corthm propthm lemthm cor prop lem defnthm asuthm conthm defn asu con remthm
exathm masuthm notathm settthm rem exa masu nota sett =10pt

# Strong well-posedness and dynamics of a nematic liquid crystal-colloidal
interaction model

Tim Binz  Technische Universität Darmstadt, Schloßgartenstraße 7, 64289
Darmstadt, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>darmstadt.de<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Felix
Brandt  ,  MATTHIAS HIEBER  and  Arnab Roy

###### Abstract.

This paper investigates the interaction of nematic liquid crystals modeled by
a simplified Ericksen-Leslie model with a colloidal particle. It is shown that
this problem is locally strongly well-posed, and that it also admits a unique
global strong solution for initial data close to constant equilibria. The
proof relies on the property of maximal regularity of the corresponding
linearized problem. In particular, for the global well-posedness, a splitting
of the director part in mean value zero and average part is employed.

###### Key words and phrases:

Liquid crystal-colloidal interaction, Ericksen-Leslie model, fluid-structure
interaction, local and global well-posedness, convergence to equilibria

###### 2020 Mathematics Subject Classification:

35Q35, 76A15, 74F10, 76D03, 35K59

##  1\. Introduction

Ericksen [10] and Leslie [30] pioneered the development of a continuum theory
for the flow of nematic liquid crystals. Their theory models the flow of
nematic crystals from a fluid dynamical point of view and describes the
evolution of the underlying system by a certain fluid equation coupled to the
configuration $d$ of rod-like crystals. The original derivation is based on
the conservation laws for mass, linear momentum as well as on the constitutive
relations given by Ericksen [10] and Leslie [30]. The system subject to
general Leslie and general Ericksen stress is quite involved, and many
analytical question related to the general model remain open until today.

A simplified system was introduced and analyzed by Lin [32] as well as by Lin
and Liu [33]. They studied first the situation where the nonlinearity in the
equation for $d$ is replaced by a Ginzburg-Landau term and proved the
existence of global weak solutions under suitable assumptions. Concerning
strong solutions and bounded domains, a rather complete understanding of the
well-posedness and the dynamics for the simplified system was obtained in
[22]. Only recently, a quite thorough understanding of the Ericksen-Leslie
system subject to general Ericksen and Leslie stresses was obtained by Hieber
and Prüss [23, 24, 25]. For a survey of existing results, we refer to [31],
[26] and [46].

In this article, we investigate the interaction problem of liquid crystals
with a colloidal particle modeled as a rigid body. Let us note that there is a
well developed theory for the interaction of a rigid body which is immersed in
an incompressible Newtonian fluid. We refer for example to the works of
Hoffmann and Starovoitov [27], Desjardins and Esteban [8, 9], Conca, San
Martín and Tucsnak [4], Galdi [13], Galdi and Silvestre [14], Takahashi [44],
Cumsille and Tucsnak [6], Cumsille and Takahashi [5], San Martín et al. [42],
Geissert, Götze and Hieber [15], Kaltenbacher et al. [29], Maity and Tucsnak
[36], Nečasová et al. [39], Feireisl, Roy and Zarnescu [11], Glass and Sueur
[18], Glass, Lacave and Sueur [17]. For related results in the context of
elastic structures, we also refer to the articles of Grandmont and Hillairet
[19] as well as Maity, Raymond and Roy [34], Grandmont, Hillairet and
Lequeurre [20].

On the other hand, the interaction problem of liquid crystals with a colloidal
particle seems not to be investigated analytically so far, although it is a
very natural question. Let us note that liquid crystal colloids are formed by
dispersion of colloidal particles in the liquid crystal host medium. For
investigations in this direction in the applied sciences, we refer e.g. to
[43, 38], where suspended colloidal particles in a nematic-fluid flow are
studied. We refer also to the work [37] for modeling and numerics of liquid
crystal microfluidic channels with a circular particle interaction.

Describing the interaction of a colloid immersed in a liquid crystal liquid,
one observes a competition between elasticity and the interaction between
liquid crystal molecules and surfaces. This makes the interaction problem
described below in detail different from the incompressible Newtonian fluid-
rigid body interaction. The present article investigates the interaction
problem of a body immersed in an Ericksen-Leslie flow for the first time.

We establish local strong well-posedness of the nematic liquid crystal-
colloidal interaction problem, and we also show that the system is globally
strongly well-posed provided the initial data are chosen close to constant
equilibria. Let us emphasize that we verify the preservation of the director
condition in the interaction problem instead of using the Ginzburg-Landau
energy functional.

More precisely, let $\mathcal{O}\subset\mathbb{R}^{3}$ denote a bounded domain
with boundary of class $\mathrm{C}^{3}$, and let $0<T\leq\infty$ represent a
positive time. We consider a colloidal particle moving inside $\mathcal{O}$ as
a rigid body in $\mathbb{R}^{3}$ which is assumed to be closed, bounded and
simply connected. Furthermore, the time-dependent domain of the colloidal
particle is denoted by $\mathcal{S}(t)$ for $t\in(0,T)$, and the remaining
part of the domain $\mathcal{F}(t)\coloneqq\mathcal{O}\setminus\mathcal{S}(t)$
is assumed to be filled by a viscous incompressible fluid with active liquid
crystals. The initial domain of the colloidal particle is denoted by
$\mathcal{S}_{0}$. Besides, we suppose that its boundary
$\partial\mathcal{S}_{0}$ is of class $\mathrm{C}^{3}$ as well. Then
$\mathcal{F}_{0}=\mathcal{O}\setminus\mathcal{S}_{0}$ is the initial fluid
domain. The solid domain $\mathcal{S}(t)$ at time $t$ is given by

(1.1) $\mathcal{S}(t)=\left\\{h(t)+Q(t)y:y\in\mathcal{S}_{0}\right\\},$

where $h(t)$ represents the center of mass of the body, and
$Q(t)\in\mathrm{SO}(3)$ is associated to the rotation of the rigid body.
Moreover, since $Q^{\prime}(t)Q^{\top}(t)$ is skew symmetric, for each $t>0$,
we can represent this matrix by a unique vector $\Omega(t)\in\mathbb{R}^{3}$
such that

$Q^{\prime}(t)Q^{\top}(t)y=\Omega(t)\times y,\enspace\text{for all}\enspace
y\in\mathbb{R}^{3}.$

By $\mathcal{F}_{T}$, $\mathcal{S}_{T}$ and $\partial\mathcal{S}_{T}$, we
denote the spacetime related to $\mathcal{F}(t)$, $\mathcal{S}(t)$ and
$\partial\mathcal{S}(t)$, respectively, i.e.,

$\mathcal{F}_{T}\coloneqq\\{(t,x):t\in(0,T),\,x\in\mathcal{F}(t)\\},\enspace\mathcal{S}_{T}\coloneqq\\{(t,x):t\in(0,T),\,x\in\mathcal{S}(t)\\},\enspace\text{and}\enspace$
$\partial\mathcal{S}_{T}\coloneqq\\{(t,x):t\in(0,T),\,x\in\partial\mathcal{S}(t)\\}.$

The velocity of the rigid body is described by

$u^{\mathcal{S}}(t,x)=h^{\prime}(t)+\Omega(t)\times(x-h(t)),\enspace\text{for
all}\enspace(t,x)\in\mathcal{S}_{T},$

where $h^{\prime}\colon(0,T)\to\mathbb{R}^{3}$ is the translational velocity
of the center of mass, and $\Omega\colon(0,T)\to\mathbb{R}^{3}$ denotes the
angular velocity of the body.

In the context of the Ericksen-Leslie model for nematic liquid crystals, the
model variables are the fluid velocity
$u^{\mathcal{F}}\colon\mathcal{F}_{T}\to\mathbb{R}^{3}$, the fluid pressure
$\pi^{\mathcal{F}}\colon\mathcal{F}_{T}\to\mathbb{R}$, and the director
$\overline{d}\colon\mathcal{F}_{T}\to\mathbb{S}^{2}$, where
$\mathbb{S}^{2}=\left\\{x\in\mathbb{R}^{3}:|x|=1\right\\}$. For
$\mathbb{D}=\nicefrac{{1}}{{2}}(\nabla u^{\mathcal{F}}+(\nabla
u^{\mathcal{F}})^{\top})$, the stress tensor in the situation of the
simplified isothermal Ericksen-Leslie model under consideration is given by
$\mathbb{T}=\mathbb{T}_{N}+\mathbb{T}_{E}$, where

$\mathbb{T}_{N}=2\mu\mathbb{D},\enspace\text{and}\enspace\mathbb{T}_{E}=-\lambda\nabla\overline{d}\left(\nabla\overline{d}\right)^{\top},\enspace\text{so}\enspace\mathbb{T}=2\mu\mathbb{D}-\lambda\nabla
d(\nabla d)^{\top},$

and $\mu>0$ and $\lambda>0$ are constants. Upon assuming that the constant
fluid density $\rho^{\mathcal{F}}$ equals one, and for another constant
$\gamma>0$, we obtain the following simplified isothermal Ericksen-Leslie
model

(1.2) $\left\\{\begin{aligned}
{\partial_{t}u^{\mathcal{F}}}+\left(u^{\mathcal{F}}\cdot\nabla\right)u^{\mathcal{F}}+\nabla\pi^{\mathcal{F}}&=\mu\Delta
u^{\mathcal{F}}-\lambda\operatorname{div}\left(\nabla\overline{d}\left(\nabla\overline{d}\right)^{\top}\right),&&\enspace\text{in}\enspace\mathcal{F}_{T},\\\
\operatorname{div}u^{\mathcal{F}}&=0,&&\enspace\text{in}\enspace\mathcal{F}_{T},\\\
\gamma\left[\partial_{t}\overline{d}+\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\right]-\lambda\Delta\overline{d}&=\lambda|\nabla\overline{d}|^{2}\overline{d},&&\enspace\text{in}\enspace\mathcal{F}_{T}.\end{aligned}\right.$

With regard to the colloidal particle, let $\rho_{\mathcal{S}}$ denote its
density. For simplicity, we assume that $\rho_{\mathcal{S}}\equiv 1$ in the
sequel, yielding that $m_{\mathcal{S}}=\int_{\mathcal{S}(t)}\,\mathrm{d}x$ is
its total mass. In addition, without loss of generality, we suppose that the
center of mass of the rigid body starts in the origin, i.e., $h(0)=0$. The
initial value of the inertial matrix $J_{0}$ and the inertia matrix $J(t)$ are
then given by

$J_{0}=\int\limits_{\mathcal{S}_{0}}\left(|x|^{2}\operatorname{Id}-x\otimes
x\right)\,\mathrm{d}x\enspace\text{and}\enspace
J(t)=\int\limits_{\mathcal{S}(t)}\left(|x-h(t)|^{2}\operatorname{Id}-(x-h(t))\otimes(x-h(t))\right)\,\mathrm{d}x,$

respectively. Consequently, $J(t)$ is symmetric positive definite and
satisfies

(1.3) $J(t)=Q(t)J_{0}Q^{\top}(t)\enspace\text{as well as}\enspace J(t)a\cdot
b=\int\limits_{\mathcal{S}_{0}}(a\times Q(t)y)\cdot(b\times
Q(t)y)\,\mathrm{d}y,\enspace\text{for all}\enspace\ a,b\in\mathbb{R}^{3}.$

For the underlying total stress tensor

(1.4)
$\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})=\left(\mathbb{T}-\pi^{\mathcal{F}}\operatorname{Id}\right)=2\mu\mathbb{D}-\lambda\nabla\overline{d}\left(\nabla\overline{d}\right)^{\top}-\pi^{\mathcal{F}}\operatorname{Id},$

the motion of the colloidal particle can be described by Newton’s law, i.e.,

(1.5)
$m_{\mathcal{S}}h^{\prime\prime}(t)=-\int\limits_{\partial\mathcal{S}(t)}\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})\nu\,\mathrm{d}\Gamma,\enspace\text{and}\enspace(J\Omega)^{\prime}(t)=-\int\limits_{\partial\mathcal{S}(t)}(x-h(t))\times\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})\nu\,\mathrm{d}\Gamma,$

where $\nu$ denotes the unit outward normal to the boundary of
$\mathcal{F}(t)$, i.e., it is directed towards $\mathcal{S}(t)$.

We assume that the fluid velocity satisfies homogeneous boundary conditions on
the outer boundary and coincides with the velocity of the colloidal particle
on their common interface, while the director is supposed to fulfill
homogeneous Neumann boundary conditions, i.e.,

(1.6)
$u^{\mathcal{F}}=0,\enspace\partial_{\nu}\overline{d}=0,\enspace\text{on}\enspace(0,T)\times\partial\mathcal{O},\enspace\text{and}\enspace
u^{\mathcal{F}}=u^{\mathcal{S}},\enspace\partial_{\nu}\overline{d}=0,\enspace\text{on}\enspace\partial\mathcal{S}_{T}.$

The system is completed by the initial conditions

(1.7)
$u^{\mathcal{F}}(0,x)=v_{0}(x),\enspace\overline{d}(0,x)=d_{0}(x)\enspace\text{in}\enspace\mathcal{F}_{0},\enspace
h(0)=0,\enspace
h^{\prime}(0)=\ell_{0},\enspace\text{and}\enspace\Omega(0)=\omega_{0}.$

Before tackling the interaction problem, we show that the director condition
$\overline{d}\equiv 1$ is also preserved in the present interaction problem by
testing the equation satisfied by the director suitably. Moreover, we show
that the energy functional is non-increasing and identify the set of
equilibrium solutions to the interaction problem as its critical points.

The first step for solving the interaction problem is then to transform the
moving domain problem to a problem on a fixed domain. To this end, we use a
local change of coordinates due to Inoue and Wakimoto [28]. This change of
coordinates only acts on a neighborhood of the colloidal particle, it
preserves the solenoidal condition on the fluid velocity, and it does not
affect the regularity of the solutions. On the other hand, the change of
coordinates leads to several additional terms in the system of equations on
the fixed reference domain.

The main results on local well-posedness and the global strong well-posedness
for initial data close to constant equilibria rely on fixed point arguments.
The maximal regularity of the associated linearized problem is obtained in
view of the upper triangular structure of the problem by combining the maximal
regularity of the Neumann Laplacian operator with the maximal regularity of
the linearized fluid-structure problem as shown in [15, Theorem 4.1], and by
using a perturbation argument. With regard to the global result, we require
maximal $\mathrm{L}^{p}([0,\infty))$-regularity of the linearization around
constant equilibria. Invoking the fluid-structure semigroup from [36], and
recalling properties of the Neumann Laplacian operator, we first get maximal
$\mathrm{L}^{p}([0,\infty))$-regularity up to a shift.

As the Neumann Laplacian is not invertible, we are using the strategy to split
the equations satisfied by the director in a part with mean value zero in the
spatial domain and the remaining average part. This yields a maximal
$\mathrm{L}^{p}([0,\infty))$-regularity type estimate with exponential decay
of the complete system of equations. It is inspired by the approach of Haak et
al. [21] to global strong well-posedness of the interaction problem of a
compressible Navier-Stokes-Fourier fluid with a rigid body for small initial
data.

The proofs are completed by Lipschitz estimates of the nonlinear terms for
which we rely on appropriate estimates of the terms associated to the
coordinate transform in the local and global setting. In particular, we verify
that no collision of the colloidal particle and the outer boundary of the
viscous incompressible fluid with active liquid crystals can occur provided
the particle starts with an initial distance from the boundary, and the
initial values are taken sufficiently close to the equilibria.

This paper is organized as follows. In Section 2, the main results on the
local strong well-posedness and the global strong well-posedness close to
constant equilibria of the liquid crystal-colloidal interaction problem are
stated. The aim of Section 3 is twofold: On the one hand, it is shown that the
director condition is preserved in the interaction problem. On the other hand,
this section discusses the energy of the interaction problem and deduces the
equilibrium solutions from there. Section 4 is dedicated to the transformation
of the interaction problem to a fixed domain by means of the local transform,
and the main results are reformulated in this reference configuration. In
Section 5, we establish the maximal regularity of the linearized problem.
Moreover, we also show that the linearization around constant equilibria
admits maximal $\mathrm{L}^{p}([0,\infty))$-regularity up to a shift. Section
6 presents the proof of the local strong well-posedness by means of a fixed
point argument. For this, estimates of the nonlinear terms based on a good
control of the terms resulting from the coordinate transform are proved. In
Section 7, we first introduce the maximal regularity type result in order to
get exponential decay and then conclude the proof of the global strong well-
posedness close to constant equilibria from the contraction mapping principle
by invoking adjusted estimates of the transform.

##  2\. Main results

For the statements of the main theorems, we require the precise notion of
solutions, so we first define function spaces on time-dependent domains. In
fact, let $X\colon\mathcal{F}_{T}\to\mathcal{F}_{0}$ be a map such that

$\varphi\colon\mathcal{F}_{T}\to(0,T)\times\mathcal{F}_{0},\enspace(t,x)\mapsto(t,X(t,x))$

is a $\mathrm{C}^{1}$-diffeomorphism, and
$X(\tau,\cdot)\colon\mathcal{F}(\tau)\to\mathcal{F}_{0}$ are
$\mathrm{C}^{2}$-diffeomorphisms for all $\tau\in[0,T]$. We then define

$\mathrm{W}^{s,p}(0,T;\mathrm{W}^{l,q}(\mathcal{F}(\cdot)))\coloneqq\\{f(t,\cdot)\colon\mathcal{F}(t)\to\mathbb{R}:f\circ\varphi\in\mathrm{W}^{s,p}(0,T;\mathrm{W}^{l,q}(\mathcal{F}_{0}))\\}$

for $p,q\in(1,\infty)$, $s\in\\{0,1\\}$ and $l\in\\{0,1,2\\}$. By
$\mathrm{L}_{0}^{q}(\mathcal{F}_{0})$, we will denote the functions in
$\mathrm{L}^{q}(\mathcal{F}_{0})$ with mean value zero. Next, we settle some
notation with regard to the initial data. For this purpose, let us recall the
space of solenoidal vector fields on $\mathrm{L}^{q}(\mathcal{F}_{0})^{3}$
defined by
$\mathrm{L}_{\sigma}^{q}(\mathcal{F}_{0}):=\mathbb{P}\mathrm{L}^{q}(\mathcal{F}_{0})^{3}$,
where $\mathbb{P}$ represents the Helmholtz projection on $\mathcal{F}_{0}$.
Denoting the principle variable by $z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})$,
we then define

(2.1) $\displaystyle\mathrm{X}_{\gamma}\coloneqq\bigl{\\{}$ $\displaystyle
z_{0}\in\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})^{3}\cap\mathrm{L}_{\sigma}^{q}(\mathcal{F}_{0})\times\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}:v_{0}=0,\enspace\partial_{\nu}d_{0}=0,\enspace\text{on}\enspace\partial\mathcal{O},\enspace\text{and}\enspace$
$\displaystyle v_{0}=\ell_{0}+\omega_{0}\times
y,\enspace\partial_{\nu}d_{0}=0,\enspace\text{on}\enspace\partial\mathcal{S}_{0}\bigr{\\}},$

where $\mathrm{B}_{qp}^{s}(\mathcal{F}_{0})$ denotes the Besov space with
parameters $p,q\in(1,\infty)$ and $s\in\mathbb{R}$. The fact that the trace
and the normal derivative are defined is guaranteed by the following
conditions. Let $p,q\in(1,\infty)$ fulfill

(2.2) $\frac{2}{p}+\frac{3}{q}<1.$

We observe that (2.2) yields the embedding
$\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})\hookrightarrow\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})$,
see [45, Theorem 4.6.1]. The first main result of this paper on the local
strong well-posedness of the interaction problem reads as follows.

######  Theorem 2.1.

Let $p,q\in(1,\infty)$ be such that (2.2) holds true, let
$\mathcal{O}\subset\mathbb{R}^{3}$ be a bounded domain of class
$\mathrm{C}^{3}$, and consider the domains $\mathcal{S}_{0}$ and
$\mathcal{F}_{0}$ occupied by the rigid body and the fluid at time zero.
Furthermore, let
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, where
$\mathrm{X}_{\gamma}$ is defined in (2.1) and $|d_{0}|=1$. If for some $r>0$,
it is valid that $\mathrm{dist}(\mathcal{S}_{0},\partial\mathcal{O})>r$, then
there are $T^{\prime}\in(0,T]$ and
$X\in\mathrm{C}^{1}\left([0,T^{\prime}];\mathrm{C}^{2}(\mathbb{R}^{3})^{3}\right)$
such that $X(\tau,\cdot)\colon\mathcal{F}(\tau)\to\mathcal{F}_{0}$ are
$\mathrm{C}^{2}$-diffeomorphisms for all $\tau\in[0,T^{\prime}]$, and the
interaction problem (1.2)–(1.7) admits a unique solution

$\displaystyle u^{\mathcal{F}}$
$\displaystyle\in\mathrm{W}^{1,p}\left(0,T^{\prime};\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right),\enspace\pi^{\mathcal{F}}\in\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{1,q}(\mathcal{F}(\cdot))\cap\mathrm{L}_{0}^{q}(\mathcal{F}(\cdot))\right),$
$\displaystyle\overline{d}$
$\displaystyle\in\mathrm{W}^{1,p}\left(0,T^{\prime};\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right),$
$\displaystyle h^{\prime}$
$\displaystyle\in\mathrm{W}^{1,p}(0,T^{\prime})^{3},\enspace
h\in\mathrm{L}^{\infty}(0,T^{\prime})^{3},\enspace\text{and}\enspace\Omega\in\mathrm{W}^{1,p}(0,T^{\prime})^{3}.$

######  Remark .

  1. (a)

We refer to a solution in the regularity class as in Theorem 2.1 as a strong
solution.

  2. (b)

We can reproduce a similar local-in-time result for forcing terms. For
instance, one can consider external forces and torques in the equations
accounting for the motion of the colloidal particle (1.5). An analogous result
as Theorem 2.1 is then obtained if the forcing terms are in
$\mathrm{L}^{p}(0,T)^{3}$.

  3. (c)

Let us observe that the uniqueness of the pressure is a consequence of the
choice of the intersection with $\mathrm{L}_{0}^{q}(\mathcal{F}(\cdot))$ in
the spatial component.

It is easy to verify that $z_{*}=(0,d_{*},0,0)$, with $|d_{*}|=1$ and $d_{*}$
constant is a stationary solution of the interaction problem (1.2)–(1.7). The
second main result in this paper asserts the global strong well-posedness of
the interaction problem for initial data close to such constant equilibria.

######  Theorem 2.2.

Let $p,q\in(1,\infty)$ be such that (2.2) holds true, let
$\mathcal{O}\subset\mathbb{R}^{3}$ be a bounded domain of class
$\mathrm{C}^{3}$, and consider the domains $\mathcal{S}_{0}$ and
$\mathcal{F}_{0}$ occupied by the rigid body and the fluid at time zero.
Moreover, assume that $\mathrm{dist}(\mathcal{S}_{0},\partial\mathcal{O})>r$
for some $r>0$, and let $d_{*}\in\mathbb{R}^{3}$ constant with $|d_{*}|=1$.
Then there is $\eta_{0}>0$ such that for all $\eta\in(0,\eta_{0})$, there are
two constants $\delta_{0}>0$ and $C>0$ such that for all
$\delta\in(0,\delta_{0})$ and for all
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, with
$\mathrm{X}_{\gamma}$ as in (2.1), $|d_{0}|=1$ and
$\|(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\|_{\mathrm{X}_{\gamma}}\leq\delta$,
the interaction problem (1.2)–(1.7) admits a unique strong solution
$(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d},h^{\prime},\Omega)$ such that

$\displaystyle\quad\left\|\mathrm{e}^{\eta(\cdot)}u^{\mathcal{F}}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{W}^{2,q}(\mathcal{F}(\cdot)))}+\left\|\mathrm{e}^{\eta(\cdot)}\partial_{t}u^{\mathcal{F}}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}(\cdot)))}+\left\|\mathrm{e}^{\eta(\cdot)}u^{\mathcal{F}}\right\|_{\mathrm{L}^{\infty}\left(0,\infty;\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}(\cdot))\right)}$
$\displaystyle+\left\|\mathrm{e}^{\eta(\cdot)}\pi^{\mathcal{F}}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{W}^{1,q}(\mathcal{F}(\cdot))\cap\mathrm{L}_{0}^{q}(\mathcal{F}(\cdot)))}+\left\|\mathrm{e}^{\eta(\cdot)}\nabla\overline{d}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}(\cdot)))}+\left\|\mathrm{e}^{\eta(\cdot)}\nabla^{2}\overline{d}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}(\cdot)))}$
$\displaystyle+\left\|\mathrm{e}^{\eta(\cdot)}\partial_{t}\overline{d}\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}(\cdot)))}+\left\|\overline{d}-d_{*}\right\|_{\mathrm{L}^{\infty}\left(0,\infty;\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}(\cdot))\right)}+\left\|\mathrm{e}^{\eta(\cdot)}h^{\prime}\right\|_{\mathrm{W}^{1,p}(0,\infty)}+\left\|h\right\|_{\mathrm{L}^{\infty}(0,\infty)}$
$\displaystyle+\left\|\mathrm{e}^{\eta(\cdot)}\Omega\right\|_{\mathrm{W}^{1,p}(0,\infty)}\leq
C\delta.$

In addition, we have
$\mathrm{dist}(\mathcal{S}(t),\partial\mathcal{O})>\nicefrac{{r}}{{2}}$ for
all $t\in[0,\infty)$.

The following result discusses the convergence of the solution to the
equilibria in the trace space at an exponential rate. It is an immediate
consequence of Theorem 2.2.

######  Corollary .

Under the assumptions of Theorem 2.2, for a constant $C>0$ independent of
$t>0$, the solution
$(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d},h^{\prime},\Omega)$ to the
interaction problem (1.2)–(1.7) especially satisfies

$\left\|u^{\mathcal{F}}(t,\cdot)\right\|_{\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}(t))}+\|h^{\prime}(t)\|_{\mathbb{R}^{3}}+\|\Omega(t)\|_{\mathbb{R}^{3}}\leq
C\delta\mathrm{e}^{-\eta t},\enspace\text{for all}\enspace t>0.$

##  3\. The director condition, energy and equilibria

We verify that the director part $\overline{d}$ of a strong solution to the
interaction problem (1.2)–(1.7) always satisfies the condition
$|\overline{d}(t,x)|=1$ in Subsection 3.1. Subsection 3.2 is dedicated to
showing that the energy of the interaction system is non-increasing, and to
determining the set of equilibrium solutions.

###  3.1. Preservation of the director condition

Due to the physical interpretation of the director $\overline{d}$, it is
natural to include the condition

(3.1) $|\overline{d}(t,x)|=1,\enspace\text{for
all}\enspace(t,x)\in\mathcal{F}_{T}.$

Assuming $|d_{0}|=1$, we can prove that (3.1) is preserved in view of the
equation (1.2)3 satisfied by $\overline{d}$.

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and let
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, for
$\mathrm{X}_{\gamma}$ as introduced in (2.1), be such that $|d_{0}|=1$.
Moreover, let
$(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d},h^{\prime},\Omega)$ be a
solution to (1.2)–(1.7) satisfying

$(u^{\mathcal{F}},\overline{d})\in\mathrm{W}^{1,p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T;\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right)\times\mathrm{W}^{1,p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T;\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right).$

Then $\left|\overline{d}(t,x)\right|\equiv 1$ holds for all
$(t,x)\in\mathcal{F}_{T}$.

###### Proof.

Let us multiply (1.2)3 by $2\overline{d}(|\overline{d}|^{2}-1)$ and integrate
over $\mathcal{F}(t)$ so that

(3.2)
$\int\limits_{\mathcal{F}(t)}\left(\gamma\left[{\partial_{t}\overline{d}}+(u^{\mathcal{F}}\cdot\nabla)\overline{d}\right]-\lambda\Delta\overline{d}-\lambda|\nabla\overline{d}|^{2}\overline{d}\right)\cdot
2\overline{d}\left(|\overline{d}|^{2}-1\right)\,\mathrm{d}x=0.$

First term of (3.2). We use integration by parts and the Reynolds transport
theorem to obtain

(3.3)
$\displaystyle\quad\int\limits_{\mathcal{F}(t)}\partial_{t}\overline{d}\cdot
2\overline{d}\left(|\overline{d}|^{2}-1\right)\,\mathrm{d}x=\int\limits_{\mathcal{F}(t)}\partial_{t}\overline{d}\cdot
2\overline{d}|\overline{d}|^{2}\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}\partial_{t}\overline{d}\cdot
2\overline{d}\,\mathrm{d}x$
$\displaystyle=\int\limits_{\mathcal{F}(t)}\frac{1}{2}\partial_{t}\left(|\overline{d}|^{2}|\overline{d}|^{2}\right)\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}\partial_{t}\left(|\overline{d}|^{2}\right)\,\mathrm{d}x$
$\displaystyle=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\mathcal{F}(t)}\left(|\overline{d}|^{2}|\overline{d}|^{2}\right)\,\mathrm{d}x-\frac{1}{2}\int\limits_{\partial\mathcal{F}(t)}\left(|\overline{d}|^{2}|\overline{d}|^{2}\right)\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma-\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\mathcal{F}(t)}|\overline{d}|^{2}\,\mathrm{d}x+\int\limits_{\partial\mathcal{F}(t)}|\overline{d}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\mathcal{F}(t)}\left(\frac{1}{2}\left(|\overline{d}|^{2}|\overline{d}|^{2}\right)-|\overline{d}|^{2}+\frac{1}{2}\right)\,\mathrm{d}x-\frac{1}{2}\int\limits_{\partial\mathcal{F}(t)}(|\overline{d}|^{2}|\overline{d}|^{2})\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma+\int\limits_{\partial\mathcal{F}(t)}|\overline{d}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma.$

Second term of (3.2). We use $\operatorname{div}u^{\mathcal{F}}=0\ \mbox{ in
}\mathcal{F}(t)$ and integrate by parts again to get

(3.4)
$\displaystyle\int\limits_{\mathcal{F}(t)}\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\cdot
2\overline{d}\left(|\overline{d}|^{2}-1\right)\,\mathrm{d}x$
$\displaystyle=\int\limits_{\mathcal{F}(t)}2\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\cdot\overline{d}|\overline{d}|^{2}\,\mathrm{d}x-2\int\limits_{\mathcal{F}(t)}\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\cdot\overline{d}\,\mathrm{d}x$
$\displaystyle=\int\limits_{\mathcal{F}(t)}(u^{\mathcal{F}}\cdot\nabla)|\overline{d}|^{2}|\overline{d}|^{2}\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}(u^{\mathcal{F}}\cdot\nabla)|\overline{d}|^{2}\,\mathrm{d}x$
$\displaystyle=\frac{1}{2}\int\limits_{\partial\mathcal{F}(t)}(|\overline{d}|^{2}|\overline{d}|^{2})(u^{\mathcal{F}}\cdot
n)\,\mathrm{d}\Gamma-\int\limits_{\partial\mathcal{F}(t)}|\overline{d}|^{2}(u^{\mathcal{F}}\cdot
n)\,\mathrm{d}\Gamma.$

Third term of (3.2). Another integration by parts leads to

(3.5)
$\displaystyle-2\lambda\int\limits_{\mathcal{F}(t)}\Delta\overline{d}\cdot\overline{d}\left(|\overline{d}|^{2}-1\right)\,\mathrm{d}x$
$\displaystyle=2\lambda\int\limits_{\mathcal{F}(t)}\nabla\overline{d}\cdot\nabla\left[\overline{d}\left(|\overline{d}|^{2}-1\right)\right]\,\mathrm{d}x$
$\displaystyle=2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}\left(|\overline{d}|^{2}-1\right)\,\mathrm{d}x+4\lambda\int\limits_{\mathcal{F}(t)}|\overline{d}|^{2}|\nabla\overline{d}|^{2}\,\mathrm{d}x.$

Fourth term of (3.2). We calculate

(3.6)
$-2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}|\overline{d}|^{2}(|\overline{d}|^{2}-1)\,\mathrm{d}x=-2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}\left(|\overline{d}|^{2}-1\right)(|\overline{d}|^{2}-1)\,\mathrm{d}x-2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}(|\overline{d}|^{2}-1)\,\mathrm{d}x.$

Adding the relations (3.3)–(3.6), we deduce from the equation (3.2) that

(3.7)
$\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\mathcal{F}(t)}\gamma\left(\frac{1}{2}\left(|\overline{d}|^{2}|\overline{d}|^{2}\right)-|\overline{d}|^{2}+\frac{1}{2}\right)\,\mathrm{d}x+4\lambda\int\limits_{\mathcal{F}(t)}|\overline{d}|^{2}|\nabla\overline{d}|^{2}\,\mathrm{d}x-2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}||\overline{d}|^{2}-1|^{2}\,\mathrm{d}x=0.$

Setting $\varphi:=|\overline{d}|^{2}-1$, we derive that

(3.8)
$|\varphi|^{2}=|\overline{d}|^{2}|\overline{d}|^{2}-2|\overline{d}|^{2}+{1},\enspace\text{and}\enspace|\nabla\varphi|^{2}=4|\overline{d}|^{2}|\nabla\overline{d}|^{2}.$

By virtue of (3.8), the equation (3.7) can be rewritten as

$\frac{\gamma}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\mathcal{F}(t)}|\varphi|^{2}\,\mathrm{d}x+\lambda\int\limits_{\mathcal{F}(t)}|\nabla\varphi|^{2}\,\mathrm{d}x=2\lambda\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}|\varphi|^{2}\,\mathrm{d}x.$

As
$\overline{d}\in\mathrm{W}^{1,p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T;\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right)$
and $\gamma,\lambda>0$, for

$\zeta(t):=\int\limits_{\mathcal{F}(t)}|\varphi(t,x)|^{2}\,\mathrm{d}x,$

we find the inequality $\zeta^{\prime}(t)\leq C\zeta(t)$ for some $C>0$. As
$|d_{0}|=1$, we obtain $\zeta(0)=0$, so Gronwall’s inequality yields
$\zeta=0$. This implies $\varphi=0$, and we conclude
$\left|\overline{d}(t,x)\right|\equiv 1$ for all $(t,x)\in\mathcal{F}_{T}$. ∎

###  3.2. Energy and equilibria of the interaction problem

Let us consider the energy of the interaction system (1.2)–(1.7) defined by

(3.9) $\displaystyle\mathrm{E}(t)$
$\displaystyle\coloneqq\mathrm{E}_{\mathrm{kin}}(t)+\mathrm{E}_{\mathrm{pot}}(t)+\mathrm{E}_{\mathrm{trans}}(t)+\mathrm{E}_{\mathrm{rot}}(t)$
$\displaystyle\coloneqq\frac{1}{2}\int\limits_{\mathcal{F}(t)}\left|u^{\mathcal{F}}(t,x)\right|^{2}\,\mathrm{d}x+\frac{1}{2}\int\limits_{\mathcal{F}(t)}\left|\nabla\overline{d}(t,x)\right|^{2}\,\mathrm{d}x+\frac{1}{2}m_{\mathcal{S}}|h^{\prime}(t)|^{2}+\frac{1}{2}(J\Omega\cdot\Omega)(t).$

We now show that $\mathrm{E}(t)$ is non-increasing and determine the set of
equilibria $\mathcal{E}$ of (1.2)–(1.7).

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2) and
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, with
$|d_{0}|=1$ and $\mathrm{X}_{\gamma}$ as defined in (2.1). Moreover, let
$(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d},h^{\prime},\Omega)$ be a
solution to (1.2)–(1.7) satisfying

$\displaystyle(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d},h^{\prime},\Omega)$
$\displaystyle\in\mathrm{W}^{1,p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T;\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right)\times\mathrm{L}^{p}\left(0,T;\mathrm{W}^{1,q}(\mathcal{F}(\cdot))\cap\mathrm{L}_{0}^{q}(\mathcal{F}(\cdot))\right)$
$\displaystyle\quad\times\mathrm{W}^{1,p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}(\cdot))^{3}\right)\cap\mathrm{L}^{p}\left(0,T;\mathrm{W}^{2,q}(\mathcal{F}(\cdot))^{3}\right)\times\mathrm{W}^{1,p}(0,T)^{3}\times\mathrm{W}^{1,p}(0,T)^{3}.$

The energy functional $\mathrm{E}$ defined in (3.9) is non-increasing.
Moreover, the set of equilibria is given by

$\mathcal{E}=\left\\{(0,d_{*},0,0):\Delta d_{*}+|\nabla
d_{*}|^{2}d_{*}=0\enspace\text{in}\enspace\mathcal{F}_{0},\enspace\partial_{\nu}d_{*}=0\enspace\text{on}\enspace\partial\mathcal{F}_{0}\right\\}.$

###### Proof.

Let us first prove that $\mathrm{E}(t)$ is non-increasing along the solutions.
We use integration by parts, the Reynold’s transport theorem, the equations
(1.2)1–(1.2)2, and the boundary conditions (1.6) to derive

(3.10)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}_{\mathrm{kin}}(t)$
$\displaystyle=\int\limits_{\mathcal{F}(t)}\partial_{t}u^{\mathcal{F}}\cdot
u^{\mathcal{F}}\,\mathrm{d}x+\frac{1}{2}\int\limits_{\partial\mathcal{S}(t)}|u^{\mathcal{F}}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma$
$\displaystyle=-\int\limits_{\mathcal{F}(t)}\left(u^{\mathcal{F}}\cdot\nabla\right)u^{\mathcal{F}}\cdot
u^{\mathcal{F}}\,\mathrm{d}x+\int\limits_{\mathcal{F}(t)}\operatorname{div}\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})\cdot
u^{\mathcal{F}}\,\mathrm{d}x+\frac{1}{2}\int\limits_{\partial\mathcal{S}(t)}|u^{\mathcal{F}}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma$
$\displaystyle=-\int\limits_{\mathcal{F}(t)}|\nabla
u^{\mathcal{F}}|^{2}\,\mathrm{d}x+\int\limits_{\mathcal{F}(t)}\nabla\overline{d}(\nabla\overline{d})^{\top}:\nabla
u^{\mathcal{F}}\,\mathrm{d}x-\int\limits_{\partial\mathcal{S}(t)}\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})n\cdot
u^{\mathcal{F}}\,\mathrm{d}\Gamma.$

Another integration by parts in conjunction with the Reynold’s transport
theorem leads to

(3.11)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}_{\mathrm{pot}}(t)=\frac{1}{2}\int\limits_{\mathcal{F}(t)}\partial_{t}|\nabla\overline{d}|^{2}\,\mathrm{d}x+\frac{1}{2}\int\limits_{\partial\mathcal{S}(t)}|\nabla\overline{d}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma.$

By using (1.2)3 and $|\overline{d}|=1$, following from Subsection 3.1, we
obtain the identity

(3.12)
$\displaystyle\quad\int\limits_{\mathcal{F}(t)}|\Delta\overline{d}+|\nabla\overline{d}|^{2}\overline{d}|^{2}\,\mathrm{d}x=\int\limits_{\mathcal{F}(t)}\left(\Delta\overline{d}+|\nabla\overline{d}|^{2}\overline{d}\right)\cdot\left(\partial_{t}\overline{d}+\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\right)\,\mathrm{d}x=-\int\limits_{\mathcal{F}(t)}\partial_{t}(\nabla\overline{d}):\nabla\overline{d}\,\mathrm{d}x$
$\displaystyle\quad+\frac{1}{2}\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}\partial_{t}|\overline{d}|^{2}\,\mathrm{d}x+\int\limits_{\mathcal{F}(t)}|\nabla\overline{d}|^{2}\left(u^{\mathcal{F}}\cdot\nabla\right)\frac{|\overline{d}|^{2}}{2}\,\mathrm{d}x+\int\limits_{\mathcal{F}(t)}\Delta\overline{d}\cdot\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\,\mathrm{d}x$
$\displaystyle=-\frac{1}{2}\int\limits_{\mathcal{F}(t)}\partial_{t}|\nabla\overline{d}|^{2}\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}\nabla\overline{d}:\nabla\left[\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\right]\,\mathrm{d}x.$

Furthermore, we use integration by parts as well as the relation
$\operatorname{div}u^{\mathcal{F}}=0$ in $\mathcal{F}(t)$ to get

(3.13)
$\displaystyle-\int\limits_{\mathcal{F}(t)}\nabla\overline{d}:\nabla\left[\left(u^{\mathcal{F}}\cdot\nabla\right)\overline{d}\right]\,\mathrm{d}x$
$\displaystyle=-\sum\limits_{i,j,k=1}^{3}\int\limits_{\mathcal{F}(t)}\partial_{k}\left(u^{\mathcal{F}}_{i}\partial_{i}\overline{d}_{j}\right)\partial_{k}\overline{d}_{j}\,\mathrm{d}x$
$\displaystyle=-\sum\limits_{i,j,k=1}^{3}\int\limits_{\mathcal{F}(t)}\left[\partial_{k}u^{\mathcal{F}}_{i}\partial_{i}\overline{d}_{j}\partial_{k}\overline{d}_{j}+u^{\mathcal{F}}_{i}\partial_{i}\partial_{k}\overline{d}_{j}\partial_{k}\overline{d}_{j}\right]\,\mathrm{d}x$
$\displaystyle=-\int\limits_{\mathcal{F}(t)}\nabla\overline{d}(\nabla\overline{d})^{\top}:\nabla
u^{\mathcal{F}}\,\mathrm{d}x-\frac{1}{2}\int\limits_{\partial\mathcal{S}(t)}|\nabla\overline{d}|^{2}\left(u^{\mathcal{F}}\cdot
n\right)\,\mathrm{d}\Gamma.$

Hence, combining (3.11)–(3.13), we infer that

(3.14)
$\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}_{\mathrm{pot}}(t)=-\int\limits_{\mathcal{F}(t)}|\Delta\overline{d}+|\nabla\overline{d}|^{2}|\overline{d}|^{2}\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}\nabla\overline{d}(\nabla\overline{d})^{\top}:\nabla
u^{\mathcal{F}}\,\mathrm{d}x.$

The interface condition
$u^{\mathcal{F}}=h^{\prime}(t)+\Omega(t)\times(x-h(t))$ on
$\partial\mathcal{S}(t)$ and the rigid body equations (1.5) yield

(3.15)
$\displaystyle\quad-\int\limits_{\partial\mathcal{S}(t)}\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})n\cdot
u^{\mathcal{F}}\,\mathrm{d}\Gamma$
$\displaystyle=-h^{\prime}(t)\cdot\int\limits_{\partial\mathcal{S}(t)}\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})n\,\mathrm{d}\Gamma-\Omega(t)\cdot\int\limits_{\partial\mathcal{S}(t)}(x-h(t))\times\Sigma(u^{\mathcal{F}},\pi^{\mathcal{F}},\overline{d})n\,\mathrm{d}\Gamma$
$\displaystyle=-h^{\prime}(t)\cdot
m_{\mathcal{S}}h^{\prime\prime}(t)-\Omega(t)\cdot(J\Omega)^{\prime}(t)=-\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}_{\mathrm{trans}}(t)-\Omega(t)\cdot(J\Omega)^{\prime}(t).$

From (1.3), it follows that

(3.16)
$J(t)\Omega(t)^{\prime}\cdot\Omega(t)=J(t)\Omega(t)\cdot\Omega^{\prime}(t),\enspace\text{for
all}\enspace t>0.$

Using Sylvester’s law from (1.3) and (3.16), we obtain

(3.17)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}_{\mathrm{rot}}(t)$
$\displaystyle=\frac{1}{2}\left(\Omega(t)\cdot(J\Omega)^{\prime}(t)+\Omega^{\prime}(t)\cdot(J\Omega)(t)\right)$
$\displaystyle=\frac{1}{2}\left[\Omega(t)\cdot(J\Omega^{\prime}+\Omega\times
J\Omega)(t)+\Omega^{\prime}(t)\cdot(J\Omega)(t)\right]=\Omega(t)\cdot(J\Omega)^{\prime}(t).$

Therefore, we concatenate (3.10), (3.14), (3.15) and (3.17) to conclude

(3.18)
$\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}(t)=-\int\limits_{\mathcal{F}(t)}|\nabla
u^{\mathcal{F}}|^{2}\,\mathrm{d}x-\int\limits_{\mathcal{F}(t)}|\Delta\overline{d}+|\nabla\overline{d}|^{2}\overline{d}|^{2}\,\mathrm{d}x.$

The equilibria $\mathcal{E}$ are now precisely the critical points of
$\mathrm{E}$, so suppose that
$\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{E}(t)\big{|}_{t=t_{0}}=0$ at $t_{0}>0$.
Then (3.18) and $u^{\mathcal{F}}=0$ on $\partial\mathcal{O}$ imply
$u^{\mathcal{F}}=0$ in $\mathcal{F}(t_{0})$, and $\overline{d}$ satisfies
$\Delta\overline{d}+|\nabla\overline{d}|^{2}\overline{d}=0$ in
$\mathcal{F}(t_{0})$. Moreover, from
$u^{\mathcal{F}}=h^{\prime}(t)+\Omega(t)\times(x-h(t))$ on
$\partial\mathcal{S}(t)$ and $u=0$ in $\mathcal{F}(t)$, we conclude by a
slight adjustment of [41, Lemma 2.1] to the 3D case that $h^{\prime}=0$ and
$\Omega=0$, and $\mathcal{E}$ has the shape as asserted. ∎

##  4\. Change of variables

The transformation to a fixed domain presented in the sequel is classical, see
for instance the work of Inoue and Wakimoto [28]. However, we provide it here
for convenience. To this end, let us denote by $m(t)$ the skew-symmetric
matrix satisfying $m(t)x=\Omega(t)\times x$. We then consider the initial
value problem

(4.1) $\left\\{\begin{aligned}
\partial_{t}X_{0}(t,y)&=m(t)(X_{0}(t,y)-h(t))+h^{\prime}(t),&&\enspace\text{on}\enspace(0,T)\times\mathbb{R}^{3},\\\
X_{0}(0,y)&=y,&&\enspace\text{for}\enspace
y\in\mathbb{R}^{3}.\end{aligned}\right.$

The solution of (4.1) is given by $X_{0}(t,y)=Q(t)y+h(t)$ for
$Q(t)\in\mathrm{SO}(3)$, where $Q\in\mathrm{W}^{2,p}(0,T)^{3\times 3}$ is
valid provided $h^{\prime}$, $\Omega\in\mathrm{W}^{1,p}(0,T)^{3}$. The inverse
$Y_{0}(t,x)=Q^{\top}(t)(x-h(t))$ of $X_{0}(t)$ solves

$\left\\{\begin{aligned}
\partial_{t}Y_{0}(t,x)&=-M(t)Y_{0}(t,x)-\ell(t),&&\enspace\text{on}\enspace(0,T)\times\mathbb{R}^{3},\\\
Y_{0}(0,x)&=x,&&\enspace\text{for}\enspace
y\in\mathbb{R}^{3},\end{aligned}\right.$

where $M(t)\coloneqq Q^{\top}(t)m(t)Q(t)$ and $\ell(t)\coloneqq
Q^{\top}(t)h^{\prime}(t)$. Next, we adjust the diffeomorphisms $X_{0}$ and
$Y_{0}$ of $\mathcal{F}(t)$ as well as $\mathcal{S}(t)$ such that the
transform only acts in a suitable open neighborhood of the moving colloidal
particle. To this end, we define a new diffeomorphism $X$ implicitly by
demanding that it solves the initial value problem

(4.2) $\left\\{\begin{aligned}
\partial_{t}X(t,y)&=b(t,X(t,y)),&&\enspace\text{on}\enspace(0,T)\times\mathbb{R}^{3},\\\
X(0,y)&=y,&&\enspace\text{for}\enspace
y\in\mathbb{R}^{3}.\end{aligned}\right.$

The right-hand side $b$ will be constructed such that the transform fulfills
the above requirements. We assume that the colloidal particle starts from a
position with strictly positive distance from the outer boundary, i.e.,
$\mathrm{dist}(\mathcal{S}_{0},\partial\mathcal{O})>r$ for some $r>0$. As the
body velocity is continuous, we only consider the solution up to a time for
which a small distance such as $\nicefrac{{r}}{{2}}$ of the rigid body and the
outer boundary is preserved. Accordingly, we define a cut-off function
$\chi\in\mathrm{C}^{\infty}(\mathbb{R}^{3};[0,1])$ by

$\chi(x)\coloneqq\left\\{\begin{aligned}
1,&\enspace\text{if}\enspace\mathrm{dist}(x,\partial\mathcal{O})\geq r,\\\
0,&\enspace\text{if}\enspace\mathrm{dist}(x,\partial\mathcal{O})\leq\nicefrac{{r}}{{2}}.\end{aligned}\right.$

Equipped with this cut-off function, we define the right-hand side
$b\colon[0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ of (4.2) by

(4.3)
$b(t,x)\coloneqq\chi(x)[m(t)(x-h(t))+h^{\prime}(t)]-B_{\mathcal{F}_{0}}(\nabla\chi(\cdot)[m(t)(\cdot-h(t))+h^{\prime}(t)])(x),$

where
$B_{\mathcal{F}_{0}}\colon\mathrm{C}_{c}^{\infty}(\mathcal{F}_{0})\to\mathrm{C}_{c}^{\infty}(\mathcal{F}_{0})^{3}$
represents the Bogovskiĭ operator associated to $\mathcal{F}_{0}$. This
operator is bounded and yields $\mathrm{div}B_{\mathcal{F}_{0}}g=g$ provided
$\int_{\mathcal{F}_{0}}g\,\mathrm{d}y=0$. We also refer to [3, 12, 16] for
further details on the Bogovskiĭ operator. In our situation, it readily
follows that $\nabla\chi(x)[m(t)(x-h(t))+h^{\prime}(t)]$ has mean value zero
in $\mathcal{F}_{0}$, so the correction by the Bogovskiĭ results in
$\operatorname{div}(b(t,\cdot))=0$ for all $t\in[0,T]$. Consequently,
$b\in\mathrm{W}^{1,p}\left(0,T;\mathrm{C}_{c,\sigma}(\mathcal{F}_{0})^{3}\right)$
as well as $b|_{\partial\mathcal{S}_{0}}=m(x-h)+h^{\prime}$.

The Picard-Lindelöf theorem yields the existence of a unique solution
$X\in\mathrm{C}^{1}\left((0,T);\mathrm{C}^{\infty}(\mathbb{R}^{3})^{3}\right)$
to (4.2) for given $h^{\prime}$, $\Omega\in\mathrm{W}^{1,p}(0,T)^{3}$. We also
observe that the solution has continuous mixed partial derivatives
$\frac{\partial^{|\alpha|+1}X}{\partial t(\partial y_{j})^{\alpha}}$ and
$\frac{\partial^{|\alpha|}X}{(\partial y_{j})^{\alpha}}$ for a multi-index
$\alpha\in\mathbb{N}_{0}^{3}$. The uniqueness of the solution implies that the
function $X(t,\cdot)$ is bijective, with inverse denoted by $Y(t,\cdot)$.
Thanks to $\operatorname{div}b=0$, Liouville’s theorem yields that $X$ and $Y$
are both volume-preserving, so
$\mathrm{J}_{X}(t,y)\mathrm{J}_{Y}(t,X(t,y))=\operatorname{Id}$ and
$\det\mathrm{J}_{X}(t,y)=\det\mathrm{J}_{Y}(t,x)=1$, where, as usual,
$\mathrm{J}_{X}$ and $\mathrm{J}_{Y}$ represent the Jacobian matrices of $X$
and $Y$. Let us observe that the inverse transform $Y$ of $X$ solves the
initial value problem

(4.4) $\left\\{\begin{aligned}
\partial_{t}Y(t,x)&=b^{(Y)}(t,Y(t,x)),&&\enspace\text{on}\enspace(0,T)\times\mathbb{R}^{3},\\\
Y(0,x)&=x,&&\enspace\text{for}\enspace
x\in\mathbb{R}^{3},\end{aligned}\right.$

with right-hand side given by

(4.5) $b^{(Y)}(t,y)\coloneqq-\mathrm{J}_{X}^{-1}(t,y)b(t,X(t,y)).$

We emphasize that $b^{(Y)}$ as well as $Y$ have the same space and time
regularity as $b$ and $X$, and that the interval of existence of the solution
is restricted by the condition
$\mathrm{dist}(\mathcal{S}(t),\partial\mathcal{O})>\nicefrac{{r}}{{2}}$ for
all $t\in(0,T)$.

As it does not affect the analysis, we will assume that $\gamma$, $\lambda$
and $\mu$ are equal to one in the remainder of this article. Furthermore, as a
preparation for the proof of the global well-posedness, we subtract an
arbitrary constant vector $d_{*}\in\mathbb{R}^{3}$ with $|d_{*}|=1$ in the
$d$-variable, which does not change the analysis of the local strong well-
posedness significantly. For $X\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$
as in (4.2) and $(t,y)\in(0,T)\times\mathbb{R}^{3}$, we define

$\left\\{\begin{aligned} v(t,y)&\coloneqq
J_{Y}(t,X(t,y))u^{\mathcal{F}}(t,X(t,y)),\enspace
p(t,y)\coloneqq\pi^{\mathcal{F}}(t,X(t,y)),\\\
d(t,y)&\coloneqq\overline{d}(t,X(t,y))-d_{*},\\\ \ell(t)&\coloneqq
Q(t)^{\top}h^{\prime}(t),\enspace\text{and}\enspace\omega(t)\coloneqq
Q(t)^{\top}\Omega(t).\end{aligned}\right.$

Moreover, we set the transformed stress tensor as

$\sigma(v(t,y),p(t,y),d(t,y))\coloneqq
Q(t)^{\top}\Sigma(Q(t)v(t,y),p(t,y),d(t,y))Q(t).$

The metric contravariant $(g^{ij})$ and covariant tensors $(g_{ij})$ and the
Christoffel symbol are given by

$g^{ij}=\sum_{k=1}^{3}(\partial_{k}Y_{i})(\partial_{k}Y_{j}),\enspace
g_{ij}=\sum_{k=1}^{3}(\partial_{i}X_{k})(\partial_{j}X_{k}),\enspace\text{and}\enspace\Gamma^{i}_{jk}=\sum_{l=1}^{3}g^{jk}(\partial_{i}g_{jl}+\partial_{j}g_{lk}-\partial_{k}g_{ij}).$

Having these terms at hand, we introduce the transformed Laplacian operators
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ corresponding to the velocity and the
director field, respectively. They take the shape

$\displaystyle(\mathcal{L}_{1}v)_{i}$
$\displaystyle=\sum_{j,k=1}^{3}\partial_{j}(g^{jk}\partial_{k}v_{i})+2\sum_{j,k,l=1}^{3}g^{kl}\Gamma_{jk}^{i}\partial_{l}v_{j}+\sum_{j,k,l=1}^{3}\left(\partial_{k}(g^{kl}\Gamma^{i}_{kl})+\sum_{m=1}^{3}g^{kl}\Gamma^{m}_{jl}\Gamma^{i}_{km}\right)v_{j},\enspace\text{and}\enspace$
$\displaystyle(\mathcal{L}_{2}d)_{i}$
$\displaystyle=\sum\limits_{j,k=1}^{3}\left((\partial_{k}\partial_{j}d_{i})g^{jk}\right)+\sum\limits_{k=1}^{3}(\Delta
Y_{k})(\partial_{k}d_{i}).$

Defining $B(d)h$ with $B(d)d=\operatorname{div}(\nabla d(\nabla d)^{\top})$ by

(4.6) $[B(d)h]_{i}\coloneqq\partial_{i}d_{l}\Delta
h_{l}+\partial_{k}d_{l}\partial_{k}\partial_{i}h_{l},$

we calculate the transformed terms associated to the gradient of the pressure,
to $B(d)h$, to the time derivative and to the convective term in the fluid
equation

$\displaystyle(\mathcal{G}p)_{i}=\sum\limits_{j=1}^{3}g^{ij}\partial_{j}p,\enspace(\mathcal{B}(d)h)_{i}$
$\displaystyle=\sum\limits_{l=1}^{3}\left((\mathcal{L}_{2}h)_{l}\sum\limits_{m=1}^{3}{\partial_{m}d_{l}}{\partial_{i}Y_{m}}\right)$
$\displaystyle\quad+\sum\limits_{k,l,m=1}^{3}\left({\partial_{m}d_{l}}{\partial_{k}Y_{m}}\right)\left(\sum\limits_{j=1}^{3}(\partial_{j}\partial_{m}h_{l})(\partial_{k}Y_{j}\partial_{i}Y_{m})+(\partial_{m}h_{l})(\partial_{k}\partial_{i}Y_{m})\right),$
$(\mathcal{M}v)_{i}=\sum\limits_{j=1}^{3}\partial_{t}Y_{j}\partial_{j}v_{i}+\sum\limits_{j,k=1}^{3}\left(\Gamma^{i}_{jk}\partial_{t}Y_{k}+(\partial_{k}Y_{i})(\partial_{j}\partial_{t}X_{k})\right)v_{j},\enspace\text{and}\enspace(\mathcal{N}(v))_{i}=\sum\limits_{j=1}^{3}v_{j}\partial_{j}v_{i}+\sum\limits_{j,k=1}^{3}(\Gamma^{i}_{jk}v_{j}v_{k}).$

It is then possible to rewrite the system (1.2)–(1.7) on the cylindrical
domain $(0,T)\times\mathcal{F}_{0}$ as

(4.7) $\left\\{\begin{aligned}
\partial_{t}v+(\mathcal{M}-\mathcal{L}_{1})v+\mathcal{N}(v)+\mathcal{G}p+\mathcal{B}(d)d&=0,&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
\operatorname{div}v&=0,&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
{\partial_{t}d}+\nabla
d\cdot\partial_{t}Y+(v\cdot\nabla)d-\mathcal{L}_{2}d&=|\nabla
d|^{2}(d+d_{*}),&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
m_{\mathcal{S}}\ell^{\prime}+m_{\mathcal{S}}(\omega\times\ell)&=-\int\limits_{\partial\mathcal{S}_{0}}\sigma(v,p,d)N\,\mathrm{d}\Gamma,&&\enspace\text{in}\enspace(0,T),\\\
J_{0}\omega^{\prime}-\omega\times(J_{0}\omega)&=-\int\limits_{\partial\mathcal{S}_{0}}y\times\sigma(v,p,d)N\,\mathrm{d}\Gamma,&&\enspace\text{in}\enspace(0,T),\end{aligned}\right.$

with the boundary conditions

(4.8)
$v=0,\enspace\partial_{\nu}d=0\enspace\text{on}\enspace(0,T)\times\partial\mathcal{O},\enspace\text{and}\enspace
v=\ell+\omega\times
y,\enspace\partial_{\nu}d=0\enspace\text{on}\enspace\partial\mathcal{S}_{0},$

and the initial conditions, which are slightly adjusted for the director part
by the transform,

(4.9) $v(0)=v_{0},\enspace
d(0)=d_{0}-d_{*},\enspace\ell(0)=\ell_{0}\enspace\text{and}\enspace\omega(0)=\omega_{0}.$

We can now reformulate the first main result, Theorem 2.1, on the reference
configuration.

######  Theorem 4.1.

Let $p,q\in(1,\infty)$ be such that (2.2) holds true, let
$\mathcal{O}\subset\mathbb{R}^{3}$ be a bounded domain of class
$\mathrm{C}^{3}$, and consider the domains $\mathcal{S}_{0}$ and
$\mathcal{F}_{0}$ occupied by the rigid body and the fluid at initial time.
Furthermore, let
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, with
$\mathrm{X}_{\gamma}$ as defined in (2.1), and $|d_{0}|=1$. If for some $r>0$,
it is valid that $\mathrm{dist}(\mathcal{S}_{0},\partial\mathcal{O})>r$, then
there are $T^{\prime}\in(0,T]$ and
$X\in\mathrm{C}^{1}\left([0,T^{\prime}];\mathrm{C}^{2}(\mathbb{R}^{3})^{3}\right)$
such that $X(\tau,\cdot)\colon\mathcal{F}(\tau)\to\mathcal{F}_{0}$ are
$\mathrm{C}^{2}$-diffeomorphisms for all $\tau\in[0,T^{\prime}]$, and the
transformed interaction problem (4.7)–(4.9) admits a unique solution

$\displaystyle v$
$\displaystyle\in\mathrm{W}^{1,p}\left(0,T^{\prime};\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\right)\cap\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\right),\enspace
p\in\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{1,q}(\mathcal{F}_{0})\cap\mathrm{L}_{0}^{q}(\mathcal{F}_{0})\right),$
$\displaystyle d$
$\displaystyle\in\mathrm{W}^{1,p}\left(0,T^{\prime};\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\right)\cap\mathrm{L}^{p}\left(0,T^{\prime};\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\right),$
$\displaystyle\ell$
$\displaystyle\in\mathrm{W}^{1,p}(0,T^{\prime})^{3},\enspace\text{and}\enspace\omega\in\mathrm{W}^{1,p}(0,T^{\prime})^{3}.$

Next, we rewrite the second main result, Theorem 2.2, on the fixed domain.

######  Theorem 4.2.

Let $p,q\in(1,\infty)$ be such that (2.2) holds true, let
$\mathcal{O}\subset\mathbb{R}^{3}$ be a bounded domain of class
$\mathrm{C}^{3}$, and consider the domains $\mathcal{S}_{0}$ and
$\mathcal{F}_{0}$ occupied by the rigid body and the fluid at time zero.
Moreover, assume that $\mathrm{dist}(\mathcal{S}_{0},\partial\mathcal{O})>r$
for some $r>0$, and let $d_{*}\in\mathbb{R}^{3}$ constant with $|d_{*}|=1$.
Then there is $\eta_{0}>0$ such that for all $\eta\in(0,\eta_{0})$, there are
two constants $\delta_{0}$ and $C>0$ such that for all
$\delta\in(0,\delta_{0})$ and for all
$z_{0}=(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, with
$\mathrm{X}_{\gamma}$ as in (2.1), $|d_{0}|=1$ and
$\|(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\|_{\mathrm{X}_{\gamma}}\leq\delta$,
then
$X\in\mathrm{C}^{1}\left([0,\infty);\mathrm{C}^{2}(\mathbb{R}^{3})^{3}\right)$
is such that $X(\tau,\cdot)\colon\mathcal{F}(\tau)\to\mathcal{F}_{0}$ are
$\mathrm{C}^{2}$-diffeomorphisms for all $\tau\in[0,\infty)$, and the
transformed interaction problem (4.7)–(4.9) admits a unique strong solution
$(v,p,d,\ell,\omega)$ with

$\displaystyle\quad\left\|\mathrm{e}^{\eta(\cdot)}v\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{W}^{2,q}(\mathcal{F}_{0}))}+\left\|\mathrm{e}^{\eta(\cdot)}\partial_{t}v\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}_{0}))}+\left\|\mathrm{e}^{\eta(\cdot)}v\right\|_{\mathrm{L}^{\infty}\left(0,\infty;\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})\right)}$
$\displaystyle+\left\|\mathrm{e}^{\eta(\cdot)}p\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{W}^{1,q}(\mathcal{F}_{0})\cap\mathrm{L}_{0}^{q}(\mathcal{F}_{0}))}+\left\|\mathrm{e}^{\eta(\cdot)}\nabla
d\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}_{0}))}+\left\|\mathrm{e}^{\eta(\cdot)}\nabla^{2}d\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}_{0}))}$
$\displaystyle+\left\|\mathrm{e}^{\eta(\cdot)}\partial_{t}d\right\|_{\mathrm{L}^{p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}_{0}))}+\left\|d\right\|_{\mathrm{L}^{\infty}\left(0,\infty;\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})\right)}+\left\|\mathrm{e}^{\eta(\cdot)}\ell\right\|_{\mathrm{W}^{1,p}(0,\infty)}+\left\|\mathrm{e}^{\eta(\cdot)}\omega\right\|_{\mathrm{W}^{1,p}(0,\infty)}\leq
C\delta.$

##  5\. Linear theory

In this section, we establish maximal regularity of the linearized interaction
problem for the fixed point procedure of the local strong well-posedness.With
regard to the global strong well-posedness, we show that the linearization at
a constant equilibrium admits maximal $\mathrm{L}^{p}([0,\infty))$-regularity
up to a shift.

In the sequel, we introduce some further spaces to shorten notation. For this
purpose, let $0<T\leq\infty$ as well as $p$, $q\in(1,\infty)$. We introduce
the space for the terms on the right-hand side, and it is given by

(5.1)
$\mathbb{F}_{T}\coloneqq\mathbb{F}_{T}^{v}\times\mathbb{F}_{T}^{d}\times\mathbb{F}_{T}^{\ell}\times\mathbb{F}_{T}^{\omega}\coloneqq\mathrm{L}^{p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\right)\times\mathrm{L}^{p}\left(0,T;\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\right)\times\mathrm{L}^{p}(0,T)^{3}\times\mathrm{L}^{p}(0,T)^{3}.$

Furthermore, for $\mathrm{L}_{\sigma}^{q}(\mathcal{F}_{0})$ denoting the
divergence free vector fields in $\mathrm{L}^{q}(\mathcal{F}_{0})^{3}$ and
$z=(v,d,\ell,\omega)$, we set

(5.2) $\displaystyle\mathrm{X}_{0}\coloneqq$
$\displaystyle\mathrm{L}_{\sigma}^{q}(\mathcal{F}_{0})\times\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3},\enspace\text{and}\enspace$
$\displaystyle\mathrm{X}_{1}\coloneqq$
$\displaystyle\bigl{\\{}z\in\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\cap\mathrm{L}_{\sigma}^{q}(\mathcal{F}_{0})\times\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}:v=0,\enspace\partial_{\nu}d=0,\enspace\text{on}\enspace\partial\mathcal{O},\enspace\text{and}\enspace$
$\displaystyle\enspace v=\ell+\omega\times
y,\enspace\partial_{\nu}d=0,\enspace\text{on}\enspace\partial\mathcal{S}_{0}\bigr{\\}}.$

Similarly as in [2, Lemma 5.3], $\mathrm{X}_{\gamma}$ can be obtained from
$\mathrm{X}_{0}$ and $\mathrm{X}_{1}$ by real interpolation. In order to
simplify the notation, for $j\in\\{v,d\\}$, we will also use
$\|\cdot\|_{\mathrm{X}_{0}^{j}}$, $\|\cdot\|_{\mathrm{X}_{1}^{j}}$ and
$\|\cdot\|_{\mathrm{X}_{\gamma}^{j}}$ to denote
$\|\cdot\|_{\mathrm{L}^{q}(\mathcal{F}_{0})}$,
$\|\cdot\|_{\mathrm{W}^{2,q}(\mathcal{F}_{0})}$ and
$\|\cdot\|_{\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})}$,
respectively, where we remark that the norms are not affected by the boundary
conditions, or by the consideration of closed subspaces. Given
$\mathrm{X}_{0}$ and $\mathrm{X}_{1}$, we further define the solution space
$\mathbb{E}_{T}$ by

(5.3)
$\mathbb{E}_{T}\coloneqq\mathrm{W}^{1,p}(0,T;\mathrm{X}_{0})\cap\mathrm{L}^{p}(0,T;\mathrm{X}_{1}).$

We employ the notation $\prescript{}{0}{\mathbb{E}_{T}}$ for the situation of
homogeneous initial values. As above, we use $\|\cdot\|_{\mathbb{E}_{T}^{j}}$
to denote
$\|\cdot\|_{\mathrm{W}^{1,p}(0,T;\mathrm{L}^{q}(\mathcal{F}_{0}))\cap\mathrm{L}^{p}(0,T;\mathrm{W}^{2,q}(\mathcal{F}_{0}))}$
for $j\in\\{v,d\\}$. In order to account for the pressure, we set

(5.4)
$\tilde{\mathbb{E}}_{T}\coloneqq\mathbb{E}_{T}\times\mathbb{E}_{T}^{p}\coloneqq\mathbb{E}_{T}\times\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0})\cap\mathrm{L}_{0}^{q}(\mathcal{F}_{0})),$

where the intersection with $\mathrm{L}_{0}^{q}(\mathcal{F}_{0}))$ is chosen
to get uniqueness of the pressure.

Concerning the linearization, we do not include the part of the stress tensor
linked to the director $d$, but we will treat this term as a nonlinear term on
the right-hand side. In that respect, we define

(5.5) $\mathrm{T}(v,p)\coloneqq 2\mathbb{D}v-p\operatorname{Id}.$

For $d^{*}\in\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})$, where
$\mathrm{BUC}([0,T])$ denotes the bounded and uniformly continuous functions
on $[0,T]$, determined precisely later and $B(d^{*})d$ as in (4.6) as well as
$d_{*}\in\mathbb{R}^{3}$ constant such that $|d_{*}|=1$, the linearization
reads as

(5.6) $\left\\{\begin{aligned}
\partial_{t}{v}-\Delta{v}+\nabla{p}+B(d^{*})d&=f_{1},\enspace\operatorname{div}{v}=0,\enspace\text{and}\enspace{\partial_{t}{d}}-\Delta{d}=f_{2},&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
m_{\mathcal{S}}({\ell})^{\prime}+\int_{\partial\mathcal{S}_{0}}\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=f_{3},&&\enspace\text{in}\enspace(0,T),\\\
J_{0}({\omega})^{\prime}+\int_{\partial\mathcal{S}_{0}}y\times\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=f_{4},&&\enspace\text{in}\enspace(0,T),\\\
{v}(t,y)&={\ell}(t)+{\omega}(t)\times
y,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{S}_{0},\\\
{v}(t,y)&=0,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{O},\\\
\partial_{\nu}d&=0,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{F}_{0},\\\
v(0)=v_{0},\enspace
d(0)&=d_{0}-d_{*},\enspace\ell(0)=\ell_{0},\enspace\text{and}\enspace\omega(0)=\omega_{0},\end{aligned}\right.$

where $(f_{1},f_{2},f_{3},f_{4})\in\mathbb{F}_{T}$ and
$(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$. First, we discuss
the existence of a so-called reference solution to capture the initial values.

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and consider
$(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$, where
$\mathrm{X}_{\gamma}$ was introduced in (2.1). Moreover, consider (5.6) with

$\partial_{t}{v}-\Delta{v}+\nabla{p}+B(d^{*})d=f_{1}\enspace\text{replaced
by}\enspace\partial_{t}{v}-\Delta{v}+\nabla{p}=f_{1}$

as well as $(f_{1},f_{2},f_{3},f_{4})=(0,0,0,0)$. Then there is a unique
solution
$(z^{*},p^{*})=(v^{*},d^{*},\ell^{*},\omega^{*},p^{*})\in\tilde{\mathbb{E}}_{T}$
to the resulting system.

###### Proof.

Let us observe that the equation for $d$ is completely decoupled. On the one
hand, this allows us to use $\mathbb{E}_{T}^{d}$ to denote the $d$-part of the
maximal regularity space. On the other hand, we can then invoke the well known
maximal regularity of the Neumann Laplacian to obtain
$d^{*}\in\mathbb{E}_{T}^{d}$ satisfying the heat equation with homogeneous
right-hand side and initial value $d_{0}-d_{*}\in\mathrm{X}_{\gamma}^{d}$. For
the remaining fluid-structure part, we refer for instance to [15, Theorem 4.1]
and its analogue in the present setting, where we observe that
$z_{0}\in\mathrm{X}_{\gamma}$ means precisely that the compatibility
conditions $(CN)$ from [15, p. 1398] adjusted to our situation are satisfied.
∎

######  Remark .

Let
$(z^{*},p^{*})=(v^{*},d^{*},\ell^{*},\omega^{*},p^{*})\in\tilde{\mathbb{E}}_{T_{0}}$
be the reference solution on $(0,T_{0})$ resulting from Section 5. The part
$d^{*}$ is obtained by a convolution of the heat semigroup with the initial
data $d_{0}-d_{*}$, so its norm tends to zero as $T\to 0$. For the fluid-
structure part, it can also be shown that the norm of the reference solution
shrinks to zero, see for example [15, Lemma 6.6]. Hence, we get

$C_{T}^{*}\coloneqq\|z^{*}\|_{\mathbb{E}_{T}}+\|p^{*}\|_{\mathbb{E}_{T}^{p}}\to
0\enspace\text{for}\enspace T\to 0.$

The following well known result provides uniform-in-time estimates of the
trace space norms of the principle variable and the $d$-part by the initial
data and the norm of the solution, where the constant in the estimate is time-
independent.

######  Lemma .

Let $z\in\mathbb{E}_{T}$. Then for a constant $C>0$ independent of $T$, it
holds that

$\sup_{t\in[0,T]}\|z(t)\|_{\mathrm{X}_{\gamma}}\leq
C\left(\|z(0)\|_{\mathrm{X}_{\gamma}}+\|z\|_{\mathbb{E}_{T}}\right),\enspace\text{and}\enspace\sup_{t\in[0,T]}\|d(t)\|_{\mathrm{X}_{\gamma}^{d}}\leq
C\left(\|d(0)\|_{\mathrm{X}_{\gamma}^{d}}+\|d\|_{\mathbb{E}_{T}^{d}}\right).$

Next, we prove the maximal regularity of the linearized problem (5.6).

######  Proposition .

Let $0<T_{0}\leq\infty$, $T\in(0,T_{0})$, $p,q\in(1,\infty)$ such that (2.2)
holds, and consider the reference solution $d^{*}$ from Section 5,
$(f_{1},f_{2},f_{3},f_{4})\in\mathbb{F}_{T}$ and
$(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})=(0,0,0,0)$. Then there is a unique
solution
$(\tilde{v},\tilde{d},\tilde{\ell},\tilde{\omega},\tilde{p})\in\prescript{}{0}{\tilde{\mathbb{E}}_{T}}$
to (5.6) such that for the maximal regularity constant
$C_{\mathrm{MR}}=C_{\mathrm{MR}}(T_{0})>0$ independent of $T$, we have the
estimate

$\left\|(\tilde{v},\tilde{d},\tilde{\ell},\tilde{\omega},\tilde{p})\right\|_{\tilde{\mathbb{E}}_{T}}\leq
C_{\mathrm{MR}}\cdot\|(f_{1},f_{2},f_{3},f_{4})\|_{\mathbb{F}_{T}}.$

###### Proof.

The proof relies on the upper triangular structure of the linearized
interaction problem and the maximal regularity of the fluid-structure system
as established in [15, Theorem 4.1] in conjunction with the maximal regularity
of the Neumann Laplacian operator. More precisely, thanks to
$f_{2}\in\mathbb{F}_{T}^{d}$ and $d(0)=0$, the maximal regularity of the
Neumann Laplacian operator first yields $\tilde{d}\in\mathbb{E}_{T}^{d}$ such
that $\|\tilde{d}\|_{\mathbb{E}_{T}}\leq C\|f_{2}\|_{\mathbb{F}_{T}}$, and the
constant $C>0$ is $T$-independent in view of the homogeneous initial values.
Next, we can insert the resulting $\tilde{d}$ into the fluid equation and
define $\tilde{f_{1}}\coloneqq f_{1}-B(d^{*})\tilde{d}\in\mathbb{F}_{T}^{v}$.
It then remains to investigate the system given by

$\left\\{\begin{aligned}
\partial_{t}{v}-\Delta{v}+\nabla{p}&=\tilde{f_{1}},\enspace\text{and}\enspace\operatorname{div}{v}=0,&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
m_{\mathcal{S}}({\ell})^{\prime}+\int_{\partial\mathcal{S}_{0}}\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=f_{3},\enspace\text{and}\enspace
J_{0}({\omega})^{\prime}+\int_{\partial\mathcal{S}_{0}}y\times\mathrm{T}(v,p)N\,\mathrm{d}\Gamma=f_{4},&&\enspace\text{in}\enspace(0,T),\\\
{v}&={\ell}+{\omega}\times
y,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{S}_{0},\\\
{v}&=0,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{O},\\\
v(0)=0,\enspace\ell(0)&=0,\enspace\text{and}\enspace\omega(0)=0.\end{aligned}\right.$

From [15, Theorem 4.1], it then follows that this system admits a unique
solution $(\tilde{v},\tilde{p},\tilde{\ell},\tilde{\omega})$ such that

(5.7)
$\left\|\tilde{v}\right\|_{\mathbb{E}_{T}^{v}}+\left\|\tilde{p}\right\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0})\cap\mathrm{L}_{0}^{q}(\mathcal{F}_{0}))}+\left\|(\tilde{\ell},\tilde{\omega})\right\|_{\mathrm{W}^{1,p}(0,T)}\leq
C\left(\left\|\tilde{f_{1}}\right\|_{\mathbb{F}_{T}^{v}}+\|f_{3}\|_{\mathbb{F}_{T}^{\ell}}+\|f_{4}\|_{\mathbb{F}_{T}^{\omega}}\right),$

where $C>0$ is independent of $T$ thanks to the homogeneous initial values. It
remains to estimate $\tilde{f_{1}}$ in
$\mathbb{F}_{T}^{v}=\mathrm{L}^{p}(0,T;\mathrm{L}^{q}(\mathcal{F}_{0})^{3})$.
From Section 5, the above maximal regularity estimate of the heat equation
with zero initial data and the existence of the reference solution $d^{*}$ up
to time $T_{0}$, we deduce the estimate

$\displaystyle\|B(d^{*})\tilde{d}\|_{\mathbb{F}_{T}^{v}}$ $\displaystyle\leq
C\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}\leq
C\left(\|d^{*}\|_{\mathbb{E}_{T}^{d}}+\|d_{0}\|_{\mathrm{X}_{\gamma}^{d}}\right)\|f_{2}\|_{\mathbb{F}_{T}^{d}}\leq
C\left(\|d^{*}\|_{\mathbb{E}_{T_{0}}^{d}}+\|d_{0}\|_{\mathrm{X}_{\gamma}^{d}}\right)\|f_{2}\|_{\mathbb{F}_{T}^{d}},$

and we observe that $\|d^{*}\|_{\mathbb{E}_{T_{0}}^{d}}$ does not depend on
$T$. The latter estimate and (5.7) as well as the above
$\tilde{d}\in\mathbb{E}_{T}^{d}$ finally yield the desired solution
$(\tilde{v},\tilde{d},\tilde{\ell},\tilde{\omega},\tilde{p})\in\prescript{}{0}{\tilde{\mathbb{E}}_{T}}$
to (5.6) with homogeneous initial values. The maximal regularity estimate is
valid with constant $C_{\mathrm{MR}}>0$ independent of $T$ in view of the
homogeneous initial values and the preceding estimate of
$\|B(d^{*})\tilde{d}\|_{\mathbb{F}_{T}^{v}}$. ∎

After establishing the maximal regularity results with regard to the local
well-posedness, we now concentrate on the situation of the global well-
posedness. As the latter also relies on a fixed point argument, we aim for
maximal $\mathrm{L}^{p}([0,\infty))$-regularity of the linearization around
constant equilibria. The latter notion of maximal
$\mathrm{L}^{p}([0,\infty))$-regularity is generally a strong property that
requires the underlying operator to be invertible, see e.g. [40, Proposition
3.5.2], which can often only be guaranteed upon introducing a shift.
Therefore, for $\mu\geq 0$, we consider the linearization

(5.8) $\left\\{\begin{aligned}
\partial_{t}{v}+(\mu-\Delta){v}+\nabla{p}=g_{1},\enspace\operatorname{div}{v}=0,\enspace\text{and}\enspace{\partial_{t}{d}}+(\mu-\Delta){d}&=g_{2},&&\enspace\text{in}\enspace(0,\infty)\times\mathcal{F}_{0},\\\
m_{\mathcal{S}}({\ell})^{\prime}+\mu\ell+\int_{\partial\mathcal{S}_{0}}\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=g_{3},&&\enspace\text{in}\enspace(0,\infty),\\\
J_{0}({\omega})^{\prime}+\mu\omega+\int_{\partial\mathcal{S}_{0}}y\times\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=g_{4},&&\enspace\text{in}\enspace(0,\infty),\\\
{v}&=\ell+\omega\times
y,&&\enspace\text{on}\enspace(0,\infty)\times\partial\mathcal{S}_{0},\\\
{v}&=0,&&\enspace\text{on}\enspace(0,\infty)\times\partial\mathcal{O},\\\
\partial_{\nu}d&=0,&&\enspace\text{on}\enspace(0,\infty)\times\partial\mathcal{F}_{0},\\\
v(0)=v_{0},\enspace
d(0)=d_{0}-d_{*},\enspace\ell(0)=\ell_{0},\enspace\text{and}\enspace\omega(0)&=\omega_{0}.\end{aligned}\right.$

Maity and Tucsnak studied in [36] the interaction problem of a rigid body
immersed in a viscous incompressible fluid. Their handling of the linearized
problem relies on a so-called “monolithic approach”, i.e., the fluid and rigid
body equations are considered as one complete system, and an equivalent
formulation in terms of the fluid-structure operator and the pressure is
provided. The investigation of the fluid-structure semigroup relies on a
decoupling method. For a decoupling based on viewing linearized fluid-
structure problems as boundary controlled fluid systems with dynamic boundary
feedback, we also refer to [35]. Another way prove the following result would
be to use the decoupling approach from [2] based on a different lifting method
and the observation that maximal regularity is preserved under similarity
transforms.

Combining the considerations in [36, Sections 3–5], and employing the maximal
$\mathrm{L}^{p}([0,\infty))$-regularity of the Neumann Laplacian up to a shift
together with the fact that the $d$-equation is decoupled in (5.8), we get the
following result.

######  Proposition .

Let $p,q\in(1,\infty)$ such that (2.2) holds true, and consider
$(g_{1},g_{2},g_{3},g_{4})\in\mathbb{F}_{\infty}$, $d_{*}\in\mathbb{R}^{3}$
constant with $|d_{*}|=1$ as well as
$(v_{0},d_{0},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}$. Then there exists
$\mu_{0}\geq 0$ such that for all $\mu>\mu_{0}$, there exists a unique
solution $(v,d,\ell,\omega,p)\in\tilde{\mathbb{E}}_{\infty}$ to (5.8).
Moreover, for $C>0$, we get the estimate

$\left\|(v,d,\ell,\omega,p)\right\|_{\tilde{\mathbb{E}}_{\infty}}\leq
C\left(\|(g_{1},g_{2},g_{3},g_{4})\|_{\mathbb{F}_{\infty}}+\|(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\|_{\mathrm{X}_{\gamma}}\right).$

##  6\. Proof of the local strong well-posedness

In this section, we prove the local strong well-posedness of the interaction
problem with the colloidal particle. We describe the underlying fixed point
argument in more details in Subsection 6.1. In Subsection 6.2, we first
discuss the procedure to determine the diffeomorphisms $X$ and $Y$ and then
show estimates of the related terms in order to establish estimates of the
nonlinear terms in a second step. Finally, Subsection 6.3 is dedicated to
concluding Theorem 4.1, and to deducing Theorem 2.1 therefrom.

###  6.1. The fixed point argument

As already done previously, we denote the principle variable by
$z=(v,d,\ell,\omega)$ and also employ the notation $\tilde{z}$, $z^{*}$ or
$z_{i}$. In the sequel, we will reformulate the question of finding a unique
strong solution $(z,p)$ to the transformed system of equations (4.7)–(4.9) as
a fixed point problem. To this end, we recall the reference solution
$(z^{*},p^{*})$ from Section 5. For a solution $(z,p)$ to (4.7)–(4.9), we set

$\widehat{v}\coloneqq v-v^{*},\enspace\widehat{d}\coloneqq
d-d^{*},\enspace\widehat{\ell}\coloneqq\ell-\ell^{*},\enspace\widehat{\omega}\coloneqq\omega-\omega^{*},\enspace\text{and}\enspace\widehat{p}:=p-p^{*}.$

Hence,
$(\widehat{z},\widehat{p})=(\widehat{v},\widehat{d},\widehat{\ell},\widehat{\omega},\widehat{p})$
is a solution to the system of equations

(6.1) $\left\\{\begin{aligned}
\partial_{t}{\widehat{v}}-\Delta{\widehat{v}}+\nabla{\widehat{p}}+B(d^{*}){\widehat{d}}&=F_{1},\enspace\operatorname{div}{\widehat{v}}=0,\enspace\text{and}\enspace{\partial_{t}{\widehat{d}}}-\Delta{\widehat{d}}=F_{2},&&\enspace\text{in}\enspace(0,T)\times\mathcal{F}_{0},\\\
m_{\mathcal{S}}({\widehat{\ell}})^{\prime}+\int_{\partial\mathcal{S}_{0}}\mathrm{T}(\widehat{v},\widehat{p})N\,\mathrm{d}\Gamma&=F_{3},&&\enspace\text{in}\enspace(0,T),\\\
J_{0}({\widehat{\omega}})^{\prime}+\int_{\partial\mathcal{S}_{0}}y\times\mathrm{T}(\widehat{v},\widehat{p})N\,\mathrm{d}\Gamma&=F_{4},&&\enspace\text{in}\enspace(0,T),\\\
{\widehat{v}}&={\widehat{\ell}}+{\widehat{\omega}}\times
y,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{S}_{0},\\\
{\widehat{v}}&=0,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{O},\\\
\partial_{\nu}\widehat{d}&=0,&&\enspace\text{on}\enspace(0,T)\times\partial\mathcal{F}_{0},\\\
\widehat{v}(0)=0,\enspace\widehat{d}(0)&=0,\enspace\widehat{\ell}(0)=0,\enspace\text{and}\enspace\widehat{\omega}(0)=0.\end{aligned}\right.$

Furthermore, for $B(d^{*})\widehat{d}$ as in (4.6), $\mathcal{B}$ from Section
4 and $\mathrm{T}(v,p)$ as in (5.5), we define

$\mathcal{Q}(d^{*},\widehat{d})\coloneqq
B(d^{*})\widehat{d}-\mathcal{B}(d^{*}+\widehat{d})(d^{*}+\widehat{d}),\enspace\text{and}\enspace\mathcal{T}(v,p)\coloneqq
Q(t)^{\top}\mathrm{T}(Q(t)v(t,y),p(t,y))Q(t).$

Additionally recalling the transformed terms from Section 4, we find that
$F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$ are given by

(6.2) $\displaystyle F_{1}(\widehat{z},\widehat{p})$
$\displaystyle\coloneqq(\mathcal{L}_{1}-\Delta)(\widehat{v}+v^{*})-\mathcal{M}(\widehat{v}+v^{*})-\mathcal{N}(\widehat{v}+v^{*})-(\mathcal{G}-\nabla)(\widehat{p}+p^{*})+\mathcal{Q}(d^{*},\widehat{d}),$
$\displaystyle F_{2}(\widehat{z})$
$\displaystyle\coloneqq(\mathcal{L}_{2}-\Delta)(\widehat{d}+d^{*})-(\nabla(\widehat{d}+d^{*})\cdot\partial_{t}Y+(v\cdot\nabla)(\widehat{d}+d^{*}))+|\nabla(\widehat{d}+d^{*})|^{2}(\widehat{d}+d^{*}+d_{*}),$
$\displaystyle F_{3}(\widehat{z},\widehat{p})$ $\displaystyle\coloneqq-
m_{\mathcal{S}}((\widehat{\omega}+\omega^{*})\times(\widehat{\ell}+\ell^{*}))$
$\displaystyle\quad+\int\limits_{\partial\mathcal{S}_{0}}\left((\mathrm{T}-\mathcal{T})(\widehat{v},\widehat{p})+Q(t)^{\top}\nabla\left(\widehat{d}+d^{*}\right)\left(\nabla\left(\widehat{d}+d^{*}\right)\right)^{\top}Q(t)\right)N\,\mathrm{d}\Gamma,\enspace\text{and}\enspace$
$\displaystyle F_{4}(\widehat{z},\widehat{p})$
$\displaystyle\coloneqq(\widehat{\omega}+\omega^{*})\times(J_{0}(\widehat{\omega}+\omega^{*}))$
$\displaystyle\quad+\int\limits_{\partial\mathcal{S}_{0}}y\times\left((\mathrm{T}-\mathcal{T})(\widehat{v},\widehat{p})+Q(t)^{\top}\nabla\left(\widehat{d}+d^{*}\right)\left(\nabla\left(\widehat{d}+d^{*}\right)\right)^{\top}Q(t)\right)N\,\mathrm{d}\Gamma.$

In the following, we fix $T_{0}$ and $R_{0}>0$, and we consider $0<T\leq
T_{0}$ as well as $0<R\leq R_{0}$. For $p,q\in(1,\infty)$ such that (2.2)
holds, and recalling $\mathbb{E}_{T}$ as well as $\mathbb{E}_{T}^{p}$ and
$\tilde{\mathbb{E}}_{T}$ from (5.3) and (5.4), we define

(6.3)
$\mathcal{K}_{T}^{R}\coloneqq\left\\{(\tilde{z},\tilde{p})\in\prescript{}{0}{\mathbb{E}_{T}}\times\mathbb{E}_{T}^{p}:\|(\tilde{z},\tilde{p})\|_{\tilde{\mathbb{E}}_{T}}\leq
R\right\\},\enspace\text{and}\enspace\Phi_{T}^{R}\colon\mathcal{K}_{T}^{R}\to\prescript{}{0}{\mathbb{E}_{T}}\times\mathbb{E}_{T}^{p},\enspace\text{with}\enspace\Phi_{T}^{R}(\tilde{z},\tilde{p})\coloneqq(\widehat{z},\widehat{p}),$

where $(\widehat{z},\widehat{p})$ denotes the solution to (6.1) with right-
hand sides $F_{1}(\tilde{z},\tilde{p})$, $F_{2}(\tilde{z})$,
$F_{3}(\tilde{z},\tilde{p})$ and $F_{4}(\tilde{z},\tilde{p})$, and
$(\tilde{z},\tilde{p})\in\mathcal{K}_{T}^{R}$. It follows from Section 5 that
$\Phi_{T}^{R}$ is well-defined provided
$(F_{1},F_{2},F_{3},F_{4})\in\mathbb{F}_{T}$, where $\mathbb{F}_{T}$ is
defined in (5.1). With regard to (6.1), the existence and uniqueness of a
solution to (4.7)–(4.9) is equivalent to $\Phi_{T}^{R}$ admitting a unique
fixed point upon adding the reference solution $(z^{*},p^{*})$.

###  6.2. Estimates of the nonlinear terms

Throughout this section, for $T_{0}>0$ and $R_{0}>0$ fixed, let
$T\in(0,T_{0}]$ and $R\in(0,R_{0}]$. We consider $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{1},\tilde{p}_{1})$,
$(\tilde{z}_{2},\tilde{p}_{2})\in\mathcal{K}_{T}^{R}$, and for the reference
solution $(z^{*},p^{*})$ resulting from Section 5, we define
$(z,p)\coloneqq(\tilde{v}+v^{*},\tilde{d}+d^{*},\tilde{\ell}+\ell^{*},\tilde{\omega}+\omega^{*},\tilde{p}+p^{*})$
and set $(z_{i},p_{i})$, $i=1,2$, likewise. It then follows that

(6.4) $\|(z,p)\|_{\tilde{\mathbb{E}}_{T}}\leq R+C_{T}^{*},$

where $C_{T}^{*}$ from Section 5 is the norm of $(z^{*},p^{*})$. We also
denote by $C_{0}\coloneqq\|z_{0}\|_{\mathrm{X}_{\gamma}}$ the norm of the
initial values. Next, we comment on the procedure to deduce $X$ and $Y$ from
Section 4.

######  Remark .

Let $\ell$, $\omega\in\mathrm{W}^{1,p}(0,T)^{3}$ be given.

  1. (i)

First, recover $Q\in\mathrm{W}^{2,p}(0,T)^{3\times 3}$ from solving the
initial value problem

(6.5) $\dot{Q}^{\top}(t)=M(t)Q^{\top}(t),\enspace
Q^{\top}(0)=\operatorname{Id},$

where the matrix-valued function $M(t)$ fulfills $M(t)x=\omega(t)\times x$ for
all $t\in(0,T)$ and $x\in\mathbb{R}^{3}$.

  2. (ii)

Equipped with $Q$, we can deduce the original body velocities from

(6.6)
$h^{\prime}(t)=Q^{\top}(t)\ell(t)\enspace\text{and}\enspace\Omega(t)=Q^{\top}(t)\omega(t).$

As we assume that the center of gravity $h$ is located at the origin at time
zero, i.e., $h(0)=0$, it can be derived from
$h(t)=\int_{0}^{t}h^{\prime}(s)\,\mathrm{d}s$.

  3. (iii)

It is then possible to insert $b$ as defined in (4.3) into (4.2), so we obtain
$X$. As in (4.5), we set $b^{(Y)}\coloneqq\mathrm{J}_{X}^{-1}b(\cdot,X)$ and
then solve (4.4) to get the inverse diffeomorphism $Y$ of $X$.

Concerning the notation, given $(\ell_{1},\omega_{1})$,
$(\ell_{2},\omega_{2})\in\mathrm{W}^{1,p}(0,T)^{6}$, the subscript
$i\in\\{1,2\\}$ indicates that the respective objects correspond to
$(\ell_{i},\omega_{i})$ and are deduced therefrom by the procedure described
in Subsection 6.2. This is especially valid for the diffeomorphisms $X_{i}$
and $Y_{i}$ related to $\ell_{i}$ and $\omega_{i}$. Moreover, we use
$\|\cdot\|_{\infty}$ and $\|\cdot\|_{\infty,\infty}$ to denote the
$\mathrm{L}^{\infty}$-norm in time and in time and space, respectively, and we
will also write $\|\cdot\|_{p,q}$ instead of
$\|\cdot\|_{\mathrm{L}^{p}(0,T;\mathrm{L}^{q}(\mathcal{F}_{0}))}$. For a proof
of the following lemma, we refer to [15, Section 6.1].

######  Lemma .

Let $(\ell_{1},\omega_{1})$,
$(\ell_{2},\omega_{2})\in\mathrm{W}^{1,p}(0,T)^{6}$.

  1. (a)

For $i=1,2$, we have $X_{i}$,
$Y_{i}\in\mathrm{C}^{1}\left(0,T;\mathrm{C}^{\infty}(\mathbb{R}^{3})^{3}\right)$
as well as the estimates

$\displaystyle\|\partial^{\alpha}X_{i}\|_{\infty,\infty}+\|\partial^{\alpha}Y_{i}\|_{\infty,\infty}$
$\displaystyle\leq C,\enspace\text{and}\enspace$
$\displaystyle\|\partial^{\beta}(X_{1}-X_{2})\|_{\infty,\infty}+\|\partial^{\beta}(Y_{1}-Y_{2})\|_{\infty,\infty}$
$\displaystyle\leq
CT(\|\ell_{1}-\ell_{2}\|_{\infty}+\|\omega_{1}-\omega_{2}\|_{\infty})$

for all multi-indices $\alpha$ and $\beta$ with $1\leq|\alpha|\leq 3$ and
$0\leq|\beta|\leq 3$. The constants only depend on
$K_{i}\coloneqq\|\ell_{i}\|_{\infty}+\|\omega_{i}\|_{\infty}$ and not directly
on $\ell_{i}$ or $\omega_{i}$. In particular, for $i,k,m\in\\{1,2,3\\}$, we
have

$\|\partial_{k}\partial_{i}Y_{m}\|_{\infty,\infty}\leq CT(R+C_{T}^{*}).$

  2. (b)

For $(h^{\prime}_{i},\Omega_{i})$ deduced from $(\ell_{i},\omega_{i})$ as
described in (6.6), the estimates

$\|h^{\prime}_{1}-h^{\prime}_{2}\|_{\infty}\leq
C(\|\ell_{1}-\ell_{2}\|_{\infty}+\|\omega_{1}-\omega_{2}\|_{\infty}),\enspace\text{and}\enspace\|\Omega_{1}-\Omega_{2}\|_{\infty}\leq
C\|\omega_{1}-\omega_{2}\|_{\infty}$

are valid. In particular, for $K_{i}$ as above the matrix $Q_{i}$ from (6.5)
satisfies the estimates

$\|Q_{1}-Q_{2}\|_{\infty}\leq CT\|M_{1}-M_{2}\|_{\infty}\leq
CT\|\omega_{1}-\omega_{2}\|_{\infty},\enspace\text{and}\enspace\|Q_{i}\|_{\infty}+\|Q_{i}^{\top}\|_{\infty}\leq
C(1+TK_{i}\mathrm{e}^{TK_{i}}).$

  3. (c)

For all multi-indices $\beta$ with $0\leq|\beta|\leq 3$, $b_{i}$ defined in
(4.3) and associated to $h^{\prime}_{i}$ and $\Omega_{i}$ fulfills

$\|\partial^{\beta}b_{i}\|_{\infty,\infty}\leq
C,\enspace\text{and}\enspace\|\partial^{\beta}(b_{1}-b_{2})\|_{\infty,\infty}\leq
C(\|h^{\prime}_{1}-h^{\prime}_{2}\|_{\infty}+\|\Omega_{1}-\Omega_{2}\|_{\infty}).$

  4. (d)

For all multi-indices $\beta$ with $0\leq|\beta|\leq 3$, $b^{(Y_{i})}$ defined
in (4.5) and associated to $h^{\prime}_{i}$ and $\Omega_{i}$ satisfies

$\left\|\partial^{\beta}(b^{(Y_{1})}-b^{(Y_{2})})\right\|_{\mathrm{L}^{\infty}(0,T;\mathrm{C}(\mathbb{R}^{3}))}\leq
C\|\partial^{\beta}(b_{1}-b_{2})\|_{\mathrm{L}^{\infty}(0,T;\mathrm{C}^{1}(\mathbb{R}^{3}))}.$

Next, we provide estimates of the covariant and contravariant metric tensors
$g^{ij}$ and $g_{ij}$ and of the Christoffel symbol $\Gamma_{jk}^{i}$ defined
in Section 4. For $\ell$, $\ell_{1}$, $\ell_{2}$ and $\omega$, $\omega_{1}$,
$\omega_{2}$ resulting from $\tilde{\ell}$, $\tilde{\ell}_{1}$,
$\tilde{\ell}_{2}$, $\ell^{*}$ and $\tilde{\omega}$, $\tilde{\omega}_{1}$,
$\tilde{\omega}_{2}$, $\omega^{*}$ as described at the beginning of the
section, we recover the diffeomorphisms $X$, $X_{1}$, $X_{2}$ and $Y$,
$Y_{1}$, $Y_{2}$ as made precise in Subsection 6.2. The proof of the following
lemma can be found in [15, p. 1417].

######  Lemma .

For all multi-indices $0\leq|\alpha|\leq 1$, the covariant and contravariant
tensor as well as the Christoffel symbol associated to $X$, $X_{1}$, $X_{2}$
and $Y$, $Y_{1}$, $Y_{2}$ satisfy the estimates

$\|\partial^{\alpha}g^{ij}\|_{\infty,\infty}+\|\partial^{\alpha}g_{ij}\|_{\infty,\infty}+\|\partial^{\alpha}\Gamma_{jk}^{i}\|_{\infty,\infty}\leq
C,\enspace\text{and}\enspace$
$\|\partial^{\alpha}((g_{1})^{ij}-(g_{2})^{ij})\|_{\infty,\infty}+\|\partial^{\alpha}((g_{1})_{ij}-(g_{2})_{ij})\|_{\infty,\infty}+\|\partial^{\alpha}((\Gamma_{1})_{jk}^{i}-(\Gamma_{2})_{jk}^{i})\|_{\infty,\infty}\leq
CT\|(\tilde{\ell}_{1}-\tilde{\ell}_{2},\tilde{\omega}_{1}-\tilde{\omega}_{2})\|_{\infty}.$

In the following lemma, we discuss further estimates in order to handle the
nonlinear terms.

######  Lemma .

Let $p,q\in(1,\infty)$ satisfy (2.2), and for the reference solution
$(z^{*},p^{*})$ from Section 5 and $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{i},\tilde{p}_{i})\in\mathcal{K}_{T}^{R}$, let
$(z,p)=(\tilde{z}+z^{*},\tilde{p}+p^{*})$ and
$(z_{i},p_{i})=(\tilde{z}_{i}+z^{*},\tilde{p}_{i}+p^{*})$, $i=1,2$. Then

$\|g^{jk}-\delta_{jk}\|_{\infty,\infty}\leq
CT(R+C_{T}^{*}),\enspace\text{and}\enspace\|\partial_{j}Y_{k}-\delta_{jk}\|_{\infty,\infty}\leq
CT(R+C_{T}^{*}).$

###### Proof.

The identity transform $X(t,y)=y$ for all $t>0$ and $y\in\mathbb{R}^{3}$
corresponds to the situation of the body at rest, i.e., $\ell=\omega=0$.
Therefore, Subsection 6.2,
$\mathrm{W}^{1,p}(0,T)\hookrightarrow\mathrm{L}^{\infty}(0,T)$ and (6.4) yield

$\|g^{jk}-\delta_{jk}\|_{\infty,\infty}\leq CT\|(\ell,\omega)\|_{\infty}\leq
CT\|(\ell,\omega)\|_{\mathrm{W}^{1,p}(0,T)}\leq CT(R+C_{T}^{*}),$

and the estimate of $\|\partial_{j}Y_{k}-\delta_{jk}\|_{\infty,\infty}$
follows likewise. ∎

The embeddings in the lemma below will be used frequently in the estimates of
the nonlinear terms.

######  Lemma .

Let $p,q\in(1,\infty)$ satisfy (2.2), and set
$\mathrm{Y}_{0}\coloneqq\mathrm{L}^{q}_{\sigma}(\mathcal{F}_{0})\times\mathrm{L}^{q}(\mathcal{F}_{0})^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$
as well as
$\mathrm{Y}_{1}\coloneqq\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\cap\mathrm{L}^{q}_{\sigma}(\mathcal{F}_{0})\times\mathrm{W}^{2,q}(\mathcal{F}_{0})^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$.

  1. (a)

For all $\beta\in(0,1)$, denoting by
$\mathrm{Y}_{\beta}\coloneqq(\mathrm{Y}_{0},\mathrm{Y}_{1})_{\beta}$ the
complex interpolation space, the embedding

$\mathbb{E}_{T}\hookrightarrow\mathrm{W}^{1,p}(0,T;\mathrm{Y}_{0})\cap\mathrm{L}^{p}(0,T;\mathrm{Y}_{1})\hookrightarrow\mathrm{H}^{\beta,p}(0,T;\mathrm{Y}_{1-\beta})\hookrightarrow\mathrm{H}^{\beta,p}\left(0,T;\mathrm{H}^{2(1-\beta),q}(\mathcal{F}_{0})^{6}\times\mathbb{R}^{6}\right)$

is valid. In particular, we have
$\mathbb{E}_{T}^{d}\hookrightarrow\mathrm{L}^{\infty}\left(0,T;\mathrm{W}^{1,2q}(\mathcal{F}_{0})^{3}\right)\hookrightarrow\mathrm{L}^{\infty}\left(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0})^{3}\right)$.

  2. (b)

For
$\mathrm{Y}_{\gamma}=(\mathrm{Y}_{0},\mathrm{Y}_{1})_{1-\nicefrac{{1}}{{p}},p}$,
we have

$\mathbb{E}_{T}\hookrightarrow\mathrm{W}^{1,p}(0,T;\mathrm{Y}_{0})\cap\mathrm{L}^{p}(0,T;\mathrm{Y}_{1})\hookrightarrow\mathrm{BUC}([0,T];\mathrm{Y}_{\gamma})\hookrightarrow\mathrm{BUC}\left([0,T];\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})^{6}\times\mathbb{R}^{6}\right).$

In particular,
$\mathbb{E}_{T}^{d}\hookrightarrow\mathrm{BUC}\left([0,T];\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})^{3}\right)\hookrightarrow\mathrm{L}^{\infty}\left(0,T;\mathrm{W}^{1,\infty}(\mathcal{F}_{0})^{3}\right)\hookrightarrow\mathrm{L}^{\infty}\left(0,T;\mathrm{L}^{\infty}(\mathcal{F}_{0})^{3}\right)$.

In general, only the embedding constants of the embeddings into
$\mathrm{H}^{\beta,p}(0,T;\mathrm{Y}_{\beta})$ and
$\mathrm{BUC}([0,T];\mathrm{Y}_{\gamma})$ depend on time. They can be chosen
time-independent if zero initial values are considered.

###### Proof.

The embeddings in (a) follow from the mixed derivative theorem, see e.g. [40,
Corollary 4.5.10], while the second embedding in (b) is a result of [1,
Theorem III.4.10.2]. Concerning the embeddings of $\mathbb{E}_{T}^{d}$ in (a),
there is $\beta\in(0,1)$ with
$\nicefrac{{1}}{{p}}<\beta<\nicefrac{{1}}{{2}}-\nicefrac{{3}}{{2q}}$ in view
of (2.2), yielding the assertion by Sobolev embeddings. In (b), we use (2.2)
and [45, Theorem 4.6.1] to deduce
$\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})\hookrightarrow\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})$.
∎

We start with the estimates of the nonlinear term $F_{2}$ as defined in (6.2).

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and for the reference solution
$(z^{*},p^{*})$ from Section 5 and $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{i},\tilde{p}_{i})\in\mathcal{K}_{T}^{R}$, let
$(z,p)=(\tilde{z}+z^{*},\tilde{p}+p^{*})$ and
$(z_{i},p_{i})=(\tilde{z}_{i}+z^{*},\tilde{p}_{i}+p^{*})$, $i=1,2$. Recall
$C_{\mathrm{MR}}>0$ from Section 5. Then there are $C_{2}(R,T)>0$ with
$C_{2}(R,T)<\nicefrac{{R}}{{8C_{\mathrm{MR}}}}$ for $T>0$ sufficiently small
and $L_{2}(R,T)>0$ with $L_{2}(R,T)\to 0$ as $R\to 0$ and $T\to 0$ such that

$\|F_{2}(\tilde{z})\|_{\mathbb{F}_{T}^{d}}\leq
C_{2}(R,T),\enspace\text{and}\enspace\|F_{2}(\tilde{z}_{1})-F_{2}(\tilde{z}_{2})\|_{\mathbb{F}_{T}^{d}}\leq
L_{2}(R,T)\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

###### Proof.

Throughout this proof, we heavily rely on
$(\tilde{z},\tilde{p})\in\mathcal{K}_{T}^{R}$, implying that the embedding
constants are time-independent in view of Subsection 6.2. Hence, we also often
split $v$ and $d$ into their respective parts with homogeneous initial values
$\tilde{v}$ and $\tilde{d}$ as well as the reference solution parts $v^{*}$
and $d^{*}$. Let us also invoke some useful estimate at this stage. For
$\tilde{d}\in\prescript{}{0}{\mathbb{E}_{T}^{d}}$, we deduce from Hölder’s
inequality as well as Subsection 6.2(a) the existence of a $T$-independent
constant $C>0$ such that

(6.7)
$\|\tilde{d}\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))}\leq
T^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))}\leq
CT^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}.$

First, we employ Subsection 6.2(a) to estimate $\|\Delta
Y_{k}\|_{\infty,\infty}$, Subsection 6.2 to handle
$\|g^{jk}-\delta_{jk}\|_{\infty,\infty}$ as well as (6.4) in order to bound
$\|d\|_{\mathbb{E}_{T}^{d}}$, so we get

(6.8) $\displaystyle\|(\mathcal{L}_{2}-\Delta)d\|_{\mathbb{F}_{T}^{d}}$
$\displaystyle\leq
C\sup_{i,j,k}(\|g^{jk}-\delta_{jk}\|_{\infty,\infty}\|\partial_{k}\partial_{j}d_{i}\|_{p,q}+\|\Delta
Y_{k}\|_{\infty,\infty}\|\partial_{k}d_{i}\|_{p,q})$ $\displaystyle\leq
CT(R+C_{T}^{*})\|d\|_{\mathbb{E}_{T}^{d}}$ $\displaystyle\leq
CT(R+C_{T}^{*})^{2}.$

Subsection 6.2(a) allows us to estimate $\|\tilde{v}\|_{\infty,\infty}$ by
$\|\tilde{z}\|_{\mathbb{E}_{T}}$. Thanks to Section 5, we obtain an estimate
of $\|z^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma})}$ by the initial values
and the reference solution. In conjunction with (6.7), we obtain

$\displaystyle\|(v\cdot\nabla)d\|_{\mathbb{F}_{T}^{d}}$ $\displaystyle\leq
C(\|\tilde{v}\|_{\infty,\infty}+\|v^{*}\|_{\infty,\infty})(\|\tilde{d}\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))}+\|d^{*}\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))})$
$\displaystyle\leq
C(\|\tilde{z}\|_{\mathbb{E}_{T}}+\|z^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma})})(T^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}+\|d^{*}\|_{\mathbb{E}_{T}^{d}})$
$\displaystyle\leq C(R+C_{0}+C_{T}^{*})(T^{\nicefrac{{1}}{{p}}}R+C_{T}^{*}).$

By virtue of Subsection 6.2(a), which yields the boundedness of
$\partial_{t}Y$, and (6.7), it follows that

$\|\nabla d\cdot\partial_{t}Y\|_{\mathbb{F}_{T}^{d}}\leq
C\|d\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))}\leq
C(T^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}+\|d^{*}\|_{\mathbb{E}_{T}^{d}})\leq
C(T^{\nicefrac{{1}}{{p}}}R+C_{T}^{*}).$

Exploiting Subsection 6.2(b) for estimating
$\|\tilde{d}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{W}^{1,\infty}(\mathcal{F}_{0}))}$
by $\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}$ as well as
$\|d^{*}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{W}^{1,\infty}(\mathcal{F}_{0}))}$
by $\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}$, and using
Section 5 with regard to
$\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}$, we infer that

(6.9) $\displaystyle\||\nabla d|^{2}d\|_{\mathbb{F}_{T}^{d}}$
$\displaystyle\leq C\|d\|_{\infty,\infty}\|\nabla d\|_{\infty,\infty}\|\nabla
d\|_{p,q}$ $\displaystyle\leq
C(\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}^{2}+\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}^{2})(T^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))}+\|d^{*}\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{1,q}(\mathcal{F}_{0}))})$
$\displaystyle\leq
C(R^{2}+(C_{T}^{*})^{2}+C_{0}^{2})(T^{\nicefrac{{1}}{{p}}}R+C_{T}^{*}),$

and $\||\nabla d|^{2}d_{*}\|_{\mathbb{F}_{T}^{d}}$ can be estimated similarly,
so the first part of the assertion follows for some $C_{2}(R,T)>0$.

As in (6.8), also making use of Subsection 6.2 and Subsection 6.2(a) to
control the differences $(g_{1})^{jk}-(g_{2})^{jk}$ and $\Delta Y_{1,k}-\Delta
Y_{2,k}$ resulting from the shape of $\mathcal{L}_{2}$ as introduced in
Section 4, we find the estimates

$\|(\mathcal{L}_{2}^{(1)}-\Delta)(\tilde{d}_{1}-\tilde{d}_{2})\|_{\mathbb{F}_{T}^{d}}\leq
C(T(R+C_{T}^{*}))\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}},\enspace\text{and}\enspace\|(\mathcal{L}_{2}^{(1)}-\mathcal{L}_{2}^{(2)})(\tilde{d}_{2}+d^{*})\|_{\mathbb{F}_{T}^{d}}\leq
CT\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

In total, we thus get

$\|(\mathcal{L}_{2}^{(1)}-\Delta)(\tilde{d}_{1}+d^{*})-(\mathcal{L}_{2}^{(2)}-\Delta)(\tilde{d}_{2}+d^{*})\|_{\mathbb{F}_{T}^{d}}\leq
C(T(1+R+C_{T}^{*}))\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

Similar arguments as in the estimate (6.9) and the addition and subtraction of
suitable terms lead to

$\|(\tilde{v}_{1}+v^{*})\cdot\nabla(\tilde{d}_{1}+d^{*})-(\tilde{v}_{2}+v^{*})\cdot\nabla(\tilde{d}_{2}+d^{*})\|_{\mathbb{F}_{T}^{d}}\leq(T^{\nicefrac{{1}}{{p}}}(R+C_{0})+C_{T}^{*})\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

Next, additionally exploiting Subsection 6.2(a) to estimate
$\partial_{t}Y_{1}-\partial_{t}Y_{2}$, and also relying on (6.7), we conclude

$\displaystyle\quad\|\nabla(\tilde{d}_{1}+d^{*})\partial_{t}Y_{1}-\nabla(\tilde{d}_{2}+d^{*})\partial_{t}Y_{2}\|_{\mathbb{F}_{T}^{d}}$
$\displaystyle\leq\|\nabla(\tilde{d}_{1}+d^{*})(\partial_{t}Y_{1}-\partial_{t}Y_{2})\|_{p,q}+\|\nabla(\tilde{d}_{1}-\tilde{d}_{2})\partial_{t}Y_{2}\|_{p,q}$
$\displaystyle\leq
C(T^{\nicefrac{{1}}{{p}}}R+C_{T}^{*})T\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}+CT^{\nicefrac{{1}}{{p}}}\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

Proceeding analogously as in (6.9) upon adding and subtracting terms, we get

$\||\nabla(\tilde{d}_{1}+d^{*})|^{2}(\tilde{d}_{1}+d^{*})-|\nabla(\tilde{d}_{2}+d^{*})|^{2}(\tilde{d}_{2}+d^{*})\|_{\mathbb{F}_{T}^{d}}\leq
C(T^{\nicefrac{{1}}{{p}}}(R^{2}+(C_{T}^{*})^{2}+C_{0}^{2})+C_{0}C_{T}^{*})\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

Again, let us observe that the term involving $d_{*}$ can be dealt with
likewise. ∎

For the treatment of $F_{1}$, it is crucial to provide estimates of the term
$\mathcal{Q}$.

######  Lemma .

Let $p,q\in(1,\infty)$ satisfy (2.2), and for the reference solution
$(z^{*},p^{*})$ from Section 5 and $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{i},\tilde{p}_{i})\in\mathcal{K}_{T}^{R}$, let
$(z,p)=(\tilde{z}+z^{*},\tilde{p}+p^{*})$ and
$(z_{i},p_{i})=(\tilde{z}_{i}+z^{*},\tilde{p}_{i}+p^{*})$, $i=1,2$. Recall
$C_{\mathrm{MR}}>0$ from Section 5. Then there are $C_{\mathcal{Q}}(R,T)>0$
with $C_{\mathcal{Q}}(R,T)<\nicefrac{{R}}{{16C_{\mathrm{MR}}}}$ for $T>0$
sufficiently small and $L_{\mathcal{Q}}(R,T)>0$ with $L_{\mathcal{Q}}(R,T)\to
0$ as $R\to 0$ and $T\to 0$ such that

$\|\mathcal{Q}(\tilde{d},d^{*})\|_{\mathbb{F}_{T}^{v}}\leq
C_{\mathcal{Q}}(R,T),\enspace\text{and}\enspace\|\mathcal{Q}^{(1)}(\tilde{d}_{1},d^{*})-\mathcal{Q}^{(2)}(\tilde{d}_{2},d^{*})\|_{\mathbb{F}_{T}^{v}}\leq
L_{\mathcal{Q}}(R,T)\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

###### Proof.

First, we expand
$\mathcal{Q}(\tilde{d},d^{*})=B(d^{*})\tilde{d}-\mathcal{B}(d)d=B(d^{*})\tilde{d}-\mathcal{B}(\tilde{d}+d^{*})(\tilde{d}+d^{*})$
as follows:

(6.10) $\displaystyle\mathcal{Q}(\tilde{d},d^{*})$
$\displaystyle=B(d^{*})\tilde{d}-\mathcal{B}(\tilde{d}+d^{*})(\tilde{d}+d^{*})$
$\displaystyle=(B(d^{*})-B(\tilde{d}+d^{*}))(\tilde{d}+d^{*})-B(d^{*})d^{*}+(B(\tilde{d}+d^{*})-\mathcal{B}(\tilde{d}+d^{*}))(\tilde{d}+d^{*})$
$\displaystyle=:\mathcal{Q}_{\mathrm{I}}+\mathcal{Q}_{\mathrm{II}}+\mathcal{Q}_{\mathrm{III}}.$

Regarding $\mathcal{Q}_{\mathrm{I}}$, using the Einstein sum convention, we
obtain

$\mathcal{Q}_{\mathrm{I}}=(B(d^{*})-B(\tilde{d}+d^{*}))(\tilde{d}+d^{*})=-\partial_{i}\tilde{d}_{l}\Delta(\tilde{d}+d^{*})_{l}-\partial_{k}\tilde{d}_{l}\partial_{k}\partial_{i}(\tilde{d}+d^{*})_{l}.$

An application of Subsection 6.2(b) providing an estimate of
$\|\partial_{i}\tilde{d}_{l}\|_{\infty,\infty}$ by
$\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}$ and the observation that both terms can
be handled similarly lead to

(6.11) $\|\partial_{i}\tilde{d}_{l}\Delta(\tilde{d}+d^{*})_{l}\|_{p,q}\leq
C\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}\|d\|_{\mathbb{E}_{T}^{d}}\leq
CR\|d\|_{\mathbb{E}_{T}^{d}},\enspace\text{and}\enspace\|\mathcal{Q}_{\mathrm{I}}\|_{p,q}\leq
CR\|d\|_{\mathbb{E}_{T}^{d}}.$

Subsection 6.2(b) enables us to estimate
$\|\partial_{i}d_{l}^{*}\|_{\infty,\infty}$ by
$\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}$, and in view of
Section 5, we can estimate the resulting norm by norm of the initial values as
well as the reference solution, so

(6.12) $\|\mathcal{Q}_{\mathrm{II}}\|_{p,q}\leq
C\|\partial_{i}d_{l}^{*}\|_{\infty,\infty}\|d^{*}\|_{\mathbb{E}_{T}^{d}}\leq
CC_{T}^{*}\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}\leq
CC_{T}^{*}(C_{T}^{*}+C_{0}).$

Concerning the estimate of $\mathcal{Q}_{\mathrm{III}}$, we again often split
$z$ into $\tilde{z}$ with homogeneous initial values and the reference
solution $z^{*}$ as seen in the proof of Subsection 6.2. Besides, we further
expand $\mathcal{Q}_{\mathrm{III}}$ as follows:

(6.13) $\displaystyle\mathcal{Q}_{\mathrm{III}}$
$\displaystyle=(B(\tilde{d}+d^{*})-\mathcal{B}(\tilde{d}+d^{*}))(\tilde{d}+d^{*})$
$\displaystyle=\bigl{[}\partial_{i}(\tilde{d}+d^{*})_{l}\Delta(\tilde{d}+d^{*})_{l}\bigr{]}+\bigl{[}\partial_{k}(\tilde{d}+d^{*})_{l}\partial_{k}\partial_{i}(\tilde{d}+d^{*})_{l}\bigr{]}-\bigl{[}(\mathcal{L}_{2}(\tilde{d}+d^{*})_{l}\partial_{m}(\tilde{d}+d^{*})_{l}\partial_{i}Y_{m})\bigr{]}$
$\displaystyle\quad-\bigl{[}(\partial_{m}(\tilde{d}+d^{*})_{l}\partial_{k}Y_{m})(\partial_{j}\partial_{m}(\tilde{d}+d^{*})_{l}(\partial_{k}Y_{j}\partial_{i}Y_{m})+2(\tilde{d}+d^{*})_{l}\partial_{k}\partial_{i}Y_{m})\bigr{]}$
$\displaystyle=:\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV}.$

Recalling $d=\tilde{d}+d^{*}$, we calculate

$\mathrm{I}+\mathrm{III}=\partial_{i}d_{l}\Delta
d_{l}-\mathcal{L}_{2}d_{l}(\partial_{m}d_{l}\partial_{i}Y_{m})=\partial_{i}d_{l}(\Delta-\mathcal{L}_{2})d_{l}-\mathcal{L}_{2}d_{l}\partial_{m}d_{l}(\partial_{i}Y_{m}-\delta_{im}).$

The next step now is to estimate $\|\mathcal{L}_{2}d_{l}\|_{p,q}$. In fact,
making use of Subsection 6.2 to treat $\|g^{ij}\|_{\infty,\infty}$, Subsection
6.2 to bound $\|\Delta Y_{k}\|_{\infty,\infty}$ as well as (6.4) to estimate
the resulting norm $\|d\|_{\mathbb{E}_{T}^{d}}$, we find that

(6.14) $\|\mathcal{L}_{2}d_{l}\|_{p,q}\leq
C\left(\|g^{ij}\|_{\infty,\infty}\|\partial_{k}\partial_{j}d\|_{p,q}+\|\Delta
Y_{k}\|_{\infty,\infty}\|\partial_{k}d_{i}\|_{p,q}\right)\leq
C\|d\|_{\mathbb{E}_{T}^{d}}\leq C(R+C_{T}^{*}).$

Furthermore, Subsection 6.2(b) in the first step and Section 5 in the second
estimate result in

(6.15) $\|\partial_{m}d_{l}\|_{\infty,\infty}\leq
C\left(\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}+\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}\right)\leq
C(C_{T}^{*}+C_{0}+R).$

Invoking Subsection 6.2 to estimate
$\|\partial_{i}Y_{m}-\delta_{im}\|_{\infty,\infty}$, we derive from the two
preceding estimates that

(6.16)
$\|\mathcal{L}_{2}d_{l}\|_{p,q}\|\partial_{m}d_{l}\|_{\infty,\infty}\|\partial_{i}Y_{m}-\delta_{im}\|_{\infty,\infty}\leq
C(C_{T}^{*}+C_{0}+R)T(R+C_{T}^{*})^{2}.$

Recalling (6.15) and (6.8) for the first term, and plugging in the estimate
from (6.16), we infer that

(6.17) $\displaystyle\|\mathrm{I}+\mathrm{III}\|_{p,q}$ $\displaystyle\leq
C(\|\partial_{i}d_{l}\|_{\infty,\infty}\|(\Delta-\mathcal{L}_{2})d_{l}\|_{p,q}+\|\mathcal{L}_{2}d_{l}\|_{p,q}\|\partial_{m}d_{l}\|_{\infty,\infty}\|\partial_{i}Y_{m}-\delta_{im}\|_{\infty,\infty})$
$\displaystyle\leq C(C_{T}^{*}+C_{0}+R)T(R+C_{T}^{*})^{2}.$

We further split $\mathrm{IV}$ into
$\mathrm{IV}=(\partial_{m}d_{l}\partial_{k}Y_{m})(\partial_{i}\partial_{m}d_{l}(\partial_{k}Y_{j}\partial_{i}Y_{m})+\partial_{m}d_{l}\partial_{k}\partial_{i}Y_{m})=:\mathrm{V}+\mathrm{VI}$
and expand

$\displaystyle\mathrm{V}$
$\displaystyle=(\partial_{m}d_{l}\partial_{k}Y_{m})(\partial_{j}\partial_{m}d_{l})(\partial_{k}Y_{j}\partial_{i}Y_{m})$
$\displaystyle=(\partial_{m}d_{l}\partial_{k}Y_{m})(\partial_{j}\partial_{m}d_{l}\delta_{im}\delta_{jk}+\partial_{j}\partial_{m}d_{l}(\partial_{j}Y_{k}-\delta_{jk})\partial_{i}Y_{m}+\partial_{j}\partial_{m}d_{l}(\partial_{i}Y_{m}-\delta_{im})\delta_{jk})$
$\displaystyle=:\mathrm{V}(1)+\mathrm{VII}.$

Besides, we expand

$\mathrm{V}(1)=(\partial_{m}d_{l}(\partial_{k}Y_{m}-\delta_{km})+\partial_{m}d_{l}\delta_{km})\partial_{j}\partial_{m}d_{l}\delta_{im}\delta_{jk}\eqqcolon\mathrm{VIII}+\mathrm{IX}.$

Observing that
$\mathrm{IX}=\partial_{m}d_{l}\delta_{km}\partial_{j}\partial_{m}d_{l}\delta_{im}\delta_{jk}=\partial_{k}d_{l}\partial_{k}\partial_{i}d_{l}$,
and recalling the shape of $\mathrm{II}$ from (6.13), we find that
$\mathrm{II}$ and $\mathrm{IX}$ cancel out. As a result, it remains to treat
the terms $\mathrm{VIII}$, $\mathrm{VII}$ and $\mathrm{VI}$ to complete the
estimates of $\mathcal{Q}_{\mathrm{III}}$. Similarly as in (6.16), we first
get

(6.18) $\|\mathrm{VIII}\|_{p,q}\leq
C\|\partial_{k}d_{l}\|_{\infty,\infty}\|\partial_{k}Y_{m}-\delta_{km}\|_{\infty,\infty}\|\partial_{j}\partial_{m}d_{l}\|_{p,q}\leq
C(C_{T}^{*}+C_{0}+R)T(R+C_{T}^{*})^{2}.$

Likewise, also using Subsection 6.2(a) to bound
$\|\partial_{i}Y_{m}\|_{\infty,\infty}$, we argue that

(6.19) $\displaystyle\|\mathrm{VII}\|_{p,q}$ $\displaystyle\leq
C(\|\partial_{j}\partial_{m}d_{l}\|_{p,q}\|\partial_{j}Y_{k}-\delta_{jk}\|_{\infty,\infty}\|\partial_{i}Y_{m}\|_{\infty,\infty}+\|\partial_{j}\partial_{m}d_{l}\|_{p,q}\|\partial_{j}Y_{k}-\delta_{im}\|_{\infty,\infty})$
$\displaystyle\leq CT(R+C_{T}^{*})^{2}.$

By virtue of Subsection 6.2(a), yielding the desired estimate of
$\|\partial_{k}\partial_{i}Y_{m}\|_{\infty,\infty}$, we finally obtain

(6.20) $\|\mathrm{VI}\|_{p,q}\leq
C\|\partial_{k}\partial_{i}Y_{m}\|_{\infty,\infty}\|\partial_{m}d_{l}\|_{p,q}\leq
CT(R+C_{T}^{*})^{2}.$

In summary, recalling the shape of $\mathcal{Q}_{\mathrm{III}}$ from (6.13)
and concatenating (6.17), (6.18), (6.19) as well as (6.20), we find that
$\mathcal{Q}_{\mathrm{III}}$ satisfies an estimate of the desired form.
Together with (6.11) and (6.12), this implies the existence of
$C_{\mathcal{Q}}(R,T)>0$ such that the first part of the assertion follows.

For the Lipschitz estimate, we start by adding and subtracting terms in order
to get a more advantageous representation. More precisely, we write

$\displaystyle\mathcal{Q}^{(1)}(\tilde{d}_{1},d^{*})-\mathcal{Q}^{(2)}(\tilde{d}_{2},d^{*})$
$\displaystyle=[B(d^{*})-B(\tilde{d}_{1}+d^{*})](\tilde{d}_{1}-\tilde{d}_{2})+[B(\tilde{d}_{1}+d^{*})$
$\displaystyle\quad-\mathcal{B}^{(1)}(\tilde{d}_{1}+d^{*})](\tilde{d}_{1}-\tilde{d}_{2})+[\mathcal{B}^{(2)}(\tilde{d}_{2}+d^{*})-\mathcal{B}^{(1)}(\tilde{d}_{1}+d^{*})](\tilde{d}_{2}+d^{*})$
$\displaystyle=:\mathcal{R}_{\mathrm{I}}+\mathcal{R}_{\mathrm{II}}+\mathcal{R}_{\mathrm{III}}.$

The terms $\mathcal{R}_{\mathrm{I}}$ and $\mathcal{R}_{\mathrm{II}}$ are
similar to $\mathcal{Q}_{\mathrm{I}}$ and $\mathcal{Q}_{\mathrm{III}}$ from
(6.10), but the operator $B$ and its transformed version $\mathcal{B}$ are
applied to $\tilde{d}_{1}-\tilde{d}_{2}$ which has zero initial values. As in
(6.11), we obtain

$\|\mathcal{R}_{\mathrm{I}}\|_{p,q}\leq
CR\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

Concerning $\mathcal{R}_{\mathrm{II}}$, we use the same splitting in
$\mathrm{I}$, $\mathrm{II}$, $\mathrm{III}$ and $\mathrm{IV}$ as seen for
$\mathcal{Q}_{\mathrm{III}}$ in (6.13), denoted by $\mathrm{i}$,
$\mathrm{ii}$, $\mathrm{iii}$ and $\mathrm{iv}$ here for distinction. Making
use of (6.8) and (6.15) for the first part, and recalling (6.14) as well as
(6.15), and Subsection 6.2 to estimate
$\|\partial_{i}Y_{m}-\delta_{im}\|_{\infty,\infty}$, we derive that

$\displaystyle\quad\|\mathrm{i}+\mathrm{iii}\|_{p,q}$ $\displaystyle\leq
C(\|\partial_{i}(d_{1})_{l}\|_{\infty,\infty}\|(\Delta-\mathcal{L}_{2}^{(1)})(\tilde{d}_{1}-\tilde{d}_{2})_{l}\|_{p,q}+\|\mathcal{L}_{2}^{(1)}(\tilde{d}_{1}-\tilde{d}_{2})_{l}\|_{p,q}\|\partial_{m}(d_{1})_{l}\|_{\infty,\infty}\|(\partial_{i}(Y_{1})_{m}-\delta_{im}\|_{\infty,\infty})$
$\displaystyle\leq
C(C_{T}^{*}+C_{0}+R)T(R+C_{T}^{*})\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

Also expanding $\mathrm{iv}=\mathrm{v}+\mathrm{vi}$,
$\mathrm{v}=\mathrm{v}(1)+\mathrm{vii}$ as well as
$\mathrm{v}(1)=\mathrm{viii}+\mathrm{ix}$ and using that $\mathrm{ii}$ and
$\mathrm{ix}$ cancel out, it remains to estimate $\mathrm{viii}$,
$\mathrm{vii}$ and $\mathrm{vi}$ given by

$\displaystyle\mathrm{viii}$
$\displaystyle=\partial_{m}(d_{1})_{l}(\partial_{k}(Y_{1})_{m}-\delta_{km})\partial_{j}(d_{1})_{m}(\tilde{d}_{1}-\tilde{d}_{2})_{l}\delta_{im}\delta_{jk},$
$\displaystyle\mathrm{vii}$
$\displaystyle=(\partial_{m}(d_{1})_{l}\partial_{k}(Y_{1})_{m})(\partial_{j}\partial_{m}(\tilde{d}_{1}-\tilde{d}_{2})_{l}(\partial_{j}(Y_{1})_{k}-\delta_{jk})\partial_{i}(Y_{1})_{m}$
$\displaystyle\quad+\partial_{j}\partial_{m}(\tilde{d}_{1}-\tilde{d}_{2})_{l}(\partial_{i}(Y_{1})_{m}-\delta_{im})\delta_{jk}),\enspace\text{and}$
$\displaystyle\mathrm{vi}$
$\displaystyle=(\partial_{m}(d_{1})_{l}\partial_{k}(Y_{1})_{m}\partial_{m}(d_{1})_{l}\partial_{k}\partial_{i}((Y_{1})_{m}-(Y_{2})_{m})).$

Arguing as in (6.18), with $d$ replaced by $\tilde{d}_{1}-\tilde{d}_{2}$, we
get

$\|\mathrm{viii}\|_{p,q}\leq
C(C_{T}^{*}+C_{0}+R)^{2}T(R+C_{T}^{*})\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

Mimicking the procedure to obtain (6.19), we infer that

$\|\mathrm{vii}\|_{p,q}\leq
C(C_{T}^{*}+C_{0}+R+1)T(R+C_{T}^{*})\|\tilde{d}_{1}-\tilde{d}_{1}\|_{\mathbb{E}_{T}^{d}}.$

Finally, using similar arguments as for (6.20), and invoking (6.15), we
conclude the estimate

$\displaystyle\|\mathrm{vi}\|_{p,q}$ $\displaystyle\leq
C\|\partial_{m}d_{1}\|_{\infty,\infty}^{2}\|\partial_{k}\partial_{i}((Y_{1})_{m}-(Y_{2})_{m})\|_{p,q}$
$\displaystyle\leq
C(C_{T}^{*}+C_{0}+R)^{2}T\|(\tilde{\ell}_{1}-\tilde{\ell}_{2},\tilde{\omega}_{1}-\tilde{\omega}_{2})\|_{\infty}$
$\displaystyle\leq
C(C_{T}^{*}+C_{0}+R)^{2}T\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

A concatenation of the previous estimates yields an estimate of
$\|\mathcal{R}_{\mathrm{II}}\|_{p,q}$ of the desired form. With respect to
$\mathcal{Q}^{(1)}(\tilde{d}_{1},d^{*})-\mathcal{Q}^{(2)}(\tilde{d}_{2},d^{*})$,
it remains to estimate $\mathcal{R}_{\mathrm{III}}$. To this end, we first
consider the difference in the first addend. Recalling that
$d_{i}=\tilde{d}_{i}+d^{*}$ and using the Einstein sum convention, we get

$\displaystyle\quad\mathcal{L}_{2}^{(2)}(d_{2})_{l}\partial_{m}(d_{2})_{l}\partial_{i}(Y_{2})_{m}-\mathcal{L}_{2}^{(1)}(d_{2})_{l}\partial_{m}(d_{1})_{l}\partial_{i}(Y_{1})_{m}$
$\displaystyle=\mathcal{L}_{2}^{(2)}(d_{2})_{l}\partial_{m}(d_{2})_{l}(\partial_{i}(Y_{2})_{m}-\partial_{i}(Y_{1})_{m})+\mathcal{L}_{2}^{(2)}(d_{2})_{l}\partial_{m}(d_{2}-d_{1})_{l}\partial_{i}(Y_{1})_{m}$
$\displaystyle\quad+(\mathcal{L}_{2}^{(2)}-\mathcal{L}_{2}^{(1)})(d_{2})_{l}\partial_{m}(d_{1})_{l}\partial_{i}(Y_{1})_{m}.$

Exploiting (6.14) for the estimate of
$\|\mathcal{L}_{2}^{(2)}(d_{2})_{l}\|_{p,q}$ and (6.15) to control
$\|\partial_{m}(d_{2})_{l}\|_{\infty,\infty}$, and estimating the difference
$(\partial_{i}(Y_{2})_{m}-\partial_{i}(Y_{1})_{m})$ by Subsection 6.2(a), we
deduce that

$\|\mathcal{L}_{2}^{(2)}(d_{2})_{l}\partial_{m}(d_{2})_{l}(\partial_{i}(Y_{2})_{m}-\partial_{i}(Y_{1})_{m})\|_{p,q}\leq
C(C_{T}^{*}+C_{0}+R)T(R+C_{T}^{*})\|\tilde{z}_{1}-\tilde{z}_{2}\|_{\mathbb{E}_{T}}.$

Thanks to Subsection 6.2(b), we further obtain

$\|\mathcal{L}_{2}^{(2)}(d_{2})_{l}\partial_{m}(d_{2}-d_{1})_{l}\partial_{i}(Y_{1})_{m}\|_{p,q}\leq
C(R+C_{T}^{*})T^{\nicefrac{{1}}{{p}}}\|\tilde{d}_{1}-\tilde{d}_{2}\|_{\mathbb{E}_{T}^{d}}.$

The last term can be estimated in a similar way. ∎

The result below on estimates of $F_{1}$ as introduced in (6.2) is a
consequence of Subsection 6.2 and the estimates of the other transformed terms
from Section 4 which can be obtained in a similar way as the estimates in
Subsection 6.2 or Subsection 6.2. We also refer to [15, Lemmas 6.6 and 6.8].

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and for the reference solution
$(z^{*},p^{*})$ from Section 5 and $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{i},\tilde{p}_{i})\in\mathcal{K}_{T}^{R}$, let
$(z,p)=(\tilde{z}+z^{*},\tilde{p}+p^{*})$ and
$(z_{i},p_{i})=(\tilde{z}_{i}+z^{*},\tilde{p}_{i}+p^{*})$, $i=1,2$. Recall
$C_{\mathrm{MR}}>0$ from Section 5. Then there are $C_{1}(R,T)>0$ with
$C_{1}(R,T)<\nicefrac{{R}}{{8C_{\mathrm{MR}}}}$ for $T>0$ sufficiently small
and $L_{1}(R,T)>0$ with $L_{1}(R,T)\to 0$ as $R\to 0$ and $T\to 0$ such that

$\|F_{1}(\tilde{z},\tilde{p})\|_{\mathbb{F}_{T}^{v}}\leq
C_{1}(R,T),\enspace\text{and}\enspace\|F_{1}(\tilde{z}_{1},\tilde{p}_{1})-F_{1}(\tilde{z}_{2},\tilde{p}_{2})\|_{\mathbb{F}_{T}^{v}}\leq
L_{1}(R,T)\|(\tilde{z}_{1},\tilde{p}_{1})-(\tilde{z}_{2},\tilde{p}_{2})\|_{\tilde{\mathbb{E}}_{T}}.$

Finally, we prove estimates of the terms $F_{3}$ and $F_{4}$ appearing in the
equations of the rigid body and defined in (6.2).

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and for the reference solution
$(z^{*},p^{*})$ from Section 5 and $(\tilde{z},\tilde{p})$,
$(\tilde{z}_{i},\tilde{p}_{i})\in\mathcal{K}_{T}^{R}$, let
$(z,p)=(\tilde{z}+z^{*},\tilde{p}+p^{*})$ and
$(z_{i},p_{i})=(\tilde{z}_{i}+z^{*},\tilde{p}_{i}+p^{*})$, $i=1,2$. Recall
$C_{\mathrm{MR}}>0$ from Section 5. Then there are $C_{3}(R,T)$,
$C_{4}(R,T)>0$ with $C_{3}(R,T)$,
$C_{4}(R,T)<\nicefrac{{R}}{{8C_{\mathrm{MR}}}}$ for $T>0$ sufficiently small
and $L_{3}(R,T)$, $L_{4}(R,T)>0$ with $L_{3}(R,T)$, $L_{4}(R,T)\to 0$ as $R\to
0$ and $T\to 0$ such that

$\displaystyle\|F_{3}(\tilde{z},\tilde{p})\|_{\mathbb{F}_{T}^{\ell}}$
$\displaystyle\leq
C_{3}(R,T),\enspace\text{and}\enspace\|F_{3}(\tilde{z}_{1},\tilde{p}_{1})-F_{3}(\tilde{z}_{2},\tilde{p}_{2})\|_{\mathbb{F}_{T}^{\ell}}\leq
L_{3}(R,T)\|(\tilde{z}_{1},\tilde{p}_{1})-(\tilde{z}_{2},\tilde{p}_{2})\|_{\tilde{\mathbb{E}}_{T}},\enspace\text{as
well as}\enspace$
$\displaystyle\|F_{4}(\tilde{z},\tilde{p})\|_{\mathbb{F}_{T}^{\ell}}$
$\displaystyle\leq
C_{4}(R,T),\enspace\text{and}\enspace\|F_{4}(\tilde{z}_{1},\tilde{p}_{1})-F_{4}(\tilde{z}_{2},\tilde{p}_{2})\|_{\mathbb{F}_{T}^{\ell}}\leq
L_{4}(R,T)\|(\tilde{z}_{1},\tilde{p}_{1})-(\tilde{z}_{2},\tilde{p}_{2})\|_{\tilde{\mathbb{E}}_{T}}.$

###### Proof.

The estimates of $\omega\times\ell$ and $\omega\times(J_{0}\omega)$ readily
follow, so it remains to investigate the surface integrals. For this, we
introduce
$\mathcal{J}\colon\mathrm{W}^{\varepsilon+\nicefrac{{1}}{{q}},q}(\mathcal{F}_{0})^{3\times
3}\to\mathbb{R}^{6}$ defined by

(6.21)
$\mathcal{J}(h):=\begin{pmatrix}\int_{\partial\mathcal{S}_{0}}hN\,\mathrm{d}\Gamma\\\
\int_{\partial\mathcal{S}_{0}}y\times
hN\,\mathrm{d}\Gamma\end{pmatrix},\enspace\text{with}\enspace|\mathcal{J}(h)|\leq
C\|h\|_{\mathrm{W}^{\varepsilon+\nicefrac{{1}}{{q}},q}(\mathcal{F}_{0})}\enspace\text{for}\enspace
h\in\mathrm{W}^{\varepsilon+\nicefrac{{1}}{{q}},q}(\mathcal{F}_{0})^{3\times
3}.$

With regard to the surface integrals in $F_{3}$ and $F_{4}$, we observe that
they consist of two parts, namely the part associated to the difference in the
linearized stress tensor, $\mathrm{T}-\mathcal{T}$, and the nonlinear term
resulting from the $d$-part in the stress tensor. Concerning the first part,
we observe that the estimates have already been established in [15, Lemmas
6.6, 6.7 and 6.8] by invoking (6.21). Therefore, we can restrict ourselves to
the respective estimates for the remaining $d$-part in the surface integrals.
Here, we use a direct approach to estimate the terms instead of (6.21). From
the continuity of the trace, we conclude

(6.22)
$\displaystyle\left|\int_{\partial\mathcal{S}_{0}}Q^{\top}(\nabla(\tilde{d}+d^{*}))((\nabla(\tilde{d}+d^{*})))^{\top}QN\,\mathrm{d}\Gamma\right|$
$\displaystyle\leq
C|Q|^{2}\|\tilde{d}+d^{*}\|_{\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0})}^{2}.$

It further follows from the mixed derivative theorem and the condition
$\nicefrac{{2}}{{p}}+\nicefrac{{3}}{{q}}<1$ that

(6.23)
$\mathbb{E}_{T}^{d}\hookrightarrow\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{2(1-\theta),q}(\mathcal{F}_{0})^{3})\hookrightarrow\mathrm{L}^{\infty}(0,T;\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0})^{3}),\enspace\text{and}\enspace\mathrm{X}_{\gamma}^{d}\hookrightarrow\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0})^{3}$

thanks to [45, Theorem 4.6.1]. Applying the above equation (6.22), Subsection
6.2(b) to bound $\|Q\|_{\infty}$, the preceding embedding (6.23) and Section 5
for the estimate of $\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}$
by the norm of the initial data and the reference solution, we derive that

(6.24)
$\displaystyle\quad\left\|\int_{\partial\mathcal{S}_{0}}Q^{\top}(\nabla(\tilde{d}+d^{*}))((\nabla(\tilde{d}+d^{*})))^{\top}QN\,\mathrm{d}\Gamma\right\|_{p}$
$\displaystyle\leq
C\|\tilde{d}+d^{*}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0}))}\|\tilde{d}+d^{*}\|_{\mathrm{L}^{p}(0,T;\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0}))}$
$\displaystyle\leq
C\left(\|\tilde{d}\|_{\mathbb{E}_{T}^{d}}+\|d^{*}\|_{\mathrm{BUC}([0,T];\mathrm{X}_{\gamma}^{d})}\right)\left(T^{\nicefrac{{1}}{{p}}}\|\tilde{d}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{H}^{\nicefrac{{3}}{{2}}}(\mathcal{F}_{0}))}+\|d^{*}\|_{\mathbb{E}_{T}^{d}}\right)$
$\displaystyle\leq C(R+C_{T}^{*}+C_{0})(T^{\nicefrac{{1}}{{p}}}R+C_{T}^{*}).$

Let us remark that the other surface integral involving the third addend of
the stress tensor can be dealt with analogously. The first part of the
assertion then follows.

With respect to the Lipschitz estimate, we can expand the terms suitably to
get a difference in each component. For the differences in $Q$, we can use a
similar estimate as (6.24) together with Subsection 6.2(b) in order to get an
estimate by the difference. Concerning differences in $d$, a slight
modification of the estimate is necessary. In fact, as a preparation, we
consider $d_{1}$, $d_{2}\in\mathbb{E}_{T}$, where $d_{1}$ additionally
satisfies $d_{1}(0)=0$. Let us observe that this corresponds to the situation
of the difference being in $d_{1}$, i.e., $d_{1}=\tilde{d}_{1}-\tilde{d}_{2}$.
The other term can be handled completely analogously. By Subsection 6.2(b), we
obtain in particular that $d_{1}$,
$d_{2}\in\mathrm{BUC}([0,T];\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})^{3})$.
Denoting by $q^{\prime}\in(1,\infty)$ the Hölder conjugate of $q$, and using
Hölder’s inequality as well as continuity of the trace in addition to the
aforementioned embedding, we get

$\displaystyle\left|\int_{\partial\mathcal{S}_{0}}\nabla d_{1}(\nabla
d_{2})^{\top}N\,\mathrm{d}\Gamma\right|$ $\displaystyle\leq C\|\nabla
d_{1}\|_{\mathrm{L}^{q^{\prime}}(\partial\mathcal{S}_{0})}\|\nabla
d_{2}\|_{\mathrm{L}^{q}(\partial\mathcal{S}_{0})}\leq
C\|d_{1}\|_{\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})}\|d_{2}\|_{\mathrm{W}^{1+\nicefrac{{1}}{{q}}+\varepsilon,q}(\mathcal{F}_{0})}$
$\displaystyle\leq
C\|d_{1}\|_{\mathrm{C}^{1}(\overline{\mathcal{F}_{0}})}\|d_{2}\|_{\mathrm{W}^{2,q}(\mathcal{F}_{0})}.$

Therefore, employing Hölder’s inequality in time in conjunction with
Subsection 6.2(b) to control $\|Q\|_{\infty}$ and Subsection 6.2(b) for an
embedding, we infer that

$\left\|\int_{\partial\mathcal{S}_{0}}Q^{\top}\nabla d_{1}(\nabla
d_{2})^{\top}QN\right\|_{p}\leq
C\|d_{1}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{C}^{1}(\overline{\mathcal{F}_{0}}))}\|d_{2}\|_{\mathrm{L}^{p}(0,T;\mathrm{W}^{2,q}(\mathcal{F}_{0}))}\leq
C\|d_{1}\|_{\mathbb{E}_{T}^{d}}\|d_{2}\|_{\mathbb{E}_{T}^{d}}.$

The embedding constant resulting from Subsection 6.2(b) is $T$-independent
thanks to the homogeneous initial values. This shows how to estimate the terms
with differences in the $d$-terms and then completes the proof as the terms
corresponding to the second surface integral can be treated by analogy. ∎

###  6.3. Proof of Theorem 2.1

We first show Theorem 4.1 and then deduce Theorem 2.1 by performing the
backward change of variables.

###### Proof of Theorem 4.1.

For $(\tilde{z},\tilde{p})\in\mathcal{K}_{T}^{R}$, in view of (6.1), it
follows from Section 5, Subsection 6.2, Subsection 6.2 and Subsection 6.2 that

$\|\Phi_{T}^{R}(\tilde{z},\tilde{p})\|_{\tilde{\mathbb{E}}_{T}}\leq
C_{\mathrm{MR}}\|(F_{1}(\tilde{z},\tilde{p}),F_{2}(\tilde{z}),F_{3}(\tilde{z},\tilde{p}),F_{4}(\tilde{z},\tilde{p}))\|_{\mathbb{F}_{T}}\leq
C_{\mathrm{MR}}C(R,T),$

where given $R>0$, we have $C(R,T)<\nicefrac{{R}}{{2C_{\mathrm{MR}}}}$ for
$T>0$ sufficiently small. On the other hand, the same arguments imply that for
$(\tilde{z}_{1},\tilde{p}_{1})$,
$(\tilde{z}_{2},\tilde{p}_{2})\in\mathcal{K}_{T}^{R}$, we get

$\|\Phi_{T}^{R}(\tilde{z}_{1},\tilde{p}_{1})-\Phi_{T}^{R}(\tilde{z}_{2},\tilde{p}_{2})\|_{\tilde{\mathbb{E}}_{T}}\leq
C_{\mathrm{MR}}L(R,T)\|(\tilde{z}_{1},\tilde{p}_{1})-(\tilde{z}_{2},\tilde{p}_{2})\|_{\tilde{\mathbb{E}}_{T}}.$

Hence, by the time-independence of $C_{\mathrm{MR}}>0$, we choose $R>0$ and
$T>0$ sufficiently small such that
$C_{\mathrm{MR}}C(R,T)\leq\nicefrac{{R}}{{2}}$ and
$C_{\mathrm{MR}}L(R,T)\leq\nicefrac{{1}}{{2}}$, so $\Phi_{T}^{R}$ is a self
map and contraction. The contraction mapping principle yields the existence of
a unique fixed point
$(\widehat{z},\widehat{p})\in\prescript{}{0}{\mathbb{E}_{T}}\times\mathbb{E}_{T}^{p}$.
Adding $(z^{*},p^{*})$ from Section 5, we get the unique solution
$(z,p)=(\widehat{z}+z^{*},\widehat{p}+p^{*})\in\tilde{\mathbb{E}}_{T}$ to
(4.7)–(4.9). Let us observe that $(z,p)\in\tilde{\mathbb{E}}_{T}$ corresponds
to $(v,d,\ell,\omega,p)$ being in the regularity class as asserted in Theorem
4.1. ∎

###### Proof of Theorem 2.1.

From $(\ell,\omega)\in\mathrm{W}^{1,p}(0,T)^{6}$ emerging from Theorem 4.1, we
derive $h^{\prime}$, $\Omega$ as well as the diffeomorphism $X$ as described
in Subsection 6.2. Performing the backward change of variables and coordinates
as pointed out in Section 4, we conclude the assertion of Theorem 2.1 from
Theorem 4.1 upon adding $d_{*}$ in the $d$-component. The uniqueness of the
solution is a consequence of the uniqueness of the diffeomorphism. Moreover,
the average zero condition of the pressure is preserved in view of the
definition of the transformed pressure $p$ on the fixed domain, see Section 4.
∎

##  7\. Proof of the global strong well-posedness close to equilibria

In this section, we prove the second main result of this paper on the global
strong well-posedness of the interaction problem (1.2)–(1.7) for initial data
close to constant equilibria. To this end, we establish a maximal
$\mathrm{L}^{p}([0,\infty))$-regularity type result in Subsection 7.1 by using
a splitting into the mean value-zero part and the average in the $d$-component
in order to get exponential decay. Subsection 7.2 is dedicated to showing
estimates of the nonlinear terms tailored to the adjusted setting. These
estimates lead to the proof of the second main result in Subsection 7.3 by
means of a fixed point argument.

###  7.1. Analysis of the linearized system

Let us recall from Section 5 that the maximal regularity of (5.8) is only
obtained up to a shift. It can be shown that we obtain maximal
$\mathrm{L}^{p}([0,\infty))$-regularity of the linearized system provided for
the associated resolvent problem, we can establish that it admits a unique
solution for all $\lambda\in\mathbb{C}_{-}\cup\\{0\\}$. The resolvent problem
takes the shape

(7.1) $\left\\{\begin{aligned} \lambda
v-\Delta{v}+\nabla{p}&=g_{1},\enspace\operatorname{div}{v}=0,\enspace\text{and}\enspace\lambda
d-\Delta d=g_{2},&&\enspace\text{in}\enspace\mathcal{F}_{0},\\\ \lambda
m_{\mathcal{S}}{\ell}+\int_{\partial\mathcal{S}_{0}}\mathrm{T}(v,p)N\,\mathrm{d}\Gamma&=g_{3},\enspace\text{and}\enspace\lambda
J_{0}{\omega}+\int_{\partial\mathcal{S}_{0}}y\times\mathrm{T}(v,p)N\,\mathrm{d}\Gamma=g_{4},&&\enspace\text{in}\enspace\mathbb{R}^{3},\\\
{v}&=\ell+\omega\times
y,\enspace\text{on}\enspace\partial\mathcal{S}_{0},\enspace
v=0,\enspace\text{on}\enspace\partial\mathcal{O},\enspace\text{and}\enspace\partial_{\nu}d=0,&&\enspace\text{on}\enspace\partial\mathcal{F}_{0}.\end{aligned}\right.$

When employing the operator theoretic formulation as in [36, Section 3]
together with the Neumann Laplacian operator for the director part, it readily
follows that the resulting operator has a compact resolvent, so the spectrum
only consists of eigenvalues, and it suffices to consider the
$\mathrm{L}^{2}$-case. Hence, considering $(v,d,\ell,\omega,p)$ solving (7.1)
with $(g_{1},g_{2},g_{3},g_{4})=(0,0,0,0)$, we test the resulting fluid
equation by $v$, integrate it over $\mathcal{F}_{0}$, and employ
$\operatorname{div}v=0$ as well as the boundary conditions $v=0$ on
$\partial\mathcal{O}$ and $v=\ell+\omega\times y$. Moreover, observing that
$-\Delta v+\nabla p=-\operatorname{div}(2\mathbb{D}v-p\operatorname{Id})$, we
deduce from integration by parts, from $a\cdot(b\times c)=b\cdot(c\times a)$
and from $\ell$ and $\omega$ being constant in space that

(7.2) $\displaystyle 0$
$\displaystyle=-\int_{\mathcal{F}_{0}}\operatorname{div}(2\mathbb{D}v-p\operatorname{Id})\cdot
v\,\mathrm{d}y=\int_{\mathcal{F}_{0}}|\nabla
v|^{2}\,\mathrm{d}y-\int_{\partial\mathcal{S}_{0}}(2\mathbb{D}v-p\operatorname{Id})\cdot
vN\,\mathrm{d}\Gamma$ $\displaystyle=\int_{\mathcal{F}_{0}}|\nabla
v|^{2}\,\mathrm{d}y-\ell\cdot\int_{\partial\mathcal{S}_{0}}(2\mathbb{D}v-p\operatorname{Id})N\,\mathrm{d}\Gamma-\omega\cdot\int_{\partial\mathcal{S}_{0}}y\times(2\mathbb{D}v-p\operatorname{Id})N\,\mathrm{d}\Gamma.$

Therefore, testing the resolvent problem with right-hand sides equal to zero
by $(v,d,\ell,\omega)$, integrating by vector parts, invoking (7.2) and
additionally using that $\partial_{\nu}d=0$ on $\partial\mathcal{F}_{0}$, we
obtain

(7.3)
$0=\lambda\left(\|v\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}+\|d\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}+|\ell|^{2}+|\omega|^{2}\right)+\|\nabla
v\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}+\|\nabla
d\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}.$

As $\|\nabla v\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}$ and $\|\nabla
d\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}$ are real and non-negative, it
follows that $\lambda\in\mathbb{R}$ and $\lambda\geq 0$. In the case
$\lambda=0$, we derive from (7.3) that $0=\|\nabla
v\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}+\|\nabla
d\|_{\mathrm{L}^{2}(\mathcal{F}_{0})}^{2}$, i.e., $v$ and $d$ are constant on
$\mathcal{F}_{0}$. In view of $v=0$ on $\partial\mathcal{O}$, we conclude that
$v=0$. As the velocity of the fluid and the rigid body coincide on their
interface, we deduce therefrom that $\ell+\omega\times y=0$ on
$\partial\mathcal{S}_{0}$. Similarly as in Subsection 3.2, using the 3D
analogue of [41, Lemma 2.1], this implies that $\ell=\omega=0$. For the
pressure $p$ with average zero, it also follows that $p=0$. On the other hand,
in order to remedy the fact that $d$ can be constant, it is natural to invoke
the space of average zero functions $\mathrm{L}_{0}^{q}(\mathcal{F}_{0})$. It
readily follows that $\mathrm{L}_{0}^{q}(\mathcal{F}_{0})$ is invariant under
the semigroup $\mathrm{e}^{\Delta_{\mathrm{N}}t}$ and that the restriction of
$\Delta_{\mathrm{N}}$ to $\mathrm{L}_{0}^{q}(\mathcal{F}_{0})$ is invertible.
This yields that when invoking $\mathrm{L}_{0}^{q}(\mathcal{F}_{0})^{3}$ for
the $d$-equation, the resolvent problem admits a unique solution for all
$\lambda\in\mathbb{C}_{-}\cup\\{0\\}$.

In order to account for the average zero condition in the $d$-part, we
introduce some further notation. In fact, for $\mathrm{X}_{0}$,
$\mathrm{X}_{1}$ and $\mathrm{X}_{\gamma}$ as made precise in (5.2) and (2.1),
respectively, and for $i\in\\{0,1,\gamma\\}$, we define

$\mathrm{X}_{i}^{m}\coloneqq\left\\{z=(v,d,\ell,\omega)\in\mathrm{X}_{i}:d\in\mathrm{L}_{0}^{q}(\mathcal{F}_{0})^{3}\right\\}.$

Accordingly, for $\mathbb{F}_{T}^{v}$, $\mathbb{F}_{T}^{\ell}$ and
$\mathbb{F}_{T}^{\omega}$ as made precise in (5.1), we set

(7.4) $\displaystyle\mathbb{F}_{T}^{m}$
$\displaystyle\coloneqq\mathbb{F}_{T}^{v}\times\mathrm{L}^{p}\left(0,T;\mathrm{L}_{0}^{q}(\mathcal{F}_{0})^{3}\right)\times\mathbb{F}_{T}^{\ell}\times\mathbb{F}_{T}^{\omega},$
$\displaystyle\mathbb{E}_{T}^{m}$
$\displaystyle\coloneqq\mathrm{W}^{1,p}(0,T;\mathrm{X}_{0}^{m})\cap\mathrm{L}^{p}(0,T;\mathrm{X}_{1}^{m}),\enspace\text{and}\enspace\tilde{\mathbb{E}}_{T}^{m}\coloneqq\mathbb{E}_{T}^{m}\times\mathbb{E}_{T}^{p}.$

We summarize the preceding discussion in the following proposition.

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), and consider $d_{*}\in\mathbb{R}^{3}$
constant such that $|d_{*}|=1$. Then for all $\mu\geq 0$,
$(g_{1},g_{2},g_{3},g_{4})\in\mathbb{F}_{\infty}^{m}$ and
$(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}^{m}$, there
exists a unique solution
$(v,d,\ell,\omega,p)\in\tilde{\mathbb{E}}_{\infty}^{m}$ to (5.8). Besides, for
$C>0$, we obtain the estimate

$\|(v,d,\ell,\omega,p)\|_{\tilde{\mathbb{E}}_{\infty}^{m}}\leq
C\left(\|(g_{1},g_{2},g_{3},g_{4})\|_{\mathbb{F}_{\infty}^{m}}+\|(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\|_{\mathrm{X}_{\gamma}^{m}}\right).$

As the resolvent set is open, we even obtain the existence of $\eta_{0}$ such
that the semigroup associated to the operator $A_{0}^{m}$ corresponding to the
linearized problem is exponentially stable, i.e., for
$z=(v,d,\ell,\omega)\in\mathrm{X}_{0}^{m}$, we have

(7.5) $\|\mathrm{e}^{-A_{0}^{m}t}(v,d,\ell,\omega)\|_{\mathrm{X}_{0}^{m}}\leq
C\mathrm{e}^{-\eta_{0}t}\|(v,d,\ell,\omega)\|_{\mathrm{X}_{0}^{m}}.$

The average zero condition is generally not preserved by the equation of the
director $d$ in the interaction problem, so we decompose the equation into two
parts. More precisely, for a function $f\in\mathrm{L}^{1}(\mathcal{F}_{0})$,
we introduce the splitting

(7.6) $f=f_{m}+f_{\mathrm{avg}},\enspace\text{where}\enspace
f_{\mathrm{avg}}\coloneqq\frac{1}{|\mathcal{F}_{0}|}\int_{\mathcal{F}_{0}}f(y)\,\mathrm{d}y\enspace\text{and}\enspace
f_{m}\coloneqq f-f_{\mathrm{avg}},$

resulting in $f_{m}\in\mathrm{L}_{0}^{1}(\mathcal{F}_{0})$. For
$\mathbb{F}_{T}$ and $\tilde{\mathbb{E}}_{T}^{m}$ as defined in (5.1) and
(7.4) as well as $z=(v,d,\ell,\omega)$, we set

$\mathrm{e}^{-\eta(\cdot)}\mathbb{F}_{\infty}\coloneqq\left\\{z\in\mathbb{F}_{\infty}:\mathrm{e}^{\eta(\cdot)}z\in\mathbb{F}_{\infty}\right\\},\enspace\text{and}\enspace\mathrm{e}^{-\eta(\cdot)}\tilde{\mathbb{E}}_{\infty}^{m}\coloneqq\left\\{(z,p)\in\tilde{\mathbb{E}}_{\infty}^{m}:\mathrm{e}^{\eta(\cdot)}(z,p)\in\tilde{\mathbb{E}}_{\infty}^{m}\right\\}.$

The maximal regularity type result concerning the linearized problem (5.8)
then reads as follows. Let us observe that we may consider (5.8) with $\mu=0$
by virtue of Subsection 7.1.

######  Proposition .

Let $p,q\in(1,\infty)$ satisfy (2.2), let $d_{*}\in\mathbb{R}^{3}$ be constant
such that $|d_{*}|=1$, and consider $\eta\in(0,\eta_{0})$, where $\eta_{0}$ is
the constant from (7.5). Then for all
$(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\in\mathrm{X}_{\gamma}^{m}$ and for
all $(g_{1},g_{2},g_{3},g_{4})\in\mathrm{e}^{-\eta(\cdot)}\mathbb{F}_{\infty}$
with $g_{2,\mathrm{avg}}\in\mathrm{L}^{1}(0,\infty)$, (5.8) with $\mu=0$ has a
unique solution $(v,d,\ell,\omega,p)$ such that

$(v,d_{m},\ell,\omega,p)\in\mathrm{e}^{-\eta(\cdot)}\tilde{\mathbb{E}}_{\infty}^{m},\enspace\text{and}\enspace
d_{\mathrm{avg}}\in\mathrm{L}^{\infty}(0,\infty),$

and there exists $C_{\mathrm{exp}}=C_{\mathrm{exp}}(p,q,\eta)>0$ such that

$\displaystyle\quad\left\|\mathrm{e}^{\eta(\cdot)}(v,d_{m},\ell,\omega,p)\right\|_{\tilde{\mathbb{E}}_{\infty}^{m}}+\left\|d_{\mathrm{avg}}\right\|_{\mathrm{L}^{\infty}(0,\infty)}+\left\|\mathrm{e}^{\eta(\cdot)}\partial_{t}d_{\mathrm{avg}}\right\|_{\mathrm{L}^{p}(0,\infty)}$
$\displaystyle\leq
C_{\mathrm{exp}}\left(\left\|(v_{0},d_{0}-d_{*},\ell_{0},\omega_{0})\right\|_{\mathrm{X}_{\gamma}^{m}}+\left\|\mathrm{e}^{\eta(\cdot)}(g_{1},g_{2},g_{3},g_{4})\right\|_{\mathbb{F}_{\infty}}+\left\|g_{2,\mathrm{avg}}\right\|_{\mathrm{L}^{1}(0,\infty)}\right).$

###### Proof.

We focus on the case $\eta=0$ and remark that the situation of $\eta>0$
readily follows by multiplying the functions by $\mathrm{e}^{\eta t}$ and
making use of the maximal regularity of $A_{0}^{m}+\eta$ as well as the
exponential stability in view of the invertibility. We also observe that the
$d$-component in (5.8) is decoupled from the remaining part, and the desired
estimates in the components $v$, $\ell$ and $\omega$ hence follow from
Subsection 7.1, so it only remains to investigate the $d$-component. We split
the problem in several parts and first consider the lifting of the initial
values. To this end, we introduce $\varphi^{1}$ solving

(7.7)
$\displaystyle\partial_{t}\varphi^{1}+\mu^{\prime}\varphi^{1}-\Delta\varphi^{1}$
$\displaystyle=0,$
$\displaystyle\enspace\text{in}\enspace(0,\infty)\times\mathcal{F}_{0},$
$\displaystyle\partial_{\nu}\varphi^{1}$ $\displaystyle=0,$
$\displaystyle\enspace\text{on}\enspace(0,\infty)\times\partial\mathcal{F}_{0},$
$\displaystyle\varphi^{1}(0)$ $\displaystyle=d_{0}-d_{*},$
$\displaystyle\enspace\text{in}\enspace\mathcal{F}_{0}.$

In (7.7), $\mu^{\prime}>0$ is sufficiently large to guarantee the existence
and uniqueness of a strong solution

$\varphi^{1}\in\mathrm{L}^{p}(0,\infty;\mathrm{W}_{\mathrm{N}}^{2,q}(\mathcal{F}_{0}))\cap\mathrm{W}^{1,p}(0,\infty;\mathrm{L}^{q}(\mathcal{F}_{0}))$

by maximal regularity of the Neumann Laplacian, see e.g. [7, Theorem 8.2].
Performing the splitting
$\varphi^{1}=\varphi_{m}^{1}+\varphi_{\mathrm{avg}}^{1}$ as introduced in
(7.6), we infer that $\varphi_{\mathrm{avg}}^{1}$ satisfies

(7.8)
$\varphi_{\mathrm{avg}}^{1}(t)=(d_{0,\mathrm{avg}}-d_{*})\mathrm{e}^{-\mu^{\prime}t}.$

This also implies $\varphi_{\mathrm{avg}}^{1}\in\mathrm{L}^{r}(0,\infty)$ for
all $r\in[1,\infty]$. Moreover, for $g_{2}=g_{2,m}+g_{2,\mathrm{avg}}$, we set

(7.9)
$\varphi^{2}(t)\coloneqq\int_{0}^{t}g_{2,\mathrm{avg}}(s)+\mu^{\prime}\varphi_{\mathrm{avg}}^{1}(s)\,\mathrm{d}s.$

The assumption $g_{2,\mathrm{avg}}\in\mathrm{L}^{1}(0,\infty)$ and the above
observation on $\varphi_{\mathrm{avg}}^{1}$ yield that $\varphi^{2}$ is well-
defined, and

(7.10) $|\varphi^{2}(t)|\leq
C\left(\|g_{2,\mathrm{avg}}\|_{\mathrm{L}^{1}(0,\infty)}+\|\varphi_{\mathrm{avg}}^{1}\|_{\mathrm{L}^{1}(0,\infty)}\right)$

for all $t\in(0,\infty)$. Let us also observe that
$\varphi^{2}=\varphi_{\mathrm{avg}}^{2}$ because $\varphi^{2}$ is constant in
space by definition. For $\varphi^{1}$ as in (7.7), $\varphi^{2}$ as in (7.9)
and $d$ solving the $d$-part associated to (5.8), i.e., $\partial_{t}d-\Delta
d=g_{2}$ in $(0,\infty)\times\mathcal{F}_{0}$, $\partial_{\nu}d=0$ on
$(0,\infty)\times\partial\mathcal{F}_{0}$ and $d(0)=d_{0}-d_{*}$ in
$\mathcal{F}_{0}$, we define $\tilde{\varphi}\coloneqq
d-\varphi^{1}-\varphi^{2}$. By construction, it follows that
$\tilde{\varphi}_{\mathrm{avg}}=0$. In addition, we obtain

$\frac{1}{|\mathcal{F}_{0}|}\int_{\mathcal{F}_{0}}(d_{0}(y)-d_{*})\,\mathrm{d}y=d_{0,\mathrm{avg}}-d_{*}.$

In view of
$\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})\hookrightarrow\mathrm{L}^{\infty}(\mathcal{F}_{0})$
thanks to (2.2), we infer that

(7.11)
$|d_{0,\mathrm{avg}}-d_{*}|\leq\|d_{0}-d_{*}\|_{\mathrm{L}^{\infty}(\mathcal{F}_{0})}\leq
C\|d_{0}-d_{*}\|_{\mathrm{B}_{qp}^{2-\nicefrac{{2}}{{p}}}(\mathcal{F}_{0})}.$