# A 2D stochastic Fluid-structure interaction problem in compliant arteries
with non-zero longitudinal displacement

Krutika Tawri

###### Abstract.

We give the existence of martingale solutions to a stochastic fluid-structure
interaction problem involving a viscous, incompressible fluid flow, modeled by
the Navier-Stokes equations, through a deformable elastic tube modeled by
shell/membrane equations. The fluid and the structure are nonlinearly coupled.
In this article we consider the case where the structure is allowed to have
non-zero longitudinal displacement.

##  1\. Introduction

This paper introduces a constructive approach for investigating solutions to a
complex problem describing the interaction between a deformable elastic
membrane or shell and a two-dimensional viscous, incompressible fluid flow,
both subject to multiplicative stochastic forces. There has been a lot of work
done in the field of deterministic FSI in the past two decades. However, even
though there is a lot of evidence pointing to the need for studying their
stochastic perturbations, the mathematical theory of stochastic FSI or, more
generally, of stochastic PDEs on randomly moving domains is completely
undeveloped.

The fluid is described using the 2D Navier-Stokes equations, while the
membrane or shell is characterized by linearly elastic equations. The fluid
and the structure are fully coupled across the moving interface through a two-
way coupling that ensures continuity of their velocities and contact forces at
their interface. We consider the more difficult case of a purely elastic
structure however our analysis can easily include a visco-elastic structure
(e.g. Koiter shell).

The main result of this paper is the establishment of the existence of weak
martingale solutions to this highly nonlinear stochastic fluid-structure
interaction problem. The recent article [11], provided the existence of weak
martingale solutions to a stochastic FSI problem where the structure
displacement is assumed only in the radial (vertical) direction. In this paper
we remove this restriction and consider a thin structure which is allowed to
be displaced longitudinally. Dealing with the non-zero longitudinal
displacement is not trivial and it introduces several mathematical
difficulties. Due to the non-zero longitudinal displacement, extra care has to
be taken to deal with degenerate fluid domains i.e. when the structure touches
a part of the fluid domain boundary during deformation. The incompressibility
condition pose more challenges in terms of the construction of test functions,
which depend on the random fluid domain configurations. To the best our
knowledge, this work is the first existence result involving non-zero
longitudinal structure displacement for a stochastic moving boundary problem.

The main new component of the present work is the treatment of the geometric
nonlinearity associated with the stochastic fluid domain motion. The main
mathematical issues are related to the following: (1) the fluid domain
boundary is a random variable, not known a priori, (2) the fluid domain can
possibly degenerate in a random fashion and (3) the incompressibility
condition give rise to the difficulties in the construction of appropriate
test functions on stochastic moving domains. While similar issues arise in the
deterministic case as well, their resolution in the stochastic case is
markably different. The use of the ALE mappings and the analysis that follows
is valid for as long as there is no loss of injectivity of the Jacobian of the
ALE transformation. Handling this injectivity condition in the stochastic case
requires a delicate approach, which in this manuscript is done using a cut-off
function and a stopping time argument. The cut-off function artificially
provides a deterministic lower bound on the Jacobian of the ALE maps and a
deterministic upper bound on the $H^{s}$-norm (large enough $s<2$) of the
structure displacement.

Furthermore, the incompressibility condition requires that we construct random
test functions that depend on the domain configurations. This creates problems
as we apply, for example, the Skorohod representation theorem and move to a
new probability space to upgrade convergence results. Hence we introduce a
system that approximates the original system by augmenting it by a singular
term that penalizes the divergence and the boundary behavior of the fluid
velocity. We show that the solutions to this approximate system indeed
converge to a desired solution of the limiting equations.

Since the stochastic forcing appears not only in the structure equations but
also in the fluid equations themselves, we come across additional
difficulties, which are associated with the construction of the appropriate
”test processes” on the approximate and limiting moving fluid domains. Namely,
along with the required divergence-free property on these domains, the test
functions also have to satisfy appropriate boundary conditions and
measurability properties. This is new. We construct these approximate test
functions by first constructing a Carathéodory function that gives the
definition of a test function for the limiting equations and then by extending
appropriate boundary values on the approximate domains in a way that desired
properties are preserved.

The article is organized as follows: In Section 2 we describe the problem,
give weak formulations in both the deterministic and stochastic cases and give
the definition of solutions. The rest of the article is devoted to the proof
of the main result Theorem, 4, that provides the existence of the desired
martingale solutions.

##  2\. Problem setup

We begin describing the problem by first considering the deterministic model.

###  2.1. The deterministic model and a weak formulation

We consider the flow of an incompressible, viscous fluid in a two-dimensional
compliant cylinder $\mathcal{O}=(0,L)\times(0,1)$ with a deformable lateral
boundary $\Gamma$. The left and the right boundary of the cylinder are the
inlet and outlet for the time-dependent fluid flow. We assume “axial symmetry”
of the data and of the flow, rendering the central horizontal line as the axis
of symmetry. This allows us to consider the flow only in the upper half of the
domain, with the bottom boundary fixed and equipped with the symmetry boundary
conditions.

Assume that the time-dependent fluid domain, whose displacement is not known a
priori is denoted by
$\mathcal{O}_{{\boldsymbol{\eta}}}(t)={\boldsymbol{\varphi}}(t,\mathcal{O})$
whereas its deformable interface is given by
$\Gamma_{\boldsymbol{\eta}}(t)={\boldsymbol{\varphi}}(t,\Gamma)$. Assume that
${\boldsymbol{\varphi}}:\mathcal{O}\rightarrow\mathcal{O}_{{\boldsymbol{\eta}}}$
is a $C^{1}$ diffeomorphism such that

${\boldsymbol{\varphi}}|_{\Gamma_{in},\Gamma_{out},\Gamma_{out}}=id,\quad
det\nabla{\boldsymbol{\varphi}}(t,{\bf x})>0,$

where the inlet, outlet and bottom boundaries of $\mathcal{O}$ are given by
$\Gamma_{\text{in}}=\\{0\\}\times(0,1),\Gamma_{\text{out}}=\\{L\\}\times(0,1),\Gamma_{b}=(0,L)\times\\{0\\}$
respectively. The displacement of the elastic structure at the top lateral
boundary $\Gamma$ can be identified by $(0,L)$ will be given by
${\boldsymbol{\eta}}(t,z)={\boldsymbol{\varphi}}(t,z)-(z,1)$ for $z\in(0,L)$
(see Fig. 1). The mapping
${{\boldsymbol{\eta}}}:[0,L]\times[0,T]\rightarrow\mathbb{R}^{2}$ such that
${\boldsymbol{\eta}}=({\eta}_{z}(z,t),{\eta}_{r}(z,t))$ is one of the unknowns
in the problem.

Figure 1. A realization of the domain

We use the convention that bold-faced letters denote spaces containing vector-
valued functions.

The fluid subproblem: The fluid flow is modeled by the incompressible Navier-
Stokes equations in the 2D time-dependent domain
$\mathcal{O}_{{\boldsymbol{\eta}}}(t)$

(1) $\begin{split}\partial_{t}{\bf u}+({\bf u}\cdot\nabla){\bf
u}&=\nabla\cdot\sigma+F^{ext}_{u}\\\ \nabla\cdot{\bf u}&=0,\end{split}$

where ${\bf u}=(u_{z},u_{r})$ is the fluid velocity. The Cauchy stress tensor
is $\sigma=-pI+2\nu{\bf D}({\bf u})$ where $p$ is the fluid pressure, $\nu$ is
the kinematic viscosity coefficient and ${\bf D}({\bf
u})=\frac{1}{2}(\nabla{\bf u}+(\nabla{\bf u})^{T})$ is the symmetrized
gradient. Here $F_{u}^{ext}$ represents any external forcing impacting the
fluid. In this work we will be assuming that this force is random, as we shall
see below. The fluid flow is driven by dynamic pressure data given at the
inlet and the outlet boundaries as follows:

(2) $\begin{split}p+\frac{1}{2}|{\bf u}|^{2}&=P_{in/out}(t),\\\
u_{r}&=0\quad\text{ on }\Gamma_{in/out}.\end{split}$

Whereas on the bottom boundary $\Gamma_{b}$ we prescribe the symmetry boundary
condition:

(3) $\displaystyle u_{r}=\partial_{r}u_{z}=0\quad\text{on }\Gamma_{b}.$

The structure sub-problem: The elastodynamics problem is given in terms of the
displacement ${\boldsymbol{\eta}}=(\eta_{z},\eta_{r})$ with respect to
$\Gamma$:

(4)
$\displaystyle\partial^{2}_{t}{\boldsymbol{\eta}}+\mathscr{L}_{e}({\boldsymbol{\eta}})=F_{\boldsymbol{\eta}}\quad\text{
in }(0,L),$

where $F_{{\boldsymbol{\eta}}}$ is the total force experienced by the
structure and $\mathscr{L}_{e}$ is a continuous, self-adjoint, coercive,
linear operator on ${\bf H}^{2}_{0}(0,L)$. The above equation is supplemented
with the following boundary conditions:

(5)
$\begin{split}{\boldsymbol{\eta}}(0)={\boldsymbol{\eta}}(L)=\partial_{z}{\boldsymbol{\eta}}(0)=\partial_{z}{\boldsymbol{\eta}}(L)=0.\end{split}$

The non-linear fluid-structure coupling: The coupling between the fluid and
the structure takes place across the current location of the fluid-structure
interface, which is simply the current location of the membrane/shell,
described above.

  * •

The kinematic coupling conditions are:

(6) $\displaystyle\partial_{t}{\boldsymbol{\eta}}(t,z)={\bf
u}({\boldsymbol{\varphi}}(t,z)),\quad(t,z)\in[0,T]\times[0,L].$

  * •

The dynamic coupling condition is:

(7) $\displaystyle
F_{\boldsymbol{\eta}}=-S_{\boldsymbol{\eta}}(t,z)(\sigma{\bf
n}^{\boldsymbol{\eta}})|_{(t,z,{\boldsymbol{\varphi}}(t,z))}+F_{\boldsymbol{\eta}}^{ext},$

where ${\bf n}^{\boldsymbol{\eta}}(t,z)$ is the unit outward normal to the top
boundary at the point, $\tau^{\boldsymbol{\eta}}$ is the tangent vector given
by $\tau^{\boldsymbol{\eta}}(t,z)=\partial_{z}{\boldsymbol{\varphi}}(t,z)$ and
$S_{\boldsymbol{\eta}}(t,z)$ is the Jacobian of the transformation from
Eulerian to Lagrangian coordinates. As earlier, $F^{ext}_{\boldsymbol{\eta}}$
denotes any external force impacting the structure.

This system is supplemented with the following initial conditions:

(8) $\displaystyle{\bf u}(t=0)={\bf
u}_{0},{\boldsymbol{\eta}}(t=0)={\boldsymbol{\eta}}_{0},\partial_{t}{\boldsymbol{\eta}}(t=0)={\bf
v}_{0}.$

### Weak formulation on moving domain

Before we derive the weak formulation of the deterministic system described in
the previous sub-section, we define the following relevant function spaces for
the fluid velocity, the structure, and the coupled FSI problem:

$\displaystyle\widetilde{\mathscr{V}}_{F}(t)=\\{{\bf u}=(u_{z},u_{r})\in{\bf
H}^{1}(\mathcal{O}_{{\boldsymbol{\eta}}}(t)):\nabla\cdot{\bf
u}=0,u_{z}=0\text{ on }\Gamma_{{\boldsymbol{\eta}}}(t),u_{r}=0\text{ on
}\partial\mathcal{O}_{{\boldsymbol{\eta}}}\setminus\Gamma_{{\boldsymbol{\eta}}}(t)\\}$
$\displaystyle\widetilde{\mathscr{W}}_{F}(0,T)=L^{\infty}(0,T;{\bf
L}^{2}(\mathcal{O}_{\boldsymbol{\eta}}(\cdot)))\cap
L^{2}(0,T;\widetilde{\mathscr{V}}_{F}(\cdot))$
$\displaystyle\widetilde{\mathscr{W}}_{S}(0,T)=W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))\cap L^{\infty}(0,T;{\bf H}^{2}_{0}(0,L))$
$\displaystyle\widetilde{\mathscr{W}}(0,T)=\\{({\bf
u},{\boldsymbol{\eta}})\in\widetilde{\mathscr{W}}_{F}(0,T)\times\widetilde{\mathscr{W}}_{S}(0,T):{{\bf
u}({\boldsymbol{\varphi}}(t,z))}=\partial_{t}{\boldsymbol{\eta}}(t,z),(t,z)\in(0,T)\times\Gamma\\}.$

Next, we derive a weak formulation of the problem on the moving domains. We
consider ${\bf q}\in C^{1}([0,T];\widetilde{\mathscr{V}}_{F}(\cdot))$ such
that ${{\bf q}}({\boldsymbol{\varphi}}(t,z))=\boldsymbol{\psi}(t,z)$ on
$(0,T)\times\Gamma$ for some $\boldsymbol{\psi}\in C^{1}([0,T];{\bf
H}^{2}_{0}(\Gamma))$. We multiply (1) by ${\bf q}$, integrate in time and
space and use Reynold’s transport theorem to obtain,

$\displaystyle({\bf u}(t),{\bf
q}(t))_{\mathcal{O}_{{\boldsymbol{\eta}}}(t)}=({\bf u}(0),{\bf
q}(0))_{\mathcal{O}_{{\boldsymbol{\eta}}}(0)}+\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(s)}{\bf
u}(s)\cdot\partial_{s}{\bf q}(s)$
$\displaystyle+\int_{0}^{t}\int_{\Gamma_{{\boldsymbol{\eta}}}(s)}({\bf
u}(s)\cdot{\bf q}(s))({\bf u}(s)\cdot{\bf
n}^{\boldsymbol{\eta}}(s))-\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(s)}({\bf
u}(s)\cdot\nabla){\bf u}(s){\bf q}(s)$
$\displaystyle-2\nu\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(s)}{\bf
D}({\bf u}(s))\cdot{\bf D}({\bf
q}(s))ds+\int_{0}^{t}\int_{\partial\mathcal{O}_{{\boldsymbol{\eta}}}(s)}(\sigma{\bf
n}^{\boldsymbol{\eta}}(s))\cdot{\bf
q}(s)+\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(s)}F^{ext}_{u}(s){\bf
q}(s).$

Let,

$b(t,{\bf u},{\bf v},{\bf
w}):=\frac{1}{2}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(t)}\left(({\bf
u}\cdot\nabla){\bf v}\cdot{\bf w}-({\bf u}\cdot\nabla){\bf w}\cdot{\bf
v}\right).$

With a slight abuse of notation, we let ${\bf n}^{\boldsymbol{\eta}}$ denote
the unit normal to $\partial\mathcal{O}_{\boldsymbol{\eta}}$ in the following
calculation:

$\displaystyle-(({\bf u}\cdot\nabla){\bf u},{\bf
q})_{\mathcal{O}_{{\boldsymbol{\eta}}}}$ $\displaystyle=-\frac{1}{2}(({\bf
u}\cdot\nabla){\bf u},{\bf
q})_{\mathcal{O}_{{\boldsymbol{\eta}}}}+\frac{1}{2}(({\bf u}\cdot\nabla){\bf
q},{\bf
u})_{\mathcal{O}_{{\boldsymbol{\eta}}}}-\frac{1}{2}\int_{\partial\mathcal{O}_{{\boldsymbol{\eta}}}}{\bf
u}\cdot{\bf q}{\bf u}\cdot{\bf n}^{\boldsymbol{\eta}}$
$\displaystyle=-b(s,{\bf u},{\bf u},{\bf
q})-\frac{1}{2}\int_{\Gamma_{{\boldsymbol{\eta}}}}{\bf u}\cdot{\bf q}{\bf
u}\cdot{\bf
n}^{\boldsymbol{\eta}}+\frac{1}{2}\int_{\Gamma_{in}}|u_{z}|^{2}q_{z}-\frac{1}{2}\int_{\Gamma_{out}}|u_{z}|^{2}q_{z}.$

Now using the divergence free property of fluid velocity ${\bf u}$ and the
fact that $u_{r}=0$ on $\Gamma_{in/out}$ we have that
$\partial_{r}u_{r}=-\partial_{z}u_{z}=0$ on $\Gamma_{in/out}$. Hence we
obtain,

$\displaystyle\int_{\Gamma_{in/out}}\sigma{\bf n}^{\boldsymbol{\eta}}\cdot{\bf
q}=\int_{\Gamma_{in/out}}\pm
p\,q_{z}=\int_{\Gamma_{in}}\left(P_{in}-\frac{1}{2}|{\bf
u}|^{2}\right)q_{z}dr-\int_{\Gamma_{out}}\left(P_{out}-\frac{1}{2}|{\bf
u}|^{2}\right)q_{z}dr,$

whereas,

$\displaystyle\int_{\Gamma_{b}}\sigma{\bf n}^{\boldsymbol{\eta}}\cdot{\bf
q}=0\quad\text{ and, }\int_{\Gamma_{\boldsymbol{\eta}}}\sigma{\bf
n}^{\boldsymbol{\eta}}\cdot{\bf
q}=\int_{\Gamma}S_{\boldsymbol{\eta}}\sigma{\bf
n}^{\boldsymbol{\eta}}\cdot\boldsymbol{\psi}.$

Next we consider the structure equation (4). We multiply (4) by
$\boldsymbol{\psi}$ and integrate in time and space to obtain

$\displaystyle(\partial_{t}{\boldsymbol{\eta}}(t),\boldsymbol{\psi}(t))$
$\displaystyle=(v_{0},\boldsymbol{\psi}(0)+\int_{0}^{t}\int_{0}^{L}\partial_{s}{\boldsymbol{\eta}}\cdot\partial_{s}\boldsymbol{\psi}dzds-\int_{0}^{t}\langle\mathscr{L}_{e}({\boldsymbol{\eta}}),\boldsymbol{\psi}\rangle
ds$ $\displaystyle-\int_{0}^{t}\int_{0}^{L}S_{\boldsymbol{\eta}}\sigma{\bf
n}^{\boldsymbol{\eta}}\cdot\boldsymbol{\psi}dzds+\int_{0}^{t}\int_{0}^{L}F_{\boldsymbol{\eta}}^{ext}\cdot\boldsymbol{\psi}dzds.$

Hence, to summarize, for any test function ${\bf Q}=({\bf
q},\boldsymbol{\psi})$ we look for a solution $({\bf
u},{\boldsymbol{\eta}})\in\widetilde{\mathscr{W}}(0,T)$, that satisfies the
following equation for almost every $t\in[0,T]$:

(9) $\begin{split}&{\int_{\mathcal{O}_{{\boldsymbol{\eta}}(t)}}{\bf u}(t){\bf
q}(t)d{\bf
x}+\int_{0}^{L}\partial_{t}{\boldsymbol{\eta}}(t)\boldsymbol{\psi}(t)dz}-\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}(s)}}{\bf
u}\cdot\partial_{s}{\bf q}d{\bf x}ds\\\ &+\int_{0}^{t}b(s,{\bf u},{\bf u},{\bf
q})ds-\frac{1}{2}\int_{0}^{t}\int_{\Gamma_{\boldsymbol{\eta}}}({\bf
u}\cdot{\bf q})({\bf u}\cdot{\bf
n}^{\boldsymbol{\eta}})dSds+2\nu\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}(s)}}{\bf
D}({\bf u})\cdot{\bf D}({\bf q})d{\bf x}ds\\\
&-\int_{0}^{t}\int_{0}^{L}\partial_{s}{\boldsymbol{\eta}}\partial_{s}\boldsymbol{\psi}dzds+\int_{0}^{t}\langle\mathscr{L}_{e}({\boldsymbol{\eta}}),\boldsymbol{\psi}\rangle
ds\\\ &=\int_{\mathcal{O}_{{\boldsymbol{\eta}}_{0}}}{\bf u}_{0}{\bf q}(0)d{\bf
x}+\int_{0}^{L}{\bf
v}_{0}\boldsymbol{\psi}(0)dz+\int_{0}^{t}P_{{in}}\int_{0}^{1}q_{z}\Big{|}_{z=0}drds-\int_{0}^{t}P_{{out}}\int_{0}^{1}q_{z}\Big{|}_{z=1}drds\\\
&+\int_{0}^{t}\int_{\mathcal{O}_{{\boldsymbol{\eta}}}(s)}{\bf q}\cdot
F_{u}^{ext}\,d{\bf x}ds+\int_{0}^{t}\int_{0}^{L}\boldsymbol{\psi}\cdot
F_{{\boldsymbol{\eta}}}^{ext}\,dzds,\end{split}$

where $F_{u}^{ext}$ is the volumetric external force and
$F_{{\boldsymbol{\eta}}}^{ext}$ is the external force applied to the
deformable boundary, chosen to be random in nature, as we explain next.

###  2.2. The stochastic framework and definition of martingale solutions

We will assume that the external forces $F_{u}^{ext}$ and
${\color[rgb]{0,0,0}F_{{\boldsymbol{\eta}}}^{ext}}$ are multiplicative
stochastic forces and that we can then write the combined stochastic forcing
$F^{ext}$ as follows:

(10) ${\color[rgb]{0,0,0}F^{ext}}:=G({\bf u},{\bf
v},{\boldsymbol{\eta}}){dW},$

where ${\bf u}$ is the fluid velocity, ${\bf v}$ is the structure velocity,
${\boldsymbol{\eta}}$ is the structure displacement, and $W$ is a Wiener
process.

We will now give the relevant probabilistic framework. The stochastic noise
term is defined on a given filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ that satisfies
the usual assumptions, i.e., $\mathcal{F}_{0}$ is complete and the filtration
is right continuous, that is, $\mathcal{F}_{t}=\cap_{s\geq t}\mathcal{F}_{s}$
for all $t\geq 0$. We assume that $W$ is a $U$-valued Wiener process with
respect to the filtration $(\mathcal{F}_{t})_{t\geq 0}$, where $U$ is a
separable Hilbert space. We denote by $Q$ the covariance operator of $W$,
which is a positive, trace class operator on $U$, and define
$U_{0}:=Q^{\frac{1}{2}}(U)$.

Introducing the notation ${\bf L}^{2}:={\bf L}^{2}(\mathcal{O})\times{\bf
L}^{2}(0,L)$ we now give assumptions on the noise coefficient $G$:

######  Assumption 2.1.

Let $L_{2}(X,Y)$ denote the space of Hilbert-Schmidt operators from a Hilbert
space $X$ to another Hilbert space $Y$. The noise coefficient $G$ is a
function $G:{\mathscr{U}}\times{\bf L}^{\infty}(0,L)\rightarrow
L_{2}(U_{0};{\bf L}^{2})$ such that the following Lipschitz continuity
assumptions hold:

(11) $\begin{split}&\|G({\bf u},{\bf
v},{\boldsymbol{\eta}})\|_{L_{2}(U_{0};{\bf
L}^{2})}\leq\|{\boldsymbol{\eta}}\|_{{\bf L}^{\infty}(0,L)}\|{{\bf u}}\|_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}\|_{{\bf L}^{2}(0,L)},\\\ &\|G({\bf u}_{1},{\bf
v}_{1},{\boldsymbol{\eta}})-G({\bf u}_{2},{\bf
v}_{2},{\boldsymbol{\eta}})\|_{L_{2}(U_{0};{\bf
L}^{2})}\leq\|{\boldsymbol{\eta}}\|_{{\bf L}^{\infty}(0,L)}\|{{\bf
u}_{1}}-{{\bf u}_{2}}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}_{1}-{\bf
v}_{2}\|_{{\bf L}^{2}(0,L)},\\\ &\|G({\bf u},{\bf
v},{\boldsymbol{\eta}}_{1})-G({\bf u},{\bf
v},{\boldsymbol{\eta}}_{2})\|_{L_{2}(U_{0};{\bf
L}^{2})}\leq\|{\boldsymbol{\eta}}_{1}-{\boldsymbol{\eta}}_{2}\|_{{\bf
L}^{\infty}(0,L)}\|{\bf u}\|_{{\bf L}^{2}(\mathcal{O})}.\end{split}$

We will use this framework below to define a martingale solution to our
stochastic FSI problem. In order to do this, we first transform the problem
defined on the moving domains onto a fixed reference domain using a family of
Arbitrary Lagrangian-Eulerian (ALE) mappings. The mappings are defined next.

####  2.2.1. ALE mappings

To deal the geometric nonlinearity arising due to the motion of the fluid
domain, we consider the arbitrary Lagrangian-Eulerian (ALE) mappings which are
a family of diffeomorphisms from the fixed domain
$\mathcal{O}=(0,L)\times(0,1)$ onto the moving domain
$\mathcal{O}_{{\boldsymbol{\eta}}}(t)$. Notice that the presence of the
stochastic forcing implies that the domains $\mathcal{O}_{\boldsymbol{\eta}}$
are themselves random and that we must consider the ALE mappings defined
pathwise, that is, for every $\omega\in\Omega$ we will consider the maps
$A_{{\boldsymbol{\eta}}}^{\omega}(t):\mathcal{O}\rightarrow\mathcal{O}_{{\boldsymbol{\eta}}}(t,\omega)$
such that
$A^{\omega}_{\boldsymbol{\eta}}(t)=\textbf{id}+{\boldsymbol{\eta}}(t,\omega)\text{
on }\Gamma$ and $A^{\omega}_{\boldsymbol{\eta}}(t)=\textbf{id}\text{ on
}\partial\mathcal{O}\setminus\Gamma.$

Assuming the existence of such a map we will give the definition of martingale
solutions (see Definition 1) on fixed domain by first introducing the required
notations. Under this transformation, the pathwise transformed gradient and
symmetrized gradient will be denoted by

$\nabla^{\boldsymbol{\eta}}f=\nabla f(\nabla
A_{\boldsymbol{\eta}})^{-1},\text{ and }{\bf D}^{\boldsymbol{\eta}}({\bf
u})=\frac{1}{2}(\nabla^{\boldsymbol{\eta}}{\bf
u}+(\nabla^{\boldsymbol{\eta}})^{T}{\bf u}).$

This gives us the definition of the transformed divergence as:

$div^{\boldsymbol{\eta}}f=tr(\nabla^{\boldsymbol{\eta}}f).$

The Jacobian of the ALE mapping is given by

(12) $J^{\omega}_{\boldsymbol{\eta}}(t)=\text{det }\nabla
A_{{\boldsymbol{\eta}}}^{\omega}(t).$

Using ${\bf w}^{\boldsymbol{\eta}}$ to denote the ALE velocity ${\bf
w}^{\boldsymbol{\eta}}=\frac{d}{dt}A_{\boldsymbol{\eta}}$, we rewrite the
advection term as follows:

$b^{\eta}({\bf u},{\bf u},{\bf
q})=\frac{1}{2}\int_{\mathcal{O}}J_{{\boldsymbol{\eta}}}\left((({\bf u}-{\bf
w}^{\boldsymbol{\eta}})\cdot\nabla^{\boldsymbol{\eta}}){\bf u}\cdot{\bf
q}-(({\bf u}-{\bf w}^{\boldsymbol{\eta}})\cdot\nabla^{\boldsymbol{\eta}}){\bf
q}\cdot{\bf u}\right).$

####  2.2.2. Definition of martingale solutions

We start by introducing the functional framework for the stochastic problem on
the fixed reference domain $\mathcal{O}=(0,L)\times(0,1)$. The following are
the functions spaces for the stochastic FSI problem defined on the fixed
domain $\mathcal{O}$:

$\displaystyle V=\\{{\bf u}=(u_{z},u_{r})\in{{\bf
H}}^{1}(\mathcal{O}):u_{z}=0\text{ on }\Gamma,u_{r}=0\text{ on
}\partial\mathcal{O}\setminus\Gamma\\},$
$\displaystyle\mathscr{W}_{F}=L^{2}(\Omega;L^{\infty}(0,T;{\bf
L}^{2}(\mathcal{O})))\cap L^{2}(\Omega;L^{2}(0,T;V)),$
$\displaystyle\mathscr{W}_{S}=L^{2}(\Omega;W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))\cap L^{\infty}(0,T;{\bf H}^{2}_{0}(0,L))),$
$\displaystyle\mathscr{W}(0,T)=\\{({\bf
u},{\boldsymbol{\eta}})\in\mathscr{W}_{F}\times\mathscr{W}_{S}:{\bf
u}(t)|_{\Gamma}=\partial_{t}{\boldsymbol{\eta}}(t),\,\,\nabla^{\boldsymbol{\eta}}\cdot{{\bf
u}}=0\,\,{\mathbb{P}}-a.s.\\}.$

We define the following spaces for test functions and for fluid and structure
velocities:

(13) $\mathscr{D}=\\{({\bf q},\boldsymbol{\psi})\in V\times{\bf
H}^{2}_{0}(0,L)\\},\qquad{\mathscr{U}}=\\{({\bf u},{\bf v})\in V\times{\bf
L}^{2}(0,L)\\}.$

######  Definition 1.

(Martingale solution) Given deterministic initial data, ${\bf u}_{0}\in{\bf
L}^{2}(\mathcal{O})$, ${\bf v}_{0}\in{\bf L}^{2}(0,L)$ and compatible initial
structure configuration ${\boldsymbol{\eta}}_{0}\in{\bf H}_{0}^{2}(0,L)$ such
that for some $\delta_{1},\delta_{2}>0$ such that, the initial configuration
satisfies

(14)
$\displaystyle{\delta_{1}<\inf_{\mathcal{O}}J_{{\boldsymbol{\eta}}_{0}},\quad\text{
and }\quad\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}_{0}(0,L)}<\frac{1}{\delta_{2}}},$

we say that $(\mathscr{S},{\bf u},{\boldsymbol{\eta}},T^{\boldsymbol{\eta}})$
is a martingale solution to the system (1)-(8) under the assumptions
(LABEL:growthG) if

  1. (1)

$\mathscr{S}=(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}},W)$
is a stochastic basis, that is, $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\mathbb{P}})$ is a filtered probability space satisfying the usual
conditions and $W$ is a $U$-valued Wiener process.

  2. (2)

$({\bf u},{\boldsymbol{\eta}})\in\mathscr{W}(0,T)$.

  3. (3)

$T^{\boldsymbol{\eta}}$ is a ${\mathbb{P}}$-a.s. strictly positive,
$\mathcal{F}_{t}-$stopping time.

  4. (4)

$({\bf u},\partial_{t}{\boldsymbol{\eta}})$ and ${\boldsymbol{\eta}}$ are
$(\mathcal{F}_{t})_{t\geq 0}-$progressively measurable.

  5. (5)

For every $(\mathcal{F}_{t})_{t\geq 0}-$adapted, essentially bounded process
${\bf Q}:=({\bf q},\boldsymbol{\psi})$ with continuous paths in $\mathscr{D}$
such that $\nabla^{\boldsymbol{\eta}}\cdot{\bf q}=0$, and ${\bf
q}|_{\Gamma}=\boldsymbol{\psi}$, the equation

(15) $\begin{split}&{\int_{\mathcal{O}}J_{\boldsymbol{\eta}}(t){\bf u}(t){\bf
q}(t)+\int_{0}^{L}\partial_{t}{\boldsymbol{\eta}}(t)\boldsymbol{\psi}(t)}=\int_{\mathcal{O}}J_{0}{\bf
u}_{0}{\bf q}(0)+\int_{0}^{L}{\bf v}_{0}\boldsymbol{\psi}(0)\\\
&+\int_{0}^{t}\int_{\mathcal{O}}J_{\boldsymbol{\eta}}\,{\bf
u}\cdot\partial_{t}{\bf q}-{\int_{0}^{t}b^{{\boldsymbol{\eta}}}({\bf u},{\bf
u},{\bf
q})}+\int_{0}^{t}\int_{\mathcal{O}}J_{\boldsymbol{\eta}}(\nabla^{\boldsymbol{\eta}}\cdot{\bf
w}){\bf u}\cdot{\bf q}\\\
&-2\nu\int_{0}^{t}\int_{\mathcal{O}}J_{\boldsymbol{\eta}}{\bf
D}^{\boldsymbol{\eta}}({\bf u})\cdot{\bf D}^{\boldsymbol{\eta}}({\bf
q})+\int_{0}^{t}\int_{0}^{L}\partial_{t}{\boldsymbol{\eta}}\partial_{t}\boldsymbol{\psi}-\int_{0}^{t}\int_{0}^{L}\langle\mathscr{L}_{e}({\boldsymbol{\eta}}),\boldsymbol{\psi}\rangle\\\
&+\int_{0}^{t}\left(P_{{in}}\int_{0}^{1}q_{z}\Big{|}_{z=0}dr-
P_{{out}}\int_{0}^{1}q_{z}\Big{|}_{z=1}dr\right)+\int_{0}^{t}({\bf Q},G({\bf
u},\partial_{t}\boldsymbol{\eta},{\boldsymbol{\eta}})\,dW),\end{split}$

holds ${\mathbb{P}}$-a.s. for almost every $t\in[0,T^{\boldsymbol{\eta}})$.

In the rest of this manuscript we will present a constructive proof of the
existence of martingale solutions. The construction is based on the following
operator splitting scheme.

##  3\. Operator splitting scheme

In this section we introduce a Lie operator splitting scheme that defines a
sequence of approximate solutions to (15) by semi-discretizing the problem in
time. The final goal is to show that up to a subsequence, approximate
solutions converge in a certain sense to a martingale solution of the
stochastic FSI problem.

###  3.1. Definition of the splitting scheme

We semidiscretize the problem in time and use operator splitting to divide the
coupled stochastic problem into two subproblems, a fluid and a structure
subproblem. We denote the time step by $\Delta t=\frac{T}{N}$ and use the
notation $t^{n}=n\Delta t$ for $n=0,1,...,N$.

Let $({\bf u}^{0},{\boldsymbol{\eta}}^{0},{\bf v}^{0})=({\bf
u}_{0},{\boldsymbol{\eta}}_{0},{\bf v}_{0})$ be the initial data. At the
$i^{th}$ time level, we update the vector $({\bf
u}^{n+\frac{i}{2}},{\boldsymbol{\eta}}^{n+\frac{i}{2}},{\bf
v}^{n+\frac{i}{2}})$, where $i=1,2$ and $n=0,2,...,N-1$, according to the
following scheme.

### The structure sub-problem

In this sub-problem we update the structure displacement and the structure
velocity while keeping the fluid velocity the same. That is, given
$({\boldsymbol{\eta}}^{n},{\bf v}^{n})\in{\bf H}^{2}_{0}(0,L)\times{\bf
L}^{2}(0,L)$ we look for a pair $({\boldsymbol{\eta}}^{n+\frac{1}{2}},{\bf
v}^{n+\frac{1}{2}})\in{\bf H}^{2}_{0}(0,L)\times{\bf H}^{2}_{0}(0,L)$ that
satisfies the following equations pathwise i.e. for each $\omega\in\Omega$:

(16) $\begin{split}{\bf u}^{n+\frac{1}{2}}&={\bf u}^{n},\\\
\int_{0}^{L}({\boldsymbol{\eta}}^{n+\frac{1}{2}}-{\boldsymbol{\eta}}^{n})\boldsymbol{\phi}dz&=(\Delta
t)\int_{0}^{L}{\bf v}^{n+\frac{1}{2}}\boldsymbol{\phi}dz,\\\
\int_{0}^{L}\left({\bf v}^{n+\frac{1}{2}}-{\bf
v}^{n}\right)\boldsymbol{\psi}dz&+(\Delta
t)\langle\mathscr{L}_{e}({\boldsymbol{\eta}}^{n+\frac{1}{2}}),\boldsymbol{\phi}\rangle=0,\end{split}$

for any $\boldsymbol{\phi}\in{\bf L}^{2}(0,L)$ and $\boldsymbol{\psi}\in{\bf
H}^{2}_{0}(0,L)$.

Before commenting on the existence of the random variables
${\boldsymbol{\eta}}^{n+\frac{1}{2}},{\bf v}^{n+\frac{1}{2}}$ and their
measurability properties, we introduce the second sub-problem.

### The fluid sub-problem

In this step we update the fluid and the structure velocities while keeping
the structure displacement unchanged. However there are two major difficulties
associated with this method in this sub-problem. The first difficulty is
associated with the fact that the fluid domains can degenerate randomly.

First observe that for any $\delta_{2}>0$, $s>\frac{3}{2}$ and
$\boldsymbol{\eta}=(\eta_{z},\eta_{r})\in{\bf H}_{0}^{2}(\Gamma)$ with
$\|\boldsymbol{\eta}\|_{{\bf H}^{s}(\Gamma)}\leq\frac{1}{\delta_{2}}$, Sobolev
embedding theorem gives us the existence of a deterministic constant $C_{*}>0$
depending only on $\Gamma$ and $s$ such that,

$\|\partial_{z}\boldsymbol{\eta}\|_{{\bf L}^{\infty}(\Gamma)}\leq
C_{*}\|\boldsymbol{\eta}\|_{{\bf H}^{s}(\Gamma)}\leq\frac{C_{*}}{\delta_{2}}.$

Hence for any $z\in(0,L)$ we have
$1-\frac{C_{*}}{\delta_{2}}\leq(1+\partial_{z}\eta_{z}(z))$. Then choosing
$\delta_{2}>0$ large enough to ensure

(17) $\displaystyle
0<1-\frac{C_{*}}{\delta_{2}}<1+\partial_{z}\eta_{z}(z),\quad\forall z\in[0,L]$

we obtain that the map $z+\eta_{z}(z)$ is injective. Moreover since
$H^{1}(\Gamma)\hookrightarrow C^{0,\frac{1}{2}}(\bar{\Gamma})$, the
intermediate value theorem implies that $id+\eta_{z}:[0,L]\to[0,L]$ is
bijective. Assuming that this is true for any ${\bf H}_{0}^{2}(\Gamma)$-valued
process $\boldsymbol{\eta}$ at all times $t\in[0,T]$ and $\omega\in\Omega$, we
can define the pathwise ALE map
$A_{\boldsymbol{\eta}}^{\omega}(t):\mathcal{O}\rightarrow\mathcal{O}_{\boldsymbol{\eta}}(t,\omega)$
as follows:

(18) $\displaystyle
A_{\boldsymbol{\eta}}^{\omega}(t)(z,r)=(z,(1+\eta_{r}(t,\varphi^{-1}_{\boldsymbol{\eta}}(t,\omega)(z),\omega))r),$

where for any $t\in[0,T],\omega\in\Omega$, the map
$\varphi_{\boldsymbol{\eta}}(t,\omega):[0,L]\to[0,L]$ is given by

(19)
$\displaystyle\varphi_{\boldsymbol{\eta}}(t,\omega)(z)=z+\eta_{z}(t,z,\omega).$

Then, given any $\omega\in\Omega$, the Jacobian of the pathwise ALE mapping is
given by

(20) $\text{det }\nabla
A_{\boldsymbol{\eta}}^{\omega}(t)=1+\eta_{r}(t,\varphi_{\boldsymbol{\eta}}(t,\omega)^{-1}(z),\omega).$

Based on the observations above, to overcome the problem of random fluid
domain degeneration, we introduce an ”artificial structure displacement”
random variable by the means of a cut-off function as follows: For
$\delta_{1},\delta_{2}>0$, let $\Theta_{\delta}$ be the step function such
that $\Theta_{\delta}(x,y)=1$ if $\delta_{1}<x,\text{}y<\frac{1}{\delta_{2}}$,
and $\Theta_{\delta}(x,y)=0$ otherwise. For brevity we define, for a fixed
$s\in(\frac{3}{2},2)$, a real-valued function
$\theta_{\delta}({\boldsymbol{\eta}}^{n})$ which tracks all the structure
displacements until the time step $n$, and is equal to 1 until the step for
which the structure quantities leave the desired bounds given in terms of
$\delta=(\delta_{1},\delta_{2})$:

(21) $\displaystyle\theta_{\delta}({\boldsymbol{\eta}}^{n}):=\min_{k\leq
n}\,\Theta_{\delta}\left(\inf_{\mathcal{O}}J^{k},\|{\boldsymbol{\eta}}^{k}\|_{{\bf
H}^{s}(\Gamma)}\right),$

where $J^{k}(\omega)=\text{det}\nabla A^{\omega}_{{\boldsymbol{\eta}}^{k}}$ is
the Jacobian of the map defined in (18) as given in (20).

Now we define the artificial structure displacement random variable as
follows,

(22)
$\displaystyle{\boldsymbol{\eta}}^{n}_{*}(z,\omega)={\boldsymbol{\eta}}^{\max_{0\leq
k\leq n}\theta_{\delta}({\boldsymbol{\eta}}^{k})k}(z,\omega)\quad\text{for
every }\omega\in\Omega,$

where the superscript in the definition above indicates the time step and not
the power of structure displacement. This artificial structure variable is
defined with care in a way that ensures the stability of our scheme.
Furthermore, for $\delta_{2}$ satisfying (17), the domain configurations
corresponding the artificial variables
$\boldsymbol{\eta}^{n}_{*}=((\eta^{n}_{*})_{z},(\eta^{n}_{*})_{r})$ are non-
degenerate and their Jacobians have a deterministic $\delta_{1}>0$ lower
bound. That is for any $n$ and $\omega$,

(23) $\displaystyle
1+(\eta^{n}_{*})_{r}\circ\varphi^{-1}_{\boldsymbol{\eta}^{n}_{*}}(z)>\delta_{1},\quad\forall
z\in[0,L]\quad\text{ and }\quad\|\boldsymbol{\eta}^{n}_{*}\|_{{\bf
H}^{s}(\Gamma)}\leq\frac{1}{\delta_{2}}.$

Observe also that for $\delta_{2}$ satisfying (17), and as earlier
$\varphi_{\boldsymbol{\eta}^{*}_{n}}(z,\omega)=z+(\eta^{n}_{*})_{z}(z,\omega)$
we have

(24)
$\displaystyle|\partial_{z}\varphi_{\boldsymbol{\eta}^{n}_{*}}^{-1}(z)|\leq\frac{1}{1-\frac{C_{*}}{\delta_{2}}}\qquad\text{
for every $z\in[0,L]$, $\omega\in\Omega$, and any $n$}.$

Thus we conclude that for some constant independent of $\omega$ and $n$, we
have

(25) $\displaystyle\|\nabla A^{\omega}_{{\boldsymbol{\eta}}_{*}^{n}}\|_{{\bf
L}^{\infty}(\bar{\mathcal{O}})}\leq C(\delta_{2}).$

We will define the fluid-subproblem on these artificial domain configurations.

The second difficulty in defining the fluid sub-problem is associated with the
fact that, through the transformed divergence-free condition, the test
functions depend on the structure displacement found in the previous sub-
problem. However it is not clear how to deal with such test functions, as in
the following section we construct and work on a new probability space. Hence,
in this sub-problem we supplement the weak formulation by a penalty term of
the form $N\int\text{div}^{\boldsymbol{\eta}}{\bf
u}\text{div}^{\boldsymbol{\eta}}{\bf q}$ to enforce the incompressibility
condition only in the limit as $N\rightarrow\infty$.

A penalized system: We introduce a penalty term and define the fluid
subproblem as follows. Let $\Delta_{n}W:=W(t^{n+1})-W(t^{n})$.
Then given ${\bf U}^{n}=({\bf u}^{n},{\bf v}^{n})\in{\mathscr{U}}$ and
$\delta_{2}$ that satisfies (17), we look for $({\bf u}^{n+1},{\bf
v}^{n+1})\in{\mathscr{U}}$ that solves

(26)
$\begin{split}&\qquad\qquad{\boldsymbol{\eta}}^{n+1}:={\boldsymbol{\eta}}^{n+\frac{1}{2}},\\\
&\int_{\mathcal{O}}J^{n}_{*}\left({\bf u}^{n+1}-{\bf
u}^{n+{\frac{1}{2}}}\right){\bf q}d{\bf
x}+\frac{1}{2}\int_{\mathcal{O}}\left(J_{*}^{n+1}-J^{n}_{*}\right){\bf
u}^{n+1}\cdot{\bf q}d{\bf x}\\\ &+\frac{1}{2}(\Delta
t)\int_{\mathcal{O}}J_{*}^{n}(({\bf u}^{n}-{\bf
w}_{*}^{{n+1}})\cdot\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf u}^{n+1}\cdot{\bf
q}-({\bf u}^{n}-{\bf
w}_{*}^{{n+1}})\cdot\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf q}\cdot{\bf
u}^{n+1})d{\bf x}\\\ &+2\nu(\Delta t)\int_{\mathcal{O}}J^{n}_{*}{\bf
D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf u}^{n+1})\cdot{\bf
D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf q})d{\bf x}+N{(\Delta
t)}\int_{\mathcal{O}}\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf
u}^{n+1}\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf q}d{\bf x}\\\
&+\int_{0}^{L}({\bf v}^{n+1}-{\bf v}^{n+\frac{1}{2}})\boldsymbol{\psi}dz\\\
&=(\Delta
t)\left(P^{n}_{{in}}\int_{0}^{1}q_{z}\Big{|}_{z=0}dr-P^{n}_{{out}}\int_{0}^{1}q_{z}\Big{|}_{z=1}dr\right)+(G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf Q}),\end{split}$

for any ${\bf Q}=({\bf q},\boldsymbol{\psi})\in{\mathscr{U}}$. Here,

$P^{n}_{in/out}:=\frac{1}{\Delta t}\int_{t^{n}}^{t^{n+1}}P_{in/out}\,dt,\quad$

and the random variables ${\bf n}^{n}_{*}={\bf
n}^{{\boldsymbol{\eta}}^{n}_{*}}$ and

${\bf w}^{n+1}=\frac{1}{\Delta
t}(A^{\omega}_{{\boldsymbol{\eta}}^{n+1}_{*}}-A^{\omega}_{{\boldsymbol{\eta}}^{n}_{*}}),\quad
J^{n}_{*}=\text{det}\nabla A^{\omega}_{{\boldsymbol{\eta}}^{n}_{*}}.$

######  Remark 1.

Observe that using the data from the second subproblem, we update
$\boldsymbol{\eta}^{n}$ in the first subproblem and not
$\boldsymbol{\eta}_{*}^{n}$. This is required to obtain a stable scheme.
However this also means that after a certain time (which will be random in
nature), we will produce solutions that are meaningless. However the
discrepensies caused by the artificial structure variable will be taken care
of by introducing a stopping time until which the limiting solutions,
corresponding to the approximations constructed in Section 3.2 using
${\boldsymbol{\eta}}^{n}$’s and ${\boldsymbol{\eta}}^{n}_{*}$’s, are equal.

We are now ready to prove the existence of solutions to the two sub-problems.
For this purpose we introduce the following discrete energy and dissipation
for $i=0,1$:

(27)
$\begin{split}E^{n+\frac{i}{2}}&=\frac{1}{2}\Big{(}\int_{\mathcal{O}}J_{*}^{n}|{\bf
u}^{n+\frac{i}{2}}|^{2}d{\bf x}+\|{\bf v}^{n+\frac{i}{2}}\|^{2}_{{\bf
L}^{2}(0,L)}+\|{\boldsymbol{\eta}}^{n+\frac{i}{2}}\|^{2}_{{\bf
H}_{0}^{2}(0,L)}\Big{)},\\\ D^{n}&=\Delta t\int_{\mathcal{O}}\left(2\nu
J_{*}^{n}|{\bf D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf
u}^{n+1})|^{2}+{N}|\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf
u}^{n+1}|^{2}\right)d{\bf x}\end{split}$

######  Lemma 3.1.

(Existence for the structure sub-problem.) Assume that
${\boldsymbol{\eta}}^{n}$ and ${\bf v}^{n}$ are ${\bf H}^{2}_{0}(0,L)$ and
${\bf L}^{2}(0,L)$ valued $\mathcal{F}_{t^{n}}$-measurable random variables,
respectively. Then there exist ${\bf H}^{2}_{0}(0,L)$\- valued
$\mathcal{F}_{t^{n}}$-measurable random variables
${\boldsymbol{\eta}}^{n+\frac{1}{2}},{\bf v}^{n+\frac{1}{2}}$ that solve (16),
and the following semidiscrete energy inequality holds:

(28) $\begin{split}E^{n+\frac{1}{2}}+C_{1}^{n}\leq E^{n},\end{split}$

where

$C^{n}_{1}:=\frac{1}{2}\|{\bf v}^{n+\frac{1}{2}}-{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\frac{1}{2}\|{\boldsymbol{\eta}}^{n+\frac{1}{2}}-{\boldsymbol{\eta}}^{n}\|_{{\bf
H}_{0}^{2}(0,L)}^{2},$

corresponds to numerical dissipation.

###### Proof.

The proof of existence and uniqueness of measurable solutions is
straightforward (see [5]). Furthermore we can write

${\bf
v}^{n+\frac{1}{2}}=\frac{{\boldsymbol{\eta}}^{n+\frac{1}{2}}-{\boldsymbol{\eta}}^{n}}{\Delta
t}.$

Using this pathwise equality while taking $\boldsymbol{\psi}={\bf
v}^{n+\frac{1}{2}}$ in (16)3 and using
$a(a-b)=\frac{1}{2}(|a|^{2}-|b|^{2}+|a-b|^{2})$, we obtain,

(29) $\begin{split}&\|{\bf v}^{n+\frac{1}{2}}\|_{{\bf L}^{2}(0,L)}^{2}+\|{\bf
v}^{n+\frac{1}{2}}-{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n+\frac{1}{2}}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n+\frac{1}{2}}-{\boldsymbol{\eta}}^{n}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}\\\ &\leq\|{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}.\end{split}$

Hence using the fact that ${\bf u}^{n}={\bf u}^{n+\frac{1}{2}}$ and adding the
relevant terms on both sides of (29) we obtain (28). ∎

######  Lemma 3.2.

(Existence for the fluid sub-problem.) For given $\delta>0$, and given
$\mathcal{F}_{t^{n}}$-measurable random variables $({\bf
u}^{n+\frac{1}{2}},{\bf v}^{n})$ taking values in ${\mathscr{U}}$ and ${\bf
v}^{n+\frac{1}{2}}$ taking values in ${\bf H}^{2}_{0}(0,L)$, there exists an
$\mathcal{F}_{t^{n+1}}$-measurable random variable $({\bf u}^{n+1},{\bf
v}^{n+1})$ taking values in ${\mathscr{U}}$ that solves (LABEL:second), and
the solution satisfies the following energy estimate

(30) $\begin{split}E^{n+1}+D^{n}+C_{2}^{n}&\leq E^{n+\frac{1}{2}}+C\Delta
t((P^{n}_{in})^{2}+(P^{n}_{out})^{2})+{C}\|\Delta_{n}W\|_{U_{0}}^{2}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf L}^{2})}^{2}\\\
&+\|(G({\bf U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf
U}^{n})\|+\frac{1}{4}\int_{0}^{L}|{\bf v}^{n+\frac{1}{2}}-{\bf
v}^{n}|^{2}dz\end{split}$

where

$C_{2}^{n}:=\frac{1}{4}\int_{\mathcal{O}}J_{*}^{n}\left(|{\bf u}^{n+1}-{\bf
u}^{n}|^{2}\right)d{\bf x}+\frac{1}{4}\int_{0}^{L}|{\bf v}^{n+1}-{\bf
v}^{n+\frac{1}{2}}|^{2}dz$

is numerical dissipation, and ${\boldsymbol{\eta}}^{n}_{*}$ is as defined in
(22).

###### Proof.

The proof of existence and uniqueness of solutions follows from the Lax-
Milgram theorem.

Next we will show that the solution satisfies energy estimate (29). For this
purpose we will derive a pathwise inequality involving the discrete energies.
We take $({\bf q},\boldsymbol{\psi})=({\bf u}^{n+1},{\bf v}^{n+1})$ in
(LABEL:second) and using the identity
$a(a-b)=\frac{1}{2}(|a|^{2}-|b|^{2}+|a-b|^{2})$, we obtain

$\displaystyle\frac{1}{2}\int_{\mathcal{O}}J_{*}^{n}\left(|{\bf
u}^{n+1}|^{2}-|{\bf u}^{n+\frac{1}{2}}|^{2}+|{\bf u}^{n+1}-{\bf
u}^{n+\frac{1}{2}}|^{2}\right)+\frac{1}{2}\int_{\mathcal{O}}\left(J_{*}^{n+1}-J_{*}^{n}\right)|{\bf
u}^{n+1}|^{2}d{\bf x}$ $\displaystyle+2\nu(\Delta
t)\int_{\mathcal{O}}J_{*}^{n}|{\bf D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf
u}^{n+1})|^{2}d{\bf x}+N{(\Delta
t)}\int_{\mathcal{O}}|\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf
u}^{n+1}|^{2}d{\bf x}$ $\displaystyle+\frac{1}{2}\int_{0}^{L}|{\bf
v}^{n+1}|^{2}-|{\bf v}^{n+\frac{1}{2}}|^{2}+|{\bf v}^{n+1}-{\bf
v}^{n+\frac{1}{2}}|^{2}dz$ $\displaystyle=(\Delta
t)\left(P^{n}_{{in}}\int_{0}^{1}u^{n+1}_{z}\Big{|}_{z=0}dr-P^{n}_{{out}}\int_{0}^{1}u^{n+1}_{z}\Big{|}_{z=1}dr\right)$
$\displaystyle+(G({\bf U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,({\bf
U}^{n+1}-{\bf U}^{n}))+(G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf U}^{n}).$

Also, observe that the discrete stochastic integral is divided into two terms.
We estimate the first term by using the Cauchy-Schwarz inequality to obtain
that for some $C(\delta)>0$ independent of $n$ the following holds:

$\displaystyle|(G({\bf U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,({\bf
U}^{n+1}-{\bf U}^{n}))|$
$\displaystyle\leq{C}\|\Delta_{n}W\|_{U_{0}}^{2}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}+\frac{1}{4}\int_{\mathcal{O}}J_{*}^{n}|{\bf u}^{n+1}-{\bf
u}^{n}|^{2}d{\bf x}+\frac{1}{8}\int_{0}^{L}|{\bf v}^{n+1}-{\bf v}^{n}|^{2}dz$
$\displaystyle\leq{C}\|\Delta_{n}W\|_{U_{0}}^{2}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}+\frac{1}{4}\int_{\mathcal{O}}J_{*}^{n}|{\bf u}^{n+1}-{\bf
u}^{n}|^{2}d{\bf x}$ $\displaystyle+\frac{1}{4}\int_{0}^{L}|{\bf v}^{n+1}-{\bf
v}^{n+\frac{1}{2}}|^{2}dz+\frac{1}{4}\int_{0}^{L}|{\bf v}^{n+\frac{1}{2}}-{\bf
v}^{n}|^{2}dz.$

We treat the terms with $P_{in/out}$ similarly to obtain (30). This completes
the proof of Lemma 3.2. ∎

Next, we will obtain uniform estimates on the expectation of the kinetic and
elastic energy and dissipation of the coupled problem.

######  Theorem 1.

(Uniform Estimates.) For any $N>0$, $\Delta t=\frac{T}{N}$, there exists a
constant $C>0$ that depends on the initial data, $\delta$, $T$, and
$P_{in/out}$ and is independent of $N$ such that

  1. (1)

${\mathbb{E}}\left(\max_{1\leq n\leq N}E^{n}\right)<C$,
${\mathbb{E}}\left(\max_{0\leq n\leq N-1}E^{n+\frac{1}{2}}\right)<C.$

  2. (2)

${\mathbb{E}}\sum_{n=0}^{N-1}D^{n}<C$.

  3. (3)

${\mathbb{E}}\sum_{n=0}^{N-1}\int_{\mathcal{O}}\left(({J}_{*}^{n})|{\bf
u}^{n+1}-{\bf u}^{n}|^{2}\right)d{\bf x}+\|{\bf v}^{n+1}-{\bf
v}^{n+\frac{1}{2}}\|_{{\bf L}^{2}(0,L)}^{2}<C.$

  4. (4)

${\mathbb{E}}\sum_{n=0}^{N-1}\|{\bf v}^{n+\frac{1}{2}}-{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n+1}-{\boldsymbol{\eta}}^{n}\|_{{\bf
H}^{2}(0,L)}^{2}<C,$

where $E^{n}$ and $D^{n}$ are defined in (27).

###### Proof.

We add the energy estimates for the two subproblems (28) and (30) to obtain:

(31) $\begin{split}E^{n+1}+D^{n}&+C^{n}_{1}+C^{n}_{2}\leq E^{n}+C\Delta
t((P^{n}_{in})^{2}+(P^{n}_{out})^{2})\\\
&+C\|\Delta_{n}W\|_{U_{0}}^{2}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}+\left|(G({\bf U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf
U}^{n})\right|.\end{split}$

Then for any $m\geq 1$, summing $0\leq n\leq m-1$ gives us

(32)
$\begin{split}&E^{m}+\sum_{n=0}^{m-1}D^{n}+\sum_{n=0}^{m-1}C_{1}^{n}+\sum_{n=0}^{m-1}C_{2}^{n}\leq
E^{0}+C\,\Delta
t\sum_{n=0}^{m-1}\left((P^{n}_{{in}})^{2}+(P^{n}_{{out}})^{2}\right)\\\
&+\sum_{n=0}^{m-1}\left|(G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf
U}^{n})\right|+\sum_{n=0}^{m-1}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}.\end{split}$

Next we take supremum over $1\leq m\leq N$ and then take expectation on both
sides of (32). We begin by treating the right hand side terms. First we have,

$\Delta
t\sum_{n=0}^{N-1}(P^{n}_{{in/out}})^{2}\leq\|P_{in/out}\|^{2}_{L^{2}(0,T)}.$

Next we apply the discrete Burkholder-Davis-Gundy inequality (see e.g. Theorem
2.7 [10]) and using (25) and ${\bf u}^{n+\frac{1}{2}}={\bf u}^{n}$, we obtain
for some $C(\delta)>0$ that,

$\displaystyle{\mathbb{E}}$ $\displaystyle\max_{1\leq m\leq
N}|\sum_{n=0}^{m-1}(G({\bf U}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf
U}^{n})|$ $\displaystyle\leq{C}{\mathbb{E}}\left[\Delta
t\sum_{n=0}^{N-1}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|^{2}_{L_{2}(U_{0},{\bf
L}^{2})}\left(\left\|(\sqrt{J_{*}^{n}}){\bf u}^{n}\right\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf
v}^{n}\|_{L^{2}(0,L)}^{2}\right)\right]^{\frac{1}{2}}$
$\displaystyle\leq{C}{}{\mathbb{E}}\left[\left(\max_{0\leq m\leq
N}\left\|(\sqrt{J_{*}^{m}}){\bf u}^{m}\right\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{m}\|^{2}_{{\bf
L}^{2}(0,L)}\right)\sum_{n=0}^{N-1}\Delta t\left(\|\sqrt{J^{n}_{*}}{\bf
u}^{n}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}\right)\right]^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{2}\|(\sqrt{J_{0}}){\bf u}_{0}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\frac{1}{2}\|{\bf v}_{0}\|^{2}_{{\bf
L}^{2}(0,L)}+\frac{1}{2}{\mathbb{E}}\max_{1\leq m\leq
N}\left[\left\|(\sqrt{J_{*}^{m}}){\bf u}^{m}\right\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{m}\|^{2}_{{\bf L}^{2}(0,L)}\right]$
$\displaystyle+{C\Delta
t}{\mathbb{E}}\left(\sum_{n=0}^{N-1}\|(\sqrt{J^{n}_{*}}){\bf
u}^{n}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}\right),$

Using the tower property and (LABEL:growthG) for each $n=0,...,m-1$ we obtain
we write,

$\displaystyle{\mathbb{E}}[\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}|\mathcal{F}_{n}]]$
$\displaystyle={\mathbb{E}}[\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}{\mathbb{E}}[\|\Delta_{n}W\|_{U_{0}}^{2}\|\mathcal{F}_{n}]]$
$\displaystyle=\Delta t(\text{Tr}{\bf Q}){\mathbb{E}}\|G({\bf
U}^{n},{\boldsymbol{\eta}}_{*}^{n})\|^{2}_{L_{2}(U_{0},{\bf L}^{2})}$ (33)
$\displaystyle\leq C\Delta t(\text{Tr}{\bf
Q}){\mathbb{E}}[\|\sqrt{J_{*}^{n}}{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}].$

Thus we obtain for some $C>0$ depending on $\delta$ and on $\text{Tr}{{\bf
Q}}$, that the following holds:

(34) $\begin{split}{\mathbb{E}}\max_{1\leq n\leq
N}E^{n}&+{\mathbb{E}}\sum_{n=0}^{N-1}D^{n}+{\mathbb{E}}\sum_{n=0}^{N-1}C_{1}^{n}+{\mathbb{E}}\sum_{n=0}^{N-1}C_{2}^{n}\leq
CE^{0}+C\|P_{in/out}\|^{2}_{L^{2}(0,T)}\\\ &+C\Delta
t{\mathbb{E}}\left[\sum_{n=0}^{N-1}\|(\sqrt{J^{n}_{*}}){\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}\right]\\\
&+\frac{1}{2}{\mathbb{E}}\max_{1\leq n\leq
N}\left[\left\|(\sqrt{J_{*}^{n}}){\bf u}^{n}\right\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|^{2}_{{\bf
L}^{2}(0,L)}\right].\end{split}$

Hence by absorbing the last term on the right of (34) we obtain

$\displaystyle{\mathbb{E}}\max_{1\leq n\leq N}$
$\displaystyle\left(\|(\sqrt{J^{n}_{*}}){\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}\right)\leq
CE^{0}+C\|P_{in/out}\|^{2}_{L^{2}(0,T)}$ $\displaystyle+\sum_{n=1}^{N-1}\Delta
t{\mathbb{E}}\max_{1\leq m\leq n}\left(\|(\sqrt{J^{m}_{*}}){\bf
u}^{m}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}^{m}\|^{2}_{{\bf
L}^{2}(0,L)}\right).$

Applying the discrete Gronwall inequality to ${\mathbb{E}}\max_{1\leq n\leq
N}(\|(\sqrt{J^{n}_{*}}){\bf u}^{n}\|_{{\bf L}^{2}(\mathcal{O})}^{2}+\|{\bf
v}^{n}\|^{2}_{{\bf L}^{2}(0,L)})$ we obtain,

${\mathbb{E}}\max_{1\leq n\leq N}\left(\|(\sqrt{J^{n}_{*}}){\bf u}^{n}\|_{{\bf
L}^{2}(\mathcal{O})}^{2}+\|{\bf v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}\right)\leq
Ce^{T},$

where $C$ depends only on the given data and in particular $\delta$.

Hence for $E^{n},D^{n}$ defined in (27) we have,

(35) $\begin{split}{\mathbb{E}}\max_{1\leq n\leq
N}E^{n}+{\mathbb{E}}\sum_{n=0}^{N-1}D^{n}&+{\mathbb{E}}\sum_{n=0}^{N-1}C_{1}^{n}+{\mathbb{E}}\sum_{n=0}^{N-1}C_{2}^{n}\\\
&\leq
C(E^{0}+\|P_{in}\|^{2}_{L^{2}(0,T)}+\|P_{out}\|^{2}_{L^{2}(0,T)}+Te^{T}).\end{split}$

∎

###  3.2. Approximate solutions

In this subsection, we use the solutions $({\bf
u}^{n+\frac{i}{2}},{\boldsymbol{\eta}}^{n+\frac{i}{2}},{\bf
v}^{n+\frac{i}{2}})$, $i=0,1$, defined for every
$N\in\mathbb{N}\setminus\\{0\\}$ at discrete times to define approximate
solutions on the entire interval $(0,T)$. We start by introducing approximate
solutions that are piece-wise constant on each sub-interval $[n\Delta
t,(n+1)\Delta t)$ as

(36) $\displaystyle{\bf u}_{N}(t,\cdot)={\bf
u}^{n},\,{\boldsymbol{\eta}}_{N}(t,\cdot)={\boldsymbol{\eta}}^{n},{\boldsymbol{\eta}}_{N}^{*}(t,\cdot)={\boldsymbol{\eta}}_{*}^{n},{\bf
v}_{N}(t,\cdot)={\bf v}^{n},{{\bf v}}^{\\#}_{N}(t,\cdot)={\bf
v}^{n+\frac{1}{2}}.$

Observe that all of the processes defined above are adapted to the given
filtration $(\mathcal{F}_{t})_{t\geq 0}$.
The following are time-shifted versions of the functions defined in (36):

$\displaystyle{\bf u}^{+}_{N}(t,\cdot)={\bf
u}^{n+1},\quad{\boldsymbol{\eta}}^{+}_{N}(t,\cdot)={\boldsymbol{\eta}}^{n+1}\quad
t\in(n\Delta t,(n+1)\Delta t].$

We also define the corresponding piece-wise linear interpolations as follows:
for $t\in[t^{n},t^{n+1}]$

(37) $\begin{split}&\widetilde{\bf u}_{N}(t,\cdot)=\frac{t-t^{n}}{\Delta
t}{\bf u}^{n+1}+\frac{t^{n+1}-t}{\Delta t}{\bf u}^{n},\quad\widetilde{\bf
v}_{N}(t,\cdot)=\frac{t-t^{n}}{\Delta t}{\bf v}^{n+1}+\frac{t^{n+1}-t}{\Delta
t}{\bf v}^{n}\\\
&\widetilde{\boldsymbol{\eta}}_{N}(t,\cdot)=\frac{t-t^{n}}{\Delta
t}{\boldsymbol{\eta}}^{n+1}+\frac{t^{n+1}-t}{\Delta
t}{\boldsymbol{\eta}}^{n},\quad\widetilde{\boldsymbol{\eta}}^{*}_{N}(t,\cdot)=\frac{t-t^{n}}{\Delta
t}{\boldsymbol{\eta}}^{n+1}_{*}+\frac{t^{n+1}-t}{\Delta
t}{\boldsymbol{\eta}}^{n}_{*}.\end{split}$

Observe that,

(38) $\displaystyle\frac{\partial\widetilde{\boldsymbol{\eta}}_{N}}{\partial
t}={\bf
v}^{\\#}_{N},\quad\frac{\partial\widetilde{\boldsymbol{\eta}}^{*}_{N}}{\partial
t}={\sum_{n=0}^{N-1}\theta_{\delta}({\boldsymbol{\eta}}^{n+1})}{\bf
v}^{\\#}_{N}\chi_{(t^{n},t^{n+1})}:={\bf v}_{N}^{*}\quad a.e.\text{ on
}(0,T),$

where $v^{\\#}_{N}$ was introduced in (36).

######  Lemma 3.3.

Given ${\bf u}_{0}\in{\bf L}^{2}(\mathcal{O})$,
${\boldsymbol{\eta}}_{0}\in{\bf H}^{2}_{0}(0,L)$, ${\bf v}_{0}\in{\bf
L}^{2}(0,L)$, for a fixed $\delta$, we have that

  1. (1)

$\\{{\boldsymbol{\eta}}_{N}\\},\\{{\boldsymbol{\eta}}_{N}^{*}\\}$ and thus
$\\{\widetilde{\boldsymbol{\eta}}_{N}\\},\\{\widetilde{\boldsymbol{\eta}}_{N}^{*}\\}$
are bounded independently of $N$ in $L^{2}(\Omega;L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L)))$.

  2. (2)

$\\{{\bf v}_{N}\\},\\{{\bf v}_{N}^{\\#}\\},\\{{\bf v}_{N}^{*}\\}$ are bounded
independently of $N$ in $L^{2}(\Omega;L^{\infty}(0,T;{\bf L}^{2}(0,L)))$.

  3. (3)

$\\{{\bf u}_{N}\\}$ is bounded independently of $N$ in
$L^{2}(\Omega;L^{\infty}(0,T;{\bf L}^{2}(\mathcal{O})))$.

  4. (4)

$\\{{\bf u}^{+}_{N}\\}$ is bounded independently of $N$ in
$L^{2}(\Omega;L^{2}(0,T;V)\cap L^{\infty}(0,T;{\bf L}^{2}(\mathcal{O}))).$

  5. (5)

$\\{{\sqrt{N}}\text{div}^{{\boldsymbol{\eta}}^{*}_{N}}{\bf u}^{+}_{N}\\}$ is
bounded independently of $N$ in $L^{2}(\Omega;L^{2}(0,T;L^{2}(\mathcal{O})))$.

###### Proof.

The proofs of the first three statements and that of (6) follow immediately
from Theorem 1. In order to prove (4) observe that for each $\omega\in\Omega$,
$\nabla{\bf u}^{n+1}=\nabla^{{\boldsymbol{\eta}}^{n}_{*}}{\bf u}^{n+1}(\nabla
A^{\omega}_{{\boldsymbol{\eta}}^{n}_{*}})$. Thus we have,

$\displaystyle\delta_{1}{\mathbb{E}}\int_{\mathcal{O}}|\nabla{\bf
u}^{n+1}|^{2}d{\bf x}$
$\displaystyle\leq{\mathbb{E}}\int_{\mathcal{O}}(J_{*}^{n})|\nabla{\bf
u}^{n+1}|^{2}d{\bf
x}={\mathbb{E}}\int_{\mathcal{O}}(J^{n}_{*})|\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf
u}^{n+1}\cdot\nabla A^{\omega}_{{\boldsymbol{\eta}}_{*}^{n}}|^{2}d{\bf x}$
$\displaystyle\leq
C(\delta){\mathbb{E}}\int_{\mathcal{O}}(J^{n}_{*})|\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf
u}^{n+1}|^{2}d{\bf
x}\leq{K}C(\delta){\mathbb{E}}\int_{\mathcal{O}}(J_{*}^{n})|{\bf
D}^{{\boldsymbol{\eta}}_{*}^{n}}{\bf u}^{n+1}|^{2}d{\bf x},$

where $K>0$ is the universal Korn constant that depends only on the reference
domain $\mathcal{O}$. This result follows from Lemma 1 in [14] and because of
uniform bounds (25) which imply that
$\\{A^{\omega}_{\boldsymbol{\eta}^{*}_{N}}(t);\,\omega\in\Omega,\,t\in[0,T]\\}$
is compact in ${\bf W}^{1,\infty}(\mathcal{O})$. Thus, there exists $C>0$,
independent of $N$, such that

(39) $\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{O}}|\nabla{\bf
u}^{+}_{N}|^{2}d{\bf x}ds={\mathbb{E}}\sum_{n=0}^{N-1}\Delta
t\int_{\mathcal{O}}|\nabla{\bf u}^{n+1}|^{2}d{\bf x}\leq C(\delta).$

∎

##  4\. Passing $N\rightarrow\infty$

In this section we will obtain almost sure convergence of the stochastic
approximate solutions, which is required to be able to pass
$N\rightarrow\infty$. We use compactness arguments and establish tightness of
the laws of the approximate random variables defined in Section 3.2.

###  4.1.  Tightness results

Given our stochastic setting, the first set of tightness results, i.e. Lemma
4.4 below, will rely on the use of the following three lemmas. The first one
is a classical result (see [12] or [9] for a brief explanation) providing
spaces for uniform bounds of the fractional time derivatives.

######  Lemma 4.1.

Let $\mathcal{Y}_{0}\subset\mathcal{Y}\subset\mathcal{Y}_{1}$ be Banach
spaces, $\mathcal{Y}_{0}$ and $\mathcal{Y}_{1}$ reflexive with compact
embedding of $\mathcal{Y}_{0}$ in $\mathcal{Y}$, then for any $q>1$ and
$\alpha\in(0,1)$, the embedding

$L^{q}(0,T;\mathcal{Y}_{0})\cap
W^{\alpha,q}(0,T;\mathcal{Y}_{1})\hookrightarrow L^{q}(0,T;\mathcal{Y}),$

is compact.

The following two lemmas provide the desired uniform bounds on the fractional
time derivatives of the stochastic approximate solutions. For that purpose we
introduce the translation in time by $h$ of a function $f$:

$T_{h}f(t,\cdot)=f(t-h,\cdot),\quad h\in\mathbb{R}.$

######  Lemma 4.2.

For a fixed $\delta$ there exists a constant $C>0$, dependent on $\delta$, but
independent of $N$ such that,

  1. (1)

${\mathbb{E}}\|T_{h}{\bf u}^{+}_{N}-{\bf u}^{+}_{N}\|^{2}_{L^{2}(h,T;{\bf
L}^{2}(\mathcal{O}))}\leq Ch$,

  2. (2)

${\mathbb{E}}\|T_{h}{\bf v}^{\\#}_{N}-{\bf v}^{\\#}_{N}\|^{2}_{L^{2}(h,T;{\bf
L}^{2}(0,L))}\leq Ch$,

  3. (3)

${\mathbb{E}}\|T_{h}{\boldsymbol{\eta}}_{N}^{+}-{\boldsymbol{\eta}}_{N}^{+}\|^{2}_{L^{2}(h,T;{\bf
H}_{0}^{2}(0,L))}\leq Ch$,
${\mathbb{E}}\|T_{h}\widetilde{\boldsymbol{\eta}}_{N}-\widetilde{\boldsymbol{\eta}}_{N}\|^{2}_{L^{2}(h,T;{\bf
H}_{0}^{2}(0,L))}\leq Ch$,

  4. (4)

${\mathbb{E}}\|T_{h}{\boldsymbol{\eta}}^{*}_{N}-{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{L^{2}(h,T;{\bf
H}_{0}^{2}(0,L))}\leq Ch$,
${\mathbb{E}}\|T_{h}\widetilde{\boldsymbol{\eta}}^{*}_{N}-\widetilde{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{L^{2}(h,T;{\bf
H}_{0}^{2}(0,L))}\leq Ch$.

###### Proof.

Write $h=l\Delta t+s$ for some non-negative integer $l$ and $0\leq s<\Delta
t$. Observe that we can write

$\displaystyle{\mathbb{E}}\|T_{h}{\bf u}^{+}_{N}-$ $\displaystyle{\bf
u}^{+}_{N}\|^{2}_{L^{2}(h,T;{\bf
L}^{2}(\mathcal{O}))}={\mathbb{E}}\int_{h}^{(l+1)\Delta
t}\int_{\mathcal{O}}|T_{h}{\bf u}^{+}_{N}-{\bf u}^{+}_{N}|^{2}d{\bf x}dt$
$\displaystyle+\sum_{k=l+1}^{N-1}\left({\mathbb{E}}\int_{k\Delta t}^{k\Delta
t+s}\int_{\mathcal{O}}|T_{h}{\bf u}^{+}_{N}-{\bf u}^{+}_{N}|^{2}d{\bf
x}dt+{\mathbb{E}}\int_{k\Delta t+s}^{{(k+1)}\Delta
t}\int_{\mathcal{O}}|T_{h}{\bf u}^{+}_{N}-{\bf u}^{+}_{N}|^{2}d{\bf
x}dt\right)$
$\displaystyle=\sum_{k=l+1}^{N-1}{\mathbb{E}}\int_{\mathcal{O}}s|{\bf
u}^{k+1}-{\bf u}^{k-l}|^{2}d{\bf
x}+\sum_{k=l}^{N-1}{\mathbb{E}}\int_{\mathcal{O}}(\Delta t-s)|{\bf
u}^{k+1}-{\bf u}^{k-l+1}|^{2}d{\bf x}.$

Applying the triangle inequality, collecting terms and using Theorem 1 (3) we
obtain

$\displaystyle{\mathbb{E}}\|T_{h}{\bf u}^{+}_{N}-$ $\displaystyle{\bf
u}^{+}_{N}\|^{2}_{L^{2}(h,T;{\bf L}^{2}(\mathcal{O}))}$
$\displaystyle\leq\sum_{k=l+1}^{N-1}\sum_{i=k-l}^{k}{\mathbb{E}}\int_{\mathcal{O}}s|{\bf
u}^{i+1}-{\bf u}^{i}|^{2}d{\bf
x}+\sum_{k=l}^{N-1}\sum_{i=k-l+1}^{k}{\mathbb{E}}\int_{\mathcal{O}}(\Delta
t-s)|{\bf u}^{i+1}-{\bf u}^{i}|^{2}d{\bf x}$
$\displaystyle=\sum_{k=l+1}^{N-1}\sum_{i=k-l}^{k}{\mathbb{E}}\int_{\mathcal{O}}s|{\bf
u}^{i+1}-{\bf u}^{i}|^{2}d{\bf
x}+\sum_{k=l+1}^{N}\sum_{i=k-l}^{k-1}{\mathbb{E}}\int_{\mathcal{O}}(\Delta
t-s)|{\bf u}^{i+1}-{\bf u}^{i}|^{2}d{\bf x}$
$\displaystyle=\sum_{k=0}^{l}(k\Delta t){\mathbb{E}}\int_{\mathcal{O}}|{\bf
u}^{k+1}-{\bf u}^{k}|^{2}d{\bf x}+\sum_{k=l+1}^{N-l-1}(l\Delta
t+s){\mathbb{E}}\int_{\mathcal{O}}|{\bf u}^{k+1}-{\bf u}^{k}|^{2}d{\bf x}$
$\displaystyle\quad\quad+\sum_{k=N-l}^{N-1}((N-k)\Delta
t){\mathbb{E}}\int_{\mathcal{O}}|{\bf u}^{k+1}-{\bf u}^{k}|^{2}d{\bf x}$
$\displaystyle\leq h\sum_{k=1}^{N-1}{\mathbb{E}}\int_{\mathcal{O}}|{\bf
u}^{k+1}-{\bf u}^{k}|^{2}d{\bf x}$
$\displaystyle\leq\frac{h}{\delta_{1}}{\mathbb{E}}\sum_{k=1}^{N-1}\int_{\mathcal{O}}(J^{k}_{*})|{\bf
u}^{k+1}-{\bf u}^{k}|^{2}d{\bf x}\leq\frac{Ch}{\delta_{1}}.$

We can prove statements (2) and (3)1 identically using the bounds on the
expectation of the numerical dissipation as observed in Theorem 1 (3)-(4) and
the Poincaré inequality. To prove (4)1, in the last step we additionally use
the fact that
$|{\boldsymbol{\eta}}^{n+1}_{*}-{\boldsymbol{\eta}}^{n}_{*}|=\theta_{\delta}({\boldsymbol{\eta}}^{n+1})|{\boldsymbol{\eta}}^{n+1}-{\boldsymbol{\eta}}^{n}|$.
The second halves of (3) and (4) also follow similarly albeit with a few extra
calculations. ∎

######  Lemma 4.3.

For any $0<\alpha<\frac{1}{2}$, there exists a constant $C>0$ that depends on
$\delta$ but is independent of $N$ such that

  1. (1)

${\mathbb{E}}\|{\bf u}^{+}_{N}\|^{2}_{H^{\alpha}(0,T;{\bf
L}^{2}(\mathcal{O}))}\leq C$,

  2. (2)

${\mathbb{E}}\|\widetilde{\boldsymbol{\eta}}_{N}\|^{2}_{H^{\alpha}(0,T;{\bf
H}_{0}^{2}(0,L))\cap H^{\alpha+1}(0,T;{\bf L}^{2}(0,L))}\leq C$,

  3. (3)

${\mathbb{E}}\|{\bf v}^{\\#}_{N}\|^{2}_{H^{\alpha}(0,T;{\bf L}^{2}(0,L))\cap
L^{2}(0,T;{\bf H}^{2\alpha}(0,L))}\leq C$,

  4. (4)

${\mathbb{E}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{H^{\alpha}(0,T;{\bf
H}_{0}^{2}(0,L))\cap H^{1}(0,T;{\bf L}^{2}(0,L))}\leq C$.

  5. (5)

${\mathbb{E}}\|{\bf v}^{*}_{N}\|^{2}_{H^{\alpha}(0,T;{\bf L}^{2}(0,L))\cap
L^{2}(0,T;{\bf H}^{2\alpha}(0,L))}\leq C.$

###### Proof.

Thanks to Theorem 1 (also Lemma 3.3) we know that $\\{{\bf u}^{+}_{N}\\}_{N}$
is bounded independently of $N$ in $L^{2}(\Omega;L^{2}(0,T;{\bf
L}^{2}(\mathcal{O})))$. A change of variables gives us,

$\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(s)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{|t-s|^{1+2\alpha}}dtds$
$\displaystyle=2{\mathbb{E}}\int_{0}^{T}\int_{s}^{T}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(s)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{(t-s)^{1+2\alpha}}dtds$
$\displaystyle=2{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\mathbbm{1}_{\\{t>s\\}}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(s)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{(t-s)^{1+2\alpha}}dsdt$
$\displaystyle=2{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\mathbbm{1}_{\\{t>h>0\\}}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(t-h)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{h^{1+2\alpha}}dhdt$
$\displaystyle=2{\mathbb{E}}\int_{0}^{T}\int_{h}^{T}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(t-h)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{h^{1+2\alpha}}dtdh.$

Pulling ${\mathbb{E}}$ inside the first integral and using result (1) from
Lemma 4.2 we obtain the following upper bounds when $0<\alpha<\frac{1}{2}$,

$\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{\|{\bf
u}^{+}_{N}(t)-{\bf u}_{N}^{+}(s)\|_{{\bf
L}^{2}(\mathcal{O})}^{2}}{|t-s|^{1+2\alpha}}dtds$ $\displaystyle\leq
2\int_{0}^{T}\frac{Ch}{h^{1+2\alpha}}dh\leq C.$

This completes the proof of (1).

The uniform $H^{\alpha}$-bounds in the first parts of (2)-(4) are proven
similarly. To prove the second parts of (2)-(4), we follow steps similar to
those used in [8]. Observe that the definition of ${\bf v}_{N}^{\\#}$ and the
$H^{\alpha}$ estimate of ${\bf v}_{N}^{\\#}$ in (3) means that
$\partial_{t}\widetilde{\boldsymbol{\eta}}_{N}$ is bounded independently of
$N$ in $L^{2}(\Omega;H^{\alpha}(0,T;{\bf L}^{2}(0,L)))$. Thus, combining it
with the $H^{\alpha}$ estimate in (2) we obtain that
$\widetilde{\boldsymbol{\eta}}_{N}$ are bounded independently of $N$ in
$L^{2}(\Omega;H^{\alpha+1}(0,T;{\bf L}^{2}(0,L)))$. This completes the proof
of (2) and similarly that of (4).

Furthermore, interpolation implies that the random variables
$\widetilde{\boldsymbol{\eta}}_{N}$ are bounded uniformly in $N$ in
$L^{2}(\Omega;H^{1}(0,T;{\bf H}^{2\alpha}(0,L)))$. This in turn implies that
${\bf v}^{\\#}_{N}$ is bounded independently of $N$ in
$L^{2}(\Omega;L^{2}(0,T;{\bf H}^{2\alpha}(0,L)))$ thus proving the second part
of (4).

To prove (5) we define

$\theta_{N}(t,\omega)=\sum_{n=0}^{N-1}\theta_{\delta}({\boldsymbol{\eta}}^{n+1}(\omega))\mathbbm{1}_{[t^{n},t^{n+1})}(t).$

Then we have ${\bf v}^{*}_{N}(t,\omega)=\theta_{N}(t,\omega){\bf
v}^{\\#}_{N}(t,\omega)$ and we can write,

$\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{\|{\bf
v}^{*}_{N}(t)-{\bf v}_{N}^{*}(s)\|_{{\bf
L}^{2}(0,L)}^{2}}{|t-s|^{1+2\alpha}}dtds$ $\displaystyle\leq
2{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{|\theta_{N}(t)|^{2}\|{\bf
v}^{\\#}_{N}(t)-{\bf v}_{N}^{\\#}(s)\|_{{\bf
L}^{2}(0,L)}^{2}}{|t-s|^{1+2\alpha}}dtds$
$\displaystyle+2{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{\|{\bf
v}^{\\#}(s)\|^{2}_{{\bf
L}^{2}(0,L)}|\theta_{N}(t)-\theta_{N}(s)|^{2}}{|t-s|^{1+2\alpha}}dtds$
$\displaystyle\leq 2{\mathbb{E}}\int_{0}^{T}\int_{0}^{T}\frac{\|{\bf
v}^{\\#}_{N}(t)-{\bf v}_{N}^{\\#}(s)\|_{{\bf
L}^{2}(0,L)}^{2}}{|t-s|^{1+2\alpha}}dtds$
$\displaystyle+2{\mathbb{E}}\left(\|{\bf v}^{\\#}\|^{2}_{L^{\infty}(0,T;{\bf
L}^{2}(0,L))}\int_{0}^{T}\int_{0}^{T}\frac{|\theta_{N}(t)-\theta_{N}(s)|^{2}}{|t-s|^{1+2\alpha}}dtds\right).$

Now observe that, for any fixed $\omega$ and ${n^{*}(\omega)}:=\max\\{n\leq
N:\theta_{\delta}({\boldsymbol{\eta}}^{n}(\omega))=1\\}$, we can write,

$\displaystyle\int_{0}^{T}\int_{0}^{T}\frac{|\theta_{N}(t)-\theta_{N}(s)|^{2}}{|t-s|^{1+2\alpha}}dtds$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\Big{(}1+\int_{1}^{\infty}\mathbbm{1}_{\\{(t,s):\frac{|\theta_{N}(t)-\theta_{N}(s)|^{2}}{|t-s|^{1+2\alpha}}>x\\}}dx\Big{)}dtds$
$\displaystyle=T^{2}+2\int_{1}^{\infty}\int_{t^{n^{*}(\omega)}-x^{\frac{-1}{2\alpha+1}}}^{t^{n^{*}(\omega)}}\int_{t^{n^{*}(\omega)}}^{s+x^{\frac{-1}{2\alpha+1}}}dtdsdx$
$\displaystyle=T^{2}+\int_{1}^{\infty}x^{\frac{-2}{2\alpha+1}}dx=T^{2}+\frac{2\alpha+1}{1-2\alpha}\quad\text{
since }\alpha<\frac{1}{2}.$

Hence using the first part of statement (3) and Lemma 3.3 (2) we obtain the
first part of (5) (i.e., the $H^{\alpha}$ bounds). Finally, the second part of
(5) follows immediately from the second part of (3). ∎

Now we are in a position to state our first tightness results. Lemma 4.1
combined with Theorem 3.3 and Lemma 4.3 gives us the following result.

######  Lemma 4.4.

(First tightness result) The following statements hold:

  1. (1)

The laws of $\\{\widetilde{\boldsymbol{\eta}}_{N}\\}_{N\in\mathbb{N}}$ and
that of $\\{\widetilde{\boldsymbol{\eta}}^{*}_{N}\\}_{N\in\mathbb{N}}$ are
tight in $C([0,T],{\bf H}^{s}(0,L))$ for any $s<2$.

  2. (2)

The laws of $\\{{\bf u}^{+}_{N}\\}_{N\in\mathbb{N}}$ are tight in
$L^{2}(0,T,{\bf H}^{k}(\mathcal{O}))$ for any $k<1$.

  3. (3)

The laws of $\\{{\bf v}^{\\#}_{N}\\}_{N\in\mathbb{N}}$ and that of $\\{{\bf
v}^{*}_{N}\\}_{N\in\mathbb{N}}$ are tight in $L^{2}(0,T,{\bf H}^{k}(0,L))$ for
any $k<1$.

  4. (4)

The laws of $\\{\|{\bf u}^{+}_{N}\|_{L^{2}(0,T;{\bf
H}^{1}(\mathcal{O}))}\\}_{N\in\mathbb{N}}$ are tight in $\mathbb{R}$.

  5. (5)

The laws of $\\{\|\boldsymbol{\eta}^{*}_{N}\|_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(\Gamma))}\\}_{N\in\mathbb{N}}$ are tight in $\mathbb{R}$.

###### Proof.

To prove the first statement we observe, thanks to Lemma 3.3, that
$\widetilde{\boldsymbol{\eta}}_{N}$ and
$\widetilde{\boldsymbol{\eta}}_{N}^{*}$ are bounded independently of $N$ in
$L^{2}(\Omega;L^{\infty}(0,T;{\bf H}^{2}(0,L))\cap W^{1,\infty}(0,T;{\bf
L}^{2}(0,L)))$. Now the Aubin-Lions theorem implies that for $0<s<2$

$L^{\infty}(0,T;{\bf H}^{2}(0,L))\cap W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))\subset\subset C([0,T];{\bf H}^{s}(0,L)).$

Hence consider

$\displaystyle\mathcal{K}_{M}:=$ $\displaystyle\\{{\boldsymbol{\eta}}\in
L^{\infty}(0,T;{\bf H}^{2}_{0}(0,L))\cap W^{1,\infty}(0,T;{\bf L}^{2}(0,L)):$
$\displaystyle\|{\boldsymbol{\eta}}\|^{2}_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L))}+\|{\boldsymbol{\eta}}\|^{2}_{W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))}\leq M\\}.$

Using the Chebyshev inequality once again we obtain for some $C>0$ independent
of $N$ that the following holds:

(40)
$\displaystyle{\mathbb{P}}\left[\widetilde{\boldsymbol{\eta}}_{N}\not\in{\mathcal{K}}_{M}\right]$
$\displaystyle\leq{\mathbb{P}}\left[{\|\widetilde{\boldsymbol{\eta}}_{N}\|}^{2}_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L))}\geq\frac{M}{2}\right]+{\mathbb{P}}\left[{\|\widetilde{\boldsymbol{\eta}}_{N}\|}^{2}_{W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))}\geq\frac{M}{2}\right]$
$\displaystyle\leq\frac{4}{M^{2}}{\mathbb{E}}\left[{\|\widetilde{\boldsymbol{\eta}}_{N}\|}^{2}_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L))}+\|\widetilde{\boldsymbol{\eta}}_{N}\|^{2}_{W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))}\right]\leq\frac{C}{M^{2}}.$

The proof of the statements (2)-(3) follow by a similar application of the
Chebyshev inequality using the bounds obtained in Lemmas 3.3 and 4.3 together
with Lemma 4.1. Finally, to prove statement (4), and similarly statement (5),
we use the Chebyshev inequality again to write for any $M>0$,

$\displaystyle{\mathbb{P}}[\|{\bf u}^{+}_{N}\|_{L^{2}(0,T;{\bf
H}^{1}(\mathcal{O}))}>M]\leq\frac{1}{M^{2}}{\mathbb{E}}[\|{\bf
u}^{+}_{N}\|^{2}_{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}]\leq\frac{C}{M^{2}}.$

This completes the proof of the tightness results stated in Lemma 4.4. ∎

To obtain almost sure convergence of the rest of the random variables we will
use the following lemma.

######  Lemma 4.5.

The following convergence results hold:

  1. (1)

$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\bf u}_{N}-{\bf
u}^{+}_{N}\|^{2}_{{\bf L}^{2}(\mathcal{O})}dt=0$,
$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\bf u}_{N}-\widetilde{\bf
u}_{N}\|^{2}_{{\bf L}^{2}(\mathcal{O})}dt=0$

  2. (2)

$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\boldsymbol{\eta}}_{N}-\widetilde{\boldsymbol{\eta}}_{N}\|^{2}_{{\bf
H}_{0}^{2}(0,L)}dt=0$,
$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\boldsymbol{\eta}}^{+}_{N}-\widetilde{\boldsymbol{\eta}}_{N}\|^{2}_{{\bf
H}_{0}^{2}(0,L)}dt=0$

  3. (3)

$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\boldsymbol{\eta}}^{*}_{N}-\widetilde{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{{\bf
H}_{0}^{2}(0,L)}dt=0$

  4. (4)

$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\bf v}_{N}-\widetilde{\bf
v}_{N}\|^{2}_{{\bf L}^{2}(0,L)}dt=0$

  5. (5)

$\lim_{N\rightarrow\infty}{\mathbb{E}}\int_{0}^{T}\|{\bf v}_{N}-{\bf
v}^{\\#}_{N}\|^{2}_{{\bf L}^{2}(0,L)}dt=0$.

###### Proof.

Statement (1)1 follows immediately from Theorem 1 (3). We prove (1)2 below.
The rest follows similarly from the uniform estimates stated in Theorem 1.

$\displaystyle{\mathbb{E}}\int_{0}^{T}\|{\bf u}_{N}-\widetilde{\bf
u}_{N}\|^{2}_{{\bf L}^{2}(\mathcal{O})}dt$
$\displaystyle={\mathbb{E}}\sum_{n=0}^{N-1}\int_{t^{n}}^{t^{n+1}}\frac{1}{\Delta
t}\|(t-t^{n}){\bf u}^{n+1}+(t^{n+1}-t-\Delta t){\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}dt$ $\displaystyle={\mathbb{E}}\sum_{n=0}^{N-1}\|{\bf
u}^{n+1}-{\bf u}^{n}\|_{{\bf
L}^{2}(\mathcal{O})}^{2}\int_{t^{n}}^{t^{n+1}}\left(\frac{t-t^{n}}{\Delta
t}\right)^{2}dt$ $\displaystyle\leq\frac{CT}{\delta_{1}N}\rightarrow 0\quad
N\rightarrow\infty.$

∎

###  4.2. Almost sure convergence

Let $\mu_{N}$ be the joint law of the random variable
$\mathcal{U}_{N}:=({\bf
u}^{+}_{N},v^{\\#}_{N},v^{*}_{N},\widetilde{\eta}^{*}_{N},\widetilde{\eta}_{N},\|{\bf
u}^{+}_{N}\|_{L^{2}(0,T;{\bf
H}^{1}(\mathcal{O}))},\|\boldsymbol{\eta}^{*}_{N}\|_{L^{\infty}(0,T;{\bf
H}^{2}(\Gamma))},W)$ taking values in the phase space

$\displaystyle\Upsilon$ $\displaystyle:=[L^{2}(0,T;{\bf
H}^{\frac{1}{2}}(\mathcal{O}))]\times[L^{2}(0,T;{\bf
H}^{\frac{3}{4}}(0,L))]^{2}\times[C([0,T],{\bf
H}^{s}(0,L))]^{2}\times\mathbb{R}^{2}\times C([0,T];U),$

for some fixed $\frac{3}{2}<s<2$.

Since $C([0,T];U)$ is separable and metrizable by a complete metric, the
sequence of Borel probability measures,
$\mu^{W}_{N}(\cdot):={\mathbb{P}}(W\in\cdot)$, that are constantly equal to
one element, is tight on $C([0,T];U)$.

Next, recalling Lemmas 4.4, 4.7 and using Tychonoff’s theorem it follows that
the sequence of the joint probability measures $\mu_{N}$ of the approximating
sequence $\mathcal{U}_{N}$ is tight on the Polish space $\Upsilon$. Hence, by
applying the Prohorov theorem and the Skorohod representation theorem we
obtain the following result. To be precise we use the version given in Theorem
1.10.4 in [13].

######  Theorem 2.

There exists a probability space
$(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})$ and random variables
$\bar{\mathcal{U}}_{N}:=(\bar{{\bf u}}^{+}_{N},\bar{\bf v}^{\\#}_{N},\bar{\bf
v}^{*}_{N},\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N},\bar{\widetilde{\boldsymbol{\eta}}}_{N},m_{N},k_{N},\bar{\widetilde{\bf
U}}_{N},\bar{W}_{N})$ and $\bar{\mathcal{U}}:=(\bar{{\bf u}},{\bar{\bf
v}},\bar{\bf
v}^{*},\bar{\boldsymbol{\eta}}^{*},\bar{\boldsymbol{\eta}},m,k,\bar{\widetilde{\bf
U}},\bar{W})$ defined on this new probability space, such that

$\bar{\mathcal{U}}_{N}\rightarrow\bar{\mathcal{U}},\qquad\quad\bar{\mathbb{P}}-a.s.\quad\text{in
the topology of }\Upsilon.$

Additionally, there exist measurable maps
$\phi_{N}:\bar{\Omega}\rightarrow\Omega$ such that

(41)
$\displaystyle\bar{\mathcal{U}}_{N}(\bar{\omega})={\mathcal{U}}_{N}(\phi_{N}(\bar{\omega}))\quad\text{
for }\bar{\omega}\in\bar{\Omega},$

and $\bar{\mathbb{P}}\circ\phi_{N}^{-1}={\mathbb{P}}$

Now we will define,

$\displaystyle\bar{\bf u}_{N}={\bf
u}_{N}\circ\phi_{N},\quad\bar{\widetilde{\bf u}}_{N}=\widetilde{\bf
u}_{N}\circ\phi_{N},$
$\displaystyle\bar{\boldsymbol{\eta}}_{N}={\boldsymbol{\eta}}_{N}\circ\phi_{N},\quad{\bar{\boldsymbol{\eta}}}^{+}_{N}={\boldsymbol{\eta}}^{+}_{N}\circ\phi_{N},\quad\bar{\boldsymbol{\eta}}^{*}_{N}={\boldsymbol{\eta}}^{*}_{N}\circ\phi_{N},$
$\displaystyle\bar{\bf v}_{N}={\bf v}_{N}\circ\phi_{N}.$

Then, from part (1), Lemmas 4.5 and an application of the Borel-Cantelli
lemma, we obtain

(42) $\displaystyle\bar{\bf u}_{N}\to\bar{\bf u},\quad\bar{\widetilde{\bf
u}}_{N}\to\bar{\bf u},\quad\bar{\mathbb{P}}-a.s.\quad\text{in }L^{2}(0,T;{\bf
L}^{2}(\mathcal{O})),$ (43)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\to\bar{\boldsymbol{\eta}},\quad{\bar{\boldsymbol{\eta}}}^{+}_{N}\to\bar{\boldsymbol{\eta}},\quad\bar{\boldsymbol{\eta}}^{*}_{N}\to\bar{\boldsymbol{\eta}}^{*},\quad\bar{\mathbb{P}}-a.s.\quad\text{in
}L^{2}(0,T;{\bf H}^{s}(0,L)),$ (44) $\displaystyle\bar{\bf v}_{N}\to\bar{\bf
v},\quad\bar{\mathbb{P}}-a.s.\quad\text{ in }L^{2}(0,T;{\bf L}^{2}(0,L)).$

Thanks to these explicit maps we can identify the real-valued random variables
${m}_{N}=\|\bar{\bf u}^{+}_{N}\|_{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}$. Hence
almost sure convergence of ${m}_{N}$ implies that $\|\bar{\bf
u}^{+}_{N}\|_{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}$ is bounded a.s. and thus,
up to a subsequence,

(45) $\displaystyle\bar{\bf u}^{+}_{N}\rightharpoonup\bar{\bf u}\quad\text{
weakly in }{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}\quad\bar{\mathbb{P}}-a.s.$

Using the same argument we also obtain that

(46)
$\displaystyle\bar{\boldsymbol{\eta}}^{*}_{N}\rightharpoonup\bar{\boldsymbol{\eta}}^{*}\quad\text{
weakly* in }{L^{\infty}(0,T;{\bf
H}_{0}^{2}(\Gamma))}\quad\bar{\mathbb{P}}-a.s.$

Observe also that, since $\mathcal{O}$ is a Lipschitz domain, the space ${{\bf
H}^{\gamma}(\mathcal{O})}$ is continuously embedded in ${{\bf
L}^{q}(\mathcal{O})}$ for any $q\in[1,\frac{2}{1-\gamma}]$ and $\gamma<1$ and
thus Theorem 2 implies that

(47) $\displaystyle\bar{\bf u}_{N}\rightarrow\bar{\bf u}\quad\text{in
}L^{2}(0,T;{\bf L}^{4}(\mathcal{O})),\quad\bar{\mathbb{P}}-a.s.$

Similarly, since $H^{\frac{3}{4}}(0,L)\hookrightarrow C^{0,\frac{1}{4}}(0,L)$
, we obtain

(48) $\displaystyle\bar{\bf v}^{*}_{N}\rightarrow\bar{\bf v}^{*}\quad\text{ in
}\,L^{2}(0,T;{\bf L}^{\infty}(0,L)),\quad\bar{\mathbb{P}}-a.s.$

Furthermore, we have,

(49) $\displaystyle\partial_{t}\bar{\boldsymbol{\eta}}=\bar{\bf v}\text{ and
}\partial_{t}\bar{\boldsymbol{\eta}}^{*}=\bar{\bf v}^{*},\quad\text{ in the
sense of distributions, almost surely.}$

This is true since we have

$-\int_{0}^{T}\int_{0}^{L}\widetilde{\boldsymbol{\eta}}_{N}\partial_{t}\boldsymbol{\phi}dzdt=\int_{0}^{T}\int_{0}^{L}{{\bf
v}^{\\#}_{N}\boldsymbol{\phi}}dzdt,\quad\forall\phi\in C_{0}^{1}(0,T;{\bf
L}^{2}(0,L)),$

which together with Theorem 2 implies,

$-\int_{0}^{T}\int_{0}^{L}\bar{\widetilde{\boldsymbol{\eta}}}_{N}\partial_{t}\boldsymbol{\phi}dzdt=\int_{0}^{T}\int_{0}^{L}{\bar{{\bf
v}}^{\\#}_{N}\boldsymbol{\phi}}dzdt,\quad\forall\boldsymbol{\phi}\in
C_{0}^{1}(0,T;{\bf L}^{2}(0,L)).$

Then passing $N\rightarrow\infty$ we come to the desired conclusion. The
second half of the statement can be proven similarly.

Observe also that the bounds obtained in Lemma 3.3 hold for the new random
variables $\bar{\mathcal{U}}_{N}$ as well. As a result, we obtain

(50) $\begin{split}&\bar{\bf u}\in L^{2}(\bar{\Omega};L^{\infty}(0,T;{\bf
L}^{2}(\mathcal{O})))\cap L^{2}(\Omega;L^{2}(0,T;V)),\\\ &\bar{\bf v},\bar{\bf
v}^{*}\in L^{2}(\bar{\Omega};L^{\infty}(0,T;{\bf L}^{2}(0,L))),\\\
&\bar{{\boldsymbol{\eta}}},\bar{\boldsymbol{\eta}}^{*}\in
L^{2}(\bar{\Omega};L^{\infty}(0,T;{\bf H}^{2}_{0}(0,L)))\cap
L^{2}(\bar{\Omega};W^{1,\infty}(0,T;{\bf L}^{2}(0,L))),\end{split}$

where $\bar{\bf u},\bar{\bf v},\bar{\bf v}^{*},\bar{{\boldsymbol{\eta}}}$, and
$\bar{\boldsymbol{\eta}}^{*}$ are the limits defined in Theorem 2.

We also have the following upgraded convergence results for the displacements.
Namely, notice that Theorem 2 statement (2) implies that for given
$\frac{3}{2}<s<2$ (see [7] Lemma 3),

(51)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\rightarrow\bar{\boldsymbol{\eta}}\text{
and
}\bar{\boldsymbol{\eta}}^{*}_{N}\rightarrow\bar{\boldsymbol{\eta}}^{*}\,\,\,\text{
in }L^{\infty}(0,T;{\bf H}^{s}(0,L))\,a.s.$

and thus the following uniform convergence result holds

(52)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\rightarrow\bar{\boldsymbol{\eta}}\text{
and
}\bar{\boldsymbol{\eta}}^{*}_{N}\rightarrow\bar{\boldsymbol{\eta}}^{*}\,\,\,\text{
in }L^{\infty}(0,T;{\bf C}^{1}[0,L])\,a.s.$

We also define $\bar{\bf n}^{n}_{*}$ to be the unit outward normal to
$\Gamma_{\bar{\boldsymbol{\eta}}^{n}_{*}}$ and $\bar{\bf
w}_{*}^{n+1}=\frac{1}{\Delta
t}(A^{\omega}_{\bar{\boldsymbol{\eta}}_{*}^{n+1}}-A^{\omega}_{\bar{\boldsymbol{\eta}}^{n}_{*}})$
and, $\bar{J}^{n}_{*}=\text{det}\nabla
A^{\omega}_{\bar{\boldsymbol{\eta}}^{n}_{*}}$ where, for any fixed $N$, we
define $\bar{\bf u}^{n}={\bf u}^{n}\circ\phi_{N},\bar{\bf v}^{n}={\bf
v}^{n}\circ\phi_{N}$,
$\bar{\boldsymbol{\eta}}^{n}={\boldsymbol{\eta}}^{n}\circ\phi_{N}$ and
$\bar{\boldsymbol{\eta}}_{*}^{n}={\boldsymbol{\eta}}_{*}^{n}\circ\phi_{N}$ for
all $n=1,2,..,N$. Using these notations, we define piecewise constant
interpolations $\bar{S}_{N}$, $\bar{\tau}_{N}$, $\bar{\bf n}^{*}_{N}$,
$\bar{\bf w}^{*}_{N},\bar{J}^{*}_{N}$, $A_{\bar{\boldsymbol{\eta}}^{*}_{N}}$,
and the piecewise linear interpolation $\widetilde{\bar{J}}^{*}_{N}$. Observe
that, we have

(53) $\begin{split}&\bar{S}_{N}\rightarrow
S_{\bar{\boldsymbol{\eta}}}\quad\text{ in
}L^{\infty}(0,T;C(\bar{\Gamma}))\,a.s.\\\
&\bar{\tau}_{N}\rightarrow\tau^{\bar{\boldsymbol{\eta}}}\quad\text{ in
}L^{\infty}(0,T;C(\bar{\Gamma}))\,a.s.\\\ &\bar{\bf n}^{*}_{N}\rightarrow{\bf
n}^{\bar{\boldsymbol{\eta}}^{*}}\quad\text{ in
}L^{\infty}(0,T;C(\bar{\Gamma}))\,a.s.\end{split}$

As defined in (19) we also introduce
$\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t,\omega)(z)=z+(\bar{\eta}^{*}_{N})_{z}(t,z,\omega)$
and
$\varphi_{\bar{\boldsymbol{\eta}}^{*}}(t,\omega)(z)=z+\bar{\eta}_{z}^{*}(t,z,\omega)$.
Note here that the subscript $z$ represents the first coordinate of the
vectors. Now since $\delta_{2}$ satisfies (17) we have

$\displaystyle|\partial_{z}(\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t,\omega))^{-1}(z)-\partial_{z}(\varphi_{\bar{\boldsymbol{\eta}}^{*}}(t,\omega))^{-1}(z)|\leq\frac{|\partial_{z}\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t,\omega)(\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}}(z))-\partial_{z}\varphi_{\bar{\boldsymbol{\eta}}^{*}}(t,\omega)(\varphi_{\bar{\boldsymbol{\eta}}^{*}}(z))|}{C(\delta_{2})}$

Hence, the mean value theorem and a change of variables give us for any
$\omega\in\bar{\Omega}$,

$\displaystyle\|\partial_{z}(\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}})^{-1}-\partial_{z}(\varphi_{\bar{\boldsymbol{\eta}}^{*}})^{-1}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\leq
C(\delta_{2})\|\partial_{zz}(\bar{\eta}^{*}_{N})_{z}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\|\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\varphi_{\bar{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}$
$\displaystyle+\|\partial_{z}(\bar{\eta}^{*}_{N})_{z}-\partial_{z}\bar{\eta}^{*}_{z}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}\|\partial_{z}\varphi_{\bar{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T;L^{2}(\Gamma))}.$

Thanks to (51) and (46) we obtain

(54)
$\displaystyle(\varphi_{\bar{\boldsymbol{\eta}}^{*}_{N}})^{-1}\rightarrow(\varphi_{\bar{\boldsymbol{\eta}}^{*}})^{-1}\quad\text{in
}L^{\infty}(0,T;H^{1}(\Gamma))\quad\bar{\mathbb{P}}-a.s.$

The convergence in (54) also holds for the inverse of
$\varphi_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(t,\omega)(z)=z+(\bar{\widetilde{\eta}}^{*}_{N})_{z}(t,z,\omega)$.
Hence we obtain,

(55) $\begin{split}&\nabla
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}\rightarrow\nabla
A_{\bar{\boldsymbol{\eta}}^{*}}\text{ and }(\nabla
A_{\bar{\boldsymbol{\eta}}^{*}_{N}})^{-1}\rightarrow(\nabla
A_{\bar{\boldsymbol{\eta}}^{*}})^{-1}\quad\text{ in }L^{\infty}(0,T;{\bf
C}(\mathcal{O}))\,a.s.\\\ &\bar{J}^{*}_{N}\rightarrow
J_{\bar{\boldsymbol{\eta}}^{*}}\quad\text{ in
}L^{\infty}(0,T;C(\bar{\mathcal{O}}))\,a.s.,\end{split}$

where for every $\omega\in\bar{\Omega}$,

$A_{\bar{\boldsymbol{\eta}}^{*}}(t)(z,r)=(z,(1+\bar{\eta}^{*}_{r}(t,\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}(t)(z)))r),\quad\text{and}\quad
J_{\bar{\boldsymbol{\eta}}^{*}}(t)=\text{det}\nabla
A_{{\bar{\boldsymbol{\eta}}^{*}}}(t).$

Next, observe that, since
$\partial_{t}\varphi_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}=\partial_{t}({\bar{\widetilde{\eta}}}_{N}^{*})_{z}=(\bar{v}_{N})_{z}^{*}$,
we can write

$\displaystyle\partial_{t}\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}=\frac{(\bar{v}_{N}^{*})_{z}\circ\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}}{\partial_{z}\varphi_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}\circ\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}},\quad\text{and
}\quad\partial_{t}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}=\frac{\bar{v}^{*}_{z}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}}{\partial_{z}\varphi_{\bar{\boldsymbol{\eta}}^{*}}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}}.$

Hence we obtain that

$\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}\rightharpoonup\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}\text{
weakly in }H^{1}(0,T;L^{2}(0,L))\quad\bar{\mathbb{P}}-a.s.$

We will need to upgrade his weak convergence. Hence, we will show the above
convergence in norm as follows.

Recall that, due to the explicit representation (41), for any $t\in[0,T]$ and
$\omega\in\Omega$, the maps
$\varphi_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(t,\omega),\varphi_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}}(t,\omega):[0,L]\to[0,L]$
, for any fixed $t\in[0,T],\omega\in\bar{\Omega}$, are bijective. Hence a
change of variables then gives us,

$\displaystyle\|\partial_{t}\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}\|_{L^{2}(0,T;L^{2}(0,L))}=\int_{0}^{T}\int_{0}^{L}\frac{|(\bar{v}^{*}_{N})_{z}(t,z)|^{2}}{1+\partial_{z}(\bar{\widetilde{\eta}}^{*}_{N})_{z}(t,z)}\,dzdt,\quad\text{
and },$
$\displaystyle\|\partial_{t}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}\|_{L^{2}(0,T;L^{2}(0,L))}=\int_{0}^{T}\int_{0}^{L}\frac{|\bar{v}^{*}_{z}(t,z)|^{2}}{1+\partial_{z}\bar{\widetilde{\eta}}^{*}_{z}(t,z)}\,dzdt.$

Since (17) holds for every artificial structure we obtain,

$\displaystyle|\|\partial_{t}\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}\|_{L^{2}(0,T;L^{2}(0,L))}-\|\partial_{t}\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}}\|_{L^{2}(0,T;L^{2}(0,L))}|$
$\displaystyle\leq\int_{0}^{T}\int_{0}^{L}\Big{|}\frac{|(\bar{v}^{*}_{N})_{z}(t,z)|^{2}-|\bar{v}^{*}_{z}(t,z)|^{2}}{1+\partial_{z}(\bar{\widetilde{\eta}}^{*}_{N})_{z}}\Big{|}\,dzdt+\int_{0}^{T}\int_{0}^{L}|\bar{v}^{*}_{z}(t,z)|^{2}\left(\frac{|\partial_{z}(\bar{\widetilde{\eta}}^{*}_{N})_{z}-\partial_{z}\bar{\widetilde{\eta}}^{*}_{z}|}{|(1+\partial_{z}(\bar{\widetilde{\eta}}^{*}_{N})_{z})(1+\partial_{z}\bar{\widetilde{\eta}}^{*}_{z})}\right)\,dzdt$
$\displaystyle\leq
C(\delta)\int_{0}^{T}\int_{0}^{L}\Big{|}{|(\bar{v}^{*}_{N})_{z}(t,z)|^{2}-|\bar{v}^{*}_{z}(t,z)|^{2}}\Big{|}\,dzdt+\int_{0}^{T}\int_{0}^{L}|\bar{v}^{*}_{z}(t,z)|^{2}\left({|\partial_{z}(\bar{\widetilde{\eta}}^{*}_{N})_{z}-\partial_{z}\bar{\widetilde{\eta}}^{*}_{z}|}\right)\,dzdt$
$\displaystyle\to 0\quad\bar{\mathbb{P}}-a.s.$

The first term above converges to 0 almost surely since $\bar{\bf
v}^{*}_{N}\to\bar{\bf v}^{*}$ a.s. in $L^{2}(0,T;{\bf L}^{2}(0,L))$ whereas
the second term converges to 0 a.s. since
$\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}\to\bar{\boldsymbol{\eta}}^{*}$ in
$C(0,T;{\bf H}^{s}(0,L))$ a.s. for $s>\frac{3}{2}$ and thus
$\partial_{z}\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}\to\partial_{z}\bar{\boldsymbol{\eta}}^{*}$
in $C(0,T;{\bf L}^{\infty}(0,L))$ a.s.

Hence we conclude that,

(56)
$\displaystyle\partial_{t}\varphi^{-1}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}\to\partial_{t}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}\quad\text{in
}{L^{2}(0,T;L^{2}(0,L))}\quad\bar{\mathbb{P}}-a.s.$

Similar calculations then lead to the convergence of the ALE velocities,

(57) $\displaystyle\bar{\bf w}^{*}_{N}\rightarrow\bar{\bf w}^{*}\text{ and
}\,\quad\text{ in }L^{2}(0,T;{\bf
L}^{2}(\mathcal{O})),\quad\,\bar{\mathbb{P}}-a.s.$

Finally, we give the definition of the required filtrations. To define
$\bar{\mathcal{F}}^{N}_{t}$, for any fixed $N$, let
$\widetilde{\mathcal{F}}^{N}_{t}$ be the $\sigma$-field generated by the
random variables $(\bar{{\bf u}}_{N}(s),\bar{\bf
v}_{N}(s)),\bar{\boldsymbol{\eta}}_{N}(s),\bar{W}_{N}(s)$ for all $s\leq t$
and define

(58) $\displaystyle\mathcal{N}$
$\displaystyle:=\\{\mathcal{A}\in\bar{\mathcal{F}}\ |\
\bar{\mathbb{P}}(\mathcal{A})=0\\},\qquad\bar{\mathcal{F}}^{N}_{t}:=\bigcap_{s\geq
t}\sigma(\widetilde{\mathcal{F}}^{N}_{s}\cup\mathcal{N}).$

Since the candidate solution for fluid and structure velocities, $\bar{\bf
U}=(\bar{\bf u},\bar{\bf v})$, is not regular enough to be a stochastic
process in the classical sense we construct the $\sigma-$algebras
$(\bar{\mathcal{F}}_{t})_{t\geq 0}$ as follows:

$\sigma_{t}(\bar{\bf U}):=\bigcap_{s\geq t}\sigma\left(\bigcup_{{\bf Q}\in
C^{\infty}_{0}([0,s];\mathscr{D})}\\{(\bar{\bf U},{\bf
Q})<1\\}\cup\mathcal{N}\right).$

Let $\bar{\mathcal{F}}_{t}^{\prime}$ be the $\sigma-$ field generated by the
random variables $\bar{\boldsymbol{\eta}}(s),\bar{W}(s)$ for all $0\leq s\leq
t$. Then we define

(59) $\displaystyle\bar{\mathcal{F}}^{0}_{t}:=\bigcap_{s\geq
t}\sigma(\bar{\mathcal{F}}_{s}^{\prime}\cup\mathcal{N}),\qquad\bar{\mathcal{F}}_{t}:=\sigma(\sigma_{t}(\bar{\bf
U})\cup\bar{\mathcal{F}}^{0}_{t}).$

######  Lemma 4.6.

There exists a stochastic process in $L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}))\times L^{2}(0,T;{\bf L}^{2}(0,L))$ a.s. which is a
$(\bar{\mathcal{F}}_{t})_{t\geq 0}$-progressively measurable representative of
$\bar{\bf U}=(\bar{\bf u},\bar{\bf v})$.

The proof of the above lemma can be found in [2]. Hence we now have a
complete, right-continuous filtration $(\bar{\mathcal{F}}_{t})_{t\geq 0}$, on
the probability space $(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})$, to
which the noise processes and candidate solutions are adapted.

We are now ready to state the main result of this section. The following
theorem establishes the desired existence result except with
$\bar{\boldsymbol{\eta}}$ replaced by the artificial structure random variable
$\bar{\boldsymbol{\eta}}^{*}$ in certain terms. Thus we obtain the desired
martingale solution in the sense of Definition 1 for as long as
$\bar{\boldsymbol{\eta}}$ and $\bar{\boldsymbol{\eta}}^{*}$ are equal.

######  Theorem 3.

(Main result.) For any fixed $\delta$ for which (17) is satisfied, the
stochastic processes $(\bar{\bf
u},\bar{\boldsymbol{\eta}},\bar{\boldsymbol{\eta}}^{*})$ constructed in
Theorem 2 satisfy

(60) $\begin{split}&(\bar{J}_{\bar{\boldsymbol{\eta}}^{*}}(t)\bar{\bf
u}(t),{\bf
q}(t))+(\partial_{t}\bar{{\boldsymbol{\eta}}}(t),\boldsymbol{\psi}(t))=((J_{0}){\bf
u}_{0},{\bf q}(0))+({\bf v}_{0},\boldsymbol{\psi}(0))\\\
&+\int_{0}^{t}\int_{\mathcal{O}}{\bar{J}}_{\bar{\boldsymbol{\eta}}^{*}}\bar{\bf
u}\cdot\partial_{t}{{\bf
q}}+\int_{0}^{t}\int_{0}^{L}\partial_{t}\bar{\boldsymbol{\eta}}\,\partial_{t}{\boldsymbol{\psi}}-\int_{0}^{t}\langle\mathscr{L}_{e}(\bar{\boldsymbol{\eta}}),\boldsymbol{\psi}\rangle\\\
&-\frac{1}{2}\int_{0}^{t}\int_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}^{*}}((\bar{\bf
u}-\bar{\bf w}^{*})\cdot\nabla^{\bar{\boldsymbol{\eta}}^{*}}\bar{\bf
u}\cdot{\bf q}-(\bar{\bf u}-\bar{\bf
w}^{*})\cdot\nabla^{\bar{\boldsymbol{\eta}}^{*}}{\bf q}\cdot\bar{\bf
u}-(\nabla^{\bar{\boldsymbol{\eta}}^{*}}\cdot\bar{\bf w}^{*}){\bf
q}\cdot\bar{\bf u})\\\
&-2\nu\int_{0}^{t}\int_{\mathcal{O}}\bar{J}_{\bar{\boldsymbol{\eta}}^{*}}\,{\bf
D}^{\bar{\boldsymbol{\eta}}^{*}}(\bar{\bf u})\cdot{\bf
D}^{\bar{\boldsymbol{\eta}}^{*}}({\bf q})\\\
&+\int_{0}^{t}\left(P_{{in}}\int_{0}^{1}q_{z}\Big{|}_{z=0}dr-
P_{{out}}\int_{0}^{1}q_{z}\Big{|}_{z=1}dr\right)ds+\int_{0}^{t}(G(\bar{\bf
u},\bar{\bf v},\bar{\boldsymbol{\eta}}^{*})d\bar{W},{\bf Q}),\end{split}$

$\bar{\mathbb{P}}$-a.s. for almost every $t\in[0,T]$ and for any
$(\bar{\mathcal{F}}_{t})_{t\geq 0}-$adapted process ${\bf Q}=({\bf
q},\boldsymbol{\psi})$ with continuous paths in $\mathscr{D}$ such that
$\nabla^{\bar{\boldsymbol{\eta}}^{*}}\cdot{\bf q}=0$ and ${\bf
q}|_{\Gamma}=\boldsymbol{\psi}$ a.s.

Before we proceed and consider the limit as $N\to\infty$, we argue that

(61)
$\displaystyle\bar{\boldsymbol{\eta}}^{*}(t)=\bar{\boldsymbol{\eta}}(t)\quad\text{
for any }t<T^{\boldsymbol{\eta}},\quad\bar{\mathbb{P}}-a.s.$

where for a given $\delta=(\delta_{1},\delta_{2})$,

$T^{\boldsymbol{\eta}}:=T\wedge\inf\\{t>0:\inf_{\mathcal{O}}(J_{\bar{\boldsymbol{\eta}}^{*}}(t,z,r))\leq\delta_{1}\text{
or }\|\bar{\boldsymbol{\eta}}(t)\|_{{\bf
H}^{s}(0,L)}\geq\frac{1}{\delta_{2}}\\}.$

Indeed, to show that this is true, let us introduce the following stopping
times. For $\frac{3}{2}<s<2$ we define

$\displaystyle T^{\boldsymbol{\eta}}_{N}$
$\displaystyle:=T\wedge\inf\\{t>0:\inf_{\mathcal{O}}(\bar{J}_{N}^{*}(t,z,r))\leq\delta_{1}\text{
or }\|\bar{\boldsymbol{\eta}}_{N}(t)\|_{{\bf
H}^{s}(0,L)}\geq\frac{1}{\delta_{2}}\\}.$

Then thanks to (52),
$T^{\boldsymbol{\eta}}\leq\liminf_{N\rightarrow\infty}T^{\boldsymbol{\eta}}_{N}$
a.s. Observe further that for almost any $\omega\in\bar{\Omega}$ and
$t<T^{\boldsymbol{\eta}}$, and for any $\epsilon>0$, there exists an $N$ such
that

$\displaystyle\|\bar{\boldsymbol{\eta}}(t)-\bar{\boldsymbol{\eta}}^{*}(t)\|_{{\bf
H}^{s}(0,L)}$
$\displaystyle<\|\bar{\boldsymbol{\eta}}(t)-\bar{\boldsymbol{\eta}}_{N}(t)\|_{{\bf
H}^{s}(0,L)}+\|\bar{\boldsymbol{\eta}}^{*}_{N}(t)-\bar{\boldsymbol{\eta}}_{N}(t)\|_{{\bf
H}^{s}(0,L)}+\|\bar{\boldsymbol{\eta}}^{*}(t)-\bar{\boldsymbol{\eta}}^{*}_{N}(t)\|_{{\bf
H}^{s}(0,L)}$ $\displaystyle<\epsilon.$

This is true because for any $\epsilon>0$ there exists an $N_{1}\in\mathbb{N}$
such that the first and the last terms on the right side of the above
inequality are each bounded by $\frac{\epsilon}{2}$ for any $N\geq N_{1}$
thanks to the uniform convergence (52). Furthermore, since
$t<T^{\boldsymbol{\eta}}_{N}$, for infinitely many $N$’s, the second term is
equal to 0. Hence we conclude that indeed

(62)
$\displaystyle\bar{\boldsymbol{\eta}}^{*}(t)=\bar{\boldsymbol{\eta}}(t)\quad\text{
for any }t<T^{\boldsymbol{\eta}},\quad\bar{\mathbb{P}}-a.s.$

###### Proof of Theorem 3.

We will begin the proof by giving a construction for the test functions $({\bf
q}_{N},\boldsymbol{\psi}_{N})\in\mathscr{D}$, satisfying appropriate boundary
conditions on $\Gamma$ such that ${\bf q}_{N}$ satisfies the divergence-free
condition. This is required so that the penalty terms in the fluid sub-problem
drop out. This dependence of test functions on $\bar{\boldsymbol{\eta}}^{*}$
leads us to construct test functions, specific to our problem, on a maximal
domain $\mathcal{O}_{\delta}=(0,L)\times(0,R_{\delta})$ (a rectangular domain
comprising of all the moving domains
$\mathcal{O}_{\bar{\boldsymbol{\eta}}_{N}^{*}}$).

We begin by constructing an appropriate test functions for the limiting
equation (LABEL:martingale2) as follows: Consider a smooth
$(\bar{\mathcal{F}}_{t})_{t\geq 0}$-adapted process ${\bf r}=(r_{1},r_{2})$ on
$\bar{\mathcal{O}}_{\delta}$ such that $\nabla\cdot{\bf r}=0$ and such that
${\bf r}$ satisfies the required boundary conditions $r_{2}=0\text{ on
}z=0,L,r=0$ and $\partial_{r}r_{1}=0\text{ on }\Gamma_{b}$. Assume also that,
for some $\boldsymbol{\psi}=(\psi_{z},\psi_{r})$, on the top lateral boundary
of the moving domain associated with $\bar{\boldsymbol{\eta}}^{*}$,
$\Gamma_{\bar{\boldsymbol{\eta}}^{*}}$, the function ${\bf r}$ satisfies ${\bf
r}(t)|_{\Gamma_{\bar{\boldsymbol{\eta}}^{*}}}=\boldsymbol{\psi}(t)$. Next, we
define

${\bf q}(t,z,r,\omega)={\bf
r}(t,\omega)\circ\,A^{\omega}_{\bar{\boldsymbol{\eta}}^{*}}(t)(z,r).$

Now, to observe that ${\bf q}$ is a suitable test function we define, for any
$t\in[0,T]$ and given process ${\bf r}$ as mentioned above, $\mathcal{C}_{\bf
r}:\bar{\Omega}\times{\bf C}([0,L])\rightarrow{\bf C}^{1}(\bar{\mathcal{O}})$
as

$\mathcal{C}_{{\bf r}}(\omega,{\boldsymbol{\eta}})=F_{\boldsymbol{\eta}}({\bf
r}(t,\omega)),$

where $F_{\boldsymbol{\eta}}({\bf f}):={\bf f}\circ
A^{\omega}_{{\boldsymbol{\eta}}}$ is a well-defined map from ${\bf
C}(\bar{\mathcal{O}}_{{\boldsymbol{\eta}}})$ to ${\bf C}(\bar{\mathcal{O}})$
for any ${\boldsymbol{\eta}}\in{\bf C}([0,L])$. Thanks to the continuity of
the composition operator $F_{\boldsymbol{\eta}}$ and the assumption that ${\bf
r}(t)$ is $\bar{\mathcal{F}}_{t}$-measurable, we obtain that, for any
${\boldsymbol{\eta}}$, the ${\bf C}^{1}(\bar{\mathcal{O}})$-valued map
$\omega\mapsto\mathcal{C}_{\bf r}(\omega,{\boldsymbol{\eta}})$ is
$\bar{\mathcal{F}}_{t}$-measurable (where ${\bf C}^{1}(\bar{\mathcal{O}})$ is
endowed with Borel $\sigma$-algebra). Note also that for any fixed $\omega$,
the map ${\boldsymbol{\eta}}\mapsto\mathcal{C}_{\bf
r}(\omega,{\boldsymbol{\eta}})$ is continuous. Hence we deduce that
$\mathcal{C}_{\bf r}$ is a Carathéodory function. Now by the construction of
the filtration $(\bar{\mathcal{F}}_{t})_{t\geq 0}$ in (59), we know that
$\bar{\boldsymbol{\eta}}^{*}$, is $(\bar{\mathcal{F}}_{t})_{t\geq 0}$-adapted.
Therefore, we conclude that the ${\bf C}^{1}(\bar{\mathcal{O}})$-valued
process ${\bf q}$, which by definition is ${\bf q}(t,\omega)=\mathcal{C}_{\bf
r}(\omega,\bar{\boldsymbol{\eta}}^{*}(t,\omega))$ is
$(\bar{\mathcal{F}}_{t})_{t\geq 0}$-adapted as well. The same conclusions for
the process $\boldsymbol{\psi}$ follow using the same argument.

Next, we will construct approximate test functions. However, since
$\int_{\Gamma_{\bar{\boldsymbol{\eta}}^{*}_{N}}}\boldsymbol{\psi}\cdot{\bf
n}^{*}_{N}=-\int_{0}^{L}\partial_{z}\boldsymbol{\psi}\times\bar{\boldsymbol{\eta}}^{*}_{N}$
need not be 0, $\boldsymbol{\psi}$ may not have an incompressible extension in
$\mathcal{O}_{\bar{\boldsymbol{\eta}}^{*}_{N}}$ for all $N$ with desired
boundary values.

Hence given any structure displacement
$\boldsymbol{\eta}=(\eta_{z},\eta_{r})$, we consider the ”almost solenoidal”
extension operator, denoted by $E_{\boldsymbol{\eta}}^{\omega}$ for any
$\omega\in\bar{\Omega}$, which was defined in Proposition 3.5 in [4] using the
time-independent universal Bogovski operator
$\mathcal{B}:C_{0}^{\infty}(\mathcal{O}_{\delta})\rightarrow{\bf
C}_{0}^{\infty}(\mathcal{O}_{\delta})$ constructed in Theorem 3.3 in [4]:

$E_{\boldsymbol{\eta}}^{\omega}(\boldsymbol{\psi})(t,z,r):=\boldsymbol{\psi}(t,\varphi_{\boldsymbol{\eta}}^{-1}(z))\chi_{\delta}(r)-\mathcal{B}(\nabla\cdot(\boldsymbol{\psi}(t,\varphi_{\boldsymbol{\eta}}^{-1}(z))\chi_{\delta}(r))-\lambda^{\boldsymbol{\eta}}(t)\sigma(z,r)),$

where $\sigma$ is a time-independent smooth function with compact support in
every $\mathcal{O}_{\boldsymbol{\eta}}(t)$ such that
$\int_{\mathcal{O}_{\delta}}\sigma=1$, $\chi_{\delta}$ is a smooth cutoff
function which is 1 when $r>\delta_{1}$, $\lambda(t)=-\int_{\Gamma}({\bf
id}+\boldsymbol{\eta}(z))\times\partial_{z}\boldsymbol{\psi}(t,z)dz$ and
$\varphi_{\boldsymbol{\eta}}(z)=z+\eta_{z}(z)$. For any Lipschitz
$\boldsymbol{\eta}$ satisfying the uniform deterministic bounds (23), the
operator $\mathcal{B}$ maps $C^{\infty}_{0}(\mathcal{O}_{\boldsymbol{\eta}})$
to ${\bf C}^{\infty}_{0}(\mathcal{O}_{\boldsymbol{\eta}})$ such that for
constant some $C>0$ depending only on $\delta$,

(63) $\displaystyle\|\mathcal{B}({\bf f})\|_{{\bf
H}^{1}(\mathcal{O}_{\boldsymbol{\eta}})}\leq C(\delta)\|{\bf f}\|_{{\bf
L}^{2}(\mathcal{O}_{\boldsymbol{\eta}})}.\quad$

Hence for any $\omega\in\bar{\Omega}$, the extension
$E^{\omega}_{\boldsymbol{\eta}}(\boldsymbol{\psi})$ satisfies the correct
boundary conditions on $\Gamma_{in/out/b}$ and,

$E^{\omega}_{\boldsymbol{\eta}}(\boldsymbol{\psi})\Big{|}_{\Gamma_{\boldsymbol{\eta}}}=\boldsymbol{\psi},\quad\text{and}\quad\text{div
}E^{\omega}_{\boldsymbol{\eta}}(\boldsymbol{\psi})=\lambda^{\boldsymbol{\eta}}(t)\sigma\quad\text{in
}\mathcal{O}_{\boldsymbol{\eta}}.$

Note here that due to the deterministic $\delta$ lower and upper bounds (23)
on the artificial structure variables $\bar{\boldsymbol{\eta}}^{*}_{N}$, the
operator $\mathcal{B}$ does not depend on $\omega$ either and $\mathcal{B}$
maps
$C^{\infty}_{0}(\mathcal{O}^{\omega}_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t))$ to
${\bf
C}^{\infty}_{0}(\mathcal{O}^{\omega}_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t))$
and $C^{\infty}_{0}(\mathcal{O}^{\omega}_{\bar{\boldsymbol{\eta}}^{*}}(t))$ to
${\bf C}^{\infty}_{0}(\mathcal{O}^{\omega}_{\bar{\boldsymbol{\eta}}^{*}}(t))$
for every $t\in[0,T]$, $\omega\in\bar{\Omega}$.

We write $({\bf r},\boldsymbol{\psi})=({\bf
r}-E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi}),0)+(E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi}),\boldsymbol{\psi})$.
In what follows we will find approximations of these two pairs that will serve
as an admissible test pair for the approximate system.

First, as in [11], we construct a suitable approximation of $({\bf
r}-E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi}),0)$ by ”squeezing it” so
that the approximate fluid test function ${\bf r}^{0}_{N}$ satisfies desired
properties. Let,

$\displaystyle(r^{0}_{z},r^{0}_{r})={\bf r}^{0}:=\begin{cases}{\bf
r}-E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi}),&\text{if
}\mathcal{O}_{\bar{\boldsymbol{\eta}}^{*}},\\\ 0,&\text{elsewhere in
}\bar{\mathcal{O}}_{\delta}.\end{cases}$

Observe that ${\bf r}^{0}\in{\bf H}^{1}(\mathcal{O}_{\delta})$ (see [6]). Now
we will scale ${\bf r}^{0}$ to construct its suitable $N$-approximations while
also preserving its desired properties. Define for any
$\omega\in\bar{\Omega}$, the random variable

$\beta_{N}(\omega)=\frac{h(\omega)}{h(\omega)-\alpha(N,\omega)},$

where
$\alpha(N,\omega)=\max\\{0,\sup_{(t,z)\in[0,T]\times[0,L]}\left(\bar{\eta}^{*}_{r}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}-(\bar{\eta}^{*}_{N})_{r}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}_{N}^{*}}\right)\\}$
and
$h(\omega)=\inf_{(t,z)\in[0,T]\times[0,L]}[1+\eta^{*}_{r}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}]$.
Thanks to the uniform convergence (52) and (54), we know that
$\beta_{N}\rightarrow 1$ almost surely as $N\to\infty$. Using this definition
we set,

(64) $\displaystyle{\bf
r}^{0}_{N}(t,z,r,\omega)=\begin{cases}(\beta_{N}(\omega)r^{0}_{z}(t,z,\beta_{N}(\omega)r,\omega),r^{0}_{r}(t,z,\beta_{N}(\omega)r,\omega)),&\text{
if }r\leq\frac{1}{\delta_{2}\beta_{N}},\\\ 0,&\text{ elsewhere in
}\bar{\mathcal{O}}_{\delta}.\end{cases}$

Notice that by scaling ${\bf r}^{0}$ in this fashion we still have that
$\nabla\cdot{\bf r}^{0}_{N}=\beta_{N}\nabla\cdot{\bf r}^{0}=0$ for every
$\omega\in\bar{\Omega}$ and that ${\bf
r}^{0}_{N}|_{\Gamma_{\bar{\boldsymbol{\eta}}^{*}_{N}}}=0$. Moreover,

(65) $\displaystyle{\bf r}^{0}_{N}\rightarrow{\bf r}^{0}\quad\text{ in
}L^{2}(0,T;{\bf H}^{1}(\mathcal{O}_{\delta}))\cap H^{1}(0,T;{\bf
L}^{2}(\mathcal{O}_{\delta}))\,a.s.$

Next, for an approximation of
$(E^{\omega}_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi}),\boldsymbol{\psi})$
for any $\omega\in\bar{\Omega}$, we define

$({\bf
r}^{1}_{N}(t),\boldsymbol{\psi}_{N}(t)):=\left(E^{\omega}_{\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}}(\boldsymbol{\psi})(t)-\frac{\lambda^{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(t)}{\lambda^{N}_{0}(t)}\zeta_{N}(t),\boldsymbol{\psi}-\frac{\lambda^{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(t)}{\lambda^{N}_{0}(t)}\xi_{0}\right),$

where we pick any $\xi_{0}\in C_{0}^{\infty}(\Gamma)$ and set
$\zeta_{N}:=E_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(\xi_{0})$ and,

$\lambda_{0}^{N}(t)=-\int_{\Gamma}({\bf
id}+\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}(t))\times\partial_{z}\xi_{0}\,dz\quad\text{and},\quad\lambda^{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}=-\int_{\Gamma}({\bf
id}+\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}(t))\times\partial_{z}\boldsymbol{\psi}(t)dz.$

Observe that for any $t\in[0,T],\omega\in\bar{\Omega}$, div ${\bf
r}^{1}_{N}=0$ and ${\bf
r}^{1}_{N}|_{\Gamma_{\bar{\boldsymbol{\eta}}^{*}_{N}}}=\boldsymbol{\psi}_{N}$.
Adaptedness of $({\bf r}^{1}_{N},\boldsymbol{\psi}_{N})$ follows by using the
same arguments as above and from the continuity of the operator
$E_{\bar{\boldsymbol{\eta}}^{*}_{N}}$.

Before moving on we will study some convergence properties. Observe that, an
application of the mean value theorem gives us,

$\displaystyle|\partial_{z}(\boldsymbol{\psi}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}})-\partial_{z}(\boldsymbol{\psi}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}})|\leq|\partial_{z}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\partial_{z}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}||\partial_{z}\boldsymbol{\psi}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}}|+|\partial_{z}\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}||\partial_{zz}\boldsymbol{\psi}||\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}}|.$

Thus combining the observation above with the convergence result in (54), we
obtain

$(\boldsymbol{\psi}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}})\chi_{\delta}\rightarrow(\boldsymbol{\psi}\circ\varphi^{-1}_{\bar{\boldsymbol{\eta}}^{*}})\chi_{\delta}\quad\text{
in }L^{\infty}(0,T;{\bf H}^{1}(\mathcal{O}_{\delta}))\,\bar{\mathbb{P}}-a.s.$

Additionally, using (63) and the fact that
$\lambda^{\bar{\boldsymbol{\eta}}^{*}_{N}}\to\lambda^{\bar{\boldsymbol{\eta}}^{*}}=0$
a.s. in $L^{\infty}(0,T)$, we obtain

$E_{\bar{\boldsymbol{\eta}}^{*}_{N}}(\boldsymbol{\psi})\to
E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi})\quad\text{ in }\quad
L^{\infty}(0,T;{\bf H}^{1}(\mathcal{O}_{\delta}))\quad\bar{\mathbb{P}}-a.s,$

and thus

(66) $\displaystyle{\bf r}^{1}_{N}\to
E_{\bar{\boldsymbol{\eta}}^{*}}(\boldsymbol{\psi})\quad\text{ in }\quad
L^{\infty}(0,T;{\bf H}^{1}(\mathcal{O}_{\delta}))\quad\bar{\mathbb{P}}-a.s,$

Observe also that

$\displaystyle\partial_{t}{\bf
r}^{1}_{N}=\partial_{t}E_{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(\boldsymbol{\psi})-\partial_{t}\left(\frac{\lambda^{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}(t)}{\lambda^{N}_{0}(t)}\zeta_{N}(t)\right)$

Now we let ${\bf r}_{N}:={\bf r}^{0}_{N}+{\bf r}^{1}_{N}$. Finally, we define
the approximate fluid test function as,

$\displaystyle{\bf q}_{N}(t,z,r,\omega)={\bf
r}_{N}(t_{n},\omega)\circ\,A^{\omega}_{\bar{\boldsymbol{\eta}}_{N}^{*}}(t)(z,r)\quad\text{
if }t\in(t_{n},t_{n+1}).$

Thanks to the aforementioned properties of ${\bf r}_{N}$ we deduce that for
any $t\in[0,T],\omega\in\bar{\Omega}$,

${\bf q}_{N}\Big{|}_{\Gamma}=\boldsymbol{\psi}_{N}\quad\text{ and
}\quad\text{div}^{\bar{\boldsymbol{\eta}}^{*}_{N}}{\bf q}_{N}=0\text{ in
}\mathcal{O}.$

Adaptedness of the process ${\bf q}_{N}$ to the filtration
$(\bar{\mathcal{F}}^{N}_{t})_{t\geq 0}$ follows using the same arguments as
above. Hence, the $\mathscr{D}$-valued process ${\bf Q}_{N}=({\bf
q}_{N},\boldsymbol{\psi}_{N})$ will serve as our random test function for the
$N^{th}$-approximation.

Now we study the convergence of ${\bf Q}_{N}$ as $N\rightarrow\infty$. We
begin by writing for any $\omega\in\bar{\Omega}$,

$\displaystyle\nabla{\bf q}_{N}-\nabla{\bf q}$ $\displaystyle=\nabla{\bf
r}\circ(A_{\bar{\boldsymbol{\eta}}^{*}}-A_{\bar{\boldsymbol{\eta}}^{*}_{N}})\nabla
A_{\bar{\boldsymbol{\eta}}^{*}}+(\nabla{\bf r}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\nabla{\bf r}_{N}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}})\nabla A_{\bar{\boldsymbol{\eta}}^{*}}$
$\displaystyle+(\nabla{\bf r}_{N}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}})(\nabla
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\nabla A_{\bar{\boldsymbol{\eta}}^{*}}).$

Since ${\bf r}$ is smooth we can apply the mean value theorem on the first
term and obtain its convergence to 0 in $L^{\infty}(0,T;{\bf
L}^{2}(\mathcal{O}))$ a.s. thanks to the convergence of the ALE maps obtained
in (55). For the second term we use the fact that the Jacobians of
$A_{\bar{\boldsymbol{\eta}}^{*}_{N}}$ and $A_{\bar{\boldsymbol{\eta}}^{*}}$
have bounds
$\delta_{1}<J_{\bar{\boldsymbol{\eta}}^{*}_{N}},J_{\bar{\boldsymbol{\eta}}^{*}}<2$
(see (21) and (25)) for every
$\omega\in\bar{\Omega},\,(z,r)\in\mathcal{O},\,t\in[0,T]$, which gives us the
equivalence of $L^{2}-$norms on the corresponding moving and the fixed
domains. Indeed, thanks to (65) and (66), we have

$\displaystyle\|\nabla{\bf r}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\nabla{\bf r}_{N}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}\|_{L^{2}(0,T{\bf L}^{2}(\mathcal{O}))}$
$\displaystyle=\int_{0}^{T}\int_{\mathcal{O}_{\delta}}|\nabla{\bf
r}-\nabla{\bf
r}_{N}|^{2}(J^{-1}_{\bar{\boldsymbol{\eta}}^{*}_{N}})\mathbbm{1}_{\mathcal{O}_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t)}dxdt$
$\displaystyle\leq C(\delta)\|{\bf r}-{\bf r}_{N}\|_{L^{\infty}(0,T;{\bf
H}^{1}(\mathcal{O}_{\delta}))}\rightarrow 0,\quad\bar{\mathbb{P}}-a.s.$

Similarly for the third term, because of (55)1, we obtain

$\displaystyle\|(\nabla{\bf r}_{N}\circ
A_{\bar{\boldsymbol{\eta}}^{*}_{N}})(\nabla
A_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\nabla
A_{\bar{\boldsymbol{\eta}}^{*}})\|_{L^{\infty}(0,T;{\bf L}^{2}(\mathcal{O}))}$
$\displaystyle\leq C(\delta)\|\nabla{\bf r}_{N}\|_{L^{\infty}(0,T;{\bf
L}^{2}(\mathcal{O}))}\|\nabla A_{\bar{\boldsymbol{\eta}}^{*}_{N}}-\nabla
A_{\bar{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T;{\bf C}^{1}(\mathcal{O}))}$
$\displaystyle\rightarrow 0\quad\bar{\mathbb{P}}-a.s.\quad\text{as}\quad
N\to\infty.$

Hence we conclude that

(67) $\displaystyle{\bf q}_{N}\rightarrow{\bf q}\quad\text{ in
}L^{\infty}(0,T;{\bf
H}^{1}(\mathcal{O}))\quad\bar{\mathbb{P}}-a.s.,\quad\text{ as }N\to\infty.$

Observe that ${\bf q}_{N}$ is constantly equal to ${\bf q}^{n}$ for
$t\in[n\frac{T}{N},(n+1)\frac{T}{N})$. We will also need the following
piecewise linear definition of test functions.

$\widetilde{\bf q}_{N}(t,\cdot)=\frac{t-t^{n}}{\Delta t}{\bf
q}^{n+1}+\frac{t^{n+1}-t}{\Delta t}{\bf q}^{n}.$

Similarly we also define $\widetilde{\boldsymbol{\psi}}_{N}$. As above, we
observe for any fixed $\omega\in\bar{\Omega}$ that,

$\displaystyle\|\partial_{t}\widetilde{\bf q}_{N}\|_{L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}))}$ $\displaystyle\leq\|\partial_{t}{\bf
r}_{N}\|_{L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}_{\delta})}\|J^{-1}_{\bar{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T;C(\mathcal{O}))}+\|\nabla^{\boldsymbol{\eta}^{*}_{N}}{\bf
q}_{N}\|_{L^{\infty}(0,T;{\bf L}^{2}(\mathcal{O}))}\|{\bf
w}^{*}_{N}\|_{L^{2}(0,T;{\bf L}^{\infty}(\Gamma))}$ $\displaystyle\leq
C(\delta).$

Hence, we obtain

(68) $\displaystyle\partial_{t}\widetilde{\bf
q}_{N}\rightharpoonup\partial_{t}{\bf q}\quad\text{ in }L^{2}(0,T;{\bf
L}^{2}(\mathcal{O})),\quad\bar{\mathbb{P}}-a.s.\,$

Similarly, since
$\lambda^{\bar{\widetilde{\boldsymbol{\eta}}}^{*}_{N}}\rightarrow 0$ in
$H^{1}(0,T)$ a.s. we have that

(69)
$\begin{split}\boldsymbol{\psi}_{N}\rightarrow\boldsymbol{\psi}\quad\text{ in
}L^{\infty}(0,T;{\bf C}^{2}(\bar{\Gamma}))\,a.s.\\\
\partial_{t}\widetilde{\boldsymbol{\psi}}_{N}\rightarrow\partial_{t}\boldsymbol{\psi}\quad\text{
in }L^{2}(0,T;{\bf L}^{2}(\Gamma))\,a.s.\end{split}$

Now we will give the exact approximate system and use ${\bf Q}_{N}=({\bf
q}_{N},\boldsymbol{\psi}_{N})$ as the test functions. Observe that
$\widetilde{\boldsymbol{\eta}}_{N}^{*}={\boldsymbol{\eta}}_{N}^{*}+{(t-t^{n})}\frac{\partial\widetilde{\boldsymbol{\eta}}_{N}^{*}}{\partial
t},$ and that the equation same holds true when ${\boldsymbol{\eta}}^{*}$ is
replaced by ${\bf u}$ and ${\bf q}$. This gives for each $\omega\in\Omega$ and
$t\in(t^{n},t^{n+1})$,

$\displaystyle\frac{\partial((\widetilde{\bar{J}}_{N}^{*})\bar{\widetilde{\bf
u}}_{N}\widetilde{\bf q}_{N})}{\partial t}$
$\displaystyle=\frac{\partial\bar{\widetilde{\bf u}}_{N}}{\partial t}{\bf
q}_{N}({\bar{J}}_{N}^{*})+(\bar{\widetilde{\bf u}}_{N}-\bar{\bf
u}_{N})\left(\frac{\partial\widetilde{\bf
q}_{N}}{dt}({\bar{J}}_{N}^{*})+\widetilde{\bf
q}_{N}\frac{\partial\widetilde{\bar{J}}^{*}_{N}}{\partial t}\right).$

Hence we can write on $(t^{n},t^{n+1})$ we have,

(70)
$\begin{split}\frac{\partial((\widetilde{\bar{J}}_{N}^{*})\bar{\widetilde{\bf
u}}_{N}\widetilde{\bf q}_{N})}{\partial t}&=\frac{\partial\bar{\widetilde{\bf
u}}_{N}}{\partial t}{\bf q}_{N}({\bar{J}}_{N}^{*})+(2\bar{\widetilde{\bf
u}}_{N}-\bar{\bf u}_{N})\left(\widetilde{\bf
q}_{N}\frac{\partial\bar{\widetilde{J}}^{*}_{N}}{\partial
t}\right)+\frac{\partial\widetilde{\bf q}_{N}}{\partial
t}((\bar{\widetilde{\bf u}}_{N}-\bar{\bf
u}_{N})({\bar{J}}_{N}^{*})+(\widetilde{\bar{J}}_{N}^{*})\bar{\widetilde{\bf
u}}_{N}).\end{split}$

Similar algebra gives for any $t\in(t^{n},t^{n+1})$,

(71) $\displaystyle\frac{\partial(\widetilde{\bar{\bf
v}}_{N}\widetilde{\boldsymbol{\psi}}_{N})}{\partial
t}=\frac{\partial\widetilde{\bar{\bf v}}_{N}}{\partial
t}\boldsymbol{\psi}_{N}+\frac{\partial\widetilde{\boldsymbol{\psi}}_{N}}{\partial
t}(2\widetilde{\bar{\bf v}}_{N}-\bar{\bf v}_{N}).$

Now, for a fixed $N$, combining (70) and (71) and integrating in time on
$(t^{n},t^{n+1})$, where $t^{j}=j\frac{T}{N}$, we obtain,

$\begin{split}&((R+\bar{\widetilde{\boldsymbol{\eta}}}_{N}^{*}(t))\bar{\widetilde{\bf
u}}_{N}(t),\widetilde{\bf q}_{N}(t))+(\widetilde{\bar{\bf
v}}_{N}(t),\boldsymbol{\psi}_{N}(t))\\\ &=({\bf
u}_{0}(R+{\boldsymbol{\eta}}_{0}),{\bf q}(0))+({\bf
v}_{0},\boldsymbol{\psi}(0))-\int_{0}^{t}\langle\mathscr{L}_{e}(\bar{\boldsymbol{\eta}}^{+}_{N}),\boldsymbol{\psi}_{N}\rangle
ds\\\
&+\sum_{n=0}^{N-1}\int_{0}^{t}\chi_{(t^{n},t^{n+1})}\left(\int_{\mathcal{O}}\frac{({\bar{\bf
u}}^{n+1}-{\bar{\bf u}}^{n})}{\Delta t}(\bar{J}^{n}_{*})\,{\bf
q}^{n}dx+\int_{0}^{L}\frac{(\bar{\bf v}^{n+\frac{1}{2}}-\bar{\bf
v}^{n})}{\Delta t}\,\boldsymbol{\psi}^{n}dz\right)ds\\\
&+\int_{0}^{t}\int_{\mathcal{O}}\frac{\partial\widetilde{\bar{J}}^{*}_{N}}{\partial
t}(2\bar{\widetilde{\bf u}}_{N}-\bar{\bf u}_{N})\widetilde{\bf
q}_{N}dxds+\int_{0}^{t}\int_{\mathcal{O}}\frac{\partial\widetilde{\bf
q}_{N}}{\partial t}((\bar{\widetilde{\bf u}}_{N}-\bar{\bf
u}_{N})(\bar{J}^{*}_{N})+(\widetilde{\bar{J}}_{N}^{*})\bar{\widetilde{\bf
u}}_{N})dxds\\\
&+\int_{0}^{t}\int_{0}^{L}\frac{\partial\widetilde{\boldsymbol{\psi}}_{N}}{\partial
t}(2\widetilde{\bar{\bf v}}_{N}-\bar{\bf v}_{N})dzds.\end{split}$

Expanding the third row in the equation above using the representation (41),
we obtain

(72) $\begin{split}&((\bar{\widetilde{J}}_{N}^{*}(t))\bar{\widetilde{\bf
u}}_{N}(t),\widetilde{\bf q}_{N}(t))+(\bar{\widetilde{\bf
v}}_{N}(t),\widetilde{\boldsymbol{\psi}}_{N}(t))\\\ &=(J_{0}{\bf u}_{0},{\bf
q}(0))+({\bf
v}_{0},\boldsymbol{\psi}(0))-\int_{0}^{t}\langle\mathscr{L}_{e}(\bar{\boldsymbol{\eta}}^{+}_{N}),\boldsymbol{\psi}_{N}\rangle\\\
&+\int_{0}^{t}\int_{\mathcal{O}}\frac{\partial\widetilde{\bar{J}}^{*}_{N}}{\partial
t}(2\bar{\widetilde{\bf u}}_{N}-{\bar{\bf u}}_{N})\widetilde{\bf
q}_{N}-\frac{1}{2}\frac{\partial\widetilde{\bar{J}}^{*}_{N}}{\partial
t}\bar{\bf u}^{+}_{N}{\bf q}_{N}\\\
&-\frac{1}{2}\int_{0}^{t}\int_{\mathcal{O}}({\bar{J}}_{N}^{*})((\bar{\bf
u}_{N}-\bar{\bf w}_{N})\cdot\nabla^{\bar{\boldsymbol{\eta}}_{N}^{*}}\bar{\bf
u}^{+}_{N}\cdot{\bf q}_{N}-(\bar{\bf u}_{N}-\bar{\bf
w}_{N})\cdot\nabla^{\bar{\boldsymbol{\eta}}_{N}^{*}}{\bf q}_{N}\cdot\bar{\bf
u}^{+}_{N})\\\ &-2\nu\int_{0}^{t}\int_{\mathcal{O}}({\bar{J}}_{N}^{*}){\bf
D}^{\bar{\boldsymbol{\eta}}_{N}^{*}}(\bar{\bf u}^{+}_{N})\cdot{\bf
D}^{\bar{\boldsymbol{\eta}}_{N}^{*}}({\bf q}_{N})\\\
&+\int_{0}^{t}\left(P_{{in}}\int_{0}^{1}(q_{N})_{z}\Big{|}_{z=0}-P_{{out}}\int_{0}^{1}(q_{N})_{z}\Big{|}_{z=1}\right)\\\
&+\int_{0}^{t}\int_{\mathcal{O}}\frac{\partial\widetilde{\bf q}_{N}}{\partial
t}((\bar{\widetilde{\bf u}}_{N}-{\bar{\bf
u}}_{N})(\bar{J}^{*}_{N})+(\widetilde{\bar{J}}_{N}^{*})\bar{\widetilde{\bf
u}}_{N})dxds+\int_{0}^{t}\int_{0}^{L}\frac{\partial\bar{\widetilde{\boldsymbol{\psi}}}_{N}}{\partial
t}(2\bar{\widetilde{\bf v}}_{N}-\bar{\bf v}_{N})\\\ &+\int_{0}^{t}(G({\bar{\bf
u}}_{N},\bar{\bf v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N})d\bar{W}_{N},{\bf
Q}_{N})+(E_{N}(t),{\bf Q}_{N}),\end{split}$

where

$E_{N}(t)=\sum_{m=0}^{N-1}\left(\frac{t-t^{m}}{\Delta t}G(\bar{\bf
u}^{m},\bar{\bf
v}^{m},\bar{\boldsymbol{\eta}}^{m}_{*})\Delta_{m}\bar{W}-\int_{t^{m}}^{t}G(\bar{\bf
u}^{m},\bar{\bf
v}^{m},\bar{\boldsymbol{\eta}}^{m}_{*})d\bar{W}\right)\chi_{[t^{m},t^{m+1})}.$

The term $E_{N}$ is an error term caused by the discretization of the
stochastic integral. We can now pass $N\to\infty$ in the deterministic terms
in (LABEL:epeq) using the convergence results obtained in Theorem 2, (52),
(LABEL:convrest1), (55), (57), (67), (68), (69). Next, we demonstrate the
passage of $N\to\infty$ in the last two terms.

######  Lemma 4.7.

The numerical error $E_{N}$ of the stochastic term has the following property:

$\bar{\mathbb{E}}\int_{0}^{T}\|E_{N}(t)\|^{2}_{{\bf
L}^{2}(\mathcal{O})\times{\bf L}^{2}(0,L)}dt\to 0\text{ as
}{N\rightarrow\infty}.$

###### Proof.

First, for any $N$, we have

$E_{N}(t)=\sum_{m=0}^{N-1}\left(\frac{t-t^{m}}{\Delta t}G(\bar{\bf
u}^{m},\bar{\bf
v}^{m},\bar{\boldsymbol{\eta}}^{m}_{*})\Delta_{m}\bar{W}_{N}-\int_{t^{m}}^{t}G(\bar{\bf
u}^{m},\bar{\bf
v}^{m},\bar{\boldsymbol{\eta}}^{m}_{*})d\bar{W}_{N}\right)\chi_{[t^{m},t^{m+1})}=:E_{N}^{1}+E_{N}^{2}$

We estimate $E^{1}_{N}$ and $E^{2}_{N}$. Recall our notation ${\bf L}^{2}={\bf
L}^{2}(\mathcal{O})\times{\bf L}^{2}(0,L)$. Observe that $E^{1}_{N}$ satisfies
(see also (33)),

$\displaystyle\bar{\mathbb{E}}\int_{0}^{T}\|E_{N}^{1}(t)\|_{{\bf
L}^{2}}^{2}dt={\mathbb{E}}\sum_{n=0}^{N-1}\|G(\bar{\bf u}^{n},\bar{\bf
v}^{n},\bar{\boldsymbol{\eta}}^{n}_{*})\Delta_{n}\bar{W}_{N}\|_{{\bf
L}^{2}}^{2}\int_{t^{n}}^{t^{n+1}}|\frac{t-t^{m}}{\Delta t}|^{2}dt$
$\displaystyle=\bar{\mathbb{E}}\sum_{n=0}^{N-1}\|G(\bar{\bf u}^{n},\bar{\bf
v}^{n},\bar{\boldsymbol{\eta}}^{n}_{*})\Delta_{n}\bar{W}_{N}\|_{{\bf
L}^{2}}^{2}\frac{\Delta t}{3}\leq\bar{\mathbb{E}}\sum_{n=0}^{N-1}\|G(\bar{\bf
u}^{n},\bar{\bf v}^{n},\bar{\boldsymbol{\eta}}^{n}_{*})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}\|\Delta_{n}\bar{W}_{N}\|_{U_{0}}^{2}{\Delta t}$
$\displaystyle=(\Delta t)^{2}\bar{\mathbb{E}}\sum_{n=0}^{N-1}\|G(\bar{\bf
u}^{n},\bar{\bf v}^{n},\bar{\boldsymbol{\eta}}^{n}_{*})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}\leq C\Delta t,$

where $C>0$ does not depend on $N$, as a consequence of Theorem 1 which still
holds true for the new random variables.

To estimate $E^{2}_{N}$ we use the Itô isometry to obtain,

$\displaystyle\bar{\mathbb{E}}\int_{0}^{T}\|E_{N}^{2}(t)\|_{{\bf
L}^{2}}^{2}dt=\bar{\mathbb{E}}\sum_{n=0}^{N-1}\int_{t^{n}}^{t^{n+1}}\|\int_{t^{n}}^{t}G(\bar{\bf
u}^{n},\bar{\bf v}^{n},\bar{\boldsymbol{\eta}}_{*}^{n})d\bar{W}\|_{{\bf
L}^{2}}^{2}dt$
$\displaystyle=\bar{\mathbb{E}}\sum_{n=0}^{N-1}\int_{t^{n}}^{t^{n+1}}\int_{t^{n}}^{t}\|G(\bar{\bf
u}^{n},\bar{\bf v}^{n},\bar{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}dsdt=\frac{1}{2}\bar{\mathbb{E}}\sum_{n=0}^{N-1}\|G(\bar{\bf
u}^{n},\bar{\bf v}^{n},\bar{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}(\Delta t)^{2}$ $\displaystyle\leq C\Delta t.$

Thus, as $N\rightarrow\infty$ we obtain,

$\bar{\mathbb{E}}\int_{0}^{T}\|E_{N}(t)\|^{2}_{{\bf
L}^{2}(\mathcal{O})\times{\bf L}^{2}(0,L)}dt\rightarrow 0.$

∎

Next, We deal with the stochastic integral in this step. Observe that, by
construction, ${\bf Q}$ is $(\bar{\mathcal{F}}_{t})_{t\geq 0}$-adapted.

######  Lemma 4.8.

The processes $\left(\int_{0}^{t}(G(\bar{{\bf u}}_{N}(s),\bar{\bf
v}_{N}(s),\bar{\boldsymbol{\eta}}^{*}_{N}(s))d\bar{W}_{N}(s),{\bf
Q}_{N}(s))\right)_{t\in[0,T]}$ converge to $\left(\int_{0}^{t}(G(\bar{{\bf
u}}(s),\bar{\bf v}(s),\bar{\boldsymbol{\eta}}^{*}(s))d\bar{W}(s),{\bf
Q}(s))\right)_{t\in[0,T]}$ in $L^{1}(\bar{\Omega};L^{1}(0,T;\mathbb{R}))$ as
$N\rightarrow\infty$.

###### Proof.

First, by using (LABEL:growthG)2,3 we observe that

$\displaystyle\int_{0}^{T}\|(G(\bar{{\bf u}}_{N},\bar{\bf
v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N}),{\bf Q}_{N})-(G(\bar{\bf u},\bar{\bf
v},\bar{\boldsymbol{\eta}}^{*}),{\bf Q})\|^{2}_{L_{2}(U_{0},\mathbb{R})}ds$
$\displaystyle=\int_{0}^{T}\|(G(\bar{{\bf u}}_{N},\bar{\bf
v}_{N},\bar{{\boldsymbol{\eta}}}^{*}_{N})-G(\bar{{\bf u}},\bar{\bf
v},\bar{{\boldsymbol{\eta}}}^{*}_{N}),{\bf
Q}_{N})\|^{2}_{L_{2}(U_{0};\mathbb{R})}+\int_{0}^{T}\|(G(\bar{{\bf
u}},\bar{\bf v},\bar{{\boldsymbol{\eta}}}^{*}_{N}),{\bf Q}_{N}-{\bf
Q})\|^{2}_{L_{2}(U_{0};\mathbb{R})}$
$\displaystyle\qquad+\int_{0}^{T}\|(G(\bar{\bf u},\bar{\bf
v},\bar{\boldsymbol{\eta}}^{*}_{N})-G(\bar{\bf u},\bar{\bf
v},\bar{\boldsymbol{\eta}}^{*}),{\bf Q})\|^{2}_{L_{2}(U_{0};\mathbb{R})}$
$\displaystyle\leq\int_{0}^{T}\left(\|\bar{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{L^{\infty}(0,L)}\|\bar{{\bf
u}}_{N}-\bar{\bf u}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|\bar{\bf
v}_{N}-\bar{\bf v}\|_{{\bf L}^{2}(0,L)}^{2}\right)\|{\bf Q}_{N}\|_{{\bf
L}^{2}}^{2}$
$\displaystyle+\int_{0}^{T}\left(\|\bar{\boldsymbol{\eta}}^{*}_{N}\|^{2}_{{\bf
L}^{\infty}(0,L)}\|\bar{\bf u}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|\bar{\bf
v}\|_{{\bf L}^{2}(0,L)}^{2}\right)\|{\bf Q}_{N}-{\bf Q}\|_{{\bf
L}^{2}}^{2}+\int_{0}^{T}\|\bar{\boldsymbol{\eta}}^{*}_{N}-\bar{\boldsymbol{\eta}}^{*}\|^{2}_{{\bf
L}^{\infty}(0,L)}\|\bar{{\bf u}}\|^{2}_{{\bf L}^{2}(\mathcal{O})}\|{\bf
Q}\|_{{\bf L}^{2}}^{2}$ $\displaystyle\leq\frac{1}{\delta}\|{\bf
Q}_{N}\|^{2}_{L^{2}(0,T;{\bf L}^{2})}\|\bar{\bf u}_{N}-\bar{\bf
u}\|^{2}_{L^{2}(0,T;{\bf L}^{2}(\mathcal{O}))}$
$\displaystyle+\frac{1}{\delta}\|{\bf Q}_{N}-{\bf Q}\|^{2}_{L^{2}(0,T;{\bf
L}^{2})}\left(\|\bar{{\bf u}}\|_{L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}))}^{2}+\|\bar{\bf v}\|^{2}_{L^{2}(0,T;{\bf
L}^{2}(0,L))}\right)$
$\displaystyle+\|\bar{\boldsymbol{\eta}}^{*}_{N}-\bar{\boldsymbol{\eta}}^{*}\|_{L^{\infty}(0,T;{\bf
L}^{\infty}(0,L))}\|\bar{{\bf u}}\|^{2}_{L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}))}\|{\bf Q}\|_{L^{2}(0,T;{\bf L}^{2})}^{2}.$

Thanks to Theorem 2, the right hand side of the inequality above converges to
0, $\bar{\mathbb{P}}$-a.s. as $N\to\infty$. Hence we have proven that

(73) $\displaystyle(G(\bar{{\bf u}}_{N},\bar{\bf
v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N}),{\bf Q}_{N})\rightarrow(G(\bar{\bf
u},\bar{\bf v},\bar{\boldsymbol{\eta}}^{*}),{\bf
Q}),\qquad\text{$\bar{\mathbb{P}}$-a.s \quad in
$L^{2}(0,T;L_{2}(U_{0},\mathbb{R})).$ }$

Now using classical ideas from [1] (see also Lemma 2.1 of [3] for a proof),
the convergence (73) implies that

(74) $\displaystyle\int_{0}^{t}(G(\bar{{\bf u}}_{N},\bar{\bf
v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N})d\bar{W}_{N},{\bf
Q}_{N})\rightarrow\int_{0}^{t}(G(\bar{\bf u},\bar{\bf
v},\bar{\boldsymbol{\eta}}^{*})d\bar{W},{\bf Q})$

in probability in $L^{2}(0,T;\mathbb{R})$.

Furthermore observe that for some $C>0$ independent of $N$ we have the
following bounds which follow from the Itô isometry:

$\displaystyle\bar{\mathbb{E}}\int_{0}^{T}|\int_{0}^{t}(G(\bar{{\bf
u}}_{N},\bar{\bf v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N})d\bar{W}_{N}(s),{\bf
Q}_{N})$
$\displaystyle|^{2}dt=\int_{0}^{T}\bar{\mathbb{E}}\int_{0}^{t}\|(G(\bar{{\bf
u}}_{N},\bar{\bf v}_{N},\bar{\boldsymbol{\eta}}^{*}_{N}),{\bf
Q}_{N})\|^{2}_{L_{2}(U_{0},\mathbb{R})}dsdt$ (75) $\displaystyle\leq
T\bar{\mathbb{E}}\left(\int_{0}^{T}\left(\|{\bar{\boldsymbol{\eta}}^{*}_{N}}\|_{{\bf
L}^{\infty}(0,L)}^{2}\|\bar{{\bf u}}_{N}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|\bar{\bf v}_{N}\|^{2}_{{\bf
L}^{2}(0,L)}\right)ds\right)$ $\displaystyle\leq C(\delta).$

Combining (74), (75) and using the Vitali convergence theorem, we thus
conclude the proof of Lemma 4.8.

∎

The rest of the convergence results follows as in [11]. This completes the
proof of Theorem 3.

∎

Notice that in the statement of Theorem 3 we still have the function
$\bar{\boldsymbol{\eta}}^{*}(t)$ in the weak formulation, which keeps the
displacement uniformly bounded. We now show that in fact
$\bar{\boldsymbol{\eta}}^{*}(t)$ can be replaced by the limiting stochastic
process $\bar{\boldsymbol{\eta}}(t)$ to obtain the desired weak formulation
and martingale solution until some stopping time $T^{\boldsymbol{\eta}}$,
which we show is strictly greater than zero almost surely.

######  Lemma 4.9.

(Stopping time.) Let the deterministic initial data ${\boldsymbol{\eta}}_{0}$
satisfy the assumptions (14). Then, for any $\delta>0$ satisfying (17) and for
a given $\frac{3}{2}<s<2$, there exists an almost surely positive stopping
time $T^{\boldsymbol{\eta}}$, given by

(76)
$T^{\boldsymbol{\eta}}:=T\wedge\inf\\{t>0:\inf_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}}(t)\leq\delta\\}\wedge\inf\\{t>0:\|\bar{\boldsymbol{\eta}}(t)\|_{{\bf
H}^{s}(0,L)}\geq\frac{1}{\delta}\\},$

such that

(77)
$\displaystyle\bar{\boldsymbol{\eta}}^{*}(t)=\bar{\boldsymbol{\eta}}(t)\quad\text{for
}t<T^{\boldsymbol{\eta}}.$

###### Proof.

To prove this lemma we write the stopping time as

$T^{\boldsymbol{\eta}}=T\wedge
T^{\boldsymbol{\eta}}_{1}+T^{\boldsymbol{\eta}}_{2},$

where $T^{\boldsymbol{\eta}}_{1}$ and $T^{\boldsymbol{\eta}}_{2}$ are defined
by:

$T^{\boldsymbol{\eta}}=T\wedge\inf\\{t>0:\inf_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}}(t)\leq\delta\\}\wedge\inf\\{t>0:\|\bar{\boldsymbol{\eta}}(t)\|_{{\bf
H}^{s}(0,L)}\geq\frac{1}{\delta}\\}=:T\wedge
T^{\boldsymbol{\eta}}_{1}+T^{\boldsymbol{\eta}}_{2}.$

We start with $T^{\boldsymbol{\eta}}_{2}$. Observe that using the triangle
inequality, for any $\delta_{0}>{\delta}$, we obtain

$\displaystyle\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}_{2}=0,\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}(0,L)}$ $\displaystyle<\frac{1}{\delta_{0}}]=\lim_{\epsilon\rightarrow
0}\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}_{2}<\epsilon,\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}(0,L)}<\frac{1}{\delta_{0}}]$
$\displaystyle\leq\limsup_{\epsilon\rightarrow
0^{+}}\bar{\mathbb{P}}[\sup_{t\in[0,\epsilon)}\|\bar{\boldsymbol{\eta}}(t)\|_{{\bf
H}^{s}(0,L)}>\frac{1}{\delta},\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}(0,L)}<\frac{1}{\delta_{0}}]$
$\displaystyle\leq\limsup_{\epsilon\rightarrow
0^{+}}\bar{\mathbb{P}}[\sup_{t\in[0,\epsilon)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{s}(0,L)}>\frac{1}{\delta}-\frac{1}{\delta_{0}}]$
$\displaystyle\leq\frac{1}{(\frac{1}{\delta}-\frac{1}{\delta_{0}})}\limsup_{N\to\infty}\bar{\mathbb{E}}[\sup_{t\in[0,\epsilon)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{s}(0,L)}]$
$\displaystyle\leq\frac{1}{(\frac{1}{\delta}-\frac{1}{\delta_{0}})}\limsup_{\epsilon\rightarrow
0}\bar{\mathbb{E}}[\sup_{t\in[0,\epsilon)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|^{1-\frac{s}{2}}_{{\bf
L}^{2}(0,L)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|^{\frac{s}{2}}_{{\bf
H}^{2}(0,L)}]$
$\displaystyle\leq\frac{1}{(\frac{1}{\delta}-\frac{1}{\delta_{0}})}\limsup_{\epsilon\rightarrow
0}\bar{\mathbb{E}}[\sup_{t\in[0,\epsilon)}\epsilon\|\bar{\bf
v}(t)\|^{1-\frac{s}{2}}_{{\bf
L}^{2}(0,L)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|^{\frac{s}{2}}_{{\bf
H}^{2}(0,L)}]$ $\displaystyle\leq\limsup_{\epsilon\rightarrow
0}\frac{\epsilon}{(\frac{1}{\delta}-\frac{1}{\delta_{0}})}\left(\bar{\mathbb{E}}[\sup_{t\in[0,\epsilon)}\|\bar{\bf
v}(t)\|^{2}_{{\bf
L}^{2}(0,L)}]\right)^{\frac{2-s}{4}}\left(\bar{\mathbb{E}}[\sup_{t\in(0,\epsilon)}\|\bar{\boldsymbol{\eta}}(t)-{\boldsymbol{\eta}}_{0}\|^{2}_{{\bf
H}^{2}(0,L)}]\right)^{\frac{s}{4}}$ $\displaystyle=0.$

Hence, by continuity from below, we deduce that for any $\delta>0$,

(78)
$\displaystyle\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}_{2}=0,\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}(0,L)}<\frac{1}{\delta}]=0.$

To estimate $T^{\boldsymbol{\eta}}_{1}$ we observe that, similarly, since for
any $t\in[0,T]$ we have that
$\inf_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}}(t)\geq\inf_{\mathcal{O}}\bar{J}_{0}-\|J_{\bar{\boldsymbol{\eta}}}(t)-J_{0}\|_{C(\mathcal{O})}$,
we write for any $\delta_{0}>\delta$

$\displaystyle\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}_{1}=0,\inf_{\mathcal{O}}J_{0}>\delta_{0}]$
$\displaystyle\leq\limsup_{\epsilon\rightarrow
0^{+}}\bar{\mathbb{P}}[\inf_{t\in[0,\epsilon)}\inf_{\bar{\mathcal{O}}}J_{\bar{\boldsymbol{\eta}}}(t)<\delta,\inf_{\mathcal{O}}\bar{J}_{0}>\delta_{0}]$
$\displaystyle\leq\limsup_{\epsilon\rightarrow
0^{+}}\bar{\mathbb{P}}[\sup_{t\in[0,\epsilon)}\|J_{\bar{\boldsymbol{\eta}}}(t)-J_{0}\|_{C(\mathcal{O})}>\delta_{0}-\delta]$
$\displaystyle\leq\frac{1}{(\delta_{0}-\delta)^{2}}\limsup_{\epsilon\rightarrow
0}\bar{\mathbb{E}}[\sup_{t\in[0,\epsilon)}\|J_{\bar{\boldsymbol{\eta}}}(t)-J_{0}\|^{2}_{C(\mathcal{O})}]$
$\displaystyle=0.$

Hence for given $\delta>0$,

(79)
$\displaystyle\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}_{1}=0,\inf_{\mathcal{O}}J_{0}>\delta]=0.$

That is we have,

(80)
$\displaystyle\bar{\mathbb{P}}[T^{\boldsymbol{\eta}}=0,\inf_{\mathcal{O}}J_{0}>\delta,\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}(0,L)}<\frac{1}{\delta}]=0.$

∎

Thus, by combining Theorem 3, Lemma 4.9, and (77), we finish the proof of the
main result:

######  Theorem 4.

(Main result.) For any given $\delta>0$ satisfying (17), if the deterministic
initial data ${\boldsymbol{\eta}}_{0}$ satisfies (14), then the stochastic
processes $(\bar{\bf u},\bar{\boldsymbol{\eta}},T^{\boldsymbol{\eta}})$
obtained in the limit specified in Theorem 2, along with the stochastic basis
constructed in Theorem 2, determine a martingale solution in the sense of
Definition 1 of the stochastic FSI problem (1)-(8), with the stochastic
forcing given by (10).

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