Stability of Time-Dependent Motions for Fluid–Rigid Ball Interaction

Abstract

We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier–Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce $L^q$–$L^r$ decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid–structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.

1 Introduction

We study the stability of nontrivial basic motions for the fluid–rigid ball interaction in 3D within the framework of $L^q$-theory. This work is inspired by Ervedoza, Maity and Tucsnak [10], who have successfully established the $L^q$-stability of the rest state. The novelty of our study is to develop analysis even for time-dependent basic motions such as time-periodic (and almost periodic) ones. We emphasize that what is done here is already new when the basic motion is a nontrivial steady state. On the other hand, the shape of a single rigid body is not allowed to be arbitrary; in fact, it is assumed to be a ball in the present paper. Later, we will explain the reason why and mention the existing literature, in particular, [17, 36] by Galdi and by Maity and Tucsnak, about the related issue for the case of arbitrary shape. To fix the idea, in what follows, we are going to discuss the self-propelled regime as a typical case, nevertheless, the approach itself can be adapted to other situations of the fluid–rigid ball interaction, in which physically relevant nontrivial basic motions could be available, as long as the system (1.5) below for disturbance is the same.

When the rigid body is a ball, it is reasonable to take the frame attached to the center of mass of the ball just by translation to reduce the problem to the one in a time-independent domain. Let B be the open ball centered at the origin with radius 1, that is identified with a rigid body whose density $\rho >0$ is assumed to be a constant. In the resultant problem after the change of variable, a viscous incompressible fluid occupies the exterior domain $\Omega ={\mathbb {R}}^3\setminus {\overline{B}}$ and the motion obeys the Navier–Stokes system with an additional nonlinear term (due to the change of variable), where the fluid velocity and pressure are denoted by $u(x,t)\in {\mathbb {R}}^3$ and $p(x,t)\in {\mathbb {R}}$, whereas $\eta (t)\in {\mathbb {R}}^3$ and $\omega (t)\in {\mathbb {R}}^3$ respectively stand for the translational and angular velocities of the rigid ball and are governed by the balance for linear and angular momentum. The fluid velocity u meets the rigid motion $\eta +\omega \times x$ (with $\times$ being the vector product) at the boundary $\partial \Omega$, where an extra velocity $u_*(x,t)\in {\mathbb {R}}^3$ generated by the body is allowed to be involved. Indeed, the velocity $u_*$ plays an important role within the self-propelled regime, in which there are no external force and torque so that the body moves due to an internal mechanism described by $u_*$ through interaction of the fluid–rigid body, see Galdi [14, 15] for the details. We assume that $u_*$ is tangential to the boundary $\partial \Omega$, that is, $\nu \cdot u_*|_{\partial \Omega }=0$. Here, $\nu$ stands for the unit normal to $\partial \Omega$ directed toward the interior of the rigid body; indeed, $\nu =-x$ at $\partial \Omega$ being the unit sphere. Consequently, the unknowns $u,\,p,\,\eta$ and $\omega$ obey ([10, 15])

$$\begin{aligned} \begin{aligned}&\partial _tu+(u-\eta )\cdot \nabla u=\Delta u-\nabla p, \qquad \text{ div }\; u =0 \quad \text { in }\; \Omega \times I, \\ {}&u|_{\partial \Omega }=\eta +\omega \times x+u_*, \qquad u\rightarrow 0 \quad \text{ as }\ |x|\rightarrow \infty , \\ {}&m\frac{d\eta }{dt}+\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =0, \\ {}&J\frac{d\omega }{dt}+\int _{\partial \Omega }x\times \mathbb S(u,p)\nu \,d\sigma =0, \end{aligned} \end{aligned}$$

(1.1)

where I is the whole or half time axis in ${\mathbb {R}}$, $\mathbb S(u,p)$ denotes the Cauchy stress tensor, that is,

$$\begin{aligned} {\mathbb {S}}(u,p)=2Du-p\,{\mathbb {I}}, \qquad Du=\frac{\nabla u+(\nabla u)^\top }{2} \end{aligned}$$

(1.2)

with ${\mathbb {I}}$ being $3\times 3$ unity matrix. Here and hereafter, $(\cdot )^\top$ stands for the transpose. The mass and tensor of inertia of the rigid ball are given by

$$\begin{aligned} m= \frac{4\pi \rho }{3}, \qquad J= \frac{2m}{5}\,{\mathbb {I}}. \end{aligned}$$

(1.3)

The model with several physical parameters can be reduced to the problem above in which both the kinematic viscosity of the fluid and the radius of the rigid ball are normalized. Then $\rho$ is regarded as the ratio of both densities of the fluid–structure, see Sect. 2.1.

In [10] Ervedoza, Maity and Tucsnak considered the case when the extra velocity $u_*$ is absent and proved that the initial value problem for (1.1) admits a unique solution globally in time with decay properties such as

$$\begin{aligned} \Vert u(t)\Vert _{\infty ,\Omega }+|\eta (t)|+|\omega (t)|=O(t^{-1/2}) \end{aligned}$$

as $t\rightarrow \infty$ under the smallness of the $L^3$-norm of the initial velocity as well as the initial rigid motion. This is a desired development of the local well-posedness in the space $L^q$ due to Wang and Xin [51] and may be also regarded as the stability of the rest state. The stability of that state in the 2D case was already studied by Ervedoza, Hillairet and Lacave [9] when the rigid body is a disk. The present paper is aiming at the study of stability criterion for nontrivial basic motions $\{u_b,p_b,\eta _b,\omega _b\}$ to (1.1), that is, we intend to deduce the large time behavior

$$\begin{aligned} \Vert u(t)-u_b(t)\Vert _{\infty ,\Omega }+\Vert \nabla u(t)-\nabla u_b(t)\Vert _{3,\Omega } +|\eta (t)-\eta _b(t)|+|\omega (t)-\omega _b(t)|=o(t^{-1/2}) \end{aligned}$$

(1.4)

of the solution to the initial value problem for (1.1) as $t\rightarrow \infty$ provided that the initial disturbance is small enough in the same sense as in [10] and that the basic motion is also small in a sense uniformly in t as well as globally Hölder continuous in t. We will discuss briefly in Sect. 2.4 possible basic motions within the self-propelled regime (1.1). Let us also mention that the asymptotic rate of $\nabla u(t)$ in (1.4) is an improvement of the corresponding result due to [10], in which less rate is deduced when $u_b=0,\, \eta _b=\omega _b=0$. Let $\{u_b,\,p_b,\,\eta _b,\,\omega _b\}$ be a solution to (1.1) on the whole time axis $I={\mathbb {R}}$. We intend to find a solution to the initial value problem for (1.1) in the form

$$\begin{aligned} u=u_b+{\widetilde{u}}, \quad p=p_b+{\widetilde{p}}, \quad \eta =\eta _b+{\widetilde{\eta }}, \quad \omega =\omega _b+{\widetilde{\omega }}. \end{aligned}$$

Omitting the tildes $\;\widetilde{(\cdot )}$, the disturbance should obey

$$\begin{aligned} \begin{aligned}&\left. \begin{array}{rl} \partial _tu+(u-\eta )\cdot \nabla u+(u_b-\eta _b)\cdot \nabla u+(u-\eta )\cdot \nabla u_b =&{}\Delta u -\nabla p \\ \hbox { div}\ u=&{}0 \end{array} \right\} \hbox { in}\ \Omega \times (s,\infty ), \\&\quad u|_{\partial \Omega }=\eta +\omega \times x, \qquad u\rightarrow 0\quad \hbox { as}\ |x|\rightarrow \infty , \\&\quad m\frac{d\eta }{dt}+\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =0, \\&\quad J\frac{d\omega }{dt}+\int _{\partial \Omega }x\times \mathbb S(u,p)\nu \,d\sigma =0, \end{aligned} \end{aligned}$$

(1.5)

endowed with the initial conditions at the initial time $s\in \mathbb R$.

Besides [10], we will mention the existing literature, that is particularly related to this study, about the original problem in the inertial frame in 3D. Serre [41] constructed a global weak solution to the initial value problem under the action of gravity. This solution is of the Leray-Hopf class and the argument is based on the energy relation as in the standard Navier–Stokes theory. Silvestre [43] also studied a weak solution under the self-propelling condition and moreover, in a specific case that the body is axisymmetric, she constructed a global strong solution with some parity conditions and discussed the attainability of the translational self-propelled steady motion found by Galdi [13], answering to celebrated Finn’s starting problem in her context. When the external force and torque act on the rigid body whose shape is arbitrary, a strong solution locally in time is constructed first by Galdi and Silvestre [20]. All of those literature [20, 41, 43] developed the $L^2$-theory and the problem was studied in a frame attached to the body of arbitrary shape (except the latter half of [43]), where, unlike the derivation of (1.1), the change of variable must involve the rotation matrix as well as translation unless the body is a ball, see Galdi [15] for the details. Then the resultant equation reads

$$\begin{aligned} \partial _t u+(u-\eta -\omega \times x)\cdot \nabla u +\omega \times u=\Delta u-\nabla p \end{aligned}$$

(1.6)

in place of the first equation of (1.1). The equations of the rigid body in (1.1) are also replaced by

$$\begin{aligned} \begin{aligned}&m\frac{d\eta }{dt}+\omega \times (m\eta ) +\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =0, \\&J\frac{d\omega }{dt}+\omega \times (J\omega ) +\int _{\partial \Omega }x\times {\mathbb {S}}(u,p)\nu \,d\sigma =0. \end{aligned} \end{aligned}$$

(1.7)

If we considered the exterior problem with a prescribed rigid motion, the difficulty caused by the drift term $(\omega \times x)\cdot \nabla u$ with spatially unbounded (possibly even time-dependent) coefficient would be more or less overcome due to efforts by several authors, see [19, 25,26,27,28] and the references therein. But this term is indeed a nonlinear term for the fluid–structure interaction under consideration and still prevents us from carrying out analysis in a successful way. This is why we are forced to restrict ourselves to the case of the ball in the present paper as well as in [10]. The linear theory of large time behavior is well established for the case of arbitrary shape in [10], while we have already a difficulty at the level of linear analysis in this paper, see Remark 3.4 in Sect. 3.7. Recently, Galdi [17] has succeeded in continuation of the solution obtained in [20] globally in time by means of energy estimates and, moreover, he has proved the large time decay

$$\begin{aligned} \lim _{t\rightarrow \infty }\Big (\Vert u(t)\Vert _{6,\Omega }+\Vert \nabla u(t)\Vert _{2,\Omega }+|\eta (t)|+|\omega (t)|\Big )=0 \end{aligned}$$

(1.8)

even if the shape of the body is arbitrary when the $H^1$-norm of the initial velocity, the initial rigid motion, and the $L^2$-norm in $t\in (0,\infty )$ of the external force and torque acting on the body are small enough. Although there is no definite decay rate in general, one can derive the rate $O(t^{-1/2})$ in (1.8) if in particular the body is a ball, see [17, Remark 4.1].

There is the other way of transformation to reduce the original problem to the one in a time-independent domain. This is a local transformation that keeps the situation far from the body as it is in order to avoid the term $(\omega \times x)\cdot \nabla u$ in (1.6) even if the shape of the body is arbitrary. Cumsille and Takahashi [5] adopted this transformation to construct a solution locally in time and then successfully derived a priori estimates in the inertial frame so that the global existence of a strong solution for small data was first established within the $L^2$-theory, however, without any information about the large time behavior. By the same transformation, Geissert, Götze and Hieber [21] developed the $L^q$-theory for the local well-posedness in the maximal regularity class; further, in the recent work [36], Maity and Tucsnak have proved even large time decay with definite rate as well as global well-posedness for small data by adapting the time-shifted method proposed by Shibata [42] in the framework of maximal regularity with time-weighted norms. Since there are several complicated terms arising from this latter transformation, it seems difficult to take a reasonable nontrivial basic motion in the resultant system and to discuss its stability, that is indeed the issue here, unless one considers the problem around the rest state. Thus the latter transformation is not preferred in the present paper.

Let us turn to our problem (1.5) around a nontrivial motion. Towards the large time behavior of the disturbance, the essential step is to develop the temporal decay estimates of solutions to the linearized system. In fact, this was successfully done by [10] when $u_b=0$ and $\eta _b=0$. They adopted the monolithic approach which is traced back to Takahashi and Tucsnak [46], see also Silvestre [43], and then derived the $L^q$–$L^r$ decay estimates of the semigroup, that they call the fluid–structure semigroup (or Takahashi-Tucsnak semigroup). Look at (1.5), then the term which is never subordinate to the fluid–structure semigroup is $\eta _b\cdot \nabla u$. This term is well known as the Oseen term in studies of the exterior Navier–Stokes problem, see [16, 34]. For the fluid–structure interaction problem, however, fine temporal behavior of the (purely) Oseen operator $-\Delta u-\eta _b\cdot \nabla u+\nabla p$ is hopeless because the term $\eta _b\cdot \nabla u$ is no longer skew-symmetric on account of the boundary condition. The idea is to combine the Oseen term with $u_b\cdot \nabla u$ since the skew-symmetry is recovered for $(u_b-\eta _b)\cdot \nabla u$, see (3.89)–(3.90). The first step is therefore to derive some decay properties of solutions to the linearized system with the term $(U_b-\eta _b)\cdot \nabla u$ in the whole space without the rigid body, see Sect. 4.1, where $U_b$ is the monolithic velocity (2.32). Here, the linear term $U_b\cdot \nabla u$ can be treated as perturbation from the purely Oseen evolution operator provided that $u_b$ is of sub-critical class such as $u_b\in L^q(\Omega )$ with some $q<3$ even though $u_b$ is time-dependent. In the scale-critical case such as $u_b\in L^{3,\infty }(\Omega )$ (weak-$L^3$ space), decay estimate (4.14) with $r<\infty$ is still available, however, it is difficult to deduce gradient estimate (4.15) of the solution when $u_b$ is time-dependent. Although there is a device to construct the Navier–Stokes flow even in this situation as proposed in [27, Sect. 5], it does not work for the other nonlinear term $\eta \cdot \nabla u$. Thus the critical case is out of reach in this paper, and let us concentrate ourselves on the sub-critical case, which covers the self-propelled motion and the motion with wake structure (due to translation). In the subcritical case, the linear term $u\cdot \nabla u_b$ can be discussed together with the nonlinear term as perturbation from the principal part of the linearized system. Moreover, the other term $\eta \cdot \nabla u_b$ is comparable to $u\cdot \nabla u_b$ since $\eta (t)$ behaves like $\Vert u(t)\Vert _{\infty ,\Omega }$ as $t\rightarrow \infty$. As for an alternative way in which the term $(u-\eta )\cdot \nabla u_b$ is also involved into the linearization, see Remark 3.3 in Sect. 3.7. The right choice of the principal part of the linearized system would be the following non-autonomous system, which we call the Oseen-structure system:

$$\begin{aligned} \begin{aligned}&\partial _tu=\Delta u+(\eta _b-u_b)\cdot \nabla u-\nabla p, \qquad \text{ div }\; u =0 \quad \text { in }\; \Omega \times (s,\infty ), \\ {}&u|_{\partial \Omega }=\eta +\omega \times x, \qquad u\rightarrow 0 \quad \text{ as }\ |x|\rightarrow \infty , \\ {}&m\frac{d\eta }{dt}+\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =0, \\ {}&J\frac{d\omega }{dt}+\int _{\partial \Omega }x\times {\mathbb {S}}(u,p)\nu \,d\sigma =0, \\ {}&u(\cdot ,s)=u_0, \quad \eta (s)=\eta _0, \quad \omega (s)=\omega _0 \end{aligned} \end{aligned}$$

(1.9)

with $s\in {\mathbb {R}}$ being the initial time. In fact, in this paper, we develop the $L^q$–$L^r$ estimates of solutions to (1.9) and apply them to the nonlinear problem (1.5). We do need the specific shape already in linear analysis, otherwise new term $(\omega \times x)\cdot \nabla u_b$ appears in (1.5), which arises from $(\omega \times x)\cdot \nabla u$ in (1.6), see Remark 3.4 for further discussion, although the other term $(\eta _b+\omega _b\times x-u_b)\cdot \nabla u$ can be handled as in [27, 28] by the present author. Since the difficult term above is absent around the rest state even if the shape of the rigid body is arbitrary, our method developed in this paper works well to provide an alternative proof of the $L^q$–$L^r$ estimates of the fluid–structure semigroup established by [10].

The difficulty to analyze (1.9) is the non-autonomous character. For the linearized problem relating to the Navier–Stokes system in exterior domains, the only results that study the temporal decay for the non-autonomous system seem to be due to the present author [27, 28] on the $L^q$–$L^r$ estimates of the Oseen evolution operator arising from (prescribed) time-dependent rigid motions. In this paper we adapt the method developed in those papers to the Oseen-structure system (1.9), where the crucial step is to deduce the uniformly boundedness of the evolution operator for large time, see (4.32)–(4.33) in Sect. 4.3, with the aid of (4.27)–(4.28) which follow from the energy relations (4.25)–(4.26). In this stage one needs to discuss the issue above simultaneously with the one of the adjoint evolution operator, that is the solution operator to the backward problem for the adjoint system, see Sect. 3.7. We then proceed to the next stage in which gradient estimates of the evolution operator are derived near the rigid body (Proposition 4.4) and near spatial infinity (Proposition 4.5). In this latter stage, asymptotic behavior (for $t-s\rightarrow 0$ and $t-s\rightarrow \infty$) of the associated pressure together with the time derivative of the evolution operator plays an important role. We first deduce the bahavior of the pressure near the initial time in Sect. 3.6, where analysis is developed with the aid of fractional powers of the Stokes-structure operator and very different from the study of the same issue in [28] for the case of prescribed time-dependent rigid motions. In doing so, we make use the representation of the Stokes-structure resolvent, which is given in Sect. 3.3. To employ the monolithic approach, in this paper unlike [10], we do not rely on the decomposition shown by Wang and Xin [51] because the associated projection is not symmetric with respect to the duality pairing which involves the constant density $\rho$ of the rigid body, see (1.3) and (2.11), unless $\rho =1$. This situation is not consistent with our argument particularly when making use of the adjoint evolution operator. For this reason, our analysis is based on the similar but different decomposion (2.18) below, then the associated projection possesses the aforementioned desired symmetry. The decomposition (2.18) was established by Silvestre [43] when $q=2$. Since the same result for the $L^q$-space does not seem to be found in the existing literature, the proof is given in Sect. 3.1.

The paper consists of five sections. In the next section we introduce the Oseen-structure operator as well as the Stokes-structure one and then give our main results: Theorem 2.1 on $L^q$–$L^r$ estimates of the evolution operator and Theorem 2.2 on the nonlinear stability. Section 3 is concerned with some preparatory results: the decomposition of the $L^q$-space mentioned above, reformulation of the Stokes-structure operator, smoothing estimate as well as generation of the Oseen-structure evolution operator, analysis of the pressure and, finally, the adjoint evolution operator. In Sect. 4 we study in detail the large time decay of the evolution operator to complete the proof of Theorem 2.1. Final section is devoted to the proof of Theorem 2.2 for the nonlinear problem (1.5).

2 Main Results

2.1 Nondimensional Variables

If all physical parameters are taken into account, the system (1.1) should read

$$\begin{aligned} \begin{aligned}&\rho _{_L}\left\{ \partial _tu+(u-\eta )\cdot \nabla u\right\} =\mu \Delta u-\nabla p, \qquad \text{ div }\; u =0 \quad \text { in }\; \Omega ^R\times I, \\ {}&u|_{\partial \Omega ^R}=\eta +\omega \times x+u_*, \qquad u\rightarrow 0 \quad \text{ as }\ |x|\rightarrow \infty , \\ {}&m_{_R}\frac{d\eta }{dt}+\int _{\partial \Omega ^R}{\mathbb {S}}_\mu (u,p)\nu \,d\sigma =0, \\ {}&J_{_R}\frac{d\omega }{dt}+\int _{\partial \Omega ^R}x\times \mathbb S_\mu (u,p)\nu \,d\sigma =0, \end{aligned} \end{aligned}$$

(2.1)

with

$$\begin{aligned} \begin{aligned}&\Omega ^R={\mathbb {R}}^3\setminus \overline{B_R}, \qquad {\mathbb {S}}_\mu (u,p)=2\mu Du-p\,{\mathbb {I}}, \\&m_{_R}=\int _{B_R}\rho _{_S}\,dx =\frac{4\pi R^3\rho _{_S}}{3}, \qquad J_{_R}=\int _{B_R}\big (|x|^2\,{\mathbb {I}}-x\otimes x\big )\rho _{_S}\,dx =\frac{8\pi R^5\rho _{_S}}{15}\,{\mathbb {I}}, \end{aligned} \end{aligned}$$

where the rigid body is a ball $B_R$ centered at the origin with radius R, $\rho _{_L}$ and $\rho _{_S}$ are respectively the densities of the liquid and solid, and $\mu$ stands for the viscosity coefficient. All of those parameters are assumed to be positive constants. Setting

$$\begin{aligned} \begin{aligned}&{\widetilde{x}}=\frac{1}{R}\,x, \qquad {\widetilde{t}}=\frac{\mu }{R^2\rho _{_L}}\,t, \\&{\widetilde{u}}({\widetilde{x}},\widetilde{t})=\frac{R\rho _{_L}}{\mu }\,u(x,t), \qquad {\widetilde{p}}({\widetilde{x}},{\widetilde{t}})=\frac{R^2\rho _{_L}}{\mu ^2}\,p(x,t), \\&{\widetilde{\eta }}({\widetilde{t}})=\frac{R\rho _{_L}}{\mu }\,\eta (t), \qquad {\widetilde{\omega }}(\widetilde{t})=\frac{R^2\rho _{_L}}{\mu }\,\omega (t), \qquad \widetilde{u}_*({\widetilde{x}},{\widetilde{t}})=\frac{R\rho _{_L}}{\mu }\,u_*(x,t) \end{aligned} \end{aligned}$$

and omitting the tildes $\;\widetilde{(\cdot )}$, we are led to (1.1) with

$$\begin{aligned} \rho =\frac{\rho _{_S}}{\rho _{_L}}, \quad m=\int _{B_1}\rho \,dx=\frac{4\pi \rho }{3}, \quad J=\int _{B_1}\big (|x|^2{\mathbb {I}}-x\otimes x\big )\rho \,dx=\frac{2m}{5}\,{\mathbb {I}}, \end{aligned}$$

(2.2)

where J is called the tensor of inertia. It is worth while mentioning

$$\begin{aligned} x\times (a\times x)=\big (|x|^2{\mathbb {I}}-x\otimes x\big )a \end{aligned}$$

(2.3)

for all $a\in {\mathbb {R}}^3$, that implies

$$\begin{aligned} \int _{B_1}(a\times x)\cdot (b\times x)\rho \,dx=(Ja)\cdot b=a\cdot (Jb) \end{aligned}$$

(2.4)

for all $a,\, b\in {\mathbb {R}}^3$. Even if the shape of the rigid body is not a ball, the symmetric matrix J defined by the latter integral of (2.2) satisfies (2.4) and thus J is positive definite.

2.2 Stokes-Structure Operator

Let us begin with introducing fundamental notation. Throughout this paper, we set

$$\begin{aligned} B:=B_1, \qquad \Omega :={\mathbb {R}}^3\setminus {\overline{B}}, \qquad \Omega _R:=\Omega \cap B_R\quad (R>1), \end{aligned}$$

where $B_R$ denotes the open ball centered at the origin with radius R. Given a domain $D\subset {\mathbb {R}}^3$, $q\in [1,\infty ]$ and integer $k\ge 0$, the standard Lebesgue and Sobolev spaces are denoted by $L^q(D)$ and $W^{k,q}(D)$. We abbreviate the norm $\Vert \cdot \Vert _{q,D}=\Vert \cdot \Vert _{L^q(D)}$. By $\langle \cdot ,\cdot \rangle _D$ (resp. $\langle \cdot ,\cdot \rangle _{\partial D}$) we denote standard duality pairings over the domain D (resp. $\partial D$ being the boundary of D) in each context. As the space for the pressure, one needs also the homogeneous Sobolev space

$$\begin{aligned} {\widehat{W}}^{1,q}(D)=\{p\in L^q_\textrm{loc}({\overline{D}});\;\nabla p\in L^q(D)\} \end{aligned}$$

with seminorm $\Vert \nabla (\cdot )\Vert _{q,D}$ for $1<q<\infty$. When $D=\Omega$, let us introduce

$$\begin{aligned} {\widehat{W}}^{1,q}_{(0)}(\Omega )=\left\{ p\in \widehat{W}^{1,q}(\Omega );\; \int _{\Omega _3}p\,dx=0\right\} , \end{aligned}$$

that is a Banach space with norm $\Vert \nabla (\cdot )\Vert _{q,\Omega }$. The class $C_0^\infty (D)$ consists of all $C^\infty$-functions with compact support in D, then $W^{k,q}_0(D)$ denotes the completion of $C_0^\infty (D)$ in $W^{k,q}(D)$, where $k>0$ is an integer. Given $q\in [1,\infty ]$, let $q^\prime \in [1,\infty ]$ be the conjugate number defined by $1/q^\prime +1/q=1$. For $q\in (1,\infty )$, we define the Sobolev space of order $(-1)$ by $W^{-1,q}(D)=W^{1,q^\prime }_0(D)^*$. In what follows we adopt the same symbols for denoting scalar and vector (even tensor) functions as long as no confusion occurs. By $C^\infty _{0,\sigma }(D)$ we denote the class of all vector fields u which are in $C_0^\infty (D)$ and satisfy $\hbox { div}\ u=0$ in D. Let X be a Banach space. Then ${{\mathcal {L}}}(X)$ stands for the Banach space consisting of all bounded linear operators from X into itself. Finally, we denote several positive constants by C, which may change from line to line.

Before stating our main results, let us also introduce the underlying space and the Stokes-structure operator to formulate (1.5) and (1.9) within the monolithic framework as in [10, 43, 46]. By $\textrm{RM}$ we denote the space of all rigid motions, that is,

$$\begin{aligned} \textrm{RM}:=\{\eta +\omega \times x;\; \eta ,\,\omega \in {\mathbb {R}}^3\}. \end{aligned}$$

For the resolvent problem, see Sect. 3.3, we have to consider the complex rigid motions $\eta +\omega \times x$ with $\eta ,\, \omega \in {\mathbb {C}}^3$. For $1<q<\infty$, we set

$$\begin{aligned} L^q_R({\mathbb {R}}^3):=\{U\in L^q({\mathbb {R}}^3)^3;\; U|_B\in \textrm{RM}\} \end{aligned}$$

(2.5)

that is closed in $L^q({\mathbb {R}}^3)^3$. The underlying space we adopt is

$$\begin{aligned} \begin{aligned} X_q({\mathbb {R}}^3):=\{U\in L^q_R({\mathbb {R}}^3);\; \text { div } U=0\; \text { in }\; {\mathbb {R}}^3\}. \end{aligned} \end{aligned}$$

(2.6)

It is convenient to define the map

$$\begin{aligned} i: L^q_R({\mathbb {R}}^3)\rightarrow L^q(\Omega )^3\times {\mathbb {R}}^3\times {\mathbb {R}}^3 \end{aligned}$$

or

$$\begin{aligned} i: L^q_R({\mathbb {R}}^3)\rightarrow L^q(\Omega )^3\times {\mathbb {C}}^3\times {\mathbb {C}}^3 \end{aligned}$$

for the associated resolvent problem, where the scalar field of the Lebesgue space is ${\mathbb {C}}$ for the latter case, by

$$\begin{aligned} \begin{aligned}&i: U\mapsto i(U):=(u,\eta ,\omega ) \;\;\text{ with } \\&u=U|_\Omega , \qquad \eta =\frac{1}{m}\int _B U(x)\rho \,dx, \qquad \omega =J^{-1}\int _B x\times U(x)\rho \,dx \end{aligned} \end{aligned}$$

(2.7)

with $\rho ,\, m$ and J being given by (2.2), then we see from (2.3) that

$$\begin{aligned} U|_B=\eta +\omega \times x. \end{aligned}$$

If in particular $U\in X_q({\mathbb {R}}^3)$, we find

$$\begin{aligned} \nu \cdot (u-\eta -\omega \times x)|_{\partial \Omega }=0 \end{aligned}$$

(2.8)

with $\nu$ being the unit normal to $\partial \Omega$ directed toward B (indeed, $\nu =-x$ since $\partial \Omega$ is the unit sphere) on account of $\hbox { div}\ U=0$ in ${\mathbb {R}}^3$. Conversely, if $u\in L^q(\Omega )$ satisfies $\hbox { div}\ u=0$ in $\Omega$ (so that the normal trace $\nu \cdot u|_{\partial \Omega }$ makes sense) and $\nu \cdot (u-\eta -\omega \times x)|_{\partial \Omega }=0$ for some pair of $\eta ,\, \omega \in {\mathbb {R}}^3$, then

$$\begin{aligned} U:=u\chi _\Omega +(\eta +\omega \times x)\chi _B\in X_q({\mathbb {R}}^3) \end{aligned}$$

(2.9)

with $i(U)=(u,\eta ,\omega )$, where $\chi _\Omega$ and $\chi _B$ denote the characteristic functions. In this way, elements of $X_q(\mathbb R^3)$ are understood through (2.7)–(2.9).

The space $X_q({\mathbb {R}}^3)$, $1<q<\infty$, is a Banach space endowed with norm

$$\begin{aligned} \Vert U\Vert _{X_q({\mathbb {R}}^3)}:=\Big (\Vert u\Vert _{q,\Omega }^q+\Vert (\eta _u+\omega _u\times x)\rho ^{1/q}\Vert _{q,B}^q\Big )^{1/q} \end{aligned}$$

(2.10)

for $U\in X_q({\mathbb {R}}^3)$ and we have the duality relation $X_q({\mathbb {R}}^3)^*=X_{q^\prime }({\mathbb {R}}^3)$, see (3.8) in Proposition 3.1 below, with the pairing

$$\begin{aligned} \begin{aligned} \langle U,V\rangle _{{\mathbb {R}}^3,\rho }&:=\int _\Omega u\cdot v\,dx +\int _B (\eta _u+\omega _u\times x)\cdot (\eta _v+\omega _v\times x)\rho \,dx \\&=\langle u,v\rangle _\Omega +m\eta _u\cdot \eta _v+(J\omega _u)\cdot \omega _v \end{aligned} \end{aligned}$$

(2.11)

(with obvious change if one should consider the complex rigid motions for the resolvent problem) for $U\in X_q({\mathbb {R}}^3)$ and $V\in X_{q^\prime }({\mathbb {R}}^3)$, where $i(U)=(u,\eta _u,\omega _u),\, i(V)=(v,\eta _v,\omega _v)$, see (2.2), (2.4) and (2.7). It is clear that the pairing (2.11) is defined for $U\in L^q_R({\mathbb {R}}^3)$ and $V\in L^{q^\prime }_R({\mathbb {R}}^3)$ as well. Notice that the constant weight $\rho$ is involved in the integral over the rigid body B, see (2.10)–(2.11), nonetheless, it is obvious from (2.7) to see that the following three quantities are equivalent for $U\in X_q({\mathbb {R}}^3),\; i(U)=(u,\eta _u,\omega _u)$:

$$\begin{aligned} \Vert U\Vert _{X_q({\mathbb {R}}^3)}\sim \Vert U\Vert _{q,{\mathbb {R}}^3}\sim \Vert u\Vert _{q,\Omega }+|\eta _u|+|\omega _u|, \end{aligned}$$

(2.12)

where the symbol $\sim$ means that inequalities in both directions hold with some constants. Thus the norm (2.10) does not play any role to discuss the asymptotic behavior of solutions to (1.5) and (1.9). Nevertheless, the reason for introducing (2.10) is to describe the energy

$$\begin{aligned} \Vert U\Vert _{X_2({\mathbb {R}}^3)}^2=\langle U,U\rangle _{{\mathbb {R}}^3,\rho } =\Vert u\Vert _{2,\Omega }^2+m|\eta _u|^2+(J\omega _u)\cdot \omega _u \end{aligned}$$

(2.13)

with $(J\omega _u)\cdot \omega _u=\frac{2m}{5}|\omega _u|^2$ when B is a ball. In fact, the energy (2.13) fulfills the identity in the desired form along time-evolution of the linearized system (1.9), see (4.25)–(4.26) in Sect. 4.2. The other reason is to ensure several duality relations, see (3.85), (3.87) and (3.97). Even for general $U,\, V\in L^q(\mathbb R^3)$, $1<q<\infty$, it is sometimes (see subsection 4.6) convenient to introduce the other norm and pairing

$$\begin{aligned} \Vert U\Vert _{q,({\mathbb {R}}^3,\rho )}:=\left( \int _{\mathbb R^3}|U(x)|^q\,(\rho \chi _B+\chi _\Omega )\,dx\right) ^{1/q}, \quad \langle U,V\rangle _{{\mathbb {R}}^3,\rho }:=\int _{{\mathbb {R}}^3}(U\cdot V) (\rho \chi _B+\chi _\Omega )\,dx, \end{aligned}$$

(2.14)

which are consistent with (2.10)–(2.11), in order that $L^q_R({\mathbb {R}}^3)$ is regarded as a subspace of $L^q(\mathbb R^3)^3$. By [10, Proposition 3.3] it is shown that the class

$$\begin{aligned} {\mathcal {E}}({\mathbb {R}}^3):=\{U\in C_{0,\sigma }^\infty ({\mathbb {R}}^3);\; DU=O\;\hbox { in}\ B\} =C_0^\infty ({\mathbb {R}}^3)^3\cap X_q(\mathbb R^3) \end{aligned}$$

(2.15)

is dense in $X_q({\mathbb {R}}^3)$, $1<q<\infty$. An alternative proof will be given in Proposition 3.1 as well.

Let $U\in X_q({\mathbb {R}}^3)$ satisfy, in particular, $u\in W^{1,q}(\Omega )$, where $i(U)=(u,\eta ,\omega )$, see (2.7). Then, $U\in W^{1,q}({\mathbb {R}}^3)$ if and only if

$$\begin{aligned} u|_{\partial \Omega }=\eta +\omega \times x. \end{aligned}$$

(2.16)

Moreover, under the condition (2.16) it holds that

$$\begin{aligned} \nabla U=(\nabla u)\chi _\Omega +M_\omega \chi _B, \qquad M_\omega :=\left( \begin{array}{ccc} 0 &{} -\omega _3 &{} \omega _2 \\ \omega _3 &{} 0 &{} -\omega _1 \\ -\omega _2 &{} \omega _1 &{} 0 \end{array} \right) . \end{aligned}$$

(2.17)

Let $1<q<\infty$. With the space $X_q({\mathbb {R}}^3)$ at hand as the one in which we are going to look for the monolithic velocity (2.9), there are two ways to eliminate the pressure. They are similar but slightly different. One is the approach within the space $L^q_R({\mathbb {R}}^3)$ by Silvestre [43, 44], who developed the case $q=2$. In Proposition 3.1 below we will establish the decomposition

$$\begin{aligned} L^q_R({\mathbb {R}}^3)=X_q({\mathbb {R}}^3)\oplus Z_q({\mathbb {R}}^3) \end{aligned}$$

(2.18)

with

$$\begin{aligned} \begin{aligned}&Z_q({\mathbb {R}}^3)=\Big \{V\in L^q_R({\mathbb {R}}^3);\; V|_\Omega =\nabla p,\; p\in {\widehat{W}}^{1,q}(\Omega ), \; V|_B=\eta +\omega \times x, \\&\qquad \qquad \qquad \qquad \qquad \eta =\frac{-1}{m}\int _{\partial \Omega }p\nu \,d\sigma ,\; \omega =-J^{-1}\int _{\partial \Omega }x\times (p\nu )\,d\sigma \Big \} \\ \end{aligned} \end{aligned}$$

(2.19)

as well as $X_q({\mathbb {R}}^3)^\perp =Z_{q^\prime }({\mathbb {R}}^3)$. The latter relation means that $Z_{q^\prime }({\mathbb {R}}^3)$ is the annihilator of $X_q({\mathbb {R}}^3)$ with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$, see (2.11). Note that, for the element of the space $Z_q({\mathbb {R}}^3)$, p is determined uniquely up to constant which, however, does not change $\eta$ and $\omega$ since $\int _{\partial \Omega }\nu \,d\sigma =\int _{\partial \Omega }x\times \nu \,d\sigma =0$. When $q=2$, (2.18) was already proved by Silvestre [43], who studied her problem within, instead of $L^q_R({\mathbb {R}}^3)$, the class $L^2(\Omega )+\textrm{RM}$ in which the flow behaves like a rigid motion at infinity in the exterior domain $\Omega$. By

$$\begin{aligned} {\mathbb {P}}={\mathbb {P}}_q: L^q_R({\mathbb {R}}^3)\rightarrow X_q({\mathbb {R}}^3) \end{aligned}$$

(2.20)

we denote the bounded projection associated with (2.18). Then we have the relation ${\mathbb {P}}_q^*={\mathbb {P}}_{q^\prime }$ with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ in which the constant weight $\rho$ is involved, that is, in the sense of (3.10), see Proposition 3.1.

The other way is to use the following decomposition developed by Wang and Xin [51, Theorem 2.2], see also Dashti and Robinson [6] for the case $q=2$:

$$\begin{aligned} L^q({\mathbb {R}}^3)=G_q^{(1)}({\mathbb {R}}^3) \oplus L^q_\sigma (\mathbb R^3) =G_q^{(1)}({\mathbb {R}}^3) \oplus G_{q}^{(2)}({\mathbb {R}}^3)\oplus X_q({\mathbb {R}}^3) \end{aligned}$$

(2.21)

where

$$\begin{aligned} \begin{aligned}&L^q_\sigma ({\mathbb {R}}^3)=\{V\in L^q({\mathbb {R}}^3);\; \text{ div }\; V =0\;\text { in } {\mathbb {R}}^3\}, \\ {}&G_q^{(1)}({\mathbb {R}}^3)=\{\nabla p_1\in L^q({\mathbb {R}}^3);\; p_1\in {\widehat{W}}^{1,q}({\mathbb {R}}^3)\}, \\ {}&G_{q}^{(2)}({\mathbb {R}}^3)=\Big \{ V\in L^q({\mathbb {R}}^3);\; \text { div } \text { V }=0\; \text{ in }\ {\mathbb {R}}^3,\; V|_\Omega =\nabla p_2,\; p_2\in {\widehat{W}}^{1,q}(\Omega ), \\ {}&\qquad \qquad \qquad \int _B V\,dx=-\int _{\partial \Omega }p_2\nu \,d\sigma ,\; \int _B x\times V\,dx=-\int _{\partial \Omega }x\times (p_2\nu )\,d\sigma \Big \}. \end{aligned} \end{aligned}$$

Here, the density $\rho$ is not involved in the conditions for $V\in G_{q}^{(2)}({\mathbb {R}}^3)$, so that $\langle U,V\rangle _{\mathbb R^3}=0$ for all $U\in X_q({\mathbb {R}}^3)$ and $V\in G_{q^\prime }^{(2)}({\mathbb {R}}^3)$. One cannot avoid this situation because $L^q_\sigma ({\mathbb {R}}^3)^\perp =G_{q^\prime }^{(1)}(\mathbb R^3)$ holds with respect to the standard pairing $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3}$. As a consequence, the bounded projection

$$\begin{aligned} {\mathbb {Q}}={\mathbb {Q}}_q: L^q({\mathbb {R}}^3)\rightarrow X_q({\mathbb {R}}^3) \end{aligned}$$

associated with (2.21) fulfills the relation $\mathbb Q_q^*={\mathbb {Q}}_{q^\prime }$ in the sense that

$$\begin{aligned} \langle {\mathbb {Q}}_qU,V\rangle _{{\mathbb {R}}^3}=\langle U,\mathbb Q_{q^\prime }V\rangle _{{\mathbb {R}}^3} \end{aligned}$$

for all $U\in L^q({\mathbb {R}}^3)$ and $V\in L^{q^\prime }(\mathbb R^3)$ with respect to the standard pairing $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3}$, which should be compared with (3.10) in Proposition 3.1. The fact that ${\mathbb {Q}}$ is not symmetric with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ unless $\rho =1$ is not consistent with duality arguments especially in Sects. 3.6, 3.7, 4.3 and 4.6. For this reason, the latter way is not convenient for us and thus the decomposition (2.18) is preferred in this paper.

The classical Fujita-Kato projection

$$\begin{aligned} {\mathbb {P}}_0={\mathbb {P}}_{0,q}: L^q({\mathbb {R}}^3)\rightarrow L^q_\sigma (\mathbb R^3) \end{aligned}$$

(2.22)

is well-known and it is described in terms of the Riesz transform ${{\mathcal {R}}}=(-\Delta )^{-1/2}\nabla$ as ${\mathbb {P}}_0=\mathcal {I}+{{\mathcal {R}}}\otimes {{\mathcal {R}}}$ with ${{\mathcal {I}}}$ being the identity operator. Notice the relation ${\mathbb {P}}_{0,q}^*=\mathbb P_{0,q^\prime }$ with respect to $\langle \cdot ,\cdot \rangle _{\mathbb R^3}$. The projection ${\mathbb {P}}_0$ is used in Sects. 4.1 and 4.5.

Let us call (1.9) with $\{u_b,\eta _b\}=\{0,0\}$ the Stokes-structure system (although it is called the fluid–structure system in the existing literature); that is, it is written as

$$\begin{aligned} \begin{aligned}&\partial _tu=\Delta u-\nabla p, \qquad \text{ div }\; u =0 \quad \text { in }\; \Omega \times (0,\infty ), \\ {}&u|_{\partial \Omega }=\eta +\omega \times x, \qquad u\rightarrow 0 \quad \text{ as }\ |x|\rightarrow \infty , \\ {}&m\frac{d\eta }{dt}+\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =0, \\ {}&J\frac{d\omega }{dt}+\int _{\partial \Omega }x\times \mathbb S(u,p)\nu \,d\sigma =0, \end{aligned} \end{aligned}$$

(2.23)

subject to the initial conditions at $s=0$ (since (2.23) is autonomous). This problem can be formulated as the evolution equation of the form

$$\begin{aligned} \frac{dU}{dt}+AU=0 \end{aligned}$$

(2.24)

in the space $X_q({\mathbb {R}}^3)$, where the velocities of the fluid and the rigid body are unified as a velocity U in the whole space ${\mathbb {R}}^3$ through (2.9). Here, the operator A, to which we refer as the Stokes-structure operator in this paper, is defined by

$$\begin{aligned} \begin{aligned} D_q(A)&=\big \{U\in X_q({\mathbb {R}}^3)\cap W^{1,q}({\mathbb {R}}^3);\; u=U|_\Omega \in W^{2,q}(\Omega )\big \}, \\ AU&={\mathbb {P}}{{\mathcal {A}}}U, \\ {{\mathcal {A}}}U&= \left\{ \begin{aligned}&-\hbox { div}\ (2Du)=-\Delta u, \qquad x\in \Omega , \\&\frac{1}{m}\int _{\partial \Omega }(2Du)\nu \,d\sigma + \left( J^{-1}\int _{\partial \Omega }y\times (2Du)\nu \,d\sigma _y\right) \times x, \qquad x\in B, \end{aligned} \right. \\ \end{aligned} \end{aligned}$$

(2.25)

where ${\mathbb {P}}$ is the projection (2.20). Notice that the domain $D_q(A)$ is dense since so is ${{\mathcal {E}}}({\mathbb {R}}^3)$ and that the boundary condition (2.16) is hidden for $U\in D_q(A)$, where $(u,\eta ,\omega )=i(U)$, see (2.7).

In [10] the other operator ${\widetilde{A}}U={\mathbb {Q}}\mathcal {A}U$ with the same domain $D_q({\widetilde{A}})=D_q(A)$ acting on the same space $X_q({\mathbb {R}}^3)$ is defined by use of the other projection ${\mathbb {Q}}$ associated with (2.21). Due to [10, Proposition 3.4], given $F\in X_q({\mathbb {R}}^3)$, the resolvent problem

$$\begin{aligned} (\lambda +{\widetilde{A}})U=F \qquad \hbox { in}\ X_q({\mathbb {R}}^3) \end{aligned}$$

(2.26)

is equivalent to the Stokes-structure resolvent system

$$\begin{aligned} \begin{aligned}&\lambda u-\Delta u+\nabla p=f, \qquad \text{ div }\; u =0 \quad \text { in }\; \Omega , \\ {}&u|_{\partial \Omega }=\eta +\omega \times x, \qquad u\rightarrow 0\quad \text{ as }\ |x|\rightarrow \infty , \\ {}&\lambda \eta +\frac{1}{m}\int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =\kappa , \\ {}&\lambda \omega +J^{-1}\int _{\partial \Omega }x\times \mathbb S(u,p)\nu \,d\sigma =\mu , \end{aligned} \end{aligned}$$

(2.27)

where $(f,\kappa ,\mu )=i(F)$, see (2.7), and the associated pressure p is appropriately determined. Likewise, as we will show in Proposition 3.2, our resolvent problem

$$\begin{aligned} (\lambda +A)U=F \qquad \hbox { in}\ X_q({\mathbb {R}}^3) \end{aligned}$$

(2.28)

is also equivalent to (2.27). Moreover, via uniqueness of solutions to the problem (2.27) with $\lambda =1$, we find that $A={\widetilde{A}}$ in Proposition 3.3 and, thereby, that several results for ${\widetilde{A}}$ established by [10] continue to hold for A as well. We note that, as observed first by Takahashi and Tucsnak [46], the evolution equation (2.24) is also shown to be equivalent to the system (2.23) in the similar fashion to Proposition 3.2.

If we denote by $A_q$ the operator A with $D_q(A)$ acting on $X_q({\mathbb {R}}^3)$, we have the duality relation

$$\begin{aligned} A_q^*=A_{q^\prime } \end{aligned}$$

(2.29)

with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ for every $q\in (1,\infty )$ and thus A is closed. The duality (2.29) is observed from (3.87) below combined with the fact that $\lambda +A$ with $\lambda >0$ is surjective. This surjectivity follows from analysis of the resolvent mentioned just below. In particular, it is a positive self-adjoint operator on $X_2({\mathbb {R}}^3)$ as shown essentially by Silvestre [44, Theorem 4.1] and by Takahashi and Tucsnak [46, Proposition 4.2]. Given $F\in X_q({\mathbb {R}}^3)$, consider the resolvent problem (2.28). Then, for every $\varepsilon \in (0,\pi /2)$, there is a constant $C_\varepsilon >0$ such that ${\mathbb {C}}{\setminus } (-\infty ,0]\subset \rho (-A)$ being the resolvent set of the operator $-A$ subject to

$$\begin{aligned} \Vert (\lambda +A)^{-1}F\Vert _{q,{\mathbb {R}}^3} \le \frac{C_\varepsilon }{|\lambda |}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(2.30)

for all $\lambda \in \Sigma _\varepsilon$ and $F\in X_q({\mathbb {R}}^3)$, which is due to Ervedoza, Maity and Tucsnak [10, Theorem 6.1] and implies that the operator $-A$ generates a bounded analytic semigroup (which is called the fluid–structure semigroup in the existing literature) $\{e^{-tA};\, t\ge 0\}$ on $X_q({\mathbb {R}}^3)$ for every $q\in (1,\infty )$, that is an improvement of Wang and Xin [51], where

$$\begin{aligned} \Sigma _\varepsilon := \{\lambda \in {\mathbb {C}}\setminus \{0\};\; |\arg \lambda |\le \pi -\varepsilon \} . \end{aligned}$$

(2.31)

Hence, the fractional powers $A^\alpha$ with $\alpha >0$ are well-defined as closed operators on $X_q({\mathbb {R}}^3)$. Estimate (2.30) especially for large $|\lambda |$ will be revisited in Sect. 3.3.

2.3 Main Results

As the basic motion, we fix a solution $\{u_b,\eta _b,\omega _b\}$ to the problem (1.1) on the whole time axis ${\mathbb {R}}$ together with the associated pressure $p_b$ and set

$$\begin{aligned} U_b(x,t)=u_b(x,t)\chi _\Omega (x)+\big (\eta _b(t)+\omega _b(t)\times x\big )\chi _B(x). \end{aligned}$$

(2.32)

Let us assume that

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} u_b\in L^\infty ({\mathbb {R}};\, L^{q_0}(\Omega )\cap L^\infty (\Omega ))\; \text { with } \text { some }\; q_0\in (1,3), \quad \text{ div }\; u_b =0\;\text { in }\; \Omega , \\ \eta _b,\,\omega _b\in L^\infty ({\mathbb {R}};\, {\mathbb {R}}^3), \\ \nu \cdot (u_b-\eta _b-\omega _b\times x)|_{\partial \Omega }=0, \quad \end{array} \right. \end{aligned} \end{aligned}$$

(2.33)

then we see that $U_b\in L^\infty ({\mathbb {R}};\, X_q({\mathbb {R}}^3))$ for every $q\in [q_0,\infty )$. Let us also mention that the assumption on $\omega _b$ is used merely in the proof of Proposition 4.1 for the whole space problem without body. We further make the assumption

$$\begin{aligned} u_b\in C^\theta ({\mathbb {R}};\,L^\infty (\Omega )), \quad \eta _b\in C^\theta ({\mathbb {R}};\, {\mathbb {R}}^3)\;\; \text{ with } \text{ some }\; \theta \in (0,1). \end{aligned}$$

(2.34)

Set

$$\begin{aligned} \begin{aligned}&\Vert U_b\Vert :=\sup _{t\in {\mathbb {R}}}\big (\Vert u_b(t)\Vert _{q_0,\Omega }+\Vert u_b(t)\Vert _{\infty ,\Omega }+|\eta _b(t)|+|\omega _b(t)|\big ), \\&[U_b]_\theta :=\sup _{t>s}\frac{\Vert u_b(t)-u_b(s)\Vert _{\infty ,\Omega }+|\eta _b(t)-\eta _b(s)|}{(t-s)^\theta }. \end{aligned} \end{aligned}$$

(2.35)

Remark 2.1

The condition at the boundary $\partial \Omega$ in (2.33) may be rewritten as $\nu \cdot (u_b-\eta _b)|_{\partial \Omega }=0$ since $\nu \cdot (\omega _b\times x)=0$ at the sphere always holds. In view of the boundary condition in (1.1), one can deal with the case of tangential velocity $u_*$, that is, $\nu \cdot u_*|_{\partial \Omega }=0$.

Examples of the basic motion for which the assumptions (2.33)–(2.34) could be met will be discussed briefly in Sect. 2.4.

To study the Oseen-structure system (1.9), let us introduce the family $\{L_\pm (t);\, t\in {\mathbb {R}}\}$ of the Oseen-structure operators on $X_q({\mathbb {R}}^3)$, $1<q<\infty$, by

$$\begin{aligned} D_q(L_\pm (t))=D_q(A), \qquad L_\pm (t)U=AU\pm B(t)U, \end{aligned}$$

(2.36)

where A is the Stokes-structure operator (2.25) and

$$\begin{aligned} B(t)U ={\mathbb {P}}\big [\{(u_b(t)-\eta _b(t))\cdot \nabla u\}\chi _\Omega \big ] \end{aligned}$$

(2.37)

for $u=U|_\Omega$ with ${\mathbb {P}}$ being the projection given by (2.20). As mentioned in the preceding subsection and as we will show in Proposition 3.3, our Stokes-structure operator A coincides with the operator ${\widetilde{A}}$ studied in [10] by Ervedoza, Maity and Tucsnak. With the aid of the elliptic estimate due to [10, Proposition 7.3]

$$\begin{aligned} \Vert u\Vert _{W^{2,q}(\Omega )}\le C\big (\Vert AU\Vert _{q,\Omega }+\Vert U\Vert _{q,\mathbb R^3}\big ), \qquad U\in D_q(A),\; u=U|_\Omega , \end{aligned}$$

(2.38)

we find

$$\begin{aligned} \begin{aligned} \Vert \nabla u\Vert _{q,\Omega }&\le C\Vert u\Vert _{W^{2,q}(\Omega )}^{1/2}\Vert u\Vert _{q,\Omega }^{1/2} \\&\le C\big (\Vert AU\Vert _{q,\Omega }+\Vert U\Vert _{q,{\mathbb {R}}^3}\big )^{1/2}\Vert u\Vert _{q,\Omega }^{1/2} \\&\le C \left( \Vert AU\Vert _{q,{\mathbb {R}}^3}^{1/2}\Vert U\Vert _{q,\mathbb R^3}^{1/2}+\Vert U\Vert _{q,{\mathbb {R}}^3}\right) \end{aligned} \end{aligned}$$

(2.39)

which together with (2.33)–(2.35) leads to

$$\begin{aligned} \Vert B(t)U\Vert _{q,{\mathbb {R}}^3}\le C\Vert (u_b(t)-\eta _b(t))\cdot \nabla u\Vert _{q,\Omega } \le C\Vert U_b\Vert \left( \Vert AU\Vert _{q,\mathbb R^3}^{1/2}\Vert U\Vert _{q,{\mathbb {R}}^3}^{1/2}+\Vert U\Vert _{q,{\mathbb {R}}^3}\right) \end{aligned}$$

(2.40)

for all $U\in D_q(A)$. Thus, one justifies the relation $D_q(L_\pm (t))=D_q(A)$, see (2.36), on which we have

$$\begin{aligned} \begin{aligned} \Vert L_\pm (t)U\Vert _{q,{\mathbb {R}}^3}&\le (1+C\Vert U_b\Vert )\Vert AU\Vert _{q,{\mathbb {R}}^3}+C\Vert U_b\Vert \Vert U\Vert _{q,{\mathbb {R}}^3}, \\ \Vert AU\Vert _{q,{\mathbb {R}}^3}&\le 2\Vert L_\pm (t)U\Vert _{q,\mathbb R^3}+C(\Vert U_b\Vert ^2+\Vert U_b\Vert )\Vert U\Vert _{q,{\mathbb {R}}^3}. \end{aligned} \end{aligned}$$

(2.41)

Moreover, it follows immediately from (2.25) that

$$\begin{aligned} \Vert AU\Vert _{q,{\mathbb {R}}^3} \le C\Vert u\Vert _{W^{2,q}(\Omega )}, \qquad U\in D_q(A),\, u=U|_\Omega , \end{aligned}$$

which combined with (2.38) tells us that $\Vert \cdot \Vert _{W^{2,q}(\Omega )}+\Vert \cdot \Vert _{q,{\mathbb {R}}^3}$ is equivalent to the graph norm of A and, therefore, also to that of $L_\pm (t)$ uniformly in $t\in {\mathbb {R}}$ on account of (2.41); in particular,

$$\begin{aligned} \Vert u\Vert _{W^{2,q}(\Omega )} \le C\big (\Vert L_\pm (t)U\Vert _{q,\mathbb R^3}+\Vert U\Vert _{q,{\mathbb {R}}^3}\big ), \qquad u=U|_\Omega , \end{aligned}$$

(2.42)

for all $U\in D(L_\pm (t))$ with some $C>0$ (involving $\Vert U_b\Vert$) independent of $t\in {\mathbb {R}}$.

The initial value problems for (1.5) as well as (1.9) are formulated as

$$\begin{aligned} \frac{dU}{dt}+L_+(t)U=0,\quad t\in (s,\infty ); \qquad U(s)=U_0 \end{aligned}$$

(2.43)

and

$$\begin{aligned} \frac{dU}{dt}+L_+(t)U=H(U), \quad t\in (s,\infty ); \qquad U(s)=U_0 \end{aligned}$$

(2.44)

where

$$\begin{aligned} U_0=u_0\chi _\Omega +(\eta _0+\omega _0\times x)\chi _B \end{aligned}$$

(2.45)

and

$$\begin{aligned} H(U)= {\mathbb {P}}\left[ \big \{(\eta -u)\cdot \nabla (u_b+u)\big \}\chi _\Omega \right] \end{aligned}$$

(2.46)

with $(u,\eta ,\omega )=i(U)$, see(2.7). Recall that the assumption $U_0\in X_q({\mathbb {R}}^3)$ involves the compatibility condition $\nu \cdot (u_0-\eta _0-\omega _0\times x)|_{\partial \Omega }=0$. The first main result of this paper is the following theorem on $L^q$–$L^r$ estimates of the evolution operator generated by $L_+(t)$.

Theorem 2.1

Suppose (2.33) and (2.34), then the operator family $\{L_+(t);\, t\in {\mathbb {R}}\}$ generates an evolution operator $\{T(t,s);\, -\infty<s\le t<\infty \}$ on $X_q({\mathbb {R}}^3)$ for every $q\in (1,\infty )$ with the following properties:

$$\begin{aligned}{} & {} T(t,\tau )T(\tau ,s)=T(t,s)\quad (s\le \tau \le t), \quad T(t,t)={{\mathcal {I}}} \quad \text{ in } {{\mathcal {L}}}(X_q(\mathbb R^3)), \end{aligned}$$

(2.47)

$$\begin{aligned}{} & {} (t,s)\mapsto T(t,s)F\in X_q({\mathbb {R}}^3)\; \text{ is } \text{ continuous } \text{ for } F\in X_q({\mathbb {R}}^3), \end{aligned}$$

(2.48)

$$\begin{aligned}{} & {} \left\{ \begin{aligned}&T(\cdot ,s)F\in C^1((s,\infty );\, X_q({\mathbb {R}}^3)) \cap C((s,\infty );\,D_q(A)), \\&\partial _tT(t,s)F+L_+(t)T(t,s)F=0 \quad \text{ for } F\in X_q(\mathbb R^3),\,t\in (s,\infty ), \end{aligned} \right. \end{aligned}$$

(2.49)

$$\begin{aligned}{} & {} \left\{ \begin{aligned}&T(t,\cdot )F\in C^1((-\infty ,t);\, X_q({\mathbb {R}}^3)), \\&\partial _sT(t,s)F=T(t,s)L_+(s)F \quad \hbox { for}\ F\in D_q(A),\, s\in (-\infty ,t). \end{aligned} \right. \end{aligned}$$

(2.50)

Furthermore, if $\Vert U_b\Vert \le \alpha _j$ being small enough, to be precise, see below about how small it is for each item $j=1,2,3,4$, then the evolution operator T(t, s) enjoys the following estimates with some constant $C=C(q,r,\alpha _j,\beta _0,\theta )>0$ whenever $[U_b]_\theta \le \beta _0$, where $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35) and $\beta _0>0$ is arbitrary.

• 1. Let $q\in (1,\infty )$ and $r\in [q,\infty ]$, then

  $$\begin{aligned} \Vert T(t,s)F\Vert _{r,{\mathbb {R}}^3} \le C(t-s)^{-(3/q-3/r)/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

  (2.51)

  for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$. To be precise, there is a constant $\alpha _1=\alpha _1(q_0)>0$ such that if $\Vert U_b\Vert \le \alpha _1$, then the assertion above holds for every $q\in (1,\infty )$ and $r\in [q,\infty ]$. Estimate (2.51) holds true for the adjoint evolution operator $T(t,s)^*$ as well under the same smallness of $\Vert U_b\Vert$ as above.

• 2. Let $1<q\le r<\infty$, then

  $$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{r,{\mathbb {R}}^3}\le C(t-s)^{-1/2-(3/q-3/r)/2}(1+t-s)^{\max \{(1-3/r)/2,\, 0\}} \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

  (2.52)

  for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$. To be precise, given $r_1\in (1,4/3]$, there is a constant $\alpha _2=\alpha _2(r_1,q_0)\in (0,\alpha _1]$ such that if $\Vert U_b\Vert \le \alpha _2$, then the assertion above holds for every $r\in [r_1,\infty )$ and $q\in (1,r]$, where $\alpha _1=\alpha _1(q_0)$ is the constant given in the previous item. Estimate (2.52) holds true for the adjoint evolution operator $T(t,s)^*$ as well under the same smallness of $\Vert U_b\Vert$ as above.

• 3. Let $q\in (1,\infty )$ and $r\in [q,\infty ]$, then

  $$\begin{aligned} \Vert T(t,s){\mathbb {P}}{\textrm{div}}\ F\Vert _{r,{\mathbb {R}}^3} \le C(t-s)^{-1/2-(3/q-3/r)/2}(1+t-s)^{\max \{(3/q-2)/2,\, 0\}} \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

  (2.53)

  for all (t, s) with $t>s$ and $F\in L^q({\mathbb {R}}^3)^{3\times 3}$ with $(F\nu )|_{\partial \Omega }=0$ as well as ${\textrm{div}}\ F\in L^p_R({\mathbb {R}}^3)$ with some $p\in (1,\infty )$. To be precise, given $r_0\in [4,\infty )$, if $\Vert U_b\Vert \le \alpha _3$ with $\alpha _3(r_0,q_0):=\alpha _2(r_0^\prime ,q_0)$, where $\alpha _2$ is the constant given in the previous item, then the assertion above holds for every $q\in (1,r_0]$ and $r\in [q,\infty ]$.

• 4. Let $1< q\le r<\infty$, then

  $$\begin{aligned} \begin{aligned}&\quad \Vert \nabla T(t,s){\mathbb {P}}{\textrm{div}}\ F\Vert _{r,{\mathbb {R}}^3} \\&\le C(t-s)^{-1-(3/q-3/r)/2}(1+t-s)^{\max \{(1-3/r)/2,\, 0\}+\max \{(3/q-2)/2,\, 0\}} \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

  (2.54)

  for all (t, s) with $t>s$ and $F\in L^q({\mathbb {R}}^3)^{3\times 3}$ with $(F\nu )|_{\partial \Omega }=0$ as well as ${{ \mathrm div}}\ F\in L^p_R({\mathbb {R}}^3)$ with some $p\in (1,\infty )$. To be precise, given $r_0\in [4,\infty )$ and $r_1\in (1,4/3]$, if $\Vert U_b\Vert \le \alpha _4$ with $\alpha _4(r_0,r_1,q_0):=\alpha _2\big (\min \{r_0^\prime ,r_1\},q_0\big )$, where $\alpha _2$ is the constant given in the item 2, then the assertion above holds for $1<q\le r<\infty$ with $q\in (1,r_0]$ as well as $r\in [r_1,\infty )$.

Remark 2.2

According to (2.52), the rate of decay is given by

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{r,{\mathbb {R}}^3}\le \left\{ \begin{array}{ll} C(t-s)^{-1/2-(3/q-3/r)/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \quad &{} (r\le 3), \\ C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} &{} (r>3), \end{array} \right. \end{aligned}$$

(2.55)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$. This rate is the same as the one for the Stokes and Oseen semigroups in exterior domains due to Iwashita [33], Kobayashi and Shibata [34], Maremonti and Solonnikov [37], see also [28] for the generalized Oseen evolution operator even with rotating effect in which the time-dependent motion of a rigid body is prescribed. It is known ([24, 37]) that the rate (2.55) is sharp for the Stokes semigroup in exterior domains, whereas the optimality is not clear for the problem under consideration. In fact, one of the points of their argument is that the steady Stokes flow in exterior domains does not possess fine summability such as $L^3(\Omega )$ at infinity even for the external force $f\in C_0^\infty (\Omega )$ unless $\int _{\partial \Omega }\mathbb S(u,p)\nu \,d\sigma =0$. Compared with that, if $f\chi _\Omega +(\kappa +\mu \times x)\chi _B={\textrm{div}}\ F$ with F satisfying the conditions in the item 3 of Theorem 2.1 for $T(t,s){\mathbb {P}}\text{ div }$ and if U is the steady Stokes-structure solution to $AU={\mathbb {P}}{\textrm{div}}\ F$, then we have

$$\begin{aligned} \int _{\partial \Omega }{\mathbb {S}}(u,p)\nu \,d\sigma =m\kappa =\int _B (\kappa +\mu \times x)\rho \,dx =-\int _{\partial \Omega }(F\nu )\rho \,dx=0, \end{aligned}$$

yielding better decay of $u=U|_\Omega$, where p is the associated pressure, and thus the desired rate (2.55)$_1$ even with $r>3$ for $e^{-(t-s)A}$ (case $U_b=0$) does not lead us to any contradiction unlike the Stokes semigroup in exterior domains.

Remark 2.3

Since we know from (2.47), (2.51) and (2.49) that $T(t,s)F\in D_r(A)\subset W^{1,r}({\mathbb {R}}^3)$ for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$, estimate of $\Vert \nabla T(t,s)F\Vert _{r,{\mathbb {R}}^3}$ makes sense. Of course, $\nabla T(t,s)F$ of the form (2.17) never belongs to $X_r({\mathbb {R}}^3)$. By (2.7) with $U=T(t,s)F$, we have

$$\begin{aligned} \begin{aligned} \Vert \nabla T(t,s)F\Vert _{r,B}&\le C|\omega (t)| \le C\Vert T(t,s)F\Vert _{1,B} \\&\le \left\{ \begin{array}{ll} C\Vert F\Vert _{q,{\mathbb {R}}^3} &{} (t-s\le 2), \\ C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3}\quad &{} (t-s>2), \end{array} \right. \end{aligned} \end{aligned}$$

for all $F\in X_q({\mathbb {R}}^3)$ on account of (2.51). Hence, estimate of $\Vert \nabla T(t,s)F\Vert _{r,\Omega }$ over the fluid region is dominant in the sense that it determines (2.52).

Remark 2.4

The adjoint evolution operator $T(t,s)^*$ is studied in Sect. 3.7. The duality argument shows (2.53) of the operator $T(t,s){\mathbb {P}}\text{ div }$, that plays a role for the proof of Theorem 2.2 below and that corresponds to Lemma 8.1 of [10] in which the additional condition $F|_B=0$ is imposed. The reason why this condition is removed in Theorem 2.1 is that the same estimate of $\nabla T(t,s)^*$ as in (2.52) is deduced over the whole space ${\mathbb {R}}^3$, as mentioned above, rather than the exterior domain $\Omega$ solely. Nevertheless, as pointed out by Ervedoza, Hillairet and Lacave [9, remark after Corollary 3.10] as well as [10], $\nabla T(t,s)^*$ is not exactly adjoint of the operator $T(t,s){\mathbb {P}}\text{ div }$ (unless $\rho =1$). This is because several duality relations hold true with respect to the pairing (2.11) involving the constant weight $\rho$, and this is why one needs the vanishing normal trace $(F\nu )|_{\partial \Omega }=0$ for (2.53) in order that the boundary integral disappears by integration by parts even though $\rho \ne 1$. Notice that $(F\nu )|_{\partial \Omega }$ from both directions coincide with each other since ${\textrm{div}}\ F\in L^p({\mathbb {R}}^3)$ for some $p\in (1,\infty )$.

To proceed to the nonlinear problem (1.5), we do need the latter estimates (2.53)–(2.54) in Theorem 2.1 to deal with the nonlinear term $\eta \cdot \nabla u$ among four terms in (2.46) as in [9, 10]. For the other terms one can discuss them by use of (2.51) and (2.52) under a bit more assumptions on $\nabla u_b$ than (2.56) below, however, if one applies (2.53)–(2.54) partly to the linear term $(\eta -u)\cdot \nabla u_b$ in (2.46), then one needs less further assumption:

$$\begin{aligned} \nabla u_b \in L^\infty ({\mathbb {R}};\, L^3(\Omega )) \cap C^{{\tilde{\theta }}}_\textrm{loc}({\mathbb {R}};\, L^3(\Omega )) \quad \text{ with } \text{ some }\; {\tilde{\theta }}\in (0,1). \end{aligned}$$

(2.56)

Accordingly, in addition to (2.35), let us introduce the quantity

$$\begin{aligned} \Vert U_b\Vert ^\prime :=\Vert U_b\Vert +\sup _{t\in {\mathbb {R}}}\Vert \nabla u_b(t)\Vert _{3,\Omega } \end{aligned}$$

(2.57)

but the Hölder seminorm of $\nabla u_b(t)$ is not needed since the Hölder condition in (2.56) is used merely to show that a mild solution (solution to (5.1) below) becomes a strong solution to the initial value problem (2.44). The second main result reads

Theorem 2.2

Suppose (2.33)–(2.34) and (2.56). There exists a constant $\alpha =\alpha (q_0,\beta _0,\theta )>0$ such that if $\Vert U_b\Vert ^\prime \le \alpha$ as well as $[U_b]_\theta \le \beta _0$, then the following statement holds, where $\Vert U_b\Vert ^\prime$ is given by (2.57) and $\beta _0>0$ is arbitrary: There is a constant $\delta =\delta (\alpha ,\beta _0,\theta )>0$ such that if $U_0\in X_3({\mathbb {R}}^3)$ satisfies $\Vert U_0\Vert _{3,{\mathbb {R}}^3}<\delta$, where $U_0$ is given by (2.45), then problem (2.44) admits a unique strong solution

$$\begin{aligned} U\in C([s,\infty );\,X_3({\mathbb {R}}^3))\cap C((s,\infty );\,D_3(A))\cap C^1((s,\infty );\,X_3({\mathbb {R}}^3)) \end{aligned}$$

which enjoys

$$\begin{aligned} \begin{aligned}&\Vert u(t)\Vert _{q,\Omega }=o\big ((t-s)^{-1/2+3/2q}\big ), \\&\Vert \nabla u(t)\Vert _{r,\Omega } +|\eta (t)|+|\omega (t)|=o\big ((t-s)^{-1/2}\big ) \end{aligned} \end{aligned}$$

(2.58)

as $(t-s)\rightarrow \infty$ for every $q\in [3,\infty ]$ and $r\in [3,\infty )$, where $(u,\eta ,\omega )=i(U)$, see (2.7).

Remark 2.5

The decay rate $(t-s)^{-1/2}$ for $\Vert \nabla u(t)\Vert _{3,\Omega }$ in (2.58) is new even when $U_b=0$, see [10] in which less rate is deduced. This improvement is due to (5.15).

Remark 2.6

In view of the proof, we see that the large time decay (2.58) with $r\in [3,\infty )$ replaced by $r\in [3,\sigma _{0*})$ of the mild solution can be obtained even if $\nabla u_b\in L^\infty ({\mathbb {R}};\,L^{\sigma _0}(\Omega ))$ with some $\sigma _0\in (3/2,3]$ instead of (2.56), where $1/\sigma _{0*}=1/\sigma _0-1/3$. This solution becomes a strong one with values in $D_3(A)+D_{\sigma _0}(A)$ under the additional condition $\nabla u_b\in C^{{\tilde{\theta }}}_\textrm{loc}({\mathbb {R}};\, L^{\sigma _0}(\Omega ))$ with some ${\tilde{\theta }}\in (3/2\sigma _0-1/2,1)$ as well as $\theta \in (3/2\sigma _0-1/2,1)$, where $\theta$ is given in (2.34).

2.4 Basic Motions

In this subsection we briefly discuss the basic motions, specifically the self-propelled motions, in the literature. Those motions are stable as long as they are small enough when applying Theorem 2.2, but the details should be discussed elsewhere.

By following the compactness argument due to Galdi [14, Theorem 5.1] who studied the steady problem attached to the body of arbitrary shape, it is possible to show the existence of a solution of the Leray class $\nabla u_b\in L^2(\Omega )$ to the steady problem associated with (1.1) when $u_*$ is independent of t and small in $H^{1/2}(\partial \Omega )$ as well as vanishing flux condition. If we assume further $u_*\in H^{3/2}(\partial \Omega )$, then we have $\nabla u_b\in H^1(\Omega )\subset L^3(\Omega )$, see (2.56), $u_b\in W^{1,6}(\Omega )\subset L^\infty (\Omega )\cap C^{1/2}(\overline{\Omega })$. The uniqueness of the solution in the small is also available when the body is a ball, while this issue remains still open for the case of arbitrary shape. If, for instance, the steady rigid motion $\eta _b+\omega _b\times x$ obtained in this way fulfills $\eta _b\cdot \omega _b\ne 0$, then the result due to Galdi and Kyed [18] concludes that the solution $u_b$ enjoys a wake structure, yielding (2.33) with $q_0\in (2,3)$ provided $u_*$ is tangential to $\partial \Omega$. Actually, we are able to deduce even more; indeed, according to Theorems 5.1 and 5.2 of the same paper [18], we have pointwise estimates

$$\begin{aligned} |u_b(x)|\le C|x|^{-1}, \qquad |\nabla u_b(x)|\le C|x|^{-3/2} \end{aligned}$$

(2.59)

for large |x| with further wake behavior so that $u_b$ and $\nabla u_b$ decays even faster outside the wake region. With (2.59)$_1$ for $u_b$ at hand, one can use Theorem 1.2 of Silvestre, Takahashi and the present author [31] to find even better summability (2.33) with $q_0\in (4/3,3)$ on account of the self-propelling condition, that is related to the asymptotic structure of the Navier–Stokes flow in exterior domains, see [19, 26].

Another observation is that the steady solution, denoted by $\{u_s,\eta _s,\omega _s\}$, constructed by Galdi [14, Theorem 5.1] to the problem (1.1) in which the equations of motions are replaced by (1.6)–(1.7) becomes a time-periodic solution to (1.1) by the change of variable

$$\begin{aligned} u_b(x,t)=Q(t)^\top u_s(Q(t)x), \quad \eta _b(t)=Q(t)^\top \eta _s, \quad \omega _b(t)=Q(t)^\top \omega _s \end{aligned}$$

with a suitable orthogonal matrix Q(t) when the body is a ball. This solution $u_b$ enjoys the desired summability properties mentioned above if $u_*\in H^{3/2}(\partial \Omega )$, $\nu \cdot u_*|_{\partial \Omega }=0$ and $\eta _s\cdot \omega _s\ne 0$. Moreover, it is seen that

$$\begin{aligned} u_b(x,t)-u_b(x,s) =\int _s^t \big [(\omega _b(\tau )\times x)\cdot \nabla u_b(x,\tau )-\omega _b(\tau )\times u_b(x,\tau )\big ]\,d\tau . \end{aligned}$$

Then the pointwise decay (2.59)$_2$ for $\nabla u_s$ implies that $u_b$ is globally Lipschitz continuous with values in $L^\infty (\Omega )$, which combined with $u_b\in L^\infty (\mathbb R;\,L^\infty (\Omega ))$ leads to (2.34) for every $\theta \in (0,1)$. One can also verify (2.56) from the globally Lipschitz continuity of $\nabla u_b$ with values in $L^3(\Omega )$ if $|x|\nabla ^2u_s\in L^3(\Omega )$ (that needs further discussion about the steady motion $u_s$ when $\eta _s\cdot \omega _s\ne 0$). Even though the Hölder condition in (2.56) is not available, the proof of Theorem 2.2 tells us that we have still the asymptotic decay (2.58) for the mild solution, that is, solution to the integral equation (5.1).

Galdi [14] discussed the direct problem, while a steady control problem was studied in [31, 32] by Silvestre, Takahashi and the present author; in fact, we found, among others, a tangential control $u_*$ (together with $u_b$ and $p_b$) which attains a target rigid motion $\eta _b+\omega _b\times x$ to be small, where the shape of the body is arbitrary. The solution obtained there also becomes a time-periodic one to (1.1) with (2.34) by the same reasoning as described in the previous paragraph. Fine decay structure caused by the self-propelling condition was already deduced in [31, Theorem 1.2], which implies (2.33) with $q_0\in (3/2,3)$, no matter what $\{\eta _b,\omega _b\}$ would be.

3 Oseen-Structure Evolution Operator

Before analyzing the Oseen-structure operator (2.36)–(2.37), we begin with some preparatory results in the first three subsections: the decomposition (2.18) to eliminate the pressure, justification of the monolithic formulation (of the resolvent problem), and the other formulation of the Stokes-structure operator (2.25) to derive the resolvent estimate (2.30). In those subsections, the shape of a rigid body is allowed to be arbitrary. Let B be a bounded domain with connected boundary of class $C^{1,1}$, that is assumed to be

$$\begin{aligned} B\subset B_1, \qquad \int _B x\,dx=0, \end{aligned}$$

and set

$$\begin{aligned} \Omega ={\mathbb {R}}^3\setminus {\overline{B}}, \qquad m=\int _B\rho \,dx, \qquad J=\int _B\big (|x|^2{\mathbb {I}}-x\otimes x\big )\rho \,dx. \end{aligned}$$

In Sect. 3.4 we show that the Oseen-structure operator generates an evolution operator on the space $X_q(\mathbb R^3)$ and, successively in Sect. 3.5, we deduce smoothing rates of the evolution operator near the initial time. Such rates of the associated pressure is studied in Sect. 3.6. Analysis of the pressure is rather technical part of this paper. Section 3.7 is devoted to investigation of the backward problem for the adjoint system.

3.1 Decomposition of $L^q_R({\mathbb {R}}^3)$

In this subsection we establish the decomposition (2.18). To this end, let us start with the following preparatory lemma on the space $L^q_R({\mathbb {R}}^3)$, see (2.5).

Lemma 3.1

Let $1<q<\infty$. Then the Banach space $L^q_R({\mathbb {R}}^3)$ is reflexive. Furthermore, the class

$$\begin{aligned} {{\mathcal {E}}}_R({\mathbb {R}}^3):=\{U\in C_0^\infty ({\mathbb {R}}^3)^3;\; DU=O\;{\textrm{in}}\ B\} \end{aligned}$$

(3.1)

is dense in $L^q_R({\mathbb {R}}^3)$.

Proof

Set

$$\begin{aligned} J^q({\mathbb {R}}^3):=\Big \{U\in L^q({\mathbb {R}}^3)^3;\; U|_\Omega =0,\;\int _B U\rho \,dx=0,\; \int _B x\times U\rho \,dx=0\Big \}. \end{aligned}$$

We first show that

$$\begin{aligned} L^q({\mathbb {R}}^3)^3=L^q_R({\mathbb {R}}^3)\oplus J^q({\mathbb {R}}^3). \end{aligned}$$

(3.2)

It is readily seen that $L^q_R({\mathbb {R}}^3)\cap J^q(\mathbb R^3)=\{0\}$. Given $U\in L^q({\mathbb {R}}^3)$, we set

$$\begin{aligned} V=U\chi _\Omega +(\eta +\omega \times x)\chi _B \end{aligned}$$

with

$$\begin{aligned} \eta =\frac{1}{m}\int _BU\rho \,dx, \qquad \omega =J^{-1}\int _B x\times U\rho \,dx. \end{aligned}$$

Then we observe

$$\begin{aligned} V\in L^q_R({\mathbb {R}}^3), \qquad U-V\in J^q({\mathbb {R}}^3), \end{aligned}$$

yielding the decomposition (3.2).

We next show that $J^q({\mathbb {R}}^3)^\perp \subset L^{q^\prime }_R({\mathbb {R}}^3)$, where the annihilator is considered with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ (but the constant weight $\rho$ does not play any role here). Suppose that $U\in L^{q^\prime }({\mathbb {R}}^3)^3$ satisfies

$$\begin{aligned} \langle U,\Psi \rangle _{{\mathbb {R}}^3,\rho }=0, \qquad \forall \Psi \in J^q({\mathbb {R}}^3). \end{aligned}$$

(3.3)

Let $\Phi \in C_0^\infty (B)^{3\times 3}$, then we find

$$\begin{aligned} \int _B\hbox { div}\ (\Phi +\Phi ^\top )\,dx=0, \qquad \int _B x\times \hbox { div}\ (\Phi +\Phi ^\top )\,dx=0, \end{aligned}$$

so that $\hbox { div}\ (\Phi +\Phi ^\top )\in J^q({\mathbb {R}}^3)$ by setting zero outside B. By (3.3) we are led to

$$\begin{aligned} 2\rho \langle DU,\Phi \rangle _B =-\int _B U\cdot \text{ div } (\Phi +\Phi ^\top )\rho \,dx =-\langle U, \text{ div } (\Phi +\Phi ^\top )\rangle _{{\mathbb {R}}^3,\rho }=0 \end{aligned}$$

for all $\Phi \in C_0^\infty (B)^{3\times 3}$, yielding $U|_B\in \textrm{RM}$. We thus obtain the desired inclusion relation. Since the opposite inclusion is obvious, we infer $J^q(\mathbb R^3)^\perp =L^{q^\prime }_R({\mathbb {R}}^3)$. This combined with (3.2) leads us to

$$\begin{aligned} L^q_R({\mathbb {R}}^3)^*=\big [L^q({\mathbb {R}}^3)^3/J^q(\mathbb R^3)\big ]^*=J^q({\mathbb {R}}^3)^\perp =L^{q^\prime }_R({\mathbb {R}}^3), \end{aligned}$$

which implies that $L^q_R({\mathbb {R}}^3)$ is reflexive.

Finally, let us show that the class (3.1) is dense in $L^q_R({\mathbb {R}}^3)$. Given $U\in L^q_R({\mathbb {R}}^3)$, we set $\eta +\omega \times x=U|_B$. Let us take the following lift of the rigid motion:

$$\begin{aligned} \begin{aligned} \ell (\eta ,\omega )(x)&:= \frac{1}{2}\,\text{ rot } \Big (\phi (x)\big (\eta \times x-|x|^2\omega \big )\Big ) \\&=\phi (x)(\eta +\omega \times x)+\nabla \phi (x)\times (\eta \times x-|x|^2\omega ), \end{aligned} \end{aligned}$$

(3.4)

where $\phi$ is a cut-off function satisfying

$$\begin{aligned} \phi \in C_0^\infty (B_3), \qquad 0\le \phi \le 1, \qquad \phi =1\;\; \hbox { on}\ B_2. \end{aligned}$$

(3.5)

Then we have

$$\begin{aligned} \ell (\eta ,\omega )\in C_0^\infty (B_3), \qquad \text{ div } \ell (\eta ,\omega )=0, \qquad \ell (\eta ,\omega )|_{{\overline{B}}}=\eta +\omega \times x. \end{aligned}$$

Since the lift (3.4) will be also used later, it is convenient to introduce the lifting operator

$$\begin{aligned} \ell : (\eta ,\omega )\mapsto \ell (\eta ,\omega ). \end{aligned}$$

(3.6)

Set $U_0=\ell (\eta ,\omega )$. One can take $u_j\in C_0^\infty (\Omega )$ satisfying $\Vert u_j-(U-U_0)\Vert _{q,\Omega }\rightarrow 0$ as $j\rightarrow \infty$, and let us denote by $u_j$ again by setting zero outside $\Omega$. We then put $V_j:=u_j+U_0\in C_0^\infty (\mathbb R^3)$ that attains $\eta +\omega \times x$ on B (thus $V_j\in {{\mathcal {E}}}_R({\mathbb {R}}^3)$) and satisfies $\Vert V_j-U\Vert _{q,\mathbb R^3}\rightarrow 0$ as $j\rightarrow \infty$, leading to the desired denseness. The proof is complete. $\square$

The decomposition (3.7) below was proved by Silvestre [43, Theorem 3.2] when $q=2$. Proposition 3.1 may be regarded as generalization of her result by following the idea due to Wang and Xin [51, Theorem 2.2], who established the other decomposition (2.21) for general $q\in (1,\infty )$. It should be emphasized that the projection ${\mathbb {P}}$ associated with the decomposition (3.7) is symmetric with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ with constant weight $\rho$ as in (3.10) below unlike the projection $\mathbb Q$ associated with the other one (2.21). In fact, this is the reason why we do need the following proposition in the present study.

Proposition 3.1

Let $1<q<\infty$. Let $X_q({\mathbb {R}}^3)$ and $Z_q({\mathbb {R}}^3)$ be the spaces given respectively by (2.6) and (2.19). Then the class ${{\mathcal {E}}}({\mathbb {R}}^n)$, see (2.15), is dense in $X_q({\mathbb {R}}^n)$ and

$$\begin{aligned} L^q_R({\mathbb {R}}^3)= & {} X_q({\mathbb {R}}^3)\oplus Z_q({\mathbb {R}}^3), \end{aligned}$$

(3.7)

$$\begin{aligned} X_q({\mathbb {R}}^3)^*= & {} Z_q({\mathbb {R}}^3)^\perp =X_{q^\prime }({\mathbb {R}}^3), \qquad Z_q({\mathbb {R}}^3)^*=X_q(\mathbb R^3)^\perp =Z_{q^\prime }({\mathbb {R}}^3), \end{aligned}$$

(3.8)

where $Z_q({\mathbb {R}}^3)^\perp$ and $X_q({\mathbb {R}}^3)^\perp$ stand for the annihilators with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$, see (2.11).

Let ${\mathbb {P}}={\mathbb {P}}_q$ be the projection from $L^q_R(\mathbb R^3)$ onto $X_q({\mathbb {R}}^3)$, then

$$\begin{aligned} \Vert \mathbb PU\Vert _{q,{\mathbb {R}}^3}\le C\Vert U\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.9)

for all $U\in L^q_R({\mathbb {R}}^3)$ with some constant $C>0$ as well as the relation ${\mathbb {P}}_q^*={\mathbb {P}}_{q^\prime }$ in the sense that

$$\begin{aligned} \langle {\mathbb {P}}_qU,V\rangle _{{\mathbb {R}}^3,\rho }=\langle U,\mathbb P_{q^\prime }V\rangle _{{\mathbb {R}}^3,\rho } \end{aligned}$$

(3.10)

for all $U\in L^q_R({\mathbb {R}}^3)$ and $V\in L^{q^\prime }_R(\mathbb R^3)$. If in particular $U\in L^q_R({\mathbb {R}}^3)$ satisfies $u=U|_\Omega \in W^{1,q}(\Omega )$, then we have $(\mathbb PU)|_\Omega \in W^{1,q}(\Omega )$ and

$$\begin{aligned} \Vert \nabla \mathbb PU\Vert _{q,\Omega }\le C\big (\Vert U\Vert _{q,\mathbb R^3}+\Vert \nabla u\Vert _{q,\Omega }\big ) \end{aligned}$$

(3.11)

with some constant $C>0$ independent of U.

Proof

Step 1. We first verify that $X_q({\mathbb {R}}^3)\cap Z_q(\mathbb R^3)=\{0\}$. Suppose $U\in X_q({\mathbb {R}}^3)\cap Z_q({\mathbb {R}}^3)$ and set

$$\begin{aligned} U|_\Omega =\nabla p, \qquad U|_B=\eta +\omega \times x, \end{aligned}$$

then we have

$$\begin{aligned} \Delta p=0\quad \hbox { in}\ \Omega , \qquad \partial _\nu p=\nu \cdot (\eta +\omega \times x)\quad \hbox { on}\ \partial \Omega \end{aligned}$$

(3.12)

with

$$\begin{aligned} \eta =\frac{-1}{m}\int _{\partial \Omega }p\nu \,d\sigma , \qquad \omega =-J^{-1}\int _{\partial \Omega }x\times (p\nu )\,d\sigma . \end{aligned}$$

(3.13)

It is observed that $\nabla p\in L^2_\textrm{loc}(\overline{\Omega })$ even though q is close to 1 and that $p-p_\infty$ with some constant $p_\infty \in {\mathbb {R}}$ (resp. $\nabla p$) behaves like the fundamental solution $\frac{-1}{4\pi |x|}$ (resp. its gradient) of the Laplacian at inifinity since $\nabla p\in L^q(\Omega )$; in fact, they go to zero even faster because $\int _{\partial \Omega }\partial _\nu p\,d\sigma \, (=0)$ is the coefficient of the leading term of the asymptotic expansion at infinity. This together with (3.12)–(3.13) justifies the equality (3.15) below when multiplying the Laplace equation in (3.12) by $(p-p_\infty )\phi _R$ with

$$\begin{aligned} \phi _R(x):=\phi (x/R), \end{aligned}$$

(3.14)

where $\phi$ is fixed as in (3.5), and then letting $R\rightarrow \infty$. Indeed, since

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _{2R<|x|<3R}|\nabla p||p-p_\infty ||\nabla \phi _R|\,dx=0, \end{aligned}$$

we obtain

$$\begin{aligned} \int _\Omega |\nabla p|^2\,dx =\int _{\partial \Omega }(\partial _\nu p)(p-p_\infty )\,d\sigma =\int _{\partial \Omega }\nu \cdot (\eta +\omega \times x)p\,d\sigma =-m|\eta |^2-(J\omega )\cdot \omega . \end{aligned}$$

(3.15)

This implies that $\nabla p=0$ as well as $\eta =\omega =0$, leading to $U=0$.

Step 2. Let us show that the class ${{\mathcal {E}}}({\mathbb {R}}^3)$ is dense in $X_q({\mathbb {R}}^3)$. Given $U\in X_q({\mathbb {R}}^3)$, we set $\eta +\omega \times x=U|_B$ and proceed as in the latter half of Lemma 3.1. Using the same lifting function $U_0=\ell (\eta ,\omega )\in C^\infty _{0,\sigma }(B_3)$ given by (3.4), we see that $(U-U_0)|_\Omega$ belongs to the space

$$\begin{aligned} L^q_\sigma (\Omega )=\{u\in L^q(\Omega );\; \hbox { div}\ u=0\;\text{ in } \Omega ,\;\nu \cdot u|_{\partial \Omega }=0\}. \end{aligned}$$

(3.16)

Since $C^\infty _{0,\sigma }(\Omega )$ is dense in $L^q_\sigma (\Omega )$ ([16, 38, 45]), the proof of the desired denseness is complete in the similar manner to descriptions in the last paragraph of the proof of Lemma 3.1. This was already shown in [10, Appendix A.2] within the context of the other decomposition (2.21).

We next prove that ${{\mathcal {E}}}({\mathbb {R}}^3)^\perp =X_q(\mathbb R^3)^\perp =Z_{q^\prime }({\mathbb {R}}^3)$ with respect to $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ by following the argument in [43] (in which the case $q=2$ is discussed). Let $U\in {{\mathcal {E}}}({\mathbb {R}}^3)$ and $V\in Z_{q^\prime }(\mathbb R^3)$, with $i(U)=(u,\eta _u,\omega _u)$ and $i(V)=(\nabla p,\eta ,\omega )$, where $\eta$ and $\omega$ are specified as (3.13). Then it is readily seen that

$$\begin{aligned} \langle U,V\rangle _{{\mathbb {R}}^3,\rho } =\int _\Omega \text{ div } (up)\,dx -\eta _u\cdot \int _{\partial \Omega }p\nu \,d\sigma -\omega _u\cdot \int _{\partial \Omega } x\times (p\nu )\,d\sigma =0. \end{aligned}$$

This relation holds even for all $U\in X_q({\mathbb {R}}^3)$ and $V\in Z_{q^\prime }({\mathbb {R}}^3)$ thanks to the denseness observed above. We are thus led to $Z_{q^\prime }({\mathbb {R}}^3)\subset X_q(\mathbb R^3)^\perp$.

Conversely, let $V\in L^{q^\prime }_R({\mathbb {R}}^3)$ satisfy $\langle \Phi ,V\rangle _{{\mathbb {R}}^3,\rho }=0$ for all $\Phi \in {{\mathcal {E}}}({\mathbb {R}}^3)$. Set $(v,\eta ,\omega )=i(V)$ and $(\phi ,\eta _\phi ,\omega _\phi )=i(\Phi )$. If, in particular,

$$\begin{aligned} \eta _\phi =\omega _\phi =0, \qquad \phi \in C_{0,\sigma }^\infty (\Omega ), \; \end{aligned}$$

(3.17)

then it follows that $v=\nabla p$ for some $p\in L^{q^\prime }_\textrm{loc}(\overline{\Omega })$. This implies that

$$\begin{aligned} \begin{aligned} m\eta _\phi \cdot \eta +\omega _\phi \cdot (J\omega )&=-\int _{\Omega }\hbox { div}\ (\phi p)\,dx \\&=-\eta _\phi \cdot \int _{\partial \Omega }p\nu \,d\sigma -\omega _\phi \cdot \int _{\partial \Omega }x\times (p\nu )\,d\sigma \end{aligned} \end{aligned}$$

for all $\Phi \in {{\mathcal {E}}}({\mathbb {R}}^3)$. Since $\eta _\phi$ and $\omega _\phi$ are arbitrary, we conclude

$$\begin{aligned} \eta =\frac{-1}{m}\int _{\partial \Omega }p\nu \,d\sigma , \qquad \omega =-J^{-1}\int _{\partial \Omega }x\times (p\nu )\,d\sigma , \end{aligned}$$

that is, $V\in Z_{q^\prime }({\mathbb {R}}^3)$. This proves ${\mathcal {E}}({\mathbb {R}}^3)^\perp \subset Z_{q^\prime }({\mathbb {R}}^3)$ and, therefore,

$$\begin{aligned} {{\mathcal {E}}}({\mathbb {R}}^3)^\perp = X_q(\mathbb R^3)^\perp =Z_{q^\prime }({\mathbb {R}}^3), \qquad Z_{q^\prime }(\mathbb R^3)^\perp =X_q({\mathbb {R}}^3). \end{aligned}$$

(3.18)

In fact, the latter follows from the former with $X_q(\mathbb R^3)^{\perp \perp }=X_q({\mathbb {R}}^3)$ since $X_q({\mathbb {R}}^3)$ is closed in the reflexive space $L^q_R({\mathbb {R}}^3)$, see Lemma 3.1. Now, (3.18) immediately leads to (3.7) when $q=2$ ([43, Theorem 3.2]).

Step 3. To complete the proof of (3.7) for the case $q\ne 2$, we use it for the case $q=2$. Given $U\in \mathcal {E}_R({\mathbb {R}}^3)$, see (3.1), there is a unique pair of $V\in X_2({\mathbb {R}}^3)$ and $W\in Z_2({\mathbb {R}}^3)$ such that $U=V+W$. By following the argument developed by Wang and Xin [51, Theorem 2.2], we will prove that $V\in X_q({\mathbb {R}}^3)$ and $W\in Z_q({\mathbb {R}}^3)$ along with estimate (3.25) below, where $(\nabla p,\eta ,\omega )=i(W)$, see (2.7). Set also $(u,\eta _u,\omega _u)=i(U)$ and $(v,\eta _v,\omega _v)=i(V)$. Since $\nu \cdot v|_{\partial \Omega }=\nu \cdot (\eta _v+\omega _v\times x)$, see (2.8), we have

$$\begin{aligned} \begin{aligned}&\Delta p=\text{ div }\; u \quad \text { in }\; \Omega , \\ {}&\nu \cdot (\nabla p-u)=-\nu \cdot (\eta _v+\omega _v\times x)\quad \text{ on }\ \partial \Omega . \end{aligned} \end{aligned}$$

Let $\ell$ be the lifting operator given by (3.4) and (3.6). Then the problem above is rewritten as

$$\begin{aligned} \begin{aligned}&\Delta p=\hbox { div}\ \big (u-\ell (\eta _v,\omega _v)\big ) \quad \hbox { in}\ \Omega , \\&\nu \cdot (\nabla p-u)=-\nu \cdot \ell (\eta _v,\omega _v) \quad \hbox { on}\ \partial \Omega . \end{aligned} \end{aligned}$$

(3.19)

One is then able to apply the theory of the Neumann problem in exterior domains, see [38, 45], to (3.19) to infer

$$\begin{aligned} \Vert \nabla p\Vert _{q,\Omega }\le C\big (\Vert u\Vert _{q,\Omega }+|\eta _v|+|\omega _v|\big ) \le C\big (\Vert U\Vert _{q,{\mathbb {R}}^3}+|\eta |+|\omega |\big ), \end{aligned}$$

(3.20)

where (2.12) is also taken into account. Let us single out a solution p in such a way that $\int _{\Omega _3}p\,dx=0$ to obtain

$$\begin{aligned} \Vert p\Vert _{q,\Omega _3}\le C\Vert \nabla p\Vert _{q,\Omega }. \end{aligned}$$

(3.21)

We show that

$$\begin{aligned} |\eta |+|\omega |\le C\Vert U\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.22)

for all $U\in {{\mathcal {E}}}_R({\mathbb {R}}^3)$. Suppose the contrary, then one can take a sequence $\{U_k\}\subset {{\mathcal {E}}}_R(\mathbb R^3)$ and the corresponding

$$\begin{aligned} \begin{aligned}&(\nabla p_k)\chi _\Omega +(\eta _k+\omega _k\times x)\chi _B\in Z_2({\mathbb {R}}^3), \qquad \int _{\Omega _3} p_k\,dx=0, \\&\eta _k=\frac{-1}{m}\int _{\partial \Omega }p_k\nu \,d\sigma , \qquad \omega _k=-J^{-1}\int _{\partial \Omega }x\times (p_k\nu )\,d\sigma , \\&v_k\chi _\Omega +(\eta _{v_k}+\omega _{v_k}\times x)\chi _B\in X_2({\mathbb {R}}^3), \end{aligned} \end{aligned}$$

(3.23)

such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert U_k\Vert _{q,{\mathbb {R}}^3}=0, \qquad |\eta _k|+|\omega _k|=1. \end{aligned}$$

(3.24)

By virtue of (3.20)–(3.21) together with (3.24), there are $\eta ,\,\omega \in {\mathbb {R}}^3$ and $p\in {\widehat{W}}^{1,q}_{(0)}(\Omega )$ as well as a subsequence, still denoted by the same symbol, such that, along the subsequence,

$$\begin{aligned} \begin{aligned}&\hbox { w-}\ \displaystyle {\lim _{k\rightarrow \infty }\nabla p_k=\nabla p}\quad \hbox { in}\ L^q(\Omega ), \qquad \lim _{k\rightarrow \infty }\Vert p_k-p\Vert _{q,\Omega _3}=0, \\&\lim _{k\rightarrow \infty }\eta _k=\eta , \qquad \lim _{k\rightarrow \infty }\omega _k=\omega . \end{aligned} \end{aligned}$$

By (3.23) and by the trace estimate

$$\begin{aligned} \Vert p_k-p\Vert _{q,\partial \Omega } \le C\Vert \nabla p_k-\nabla p\Vert _{q,\Omega _3}^{1/q}\Vert p_k-p\Vert _{q,\Omega _3}^{1-1/q}+C\Vert p_k-p\Vert _{q,\Omega _3}, \end{aligned}$$

we obtain

$$\begin{aligned} \eta =\frac{-1}{m}\int _{\partial \Omega }p\nu \,d\sigma , \qquad \omega =-J^{-1}\int _{\partial \Omega }x\times (p\nu )\,d\sigma . \end{aligned}$$

Since $p_k$ obeys (3.19) with $u_k=U_k|_\Omega$, we find $\Delta p=0$ in $\Omega$, so that $\partial _\nu p|_{\partial \Omega }$ makes sense, and

$$\begin{aligned} \begin{aligned}&\langle \nu \cdot (\nabla p_k-u_k-\nabla p), \psi \rangle _{\partial \Omega } =\langle \nabla p_k-u_k-\nabla p, \nabla \psi \rangle _{\Omega _3} \rightarrow 0, \\&-\nu \cdot (\eta _{v_k}+\omega _{v_k}\times x)=\nu \cdot (\eta _k+\omega _k\times x-\eta _{u_k}-\omega _{u_k}\times x) \rightarrow \nu \cdot (\eta +\omega \times x), \end{aligned} \end{aligned}$$

as $k\rightarrow \infty$ for all $\psi \in C_0^\infty (B_3)$ on account of (3.24), where $\eta _{u_k}+\omega _{u_k}\times x=U_k|_B$. As a consequence, $p,\,\eta$ and $\omega$ solve (3.12)–(3.13), leading to $\eta =\omega =0$ as explained in Step 1. This contradicts $|\eta |=|\omega |=1$ and thus concludes (3.22), which combined with (3.20) proves

$$\begin{aligned} \Vert V\Vert _{q,{\mathbb {R}}^3}+\Vert W\Vert _{q,{\mathbb {R}}^3}\sim \Vert V\Vert _{q,\mathbb R^3}+ \Vert \nabla p\Vert _{q,\Omega }+|\eta |+|\omega | \le C\Vert U\Vert _{q,\mathbb R^3} \end{aligned}$$

(3.25)

for all $U\in {{\mathcal {E}}}_R({\mathbb {R}}^3)$. With (3.25) at hand, given $U\in L^q_R({\mathbb {R}}^3)$, one can construct $V\in X_q({\mathbb {R}}^3)$ and $W\in Z_q({\mathbb {R}}^3)$ such that $U=V+W$ along with the same estimate (3.25) since $\mathcal {E}_R({\mathbb {R}}^3)$ is dense in $L^q_R({\mathbb {R}}^3)$ by Lemma 3.1. This completes the proof of (3.7) and (3.9). From (3.7) we find

$$\begin{aligned} X_q({\mathbb {R}}^3)^*=\big [L^q_R({\mathbb {R}}^3)/Z_q(\mathbb R^3)\big ]^*=Z_q({\mathbb {R}}^3)^\perp , \qquad Z_q(\mathbb R^3)^*=\big [L^q_R({\mathbb {R}}^3)/X_q({\mathbb {R}}^3)\big ]^*=X_q(\mathbb R^3)^\perp , \end{aligned}$$

which along with (3.18) implies (3.8) and (3.10) as well.

Step 4. Finally, let us show (3.11) for $U\in L^q_R({\mathbb {R}}^3)$ with $(u,\eta _u,\omega _u)=i(U)$, see (2.7), when additionally assuming $u\in W^{1,q}(\Omega )$. Since $\mathbb PU=U-W$ with

$$\begin{aligned} W=(\nabla p)\chi _\Omega +(\eta +\omega \times x)\chi _B\in Z_q(\mathbb R^3), \end{aligned}$$

it suffices to prove that

$$\begin{aligned} \Vert \nabla ^2p\Vert _{q,\Omega }\le C\big (\Vert U\Vert _{q,{\mathbb {R}}^3}+\Vert \nabla u\Vert _{q,\Omega }\big ), \end{aligned}$$

(3.26)

where p should obey

$$\begin{aligned} \begin{aligned}&\Delta p=\text{ div }\; u \quad \text { in }\; \Omega , \\ {}&\partial _\nu p=\nu \cdot (u-\eta _u-\omega _u\times x+\eta +\omega \times x)\quad \text{ on }\ \partial \Omega , \end{aligned} \end{aligned}$$

with $\eta$ and $\omega$ given by (3.13). As in (3.19), we utilize the lift

$$\begin{aligned} U_0:=\ell (\eta _u-\eta , \omega _u-\omega ) \end{aligned}$$

to rewrite the problem above as

$$\begin{aligned} \begin{aligned}&\Delta p=\text{ div } \big (u-U_0\big ) \quad \text { in }\; \Omega , \\ {}&\partial _\nu p=\nu \cdot \big (u-U_0\big ) \quad \text { on } \partial \Omega . \end{aligned} \end{aligned}$$

By [23, Lemma 2.3] we have

$$\begin{aligned} \begin{aligned} \Vert \nabla ^2p\Vert _{q,\Omega }&\le C\Vert u-U_0\Vert _{W^{1,q}(\Omega )} \\&\le C\big (\Vert u\Vert _{W^{1,q}(\Omega )}+|\eta _u|+|\omega _u|+|\eta |+|\omega |\big ) \\&\le C\big (\Vert U\Vert _{q,{\mathbb {R}}^3}+\Vert \nabla u\Vert _{q,\Omega }+|\eta |+|\omega |\big ) \end{aligned} \end{aligned}$$

which combined with (3.25) concludes (3.26). The proof is complete. $\square$

3.2 (2.28) is Equivalent to (2.27)

This subsection claims that the resolvent equation (2.28) in $X_q({\mathbb {R}}^3)$ with the Stokes-structure operator A defined by (2.25) is equivalent to the boundary value problem (2.27). This can be proved in the same way as in [10, Proposition 3.4] on the same issue for (2.26) with the other operator ${\widetilde{A}}$, see also [46], however, one has to prove the equivalence independently of the existing literature about the operator ${\widetilde{A}}$. In fact, with the following proposition at hand, we will then verify $A=\widetilde{A}$ afterwards by use of uniqueness of solutions to the boundary value problem (2.27).

Proposition 3.2

Let $1<q<\infty$ and $\lambda \in {\mathbb {C}}$. Suppose that $F\in X_q({\mathbb {R}}^3)$ and $(f,\kappa ,\mu )=i(F)$ through (2.7). If

$$\begin{aligned} u\in W^{2,q}(\Omega ), \quad p\in {\widehat{W}}^{1,q}(\Omega ), \; \quad (\eta ,\omega )\in {\mathbb {C}}^3\times {\mathbb {C}}^3 \end{aligned}$$

(3.27)

fulfill (2.27), then (2.28) holds with $U=u\chi _\Omega +(\eta +\omega \times x)\chi _B$. Conversely, assume that $U\in D_q(A)$ satisfies (2.28). Then there exists a pressure $p\in {\widehat{W}}^{1,q}(\Omega )$ such that $(u,\eta ,\omega )=i(U)$ together with p enjoys (2.27).

Proof

Concerning the first half, it is obvious that

$$\begin{aligned} \begin{aligned}&U=u\chi _\Omega +(\eta +\omega \times x)\chi _B\in D_q(A), \\&(\nabla p)\chi _\Omega +\left[ \frac{-1}{m}\int _{\partial \Omega }p\nu \,d\sigma +\left( -J^{-1}\int _{\partial \Omega }y\times (p\nu )\,d\sigma _y\right) \times x\right] \chi _B \in Z_q({\mathbb {R}}^3) \end{aligned} \end{aligned}$$

and that, in view of (2.18)–(2.19), applying the projection ${\mathbb {P}}$ to (2.27) yields (2.28).

To show the second half, suppose that $U\in D_q(A)$ fulfills (2.28), then u possesses the desired regularity together with the boundary condition $u|_{\partial \Omega }=\eta +\omega \times x$. We also find

$$\begin{aligned} \langle (\lambda +A)U,\Phi \rangle _{{\mathbb {R}}^3,\rho }=\langle F,\Phi \rangle _{{\mathbb {R}}^3,\rho } \end{aligned}$$

which is rewritten as

$$\begin{aligned} \begin{aligned}&\lambda \langle u,\phi \rangle _\Omega +\lambda m\eta \cdot \eta _\phi +\lambda (J\omega )\cdot \omega _\phi \\&\qquad -\langle \Delta u,\phi \rangle _\Omega +\int _{\partial \Omega }(2Du)\nu \,d\sigma \cdot \eta _\phi +\int _{\partial \Omega }y\times (2Du)\nu \,d\sigma _y\cdot \omega _\phi \\&\quad =\langle f,\phi \rangle _\Omega +m\kappa \cdot \eta _\phi +(J\mu )\cdot \omega _\phi \end{aligned} \end{aligned}$$

(3.28)

for all $\Phi \in {{\mathcal {E}}}({\mathbb {R}}^3)$ with $(\phi ,\eta _\phi ,\omega _\phi )=i(\Phi )$ on account of (2.11) and (3.10). Let us, in particular, choose (3.17), then we get

$$\begin{aligned} \langle \lambda u-\Delta u-f, \phi \rangle _\Omega =0 \end{aligned}$$

for all $\phi \in C_{0,\sigma }^\infty (\Omega )$. Since $\lambda u-\Delta u-f\in L^q(\Omega )$, there is a function $p\in L^q_\textrm{loc}(\overline{\Omega })$ with $\nabla p\in L^q(\Omega )$ such that

$$\begin{aligned} \lambda u-\Delta u-f=-\nabla p \end{aligned}$$

in $\Omega$. By taking into account this equation in (3.28), we are led to

$$\begin{aligned} \begin{aligned}&\left( \lambda m\eta +\int _{\partial \Omega }(2Du)\nu \,d\sigma -\int _{\partial \Omega }p\nu \,d\sigma -m\kappa \right) \cdot \eta _\phi \\&+\left( \lambda J\omega +\int _{\partial \Omega }y\times (2Du)\nu \,d\sigma _y-\int _{\partial \Omega }y\times (p\nu )\,d\sigma _y-J\mu \right) \cdot \omega _\phi =0 \end{aligned} \end{aligned}$$

for all $\eta _\phi ,\,\omega _\phi \in {\mathbb {C}}^3$, which yields the equations for the rigid body in (2.27). The proof is complete. $\square$

Proposition 3.3

The operator A defined by (2.25) coincides with ${\widetilde{A}}$, where the latter operator is defined by (2.25) in which ${\mathbb {P}}$ is replaced by the other projection ${\mathbb {Q}}: L^q({\mathbb {R}}^3)\rightarrow X_q({\mathbb {R}}^3)$ associated with (2.21).

Proof

By Proposition 3.2 and by [10, Proposition 3.4], both equations (2.26) and (2.28) are equivalent to the boundary value problem (2.27). Hence, it suffices to show that the only solution to (2.27) with $\lambda =1$ and $F=0$ within the class (3.27) is the trivial one. In fact, given $U\in D_q(A)$, one can see from [10, Theorem 6.1] that there is a unique ${\widetilde{U}}\in D_q(\widetilde{A})=D_q(A)$ satisfying $(1+{\widetilde{A}}){\widetilde{U}}=(1+A)U$ in $X_q({\mathbb {R}}^3)$. Then the uniqueness for (2.27) (which we will show below) implies that ${\widetilde{U}}=U$ and, thereby, ${\widetilde{A}}U={\widetilde{A}}{\widetilde{U}}=AU$.

Consider the problem (2.27) with $\lambda =1,\, f=0$ and $\kappa =\mu =0$. Then we have $u\in W^{1,2}_\textrm{loc}(\overline{\Omega })$ even if q is close to 1. Moreover, from (3.27) it follows that $\{u,p-p_\infty \}$ with some constant $p_\infty \in {\mathbb {R}}$ and $\nabla u$ behave like the fundamental solution and its gradient to the Stokes resolvent system even though q is large. Let $\phi _R$ be the same cut-off function as in (3.14). Computing

$$\begin{aligned} 0=\int _\Omega \big (u-\Delta u+\nabla (p-p_\infty )\big )\cdot u\phi _R\, dx \end{aligned}$$

and then letting $R\rightarrow \infty$, where

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _{2R<|x|<3R}|\mathbb S(u,p-p_\infty )||u||\nabla \phi _R|\,dx=0, \end{aligned}$$

we are led to

$$\begin{aligned} \Vert u\Vert _{2,\Omega }^2+m|\eta |^2+(J\omega )\cdot \omega +2\Vert Du\Vert _{2,\Omega }^2=0, \end{aligned}$$

which concludes that $u=0$ and $\eta =\omega =0$. The proof is complete. $\square$

3.3 Stokes-Structure Resolvent

Estimate (2.30) for large $|\lambda |$ is established in [10, Proposition 5.1], where the authors of [10] however omit the proof since it is a slight variation of the proof given by Maity and Tucsnak [35, Theorem 3.1] on the same issue for the Stokes-structure system in a bounded container. In fact, their idea based on the reformulation below rather than (2.28) or (2.26) is fine, but the variation needs a couple of nontrivial modifications since $\Omega$ is unbounded. For completeness, this subsection is devoted to the details of reconstruction of the proof of (2.30) for large $|\lambda |$. Once we have that, a contradiction argument performed in [10, Sect. 6] leads to (2.30) even for $\lambda \in \Sigma _\varepsilon$ close to the origin $\lambda =0$. The reason why the details are provided here is that we do need a representation of the resolvent, see (3.48) together with (3.44) below, to deduce a useful estimate of the associated pressure near the initial time in Sect. 3.6.

The underlying space of the other formulation of the resolvent problem (2.27) is

$$\begin{aligned} Y_q:=L^q_\sigma (\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3, \end{aligned}$$

where $L^q_\sigma (\Omega )$ given by (3.16) is the standard underlying space when considering the Stokes resolvent in the exterior domain $\Omega$. By $P_\Omega : L^q(\Omega )\rightarrow L^q_\sigma (\Omega )$ we denote the classical Fujita-Kato projection associated with the Helmholtz decomposition [38, 45]

$$\begin{aligned} L^q(\Omega )=L^q_\sigma (\Omega )\oplus \{\nabla p\in L^q(\Omega );\; p\in {\widehat{W}}^{1,q}(\Omega )\}. \end{aligned}$$

Then the Stokes operator $A_\Omega$ in exterior domains is defined by

$$\begin{aligned} D_q(A_\Omega )=L^q_\sigma (\Omega )\cap W^{1,q}_0(\Omega )\cap W^{2,q}(\Omega ), \qquad A_\Omega =-P_\Omega \Delta . \end{aligned}$$

Let us reformulate the resolvent system (2.27) by following the procedure due to Maity and Tucsnak [35]. Let $F\in X_q({\mathbb {R}}^3),\, 1<q<\infty$, and consider (2.27) with $(f,\kappa ,\mu )=i(F)$, see (2.7). Given $(\eta ,\omega )\in {\mathbb {C}}^3\times {\mathbb {C}}^3$, we take a lifting function

$$\begin{aligned} U_0=\ell (\eta ,\omega ) \end{aligned}$$

of the rigid motion $\eta +\omega \times x$, where $\ell$ is given by (3.4) and (3.6). Obviously, we have

$$\begin{aligned} \Vert \ell (\eta ,\omega )\Vert _{W^{2,q}(\Omega )}\le C|(\eta ,\omega )|. \end{aligned}$$

(3.29)

The fluid part of (2.27) is reduced to finding ${\widetilde{u}}:=u-U_0\in D(A_\Omega )$ that obeys

$$\begin{aligned} \lambda ({\widetilde{u}}+P_\Omega U_0)+A_\Omega \widetilde{u}-P_\Omega \Delta U_0=P_\Omega f \end{aligned}$$

(3.30)

in $L^q_\sigma (\Omega )$, while the associated pressure consists of

$$\begin{aligned} \begin{aligned}&p=p_1+p_2-\lambda p_3+p_4 \quad \text{ with } \\&p_1=N(\Delta {\widetilde{u}}), \quad p_2=N(\Delta U_0), \quad p_3=N(U_0), \quad p_4=N(\ell (\kappa ,\mu )) \end{aligned} \end{aligned}$$

(3.31)

in terms of the Neumann operator $N: E_q(\Omega )\ni h\mapsto N(h):=\psi \in {\widehat{W}}^{1,q}_{(0)}(\Omega )$ which singles out a unique solution to

$$\begin{aligned} \begin{aligned} \Delta \psi =\text{ div }\; h \quad \text { in }\; \Omega , \qquad \partial _\nu \psi =\nu \cdot h\quad \text{ on }\ \partial \Omega , \qquad \int _{\Omega _3}\psi \,dx=0, \end{aligned} \end{aligned}$$

where $E_q(\Omega ):=\{h\in L^q(\Omega )^3;\; \hbox { div}\ h\in L^q(\Omega )\}$. Then we have

$$\begin{aligned} \Vert N(h)\Vert _{q,\Omega _3}\le C\Vert \nabla N(h)\Vert _{q,\Omega _3} \le C\Vert \nabla N(h)\Vert _{q,\Omega }\le C\Vert h\Vert _{q,\Omega }. \end{aligned}$$

(3.32)

Note that

$$\begin{aligned} \nu \cdot f=\nu \cdot (\kappa +\mu \times x)=\nu \cdot \ell (\kappa ,\mu ) \qquad \hbox { on}\ \partial \Omega , \end{aligned}$$

where the function $\ell (\kappa ,\mu )$ is introduced for description of $p_4$ in (3.31) since $\kappa +\mu \times x\notin E_q(\Omega )$. In (3.30) one can not write $A_\Omega U_0$ because $U_0$ never belongs to $D(A_\Omega )$ on account of $U_0|_{\partial \Omega }=\eta +\omega \times x$.

We rewrite the equation of balance for linear momentum

$$\begin{aligned} \lambda m\eta +\int _{\partial \Omega }{\mathbb {S}}(\widetilde{u}+U_0,p_1+p_2-\lambda p_3+p_4)\nu \,d\sigma =m\kappa \end{aligned}$$

in (2.27) as

$$\begin{aligned} \lambda \left( m\eta +\int _{\partial \Omega }p_3\nu \,d\sigma \right) +\int _{\partial \Omega }{\mathbb {S}}({\widetilde{u}},p_1)\nu \,d\sigma +\int _{\partial \Omega }{\mathbb {S}}(U_0,p_2)\nu \,d\sigma =m\kappa +\int _{\partial \Omega }p_4\nu \,d\sigma . \end{aligned}$$

(3.33)

Likewise, the equation of balance for angular momentum is described as

$$\begin{aligned} \begin{aligned} \lambda \left( J\omega +\int _{\partial \Omega }x\times (p_3\nu )\,d\sigma \right) +\int _{\partial \Omega }x\times \mathbb S({\widetilde{u}},p_1)\nu \,d\sigma&+\int _{\partial \Omega }x\times {\mathbb {S}}(U_0,p_2)\nu \,d\sigma \\&=J\mu +\int _{\partial \Omega }x\times (p_4\nu )\,d\sigma . \end{aligned} \end{aligned}$$

(3.34)

It is convenient to introduce

$$\begin{aligned} K \left( \begin{array}{c} \eta \\ \omega \end{array} \right)= & {} \left( \begin{aligned}&m\eta +\int _{\partial \Omega }N(\ell (\eta ,\omega ))\nu \,d\sigma \\&J\omega +\int _{\partial \Omega }x\times N(\ell (\eta ,\omega ))\nu \,d\sigma \end{aligned} \right) \end{aligned}$$

(3.35)

$$\begin{aligned} Q_1{\widetilde{u}}= & {} \left( \begin{aligned}&\int _{\partial \Omega }{\mathbb {S}}({\widetilde{u}},N(\Delta {\widetilde{u}}))\nu \,d\sigma \\&\int _{\partial \Omega }x\times {\mathbb {S}}(\widetilde{u},N(\Delta {\widetilde{u}}))\nu \,d\sigma \end{aligned} \right) \end{aligned}$$

(3.36)

$$\begin{aligned} Q_2(\eta ,\omega )= & {} \left( \begin{aligned}&\int _{\partial \Omega }{\mathbb {S}}(\ell (\eta ,\omega ),N(\Delta \ell (\eta ,\omega ))\nu \,d\sigma \\&\int _{\partial \Omega }x\times \mathbb S(\ell (\eta ,\omega ),N(\Delta \ell (\eta ,\omega ))\nu \,d\sigma \end{aligned} \right) . \end{aligned}$$

(3.37)

Note that $K\in {\mathbb {C}}^{6\times 6}$ is independent of the choice of lift of the rigid motion $\eta +\omega \times x$ since $\psi =N(\ell (\eta ,\omega ))$ solves

$$\begin{aligned} \Delta \psi =0\quad \hbox { in}\ \Omega , \qquad \partial _\nu \psi =\nu \cdot (\eta +\omega \times x)\quad \hbox { on}\ \partial \Omega , \qquad \int _{\Omega _3}\psi \,dx=0, \end{aligned}$$

and that K is invertible, see [35, Lemma 3.8]. Hence, the equations (3.33)–(3.34) read

$$\begin{aligned} \lambda \left( \begin{array}{c} \eta \\ \omega \end{array} \right) +K^{-1}Q_1{\widetilde{u}}+K^{-1}Q_2(\eta ,\omega ) =\left( \begin{array}{c} \kappa \\ \mu \end{array} \right) . \end{aligned}$$

(3.38)

By (3.38) one can describe the term $\lambda P_\Omega U_0$ of (3.30) as

$$\begin{aligned} \lambda P_\Omega U_0= P_\Omega \ell (\lambda \eta ,\lambda \omega )=P_\Omega \ell \left( (\kappa ,\mu )-K^{-1}Q_1\widetilde{u}-K^{-1}Q_2(\eta ,\omega )\right) , \end{aligned}$$

from which (3.30) is reduced to

$$\begin{aligned} \begin{aligned} \lambda {\widetilde{u}}+A_\Omega {\widetilde{u}}-P_\Omega \Delta U_0 -P_\Omega \ell \left( K^{-1}Q_1\widetilde{u}+K^{-1}Q_2(\eta ,\omega )\right)&=P_\Omega f-P_\Omega \ell (\kappa ,\mu ) \\&=f-\ell (\kappa ,\mu ), \end{aligned} \end{aligned}$$

(3.39)

where the last equality above follows from $f-\ell (\kappa ,\mu )\in L^q_\sigma (\Omega )$.

Let us introduce the other Stokes-structure operator ${\mathbb {A}}$ acting on $Y_q$ by

$$\begin{aligned} \begin{aligned}&D_q({\mathbb {A}})=D_q(A_\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3, \\&{\mathbb {A}} \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) = \left( \begin{array}{cc} 1 &{} -P_\Omega \ell \\ 0 &{} 1 \end{array} \right) \left( \begin{array}{cc} A_\Omega &{} -P_\Omega \Delta \ell \\ K^{-1}Q_1 &{} K^{-1}Q_2 \end{array} \right) \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) , \end{aligned} \end{aligned}$$

(3.40)

then, in view of (3.38)–(3.39), the resolvent system (2.27) is reformulated as

$$\begin{aligned} (\lambda +{\mathbb {A}})\left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) = \left( \begin{array}{c} f -\ell (\kappa ,\mu ) \\ (\kappa ,\mu ) \end{array} \right) \qquad \hbox { in}\ Y_q. \end{aligned}$$

(3.41)

The operator ${\mathbb {A}}$ is splitted into

$$\begin{aligned} {\mathbb {A}}={\mathbb {A}}_0+{\mathbb {A}}_1 \end{aligned}$$

with

$$\begin{aligned} {\mathbb {A}}_0\left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right)= & {} \left( \begin{array}{cc} 1 &{} -P_\Omega \ell \\ 0 &{} 1 \end{array} \right) \left( \begin{array}{cc} A_\Omega &{} -P_\Omega \Delta \ell \\ 0 &{} 0 \end{array} \right) \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) , \end{aligned}$$

(3.42)

$$\begin{aligned} {\mathbb {A}}_1 \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right)= & {} \left( \begin{array}{cc} 1 &{} -P_\Omega \ell \\ 0 &{} 1 \end{array} \right) \left( \begin{array}{cc} 0 &{} 0 \\ K^{-1}Q_1 &{} K^{-1}Q_2 \end{array} \right) \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) . \end{aligned}$$

(3.43)

Let $\lambda \in {\mathbb {C}}\setminus (-\infty ,0]$, then $\lambda +{\mathbb {A}}_0$ is invertible and

$$\begin{aligned} (\lambda +{\mathbb {A}}_0)^{-1}=\left( \begin{array}{cc} (\lambda +A_\Omega )^{-1} &{} \lambda ^{-1}(\lambda +A_\Omega )^{-1}P_\Omega \Delta \ell \\ 0 &{} \lambda ^{-1} \end{array} \right) \end{aligned}$$

(3.44)

in ${{\mathcal {L}}}(Y_q)$. Moreover, we find

$$\begin{aligned} \Vert (\lambda +{\mathbb {A}}_0)^{-1}\Vert _{{{\mathcal {L}}}(Y_q)}\le C_\varepsilon |\lambda |^{-1} \end{aligned}$$

(3.45)

for all $\lambda \in \Sigma _\varepsilon$ with $|\lambda |\ge 1$ (say), see (2.31), where $\varepsilon \in (0,\pi /2)$ is fixed arbitrarily, because of the parabolic resolvent estimate of the Stokes operator $A_\Omega$.

We will show that ${\mathbb {A}}_1$ is subordinate to ${\mathbb {A}}_0$. To this end, let us deduce estimates of $Q_1$ and $Q_2$ given by (3.36)–(3.37). By the trace estimate together with (3.29) and (3.32), we have

$$\begin{aligned} |Q_1{\widetilde{u}}|_{{\mathbb {C}}^3\times {\mathbb {C}}^3} \le C\big (\Vert A_\Omega {\widetilde{u}}\Vert _{q,\Omega }+\Vert \widetilde{u}\Vert _{q,\Omega }\big ) \end{aligned}$$

(3.46)

as well as

$$\begin{aligned} |Q_2(\eta ,\omega )|_{{\mathbb {C}}^3\times {\mathbb {C}}^3}\le C|(\eta ,\omega )|. \end{aligned}$$

Those estimates together with (3.29) lead us to

$$\begin{aligned} \begin{aligned}&\quad \Vert \ell \left( K^{-1}Q_1\widetilde{u}+K^{-1}Q_2(\eta ,\omega )\right) \Vert _{W^{1,q}(\Omega )} +\left| K^{-1}Q_1{\widetilde{u}}+K^{-1}Q_2(\eta ,\omega )\right| _{{\mathbb {C}}^3\times {\mathbb {C}}^3} \\&\le C\big (\Vert A_\Omega {\widetilde{u}}\Vert _{q,\Omega }+\Vert \widetilde{u}\Vert _{q,\Omega } +|(\eta ,\omega )|\big ). \end{aligned} \end{aligned}$$

Since the support of the lifting function (3.4) is bounded, the Rellich theorem implies that, for any sequence $\{\widetilde{u}_j, (\eta _j,\omega _j)\}$ being bounded in $D_q(A_\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3$, one can subtract a convergent subsequence in $L^q(\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3$ from $\left\{ \ell \big (K^{-1}Q_1\widetilde{u}_j+K^{-1}Q_2(\eta _j,\omega _j)\big ),\, K^{-1}Q_1\widetilde{u}_j+K^{-1}Q_2(\eta _j,\omega _j)\right\}$. Along the same subsequence, ${\mathbb {A}}_1\left( \begin{array}{c} {\widetilde{u}}_j \\ (\eta _j,\omega _j) \end{array} \right)$ is also convergent in $Y_q$ as $P_\Omega$ is bounded on $L^q(\Omega )$. Hence, ${\mathbb {A}}_1$ is a compact operator from $D_q({\mathbb {A}}_0)=D_q(A_\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3$ (endowed with the graph norm) into $Y_q$. One can then employ a perturbation theorem [7, Chapter III, Lemma 2.16] to conclude that ${\mathbb {A}}_1$ is ${\mathbb {A}}_0$-bounded and its $\mathbb A_0$-bound is zero; that is, for every small $\delta >0$ there is a constant $C_\delta >0$ satisfying

$$\begin{aligned}{} & {} \left\| {\mathbb {A}}_1\left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) \right\| _{Y_q} \\{} & {} \quad \le \delta \left\| {\mathbb {A}}_0\left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) \right\| _{Y_q}+C_\delta \left\| \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) \right\| _{Y_q} \end{aligned}$$

where $\Vert \cdot \Vert _{Y_q}$ is obviously given by the sum of $\Vert \cdot \Vert _{q,\Omega }$ and $|(\cdot ,\cdot )|_{{\mathbb {C}}^3\times {\mathbb {C}}^3}$. This combined with (3.45) allows us to take $\Lambda _\varepsilon >0$ for each $\varepsilon \in (0,\pi /2)$ such that

$$\begin{aligned} \Vert (\lambda +{\mathbb {A}})^{-1}\Vert _{{{\mathcal {L}}}(Y_q)}\le C_\varepsilon |\lambda |^{-1} \end{aligned}$$

(3.47)

for all $\lambda \in \Sigma _\varepsilon$ with $|\lambda |\ge \Lambda _\varepsilon$, where the resolvent is described as the Neumann series in ${{\mathcal {L}}}(Y_q)$ for such $\lambda$:

$$\begin{aligned} \begin{aligned}&(\lambda +{\mathbb {A}})^{-1}=(\lambda +{\mathbb {A}}_0)^{-1} \sum _{j=0}^\infty \left[ -{\mathbb {A}}_1(\lambda +{\mathbb {A}}_0)^{-1}\right] ^j \\&\text{ with }\quad \sum _{j=0}^\infty \left\| \mathbb A_1(\lambda +{\mathbb {A}}_0)^{-1}\right\| _{{{\mathcal {L}}}(Y_q)}^j \le 2. \end{aligned} \end{aligned}$$

(3.48)

With (3.47) at hand, we immediately obtain

$$\begin{aligned} \Vert {\widetilde{u}}\Vert _{q,\Omega }+|(\eta ,\omega )|\le C_\varepsilon |\lambda |^{-1}\big (\Vert f\Vert _{q,\Omega }+|(\kappa ,\mu )|\big ) \end{aligned}$$

for (3.41), from which together with (3.29) we are led to

$$\begin{aligned} \Vert u\Vert _{q,\Omega }+|(\eta ,\omega )|\le C_\varepsilon |\lambda |^{-1}\big (\Vert f\Vert _{q,\Omega }+|(\kappa ,\mu )|\big ) \end{aligned}$$

for the solution to (2.27). On account of Proposition 3.2, we conclude

$$\begin{aligned} \Vert U\Vert _{q,{\mathbb {R}}^3}\le C_\varepsilon |\lambda |^{-1}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

for (2.28) with $C_\varepsilon >0$ independent of $\lambda \in \Sigma _\varepsilon$ satisfying $|\lambda |\ge \Lambda _\varepsilon$, where $\varepsilon \in (0,\pi /2)$ is arbitrary.

3.4 Generation of the Evolution Operator

In this subsection we show the generation of the evolution operator by the family $\{L_+(t);\, t\in {\mathbb {R}}\}$ of the Oseen-structure operators (2.36)–(2.37). It is of parabolic type (in the sense of Tanabe and Sobolevskii [48, 52]) with the properties (2.47)–(2.50).

Let $1<q<\infty$. We first see that, for each fixed $t\in {\mathbb {R}}$, the operator $-L_+(t)$ generates an analytic semigroup on the space $X_q({\mathbb {R}}^3)$. This is readily verified as follows by a simple perturbation argument. We fix $\varepsilon \in (0,\pi /2)$. Given $F\in X_q({\mathbb {R}}^3)$, let us take $U=(\lambda +A)^{-1}F$ with $\lambda \in \Sigma _\varepsilon$, see (2.31), in (2.40) and employ (2.30) to obtain

$$\begin{aligned} \Vert B(t)(\lambda +A)^{-1}F\Vert _{q,{\mathbb {R}}^3} \le C\big (|\lambda |^{-1/2}+|\lambda |^{-1}\big ) \Vert U_b\Vert \Vert F\Vert _{q,\mathbb R^3}. \end{aligned}$$

(3.49)

Hence, there is a constant $c_0>0$ dependent on $\Vert U_b\Vert$ but independent of t such that if $\lambda \in \Sigma _\varepsilon$ fulfills $|\lambda |\ge c_0$, then the right-hand side of (3.49) is bounded from above by $\frac{1}{2}\Vert F\Vert _{q,{\mathbb {R}}^3}$, yielding $\lambda \in \rho (-L_+(t))$ subject to

$$\begin{aligned} \Vert (\lambda +L_+(t))^{-1}\Vert _{{{\mathcal {L}}}(X_q({\mathbb {R}}^3))}\le 2\Vert (\lambda +A)^{-1}\Vert _{{{\mathcal {L}}}(X_q({\mathbb {R}}^3))} \le \frac{C}{|\lambda |} \end{aligned}$$

for every $t\in {\mathbb {R}}$ by the Neumann series argument.

We next verify the regularity of $L_+(t)$ in t that allows us to apply the theory of parabolic evolution operators, see [48, Chapter 5]. In fact, by the same computations as above with use of (2.39), it follows from (2.34)–(2.35) that

$$\begin{aligned} \begin{aligned}&\Vert (L_+(t)-L_+(s))(\lambda +L_+(\tau ))^{-1}F\Vert _{q,{\mathbb {R}}^3} \\ {}&\quad =\big \Vert (B(t)-B(s))(\lambda +A)^{-1}\big [1+B(\tau )(\lambda +A)^{-1}\big ]^{-1}F\big \Vert _{q,{\mathbb {R}}^3} \\ {}&\quad \le C(c_0^{-1/2}+c_0^{-1})(t-s)^\theta \, [U_b]_\theta \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

(3.50)

for all $F\in X_q({\mathbb {R}}^3)$, $\lambda \in \Sigma _\varepsilon$ with $|\lambda |\ge c_0$, and $t,\,s,\,\tau \in {\mathbb {R}}$ with $t>s$. As a consequence, the family $\{L_+(t);\, t\in {\mathbb {R}}\}$ generates an evolution operator $\{T(t,s);\, s,t\in I,\, s\le t\}$ on $X_q({\mathbb {R}}^3)$ for every compact interval $I\subset {\mathbb {R}}$, which provides the family $\{T(t,s);\, -\infty<s\le t<\infty \}$ with the properties (2.47)–(2.50) by uniqueness of evolution operators. Notice that the locally Hölder continuity in t of $u_b(t)$ (with values in $L^\infty (\Omega )$) as well as $\eta _b(t)$ is enough just for generation of the evolution operator, however, the globally Hölder continuity (2.34) is needed for further studies. This point is indeed the issue of the next subsection.

3.5 Smoothing Estimates of the Evolution Operator

This subsection is devoted to the $L^q$–$L^r$ smoothing estimates (2.51)–(2.52) for all (t, s) with $t-s\in (0,\tau _*]$, where $\tau _*\in (0,\infty )$ is fixed arbitrarily. Those rates themselves are quite standard, nevertheless, the point is to show that the constants in (2.51)–(2.52) may be dependent on $\tau _*$, however, can be taken uniformly with respect to such $t,\,s$ under the globally Hölder condition (2.34) on $u_b(t)$ and $\eta _b(t)$. The similar studies are found in [27, Lemma 3.2].

Proposition 3.4

Suppose (2.33) and (2.34). Let $1<q<\infty$, $r\in [q,\infty ]$ (except $r=\infty$ for (2.52)) and $\tau _*,\, \alpha _0,\, \beta _0\in (0,\infty )$. Then there is a constant $C=C(q,r,\tau _*,\alpha _0,\beta _0,\theta )>0$ such that both (2.51) and (2.52) hold for all (t, s) with $t-s\in (0,\tau _*]$ and $F\in X_q({\mathbb {R}}^3)$ whenever $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, where $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35). Moreover, we have

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow s}\; (t-s)^{(3/q-3/r)/2}\Vert T(t,s)F\Vert _{r,{\mathbb {R}}^3}&=0, \qquad r\in (q,\infty ], \\ \lim _{t\rightarrow s}\; (t-s)^{1/2+(3/q-3/r)/2}\Vert \nabla T(t,s)F\Vert _{r,\mathbb R^3}&=0, \qquad r\in [q,\infty ), \end{aligned} \end{aligned}$$

(3.51)

for every $F\in X_q({\mathbb {R}}^3)$, where the convergence is uniform on each precompact set of $X_q({\mathbb {R}}^3)$.

Proof

Let us fix $\tau _*>0$. In the proof of construction of the evolution operator, an important step is to show that

$$\begin{aligned} \Vert L_+(t)T(t,s)\Vert _{{{\mathcal {L}}}(X_q({\mathbb {R}}^3))}\le C(t-s)^{-1} \end{aligned}$$

(3.52)

as well as

$$\begin{aligned} \Vert T(t,s)\Vert _{{{\mathcal {L}}}(X_q({\mathbb {R}}^3))}\le C \end{aligned}$$

(3.53)

for $t-s\le \tau _*$. If we look into details of deduction of (3.52)–(3.53), see Tanabe [48, Chapter 5], we find that both constants $C=C(q,\tau _*,\alpha _0,\beta _0,\theta )>0$ can be taken uniformly in (t, s) with $t-s\le \tau _*$ on account of the globally Hölder continuity (3.50). See also the discussions in [27, Lemma 3.2]. Based on (3.52)–(3.53), we take $U(t)=T(t,s)F$ with $F\in X_q({\mathbb {R}}^3)$ in (2.41)$_2$ and use (2.39) to infer

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{q,\Omega }\le C(t-s)^{-1/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.54)

for $t-s\le \tau _*$ with some constant $C=C(q,\tau _*,\alpha _0,\beta _0,\theta )>0$. It follows from (3.53)–(3.54) together with the Gagliardo-Nirenberg inequality that

$$\begin{aligned} \Vert T(t,s)F\Vert _{r,\Omega }\le C(t-s)^{-(3/q-3/r)/2}\Vert F\Vert _{q,\mathbb R^3} \end{aligned}$$

(3.55)

for $t-s\le \tau _*$ provided $1/q-1/r< 1/3$ (actually, $1/q-1/r\le 1/3$ if $q\ne 3$). From the relation (2.7) with $U(t)=T(t,s)F$, it is readily seen that

$$\begin{aligned} \Vert U(t)\Vert _{r,B}\le C(|\eta (t)|+|\omega (t)|)\le C\int _B|U(y,t)|\,dy\le C\Vert U(t)\Vert _{q,{\mathbb {R}}^3}\le C\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.56)

and, from (2.17), that

$$\begin{aligned} \Vert \nabla U(t)\Vert _{r,B}\le C|\omega (t)|\le C\Vert F\Vert _{q,{\mathbb {R}}^3}. \end{aligned}$$

(3.57)

Estimates (3.55) and (3.56) imply (2.51) for $t-s\le \tau _*$ if $1<q\le r\le \infty \;(q\ne \infty )$ and $1/q-1/r< 1/3$, but the latter restriction can be eventually removed by using the semigroup property (2.47). Then (3.54) together with (2.51) for $t-s\le \tau _*$ leads us to

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{r,\Omega }\le C(t-s)^{-1/2-(3/q-3/r)/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

for $t-s\le \tau _*$, which along with (3.57) concludes (2.52) for such $t,\, s$.

It remains to show (3.51). If $F\in {{\mathcal {E}}}(\mathbb R^3)\subset D_q(A),\, 1<q<\infty$, then it follows from (2.39), (2.41) and the boundedness

$$\begin{aligned} \Vert L_+(t)T(t,s)(k+L_+(s))^{-1}\Vert _{{{\mathcal {L}}}(X_q({\mathbb {R}}^3))}\le C \end{aligned}$$

near $t=s$ ([48, Chapter 5, Theorem 2.1]), where $k>0$ is fixed large enough, that

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{q,\Omega }\le C\big (\Vert AF\Vert _{q,\mathbb R^3}+\Vert F\Vert _{q,{\mathbb {R}}^3}\big ) \end{aligned}$$

near $t=s$. This combined with (3.57) (with $r=q$) implies that

$$\begin{aligned} \lim _{t\rightarrow s}\; (t-s)^{1/2}\Vert \nabla T(t,s)F\Vert _{q,{\mathbb {R}}^3}=0 \end{aligned}$$

for every $F\in {{\mathcal {E}}}({\mathbb {R}}^3)$ and, therefore, every $F\in X_q({\mathbb {R}}^3)$ since ${{\mathcal {E}}}({\mathbb {R}}^3)$ is dense in $X_q({\mathbb {R}}^3)$, see Proposition 3.1. The other behaviors in (3.51) are verified easier. The proof is complete. $\square$

We next derive the Hölder estimate of the evolution operator. This plays a role to verify that a mild solution becomes a strong one to the nonlinear initial value problem in Sect. 5. The argument is similar to the one for the autonomous case based on the theory of analytic semigroups, but not completely the same. It can be found, for instance, in the paper [49, Lemma 2.8] by Teramoto who used the fractional powers of generators, however, in order to justify the argument there, one has to deduce estimate of $\Vert (k+L_+(t))^\alpha (k+L_+(s))^{-\alpha }\Vert _{\mathcal {L}(X_q({\mathbb {R}}^3))}$ independent of (t, s), where $k>0$ is fixed large enough, as pointed out by Farwig and Tsuda [11, Lemma 3.6]. In the latter literature, the authors discuss the desired estimate of the fractional powers by use of bounded ${{\mathcal {H}}}^\infty$-calculus of generators uniformly in t. Instead, in this paper, we take easier way to deduce the following result, which is enough for later use in Sect. 5.

Proposition 3.5

Suppose (2.33) and (2.34). Let $j\in \{0,\,1\}$, $1<q<\infty$, $r\in [q,\infty ]$ (except $r=\infty$ for $j=1$) and $\tau _*,\alpha _0,\beta _0\in (0,\infty )$. Assume that q and r satisfy

$$\begin{aligned} \frac{1}{q}-\frac{1}{r}<\frac{2-j}{3}\qquad (j=0,\,1). \end{aligned}$$

(3.58)

Given $\mu$ satisfying

$$\begin{aligned} 0<\mu <1-\frac{j}{2}-\frac{3}{2}\left( \frac{1}{q}-\frac{1}{r}\right) , \end{aligned}$$

(3.59)

set

$$\begin{aligned} \kappa =\max \left\{ \frac{j}{2}+\frac{3}{2}\left( \frac{1}{q}-\frac{1}{r}\right) +\mu ,\;\frac{1}{2}\right\} . \end{aligned}$$

Then there is a constant $C=C(\mu ,q,r,\tau _*,\alpha _0,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert \nabla ^j T(t+h,s)F-\nabla ^j T(t,s)F\Vert _{r,{\mathbb {R}}^3} \le C(t-s)^{-\kappa }h^\mu \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.60)

for all (t, s) with $t-s\in (0,\tau _*]$, $h\in (0,1)$ and $F\in X_q({\mathbb {R}}^3)$, whenever $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, where $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35).

Proof

Set $U(t)=T(t,s)F$, that satisfies the equation

$$\begin{aligned} U(t)=e^{-(t-s)A}F-V(t), \qquad V(t)=\int _s^t e^{-(t-\tau )A}B(\tau )U(\tau )\,d\tau , \end{aligned}$$

in terms of the fluid–structure semigroup $e^{-tA}$. By $L^q$–$L^r$ estimates of the semigroup $e^{-tA}$ due to [10] we get

$$\begin{aligned} \Vert \nabla ^j \big (e^{-(t+h-s)A}-e^{-(t-s)A}\big )F\Vert _{r,{\mathbb {R}}^3} \le C(t-s)^{-j/2-(3/q-3/r)/2-\mu }h^\mu \Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.61)

for every $\mu \in (0,1)$. Proposition 3.4 implies that

$$\begin{aligned} \Vert B(t)U(t)\Vert _{q,{\mathbb {R}}^3}\le C\Vert U_b\Vert (t-s)^{-1/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

which together with (3.61) leads to

$$\begin{aligned} \begin{aligned}&\Vert \nabla ^j V(t+h)-\nabla ^j V(t)\Vert _{r,{\mathbb {R}}^3} \\&\quad \le C\Vert U_b\Vert \Vert F\Vert _{q,{\mathbb {R}}^3} \Big [(t-s)^{1/2-j/2-(3/q-3/r)/2-\mu }\,h^\mu +(t-s)^{-1/2}\,h^{1-j/2-(3/q-3/r)/2} \Big ] \end{aligned} \end{aligned}$$

with $\mu$ satisfying (3.59). In view of this and (3.61), we conclude (3.60). $\square$

3.6 Estimate of the Pressure

In this subsection we study a smoothing rate near the initial time of the pressure associated with T(t, s)F. As in [28], this must be an important issue at the final stage of the proof of decay estimates of T(t, s)F because of the non-autonomous character. Indeed, this circumstance differs from that for the autonomous case [10] in which several advantages of analytic semigroups are used. For our purpose, we need to discuss the domains of fractional powers of the Stokes-structure operator A through the behavior of the resolvent of the other operator ${\mathbb {A}}$. We start with the following lemma.

Lemma 3.2

Suppose that ${\mathbb {A}}$ is the operator given by (3.40). Let $1<q<\infty$ and $\vartheta \in (0,1/2q)$. Then there is a constant $C=C(q,\vartheta )>0$ such that

$$\begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}})^{-1}G\Vert _{Y_q}\le C\lambda ^{-\vartheta }\big (\Vert g\Vert _{W^{1,q}(\Omega )}+\Vert G\Vert _{Y_q}\big ) \end{aligned}$$

(3.62)

for all $\lambda \ge 1$ and $G=(g,\kappa ,\mu )\in Y_q=L^q_\sigma (\Omega )\times {\mathbb {C}}^3\times {\mathbb {C}}^3$ with $g\in W^{1,q}(\Omega )$.

Proof

Since we are going to discuss the behavior of the resolvent for $\lambda \rightarrow \infty$ only along the real half line, we may fix, for instance, $\varepsilon =\pi /4$ and assume that $\lambda \ge \Lambda _{\pi /4}$. Then we know the representation (3.48) of $(\lambda +{\mathbb {A}})^{-1}$, which leads to

$$\begin{aligned} {\mathbb {A}}(\lambda +{\mathbb {A}})^{-1}G={\mathbb {A}}(\lambda +\mathbb A_0)^{-1}(G+H(\lambda )) \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} H(\lambda )= \left( \begin{array}{c} h(\lambda ) \\ \big (\alpha (\lambda ),\beta (\lambda )\big ) \end{array} \right)&:=\sum _{j=1}^\infty \big [-{\mathbb {A}}_1(\lambda +{\mathbb {A}}_0)^{-1}\big ]^j\,G \\&=-{\mathbb {A}}_1(\lambda +{\mathbb {A}}_0)^{-1}(G+H(\lambda )) \end{aligned} \end{aligned}$$

(3.63)

and

$$\begin{aligned} \Vert G+H(\lambda )\Vert _{Y_q}\le \sum _{j=0}^\infty \Vert \mathbb A_1(\lambda +{\mathbb {A}}_0)^{-1}\Vert _{{{\mathcal {L}}}(Y_q)}^j\Vert G\Vert _{Y_q} \le 2\Vert G\Vert _{Y_q}. \end{aligned}$$

(3.64)

In view of (3.40) and (3.44) we find

$$\begin{aligned} \begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}}_0)^{-1}G\Vert _{Y_q}&\le \Vert A_\Omega w(\lambda )\Vert _{q,\Omega } +\lambda ^{-1}\Vert P_\Omega \Delta \ell (\kappa ,\mu )\Vert _{q,\Omega } \\&\quad +\Vert P_\Omega \ell \big (K^{-1}Q_1w(\lambda )+\lambda ^{-1}K^{-1}Q_2(\kappa ,\mu )\big )\Vert _{q,\Omega } \\&\quad +|K^{-1}Q_1w(\lambda )|_{{\mathbb {C}}^3\times \mathbb C^3}+\lambda ^{-1}|K^{-1}Q_2(\kappa ,\mu )|_{{\mathbb {C}}^3\times \mathbb C^3} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} w(\lambda ):=(\lambda +A_\Omega )^{-1}g+\lambda ^{-1}(\lambda +A_\Omega )^{-1}P_\Omega \Delta \ell (\kappa ,\mu ). \end{aligned}$$

From (3.29), (3.46) and the resolvent estimate of $A_\Omega$ it follows that

$$\begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}}_0)^{-1}G\Vert _{Y_q} \le C\Vert A_\Omega (\lambda +A_\Omega )^{-1}g\Vert _{q,\Omega }+C\lambda ^{-1}\big (\Vert g\Vert _{q,\Omega }+|(\kappa ,\mu )|\big ). \end{aligned}$$

(3.65)

We here recall the fact that, except the vanishing normal trace, the space $D_q(A_\Omega ^\vartheta )$ does not involve any boundary condition (Noll and Saal [40, Sect. 2.3], Giga and Sohr [23]) and, thereby,

$$\begin{aligned} g\in L^q_\sigma (\Omega )\cap W^{1,q}(\Omega )\subset L^q_\sigma (\Omega )\cap H_q^{2\vartheta }(\Omega ) =[L^q_\sigma (\Omega ), D_q(A_\Omega )]_\vartheta =D_q(A_\Omega ^\vartheta ) \end{aligned}$$

provided $\vartheta \in (0,1/2q)$, where $[\cdot ,\cdot ]_\vartheta$ stands for the complex interpolation functor and $H^{2\vartheta }_q(\Omega ):=[L^q(\Omega ), W^{2,q}(\Omega )]_\vartheta$ is the Bessel potential space. For such $\vartheta$, we infer

$$\begin{aligned} \begin{aligned} \Vert A_\Omega (\lambda +A_\Omega )^{-1}g\Vert _{q,\Omega }&=\Vert A_\Omega ^{1-\vartheta }(\lambda +A_\Omega )^{-1}A_\Omega ^\vartheta g\Vert _{q,\Omega } \\&\le C\Vert A_\Omega (\lambda +A_\Omega )^{-1}A_\Omega ^\vartheta g\Vert _{q,\Omega }^{1-\vartheta } \Vert (\lambda +A_\Omega )^{-1}A_\Omega ^\vartheta g\Vert _{q,\Omega }^\vartheta \\&\le C\lambda ^{-\vartheta }\Vert A_\Omega ^\vartheta g\Vert _{q,\Omega } \\&\le C\lambda ^{-\vartheta }\Vert g\Vert _{W^{1,q}(\Omega )} \end{aligned} \end{aligned}$$

which combined with (3.65) implies that

$$\begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}}_0)^{-1}G\Vert _{Y_q} \le C\lambda ^{-\vartheta }\big (\Vert g\Vert _{W^{1,q}(\Omega )}+\Vert G\Vert _{Y_q}\big ) \end{aligned}$$

(3.66)

for all $\lambda \ge \Lambda _{\pi /4}$.

It remains to show the other estimate

$$\begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}}_0)^{-1}H(\lambda )\Vert _{Y_q}\le C\lambda ^{-\vartheta }\Vert G\Vert _{Y_q} \end{aligned}$$

(3.67)

in which the additional condition $g\in W^{1,q}(\Omega )$ is not needed. In fact, exactly by the same argument as above for deduction of (3.66), we furnish

$$\begin{aligned} \begin{aligned} \Vert {\mathbb {A}}(\lambda +{\mathbb {A}}_0)^{-1}H(\lambda )\Vert _{Y_q}&\le C\lambda ^{-\vartheta }\big (\Vert h(\lambda )\Vert _{W^{1,q}(\Omega )}+\Vert H(\lambda )\Vert _{Y_q}\big ) \\&\le C\lambda ^{-\vartheta }\big (\Vert h(\lambda )\Vert _{W^{1,q}(\Omega )}+\Vert G\Vert _{Y_q}\big ). \end{aligned} \end{aligned}$$

(3.68)

By the representation (3.63) of $H(\lambda )$ along with (3.43)–(3.44) we find

$$\begin{aligned} \Vert h(\lambda )\Vert _{W^{1,q}(\Omega )} =\Vert P_\Omega \ell \big (K^{-1}Q_1 v(\lambda )+\lambda ^{-1}K^{-1}Q_2(\kappa +\alpha (\lambda ),\mu +\beta (\lambda ))\big )\Vert _{W^{1,q}(\Omega )} \end{aligned}$$

where

$$\begin{aligned} v(\lambda ):=(\lambda +A_\Omega )^{-1}(g+h(\lambda ))+\lambda ^{-1}(\lambda +A_\Omega )^{-1}P_\Omega \Delta \ell (\kappa +\alpha (\lambda ),\mu +\beta (\lambda )). \end{aligned}$$

Since $P_\Omega$ is bounded on $W^{1,q}(\Omega )$, we use (3.29), (3.46) and (3.64) to observe

$$\begin{aligned} \Vert h(\lambda )\Vert _{W^{1,q}(\Omega )} \le C\Vert G+H(\lambda )\Vert _{Y_q} \le C\Vert G\Vert _{Y_q} \end{aligned}$$

with some $C>0$ independent of $\lambda \ge \Lambda _{\pi /4}$. In this way, we conclude (3.67). The proof is complete. $\square$

The behavior of the resolvent with respect to $\lambda$ obtained in Lemma 3.2 is inherited by the Stokes-structure operator to conclude the following lemma.

Lemma 3.3

Suppose that A is the operator given by (2.25). Let $1<q<\infty$ and $\vartheta \in (0,1/2q)$. Then there is a constant $C=C(q,\vartheta )>0$ such that

$$\begin{aligned} \Vert A(\lambda +A)^{-1}F\Vert _{q,{\mathbb {R}}^3}\le C\lambda ^{-\vartheta }\big (\Vert f\Vert _{W^{1,q}(\Omega )}+\Vert F\Vert _{q,\mathbb R^3}\big ) \end{aligned}$$

(3.69)

for all $\lambda \ge 1$ and $F\in X_q({\mathbb {R}}^3)$ with $f=F|_\Omega \in W^{1,q}(\Omega )$.

Proof

Set $(f,\kappa ,\mu )=i(F)$, see (2.7), and

$$\begin{aligned} U=(\lambda +A)^{-1}F, \qquad (u,\eta ,\omega )=i(U). \end{aligned}$$

Following Sect. 3.3, we take the lifting function $U_0=\ell (\eta ,\omega )$ as in (3.4). Then we have

$$\begin{aligned} A(\lambda +A)^{-1}F =F-\lambda U=(f-\lambda u)\chi _\Omega +\big \{(\kappa -\lambda \eta )+(\mu -\lambda \omega )\times x\big \}\chi _B \end{aligned}$$

so that

$$\begin{aligned} \begin{aligned} \quad \Vert A(\lambda +A)^{-1}F\Vert _{q,{\mathbb {R}}^3}&\le C\Vert f-\lambda u\Vert _{q,\Omega }+C|(\kappa ,\mu )-\lambda (\eta ,\omega )| \\&\le C\Vert f-\ell (\kappa ,\mu )-\lambda \big (u-\ell (\eta ,\omega )\big )\Vert _{q,\Omega } +C|(\kappa ,\mu )-\lambda (\eta ,\omega )| \end{aligned} \end{aligned}$$

(3.70)

on account of (3.29). By virtue of (3.41) in the other formulation, we obtain

$$\begin{aligned} \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) =(\lambda +{\mathbb {A}})^{-1} \left( \begin{array}{c} g \\ (\kappa ,\mu ) \end{array} \right) \end{aligned}$$

with

$$\begin{aligned} {\widetilde{u}}=u-\ell (\eta ,\omega ), \qquad g=f-\ell (\kappa ,\mu ). \end{aligned}$$

By the assumption $f\in W^{1,q}(\Omega )$ we have $g\in W^{1,q}(\Omega )$ with

$$\begin{aligned} \Vert \nabla g\Vert _{q,\Omega }\le \Vert \nabla f\Vert _{q,\Omega }+C|(\kappa ,\mu )| \end{aligned}$$

and, therefore, we know (3.62) for

$$\begin{aligned} {\mathbb {A}}(\lambda +{\mathbb {A}})^{-1}G=G-\lambda \left( \begin{array}{c} {\widetilde{u}} \\ (\eta ,\omega ) \end{array} \right) , \qquad G=\left( \begin{array}{c} g \\ (\kappa ,\mu ) \end{array} \right) . \end{aligned}$$

Thus, in view of (3.70), we conclude (3.69). The proof is complete. $\square$

In the following lemma, we note that ${\mathbb {P}}\psi$ never fulfills the boundary condition at $\partial \Omega$ which is involved in $D_r(A)$, see (2.25), even if $\psi \in C_0^\infty (\Omega )\subset L^r_R({\mathbb {R}}^3)$ (by setting zero outside $\Omega$); thus, $\delta >0$ must be small in order that (3.71) holds true.

Lemma 3.4

Let $1<r<\infty$ and $\delta \in (0,1/2r)$. Then there is a constant $C=C(r,\delta )>0$ such that ${\mathbb {P}} \psi \in D_r(A^\delta )$ subject to

$$\begin{aligned} \Vert A^\delta {\mathbb {P}}\psi \Vert _{r,{\mathbb {R}}^3}\le C \big (\Vert \psi \Vert _{W^{1,r}(\Omega )}+\Vert \psi \Vert _{r,{\mathbb {R}}^3}\big ) \end{aligned}$$

(3.71)

for all $\psi \in L^r_R({\mathbb {R}}^3)$ with $\psi |_{\Omega }\in W^{1,r}(\Omega )$.

Proof

Given $\delta \in (0,1/2r)$, we take $\vartheta \in (\delta , 1/2r)$. Let $F\in X_r({\mathbb {R}}^3)$ with $f=F|_\Omega \in W^{1,r}(\Omega )$, then we have (3.69) with such $\vartheta$, which implies that $F\in D_r(A^\delta )$ subject to

$$\begin{aligned} \Vert A^\delta F\Vert _{r,{\mathbb {R}}^3} \le C\big (\Vert f\Vert _{W^{1,r}(\Omega )}+\Vert F\Vert _{r,{\mathbb {R}}^3}\big ) \end{aligned}$$

(3.72)

by following the argument in [52, Theorem 2.24]. In this literature, $0\in \rho (A)$ is assumed, however, it is not needed by considering $1+A$ instead of A as follows: In fact, by (3.69) with $\vartheta \in (\delta ,1/2r)$ we obtain

$$\begin{aligned} \int _1^\infty \lambda ^{-1+\delta }\Vert (1+A)(\lambda +1+A)^{-1}F\Vert _{r,\mathbb R^3}\,d\lambda \le C\big (\Vert f\Vert _{W^{1,r}(\Omega )}+\Vert F\Vert _{r,\mathbb R^3}\big ), \end{aligned}$$

while the integral over the interval (0, 1) of the same integrand always convergent. As a consequence,

$$\begin{aligned} \begin{aligned} (1+A)^{-1+\delta }F&=\frac{\sin \pi (1-\delta )}{\pi }\int _0^\infty \lambda ^{-1+\delta }(\lambda +1+A)^{-1}F\,d\lambda \\&=\frac{\sin \pi (1-\delta )}{\pi }(1+A)^{-1}\int _0^\infty \lambda ^{-1+\delta }(1+A)(\lambda +1+A)^{-1}F\,d\lambda \end{aligned} \end{aligned}$$

belongs to $D_r(A)$, yielding $F\in D_r(A^\delta )$ along with (3.72).

By (3.11) together with (3.9) we know that the projection ${\mathbb {P}}$ is bounded from $L^r_R({\mathbb {R}}^3)\cap W^{1,r}(\Omega )$ to $X_r({\mathbb {R}}^3)\cap W^{1,r}(\Omega )$. Thus, (3.72) implies (3.71). $\square$

We provide estimate of the pressure near the initial time, that would be of independent interest.

Proposition 3.6

Suppose (2.33) and (2.34). Let $1<q<\infty$ and $\tau _*,\, \alpha _0,\, \beta _0\in (0,\infty )$. Then, for every $\gamma \in \big ((1+1/q)/2,\, 1\big )$, there is a constant $C=C(\gamma ,q,\tau _*,\alpha _0,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert p(t)\Vert _{q,\Omega _3}+\Vert \partial _tT(t,s)F\Vert _{W^{-1,q}(\Omega _3)}\le C(t-s)^{-\gamma }\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.73)

for all (t, s) with $t-s\in (0,\tau _*]$ and $F\in X_q({\mathbb {R}}^3)$ whenever $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, where p(t) denotes the pressure associated with T(t, s)F and it is singled out in such a way that $\int _{\Omega _3}p(t)\,dx=0$ for each $t\in (s,\infty )$, while $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35).

Proof

Set $U(t)=T(t,s)F$ and $u(t)=U(t)|_\Omega$. We follow the approach developed by Noll and Saal [40, Lemma 13]. Given $\phi \in C_0^\infty (\Omega _3)$, we take $\psi :=\mathbb B[\phi -{\overline{\phi }}]\in W^{1,q^\prime }_0(\Omega _3)$ that solves

$$\begin{aligned} \begin{aligned} \text{ div }\; \psi =\phi -{\overline{\phi }} \quad \text { in }\; \Omega _3, \qquad \psi =0\quad \text{ on }\ \partial \Omega _3, \end{aligned} \end{aligned}$$

(3.74)

where ${\mathbb {B}}$ denotes the Bogovskii operator for the domain $\Omega _3$ and ${\overline{\phi }}:=\frac{1}{|\Omega _3|}\int _{\Omega _3}\phi (x)\,dx$. The boundary value problem (3.74) (in which $\phi -\overline{\phi }$ is replaced by g with the compatibility condition) admits many solutions, among which the Bogovskii operator

$$\begin{aligned} {\mathbb {B}}: W_0^{k,r}(\Omega _3)\rightarrow W_0^{k+1,r}(\Omega _3)^3, \qquad r\in (1,\infty ),\; k=0,1,2,\cdots \end{aligned}$$

(3.75)

specifies a particular solution discovered by Bogovskii [1] with fine regularity properties

$$\begin{aligned} \Vert \mathbb \nabla ^{k+1}\mathbb Bg\Vert _{r,\Omega _3} \le C\Vert \nabla ^k g\Vert _{r,\Omega _3}, \qquad \Vert \mathbb Bg\Vert _{r,\Omega _3} \le C\Vert g\Vert _{W^{1,r^\prime }(\Omega _3)^*} \end{aligned}$$

(3.76)

with some $C>0$ (dependent on the same $k,\, r$ as above) as well as $\hbox { div}\ \mathbb Bg=g$ provided $\int _{\Omega _3}g\,dx=0$, see [2, 16, 22] for the details. Note that $\mathbb Bg\in C_0^\infty (\Omega _3)^3$ for every $g\in C_0^\infty (\Omega _3)$ and that the right-hand side of the latter estimate of (3.76) cannot be replaced by $\Vert g\Vert _{W^{-1,r}(\Omega _3)}$.

We may understand $\psi \in W^{1,q^\prime }({\mathbb {R}}^3)\cap L^{q^\prime }_R({\mathbb {R}}^3)$ by setting zero outside $\Omega _3$ and use (3.76) to get

$$\begin{aligned} \Vert \psi \Vert _{W^{1,q^\prime }(\mathbb R^3)}=\Vert \psi \Vert _{W^{1,q^\prime }_0(\Omega _3)} \le C\Vert \phi -{\overline{\phi }}\Vert _{q^\prime ,\Omega _3} \le C\Vert \phi \Vert _{q^\prime ,\Omega _3}. \end{aligned}$$

(3.77)

By virtue of $\int _{\Omega _3}p(t)\,dx=0$, we obtain

$$\begin{aligned} \begin{aligned} \langle p(t),\phi \rangle _{\Omega _3}&=\langle p(t),\phi -{\overline{\phi }}\rangle _{\Omega _3} =\langle p(t),\text{ div } \psi \rangle _{\Omega _3} =-\langle \nabla p(t),\psi \rangle _{\Omega _3} \\&=\langle \partial _tu-\Delta u+(u_b-\eta _b)\cdot \nabla u, \psi \rangle _{\Omega _3} \end{aligned} \end{aligned}$$

for all $\phi \in C_0^\infty (\Omega _3)$. Since $\psi =0$ outside $\Omega _3$, we have

$$\begin{aligned} \langle \partial _tu,\psi \rangle _{\Omega _3}=\langle \partial _tU, \psi \rangle _{{\mathbb {R}}^3,\rho } =-\langle AU+B(t)U, \psi \rangle _{{\mathbb {R}}^3,\rho } \end{aligned}$$

and, thereby,

$$\begin{aligned} \begin{aligned} \langle p(t), \phi \rangle _{\Omega _3}&=-\langle AU,\psi \rangle _{{\mathbb {R}}^3,\rho }-\langle \Delta u,\psi \rangle _\Omega +\langle (1-{\mathbb {P}})\big [\{(u_b-\eta _b)\cdot \nabla u\}\chi _\Omega \big ], \psi \rangle _{{\mathbb {R}}^3,\rho } \\&=:I+II+III \end{aligned} \end{aligned}$$

(3.78)

for all $\phi \in C_0^\infty (\Omega _3)$. Note that the pairing over ${\mathbb {R}}^3$ should involve the constant weight $\rho$, otherwise one can use neither (2.29) nor (3.10). It follows from Proposition 3.4, (2.33) and (3.77) that

$$\begin{aligned} |III|\le C\Vert U_b\Vert \Vert \nabla u\Vert _{q,\Omega }\Vert \psi \Vert _{q^\prime ,\mathbb R^3} \le C(t-s)^{-1/2}\Vert U_b\Vert \Vert F\Vert _{q,\mathbb R^3}\Vert \phi \Vert _{q^\prime ,\Omega _3} \end{aligned}$$

(3.79)

and that

$$\begin{aligned} |II|\le \Vert \nabla u\Vert _{q,\Omega }\Vert \nabla \psi \Vert _{q^\prime ,\Omega } \le C(t-s)^{-1/2}\Vert F\Vert _{q,{\mathbb {R}}^3}\Vert \phi \Vert _{q^\prime ,\Omega _3} \end{aligned}$$

(3.80)

for all (t, s) with $t-s\le \tau _*$ and $F\in X_q({\mathbb {R}}^3)$.

By virtue of the momentum inequality for fractional powers with $\gamma \in (0,1)$ together with (2.41) and (3.52)–(3.53) we find

$$\begin{aligned} \Vert A^\gamma U\Vert _{q,{\mathbb {R}}^3} \le C\Vert AU\Vert _{q,{\mathbb {R}}^3}^\gamma \Vert U\Vert _{q,{\mathbb {R}}^3}^{1-\gamma } \le C(t-s)^{-\gamma }\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.81)

for all (t, s) with $t-s\le \tau _*$ and $F\in X_q({\mathbb {R}}^3)$. If in particular $\gamma \in \big ((1+1/q)/2,\, 1\big )$, then Lemma 3.4 with $r=q^\prime$, $\delta =1-\gamma \in (0,1/2q^\prime )$ and (3.77) for $\psi \in W^{1,q^\prime }({\mathbb {R}}^3)\cap L^{q^\prime }_R({\mathbb {R}}^3)$ imply

$$\begin{aligned} \Vert A^{1-\gamma }{\mathbb {P}}\psi \Vert _{q^\prime ,{\mathbb {R}}^3} \le C\big (\Vert \psi \Vert _{W^{1,q^\prime }(\Omega )}+\Vert \psi \Vert _{q^\prime ,\mathbb R^3}\big ) \le C\Vert \phi \Vert _{q^\prime ,\Omega _3} \end{aligned}$$

for all $\phi \in C_0^\infty (\Omega _3)$, which combined with (3.81) yields

$$\begin{aligned} |I|=|\langle A^\gamma U, A^{1-\gamma }{\mathbb {P}} \psi \rangle _{\mathbb R^3,\rho }| \le C(t-s)^{-\gamma }\Vert F\Vert _{q,\mathbb R^3}\Vert \phi \Vert _{q^\prime ,\Omega _3} \end{aligned}$$

(3.82)

for all (t, s) with $t-s\le \tau _*$ and $F\in X_q({\mathbb {R}}^3)$ by taking into account (2.29) and (3.10). We collect (3.78)–(3.80) and (3.82) to conclude (3.73) for the pressure.

It remains to show (3.73) for $\partial _tU(t)|_{\Omega _3}=\partial _tu(t)|_{\Omega _3}$, that readily follows from

$$\begin{aligned} \begin{aligned} |\langle \partial _tu, \Phi \rangle _{\Omega _3}|&=|\langle \Delta u+(\eta _b-u_b)\cdot \nabla u-\nabla p, \Phi \rangle _{\Omega _3}| \\&\le C\big (\Vert \nabla u\Vert _{q,\Omega _3}+\Vert U_b\Vert \Vert u\Vert _{q,\Omega _3}+\Vert p\Vert _{q,\Omega _3}\big )\Vert \nabla \Phi \Vert _{q^\prime ,\Omega _3} \end{aligned} \end{aligned}$$

for all $\Phi \in C_0^\infty (\Omega _3)^3$ together with estimate for the pressure obtained above and Proposition 3.4. The proof is complete. $\square$

Remark 3.1

Tolksdorf [50, Proposition 3.4] has made it clear that the smoothing rate $(t-s)^{-3/4}$ of the pressure is sharp within the $L^2$ theory of the Stokes semigroup in bounded domains through the behavior of the resolvent for $|\lambda |\rightarrow \infty$. This suggests that (3.73) would be almost optimal. The same rate as in (3.73) was discovered first by Noll and Saal [40] and it was slightly improved as $\gamma =(1+1/q)/2$ by [28, 30] for the exterior problem with prescribed rigid motions.

3.7 Adjoint Evolution Operator and Backward Problem

Let $t\in {\mathbb {R}}$ be a parameter as the final time of the problem below. For better understanding of the initial value problem (1.9) for the linearized system, it is useful to study the backward problem for the adjoint system subject to the final conditions at t:

$$\begin{aligned} \begin{aligned}&-\partial _sv=\Delta v-(\eta _b(s)-u_b(s))\cdot \nabla v+\nabla p_v, \qquad \text{ div }\; v =0 \quad \text { in }\; \Omega \times (-\infty ,t), \\ {}&v|_{\partial \Omega }=\eta +\omega \times y, \qquad v\rightarrow 0 \quad \text{ as }\ |y|\rightarrow \infty , \\ {}&-m\,\frac{d\eta }{ds}+\int _{\partial \Omega }{\mathbb {S}}(v,-p_v)\nu \,d\sigma =0, \\ {}&-J\,\frac{d\omega }{ds}+\int _{\partial \Omega }y\times {\mathbb {S}}(v,-p_v)\nu \,d\sigma =0, \\ {}&v(\cdot ,t)=v_0, \quad \eta (t)=\eta _0, \quad \omega (t)=\omega _0, \end{aligned} \end{aligned}$$

(3.83)

where v(y, s), $p_v(y,s)$, $\eta (s)$ and $\omega (s)$ are unknown functions. By using the Oseen-structure operator (2.36) together with the description (2.7)–(2.9), the backward problem (3.83) is formulated as

$$\begin{aligned} -\frac{dV}{ds}+L_-(s)V=0, \quad s\in (-\infty ,t);\qquad V(t)=V_0 \end{aligned}$$

(3.84)

for the monolithic velocity $V=v\chi _\Omega +(\eta +\omega \times y)\chi _B$, where $V_0=v_0\chi _\Omega +(\eta _0+\omega _0\times y)\chi _B$. Conversely, once we have a solution to (3.84), there exists a pressure $p_v$ which together with $(v,\eta ,\omega )$ solves (3.83) as in Sect. 3.2.

We begin with justification of the duality relation between the operators $L_\pm (t)$ within $X_q({\mathbb {R}}^3)^*=X_{q^\prime }(\mathbb R^3)$, see (3.8), and the dissipative structure of each of those operators. From Sect. 3.4 we know that $k+L_\pm (t)$ is bijective for large $k>0$, which combined with (3.85) below implies that $L_\pm (t)^*=L_\mp (t)$. The latter property (3.86) also plays a crucial role in Sect. 4.

Lemma 3.5

Suppose (2.33) and (2.34). Let $1<q<\infty$, then

$$\begin{aligned} \langle L_\pm (t)U,V\rangle _{{\mathbb {R}}^3,\rho }=\langle U,L_\mp (t)V\rangle _{{\mathbb {R}}^3,\rho } \end{aligned}$$

(3.85)

for all $U\in D_q(A)$, $V\in D_{q^\prime }(A)$ and $t\in {\mathbb {R}}$, where $\langle \cdot ,\cdot \rangle _{{\mathbb {R}}^3,\rho }$ is given by (2.11). Moreover, we have

$$\begin{aligned} \langle L_\pm (t)U, U\rangle _{{\mathbb {R}}^3,\rho }=2\Vert Du\Vert _{2,\Omega }^2 \end{aligned}$$

(3.86)

for all $U\in D_2(A)$ and $t\in {\mathbb {R}}$, where $u=U|_\Omega$.

Proof

The following fine properties of the Stokes-structure operator A is well-known, see Takahashi and Tucsnak [46, Sect. 4]:

$$\begin{aligned} \langle AU,V\rangle _{{\mathbb {R}}^3,\rho }=\langle U,AV\rangle _{\mathbb R^3,\rho } =2\langle Du, Dv\rangle _\Omega \end{aligned}$$

(3.87)

for all $U\in D_q(A)$ and $V\in D_{q^\prime }(A)$, where $u=U|_\Omega ,\, v=V|_\Omega$; in particular,

$$\begin{aligned} \langle AU,U\rangle _{{\mathbb {R}}^3,\rho }=2\Vert Du\Vert _{2,\Omega }^2 \end{aligned}$$

(3.88)

for all $U\in D_2(A)$. Computation of (3.88) is already done in the latter half of the proof of Proposition 3.3 of the present paper as well. We stress that the constant weight $\rho >0$ is needed in order that (3.87) and (3.88) hold true. It thus suffices to show that

$$\begin{aligned} \langle B(t)U,V\rangle _{{\mathbb {R}}^3,\rho }+\langle U,B(t)V\rangle _{{\mathbb {R}}^3,\rho }=0 \end{aligned}$$

(3.89)

for all U and V as above. We see from (3.10) that

$$\begin{aligned} \langle B(t)U,V\rangle _{{\mathbb {R}}^3,\rho } =\langle (u_b(t)-\eta _b(t))\cdot \nabla u, v\rangle _\Omega \qquad \end{aligned}$$

(3.90)

and the same relation for $\langle U,B(t)V\rangle _{\mathbb R^3,\rho }$. Let us use the same cut-off function $\phi _R$ as in (3.14). Since

$$\begin{aligned} \nu \cdot (u_b(t)-\eta _b(t))=x\cdot (\omega _b(t)\times x)=0, \qquad x\in \partial \Omega \end{aligned}$$

(3.91)

by (2.33), we find

$$\begin{aligned} \int _{\Omega }\text{ div } \big [(u\cdot v)(u_b(t)-\eta _b(t))\phi _R\big ]\,dx =\int _{\partial \Omega }(u\cdot v)\nu \cdot (u_b(t)-\eta _b(t))\,d\sigma =0 \end{aligned}$$

for all $u=U|_\Omega$ and $v=V|_\Omega$. By (2.33) again it is readily seen that

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _{2R<|x|<3R}|u||v||(u_b(t)-\eta _b(t))\cdot \nabla \phi _R|\,dx= 0, \end{aligned}$$

which implies

$$\begin{aligned} \int _\Omega (u_b(t)-\eta _b(t))\cdot \nabla (u\cdot v)\,dx=0. \end{aligned}$$

This combined with (3.90) concludes (3.89), which together with (3.87) leads us to (3.85). The relation (3.86) follows from (3.88) and (3.89) with $V=U\in D_2(A)$. The proof is complete. $\square$

Several remarks are in order.

Remark 3.2

It should be emphasized that Lemma 3.5 is not accomplished if the drift term in (1.9)/(3.83) is replaced by the purely Oseen term $\eta _b\cdot \nabla u$, see (3.91). This is why the other drift term $u_b\cdot \nabla u$ must be additionally involved into the right linearization. If the shape of the body is arbitrary, we see from (1.6) that the corresponding drift term is given by $(u_b-\eta _b-\omega _b\times x)\cdot \nabla u$, so that the boundary integral from this term vanishes as in the proof of Lemma 3.5.

Remark 3.3

If we wish to involve the term $(u-\eta )\cdot \nabla u_b$ as well into the linearization, we employ

$$\begin{aligned} |\langle (u-\eta )\cdot \nabla u_b(t), u\rangle _\Omega | \le C\left( \Vert u_b(t)\Vert _{L^{3,\infty }(\Omega )}+\Vert u_b(t)\Vert _{2,\Omega }\right) \Vert \nabla u\Vert _{2,\Omega }^2 \end{aligned}$$

with the aid of $|\eta |\le C\Vert Du\Vert _{2,\Omega }$, see for instance Galdi [15, Lemma 4.9], where $L^{3,\infty }(\Omega )$ stands for the weak-$L^3$ space. Hence the smallness of $u_b$ in $L^2(\Omega )$, that is more restrictive (from the viewpoint of summability at infinity) than (2.33), is needed to ensure the desired energy relation in Sect. 4.2. Indeed this is possible under the self-propelling condition, see Sect. 2.4, but it does not follow solely from the wake structure. Since we want to develop the theory under less assumption (2.33) on the basic motion, it is better not to put the term $(u-\eta )\cdot \nabla u_b$ in the linearization but to treat this term together with the nonlinear term.

Remark 3.4

If the shape of the body is arbitrary, the corresponding term to the one mentioned in Remark 3.3 is given by $(u-\eta -\omega \times x)\cdot \nabla u_b$ in view of (1.6). In order that the desired energy relation is available even if this term is involved into the linearization, we have to ask it to be $|x|u_b\in L^2(\Omega )$, which is too restrictive. In fact, $(\omega \times x)\cdot \nabla u_b$ is the worst term among all linear terms and thus we do need the specific shape already in linear analysis unlike [10].

Let us consider the auxiliary initial value problem

$$\begin{aligned} \frac{dW}{d\tau }+L_-(t-\tau )W=0, \quad \tau \in (s,\infty );\qquad W(s)=V_0, \end{aligned}$$

(3.92)

where $t\in {\mathbb {R}}$ is a parameter involved in the coefficient operator. By exactly the same argument as in Sects. 3.4 and 3.5, we see that the operator family $\{L_-(t-\tau );\,\tau \in {\mathbb {R}}\}$ generates an evolution operator $\{{\widetilde{T}}(\tau ,s;\,t);\, -\infty<s\le \tau <\infty \}$ on $X_q({\mathbb {R}}^3)$ for every $q\in (1,\infty )$ and that it satisfies the similar smoothing estimates to (2.51)–(2.52), in which the constants $C=C(\tau _*)$ are taken uniformly in $(\tau ,s)$ with $\tau -s\in (0,\tau _*]$ for given $\tau _*>0$ and do not depend on $t\in \mathbb R$. This implies (3.98) below on the evolution operator defined by

$$\begin{aligned} S(t,s):={\widetilde{T}}(t-s,0;\,t),\qquad -\infty<s\le t<\infty , \end{aligned}$$

(3.93)

which coincides with the adjoint of T(t, s), see (3.95) in the next lemma.

For every $V_0\in X_q({\mathbb {R}}^3)$, the function $V(s)=S(t,s)V_0$ is a solution to the backward problem (3.84), that is,

$$\begin{aligned} -\partial _sS(t,s)+L_-(s)S(t,s)=0, \quad s\in (-\infty ,t); \qquad S(t,t)={{\mathcal {I}}} \end{aligned}$$

(3.94)

in ${{\mathcal {L}}}(X_q({\mathbb {R}}^3))$. As we would expect, the following duality relation (3.95) holds true. Note that the latter assertion (3.96) does not follow directly from (3.93).

Lemma 3.6

Let $1<q<\infty$, then

$$\begin{aligned} T(t,s)^*=S(t,s), \qquad S(t,s)^*=T(t,s) \qquad \text{ in } \mathcal {L}(X_q({\mathbb {R}}^3)) \end{aligned}$$

(3.95)

for all (t, s) with $-\infty<s\le t<\infty$ in the sense of (3.97) below. Moreover, we have the backward semigroup property

$$\begin{aligned} S(\tau ,s)S(t,\tau )=S(t,s)\qquad (-\infty<s\le \tau \le t<\infty ) \end{aligned}$$

(3.96)

in ${{\mathcal {L}}}(X_q({\mathbb {R}}^3))$.

Proof

We fix (t, s) with $-\infty<s<t<\infty$. For every $\tau \in (s,t)$, it follows from (2.49), (3.85) and (3.94) that

$$\begin{aligned} \begin{aligned}&\partial _\tau \langle T(\tau ,s)F, S(t,\tau )G\rangle _{{\mathbb {R}}^3,\rho } \\&=\langle -L_+(\tau )T(\tau ,s)F, S(t,\tau )G\rangle _{{\mathbb {R}}^3,\rho } +\langle T(\tau ,s)F, L_-(\tau )S(t,\tau )G\rangle _{{\mathbb {R}}^3,\rho }=0 \end{aligned} \end{aligned}$$

for all $F\in X_q({\mathbb {R}}^3)$ and $G\in X_{q^\prime }(\mathbb R^3)$, which implies

$$\begin{aligned} \langle T(t,s)F, G\rangle _{{\mathbb {R}}^3,\rho } =\langle F, S(t,s)G\rangle _{{\mathbb {R}}^3,\rho } \end{aligned}$$

(3.97)

yielding (3.95). Then the semigroup property (2.47) of T(t, s) leads to (3.96), which completes the proof. $\square$

In the following proposition we provide the estimate of the associated pressure as well as

$$\begin{aligned} \Vert \nabla ^jT(t,s)^*G\Vert _{r,{\mathbb {R}}^3}\le C(t-s)^{-j/2-(3/q-3/r)/2} \Vert G\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.98)

near $s=t$. Indeed (3.98) with $j=0$ and $q,\,r\in (1,\infty )$ follows from the duality, but the other cases are obtained via estimates of ${\widetilde{T}}(\tau ,s;t)$, see (3.93). The proof of the estimates of the pressure and $\partial _sT(t,s)^*$ are exactly the same as in Proposition 3.6 and may be omitted.

Proposition 3.7

Suppose (2.33) and (2.34). Let

$$\begin{aligned} \begin{array}{ll} 1<q<\infty , \;\; q\le r\le \infty \qquad &{}\hbox { for}\ j=0, \\ 1<q\le r<\infty &{}\hbox { for}\ j=1. \end{array} \end{aligned}$$

Given $\tau _*,\, \alpha _0,\, \beta _0\in (0,\infty )$, estimate (3.98) holds with some constant $C=C(j,q,r,\tau _*,\alpha _0,\beta _0,\theta )>0$ for all (t, s) with $t-s\in (0,\tau _*]$ and $G\in X_q({\mathbb {R}}^3)$ whenever $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, where $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35).

Moreover, for every $\gamma \in \big ((1+1/q)/2,\, 1\big )$, there is a constant $C=C(\gamma ,q,\tau _*,\alpha _0,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert p_v(s)\Vert _{q,\Omega _3}+\Vert \partial _sT(t,s)^*G\Vert _{W^{-1,q}(\Omega _3)} \le C(t-s)^{-\gamma }\Vert G\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(3.99)

for all (t, s) with $t-s\in (0,\tau _*]$ and $G\in X^q({\mathbb {R}}^3)$ whenever $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, where $p_v(s)$ denotes the pressure associated with $v(s)=T(t,s)^*G$ and it is singled out in such a way that $\int _{\Omega _3}p_v(s)\,dy=0$ for each $s\in (-\infty ,t)$.

4 Decay Estimates of the Evolution Operator

In this section we study the large time behavior of the Oseen-structure evolution operator. Our argument is based on the following two ingredients: One is the decay estimates of the related evolution operator without the rigid body, the other is a consequence from the energy relation. The former is discussed in Sect. 4.1, whereas the latter is deduced in Sect. 4.2. Then the proof of (2.51) (except for the $L^\infty$-estimate) is given in Sect. 4.3. Toward the gradient estimate (as well as the $L^\infty$-estimate), the local energy decay property is established in Sect. 4.4, and successively the large time behavior near spatial infinity is studied in Sect. 4.5. The final subsection is devoted to completion of the linear theory.

4.1 Evolution Operator Without Rigid Body

In this subsection we consider the initial value problem for the fluid equation relating to (1.9) in the whole space without the rigid body, that is,

$$\begin{aligned} \begin{aligned}&\partial _tu=\Delta u+(\eta _b(t)-U_b(t))\cdot \nabla u-\nabla p, \quad \hbox { div}\ u=0, \quad (x,t)\in {\mathbb {R}}^3\times (s,\infty ), \\&u\rightarrow 0\quad \hbox { as}\ |x|\rightarrow \infty , \\&u(\cdot ,s)=\psi , \end{aligned} \end{aligned}$$

(4.1)

and the associated backward problem for the adjoint system

$$\begin{aligned} \begin{aligned}&-\partial _sv=\Delta v-(\eta _b(s)-U_b(s))\cdot \nabla v+\nabla p_v, \quad \hbox { div}\ v=0, \quad (y,s)\in {\mathbb {R}}^3\times (-\infty ,t), \\&v\rightarrow 0\quad \hbox { as}\ |y|\rightarrow \infty , \\&v(\cdot ,t)=\phi , \end{aligned} \end{aligned}$$

(4.2)

where the initial and final velocities are taken from the space $L^q_\sigma ({\mathbb {R}}^3)$ with some $q\in (1,\infty )$, see (2.21), while $U_b$ and $\eta _b$ are as in (2.32)–(2.34). The family $\{L_{0,\pm }(t);\, t\in {\mathbb {R}}\}$ of the modified Oseen operators on $L^q_\sigma ({\mathbb {R}}^3)$ is given by

$$\begin{aligned} \begin{aligned}&D_q(L_{0,\pm }(t))=L^q_\sigma ({\mathbb {R}}^3)\cap W^{2,q}({\mathbb {R}}^3), \\ {}&L_{0,\pm }(t)u=-{\mathbb {P}}_0\big [\Delta u\pm \big (\eta _b(t)-U_b(t)\big )\cdot \nabla u\big ], \end{aligned} \end{aligned}$$

(4.3)

where ${\mathbb {P}}_0$ denotes the classical Fujita-Kato projection, see (2.22). Then the same arguments as in subsections 3.4 and 3.7 show that the operator families $\{L_{0,+}(t);\, t\in {\mathbb {R}}\}$ and $\{L_{0,-}(t-\tau );\, \tau \in {\mathbb {R}}\}$ generate parabolic evolution operators $\{T_0(t,s);\, -\infty<s\le t<\infty \}$ and $\{\widetilde{T}_0(\tau ,s;\,t);\, -\infty<s\le \tau <\infty \}$, respectively, on $L^q_\sigma ({\mathbb {R}}^3)$ for every $q\in (1,\infty )$ and that the duality relation between the operators $L_{0,\pm }(t)$ as in Lemma 3.5 implies that $T_0(t,s)^*={\widetilde{T}}_0(t-s,0;\,t)$ is the solution operator to the backward problem (4.2).

Let $q\in (1,\infty )$ and $\tau _*,\, \alpha _0,\, \beta _0\in (0,\infty )$. On account of the globally Hölder condition (2.34), there is a constant $C=C(q,\tau _*,\alpha _0,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert \nabla T_0(t,s)\psi \Vert _{q,{\mathbb {R}}^3}\le & {} C(t-s)^{-1/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.4)

$$\begin{aligned} \Vert \nabla T_0(t,s)^*\phi \Vert _{q,{\mathbb {R}}^3}\le & {} C(t-s)^{-1/2}\Vert \phi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.5)

$$\begin{aligned} \Vert T_0(t,s)^*{\mathbb {P}}_0\hbox { div}\ \Phi \Vert _{q,{\mathbb {R}}^3}\le & {} C(t-s)^{-1/2} \Vert {\mathbb {P}}_0\Vert _{{{\mathcal {L}}}(L^q(\mathbb R^3))}\Vert \Phi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.6)

for all (t, s) with $t-s\in (0,\tau _*]$, $\psi ,\,\phi \in L^q_\sigma ({\mathbb {R}}^3)$ and $\Phi \in L^q({\mathbb {R}}^3)^{3\times 3}$ with $\hbox { div}\ \Phi \in \cup _{1<\sigma <\infty }L^\sigma (\mathbb R^3)$ as long as $\Vert U_b\Vert \le \alpha _0$ and $[U_b]_\theta \le \beta _0$, see (2.35), by the same argument as in Proposition 3.4 and by duality as to (4.6). Our task here is to deduce the large time behavior of $T_0(t,s)$ and $T_0(t,s)^*$. This is the stage in which the smallness of $\Vert U_b\Vert$ as well as the condition $q_0<3$ in (2.33) is needed.

The idea is to regard the problem (4.1) as perturbation from the Oseen evolution operator

$$\begin{aligned} \big (E(t,s)f\big )(x) =\int _{\mathbb R^3}G\left( x-y+\int _s^t\eta _b(\sigma )\,d\sigma ,\; t-s\right) f(y)\,dy \end{aligned}$$

(4.7)

which solves the non-autonomous Oseen initial problem

$$\begin{aligned} \begin{aligned}&\partial _tu=\Delta u+\eta _b(t)\cdot \nabla u-\nabla p, \quad \hbox { div}\ u=0, \quad (x,t) \in {\mathbb {R}}^3\times (s,\infty ), \\&u \rightarrow 0 \quad \hbox { as}\ |x|\rightarrow \infty , \\&u(\cdot ,s)=f, \end{aligned} \end{aligned}$$

(4.8)

provided that f is a solenoidal vector field, where

$$\begin{aligned} G(x,t)=(4\pi t)^{-3/2}e^{-|x|^2/4t} \end{aligned}$$

so that

$$\begin{aligned} e^{t\Delta }f=G(t)*f \end{aligned}$$

with $*$ being the convolution denotes the heat semigroup. Likewise, the problem (4.2) is viewed as perturbation from

$$\begin{aligned} \big (E(t,s)^*g\big )(y) =\int _{\mathbb R^3}G\left( x-y+\int _s^t\eta _b(\sigma )\,d\sigma ,\; t-s\right) g(x)\,dx. \end{aligned}$$

(4.9)

The heat semigroup enjoys the $L^q$–$L^r$ estimates ($q\le r$)

$$\begin{aligned} \Vert e^{t\Delta }f\Vert _{r,{\mathbb {R}}^3}\le & {} (4\pi t)^{-(3/q-3/r)/2}\Vert f\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.10)

$$\begin{aligned} \Vert \nabla e^{t\Delta }f\Vert _{r,{\mathbb {R}}^3}\le & {} 4(2\pi t)^{-1/2-(3/q-3/r)/2}\Vert f\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.11)

$$\begin{aligned} \Vert e^{t\Delta }{\mathbb {P}}_0\hbox { div}\ F\Vert _{r,{\mathbb {R}}^3}\le & {} 4(2\pi t)^{-1/2-(3/q-3/r)/2}\Vert {\mathbb {P}}_0\Vert _{{{\mathcal {L}}}(L^r(\mathbb R^3))}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.12)

for all $t>0$, $f\in L^q({\mathbb {R}}^3)$ and $F\in L^q(\mathbb R^3)^{3\times 3}$ with $\hbox { div}\ F\in \cup _{1<\sigma <\infty }L^\sigma ({\mathbb {R}}^3)$. In fact, (4.11) follows from $\Vert \nabla G(t)\Vert _{1,\mathbb R^3}=4(4\pi t)^{-1/2}$ and (4.10). As described in the right-hand sides of (4.10)–(4.11), the constants can be taken uniformly in q and r since those with the case $q=r$ give upper bounds (although the upper bound $\sqrt{8/\pi }$ for (4.11) is not sharp). This will be taken into account in the proof of the following proposition; for instance, the specific constant $c_0$ in (4.20) below is independent of q. Gradient estimate (4.11) implies (4.12) by duality and by using ${\mathbb {P}}_0e^{t\Delta }=e^{t\Delta }\mathbb P_0$, where the end-point case ($r=1,\,\infty$) is missing (although $q=1$ is allowed); indeed, this case can be actually covered and the additional condition $\text{ div } F\in \cup _{1<\sigma <\infty }L^\sigma ({\mathbb {R}}^3)$ is redundant if one makes use of estimate of the kernel function of the composite operator $e^{t\Delta }{\mathbb {P}}_0\text{ div }$ as in [39], however, this is not useful here because we wish to replace $e^{t\Delta }$ by the Oseen evolution operators (4.7) and (4.9). We also use the fact: For every $r_1,\, r_2$ with $1<r_1<r_2<\infty$, there is a constant $C(r_1,r_2)>0$ independent of $q\in [r_1,r_2]$ such that the Riesz transform satisfies

$$\begin{aligned} \sup _{r_1\le q\le r_2}\Vert {{\mathcal {R}}}\Vert _{{{\mathcal {L}}}(L^q(\mathbb R^3))} \le C(r_1,r_2), \end{aligned}$$

(4.13)

which follows simply from the Riesz-Thorin theorem, and thereby, the Fujita-Kato projection ${\mathbb {P}}_0={{\mathcal {I}}}+{{\mathcal {R}}}\otimes {{\mathcal {R}}}$ possesses the same property. It is obvious that both evolution operators E(t, s) and $E(t,s)^*$ fulfill the same estimates as in (4.10)–(4.12) with the same constants, in the right-hand sides of which t is of course replaced by $t-s$.

The following proposition provides us with (4.14) for $1<q\le r\le \infty \; (q\ne \infty )$ and (4.15) for $1<q\le r<\infty$ when the basic motion $U_b$ is small enough, however, the smallness is not uniform near the end-point $q=1$. The smallness of the basic motions together with the condition $q_0<3$ in (2.33) to develop the linear analysis is needed merely at the present stage. In fact, the small constant $\alpha _2$ in Theorem 2.1 is determined by the following proposition.

Proposition 4.1

Suppose (2.33) and (2.34). Given $\beta _0>0$, assume that $[U_b]_\theta \le \beta _0$. Given $r_1\in (1,4/3]$, there exist constants $\alpha _2(r_1,q_0)>0$ and $C=C(q,r,\alpha _2,\beta _0,\theta )>0$ such that if $\Vert U_b\Vert \le \alpha _2$, then the evolution operator $T_0(t,s)$ and the pressure p(t) associated with $u(t)=T_0(t,s)\psi$ enjoy

$$\begin{aligned} \Vert T_0(t,s)\psi \Vert _{r,{\mathbb {R}}^3}\le & {} C(t-s)^{-(3/q-3/r)/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.14)

$$\begin{aligned} \Vert \nabla T_0(t,s)\psi \Vert _{r,{\mathbb {R}}^3}\le & {} C(t-s)^{-1/2-(3/q-3/r)/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.15)

$$\begin{aligned} \Vert \nabla p(t)\Vert _{r,{\mathbb {R}}^3}\le & {} C(t-s)^{-1/2-(3/q-3/r)/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.16)

for all (t, s) with $t>s$ and $\psi \in L^q_\sigma ({\mathbb {R}}^3)$ as long as

$$\begin{aligned} \begin{aligned} \begin{array}{ll} r_1\le q<\infty ,\quad q\le r\le \infty \qquad &{}{}\text { for } (4.14), \\ r_1\le q\le r<\infty &{}{}\text { for } (4.15)-(4.16), \end{array} \end{aligned} \end{aligned}$$

where $\Vert U_b\Vert$ is given by (2.35).

The same assertions hold true for the adjoint $T_0(t,s)^*$ and the associated pressure $p_v(s)$ to (4.2) under the same smallness condition on $\Vert U_b\Vert$ as above.

Proof

The solution $u(t)=T_0(t,s)\psi$ of (4.1) with $\psi \in L^q_\sigma ({\mathbb {R}}^3)$ obeys

$$\begin{aligned} u(t)=E(t,s)\psi -\int _s^tE(t,\tau ){\mathbb {P}}_0(U_b\cdot \nabla u)(\tau )\,d\tau . \end{aligned}$$

(4.17)

Let $-\infty<s<t<\infty$ and set

$$\begin{aligned} M(t,s):=\sup _{\tau \in (s,t)}(\tau -s)^{1/2}\Vert \nabla u(\tau )\Vert _{q,{\mathbb {R}}^3}. \end{aligned}$$

From (4.4) we know $M(t,s)<\infty$ and

$$\begin{aligned} M(t,s)\le C\Vert \psi \Vert _{q,{\mathbb {R}}^3} \qquad \hbox { for}\ t-s\le 2. \end{aligned}$$

We are going to show that

$$\begin{aligned} M(t,s)\le C\Vert \psi \Vert _{q,{\mathbb {R}}^3} \qquad \hbox { for}\ t-s>2 \end{aligned}$$

(4.18)

with some constant $C>0$ independent of such (t, s) when $\Vert U_b\Vert$ is small enough. Let $t-s>2$. Since $u_b(t)\in L^\infty (\Omega )$, one may assume that $q_0\in [8/3,3)$ in (2.33). Then we have $U_b(t)\in X_{q_0}({\mathbb {R}}^3)\cap X_{6}({\mathbb {R}}^3)$ with

$$\begin{aligned} \Vert U_b(t)\Vert _{q_0,{\mathbb {R}}^3}+\Vert U_b(t)\Vert _{6,{\mathbb {R}}^3}\le C\Vert U_b\Vert \end{aligned}$$

by (2.35). The following argument is nowadays standard and it was traced back to Chen [4] in the nonlinear context (where the case $q=3$ is important), but it works merely for $q\in (3/2,\infty )$. We thus consider the case $q\in [2,\infty )$ so that

$$\begin{aligned} \frac{1}{q}+\frac{1}{q_0}\le \frac{7}{8} \end{aligned}$$

(4.19)

on account of $q_0\in [8/3,3)$ and, given $r_1\in (1,4/3]$, the other case $q\in [r_1,2]$ will be discussed later by duality. We use (4.17) and apply (4.11) in which $e^{t\Delta }$ is replaced by E(t, s) to find

$$\begin{aligned} \begin{aligned} \Vert \nabla u(t)\Vert _{q,{\mathbb {R}}^3}&\le C(t-s)^{-1/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \\&\quad +C\int _s^{t-1}(t-\tau )^{-1/2-3/2q_0}\Vert U_b(\tau )\Vert _{q_0,{\mathbb {R}}^3}\Vert \nabla u(\tau )\Vert _{q,{\mathbb {R}}^3}\,d\tau \\&\quad +C\int _{t-1}^t(t-\tau )^{-3/4}\Vert U_b(\tau )\Vert _{6,{\mathbb {R}}^3}\Vert \nabla u(\tau )\Vert _{q,{\mathbb {R}}^3}\,d\tau \\&\le C(t-s)^{-1/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} +c_0\Vert U_b\Vert (t-s)^{-1/2}M(t,s) \end{aligned} \end{aligned}$$

(4.20)

with some constant $c_0=c_0(q_0)>0$, which involves $\sup _{8/7\le r\le 6}\Vert {\mathbb {P}}_0\Vert _{{{\mathcal {L}}}(L^r({\mathbb {R}}^3))}$, see (4.13) and (4.19), and is independent of $q\in [2,\infty ]$. This readily follows by splitting the former integral into $\int _s^{(t+s)/2}+\int _{(t+s)/2}^{t-1}$ and by using $q_0<3$, see (2.33); in fact,

$$\begin{aligned} \begin{aligned}&\int _s^{(t+s)/2}\le \Vert U_b\Vert M(t,s)\left( \frac{t-s}{2}\right) ^{-1/2-3/2q_0}\int _s^{(t+s)/2}(\tau -s)^{-1/2}\,d\tau , \\&\int _{(t+s)/2}^{t-1}\le \Vert U_b\Vert M(t,s)\left( \frac{t-s}{2}\right) ^{-1/2}\int _1^\infty \tau ^{-1/2-3/2q_0}\,d\tau . \end{aligned} \end{aligned}$$

As a consequence, we obtain

$$\begin{aligned} M(t,s)\le C\Vert \psi \Vert _{q,{\mathbb {R}}^3}+c_0\Vert U_b\Vert M(t,s) \end{aligned}$$

for all (t, s) with $t-s>2$, which implies (4.18) and, therefore,

$$\begin{aligned} \Vert \nabla T_0(t,s)\psi \Vert _{q,{\mathbb {R}}^3}\le C(t-s)^{-1/2}\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.21)

for all $t>s$, $\psi \in L^q_\sigma ({\mathbb {R}}^3)$ and $q\in [2,\infty )$ when $\Vert U_b\Vert \le 1/2c_0$.

Let $1<r_1\le 4/3$. We turn to the case $q\in [r_1,2]$, so that $q^\prime \in [2,r_1^\prime ]$. To this end, by use of $\text{ div } U_b=0$, we consider the solution $v(s)=T_0(t,s)^*\phi$ to (4.2) in the form

$$\begin{aligned} v(s)=E(t,s)^*\phi +\int _s^t E(\tau ,s)^* {\mathbb {P}}_0\,\text{ div } (v\otimes U_b)(\tau )\,d\tau \end{aligned}$$

with the final velocity field of the specific form $\phi =\mathbb P_0\hbox { div}\ \Phi$, where $\Phi \in C_0^\infty (\mathbb R^3)^{3\times 3}$. Set

$$\begin{aligned} {\widetilde{M}}(t,s):=\sup _{\tau \in (s,t)}(t-\tau )^{1/2}\Vert v(\tau )\Vert _{q^\prime ,{\mathbb {R}}^3} \end{aligned}$$

which is finite and

$$\begin{aligned} {\widetilde{M}}(t,s)\le C\Vert \Phi \Vert _{q^\prime ,\mathbb R^3}\qquad \hbox { for}\ t-s\le 2 \end{aligned}$$

on account of (4.6). Let $t-s>2$. Applying (4.12) in which $e^{t\Delta }$ is replaced by $E(t,s)^*$, we find

$$\begin{aligned} \Vert v(s)\Vert _{q^\prime ,{\mathbb {R}}^3}\le C(t-s)^{-1/2}\Vert \Phi \Vert _{q^\prime ,{\mathbb {R}}^3} +c_0^\prime \Vert U_b\Vert (t-s)^{-1/2}{\widetilde{M}}(t,s) \end{aligned}$$

with some constant $c_0^\prime =c_0^\prime (r_1,q_0)>0$, which involves $\sup _{2\le r\le r_1^\prime }\Vert {\mathbb {P}}_0\Vert _{\mathcal {L}(L^r({\mathbb {R}}^3))}$ and does not depend on $q^\prime \in [2,r_1^\prime ]$, by the same splitting of the integral as in (4.20). Note that $c_0^\prime$ is increasing to $\infty$ when $r_1\rightarrow 1$. We thus obtain

$$\begin{aligned} \Vert T_0(t,s)^*{\mathbb {P}}_0\hbox { div}\ \Phi \Vert _{q^\prime ,{\mathbb {R}}^3} \le C(t-s)^{-1/2}\Vert \Phi \Vert _{q^\prime ,{\mathbb {R}}^3} \end{aligned}$$

(4.22)

for all $t>s$, $\Phi \in C_0^\infty ({\mathbb {R}}^3)^{3\times 3}$ and $q^\prime \in [2,r_1^\prime ]$ when $\Vert U_b\Vert \le 1/2c_0^\prime$. By continuity the composite operator $T_0(t,s)^*{\mathbb {P}}_0\text{ div }$ extends to a bounded operator on $L^{q^\prime }({\mathbb {R}}^3)^{3\times 3}$ with (4.22). By duality, we are led to (4.21) for $q\in [r_1,2]$ as well under the same smallness condition. Set

$$\begin{aligned} \alpha _2(r_1,q_0):=\min \left\{ \frac{1}{2c_0(q_0)},\; \frac{1}{2c_0^\prime (r_1,q_0)}\right\} , \end{aligned}$$

(4.23)

then we furnish (4.21) for all $t>s$ and $q\in [r_1,\infty )$ when $\Vert U_b\Vert \le \alpha _2$. By the aforementioned dependence of $c_0^\prime$ on $r_1$, we see that $\alpha _2$ is decreasing to zero when $r_1\rightarrow 1$.

Suppose still this smallness of $\Vert U_b\Vert$. We then use (4.17) and (4.21) to see immediately that

$$\begin{aligned} \begin{aligned} \Vert u(t)\Vert _{q,{\mathbb {R}}^3}&\le C\Vert \psi \Vert _{q,\mathbb R^3}+C\int _s^t(t-\tau )^{-1/2}\Vert U_b(\tau )\Vert _{3,{\mathbb {R}}^3} \Vert \nabla u(\tau )\Vert _{q,{\mathbb {R}}^3}\,d\tau \\&\le C(1+\Vert U_b\Vert )\Vert \psi \Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

(4.24)

yielding (4.14) with $r=q\in [r_1,\infty )$ for all $t>s$. By the interpolation inequality together with (4.24) as well as (4.21) for $q\in [r_1,\infty )$, we conclude (4.14) for $r\in [q,\infty ]$. Then, (4.15) for $r_1\le q\le r<\infty$ follows from (4.14) and (4.21) for $q\in [r_1,\infty )$ with the aid of the semigroup property.

By (4.1) we have

$$\begin{aligned} -\Delta p=\hbox { div}\ (U_b\cdot \nabla u) \end{aligned}$$

in ${\mathbb {R}}^3$, which leads us to

$$\begin{aligned} \nabla p=({{\mathcal {R}}}\otimes {{\mathcal {R}}})(U_b\cdot \nabla u), \end{aligned}$$

where ${{\mathcal {R}}}$ denotes the Riesz transform. From (2.35) and (4.13) it follows that

$$\begin{aligned} \Vert \nabla p(t)\Vert _{r,{\mathbb {R}}^3}\le C\Vert U_b\Vert \Vert \nabla T_0(t,s)\psi \Vert _{r,{\mathbb {R}}^3} \end{aligned}$$

which combined with (4.15) concludes (4.16).

Finally, one can deduce the same estimates of $T_0(t,s)^*$ and $T_0(t,s){\mathbb {P}}_0\text{ div }$ as in (4.21)–(4.22) to conclude the same result for the adjoint $T_0(t,s)^*$. The proof is complete. $\square$

4.2 Useful Estimates from Energy Relations

By the dissipative structure (3.86) along with (2.49) we find

$$\begin{aligned} \frac{1}{2}\,\partial _t\Vert T(t,s)F\Vert _{X_2(\mathbb R^3)}^2+2\Vert DT(t,s)F\Vert _{2,\Omega }^2=0 \end{aligned}$$

(4.25)

for every $F\in X_2({\mathbb {R}}^3)$. Recall that the energy (2.13) is written as

$$\begin{aligned} \Vert T(t,s)F\Vert _{X_2({\mathbb {R}}^3)}^2 =\Vert u(t)\Vert _{2,\Omega }^2 +m|\eta (t)|^2+\frac{2m}{5}|\omega (t)|^2 \end{aligned}$$

where $(u,\eta ,\omega )=i(U)$ with $U=T(t,s)F$, see (2.7). The $L^2$-norm we should adopt is not the usual one but the norm above to describe exactly the energy relation (4.25).

Likewise, we use (3.94) to observe

$$\begin{aligned} \frac{1}{2}\,\partial _s\Vert T(t,s)^*G\Vert _{X_2(\mathbb R^3)}^2=2\Vert DT(t,s)^*G\Vert _{2,\Omega }^2 \end{aligned}$$

(4.26)

for every $G\in X_2({\mathbb {R}}^3)$, where

$$\begin{aligned} \Vert T(t,s)^*G\Vert _{X_2({\mathbb {R}}^3)}^2=\Vert v(s)\Vert _{2,\Omega }^2 +m|\eta _v(s)|^2+\frac{2m}{5}|\omega _v(s)|^2 \end{aligned}$$

where $(v,\eta _v,\omega _v)=i(V)$ with $V=T(t,s)^*G$. This subsection shows that the energy relations (4.25)–(4.26) imply key inequalities (4.27)–(4.28) below, which are very useful in the next subsection, see (4.48) and (4.50).

Proposition 4.2

Suppose (2.33) and (2.34). Then there is a constant $C>0$ such that

$$\begin{aligned} \int _\sigma ^t\Vert T(\tau ,s)F\Vert _{2,\Omega _3}^2\,d\tau\le & {} C\Vert T(\sigma ,s)F\Vert _{2,{\mathbb {R}}^3}^2 \end{aligned}$$

(4.27)

$$\begin{aligned} \int _s^\sigma \Vert T(t,\tau )^*G\Vert _{2,\Omega _3}^2\,d\tau\le & {} C\Vert T(t,\sigma )^*G\Vert _{2,{\mathbb {R}}^3}^2 \end{aligned}$$

(4.28)

for all $F,\, G\in X_2({\mathbb {R}}^3)$ and $s<\sigma <t$.

Proof

The proof is easy, but we briefly show (4.28). Fix (t, s) with $s<t$ and set $V(\tau )=T(t,\tau )^*G$ for $\tau \in (s,t)$, then we observe

$$\begin{aligned} \Vert V(\tau )\Vert _{2,\Omega _3}\le C\Vert V(\tau )\Vert _{6,{\mathbb {R}}^3} \le C\Vert \nabla V(\tau )\Vert _{2,{\mathbb {R}}^3}. \end{aligned}$$

(4.29)

Since V is solenoidal, we have $\Delta V=\hbox { div}\ (2DV)$ in ${\mathbb {R}}^3$, which together with $V|_B\in \textrm{RM}$ yields

$$\begin{aligned} \Vert \nabla V(\tau )\Vert _{2,{\mathbb {R}}^3}^2=2\Vert Dv(\tau )\Vert _{2,\Omega }^2, \end{aligned}$$

(4.30)

where $v=V|_{\Omega }$. By (4.29) and (4.30) we are led to

$$\begin{aligned} C\Vert V(\tau )\Vert ^2_{2,\Omega _3}\le 2\Vert Dv(\tau )\Vert ^2_{2,\Omega }. \end{aligned}$$

(4.31)

We now employ the energy relation (4.26) and (2.12) to conclude (4.28) for every $\sigma \in (s,t)$, where $C>0$ is dependent only on $\rho$. The other estimate (4.27) is proved similarly. The proof is complete. $\square$

4.3 Proof of (2.51) Except for the Case $r=\infty$

This subsection is devoted to the proof of (2.51) for all $t>s$ when $1<q\le r<\infty$. This can be proved simultaneously with (3.98) for all $t>s$. $L^\infty$-estimate will be studied in Sects. 4.4 and 4.5. The essential step is to show the uniformly boundedness of both T(t, s) and $T(t,s)^*$ for every $r\in (2,\infty )$ and all (t, s) with $t>s$:

$$\begin{aligned}{} & {} \Vert T(t,s)F\Vert _{r,{\mathbb {R}}^3}\le C\Vert F\Vert _{r,{\mathbb {R}}^3}, \end{aligned}$$

(4.32)

$$\begin{aligned}{} & {} \Vert T(t,s)^*G\Vert _{r,{\mathbb {R}}^3}\le C\Vert G\Vert _{r,{\mathbb {R}}^3}. \end{aligned}$$

(4.33)

Since we know (4.32)–(4.33) for $t-s\le 3$ by Propositions 3.4 and 3.7, it suffices to derive them for $t-s>3$. In fact, we have the following lemma.

Lemma 4.1

Assume (2.33) and (2.34).

• 1. Suppose that, for some $r_0\in (2,\infty )$, estimate (4.32) with $r=r_0$ holds for all (t, s) with $t>s$ and $F\in \mathcal {E}({\mathbb {R}}^3)$, see (2.15).

  • (a) Let $2\le q\le r\le r_0$. Then we have (2.51) for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$.

  • (b) Let $r_0^\prime \le q\le r\le 2$. Then we have (3.98) with $j=0$ for all (t, s) with $t>s$ and $G\in X_q({\mathbb {R}}^3)$.

• 2. Suppose that, for some $r_0\in (2,\infty )$, estimate (4.33) with $r=r_0$ holds for all (t, s) with $t>s$ and $G\in {{\mathcal {E}}}({\mathbb {R}}^3)$.

  • (a) Let $2\le q\le r\le r_0$. Then we have (3.98) with $j=0$ for all (t, s) with $t>s$ and $G\in X_q({\mathbb {R}}^3)$.

  • (b) Let $r_0^\prime \le q\le r\le 2$. Then we have (2.51) for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$.

Proof

We show (b) of the first item, from which (a) follows by duality and by (2.12). The other item is proved in the same way.

The case $q=r=2$ is obvious by taking into account (2.12)–(2.13) since we have the energy relation (4.26). Fix $q\in [r_0^\prime ,2)$, then we obtain from the assumption together with (4.26) that

$$\begin{aligned} \Vert T(t,s)^*G\Vert _{q,{\mathbb {R}}^3}\le C\Vert G\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.34)

for all $t>s$ and $G\in X_q({\mathbb {R}}^3)$, in which $\mathcal {E}({\mathbb {R}}^3)$ is dense, see Proposition 3.1. From (4.34) and (4.30) with the aid of the interpolation inequality it follows that

$$\begin{aligned} \Vert T(t,s)^*G\Vert _{X_2({\mathbb {R}}^3)} \le C\Vert \nabla T(t,s)^*G\Vert _{2,{\mathbb {R}}^3}^\mu \Vert T(t,s)^*G\Vert _{q,{\mathbb {R}}^3}^{1-\mu } \le C\Vert DT(t,s)^*G\Vert _{2,\Omega }^\mu \Vert G\Vert _{q,{\mathbb {R}}^3}^{1-\mu } \end{aligned}$$

for all $G\in {{\mathcal {E}}}({\mathbb {R}}^3)\setminus \{0\}$, where $1/2=\mu /6+(1-\mu )/q$. We fix $t\in {\mathbb {R}}$ and combine the inequality above with (4.26) to find that $v(s)=T(t,s)^*G$ enjoys

$$\begin{aligned} \frac{d}{ds}\Vert v(s)\Vert _{X_2({\mathbb {R}}^3)}^2\ge \frac{C\Vert v(s)\Vert _{X_2({\mathbb {R}}^3)}^{2/\mu }}{\Vert G\Vert _{q,\mathbb R^3}^{2(1/\mu -1)}} \end{aligned}$$

for $s\in (-\infty , t)$, which implies that

$$\begin{aligned} \Vert v(s)\Vert _{2,{\mathbb {R}}^3}\le C\Vert v(s)\Vert _{X_2({\mathbb {R}}^3)} \le C(t-s)^{-\frac{\mu }{2(1-\mu )}}\Vert G\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

with

$$\begin{aligned} \frac{\mu }{2(1-\mu )}=\frac{3}{2}\left( \frac{1}{q}-\frac{1}{2}\right) . \end{aligned}$$

This together with (4.34) leads us to (2.51) for $r_0^\prime \le q\le r\le 2$. The proof is complete. $\square$

Proposition 4.3

Suppose (2.33) and (2.34). Given $\beta _0>0$, assume that $[U_b]_\theta \le \beta _0$. There is a constant $\alpha _1=\alpha _1(q_0)$ such that if $\Vert U_b\Vert \le \alpha _1$, then (3.98) with $j=0$ as well as (2.51) holds for all (t, s) with $t>s$ and $F,\, G\in X_q({\mathbb {R}}^3)$ provided that $1< q\le r<\infty$, where the constants C in those estimates depend on $q,r,\alpha _1,\beta _0,\theta$.

Proof

We follow the argument developed by [27, 29]. Set

$$\begin{aligned} \alpha _1=\alpha _1(q_0):=\alpha _2(4/3,q_0), \end{aligned}$$

(4.35)

where $\alpha _2$ is the constant given in Proposition 4.1, see also (4.23). In what follows we assume $\Vert U_b\Vert \le \alpha _1$. Let $2<r<\infty$. Given $F\in \mathcal {E}({\mathbb {R}}^3)$, see (2.15), we set $(f,\eta _f,\omega _f)=i(F)$, see (2.7). For the proof of (4.32), let us take $u_0(t)=T_0(t,s)F$ as the approximation near spatial infinity of $U(t)=T(t,s)F$. We fix a cut-off function $\phi \in C_0^\infty (B_3)$ as in (3.5). Let ${\mathbb {B}}$ be the Bogovskii operator for the domain $\Omega _3$, see (3.75). By (4.14) together with (3.76) we see that

$$\begin{aligned} U_0(t):=(1-\phi )u_0(t)+{\mathbb {B}}\left[ u_0(t)\cdot \nabla \phi \right] \quad \hbox { with}\ u_0(t)=T_0(t,s)F \end{aligned}$$

(4.36)

belongs to $X_r({\mathbb {R}}^3)$ and satisfies

$$\begin{aligned} \Vert U_0(t)\Vert _{r,{\mathbb {R}}^3}\le C\Vert F\Vert _{r,{\mathbb {R}}^3} \end{aligned}$$

(4.37)

for all (t, s) with $t>s$ since $\Vert U_b\Vert \le \alpha _1$. We fix the pressure p(t) associated with $U(t)=T(t,s)F$ and also choose the pressure $p_0(t)$ associated with $u_0(t)=T_0(t,s)F$ such that $\int _{\Omega _3}p_0(t)\,dx=0$, yielding

$$\begin{aligned} \Vert p_0(t)\Vert _{r,\Omega _3}\le C\Vert \nabla p_0(t)\Vert _{r,\Omega _3}. \end{aligned}$$

(4.38)

Let us define V and $p_v$ by

$$\begin{aligned} U(t)=U_0(t)+V(t), \qquad p(t)=(1-\phi )p_0(t)+p_v(t) \end{aligned}$$

(4.39)

and set $(v,\eta ,\omega )=i(V)$, see (2.7). Then the rigid motion $\eta +\omega \times x$ associated with V(t) is exactly the same as the one determined by $U(t)=T(t,s)F$ through (2.7) since $U(t)|_B=V(t)|_B$. We see that v, $p_v$, $\eta$ and $\omega$ obey

$$\begin{aligned} \begin{aligned}&\partial _tv=\Delta v+(\eta _b(t)-u_b(t))\cdot \nabla v-\nabla p_v+H(t), \qquad \text{ div }\; v =0 \quad \text { in }\; \Omega \times (s,\infty ), \\ {}&v|_{\partial \Omega }=\eta +\omega \times x, \qquad v\rightarrow 0 \quad \text{ as }\ |x|\rightarrow \infty , \\ {}&m\frac{d\eta }{dt}+\int _{\partial \Omega }{\mathbb {S}}(v,p_v)\nu \,d\sigma =0, \\ {}&J\frac{d\omega }{dt}+\int _{\partial \Omega } x\times {\mathbb {S}}(v,p_v)\nu \,d\sigma =0, \\ {}&v(\cdot ,s)=\phi f-{\mathbb {B}}[f\cdot \nabla \phi ], \quad \eta (s)=\eta _f, \quad \omega (s)=\omega _f, \end{aligned} \end{aligned}$$

(4.40)

where

$$\begin{aligned} \begin{aligned} H(t)&=-2\nabla \phi \cdot \nabla u_0(t)-\big [\Delta \phi +(\eta _b(t)-u_b(t))\cdot \nabla \phi \big ]\,u_0(t) \\&\quad +(\nabla \phi )p_0(t) +\big \{-\partial _t+\Delta +(\eta _b(t)-u_b(t))\cdot \nabla \big \}\mathbb B\left[ u_0(t)\cdot \nabla \phi \right] . \end{aligned} \end{aligned}$$

(4.41)

By the same symbol H(t) we denote its extension on ${\mathbb {R}}^3$ by setting zero outside $\Omega _3$, then $H(t)\in L^r_R(\mathbb R^3)$ for every $r\in (1,\infty )$. Furthermore, we find

$$\begin{aligned} \Vert H(t)\Vert _{r,\Omega _3} \le C(t-s)^{-1/2}(1+t-s)^{-3/2r+1/2}\Vert F\Vert _{r,{\mathbb {R}}^3} \end{aligned}$$

(4.42)

for every $r\in [4/3,\infty )$ (we are considering the case $r>2$) and all (t, s) with $t>s$ owing to Proposition 4.1 together with (3.76) since $\Vert U_b\Vert \le \alpha _1$. In fact, it follows from (4.16) and (4.38) that

$$\begin{aligned} \Vert p_0(t)\Vert _{r,\Omega _3}\le \left\{ \begin{array}{ll} \Vert \nabla p_0(t)\Vert _{r,{\mathbb {R}}^3}\le C(t-s)^{-1/2}\Vert F\Vert _{r,\mathbb R^3}\quad &{} (t-s\le 1), \\ \Vert \nabla p_0(t)\Vert _{\max \{r,3\},{\mathbb {R}}^3}\le C(t-s)^{-3/2r}\Vert F\Vert _{r,{\mathbb {R}}^3} &{} (t-s>1), \end{array} \right. \end{aligned}$$

(4.43)

and that

$$\begin{aligned} \begin{aligned}&\quad \Vert \partial _t{\mathbb {B}}\left[ u_0(t)\cdot \nabla \phi \right] \Vert _{r,\Omega _3} \\ {}&\qquad =\Vert {\mathbb {B}}\left[ \{\Delta u_0+(\eta _b-U_b)\cdot \nabla u_0-\nabla p_0\}\cdot \nabla \phi \right] \Vert _{r,\Omega _3} \\ {}&\qquad \le C\Vert \{\Delta u_0+(\eta _b-U_b)\cdot \nabla u_0-\nabla p_0\}\cdot \nabla \phi \Vert _{W^{1,r^\prime }(\Omega _3)^*} \\ {}&\qquad \le C\Vert \nabla u_0(t)\Vert _{r,\Omega _3}+C\Vert p_0(t)\Vert _{r,\Omega _3} \end{aligned} \end{aligned}$$

to which one can apply (4.15) and (4.43). The other terms are harmless. In this way, we obtain (4.42). Note that the smallness $\Vert U_b\Vert \le \alpha _1$ is needed only for large $(t-s)$, see Proposition 4.1; indeed, we have used (4.14) with $r=\infty$ and (4.15)–(4.16) with r replaced by $\max \{r,3\}$.

We formulate the problem (4.40) as

$$\begin{aligned} \frac{dV}{dt}+L_+(t)V=\mathbb PH(t), \quad t\in (s,\infty ); \qquad V(s)={\widetilde{F}} \end{aligned}$$

where

$$\begin{aligned} {\widetilde{F}}=\big (\phi f-\mathbb B[f\cdot \nabla \phi ]\big )\chi _\Omega +(\eta _f+\omega _f\times x)\chi _B \end{aligned}$$

(4.44)

whose support is contained in $B_3$ and which belongs to $X_q({\mathbb {R}}^3)$ for every $q\in (1,\infty )$. By use of the evolution operator T(t, s), we convert the problem above into

$$\begin{aligned} V(t)=T(t,s){\widetilde{F}}+\int _s^tT(t,\tau )\mathbb PH(\tau )\,d\tau . \end{aligned}$$

(4.45)

To make full use of the advantage that $H(\tau )$ is compactly supported, it is better to deal with the weak form

$$\begin{aligned} \langle V(t),\psi \rangle _{{\mathbb {R}}^3,\rho }=\langle {\widetilde{F}}, T(t,s)^*\psi \rangle _{{\mathbb {R}}^3,\rho } +\int _s^t\langle H(\tau ), T(t,\tau )^*\psi \rangle _{\Omega _3}\,d\tau \end{aligned}$$

for $\psi \in {{\mathcal {E}}}({\mathbb {R}}^3)$, where we have used

$$\begin{aligned} \langle T(t,\tau ){\mathbb {P}} H(\tau ), \psi \rangle _{{\mathbb {R}}^3,\rho } =\langle H(\tau ), T(t,\tau )^*\psi \rangle _{{\mathbb {R}}^3,\rho } =\langle H(\tau ), T(t,\tau )^*\psi \rangle _{\Omega _3} \end{aligned}$$

on account of (3.10) and (3.97), and employ the duality argument. Let $t-s>3$ and recall that $2<r<\infty$. It is readily seen from Proposition 3.7, (4.26) and (4.42) that

$$\begin{aligned} \begin{aligned}&|\langle {\widetilde{F}}, T(t,s)^*\psi \rangle _{\mathbb R^3,\rho }|+ \left| \left( \int _s^{s+1}+\int _{t-1}^t\right) \langle H(\tau ), T(t,\tau )^*\psi \rangle _{\Omega _3}\,d\tau \right| \\&\quad \le \Vert {\widetilde{F}}\Vert _{X_2(\mathbb R^3)}\Vert T(t,s)^*\psi \Vert _{X_2({\mathbb {R}}^3)} +\int _s^{s+1}\Vert H(\tau )\Vert _{2,\Omega _3}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}\,d\tau \\&\qquad +\int _{t-1}^t\Vert H(\tau )\Vert _{r,\Omega _3}\Vert T(t,\tau )^*\psi \Vert _{r^\prime ,\Omega _3}\,d\tau \\&\quad \le C\left( \Vert {\widetilde{F}}\Vert _{r,\Omega _3} +\int _s^{s+1}\Vert H(\tau )\Vert _{r,\Omega _3}\,d\tau \right) \Vert T(t,t-1)^*\psi \Vert _{X_2({\mathbb {R}}^3)} \\&\qquad +C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3}\int _{t-1}^t (\tau -s)^{-3/2r}\,d\tau \\&\quad \le C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3} \\&\quad \le C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{X_{r^\prime }({\mathbb {R}}^3)}. \end{aligned} \end{aligned}$$

(4.46)

Our main task is thus to discuss the term

$$\begin{aligned} J:=\int _{s+1}^{t-1}\langle H(\tau ), T(t,\tau )^*\psi \rangle _{\Omega _3}\,d\tau , \end{aligned}$$

for which we have

$$\begin{aligned} |J|\le \int _{s+1}^{t-1}\Vert H(\tau )\Vert _{2,\Omega _3}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}\,d\tau . \end{aligned}$$

(4.47)

We make use of (4.28) with $\sigma =t-1$ along with (4.42) for $r>2$ to find

$$\begin{aligned} |J|\le C\left( \int _{s+1}^{t-1}(\tau -s)^{-3/r}\,d\tau \right) ^{1/2}\Vert F\Vert _{r,\mathbb R^3}\Vert T(t,t-1)^*\psi \Vert _{2,{\mathbb {R}}^3} \end{aligned}$$

(4.48)

which combined with (3.98) with $\tau _*=1$ yields

$$\begin{aligned} |J|\le C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3} \le C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{X_{r^\prime }({\mathbb {R}}^3)} \end{aligned}$$

for $t-s>3$ and $\psi \in {{\mathcal {E}}}({\mathbb {R}}^3)$ provided $2<r<3$. This together with (4.46) and (4.37) imply (4.32) for $t-s>3$ and, therefore, for all (t, s) with $t>s$ since we already know (4.32) for $t-s\le 3$ from Proposition 3.4. Hence, by virtue of Lemma 4.1 we obtain (3.98) with $j=0$ for all (t, s) with $t>s$ and $G\in X_q({\mathbb {R}}^3)$ provided $3/2< q\le r\le 2$.

With this at hand, we proceed to the next step in which the case $r\in (3,6)$ is discussed. Given such r, we set

$$\begin{aligned} \mu =1-\frac{3}{r}, \qquad \frac{1}{q}-\frac{1}{2}=\frac{\mu }{3}, \end{aligned}$$

(4.49)

where $\mu$ comes from the growth rate $(t-s-1)^\mu$ of the integral of (4.48), which we intend to overcome by use of the decay property of the adjoint evolution operator. Then we have $\mu \in (0,1/2)$ and $q\in (3/2,2)$. We use (4.28) with $\sigma =(s+t)/2$, apply the result obtained in the previous step and recall (3.98) with $\tau _*=1$ to furnish

$$\begin{aligned} \begin{aligned} \int _{s+1}^{(s+t)/2}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}^2\,d\tau&\le C\Vert T(t,(s+t)/2)^*\psi \Vert _{2,{\mathbb {R}}^3}^2 \\&\le C(t-s-2)^{-\mu }\Vert T(t,t-1)^*\psi \Vert _{q,{\mathbb {R}}^3}^2 \\&\le C(t-s-2)^{-\mu }\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3}^2 \end{aligned} \end{aligned}$$

(4.50)

for $t-s>2$. Following (3.93), we set $W(t-\tau ):=T(t,\tau )^*\psi$ and $\sigma :=(t-s-2)/2$. We then rewrite (4.50) as

$$\begin{aligned} \int _{1+\sigma }^{1+2\sigma }\Vert W(\tau )\Vert _{2,\Omega _3}^2\,d\tau \le c_1\,\sigma ^{-\mu }\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3}^2 \end{aligned}$$

with some $c_1>0$ for all $\sigma >0$, from which one can deduce the following optimal growth estimate by dyadic splitting method developed in [27, Lemma 3.4]:

$$\begin{aligned} \begin{aligned} \int _{(s+t)/2}^{t-1}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}\,d\tau&=\int _1^{1+\sigma }\Vert W(\tau )\Vert _{2,\Omega _3}\,d\tau \\&=\sum _{j=0}^\infty \int _{1+\sigma /2^{j+1}}^{1+\sigma /2^j}\Vert W(\tau )\Vert _{2,\Omega _3}\,d\tau \\&\le \sqrt{c_1}\,\sigma ^{(1-\mu )/2}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3}\sum _{j=0}^\infty 2^{-(1-\mu )(j+1)/2} \\&=C(t-s-2)^{(1-\mu )/2}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

(4.51)

for $t-s>2$. We now split (4.47) into two parts as below which should be comparable with each other and then use (4.50)–(4.51) together with (4.42) to find

$$\begin{aligned} \begin{aligned} |J|&\le \int _{s+1}^{(s+t)/2}+\int _{(s+t)/2}^{t-1} \\&\le C(t-s)^{(1-3/r)/2}\Vert F\Vert _{r,{\mathbb {R}}^3}\left( \int _{s+1}^{(s+t)/2}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}^2\,d\tau \right) ^{1/2} \\&\quad +C(t-s)^{-3/2r}\Vert F\Vert _{r,{\mathbb {R}}^3}\int _{(s+t)/2}^{t-1}\Vert T(t,\tau )^*\psi \Vert _{2,\Omega _3}\,d\tau \\&\le C\Vert F\Vert _{r,{\mathbb {R}}^3}\Vert \psi \Vert _{r^\prime ,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for $t-s>3$, yielding (4.32) for all $t>s$ provided $3<r<6$. Then Lemma 4.1 concludes (3.98) with $j=0$ for all (t, s) with $t>s$ provided $6/5< q\le r\le 2$.

At the final stage, given $r\in (6,\infty )$, we take the same $\mu$ and q as in (4.49), then we observe $\mu \in (1/2,1)$ and $q\in (6/5,3/2)$. The same argument as above with better decay property of $T(t,s)^*$ obtained in the previous step implies (4.32) for all (t, s) with $t>s$ and, thereby, (2.51) for such (t, s) provided $2\le q\le r<\infty$ as well as (3.98) with $j=0$ for the same (t, s) provided $1< q\le r\le 2$.

The opposite case, that is, (2.51) for $1< q\le r\le 2$ and (3.98) with $j=0$ for $2\le q\le r<\infty$ can be discussed with the aid of (4.27) under the same condition $\Vert U_b\Vert \le \alpha _1$ in the similar fashion, see also the last part of [27, Sect. 4]. Finally, the remaining case $1< q<2<r<\infty$ for both estimates is obvious because of the semigroup properties (2.47) and (3.96). The proof is complete. $\square$

Remark 4.1

It would be remarkable that Proposition 4.3 is established under the smallness of $\Vert U_b\Vert$ uniformly in (q, r) regardless of the circumstances in Proposition 4.1. This is because one needs (4.14)–(4.16) only with the case $q>2$. The other remark concerns the constants in (2.51) and (3.98) with $j=0$. Let $\Vert U_b\Vert \le \alpha _0$, then, according to Proposition 3.4, the constant C in (2.51) near $t=s$ depends on $\alpha _0$. This is why the constant C in Proposition 4.3 is also dependent on $\alpha _1$. It is also the case in Propsitions 4.4 and 4.5 below.

4.4 Local Energy Decay Estimates

For the proof of (2.52), it suffices to show that

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{q,{\mathbb {R}}^3}\le C(t-s)^{-\min \{1/2,\, 3/2q\}}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.52)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$ on account of the semigroup property, (2.51) with $r<\infty$ and Proposition 3.4. As in [8, 10, 28,29,30, 33, 34], let us split (4.52) into estimates of $\Vert \nabla T(t,s)F\Vert _{q,B_3}$ and $\Vert \nabla T(t,s)F\Vert _{q,{\mathbb {R}}^3\setminus B_3}$. The former is given by the following proposition and it is called the local energy decay property, whereas the latter is studied in the next subsection. To discuss the latter, the local energy decay of $\partial _tT(t,s)F$ is also needed, see (4.54) below. The following proposition gives us (4.53)–(4.54) for $1<q<\infty$ when $\Vert U_b\Vert$ is small enough, however, the smallness is not uniform near $q=1$; in fact, this circumstance arises from Proposition 4.1.

Proposition 4.4

Suppose (2.33) and (2.34). Given $r_1\in (1,4/3]$ and $\beta _0\in (0,\infty )$, assume that $\Vert U_b\Vert \le \alpha _2\in (0,\alpha _1]$ and $[U_b]_\theta \le \beta _0$, where $\alpha _2=\alpha _2(r_1,q_0)$ and $\alpha _1=\alpha _1(q_0)=\alpha _2(4/3,q_0)$, see (4.35), are respectively the constants given in Propositions 4.1 and 4.3, while $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35). Then there is a constant $C=C(q,\alpha _2,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert T(t,s)F\Vert _{W^{1,q}(B_3)}\le & {} C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.53)

$$\begin{aligned} \Vert \partial _tT(t,s)F\Vert _{W^{-1,q}(\Omega _3)}+\Vert p(t)\Vert _{q,\Omega _3}\le & {} C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.54)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$ as long as $r_1\le q<\infty$, where p(t) denotes the pressure associated with T(t, s)F and it is singled out in such a way that $\int _{\Omega _3}p(t)\,dx=0$ for each $t\in (s,\infty )$.

Under less smallness condition $\Vert U_b\Vert \le \alpha _1$ than described above, that is, the same one as in Proposition 4.3, there is a constant $C=C(q,\alpha _1,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert T(t,s)F\Vert _{\infty ,B_3}\le C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.55)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$ as long as $1<q<\infty$.

The same assertions hold true for the adjoint $T(t,s)^*$ and the associated pressure $p_v(s)$ to (3.83) under the same smallness conditions on $\Vert U_b\Vert$ as above.

Proof

Given $q\in (1,\infty )$, let us take $r_0$ so large that

$$\begin{aligned} \max \left\{ \frac{2q}{q-1},\; q,\, 6\right\}<r_0<\infty , \end{aligned}$$

which yields

$$\begin{aligned} q>r_0^\prime , \qquad \kappa :=\frac{3}{2}\left( 1-\frac{2}{r_0}\right) >\max \left\{ \frac{3}{2q},\; 1\right\} . \end{aligned}$$

(4.56)

Let $\Vert U_b\Vert \le \alpha _1(q_0)$, then we have (2.51) except for the case $r=\infty$ by Proposition 4.3. If $H\in L^q_R({\mathbb {R}}^3)$ satisfies $H(x)=0$ a.e. ${\mathbb {R}}^3{\setminus } B_3$, then it follows from Proposition 3.4 and (3.52) with $\tau _*=1$ as well as Proposition 4.3 that

$$\begin{aligned} \begin{aligned} \quad \Vert T(t,s){\mathbb {P}} H\Vert _{W^{1,r_0}({\mathbb {R}}^3)} +\Vert L_+(t)T(t,s){\mathbb {P}} H\Vert _{r_0,{\mathbb {R}}^3}&\le C\Vert T(t-1,s){\mathbb {P}} H\Vert _{r_0,{\mathbb {R}}^3} \\&\le C(t-s-1)^{-(3/r_0^\prime -3/r_0)/2}\Vert {\mathbb {P}} H\Vert _{r_0^\prime ,{\mathbb {R}}^3} \\&\le C(t-s)^{-\kappa }\Vert H\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for $t-s>2$. This along with Propositions 3.4 and 3.6 implies that

$$\begin{aligned} \Vert T(t,s){\mathbb {P}} H\Vert _{W^{1,q}(B_3)}\le C(t-s)^{-1/2}(1+t-s)^{-\kappa +1/2}\Vert H\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.57)

and that

$$\begin{aligned} \Vert \partial _tT(t,s){\mathbb {P}} H\Vert _{W^{-1,q}(\Omega _3)} \le C(t-s)^{-\gamma }(1+t-s)^{-\kappa +\gamma }\Vert H\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.58)

for all (t, s) with $t>s$, where $\gamma \in \big ((1+1/q)/2,\,1\big )$ is fixed arbitrarily.

We fix $r_1\in (1,4/3]$ arbitrarily, and let $q\in [r_1,\infty )$. Given $F\in {{\mathcal {E}}}({\mathbb {R}}^3)$, which is dense in $X_q({\mathbb {R}}^3)$ (see Proposition 3.1), the function (4.36) and its temporal derivative $\partial _t U_0(t)$ enjoy the desired estimates as in (4.53)–(4.54) due to (4.14)–(4.16) and (3.76) (together with the equation (4.1) for $\partial _tT_0(t,s)F$) under the condition

$$\begin{aligned} \Vert U_b\Vert \le \alpha _2(r_1,q_0)\le \alpha _2(4/3,q_0)=\alpha _1(q_0), \end{aligned}$$

(4.59)

see (4.35) as well as (4.23). Our task is thus to estimate V(t) defined by (4.39). We use the integral equation (4.45) and its temporal derivative

$$\begin{aligned} \partial _tV(t)=\partial _tT(t,s){\widetilde{F}}+{\mathbb {P}} H(t)+\int _s^t \partial _tT(t,\tau ){\mathbb {P}} H(\tau )\,d\tau , \end{aligned}$$

(4.60)

in $W^{-1,q}(\Omega _3)$, where we can apply (4.57)–(4.58) to ${\widetilde{F}}$ given by (4.44) and also to $H(\tau )$ given by (4.41) since they vanish outside $B_3$. In view of the relation (4.56), we see at once that $T(t,s){\widetilde{F}}$ and $\partial _tT(t,s){\widetilde{F}}$ fulfill the desired estimates; moreover, so does the second term ${\mathbb {P}} H(t)$ of (4.60) already by (4.42), which holds for $r\in [r_1,\infty )$ under (4.59). From (4.57)–(4.58) together with (4.42) it follows that

$$\begin{aligned} \begin{aligned}&\int _s^t\Vert T(t,\tau ){\mathbb {P}} H(\tau )\Vert _{W^{1,q}(B_3)}\,d\tau \\&\quad \le C\Vert F\Vert _{q,{\mathbb {R}}^3}\left( \int _s^{(s+t)/2}+\int _{(s+t)/2}^t\right) \\&\qquad (t-\tau )^{-1/2}(1+t-\tau )^{-\kappa +1/2} (\tau -s)^{-1/2}(1+\tau -s)^{-3/2q+1/2}\,d\tau \\&\quad \le C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

and, similarly, that

$$\begin{aligned} \begin{aligned}&\int _s^t \Vert \partial _t T(t,\tau ){\mathbb {P}} H(\tau )\Vert _{W^{-1,q}(\Omega _3)}\,d\tau \\&\quad \le C\Vert F\Vert _{q,{\mathbb {R}}^3}\left( \int _s^{(s+t)/2}+\int _{(s+t)/2}^t\right) \\&\qquad (t-\tau )^{-\gamma }(1+t-\tau )^{-\kappa +\gamma } (\tau -s)^{-1/2}(1+\tau -s)^{-3/2q+1/2}\,d\tau \\&\quad \le C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for all (t, s) with $t-s>2$, which conclude (4.54) for $\partial _tT(t,s)F$ as well as (4.53) provided (4.59) is satisfied.

Since $\int _{\Omega _3}p(t)\,dx=0$, we see that

$$\begin{aligned} \Vert p(t)\Vert _{q,\Omega _3}\le C\Vert \nabla p(t)\Vert _{W^{-1,q}(\Omega _3)} \end{aligned}$$

from which together with the first equation of (1.9), (4.53), (4.54) for $\partial _tT(t,s)F$ and (2.33), we obtain (4.54) for the pressure under the same smallness (4.59).

Let $\Vert U_b\Vert \le \alpha _1(q_0)=\alpha _2(4/3,q_0)$, see (4.35), then (4.53) with $q>3$ implies (4.55) for the same q, which combined with Proposition 4.3 gives (4.55) for the other case $q\in (1,3]$ by the semigroup property (2.47). Finally, the argument for the adjoint $T(t,s)^*$ works essentially in the same manner. The proof is complete. $\square$

Propositions 3.4, 3.6 and 4.4 immediately lead us to the following corollary, that plays an important role in the next subsection.

Corollary 4.1

Assume the same conditions as in the first half of Proposition 4.4. Then, for every $\gamma \in \big ((1+1/q)/2,\,1\big )$, there are constants $C_1=C_1(\gamma ,q,\alpha _2,\beta _0,\theta )>0$ and $C_2=C_2(q,\alpha _2,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert \partial _tT(t,s)F\Vert _{W^{-1,q}(\Omega _3)}+\Vert p(t)\Vert _{q,\Omega _3}\le & {} C_1(t-s)^{-\gamma }(1+t-s)^{-3/2q+\gamma }\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.61)

$$\begin{aligned} \Vert T(t,s)F\Vert _{W^{1,q}(B_3)}\le & {} C_2(t-s)^{-1/2}(1+t-s)^{-3/2q+1/2}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.62)

for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$ as long as $r_1\le q<\infty$, where the associated pressure p(t) is chosen as in Proposition 4.4.

The same assertions hold true for the adjoint $T(t,s)^*$ and the associated pressure $p_v(s)$ to (3.83) under the same condition $\Vert U_b\Vert \le \alpha _2(r_1,q_0)$.

4.5 Large Time Behavior near Spatial Infinity

In this subsection we deduce the decay property of $\nabla T(t,s)$ near spatial infinity under the same conditions as in Proposition 4.4 to complete the proof of (4.52).

Proposition 4.5

Suppose (2.33) and (2.34). Given $r_1\in (1,4/3]$ and $\beta _0\in (0,\infty )$, assume that $\Vert U_b\Vert \le \alpha _2\in (0,\alpha _1]$ and $[U_b]_\theta \le \beta _0$, where $\alpha _2=\alpha _2(r_1,q_0)$ and $\alpha _1=\alpha _1(q_0)$ are the constants given in Propositions 4.1 and 4.3, while $\Vert U_b\Vert$ and $[U_b]_\theta$ are given by (2.35). Then there is a constant $C=C(q,\alpha _2,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{q,{\mathbb {R}}^3\setminus B_3}\le C(t-s)^{-\min \{1/2,\, 3/2q\}}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.63)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$ as long as $r_1\le q<\infty$.

Under the same condition $\Vert U_b\Vert \le \alpha _1$ as for (4.55), there is a constant $C=C(q,\alpha _1,\beta _0,\theta )>0$ such that

$$\begin{aligned} \Vert T(t,s)F\Vert _{\infty ,{\mathbb {R}}^3\setminus B_3}\le C(t-s)^{-3/2q}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.64)

for all (t, s) with $t-s>2$ and $F\in X_q({\mathbb {R}}^3)$ as long as $1<q<\infty$.

The same assertions hold true for the adjoint $T(t,s)^*$ under the same smallness conditions on $\Vert U_b\Vert$ as above.

Proof

Given $F\in {{\mathcal {E}}}({\mathbb {R}}^3)$, see (2.15), we set $U(t)=T(t,s)F$, $u(t)=U(t)|_\Omega$ and take the associated pressure p(t) such that $\int _{\Omega _3}p(t)\,dx=0$. Using the same cut-off function $\phi$ and the Bogovskii operator ${\mathbb {B}}$, see (3.5) and (3.75), as in the proof of Propositions 4.3 and 4.4, we consider

$$\begin{aligned} v(t)=(1-\phi )u(t)+{\mathbb {B}}[u(t)\cdot \nabla \phi ], \qquad p_v(t)=(1-\phi )p(t). \end{aligned}$$

Then v(t) obeys

$$\begin{aligned} v(t)=T_0(t,s){\widetilde{F}}+\int _s^tT_0(t,\tau )\mathbb P_0K(\tau )\,d\tau \end{aligned}$$

(4.65)

in terms of the evolution operator $T_0(t,s)$ without the rigid body studied in Sect. 4.1, where

$$\begin{aligned} {\widetilde{F}}=(1-\phi )F+{\mathbb {B}}[F\cdot \nabla \phi ]\in C^\infty _{0,\sigma }({\mathbb {R}}^3) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} K(t)&=2\nabla \phi \cdot \nabla u(t)+\left[ \Delta \phi +\big (\eta _b(t)-u_b(t)\big )\cdot \nabla \phi \right] u(t) \\&\quad -(\nabla \phi )p(t)+\left\{ \partial _t-\Delta -\big (\eta _b(t)-u_b(t)\big )\cdot \nabla \right\} {\mathbb {B}}[u(t)\cdot \nabla \phi ]. \end{aligned} \end{aligned}$$

By the same symbol K(t) we denote its extension on ${\mathbb {R}}^3$ by setting zero outside $\Omega _3$. Given $r_1\in (1,4/3]$, we assume the smallness (4.59) on $\Vert U_b\Vert$ as in Proposition 4.4 and Corollary 4.1, then it follows from (4.61)–(4.62) and (2.33) together with (3.76) that

$$\begin{aligned} \Vert K(t)\Vert _{r,{\mathbb {R}}^3}\le C(t-s)^{-\gamma }(1+t-s)^{-3/2q+\gamma }\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.66)

for all (t, s) with $t>s$ and $r\in (1,q]$ as long as $q\in [r_1,\infty )$, where $\gamma \in \big ((1+1/q)/2,\,1\big )$ is fixed arbitrarily.

Let $t-s>2$. For the proof of (4.63)–(4.64), our task is to deduce the desired eatimates of $\Vert \nabla v(t)\Vert _{q,{\mathbb {R}}^3}$ and $\Vert v(t)\Vert _{\infty ,{\mathbb {R}}^3}$ by using (4.65). It is obvious that the term $T_0(t,s)\widetilde{F}$ fulfills those thanks to (4.14)–(4.15) provided $q\ge r_1$. By (4.66) along with (4.15) (for $q\ge r_1$) and by choosing, for instance,

$$\begin{aligned} \left\{ \begin{array}{ll} r=q \qquad &{} \hbox { if}\ q\in [r_1,3/2), \\ r=4/3 &{} \hbox { if}\ q\in [3/2,\infty ), \end{array} \right. \end{aligned}$$

(note that $r\in [r_1,q]$ is required to apply Proposition 4.1), we find

$$\begin{aligned} \begin{aligned}&\int _s^t\Vert \nabla T_0(t,\tau ){\mathbb {P}}_0K(\tau )\Vert _{q,{\mathbb {R}}^3}\,d\tau \\&\quad \le C\Vert F\Vert _{q,{\mathbb {R}}^3}\left( \int _s^{(s+t)/2}+\int _{(s+t)/2}^t\right) \\&\qquad (t-\tau )^{-1/2}(1+t-\tau )^{-(3/r-3/q)/2}(\tau -s)^{-\gamma }(1+\tau -s)^{-3/2q+\gamma }\,d\tau \\&\quad \le C(t-s)^{-\min \{1/2,\, 3/2q\}}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

(4.67)

which concludes (4.63) under the condition $\Vert U_b\Vert \le \alpha _2(r_1,q_0)$.

For the $L^\infty$-estimate, the integrand of (4.67) is replaced by

$$\begin{aligned} (t-\tau )^{-3/2q}(1+t-\tau )^{-(3/r-3/q)/2}(\tau -s)^{-\gamma }(1+\tau -s)^{-3/2q+\gamma }. \end{aligned}$$

Let $3/2<q<\infty$. We then take $r=4/3$ and compute the integral in the same way as above to deduce (4.64) provided

$$\begin{aligned} \Vert U_b\Vert \le \alpha _1(q_0)=\alpha _2(4/3,q_0), \end{aligned}$$

which ensures (4.66) with $q\in (3/2,\infty )$ and allows us to apply (4.14) with $r=\infty$ and $q=4/3$. In order to derive this for the other case $q\in (1,3/2]$ as well, we have only to combine (4.64) for $q=2$ (say) obtained above with (2.51) for $q\in (1,3/2]$ under the condition $\Vert U_b\Vert \le \alpha _1(q_0)$ by taking into account the semigroup property (2.47). The adjoint $T(t,s)^*$ is discussed similarly. The proof is complete. $\square$

4.6 Proof of Theorem 2.1

We collect Proposition 3.4 (case $r=\infty$), Proposition 4.3, (4.55) and (4.64) to furnish (2.51) provided that $\Vert U_b\Vert \le \alpha _1(q_0)$.

Let $r_1\in (1,4/3]$ and suppose $\Vert U_b\Vert \le \alpha _2(r_1,q_0)\le \alpha _1(q_0)$. Then it follows from (4.53) and (4.63) along with Proposition 3.4 that

$$\begin{aligned} \Vert \nabla T(t,s)F\Vert _{q,{\mathbb {R}}^3}\le C(t-s)^{-1/2}(1+t-s)^{\max \{(1-3/q)/2,\,0\}}\Vert F\Vert _{q,{\mathbb {R}}^3} \end{aligned}$$

(4.68)

for all (t, s) with $t>s$ and $F\in X_q({\mathbb {R}}^3)$ as long as $r_1\le q<\infty$. One may combine (4.68) with (2.51) to conclude (2.52) for $r\in [r_1,\infty )$ and $q\in (1,r]$.

With the same estimates for the adjoint $T(t,s)^*$ under the same smallness of $\Vert U_b\Vert$ at hand, let us show (2.53). Let $1< q<\infty$ and $\phi \in {{\mathcal {E}}}({\mathbb {R}}^3)$. Then, in view of (3.97) together with (3.10), see also (2.14), we have

$$\begin{aligned} \begin{aligned}&\big |\langle T(t,s){\mathbb {P}} \text{ div }\ F,\; \phi \rangle _{{\mathbb {R}}^3,\rho }\big | \\ {}&\quad =\left| -\langle F,\; \nabla T(t,s)^*\phi \rangle _{{\mathbb {R}}^3,\rho }+(1-\rho )\int _{\partial \Omega }(F\nu )\cdot \big (T(t,s)^*\phi \big )\,d\sigma \right| \\ {}&\quad \le \Vert F\Vert _{q,({\mathbb {R}}^3,\rho )}\Vert \nabla T(t,s)^*\phi \Vert _{q^\prime ,({\mathbb {R}}^3,\rho )} \\ {}&\quad \le C\Vert F\Vert _{q,{\mathbb {R}}^3}\Vert \nabla T(t,s)^*\phi \Vert _{q^\prime ,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for all $F\in L^q({\mathbb {R}}^3)^{3\times 3}$ with $F\nu =0$ at $\partial \Omega$ as well as $\hbox { div}\ F$ belonging to $L^p({\mathbb {R}}^3)$ for some $p\in (1,\infty )$ (so that $(F\nu )|_{\partial \Omega }$ from both directions coincide with each other) and fulfilling $(\hbox { div}\ F)|_B\in \textrm{RM}$. Given $r_0\in [4,\infty )$, suppose that

$$\begin{aligned} \Vert U_b\Vert \le \alpha _3(r_0,q_0):=\alpha _2(r_0^\prime ,q_0)\le \alpha _1(q_0). \end{aligned}$$

(4.69)

Then we apply (2.52) for $T(t,s)^*$ to conclude (2.53) by duality provided that $q\in (1,r_0]$ and $r\in [q,\infty )$. This together with (2.51) with $r=\infty$ leads us to (2.53) with $r=\infty$ as well.

Finally, we deduce (2.54). Given $r_0\in [4,\infty )$ and $r_1\in (1,4/3]$, assume that

$$\begin{aligned} \Vert U_b\Vert \le \alpha _4(r_0,r_1,q_0):=\alpha _2\big (\min \{r_0^\prime ,r_1\}, q_0\big ). \end{aligned}$$

(4.70)

Then we use (2.52) and (2.53) to obtain (2.54) for $1<q\le r<\infty$ with $q\in (1,r_0]$ as well as $r\in [r_1,\infty )$. The proof is complete.

5 Stability of the Basic Motion

The initial value problem (2.44) is transformed into

$$\begin{aligned} U(t)=T(t,s)U_0+\int _s^t T(t,\tau )H(U(\tau ))\,d\tau =:{\overline{U}}(t)+(\Lambda U)(t). \end{aligned}$$

(5.1)

Look at (2.46); since $(\eta -u)\cdot \nu =0$ at $\partial \Omega$, we observe

$$\begin{aligned} H(U) ={\mathbb {P}}\left[ \big \{\hbox { div}\ \big ((u_b+u)\otimes (\eta -u)\big )\big \}\chi _\Omega \right] ={\mathbb {P}}\, \text{ div } \big \{(u_b+u)\otimes (\eta -u)\chi _\Omega \big \}. \end{aligned}$$

(5.2)

We do need this divergence form especially for the nonlinear term $\eta \cdot \nabla u$ in the first integral of (5.15) below (as in [9, 10]), while the other terms can be discussed anyway if we impose more assumptions on $\nabla u_b$ than (2.56). For finding a solution to (5.1) we adopt the function space

$$\begin{aligned} \begin{aligned} E:=\big \{U\in&C\big ((s,\infty );\,W^{1,3}({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)\big );\; \\&U(t)\in X_3({\mathbb {R}}^3)\;\forall t\in (s,\infty ),\;\lim _{t\rightarrow s}\Vert U\Vert _{E(t)}=0,\;\Vert U\Vert _E<\infty \big \} \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \Vert U\Vert _{E(t)}&:=\sup _{\tau \in (s,t)} (\tau -s)^{1/2}\big (\Vert \nabla U(\tau )\Vert _{3,{\mathbb {R}}^3}+\Vert U(\tau )\Vert _{\infty ,{\mathbb {R}}^3}\big )\quad \hbox { for}\ t\in (s,\infty ), \\ \Vert U\Vert _E&:=\sup _{t\in (s,\infty )}\left( \Vert U\Vert _{E(t)}+\Vert U(t)\Vert _{3,{\mathbb {R}}^3}\right) . \end{aligned} \end{aligned}$$

Then E is a Banach space endowed with norm $\Vert \cdot \Vert _E$. Let us remark that $U\in E$ already involves the boundary condition (2.16) with $(u(t),\eta (t),\omega (t))=i(U(t))$ for each $t>s$, see (2.7), since $U(t)\in W^{1,3}({\mathbb {R}}^3)$. In view of (2.7), $U\in E$ implies that

$$\begin{aligned} (t-s)^{1/2}\big (\Vert \nabla u(t)\Vert _{3,\Omega }+\Vert u(t)\Vert _{\infty ,\Omega }+|\eta (t)|+|\omega (t)|\big )\le C\Vert U\Vert _{E(t)}\rightarrow 0 \end{aligned}$$

(5.3)

as $t\rightarrow s$ and that

$$\begin{aligned} (t-s)^{1/2}\big (\Vert \nabla u(t)\Vert _{3,\Omega }+\Vert u(t)\Vert _{\infty ,\Omega }\big )+(1+t-s)^{1/2}\big (|\eta (t)|+|\omega (t)|\big ) +\Vert u(t)\Vert _{3,\Omega } \le C\Vert U\Vert _E \end{aligned}$$

(5.4)

for all (t, s) with $t>s$.

By $\Lambda U$ we denote the Duhamel term in (5.1):

$$\begin{aligned} (\Lambda U)(t):=\int _s^t T(t,\tau )H(U(\tau ))\,d\tau . \end{aligned}$$

Then we see the following lemma.

Lemma 5.1

Suppose (2.33)–(2.34) and (2.56). If $\Vert U_b\Vert \le \alpha _1$ with $\alpha _1=\alpha _1(q_0)$ being the constant given in Proposition 4.3, see (4.35), then we have $\Lambda U\in E$ as well as

$$\begin{aligned} \lim _{t\rightarrow s}\Vert (\Lambda U)(t)\Vert _{3,{\mathbb {R}}^3}=0 \end{aligned}$$

(5.5)

for every $U\in E$ and

$$\begin{aligned} \Vert \Lambda U\Vert _E\le & {} c_1\Vert U_b\Vert ^\prime \Vert U\Vert _E+c_2\Vert U\Vert _E^2 \end{aligned}$$

(5.6)

$$\begin{aligned} \Vert \Lambda U-\Lambda V\Vert _E\le & {} \big (c_1\Vert U_b\Vert ^\prime +c_2\Vert U\Vert _E+c_2\Vert V\Vert _E\big )\Vert U-V\Vert _E \end{aligned}$$

(5.7)

for all $U,\, V\in E$ with some constants $c_1=c_1(q_0,\alpha _1,\beta _0,\theta )>0$ and $c_2=c_2(\alpha _1,\beta _0,\theta )$, where $\Vert U_b\Vert$ and $\Vert U_b\Vert ^\prime$ are the constants given by (2.35) and (2.57), respectively. Furthermore, under the condition above, the following additional properties hold for every $U\in E$: (i) Let $r\in (3,\infty )$, then

$$\begin{aligned} \Vert \nabla (\Lambda U)(t)\Vert _{r,{\mathbb {R}}^3}=O\big ((t-s)^{-1/2}\big ) \end{aligned}$$

(5.8)

as $(t-s)\rightarrow \infty$. (ii) $\Lambda U$ is locally Hölder continuous on $(s,\infty )$ with values in $W^{1,3}({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)$, to be precise,

$$\begin{aligned} \begin{aligned}&\Lambda U\in C^{\theta _0}_\textrm{loc}\big ((s,\infty );\, X_3({\mathbb {R}}^3)\big )\cap C^{\theta _1}_\textrm{loc}\big ((s,\infty );\, L^\infty ({\mathbb {R}}^3)\big ), \\&\nabla \Lambda U\in C^{\theta _1}_\textrm{loc}\big ((s,\infty );\, L^3({\mathbb {R}}^3)\big ), \end{aligned} \end{aligned}$$

(5.9)

for every $\theta _0\in (0,3/4)$ and $\theta _1\in (0,1/4)$.

Proof

We are concerned only with (5.5)–(5.6) since the other estimate (5.7) is shown similarly. Set

$$\begin{aligned}{} & {} \begin{aligned} (\Lambda _1U)(t)&=\int _s^tT(t,\tau ){\mathbb {P}}[\{(\eta -u)\cdot \nabla u_b\}\chi _\Omega ](\tau )\,d\tau \\&=\int _s^tT(t,\tau ){\mathbb {P}}\text{ div } \{u_b\otimes (\eta -u)\chi _\Omega \}(\tau )\,d\tau , \end{aligned} \end{aligned}$$

(5.10)

$$\begin{aligned}{} & {} \begin{aligned} (\Lambda _2U)(t)&=\int _s^tT(t,\tau ){\mathbb {P}}[\{(\eta -u)\cdot \nabla u\}\chi _\Omega ](\tau )\,d\tau \\&=\int _s^tT(t,\tau ){\mathbb {P}}\text{ div } \{u\otimes (\eta -u)\chi _\Omega \}(\tau )\,d\tau , \end{aligned} \end{aligned}$$

(5.11)

where (5.2) is taken into account. Let us note that both $u_b\otimes (\eta -u)\chi _\Omega$ and $u\otimes (\eta -u)\chi _\Omega$ satisfy the conditions imposed on F for (2.53)–(2.54) since $(\eta -u)\cdot \nu =0$ at $\partial \Omega$. In what follows we assume that

$$\begin{aligned} \Vert U_b\Vert \le \alpha _1(q_0)=\alpha _2(4/3,q_0)=\alpha _3(4,q_0)=\alpha _4(4,4/3,q_0), \end{aligned}$$

see (4.35) and (4.69)–(4.70). This condition allows us to employ all the estimates obtained in Theorem 2.1 with the exponents needed below. Let $U\in E$ and let us begin with estimate of (5.10). Since $u_b(t)\in L^\infty (\Omega )$, one may assume that $q_0\in (3/2,3)$ in (2.33). We make use of (2.51)–(2.54) along with (5.3)–(5.4) to find that

$$\begin{aligned} \begin{aligned}&\Vert \nabla (\Lambda _1U)(t)\Vert _{3,{\mathbb {R}}^3}+\Vert (\Lambda _1U)(t)\Vert _{\infty ,{\mathbb {R}}^3} \\&\quad \le C\int _s^{t-1}(t-\tau )^{-3/2q_0-1/2}\Vert u_b(\tau )\Vert _{q_0,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\qquad +C\int _{t-1}^t(t-\tau )^{-1/2}\Vert \nabla u_b(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\quad \le C(t-s)^{-1/2}\Vert U_b\Vert ^\prime \Vert U\Vert _{E(t)} \end{aligned} \end{aligned}$$

(5.12)

for $t-s>2$ by splitting the first integral further into $\int _s^{(s+t)/2}+\int _{(s+t)/2}^{t-1}$, where $q_0\in (3/2,3)$ allows us to obtain sharp decay rate in (2.54), and that

$$\begin{aligned} \begin{aligned}&\Vert \nabla (\Lambda _1U)(t)\Vert _{3,{\mathbb {R}}^3}+\Vert (\Lambda _1U)(t)\Vert _{\infty ,{\mathbb {R}}^3} \\&\quad \le C\int _s^t(t-\tau )^{-1/2}\Vert \nabla u_b(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\quad \le C \Vert U_b\Vert ^\prime \Vert U\Vert _{E(t)} \end{aligned} \end{aligned}$$

for $t-s\le 2$. Also, it is readily seen that

$$\begin{aligned} \begin{aligned} \Vert (\Lambda _1U)(t)\Vert _{3,{\mathbb {R}}^3}&\le C\int _s^t(t-\tau )^{-1/2}\Vert u_b(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\le C\Vert U_b\Vert \Vert U\Vert _{E(t)} \end{aligned} \end{aligned}$$

(5.13)

for all $t>s$. Collecting estimates above, we infer

$$\begin{aligned} \begin{aligned}&\Vert \Lambda _1U\Vert _E\le c_1\Vert U_b\Vert ^\prime \Vert U\Vert _E, \\&\lim _{t\rightarrow s}\big (\Vert \Lambda _1U\Vert _{E(t)}+\Vert (\Lambda _1U)(t)\Vert _{3,\mathbb R^3}\big )=0, \end{aligned} \end{aligned}$$

(5.14)

for all $U\in E$ with some constant $c_1=c_1(q_0,\alpha _1,\beta _0,\theta )$.

We turn to the other integral $\Lambda _2U$ given by (5.11). The computation with splitting below was not done by [10] and this was why the decay rate of $\Vert \nabla u(t)\Vert _{3,\Omega }$ was not sharp in that literature. From (2.51)–(2.54) and (5.3)–(5.4) it follows that

$$\begin{aligned} \begin{aligned}&\Vert \nabla (\Lambda _2U)(t)\Vert _{3,{\mathbb {R}}^3}+\Vert (\Lambda _2U)(t)\Vert _{\infty ,{\mathbb {R}}^3} \\&\quad \le C\int _s^{(s+t)/2}(t-\tau )^{-1}\Vert u(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\qquad +C\int _{(s+t)/2}^t(t-\tau )^{-1/2}\Vert \nabla u(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\quad \le C(t-s)^{-1/2}\big (\Vert U\Vert _E\Vert U\Vert _{E(t)}+\Vert U\Vert _{E(t)}^2\big ) \end{aligned} \end{aligned}$$

(5.15)

for all $t>s$ and that

$$\begin{aligned} \begin{aligned} \Vert (\Lambda _2U)(t)\Vert _{3,{\mathbb {R}}^3}&\le C\int _s^t(t-\tau )^{-1/2}\Vert u(\tau )\Vert _{3,\Omega }\big (|\eta (\tau )|+\Vert u(\tau )\Vert _{\infty ,\Omega }\big )\,d\tau \\&\le c_0B(1/2,1/2)\Vert U\Vert _E\Vert U\Vert _{E(t)} \end{aligned} \end{aligned}$$

(5.16)

for all $t>s$ with some constant $c_0>0$, where $B(\cdot ,\cdot )$ denotes the beta function and $B(\frac{1}{2},\frac{1}{2})=\pi$. Those estimates lead us to

$$\begin{aligned} \begin{aligned}&\Vert \Lambda _2U\Vert _E\le c_2\Vert U\Vert _E^2, \\&\lim _{t\rightarrow s}\big (\Vert \Lambda _2U\Vert _{E(t)}+\Vert (\Lambda _2U)(t)\Vert _{3,\mathbb R^3}\big )=0, \end{aligned} \end{aligned}$$

(5.17)

for all $U\in E$ with some constant $c_2=c_2(\alpha _1,\beta _0,\theta )$, which together with (5.14) completes the proof of (5.5)–(5.6).

It remains to show the additional properties:

• (i) Look at (5.12) and (5.15) in which $\Vert \nabla (\cdot )\Vert _{3,{\mathbb {R}}^3}$ is replaced by $\Vert \nabla (\cdot )\Vert _{r,{\mathbb {R}}^3}$ with $r\in (3,\infty )$, then the computations still work, where further splitting $\int _{(s+t)/2}^{t-1}+\int _{t-1}^t$ is needed in (5.15) and, in the latter integral, $(t-\tau )^{-1/2}$ must be replaced by $(t-\tau )^{-1+3/2r}$.

• (ii) Let us recall the Hölder estimate of the evolution operator obtained in Proposition 3.5. Since the issue is merely local in time, we do not have to take care of the large time behavior and thus we have only to use the first form of each of (5.10)–(5.11), respectively. We employ (3.60) with $(j,q,r)=(0,3,3),(0,3,\infty )$ and (1, 3, 3) for $\Lambda _1 U$. Concerning $\Lambda _2U$, we use (3.60) with $(j,q,r)=(0,2,3),(0,2,\infty )$ and (1, 2, 3) for $u\cdot \nabla u$, while $(j,q,r)=(0,3,3),(0,3,\infty )$ and (1, 3, 3) for $\eta \cdot \nabla u$. Note that (3.58) is fulfilled for all of those (j, q, r). Then the computations are the same as in the autonomous case with analytic semigroups. The proof is complete.

$\square$

Set ${\overline{U}}(t):=T(t,s)U_0$ with $U_0\in X_3({\mathbb {R}}^3)$, then (3.51) implies that $\Vert {\overline{U}}\Vert _{E(t)}\rightarrow 0$ as $t\rightarrow s$. By taking into account (3.60) as well, it is clear to see that ${\overline{U}}\in E$ together with

$$\begin{aligned} \Vert {\overline{U}}\Vert _E\le c_*\Vert U_0\Vert _{3,{\mathbb {R}}^3}, \end{aligned}$$

with some constant $c_*>0$, which follows from (2.51)–(2.52) provided $\Vert U_b\Vert \le \alpha _1$. With Lemma 5.1 at hand, we easily find that the map

$$\begin{aligned} U\mapsto {\overline{U}}+\Lambda U \end{aligned}$$

is contractive from the closed ball $E_R=\{U\in E;\; \Vert U\Vert _E\le R\}$ with radius

$$\begin{aligned} R=\frac{1}{2c_2}\left( \frac{1}{2}-\sqrt{\frac{1}{4}-4c_2c_*\Vert U_0\Vert _{3,\mathbb R^3}}\right) <4c_*\Vert U_0\Vert _{3,{\mathbb {R}}^3} \end{aligned}$$

(5.18)

into itself provided that

$$\begin{aligned} \Vert U_b\Vert ^\prime \le \frac{1}{2c_1}, \qquad \Vert U_0\Vert _{3,{\mathbb {R}}^3} <\delta :=\frac{1}{16c_2c_*}. \end{aligned}$$

(5.19)

The smallness of the basic motion thus reads

$$\begin{aligned} \Vert U_b\Vert ^\prime \le \alpha =\alpha (q_0,\beta _0,\theta ):=\min \left\{ \alpha _1,\,\frac{1}{2c_1}\right\} , \end{aligned}$$

where $\alpha _1=\alpha _1(q_0)$ is the constant given in Proposition 4.3, see (4.35). Then the fixed point $U\in E_R$ provides a solution to (5.1) and also the initial condition

$$\begin{aligned} \lim _{t\rightarrow s}\Vert U(t)-U_0\Vert _{3,{\mathbb {R}}^3}=0 \end{aligned}$$

holds on account of (5.5).

Let us remark that uniqueness of solutions to (5.1) still holds within the class E rather than the ball $E_R$ with small radius (5.18) by means of standard argument as in Fujita and Kato [12], where the behavior $\Vert U\Vert _{E(t)}\rightarrow 0$ for $t\rightarrow s$ plays a role. Actually, even this behavior near the initial time is redundant for uniqueness of the solution constructed above, as pointed out by Brezis [3], where the last assertion on the uniformly convergence (3.51) in Proposition 3.4 is employed. Note, however, that uniqueness within the class E is independent of the existence of solutions, while it is not the case for the latter.

All the desired properties of the solution obtained above (except for the sharp large time behavior that will be seen below) follow from Lemma 5.1 as well as several properties of the evolution operator. By (ii) of Lemma 5.1 together with (2.56), the term H(U) given by (2.46) is locally Hölder continuous with values in $X_3({\mathbb {R}}^3)$, so that the solution U(t) is a strong one ([48, Chapter 5, Theorem 2.3], [52, Theorem 3.9]) as described in Theorem 2.2.

Finally, let us close the paper with verification of the sharp large time behavior, such as $\Vert U(t)\Vert _{\infty , \mathbb R^3}=o\big ((t-s)^{-1/2}\big )$, although this is common as long as we work with underlying space in which the class of nice functions is dense. Suppose $U_0\in X_3({\mathbb {R}}^3)\cap X_p({\mathbb {R}}^3)$ with some $p\in (1,3)$, then both ${\overline{U}}(t)$ and $(\Lambda _1U)(t)$ dacays to zero in $X_3({\mathbb {R}}^3)$ with definite rate, say, $(t-s)^{-\gamma }$. In fact, we have only to replace $\Vert u_b\Vert _{3,\Omega }$ by $\Vert u_b\Vert _{q_0,\Omega }$ with $q_0\in (3/2,3)$ in (5.13) as for $\Lambda _1U$. One the other hand, it is readily seen that

$$\begin{aligned} \Vert (\Lambda _2U)(t)\Vert _{3,{\mathbb {R}}^3} \le c_0B(1/2,1/2-\gamma )(t-s)^{-\gamma }\Vert U\Vert _E\sup _{\tau \in (s,t)}(\tau -s)^\gamma \Vert U(\tau )\Vert _{3,{\mathbb {R}}^3} \end{aligned}$$

for all $t>s$, where $c_0$ is the same constant as in (5.16). Note that the constant $c_2$ in (5.17) should satisfy $c_2\ge \pi c_0=c_0B(\frac{1}{2},\frac{1}{2})$. By the continuity of the beta function, one can take $\gamma >0$ so small that $c_0B(\frac{1}{2},\frac{1}{2}-\gamma )\le 2c_2$, which together with (5.18) leads to

$$\begin{aligned} c_0B(1/2,1/2-\gamma )\Vert U\Vert _E\le 2c_2R<8c_2c_*\Vert U_0\Vert _{3,\mathbb R^3}<\frac{1}{2} \end{aligned}$$

under the condition (5.19). We thus find

$$\begin{aligned} \Vert U(t)\Vert _{3,{\mathbb {R}}^3}\le C(t-s)^{-\gamma }\big (\Vert U_0\Vert _{3,\mathbb R^3}+\Vert U_0\Vert _{p,{\mathbb {R}}^3}\big ), \end{aligned}$$

from which combined with the continuity of the solution map $U(s)\mapsto U$ in the sense that

$$\begin{aligned} \sup _{t\in [s,\infty )}\Vert U(t)-V(t)\Vert _{3,{\mathbb {R}}^3}\le C\Vert U(s)-V(s)\Vert _{3,{\mathbb {R}}^3} \end{aligned}$$

as well as denseness of $X_3({\mathbb {R}}^3)\cap X_p({\mathbb {R}}^3)$ in $X_3({\mathbb {R}}^3)$, we conclude that

$$\begin{aligned} \lim _{t-s\rightarrow \infty }\Vert U(t)\Vert _{3,{\mathbb {R}}^3}=0. \end{aligned}$$

(5.20)

Several papers (including mine) on the Navier–Stokes claim that (5.20) is accomplished provided initial data are still smaller, however, if we look carefully into estimates as above, then we see that further smallness than (5.19) is not needed. This observation is due to Tomoki Takahashi, who is the author of [47].

Once we have (5.20), we can deduce the other decay properties by following the argument as in [8], see also [47]. Let $\tau _*>s$. Using the equation

$$\begin{aligned} U(t)=T(t,\tau _*)U(\tau _*)+\int _{\tau _*}^t T(t,\tau )H(U(\tau ))\,d\tau \end{aligned}$$

(5.21)

and performing exactly the same computations as in the proof of Lemma 5.1, we infer

$$\begin{aligned} \begin{aligned}&(t-\tau _*)^{1/2}\big (\Vert U(t)\Vert _{\infty ,{\mathbb {R}}^3}+\Vert \nabla U(t)\Vert _{3,{\mathbb {R}}^3}\big ) \\&\quad \le C\Vert U(\tau _*)\Vert _{3,{\mathbb {R}}^3} +\big (c_1\Vert U_b\Vert ^\prime +c_2\Vert U\Vert _E\big )\sup _{\tau \in (\tau _*,t)}(\tau -\tau _*)^{1/2}\Vert U(\tau )\Vert _{\infty ,{\mathbb {R}}^3} \\&\quad \le C\Vert U(\tau _*)\Vert _{3,{\mathbb {R}}^3} +\frac{3}{4}\sup _{\tau \in (\tau _*,t)}(\tau -\tau _*)^{1/2}\Vert U(\tau )\Vert _{\infty ,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for all $t>\tau _*$ in view of (5.18)–(5.19), where the constants $c_1$ and $c_2$ are the same as those in (5.6)–(5.7). Given $\varepsilon >0$ arbitrarily, we take $\tau _*-s$ so large that the first term of the right-hand side above is less than $\varepsilon$, which is indeed possible because of (5.20). We then have $(t-\tau _*)^{1/2}\big (\Vert U(t)\Vert _{\infty ,{\mathbb {R}}^3}+\Vert \nabla U(t)\Vert _{3,{\mathbb {R}}^3}\big )\le 4\varepsilon$ for all $t>\tau _*$. If $t-s>2(\tau _*-s)$, then we get

$$\begin{aligned} \left( \frac{t-s}{2}\right) ^{1/2} \big (\Vert U(t)\Vert _{\infty ,\mathbb R^3}+\Vert \nabla U(t)\Vert _{3,{\mathbb {R}}^3}\big )\le 4\varepsilon , \end{aligned}$$

which concludes (2.58) except for $\Vert \nabla u(t)\Vert _{r,\Omega }$ with $r\in (3,\infty )$. For such r, finally, we use (5.21) with $\tau _*$ replaced by $\frac{t+s}{2}$ and compute it as in deduction of (5.8) to find

$$\begin{aligned} \begin{aligned} \Vert \nabla U(t)\Vert _{r,{\mathbb {R}}^3}&\le C(t-s)^{-1/2}\Vert U((t+s)/2)\Vert _{3,{\mathbb {R}}^3} \\&\quad +C(t-s)^{-1/2}\big (\Vert U_b\Vert ^\prime +\Vert U\Vert _E\big )\sup _{\tau >(t+s)/2}(\tau -s)^{1/2}\Vert U(\tau )\Vert _{\infty ,{\mathbb {R}}^3} \end{aligned} \end{aligned}$$

for $t-s>2$. Hence, (2.58) with $q=\infty$ and (5.20) yield (2.58) with $r\in (3,\infty )$. The proof of Theorem 2.2 is complete.