nabla_{x}\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle-\int_{\tau}^{\tau+1}\int_{\Omega}\varrho\nabla_{x}G\cdot\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\left[\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\right]_{t=\tau}^{t=\tau+1}$
$\displaystyle-\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\partial_{t}\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t.$ (4.10)

In addition, as $\varrho$ satisfies the renormalized equation of continuity,
we obtain

$\displaystyle\int_{\tau}^{\tau+1}$ $\displaystyle\int_{\Omega}\varrho{\bf
u}\cdot\partial_{t}\mathcal{B}\left[b(\varrho)-\frac{1}{|\Omega|}\int_{\Omega}b(\varrho)\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle=-\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}[{\rm div}_{x}(b(\varrho){\bf u}]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[(b(\varrho)-b^{\prime}(\varrho)\varrho){\rm
div}_{x}{\bf
u}-\frac{1}{|\Omega|}\int_{\Omega}(b(\varrho)-b^{\prime}(\varrho)\varrho){\rm
div}_{x}{\bf u}\ \,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t.$ (4.11)

The unit length of the time interval has been chosen just for simplicity.

###  4.6 Uniform bounds

In view of the structural restrictions imposed through hypotheses (2.11),
(2.13) and (2.16), for any $\lambda>1$ there exist two constants
$c_{1}(\lambda)$, $c_{2}(\lambda)$ such that

$c_{1}(\lambda,{\rm data})+\frac{1}{\lambda}E(\varrho,\vartheta,{\bf u})\leq
E_{\tilde{\vartheta}}(\varrho,\vartheta,{\bf u})-\varrho G\leq\lambda
E(\varrho,\vartheta,{\bf u})+c_{2}(\lambda,{\rm data}).$ (4.12)

The following result is crucial for showing the existence of a bounded
absorbing set.

######  Lemma 4.1.

Suppose that

$\int_{\Omega}\left[E_{\tilde{\vartheta}}(\tau,\cdot)-\varrho(\tau,\cdot)G\right]\
\,{\rm
d}{x}-\int_{\Omega}\left[E_{\tilde{\vartheta}}(\tau+1,\cdot)-\varrho(\tau+1,\cdot)G\right]\
\,{\rm d}{x}\leq K.$ (4.13)

Then there exists $L=L(K,M,{\rm data})$ such that

${\rm ess}\sup_{\tau\leq t\leq\tau+1}\int_{\Omega}E(t,\cdot)\ \,{\rm d}{x}\leq
L.$ (4.14)

######  Remark 4.2.

Strictly speaking, the pointwise values of the ballistic energy appearing in
(4.13) are defined only for a.a. $\tau\in(T;\infty)$. However, thanks to the
inequality (4.8), we may identify $E_{\tilde{\vartheta}}$ with its, say,
càglàd representative defined for any $\tau>T$.

The rest of this subsection is devoted to the proof of Lemma 4.1. If (4.13)
holds, it follows from (4.8) that

$\displaystyle\int_{\tau}^{\tau+1}$ $\displaystyle\left(\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}+\|\vartheta^{\frac{\beta}{2}}\|_{W^{1,2}(\Omega)}+\|\log(\vartheta)\|^{2}_{W^{1,2}(\Omega)}\right)\,{\rm
d}t$ $\displaystyle\leq c({\rm
data},K)\left(1+\mathcal{S}(r)\int_{\tau}^{\tau+1}\int_{\Omega}\varrho|{\bf
u}|\ \,{\rm d}{x}\,{\rm d}t\right)+\Lambda({\rm data},K,r).$ (4.15)

####  4.6.1 Pressure estimates revisited

At this stage, we use the pressure estimates (4.10), with

$b(\varrho)=\varrho^{\alpha},\ \alpha>0.$

In view of the hypotheses (2.10), (2.12) imposed on the equation of state, we
have

$c_{1}\Big{(}\varrho^{\frac{5}{3}}+\vartheta^{4}\Big{)}\leq
p(\varrho,\vartheta)\leq
c_{2}\Big{(}\varrho^{\frac{5}{3}}+\vartheta^{4}+1\Big{)},\ c_{1},c_{2}>0.$
(4.16)

Moreover, as the total mass is constant via (4.2), the smoothing properties of
$\mathcal{B}$ stated in (4.9) imply

$\left|\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\right|\leq c(M)\ \mbox{as soon as}\ \alpha<\frac{1}{3}$

to provide that $W^{1,\frac{1}{\alpha}}(\Omega)\subset L^{\infty}(\Omega)$.
Thus inequality (4.10) gives rise to

$\displaystyle\int_{\tau}^{\tau+1}$
$\displaystyle\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\ \,{\rm d}{x}\,{\rm
d}t\leq c(M)\left(1+\int_{\tau}^{\tau+1}\int_{\Omega}\vartheta^{4}\ \,{\rm
d}{x}\,{\rm d}t\right.$
$\displaystyle-\int_{\tau}^{\tau+1}\int_{\Omega}\varrho({\bf u}\otimes{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\int_{\tau}^{\tau+1}\int_{\Omega}\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\left[\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\right]_{t=\tau}^{t=\tau+1}$
$\displaystyle-\left.\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\partial_{t}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t\right).$ (4.17)

Next, using again the smoothing properties (4.9) of $\mathcal{B}$ we get

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho({\bf
u}\otimes{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t\right|$
$\displaystyle\quad\leq\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\gamma}(\Omega)}\|{\bf
u}\|^{2}_{L^{6}(\Omega;R^{3})}\|\varrho^{\alpha}\|_{L^{q}(\Omega)}\,{\rm d}t$
$\displaystyle\quad\leq\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\gamma}(\Omega)}\int_{\tau}^{\tau+T}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\sup_{t\in(\tau,\tau+1)}\|\varrho^{\alpha}\|_{L^{q}(\Omega)}\,{\rm
d}t,$ (4.18)

where

$q=\frac{3\gamma}{2\gamma-3}>1\ \mbox{provided}\ \gamma>\frac{3}{2}.$

Thus setting

$\gamma=\frac{5}{3},\
\alpha=\frac{2\gamma-3}{3\gamma}=\frac{1}{15}<\frac{1}{3},$ (4.19)

we may use the total mass conservation (4.2) to conclude

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho({\bf
u}\otimes{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t\right|$ $\displaystyle\quad\leq
c(M)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t,$ (4.20)

Similarly, going back to (4.11) we have

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}[{\rm div}_{x}(\varrho^{\alpha}{\bf u})]\ \,{\rm
d}{x}\right|\leq\int_{0}^{\tau+1}\|\varrho\|_{L^{\gamma}(\Omega)}\|{\bf
u}\|_{L^{6}(\Omega;R^{3})}\|\varrho^{\alpha}{\bf
u}\|_{L^{q}(\Omega;R^{3})}\,{\rm d}t,$

where

$\frac{1}{\gamma}+\frac{1}{6}+\frac{1}{q}=1.$

Moreover,

$\|\varrho^{\alpha}{\bf u}\|_{L^{q}(\Omega;R^{3})}\leq\|{\bf
u}\|_{L^{6}(\Omega;R^{3})}\|\varrho^{\alpha}\|_{L^{p}(\Omega)},\mbox{where}\
\frac{1}{q}=\frac{1}{6}+\frac{1}{p};$

whence

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}[{\rm div}_{x}(\varrho^{\alpha}{\bf u})]\ \,{\rm
d}{x}\right|\leq
c(M)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t$ (4.21)

as soon as (4.19) holds.

Finally,

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[\varrho^{\alpha}{\rm div}_{x}{\bf
u}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}{\rm div}_{x}{\bf u}\ \,{\rm
d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t\right|$
$\displaystyle\quad\leq\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\gamma}(\Omega)}\|{\bf
u}\|_{L^{6}(\Omega;R^{3})}\left\|\mathcal{B}\left[\varrho^{\alpha}{\rm
div}_{x}{\bf u}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}{\rm
div}_{x}{\bf u}\ \,{\rm d}{x}\right]\right\|_{L^{q}(\Omega;R^{3})}\,{\rm d}t,$

where

$\frac{1}{\gamma}+\frac{1}{6}+\frac{1}{q}=1.$

Furthermore,

$\left\|\mathcal{B}\left[\varrho^{\alpha}{\rm div}_{x}{\bf
u}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}{\rm div}_{x}{\bf u}\ \,{\rm
d}{x}\right]\right\|_{L^{q}(\Omega;R^{3})}\stackrel{{\scriptstyle<}}{{\sim}}\|\varrho^{\alpha}{\rm
div}_{x}{\bf u}\|_{L^{r}(\Omega;R^{3})},\ q=\frac{3r}{3-r},$

and

$\|\varrho^{\alpha}{\rm div}_{x}{\bf u}\|_{L^{r}(\Omega;R^{3})}\leq\|{\bf
u}\|_{W^{1,2}(\Omega;R^{3})}\|\varrho^{\alpha}\|_{L^{p}(\Omega)},\
\mbox{with}\ \frac{1}{2}+\frac{1}{p}=\frac{1}{r}.$

Consequently, condition (4.19) yields

$\displaystyle\left|\int_{\tau}^{\tau+1}\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[\varrho^{\alpha}{\rm div}_{x}{\bf
u}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}{\rm div}_{x}{\bf u}\ \,{\rm
d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t\right|$ $\displaystyle\quad\leq
c(M)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t.$ (4.22)

Summing up the previous inequalities and going back to (4.17), we get

$\displaystyle\int_{\tau}^{\tau+1}$
$\displaystyle\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\ \,{\rm d}{x}\,{\rm
d}t\leq c(M)\left(1+\int_{\tau}^{\tau+1}\int_{\Omega}\vartheta^{4}\ \,{\rm
d}{x}\,{\rm d}t\right.$
$\displaystyle+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t$
$\displaystyle+\int_{\tau}^{\tau+T}\int_{\Omega}\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\,{\rm d}t$
$\displaystyle+\left.\left[\int_{\Omega}\varrho{\bf
u}\cdot\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}\right]_{t=\tau}^{t=\tau+1}\right),\
\alpha=\frac{1}{15}.$ (4.23)

Now,

$\displaystyle\int_{\Omega}\mathbb{S}(\vartheta,\mathbb{D}_{x}{\bf
u}):\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\ \,{\rm d}{x}$
$\displaystyle\quad\leq(1+\|\vartheta\|_{L^{4}(\Omega)})\|{\bf
u}\|_{W^{1,2}(\Omega;R^{3})}\left\|\nabla_{x}\mathcal{B}\left[\varrho^{\alpha}-\frac{1}{|\Omega|}\int_{\Omega}\varrho^{\alpha}\
\,{\rm d}{x}\right]\right\|_{L^{4}(\Omega;R^{3})}$ $\displaystyle\quad\leq
c(M)(1+\|\vartheta\|_{L^{4}(\Omega)})\|{\bf u}\|_{W^{1,2}(\Omega;R^{3})}.$

We therefore conclude

$\displaystyle\int_{\tau}^{\tau+1}$
$\displaystyle\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\ \,{\rm d}{x}\,{\rm
d}t\leq c(M)\left[1+\int_{\tau}^{\tau+1}\int_{\Omega}\vartheta^{4}\ \,{\rm
d}{x}\,{\rm d}t\right.$
$\displaystyle+\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm
d}t+\left.\sup_{t\in(\tau,\tau+1)}\int_{\Omega}\varrho|{\bf u}|\ \,{\rm
d}{x}\right],\ \alpha=\frac{1}{15}.$ (4.24)

####  4.6.2 Proof of Lemma 4.1

Now, in accordance with (4.15),

$\displaystyle\int_{\tau}^{\tau+1}\int_{\Omega}\vartheta^{4}\ \,{\rm d}{x}$
$\displaystyle\leq c({\rm
data})\left(1+\int_{\tau}^{\tau+1}\|\vartheta^{\frac{\beta}{2}}\|^{2}_{W^{1,2}(\Omega)}\,{\rm
d}t\right)$ $\displaystyle\leq c({\rm
data},K)\left(1+\mathcal{S}(r)\int_{\tau}^{\tau+1}\int_{\Omega}\varrho|{\bf
u}|\ \,{\rm d}{x}\,{\rm d}t\right)+\Lambda({\rm data},K,r),$

where we may fix $r=1$. Consequently, inequality (4.24) reduces to

$\displaystyle\int_{\tau}^{\tau+1}\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\
\,{\rm d}{x}\,{\rm d}t$ $\displaystyle\leq c(K,M,{\rm
data})\left[\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t\right.$
$\displaystyle+\left.\sup_{t\in(\tau,\tau+1)}\int_{\Omega}\varrho|{\bf u}|\
\,{\rm d}{x}+1\right],\ \alpha=\frac{1}{15}.$ (4.25)

Next, it follows from the hypotheses (4.13), (4.15), and (4.8) that

$\sup_{t\in(\tau,\tau+1)}\int_{\Omega}E(t,\cdot)\ \,{\rm d}{x}\leq c({\rm
data})\left(1+\int_{\tau}^{\tau+1}E(s,\cdot){\rm d}s\right).$ (4.26)

Moreover, relation (4.15) yields

$\int_{\tau}^{\tau+1}\|{\bf u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\leq c({\rm
data},K)\mathcal{S}(r)\int_{\tau}^{\tau+1}\int_{\Omega}\varrho|{\bf u}|\
\,{\rm d}{x}\,{\rm d}t+\Lambda({\rm data},K,r),$

where, by Hölder’s inequality and Sobolev embedding theorem,

$\int_{\Omega}\varrho|{\bf u}|\ \,{\rm
d}{x}\leq\|\sqrt{\varrho}\|_{L^{2}(\Omega)}\|\sqrt{\varrho}\|_{L^{3}(\Omega)}\|{\bf
u}\|_{L^{6}(\Omega;R^{3})}\leq
c\sqrt{M}\|\sqrt{\varrho}\|_{L^{3}(\Omega)}\|{\bf
u}\|_{W^{1,2}(\Omega;R^{3})}.$

Thus we may infer that

$\int_{\tau}^{\tau+1}\|{\bf u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\leq c({\rm
data},K,M)\mathcal{S}(r)\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}+\Lambda({\rm
data},K,r).$ (4.27)

Finally, using (4.27) we may estimate the kinetic energy,

$\displaystyle\int_{\tau}^{\tau+1}$ $\displaystyle\int_{\Omega}\varrho|{\bf
u}|^{2}\ \,{\rm d}{x}\,{\rm
d}t\leq\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{L^{6}(\Omega;R^{3})}\,{\rm d}t$ $\displaystyle\leq
c\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\int_{\tau}^{\tau+1}\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}\,{\rm d}t$ $\displaystyle\leq\Lambda({\rm
data},K,r)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}+c({\rm
data},K,M)\mathcal{S}(r)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\,{\rm
d}t.$

Now, by interpolation,

$\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\leq\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}^{\frac{5}{6}}\|\varrho\|_{L^{1}(\Omega)}^{\frac{1}{6}};$

whence

$\displaystyle\int_{\tau}^{\tau+1}\int_{\Omega}\varrho|{\bf u}|^{2}\ \,{\rm
d}{x}$ $\displaystyle\quad\leq\Lambda({\rm
data},K,M,r)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}^{\frac{5}{6}}+c({\rm
data},K,M)\mathcal{S}(r)\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}^{\frac{5}{6}}\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}^{\frac{5}{6}}\,{\rm
d}t.$ (4.28)

Going back to (4.25) and using (4.27) we get

$\displaystyle\int_{\tau}^{\tau+1}\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\
\,{\rm d}{x}\,{\rm d}t$ $\displaystyle\leq\Lambda(K,M,{\rm
data},r)\left[\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\,{\rm
d}t\right.$ $\displaystyle+c(K,M,{\rm
data})\mathcal{S}(r)\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\frac{3}{2}}(\Omega)}\,{\rm
d}t$ $\displaystyle+\left.\sup_{t\in(\tau,\tau+1)}\int_{\Omega}\varrho|{\bf
u}|\ \,{\rm d}{x}+1\right]$ $\displaystyle\leq\Lambda(K,M,{\rm
data},r)\left[\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\,{\rm
d}t\right.$ $\displaystyle+c(K,M,{\rm
data})\mathcal{S}(r)\left(1+\sup_{t\in(\tau,\tau+1)}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}\right)\int_{\tau}^{\tau+1}\|\varrho\|_{L^{\frac{5}{3}}(\Omega)}^{\frac{5}{6}}\,{\rm
d}t$ $\displaystyle+\left.\sup_{t\in(\tau,\tau+1)}\int_{\Omega}\varrho|{\bf
u}|\ \,{\rm d}{x}+1\right],\ \alpha=\frac{1}{15}.$ (4.29)

Now, interpolating $L^{1}$ and $L^{\frac{5}{3}+\alpha}$, we get

$\int_{\tau}^{\tau+1}\int_{\Omega}\varrho^{\frac{5}{3}}\ \,{\rm d}{x}\,{\rm
d}t\leq
c(M)\left(\int_{\tau}^{\tau+1}\int_{\Omega}\varrho^{\frac{5}{3}+\alpha}\
\,{\rm d}{x}\,{\rm d}t\right)^{\frac{10}{11}}\ \mbox{provided}\
\alpha=\frac{1}{15}.$

Gathering the available bounds we conclude

$\displaystyle\sup_{t\in(\tau,\tau+1)}$ $\displaystyle\int_{\Omega}E(t,\cdot)\
\,{\rm d}{x}\leq c({\rm data})\left(1+\int_{\tau}^{\tau+1}E(s,\cdot){\rm
d}s\right)$ $\displaystyle\leq c({\rm data})$
$\displaystyle\left(1+\int_{\tau}^{\tau+1}\left(\|{\bf
u}\|^{2}_{W^{1,2}(\Omega;R^{3})}+\|\vartheta^{\frac{\beta}{2}}\|_{W^{1,2}(\Omega)}+\|\log(\vartheta)\|^{2}_{W^{1,2}(\Omega)}\right)\,{\rm
d}t\right)$ $\displaystyle+c({\rm data})$
$\displaystyle\left(\int_{\tau}^{\tau+1}\int_{\Omega}\varrho|{\bf u}|^{2}\
\,{\rm d}{x}\,{\rm d}t+\int_{\tau}^{\tau+1}\int_{\Omega}\varrho^{\frac{5}{3}}\
\,{\rm d}{x}\,{\rm d}t\right)$ $\displaystyle\leq\Lambda({\rm data},K,r)$
$\displaystyle\left[1+\left(\sup_{t\in(\tau,\tau+1)}\int_{\Omega}E\ \,{\rm
d}{x}\,{\rm d}t\right)^{\lambda}\right]+c({\rm
data},K,M)\mathcal{S}(r)\sup_{t\in(\tau,\tau+1)}\int_{\Omega}E\ \,{\rm
d}{x}\,{\rm d}t$ (4.30)

for certain $0<\lambda<1$. Consequently, choosing $r=r({\rm data},K,M)$ large
enough, the desired conclusion follows since $\mathcal{S}(r)\to 0$ as
$r\to\infty$.

We have proved Lemma 4.1.

###  4.7 Bounded absorbing sets

The existence of a bounded absorbing set follows easily from Lemma 4.1. Indeed
consider a global–in–time solution as in Theorem 3.1 satisfying

${\rm ess}\limsup_{t\to T-}\int_{\Omega}E(\varrho,\vartheta,{\bf u})(t,\cdot)\
\,{\rm d}{x}\leq\mathcal{E}_{0}.$

Consider $K=1$ in Lemma 4.1. In view of (4.12), there exists
$\tau=\tau(\mathcal{E}_{0})$ such that (4.12) holds. Repeating the same
argument with $\mathcal{E}_{0}$ replaced by $L$ given by (4.14) we deduce that
there exists $H=H(L)$ such that any time interval $(s,s+H)$,
$s\geq\tau(\mathcal{E}_{0})$ contains $\tau$ such that (4.13) holds yielding
(4.14). Consequently, we may consider

$\mathcal{E}_{\infty}\leq L\exp H(L).$

We have proved Theorem 3.1. As pointed out in Section 3, Theorem 3.1 yields
Theorem 3.3 via the existing arguments presented e.g. in [14].

##  5 Applications, long–time behavior

We finish the paper by discussing the impact of Theorems 3.1, 3.3 on the long
time behavior of solutions to the Rayleigh–Bénard problem in the framework of
compressible viscous and heat conducting fluids.

###  5.1 Trajectory space

We start by introducing a suitable _trajectory space_ $\mathcal{T}$. In view
of the framework of Theorems 3.1, 3.3, the “natural” trajectory space should
be based on the standard phase variables $(\varrho,\vartheta,{\bf u})$.
Unfortunately, neither $\vartheta$ nor ${\bf u}$ admit well defined
_instantaneous values_ at any time $t\in R$. It is therefore more convenient
to consider the _conservative entropy_ variables $(\varrho,S,{\bf m})$, with

$\mbox{momentum}\ {\bf m}=\varrho{\bf u},\ \mbox{and total entropy}\ S=\varrho
s(\varrho,\vartheta).$

On the one hand, the state variables $(\varrho,S,{\bf m})$ are uniquely
determined by $(\varrho,\vartheta,{\bf u})$. On the other hand, knowing
$(\varrho,S,{\bf m})$ we first obtain $\vartheta$ as $\vartheta\mapsto
s(\varrho,\vartheta)$ is a strictly increasing function. The velocity ${\bf
u}$ is a priori not well defined on the hypothetical vacuum zone, however, it
can be recovered in terms of $(\varrho,\vartheta,{\bf m})$ from the momentum
equation (2.6).

The phase variables $(\varrho,S,{\bf m})$ admit well defined instantaneous
values understood in the weak sense. Specifically, it follows from the weak
formulation (2.4), (2.6), (2.7) that the one sided limits

$\displaystyle\left<\varrho(\tau-,\cdot);\phi\right>$
$\displaystyle\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau-\delta}^{\tau}\int_{\Omega}\varrho(t,\cdot)\phi\
\,{\rm d}{x}\,{\rm d}t,\
\left<\varrho(\tau+,\cdot);\phi\right>\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}\varrho(t,\cdot)\phi\
\,{\rm d}{x}\,{\rm d}t$ $\displaystyle\left<{\bf
m}(\tau-,\cdot);\boldsymbol{\varphi}\right>$
$\displaystyle\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau-\delta}^{\tau}\int_{\Omega}{\bf
m}(t,\cdot)\cdot\boldsymbol{\varphi}\ \,{\rm d}{x}\,{\rm d}t,\ \left<{\bf
m}(\tau+,\cdot);\boldsymbol{\varphi}\right>\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}{\bf
m}(t,\cdot)\cdot\boldsymbol{\varphi}\ \,{\rm d}{x}\,{\rm d}t,$
$\displaystyle\left<S(\tau-,\cdot);\phi\right>$
$\displaystyle\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau-\delta}^{\tau}\int_{\Omega}\varrho
s(t,\cdot)\phi\ \,{\rm d}{x}\,{\rm d}t,\
\left<S(\tau+,\cdot);\phi\right>\equiv\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}\varrho
s(t,\cdot)\phi\ \,{\rm d}{x}\,{\rm d}t$

exist for _any_ $\tau\in R$ and any $\phi\in C^{1}_{c}(\Omega)$,
$\boldsymbol{\varphi}\in C^{1}_{c}(\Omega;R^{3})$. In addition,

$\displaystyle\left<\varrho(\tau-,\cdot);\phi\right>$
$\displaystyle=\left<\varrho(\tau+,\cdot);\phi\right>,\ \mbox{and}\
\tau\mapsto\left<\varrho(\tau,\cdot);\phi\right>\in BC(R),$
$\displaystyle\left<{\bf m}(\tau-,\cdot);\boldsymbol{\varphi}\right>$
$\displaystyle=\left<{\bf m}(\tau+,\cdot);\boldsymbol{\varphi}\right>,\
\mbox{and}\ \tau\mapsto\left<{\bf
m}(\tau,\cdot);\boldsymbol{\varphi}\right>\in BC(R),$

and

$\displaystyle\left<S(\tau-,\cdot);\phi\right>$
$\displaystyle\leq\left<S(\tau+,\cdot);\phi\right>\ \mbox{whenever}\ \phi\geq
0,$ $\displaystyle\mbox{and}\ \tau$
$\displaystyle\mapsto\left<S(\tau-,\cdot);\phi\right>=h_{\phi}(\tau)+g_{\phi}(\tau),\
g_{\phi}\in C_{\rm loc}(R),\ h_{\phi}\ \mbox{non--decreasing
c\\`{a}gl\\`{a}d.}$ (5.1)

The trajectory space can be therefore identified with “weakly càglàd” and
bounded functions defined on $R$. To this end, consider the Hilbert space

$W^{k,2}_{0}(\Omega),\ k>\frac{3}{2}\ \mbox{so that}\
W^{k,2}_{0}\hookrightarrow C(\overline{\Omega})$

with an orthonormal basis of smooth functions $\\{\phi_{n}\\}_{n=1}^{\infty}$.
Similarly, we consider the same space of vector valued functions
$W^{1,2}_{0}(\Omega;R^{3})$ with a basis
$\\{\boldsymbol{\varphi}_{n}\\}_{n=1}^{\infty}$. Finally, we define a metrics

$\displaystyle d_{\mathcal{T}}$ $\displaystyle\left[(\varrho^{1},S^{1},{\bf
m}^{1});(\varrho^{2},S^{2},{\bf m}^{2})\right]$
$\displaystyle=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)G\left(\|\left<\varrho^{1}-\varrho^{2};\phi_{n}\right>\|_{C[-t,t]}\right)\,{\rm
d}t$
$\displaystyle+\sum_{n=1}^{\infty}\frac{1}{2^{n}}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)G\left(\|\left<{\bf
m}^{1}-{\bf
m}^{2};\boldsymbol{\varphi}_{n}\right>\|_{C([-t,t];R^{3})}\right)\,{\rm d}t$
$\displaystyle=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)G\left(\left[\left<S^{1};\phi_{n}\right>;\left<S^{2};\phi_{n}\right>\right]_{D[-t,t]}\right)\,{\rm
d}t,$ (5.2)

where

$G(Z)=\frac{Z}{1+Z}$

and $D[-t,t]$ denotes the Skorokhod space of of càglàd functions defined on
$[-t,t]$ with the associated complete metrics $[\cdot;\cdot]_{D[-t,t]}$, see
e.g. Whitt [38].

The trajectory space is defined as

$\mathcal{T}=\cup_{L=1}^{\infty}\mathcal{T}_{L},$

where

$\displaystyle\mathcal{T}_{L}=\Big{\\{}(\varrho,S,{\bf m})\ \Big{|}\ $
$\displaystyle\varrho\in L^{\infty}(R;W^{-k,2}(\Omega)),\
\left<\varrho;\phi_{n}\right>\in C(R),\ n=1,2,\dots,$ $\displaystyle\sup_{t\in
R}\|\varrho(t,\cdot)\|_{W^{-k,2}(\Omega)}\leq L,$ $\displaystyle{\bf m}\in
L^{\infty}(R;W^{-k,2}(\Omega;R^{3})),\ \left<{\bf
m};\boldsymbol{\varphi}_{n}\right>\in C(R),\ n=1,2,\dots,$
$\displaystyle\sup_{t\in R}\|{\bf m}(t,\cdot)\|_{W^{-k,2}(\Omega;R^{3})}\leq
L,$ $\displaystyle S\in L^{\infty}(R;W^{-k,2}(\Omega)),\
\left<S;\phi_{n}\right>\ \mbox{c\\`{a}gl\\`{a}d in}\ R,\ n=1,2,\dots,$
$\displaystyle\sup_{t\in R}\|S(t,\cdot)\|_{W^{-k,2}(\Omega)}\leq L\Big{\\}}.$

Note that the trajectory space is larger then the set of entire solutions to
the Navier–Stokes–Fourier system and consists of time dependent functionals
ranging in the space of distributions on $\Omega$.

Each set $\mathcal{T}_{L}$ endowed with the metrics $d_{\mathcal{T}}$ is a
Polish space. We define inductive topology on $\mathcal{T}$:

$\displaystyle(\varrho_{n},S_{n},{\bf m}_{n})\to(\varrho,S,{\bf m})\ \
\mbox{in}\ \mathcal{T}\ \Leftrightarrow\ $ $\displaystyle\mbox{{\bf(i)} there
exists}\ L\ \mbox{such that}\ (\varrho_{n},S_{n},{\bf
m}_{n})\in\mathcal{T}_{L}\ \mbox{for all}\ n=1,2,\dots$
$\displaystyle\mbox{{\bf(ii)}}\ d_{\mathcal{T}}\left[(\varrho_{n},S_{n},{\bf
m}_{n});(\varrho,S,{\bf m})\right]\to 0.$

Our choice of the topology of the trajectory space may seem a bit awkward at
the first glance but accommodates the instantaneous convergence of the state
variables. Alternatively, a weaker $L^{p}$ topology can be used being
equivalent on the attractor $\mathcal{A}$ to $d_{\mathcal{T}}$.

###  5.2 Attractor

As the weak solutions are not (known to be) uniquely determined by the
initial/boundary data, we adopt the approach of Sell [35] and Málek and Nečas
[27] replacing the standard phase space by the trajectory space $\mathcal{T}$.
Here and hereafter, we always assume that the principal hypotheses of Theorem
3.1 concerning the constitutive relations are satisfied. Moreover, we fix the
total mass of the fluid,

$\int_{\Omega}\varrho(t,\cdot)\ \,{\rm d}{x}=M>0.$ (5.3)

Accordingly, the set $\mathcal{A}$,

$\displaystyle\mathcal{A}=\Big{\\{}$ $\displaystyle(\varrho,S,{\bf m})\
\Big{|}\ (\varrho,S,{\bf m})\ \mbox{a weak solution of the Navier--Stokes--
Fourier system}$ $\displaystyle\mbox{in the sense of Definition \ref{Dw1} on
the time interval}\ t\in R\Big{\\}},$ (5.4)

is a natural candidate to be _global attractor_ in the trajectory space
$\mathcal{T}$. Indeed, as shown in Theorem 3.1,

$\sup_{t\in R}\int_{\Omega}E\left(\varrho,S,{\bf m}\right)(t,\cdot)\ \,{\rm
d}{x}\leq\mathcal{E}_{\infty}<\infty.$ (5.5)

In particular, $\mathcal{A}\subset\mathcal{T}_{L}\subset\mathcal{T}$ for a
sufficiently large $L$.

######  Lemma 5.1.

Under the hypotheses of Theorem 3.1, the set $\mathcal{A}$ is

  * •

non–empty;

  * •

time–shift invariant,

$(\varrho,S,{\bf m})\in\mathcal{A}\ \Rightarrow\ (\varrho,S,{\bf
m})(\cdot+T)\in\mathcal{A}\ \mbox{for any}\ T\in R;$

  * •

compact in the metric topology $(\mathcal{T}_{L},d_{\mathcal{T}})$ for a
sufficiently large $L$.

###### Proof.

As shown in [9, Theorem 4.2], the Navier–Stokes–Fourier system with the
boundary conditions (1.5), (1.6) admits a global–in–time weak solution
$(\varrho,\vartheta,{\bf u})$ on the time interval $[0,\infty)$ for any
initial data with finite energy. It follows from Theorem 3.3 that there exists
a sequence of times $T_{n}\to\infty$ such that $((\varrho,\vartheta,{\bf
u})(\cdot+T_{n}))_{n=1}^{\infty}$ converge to a weak solution
$(\tilde{\varrho},\tilde{\vartheta},{\tilde{\bf u}})$ of the same problem
defined for all $t\in R$ with globally bounded energy. Obviously,

$(\varrho,S,{\bf
m})=(\tilde{\varrho},\tilde{\varrho}s(\tilde{\varrho},\tilde{\vartheta}),\tilde{\varrho}{\tilde{\bf
u}})\in\mathcal{A}.$

Moreover, as the underlying system is autonomous, the set $\mathcal{A}$ is
time–shift invariant.

Compactness of the set $\mathcal{A}$ follows again from Theorem 3.3, where we
consider $T_{n}=-\infty$. As pointed out, $\mathcal{A}\subset\mathcal{T}_{L}$,
where the latter is a metric space; whence compactness is equivalent to
sequential compactness. At the level of the density, convergence in the metric
$d_{\mathcal{T}}$ follows from (3.2). Moreover, since the momenta $\varrho{\bf
u}$ satisfy equation (2.6), they are precompact in the topology of the space

$C_{\rm weak,loc}(R;L^{\frac{5}{4}}(\Omega;R^{3})),$

which implies compactness in the momentum component of $\mathcal{T}_{L}$ with
the $d_{\mathcal{T}}$ metrics.

Thus it remains to show compactness at the level of the total entropy $S$.
First observe that

$\tau\mapsto\int_{\Omega}S(\tau,\cdot)\phi\ \,{\rm
d}{x}-\int_{0}^{\tau}\int_{\Omega}\varrho s(\varrho,\vartheta){\bf
u}\cdot\nabla_{x}\phi\ \,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\frac{\kappa(\vartheta)}{\vartheta}\cdot\nabla_{x}\phi\
\,{\rm d}{x}\,{\rm d}t$

is a non–decreasing function of $\tau$ for any test function $\phi\in
C^{1}_{c}(\Omega)$, $\phi\geq 0$. In view of boundedness of the total energy,
the family

$\tau\mapsto\int_{0}^{\tau}\int_{\Omega}\varrho s(\varrho,\vartheta){\bf
u}\cdot\nabla_{x}\phi\ \,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\frac{\kappa(\vartheta)}{\vartheta}\cdot\nabla_{x}\phi\
\,{\rm d}{x}\,{\rm d}t$

is precompact in $C_{\rm loc}(R)$; whence precompactness of $S$ in
$d_{\mathcal{T}}$ reduces to precompactness of a _non–decreasing (in time)_
sequence of functions

$\tau\mapsto\left<\widetilde{S}_{n};\phi\right>\equiv\int_{\Omega}S_{n}(\tau,\cdot)\phi\
\,{\rm
d}{x}-\int_{0}^{\tau}\int_{\Omega}\varrho_{n}s(\varrho_{n},\vartheta_{n}){\bf
u}_{n}\cdot\nabla_{x}\phi\ \,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\frac{\kappa(\vartheta_{n})}{\vartheta_{n}}\cdot\nabla_{x}\phi\
\,{\rm d}{x}\,{\rm d}t$

with respect to the metrics $d$ of the Skorokhod space $D[-M,M]$ of càglàd
functions defined on compact time intervals $[-M,M]$. To this end, we recall
the criterion due to Whitt [38, Chapter 12, Corollary 12.5.1]:

Let

$h_{n}:[-M,M]\to R$

be a sequence of monotone functions.

Then

$d[h_{n};h]\to 0\ \mbox{as}\ n\to\infty\ \mbox{for some}\ h\in D[-M,M]$

if and only if

$h_{n}(t)\to h(t)\ \mbox{for all}\ t\ \mbox{belonging to a dense set in}\
[-M,M]\ \mbox{including the end points}\ -M\ \mbox{and}\ M.$

Now, as $S_{n}$ are the total entropies generated by a family of uniformly
bounded weak solutions of the Navier–Stokes–Fourier system, we have

$\displaystyle\left<\widetilde{S}_{n}(\tau,\cdot);\phi\right>$
$\displaystyle\to\left<\widetilde{S}(\tau,\cdot);\phi\right>$
$\displaystyle\equiv\int_{\Omega}S(\tau,\cdot)\phi\ \,{\rm
d}{x}-\int_{0}^{\tau}\int_{\Omega}\varrho s(\varrho,\vartheta){\bf
u}\cdot\nabla_{x}\phi\ \,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\frac{\kappa(\vartheta)}{\vartheta}\cdot\nabla_{x}\phi\
\,{\rm d}{x}\,{\rm d}t$

for a.a. $\tau\in R$ at least for a suitable subsequence, where
$(\varrho,S=\varrho s(\varrho,\vartheta),{\bf m}=\varrho{\bf
u})\in\mathcal{A}$, is another entire solution of the same problem. In
particular,

$\left<\widetilde{S}_{n}(\tau,\cdot);\phi\right>\to\left<\widetilde{S}(\tau,\cdot);\phi\right>\
\mbox{for a dense set of times for any compact interval}\ [-M,M];$

whence, in accordance with the above convergence criterion,

$\left<\widetilde{S}_{n}(\tau,\cdot);\phi\right>\to\left<\widetilde{S}(\tau,\cdot);\phi\right>\
\mbox{in}\ D[-M,M]\ \mbox{for a.a.}\ M>0$

yielding the desired conclusion

$\left<S_{n}(\tau,\cdot);\phi\right>\to\left<S(\tau,\cdot);\phi\right>\
\mbox{in}\ D[-M,M]\ \mbox{for a.a.}\ M>0.$

∎

In accordance with Theorems 3.1, 3.3, and Lemma 5.1, we may state our main
result concerning the existence of a trajectory attractor for the
Rayleigh–Bénard problem:

######  Theorem 5.2 (Trajectory attractor).

Let $M>0$, $\mathcal{E}_{0}$ be given. Let $\mathcal{F}[M,\mathcal{E}_{0}]$ be
a family of weak solutions to the Rayleigh–Bénard problem for the
Navier–Stokes–Fourier system on the time interval $(0,\infty)$ satisfying
$\int_{\Omega}\varrho\ \,{\rm d}{x}=M,\ {\rm ess}\limsup_{\tau\to
0+}\int_{\Omega}E(\varrho,S,{\bf m})(\tau,\cdot)\ \,{\rm
d}{x}\leq\mathcal{E}_{0}.$ We identify the set
$\mathcal{F}[M,\mathcal{E}_{0}]$ with a subset of the trajectory space
$\mathcal{T}$ extending $\varrho(\tau,\cdot)=\lim_{t\to 0+}\varrho(t,\cdot),\
{\bf m}(\tau,\cdot)=\lim_{t\to 0+}{\bf m}(t,\cdot),\ 0\leq
S(\tau,\cdot)\leq\lim_{t\to 0+}S(t,\cdot)\ \mbox{for}\ \tau<0,$ where the
limits are understood in the weak (distributional) sense. Then for any
$\varepsilon>0$, there exists a time $T(\varepsilon)$ such that
$d_{\mathcal{T}}[(\varrho,S,{\bf m})(\cdot+T);\mathcal{A}]<\varepsilon\
\mbox{for any}\ (\varrho,S,{\bf m})\in\mathcal{F}[M,\mathcal{E}_{0}]\
\mbox{and any}\ T>T(\varepsilon).$

###  5.3 Stationary statistical solutions

Following the ideas of the preceding section we are ready to identify
statistical solutions with shift invariant probability measures on
${\mathcal{T}}$, which are supported by solutions to the Navier-Stokes-Fourier
system. In accordance with Theorem 5.2, these shift invariant probability
measures supported by the global trajectory attractor $\mathcal{A}$.

The construction of a bounded invariant measure is the same as in [14], note
that a similar approach in the incompressible setting was used by Foias, Rosa
and Temam [23], [24]. Given a trajectory $(\varrho,S,{\bf m})\in\mathcal{T}$
we consider a probability measure

$\mathcal{V}_{T}\equiv\frac{1}{T}\int_{0}^{T}\delta_{(\varrho,S,{\bf
m})(\cdot+t)}\,{\rm d}t,$

where $\delta$ denotes the Dirac mass. Obviously, $\mathcal{V}_{T}$ is a
probability measure. If, in addition,

$(\varrho,S,{\bf m})\in\mathcal{A},$

then $\mathcal{V}_{T}\in\mathfrak{P}(\mathcal{A})$, where
$\mathfrak{P}(\mathcal{A})$ denotes the set of all probablity measures on a
compact Polish space $\mathcal{A}$. In particular, the family

$\\{\mathcal{V}_{T}\\}_{T\geq 0}\ \mbox{is tight.}$

By Prokhorov theorem, there is a sequence $T_{n}\to\infty$ such that

$\mathcal{V}_{T_{n}}\to\mathcal{V}\ \mbox{narowly in}\
\mathfrak{P}(\mathcal{A}).$

Finally, exactly as in [14, Section 5.1], we may show that the measure
$\mathcal{V}$ is time–shift invariant, meaning

$\mathcal{V}[\mathfrak{B}(\cdot+T)]=\mathcal{V}[\mathfrak{B}]\ \mbox{for any
Borel set}\ \mathfrak{B}\subset\mathcal{T}.$ (5.6)

A Borel probability measure $\mathcal{V}\in\mathfrak{P}(\mathcal{A})$ enjoying
the property (5.6) is called _statistical stationary solution_ of the
Rayleigh–Bénard problem for the Navier–Stokes–Fourier system.

Finally, observe that the above construction may be restricted to any
shift–invariant subset $\mathcal{U}\subset\mathcal{A}$.

######  Theorem 5.3.

Let ${\mathcal{U}}\subset\mathcal{A}$ be a non-empty time–shift invariant set,
meaning $(\varrho,S,{\bf m})\in\mathcal{U}\ \Rightarrow\ (\varrho,S,{\bf
m})(\cdot+T)\in\mathcal{U}\ \mbox{for any}\ T\in R.$ Then there exists a
stationary statistical solution ${\mathcal{V}}$ supported by
$\overline{\mathcal{U}}$: • ${\mathcal{V}}$ is a Borel probability measure,
$\mathcal{V}\in\mathfrak{P}(\overline{\mathcal{U}})$; •
$\operatorname{supp}{\mathcal{V}}\subset\overline{\mathcal{U}}$, where the
closure of a ${\mathcal{U}}$ is a compact invariant set; • ${\mathcal{V}}$ is
shift invariant, i.e.,
${\mathcal{V}}[\mathfrak{B}]={\mathcal{V}}[{\mathfrak{B}(\cdot+T)}]$ for any
Borel set $\mathfrak{B}\subset{\mathcal{T}}$ and any $T\in R$.

###  5.4 Convergence of ergodic means

We conclude this section by a direct application of Birkhoff–Khinchin ergodic
theorem. Similarly to [14, Section 5], we may consider the state space

$H=W^{-k,2}(\Omega)\times W^{-k,2}(\Omega)\times W^{-k,2}(\Omega;R^{3}).$

Let $\mathcal{V}\in\mathfrak{P}(\mathcal{T})$ be a statistical stationary
solution, and thus a Borel probability measure. Consider a probability basis
$(\mathcal{T},\mathfrak{B}[\mathcal{T}],\mathcal{V})$ where $\mathcal{T}$ is
the trajectory space and $\mathfrak{B}[\mathcal{T}]$ is the family of Borel
sets. Given a trajectory $(\varrho,S,{\bf m})\in\mathcal{T}$ and $\tau\in R$
we may consider the associated canonical process

$(\varrho,S,{\bf m})\times\tau\mapsto(\varrho,S,{\bf m})(\tau,\cdot)\in H.$

As $\mathcal{V}$ is shift invariant, the above process is a stationary process
with respect to the probability basis
$(\mathcal{T},\mathfrak{B}[\mathcal{T}],\mathcal{V})$ defined for $\tau\in R$,
with càglàd paths ranging in $H$.

Exactly as in [14, Theorem 6.4, Section 6] we can establish the following
result.

######  Theorem 5.4 (Convergence of ergodic averages).

Let $\mathcal{V}$ be a stationary statistical solution and $(\varrho,S,{\bf
m})$ the associated stationary process. Let $F:H\to R$ be a Borel measurable
function such that $\int_{\mathcal{T}}|F(\varrho(0,\cdot),S(0,\cdot),{\bf
m}(0,\cdot)|\ {\rm d}\mathcal{V}<\infty.$ Then there exists a measurable
function $\overline{F}$, $\overline{F}:(\mathcal{T},\mathcal{V})\to R$ such
that $\frac{1}{T}\int_{0}^{T}F(\varrho(t,\cdot),S(t,\cdot),{\bf
m}(t,\cdot))\,{\rm d}t\to\overline{F}\ \mbox{as}\ T\to\infty$
$\mathcal{V}-$a.s. and in $L^{1}(\mathcal{T},\mathcal{V})$.

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