# Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

Yachun Li  School of Mathematical Sciences, CMA-Shanghai, MOE-LSC, and SHL-
MAC, Shanghai Jiao Tong University, China<EMAIL_ADDRESS>,  Zirong Zeng
School of Mathematical Sciences, Shanghai Jiao Tong University, China.
<EMAIL_ADDRESS>and  Deng Zhang  School of Mathematical Sciences, CMA-
Shanghai, Shanghai Jiao Tong University, China<EMAIL_ADDRESS>
###### Abstract.

We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD
for short) equations. The constructed weak solutions do not conserve the
magnetic helicity and can be close to any given smooth, divergence-free and
mean-free velocity and magnetic fields. Furthermore, we prove that the weak
solutions constructed by Beekie-Buckmaster-Vicol [2] for the ideal MHD can be
obtained as a strong vanishing viscosity and resistivity limit of a sequence
of weak solutions to MHD equations. This shows that, in contrast to the weak
ideal limits, Taylor’s conjecture does not hold along the vanishing viscosity
and resistivity limits. Unlike in the context of the NSE [13] and the ideal
MHD [2], new types of velocity and magnetic flows, featuring both the refined
spatial and temporal intermittency, are constructed to respect the geometry of
MHD and to control the strong viscosity and resistivity. Compatible algebraic
structure is derived in the convex integration scheme. More interestingly, the
new intermittent flows indeed enable us to prove the aforementioned results
for the hyper-viscous and hyper-resistive MHD equations up to the sharp
exponent $5/4$, which coincides exactly with the Lions exponent for 3D hyper-
viscous NSE.

###### Key words and phrases:

Convex integration; MHD equations; intermittency; non-uniqueness; Taylor’s
conjecture.

###### 2010 Mathematics Subject Classification:

35A02, 35D30, 35Q30, 76D05, 76W05.

##  1\. Introduction and main results

###  1.1. Introduction

The problem of non-uniqueness of weak solutions to hydrodynamic models attract
a lot of attention in the last decade. Significant progresses have been made
towards the Euler equations and Navier-Stokes equations based on the convex
integration method, which originates from the work by Nash [59] concerning the
construction of $C^{1}$ isometric immersions of $S^{n}$ into
${\mathbb{R}}^{m}$ $(m\geq n+2)$ and was introduced by De Lellis-Székelyhidi
in the pioneering paper [27] to 3D Euler equations.

Among various hydrodynamic models, the viscous and resistive MHD system is a
canonical macroscopic model to describe the motion of conductive fluid, such
as plasma or liquid metals, under a complicated interaction between the
electromagnetic phenomenon and fluid dynamical phenomenon (see [50]). We refer
the reader to [4, 5, 26, 61] for more physical interpretations of the MHD
system.

The applicability of convex integration scheme to MHD equations is at an early
stage (see [15] and the literature review below), and it is the objective of
the present article. As a matter of fact, because of the strong coupling of
the velocity and magnetic fields, the convex integration method requires the
construction of both the velocity and magnetic fields in an appropriate way,
in order to respect the geometry of MHD. This particular structural
requirement, however, limits the oscillation directions and thus the spatial
intermittency of flows, which results in a substantial difficulty to control
the strong viscosity and resistivity.

More precisely, we are concerned with the three-dimensional MHD system on the
torus ${\mathbb{T}}^{3}:=[-\pi,\pi]^{3}$,

$\left\\{\begin{aligned} &\partial_{t}u-\nu_{1}\Delta
u+(u\cdot\nabla)u-(B\cdot\nabla)B+\nabla P=0,\\\ &\partial_{t}B-\nu_{2}\Delta
B+(u\cdot\nabla)B-(B\cdot\nabla)u=0,\\\
&{\mathrm{div}}u=0,\quad{\mathrm{div}}B=0,\end{aligned}\right.$ (1.1)

where $u=(u_{1},u_{2},u_{3})^{\top}(t,x)\in{\mathbb{R}}^{3}$,
$B=(B_{1},B_{2},B_{3})^{\top}(t,x)\in{\mathbb{R}}^{3}$ and
$P=P(t,x)\in{\mathbb{R}}$ correspond to the velocity field, magnetic field and
pressure of the fluid, respectively, and $\nu_{1},\nu_{2}\geq 0$ are the
viscous and resistive coefficients, respectively. We note that, in the case
without magnetic fields, (1.1) reduces to the incompressible Navier-Stokes
equation (NSE for short).

Another important model is the ideal MHD system, where the viscous and
resistive coefficients vanish, namely,

$\left\\{\begin{aligned} &\partial_{t}u+(u\cdot\nabla)u-(B\cdot\nabla)B+\nabla
P=0,\\\ &\partial_{t}B+(u\cdot\nabla)B-(B\cdot\nabla)u=0,\\\
&{\mathrm{div}}u=0,\quad{\mathrm{div}}B=0.\end{aligned}\right.$ (1.2)

The incompressible ideal MHD system is the classical macroscopic model
coupling the Maxwell equations to the evolution of an electrically conducting
incompressible fluid (cf. [4, 26]). In the case $B\equiv 0$, (1.2) then
reduces to the Euler equations.

The well-posedness problem of NSE and MHD has been extensively studied in the
literature. For the initial data with finite energy, the existence of global
weak solution $u$ to NSE was first proved by Leray [49] in 1934 and later by
Hopf [41] in 1951 in bounded domains, which satisfies $u\in
C_{weak}\big{(}[0,+\infty);L^{2}(\Omega)\big{)}\cap
L^{2}\big{(}[0,+\infty);\dot{H}^{1}(\Omega)\big{)}$, $\Omega={\mathbb{R}}^{3}$
or $\mathbb{T}^{3}$, and obeys the following energy inequality

$\displaystyle\|u(t)\|_{L^{2}}^{2}+2\nu\int_{t_{0}}^{t}\|{\nabla}u(s)\|_{L^{2}}^{2}{\rm
d}s\leq\|u(t_{0})\|_{L^{2}}^{2}$ (1.3)

for any $t>0$ and a.e. $t_{0}\geq 0$. The existence of Leray-Hopf solutions to
generalized MHD equations was proved by Wu [67].

The uniqueness of Leray-Hopf solutions is considered as one of the most
important issues in the mathematical fluid mechanics. The well known
Ladyžhenskaya-Prodi-Serrin condition states that, the Leray-Hopf solution to
NSE is regular and thus unique in the space $L^{\gamma}_{t}L^{p}_{x}$ for
${2}/{\gamma}+{3}/{p}\leq 1$ with $p>3$ (see [46, 60, 62]). The endpoint case
$L^{\infty}_{t}L^{3}_{x}$ was solved by Escauriaza-Seregin-Šverák [33]. The
well-posedness of NSE for small initial data in the space ${\rm BMO^{-1}}$ was
proved by Koch-Tataru [48]. Moreover, Lions [52] obtained the well-posedness
for the hyper-viscous NSE where the exponent of viscosity is bigger than
$5/4$. Concerning the MHD type equations, the uniqueness of global classical
solutions was proved by Wu [67], when the exponents of viscosity and
resistivity are bigger than ${1}/{2}+{n}/{4}$, which is exactly the Lions
exponent for the hyper-viscous NSE in dimension three $n=3$. Moreover, the
uniqueness in the case analogous to the Ladyžhenskaya-Prodi-Serrin condition
was proved by Zhou [69]. See also [51, 61, 63, 66, 68] and references therein.

Concerning the non-uniqueness problem of weak solutions, in the pioneering
paper [27] De Lellis-Székelyhidi introduced the convex integration scheme to
the 3D Euler equations and proved the existence of bounded solutions with
compact support in space-time (see also [28]). Afterwards, significant
progress towards the famous Onsager conjecture for the 3D Euler equations was
made by De Lellis-Székelyhidi [29, 30]. The resolution of the Onsager
conjecture was finally solved in $C_{x,t}^{\beta}\,(0<\beta<1/3)$ by Isett
[42], and by Buckmaster-De Lellis-Székelyhidi-Vicol [11] for dissipative
solutions. We also refer the reader to the papers [17, 21, 55] for Euler
equations, [9, 19, 58, 57] for transport equations, and [12] for SQG
equations. See also the surveys [31, 14, 15] on convex integration methods.

A recent break-through was obtained by Buckmaster-Vicol [13] for the 3D
incompressible NSE. The non-uniqueness of weak solutions to NSE was proved in
[13] by using an $L_{x}^{2}$-based intermittent convex integration scheme,
which is efficient to handle the viscosity of NSE. Later, Luo-Titi [54] proved
the non-uniqueness results for the 3D hyper-viscous NSE, whenever the exponent
of viscosity is less than the Lions exponent $5/4$. Moreover, Luo-Qu [53]
proved the non-uniqueness for the 2D hypoviscous NSE. The non-uniqueness
results for the stationary NSE were proved by Cheskidov-Luo [18, 56]. We also
refer to the works by Jia-Šverák [43, 44] for another method for the non-
uniqueness issue of Leray-Hopf solutions in $L^{\infty}_{t}L^{3}_{x}$ under a
certain assumption for the linearized Navier-Stokes operator, and Colombo-De
Lellis-De Rosa [22] and De Rosa [32] for the non-uniqueness of Leray weak
solutions to hypoviscous NSE with small viscosity.

Regarding the ideal MHD system (1.2), it posses a number of global invariants:

  1. $\bullet$

The total energy:
$\displaystyle\mathcal{E}(t)=\frac{1}{2}\int_{{\mathbb{T}}^{3}}|u(t,x)|^{2}+|B(t,x)|^{2}{\rm
d}x$;

  2. $\bullet$

The cross helicity:
$\displaystyle\mathcal{H}_{\omega,B}(t)=\int_{{\mathbb{T}}^{3}}u(t,x)\cdot
B(t,x){\rm d}x$;

  3. $\bullet$

The magnetic helicity:
$\displaystyle\mathcal{H}_{B,B}(t):=\int_{{\mathbb{T}}^{3}}A(t,x)\cdot
B(t,x){\rm d}x$.

Here $A$ is a mean-free periodic vector field satisfying ${\mathrm{curl}}A=B$.

In analogy with the Onsager conjecture for Euler equations, it was conjectured
in [15] that, $L^{3}_{t,x}$ (resp. $L^{3}_{t}B^{0}_{3,c_{0},x}$) is the
critical space for the conservation of the magnetic helicity. More precisely,
any weak solutions belonging to $L^{3}_{t,x}$ (resp.
$L^{3}_{t}B^{0}_{3,c_{0},x}$) conserve the magnetic helicity (the rigidity
part), while below the threshold there exist weak solutions violating the
magnetic helicity conservation (the flexible part). Moreover, the space
$C^{1/3}_{t,x}$ (resp. $L^{3}_{t}B^{1/3}_{3,{\infty},x}$) was conjectured to
be the threshold for the conservation of the total energy and the cross
helicity (see Remark 1.6 (ii) below).

These conjectures indeed lie in the general scope of the important issue to
identify the critical/threshold regularity for the conservation laws. We refer
to Klainerman [47] for the general formulations of different thresholds for
supercritical Hamiltonian evolution equations, including the 3D MHD equations.

On the rigidity side, the magnetic helicity conservation for the 3D ideal MHD
was proved in [45, 1, 34] and [16], respectively, for the critical spaces
$L^{3}_{t,x}$ and $L^{3}_{t}B^{\alpha}_{3,{\infty},x}$ with ${\alpha}>0$. The
corresponding rigidity results for the total energy conservation were proved
in [16]. On the flexible side, weak solutions with non-trivial energy and
vanishing magnetic helicity were first given in [8], by embedding the ideal
MHD into the $2\frac{1}{2}$D Euler flow via a symmetry assumption. Wild
solutions with compact support in space-time were first constructed by Faraco-
Lindberg-Székelyhidi [36], via the $L^{\infty}_{t,x}$ convex integration
scheme. The constructed solutions in [36] violate the energy conservation,
while the magnetic helicity is still conserved and thus vanishes due to the
compact support in time of solutions.

The delicate point, as observed by Beekie-Buckmaster-Vicol [2], is that the
magnetic helicity is conserved under much milder regularity conditions. As a
matter of fact, the magnetic helicity is commonly expected in the plasma
physics to be conserved in the infinite conductivity limit, known as Taylor’s
conjecture ([64, 65]). This conjecture was proved by Faraco and Lindberg [35]
under the weak ideal limits, namely, the weak limits of Leray solutions to MHD
(1.1).

For general weak solutions, Beekie-Buckmaster-Vicol [2] first constructed weak
solutions in $C_{t}L^{2}_{x}$ violating the magnetic helicity conservation.
Unlike in [36], the proof of [2] relies on the intermittent convex integration
scheme, in which the intermittent shear flows are constructed to respect the
geometry of MHD. The flexible part of the Onsager-type conjecture for the
magnetic helicity was recently solved by Faraco-Lindberg-Székelyhidi [37],
based on the convex integration via staircase laminates.

See also Dai [25] for the Hall MHD system where the Hall nonlinearity takes
the dominant effect, and Feireisl-Li [38] for the ill-posedness of the MHD
system where the fluid is compressible, inviscid and magnetically resistive.

As mentioned above, the strong coupling of the velocity and magnetic fields in
MHD equations limits the intermittency of flows in the convex integration
scheme. More precisely, for the NSE, the flows constructed in [13] have 3D
spatial intermittency, which is essential to control the viscosity in NSE.
Regarding the ideal MHD, the intermittent shear flows constructed in [2] have
1D intermittency which only permit to control the hypo-viscosity and hypo-
resistivity $(-\Delta)^{\alpha}$ with $\alpha\in(0,1/2)$. Hence, both the
above intermittent flows are not applicable to the viscous and resistive MHD
(1.1) under consideration here.

The main purpose of this paper is to address the non-uniqueness problem of
weak solutions for the canonical MHD system (1.1). By constructing a new class
of intermittent velocity and magnetic flows adapted to the geometry of MHD, we
are able to control the strong viscosity and resistivity. More interestingly,
the new class of intermittent flows even enables us to achieve the non-
uniqueness of weak solutions to a more general class of MHD type equations,
for all the exponents of viscosity and resistivity less than the sharp Lions
exponent, namely,

$\left\\{\begin{aligned}
&\partial_{t}u+\nu_{1}(-\Delta)^{\alpha_{1}}u+(u\cdot\nabla)u-(B\cdot\nabla)B+\nabla
P=0,\\\
&\partial_{t}B+\nu_{2}(-\Delta)^{\alpha_{2}}B+(u\cdot\nabla)B-(B\cdot\nabla)u=0,\\\
&{\mathrm{div}}u=0,\quad{\mathrm{div}}B=0,\end{aligned}\right.$ (1.4)

where $\alpha_{1},\alpha_{2}\in(0,5/4)$, $\nu_{1},\nu_{2}\geq 0$. We note
that, the value $5/4$ coincides with the Lions exponent for the well-posedness
of the hyper-viscous NSE [52], and with the exponent $1/2+n/4$ for the well-
posedness of generalized MHD in dimension three $n=3$ [67].

The constructed weak solutions can be close to any given divergence-free,
mean-free velocity and magnetic fields. In particular, the weak solutions live
in the space $L^{\gamma}_{t}W^{s,p}_{x}$ with $0\leq
s<{2}/{p}+{1}/{\gamma}-3/2$, $1\leq p,\gamma<+{\infty}$, and do not conserve
the magnetic helicity. Thus, this provides more examples in the spaces
$L^{\gamma}_{t}W^{s,p}_{x}$ for the flexible part of the Onsager-type
conjecture for the ideal MHD (1.2).

Another aim of the present work is to make some progress towards the
understanding of the relationship between the non-uniqueness of solutions to
the ideal MHD (1.2) and MHD equations (1.1) or (1.4), which in turn relates to
the physical Taylor conjecture.

As pointed out in [2, 15], the weak solutions to the ideal MHD (1.2) with non-
trivial magnetic helicity cannot be obtained as the weak ideal limits of MHD
Leray-Hopf solutions. Instead, we prove that the weak solutions constructed by
Beekie-Buckmaster-Vicol [2] can be obtained as a strong vanishing viscosity
and resistivity limit of a sequence of weak solutions to the MHD equations
(1.4), where the exponents ${\alpha}_{i}$ can be even larger than one. An
interesting outcome is that, together with the non-conservative results in the
work [2], this shows that Taylor’s conjecture does not hold along the
vanishing viscosity and resistivity limits of general weak solutions, in
contrast to the weak ideal limits of Leray weak solutions.

One of the main novelties of our proof is the construction of the velocity and
magnetic flows with the refined spatial intermittency, which are designed
specifically to have two oscillation directions and concentrate on small
spatial cuboids. This structure enables us to gain an additional 1D spatial
intermittency than that in the ideal MHD case [2], and thus to control the
viscosity and resistivity $(-\Delta)^{{\alpha}_{i}}$ with
${\alpha}_{i}\in[0,1)$, $i=1,2$.

Another nice feature is that, the new intermittent flows have almost mutually
disjoint supports. Actually, the intermittent flows have much smaller
intersections, which provide almost 2D spatial intermittency and thus
contribute negligible errors in the convex integration scheme. This in
particular simplifies the control of the nonlinearities in MHD equations.

In order to pass beyond the border line ${\alpha}_{i}=1$, $i=1,2$, another key
novelty of proof is the construction of the $L_{t}^{2}$-based temporal
intermittent flows adapted to the structure of MHD. We introduce the high
temporal oscillations in the velocity and magnetic flows. Rather than the
pointwise analysis in time, we measure the solutions on average, namely, in
the space $L^{2}_{t}L^{2}_{x}$, which is in spirit close to the works [13,
19]. In particular, this new temporal intermittency gains an extra 1D
intermittency, which permits to control the much stronger viscosity and
resistivity $(-\Delta)^{{\alpha}_{i}}$, ${\alpha}_{i}\geq 1$, and, more
interestingly, even to achieve all the exponents ${\alpha}_{i}\in[1,5/4)$ up
to the sharp Lions exponent.

Let us also mention that, besides the incompressibility correctors in [2], two
new types of temporal correctors adapted to MHD are also introduced for the
velocity and magnetic perturbations, in order to balance both the high spatial
and temporal oscillations arising from the corresponding concentration
functions. The new velocity flows and magnetic flows indeed exhibit a quite
compatible algebraic structure in the convex integration scheme. We expect the
new method developed here would be also of interest in the further
understanding of MHD equations.

Notations. For $p\in[1,+\infty]$ and $s\in{\mathbb{R}}$, we use the following
short notations

$\displaystyle L^{p}_{x}:=L^{p}_{x}({\mathbb{T}}^{3}),\quad
H^{s}_{x}:=H^{s}_{x}({\mathbb{T}}^{3}),\quad
W^{s,p}_{x}:=W^{s,p}_{x}({\mathbb{T}}^{3}).$

The mean of $u\in L^{1}({\mathbb{T}}^{n})$ is given by
${\fint}_{{\mathbb{T}}^{n}}u{\rm
d}x={|{\mathbb{T}}^{n}|}^{-1}\int_{{\mathbb{T}}^{n}}u{\rm d}x,$ where
$|\cdot|$ denotes the Lebesgue measure. For any $p,\gamma\in[1,+\infty]$,
$L^{\gamma}_{t}L^{p}_{x}$ denotes the usual Banach space
$L^{\gamma}_{t}({\mathbb{T}};L^{p}({\mathbb{T}}^{3}))$. Moreover, given any
Banach space $X$, $C({\mathbb{T}};X)$ denotes the space of continuous
functions from $\mathbb{T}$ to $X$, equipped with the norm
$\|u\|_{C_{t}X}:=\sup_{t\in\mathbb{T}}\|u(t)\|_{X}$. In particular, we write
$L^{p}_{t,x}:=L^{p}_{t}L^{p}_{x}$ and $C_{t,x}:=C_{t}C_{x}$ for simplicity.
Let

$\displaystyle\lVert u\rVert_{W^{N,p}_{t,x}}:=\sum_{0\leq m+|\zeta|\leq
N}\lVert\partial_{t}^{m}{\nabla}^{\zeta}u\rVert_{L^{p}_{t,x}},\ \ \lVert
u\rVert_{C_{t,x}^{N}}:=\sum_{0\leq m+|\zeta|\leq
N}\lVert\partial_{t}^{m}{\nabla}^{\zeta}u\rVert_{C_{t,x}},$

where $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})$ is the multi-index and
${\nabla}^{\zeta}:=\partial_{x_{1}}^{\zeta_{1}}\partial_{x_{2}}^{\zeta_{2}}\partial_{x_{3}}^{\zeta_{3}}$.
For any $A\subseteq{\mathbb{T}}$, set

$\displaystyle N_{\varepsilon_{*}}(A):=\\{t\in{\mathbb{T}}:\ \exists s\in A,\
s.t.\ |t-s|\leq\varepsilon_{*}\\}.$

We also adapt the notations from [50]. Let $u,\,v$ be two vector fields, the
corresponding second order tensor product is defined by

$\displaystyle u\otimes v:=(u_{i}v_{j})_{1\leq i,j\leq 3}.$

For any second-order tensor $A=(a_{ij})_{1\leq i,j\leq 3}$, set

$\displaystyle{\mathrm{div}}A:=\left(\sum_{j=1}^{3}{\partial_{x_{j}}a_{1j}},\sum_{j=1}^{3}{\partial_{x_{j}}a_{2j}},\sum_{j=1}^{3}{\partial_{x_{j}}a_{3j}}\right)^{\top}.$

The right product of a vector field $v=(v_{1},v_{2},v_{3})^{\top}$ to a
second-order tensor $A=(a_{ij})_{1\leq i,j\leq 3}$ is defined by

$\displaystyle
Av:=\left(\sum_{j=1}^{3}a_{1j}v_{j},\sum_{j=1}^{3}a_{2j}v_{j},\sum_{j=1}^{3}a_{3j}v_{j}\right)^{\top}.$

In particular, for any scalar function $f$ and second-order tensor
$A=(a_{ij})_{1\leq i,j\leq 3}$, one has the Leibniz rule

$\displaystyle{\mathrm{div}}(fA)=f{\mathrm{div}}A+A\nabla f.$

We use the notation $a\lesssim b$, which means that $a\leq Cb$ for some
constant $C>0$.

###  1.2. Formulation of main results

To begin with, let us formulate precisely the definition of weak solutions to
(1.4).

######  Definition 1.1 (Weak solution).

We say that $(u,B)\in L^{2}({\mathbb{T}};L^{2}({\mathbb{T}}^{3}))$ is a weak
solution to the MHD equations (1.4) if

  * •

For all $t\in{\mathbb{T}}$, $(u(t,\cdot),B(t,\cdot))$ are divergence free in
the sense of distributions and have zero spatial mean.

  * •

Equation (1.4) holds in the sense of distributions, i.e., for any divergence-
free test functions $\varphi\in
C_{0}^{\infty}({\mathbb{T}}\times\mathbb{T}^{3})$,

$\displaystyle\int_{\mathbb{T}^{3}}\partial_{t}\varphi\cdot
u-\nu_{1}(-\Delta)^{\alpha_{1}}\varphi\cdot u+\nabla\varphi:(u\otimes
u-B\otimes B){\rm d}x$ $\displaystyle=0,$
$\displaystyle\int_{\mathbb{T}^{3}}\partial_{t}\varphi\cdot
B-\nu_{2}(-\Delta)^{\alpha_{2}}\varphi\cdot B+\nabla\varphi:(B\otimes
u-u\otimes B){\rm d}x$ $\displaystyle=0.$

Here, for any $3\times 3$ matrices $A=({A_{ij}})$ and $S=({S_{ij}})$,
$A:S=\sum_{i,j=1}^{3}A_{ij}S_{ij}$.

For simplicity, we consider the solutions on the temporal interval
$\mathbb{T}$ to be consistent with the spatial torus $\mathbb{T}^{3}$. The
arguments of this paper can be also extended to any bounded temporal
intervals.

Below we mainly formulate the main results for the more general MHD type
equations (1.4), which in particular includes the canonical MHD system (1.1)
that is the main model of the present work.

######  Theorem 1.2.

Let $\alpha_{1},\alpha_{2}\in[0,5/4)$, $(\tilde{u},\tilde{B})$ be any smooth,
divergence-free and mean-free vector fields on
${\mathbb{T}}\times{\mathbb{T}}^{3}$ , and $(s,p,\gamma)\in\mathcal{A}$ where

$\displaystyle\mathcal{A}:=\left\\{(s,p,\gamma)\in[0,\frac{3}{2})\times[1,+{\infty})\times[1,+{\infty}):0\leq
s<\frac{2}{p}+\frac{1}{\gamma}-\frac{3}{2}\right\\}.$ (1.5)

(See Figure 1. for the admissible set $\mathcal{A}$ of regularities when
$s=0$.)

Then, there exists $\beta^{\prime}\in(0,1)$, such that for any given
$\varepsilon_{*}>0$, there exist a velocity field $u$ and a magnetic field $B$
such that the following holds:

  1. (i)

Weak solution: $(u,B)$ is a weak solution to (1.4) in the sense of Definition
1.1 with zero spatial mean.

  2. (ii)

Regularity: $u,B\in H^{\beta^{\prime}}_{t,x}\cap L^{\gamma}_{t}W^{s,p}_{x}$.

  3. (iii)

Temporal support: $\mathrm{supp\,}_{t}u\cup\mathrm{supp\,}_{t}B\subseteq
N_{\varepsilon_{*}}(\mathrm{supp\,}_{t}\tilde{u}\cup\mathrm{supp\,}_{t}\tilde{B})$.

  4. (iv)

Small deviations on average:
$\|u-\tilde{u}\|_{L^{1}_{t}L^{2}_{x}}\leq\varepsilon_{*}$,
$\|B-\tilde{B}\|_{L^{1}_{t}L^{2}_{x}}\leq\varepsilon_{*}$.

  5. (v)

Small deviations of magnetic helicity:
$\|\mathcal{H}_{B,B}-\mathcal{H}_{\widetilde{B},\widetilde{B}}\|_{L_{t}^{1}}\leq\varepsilon_{*}$.

Figure 1.

(The red point is the conjectured regularity corresponding to $L^{3}_{t,x}$,

the blue region is the admissible set $\mathcal{A}$ of regularities when
$s=0$.)

As a consequence, Theorem 1.2 gives the following result concerning the non-
uniqueness of weak solutions and the non-conservation of magnetic helicity.

######  Corollary 1.3.

For any ${\alpha}_{1},{\alpha}_{2}\in[0,5/4)$ and any
$(s,p,\gamma)\in\mathcal{A}$, where the regularity admissible set
$\mathcal{A}$ is as in (1.5), there exist infinitely many weak solutions to
(1.4) with the same data at time zero, which live in the space
$H^{\beta^{\prime}}_{t,x}\cap L^{\gamma}_{t}W^{s,p}_{x}$ for some
$\beta^{\prime}>0$ and do not conserve the magnetic helicity.

######  Remark 1.4.

(i). The non-uniqueness results for the ideal MHD (1.2) were proved by Faraco-
Lindberg-Székelyhidi [36] for weak solutions with compact support in space-
time, and by Beekie-Buckmaster-Vicol [2] for weak solutions with non-trivial
magnetic helicity.

To the best of our knowledge, Corollary 1.3 provides the first non-uniqueness
result for the weak solutions to the MHD equations (1.4) with
$\alpha_{i}\in(0,5/4)$, i.e., less than the sharp Lions exponent.

(ii). By the Onsager-type conjecture for the magnetic helicity in [15], it was
expected that $L^{3}_{t,x}$ is the threshold for the magnetic helicity
conservation for the ideal MHD. This conjecture was solved recently by Faraco-
Lindberg-Székelyhidi [37], by constructing weak solutions uniformly-in-time in
the spatial Lorentz space $L^{3,\infty}_{x}$ with non-trivial magnetic
helicity, based on the convex integration via staircase laminates.

In view of Corollary 1.3, one has more examples in the spaces
$H^{\beta^{\prime}}_{t,x}\cap L^{\gamma}_{t}W^{s,p}_{x}$ for the flexible part
of the conjecture for the magnetic helicity, where $\beta^{\prime}$ is a
positive small constant and $(s,p,\gamma)\in\mathcal{A}$ with $\mathcal{A}$ as
in Theorem 1.2.

(iii). In the higher dimensions $d\geq 4$, one may prove the non-uniqueness of
weak solutions in $C_{t}L^{2}_{x}$ for the viscous and resistive MHD (1.1).
The solutions can be continuous in time. This is possible because, in high
dimensions $d\geq 4$, there is more freedom to choose the oscillation
directions in the velocity and magnetic flows, which permit to gain the 3D
spatial intermittency to control the Laplacian $(-\Delta)$.

(iv). We expect that the proof of this paper applies to the non-uniqueness in
law for the stochastic generalized MHD equations driven by Wiener processes,
which will be done in a forthcoming work. We would like to refer the reader to
the papers [6, 20, 40, 39] for the recent developments in the stochastic
contexts.

Our next result is concerned with the vanishing viscosity and resistivity
limits of the weak solutions to (1.4).

######  Theorem 1.5.

Let ${\alpha}_{1},{\alpha}_{2}\in(0,5/4)$ and $(u,B)\in
H^{\widetilde{\beta}}_{t,x}\times H^{\widetilde{\beta}}_{t,x}$ be any mean-
free weak solution to the ideal MHD (1.2), where $\widetilde{\beta}>0$. Then,
there exist $\beta^{\prime}\in(0,\widetilde{\beta})$ and a sequence of weak
solutions $(u^{(\nu_{n})},B^{(\nu_{n})})\in H^{\beta^{\prime}}_{t,x}\times
H^{\beta^{\prime}}_{t,x}$ to the MHD equations (1.4), where
$\nu_{n}=(\nu_{1,n},\nu_{2,n})$ and $\nu_{1,n}$, $\nu_{2,n}$ are the viscosity
and resistivity coefficients, respectively, such that as $\nu_{n}\rightarrow
0$,

$\displaystyle u^{(\nu_{n})}\rightarrow u,\quad B^{(\nu_{n})}\rightarrow
B\quad\text{strongly in}\ H^{\beta^{\prime}}_{t,x}.$ (1.6)

######  Remark 1.6.

(i). By Theorem 1.5, the set of the accumulation points of weak solutions to
the MHD equations (1.4), in the $H^{\beta^{\prime}}_{t,x}$ topology, contains
the weak solutions to the ideal MHD equations (1.2). This extends the
vanishing viscosity result in the context of the NSE [13] to the MHD equations
up to the Lions exponent.

Let us mention that, the Sobolev space $H^{\widetilde{\beta}}_{t,x}$ is less
regular than $C^{\widetilde{\beta}}_{t,x}$ used in the vanishing viscosity
limit result for the NSE in [13]. In order to get the temporal regularity of
solutions, we use the Slobodetskii-type norm of $H^{\widetilde{\beta}}_{t,x}$
to gain the regularity on average, which turns out to be efficient to prove
the vanishing viscosity limit in the Sobolev space $H^{\beta^{\prime}}_{t,x}$
with $\beta^{\prime}<\widetilde{\beta}$. See the proof of (6.35) and (6.3)
below.

(ii). Another interesting outcome of Theorem 1.5 is related to the weak
solutions constructed in [2] and to the Taylor conjecture concerning the
conservation of magnetic helicity.

The Taylor conjecture is mathematically formulated in the concept of weak
ideal limits, i.e., the weak limits of Leray weak solutions, and has been
recently proved by Faraco-Lindberg [35]. The crucial ingredient of the proof
is the regularity of Leray approximating solutions. Hence, the weak solutions
constructed in [2] with non-trivial magnetic helicity cannot be obtained as
the weak ideal limits of Leray-Hopf solutions to the MHD (1.1) (cf. [2, 15]).

Let us mention that, the weak solutions constructed in [2] has the regularity
$H^{\widetilde{\beta}}_{t,x}$ for some $\widetilde{\beta}>0$ (see Remark 6.1
below). Hence, by virtue of Theorem 1.5, we note that, in contrast to the weak
ideal limits, the weak solutions in [2] actually can be obtained as the
vanishing viscosity and resistivity limits of those to the MHD equations
(1.4), where the exponents ${\alpha}_{i}$ can be even larger than one. In
particular, this yields that Taloy’s conjecture does not hold along the
vanishing viscosity and resistivity limits of the weak solutions to (1.4).

###  1.3. Outline of the proof

Our strategy of proof is based on the intermittent convex integration scheme,
inspired by the works [2, 13, 19].

More precisely, for each integer $q\in\mathbb{N}$, we consider the following
relaxation system

$\left\\{\begin{aligned}
&\partial_{t}{u_{q}}+\nu_{1}(-\Delta)^{\alpha_{1}}{u_{q}}+{\mathrm{div}}({u_{q}}\otimes{u_{q}}-{B_{q}}\otimes{B_{q}})+\nabla
P_{q}={\mathrm{div}}\mathring{R}_{q}^{u},\\\
&\partial_{t}{B_{q}}+\nu_{2}(-\Delta)^{\alpha_{2}}{B_{q}}+{\mathrm{div}}({B_{q}}\otimes{u_{q}}-{u_{q}}\otimes{B_{q}})={\mathrm{div}}\mathring{R}_{q}^{B},\\\
&{\mathrm{div}}{u_{q}}=0,\quad{\mathrm{div}}{B_{q}}=0,\end{aligned}\right.$
(1.7)

where the Reynolds stress $\mathring{R}_{q}^{u}$ is a symmetric traceless
$3\times 3$ matrix, and the magnetic stress $\mathring{R}_{q}^{B}$ is a skew-
symmetric $3\times 3$ matrix.

The quantitative estimates of the relaxation solutions
$(u_{q},B_{q},\mathring{R}^{u}_{q},\mathring{R}^{B}_{q})$ are specified in the
main iteration result below.

More precisely, let $\alpha:=\max\\{\alpha_{1},\alpha_{2}\\}$,
$\alpha_{*}:=\max\\{2\alpha-1,0\\}$. Take $\varepsilon\in\mathbb{Q}_{+}$
sufficiently small such that

$\varepsilon\leq\min\left\\{\frac{3-2\alpha_{*}}{20},\frac{1}{10}\right\\},$
(1.8)

and

$\displaystyle s+\frac{3}{2}-\frac{2}{p}-\frac{1}{\gamma}+8{\varepsilon}<0.$
(1.9)

For $q\in\mathbb{N}$, the frequency parameter $\lambda_{q}$ and the amplitude
parameter $\delta_{q+2}$ are defined by

$\lambda_{q}=a^{(b^{q})},\ \ \delta_{q+2}=\lambda_{q+2}^{-2\beta}.$ (1.10)

Here, $a\in 5\mathbb{N}$ is a large integer of multiple $5$ such that
$a^{\varepsilon}\in 5\mathbb{N}$, $b\in 2\mathbb{N}$ is a large integer of
multiple $2$ such that $\lambda_{q+1}^{2\varepsilon}=\lambda_{q}^{2\varepsilon
b}\in\mathbb{N}$ and

$\displaystyle b>\frac{1000}{\varepsilon},\ \ 0<\beta<\frac{1}{100b^{2}}.$
(1.11)

Let $q\in\mathbb{N}$, we assume the following inductive estimates of the
relexation solutions to (1.7) at level $q$:

$\displaystyle\|{u_{q}}\|_{C_{t,x}^{1}}\leq\lambda_{q}^{7},\quad\|{B_{q}}\|_{C_{t,x}^{1}}\leq\lambda_{q}^{7},$
(1.12)
$\displaystyle\|\mathring{R}_{q}^{u}\|_{C^{1}_{t,x}}\leq\lambda_{q}^{14},\quad\|\mathring{R}_{q}^{B}\|_{C^{1}_{t,x}}\leq\lambda_{q}^{14},$
(1.13)
$\displaystyle\|\mathring{R}_{q}^{u}\|_{L^{1}_{t,x}}\leq\delta_{q+1},\quad\|\mathring{R}_{q}^{B}\|_{L^{1}_{t,x}}\leq\delta_{q+1}.$
(1.14)

The main iteration result is contained in the following theorem.

######  Theorem 1.7 (Main iteration).

Let $\alpha_{1},\alpha_{2}\in[0,5/4)$. Then, there exist $\beta\in(0,1)$,
$M>0$ large enough and $a_{0}=a_{0}(\beta,M)$, such that for any integer
$a\geq a_{0}$, the following holds:

Suppose that $({u_{q}},{B_{q}},\mathring{R}_{q}^{u},\mathring{R}_{q}^{B})$
solves (1.7) and satisfies (1.12)-(1.14). Then, there exists
$(u_{q+1},B_{q+1},\mathring{R}^{u}_{q+1},\mathring{R}^{B}_{q+1})$ solving
(1.7) and satisfying (1.12)-(1.14) with $q+1$ replacing $q$. In addition, we
have

$\left\|u_{q+1}-u_{q}\right\|_{L^{2}_{t,x}}\leq
M\delta_{q+1}^{\frac{1}{2}},\quad\left\|B_{q+1}-B_{q}\right\|_{L^{2}_{t,x}}\leq
M\delta_{q+1}^{\frac{1}{2}},$ (1.15)
$\left\|u_{q+1}-u_{q}\right\|_{L^{1}_{t}L^{2}_{x}}\leq\delta_{q+2}^{\frac{1}{2}},\quad\left\|B_{q+1}-B_{q}\right\|_{L^{1}_{t}L^{2}_{x}}\leq\delta_{q+2}^{\frac{1}{2}},$
(1.16)

and

$\displaystyle\mathrm{supp\,}_{t}u_{q+1}\cup\mathrm{supp\,}_{t}B_{q+1}\subseteq
N_{\delta_{q+2}^{\frac{1}{2}}}\left(\mathrm{supp\,}_{t}u_{q}\cup\mathrm{supp\,}_{t}B_{q}\cup\mathrm{supp\,}_{t}\mathring{R}_{q}^{u}\cup\mathrm{supp\,}_{t}\mathring{R}_{q}^{B}\right),$
(1.17)
$\displaystyle\mathrm{supp\,}_{t}\mathring{R}_{q+1}^{u}\cup\mathrm{supp\,}_{t}\mathring{R}_{q+1}^{B}\subseteq
N_{\delta_{q+2}^{\frac{1}{2}}}\left(\mathrm{supp\,}_{t}u_{q}\cup\mathrm{supp\,}_{t}B_{q}\cup\mathrm{supp\,}_{t}\mathring{R}_{q}^{u}\cup\mathrm{supp\,}_{t}\mathring{R}_{q}^{B}\right).$
(1.18)

The heart of the proof of Theorem 1.7 is to construct suitable perturbations

$\displaystyle w_{q+1}\simeq u_{q+1}-u_{q},\ \ d_{q+1}\simeq B_{q+1}-B_{q},$

module small mollification errors, such that the corresponding nonlinear
effects decrease the amplitudes of the Reynolds and magnetic stresses, in
order to fulfill the above iteration procedure.

###  1.4. New ingredients of proof

As mentioned in §1.1, besides the intermittent velocity flows $\\{W_{(k)}\\}$,
the intermittent magnetic flows $\\{D_{(k)}\\}$ are also required and shall be
constructed in an appropriate way, in order to respect the geometry of MHD.
This forces to limit the oscillation directions, and thus the intermittent
velocity and magnetic flows do not have the 3D spatial intermittency as in the
NSE case [13].

In the context of the ideal MHD (1.2), the key intermittent shear velocity
flows and the intermittent shear magnetic flows were introduced of the form

$\displaystyle W_{(k)}=\phi_{(k)}k_{1},\quad D_{(k)}=\phi_{(k)}k_{2},$ (1.19)

where $\\{\phi_{(k)}\\}$ are the spatial concentration functions with one
oscillation direction $k$, orthogonal to the other two directions
$k_{1},k_{2}$. The shear flows have the almost 1D spatial intermittency, i.e.,

$\displaystyle\|W_{(k)}\|_{C_{t}L^{1}_{x}}+\|D_{(k)}\|_{C_{t}L^{1}_{x}}\lesssim\lambda^{-\frac{1}{2}+{\varepsilon}},$
(1.20)

which permits to control the fractional viscosity and resistivity
$(-\Delta)^{{\alpha}_{i}}$ with ${\alpha}_{i}\in[0,1/2)$, $i=1,2$.

One of the novelties of our proof is the construction of a new class of
intermittent velocity and magnetic flows adapted to the geometry of MHD, which
feature the refined intermittency in both the space and time and, in
particular, enable us to control the stronger viscosity and resistivity
$(-\Delta)^{{\alpha}_{i}}$ with ${\alpha}_{i}\geq 1/2$, $i=1,2$.

(i) Spatial intermittent building blocks. We construct the refined
intermittent velocity and magnetic flows

$W_{(k)}=\psi_{(k_{1})}\phi_{(k)}k_{1},\quad
D_{(k)}=\psi_{(k_{1})}\phi_{(k)}k_{2},$ (1.21)

which are $({{\mathbb{T}}}/({\lambda r_{\perp}}))^{3}$-periodic and
concentrate on smaller cuboids with volume
$\lambda^{-3}r_{\perp}^{-3}r_{\parallel}$ in each of the cubes with side
length ${2\pi}/{(\lambda r_{\perp})}$. Here, $\psi_{(k_{1})}$ and $\phi_{(k)}$
are suitable concentration functions (see (3.8) below), $r_{\parallel}$ and
$r_{\perp}$ are the small constants to parameterize the spatial concentration
of the building blocks, as in the intermittent jets for the NSE [14].

One nice feature of the refined flows is that, they have the almost 2D
intermittency

$\displaystyle\|W_{(k)}\|_{C_{t}L^{1}_{x}}+\|D_{(k)}\|_{C_{t}L^{1}_{x}}\lesssim\lambda^{-1+{\varepsilon}},$
(1.22)

and thus enable one to control the viscosity and resistivity
$(-\Delta)^{{\alpha}_{i}}$ with ${\alpha}_{i}\in[1/2,1)$, $i=1,2$. Another
nice feature is that, the intersections between distinct intermittent flows
concentrate on much smaller cuboids, which provide the almost 2D
intermittency, i.e.,

$\displaystyle\|W_{(k)}\otimes
W_{(k^{\prime})}\|_{C_{t}L^{1}_{x}}\lesssim\lambda^{-1+{\varepsilon}},\quad
k\neq k^{\prime},$ (1.23)

and thus simplify the control of the nonlinearities of MHD.

Let us also mention that, unlike the one oscillation direction $k$ in (1.19)
in the ideal MHD case, the building blocks $W_{(k)}$ and $D_{(k)}$ in (1.21)
oscillate in two orthogonal directions $k_{1}$ and $k$. This new oscillation
direction $k_{1}$ causes extra high spatial oscillations, which need to be
balanced by a new temporal parameter $\mu$ in the flows and the temporal
correctors $w^{(t)}_{q+1}$ and $d^{(t)}_{q+1}$. See (4.37a), (4.37b) and the
important indentities (4.4.1) and (4.4.1) below.

It might be tempting to introduce one more oscillation direction $k_{2}$ in
the building blocks, i.e., with the oscillation directions $(k,k_{1},k_{2})$
as in the NSE context [13, 14]. However, this causes even more high spatial
oscillations, which seem not possible to be balanced by the temporal parameter
$\mu$.

(ii) Temporal intermittent building blocks. The next major difficulty is to
pass beyond the border line ${\alpha}_{i}=1$, $i=1,2$.

The key idea here is to exploit the new intermittency from the temporal
oscillations. Thus, rather than performing the pointwise analysis in time, we
measure the Reynolds and magnetic stresses on average, i.e., in the space
$L^{1}_{t,x}$.

Let us mention that, this is in spirit close to the works [13, 18]. In [18]
the temporal intermittency is crucial to achieve the non-uniqueness of weak
solutions for transport equations at the critical space regularity. In the NSE
context [13], the key ingredient is the $L^{2}_{x}$-based spatial
intermittency, instead of the previous pointwise analysis in space as in the
Euler settings.

We introduce the temporal concentration function $g_{(\tau)}$ in the principal
part of perturbations $(w_{q+1}^{(p)}$, $d_{q+1}^{(p)})$, and control the
oscillation errors in the space $L^{1}_{t}$. Here, $\tau$ is the constant to
parameterize the temporal concentration. In particular, this permits to gain
an additional 1D intermittency, i.e.,

$\displaystyle\|g_{(\tau)}\|_{L^{1}_{t}}\lesssim\lambda^{-\frac{1}{2}+{\varepsilon}}.$
(1.24)

More interestingly, the gained temporal intermittency enables to control the
much stronger viscosity and resistivity $(-\Delta)^{{\alpha}_{i}}$ for all the
exponents ${\alpha}_{i}$ less than $5/4$, $i=1,2$, that is, exactly the Lions
exponent.

It should be mentioned that, extra high temporal oscillations also arise due
to the presence of the temporal concentration function $g_{(\tau)}$. In order
to balance these high oscillations, another new type of temporal correctors
$w_{q+1}^{(o)},d_{q+1}^{(o)}$ will be introduced, respectively, for the
velocity and magnetic flows. See (4.40a), (4.40b) and the key identities
(4.4.2) and (4.4.2) below.

The remainder of this paper is organized as follows. We first regularize the
relexation solutions to (1.7) by the standard mollification procedure in
Section 2. Section 3 is devoted to the construction of the crucial
intermittent velocity and magnetic flows adapted to MHD, which feature the
intermittency in both the space and the time. Then, in Section 4 we construct
the velocity and magnetic perturbations, consisting of the principal parts to
decrease the Reynolds and magnetic stresses, the incompressibility correctors
and two types of temporal correctors. Several key algebraic identities and
analytic estimates also will be given there, which lead to the verification of
the inductive estimates for the velocity and magnetic fields. Then, the
inductive estimates of Reynolds and magnetic stresses are verified in Section
5. Consequently, the proof of the main results is presented in Section 6.

##  2\. Mollification

In order to avoid the loss of derivatives, we mollify the velocity and
magnetic fields. Let $\phi_{\varepsilon}$ and $\varphi_{\varepsilon}$ be the
mollifiers on ${\mathbb{T}}^{3}$ and ${\mathbb{T}}$, respectively,
${\varepsilon}>0$, and
$\mathrm{supp\,}{\varphi}_{\varepsilon}\subseteq(-{\varepsilon},{\varepsilon})$.

The mollifications of
$({u_{q}},{B_{q}},\mathring{R}_{q}^{u},\mathring{R}_{q}^{B})$ in space and
time are defined by

$\begin{array}[]{ll}&u_{\ell}:=\left(u_{q}*_{x}\phi_{\ell}\right)*_{t}\varphi_{\ell},\quad
B_{\ell}:=\left({B_{q}}*_{x}\phi_{\ell}\right)*_{t}\varphi_{\ell},\\\
&\mathring{R}_{\ell}^{u}:=\left(\mathring{R}_{q}^{u}*_{x}\phi_{\ell}\right)*_{t}\varphi_{\ell},\quad\mathring{R}_{\ell}^{B}:=\left(\mathring{R}_{q}^{B}*_{x}\phi_{\ell}\right)*_{t}\varphi_{\ell},\end{array}$
(2.1)

where the scale of mollification is parameterized by

$\displaystyle\ell:=\lambda_{q}^{-20}.$ (2.2)

Then, by equation (1.7),
$(u_{\ell},B_{\ell},\mathring{R}_{\ell}^{u},\mathring{R}_{\ell}^{B})$
satisfies

$\left\\{\begin{aligned}
&\partial_{t}u_{\ell}+\nu_{1}(-\Delta)^{\alpha_{1}}u_{\ell}+{\mathrm{div}}\left(u_{\ell}\otimes
u_{\ell}-B_{\ell}\otimes B_{\ell}\right)+\nabla
P_{\ell}={\mathrm{div}}(\mathring{R}_{\ell}^{u}+\mathring{R}_{com}^{u}),\\\
&\partial_{t}B_{\ell}+\nu_{2}(-\Delta)^{\alpha_{2}}B_{\ell}+{\mathrm{div}}\left(B_{\ell}\otimes
u_{\ell}-u_{\ell}\otimes
B_{\ell}\right)={\mathrm{div}}(\mathring{R}_{\ell}^{B}+\mathring{R}_{com}^{B}),\\\
&{\mathrm{div}}u_{\ell}=0,\quad{\mathrm{div}}B_{\ell}=0,\\\
\end{aligned}\right.$ (2.3)

where the traceless symmetric commutator stress $\mathring{R}_{com}^{u}$ and
the skew-symmetric commutator stress $\mathring{R}_{com}^{B}$ are of form

$\displaystyle\mathring{R}_{com}^{u}:=u_{\ell}\mathring{\otimes}u_{\ell}-B_{\ell}\mathring{\otimes}B_{\ell}-\left({u_{q}}\mathring{\otimes}{u_{q}}-{B_{q}}\mathring{\otimes}{B_{q}}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell},$
(2.4) $\displaystyle\mathring{R}_{com}^{B}:=B_{\ell}\otimes
u_{\ell}-u_{\ell}\otimes
B_{\ell}-\left({B_{q}}\otimes{u_{q}}-{u_{q}}\otimes{B_{q}}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell},$

and the pressure $P_{\ell}$ is given by

$\displaystyle
P_{\ell}:=\left(P_{q}*_{x}\phi_{\ell}\right)*_{t}\varphi_{\ell}-\left|u_{\ell}\right|^{2}+\left|B_{\ell}\right|^{2}+\left(\left|{u_{q}}\right|^{2}-\left|{B_{q}}\right|^{2}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell}.$
(2.5)

By the standard mollification estimates and the inductive estimates
(1.12)-(1.14), for any $N\in\mathbb{N}_{+}$,

$\displaystyle\left\|u_{\ell}\right\|_{C_{t,x}^{N}}+\left\|B_{\ell}\right\|_{C_{t,x}^{N}}\lesssim\ell^{-N+1}(\left\|u_{q}\right\|_{C_{t,x}^{1}}+\left\|B_{q}\right\|_{C_{t,x}^{1}})\lesssim\ell^{-N+1}\lambda_{q}^{7}\lesssim\ell^{-N},$
(2.6)
$\displaystyle\|\mathring{R}_{\ell}^{u}\|_{C_{t,x}^{N}}+\|\mathring{R}_{\ell}^{B}\|_{C_{t,x}^{N}}\lesssim\ell^{-N+1}(\|\mathring{R}_{q}^{u}\|_{C_{t,x}^{1}}+\|\mathring{R}_{q}^{B}\|_{C_{t,x}^{1}})\lesssim\ell^{-N+1}\lambda_{q}^{14}\lesssim\ell^{-N},$
(2.7)
$\displaystyle\|\mathring{R}_{\ell}^{u}\|_{L^{1}_{t,x}}\lesssim\delta_{q+1},\quad\|\mathring{R}_{\ell}^{B}\|_{L^{1}_{t,x}}\lesssim\delta_{q+1}.$
(2.8)

Moreover, by the inductive estimate (1.12) and the double commutator estimate
(see, e.g., [56, Proposition B.1], [24, Lemma 1], [23]), for $1<p<+{\infty}$,

$\|\mathring{R}_{com}^{u}\|_{L^{1}_{t}L^{p}_{x}}\lesssim\|\mathring{R}_{com}^{u}\|_{C_{t,x}}\lesssim\ell(\|u_{q}\|_{C_{t,x}^{1}}^{2}+\|B_{q}\|_{C_{t,x}^{1}}^{2})\lesssim\ell\lambda_{q}^{14},$
(2.9)

and, similarly,

$\|\mathring{R}_{com}^{B}\|_{L^{1}_{t}L^{p}_{x}}\lesssim\|\mathring{R}_{com}^{B}\|_{C_{t,x}}\lesssim\ell\lambda_{q}^{14}.$
(2.10)

##  3\. Intermittent velocity and magnetic flows

This section contains the crucial intermittent velocity and magnetic flows,
which are indeed the fundamental building blocks in the convex integration
scheme.

We set the parameters $r_{\perp}$, $r_{\parallel}$, $\lambda$, $\mu$, $\tau$
and $\sigma$ as follows

$r_{\perp}:=\lambda_{q+1}^{-1+2\varepsilon},\
r_{\parallel}:=\lambda_{q+1}^{-1+6\varepsilon},\ \lambda:=\lambda_{q+1},\
\mu:=\lambda_{q+1}^{\frac{3}{2}-6\varepsilon},\
\tau:=\lambda_{q+1}^{1-6\varepsilon},\ \sigma:=\lambda_{q+1}^{2\varepsilon},$
(3.1)

where $\varepsilon$ is the small constant satisfying (1.8).

###  3.1. Geometric lemmas.

To begin with, let us first recall two geometric lemmas in [2].

######  Lemma 3.1.

(First Geometric Lemma, [2, Lemma 4.1]) There exists a set $\Lambda_{B}\subset
S^{2}\cap\mathbb{Q}^{3}$ that consists of vectors $k$ with associated
orthonormal bases $(k,k_{1},k_{2})$, $\varepsilon_{B}>0$, and smooth positive
functions $\gamma_{(k)}:B_{\varepsilon_{B}}(0)\to\mathbb{R}$, where
$B_{\varepsilon_{B}}(0)$ is the ball of radius $\varepsilon_{B}$ centered at 0
in the space of $3\times 3$ skew-symmetric matrices, such that for $A\in
B_{\varepsilon_{B}}(0)$ we have the following identity:

$A=\sum_{k\in\Lambda_{B}}\gamma_{(k)}^{2}(A)(k_{2}\otimes k_{1}-k_{1}\otimes
k_{2}).$ (3.2)

######  Lemma 3.2.

(Second Geometric Lemma, [2, Lemma 4.2]) There exists a set
$\Lambda_{u}\subset S^{2}\cap\mathbb{Q}^{3}$ that consists of vectors $k$ with
associated orthonormal bases $(k,k_{1},k_{2})$, $\varepsilon_{u}>0$, and
smooth positive functions
$\gamma_{(k)}:B_{\varepsilon_{u}}(\mathrm{Id})\to\mathbb{R}$, where
$B_{\varepsilon_{u}}(\mathrm{Id})$ is the ball of radius $\varepsilon_{u}$
centered at the identity in the space of $3\times 3$ symmetric matrices, such
that for $S\in B_{\varepsilon_{u}}(\mathrm{Id})$ we have the following
identity:

$S=\sum_{k\in\Lambda_{u}}\gamma_{(k)}^{2}(S)k_{1}\otimes k_{1}.$ (3.3)

Furthermore, we may choose $\Lambda_{u}$ such that
$\Lambda_{B}\cap\Lambda_{u}=\emptyset$.

As pointed out in [2], there exists $N_{\Lambda}\in\mathbb{N}$ such that

$\\{N_{\Lambda}k,N_{\Lambda}k_{1},N_{\Lambda}k_{2}\\}\subset
N_{\Lambda}\mathbb{S}^{2}\cap\mathbb{Z}^{3}.$ (3.4)

For instance, $N_{\Lambda}=65$ suffices. We denote by $M_{*}$ the geometric
constant such that

$\displaystyle\sum_{k\in\Lambda_{u}}\lVert\gamma_{(k)}\rVert_{C^{4}(B_{\varepsilon_{u}}(\mathrm{Id}))}+\sum_{k\in\Lambda_{B}}\lVert\gamma_{(k)}\rVert_{{C^{4}(B_{\varepsilon_{B}}}(0))}\leq
M_{*}.$ (3.5)

This parameter is universal and will be used later in the estimates of the
size of perturbations.

According to the proof of [2, Lemmas 4.1 and 4.2], we can choose the wave
vectors satisfying $k_{2}\not=k_{2}^{\prime}$ if $k\not=k^{\prime}$, so that
the intersections between distinct intermittent flows concentrate on much
smaller cuboids and have 2D spatial intermittency (see Lemma 3.4 below).

###  3.2. Spatial building blocks.

Let $\Phi:\mathbb{R}\to\mathbb{R}$ be a smooth cut-off function supported on
the interval $[-1,1]$. We normalize $\Phi$ such that
$\phi:=-\frac{d^{2}}{dx^{2}}\Phi$ satisfies

$\frac{1}{2\pi}\int_{\mathbb{R}}\phi^{2}(x){\rm d}x=1.$ (3.6)

Moreover, let $\psi:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth and mean zero
function, supported on the interval $[-1,1]$, satisfying

$\frac{1}{2\pi}\int_{\mathbb{R}}\psi^{2}\left(x\right){\rm d}x=1.$ (3.7)

The corresponding rescaled cut-off functions are defined by

$\phi_{r_{\perp}}(x):={r_{\perp}^{-\frac{1}{2}}}\phi\left(\frac{x}{r_{\perp}}\right),\quad\Phi_{r_{\perp}}(x):={r_{\perp}^{-\frac{1}{2}}}\Phi\left(\frac{x}{r_{\perp}}\right),\quad\psi_{r_{\parallel}}\left(x\right):={r_{\|}^{-\frac{1}{2}}}\psi\left(\frac{x}{r_{\|}}\right).$

Note that,
$\phi_{r_{\perp}}=-r_{\perp}^{2}\frac{d^{2}}{dx^{2}}\Phi_{r_{\perp}}$, and
$\phi_{r_{\perp}},\psi_{r_{\parallel}}$ are supported in the ball of radius
$r_{\perp}$ and $r_{\parallel}$, respectively, in ${\mathbb{R}}$. By an abuse
of notation, we periodize $\phi_{r_{\perp}}$, $\Phi_{r_{\perp}}$ and
$\psi_{r_{\parallel}}$ so that they are treated as periodic functions defined
on $\mathbb{T}$.

The intermittent velocity flows are defined by

$W_{(k)}:=\psi_{r_{\parallel}}(\lambda r_{\perp}N_{\Lambda}(k_{1}\cdot x+\mu
t))\phi_{r_{\perp}}(\lambda r_{\perp}N_{\Lambda}k\cdot x)k_{1},\ \
k\in\Lambda_{u}\cup\Lambda_{B},$

and the intermittent magnetic flows are defined by

$D_{(k)}:=\psi_{r_{\parallel}}(\lambda r_{\perp}N_{\Lambda}(k_{1}\cdot x+\mu
t))\phi_{r_{\perp}}(\lambda r_{\perp}N_{\Lambda}k\cdot x)k_{2},\ \
k\in\Lambda_{B}.$

Here, $N_{\Lambda}$ is given by (3.4) above, $(k,k_{1},k_{2})$ are the
orthonormal bases in ${\mathbb{R}}^{3}$ in Lemmas 3.1 and 3.2. The parameters
$r_{\parallel}$ and $r_{\perp}$ parameterize the concentration of the flows,
and $\mu$ is the temporal oscillation parameter.

By definition, $\\{W_{(k)},D_{(k)}\\}$ are $(\mathbb{T}/(\lambda
r_{\perp}))^{3}$-periodic, supported on thin cuboids with length
$\sim{1}/{\lambda r_{\perp}}$, width $\sim{r_{\parallel}}/{(\lambda
r_{\perp})}$ and height $\sim{1}/{\lambda}$. See the intermittent flows in
Figure 2. See also the analytic properties in Lemma 3.3 below.

Figure 2. Intermittent flows

Moreover, as there are only finite number of vectors in the wavevector sets
$\Lambda_{u}$ and $\Lambda_{B}$, by choosing $k_{2}\neq k_{2}^{\prime}$ if
$k\neq k^{\prime}$, we note that the intersections of distinct intermittent
flows have much smaller volume. See Figure 3 below for the typical cases of
intersections, and Lemma 3.4 below for the corresponding analytic properties.

(a) Case 1

(b) Case 2

Figure 3. Typical cases of intersections

To ease notations, we set

$\begin{array}[]{ll}&\phi_{(k)}(x):=\phi_{r_{\perp}}(\lambda
r_{\perp}N_{\Lambda}k\cdot x),\ \ \Phi_{(k)}(x):=\Phi_{r_{\perp}}(\lambda
r_{\perp}N_{\Lambda}k\cdot x),\\\
&\psi_{(k_{1})}(x):=\psi_{r_{\parallel}}(\lambda
r_{\perp}N_{\Lambda}(k_{1}\cdot x+\mu t)),\end{array}$ (3.8)

and thus

$W_{(k)}=\psi_{(k_{1})}\phi_{(k)}k_{1},\ k\in\Lambda_{u}\cup\Lambda_{B},\ \ \
D_{(k)}=\psi_{(k_{1})}\phi_{(k)}k_{2},\ k\in\Lambda_{B}.$ (3.9)

Note that, $W_{(k)}$ and $D_{(k)}$ are mean zero on ${\mathbb{T}}^{3}$.
Moreover, since

${\fint}_{\mathbb{T}^{3}}\phi_{(k)}^{2}(x){\rm
d}x=1,\quad\text{and}\quad{\fint}_{\mathbb{T}^{3}}\psi_{(k_{1})}^{2}(x){\rm
d}x=1,$ (3.10)

it follows that

$\begin{array}[]{ll}&\displaystyle{\fint}_{\mathbb{T}^{3}}W_{(k)}\otimes
W_{(k)}{\rm d}x=k_{1}\otimes k_{1},\ \ {\fint}_{\mathbb{T}^{3}}D_{(k)}\otimes
D_{(k)}{\rm d}x=k_{2}\otimes k_{2},\end{array}$ (3.11)
$\begin{array}[]{ll}&\displaystyle{\fint}_{\mathbb{T}^{3}}W_{(k)}\otimes
D_{(k)}{\rm d}x=k_{1}\otimes k_{2},\ \ {\fint}_{\mathbb{T}^{3}}D_{(k)}\otimes
W_{(k)}{\rm d}x=k_{2}\otimes k_{1}.\end{array}$ (3.12)

It is worth noting that, the building blocks in [2] have one oscillation
direction $k$, orthogonal to the direction vectors $k_{1}$ and $k_{2}$, and
have the 1D intermittency.

In order to control the stronger viscosity and resistivity
$(-\Delta)^{{\alpha}_{i}}$ when ${\alpha}_{i}\geq 1/2$, $i=1,2$, the idea here
to explore more intermittency, inspired by the work [13] in the NSE context,
is to introduce the new concentration function $\psi_{(k_{1})}$ in (3.9). Two
new parameters $r_{\perp}$ (the rescaling parameter in the direction vertical
to the flow) and $r_{\parallel}$ (the rescaling parameter in the direction
parallel to the flow) parameterize the concentration of the intermittent flows
$W_{(k)}$ and $D_{(k)}$. In particular, one has

$\displaystyle\sum\limits_{k\in\Lambda_{u}\cup\Lambda_{B}}\|W_{(k)}\|_{C_{t}L^{1}_{x}}+\sum\limits_{k\in\Lambda_{B}}\|D_{(k)}\|_{C_{t}L^{1}_{x}}\lesssim
r_{\perp}^{\frac{1}{2}}r_{\parallel}^{\frac{1}{2}}=\lambda_{q+1}^{-1+4{\varepsilon}},$
(3.13)

which gives the almost $2D$ intermittency and thus permits to control the
fractional viscosity and resistivity $(-\Delta)^{{\alpha}_{i}}$ in (1.4) for
any ${\alpha}_{1},{\alpha}_{2}\in[0,1)$.

Furthermore, the new parameter $\mu$ permits to control the temporal
correctors later to balance the high spatial oscillations, arising from the
concentration function $\psi_{(k)}.$ More precisely, using the orthogonality

$\begin{array}[]{ll}&\nabla\phi_{(k)}\cdot k_{1}=\nabla\phi_{(k)}\cdot
k_{2}=0,\\\ &\nabla\psi_{(k_{1})}\cdot k=\nabla\psi_{(k_{1})}\cdot
k_{2}=0,\end{array}$ (3.14)

we have the important algebraic identities

$\begin{array}[]{ll}&\displaystyle{\mathrm{div}}(W_{(k)}\otimes
W_{(k)})=2(W_{(k)}\cdot\nabla\psi_{(k_{1})})\phi_{(k)}k_{1}=\frac{1}{\mu}\partial_{t}\left(\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}\right),\\\
&\displaystyle{\mathrm{div}}(D_{(k)}\otimes
D_{(k)})=2(D_{(k)}\cdot\nabla\psi_{(k_{1})})\phi_{(k)}k_{2}=0,\end{array}$
(3.15)

and

$\begin{array}[]{ll}&\displaystyle{\mathrm{div}}(D_{(k)}\otimes
W_{(k)})=({\mathrm{div}}W_{(k)})D_{(k)}+(W_{(k)}\cdot\nabla)D_{(k)}=\frac{1}{\mu}\partial_{t}\left(\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}\right),\\\
&\displaystyle{\mathrm{div}}(W_{(k)}\otimes
D_{(k)})=({\mathrm{div}}D_{(k)})W_{(k)}+(D_{(k)}\cdot\nabla)W_{(k)}=0.\end{array}$
(3.16)

Thus, when combining with the temporal correctors, module a pressure term, the
time derivative can be moved onto the low oscillating amplitudes (see (4.4.1),
(4.4.1) below).

Because the intermittent flow $W_{(k)}$ is not divergence-free, we also need
the corrector

$\displaystyle\widetilde{W}_{(k)}^{c}:=\frac{1}{\lambda^{2}N_{\Lambda}^{2}}\nabla\psi_{(k_{1})}\times{\mathrm{curl}}(\Phi_{(k)}k_{1}).$
(3.17)

Then, straightforward computations show that

$W_{(k)}+\widetilde{W}_{(k)}^{c}={\mathrm{curl}}{\mathrm{curl}}\left(\frac{1}{\lambda^{2}N_{\Lambda}^{2}}\psi_{(k_{1})}\Phi_{(k)}k_{1}\right)={\mathrm{curl}}{\mathrm{curl}}W^{c}_{(k)},$
(3.18)

where $W^{c}_{(k)}$ is given by

$\displaystyle
W^{c}_{(k)}:=\frac{1}{\lambda^{2}N_{\Lambda}^{2}}\psi_{(k_{1})}\Phi_{(k)}k_{1}.$
(3.19)

Thus, it follows that

$\displaystyle{\mathrm{div}}(W_{(k)}+\widetilde{W}^{c}_{(k)})=0.$ (3.20)

Regarding the magnetic flows, we introduce the correctors

$D_{(k)}^{c}:=\frac{1}{\lambda^{2}N_{\Lambda}^{2}}\psi_{(k_{1})}\Phi_{(k)}k_{2},\quad\widetilde{D}_{(k)}^{c}:=-\frac{1}{\lambda^{2}N_{\Lambda}^{2}}\Delta\psi_{(k_{1})}\Phi_{(k)}k_{2},\quad
k\in\Lambda_{B}.$ (3.21)

It holds that

$\displaystyle
D_{(k)}+\widetilde{D}_{(k)}^{c}={\mathrm{curl}}{\mathrm{curl}}D^{c}_{(k)}.$
(3.22)

Lemma 3.3 below contains the key estimates of the intermittent velocity and
magnetic flows.

######  Lemma 3.3 (Estimates of spatial intermittency).

For $p\in[1,+\infty]$, $N,\,M\in\mathbb{N}$, we have

$\displaystyle\left\|\nabla^{N}\partial_{t}^{M}\psi_{(k_{1})}\right\|_{C_{t}L^{p}_{x}}\lesssim
r_{\|}^{\frac{1}{p}-\frac{1}{2}}\left(\frac{r_{\perp}\lambda}{r_{\|}}\right)^{N}\left(\frac{r_{\perp}\lambda\mu}{r_{\|}}\right)^{M},$
(3.23)
$\displaystyle\left\|\nabla^{N}\phi_{(k)}\right\|_{L^{p}_{x}}+\left\|\nabla^{N}\Phi_{(k)}\right\|_{L^{p}_{x}}\lesssim
r_{\perp}^{\frac{1}{p}-\frac{1}{2}}\lambda^{N},$ (3.24)

where the implicit constants are independent of
$r_{\perp},\,r_{\parallel},\,\lambda$ and $\mu$. In particular, we have

$\displaystyle\displaystyle\left\|\nabla^{N}\partial_{t}^{M}W_{(k)}\right\|_{C_{t}L^{p}_{x}}+\frac{r_{\|}}{r_{\perp}}\left\|\nabla^{N}\partial_{t}^{M}\widetilde{W}_{(k)}^{c}\right\|_{C_{t}L^{p}_{x}}+\lambda^{2}\left\|\nabla^{N}\partial_{t}^{M}W_{(k)}^{c}\right\|_{C_{t}L^{p}_{x}}\displaystyle$
$\displaystyle\qquad\lesssim
r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\|}^{\frac{1}{p}-\frac{1}{2}}\lambda^{N}\left(\frac{r_{\perp}\lambda\mu}{r_{\|}}\right)^{M},\
\ k\in\Lambda_{u}\cup\Lambda_{B},$ (3.25)
$\displaystyle\displaystyle\left\|\nabla^{N}\partial_{t}^{M}D_{(k)}\right\|_{C_{t}L^{p}_{x}}+\frac{r_{\|}}{r_{\perp}}\left\|\nabla^{N}\partial_{t}^{M}\widetilde{D}_{(k)}^{c}\right\|_{C_{t}L^{p}_{x}}+\lambda^{2}\left\|\nabla^{N}\partial_{t}^{M}D_{(k)}^{c}\right\|_{C_{t}L^{p}_{x}}$
$\displaystyle\qquad\lesssim
r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\|}^{\frac{1}{p}-\frac{1}{2}}\lambda^{N}\left(\frac{r_{\perp}\lambda\mu}{r_{\|}}\right)^{M},\
\ k\in\Lambda_{B}.$ (3.26)

###### Proof.

By the definitions of $\psi_{(k_{1})}$, $\phi_{(k)}$ and $\Phi_{(k)}$, the
$L^{\infty}$-norms can be bounded by

$\displaystyle\left\|\nabla^{N}\partial_{t}^{M}\psi_{(k_{1})}\right\|_{C_{t,x}}\lesssim
r_{\|}^{-\frac{1}{2}}\left(\frac{r_{\perp}\lambda}{r_{\|}}\right)^{N}\left(\frac{r_{\perp}\lambda\mu}{r_{\|}}\right)^{M},$
(3.27)
$\displaystyle\left\|\nabla^{N}\phi_{(k)}\right\|_{L^{\infty}_{x}}+\left\|\nabla^{N}\Phi_{(k)}\right\|_{L^{\infty}_{x}}\lesssim\lambda^{N}r_{\perp}^{-\frac{1}{2}}.$
(3.28)

In order to control the $L^{p}$-norm, $1\leq p<+{\infty}$, let us estimate the
volume of the support of $\psi_{(k_{1})}$ below. As $\psi_{(k_{1})}$ is
$(\mathbb{T}/(\lambda r_{\perp})^{3})$-periodic, it suffices to estimate the
volume of the support of $\psi_{(k_{1})}$ on the cubes with side length
$2\pi/(\lambda r_{\perp})$ and then multiply the resulting estimate by
$(\lambda r_{\perp})^{3}$. One may let the oscillation parameter $\mu=0$
without changing the volume of the support of $\psi_{(k_{1})}$.

In each one of these cubes, $\psi_{(k_{1})}$ is supported on at most
$2N_{\Lambda}$ thickened planes with thickness $\sim{r_{\parallel}}/{(\lambda
r_{\perp})}$, which are parallel to each other and perpendicular to the
direction $k_{1}$. Hence, the volume of the support of $\psi_{(k_{1})}$ on
$\mathbb{T}^{3}$ can be bounded by

$|\mathrm{supp\,}\psi_{(k_{1})}|\lesssim(\lambda
r_{\perp})^{-2}\frac{r_{\parallel}}{\lambda r_{\perp}}(\lambda
r_{\perp})^{3}\lesssim r_{\parallel}.$ (3.29)

This along with (3.27) yields that

$\displaystyle\|\nabla^{N}\partial_{t}^{M}\psi_{(k_{1})}\|_{C_{t}L^{p}_{x}}\lesssim$
$\displaystyle|\mathrm{supp\,}\psi_{(k_{1})}|^{\frac{1}{p}}\,\left\|\nabla^{N}\partial_{t}^{M}\psi_{(k_{1})}\right\|_{C_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim$ $\displaystyle
r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\left(\frac{r_{\perp}\lambda}{r_{\parallel}}\right)^{N}\left(\frac{r_{\perp}\lambda\mu}{r_{\parallel}}\right)^{M}.$
(3.30)

Since there are only finite wavevectors in $\Lambda_{u}$ and $\Lambda_{B}$,
the implicit constant can be taken independent of $k_{1}$ by taking the
maximum over the wavevector sets $\Lambda_{u}$ and $\Lambda_{B}$. Thus, (3.23)
is verified.

Estimate (3.24) can be proved similarly. Actually, in each of the cubic with
length $2\pi/(\lambda r_{\perp})$, $\phi_{(k)}$ is supported on at most
$2N_{\Lambda}$ thickened planes with thickness $\sim\lambda^{-1}$,
perpendicular to the direction $k$. Hence,

$\displaystyle|\mathrm{supp\,}\phi_{(k)}|\lesssim(\lambda
r_{\perp})^{-2}\lambda^{-1}(\lambda r_{\perp})^{3}\lesssim r_{\perp},$

which along with (3.28) yields that

$\displaystyle\|{\nabla}^{N}\phi_{(k)}\|_{L^{p}_{x}}\lesssim|\mathrm{supp\,}\phi_{(k)}|^{\frac{1}{p}}\|{\nabla}^{N}\phi_{(k)}\|_{L^{\infty}_{x}}\lesssim
r_{\perp}^{\frac{1}{p}}\lambda^{N}r_{\perp}^{-\frac{1}{2}}\lesssim
r_{\perp}^{\frac{1}{p}-\frac{1}{2}}\lambda^{N}.$

Thus, (3.24) is verified.

Finally, estimates (3.25) and (3.26) follow from (3.23), (3.24) and Fubini’s
theorem. ∎

The following lemma illustrates that, more intermittency can be gained for the
interactions betweens different intermittent building blocks. Intuitively, the
supports of intermittent building blocks can be regarded as thin cuboids,
because $k_{2}\not=k_{2}^{\prime}$ for different wavevectors, the intersection
of two nonparallel thin cuboids has much smaller support.

######  Lemma 3.4 (Product estimate).

For $k\neq k^{\prime}\in\Lambda_{u}\cup\Lambda_{B}$ and $p\in[1,\infty]$, we
have

$\|\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}\|_{C_{t}L^{p}_{x}}\lesssim
r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{2}{p}-1},$ (3.31)

where the implicit constants are independent of the parameters
$r_{\perp},\,r_{\parallel}$ and $\lambda$.

###### Proof.

As in the proof of Lemma 3.3, let us start with the $L^{\infty}$-estimate. By
(3.8),

$\|\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}\|_{C_{t,x}}\lesssim
r_{\perp}^{-1}r_{\parallel}^{-1}.$ (3.32)

Next, we estimate the volume of the support of
$\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}$. As
$\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}$ is
$(\mathbb{T}/\lambda r_{\perp})^{3}$-periodic, it suffices to consider the
support on the small cubes with side length $2\pi/\lambda r_{\perp}$ and then
multiply the result by $(\lambda r_{\perp})^{3}$.

In each of these cubes, the support of $\psi_{(k_{1})}\phi_{(k)}$ consists of
at most $2N_{\Lambda}$ parallel thin cuboids with length $\sim(\lambda
r_{\perp})^{-1}$, width $\displaystyle\sim{r_{\parallel}}{(\lambda
r_{\perp})^{-1}}$ and height $\sim\lambda^{-1}$. Note that, the width and
height is much smaller than the length. Since $k_{2}\not=k_{2}^{\prime}$, the
intersections between different supports of $\psi_{(k_{1})}\phi_{(k)}$ and
$\psi_{(k_{1}^{\prime})}\phi_{(k^{\prime})}$ are contained in much smaller
cuboids, with the length and width bounded by
$\displaystyle\sim{r_{\parallel}}{(\lambda r_{\perp})^{-1}}$ and with the
height $\sim\lambda^{-1}$. As there are at most $2N_{\Lambda}$ cuboids in the
support of $\psi_{(k_{1})}\phi_{(k)}$, the number of intersections for two
distinct cuboids is at most $4N_{\Lambda}^{2}$. Thus, the volume of the
support of
$\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}$ is
bounded by

$|\mathrm{supp\,}(\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})})|\lesssim
r_{\parallel}^{2}(\lambda r_{\perp})^{-2}\lambda^{-1}(\lambda
r_{\perp})^{3}\lesssim r_{\parallel}^{2}r_{\perp}.$ (3.33)

The implicit constant can be taken independent of $k_{1}$ and
$k_{1}^{\prime}$, as there are only finitely many vectors in the wavevector
sets $\Lambda_{u}$ and $\Lambda_{B}$.

Therefore, by virtue of (3.32) and (3.33), we arrive at

$\displaystyle\|\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}\|_{C_{t}L^{p}_{x}}\leq$
$\displaystyle\sup_{t}|\mathrm{supp\,}\,(\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})})|^{\frac{1}{p}}\,\|\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}\|_{C_{t,x}}$
$\displaystyle\lesssim$ $\displaystyle
r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{2}{p}-1},$ (3.34)

which yields (3.31) and finishes the proof. ∎

###  3.3. Temporal building blocks.

As mentioned in Section 1, the previous construction of intermittent velocity
and magnetic flows cannot provide us with enough intermittency to handle the
stronger viscosity and resistivity $(-\Delta)^{{\alpha}_{i}}$ when
${\alpha}_{i}\in[1,5/4)$, $i=1,2$.

The key idea here is to exploit more intermittency from temporal oscillations.
Two new parameters $\tau$ and $\sigma$ will be introduced to parameterize the
concentrations of the temporal function.

More precisely, let $g\in C_{c}^{\infty}({\mathbb{T}})$ be any cut-off
function such that

$\displaystyle{\fint}_{{\mathbb{T}}}g^{2}(t){\rm d}t=1.$

Let $\tau\in\mathbb{N}_{+}$ and rescale the cut-off function $g$ by

$\displaystyle g_{\tau}(t)=\tau^{\frac{1}{2}}g(\tau t).$ (3.35)

Then, we periodize $g_{\tau}$ such that the resulting functions (by an abuse
of notion, still denoted by $g_{\tau}$) are ${\mathbb{T}}-$periodic functions.

In order to construct the temporal correctors $w_{q+1}^{(o)}$ and
$d_{q+1}^{(o)}$ (see (4.40a)-(4.40b)) to balance the high temporal
oscillations arising from the concentration function $g_{(\tau)}$, we also
need the function $h_{\tau}:\mathbb{T}\to{\mathbb{R}}$, defined by

$\displaystyle h_{\tau}(t):=\int_{-\pi}^{t}\left(g_{\tau}^{2}(s)-1\right)\
ds.$ (3.36)

Set

$\displaystyle g_{(\tau)}:=g_{\tau}(\sigma t),\ \
h_{(\tau)}(t):=h_{\tau}(\sigma t).$ (3.37)

Note that, $h_{(\tau)}$ has the uniform $L_{t}^{\infty}$-bound

$\displaystyle\|h_{(\tau)}\|_{L_{t}^{\infty}({\mathbb{T}})}\leq 1.$ (3.38)

Moreover, we have the following temporal intermittent estimates of
$g_{(\tau)}$.

######  Lemma 3.5 (Estimates of temporal intermittency).

For $\gamma\in[1,+\infty]$, $M\in\mathbb{N}$, we have

$\displaystyle\left\|\partial_{t}^{M}g_{(\tau)}\right\|_{L^{\gamma}_{t}}\lesssim\sigma^{M}\tau^{M+\frac{1}{2}-\frac{1}{\gamma}},$
(3.39)

where the implicit constants are independent of $\tau$ and $\sigma$.

###### Proof.

As in the proof of Lemma 3.3, we first estimate the $L^{\infty}$-norm of
$g_{(\tau)}$. By definition,

$\displaystyle\left\|\partial_{t}^{M}g_{(\tau)}\right\|_{L^{{\infty}}_{t}({\mathbb{T}})}\lesssim\sigma^{M}\tau^{M+\frac{1}{2}}.$
(3.40)

Next, we give a bound on the volume of the support of $g_{(\tau)}$ on
${\mathbb{T}}$. Since $g_{(\tau)}$ is ${\mathbb{T}}/\sigma$-periodic, it is
sufficient to obtain the estimate on the interval $[-\pi/\sigma,\pi/\sigma]$
and then multiply the result by $\sigma$. Note that, in the interval
$[-\pi/\sigma,\pi/\sigma]$, $g_{(\tau)}$ is supported on an interval with
length $\sim(\sigma\tau)^{-1}$, and thus the volume of the support of
$g_{(\tau)}$ on ${\mathbb{T}}$ is bounded by

$\displaystyle|\mathrm{supp\,}g_{(\tau)}|\lesssim(\sigma\tau)^{-1}\sigma\lesssim\tau^{-1}.$
(3.41)

Combining (3.40) and (3.41) together we obtain

$\displaystyle\|\partial_{t}^{M}g_{(\tau)}\|_{L^{\gamma}_{t}}\lesssim|\mathrm{supp\,}g_{(\tau)}|^{\frac{1}{\gamma}}\left\|\partial_{t}^{M}g_{(\tau)}\right\|_{L^{\infty}_{t}({\mathbb{T}})}\lesssim\sigma^{M}\tau^{M+\frac{1}{2}-\frac{1}{\gamma}}.$
(3.42)

Therefore, estimate (3.39) is proved. ∎

##  4\. Velocity and magnetic perturbations

This section is devoted to the construction of the key velocity and magnetic
perturbations, including the principal parts, the incompressibility
correctors, the spatial and temporal correctors. To begin with, let us first
define suitable amplitudes of perturbations, which are the key to decrease the
effects of old velocity and magnetic stresses.

###  4.1. Amplitudes

####  4.1.1. The magnetic amplitudes.

Let $\chi:[0,+\infty)\to\mathbb{R}$ be a smooth cut-off function such that

$\chi(z)=\left\\{\begin{aligned} &1,\quad 0\leq z\leq 1,\\\ &z,\quad z\geq
2,\end{aligned}\right.$ (4.1)

and

$\frac{1}{2}z\leq\chi(z)\leq 2z\quad\text{for}\quad z\in(1,2).$ (4.2)

Set

$\rho_{B}(t,x):=2\varepsilon_{B}^{-1}\delta_{q+1}\chi\left(\frac{|\mathring{R}_{\ell}^{B}(t,x)|}{\delta_{q+1}}\right),$
(4.3)

where $\varepsilon_{B}$ is the small radius in the geometric Lemma 3.1. Then,
by (4.1), (4.2) and (4.3),

$\left|\frac{\mathring{R}_{\ell}^{B}}{\rho_{B}}\right|=\left|\frac{\mathring{R}_{\ell}^{B}}{2\varepsilon_{B}^{-1}\delta_{q+1}\chi(\delta_{q+1}^{-1}|\mathring{R}_{\ell}^{B}|)}\right|\leq\varepsilon_{B},$
(4.4)

and for any $p\in[1,+\infty]$,

$\displaystyle\rho_{B}\geq\varepsilon_{B}^{-1}\delta_{q+1},$ (4.5)
$\displaystyle\lVert\rho_{B}\rVert_{L^{p}_{t,x}}\leq
8\varepsilon_{B}^{-1}\left((16\pi^{4})^{\frac{1}{p}}\delta_{q+1}+\lVert\mathring{R}_{\ell}^{B}\rVert_{L^{p}_{t,x}}\right).$
(4.6)

Furthermore, by the standard Hölder estimate (see [10, (130)]), the iterative
estimate (1.13) and (4.5), for $1\leq N\leq 4$,

$\displaystyle\lVert\rho_{B}\rVert_{C_{t,x}}\lesssim\ell^{-1},\quad\lVert\rho_{B}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-N}\delta^{-N+1}_{q+1},$
(4.7)
$\displaystyle\lVert\rho_{B}^{1/2}\rVert_{C_{t,x}}\lesssim\ell^{-1},\quad\lVert\rho_{B}^{1/2}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-N}\delta^{-2N}_{q+1},$
(4.8)
$\displaystyle\lVert\rho_{B}^{-1}\rVert_{C_{t,x}}\lesssim\delta_{q+1}^{-1},\quad\lVert\rho_{B}^{-1}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-N}\delta^{-2N}_{q+1}.$
(4.9)

Moreover, we choose the smooth temporal cut-off function $f_{B}$ such that

  * •

$0\leq f_{B}\leq 1$, $f_{B}\equiv 1$ on
$\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{B}$;

  * •

$\mathrm{supp\,}_{t}f_{B}\subseteq
N_{l}(\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{B})$;

  * •

$\|f_{B}\|_{C_{t}^{N}}\lesssim\ell^{-N}$, $1\leq N\leq 4$.

The amplitudes of the magnetic perturbations are defined by

$a_{(k)}(t,x):=a_{k,B}(t,x)=\rho_{B}^{\frac{1}{2}}(t,x)f_{B}(t)\gamma_{(k)}\left(\frac{-\mathring{R}_{\ell}^{B}(t,x)}{\rho_{B}(t,x)}\right),\quad
k\in\Lambda_{B},$ (4.10)

where $\gamma_{(k)}$ is the smooth function in the geometric Lemma 3.1.

Note that, by virtue of the geometric Lemma 3.1, the identity (3.12) and the
expression (4.10) of $a_{(k)}$, the following algebraic identity holds:

$\displaystyle\sum\limits_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)})$ $\displaystyle=$
$\displaystyle-\mathring{R}_{\ell}^{B}+\sum\limits_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}\mathbb{P}_{\neq
0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)})$
$\displaystyle+\sum_{k\in\Lambda_{B}}a_{(k)}^{2}(g_{(\tau)}^{2}-1){\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x,$ (4.11)

where $\mathbb{P}_{\neq 0}$ denotes the spatial projection onto nonzero
Fourier modes.

Moreover, in view of (4.5)-(4.9), we also have the following analytic
estimates.

######  Lemma 4.1 (Estimates of magnetic amplitudes).

For $1\leq N\leq 4$, $k\in\Lambda_{B}$, we have

$\displaystyle\lVert
a_{(k)}\rVert_{L^{2}_{t,x}}\lesssim\delta_{q+1}^{\frac{1}{2}},$ (4.12)
$\displaystyle\lVert a_{(k)}\rVert_{C_{t,x}}\lesssim\ell^{-1},\ \ \lVert
a_{(k)}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-4N}.$ (4.13)

###### Proof.

To obtain the $L^{2}_{t,x}$-estimate of $a_{(k)}$, $k\in\Lambda_{B}$, we use
(1.14) and (4.6) to bound

$\displaystyle\lVert a_{(k)}\rVert_{L^{2}_{t,x}}$
$\displaystyle\leq\lVert\rho_{B}\rVert_{L^{1}_{t,x}}^{\frac{1}{2}}\lVert\gamma_{(k)}\rVert_{C_{x}(B_{\varepsilon_{B}}(0))}\|f_{B}\|_{C_{t}}$
$\displaystyle\leq
M_{*}(8\varepsilon_{B}^{-1})^{\frac{1}{2}}(16\pi^{4}\delta_{q+1}+\lVert\mathring{R}_{\ell}^{B}\rVert_{L^{1}_{t,x}})^{\frac{1}{2}}$
$\displaystyle\leq
M_{*}(8\varepsilon_{B}^{-1})^{\frac{1}{2}}(16\pi^{4}+1)^{\frac{1}{2}}\delta_{q+1}^{\frac{1}{2}},$
(4.14)

which yields (4.12).

Moreover, for $N\geq 1$, by (4.8) and the standard Hölder estimate (cf. [10,
(129)]),

$\displaystyle\lVert a_{(k)}\rVert_{C^{N}_{t,x}}$
$\displaystyle\lesssim\sum_{0\leq N_{1}+N_{2}+N_{3}\leq
N}\lVert\rho_{B}^{\frac{1}{2}}\rVert_{C^{N_{1}}_{t,x}}\lVert\gamma_{(k)}(-\rho_{B}^{-1}\mathring{R}^{B}_{\ell})\rVert_{C^{N_{2}}_{t,x}}\lVert
f_{B}\rVert_{C_{t}^{N_{3}}}$ $\displaystyle\lesssim\sum_{0\leq
N_{1}+N_{2}+N_{3}\leq
N}\ell^{-N_{1}-1}\ell^{-N_{3}}\left(1+\lVert\rho_{B}^{-1}\mathring{R}^{B}_{\ell}\rVert_{C^{N_{2}}_{t,x}}+\lVert\rho_{B}^{-1}\mathring{R}^{B}_{\ell}\rVert_{C_{t,x}}^{N_{2}-1}\lVert\rho_{B}^{-1}\mathring{R}^{B}_{\ell}\rVert_{C^{N_{2}}_{t,x}}\right).$
(4.15)

Since by (1.13), (2.7) and (4.9),

$\displaystyle\lVert\rho_{B}^{-1}\mathring{R}^{B}_{\ell}\rVert_{C^{N_{2}}_{t,x}}$
$\displaystyle\lesssim\sum_{0\leq N_{21}+N_{22}\leq
N_{2}}\lVert\rho_{B}^{-1}\rVert_{C^{N_{21}}_{t,x}}\lVert\mathring{R}^{B}_{\ell}\rVert_{C^{N_{22}}_{t,x}}$
$\displaystyle\lesssim\sum_{0\leq N_{21}+N_{22}\leq
N_{2}}\ell^{-N_{21}-1}\ell^{-N_{22}}\lesssim\ell^{-N_{2}-1},$ (4.16)

we get

$\displaystyle\lVert a_{(k)}\rVert_{C^{N}_{t,x}}$
$\displaystyle\lesssim\sum_{0\leq N_{1}+N_{2}+N_{3}\leq
N}\ell^{-N_{1}-1}\ell^{-N_{3}}(\ell^{-N_{2}-1}+\ell^{-2N_{2}})\lesssim\ell^{-4N}.$
(4.17)

The $C_{t,x}$-estimate of $a_{(k)}$ can be bounded by, via (4.8),

$\displaystyle\lVert a_{(k)}\rVert_{C_{t,x}}$
$\displaystyle\lesssim\lVert\rho_{B}^{\frac{1}{2}}\rVert_{C_{t,x}}\lVert
f_{B}\rVert_{C_{t}}\lVert\gamma_{(k)}\rVert_{C_{x}(B_{\varepsilon_{B}}(0))}\lesssim\ell^{-1}.$

Therefore, the estimates in (4.13) are verified. The proof is complete. ∎

####  4.1.2. The velocity amplitudes.

Below we define the velocity amplitudes. Unlike in the previous magnetic case,
because of the strong coupling between the velocity and magnetic fields, a new
matrix $\mathring{G}^{B}$ is needed in order to maintain the cancellation
between the perturbations and the old stresses,

$\mathring{G}^{B}:=\sum_{k\in\Lambda_{B}}a_{(k)}^{2}{\fint}_{\mathbb{T}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x.$ (4.18)

In view of estimates (4.12) and (4.13), we have that for $N\geq 1$,

$\displaystyle\lVert\mathring{G}^{B}\rVert_{C_{t,x}}\lesssim\ell^{-2},\quad\lVert\mathring{G}^{B}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-4N-1},\quad\lVert\mathring{G}^{B}\rVert_{L^{1}_{t,x}}$
$\displaystyle\lesssim\delta_{q+1}.$ (4.19)

Set

$\displaystyle\rho_{u}(t,x):=2\varepsilon_{u}^{-1}\delta_{q+1}\chi\left(\frac{|\mathring{R}_{\ell}^{u}(t,x)+\mathring{G}^{B}(t,x)|}{\delta_{q+1}}\right).$
(4.20)

Then, by (4.1) and (4.2),

$\displaystyle\left|\frac{\mathring{R}_{\ell}^{u}+\mathring{G}^{B}}{\rho_{u}}\right|\leq\varepsilon_{u}.$
(4.21)

We choose the smooth temporal cut-off function $f_{u}$ such that

  * •

$0\leq f_{u}\leq 1$, $f_{u}\equiv 1$ on
$\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{u}\cup\mathrm{supp\,}_{t}\mathring{G}^{B}$;

  * •

$\mathrm{supp\,}_{t}f_{u}\subseteq
N_{l}(\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{u}\cup\mathrm{supp\,}_{t}\mathring{G}^{B})\subseteq
N_{2l}(\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{u}\cup\mathrm{supp\,}_{t}\mathring{R}_{\ell}^{B})$;

  * •

$\|f_{u}\|_{C_{t}^{N}}\lesssim\ell^{-N}$, $1\leq N\leq 4$.

The velocity amplitudes are defined by

$\displaystyle
a_{(k)}(t,x):=a_{k,u}(t,x)=\rho_{u}^{\frac{1}{2}}f_{u}(t)\gamma_{(k)}\left(\mathrm{Id}-\frac{\mathring{R}_{\ell}^{u}(t,x)+\mathring{G}^{B}(t,x)}{\rho_{u}(t,x)}\right),\quad
k\in\Lambda_{u}.$ (4.22)

In view of the normalization (3.11), the geometric Lemma 3.2 and the
expression (4.22) of $a_{(k)}$, we get the following algebraic identity:

$\displaystyle\sum\limits_{k\in\Lambda_{u}}a_{(k)}^{2}g_{(\tau)}^{2}W_{(k)}\otimes
W_{(k)}$
$\displaystyle=\rho_{u}f^{2}_{u}\mathrm{Id}-\mathring{R}_{\ell}^{u}-\mathring{G}^{B}+\sum\limits_{k\in\Lambda_{u}}a_{(k)}^{2}g_{(\tau)}^{2}\mathbb{P}_{\neq
0}(W_{(k)}\otimes W_{(k)})$
$\displaystyle\quad+\sum_{k\in\Lambda_{u}}a_{(k)}^{2}\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}{\rm d}x.$ (4.23)

Moreover, similarly to (4.5)-(4.9), we have for any $p\in[1,+\infty)$,

$\displaystyle\rho_{u}\geq\varepsilon_{u}^{-1}\delta_{q+1},$ (4.24)
$\displaystyle\lVert\rho_{u}\rVert_{L^{p}_{t,x}}\leq
8\varepsilon_{u}^{-1}\left((16\pi^{4})^{\frac{1}{p}}\delta_{q+1}+\lVert\mathring{R}_{\ell}^{u}+\mathring{G}^{B}\rVert_{L^{p}_{t,x}}\right),$
(4.25)

and for $1\leq N\leq 4$,

$\displaystyle\lVert\rho_{u}\rVert_{C_{t,x}}\lesssim\ell^{-2},\quad\lVert\rho_{u}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-5N}\delta_{q}^{-N+1},$
(4.26)
$\displaystyle\lVert\rho_{u}^{1/2}\rVert_{C_{t,x}}\lesssim\ell^{-1},\quad\lVert\rho_{u}^{1/2}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-5N}\delta_{q+1}^{-2N},$
(4.27)
$\displaystyle\lVert\rho_{u}^{-1}\rVert_{C_{t,x}}\lesssim\delta_{q+1}^{-1},\quad\lVert\rho_{u}^{-1}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-5N}\delta_{q+1}^{-2N}.$
(4.28)

Arguing as in the proof of Lemma 4.1, using the standard Hölder estimate (cf.
[10, (129)]) and estimates (1.14), (4.19), (4.24)-(4.28) we have the following
analytic estimates for the velocity amplitude functions.

######  Lemma 4.2 (Estimates of velocity amplitudes).

For $1\leq N\leq 4$, $k\in\Lambda_{u}$, we have

$\displaystyle\lVert
a_{(k)}\rVert_{L^{2}_{t,x}}\lesssim\delta_{q+1}^{\frac{1}{2}},$ (4.29)
$\displaystyle\lVert a_{(k)}\rVert_{C_{t,x}}\lesssim\ell^{-1},\quad\lVert
a_{(k)}\rVert_{C_{t,x}^{N}}\lesssim\ell^{-8N}.$ (4.30)

###  4.2. Principal parts of perturbations

We define the principal parts $w_{q+1}^{(p)}$ and $d_{q+1}^{(p)}$,
respectively, of the velocity and magnetic perturbations by

$\displaystyle w_{q+1}^{(p)}$
$\displaystyle:=\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}a_{(k)}g_{(\tau)}W_{(k)},$
(4.31a) $\displaystyle d_{q+1}^{(p)}$
$\displaystyle:=\sum_{k\in\Lambda_{B}}a_{(k)}g_{(\tau)}D_{(k)}.$ (4.31b)

By the algebraic identity (4.1.1),

$\displaystyle d_{q+1}^{(p)}\otimes w_{q+1}^{(p)}-w_{q+1}^{(p)}\otimes
d_{q+1}^{(p)}+\mathring{R}^{B}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)})+\mathring{R}^{B}_{\ell}$
$\displaystyle+\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}(D_{(k^{\prime})}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k^{\prime})})$ $\displaystyle=$
$\displaystyle\sum_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}\mathbb{P}_{\neq
0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)})$
$\displaystyle+\sum_{k\in\Lambda_{B}}a_{(k)}^{2}(g_{(\tau)}^{2}-1){\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x$ $\displaystyle+\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}(D_{(k^{\prime})}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k^{\prime})}).$ (4.32)

Moreover, for the nonlinearity in the velocity equation, by (4.18) and
(4.1.2),

$\displaystyle w_{q+1}^{(p)}\otimes w_{q+1}^{(p)}-d_{q+1}^{(p)}\otimes
d_{q+1}^{(p)}+\mathring{R}^{u}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{k\in\Lambda_{u}}a_{(k)}^{2}g_{(\tau)}^{2}W_{(k)}\otimes
W_{(k)}+\sum_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}(W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)})+\mathring{R}^{u}_{\ell}$
$\displaystyle+\sum_{k\neq
k^{\prime}\in\Lambda_{u}\cup\Lambda_{B}}a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}W_{(k)}\otimes
W_{(k^{\prime})}-\sum_{k\neq
k^{\prime}\in\Lambda_{B}}a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}D_{(k)}\otimes
D_{(k^{\prime})}$ $\displaystyle=$
$\displaystyle\rho_{u}f^{2}_{u}Id+\sum_{k\in\Lambda_{u}}a_{(k)}^{2}g_{(\tau)}^{2}\mathbb{P}_{\neq
0}(W_{(k)}\otimes W_{(k)})$
$\displaystyle+\sum_{k\in\Lambda_{B}}a_{(k)}^{2}g_{(\tau)}^{2}\mathbb{P}_{\neq
0}(W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)})$
$\displaystyle+\sum_{k\in\Lambda_{u}}a_{(k)}^{2}\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}{\rm d}x$
$\displaystyle+\sum_{k\in\Lambda_{B}}a_{(k)}^{2}\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x$ $\displaystyle+\sum_{k\neq
k^{\prime}\in\Lambda_{u}\cup\Lambda_{B}}a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}W_{(k)}\otimes
W_{(k^{\prime})}-\sum_{k\neq
k^{\prime}\in\Lambda_{B}}a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}D_{(k)}\otimes
D_{(k^{\prime})}.$ (4.33)

######  Remark 4.3.

The key fact here is that, module the high frequency part and negligible
intersections, the nonlinearities of the principal parts cancel with the old
magnetic stresses $\mathring{R}_{\ell}^{B}$ and $\mathring{R}_{\ell}^{u}$,
which enables to decrease the amplitudes of the old stresses and to yield the
inductive estimate (1.14).

###  4.3. Incompressibility correctors

Because the amplitude functions $\\{a_{(k)},k\in\Lambda_{u}\cup\Lambda_{B}\\}$
depend on the space, the principal parts of perturbations are not divergence
free. This leads to the introduction of the incompressibility correctors

$\displaystyle w_{q+1}^{(c)}$
$\displaystyle:=\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}g_{(\tau)}\left({\mathrm{curl}}(\nabla
a_{(k)}\times W^{c}_{(k)})+\nabla
a_{(k)}\times{\mathrm{curl}}W^{c}_{(k)}+a_{(k)}\widetilde{W}_{(k)}^{c}\right),$
(4.34a) $\displaystyle d_{q+1}^{(c)}$
$\displaystyle:=\sum_{k\in\Lambda_{B}}g_{(\tau)}\left({\mathrm{curl}}(\nabla
a_{(k)}\times D_{(k)}^{c})+\nabla
a_{(k)}\times{\mathrm{curl}}D_{(k)}^{c}+a_{(k)}\widetilde{D}_{(k)}^{c}\right),$
(4.34b)

where $W^{c}_{(k)}$ and $\widetilde{W}_{k}^{c}$ are given by (3.19) and
(3.17), respectively, and $D_{(k)}^{c}$ and $\widetilde{D}_{(k)}^{c}$ are as
in (3.21). Then, it follows from (3.18) and (3.22) that

$\displaystyle
w_{q+1}^{(p)}+w_{q+1}^{(c)}={\mathrm{curl}}{\mathrm{curl}}\left(\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}a_{(k)}g_{(\tau)}W^{c}_{(k)}\right),$
(4.35a) $\displaystyle
d_{q+1}^{(p)}+d_{q+1}^{(c)}={\mathrm{curl}}{\mathrm{curl}}\left(\sum_{k\in\Lambda_{B}}a_{(k)}g_{(\tau)}D_{(k)}^{c}\right).$
(4.35b)

In particular,

$\displaystyle{\mathrm{div}}(w_{q+1}^{(p)}+w_{q+1}^{(c)})={\mathrm{div}}(d_{q+1}^{(p)}+d_{q+1}^{(c)})=0,$
(4.36)

which justifies the definition of the incompressibility correctors.

###  4.4. Temporal correctors

Two new types of temporal correctors will be introduced in order to balance
the high spatial and temporal oscillations in (4.2) and (4.2).

####  4.4.1. Temporal correctors to balance spatial oscillations.

In order to balance the high spatial oscillations in (4.2) and (4.2) caused by
the spatial concentration function $\psi_{(k_{1})}$, we introduce the temporal
correctors $w_{q+1}^{(t)}$ and $d_{q+1}^{(t)}$, defined by

$\displaystyle
w_{q+1}^{(t)}:=-{\mu}^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}(a_{(k)}^{2}g_{(\tau)}^{2}\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}),$ (4.37a)
$\displaystyle
d_{q+1}^{(t)}=-{\mu}^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}(a_{(k)}^{2}g_{(\tau)}^{2}\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}),$ (4.37b)

where $\mathbb{P}_{H}$ denotes the Helmholtz-Leray projector, i.e.,
$\mathbb{P}_{H}=\mathrm{Id}-\nabla\Delta^{-1}{\mathrm{div}}.$

Then, by the Leibniz rule, (3.15) and (3.16),

$\displaystyle\partial_{t}w_{q+1}^{(t)}+\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{\neq
0}\left(a_{(k)}^{2}g_{(\tau)}^{2}{\mathrm{div}}(W_{(k)}\otimes
W_{(k)})\right)$ $\displaystyle=$
$\displaystyle-{\mu}^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\partial_{t}\left(a_{(k)}^{2}g_{(\tau)}^{2}\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}\right)$
$\displaystyle+{\mu}^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{\neq
0}\left(a_{(k)}^{2}g_{(\tau)}^{2}\partial_{t}(\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1})\right)$
$\displaystyle=$
$\displaystyle(\nabla\Delta^{-1}{\mathrm{div}}){\mu}^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{\neq
0}\partial_{t}\left(a_{(k)}^{2}g_{(\tau)}^{2}\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}\right)$
$\displaystyle-{\mu}^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathbb{P}_{\neq
0}(\partial_{t}\left(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}\right),$
(4.38)

and

$\displaystyle\partial_{t}d_{q+1}^{(t)}+\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\left(a_{(k)}^{2}g_{(\tau)}^{2}{\mathrm{div}}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)})\right)$ $\displaystyle=$
$\displaystyle(\nabla\Delta^{-1}{\mathrm{div}}){\mu}^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\partial_{t}\left(a_{(k)}^{2}g_{(\tau)}^{2}\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}\right)$
$\displaystyle-{\mu}^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}(\partial_{t}\left(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}\right).$
(4.39)

######  Remark 4.4.

The important fact here is that, the first terms on the R.H.S. of (4.4.1) and
(4.4.1) can be treated as the pressures and can be removed by using the
Helmholtz-Leray projector, while the remaining terms are of low spatial
oscillations and thus can be controlled by the large parameter $\mu$. See
(5.3) and (5.4) below.

####  4.4.2. Temporal correctors to balance temporal oscillations.

Another new type of temporal correctors is introduced below, in order to
balance the high temporal oscillations in (4.2) and (4.2) caused by the
temporal concentration function $g_{(\tau)}$,

$\displaystyle
w_{q+1}^{(o)}:=-\sigma^{-1}\sum_{k\in\Lambda_{u}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes W_{(k)}{\rm
d}x\nabla(a_{(k)}^{2})\right)$ $\displaystyle\qquad\quad\
-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right),$ (4.40a)
$\displaystyle
d_{q+1}^{(o)}:=-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}){\fint}_{{\mathbb{T}}^{3}}D_{(k_{)}}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right),$ (4.40b)
where we recall that $h_{(\tau)}$ is given by (3.36).

Then, by the Leibniz rule and the fact that
$\frac{d}{dt}h_{(\tau)}=\sigma(g^{2}_{(\tau)}-1)$,

$\displaystyle\partial_{t}w_{q+1}^{(o)}+\sum_{k\in\Lambda_{u}}\mathbb{P}_{\neq
0}\left(\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle+\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\left(\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle=$
$\displaystyle\left(\nabla\Delta^{-1}{\mathrm{div}}\right)\sigma^{-1}\sum_{k\in\Lambda_{u}}\mathbb{P}_{\neq
0}\partial_{t}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle+\left(\nabla\Delta^{-1}{\mathrm{div}}\right)\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\partial_{t}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle-\sigma^{-1}\sum_{k\in\Lambda_{u}}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes W_{(k)}{\rm
d}x\partial_{t}\nabla(a_{(k)}^{2})\right)$
$\displaystyle-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial_{t}\nabla(a_{(k)}^{2})\right),$
(4.41)

and

$\displaystyle\partial_{t}d_{q+1}^{(o)}+\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\left(\left(g_{(\tau)}^{2}-1\right){\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle=$
$\displaystyle\left(\nabla\Delta^{-1}{\mathrm{div}}\right)\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\partial_{t}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla(a_{(k)}^{2})\right)$
$\displaystyle-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\partial_{t}\nabla(a_{(k)}^{2})\right).$
(4.42)

######  Remark 4.5.

As in the previous spatial oscillation case in Remark 4.4, the first two terms
on the R.H.S. of (4.4.2) are the harmless pressures, and the remaining terms
are of low temporal oscillations which can be controlled by the large
parameter $\sigma$. Similar structure also appear in the formula (4.4.2).

###  4.5. Velocity and magnetic perturbations

We are now in position to define the velocity and magnetic perturbations
$w_{q+1}$ and $d_{q+1}$ at level $q+1$:

$\displaystyle w_{q+1}$
$\displaystyle:=w_{q+1}^{(p)}+w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)},$
(4.43a) $\displaystyle d_{q+1}$
$\displaystyle:=d_{q+1}^{(p)}+d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)}.$
(4.43b)

Then, the velocity and magnetic fields at level $q+1$ are defined by

$\displaystyle u_{q+1}:=u_{\ell}+w_{q+1},$ (4.44a) $\displaystyle
B_{q+1}:=B_{\ell}+d_{q+1}.$ (4.44b)

The estimates of velocity and magnetic perturbations are summarized in Lemma
4.6.

######  Lemma 4.6 (Estimates of perturbations).

For any $\rho\in(1,{\infty}),\gamma\in[1,+\infty]$ and integer $0\leq N\leq
4$, the following estimates hold:

$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(p)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}+\lVert{\nabla}^{N}d_{q+1}^{(p)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-1}\lambda^{N}r_{\perp}^{\frac{1}{\rho}-\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{1}{2}}\tau^{\frac{1}{2}-\frac{1}{\gamma}},$
(4.45)
$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(c)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}+\lVert{\nabla}^{N}d_{q+1}^{(c)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-1}\lambda^{N}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}}\tau^{\frac{1}{2}-\frac{1}{\gamma}},$
(4.46)
$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(t)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}+\lVert{\nabla}^{N}d_{q+1}^{(t)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-2}\lambda^{N}\mu^{-1}r_{\perp}^{\frac{1}{\rho}-1}r_{\parallel}^{\frac{1}{\rho}-1}\tau^{1-\frac{1}{\gamma}},$
(4.47)
$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(o)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-8N-10}\sigma^{-1},\
\
\lVert{\nabla}^{N}d_{q+1}^{(o)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-4N-6}\sigma^{-1},$
(4.48) $\displaystyle\lVert w_{q+1}^{(p)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(c)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(t)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(o)}\rVert_{C_{t,x}^{N}}\lesssim\lambda^{\frac{5N}{2}+3},$ (4.49)
$\displaystyle\lVert d_{q+1}^{(p)}\rVert_{C_{t,x}^{N}}+\lVert
d_{q+1}^{(c)}\rVert_{C_{t,x}^{N}}+\lVert
d_{q+1}^{(t)}\rVert_{C_{t,x}^{N}}+\lVert
d_{q+1}^{(o)}\rVert_{C_{t,x}^{N}}\lesssim\lambda^{\frac{5N}{2}+3}.$ (4.50)

###### Proof.

By Lemmas 3.3, 3.5, 4.1 and 4.2 and $\ell^{-6N_{1}}\lesssim\lambda^{N_{1}}$,
for any $\rho\in(1,+\infty)$,

$\displaystyle\lVert\nabla^{N}w_{q+1}^{(p)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}+\lVert\nabla^{N}d_{q+1}^{(p)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\sum\limits_{N_{1}+N_{2}=N}\|a_{(k)}\|_{C^{N_{1}}_{t,x}}\|g_{(\tau)}\|_{L_{t}^{\gamma}}\lVert\nabla^{N_{2}}W_{(k)}\rVert_{C_{t}L^{\rho}_{x}}$
$\displaystyle+\sum_{k\in\Lambda_{B}}\sum\limits_{N_{1}+N_{2}=N}\|a_{(k)}\|_{C^{N_{1}}_{t,x}}\|g_{(\tau)}\|_{L_{t}^{\gamma}}\lVert\nabla^{N_{2}}D_{(k)}\rVert_{C_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\ell^{-1}\lambda^{N}r_{\perp}^{\frac{1}{\rho}-\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{1}{2}}\tau^{\frac{1}{2}-\frac{1}{\gamma}},$
(4.51)

which verifies (4.45).

Moreover, using (1.11), (3.25), (4.34a) and Lemmas 4.1 and 4.2 we have

$\displaystyle\quad\lVert{\nabla}^{N}w_{q+1}^{(c)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\left\|g_{(\tau)}{\nabla}^{N}\left({\mathrm{curl}}(\nabla
a_{(k)}\times W^{c}_{(k)})+\nabla
a_{(k)}\times{\mathrm{curl}}W_{(k)}^{c}+a_{(k)}\widetilde{W}^{c}_{(k)}\right)\right\|_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\sum\limits_{k\in\Lambda_{u}\cup\Lambda_{B}}\|g_{(\tau)}\|_{L^{\gamma}_{t}}\sum_{N_{1}+N_{2}=N}\left(\lVert
a_{(k)}\rVert_{C_{t,x}^{N_{1}+2}}\lVert{\nabla}^{N_{2}}W^{c}_{(k)}\rVert_{C_{t}W^{1,\rho}_{x}}+\lVert
a_{(k)}\rVert_{C_{t,x}^{N_{1}}}\lVert{\nabla}^{N_{2}}\widetilde{W}^{c}_{(k)}\rVert_{C_{t}L^{\rho}_{x}}\right)$
$\displaystyle\lesssim$
$\displaystyle\sum_{N_{1}+N_{2}=N}(\ell^{-8N_{1}-17}r_{\perp}^{\frac{1}{\rho}-\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{1}{2}}\lambda^{N_{2}-1}+\ell^{-8N_{1}-1}\lambda^{N_{2}}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}})\tau^{\frac{1}{2}-\frac{1}{\gamma}}$
$\displaystyle\lesssim$
$\displaystyle(\ell^{-17}r_{\perp}^{\frac{1}{\rho}-\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{1}{2}}\lambda^{N-1}+\ell^{-1}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}}\lambda^{N})\tau^{\frac{1}{2}-\frac{1}{\gamma}}$
$\displaystyle\lesssim$
$\displaystyle\ell^{-1}\lambda^{N}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}}\tau^{\frac{1}{2}-\frac{1}{\gamma}}.$
(4.52)

Similarly, by (3.26), (4.13) and (4.34b),

$\displaystyle\quad\
\lVert{\nabla}^{N}d_{q+1}^{(c)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\leq\sum_{k\in\Lambda_{B}}\left\|g_{(\tau)}{\nabla}^{N}\left({\mathrm{curl}}(\nabla
a_{(k)}\times D^{c}_{(k)})+\nabla
a_{(k)}\times{\mathrm{curl}}D^{c}_{(k)}+a_{(k)}\widetilde{D}_{(k)}^{c}\right)\right\|_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim\sum\limits_{k\in\Lambda_{B}}\|g_{(\tau)}\|_{L^{\gamma}_{t}}\sum_{N_{1}+N_{2}=N}(\lVert
a_{(k)}\rVert_{C_{t,x}^{N_{1}+2}}\lVert{\nabla}^{N_{2}}D^{c}_{(k)}\rVert_{C_{t}W^{1,\rho}_{x}}+\|a_{(k)}\|_{C^{N_{1}}_{t,x}}\|{\nabla}^{N_{2}}\widetilde{D}^{c}_{(k)}\|_{C_{t}L^{\rho}_{x}})$
$\displaystyle\lesssim\sum_{N_{1}+N_{2}=N}(\ell^{-4N_{1}-9}\lambda^{N_{2}-1}r_{\perp}^{\frac{1}{\rho}-\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{1}{2}}+\ell^{-4N_{1}-1}\lambda^{N_{2}}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}})\tau^{\frac{1}{2}-\frac{1}{\gamma}}$
$\displaystyle\lesssim\ell^{-1}\lambda^{N}r_{\perp}^{\frac{1}{\rho}+\frac{1}{2}}r_{\parallel}^{\frac{1}{\rho}-\frac{3}{2}}\tau^{\frac{1}{2}-\frac{1}{\gamma}}.$
(4.53)

Thus, we obtain (4.46).

Regarding the temporal correctors, by (4.37a), Lemmas 3.3, 3.5, 4.1 and 4.2
and the boundedness of operators $P_{\not=0}$ and $\mathbb{P}_{H}$ in
$L^{\rho}$,

$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(t)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\,\mu^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\lVert
g_{(\tau)}\rVert_{L^{2\gamma}_{t}}^{2}\sum_{N_{1}+N_{2}+N_{3}=N}\|\nabla^{N_{1}}(a_{(k)}^{2})\|_{C_{t,x}}\lVert{\nabla}^{N_{2}}(\psi_{(k_{1})}^{2})\rVert_{C_{t}L^{\rho}_{x}}\lVert{\nabla}^{N_{3}}(\phi_{(k)}^{2})\rVert_{L^{\rho}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\,\mu^{-1}\tau^{1-\frac{1}{\gamma}}\sum_{N_{1}+N_{2}+N_{3}=N}\ell^{-8N_{1}-2}\lambda^{N_{2}}r_{\parallel}^{\frac{1}{\rho}-1}\lambda^{N_{3}}r_{\perp}^{\frac{1}{\rho}-1}$
$\displaystyle\lesssim$
$\displaystyle\,\ell^{-2}\lambda^{N}\mu^{-1}r_{\perp}^{\frac{1}{\rho}-1}r_{\parallel}^{\frac{1}{\rho}-1}\tau^{1-\frac{1}{\gamma}},$
(4.54)

and similarly,

$\displaystyle\lVert{\nabla}^{N}d_{q+1}^{(t)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}$
$\displaystyle\lesssim\ell^{-2}\lambda^{N}\mu^{-1}r_{\perp}^{\frac{1}{\rho}-1}r_{\parallel}^{\frac{1}{\rho}-1}\tau^{1-\frac{1}{\gamma}},$
(4.55)

which yields (4.47).

Concerning the estimates of temporal correctors $w_{q+1}^{(o)}$ and
$d_{q+1}^{(o)}$, by (3.38), Lemmas 4.1 and 4.2,

$\displaystyle\lVert{\nabla}^{N}d_{q+1}^{(o)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\sigma^{-1}\sum_{k\in\Lambda_{B}}\|h_{(\tau)}\|_{C_{t}}\|\nabla^{N+1}(a^{2}_{(k)})\|_{C_{t,x}}\lesssim\ell^{-4N-6}\sigma^{-1},$
(4.56)

and

$\displaystyle\lVert{\nabla}^{N}w_{q+1}^{(o)}\rVert_{L^{\gamma}_{t}L^{\rho}_{x}}\lesssim\ell^{-8N-10}\sigma^{-1}.$
(4.57)

It remains to prove the $C^{N}$-estimates (4.49) and (4.50) of perturbations.
By Lemmas 3.3, 3.5, 4.1 and 4.2,

$\displaystyle\lVert w_{q+1}^{(p)}\rVert_{C_{t,x}^{N}}\lesssim$
$\displaystyle\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\|a_{(k)}\|_{C_{t,x}^{N}}\sum_{0\leq
N_{1}+N_{2}\leq N}\lVert g_{(\tau)}\rVert_{C_{t}^{N_{1}}}\lVert
W_{(k)}\rVert_{C_{t,x}^{N_{2}}}$ $\displaystyle\lesssim$
$\displaystyle\sum_{0\leq N_{1}+N_{2}\leq
N}\sum_{N_{21}+N_{22}=N_{2}}\ell^{-8N-1}\sigma^{N_{1}}\tau^{N_{1}+\frac{1}{2}}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}\lambda^{N_{21}}\left(\frac{r_{\perp}\lambda\mu}{r_{\parallel}}\right)^{N_{22}}$
$\displaystyle\lesssim$ $\displaystyle\lambda^{\frac{5N+3}{2}},$ (4.58)

where we also used (1.11) and (3.1) in the last step.

Similarly, we have

$\displaystyle\quad\lVert w_{q+1}^{(c)}\rVert_{C_{t,x}^{N}}$
$\displaystyle\lesssim\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\left\|g_{(\tau)}\left({\mathrm{curl}}(\nabla
a_{(k)}\times W^{c}_{(k)})+\nabla
a_{(k)}\times{\mathrm{curl}}W_{(k)}^{c}+a_{(k)}\widetilde{W}^{c}_{(k)}\right)\right\|_{C_{t,x}^{N}}$
$\displaystyle\lesssim\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\|a_{(k)}\|_{C_{t,x}^{N+2}}\sum_{0\leq
N_{1}+N_{2}\leq N}\lVert g_{(\tau)}\rVert_{C_{t}^{N_{1}}}\left(\lVert
W^{c}_{(k)}\rVert_{C_{t,x}^{N_{2}}}+\lVert\nabla
W^{c}_{(k)}\rVert_{C_{t,x}^{N_{2}}}+\lVert\widetilde{W}^{c}_{(k)}\rVert_{C^{N_{2}}_{t,x}}\right)$
$\displaystyle\lesssim\sum_{0\leq N_{1}+N_{2}\leq
N}\ell^{-8N-1}\sigma^{N_{1}}\tau^{N_{1}+\frac{1}{2}}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}\left(\frac{r_{\perp}\lambda\mu}{r_{\parallel}}\right)^{N_{2}}\left(\ell^{-16}\lambda^{-1}+\ell^{-16}\frac{r_{\perp}}{r_{\parallel}}\right)$
$\displaystyle\lesssim\lambda^{\frac{5N+3}{2}}.$ (4.59)

Moreover, by Sobolev’s embedding $W^{1,6}(\mathbb{T}^{3})\hookrightarrow
L^{\infty}(\mathbb{T}^{3})$ and the boudedness of operators
$\mathbb{P}_{H}\mathbb{P}_{\not=0}$ in the space $L^{6}_{x}$,

$\displaystyle\lVert w_{q+1}^{(t)}\rVert_{C_{t,x}^{N}}$
$\displaystyle\lesssim\mu^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\|a_{(k)}^{2}g_{(\tau)}^{2}\psi^{2}_{(k_{1})}\phi^{2}_{(k)}\|_{C^{N}_{t}W^{N+1,6}_{x}}$
$\displaystyle\lesssim\mu^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\sum_{0\leq
N_{1}+N_{2}\leq
N+1}\|a_{(k)}^{2}\|_{C_{t,x}^{N}}\|g_{(\tau)}^{2}\psi^{2}_{(k_{1})}\phi^{2}_{(k)}\|_{C_{t}^{N_{1}}C^{N_{2}}_{x}}$
$\displaystyle\leq\frac{1}{3}\lambda^{\frac{5N}{2}+3},$ (4.60)

where we also used Lemmas 3.3, 3.5, 4.1 and 4.2 in the last step.

Finally, arguing as above we get

$\displaystyle\lVert w_{q+1}^{(o)}\rVert_{C_{t,x}^{N}}$
$\displaystyle\lesssim\sigma^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\|h_{(\tau)}{\nabla}(a_{(k)}^{2})\|_{C^{N}_{t}W^{N+1,6}_{x}}$
$\displaystyle\lesssim\sigma^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\|h_{(\tau)}\|_{C_{t}^{N+1}}\|\nabla(a^{2}_{(k)})\|_{C_{t,x}^{N+1}}$
$\displaystyle\lesssim\sigma^{N}\tau^{N+1}\ell^{-8(N+1)-2}$
$\displaystyle\lesssim\lambda^{2{\varepsilon}N}\lambda^{(1-8{\varepsilon})(N+1)-2}\lambda^{\frac{84}{100}{\varepsilon}}\lesssim\lambda^{N+1},$
(4.61)

where the last two steps were due to (1.11) and (3.1).

Thus, combining (4.5)-(4.5) altogether and using (3.1) we conclude that

$\displaystyle\lVert w_{q+1}^{(p)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(c)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(t)}\rVert_{C_{t,x}^{N}}+\lVert
w_{q+1}^{(o)}\rVert_{C_{t,x}^{N}}\leq\frac{1}{2}\lambda^{\frac{5N}{2}+3}.$
(4.62)

The $C^{N}$-estimates of the magnetic perturbations in (4.50) can be proved
similarly. ∎

###  4.6. Verification of inductive estimates (1.12), (1.15) and (1.16)

We are now in position to verify the inductive estimates (1.12), (1.15) and
(1.16) for the velocity and magnetic fields.

Note that, the previous estimate (4.45) cannot yield the decay properties
required by the inductive estimates (1.15) and (1.16). An important ingredient
here is the $L^{p}$ decorrelation Lemma 4.7 below, which permits to derive the
decay of the $L^{2}_{t,x}$-norm of principal parts.

######  Lemma 4.7 ([19], Lemma 2.4; see also [13], Lemma 3.7).

Let $\sigma\in\mathbb{N}$ and $f,g:\mathbb{T}^{d}\rightarrow{\mathbb{R}}$ be
smooth functions. Then for every $p\in[1,+{\infty}]$,

$\big{|}\|fg(\sigma\cdot)\|_{L^{p}({\mathbb{T}}^{d})}-\|f\|_{L^{p}({\mathbb{T}}^{d})}\|g\|_{L^{p}({\mathbb{T}}^{d})}\big{|}\lesssim\sigma^{-\frac{1}{p}}\|f\|_{C^{1}({\mathbb{T}}^{d})}\|g\|_{L^{p}({\mathbb{T}}^{d})}.$
(4.63)

We apply Lemma 4.7 with $f=a_{(k)}$, $g=g_{(\tau)}\psi_{(k_{1})}\phi_{(k)}$
and $\sigma=\lambda^{2{\varepsilon}}$. By (1.10), (1.11) and Lemmas 4.1 and
4.2,

$\displaystyle\lVert w^{(p)}_{q+1}\rVert_{L^{2}_{t,x}}$
$\displaystyle\lesssim\sum\limits_{k\in\Lambda_{u}\cup\Lambda_{B}}\Big{(}\|a_{(k)}\|_{L^{2}_{t,x}}\lVert
g_{(\tau)}\rVert_{L^{2}_{t}}\lVert\psi_{(k_{1})}\phi_{(k)}\rVert_{C_{t}L^{2}_{x}}$
$\displaystyle\qquad\qquad\qquad+\sigma^{-\frac{1}{2}}\|a_{(k)}\|_{C^{1}_{t,x}}\lVert
g_{(\tau)}\rVert_{L^{2}_{t}}\lVert\psi_{(k_{1})}\phi_{(k)}\rVert_{C_{t}L^{2}_{x}}\Big{)}$
$\displaystyle\lesssim\delta_{q+1}^{\frac{1}{2}}+\ell^{-8}\lambda^{-{\varepsilon}}_{q+1}\lesssim\delta_{q+1}^{\frac{1}{2}},$
(4.64) $\displaystyle\lVert d^{(p)}_{q+1}\rVert_{L^{2}_{t,x}}$
$\displaystyle\lesssim\sum\limits_{k\in\Lambda_{B}}\Big{(}\|a_{(k)}\|_{L^{2}_{t,x}}\lVert
g_{(\tau)}\rVert_{L^{2}_{t}}\lVert\psi_{(k_{1})}\phi_{(k)}\rVert_{C_{t}L^{2}_{x}}$
$\displaystyle\qquad\qquad\quad+\sigma^{-\frac{1}{2}}\|a_{(k)}\|_{C^{1}_{t,x}}\lVert
g_{(\tau)}\rVert_{L^{2}_{t}}\lVert\psi_{(k_{1})}\phi_{(k)}\rVert_{C_{t}L^{2}_{x}}\Big{)}$
$\displaystyle\lesssim\delta_{q+1}^{\frac{1}{2}}+\ell^{-4}\lambda^{-{\varepsilon}}_{q+1}\lesssim\delta_{q+1}^{\frac{1}{2}}.$
(4.65)

Below we verify the iterative estimates for $u_{q+1}$ and $B_{q+1}$. For the
velocity perturbation $w_{q+1}$, using (4.43a), (4.6) and Lemma 4.6 we have

$\displaystyle\lVert w_{q+1}\rVert_{L^{2}_{t,x}}$ $\displaystyle\lesssim\lVert
w_{q+1}^{(p)}\rVert_{L^{2}_{t,x}}+\lVert
w_{q+1}^{(c)}\rVert_{L^{2}_{t,x}}+\lVert
w_{q+1}^{(t)}\rVert_{L^{2}_{t,x}}+\lVert w_{q+1}^{(o)}\rVert_{L^{2}_{t,x}}$
$\displaystyle\lesssim\delta_{q+1}^{\frac{1}{2}}+\ell^{-1}r_{\perp}r_{\parallel}^{-1}+\ell^{-2}\mu^{-1}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}\tau^{\frac{1}{2}}+\ell^{-10}\sigma^{-1}\lesssim\delta_{q+1}^{\frac{1}{2}},$
(4.66)

and

$\displaystyle\lVert w_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\lVert w_{q+1}^{(p)}\rVert_{L^{1}_{t}L^{2}_{x}}+\lVert
w_{q+1}^{(c)}\rVert_{L^{1}_{t}L^{2}_{x}}+\lVert
w_{q+1}^{(t)}\rVert_{L^{1}_{t}L^{2}_{x}}+\lVert
w_{q+1}^{(o)}\rVert_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\ell^{-1}\tau^{-\frac{1}{2}}+\ell^{-1}r_{\perp}r_{\parallel}^{-1}\tau^{-\frac{1}{2}}+\ell^{-2}\mu^{-1}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}+\ell^{-10}\sigma^{-1}\lesssim\lambda_{q+1}^{-{\varepsilon}},$
(4.67)

where we also used (1.11) in the last steps.

Then, in view of (1.12), (4.49), (4.44a) and the standard mollification
estimates, we get for $a$ sufficiently large,

$\displaystyle\lVert u_{q+1}\rVert_{C^{1}_{t,x}}\lesssim\lVert
u_{\ell}\rVert_{C^{1}_{t,x}}+\lVert
w_{q+1}\rVert_{C^{1}_{t,x}}\lesssim\lambda_{q}^{7}+\lambda_{q+1}^{\frac{13}{2}}\leq\lambda_{q+1}^{7},$
(4.68) $\displaystyle\lVert u_{q}-u_{q+1}\rVert_{L^{2}_{t,x}}\leq\lVert
u_{q}-u_{\ell}\rVert_{L^{2}_{t,x}}+\lVert
u_{\ell}-u_{q+1}\rVert_{L^{2}_{t,x}}$ $\displaystyle\qquad\qquad\qquad\
\lesssim\lVert u_{q}-u_{\ell}\rVert_{C_{t,x}}+\lVert
w_{q+1}\rVert_{L^{2}_{t,x}}$ $\displaystyle\qquad\qquad\qquad\
\lesssim\ell\lVert u_{q}\rVert_{C_{t,x}^{1}}+\delta_{q+1}^{\frac{1}{2}}\leq
M\delta_{q+1}^{\frac{1}{2}},$ (4.69)

and

$\displaystyle\lVert u_{q}-u_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}\lesssim$
$\displaystyle\lVert u_{q}-u_{\ell}\rVert_{C_{t,x}}+\lVert
w_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\lesssim$
$\displaystyle\ell\lambda_{q}^{7}+\lambda_{q+1}^{-{\varepsilon}}\leq\delta_{q+2}^{\frac{1}{2}},$
(4.70)

where we also used the inequalities
$\lambda_{q}^{-13}\ll\delta_{q+1}^{\frac{1}{2}}$ and
$\ell\lambda_{q}^{7}\ll\delta_{q+2}^{\frac{1}{2}}$, due to (1.11) and $M$ is a
fixed large universal constant.

Similarly, for the magnetic perturbation $d_{q+1}$, by (4.43b), (4.6) and
Lemma 4.6,

$\displaystyle\lVert d_{q+1}\rVert_{L^{2}_{t,x}}$ $\displaystyle\leq\lVert
d_{q+1}^{(p)}\rVert_{L^{2}_{t,x}}+\lVert
d_{q+1}^{(c)}\rVert_{L^{2}_{t,x}}+\lVert
d_{q+1}^{(t)}\rVert_{L^{2}_{t,x}}+\lVert d_{q+1}^{(o)}\rVert_{L^{2}_{t,x}}$
$\displaystyle\lesssim\delta_{q+1}^{\frac{1}{2}}+\ell^{-1}r_{\perp}r_{\parallel}^{-1}+\ell^{-2}\mu^{-1}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}\tau^{\frac{1}{2}}+\ell^{-6}\sigma^{-1}\lesssim\delta_{q+1}^{\frac{1}{2}},$
(4.71)

and

$\displaystyle\lVert d_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\lVert d_{q+1}^{(p)}\rVert_{L^{1}_{t}L^{2}_{x}}+\lVert
d_{q+1}^{(c)}\rVert_{L^{1}_{t}L^{2}_{x}}+\lVert
d_{q+1}^{(t)}\rVert_{L^{1}_{t}L^{2}_{x}}+\|d_{q+1}^{(o)}\|_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\ell^{-1}\tau^{-\frac{1}{2}}+\ell^{-1}r_{\perp}r_{\parallel}^{-1}\tau^{-\frac{1}{2}}+\ell^{-2}\mu^{-1}r_{\perp}^{-\frac{1}{2}}r_{\parallel}^{-\frac{1}{2}}+\ell^{-6}\sigma^{-1}\leq\lambda_{q+1}^{-{\varepsilon}},$
(4.72)

which along with (1.12), (4.44b) and Lemma 4.6 yield that

$\displaystyle\lVert B_{q+1}\rVert_{C^{1}_{t,x}}$ $\displaystyle\lesssim\lVert
B_{\ell}\rVert_{C^{1}_{t,x}}+\lVert
d_{q+1}\rVert_{C^{1}_{t,x}}\lesssim\lambda_{q}^{7}+\lambda_{q+1}^{\frac{13}{2}}\leq\lambda_{q+1}^{7},$
(4.73) $\displaystyle\lVert B_{q}-B_{q+1}\rVert_{L^{2}_{t,x}}$
$\displaystyle\lesssim\lVert B_{q}-B_{\ell}\rVert_{C_{t,x}}+\lVert
d_{q+1}\rVert_{L^{2}_{t,x}}$ $\displaystyle\lesssim\ell\lVert
B_{q}\rVert_{C_{t,x}^{1}}+\delta_{q+1}^{\frac{1}{2}}\leq
M\delta_{q+1}^{\frac{1}{2}},$ (4.74)

and

$\displaystyle\lVert B_{q}-B_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\lVert B_{q}-B_{\ell}\rVert_{C_{t,x}}+\lVert
d_{q+1}\rVert_{L^{1}_{t}L^{2}_{x}}\lesssim\ell\lambda_{q}^{7}+\lambda_{q+1}^{-{\varepsilon}}\leq\delta_{q+2}^{\frac{1}{2}}.$
(4.75)

Therefore, the inductive estimates in (1.12), (1.15) and (1.16) are verified.

##  5\. Reynolds and magnetic stresses

The main purpose of this section is to determine the Reynolds and magnetic
stresses and to prove the main iteration in Theorem 1.7.

The important roles here are played by the inverse-divergence operators
$\mathcal{R}^{u}$ and $\mathcal{R}^{B}$, defined by

$\displaystyle(\mathcal{R}^{u}v)^{kl}:=\partial_{k}\Delta^{-1}v^{l}+\partial_{l}\Delta^{-1}v^{k}-\frac{1}{2}(\delta_{kl}+\partial_{k}\partial_{l}\Delta^{-1}){\mathrm{div}}\Delta^{-1}v,$
(5.1)
$\displaystyle(\mathcal{R}^{B}f)_{ij}:=\varepsilon_{ijk}(-\Delta)^{-1}({\mathrm{curl}}f)_{k},$
(5.2)

where $\int_{\mathbb{T}^{3}}vdx=0$, ${\mathrm{div}}f=0$, and
$\varepsilon_{ijk}$ is the Levi-Civita tensor, $i,j,k,l\in\\{1,2,3\\}$.

The operator $\mathcal{R}^{u}$ returns symmetric and trace-free matrices,
while the operator $\mathcal{R}^{B}$ returns skew-symmetric matrices.
Moreover, one has the algebraic identities

$\displaystyle{\mathrm{div}}\mathcal{R}^{u}(v)=v,\ \
{\mathrm{div}}\mathcal{R}^{B}(f)=f.$

Both $|\nabla|\mathcal{R}^{u}$ and $|\nabla|\mathcal{R}^{B}$ are Calderon-
Zygmund operators and thus they are bounded in the spaces $L^{p}$,
$1<p<+\infty$. See [2, 29] for more details.

###  5.1. Decomposition of magnetic stress

Using (1.7) with $q+1$ replacing $q$, (2.3), (4.43), and (4.44) we derive the
equation for the magnetic stress $\mathring{R}^{B}_{q+1}$:

$\displaystyle\displaystyle{\mathrm{div}}\mathring{R}_{q+1}^{B}$
$\displaystyle=\underbrace{\partial_{t}(d^{(p)}_{q+1}+d^{(c)}_{q+1})+\nu_{2}(-\Delta)^{\alpha_{2}}d_{q+1}+{\mathrm{div}}(d_{q+1}\otimes
u_{\ell}-u_{\ell}\otimes d_{q+1}+B_{\ell}\otimes w_{q+1}-w_{q+1}\otimes
B_{\ell})}_{{\mathrm{div}}\mathring{R}_{lin}^{B}}$
$\displaystyle\quad+\underbrace{{\mathrm{div}}(d_{q+1}^{(p)}\otimes
w_{q+1}^{(p)}-w_{q+1}^{(p)}\otimes
d_{q+1}^{(p)}+\mathring{R}_{\ell}^{B})+\partial_{t}d_{q+1}^{(t)}+\partial_{t}d_{q+1}^{(o)}}_{{\mathrm{div}}\mathring{R}_{osc}^{B}}$
$\displaystyle\quad+{\mathrm{div}}\Big{(}d_{q+1}^{(p)}\otimes(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})-(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})\otimes
d_{q+1}$
$\displaystyle\qquad\underbrace{\qquad\quad+(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes
w_{q+1}-w_{q+1}^{(p)}\otimes(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\Big{)}}_{{\mathrm{div}}\mathring{R}_{cor}^{B}}$
$\displaystyle\quad+\underbrace{{\mathrm{div}}\left(B_{\ell}\otimes
u_{\ell}-u_{\ell}\otimes
B_{\ell}-\left({B_{q}}\otimes{u_{q}}-{u_{q}}\otimes{B_{q}}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell}\right)}_{{\mathrm{div}}\mathring{R}_{com}^{B}}.$
(5.3)

Using the inverse divergence operator $\mathcal{R}^{B}$ we define the magnetic
stress by

$\displaystyle\mathring{R}_{q+1}^{B}:=\mathring{R}_{lin}^{B}+\mathring{R}_{osc}^{B}+\mathring{R}_{cor}^{B}+\mathring{R}_{com}^{B},$
(5.4)

where

$\displaystyle\mathring{R}_{lin}^{B}:=$
$\displaystyle\mathcal{R}^{B}\left(\partial_{t}(d^{(p)}_{q+1}+d^{(c)}_{q+1})\right)+\nu_{2}\mathcal{R}^{B}(-\Delta)^{{\alpha}_{2}}d_{q+1}$
$\displaystyle+\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\left(d_{q+1}\otimes
u_{\ell}-u_{\ell}\otimes d_{q+1}+B_{\ell}\otimes w_{q+1}-w_{q+1}\otimes
B_{\ell}\right),$ (5.5)

the oscillation error

$\displaystyle\mathring{R}_{osc}^{B}$
$\displaystyle:=\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(g_{(\tau)}^{2}\mathbb{P}_{\neq 0}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)})\nabla(a_{(k)}^{2})\right)$
$\displaystyle\quad-\mu^{-1}\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}(\partial_{t}(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2})$
$\displaystyle\quad-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\partial_{t}\nabla(a_{(k)}^{2})\right)$
$\displaystyle\quad+\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\left(a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}(D_{(k^{\prime})}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k^{\prime})})\right),$ (5.6)

the corrector error

$\displaystyle\mathring{R}_{cor}^{B}:=$
$\displaystyle\,\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\bigg{(}d_{q+1}^{(p)}\otimes(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})-(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})\otimes
d_{q+1}$
$\displaystyle\qquad\qquad+(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes
w_{q+1}-w_{q+1}^{(p)}\otimes(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\bigg{)},$
(5.7)

and the commutator error

$\displaystyle\mathring{R}_{com}^{B}=\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\left(B_{\ell}\otimes
u_{\ell}-u_{\ell}\otimes
B_{\ell}-\left({B_{q}}\otimes{u_{q}}-{u_{q}}\otimes{B_{q}}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell}\right).$
(5.8)

######  Remark 5.1.

Since

$\displaystyle\mathbb{P}_{H}\mathbb{P}_{\not=0}=\mathbb{P}_{\not=0}\mathbb{P}_{H},\
\
\mathbb{P}_{H}{\mathrm{div}}=\mathbb{P}_{\not=0}\mathbb{P}_{H}{\mathrm{div}},$
(5.9)

by (4.36) and the definitions (4.37b) and (4.40b),

$\displaystyle{\mathrm{div}}(d^{(p)}_{q+1}+d^{(c)}_{q+1})=0,\ \
{\mathrm{div}}(-\Delta)^{{\alpha}_{2}}d_{q+1}=0.$ (5.10)

Taking into account ${\mathrm{div}}\mathbb{P}_{H}=0$ we see that the terms in
(5.5)-(5.8) are in the domain of the operator $\mathcal{R}^{B}$.

By (5.10),

$\displaystyle
d^{(p)}_{q+1}+d^{(c)}_{q+1}=\mathbb{P}_{H}(d^{(p)}_{q+1}+d^{(c)}_{q+1}),\ \
(-\Delta)^{{\alpha}_{2}}d_{q+1}=\mathbb{P}_{H}(-\Delta)^{{\alpha}_{2}}d_{q+1}.$
(5.11)

We also note that, in contrast to the nonlinearities in the velocity equation,
the nonlinear terms in the magnetic equation is skew symmetric, which, in
particular, yields that

$\displaystyle{\mathrm{div}}({\mathrm{div}}\mathring{R}^{B}_{q+1})=0.$ (5.12)

Thus, by virtue of (5.11) and (5.12), we obtain

$\displaystyle{\mathrm{div}}\mathring{R}^{B}_{q+1}=\mathbb{P}_{H}{\mathrm{div}}\mathring{R}^{B}_{q+1}.$
(5.13)

Then, using the algebraic identities
${\mathrm{div}}\mathcal{R}^{B}=\mathrm{Id}$, (4.2), (4.4.1) and (4.4.2),
together with the facts that
${\mathrm{div}}={\mathrm{div}}\mathbb{P}_{\not=0}=\mathbb{P}_{\not=0}{\mathrm{div}}$,
we obtain that $\mathring{R}^{B}_{q+1}$ satisfies the relaxation magnetic
equation in (1.7) and the following identity holds

$\displaystyle\mathring{R}^{B}_{q+1}=\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\mathring{R}^{B}_{q+1}.$
(5.14)

###  5.2. Decomposition of Reynolds stress

Concerning the Reynolds stress we compute

$\displaystyle\displaystyle{\mathrm{div}}\mathring{R}_{q+1}^{u}-\nabla
P_{q+1}$
$\displaystyle\displaystyle=\underbrace{\partial_{t}(w_{q+1}^{(p)}+w_{q+1}^{(c)})+\nu_{1}(-\Delta)^{\alpha_{1}}w_{q+1}+{\mathrm{div}}\big{(}u_{\ell}\otimes
w_{q+1}+w_{q+1}\otimes u_{\ell}-B_{\ell}\otimes d_{q+1}-d_{q+1}\otimes
B_{\ell}\big{)}}_{{\mathrm{div}}\mathring{R}_{lin}^{u}+\nabla P_{lin}}$
$\displaystyle\displaystyle\quad+\underbrace{{\mathrm{div}}(w_{q+1}^{(p)}\otimes
w_{q+1}^{(p)}-d_{q+1}^{(p)}\otimes
d_{q+1}^{(p)}+\mathring{R}_{\ell}^{u})+\partial_{t}w_{q+1}^{(t)}+\partial_{t}w_{q+1}^{(o)}}_{{\mathrm{div}}\mathring{R}_{osc}^{u}+\nabla
P_{osc}}$
$\displaystyle\displaystyle\quad+{\mathrm{div}}\Big{(}(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})\otimes
w_{q+1}+w_{q+1}^{(p)}\otimes(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})$
$\displaystyle\qquad\underbrace{\qquad-(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes
d_{q+1}-d_{q+1}^{(p)}\otimes(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\Big{)}}_{{\mathrm{div}}\mathring{R}_{cor}^{u}+\nabla
P_{cor}}$
$\displaystyle\displaystyle\quad+\underbrace{{\mathrm{div}}\left(u_{\ell}\mathring{\otimes}u_{\ell}-B_{\ell}\mathring{\otimes}B_{\ell}-\left({u_{q}}\mathring{\otimes}{u_{q}}-{B_{q}}\mathring{\otimes}{B_{q}}\right)*_{x}\phi_{\ell}*_{t}\varphi_{\ell}\right)}_{{\mathrm{div}}\mathring{R}_{com}^{u}}-\nabla
P_{\ell}.$ (5.15)

Then, using the inverse divergence operator $\mathcal{R}^{u}$ we define the
Reynolds stress by

$\displaystyle\mathring{R}_{q+1}^{u}:=\mathring{R}_{lin}^{u}+\mathring{R}_{osc}^{u}+\mathring{R}_{cor}^{u}+\mathring{R}_{com}^{u},$
(5.16)

where

$\displaystyle\mathring{R}_{lin}^{u}$
$\displaystyle:=\mathcal{R}^{u}\left(\partial_{t}(w_{q+1}^{(p)}+w_{q+1}^{(c)})\right)+\nu_{1}\mathcal{R}^{u}(-\Delta)^{{\alpha}_{1}}w_{q+1}$
$\displaystyle\quad+\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\left(u_{\ell}\mathring{\otimes}w_{q+1}+w_{q+1}\mathring{\otimes}u_{\ell}-B_{\ell}\mathring{\otimes}d_{q+1}-d_{q+1}\mathring{\otimes}B_{\ell}\right),$
(5.17)

the oscillation error

$\displaystyle\mathring{R}_{osc}^{u}:=$
$\displaystyle\sum_{k\in\Lambda_{u}}\mathcal{R}^{u}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(g_{(\tau)}^{2}\mathbb{P}_{\neq 0}(W_{(k)}\otimes
W_{(k)})\nabla(a_{(k)}^{2})\right)$
$\displaystyle+\sum_{k\in\Lambda_{B}}\mathcal{R}^{u}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(g_{(\tau)}^{2}\mathbb{P}_{\neq 0}(W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)})\nabla(a_{(k)}^{2})\right)$
$\displaystyle-\mu^{-1}\sum_{k\in\Lambda_{u}\cup\Lambda_{B}}\mathcal{R}^{u}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(\partial_{t}(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{1}\right)$
$\displaystyle-\sigma^{-1}\sum_{k\in\Lambda_{u}}\mathcal{R}^{u}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes W_{(k)}{\rm
d}x\partial_{t}\nabla(a_{(k)}^{2})\right)$
$\displaystyle-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathcal{R}^{u}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}W_{(k)}\otimes
W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial_{t}\nabla(a_{(k)}^{2})\right)$
$\displaystyle+\sum_{k\neq
k^{\prime}\in\Lambda_{u}\cup\Lambda_{B}}\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\left(a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}W_{(k)}\otimes
W_{(k^{\prime})}\right)$ $\displaystyle-\sum_{k\neq
k^{\prime}\in\Lambda_{B}}\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\left(a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}D_{(k)}\otimes
D_{(k^{\prime})}\right),$ (5.18)

the corrector error

$\displaystyle\mathring{R}_{cor}^{u}$
$\displaystyle:=\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\bigg{(}w^{(p)}_{q+1}\mathring{\otimes}(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})+(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})\mathring{\otimes}w_{q+1}$
$\displaystyle\qquad\qquad\qquad-d^{(p)}_{q+1}\mathring{\otimes}(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})-(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\mathring{\otimes}d_{q+1}\bigg{)},$
(5.19)

and the commutator error

$\displaystyle\mathring{R}^{u}_{com}:=\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\left(u_{\ell}\mathring{\otimes}u_{\ell}-B_{\ell}\mathring{\otimes}B_{\ell}-({u_{q}}\mathring{\otimes}{u_{q}}-{B_{q}}\mathring{\otimes}{B_{q}})*_{x}\phi_{\ell}*_{t}\varphi_{\ell}\right).$
(5.20)

######  Remark 5.2.

We infer from (5.9) that the above terms in (5.17)-(5.20) are in the domain of
the operator $\mathcal{R}^{u}$.

As in (5.10), by (4.36), the definitions (4.37a) and (4.40a), and
${\mathrm{div}}\mathbb{P}_{H}=0$,

$\displaystyle{\mathrm{div}}(w^{(p)}_{q+1}+w^{(c)}_{q+1})=0,\ \
{\mathrm{div}}(w^{(t)}_{q+1}+w^{(o)}_{q+1})=0,$ (5.21)

which yield that

$\displaystyle
w^{(p)}_{q+1}+w^{(c)}_{q+1}=\mathbb{P}_{H}(w^{(p)}_{q+1}+w^{(c)}_{q+1}),\ \
(-\Delta)^{{\alpha}_{1}}w_{q+1}=\mathbb{P}_{H}(-\Delta)^{{\alpha}_{1}}w_{q+1}.$
(5.22)

Then, using the algebraic identities
${\mathrm{div}}\mathcal{R}^{u}=\mathrm{Id}$, (4.2), (4.4.1) and (4.4.2) we
obtain

$\displaystyle{\mathrm{div}}\mathring{R}^{u}_{q+1}=\mathbb{P}_{H}\left(\partial_{t}{u_{q}}+\nu_{1}(-\Delta)^{\alpha_{1}}{u_{q}}+{\mathrm{div}}({u_{q}}\otimes{u_{q}}-{B_{q}}\otimes{B_{q}})\right),$
(5.23)

which yields that $\mathring{R}^{u}_{q+1}$ satisfies the relaxation velocity
equation in (1.7).

Moreover, since $\mathbb{P}^{2}_{H}=\mathbb{P}_{H}$, it also holds that

$\displaystyle\mathring{R}^{u}_{q+1}=\mathcal{R}^{u}\mathbb{P}_{H}{\mathrm{div}}\mathring{R}^{u}_{q+1}.$
(5.24)

###  5.3. Estimates of magnetic stress

The purpose of this subsection is to estimate the magnetic stress
$\mathring{R}^{B}_{q+1}$ given by (5.4) at level $q+1$.

Since the Calderón-Zygmund operators are bounded in the space $L^{p}_{x}$ for
any $1<p<+{\infty}$, we choose

$\displaystyle p:=\frac{2-8\varepsilon}{2-9\varepsilon}\in(1,2),$ (5.25)

where ${\varepsilon}$ is given by (1.8). In particular,

$(1-4\varepsilon)(1-\frac{1}{p})=\frac{\varepsilon}{2},$ (5.26)

which, via (3.1), yields that

$\displaystyle
r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{1}{p}-1}=\lambda^{\varepsilon},\quad
r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}=\lambda^{-1+5\varepsilon}.$
(5.27)

Let us estimate the four parts in the decomposition of
$\mathring{R}^{B}_{q+1}$ separately.

##### Linear error.

Note that, by (4.35b),

$\displaystyle\|\mathcal{R}^{B}\partial_{t}(d_{q+1}^{(p)}+d_{q+1}^{(c)})\|_{L_{t}^{1}L_{x}^{p}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k\in\Lambda_{B}}\|\mathcal{R}^{B}{\mathrm{curl}}{\mathrm{curl}}\partial_{t}(g_{(\tau)}a_{(k)}D^{c}_{(k)})\|_{L_{t}^{1}L_{x}^{p}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k\in\Lambda_{B}}\|{\mathrm{curl}}\partial_{t}(g_{(\tau)}a_{(k)}D^{c}_{(k)})\|_{L_{t}^{1}L_{x}^{p}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k\in\Lambda_{B}}\Big{(}\|g_{(\tau)}\|_{L^{1}_{t}}\left(\|a_{(k)}\|_{C_{t,x}^{2}}\|D^{c}_{(k)}\|_{C_{t}W_{x}^{1,p}}+\|a_{(k)}\|_{C_{t,x}^{1}}\|\partial_{t}D^{c}_{(k)}\|_{C_{t}W^{1,p}_{x}}\right)$
$\displaystyle\qquad\qquad+\|\partial_{t}g_{(\tau)}\|_{L_{t}^{1}}\|a_{(k)}\|_{C_{t,x}^{1}}\|D^{c}_{(k)}\|_{C_{t}W_{x}^{1,p}}\Big{)}.$
(5.28)

Then, by Lemmas 3.3 and 4.1, (1.8), (3.1) and (5.27),

$\displaystyle\|\mathcal{R}^{B}\partial_{t}(d_{q+1}^{(p)}+d_{q+1}^{(c)})\|_{L_{t}^{1}L_{x}^{p}}$
$\displaystyle\lesssim$
$\displaystyle\tau^{-\frac{1}{2}}(\ell^{-8}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\lambda^{-1}+\ell^{-4}r_{\perp}^{\frac{1}{p}+\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{3}{2}}\mu)+\sigma\tau^{\frac{1}{2}}\ell^{-4}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\lambda^{-1}$
$\displaystyle\lesssim$
$\displaystyle\ell^{-8}(\lambda^{-\frac{3}{2}+4{\varepsilon}}+\lambda^{-2{\varepsilon}})\lesssim\ell^{-8}\lambda^{-2{\varepsilon}}.$
(5.29)

The control of the fractional viscosity $(-\Delta)^{{\alpha}_{2}}$ requires
both the temporal and spatial intermittency of the magnetic flows. More
precisely, by Lemma 4.6, for ${\alpha}_{2}\in[0,1/2]$,

$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}\rVert_{L_{t}^{1}L^{p}_{x}}\lesssim$
$\displaystyle\|d_{q+1}\|_{L_{t}^{1}L^{p}_{x}}$ $\displaystyle\lesssim$
$\displaystyle\ell^{-1}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\tau^{-\frac{1}{2}}+\ell^{-6}\sigma^{-1}$
$\displaystyle\lesssim$
$\displaystyle\ell^{-6}\lambda^{-2{\varepsilon}}\lesssim\ell^{-1}\lambda^{-{\varepsilon}}.$
(5.30)

For the stronger viscosity $(-\Delta)^{{\alpha}_{2}}$ with
${\alpha}_{2}\in(1/2,{5}/{4})$, by (4.43b),

$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}\rVert_{L_{t}^{1}L^{p}_{x}}\lesssim$
$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(p)}\rVert_{L_{t}^{1}L^{p}_{x}}+\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(c)}\rVert_{L_{t}^{1}L^{p}_{x}}$
$\displaystyle+\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(t)}\rVert_{L_{t}^{1}L^{p}_{x}}+\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(o)}\rVert_{L_{t}^{1}L^{p}_{x}}.$
(5.31)

In order to control the R.H.S. of (5.31), using the interpolation inequality
(cf. [7]) and (4.45),

$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(p)}\rVert_{L_{t}^{1}L^{p}_{x}}$
$\displaystyle\lesssim\lVert|{\nabla}|^{2{\alpha}_{1}-1}d_{q+1}^{(p)}\rVert_{L_{t}^{1}L^{p}_{x}}$
$\displaystyle\lesssim\lVert
d_{q+1}^{(p)}\rVert_{L_{t}^{1}L^{p}_{x}}^{\frac{3-2{\alpha}_{2}}{2}}\lVert
d_{q+1}^{(p)}\rVert_{L_{t}^{1}W^{2,p}_{x}}^{\frac{2{\alpha}_{2}-1}{2}}$
$\displaystyle\lesssim\ell^{-1}\lambda^{2\alpha_{2}-1}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\tau^{-\frac{1}{2}}.$
(5.32)

Similarly, by Lemma 4.6,

$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(c)}\rVert_{L_{t}^{1}L^{p}_{x}}\lesssim\ell^{-1}\lambda^{2\alpha_{2}-1}r_{\perp}^{\frac{1}{p}+\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{3}{2}}\tau^{-\frac{1}{2}},$
(5.33)
$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(t)}\rVert_{L_{t}^{1}L^{p}_{x}}\lesssim\ell^{-2}\lambda^{2\alpha_{2}-1}\mu^{-1}r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{1}{p}-1},$
(5.34)
$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}^{(o)}\rVert_{L_{t}^{1}L^{p}_{x}}\lesssim\ell^{-14}\sigma^{-1}.$
(5.35)

Hence, we conclude from (5.31)-(5.35) and the fact that
$2\alpha_{2}-1\leq{3}/{2}-10\varepsilon$ that

$\displaystyle\lVert\mathcal{R}^{B}(-\Delta)^{\alpha_{2}}d_{q+1}\rVert_{L_{t}^{1}L^{p}_{x}}$
$\displaystyle\lesssim\ell^{-1}\lambda^{2\alpha_{2}-1}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\tau^{-\frac{1}{2}}+\ell^{-1}\lambda^{2\alpha_{2}-1}r_{\perp}^{\frac{1}{p}+\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{3}{2}}\tau^{-\frac{1}{2}}$
$\displaystyle\quad+\ell^{-2}\lambda^{2\alpha_{2}-1}\mu^{-1}r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{1}{p}-1}+\ell^{-14}\sigma^{-1}$
$\displaystyle\lesssim\ell^{-1}\lambda^{-{\varepsilon}}.$ (5.36)

Regarding the nonlinear terms in (5.5), estimating as in (5.3) we have

$\displaystyle\lVert\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\left(d_{q+1}\otimes
u_{\ell}-u_{\ell}\otimes d_{q+1}+B_{\ell}\otimes w_{q+1}-w_{q+1}\otimes
B_{\ell}\right)\rVert_{L_{t}^{1}L^{p}_{x}}$ $\displaystyle\lesssim\,$
$\displaystyle\lVert d_{q+1}\otimes u_{\ell}-u_{\ell}\otimes
d_{q+1}+B_{\ell}\otimes w_{q+1}-w_{q+1}\otimes
B_{\ell}\rVert_{L_{t}^{1}L^{p}_{x}}$ $\displaystyle\lesssim\,$
$\displaystyle\lVert u_{\ell}\rVert_{C_{t,x}}\lVert
d_{q+1}\rVert_{L_{t}^{1}L^{p}_{x}}+\lVert B_{\ell}\rVert_{C_{t,x}}\lVert
w_{q+1}\rVert_{L_{t}^{1}L^{p}_{x}}$ $\displaystyle\lesssim\,$
$\displaystyle\lambda^{7}_{q}(\ell^{-1}r_{\perp}^{\frac{1}{p}-\frac{1}{2}}r_{\parallel}^{\frac{1}{p}-\frac{1}{2}}\tau^{-\frac{1}{2}}+\ell^{-6}\sigma^{-1}+\ell^{-10}\sigma^{-1})\lesssim\ell^{-10}\lambda^{-2{\varepsilon}}.$
(5.37)

Therefore, combining (5.3), (5.3), (5.3), (5.3) together and using (1.11) and
the fact that $2\alpha_{2}-1\leq{3}/{2}-10\varepsilon$ we arrive at

$\displaystyle\lVert\mathring{R}_{lin}^{B}\rVert_{L_{t}^{1}L^{p}_{x}}$
$\displaystyle\lesssim\ell^{-8}\lambda^{-2{\varepsilon}}+\ell^{-1}\lambda^{-{\varepsilon}}+\ell^{-10}\lambda^{-2{\varepsilon}}\lesssim\ell^{-1}\lambda^{-{\varepsilon}}.$
(5.38)

##### Oscillation error.

In order to treat the delicate magnetic oscillations, we decompose

$\displaystyle\mathring{R}_{osc}^{B}=\mathring{R}_{osc.1}^{B}+\mathring{R}_{osc.2}^{B}+\mathring{R}_{osc.3}^{B}+\mathring{R}_{osc.4}^{B},$

where $\mathring{R}_{osc.1}^{B}$ contains the low-high spatial oscillations

$\displaystyle\mathring{R}_{osc.1}^{B}$
$\displaystyle:=\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(g_{(\tau)}^{2}\mathbb{P}_{\neq 0}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)})\nabla(a_{(k)}^{2})\right),$

$\mathring{R}_{osc.2}^{B}$ contains the high temporal oscillation

$\displaystyle\mathring{R}_{osc.2}^{B}$
$\displaystyle:=-\mu^{-1}\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(\partial_{t}(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}\right),$

$\mathring{R}_{osc.3}^{B}$ is of low frequency

$\displaystyle\mathring{R}_{osc.3}^{B}$
$\displaystyle:=-\sigma^{-1}\sum_{k\in\Lambda_{B}}\mathcal{R}^{B}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(h_{(\tau)}{\fint}_{{\mathbb{T}}^{3}}D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\partial_{t}\nabla(a_{(k)}^{2})\right),$

and $\mathring{R}_{osc.4}^{B}$ contains the interaction osillation

$\displaystyle\mathring{R}_{osc.4}^{B}$ $\displaystyle:=\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)\mathcal{R}^{B}\mathbb{P}_{H}{\mathrm{div}}\left(a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}(D_{(k^{\prime})}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k^{\prime})})\right).$

In order to estimate $\mathring{R}_{osc.1}^{B}$, the key fact is that, the
velocity and magnetic flows are of high oscillations

$\displaystyle\mathbb{P}_{\not=0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes
D_{(k)})=\mathbb{P}_{\geq\frac{1}{2}\lambda r_{\perp}}(D_{(k)}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k)}),$

while the amplitude function $a_{(k)}$ is slowly varying. Hence, intuitively,
the frequency of $\mathring{R}^{B}_{osc.1}$ concentrates at the high mode, and
thus the inverse-divergence operator $\mathcal{R}^{B}$ permits to gain a small
factor $(\lambda r_{\perp})^{-1}$. This is the content of Lemma 5.3 below.

######  Lemma 5.3 ([54], Lemma 6; see also [13], Lemma B.1).

Let $a\in C^{2}\left(\mathbb{T}^{3}\right)$. For all $1<p<+\infty$ we have

$\left\||\nabla|^{-1}\mathbb{P}_{\neq 0}\left(a\mathbb{P}_{\geq
k}f\right)\right\|_{L^{p}\left(\mathbb{T}^{3}\right)}\lesssim
k^{-1}\left\|\nabla^{2}a\right\|_{L^{\infty}\left(\mathbb{T}^{3}\right)}\|f\|_{L^{p}\left(\mathbb{T}^{3}\right)},$

holds for any smooth function $f\in L^{p}\left(\mathbb{T}^{3}\right)$.

We apply Lemma 5.3 with $a=\nabla(a_{(k)}^{2})$ and
$f=\psi_{(k_{1})}^{2}\phi_{(k)}^{2}$ to get

$\displaystyle\lVert\mathring{R}_{osc.1}^{B}\rVert_{L^{1}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\sum_{k\in\Lambda_{B}}\|g_{(\tau)}\|_{L^{2}_{t}}^{2}\lVert|\nabla|^{-1}\mathbb{P}_{\not=0}\left(\mathbb{P}_{\geq(\lambda
r_{\perp}/2)}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes
D_{(k)})\nabla(a_{(k)}^{2})\right)\rVert_{C_{t}L^{p}_{x}}$
$\displaystyle\lesssim\sum_{k\in\Lambda_{B}}\lambda^{-1}r_{\perp}^{-1}\lVert{\nabla}^{3}(a^{2}_{(k)})\rVert_{C_{t,x}}\lVert\psi_{(k_{1})}^{2}\phi_{(k)}^{2}\rVert_{C_{t}L^{p}_{x}}$
$\displaystyle\lesssim\sum_{k\in\Lambda_{B}}\ell^{-14}\lambda^{-1}r_{\perp}^{-1}\lVert\psi^{2}_{(k_{1})}\rVert_{C_{t}L^{p}_{x}}\lVert\phi^{2}_{(k)}\rVert_{C_{t}L^{p}_{x}}$
$\displaystyle\lesssim\ell^{-14}\lambda^{-1}r_{\perp}^{\frac{1}{p}-2}r_{\parallel}^{\frac{1}{p}-1},$
(5.39)

where we also used Lemmas 3.3 and 4.1 in the third step.

Regarding $\mathring{R}_{osc.2}^{B}$, as mentioned in Remark 4.4, the high
temporal oscillations arising from $g_{(\tau)}$ can be balanced by the large
parameter $\mu$, namely, we have

$\displaystyle\lVert\mathring{R}_{osc.2}^{B}\rVert_{L^{1}_{t}L_{x}^{p}}$
$\displaystyle\lesssim{\mu}^{-1}\sum_{k\in\Lambda_{B}}\lVert|\nabla|^{-1}\mathbb{P}_{H}\mathbb{P}_{\neq
0}\left(\partial_{t}(a_{(k)}^{2}g_{(\tau)}^{2})\psi_{(k_{1})}^{2}\phi_{(k)}^{2}k_{2}\right)\rVert_{L^{1}_{t}L_{x}^{p}}$
$\displaystyle\lesssim{\mu}^{-1}\sum_{k\in\Lambda_{B}}\left(\lVert\partial_{t}(a_{(k)}^{2})\rVert_{C_{t,x}}\lVert
g_{(\tau)}^{2}\rVert_{L^{1}_{t}}+\lVert
a_{(k)}\rVert_{C_{t,x}}^{2}\lVert\partial_{t}(g_{(\tau)}^{2})\rVert_{L_{t}^{1}}\right)\lVert\psi_{(k_{1})}\rVert_{C_{t}L^{2p}_{x}}^{2}\lVert\phi_{(k)}\rVert_{L^{2p}_{x}}^{2}$
$\displaystyle\lesssim(\ell^{-6}+\ell^{-2}\tau\sigma)\mu^{-1}r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{1}{p}-1}\lesssim\ell^{-6}\tau\sigma\mu^{-1}r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{1}{p}-1}.$
(5.40)

The low frequency part $\mathring{R}_{osc.3}^{B}$ can be estimated easily by
(3.38) and (4.13),

$\displaystyle\lVert\mathring{R}_{osc.3}^{B}\rVert_{L^{1}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\sigma^{-1}\sum_{k\in\Lambda_{B}}\left\|h_{(\tau)}(k_{2}\otimes
k_{1}-k_{1}\otimes
k_{2})\partial_{t}\nabla(a_{(k)}^{2})\right\|_{L^{1}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\sigma^{-1}\sum_{k\in\Lambda_{B}}\|h_{(\tau)}\|_{C_{t}}\left(\lVert
a_{(k)}\rVert_{C_{t,x}}\lVert a_{(k)}\rVert_{C_{t,x}^{2}}+\lVert
a_{(k)}\rVert_{C_{t,x}^{1}}^{2}\right)$
$\displaystyle\lesssim\ell^{-9}\sigma^{-1}.$ (5.41)

Finally, thanks to the small interactions between different intermittent
flows, the interaction oscillation $\mathring{R}_{osc.4}^{B}$ can be
controlled by the product estimate in Lemma 3.4,

$\displaystyle\lVert\mathring{R}_{osc.4}^{B}\rVert_{L^{1}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)\lVert
a_{(k)}a_{(k^{\prime})}g_{(\tau)}^{2}(D_{(k^{\prime})}\otimes
W_{(k)}-W_{(k)}\otimes D_{(k^{\prime})})\rVert_{L^{1}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\left(\sum_{k\neq
k^{\prime}\in\Lambda_{B}}+\sum_{k\in\Lambda_{u},k^{\prime}\in\Lambda_{B}}\right)\lVert
a_{(k)}\rVert_{C_{t,x}}\lVert a_{(k^{\prime})}\rVert_{C_{t,x}}\lVert
g_{(\tau)}^{2}\rVert_{L^{1}_{t}}\|\psi_{(k_{1})}\phi_{(k)}\psi_{(k^{\prime}_{1})}\phi_{(k^{\prime})}\|_{C_{t}L^{p}_{x}}$
$\displaystyle\lesssim\ell^{-2}r_{\perp}^{\frac{1}{p}-1}r_{\parallel}^{\frac{2}{p}-1}\,.$
(5.42)

Therefore, putting estimates (5.3)-(5.3) altogether and using (3.1), (5.27) we
obtain

$\displaystyle\lVert\mathring{R}_{osc}^{B}\rVert_{L_{t}^{1}L^{p}_{x}}$