# Martingale solutions in stochastic fluid-structure interaction

Dominic Breit  Institute of Mathematics, TU Clausthal, Erzstraße 1, 38678
Clausthal-Zellerfeld, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>,
Prince Romeo Mensah  and  Thamsanqa Castern Moyo

###### Abstract.

We consider a viscous incompressible fluid interacting with a linearly elastic
shell of Koiter type which is located at some part of the boundary. Recently
models with stochastic perturbation in the shell equation have been proposed
in the literature but only analysed in simplified cases. We investigate the
full model with transport noise, where (a part of) the boundary of the fluid
domain is randomly moving in time. We prove the existence of a weak martingale
solution to the underlying system.

###### Key words and phrases:

Incompressible Navier–Stokes equation, Transport noise, Fluid-Structure
interaction

###### 2020 Mathematics Subject Classification:

76D05; 76D09; 74F10; 60H15

##  1\. Introduction

The mathematical analysis of systems of partial differential equations arising
from fluid-structure interaction has seen a vast progress in the last two
decades. This is motivated by a variety of applications, for instance in
biomechanics [1], hydro-dynamics [6], aero-elasticity [10] and hemo-mechanics
[14].

###  1.1. Deterministic models

We are interested in the case where a viscous incompressible fluid interacts
with an elastic structure located at a part of the boundary of the fluid’s
domain $\mathcal{O}\subset\mathbb{R}^{3}$ denoted by $\Gamma$. The structure
reacts to the forces imposed by the fluid at the boundary. Assuming that this
deformation only acts in the normal direction and denoting by $\mathbf{n}$ the
outer unit normal at the reference domain, $\mathcal{O}$ is deformed to the
domain $\mathcal{O}_{\eta(t)}$ defined through its boundary

$\displaystyle\partial\mathcal{O}_{\eta(t)}:=\\{\mathbf{y}+\eta(t,\mathbf{y})\mathbf{n}(\mathbf{y}):\,\mathbf{y}\in\Gamma\\}.$
(1.1)

Here $\eta:(t,\mathbf{y}):I\times\Gamma\mapsto\eta(t,\mathbf{y})\in\mathbb{R}$
describes the deformation of the structure and $I:=(0,T)$, for some $T>0$
denotes a time interval. For technical simplification we will suppose that
$\Gamma$ is the whole boundary and identify it with the two-dimensional torus
(the precise geometric set-up is presented in Section 2.2).

As a prototype let us consider the following problem where the equation for
the shell can be seen as a linearised version of Koiter’s model (neglecting
lower order terms for simplicity and setting all positive physical constants
to 1). In the unknowns

$\displaystyle\mathbf{u}:(t,\mathbf{x}):I\times\mathcal{O}_{\eta}\mapsto\mathbf{u}(t,\mathbf{x})\in\mathbb{R}^{3},$
$\displaystyle\pi:(t,\mathbf{x}):I\times\mathcal{O}_{\eta}\mapsto\pi(t,\mathbf{x})\in\mathbb{R},$

accounting for the fluid’s velocity field and pressure, respectively (defined
on a moving space-time cylinder ), it reads as (for simplicity we neglect
volume forces in the fluid equations)

$\displaystyle\mathrm{div}_{\mathbf{x}}\mathbf{u}=0,$ (1.2)
$\displaystyle\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\nabla_{\mathbf{x}})\mathbf{u}=\Delta_{\mathbf{x}}\mathbf{u}-\nabla_{\mathbf{x}}\pi,$
(1.3)
$\displaystyle\partial_{t}^{2}\eta+\Delta_{\mathbf{y}}^{2}\eta=-\mathbf{n}^{\top}(\mathbb{T}(\mathbf{u},\pi)\mathbf{n}_{\eta})\circ\bm{\varphi}_{\eta}|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\eta})|.$
(1.4)

The system is complemented by the kinetic boundary condition

$\displaystyle\mathbf{u}\circ\bm{\varphi}_{\eta}=\mathbf{n}\partial_{t}\eta$
$\displaystyle\quad\text{on }I\times\Gamma$ (1.5)

at the fluid-structure interface as well as initial conditions for (1.3)-(1.4)
and periodic boundary conditions for (1.4). Here
$\mathbb{T}(\mathbf{u},\pi)=(\nabla_{\mathbf{x}}\mathbf{u}+\nabla_{\mathbf{x}}\mathbf{u}^{\top})-\pi\mathbb{I}_{3\times
3}$ is the stress tensor of the fluid. The vectors $\mathbf{n}$ and
$\mathbf{n}_{\eta}$ denote the normal vectors on $\mathcal{O}$ and
$\mathcal{O}_{\eta}$, respectively. The function $\bm{\varphi}_{\eta}$ gives
the coordinate transform from $\Gamma\rightarrow\partial\mathcal{O}_{\eta}$.
The existence of a weak solution to (1.2)–(1.5) has been shown in [25] (see
also [26] for the case of a cylindrical shell model). It satisfies the energy
balance

$\displaystyle\begin{aligned}
\tfrac{1}{2}\int_{\mathcal{O}_{\eta}}|\mathbf{u}(t)|^{2}\,\mathrm{d}\mathbf{x}&+\int_{0}^{t}\int_{\mathcal{O}_{\eta}}|\nabla_{\mathbf{x}}\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta(t)|^{2}\,\mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_{\Gamma}|\Delta_{\mathbf{y}}\eta(t)|^{2}\,\mathrm{d}\mathbf{y}\\\
&\leq\tfrac{1}{2}\int_{\mathcal{O}_{\eta(0)}}|\mathbf{u}(0)|^{2}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta(0)|^{2}\,\mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_{\Gamma}|\Delta_{\mathbf{y}}\eta(0)|^{2}\,\mathrm{d}\mathbf{y}\end{aligned}$
(1.6)

for a.a. $t\in I$, from which one can easily deduce the function spaces in
which the weak solution lives. The easier case of an elastic plate (where the
reference geometry is flat) has been studied before in [16]. The main
advancement in [25] is a new compactness method which eventually allows to
establish compactness of the velocity field. On account of the deformed space-
time cylinder on which the problem is posed it is impossible to apply the
standard Aubin-Lions compactness lemma in order to pass to the limit in the
convective term of approximate solutions. Interestingly, this issue is
ultimately linked to the divergence-free constraint (1.2). Without it, the
compactness can simply be localised thus completely removing the difficulties
posed by the moving boundary, see [5] where the compressible Navier–Stokes
equations are studied. In [27] (where even the fully nonlinear Koiter model is
considered) the compactness argument from [25] has been replaced by an
abstract compactness criterion which is more in the spirit of the classical
Aubin-Lions result and thus allows for wider applications. Let us finally
remark that all the results just mentioned hold under the assumption that
there is no self-intersection of the structure (which can always be avoided if
$\|\eta\|_{L^{\infty}_{\mathbf{y}}}$ is not too large).

###  1.2. Stochastic models

It was recently suggested in [23] to consider a stochastic perturbation in
(1.4) to account for random effects in real-life problems and uncertainty in
the data. A first step towards a well-posedness theory for such stochastic
fluid-structure interaction models is done in [24], where the 2D time-
dependent Stokes equations are linearly coupled to a structure described by a
stochastic 1D wave equation. Although this is only a simplified model (and the
boundary is not moving in time) the analysis is already quite advanced. As
already indicated above, the geometry breaks down if $\eta$ causes a self-
intersection of the domain. In the simplified case, where the reference domain
is a box and the deformation only occurs in the vertical direction, this
happens exactly when the value of $-\eta$ coincides with the height of the
box. If $\eta$ has a Gaussian distribution as in [24], this can always happen
(though maybe only with a low probability) no matter how short the time
horizon is. This issue may be circumvented by studying the local-in-time well-
posedness of the problem which is done in [31]. The authors of [31] study the
interaction of an elastic plate (the reference geometry is flat) with the 2D
Navier–Stokes equation. The existence time is a random variable about which
the only available information is $\mathbb{P}$-a.s. positivity.

In this paper we aim for the natural next step by considering the full model
(1.2)–(1.5) globally in time, where (1.4) is subject to some Gaussian
noise.111One can very well add a suitable stochastic term in (1.3). This is
covered by our analysis as long as it is energy conservative. We take a
different perspective to [23, 24, 31] and do not consider stochasticity
entering as an external force but as an intrinsic property of the system. Thus
we consider transport noise in the shell equation (see (1.9) below). It has
the very appealing feature of being energy conservative. If the initial data
is deterministic (or simply bounded in probability) we have a pathwise control
over the energy and the restrictions on the time interval are the same as in
previous deterministic papers such as [16, 25, 27]. Transport noise has a
clear physical meaning in fluid mechanical transport processes, see [7, 20,
21] as well as [8, 13]. Also, it has been observed that transport noise has
regularising effects on certain ill-posed PDEs, see [11, 12]. Nevertheless,
the role of transport noise for elastic materials must be further explored.

Our goal is to construct on random space-time cylinders $\Omega\times
I\times\mathcal{O}_{\eta}$ and $\Omega\times I\times\Gamma$, a global weak
solution triple (note that the pressure does not enter the weak formulation)

$\displaystyle\mathbf{u}:(\omega,t,\mathbf{x}):\Omega\times
I\times\mathcal{O}_{\eta}\mapsto\mathbf{u}(\omega,t,\mathbf{x})\in\mathbb{R}^{3},$
$\displaystyle\pi:(\omega,t,\mathbf{x}):\Omega\times
I\times\mathcal{O}_{\eta}\mapsto\pi(\omega,t,\mathbf{x})\in\mathbb{R},$
$\displaystyle\eta:(\omega,t,\mathbf{y}):\Omega\times
I\times\Gamma\mapsto\eta(\omega,t,\mathbf{y})\in\mathbb{R},$

representing the fluid’s velocity, the fluid’s pressure and the structure
displacement of the coupled fluid-structure system given by

$\displaystyle\mathrm{div}_{\mathbf{x}}\mathbf{u}=0,$ (1.7)
$\displaystyle\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\nabla_{\mathbf{x}})\mathbf{u}=\Delta_{\mathbf{x}}\mathbf{u}-\nabla_{\mathbf{x}}\pi,$
(1.8)
$\displaystyle\,\mathrm{d}\partial_{t}\eta+(\Delta_{\mathbf{y}}^{2}\eta+g_{\eta})\,\mathrm{d}t+((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\circ\,\mathrm{d}B_{t}=0,$
(1.9)

with
$g_{\eta}=-\mathbf{n}^{\top}(\mathbb{T}(\mathbf{u},\pi)\mathbf{n}_{\eta})\circ\bm{\varphi}_{\eta}|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\eta})|$.
Here, $\Omega$ is a sample space of a filtered probability space
$(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P})$ with
associated expectation $\mathbb{E}(\cdot)$. Equation (1.9) contains a
Stratonovich differential of a real-valued Brownian motion $(B_{t})$ and
$\bm{\varkappa}$ is a given solenoidal (incompressible) vector field (i.e.
$\mathrm{div}_{\mathbf{y}}\bm{\varkappa}=0$)222One can allow a (possibly
infinite) sum of stochastic transport terms in (1.9) without affecting the
analysis. in $\mathbb{R}^{2}$. The initial conditions for (1.7)–(1.9) are

$\displaystyle\eta(\cdot,0,\cdot)=\eta_{0}(\cdot,\cdot),\quad\partial_{t}\eta(\cdot,0,\cdot)=\eta_{1}(\cdot,\cdot)$
$\displaystyle\quad\text{in }\Omega\times\Gamma,$ (1.10)
$\displaystyle\mathbf{u}(\cdot,0,\cdot)=\mathbf{u}_{0}(\cdot,\cdot)$
$\displaystyle\quad\text{in }\Omega\times\mathcal{O}_{\eta_{0}}.$ (1.11)

With regards to boundary conditions, we supplement the shell equation (1.9)
with periodic boundary conditions and impose

$\displaystyle\mathbf{u}\circ\bm{\varphi}_{\eta}=\mathbf{n}\partial_{t}\eta$
$\displaystyle\quad\text{on }\Omega\times I\times\Gamma$ (1.12)

at the fluid-structure interface. Note that (1.7)–(1.12) is a free-boundary
problem where the boundary of the fluid domain is moving randomly in time.

In the 3D case regularity and uniqueness of solutions to (1.7)–(1.12) is
certainly out of reach (at least globally in time) so that one can only hope
for the existence of weak martingale solutions. Here weak refers to the
analytical concept of distributional derivatives, whereas martingale solutions
refers to solutions which are weak in the probabilistic sense (they do not
exist on a given stochastic basis; the latter becomes an integral part of the
solution). Such a concept is very common in stochastic evolutionary problems
(even on the level of ordinary stochastic differential equations), whenever
uniqueness of the underlying system is unavailable.

###  1.3. The weak formulation

A first rather philosophical question is to come up with an analytically weak
formulation for the problem. In fluid-structure interaction problems the space
of test-functions typically depends on the structure displacement $\eta$ (the
test-function for the fluid sub-problem and the structure sub-problem must
match at the interface as in (1.5) and (1.12)). On the other hand, in
stochastic PDEs it is common to work with spatial test-functions. This is also
our preferred point of view as an $\eta$-dependence of the test-functions in
our case means that they depend on time and are also random. The idea now is
to start with a pair of test-functions $(\phi,\bm{\phi})$ on the reference
domain (that is $\phi:\Gamma\rightarrow\mathbb{R}$ and
$\bm{\phi}:\mathcal{O}\rightarrow\mathbb{R}^{3}$) with the correct boundary
condition and transform $\bm{\phi}$ to the moving domain. An obvious choice,
therefore, is the Hanzawa transform
$\bm{\Psi}_{\eta}:\mathcal{O}\rightarrow\mathcal{O}_{\eta}$ which we formally
introduce in Section 2.2. Unfortunately, it has the disadvantage of destroying
the divergence-free constraint on the test-functions. At the level of weak
solutions this cannot be remedied through the recovery of the pressure
function as the latter only exists as a distribution on the solenoidal test-
functions. Thus, we use instead the Piola transform

$\displaystyle\mathcal{J}_{\eta}\mathbf{v}=\big{(}\nabla_{\mathbf{x}}\bm{\Psi}_{\eta}(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\eta})^{-1}\mathbf{v}\big{)}\circ\bm{\Psi}_{\eta}^{-1},$
(1.13)

which preserves the solenoidability of a functions. Using now
$(\iota_{\eta}\phi,\mathcal{J}_{\eta(t)}\bm{\phi})$ with
$\iota_{\eta}=(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\eta})^{-1}$ we
obtain the following weak formulation

$\displaystyle\,\mathrm{d}\bigg{(}\int_{\mathcal{O}_{\eta}}$
$\displaystyle\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\partial_{t}\eta\,\iota_{\eta}\phi\,\mathrm{d}\mathbf{y}\bigg{)}=\int_{\Gamma}\big{(}\partial_{t}\eta\,\partial_{t}(\iota_{\eta}\phi)-\Delta_{\mathbf{y}}(\iota_{\eta}\phi)\,\Delta_{\mathbf{y}}\eta\big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\eta}}\bigg{(}\mathbf{u}\cdot\partial_{t}(\mathcal{J}_{\eta(t)}\bm{\phi})+((\mathbf{u}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\eta(t)}\bm{\phi})\cdot\mathbf{u}-\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}\mathcal{J}_{\eta(t)}\bm{\phi}\bigg{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle-\int_{\Gamma}\partial_{t}\eta((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\eta}\phi)\,\mathrm{d}\mathbf{y}\circ\,\mathrm{d}B_{t}$
(1.14)

for all test-functions $(\phi,\bm{\phi})$ (which clearly depend only on
space), see Section 2.4 for the derivation. One easily shows that a dense set
of pairs of test-functions $(\phi,\bm{\phi})$ with the correct boundary
condition leads to a dense set of pairs of test-functions
$(\iota_{\eta}\phi,\mathcal{J}_{\eta(t)}\bm{\phi})$ on the moving domain with
the right boundary condition, see [25, page 237]. Thus our weak formulation is
consistent with the strong formulation. However, the Piola transform behaves
like $\nabla_{\mathbf{x}}\bm{\Psi}_{\eta}$ and inherits the regularity of
$\nabla_{\mathbf{y}}\eta$ so that we require more regularity on $\eta$. Since
the embedding $W^{2,2}(\Gamma)\hookrightarrow W^{1,\infty}(\Gamma)$ fails in
two dimensions, $\nabla_{\mathbf{x}}\bm{\Psi}_{\eta}$ is not bounded. Thus the
information from (1.6) is not sufficient to give a meaning to all the terms in
(1.14). Hence we need additional regularity. A crucial point in our approach
is, therefore, to establish an estimate for the
$L^{2}(I;W^{2+s,2}(\Gamma))$-norm of $\eta$ for all $s\in(0,1/2)$. One can
easily check by using Hölder’s inequality and Sobolev’s embedding that this
information, together with (1.6), is sufficient to define all integrals
appearing in (1.14). Different to (1.6) this estimate is not independent of
the transport noise and hence only holds in expectation. As it turns out, the
regularity of the terms arising from the transport noise have just enough
regularity to close the estimate. Details can be found in Section 5.1.
Concluding this discussion, the additional fractional differentiability of the
shell displacement must be included in the definition of a solution, see
Definition 2.6.

###  1.4. Plan

This work straddles different fields of mathematics including fluid mechanics,
partial differential equations, differential geometry and stochastic analysis.
In order to make this work as self-contained as possible, we collect in
Section 2 useful results in the different fields of mathematics that are
essential in establishing our result. We begin by giving a rigorous
interpretation of the Stratonovich integral in (1.9) after which we introduce
the geometric setup for the fluid-structure system (1.7)–(1.9). We also
present the functional analytic framework (function spaces on moving domains)
and present some key results necessary for our analysis (extension operators).
Finally, we make precise, the notion of a solution that we are interested in
(Definition 2.6) and state our main result (Theorem 2.7).

The proof of our main result can be summarized into three main steps. In
Section 3, we consider an extension of the fluid-structure system that
incorporate ‘artificial’ regularising terms in the shell equation (1.9). We
then construct a solution to the linearised version of this extended system
using a Galerkin approximation and stochastic compactness tools. We then move
to Section 4 where we use a fixed-point argument to remove the linearisation
performed in the previous section and obtain a solution to the fully nonlinear
system (with the extra regularising terms in the shell equation). To complete
the proof of the main result, we pass to the limit in the regularisation
parameter in Section 5 to finally obtain a solution for (1.7)–(1.12). In each
stage we apply a refined stochastic compactness method which is based on
Jakubowski’s extension of the Skorokhod representation theorem [22]. In our
case it is crucial to re-interpret the compactness lemma from [27] in the
context of tightness of probability measures, see Sections 3.2 and 4.1.

######  Remark 1.1 (The 2D case).

One might wonder whether it is possible to show the global-in-time existence
of strong pathwise solutions to (1.7)–(1.12) (solutions existing on a given
stochastic basis). In the 2D case the existence of global-in-time strong
solutions to (1.2)–(1.5) has been shown in [2] with additional dissipation
(see also for [17] for previous results for elastic plates) and in [30] for
the case of plates (but without dissipation). In all cases the approach is
heavily based on taking temporal derivatives of $\mathbf{u}$ and
$\partial_{t}\eta$, which is not possible for (1.7)–(1.12). Hence its is
unclear if one should even expect such a result here.

##  2\. Mathematical framework and the main result

###  2.1. Stratonovich integrals

Let $(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P})$ be a
stochastic basis with a complete, right-continuous filtration and let
$(B_{t})$ represent a real-valued Brownian motion relative to
$(\mathfrak{F}_{t})$. We consider a smooth solenoidal vector field
$\bm{\varkappa}:\Gamma\rightarrow\mathbb{R}^{2}$. If $\xi\in
L^{2}(\Omega;C(\overline{I};W^{1,2}(\Gamma)))$ is
$(\mathfrak{F}_{t})$-adapted, the stochastic integral

$\displaystyle\int_{0}^{t}\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\xi\,\,\mathrm{d}B_{s}$

is well-defined in the sense of Itô with values in $L^{2}(\Gamma)$. If we only
have $\xi\in L^{2}(\Omega;C(\overline{I};L^{2}(\Gamma)))$ one can use the
identity
$(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\xi=\mathrm{div}_{\mathbf{y}}(\bm{\varkappa}\xi)$
and define the stochastic integral

$\displaystyle\int_{0}^{t}\mathrm{div}_{\mathbf{y}}(\xi\bm{\varkappa})\,\,\mathrm{d}B_{s}$

with values in $W^{-1,2}(\Gamma)$. We define the Stratonovich integrals in
(1.9) by means of the Itô-Stratonovich correction, that is

$\displaystyle\int_{0}^{t}\int_{\Gamma}\xi\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\phi\,\mathrm{d}\mathbf{y}\,\circ\,\mathrm{d}B_{s}$
$\displaystyle=\int_{0}^{t}\int_{\Gamma}\xi\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\phi\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}B_{s}$
(2.1)
$\displaystyle+\frac{1}{2}\Big{\langle}\Big{\langle}\int_{\Gamma}\xi\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\phi\,\mathrm{d}\mathbf{y},B_{t}\Big{\rangle}\Big{\rangle}_{t}$

for $\phi\in W^{1,2}(\Gamma)$. Here
$\langle\langle\cdot,\cdot\rangle\rangle_{t}$ denotes the cross variation. We
compute now the cross variations by means of (1.9). If $\xi=\partial_{t}\eta$,
where $\eta$ solves (1.9), we have for all $\phi\in W^{2,2}(\Gamma)$ and $t\in
I$,

$\displaystyle\int_{\Gamma}\partial_{t}\eta\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\phi\,\mathrm{d}\mathbf{y}=$
$\displaystyle\int_{\Gamma}\partial_{t}\eta(0)\bm{\varkappa}\cdot\nabla_{\mathbf{y}}\phi\,\mathrm{d}\mathbf{y}-\int_{0}^{t}\int_{\Gamma}\Delta_{\mathbf{y}}\eta\Delta_{\mathbf{y}}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\phi)\,\mathrm{d}\mathbf{y}\,\mathrm{d}s$
$\displaystyle-\int_{0}^{t}\int_{\Gamma}g_{\eta}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\phi)\,\mathrm{d}\mathbf{y}\,\mathrm{d}s+\frac{1}{2}\Big{\langle}\Big{\langle}\int_{0}^{t}\int_{\Gamma}\partial_{t}\eta\,((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\phi)\,\mathrm{d}\mathbf{y},B_{t}\Big{\rangle}\Big{\rangle}_{t}$
$\displaystyle+\int_{0}^{t}\int_{\Gamma}\partial_{t}\eta\,((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\phi)\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{s}.$

Only the last term on the right contributes to the cross variation when
inserted in (2.1). Plugging the previous considerations together we obtain

$\displaystyle\int_{0}^{t}(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta\,\circ\,\mathrm{d}B_{s}=\int_{0}^{t}(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta\,\,\mathrm{d}B_{s}-\frac{1}{2}\int_{0}^{t}(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta\,\mathrm{d}s.$

to be understood in $W^{-1,2}(\Gamma)$ or $W^{-2,2}(\Gamma)$, respectively,
depending on the regularity of $\partial_{t}\eta$.

###  2.2. Geometric setup

The spatial domain $\mathcal{O}$ is assumed to be an open bounded subset of
$\mathbb{R}^{3}$ with smooth boundary $\partial\mathcal{O}$ and an outer unit
normal ${\mathbf{n}}$. We assume that $\partial\mathcal{O}$ can be
parametrised by an injective mapping ${\bm{\varphi}}\in
C^{k}(\Gamma;\mathbb{R}^{3})$ for some sufficiently large $k\in\mathbb{N}$,
where $\Gamma$ is the two-dimensional torus. We suppose for all points
$\mathbf{y}=(y_{1},y_{2})\in\Gamma$ that the pair of vectors
$\partial_{i}{\bm{\varphi}}(\mathbf{y})$, $i=1,2,$ is linearly independent.
For a point $\mathbf{x}$ in the neighbourhood of $\partial\mathcal{O}$, we
define the functions $\mathbf{y}$ and $s$ by

$\displaystyle\mathbf{y}(\mathbf{x})=\arg\min_{\mathbf{y}\in\Gamma}|\mathbf{x}-\bm{\varphi}(\mathbf{y})|,\quad
s(\mathbf{x})=(\mathbf{x}-\mathbf{y}(\mathbf{x}))\cdot\mathbf{n}(\mathbf{y}(\mathbf{x})).$

Moreover, we define the projection
$\mathbf{p}(\mathbf{x})=\bm{\varphi}(\mathbf{y}(\mathbf{x}))$. We define $L>0$
to be the largest number such that $s,\mathbf{y}$ and $\mathbf{p}$ are well-
defined on $S_{L}$, where

$\displaystyle
S_{L}=\\{\mathbf{x}\in\mathbb{R}^{n}:\,\mathrm{dist}(\mathbf{x},\partial\mathcal{O})<L\\}.$
(2.2)

Due to the smoothness of $\partial\mathcal{O}$ for $L$ small enough we have
$|s(\mathbf{x})|=\min_{\mathbf{y}\in\Gamma}|\mathbf{x}-\bm{\varphi}(\mathbf{y})|$
for all $\mathbf{x}\in S_{L}$. This implies that
$S_{L}=\\{s\mathbf{n}(\mathbf{y})+\mathbf{y}:(s,\mathbf{y})\in(-L,L)\times\omega\\}$.
For a given function $\eta:I\times\Gamma\rightarrow\mathbb{R}$ we parametrise
the deformed boundary by

$\displaystyle{\bm{\varphi}}_{\eta}(t,\mathbf{y})={\bm{\varphi}}(\mathbf{y})+\eta(t,\mathbf{y}){\mathbf{n}}(\mathbf{y}),\quad\,\mathbf{y}\in\Gamma,\,t\in
I.$ (2.3)

With an abuse of notation we define the deformed space-time cylinder as
$I\times\mathcal{O}_{\eta}=\bigcup_{t\in
I}\\{t\\}\times\mathcal{O}_{\eta(t)}\subset\mathbb{R}^{4}$. The corresponding
function spaces for variable domains are defined as follows.

######  Definition 2.1.

(Function spaces) For $I=(0,T)$, $T>0$, and $\eta\in
C(\overline{I}\times\omega)$ with $\|\eta\|_{L^{\infty}(I\times\Gamma)}<L$ we
define for $1\leq p,r\leq\infty$

$\displaystyle L^{p}(I;L^{r}(\mathcal{O}_{\eta}))$
$\displaystyle:=\big{\\{}v\in
L^{1}(I\times\mathcal{O}_{\eta}):\,\,v(t,\cdot)\in
L^{r}(\mathcal{O}_{\eta(t)})\,\,\text{for a.e.
}t,\,\,\|v(t,\cdot)\|_{L^{r}(\mathcal{O}_{\eta(t)})}\in L^{p}(I)\big{\\}},$
$\displaystyle L^{p}(I;W^{1,r}(\mathcal{O}_{\eta}))$
$\displaystyle:=\big{\\{}v\in
L^{p}(I;L^{r}(\mathcal{O}_{\eta})):\,\,\nabla_{\mathbf{x}}v\in
L^{p}(I;L^{r}(\mathcal{O}_{\eta}))\big{\\}}.$

To establish a relationship between the fixed domain and the time-dependent
domain, we introduce the Hanzawa transform
$\bm{\Psi}_{\eta}:\mathcal{O}\rightarrow\mathcal{O}_{\eta}$ defined by

$\bm{\Psi}_{\eta}(\mathbf{x})=\left\\{\begin{array}[]{lr}\mathbf{p}(\mathbf{x})+\big{(}s(\mathbf{x})+\eta(\mathbf{y}(\mathbf{x}))\phi(s(\mathbf{x}))\big{)}\mathbf{n}(\mathbf{y}(\mathbf{x}))&\text{if
dist}(\mathbf{x},\partial\mathcal{O})<L,\\\
\mathbf{x}&\text{elsewhere}.\end{array}\right.$ (2.4)

for any $\eta:\omega\rightarrow(-L,L)$. Here $\phi\in C^{\infty}(\mathbb{R})$
is such that $\phi\equiv 0$ in a neighborhood of $-L$ and $\phi\equiv 1$ in a
neighborhood of $0$. The other variables $\mathbf{p}$, $s$ and $\mathbf{n}$
are as defined earlier in this section. A straightforward verification shows
that the inverse of $\bm{\Psi}_{\eta(t)}$ is $\bm{\Psi}_{-\eta(t)}$.

In order to obtain a weak formulation for the fluid-structure system, we also
introduce the Piola transform

$\displaystyle\mathcal{J}_{\zeta}\mathbf{v}=\big{(}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta}(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta})^{-1}\mathbf{v}\big{)}\circ\bm{\Psi}_{\zeta}^{-1}$
(2.5)

of a vector field $\mathbf{v}:\mathcal{O}\rightarrow\mathbb{R}^{3}$ with
respect to a mapping $\zeta:\Gamma\rightarrow\mathbb{R}$. The Piola transform
is invertible with inverse

$\displaystyle\mathcal{J}_{\zeta}^{-1}\mathbf{v}=\big{(}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta})^{-1}(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta})\mathbf{v}\big{)}\circ\bm{\Psi}_{\zeta}.$
(2.6)

It preserves vanishing boundary values as well as the divergence-free property
of a function. In order to compensate for the additional factor
$(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta})^{-1}$ in the trace of
$\mathcal{J}_{\zeta}\mathbf{v}$ we define the mapping

$\displaystyle\iota_{\zeta}\phi:=(\mathrm{det}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta})^{-1}\phi$

for a function $\phi:\Gamma\rightarrow\mathbb{R}$. If
$\bm{\phi}\circ\bm{\varphi}=\phi$ on $\Gamma$ it follows that
$(\mathcal{J}_{\eta}\bm{\phi})\circ\bm{\varphi}_{\eta}=\iota_{\eta}\phi$ on
$\Gamma$. Thus a pair of test-functions $(\phi,\bm{\phi})$ with the correct
boundary condition leads to a pair of test-functions
$(\iota_{\eta}\phi,\mathcal{J}_{\eta(t)}\bm{\phi})$ on the moving domain with
the right boundary condition. Also, a dense set of test-functions on the
reference domain leads to a dense set of test-functions on the moving domain,
see [25, page 237].

We finish this section by recalling the following Aubin-Lions type lemma which
is shown in [27, Theorem 5.1. & Remark 5.2.] and slightly reformulated for our
purposes.

######  Theorem 2.2.

Let $X,Z$ be two Banach spaces, such that $X^{\prime}\subset Z^{\prime}$.
Assume that $f_{n}:I\to X$ and $g_{n}:I\to X^{\prime}$, such that $g_{n}\in
L^{\infty}(I;Z^{\prime})$ uniformly. Moreover assume the following:

  1. (a)

The boundedness: for some $s\in[1,\infty]$ we have that $(f_{n})$ is bounded
in $L^{s}(X)$ and $(g_{n})$ in $L^{s^{\prime}}(X^{\prime})$.

  2. (b)

The approximability-condition is satisfied: For every $\kappa\in(0,1]$ there
exists a $f_{n,\kappa}\in L^{s}(I;X)\cap L^{1}(I;Z)$, such that for every
$\epsilon\in(0,1)$ there exists some $\kappa_{\epsilon}\in(0,1)$ (depending
only on $\epsilon$) such that

$\|f_{n}-f_{n,\kappa}\|_{L^{s}(I;X)}\leq\epsilon\text{ for all
}\kappa\in(0,\kappa_{\epsilon}]$

and for every $\kappa\in(0,1]$ there is some $C(\kappa)$ such that

$\|f_{n,\kappa}\|_{L^{1}(I;Z)}\,\mathrm{d}t\leq C(\kappa).$

  3. (c)

The equi-continuity of $g_{n}$: We require that there exists some
$\alpha\in(0,1]$, functions $A_{n}$ with $A_{n}\in L^{1}(I)$ uniformly, such
that for every $\kappa>0$ that there exist some $C(\kappa)>0$ and some
$n_{\kappa}\in\mathbb{N}$ such that for $\tau>0$, $n\geq n_{\kappa}$ and a.e.
$t\in[0,T-\tau]$

$\Big{|}\tau^{-1}\int_{0}^{\tau}\langle
g_{n}(t)-g_{n}(t+s),f_{n,\kappa}(t)\rangle_{X^{\prime},X}\,\,\mathrm{d}s\Big{|}\leq
C(\kappa)\tau^{\alpha}(A_{n}(t)+1).$

  4. (d)

The compactness assumption is satisfied:
$X^{\prime}\hookrightarrow\hookrightarrow Z^{\prime}$. More precisely, every
uniformly bounded sequence in $X^{\prime}$ has a strongly converging
subsequence in $Z^{\prime}$.

Then there is a subsequence, such that

$\int_{0}^{T}\langle
f_{n},g_{n}\rangle_{X,X^{\prime}}\,\mathrm{d}t\rightarrow\int_{0}^{T}\langle
f,g\rangle_{X,X^{\prime}}\,\mathrm{d}t.$

###  2.3. Solenoidal extension

In this section, we present a linear solenoidal extension operator that maps
boundary elements of a spatial domain into the interior. For this end, we
first consider the corrector map

$\displaystyle\mathscr{K}_{\eta}:L^{1}(\Gamma)\rightarrow\mathbb{R},\qquad\mathscr{K}_{\eta}(\xi)=\frac{\int_{\mathcal{A}_{\kappa}}{\xi}(\mathbf{y}(\mathbf{x}))\lambda_{\eta}(t,\mathbf{x})\,\mathrm{d}\mathbf{x}}{\int_{\mathcal{A}_{\kappa}}\lambda_{\eta}(t,\mathbf{x})\,\mathrm{d}\mathbf{x}},$

where $\lambda_{\eta}\geq 0$ for $(t,\mathbf{x})\in
I\times\mathcal{A}_{\kappa}$ is an appropriately chosen weight function, cf.
[27, equ. (3.3)], and $\mathcal{A}_{\kappa}:=S_{\kappa/2}\setminus
S_{\kappa}$. It satisfies

$\displaystyle\|\mathscr{K}_{\eta}(\xi)\|_{L^{q}(I)}\lesssim\|\xi\|_{L^{q}(I;L^{1}(\Gamma))}$
$\displaystyle\|\partial_{t}\mathscr{K}_{\eta}(\xi)\|_{L^{q}(I)}\lesssim\|\partial_{t}\xi\|_{L^{q}(I;L^{1}(\Gamma))}+\|\xi\partial_{t}\eta\|_{L^{q}(I;L^{1}(\Gamma))}$

for all $q\in[1,\infty]$. The corrector $\mathscr{K}_{\eta}$ above
preconditions the boundary data to be compatible with the interior
solenoidality. The following is proved in [27, Prop. 3.3] and it provides a
solenoidal extension. For that we introduce the solenoidal space
$W^{1,1}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O}\cup
S_{\alpha}):=\\{\mathbf{w}\in W^{1,1}(\mathcal{O}\cup
S_{\alpha})\,:\,\mathrm{div}_{\mathbf{x}}\mathbf{w}=0\\}$.

######  Proposition 2.3.

For a given $\eta\in L^{\infty}(I;W^{1,2}(\Gamma))$ with
$\|\eta\|_{L^{\infty}(I\times\Gamma)}<\alpha<L$, there is a linear operator

$\displaystyle{\mathscr{F}^{\eta}}:\\{\xi\in
L^{1}(I;W^{1,1}(\Gamma)):\,\mathscr{K}_{\eta}(\xi)=0\\}\rightarrow
L^{1}(I;W^{1,1}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O}\cup S_{\alpha})),$

such that the tuple
$({\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}(\xi)),\xi-\mathscr{K}_{\eta}(\xi))$
satisfies

$\displaystyle{\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}(\xi))$
$\displaystyle\in L^{\infty}(I;L^{2}({\mathcal{O}}_{\eta}))\cap
L^{2}(I;W^{1,2}_{\mathrm{div}_{\mathbf{x}}}({\mathcal{O}}_{\eta})),$
$\displaystyle\xi-\mathscr{K}_{\eta}(\xi)$ $\displaystyle\in
L^{\infty}(I;W^{2,2}(\Gamma))\cap W^{1,\infty}(I;L^{2}(\Gamma)),$
$\displaystyle({\mathscr{F}^{\eta}}$
$\displaystyle(\xi-\mathscr{K}_{\eta}(\xi))\circ\bm{\varphi}_{\eta}=\mathbf{n}(\xi-\mathscr{K}_{\eta}(\xi)),$
$\displaystyle\partial_{t}({\mathscr{F}^{\eta}}$
$\displaystyle(\xi-\mathscr{K}_{\eta}(\xi))\in
L^{2}(I;L^{2}({\mathcal{O}}_{\eta})),$
$\displaystyle{\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}$
$\displaystyle(\xi))(t,x)=0\text{ for }(t,x)\in I\times({\mathcal{O}}\setminus
S_{\alpha})$

provided we have $\xi\in L^{\infty}(I;W^{2,2}(\Gamma))\cap
W^{1,\infty}(I;L^{2}(\Gamma))$. In particular, we have the estimates

$\displaystyle\|{\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}(\xi))\|_{L^{q}(I;W^{1,p}({\mathcal{O}}\cup
S_{\alpha}))}\lesssim\|\xi\|_{L^{q}(I;W^{1,p}(\Gamma))}+\|\xi\nabla_{\mathbf{y}}\eta\|_{L^{q}(I;L^{p}(\Gamma))},$
(2.7)
$\displaystyle\|\partial_{t}{\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}(\xi))\|_{L^{q}(I;L^{p}({\mathcal{O}}\cup
S_{\alpha}))}\lesssim\|\partial_{t}\xi\|_{L^{q}(I;L^{p}(\Gamma))}+\|\xi\partial_{t}\eta\|_{L^{q}(I;L^{p}(\Gamma))},$
(2.8)

for any $p\in(1,\infty),q\in[1,\infty]$.

The following result is a consequence of Proposition 2.3.

######  Corollary 2.4.

Let the assumptions of Proposition 2.3 be satisfied and in addition, let
$a,r\in[2,\infty]$, $p,q\in(1,\infty)$ and $s\in[0,1]$, and assume that
$\eta\in L^{r}(I;W^{2,a}(\Gamma))\cap W^{1,r}(I;L^{a}(\Gamma))$. Let $\xi\in
W^{s,p}(\Gamma)$ and let $\xi_{\delta}$ be a smooth approximation of $\xi$ in
$\Gamma$. Then
$\mathcal{E}^{\eta}_{\delta}(\xi):={\mathscr{F}^{\eta}}(\xi_{\delta}-\mathscr{K}_{\eta}(\xi_{\delta}))$
satisfies all the conclusions in Proposition 2.3. In particular,

$\displaystyle\|\partial_{t}\mathcal{E}^{\eta}_{\delta}(\xi)\|_{L^{r}(I;L^{a}(\mathcal{O}\cup
S_{\alpha}))}\lesssim\|(\xi_{\delta})\partial_{t}\eta\|_{L^{r}(I;L^{a}(\Gamma))}$

and

$\displaystyle\|\mathcal{E}^{\eta}_{\delta}(\xi)-{\mathscr{F}^{\eta}}(\xi-\mathscr{K}_{\eta}(\xi))\|_{L^{p}(\mathcal{O}\cup
S_{\alpha})}\lesssim\|\xi_{\delta}-\xi\|_{L^{p}(\Gamma)}$

holds uniformly in $t\in I$.

For the final statement of this subsection, borrowed from [27, Lemma 3.5], we
first introduce the following fractional difference quotient in space in the
direction $\mathbf{e}_{i}$ given by
$\Delta_{h}^{s}f(\mathbf{y})=h^{-s}(f(\mathbf{y}+\mathbf{e}_{i}h)-f(\mathbf{y}))$
for some $h>0$. Now, we define

$\displaystyle
D^{s,\mathscr{K}}_{-h,h}\eta:=\Delta_{-h}^{s}\Delta_{h}^{s}\eta-\mathscr{K}_{\eta}(\Delta_{-h}^{s}\Delta_{h}^{s}\eta),$

where $s\in(0,\frac{1}{2})$ and the result is as follows:

######  Lemma 2.5.

Let the assumptions of Proposition 2.3 be satisfied and in addition, let
$p,\tilde{a}\in(1,\infty)$ be such that
$p^{\prime}<\tilde{a}\leq\frac{3p^{\prime}}{3-p^{\prime}}$ if $p^{\prime}<3$,
and $p^{\prime}<\tilde{a}<\infty$ otherwise. Furthermore, assume that $\eta\in
C^{0,\theta}(\Gamma)\cap
W^{1,\frac{\tilde{a}p}{\tilde{a}p-\tilde{a}-p}}(\Gamma)$ and $\mathbf{u}\in
W^{1,p^{\prime}}(\mathcal{O}_{\eta})$. Then

$\displaystyle\bigg{|}\int_{\mathcal{O}_{\eta}}\mathbf{u}\cdot{\mathscr{F}^{\eta}}(D^{s,\mathscr{K}}_{-h,h}\xi)\,\mathrm{d}\mathbf{x}\bigg{|}\leq\Big{(}h^{\theta-s}+\|\Delta_{h}^{s}\eta\|_{W^{1,\frac{\tilde{a}p}{\tilde{a}p-\tilde{a}-p}}(\Gamma)}\Big{)}\|\mathbf{u}\|_{W^{1,p^{\prime}}(\mathcal{O}_{\eta})}\|\xi\|_{L^{p}(\Gamma)}$

and when $\partial_{t}\xi\in L^{p}(\Gamma)$,

$\displaystyle\bigg{|}\int_{\mathcal{O}_{\eta}}\mathbf{u}\cdot\partial_{t}{\mathscr{F}^{\eta}}(D^{s,\mathscr{K}}_{-h,h}\xi)\,\mathrm{d}\mathbf{x}\bigg{|}\lesssim$
$\displaystyle\Big{(}h^{\theta-s}+\|\Delta_{h}^{s}\eta\|_{W^{1,\frac{\tilde{a}p}{\tilde{a}p-\tilde{a}-p}}(\Gamma)}\Big{)}\|\mathbf{u}\|_{W^{1,p^{\prime}}(\mathcal{O}_{\eta})}\|\partial_{t}\xi\|_{L^{p}(\Gamma)}$
$\displaystyle+\Big{(}\big{\|}\,|\Delta_{h}^{s}\xi(t)|\,|\partial_{t}\eta|\,\big{\|}_{L^{\tilde{a}}(\Gamma)}+\big{\|}\Delta_{h}^{s}\xi(t)\big{\|}_{L^{1}(\Gamma)}\big{\|}\partial_{t}\eta|\big{\|}_{L^{1}(\Gamma)}\Big{)}\|\mathbf{u}\|_{W^{1,p^{\prime}}(\mathcal{O}_{\eta})}.$

Here, the constants only depends on $\alpha,L$ and
$\|\eta\|_{C^{0,\theta}(\Gamma)}$.

###  2.4. Weak martingale solutions

We are interested in a solution to (1.7)–(1.9) that is weak in the
probabilistic sense and also weak in the deterministic sense. From the
probabilistic point of view, this means that the stochastic basis is also an
unknown of the system and from the deterministic angle, we want a
distributional solution of the system integrated against a deterministic test
function pair $(\phi,\bm{\phi})\in W^{2,2}(\Gamma)\times
W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$ that satisfies
$\bm{\phi}\circ\bm{\varphi}=\phi\mathbf{n}$ at the fluid-structure interface
$\Gamma$.

We are now deriving the weak formulation of the coupled system assuming we
have a sufficiently regular solution at hand. Since the momentum equation
(1.8) is merely a random PDE rather than a SPDE, and advected by the large-
scale incompressible vector field, we can directly apply Reynolds transport
theorem [19] to obtain for $(\iota_{\eta}\phi,\mathcal{J}_{\eta(t)}\bm{\phi})$
(recalling the definitions from Section 2.2)

$\displaystyle\,\mathrm{d}\int_{\mathcal{O}_{\eta}}\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}$
$\displaystyle=\int_{\mathcal{O}_{\eta}}\partial_{t}\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{\mathcal{O}_{\eta}}\mathbf{u}\cdot\partial_{t}(\mathcal{J}_{\eta(t)}\bm{\phi})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\eta}}(\mathbf{u}\cdot\nabla_{\mathbf{x}})(\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t.$

We can now use the momentum equation (1.8) and the divergence-free condition
on $\bm{\phi}$ (which transfers to $\mathcal{J}_{\eta(t)}\bm{\phi}$) to obtain

$\displaystyle\int_{\mathcal{O}_{\eta}}\partial_{t}\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=-\int_{\mathcal{O}_{\eta}}((\mathbf{u}\cdot\nabla_{\mathbf{x}})\mathbf{u})\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t-\int_{\mathcal{O}_{\eta}}\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\eta}}\mathrm{div}_{\mathbf{x}}(\mathbb{T}(\mathbf{u},\pi)\mathcal{J}_{\eta(t)}\bm{\phi})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$

with the latter satisfying

$\displaystyle\int_{\mathcal{O}_{\eta}}\mathrm{div}_{\mathbf{x}}(\mathbb{T}(\mathbf{u},\pi)\mathcal{J}_{\eta(t)}\bm{\phi})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=\int_{\partial\mathcal{O}_{\eta}}\mathbf{n}_{\eta}\cdot(\mathbb{T}(\mathbf{u},\pi)\mathcal{J}_{\eta(t)}\bm{\phi})\,\mathrm{d}\mathcal{H}^{2}\,\mathrm{d}t$
$\displaystyle=\int_{\Gamma}g_{\eta}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}t.$

To obtain a distributional formulation for the shell equation (1.9), we first
transform it into the Itô equation

$\displaystyle\,\mathrm{d}\partial_{t}\eta+\big{[}\Delta_{\mathbf{y}}^{2}\eta+g_{\eta}-\tfrac{1}{2}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\big{]}\,\mathrm{d}t+((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\,\mathrm{d}B_{t}=0,$

cf. the discussion in Section 2.1. If we now use Itô’s formula, we obtain

$\displaystyle\,\mathrm{d}\int_{\Gamma}\partial_{t}\eta\,\iota_{\eta}\phi\,\mathrm{d}\mathbf{y}$
$\displaystyle=-\int_{\Gamma}\iota_{\eta}\phi\bigg{[}\Delta_{\mathbf{y}}^{2}\eta+g_{\eta}-\tfrac{1}{2}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\bigg{]}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle\quad+\int_{\Gamma}\partial_{t}\eta\,\partial_{t}(\iota_{\eta}\phi)\,\mathrm{d}\mathbf{y}+\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\eta}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t},$

where due to the periodicity of the boundary of $\Gamma$,

$\displaystyle\int_{\Gamma}\iota_{\eta}\phi\Delta_{\mathbf{y}}^{2}\eta\,\mathrm{d}\mathbf{y}\,\mathrm{d}t=\int_{\Gamma}\Delta_{\mathbf{y}}\iota_{\eta}\phi\Delta_{\mathbf{y}}\eta\,\mathrm{d}\mathbf{y}\,\mathrm{d}t.$

If we now use the identity
$(\mathbf{v}_{1}\cdot\nabla_{\mathbf{x}})(\mathbf{v}_{2}\cdot\mathbf{v}_{3})-((\mathbf{v}_{1}\cdot\nabla_{\mathbf{x}})\mathbf{v}_{2})\cdot\mathbf{v}_{3}=((\mathbf{v}_{1}\cdot\nabla_{\mathbf{x}})\mathbf{v}_{3})\cdot\mathbf{v}_{2}$,
it follows that

$\displaystyle\,\mathrm{d}\bigg{(}\int_{\mathcal{O}_{\eta}}$
$\displaystyle\mathbf{u}\cdot\mathcal{J}_{\eta(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\partial_{t}\eta\,\iota_{\eta}\phi\,\mathrm{d}\mathbf{y}\bigg{)}=\int_{\Gamma}\big{(}\partial_{t}\eta\,\partial_{t}(\iota_{\eta})\phi-\Delta_{\mathbf{y}}\iota_{\eta}\phi\Delta_{\mathbf{y}}\eta\big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
(2.9)
$\displaystyle+\int_{\mathcal{O}_{\eta}}\bigg{(}\mathbf{u}\cdot\partial_{t}(\mathcal{J}_{\eta(t)})\bm{\phi}+((\mathbf{u}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\eta(t)}\bm{\phi})\cdot\mathbf{u}-\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}\mathcal{J}_{\eta(t)}\bm{\phi}\bigg{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\frac{1}{2}\int_{\Gamma}\partial_{t}\eta\,((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\iota_{\eta}\phi))\,\mathrm{d}\mathbf{y}\,\mathrm{d}t-\int_{\Gamma}\partial_{t}\eta\,((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\eta}\phi)\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t}.$

Note that $\mathrm{div}_{\mathbf{x}}(\mathcal{J}_{\eta(t)}\bm{\phi})=0$ and
thus, no pressure term appears in the weak formulation. With this preparation,
we now give the precise notion of a solution.

######  Definition 2.6 (Weak martingale solution).

Let $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ be a dataset such that

$\displaystyle\eta_{0}\in W^{2,2}(\Gamma)\text{ with
}\|\eta_{0}\|_{L^{\infty}(\Gamma)}<L,\quad\eta_{1}\in L^{2}(\Gamma),$ (2.10)
$\displaystyle\mathbf{u}_{0}\in
L^{2}_{\mathrm{\mathrm{div}_{\mathbf{x}}}}(\mathcal{O}_{\eta_{0}})\text{ is
such that }\mathbf{u}_{0}\circ\bm{\varphi}_{\eta_{0}}=\eta_{1}\mathbf{n}\text{
on }\Omega\times\Gamma,$
$\displaystyle\|\bm{\varkappa}\|_{W^{1,\infty}(\Gamma)}\lesssim 1.$

We call $((\Omega,\mathfrak{F},(\mathfrak{F})_{t\geq
0},\mathbb{P}),\eta,\mathbf{u})$ a weak martingale solution of (1.7)–(1.9)
with data $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ provided that
the following holds:

  * (a)

$(\Omega,\mathfrak{F},(\mathfrak{F})_{t\geq 0},\mathbb{P})$ is a stochastic
basis with a complete right-continuous filtration;

  * (b)

$B_{t}$ is an $(\mathfrak{F}_{t})$-Brownian motion;

  * (c)

the shell function $\eta$ is $(\mathfrak{F}_{t})$-adapted with
$\|\eta\|_{L^{\infty}(I\times\Gamma)}<L$ a.s. and

$\displaystyle\eta\in
L^{\infty}\big{(}I;W^{2,2}(\Gamma)\big{)},\qquad\partial_{t}\eta\in
C_{w}\big{(}\overline{I};L^{2}(\Gamma)\big{)}\qquad a.s.;$

  * (d)

the velocity $\mathbf{u}$ is $(\mathfrak{F}_{t})$-adapted with
$\mathbf{u}\circ\bm{\varphi}_{\eta}=\mathbf{n}\partial_{t}\eta$ on
$I\times\Gamma$ a.s. and

$\displaystyle\mathbf{u}\in
C_{w}\big{(}\overline{I};L^{2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O}_{\eta})\big{)}\cap
L^{2}\big{(}I;W^{1,2}(\mathcal{O}_{\eta})\big{)}\qquad a.s.;$

  * (e)

equation (2.9) holds a.s. for all $(\phi,\bm{\phi})\in W^{2,2}(\Gamma)\times
W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$ with
$\bm{\phi}\circ\bm{\varphi}=\phi\mathbf{n}$ on $\Gamma$.

  * (e)

The energy inequality holds in the sense that

$\displaystyle\begin{aligned}
\tfrac{1}{2}\int_{\mathcal{O}_{\eta(t)}}|\mathbf{u}(t)|^{2}\,\mathrm{d}\mathbf{x}&+\int_{0}^{t}\int_{\mathcal{O}_{\eta(s)}}|\nabla_{\mathbf{x}}\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta(t)|^{2}\,\mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_{\Gamma}|\Delta_{\mathbf{y}}\eta(t)|^{2}\,\mathrm{d}\mathbf{y}\\\
&\leq\tfrac{1}{2}\int_{\mathcal{O}_{\eta_{0}}}|\mathbf{u}_{0}|^{2}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\eta_{1}|^{2}\,\mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_{\Gamma}|\Delta_{\mathbf{y}}\eta_{0}|^{2}\,\mathrm{d}\mathbf{y}\end{aligned}$
(2.11)

a.s. for a.a. $t\in I$.

The following is our main result.

######  Theorem 2.7.

Let $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ be a dataset such that
(LABEL:datasetAlone) holds. Then there is a weak martingale solution of
(1.7)–(1.9) with data $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ in
the sense of Definition 2.6. The interval of existence is of the form
$\overline{I}=(0,t)$, where $t<T$ only if $\lim_{s\rightarrow
t}\|\eta(s)\|_{L^{\infty}(\Gamma)}=L$ a.s. in $\Omega_{0}$ for some
$\Omega_{0}\subset\Omega$ with $\mathbb{P}(\Omega_{0})>0$.

##  3\. The linearised problem

In the first instant, we wish to construct a weak solution to a system with a
regularized geometry and a regularized convection term. Here, by a regularized
geometry, we mean a regularization of a solution to a given shell equation and
not the solution to our anticipated shell equation (1.9). Thus, we aim at
solving the system

$\displaystyle\mathrm{div}_{\mathbf{x}}\mathbf{u}=0,$ (3.1)
$\displaystyle\partial_{t}\mathbf{u}+(\mathbf{u}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u}=\Delta_{\mathbf{x}}\mathbf{u}-\nabla_{\mathbf{x}}\pi,$
(3.2)
$\displaystyle\,\mathrm{d}\partial_{t}\eta+(\epsilon\mathcal{L}^{\prime}(\eta)+\epsilon\partial_{t}\Delta_{\mathbf{y}}^{2}\eta+\Delta_{\mathbf{y}}^{2}\eta+g_{\eta_{\epsilon}})\,\mathrm{d}t+((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\circ\,\mathrm{d}B_{t}=0,$
(3.3)

in $I\times\mathcal{O}_{\eta_{\epsilon}}$ where

$\displaystyle
g_{\eta_{\epsilon}}=\mathbf{n}^{\top}(\mathbb{T}(\mathbf{u},\pi)\mathbf{n}_{\eta_{\epsilon}})\circ\bm{\varphi}_{\eta_{\epsilon}}|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\eta_{\epsilon}})|,\quad\mathbb{T}(\mathbf{u},\pi)=(\nabla_{\mathbf{x}}\mathbf{u}+\nabla_{\mathbf{x}}\mathbf{u}^{\top})-\pi\mathbb{I}_{3\times
3},$

$\mathcal{L}^{\prime}$ is the operator given
$\int_{\Gamma}\mathcal{L}^{\prime}(\eta)\phi\,\mathrm{d}\mathbf{y}=\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\eta:\nabla_{\mathbf{y}}^{3}\phi\,\mathrm{d}\mathbf{y}$
for all $\phi\in W^{3,2}(\Gamma)$, and $\epsilon>0$ is a fixed regularisation
parameter. With some slight abuse of notation we denote by $f_{\epsilon}$ the
regularisation of a function on the fluid domain (which is previously extended
by zero to the whole space) as well as the regularization of a function
defined on $I\times\Gamma$. The regularisation is taken with respect to space
and time, where the temporal regularization is taken backwards in extending
functions to $(-\infty,T)$ by their values at time 0. A martingale solution to
(3.1)–(3.3) can be defined analogously to Definition 2.6. We aim to show the
following result (the proof of Theorem 3.1 can be found in the next section).

######  Theorem 3.1.

Let $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ be a dataset such that
(LABEL:datasetAlone) holds and we have additionally $\eta_{0}\in
W^{3,2}(\Gamma)$. Then there is a weak martingale solution of (3.1)–(3.3) with
data $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$. The interval of
existence is of the form $\overline{I}=(0,t)$, where $t<T$ only if
$\lim_{s\rightarrow t}\|\eta(s)\|_{L^{\infty}(\Gamma)}=L$ a.s. in $\Omega_{0}$
for some $\Omega_{0}\subset\Omega$ with $\mathbb{P}(\Omega_{0})>0$.

In order to solve (3.1)–(3.3) we linearize the problem by replacing the
regularized velocity in the convective term with a regularization of a given
velocity field $\mathbf{v}\in\mathbb{R}^{3}$. We also replace the regularized
geometry with a regularized geometry with respect to a given structure
displacement $\zeta$ with an initial state $\zeta(0,\cdot)=\eta_{0}$. The
corresponding regularization of the pair $(\zeta,\mathbf{v})$ is denoted by
$(\zeta_{\epsilon},\mathbf{v}_{\epsilon})$. The solution we seek will be
constructed as the limit $N\rightarrow\infty$ of the solution
$(\eta^{N},\mathbf{u}^{N})$ to a finite dimensional Galerkin approximation
system incorporating these regularizing terms. Since this is a linear system
we aim to construct a probabilistically strong solution defined on a
stochastic basis $(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P})$ and driven by a given Brownian motion $(B_{t})$ relative to
$(\mathfrak{F}_{t})$. Suppose that $(\zeta,\mathbf{v})$ (and thus its
regularization $(\zeta_{\epsilon},\mathbf{v}_{\epsilon})$) are a given pair of
$(\mathfrak{F}_{t})$-progressively measurable333To be understood in the sense
of random distributions, cf. [3, Chapter 2.8]. random variables with values in
$C(\overline{I}\times\Gamma)\times L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))$
belonging to $L^{p}(\Omega)$ for some sufficiently large $p$. where we suppose
that $\epsilon$ is small enough such that
$\|\zeta_{\epsilon}\|_{L^{\infty}(I\times\Gamma)}<\alpha<L$ a.s. We now look
for an $(\mathfrak{F}_{t})$-progressively measurable process
$(\eta,\mathbf{u})$ with values in the space

$\displaystyle W^{1,2}(I;W^{2,2}(\Gamma))\times
L^{\infty}(I;W^{3,2}(\Gamma))\cap W^{1,\infty}(I;L^{2}(\Gamma))\times
L^{\infty}(I;L^{2}(\Omega_{\zeta_{\epsilon}}))\cap
L^{2}(I;W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\Omega_{\zeta_{\epsilon}}))$ (3.4)

such that

$\displaystyle\,\mathrm{d}\bigg{(}\int_{\mathcal{O}_{\zeta_{\epsilon}}}$
$\displaystyle\mathbf{u}\cdot\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\partial_{t}\eta\,\iota_{\zeta_{\epsilon}}\phi\,\mathrm{d}\mathbf{y}\bigg{)}$
(3.5)
$\displaystyle=\int_{\Gamma}\big{(}\partial_{t}\eta\,\partial_{t}(\iota_{\zeta_{\epsilon}}\phi)-\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}\phi\,\Delta_{\mathbf{y}}\eta-\epsilon\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}\phi\,\partial_{t}\Delta_{\mathbf{y}}\eta\big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}\phi:\nabla_{\mathbf{y}}^{3}\eta\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\Gamma}\bigg{(}\frac{1}{2}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}\iota_{\zeta_{\epsilon}}\phi\,\partial_{t}\zeta_{\epsilon}\,\partial_{t}\eta\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\bigg{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\mathbf{u}\cdot\partial_{t}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})\cdot\mathbf{u}-\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\frac{1}{2}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\zeta_{\epsilon}}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\zeta_{\epsilon}}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t}$

for all $(\phi,\bm{\phi})\in W^{3,2}(\Gamma)\times
W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$ with
$\bm{\phi}\circ\bm{\varphi}=\phi\mathbf{n}$ on $\Gamma$. Moreover, we require
$\mathbf{u}\circ\bm{\varphi}_{\zeta_{\epsilon}}=\mathbf{n}\partial_{t}\eta$ on
$I\times\Gamma$.

######  Theorem 3.2.

Let $(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$ be a dataset such that
(LABEL:datasetAlone) holds and we have additionally $\eta_{0}\in
W^{3,2}(\Gamma)$. Let $(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P})$ be a stochastic basis with a complete, right-continuous
filtration and let $(B_{t})$ be an $(\mathfrak{F}_{t})$-Brownian motion. Then
there is a unique probabilistically strong solution of (3.5) with data
$(\eta_{0},\eta_{1},\mathbf{u}_{0},\bm{\varkappa})$. The interval of existence
is of the form $\overline{I}=(0,t)$, where $t<T$ only if $\lim_{s\rightarrow
t}\|\eta(s)\|_{L^{\infty}(\Gamma)}=L$ a.s. in $\Omega_{0}$ for some
$\Omega_{0}\subset\Omega$ with $\mathbb{P}(\Omega_{0})>0$.

It will turn out that the solution satisfies the energy equality

$\displaystyle\begin{aligned}
\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}&|\mathbf{u}(t)|^{2}\,\mathrm{d}\mathbf{x}+\int_{0}^{t}\int_{\mathcal{O}_{\zeta_{\epsilon}}}|\nabla_{\mathbf{x}}\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\partial_{t}\eta(t)|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{y}}\eta(t)|^{2}+\epsilon|\nabla_{\mathbf{y}}^{3}\eta(t)|^{2}\Big{)}\,\mathrm{d}\mathbf{y}\\\
&+\epsilon\int_{0}^{t}\int_{\Gamma}|\partial_{s}\Delta_{\mathbf{y}}\eta|^{2}\,\mathrm{d}\mathbf{y}\,\mathrm{d}s=\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}(0)}}|\mathbf{u}_{0}|^{2}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\eta_{1}|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{y}}\eta_{0}|^{2}+\epsilon|\nabla_{\mathbf{y}}^{3}\eta_{0}|\Big{)}\,\mathrm{d}\mathbf{y}\end{aligned}$
(3.6)

a.s. for a.a. $t\in I$.

The aim of the following subsection is to construct a Galerkin approximation
of (3.1)–(3.3) on a given stochastic basis
$(\Omega,\mathfrak{F},(\mathfrak{F})_{t\geq 0},\mathbb{P})$, while its limit
passage (and thus the proof of Theorem 3.1) can be found in the next section.

###  3.1. The linearised Galerkin problem

Let us now explain in which function spaces we seek the finite dimensional
objects $(\eta^{N},\mathbf{n}^{N})$. Let $(Y_{i})_{i\in\mathbb{N}}$ be a basis
of $W^{2,2}(\Gamma)$ and let $(\mathbf{X}_{i})_{i\in\mathbb{N}}$ be a basis of
$W^{1,2}_{0,\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$. Clearly, there exists
divergence-free vectors fields $\mathbf{Y}_{i}$ that are solving Stokes
systems in the reference domain $\mathcal{O}$ with boundary data
$(Y_{i}\mathbf{n})\circ\bm{\varphi}^{-1}$. We then set

$\displaystyle\bm{w}_{i}=\left\\{\begin{array}[]{lr}\mathbf{X}_{i}&:i\text{
even},\\\ \mathbf{Y}_{i}&:i\text{ odd}\end{array}\right.$ (3.9)

and set $w_{i}=\bm{w}_{i}\circ\bm{\varphi}|_{\Gamma}\cdot\mathbf{n}$.

Now we take the pair $(\zeta,\mathbf{v})$ from

$L^{\infty}(\Omega;C(\overline{I}\times\Gamma))\times
L^{\infty}(\Omega;L^{2}(I;L^{2}(\mathbb{R}^{3})))$

being $(\mathfrak{F}_{t})$-progressively measurable. We search for
$(\alpha^{N}_{i})_{k,N\in\mathbb{N}}:\Omega\times\overline{I}\rightarrow\mathbb{R}^{n}$
such that

$\mathbf{u}^{N}=\sum_{i=1}^{N}\alpha^{N}_{i}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})\quad\text{and}\quad\eta^{N}(t,\cdot)=\sum_{i=1}^{N}\int_{0}^{t}\alpha^{N}_{i}\iota_{\zeta_{\epsilon}}w_{i}\,\mathrm{d}s+\eta_{0}$

solve the equations444We neglect the dependency of the unknown
$(\eta^{N},\mathbf{u}^{N})$ on $\epsilon$ at this point for simplicity.

$\displaystyle\,\mathrm{d}\bigg{(}\int_{\mathcal{O}_{\zeta_{\epsilon}}}$
$\displaystyle\mathbf{u}^{N}\cdot\mathcal{J}_{{\zeta_{\epsilon}}(t)}\bm{w}_{j}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\partial_{t}\eta^{N}\,\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\bigg{)}$
(3.10)
$\displaystyle=\int_{\Gamma}\big{(}\partial_{t}\eta^{N}\,\partial_{t}(\iota_{\zeta_{\epsilon}}w_{j})-\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\Delta_{\mathbf{y}}\eta^{N}-\epsilon\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\,\partial_{t}\Delta_{\mathbf{y}}\eta^{N}\big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{j}:\nabla_{\mathbf{y}}^{3}\eta^{N}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\Gamma}\bigg{(}\frac{1}{2}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}\iota_{\zeta_{\epsilon}}w_{j}\,\partial_{t}{\zeta_{\epsilon}}\,\partial_{t}\eta^{N}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\bigg{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\mathbf{u}^{N}\cdot\partial_{t}(\mathcal{J}_{{\zeta_{\epsilon}}(t)}\bm{w}_{j})-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u}^{N})\cdot(\mathcal{J}_{{\zeta_{\epsilon}}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{{\zeta_{\epsilon}}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}-\nabla_{\mathbf{x}}\mathbf{u}^{N}:\nabla_{\mathbf{x}}(\mathcal{J}_{{\zeta_{\epsilon}}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\frac{1}{2}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{N})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{N})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t}$

for $1\leq j\leq N$ with an initial condition $\alpha^{N}_{i}(0)$ which is
such that

$\displaystyle\partial_{t}\eta^{N}(0,\cdot)\rightarrow\eta_{1}\qquad\text{in}\qquad
L^{2}(\Gamma),$ (3.11)
$\displaystyle\mathbf{u}^{N}(0,\cdot)\rightarrow\mathbf{u}_{0}\qquad\text{in}\qquad
L^{2}(\mathcal{O}_{\zeta_{\epsilon}(0)}).$ (3.12)

Note that the derivation of the weak formulation (3.10) is slightly different
to the derivation of (2.9) due to the differences in their respective
advective terms. The treatment of the former advection term goes as follows.
By using the trivial identity

$\displaystyle\int_{\mathcal{O}_{\zeta_{\epsilon}}}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=\frac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\frac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$

we rewrite the last term as follows

$\displaystyle\int_{\mathcal{O}_{\zeta_{\epsilon}}}$
$\displaystyle((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t=\int_{\mathcal{O}_{\zeta_{\epsilon}}}\mathrm{div}_{\mathbf{x}}(\mathbf{v}_{\epsilon}\otimes\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=\int_{\partial\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{n}_{\zeta_{\epsilon}}\cdot([\mathbf{v}_{\epsilon}\otimes\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}]\mathbf{u}^{N})\,\mathrm{d}\mathcal{H}^{2}\,\mathrm{d}t-\int_{\mathcal{O}_{\zeta_{\epsilon}}}(\mathbf{v}_{\epsilon}\otimes\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}):\nabla_{\mathbf{x}}\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=\int_{\partial\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{n}_{\zeta_{\epsilon}}\cdot([\mathbf{v}_{\epsilon}\circ\bm{\varphi}_{\zeta_{\epsilon}}\circ\bm{\varphi}_{\zeta_{\epsilon}}^{-1}((\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\circ\bm{\varphi}_{\zeta_{\epsilon}}\circ\bm{\varphi}_{\zeta_{\epsilon}}^{-1})^{\top}]\mathbf{u}^{N}\circ\bm{\varphi}_{\zeta_{\epsilon}}\circ\bm{\varphi}_{\zeta_{\epsilon}}^{-1})\,\mathrm{d}\mathcal{H}^{2}\,\mathrm{d}t$
$\displaystyle-\int_{\mathcal{O}_{\zeta_{\epsilon}}}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u}^{N})\cdot\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle=\int_{\Gamma}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}\iota_{\zeta_{\epsilon}}w_{j}\,\partial_{t}\zeta_{\epsilon}\,\partial_{t}\eta^{N}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\mathcal{O}_{\zeta_{\epsilon}}}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u}^{N})\cdot\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$

where we have used
$\mathbf{u}^{N}\circ\bm{\varphi}_{\zeta_{\epsilon}}=\mathbf{n}\partial_{t}\eta^{N}$
and
$\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}\circ\bm{\varphi}_{\zeta_{\epsilon}}=\iota_{\zeta_{\epsilon}}w_{j}\mathbf{n}$
in the last step. This explains the presence of the Jacobian determinant in
(3.10).
Moving on, we note that equation (3.10) is equivalent to

$\displaystyle\,\mathrm{d}\bigg{[}$
$\displaystyle\sum_{i=1}^{N}\alpha^{N}_{i}\bigg{(}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i}\cdot\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\iota_{\zeta_{\epsilon}}w_{i}\,\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\bigg{)}\bigg{]}$
(3.13)
$\displaystyle=\sum_{i=1}^{N}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\alpha^{N}_{i}\Big{(}\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i}\cdot\partial_{t}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\sum_{i=1}^{N}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\alpha^{N}_{i}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})-\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i}):\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\sum_{i=1}^{N}\frac{\alpha^{N}_{i}}{2}\int_{\Gamma}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}\iota_{\zeta_{\epsilon}}w_{j}\,\partial_{t}\zeta_{\epsilon}\,\iota_{\zeta_{\epsilon}}w_{i}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\,\mathrm{d}\mathbf{y}\,\mathrm{d}t-\int_{\Gamma}\Delta_{\mathbf{y}}\eta_{0}\,\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle-\sum_{i=1}^{N}\int_{\Gamma}\int_{0}^{t}\alpha^{N}_{i}(s)\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{i}(s)\,\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}(t)\,\mathrm{d}s\,\mathrm{d}\mathbf{y}\,\mathrm{d}t-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\eta_{0}:\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle-\epsilon\sum_{i=1}^{N}\int_{\Gamma}\int_{0}^{t}\alpha^{N}_{i}(s)\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{i}(s):\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{j}(t)\,\mathrm{d}s\,\mathrm{d}\mathbf{y}\,\mathrm{d}t-\epsilon\sum_{i=1}^{N}\int_{\Gamma}\alpha^{N}_{i}\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{i}\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle+\sum_{i=1}^{N}\frac{\alpha^{N}_{i}}{2}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\zeta_{\epsilon}}w_{i})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\sum_{i=1}^{N}\alpha^{N}_{i}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\zeta_{\epsilon}}w_{i})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t}.$

To simplify notations, we drop the summation signs by employing Einstein’s
summation convention. Then for

$\displaystyle(a_{ij}(t))$
$\displaystyle:=\int_{\mathcal{O}_{\zeta_{\epsilon}}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\iota_{\zeta_{\epsilon}}w_{i}\,\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y},$
$\displaystyle(b_{ij}(t))$
$\displaystyle:=\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})\cdot\partial_{t}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x},$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i})-\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{i}):\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{w}_{j})\Big{)}\,\mathrm{d}\mathbf{x}$
$\displaystyle+\frac{1}{2}\int_{\Gamma}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}\iota_{\zeta_{\epsilon}}w_{j}\,\partial_{t}\zeta_{\epsilon}\,\iota_{\zeta_{\epsilon}}w_{i}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\,\mathrm{d}\mathbf{y}-\epsilon\int_{\Gamma}\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{i}\,\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}$
$\displaystyle+\frac{1}{2}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\zeta_{\epsilon}}w_{i})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y},$
$\displaystyle(c_{ij}(t,s))$
$\displaystyle:=-\int_{\Gamma}\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{i}(s)\,\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}(t)\,\mathrm{d}\mathbf{y}-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{i}(s):\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{j}(t)\,\mathrm{d}\mathbf{y},$
$\displaystyle(d_{j}(t))$
$\displaystyle:=-\int_{\Gamma}\Delta_{\mathbf{y}}\eta_{0}\,\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y}-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\eta_{0}:\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y},$
$\displaystyle(e_{ij}(t))$
$\displaystyle:=\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\iota_{\zeta_{\epsilon}}w_{i})\iota_{\zeta_{\epsilon}}w_{j}\,\mathrm{d}\mathbf{y},$

we can rewrite the above as the following system of SDEs

$\displaystyle\int_{I}\,\mathrm{d}\big{[}\alpha^{N}_{i}(t)(a_{ij}(t))\big{]}$
$\displaystyle=\int_{I}\alpha^{N}_{i}(t)(b_{ij}(t))\,\mathrm{d}t-\int_{I}\int_{0}^{t}\alpha^{N}_{i}(s)(c_{ij}(t,s))\,\mathrm{d}s\,\mathrm{d}t-\int_{I}(d_{j}(t))\,\mathrm{d}t$
(3.14) $\displaystyle+\int_{I}\alpha^{N}_{i}(t)(e_{ij}(t))\,\mathrm{d}B_{t}.$

Since the coefficient matrix $(a_{ij}(t))$ is symmetric and positive definite,
it is invertible. As the problem is linear, we can infer the existence of a
unique global solution, cf. [29, Theorem 3.1.1.]. Moreover, we have the
following energy estimate which is obtained by applying Itô’s formula to the
process
$t\mapsto\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}|\mathbf{u}^{N}(t)|^{2}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta^{N}(t)|^{2}\,\mathrm{d}\mathbf{y}$:
it holds

$\displaystyle\begin{aligned}
\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}&|\mathbf{u}^{N}(t)|^{2}\,\mathrm{d}\mathbf{x}+\int_{0}^{t}\int_{\mathcal{O}_{\zeta_{\epsilon}}}|\nabla_{\mathbf{x}}\mathbf{u}^{N}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\partial_{t}\eta^{N}(t)|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{y}}\eta^{N}(t)|^{2}+\epsilon|\nabla_{\mathbf{y}}^{3}\eta^{N}(t)|^{2}\Big{)}\,\mathrm{d}\mathbf{y}\\\
&+\epsilon\int_{0}^{t}\int_{\Gamma}|\partial_{s}\Delta_{\mathbf{y}}\eta^{N}|^{2}\,\mathrm{d}\mathbf{y}\,\mathrm{d}s=\tfrac{1}{2}\int_{\mathcal{O}_{\eta_{0}}}|\mathbf{u}_{0}|^{2}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\eta_{1}|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{y}}\eta_{0}|^{2}+\epsilon|\nabla_{\mathbf{y}}^{3}\eta_{0}|\Big{)}\,\mathrm{d}\mathbf{y}\end{aligned}$
(3.15)

a.s. for a.a. $t\in I$. From the above, we then obtain

$\displaystyle\sup_{I}\Big{(}\|\eta^{N}\|_{W^{2,2}(\Gamma)}^{2}+\epsilon\|\eta^{N}\|_{W^{3,2}(\Gamma)}^{2}\Big{)}\lesssim
1,$ (3.16)
$\displaystyle\sup_{I}\|\partial_{t}\eta^{N}\|_{L^{2}(\Gamma)}^{2}\lesssim 1,$
(3.17)
$\displaystyle\epsilon\int_{I}\|\partial_{t}\eta^{N}\|_{W^{2,2}(\Gamma)}^{2}\,\mathrm{d}t\lesssim
1,$ (3.18)
$\displaystyle\sup_{I}\|\mathbf{u}^{N}\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}})}^{2}\lesssim
1,$ (3.19)
$\displaystyle\int_{I}\|\nabla_{\mathbf{x}}\mathbf{u}^{N}\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}})}^{2}\,\mathrm{d}t\lesssim
1.$ (3.20)

In addition, for any $s\in(0,\frac{1}{2})$, it follows from
$\mathbf{u}^{N}\circ\bm{\varphi}_{\zeta_{\epsilon}}=\mathbf{n}\partial_{t}\eta^{N}$,
(3.20) and the trace theorem that

$\displaystyle\int_{I}\|\partial_{t}\eta^{N}\|_{W^{s,2}(\Gamma)}^{2}\,\mathrm{d}t\lesssim
1$ (3.21)

holds.

###  3.2. Tightness of $\partial_{t}\eta^{N}$

The effort of this subsection is to prove tightness of the law of
$\partial_{t}\eta^{N}$ on $L^{2}$ in order to pass to the limit in the
stochastic integral. We define the projection $\mathcal{P}^{N}$ and the
extension $\mathscr{F}_{N}^{\zeta_{\epsilon}}$ (for a given
$\zeta:\omega\rightarrow(-L,L)$)

$\displaystyle\mathcal{P}^{N}b=\sum_{k=1}^{N}\alpha_{k}(b)\iota_{\zeta_{\epsilon}}w_{k},\quad\mathscr{F}_{N}^{\zeta_{\epsilon}}b=\sum_{k=1}^{N}\alpha_{k}(b)\mathcal{J}_{\zeta_{\epsilon}}\bm{w}_{k},$

where $\alpha_{k}(b)=\langle
b,\iota_{\zeta_{\epsilon}}w_{k}\rangle_{W^{3,2}(\Gamma)}$ if $w_{k}=Y_{\ell}$
for some $\ell\in\mathbb{N}$ and $\alpha_{k}(b)=0$ otherwise. Obviously, we
have
$\mathscr{F}_{N}^{\zeta_{\epsilon}}b\circ\bm{\varphi}_{\zeta_{\epsilon}}=\mathbf{n}\mathcal{P}^{N}b$
for any $b\in W^{3,2}(\Gamma)$. We have by definition,

$\displaystyle\|\mathcal{P}^{N}b\|_{W^{3,2}(\Gamma)}^{2}\leq\,\|b\|_{W^{3,2}(\Gamma)}^{2}\quad\forall
b\in W^{3,2}(\Gamma).$ (3.22)

The eigenvalue equation for the basis vectors implies additionally that

$\displaystyle\|\mathcal{P}^{N}b\|_{L^{2}(\Gamma)}^{2}\lesssim\|b\|_{L^{2}(\Gamma)}^{2}\quad\forall
b\in L^{2}(\Gamma).$ (3.23)

By interpolation we obtain an estimate on $W^{s,2}(\Gamma)$ for any
$s\in[0,3]$, that is

$\displaystyle\|\mathcal{P}^{N}b\|_{W^{s,2}(\Gamma)}\lesssim\|b\|_{W^{s,2}(\Gamma)}.$
(3.24)

Finally, for ${\zeta_{\epsilon}}\in W^{3,2}(\Gamma)$ with
$\|\zeta_{\epsilon}\|_{L^{\infty}(\Gamma)}<\alpha$ for $\alpha\in(0,L)$ we
have

$\displaystyle\begin{aligned}
\|\mathscr{F}^{\zeta_{\epsilon}}_{N}b\|_{W^{s+1/2,2}(\mathcal{O}\cup
S_{\alpha})}^{2}&=\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)\mathcal{J}_{\zeta_{\epsilon}}\bm{w}_{k}\bigg{\|}_{W^{s+1/2,2}(\mathcal{O}\cup
S_{\alpha})}^{2}\\\
&\lesssim\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)\iota_{\zeta_{\epsilon}}w_{k}\bigg{\|}_{W^{s,2}(\Gamma)}^{2}\\\
&\leq\,\|b\|_{W^{s,2}(\Gamma)}^{2}\end{aligned}$ (3.25)

for all $b\in W^{s,2}(\Gamma)$ and for any $s\in[0,3]$. Here we used that
$\sum_{k=1}^{N}\alpha_{k}(b)\bm{w}_{k}$ solves the homogeneous Stokes problem
with boundary datum $\sum_{k=1}^{N}\alpha_{k}(b)w_{k}\mathbf{n}$ in
$\mathcal{O}$ and well-known elliptic estimates (see [15, Chapter IV]).
Similarly, we obtain for $\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2}\in
W^{1,2}(\Gamma)$

$\displaystyle\begin{aligned}
\|\mathscr{F}^{\zeta_{\epsilon}^{1}}_{N}b-\mathscr{F}^{\zeta_{\epsilon}^{2}}_{N}b\|_{L^{2}(\mathcal{O}\cup
S_{\alpha})}^{2}&=\bigg{\|}(\mathcal{J}_{\zeta_{\epsilon}^{1}}-\mathcal{J}_{\zeta_{\epsilon}^{2}})\sum_{k=1}^{N}\alpha_{k}(b)\bm{w}_{k}\bigg{\|}_{L^{2}(\mathcal{O}\cup
S_{\alpha})}^{2}\\\
&\lesssim\|(\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2})\|^{4}_{W^{1,\infty}(\Gamma)}\|\zeta_{\epsilon}^{1}-\zeta_{\epsilon}^{2}\|_{L^{2}(\Gamma)}^{2}\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)\bm{w}_{k}\bigg{\|}_{W^{1,\infty}(\mathcal{O})}^{2}\\\
&+\|(\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2})\|^{2}_{W^{1,\infty}(\Gamma)}\|\nabla_{\mathbf{y}}(\zeta_{\epsilon}^{1}-\zeta_{\epsilon}^{2})\|_{L^{2}(\Gamma)}^{2}\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)\bm{w}_{k}\bigg{\|}_{L^{\infty}(\mathcal{O})}^{2}\\\
&\lesssim(1+\|(\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2})\|^{4}_{W^{1,\infty}(\Gamma)})\|\zeta_{\epsilon}^{1}-\zeta_{\epsilon}^{2}\|_{W^{1,2}(\Gamma)}^{2}\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)\bm{w}_{k}\bigg{\|}_{W^{3,2}(\mathcal{O})}^{2}\\\
&\lesssim(1+\|(\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2})\|^{4}_{W^{1,\infty}(\Gamma)})\|\zeta_{\epsilon}^{1}-\zeta_{\epsilon}^{2}\|_{W^{1,2}(\Gamma)}^{2}\bigg{\|}\sum_{k=1}^{N}\alpha_{k}(b)w_{k}\bigg{\|}_{W^{3,2}(\Gamma)}^{2}\\\
&\leq\,(1+\|(\zeta_{\epsilon}^{1},\zeta_{\epsilon}^{2})\|^{4}_{W^{1,\infty}(\Gamma)})\|\zeta_{\epsilon}^{1}-\zeta_{\epsilon}^{2}\|_{W^{1,2}(\Gamma)}^{2}\|b\|_{W^{3,2}(\Gamma)}^{2}\end{aligned}$
(3.26)

for all $b\in W^{3,2}(\Gamma)$. Following [25], we write

$\displaystyle\begin{aligned}
\int_{I}\int_{\mathcal{O}_{\zeta_{\epsilon}}}|\mathbf{u}^{N}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t&+\int_{I}\int_{\Gamma}|\partial_{t}\eta^{N}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\\\
&=\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}\cdot\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\mathcal{\eta}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\eta^{N}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\\\
&+\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}\cdot(\mathbf{u}^{N}-\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\mathcal{\eta}^{N})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t.\end{aligned}$
(3.27)

We consider the space $X:=L^{2}(\Gamma)\times W^{-s,2}(\mathcal{O}\cup
S_{\alpha})$ with $s\in(0,1/4)$. In order to apply Theorem 2.2 yielding
tightness of the corresponding laws we need to equip $L^{2}(I;X^{\prime}\times
X)$ with an unconventional topology which we denote by $\tau_{\sharp}$ and
define the convergence $\rightarrow^{\sharp}$ as follows. We say that

$((\zeta^{N},\mathbf{v}^{N}),(\xi^{N},\mathbf{w}^{N}))\rightarrow^{\sharp}((\zeta,\mathbf{v}),(\xi,{\mathbf{w}}))\,\,\text{in}\,\,L^{2}(I;X^{\prime}\times
X)$

provided that

$((\zeta^{N},\mathbf{v}^{N}),(\xi^{N},\mathbf{w}^{N}))\rightharpoonup((\zeta,\mathbf{v}),(\xi,{\mathbf{w}}))\,\,\text{in}\,\,L^{2}(I;X^{\prime}\times
X)$

and it holds

$\displaystyle\begin{aligned}
\int_{I}\langle\mathbf{v}^{N},\mathbf{w}^{N}\rangle_{W^{s,2},W^{-s,2}}\,\mathrm{d}t&+\int_{I}\int_{\Gamma}\zeta^{N}\,\xi^{N}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\\\
&\longrightarrow\int_{I}\langle\mathbf{v},{\mathbf{w}}\rangle_{W^{s,2},W^{-s,2}}\,\mathrm{d}t+\int_{I}\int_{\Gamma}\zeta\,\xi\,\mathrm{d}\mathbf{y}\,\mathrm{d}t.\end{aligned}$
(3.28)

Since this topology is finer than the weak topology on
$L^{2}(I;X^{\prime}\times X)$ it is clear that $(L^{2}(I;X^{\prime}\times
X),\tau_{\sharp})$ is a quasi-Polish space such that Jakubowski’s version of
the Skorokhod representation theorem applies. We obtain the following result
concerning tightness.

######  Lemma 3.3.

The laws of

$\displaystyle((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}))\quad\text{and}\quad((\partial_{t}\eta_{n},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(0,\mathbf{u}^{N}-\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}))$

on $(L^{2}(I;X^{\prime}\times X),\tau_{\sharp})$ are tight.

###### Proof.

According to Theorem 2.2 we first need boundedness in
$L^{2}(I;X^{\prime}\times X)$. By (3.16)–(3.21) and the properties of the
projection and extension operators above this follows immediately (even
uniformly in probability). Note that the extension by zero is a bounded
operator on $W^{s,2}$ for $s<\frac{1}{4}$. For $(b)$ we observe that we may
assume that a regularizer $b\mapsto(b)_{\kappa}$ exists such that for any
$s,a\in\mathbb{R}$

$\displaystyle\|b-(b)_{\kappa}\|_{W^{a,2}(\Gamma)}\lesssim\kappa^{s-a}\|b\|_{W^{s,2}(\Gamma)},\quad
b\in W^{s,2}(\Gamma).$ (3.29)

The estimate is well known for $a,s\in\mathbb{N}_{0}$, while the general case
follows by interpolation and duality. Next we introduce the mollification
operator on $\partial_{t}\eta^{N}$ by considering for $\kappa>0$ and
$N\in\mathbb{N}$,
$f_{N}(t):=(\mathcal{P}^{N}(\partial_{t}\eta^{N}(t)),\mathscr{F}_{N}^{\zeta_{\epsilon}(t)}(\mathcal{P}^{N}(\partial_{t}\eta^{N}(t))))$
and set

$f_{N,\kappa}(t):=(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}),\mathscr{F}_{N}^{\zeta_{\epsilon}(t)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))).$

We find by the continuity of the mollification operator from (3.29), the
continuity of the projection operator from (3.24) and the estimate for the
extension operator (3.25) that for a.e. $t\in I$ and $s<s_{0}<1/2$

$\displaystyle\|f_{N,\kappa}-f_{N}\|_{L^{2}(\Gamma)\times
W^{s,2}(\mathcal{O}\cup
S_{L/2})}\lesssim\kappa^{s_{0}-s}\|\partial_{t}\eta^{N}\|_{W^{s_{0},2}(\Gamma)},$
(3.30)

which can be made arbitrarily small in $L^{2}$ by choosing $\kappa$
appropriately, cf. (3.21). Similarly, we have

$\displaystyle\|f_{N,\kappa}\|_{W^{1,2}(\Gamma)\times W^{1,2}(\mathcal{O}\cup
S_{L/2})}\lesssim\kappa^{-1}\|\partial_{t}\eta^{N}\|_{L^{2}(\Gamma)}.$

Clearly, if $f_{N}\rightharpoonup f$ in $L^{2}(I;X)$ then we can deduce a
converging subsequence such that $f_{N,\kappa}\rightharpoonup f_{\kappa}$ (for
some $f_{\kappa}$) in $L^{2}(I;X)$ for any $\kappa>0$, which implies (b).

For (c) we have to control $\langle g_{N}(t)-g_{N}(s),f_{N,\kappa}(t)\rangle$,
where
$g_{N}(t):=(\partial_{t}\eta^{N}(t),\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}(t)}}\mathbf{u}^{N}(t))$
and hence we decompose

$\displaystyle\langle g_{N}(t)-g_{N}(s),f_{N,\kappa}(t)\rangle$
$\displaystyle=\big{(}\langle
g_{N}(t),(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}),\mathscr{F}_{N}^{\zeta_{\epsilon}(t)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})))\rangle$
$\displaystyle\quad-\langle
g_{N}(s),(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}),\mathscr{F}_{N}^{\zeta_{\epsilon}(s)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})))\rangle\big{)}$
$\displaystyle\quad+\langle
g_{N}(s),(0,\mathscr{F}_{N}^{\zeta_{\epsilon}(s)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))-\mathscr{F}_{N}^{\zeta_{\epsilon}(t)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))\rangle$
$\displaystyle=:(I)+(II).$

We begin estimating $(II)$ to find that

$\displaystyle(II)$
$\displaystyle=\int_{\mathcal{O}_{\zeta_{\epsilon}(s)}}\mathbf{u}^{N}(s)\cdot\Big{(}\mathscr{F}_{N}^{\zeta_{\epsilon}(s)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))-\mathscr{F}_{N}^{\zeta_{\epsilon}(t)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\big{)}\,\mathrm{d}\mathbf{x}$
$\displaystyle\lesssim\|\mathbf{u}^{N}(s)\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}(s)})}(1+\|\zeta_{\epsilon}\|^{2}_{L^{\infty}(I;W^{1,\infty}(\Gamma))})\|\zeta_{\epsilon}(t)-\zeta_{\epsilon}(s)\|_{W^{1,2}(\Gamma)}\|\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\|_{W^{3,2}(\Gamma)}$

using (3.26). By (3.29) and (3.24) the last term can be estimated by

$\displaystyle\|\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\|_{W^{3,2}(\Gamma)}\leq
c\|(\partial_{t}\eta^{N}(t))_{\kappa}\|_{W^{3,2}(\Gamma)}\leq
c\kappa^{-3}\|\partial_{t}\eta^{N}(t)\|_{L^{2}(\Gamma)},$

which is bounded by (3.17). Due to this as well as (3.17) and (3.19) we
further have that

$\displaystyle|(II)|$ $\displaystyle\leq
c(\kappa)|t-s|(1+\|\zeta_{\epsilon}\|^{3}_{W^{1,\infty}(I;W^{1,\infty}(\Gamma))})$
$\displaystyle\leq
c(\kappa,\epsilon)|t-s|(1+\|\zeta\|^{3}_{C(\overline{I}\times\Gamma)}).$

In teh last step we used standard properties of the mollification recalling
that $\epsilon$ is a fixed parameter on this level. It follows by Markov’s
inequality for $K>0$

$\displaystyle\mathbb{P}\big{(}|t-s|^{-1}|(II)|\geq K\big{)}$
$\displaystyle\leq\,\frac{c(\kappa,\epsilon)}{K}\mathbb{E}\Big{[}1+\|\zeta\|^{3}_{C(\overline{I}\times\Gamma)}\Big{]}\leq\,\frac{c(\kappa,\epsilon)}{K},$

which can be amde arbitrarily small for large $K$ provided $\zeta\in
L^{3}(\Omega;C(\overline{I}\times\Gamma))$. The term $(I)$ is estimated using
the test function $f_{N,\kappa}$ obtaining

$\displaystyle\int_{t}^{s}(I)\,\mathrm{d}r=$
$\displaystyle-\int_{t}^{s}\int_{\Gamma}\Delta_{\mathbf{y}}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\Delta_{\mathbf{y}}\eta^{N}\,\mathrm{d}\mathbf{y}\,\mathrm{d}r$
$\displaystyle-\epsilon\int_{t}^{s}\int_{\Gamma}\Big{(}\Delta_{\mathbf{y}}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\partial_{t}\Delta_{\mathbf{y}}\eta^{N}+\nabla_{\mathbf{y}}^{3}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}):\nabla_{\mathbf{y}}^{3}\eta^{N}\Big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\Gamma}\bigg{(}\frac{1}{2}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\partial_{t}{\zeta_{\epsilon}}\,\partial_{t}\eta^{N}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\bigg{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}\cdot\partial_{r}\mathscr{F}_{N}^{\zeta_{\epsilon}(r)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))\,\mathrm{d}\mathbf{x}\,\mathrm{d}r$
$\displaystyle-\int_{t}^{s}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u}^{N})\cdot\mathscr{F}_{N}^{\zeta_{\epsilon}}(\mathcal{P}^{N}((\partial_{r}\eta^{N}(t))_{\kappa}))\,\mathrm{d}\mathbf{x}\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathscr{F}_{N}^{\zeta_{\epsilon}}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})))\cdot\mathbf{u}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}r$
$\displaystyle-\int_{t}^{s}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\nabla_{\mathbf{x}}\mathbf{u}^{N}:\nabla_{\mathbf{x}}\mathscr{F}_{N}^{\zeta_{\epsilon}}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))\,\mathrm{d}\mathbf{x}\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{N})(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\,\mathrm{d}\mathbf{y}\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{N})(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa})\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{r}.$
(3.31)

Due to (3.29), (3.24) and (3.25) and the uniform estimates (3.16)–(3.21), it
is clear that all terms can be estimated by
$c(\kappa,\epsilon)|t-s|^{1/2}(1+\|\zeta\|_{C(\overline{I}\times\Gamma)}+\|\nabla_{\mathbf{x}}\mathbf{u}^{N}\|_{L^{2}(I\times\mathcal{O}_{\zeta_{\epsilon}})})$
except for the last one. Here we have by Markov’s inequality and Itô-isometry
(using also $\mathrm{div}_{\mathbf{y}}\bm{\varkappa}=0$)

$\displaystyle\mathbb{P}\bigg{(}|t-s|^{-1/2}\bigg{|}$
$\displaystyle\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{N})\mathcal{P}^{N}(\partial_{t}\eta^{N}(t))_{\kappa}\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{r}\bigg{|}\geq
K\bigg{)}$
$\displaystyle\leq\,\frac{1}{K^{2}}\mathbb{E}\bigg{|}|t-s|^{-1/2}\int_{t}^{s}\int_{\Gamma}\partial_{t}\eta^{N}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\mathcal{P}^{N}(\partial_{t}\eta^{N}(t))_{\kappa})\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{r}\bigg{|}^{2}$
$\displaystyle=\,\frac{1}{K^{2}}\mathbb{E}\bigg{[}|t-s|^{-1}\int_{t}^{s}\bigg{|}\int_{\Gamma}\partial_{t}\eta^{N}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\mathcal{P}^{N}(\partial_{t}\eta^{N}(t))_{\kappa})\,\mathrm{d}\mathbf{y}\bigg{|}^{2}\,\,\mathrm{d}r\bigg{]}$
$\displaystyle\lesssim\,\frac{1}{K^{2}}\mathbb{E}\bigg{[}|t-s|^{-1}\int_{t}^{s}\|\partial_{t}\eta^{N}\|_{L^{2}(\Gamma)}\|\nabla_{\mathbf{y}}\mathcal{P}^{N}(\partial_{t}\eta^{N}(t))_{\kappa}\|_{L^{2}(\Gamma)}\,\,\mathrm{d}r\bigg{]}$
$\displaystyle\lesssim\,\frac{\kappa^{-1}}{K^{2}}\mathbb{E}\bigg{[}|t-s|^{-1}\int_{t}^{s}\|\partial_{t}\eta^{N}(r)\|_{L^{2}(\Gamma)}\|\partial_{t}\eta^{N}(t)\|_{L^{2}(\Gamma)}\,\,\mathrm{d}r\bigg{]}\lesssim\,\frac{\kappa^{-1}}{K^{2}}$

using (3.17). This can be made arbitrarily small for $K$ large. Now we set
$Z:=W^{s_{0},2}(\Gamma)\times W^{s_{0},2}(\mathcal{O}\cup S_{\alpha})$, where
$s_{0}\in(s,\frac{1}{4})$. Noticing that property (d) from Theorem 2.2 follows
by the usual compactness in (negative) Sobolev spaces, we conclude tightness
of the law of
$((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}))$
on $(L^{2}(I;X^{\prime}\times X),\tau_{\sharp})$.

The tightness for
$((0,\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(0,\mathbf{u}^{N}-\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}))$
follows along the same line, the only difference being the regularisation of

$f_{N}:=\big{(}0,\mathbf{u}^{N}-\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}\big{)}.$

While $\mathscr{F}_{N}^{\zeta_{\epsilon}}\partial_{t}\eta^{N}$ can be replaced
by
$\mathscr{F}_{N}^{\zeta_{\epsilon}(s)}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))$
as above, we need to regularise $\mathbf{u}^{N}$ accordingly to preserve the
homogeneous boundary conditions of $f_{N}$. Recalling the definition of
$\bm{w}_{i}$ from (3.9) we define $\mathbf{X}_{i}^{\kappa}\in
W^{1,2}_{0,\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$ as a spatial
regularisation of $\mathbf{X}_{i}$ and $\mathbf{Y}_{i}^{\kappa}$ as the
solution to the Stokes problem with boundary datum $Y_{i}^{\kappa}\mathbf{n}$
(which inherits the regularity of $Y_{i}^{\kappa}$). Then we set

$\displaystyle\bm{w}_{i}^{\kappa}=\left\\{\begin{array}[]{lr}\mathbf{X}_{i}^{\kappa}&:i\text{
even},\\\ \mathbf{Y}_{i}^{\kappa}&:i\text{ odd}.\end{array}\right.$ (3.34)

If
$\mathbf{u}^{N}=\sum_{i=1}^{N}\alpha_{i}^{N}\mathcal{J}_{\zeta_{\epsilon}}\bm{w}_{i}$
we set

$\displaystyle
f_{N,\kappa}=\bigg{(}0,\sum_{i=1}^{N}\alpha_{i}^{N}\mathcal{J}_{\zeta_{\epsilon}}\bm{w}_{i}^{\kappa}-\mathscr{F}_{N}^{\zeta_{\epsilon}}(\mathcal{P}^{N}((\partial_{t}\eta^{N}(t))_{\kappa}))\bigg{)}.$

Hence $f_{N,\kappa}$ has zero boundary conditions and thus only the fluid part
is seen in the expression $\langle g_{N}(t)-g_{N}(s),f_{N,\kappa}(t)\rangle$
(where
$g_{N}:=(\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N})$
as above). In particular, the noise is not seen and we conclude

$\displaystyle\langle
g_{N}(t)-g_{N}(s),f_{N,\kappa}(t)\rangle\leq\,c(\kappa,\epsilon)|t-s|^{1/6}(1+\|\zeta\|_{C(\overline{I}\times\Gamma)}+\|\nabla_{\mathbf{x}}\mathbf{u}^{N}\|_{L^{2}(I\times\mathcal{O}_{\zeta_{\epsilon}})})$

obtaining again the claimed tightness. ∎

###  3.3. Stochastic compactness

With the bounds from (3.16)–(3.21) in hand, we wish to obtain compactness. For
this, we define the path space

$\displaystyle\chi=\chi_{B}\times\chi_{{\mathbf{u}}}^{2}\times\chi_{\nabla{\mathbf{u}}}\times\chi_{\eta}\times\chi_{\zeta}\times\chi_{f,g}^{2}$

where

$\displaystyle\chi_{B}=C(\overline{I}),\quad\chi_{\mathbf{u}}=\big{(}L^{2}(I;L^{2}(\mathcal{O}\cup
S_{\alpha})),w\big{)},\quad\chi_{\nabla{\mathbf{u}}}=\big{(}L^{2}(I;L^{2}(\mathcal{O}\cup
S_{\alpha})),w\big{)},$
$\displaystyle\chi_{\eta}=\big{(}W^{1,\infty}(I;L^{2}(\Gamma)),w^{\ast}\big{)}\cap\big{(}L^{\infty}(I;W^{3,2}(\Gamma)),w^{\ast}\big{)}\cap\big{(}W^{1,2}(I;W^{2,2}(\Gamma)),w\big{)},$
$\displaystyle\chi_{\zeta}=C(\overline{I}\times\Gamma),\quad\chi_{f,g}=\big{(}L^{2}(I;X^{\prime}\times
X),\tau_{\sharp}\big{)}.$

From (3.16)–(3.21) (together with Alaoglu-Bourbaki Theorem) we obtain the
following.

######  Lemma 3.4.

For fixed $\epsilon>0$, the joint law

$\displaystyle\bigg{\\{}\mathcal{L}\bigg{[}B_{t},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N},\mathbf{v},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\nabla_{\mathbf{x}}\mathbf{u}^{N},\eta^{N},\zeta,\begin{pmatrix}((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\\\
((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(0,\mathbf{u}^{N}-\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\end{pmatrix}\bigg{]}\bigg{]};\,N\in\mathbb{N}\bigg{\\}}$

is tight on $\chi$.

Now we use Jakubowski’s version of the Skorokhod representation theorem, see
[22], to infer the following result (we refer to [28, Theorem A.1] for a
statement which combines Prokhorov’s and Skorokhod’s theorem for quasi-Polish
spaces) which one obtains after taking a non-relabelled subsequence.

######  Proposition 3.5.

There exists a complete probability space
$(\tilde{\Omega},\tilde{\mathfrak{F}},\tilde{\mathbb{P}})$ with $\chi$-valued
random variables

$\tilde{\bm{\Theta}}^{N}:=\bigg{[}\tilde{B}_{t}^{N},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{N}_{\epsilon}}}\tilde{\mathbf{u}}^{N},\tilde{\mathbf{v}}^{N},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{N}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}^{N},\tilde{\eta}^{N},\tilde{\zeta}^{N},\begin{pmatrix}((\partial_{t}\tilde{\eta}^{N},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{N}_{\epsilon}}}\tilde{\mathbf{u}}^{N}),(\partial_{t}\tilde{\eta}^{N},\mathscr{F}^{\tilde{\zeta}^{N}_{\epsilon}}_{N}\partial_{t}\tilde{\eta}^{N}))\\\
((\partial_{t}\tilde{\eta}^{N},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{N}_{\epsilon}}}\tilde{\mathbf{u}}^{N}),(0,\tilde{\mathbf{u}}^{N}-\mathscr{F}^{\tilde{\zeta}^{N}_{\epsilon}}_{N}\partial_{t}\tilde{\eta}^{N}))\end{pmatrix}\bigg{]},\quad
N\in\mathbb{N},$

and

$\tilde{\bm{\Theta}}:=\bigg{[}\tilde{B}_{t},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}},\tilde{\mathbf{v}},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}},\tilde{\eta},\tilde{\zeta},\begin{pmatrix}((\partial_{t}\tilde{\eta},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}),(\partial_{t}\tilde{\eta},\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta}))\\\
((\partial_{t}\tilde{\eta},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}),(0,\tilde{\mathbf{u}}-\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta}))\end{pmatrix}\bigg{]}$

such that

  1. (a)

For all $n\in\mathbb{N}$ the law of $\tilde{\bm{\Theta}}^{N}$ on $\chi$ is
given by

$\displaystyle\mathcal{L}\bigg{[}B_{t},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N},\mathbf{v},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\nabla_{\mathbf{x}}\mathbf{u}^{N},\eta^{N},\zeta,\begin{pmatrix}((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\\\
((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(0,{\mathbf{u}}^{N}-\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\end{pmatrix}\bigg{]}$

  2. (b)

$\tilde{\bm{\Theta}}^{N}$ converges $\,\tilde{\mathbb{P}}$-almost surely to
$\tilde{\bm{\Theta}}$ in the topology of $\chi$, i.e.

$\displaystyle\begin{aligned}
\tilde{B}_{t}^{N}&\rightarrow\tilde{B}_{t}\quad\mbox{in}\quad C(\overline{I})\
\tilde{\mathbb{P}}\mbox{-a.s.},\\\
\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{N}}}\tilde{\mathbf{u}}^{N},\,\tilde{\mathbf{v}}^{N}&\rightharpoonup\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}},\,\tilde{\mathbf{v}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\mathbb{I}_{\mathcal{O}_{\zeta^{N}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}^{N}&\rightharpoonup\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{N}&\rightharpoonup^{\ast}\tilde{\eta}\quad\mbox{in}\quad
L^{\infty}(I;W^{3,2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{N}&\rightharpoonup^{\ast}\tilde{\eta}\quad\mbox{in}\quad
W^{1,\infty}(I;L^{2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{N}&\rightharpoonup\tilde{\eta}\quad\mbox{in}\quad
W^{1,2}(I;W^{2,2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\zeta}^{N}&\rightarrow\tilde{\zeta}\quad\mbox{in}\quad
C(\overline{I}\times\Gamma)\ \tilde{\mathbb{P}}\mbox{-a.s.},\end{aligned}$
(3.35)

as well as (recalling the definition of $\tau_{\sharp}$ from (3.28))

$\displaystyle\begin{aligned} \int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}&\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{N}}}\mathbf{u}^{N}\cdot\mathscr{F}^{\tilde{\zeta}^{N}_{\epsilon}}_{N}\partial_{t}\tilde{\eta}^{N}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}^{N}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\\\
&\longrightarrow\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}\cdot\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\end{aligned}$
(3.36)

and

$\displaystyle\begin{aligned} \int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}&\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{N}}}\tilde{\mathbf{u}}^{N}\cdot(\tilde{\mathbf{u}}^{N}-\mathscr{F}^{\tilde{\zeta}^{N}_{\epsilon}}_{N}\partial_{t}\tilde{\eta}^{N})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\\
&\longrightarrow\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}\cdot(\tilde{\mathbf{u}}-\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\end{aligned}$
(3.37)

$\tilde{\mathbb{P}}$-a.s.

Now we introduce the filtration on the new probability space, which ensures
the correct measurabilities of the new random variables. Let
$(\tilde{\mathfrak{F}}_{t})_{t\geq 0}$ and
$(\tilde{\mathfrak{F}}_{t}^{N})_{t\geq 0}$ be the
$\tilde{\mathbb{P}}$-augmented canonical filtration on the variables
$\tilde{\bm{\Theta}}$ and $\tilde{\bm{\Theta}}^{N}$, respectively, that
is555Some of the variables are not continuous in time, for those one can
deifne $\sigma_{t}$ as the history of a random distribution, cf. [3, Chapter
2.8]

$\displaystyle\tilde{\mathfrak{F}}_{t}=\sigma\big{[}\sigma_{t}(\tilde{B}_{t})\cup\sigma_{t}(\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}})\cup\sigma_{t}(\tilde{\mathbf{v}})\cup\sigma_{t}(\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}})\cup\sigma_{t}(\tilde{\eta})\cup\sigma_{t}(\tilde{\zeta})\big{]}$

for $t\in I$ and similarly for $\tilde{\mathfrak{F}}_{t}^{N}$. By [3, Theorem
2.9.1] the weak equation continuous to hold on the new probability space.
Combining (5.13) and (5.14) we have

$\displaystyle\int_{I}\int_{\mathcal{O}_{\tilde{\zeta}^{N}_{\epsilon}}}$
$\displaystyle|\tilde{\mathbf{u}}^{N}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}^{N}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle\longrightarrow\int_{I}\int_{\mathcal{O}_{\tilde{\zeta}}}|\tilde{\mathbf{u}}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\tilde{\partial}_{t}\eta|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$

$\tilde{\mathbb{P}}$-a.s. By uniform convexity of the $L^{2}$-norm this
implies

$\displaystyle\partial_{t}\tilde{\eta}^{N}$
$\displaystyle\rightarrow\partial_{t}\tilde{\eta}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.}$

This is sufficient to pass to the limit in the weak formulation of the
equations (note that all terms except for the stochastic integral can be
treated by using (3.35)).

Since we have a linear problem at hand, we may now take appropriate
subsequence and pass to the limit in (3.10). Thus we obtain a martingale
solution to (3.5). In order to obtain a probabilistically strong solution we
prove in the following subsections pathwise uniqueness as well as convergence
in probability of the original sequence.

###  3.4. Pathwise uniqueness

We are now going to prove that any martingale solution satisfies the energy
equality (3.6). Pathwise uniqueness is then a direct consequence of the
linearity of the problem.

######  Proposition 3.6.

Suppose that $(\eta,\mathbf{u})$ is a weak martingale solution to (3.5). Then
it holds $\mathbb{P}$-a.s.

$\displaystyle\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}}}$
$\displaystyle|\mathbf{u}(t)|^{2}\,\mathrm{d}\mathbf{x}+\int_{0}^{t}\int_{\mathcal{O}_{\zeta_{\epsilon}}}|\nabla_{\mathbf{x}}\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\partial_{t}\eta(t)|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{y}}\eta(t)|^{2}+\epsilon|\nabla_{\mathbf{y}}^{3}\eta(t)|^{2}\Big{)}\,\mathrm{d}\mathbf{y}$
$\displaystyle+\epsilon\int_{0}^{t}\int_{\Gamma}|\partial_{s}\Delta_{\mathbf{y}}\eta|^{2}\,\mathrm{d}\mathbf{y}\,\mathrm{d}s=\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}(0)}}|{\mathbf{u}}_{0}|^{2}\,\mathrm{d}\mathbf{x}+\int_{\Gamma}\Big{(}\tfrac{1}{2}|\eta_{1}|^{2}+\tfrac{1}{2}|\Delta_{\mathbf{x}}\eta_{0}|^{2}+\epsilon|\nabla_{\mathbf{x}}^{3}\eta_{0}|\Big{)}\,\mathrm{d}\mathbf{y}$

for a.a. $t\in I$.

###### Proof.

We rewrite (3.5) as an equation for
$(\partial_{t}\bar{\eta},\bar{\mathbf{v}})$, where
$\bar{\eta}:=\int_{0}^{t}\iota_{\zeta_{\epsilon}}\partial_{t}\eta\,\,\mathrm{d}s$
and

$\bar{\mathbf{v}}:=\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}(|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|^{-1}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})\bar{\mathbf{u}}$

with $\bar{\mathbf{u}}:=\mathcal{J}_{\zeta_{\epsilon}}^{-1}\mathbf{u}$ which
reads as

$\displaystyle\,\mathrm{d}\bigg{\langle}\begin{pmatrix}\partial_{t}\bar{\eta}\\\
\bar{\mathbf{v}}\end{pmatrix}\cdot\begin{pmatrix}\phi\\\
\bm{\phi}\end{pmatrix}\bigg{\rangle}_{\mathbb{H}}$
$\displaystyle=\mathscr{L}\begin{pmatrix}\phi\\\
\bm{\phi}\end{pmatrix}+\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\zeta_{\varepsilon}}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}B$
(3.38)
$\displaystyle:=\int_{\Gamma}\big{(}\partial_{t}\eta\,\partial_{t}(\iota_{\zeta_{\epsilon}})\phi-\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}\phi\Delta_{\mathbf{y}}\eta-\epsilon\Delta_{\mathbf{y}}\iota_{\zeta_{\epsilon}}\phi\,\partial_{t}\Delta_{\mathbf{y}}\eta\big{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle-\epsilon\int_{\Gamma}\nabla_{\mathbf{y}}^{3}\iota_{\zeta_{\epsilon}}\phi:\nabla_{\mathbf{y}}^{3}\eta\,\mathrm{d}\mathbf{y}\,\mathrm{d}t+\int_{\Gamma}\bigg{(}\frac{1}{2}\mathbf{n}_{\zeta_{\epsilon}}\cdot\mathbf{n}^{\top}\iota_{\zeta_{\epsilon}}\phi\,\partial_{t}{\zeta_{\epsilon}}\,\partial_{t}\eta\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\bigg{)}\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}(t)}}\Big{(}\mathbf{u}\cdot\partial_{t}(\mathcal{J}_{\zeta_{\epsilon}(t)})\bm{\phi}-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u})\cdot(\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\int_{\mathcal{O}_{\zeta_{\epsilon}(t)}}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{J}_{\zeta_{\epsilon}(t)}\bm{\phi})\cdot\mathbf{u}-\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\varepsilon}(t)}\bm{\phi})\Big{)}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$
$\displaystyle+\frac{1}{2}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\zeta_{\varepsilon}}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}t-\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\iota_{\zeta_{\varepsilon}}\phi\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{t}$

where

$\displaystyle\mathbb{H}:=\big{\\{}(\xi,\mathbf{w})\in W^{2,2}(\Gamma)\times
W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O}):\,\mathbf{w}\circ\bm{\varphi}=\xi\mathbf{n}\,\,\text{on}\,\Gamma\big{\\}}.$

Note that $\mathbb{H}$ is a Hilbert space (as closed subset of the Hilbert
space $W^{2,2}(\Gamma)\times
W^{1,2}_{\mathrm{div}_{\mathbf{x}}}(\mathcal{O})$) and
$\mathscr{L}\in\mathbb{H}^{\prime}$. Hence we get from Itô’s formula in
Hilbert spaces (see [9, Theorem 4.17]) applied to the mapping

$\displaystyle
t\mapsto\tfrac{1}{2}\int_{\mathcal{O}}\bar{\mathbf{v}}\cdot|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top})^{-1}\bar{\mathbf{v}}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}\Big{|}\tfrac{\partial_{t}\bar{\eta}}{\iota_{\zeta_{\epsilon}}}\Big{|}^{2}\,\,\mathrm{d}\mathbf{y},$

noticing that
$|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top})^{-1}\bar{\mathbf{v}}=\bar{\mathbf{u}}$,

$\displaystyle\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}(t)}}|\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}$
$\displaystyle+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta|^{2}\,\mathrm{d}\mathbf{y}$
$\displaystyle=\tfrac{1}{2}\int_{\mathcal{O}}\bar{\mathbf{v}}\cdot|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top})^{-1}\bar{\mathbf{v}}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}\Big{|}\tfrac{\partial_{t}\bar{\eta}}{\iota_{\zeta_{\epsilon}}}\Big{|}^{2}\,\mathrm{d}\mathbf{y}$
$\displaystyle=\tfrac{1}{2}\int_{\mathcal{O}}\bar{\mathbf{v}}(0)\cdot\bar{\mathbf{u}}(0)\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\eta_{1}|^{2}\,\,\mathrm{d}\mathbf{y}+\int_{0}^{t}\mathscr{L}\begin{pmatrix}\iota_{\zeta_{\epsilon}}^{-2}\partial_{t}\bar{\eta}\\\
\bar{\mathbf{u}}\end{pmatrix}\,\,\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)\partial_{t}\eta\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{s}+\tfrac{1}{2}\int_{0}^{t}\int_{\Gamma}|(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta|^{2}\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}s$
$\displaystyle+\tfrac{1}{2}\int_{\mathcal{O}}\bar{\mathbf{v}}\cdot\partial_{t}\big{(}|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top})^{-1}\big{)}\bar{\mathbf{v}}\,\mathrm{d}\mathbf{x}-\int_{\Gamma}\tfrac{|\partial_{t}\bar{\eta}|^{2}}{\iota_{\zeta_{\epsilon}}^{3}}\partial_{t}\iota_{\zeta_{\epsilon}}\,\,\mathrm{d}\mathbf{y}.$

Noticing various cancellations such as

$\displaystyle\int_{\mathcal{O}}\bar{\mathbf{v}}\cdot\partial_{t}\big{(}|\mathrm{det}(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}})|(\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top}\nabla_{\mathbf{x}}\bm{\Psi}_{\zeta_{\epsilon}}^{\top})^{-1}\big{)}\bar{\mathbf{v}}\,\mathrm{d}\mathbf{x}$
$\displaystyle=\int_{\mathcal{O}_{\zeta_{\epsilon}(t)}}\mathbf{u}\cdot(\partial_{t}\mathcal{J}_{\zeta_{\epsilon}(t)}\bar{\mathbf{u}})\,\mathrm{d}\mathbf{x}$
$\displaystyle\int_{\Gamma}\tfrac{|\partial_{t}\bar{\eta}|^{2}}{\iota_{\zeta_{\epsilon}}^{3}}\partial_{t}\iota_{\zeta}\,\,\mathrm{d}\mathbf{y}$
$\displaystyle=\int_{\Gamma}\partial_{t}\eta\,(\partial_{t}\iota_{\zeta_{\varepsilon}})\iota_{\zeta_{\epsilon}}^{-2}\partial_{t}\bar{\eta}\,\mathrm{d}\mathbf{y}$

we obtain

$\displaystyle\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}(t)}}|\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\partial_{t}\eta|^{2}\,\mathrm{d}\mathbf{y}$
$\displaystyle=\tfrac{1}{2}\int_{\mathcal{O}_{\zeta_{\epsilon}(0)}}|\mathbf{u}_{0}|^{2}\,\mathrm{d}\mathbf{x}+\tfrac{1}{2}\int_{\Gamma}|\eta_{1}|^{2}\,\,\mathrm{d}\mathbf{y}$
$\displaystyle-\int_{\Gamma}\Delta_{\mathbf{y}}\iota_{\zeta_{\varepsilon}}\partial_{t}\bar{\eta}\,\Delta_{\mathbf{y}}\eta\,\mathrm{d}\mathbf{y}-\int_{0}^{t}\int_{\mathcal{O}_{\zeta}}\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}(\mathcal{J}_{\zeta_{\varepsilon}(t)}\bar{\mathbf{u}})\,\mathrm{d}\mathbf{x}\,\mathrm{d}s.$

The claim follows on noticing that
$\mathbf{u}=\mathcal{J}_{\zeta_{\varepsilon}(t)}\bar{\mathbf{u}}$ and
$\Delta_{\mathbf{y}}(\iota_{\zeta_{\varepsilon}}\partial_{t}\eta)=\partial_{t}\Delta_{\mathbf{y}}\bar{\eta}$.
∎

###  3.5. Convergence in probability

In order to complete the proof of Theorem 3.1, we make use of [3, Chapter 2,
Theorem 2.10.3] which is a generalization of the Gyöngy–Krylov
characterization of convergence in probability introduced in [18] to quasi-
Polish spaces. It applies to situations where pathwise uniqueness and
existence of a martingale solution are valid and allows to establish existence
of a probabilistically strong solution. We consider two sequences
$(N_{n}),(N_{m})\subset\mathbb{N}$ diverging to infinity. Let

${\bm{\vartheta}}^{N}:=\bigg{[}\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\nabla_{\mathbf{x}}{\mathbf{u}}^{N},\eta^{N},\begin{pmatrix}((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\\\
((\partial_{t}\eta^{N},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{N}),(\partial_{t}\eta^{N},\mathbf{u}^{N}-\mathscr{F}^{\zeta_{\epsilon}}_{N}\partial_{t}\eta^{N}))\end{pmatrix}\bigg{]},\quad
N\in\mathbb{N},$

and set

${\bm{\vartheta}}^{n}:={\bm{\vartheta}}^{N_{n}},\quad{\bm{\vartheta}}^{m}:={\bm{\vartheta}}^{N_{m}}.$

We also set $\mathbf{u}^{n}:=\mathbf{u}^{N_{n}}$, $\eta^{n}:=\eta^{N_{n}}$ and
$\mathbf{u}^{m}:=\mathbf{u}^{N_{m}},\eta^{m}:=\eta^{N_{m}}$. We consider the
collection of joint laws of
$({\bm{\vartheta}}^{n},{\bm{\vartheta}}^{m},\mathbf{v},\zeta,B_{t})$ on the
extended path space

$\chi_{\tt
ext}:=\big{(}\chi_{\mathbf{u}}\times\chi_{\nabla\mathbf{u}}\times\chi_{\eta}\times\chi_{f,g}^{2}\big{)}^{2}\times\chi_{\mathbf{u}}\times\chi_{\eta}\times\chi_{B}.$

Similarly to Lemma 3.4 we obtain tightness of

$\\{\mathcal{L}[{\bm{\vartheta}}^{n},{\bm{\vartheta}}^{m},\mathbf{v},\zeta,B_{t});\,n,m\in\mathbb{N}\\}$

on $\chi_{\tt ext}$.

Let us take any subsequence
$({\bm{\vartheta}}^{n_{k}},{\bm{\vartheta}}^{m_{k}},\mathbf{v},\zeta,B_{t})$.
By the Jakubowski-Skorokhod representation theorem we infer for a non-
relabelled subsequence the existence of a probability space
$(\bar{\Omega},\bar{\mathfrak{F}},\bar{\mathbb{P}})$ with a sequence of random
variables
$(\hat{\bm{\vartheta}}^{n_{k}},\check{\bm{\vartheta}}^{m_{k}},\overline{\mathbf{v}}^{k},\overline{\zeta}^{k},\overline{B}_{t}^{k})$
converging almost surely in $\chi_{\tt ext}$ to a random variable
$(\hat{\bm{\vartheta}},\check{\bm{\vartheta}},\overline{\mathbf{v}},\overline{\zeta},\overline{B}_{t})$.
Moreover,

$\mathcal{L}[\hat{\bm{\vartheta}}^{n_{k}},\check{\bm{\vartheta}}^{m_{k}},\overline{\mathbf{v}}^{k},\overline{\zeta}^{k},\overline{B}_{t}^{k}]=\mathcal{L}[{\bm{\vartheta}}^{n_{k}},{\bm{\vartheta}}^{m_{k}},\mathbf{v}^{k},\zeta^{k},B_{t}^{k}]$

on $\chi_{\tt ext}$ for all $k\in\mathbb{N}$. Observe that, in particular,
$\mathcal{L}[\hat{\bm{\vartheta}}^{n_{k}},\check{\bm{\vartheta}}^{m_{k}},\overline{\mathbf{v}}^{k},\overline{\zeta}^{k},\overline{B}_{t}^{k}]$
converges weakly to the measure
$\mathcal{L}[\hat{\bm{\vartheta}},\check{\bm{\vartheta}},\overline{\mathbf{v}},\overline{\zeta},\overline{B}_{t}]$.
As in Section 3.3 we can show that the limit objects are martingale solutions
to (3.5) defined on the same stochastic basis
$(\bar{\Omega},\bar{\mathfrak{F}},(\bar{\mathfrak{F}}_{t}),\bar{\mathbb{P}})$,
where $(\bar{\mathfrak{F}}_{t})_{t\geq 0}$ is the $\bar{\mathbb{P}}$-augmented
canonical filtration of
$(\hat{\bm{\vartheta}},\check{\bm{\vartheta}},\overline{\mathbf{v}},\overline{\zeta},\overline{B}_{t})$.
We employ the pathwise uniqueness result from Proposition 3.6. Therefore, the
solutions $(\hat{\eta},\hat{\mathbf{u}})$ and
$(\check{\eta},\check{\mathbf{u}})$ coincide $\bar{\mathbb{P}}$-a.s. and we
have

$\displaystyle\mathcal{L}$
$\displaystyle[\hat{\bm{\vartheta}},\check{\bm{\vartheta}},\overline{\mathbf{v}},\overline{\zeta},\overline{B}_{t}]\Big{(}({\bm{\vartheta}}_{1},{\bm{\vartheta}}_{2},\mathbf{v},\zeta,B_{t})\in\chi_{\tt
ext}:\;(\eta_{1},\mathbf{u}_{1})=(\eta_{2},\mathbf{u}_{2})\Big{)}=\bar{\mathbb{P}}((\hat{\eta},\hat{\mathbf{u}})=(\check{\eta},\check{\mathbf{u}}))=1.$

Now, we have all in hand to apply the Gyöngy–Krylov theorem from [3, Chapter
2, Theorem 2.10.3]. It implies that the original sequence
${\bm{\vartheta}}^{N}$ defined on the initial probability space
$(\Omega,\mathfrak{F},\mathbb{P})$ converges in probability in the topology of
$\chi_{\mathbf{u}}\times\chi_{\nabla\mathbf{u}}\times\chi_{\eta}\times\chi_{f,g}^{2}$
to the random variable

${\bm{\vartheta}}:=\bigg{[}\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\nabla_{\mathbf{x}}{\mathbf{u}},\eta,\begin{pmatrix}((\partial_{t}\eta,\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}),(\partial_{t}\eta,\mathscr{F}^{\zeta_{\epsilon}}\partial_{t}\eta))\\\
((\partial_{t}\eta,\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}),(0,\mathbf{u}-\mathscr{F}^{\zeta_{\epsilon}}\partial_{t}\eta))\end{pmatrix}\bigg{]}.$

Therefore, we finally deduce that $(\eta,\mathbf{u})$ is a probabilistically
strong solution to (3.5). This completes the proof of Theorem 3.2.

##  4\. The nonlinear regularised problem

The aim of this section is to obtain a solution of the regularised problem
thus completing the proof of Theorem 3.1. This is done via a fixed point
argument for which the main point is proving compactness of the mapping
$(\zeta,\mathbf{v})\mapsto(\eta,\mathbf{u})$. Given a bounded sequence
$(\zeta_{n},\mathbf{v}_{n})$ this is achieved in three steps:

  * •

We first need to establish tightness of the probability laws, where the main
difficulty arises for the velocity field.

  * •

We apply the stochastic compactness method based on Jakubowski’s extension of
the Skorokhod representation theorem. This yields a.s. convergence on a new
probability space.

  * •

We apply a Gyöngy-Krylov type argument to obtain convergence in probability of
the original sequence on the original probability space. This requires the
pathwise uniqueness from Proposition 3.6.

Suppose there is a sequence of processes $(\zeta^{n},\mathbf{v}^{n})$ which
are $(\mathfrak{F}_{t})$-progressively measurable and bounded in

$L^{p}(\Omega;C(\overline{I}\times\Gamma))\times
L^{p}(\Omega;L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha})))$

for some sufficiently large $p$. Now apply Theorem 3.2 yielding a sequence
$(\eta^{n},\mathbf{u}^{n})$ of solutions to (3.5). By the energy equality from
Proposition 3.6 we obtain

$\displaystyle\sup_{I}\|\eta^{n}\|_{W^{2,2}(\Gamma)}^{2}+\epsilon\sup_{I}\|\eta^{n}\|_{W^{3,2}(\Gamma)}^{2}\lesssim
1,$ (4.1)
$\displaystyle\sup_{I}\|\partial_{t}\eta^{n}\|_{L^{2}(\Gamma)}^{2}\lesssim 1,$
(4.2)
$\displaystyle\epsilon\int_{I}\|\partial_{t}\eta^{n}\|_{W^{2,2}(\Gamma)}^{2}\,\mathrm{d}t\lesssim
1,$ (4.3)
$\displaystyle\sup_{I}\|\mathbf{u}^{n}\|_{L^{2}(\mathcal{O}_{\zeta^{n}_{\epsilon}})}^{2}\lesssim
1,$ (4.4)
$\displaystyle\int_{I}\|\nabla_{\mathbf{x}}\mathbf{u}^{n}\|_{L^{2}(\mathcal{O}_{\zeta^{n}_{\epsilon}})}^{2}\,\mathrm{d}t\lesssim
1.$ (4.5)

In addition, for any $s\in(0,\frac{1}{2})$, it follows from
$\mathbf{u}^{n}\circ\bm{\varphi}_{\zeta_{\epsilon}^{n}}=\mathbf{n}\partial_{t}\eta^{n}$,
(4.5) and the trace theorem that

$\displaystyle\int_{I}\|\partial_{t}\eta^{n}\|_{W^{s,2}(\Gamma)}^{2}\,\mathrm{d}t\lesssim
1$ (4.6)

holds.

###  4.1. Tightness of the velocity sequence

The effort of this subsection is to prove tightness of the law of
$\mathbf{u}^{n}$. Similarly to Lemma 3.3 we obtain the following result.

######  Lemma 4.1.

The laws of

$\displaystyle((\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n}),(\partial_{t}\eta^{n},\mathscr{F}^{\zeta^{n}_{\epsilon}}\partial_{t}\eta^{n}))\quad\text{and}\quad((\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}^{n}}}\mathbf{u}^{n}),(0,\mathbf{u}^{n}-\mathscr{F}^{\zeta_{\epsilon}^{n}}\partial_{t}\eta^{n}))$

on $(L^{2}(I;X^{\prime}\times X),\tau_{\sharp})$ are tight.

###### Proof.

As in Lemma 3.3 we must verify the assumptions from Theorem 2.2. First of all,
boundedness in $L^{2}(I;X^{\prime}\times X)$ follows from By (4.1)–(4.6) and
the properties of the extension $\mathscr{F}^{\zeta_{n}}$ from Proposition
2.3. For $(b)$ we consider again the regularisation of $\partial_{t}\eta^{n}$
with parameter $\kappa>0$ and set for $n\in\mathbb{N}$

$f_{n,\kappa}(t):=((\partial_{t}\eta^{n}(t))_{\kappa},\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}}(\partial_{t}\eta^{n}(t))),$

where $\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}}$ is given in Corollary 2.4.
We find by the continuity of the mollification operator from (3.29) and the
continuity of $\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}}$ from Corollary 2.4
that for a.e. $t\in(0,T)$ and $s<1/2$

$\displaystyle\|f_{n,\kappa}-f_{n}\|_{L^{2}(\Gamma)\times
L^{2}(\mathcal{O}\cup
S_{\alpha})}\leq\,c\kappa^{s}\|\partial_{t}\eta^{n}\|_{W^{s,2}(\Gamma)},$
(4.7)

which can be made arbitrarily small in $L^{2}$ by choosing $\kappa$
appropriately, cf. (4.6). Similarly, we have

$\displaystyle\|f_{n,\kappa}\|_{W^{1,2}(\Gamma)\times W^{1,2}(\Omega\cup
S_{\alpha})}\leq\,c\kappa^{-1}\|\partial_{t}\eta^{n}\|_{L^{2}(\Gamma)}.$

Clearly, if $f_{N}\rightharpoonup f$ in $L^{2}(I;X)$ then we can deduce a
converging subsequence such that $f_{N,\kappa}\rightharpoonup f_{\kappa}$ (for
some $f_{\kappa}$) in $L^{2}(I;X)$ for any $\kappa>0$, which implies (b).

For (c) we have to control $\langle g_{n}(t)-g_{n}(s),f_{n,\kappa}(t)\rangle$,
where
$g_{n}:=(\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n})$
and hence decompose

$\displaystyle\langle g_{n}(t)-g_{n}(s),f_{n,\kappa}(t)\rangle$
$\displaystyle=\big{(}\langle
g_{n}(t),(\partial_{t}\eta^{n}(t))_{\kappa},\mathcal{E}_{\kappa}^{\zeta^{n}_{\epsilon}(t)}(\partial_{t}\eta^{n}(t)))\rangle$
$\displaystyle\quad-\langle
g_{n}(s),(\partial_{t}\eta_{n}(t))_{\kappa},\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(s)}(\partial_{t}\eta^{n}(t))\rangle\big{)}$
$\displaystyle\quad+\langle
g_{N}(s),(0,\mathcal{E}_{\kappa}^{\zeta^{n}_{\epsilon}(t)}(\partial_{t}\eta^{n}(t)))-\mathcal{E}_{\kappa}^{\zeta^{n}_{\epsilon}(s)}(\partial_{t}\eta^{n}(t))\rangle=:(I)+(II).$

We begin estimating $(II)$ to find that

$\displaystyle(II)$
$\displaystyle=\int_{\mathcal{O}_{\zeta_{\epsilon}^{n}(s)}}\mathbf{u}^{n}(s)\cdot\int_{t}^{s}\partial_{r}\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(r)}(\partial_{t}\eta^{n}(t))\,\,\mathrm{d}r\,\mathrm{d}\mathbf{x}$
$\displaystyle\leq\,c\|\mathbf{u}^{n}(s)\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}^{n}(s)})}\|\partial_{t}(\zeta_{\epsilon}^{n})_{\kappa}\|_{L^{\infty}(I\times\Gamma)}\|\partial_{t}\eta^{n}(t)\|_{L^{\infty}(I;L^{2}(\Gamma))}|t-s|$
$\displaystyle\leq\,c(\kappa)\|\mathbf{u}^{n}(s)\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}^{n}(s)})}\|\partial_{t}\zeta_{\epsilon}^{n}\|_{L^{\infty}(I;L^{2}(\Gamma)}\|\partial_{t}\eta^{n}(t)\|_{L^{\infty}(I;L^{2}(\Gamma))}|t-s|.$

using Corollary 2.4. By (4.2) and (4.4) we thus get

$\displaystyle|(II)|$
$\displaystyle\leq\,c(\kappa)\|\partial_{t}\zeta_{\epsilon}^{n}\|_{L^{\infty}(I;L^{2}(\Gamma)}|t-s|$
$\displaystyle\leq\,c(\kappa,\epsilon)\|\zeta^{n}\|_{L^{\infty}(I;L^{2}(\Gamma)}|t-s|.$

As in the proof of Lemma 3.3 we conclude for $K>0$ that

$\displaystyle\mathbb{P}\big{(}|t-s|^{-1}|(II)|\geq K\big{)}$
$\displaystyle\leq\,\frac{c(\kappa,\epsilon)}{K},$

provided $\zeta\in L^{1}(\Omega;L^{\infty}(I;L^{2}(\Gamma))$. The term $(I)$
is estimated using (3.5) obtaining (this can be justified by Itô’s formula
similarly to the proof of Proposition 3.6)

$\displaystyle(I)$
$\displaystyle=-\int_{t}^{s}\int_{\Gamma}\big{(}\epsilon\Delta_{\mathbf{y}}(\partial_{t}\eta^{n}(t))_{\kappa}\,\partial_{t}\Delta_{\mathbf{y}}\eta^{n}+\Delta_{\mathbf{y}}(\partial_{t}\eta^{n}(t))_{\kappa}\,\Delta_{\mathbf{y}}\eta^{n}\big{)}\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}r$
(4.8)
$\displaystyle-\epsilon\int_{t}^{s}\int_{\Gamma}\nabla_{\mathbf{y}}^{3}(\partial_{t}\eta^{n}(t))_{\kappa}:\nabla_{\mathbf{y}}^{3}\eta^{n}\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}r+\int_{t}^{s}\int_{\Gamma}\frac{1}{2}\mathbf{n}_{\zeta^{n}_{\epsilon}}\cdot\mathbf{n}(\partial_{t}\eta^{n}(t))_{\kappa}\,\partial_{t}\zeta_{\epsilon}\,\partial_{t}\eta^{n}\,|\det(\nabla_{\mathbf{y}}\bm{\varphi}_{\zeta_{\epsilon}})|\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\mathcal{O}_{\zeta_{\epsilon}}}\Big{(}\mathbf{u}\cdot\partial_{r}\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(r)}(\partial_{t}\eta^{n}(t)))-\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathbf{u})\cdot\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(r)}(\partial_{t}\eta^{n}(t))\Big{)}\,\mathrm{d}\mathbf{x}\,\,\mathrm{d}r$
$\displaystyle+\int_{t}^{s}\int_{\mathcal{O}_{\zeta}}\Big{(}\frac{1}{2}((\mathbf{v}_{\epsilon}\cdot\nabla_{\mathbf{x}})\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(r)}(\partial_{t}\eta^{n}(t)))\cdot\mathbf{u}-\nabla_{\mathbf{x}}\mathbf{u}:\nabla_{\mathbf{x}}\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(r)}(\partial_{t}\eta^{n}(t)))\Big{)}\,\mathrm{d}\mathbf{x}\,\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})(\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta)(\partial_{t}\eta^{n}(t))_{\kappa}\,\mathrm{d}\mathbf{y}\,\,\mathrm{d}r-\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{n})(\partial_{t}\eta^{n}(t))_{\kappa}\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{r}.$

Due to (3.29), Corollary 2.4 and the uniform estimates (4.1)–(4.6), it is
clear that all terms can be estimated by
$c(\kappa)|t-s|^{1/2}(1+\|\zeta^{n}\|_{C(I\times\Gamma)}+\|\nabla_{\mathbf{x}}\mathbf{u}^{n}\|_{L^{2}(\mathcal{O}_{\zeta^{n}_{\epsilon}})})$
except for the last one. As in the proof of Lemma 3.3 it can be controlled via

$\displaystyle\mathbb{P}\bigg{(}|t-s|^{-1/2}\bigg{|}$
$\displaystyle\int_{t}^{s}\int_{\Gamma}((\bm{\varkappa}\cdot\nabla_{\mathbf{y}})\partial_{t}\eta^{n})(\partial_{t}\eta^{n})_{\kappa}\,\mathrm{d}\mathbf{y}\,\mathrm{d}B_{r}\bigg{|}>K\bigg{)}$
$\displaystyle\lesssim\,\frac{\kappa^{-1}}{K^{2}}\mathbb{E}\bigg{[}|t-s|^{-1/2}\int_{t}^{s}\|\partial_{t}\eta^{n}(r)\|_{L^{2}(\Gamma)}\|\nabla_{\mathbf{y}}(\partial_{t}\eta^{n}(t))_{\kappa}\|_{L^{2}(\Gamma)}\,\,\mathrm{d}r\bigg{]}$
$\displaystyle\lesssim\,\frac{1}{K^{2}}\mathbb{E}\bigg{[}|t-s|^{-1/2}\int_{t}^{s}\|\partial_{t}\eta^{n}(r)\|_{L^{2}(\Gamma)}\|\partial_{t}\eta^{n}(t)\|_{L^{2}(\Gamma)}\,\,\mathrm{d}r\bigg{]}\lesssim\,\frac{\kappa^{-1}}{K^{2}}$

using (4.2). This can be made arbitrarily small for $K$ large. Noticing that
property $(4)$ from Theorem 2.2 follows by the usual compactness in (negative)
Sobolev spaces, we conclude tightness of the law of
$((\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n}),(\partial_{t}\eta^{n},\mathscr{F}^{\zeta^{n}_{\epsilon}}\eta^{N}))$
on $(L^{2}(I;X^{\prime}\times X),\tau_{\sharp})$.

The tightness for
$((\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n}),(0,\mathbf{u}^{n}-\mathscr{F}^{\zeta_{\epsilon}^{n}}\partial_{t}\eta^{n}))$
follows along the same lines, the only difference being the regularisation of

$f_{n}:=(0,\mathbf{u}^{n}-\mathscr{F}^{\zeta_{\epsilon}^{n}}\partial_{t}\eta^{n}).$

While $\mathscr{F}^{\zeta_{\epsilon}^{n}}\partial_{t}\eta^{n}$ can be replaced
by $\mathcal{E}_{\kappa}^{\zeta_{\epsilon}^{n}(s)}(\partial_{t}\eta^{n}(t))$
as above, we need to regularise $\mathbf{u}^{n}$ accordingly to preserve the
homogeneous boundary conditions on $f_{n}$, which can by done by using [25,
Proposition 2.28 and Lemma A.13]. This leads to a function $f_{n,\kappa}$
which has zero boundary conditions and thus only the fluid part is seen in the
expression $\langle g_{n}(t)-g_{n}(s),f_{n,\kappa}(t)\rangle$ (where
$g_{n}:=(\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n})$
as above). In particular, the noise is not seen and we conclude that

$\displaystyle\langle
g_{n}(t)-g_{n}(s),f_{n,\kappa}(t)\rangle\leq\,c(\kappa)|t-s|^{1/2}(1+\|\zeta^{n}\|_{C(\overline{I}\times\Gamma)}+\|\nabla_{\mathbf{x}}\mathbf{u}^{n}\|_{L^{2}(\mathcal{O}_{\zeta_{\epsilon}^{n}})})$

and we obtain again the claimed tightness. ∎

###  4.2. Passage to the limit

With the bounds from (4.1)–(4.6) at hand, we wish to obtain compactness. We
consider the same path space $\chi$ as in Section 3.3. From (3.16)–(3.21)
(together the Alaoglu-Burbaki Theorem) and Lemma 3.3 we obtain similarly to
Proposition 3.5:

######  Proposition 4.2.

There exists a complete probability space
$(\tilde{\Omega},\tilde{\mathfrak{F}},\tilde{\mathbb{P}})$ with $\chi$-valued
random variables

$\tilde{\bm{\Theta}}^{n}:=\bigg{[}\tilde{B}_{t}^{n},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\tilde{\mathbf{u}}^{n},\tilde{\mathbf{v}}^{n},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}^{n},\tilde{\eta}^{n},\tilde{\zeta}^{n},\begin{pmatrix}((\partial_{t}\tilde{\eta}^{n},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\tilde{\mathbf{u}}^{n}),(\partial_{t}\tilde{\eta}^{n},\mathscr{F}^{\tilde{\zeta}^{n}_{\epsilon}}\partial_{t}\tilde{\eta}^{n}))\\\
((0,\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\tilde{\mathbf{u}}^{n}),(\partial_{t}\tilde{\eta}^{n},\tilde{\mathbf{u}}^{n}-\mathscr{F}^{\tilde{\zeta}^{n}_{\epsilon}}\partial_{t}\tilde{\eta}^{n}))\end{pmatrix}\bigg{]},\quad
n\in\mathbb{N},$

and

$\tilde{\bm{\Theta}}:=\bigg{[}\tilde{B}_{t},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}^{n},\tilde{\mathbf{v}},\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}},\tilde{\eta},\tilde{\zeta},\begin{pmatrix}((\partial_{t}\eta,\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}\mathbf{u}^{n}),(\partial_{t}\tilde{\eta},\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta}))\\\
((0,\mathbb{I}_{\mathcal{O}_{\tilde{\eta}}}\tilde{\mathbf{u}}),(\partial_{t}\tilde{\eta},\tilde{\mathbf{u}}-\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta}))\end{pmatrix}\bigg{]}$

such that

  1. (a)

for all $n\in\mathbb{N}$ the law of $\tilde{\bm{\Theta}}^{n}$ on $\chi$ is
given by

$\displaystyle\mathcal{L}\bigg{[}B_{t},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n},\mathbf{v}^{n},\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}^{n}}}\nabla_{\mathbf{x}}\mathbf{u}^{n},\eta^{n},\zeta^{n},\begin{pmatrix}((\partial_{t}\eta^{n},\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n}),(\partial_{t}\eta^{n},\mathscr{F}^{\zeta^{n}}\partial_{t}\eta^{n}))\\\
((0,\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}\mathbf{u}^{n}),(\partial_{t}\eta^{n},\mathbf{u}^{n}-\mathscr{F}^{\zeta^{n}_{\epsilon}}\partial_{t}\eta^{n}))\end{pmatrix}\bigg{]}$

  2. (b)

$\tilde{\bm{\Theta}}^{n}$ converges $\,\tilde{\mathbb{P}}$-almost surely to
$\tilde{\bm{\Theta}}$ in the topology of $\chi$, i.e.

$\displaystyle\begin{aligned}
\tilde{B}_{t}^{n}&\rightarrow\tilde{B}_{t}\quad\mbox{in}\quad C(\overline{I})\
\tilde{\mathbb{P}}\mbox{-a.s.},\\\
\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{n}}}\tilde{\mathbf{u}}^{n},\,\tilde{\mathbf{v}}^{n}&\rightharpoonup\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}},\,\tilde{\mathbf{v}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}^{n}&\rightharpoonup\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{n}&\rightharpoonup^{\ast}\tilde{\eta}\quad\mbox{in}\quad
L^{\infty}(I;W^{3,2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{n}&\rightharpoonup^{\ast}\tilde{\eta}\quad\mbox{in}\quad
W^{1,\infty}(I;L^{2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\eta}^{n}&\rightharpoonup\tilde{\eta}\quad\mbox{in}\quad
W^{1,2}(I;W^{2,2}(\Gamma))\ \tilde{\mathbb{P}}\mbox{-a.s.},\\\
\tilde{\zeta}^{n}&\rightarrow\tilde{\zeta}\quad\mbox{in}\quad
W^{1,2}(I\times\Gamma)\ \tilde{\mathbb{P}}\mbox{-a.s.},\end{aligned}$ (4.9)

as well as (recalling the definition of $\tau_{\sharp}$ from (3.28))

$\displaystyle\begin{aligned} \int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}&\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{n}}}\mathbf{u}^{n}\cdot\mathscr{F}^{\tilde{\zeta}^{n}_{\epsilon}}\partial_{t}\tilde{\eta}^{n}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}^{n}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\\\
&\longrightarrow\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}\cdot\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\eta\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t\end{aligned}$
(4.10)

and

$\displaystyle\begin{aligned} \int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}&\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}^{n}}}\tilde{\mathbf{u}}^{n}\cdot(\tilde{\mathbf{u}}^{n}-\mathscr{F}^{\tilde{\zeta}^{n}_{\epsilon}}\partial_{t}\tilde{\eta}^{n})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\\
&\longrightarrow\int_{I}\int_{\mathcal{O}\cup
S_{\alpha}}\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}\cdot(\tilde{\mathbf{u}}-\mathscr{F}^{\tilde{\zeta}_{\epsilon}}\partial_{t}\tilde{\eta})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\end{aligned}$
(4.11)

$\tilde{\mathbb{P}}$-a.s.

Now we introduce the filtration on the new probability space, which ensures
the correct measurabilities of the new random variables. Let
$(\tilde{\mathfrak{F}}_{t})_{t\geq 0}$ and
$(\tilde{\mathfrak{F}}_{t}^{n})_{t\geq 0}$ be the
$\tilde{\mathbb{P}}$-augmented canonical filtration on the variables
$\tilde{\bm{\Theta}}$ and $\tilde{\bm{\Theta}}^{n}$, respectively, that
is666Some of the variables are not continuous in time. For those, one can
define $\sigma_{t}$ as the history of a random distribution, cf. [3, Chapter
2.8]

$\displaystyle\tilde{\mathfrak{F}}_{t}=\sigma\big{[}\sigma_{t}(\tilde{B}_{t})\cup\sigma_{t}(\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}})\cup\sigma_{t}(\tilde{\mathbf{v}})\cup\sigma_{t}(\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\nabla_{\mathbf{x}}\tilde{\mathbf{u}})\cup\sigma_{t}(\tilde{\eta})\cup\sigma_{t}(\tilde{\zeta})\big{]}$

for $t\in I$ and similarly for $\tilde{\mathfrak{F}}_{t}^{n}$.

By [3, Theorem 2.9.1] the weak equation continuous to hold on the new
probability space. Combining (4.10) and (4.11) we have

$\displaystyle\int_{I}\int_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}$
$\displaystyle|\tilde{\mathbf{u}}^{n}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}^{n}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$
$\displaystyle\longrightarrow\int_{I}\int_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}|\tilde{\mathbf{u}}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_{I}\int_{\Gamma}|\partial_{t}\tilde{\eta}|^{2}\,\,\mathrm{d}\mathbf{y}\,\mathrm{d}t$

$\tilde{\mathbb{P}}$-a.s. By uniform convexity of the $L^{2}$-norm this
implies

$\displaystyle\tilde{\eta}^{n}$
$\displaystyle\rightarrow\tilde{\eta}\quad\mbox{in}\quad
W^{1,2}(I;L^{2}(\Gamma)),$
$\displaystyle\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}^{n}_{\epsilon}}}\tilde{\mathbf{u}}^{n}$
$\displaystyle\rightarrow\mathbb{I}_{\mathcal{O}_{\tilde{\zeta}_{\epsilon}}}\tilde{\mathbf{u}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\ \tilde{\mathbb{P}}\mbox{-a.s.}.$

We can now apply Proposition 3.6 and argue as in Section 3.5 to obtain
convergence of the original sequence, in particular,

$\displaystyle\eta^{n}$ $\displaystyle\rightarrow\eta\quad\mbox{in}\quad
C(\overline{I}\times\Gamma)\quad\mbox{in probability},$
$\displaystyle\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}{\mathbf{u}}^{n}$
$\displaystyle\rightarrow\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}{\mathbf{u}}\quad\mbox{in}\quad
L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))\quad\mbox{in probability}.$

This yields due the uniform-in-probability estimates (for arbitrary
$p<\infty$)

$\displaystyle\eta^{n}$ $\displaystyle\rightarrow\eta\quad\mbox{in}\quad
L^{p}(\Omega;C(\overline{I}\times\Gamma)),$
$\displaystyle\mathbb{I}_{\mathcal{O}_{\zeta^{n}_{\epsilon}}}{\mathbf{u}}^{n}$
$\displaystyle\rightarrow\mathbb{I}_{\mathcal{O}_{\zeta_{\epsilon}}}{\mathbf{u}}\quad\mbox{in}\quad
L^{p}(\Omega;L^{2}(I;L^{2}(\mathcal{O}\cup S_{\alpha}))),$