# A stochastic fluid-structure interaction problem with the Navier slip
boundary condition

Krutika Tawri1
1 Department of Mathematics, University of California Berkeley, CA, USA.
<EMAIL_ADDRESS>
###### Abstract.

We prove 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
via the kinematic and dynamic coupling conditions at the fluid-structure
interface. This article considers the case where the structure can have
unrestricted displacement and explores the Navier-slip boundary condition
imposed at the fluid-structure interface, displacement of which is not known a
priori and is itself a part of the solution. The proof takes a constructive
approach based on a Lie splitting scheme. The geometric nonlinearity stemming
from the nonlinear coupling, the possibility of random fluid domain
degeneracy, the potential jumps in the tangential components of the fluid and
structure velocities at the moving interface and the low regularity of the
structure velocity require the development of new techniques that lead to the
existence of martingale solutions.

##  1\. Introduction

This paper introduces a constructive approach for investigating solutions to a
complex problem describing the interaction between a deformable (purely)
elastic membrane and a two-dimensional viscous, incompressible fluid flow,
under the influence of multiplicative stochastic forces. The fluid flow is
described by the 2D Navier-Stokes equations, while the membrane is
characterized by shell equations. The fluid and the structure are fully
coupled across the moving interface through a two-way coupling that ensures
continuity of the normal components of their velocities and contact forces at
the interface. There has been a lot of work done in the field of deterministic
FSI in the past two decades (see e.g. [5, 12, 16, 17, 13] and the references
therein), 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 main result of this paper is the establishment of the existence of weak
martingale solutions to this highly nonlinear stochastic fluid-structure
interaction problem via a Lie operator splitting scheme. The recent articles
[23, 22], which represent the only work addressing stochastic moving boundary
problems, have demonstrated the existence of weak martingale solutions to FSI
problems with scalar and unrestricted structural deformations, respectively.
They also consider the no-slip boundary conditions imposed at the fluid-
structure interface. While this is a common assumption in the blood flow
literature (see e.g. [20], [26]), the slip condition is considered to be a
more realistic boundary condition in modeling near-contact dynamics, such as
the closure of heart valves, as it allows for the possibility of collisions
(see e.g. [19], [18]). In this work, we provide the first existence result for
a stochastic moving boundary problem involving the Navier-slip boundary
condition. It also provides, for the first time, a compactness argument, in
the context of FSI involving elastic structures, by constructing test
functions that are allowed to have possible jumps in the tangential direction
at the fluid-structure interface, which is a key feature of the slip
condition. Our compactness result thus generalizes the existing results while
also revealing hidden regularity of the structure.

The first mathematical issues that we come across are related to the facts
that the fluid domain boundary is a random variable, not known a priori, which
can possibly degenerate in a random fashion and that the incompressibility
condition and the Navier-slip boundary condition lead to the dependence of the
test functions on the randomly moving domains and thus require us to consider
random test processes which is highly unusual for typical SPDEs on fixed
domains. Due to the possibility of non-zero longitudinal structural
displacement, extra care has to be taken in dealing with degenerate fluid
domains i.e. when the structure touches a part of the fluid domain boundary
during deformation.

First, using the Arbitrary Lagrangian Eulerian (ALE) transformations, we map
the fluid equations onto a fixed domain. 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 ALE transformation. To deal with this injectivity condition in the
stochastic case we use a cut-off function and a stopping time argument.
Furthermore, via the ALE maps, additional nonlinearities appear in the weak
formulation of the problem that track several geometric quantities such as the
fluid-structure interface tangent and normal.

The dependence of the test functions on the domain configurations (via the ALE
maps) creates issues as we apply move to a new probability space in search of
martingale solutions. 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. However, addition
of this penalty term and the low temporal regularity of the solutions create
further difficulties in establishing compactness which we overcome by
employing non-standard compactness arguments. In establishing tightness of the
laws of the approximate solutions, we also do not have the extra regularity
for the structure velocity obtained from the fluid dissipation in the no-slip
case. We finally show that the solutions to the approximate systems indeed
converge to a desired martingale 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 (time-dependent and random)
fluid domains. Namely, along with the required divergence-free property on
these domains, the test functions have to satisfy appropriate boundary and
measurability conditions. 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 transforming this limiting
test function in a way that preserves its desired properties on the
approximate domains.

##  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, which 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

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

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 fluid domain

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

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

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 and we prescribe the symmetry boundary
condition on the botton boundary as follows:

(2.2) $\displaystyle p+\frac{1}{2}|{\bf u}|^{2}$
$\displaystyle=P_{in/out}(t),\qquad u_{r}=0\quad\text{ on }\Gamma_{in/out}.$
(2.3) $\displaystyle u_{r}$ $\displaystyle=\partial_{r}u_{z}=0\quad\text{on
}\Gamma_{b}.$

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

(2.4)
$\displaystyle\partial^{2}_{t}{\boldsymbol{\eta}}+\mathscr{L}_{e}({\boldsymbol{\eta}})=F_{\boldsymbol{\eta}}\quad\text{
in }(0,L),$

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 ${\bf H}^{2}_{0}(0,L)$. This equation is supplemented with
the following boundary conditions:

(2.5)
$\begin{split}{\boldsymbol{\eta}}(0)={\boldsymbol{\eta}}(L)=\partial_{z}{\boldsymbol{\eta}}(0)=\partial_{z}{\boldsymbol{\eta}}(L)=0.\end{split}$

The non-linear fluid-structure coupling: The coupling between the fluid and
the structure takes place across the moving fluid-structure interface.

  * •

The kinematic coupling conditions in the Navier-slip case are:

(2.6) $\displaystyle\partial_{t}{\boldsymbol{\eta}}(t,z)\cdot{\bf
n}^{\boldsymbol{\eta}}={\bf u}({\boldsymbol{\varphi}}(t,z))\cdot{\bf
n}^{\boldsymbol{\eta}},\quad(t,z)\in[0,T]\times[0,L],$ (2.7)
$\displaystyle(\partial_{t}{\boldsymbol{\eta}}-{\bf
u}({\boldsymbol{\varphi}}(t,z)))\cdot\boldsymbol{\tau}^{\boldsymbol{\eta}}=\alpha\sigma({\boldsymbol{\varphi}}(t,z)){\bf
n}^{\boldsymbol{\eta}}\cdot\boldsymbol{\tau}^{\boldsymbol{\eta}},\quad(t,z)\in[0,T]\times[0,L].$

  * •

The dynamic coupling condition is:

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

where ${\bf n}^{\boldsymbol{\eta}}(t,z)$ is the unit outward normal to the top
boundary at the point, $\boldsymbol{\tau}^{\boldsymbol{\eta}}$ is the tangent
vector given by
$\boldsymbol{\tau}^{\boldsymbol{\eta}}(t,z)=\partial_{z}{\boldsymbol{\varphi}}(t,z)$
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:

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

### Weak formulation on moving domain

Using the convention that bold-faced letters denote spaces containing vector-
valued functions, we define the following relevant function spaces for the
fluid velocity and the structure displacement:

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

Next, we derive a deterministic 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 ${{\bf
q}}({\boldsymbol{\varphi}}(t,z))\cdot{\bf
n}^{\boldsymbol{\eta}}=\boldsymbol{\psi}(t,z)\cdot{\bf n}^{\boldsymbol{\eta}}$
on $(0,T)\times\Gamma$ for some $\boldsymbol{\psi}\in C^{1}([0,T];{\bf
H}^{2}_{0}(\Gamma))$. We multiply (2.1) by ${\bf q}$, integrate in time and
space and use Reynold’s transport theorem to obtain,

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

Set,

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

and observe that,

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

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

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

whereas, $\int_{\Gamma_{b}}\sigma{\bf n}^{\boldsymbol{\eta}}\cdot{\bf q}=0.$
We also write $\int_{\Gamma_{\boldsymbol{\eta}}}\sigma{\bf
n}^{\boldsymbol{\eta}}\cdot{\bf q}$ as,

$\int_{\Gamma_{\boldsymbol{\eta}}}\sigma{\bf n}^{\boldsymbol{\eta}}\cdot(({\bf
q}\cdot{\bf n}^{\boldsymbol{\eta}}){\bf n}^{\boldsymbol{\eta}}+({\bf
q}\cdot\boldsymbol{\tau}^{\boldsymbol{\eta}})\boldsymbol{\tau}^{\boldsymbol{\eta}})=\int_{\Gamma_{\boldsymbol{\eta}}}\sigma{\bf
n}^{\boldsymbol{\eta}}\cdot{\bf
n}^{\boldsymbol{\eta}}(\boldsymbol{\psi}\cdot{\bf
n}^{\boldsymbol{\eta}})+\frac{1}{\alpha}(\partial_{t}{\boldsymbol{\eta}}-{\bf
u})\cdot\boldsymbol{\tau}^{\boldsymbol{\eta}}({\bf
q}\cdot\boldsymbol{\tau}^{\boldsymbol{\eta}}).$

Next we multiply the structure equation (2.4) by $\boldsymbol{\psi}$ and
integrate in time and space to obtain

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

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

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

Here $F_{u}^{ext}$ is the volumetric external force applied to the fluid and
$F_{{\boldsymbol{\eta}}}^{ext}$ is the external force applied to the
deformable boundary.

###  2.2. Stochastic framework on fixed domain

We will take $F^{ext}_{u},F^{ext}_{\boldsymbol{\eta}}$ to be random forces.
For that purpose, consider a 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 $t\geq 0$.

####  2.2.1. ALE mappings

To deal the geometric nonlinearity arising due to the motion of the fluid
domain, we work with 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 define the ALE mappings 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 $A^{\omega}_{\boldsymbol{\eta}}(t)=\textbf{id}\text{ on
}\partial\mathcal{O}\setminus\Gamma.$

The pathwise transformed gradient, symmetrized gradient and divergence under
this transformation will be denoted by

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

The Jacobian of the ALE mapping is given by
$J^{\omega}_{\boldsymbol{\eta}}(t)=\text{det }\nabla
A_{{\boldsymbol{\eta}}}^{\omega}(t).$ Using ${\bf w}^{\boldsymbol{\eta}}$ to
denote the ALE velocity ${\bf
w}^{\boldsymbol{\eta}}=\frac{d}{dt}A_{\boldsymbol{\eta}}$, we note that
$\partial_{t}J_{\boldsymbol{\eta}}=J_{\boldsymbol{\eta}}\nabla^{\boldsymbol{\eta}}\cdot{\bf
w}^{\boldsymbol{\eta}}.$ We also rewrite the advection term as follows:

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

We will transform (LABEL:origweakform) using these ALE maps and give the
definition of martingale solutions on the fixed domain $\mathcal{O}$.

We begin by describing the noise. 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:

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

where $W$ 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 $U_{0}:=Q^{\frac{1}{2}}(U)$. Letting ${\bf
L}^{2}={\bf L}^{2}(\mathcal{O})\times{\bf L}^{2}(0,L)$, we now give
assumptions on the noise coefficient $G$:

######  Assumption 2.1.

The noise coefficient $G$ is a function $G:{\bf L}^{2}(\mathcal{O})\times{\bf
H}^{2}_{0}(0,L)\rightarrow L_{2}(U_{0};{\bf L}^{2})$, such that for any
$\frac{3}{2}\leq s<2$ the following conditions hold true:

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

Here $L_{2}(X,Y)$ denotes the space of Hilbert-Schmidt operators from Hilbert
spaces $X$ to $Y$.

####  2.2.2. Definition of martingale solutions

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

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

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

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

######  Definition 1.

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

(2.14)
$\displaystyle{\delta_{1}<\inf_{\mathcal{O}}J_{{\boldsymbol{\eta}}_{0}},\quad\text{
and }\quad\|{\boldsymbol{\eta}}_{0}\|_{{\bf
H}^{2}_{0}(0,L)}<\frac{1}{\delta_{2}}},$

we say that $(\mathscr{S},{\bf u},{\boldsymbol{\eta}},T^{\boldsymbol{\eta}})$
is a martingale solution to the system (2.1)-(2.9) 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)

${\bf u}$ 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 $C^{1}$ paths in $\mathscr{D}$
such that $\nabla^{\boldsymbol{\eta}}\cdot{\bf q}=0$, and ${\bf
q}_{\Gamma}\cdot{\bf n}^{\boldsymbol{\eta}}=\boldsymbol{\psi}\cdot{\bf
n}^{\boldsymbol{\eta}}$ the equation

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

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

We will next present a constructive proof of the existence of martingale
solutions based on the operator splitting scheme constructed in the following
section.

##  3\. Operator splitting scheme

In this section we introduce a Lie operator splitting scheme that defines a
sequence of approximate solutions to (2.15) by semi-discretizing the problem
in time. Our aim 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 discretize the problem in time and use an operator splitting to decouple
the stochastic problem into two subproblems, viz the structure and the fluid
subproblems. 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. Then at the
$i^{th}$ time level, we update the vector $({\bf
u}^{n+\frac{i}{2}},{\boldsymbol{\eta}}^{n+\frac{i}{2}},{\bf
v}^{n+\frac{i}{2}})$, for $i=1,2$ and $n=0,1,2,...,N-1$, as follows.

### The structure sub-problem

In this sub-problem we update the structure displacement and the structure
velocity while keeping the fluid velocity unchanged. 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 pathwise solution
$({\boldsymbol{\eta}}^{n+\frac{1}{2}},{\bf v}^{n+\frac{1}{2}})\in{\bf
H}^{2}_{0}(0,L)\times{\bf H}^{2}_{0}(0,L)$ to the following equations: For any
$\boldsymbol{\phi}\in{\bf L}^{2}(0,L)$ and $\boldsymbol{\psi}\in{\bf
H}^{2}_{0}(0,L)$,

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

For each $\omega\in\Omega$ and $n$, we define the ALE map associated with the
structure variable $\boldsymbol{\eta}^{n}$ as the solution to:

(3.2) $\begin{split}\Delta A^{\omega}_{{\boldsymbol{\eta}}^{n}}&=0,\quad\text{
in }\mathcal{O},\\\
A^{\omega}_{{\boldsymbol{\eta}}^{n}}=\textbf{id}+{\boldsymbol{\eta}}^{n}(\omega)\text{
on }\Gamma,&\quad\text{ and }\quad
A^{\omega}_{{\boldsymbol{\eta}}^{n}}=\textbf{id}\text{ on
}\partial\mathcal{O}\setminus\Gamma.\end{split}$

Note that we have added the last term in (3.1)3 to regularize the structure
velocity. This is required for the time derivative of the Jacobian in
(LABEL:martingale1) to make sense and to circumvent the issues associated with
the ”very weak” solutions to the Poisson equation on polygonal domains with
corners. We will first pass $N\to\infty$ and then $\varepsilon\to 0$.

######  Lemma 3.1.

Consider $v\in H^{2}_{0}(\Gamma)$ and $w\in H^{2}(\mathcal{O})$ that solve,

(3.3) $\displaystyle-\Delta w=0\text{ in }\mathcal{O},\quad w=v\text{ on
}\Gamma,\quad w=0\text{ on }\partial\mathcal{O}\setminus\Gamma.$

Then,

(3.4)
$\displaystyle\|w\|_{L^{2}(\mathcal{O})}\leq\|v\|_{H^{-\frac{1}{2}}(\Gamma)}.$

###### Proof.

We introduce the following system:

$\displaystyle-\Delta g=w\text{ in }\mathcal{O},\quad g=0\text{ on
}\partial\mathcal{O}.$

Since $w,g\in H^{2}(\mathcal{O})$, thanks to Theorem 1.5.3.3. in [14] (compare
with Remark 1.5.3.5 in [14]), the following holds,

$\displaystyle(w,w)=-(\Delta g,w)=-(g,\Delta
w)+\int_{\Gamma}v\,\partial_{n}g=\int_{\Gamma}v\,\partial_{n}g.$

Hence, thanks to Theorem 1.5.2.1 in [14] we then obtain

$\displaystyle\|w\|_{L^{2}(\mathcal{O})}^{2}\leq\|v\|_{H^{-\frac{1}{2}}(\Gamma)}\|\partial_{n}g\|_{H^{\frac{1}{2}}(\Gamma)}\leq\|v\|_{H^{-\frac{1}{2}}(\Gamma)}\|g\|_{H^{2}(\mathcal{O})}\leq\|v\|_{H^{-\frac{1}{2}}(\Gamma)}\|w\|_{L^{2}(\mathcal{O})}.$

∎

### The fluid sub-problem

In this sub-problem we update the fluid and the structure velocities while
keeping the structure displacement the same. As notes in the introduction,
there are two major difficulties associated with constructing this half of the
scheme. The first difficulty arises because the fluid domains can degenerate
randomly. Hence, we introduce an ”artificial structure displacement” random
variable by the means of a cut-off function as follows: For
$\delta=(\delta_{1},\delta_{2})$, let $\Theta_{\delta}$ be the step function
that satisfies $\Theta_{\delta}(x,y)=1$ if
$\delta_{1}<x,\text{}y<\frac{1}{\delta_{2}}$, and $\Theta_{\delta}(x,y)=0$
otherwise. For brevity we define,

(3.5) $\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)\quad\text{for a fixed $s\in(\frac{3}{2},2)$},$

where $J^{k}(\omega)=\text{det}\nabla A^{\omega}_{{\boldsymbol{\eta}}^{k}}$ is
the Jacobian of the map defined in (3.2). Note that, $\theta_{\delta}$ is a
real-valued function which tracks all the structure displacements, and is
equal to 1 until the step for which the structure quantities leave the desired
bounds given in terms of $\delta$. Now we define the artificial structure
displacement random variable as the following stopped process,

(3.6)
$\displaystyle{\boldsymbol{\eta}}^{n}_{*}(z,\omega)={\boldsymbol{\eta}}^{\max_{0\leq
k\leq n}\theta_{\delta}({\boldsymbol{\eta}}^{k})k}(z,\omega)\quad\text{for
every }\omega\in\Omega,\,\,z\in[0,L].$

Observe that, for any $p>2$ and $s\geq\frac{5}{2}-\frac{2}{p}$, we have the
following regularity result for the harmonic extension of the boundary data
associated with $\boldsymbol{\eta}^{n}_{*}$ on a square (see Section 5 in
[14]):

(3.7) $\displaystyle\|A^{\omega}_{{\boldsymbol{\eta}}_{*}^{n}}-{\bf
id}\|_{{\bf W}^{2,p}(\mathcal{O})}\leq C\|{\boldsymbol{\eta}}_{*}^{n}\|_{{\bf
W}^{2-\frac{1}{p},p}(\Gamma)}\leq C\|{\boldsymbol{\eta}}_{*}^{n}\|_{{\bf
H}^{s}(\Gamma)}.$

Then Morrey’s inequality for some $p<4$, gives us a constant $C_{*}>0$ (see
Theorem 7.26 in [10]) such that,

(3.8) $\displaystyle\|\nabla(A^{\omega}_{{\boldsymbol{\eta}}_{*}^{n}}-{\bf
id})\|_{{\bf C}^{0,\frac{1}{p}}(\bar{\mathcal{O}})}\leq
C\|{\boldsymbol{\eta}}_{*}^{n}\|_{{\bf
H}^{s}(\Gamma)}\leq\frac{C_{*}}{\delta_{2}},\quad\frac{5}{2}-\frac{2}{p}\leq
s<2.$

Theorem 5.5-1 (B) of [6], then ensures that the map
$A^{\omega}_{{\boldsymbol{\eta}}^{n}_{*}}\in{\bf
C}^{1,\frac{1}{p}}(\bar{\mathcal{O}})$ is injective for any $n$ if
$\delta_{2}$ satisfies

(3.9) $\displaystyle C_{*}<\delta_{2}.$

Hence, such $\delta_{2}$, the domain configurations corresponding the the
artificial variables $\boldsymbol{\eta}^{n}_{*}$ are non-degenerate and their
Jacobians have a deterministic lower bound of $\delta_{1}$. These artificial
domain configurations will be used to define the fluid sub-problem.

The second difficulty in constructing the dependence of the fluid test
functions, through the transformed divergence-free condition and the kinematic
coupling condition, on the structure displacement found in the previous sub-
problem. Hence to avoid dealing with random test function we supplement the
weak formulation in this sub-problem by penalty terms, via the parameter
$\varepsilon>0$, that enforce the incompressibility condition and the
continuity of velocities in the normal direction, only in the limit
$\varepsilon\rightarrow 0$.

A penalized system on artificial domains: Let
$\Delta_{n}W:=W(t^{n+1})-W(t^{n})$. Then we look for $({\bf u}^{n+1},{\bf
v}^{n+1})\in{\mathscr{U}}$ that solves

(3.10)
$\begin{split}&\qquad\qquad{\boldsymbol{\eta}}^{n+1}:={\boldsymbol{\eta}}^{n+\frac{1}{2}},\\\
&\int_{\mathcal{O}}J^{n}_{*}\left({\bf u}^{n+1}-{\bf
u}^{n+{\frac{1}{2}}}\right){\bf q}d{\bf
x}+\frac{1}{2}\int_{\mathcal{O}}\left(J_{*}^{n+1}-J^{n}_{*}\right){\bf
u}^{n+1}\cdot{\bf q}d{\bf x}\\\ &+\frac{1}{2}(\Delta
t)\int_{\mathcal{O}}J_{*}^{n}(({\bf u}^{n+1}-{\bf
w}_{*}^{{n}})\cdot\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf u}^{n+1}\cdot{\bf
q}-({\bf u}^{n+1}-{\bf
w}_{*}^{{n}})\cdot\nabla^{{\boldsymbol{\eta}}_{*}^{n}}{\bf q}\cdot{\bf
u}^{n+1})d{\bf x}\\\ &+2\nu(\Delta t)\int_{\mathcal{O}}J^{n}_{*}{\bf
D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf u}^{n+1})\cdot{\bf
D}^{{\boldsymbol{\eta}}_{*}^{n}}({\bf q})d{\bf x}\\\ &+\frac{(\Delta
t)}{\varepsilon}\int_{\mathcal{O}}\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf
u}^{n+1}\text{div}^{{\boldsymbol{\eta}}^{n}_{*}}{\bf q}d{\bf x}+\frac{(\Delta
t)}{\varepsilon}\int_{\Gamma}({\bf u}^{n+1}-{\bf v}^{n+1})\cdot{\bf
n}^{n}_{*}({\bf q}-\boldsymbol{\psi})\cdot{\bf n}^{n}_{*}\\\ &+\frac{(\Delta
t)}{\alpha}\int_{\Gamma}S^{n}({\bf u}^{n+1}-{\bf v}^{n+1})({\bf
q}-\boldsymbol{\psi})dz+\int_{0}^{L}({\bf v}^{n+1}-{\bf
v}^{n+\frac{1}{2}})\boldsymbol{\psi}dz\\\
&=\int_{0}^{t}\left(P^{n}_{{in}}\int_{0}^{1}q_{z}\Big{|}_{z=0}dr-P^{n}_{{out}}\int_{0}^{1}q_{z}\Big{|}_{z=1}dr\right)+(G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf Q})_{{\bf
L}^{2}},\end{split}$

for any $({\bf q},\boldsymbol{\psi})\in{\mathscr{U}}$. Here we set,
$P^{n}_{in/out}:=\frac{1}{\Delta t}\int_{t^{n}}^{t^{n+1}}P_{in/out}\,dt$.
Moreover, the random variable ${\bf n}^{n}_{*}$ is the unit normal to
$\Gamma_{\boldsymbol{\eta}^{n}_{*}}$ and

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

######  Remark 1.

Note that, to obtain a stable scheme, we update $\boldsymbol{\eta}^{n}$ in the
first sub-problem using the data from the second sub-problem and not
$\boldsymbol{\eta}_{*}^{n}$. However this means that after a certain random
time, we will produce solutions that are meaningless. These discrepensies will
be handled 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.

Now we introduce the following discrete energy and dissipation for $i=0,1$:

(3.11)
$\begin{split}&E^{n+\frac{i}{2}}=\frac{1}{2}\Big{(}\int_{\mathcal{O}}J^{n}_{*}|{\bf
u}^{n+\frac{i}{2}}|^{2}d{\bf x}+\|{\bf v}^{n+\frac{i}{2}}\|^{2}_{{\bf
L}^{2}(0,L)}+\|{\boldsymbol{\eta}}^{n+\frac{i}{2}}\|^{2}_{{\bf
H}_{0}^{2}(0,L)}\Big{)},\\\ &D^{n}_{1}=\varepsilon{(\Delta
t)}\int_{0}^{L}|\partial_{zz}{\bf v}^{n+\frac{1}{2}}|^{2},\\\
&D^{n}_{2}=\Delta t\int_{\mathcal{O}}\left(2\nu J^{n}_{*}|{\bf D}({\bf
u}^{n+1})|^{2}\right)d{\bf x}+\frac{\Delta t}{\alpha}\int_{\Gamma}|{\bf
u}^{n+1}-{\bf v}^{n+1}|^{2}\\\ &+\frac{(\Delta
t)}{\varepsilon}\int_{\mathcal{O}}|\text{div}^{\boldsymbol{\eta}^{n}_{*}}{\bf
u}^{n+1}|^{2}dx+\frac{\Delta t}{\varepsilon}\int_{\Gamma}|({\bf u}^{n+1}-{\bf
v}^{n+1})\cdot{\bf n}^{n}_{*}|^{2}.\end{split}$

######  Lemma 3.2.

(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
(3.1), and the following semidiscrete energy inequality holds:

(3.12) $\begin{split}E^{n+\frac{1}{2}}+D^{n}_{1}+C_{1}^{n}=E^{n},\end{split}$

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 straight-
forward and the reader is referred to [15] for details. This allows us to
write:

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

We now take $\boldsymbol{\psi}={\bf v}^{n+\frac{1}{2}}$ in (3.1)3 and using
$a(a-b)=\frac{1}{2}(|a|^{2}-|b|^{2}+|a-b|^{2})$, we obtain,

(3.13) $\begin{split}&\|{\bf v}^{n+\frac{1}{2}}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\bf v}^{n+\frac{1}{2}}-{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n+\frac{1}{2}}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n+\frac{1}{2}}-{\boldsymbol{\eta}}^{n}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}\\\ &+\varepsilon(\Delta t)\|\partial_{zz}{\bf
v}^{n+\frac{1}{2}}\|^{2}_{{\bf L}^{2}(0,L)}=\|{\bf v}^{n}\|_{{\bf
L}^{2}(0,L)}^{2}+\|{\boldsymbol{\eta}}^{n}\|_{{\bf
H}_{0}^{2}(0,L)}^{2}.\end{split}$

Recalling that ${\bf u}^{n}={\bf u}^{n+\frac{1}{2}}$ and adding the relevant
terms on both sides of (3.13) we obtain (3.12). ∎

######  Lemma 3.3.

(Existence for the fluid sub-problem.) For a given
$\delta=(\delta_{1},\delta_{2})$ satisfying (3.9), 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 ${\bf U}^{n+1}=({\bf
u}^{n+1},{\bf v}^{n+1})$ taking values in ${\mathscr{U}}$ that solves
(LABEL:second), and the solution satisfies the following estimate:

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

where

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

is numerical dissipation, and ${\boldsymbol{\eta}}^{n}_{*}$ is as defined in
(3.6).

###### Proof.

The proof of existence and measurability of the solutions is given using
Brouwer’s fixed point theorem and the Kuratowski selection theorem in [23].

To obtain (3.13), we take $({\bf q},\boldsymbol{\psi})=({\bf u}^{n+1},{\bf
v}^{n+1})$ in (LABEL:second) and use the identity
$a(a-b)=\frac{1}{2}(|a|^{2}-|b|^{2}+|a-b|^{2})$,:

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

Here we split the discrete stochastic integral into two terms. We estimate the
first term by using the Cauchy-Schwarz inequality: for some $C(\delta)>0$
independent of $n$ the following holds:

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

We treat the terms with $P_{in/out}$ similarly to obtain (3.14). ∎

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

######  Theorem 3.1.

(Uniform Estimates.) There exists a constant $C>0$ that depends on the initial
data, $\delta$, $T$, and $P_{in/out}$ and is independent of $N$ and
$\varepsilon$ 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)

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

  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 $D^{n}=D^{n}_{1}+D^{n}_{2}$ (see Definitions (3.11)).

###### Proof.

We first add (3.12) and (3.14) to obtain:

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

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

(3.16)
$\begin{split}&E^{m}+\sum_{n=0}^{m-1}D^{n}+\sum_{n=0}^{m-1}C_{1}^{n}+\sum_{n=0}^{m-1}C_{2}^{n}\leq
E^{0}+C\,\Delta
t\sum_{n=0}^{m-1}\left((P^{n}_{{in}})^{2}+(P^{n}_{{out}})^{2}\right)\\\
&+\sum_{n=0}^{m-1}\left|(G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\Delta_{n}W,{\bf
U}^{n})\right|+\sum_{n=0}^{m-1}\|G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}.\end{split}$

We take supremum over $1\leq m\leq N$ and then take expectation on both sides
of (3.16). The discrete Burkholder-Davis-Gundy inequality, (3.6) and (3.8)
give us for some $C(\delta)>0$ that,

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

We use the tower property and (LABEL:growthG) for each $n=0,...,m-1$ to
obtain,

$\displaystyle{\mathbb{E}}[\|G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[\|G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}\|\Delta_{n}W\|_{U_{0}}^{2}|\mathcal{F}_{n}]]$
$\displaystyle={\mathbb{E}}[\|G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\|_{L_{2}(U_{0},{\bf
L}^{2})}^{2}{\mathbb{E}}[\|\Delta_{n}W\|_{U_{0}}^{2}\|\mathcal{F}_{n}]]$
$\displaystyle=\Delta t(\text{Tr}{\bf Q}){\mathbb{E}}\|G({\bf
u}^{n},{\boldsymbol{\eta}}_{*}^{n})\|^{2}_{L_{2}(U_{0},{\bf L}^{2})}$ (3.17)
$\displaystyle\leq C\Delta t(\text{Tr}{\bf
Q}){\mathbb{E}}\left(\|\sqrt{J_{*}^{n}}{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|\boldsymbol{\eta}^{n}_{*}\|^{2}_{{\bf
H}^{2}_{0}(0,L)}\right).$

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

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

The discrete Gronwall inequality finally gives us,

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

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

Hence for $E^{n},D^{n}$ defined in (3.11) we have,

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

∎

###  3.2. Approximate solutions

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 $(0,T)$. First we introduce approximate
solutions that are piece-wise constant on each sub-interval $[n\Delta
t,(n+1)\Delta t)$:

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

These processes are adapted to the given filtration $(\mathcal{F}_{t})_{t\geq
0}$. The following are their time-shifted versions which are commonly used in
deterministic settings:

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

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

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

Observe that,

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

where ${\bf v}^{\\#}_{N}$ is introduced in (3.19). In the following Lemma we
summarize the results that are immediate consequences of the estimates obtain
in Theorem 3.1.

######  Lemma 3.4.

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 $\delta=(\delta_{1},\delta_{2})$ satisfying (3.9),
we have that

  1. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

$\\{{\bf u}^{+}_{N}\\}$ is bounded independently of $N$ and $\varepsilon$ in
$L^{2}(\Omega;L^{2}(0,T;{\bf H}^{1}(\mathcal{O})).$

  5. (5)

$\\{\frac{1}{\sqrt{\varepsilon}}\text{div}^{{\boldsymbol{\eta}}^{*}_{N}}{\bf
u}^{+}_{N}\\}$ is bounded independently of $N$ and $\varepsilon$ in
$L^{2}(\Omega;L^{2}(0,T;L^{2}(\mathcal{O})))$.

  6. (6)

$\\{\frac{1}{\sqrt{\varepsilon}}({\bf u}^{+}_{N}-{\bf v}^{+}_{N})\cdot{\bf
n}^{*}_{N}\\}$ is bounded independently of $N,\varepsilon$ in
$L^{2}(\Omega;L^{2}(0,T;L^{2}(\Gamma)))$

  7. (7)

$\\{\sqrt{\varepsilon}{\bf v}^{\\#}_{N}\\}$ is bounded independently of $N$
and $\varepsilon$ in $L^{2}(\Omega;L^{2}(0,T;{\bf H}_{0}^{2}(0,L)))$.

###### Proof.

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 [27] and because of
uniform bounds (3.8) 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, thanks to Theorem 3.1,
there exists $C>0$, independent of $N$, such that

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

The proofs of the rest of the statements follow immediately from Theorem 3.1.
∎

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

In this section we will pass $N\to\infty$ by first establishing tightness of
the laws of the approximate random variables defined in Section 3.2.

###  4.1.  Tightness results

Since we do not expect our candidate solutions to be differentiable in time,
the tightness results, i.e. Lemmas 4.2 and 4.3 below, will rely on the
following theorem; see [24, 21]:

######  Lemma 4.1.

Denote the translation in time by $h$ of a function $f$ by:

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

Assume that $\mathcal{Y}_{0}\subset\mathcal{Y}\subset\mathcal{Y}_{1}$ are
reflexive Banach spaces such that the embedding of $\mathcal{Y}_{0}$ in
$\mathcal{Y}$ is compact. Then for any $m>0$, the embedding

$\\{{\bf u}\in
L^{2}(0,T;\mathcal{Y}_{0}):\sup_{0<h<T}\frac{1}{h^{m}}\|T_{h}{\bf u}-{\bf
u}\|_{L^{2}(h,T;\mathcal{Y}_{1})}<\infty\\}\hookrightarrow
L^{2}(0,T;\mathcal{Y}),$

is compact.

We will now obtain our first tightness result for the fluid and structure
velocities by bypassing the need for higher moment estimates (see also [11] in
this context).

######  Lemma 4.2.

The laws of ${\bf u}^{+}_{N}$are tight in $L^{2}(0,T;{\bf
H}^{{\alpha}}(\mathcal{O}))$, and that of ${\bf v}^{+}_{N}$ are tight in
$L^{2}(0,T;{\bf H}^{-{\beta}}(0,L))$ for any $0\leq\alpha<1$,
$0<\beta<\frac{1}{2}$.

###### Proof.

The aim of this proof is to apply Lemma 4.1 by obtaining bounds for

$\displaystyle\int_{h}^{T}\|T_{h}{\bf u}^{+}_{N}-{\bf u}^{+}_{N}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}$ $\displaystyle+\|T_{h}{\bf v}^{+}_{N}-{\bf
v}^{+}_{N}\|^{2}_{{\bf H}^{-\beta}(0,L)}$ (4.1) $\displaystyle=(\Delta
t)\sum_{n=j}^{N}\|{\bf u}^{n}-{\bf u}^{n-j}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|{\bf v}^{n}-{\bf v}^{n-j}\|^{2}_{{\bf
H}^{-\beta}(0,L)},$

for any $N$ and $h$. Here $1\leq j\leq N$ such that $h=j\Delta t-l$ for some
$l<\Delta t$. For simplicity we take $l=0$.

To achieve this goal, we will construct appropriate test functions for
equations (3.1) and (LABEL:second) that will result into the term on the right
side of (4.1). This is done by transforming ${\bf u}^{k}$ and ${\bf v}^{k}$ in
a way that they can be used as test functions for the equations for ${\bf
u}^{n}$ and ${\bf v}^{n}$. This has to be done carefully since ${\bf u}^{k}$
and ${\bf u}^{n}$ are defined on different physical domains. Observe also that
we can not test (3.1)3 directly with ${\bf v}^{k}$, as it does not have the
required $H^{2}_{0}$-bounds independent of $\varepsilon$. Thus, we will use a
space regularization of ${\bf v}^{k}$ to arrive at the desired test function
for (3.1) and (LABEL:second). First, for any ${\bf g}\in{\bf
H}^{1}(\mathcal{O})$, we extend ${\bf g}$ by its boundary values at
$\partial\mathcal{O}$ constantly in the normal direction outside of
$\mathcal{O}$ and define it to be 0 elsewhere. Then, denoting the 1D bump
function by $\rho$, we define its horizontally mollified version as follows:

$\displaystyle{\bf g}_{\sigma}(z,r)=\int_{-1}^{1}{\bf g}(z+\sigma
y,r)\rho(y)dy.$

Now let $P_{M}$ denote the orthonormal projector in $L^{2}(\Gamma)$ onto the
space $\text{span}_{1\leq i\leq M}\\{\varphi_{i}\\}$, where $\varphi_{i}$
satisfies $-\Delta\varphi_{i}=\gamma_{i}\varphi_{i}$ and $\varphi_{i}(z)=0$
when $z=0,L$. For any ${\bf v}\in{\bf L}^{2}(\Gamma)$ we notate ${\bf
v}_{M}=P_{M}{\bf v}$ and denote by ${\bf w}_{M}$ the harmonic extension of
${\bf v}_{M}$ in $\mathcal{O}$, such that ${\bf w}_{M}=0$ on
$\partial\mathcal{O}\setminus\Gamma$ (cf. (3.3)). Similarly, let ${\bf w}^{k}$
be the harmonic extension of ${\bf v}^{k}$ in $\mathcal{O}$ with 0 boundary
values on $\partial\mathcal{O}\setminus\Gamma$. For the rest of the proof we
fix, $\sigma=\frac{L}{4}h^{\frac{1}{12}}$ and since
$\gamma_{M}\sim\gamma_{1}M^{2}$, we choose $M$ such that
$\gamma_{M}=ch^{-\frac{1}{6}}$.

Now we define the following function that will lead to a suitable test
function for the fluid subproblem (see (4.8)):

$\displaystyle{\bf u}^{k,n}_{\sigma,M}:=(J^{n}_{*})^{-1}\nabla
A_{\boldsymbol{\eta}^{n}_{*}}\left(J^{k}_{*}(\nabla
A_{\boldsymbol{\eta}^{k}_{*}})^{-1}({\bf u}^{k}-{\bf
w}^{k})\right)_{\sigma}+{\bf
w}^{k}_{M}+\left(\frac{\lambda^{k}_{\sigma}-\lambda^{k,n}_{M}}{\lambda^{n}_{0}}\right)\xi_{0}\chi$
$\displaystyle-\frac{1}{J^{n}_{*}}\mathcal{B}\left(\text{div}\left((J^{k}_{*}(\nabla
A_{\boldsymbol{\eta}^{k}_{*}})^{-1}{\bf w}^{k})_{\sigma}-J^{n}_{*}(\nabla
A_{\boldsymbol{\eta}^{n}_{*}})^{-1}{\bf
w}^{k}_{M}\right)-\left(\frac{\lambda^{k}_{\sigma}-\lambda^{k,n}_{M}}{\lambda^{n}_{0}}\right)\text{div}((J^{n}_{*})^{-1}\nabla
A_{\boldsymbol{\eta}^{n}_{*}}\xi_{0}\chi)\right),$

where

  * •

$\chi(r)$ is a smooth function on $\mathcal{O}$ such that $\chi(1)=1$ and
$\chi(0)=0$ ,

  * •

$\xi_{0}\in{\bf C}^{\infty}_{0}(0,L)$ that satisfies
$\lambda_{0}^{n}:=\int_{0}^{L}(\partial_{z}({\bf
id}+\boldsymbol{\eta}^{n}_{*})\times\xi_{0})\neq 0$ for any $n$,

  * •

The constants $\lambda_{\sigma}^{k}=\int_{0}^{L}(\partial_{z}({\bf
id}+\boldsymbol{\eta}^{k}_{*})\times{\bf v}^{k})_{\sigma}$ and
$\lambda^{k,n}_{M}=\int_{0}^{L}(\partial_{z}({\bf
id}+\boldsymbol{\eta}^{n}_{*})\times{\bf v}_{M}^{k})$,

  * •

Finally, $\mathcal{B}:L^{2}(\mathcal{O})\to{\bf H}^{1}_{0}(\mathcal{O})$ is
Bogovski’s operator (see [8]). $\mathcal{B}$ along with the constants in the
previous point are used to correct the divergence of the extra terms appearing
in the definition of ${\bf u}^{k,n}_{\sigma,M}$ due to the extension of
structure velocities in the fluid domains.

Observe that to preserve 0 boundary conditions on
$\partial\mathcal{O}\setminus\Gamma$ we must also ”squeeze” the mollified
function by $\sim 1+\sigma$ as in [23, 22]. However, we skip that for a
cleaner presentation as it does not change the following estimates and
argument. Next let,

${\bf v}^{k,n}_{\sigma,M}:={\bf
v}^{k}_{M}-\left(\frac{\lambda^{k}_{\sigma}-\lambda^{k,n}_{M}}{\lambda_{0}^{n}}\right)\xi_{0}.$

Observe that we used Piola transformations to define the fluid test functions
${\bf u}^{k,n}_{\sigma,M}$. Thanks to (3.8) and Theorem 1.7-1 in [6] we thus
observe that,

$\displaystyle{J^{n}_{*}}(\text{div}^{\boldsymbol{\eta}^{n}_{*}}{\bf
u}^{k,n}_{\sigma,M})=\text{div}\left(J^{k}_{*}(\nabla
A_{\boldsymbol{\eta}^{k}_{*}})^{-1}{\bf
u}^{k}\right)_{\sigma}=\left(\text{div}\left(J^{k}_{*}(\nabla
A_{\boldsymbol{\eta}^{k}_{*}})^{-1}{\bf
u}^{k}\right)\right)_{\sigma}={(J^{k}_{*}\text{div}^{\boldsymbol{\eta}^{k}_{*}}{\bf
u}^{k})_{\sigma}}$

Hence,

(4.2) $\begin{split}&\|\text{div}^{\boldsymbol{\eta}^{n}_{*}}{\bf
u}^{k,n}_{\sigma,M}\|_{L^{2}(\mathcal{O})}\leq
C(\delta)\|\text{div}^{\boldsymbol{\eta}^{k}_{*}}{\bf
u}^{k}\|_{L^{2}(\mathcal{O})},\\\ &\|({\bf u}^{k,n}_{\sigma,M}-{\bf
v}^{k,n}_{\sigma,M})\cdot{\bf n}^{n}_{*}\|_{L^{2}(\Gamma)}\leq
C(\delta)\|({\bf u}^{k}-{\bf v}^{k})\cdot{\bf
n}^{k}_{*}\|_{L^{2}(\Gamma)}.\end{split}$

Note that, to ensure that (LABEL:kton)2 holds, we mollified only in the
tangential direction i.e. along $\Gamma_{\boldsymbol{\eta}^{k}_{*}}$ in the
definition of ${\bf u}^{k,n}_{\sigma,M}$.

Observe that for any $\beta<\frac{1}{2}$, we have,

(4.3) $\displaystyle\|{\bf v}_{M}-{\bf v}\|_{{\bf
H}^{-\beta}(0,L)}\leq\gamma_{M}^{-\frac{\beta}{2}}\|{\bf v}\|_{{\bf
L}^{2}(0,L)},\quad\|{\bf v}_{\sigma}-{\bf v}\|_{{\bf
H}^{-\beta}(0,L)}\leq\sigma^{\beta}\|{\bf v}\|_{{\bf L}^{2}(0,L)}.$

Observe also that
$\|T_{h}\boldsymbol{\eta}^{*}_{N}-\boldsymbol{\eta}^{*}_{N}\|_{L^{\infty}(0,T;{\bf
H}^{s}(0,L))}\leq\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}h^{\frac{1}{4}}$ for any $s>\frac{3}{2}$. Thanks to these
observations and (3.4), for any $s\geq\frac{3}{2}$ and $\beta<\frac{1}{2}$, we
calculate

$\displaystyle|\lambda_{\sigma}^{k}-\lambda^{k,n}_{M}|\leq\int_{0}^{L}\int_{-1}^{1}\left(\int^{z+\sigma
y}_{z}|\partial_{zz}\boldsymbol{\eta}^{k}_{*}(w)|dw\right)\,|{\bf
v}^{k}(z+\sigma y)|dydz$ $\displaystyle+\|\boldsymbol{\eta}^{k}_{*}\|_{{\bf
H}^{s}(0,L)}\|{\bf v}^{k}_{\sigma}-{\bf v}^{k}\|_{{\bf
H}^{-\frac{1}{2}}(0,L)}+\|\boldsymbol{\eta}^{k}_{*}-\boldsymbol{\eta}^{n}_{*}\|_{{\bf
H}^{s}(0,L)}\|{\bf v}^{k}\|_{{\bf L}^{2}(0,L)}$
$\displaystyle+\|\boldsymbol{\eta}^{n}_{*}\|_{{\bf H}^{s}(0,L)}\|{\bf
v}^{k}_{M}-{\bf v}^{k}\|_{{\bf H}^{-\frac{1}{2}}(0,L)}$ (4.4)
$\displaystyle\leq C(\delta)\left(\|\boldsymbol{\eta}^{n}_{*}\|_{{\bf
H}_{0}^{2}(0,L)}\sigma^{\frac{1}{2}}+\sigma^{\beta}+{\gamma_{M}^{-\frac{\beta}{2}}}+h^{\frac{1}{4}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}\right)\|{\bf v}^{k}\|_{{\bf L}^{2}(0,L)}.$

Observe that, due to the properties of mollification, $\partial_{zz}{\bf
w}_{\sigma}=(\partial_{zz}{\bf w})_{\sigma}$ and $\partial_{rr}{\bf
w}_{\sigma}=(\partial_{rr}{\bf w})_{\sigma}$. Hence ${\bf w}^{k}_{\sigma}$ is
harmonic with 0 boundary values on $\partial\mathcal{O}\setminus\Gamma$ and
such that ${\bf w}^{k}_{\sigma}(z,1)={\bf v}^{k}_{\sigma}(z)$.

Moreover, the operator $\mathcal{B}$ can be continuously extended to a bounded
operator from $W^{l,p}_{0}(\mathcal{O})$ to ${\bf W}^{l+1,p}_{0}(\mathcal{O})$
for any $l>-2+\frac{1}{p}$ (see [9], [8]). Combining these observations with
the bounds (3.8) and (3.4), we obtain for $s>\frac{3}{2}$, $\beta<\frac{1}{2}$
and $n-j\leq k\leq n$ that

$\displaystyle\|{\bf u}^{k,n}_{\sigma,M}-{\bf u}^{k}$ $\displaystyle\|_{{\bf
L}^{2}(\mathcal{O})}\lesssim\|\nabla A_{\boldsymbol{\eta}_{*}^{n}}-\nabla
A_{\boldsymbol{\eta}_{*}^{k}}\|_{{\bf L}^{\infty}(\mathcal{O})}\|{\bf
u}^{k}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf u}^{k}-{\bf u}^{k}_{\sigma}\|_{{\bf
L}^{2}(\mathcal{O})}+$ $\displaystyle+\|{\bf w}^{k}-{\bf
w}^{k}_{\sigma}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf w}^{k}-{\bf
w}^{k}_{M}\|_{{\bf
L}^{2}(\mathcal{O})}+|\lambda^{k}_{\sigma}-\lambda_{M}^{k,n}|$
$\displaystyle\lesssim\|{\boldsymbol{\eta}_{*}^{n}}-{\boldsymbol{\eta}_{*}^{k}}\|_{{\bf
H}^{s}(0,L)}\|{\bf u}^{k}\|_{{\bf L}^{2}(\mathcal{O})}+\sigma\|{\bf
u}^{k}\|_{{\bf H}^{1}(\mathcal{O})}$ $\displaystyle+\|{\bf v}^{k}-{\bf
v}^{k}_{\sigma}\|_{{\bf H}^{-1/2}(0,L)}+\|{\bf v}^{k}-{\bf v}^{k}_{M}\|_{{\bf
H}^{-1/2}(0,L)}+|\lambda^{k}_{\sigma}-\lambda^{k,n}_{M}|$
$\displaystyle\lesssim
h^{\frac{1}{4}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}(\|{\bf u}^{k}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}^{k}\|_{{\bf
L}^{2}(0,L)})+\sigma\|{\bf u}^{k}\|_{{\bf H}^{1}(\mathcal{O})}$ (4.5)
$\displaystyle+(\sigma^{\beta}+\gamma_{M}^{-\frac{\beta}{2}}+h^{\frac{1}{4}})\|{\bf
v}^{k}\|_{{\bf L}^{2}(0,L)}$

where the hidden constants depend only on $\delta$. Observe, due to (3.7),
that,

$\displaystyle\|{\bf u}^{k,n}_{\sigma,M}\|_{{\bf
H}^{1}(\mathcal{O})}\leq\|A_{\boldsymbol{\eta}^{k}_{*}}\|_{{\bf
W}^{2,3}(\mathcal{O})}(\|{\bf u}^{k}\|_{{\bf H}^{1}(\mathcal{O})}+\|{\bf
v}_{\sigma}^{k}\|_{{\bf H}^{1}(0,L)})+\|{\bf v}_{M}^{k}\|_{{\bf
H}^{1}(0,L)}+|\lambda^{k}_{\sigma}-\lambda_{M}^{k,n}|$ $\displaystyle\leq
C(\delta)(\|{\bf u}^{k}\|_{{\bf H}^{1}(\mathcal{O})}+{\sigma^{-1}}\|{\bf
v}^{k}\|_{{\bf L}^{2}(\mathcal{O})})+\gamma_{M}^{\frac{1}{2}}\|{\bf
v}^{k}\|_{{\bf L}^{2}(0,L)}+C(\delta)\|{\bf v}^{k}\|_{{\bf L}^{2}(0,L)}.$

Therefore, we obtain for some $C>0$ that depends only on $T$ and $\delta$
that,

$\displaystyle{\mathbb{E}}[(\Delta t)\sup_{0\leq n\leq N}$
$\displaystyle\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}]$ $\displaystyle\leq C(\delta,T){\mathbb{E}}\left((\Delta
t)\sum_{n=0}^{N}\|{\bf u}^{k}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}+({\sigma^{-2}}+\gamma_{M})\sup_{0\leq n\leq N}\|{\bf
v}^{k}\|^{2}_{{\bf L}^{2}(0,L)}\right)$ (4.6) $\displaystyle\leq
C(\delta,T)(1+{\sigma^{-2}}+\gamma_{M}).$

Similarly (see also (4.4)),

(4.7) $\displaystyle\|{\bf v}^{k,n}_{\sigma,M}\|_{{\bf
H}^{2}(0,L)}\leq\gamma_{M}\|{\bf v}^{k}\|_{{\bf L}^{2}(0,L)}+C(\delta)\|{\bf
v}^{k}\|_{{\bf L}^{2}(0,L)}.$

Then, as in [12], for any $n\leq N$ we ”test” (3.1)3 and (LABEL:second)2 with

(4.8) $\displaystyle{\bf Q}_{n}:=({\bf
q}_{n},\boldsymbol{\psi}_{n})=\left((\Delta t)\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M},\,\,(\Delta t)\sum_{k=n-j+1}^{n}{\bf
v}_{\sigma,M}^{k,n}\right).$

This gives us

$\displaystyle\int_{\mathcal{O}}\left((J^{n+1}_{*}){\bf
u}^{n+1}-(J^{n}_{*}){\bf u}^{n}\right)\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}\right)+\int_{0}^{L}({\bf v}^{n+1}-{\bf v}^{n})\left(\Delta
t\sum_{k=n-j+1}^{n}{\bf v}_{\sigma,M}^{k,n}\right)$
$\displaystyle\frac{-1}{2}\int_{\mathcal{O}}\left(J_{*}^{n+1}-J_{*}^{n}\right){\bf
u}^{n+1}\cdot\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}\right)+(\Delta t)b^{\boldsymbol{\eta}_{n}^{*}}({\bf
u}^{n+1},{\bf w}^{n}_{*},\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}\right))$ $\displaystyle+\frac{(\Delta
t)}{\alpha}\int_{\Gamma}S^{n}_{*}({\bf u}^{n+1}-{\bf v}^{n+1})\left((\Delta
t)\sum_{k=n-j+1}^{n}{\bf u}^{k,n}_{\sigma,M}-{\bf v}_{\sigma,M}^{k,n}\right)$
$\displaystyle+\frac{(\Delta
t)}{\varepsilon}\int_{\mathcal{O}}\text{div}^{\boldsymbol{\eta}^{n}_{*}}{\bf
u}^{n+1}\text{div}^{\boldsymbol{\eta}^{n}_{*}}\left(\Delta
t\sum_{k=n-j+1}^{n}{\bf u}^{k,n}_{\sigma,M}\right)d{\bf x}$
$\displaystyle+\frac{(\Delta t)}{\varepsilon}\int_{\Gamma}({\bf u}^{n+1}-{\bf
v}^{n+1})\cdot{\bf n}^{n}_{*}\left((\Delta t)\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}-{\bf v}_{\sigma,M}^{k,n}\right)\cdot{\bf n}^{n}_{*}$
$\displaystyle+2\nu(\Delta t)\int_{\mathcal{O}}(J^{n}_{*}){\bf
D}^{\eta_{*}^{n}}({\bf u}^{n+1})\cdot{\bf D}^{\eta_{*}^{n}}\left(\Delta
t\sum_{k=n-j+1}^{n}{\bf u}^{k,n}_{\sigma,M}\right)$ $\displaystyle+(\Delta
t)\left(\mathscr{L}_{e}(\boldsymbol{\eta}^{n+\frac{1}{2}}),\left(\Delta
t\sum_{k=n-j+1}^{n}{\bf
v}_{\sigma,M}^{k,n}\right)+\varepsilon\int_{0}^{L}\partial_{zz}{\bf
v}^{n+\frac{1}{2}}\cdot\partial_{zz}\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
v}^{k,n}_{\sigma,M}\right)\right)$ $\displaystyle=(\Delta
t)\left(P^{n}_{{in}}\int_{0}^{1}q^{n}_{z}\Big{|}_{z=0}dr-P^{n}_{{out}}\int_{0}^{1}q^{n}_{z}\Big{|}_{z=1}dr\right)+(G({\bf
u}^{n},\boldsymbol{\eta}_{*}^{n})\Delta_{n}W,{\bf Q}_{n}).$

Observe that summation by parts formula gives us for the first two terms,

$\displaystyle=(\Delta t)\sum_{n=1}^{N}\left(\int_{\mathcal{O}}(J^{n}_{*}){\bf
u}^{n}({\bf u}^{n,n}_{\sigma,M}-{\bf u}_{\sigma,M}^{n-j,n})d{\bf
x}+\int_{0}^{L}{\bf v}^{n}({\bf v}_{\sigma,M}^{n,n}-{\bf
v}_{\sigma,M}^{n-j,n})dz\right)$
$\displaystyle-\int_{\mathcal{O}}(J^{N+1}_{*}){\bf u}^{N+1}\left(\Delta
t\sum_{k=N-j+1}^{N}{\bf u}^{k,N}_{\sigma,M}\right)-\int_{0}^{L}{\bf
v}^{N}\left(\Delta t\sum_{k=N-j+1}^{N}{\bf v}_{\sigma,M}^{k,n}\right),$

where the first term on the right side above can be written as,

$\displaystyle(\Delta t)\sum_{n=0}^{N}\left(\int_{\mathcal{O}}(J^{n}_{*}){\bf
u}^{n}({\bf u}^{n}-{\bf u}^{n-j})d{\bf x}+\int_{\mathcal{O}}(J^{n}_{*}){\bf
u}^{n}({\bf u}^{n,n}_{\sigma,M}-{\bf u}^{n}-({\bf u}_{\sigma,M}^{n-j,n}-{\bf
u}^{n-j}))d{\bf x}\right)$ $\displaystyle+\int_{0}^{L}{\bf v}^{n}\left({\bf
v}_{M}^{n}-\left(\frac{\lambda^{n}_{\sigma}-\lambda^{n,n}_{M}}{\lambda_{0}^{n}}\right)\xi_{0}-{\bf
v}_{M}^{n-j}+\left(\frac{\lambda^{n-j}_{\sigma}-\lambda^{n-j,n}_{M}}{\lambda_{0}^{n}}\right)\xi_{0}\right)dz$
$\displaystyle=(\Delta
t)\sum_{n=0}^{N}\left(\frac{1}{2}\int_{\mathcal{O}}(J^{n}_{*})(|{\bf
u}^{n}|^{2}-|{\bf u}^{n-j}|^{2}+|{\bf u}^{n}-{\bf u}^{n-j}|^{2})d{\bf
x}\right)$ $\displaystyle+(\Delta
t)\sum_{n=0}^{N}\left(\int_{\mathcal{O}}(J^{n}_{*}){\bf u}^{n}({\bf
u}^{n,n}_{\sigma,M}-{\bf u}^{n}-({\bf u}_{\sigma,M}^{n-j,n}-{\bf
u}^{n-j}))d{\bf x}\right)$ $\displaystyle+\frac{(\Delta
t)}{2}\sum_{n=0}^{N}\int_{0}^{L}|{\bf v}_{M}^{n}|^{2}-|{\bf
v}_{M}^{n-j}|^{2}+|{\bf v}_{M}^{n}-{\bf v}_{M}^{n-j}|^{2}-{\bf
v}^{n}(\frac{\lambda^{n}_{\sigma}-\lambda^{n-j}_{\sigma}+\lambda^{n-j,n}_{M}-\lambda^{n,n}_{M}}{\lambda_{0}^{n}})\xi_{0}$

where we set ${\bf u}^{n}=0$ and ${\bf v}^{n}=0$ for $n<0$ and $n>N$. Observe
that since

$\displaystyle\|{\bf v}^{n}-{\bf v}^{n}_{M}-{\bf v}^{n-j}-{\bf
v}^{n-j}_{M}\|_{{\bf H}^{-\beta}(0,L)}\leq\gamma_{M}^{-\frac{\beta}{2}}(\|{\bf
v}^{n}\|_{{\bf L}^{2}(0,L)}+\|{\bf v}^{n-j}\|_{{\bf L}^{2}(0,L)}),$

for any $0\leq\beta<\frac{1}{2}$, the right hand side of the preceding
equation will give us the desired terms (4.1). In what follows we will treat
rest of the terms. First,

$\displaystyle I_{0}:=\Delta t\sum_{n=0}^{N}\int_{0}^{L}|{\bf
v}^{n}_{M}|^{2}-|{\bf v}^{n-j}_{M}|^{2}=\Delta t\sum_{n=N-j+1}^{N}\|{\bf
v}^{n}_{M}\|^{2}_{{\bf L}^{2}(0,L)}\leq h\max_{0\leq n\leq N}\|{\bf
v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}.$

Hence, ${\mathbb{P}}(|I_{0}|>R)\leq\frac{h}{R}{\mathbb{E}}[\max_{0\leq n\leq
N}\|{\bf v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}]\leq\frac{Ch}{R}.$

Next, we recall that for two matrices $A$ and $B$, the derivative of the
determinant $D(det)(A)B=det(A)tr(BA^{-1})$. Hence applying the mean value
theorem to $det(\nabla A_{\boldsymbol{\eta}^{n}_{*}})-det(\nabla
A_{\boldsymbol{\eta}^{n+j}_{*}})$, using (3.8) and that $det(A)\leq(\max
A_{ij})^{2}$, we obtain, for some $\beta\in[0,1]$, that (for more details see
(73) in [18]):

(4.9)
$\begin{split}\|{J^{n+j}_{*}-J^{n}_{*}}\|_{L^{\infty}(\mathcal{O})}&=\|\text{det}(\nabla
A^{n,\beta})\nabla^{n,\beta}\cdot(A_{\boldsymbol{\eta}^{n+j}_{*}}-A_{\boldsymbol{\eta}^{n}_{*}})\|_{L^{\infty}(\mathcal{O})}\\\
&\leq
C(\delta)\|A_{\boldsymbol{\eta}^{n+j}_{*}}-A_{\boldsymbol{\eta}^{n}_{*}}\|_{{\bf
C}^{1}(\mathcal{O})},\end{split}$

where
$\nabla^{n,\beta}=\nabla^{\boldsymbol{\eta}^{n}_{*}}+\beta(\nabla^{\boldsymbol{\eta}^{n+j}_{*}}-\nabla^{\boldsymbol{\eta}^{n}_{*}})$
and $\nabla A^{n,\beta}=\nabla A_{\boldsymbol{\eta}^{n}_{*}}+\beta(\nabla
A_{\boldsymbol{\eta}^{n+j}_{*}}-\nabla A_{\boldsymbol{\eta}^{n}_{*}})$.
Using (3.8) again, we find the following bounds for any $s>\frac{3}{2}$,

$\displaystyle I_{1}$ $\displaystyle:=(\Delta
t)\sum_{n=0}^{N}\left(\int_{\mathcal{O}}(J^{n}_{*})(|{\bf u}^{n}|^{2}-|{\bf
u}^{n-j}|^{2})d{\bf x}\right)$ $\displaystyle=(\Delta
t)\left(\sum_{n=N-j+1}^{N}\int_{\mathcal{O}}(J^{n}_{*})|{\bf u}^{n}|^{2}d{\bf
x}+\sum_{n=0}^{N-j}\int_{\mathcal{O}}(J^{n}_{*}-J^{n+j}_{*})|{\bf
u}^{n}|^{2}d{\bf x}\right)$ $\displaystyle\leq(\Delta
t)\sum_{n=N-j+1}^{N}\int_{\mathcal{O}}(J^{n}_{*})|{\bf u}^{n}|^{2}d{\bf
x}+(\Delta
t)\sum_{n=0}^{N-j}\|A_{\boldsymbol{\eta}^{n+j}_{*}}-A_{\boldsymbol{\eta}^{n}_{*}}\|_{{\bf
C}^{1}(\mathcal{O})}\|{\bf u}^{n}\|^{2}_{{\bf L}^{2}(\mathcal{O})}$
$\displaystyle\leq\left(h\sup_{1\leq n\leq
N}\int_{\mathcal{O}}(J^{n}_{*})|{\bf u}^{n}|^{2}d{\bf x}+\sup_{0\leq k\leq
N-1}\|{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}h^{\frac{1}{4}}\|\boldsymbol{\eta}^{n+j}_{*}-\boldsymbol{\eta}^{n}_{*}\|_{{\bf
H}^{s}(0,L)}\right)$ $\displaystyle\leq h^{\frac{1}{4}}\sup_{0\leq k\leq
N-1}\|{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}\left(1+\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}\right).$

In what follows, we will repeatedly make use of the fact that for any two
positive random variables $X$ and $Y$,
$\\{\omega:X+Y>R\\}\subseteq\\{\omega:X>\frac{R}{2}\\}\cup\\{\omega:Y>\frac{R}{2}\\}$
which further implies that

$\displaystyle{\mathbb{P}}(|X|+|Y|>R)\leq{\mathbb{P}}(|X|>\frac{R}{2})+{\mathbb{P}}(|Y|>\frac{R}{2}),$
and similarly,
$\displaystyle{\mathbb{P}}(|XY|>R)\leq{\mathbb{P}}(|X|>\sqrt{R})+{\mathbb{P}}(|Y|>\sqrt{R}).$

The embedding $W^{1,\infty}(0,T;L^{2}(0,L))\cap
L^{2}(0,T;H^{2}_{0}(0,L))\hookrightarrow
C^{0,\frac{1}{4}}(0,T;H^{\frac{3}{2}}(0,L))$, then gives us for any
$s\geq\frac{3}{2}$ that

$\displaystyle{\mathbb{P}}$
$\displaystyle\left(|I_{1}|>R\right)\leq{\mathbb{P}}\left(h^{\frac{1}{8}}\sup_{1\leq
n\leq N}\|{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}\geq\sqrt{R}\right)+{\mathbb{P}}\left(h^{\frac{1}{8}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}\geq\sqrt{R}\right)$
$\displaystyle\leq\frac{{h}^{\frac{1}{8}}}{\sqrt{R}}{\mathbb{E}}\left(\sup_{1\leq
n\leq N}\|{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}\right)$
$\displaystyle\leq\frac{{h}^{\frac{1}{8}}}{\sqrt{R}}{\mathbb{E}}\left(\sup_{1\leq
n\leq N}\|{\bf u}^{n}\|^{2}_{{\bf
L}^{2}(\mathcal{O})}+\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{W^{1,\infty}(0,T;{\bf
L}^{2}(0,L))\cap L^{2}(0,T;{\bf H}^{2}_{0}(0,L))}\right)\leq
C\frac{{h}^{\frac{1}{8}}}{\sqrt{R}}.$

Next, thanks to (4.5), we see that

$\displaystyle I_{2}=|(\Delta
t)\sum_{n=0}^{N}\left(\int_{\mathcal{O}}(J^{n}_{*}){\bf u}^{n}({\bf
u}^{n,n}_{\sigma,M}-{\bf u}^{n}-({\bf u}_{\sigma,M}^{n-j,n}-{\bf
u}^{n-j}))d{\bf x}\right)|$ $\displaystyle\leq C(\delta)(\Delta
t)\sum_{n=0}^{N}\|{\bf u}^{n}\|_{{\bf L}^{2}(\mathcal{O})}(\|{\bf
u}^{n,n}_{\sigma,M}-{\bf u}^{n}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf
u}^{n-j,n}_{\sigma,M}-{\bf u}^{n-j}\|_{{\bf L}^{2}(\mathcal{O})})$
$\displaystyle\lesssim\sup_{1\leq n\leq N}\|{\bf u}^{n}\|_{{\bf
L}^{2}(\mathcal{O})}\sum_{k=0}^{N}\Big{(}h^{\frac{1}{4}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}(\|{\bf u}^{k}\|_{{\bf L}^{2}(\mathcal{O})}+\|{\bf v}^{k}\|_{{\bf
L}^{2}(0,L)})$ $\displaystyle+\sigma\|{\bf u}^{k}\|_{{\bf
H}^{1}(\mathcal{O})}+(\sigma^{\beta}+\gamma_{M}^{-\frac{\beta}{2}}+h^{\frac{1}{4}})\|{\bf
v}^{k}\|_{{\bf L}^{2}(0,L)}\Big{)}.$

Similarly, thanks to (4.4), we see for any $0\leq\beta<\frac{1}{2}$ that

$\displaystyle I_{3}$ $\displaystyle:=|(\Delta
t)\sum_{n=0}^{N}\int_{0}^{L}{\bf
v}^{n}(\frac{\lambda^{n,n}_{\sigma}-\lambda^{n-j,n}_{\sigma}+\lambda^{n-j,n}_{M}-\lambda^{n,n}_{M}}{\lambda_{0}^{n}})\xi_{0}|$
$\displaystyle\leq C(\delta)\left(\|\boldsymbol{\eta}^{n}_{*}\|_{{\bf
H}_{0}^{2}(0,L)}\sigma^{\frac{1}{2}}+\sigma^{\beta}+{\gamma_{M}^{-\frac{\beta}{2}}}+h^{\frac{1}{4}}\|\widetilde{\boldsymbol{\eta}}^{*}_{N}\|_{C^{0,\frac{1}{4}}(0,T;{\bf
H}^{s}(0,L))}\right)\|{\bf v}^{n}\|_{{\bf L}^{2}(0,L)}.$

Hence, we obtain that,

${\mathbb{P}}(|I_{2}+I_{3}|>R)\leq\frac{C}{R}(h^{\frac{1}{4}}+\sigma^{\beta}+\gamma_{M}^{-\frac{\beta}{2}})\leq\frac{C}{R}h^{\frac{\beta}{12}}.$

For the penalty term, thanks to (LABEL:kton)1, we have,

$\displaystyle{\mathbb{E}}[I_{4}]:={\mathbb{E}}|\frac{(\Delta
t)}{\varepsilon}\sum_{n=0}^{N}\int_{\mathcal{O}}\text{div}^{\boldsymbol{\eta}^{n}_{*}}{\bf
u}^{n+1}\left(\Delta
t\sum_{k=n-j+1}^{n}\text{div}^{\boldsymbol{\eta}^{n}_{*}}({\bf
u}^{k,n}_{\sigma,\lambda})\right)d{\bf x}|$
$\displaystyle\leq{\mathbb{E}}\frac{(\Delta
t)}{{\varepsilon}}\sum_{n=0}^{N}\|\text{div}^{\eta^{n}_{*}}{\bf
u}^{n+1}\|_{L^{2}(\mathcal{O})}\left({(\Delta
t)}{{}}\sum_{k=n-j+1}^{n}\|\text{div}^{\boldsymbol{\eta}^{n}_{*}}({\bf
u}^{k,n}_{\sigma,\lambda})\|^{2}_{L^{2}(\mathcal{O})}\right)^{\frac{1}{2}}\sqrt{h}$
$\displaystyle\leq C\sqrt{hT}{\mathbb{E}}\left(\frac{(\Delta
t)}{\varepsilon}\sum_{n=0}^{N}\|\text{div}^{\boldsymbol{\eta}^{n}_{*}}({\bf
u}^{n+1})\|^{2}_{L^{2}(\mathcal{O})}\right)\leq Ch^{\frac{1}{2}}.$

Notice that, due to Lemma 3.4 (5), the constant $C$ in the estimate above does
not depend on $\varepsilon$. The other penalty term is handled identically
thanks to (LABEL:kton)2 and Lemma 3.4 (6). Similarly, due to (4.6), we have

$\displaystyle{\mathbb{E}}[I_{5}]=$ $\displaystyle{\mathbb{E}}|{(\Delta
t)}{}\sum_{n=0}^{N}\int_{\mathcal{O}}{\bf D}^{\eta^{n}_{*}}{\bf
u}^{n+1}\left(\Delta t\sum_{k=n-j+1}^{n}{\bf D}^{\eta^{n}_{*}}({\bf
u}^{k,n}_{\sigma,M})\right)d{\bf x}|$
$\displaystyle\leq\sqrt{h}{\mathbb{E}}((\Delta t)\sum_{n=0}^{N}\|{\bf
u}^{n+1}\|_{{\bf H}^{1}(\mathcal{O})}\left({(\Delta
t)}\sum_{k=n-j+1}^{n}\|({\bf u}^{k,n}_{\sigma,M})\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)^{\frac{1}{2}})$
$\displaystyle\leq\sqrt{h}{\mathbb{E}}\sum_{n=0}^{N}(\Delta t)\|{\bf
u}^{n+1}\|^{2}_{{\bf H}^{1}(\mathcal{O})}+\sqrt{h}{\mathbb{E}}\left({(\Delta
t)}\sup_{0\leq n\leq N}\sum_{k=0}^{N}\|({\bf u}^{k,n}_{\sigma,M})\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)$ $\displaystyle\leq
C(\delta)h^{\frac{1}{4}}(1+{\sigma^{-2}}+\gamma_{M})\leq Ch^{\frac{1}{12}}.$

Next, using the embedding $H^{\frac{1}{2}}(\mathcal{O})\hookrightarrow
L^{4}(\mathcal{O})$ we obtain, for some $C>0$ which depends only on $\delta$,
that

$\displaystyle I_{6}:=|(\Delta
t)\sum_{n=0}^{N}b^{\boldsymbol{\eta}^{n}_{*}}\left({\bf u}^{n+1},{\bf
w}_{*}^{n},\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}\right)\right)|$ $\displaystyle\leq\sqrt{h}(\Delta
t)\sum_{n=0}^{N}\|{\bf u}^{n+1}-{\bf w}_{*}^{n}\|_{{\bf
L}^{4}(\mathcal{O})}\|{\bf u}^{n+1}\|_{{\bf L}^{4}(\mathcal{O})}\left(\Delta
t\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)^{\frac{1}{2}}$ $\displaystyle\leq\sqrt{h}(\Delta
t)\sum_{n=0}^{N}\left(\|{\bf u}^{n+1}\|_{{\bf H}^{1}(\mathcal{O})}+\|{\bf
w}_{*}^{n}\|_{{\bf H}^{\frac{1}{2}}(0,L)}\right)\|{\bf u}^{n+1}\|_{{\bf
H}^{1}(\mathcal{O})}$ $\displaystyle\qquad\qquad\left(\Delta
t\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)^{\frac{1}{2}}.$

Again, thanks to (4.6), for any $R>0$, we have

$\displaystyle{\mathbb{P}}\left(|I_{6}|>R\right)$
$\displaystyle\leq\frac{h^{\frac{1}{4}}}{\sqrt{R}}{\mathbb{E}}[(\Delta
t)\sum_{n=0}^{N}\|{\bf u}^{n+1}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}]+\frac{h^{\frac{1}{4}}}{\sqrt{R}}{\mathbb{E}}[\sup_{0\leq
n\leq N-1}\|{\bf v}^{n+\frac{1}{2}}\|^{2}_{{\bf L}^{2}(0,L)}]$
$\displaystyle+\frac{h^{\frac{1}{2}}}{R}{\mathbb{E}}[(\Delta
t)^{2}\sum_{n=0}^{N}\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}]$
$\displaystyle\leq\frac{C(\delta)h^{\frac{1}{4}}}{\sqrt{R}}+\frac{C(\delta)h^{\frac{1}{2}}}{{R}}(1+{\sigma^{-2}}+\gamma_{M})\leq\frac{C(\delta)}{\sqrt{R}}h^{\frac{1}{4}}.$

Next, calculations similar to (4.9), give us

(4.10) $\displaystyle\|\frac{J^{n+1}_{*}-J^{n}_{*}}{\Delta
t}\|_{L^{2}(\mathcal{O})}\leq C(\delta)\|{\bf w}^{n+1}_{*}\|_{{\bf
H}^{1}(\mathcal{O})}\leq C(\delta)\|{\bf v}_{*}^{n+1}\|_{{\bf
H}^{\frac{1}{2}}(0,L)}.$

Hence,

$\displaystyle
I_{7}:=|\sum_{n=0}^{N}\int_{\mathcal{O}}\left(J_{*}^{n+1}-J_{*}^{n}\right){\bf
u}^{n+1}\cdot\left(\Delta t\sum_{k=n-j+1}^{n}{\bf
u}^{k,n}_{\sigma,M}\right)d{\bf x}|$ $\displaystyle\leq(\Delta
t)\sum_{n=0}^{N}\|{\bf v}^{n+\frac{1}{2}}\|_{{\bf H}^{\frac{1}{2}}(0,L)}\|{\bf
u}^{n+1}\|_{{\bf L}^{6}(\mathcal{O})}\left((\Delta t)\sum_{k=n-j+1}^{n}\|{\bf
u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
L}^{3}(\mathcal{O})}\right)^{\frac{1}{2}}\sqrt{h}$ $\displaystyle\leq
C\sqrt{h}\left((\Delta t)\sum_{n=0}^{N}\|{\bf v}^{n+\frac{1}{2}}\|^{2}_{{\bf
H}^{\frac{1}{2}}(0,L)}\right)^{\frac{1}{2}}\left((\Delta
t)\sum_{n=0}^{N}\|{\bf u}^{n}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)^{\frac{1}{2}}$ $\displaystyle\qquad\left((\Delta
t)\sup_{0\leq n\leq N}\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|^{2}_{{\bf
H}^{1}(\mathcal{O})}\right)^{\frac{1}{2}}.$

This implies, thanks to (4.6), that,

(4.11) $\displaystyle{\mathbb{P}}\left(|I_{7}|\geq R\right)$
$\displaystyle\leq\frac{C}{\varepsilon}\frac{h^{\frac{1}{3}}}{R^{\frac{2}{3}}}+\frac{C(\delta)h^{\frac{1}{3}}}{R^{\frac{2}{3}}}(1+{\sigma^{-2}}+\gamma_{M})\leq{C(\varepsilon)}\frac{h^{\frac{1}{6}}}{R^{\frac{2}{3}}}.$

Note that this estimate depends on $\varepsilon$. Next, using (4.7), we obtain
that,

$\displaystyle{\mathbb{E}}[I_{8}]:={\mathbb{E}}|(\Delta
t)\sum_{n=0}^{N}\left(\mathscr{L}_{e}(\boldsymbol{\eta}^{n+1}),\left(\Delta
t\sum_{k=n-j+1}^{n}{\bf v}^{k}_{\sigma,M}\right)\right)|$
$\displaystyle\qquad\leq C(\delta)(\Delta
t)^{2}{\mathbb{E}}\sum_{n=0}^{N}\left(\|\partial_{zz}\boldsymbol{\eta}^{n}\|_{{\bf
L}^{2}(0,L)}\sum_{k=n-j-1}^{n}\|\partial_{zz}{\bf v}^{k,n}_{\sigma,M}\|_{{\bf
L}^{2}(0,L)}\right)$ $\displaystyle\qquad\leq{\mathbb{E}}(\Delta
t)^{2}\sum_{n=0}^{N}\left(\|\partial_{zz}\boldsymbol{\eta}^{n}\|_{{\bf
L}^{2}(0,L)}\sum_{k=n-j-1}^{n}\gamma_{M}\|{\bf v}^{k}\|_{{\bf
L}^{2}(0,L)}\right)$
$\displaystyle\qquad\leq{CT}{h}\gamma_{M}{\mathbb{E}}\max_{0\leq n\leq
N}\left(\|\boldsymbol{\eta}^{n}\|^{2}_{{\bf H}^{2}(0,L)}+\|{\bf
v}^{n}\|^{2}_{{\bf L}^{2}(0,L)}\right)\leq Ch^{\frac{5}{6}}.$

Finally, we treat the stochastic term (see also (3.17)) using Young’s
inequality:

$\displaystyle{\mathbb{E}}\left(\sum_{n=0}^{N}|(G({\bf
u}^{n},\boldsymbol{\eta}_{*}^{n})\Delta_{n}W,{\bf
Q}_{n})|\right)\leq{\mathbb{E}}\Big{(}\sum_{n=0}^{N}\|G({\bf
u}^{n},\boldsymbol{\eta}_{*}^{n})\|_{L_{2}(U_{0};{\bf
L}^{2})}\|\Delta_{n}W\|_{U_{0}}$ $\displaystyle\qquad\times\left((\Delta
t)\sum_{k=n-j+1}^{n}\|{\bf u}^{k,n}_{\sigma,M}\|_{{\bf
L}^{2}(\mathcal{O})}^{2}+\|{\bf v}_{\sigma,M}^{k}\|_{{\bf
L}^{2}(0,L)}^{2}\right)^{\frac{1}{2}}h^{\frac{1}{2}}\Big{)}$
$\displaystyle\qquad\leq h^{\frac{1}{2}}{\mathbb{E}}\sum_{n=0}^{N}\|G({\bf
u}^{n},\boldsymbol{\eta}_{*}^{n})\|^{2}_{L_{2}(U_{0};{\bf
L}^{2})}\|\Delta_{n}W\|^{2}_{U_{0}}$
$\displaystyle\qquad+{\mathbb{E}}\left((\Delta t)\sum_{k=n-j+1}^{n}\|{\bf
u}^{k,n}_{\sigma,M}\|_{{\bf L}^{2}(\mathcal{O})}^{2}+\|{\bf
v}_{\sigma,M}^{k}\|_{{\bf L}^{2}(0,L)}^{2}\right)\leq Ch^{\frac{1}{2}}.$

Now to show that the laws of the random variables mentioned in the statement
of the theorem are tight, we will consider the following sets for
$0\leq\alpha<1$ and $0<\beta<\frac{1}{2}$ and any $R>0$,

$\displaystyle{\mathcal{B}}_{R}:=\\{({\bf u},{\bf v})\in L^{2}(0,T;{\bf
H}^{1}(\mathcal{O}))\times L^{2}(0,T;{\bf L}^{2}(0,L)):\|{{\bf
u}}\|^{2}_{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}+$ $\displaystyle\|{{\bf
v}}\|^{2}_{L^{2}(0,T;{\bf
L}^{2}(0,L))}+\sup_{0<h<1}{h^{-\frac{\beta}{12}}}\int_{h}^{T}\left(\|T_{h}{\bf
u}-{\bf u}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|T_{h}{\bf v}-{\bf v}\|^{2}_{{\bf
H}^{-\beta}(0,L)}\right)\leq R\\}.$

Thanks to Lemma 4.1, $\mathcal{B}_{R}$ is compact in $L^{2}(0,T;{\bf
H}^{\alpha}(\mathcal{O}))\times L^{2}(0,T;{\bf H}^{-\beta}(0,L))$ for each
$R>0$, $0\leq\alpha<1$ and $0<\beta<\frac{1}{2}$. Now we apply Chebyshev’s
inequality to obtain the desired result:

$\displaystyle{\mathbb{P}}(({\bf u}^{+}_{N},$ $\displaystyle{\bf
v}^{+}_{N})\notin\mathcal{B}_{R})\leq{\mathbb{P}}\left(\|{{\bf
u}^{+}_{N}}\|^{2}_{L^{2}(0,T;{\bf H}^{1}(\mathcal{O}))}+\|{{\bf
v}^{+}_{N}}\|^{2}_{L^{2}(0,T;{\bf L}^{2}(0,L))}>\frac{R}{2}\right)$
$\displaystyle+{\mathbb{P}}\left(\sup_{0<h<1}{h^{-\frac{\beta}{12}}}\int_{h}^{T}\left(\|T_{h}{\bf
u}^{+}_{N}-{\bf u}^{+}_{N}\|^{2}_{{\bf L}^{2}(\mathcal{O})}+\|T_{h}{\bf
v}^{+}_{N}-{\bf v}^{+}_{N}\|^{2}_{{\bf
H}^{-\beta}(0,L)}\right)>\frac{R}{2}\right)$
$\displaystyle\leq\frac{C}{\sqrt{R}},$

where $C>0$ depends only on $\delta$, Tr$Q$, $\varepsilon$ and the given data
and is independent of $N$. ∎

Next we give the tightness results for the structure displacements and
velocities.

######  Lemma 4.3.

For fixed $\delta$, 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 $\\{\|{\bf u}^{+}_{N}\|_{L^{2}(0,T;V)}\\}_{N\in\mathbb{N}}$ are
tight in $\mathbb{R}$.

  3. (3)

For a fixed $\varepsilon$, the laws of $\\{\|{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf
H}_{0}^{1}(0,L))}\\}_{N\in\mathbb{N}}$ are tight in $\mathbb{R}$.

###### Proof.

The Aubin-Lions theorem gives us: For $0<s<2$,

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

Hence for any $R>0$ we consider

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

Using the Chebyshev inequality and Lemma 3.4 we obtain for some $C>0$
independent of $N$ and $\varepsilon$ that the following holds:

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

The proof of the statements (2), (3) follow by an identical application of the
Chebyshev inequality and the bounds obtained in Lemma 3.4. For any $R>0$,

$\displaystyle{\mathbb{P}}[\|{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf
H}_{0}^{1}(0,L))}>R]\leq\frac{1}{R^{2}}{\mathbb{E}}[\|{\bf
v}^{*}_{N}\|^{2}_{L^{2}(0,T;{\bf
H}_{0}^{1}(0,L))}]\leq\frac{C(\varepsilon)}{R^{2}}.$

This completes the proof of Lemma 4.3. ∎

To obtain almost sure convergence of the rest of the random variables we will
use the following lemma which is a consequence of the bounds on numerical
dissipation.

######  Lemma 4.4.

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)

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

  3. (3)

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

###### Proof.

Statement (1)1 is true thanks to Theorem 3.1 (3). We prove (1)2 below exactly
as in [23].

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

The rest follows identically from the uniform estimates stated in Theorem 3.1.
∎

To pass $N\to\infty$ in the weak formulation, we consider the following random
variable:

(4.12) $\displaystyle\widetilde{\bf
U}_{N}(t):=((\widetilde{J}_{N}^{*}(t))\widetilde{\bf u}_{N}(t),\widetilde{\bf
v}_{N}(t))-E_{N}(t),$

where $E_{N}$ is an error term that appears due to discretizing the stochastic
integral (see (LABEL:approxsystem)) given by:

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

Let ${\mathscr{U}}_{1}:={\mathscr{U}}\cap({\bf H}^{2}(\mathcal{O})\times{\bf
H}^{3}_{0}(0,L)).$ For any $\mathscr{V}_{1}\subset\subset{\mathscr{U}}_{1}$,
we denote by $\mu^{u,v}_{N}$ the probability measure of $\widetilde{\bf
U}_{N}$:

$\mu^{u,v}_{N}:={\mathbb{P}}\left(\widetilde{\bf U}_{N}\in\cdot\right)\in
Pr(C([0,T];\mathscr{V}^{\prime}_{1})),$

where $Pr(S)$, denotes the set of probability measures on a metric space $S$.
Then we have the following tightness result which is proven identically as
Lemma 4.6 in [23].

######  Lemma 4.5.

For a fixed $\varepsilon>0$ and $\delta$, the laws $\\{\mu_{N}^{u,v}\\}_{N}$
of the random variables $\\{\widetilde{\bf U}_{N}\\}_{N}$ are tight in
$C([0,T];\mathscr{V}_{1}^{\prime})$.

We note here that this result is available only in the case of fixed
$\varepsilon>0$ and that we will not have this result in the next section. In
the following lemma, we will show that the stochastic error term vanishes as
$N\to\infty$.

######  Lemma 4.6.

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

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

###### Proof.

For any $N$, we begin by writing $E_{N}(t)=:E_{N}^{1}+E_{N}^{2}$. Observe that
$E^{1}_{N}$ satisfies,

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

where, as a consequence of Theorem 3.1, $C>0$ is independent of $N$ and
$\varepsilon$.

To estimate $E^{2}_{N}$ we use the Itô isometry as follows

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

∎

###  4.2. Almost sure convergence

In this section we will obtain almost sure convergence results for our
approximate solutions. To that end, let $\mu_{N}$ be the joint law of the
approximate random variables $\mathcal{U}_{N}=({\bf u}^{+}_{N},{\bf
v}^{+}_{N},\widetilde{\boldsymbol{\eta}}^{*}_{N},\widetilde{\boldsymbol{\eta}}_{N},\|{\bf
u}^{+}_{N}\|_{L^{2}(0,T;V)},\|{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf
H}_{0}^{1}(0,L))},\widetilde{\bf U},W)$ taking values in the phase space

$\displaystyle\Upsilon$ $\displaystyle:=[L^{2}(0,T;{\bf
L}^{2}(\mathcal{O}))]\times[L^{2}(0,T;{\bf
H}^{-\beta}(0,L))]\times[C([0,T],{\bf H}^{s}(0,L))]^{2}\times\mathbb{R}^{2}$
$\displaystyle\times C([0,T];\mathscr{V}_{1}^{\prime})\times C([0,T];U),$

for some fixed $\frac{3}{2}<s<2$ and $0<\beta<\frac{1}{2}$.

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 $C([0,T];U)$. Thus, recalling Lemmas 4.2 4.3, 4.6 and
using Tychonoff’s theorem it follows that the sequence of the 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 a
variant of the Skorohod representation theorem (Theorem 1.10.4 in [25]) we
obtain the following convergence result.

######  Theorem 4.1.

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

(4.13)
$\displaystyle\bar{\mathcal{U}}_{N}\rightarrow\bar{\mathcal{U}},\qquad\quad\bar{\mathbb{P}}-a.s.\quad\text{in
the topology of }\Upsilon.$

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

(4.14)
$\displaystyle\bar{\mathcal{U}}_{N}(\bar{\omega})={\mathcal{U}}_{N}(\phi_{N}(\bar{\omega}))\quad\text{
for }\bar{\omega}\in\bar{\Omega},$

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

Now we define rest of the approximate random variables:

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

Then, from part (1), Lemmas 4.4 and an application of the Borel-Cantelli
lemma, we obtain for $\frac{3}{2}<s<2$ and $0<\beta<\frac{1}{2}$ that

(4.15) $\displaystyle\bar{\bf u}_{N}\to\bar{\bf u},\quad\bar{\widetilde{\bf
u}}_{N}\to\bar{\bf u},\quad\bar{\mathbb{P}}-a.s.\quad\text{in }L^{2}(0,T;{\bf
L}^{2}(\mathcal{O})),$ (4.16)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\to\bar{\boldsymbol{\eta}},\quad{\bar{\boldsymbol{\eta}}}^{+}_{N}\to\bar{\boldsymbol{\eta}},\quad\bar{\boldsymbol{\eta}}^{*}_{N}\to\bar{\boldsymbol{\eta}}^{*},\quad\bar{\mathbb{P}}-a.s.\quad\text{in
}L^{2}(0,T;{\bf H}^{s}(0,L)),$ (4.17) $\displaystyle\bar{\bf v}_{N}\to\bar{\bf
v},\quad\bar{\widetilde{\bf v}}_{N}\to\bar{\bf
v},\quad\bar{\mathbb{P}}-a.s.\quad\text{ in }L^{2}(0,T;{\bf
H}^{-\beta}(0,L)).$

Thanks to these explicit maps we can identify the real-valued random variables
${k}_{N}=\|\bar{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf H}_{0}^{1}(0,L))}$. Thus,
a.s. convergence of ${k}_{N}$ implies that, for a fixed $\varepsilon>0$,
$\|\bar{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf H}_{0}^{1}(0,L))}$ is bounded a.s.
and thus, up to a subsequence,

(4.18) $\displaystyle\bar{\bf v}^{*}_{N}\rightharpoonup\bar{\bf
v}^{*}\quad\text{ weakly in }{L^{2}(0,T;{\bf
H}_{0}^{1}(0,L))}\quad\bar{\mathbb{P}}-a.s.$

Similarly $\bar{\bf u}^{+}_{N}\rightharpoonup\bar{\bf u}$ weakly in
$L^{2}(0,T;V)$ a.s.

Observe also that the bounds obtained in Lemma 3.4 hold for the new random
variables $\bar{\mathcal{U}}_{N}$ as well. Particularly, thanks to (4.13) and
(4.14), we have the same deterministic bounds
$\|\bar{\boldsymbol{\eta}}^{*}\|_{C(0,T;{\bf
H}^{s}(0,L))}\leq\frac{1}{\delta_{2}}$ for $\frac{3}{2}<s<2$. We also have the
following convergence results for the displacements. Namely, notice that
Theorem 4.1 implies that for given $\frac{3}{2}<s<2$ (see [17] Lemma 3),

(4.19)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\rightarrow\bar{\boldsymbol{\eta}}\text{
and
}\bar{\boldsymbol{\eta}}^{*}_{N}\rightarrow\bar{\boldsymbol{\eta}}^{*}\,\,\,\text{
in }L^{\infty}(0,T;{\bf H}^{s}(0,L))\,a.s.$

and thus,

(4.20)
$\displaystyle\bar{\boldsymbol{\eta}}_{N}\rightarrow\bar{\boldsymbol{\eta}}\text{
and
}\bar{\boldsymbol{\eta}}^{*}_{N}\rightarrow\bar{\boldsymbol{\eta}}^{*}\,\,\,\text{
in }L^{\infty}(0,T;{\bf C}^{1}[0,L])\,a.s.$

Next we define piecewise constant interpolations of the ALE maps and Jacobians
$A_{\bar{\boldsymbol{\eta}}^{*}_{N}},J_{\bar{\boldsymbol{\eta}}^{*}_{N}}$, and
their piecewise linear interpolation
$\widetilde{A}_{\bar{\boldsymbol{\eta}}^{*}_{N}},\widetilde{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}$.
Then (4.20), (3.8) and (3.7), yield

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

Furthermore, $A_{\bar{\boldsymbol{\eta}}^{*}}\in L^{\infty}(0,T;{\bf
W}^{2,p}(\mathcal{O})$ for $p<4$ and is the solution to (3.2) corresponding to
the boundary data $\text{\bf id}+\bar{\boldsymbol{\eta}}^{*}$ on $\Gamma$.
Next let $\bar{\bf
w}^{*}_{N}=\partial_{t}\widetilde{A}_{\bar{\boldsymbol{\eta}}^{*}_{N}}=\sum_{n=0}^{N}\frac{1}{\Delta
t}(A^{\omega}_{\bar{\boldsymbol{\eta}}_{*}^{n}}-A^{\omega}_{\bar{\boldsymbol{\eta}}^{n-1}_{*}})\chi_{(t_{n},t_{n+1})}$.
Note that for every $\omega\in\bar{\Omega}$, we have (see [14]):

$\displaystyle\|\bar{\bf w}^{*}_{N}\|_{L^{2}(0,T;{\bf
H}^{k+\frac{1}{2}}(\mathcal{O}))}\leq C\|\bar{\bf v}^{*}_{N}\|_{L^{2}(0,T;{\bf
H}^{k}(0,L))},\quad\text{ for any }0\leq k\leq 1,$

where $C$ depends only on $k$. Thus using (4.18), for a fixed $\varepsilon>0$,
we obtain that,

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

where $\bar{\bf w}^{*}$ satisfies (3.3) with values $\bar{\bf v}^{*}$ on
$\Gamma$. Similarly, (4.10) gives us

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

Finally, we give the definition of the required filtrations. we denote by
$\mathcal{F}_{t}^{\prime}$ the $\sigma$-field generated by the random
variables $(\bar{{\bf u}}(s),\bar{\bf
v}(s)),\bar{\boldsymbol{\eta}}(s),\bar{W}(s)$ for all $s\leq t$. Then we
define

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

We note here that the stochastic processes
$(J_{\bar{\boldsymbol{\eta}}^{*}}\bar{\bf u},\bar{\boldsymbol{\eta}})$ that
are $(\bar{\mathcal{F}}_{t})_{t\geq 0}-$progressively measurable. For each $N$
we also define a filtration $\left(\bar{\mathcal{F}}^{N}_{t}\right)_{t\geq 0}$
on $(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})$ the same way. Moreover,
using usual arguments we can see that $\bar{W}_{N}$ is an
$\bar{\mathcal{F}}^{N}_{t}$-Wiener process (see e.g. [3]). Next, relative to
the new stochastic basis
$(\bar{\Omega},\bar{\mathcal{F}},(\bar{\mathcal{F}}^{N}_{t})_{t\geq
0},\bar{\mathbb{P}},\bar{W}_{N})$, thanks to (4.14) we can see that for each
$N$, the following equation holds

(4.24)
$\begin{split}&((\widetilde{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}(t),{\bf
q})+(\bar{\widetilde{\bf v}}_{N}(t),\boldsymbol{\psi})=(J_{0}{\bf u}_{0},{\bf
q})+({\bf
v}_{0},\boldsymbol{\psi})-\int_{0}^{t}\langle\mathscr{L}_{e}(\bar{\boldsymbol{\eta}}^{+}_{N}),\boldsymbol{\psi}\rangle\\\
&+\int_{0}^{t}\int_{\mathcal{O}}\frac{\partial\widetilde{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}}{\partial
t}(2\bar{\widetilde{\bf u}}_{N}-{\bar{\bf u}}_{N})\cdot{\bf
q}-\frac{1}{2}\frac{\partial\widetilde{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}}{\partial
t}\bar{\bf u}^{+}_{N}\cdot{\bf
q}-\varepsilon\int_{0}^{t}\int_{0}^{L}\partial_{zz}\bar{{\bf
v}}^{\\#}_{N}\cdot\partial_{zz}\boldsymbol{\psi}\\\
&-\frac{1}{2}\int_{0}^{t}\int_{\mathcal{O}}(J_{\bar{\boldsymbol{\eta}}^{*}_{N}})((\bar{\bf
u}^{+}_{N}-\bar{\bf
w}^{*}_{N})\cdot\nabla^{\bar{\boldsymbol{\eta}}_{N}^{*}}\bar{\bf
u}^{+}_{N}\cdot{\bf q}-(\bar{\bf u}^{+}_{N}-\bar{\bf
w}^{*}_{N})\cdot\nabla^{\bar{\boldsymbol{\eta}}_{N}^{*}}{\bf q}\cdot\bar{\bf
u}^{+}_{N})\\\
&-2\nu\int_{0}^{t}\int_{\mathcal{O}}(J_{\bar{\boldsymbol{\eta}}^{*}_{N}}){\bf
D}^{\bar{\boldsymbol{\eta}}_{N}^{*}}(\bar{\bf u}^{+}_{N})\cdot{\bf
D}^{\bar{\boldsymbol{\eta}}_{N}^{*}}({\bf
q})-\frac{1}{\alpha}\int_{0}^{t}\int_{\Gamma}S_{\bar{\boldsymbol{\eta}}_{N}}(\bar{\bf
u}^{+}_{N}-\bar{\bf v}^{+}_{N})({\bf q}-\boldsymbol{\psi})\\\
&-\frac{1}{\varepsilon}\int_{0}^{t}\int_{\mathcal{O}}\text{div}^{{\bar{\boldsymbol{\eta}}}^{*}_{N}}\bar{\bf
u}^{+}_{N}\text{div}^{{\bar{\boldsymbol{\eta}}}^{*}_{N}}{\bf q}d{\bf
x}-\frac{1}{\varepsilon}\int_{0}^{t}\int_{\Gamma}(\bar{\bf u}^{+}_{N}-\bar{\bf
v}^{+}_{N})\cdot\bar{\bf n}^{*}_{N}({\bf q}-\boldsymbol{\psi})\cdot\bar{\bf
n}^{*}_{N}\\\
&\pm\int_{0}^{t}\left(P_{{in/out}}\int_{0}^{1}q_{z}\Big{|}_{z=0/1}\right)+\int_{0}^{t}(G({\bar{\bf
u}}_{N},\bar{\boldsymbol{\eta}}^{*}_{N})d\bar{W}_{N},{\bf Q})+(E_{N}(t),{\bf
Q}),\end{split}$

$\bar{\mathbb{P}}$-a.s. for every $t\in[0,T]$ and any ${\bf
Q}\in\mathscr{D}\cap{\mathscr{U}}_{1}$.

Using the convergence results stated in Theorem 4.1 we can then pass
$N\to\infty$ in the deterministic terms in (LABEL:approxsystem). For the
stochastic integral see Lemma 5.4. We mention here how we treat the convective
term which is the only term that needs an explanation. By integrating by parts
we obtain

$\displaystyle\int_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}^{*}_{N}}\bar{\bf
w}_{N}^{*}\cdot\nabla^{\bar{\boldsymbol{\eta}}_{N}^{*}}\bar{\bf
u}^{+}_{N}\cdot{\bf q}$
$\displaystyle=-\int_{\mathcal{O}}{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}\nabla^{\bar{\boldsymbol{\eta}}^{*}_{N}}\cdot\bar{\bf
w}_{N}^{*}\bar{{\bf u}}^{+}_{N}\cdot{\bf
q}-\int_{\mathcal{O}}{J}_{\bar{\boldsymbol{\eta}}^{*}_{N}}\bar{\bf
w}^{*}_{N}\cdot\nabla^{\bar{\boldsymbol{\eta}}^{*}_{N}}{\bf q}\cdot\bar{{\bf
u}}^{+}_{N}$ (4.25)
$\displaystyle+\int_{\Gamma}S_{\bar{\boldsymbol{\eta}}_{N}^{*}}(\bar{\bf
v}_{N}^{*}\cdot{\bf n}_{N}^{*})(\bar{\bf u}^{+}_{N}\cdot{\bf q}),$

where $S_{\bar{\boldsymbol{\eta}}^{*}_{N}}$ is the Jacobian of the
transformation from Eulerian to Lagrangian coordinates. Thus, using the weak
and strong convergence results in Theorem 4.1, (4.18), (4.22) and (4.20) we
can pass $N\to\infty$ in (4.25). Then using the fact that,

(4.26)
$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\partial_{t}J_{\bar{\boldsymbol{\eta}}^{*}}\bar{{\bf
u}}\cdot{\bf
q}=\int_{0}^{t}\int_{\mathcal{O}}J_{\bar{\boldsymbol{\eta}}^{*}}\nabla^{\boldsymbol{\eta}^{*}}\cdot{\bf
w}^{*}\bar{{\bf u}}\cdot{\bf q},$

we arrive at the following weak formulation.

######  Theorem 4.2.

For the stochastic basis
$(\bar{\Omega},\bar{\mathcal{F}},(\bar{\mathcal{F}}_{t})_{t\geq
0},\bar{\mathbb{P}},\bar{W})$ constructed in Theorem 4.1, given any fixed
$\varepsilon>0$ and $\delta=(\delta_{1},\delta_{2})$ satisfying (3.9), the
processes $(\bar{\bf u},\bar{\boldsymbol{\eta}},\bar{\boldsymbol{\eta}}^{*})$
obtained in Theorem 4.1 are such that $\bar{\boldsymbol{\eta}}^{*}$ and
$(\bar{J}_{\boldsymbol{\eta}^{*}}\bar{\bf
u},\partial_{t}\bar{\boldsymbol{\eta}})$ are $(\bar{\mathcal{F}}_{t})_{t\geq
0}$-progressively measurable with $\bar{\mathbb{P}}$-a.s. continuous paths in
${\bf H}^{s}(0,L)$, $s<2$ and $\mathscr{V}_{1}^{\prime}$ respectively and the
following weak formulation holds $\bar{\mathbb{P}}$-a.s. for every $t\in[0,T]$
and ${\bf Q}\in\mathscr{D}$:

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

Next, we argue that

(4.28)
$\displaystyle\bar{\boldsymbol{\eta}}^{*}(t)=\bar{\boldsymbol{\eta}}(t)\quad\text{
for any }t<T^{\boldsymbol{\eta}},\quad\bar{\mathbb{P}}-a.s.$

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

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

Indeed, to prove (4.28), we introduce the following stopping times. For
$\frac{3}{2}<s<2$ we define

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

Then (4.20) implies that
$T^{\boldsymbol{\eta}}\leq\liminf_{N\rightarrow\infty}T^{\boldsymbol{\eta}}_{N}$
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

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

This is true because, the uniform convergence (4.20) implies that for any
$\epsilon>0$ there exists an $N_{1}\in\mathbb{N}$ such that the first two
terms on the right side of the above inequality are each bounded by
$\frac{\epsilon}{2}$ for all $N\geq N_{1}$. Moreover, since
$t<T^{\boldsymbol{\eta}}_{N}$ for infinitely many $N$’s, the third term is
equal to 0. This concludes the proof of (4.28).

##  5\. Passing to the limit $\varepsilon\to 0$

To emphasize the dependence on the parameter $\varepsilon>0$, hereon we will
denote the martingale solution found in the previous section by $(\bar{\bf
u}_{\varepsilon},\bar{\bf v}_{\varepsilon},\bar{\bf
v}^{*}_{\varepsilon},\bar{\boldsymbol{\eta}}_{\varepsilon},\bar{\boldsymbol{\eta}}^{*}_{\varepsilon},\bar{W}_{\varepsilon})$
and
$(\Omega^{\varepsilon},\mathcal{F}^{\varepsilon},(\mathcal{F}^{\varepsilon}_{t})_{t\geq
0},{\mathbb{P}}^{\varepsilon})$. The aim of this section is to pass
$\varepsilon\rightarrow 0$ in (LABEL:martingale1) by constructing appropriate
test functions. Most of the results in the first half of this section can be
proven as in the previous section and so we will only summarize the important
theorems without proof. Observe that, thanks to the weak lower-semicontinuity
of norm, the uniform estimates obtained in the previous section still hold. As
a consequence of Lemmas 3.4 and Theorem 4.1, we thus have the following
uniform bounds.

######  Lemma 5.1 (Uniform boundedness).

For a fixed $\delta=(\delta_{1},\delta_{2})$ that satisfies (3.9), we have for
some $C>0$ independent of $\varepsilon$ that

  1. (1)

${\mathbb{E}}^{\varepsilon}\|\bar{\bf
u}_{\varepsilon}\|^{2}_{L^{\infty}(0,T;{\bf L}^{2}(\mathcal{O}))\cap
L^{2}(0,T;V)}<C.$

  2. (2)

${\mathbb{E}}^{\varepsilon}\|\bar{\bf
v}^{*}_{\varepsilon}\|^{2}_{L^{\infty}(0,T;{\bf L}^{2}(0,L))}<C$.

  3. (3)

${\mathbb{E}}^{\varepsilon}\|\bar{\boldsymbol{\eta}}_{\varepsilon}\|^{2}_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L)\,\cap\,W^{1,\infty}(0,T;{\bf L}^{2}(0,L)))}<C.$

  4. (4)

${\mathbb{E}}^{\varepsilon}\|\bar{\boldsymbol{\eta}}^{*}_{\varepsilon}\|^{2}_{L^{\infty}(0,T;{\bf
H}^{2}_{0}(0,L)\,\cap\,W^{1,\infty}(0,T;{\bf L}^{2}(0,L)))}<C.$

  5. (5)

${\mathbb{E}}^{\varepsilon}\|\text{div}^{\bar{\boldsymbol{\eta}}^{*}_{\varepsilon}}\bar{\bf
u}_{\varepsilon}\|^{2}_{L^{2}(0,T;L^{2}(\mathcal{O}))}<C{\varepsilon}$,
${\mathbb{E}}^{\varepsilon}\|(\bar{\bf u}_{\varepsilon}|_{\Gamma}-\bar{\bf
v}_{\varepsilon})\cdot{\bf
n}^{*}_{\varepsilon}\|^{2}_{L^{2}(0,T;L^{2}(0,L))}<C{\varepsilon}$.

  6. (6)

$\sqrt{\varepsilon}{\mathbb{E}}^{\varepsilon}\|\partial_{zz}\bar{\bf
v}_{\varepsilon}\|^{2}_{L^{2}(0,T;{\bf L}^{2}(0,L))}<C$.

  7. (7)

$\|\bar{\boldsymbol{\eta}}_{\varepsilon}^{*}\|_{C(0,T;{\bf
H}^{s}(0,L))}\leq\frac{1}{\delta_{2}}$ for $\frac{3}{2}<s<2$, for almost every
$\omega\in\Omega^{\varepsilon}$.

Next we have the following tightness results:

######  Lemma 5.2 (Tightness of the laws).

  1. (1)

The laws of $\bar{\bf u}_{\varepsilon}$ and $\bar{\bf v}_{\varepsilon}$ are
tight in $L^{2}(0,T;{\bf H}^{\alpha}(\mathcal{O}))$ and $L^{2}(0,T;{\bf
H}^{-{\beta}}(0,L))$ respectively for any $0\leq\alpha<1$,
$0<\beta<\frac{1}{2}$.

  2. (2)

The laws of $\bar{\boldsymbol{\eta}}_{\varepsilon}$ and that of
$\bar{\boldsymbol{\eta}}^{*}_{\varepsilon}$ are tight in $C([0,T];{\bf
H}^{s}(0,L))$ for $\frac{3}{2}<s<2$.

  3. (3)

The laws of $\|\bar{\bf u}_{\varepsilon}\|_{L^{2}(0,T;V)}$ are tight in
$\mathbb{R}$.

  4. (4)

The laws of $\|\bar{\bf v}^{*}_{\varepsilon}\|_{L^{2}(0,T;{\bf L}^{2}(0,L))}$
are tight in $\mathbb{R}$.

###### Proof.

We describe how to prove the first statement, which follows from the proof of
Lemma 4.2 almost identically. Construction of suitable test functions $({\bf
q}_{\varepsilon},\boldsymbol{\psi}_{\varepsilon})$ is the same as (4.8) and we
apply the variant of Itô’s formula stated in Lemma 5.1 in [4], which justifies
testing (LABEL:martingale1) with the continuous-in-time versions of the random
test functions (4.8) (i.e. where $(\Delta t)\sum_{n-j+1}^{n}$ is replaced by
$\int_{t-h}^{t}dt$). Recall that all the bounds obtained in the proof of Lemma
4.2, except in (4.11), do not depend on $\varepsilon$. However, due to
integrating by parts in (4.25) and applying (4.26), the weak formulation
(LABEL:martingale1) now contains the boundary integral
$\int_{0}^{t}\int_{\Gamma}S_{\bar{\boldsymbol{\eta}}_{\varepsilon}^{*}}(\bar{\bf
v}_{\varepsilon}^{*}\cdot{\bf n}_{\varepsilon}^{*})(\bar{\bf
u}_{\varepsilon}\cdot{\bf q}_{\varepsilon})$ instead of the aforementioned
term involving the derivatives of $\bar{\bf w}_{\varepsilon}^{*}$. Then, for
the process ${\bf q}_{\varepsilon}$ taking values in ${\bf
H}^{1}(\mathcal{O})$, described above and constructed as in (4.8), we can
bound this boundary integral independently of $\varepsilon$, by using the fact
that $\|{\bf n}^{*}_{\varepsilon}\|_{{\bf
L}^{\infty}((0,T)\times(0,L))}<C(\delta)$ together with the bounds for the
trace $\bar{\bf u}_{\varepsilon}|_{\Gamma}$ in
$L^{2}(\hat{\Omega};L^{2}(0,T;{\bf H}^{\frac{1}{2}}(\Gamma)))$ (see Theorem
1.5.2.1 in [14]) and that for $\bar{\bf v}^{*}_{\varepsilon}$ in
$L^{2}(\hat{\Omega};L^{2}(0,T;{\bf L}^{2}(0,L)))$ which are independent of
$\varepsilon$. ∎

Now for an infinite denumerable set of indices $\Lambda$, we denote by
$\mu_{\varepsilon}$ the joint law of the random variable
$\bar{\mathcal{U}}_{\varepsilon}:=(\bar{{\bf u}}_{\varepsilon},\bar{\bf
v}_{\varepsilon},\bar{\boldsymbol{\eta}}_{\varepsilon},\bar{\boldsymbol{\eta}}^{*}_{\varepsilon},\|\bar{\bf
u}_{\varepsilon}\|_{L^{2}(0,T;V)},\|\bar{\bf
v}^{*}_{\varepsilon}\|_{L^{2}(0,T;{\bf L}^{2}(0,L))},\bar{W}_{\varepsilon})$
taking values in the phase space

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

for some $0<\beta<\frac{1}{2}$, $\frac{3}{2}<s<2$.

Then the tightness of the laws $\mu_{\varepsilon}$ on $\mathcal{S}$ and an
application of the Prohorov theorem and the almost sure representation in [25]
give us the following result.

######  Theorem 5.1.

There exists a probability space
$(\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})$ and random variables
$\hat{\mathcal{U}}_{\varepsilon}=(\hat{{\bf u}}_{\varepsilon},\hat{\bf
v}_{\varepsilon},\hat{\boldsymbol{\eta}}_{\varepsilon},\hat{\boldsymbol{\eta}}^{*}_{\varepsilon},m_{\varepsilon},k_{\varepsilon},\hat{W}_{\varepsilon})$
and $\hat{\mathcal{U}}=(\hat{{\bf u}},\hat{\bf
v},\hat{\boldsymbol{\eta}},\hat{\boldsymbol{\eta}}^{*},{m},{k},\hat{W})$ such
that

  1. (1)

$\hat{\mathcal{U}}_{\varepsilon}=^{d}\bar{\mathcal{U}}_{\varepsilon}$ for
every $\varepsilon\in\Lambda$.

  2. (2)

$\hat{\mathcal{U}}_{\varepsilon}\rightarrow\hat{\mathcal{U}}$
$\hat{\mathbb{P}}$-a.s. in the topology of $\mathcal{S}$ as
$\varepsilon\rightarrow 0$.

  3. (3)

$\partial_{t}\hat{\boldsymbol{\eta}}=\hat{\bf v}$ and
$\partial_{t}\hat{\boldsymbol{\eta}}^{*}=\hat{\bf v}^{*}$, in the sense of
distributions, almost surely.

We recall again that Theorem 1.10.4 in [25] tells us that the random variables
$\hat{\mathcal{U}}_{\varepsilon}$ can be chosen such that for every
$\varepsilon\in\Lambda$,

(5.1)
$\displaystyle\hat{\mathcal{U}}_{\varepsilon}(\omega)=\bar{\mathcal{U}}_{\varepsilon}(\phi_{\varepsilon}(\omega)),\quad\omega\in\hat{\Omega},$

and $\hat{\mathbb{P}}\circ\phi_{\varepsilon}^{-1}={\mathbb{P}}^{\varepsilon}$,
where $\phi_{\varepsilon}:\hat{\Omega}\rightarrow\Omega^{\varepsilon}$ is
measurable.

Thanks to these explicit maps we identify the real-valued random variables
$m_{\varepsilon}$ as $m_{\varepsilon}=\|\hat{\bf
u}_{\varepsilon}\|_{L^{2}(0,T;V)}$ and notice that $m_{\varepsilon}$ converge
almost surely due to Theorem 5.1. Hence as in Theorem 4.1, we obtain, up to a
subsequence, that

(5.2) $\displaystyle\hat{\bf u}_{\varepsilon}\rightharpoonup\hat{\bf
u}\quad\text{ weakly in }{L^{2}(0,T;V)}\quad\hat{\mathbb{P}}-a.s.$

Similarly,

(5.3) $\displaystyle\hat{\bf v}^{*}_{\varepsilon}\rightharpoonup\hat{\bf
v}^{*}\quad\text{ weakly in }{L^{2}(0,T;{\bf
L}^{2}(0,L))}\quad\hat{\mathbb{P}}-a.s.$

As in the previous section, we also have that

(5.4)
$\displaystyle\hat{\boldsymbol{\eta}}_{\varepsilon}\rightarrow\hat{\boldsymbol{\eta}}\text{
and
}\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}\rightarrow\hat{\boldsymbol{\eta}}^{*}\,\,\,\text{
in }L^{\infty}(0,T;{\bf C}^{1}[0,L])\,a.s.,$

and that,

(5.5) $\begin{split}&A_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}\rightarrow
A_{\hat{\boldsymbol{\eta}}^{*}},(A_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}})^{-1}\rightarrow(A_{\hat{\boldsymbol{\eta}}^{*}})^{-1}\text{
in }L^{\infty}(0,T;{\bf W}^{2,p}(\mathcal{O}))\,\text{ for any
$p<4$}\,\,\,a.s.,\\\ &J_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}\rightarrow
J_{\hat{\boldsymbol{\eta}}^{*}}=\text{det}(\nabla
A_{\hat{\boldsymbol{\eta}}^{*}})\quad\text{ in
}L^{\infty}(0,T;C(\bar{\mathcal{O}}))\,\quad\,\hat{\mathbb{P}}-a.s,\\\
&S_{\hat{\boldsymbol{\eta}}_{\varepsilon}}\rightarrow
S_{\hat{\boldsymbol{\eta}}}\quad\text{ in
}L^{\infty}(0,T;C(\bar{\Gamma}))\,\quad\,\hat{\mathbb{P}}-a.s,\\\ &\hat{\bf
w}^{*}_{\varepsilon}\rightharpoonup\hat{\bf w}^{*}\,\quad\text{ weakly in
}L^{2}(0,T;{\bf H}^{\frac{1}{2}}(\mathcal{O})),\quad\,\hat{\mathbb{P}}-a.s.\\\
&\hat{\bf n}^{*}_{\varepsilon}\to\hat{\bf n}^{*}\,\quad\text{ in
}L^{\infty}(0,T;C(\bar{\Gamma})),\quad\,\hat{\mathbb{P}}-a.s.\end{split}$

Due to the lack of the equivalent of Lemma 4.5, we have one more obstacle to
deal with. Namely, that the candidate solution for fluid velocity, $\hat{\bf
u}$, does not have the desired temporal regularity to be a stochastic process
in the classical sense. Hence, we construct an appropriate filtration as
follows: first define the $\sigma-$fields

$\sigma_{t}(\hat{\bf u}):=\bigcap_{s\geq t}\sigma\left(\bigcup_{{\bf Q}\in
C^{\infty}_{0}((0,s);\mathscr{D})}\\{(\hat{\bf u},{\bf
q})<1\\}\cup\mathcal{N}\right),\quad\mathcal{N}=\\{\mathcal{A}\in\hat{\mathcal{F}}\
|\hat{\mathbb{P}}(\mathcal{A})=0\\}.$

Let $\hat{\mathcal{F}}_{t}^{\prime}$ be the $\sigma-$ field generated by the
random variables $\hat{\boldsymbol{\eta}}(s),\hat{W}(s)$ for all $0\leq s\leq
t$. Then we define the history of the random distributions $\hat{\bf u}$,
$\hat{\mathcal{F}}_{t}$, as follows

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

This gives a complete, right-continuous filtration
$(\hat{\mathcal{F}}_{t})_{t\geq 0}$, on the probability space
$(\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})$, to which the noise and
the candidate solution $\hat{\bf u}$ are adapted. Now we state the following
result from [3].

######  Lemma 5.3.

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

######  Theorem 5.2.

For any fixed $\delta=(\delta_{1},\delta_{2})$ that satisfies (3.9), the
randome variables $(\hat{\bf
u},\hat{\boldsymbol{\eta}},\hat{\boldsymbol{\eta}}^{*})$ constructed in
Theorem 5.1 satisfy the following

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

$\hat{\mathbb{P}}$-a.s. for almost every $t\in[0,T]$ and for any
$(\hat{\mathcal{F}}_{t})_{t\geq 0}-$adapted process ${\bf Q}=({\bf
q},\boldsymbol{\psi})$ with $C^{1}$\- paths in $\mathscr{D}$ such that
$\nabla^{\hat{\boldsymbol{\eta}}^{*}}\cdot{\bf q}=0$ and ${\bf
q}|_{\Gamma}\cdot{\bf
n}^{\hat{\boldsymbol{\eta}}^{*}}=\boldsymbol{\psi}\cdot{\bf
n}^{\hat{\boldsymbol{\eta}}^{*}}$ a.s. Moreover,
$\nabla^{\hat{\boldsymbol{\eta}}^{*}}\cdot\hat{\bf u}=0$.

###### Proof of Theorem 5.2.

First we must construct $\mathscr{D}$-valued test processes $({\bf
q}_{\varepsilon},\boldsymbol{\psi}_{\varepsilon})$, satisfying the kinematic
coupling condition and such that ${\bf q}_{\varepsilon}$ satisfies the
transformed divergence-free condition. This is required so that the two
penalty terms in the approximate weak formulation drop out.

We first construct an appropriate test functions for the limiting equation
(LABEL:martingale2) as follows: Recall that the maximal domain
$\mathcal{O}_{\delta}=(0,L)\times(0,R_{\delta})$ is a rectangular domain
comprising of all the moving domains
$\mathcal{O}_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}}$. Consider a smooth
$(\hat{\mathcal{F}}_{t})_{t\geq 0}$-adapted process ${\bf g}=(g_{z},g_{r})$ on
$\bar{\mathcal{O}}_{\delta}$ such that $\nabla\cdot{\bf g}=0$ and such that
${\bf g}$ satisfies the required boundary conditions $g_{r}=0\text{ on
}z=0,L,r=0$ and $\partial_{r}g_{z}=0\text{ on }\Gamma_{b}$. Assume also that,
on the top lateral boundary of the moving domain associated with
$\hat{\boldsymbol{\eta}}^{*}$, $\Gamma_{\hat{\boldsymbol{\eta}}^{*}}$, the
function ${\bf g}$ satisfies ${\bf
g}(t)|_{\Gamma_{\hat{\boldsymbol{\eta}}^{*}(t)}}\cdot\hat{\bf
n}^{*}(t)=\boldsymbol{\psi}(t)\cdot\hat{\bf n}^{*}(t)$ for some smooth
$(\hat{\mathcal{F}}_{t})_{t\geq 0}$-adapted process
$\boldsymbol{\psi}=(\psi_{z},\psi_{r})$. We define

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

To observe that ${\bf q}$ is a suitable test function we consider, for any
$t\in[0,T]$ and given process ${\bf g}$, the map $\mathcal{C}_{\bf
g}:\hat{\Omega}\times{\bf C}([0,L])\rightarrow{\bf C}^{1}(\bar{\mathcal{O}})$,

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

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 ${\boldsymbol{\eta}}\in{\bf C}([0,L])$. Due to the continuity of the
composition operator $F_{\boldsymbol{\eta}}$, the assumption that ${\bf g}(t)$
is $\hat{\mathcal{F}}_{t}-$measurable, implies for any ${\boldsymbol{\eta}}$,
that the ${\bf C}^{1}(\bar{\mathcal{O}})$-valued map
$\omega\mapsto\mathcal{C}_{\bf g}(\omega,{\boldsymbol{\eta}})$ is
$\hat{\mathcal{F}}_{t}$-measurable (where ${\bf C}^{1}(\bar{\mathcal{O}})$ is
endowed with Borel $\sigma$-algebra). Note also that for a fixed $\omega$, the
map ${\boldsymbol{\eta}}\mapsto\mathcal{C}_{\bf
g}(\omega,{\boldsymbol{\eta}})$ is continuous. Hence, we infer that
$\mathcal{C}_{\bf g}$ is a Carathéodory function. Recall also that
$\hat{\boldsymbol{\eta}}^{*}$ is $(\hat{\mathcal{F}}_{t})_{t\geq 0}$-adapted.
Therefore, we deduce that the ${\bf C}^{1}(\bar{\mathcal{O}})$-valued process
${\bf q}(t,\omega)=\mathcal{C}_{\bf
g}(\omega,\hat{\boldsymbol{\eta}}^{*}(t,\omega))$ is
$(\hat{\mathcal{F}}_{t})_{t\geq 0}$-adapted as well.

In summary, we have $\\{\hat{\mathcal{F}}_{t}\\}_{t\geq 0}$-adapted processes
$({\bf q},\boldsymbol{\psi})$ with continuous paths in $\mathscr{D}$ such that

$\nabla^{\hat{\boldsymbol{\eta}}^{*}}\cdot{\bf q}=0\text{ and }{\bf
q}\Big{|}_{\Gamma}\cdot{\bf
n}^{\hat{\boldsymbol{\eta}}^{*}}=\boldsymbol{\psi}\cdot{\bf
n}^{\hat{\boldsymbol{\eta}}^{*}}.$

Moreover, for any $\omega\in\hat{\Omega}$ that ${\bf q}\in L^{\infty}(0,T;{\bf
H}^{2+k}(\mathcal{O}))\cap H^{1}(0,T;{\bf H}^{k}(\mathcal{O}))$ for any
$k\leq\frac{1}{2}$. Now we define the approximate test functions $({\bf
q}_{\varepsilon},\boldsymbol{\psi}_{\varepsilon})$, with the aid of the Piola
transformation as done in the proof of Lemma 4.2:

$\displaystyle{\bf
q}_{\varepsilon}={J^{-1}_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}}\nabla
A_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}{J_{\hat{\boldsymbol{\eta}}^{*}}}\nabla
A^{-1}_{\hat{\boldsymbol{\eta}}^{*}}\left({\bf
q}-\boldsymbol{\psi}\chi\right)+\boldsymbol{\psi}\chi-\frac{\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}-\lambda^{\hat{\boldsymbol{\eta}}^{*}}}{\lambda^{\varepsilon}_{0}}(\xi_{0}\chi)$
$\displaystyle+{J^{-1}_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}}\mathcal{B}\left(\text{div}\left((J_{\hat{\boldsymbol{\eta}}^{*}}(\nabla
A_{\hat{\boldsymbol{\eta}}^{*}})^{-1}-J_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}}(\nabla
A_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}})^{-1})\boldsymbol{\psi}\chi-\frac{\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}-\lambda^{\hat{\boldsymbol{\eta}}^{*}}}{\lambda^{\varepsilon}_{0}}J_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}}(\nabla
A_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}})^{-1}\xi_{0}\chi\right)\right)$

and,

$\displaystyle\boldsymbol{\psi}_{\varepsilon}=\boldsymbol{\psi}-\frac{\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}-\lambda^{\hat{\boldsymbol{\eta}}^{*}}}{\lambda^{\varepsilon}_{0}}\xi_{0},$

where we pick an appropriate $\xi_{0}\in{\bf C}_{0}^{\infty}(\Gamma)$ such
that $\lambda_{0}^{\varepsilon}$ defined below is not 0 for any
$\varepsilon>0$,

$\lambda_{0}^{\varepsilon}(t)=-\int_{\Gamma}({\bf
id}+\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}(t))\times\partial_{z}\xi_{0}\,dz.\quad\text{}$

We also define the real-valued corrector functions,

$\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}(t)=-\int_{\Gamma}({\bf
id}+\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}(t))\times\partial_{z}\boldsymbol{\psi}(t)dz,\quad\lambda^{\hat{\boldsymbol{\eta}}^{*}}(t)=-\int_{\Gamma}({\bf
id}+\hat{\boldsymbol{\eta}}^{*}(t))\times\partial_{z}\boldsymbol{\psi}(t)dz.$

As earlier, $\chi(r)$ is a smooth function on $\mathcal{O}$ such that
$\chi(1)=1$ and $\chi(0)=0$. Observe that the properties of the Piola
transformation (see e.g. Theorem 1.7 in [6]), imply that

$\nabla^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}\cdot{\bf
q}_{\varepsilon}=J_{\hat{\boldsymbol{\eta}}^{*}}J^{-1}_{\hat{\boldsymbol{\eta}}_{\varepsilon}^{*}}\nabla^{\hat{\boldsymbol{\eta}}^{*}}\cdot{\bf
q}=0,\qquad\text{and}\qquad{\bf q}_{\varepsilon}|_{\Gamma}\cdot{\bf
n}_{\varepsilon}^{*}=\boldsymbol{\psi}_{\varepsilon}\cdot{\bf
n}_{\varepsilon}^{*}.$

Furthermore, we have

$|\frac{d}{dt}\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}|\leq\|\hat{\bf
v}^{*}_{\varepsilon}\|_{{\bf L}^{2}(0,L)}\|\boldsymbol{\psi}\|_{{\bf
H}_{0}^{1}(0,L)}+\|\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}\|_{{\bf
L}^{\infty}(0,L)}\|\partial_{t}\partial_{z}\boldsymbol{\psi}\|_{{\bf
L}^{\infty}(0,L)}.$

Hence
$\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}\to\lambda^{\hat{\boldsymbol{\eta}}^{*}}$
strongly in $L^{\infty}(0,T)$ and weakly in $H^{1}(0,T)$ a.s. Additionally,
thanks to (5.5) we obtain that

$\displaystyle\|{\bf q}_{\varepsilon}-{\bf q}\|_{L^{\infty}(0,T;{\bf
H}^{1}(\mathcal{O}))}$
$\displaystyle\leq\|A_{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}-A_{\hat{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T;{\bf
W}^{2,3}(\mathcal{O}))}\left(\|{\bf q}\|_{L^{\infty}(0,T;{\bf
H}^{1}(\mathcal{O}))}+\|\boldsymbol{\psi}\|_{L^{\infty}(0,T;{\bf
H}^{1}_{0}(0,L))}\right)$ (5.8)
$\displaystyle\qquad+\|\lambda^{\hat{\boldsymbol{\eta}}^{*}_{\varepsilon}}-\lambda^{\hat{\boldsymbol{\eta}}^{*}}\|_{L^{\infty}(0,T)}\to
0\qquad\hat{\mathbb{P}}-\text{ a.s.}$