Large-Time Behavior of a Rigid Body of Arbitrary Shape in a Viscous Fluid Under the Action of Prescribed Forces and Torques

Abstract

Let ${\mathcal B}$ be a sufficiently smooth rigid body (compact set of ${\mathbb R}^3$) of arbitrary shape moving in an unbounded Navier–Stokes liquid under the action of prescribed external force, $F$, and torque, $M$. We show that if the data are suitably regular and small, and $F$ and $M$ vanish for large times in the $L^2$-sense, there exists at least one global strong solution to the corresponding initial-boundary value problem. Moreover, this solution converges to zero as time approaches infinity. This type of results was known, so far, only when ${\mathcal B}$ is a ball.

1 Introduction

The motion of a (finite) rigid body, ${\mathcal B}$, in an unbounded Navier–Stokes liquid has been the object of a number of deep researches. Particularly intriguing is the case when the motion of the body is not given and, in general, one prescribes total force, $F$, and torque $M$, acting on it. Since the presence of the body affects the flow of the liquid, and this, in turn, affects the motion of the body, the problem of determining the flow characteristics thus becomes highly coupled. It is this distinctive property that makes any mathematical problem related to body-liquid interaction especially interesting and challenging.

In this paper we are interested in the study of two basic questions related to the situation just described when the shape of ${\mathcal B}$ is not specified, and precisely: (i) existence of a global-in-time strong solutions to the relevant initial-boundary value problem, and (ii) their asymptotic behavior for all large times. Before stating our results, we would like to recall all known major contributions related to this type of investigation, which will also furnish the motivation for the present study.¹ The first existence result is due to Serre [16], who proves global existence of weak solutions a la Leray-Hopf. As the author himself observes, the proof is exactly the same as the classical one for the Navier–Stokes problem and presents no challenges. Instead, a less obvious task is to show existence of strong solutions having enough regularity as to solve the given equations (at least) at almost every point of the space-time. This question was first successfully tackled by Galdi and Silvestre [8] who proved existence of strong solutions, in the sense of Prodi-Ladyzhenskaya [14, 15], for data of arbitrary “size” –in a suitable class– at least in a time-interval [0, T) for some $T>0$. Successively, a similar result, but with a different approach, was established by Cumsille and Tucsnak [4], when the body is allowed to rotate but not to translate, and $F \equiv M \equiv 0$. In both papers [4, 8] the functional framework is the $L^2$ Hilbert-setting. The study of existence of strong solutions in the $L^q$ setting, $q\in (1,\infty )$, was initiated by Wang and Xin [18], who established a local in time result with $F \equiv M \equiv 0$, in the special case when ${\mathcal B}$ is a ball. Since the general shape of the body is a most relevant feature of our result here, let us briefly comment on how the hypothesis of ${\mathcal B}$ being a ball brings in some basic simplifications and mathematical properties that are lost in the general case. In the first place, this assumption eliminates the presence, in the linear momentum equation, of a term whose coefficient becomes unbounded at large spatial distances. Furthermore, as shown in [3, 18], the relevant linear operator, suitably defined, is the generator of an analytic semigroup, a property that is no longer valid for bodies of arbitrary shape, just because of the occurrence of the unbounded term [12, 17]. Local $L^q$ existence for ${\mathcal B}$ of arbitrary shape was successively established by Geissert et al. [11], by maximal regularity theory, again with $F \equiv M \equiv 0$. Concerning the question of global existence, it was first studied and positively answered by Cumsille and Takahashi [3]. In particular, they showed that if, in appropriate norms, the initial data are “small” and the external forces are summable over the whole half-line $(0,\infty )$ and “small,” there exists a (unique) corresponding solution defined for all times and belonging to a functional class similar to that considered in [8]. The method used in [3] is based on a particular cut-off technique that, on one hand, eliminates the difficulty due to the unbounded coefficient, but, on the other hand, is not able to provide any information on the large-time behavior of the solutions that, under the given assumption, are expected to reach, eventually, the rest-state. The question of the asymptotic behavior of solutions (along with their global existence) has been analyzed very recently in a remarkable paper by Ervedoza et al. [5], when $F \equiv M \equiv 0$. The main tool is new $L^p- L^q$ estimates for the body-liquid semigroup. Even though the estimates are proved for bodies of arbitrary shape, their use in showing global existence of solutions (for small data) along with their asymptotic decay to rest-state requires ${\mathcal B}$ to be a ball. In such a case, the authors also provide a sharp decay rate that implies that the center of mass of ${\mathcal B}$ can only cover a finite distance from its initial position, as expected on physical grounds.

In view of all the above, the following basic question –brought to my attention by Professor Toshiaki Hishida– remains still open: Let ${\mathcal B}$ be of arbitrary shape, subject to prescribed force and torque that vanish (in suitable sense) as time goes to infinity. Does the body-liquid problem have a global solution that, in addition, ultimately tends to the rest-state?

Objective of this note is to give a positive answer to this question, under the assumption of “small” data. More precisely, we show (see Theorem 2.1) that the local solution constructed in [8] can, in fact, be extended to arbitrarily positive times, if the data are small enough. This is accomplished by means of a generalized Gronwall’s lemma, proved in Lemma 4.3. By the same tool and under the same hypotheses, we then prove that solutions must eventually converge to the state of rest. Unfortunately, we are not able to furnish a rate of decay, which thus leaves room to further investigation. However, in the case when ${\mathcal B}$ is a ball, we do provide such a decay that, in the $L^2$ framework considered here, appears to be rather sharp; see Remark 4.1.

The plan of the paper is as follows. In Sect. 2 we formulate the problem and state our main result in Theorem 2.1. Successively, in Sect. 3 we prove two basic “energy equations” valid in the class of solutions considered in [8]; see Lemma 3.3. This requires some estimates on the time derivative of the velocity field and on the pressure field that are carried out in Lemmas 3.1 and 3.2. Finally, in Sect. 4 we give a proof of Theorem 2.1, by combining estimates obtained from the energy equations with the Gronwall-like lemma showed in Lemma 4.3.

2 Mathematical Formulation and Main Result

A rigid body ${\mathcal B}$ –that is, a sufficiently smooth, compact and connected set of ${\mathbb R}^3$– is fully immersed in a quiescent Navier–Stokes liquid, ${\mathcal L}$, that fills the entire three-dimensional space exterior to ${\mathcal B}$. We suppose that, with respect to an inertial frame, , the body is subject to prescribed force, $F =F (t)$, and torque, $M =M (t)$, $t\ge 0$. Following a standard procedure, we shall describe the motion of the coupled system ${\mathcal S}:=\{{\mathcal B},{\mathcal L}\}$ with respect to a frame, , attached to ${\mathcal B}$ and with its origin at the center of mass, G, of ${\mathcal B}$. In such a way, in particular, the domain occupied by ${\mathcal L}$ becomes time-independent, and we will denote it by ${\mathcal D}$ ($:={\mathbb R}^3\backslash {\mathcal B}$) and by $\Sigma$ its boundary. Assuming, without loss of generality, that and coincide at $t=0$, the equations governing the motion of ${\mathcal S}$ in are given by (see [6])

$$\begin{aligned} \left. \begin{array}{c} \left. \displaystyle \begin{array}{c} \varrho \partial _t{u} = {\textrm{div}\,}T(u,p) - \varrho [ (u-V)\cdot {\nabla }u + \omega \times u ] \\ {\textrm{div}\,}u=0 \end{array} \displaystyle \right\} \textrm{in }\,\ {\mathcal {D}}\times (0,\infty ) \\ \displaystyle u = V \ \ \textrm{at }\ \Sigma \times (0,\infty ) \\ \displaystyle \lim _{|x| \rightarrow \infty }u(x,t)=0,{ }\,\ t\in (0,\infty ) \\ \displaystyle m\dot{\xi }+ m \omega \times \xi + \int _{\Sigma } T(u,p)\cdot n = \textsf {F} \\ \displaystyle \textsf{I}\cdot \dot{\omega }+ \omega \times (\textsf{I} \cdot \omega )+ \int _{\Sigma }x \times T(u,p)\cdot n = \textsf {M} \\ \displaystyle \xi (0)=\xi _{0},\quad \ \omega (0)=\omega _{0} \\ \displaystyle u(x,0) = u_{0}(x),{ }\, x\in \mathcal {D}\,. \end{array} \right. \end{aligned}$$

(2.1)

Here, u and p are velocity and pressure fields of ${\mathcal L}$, $\varrho$ its (constant) density, and $V(x,t):= \xi (t) + \omega (t) \times x,$ where $\xi$ is the velocity of the center of mass of ${\mathcal B}$ and $\omega$ its angular velocity. Also, T is the Cauchy stress tensor given by

$$\begin{aligned} T(u,p)=2\mu \,D(u)-p\,{\text {I}},\ \ D(u):={\frac{1}{2}}\, \big ({\nabla }u+({\nabla }u)^\top \big ), \end{aligned}$$

with $\mu$ shear-viscosity coefficient and ${\text {I}}$ identity. Moreover, m is the mass of ${\mathcal B}$ and $\textsf{I}$ its inertia tensor relative to G. Furthermore,

$$\begin{aligned} \left\{ \begin{array}{c} \textsf {F}(t)=Q^{\top }(t)\cdot F (t), \\ \textsf {M}(t)=Q^{\top }(t)\cdot M (t)\,, \end{array} \right. \end{aligned}$$

(2.2)

with the tensor Q satisfying the following equation

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{c} \displaystyle \dot{Q} = -Q\cdot \Omega (\omega )\\ \displaystyle Q(0)={\text{ I }}\end{array} \right. \qquad \qquad \qquad \Omega (\omega )= \left[ \begin{array}{ccc}0&{}{} \omega _3&{}{}-\omega _2\\ \omega _3&{}{}\,\,\, 0 &{}{}\,\,\,\omega _1\\ \omega _2&{}{}\,\,\, -\omega _1&{}{}\,\,\, 0 \end{array}\right] \end{aligned} \end{aligned}$$

(2.3)

In particular, Q is proper orthogonal, that is,

$$\begin{aligned} \displaystyle Q^{\top }(t)\cdot Q(t)=Q(t)\cdot Q^{\top }(t)={\text {I}}, \ \ \ \displaystyle \det Q(t)=1, \quad \text{ for } \text{ all }\, t \in {\mathbb R}. \end{aligned}$$

In order to state our main result, we need a suitable function space. Let

$$\begin{aligned} {\mathcal R}:=\{\overline{u}\in C^{\infty }({{\mathbb R}}^3): \overline{u}(x)=\overline{u}_1 + \overline{u}_2 \times x, \quad \overline{u}_1,\overline{u}_2 \in {{\mathbb R}}^3\}, \end{aligned}$$

and define ²

$$\begin{aligned} {\mathcal V}({\mathcal {D}}) = \{ u \in W^{1,2}({\mathcal {D}}): {\textrm{div}\,}u =0 \text { in }{\mathcal {D}}, \text { }\text { } u\left| _{\Sigma }\right. = \overline{u}, \ \text{ for } \text{ some } \ \overline{u} \in {\mathcal R}\}. \end{aligned}$$

We also set

$$\begin{aligned} \begin{array}{ll} B_R:=\{x\in {\mathbb R}^3:\,|x|< R\};\ \ R_*:= 2\inf \,\{R\in (0,\infty ): {\mathcal B}\cap B_R\supset {\mathcal B}\};\\ {\mathcal D}_R:={\mathcal D}\cap B_R,\ \ {\mathcal D}^R={\mathcal D}\backslash \overline{{\mathcal D}}_{R},\ \ R>R_*. \end{array} \end{aligned}$$

The main objective of this paper is to show the following result.

Theorem 2.1

Let ${{\mathcal D}}$ be of class $C^2.$ Let $F ,M \in L^2(0,\infty )$ and $u_0 \in {\mathcal V}({\mathcal {D}})$ with $u_0|_{\Sigma }=\xi _0+\omega _0\times x$. Then, there is $\delta >0$ such that if

$$\begin{aligned} \Vert u_0\Vert _{1,2}+|\xi _0|+|\omega _0|+\Vert F \Vert _{L^2(0,\infty )}+\Vert M \Vert _{L^2(0,\infty )}\le \delta \,, \end{aligned}$$

(2.4)

there exist functions $u=u(x,t),$ $p=p(x,t),$ $\xi =\xi (t),$ $\omega =\omega (t)$, and $Q=Q(t)$ satisfying (2.1)–(2.3) a.e., such that

$$\begin{aligned} \begin{array}{ll} u \in L^{\infty }(0,\infty ;W^{1,2}({\mathcal {D}})),\ \ \ {\nabla }u \in L^2(0,\infty ;W^{1,2}({\mathcal {D}}))\\ \xi ,\text { } \omega \in W^{1,2}(0,\infty ), \ \ \ \nabla p\in L^2(0,\infty ;L^2({\mathcal D}))\,, \ \ \ Q\in W^{2,2}(0,\infty )\\ \partial _t u,\text { } p\in L^2(0,\infty ;L^2({\mathcal {D}}_R)), \ \text{ for } \text{ all }\, R \ge R_*\,. \end{array} \end{aligned}$$

(2.5)

Moreover, for all $T>0$,

$$\begin{aligned}{} & {} \xi ,\text { } \omega ,\, Q \in C([0,T]), \ \text { with }\xi (0)=\xi _0,\text { } \omega (0)=\omega _0,\text { } Q(0)=\textrm{I}\\{} & {} \quad u \in C([0,T];W^{1,2}({\mathcal {D}}_R)), \ \text{ for } \text{ all }\, R \ge R_*, \ \text { with } u(.,0)=u_0(.). \end{aligned}$$

Finally,

$$\begin{aligned} \lim _{t\rightarrow \infty }\big (\Vert u(t)\Vert _6+\Vert \nabla u(t)\Vert _2+|\xi (t)|+|\omega (t)|\big )=0\,. \end{aligned}$$

(2.6)

Before carrying out, in the following sections, the proof of the theorem, we would like to make some comments. The major aspect of our results is expressed by the asymptotic property (2.6), which states that, eventually, the coupled system ${\mathcal S}$ will go to rest, independently of the shape of ${\mathcal B}$. In fact, to date, this property was known only when ${\mathcal B}$ is a ball [5] and $F \equiv M \equiv 0$. However, unlike [5], even with the additional assumption on $F$ and $M$, we are not able to furnish a rate of decay. We may guess that it is $O(t^{-\frac{1}{2}})$, but a proof seems to be currently out of reach; see also Remark 4.1.

On the other side, in the case when either $F$ or $M$ is time-independent the existence of global strong solutions and, more intriguingly, the assessment of their asymptotic behavior represents a formidable open question. A remarkable example is the free-falling body problem where $M =0$ and $F =m_e g$, with $m_e$ buoyant mass of ${\mathcal B}$ and g acceleration of gravity. In such a case it is expected that, at least for small $m_e$, the coupled system ${\mathcal S}$ will tend, as $t\rightarrow \infty$, to a steady-state configuration. However, as shown in [10], the steady-state problem may have multiple solutions, even for vanishingly small $m_e$. One may thus conjecture that ${\mathcal S}$ will approach, eventually, one of the locally unique, stable configuration that are experimentally observed, at least when ${\mathcal B}$ has fore-and-aft symmetry, like homogeneous cylinder [13]. Nevertheless, even in this case, a rigorous proof is far from obvious.³

3 Preliminary Results

The goal of this section is to derive a number of a priori estimates for solutions to (2.1)–(2.3) in a suitable function class that we define next.

Definition 3.1

We say that $(u,p,\xi ,\omega ,Q)$ is in the class ${\mathscr {C}}_T$, some $T\in (0,\infty ]$, if, for all $\tau \in (0,T)$,

$$\begin{aligned} \begin{array}{ll} u \in L^{\infty }(0,\tau ;W^{1,2}({\mathcal {D}})),\ \ \ {\nabla }u \in L^2(0,\tau ;W^{1,2}({\mathcal {D}}))\\ \xi ,\text { } \omega \in W^{1,2}(0,\tau ), \ \ \ \nabla p\in L^2(0,\tau ;L^2({\mathcal D})), \ \ \ Q\in W^{2,2}(0,\tau )\\ u \in C([0,\tau ];W^{1,2}({\mathcal {D}}_R)) ,\ \ \partial _t u,\text { } p\in L^2(0,\tau ;L^2({\mathcal {D}}_R)),\ \ \text{ for } \text{ all } \, R \ge R_*. \end{array} \end{aligned}$$

The following results hold.

Lemma 3.1

Let $(u,p,\xi ,\omega ,Q)$ be a solution to (2.1)–(2.3) in the class ${\mathscr {C}}_T$. Then for a.a. $t\in (0,T)$

$$\begin{aligned} \partial _tu/r\in L^2({\mathcal D}),\ \ r:=(x_ix_i)^{\frac{1}{2}}. \end{aligned}$$

Proof

From the assumption, we immediately show

$$\begin{aligned} \big (\textrm{div}\,T(u,p)+\varrho \,(V\cdot \nabla u-\omega \times u\big )/r\in L^2({\mathcal D})\,. \end{aligned}$$

(3.7)

Moreover, by Schwarz and Sobolev inequalities

$$\begin{aligned} \Vert u\cdot \nabla u\Vert _2\le \Vert u\Vert _4\Vert \nabla u\Vert _4\le c\,\Vert u\Vert _{2,2}^2\,. \end{aligned}$$

(3.8)

Thus, since $r^{-1}\in L^\infty ({\mathcal D})$, the lemma follows from (3.7), (3.8) and (2.1)$_1$. $\square$

Lemma 3.2

Let $(u,p,\xi ,\omega ,Q)$ be a solution to (2.1)–(2.3) in the class ${\mathscr {C}}_T$. Then for a.a. $t\in (0,T)$

$$\begin{aligned} \nabla p\in L^{q_1}({\mathcal D}^{2R_*}),\ \ p\in L^{q_2}({\mathcal D}^{2R_*}),\ \ \text{ for } \text{ all } q_1\in (1,6],\ q_2\in (\frac{3}{2},\infty ]. \end{aligned}$$

Proof

Observing that, in the sense of distribution,

$$\begin{aligned} \textrm{div}\,[\varrho \big (\partial u_t-(V-u)\cdot \nabla u\big )-\mu \Delta u]=0, \end{aligned}$$

from (2.1)$_1$ we get for a.a. $t\in (0,T)$

$$\begin{aligned} \Delta p=\textrm{div}\,f\,\ \text {in}\ {\mathcal D}\,,\ \ f:=\varrho \,u\cdot \nabla u\,. \end{aligned}$$

(3.9)

Let $\psi =\psi (|x|)$ be a smooth, non-decreasing function such that $\psi (|x|)=0$, if $|x|\le R_*)$, while $\psi (|x|)=1$ if $|x|\ge 2R_*$. Setting $\textsf{p}:=\psi \, p$, and extending $\textsf{p}$ to zero outside ${\mathcal D}^{R_*}$, from (3.9) we find

$$\begin{aligned} \Delta \textsf{p}=F\ \ \text{ in } {\mathbb R}^3 \,, \end{aligned}$$

(3.10)

where

$$\begin{aligned} F:=\textrm{div}\,(2 p\,\nabla \psi +\psi \,f)-\nabla \psi \cdot f-p\,\Delta \psi \,. \end{aligned}$$

(3.11)

Since $u,p\in {\mathscr {C}}_T$, from Sobolev embedding theorem it follows that

$$\begin{aligned} u\in L^q({\mathcal D}),\ \ \nabla u\in L^s({\mathcal D}),\ \ p\in L^r_\textrm{loc}(\overline{{\mathcal D}}),\ \ \text{ for } \text{ all } q\in [2,\infty ), s\in [2,6], r\in [1,6]. \end{aligned}$$

This implies, in particular,

$$\begin{aligned} f\in L^r({\mathcal D}),\ \ \text{ for } \text{ all } r\in [1,6]. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert F\Vert _{-1,r}\le c\,\left( \Vert f\Vert _{r}+\Vert p\Vert _{r,{\mathcal D}_{2R_*}}\right) ,\ \ \text{ for } \text{ all } r\in [1,6] \end{aligned}$$

Problem (3.10)–(3.11) formally coincides with problem (III.1.20) studied in [7, pp. 149–150], for which, observing that $\nabla \textsf{p}\in L^2({\mathbb R}^3)$ and recalling that $\psi _R\equiv 1$ in ${\mathcal D}^{2R_*}$, one proves (see [7, eq. (III.1.23)]) that $\nabla p\in L^{r}({\mathcal D}^{2R_*})$ for all $r\in (1,6]$. By [7, Theorems II.2.1(i) and II.9.1] the latter in turn implies $p\in L^\sigma ({\mathcal D}^{{2R_*}})$, for all $\sigma \in (\frac{3}{2},\infty ]$, which completes the proof. $\square$

Lemma 3.3

Let $(u,p,\xi ,\omega ,Q)$ be a solution to (2.1)–(2.3) in the class ${\mathscr {C}}_T$. Then, the following relations hold, for all $t\in (0,T)$

$$\begin{aligned} \begin{array}{ll} \frac{1}{2}\displaystyle {\frac{\displaystyle {d}}{\displaystyle {dt}}}\bigg ( \varrho \Vert u \Vert _2^2 + m \left| \xi \right| ^2 + \omega \cdot I \cdot \omega \bigg ) + 2 \mu \Vert D(u) \Vert _2^2=\textsf{F}\cdot \xi + \textsf{M}\cdot \omega \\ \displaystyle \mu \frac{d}{dt}\Vert D(u)\Vert _{2}^2 + m |\dot{\xi }|^2 + \dot{\omega } \cdot \textsf{I} \cdot \dot{\omega } + \Vert {\textrm{div}\,}T(u,p)\Vert _{2}^2 = -m \omega \times \xi \cdot \dot{\xi } - \omega \times (\textsf{I} \cdot \omega )\cdot \dot{\omega }- \textsf{F} \cdot \dot{\xi } - \textsf{M} \cdot \dot{\omega } \\ \quad \,+ \varrho \displaystyle \int _{{\mathcal {D}}} u\cdot {\nabla }u \cdot {\textrm{div}\,}T(u,p) + \varrho \,\mu \displaystyle \int _{{\mathcal {D}}} ( \omega \times {\nabla }u_i\cdot \nabla u_i - \nabla (\omega \times u):\nabla u ) -\varrho \,\mu {\displaystyle \int _{ \Sigma }^{ }}\left( n\cdot \nabla u\cdot \Phi -{\frac{1}{2}}V\cdot n|\nabla u|^2\right) , \end{array} \end{aligned}$$

(3.12)

where $\Phi :=V\cdot \nabla u-\omega \times u$ .

Proof

Let

$$\begin{aligned} \displaystyle T_1:= \frac{1}{m} \int _{\Sigma }T(u,p)\cdot n, \ \ T_2:= \textsf{I}^{-1}\cdot \int _{\Sigma }x \times T(u,p)\cdot n. \end{aligned}$$

We test both sides of (2.1)$_1$ by u, integrate by parts over ${\mathcal D}_R:={\mathcal D}\cap \{|x|<R\}$, $R>R_*$, and use (2.1)$_{2,3}$ to get

$$\begin{aligned} \begin{aligned} {\frac{1}{2}}\varrho {\displaystyle \frac{{ d } }{{d} {t }}}\Vert u(t)\Vert _{2,{\mathcal D}_R}^2+2\mu \Vert D(u)\Vert ^2_{2,{\mathcal D}_R}=m\xi \cdot T_1+\omega \cdot \textsf {I}\cdot T_2 +\int _{\partial B_R}\left( u\cdot T(u,p)\cdot n-{\frac{1}{2}}\varrho u^2(u-\xi )\cdot n\right) \,, \end{aligned} \end{aligned}$$

(3.13)

where we observed that $\omega \times x\cdot n=0$ at $\partial B_R$. Since $u\in {\mathscr {C}}_T$, and also with the help of Lemma 3.2, it is readily seen that the surface integral in (3.13) is in $L^1(R_*,\infty )$, so that we may let $R\rightarrow \infty$ along a sequence to get

$$\begin{aligned} \begin{aligned} {\frac{1}{2}}\varrho {\displaystyle \frac{{ d } }{{d} {t}}}\Vert u(t)\Vert _{2}^2+2\mu \Vert D(u)\Vert ^2_{2}=m\xi \cdot T_1+\omega \cdot \textsf {I}\cdot T_2 \,. \end{aligned} \end{aligned}$$

(3.14)

As a result, employing (2.1)$_{5,6}$ in (3.14) we deduce (3.12)$_1$. In order to show (3.12)$_2$, we begin to observe that [8, Lemma 2.4(b)]

$$\begin{aligned} \Phi \cdot n=0\ \ \text{ at } \Sigma \,. \end{aligned}$$

(3.15)

Moreover, for any $R>R_*$, let $\psi _R=\psi _R(|x|)$ be a non-decreasing, smooth function such that $\psi _R(|x|)=1$, if $|x|\le R$ and $\psi _R(|x|)=0$, if $|x|\ge 2R$, and

$$\begin{aligned} |\nabla \psi _R(|x|)|\le C\,R^{-1}\,, \end{aligned}$$

(3.16)

with C independent of x and R. We next test (2.1)$_1$ by $\psi _R\,\textrm{div}\,T(u,p)$ to get

$$\begin{aligned} \int _{\mathcal D}\psi _R\partial u_t\cdot \textrm{div}\,T=\Vert \sqrt{\psi _R}\textrm{div}\,T\Vert _2^2-\varrho \int _{\mathcal D}\left( \psi _Ru\cdot \nabla u\cdot \textrm{div}\,T-\psi _R\Phi \cdot \textrm{div}\,T\right) \,. \end{aligned}$$

(3.17)

By integration by parts, we show

$$\begin{aligned} \begin{aligned} \begin{array}{rl}\displaystyle \int _{\mathcal D}\psi _R\partial u_t\cdot \text {div}\,T&{}{}=\displaystyle \int _{\mathcal D}\left[ \text {div}\,(\psi _R\,\partial _tu\cdot T)-2\mu \,\psi _R\,D(\partial _tu):D(u)\right] \\ {} &{}{}=\displaystyle \int _\Sigma \dot{V}\cdot T\cdot n-\mu {\displaystyle \frac{{ d } }{{d}{ t }}} \Vert \sqrt{\psi _R}D(u)\Vert _2^2-\int _{\mathcal D}\nabla \psi _R\cdot T\cdot \partial _tu. \end{array} \end{aligned} \end{aligned}$$

Using (2.1)$_{5,6}$ in the surface integral, we deduce

$$\begin{aligned} \begin{aligned} \begin{array}{rl}\displaystyle \int _{\mathcal D}\psi _R\partial u_t\cdot \text {div}\,T=&{}{}\!\!-m\dot{\xi }^2-\dot{\omega }\cdot \textsf {I}\cdot \dot{\omega }-m\omega \times \xi \cdot \dot{\xi }-\omega \times (\textsf {I}\cdot \omega )\dot{\omega }-\textsf {F}\cdot \dot{\xi }-\textsf {M}\cdot \dot{\omega }\\ {} &{}{}\displaystyle -\mu {\displaystyle \frac{{d} }{{d} {t}}} \Vert \sqrt{\psi _R}D(u)\Vert _2^2-\int _{\mathcal D}\nabla \psi _R\cdot T\cdot \partial _tu\,. \end{array} \end{aligned} \end{aligned}$$

(3.18)

Next, integrating by parts and with the help of (3.15) we show

$$\begin{aligned} {\displaystyle \int _{ {\mathcal D}}^{ }}\psi _R\Phi \cdot \textrm{div}\,T=2\mu {\displaystyle \int _{ \Sigma }^{ }}\Phi \cdot D(u)\cdot n-2\mu {\displaystyle \int _{ {\mathcal D}}^{ }}\psi _R\partial _i\Phi _jD_{ij}(u)-{\displaystyle \int _{ {\mathcal D}}^{ }}\nabla \psi _R\cdot T\cdot \Phi \,. \end{aligned}$$

(3.19)

Now, using $\textrm{div}\,u=\textrm{div}\,V=0$,

$$\begin{aligned} \begin{array}{rl} 2\partial _i\Phi _jD_{ij}(u)&{}=\partial _i(\Phi _j\partial _j u_i)+(\partial _iV)\cdot \nabla u_j\partial _iu_j+{\frac{1}{2}}V\cdot \nabla (|\nabla u|^2)-\nabla (\omega \times u):\nabla u\\ &{}=\textrm{div}\,(\Phi \cdot \nabla u+{\frac{1}{2}}V\,|\nabla u|^2)+\omega \times \nabla u_i\cdot \nabla u_i-\nabla (\omega \times u):\nabla u. \end{array} \end{aligned}$$

Substituting the latter in (3.19) and using Gauss theorem, we infer

$$\begin{aligned} \begin{array}{ll} {\displaystyle \int _{ {\mathcal D}}^{ }}\psi _R\Phi \cdot \textrm{div}\,T=&{}\!\! -\mu {\displaystyle \int _{ {\mathcal D}}^{ }}\psi _R\left( \omega \times \nabla u_i\cdot \nabla u_i-\nabla (\omega \times u):\nabla u\right) +\mu {\displaystyle \int _{ \Sigma }^{ }}\left( n\cdot {\nabla }u\cdot \Phi -{\frac{1}{2}}V\cdot n|\nabla u|^2\right) \\ &{}+{\displaystyle \int _{ {\mathcal D}}^{ }}\nabla \psi _R\cdot \left[ 2\mu (\nabla u^\top \cdot \Phi +{\frac{1}{2}}V|\nabla u|^2)-T\cdot \Phi \right] \,. \end{array} \end{aligned}$$

(3.20)

Collecting (3.17), (3.18) and (3.20) we deduce

$$\begin{aligned} \begin{aligned} \begin{array}{ll} \mu {\displaystyle \frac{{ d } }{{d} {t}}} \Vert \sqrt{\psi _R}D(u)\Vert _2^2 +\Vert \sqrt{\psi _R}\text {div}\,T\Vert _2^2+m\dot{\xi }^2+\dot{\omega }\cdot \textsf {I}\cdot \dot{\omega }=-m\omega \times \xi \cdot \dot{\xi }-\omega \times (\textsf {I}\cdot \omega )\,\dot{\omega }-\textsf {F}\cdot \dot{\xi }-\textsf {M}\cdot \dot{\omega }\\ \displaystyle +\varrho {\displaystyle \int _{ {\mathcal D}}^{ }}\psi _R\left[ u\cdot \nabla u\cdot \text {div}\,T +\mu \left( \omega \times \nabla u_i\cdot \nabla u_i-\nabla (\omega \times u):\nabla u\right) \right] -\varrho \mu {\displaystyle \int _{ \Sigma }^{ }}\left( n\cdot {\nabla }u\cdot \Phi -{\frac{1}{2}}V\cdot n|\nabla u|^2\right) \\ -\varrho {\displaystyle \int _{ {\mathcal D}}^{ }}\nabla \psi _R\cdot \left[ T\cdot \partial _tu+2\mu (\nabla u^\top \cdot \Phi +{\frac{1}{2}}V|\nabla u|^2)-T\cdot \Phi \right] \end{array} \end{aligned} \end{aligned}$$

(3.21)

Let us denote by ${\mathcal F}(R)$ the last integral on the right-hand side of (3.21). Observing that $\textrm{supp}\,(\psi _R)\subseteq \{R\le |x|\le 2R\}$ and recalling (3.16), we obtain

$$\begin{aligned} |{\mathcal F}(R)|\le & {} c{\displaystyle \int _{ R\le |x|\le 2R}^{ }}|x|^{-1}\left[ (|T|(|\partial _tu|+|\Phi |)+|\nabla u||\Phi |+|V||\nabla u|^2\right] \\\le & {} c {\displaystyle \int _{ {\mathcal D}^R}^{ }}\left( |x|^{-2}|\partial _tu|^2+|\nabla u|^2+|u|^2+|p|^2\right) . \end{aligned}$$

As a result, by assumption, Lemma 3.1, and Lemma 3.2, we show

$$\begin{aligned} \lim _{R\rightarrow \infty }{\mathcal F}(R)=0\,. \end{aligned}$$

(3.22)

Furthermore, since $(u,p)\in {\mathscr {C}}_T$ it readily checked that

$$\begin{aligned} \left[ u\cdot \nabla u\cdot \textrm{div}\,T +\mu \left( \omega \times \nabla u_i\cdot \nabla u_i-\nabla (\omega \times u):\nabla u\right) \right] \in L^1({\mathcal D})\,. \end{aligned}$$

(3.23)

We integrate both sides of (3.20) over (0, t), $t\in (0,\tau ]$, let $R\rightarrow \infty$ and employ (3.22), (3.23) along with Lebesgue dominated convergence theorem. If we differentiate with respect to t the resulting equation, we then end up with (3.12)$_2$, which completes the proof of the lemma. $\square$

4 Proof of the Main Result

In this section we shall prove Theorem 2.1. Before beginning the proof, however, we premise further lemmas.

Lemma 4.1

Let $u\in {\mathcal V}(\mathcal D)$. Then,

$$\begin{aligned} \Vert \nabla u\Vert _2=\sqrt{2}\Vert D(u)\Vert _2. \end{aligned}$$

Moreover, there is $c_1=c_1({\mathcal D})$ such that

$$\begin{aligned} |\overline{u}_1|+|\overline{u}_2|\le c\,\Vert D(u)\Vert _2. \end{aligned}$$

Finally, $u\in L^6({\mathcal D})$ and there is $c_2=c_2({\mathcal D})$ such that

$$\begin{aligned} \Vert u\Vert _6\le c_2\,\Vert D(u)\Vert _2. \end{aligned}$$

Proof

See [6, Section 4.2.1]. $\square$

Lemma 4.2

Let $u\in {\mathcal V}({\mathcal D})\cap W^{2,2}({\mathcal D})$, $\nabla p\in L^2({\mathcal D})$. Then, there is $C=C({\mathcal D})$ such that

$$\begin{aligned} \Vert D^2u\Vert _2\le C\,(\Vert \textrm{div}\,T\Vert _2+\Vert D(u)\Vert _2). \end{aligned}$$

Proof

The lemma follows from [7, Lemma V.4.3] and Lemma 4.1. $\square$

Lemma 4.3

Let $y:[0,T)\mapsto [0,\infty )$, $T>0$, be absolutely continuous, such that

$$\begin{aligned} y'(t)\le G(t)+c_1y(t)+c_2y^\alpha (t)\,,\ \ \alpha >1,\ \ t\in (0,T)\,, \end{aligned}$$

(4.1)

where $G\in L^1(0,T)$, $G(t)\ge 0$, and $c_i\in [0,\infty )$, $i=1,2$. Then, if –in case at least one of the constants $c_i$ is not zero– it is also $y\in L^1(0,T)$, there exists $\eta >0$, such that from

$$\begin{aligned} y(0)+\int _0^T G(s)\,ds+\int _0^T y(s)\,ds\le \eta \end{aligned}$$

(4.2)

it follows $y\in L^\infty (0,T)$ and

$$\begin{aligned} y(t)< M\,\eta \,,\ \ M:=2\max \{1,c_1,c_2\}, \ \ t\in [0,T)\,. \end{aligned}$$

(4.3)

In the case $T=\infty$, we also have

$$\begin{aligned} \lim _{t\rightarrow \infty }y(t)=0\,, \end{aligned}$$

(4.4)

and, if $t\,G\in L^1(0,\infty )$, $c_1=0$, and $\alpha \ge 2$, even

$$\begin{aligned} \sup _{t\in (0,\infty )}\left( t\,y(t)\right) \le A<\infty \,. \end{aligned}$$

(4.5)

Proof

Since $y(0)\le \eta$, contradicting (4.3) means that there exists $t_0\in (0,T)$ such that $y(t)<M\,\eta$ for all $t\in [0,t_0)$ and $y(t_0)=M\,\eta$. Integrating both sides of (4.1) from 0 to $t_0$, we deduce, in particular

$$\begin{aligned} y(t_0)\le y(0) +\int _0^{T}G(s)\,\textrm{d}s+c_1\int _0^T y(s)\,\textrm{d}s +c_2\int _0^{t_0}y^\alpha (s)\,\textrm{d}s. \end{aligned}$$

Therefore, setting $\mu =\max \{1,c_1,c_2\}$, from this inequality and (4.2) we find

$$\begin{aligned} y(t_0)\le \mu \,\eta \left[ 1 + (M\eta )^{\alpha -1}\right] , \end{aligned}$$

so that, choosing $\eta \in (0,(1/M)^{1/(\alpha -1)})$, we obtain $y(t_0)<M\,\eta$, a contradiction that proves (4.3). In order to show the property (4.4), we observe that, in view of assumption (4.2) and being $T=\infty$, there exists an unbounded sequence $\{t_k\}\subset (0,\infty )$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }y(t_k)=0 \,. \end{aligned}$$

(4.6)

We then integrate (4.1) from $t_k$ to arbitrary $t>t_k$, and recall (4.3), to deduce, in particular,

$$\begin{aligned} y(t)\le y(t_k) +\int _{t_k}^{\infty }G(s)\,\textrm{d}s+c_3\int _{t_k}^\infty y(s)\,\textrm{d}s, \end{aligned}$$

for some $c_3>0$ and all $t>t_k$. In view of (4.2) with $T=\infty$ and (4.6), the right-hand side of this inequality can be made as small as we please, by taking sufficiently large k, and property (4.4) follows. Finally, take in (4.1) $c_1=0$ and $\alpha \ge 2$. Multiplying both sides of the resulting inequality by $t>0$ and setting $Y(t):= t y(t)$, $g(t):=t G(t)$, we get

$$\begin{aligned} Y^\prime (t)\le g(t)+y(t)+c_2y^{\alpha -1}(t)Y(t), \end{aligned}$$

which entails

$$\begin{aligned} Y(t)\le \beta +\int _0^th(s)Y(s)\textrm{d}s,\ \ t\in (0,\infty ), \end{aligned}$$

with

$$\begin{aligned} \beta :=\int _0^\infty (g(s)+y(s))\textrm{d}s,\ \ h(t):=c_2y^{\alpha -1}. \end{aligned}$$

Using Gronwall’s lemma, we show

$$\begin{aligned} Y(t)\le \beta \,\textrm{exp}\left( \int _0^t h(s)\textrm{d}s\right) \,. \end{aligned}$$

(4.7)

By assumption and (4.3) it follows that $\beta <\infty$ and $h\in L^1(0,\infty )$, so that the lemma follows from (4.7). $\square$

Proof of Theorem 2.1

In [8, Theorem 4.1], it is shown the existence of a solution $(u,p,\xi ,\omega ,Q)$ to (2.1)–(2.3) in the class ${\mathscr {C}}_T$,,⁴ where T is maximal, namely, either $T=\infty$, or else there is $\{t_k\}\in (0,T)$ such that

$$\begin{aligned} \lim _{t_k\rightarrow T}\Vert D(u(t_k))\Vert _2=+\infty \,. \end{aligned}$$

(4.8)

We shall show that, in fact, (4.8) cannot occur, provided the data satisfy (2.4) for suitable $\delta >0$, thus implying that $(u,p,\xi ,\omega ,Q)$ exists for all times and is in ${\mathscr {C}}_\infty$. We begin to observe that, clearly, $(u,p,\xi ,\omega ,Q)$ satisfies (3.12) for all $t\in (0,T)$. Thus, using Cauchy–Schwarz on the right-hand side of (3.12)$_1$ and integrating over $t\in (0,T)$ we get

$$\begin{aligned} \begin{array}{ll}\displaystyle \sup _{t\in [0,T]}\bigg ( \varrho \Vert u (t)\Vert _2^2 + m \left| \xi (t) \right| ^2 + \omega (t) \cdot I \cdot \omega (t) \bigg ) + {\frac{1}{2}}\mu \int _0^T \Vert D(u(s)) \Vert _2^2\,\textrm{d}s\\ \qquad \qquad \le \varrho \Vert u_0\Vert _2^2 + m \left| \xi _0 \right| ^2 + \omega _0 \cdot I \cdot \omega _0+c(\Vert F \Vert _{L^2(0,\infty )}+\Vert M \Vert _{L^2(0,\infty )})\,. \end{array} \end{aligned}$$

(4.9)

Moreover, again by Cauchy–Schwarz inequality and Lemma 4.1,

$$\begin{aligned} \begin{array}{ll} \big |m \omega \times \xi \cdot \dot{\xi } + \omega \times (\textsf{I} \cdot \omega )\cdot \dot{\omega }+ \textsf{F} \cdot \dot{\xi } + \textsf{M} \cdot \dot{\omega }\big |\le \frac{m}{2}|\dot{\xi }|^2+\frac{1}{2}\dot{\omega }\cdot \textsf{I}\cdot \dot{\omega }+c\left( \Vert D(u)\Vert _2^4+|F |^2+|M |^2\right) \\ \displaystyle \left| \int _{{\mathcal {D}}} ( \omega \times {\nabla }u_i\cdot \nabla u_i - \nabla (\omega \times u):\nabla u )\right| \le c\,\Vert D(u)\Vert _2^3\,. \end{array} \end{aligned}$$

(4.10)

Using the well-known trace inequality

$$\begin{aligned} \Vert w\Vert _{2,\Sigma }\le c(\Vert w\Vert _2+\Vert w\Vert _2^\frac{1}{2}\Vert \nabla w\Vert _2^\frac{1}{2}),\ \ w\in W^{1,2}({\mathcal D}), \end{aligned}$$

along with Lemmas 4.1 and 4.2, we show

$$\begin{aligned} \begin{array}{rl} \left| {\displaystyle \int _{ \Sigma }^{ }}\left( n\cdot \nabla u\cdot \Phi -{\frac{1}{2}}V\cdot n|\nabla u|^2\right) \right| &{}\le c\,(|\xi |+|\omega |)\Vert \nabla u\Vert ^2_{2,\Sigma }\le c\, \big (\Vert D(u)\Vert _2^3+\Vert D(u)\Vert _2^2\Vert D^2u\Vert _2\big )\\ &{}\le c\, \big (\Vert D(u)\Vert _2^3+\Vert D(u)\Vert _2^4\big )+\frac{1}{4}\Vert \textrm{div}\,T(u,p)\Vert _2^2\,. \end{array} \end{aligned}$$

(4.11)

Finally, employing the embedding inequality

$$\begin{aligned} \Vert w\Vert _3\le c\,(\Vert \nabla w\Vert _2^\frac{1}{2}\Vert w\Vert _2^\frac{1}{2}+\Vert w\Vert _2),\ \ w\in W^{1,2}({\mathcal D}), \end{aligned}$$

Hölder inequality, Lemmas 4.1 and 4.2, we show

$$\begin{aligned} \begin{array}{rl}\displaystyle \left| \int _{{\mathcal {D}}} u\cdot {\nabla }u \cdot {\textrm{div}\,}T(u,p)\right| &{}\!\!\le c\,\Vert u\Vert _6\Vert D(u)\Vert _3\Vert {\textrm{div}\,}T(u,p)\Vert _2\le c\,\Vert D(u)\Vert _2(\Vert D^2u\Vert _2^\frac{1}{2}\Vert D(u)\Vert _2^\frac{1}{2}+\Vert D(u)\Vert _2)\Vert \textrm{div}\,T\Vert _2\\ &{}\!\!\le c\,(\Vert D(u)\Vert _2^4+\Vert D(u)\Vert _2^6)+\frac{1}{4}\Vert \textrm{div}\,T\Vert _2^2\,. \end{array} \end{aligned}$$

(4.12)

Using (4.10)–(4.12) in (3.12)$_2$ we infer

$$\begin{aligned} \mu \frac{d}{dt}\Vert D(u)\Vert _{2}^2 + {\frac{1}{2}}m |\dot{\xi }|^2 + {\frac{1}{2}}\dot{\omega } \cdot \textsf{I} \cdot \dot{\omega } + {\frac{1}{2}}\Vert {\textrm{div}\,}T(u,p)\Vert _{2}^2 \le c\,(\Vert D(u)\Vert _{2}^3+\Vert D(u)\Vert _{2}^4+\Vert D(u)\Vert _{2}^6+|F |^2+|M |^2)\,, \end{aligned}$$

(4.13)

and so, setting $y:=\Vert D(u)\Vert _2^2$, $G:=(c/\mu )(|F |^2+|M |^2)$ (4.13) furnishes, in particular,

$$\begin{aligned} y^\prime (t)\le G(t)+ c\left( y^\frac{3}{2}(t)+y^2(t)+y^3(t)\right) \,,\ \ t\in (0,T)\,, \end{aligned}$$

(4.14)

Using multiple times Cauchy–Schwarz inequality, we show that (4.14) implies (4.1) with $\alpha =3$. Moreover, from (4.9), we can find $\delta >0$ such that if (2.4) holds, then assumption (4.2) of Lemma 4.2 is satisfied.⁵ Thus, by that lemma, it follows that

$$\begin{aligned} \sup _{t\in [0,T]}\Vert D(u(t))\Vert _2\le C\,, \end{aligned}$$

(4.15)

which contradicts (4.8). As a result, $T=\infty$ and therefore, by the second part of Lemma 4.3, we get

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert D(u(t))\Vert _2=0. \end{aligned}$$

The latter, in conjunction with Lemma 4.1, entails (2.6). We next observe that, from (4.9) and Lemma 4.1, it follows that

$$\begin{aligned} \int _0^\infty \left( |{\xi }(t)|^2+|{\omega }(t)|^2+\Vert \nabla u(t)\Vert _2^2\right) \textrm{d}t\le C, \end{aligned}$$

while, by integrating both sides of (4.13) over $t\in (0,\infty )$ and with the help of Lemma 4.2 and (4.15), we get

$$\begin{aligned} \int _0^\infty \left( |\dot{\xi }(t)|^2+|\dot{\omega }(t)|^2+\Vert D^2u(t)\Vert _2^2\right) \textrm{d}t\le C, \end{aligned}$$

where the constant C depends only on the data. Finally, from the latter, (4.12) and Sobolev inequality⁶ we deduce

$$\begin{aligned} \int _0^\infty \left( \Vert p(t)\Vert _6^2+\Vert \nabla p(t)\Vert _2^2\right) \textrm{d}t\le C, \end{aligned}$$

which completes the proof of the theorem. $\square$

Remark 4.1

Theorem 2.1 shows that, under the given assumptions on the data, the coupled system will eventually go to a state of rest in the sense specified in (2.6). However, as also mentioned earlier on, we are not able to provide a rate of decay. In fact, we cannot apply the result stated in the last part of Lemma 4.3 to the general case studied here, namely, a body of arbitrary shape. What prevents us from doing so is the presence of $\Vert D(u)\Vert _2^3$ in (4.13) or, equivalently, $y^\frac{3}{2}$ in (4.14). However, if ${\mathcal B}$ is a ball, that term does not occur. To show this, we notice that in this situation the term $(\omega \times x\cdot \nabla u-\omega \times u)$ is no longer present in (2.1)$_1$ as well as are not the terms $m\omega \times \xi$ and $\omega \times (\textsf{I}\cdot \omega )$ in (2.1)$_{5}$, (2.1)$_{6}$, respectively (see, e.g., [5]). Therefore, (3.12)$_2$ becomes

$$\begin{aligned} \mu \frac{d}{dt}\Vert D(u)\Vert _{2}^2 + m |\dot{\xi }|^2+\dot{\omega } \cdot \textsf{I} \cdot \dot{\omega } + \Vert {\textrm{div}\,}T(u,p)\Vert _{2}^2 = - \textsf{F} \cdot \dot{\xi } - \textsf{M} \cdot \dot{\omega } + \varrho \displaystyle \int _{{\mathcal {D}}} (u-\xi )\cdot {\nabla }u \cdot {\textrm{div}\,}T(u,p). \end{aligned}$$

Arguing as in the proof of Theorem 2.1, one then shows

$$\begin{aligned} \begin{array}{rl} \left| - \textsf{F} \cdot \dot{\xi } - \textsf{M} \cdot \dot{\omega } + \varrho \displaystyle \int _{{\mathcal {D}}} (u-\xi )\cdot {\nabla }u \cdot {\textrm{div}\,}T(u,p)\right| &{}\le {\frac{1}{2}}\big ( m |\dot{\xi }|^2+ \dot{\omega } \cdot \textsf{I} \cdot \dot{\omega } + \Vert {\textrm{div}\,}T\Vert _2^2\big )\\ {} &{}\quad +\,c\,(|F |^2+|M |^2+\Vert D(u)\Vert _2^4+\Vert D(u)\Vert _2^6). \end{array} \end{aligned}$$

Consequently, combining the last two displayed relations and recalling that $\Vert D(u(t))\Vert _2$ is uniformly bounded in t by the data, we deduce

$$\begin{aligned} \frac{d}{dt}\Vert D(u)\Vert _{2}^2 \le c\,(|F |^2+|M |^2+\Vert D(u)\Vert _{2}^4), \end{aligned}$$

and by applying Lemmas 4.1 and 4.3 we conclude

$$\begin{aligned} \Vert u(t)\Vert _6+\Vert \nabla u(t)\Vert _2+|\xi (t)|+|\omega (t)|=O(t^{-\frac{1}{2}})\ \ \text{ as } t\rightarrow \infty , \end{aligned}$$

provided $t^\frac{1}{2}\textrm{F},\, t^\frac{1}{2}\textrm{M}\in L^2(0,\infty )$.