# On the Motion of a Pendulum with a Cavity
Filled with a Compressible Fluid

G. P. Galdi111University of Pittsburgh, email<EMAIL_ADDRESS>and V.
Mácha222Institute of Mathematics of the Academy of Sciences of the Czech
Republic, email<EMAIL_ADDRESS>and Š. Nečasová333Institute of Mathematics
of the Academy of Sciences of the Czech Republic, email<EMAIL_ADDRESS>and
B. She444Institute of Mathematics of the Academy of Sciences of the Czech
Republic and Department of Mathematical Analysis of the Charles University in
Prague, email<EMAIL_ADDRESS>
###### Abstract

We study the motion of the coupled system, $\mathscr{S}$, constituted by a
physical pendulum, $\mathscr{B}$, with an interior cavity entirely filled with
a viscous, compressible fluid, $\mathscr{F}$. The presence of the fluid may
strongly affect on the motion of $\mathscr{B}$. In fact, we prove that, under
appropriate assumptions, the fluid acts as a damper, namely, $\mathscr{S}$
must eventually reach a rest-state. Such a state is characterized by a
suitable time-independent density distribution of $\mathscr{F}$ and a
corresponding equilibrium position of the center of mass of $\mathscr{S}$.
These results are proved in the very general class of weak solutions and do
not require any restriction on the initial data, other than having a finite
energy. We complement our findings with some numerical tests. The latter show,
among other things, the interesting property that “large” compressibility
favors the damping effect, since it drastically reduces the time that
$\mathscr{S}$ takes to go to rest.

##  1 Introduction

The general problem of the motion of a rigid body with an interior, hollow
cavity entirely filled with a fluid has all along attracted the attention of
engineers and applied mathematicians. The list of major contributions only
would be too long to include here, and for this we refer the reader to the
monographs [3, 4] and the references therein.

One of the remarkable phenomena that motivated this study traces back to the
famous experiments of Lord Kelvin [29]. His tests unequivocally showed that
the presence of the fluid in the cavity substantially influences the motion of
the body by producing a significant stabilizing effect. Modern primary
applications of this distinctive property are, for example, liquid sloshing
dampers for vibration control of tall buildings [6] and oscillations
suppressors in spacecraft and artificial satellites [1].

In spite of its relevance, a rigorous and systematic mathematical analysis of
the motion of a body with a fluid-filled cavity has started only a few years
ago [5, 12, 15, 16, 17, 18, 19, 24, 25, 26]. These works have, on the one
hand, produced a full explanation of experimental observations and, on the
other hand, hinted at other, new interesting features that might be supported
by numerical or lab tests. In particular, a remarkable result proved in [15,
19] shows that, under certain conditions, the presence of fluid can even bring
the coupled system body-fluid to full rest.

At this point, it must be emphasized that in all the papers indicated above,
the fluid is supposed to be viscous and incompressible. Thus, more recently,
in [13, 14] we began to investigate the case where the fluid is still viscous
but compressible. This study has a two-fold motivation. In the first place, to
answer the natural question of the influence that compressibility may have on
the characteristics of the terminal state. Secondly, the mathematical
challenge constituted by the fact that, being the density no longer a
constant, a much richer set of terminal states may occur and, therefore, the
problem of their attainability can become of primary importance. In [13, 14],
we limited ourselves to the problem where the coupled system body-fluid,
$\mathscr{S}$, moves in absence of external forces (inertial motions). In
particular, we proved, that for “small” Mach numbers and for initial data of
restricted magnitude, the system will reach a terminal state where the body
rotates with constant angular velocity and the fluid is at rest with respect
to the body. Notice that this result is in sharp contrast with the analogous
one in absence of fluid, where the generic motion is a complicated motion a la
Poinsot. This shows, in particular, the stabilizing effect of the fluid
mentioned earlier on.

In the current article we begin to analyze the situation when the coupled
system $\mathscr{S}$ is subject to external forces. To this end, we have
chosen the classical example where the body is a physical pendulum whose
interior is filled up with a viscous barotropic fluid with a classical
constitutive law; see (2.9). Our main findings will be described next. In the
first place, we formulate the problem in the wide class of weak solutions,
namely, suitably renormalized, distributional solutions satisfying the “energy
inequality” and corresponding to initial data that are only requested to have
a finite energy; see Definition 2.1. Our objective is to investigate the
behavior of these solutions as time goes to infinity and determine all
possible terminal states. It comes then natural to consider the class of
steady-state solutions, $\mathscr{C}$, as significant candidates. We thus show
that, in such states, $\mathscr{S}$ must be at rest with a corresponding
(time-independent) distribution of fluid density compatible with the vanishing
of the axial component (that is, along the axis of rotation) of the total
angular momentum. These states represent all allowed equilibrium
configurations for $\mathscr{S}$ and are characterized by having their center
of mass, $C$, belonging to the vertical plane containing the axis of rotation;
see Theorem 3.1. However, unlike the incompressible case, there could be more
than two configurations of $\mathscr{S}$ that could furnish the same location
of $C$, due to the fact that the density of the fluid is not constant, thus
leading to the circumstance of multiple solutions; see Subsections 3.2 and
3.4. This fact makes the problem of attainability of steady-state solutions
more complicated, also due to the lack of uniqueness of weak solutions. In any
case, we are able to prove that, provided the cavity is convex, $\mathscr{C}$
is not empty, since it contains the non-empty class of minimizers of the total
energy; see Theorem 3.5. We then address the question of the asymptotic in
time behavior in the class of weak solutions. While their existence can be
obtained by a rather standard method (Theorem 4.1), their behavior for large
times requires some efforts, especially for the proof of appropriate
convergence of the pressure field; see Subsections 4.1 and 4.2. As a result,
we are able to show that every weak solution tends to a steady state
(equilibrium configuration), on condition that there is only one of them with
total energy not greater than that of the initial data; see Theorem 4.4. We
then check that this condition is certainly satisfied if $\mathscr{S}$
possesses suitable symmetry properties. Precisely, we prove that if the cavity
is a sphere with its center on the line passing through the center of mass of
the body and its projection on the rotation axis, then whenever $\mathscr{S}$
is released from rest and in any position other than the straight-down and
straight-up ones, it will eventually reach the equilibrium where its center of
mass in the straight-down position.

The above analytical findings are supported and complemented by several two-
dimensional numerical tests. Here the coupled system $\mathscr{S}$ consists of
two concentric circles $C_{1}$ and $C_{2}\subset C_{1}$, where
$C_{1}\backslash C_{2}$ is “the body” and $C_{2}$ the “cavity”. The objective
is to study the behavior in time of $\mathscr{S}$, for different values of the
physical quantities involved and, in particular, in the limit of very large
values of the gas parameter $a$, that is, small Mach number (incompressible
limit). The tests show, among other things, a surprising property, namely,
that compressibility acts in favor of stability. In other words, all other
parameters being fixed, $\mathscr{S}$ will reach the rest in a shorter time
for “large” $a$, rather than “small” $a$.

The plan of the paper is as follows. After formulating the problem in Section
2, including the definition of weak solution, in Section 2 we prove a
characterization (Subsection 3.1) and the existence (Subsection 3.3) of
steady-state solutions, along with some comments about their uniqueness
(Subsections 3.2 and 3.4). Successively, in Section 4, we study the large-time
behavior of weak solutions and prove there our main result on the
attainability of steady states. Finally, Section 5 is dedicated to the
numerical tests mentioned previously.

##  2 Formulation of the Problem

Let $\mathscr{B}$ be a finite rigid body, with an interior hollow cavity
${\cal C}$ filled with a viscous fluid. In mathematical terms, ${\cal C}$ is
an open simply connected domain of 3 completely surrounded by a domain
$\mathscr{B}$ in such a way that $\partial\mathcal{C}\subset\mathscr{B}$,
$\mathcal{C}\cap\mathscr{B}=\emptyset$, and $\mathcal{C}\cup\mathscr{B}$ is
bounded, simply connected and open.

The body $\mathscr{B}$ is constrained at all times to rotate around a
horizontal axis, ${\sf a}$, and we indicate by $O$ the orthogonal projection
of the center of mass $G$ of $\mathscr{B}$ on ${\sf a}$. Our objective is to
study the motion of coupled system body-fluid and, in particular, its behavior
for large times. To this end, let ${\cal F}=\\{O,{\mbox{\boldmath$e$}}_{i}\\}$
be the fixed (inertial) frame with ${\mbox{\boldmath$e$}}_{3}$ and
${\mbox{\boldmath$e$}}_{1}$ directed, respectively, along ${\sf a}$ and the
downward vertical, so that, indicating by $\widehat{{\mbox{\boldmath$g$}}}$
the acceleration of gravity, in the frame ${\cal F}$ we have

$\widehat{{\mbox{\boldmath$g$}}}=g\,{\mbox{\boldmath$e$}}_{1}\,,\ \
g=|\widehat{{\mbox{\boldmath$g$}}}|.$ (2.1)

Further, let $\mbox{\boldmath$\omega$}=\omega(t)\,{\mbox{\boldmath$e$}}_{3}$
be the angular velocity of $\mathscr{B}$, set

$\mathbb{A}(\omega):=\left(\begin{array}[]{ccc}0&-\omega&0\\\ \omega&0&0\\\
0&0&0\end{array}\right)$ (2.2)

and denote by $\mathbb{Q}=\mathbb{Q}(t)$, $t\geq 0$, the family of proper
orthogonal transformations solving the following IVP:

$\dot{\mathbb{Q}}=\mathbb{A}\cdot\mathbb{Q}\,,\ \
\mathbb{Q}(0)=\mathbb{Q}_{0}\,,$

with

$\mathbb{Q}_{0}=\left(\begin{array}[]{ccc}\cos\vartheta_{0}&-\sin\vartheta_{0}&0\\\
\sin\vartheta_{0}&\cos\vartheta_{0}&0\\\ 0&0&1\end{array}\right)\,,\ \
\mbox{some}\ \vartheta_{0}\in[0,2\pi).$

Putting

$\vartheta(t):=\int_{0}^{t}\omega(s){\rm d}s+\vartheta_{0}\,,\ \ t\geq 0\,,$

we obtain

$\mathbb{Q}(t)=\left(\begin{array}[]{ccc}\cos\vartheta(t)&-\sin\vartheta(t)&0\\\
\sin\vartheta(t)&\cos\vartheta(t)&0\\\ 0&0&1\end{array}\right)\,.$ (2.3)

Let $\mathscr{B}_{0}$, ${\cal C}_{0}$ be arbitrarily fixed reference
configurations of $\mathscr{B}$ and ${\cal C}$, respectively, and set, for all
$t\geq 0$,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt{\cal
C}(t):=\\{\mbox{\boldmath$y$}\in^{3}:\
\mbox{\boldmath$y$}=\mathbb{Q}(t)\cdot\mbox{\boldmath$x$},\
\mbox{\boldmath$x$}\in{\cal C}_{0}\\}\,,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\mathscr{B}(t):=\\{\mbox{\boldmath$y$}\in^{3}:\
\mbox{\boldmath$y$}=\mathbb{Q}(t)\cdot\mbox{\boldmath$x$},\
\mbox{\boldmath$x$}\in\mathscr{B}_{0}\\}\,,\\\
\mathcal{S}(t):=\mathscr{B}(t)\cup{\cal C}(t)\,.\end{array}$ (2.4)

Then, the equations of motion of the coupled system body-liquid in the frame
$\mathcal{F}$ (that is, in the $y$-variable) are given by [22]

$\begin{array}[]{cc}\vskip 3.0pt plus 1.0pt minus
1.0pt\left.\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\partial_{t}(r{\mbox{\boldmath$w$}})+\mbox{\rm
div}\,(r{\mbox{\boldmath$w$}}\otimes{\mbox{\boldmath$w$}})=\mbox{\rm
div}\,\widehat{\mathbb{S}}({\mbox{\boldmath$w$}})-\nabla
p(r)+r\,g\,{\mbox{\boldmath$e$}}_{1}\\\ \partial_{t}r+\mbox{\rm
div}\,(r{\mbox{\boldmath$w$}})=0\end{array}\right\\}\ \
(\mbox{\boldmath$y$},t)\in\cup_{t>0}\,{\cal C}(t)\times\\{t\\}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt{\mbox{\boldmath$w$}}=\omega(t){\mbox{\boldmath$e$}}_{3}\times\mbox{\boldmath$y$}\,\
\ (\mbox{\boldmath$y$},t)\in\cup_{t>0}\,\partial{\cal C}(t)\times\\{t\\}\\\
{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\left(J(t)\omega+{\mbox{\boldmath$e$}}_{3}\cdot{\displaystyle\int_{{\cal
C}(t)}}r\,\mbox{\boldmath$y$}\times{\mbox{\boldmath$w$}}\,{\rm
d}y\right)={\mbox{\boldmath$e$}}_{3}\cdot\left[\left({\displaystyle\int_{\mathcal{S}(t)}}\widehat{r}\,\mbox{\boldmath$y$}\,{\rm
d}y\right)\times g{\mbox{\boldmath$e$}}_{1}\right]\equiv
g{\displaystyle\int_{\mathcal{S}(t)}}\widehat{r}y_{2}\,{\rm d}y\,.\end{array}$
(2.5)

Here, $r=r(\mbox{\boldmath$y$},t)$,
${\mbox{\boldmath$w$}}={\mbox{\boldmath$w$}}(\mbox{\boldmath$y$},t)$ are
density and velocity fields of the fluid, while $\widehat{\mathbb{S}}$ is the
viscous part of Cauchy stress tensor. Moreover, denoting by
$r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})$ the density of $\mathscr{B}$, we
set

$\widehat{r}:=\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0ptr(\mbox{\boldmath$y$},t)&\ \mbox{if $y\in{\cal C}(t)$}\\\ \vskip 3.0pt
plus 1.0pt minus 1.0ptr_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})&\ \mbox{if
$y\in\mathscr{B}(t)$}\end{array}\right.$

and

$J(t):=\int_{\mathscr{B}(t)}r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})\delta^{2}(\mbox{\boldmath$y$})\,{\rm
d}y$

where $\delta(\mbox{\boldmath$y$})=\mbox{\rm dist\,}(\mbox{\boldmath$y$},{\sf
a})$.

In order to convert the problem into an equivalent one where the domain of the
fluid does not change with time, we define

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\varrho(\mbox{\boldmath$x$},t):=r(\mathbb{Q}(t)\cdot\mbox{\boldmath$x$},t),\,{\mbox{\boldmath$u$}}(\mbox{\boldmath$x$},t):=\mathbb{Q}^{\top}(t)\cdot{\mbox{\boldmath$w$}}(\mathbb{Q}(t)\cdot\mbox{\boldmath$x$},t),\
(\mbox{\boldmath$x$},t)\in{\cal C}_{0}\times(0,\infty),\\\
\mathbb{S}({\mbox{\boldmath$u$}}):=\mathbb{Q}^{\top}(t)\cdot\widehat{\mathbb{S}}(\mathbb{Q}(t)\cdot{\mbox{\boldmath$u$}})\cdot\mathbb{Q}(t),\,{\mbox{\boldmath$g$}}(t):=\mathbb{Q}^{\top}(t)\cdot\widehat{{\mbox{\boldmath$g$}}},\,\varrho_{{}_{\mathscr{B}}}(\mbox{\boldmath$x$},t):=r_{{}_{\mathscr{B}}}(\mathbb{Q}(t)\cdot\mbox{\boldmath$x$}),\
t\in(0,\infty)\,,\end{array}$

so that, recalling that

$\mathbb{Q}(t)\cdot{\mbox{\boldmath$e$}}_{3}={\mbox{\boldmath$e$}}_{3}\,,\ \
\mbox{ for all $t\geq 0$},$ (2.6)

the system (2.5), in terms of the $x$-variable and fields $\varrho$ and $u$,
thus becomes [22]

$\begin{array}[]{cc}\vskip 3.0pt plus 1.0pt minus
1.0pt\left.\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\partial_{t}(\varrho{\mbox{\boldmath$u$}})+\mbox{\rm
div}\,(\varrho{\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}})+\varrho\,\omega{\mbox{\boldmath$e$}}_{3}\times{\mbox{\boldmath$u$}}=\mbox{\rm
div}\,{\mathbb{S}}({\mbox{\boldmath$u$}})-\nabla
p(\varrho)+\varrho\,{\mbox{\boldmath$g$}}\\\ \partial_{t}\varrho+\mbox{\rm
div}\,(\varrho{\mbox{\boldmath$v$}})=0\end{array}\right\\}\ \
(\mbox{\boldmath$x$},t)\in{\cal C}_{0}\times(0,\infty)\\\ \vskip 6.0pt plus
2.0pt minus
2.0pt{\mbox{\boldmath$u$}}=\omega(t){\mbox{\boldmath$e$}}_{3}\times\mbox{\boldmath$x$}\,\
\ (\mbox{\boldmath$x$},t)\in\partial{\cal C}_{0}\times(0,\infty)\\\
{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\left(I\,\omega+{\mbox{\boldmath$e$}}_{3}\cdot{\displaystyle\int_{{\cal
C}_{0}}}\varrho\,\mbox{\boldmath$x$}\times{\mbox{\boldmath$u$}}\,\,\mathrm{d}x\right)={\mbox{\boldmath$e$}}_{3}\cdot\left[\left({\displaystyle\int_{\mathcal{S}_{0}}}\widehat{\varrho}\,\mbox{\boldmath$x$}\,\,\mathrm{d}x\right)\times{\mbox{\boldmath$g$}}\right]\,,\end{array}$
(2.7)

where

${\mbox{\boldmath$v$}}:={\mbox{\boldmath$u$}}-\omega{\mbox{\boldmath$e$}}_{3}\times\mbox{\boldmath$x$}\,,$
(2.8)

and

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\widehat{\varrho}:=\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt
minus 1.0pt\varrho(\mbox{\boldmath$x$},t)&\ \mbox{if $x\in{\cal C}_{0}$}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\varrho_{{}_{\mathscr{B}}}(\mbox{\boldmath$x$})&\ \mbox{if
$x\in\mathscr{B}_{0}$}\end{array}\right.\,,\\\
I:={\displaystyle\int_{\mathscr{B}_{0}}}\varrho_{{}_{\mathscr{B}}}(\mbox{\boldmath$x$})\delta^{2}(\mbox{\boldmath$x$})\,\,\mathrm{d}x\,.\end{array}$

Moreover,

$\mathbb{S}({\mbox{\boldmath$u$}})=2\mu\mathbb{D}({{\mbox{\boldmath$u$}}})+\left(\lambda-\frac{2}{3}\mu\right){\mathbb{I}}\operatorname{div}{\mbox{\boldmath$u$}}$

where $\mathbb{D}$ denotes the symmetric part of
$\nabla{\mbox{\boldmath$u$}}$, $\mathbb{I}$ the identity matrix, while $\mu>0$
and $\lambda\geq 0$ are (constant) shear and bulk viscosity coefficients.
Also, observing that, by (2.2) and (2.3),
$\dot{\mathbb{Q}}^{\top}\cdot\mathbb{Q}=\mathbb{A}(\omega)$. we derive

$\dot{{\mbox{\boldmath$g$}}}=\dot{\mathbb{Q}}^{\top}\cdot\mathbb{Q}\cdot{\mbox{\boldmath$g$}}=\mathbb{A}(\omega)\cdot{\mbox{\boldmath$g$}}\,,$

namely,

$\dot{{\mbox{\boldmath$g$}}}+\omega\,{\mbox{\boldmath$e$}}_{3}\times{\mbox{\boldmath$g$}}={\mbox{\boldmath$0$}}\,,\
\ t\in(0,\infty)\,.$

For the pressure $p$ we assume the following constitutive law

$p(\varrho)=a\varrho^{\gamma}\,,$ (2.9)

for some $a>0$ and $\gamma>3/2$. Further, we endow (2.7) with the initial
conditions

$\varrho(0,\mbox{\boldmath$x$})=\varrho_{0}(\mbox{\boldmath$x$}),\qquad\varrho(0,\mbox{\boldmath$x$}){\mbox{\boldmath$u$}}(0,\mbox{\boldmath$x$})=(\varrho{\mbox{\boldmath$u$}})_{0}(\mbox{\boldmath$x$})\,$

so that, integrating (2.7)2 over $(0,t)\times\mathcal{C}_{0}$ for arbitrary
$t\in\mathbb{R}$ we deduce the equation of conservation of mass for the fluid

$\int_{\mathcal{C}_{0}}\varrho(t,\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{\mathcal{C}_{0}}\varrho_{0}(\mbox{\boldmath$x$})\,\mathrm{d}x.$
(2.10)

The unknowns of (2.7) are ${\mbox{\boldmath$u$}}:(0,T)\times{\cal
C}_{0}\to\mathbb{R}^{3}$,
$\varrho:(0,T)\times(\mathscr{B}_{0}\cup\overline{\cal C}_{0})\to\mathbb{R}$
and ${{\mbox{\boldmath$g$}}}:(0,T)\to\mathbb{R}^{3}$, while we assume that the
density $\varrho_{\mathscr{B}}$ of $\mathscr{B}$ is prescribed. However,
instead of the unknown $u$, sometime we may find it more appropriate to use
the velocity $v$ defined in (2.8).

If we formally multiply (2.7)1 by $u$, (2.7)4 by $\omega$ and integrate by
parts, we deduce the energy inequality:

$\frac{1}{2}\left(I\,{\rm
d}_{t}|\omega|^{2}+\partial_{t}\int_{\mathcal{C}_{0}}\varrho|{{\mbox{\boldmath$u$}}}|^{2}\,\,\mathrm{d}x\right)+\int_{\mathcal{C}_{0}}\mathbb{S}({\mbox{\boldmath$v$}}):\nabla{\mbox{\boldmath$v$}}\,\,\mathrm{d}x+\partial_{t}\int_{\mathcal{C}_{0}}P(\varrho)\,\,\mathrm{d}x\leq\partial_{t}\int_{\mathcal{S}_{0}}\widehat{\varrho}\,\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}\,\,\mathrm{d}x$

which, after integration, leads to

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\left[I\,\displaystyle{\frac{\displaystyle{|\omega(t)|^{2}}}{\displaystyle{2}}}+\frac{1}{2}{\displaystyle\int_{\mathcal{C}_{0}}}\varrho(t)|{{\mbox{\boldmath$u$}}}(t)|^{2}\,\mathrm{d}x+{\displaystyle\int_{\mathcal{C}_{0}}}P(\varrho(t))\,\mathrm{d}x-{\displaystyle\int_{\mathcal{S}_{0}}}\widehat{\varrho}\,\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}\,\mathrm{d}x\right]_{t=0}^{\tau}\\\
\hskip
184.9429pt+{\displaystyle\int_{0}^{\tau}}{\displaystyle\int_{\mathcal{C}_{0}}}\mathbb{S}({\mbox{\boldmath$v$}}):\nabla{\mbox{\boldmath$v$}}\,\mathrm{d}x{\rm
d}t\leq 0,\end{array}$

where

$P(\varrho)=\frac{a}{\gamma-1}\varrho^{\gamma}.$

Our primary objective is to investigate the long-time behavior of the system
(2.7)–(2.9) in the class of weak solutions, which we defined next.

######  Definition 2.1.

A quadruple $(\varrho,{\mbox{\boldmath$v$}},\omega,{{\mbox{\boldmath$g$}}})$
is a renormalized weak solution to (2.7) on time interval $(0,T)$ if 555Recall
(2.8) .

  * •

The momentum equation (2.7)1 is fulfilled in a weak sense, i.e.

$\int_{0}^{T}\int_{\mathcal{C}_{0}}\varrho{\mbox{\boldmath$u$}}\cdot\partial_{t}\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t+\int_{0}^{T}\int_{\mathcal{C}_{0}}\varrho{\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}}:\nabla\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t-\int_{0}^{T}\int_{\mathcal{C}_{0}}\varrho\omega{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t\\\
+\int_{0}^{T}\int_{\mathcal{C}_{0}}p(\varrho)\operatorname{div}\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t-\int_{0}^{T}\int_{\mathcal{C}_{0}}\mathbb{S}({\mbox{\boldmath$u$}}):\nabla\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t\\\
=-\int_{0}^{T}\int_{\mathcal{C}_{0}}\varrho\,{\mbox{\boldmath$g$}}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x{\rm
d}t-\int_{\mathcal{C}_{0}}(\varrho{\mbox{\boldmath$u$}})_{0}\cdot\mbox{\boldmath$\varphi$}(0)\,\mathrm{d}x{\rm
d}t$ (2.11)

for all $\mbox{\boldmath$\varphi$}\in
C_{c}^{\infty}([0,T)\times\mathcal{C}_{0})$,
$\mbox{\boldmath$\varphi$}|_{\partial\mathcal{C}_{0}}=0$.

  * •

The continuity equation is fulfilled in a renormalized weak sense, i.e.

$\int_{0}^{T}\int_{\mathcal{C}_{0}}b(\varrho)\partial_{t}\varphi\,\mathrm{d}x{\rm
d}t+\int_{0}^{T}\int_{\mathcal{C}_{0}}b(\varrho){\mbox{\boldmath$v$}}\cdot\nabla\varphi\,\mathrm{d}x{\rm
d}t\\\
+\int_{0}^{T}\int_{\mathcal{C}_{0}}(b(\varrho)-b^{\prime}(\varrho)\varrho)\operatorname{div}{\mbox{\boldmath$v$}}\,\varphi\,\mathrm{d}x{\rm
d}t=-\int_{\mathcal{C}_{0}}\varrho_{0}\varphi(0)\,\mathrm{d}x$ (2.12)

for all $\varphi\in C^{\infty}_{c}([0,T)\times\overline{\mathcal{C}_{0}})$ and
any $b\in C^{1}[0,\infty)$, $|b^{\prime}(z)z|\leq c\sqrt{|z|}$.

  * •

The equations (2.7)3,4 are fulfilled.

  * •

The energy inequality

$\mathcal{E}(\varrho(\tau),{{\mbox{\boldmath$u$}}}(\tau),\omega(\tau),{{\mbox{\boldmath$g$}}}(\tau))\leq\mathcal{E}\left(\varrho_{0},\frac{(\varrho{\mbox{\boldmath$u$}})_{0}}{\varrho_{0}},\omega_{0},{{\mbox{\boldmath$g$}}}_{0}\right)-\int_{0}^{\tau}\int_{\mathcal{C}_{0}}\mathbb{S}({\mbox{\boldmath$v$}}):\nabla{\mbox{\boldmath$v$}}\,\mathrm{d}x{\rm
d}t$

is fulfilled for almost all $\tau\in[0,T)$, where

$\begin{array}[]{rl}\vskip 6.0pt plus 2.0pt minus
2.0pt\mathcal{E}(\varrho,{{\mbox{\boldmath$u$}}},\omega,{{\mbox{\boldmath$g$}}}):=&\\!\\!\\!I\,\displaystyle{\frac{|\omega|^{2}}{2}+\frac{1}{2}\int_{\mathcal{C}_{0}}\varrho|{{\mbox{\boldmath$u$}}}|^{2}\,\mathrm{d}x+\int_{\mathcal{C}_{0}}P(\varrho)-P^{\prime}(\overline{\varrho})(\varrho-\overline{\varrho})-P(\overline{\varrho})\,\mathrm{d}x}\\\
&\\!\\!\\!\displaystyle{-\int_{\mathcal{S}_{0}}\widehat{\varrho}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}\,\mathrm{d}x\,,}\end{array}$
(2.13)

and $\overline{\varrho}=\frac{1}{|{\cal
C}_{0}|}\int_{\mathcal{C}_{0}}\varrho\,\mathrm{d}x$ is constant in time due to
(2.10).

######  Remark 2.2.

As shown in [13], a sufficiently smooth weak solution defined as above solves,
in fact, (2.7) pointwise.

######  Remark 2.3.

Our definition of weak solution allows us to deduce a weak formulation for a
larger class of test functions. Precisely, take

$\mbox{\boldmath$\varphi$}=\mbox{\boldmath$\varphi$}_{0}+\eta{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$}\
\mbox{ where }\mbox{\boldmath$\varphi$}_{0}\in
C^{\infty}_{c}([0,T)\times\mathcal{C}_{0})\ \mbox{and }\eta\in
C^{\infty}_{c}([0,T)).$ (2.14)

We multiply (2.7)4 by $\eta$ to get

$I\dot{\omega}\,\eta-\int_{{\cal
C}_{0}}\eta\,\partial_{t}(\varrho{\mbox{\boldmath$u$}})\times\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$e$}}}_{3}\,\mathrm{d}x=\int_{\mathcal{S}_{0}}\eta\,\widehat{\varrho}\mbox{\boldmath$x$}\,\mathrm{d}x\times{\mbox{\boldmath$g$}}\cdot{{\mbox{\boldmath$e$}}}_{3}\,.$
(2.15)

Since, by definition,

$I\,{{\mbox{\boldmath$e$}}}_{3}=\int_{\mathscr{B}}\varrho_{\mathscr{B}}\mbox{\boldmath$x$}\times({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x\,,$

the term on the left hand side of (2.15) is equal to

$\int_{{\cal
S}_{0}}\partial_{t}\eta(\widehat{\varrho}{\mbox{\boldmath$u$}})\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x\,,$

where $u$ is extended to
$\omega{\mbox{\boldmath$e$}}_{3}\times\mbox{\boldmath$x$}$ on
$\mathscr{B}_{0}$. Further, the right hand side of (2.15) can be rewritten as

$\int_{\mathcal{S}_{0}}\eta\,\widehat{\varrho}{{\mbox{\boldmath$g$}}}\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x\,,$

so that, (2.15) can be equivalently formulated as follows

$\partial_{t}\int_{{\cal
S}_{0}}\eta(\widehat{\varrho}{\mbox{\boldmath$u$}})\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x-\int_{{\cal
S}_{0}}\partial_{t}\eta\,(\widehat{\varrho}{\mbox{\boldmath$u$}})\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{{\cal
S}_{0}}\eta\,\widehat{\varrho}\,{{\mbox{\boldmath$g$}}}\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x$
(2.16)

Next, we observe that

$\int_{{\cal
S}_{0}}\eta\,\widehat{\varrho}\omega{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{{\cal
S}_{0}}\eta\,\widehat{\varrho}\,\omega{{\mbox{\boldmath$e$}}}_{3}\times({\mbox{\boldmath$v$}}+\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\cdot({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x\\\
=\int_{{\cal
S}_{0}}\eta\,\widehat{\varrho}({{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$v$}})\cdot(\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{{\cal
C}_{0}}\eta\,{\varrho}(v_{2},-v_{1},0)\cdot(\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x.$

Furthermore,

$\int_{{\cal
S}_{0}}\widehat{\varrho}({\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}}):\nabla(\eta{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{{\cal
C}_{0}}\varrho\eta({\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}}):\nabla({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\,\mathrm{d}x=\int_{{\cal
C}_{0}}\varrho\eta
v_{i}u_{j}\partial_{i}\left(x_{2},-x_{1},0\right)_{j}\,\,\mathrm{d}x\\\
=\int_{{\cal
C}_{0}}\varrho\eta(-v_{1}u_{2}+v_{2}u_{1})\,\,\mathrm{d}x=\int_{{\cal
C}_{0}}\varrho\eta(v_{2},-v_{1},0)\cdot(\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\,\mathrm{d}x\,,$

from which we deduce

$-\int_{{\cal
S}_{0}}\widehat{\varrho}({\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}}):\nabla(\eta{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\,\mathrm{d}x+\int_{{\cal
S}_{0}}\widehat{\varrho}\omega{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}\cdot(\eta{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\,\mathrm{d}x=0$
(2.17)

Also, we observe that since $\mathbb{S}$ is a symmetric tensor whereas
$\nabla({{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})$ is
antisymmetric, we get

$\int_{{\cal
C}_{0}}\mathbb{S}({\mbox{\boldmath$v$}}):\nabla(\eta{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$})\,\,\mathrm{d}x=0.$

Thus, adding (2.11) with
$\mbox{\boldmath$\varphi$}=\mbox{\boldmath$\varphi$}_{0}$, (2.16) and (2.17),
we obtain

$\partial_{t}{\displaystyle\int_{{\cal
S}_{0}}}\widehat{\varrho}{\mbox{\boldmath$u$}}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x-{\displaystyle\int_{{\cal
S}_{0}}}\widehat{\varrho}{\mbox{\boldmath$u$}}\cdot\partial_{t}\mbox{\boldmath$\varphi$}\,\mathrm{d}x-{\displaystyle\int_{{\cal
S}_{0}}}\widehat{\varrho}({\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}}):\nabla\varphi\,\mathrm{d}x\\\
+{\displaystyle\int_{{\cal
S}_{0}}}\widehat{\varrho}\omega{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x+{\displaystyle\int_{{\cal
S}_{0}}}\mathbb{S}({\mbox{\boldmath$v$}}):\nabla\mbox{\boldmath$\varphi$}\,\mathrm{d}x={\displaystyle\int_{{\cal
S}_{0}}}\widehat{\varrho}{{\mbox{\boldmath$g$}}}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x,$
(2.18)

where $\varphi$ is a test function of the form (2.14). Note that the energy
inequality may be deduced formally from (2.18) by taking
$\mbox{\boldmath$\varphi$}={\mbox{\boldmath$u$}}$.

##  3 Steady states

One may expect that, for sufficiently large times, the generic weak solution
may approach some steady state (namely, a time-independent solution of (2.7))
in a suitable topology. This will be investigated in Section 4. Therefore, the
main goal of this section is to find and characterize all possible steady
states, in the class of renormalized weak solutions. Before performing this
study, however, we would like to make some simple but important remarks
concerning the class of irrotational solutions to (2.7), that is, those for
which $\omega(t)=0$ for all $t\geq 0$.

From what we presented at the beginning of the previous section, in those
motions where $\omega(t)\equiv 0$, we have

$\mathbb{Q}(t)=\mathbb{Q}_{0}\,,\ \mbox{for all $t\geq 0$},$ (3.1)

implying that

$\mbox{\boldmath$y$}=\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$},$ (3.2)

and, moreover,

$\varrho=r(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}),\,{\mbox{\boldmath$u$}}=\mathbb{Q}_{0}^{\top}\cdot{\mbox{\boldmath$w$}}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}),\,\varrho_{{}_{\mathscr{B}}}=r_{{}_{\mathscr{B}}}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}),\,{\mbox{\boldmath$g$}}=\mathbb{Q}^{\top}_{0}\cdot\widehat{{\mbox{\boldmath$g$}}}\,.$
(3.3)

We notice that, by (2.4) and (3.1), in such a case the position of the body
(as well as that of the cavity) is time independent in the original frame
$\mathcal{F}$. We also notice that the system of equations (2.5) (or,
equivalently (2.7)) might seem overdetermined. However, this is not the case,
because, in general, we cannot expect that motions with $\omega(t)\equiv 0$
may occur for any $\mathbb{Q}_{0}$ (that is, any orientation of
$\mathscr{B}$). Therefore, $\mathbb{Q}_{0}$ (namely, $\vartheta_{0}$) becomes
a further unknown, which thus makes the problem well-defined.

###  3.1 Characterization

With these premises, we now turn to the characterization of steady-state
solutions. From (2.7) we derive that they must satisfy the following set of
equations

$\begin{split}\operatorname{div}(\varrho{\mbox{\boldmath$v$}}\otimes{\mbox{\boldmath$u$}})+\varrho\omega{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}+\nabla
p(\varrho)-\operatorname{div}\mathbb{S}({\mbox{\boldmath$u$}})&=\varrho\,{\mbox{\boldmath$g$}}\\\
\operatorname{div}(\varrho{\mbox{\boldmath$v$}})&=0\\\
{\mbox{\boldmath$u$}}&=\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$}\\\
{{\mbox{\boldmath$e$}}}_{3}\cdot\left(\int_{{\cal
S}_{0}}\widehat{\varrho}\mbox{\boldmath$x$}\,\mathrm{d}x\right)\times{{\mbox{\boldmath$g$}}}&=0\\\
\omega{{\mbox{\boldmath$e$}}}_{3}\times{{\mbox{\boldmath$g$}}}&=0.\end{split}$
(3.4)

We work with a renormalized weak solution, i.e. a quadruple
$(\varrho,{{\mbox{\boldmath$v$}}},\omega,{{\mbox{\boldmath$g$}}})$ (recall
${\mbox{\boldmath$v$}}={{\mbox{\boldmath$u$}}}-\omega{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$}$)
which satisfies (3.4) and

$\operatorname{div}(b(\varrho){\mbox{\boldmath$v$}})+(\varrho
b^{\prime}(\varrho)-b(\varrho))\operatorname{div}{\mbox{\boldmath$v$}}=0,\
\forall\;b\in C^{1}({\mathbb{R}})$

in distributional sense. The system (3.4) is complemented with the
conservation of mass (2.10):

$\int_{{\cal C}_{0}}{\varrho}(\mbox{\boldmath$x$})\,\mathrm{d}x=\int_{{\cal
C}_{0}}\varrho_{0}(\mbox{\boldmath$x$})\,\mathrm{d}x:=M\,.$ (3.5)

We also recall that the gravity has prescribed magnitude, i.e.,

$|{{\mbox{\boldmath$g$}}}|=|{{\mbox{\boldmath$g$}}}_{0}|\equiv g\,.$ (3.6)

Now, since ${\mbox{\boldmath$g$}}\cdot{{\mbox{\boldmath$e$}}}_{3}=0$, from the
last equation in (3.4) we get $\omega=0$, and so, arguing exactly as in [13,
Lemma 1], we show that ${\mbox{\boldmath$v$}}=0$. Consequently, (3.4) reduces
to a system of only two relevant equations:

$\begin{split}\nabla
p(\varrho(\mbox{\boldmath$x$}))&=\varrho(\mbox{\boldmath$x$}){{\mbox{\boldmath$g$}}}\
\ \mbox{in}\ {\cal C}_{0},\\\ {\mbox{\boldmath$e$}}_{3}\cdot\left(\int_{{\cal
S}_{0}}\widehat{\varrho}(\mbox{\boldmath$x$})\mbox{\boldmath$x$}\,\mathrm{d}x\right)\times{{\mbox{\boldmath$g$}}}&=0\,.\end{split}$
(3.7)

Since $\omega=0$, by what we just proved and what we remarked at the beginning
of this section, by (3.1)–(3.3) we deduce

$\varrho:=r_{s}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}),\ \
\widehat{\varrho}:=\widehat{r}_{s}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}),\ \
{\mbox{\boldmath$u$}}\equiv{\mbox{\boldmath$w$}}\equiv 0,\ \
{\mbox{\boldmath$g$}}=\mathbb{Q}^{\top}_{0}\cdot\widehat{{\mbox{\boldmath$g$}}}\,,$
(3.8)

for some $\mathbb{Q}_{0}$ to be found, where, from (3.7)1, $r_{s}$ satisfies

$\nabla_{\mbox{\footnotesize$\mbox{\boldmath$x$}$}}p(r_{s}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}))=r_{s}(\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$})\,{\mbox{\boldmath$g$}},$

and where we have emphasized that the derivatives are taken with respect to
the $x$-variable. Employing (3.2), (3.8) and (2.1) in the latter, we show
(derivatives now taken with respect to the $y$-variable)

$\mathbb{Q}_{0}^{\top}\cdot[\nabla_{\mbox{\footnotesize$\mbox{\boldmath$y$}$}}p(r_{s}(\mbox{\boldmath$y$}))-r_{s}(\mbox{\boldmath$y$})\,g\,{\mbox{\boldmath$e$}}_{1}]=0\,,$

which, recalling that $p(r_{s})=a\,r_{s}^{\gamma}$, is in turn equivalent to

${\displaystyle\frac{\mbox{$dr_{s}^{\gamma-1}$}}{\mbox{$dy_{1}$}}}=\frac{\gamma-1}{a\,\gamma}\,g\,.$

Integrating both sides of this equation, and assuming that the cavity is
convex we conclude

$r_{s}=r_{s}(y_{1})=\left[\left(\frac{a\,\gamma}{\gamma-1}\,g\,y_{1}+c\right)_{+}\right]^{\frac{1}{\gamma-1}}\,,$
(3.9)

for some constant $c\in$.

We next investigate the class of all possible $\mathbb{Q}_{0}$ compatible with
steady-state solutions, that is, the equilibrium configurations of the
pendulum. From (3.7)2 and (3.8) we obtain

${\mbox{\boldmath$e$}}_{3}\cdot\left[{\displaystyle\int_{\mathcal{S}_{0}}}\widehat{r}(\mathbb{Q}_{0}\cdot\,\mbox{\boldmath$x$})\,\mbox{\boldmath$x$}\times{\mbox{\boldmath$g$}}\,\,\mathrm{d}x\right]=0\,.$
(3.10)

In this integral we now perform the change of variable (3.2). Thus, taking
into account (3.8)4, (2.6), (3.9) and that

$\mbox{\boldmath$x$}\times{\mbox{\boldmath$g$}}=(\mathbb{Q}_{0}^{\top}\cdot\mbox{\boldmath$y$})\times(\mathbb{Q}_{0}^{\top}\cdot\widehat{{\mbox{\boldmath$g$}}})=\mathbb{Q}_{0}^{\top}\cdot(\mbox{\boldmath$y$}\times\widehat{{\mbox{\boldmath$g$}}})\,,$

we show that (3.10) is equivalent to

${\mbox{\boldmath$e$}}_{3}\cdot\left[{\displaystyle\int_{\mathcal{S}_{eq}}}\widehat{r}(\mbox{\boldmath$y$})\,\mbox{\boldmath$y$}\times\widehat{{\mbox{\boldmath$g$}}}\,{\rm
d}y\right]=0\,,$ (3.11)

where

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\mathcal{S}_{eq}:=\mathscr{B}_{eq}\cup\mathcal{C}_{eq}\,,\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\mathscr{B}_{eq}:=\\{\mbox{\boldmath$y$}\in^{3}:\
\mbox{\boldmath$y$}=\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$},\
\mbox{\boldmath$x$}\in\mathscr{B}_{0}\\}\,,\\\
\mathcal{C}_{eq}:=\\{\mbox{\boldmath$y$}\in^{3}:\
\mbox{\boldmath$y$}=\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$},\
\mbox{\boldmath$x$}\in\mathcal{C}_{0}\\}\,.\end{array}$ (3.12)

The relation (3.11) expresses the vanishing of the axial component of the
angular moment of the coupled system $\mathscr{S}$ at equilibrium in the fixed
frame $\mathcal{F}$. By keeping in mind (3.9) and (2.1), we show that (3.11),
in turn, is equivalent to the following one

${\displaystyle\int_{\mathcal{C}_{eq}}}r_{s}(y_{1})y_{2}\,{\rm
d}y+{\displaystyle\int_{\mathscr{B}_{eq}}}r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})y_{2}\,{\rm
d}y=0\,,$ (3.13)

which tells us that the center of mass $C$ of the coupled system must belong
to the vertical plane containing the rotation axis ${\sf a}$. Now, in any such
equilibrium configurations $\mathcal{S}_{eq}$, the position
$\mbox{\boldmath$y$}_{C}$ $(\equiv\overrightarrow{OC})$ in the fixed
(inertial) frame is given by

$\mbox{\boldmath$y$}_{C}=\frac{1}{{\cal
M}}\left(\int_{\mathcal{C}_{eq}}r_{s}(y_{1})\mbox{\boldmath$y$}\,{\rm
d}y+\int_{\mathscr{B}_{eq}}r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})\,\mbox{\boldmath$y$}\,{\rm
d}y\right)\,,$

with $\mathcal{M}=M+m$, and $m$ mass of $\mathscr{B}$. Since we chose $O$ as
the orthogonal projection of the center of mass $G$ of $\mathscr{B}$ on ${\sf
a}\equiv{\mbox{\boldmath$e$}}_{3}$, we have

$\int_{\mathscr{B}_{eq}}r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})\,y_{3}\,{\rm
d}y=0\,.$

Therefore, collecting the above results we conclude with the following
characterization of steady states.

######  Theorem 3.1.

The quadruple
$(\varrho_{s},{\mbox{\boldmath$u$}}_{s}:={\mbox{\boldmath$v$}}_{s}+\omega_{s}{\mbox{\boldmath$e$}}_{3}\times\mbox{\boldmath$x$},\omega_{s},{\mbox{\boldmath$g$}}_{s})$
is a renormalized weak solution to (3.4) if and only if the following
conditions (i)–(iii) are met:

  * (i)

$\omega_{s}=0$, ${\mbox{\boldmath$u$}}_{s}\equiv{\mbox{\boldmath$0$}}$.

Setting $\mbox{\boldmath$y$}=\mathbb{Q}_{0}\cdot\mbox{\boldmath$x$}$:

  * (ii)

$\varrho_{s}=r_{s}(y_{1})$, where $r_{s}(y_{1})$ is given in (3.9), and
$\widehat{{\mbox{\boldmath$g$}}}_{s}=g{\mbox{\boldmath$e$}}_{1}$ ;

  * (iii)

The rotation matrix $\mathbb{Q}_{0}$ is determined by the request that the
center of mass $C$ of ${\cal S}$ is located in the vertical plane, ${\sf V}$,
containing the rotation axis ${\sf a}$, and precisely at the point
$\mbox{\boldmath$y$}_{C}$ that, in the fixed (inertial) frame, is given by

$\mbox{\boldmath$y$}_{C}=\frac{1}{{\cal
M}}\left[\left(\int_{\mathcal{C}_{eq}}r_{s}(y_{1})y_{1}\,{\rm
d}y+\int_{\mathscr{B}_{eq}}r_{{}_{\mathscr{B}}}(\mbox{\boldmath$y$})\,y_{1}\,{\rm
d}y\right){\mbox{\boldmath$e$}}_{1}+\left(\int_{\mathcal{C}_{eq}}r_{s}(y_{1})y_{3}\,{\rm
d}y\right){\mbox{\boldmath$e$}}_{3}\right]\,,$ (3.14)

with $\mathscr{B}_{eq}$ and ${\cal C}_{eq}$ given in (3.12).

Finally, the constant $c$ in (3.9) is obtained by the condition

$\int_{\mathcal{C}_{eq}}r_{s}(y_{1})\,{\rm d}y=M\,.$ (3.15)

###  3.2 Some relevant consequences of Theorem 3.1

We would like to analyze some interesting conclusions that can be drawn as
corollary to the previous theorem.

We begin to observe that from (3.14) it follows that, in general, in the
equilibrium configuration, $\mbox{\boldmath$y$}_{C}\equiv\overrightarrow{OC}$
is not aligned with $\widehat{{\mbox{\boldmath$g$}}}$ (namely,
${\mbox{\boldmath$e$}}_{1}$). In fact, this alignment occurs if and only if

$\int_{\mathcal{C}_{eq}}r_{s}(y_{1})y_{3}\,{\rm d}y=0\,.$ (3.16)

The validity or invalidity of (3.16) may depend on the location of the cavity
with respect to $O$ (or, equivalently, $G$) and its shape. In particular,
(3.16) holds if $G$ and $\mathcal{C}_{eq}$ are such that
$(\cdot,y_{3})\in\mathcal{C}_{eq}\,\Rightarrow\,(\cdot,-y_{3})\in\mathcal{C}_{eq}$,
but may not hold otherwise. A simple example is shown in the following figure.

> Figure 1.: (a) $\mathscr{B}$ is a homogeneous spherical shell with the
> cavity (blue) being the inner sphere. In this case $\mbox{\boldmath$y$}_{C}$
> and $\widehat{{\mbox{\boldmath$g$}}}$ are parallel with same orientation;
> (b) $\mathscr{B}$ is a homogeneous sphere with an off-centered interior
> spherical cavity (blue). $\mbox{\boldmath$y$}_{C}$ and
> $\widehat{{\mbox{\boldmath$g$}}}$ are not parallel.

We shall next consider the case when the system $\mathscr{S}$ possesses some
suitable symmetries. Precisely, let $G\neq O$, and suppose the cavity ${\cal
C}$ is a body of revolution around the axis ${\sf
e}:=\overrightarrow{OG}/|\overrightarrow{OG}|$. Moreover, denote by
$\alpha\in[0,2\pi)$ the angle between ${\sf e}$ and
$\widehat{{\mbox{\boldmath$g$}}}$.

(1) Consider the configurations of $\mathscr{B}$ (in the inertial frame) where
${\sf e}$ is parallel to $\widehat{{\mbox{\boldmath$g$}}}$, namely, ${\sf
e}=\pm{\mbox{\boldmath$e$}}_{1}$ corresponding to $\alpha=0,\pi$. Clearly,
these are (the only two) equilibrium configurations, $\mathscr{B}_{eq}^{\pm}$,
for $\mathscr{B}$, since we have

$\int_{\mathscr{B}_{eq}^{\pm}}r_{\mathscr{B}}(\mbox{\boldmath$y$})y_{i}\,{\rm
d}y=0\,,\ \ i=2,3\,.$ (3.17)

However, when $\mathscr{B}$ is in either one of the configurations
$\mathscr{B}_{eq}^{\pm}$, the coupled system $\mathscr{S}$ is in equilibrium
as well. In fact, denoted by ${\cal C}_{eq}^{\pm}$ the positions of ${\cal C}$
when $\mathscr{B}=\mathscr{B}_{eq}^{\pm}$, owing to the symmetry properties of
${\cal C}$, we deduce

$\int_{{\cal C}_{eq}^{\pm}}r_{s}(y_{1})y_{i}\,{\rm d}y=0\,,\ \ i=2,3\,.$
(3.18)

Therefore, (3.14) follows from (3.17) and (3.18).

(2) Let us comment about the possible uniqueness of the above “vertical”
configurations. To this end, denote by ${\sf G}={\sf G}(\alpha)$ the center of
mass of ${\cal C}$ with the density distribution given in (3.9). In view of
Theorem 3.1(iii), equilibrium configurations for $\alpha\in(0,2\pi)$ may exist
if and only if $C(\alpha)\in{\sf V}$, namely,

${\sf w}_{\alpha}:=[M\,\overrightarrow{O{\sf
G}}(\alpha)+m_{\mathscr{B}}\,\overrightarrow{OG}]\cdot{\mbox{\boldmath$e$}}_{2}=0\,,$
(3.19)

with $m_{\mathscr{B}}$ mass of $\mathscr{B}$. Since ${\sf w}_{0}={\sf
w}_{\pi}=0$, this suggests that, for suitable $r_{s}$ (that is, $a,\gamma$ and
$M$) and ${\cal C}$, the location of ${\sf G}(\alpha)$ may vary with $\alpha$
in such a way that (3.19) is satisfied also for $\alpha$ close to $0,\pi$,
thus entailing the existence of some other equilibria, around $\alpha=0,\pi$;
see Remark 3.7. However, the latter circumstance is ruled out if ${\cal C}$
has suitable symmetry. For example, assume that ${\cal C}$ is a ball centered
at $O^{\prime}$, and set ${\sf R}=|O^{\prime}O|$. Because $r_{s}=r_{s}(y_{1})$
and of the symmetry properties of ${\cal C}$, it follows that ${\sf
G}(\alpha)$ belongs to the straight line parallel to
${\mbox{\boldmath$e$}}_{1}$ and passing through $O^{\prime}$, for any
$\alpha\in[0,2\pi)$. So, denoting by
$\\{O^{\prime},{\mbox{\boldmath$e$}}_{1}^{\prime}\\}$ the frame with origin at
$O^{\prime}$ and ${\mbox{\boldmath$e$}}_{1}^{\prime}$ parallel to and oriented
as ${\mbox{\boldmath$e$}}_{1}$, we have $\overrightarrow{O^{\prime}{\sf
G}}=\ell^{\prime}{\mbox{\boldmath$e$}}_{1}^{\prime}$
($\equiv\ell^{\prime}{\mbox{\boldmath$e$}}_{1}$), for some $\ell^{\prime}\in$.
Notice that $\ell^{\prime}$ is independent of $\alpha$. Setting $\ell:=|OG|$,
we then infer

$\overrightarrow{O{\sf
G}}=\overrightarrow{O{O^{\prime}}}+\overrightarrow{O^{\prime}{\sf G}}=({\sf
R}\cos\alpha+\ell^{\prime}){\mbox{\boldmath$e$}}_{1}-{\sf
R}\sin\alpha{\mbox{\boldmath$e$}}_{2}\,,\ \
\overrightarrow{O{G}}=\ell(\cos\alpha{\mbox{\boldmath$e$}}_{1}-\sin\alpha{\mbox{\boldmath$e$}}_{2})\,,$

and, consequently, condition (3.19) becomes

$(M\,{\sf R}+m_{\mathscr{B}}\,{\ell})\sin\alpha=0\,,$ (3.20)

that is satisfied if and only if $\alpha=0,\pi$, which means that the
configurations discussed in (1) are the only possible equilibria for
$\mathscr{S}$.

(3) If $G$ lies outside the cavity ${\cal C}$, there exists a “critical
angle,” $\alpha_{c}>0$, such that no equilibrium is allowed for
$\alpha\in(\alpha_{c},\pi-\alpha_{c})$. In fact, let $\Gamma$ be the smallest
cone having vertex at $G$ and containing ${\cal C}$. Then, on the one hand,
$C$ must belong to the segment $\overline{G{\sf G}}$ which, on the other hand,
must be in the interior of $\Gamma$ or, at most, overlap with one of its
generatrices. Thus, $\alpha_{c}$ is precisely the least value of $\alpha$ for
which $\Gamma\cap{\sf V}=\emptyset$.

Suppose now $G=O$ and let ${\cal C}$ be a ball centered at $O$. Then, clearly,

$\int_{\mathscr{B}_{eq}}r_{\mathscr{B}}(\mbox{\boldmath$y$})\mbox{\boldmath$y$}\,{\rm
d}y=0\,,$ (3.21)

for all $\mathscr{B}_{eq}$. Moreover, by symmetry,

$\int_{\mathcal{C}_{eq}}r_{s}(y_{1})y_{i}\,{\rm d}y=0\,,\ \ i=2,3\,,$ (3.22)

for all $\mathcal{C}_{eq}$. From (3.21) and (3.22) we deduce that (3.13) is
satisfied in every position of $\mathscr{S}$, implying the existence of a
continuum of equilibrium configurations.

Throughout this section, we have provided explicit examples of solutions to
the steady-state problem (3.7), as a consequence of their characterization
furnished in Theorem 3.1. However, at this stage, we do not know if
(3.5)–(3.7) admits a solution for every given $\mathscr{B}$ and $\mathcal{C}$.
The (positive) answer to this question will be furnished in the following
subsection.

###  3.3 Existence

Objective of this subsection is to show existence to the problem (3.5)–(3.7).
The main question to address here is not to find the distribution of density
(this was already done in the previous subsection) but, instead, to provide
the existence of an orientation of $\mathscr{S}$ with respect to $g$
compatible with a steady state (equilibrium configurations) or, equivalently,
the matrix $\mathbb{Q}_{0}$ introduced in the previous section.

To reach this objective, we notice, as before, that the first equation in
(3.7) entails

$\varrho=\sqrt[\gamma-1]{\left(\frac{\gamma-1}{a\gamma}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}+c\right)_{+}}$
(3.23)

for some constant $c\in$. We also recall that we are assuming ${\cal C}$
convex, and thus $\mbox{\rm supp}\,\varrho$ has just one connected component;
see [10].

Denote by $\mathcal{P}$ the projection $\mathbb{R}^{3}\to\mathbb{R}^{3}$,
$\mathcal{P}:(x_{1},x_{2},x_{3})\mapsto(x_{1},x_{2},0)$. The second equation
in (3.7) then yields that $g$ is parallel to
$\mathcal{P}\left(\int_{\mathcal{S}}\widehat{\varrho}(\mbox{\boldmath$x$}){\mbox{\boldmath$x$}}\,\,\mathrm{d}x\right)$.666For
simplicity of notation, in what follows we set
$\mathscr{S}_{0}\equiv\mathscr{S}$, $\mathscr{B}\equiv\mathscr{B}_{0}$, and
${\cal C}\equiv{\cal C}_{0}$. Thus, setting

${\bf
l}:=\mathcal{P}\left(\int_{\mathscr{B}}\varrho_{\mathscr{B}}(\mbox{\boldmath$x$})\mbox{\boldmath$x$}\,\,\mathrm{d}x\right)\,,$

we infer that (3.5)–(3.7) is equivalent to the following system of four
equations

$\begin{split}\int_{{\cal
C}}\sqrt[\gamma-1]{\left(\frac{\gamma-1}{a\gamma}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}+c\right)_{+}}\,\,\mathrm{d}x-\int_{{\cal
C}}\varrho_{0}\,\,\mathrm{d}x&=0\\\
d{{\mbox{\boldmath$g$}}}-\mathcal{P}\left(\int_{{\cal
C}}\sqrt[\gamma-1]{\left(\frac{\gamma-1}{a\gamma}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}+c\right)_{+}}\mbox{\boldmath$x$}\,\,\mathrm{d}x\right)-{\bf
l}&=0\\\
|{{\mbox{\boldmath$g$}}}|^{2}-|{{\mbox{\boldmath$g$}}}_{0}|^{2}&=0\end{split}$
(3.24)

for four unknowns: ${{\mbox{\boldmath$g$}}}=(g_{1},g_{2},0)\in\mathbb{R}^{3}$,
$c\in\mathbb{R}$, and $d\in\mathbb{R}$.

In view of the above and of what established in the previous subsection, we
can then state the following lemma.

######  Lemma 3.2.

Let $v$, $\varrho$, $\omega$ and $g$ be a renormalized weak solution to (3.4).
Then ${{\mbox{\boldmath$v$}}}=0,\ \omega=0,\ \varrho$ is given by (3.23) and
$g$ satisfies (3.24).

######  Remark 3.3.

We would like to explain the meaning of the parameter $d$. Equation (3.24)2
can be rearranged as

$d{\bf
g}=\mathcal{P}\left(\int_{\mathcal{C}}\sqrt[\gamma-1]{\left(\frac{\gamma-1}{2\gamma}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}+c\right)_{+}}\mbox{\boldmath$x$}\,\mathrm{d}x\right)+{\bf
l}.$

As we know from previous subsection (see (3.14)), the right-hand side of this
equation is the vector $\overrightarrow{OC}$, with $C$ center of mass of the
whole system at equilibrium. Thus, $d\neq 0$ means that $\overrightarrow{OC}$
and $g$ must be parallel. Moreover, $d$ positive means that the $g$ and
$\overrightarrow{OC}$ have the same orientation ($C$ is below the hinge),
whereas $d$ negative means the opposite ($C$ is above the hinge).

By using the standard theory associated to the Euler–Lagrange equations, we
can show the following result.

######  Lemma 3.4.

Let $\varrho_{s}\in L^{\gamma}({\cal C})$ and
${{\mbox{\boldmath$g$}}}_{s}\in\mathbb{R}^{2}\times\\{0\\}$ be a minimizer of
the functional

$\mathcal{I}:(\varrho,{{\mbox{\boldmath$g$}}})\mapsto\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}})$

with $\mathcal{I}$ defined on the set

$\\{\varrho\in
L^{\gamma},{{\mbox{\boldmath$g$}}}\in\mathbb{R}^{2}\times\\{0\\},\ \varrho\geq
0,\ |{{\mbox{\boldmath$g$}}}|=|{{\mbox{\boldmath$g$}}}_{0}|,\ \int_{{\cal
C}}\varrho(\mbox{\boldmath$x$})\,\,\mathrm{d}x=\int_{\cal
C}\varrho_{0}(\mbox{\boldmath$x$})\,\,\mathrm{d}x\\}.$

Then $\varrho_{s}$ and ${{\mbox{\boldmath$g$}}}_{s}$ solve the system (3.23)
and (3.24).

###### Proof.

In particular, for fixed $\varrho_{s}$, ${{\mbox{\boldmath$g$}}}_{s}$ is a
minimizer of the smooth functional

$\mathcal{I}_{\varrho_{s}}:{{\mbox{\boldmath$g$}}}\mapsto\mathcal{I}(\varrho_{s},{{\mbox{\boldmath$g$}}})$

where, for simplicity, we assume ${{\mbox{\boldmath$g$}}}\in\mathbb{R}^{2}$ as
only the first and second components of $g$ matter. We also assume that $g$
satisfies the constraint (3.6). Then the standard results for Lagrangian
multipliers yields

$-\mathcal{P}\left(\int_{\mathcal{S}}\widehat{\varrho}(\mbox{\boldmath$x$})\mbox{\boldmath$x$}\,\,\mathrm{d}x\right)-2\lambda{{\mbox{\boldmath$g$}}}_{s}=0$

for some $\lambda\in\mathbb{R}$. We get (3.24)2 assuming $\varrho$ is given by
(3.23) and $d=-2\lambda$. Likewise, for fixed ${{\mbox{\boldmath$g$}}}_{s}$,
$\varrho_{s}$ is a minimizer of the functional

$\mathcal{I}_{{\footnotesize{{\mbox{\boldmath$g$}}}_{s}}}:\varrho\mapsto\mathcal{I}(\varrho,{{\mbox{\boldmath$g$}}}_{s})$

where $\varrho$ ranges in the nonempty closed convex set

$K:=\left\\{\varrho\in L^{\gamma},\
\int_{\mathcal{C}}\varrho\,\,\mathrm{d}x=\int_{\mathcal{C}}\varrho_{0}\,\,\mathrm{d}x,\
\varrho\geq 0\ \mbox{on }\mathcal{C}\right\\}.$

According to [2, Corollary 2.184 $\&$ Example 2.186], the minimizer
$\varrho_{s}$ satisfies

$0\ni\partial\mathcal{I}_{{{\mbox{\boldmath$g$}}}_{s}}(\varrho_{s})+N_{K}(\varrho_{s})$
(3.25)

where $\partial$ denotes Frechét derivative and $N_{K}(\varrho_{s})$ is the
normal cone defined by

$N_{K}(\varrho_{s}):=\left\\{\eta\in L^{\gamma^{\prime}},\ \forall f\in K\
\int_{\mathcal{C}}\eta(\mbox{\boldmath$x$})(f(\mbox{\boldmath$x$})-\varrho_{0}(\mbox{\boldmath$x$}))\,\,\mathrm{d}x\leq
0\right\\}.$

We now analyze the structure of $N_{K}$. First, let $\eta$ be a constant. Then

$\int_{\mathcal{C}}\eta(\mbox{\boldmath$x$})(f(\mbox{\boldmath$x$})-\varrho_{s}(\mbox{\boldmath$x$}))\,\,\mathrm{d}x=0\
\ \mbox{for all $f\in K$}\,,$

which implies that every constant function belongs to $N_{K}$. Next, let
$\eta\in N_{K}$, and let us show that $\eta|_{\mbox{\rm supp}\,\varrho_{s}}$
must be a constant. Suppose otherwise. Without loss of generality, we may
assume $\int_{\mbox{\rm supp}\,\varrho_{s}}\eta\,\mathrm{d}x=0$. Consequently,
there exist $\varepsilon>0$ and sets $A,B\subset\mbox{\rm supp}\,\varrho_{s}$
of positive measure such that $\eta|_{A}>\varepsilon$,
$\eta|_{B}<-\varepsilon$, $\varrho_{s}|_{A\cup B}>\varepsilon$ and $|A|=|B|$.
Take $f=\varrho_{s}+\varepsilon(\chi_{A}-\chi_{B})$ where $\chi_{S}$ is the
characteristic function of the set $S$. Then

$\int_{\mathcal{C}}\eta(\mbox{\boldmath$x$})(f(\mbox{\boldmath$x$})-\varrho(\mbox{\boldmath$x$}))\,\,\mathrm{d}x>0\,,$

which yields that $\eta$ is not in $N_{K}$: a contraddiction. Summing up, we
can thus state that every function in $N_{K}$ is constant on $\mbox{\rm
supp}\,\varrho_{s}$ and every constant belongs to $N_{K}$. Consequently,
(3.25) yields

$P^{\prime}(\varrho_{s})-P^{\prime}(\overline{\varrho})-\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}_{s}+\lambda=0\
\mbox{on }\mbox{\rm supp}\,\varrho_{s}\,,$

for some $\lambda\in\mathbb{R}$. Since
$P^{\prime}(\varrho_{s})=\frac{a\gamma}{\gamma-1}\varrho^{\gamma-1}$, this
equation implies (3.23) with
$c=\frac{(P^{\prime}(\overline{\varrho})-\lambda)(\gamma-1)}{a\gamma}$.

∎

We are now in a position to show our existence result.

######  Theorem 3.5.

Suppose ${\cal C}$ convex. Then (3.5)–(3.7) has at least one solution.

###### Proof.

In view of Lemmas 3.2 and 3.4, we only have to prove that the functional

$\varrho,{{\mbox{\boldmath$g$}}}\mapsto\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}})$

has at least one minimizer in the set

$\mathcal{A}:=\\{(\varrho,{{\mbox{\boldmath$g$}}})\in L^{\gamma}({\cal
C})\times(\mathbb{R}^{2}\times\\{0\\}),\varrho\geq 0,\int_{\cal
C}\varrho\,\,\mathrm{d}x=\int_{\cal
C}\varrho_{0}\,\,\mathrm{d}x,|{{\mbox{\boldmath$g$}}}|=|{{\mbox{\boldmath$g$}}}_{0}|\\}.$

We begin to show that for a fixed $g$, the functional

$\mathcal{I}_{{\footnotesize{\mbox{\boldmath$g$}}}}:\varrho\mapsto\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}})$

defined on

${\cal A}_{0}:=\\{\varrho\in L^{\gamma}({\cal C}),\varrho\geq 0,\int_{\cal
C}\varrho\,\mathrm{d}x=\int_{\cal C}\varrho_{0}\,\mathrm{d}x\\}$

attains there a minimum. Consider the function

$P^{\infty}(z)=\left\\{\begin{array}[]{l}P(z)-P^{\prime}(\overline{\varrho})(z-\overline{\varrho})-P(\overline{\varrho}),\
\mbox{for }z\in[0,\infty),\\\ +\infty,\ \mbox{for
}z\in(-\infty,0).\end{array}\right.$

and redefine $\mathcal{I}_{\footnotesize{\mbox{\boldmath$g$}}}$ in the
following way

$\mathcal{I}_{\footnotesize{\mbox{\boldmath$g$}}}:\varrho\mapsto\int_{\cal
C}\left(P^{\infty}(\varrho)+\varrho\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}\right)\,\,\mathrm{d}x.$

Owing to [11, Theorem 6.54], this functional is lower semicontinuous, and
since ${\cal A}_{0}$ is convex and closed, we obtain the existence of a
minimizer by the direct method of calculus of variations; see [11, Section
3.2]. Next, consider the function

$f:{{\mbox{\boldmath$g$}}}\mapsto\min_{\varrho}\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}})\,,$

defined in
$E:=\\{{{\mbox{\boldmath$g$}}}\in\mathbb{R}^{2}\times\\{0\\},|{{\mbox{\boldmath$g$}}}|=|{{\mbox{\boldmath$g$}}}_{0}|\\}$
with values in $\mathbb{R}$. In order to show the theorem, it remains to prove
that $f$ attains a minimum in $E$. Since $E$ is compact, it suffices to check
that $f$ is continuous there. The definition of $\mathcal{E}$ yields

$|\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}}_{1})-\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}}_{2})|\leq
c|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|$

with $c$ independent of $\varrho$ (but dependent on $\varrho_{0}$), from which
it follows that

$\min_{\varrho}\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}}_{1})\leq\min_{\varrho}\mathcal{E}(\varrho,0,0,{{\mbox{\boldmath$g$}}}_{2})+c|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|.$

Interchanging the role of ${{\mbox{\boldmath$g$}}}_{1}$ and
${{\mbox{\boldmath$g$}}}_{2}$, we deduce the opposite inequality, which
furnishes the desired continuity and thus completes the proof of the theorem.

∎

###  3.4 Further comments about uniqueness

We shall now provide a result regarding the uniqueness of the “vertical”
equilibrium configurations that relates to what discussed in Section 3.2(1).
To this end, set

$\Pi({{\mbox{\boldmath$g$}}})=\mathcal{P}\left(\int_{\mathcal{C}}\sqrt[\gamma-1]{\left(\frac{\gamma-1}{a\gamma}\mbox{\boldmath$x$}\cdot{{\mbox{\boldmath$g$}}}+c\right)_{+}}\mbox{\boldmath$x$}\,\,\mathrm{d}x\right)$

where the constant $c$ is determined uniquely by (3.24)1. The following result
holds.

######  Theorem 3.6.

Assume the cavity ${\cal C}$ is such that, for any
${{\mbox{\boldmath$g$}}}\in\mathbb{R}^{2}\times\\{0\\}$,
$|{{\mbox{\boldmath$g$}}}|=|{{\mbox{\boldmath$g$}}}_{0}|$,

$\left\langle\Pi({{\mbox{\boldmath$g$}}}),{\bf l}\right\rangle>0\,.$ (3.26)

Then, the corresponding $d$ is not 0. Assume, further, we are in a class of
solutions such that

$|d|>\delta_{2}>0\,,$ (3.27)

and

$|\Pi({{\mbox{\boldmath$g$}}}_{1})-\Pi({{\mbox{\boldmath$g$}}}_{2})|\leq\delta_{1}|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|\,,$
(3.28)

for some $\delta_{2}>2\delta_{1}>0$. Then, there are at most two solutions to
(3.24), one with $d<0$ and the other with $d>0$.

###### Proof.

We begin to notice that condition (3.26) is guaranteed once we know, for
example, that every $\mbox{\boldmath$x$}\in\mathcal{C}$ satisfies
$\langle\mbox{\boldmath$x$},{\bf l}\rangle>0$. From (3.24) we get

$|d|=\frac{|\Pi({{\mbox{\boldmath$g$}}})+{\bf
l}|}{|{{\mbox{\boldmath$g$}}}_{0}|}\,,$

which, by (3.26), implies $|d|>0$. We distinguish the two cases $d>0$ and
$d<0$, and begin to treat the case $d>0$ first. Let
${{\mbox{\boldmath$g$}}}_{1},c_{1},d_{1}$ and
${{\mbox{\boldmath$g$}}}_{2},c_{2},d_{2}$ be two solutions to (3.24).
Employing (3.28), we infer

$|d_{1}-d_{2}|=\frac{||\Pi({{\mbox{\boldmath$g$}}}_{1})+{\bf
l}|-|\Pi({{\mbox{\boldmath$g$}}}_{2})+{\bf
l}||}{|{{\mbox{\boldmath$g$}}}_{0}|}\leq\frac{|\Pi({{\mbox{\boldmath$g$}}}_{1})-\Pi({{\mbox{\boldmath$g$}}}_{2})|}{|{{\mbox{\boldmath$g$}}}_{0}|}\leq\frac{\delta_{1}}{|{{\mbox{\boldmath$g$}}}_{0}|}|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|\,.$

On the other hand, from (3.24)2 we show

$0=\langle
d_{1}{{\mbox{\boldmath$g$}}}_{1}-d_{2}{{\mbox{\boldmath$g$}}}_{2}-(\Pi({{\mbox{\boldmath$g$}}}_{1})-\Pi({{\mbox{\boldmath$g$}}}_{2})),{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}\rangle=\\\
d_{1}|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|^{2}+(d_{1}-d_{2})\langle{{\mbox{\boldmath$g$}}}_{2},{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}\rangle-\langle\Pi({{\mbox{\boldmath$g$}}}_{1})-\Pi({{\mbox{\boldmath$g$}}}_{2}),{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}\rangle\\\
\geq(\delta_{2}-2\delta_{1})|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|^{2}\geq(\delta_{2}-2\delta_{1})|{{\mbox{\boldmath$g$}}}_{1}-{{\mbox{\boldmath$g$}}}_{2}|^{2}$

and thus, assuming $d>0$ and $\delta_{2}-2\delta_{1}>0$, there is at most one
solution. Note that the same conclusion holds also in the case $d<0$.

∎

######  Remark 3.7.

The assumptions of Theorem 3.6 are rather significant. In fact, they ensure
that the center of mass of the whole system does not vary too much for
different directions of gravity. We now show that these assumptions are
somehow also necessary, by bringing an example that shows that, if they are
violated, the conclusion of the theorem is not true. Let consider
$\mathcal{C}=(-1,1)\times(-1,1)\times(-1,1)$, $\gamma=2$, and $a=\frac{1}{2}$.
The total mass is assumed to be 4. Furthermore, the body is such that ${\bf
l}=(1,0,0)$. Then ${{\mbox{\boldmath$g$}}}_{1}=(1,0,0)$ and
${{\mbox{\boldmath$g$}}}_{2}=(-1,0,0)$ are two solutions for which the
appropriate $c_{1}$ and $c_{2}$ is both equal to $1$. However, for
${{\mbox{\boldmath$g$}}}_{1}$ we have

$\Pi({{\mbox{\boldmath$g$}}}_{1})+{\bf
l}=(11/3,0,0)=\frac{11}{3}{{\mbox{\boldmath$g$}}}_{1}$

and for ${{\mbox{\boldmath$g$}}}_{2}$ we have

$\Pi({{\mbox{\boldmath$g$}}}_{2})+{\bf
l}=(-5/3,0,0)=\frac{5}{3}{{\mbox{\boldmath$g$}}}_{2}$

and we have two solutions for which $d>0$ (namely, $d_{1}=\frac{11}{3}$ and
$d_{2}=\frac{5}{3}$). Notice that in this case, $\delta_{1}=2$, $|d|=11/3$, so
that $|d|<2\delta_{1}$, and (3.28) is violated, for all
$\delta_{2}>2\delta_{1}$.

##  4 Global behavior of weak solutions

For simplicity, in what follow we set $\mathcal{C}_{0}\equiv\mathcal{C}$. We
begin by stating an existence result of weak solutions.

######  Theorem 4.1.

Let $\mathcal{C}$ be of class $C^{2+\nu}$, for some $\nu>0$, and let
$\varrho_{0}\in L^{\gamma}(\mathcal{C})$, $\gamma>3/2$, with
$\varrho_{0}|_{\mathscr{B}}=\varrho_{c}$, $\varrho_{c}\in\mathbb{R}$. Further,
let ${{\mbox{\boldmath$u$}}}_{0}:\mathcal{S}\to\mathbb{R}^{3}$ be such that
$\varrho_{0}|{{\mbox{\boldmath$u$}}}_{0}|^{2}\in L^{1}(\mathcal{C})$ and
${{\mbox{\boldmath$u$}}}_{0}|_{\mathscr{B}}=\omega{{\mbox{\boldmath$e$}}}_{1}\times{\mbox{\boldmath$x$}}$
for some $\omega\in\mathbb{R}$. Then there exists a weak solution to (2.7) in
the sense of Definition 5 on the time interval $(0,T)$, arbitrary $T>0$.

######  Remark 4.2.

The proof of this theorem is omitted, since it can be obtained by simply
combining the arguments used in [7] in the case when the motion of
$\mathscr{B}$ is prescribed with those of [14], where the motion of
$\mathscr{B}$ is a further unknown. The crucial point is to show uniform
estimates to derive the regularity of the pressure by using the Bogovski
operator that allows for the passage to the limit in the pressure term. A
detailed treatment of this issue can be found in [14] and [8].

######  Remark 4.3.

The regularity on $\mathcal{C}$ stated in the theorem could be relaxed to
assume ${\cal C}$ to be just of class $C^{0,1}$ (or even less regular). The
method may be found in [23].

###  4.1 Global estimates

Hereinafter we assume that $\varrho$ and $u$ is a weak solution in the sense
of Definition 5. Moreover, we assume $\mathcal{C}$ is of class $C^{0,1}$ and
$\gamma>\frac{3}{2}$. We recall that, from the energy inequality, we deduce
the following estimates

$\begin{split}\mbox{ess
sup}_{t\in(0,\infty)}\|\varrho(t,\cdot)\|_{\gamma}&\leq c\\\
\sup_{t\in(0,\infty)}|\omega(t)|&\leq c\\\ \mbox{ess
sup}_{t\in(0,\infty)}\|\varrho(t,\cdot)|{\mbox{\boldmath$u$}}|^{2}(t,\cdot)\|_{1}&\leq
c\\\ \|{{\mbox{\boldmath$u$}}}\|_{L^{2}((0,\infty)\times\mathcal{C})}&\leq
c\\\ \|\nabla{\mbox{\boldmath$u$}}\|_{L^{2}((0,\infty)\times\mathcal{C})}&\leq
c\end{split}$

for some $c>0$. In order to perform the long-time analysis of our solutions we
need some other uniform bounds that we are going to derive. First, we observe
that from (2.7)5 we get at once

$|{{\mbox{\boldmath$g$}}}(t)|=|{{\mbox{\boldmath$g$}}}(0)|\ \mbox{for all
}t\in(0,\infty).$ (4.1)

We next define the sequence

$(\varrho_{n}(t),{\mbox{\boldmath$v$}}_{n}(t),\omega_{n}(t)):=(\varrho(n+t),{\mbox{\boldmath$v$}}(n+t),\omega(n+t))$

and investigate its behavior as $n\to\infty$. Throughout, we shall use the
letter $c$ to denote an arbitrary constant independent of $n$. We begin to
show higher integrability properties of the density, by adapting a method from
[20, Section 7.9.5]. Consider the test function

$\varphi(t,\mbox{\boldmath$x$})=\psi(t)\Phi(t,\mbox{\boldmath$x$}),\
\Phi={\mathfrak{B}}\left(S_{\alpha}(b_{k}(\varrho_{n}))-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathcal{C}}S_{\alpha}(b_{k}(\varrho_{n}))\,{\rm
d}t\right)$

where $\psi\in C^{\infty}_{c}(-1,2)$, ${\mathfrak{B}}$ is the Bogovski
operator, $S_{\alpha}$ is a mollifying operator with respect to time and

$b_{k}(\varrho)=\left\\{\begin{array}[]{l}\varrho^{\nu}\ \mbox{for
}\varrho\in[0,k)\\\ k^{\nu}\ \mbox{for
}\varrho\in[k,\infty)\end{array}\right.$

for some $\nu\in(0,\frac{2}{3}\gamma-1]$. Such a $\varphi$ is an admissible
test function for (2.7). We thus obtain

$\int_{-1}^{2}\psi\int_{\mathcal{C}}p(\varrho_{n})S_{\alpha}(b_{k}(\varrho_{n}))\,\,\mathrm{d}x{\rm
d}t=\int_{-1}^{2}\int_{\mathcal{C}}\psi
p(\varrho_{n})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathcal{C}}S_{\alpha}(b_{k}(\varrho_{n})\right)\,\,\mathrm{d}x{\rm
d}t\\\
+\int_{-1}^{2}\int_{\mathcal{C}}\psi\mathbb{S}({\mbox{\boldmath$v$}}_{n}):\nabla\Phi\,\,\mathrm{d}x{\rm
d}t+\int_{-1}^{s}\int_{\mathcal{C}}\psi\varrho_{n}{{\mbox{\boldmath$g$}}}\cdot\Phi\,\,\mathrm{d}x{\rm
d}t-\int_{-1}^{2}\int_{\mathcal{C}}\psi\varrho_{n}{\mbox{\boldmath$v$}}_{n}\otimes{\mbox{\boldmath$u$}}_{n}:\nabla\Phi\,\,\mathrm{d}x{\rm
d}t\\\
-\int_{-1}^{2}\int_{\mathcal{C}}\varrho_{n}{\mbox{\boldmath$u$}}_{n}\cdot\Phi\partial_{t}\psi\,\,\mathrm{d}x{\rm
d}t-\int_{-1}^{2}\int_{\mathcal{C}}\varrho_{n}{\mbox{\boldmath$u$}}_{n}\cdot\partial_{t}\Phi\psi\,\,\mathrm{d}x{\rm
d}t+\int_{-1}^{2}\int_{\mathcal{C}}\psi\varrho_{n}\omega_{n}{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}_{n}\cdot\Phi\,\,\mathrm{d}x{\rm
d}t$

Every term above, except for the last one, may be estimated similarly as it is
done in [20, Section 7.9.5.2]. The last term may be estimated as follows
(compare with the estimate of term $J_{5}$ in [20, Section 7.9.5.2])

$\left|\int_{-1}^{2}\int_{\mathcal{C}}\varrho_{n}\omega_{n}{{\mbox{\boldmath$e$}}}_{3}\times{\mbox{\boldmath$u$}}_{n}\Phi\,\,\mathrm{d}x{\rm
d}t\right|\leq
c\int_{-1}^{2}\int_{\mathcal{C}}|\psi|\varrho_{n}|\omega_{n}|^{2}|\Phi|\,\,\mathrm{d}x{\rm
d}t+c\int_{-1}^{2}\int_{\mathcal{C}}|\psi|\varrho_{n}|\omega_{n}||{\mbox{\boldmath$v$}}_{n}||\Phi|\,\,\mathrm{d}x{\rm
d}t\\\ \leq
c\|\psi\|_{L^{1}}\|S_{\alpha}(b_{k}(\varrho_{n}))\|_{L^{\infty}(L^{\frac{6\gamma}{5\gamma-3}})}.$

Thus, one may let $\alpha\to 0$ and $k\to\infty$ and, in the same fashion as
[20], to deduce

$\int_{0}^{1}\int_{\mathcal{C}}\varrho_{n}^{\gamma+\nu}\,\,\mathrm{d}x{\rm
d}t\leq c.$

Furthermore, from the energy and Korn’s inequalities we easily derive

$\int_{\tau}^{\tau+1}\int_{\cal
C}|\nabla{\mbox{\boldmath$v$}}_{n}|^{2}\,\,\mathrm{d}x{\rm d}t\to 0$

as $\tau\to\infty$. As a result, along a subsequence,

$\begin{split}\varrho_{n}&\to\varrho_{s}\ \mbox{weakly in
}L^{\gamma+\nu}((0,1)\times{\cal C})\\\
{\mbox{\boldmath$v$}}_{n}&\to{\mbox{\boldmath$v$}}_{s}\equiv 0\ \mbox{weakly
in }L^{2}(0,1;W^{1,2}({\cal C}))\\\ \omega_{n}&\to\omega_{s}\
\mbox{weakly}^{*}\ \mbox{in }L^{\infty}(0,1)\\\ p(\varrho_{n})&\to
p(\varrho)_{s}\ \mbox{weakly in }L^{1+\gamma/\nu}((0,1)\times{\cal
C}).\end{split}$ (4.2)

The functions $\varrho_{s},\ {\mbox{\boldmath$v$}}_{s},\ \omega_{s}$ and
$p(\varrho)_{s}$ solve (3.4) and thus $\omega_{s}=0$. Notice that
$p(\varrho)_{s}$ denotes a weak limit of $p(\varrho_{n})$ and since $p$ is
nonlinear, it is not necessarily true that $p(\varrho)_{s}=p(\varrho_{s})$. We
shall address this issue in the next subsection.

###  4.2 Limit of the pressure term

We will prove that $p(\varrho)_{s}=p(\varrho_{s})$. To this end, it is
sufficient to adapt the method from [9, Section 4]. Let

$G(z)=z^{\alpha},\
0<\alpha<\min\left\\{\frac{1}{2\gamma},\frac{\nu}{2(\nu+\gamma)}\right\\}$

and consider a function $b(z)=G(p(z))$ in (2.12) to deduce

$|\langle\partial_{t}G(p(\varrho_{n})),\varphi\rangle|=\\\
\left|\int_{0}^{1}\int_{\mathcal{C}}G(p(\varrho_{n})){\mbox{\boldmath$v$}}_{n}\nabla\varphi\,\,\mathrm{d}x{\rm
d}t+\int_{0}^{1}\int_{\mathcal{C}}(G(p(\varrho_{n}))-G^{\prime}(p(\varrho_{n}))\varrho_{n})\varphi{\mbox{\boldmath$v$}}\,\,\mathrm{d}x{\rm
d}t\right|\leq c\|\varphi\|_{1,q_{1}},$

for some $q_{1}>1$ and for $\varphi\in
C^{\infty}_{c}((0,1)\times\mathcal{C})$. Consequently

$\mbox{Div}_{t,x}(G(p(\varrho_{n}),0,0,0)\ \mbox{is precompact in
}W^{-1,q_{1}}_{\operatorname{loc}}((0,1)\times{\cal C}).$

We know that

$|\langle\nabla
p(\varrho_{n}),\varphi\rangle|=\left|-\int_{0}^{1}\int_{\mathcal{C}}p(\varrho_{n})\operatorname{div}\varphi\,\,\mathrm{d}x{\rm
d}t\right|\leq c\|\varphi\|_{1,q_{2}}.$

for some $q_{2}>1$ and for $\varphi\in
C^{\infty}_{c}((0,1)\times\mathcal{C})$. Thus

$\mbox{Curl}_{t,x}(p(\varrho_{n}),0,0,0)\ \mbox{is precompact in
}W^{-1,q_{2}}_{\operatorname{loc}}((0,1)\times{\cal C})$

The well known div-curl lemma (see [28]) yields

$G(p(\varrho_{n}))p(\varrho_{n})\to G(p(\varrho)_{s})p(\varrho)_{s}.$ (4.3)

According to [27, Theorem 6.2] there exists a parametrized family of
probabilistic measures $\nu_{t,x}$ on $[0,\infty)$ such that

$\varrho_{s}(t,\mbox{\boldmath$x$})=\int_{0}^{\infty}\rho\,{\rm
d}\nu_{t,x}(\rho).$

and, according to (4.3), we also have

$\int_{0}^{\infty}\rho^{\alpha\gamma+\gamma}\,{\rm
d}\nu_{t,x}(\rho)=\int_{0}^{\infty}\rho^{\alpha\gamma}\,{\rm
d}\nu_{t,x}(\rho)\int_{0}^{\infty}\rho^{\gamma}\,{\rm d}\nu_{t,x}(\rho).$
(4.4)

where we assume for simplicity that $p(\varrho)=\varrho^{\gamma}$. Fix
$(t,\mbox{\boldmath$x$})$ and set
$\theta^{\alpha\gamma}:=\int_{0}^{\infty}\rho^{\alpha\gamma}\ {\rm
d}\nu_{t,x}(\rho)$. Then (4.4) yields

$\int_{0}^{\infty}\left(\rho^{\gamma\alpha+\gamma}-\theta^{\gamma\alpha}\rho^{\gamma}-\theta^{\gamma}(\theta^{\alpha\gamma}-\rho^{\alpha\gamma})\right)\,{\rm
d}\nu_{t,x}(\rho)=0,$

which transforms into

$\int_{0}^{\infty}(\rho^{\gamma}-\theta^{\gamma})(\rho^{\alpha\gamma}-\theta^{\alpha\gamma})\,{\rm
d}\nu_{t,x}(\rho)=0.$

The integrand is strictly positive for all $\rho\neq\theta$ and since
$\nu_{t,x}$ is a probabilistic measure, we get
$\nu_{t,x}=\delta_{\varrho_{s}(t,\mbox{\boldmath$x$})}$ where
$\delta_{\alpha}$ is a Dirac mass at point $\alpha$. Consequently,

$\varrho_{n}\to\varrho_{s}\ \mbox{strongly in }L^{q}((0,1)\times{\cal C}),$

for all $q\in[1,\gamma+\nu)$ yielding $p(\varrho)_{s}=p(\varrho_{s})$.

###  4.3 Large-time behavior

In view of what we have proved in the previous subsections, we may now proceed
to the limit in (2.7) and deduce that $\varrho_{s},\ \omega_{s},\
{{\mbox{\boldmath$g$}}}_{s}$ solve (3.4). Furthermore, (4.2) allows us to pass
to a limit also in the energy as follows

$\lim_{t_{n}\to\infty}\int_{t_{n}}^{t_{n}+1}\mathcal{E}(\varrho(t),{{\mbox{\boldmath$u$}}}(t),\omega(t),{{\mbox{\boldmath$g$}}}(t))\
{\rm d}t=\mathcal{E}(\varrho_{s},0,0,{{\mbox{\boldmath$g$}}}_{s})\,.$

We will assume that there is only one solution to (3.7) fulfilling the
condition

$\mathcal{E}(\varrho_{s},0,0,{{\mbox{\boldmath$g$}}}_{s})\leq\mathcal{E}(\varrho_{0},{{\mbox{\boldmath$u$}}}_{0},\omega_{0},{{\mbox{\boldmath$g$}}}_{0})\,.$

In such a case, as there is only one possible limit, we immediately get
${{\mbox{\boldmath$g$}}}(t)\to{{\mbox{\boldmath$g$}}}_{s}$ as $t\to\infty$.
Due to (2.7)2 we have $\partial_{t}\varrho\in L^{2}(W^{-1,2})$ and,
consequently $\varrho(t)\to\varrho_{s}$ as $t\to\infty$.

We have just proved the following theorem.

######  Theorem 4.4.

Let ${\cal C}$ be a Lipschitz domain and let the initial conditions
$\varrho_{0},{\mbox{\boldmath$v$}}_{0},\omega_{0}$ and
${{\mbox{\boldmath$g$}}}_{0}$ be the same as in Theorem 4.1. Assume that there
is just one solution to (3.4) for which
$\mathcal{E}(\varrho_{s},{\mbox{\boldmath$0$}},0,{{\mbox{\boldmath$g$}}}_{s})\leq\mathcal{E}(\varrho_{0},{{\mbox{\boldmath$u$}}}_{0},\omega_{0},{{\mbox{\boldmath$g$}}}_{0})$.
Then every renormalized weak solution to (2.7) tends to
$(\varrho_{s},{\mbox{\boldmath$0$}},0,{{\mbox{\boldmath$g$}}}_{s})$. More
precisely,

$\begin{split}\varrho(t)&\to\varrho_{s}\ \mbox{weakly in }L^{\gamma}\ \mbox{as
}t\to\infty,\\\ {\mbox{\boldmath$v$}}(t_{n}+t)&\to 0\ \mbox{strongly in
}L^{2}(0,1;W^{1,2}({\cal C}))\ \mbox{as}\ t_{n}\to\infty,\\\ \omega(t)&\to
0\mbox{ as }t\to\infty,\\\
{{\mbox{\boldmath$g$}}}(t)&\to{{\mbox{\boldmath$g$}}}_{s}\ \mbox{ as
}t\to\infty.\end{split}$

As a simple application of this theorem, consider the case when the cavity
${\cal C}$ is a sphere $S$ with its center $O^{\prime}$ belonging to the
straight line $\overline{OG}$, $G\neq O$. Then, from Section 3.2(3), we know
that there are two and only two equilibrium configurations, namely, with the
pendulum either in the straight-down or straight-up position. More precisely,
these configurations are characterized by two numbers $\sigma^{+}>0$ and
$\sigma^{-}<0$, such that

$\overrightarrow{OC^{\pm}}=\sigma^{\pm}\,{\mbox{\boldmath$e$}}_{1},$ (4.5)

corresponding to the case when the center of mass $C$ of the coupled system
$\mathscr{S}$ is below $(C^{+})$ or above $(C^{-})$ the hinge. Let us denote
by $(\varrho_{s}^{+},{\mbox{\boldmath$g$}}_{s}^{+})$ and
$(\varrho_{s}^{-},{\mbox{\boldmath$g$}}_{s}^{-})$ the two associated steady-
state solutions, and set ${\cal E}^{\pm}:={\cal
E}(\varrho_{s}^{\pm},{\mbox{\boldmath$0$}},0,{{\mbox{\boldmath$g$}}}_{s}^{\pm})$.
Thus, $\varrho_{s}^{+}=\varrho_{s}^{-}\equiv r_{s}$, with $r_{s}$ given in
(3.9), and

$\int_{S^{+}}P(\varrho_{s}^{+})-P^{\prime}(\overline{\varrho})(\varrho_{s}^{+}-\overline{\varrho})-P(\overline{\varrho})\,\mathrm{d}x=\int_{S^{-}}P(\varrho_{s}^{-})-P^{\prime}(\overline{\varrho})(\varrho_{s}^{-}-\overline{\varrho})-P(\overline{\varrho})\,\mathrm{d}x\,,$

where $S^{+}$ [resp. $S_{-}$] denotes the position of the sphere in the
straight-down [resp. straight-up] configuration of $\mathscr{S}$. Moreover,

$\int_{\mathscr{B}}\widehat{\varrho}\,\mbox{\boldmath$x$}\cdot{\mbox{\boldmath$g$}}=g{\cal
M}\,\overrightarrow{OC}\cdot{\mbox{\boldmath$e$}}_{1}\,.$

Collecting all the above, using (4.5) and recalling (2.13), we show

$\mathcal{E}^{+}-\mathcal{E}^{-}=-g\,{\cal M}\,(\sigma_{+}+\sigma_{-})<0\,.$
(4.6)

Therefore, if we choose the initial data in such a way that

${\cal
E}^{+}<\mathcal{E}(\varrho_{0},{{\mbox{\boldmath$u$}}}_{0},\omega_{0},{{\mbox{\boldmath$g$}}}_{0})<{\cal
E}^{-}\,,$ (4.7)

then every (renormalized) weak solution will converge for large times to
$(\varrho_{s}^{+},{\mbox{\boldmath$g$}}^{+})$, namely, the pendulum will
eventually reach the configuration with its center of mass in its lowest
position. This will certainly happen, if we start the pendulum from rest
(${\mbox{\boldmath$u$}}_{0}\equiv{\bf 0},\omega_{0}=0)$ and pick
$(\varrho_{0},{\mbox{\boldmath$g$}}_{0})\neq(\varrho^{\pm},{\mbox{\boldmath$g$}}^{\pm})$,
that is, the pendulum is initially away from either straight-down and
straight-up configurations.777Actually, if $\mathscr{S}$ is initially in one
of these two positions with ${\mbox{\boldmath$u$}}_{0}$ and $\omega_{0}$ both
vanishing, it will stay there for all times. In fact, from Lemma 3.4 we know
that any minimizer of ${\cal E}(\varrho,{\bf 0},0,{\mbox{\boldmath$g$}})$ is a
solution to (3.5)–(3.7) and, by Theorem 3.5 that the set of minimizers is not
empty. However, from the results of Section 3.2(3) and (4.6), we deduce that
$(\varrho^{+},{\mbox{\boldmath$g$}}^{+})$ is the only minimizer, which proves
our claim.

##  5 Numerical results

As we mentioned in the introductory section, in [15] a problem analogous to
the one treated here was investigated under the assumption that the fluid
filling the cavity was incompressible. One interesting point to investigate is
whether there is any quantitative difference between the two problems. For
example, how the characteristic time taken to reach the terminal state (the
rest) depends on the compressibility of the fluid. Unfortunately, an analytic
study of such a question is, to date, beyond our grasp. However, we have
performed numerical tests that may suggest the answer. Objective of this
section is to present these findings.

For simplicity, we assume that the flow is two-dimensional – this is a
reasonable assumption as the physical phenomenon may hint to neglect the third
dimension.

We propose a mixed finite volume – finite element method for the approximation
of the system (2.7) that we are going to describe next.

###  5.1 The mixed finite volume – finite element scheme

To begin, let ${\cal C}_{h}$ be a regular and quasi-uniform triangulation of
the cavity ${\cal C}$ and $\mathcal{F}_{h}$ be the set of all interior faces
of ${\cal C}_{h}$. Further, we write $h=\max_{K\in{\cal C}_{h}}h_{K}$ as the
mesh size, where $h_{K}$ is the diameter of an element $K\in{\cal C}_{h}$. We
denote by $Q_{h}$ the space of piecewise constant functions and by
$\mathbf{V}_{h}$ the piecewise linear Crouzeix–Raviart element space:

$Q_{h}=\left\\{v\in L^{1}({\cal C})|\;{\mbox{\boldmath$v$}}_{K}\text{ is a
constant}\;\forall\;K\in{\cal C}_{h}\right\\},$
$\mathbf{V}_{h}=\left\\{{\mbox{\boldmath$v$}}\in L^{2}({\cal
C})\;|\;{\mbox{\boldmath$v$}}_{K}\text{ is a piecewise affine
function}\;\forall\;K\in{\cal
C}_{h};\;\int_{\sigma}\left[\left[{\mbox{\boldmath$v$}}\right]\right]{\rm
d}S_{x}=0\;\forall\;\sigma\in\mathcal{F}_{h}\right\\},$

where $\left[\left[\cdot\right]\right]|_{\sigma}$ represents the jump over the
interface $\sigma$. To specify the homogeneous Dirichlet boundary condition,
we define

$\mathbf{V}_{0,h}=\left\\{\mbox{\boldmath$\varphi$}\in\mathbf{V}_{h}\;|\;\int_{\sigma}\mbox{\boldmath$\varphi$}{\rm
d}S_{x}=0\;\forall\;\sigma\in\partial{\cal C}\right\\}.$

Now we are ready to introduce the following mixed finite volume – finite
element method.

##### Numerical method (compressible solver).

Let $\Delta t$ be the time increment,
${{\mbox{\boldmath$g$}}}_{h}^{0}={{\mbox{\boldmath$g$}}}(0)$,
$\omega_{h}^{0}=\omega(0)$, and let
$(\varrho_{h}^{0},{{\mbox{\boldmath$u$}}}_{h}^{0})$ be the projection of the
initial data $(\varrho,{{\mbox{\boldmath$u$}}})(0)$ onto the space
$Q_{h}\times\mathbf{V}_{h}$. Then, for $k=1,\dots,N_{t}=T/\Delta t$ we seek
$(\varrho_{h}^{k},{{\mbox{\boldmath$u$}}}_{h}^{k},{{\mbox{\boldmath$g$}}}_{h}^{k},\omega_{h}^{k})\in
Q_{h}\times\mathbf{V}_{h}\times\mathbb{R}^{2}\times\mathbb{R}$ as solutions to
the following system of algebraic equations

$D_{t}{{\mbox{\boldmath$g$}}}_{h}^{k}+\omega_{h}^{k-1}{{\mbox{\boldmath$e$}}}_{3}\times{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}=0,$
(5.1a) where
${{\mbox{\boldmath$g$}}}_{h}^{k+1/2}=\frac{{{\mbox{\boldmath$g$}}}_{h}^{k-1}+{{\mbox{\boldmath$g$}}}_{h}^{k}}{2}$
and $D_{t}v_{h}^{k}=\frac{v_{h}^{k}-v_{h}^{k-1}}{\Delta t}$;
$\int_{K}D_{t}\varrho_{h}^{k}\,\mathrm{d}x+\int_{\partial K}\varrho_{h}^{k,\rm
up}{\mbox{\boldmath$v$}}_{h}^{k-1}\cdot{{\mbox{\boldmath$n$}}}{\rm
d}S_{x}=0\text{ for all }K\in{\cal C}_{h}\,,$ (5.1b) where
${\mbox{\boldmath$v$}}_{h}={{\mbox{\boldmath$u$}}}_{h}-{{\mbox{\boldmath$u$}}}_{B},{{\mbox{\boldmath$u$}}}_{B}=\omega_{h}{{\mbox{\boldmath$e$}}}_{3}\times\mbox{\boldmath$x$}$,
$n$ is the outer normal vector, and $\varrho_{h}^{up}$ is the so-called upwind
value of the density given by $\varrho_{h}^{up}=\begin{cases}\lim_{\delta\to
0}\varrho_{h}(\mbox{\boldmath$x$}+\delta{{\mbox{\boldmath$n$}}})&\text{if
}{\mbox{\boldmath$v$}}_{h}\cdot{{\mbox{\boldmath$n$}}}\geq 0,\\\
\lim_{\delta\to
0}\varrho_{h}(\mbox{\boldmath$x$}-\delta{{\mbox{\boldmath$n$}}})&\text{otherwise};\end{cases}$
$\frac{1}{2}\int_{{\cal
C}}\Big{(}D_{t}(\varrho_{h}^{k}{{\mbox{\boldmath$u$}}}_{h}^{k})\cdot\mbox{\boldmath$\varphi$}+\varrho_{h}^{k-1}D_{t}{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}+\varrho_{h}^{k}{\mbox{\boldmath$v$}}_{h}^{k-1}\cdot\nabla{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}-\varrho_{h}^{k}{\mbox{\boldmath$v$}}_{h}^{k-1}\cdot\nabla\mbox{\boldmath$\varphi$}\cdot{{\mbox{\boldmath$u$}}}_{h}^{k}\Big{)}\,\mathrm{d}x\\\
+\int_{{\cal
C}}\varrho_{h}^{k}w_{h}^{k-1}{{\mbox{\boldmath$e$}}}_{3}\times{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x+\int_{{\cal
C}}\Big{(}\mathbb{S}({{\mbox{\boldmath$u$}}}_{h}^{k}):\nabla\mbox{\boldmath$\varphi$}-p(\varrho_{h}^{k})\operatorname{div}\mbox{\boldmath$\varphi$}\Big{)}\,\mathrm{d}x=\int_{{\cal
C}}\varrho_{h}^{k}{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x,\\\
\mbox{ for all }\;\mbox{\boldmath$\varphi$}\in\mathbf{V}_{0,h};$ (5.1c)
$I_{B33}D_{t}\omega_{h}^{k}+D_{t}\left(\int_{{\cal
C}}\varrho_{h}^{k}\mbox{\boldmath$x$}\times{{\mbox{\boldmath$u$}}}_{h}^{k}\,\mathrm{d}x\right)\cdot{{\mbox{\boldmath$e$}}}_{3}=\int_{\mathscr{B}+{\cal
C}}\varrho_{h}^{k}\mbox{\boldmath$x$}\,\mathrm{d}x\times{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}\cdot{{\mbox{\boldmath$e$}}}_{3}.$

######  Remark 5.1.

The scheme (5.1) enjoys the following properties for all $k=1,\dots,N_{t}$:

  * •

Conservation of mass. Indeed, summing up over all elements leads to the mass
conservation.

$\int_{{\cal C}}\varrho_{h}^{k}\,\mathrm{d}x=\int_{{\cal
C}}\varrho_{h}^{k-1}\,\mathrm{d}x=\cdots=\int_{{\cal
C}}\varrho_{h}^{0}\,\mathrm{d}x.$

  * •

Conservation of gravity in the sense of (4.1), i.e.,

$|{{\mbox{\boldmath$g$}}}_{h}^{k}|=|{{\mbox{\boldmath$g$}}}_{h}^{k-1}|=\cdots=|{{\mbox{\boldmath$g$}}}_{h}^{0}|,$

which can be easily obtained by multiplying (5.1a) with
${{\mbox{\boldmath$g$}}}_{h}^{k+1/2}=\frac{{{\mbox{\boldmath$g$}}}_{h}^{k-1}+{{\mbox{\boldmath$g$}}}_{h}^{k}}{2}$.

  * •

Positivity preserving of density. We have $\varrho_{h}^{k}>0$ provided
$\varrho_{h}^{0}>0$, for which we refer the proof to [21, Lemma 4.1].

###  5.2 Numerical experiments

We take the pendulum as a circular plate with a circular cavity in the center

$\mathscr{B}=\left\\{\mbox{\boldmath$x$}\;|R_{0}\leq\sqrt{(x_{1}-L)^{2}+x_{2}^{2}}\leq
R_{1}\right\\},\quad{\cal
C}=\left\\{\mbox{\boldmath$x$}\;|\sqrt{(x_{1}-L)^{2}+x_{2}^{2}}\leq
R_{0}\right\\},$

with $R_{0}=0.1$, $R_{1}=0.2$, and $L$ be the length of the pendulum, see
Figure 2. In our numerical experiments we set $\gamma=5/3$, $\mu=100$, and
$\eta=0$ if not otherwise mentioned. Further, we denote
$\varrho_{\mathscr{B}}$ as the density of the body $\mathscr{B}$,
$\overline{\varrho_{0}}=\frac{1}{|{\cal C}|}\int_{{\cal
C}}\varrho(0)\,\mathrm{d}x$ as the averaged initial density of the fluid in
the cavity ${\cal C}$ and
$R_{\varrho}=\varrho_{\mathscr{B}}/\overline{\varrho_{0}}$ as the ratio of the
densities. The initial data are set as
$\varrho(0)=1,{{\mbox{\boldmath$u$}}}(0)={\bf
0},\omega(0)=0,{{\mbox{\boldmath$g$}}}(0)=(\cos{\vartheta_{0}},\sin{\vartheta_{0}})$
with $\vartheta_{0}=\pi/45$.

$x_{1}$$x_{2}$$g$$L$$R_{0}$$R_{1}$$\vartheta$ Figure 2: Pendulum with a
cavity.

####  5.2.1 Experiment 1: influence of gas parameters of the compressible
solver (5.1)

We show in Figure 3 \- Figure 5 the evolution of pendulum position (angle
$\vartheta$) for different values of density ratio $R_{\varrho}$, gas
parameter $a$, and pendulum length $L$. First of all, in all these numerical
experiments we observe the effect of the dissipation due to the viscosity of
the fluid. Moreover, we see _larger_ dissipation effects for:

  1. 1.

smaller density ratio $R_{\varrho}$ (fixed gas parameter $a$ and pendulum
length $L$) in Figure 3;

  2. 2.

smaller gas parameter $a$ (fixed density ratio $R_{\varrho}$ and pendulum
length $L$) in Figure 4;

  3. 3.

smaller pendulum length $L$ (fixed density ratio $R_{\varrho}$ and gas
parameter $a$) in Figure 5.

Figure 3: Evolution of pendulum position (angle $\vartheta$) for different
density ratio $R_{\varrho}$ with fixed gas parameter $a=10$ and pendulum
length $L=0.4$. Figure 4: Evolution of pendulum position (angle $\vartheta$)
for different gas parameter $a$ with fixed density ratio $r=1$ and length
$L=0.4$. Figure 5: Evolution of pendulum position (angle $\vartheta$) for
different pendulum length $L$ with fixed density ratio $r=1$ and gas parameter
$a=10$.

####  5.2.2 Experiment 2: comparison with an incompressible solver

To compare the damping effects of compressible and incompressible fluids, we
also introduce an incompressible solver which replaces the Navier–Stokes part
of the compressible solver (5.1), that is (5.1b)–(5.1c), by it incompressible
counterpart, while keeping the method of ${{\mbox{\boldmath$g$}}}_{h}$ and
$\omega_{h}$ unchanged.

##### Incompressible solver.

Let
${{\mbox{\boldmath$u$}}}_{h}^{0},{{\mbox{\boldmath$g$}}}_{h}^{0},\omega_{h}^{0}$
be given in the same way as the compressible solver. Let $\varrho_{\cal C}$ be
the density of the incompressible fluid in the cavity ${\cal C}$. For
$k=1,\ldots,N_{t}$ we seek
$(p_{h}^{k},{{\mbox{\boldmath$u$}}}_{h}^{k},{{\mbox{\boldmath$g$}}}_{h}^{k},\omega_{h}^{k})\in
X_{h}\times\mathbf{V}_{h}\times\mathbb{R}^{2}\times\mathbb{R}$ such that the
following system of algebraic equations hold:

$D_{t}{{\mbox{\boldmath$g$}}}_{h}^{k}+\omega_{h}^{k-1}{{\mbox{\boldmath$e$}}}_{3}\times{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}=0,$
(5.2) $\displaystyle\varrho_{\cal C}\int_{{\cal
C}}\Big{(}D_{t}{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}+\frac{1}{2}{\mbox{\boldmath$v$}}_{h}^{k-1}\cdot\nabla{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}-\frac{1}{2}{\mbox{\boldmath$v$}}_{h}^{k-1}\cdot\nabla\mbox{\boldmath$\varphi$}\cdot{{\mbox{\boldmath$u$}}}_{h}^{k}\Big{)}\,\mathrm{d}x+\varrho_{\cal
C}\int_{{\cal
C}}w_{h}^{k-1}{{\mbox{\boldmath$e$}}}_{3}\times{{\mbox{\boldmath$u$}}}_{h}^{k}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x$
$\displaystyle+\int_{{\cal
C}}\Big{(}\mathbb{S}({{\mbox{\boldmath$u$}}}_{h}^{k}):\nabla\mbox{\boldmath$\varphi$}-p_{h}^{k}\operatorname{div}\mbox{\boldmath$\varphi$}-q_{h}\operatorname{div}{{\mbox{\boldmath$u$}}}_{h}^{k}\Big{)}\,\mathrm{d}x=\int_{{\cal
C}}\varrho_{\cal
C}{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}\cdot\mbox{\boldmath$\varphi$}\,\mathrm{d}x,\quad\forall\;q_{h}\in
X_{h},\;\mbox{\boldmath$\varphi$}\in\mathbf{V}_{0,h},$

$I_{B33}D_{t}\omega_{h}^{k}+D_{t}\left(\int_{{\cal C}}\varrho_{\cal
C}\mbox{\boldmath$x$}\times{{\mbox{\boldmath$u$}}}_{h}^{k}\,\mathrm{d}x\right)\cdot{{\mbox{\boldmath$e$}}}_{3}=\left(\int_{\mathscr{B}}\varrho_{\mathscr{B}}\mbox{\boldmath$x$}\,\mathrm{d}x+\int_{{\cal
C}}\varrho_{\cal
C}\mbox{\boldmath$x$}\,\mathrm{d}x\right)\times{{\mbox{\boldmath$g$}}}_{h}^{k+1/2}\cdot{{\mbox{\boldmath$e$}}}_{3},$

where $X_{h}:=\\{v\in L^{2}({\cal C})|\;v_{K}\text{ is a constant
}\;\forall\;K\in{\cal C}_{h};\int_{{\cal C}}v\,\mathrm{d}x=0\\}$.

We show in Figure 6 the evolution of pendulum positions (represented by the
angle $\vartheta$) obtained by the compressible solver and the incompressible
solver. Here we have used _same parameters_ for both solvers: $L=0.4$,
$\mu=100$, $\eta=0$, $\varrho_{\mathscr{B}}=1$, and initial fluid density
$\varrho_{\cal C}=1.0$. Note that the only difference relies on the gas
parameter $a(=0.1,\;20,\;100)$ in the compressible solver, which is not needed
in the incompressible solver. Here, let us point out that larger gas parameter
$a$ means smaller Mach number. Obviously, Figure 6 tells that compressible
fluids brings more damping.

Figure 6: Comparison of numerical solutions of compressible solver ($a=0.1$
$a=20$ and $a=100$) and incompressible solver.

## Acknowledge

The work of Václav Mácha and Šárka Nečasová are supported by Praemium Academiæ
of Š. Nečasová and by grant GAČR GA22-01591S in the framework of RVO:67985840.
The work of B.S. were supported by ERC-CZ grant LL2105 of the Ministry of
Education, Youth and Sport of the Czech Republic and Charles University
Research program No. UNCE/SCI/023

## Statements and Declarations

Conflict of interests: On behalf of all authors, the corresponding author
states that there is no conflict of interest.
Data availability: All data generated or analysed during this study are
included in this article.

## References

  * [1] Bhuta, P.G., Koval, L.R., A viscous ring damper for a freely precessing satellite. Intern. J. Mech. Sci. 8 (1966) 383–395
  * [2] Carl, S., Le, V., K., Montreanu, K.: Nonsmooth variational problems and their inequalities, Comparison principles and applications. Springer, 2007
  * [3] Chernousko, F.L.,Motion of a rigid body with cavities containing a viscous fluid (1968). NASA Technical Translations, Moscow, 1972
  * [4] Chernousko, F.L., Akulenko, L.D., Leshchenko, D.D., Evolution of Motions of a Rigid Body About its Center of Mass, Springer, Berlin 2017
  * [5] Disser, K., Galdi, G.P., Mazzone, G., Zunino, P.: Inertial Motions of a Rigid Body with a cavity filled with a viscous liquid. Arch. Ration. Mech. Anal. 221, 487–526 2016
  * [6] Eswaran, M., Saha, U.K., Sloshing of liquids in partially filled tanks – a review of experimental investigations, Ocean Systems Engineering, 1 (2011) 131–155
  * [7] Feireisl, E., Novotný, A.: Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel, second edition, 2017
  * [8] Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3 (2001), no. 4, 358–392
  * [9] Feireisl, E., Petzeltová, H.: Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow. Arch. Rational Mech. Anal 150 (1999), 77–96
  * [10] Feireisl, E., Petzeltová, H.: On the zero-velocity-limit solutions to the Navier-Stokes equations of compressible flow. Manuscripta Math. 97 (1998), no. 1, 109–116
  * [11] Fonseca, I., Leoni, G.: Modern methods in the calculus of variations: $L^{p}$ spaces. Springer, New York, 2007
  * [12] Galdi, G.P.: Stability of permanent rotations and long-time behavior of inertial motions of a rigid body with an interior liquid-filled cavity. Particles in flows, 217–253, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, 2017
  * [13] Galdi, G., P., Mácha, V., Nečasová, Š.: On the motion of a body with a cavity filled with compressible fluid. Arch. Ration. Mech. Anal. 232 (2019), no. 3, 1649–1683
  * [14] Galdi, G., P., Mácha, V., Nečasová, Š.: On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior. International Journal of Non-Linear Mechanics 121, 2020, 103431
  * [15] Galdi, G., P., Mazzone, G.: On the motion of a pendulum with a cavity entirely filled with a viscous liquid in Recent progress in the theory of the Euler and Navier-Stokes equations, 37–56, London Math. Soc. Lecture Note Ser., 430, Cambridge Univ. Press, Cambridge, 2016
  * [16] Galdi, G.P.; Mazzone, G.: Nonlinear stability analysis of a spinning top with an interior liquid-filled cavity. Math. Model. Nat. Phenom. 16 (2021), Paper No. 22, 21 pp.
  * [17] Galdi, G.P., Mazzone, G., Zunino P.: Inertial motions of a rigid body with a cavity filled with a viscous liquid, Comptes Rendus Mécanique 341, 760–765 2013
  * [18] Galdi, G.P., Mazzone, G., Mohebbi, M.: On the motion of a liquid-filled rigid body subject to a time-periodic torque, Recent developments of mathematical fluid mechanics, Adv. Math. Fluid Mech., Birkhäuser/Springer, Basel, pp. 233–255 2016
  * [19] Galdi, G.P., Mazzone, G., Mohebbi, M.: On the motion of a liquid-filled heavy body around a fixed point. Q. Appl. Math.  76, 113–145 2018
  * [20] Novotný, A., Straškraba, I.:Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications 27, Oxford University Press, Oxford, 2004
  * [21] Karlsen K. H., Karper T. K.: Convergence of a mixed method for a semi-stationary compressible Stokes system, Math. Comp. 80 (2011), 1459–1498
  * [22] Kračmar, S., Nečasová, Š., Novotný, A., The motion of a compressible viscous fluid around rotating body. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60, (2014) 189–208
  * [23] Kukučka, P.: On the existence of finite energy weak solutions to the Navier-Stokes equations in irregular domains, Math. Methods Appl. Sci. 32 (2009)
  * [24] Mazzone, G.; Prüss, J.; Simonett, G., A maximal regularity approach to the study of motion of a rigid body with a fluid-filled cavity. J. Math. Fluid Mech. 21 (2019), Paper No. 44, 20 pp
  * [25] Mazzone, G.; Prüss, J., Simonett, G., On the motion of a fluid-filled rigid body with Navier boundary conditions. SIAM J. Math. Anal. 51 (2019) 1582–1606
  * [26] Mazzone, G.: On the free rotations of rigid bodies with a liquid-filled gap. J. Math. Anal. Appl. 496 (2021), no. 2, Paper No. 124826, 37 pp.
  * [27] Pedregal, P.: Parametrized measures and Variational principles, Birkhäuser Verlag, Basel, 1997
  * [28] Tartar, L.: Compensated compactness and applications to partial differential equations in Nonlinear analysis and mechanics: Heriot–Watt Symposium, Vol. IV, pp. 136–212, Res. Notes in Math., 39, Pitman, Boston, Mass.–London, 1979.
  * [29] Thomson, W. (Lord Kelvin): On an experimental illustration of minimum energy. Nature. 23, 69–70 (1880)