# Mean-field Magnetohydrodynamics Models as Scaling Limits of Stochastic
Induction Equations

Federico Butori  Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126
Pisa, Italia  federico.butori at sns.it and  Eliseo Luongo  Scuola Normale
Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italia  eliseo.luongo at
sns.it

###### Abstract.

We study the asymptotic properties of a stochastic model for the induction
equations of the magnetic field in a three dimensional periodic domain. The
turbulent velocity field driving the electromotive force on the magnetic field
is modeled by a noise white in time. For this model we rigorously take a
scaling limit leading to a deterministic model. While in case of isotropic
turbulence this produces an additional dissipation in the limit model which
influences also the decay rate of the Magnetic field in the stochastic model,
the case of turbulence devoloped in a preferential direction allows us to find
a dynamo effect.

###### Acknowledgements.

The authors thank Professor Franco Flandoli for useful discussions and
valuable insight into the subject.

###### Key words and phrases:

Roberts Flow, Magnetohydrodynamics, Transport Noise, Eddy Viscosity Model,
Alpha Effect, Beta Effect

###### 1991 Mathematics Subject Classification:

60H15, 76F25

##  1\. Introduction

The evolution of the magnetic field induced by the motion of a (non-
relativistic) conducting fluid with resistivity $\eta$ (or conductivity
$1/\eta$) is governed by the Maxwell equation and Ohm’s law. Together, these
laws allow to derive the so called _Induction Equation_

$\displaystyle\partial_{t}B_{t}$ $\displaystyle=\eta\Delta
B_{t}+\nabla\times(u_{t}\times B_{t}),\qquad\operatorname{div}B_{t}=0.$

The two terms on the right hand side represent respectively the Ohmic
dissipation of magnetic energy due to the resistivity of the media, and the
magnetic field induced by the electromotive force $u_{t}\times B_{t}$,
generated by the motion of the fluid. This equation is then coupled with some
equation from fluid dynamics describing the evolution of the conducting fluid
$u_{t}$. A correct and complete description of the system then shall include
also the back reaction of $B_{t}$ on $u_{t}$ resulting in a system of coupled
equations. This set of equations is at the core of Magneto-Hydrodynamics (MHD)
and at that of the kinematic and dynamical aspects of electrically conducting
fluids. In this field, an aspect that has remained obscure for many years is
the mechanism with which the turbulent motion of astrophysical fluids is able
to sustain large magnetic fields for timescales much longer than the
dissipative one, think for example to the Sun, Earth or galaxies. Whether the
fluid is in the core of a planet or on the surface of a star, it is an almost
universal fact that when it is involved in rotating motion, there is the
appearance of a magnetic field. The capability of these system to self excite
and sustain their magnetic fields was phenomenologically explained through
Dynamo Theory. However, the derivation of a precise Dynamo Theory from the
induction equation above, has eluded physicists for many years. It was not
until the ’60s, with the work of Steenback, Krause and Radler, that the
situation was substantially transformed, see the seminal paper [28] or the
books [19], [23] and the references therein. The development of their two
scales method, and in turn of the mean-field description of MHD allowed for a
systematic treatment and understanding of the dynamo problem in turbulently
moving electrically conducting media. Their approach was based on splitting
the equations and the two fields in large scales
$\overline{B_{t}},\overline{u_{t}}$ and small scales components
$B^{\prime}_{t},\ u^{\prime}_{t}$. When the large scales are produced by some
type of averaging operation, taking the average of a product is not the same
as taking the product of the averages, thus the equation for the large scales
produces a coupling between large and small scales:

$\partial_{t}\overline{B_{t}}=\eta\Delta\overline{B_{t}}+\nabla\times(\overline{u}_{t}\times\overline{B_{t}})+\nabla\times(\overline{u^{\prime}_{t}\times
B^{\prime}_{t}})$

The term $\overline{u^{\prime}_{t}\times B^{\prime}_{t}}$, often referred to
as _Turbulent electromotive force_ describes the effect of the interaction of
the small scales of $u$ and $B$ on the large scale components. In order to
close the equation for the large scales, at least in the case where the back
reaction of $B_{t}$ on $u$ is neglected, Krause and Radler proposed as a first
approximation to model this interaction by a mean-field effective term of the
form

$\mathcal{A}\overline{B_{t}}-\eta_{T}\operatorname{\nabla}\times\overline{B_{t}}$

where $\mathcal{A}$ and $\eta_{T}$ are quantities that must be derived from
the statistics of $u^{\prime}$. This two new terms are usually called _alpha_
and _beta_ effect terms. The mean field equation then takes the form

$\partial_{t}\overline{B_{t}}=(\eta+\eta_{T})\Delta\overline{B_{t}}+\nabla\times(\overline{u}_{t}\times\overline{B_{t}})+\nabla\times(\mathcal{A}\overline{B_{t}}).$

From this equation we see that the role of the _beta_ term
$\eta_{T}\nabla\times\overline{B_{t}}$ is that of enhancing dissipation, that
is, it plays against the development of a dynamo; it is instead the _alpha_
term $\mathcal{A}\overline{B_{t}}$ that, being of the first order, can provide
the mechanism necessary to sustain the magnetic field. It is the relative
magnitude of these two terms that in the end determines if the large scale
magnetic field $\overline{B_{t}}$ is able to self-sustain or if it dissipates.
Thus, the precise derivation of the quantities $\eta_{T}$ and $\mathcal{A}$,
is one of the main challenges of applied mean-field MHD, see [2] for a review
of numerical studies on the dynamo problem.
In view of the discussion above, the aim of this work is twofold: on one hand
we propose a mathematically rigorous and complete procedure to derive the mean
field effective system of Krause and Radler, in the case of negligible back
reaction of $B_{t}$ onto $u_{t}$, from a scaling limit of a stochastic
induction equation, in which the turbulent velocity field is replaced by a
white noise and the product $u_{t}\times B_{t}$ is interpreted in Stratonovich
form, according to a Wong-Zakai principle (_cf._ [6] and the references
therein). We refer to Theorem 2.17 for the precise convergence result. On the
other hand, and as a byproduct of our scheme, we are able to directly link the
_alpha_ and _beta_ terms to the covariance of the noise, thus to the
statistics of the turbulent field, so that we bypass the difficulty of having
to deriving them, as they can be prescribed a priori by an appropriate choice
of the noise. In this respect, the modelling choice of Gaussian noise can be
seen as parallel to the second order correlation approximation (SOCA) of
Krause and Radler, _cf._[19, Chapter 4], in which the _alpha_ and _beta_ terms
are deduced dropping correlations of order higher than two of $u^{\prime}$.
Ultimately, the Gaussian assumption, as well as the SOCA one, are too
restrictive and naïve for the description of fully developed turbulence.
However, we can content ourselves by dealing with this incomplete description,
as doing differently would require possessing the solution of the Turbulence
problem.
In order to stress the flexibility of our approach, we propose several choices
of the shape of the noise that account for different turbulent regimes. In
particular, we are able to explicitly identify the key role that helicity of
$u_{t}$ has in the appearance of the _alpha_ -effect and how it is naturally
linked to the size of the coefficient, in a way that is consistent with the
well established models of Mean Field MHD (see for instance chapters $3$ and
$4$ in [19]).

Our scheme is in spirit the same of our previous work [3] and falls into the
category of so called Itô-Stratonovich diffusion limits initiated in [17]: the
Itô-Stratonovich corrector appearing when we rewrite the term
$\nabla\times\left(u_{t}\times B_{t}\right)$ from the Stratonovich form to the
Itô one, corresponds precisely to the mean field term (or sum of terms)
discovered in the literature under the most natural closure formulae. But in
our rigorous derivation there is still the Itô term, which should disappear in
the scaling limit limit. There is no obvious reason why it should disappear.
Indeed, previous results in such direction where limited to consider scalar
and 2D models, where stretching of vector quantities does not appear. In such
case it is relatively easy to prove that the Itô term is negligible in the
limit, see for example [9], [10], [4], [13], [16], [1]. In 3D, the problem was
left open for a long time. Previous results in the 3D framework have drastic
unphysical simplifactions as neglecting the stretching term [11], or dealing
with 2D3C models [12] and their variants [3]. This work, therefore, represents
the first example where the techniques of the Itô-Stratonovich diffusion limit
are applied successfully in a completely 3D model. In particular it was
conjectured that only a suitable geometry of the involved vector fields could
remedy the excess of fluctuations in the stretching term [12]. Our approach
seems to go in the opposite direction, indeed we are able to consider random
vector fields $u_{t}$ either in case of an isotropic structure of the
covariance and in case of a preferential direction.
Although they are similar in spirit, from the physical standpoint one main
differences arises between this work and [3]: in our previous work, as
expected by the well known no-dynamo theorems [18, Section 3.5], since our
limit equations lose any dependence from one coordinate (becomes 2D3C), the
_alpha_ -effect in the mean field equation cannot provide any dynamo activity
for the slow-varying large scales, thus it might be classified as perturbed
version of a no-dynamo theorem, displaying that a ‘too simple’, almost two
dimensional, turbulent motion of the fluid cannot sustain a dynamo (see
instead [26] for the study of the dynamo problem in 3D flows independent by
one component and [25] for some others paradigmatic examples of dynamos). Here
instead, even when we deduce weaker convergence results, these are enough to
provide dynamo activity in the mean field equations arising from the _alpha_
-effect if the covariance of the noise is sufficiently anisotropic _cf._ 2.20.
On the contrary, if the breaking of the symmetry of the covariance is weak
enough, the _beta_ -effect is dominant, providing an enhanced decaying rate
_cf._ 2.21. This includes in particular the case of isotropic mirror symmetric
turbulence. In this regard, our results are in accordance with the general
principle that it is really the (anisotropic) 3D features of the fluid motion
that provide for dynamo activity and we confirm the famous prediction of [19,
Page 12] that “Homogeneous isotropic mirror symmetric turbulence only
influences the decay rate of the mean magnetic fields, which is enhanced in
almost all cases of physical interest”.
From the mathematical standpoint instead, the regularising effect of the thin
domain is here lost, resulting in either convergence in weaker norms or in
more restrictive choices of the noise coefficients. It is however important to
keep in mind that, even though the choice of the noise might seems tailored to
the needs of the mathematical treatise of the equation, it actually stems from
physical considerations, and is tuned as to keep non-trivial and non-vanishing
mean helicity of the flow throughout the limiting procedure.

The content of the paper is the following one. In section 2 we introduce
rigorously our framework. In particular, we define the structure of the noise
in subsection 2.3, while we state our main result and its consequences in
subsection 2.4. The remainder of the paper is dedicated to proving Theorem
2.17. In section 3 we prove some estimates for $B_{t}$ uniformly in the
scaling limit of our noise, then we show the required convergence in section
4.

##  2\. Functional Setting and Main Results

In subsection 2.1 we introduce the notation we follow along the paper. In
subsection 2.2 we introduce some functional analysis tools we employ in order
to carry on rigorously our analysis described in section 1. We describe the
noise we employ in subsection 2.3 providing some insights in the meaning of
the assumptions and comparisons with the existing literature. Lastly
subsection 2.4 is devoted to present our main results.

###  2.1. Notation

In the following we denote by
$\mathbb{T}^{3}=\mathbb{R}^{3}/(2\pi\mathbb{Z})^{3}$the three dimensional
torus and by $\mathbb{Z}^{3}_{0}=\mathbb{Z}^{3}\smallsetminus\\{0\\}$ (resp.
$\mathbb{Z}^{2}_{0}=\mathbb{Z}^{2}\smallsetminus\\{0\\}$) the three (resp.
two) dimensional lattice made by points with integer coordinates. We introduce
a partition of $\mathbb{Z}^{3}_{0}=\Gamma_{+}\cup\Gamma_{-}$ such that

$\displaystyle\Gamma_{+}=\\{k\in\mathbb{Z}^{3}_{0}:k_{3}>0\vee(k_{3}=0\land
k_{2}>0)\vee(k_{3}=k_{2}=0\land k_{1}>0)\\},\quad\Gamma_{-}=-\Gamma_{+}.$

We denote by $\\{e_{1},e_{2},e_{3}\\}$ the canonical basis of
$\mathbb{R}^{3}$. If $v=(v^{1},v^{2},v^{3})^{t}$ and
$w=(w^{1},w^{2},w^{3})^{t}$ are two vector fields we will denote by
$v^{H}=(v^{1},v^{2})^{t}$ and $\mathcal{L}_{v}w$ for the Lie derivative, i.e.
$\mathcal{L}_{v}w=v\cdot\nabla w-w\cdot\nabla v.$
For each $j\geq 0,\ l>1$ we denote by

$\displaystyle\zeta^{n}_{s,j}$
$\displaystyle=n^{j-3}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0},\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{j}},\quad\zeta_{s,j}=4\pi\begin{cases}\log 2&\text{ if }j=3,\\\
\frac{2^{3-j}-1}{3-j}&\text{ if }j\neq 3\end{cases},$
$\displaystyle\quad\zeta^{n}_{H,j}$
$\displaystyle=n^{j-2}\sum_{\begin{subarray}{c}k_{H}\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k_{H}\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k_{H}\rvert^{j}},\quad\zeta_{H,j}=\int_{1\leq\lvert x\rvert\leq
2}\frac{dx_{1}dx_{2}}{\lvert x\rvert^{j}}=2\pi\begin{cases}\log 2&\text{ if
}j=2,\\\ \frac{2^{2-j}-1}{2-j}&\text{ if }j\neq 2\end{cases}.$

In particular, by the approximation properties of Riemann sums, for each
$j\geq 0$ it holds

$\displaystyle\zeta_{s,j}^{n}=\zeta_{s,j}+O(n^{-1}),\quad\zeta_{H,j}^{n}=\zeta_{H,j}+O(n^{-1}).$

Moreover, we introduce the following function of the positive octant

$\displaystyle\chi:\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R},\quad\chi(\alpha,\beta,\gamma)=\begin{cases}1\quad&\text{if
}\alpha=\beta=\gamma=3;\\\ (\gamma-3)\wedge 1\quad&\text{if }\alpha=2,\
\beta=4,\ \gamma>3;\\\ (\alpha-2)\wedge(\beta-2)\wedge(\gamma-3)\quad&\text{if
}\alpha>2,\ \beta>2,\ \gamma>3;\\\ 0\quad&\text{otherwise}.\end{cases}$

We denote by $[M,N]$ the quadratic covariation process, which is defined for
any couple $M,\ N$ of square integrable real semimartingales and we extend it
by bilinearity to the analogue complex valued processes.
Let $a,b$ be two positive numbers, then we write $a\lesssim b$ if there exists
a positive constant $C$ such that $a\leq Cb$ and $a\lesssim_{\xi}b$ when we
want to highlight the dependence of the constant $C$ on a parameter $\xi$. If
$x\in\mathbb{C}$ we denote by $\overline{x}$ its complex conjugate. If
$\mathcal{Y}$ is a second-order tensor we denote by
$\left\|\mathcal{Y}\right\|_{HS}$ its Hilbert-Schmidt norm. Lastly, if
$x\in\mathbb{R}^{2}$ we denote by $x^{\perp}=(-x_{2},x_{1})^{t}.$

###  2.2. Preliminaries

Let us start setting some classical notation and properties of operators in
the periodic setting before describing the main contributions of this work. We
refer to monographs [7, 15, 22, 24, 29, 30, 32] for a complete discussion on
the topics shortly recalled in this section.

####  2.2.1. Function Spaces in the Periodic Setting

Let $\left(H^{s,p}(\mathbb{T}^{3}),\left\|\cdot\right\|_{H^{s,p}}\right),\
s\in\mathbb{R},\ p\in(1,+\infty)$ be the Bessel spaces of periodic functions.
In case of $p=2$, we simply write $H^{s}(\mathbb{T}^{3})$ in place of
$H^{s,2}(\mathbb{T}^{3})$ and we denote by $\langle\cdot,\cdot\rangle_{H^{s}}$
the corresponding scalar products. In case of $s=0$, we write
$L^{2}(\mathbb{T}^{3})$ instead of $H^{0}(\mathbb{T}^{3})$ and we neglect the
subscript in the notation for the norm and the inner product. With some abuse
of notation, for $s>0$, we denote the duality between $H^{-s}$ and $H^{s}$ by
$\langle\cdot,\cdot\rangle$.
We denote by $\dot{H}^{s,p}(\mathbb{T}^{3})$ (resp.
$\dot{H}^{s}(\mathbb{T}^{3}),\ \dot{L^{2}}(\mathbb{T}^{3})$) the closed
subspace of $H^{s,p}(\mathbb{T}^{3})$ (resp. $H^{s}(\mathbb{T}^{3}),\
L^{2}(\mathbb{T}^{3})$) made by zero mean functions with norm inducted by
$H^{s,p}(\mathbb{T}^{3})$ (resp. $H^{s}(\mathbb{T}^{3}),\
L^{2}(\mathbb{T}^{3})$) and by $Q$ the orthogonal projection of
$L^{2}(\mathbb{T}^{3})$ on $\dot{L}^{2}(\mathbb{T}^{3})$. The restriction
(resp. extension) of $Q$ to $H^{s}(\mathbb{T}^{3})$ (resp.
$H^{-s}(\mathbb{T}^{3})$) for $s>0$ is the orthogonal projection on
$\dot{H}^{s}(\mathbb{T}^{3})$ (resp. $\dot{H}^{-s}(\mathbb{T}^{3})$). A
different characterization of the fractional Sobolev spaces can be given in
terms of powers of the Laplacian. Denoting by

$\displaystyle\Delta:\mathsf{D}(\Delta)\subseteq\dot{L^{2}}(\mathbb{T}^{3})\rightarrow\dot{L}^{2}(\mathbb{T}^{3}),$

where $\mathsf{D}(\Delta)=\dot{H}^{2}(\mathbb{T}^{3})$, it is well known that
${\Delta}$ is the infinitesimal generator of analytic semigroup of negative
type that we denote by
$e^{t\Delta}:\dot{L}^{2}(\mathbb{T}^{3})\rightarrow\dot{L}^{2}(\mathbb{T}^{3})$
and moreover for each $s>$0, $\mathsf{D}((-\Delta)^{s})$ (resp.
$\left(\mathsf{D}((-\Delta)^{s})\right)^{\prime}$) can be identified with
$\dot{H}^{2s}(\mathbb{T}^{3})$ (resp. $\dot{H}^{-2s}(\mathbb{T}^{3})$).
Similarly, we introduce the Bessel spaces of vector fields

$\displaystyle{H}^{s,p}(\mathbb{T}^{3};\mathbb{R}^{3})$
$\displaystyle=\\{u=(u^{1},u^{2},u^{3})^{t}:\ u^{1},u^{2},u^{3}\in
H^{s,p}(\mathbb{T}^{3})\\},$ $\displaystyle\langle u,v\rangle_{{H}^{s}}$
$\displaystyle=\sum_{j=1}^{3}\langle u^{j},v^{j}\rangle_{H^{s}},\quad\text{for
}s\in\mathbb{R}.$

Again, in case of $s=0$ we write ${L}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$
instead of ${H}^{0}(\mathbb{T}^{3};\mathbb{R}^{3})$ and we neglect the
subscript in the notation for the norm and the scalar product. Similarly we
denote by $\dot{H}^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$ (resp.
$\dot{L}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$) the closed subspace of
$H^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$ (resp.
$L^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$) made by zero mean functions. With some
abuse of notation we usually employ the same symbols that we introduced for
norms and inner products of scalar functions also for the corresponding vector
fields and we continue to denote the orthogonal projection between
$H^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$ and
$\dot{H}^{s}(\mathbb{T}^{3};\mathbb{R}^{3}),\ s\in\mathbb{R}$, by $Q$, since
it acts component-wise.

Lastly we denote by $\mathbf{H}^{s}$ (resp. $\mathbf{L}^{2}$) the closed
subspace of $\dot{H}^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$ (resp.
$\dot{L}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$) made by zero mean, divergence
free vector fields with norm induces by $H^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$
(resp. $L^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$) and by

$\displaystyle
P:\dot{L}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})\rightarrow\mathbf{L}^{2}$

the Leray projection which, in the periodic setting, is an orthogonal
projection even when acting between
$\dot{H}^{s}(\mathbb{T}^{3};\mathbb{R}^{3})$ and $\mathbf{H}^{s}.$ If
$\\{\frac{k}{\lvert k\rvert},a_{k,1},a_{k,2}\\}$ is an orthonormal system of
$\mathbb{R}^{3}$ for each $k\in\mathbb{Z}^{3}_{0}$ then, it is well-known that
the family $\\{\frac{1}{(2\pi)^{3/2}}a_{k,j}e^{ik\cdot
x}\\}_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}$ is an orthonormal system of $\mathbf{L}^{2}$ and
is orthogonal in $\mathbf{H}^{s}$ for each $s\in\mathbb{R}$.
We conclude this subsection introducing some classical notation when dealing
with stochastic processes taking values in separable Hilbert spaces. Let $Z$
be a separable Hilbert space, with associated norm $\|\cdot\|_{Z}$. We denote
by $C_{\mathcal{F}}\left(\left[0,T\right];Z\right)$ the space of weakly
continuous adapted processes $\left(X_{t}\right)_{t\in\left[0,T\right]}$ with
values in $Z$ such that

$\mathbb{E}\bigg{[}\sup_{t\in\left[0,T\right]}\left\|X_{t}\right\|_{Z}^{2}\bigg{]}<\infty$

and by $L_{\mathcal{F}}^{p}\left(0,T;Z\right),\ p\in[1,\infty),$ the space of
progressively measurable processes $\left(X_{t}\right)_{t\in\left[0,T\right]}$
with values in $Z$ such that

$\mathbb{E}\bigg{[}\int_{0}^{T}\left\|X_{t}\right\|_{Z}^{p}dt\bigg{]}<\infty.$

####  2.2.2. Operators in the Periodic Setting

In this section we are interested to the properties of differential operators
of the form

$\displaystyle\mu\partial_{11}+\mu\partial_{22}+\nu\partial_{33},\
\mu\geq\nu>0,$

the case $\mu=\nu$ corresponding to the Laplacian. It is well known that for
each $\mu\geq\nu>0$, the operator

$\displaystyle
A_{\mu,\nu}:\dot{H}^{2}(\mathbb{T}^{3})\subseteq\dot{L}^{2}(\mathbb{T}^{3})\rightarrow\dot{L}^{2}(\mathbb{T}^{3}),\quad
A_{\mu,\nu}f=\mu\partial_{11}f+\mu\partial_{22}f+\nu\partial_{33}f$

is closed and generates an analytic semigroup of negative type which we denote
by
$S^{\mu,\nu}(t):\dot{L}^{2}(\mathbb{T}^{3})\rightarrow\dot{L}^{2}(\mathbb{T}^{3})$.
The family $\\{\frac{1}{(2\pi)^{3/2}}e^{ik\cdot
x}\\}_{k\in\mathbb{Z}^{3}_{0}}$ is a complete orthogonal systems of
$\dot{H}^{s}(\mathbb{T}^{3})$ made by eigengunctions of $A_{\mu,\nu}$ for each
$\mu\geq\nu>0$. Moreover it is orthonormal in $\dot{L}^{2}(\mathbb{T}^{3})$.
This allows us to write the operators $A_{\mu,\nu},\ S^{\mu,\nu}(t)$ in
Fourier basis as

(2.1) $\displaystyle
A_{\mu,\nu}f=-\frac{1}{(2\pi)^{3}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\end{subarray}}\left(\mu(k_{1}^{2}+k_{2}^{2})+\nu
k_{3}^{2}\right)\langle f,e^{-ik\cdot x}\rangle e^{ik\cdot x},\
S^{\mu,\nu}(t)=\frac{1}{(2\pi)^{3}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\end{subarray}}e^{-\left(\mu(k_{1}^{2}+k_{2}^{2})+\nu
k_{3}^{2}\right)t}\langle g,e^{-ik\cdot x}\rangle e^{ik\cdot x}$

for $f\in\dot{H}^{2}(\mathbb{T}^{3}),\ g\in\dot{L}^{2}(\mathbb{T}^{3}).$ The
expression in Fourier basis has as immediate consequences the following facts
which we recall here for the convenience of the reader

######  Lemma 2.1.

For each $s\geq 0$, the extension (resp. restriction) of
$S^{\mu,\nu}(t):\dot{H}^{-s}(\mathbb{T}^{3})\rightarrow\dot{H}^{-s}(\mathbb{T}^{3})$
(resp.
$S^{\mu,\nu}(t):\dot{H}^{s}(\mathbb{T}^{3})\rightarrow\dot{H}^{s}(\mathbb{T}^{3})$)
is an analytic semigroup of negative type. Moreover, for each
$\mu_{1}\geq\nu_{1}>0,\ \mu_{2}\geq\nu_{2}>0$ the operators
$S^{\mu_{1},\nu_{1}}(t)$ and $S^{\mu_{1},\nu_{1}}(t)$ commute. In particular

$\displaystyle
S^{\mu_{1},\nu_{1}}(t)S^{\mu_{2},\nu_{2}}(t)=S^{\mu_{2},\nu_{2}}(t)S^{\mu_{1},\nu_{1}}(t)=S^{\mu_{1}+\mu_{2},\nu_{1}+\nu_{2}}(t).$

######  Lemma 2.2.

Let $q\in\dot{H}^{s}(\mathbb{T}^{3}),\ s\in\mathbb{R}$. Then:

  1. i)

for any $\varphi\geq 0,$ it holds $\lVert
S^{\mu,\nu}(t)q\rVert_{\dot{H}^{s+\varphi}}\lesssim_{\varphi}(t\nu)^{-\varphi/2}\lVert
q\rVert_{\dot{H}^{s}}$, the hidden constant being increasing in $\varphi$;

  2. ii)

for any $\varphi\in[0,2],$ it holds
$\lVert\left(I-S^{\mu,\nu}(t)\right)q\rVert_{\dot{H}^{s-\varphi}}\lesssim(t\mu)^{\varphi/2}\lVert
q\rVert_{\dot{H}^{s}}$. The hidden constant being uniformly bounded for
$\varphi\in[0,2]$.

Lastly we need a closedness results on operators of this form, which has some
similarities with the second item of 2.2.

######  Lemma 2.3.

Let $q\in\dot{H}^{s}(\mathbb{T}^{3}),\ s\in\mathbb{R}$. If
$\mu_{1}\geq\nu_{1}>0$ and $\mu_{2}\geq\nu_{2}>0$, for any $\varphi\in[0,2],$
it holds

$\displaystyle\lVert\left(S^{\mu_{1},\nu_{1}}(t)-S^{\mu_{2},\nu_{2}}(t)\right)q\rVert_{\dot{H}^{s-\varphi}}\lesssim\left(\lvert\mu_{1}-\mu_{2}\rvert+\lvert\nu_{1}-\nu_{2}\rvert\right)^{\varphi/2}t^{\varphi/2}\lVert
q\rVert_{\dot{H}^{s}}.$

The hidden constant being uniformly bounded for $\varphi\in[0,2]$.

###### Proof.

Thanks to the Fourier expansion formula (2.1) we have

$\displaystyle\lVert\left(S^{\mu_{1},\nu_{1}}(t)-S^{\mu_{2},\nu_{2}}(t)\right)q\rVert^{2}_{\dot{H}^{s-\varphi}}$
$\displaystyle=\sum_{k\in\mathbb{Z}^{3}_{0}}\left(e^{-\left(\mu_{1}(k_{1}^{2}+k_{2}^{2})+\nu_{1}k_{3}^{2}\right)t}-e^{-\left(\mu_{2}(k_{1}^{2}+k_{2}^{2})+\nu_{2}k_{3}^{2}\right)t}\right)^{2}\langle
q,\frac{e^{-ik\cdot x}}{(2\pi)^{3/2}}\rangle^{2}\frac{\lvert
k\rvert^{2s}}{\lvert k\rvert^{2\varphi}}$
$\displaystyle\leq\left\|q\right\|_{\dot{H}^{s}}^{2}\operatorname{sup_{k\in\mathbb{Z}^{3}_{0}}}\frac{\left(e^{-\left(\mu_{1}(k_{1}^{2}+k_{2}^{2})+\nu_{1}k_{3}^{2}\right)t}-e^{-\left(\mu_{2}(k_{1}^{2}+k_{2}^{2})+\nu_{2}k_{3}^{2}\right)t}\right)^{2}}{\lvert
k\rvert^{2\varphi}}.$

Therefore, the proof is complete if we are able to show that

$\displaystyle\operatorname{sup_{k\in\mathbb{Z}^{3}_{0}}}\frac{\left(e^{-\left(\mu_{1}(k_{1}^{2}+k_{2}^{2})+\nu_{1}k_{3}^{2}\right)t}-e^{-\left(\mu_{2}(k_{1}^{2}+k_{2}^{2})+\nu_{2}k_{3}^{2}\right)t}\right)^{2}}{\lvert
k\rvert^{2\varphi}}\lesssim_{\varphi}\left(\lvert\mu_{1}-\mu_{2}\rvert+\lvert\nu_{1}-\nu_{2}\rvert\right)^{\varphi}t^{\varphi}.$

This is indeed the case. If $\mu_{1}<\mu_{2}$ and $\nu_{1}<\nu_{2}$

$\displaystyle\frac{\left(e^{-\left(\mu_{1}(k_{1}^{2}+k_{2}^{2})+\nu_{1}k_{3}^{2}\right)t}-e^{-\left(\mu_{2}(k_{1}^{2}+k_{2}^{2})+\nu_{2}k_{3}^{2}\right)t}\right)^{2}}{\lvert
k\rvert^{2\varphi}}$
$\displaystyle\leq\frac{\left(1-e^{-\left((\mu_{2}-\mu_{1})(k_{1}^{2}+k_{2}^{2})+(\nu_{2}-\nu_{1})k_{3}^{2}\right)t}\right)^{2}}{\lvert
k\rvert^{2\varphi}}$ $\displaystyle\leq
t^{\varphi}(\lvert\mu_{2}-\mu_{1}\rvert\wedge\lvert\nu_{2}-\nu_{1}\rvert)^{\varphi}\frac{\left(1-e^{-\left((\mu_{2}-\mu_{1})(k_{1}^{2}+k_{2}^{2})+(\nu_{2}-\nu_{1})k_{3}^{2}\right)t}\right)^{2}}{t^{\varphi}\left(\lvert\mu_{1}-\mu_{2}\rvert(k_{1}^{2}+k_{2}^{2})+\lvert\nu_{1}-\nu_{2}\rvert
k_{3}^{2}\right)^{\varphi}}$ $\displaystyle\lesssim
t^{\varphi}(\lvert\mu_{2}-\mu_{1}\rvert\wedge\lvert\nu_{2}-\nu_{1}\rvert)^{\varphi}$

since $\frac{1-e^{-x}}{x^{\varphi/2}}$ is bounded for $x>0$, uniformly in
$\varphi\in[0,2]$. In case of $\mu_{1}<\mu_{2}$ and $\nu_{2}<\nu_{1}$ and
either $(k_{1},k_{2})\neq 0$ and $k_{3}\neq 0$

$\displaystyle\frac{\left(e^{-\left(\mu_{1}(k_{1}^{2}+k_{2}^{2})+\nu_{1}k_{3}^{2}\right)t}-e^{-\left(\mu_{2}(k_{1}^{2}+k_{2}^{2})+\nu_{2}k_{3}^{2}\right)t}\right)^{2}}{\lvert
k\rvert^{2\varphi}}$
$\displaystyle\lesssim\frac{\left(1-e^{-\left((\mu_{2}-\mu_{1})(k_{1}^{2}+k_{2}^{2})\right)t}\right)^{2}}{\left(k_{1}^{2}+k_{2}^{2}\right)^{\varphi}}+\frac{\left(1-e^{-\left((\nu_{1}-\nu_{2})k_{3}^{2}\right)t}\right)^{2}}{\lvert
k_{3}\rvert^{2\varphi}}$ $\displaystyle\leq
t^{\varphi}\lvert\mu_{2}-\mu_{1}\rvert^{\varphi}\frac{\left(1-e^{-(\mu_{2}-\mu_{1})(k_{1}^{2}+k_{2}^{2})t}\right)^{2}}{\left(t\lvert\mu_{1}-\mu_{2}\rvert(k_{1}^{2}+k_{2}^{2})\right)^{\varphi}}$
$\displaystyle+t^{\varphi}\lvert\nu_{2}-\nu_{1}\rvert^{\varphi}\frac{\left(1-e^{-(\nu_{1}-\nu_{2})k_{3}^{2}t}\right)^{2}}{\left(t\lvert\nu_{1}-\nu_{2}\rvert
k_{3}^{2}\right)^{\varphi}}$ $\displaystyle\lesssim
t^{\varphi}\left(\lvert\mu_{2}-\mu_{1}\rvert^{\varphi}+\lvert\nu_{2}-\nu_{1}\rvert^{\varphi}\right).$

The case $(k_{1},k_{2})=0$ or $k_{3}=0$ easily reconducts to the one treated.
This completes the proof. ∎

With some abuse of notation we denote by

$\displaystyle
A_{\mu,\nu}:\mathbf{H}^{2}\subseteq\mathbf{L}^{2}\rightarrow\mathbf{L}^{2},\quad
A_{\mu,\nu}f=(A_{\mu,\nu}f^{1},A_{\mu,\nu}f^{2},A_{\mu,\nu}f^{3})^{t}.$

The Fourier expansion of (2.1), then allows us to write for
$f\in\mathbf{H}^{s}$

$\displaystyle
A_{\mu,\nu}f=-\frac{1}{(2\pi)^{3}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\left(\mu(k_{1}^{2}+k_{2}^{2})+\nu
k_{3}^{2}\right)\langle f,a_{k,j}e^{-ik\cdot x}\rangle e^{ik\cdot x}a_{k,j}.$

This implies that $A_{\mu,\nu}$ generates an analytic semigroup on
$\mathbf{L}^{2}$ which we still denote by $S^{\mu,\nu}(t)$ and 2.1, 2.2, 2.3
continue to hold in this vector valued framework.

###  2.3. Description of the Stochastic System

We begin with the description of the noise. We give a general description
which cover all the three regimes we are interested in: we call them the
Helical noise, the perturbed 2D noise and the Isotropic noise. Choose
$\rho\in[-1,1]$, a correlation factor whose role will be that of introducing
some correlation between the vertical and horizontal components of the ‘2D’
modes of the random field. Then assume
$\left(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}\right)$ 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 continuous
filtration and $\mathcal{F}_{0}$ contains every $\mathbb{P}$ null subset of
$\Omega$. Now, for each $n\in\mathbb{N}$, $\\{W_{t}^{k,j}\\},\
{{k\in\mathbb{Z}^{3}_{0},\ j\in\\{1,2\\}}}$ is a sequence of complex-valued
Brownian motions adapted to $\mathcal{F}_{t}$ such that
$W^{-k,j}_{t}=(W^{k,j}_{t})^{*}$ and

$\displaystyle\mathbb{E}\left[W^{k,j}_{1},{W^{l,m}_{1}}^{*}\right]$
$\displaystyle=\begin{cases}2&\textit{if }k=l,\ m=j\\\ 2\rho&\textit{if }k=l,\
k_{3}=0,\ m\neq
j;\end{cases}\quad\left[W^{k,j}_{\cdot},W^{l,m}_{\cdot}\right]_{t}=\begin{cases}2t&\textit{if
}k=-l,\ m=j\\\ 2\rho t&\textit{if }k=-l,\ k_{3}=0,\ m\neq j.\end{cases}$

Then, for each $k\in\mathbb{Z}^{3}_{0},\ j\in\\{1,2\\}$ we denote by
$\sigma_{k,j}^{n}(x)=\theta_{k,j}^{n}a_{k,j}e^{ik\cdot x}$, where
$\\{\frac{k}{\lvert k\rvert},a_{k,1},a_{k,2}\\}$ is an orthonormal system of
$\mathbb{R}^{3}$ for $k_{3}>0$ and $a_{k,j}=a_{-k,j}$ if $k_{3}<0$, while the
(complex) coefficients $\theta_{k,j}^{\varepsilon}$ will be defined later. It
is well known that the family $\\{a_{k,j}\frac{1}{2\pi\sqrt{2\pi}}e^{ik\cdot
x}\\},\ {{k\in\mathbb{Z}^{3}_{0},\ j\in\\{1,2\\}}}$ is a complete orthogonal
systems of $\mathbf{H}^{s}$ made by eigenfunctions of $-\Delta$. Moreover it
is orthonormal in $\mathbf{L}^{2}$. Without losing of generality, we make a
choice that will simplify some of the computations: when $k_{3}=0$ we choose
$a_{k,1}=(-k_{2}/\lvert k\rvert,k_{1}/\lvert k\rvert,0),\ a_{k,2}=(0,0,1)$ if
$k\in\Gamma_{+}$, $a_{k,j}=a_{-k,j}$ otherwise. For a physical intuition
behind our construction see our previous work [3] and the discussion therein.
Then we define

$\displaystyle W^{n}_{t}=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\sigma_{k,j}^{n}W^{k,j}_{t},$

with $\sigma_{k,j}^{n}=\theta_{k,j}^{n}a_{k,j}e^{ik\cdot x}$. With our
choices, to ensure that $W^{n}$ takes values in a space of real valued
functions, the coefficients must satisfy the condition

(2.2) ${\theta_{k,j}^{n}}^{*}=\theta_{-k,j}^{n}.$

The explicit choices of the coefficients $\theta_{k,j}^{n}$:

(2.3) $\displaystyle\theta_{k,j}^{n}$ $\displaystyle=\bm{1}_{\\{n\leq\lvert
k\rvert\leq 2n\\}}\begin{cases}\frac{i}{C_{1,H}\lvert
k\rvert^{\alpha/2}}&\text{if }k_{3}=0,k\in\Gamma_{+},\ j=1\\\
\frac{-i}{C_{1,H}\lvert k\rvert^{\alpha/2}}&\text{if }k_{3}=0,k\in\Gamma_{-},\
j=1\\\ \frac{1}{C_{2,H}\lvert k\rvert^{\beta/2}}&\text{if }k_{3}=0,\ j=2\\\
\frac{1}{C_{V}\lvert k\rvert^{\gamma/2}}&\text{if }k_{3}\neq 0.\end{cases}$

allows us to distinguish our three regimes:

######  Hypothesis 2.4 (Helical).

  * •

$\alpha\geq 2,\ \beta>2,\ \gamma>3,\ \alpha+\beta=6$.

  * •

$C_{1,H},\ C_{2,H},\ C_{V}>0$ and $C_{1,H}>\sqrt{\frac{\zeta_{H,2}}{\eta}}$ if
$\alpha=2$.

######  Hypothesis 2.5 (Perturbed 2D).

  * •

$\alpha=2,\ \beta=4,\ \gamma\geq 5$.

  * •

$C_{1,H}>\sqrt{\frac{\zeta_{H,2}}{\eta}},\ C_{2,H},\ C_{V}>0$.

######  Hypothesis 2.6 (Isotropic).

  * •

$\alpha=\beta=\gamma=3$.

  * •

$C_{1,H}=C_{2,H}=C_{V}>\sqrt{\frac{2\zeta_{s,3}}{3\eta}}$.

######  Remark 2.7.

The term ‘isotropic’ in 2.6 is understood with a slight abuse, indeed it
refers to the fact that the scaling parameters are all equal, but not to the
fact that the fields $W^{n}_{t}$ are actually isotropic, a condition that is
met only in the special case of $\rho=0$, and if one sets all the coefficients
to be real. However, since our analysis and our main result remains unchanged
whether if $\rho$ is zero or not in this regime, we included also the latter
case for generality.
The Perturbed 2D regime, instead, is a sub case of the Helical regime. Its
relevance comes from the fact that due to the strong suppression of true 3D
modes ($\gamma\geq 5$), this model represents a small 3D perturbation of a
2D-3C flow with weak vertical components ($\beta=4$), and akin to the works
[12], [3] which previously employed similar noises, this choice is the only
one that allows us to obtain a slightly stronger convergence result.

######  Remark 2.8.

Due to our choice of the coefficients $\theta_{k,j}^{n}$ and the vectors
$a_{k,j}$ it follows that for each $k$ such that $k_{3}=0,\ n\leq\lvert
k\rvert\leq 2n$

$\displaystyle\theta_{k,1}^{n}a_{k,1}=\frac{i}{C_{1,H}}\frac{k^{\perp}}{\lvert
k\rvert^{\alpha/2+1}},\
\theta_{k,2}^{n}a_{k,2}=\frac{1}{C_{2,H}}\frac{e_{3}}{\lvert
k\rvert^{\beta/2+1}}.$

The space covariance function associated to our noise is

(2.4) $\displaystyle
Q^{n}(x,y)=\mathbb{E}\left[W^{n}_{1}(x)\otimes{W^{n}_{1}(y)}^{*}\right]=Q^{n}_{0}(x,y)+\overline{Q^{n}_{\rho}}(x,y)=\overline{Q^{n}}(x,y)+\left(Q^{n}\right)^{\prime}(x,y)+\overline{Q^{n}_{\rho}}(x,y)$

where we denoted by

$\displaystyle\overline{Q^{n}}(x,y)$
$\displaystyle=2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0,\quad
j\in\\{1,2\\}\end{subarray}}\left\lvert\theta_{k,j}^{n}\right\rvert^{2}a_{k,j}\otimes
a_{k,j}e^{ik\cdot(x-y)},\quad\left(Q^{n}\right)^{\prime}(x,y)=2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}\neq 0,\quad
j\in\\{1,2\\}\end{subarray}}\left\lvert\theta_{k,j}^{n}\right\rvert^{2}a_{k,j}\otimes
a_{k,j}e^{ik\cdot(x-y)},$ $\displaystyle\overline{Q^{n}_{\rho}}(x,y)$
$\displaystyle=2\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\left(\theta_{k,1}^{n}\left(\theta_{k,2}^{n}\right)^{*}a_{k,1}\otimes
a_{k,2}+\theta_{k,2}^{n}\left(\theta_{k,1}^{n}\right)^{*}a_{k,2}\otimes
a_{k,1}\right)e^{ik\cdot(x-y)}.$

Due to our choice, we have that $\overline{Q^{n}}(x,y),$
$\left(Q^{n}\right)^{\prime}(x,y)$ and $\overline{Q^{n}_{\rho}}(x,y)$ are all
translation invariant, therefore we simply write $Q^{n}(x-y)$ (resp.
$Q^{n}_{0}(x-y),\ \overline{Q^{n}}(x-y),\ \left(Q^{n}\right)^{\prime}(x-y),\
\overline{Q^{n}_{\rho}}(x-y)$) in place of $Q^{n}(x,y)$ (resp.
$Q^{n}_{0}(x,y),\ \overline{Q^{n}}(x,y),\ \left(Q^{n}\right)^{\prime}(x,y),\
\overline{Q^{n}_{\rho}}(x,y)$). In particular $Q^{n}_{0}(x),\
\overline{Q^{n}}(x),\ \left(Q^{n}\right)^{\prime}(x)$ are mirror symmetric,
namely it holds

(2.5) $\displaystyle Q^{n}_{0}(x)=Q^{n}_{0}(-x),\
\overline{Q^{n}}(x)=\overline{Q^{n}}(-x),\
\left(Q^{n}\right)^{\prime}(x)=\left(Q^{n}\right)^{\prime}(-x).$

On the contrary $\overline{Q^{n}_{\rho}}(x)$ is an odd function, in particular
$\overline{Q^{n}_{\rho}}(0)=0$. As a consequence, our noise satisfies the
mirror symmetry property if and only if $\rho=0$.

We recall here some properties of the covariance functions
$\overline{Q^{n}_{0}}(x),(Q^{n}_{0})^{\prime}(x),\overline{Q^{n}_{\rho}}(x)$.
Their proofs are a simple adaptation of the arguments of [3, Section 2.3] to
which we refer for details.

######  Lemma 2.9.

If either 2.4 or 2.5 hold, then

$\displaystyle\overline{Q^{n}_{0}}(0)=2\eta^{n}_{T}(e_{1}\otimes
e_{1}+e_{2}\otimes e_{2})+2\eta^{n}_{R}\left(e_{3}\otimes
e_{3}\right),\quad(Q^{n}_{0})^{\prime}(0)=2(\eta^{n}_{V}-\eta^{n}_{R,V})I+\eta^{n}_{R,V}I_{2}$

where we set

$\displaystyle\eta^{n}_{T}$
$\displaystyle=\frac{\zeta_{H,\alpha}^{n}}{2C_{1,H}^{2}n^{\alpha-2}}=O(n^{2-\alpha}),\quad\eta^{n}_{R}=\frac{\zeta^{n}_{H,\beta}}{C_{2,H}^{2}n^{\beta-2}}=O(n^{2-\beta}),\quad\eta^{n}_{V}=\frac{2\zeta^{n}_{s,\gamma-3}}{3C_{V}^{2}n^{\gamma-3}}=O(n^{3-\gamma}),$
$\displaystyle\eta^{n}_{R,V}$
$\displaystyle=\frac{\zeta^{n}_{H,\gamma}}{C_{V}^{2}n^{\gamma-2}}=O(n^{2-\gamma}).$

In particular, if $\alpha=2$

$\displaystyle\eta^{n}_{T}=\frac{\zeta_{H,2}}{2C_{1,H}^{2}}+O(n^{-1}).$

######  Lemma 2.10.

If 2.6 holds, then

$\displaystyle{Q^{n}_{0}}(0)=2\eta^{n}_{iso}I$

where we set

$\displaystyle\eta^{n}_{iso}=\frac{2\zeta^{n}_{s,3}}{3C_{V}^{2}}=\frac{2\zeta_{s,3}}{3C_{V}^{2}}+O(n^{-1}).$

######  Lemma 2.11.

It holds

(2.6) $\displaystyle\left(\partial_{l}\overline{Q^{n}_{0}}\right)(0)$
$\displaystyle=2i\sum_{\begin{subarray}{c}k\in\mathbb{Z}_{0}^{3},\\\ k_{3}=0,\
j\in\\{1,2\\}\end{subarray}}\left\lvert\theta_{k,j}^{n}\right\rvert^{2}k_{l}a_{k,j}\otimes
a_{k,j}=0,\quad\left(\partial_{l}{Q^{n}_{0}}\right)^{\prime}(0)$
$\displaystyle=2i\sum_{\begin{subarray}{c}k\in\mathbb{Z}_{0}^{3},\\\ k_{3}\neq
0,\
j\in\\{1,2\\}\end{subarray}}\left\lvert\theta_{k,j}^{n}\right\rvert^{2}k_{l}a_{k,j}\otimes
a_{k,j}=0.$

As a consequence, for each $l,r,s\in\\{1,2,3\\}$ and $f\in
L^{2}(\mathbb{T}^{3}),\ g\in H^{1}(\mathbb{T}^{3})$ it holds

(2.7) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}_{0}^{3},\\\
k_{3}=0,\
j\in\\{1,2\\}\end{subarray}}\langle\partial_{l}g\sigma_{k,j}^{n,l},f\partial_{r}\sigma_{-k,j}^{n,s}\rangle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}_{0}^{3},\\\
k_{3}\neq 0,\
j\in\\{1,2\\}\end{subarray}}\langle\partial_{l}g\sigma_{k,j}^{n,l},f\partial_{r}\sigma_{-k,j}^{n,s}\rangle=0.$

######  Lemma 2.12.

We have

(2.8)
$\displaystyle\nabla\overline{Q^{n}_{\rho}}(0)=\frac{2\rho}{C_{1,H}C_{2,H}}\sum_{\begin{subarray}{c}k_{H}\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k_{H}\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k_{H}\rvert^{2}}\left(e_{3}\otimes\frac{k_{H}^{\perp}}{\lvert
k_{H}\rvert}\otimes\frac{k_{H}}{\lvert
k_{H}\rvert}-\frac{k_{H}^{\perp}}{\lvert k_{H}\rvert}\otimes
e_{3}\otimes\frac{k_{H}}{\lvert k_{H}\rvert}\right).$

In particular, it holds

$\displaystyle\left\|\nabla\overline{Q^{n}_{\rho}}(0)\right\|_{HS}\leq\frac{2\sqrt{2}\lvert\rho\rvert\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}$

and

$\displaystyle\\{\mathcal{R}_{n}^{r,s}\\}_{r,s\in\\{1,2\\}}$
$\displaystyle:=\partial_{s}\overline{Q^{n,3,r}_{\rho}}(0)=\frac{\rho\zeta^{N}_{H,2}}{C_{1,H}C_{2,H}}\begin{bmatrix}0&-1\\\
1&0\end{bmatrix}=\frac{\rho\zeta_{H,2}}{C_{1,H}C_{2,H}}\begin{bmatrix}0&-1\\\
1&0\end{bmatrix}+O(n^{-1}),$
$\displaystyle\\{\mathcal{V}_{n}^{r,s}\\}_{r,s\in\\{1,2\\}}:=\partial_{s}\overline{Q^{n,r,3}_{\rho}}(0)=-\partial_{s}\overline{Q^{n,3,r}_{\rho}}(0).$

As discussed in [15, Chapter 3], the quantity $Q^{n}(0)$ is the core in order
to obtain a nontrivial second order operator in our Itô-Stratonovich diffusion
limit, i.e. the _beta_ effect we discussed in section 1. In particular, the
choice of 2.4 and $\alpha=2$ or 2.6 allows us to obtain a non trivial _beta_
effect in the scaling limit. As discussed in section 1, the _alpha_ effect is
instead linked to the helicity of the velocity field. The role of the helicity
is usually understood in terms of distortion of the axial magnetic field lines
that generates currents in the rotating direction, which in turn induce
magnetic fields in the axial direction, see for example [19, Chapter 3, figure
3.2] and related comments. In our stochastic framework we denote the time
average of the helicity of our random velocity field

$\displaystyle\mathcal{H}^{n}(x):=\frac{1}{2T}\mathbb{E}\left[W^{n}_{T}(x)\cdot(\operatorname{\nabla}\times
W^{n}_{T}(x))\right].$

Exploiting the definition of our random vector field and arguing as in [3,
Appendix B] we can easily see that $\mathcal{H}^{n}(x)$ is independent on $x$
and it satisfies

$\displaystyle\mathcal{H}^{n}(x)=-\frac{2\rho\zeta^{n}_{H,\frac{\alpha+\beta-2}{2}}}{C_{1,H}C_{2,H}n^{\frac{\alpha+\beta-6}{2}}}\longrightarrow\mathcal{H}:=-\frac{4\pi\rho\log
2}{C_{1,H}C_{2,H}},$

i.e. the choice of $\alpha+\beta=6$ is linked to reach a finite and nontrivial
_alpha_ term in the limit.

###  2.4. Main Results

Following the ideas introduced in section 1, we fix $T\in(0,+\infty)$ and we
are interested in studying the properties of the following stochastic model on
$\mathbb{T}^{3}$:

(2.9) $\begin{cases}dB^{n}_{t}&=\eta\Delta
B^{n}_{t}dt+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}\circ
dW^{k,j}_{t},\\\ \operatorname{div}B^{n}_{t}&=0,\\\
B^{n}_{t}|_{t=0}&=B^{n}_{0},\end{cases}$

with the coefficients $\sigma_{k,j}^{n}$ defined in subsection 2.3. We start
rewriting it in Itô form. First, let us observe that for each
$k\in\mathbb{Z}^{3}_{0},\ j\in\\{1,2\\}$ we have due to the linearity of
$\mathcal{L}_{\sigma_{k,j}^{n}}$

$\displaystyle\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}\circ dW^{k,j}_{t}$
$\displaystyle=\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}dW^{k,j}_{t}+\frac{1}{2}\left[\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{\cdot},W^{k,j}_{\cdot}\right]_{t}$
$\displaystyle=\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}dW^{k,j}_{t}+\frac{1}{2}\mathcal{L}_{\sigma_{k,j}^{n}}\left[B^{n}_{\cdot},W^{k,j}_{\cdot}\right]_{t}$
$\displaystyle=\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}dW^{k,j}_{t}+\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}B^{n}_{t}dt+\rho\bm{1}_{\\{k_{3}=0,\
l\neq
j\\}}\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,l}^{n}}B^{n}_{t}dt.$

Therefore the equation for $B^{n}_{t}$ can be rewritten as

$\displaystyle\begin{cases}dB^{n}_{t}&=\left(\eta\Delta+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}+\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\mathcal{L}_{\sigma_{k,1}^{n}}\mathcal{L}_{\sigma_{-k,2}^{n}}+\mathcal{L}_{\sigma_{k,2}^{n}}\mathcal{L}_{\sigma_{-k,1}^{n}}\right)B^{n}_{t}dt\\\
&+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}dW^{k,j}_{t},\\\
\operatorname{div}B^{n}_{t}&=0,\\\ B^{n}_{t}|_{t=0}&=B^{n}_{0}.\end{cases}$

We are left to identify $\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}+\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\mathcal{L}_{\sigma_{k,1}^{n}}\mathcal{L}_{\sigma_{-k,2}^{n}}+\mathcal{L}_{\sigma_{k,2}^{n}}\mathcal{L}_{\sigma_{-k,1}^{n}}$.
We sketch the proof of the following result for the convenience of the reader,
in order to highlights the role of $Q^{n}_{0}(0)$ and
$\nabla\overline{Q^{n}_{\rho}}(0)$ in our procedure.

######  Lemma 2.13.

If $F:\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ is a smooth, zero mean
divergence free vector field, then it holds

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}F+\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\left(\mathcal{L}_{\sigma_{k,1}^{n}}\mathcal{L}_{\sigma_{-k,2}^{n}}+\mathcal{L}_{\sigma_{k,2}^{n}}\mathcal{L}_{\sigma_{-k,1}^{n}}\right)F$
$\displaystyle=\Lambda^{n}F+\Lambda_{\rho}^{n}F,\quad$

where in the formula above we denoted by $\Lambda^{n}$ the differential
operator

$\displaystyle\Lambda^{n}F=\begin{cases}(\eta^{n}_{T}+\eta^{n}_{V}-\eta^{n}_{R,V}/2)(\partial^{2}_{1}+\partial^{2}_{2})F+(\eta^{n}_{R}+\eta^{n}_{V}-\eta^{n}_{R,V})\partial^{2}_{3}F&\quad\text{
if \autoref{HP noise 1} holds},\\\ \eta^{n}_{iso}\Delta F&\quad\text{ if
\autoref{HP noise isotropo} holds}\end{cases}$

and by $\Lambda_{\rho}^{n}$ the differential operator

$\displaystyle\Lambda_{\rho}^{n}F$
$\displaystyle=-\sum_{l\in\\{1,2\\}}\partial_{l}\overline{Q^{n}_{\rho}}(0)\cdot\nabla
F_{l}=-\sum_{\begin{subarray}{c}j\in\\{1,2,3\\}\\\
l\in\\{1,2\\}\end{subarray}}\partial_{l}\overline{Q^{n,\cdot,j}_{\rho}}(0)\partial_{j}F_{l}$
$\displaystyle=\nabla\times(\mathcal{A}^{n}_{\rho}F),$

where

$\displaystyle\mathcal{A}^{n}_{\rho}=\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{bmatrix}.$

In particular it holds

$\displaystyle(\Lambda_{\rho}^{n}F)_{H}=\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\partial_{3}F_{H}^{\perp},\quad(\Lambda_{\rho}^{n}F)_{3}=-\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\operatorname{div}_{H}(F_{H}^{\perp}).$

###### Proof.

All the terms involved are Lie brackets, thus we begin with some preliminary
computations. By recalling that we have
$\nabla\sigma_{k,j}^{n}=i\theta_{k,j}^{n}k\otimes a_{k,j}e^{ikx}$, and also
that $a_{-k,j}=a_{k,j}$ for all $k\in\mathbb{Z}_{0}^{3},\ j=1,2$ we write

(2.10) $\displaystyle\sigma_{k,l}^{n}\cdot\nabla(\sigma_{-k,j}^{n}\cdot\nabla
F)$ $\displaystyle=(\sigma_{k,l}^{n}\cdot\nabla\sigma_{-k,j}^{n})\cdot\nabla
F+\sigma_{k,l}^{n}\otimes\sigma_{-k,j}^{n}:\nabla^{2}F$
$\displaystyle=-i\theta_{k,l}^{n}(a_{k,l}\cdot k)a_{-k,l}\cdot\nabla
F+\sigma_{k,l}^{n}\otimes\sigma_{-k,j}^{n}:\nabla^{2}F$
$\displaystyle=\sigma_{k,l}^{n}\otimes\sigma_{-k,j}^{n}:\nabla^{2}F$

since $a_{k,l}\cdot k=0$. On the other hand we have

(2.11)
$\displaystyle(F\cdot\nabla\sigma_{-k,j}^{n})\cdot\nabla\sigma_{k,l}^{n}=\theta_{-k,j}^{n}\theta_{k,l}^{n}(F\cdot
k)(a_{k,j}\cdot k)a_{k,l}=0$

while

(2.12)
$\displaystyle\sigma_{-k,j}^{n}\cdot\nabla\left(F\cdot\nabla\sigma_{k,l}^{n}\right)$
$\displaystyle=i\theta_{k,l}^{n}\theta_{-k,j}^{n}e^{-ikx}a_{k,j}^{r}\partial_{r}((F\cdot
k)a_{k,l}e^{ikx})$
$\displaystyle=i\theta_{k,l}^{n}\theta_{-k,j}^{n}e^{-ikx}a_{k,j}^{r}((\partial_{r}F\cdot
k)a_{k,l}e^{ikx}+(F\cdot k)a_{k,l}k_{r}e^{ikx})$
$\displaystyle=i\theta_{k,l}^{n}\theta_{-k,j}^{n}a_{k,j}^{r}(\partial_{r}F\cdot
k)a_{k,l}$

and

(2.13) $\displaystyle\left(\sigma_{k,l}^{n}\cdot\nabla
F\right)\cdot\nabla\sigma_{-k,j}^{n}$
$\displaystyle=-i\theta_{k,l}^{n}\theta_{-k,j}^{n}a^{r}_{k,l}(\partial_{r}F\cdot
k)a_{k,j}.$

In particular we see that if $l=j$ the sum of (2.12) and (2.13) is zero.
We are now ready to compute the correctors. We have

$\displaystyle\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}F$
$\displaystyle=\sigma_{k,j}^{n}\cdot\operatorname{\nabla}(\sigma_{-k,j}^{n}\cdot\nabla
F)-\sigma_{k,j}^{n}\cdot\operatorname{\nabla}(F\cdot\nabla\sigma_{-k,j}^{n})-(\sigma_{-k,j}^{n}\cdot\nabla
F)\cdot\nabla\sigma_{k,j}^{n}+(F\cdot\nabla\sigma_{-k,j}^{n})\cdot\nabla\sigma_{k,j}^{n}.$

By (2.11) the last term is zero, while by (2.12) and (2.13), the sum of the
two middle terms is zero. Summing over $k\in\mathbb{Z}_{0}^{3}$ and $j=1,2$ we
get

(2.14) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}F$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\sigma_{k,l}^{n}\otimes\sigma_{-k,j}^{n}:\nabla^{2}F$
$\displaystyle=\operatorname{div}(Q_{0}^{n}(0)\operatorname{\nabla}F)$
$\displaystyle=\Lambda^{n}F$

where we used again the fact that
$\sigma_{k,j}^{n}\cdot\operatorname{\nabla}\sigma_{-k,j}^{n}=0$ in the second
line and 2.9, 2.10 in the last equality.
We now look at the part of the corrector relative to the correlation $\rho$,
that is

(2.15) $\displaystyle\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\left(\mathcal{L}_{\sigma_{k,1}^{n}}\mathcal{L}_{\sigma_{-k,2}^{n}}+\mathcal{L}_{\sigma_{k,2}^{n}}\mathcal{L}_{\sigma_{-k,1}^{n}}\right)$

We write the general form of the two summands, for $l\neq j$, as

$\displaystyle\mathcal{L}_{\sigma_{k,l}^{n}}\mathcal{L}_{\sigma_{-k,j}^{n}}F$
$\displaystyle=\sigma_{k,l}^{n}\cdot\operatorname{\nabla}(\sigma_{-k,j}^{n}\cdot\nabla
F)-\sigma_{k,l}^{n}\cdot\operatorname{\nabla}(F\cdot\nabla\sigma_{-k,j}^{n})-(\sigma_{-k,j}^{n}\cdot\nabla
F)\cdot\nabla\sigma_{k,l}^{n}+(F\cdot\nabla\sigma_{-k,j}^{n})\cdot\nabla\sigma_{k,l}^{n}.$

Again, by (2.11) the last term is always zero. Now summing over $k$ and $l\neq
j$ we see, using first (2.10) and reasoning as in the manipulation of (2.14)

(2.16) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\sigma^{n}_{k,1}\cdot\operatorname{\nabla}(\sigma^{n}_{-k,2}\cdot\nabla
F)+\sigma^{n}_{k,2}\cdot\operatorname{\nabla}(\sigma^{n}_{-k,1}\cdot\nabla F)$
$\displaystyle=\operatorname{div}(\overline{Q_{\rho}^{n}}(0)\nabla F)=0.$

We are left with the contributions of the middle terms which no longer sum to
zero as in the first corrector. Recalling our choice of the coefficients
(2.3), we see that if $l\neq j$ and $n\leq\lvert k\rvert\leq 2n,\ k_{3}=0$

$\theta_{k,j}^{n}\theta_{-k,l}^{n}=-\theta_{k,l}^{n}\theta_{-k,j}^{n}.$

Thus, when we sum over $l\neq j$, we get from (2.12) and (2.13) that

(2.17)
$\displaystyle-\sigma^{n}_{k,1}\cdot\operatorname{\nabla}(F\cdot\nabla\sigma^{n}_{-k,2})-(\sigma^{n}_{-k,1}\cdot\nabla
F)\cdot\nabla\sigma^{n}_{k,2}$
$\displaystyle=2i\theta_{k,1}\theta_{-k,2}a^{r}_{k,1}(\partial_{r}F\cdot
k)a_{k,2}$
$\displaystyle=-\frac{2\text{sgn}(k)}{|k|^{\alpha/2+\beta/2}}a^{r}_{k,1}(\partial_{r}F\cdot
k)a_{k,2}$

and similarly

(2.18) $\displaystyle-(\sigma^{n}_{-k,2}\cdot\nabla
F)\cdot\nabla\sigma^{n}_{k,1}-\sigma^{n}_{k,2}\cdot\operatorname{\nabla}(F\cdot\nabla\sigma^{n}_{-k,1})$
$\displaystyle=-2i\theta^{n}_{-k,2}\theta^{n}_{k,1}a^{r}_{k,2}(\partial_{r}F\cdot
k)a_{k,1}$
$\displaystyle=\frac{2\text{sgn}(k)}{|k|^{\alpha/2+\beta/2}}a^{r}_{k,2}(\partial_{r}F\cdot
k),a_{k,1}$

where we indicated con $\text{sgn}(k)$ the function that takes value $1$ if
$k\in\Gamma^{+}$ and $-1$ if $k\in\Gamma^{-}$. Summing the last two
expressions and then summing over $k\in\mathbb{Z}_{0}^{3},\ k_{3}=0$ it is
straightforward to see thanks to 2.12 that

$\displaystyle\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0\end{subarray}}\left(\mathcal{L}_{\sigma_{k,1}^{n}}\mathcal{L}_{\sigma_{-k,2}^{n}}+\mathcal{L}_{\sigma_{k,2}^{n}}\mathcal{L}_{\sigma_{-k,1}^{n}}\right)$
$\displaystyle=-\sum_{l\in\\{1,2\\}}\partial_{l}\overline{Q^{n}_{\rho}}(0)\cdot\nabla
F_{l}$ $\displaystyle=\Lambda^{n}_{\rho}F.$

∎

According to 2.13, equation (2.9) can be rewritten in Itô form as:

(2.19)
$\begin{cases}dB^{n}_{t}&=\left(\eta\Delta+\Lambda^{n}+\Lambda^{n}_{\rho}\right)B^{n}_{t}dt+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}dW^{k,j}_{t},\\\
\operatorname{div}B^{n}_{t}&=0,\\\ B^{n}_{t}|_{t=0}&=B^{n}_{0},\end{cases}$

We give now the definition of weak solution for the Stochastic equation of the
magnetic field (2.19).

######  Definition 2.14.

A stochastic process

$\displaystyle B^{n}\in C_{\mathcal{F}}([0,T];\mathbf{L}^{2})\cap
L^{2}_{\mathcal{F}}([0,T];\mathbf{H}^{1})$

is a weak solution of equation (2.19) if, for every $\phi\in\mathbf{H}^{2}$,
we have

(2.20) $\displaystyle\langle B^{n}_{t},\phi\rangle$ $\displaystyle=\langle
B^{n}_{0},\phi\rangle+\eta\int_{0}^{t}\langle
B^{n}_{s},\Delta\phi\rangle\,ds+\int_{0}^{t}\langle
B^{n}_{s},\Lambda^{n}\phi\rangle\,ds+\int_{0}^{t}\langle\Lambda^{n}_{\rho}B^{n}_{s},\phi\rangle\,ds+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{s},\phi\rangle
dW^{k,j}_{s}.$

for every $t\in[0,T],\ \mathbb{P}-a.s.$

######  Remark 2.15.

Due to the definition of the operator $P$, each vector field
$\phi\in\dot{H}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$ can be decomposed in a
divergence free part, i.e. $P\phi\in\mathbf{H}^{2}$, and a part orthogonal to
$\mathbf{L}^{2}$, i.e.
$(I-P)\phi\in\dot{H}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$. Therefore for a
generic $\phi\in\dot{H}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$ we get

$\displaystyle\langle B^{n}_{t},\phi\rangle-\langle B^{n}_{0},\phi\rangle$
$\displaystyle=\langle B^{n}_{t},P\phi\rangle-\langle B^{n}_{0},P\phi\rangle$
$\displaystyle=\eta\int_{0}^{t}\langle B^{n}_{s},\Delta
P\phi\rangle\,ds+\int_{0}^{t}\langle
B^{n}_{s},\Lambda^{n}P\phi\rangle\,ds+\int_{0}^{t}\langle\Lambda^{n}_{\rho}B^{n}_{s},P\phi\rangle\,ds$
$\displaystyle+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{s},P\phi\rangle
dW^{k,j}_{s}$ $\displaystyle=\eta\int_{0}^{t}\langle
B^{n}_{s},\Delta\phi\rangle\,ds+\int_{0}^{t}\langle
B^{n}_{s},\Lambda^{n}\phi\rangle\,ds+\int_{0}^{t}\langle\Lambda^{n}_{\rho}B^{n}_{s},\phi\rangle\,ds+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{s},\phi\rangle
dW^{k,j}_{s}$

for every $t\in[0,T],\ \mathbb{P}-a.s.$, due to the fact that $B^{n}\in
C_{\mathcal{F}}([0,T];\mathbf{L}^{2})\cap
L^{2}_{\mathcal{F}}([0,T];\mathbf{H}^{1})$ and the structure of the operators
$\Lambda^{n},\ \Lambda^{n}_{\rho}$. In particular, if we choose
$\phi=(0,0,\varphi)\in\dot{H}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$ we obtain

$\displaystyle\langle B^{n,3}_{t},\varphi\rangle-\langle
B^{n,3}_{0},\varphi\rangle$ $\displaystyle=\eta\int_{0}^{t}\langle
B^{n,3}_{s},\Delta\varphi\rangle\,ds+\int_{0}^{t}\langle
B^{n,3}_{s},\Lambda^{n}\varphi\rangle\,ds+\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}\langle{B^{n,H}_{s}}^{\perp},\nabla_{H}\varphi\rangle\,ds$
$\displaystyle+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{s}-B^{n,3}\cdot\nabla\sigma_{k,j}^{n,3},\varphi\rangle dW^{k,j}_{s}$

for every $t\in[0,T],\ \mathbb{P}-a.s.$

Due to the fact that equation (2.19) is linear, the existence and uniqueness
of solutions in the sense of 2.14 is a standard fact which follows by the
abstract theory of [8, Chapters 3-5], see also [14, Section 3.1]. Indeed the
following holds.

######  Theorem 2.16.

For each $B^{n}_{0}\in\mathbf{L}^{2}$ there exists a unique weak solution of
system (2.19) in the sense of 2.14.

As discussed in section 1, we study the convergence of ${B^{n}_{t}}$ in the
scaling limit of the separation of scales, i.e. considering a noise which
concentrates on smaller and smaller scales as described in subsection 2.3.
Lastly we introduce our limit objects. Following the idea first introduced in
[17], we expect that our limit objects satisfy a PDE with an additional second
order operator with intensity related to

$\displaystyle\operatorname{lim}_{n\rightarrow+\infty}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\left\|\sigma_{k,j}^{n}\right\|^{2}.$

The quantity above is different from $0$ if $\alpha=2$ in 2.4, 2.5 or if it
holds 2.6. Besides to the fact that the limit above may be $0$ under our
assumptions, the final deterministic objects keep memory of our noise thanks
to the fact that $\alpha+\beta=6$ for each $n$, which implies that our noise
has a nontrivial limit helicity. Denoting by $\overline{B_{t}}$ the unique
weak solutions of the following linear 3D PDE

(2.21)
$\displaystyle\begin{cases}\partial_{t}\overline{B_{t}}&=(\eta\Delta+\Lambda_{\alpha,\beta,\gamma})\overline{B_{t}}+\nabla\times(\mathcal{A}_{\rho}\overline{B_{t}})\quad
x\in\mathbb{T}^{3},\ t\in(0,T)\\\ \operatorname{div}\overline{B}&=0\quad
x\in\mathbb{T}^{3},\ t\in(0,T)\\\
\overline{B_{t}}|_{t=0}&=\overline{B_{0}},\end{cases}$

where

$\displaystyle\Lambda_{\alpha,\beta,\gamma}$
$\displaystyle=\begin{cases}0\quad&\text{if \autoref{HP noise 1} holds and
$\alpha>2$,}\\\ \eta_{T}\Delta_{H}\quad&\text{if \autoref{HP noise 1} holds
and $\alpha=2$,}\\\ \eta_{iso}\Delta\quad&\text{if \autoref{HP noise isotropo}
holds},\end{cases}\quad\mathcal{A}_{\rho}=\frac{2\pi\rho\log{2}}{C_{1,H}C_{2,H}}\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{bmatrix},$

$\eta_{T}=\frac{\zeta_{H,2}}{2C_{1,H}^{2}},\
\eta_{iso}=\frac{2\zeta_{s,3}}{3C_{V}^{2}}$. Our main result reads as follows:

######  Theorem 2.17.

Let $B_{0}^{n}\in\mathbf{L}^{2}$, assuming 2.4 or 2.6 and
$B^{n}_{0}\rightharpoonup\overline{B_{0}}\text{ in }\mathbf{H}^{-1},$ for each
$\kappa\in[1,2),\ \theta\in(0,2),\ \delta\in(0,\theta)$ we have

(2.22)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|B^{n}_{t}-\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right]$
$\displaystyle\lesssim\begin{cases}\frac{1}{n^{\kappa\theta\chi(\alpha,\beta,\gamma)/2}}+\frac{1}{n^{\frac{\kappa(\alpha\wedge\beta\wedge\gamma)(\theta-\delta)}{6+\delta}}}+\left\|\overline{B_{0}}-B^{n}_{0}\right\|^{\kappa}_{\mathbf{H}^{-1-\theta}}&\text{in
case of \autoref{HP noise 1} },\\\
\frac{1}{n^{\frac{3\kappa(\theta-\delta)}{6+\delta}}}+\left\|\overline{B_{0}}-B^{n}_{0}\right\|^{\kappa}_{\mathbf{H}^{-1-\theta}}&\text{in
case of \autoref{HP noise isotropo} }.\end{cases}$

Moreover, assuming also $B^{n,3}\rightharpoonup\overline{B^{3}_{0}}$ in
$\dot{L}^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$, in case of 2.5 we have also for
each $\vartheta\in(\theta,\frac{3}{2}]$

(2.23)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|B^{n,3}_{t}-\overline{B^{3}_{t}}\right\|_{\dot{H}^{-\vartheta}}^{\kappa}\right]$
$\displaystyle\lesssim\frac{1}{n^{\frac{\kappa(\vartheta-\theta)(\theta-\delta)}{6+\delta}}}+\frac{1}{(\vartheta-\theta)^{\kappa}n^{\kappa\theta/2}}+\left\|\overline{B^{3}_{0}}-B^{n,3}_{0}\right\|^{\kappa}_{\dot{H}^{-\vartheta}}+\frac{\left\|\overline{B_{0}}-B^{n}_{0}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}}{(\vartheta-\theta)^{\kappa}}.$

In particular either the right hand side of (2.22) and of (2.23) converge to
$0$ as $n\rightarrow+\infty$.

######  Remark 2.18.

The hidden constant depends on all the parameters of the model and blows-up as
$\delta\rightarrow 0$ or if either $\eta_{iso}\nearrow\eta$ in case of 2.6 or
$2\eta_{T}\nearrow\eta$ in case of 2.4 and $\alpha=2.$

######  Remark 2.19.

Due to 2.6 $\eta_{iso}<\eta$. In case of 2.4 with $\alpha=2$ we have
$\eta_{T}<\frac{\eta}{2}$.

Contrary to our previous work [3], here also a compactness approach inspired
to [12, Section 5] works. Indeed, arguing similarly to 4.6, it is possible to
study the behaviour of the time increments of $B^{n}_{t}$ in some Sobolev
spaces of negative order. Combining this control and the uniform estimates of
3.4 allows to prove the tightness of the laws of
$\\{B^{n}\\}_{n\in\mathbb{N}}$ in $L^{2}(0,T;H^{-\delta})$ for each
$\delta>0$. Then, arguing by Prokhorov’s theorem and Skorokhod’s
representation theorem, it is possible to find an auxiliary probability space
where, up to passing to subsequences, $B^{n}\rightarrow\overline{B}$ in
$L^{2}(0,T;H^{-\delta})\quad\mathbb{P}-a.s.$ Arguing similarly to section 4,
we can identify $\overline{B}$ as the unique weak solution of (2.21). This
implies that actually the full sequence $B^{n}\rightarrow\overline{B}$ in
$L^{2}(0,T;H^{-\delta})$ in probability. Similar arguments applies also to
$B^{n,3}$ in case of 2.5. We preferred to rely on semigroup techniques and
state our main result in the form of Theorem 2.17 for a twofold reason. First,
this quantitative result gives explicit information on the rate of convergence
which we think are of independent interest. Secondly, the convergence in
$C([0,T];L^{2}(\Omega;\mathbf{H}^{-1-\theta}))$ guaranteed by Theorem 2.17
gives us more information about dynamo action of the stochastic model. Indeed,
as a corollary of Theorem 2.17 we can easily obtain 2.20 and 2.21.

######  Corollary 2.20.

Let $k=\lambda e_{3},$ for $\lambda\in\mathbb{Z}_{0}$ and
$B_{0}^{n}=\overline{B_{0}}=(\sin(k\cdot x),\cos(k\cdot x),0)^{t}$. Assuming
either 2.4 or 2.5, if $\lambda\rho>0$ and $C_{2,H}<\frac{2\pi\rho\log
2}{\lambda C_{1,H}{{\eta}}},$ then for each $\theta\in(0,2),\
\delta\in(0,\theta)$

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|^{2}\right]\gtrsim\frac{1}{\lambda^{2(1+\theta)}}e^{-(\lambda{\mathcal{H}}+2\lambda^{2}{\eta})t}-\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)}}-\frac{1}{n^{\frac{2(\alpha\wedge\beta\wedge\gamma)(\theta-\delta)}{6+\delta}}}.$

In particular, the energy of $B^{n}$ is increasing in time in mean.

###### Proof.

the unique solution of equation (2.21) with initial condition $(\sin(k\cdot
x),\cos(k\cdot x),0)^{t}$ is given by

$\displaystyle\overline{B_{t}}=e^{-(\lambda\frac{\mathcal{H}}{2}+\lambda^{2}\eta)t}(\sin(k\cdot
x),\cos(k\cdot x),0)^{t}.$

In particular, for each $\theta\in(0,2)$

$\displaystyle\left\|\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{2}=\frac{8\pi^{3}}{\lambda^{2(1+\theta)}}e^{-(\lambda{\mathcal{H}}+2\lambda^{2}\eta)t}.$

Under our assumptions $\lambda\frac{\mathcal{H}}{2}+\lambda^{2}\eta<0$.
Therefore, since for each $\kappa\in[1,2)$

$\displaystyle\left\|\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\leq
2^{\kappa-1}\left(\left\|{B^{n}_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}+\left\|B^{n}_{t}-\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right),$

thanks to Theorem 2.17 and Jensen’s inequality we obtain

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|^{2}\right]^{\frac{\kappa}{2}}$
$\displaystyle\geq\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right]$
$\displaystyle\gtrsim\frac{1}{\lambda^{\kappa(1+\theta)}}e^{-\kappa(\lambda\frac{\mathcal{H}}{2}+\lambda^{2}\eta)t}-\frac{1}{n^{\kappa\theta\chi(\alpha,\beta,\gamma)/2}}-\frac{1}{n^{\frac{\kappa(\alpha\wedge\beta\wedge\gamma)(\theta-\delta)}{6+\delta}}}$

which implies the claim. ∎

######  Corollary 2.21.

If 2.6 holds and $B_{0}^{n}=\overline{B_{0}}\in\mathbf{L^{2}}$, then for each
$\kappa\in[1,2),\ \theta\in(0,2),\ \delta\in(0,\theta)$ we have

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H^{-1-\theta}}}^{\kappa}\right]\leq\frac{C}{n^{\frac{3\kappa(\theta-\delta)}{6+\delta}}}+2^{\kappa-1}\left\|\overline{B_{0}}\right\|^{\kappa}e^{-\kappa\left(\eta+\frac{(8-6\lvert\rho\rvert)\pi\log
2}{3C_{V}^{2}}\right)t},$

where $C$ is a constant depending on
$\overline{B_{0}},C_{V},\rho,\theta,\delta,\kappa$.

###### Proof.

Thanks to simple energy estimates on (2.21) and Poincaré inequality we obtain

$\displaystyle\left\|\overline{B_{t}}\right\|^{2}+2\left(\eta+\frac{8\pi\log
2}{3C_{V}^{2}}\right)\int_{0}^{t}\left\|\nabla\overline{B_{s}}\right\|^{2}ds$
$\displaystyle\leq\left\|\overline{B_{0}}\right\|^{2}+\frac{4\pi\lvert\rho\rvert\log
2}{C_{V}^{2}}\int_{0}^{t}\left\|\nabla\overline{B_{s}}\right\|^{2}ds.$

Therefore, by Grönwall’s inequality

$\displaystyle\left\|\overline{B_{t}}\right\|^{\kappa}\leq\left\|\overline{B_{0}}\right\|^{\kappa}e^{-\kappa\left(\eta+\frac{(8-6\lvert\rho\rvert)\pi\log
2}{3C_{V}^{2}}\right)t}.$

As a consequence of the inequality above and Theorem 2.17 we obtain

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right]$
$\displaystyle\leq
2^{\kappa-1}\left(\mathbb{E}\left[\left\|B^{n}_{t}-\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right]+\left\|\overline{B_{t}}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right)$
$\displaystyle\leq\frac{C_{B_{0},C_{V},\rho,\theta,\delta,\kappa}}{n^{\frac{3\kappa(\theta-\delta)}{6+\delta}}}+2^{\kappa-1}\left\|\overline{B_{0}}\right\|^{\kappa}e^{-\kappa\left(\eta+\frac{(8-6\lvert\rho\rvert)\pi\log
2}{3C_{V}^{2}}\right)t}.$

∎

The two corollaries above have a clear interpretation in terms of the
discussion of section 1 and subsection 2.3. Indeed, 2.20 shows that if the
helicity of our velocity field is large enough and the turbulent fluid is
sufficiently anisotropic, the energy of our stochastic models increases in
time. Therefore, a dynamo effect is active on our stochastic models in case of
2.4. On the contrary, in case of 2.6, due to 2.21, the dissipation acting on
magnetic field increases due to the presence of the turbulent fluid. This is
accordance to [19]. Moreover, 2.21 implies that it is not needed that the
fluid is either completely isotropic or mirror symmetric in order to increase
the dissipative effects acting on the magnetic field.

######  Remark 2.22.

Contrary to 2.6, the matrix $Q^{n}(0)$ converge to a degenerate matrix as
$n\rightarrow+\infty$ in case of 2.4. This phenomena produces a degenerate
second order operator in the scaling limit. It is possible to avoid such a
phenomena in case of 2.4 and $\alpha=2$ considering a more general noise

$\displaystyle
W^{n}_{t}=\frac{1}{3}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}a_{k,j}e^{ik\cdot
x}\left(\theta^{n}_{1,k,j}W^{1,k,j}_{t}+\theta^{n}_{2,k,j}W^{2,k,j}_{t}+\theta^{n}_{3,k,j}W^{3,k,j}_{t}\right),$

where $W^{h,k,j}_{t}$ are complex Brownian motions satisfying

$\displaystyle\mathbb{E}\left[W^{h,k,j}_{1},{W^{h^{\prime},l,m}_{1}}^{*}\right]$
$\displaystyle=\begin{cases}0&\textit{if }h\neq h^{\prime}\\\ 2&\textit{if
}h=h^{\prime},\ k=l,\ m=j\\\ 2\rho_{h}&\textit{if }h=h^{\prime},\ k=l,\
k_{h}=0,\ m\neq j\end{cases}$

and

$\displaystyle\theta_{h,k,j}^{n}$ $\displaystyle=\bm{1}_{\\{n\leq\lvert
k\rvert\leq 2n\\}}\begin{cases}\frac{i}{C_{1,H}\lvert
k\rvert^{\alpha/2}}&\text{if }k_{h}=0,k\in\Gamma_{+},\ j=1\\\
\frac{-i}{C_{1,H}\lvert k\rvert^{\alpha/2}}&\text{if }k_{h}=0,k\in\Gamma_{-},\
j=1\\\ \frac{1}{C_{2,H}\lvert k\rvert^{\beta/2}}&\text{if }k_{h}=0,\ j=2\\\
\frac{1}{C_{V}\lvert k\rvert^{\gamma/2}}&\text{if }k_{h}\neq 0.\end{cases}$

In this case we get

$\displaystyle\Lambda_{\alpha,\beta,\gamma}=\frac{2}{3}\eta_{T}\Delta,\quad\mathcal{A}_{\rho_{1},\rho_{2},\rho_{3}}=\frac{2\pi\log
2}{3C_{1,H}C_{2,H}}\left(\rho_{1}\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}+\rho_{2}\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}+\rho_{3}\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}\right)$

and we can prove an analogous result to Theorem 2.17. Since this change would
drastically complicate the notation without any change either in the proof of
Theorem 2.17 and in the physical understanding of the dynamo and the
dissipative properties of our stochastic systems, i.e. 2.20 and 2.21, we
prefer to limit ourselves to the noise introduced in subsection 2.3. In this
spirit the _helical_ regime could be regarded as a stochastic perturbation of
a Robert’s flow [26] in which we can also allow dependence on the third space
variable. This kind of flows are extensively studied in the physical and
numerical literature as paradigmatic models for dynamo action (see for
instance [27], [31], [5] and references therein)

##  3\. A Priori Estimates

The goal of this section is to provide uniform bounds for the
$\mathbf{H}^{-1}$ norm of our vector fields $B^{n}_{t}$. In order to complete
our plan it is useful to recall a well-known result about equivalence of norms
for divergence free vector fields, see for example [22].

######  Proposition 3.1.

Let $X\in\mathbf{L}^{2},\ Y\in\mathbf{H}^{1}$. Then

(3.1) $\displaystyle\left\|X\right\|^{2}$
$\displaystyle=\left\|(-\Delta)^{-1/2}\nabla\times
X\right\|^{2},\quad\left\|\nabla Y\right\|^{2}=\left\|\nabla\times
Y\right\|^{2}.$

We start providing a control to some quantities related to the noise appearing
in equation (2.19).

######  Lemma 3.2.

Assuming 2.4, then

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}\leq\left(\frac{\zeta^{n}_{H,\alpha}}{C_{1,H}^{2}n^{2-\alpha}}+\frac{\zeta^{n}_{H,\beta}}{C_{2,H}^{2}n^{2-\beta}}+\frac{2\zeta^{n}_{s,\gamma}}{C_{V}^{2}n^{3-\gamma}}\right)\lVert{B^{n}_{t}}\rVert^{2}.$

In case of 2.6, then

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}=\frac{4\zeta^{n}_{s,3}}{3C_{V}^{2}}\lVert{B^{n}_{t}}\rVert^{2}.$

In particular, assuming 2.5 we have also

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{t}\rVert^{2}$
$\displaystyle\leq\frac{\zeta_{H,\alpha}^{n}\lVert\nabla_{H}B^{n,3}_{t}\rVert^{2}}{2C_{1,H}^{2}n^{\alpha-2}}+\frac{\zeta^{n}_{H,\beta}\lVert\partial_{3}B^{n,3}_{t}\rVert^{2}}{C_{2,H}^{2}n^{\beta-2}}+\frac{2\zeta^{n}_{s,\gamma}\lVert\nabla
B^{n,3}_{t}\rVert^{2}}{3C_{V}^{2}n^{\gamma-3}},$
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert
B^{n}_{t}\cdot\nabla\sigma_{k,j}^{n,3}\rVert^{2}$
$\displaystyle\leq\left(\frac{\zeta^{n}_{H,2}}{C_{2,H}^{2}}+\frac{\zeta^{n}_{s,\gamma-2}}{C_{V}^{2}n^{\gamma-5}}\right)\lVert
B^{n}_{t}\rVert^{2}.$

###### Proof.

By Hölder’s inequality we have

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}\leq\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}\lVert{B^{n}_{t}}\rVert^{2}.$

In the first case, due to the relation above we get

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0,\quad
j=1\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}=0,\quad
j=2\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
k_{3}\neq 0,\quad
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}\right)\lVert{B^{n}_{t}}\rVert^{2}$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{C_{1,H}^{2}\lvert
k\rvert^{\alpha}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{C_{2,H}^{2}\lvert
k\rvert^{\beta}}+2\sum_{\begin{subarray}{c}k\in\mathbb{Z}_{0}^{3}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{C_{V}^{2}\lvert
k\rvert^{\gamma}}\right)\lVert{B^{n}_{t}}\rVert^{2}$
$\displaystyle\leq\left(\frac{\zeta^{n}_{H,\alpha}}{C_{1,H}^{2}n^{2-\alpha}}+\frac{\zeta^{n}_{H,\beta}}{C_{2,H}^{2}n^{2-\beta}}+\frac{2\zeta^{n}_{s,\gamma}}{C_{V}^{2}n^{3-\gamma}}\right)\lVert{B^{n}_{t}}\rVert^{2}.$

However, in the second case, due to the symmetries of the coefficients, we can
obtain a sharp relation which is our claim. Let us denote by

$\displaystyle b^{n,h,l}_{t}:=\langle{B^{n}_{t}},a_{h,l}e^{-ih\cdot
x}\rangle,$

therefore by definition

$\displaystyle{B^{n}_{t}}$
$\displaystyle=\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l\in\\{1,2\\}\end{subarray}}b^{n,h,l}_{t}a_{h,l}e^{ih\cdot x},$
$\displaystyle\sigma^{n}_{k,j}\times{B^{n}_{t}}$
$\displaystyle=\theta_{k,j}^{n}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l\in\\{1,2\\}\end{subarray}}b^{n,h,l}_{t}a_{k,j}\times a_{h,l}e^{i(h+k)\cdot
x}$

and for each $k,j$ it holds

$\displaystyle\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}$
$\displaystyle=\lvert\theta_{k,j}^{n}\rvert^{2}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l,l^{\prime}\in\\{1,2\\}\end{subarray}}\left(a_{k,j}\times
a_{h,l}\right)\cdot\left(a_{k,j}\times
a_{h,l^{\prime}}\right)b^{n,h,l}_{t}\overline{b^{n,h,l^{\prime}}_{t}}$
$\displaystyle=\lvert\theta_{k,j}^{n}\rvert^{2}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l,l^{\prime}\in\\{1,2\\}\end{subarray}}\left(\delta_{l,l^{\prime}}-a_{k,j}\otimes
a_{k,j}:a_{h,l}\otimes
a_{h,l^{\prime}}\right)b^{n,h,l}_{t}\overline{b^{n,h,l^{\prime}}_{t}}$ (3.2)
$\displaystyle=\lvert\theta_{k,j}^{n}\rvert^{2}\lVert{B^{n}_{t}}\rVert^{2}-\lvert\theta_{k,j}^{n}\rvert^{2}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l,l^{\prime}\in\\{1,2\\}\end{subarray}}a_{k,j}\otimes a_{k,j}:a_{h,l}\otimes
a_{h,l^{\prime}}b^{n,h,l}_{t}\overline{b^{n,h,l^{\prime}}_{t}}.$

Therefore, in case of 2.6 we obtain

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}$
$\displaystyle=\frac{2\zeta^{n}_{s,3}}{C_{V}^{2}}\lVert{B^{n}_{t}}\rVert^{2}-\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l,l^{\prime}\in\\{1,2\\}\end{subarray}}\frac{2\zeta^{n}_{s,3}}{3C_{V}^{2}}I:\left(a_{h,l}\otimes
a_{h,l^{\prime}}\right)b^{n,h,l}_{t}\overline{b^{n,h,l^{\prime}}_{t}}$
$\displaystyle=\frac{4\zeta^{n}_{s,3}}{3C_{V}^{2}}\lVert{B^{n}_{t}}\rVert^{2}.$

Lastly we assume 2.5, in this case only few terms in the sum must be
considered due to our choice of the $\sigma_{k,j}^{n}$’s. Then, we obtain

$\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{t}\rVert^{2}\\\ =\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq
2n\end{subarray}}\left(\frac{\lVert\nabla_{H}B^{n,3}_{t}\rVert^{2}}{2C_{1,H}^{2}\lvert
k\rvert^{\alpha}}+\frac{\lVert\partial_{3}B^{n,3}_{t}\rVert^{2}}{C_{2,H}^{2}\lvert
k\rvert^{\beta}}\right)+\frac{1}{C_{V}^{2}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0},\
k_{3}\neq 0\\\ n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{\gamma}}\left\langle\left(I-\frac{k\otimes k}{\lvert
k\rvert^{2}}\right)\nabla B^{n,3}_{t},\nabla B^{n,3}_{t}\right\rangle\\\
\leq\frac{\zeta_{H,\alpha}^{n}\lVert\nabla_{H}B^{n,3}_{t}\rVert^{2}}{2C_{1,H}^{2}n^{\alpha-2}}+\frac{\zeta^{n}_{H,\beta}\lVert\partial_{3}B^{n,3}_{t}\rVert^{2}}{C_{2,H}^{2}n^{\beta-2}}+\frac{1}{C_{V}^{2}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{\gamma}}\left\langle\left(I-\frac{k\otimes k}{\lvert
k\rvert^{2}}\right)\nabla B^{n,3}_{t},\nabla B^{n,3}_{t}\right\rangle\\\
=\frac{\zeta_{H,\alpha}^{n}\lVert\nabla_{H}B^{n,3}_{t}\rVert^{2}}{2C_{1,H}^{2}n^{\alpha-2}}+\frac{\zeta^{n}_{H,\beta}\lVert\partial_{3}B^{n,3}_{t}\rVert^{2}}{C_{2,H}^{2}n^{\beta-2}}+\frac{2\zeta^{n}_{s,\gamma}\lVert\nabla
B^{n,3}_{t}\rVert^{2}}{3C_{V}^{2}n^{\gamma-3}}$

and by Hölder’s inequality

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert
B^{n}_{t}\cdot\nabla\sigma_{k,j}^{n,3}\rVert^{2}$
$\displaystyle\leq\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{\lVert
B^{n}_{t}\rVert^{2}}{C_{2,H}^{2}\lvert
k\rvert^{\beta-2}}+\sum_{\begin{subarray}{c}\lvert k\in Z^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{\lVert
B^{n}_{t}\rVert^{2}}{C_{V}^{2}\lvert k\rvert^{\gamma-2}}$
$\displaystyle\leq\left(\frac{\zeta^{n}_{H,2}}{C_{2,H}^{2}}+\frac{\zeta^{n}_{s,\gamma-2}}{C_{V}^{2}n^{\gamma-5}}\right)\lVert
B^{n}_{t}\rVert^{2}.$

∎

######  Remark 3.3.

Note that the simple Hölder estimate in case of 2.6 gives us

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times{B^{n}_{t}}\rVert^{2}$
$\displaystyle\leq 2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{3}C_{V}^{2}}\lVert{B^{n}_{t}}\rVert^{2}$
$\displaystyle=\frac{2\zeta^{n}_{s,3}}{C_{V}^{2}}\lVert{B^{n}_{t}}\rVert^{2},$

which is worse of the second statement of 3.2 due to the fact that we are
neglecting the second term of (3) which produce some cancellations.

Now we are ready to prove the main result of this section.

######  Proposition 3.4.

Assuming 2.4 or 2.6 then

(3.3)
$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]+\mathbb{E}\left[\int_{0}^{T}\left\|B^{n}_{t}\right\|^{2}\right]dt\lesssim\left\|B_{0}\right\|_{\mathbf{H}^{-1}}^{2}.$

In particular if 2.5 is satisfied then also

(3.4)
$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\|B^{n,3}_{t}\right\|^{2}\right]+\int_{0}^{T}\mathbb{E}\left[\left\|\nabla
B^{n,3}_{t}\right\|^{2}\right]dt\lesssim\left\|B_{0}\right\|_{\mathbf{H}^{-1}}^{2}+\left\|B^{3}_{0}\right\|^{2}.$

###### Proof.

Since $\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{t})=\mathcal{L}_{\sigma_{k,j}^{n}}B^{n}_{t}$, the Itô formula for
$\|B^{n}_{t}\|^{2}_{\mathbf{H}^{-1}}$ reads as

$\displaystyle d\|(-\Delta)^{-1/2}B^{n}_{t}\|^{2}+2\eta\|B^{n}_{t}\|^{2}dt$
$\displaystyle=2\left\langle(-\Delta)^{-1}\Lambda^{n}B^{n}_{t},B^{n}_{t}\right\rangle
dt$
$\displaystyle+2\left\langle(-\Delta)^{-1/2}\Lambda_{\rho}^{n}B^{n}_{t},(-\Delta)^{-1/2}B^{n}_{t}\right\rangle
dt$ $\displaystyle+2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\|(-\Delta)^{-1/2}\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{t})\|^{2}dt$
$\displaystyle+2\rho\sum_{k\in\mathbb{Z}^{2}_{0}}\left\langle\nabla\times(\sigma_{k,1}^{n}\times
B^{n}_{t}),(-\Delta)^{-1}\nabla\times(\sigma_{-k,2}^{n}\times
B^{n}_{t})\right\rangle dt$
$\displaystyle+2\rho\sum_{k\in\mathbb{Z}^{2}_{0}}\left\langle\nabla\times(\sigma_{k,2}^{n}\times
B^{n}_{t}),(-\Delta)^{-1}\nabla\times(\sigma_{-k,1}^{n}\times
B^{n}_{t})\right\rangle dt$ (3.5) $\displaystyle+dM^{n}_{t},$

where

$\displaystyle M^{n}_{t}$
$\displaystyle=2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle(-\Delta)^{-1}\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s}),B^{n}_{s}\rangle dW^{k,j}_{s}.$

We treat first the case of 2.6 being easier. By the definition of
$\Lambda^{n}$ in this case we get easily

(3.6)
$\displaystyle\left\langle(-\Delta)^{-1}\Lambda^{n}B^{n}_{t},B^{n}_{t}\right\rangle=-\eta^{n}_{iso}\left\|B^{n}_{t}\right\|^{2}.$

Concerning the Itô correctors we have

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\|(-\Delta)^{-1/2}\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{t})\|^{2}$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\langle(-\Delta)^{-1}\nabla\times\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{t}),\overline{\sigma_{k,j}^{n}\times B^{n}_{t}}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\langle(-\Delta)^{-1}\nabla\times\nabla\times
P[Q[\sigma_{k,j}^{n}\times B^{n}_{t}]],\overline{\sigma_{k,j}^{n}\times
B^{n}_{t}}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\langle P[Q[\sigma_{k,j}^{n}\times
B^{n}_{t}]],\overline{P[Q[\sigma_{k,j}^{n}\times B^{n}_{t}]]}\rangle$ (3.7)
$\displaystyle\leq\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert\sigma_{k,j}^{n}\times
B^{n}_{t}\rVert^{2}=2\eta^{n}_{iso}\lVert{B^{n}_{t}}\rVert^{2},$

due to the properties of the projections $P,\ Q$, relation (3.1) and 3.2. The
other terms appearing in (3) can be treated easily by Hölder and Young’s
inequalities. Indeed, again thanks to relation (3.1)

$\displaystyle\left\langle(-\Delta)^{-1/2}\Lambda_{\rho}^{n}B^{n}_{t},(-\Delta)^{-1/2}B^{n}_{t}\right\rangle$
$\displaystyle\leq\frac{\lvert\rho\rvert\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\lVert
B^{n}_{t}\rVert_{\mathbf{H}^{-1}}\left\|(-\Delta)^{-1/2}\nabla\times
B^{n,H}_{t}\right\|$
$\displaystyle\leq\frac{\lvert\rho\rvert\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\lVert
B^{n}_{t}\rVert_{\mathbf{H}^{-1}}\left\|B^{n}_{t}\right\|$ (3.8)
$\displaystyle\leq\frac{(\eta-\eta_{iso})}{2}\left\|B^{n}_{t}\right\|^{2}+\frac{\lvert\rho\rvert^{2}\left(\zeta^{n}_{H,2}\right)^{2}}{2C_{1,H}^{2}C_{2,H}^{2}(\eta-\eta_{iso})}\lVert
B^{n}_{t}\rVert_{\mathbf{H}^{-1}}^{2}$

since by assumptions $\eta>\eta_{iso}$ and

$\displaystyle\rho\sum_{k\in\mathbb{Z}^{2}_{0}}\left\langle\nabla\times(\sigma_{k,1}^{n}\times
B^{n}_{t}),(-\Delta)^{-1}\nabla\times(\sigma_{-k,2}^{n}\times
B^{n}_{t})\right\rangle$
$\displaystyle\leq\lvert\rho\rvert\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{C_{1,H}C_{2,H}\lvert
k\rvert^{\alpha/2+\beta/2}}\lVert B^{n}_{t}\rVert^{2}$ (3.9)
$\displaystyle\leq\lvert\rho\rvert\frac{\zeta^{n}_{H,\alpha/2+\beta/2-2}}{C_{1,H}C_{2,H}n^{\alpha/2+\beta/2-2}}\lVert
B^{n}_{t}\rVert^{2}$

and analogously for the other. Therefore, integrating (3) between $0$ and $t$
and considering its expected value, thanks to (3.6), (3),(3),(3) we obtain

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]+\eta\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\right\|^{2}ds\right]$
$\displaystyle\leq\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}+\left(\eta_{iso}+O(n^{-1})\right)\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\right\|^{2}ds\right]$
$\displaystyle+\left(\frac{\lvert\rho\rvert^{2}\left(\zeta_{H,2}\right)^{2}}{C_{1,H}^{2}C_{2,H}^{2}(\eta-\eta_{iso})}+O(n^{-1})\right)\mathbb{E}\left[\int_{0}^{t}\lVert
B^{n}_{s}\rVert_{\mathbf{H}^{-1}}^{2}ds\right].$

As a consequence, if $n$ is large enough such that $\eta_{iso}+O(n^{-1})<\eta$
we obtain, by Grönwall’s lemma

(3.10)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]+\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\right\|^{2}ds\right]\lesssim\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}.$

Restarting again from (3) and considering the expected value of the supremum
in time between $[0,T]$, thanks to (3.6), (3),(3),(3), for $n$ large enough
such that $\eta_{iso}+O(n^{-1})<\eta$ we obtain

(3.11)
$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]$
$\displaystyle\leq\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}+\left(\eta_{iso}+O(n^{-1})\right)T\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]+2\mathbb{E}\left[\sup_{t\in[0,T]}\left\lvert
M^{n}_{t}\right\rvert\right].$

The last term can be estimated by Burkholder-Davis-Gundy inequality obtaining

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\lvert
M^{n}_{t}\right\rvert\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\mathbb{E}\left[\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{T}\langle(-\Delta)^{-1}\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s}),B^{n}_{s}\rangle^{2}ds\right)^{1/2}\right]$
$\displaystyle\leq\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}\int_{0}^{T}\left\|B^{n}_{s}\right\|^{2}ds\right)^{1/2}\right]$
(3.12)
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\left\|B^{n}_{t}\right\|^{2}_{\mathbf{H}^{-1}}\right]+C_{\lvert\rho\rvert,\eta_{iso}}\mathbb{E}\left[\int_{0}^{T}\left\|B^{n}_{s}\right\|^{2}ds\right].$

Combining (3.10),(3.11), (3) we get immediately (3.3) in case of 2.6.
Let us now consider the case of 2.4. We start by noticing that

$\displaystyle\Lambda^{n}=\left(\eta_{T}^{n}+\eta^{n}_{V}-\frac{\eta^{n}_{R,V}}{2}\right)\Delta+\left(\eta_{R}^{n}-\frac{\eta^{n}_{R,V}}{2}-\eta_{T}^{n}\right)\partial_{33}.$

As a consequence we have

$\displaystyle\left\langle(-\Delta)^{-1}\Lambda^{n}B^{n}_{t},B^{n}_{t}\right\rangle$
$\displaystyle=-\left(\eta_{T}^{n}+\eta^{n}_{V}-\frac{\eta^{2}_{R,V}}{2}\right)\|B_{t}\|^{2}+\left(\eta_{T}^{n}+\frac{\eta^{n}_{R,V}}{2}-\eta_{R}^{n}\right)\|(-\Delta)^{-1/2}\partial_{3}B_{t}\|^{2}$
(3.13)
$\displaystyle\leq-\eta^{n}_{V}\|B^{n}_{t}\|^{2}+\eta^{n}_{R}\|B^{n}_{t}\|^{2}+\eta^{n}_{R,V}\|B^{n}_{t}\|^{2}.$

While, arguing as above, due to 3.2

(3.14) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\|(-\Delta)^{-1/2}\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{t})\|^{2}$
$\displaystyle\leq\left(2\eta^{n}_{T}+\eta^{n}_{R}+3\eta^{n}_{V}\right)\lVert{B^{n}_{t}}\rVert^{2}.$

Estimate (3) and (3) continue to hold, possibly replacing $\eta_{iso}$ with
$2\eta_{T}$. In conclusion, combining (3),(3), (3.14),(3) and (3) we obtain

$\displaystyle\mathbb{E}\left[\left\|B^{n}_{t}\right\|_{\mathbf{H}^{-1}}^{2}\right]+\eta\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\right\|^{2}ds\right]$
$\displaystyle\leq\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}+\left(4\eta^{n}_{T}-2\eta_{T}+O(n^{2-\beta}+n^{3-\gamma})\right)\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\right\|^{2}ds\right]$
(3.15)
$\displaystyle+\left(\frac{\lvert\rho\rvert^{2}\left(\zeta_{H,2}\right)^{2}}{C_{1,H}^{2}C_{2,H}^{2}(\eta-2\eta_{T})}+O(n^{-1})\right)\mathbb{E}\left[\int_{0}^{t}\lVert
B^{n}_{s}\rVert_{\mathbf{H}^{-1}}^{2}ds\right].$

Arguing as above, by Grönwall’s lemma and Burkholder-Davis-Gundy inequality we
can prove (3.3) starting from (3). We omit the easy details. We are left to
show the validity of the additional estimate in case of 2.5. Due to 2.15 and
(2.7), the Itô formula for $\|B_{t}^{n,3}\|^{2}$ reads

$\displaystyle d\|B_{t}^{n,3}\|^{2}$
$\displaystyle+2\left(\eta+\eta^{n}_{V}-\frac{\eta^{n}_{R,V}}{2}\right)\|\operatorname{\nabla}B_{t}^{n,3}\|^{2}dt+2\eta^{n}_{T}\|\nabla_{H}B_{t}^{n,3}\|^{2}dt+2\left(\eta^{n}_{R}-\frac{\eta^{n}_{RV}}{2}\right)\|\partial_{3}B_{t}^{n,3}\|^{2}dt$
$\displaystyle=\frac{2\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\left\langle(B^{n,H}_{t})^{\perp},\nabla_{H}B_{t}^{n,3}\right\rangle
dt+2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\left(\|\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{t}\|^{2}+\|B^{n}_{t}\cdot\nabla\sigma_{k,j}^{n,3}\|^{2}\right)dt$
(3.16) $\displaystyle+2\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq
2n\end{subarray}}\left(\left\langle\sigma_{k,1}^{n}\cdot\operatorname{\nabla}_{H}B_{t}^{n,3},B_{t}^{n,H}\cdot\operatorname{\nabla}\sigma_{-k,2}^{n,3}\right\rangle+\left\langle
B_{t}^{n,H}\cdot\operatorname{\nabla}\sigma_{k,2}^{n,3},\sigma_{-k,1}^{n}\cdot\operatorname{\nabla}_{H}B_{t}^{n,3}\right\rangle\right)dt+dM_{t}^{n,3},$

where

$\displaystyle M_{t}^{n,3}=2\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\left\langle\sigma_{k,j}^{n}\cdot\operatorname{\nabla}B_{s}^{3}-B_{s}\cdot\operatorname{\nabla}\sigma_{k,j}^{n,3},B_{s}^{n,3}\right\rangle
dW_{s}^{k,j}.$

We can treat the different terms in the right hand side of (3) similarly to
previous case. Thanks to 3.2 we obtain

(3.17) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\left(\|\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{t}\|^{2}+\|B^{n}_{t}\cdot\nabla\sigma_{k,j}^{n,3}\|^{2}\right)$
$\displaystyle\leq\eta^{n}_{T}\lVert\nabla_{H}B^{n,3}_{t}\rVert^{2}+O(n^{-2})\lVert\nabla
B^{n,3}_{t}\rVert^{2}+\left(\frac{\zeta^{n}_{H,2}}{C_{2,H}^{2}}+\frac{\zeta^{n}_{s,\gamma-2}}{C_{V}^{2}n^{\gamma-5}}\right)\lVert
B^{n}_{t}\rVert^{2}.$

The term associated to the correlation of our noise can be treated easily by
Holder and Young’s inequality

$\displaystyle\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\left\langle(B^{n,H}_{t})^{\perp},\nabla_{H}B_{t}^{n,3}\right\rangle$
$\displaystyle\leq\frac{\lvert\rho\rvert\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\lVert
B^{n,H}_{t}\rVert\lVert\nabla_{H}B_{t}^{n,3}\rVert$ (3.18)
$\displaystyle\leq\frac{\eta}{2}\lVert\nabla
B^{n,3}\rVert^{2}+\frac{\lvert\rho\rvert^{2}\left(\zeta^{n}_{H,2}\right)^{2}}{2C_{1,H}^{2}C_{2,H}^{2}\eta}\lVert
B^{n}_{t}\rVert^{2},$
$\displaystyle\rho\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq
2n\end{subarray}}\left(\left\langle\sigma_{k,1}^{n}\cdot\operatorname{\nabla}_{H}B_{t}^{n,3},B_{t}^{n,H}\cdot\operatorname{\nabla}\sigma_{-k,2}^{n,3}\right\rangle\right)$
$\displaystyle\leq\lvert\rho\rvert\frac{\zeta_{H,3}^{n}\lVert\nabla_{H}B^{n,3}_{t}\rVert\lVert
B^{n}_{t}\rVert}{C_{1,H}C_{2,H}n}$ (3.19) $\displaystyle\leq\frac{\lVert\nabla
B^{n,3}_{t}\rVert^{2}}{2n}+\frac{\lvert\rho\rvert^{2}\left(\zeta^{n}_{H,3}\right)^{2}\lVert
B^{n}_{t}\rVert^{2}}{2C_{1,H}^{2}C_{2,H}^{2}n}$

and analogously for the other. Therefore, integrating (3) between $0$ and $t$
and considering its expected value, thanks to (3.17), (3),(3) we obtain

$\displaystyle\mathbb{E}\left[\|B_{t}^{n,3}\|^{2}\right]$
$\displaystyle+\left(\eta-\frac{2}{n}+O(n^{-2})\right)\mathbb{E}\left[\int_{0}^{t}\|\operatorname{\nabla}B_{s}^{n,3}\|^{2}ds\right]$
$\displaystyle\leq\left\|B^{n,3}_{0}\right\|^{2}+2\left(\frac{\zeta^{n}_{H,2}}{C_{2,H}^{2}}+\frac{\zeta^{n}_{s,\gamma-2}}{C_{V}^{2}n^{\gamma-5}}+\frac{\lvert\rho\rvert^{2}\left(\zeta^{n}_{H,2}\right)^{2}}{2C_{1,H}^{2}C_{2,H}^{2}\eta}+O(n^{-1})\right)\mathbb{E}\left[\int_{0}^{T}\lVert
B^{n}_{s}\rVert^{2}ds\right].$

Therefore, if $n$ is large enough such that
$\eta-\frac{2}{n}+O(n^{-2})>\frac{\eta}{2}$, thanks to (3.3) we have
immediately

(3.20)
$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[\|B_{t}^{n,3}\|^{2}\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{T}\|\operatorname{\nabla}B_{s}^{n,3}\|^{2}ds\right]\lesssim\left\|B^{n,3}_{0}\right\|^{2}+\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}.$

Restarting again from (3) and considering the expected value of the supremum
in time between $[0,T]$, thanks to (3.17), (3),(3), we obtain for $n$ large
enough such that $\eta-\frac{2}{n}+O(n^{-2})>\frac{\eta}{2}$

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\|B_{t}^{n,3}\|^{2}\right]$
$\displaystyle\leq\left\|B^{n,3}_{0}\right\|^{2}+2\left(\frac{\zeta^{n}_{H,2}}{C_{2,H}^{2}}+\frac{\zeta^{n}_{s,\gamma-2}}{C_{V}^{2}n^{\gamma-5}}+\frac{\lvert\rho\rvert^{2}\left(\zeta^{n}_{H,2}\right)^{2}}{2C_{1,H}^{2}C_{2,H}^{2}\eta}+O(n^{-1})\right)\mathbb{E}\left[\int_{0}^{T}\lVert
B^{n}_{s}\rVert^{2}ds\right]$ (3.21)
$\displaystyle+\mathbb{E}\left[\sup_{t\in[0,T]}\lvert
M_{t}^{n,3}\rvert\right].$

The first term can be treated easily thanks to (3.3). The second one is
treated via Burkholder-Davis-Gundy inequality, (2.7), Young’s inequality and
(3.17) obtaining

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\lvert M_{t}^{n,3}\rvert\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\mathbb{E}\left[\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{T}\left\langle\sigma_{k,j}^{n}\cdot\operatorname{\nabla}B_{s}^{3}-B_{s}\cdot\operatorname{\nabla}\sigma_{k,j}^{n,3},B_{s}^{n,3}\right\rangle^{2}ds\right)^{1/2}\right]$
$\displaystyle\leq\mathbb{E}\left[\sup_{t\in[0,T]}\left\|B^{n,3}_{s}\right\|\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{T}\left\|\sigma_{k,j}^{n}\cdot\operatorname{\nabla}B_{s}^{3}\right\|+\left\|B_{s}\cdot\operatorname{\nabla}\sigma_{k,j}^{n,3}\right\|^{2}ds\right)^{1/2}\right]$
(3.22) $\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\sup_{t\in[0,T]}\lVert
B_{t}^{n,3}\rVert\right]+C_{\lvert\rho\rvert,C_{1,H},C_{2,H},C_{V},\gamma}\left(\mathbb{E}\left[\int_{0}^{T}\lVert\nabla
B^{n,3}_{s}\rVert^{2}ds\right]+\mathbb{E}\left[\int_{0}^{T}\lVert
B^{n}_{s}\rVert^{2}ds\right]\right).$

Combining (3), (3.20), (3) and (3.3) we obtain relation (3.4). ∎

##  4\. Proof of Theorem 2.17

Before exploiting the convergence properties of $B^{n}$, we start with a well-
posedness result for our limit object $\overline{B}$ solution of (2.21).

######  Definition 4.1.

We say that $\overline{B}$ is a weak solution of equation (2.21) if

$\displaystyle\overline{B}\in C(0,T;\mathbf{H}^{-1})\cap
L^{2}(0,T;\mathbf{L}^{2})$

and for each $\phi\in\mathbf{H}^{2}$, and all $t\in[0,T]$, one has

$\displaystyle\langle{\overline{B}}_{t},\phi\rangle-\langle{\overline{B}}_{0},\phi\rangle$
$\displaystyle=\eta\int_{0}^{t}\langle{\overline{B}}_{s},\Delta\phi\rangle\,ds+\int_{0}^{t}\langle{\overline{B}}_{s},\Lambda_{\alpha,\beta,\gamma}\phi\rangle\,ds$
$\displaystyle\quad+\int_{0}^{t}\langle\mathcal{A}_{\rho}{\overline{B}}_{s},\nabla\times\phi\rangle\,ds.$

Since the operator $\eta\Delta+\Lambda_{\alpha,\beta,\gamma}$ is uniformly
elliptic and equation (4.1) is linear, by standard theory of parabolic PDE we
have existence and uniqueness of equation (4.1) in the sense 4.1, see for
example [20, Chapter 3], [8, Chapter 5].

######  Theorem 4.2.

For each $B_{0}\in\mathbf{H}^{-1}$, there exists a unique solution of (2.21)
in the sense of 4.1. Moreover if $B_{0}\in\mathbf{L}^{2}$, then

$\displaystyle\overline{B}\in C(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}^{1}).$

######  Remark 4.3.

Arguing as in 2.15 we obtain easily that, given $\overline{B_{0}}$ in
$\mathbf{L}^{2},$ then $\overline{B^{3}_{t}}$ satisfies

$\displaystyle\langle\overline{B^{3}_{t}},\varphi\rangle-\langle\overline{B^{3}_{0}},\varphi\rangle$
$\displaystyle=\eta\int_{0}^{t}\langle\overline{B^{3}_{s}},\Delta\varphi\rangle\,ds+\int_{0}^{t}\langle\overline{B^{3}_{s}},\Lambda_{\alpha,\beta,\gamma}\varphi\rangle\,ds+\frac{\rho\zeta_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}\langle{\overline{B^{H}_{s}}}^{\perp},\nabla_{H}\varphi\rangle\,ds$

for every $t\in[0,T]$ and $\varphi\in\dot{H}^{2}$.

Now we are ready to provide the proof of our main theorem. First we start with
some notation. Let us denote by $S^{n},\ S$ the semigroups generated by
$\eta\Delta+\Lambda^{n}$ and $\eta\Delta+\Lambda_{\alpha,\beta,\gamma}$. Since
their infinitesimal generators have the form of
$\mu_{n}(\partial_{11}+\partial_{22})+\nu_{n}\partial_{33},\
\mu_{n}\geq\nu_{n}>0$ we are in the framework of subsubsection 2.2.2. In
particular, we recall that $\mu_{n},\nu_{n}\geq\eta$ for each
$n\in\mathbb{N}$.
Secondly, we denote by

$\displaystyle Z^{n}_{t}$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}S^{n}(t-s)\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s})dW^{k,j}_{s},$ $\displaystyle H^{n}_{t}$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}S^{n}(t-s)\left(\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{t}-B^{n}_{t}\cdot\nabla\sigma_{k,j}^{n,3}\right)dW^{k,j}_{s}.$

The stochastic integrals above are well defined thanks to the regularity
properties of $B^{n}_{t}$ and the fact that for each $n$ only a finite number
of $\sigma_{k,j}^{n}\neq 0$. The first step in order to prove Theorem 2.17 is
to rewrite $B^{n}_{t},\ B^{n,3}_{t}$ in mild form. Indeed, recalling the
definition of $\mathcal{A}^{n}_{\rho}$, see 2.13, the following lemma holds
true.

######  Lemma 4.4.

$B^{n}_{t}$ and $B^{n,3}_{t}$ satisfy the mild formula

(4.1) $\displaystyle B^{n}_{t}$
$\displaystyle=S^{n}(t)B^{n}_{0}+\int_{0}^{t}S^{n}(t-s)\nabla\times(\mathcal{A}^{n}_{\rho}B^{n}_{s})ds+Z^{n}_{t},$
(4.2) $\displaystyle B^{n,3}_{t}$
$\displaystyle=S^{n}(t)B^{n,3}_{0}-\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}S^{n}(t-s)\operatorname{div}_{H}\left((B^{n,H}_{s})^{\perp}\right)ds+H^{n}_{t},$

the former seen as an equality in $\mathbf{H}^{-1}$ and the latter in
$\dot{L}^{2}$.

###### Proof.

Let us take $\phi=\psi_{h,l},\ \psi_{h,l}=a_{h,l}e^{ih\cdot x}$ as test
function in (2.20) obtaining

$\displaystyle\langle B^{n}_{t},\psi_{h,l}\rangle$ $\displaystyle=\langle
B^{n}_{0},\psi_{h,l}\rangle-\left(\mu_{n}\left(h_{1}^{2}+h_{2}^{2}\right)+\nu_{n}h_{3}^{2}\right)\int_{0}^{t}\langle
B^{n}_{s},\psi_{h,l}\rangle\,ds$ (4.3)
$\displaystyle+\int_{0}^{t}\langle\nabla\times(\mathcal{A}^{n}_{\rho}B^{n}_{s}),\psi_{h,l}\rangle\,ds+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s}),\psi_{h,l}\rangle\,dW^{k,j}_{s}.$

Therefore, applying Itô formula to
$e^{t\left(\mu_{n}\left(h_{1}^{2}+h_{2}^{2}\right)+\nu_{n}h_{3}^{2}\right)}\langle
B^{n}_{t},\psi_{h,l}\rangle$ we have

$\displaystyle\langle B^{n}_{t},\psi_{h,l}\rangle$
$\displaystyle=e^{-t\left(\mu_{n}\left(h_{1}^{2}+h_{2}^{2}\right)+\nu_{n}h_{3}^{2}\right)}\langle
B^{n}_{0},\psi_{h,l}\rangle+\int_{0}^{t}e^{-(t-s)\left(\mu_{n}\left(h_{1}^{2}+h_{2}^{2}\right)+\nu_{n}h_{3}^{2}\right)}\langle\nabla\times(\mathcal{A}^{n}_{\rho}B^{n}_{s}),\psi_{h,l}\rangle
ds$ (4.4) $\displaystyle+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}e^{-(t-s)\left(\mu_{n}\left(h_{1}^{2}+h_{2}^{2}\right)+\nu_{n}h_{3}^{2}\right)}\langle\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s}),\psi_{h,l}\rangle dW^{k,j}_{s}\quad\mathbb{P}-a.s.\quad\forall
t\in[0,T].$

We can find $\Gamma\subseteq\Omega$ of full probability such that the above
equality holds for all $t\in[0,T]$ and all $h\in\mathbb{Z}^{3}_{0},\
l\in\\{1,2\\}$. But this is exactly (4.1) written in Fourier modes. The proof
of (4.2) is analogous, just replacing relation (2.20) with 2.15 and choosing
as test function $\phi=(0,0,\psi_{h})^{t},\ \psi_{h}=e^{ih\cdot x},\
h\in\mathbb{Z}^{3}_{0}$. We omit the easy details. ∎

Secondly we need to study some properties of the stochastic convolutions
$Z^{n}_{t},H^{n}_{t}$. 4.6 and 4.7 below are the analogous of [10, Lemma 2.5]
in our framework. In particular it is important to point out that [10,
Assumption 2.4] is false in our case and we have to deal also with the
stochastic stretching, which was neglected in previous results. In order to
reach our plan, we start with a uniform bound on our coefficients in a
distributional norm.

######  Lemma 4.5.

Assuming either 2.4 or 2.6, for each $\delta>0$

(4.5)
$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lVert P[Q[\sigma_{k,j}^{n}\times
B^{n}_{t}]]\rVert_{\mathbf{H}^{-1-\frac{\delta}{2}}}^{2}\right]$
$\displaystyle\lesssim\lVert B_{0}^{n}\rVert_{\mathbf{H}^{-1}}^{2}.$

###### Proof.

Arguing as in 3.2, if we denote by

$\displaystyle b^{n,h,l}_{t}:=\langle{B^{n}_{t}},a_{h,l}e^{-ih\cdot
x}\rangle,$

then

$\displaystyle\sigma^{n}_{k,j}\times{B^{n}_{t}}$
$\displaystyle=\theta_{k,j}^{n}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l\in\\{1,2\\}\end{subarray}}b^{n,h,l}_{t}a_{k,j}\times a_{h,l}e^{i(h+k)\cdot
x}.$

Thanks to 3.4 we already know that

(4.6)
$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0},\\\
l\in\\{1,2\\}\end{subarray}}\frac{\lvert b^{n,h,l}_{t}\rvert^{2}}{\lvert
h\rvert^{2}}\right]\lesssim\lVert{B^{n}_{0}}\rVert_{\mathbf{H}^{-1}}^{2}.$

For each $k\in\mathbb{Z}^{3}_{0}$ we have

$\displaystyle\left\|P[Q[\sigma_{k,j}^{n}\times{B^{n}_{t}}]]\right\|_{\mathbf{H}^{-1-\frac{\delta}{2}}}^{2}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}h\in\mathbb{Z}_{0}^{3}\\\
h\neq-k,\
l\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}\lvert b^{n,h,l}_{t}\rvert^{2}.$

We are left to study

(4.7)
$\displaystyle\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\sum_{\begin{subarray}{c}h\in\mathbb{Z}_{0}^{3}\\\
h\neq-k,\
l\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}b^{n,h,l}_{t}\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}\right]$
$\displaystyle=\mathbb{E}\left[\operatorname{sup}_{t\in[0,T]}\sum_{\begin{subarray}{c}h\in\mathbb{Z}_{0}^{3}\\\
l\in\\{1,2\\}\end{subarray}}\frac{\lvert b^{n,h,l}_{t}\rvert^{2}}{\lvert
h\rvert^{2}}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ h\neq-k,\\\
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}h\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}\right].$

Due to relation (4.6), it is enough to show that

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ h\neq-k,\\\
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert h+k\rvert^{2+\delta}}\lesssim 1.$

uniformly in $h\in\mathbb{Z}^{3}_{0},\ l\in\\{1,2\\}$. By splitting the
summation domain, we have

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ h\neq-k,\\\
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert h+k\rvert^{2+\delta}}$
$\displaystyle\leq\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\\\ j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\\\ j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
\lvert h+k\rvert\geq\frac{\lvert h\rvert}{2},\\\ \frac{\lvert h\rvert}{2}\vee
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert h+k\rvert^{2+\delta}}$ (4.8)
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\\\ j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert
h+k\rvert^{2+\delta}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
\lvert h+k\rvert\geq\frac{\lvert h\rvert}{2},\\\ \frac{\lvert h\rvert}{2}\vee
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}$

We start considering the case of 2.6 which is the easier one. In this case,
due to (4)

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ h\neq-k,\\\
n\leq\lvert k\rvert\leq 2n\\\
j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert h+k\rvert^{2+\delta}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{3}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ 1\leq\lvert
h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\end{subarray}}\frac{1}{\lvert k\rvert\lvert h+k\rvert^{2+\delta}}$
$\displaystyle\lesssim
1+\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ n\leq\lvert
k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{3}}\right)^{1/3}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert
h\rvert}{2}\end{subarray}}\frac{1}{\lvert
h+k\rvert^{3+\frac{3}{2}\delta}}\right)^{2/3}\lesssim_{\delta}1,$

where the first inequality follows from the fact that $\lvert
k\rvert\geq\lvert(k+h)-h\rvert\geq\frac{\lvert h\rvert}{2}$ if $1\leq\lvert
h+k\rvert\leq\frac{\lvert h\rvert}{2}$ and the second by Hölder’s inequality.
This completes the proof of (4.5) in case of 2.6. In case of 2.4 we argue
similarly. Due to the summability properties of the $\theta_{k,j}^{n}$, it is
enough to consider the second term in the right hand side of (4), i.e.

(4.9) $\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\\\ j\in\\{1,2\\}\end{subarray}}\frac{\lvert\theta_{k,j}^{n}\rvert^{2}\lvert
h\rvert^{2}}{\lvert h+k\rvert^{2+\delta}}$
$\displaystyle=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\ 1\leq\lvert
h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\end{subarray}}\frac{\lvert h\rvert^{2}}{\lvert k\rvert^{\alpha}\lvert
h+k\rvert^{2+\delta}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\end{subarray}}\frac{\lvert h\rvert^{2}}{\lvert k\rvert^{\beta}\lvert
h+k\rvert^{2+\delta}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2}\\\ k_{3}\neq 0,\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{\lvert h\rvert^{2}}{\lvert
k\rvert^{\gamma}\lvert h+k\rvert^{2+\delta}}.$

Since $\gamma>3$ the third term is a simplified version of the one treated
before. We are left to study the other two. It is clear that if we are able to
control $\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\ 1\leq\lvert
h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\end{subarray}}\frac{\lvert h\rvert^{2}}{\lvert k\rvert^{2}\lvert
h+k\rvert^{2+\delta}}$ we can control relation (4.9). Therefore we study only
this case. We observe that by triangular inequality, the constraints
$1\leq\lvert h+k\rvert\leq\frac{\lvert h\rvert}{2},n\leq\lvert k\rvert\leq 2n$
imply

$\displaystyle\lvert h\rvert=\lvert(h+k)+(-k)\rvert\leq\lvert
k\rvert+\frac{\lvert h\rvert}{2}\leq 2n+\frac{\lvert h\rvert}{2}.$

As a consequence

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\ 1\leq\lvert
h+k\rvert\leq\frac{\lvert h\rvert}{2},\\\ n\leq\lvert k\rvert\leq
2n\end{subarray}}\frac{\lvert h\rvert^{2}}{\lvert k\rvert^{2}\lvert
h+k\rvert^{2+\delta}}\lesssim\sum_{\begin{subarray}{c}z\in
h+\mathbb{Z}^{2}_{0}\\\ 1\leq\lvert z\rvert\leq
2n\end{subarray}}\frac{1}{\lvert z\rvert^{2+\delta}}\lesssim_{\delta}1.$

This concludes the proof of relation (4.5). ∎

4.5 was a necessary toolds in order to prove the following estimates on
$Z^{n}_{t},\ H^{n}_{t}$.

######  Lemma 4.6.

For each $\delta>0$

(4.10)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|Z^{n}_{t}\right\|^{2}_{\mathbf{H}^{-1-\delta}}\right]\lesssim\lVert
B_{0}^{n}\rVert_{\mathbf{H}^{-1}}^{2}.$

In case of 2.5 we have also

(4.11)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|H^{n}_{t}\right\|^{2}_{\dot{H}^{-\delta}}\right]\lesssim\lVert
B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2}+\lVert B^{n,3}_{0}\rVert^{2}.$

###### Proof.

By Burkholder-Davis-Gundy inequality and the regularization properties of the
semigroup, see 2.2, we have

$\displaystyle\mathbb{E}\left[\left\|Z^{n}_{t}\right\|^{2}_{\mathbf{H}^{-1-\delta}}\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|S^{n}(t-s)\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s})\right\|_{\mathbf{H}^{-1-\delta}}^{2}ds\right]$
$\displaystyle\lesssim_{\eta}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\frac{\delta}{2}}}\left\|\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s})\right\|_{\mathbf{H}^{-2-\frac{\delta}{2}}}^{2}ds\right]$
$\displaystyle\lesssim\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\frac{\delta}{2}}}\left\|P[Q[\sigma_{k,j}^{n}\times
B^{n}_{s}]]\right\|_{\mathbf{H}^{-1-\frac{\delta}{2}}}^{2}ds\right]$
$\displaystyle\lesssim_{\delta}\lVert B_{0}^{n}\rVert_{\mathbf{H}^{-1}}^{2}$

where the last inequality follows from (4.5) and is uniform for $t\in[0,T].$
Assuming 2.5, similarly we can prove (4.11). Again, by Burkholder-Davis-Gundy
inequality and the regularization properties of the semigroup, 2.2, we get

$\displaystyle\mathbb{E}\left[\left\|H^{n}_{t}\right\|^{2}_{\dot{H}^{-\delta}}\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|S^{n}(t-s)\left[\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{s}\right]-S^{n}(t-s)\left[B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right]\right\|_{\dot{H}^{-\delta}}^{2}ds\right]$
$\displaystyle\lesssim_{\eta}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{s}\right\|_{\dot{H}^{-1}}^{2}ds\right]+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right\|^{2}ds\right]$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}B^{n,3}_{s}\right\|^{2}ds\right]+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right\|^{2}ds\right].$

Both the two terms can be treated easily. Indeed,

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}B^{n,3}_{s}\right\|^{2}ds\right]$
$\displaystyle\leq\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\operatorname{sup}_{s\in[0,T]}\left\|B^{n,3}_{s}\right\|^{2}ds\right]$
$\displaystyle\lesssim_{\delta}\lVert
B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2}+\lVert B^{n,3}_{0}\rVert^{2}.$

due to 3.4. Similarly

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right\|^{2}ds\right]$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{\beta-2}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{\gamma-2}}\right)\mathbb{E}\left[\int_{0}^{T}\lVert
B^{n}_{s}\rVert^{2}ds\right]$ $\displaystyle\lesssim\lVert
B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2},$

again by 3.4. ∎

######  Lemma 4.7.

Assuming either 2.4, 2.6 then for each $\delta>0$

(4.12)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|Z^{n}_{t}\right\|^{2}_{\mathbf{H}^{-4-\frac{3}{2}\delta}}\right]\lesssim\frac{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}}{n^{\alpha\wedge\beta\wedge\gamma}}.$

In case of 2.5 we have also

(4.13)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|H^{n}_{t}\right\|^{2}_{\dot{H}^{-\frac{3}{2}-\frac{3}{2}\delta}}\right]\lesssim\frac{\left\|B^{n,3}_{0}\right\|^{2}+\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}}{n^{2}}.$

###### Proof.

Let us consider for a smooth divergence free vector field $\psi$

$\displaystyle\mathbb{E}\left[\langle
Z^{n}_{t},\psi\rangle^{2}\right]=\mathbb{E}\left[\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle
S^{n}(t-s)\nabla\times(\sigma_{k,j}^{n}\times B^{n}_{s}),\psi\rangle
dW^{k,j}_{s}\right)^{2}\right].$

Therefore, by Burkholder-Davis-Gundy inequality, Sobolev embedding theorem and
the contractivity property of our semigroup we obtain

$\displaystyle\mathbb{E}\left[\langle Z^{n}_{t},\psi\rangle^{2}\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\mathbb{E}\left[\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle
S^{n}(t-s)\nabla\times(\sigma_{k,j}^{n}\times
B^{n}_{s}),\psi\rangle^{2}ds\right]$
$\displaystyle=\mathbb{E}\left[\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\lvert\theta_{k,j}^{n}\rvert^{2}\int_{0}^{t}\langle
a_{k,j}e^{ik\cdot x},B^{n}_{s}\times\left(\nabla\times
S^{n}(t-s)\psi\right)\rangle^{2}ds\right]$
$\displaystyle\leq\frac{1}{n^{\alpha\wedge\beta\wedge\gamma}}\mathbb{E}\left[\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\int_{0}^{t}\langle a_{k,j}e^{ik\cdot
x},B^{n}_{s}\times\left(\nabla\times
S^{n}(t-s)\psi\right)\rangle^{2}ds\right]$
$\displaystyle\leq\frac{1}{n^{\alpha\wedge\beta\wedge\gamma}}\mathbb{E}\left[\int_{0}^{t}\lVert
B^{n}_{s}\times\left(\nabla\times S^{n}(t-s)\psi\right)\rVert^{2}ds\right]$
$\displaystyle\lesssim_{\eta,\delta}\frac{1}{n^{\alpha\wedge\beta\wedge\gamma}}\lVert\psi\rVert_{H^{\frac{5}{2}+\delta}}^{2}\mathbb{E}\left[\int_{0}^{T}\lVert
B^{n}_{s}\rVert^{2}ds\right],$

uniformly in $t\in[0,T]$. Therefore, choosing $\psi=a_{h,l}e^{ih\cdot x},\
h\in Z^{3}_{0},\ l\in\\{1,2\\}$ we obtain, due to 3.4

(4.14) $\displaystyle\mathbb{E}\left[\langle Z^{n}_{t},a_{h,l}e^{ih\cdot
x}\rangle^{2}\right]\lesssim\frac{1}{n^{\alpha\wedge\beta\wedge\gamma}}\lvert
h\rvert^{5+2\delta}{\lVert B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2}}.$

Relation (4.12) then follows from (4.14). Indeed, for each $t\in[0,T]$

$\displaystyle\mathbb{E}\left[\left\|Z^{n}_{t}\right\|^{2}_{\mathbf{H}^{-4-\frac{3}{2}\delta}}\right]$
$\displaystyle=\sum_{\begin{subarray}{c}h\in\mathbb{Z}^{3}_{0}\\\
l\in\\{1,2\\}\end{subarray}}\frac{\mathbb{E}\left[\langle
Z^{n}_{t},a_{h,l}e^{ih\cdot x}\rangle^{2}\right]}{\lvert h\rvert^{8+3\delta}}$
$\displaystyle\lesssim\frac{1}{n^{\alpha\wedge\beta\wedge\gamma}}\sum_{h\in\mathbb{Z}^{3}_{0}}\frac{\lVert
B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2}}{\lvert h\rvert^{3+\delta}}$
$\displaystyle\lesssim_{\delta}\frac{\lVert
B^{n}_{0}\rVert_{\mathbf{H}^{-1}}^{2}}{n^{\alpha\wedge\beta\wedge\gamma}}.$

In order to get (4.13) we argue slightly differently. By Burkholder-Davis-
Gundy inequality and the regularization properties of the semigroup, see 2.2,
we have

$\displaystyle\mathbb{E}\left[\left\|H^{n}_{t}\right\|^{2}_{\mathbf{H}^{-\frac{3}{2}-\frac{3}{2}\delta}}\right]$
$\displaystyle\lesssim_{\lvert\rho\rvert}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|S^{n}(t-s)\left[\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{s}\right]-S^{n}(t-s)\left[B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right]\right\|_{\dot{H}^{-\frac{3}{2}-\frac{3}{2}\delta}}^{2}ds\right]$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}\cdot\nabla
B^{n,3}_{s}\right\|_{\dot{H}^{-\frac{5}{2}-\frac{1}{2}\delta}}^{2}ds\right]$
$\displaystyle+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\cdot\nabla\sigma_{k,j}^{n,3}\right\|_{\dot{H}^{-\frac{3}{2}-\frac{3}{2}\delta}}^{2}ds\right]$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}B^{n,3}_{s}\right\|_{\dot{H}^{-\frac{3}{2}-\frac{1}{2}\delta}}^{2}ds\right]+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\sigma_{k,j}^{n,3}\right\|^{2}ds\right].$

The second term can be analyzed easily by 3.4 obtaining

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\left\|B^{n}_{s}\sigma_{k,j}^{n,3}\right\|^{2}ds\right]$
$\displaystyle\lesssim\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}_{0}\\\
n\leq\lvert k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{4}}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\ n\leq\lvert
k\rvert\leq 2n\end{subarray}}\frac{1}{\lvert
k\rvert^{\gamma}}\right)\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}$
$\displaystyle\lesssim\frac{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}}{n^{2}}.$

In order to treat the other one we need to argue in a more precise way.
Indeed, by definition of the Sobolev norm on the 3D torus, we have

$\displaystyle\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{3}_{0}\\\
j\in\\{1,2\\}\end{subarray}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|\sigma_{k,j}^{n}B^{n,3}_{s}\right\|_{\dot{H}^{-\frac{3}{2}-\frac{1}{2}\delta}}^{2}ds\right]$
$\displaystyle\lesssim\frac{1}{n^{2}}\sum_{k\in\mathbb{Z}^{3}_{0}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\left\|e^{ik\cdot
x}B^{n,3}_{s}\right\|_{\dot{H}^{-\frac{3}{2}-\frac{1}{2}\delta}}^{2}ds\right]$
$\displaystyle=\frac{1}{n^{2}}\sum_{h\in\mathbb{Z}^{3}_{0}}\frac{1}{\lvert
h\rvert^{3+\delta}}\sum_{k\in\mathbb{Z}^{3}_{0}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\langle
e^{i(k-h)\cdot x},B^{n,3}_{s}\rangle^{2}ds\right]$
$\displaystyle\leq\frac{1}{n^{2}}\sum_{h\in\mathbb{Z}^{3}_{0}}\frac{1}{\lvert
h\rvert^{3+\delta}}\mathbb{E}\left[\int_{0}^{t}\frac{1}{(t-s)^{1-\delta}}\operatorname{sup}_{t\in[0,T]}\left\|B^{n,3}_{t}\right\|^{2}ds\right]$
$\displaystyle\lesssim_{\delta}\frac{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}+\left\|B^{n,3}_{0}\right\|^{2}}{n^{2}}$

uniformly in $t\in[0,T]$. This concludes the proof of (4.13). ∎

Combining 4.6 and 4.7, by interpolation we get the following result.

######  Corollary 4.8.

Assuming either 2.4 or 2.6, for each $\theta\in(0,3]$ and
$\delta\in(0,\theta]$ we have

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[\lVert
Z^{n}_{t}\rVert_{\mathbf{H}^{-1-\theta}}^{2}\right]\lesssim\frac{1}{n^{\frac{(\alpha\wedge\beta\wedge\gamma)(\theta-\delta)}{3+\frac{\delta}{2}}}}\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}.$

In case of 2.5 we have also for each $\vartheta\in(0,\frac{3}{2}]$ and
$\delta\in(0,\vartheta]$

$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|H^{n}_{t}\right\|^{2}_{\dot{H}^{-\vartheta}}\right]\lesssim\frac{\left\|B^{n,3}_{0}\right\|^{2}+\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}}{n^{\frac{4(\vartheta-\delta)}{3}}}.$

Moreover, we recall that by classical theory of evolution equations, see for
example [21], or arguing as in 4.4, the unique weak solutions of (2.21) and
its third component can be written in mild form as

$\displaystyle\overline{B_{t}}=S(t)\overline{B_{0}}+\int_{0}^{t}S(t-s)\nabla\times(\mathcal{A}_{\rho}\overline{B_{s}})ds,\quad\overline{B_{t}^{3}}=S(t-s)\overline{B^{3}_{0}}-\frac{\rho\zeta_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}S(t-s)\operatorname{div}_{H}\left(\overline{B^{H}_{s}}^{\perp}\right)ds.$

Now we introduce some intermediate vector fields between $B^{n}_{t}$ and
$\overline{B_{t}}$. Let $\widehat{B_{t}^{n}}$ the unique weak solution of the
linear system

(4.15)
$\displaystyle\begin{cases}\partial_{t}\widehat{B^{n}_{t}}&=(\eta\Delta+\Lambda^{n})\widehat{B^{n}_{t}}+\nabla\times(\mathcal{A}^{n}_{\rho}\widehat{B^{n}_{t}})\quad
x\in\mathbb{T}^{3},\ t\in(0,T)\\\
\widehat{B^{n}_{t}}|_{t=0}&=B^{n}_{0}.\end{cases}$

Again, by classical theory of evolution equations, see for example [20, 8],
previous equation is well-posed either in $\mathbf{H}^{-1}$ and
$\mathbf{L}^{2}$. Moreover the unique weak solution and its third component
can be written in mild form as

$\displaystyle\widehat{B_{t}^{n}}$
$\displaystyle=S^{n}(t-s)B^{n}_{0}+\int_{0}^{t}S^{n}(t-s)\nabla\times(\mathcal{A}^{n}_{\rho}\widehat{B^{n}_{s}})ds,$
$\displaystyle\widehat{B_{t}^{3,n}}$
$\displaystyle=S^{n}(t-s)B^{n,3}_{0}-\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}S^{n}(t-s)\operatorname{div}_{H}\left(\widehat{B^{n,H}_{s}}^{\perp}\right)ds.$

Now we prove that $\widehat{B^{n}_{t}}$ and $\overline{B_{t}}$ are close. We
recall that the function $\chi$ introduced in subsection 2.1 satisfies

$\displaystyle 0<\chi(\alpha,\beta,\gamma)\leq 1$

for each choice of the parameters in the range described by 2.4, 2.5 and 2.6.

######  Lemma 4.9.

Under the same assumptions of Theorem 2.17, for each $\theta\in(0,2)$ we have

(4.16)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\left\|\widehat{B^{n}_{t}}-\overline{B_{t}}\right\|_{\dot{H}^{-1-\theta}}$
$\displaystyle\lesssim_{\eta,T}\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)/2}}+\left\|\overline{B_{0}}-B^{n}_{0}\right\|_{\mathbf{H}^{-1-\theta}}.$

In case of 2.5 we have also for each $\vartheta\in(\theta,\frac{3}{2}]$

(4.17)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\left\|\widehat{B^{n,3}_{t}}-{\overline{B^{3}_{t}}}\right\|_{\dot{H}^{-\vartheta}}$
$\displaystyle\lesssim_{\eta,T}\frac{1}{(\vartheta-\theta)n^{\theta/2}}+\left\|\overline{B^{3}_{0}}-B^{n,3}_{0}\right\|_{\dot{H}^{-\vartheta}}+\frac{\left\|\overline{B_{0}}-B^{n}_{0}\right\|_{\mathbf{H}^{-1-\theta}}}{\vartheta-\theta}.$

###### Proof.

First let us observe that by assumptions in case of 2.4 and 2.6 (resp. 2.5)

$\displaystyle B^{n}_{0}\rightharpoonup\overline{B_{0}}\text{ in
}\mathbf{H}^{-1}\quad(\text{resp.
}B^{n}_{0}\rightharpoonup\overline{B_{0}}\text{ in }\mathbf{H}^{-1},\
B^{n,3}_{0}\rightharpoonup\overline{B^{3}_{0}}\text{ in
}\dot{L}^{2}(\mathbb{T}^{3}))$

we have in particular that the family
$\left\\{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}\right\\}_{n\in\mathbb{N}}$
(resp.
$\left\\{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{2}\right\\}_{n\in\mathbb{N}},\
\left\\{\left\|B^{n,3}_{0}\right\|^{2}\right\\}_{n\in\mathbb{N}}$) is bounded.
Moreover, $\mathcal{A}^{n}_{\rho}=\mathcal{A}_{\rho}+O(\frac{1}{n})$, then in
particular we have

(4.18)
$\displaystyle\operatorname{sup}_{n\in\mathbb{N}}\left\|\mathcal{A}^{n}_{\rho}\right\|_{HS}<+\infty.$

Lastly, by simple energy estimates on (4.15) we have for each $n\in\mathbb{N}$

(4.19)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\left\|\widehat{B^{n}_{t}}\right\|_{\mathbf{H}^{-1}}^{2}+\int_{0}^{T}\left\|\widehat{B^{n}_{s}}\right\|^{2}ds\lesssim_{\eta}\operatorname{sup}_{n\in\mathbb{N}}\left\|\widehat{B^{n}_{0}}\right\|_{\mathbf{H}^{-1}}^{2}<+\infty,$
(4.20)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\left\|\widehat{B^{n,3}_{t}}\right\|^{2}+\int_{0}^{T}\left\|\nabla\widehat{B^{n,3}_{s}}\right\|^{2}ds\lesssim_{\eta}\operatorname{sup}_{n\in\mathbb{N}}\left\|\widehat{B^{n}_{0}}\right\|_{\mathbf{H}^{-1}}^{2}+\operatorname{sup}_{n\in\mathbb{N}}\left\|\widehat{B^{n,3}_{0}}\right\|^{2}<+\infty.$

The convergence of $\widehat{B^{n}_{t}}$ to $\overline{B_{t}}$ then follows by
triangle inequality, 2.2, 2.3 and the uniform bounds on the initial
conditions. Indeed for each $t\in[0,T]$ it holds

$\displaystyle\left\|\widehat{B^{n}_{t}}-\overline{B_{t}}\right\|_{\dot{H}^{-1-\theta}}$
$\displaystyle\leq\left(\left\|\left(S^{n}(t)-S(t)\right)\overline{B_{0}}\right\|_{\mathbf{H}^{-1-\theta}}+\left\|S^{n}(t)\left(B^{n}_{0}-\overline{B_{0}}\right)\right\|_{\mathbf{H}^{-1-\theta}}\right)$
$\displaystyle+\left\|\int_{0}^{t}\left(S^{n}(t-s)-S(t-s)\right)\nabla\times\left(\mathcal{A}^{n}_{\rho}\widehat{B^{n}_{s}}\right)ds\right\|_{\mathbf{H}^{-1-\theta}}$
$\displaystyle+\left\|\int_{0}^{t}S(t-s)\nabla\times\left(\left(\mathcal{A}^{n}_{\rho}-\mathcal{A}_{\rho}\right)\widehat{B^{n}_{s}}\right)ds\right\|_{\mathbf{H}^{-1-\theta}}$
$\displaystyle+\left\|\int_{0}^{t}S(t-s)\nabla\times\left(\mathcal{A}_{\rho}\left(\widehat{B^{n}_{s}}-\overline{B_{s}}\right)\right)ds\right\|_{\mathbf{H}^{-1-\theta}}=I_{1}+I_{2}+I_{3}+I_{4}.$

$I_{1}$ can be treated by 2.3 :

(4.21) $\displaystyle
I_{1}\lesssim_{\eta,T}\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)/2}}+\left\|\overline{B_{0}}-B^{n}_{0}\right\|_{\mathbf{H}^{-1-\theta}}.$

Thanks to 2.3, (4.18) and (4.19) it holds

$\displaystyle I_{2}$
$\displaystyle\lesssim\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)/2}}\int_{0}^{t}\lvert
t-s\rvert^{\theta/2}\left\|\nabla\times\left(\mathcal{A}^{n}_{\rho}\widehat{B^{n}_{s}}\right)ds\right\|_{\mathbf{H}^{-1}}ds$
$\displaystyle\lesssim\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)/2}}\int_{0}^{t}\lvert
t-s\rvert^{\theta/2}\left\|\widehat{B^{n}_{s}}\right\|ds$ (4.22)
$\displaystyle\lesssim_{T}\frac{1}{n^{\theta\chi(\alpha,\beta,\gamma)/2}}.$

Due to 2.2 and (4.19) we can treat $I_{3}$ obtaining easily:

$\displaystyle I_{3}$
$\displaystyle\lesssim_{\eta,\theta}\int_{0}^{t}\left\|\nabla\times\left(\left(\mathcal{A}^{n}_{\rho}-\mathcal{A}_{\rho}\right)\widehat{B^{n}_{s}}\right)\right\|_{\mathbf{H}^{-1}}ds$
$\displaystyle\lesssim\left\|\mathcal{A}^{n}_{\rho}-\mathcal{A}_{\rho}\right\|_{HS}\int_{0}^{T}\left\|\widehat{B^{n}_{s}}\right\|ds$
(4.23) $\displaystyle\lesssim_{T}\frac{1}{n}.$

Lastly we treat $I_{4}$ by 2.2:

$\displaystyle I_{4}$
$\displaystyle\lesssim_{\eta}\int_{0}^{t}\frac{1}{(t-s)^{1/2}}\left\|\nabla\times\left(\mathcal{A}_{\rho}\left(\widehat{B^{n}_{s}}-\overline{B_{s}}\right)\right)\right\|_{\mathbf{H}^{-2-\theta}}ds$
(4.24)
$\displaystyle\lesssim\int_{0}^{t}\frac{1}{(t-s)^{1/2}}\left\|\widehat{B^{n}_{s}}-\overline{B_{s}}\right\|_{\mathbf{H}^{-1-\theta}}ds.$

Combining (4.21),(4),(4),(4) we get (4.16) by Gronwall’s lemma. We move now to
$\widehat{B^{n,3}_{t}}-\overline{B^{n,3}_{t}}$. First we recall that in case
of 2.5 $\chi(\alpha,\beta,\gamma)\equiv 1$ and we have

$\displaystyle\left\|\widehat{B^{n,3}_{t}}-\overline{B^{3}_{t}}\right\|_{\dot{H}^{-\vartheta}}$
$\displaystyle\lesssim\left(\left\|\left(S^{n}(t)-S(t)\right)\overline{B^{3}_{0}}\right\|_{\dot{H}^{-\vartheta}}+\left\|S^{n}(t)\left(B^{n,3}_{0}-\overline{B^{3}_{0}}\right)\right\|_{\dot{H}^{-\vartheta}}\right)$
$\displaystyle+\zeta^{n}_{H,2}\left\|\int_{0}^{t}\left(S^{n}(t-s)-S(t-s)\right)\operatorname{div}_{H}\left(\widehat{B^{n,H}_{s}}^{\perp}\right)ds\right\|_{\dot{H}^{-\vartheta}}$
$\displaystyle+\lvert\zeta^{n}_{H,2}-\zeta_{H,2}\rvert\left\|\int_{0}^{t}S(t-s)\operatorname{div}_{H}\left(\widehat{B^{n,H}_{s}}^{\perp}\right)ds\right\|_{\dot{H}^{-\vartheta}}$
$\displaystyle+\zeta_{H,2}\left\|\int_{0}^{t}S(t-s)\operatorname{div}\left(\widehat{B^{n,H}_{s}}^{\perp}-\overline{B^{H}_{s}}^{\perp}\right)ds\right\|_{\dot{H}^{-\vartheta}}=J_{1}+J_{2}+J_{3}+J_{4}.$

$J_{1}$ can be treated as in the previous case obtaining

(4.25) $\displaystyle
J_{1}\lesssim\frac{1}{n^{\vartheta/2}}+\left\|\overline{B^{3}_{0}}-B^{n,3}_{0}\right\|_{\dot{H}^{-\vartheta}}.$

Thanks to 2.1, 2.3, 2.2 and (4.19) it holds

$\displaystyle J_{2}$
$\displaystyle\lesssim\frac{1}{n^{{\vartheta/2}}}\int_{0}^{t}\lvert
t-s\rvert^{\vartheta/2}\left\|e^{(t-s)\frac{\eta}{2}\Delta}\operatorname{div}_{H}\left(\widehat{B^{n,H}_{s}}\right)\right\|ds$
$\displaystyle\lesssim_{\eta}\frac{1}{n^{{\vartheta/2}}}\int_{0}^{t}\frac{1}{(t-s)^{1-\vartheta}}\left\|\widehat{B^{n}_{s}}\right\|_{\mathbf{H}^{-1}}ds$
(4.26) $\displaystyle\lesssim_{T}\frac{1}{\vartheta n^{{\vartheta/2}}}.$

Due to 2.2 we can treat $J_{3}$ obtaining:

$\displaystyle J_{3}$
$\displaystyle\lesssim_{\eta,\vartheta}\frac{1}{n}\int_{0}^{t}\frac{1}{(t-s)^{1-\vartheta/2}}\left\|\operatorname{div}_{H}\left(\widehat{B^{n,H}_{s}}\right)\right\|_{\dot{H}^{-2}}ds$
(4.27) $\displaystyle\lesssim\frac{1}{\vartheta n}.$

Lastly we treat $J_{4}$ combining 2.2 and (4.16):

$\displaystyle J_{4}$
$\displaystyle\lesssim_{\eta}\int_{0}^{t}\frac{1}{(t-s)^{1+\theta/2-\vartheta/2}}\left\|\widehat{B^{n}_{s}}-\overline{B_{s}}\right\|_{\dot{H}^{-1-\theta}}ds$
$\displaystyle\lesssim_{T}\frac{1}{\vartheta-\theta}\operatorname{sup}_{t\in[0,T]}\left\|\widehat{B^{n}_{t}}-\overline{B_{t}}\right\|_{\dot{H}^{-1-\theta}}$
(4.28)
$\displaystyle\lesssim_{\eta,T}\frac{n^{-\theta/2}+\left\|\overline{B_{0}}-B^{n}_{0}\right\|_{\mathbf{H}^{-1-\theta}}}{\vartheta-\theta}.$

Combining (4.25),(4),(4),(4) we obtain relation (4.17) ∎

Secondly we provide a quantitative result on the closeness of $B^{n}_{t}$
(resp. $B^{n}_{t},B^{n,3}_{t}$) and $\widehat{B^{n}_{t}}$ (resp.
$\widehat{B^{n}_{t}},\widehat{B^{n,3}_{t}}$).

######  Lemma 4.10.

Under the same assumptions of Theorem 2.17, for each $\kappa\in[1,2),\
\theta\in(0,2),$ and $\delta\in(0,\theta)$ we have

(4.29)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|\widehat{B^{n}_{t}}-B^{n}_{t}\right\|^{\kappa}_{\mathbf{H}^{-1-\theta}}\right]$
$\displaystyle\lesssim\frac{1}{n^{\frac{\kappa(\alpha\wedge\beta\wedge\gamma)(\theta-\delta)}{6+\delta}}}\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{\kappa}$

In case of 2.5 we have also for each $\vartheta\in(\theta,\frac{3}{2}]$

(4.30)
$\displaystyle\operatorname{sup}_{t\in[0,T]}\mathbb{E}\left[\left\|\widehat{B^{n,3}_{t}}-B^{n,3}_{t}\right\|^{\kappa}_{\dot{H}^{-\vartheta}}\right]$
$\displaystyle\lesssim\frac{\left\|B^{n,3}_{0}\right\|^{\kappa}}{n^{\frac{2\kappa(\vartheta-\delta)}{3}}}+\frac{\left\|B^{n}_{0}\right\|_{\mathbf{H}^{-1}}^{\kappa}}{n^{\frac{\kappa(\vartheta-\theta)(\theta-\delta)}{6+\delta}}}.$

###### Proof.

From 4.4, we already know that $B^{n}_{t}$ and $B^{n,3}_{t}$ can be rewritten
in mild form as

$\displaystyle B^{n}_{t}$
$\displaystyle=S^{n}(t)B^{n}_{0}+\int_{0}^{t}S^{n}(t-s)\nabla\times(\mathcal{A}^{n}_{\rho}B^{n}_{s})ds+Z^{n}_{t},$
$\displaystyle B^{n,3}_{t}$
$\displaystyle=S^{n}(t)B^{n,3}_{0}-\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}S^{n}(t-s)\operatorname{div}_{H}\left((B^{n,H}_{s})^{\perp}\right)ds+H^{n}_{t}.$

Therefore, in case of 2.4 and 2.6 we introduce

$\displaystyle d^{n}_{t}=B^{n}_{t}-\widehat{B^{n}_{t}},$

while in case of 2.5 we introduce also

$\displaystyle D^{n}_{t}=B^{n,3}_{t}-\widehat{B^{n,3}_{t}}.$

we have

$\displaystyle
d^{n}_{t}=\int_{0}^{t}S^{n}(t-s)\nabla\times(\mathcal{A}^{n}_{\rho}d^{n}_{s})ds+Z^{n}_{t},\quad
D^{n}_{t}=H^{n}_{t}-\frac{\rho\zeta^{n}_{H,2}}{C_{1,H}C_{2,H}}\int_{0}^{t}S^{n}(t-s)\operatorname{div}_{H}\left((d^{n,H}_{s})^{\perp}\right)ds.$

Due to 4.8 and Grönwall’s inequality we can obtain (4.29). Indeed,

$\displaystyle\mathbb{E}\left[\left\|d^{n}_{t}\right\|_{\mathbf{H}^{-1-\theta}}^{\kappa}\right]$
$\displaystyle\lesssim_{\eta}\mathbb{E}\left[\left(\int_{0}^{t}\frac{1}{\sqrt{t-s}}\left\|d^{n}_{s}\right\|_{\mathbf{H}^{-1-\theta}}ds\right)^{\kappa}\right]+\mathbb{E}\left[\left\|Z^{n}_{t}\right\|_{\mathbf{H}^{-1-\theta}}^{2}\right]^{\frac{\kappa}{2}}$