# Inviscid Limit for Stochastic Second-Grade Fluid Equations

Eliseo Luongo1
1 Scuola Normale Superiore, Piazza dei Cavalieri,7, 56126 Pisa, Italy
Email<EMAIL_ADDRESS>
###### Abstract

We consider in a smooth and bounded two dimensional domain the convergence in
the $L^{2}$ norm, uniformly in time, of the solution of the stochastic second-
grade fluid equations with transport noise and no-slip boundary conditions to
the solution of the corresponding Euler equations. We prove, that assuming
proper regularity of the initial conditions of the Euler equations and a
proper behavior of the parameters $\nu$ and $\alpha$, then the inviscid limit
holds without requiring a particular dissipation of the energy of the
solutions of the second-grade fluid equations in the boundary layer.

Keywords: Inviscid limit; Turbulence; Transport noise; No-slip boundary
conditions; Boundary layer; Additive noise; Second-grade complex fluid

##  1 Introduction

The second-grade fluid equations are a model for viscoelastic fluids, with two
parameters: $\alpha>0$, corresponding to the elastic response, and $\nu>0$,
corresponding to viscosity. Considering a constant density, $\rho=1$, their
stress tensor is given by

$T=-pI+\nu A_{1}+\alpha^{2}A_{2}-\alpha^{2}A_{1}^{2},$

where

$\displaystyle A_{1}=\frac{\nabla u+\nabla u^{T}}{2},$ $\displaystyle
A_{2}=\partial_{t}A_{1}+A_{1}\nabla u+\nabla u^{T}A_{1},$

being $p$ the pressure and $u$ the velocity field. Given this stress tensor,
the equations of motion for an incompressible homogeneous fluid of grade 2 are
given by

$\displaystyle\begin{cases}\partial_{t}v=\nu\Delta
u-\operatorname{curl}(v)\times u+\nabla p+f\\\ \operatorname{div}u=0\\\
v=u-\alpha^{2}\Delta u\\\ u|_{\partial D}=0\\\ u(0)=u_{0}.\end{cases}$ (1)

where $f$ describes some external forces, possibly stochastic, acting on the
fluid, see [9], [34] for further details on the physics behind this system.
The analysis of the deterministic system started with [6]. They proved global
existence and uniqueness without restricting the problem to the two
dimensional case. Setting, formally, $\alpha=0$ in equations (1) we can reduce
the system to the well-known Navier-Stokes one:

$\displaystyle\begin{cases}\partial_{t}u=\nu\Delta u-u\cdot\nabla u+\nabla
p+f\\\ \operatorname{div}u=0\\\ u|_{\partial D}=0\\\ u(0)=u_{0}.\end{cases}$
(2)

Thus (1) can be seen as a generalization of (2). Moreover, in [17], it has
been shown that second-grade fluid equations are a good approximation of the
Navier-Stokes system. Due to these good properties of the system it is a
legitimate question trying to understand if the second-grade fluid equations
behave better than the Navier-Stokes ones in problems related to turbulence,
like the inviscid limit for domain with boundary and no-slip boundary
conditions. In fact, such question is far for being solved for system (2) also
in the deterministic framework. Partial results are available:

  1. 1.

Unconditioned results. They are based on strong assumptions about the flows.
For example flows with radial symmetry [23], [24], or flows with analytic
boundary layers [27], [35].

  2. 2.

Conditioned results. They are based on stating some criteria about the
behavior of the solutions of the Navier-Stokes equations in the boundary layer
in order to prove the inviscid limit. This line of research started with the
famous work by Kato [19], see [7], [41], [42] for other results. For what
concern the Stochastic framework few results are available, see for example
[26] for a generalization of the Kato’s results to the additive noise case and
a wilder set of initial conditions.

The analysis of the inviscid limit for the deterministic second-grade fluid
equations is a partially well-understood topic. In particular, in [25], the
authors show that the behavior of the system changes considering different
scaling between $\nu$ and $\alpha^{2}$.

If we set, formally, $\nu=0$ in system (1) second-grade fluid equations reduce
to the so-called Euler-$\alpha$ equations:

$\displaystyle\begin{cases}\partial_{t}v=-\operatorname{curl}(v)\times
u+\nabla p+f\\\ \operatorname{div}u=0\\\ v=u-\alpha^{2}\Delta u\\\
u|_{\partial D}=0\\\ u(0)=u_{0}.\end{cases}$ (3)

This system models the averaged motion of an ideal incompressible fluid when
filtering over spatial scales smaller than $\alpha$ and its well-posedness has
been treated in [28],[37]. Euler-$\alpha$ equations, formally, satisfies the
condition of [25, Theorem 3]. Therefore we can expect that the inviscid limit
holds also in this framework. Indeed, this is true as has been showed in [22].

In this work, we will consider stochastic second-grade fluid equations and
stochastic Euler-$\alpha$ equations with transport noise which scales with
respect to the elasticity. We want to understand if the good behavior proved
in [25] if $\nu=O(\alpha^{2})$ and in [22] if $\nu=0$ is preserved also in
this case. There are several motivations to consider transport noise, as the
effect of small scales on large scales in fluid dynamics problems, see [8],
[12], [13], [16] for several discussions on this topic. A first issue related
to the analysis of the inviscid limit in the case of the transport noise is
the well-posedness of the systems. In fact the existence of strong
probabilistic solution of such systems is outside the framework treated in
[30] and [32], thus we need to improve slightly these results thanks to the
properties of the transport noise. In the following $\nu\geq 0$ and we will
always speak of second-grade fluid equations even if $\nu=0.$

The paper is organized as follows. In Section 2 we introduce the mathematical
problem, we state our main theorems and we give some well-known results for
the the Euler equations and the analysis of the stochastic second-grade fluid
equations. In Sections 3 we prove that the stochastic second-grade fluid
equations with transport noise and no-slip boundary conditions are well posed.
In Section 4, thanks to the already proven well-posedness and Hypothesis 6
below we improve the energy estimates obtained in Section 3 in order to get
some estimates crucial for the proof of Theorem 9. The proof of our main
theorem on the inviscid limit occupies Section 5. Lastly in Section 6 we add
some remarks for the analysis of the additive noise case.

##  2 Main Results

Let us start this section introducing some general assumptions which will be
always adopted under our analysis even if not recalled.

###### Hypothesis 1

  * •

$0<T<+\infty$.

  * •

$D$ is a bounded, smooth, simply connected domain.

  * •

$\left(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}\right)$ is a filtered
probability space such that $(\Omega,\mathcal{F},\mathbb{P})$ is a complete
probability space, $(\mathcal{F}_{t})_{t\in[0,T]}$ is a right continuos
filtration and $\mathcal{F}_{0}$ contains every $\mathbb{P}$ null subset of
$\Omega$.

For square integrable semimartingales taking value in separable Hilbert spaces
$U_{1},\ U_{2}$ we will denote by $[M,N]_{t}$ the quadratic covariation
process. If $M,N$ take values in the same separable Hilbert space $U$ with
orthonormal basis $u_{i}$, we will denote by $\langle\langle
M,N\rangle\rangle_{t}=\sum_{i\in\mathbb{N}}[\langle M,u_{i}\rangle_{U},\langle
N,u_{i}\rangle_{U}]_{t}$. For each $k\in\mathbb{N},\ 1\leq p\leq\infty$ we
will denote by $L^{p}(D)$ and $W^{k,p}(D)$ the well-known Lebesgue and Sobolev
spaces. We will denote by $C_{c}^{\infty}(D)$ the space of smooth functions
with compact support and by $W^{k,p}_{0}(D)$ their closure with respect to the
$W^{k,p}(D)$ topology. If $p=2$, we will write $H^{k}(D)$ (resp.
$H^{k}_{0}(D)$ ) instead of $W^{k,2}(D)$ (resp. $W^{k,2}_{0}(D)$). Let $X$ be
a separable Hilbert space, denote by $L^{p}(\mathcal{F}_{t_{0}},X)$ the space
of $p$ integrable random variables with values in $X$, measurable with respect
to $\mathcal{F}_{t_{0}}$. We will denote by $L^{p}(0,T;X)$ the space of
measurable functions from $[0,T]$ to $X$ such that

$\lVert u\rVert_{L^{p}(0,T;X)}:=\left(\int_{0}^{T}\lVert u(t)\rVert_{X}^{p}\
dt\right)^{1/p}<+\infty,\ \ 1\leq p<\infty$

and obvious generalization for $p=\infty.$ For any $r,\ p\geq 1$, we will
denote by $L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{r}(0,T;X))$ the space of
processes with values in $X$ such that

  1. 1.

$u(\cdot,t)$ is progressively measurable.

  2. 2.

$u(\omega,t)\in X$ for almost all $(\omega,t)$ and

$\displaystyle\mathbb{E}\left[\lVert
u(\omega,\cdot)\rVert_{L^{r}(0,T;X)}^{p}\right]<+\infty.$

Obvious generalizations for $p=\infty$ or $r=\infty$.

Set

$H=\\{f\in L^{2}(D;\mathbb{R}^{2}),\ \operatorname{div}f=0,\ f\cdot
n|_{\partial D}=0\\},\ V=H_{0}^{1}(D;\mathbb{R}^{2})\cap H,\
D(A)=H^{2}(D;\mathbb{R}^{2})\cap V.$

Moreover we introduce the vector space

$W=\\{u\in V:\ \operatorname{curl}(u-\alpha^{2}\Delta u)\in
L^{2}(D;\mathbb{R}^{2})\\}$

with norm $\lVert{u}\rVert_{W}^{2}=\lVert u\rVert^{2}+\alpha^{2}\lVert\nabla
u\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}+\lVert\operatorname{curl}(u-\Delta
u)\rVert_{L^{2}(D)}^{2}.$ It is well-known, see for example [6], that we can
identify $W$ with the space

$\hat{W}=\\{u\in H^{3}(D;\mathbb{R}^{2})\cap V\\}.$

Moreover there exists a constant such that

$\displaystyle\lVert u\rVert_{H^{3}}^{2}\leq C\left(\lVert
u\rVert^{2}+\lVert\nabla
u\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}+\lVert\operatorname{curl}(u-\Delta
u)\rVert^{2}_{L^{2}(D)}\right).$ (4)

We denote by $\langle\cdot,\cdot\rangle$ and $\lVert\cdot\rVert$ the inner
product and the norm in $H$ respectively. Other norms and scalar products will
be denoted with the proper subscript. On $V$ we introduce the norm $\lVert
u\rVert_{V}^{2}=\lVert u\rVert^{2}+\alpha^{2}\lVert\nabla
u\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}.$ We will shortly denote by $\lVert
u\rVert_{*}=\lVert\operatorname{curl}(u-\alpha^{2}\Delta u)\rVert_{L^{2}(D)}.$
Obviously the following inequality holds for $u\in V$, where $C_{p}$ is the
Poincarè constant associated to $D$,

$\displaystyle\frac{\lVert
u\rVert_{V}^{2}}{\alpha^{2}+C_{p}^{2}}\leq\lVert\nabla
u\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}\leq\frac{\lVert
u\rVert_{V}^{2}}{\alpha^{2}}$ (5)

Denote by $P$ the linear projector of $L^{2}\left(D;\mathbb{R}^{2}\right)$ on
$H$ and define the unbounded linear operator $A:D(A)\subseteq H\rightarrow H$
by the identity

$\displaystyle\langle Av,w\rangle=\langle\Delta
v,w\rangle_{L^{2}(D;\mathbb{R}^{2})}$ (6)

for all $v\in D(A),\ w\in H$. $A$ will be called the Stokes operator. It is
well-known (see for example [40]) that $A$ is self-adjoint, generates an
analytic semigroup of negative type on $H$ and moreover
$V=D\left(\left(-A)^{1/2}\right)\right).$ Denote by $\mathbb{L}^{4}$ the space
$L^{4}\left(D,\mathbb{R}^{2}\right)\cap H$, with the usual topology of
$L^{4}\left(D,\mathbb{R}^{2}\right)$. Define the trilinear, continuous form
$b:\mathbb{L}^{4}\times V\times\mathbb{L}^{4}\rightarrow\mathbb{R}$ as

$\displaystyle b\left(u,v,w\right)=\langle u,P(\nabla vw)\rangle.$ (7)

Now we introduce some assumptions on the stochastic part of the system.

###### Hypothesis 2

  * •

$K$ is a (possiblly countable) set of indexes.

  * •

$\sigma_{k}\in W^{1,\infty}(D;\mathbb{R}^{2})\cap V$ satisfying

$\displaystyle\sum_{k\in K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}<+\infty.$

  * •

$u_{0}\in\cap_{p\geq 2}L^{p}(\mathcal{F}_{0},W)$.

  * •

$\\{W^{k}_{t}\\}_{k\in K}$ is a sequence of real, independent Brownian motions
adapted to $\mathcal{F}_{t}$.

Let us consider the stochastic second-grade fluid equations below. Some
physical motivations for the introduction of transport noise in fluid dynamic
models can be found in [12], [16].

$\displaystyle\begin{cases}dv=(\nu\Delta u-\operatorname{curl}(v)\times
u+\nabla p)dt+\sum_{k\in K}(\sigma_{k}\cdot\nabla u+\nabla\tilde{p}_{k})\circ
dW^{k}_{t}\\\ \operatorname{div}u=0\\\ v=u-\alpha^{2}\Delta u\\\ u|_{\partial
D}=0\\\ u(0)=u_{0}.\end{cases}$

We need to add the additional pressure term $\sum_{k\in
K}\nabla\tilde{p}_{k}\circ dW^{k}_{t}$, the so-called turubulent pressure, in
the system above in order to deal with the fact that $\sum_{k\in
K}\sigma_{k}\cdot\nabla u\circ dW^{k}_{t}$ is not divergence free, therefore
an additional martingale term orthogonal to $H$ must be added to make the
system feasible.
Introducing the Stokes operator, the previous equation can be rewritten as

$\displaystyle\begin{cases}d(u-\alpha^{2}Au)=(\nu
Au-P(\operatorname{curl}(v)\times u))dt+\sum_{k\in K}P(\sigma_{k}\cdot\nabla
u)\circ dW^{k}_{t}\\\ v=u-\alpha^{2}\Delta u\\\ u(0)=u_{0}\end{cases}$ (8)

or the corresponding Itô form

$\displaystyle\begin{cases}d(u-\alpha^{2}Au)&=(\nu
Au-P(\operatorname{curl}(v)\times u))dt+\sum_{k\in K}P(\sigma_{k}\cdot\nabla
u)dW^{k}_{t}\\\ &+\frac{1}{2}\sum_{k\in
K}P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)))dt\\\ v=u-\alpha^{2}\Delta u\\\ u(0)=u_{0}.\end{cases}$ (9)

Indeed each of the Stratonovich integrals in equation (8) can be rewritten,
thanks to the Stratonovich-Itô corrector associated to previous equation, in
the following form:

$\displaystyle P(\sigma_{k}\cdot\nabla u)\circ dW^{k}_{t}$
$\displaystyle=P(\sigma_{k}\cdot\nabla
u)dW^{k}_{t}+\frac{1}{2}d[P(\sigma_{k}\cdot\nabla u),W^{k}]_{t}$
$\displaystyle=P(\sigma_{k}\cdot\nabla
u)dW^{k}_{t}+\frac{1}{2}P\left(\sigma_{k}\cdot\nabla d[u,W^{k}]_{t}\right)$
$\displaystyle=P(\sigma_{k}\cdot\nabla
u)dW^{k}_{t}+\frac{1}{2}P\left(\sigma_{k}\cdot\nabla
d\left[\int_{0}^{.}(I-\alpha^{2}A)^{-1}\sum_{j\in K}P(\sigma_{j}\cdot\nabla
u)dW^{j}_{s},W^{k}\right]_{t}\right)$ $\displaystyle=P(\sigma_{k}\cdot\nabla
u)dW^{k}_{t}+\frac{1}{2}P\left(\sigma_{k}\cdot\nabla\left((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)\right)\right)dt.$ (10)

We denote by $F(u)=\frac{1}{2}\sum_{k\in
K}P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla u)))$
and $G^{k}(u)=P(\sigma_{k}\cdot\nabla u)$. By Corollary 22 below

$\displaystyle F\in\mathcal{L}(V;H\cap H^{1}(D;\mathbb{R}^{2})),\ \
G^{K}\in\mathcal{L}(V;H).$

###### Definition 3

A stochastic process with weakly continuous trajectories with values in $W$ is
a weak solution of equation (9) if

$u\in L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{\infty}(0,T;W)),\ \forall p\geq
2.$

and $\mathbb{P}-a.s.$ for every $t\in[0,T]$ and $\phi\in W$ we have

$\displaystyle\langle u(t)-u_{0},\phi\rangle_{V}+\int_{0}^{t}\nu\langle\nabla
u(s),\nabla\phi\rangle_{L^{2}(D;\mathbb{R}^{2})}+\langle\operatorname{curl}(u(s)-\alpha^{2}\Delta
u(s))\times u(s),\phi\rangle_{L^{2}(D)}ds$ $\displaystyle=\int_{0}^{t}\langle
F(u(s)),\phi\rangle ds+\sum_{k\in K}\int_{0}^{t}\langle
G^{k}(u(s)),\phi\rangle dW^{k}_{s}.$

###### Theorem 4

Under Hypothesis 1-2, equation (9) has a unique solution in the sense of
definition 3. Moreover, almost surely, the stochastic process $u$ has $V$
continuous paths.

###### Remark 5

Actually we can weaken the integrability assumption of $u_{0}$ with respect to
$\mathbb{P}$ in order to get less integrable solution, but regular enough to
prove that the inviscid limit holds. Indeed $u_{0}\in
L^{4}(\mathcal{F}_{0},W)$ is the minimal assumption to prove either the well-
posedness, see [2] and Section 3.2 below, and the the inviscid limit, see
Sections 4 and 5. However, we prefer to not stress this assumption in order to
make our results comparable to [32].

As stated in Section 1, the proof of Theorem 4 will be the object of Section
3. Usually, in stochastic analysis, the well-posedness of a stochastic partial
differential equation is obtained considering some approximating sequence,
$\\{u^{N}\\}_{N\in\mathbb{N}}$, which solves an approximate equation in the
original probability space and showing the tightness of their law in some
spaces of functions. Then, by Prokhorov’s theorem and Skorokhod’s
representation theorem, one can find an auxiliary probability space and a
solution of the limit equation in this auxiliary probability space, $u$.
Lastly, by a Gyongy-Krylov argument, one can recover that the limit process
belongs to the original probability space and that the approximating sequence
converge in probability to $u$. See [1], [3], [12] for some examples of this
method. Here, we follow a different, perhaps, more direct approach introduced
by Breckner in [2] for Navier-Stokes equations with multiplicative noise with
particular regularity properties, but well-suited to treat transport noise,
which a priori does not satisfy the general assumptions of [2, Section 2].
This approach uses, in particular, the properties of stopping times, some
basic convergence principles from functional analysis and some properties of
fluid dynamic non-linearities. Therefore, even if the results of [2] were
related to Navier-Stokes equations, this approach can be applied also to other
fluid dynamic models, see [32], [4] for some examples to different fluid
dynamic systems. An important byproduct of this way of proceed is that the
approximations converge in mean square to the solution of the second-grade
fluid equations, see Theorem 33 below. This fact will be crucial in order to
obtain some apriori estimates on the solution, see Lemma 35 below.

Now we move to consider the inviscid limit problem and introduce a new set of
hypotheses.

###### Hypothesis 6

  * •

$\nu=O(\alpha^{2}),\ \tilde{\nu}=O(\alpha^{2})$.

  * •

$\overline{u}_{0}\in H^{s}(D;\mathbb{R}^{2})\cap H$ for some $s\geq 3.$

  * •

$\displaystyle\mathbb{E}\left[\lVert
u^{\alpha}_{0}-\bar{u}_{0}\rVert^{2}\right]\rightarrow 0;$ (11)
$\displaystyle\mathbb{E}\left[\alpha^{2}\lVert\nabla
u^{\alpha}_{0}\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}\right]=o(1);$ (12)
$\displaystyle\mathbb{E}\left[\alpha^{6}\lVert
u^{\alpha}_{0}\rVert_{H^{3}(D;\mathbb{R}^{2})}^{2}\right]=O(1).$ (13)

Let us consider the family of equations

$\displaystyle\begin{cases}d(u^{\alpha}-\alpha^{2}Au^{\alpha})&=(\nu
Au^{\alpha}-P(\operatorname{curl}(v^{\alpha})\times
u^{\alpha}))dt+\sqrt{\tilde{\nu}}\sum_{k\in K}P(\sigma_{k}\cdot\nabla
u^{\alpha})dW^{k}_{t}\\\ &+\frac{\tilde{\nu}}{2}\sum_{k\in
K}P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{\alpha})))dt\\\ v^{\alpha}=u^{\alpha}-\alpha^{2}\Delta u^{\alpha}\\\
u^{\alpha}(0)=u_{0}^{\alpha},\end{cases}$ (14)

where $\sigma_{k}$ are independent from $\nu,\ \tilde{\nu},\ \alpha$ and
$u^{\alpha}_{0}$ are random variable satisfying the assumptions of Theorem 4.
Energy relations and the behavior of the $H^{3}$ norm of $u^{\alpha}$ play a
crucial role in the analysis of the inviscid limit in the deterministic
framework, see Equation (3.2) and Equation (3.7) in [25]. If we want to have
some hope of replicating the approach of [25] we need some estimates in that
direction. This is exactly what happens. Indeed, under Hypothesis 6, Equation
(3.2) and Equation (3.7) in [25] continue to hold in the stochastic framework,
see Lemma 35 below. Therefore there is some hope to generalize the results of
[25], [22] to our stochastic framework. Now, let us consider the Euler
equations

$\displaystyle\begin{cases}\partial_{t}\bar{u}+\nabla\bar{u}\cdot\bar{u}+\nabla
p=0\ (x,t)\in D\times(0,T)\\\ \operatorname{div}\bar{u}=0\\\ \bar{u}\cdot
n|_{\partial D}=0\\\ \bar{u}(0)=\bar{u}_{0}.\end{cases}$ (15)

###### Definition 7

Given $\bar{u}_{0}\in H,$ we say that $\bar{u}\in C(0,T;H)$ is a weak solution
of equation (15) if for every $\phi\in C^{\infty}_{0}([0,T]\times D)\cap
C^{1}([0,T];H)$

$\displaystyle\langle\bar{u}(t),\phi(t)\rangle=\langle\bar{u}_{0},\phi(0)\rangle+\int_{0}^{t}\langle\bar{u}(s),\partial_{s}\phi(s)\rangle
ds+\int_{0}^{t}b(\bar{u}(s),\phi(s),\bar{u}(s))ds$

for every $t\in[0,T]$ and the energy inequality

$\displaystyle\lVert{\bar{u}(t)}\rVert^{2}\leq\lVert{\bar{u}_{0}}\rVert^{2}$

holds.

For what concern the well posedness of the Euler equations the following
results hold true, see [20], [39].

###### Theorem 8

Fix $T>0,\ s\geq 3$. Let $\bar{u}_{0}\in H^{s}(D;\mathbb{R}^{2})\cap H$. Then
there exist a unique weak solution of (15) with initial condition
$\bar{u}_{0}$ such that

$\bar{u}\in C([0,T];H^{s}(D;\mathbb{R}^{2}))\cap
C^{1}([0,T];H^{s-1}(D;\mathbb{R}^{2}))$

and $\lVert{\bar{u}(t)}\rVert=\lVert\bar{u}_{0}\rVert,\ \forall t\in[0,T].$

Now we can state our main Theorem. According to the analysis started in [15]
and continued, recently, in [10], [11] the influence of the transport noise on
the averaged solution is related to the $\ell^{2}$ norm of its coefficients,
therefore we expect that the solution of equation (14) converges to the
solution of the Euler equations with null forcing term.

###### Theorem 9

Under Hypotheses 1-2-6, calling $u^{\alpha}$ the solution of (14) and
$\bar{u}$ the solution of (15), then

$\displaystyle\lim_{\alpha\rightarrow 0}\mathbb{E}\left[\sup_{t\in[0,T]}\lVert
u^{\alpha}(t)-\bar{u}(t)\rVert^{2}\right]=0.$

###### Remark 10

If $\overline{u}_{0}\in H\cap H^{1}(D;\mathbb{R}^{2})$, the existence of a
family $u_{0}^{\alpha}$ satisfying equations (11),(12),(13) is guaranteed by
Proposition 1 of [22].

###### Remark 11

Due to the poor regularity of the coefficients $F$ and $G^{k}$, equations (9)
are not guaranteed to be well-posed for grant from the results of [32].
Indeed, neither $F$ nor $G^{k}$ satisfy the assumptions of [31] or [32].
However, due to relation (47) and the good estimates of Corollary 22, we will
be able to prove in Lemma 25 and Lemma 26 the same, actually stronger, energy
estimates that are available in [32]. These and Lemma 28 are the main
ingredients in order to prove the well-posedness of system (9). On the
contrary the well-posedness in the case of additive noise is completely solved
by the results of [32], thus in Section 6 we will only explain some remarks
about the inviscid limit and the well-posedness in the additive noise
framework.

###### Remark 12

Either Theorem 4 and Theorem 9 continue to hold considering $\nu=0$. We will
give the proof of all the statements below in full details considering the
case $\nu>0$. However if something in the proof changes considering $\nu=0$ we
will explain in a remark at the end of each proof what we need to change in
order to deal to the other case.

###### Remark 13

The arbitrariness in the choice of the parameters $\nu$ and $\tilde{\nu}$
allows us to generalize to this stochastic framework, via Theorem 9, some
results of [22] and [25]. As a byproduct of its proof we obtain that under
Hypotheses 1-2-6

$\displaystyle\alpha^{2}\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}(D;\mathbb{R}^{2})}^{2}\right]\rightarrow 0.$

Moreover, considering $\tilde{\nu}=\nu>0$ we recover the scaling introduced by
Kuksin in [21] which is relevant for the inviscid limit at the level of
invariant measures.

###### Remark 14

The results of these notes are in a certain sense complementary to what we
obtained in [26]. In [26] we required poor regularity on the initial
conditions of the Euler equations and the Navier-Stokes equations but we got a
conditioned result. On the contrary, in these notes we require strong
regularity on the initial conditions of the two problems and a special type of
convergence of the initial conditions but we arrive at a not conditioned
result.

###### Remark 15

The assumption on $\nu=O(\alpha^{2})$ is hidden in equation (14). For high
frequencies $\Delta u$ is a damping term in equation (14). In fact, for high
frequencies $v\approx-\alpha^{2}\Delta u$, thus the equation becomes,
formally,

$\displaystyle-\alpha^{2}\partial_{t}\Delta u-\nu\Delta u+\dots=0.$

Asking $\nu=O(\alpha^{2})$, means requiring that the damping coefficient does
not blow-up.

We conclude this section with few notations that will be adopted: by $C$ we
will denote several constant independent from $\nu$, $\alpha^{2}$ and
$\sigma_{k}$, perhaps changing value line by line. In the case $C$ will
depends by $\nu,\alpha$ or $\sigma_{k}$ we will add the dependence as a
subscript. Sometimes we will use the notation $a\lesssim b$, if it exists a
constant independent from $\nu$ and $\alpha^{2}$ such that $\alpha\leq Cb$. In
order to simplify the notation we will denote Sobolev spaces by $H^{s}$,
forgetting domain and range.

##  3 Well-Posedness

###  3.1 Preliminaries

Before starting with the analysis of equation (9), we need to recall some
preliminaries results on the nonlinear term in the second-grade fluid
equations, the Stokes operator $A$ and the embedding between $W$ and $V$. We
will consider the Hilbert triple

$W\hookrightarrow V\hookrightarrow W^{*}$

We start recalling in a single lemma some classical facts on the nonlinear
part of equation (1). We refer to [6],[31], [32] for the proof of the various
statements.

###### Lemma 16

For any smooth, divergence free $\phi,\ v,\ w$ the following relation holds

$\displaystyle\langle\operatorname{curl}\phi\times
v,w\rangle_{L^{2}}=b(v,\phi,w)-b(w,\phi,v).$ (16)

Moreover for $u,\ v,\ w$ the following inequalities hold

$\displaystyle\lvert\langle\operatorname{curl}(u-\alpha^{2}\Delta u)\times
v,w\rangle_{L^{2}}\rvert\leq C\lVert u\rVert_{H^{3}}\lVert v\rVert_{V}\lVert
w\rVert_{W}$ (17)
$\displaystyle\lvert\langle\operatorname{curl}(u-\alpha^{2}\Delta u)\times
u,w\rangle_{L^{2}}\rvert\leq C\lVert u\rVert^{2}_{V}\lVert w\rVert_{W}$ (18)

Therefore there exists a bilinear operator $\hat{B}:W\times V\rightarrow
W^{*}$ such that

$\displaystyle\langle\hat{B}(u,v),w\rangle_{W^{*},W}=\langle
P(\operatorname{curl}(u-\alpha^{2}\Delta u)\times v),w\rangle$ (19)

which satisfies for $u\in V,\ v\in W$

$\displaystyle\lVert\hat{B}(v,u)\rVert_{W^{*}}\leq C\lVert u\rVert_{V}\lVert
v\rVert_{W}$ (20) $\displaystyle\lVert\hat{B}(u,u)\rVert_{W^{*}}\leq C\lVert
u\rVert_{V}^{2}.$ (21)

Lastly, for $u\in W,\ v\in V,\ w\in W$

$\displaystyle\langle\hat{B}(u,v),w\rangle_{W^{*},W}=-\langle\hat{B}(u,w),v\rangle_{W^{*},W}.$
(22)

We need a basis orthonormal either in $W$ and in $V$ in order to deal with the
Galerkin approximation of equation (40). The existence of such basis is
guaranteed by the lemma below. The first part is a consequence of spectral
theorem for self-adjoint compact operators stated in [33], we refer to [5] for
the proof of the second part.

###### Lemma 17

The injection of $W$ into $V$ is compact. Let $I$ be the isomorphism of
$W^{*}$ onto $W$, then the restriction of $I$ to $V$ is a continuous compact
operator into itself. Thus, there exists a sequence $e_{i}$ of elements of $W$
which forms an orthonormal basis in $W$, and an orthogonal basis in $V$. This
sequence verifies:

$\displaystyle\textit{for any }v\in W\ \ \langle
v,e_{i}\rangle_{W}=\lambda_{i}\langle v,e_{i}\rangle_{V}$ (23)

where $\lambda_{i+1}>\lambda_{i}>0,\ i=1,2,\dots$. Thus
$\sqrt{\lambda_{i}}e_{i}$ is an orthonormal basis of $V$. Moreover $e_{i}$
belong to $H^{4}(D;\mathbb{R}^{2})$.

We will use also some properties of the projection operator $P$ and the
solution map of the Stokes operator. We refer to [40] for the proof of the
lemmas below.

###### Lemma 18

The restriction of the projection operator
$P:L^{2}(D;\mathbb{R}^{2})\rightarrow H$ to $H^{r}(D;\mathbb{R}^{2})$ is a
continuous and linear map between $H^{r}(D;\mathbb{R}^{2})$ and itself.

###### Lemma 19

Let $f\in H^{m}(D;\mathbb{R}^{2})$. Then, there exists a unique couple
$(u,p)$, with $p$ defined up to an additive constant, solution of

$\displaystyle u-\alpha^{2}\Delta u+\nabla p=f$
$\displaystyle\operatorname{div}u=0$ $\displaystyle u|_{\partial D}=0.$

Moreover $u=(I-\alpha^{2}A)^{-1}f\in H^{m+2}(D;\mathbb{R}^{2}),\ p\in
H^{m+1}(D)$,

$\displaystyle\lVert u\rVert_{H^{m+2}}+\lVert p\rVert_{H^{m+1}}\leq C\lVert
f\rVert_{H^{m}}.$

###### Lemma 20

The injection of $V$ in $H$ is compact. Thus there exists a sequence
$\tilde{e}_{i}$ of elements of $H$ which forms an orthonormal basis in $H$ and
an orthogonal basis in $V$. This sequence verifies

$\displaystyle-A\tilde{e}_{i}=\tilde{\lambda}_{i}\tilde{e}_{i}$

where $\tilde{\lambda}_{i+1}>\tilde{\lambda}_{i}>0,\ i=1,2,\dots$. Moreover
$\tilde{\lambda}_{i}\rightarrow+\infty$. Lastly $\tilde{e}_{i}\in
C^{\infty}(\overline{D};\mathbb{R}^{2})$ under our assumptions on $D$

Combining Lemma 17 and Lemma 19 above, it follows that for each $f\in H^{1}$,
$i\in\mathbb{N}$

$\displaystyle\langle(I-\alpha^{2}A)^{-1}f,e_{i}\rangle_{W}=\lambda_{i}\langle(I-\alpha^{2}A)^{-1}f,e_{i}\rangle_{V}=\lambda_{i}\langle
f,e_{i}\rangle.$ (24)

Moreover, Lemmas 18, 19, 20 above allow us to prove some useful estimates that
will be exploited along the paper. We will need Corollary 22 in order to
evaluate the regularity of the linear operators appearing in equation (40).
Instead we will need Lemma 21 in order to quantify explicitly the dependence
from $\alpha$ in several embeddings and operators. This will be crucial in
Section 4 and Section 5.
We recall first that by Poincaré inequality, equation (4), triangle inequality
and equation (5) the following relations hold:

$\displaystyle\lVert u\rVert_{H^{3}}^{2}\leq C(\lVert\nabla
u\rVert_{L^{2}}^{2}+\lVert\operatorname{curl}(u-\Delta
u)\rVert_{L^{2}}^{2})\leq
C\left(\frac{\alpha^{2}+1}{\alpha^{2}}\right)^{2}\lVert\nabla
u\rVert_{L^{2}}^{2}+\frac{C}{\alpha^{4}}\lVert\operatorname{curl}(u-\alpha^{2}\Delta
u)\rVert_{L^{2}}^{2}$ (25) $\displaystyle\lVert\nabla
u\rVert_{L^{2}}^{2}\leq\frac{\lVert u\rVert_{V}^{2}}{\alpha^{2}}.$ (26)

###### Lemma 21

Let $h\in H,\ u\in V,\ w\in V\cap H^{2}$, then

$\displaystyle\lVert G^{k}(u)\rVert$
$\displaystyle\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\nabla
u\rVert_{L^{2}}\leq\lVert\sigma\rVert_{L^{\infty}}\frac{\lVert
u\rVert_{V}}{\alpha},$ (27) $\displaystyle\lVert\nabla G^{k}(w)\rVert_{L^{2}}$
$\displaystyle\leq C\lVert\sigma_{k}\rVert_{W^{1,\infty}}{\lVert
w\rVert_{H^{2}}},$ (28) $\displaystyle\lVert(I-\alpha^{2}A)^{-1}h\rVert$
$\displaystyle\leq\lVert h\rVert,$ (29)
$\displaystyle\lVert(-A)^{1/2}(I-\alpha^{2}A)^{-1}h\rVert$
$\displaystyle\leq\frac{1}{2\alpha}\lVert h\rVert,$ (30)
$\displaystyle\lVert-A(I-\alpha^{2}A)^{-1}h\rVert$
$\displaystyle\leq\frac{1}{\alpha^{2}}\lVert h\rVert,$ (31)
$\displaystyle\lVert(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\rVert_{W}$ $\displaystyle\leq C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert
w\rVert_{H^{2}}.$ (32)

Therefore, if $u\in V$ the following inequalities hold true

$\displaystyle\lVert(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla u)\rVert$
$\displaystyle\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\nabla
u\rVert_{L^{2}}\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\frac{\lVert
u\rVert_{V}}{\alpha},$ (33)
$\displaystyle\lVert\nabla(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)\rVert_{L^{2}}$
$\displaystyle\leq\frac{\lVert\sigma_{k}\rVert_{L^{\infty}}}{2\alpha}\lVert\nabla
u\rVert_{L^{2}},$ (34) $\displaystyle\lVert
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)))\rVert$
$\displaystyle\leq\frac{\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}}{2\alpha}\lVert\nabla
u\rVert_{L^{2}}.$ (35) $\displaystyle\lVert
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)))\rVert_{H^{1}}$
$\displaystyle\leq\frac{C\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert\nabla
u\rVert_{L^{2}}}{\alpha^{2}}.$ (36)

Proof. Inequalities (27), (28) are trivial. Indeed, by Lemma 18 it holds

$\displaystyle\lVert G^{k}(u)\rVert=\lVert P(\sigma_{k}\cdot\nabla
u)\rVert\leq\lVert\sigma_{k}\cdot\nabla
u\rVert_{L^{2}}\leq\lVert\sigma_{k}\rVert_{L^{\infty}}{\lVert\nabla
u\rVert_{L^{2}}}\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\frac{\lVert
u\rVert_{V}}{\alpha}.$ $\displaystyle\lVert\nabla G^{k}(w)\rVert_{L^{2}}$
$\displaystyle=\lVert\nabla P(\sigma_{k}\cdot\nabla w)\rVert_{L^{2}}$
$\displaystyle\leq C\lVert\sigma_{k}\cdot\nabla w\rVert_{H^{1}}$
$\displaystyle\leq C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert\nabla
w\rVert_{H^{1}}$ $\displaystyle\leq
C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert w\rVert_{H^{2}}.$

In order to prove inequalities (29), (30), (31) we exploit the Fourier
decomposition $h=\sum_{i\in\mathbb{N}}\langle
h,\tilde{e}_{i}\rangle\tilde{e}_{i}$. Therefore it holds

$\displaystyle\lVert(I-\alpha^{2}A)^{-1}h\rVert^{2}=\sum_{i\in\mathbb{N}}\frac{\langle
h,\tilde{e}_{i}\rangle^{2}}{(1+\alpha^{2}\tilde{\lambda}_{i})^{2}}\leq\lVert
h\rVert^{2},$
$\displaystyle\lVert(-A)^{1/2}(I-\alpha^{2}A)^{-1}h\rVert^{2}=\sum_{i\in\mathbb{N}}\frac{\tilde{\lambda}_{i}}{(1+\alpha^{2}\tilde{\lambda}_{i})}\langle
h,\tilde{e}_{i}\rangle^{2}\leq\frac{1}{4\alpha^{2}}\lVert h\rVert^{2},$
$\displaystyle\lVert-A(I-\alpha^{2}A)^{-1}h\rVert=\sum_{i\in\mathbb{N}}\frac{\tilde{\lambda}_{i}^{2}}{(1+\alpha^{2}\tilde{\lambda}_{i})}\langle
h,\tilde{e}_{i}\rangle^{2}\leq\frac{1}{\alpha^{4}}\lVert h\rVert^{2}.$

For what concerns inequality (32), by definition of the norm in the space $W$
it holds

$\displaystyle\lVert(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\rVert_{W}^{2}=\lVert(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\rVert_{V}^{2}+\lVert\operatorname{curl}((I-\alpha^{2}\Delta)(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w)))\rVert_{L^{2}}^{2}.$

From Lemma 19, we know that

$\displaystyle\lVert(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\rVert_{V}^{2}$ $\displaystyle=\langle P(\sigma_{k}\cdot\nabla
w),(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla w))\rangle$
$\displaystyle\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\nabla
w\rVert_{L^{2}}\lVert(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\rVert_{V}$
$\displaystyle\leq\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\lVert\nabla
w\rVert_{L^{2}}^{2},$ (37)
$\displaystyle\lVert\operatorname{curl}\left((I-\alpha^{2}\Delta)(I-\alpha^{2}A)^{-1}(P(\sigma_{k}\cdot\nabla
w))\right)\rVert_{L^{2}}=\lVert\operatorname{curl}(P(\sigma_{k}\cdot\nabla
w))\rVert_{L^{2}}\leq C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert
w\rVert_{H^{2}}$ (38)

Combining (3.1) and (38), inequality (32) follows.

Combining relation (27) with relations (29) and (30), inequalities (33) and
(34) follow immediately. Let us now prove equation (35). By Hölder’s
inequality and relation (28) we have

$\displaystyle\lVert
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)))\rVert$
$\displaystyle\leq\lVert\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))\rVert_{L^{2}}$
$\displaystyle\leq\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))\rVert_{L^{2}}$
$\displaystyle\leq\frac{\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}}{2\alpha}\lVert\nabla
u\rVert_{L^{2}}.$

For what concerns the last one, by Lemma 18, 19 and relations (31) it holds

$\displaystyle\lVert
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)))\rVert_{H^{1}}$ $\displaystyle\leq
C\lVert\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))\rVert_{H^{1}}$ $\displaystyle\leq
C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))\rVert_{H^{1}}$ $\displaystyle\leq
C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u)\rVert_{H^{2}}$ $\displaystyle\leq
C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert
A(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla u)\rVert$
$\displaystyle\leq\frac{C}{\alpha^{2}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert
P(\sigma_{k}\cdot\nabla u)\rVert$
$\displaystyle\leq\frac{C\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert\nabla
u\rVert_{L^{2}}}{\alpha^{2}}.$

###### Corollary 22

$\displaystyle G^{k}\in\mathcal{L}(V;H),\quad F\in\mathcal{L}(V;H\cap
H^{1}(D;\mathbb{R}^{2})).$

In particular

$\displaystyle\lVert G^{k}(u)\rVert$
$\displaystyle\leq\lVert\sigma\rVert_{L^{\infty}}\frac{\lVert
u\rVert_{V}}{\alpha},$ (39) $\displaystyle\lVert F(u)\rVert_{H}$
$\displaystyle\leq\frac{1}{2\alpha}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\lVert\nabla u\rVert_{L^{2}}$ (40)
$\displaystyle\lVert F(u)\rVert_{H^{1}}$
$\displaystyle\leq\frac{C}{\alpha^{2}}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert\nabla
u\rVert_{L^{2}}.$ (41)

Lastly we recall two technical tools used in the proof of Theorem 9. We refer
to [14] for the proof of the interpolation inequality and to [36] for the
proof of the stochastic Grönwall’s Lemma.

###### Theorem 23

Each function $f\in H^{2}$ satisfies the following inequality:

$\displaystyle\lVert f\rVert_{H^{1}}\leq C\lVert f\rVert_{L^{2}}^{1/2}\lVert
f\rVert_{H^{2}}^{1/2}.$ (42)

###### Theorem 24

Let $Z(t)$ and $H(t)$ be continuous, nonnegative, adapted processes, $\psi(t)$
a nonnegative deterministic function and $M(t)$ a continuous local martingale
such that

$\displaystyle Z(t)\leq\int_{0}^{t}\psi(s)Z(s)ds+M(t)+H(t)\quad\forall
t\in[0,T].$

Then $Z(t)$ satisfies the following inequality

$\displaystyle\mathbb{E}[Z(t)]\leq
exp\left(\int_{0}^{t}\psi(s)ds\right)\mathbb{E}\left[\operatorname{sup}_{r\in[0,s]}H(s)\right].$
(43)

###  3.2 Galerkin Approximation and Limit Equations

Let $W^{N}=\operatorname{span}\\{e_{1},\dots,\ e_{N}\\}\subseteq W$ and
$P^{N}:W\rightarrow W^{N}$ the orthogonal projector. We start looking for a
finite dimension approximation of the solution of equation (9). We define

$u^{N}(t)=\sum_{i=1}^{N}c_{i,N}(t)e_{i}.$

The $c_{i,N}$ have been chosen in order to satisfy $\forall e_{i},\ \ 1\leq
i\leq N$

$\displaystyle\langle u^{N}(t),e_{i}\rangle_{V}-\langle
u^{N}_{0},e_{i}\rangle_{V}$ $\displaystyle=\nu\int_{0}^{t}\langle\nabla
u^{N}(s),\nabla
e_{i}\rangle_{L^{2}}ds-\int_{0}^{t}b(u^{N}(s),u^{N}(s)-\alpha^{2}\Delta
u^{N}(s),e_{i})ds$ $\displaystyle-\alpha^{2}\int_{0}^{t}b(e_{i},\Delta
u^{N}(s),u^{N}(s))ds+\int_{0}^{t}\langle F^{N}(s),e_{i}\rangle ds$
$\displaystyle+\sum_{k\in K}\int_{0}^{t}\langle G^{k,N}(s),e_{i}\rangle
dW^{k}_{s}\ \ \mathbb{P}-a.s.$

where $u_{0}^{N}=\sum_{i=1}^{N}\langle u_{0},e_{i}\rangle_{W}e_{i}$,
$F^{N}(s)=F(u^{N}(s))$ and $G^{k,N}(s)=G^{k}(u^{N}(s))$. The local well-
posedness of this equation follows from classical results about stochastic
differential equations with locally Lipshitz coefficients, see for example
[18],[38]. The global well-posedness follows from the a priori estimates in
Lemma 25, 26.

###### Lemma 25

Assuming Hypothesis 2, the following relations hold:

  * •

The Itô’s formula

$\displaystyle d\lVert u^{N}\rVert_{V}^{2}=-2\nu\lVert\nabla
u^{N}\rVert_{L^{2}}^{2}dt-\sum_{k\in
K}b(\sigma_{k},u^{N},(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}))dt.$ (44)

  * •

The inequality below holds uniformly in $N$

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{N}(t)\rVert_{V}^{p}\right]\leq C_{p,\alpha,\\{\sigma_{k}\\}_{k\in K}},\ \
\forall p\geq 1.$ (45)

Proof. If we apply the Itô’s formula to $\sum_{i=1}^{N}\lambda_{i}\langle
u^{N}(t),e_{i}\rangle_{V}^{2}$, we get

$\displaystyle\lVert u^{N}(t)\rVert_{V}^{2}+2\nu\int_{0}^{t}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds$ $\displaystyle=\lVert
u^{N}_{0}\rVert_{V}^{2}+2\int_{0}^{t}\langle F^{N}(s),u^{N}(s)\rangle
ds+\sum_{i=1}^{N}\sum_{k\in K}\lambda_{i}\int_{0}^{t}\langle
G^{k,N}(s),e_{i}\rangle^{2}ds$
$\displaystyle+2\sum_{k=1}^{N}\int_{0}^{t}\langle G^{k,N}(s),u^{N}(s)\rangle
dW^{k}_{s}.$ (46)

In the last relation we exploited the fact that
$b(u^{N}(s),u^{N}(s),u^{N}(s))=b(u^{N}(s),\Delta u^{N}(s),u^{N}(s))=0$. Now we
observe that for each $k$, $\langle G^{k,N}(s),u^{N}(s)\rangle=0$. In fact

$\displaystyle\langle
G^{k,N}(s),u^{N}(s)\rangle=b(\sigma_{k},u^{N}(s),u^{N}(s))=0.$ (47)

Moreover we have

$\displaystyle 2\int_{0}^{t}\langle F^{N}(s),u^{N}(s)\rangle
ds+\sum_{i=1}^{N}\sum_{k\in K}\lambda_{i}\int_{0}^{t}\langle
G^{k,N}(s),e_{i}\rangle^{2}ds=-\sum_{k\in
K}b(\sigma_{k},u^{N}(s),(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))).$ (48)

In fact,

$\displaystyle\sum_{i=1}^{N}\lambda_{i}\langle G^{k,N}(s),e_{i}\rangle^{2}\
=b(\sigma_{k},u^{N}(s),\sum_{i=1}^{N}\lambda_{i}e_{i}b(\sigma_{k},u^{N}(s),e_{i})).$

It remains to show that

$\displaystyle
b(\sigma_{k},u^{N}(s),\sum_{i=1}^{N}\lambda_{i}e_{i}b(\sigma_{k},u^{N}(s),e_{i}))+b(\sigma_{k},(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))),u^{N})=$
$\displaystyle-b(\sigma_{k},u^{N}(s),(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))).$

Thus it is enough to show that
$\langle\sum_{i=1}^{N}\lambda_{i}e_{i}b(\sigma_{k},u^{N}(s),e_{i}),v\rangle_{V}=\langle(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))),v\rangle_{V}$ for all $v\in V_{N}$, where
$V_{N}=\operatorname{span}\\{e_{i}\\}_{i=1}^{N}$. The last claim is true, in
fact

$\displaystyle\langle(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))),v\rangle_{V}=b(\sigma_{k},u^{N}(s),v)$ $\displaystyle
b(\sigma_{k},u^{N}(s),\sum_{i=1}^{N}\lambda_{i}e_{i}\langle
e_{i},v\rangle_{v})=\langle\sum_{i=1}^{N}\lambda_{i}e_{i}b(\sigma_{k},u^{N}(s),e_{i}),v\rangle_{V}.$

Therefore, combining equation (3.2) and equations (47), (48) we obtain

$\displaystyle\lVert u^{N}(t)\rVert_{V}^{2}+2\nu\int_{0}^{t}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds$ $\displaystyle=\lVert
u^{N}_{0}\rVert_{V}^{2}-\sum_{k\in
K}\int_{0}^{t}b(\sigma_{k},u^{N}(s),(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s)))ds$ $\displaystyle\leq\lVert u_{0}\rVert_{V}^{2}+\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\int_{0}^{t}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}\lVert(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))\rVert ds$ $\displaystyle\leq\lVert u_{0}\rVert_{V}^{2}+\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\int_{0}^{t}\lVert
u^{N}(s)\rVert_{V}\lVert(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))\rVert_{V}ds$ $\displaystyle\leq\lVert u_{0}\rVert_{V}^{2}+\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\int_{0}^{t}\lVert
u^{N}(s)\rVert_{V}\lVert P(\sigma_{k}\cdot\nabla u^{N}(s))\rVert ds$
$\displaystyle\leq\lVert u_{0}\rVert_{V}^{2}+\frac{1}{\alpha}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert
u^{N}(s)\rVert_{V}^{2}ds.$

Thus, by Grönwall,

$\displaystyle\operatorname{sup}_{t\in[0,T]}\lVert u^{N}(t)\rVert_{V}^{2}\leq
C_{\alpha,\\{\sigma_{k}\\}_{k\in K}}\lVert u_{0}\rVert_{V}^{2}.$ (49)

Taking the expected value of equation (49) we get the thesis for $p\leq 2$. If
$p>2$, raising to the power $p/2$ both sides of equation (49) the thesis
follows easily.

###### Lemma 26

Assuming Hypothesis 2, the following relation holds:

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{N}(t)\rVert_{W}^{p}\right]\leq C_{p,\nu,\alpha,\\{\sigma_{k}\\}_{k\in K}},\
\ \forall p\geq 1$ (50)

where $C_{p,\nu,\alpha,\\{\sigma_{k}\\}_{k\in K}}$ is a constant independent
from $N$.

Proof. This proof is similar to Lemma 2.4-2.5 of [31]. We will need some
changes due to the poor regularity of the coefficients $F$ and $G^{k}$. In the
part where we will not need any changes, we will refer to the equations in
[31]. Let

$\displaystyle\tau_{M}^{N}=\inf\\{t:\ \lVert u^{N}(t)\rVert_{V}+\lVert
u^{N}(t)\rVert_{*}\geq M\\}\wedge T$

and $\tilde{G}^{k,N}=(I-\alpha^{2}A)^{-1}G^{k,N}$ the solution of Stokes
problem defined in Lemma 19. From the regularity of the eigenvectors $e_{i}$,
$G^{k,N}\in H^{1}$, thus $\tilde{G}^{k,N}\in W$ and by equations (24) and (32)
the following relations hold true

$\displaystyle\langle\tilde{G}^{k,N},e_{i}\rangle_{W}$
$\displaystyle=\lambda_{i}\langle G^{k,N},e_{i}\rangle$ (51)
$\displaystyle\lVert\tilde{G}^{k,N}\rVert_{W}$ $\displaystyle\leq
C\lVert\sigma_{k}\rVert_{W^{1,\infty}}\lVert u^{N}\rVert_{H^{2}}.$ (52)

Let us call

$\phi^{N}=-\nu\Delta u^{N}+\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\times u^{N}-F^{N}.$

From the regularity of the $e_{i}$, we have that $\phi^{N}\in H^{1}$. Thus we
can find a $v^{N}\in W$ such that $v^{N}=(I-\alpha^{2}A)^{-1}\phi^{N}$. We
rewrite shortly the weak formulation satisfied by $u^{N}$

$\displaystyle d\langle u^{N},e_{i}\rangle_{V}+\langle\phi^{N},e_{i}\rangle
dt=d\langle u^{N},e_{i}\rangle_{V}+\langle v^{N},e_{i}\rangle_{V}dt=\sum_{k\in
K}\langle G^{k,N},e_{i}\rangle dW^{k}_{t}$

Multiplying each equation by $\lambda_{i}$ we get

$\displaystyle d\langle u^{N},e_{i}\rangle_{W}+\langle
v^{N},e_{i}\rangle_{W}dt=\sum_{k\in
K}\langle\tilde{G}^{k,N},e_{i}\rangle_{W}dW^{k}_{t}.$

Now we apply the Itô’s formula to $\sum_{i=1}^{N}\langle
u^{N},e_{i}\rangle_{W}^{2}$ and we obtain

$\displaystyle d(\lVert u^{N}\rVert_{V}^{2}+\lVert
u^{N}\rVert_{*}^{2})+2\left(\langle
v^{N},u^{N}\rangle_{V}+\langle\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N}),\operatorname{curl}(v^{N}-\alpha^{2}\Delta
v^{N})\rangle_{L^{2}}\right)dt=$ $\displaystyle 2\sum_{k\in
K}\langle\operatorname{curl}(\tilde{G}^{k,N}-\alpha^{2}\Delta\tilde{G}^{k,N}),\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}dW^{k}_{t}$ $\displaystyle+\sum_{k\in
K}\sum_{i=1}^{N}\lambda_{i}^{2}\langle\tilde{G}^{k,N},e_{i}\rangle_{V}^{2}\
dt+2\sum_{k\in K}\langle\tilde{G}^{k,N},u^{N}\rangle_{V}dW^{k}_{t}.$

Exploiting the definition of $v^{N},\ \tilde{G}^{k,N}$, equation (47) and the
classical fact that $\operatorname{curl}\nabla=0$ we get

$\displaystyle d(\lVert u^{N}\rVert_{V}^{2}+\lVert
u^{N}\rVert_{*}^{2})+2\left(\langle\phi^{N},u^{N}\rangle+\langle\operatorname{curl}(\phi^{N}),\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}\right)dt=$ $\displaystyle 2\sum_{k\in
K}\langle\operatorname{curl}({G}^{k,N}),\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}dW^{k}_{t}+\sum_{k\in
K}\sum_{i=1}^{N}\lambda_{i}^{2}\langle{G}^{k,N},e_{i}\rangle^{2}\ dt.$ (53)

From Lemma 25 we already know that

$\displaystyle d\lVert u^{N}\rVert_{V}^{2}=-2\nu\lVert\nabla
u^{N}\rVert_{L^{2}}^{2}dt-\sum_{k\in
K}b(\sigma_{k},u^{N},(I-P^{N})(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}))dt.$

Substituting this relation in the Itô’s formula (3.2) we get

$\displaystyle d(\lVert
u^{N}\rVert_{*}^{2})+2\left(\langle\operatorname{curl}(\phi^{N}),\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}\right)dt=$ $\displaystyle 2\sum_{k\in
K}\langle\operatorname{curl}({G}^{k,N}),\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}dW^{k}_{t}+\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\langle{G}^{k,N},e_{i}\rangle^{2}\
dt.$ (54)

Analogously to equation (4.48) in [31], the relation below holds true

$\displaystyle\langle\operatorname{curl}\phi^{N},\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}=\frac{\nu}{\alpha^{2}}\lVert
u^{N}\rVert_{*}^{2}-\frac{\nu}{\alpha^{2}}\langle\operatorname{curl}u^{N}+\frac{\alpha^{2}}{\nu}\operatorname{curl}F^{N},\operatorname{curl}(u^{N}-\alpha^{2}\Delta
u^{N})\rangle_{L^{2}}.$

Using this relation in the Itô’s formula (3.2) and integrating between $0$ and
$t\leq\tau_{M}^{N}$ we get

$\displaystyle\lVert
u^{N}(t)\rVert_{*}^{2}+\frac{2\nu}{\alpha^{2}}\int_{0}^{t}\lVert
u^{N}(s)\rVert_{*}^{2}-\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds$
$\displaystyle=\lVert
u^{N}_{0}\rVert_{*}^{2}+\int_{0}^{t}\frac{2\nu}{\alpha^{2}}\langle\operatorname{curl}u^{N}(s)+\frac{\alpha^{2}}{\nu}\operatorname{curl}F^{N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}ds$ $\displaystyle+2\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}$ $\displaystyle\leq\lVert
u^{N}_{0}\rVert_{*}^{2}+\int_{0}^{t}\frac{2\nu}{\alpha^{2}}\lVert\operatorname{curl}u^{N}(s)\rVert_{L^{2}}\lVert
u^{N}(s)\rVert_{*}ds+2\int_{0}^{t}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}\lVert
u^{N}(s)\rVert_{*}ds$ $\displaystyle+2\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert.$ (55)

Taking the supremum between $0$ and $r\wedge\tau_{M}^{N}$ in relation (3.2)
and, then, the expected value we get

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\frac{2\nu}{\alpha^{2}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle\leq 2\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{4\nu}{\alpha^{2}}\lVert\operatorname{curl}u^{N}(s)\rVert_{L^{2}}\lVert
u^{N}(s)\rVert_{*}ds\right]+4\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}\lVert
u^{N}(s)\rVert_{*}ds\right]$
$\displaystyle+4\mathbb{E}\left[\left\lvert\sum_{k\in
K}\int_{0}^{r\wedge\tau_{M}^{N}}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$ $\displaystyle\leq
2\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+4\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\left(\frac{2\nu\epsilon_{1}}{\alpha^{2}}+2\epsilon_{2}\right)\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{2\nu}{\alpha^{2}\epsilon_{1}}\lVert\operatorname{curl}u^{N}(s)\rVert_{L^{2}}^{2}ds\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{2}{\epsilon_{2}}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}^{2}ds\right]+2\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right].$
(56)

Choosing $\epsilon_{1}=\frac{1}{4}$ and $\epsilon_{2}=\frac{\nu}{4\alpha^{2}}$
we arrive at

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\frac{\nu}{\alpha^{2}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle\leq 2\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+4\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{8\nu}{\alpha^{2}}\lVert\operatorname{curl}u^{N}(s)\rVert_{L^{2}}^{2}ds\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{8\alpha^{2}}{\nu}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}^{2}ds\right]+2\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right]$
(57)

From equations (48) and (33) we know that

$\displaystyle\sum_{k\in
K}\sum_{i=1}^{N}\lambda_{i}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right]=$
$\displaystyle\sum_{k\in
K}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}b(\sigma_{k},u^{N}(s),P^{N}(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s)))ds\right]$ $\displaystyle\leq\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}\lVert(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u^{N}(s))\rVert_{V}ds\right]$ $\displaystyle\leq\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}\lVert(I-\alpha^{2}A)^{-1/2}P(\sigma_{k}\cdot\nabla
u^{N}(s))\rVert ds\right]$ $\displaystyle\leq\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds\right].$ (58)

Thanks to equations (51), (52), the interpolation estimate (42) and relation
(25) we have

$\displaystyle\sum_{k\in
K}\sum_{i=1}^{N}\lambda_{i}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right]=$
$\displaystyle\sum_{k\in
K}\sum_{i=1}^{N}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle\tilde{G}^{k,N}(s),e_{i}\rangle_{W}^{2}ds\right]$
$\displaystyle\leq\sum_{k\in
K}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\tilde{G}^{k,N}(s)\rVert_{W}^{2}ds\right]$
$\displaystyle\leq C\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{H^{2}}^{2}ds\right]$ $\displaystyle\leq C\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}\lVert u^{N}(s)\rVert_{H^{3}}ds\right]$
$\displaystyle\leq C\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}\left(\frac{\alpha^{2}+1}{\alpha^{2}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}+\frac{1}{\alpha^{2}}\lVert
u^{N}(s)\rVert_{*}\right)ds\right]$ $\displaystyle\leq\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}C\frac{\alpha^{2}+1+\frac{1}{\epsilon}}{\alpha^{2}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}+\frac{\epsilon}{\alpha^{2}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right].$ (59)

Thanks to Burkholder-Davis-Gundy inequality, equation (28), the interpolation
inequality (42) and relation (25) we get

$\displaystyle 4\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$ $\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{k\in
K}\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}G^{k,N}(s)\rVert_{L^{2}}^{2}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right)^{1/2}\right]$ $\displaystyle\leq
C\mathbb{E}\left[\operatorname{sup}_{t\leq r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)(s)\rVert_{*}\left(\sum_{k\in
K}\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}G^{k,N}(s)\rVert_{L^{2}}^{2}ds\right)^{1/2}\right]$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+C\mathbb{E}\left[\sum_{k\in
K}\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}G^{k,N}(s)\rVert_{L^{2}}^{2}ds\right]$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]+C\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{H^{2}}^{2}ds\right]$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]+\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}C\frac{\alpha^{2}+1+\frac{1}{\epsilon}}{\alpha^{2}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}+\frac{\epsilon}{\alpha^{2}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right].$ (60)

Lastly, thanks to equation (41) we have

$\displaystyle\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\frac{8\alpha^{2}}{\nu}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}^{2}ds\right]$
$\displaystyle\leq\frac{C}{\nu\alpha^{2}}\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\right)^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds\right].$ (61)

Combining estimates (3.2),(3.2),(3.2),(61) above we obtain

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\frac{\nu-\epsilon\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}}{\alpha^{2}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle\leq
C\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\frac{\alpha^{2}+1+\frac{1}{\epsilon}}{\alpha^{2}}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}\right]\right)$
$\displaystyle+\frac{C}{\nu\alpha^{2}}\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\right)^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds\right].$ (62)

Therefore, choosing $\epsilon$ small enough, by equation (49) we have

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\frac{\nu}{2\alpha^{2}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]\leq C_{\nu,\alpha,\\{\sigma_{k}\\}_{k\in K}}\
\text{indipendent from }M,\ N.$

Last inequality proves the Lemma for $p=2$, letting $M$ to $+\infty$ thanks to
monotone convergence Theorem. Now we consider $p\geq 4$ and we restart from
equation (4.79) in [31].

$\displaystyle\lVert u^{N}(t)\rVert_{*}^{p}$ $\displaystyle\leq C\lVert
u^{N}_{0}\rVert_{*}^{p}+C\bigg{(}\int_{0}^{t}\lVert
u^{N}(s)\rVert_{*}^{p/2-2}$
$\displaystyle\times\bigg{(}2\langle\operatorname{curl}F^{N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}$ $\displaystyle+\frac{1}{2}\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\langle
G^{k,N}(s),e_{i}\rangle^{2}$
$\displaystyle+\frac{2\nu}{\alpha^{2}}\langle\operatorname{curl}u^{N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}-\frac{2\nu}{\alpha^{2}}\lVert u^{N}(s)\rVert_{*}^{2}$
$\displaystyle+\frac{p-4}{p}\sum_{k\in
K}\frac{\langle\operatorname{curl}G^{k,N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}^{2}}{\lVert
u^{N}(s)\rVert_{*}^{2}}\bigg{)}ds\bigg{)}^{2}$
$\displaystyle+\left(\int_{0}^{t}\lVert u^{N}(s)\rVert_{*}^{p/2-2}\sum_{k\in
K}\langle\operatorname{curl}G^{k,N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right)^{2}.$

Let us consider all the terms, one by one. Arguing as before we have

$\displaystyle\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\langle{G}^{k,N}(s),e_{i}\rangle^{2}\leq
C_{\epsilon,\alpha,\\{\sigma_{k}\\}_{k\in K}}\lVert
u^{N}(s)\rVert_{V}^{2}+\epsilon\lVert u^{N}(s)\rVert_{*}^{2},$
$\displaystyle\lvert\langle\operatorname{curl}u^{N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}\rvert$ $\displaystyle\leq C_{\alpha}(1+\lVert
u^{N}(s)\rVert_{V})(1+\lVert u^{N}(s)\rVert_{W}),$
$\displaystyle\lvert\langle\operatorname{curl}F^{N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}\leq C_{\alpha,\\{\sigma_{k}\\}_{k\in K}}(1+\lVert
u^{N}(s)\rVert_{V})(1+\lVert u^{N}(s)\rVert_{W}),$
$\displaystyle\left\lvert\frac{\langle\operatorname{curl}G^{k,N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}^{2}}{\lVert u^{N}(s)\rVert_{*}^{2}}\right\rvert$
$\displaystyle\leq\lVert\operatorname{curl}G^{k,N}(s)\rVert_{L^{2}}^{2}$
$\displaystyle\leq C_{\epsilon,\alpha,\\{\sigma_{k}\\}_{k\in K}}(1+\lVert
u^{N}(s)\rVert_{V})^{2}+\epsilon\rVert u^{N}(s)\rVert_{W}^{2}.$

Exploiting the relations above and the continuous embedding $W\hookrightarrow
V$ we get

$\displaystyle\lVert u^{N}(t)\rVert_{*}^{p}$ $\displaystyle\leq C\lVert
u^{N}_{0}\rVert_{*}^{p}+C_{\epsilon,\nu,\alpha,\\{\sigma_{k}\\}_{k\in
K}}\left(\int_{0}^{t}\lVert u^{N}(s)\rVert_{*}^{p/2-2}(1+\lVert
u^{N}(s)\rVert_{W})^{2}ds\right)^{2}$ $\displaystyle+\left(\int_{0}^{t}\lVert
u^{N}(s)\rVert_{*}^{p/2-2}\sum_{k\in
K}\langle\operatorname{curl}G^{k,N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right)^{2}.$

Thus taking the supremum in time for $t\leq r$ and the expected value of this
we get the thesis via Grownall’s Lemma arguing exactly as in the proof of
Lemma 4.3 in [31] and exploiting previous estimate (3.2) on
$\langle\operatorname{curl}G^{k,N}(s),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}$.

###### Remark 27

In case of $\nu=0$, arguing as above we get

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]$ $\displaystyle\leq
2\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$
$\displaystyle+4\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$
$\displaystyle+4\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}^{2}ds\right]+2\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right].$
(63)

Therefore, thanks to Lemma 25 and estimates (3.2),(3.2),(3.2),(61) we obtain

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]$
$\displaystyle\leq\frac{\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}}{\alpha^{2}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle+C\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\frac{\alpha^{2}+1}{\alpha^{2}}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}\right]\right)$
$\displaystyle+\frac{C}{\alpha^{4}}\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}\lVert\sigma_{k}\rVert_{W^{1,\infty}}\right)^{2}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds\right]$ $\displaystyle\leq
C_{\alpha,\\{\sigma_{k}\\}_{k\in K}}+C_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle\leq
C_{\alpha,\\{\sigma_{k}\\}_{k\in K}}+C_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\int_{0}^{r}\mathbb{E}\left[\lVert
u^{N}(s)\rVert_{*}^{2}1_{[0,\tau_{M}^{N}]}(s)ds\right]ds.$ (64)

Since $\mathbb{E}\left[\operatorname{sup}_{t\leq r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]=\mathbb{E}\left[\operatorname{sup}_{t\leq
r}\lVert u^{N}(t)\rVert_{*}^{2}1_{[0,\tau_{M}^{N}]}(t)\right],$ by Grönwall’s
Lemma

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T\wedge\tau_{M}^{N}]}\lVert
u^{N}(t)\rVert_{W}^{2}\right]\leq C_{\alpha,\\{\sigma_{k}\\}_{k\in K}}\textit{
indipendent from }M,\ N.$

Last inequality proves the Lemma for $p=2$, letting $M$ to $\infty$ thanks to
monotone convergence Theorem. The case $p\geq 4$ can be treated as in the case
$\nu>0$, therefore we do not add other details.

Let us now introduce the operator $\hat{A}=(I-\alpha^{2}A)^{-1}A$. By Lemma 16
and Lemma 19 the weak formulation satisfied by the Galerkin approximations can
be rewritten as

$\displaystyle\langle u^{N}(t),e_{i}\rangle_{V}-\langle
u^{N}_{0},e_{i}\rangle_{V}$
$\displaystyle=\nu\int_{0}^{t}\langle\hat{A}u^{N}(s),e_{i}\rangle_{V}ds-\int_{0}^{t}\langle\hat{B}(u^{N}(s),u^{N}(s)),e_{i}\rangle_{W^{*},W}ds+\int_{0}^{t}\langle
F^{N}(s),e_{i}\rangle ds$ $\displaystyle+\sum_{k\in K}\int_{0}^{t}\langle
G^{k,N}(s),e_{i}\rangle dW^{k}_{s}\ \ \mathbb{P}-a.s.$

Thanks to relations (45),(50) and the continuity of $B,\ F$ and $G^{k}$, we
know that exists a subsequence of the Galerkin approximations, which we will
denote again by $u^{N}$ just for simplicity, and processes $u$ and
$\hat{B}^{*}$ such that

$\displaystyle\begin{cases}u^{N}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}u\text{
in }L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{\infty}(0,T;W)),\ \ p\geq 2\\\
u^{N}{\rightharpoonup}u\text{ in
}L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{q}(0,T;V)),\ \ \ p,\ q\geq 2\\\
\hat{A}u^{N}{\rightharpoonup}\hat{A}u\text{ in
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;V))\\\
\hat{B}(u^{N},u^{N})\rightharpoonup\hat{B}^{*}\text{ in
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;W^{*}))\\\
F(u^{N})\rightharpoonup F(u)\text{ in
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;H\cap H^{1}))\\\
G^{k}(u^{N})\rightharpoonup G^{k}(u)\text{ in
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;H))\end{cases}$ (65)

Next step will be showing that $\hat{B}^{*}=\hat{B}(u,u)$. In this way the
existence of a solution of equation (9) will follow. In fact, we know that
$\mathbb{P}-a.s.$ for each $i\in\mathbb{N}$, for each $t\in[0,T]$

$\displaystyle\langle u(t),e_{i}\rangle_{V}-\langle u_{0},e_{i}\rangle_{V}$
$\displaystyle=\nu\int_{0}^{t}\langle\hat{A}u(s),e_{i}\rangle_{V}ds-\int_{0}^{t}\langle\hat{B}^{*}(s),e_{i}\rangle_{W^{*},W}ds+\int_{0}^{t}\langle
F(u(s)),e_{i}\rangle ds$ $\displaystyle+\sum_{k\in K}\int_{0}^{t}\langle
G^{k}(u(s)),e_{i}\rangle dW^{k}_{s}.\ \ $

For what concern the continuity in $V$ we can argue in the following way via
Itô’s formula and Kolmogorov continuity Theorem. From the weak formulation
above we get the weak continuity in $V$ of $u$ applying the Kolmogorov
continuity Theorem for the SDE satisfied by $\langle u(t),e_{i}\rangle_{V}$,
applying the Itô’s formula to $\lVert u\rVert_{V}^{2}$ we get

$\displaystyle d\lVert u\rVert_{V}^{2}=-2\nu\lVert\nabla
u\rVert_{L^{2}}^{2}dt-\langle\hat{B}^{*},u\rangle_{W^{*},W}dt.$

From this, we get the continuity of $\lVert u\rVert_{V}^{2}$ thanks to the
integrability properties of $u$. Weak continuity and continuity of the norm
implies strong continuity, thus we have the strong continuity of $u$ as a
process taking values in $V$. Weak continuity of $u$ as a process taking
values in $W$ follows from Lemma 1.4, pag. 263 in [40]. Alternatively the
strong continuity in $V$ of $u$ follows arguing as in [2] or [29].

###  3.3 Existence, Uniqueness and Further Results

To prove the existence of the solutions of equations (9) we need the following
Lemma. As stated in Section 2, this way of proceed has been introduced in [2]
for Navier-Stokes equations.

###### Lemma 28

Let

$\tau_{M}=\operatorname{inf}\\{t\in[0,T]:\lVert u(t)\rVert_{V}+\lVert
u(t)\rVert_{*}\geq M\\}\wedge T$

then

$\displaystyle 1_{t\leq\tau_{M}}(u^{N}-u)\rightarrow 0\text{\ in\
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;V)).$

Proof. Let $P^{N}$ be the projection of $W$ on
$W^{N}=\operatorname{span}\\{e_{1},\dots,e_{N}\\}.$ Thanks to dominated
convergence Theorem,

$\displaystyle P^{N}w\rightarrow w\quad\text{in
}L^{r}(\Omega,L^{q}(0,T;W))\quad\text{ if }r,\ q\in[1,+\infty)\text{ and }w\in
L^{r}(\Omega,L^{q}(0,T;W)).$ (66)

Consequently we have also convergence in $L^{r}(\Omega,L^{q}(0,T;V))$.
Moreover, if $w\in W,\ i\leq N,\ \langle P^{N}w,e_{i}\rangle_{V}=\langle
w,e_{i}\rangle_{V}$. Let $\hat{F}(u)=(I-\alpha^{2}A)^{-1}F(u)$,
$\hat{G}^{k}(u)=(I-\alpha^{2}A)^{-1}G^{k}(u)$. From the weak formulation
satisfied by $u$, for each $i\leq N$, we get

$\displaystyle\langle P^{N}u(t),e_{i}\rangle_{V}-\langle
u^{N}_{0},e_{i}\rangle_{V}$ $\displaystyle=\nu\int_{0}^{t}\langle
P^{N}\hat{A}u(s),e_{i}\rangle_{V}ds-\int_{0}^{t}\langle\hat{B}^{*}(s),e_{i}\rangle_{W^{*},W}ds+\int_{0}^{t}\langle\hat{F}(u(s)),e_{i}\rangle_{V}ds$
$\displaystyle+\sum_{k\in K}\int_{0}^{t}\langle\hat{G}^{k}(u(s)),e_{i}\rangle
dW^{k}_{s}\ \ \mathbb{P}-a.s.$

Exploiting the relation satisfied by $u^{N}$, we get

$\displaystyle\langle P^{N}u(t)-u^{N}(t),e_{i}\rangle_{V}$
$\displaystyle=\nu\int_{0}^{t}\langle
P^{N}\hat{A}(u(s)-u^{N}(s)),e_{i}\rangle_{V}ds+\int_{0}^{t}\langle\hat{B}(u^{N}(s),u^{N}(s))-\hat{B}^{*}(s),e_{i}\rangle_{W^{*},W}ds$
$\displaystyle+\int_{0}^{t}\langle\hat{F}(u(s))-\hat{F}^{N}(s),e_{i}\rangle_{V}ds+\sum_{k\in
K}\int_{0}^{t}\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),e_{i}\rangle
dW^{k}_{s}\quad\mathbb{P}-a.s.$ (67)

Thanks to (3.3), applying the Itô’s formula to $\sigma(t)\lVert
P^{N}u(t)-u^{N}(t)\rVert_{V}^{2}$, where
$\sigma(t)=exp(-\eta_{1}t-\eta_{2}\int_{0}^{t}\lVert u(s)\rVert_{W}^{2}ds)$,
we obtain

$\displaystyle\sigma(t)\lVert
P^{N}u(t)-u^{N}(t)\rVert_{V}^{2}+2\nu\int_{0}^{t}\sigma(s)\langle\hat{A}(u(s)-u^{N}(s)),P^{N}u(s)-u^{N}(s)\rangle_{V}ds$
$\displaystyle=\sum_{i=1}^{N}\sum_{k\in
K}\lambda_{i}\int_{0}^{t}\sigma(s)\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),e_{i}\rangle_{V}^{2}ds$
$\displaystyle+2\sum_{k\in
K}\int_{0}^{t}\sigma(s)\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),P^{N}u(s)-u^{N}(s)\rangle_{V}dW^{k}_{s}$
$\displaystyle-\eta_{1}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds-\eta_{2}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}\lVert u(s)\rVert_{W}^{2}ds$
$\displaystyle+2\int_{0}^{t}\sigma(s)\langle\hat{B}(u^{N}(s),u^{N}(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds$
$\displaystyle+2\int_{0}^{t}\sigma(s)\langle\hat{F}(u(s))-\hat{F}^{N}(s),P^{N}u(s)-u^{N}(s)\rangle_{V}ds.$
(68)

Let us analyze the terms in (3.3) one by one. We will not add details where
the computations are analogous to Lemma 3.9 in [32].

$\displaystyle\langle\hat{A}(u(s)-u^{N}(s)),P^{N}u(s)-u^{N}(s)\rangle_{V}=\langle\hat{A}(u(s)-u^{N}(s)),u(s)-u^{N}(s)\rangle_{V}-\langle\hat{A}(u(s)-u^{N}(s)),u(s)-P^{N}u(s)\rangle_{V},$
$\displaystyle\frac{1}{C_{p}^{2}+\alpha^{2}}\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}\leq\langle\hat{A}(u(s)-u^{N}(s)),u(s)-u^{N}(s)\rangle_{V},$
$\displaystyle\langle\hat{F}(u(s))-\hat{F}^{N}(s),P^{N}u(s)-u^{N}(s)\rangle_{V}$
$\displaystyle\stackrel{{\scriptstyle\text{equation }\eqref{behave F V
H}}}{{\leq}}C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in K}}\lVert
u(s)-u^{N}(s)\rVert_{V}^{2}\leq 2C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert
u^{N}(s)-P^{N}u(s)\rVert_{V}^{2}+2C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert u(s)-P^{N}u(s)\rVert_{V}^{2},$
$\displaystyle\sum_{i=1}^{N}\sum_{k\in
K}\lambda_{i}\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),e_{i}\rangle_{V}^{2}$
$\displaystyle=\sum_{k\in K}\lVert
P^{N}(\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s))\rVert_{V}^{2}$
$\displaystyle\stackrel{{\scriptstyle\text{equation }\eqref{behave G^k V
H}}}{{\leq}}2C^{G}_{\alpha,\\{\sigma_{k}\\}_{k\in K}}\lVert
u^{N}(s)-P^{N}u(s)\rVert_{V}^{2}+2C^{G}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert u(s)-P^{N}u(s)\rVert_{V}^{2},$

$\displaystyle
2\langle\hat{B}(u^{N}(s),u^{N}(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}$
$\displaystyle\leq C_{B}^{2}\lVert P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}\lVert
P^{N}(s)u(s)\rVert_{W}^{2}+\lVert P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}$
$\displaystyle+2\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}.$

Inserting these relations in equality (3.3) we obtain

$\displaystyle\sigma(t)\lVert
P^{N}u(t)-u^{N}(t)\rVert_{V}^{2}+\frac{2\nu}{C_{p}^{2}+\alpha^{2}}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds$ $\displaystyle\leq
2\nu\int_{0}^{t}\sigma(s)\langle\hat{A}(u(s)-u^{N}(s)),u(s)-P^{N}u(s)\rangle_{V}ds$
$\displaystyle+\int_{0}^{t}ds\sigma(s)(2C^{G}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert
u^{N}(s)-P^{N}u(s)\rVert_{V}^{2}+2C^{G}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert u(s)-P^{N}u(s)\rVert_{V}^{2})$ $\displaystyle+2\sum_{k\in
K}\int_{0}^{t}\sigma(s)\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),P^{N}u(s)-u^{N}(s)\rangle_{V}dW^{k}_{s}$
$\displaystyle-\eta_{1}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds-\eta_{2}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}\lVert u(s)\rVert_{W}^{2}ds$
$\displaystyle+\int_{0}^{t}\sigma(s)\bigg{(}C_{B}^{2}\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}\lVert P^{N}u(s)\rVert_{W}^{2}+\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}$
$\displaystyle+2\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}\bigg{)}ds$
$\displaystyle+\int_{0}^{t}\sigma(s)(4C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert
u^{N}(s)-P^{N}u(s)\rVert_{V}^{2}+4C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}\lVert u(s)-P^{N}u(s)\rVert_{V}^{2})ds.$

Taking $\eta_{1}=2C^{G}_{\alpha,\\{\sigma_{k}\\}_{k\in
K}}+4C^{F}_{\alpha,\\{\sigma_{k}\\}_{k\in K}}+1$, $\eta_{2}=C_{B}^{2}$ we get

$\displaystyle\sigma(t)\lVert
P^{N}u(t)-u^{N}(t)\rVert_{V}^{2}+\frac{2\nu}{C_{p}^{2}+\alpha^{2}}\int_{0}^{t}\sigma(s)\lVert
P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds$ $\displaystyle\leq
2\nu\int_{0}^{t}\sigma(s)\langle\hat{A}(u(s)-u^{N}(s)),u(s)-P^{N}u(s)\rangle_{V}ds+C\int_{0}^{t}\
\sigma(s)\lVert u(s)-P^{N}u(s)\rVert_{V}^{2}ds$ $\displaystyle+2\sum_{k\in
K}\int_{0}^{t}\sigma(s)\langle\hat{G}^{k}(u(s))-\hat{G}^{k,N}(s),P^{N}u(s)-u^{N}(s)\rangle_{V}dW^{k}_{s}$
$\displaystyle+2\int_{0}^{t}\sigma(s)\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds.$
(69)

Considering the expected value of (3.3) for $t=\tau_{M}\wedge r$, $r\in[0,T]$,
the stochastic integral cancel out, thus we arrive at

$\displaystyle\mathbb{E}\left[\sigma(\tau_{M}\wedge r)\lVert
P^{N}u(\tau_{M}\wedge r)-u^{N}(\tau_{M}\wedge
r)\rVert_{V}^{2}\right]+\frac{2\nu}{C_{p}^{2}+\alpha^{2}}\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\lVert P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds\right]$
$\displaystyle\leq 2\nu\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{A}(u(s)-u^{N}(s)),u(s)-P^{N}u(s)\rangle_{V}ds\right]+C\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\ \sigma(s)\lVert u(s)-P^{N}u(s)\rVert_{V}^{2}ds\right]$
$\displaystyle+2\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds\right].$
(70)

We want understand the behavior of the last term in the inequality above. From
Lemma 26 and relation (66) we have

$\displaystyle P^{N}u-u^{N}=(P^{N}u-u)+(u-u^{N})\rightharpoonup 0\text{ in
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;W)).$ (71)

Instead we have

$\displaystyle\left\lVert 1_{[0,\tau_{M}\wedge
r]}\sigma\left(\hat{B}(P^{N}u,P^{N}u)-\hat{B}(u,u)\right)\right\rVert_{L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;W^{*}))}\rightarrow
0.$ (72)

In fact thanks to relation (66) and the boundedness properties of $\hat{B}$
(20),(21), $\mathbb{P}-a.s.$ for each $t\in[0,T]$ it holds

$\displaystyle\lVert(1_{[0,\tau_{M}\wedge
r]}\sigma)\hat{B}(P^{N}u,P^{N}u)-\hat{B}(u,u)\rVert_{L^{2}(0,T;W^{*})}\leq
C\lVert u\rVert_{L^{4}(0,T;W)}\lVert(P^{N}-I)u\rVert_{L^{4}(0,T;W)}\rightarrow
0.$

Moreover

$\displaystyle\lVert(1_{[0,\tau_{M}\wedge
r]}\sigma)\hat{B}(P^{N}u,P^{N}u)-\hat{B}(u,u)\rVert_{L^{2}(0,T;W^{*})}\leq
C\lVert u(t)\rVert_{W}^{2}\in
L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T)).$

By dominated convergence Theorem we have the validity of relation (72).
Combing the weak convergence guaranteed by relation (71) and the strong
convergence guaranteed by (72) we obtain

$\displaystyle 2\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}(u(s),u(s)),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds\right]\rightarrow
0.$

From this relation, by triangle inequality, we can analyze easily the last
term in (3.3)

$\displaystyle\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{B}(P^{N}u(s),P^{N}u(s))-\hat{B}(u(s),u(s)),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{B}(u(s),u(s))-\hat{B}^{*}(s),P^{N}u(s)-u^{N}(s)\rangle_{W^{*},W}ds\right]\rightarrow
0.$ (73)

Thanks to the boundedness of $u^{N}$ and relation (66)

$\displaystyle\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\langle\hat{A}(u(s)-u^{N}(s)),u(s)-P^{N}u(s)\rangle_{V}ds\right]$
$\displaystyle\leq\lVert
u(s)-P^{N}u(s)\rVert_{L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;V))}\lVert\hat{A}(u-u^{N})\rVert_{L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;V))}$
$\displaystyle\leq C\lVert
u-P^{N}u\rVert_{L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;V))}\rightarrow
0.$ (74)

Combining (3.3) and (3.3) in relation (3.3) we obtain

$\displaystyle\mathbb{E}\left[\sigma(\tau_{M}\wedge r)\lVert
P^{N}u(\tau_{M}\wedge r)-u^{N}(\tau_{M}\wedge
r)\rVert_{V}^{2}\right]+\frac{2\nu}{C_{p}^{2}+\alpha^{2}}\mathbb{E}\left[\int_{0}^{\tau_{M}\wedge
r}\sigma(s)\lVert P^{N}u(s)-u^{N}(s)\rVert_{V}^{2}ds\right]\rightarrow 0.$
(75)

From relation (75), $\sigma(t)\geq C_{M}>0\ \forall t\leq\tau_{M}$ and the
properties of $P^{N}$ via triangle inequality the thesis follows considering
$r=T$.

###### Remark 29

The proof presented above works only in the case $\nu>0.$ In order to treat
the case $\nu=0$ we start from relation (75). Then, triangle inequality allows
to prove

$\displaystyle\mathbb{E}\left[\sigma(\tau_{M}\wedge r)\lVert u(\tau_{M}\wedge
r)-u^{N}(\tau_{M}\wedge r)\rVert_{V}^{2}\right]\rightarrow 0\quad\forall
r\in[0,T].$ (76)

By dominated convergence theorem we can improve the pointwise convergence of
relation (76) in order to obtain Lemma 28. We omit the easy details at this
stage, since this argument will be described in full details in the proof of
Corollary 30 below.

Combing Lemma 28 and the moment estimates for $u$ and $u^{N}$ we get the
following Corollary.

###### Corollary 30

The subsequence $u^{N}$ satisfies

$\displaystyle\lim_{N\rightarrow+\infty}\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]=0,$ (77)
$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{T}\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]\ dt=0.$ (78)

Proof. By relation (75) and triangle inequality we already know that

$\displaystyle\lim_{N\rightarrow+\infty}\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}\right]=0,$ (79)
$\displaystyle\lim_{N\rightarrow+\infty}\mathbb{E}\left[\int_{0}^{T\wedge\tau_{M}}\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}dt\right]=0.$ (80)

We start proving convergence (77). By definition of $\tau_{M}$, Lemma 26 and
the weak-$*$ convergence of $u^{N}$ to $u$ described by relation (65) and
Markov’s inequality it follows that

$\displaystyle\mathbb{E}\left[\lVert u^{N}(t)-u(t)\rVert_{V}^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}1_{\tau_{M}\geq t}\right]+\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}1_{\tau_{M}<t}\right]$
$\displaystyle=\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}1_{\tau_{M}\geq
t}\right]+\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}1_{\tau_{M}<t}\right]$
$\displaystyle\leq\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}\right]+\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{4}\right]^{1/2}\mathbb{P}(\tau_{M}<t)^{1/2}$
$\displaystyle\leq\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}\right]+C\operatorname{sup}_{N\in\mathbb{N}}\mathbb{E}\left[\lVert
u^{N}(t)\rVert_{W}^{4}\right]^{1/2}\mathbb{P}(\operatorname{sup}_{t\in[0,T]}\lVert
u(t)\rVert_{W}>M)^{1/2}$ $\displaystyle\leq\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}\right]+\frac{C}{M^{2}}\operatorname{sup}_{N\in\mathbb{N}}\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{N}(t)\rVert_{W}^{4}\right]$ $\displaystyle\leq\mathbb{E}\left[\lVert
u^{N}(t\wedge\tau_{M})-u(t\wedge\tau_{M})\rVert_{V}^{2}\right]+\frac{C_{\nu,\alpha,\\{\sigma_{k}\\}_{k\in
K}}}{M^{2}},$

where $C_{\nu,\alpha,\\{\sigma_{k}\\}_{k\in K}}$ is a constant independent
from $M$ and $N$. If we fix $\epsilon>0$ and choose $M$ large enough such that
$\frac{C_{\nu,\alpha,\\{\sigma_{k}\\}_{k\in K}}}{M^{2}}\leq\epsilon$ then by
relation (79) we have

$\displaystyle\operatorname{limsup}_{N\rightarrow+\infty}\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]\leq\epsilon.$

From the arbitrariness of $\epsilon$, the first thesis follows. In order to
obtain the other convergence we apply dominated convergence Theorem. Indeed,
by relation (77) we already know that for each $t\in[0,T]$

$\displaystyle\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]\rightarrow 0.$

Moreover, by Lemma (25), for each $N$

$\displaystyle\mathbb{E}\left[\lVert u^{N}(t)-u(t)\rVert_{V}^{2}\right]$
$\displaystyle\leq 2\mathbb{E}\left[\lVert
u^{N}(t)\rVert_{V}^{2}\right]+2\mathbb{E}\left[\lVert
u(t)\rVert_{V}^{2}\right]$ $\displaystyle\leq
C_{p,\alpha,\\{\sigma_{k}\\}_{k\in K}}+2\mathbb{E}\left[\lVert
u(t)\rVert_{V}^{2}\right]\in L^{1}(0,T).$

Therefore convergence (78) follows.

From Lemma 28, without any change with respect to the proof of Lemma 3.8 in
[32], we have that the Lemma below holds, thus $u$ is a solution of problem
(9) in the sense of Definition 3.

###### Lemma 31

$\hat{B}^{*}=\hat{B}(u,u)\textit{\ in\
}L^{2}(\Omega,\mathcal{F},\mathbb{P};L^{2}(0,T;W^{*}))$

Now we can prove the uniqueness.

###### Theorem 32

The solution of problem (9) in the sense of Definition 3 is unique.

Proof. Let $u_{1}$ and $u_{2}$ be two solutions. Let $w$ be their difference,
then for each $\phi\in W$ and $t>0$

$\displaystyle\langle w(t),\phi\rangle_{V}$
$\displaystyle=\nu\int_{0}^{t}\langle\nabla
w(s),\nabla\phi\rangle_{L^{2}}ds-\int_{0}^{t}b(u_{1}(s),u_{1}(s)-\alpha^{2}\Delta
u_{1}(s),\phi)ds$ $\displaystyle-\alpha^{2}\int_{0}^{t}b(\phi,\Delta
u_{1}(s),u_{1}(s))ds+\int_{0}^{t}b(u_{2}(s),u_{2}(s)-\alpha^{2}\Delta
u_{2}(s),\phi)ds$ $\displaystyle+\alpha^{2}\int_{0}^{t}b(\phi,\Delta
u_{2}(s),u_{2}(s))ds+\int_{0}^{t}\langle F(w(s)),\phi\rangle ds$
$\displaystyle+\sum_{k\in K}\int_{0}^{t}\langle G^{k}(w(s)),\phi\rangle
dW^{k}_{s}\ \ \mathbb{P}-a.s.$

Now we apply the Itô’s formula to compute $\lVert w\rVert_{V}^{2}$. Arguing as
in the first part of the proof of Lemma 35 we obtain

$\displaystyle d\lVert w\rVert_{V}^{2}=-2\nu\lVert\nabla
w\rVert_{L^{2}}^{2}dt+(b(w,w-\alpha\Delta w,u_{2})-b(u_{2},w-\alpha\Delta
w,w))dt.$

Let us consider $exp(-\int_{0}^{t}\lVert u_{2}(s)\rVert_{W}^{2}ds)\lVert
w(t)\rVert_{V}^{2}:=\sigma(t)\lVert w(t)\rVert_{V}^{2}$, via Itô’s formula we
get

$\displaystyle d(\sigma\lVert w\rVert_{V}^{2})+2\nu\sigma\lVert\nabla
w\rVert_{L^{2}}^{2}dt=-\sigma\lVert u_{2}\rVert_{W}^{2}\lVert
w\rVert_{V}^{2}dt+\sigma(b(w,w-\alpha\Delta w,u_{2})-b(u_{2},w-\alpha\Delta
w,w))dt$

Combining relations (16) and (18) it follows that

$\displaystyle\lvert b(w,w-\alpha^{2}\Delta
w,u_{2})-b(u_{2},w-\alpha^{2}\Delta w,w)\rvert\leq C\lVert
w\rVert_{V}^{2}\lVert u_{2}\rVert_{W}.$

Therefore

$\displaystyle d(\sigma\lVert w\rVert_{V}^{2})\leq-\sigma\lVert
u_{2}\rVert_{W}^{2}\lVert w\rVert_{V}^{2}dt+C\sigma\lVert
w\rVert_{V}^{2}\lVert u_{2}\rVert_{W}dt\leq C_{\epsilon}\sigma\lVert
w\rVert_{V}^{2},$

where in the last step we applied Young’s inequality. From the last chain of
inequalities, via Grönwall’s Lemma we get the thesis.

###### Theorem 33

The entire Galerkin’s sequence $u^{N}$ satisfies

$\displaystyle\lim_{N\rightarrow+\infty}\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]=0,$
$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{T}\mathbb{E}\left[\lVert
u^{N}(t)-u(t)\rVert_{V}^{2}\right]\ dt=0.$

Proof. Each subsequence $u^{N_{k}}$ has a converging sub-subsequence
$u^{N_{k,k}}$ which satisfies all previous Lemmas. By uniqueness of the
solution of equation (9) and Corollary 30 then the thesis follows.

###### Remark 34

Theorem 33 plays no role concerning the well-posedness of equation (9), but it
will be crucial for obtaining the energy estimates of Section 4, and thus for
proving Theorem 9.

##  4 Energy Estimates

Now we start considering equations (14) and assuming also Hypothesis 6. The
goal of this Section is to prove the following lemma:

###### Lemma 35

Under Hypothesis 2-6, if $u^{\alpha}$ is the solution of problem (14) in the
sense of Definition 3, then

$\displaystyle\lVert u^{\alpha}(t)\rVert^{2}+\alpha^{2}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}+\nu\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds=\lVert
u^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u^{\alpha}_{0}\rVert_{L^{2}}^{2};$ (81)
$\displaystyle\mathbb{E}\left[\alpha^{6}\operatorname{sup}_{t\in[0,T]}\lVert
u^{\alpha}(t)\rVert_{H^{3}}^{2}\right]=O(1).$ (82)

Proof. For the sake of simplicity we write $u$ and $u_{0}$ instead of
$u^{\alpha},\ u^{\alpha}_{0}$ since $\alpha$ is fixed in this proof. Therefore
all the asymptotic expansions and limits will be considering
$N\rightarrow+\infty.$

  * •

Let $\tilde{e}_{i}$ be the eigenfunctions of the Stokes operator $-A$, and
$\tilde{\lambda}_{i}$ the corresponding eigenvalues introduced in Lemma 20.
Let, moreover, $\tilde{u}^{N}=\sum_{i=1}^{N}\langle
u,\tilde{e}_{i}\rangle\tilde{e}_{i}=\tilde{P}^{N}u$. Exploiting the weak
formulation with test functions $\tilde{e}_{i}$ we get

$\displaystyle\langle u(t),\tilde{e}_{i}\rangle-\alpha^{2}\langle
u(t),A\tilde{e}_{i}\rangle-\langle
u_{0},\tilde{e}_{i}\rangle+\alpha^{2}\langle u_{0},A\tilde{e}_{i}\rangle$
$\displaystyle=\nu\int_{0}^{t}\langle u(s),A\tilde{e}_{i}\rangle
ds-\int_{0}^{t}b(u(s),u(s)-\alpha^{2}\Delta u(s),\tilde{e}_{i})ds$
$\displaystyle-\alpha^{2}\int_{0}^{t}b(\tilde{e}_{i},\Delta
u(s),u(s))ds+\tilde{\nu}\int_{0}^{t}\langle F(u),\tilde{e}_{i}\rangle ds$
$\displaystyle+\sqrt{\tilde{\nu}}\sum_{k\in K}\int_{0}^{t}\langle
G^{k}(u(s)),\tilde{e}_{i}\rangle dW^{k}_{s}\ \ \mathbb{P}-a.s.$

Multiplying each equation by $\tilde{e}_{i}$ and summing up, we get

$\displaystyle d(\tilde{u}^{N}-\alpha^{2}A\tilde{u}^{N})$ $\displaystyle=\nu
A\tilde{u}^{N}dt-\sum_{i=1}^{N}b(u,u-\alpha^{2}\Delta u,\tilde{e}_{i})\ dt$
$\displaystyle-\alpha^{2}\sum_{i=1}^{N}\int_{0}^{t}b(\tilde{e}_{i},\Delta
u,u)\tilde{e}_{i}\ dt+\tilde{\nu}\sum_{i=1}^{N}\langle
F(u),\tilde{e}_{i}\rangle\tilde{e}_{i}\ dt$
$\displaystyle+\sqrt{\tilde{\nu}}\sum_{k\in K}\sum_{i=1}^{N}\langle
G^{k}(u),\tilde{e}_{i}\rangle\tilde{e}_{i}\ dW^{k}_{t}$

Now we can apply the Itô’s formula to the process

$\displaystyle\frac{1}{2}(\lVert\tilde{u}^{N}(t)\rVert^{2}+\alpha^{2}\lVert\nabla\tilde{u}^{N}(t)\rVert_{L^{2}}^{2})=\frac{1}{2}\langle(I-\alpha^{2}A)\tilde{u}^{N}(t),\tilde{u}^{N}(t)\rangle$

obtaining

$\displaystyle\frac{\lVert\tilde{u}^{N}(t)\rVert^{2}+\alpha^{2}\lVert\nabla\tilde{u}^{N}(t)\rVert}{2}$
$\displaystyle=\frac{\lVert\tilde{u}^{N}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla\tilde{u}^{N}_{0}\rVert}{2}-\nu\int_{0}^{t}\langle\nabla\tilde{u}^{N}(s),\nabla
u^{N}(s)\rangle_{L^{2}}ds-\int_{0}^{t}b(u(s),u(s)$
$\displaystyle-\alpha^{2}\Delta
u(s),\tilde{u}^{N}(s))-\alpha^{2}\int_{0}^{t}b(\tilde{u}^{N}(s),\Delta
u(s),u(s))ds$ $\displaystyle+\frac{\tilde{\nu}}{2}\int_{0}^{t}\sum_{k\in
K}\langle P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u(s)))),\tilde{u}^{N}(s)\rangle ds$
$\displaystyle+\sqrt{\tilde{\nu}}\sum_{k\in K}\int_{0}^{t}\langle
P(\sigma_{k}\cdot\nabla u(s)),\tilde{u}^{N}\rangle dW^{K}_{s}$
$\displaystyle+\frac{\tilde{\nu}}{2}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{N}\langle P(\sigma_{k}\cdot\nabla
u(s)),\tilde{e}_{i}\rangle^{2}\langle\tilde{e}_{i},(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rangle
ds.$

Thanks to the properties of the projector $\tilde{P}^{N}$ we get easily the
first relation. The only thing we need to prove is that

$\displaystyle\sum_{i=1}^{N}\langle P(\sigma_{k}\cdot\nabla
u),\tilde{e}_{i}\rangle^{2}\langle\tilde{e}_{i},(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rangle+\langle
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))),\tilde{u}^{N}\rangle\rightarrow 0.$

The last relation is true, in fact

$\displaystyle\sum_{i=1}^{N}\langle P(\sigma_{k}\cdot\nabla
u),\tilde{e}_{i}\rangle^{2}\langle\tilde{e}_{i},(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rangle+\langle
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))),\tilde{u}^{N}\rangle=$ $\displaystyle\sum_{i=1}^{N}\langle
P(\sigma_{k}\cdot\nabla
u),(I-\alpha^{2}A)^{-1/2}\tilde{e}_{i}\rangle^{2}+\langle
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))),\tilde{u}^{N}\rangle$
$\displaystyle\rightarrow\langle(I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u),P(\sigma_{k}\cdot\nabla u)\rangle+\langle
P(\sigma_{k}\cdot\nabla((I-\alpha^{2}A)^{-1}P(\sigma_{k}\cdot\nabla
u))),u\rangle=0.$

  * •

From Theorem 33 and equation (5), we know that

$\displaystyle\int_{0}^{T}\mathbb{E}\left[\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}\right]ds$
$\displaystyle\leq\frac{1}{\alpha^{2}}\int_{0}^{T}\mathbb{E}\left[\lVert
u^{N}(s)\rVert_{V}^{2}\right]ds=\frac{1}{\alpha^{2}}\int_{0}^{T}\mathbb{E}\left[\lVert
u(s)\rVert_{V}^{2}\right]ds+o(1).$

Thus, from the Itô formula (81) the following relations hold true:

$\displaystyle\int_{0}^{T}\mathbb{E}\left[\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}\right]ds\leq\frac{C}{\alpha^{2}}\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+C\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+o(1)$ (83)
$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert\nabla
u(t)\rVert_{L^{2}}^{2}\right]\leq\frac{1}{\alpha^{2}}\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]$ (84)

According to inequality (25), in order to prove relation (82), it remains to
study

$\displaystyle\ \mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{N}(t)\rVert_{*}^{2}\right].$

Before going on we recall some notation. For each $N\in\mathbb{N}$

$\displaystyle\tau_{M}^{N}=\inf\\{t:\ \lVert u^{N}(t)\rVert_{V}+\lVert
u^{N}(t)\rVert_{*}\geq M\\}\wedge T,$

Thanks to the scaling factor $\sqrt{\tilde{\nu}}$ appearing in front of the
noise and exploiting the asymptotic relation between $\nu,\ \tilde{\nu}$ and
$\alpha^{2}$ described by Hypothesis 6 , if we choose

$\displaystyle\epsilon=\frac{1}{2\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}},$

equation (3.2) in Lemma 26 becomes

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]\leq C\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\left(\alpha^{2}+1\right)\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}\right]\right).$ (85)

Therefore, thanks to equation (83), we have

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert
u^{N}(t)\rVert_{*}^{2}\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle\leq
C\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\left(\alpha^{2}+1\right)\left(\mathbb{E}\left[\frac{\lVert
u_{0}\rVert^{2}}{\alpha^{2}}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]\right)\right)+o(1).$ (86)

So far we showed that $u^{N}\in L^{2}(\Omega;L^{2}([0,T];H^{1}))$,
$\operatorname{curl}(u^{N}-\alpha^{2}\Delta u^{N})\in
L^{2}(\Omega,L^{\infty}([0,T];L^{2}))$. By monotone convergence Theorem, we
can remove the dependence from $M$ in relation (• ‣ 4). Therefore

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq T}\lVert
u^{N}(t)\rVert_{*}^{2}\right]\leq C\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\left(\alpha^{2}+1\right)\left(\mathbb{E}\left[\frac{\lVert
u_{0}\rVert^{2}}{\alpha^{2}}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]\right)\right)+o(1).$ (87)

Thus, by Theorem 33 and the uniform bound (87) there exists a subsequence
$N_{k}$ such that

$\displaystyle u^{N_{k}}\rightarrow u\text{ in
}L^{2}(\Omega;L^{2}([0,T];H^{1}))$
$\displaystyle\operatorname{curl}(u^{N_{k}}-\alpha^{2}\Delta
u^{N_{k}})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}g\text{ in
}L^{2}(\Omega,L^{\infty}([0,T];L^{2})).$

If we take a test function $\phi\in
L^{2}(\Omega;L^{2}(0,T;C^{\infty}_{c}(D)))$, we get easily

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\langle\phi(s),g(s)\rangle_{L^{2}}ds\right]$
$\displaystyle=\lim_{k\rightarrow+\infty}\mathbb{E}\left[\int_{0}^{T}\langle\phi(s),\operatorname{curl}(u^{N_{k}}(s)-\alpha^{2}\Delta
u^{N_{k}}(s))\rangle_{L^{2}}ds\right]$
$\displaystyle=\lim_{k\rightarrow+\infty}\mathbb{E}\left[\int_{0}^{T}\langle(I-\alpha^{2}\Delta)\nabla^{\perp}\phi(s),u^{N_{k}})(s)\rangle_{L^{2}}ds\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\langle(I-\alpha^{2}\Delta)\nabla^{\perp}\phi(s),u(s))\rangle_{L^{2}}ds\right].$

Therefore $g=\operatorname{curl}(u-\alpha^{2}\Delta u)\in
L^{2}(\Omega,L^{\infty}([0,T];L^{2}))$ and the following inequality holds true

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq T}\lVert
u(t)\rVert_{*}^{2}\right]\leq
C\left(\liminf_{k\rightarrow+\infty}\mathbb{E}\left[\lVert
u^{N_{k}}_{0}\rVert_{*}^{2}\right]+\left(\alpha^{2}+1\right)\left(\mathbb{E}\left[\frac{\lVert
u_{0}\rVert^{2}}{\alpha^{2}}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]\right)\right).$ (88)

Let us analyze better the first term. We denote by
$u_{0}^{N,\infty}=u_{0}-u_{0}^{N}$.

$\displaystyle\lVert u^{N_{k}}_{0}\rVert_{*}^{2}$ $\displaystyle=\lVert
u^{N_{k}}_{0}\rVert_{W}^{2}-\lVert u^{N_{k}}_{0}\rVert_{V}^{2}$
$\displaystyle\leq\lVert u_{0}\rVert_{W}^{2}-\lVert
u^{N_{k}}_{0}\rVert_{V}^{2}$ $\displaystyle\leq\lVert
u_{0}\rVert_{*}^{2}+\lVert u^{{N_{k}},\infty}_{0}\rVert_{V}^{2}$
$\displaystyle\leq\lVert u_{0}\rVert_{*}^{2}+\lVert u_{0}\rVert_{V}^{2}$
$\displaystyle\leq C(\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}+\alpha^{4}\lVert\operatorname{curl}\Delta
u_{0}\rVert_{L^{2}}^{2}+\lVert u_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u_{0}\rVert_{L^{2}}^{2})$ $\displaystyle\leq C(\lVert
u_{0}\rVert^{2}+(1+\alpha^{2})\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}+\alpha^{4}\lVert u_{0}\rVert_{H^{3}}^{2})$

In conclusion, combining the observation above, relations (25), (84) and (88)
we get

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq T}\lVert
u(t)\rVert_{H^{3}}^{2}\right]$ $\displaystyle\leq
C\left(\frac{\alpha^{4}+\alpha^{2}+1}{\alpha^{6}}\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\frac{1+\alpha^{2}+\alpha^{4}}{\alpha^{4}}\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+\mathbb{E}\left[\lVert
u_{0}\rVert_{H^{3}}^{2}\right]\right).$ (89)

Thanks to the assumptions on $u_{0}^{\alpha}$, see Hypothesis 6, the thesis
follows.

###### Remark 36

In the case $\nu=0$, relation (81) follows without any change with respect to
the main proof. For what concerns relation (82), equation (85) above is false
in this framework. However, introducing the proper scaling in front of the
noise we can restart from relation (63) obtaining

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]$ $\displaystyle\leq
2\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$
$\displaystyle+4\sqrt{\tilde{\nu}}\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\left\lvert\sum_{k\in
K}\int_{0}^{t}\langle\operatorname{curl}({G}^{k,N}(s)),\operatorname{curl}(u^{N}(s)-\alpha^{2}\Delta
u^{N}(s))\rangle_{L^{2}}dW^{k}_{s}\right\rvert\right]$
$\displaystyle+4\tilde{\nu}\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\operatorname{curl}F^{N}(s)\rVert_{L^{2}}^{2}ds\right]+2\tilde{\nu}\sum_{k\in
K}\sum_{i=1}^{N}(\lambda_{i}+\lambda_{i}^{2})\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\langle{G}^{k,N}(s),e_{i}\rangle^{2}ds\right].$
(90)

Therefore, combining estimates (3.2),(3.2),(3.2),(61), exploiting the
asymptotic relation between $\tilde{\nu}$ and $\alpha^{2}$ described by
Hypothesis 6 and choosing $\epsilon=\frac{1}{\sum_{k\in
K}\lVert\sigma_{k}\rVert_{W^{1,\infty}}^{2}}$, we obtain

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]$ $\displaystyle\leq
4\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+6\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]+C_{\\{\sigma_{k}\\}_{k\in
K}}\left(\alpha^{2}+1\right)\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert\nabla
u^{N}(s)\rVert_{L^{2}}^{2}ds\right].$ (91)

Therefore, thanks to equation (83), we have

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]$ $\displaystyle\leq
4\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+6\mathbb{E}\left[\int_{0}^{r\wedge\tau_{M}^{N}}\lVert
u^{N}(s)\rVert_{*}^{2}ds\right]$ $\displaystyle+C_{\\{\sigma_{k}\\}_{k\in
K}}\left(\alpha^{2}+1\right)\left(\mathbb{E}\left[\frac{\lVert
u_{0}\rVert^{2}}{\alpha^{2}}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]\right)+o(1).$ (92)

Arguing as in Remark 27, we can apply Grönwall’s Lemma in inequality (92)
obtaining

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\leq
r\wedge\tau_{M}^{N}}\lVert u^{N}(t)\rVert_{*}^{2}\right]\leq
C_{\\{\sigma_{k}\\}_{k\in K}}\left(\mathbb{E}\left[\lVert
u^{N}_{0}\rVert_{*}^{2}\right]+\left(\alpha^{2}+1\right)\left(\mathbb{E}\left[\frac{\lVert
u_{0}\rVert^{2}}{\alpha^{2}}\right]+\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]\right)\right)+o(1).$ (93)

Relation (93) is completely analogous to relation (• ‣ 4) above. Therefore we
can follow the same argument of the main proof in order to obtain estimate
(82) and we omit the details.

##  5 Proof of Theorem 9

In order to prove Theorem 9, we will follow the ideas of [25] and [26]. We
will start with a weaker result with the supremum in time outside the expected
value and then we will move to the stronger one with the supremum in time
inside the expected value.

Proof of Theorem 9. Let $W^{\alpha}=u^{\alpha}-\bar{u}$, it satisfies
$\mathbb{P}-a.s.$ for each $\phi\in H$ and $t\in[0,T]$

$\displaystyle\langle W^{\alpha}(t),\phi\rangle-\langle
W^{\alpha}_{0},\phi\rangle$ $\displaystyle=\alpha^{2}\langle
Au^{\alpha}(t),\phi\rangle-\alpha^{2}\langle
Au^{\alpha}_{0},\phi\rangle+\nu\int_{0}^{t}\langle Au^{\alpha}(s),\phi\rangle
ds$
$\displaystyle-\int_{0}^{t}b(u^{\alpha}(s),W^{\alpha}(s),\phi)ds-\int_{0}^{t}b(W^{\alpha}(s),\bar{u}(s),\phi)ds$
$\displaystyle-\alpha^{2}\int_{0}^{t}b(\phi,\Delta
u^{\alpha}(s),u^{\alpha}(s))ds+\alpha^{2}\int_{0}^{t}b(u^{\alpha}(s),\Delta
u^{\alpha}(s),\phi)ds$ $\displaystyle\tilde{\nu}\int_{0}^{t}\langle
F(u^{\alpha})(s),\phi\rangle ds+\sqrt{\tilde{\nu}}\sum_{k\in
K}\int_{0}^{t}\langle G^{k}(u^{\alpha}(s)),\phi\rangle dW^{k}_{s}.$

Following the idea of [19], let $v$ the corrector of the boundary layer of
width $\delta$, i.e. a divergence free vector field with support in a strip of
the boundary of width $\delta$ such that $\bar{u}-v\in V$ and

$\displaystyle\operatorname{sup}_{t\in[0,T]}\lVert\partial_{t}^{l}v\rVert\lesssim\delta^{1/2},\
\ \ \operatorname{sup}_{t\in[0,T]}\lVert\partial_{t}^{l}\nabla
v\rVert\lesssim\delta^{-1/2},\ l\in\\{0,1\\}.$ (94)

Let $\delta=\delta(\alpha)$ such that

$\displaystyle\lim_{\alpha\rightarrow 0}\delta=0,\ \ \lim_{\alpha\rightarrow
0}\frac{\alpha^{2}}{\delta}=0.$ (95)

We want to write the Itô’s formula for $\lVert W^{\alpha}(t)\rVert^{2}$. Let
us take an orthonormal basis of $H$, $\\{\tilde{e}_{i}\\}$ made by
eigenvectors of $A$, let $\\{-\tilde{\lambda}_{i}\\}$ the corresponding
eigenvalues. Let us consider the weak formulation with test functions
$\phi=\tilde{e}_{i}$, let us call $W^{\alpha,n}=\sum_{i=1}^{n}\langle
W^{\alpha},\tilde{e}_{i}\rangle\tilde{e}_{i}$,
$u^{\alpha,n}=\sum_{i=1}^{n}\langle
u^{\alpha},\tilde{e}_{i}\rangle\tilde{e}_{i}$,
$\bar{u}^{n}=\sum_{i=1}^{n}\langle\bar{u},\tilde{e}_{i}\rangle\tilde{e}_{i}$ e
$v^{n}=\sum_{i=1}^{n}\langle v,\tilde{e}_{i}\rangle\tilde{e}_{i}$, then,
arguing as in the proof of Lemma 35, we get

$\displaystyle W^{\alpha,n}(t)-W^{\alpha,n}_{0}$
$\displaystyle=\alpha^{2}Au^{\alpha,n}(t)-\alpha^{2}Au^{\alpha,n}_{0}+\nu\int_{0}^{t}Au^{\alpha,n}(s)ds$
$\displaystyle-\int_{0}^{t}\sum_{i=1}^{n}b(u^{\alpha},W^{\alpha}(s),\tilde{e}_{i})\tilde{e}_{i}ds-\int_{0}^{t}\sum_{i=1}^{n}b(W^{\alpha}(s),\bar{u}(s),\tilde{e}_{i})\tilde{e}_{i}ds$
$\displaystyle-\alpha^{2}\int_{0}^{t}\sum_{i=1}^{n}b(\tilde{e}_{i},\Delta
u^{\alpha}(s),u^{\alpha}(s))ds+\alpha^{2}\int_{0}^{t}\sum_{i=1}^{n}b(u^{\alpha}(s),\Delta
u^{\alpha}(s),\tilde{e}_{i})ds$
$\displaystyle+\tilde{\nu}\int_{0}^{t}\sum_{i=1}^{n}\langle
F(u^{\alpha}(s)),\tilde{e}_{i}\rangle\tilde{e}_{i}ds+\sqrt{\tilde{\nu}}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle\tilde{e}_{i}dW^{k}_{s}.$ (96)

Therefore

$\displaystyle d\lVert W^{\alpha,n}\rVert^{2}$ $\displaystyle=2\langle
W^{\alpha,n},dW^{\alpha,n}\rangle+\alpha^{4}d\langle\langle
Au^{\alpha,n},Au^{\alpha,n}\rangle\rangle_{t}+\tilde{\nu}\sum_{k\in
K}\sum_{i=1}^{n}\langle G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle^{2}dt$
$\displaystyle+\alpha^{2}\sqrt{\tilde{\nu}}\sum_{k\in K}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle
d\langle\langle\tilde{e}_{i}W^{k},Au^{\alpha,n}\rangle\rangle_{t}.$ (97)

In the same way, considering the weak formulation satisfied by $u^{\alpha}$,
we get

$\displaystyle dAu^{\alpha,n}$ $\displaystyle=\left(\nu
A(I-\alpha^{2}A)^{-1}Au^{\alpha,n}-\sum_{i=1}^{n}b(u^{\alpha,n},u^{\alpha,n},\tilde{e}_{i})A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\right)dt$
$\displaystyle-\alpha^{2}\sum_{i=1}^{n}b(\tilde{e}_{i},\Delta
u,u)A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}dt$
$\displaystyle+\tilde{\nu}\sum_{i=1}^{n}\langle
F(u^{\alpha}),\tilde{e}_{i}\rangle A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}dt$
$\displaystyle+\sqrt{\tilde{\nu}}\sum_{k\in K}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle
A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}dW^{k}_{t}.$ (98)

Combining relation (5), (5), (5) we obtain

$\displaystyle d\lVert W^{\alpha,n}\rVert^{2}$
$\displaystyle=2\alpha^{2}\langle
W^{\alpha,n},dAu^{\alpha,n}\rangle+2\nu\langle
W^{\alpha,n},Au^{\alpha,n}\rangle dt$ $\displaystyle-2\langle
W^{\alpha,n},\sum_{i=1}^{n}b(W^{\alpha},\bar{u},\tilde{e}_{i})\tilde{e}_{i}\rangle
dt-2\langle
W^{\alpha,n},\sum_{i=1}^{n}b(u^{\alpha},W^{\alpha},\tilde{e}_{i})\tilde{e}_{i}\rangle
dt$ $\displaystyle-2\alpha^{2}\langle
W^{\alpha,n},\sum_{i=1}^{n}b(\tilde{e}_{i},\Delta
u^{\alpha},u^{\alpha})\tilde{e}_{i}\rangle dt+2\alpha^{2}\langle
W^{\alpha,n},\sum_{i=1}^{n}b(u^{\alpha},\Delta
u^{\alpha},\tilde{e}_{i})\tilde{e}_{i}\rangle\ dt$
$\displaystyle+2\tilde{\nu}\sum_{i=1}^{n}\langle
F(u^{\alpha}),\tilde{e}_{i}\rangle\langle W^{\alpha,n},\tilde{e}_{i}\rangle
dt+2\sqrt{\tilde{\nu}}\sum_{k\in K}\sum_{i=1}^{n}\langle
G^{k}(u),\tilde{e}_{i}\rangle\langle W^{\alpha,n},\tilde{e}_{i}\rangle
dW^{k}_{t}$ $\displaystyle+\alpha^{4}\tilde{\nu}\sum_{k\in
K}\sum_{i=1}^{n}\langle G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle^{2}\lVert
A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rVert^{2}\ dt$
$\displaystyle+\tilde{\nu}\sum_{k\in K}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle^{2}dt+\alpha^{2}\tilde{\nu}\sum_{k\in
K}\sum_{i=1}^{n}\langle G^{k}(u^{\alpha}),\tilde{e}_{i}\rangle^{2}\langle
A(I-\alpha^{2}A)^{-1}\tilde{e}_{i},\tilde{e}_{i}\rangle dt.$

Let us rewrite $\langle W^{\alpha,n},dAu^{\alpha,n}\rangle$ in a different way

$\displaystyle\langle W^{\alpha,n},dAu^{\alpha,n}\rangle$
$\displaystyle=\langle u^{\alpha,n}-\bar{u}^{n},dAu^{\alpha,n}\rangle=\langle
u^{\alpha,n},dAu^{\alpha,n}\rangle-\langle\bar{u}^{n}-v^{n},dAu^{\alpha,n}\rangle-\langle
v^{n},dAu^{\alpha,n}\rangle$
$\displaystyle=-\langle(-A)^{1/2}u^{\alpha,n},d(-A)^{1/2}u^{\alpha,n}\rangle+\langle(-A)^{1/2}(\bar{u}^{n}-v^{n}),d(-A)^{1/2}u^{\alpha,n}\rangle-\langle
v^{n},dAu^{\alpha,n}\rangle$
$\displaystyle=-\frac{d\lVert(-A)^{1/2}u^{\alpha,n}\rVert^{2}}{2}+\frac{d\langle\langle(-A)^{1/2}u^{\alpha,n},(-A)^{1/2}u^{\alpha,n}\rangle\rangle_{t}}{2}+d\langle(-A)^{1/2}(\bar{u}^{n}-v^{n}),(-A)^{1/2}u^{\alpha,n}\rangle$
$\displaystyle-\langle(-A)^{1/2}\partial_{t}(\bar{u}^{n}-v^{n}),(-A)^{1/2}u^{\alpha,n}\rangle-d\langle
v^{n},Au^{\alpha,n}\rangle+\langle\partial_{t}v^{n},Au^{\alpha,n}\rangle.$

Therefore, we arrive to this final expression

$\displaystyle\lVert W^{\alpha,n}(t)\rVert^{2}$ $\displaystyle=\lVert
W^{\alpha,n}_{0}\rVert^{2}-\alpha^{2}\lVert\nabla
u^{\alpha,n}(t)\rVert_{L^{2}}^{2}+\alpha^{2}\lVert\nabla
u^{\alpha,n}_{0}\rVert_{L^{2}}^{2}$
$\displaystyle+\tilde{\nu}\alpha^{2}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle^{2}\lVert(-A)^{1/2}(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rVert^{2}ds$
$\displaystyle+2\alpha^{2}\langle\nabla(\bar{u}^{n}-v^{n})(t),\nabla
u^{\alpha,n}(t)\rangle_{L^{2}}-2\alpha^{2}\langle\nabla(\bar{u}^{n}-v^{n})_{0},\nabla
u^{\alpha,n}_{0}\rangle_{L^{2}}$
$\displaystyle-2\alpha^{2}\int_{0}^{t}\langle\nabla\partial_{s}(\bar{u}^{n}(s)-v^{n}(s)),\nabla
u^{\alpha,n}(s)\rangle_{L^{2}}ds$ $\displaystyle-2\alpha^{2}\langle
v^{n}(t),\Delta u^{\alpha,n}(t)\rangle_{L^{2}}+2\alpha^{2}\langle
v^{n}_{0},\Delta u^{\alpha,n}_{0}\rangle_{L^{2}}$
$\displaystyle+2\alpha^{2}\int_{0}^{t}\langle\partial_{s}v^{n}(s),\Delta
u^{\alpha,n}(s)\rangle_{L^{2}}ds+2\nu\int_{0}^{t}\langle
W^{\alpha,n}(s),Au^{\alpha,n}(s)\rangle ds$
$\displaystyle-2\int_{0}^{t}\langle
W^{\alpha,n}(s),\sum_{i=1}^{n}b(W^{\alpha}(s),\bar{u}(s),\tilde{e}_{i})\tilde{e}_{i}\rangle
ds-2\int_{0}^{t}\langle
W^{\alpha,n}(s),\sum_{i=1}^{n}b(u^{\alpha}(s),W^{\alpha}(s),\tilde{e}_{i})\tilde{e}_{i}\rangle
ds$ $\displaystyle-2\alpha^{2}\int_{0}^{t}\langle
W^{\alpha,n}(s),\sum_{i=1}^{n}b(\tilde{e}_{i},\Delta
u^{\alpha}(s),u^{\alpha}(s))\tilde{e}_{i}\rangle
ds+2\alpha^{2}\int_{0}^{t}\langle
W^{\alpha,n}(s),\sum_{i=1}^{n}b(u^{\alpha}(s),\Delta
u^{\alpha}(s),\tilde{e}_{i})\tilde{e}_{i}\rangle ds$
$\displaystyle+2\tilde{\nu}\sum_{i=1}^{n}\int_{0}^{t}\langle
F(u^{\alpha}(s)),\tilde{e}_{i}\rangle\langle
W^{\alpha,n}(s),\tilde{e}_{i}\rangle ds+2\sqrt{\tilde{\nu}}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle\langle
W^{\alpha,n}(s),\tilde{e}_{i}\rangle dW^{k}_{s}$
$\displaystyle+\alpha^{4}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle^{2}\lVert
A(I-\alpha^{2}A)^{-1}\tilde{e}_{i}\rVert^{2}ds$
$\displaystyle+\tilde{\nu}\sum_{k\in K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle^{2}ds+\alpha^{2}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\sum_{i=1}^{n}\langle
G^{k}(u^{\alpha}(s)),\tilde{e}_{i}\rangle^{2}\langle
A(I-\alpha^{2}A)^{-1}\tilde{e}_{i},\tilde{e}_{i}\rangle ds.$

Now, letting $n\rightarrow+\infty$, exploiting the regularity of $u^{\alpha},\
\bar{u},\ v$ and the continuity of the trilinear form $b$ we arrive to the
formula below

$\displaystyle\lVert W^{\alpha}(t)\rVert^{2}+\alpha^{2}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}$ $\displaystyle=\lVert
W^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u^{\alpha}_{0}\rVert_{L^{2}}^{2}$
$\displaystyle+\alpha^{2}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\lVert(-A)^{1/2}(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha}(s))\rVert^{2}ds$
$\displaystyle+2\alpha^{2}\langle\nabla(\bar{u}-v)(t),\nabla
u^{\alpha}(t)\rangle_{L^{2}}-2\alpha^{2}\langle\nabla(\bar{u}-v)_{0},\nabla
u^{\alpha}_{0}\rangle_{L^{2}}$
$\displaystyle-2\alpha^{2}\int_{0}^{t}\langle\nabla\partial_{s}(\bar{u}(s)-v(s)),\nabla
u^{\alpha}(s)\rangle_{L^{2}}ds$ $\displaystyle-2\alpha^{2}\langle v(t),\Delta
u^{\alpha}(t)\rangle_{L^{2}}+2\alpha^{2}\langle v_{0},\Delta
u^{\alpha}_{0}\rangle_{L^{2}}$
$\displaystyle+2\alpha^{2}\int_{0}^{t}\langle\partial_{s}v(s),\Delta
u^{\alpha}(s)\rangle_{L^{2}}ds+2\nu\int_{0}^{t}\langle
W^{\alpha}(s),Au^{\alpha}(s)\rangle ds$
$\displaystyle-2\int_{0}^{t}b(W^{\alpha}(s),\bar{u}(s),W^{\alpha}(s))\rangle
ds$ $\displaystyle-2\alpha^{2}\int_{0}^{t}b(W^{\alpha}(s),\Delta
u^{\alpha}(s),u^{\alpha}(s))ds+2\alpha^{2}\int_{0}^{t}b(u^{\alpha}(s),\Delta
u^{\alpha}(s),W^{\alpha}(s))ds$ $\displaystyle+2\tilde{\nu}\int_{0}^{t}\langle
F(u^{\alpha}(s)),W^{\alpha}(s)\rangle ds+2\sqrt{\tilde{\nu}}\sum_{k\in
K}\int_{0}^{t}\langle G^{k}(u^{\alpha}(s)),W^{\alpha}(s)\rangle dW^{k}_{s}$
$\displaystyle+\alpha^{4}\tilde{\nu}\sum_{k\in K}\int_{0}^{t}\lVert
A(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha}(s))\rVert^{2}ds$
$\displaystyle+\tilde{\nu}\sum_{k\in K}\int_{0}^{t}\lVert
G^{k}(u^{\alpha}(s))\rVert^{2}ds+\alpha^{2}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\langle
A(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha}(s)),G^{k}(u^{\alpha}(s))\rangle ds$
$\displaystyle=I_{1}(t)+I_{2}(t)+I_{3}(t)+I_{4}(t)+I_{5}(t)+I_{6}(t)+M(t),$

where:

$\displaystyle I_{1}(t)$ $\displaystyle=\lVert
W^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u^{\alpha}_{0}\rVert_{L^{2}}^{2}+2\alpha^{2}\langle\nabla(\bar{u}-v)(t),\nabla
u^{\alpha}(t)\rangle_{L^{2}}-2\alpha^{2}\langle\nabla(\bar{u}-v)_{0},\nabla
u^{\alpha}_{0}\rangle_{L^{2}}$ $\displaystyle-2\alpha^{2}\langle v(t),\Delta
u^{\alpha}(t)\rangle_{L^{2}}+2\alpha^{2}\langle v_{0},\Delta
u^{\alpha}_{0}\rangle_{L^{2}},$ $\displaystyle I_{2}(t)$
$\displaystyle=\alpha^{2}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\lVert(-A)^{1/2}(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha})\rVert^{2}ds+\alpha^{4}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\lVert A(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha}(s))\rVert^{2}ds$
$\displaystyle+\tilde{\nu}\sum_{k\in K}\int_{0}^{t}\lVert
G^{k}(u^{\alpha}(s))\rVert^{2}ds+\alpha^{2}\tilde{\nu}\sum_{k\in
K}\int_{0}^{t}\langle
A(I-\alpha^{2}A)^{-1}G^{k}(u^{\alpha}(s)),G^{k}(u^{\alpha}(s))\rangle ds$
$\displaystyle+2\tilde{\nu}\int_{0}^{t}\langle
F(u^{\alpha}(s)),W^{\alpha}(s)\rangle ds,$ $\displaystyle I_{3}(t)$
$\displaystyle=-2\alpha^{2}\int_{0}^{t}\langle\nabla\partial_{s}(\bar{u}(s)-v(s)),\nabla
u^{\alpha}(s)\rangle_{L^{2}}ds+2\alpha^{2}\int_{0}^{t}\langle\partial_{s}v(s),\Delta
u^{\alpha}(s)\rangle_{L^{2}}ds,$ $\displaystyle I_{4}(t)$
$\displaystyle=2\nu\int_{0}^{t}\langle W^{\alpha}(s),Au^{\alpha}(s)\rangle
ds,$ $\displaystyle I_{5}(t)$
$\displaystyle=-2\int_{0}^{t}b(W^{\alpha}(s),\bar{u}(s),W^{\alpha}(s))\rangle
ds,$ $\displaystyle I_{6}(t)$
$\displaystyle=-2\alpha^{2}\int_{0}^{t}b(W^{\alpha}(s),\Delta
u^{\alpha}(s),u^{\alpha}(s))ds+2\alpha^{2}\int_{0}^{t}b(u^{\alpha}(s),\Delta
u^{\alpha}(s),W^{\alpha}(s))ds,$ $\displaystyle
M(t)=2\sqrt{\tilde{\nu}}\sum_{k\in K}\int_{0}^{t}\langle
G^{k}(u^{\alpha}(s)),W^{\alpha}(s)\rangle dW^{k}_{s}.$

Our approach is almost completely pathwise. Therefore we need to estimate the
terms $I_{i}(t),\ t\in\\{1,\dots,6\\}$. The analysis of $I_{1}(t)$ follows by
Young’s inequality, the estimates on the boundary layer corrector (94) and the
interpolation estimate (42)

$\displaystyle I_{1}(t)$ $\displaystyle\leq\lVert
W_{0}^{\alpha}\rVert^{2}+\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+C\alpha^{2}\delta^{1/2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{1/2}\lVert u_{0}^{\alpha}\rVert_{H^{3}}^{1/2}$
$\displaystyle+C\alpha^{2}(1+\delta^{-1/2})\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}+C\alpha^{2}\delta^{1/2}\lVert\nabla
u(t)^{\alpha}\rVert^{1/2}_{L^{2}}\lVert u(t)^{\alpha}\rVert_{H^{3}}^{1/2}$
$\displaystyle+C\alpha^{2}(1+\delta^{-1/2})\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}$ $\displaystyle\leq\lVert
W_{0}^{\alpha}\rVert^{2}+C\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+C\alpha^{2}(1+\delta^{-1})+C\alpha^{6}\delta(\lVert
u_{0}^{\alpha}\rVert_{H^{3}}^{2}+\lVert
u(t)^{\alpha}\rVert_{H^{3}}^{2})+C\delta^{1/2}+\frac{\alpha^{2}}{2}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}.$ (99)

The analysis of $I_{2}(t)$ follows by Young’s inequality and the results of
Lemma 21, Corollary 22. Indeed it holds

$\displaystyle I_{2}(t)$ $\displaystyle\leq C\tilde{\nu}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+\frac{\tilde{\nu}}{\alpha}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert\lVert\nabla u^{\alpha}(s)\rVert_{L^{2}}ds$
$\displaystyle\leq C\tilde{\nu}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert^{2}ds+\left(\frac{\tilde{\nu}}{\alpha}\right)^{2}\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds.$ (100)

The analysis of $I_{3}(t)$ follows by Young’s inequality, the estimates on the
boundary layer corrector (94) and the interpolation estimate (42)

$\displaystyle I_{3}(t)$ $\displaystyle\leq
C\alpha^{2}(1+\delta^{-1/2})\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}ds+C\alpha^{2}\delta^{1/2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{1/2}\lVert u^{\alpha}(s)\rVert_{H^{3}}^{1/2}$
$\displaystyle\leq
C\delta^{1/2}+C\alpha^{2}(1+\delta^{-1})+C\alpha^{2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+C\alpha^{6}\delta\int_{0}^{t}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds.$ (101)

The analysis of $I_{4}(t)$ is analogous to equations (3.20)-(3.21)-(3.22) in
[25], it implies:

$\displaystyle 2\nu\int_{0}^{t}\langle W^{\alpha}(s),Au^{\alpha}(s)\rangle ds$
$\displaystyle\leq-2\nu\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+C\frac{\nu}{\alpha}(1+\delta^{-1/2})\int_{0}^{t}\alpha\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}ds$
$\displaystyle+\frac{C\nu\delta^{1/2}}{\alpha^{2}}\int_{0}^{t}\alpha^{2}\lVert\Delta
u^{\alpha}(s)\rVert_{L^{2}}ds.$

Therefore by the interpolation inequality (42) and Young’s inequality we have

$\displaystyle 2\nu\int_{0}^{t}\langle W^{\alpha}(s),Au^{\alpha}(s)\rangle ds$
$\displaystyle\leq-2\nu\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+C\alpha^{2}\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds$
$\displaystyle+C\alpha^{6}\delta\int_{0}^{t}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds+C\left(\frac{\nu}{\alpha^{2}}\right)^{2}\delta^{1/2}+C\left(\frac{\nu}{\alpha}\right)^{2}(1+\delta^{-1}).$
(102)

The analysis of $I_{5}(t)$ follows immediately by Hölder’s inequality:

$\displaystyle
I_{5}(t)\leq\lVert\bar{u}\rVert_{L^{\infty}(0,T;H^{3})}\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert^{2}ds.$ (103)

For what concerns the analysis of $I_{6}(t)$, preliminary we observe that

$\displaystyle-2\alpha^{2}b(W^{\alpha},\Delta
u^{\alpha},u^{\alpha})+2\alpha^{2}b(u^{\alpha},\Delta u^{\alpha},W^{\alpha})$
$\displaystyle=2\alpha^{2}b(\bar{u},\Delta
u^{\alpha},u^{\alpha})-2\alpha^{2}b(u^{\alpha},\Delta u^{\alpha},u^{\alpha})$
$\displaystyle+2\alpha^{2}b(u^{\alpha},\Delta
u^{\alpha},u^{\alpha})-2\alpha^{2}b(u^{\alpha},\Delta u^{\alpha},\bar{u}).$

Arguing as in [22], equations (4.18)-(4.19) we get

$\displaystyle I_{6}(t)$ $\displaystyle\leq
C\alpha^{2}\left(1+\lVert\bar{u}\rVert_{L^{\infty}(0,T;H^{3})}\right)\int_{0}^{t}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+C\alpha^{2}\lVert\bar{u}\rVert_{L^{\infty}(0,T;H^{3})}^{4}\int_{0}^{t}\lVert
u^{\alpha}(s)\rVert^{2}ds.$ (104)

Combining equations (5),(5),(5),(5),(103),(104) and exploiting our assumptions
on the behavior of $\nu,\ \tilde{\nu},\alpha^{2}$, see Hypothesis 6, we have
the integral relation below:

$\displaystyle\lVert W^{\alpha}(t)\rVert^{2}+\frac{\alpha^{2}}{2}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}$ $\displaystyle\leq
M(t)+C\alpha^{2}(1+\delta^{-1})+C\delta^{1/2}+\lVert
W^{\alpha}_{0}\rVert^{2}+C\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+C\alpha^{6}\delta(\lVert
u_{0}^{\alpha}\rVert_{H^{3}}^{2}+\lVert u(t)^{\alpha}\rVert_{H^{3}}^{2})$
$\displaystyle+C_{\\{\sigma_{k}\\}_{k\in K}}\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert^{2}+{\alpha^{2}}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds+C\alpha^{6}\delta\int_{0}^{t}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds+C\alpha^{2}\int_{0}^{t}\lVert
u^{\alpha}(s)\rVert^{2}ds.$ (105)

By the stochastic Grönwall’s Lemma 24 above we have:

$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\lVert
W^{\alpha}(t)\rVert^{2}\right]+\alpha^{2}\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}\right]$ $\displaystyle\leq
C_{\\{\sigma_{k}\\}_{k\in
K}}\left(\alpha^{2}\mathbb{E}\left[\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert^{2}ds\right]+\alpha^{6}\delta\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds+\alpha^{6}\delta\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{\alpha}(t)\rVert_{H^{3}}^{2}\right]\right)$
$\displaystyle+C_{\\{\sigma_{k}\\}_{k\in
K}}\left(\alpha^{2}(1+\delta^{-1})+\delta^{1/2}+\mathbb{E}\left[\lVert
W^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+\alpha^{6}\delta\lVert
u_{0}^{\alpha}\rVert_{H^{3}}^{2}\right]\right).$ (106)

Thanks to Hypothesis 6 and our assumptions on $\delta$, see equation (95), we
have that

$\displaystyle\alpha^{2}(1+\delta^{-1})+\delta^{1/2}+\mathbb{E}\left[\lVert
W^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+\alpha^{6}\delta\lVert
u_{0}^{\alpha}\rVert_{H^{3}}^{2}\right]\rightarrow 0.$ (107)

Thanks to Lemma 35, we have that

$\displaystyle\alpha^{2}\mathbb{E}\left[\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert^{2}ds\right]+\alpha^{6}\delta\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds+\alpha^{6}\delta\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{\alpha}(t)\rVert_{H^{3}}^{2}\right]\rightarrow 0.$ (108)

Therefore

$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\lVert
W^{\alpha}(t)\rVert^{2}\right]+\alpha^{2}\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}\right]\rightarrow 0.$ (109)

Restarting from equation (5) and considering the expected value of the
supremum of both the terms in the left hand side we have

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
W^{\alpha}(t)\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}\right]$ $\displaystyle\leq
C\left(\alpha^{2}\mathbb{E}\left[\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert^{2}ds\right]+\alpha^{6}\delta\int_{0}^{T}\lVert
u^{\alpha}(s)\rVert_{H^{3}}^{2}ds+\alpha^{6}\delta\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u^{\alpha}(t)\rVert_{H^{3}}^{2}\right]\right)$
$\displaystyle+C\left(\alpha^{2}(1+\delta^{-1})+\delta^{1/2}+\mathbb{E}\left[\lVert
W^{\alpha}_{0}\rVert^{2}+\alpha^{2}\lVert\nabla
u_{0}^{\alpha}\rVert_{L^{2}}^{2}+\alpha^{6}\delta\lVert
u_{0}^{\alpha}\rVert_{H^{3}}^{2}\right]\right)$
$\displaystyle+C\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}M(t)\right]+C_{\\{\sigma_{k}\\}_{k\in
K}}\mathbb{E}\left[\int_{0}^{T}\lVert
W^{\alpha}(s)\rVert^{2}+{\alpha^{2}}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds\right].$ (110)

We already proved that almost all the terms in the right hand side of equation
(5) go to $0$. Therefore in order to complete the proof we left to show that

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}M(t)\right]+\mathbb{E}\left[\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert^{2}+{\alpha^{2}}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds\right]\rightarrow 0.$

By the weaker convergence described by equation (109) and Fubini Theorem

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\lVert
W^{\alpha}(s)\rVert^{2}+{\alpha^{2}}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}ds\right]\rightarrow 0.$

For what concerns the other, the convergence follows by Burkholder-Davis-Gundy
inequality, Hypothesys 6, equation (109), Fubini Theorem and Lemma (4). Indeed

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}M(t)\right]$
$\displaystyle\leq C\sqrt{\tilde{\nu}}\mathbb{E}\left[\left(\sum_{k\in
K}\int_{0}^{T}\lVert G^{k}(u^{\alpha}(s))\rVert^{2}\lVert
W^{\alpha}(s)\rVert^{2}ds\right)^{1/2}\right]$ $\displaystyle\leq
C\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\right)^{1/2}\sqrt{\tilde{\nu}}\mathbb{E}\left[\left(\int_{0}^{T}\lVert\nabla
u^{\alpha}(s)\rVert_{L^{2}}^{2}\lVert
W^{\alpha}(s)\rVert^{2}ds\right)^{1/2}\right]$ $\displaystyle\leq
C\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\right)^{1/2}\sqrt{\tilde{\nu}}\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}\left(\int_{0}^{T}\lVert
W^{\alpha}(s)\rVert^{2}ds\right)^{1/2}\right]$ $\displaystyle\leq
C\left(\sum_{k\in
K}\lVert\sigma_{k}\rVert_{L^{\infty}}^{2}\right)^{1/2}\mathbb{E}\left[\int_{0}^{T}\lVert
W^{\alpha}(s)\rVert^{2}ds\right]^{1/2}\left(\mathbb{E}\left[\alpha^{2}\operatorname{sup}_{t\in[0,T]}\lVert\nabla
u^{\alpha}(t)\rVert_{L^{2}}^{2}\right]\right)^{1/2}$ $\displaystyle\rightarrow
0.$

Now the proof is complete.

###### Remark 37

Combining Lemma 35 and Theorem 9 we understand that, if $\nu=O(\alpha^{2})$
and $\tilde{\nu}=O(\alpha^{2})$, the assumptions on the behavior of the
initial conditions $u_{0}^{\alpha}$ in norm $H,\ H^{1}$ and $H^{3}$ are
satisfied also for $t\in[0,T]$.

##  6 The Case of Additive Noise

For what concerns the case with additive noise, as stated in Section 2, the
well-posedness is a well-known fact in case of $\nu>0$ and we can prove a
result completely analogous to Theorem 9, following exactly the same argument.
However, the restriction $\nu>0$ can be omitted modifying slightly the proof
of [32] as described in Remarks 27, 29 and 36. However, we do not stress this
assumption in this Section, therefore $\nu>0$ in what follows. What was
crucial for the proof of Theorem 9 were the energy estimates of Lemma 35. Thus
in this Section we want to explain a different approach to prove these energy
estimates in the case of additive noise. These computations are more similar
to what happens in the deterministic framework. We keep previous assumptions
on the coefficients $\sigma_{k}$ and the Brownian motions $W^{k}$. For
generality reasons we consider the equations without any scaling factor on the
noise. Thus we consider

$\displaystyle\begin{cases}dv=(\nu\Delta u-\operatorname{curl}(v)\times
u+\nabla p)dt+\sum_{k\in K}\sigma_{k}dW^{k}_{t}\\\ \operatorname{div}u=0\\\
v=u-\alpha^{2}\Delta u\\\ u|_{\partial D}=0\\\ u(0)=u_{0}\end{cases}$ (111)

Before going on, we need to recall a result of [22].

###### Lemma 38

Let $q\in L^{2}(D)$, there exists a unique $\phi\in H^{2}_{0}(D)$ solution of

$\displaystyle\begin{cases}\Delta\phi-\alpha^{2}\Delta^{2}\phi=q\\\
\phi|_{\partial D}=\partial_{n}\phi|_{\partial D}=0\end{cases}$

which satisfies

$\langle\nabla\phi,\nabla v\rangle_{L^{2}}+\alpha^{2}\langle\Delta\phi,\Delta
v\rangle_{L^{2}}=-\langle q,v\rangle\text{ for each }v\in H^{2}_{0}.$

Moreover, the solution map is continuous from $L^{2}(D)$ to $H^{2}_{0}(D)\cap
H^{4}(D).$

Thanks to this Lemma, we can define an operator
$\mathbb{K}:L^{2}(D)\rightarrow H^{3}(D)\cap W^{1,\infty}_{0}(D)$ which
associates to each $q\in L^{2}(D)$ the vector field $u=\nabla^{\perp}\phi$,
where $\phi$ is the solution of the equation of Lemma 38.

###### Definition 39

A stochastic process $u$ weakly continuous with values in $W$ and continuous
with values in $V$ is a weak solution of equation (111) if

$u\in L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{\infty}(0,T;W)),\ \forall p\geq
2.$

and $\mathbb{P}-a.s.$ for every $t\in[0,T]$ and $\phi\in D(A)$ we have

$\displaystyle\langle u(t),(I-\alpha^{2}A)\phi\rangle-\langle
u_{0},(I-\alpha^{2}A)\phi\rangle$ $\displaystyle=\nu\int_{0}^{t}\langle
u(s),A\phi\rangle ds-\int_{0}^{t}b(u(s),u(s)-\alpha^{2}\Delta u(s),\phi)ds$
$\displaystyle-\alpha^{2}\int_{0}^{t}b(\phi,\Delta u(s),u(s))ds+\sum_{k\in
K}\langle\sigma_{k},\phi\rangle W^{k}_{t}.$

Arguing as in the first part of the proof of Lemma 35 we can prove the
following result.

###### Lemma 40

Let $u$ be a weak solution of problem (111) in the sense of Definition 39,
then the following relations hold true

  1. 1.

$d\lVert u\rVert^{2}+\alpha^{2}d\lVert\nabla
u\rVert_{L^{2}}^{2}=(-2\nu\lVert\nabla u\rVert_{L^{2}}^{2}+\sum_{k\in
K}\langle\sigma_{k},(I-\alpha^{2}A)^{-1}\sigma_{k}\rangle)dt+2\sum_{k\in
K}\langle\sigma_{k},u\rangle dW^{k}_{t}$

  2. 2.

$\displaystyle\mathbb{E}\left[\lVert
u(t)\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\lVert\nabla
u(t)\rVert_{L^{2}}^{2}\right]+2\nu\int_{0}^{t}\mathbb{E}\left[\lVert\nabla
u(s)\rVert_{L^{2}}^{2}\right]ds=$ $\displaystyle\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+t\sum_{k\in
K}\langle\sigma_{k},(I-\alpha^{2}A)^{-1}\sigma_{k}\rangle$

  3. 3.

$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert
u(t)\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\lVert\nabla
u(t)\rVert_{L^{2}}^{2}\right]+2\nu\int_{0}^{T}\mathbb{E}\left[\lVert\nabla
u(s)\rVert_{L^{2}}^{2}\right]ds\leq$ $\displaystyle
C\left(\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+T\sum_{k\in
K}\langle\sigma_{k},(I-\alpha^{2}A)^{-1}\sigma_{k}\rangle+\mathbb{E}\left[\left(\int_{0}^{T}\sum_{k\in
K}\langle\sigma_{k},u(s)\rangle^{2}ds\right)^{1/2}\right]\right)$

Let us introduce the vorticity formulation of (111), we denote
$s_{k}=\operatorname{curl}\sigma_{k}$

$\displaystyle\begin{cases}dq+\left(\frac{\nu}{\alpha^{2}}(q-\operatorname{curl}u)+u\cdot\nabla
q\right)dt=\sum_{k\in K}s_{k}dW^{k}_{t}\\\ \operatorname{div}u=0\\\
q=\operatorname{curl}(u-\alpha^{2}\Delta u)\\\
q(0)=q_{0}:=\operatorname{curl}(u_{0}-\alpha^{2}\Delta u_{0})\\\ u|_{\partial
D}=0\end{cases}$ (112)

###### Definition 41

A stochastic process $q$, which is weakly continuous with values in $L^{2}(D)$
and continuous with values in $H^{-1}(D)$, is a weak solution of equation
(112) if

$q\in L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{\infty}(0,T;L^{2}(D))),\ \forall
p\geq 2.$

and $\mathbb{P}-a.s.$ for every $t\in[0,T]$ and $\phi\in H^{2}_{0}(D)$ we have

$\displaystyle\langle q(t),\phi\rangle-\langle q_{0},\phi\rangle$
$\displaystyle=\int_{0}^{t}\int_{D}u(s)\cdot\nabla\phi q(s)\
dxds-\frac{\nu}{\alpha^{2}}\int_{0}^{t}\int_{D}(q(s)-\operatorname{curl}u(s))\phi\
dxds$ $\displaystyle+\sum_{k\in K}\langle s_{k},\phi\rangle W^{k}_{t}\ \
\mathbb{P}-a.s.$

$u=\nabla^{\perp}\varphi,$ $\varphi$ obtained by Lemma 38, $u\in W.$

Let us obtain a result about the equivalence between the solutions of these
two problems. Since we know from the results of [32] that problem (111) is
well-posed, then problem (112) is well-posed as well.

###### Proposition 42

Let $u$ be a solution of (111) in the sense of Definition 39, then
$q:=\operatorname{curl}(u-\alpha^{2}\Delta u)$ is a solution of (112) in the
sense of Definition 41. Conversely, if $q$ is a solution of (112) in the sense
of Definition 41 then $u=\nabla^{\perp}\varphi$, $\varphi$ obtained by Lemma
38, is a solution of (111) in the sense of Definition 39.

Proof.

  * •

$Def.\ref{weak Solution system}\implies Def.\ref{well posed vorticity non
linear}$ is immediate taking as test function for problem (111)
$\phi=-\nabla^{\perp}\tilde{\phi}$, $\tilde{\phi}\in H^{2}_{0}(D)$.

  * •

$Def.\ref{well posed vorticity non linear}\implies Def.\ref{weak Solution
system}$ we take $u=\nabla^{\perp}\varphi,\ v=u-\alpha^{2}\Delta u,$ where
$\varphi$ is obtained by Lemma 38 and $\phi=-\nabla^{\perp}\tilde{\phi},$
where $\tilde{\phi}\in H^{2}_{0}(D)$. Then integrating by parts and exploiting
that $\operatorname{curl}\nabla^{\perp}=\Delta$,
$\Delta\varphi-\alpha^{2}\Delta^{2}\varphi=q$ and $q$ is a solution of (112)
in the sense of Definition 41 we get

$\displaystyle-\langle(I-\alpha^{2}\Delta)u(t),\nabla^{\perp}\tilde{\phi}\rangle_{L^{2}}+\langle(I-\alpha^{2}\Delta)u_{0},\nabla^{\perp}\tilde{\phi}\rangle_{L^{2}}-\frac{\nu}{\alpha^{2}}\int_{0}^{t}\langle
v(s)-u(s),\nabla^{\perp}\tilde{\phi}\rangle ds$
$\displaystyle+\int_{0}^{t}\int_{D}(u(s)\cdot\nabla)\nabla^{\perp}\tilde{\phi}v(s)\
dxds-\alpha^{2}\int_{0}^{t}\int_{D}(\nabla^{\perp}\tilde{\phi}\cdot\nabla)\Delta
u(s)u(s)dxds+\sum_{k\in
K}\langle\sigma_{k},\nabla^{\perp}{\tilde{\phi}}\rangle W^{k}_{t}=0\ \
\mathbb{P}-a.s.$

From the last relation the thesis follows if we are able to prove the
continuity properties of $u$. The weak continuity of $u$ with values in $W$
follows immediately from the regularity of $q$ and Lemma 38. Again by Lemma 38
we get the strong continuity of $u$ with values in $V$. In fact, via Lax-
Milgram Lemma we get the regularity of the solution mapping of the problem
described in Lemma 38 between $H^{-2}(D)$ and $H^{2}_{0}(D)$. Via
interpolation techniques we recover the regularity of the solution mapping
between $H^{-1}(D)$ and $H^{3}(D)\cap H^{2}_{0}(D)$, therefore the required
regularity for $u$.

Approximating the process $q(t)$ solution of (112) by the eigenvectors of the
Laplacian with Dirichlet boundary conditions and then arguing as in the first
part of the proof of Lemma 35, we can obtain some Itô’s formula and energy
estimates. Moreover, if $u\in V$ we have $\lVert\nabla
u\rVert_{L^{2}}^{2}=\lVert\operatorname{curl}u\rVert_{L^{2}}^{2}$. Thanks to
Proposition 42, we know that $u$ appearing in problem (112) is a solution of
problem (111). Therefore, thanks to Lemma 40 we know that

$\displaystyle 2\nu\int_{0}^{t}\mathbb{E}\left[\lVert\nabla
u(s)\rVert_{L^{2}}^{2}\right]ds\leq\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+t\sum_{k\in
K}\langle\sigma_{k},(I-\alpha^{2}A)^{-1}\sigma_{k}\rangle,$ (113)
$\displaystyle\alpha^{2}\mathbb{E}\left[\lVert\nabla
u(t)\rVert_{L^{2}}^{2}\right]\leq\mathbb{E}\left[\lVert
u_{0}\rVert^{2}\right]+\alpha^{2}\mathbb{E}\left[\lVert\nabla
u_{0}\rVert_{L^{2}}^{2}\right]+t\sum_{k\in
K}\langle\sigma_{k},(I-\alpha^{2}A)^{-1}\sigma_{k}\rangle$ (114)