# Non-uniqueness of weak solutions for a logarithmically supercritical
hyperdissipative Navier-Stokes system

Marco Romito  Dipartimento di Matematica, Università di Pisa, Largo Bruno
Pontecorvo 5, I–56127 Pisa, Italia<EMAIL_ADDRESS>and  Francesco
Triggiano  Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa,
Italia<EMAIL_ADDRESS>
(Date: June 11, 2024)

###### Abstract.

The existence of non-unique solutions of finite kinetic energy for the three
dimensional Navier-Stokes equations is proved in the slightly supercritical
hyper-dissipative setting introduced by Tao [Tao10]. The result is based on
the convex integration techniques of Buckmaster and Vicol [BV19], and extends
Luo and Titi [LT20] in the slightly supercritical setting. To be able to be
closer to the threshold identified by Tao, we introduce the impulsed Beltrami
flows, a variant of the intermittent Beltrami flows of Buckmaster and Vicol.

###### Key words and phrases:

Navier-Stokes equations, convex integration, non-uniqueness, slightly
supercritical hyper-dissipation

###### 2020 Mathematics Subject Classification:

Primary 35Q30; secondary 76D03, 76D05

##  1\. Introduction

In this work we consider the following hyperdissipative Navier-Stokes system
on the 3-dimensional torus

(1.1) $\begin{cases}\partial_{t}u+\operatorname{div}(u\otimes
u)+\nu(-\Delta)^{\theta,\beta}u+\nabla p=0,\\\
\operatorname{div}(u)=0,\end{cases}$

with periodic boundary conditions and zero spatial mean, where $\beta$ and
$\theta$ are non-negative parameters, and the dissipative term is defined via
Fourier transform $\mathcal{F}$ by

$\mathcal{F}[(-\Delta)^{\theta,\beta}u](k)=\mathbf{1}_{\\{k\neq
0\\}}(k)\frac{|k|^{2\theta}}{\log^{\beta}(10+|k|_{1})}\mathcal{F}(u)(k),\qquad\text{for
all }k\in\mathbb{Z}^{3},$

where $|k|$ and $|k|_{1}$ are the 2-norm and the 1-norm of $k$, respectively.

Well-posedness of a similar system has been widely studied in the last
century. In particular, it is well known that global existence and uniqueness
of sufficiently smooth solutions holds when $\beta=0$ and
$\theta\geq\frac{5}{4}$ [Lio69, KP02].

On the critical threshold $\theta=\frac{5}{4}$, the same result has been
proved by Tao [Tao10] and Barbato et al. [BMR15] when the dissipative term is
slightly weakened by a Fourier multiplier that mimics a logarithmic behaviour
raised to a sufficiently small power $\beta$.

When $\theta<\frac{5}{4}$ it is well known that there are various open
problems, such as uniqueness of Leray-Hopf solutions for classical NSE, i.e.
$\theta=1$ and $\beta=0$.

Furthermore, various negative results appeared during the last decade. In
particular, Luo et al. [LT20] proved non-uniqueness of global distributional
solutions, less regular than Leray weak solutions [Ler34], by means of the
convex integration theory [BV19, BV20, DLS14, Ise18, MS18]. The convex
integration machinery has been used by Li et al. [LQZZ22] in order to prove
non-uniqueness in suitable functional spaces also above the Lions exponent,
i.e. $\theta\geq\frac{5}{4}$ and $\beta=0$, but non-uniqueness is proved in
the space $C_{t}L^{p}$ for all $1\leq p<2$, thus in larger spaces than energy
(in general below the Ladyzenskaja-Prodi-Serrin threshold).

In this paper we work at the critical threshold $\theta=\frac{5}{4}$, whose
relevance can be also highlighted by making some scaling considerations. See
[Kla10] for a more complete discussion on the topic. Under the assumption
$\beta=0$, we recall that a functional space $\mathbf{X}$ is said to be a
critical space if $\|u\|_{\mathbf{X}}$ is invariant under the following
scaling transformation:

$u(t,x)\to\lambda^{2\theta-1}u(\lambda^{2\theta}t,\lambda x),\qquad
p(t,x)\to\lambda^{4\theta-2}p(\lambda^{2\theta}t,\lambda x).$

Since its norm represents the kinetic energy, one important functional space
is $\mathbf{X}=C_{t}L^{2}_{x}$, which is a critical space only if
$\theta=\frac{5}{4}$.

In this work we show that the choice $\theta=\frac{5}{4}$ and $\beta$ large
enough, that is weakening the dissipation by a logarithmic corrector, allows
to prove a non-uniqueness result for distributional solutions to Eq. 1.1 that
are in the relevant functional space $C_{t}L^{2}_{x}$. This non-uniqueness
result in a logarithmic supercritical case is obtained by stretching as much
as possible the convex integration scheme proposed in [BV19, LT20]. In
particular from a technical point of view, the proof is mainly based on
combining new estimates for various Fourier multipliers, which allow to
exploit the logarithmic corrector, and classical harmonic analysis results in
some Orlicz spaces. Moreover, we also propose a new modification of the
classical Beltrami flows to achieve the result by assuming a weaker condition
on $\beta$. For more details see the discussion in 2.4.

The paper is organized as follows. In Section 2 we state and prove the main
result by means of the iteration lemma. In Section 3 we introduce and study
various Fourier multipliers. Section 4 is devoted to the proof of the
iteration lemma, namely the inductive step which solves Eq. 1.1 in the limit.

###  1.1. Notations

To ease the notations, for any $\alpha\geq 0$ and $p\in[1,\infty]$ we denote

$\displaystyle C_{t}L^{p}_{x}$
$\displaystyle=C([0,\infty),L^{p}(\mathbf{T}^{3},\mathbb{R}^{3})),$
$\displaystyle C_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}$
$\displaystyle=C([0,\infty),\mathrm{L}(\mathrm{logL})^{\alpha}(\mathbf{T}^{3},\mathbb{R}^{3})),$
$\displaystyle L^{\infty}_{t}L^{p}_{x}$
$\displaystyle=L^{\infty}([0,\infty),L^{p}(\mathbf{T}^{3},\mathbb{R}^{3})),$
$\displaystyle L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}$
$\displaystyle=L^{\infty}([0,\infty),\mathrm{L}(\mathrm{logL})^{\alpha}(\mathbf{T}^{3},\mathbb{R}^{3})),$

where $\mathbf{T}^{3}$ denotes the 3-dimensional torus and
$\mathrm{L}(\mathrm{logL})^{\alpha}$ is the Orlicz space associated with the
Young function $A(s)=s\log^{\alpha}(2+s)$ (see 3.1 for the detailed
definition). We denote by
$C_{w}(\mathbb{R},L^{2}(\mathbf{T}^{3},\mathbb{R}^{3}))$ the space of weakly
continuous functions with values within the Banach space
$L^{2}(\mathbf{T}^{3},\mathbb{R}^{3})$, while
$C^{\infty}_{c}(\mathbb{R},C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3}))$ is the
space of smooth compactly supported and spatially periodic functions. In
addition, if $v\in
C^{\infty}_{c}(\mathbb{R},C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3}))$ we
denote its time support by $\operatorname{Supp}_{t}(v)$. For any sub-interval
$I\subset\mathbb{R}_{+}$, let $\mathbf{P}_{I}$ be the operator defined by

$\mathbf{P}_{I}g(x)=\sum_{|k|_{1}\in I}\mathcal{F}(g)(k)e^{ik\cdot x}.$

We denote by $P_{H}$ the Leray-Helmholtz projector, namely the projector onto
the space of divergence-free vector fields. For any $A\subset\mathbb{R}$ and
$\delta>0$, let $N_{\delta}(A)$ be the $\delta$-neighborhood of A, namely

$N_{\delta}(A)=\\{z\in\mathbb{R}:\text{ there is }y\in A\text{ s.t.
}|z-y|<\delta\\}.$

If $x,y\in\mathbb{R}^{3}$, then $x\times y$ will denote the standard vector
cross-product between $x$ and $y$. Given $a,b\in\mathbb{R}$, we use the
notations $a\lesssim b$ and $a\approx b$ if there exist $C,C_{1},C_{2}>0$ such
that $a\leq Cb$ and $C_{1}a\leq b\leq C_{2}a$, respectively.

##  2\. Main result

We first recall the definition of weak solution to Eq. 1.1.

######  Definition 2.1.

A vector field $v\in C_{w}(\mathbb{R},L^{2}(\mathbf{T}^{3},\mathbb{R}^{3}))$
is a weak solution to Eq. 1.1 if it is satisfied in a distributional sense,
namely,

$\int_{\mathbb{R}}\int_{\mathbf{T}^{3}}(\partial_{t}\phi)\cdot
v\,dx\,dt=\int_{\mathbb{R}}\int_{\mathbf{T}^{3}}v\cdot(v\cdot\nabla)\phi-\big{(}\nu(-\Delta)^{\theta,\beta}\phi\big{)}\cdot
v\,dx\,dt,$

for all $\phi\in
C^{\infty}_{c}(\mathbb{R},C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3}))$ such
that $\operatorname{div}(\phi(t,\cdot))=0$ for all $t\in\mathbb{R}$.

######  Theorem 2.2.

For $\theta=\frac{5}{4}$ and $\beta>\frac{29}{2}$, consider a smooth
divergence-free vector field
$u:[0,\infty)\times\mathbf{T}^{3}\to\mathbb{R}^{3}$, with compact support in
time and zero spatial-mean,

$\int_{\mathbf{T}^{3}}u(t,x)dx=0.$

Then, for all $\epsilon>0$, there exists a weak solution $v\in
C(\mathbb{R},L^{2}(\mathbf{T}^{3},\mathbb{R}^{3}))$ of Eq. 1.1 with compact
support in time and such that

$\||\nabla|^{\frac{3}{2}}T_{M}(v-u)\|_{L^{\infty}_{t}L^{1}_{x}}\leq\epsilon,$

where $T_{M}$ and $|\nabla|^{\frac{3}{2}}$ are the operators corresponding to
the Fourier multipliers $M(k)=\mathbf{1}_{\\{k\neq
0\\}}(k)\log^{-\beta}(|k|_{1}+10)$ and $m^{\prime}(k)=|k|^{\frac{3}{2}}$ (see
3.3). In particular, there are infinitely many weak solutions with zero
initial condition.

The result is shown by using the classical iterative procedure of the convex
integration theory, i.e. following the groundbreaking work of Buckmaster and
Vicol [BV19]. Therefore, the proof is strongly based on the following lemma.

######  Lemma 2.3 (Iteration Lemma).

Fix $\theta=\frac{5}{4}$ and $\beta>\frac{29}{2}$. let $(v_{q},p_{q},R_{q})$
be a smooth solution to the approximate system

(2.1) $\begin{cases}\partial_{t}u+\operatorname{div}(u\otimes
u)+\nu(-\Delta)^{\frac{5}{4},\beta}u+\nabla p=\operatorname{div}(R),\\\
\operatorname{div}(u)=0,\\\ \operatorname{Tr}(R(t,x))=0,\qquad
R(t,x)=R(t,x)^{T}\quad\text{for all }t,x,\end{cases}$

such that

$\|R_{q}\|_{L^{\infty}_{t}L^{1}_{x}}\leq\delta_{q+1}.$

Then for every $\delta_{q+2}>0$, there exists a smooth solution
$(v_{q+1},p_{q+1},R_{q+1})$ of Eq. 2.1 such that

(2.2)
$\begin{gathered}\|R_{q+1}\|_{L^{\infty}_{t}L^{1}_{x}}\leq\delta_{q+2},\\\
\operatorname{Supp}_{t}(v_{q+1})\cup\operatorname{Supp}_{t}(R_{q+1})\subset
N_{\delta_{q+1}}(\operatorname{Supp}_{t}(v_{q})\cup\operatorname{Supp}_{t}(R_{q})).\end{gathered}$

Moreover, the following estimates hold

(2.3) $\displaystyle\|v_{q+1}-v_{q}\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\leq C\delta_{q+1}^{\frac{1}{2}},$
$\displaystyle\||\nabla|^{\frac{3}{2}}T_{M}(v_{q+1}-v_{q})\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\leq\delta_{q+2}.$

######  Remark 2.4 (On the parameter $\beta$).

The requirement on the parameter $\beta$ will be clear in the proof of Lemma
2.3. In particular, it is required that $\beta>\frac{29}{2}$. This specific
value is due to various harmonic analysis estimates (in negative, forcing a
larger value of $\beta$), which are mainly shown in Section 3, and to a
suitable construction of the building blocks of the convex integration scheme
(in positive, lowering the minimal threshold we impose on $\beta$), which
differs from the intermittent Beltrami flows [BV19] by a logarithmic factor
(see 4.3).

As already mentioned in the introduction, Tao [Tao10] and Barbato et al.
[BMR15] proved the existence and uniqueness of smooth solutions to an equation
similar to Eq. 1.1, where the dissipative term $D(u)$ is defined via Fourier
transform as

$\mathcal{F}[D(u)](k)=\mathbf{1}_{\\{k\neq
0\\}}(k)\frac{|k|^{2\theta}}{g(|k|)}\mathcal{F}(u)(k),\qquad\text{for all
}k\in\mathbb{Z}^{3},$

for any positive, non-decreasing function $g$ such that
$\int_{1}^{\infty}\frac{1}{sg(s)}ds=\infty$. This operator differs from our
dissipative term in two aspects. Firstly, it presents a corrector computed on
$|k|$, while our works on $|k|_{1}$. This difference is due to the fact that
we need to estimate the $\mathcal{L}(L^{1},L^{1})$-norm of the operator
associated with the Fourier multiplier
$log^{-\beta}(10+|k|_{1})\mathbf{P}_{\geq N}$ (see 3.4 and 3.5) in order to
exploit completely the Fourier properties of our building blocks. The second
difference is the presence of the function $g$. Even if their result covers a
larger family of dissipative terms, the most interesting one is clearly given
by $g(s)=log^{\beta}(10+|k|)$ with $\beta\leq 1$.

Therefore, apart from the structural differences between our dissipative term
and the one adopted in [Tao10, BMR15], it can be interesting to study why
their constraint on $\beta$ is adimensional while ours depends on the
dimension (see 3.4 and 3.5) and to understand if additional technical
novelties could allow to weaken our assumption on the $\beta$ parameter in
order to get closer to the threshold identified by their results.

Assuming the previous Lemma, the proof of 2.2 basically follows the lines of
Luo and Titi [LT20]. We report it for completeness.

###### Proof of 2.2.

Consider the triple

$\displaystyle v_{0}=u,\qquad p_{0}=-\frac{1}{3}|v_{0}|^{2},$ $\displaystyle
R_{0}=\mathcal{R}(\partial_{t}v_{0}+\nu(-\Delta)^{\frac{5}{4},\beta}v_{0})+v_{0}\otimes
v_{0}+p_{0}I,$

where $\mathcal{R}$ is the anti-divergence operator, defined below in 4.10.

Given $\delta_{1}=\|R_{0}\|_{L^{\infty}_{t}L^{1}_{x}}$ and
$\delta_{q}=2^{-q}\epsilon$, consider the sequence
$(v_{q},p_{q},R_{q})_{q\in\mathbb{N}}$ coming from Lemma 2.3. Then,

$\displaystyle\sum_{q}\|v_{q+1}-v_{q}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\sum_{q}\delta_{q+1}^{\frac{1}{2}}<\infty,$
$\displaystyle\|R_{q}\|_{L^{\infty}_{t}L^{1}_{x}}\to 0\qquad\text{as
}q\to\infty.$

Therefore, $(v_{q})_{q}$ converges strongly to a weak solution $v\in
C_{T}L^{2}_{x}$. Moreover,

$\displaystyle\||\nabla|^{\frac{3}{2}}T_{M}(v-u)\|_{L^{\infty}_{t}L^{1}_{x}}\leq\sum\||\nabla|^{\frac{3}{2}}T_{M}(v_{q+1}-v_{q})\|_{L^{\infty}_{t}L^{1}_{x}}\leq\sum_{q}\delta_{q+1}\leq\epsilon,$
$\displaystyle\operatorname{Supp}_{t}(v)\subset\cup_{q}\operatorname{Supp}_{t}(v_{q})\subset
N_{\sum_{q}\delta_{q+1}}(\operatorname{Supp}_{t}(u))\subset
N_{\delta_{1}+\epsilon}\operatorname{Supp}_{t}(u).$

Finally, infinitely many weak solutions with zero initial condition can be
obtained by considering $u(t,x)=\phi(t)\sum_{|k|\leq N}a_{k}e^{ik\cdot x}$
with $a_{0}=0$, $a_{k}^{*}=a_{-k}$, $a_{k}\cdot k=0$ and $\phi\in
C^{\infty}(\mathbb{R}_{+})$, with $\phi(0)=0$. ∎

##  3\. Preliminaries on Fourier multipliers

In this section we mainly prove that the operators associated with various
Fourier multipliers are bounded from $\mathrm{L}(\mathrm{logL})^{\beta}$ to
$\mathrm{L}(\mathrm{logL})^{\alpha}$ for suitable $\alpha,\beta\geq 0$. Let us
start by defining the functional space $\mathrm{L}(\mathrm{logL})^{\alpha}$
and by proving an elementary inequality.

######  Definition 3.1.

Given $0\leq\alpha<\infty$ and $d\in\mathbb{N}$, consider the space

$\mathrm{L}(\mathrm{logL})^{\alpha}=\Bigl{\\{}f:\mathbf{T}^{3}\to\mathbb{R}^{d}\text{
measurable
}:\int_{\mathbf{T}^{3}}|f(x)|\log^{\alpha}(2+|f(x)|)\,dx<\infty\Bigr{\\}}.$

This is a Banach space when equipped with the so-called _Luxemburg norm_

(3.1)
$\|f\|_{\mathrm{L}(\mathrm{logL})^{\alpha}}=\inf\Bigl{\\{}\lambda>0:\int_{\mathbf{T}^{3}}A(|f(x)|/\lambda)dx\leq
1\Bigr{\\}},$

where $A(s)=s\log^{\alpha}(2+s)$ for all $s\geq 0$.

######  Lemma 3.2.

Consider a scalar function $f\in L^{\infty}(\mathbf{T}^{3})$ and
$g\in\mathrm{L}(\mathrm{logL})^{\alpha}(\mathbf{T}^{3},\mathbb{R}^{d})$ for
$\alpha\geq 0$. Then

$\|fg\|_{\mathrm{L}(\mathrm{logL})^{\alpha}}\leq\|f\|_{L^{\infty}}\|g\|_{\mathrm{L}(\mathrm{logL})^{\alpha}}.$

###### Proof.

If $A(s)=s\log^{\alpha}(2+s)$, then

$\int_{\mathbf{T}^{3}}A\Big{(}\frac{|fg|}{\|f\|_{L^{\infty}}\|g\|_{\mathrm{L}(\mathrm{logL})^{\alpha}}}\Big{)}dx\leq\int_{\mathbf{T}^{3}}A\Big{(}\frac{|g|}{\|g\|_{\mathrm{L}(\mathrm{logL})^{\alpha}}}\Big{)}dx\leq
1,$

where the first inequality follows since $A$ is non-decreasing, and the second
follows from the definition of the Luxemburg norm. ∎

Now, we first define the operator corresponding to a Fourier multiplier. Then
we will focus on specific multipliers.

######  Definition 3.3.

Given a family $m=(m(k))_{k\in\mathbb{Z}^{3}}$, its corresponding operator
$T_{m}$ is defined by

$T_{m}(f)(x)=\sum_{k\in\mathbb{Z}^{3}}m(k)\mathcal{F}(f)(k)e^{ik\cdot
x},\qquad\text{ for }f\in C^{\infty}(\mathbf{T}^{3}),$

where $\mathcal{F}(f)(k)$ is the $k^{\text{th}}$ Fourier coefficient of $f$.

######  Proposition 3.4.

Consider $m(k)=\mathbf{1}_{\\{k\neq 0\\}}(k)\frac{1}{|k|_{1}}$, then $T_{m}$
is bounded from $L^{1}(\mathbf{T}^{3})$ to itself.

Moreover, for all $N\in\mathbb{N}$,

(3.2)
$\|\mathbf{P}_{|k|_{1}>N}T_{m}\|_{\mathcal{L}(L^{1},L^{1})}\lesssim\frac{1}{N}\log^{3}(N),$

where $\mathcal{L}(L^{1},L^{1})$ is the space of bounded and linear operators
from $L^{1}(\mathbf{T}^{3})$ to itself.

###### Proof.

Consider $f_{N}(x)=\sum_{|k|_{1}\leq N}\frac{1}{|k|_{1}}e^{ik\cdot x},$ then
the following formula holds,

$f_{N}(x)=\sum_{j=1}^{N-1}D^{1}_{j}(x)\Bigl{(}\frac{1}{j}-\frac{1}{j+1}\Bigr{)}+D^{1}_{N}(x)\frac{1}{N},$

where $D^{1}_{j}(x)=\sum_{|k|_{1}\leq j}e^{ik\cdot x}$.

Therefore, for $N<M$,

$\|f_{N}-f_{M}\|_{L^{1}}\leq\frac{1}{N}\|D_{N}^{1}\|_{L^{1}}+\frac{1}{M}\|D_{M}^{1}\|_{L^{1}}+\sum_{j=N}^{M-1}\|D^{1}_{j}\|_{L^{1}}\big{(}\frac{1}{j}-\frac{1}{j+1}\big{)}.$

It is known that $\|D^{1}_{j}\|_{L^{1}}\lesssim\log^{3}(j)$ (see [KL21]), so

$\|f_{N}-f_{M}\|_{L^{1}}\leq\frac{1}{N}\log^{3}(N)+\frac{1}{M}\log^{3}(M)+\sum_{j=N}^{M-1}\frac{\log^{3}(j)}{j(j+1)}\lesssim\frac{1}{N}\log^{3}(N).$

The sequence $(f_{N})_{N}$ is a Cauchy sequence in $L^{1}$ and converges to
$f$ such that $\nu(A)=\int_{A}f(x)\,dx$ is a finite Borel measure on the torus
which has $m(k)$ as Fourier coefficients. The existence of the measure $\nu$
implies that $m$ is a $(1,1)$-multiplier and that
$\|T_{m}\|_{\mathcal{L}(L^{1},L^{1})}\leq|\nu|=\|f\|_{L^{1}}$, where $|\nu|$
is the total variation of the measure $\nu$ (see [G+08]). At last, by the same
argument, the estimate on
$\|\mathbf{P}_{|k|_{1}>N}T_{m}\|_{\mathcal{L}(L^{1},L^{1})}$ follows from

$\|f-f_{N}\|_{L^{1}}=\lim_{M\to\infty}\|f_{M}-f_{N}\|_{L^{1}}\leq\frac{1}{N}\log^{3}(N).\qed$

######  Remark 3.5.

By repeating the same argument above, the conclusions of 3.4 hold for
$M(k)=\mathbf{1}_{k\neq 0}(k)\frac{1}{\log^{\beta}(|k|_{1}+10)}$, with
$\beta>3$, and

(3.3)
$\|\mathbf{P}_{|k|_{1}>N}T_{M}\|_{\mathcal{L}(L^{1},L^{1})}\lesssim\frac{\log^{3}(N)}{\log^{\beta}(10+N)}.$

We notice though that the same machinery works for
$m(k)=\frac{1}{|k|_{\infty}}$, but not for $m(k)=\frac{1}{|k|}$ because of
$\|D_{j}^{2}\|_{L^{1}}\approx j^{\frac{3-1}{2}}$ where
$D^{2}_{j}(x)=\sum_{|k|\leq j}e^{ik\cdot x}$, see for instance [Sha75].

These results clearly hold in any dimension $d$ and make use of a
multidimensional version of Abel summation formula. In addition, the
appearance of a factor $\log^{d}(N)$ is coherent with various results obtained
for $L^{1}$-convergence of double Fourier series [Mor94].

######  Proposition 3.6.

Given $j\in\\{1,2,3\\}$ and the multiplier $m(k)=\mathbf{1}_{\\{k\neq
0\\}}(k)i[\operatorname{sgn}(k_{j})]$, then $T_{m}$ is bounded from
$\mathrm{L}(\mathrm{logL})^{\alpha+1}$ to $\mathrm{L}(\mathrm{logL})^{\alpha}$
for all $\alpha\geq 0$.

###### Proof.

It is known that if an operator $T$ is bounded from $L^{2}$ to itself and from
$L^{1}$ to the weak $L^{1}$ space $L^{1,\infty}$, then it is bounded from
$\mathrm{L}(\mathrm{logL})^{\alpha+1}$ to $\mathrm{L}(\mathrm{logL})^{\alpha}$
for all $\alpha\geq 0$ (see Theorem 4.4.11 and Theorem 4.6.16 in [BS88]).

Since $m$ is bounded, then $T_{m}$ is bounded from $L^{2}(\mathbf{T}^{3})$ to
itself. Now, we will conclude the proof by proving that $T_{m}$ is bounded
from $L^{1}$ to $L^{1,\infty}$.

In the remaining part of the proof $|A|$ denotes the Lebesgue measure of any
Borel set $A$. It can be easily proved that

$T_{m}(f)(t)=\frac{1}{4\pi}\,\text{p.v.}\int_{-\pi}^{\pi}f(t-\theta
e_{j})\arctan\Big{(}\frac{\theta}{2}\Big{)}\,d\theta\\\
:=\frac{1}{4\pi}\lim_{\epsilon\to 0}\int_{\epsilon<|\theta|<\pi}f(t-\theta
e_{j})\arctan\Big{(}\frac{\theta}{2}\Big{)}\,d\theta,$

where $e_{j}$ denotes the $j^{\text{th}}$ element of the canonical basis.

By a density argument, we can assume that $f$ is a finite linear combination
of characteristic functions of disjoint dyadic intervals in $[-\pi,\pi]^{3}$.
Given $\alpha>0$, then by the Calderon-Zygmund decomposition (see [G+08]),
there exist $g,b$ such that $\hat{f}=g+b$, where $\hat{f}$ is an extension of
$f$ at $[-\pi,\pi]^{3}\cup([-\pi,\pi]^{3}+2\pi e_{j})\cup([-\pi,\pi]^{3}-2\pi
e_{j})$ by periodicity. In addition, the following properties hold,

  1. (1)

$\|g\|_{L^{1}}\lesssim\|f\|_{L^{1}}$, $\|g\|_{L^{\infty}}\lesssim\alpha$,

  2. (2)

$b=\sum_{j}b_{j}$, where the $b_{j}$ functions are supported on pairwise
disjoint dyadic cubes $Q_{j}$,

  3. (3)

$\int_{Q_{j}}b_{j}(x)dx=0,$ $\|b_{j}\|_{L^{1}}\lesssim\alpha|Q_{j}|$ and
$\sum_{j}|Q_{j}|\lesssim\frac{1}{\alpha}\|f\|_{L^{1}}$,

  4. (4)

The particular structure of $f$ implies that there is a finite number of
$b_{j}$’s functions.

For all $j$, define

$\overline{Q_{j}}=\\{x\in\mathbb{R}^{3}:\text{there is
}\epsilon\in[-l(Q_{j}),l(Q_{j})]\text{ s.t. }x+\epsilon e_{j}\in Q_{j}\\},$

where $l(Q_{j})=|Q_{j}|^{\frac{1}{3}}$ is the length of the side of $Q_{j}$.
Now,

$\displaystyle|\\{x\in[-\pi,\pi]^{3}:|T_{m}(f)(x)|>\alpha\\}|\leq$
$\displaystyle\leq|\\{x\in[-\pi,\pi]^{3}:|T_{m}(g)(x)|>\tfrac{1}{2}\alpha\\}|+|\cup_{j}\overline{Q_{j}}|$
$\displaystyle\quad+|\\{x\in[-\pi,\pi]^{3}\cap(\cup_{j}\overline{Q_{j}})^{c}:|T_{m}(b)(x)|>\tfrac{1}{2}\alpha\\}|$
$\displaystyle\lesssim\alpha^{-1}\|f\|_{L^{1}}+|\\{x\in[-\pi,\pi]^{3}\cap(\cup_{j}\overline{Q_{j}})^{c}:|T_{m}(b)(x)|>\tfrac{1}{2}\alpha\\}|,$

where the second inequality follows from Markov’s inequality and the
properties satisfied by $b$ and $g$.

The last term on the right-hand side can be estimated by Markov’s inequality
and

$\int_{[-\pi,\pi]^{3}\cap(\cup_{j}\overline{Q_{j}})^{c}}|T_{m}(b)(x)|\,dx\leq\sum_{j}\int_{[-\pi,\pi]^{3}\cap(\overline{Q_{j}})^{c}}|T_{m}(b_{j})(x)|\,dx\\\
\leq\sum_{j}\int_{[-\pi,\pi]^{3}\cap(\overline{Q_{j}})^{c}}\int_{l(Q_{j})<|\theta|<\pi}\big{|}\arctan(\tfrac{\theta}{2})b_{j}(x-\theta
e_{j})\big{|}\,d\theta\,dx.$

Now if $l(Q_{j})\geq\frac{\pi}{4}$, then

$\int_{[-\pi,\pi]^{3}\cap(\overline{Q_{j}})^{c}}\int_{l(Q_{j})<|\theta|<\pi}\big{|}\arctan(\tfrac{\theta}{2})b_{j}(x-\theta
e_{j})\big{|}\,d\theta\,dx\lesssim\|b_{j}\|_{L^{1}}.$

Otherwise,

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\int_{[-\pi,\pi]^{3}\cap(\overline{Q_{j}})^{c}}\int_{l(Q_{j})<|\theta|<\pi}|\arctan(\tfrac{\theta}{2})b_{j}(x-\theta
e_{j})|\,d\theta\,dx\leq$\mbox{}\hfil\qquad&\\\
&=\int_{[-\pi,\pi]^{3}\cap(\overline{Q_{j}})^{c}}\Bigl{(}\int_{l(Q_{j})<|\theta|<\frac{\pi}{4}+l(Q_{j})}|\arctan(\tfrac{\theta}{2})b_{j}(x-\theta
e_{j})|\,d\theta\\\
&\quad+\int_{\frac{\pi}{4}+l(Q_{j})<|\theta|<\pi}|\arctan(\tfrac{\theta}{2})b_{j}(x-\theta
e_{j})|\,d\theta\Bigr{)}\,dx\\\
&\lesssim\|b_{j}\|_{L^{1}}\Big{(}\int_{l(Q_{j})}^{\frac{\pi}{4}+l(Q_{j})}\arctan(\tfrac{\theta}{2})\,d\theta+\pi\arctan(\tfrac{\pi}{8})\Big{)}\\\
&\leq\|b_{j}\|_{L^{1}}\log(\frac{\sin(\frac{1}{2}(\frac{\pi}{4}+l(Q_{j})))}{\sin(\frac{1}{2}l(Q_{j}))})\\\
&\leq C\|b_{j}\|_{L^{1}};\end{split}$

since
$x\mapsto\log(\frac{\sin(\frac{1}{2}(\frac{\pi}{4}+x))}{\sin(\frac{1}{2}x)})$
is bounded on $[0,\frac{\pi}{4}]$. In conclusion, by the properties of the
$b_{j}$ functions, it follows that the remaining term is bounded from below by
$\frac{C}{\alpha}|f|_{L^{1}}$. ∎

######  Proposition 3.7.

Given $\alpha\in\mathbb{N}^{*}$ and
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{N}$ such that
$\alpha_{1}+\alpha_{2}+\alpha_{3}=\alpha$, consider the iterated Riesz
multiplier,

$m(k)=\mathbf{1}_{\\{k\neq
0\\}}(k)k_{1}^{\alpha_{1}}k_{2}^{\alpha_{2}}k_{3}^{\alpha_{3}}/|k|^{\alpha}.$

Then, $T_{m}$ is bounded from $\mathrm{L}(\mathrm{logL})^{\alpha+1}$ to
$\mathrm{L}(\mathrm{logL})^{\alpha}$ for all $\alpha\geq 0$.

###### Proof.

Similarly to the proof of the previous result, it is enough to show that $m$
is bounded and that $T_{m}$ is bounded from $L^{1}$ to $L^{1,\infty}$.

While the first requirement is trivial, i.e. the boundedness of $m$, instead,
the second one follows directly from Mihlin-Hormander theorem (see Remark 2.9
in [RW15]). ∎

##  4\. Proof of the Iteration Lemma

This section is devoted to the proof of Lemma 2.3, namely we modify a solution
$(v_{q},p_{q},R_{q})$ to Eq. 2.1 in order to obtain a new solution
$(v_{q+1},p_{q+1},R_{q+1})$ with $\|R_{q+1}\|_{L^{\infty}_{t}L^{1}_{x}}$
sufficiently small.

###  4.1. Impulsed Beltrami flows

In this section we build the impulsed Beltrami flows, a modification of the
intermittent Beltrami flows used by Buckmaster and Vicol [BV19], which will
have a crucial role in the construction of $v_{q+1}$.

Consider $\Lambda=\Lambda^{+}\cup\Lambda^{-}\subset\mathbb{Q}^{3}$, where

$\Lambda^{+}=\Bigl{\\{}\frac{1}{5}(3e_{1}\pm 4e_{2}),\frac{1}{5}(3e_{2}\pm
4e_{3}),\frac{1}{5}(3e_{3}\pm 4e_{1})\Bigr{\\}}.$

and $\Lambda^{-}:=-\Lambda^{+}$. Here $e_{1}$, $e_{2}$, $e_{3}$ is the
canonical basis of $\mathbb{R}^{3}$. Then, by construction, $\Lambda$ is a
finite symmetric subset of $\mathbb{Q}^{3}$, with vectors of unit length and
such that

$5\Lambda\subset\mathbb{Z}^{3},\qquad\min_{\xi,\xi^{\prime}\in\Lambda,\xi+\xi^{\prime}\neq
0}|\xi+\xi^{\prime}|\geq\frac{1}{5}.$

Moreover, the following geometric lemma holds:

######  Proposition 4.1.

$($[BV19, BV20]$)$ Let $B_{1}(I_{3})$ be the ball of symmetric $3\times 3$
matrices centered at $I_{3}$ with radius $1$. Then, there exist
$\epsilon_{\Lambda}>0$ and a family $(\gamma_{\xi})_{\xi\in\Lambda}\subset
C^{\infty}(B_{\epsilon_{\Lambda}}(I_{3}))$ such that

  1. (1)

$\gamma_{\xi}=\gamma_{-\xi}$;

  2. (2)

for each $R\in B_{\epsilon_{\Lambda}}(I_{3})$,

(4.1)
$R=\sum\limits_{\xi}\frac{1}{2}\gamma_{\xi}(R)^{2}(I_{3}-\xi\otimes\xi).$

Now, for any $\xi\in\Lambda$, consider $A_{\xi}\in\mathbb{Q}^{3}$ such that

$A_{\xi}\cdot\xi=0,\qquad A_{-\xi}=A_{\xi},\qquad|A_{\xi}|=1,$

and define $B_{\xi}=\frac{1}{\sqrt{2}}(A_{\xi}+i\xi\times A_{\xi})$.

Therefore for any $(a_{\xi})_{\xi\in\Lambda}\subset\mathbb{C}$ such that
$a_{-\xi}=\overline{a_{\xi}}$, we can define $W_{\xi}=B_{\xi}e^{i\lambda
x\cdot\xi}$ and consider the classical Beltrami flows
$W=\sum_{\xi}a_{\xi}W_{\xi}$ introduced in [BV19, BV20], which have various
interesting properties such as

$\fint_{\mathbf{T}^{3}}W\otimes
Wdx=\sum_{\xi}\frac{|a_{\xi}|^{2}}{2}(I_{3}-\xi\otimes\xi).$

The impulsed Beltrami flows are obtained by modifying the classical Beltrami
flows with suitable spatial and temporal intermittency. In particular, given
$r\in\mathbb{N}$, consider the Fejer kernel

$F(y)=\frac{1}{r+1}\Bigg{(}\frac{\sin((r+1)\frac{y}{2})}{\sin\frac{y}{2}}\Bigg{)}^{2}=\sum_{j=-r}^{r}\Bigg{(}1-\frac{|j|}{r+1}\Bigg{)}e^{ijy},$

for all $y\in\mathbf{T}$, and define the $3D$-Fejer kernel as

(4.2) $F_{r}(x)=(2\pi)^{\frac{3}{2}}\prod_{i=1}^{3}F(x_{i}).$

Let us summarise some properties of the $1D$-Fejer kernels, which can be
easily extended to $3D$-Fejer kernels using Fubini-Tonelli theorem. We point
out the crucial fact that, unlike the Dirichlet kernels, the $L^{1}$ estimate
does not contain logarithmic corrections.

######  Proposition 4.2.

Let r $\in 2\mathbb{N}^{*}$ and $n\in\mathbb{N}$, then

$\displaystyle\|F\|_{L^{1}}$ $\displaystyle=1,$ $\displaystyle\|F\|_{L^{2}}$
$\displaystyle\approx r^{1/2},$ $\displaystyle\|F\|_{L^{\infty}}$
$\displaystyle\lesssim r,$ $\displaystyle\|\tfrac{d^{n}}{dy^{n}}F\|_{L^{1}}$
$\displaystyle\lesssim r^{n},$
$\displaystyle\|\tfrac{d^{n}}{dy^{n}}F\|_{L^{2}}$ $\displaystyle\lesssim
r^{n+\frac{1}{2}},$ $\displaystyle\|\tfrac{d^{n}}{dy^{n}}F\|_{L^{\infty}}$
$\displaystyle\lesssim r^{n+1}.$

###### Proof.

The proof of the first three inequalities can be found in [G+08]. Consider
then the case $n>0$. The $L^{2}$ and $L^{\infty}$ norms can be easily
estimated, respectively, by Fourier coefficients and by the formula

$\sup_{[-\pi,\pi]}\Big{|}\frac{d^{n}}{dy^{n}}F(y)\Big{|}\lesssim\sum_{j=1}^{r}(1-\frac{j}{r+1})j^{n}.$

Instead, the estimate of the $L^{1}$ norm follows from the equalities:

$\displaystyle\frac{d}{dy}F(y)=\frac{2}{r+1}\frac{\sin((r+1)\frac{y}{2})}{\sin\frac{y}{2}}\frac{d}{dy}\bigg{[}\frac{\sin((r+1)\frac{y}{2})}{\sin\frac{y}{2}}\bigg{]}=\frac{2}{r+1}D_{k}(y)\frac{d}{dy}D_{k}(y),$
$\displaystyle\frac{d^{n}}{dy^{n}}F(y)=\frac{1}{r+1}\sum_{j=0}^{n-1}C(n,j)\Bigl{(}\frac{d^{j}}{dy^{j}}D_{k}(y)\Bigr{)}\Bigl{(}\frac{d^{n-j}}{dy^{n-j}}D_{k}(y)\Bigr{)},$

where $C(n,j)$ are constants depending on $n$ and $j$, $k=\frac{r}{2}$ and
$D_{k}$ is the Dirichlet kernel, i.e.

$D_{k}(y)=\frac{\sin((k+\frac{1}{2})y)}{\sin(\frac{y}{2})}=\sum_{j=-k}^{k}e^{ijy}.$

Indeed, the $L^{1}$ norm of the derivatives can be estimated by means of the
previous equalities, Hölder inequality and since
$\|\frac{d^{n}}{dy^{n}}D_{k}(y)\|_{L^{2}}\approx k^{n}k^{1/2}$. ∎

For any $\xi\in\Lambda$, its corresponding impulsed Beltrami flow is defined
by

(4.3) $\mathbf{W}_{\xi}=\eta_{\xi}W_{\xi},$

where for every $x\in\mathbf{T}^{3}$, $t\geq 0$ and $\xi\in\Lambda^{+}$,

(4.4) $\displaystyle M_{r}^{\xi}(x,t)$
$\displaystyle=F_{r}(\lambda\sigma(x\cdot\xi+\mu t,x\cdot
A_{\xi},x\cdot\xi\times A_{\xi})),$ $\displaystyle M^{\xi}_{r}$
$\displaystyle=M^{-\xi}_{r},\qquad\text{for every }\xi\in\Lambda^{-},$
$\displaystyle\eta_{\xi}(x,t)$
$\displaystyle=\frac{1}{\|F\|_{L^{2}}^{3}}M_{r}^{\xi}(x,t).$

The parameters $\lambda,\sigma,\mu$ and $r$ will be chosen later. In
particular, they must satisfy the following conditions,

(4.5) $\displaystyle\lambda\sigma r$ $\displaystyle>1,$ $\displaystyle\sigma
r$ $\displaystyle<\frac{1}{2},$ $\displaystyle\sigma\lambda$
$\displaystyle\in\mathbb{N},$ $\displaystyle r^{\frac{3}{2}}<\mu$
$\displaystyle<\lambda^{2},$ $\displaystyle\lambda$ $\displaystyle\in
10\mathbb{N}^{*},$ $\displaystyle r$ $\displaystyle\in 2\mathbb{N}^{*},$

in order to control the $L^{\infty}_{t}L^{1}_{x}$ norm of the stress tensor
$R_{q+1}$ by means of spatial and temporal intermittency, see Section 4.3.

######  Remark 4.3.

The main difference between our building block and the intermittent Beltrami
waves presented in [BV19] is that we introduce spatial intermittency by means
of Fejer kernels instead of Dirichlet kernels.

This modification does not affect the most important properties of the
building blocks, such as the representation formula or the active Fourier
coefficients. At the same time, it allows to prove the non-uniqueness result
assuming a weaker condition on the parameter $\beta$.

######  Remark 4.4.

The time intermittency has been introduced in order to estimate a specific
term in $\operatorname{div}(R_{q+1})$, which is related to

$\operatorname{div}(\mathbf{W}_{\xi}\otimes\textbf{W}_{-\xi}+\textbf{W}_{-\xi}\otimes\mathbf{W}_{\xi})=\nabla\eta_{\xi}^{2}-((\xi\cdot\nabla)\eta_{\xi}^{2})\xi=\nabla\eta_{\xi}^{2}-\frac{\xi}{\mu}\partial_{t}\eta_{\xi}^{2}.$

######  Proposition 4.5.

Assume that the conditions in Eq. 4.5 on the parameters hold, then:

(4.6) $\displaystyle\mathbf{P}_{\leq
2\lambda}\eta_{\xi}=\eta_{\xi},\qquad\mathbf{P}_{[\frac{\lambda}{4},3\lambda)}\mathbf{W}_{\xi}=\mathbf{W}_{\xi},$
(4.7)
$\displaystyle\frac{1}{\mu}\partial_{t}\eta_{\xi}=(\xi\cdot\nabla)\eta_{\xi},\qquad\text{for
all }\xi\in\Lambda^{+},$ (4.8)
$\displaystyle\mathbf{P}_{[\frac{\lambda}{10},4\lambda)}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}=\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}},\qquad\text{for
}\xi\neq-\xi^{\prime}.$

Moreover, for all $N,M\in\mathbb{N}$ and $p\in[1,\infty]$,

(4.9)
$\displaystyle\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}L^{p}_{x}}$
$\displaystyle\lesssim(\lambda\sigma r)^{N}(\lambda\sigma
r\mu)^{M}r^{\frac{3}{2}-\frac{3}{p}},$ (4.10)
$\displaystyle\|\nabla^{N}\partial_{t}^{M}\mathbf{W}_{\xi}\|_{L^{\infty}_{t}L^{p}_{x}}$
$\displaystyle\lesssim\lambda^{N}(\lambda\sigma
r\mu)^{M}r^{\frac{3}{2}-\frac{3}{p}}.$

Finally, for all $N,M\in\mathbb{N}$ and $\alpha\geq 0$,

(4.11)
$\displaystyle\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}$
$\displaystyle\lesssim(\lambda\sigma r)^{N}(\lambda\sigma
r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda),$ (4.12)
$\displaystyle\|\nabla^{N}\partial_{t}^{M}\mathbf{W}_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}$
$\displaystyle\lesssim\lambda^{N}(\lambda\sigma
r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda).$

Here the Banach spaces $\mathrm{L}(\mathrm{logL})^{\alpha}$ are defined in
3.1.

###### Proof.

The equalities in Eq. 4.6, Eq. 4.7, Eq. 4.8 are true by construction. The
inequalities in Eq. 4.9, Eq. 4.10 follow from Fubini-Tonelli theorem, 4.2 and
the fact that the map

$x\mapsto(\lambda\sigma(x\cdot\xi+\mu t,A_{\xi}\cdot x,\xi\times A_{\xi}\cdot
x))$

is the composition of three volume preserving maps of $\mathbf{T}^{3}$.

To prove Eq. 4.11, recall that $\eta_{\xi}\approx r^{-\frac{3}{2}}M^{\xi}_{r}$
(Eq. 4.4 and 4.2), so

(4.13)
$\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\approx
r^{-\frac{3}{2}}\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}.$

Now, let us define

$C:=\int_{\mathbf{T}^{3}}|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}|\log^{\alpha}(2+|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}|)\,dx$

and distinguish two cases:

  * •

If $C\leq 1$, then Eq. 3.1 implies that

$\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\leq
1\lesssim(\lambda\sigma r)^{N}(\lambda\sigma r\mu)^{M}\log^{\alpha}(\lambda),$

therefore,

$\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\lesssim(\lambda\sigma
r)^{N}(\lambda\sigma r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda).$

  * •

If $C>1$, since $A(s)=s\log^{\alpha}(2+s)$ is a convex function and $A(0)=0$,
then, by the definition of Luxemburg norm, we have that

$\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\lesssim\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}L^{1}_{x}}\log^{\alpha}(\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}L^{\infty}_{x}}),$

then,

$\displaystyle\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}$
$\displaystyle\lesssim
r^{-\frac{3}{2}}\|\nabla^{N}\partial_{t}^{M}M^{\xi}_{r}\|_{L^{\infty}_{t}L^{1}_{x}},$
$\displaystyle\log^{\alpha}(r^{\frac{3}{2}}\|\nabla^{N}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}L^{\infty}_{x}})$
$\displaystyle\lesssim(\lambda\sigma r)^{N}(\lambda\sigma
r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda),$

where the last inequality is due to Eq. 4.13 and to Eq. 4.9.

We finally turn to the proof of Eq. 4.12. Eq. 4.11 and Lemma 3.2 imply that

$\|\nabla^{N}\partial_{t}^{M}\mathbf{W}_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\lesssim\sum_{k=0}^{N}\|\nabla^{k}\partial_{t}^{M}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{\alpha}_{x}}\|\nabla^{N-k}W_{\xi}\|_{L^{\infty}_{t}L^{\infty}_{x}}\lesssim\\\
\lesssim\sum_{k=0}^{N}(\lambda\sigma r)^{k}(\lambda\sigma
r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda)\lambda^{N-k}\leq\lambda^{N}(\lambda\sigma
r\mu)^{M}r^{-\frac{3}{2}}\log^{\alpha}(\lambda),$

which concludes the proof. ∎

######  Proposition 4.6.

Consider the family of vector fields $(\mathbf{W}_{\xi})_{\xi\in\Lambda}$ and
a family of complex values $(a_{\xi})_{\xi\in\Lambda}$ such that
$\overline{a_{-\xi}}=a_{\xi}$. Then,
$\sum_{\xi\in\Lambda}a_{\xi}\mathbf{W}_{\xi}$ is a real-valued vector field
and for all $R\in B_{\epsilon_{\Lambda}}(I_{d})$,

$\sum_{\xi\in\Lambda}\gamma_{\xi}(R)^{2}\fint_{\mathbf{T}^{3}}\mathbf{W}_{\xi}\otimes\mathbf{W}_{-\xi}dx=R,$

where $\epsilon$ and the functions $\gamma_{\xi}(R)$ are given in 4.1.

###### Proof.

We have that

$\sum_{\xi\in\Lambda}\gamma_{\xi}(R)^{2}\fint_{\mathbf{T}^{3}}\mathbf{W}_{\xi}\otimes\textbf{W}_{-\xi}dx=\sum_{\xi\in\Lambda}\gamma_{\xi}(R)^{2}B_{\xi}\otimes
B_{-\xi}\fint_{\mathbf{T}^{3}}\eta_{\xi}(x,t)^{2}dx=\\\
=\sum_{\xi\in\Lambda}\gamma_{\xi}(R)^{2}B_{\xi}\otimes
B_{-\xi}=\sum_{\xi\in\Lambda^{+}}\gamma_{\xi}(R)^{2}(I_{3}-\xi\otimes\xi)=\\\
=\frac{1}{2}\sum_{\xi\in\Lambda}\gamma_{\xi}(R)^{2}(I_{3}-\xi\otimes\xi)=R,$

where the second equality follows from the normalization introduced in Eq. 4.2
and Eq. 4.4, the fifth one follows from the symmetry of $\Lambda$ and 4.1, and
the last one follows from Eq. 4.1. ∎

###  4.2. The perturbation

Following the idea proposed by Luo et al. [LT20], consider a cut-off function
$\psi$ that controls the time support of the perturbation, i.e. a function
that has the following properties

$\displaystyle\psi(t)=1\text{ on
}\operatorname{Supp}_{t}(R_{q}),\quad\operatorname{Supp}_{t}(\psi)\subset
N_{\delta_{q+1}}(\operatorname{Supp}_{t}(R_{q})),\quad\text{ and
}\quad|\psi^{\prime}(t)|\leq 2\delta_{q+1}^{-1}.$

Given a smooth non-decreasing function such that

$\chi(s)=\begin{cases}1,&\qquad 0\leq s\leq 1,\\\ s,&\qquad s\geq
2,\end{cases}$

define a function that smoothly recovers the magnitude of the tensor stress
$R_{q}$,

(4.14)
$\rho(t,x):=\frac{2}{\epsilon_{\Lambda}}\chi(\delta_{q+1}^{-1}|R_{q}(t,x)|)\delta_{q+1}\psi^{2}(t),$

where $\epsilon_{\Lambda}$ is the value from 4.1.

Now, given $\xi\in\Lambda$, we define the magnitude of its impulsed Beltrami
flow, in order to recover $R_{q}$ through 4.6, as

(4.15)
$a_{\xi}(x,t)=\rho(x,t)^{\frac{1}{2}}\gamma_{\xi}\bigg{(}I_{3}-\frac{R_{q}(x,t)}{\rho(x,t)}\bigg{)}.$

Indeed, since

$\frac{|R_{q}|}{\rho}\leq\frac{\epsilon_{\Lambda}}{2}\qquad\text{on
}\operatorname{Supp}_{t}(R_{q}),$

then

(4.16)
$\sum_{\xi\in\Lambda}a_{\xi}^{2}\fint\mathbf{W}_{\xi}\otimes\mathbf{W}_{-\xi}dx=\rho
I_{3}-R_{q}.$

We can finally define the new velocity field as $v_{q+1}=v_{q}+w$, where

(4.17) $\displaystyle w$ $\displaystyle=w^{(p)}+w^{(c)}+w^{(t)},$
$\displaystyle w^{(p)}$
$\displaystyle=\sum_{\xi\in\Lambda}a_{\xi}\mathbf{W}_{\xi},$ $\displaystyle
w^{(c)}$
$\displaystyle=\frac{1}{\lambda}\sum_{\xi\in\Lambda}\nabla(a_{\xi}\eta_{\xi})\times
W_{\xi},$ $\displaystyle w^{(t)}$
$\displaystyle=\frac{1}{\mu}\sum_{\xi\in\Lambda^{+}}P_{H}\mathbf{P}_{\neq
0}(a_{\xi}^{2}\eta_{\xi}^{2}\xi).$

Notice that the function $\psi$ allows to identify the temporal support of the
perturbation $w$. Indeed,

(4.18)
$\operatorname{Supp}_{t}(w)\subset\operatorname{Supp}_{t}(\rho)\subset\operatorname{Supp}_{t}(\psi)\subset
N_{\delta_{q+1}}(\operatorname{Supp}_{t}R_{q+1}).$

Now, let us recall various estimates regarding the perturbation $w$.

######  Lemma 4.7 (Lemma 2, [LT20]).

Let $\rho$, $a_{\xi}$ be defined as in Eq. 4.14, Eq. 4.15, then

$\displaystyle\|\rho\|_{L^{\infty}_{t}L^{1}_{x}}\leq
C\delta_{q+1},\qquad\|\rho\|_{C^{0}(\operatorname{Supp}_{t}(R_{q}))}\lesssim\delta_{q+1}^{-1},\qquad\|\rho\|_{C^{N}_{t,x}}\lesssim
C(\delta_{q+1},|R_{q}|_{C^{N}_{t,x}}),$
$\displaystyle\|a_{\xi}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\delta_{q+1}^{\frac{1}{2}},\qquad\|a_{\xi}\|_{C^{N}_{t,x}}\lesssim
C(\delta_{q+1},|R_{q}|_{C^{N}_{t,x}}).$

######  Lemma 4.8 (Lemma 2.1, [MS18]).

Given $1\leq p\leq\infty$, if $f,g\in C^{\infty}(\mathbf{T}^{3})$, with $g$
$(\frac{\mathbf{T}}{N})^{3}$-periodic and $N\in\mathbb{N}$, then,

$\|fg\|_{L^{p}}\lesssim\|f\|_{L^{p}}\|g\|_{L^{p}}+N^{-\frac{1}{p}}\|f\|_{C^{1}}\|g\|_{L^{p}}.$

In the rest of the section we will denote by $C_{a}(N)$ any value that depends
polynomially on $\sup_{n\leq N}\|a_{\xi}\|_{C^{n}_{t,x}}$.

######  Lemma 4.9.

If $w$, $w^{(p)}$, $w^{(c)}$, $w^{(t)}$ are defined as in Eq. 4.17, then

$\displaystyle\|w^{(p)}\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\delta_{q+1}^{\frac{1}{2}}+C_{a}(1)(\lambda\sigma)^{-\frac{1}{2}},$
$\displaystyle\|w^{(c)}\|_{L^{\infty}_{t}L^{2}_{x}}$ $\displaystyle\lesssim
C_{a}(1)\Bigl{(}\frac{1}{\lambda}+\sigma r\Bigr{)},$
$\displaystyle\|w^{(t)}\|_{L^{\infty}_{t}L^{2}_{x}}$ $\displaystyle\lesssim
C_{a}(0)\frac{r^{\frac{3}{2}}}{\mu},$
$\displaystyle\|w\|_{L^{\infty}_{t}L^{s}_{x}}$ $\displaystyle\lesssim
C_{a}(0)r^{\frac{3}{2}-\frac{3}{s}},\qquad\text{ for }s\in(1,2].$

###### Proof.

Since $\mathbf{W}_{\xi}$ is $(\frac{\mathbf{T}}{\lambda\sigma})^{3}$-periodic
(this follows from Eq. 4.6), Lemma 4.8 can be applied to any term
$a_{\xi}\mathbf{W}_{\xi}$. Then, the first inequality follows from the
estimates shown in Lemma 4.7, and from Eq. 4.10.

The other inequalities from 4.5 and the fact that the Leray-Helmholtz
projector $P_{H}$ is bounded from $L^{q}$ to itself for $1<q<\infty$. ∎

###  4.3. Estimating the stress

In the previous section we have defined the perturbation $w$ and the new
velocity field $v_{q+1}$. Now, we complete the proof of Lemma 2.3 showing how
to properly build the pressure $p_{q+1}$ and the tensor stress $R_{q+1}$.

Let us first introduce an anti-divergence operator, namely a right-inverse of
the divergence operator, which maps vector fields into trace-free matrix-
valued functions.

######  Definition 4.10.

Let $u\in C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3})$ be a smooth vector field.
A _anti-divergence_ is the linear operator defined by

$\mathcal{R}(u)=\frac{1}{4}\big{(}\nabla P_{H}v+(\nabla
P_{H}v)^{T})+\frac{3}{4}(\nabla v+(\nabla
v)^{T}\big{)}-\frac{1}{2}\operatorname{div}(v)I_{d},$

where $v\in C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3})$ is the unique solution
to the problem

$\begin{cases}\Delta v=u-\fint u\,dx,\\\ \fint v\,dx=0,\end{cases}$

with periodic boundary conditions.

The next proposition fully justifies the name given to the operator
$\mathcal{R}$.

######  Proposition 4.11.

Consider $u\in C^{\infty}(\mathbf{T}^{3},\mathbb{R}^{3})$, then,

(4.19)
$\operatorname{div}(\mathcal{R}(u))=u-\fint_{\mathbf{T}^{3}}u,\qquad\operatorname{Tr}(\mathcal{R}(u))=0.$

Moreover, for all $k\in\mathbb{Z}^{3}$, with $k\neq 0$,

(4.20) $\displaystyle{}_{j,h=1}^{3}$
$\displaystyle=\frac{i}{2|k|^{2}}(\mathcal{F}(u)(k)\cdot
k)\mathbf{1}_{j=h}+\frac{i\,k_{h}k_{j}}{2|k|^{4}}(\mathcal{F}(u)(k)\cdot
k)+{}$
$\displaystyle\quad-\frac{ik_{h}}{|k|^{2}}\mathcal{F}(u)(k)_{j}-\frac{ik_{j}}{|k|^{2}}\mathcal{F}(u)(k)_{h},$

where $\mathcal{F}(v)(k)_{j}$ is the $j^{\text{th}}$ component of the
$k^{\text{th}}$ Fourier coefficient of $v$.

###### Proof.

This result follows from straightforward computations. ∎

######  Remark 4.12.

Eq. 4.20 shows that $|\nabla|\mathcal{R}(u)$ is obtained by applying three or
one Riesz transforms to the components of the vector field $u$.

Since $(v_{q},p_{q},R_{q})$ solves Eq. 2.1, then

$\displaystyle\partial_{t}v_{q+1}+\operatorname{div}(v_{q+1}\otimes
v_{q+1})+(-\Delta)^{\frac{5}{4},\beta}v_{q+1}=$
$\displaystyle=\underbrace{\operatorname{div}(R_{q})-\nabla
p_{q}+\operatorname{div}(w^{(p)}\otimes
w^{(p)})+\partial_{t}w^{(t)}}_{\text{oscillation error}}+{}$
$\displaystyle\quad+\underbrace{\operatorname{div}(v_{q}\otimes w+w\otimes
v_{q})+(-\Delta)^{\frac{5}{4},\beta}w+\partial_{t}(w^{(c)}+w^{(p)})}_{\text{linear
error}}+{}$
$\displaystyle\quad+\underbrace{\operatorname{div}(w^{(p)}\otimes(w^{(c)}+w^{(t)})+(w^{(c)}+w^{(t)})\otimes
w)}_{\text{corrector error}}.$

Now, we show that the terms on the RHS can be expressed as the gradient of a
scalar function $p_{q+1}$ and as the divergence of a matrix-valued function
$R_{q+1}$. In addition, we shall prove that
$\|R_{q+1}\|_{L^{\infty}_{t}L^{1}_{x}}\leq\delta_{q+2}$.

Before estimating $\|R_{q+1}\|_{L^{\infty}_{t}L^{1}_{x}}$, let us prove a
useful lemma.

######  Lemma 4.13.

Consider $a,b\in C^{\infty}(\mathbf{T}^{3})$ such that $\mathbf{P}_{\geq
N}b=b$.

  1. (1)

Let $\beta>3$ and $N\in 2\mathbb{N}^{*}$. Then,

(4.21)
$\|\mathcal{R}|\nabla|T_{M}(ab)\|_{L^{1}}\leq\log^{3-\beta}(N)\|a\|_{H^{3}}\|b\|_{\mathrm{L}(\mathrm{logL})^{1}},$

where $|\nabla|$ is the operator associated with the multiplier $m(k)=|k|$.

  2. (2)

Moreover,

(4.22) $\|\mathcal{R}\mathbf{P}_{\neq
0}(ab)\|_{L^{1}}\lesssim\frac{1}{N}\log^{3}(N)\|a\|_{H^{4}}\|b\|_{\mathrm{L}(\mathrm{logL})^{2}},$

where the space $\mathrm{L}(\mathrm{logL})^{\alpha}$ is defined in 3.1.

###### Proof.

Define $a_{1}=\mathbf{P}_{<\frac{N}{2}}(a)$ and
$a_{2}=\mathbf{P}_{\geq\frac{N}{2}}(a)$. The assumption on $b$ implies that

$ab=\mathbf{P}_{\geq\frac{N}{2}}(a_{1}b)+(a_{2}b).$

Notice that Eq. 4.20 implies that the operator $\mathcal{R}|\nabla|T_{M}$
corresponds to the multiplier

$\frac{m_{1}(k)}{|k|}M(k)|k|,$

where $m_{1}$ is a suitable linear combination of iterated Riesz multipliers
(see 3.7). Now,

$\|\mathcal{R}|\nabla|T_{M}(ab)\|_{L^{1}}\lesssim\log^{3-\beta}(N)\|a_{1}b\|_{\mathrm{L}(\mathrm{logL})^{1}}+\|a_{2}b\|_{\mathrm{L}(\mathrm{logL})^{1}}\leq\\\
\leq\log^{3-\beta}(N)\|a_{1}\|_{L^{\infty}}\|b\|_{\mathrm{L}(\mathrm{logL})^{1}}+\|a_{2}\|_{L^{\infty}}\|b\|_{\mathrm{L}(\mathrm{logL})^{1}}\leq\\\
\leq\bigl{(}\log^{3-\beta}(N)\|a\|_{H^{2}}+\tfrac{1}{N}\|a\|_{H^{3}}\bigr{)}\|b\|_{\mathrm{L}(\mathrm{logL})^{1}},$

where the first inequality is due to 3.7 and Eq. 3.3, the second one follows
from Lemma 3.2 and the third one can be justified by Sobolev embeddings.

The second inequality, Eq. 4.22, follows from a slight modification of the
previous proof. In particular, recall that $\mathcal{R}\mathbf{P}_{\neq 0}$
can be seen as corresponding to the Fourier multiplier given by Eq. 4.20,
which is equal to

$\frac{m_{1}(k)}{|k|_{1}}\frac{\sum_{j=1}^{3}|k_{j}|}{|k|}=\frac{m_{1}(k)}{|k|_{1}}\frac{\sum_{j=1}^{3}\operatorname{sgn}(k_{j})k_{j}}{|k|}=\frac{1}{|k|_{1}}\sum_{j}m_{2,j}(k)\operatorname{sgn}(k_{j}),$

where $m_{1}$ and $m_{2,j}$ are linear combinations of iterated Riesz
transforms. Therefore, $\mathcal{R}\mathbf{P}_{\neq 0}$ is a finite linear
combination of operators given by the composition of $T_{m}$ with
$m(k)=\mathbf{1}_{\\{k\neq 0\\}}(k)\frac{1}{|k|_{1}}$, iterated Riesz
transforms and directional Hilbert transforms. Then, the result is easily
achieved by using 3.4, 3.7 and 3.6. ∎

####  4.3.1. Corrector error

Given its formulation, it produces two addends of $R_{q+1}$ whom
$L^{\infty}_{t}L^{1}_{x}$ norm can be estimated thanks to Lemma 4.9. Indeed,

$\displaystyle\|w^{(p)}\otimes(w^{(c)}+w^{(t)})+(w^{(c)}+w^{(t)})\otimes
w\|_{L^{\infty}_{t}L^{1}_{x}}\leq$
$\displaystyle\leq\|w+w^{(p)}\|_{L^{\infty}_{t}L^{2}_{x}}+\|w^{(c)}+w^{(t)}\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\leq C_{a}(1)\Bigl{(}\frac{1}{\lambda}+\sigma
r+\frac{r^{\frac{3}{2}}}{\mu}\Bigr{)}\Bigl{(}\frac{1}{\lambda}+\sigma
r+\frac{r^{\frac{3}{2}}}{\mu}+\delta_{q+1}^{\frac{1}{2}}+(\lambda\sigma)^{-\frac{1}{2}}\Bigr{)}.$

####  4.3.2. Linear error

Since $\mathcal{R}$ is the right inverse to the divergence operator (Eq.
4.19), the terms $v_{q}\otimes w+w\otimes v_{q}$ and
$\mathcal{R}((-\Delta)^{\frac{5}{4},\beta}w+\partial_{t}(w^{(c)}+w^{(p)}))$
are part of $R_{q+1}$. In addition, Lemma 4.9 implies that

$\|v_{q}\otimes
w\|_{L^{\infty}_{t}L^{1}_{x}}\leq\|v_{q}\|_{L^{\infty}_{t}L^{\infty}_{x}}\|w\|_{L^{\infty}_{t}L^{s}_{x}}\lesssim
C_{a}(0)\|v_{q}\|_{L^{\infty}_{t}L^{\infty}_{x}}r^{\frac{3}{2}-\frac{3}{s}},$

for $s\in(1,2)$.

The remaining terms are estimated through Lemma 4.13.

######  Proposition 4.14.

The following estimates hold:

  1. (1)

$\|\mathcal{R}(\partial_{t}(w^{(c)}+w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim
C_{a}(1)\sigma\mu r^{-\frac{1}{2}}\log(\lambda)$,

  2. (2)

$\displaystyle\|\mathcal{R}(-\Delta)^{\frac{5}{4},\beta}(w^{(p)}+w^{(c)})\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim
C_{a}(2)\Bigl{(}\frac{\lambda}{r}\Bigr{)}^{\frac{3}{2}}\log^{4-\beta}(\lambda)$;

  3. (3)

$\displaystyle\|\mathcal{R}(-\Delta)^{\frac{5}{4},\beta}(w^{(t)})\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim
C_{a}(2)\frac{(\lambda\sigma r)^{\frac{3}{2}}}{\mu}\log(\lambda)$.

###### Proof.

To prove the first inequality, notice that
$w^{(p)}+w^{(c)}=\frac{1}{\lambda}\nabla\times w^{(p)}$, so

$\|\mathcal{R}(\partial_{t}(w^{(c)}+w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}=\frac{1}{\lambda}\|\mathcal{R}(\nabla\times(\partial_{t}w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim\\\
\lesssim\frac{1}{\lambda}\|\partial_{t}w^{(p)}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}\leq\frac{1}{\lambda}(C_{a}(1)\|\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}+C_{a}(0)\|\partial_{t}\eta_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}),$

where the first inequality is due to Eq. 4.20 and 3.7 and the second one
follows from Lemma 3.2. Now, Eq. 4.11 implies the required inequality.

We turn to the proof of the second inequality. Recall that $T_{M}$ is the
operator associated with the multiplier

$M(k)=\mathbf{1}_{\\{k\neq 0\\}}(k)\log^{-\beta}(|k|_{1}+10).$

Then by a classical interpolation argument, we have

$\|\mathcal{R}(-\Delta)^{\frac{5}{4},\beta}(w^{(p)}+w^{(c)})\|_{L^{\infty}_{t}L^{1}_{x}}=\frac{1}{\lambda}\||\nabla|^{\frac{3}{2}}(\mathcal{R}|\nabla|T_{M}(\nabla\times
w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}\leq\\\
\leq\frac{1}{\lambda}\|\mathcal{R}|\nabla|T_{M}\nabla(\nabla\times
w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{1}{2}}\|\mathcal{R}|\nabla|T_{M}\nabla^{2}(\nabla\times
w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{1}{2}}.$

Now we estimate the first factor
$\|\mathcal{R}|\nabla|T_{M}\nabla(\nabla\times
w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}$, the second one can be estimated by a
similar procedure.

First of all,

$\nabla(\nabla\times(a_{\xi}\mathbf{W}_{\xi}))=\nabla(a_{\xi}(\nabla\times\mathbf{W}_{\xi})+\nabla
a_{\xi}\times\mathbf{W}_{\xi})=\\\ =(\nabla
a_{\xi})(\nabla\times\mathbf{W}_{\xi})+a_{\xi}\nabla(\nabla\times\mathbf{W}_{\xi})+\nabla^{2}a_{\xi}\times\mathbf{W}_{\xi}+\nabla
a_{\xi}\times\nabla\mathbf{W}_{\xi}.$

Notice that Eq. 4.6 implies that all the terms are composed by a suitable
derivative of $a_{\xi}$ and a $(\frac{\mathbf{T}}{\lambda/5})^{3}$-periodic
function. Therefore,

$\displaystyle\|\mathcal{R}|\nabla|T_{M}\nabla(\nabla\times
w^{(p)}))\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\lesssim\log^{3-\beta}(\lambda)C_{a}(5)\|\nabla^{2}\mathbf{W}_{\xi}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}$
$\displaystyle\lesssim
C_{a}(5)\lambda^{2}r^{-\frac{3}{2}}\log^{4-\beta}(\lambda),$

where the first inequality is due to Eq. 4.21, while Eq. 4.12 implies the
second one.

Finally, we prove the third inequality. Following the same strategy used
above, we have that

$\displaystyle\|\mathcal{R}(-\Delta)^{\frac{5}{4},\beta}(w^{(t)})\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle=\||\nabla|^{\frac{3}{2}}(\mathcal{R}|\nabla|T_{M}(w^{(t)}))\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\leq\|\mathcal{R}|\nabla|T_{M}\nabla(w^{(t)}))\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{1}{2}}\|\mathcal{R}|\nabla|T_{M}\nabla^{2}(w^{(t)}))\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{1}{2}}.$

Let us show how we can estimate the first factor. Recall that the Leray-
Helmholtz projector $P_{H}$ corresponds to the multiplier $I-\frac{k\otimes
k}{|k|^{2}}$, therefore $\mathcal{R}|\nabla|P_{H}$ is associated with a linear
combination of iterated Riesz transforms. So, by 3.7 and 3.5, we have that
$\mathcal{R}|\nabla|P_{H}T_{M}$ maps $\mathrm{L}(\mathrm{logL})$ into $L^{1}$.
Hence,

$\displaystyle\|\mathcal{R}|\nabla|T_{M}\nabla(w^{(t)}))\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\lesssim\frac{1}{\mu}\|\mathcal{R}|\nabla|P_{H}T_{M}(\nabla(a_{\xi}^{2})\eta_{\xi}^{2}+a^{2}_{\xi}\nabla(\eta_{\xi}^{2}))\|_{L^{\infty}_{t}L^{1}_{x}}$
$\displaystyle\lesssim\frac{1}{\mu}\|\nabla(a_{\xi}^{2})\eta_{\xi}^{2}+a^{2}_{\xi}\nabla(\eta_{\xi}^{2})\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}$
$\displaystyle\lesssim\frac{1}{\mu}C_{a}(1)\bigl{(}\|\eta_{\xi}^{2}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}+\|\nabla\eta_{\xi}^{2}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{1}_{x}}\bigr{)}$
$\displaystyle\lesssim\frac{1}{\mu}C_{a}(1)\lambda\sigma r\log(\lambda),$

where the third inequality follows from Lemma 3.2 and the last one follows
from Eq. 4.11. ∎

####  4.3.3. Oscillation error

Finally, the oscillation error is treated as in [BV19]. So,

$\displaystyle\operatorname{div}(R_{q}-w^{(p)}\otimes w^{(p)})=$
$\displaystyle=\operatorname{div}\bigl{(}R_{q}-\sum_{\xi,\xi^{\prime}}a_{\xi}a_{\xi^{\prime}}\fint_{\mathbf{T}^{3}}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}dx+\sum_{\xi,\xi^{\prime}}a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\neq
0}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}\bigr{)}$
$\displaystyle=\operatorname{div}\bigl{(}R_{q}-\sum_{\xi}a_{\xi}^{2}\fint_{\mathbf{T}^{3}}\mathbf{W}_{\xi}\otimes\mathbf{W}_{-\xi}dx+\sum_{\xi,\xi^{\prime}}a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\neq
0}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}\bigr{)}$
$\displaystyle=\operatorname{div}\bigl{(}\rho
I_{3}+\sum_{\xi,\xi^{\prime}}a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\neq
0}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}\bigr{)},$

where the second equality follows from Eq. 4.8 and the third one from Eq.
4.16. Since $\operatorname{div}(\rho I_{3})=\nabla\rho$, then $\rho$ goes into
$p_{q+1}$ as well as $p_{q}$.

Now, we can treat
$\operatorname{div}(\sum_{\xi,\xi^{\prime}}a_{\xi}a_{\xi^{\prime}}P_{\neq
0}\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}})=:\sum_{\xi,\xi^{\prime}}E_{\xi,\xi^{\prime}}$
by taking into account each term $E_{\xi,\xi^{\prime}}+E_{\xi^{\prime},\xi}$.
In particular, since $E_{\xi,\xi^{\prime}}+E_{\xi^{\prime},\xi}$ has zero
mean, then

$\displaystyle E_{\xi,\xi^{\prime}}+E_{\xi^{\prime},\xi}$
$\displaystyle=\mathbf{P}_{\neq
0}\bigl{(}\nabla(a_{\xi}a_{\xi^{\prime}})\cdot\mathbf{P}_{\geq\lambda\sigma/2}(\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}+\mathbf{W}_{\xi^{\prime}}\otimes\mathbf{W}_{\xi})\bigr{)}$
$\displaystyle\quad+\mathbf{P}_{\neq
0}\big{(}a_{\xi}a_{\xi^{\prime}}\operatorname{div}(\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}+\mathbf{W}_{\xi^{\prime}}\otimes\mathbf{W}_{\xi})\big{)}$
$\displaystyle=:E_{\xi,\xi^{\prime},1}+E_{\xi,\xi^{\prime},2},$

The term $\mathcal{R}E_{\xi,\xi^{\prime},1}$ goes into $R_{q+1}$, Eq. 4.22 and
Lemma 3.2 imply that

$\|\mathcal{R}E_{\xi,\xi^{\prime},1}\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim\frac{\log^{3}(\lambda\sigma)}{\lambda\sigma}C_{a}(5)\|\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{2}_{x}}\lesssim\\\
\lesssim\frac{\log^{3}(\lambda\sigma)}{\lambda\sigma}C_{a}(5)\|\eta_{\xi}\eta_{\xi^{\prime}}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{2}_{x}}\lesssim
C_{a}(5)\frac{\log^{3}(\lambda\sigma)}{\lambda\sigma}\log^{2}(\lambda),$

where
$\|\eta_{\xi}\eta_{\xi^{\prime}}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{2}_{x}}$
can be estimated arguing as in Eq. 4.11.

Instead, we need to distinguish two cases in order to deal with
$E_{\xi,\xi^{\prime},2}$.

_First case: $\xi\neq-\xi^{\prime}$_. If $\xi\neq-\xi^{\prime}$, then
$\mathbf{P}_{>\lambda/10}(\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}})=\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}$.
Therefore,

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
a_{\xi}a_{\xi^{\prime}}\operatorname{div}((\mathbf{W}_{\xi}\otimes\mathbf{W}_{\xi^{\prime}}+\mathbf{W}_{\xi^{\prime}}\otimes\mathbf{W}_{\xi})=$\mbox{}\hfil\qquad&\\\
&=a_{\xi}a_{\xi^{\prime}}\operatorname{div}(\mathbf{P}_{\geq\lambda/10}\big{(}\eta_{\xi}\eta_{\xi^{\prime}}(W_{\xi}\otimes
W_{\xi^{\prime}}+W_{\xi^{\prime}}\otimes W_{\xi})\big{)}\\\
&=a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\geq\lambda/10}\big{(}\nabla(\eta_{\xi}\eta_{\xi^{\prime}})\cdot(W_{\xi}\otimes
W_{\xi^{\prime}}+W_{\xi^{\prime}}\otimes W_{\xi})\big{)}+{}\\\
&\quad+a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\geq\lambda/10}\big{(}\eta_{\xi}\eta_{\xi^{\prime}}\nabla(W_{\xi}\cdot
W_{\xi^{\prime}})\big{)}\\\
&=a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\geq\lambda/10}\big{(}\nabla(\eta_{\xi}\eta_{\xi^{\prime}})\cdot(W_{\xi}\otimes
W_{\xi^{\prime}}+W_{\xi^{\prime}}\otimes
W_{\xi})\big{)}+\nabla(a_{\xi}a_{\xi^{\prime}}\mathbf{W}_{\xi}\cdot\mathbf{W}_{\xi^{\prime}})+{}\\\
&\quad-\nabla(a_{\xi}a_{\xi^{\prime}})\mathbf{P}_{\geq\lambda/10}(\mathbf{W}_{\xi}\cdot\mathbf{W}_{\xi^{\prime}})-a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\geq\lambda/10}\big{(}W_{\xi}\cdot
W_{\xi^{\prime}}\nabla(\eta_{\xi}\eta_{\xi^{\prime}})\big{)},\end{split}$

where the second term produces a pressure, the third one can be estimated as
$E_{\xi,\xi^{\prime},1}$, the first and the fourth produce two new terms of
$R_{q+1}$ whose $L^{\infty}_{t}L^{1}_{x}$ norms can be estimated in the same
way. Namely,

$\|\mathcal{R}\mathbf{P}_{\neq
0}\Big{(}a_{\xi}a_{\xi^{\prime}}\mathbf{P}_{\geq\lambda/10}\big{(}W_{\xi}\cdot
W_{\xi^{\prime}}\nabla(\eta_{\xi}\eta_{\xi^{\prime}})\big{)}\Big{)}\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim\\\
\lesssim
C_{a}(5)\frac{\log^{3}(\lambda)}{\lambda}\|\nabla(\eta_{\xi}\eta_{\xi^{\prime}})\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{2}_{x}}\lesssim
C_{a}(5)\frac{\log^{3}(\lambda)}{\lambda}\|\nabla(\eta_{\xi})\eta_{\xi^{\prime}}\|_{L^{\infty}_{t}\mathrm{L}(\mathrm{logL})^{2}_{x}}\lesssim\\\
\lesssim
C_{a}(5)\frac{\log^{3}(\lambda)}{\lambda}\log^{2}(\lambda)\lambda\sigma r,$

where we have used Eq. 4.22.

_Second case: $\xi=-\xi^{\prime}$_. If $\xi^{\prime}=-\xi$, then
$E_{\xi,-\xi,2}=\mathbf{P}_{\neq
0}\big{(}a_{\xi}^{2}(\nabla\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2})-\frac{\xi}{\mu}\partial_{t}\eta_{\xi}^{2})\big{)}$.
Now, let us recall that $\partial_{t}w^{(t)}$ is part of the oscillation term,
i.e. we need to deal with $\sum_{\xi\in\Lambda^{+}}[\mathbf{P}_{\neq
0}\big{(}a_{\xi}^{2}(\nabla\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2})-\frac{\xi}{\mu}\partial_{t}\eta_{\xi}^{2})\big{)}+\frac{1}{\mu}\partial_{t}P_{H}\mathbf{P}_{\neq
0}(a_{\xi}^{2}\eta_{\xi}^{2}\xi)]$.
Now,

$\mathbf{P}_{\neq
0}\big{(}a_{\xi}^{2}(\nabla\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2}))\big{)}=\nabla(a_{\xi}^{2}\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2}))-\mathbf{P}_{\neq
0}(\nabla a_{\xi}^{2}\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2})),$

where the first term goes into the pressure $p_{q+1}$, while the second
produces a term of $R_{q+1}$ with $L^{\infty}_{t}L^{1}_{x}$ estimated using
Eq. 4.22. Indeed,

$\|\mathcal{R}\mathbf{P}_{\neq 0}(\nabla
a_{\xi}^{2}\mathbf{P}_{\geq\lambda\sigma/2}(\eta_{\xi}^{2}))\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim
C_{a}(5)\frac{\log^{3}(\lambda\sigma)}{\lambda\sigma}\log^{2}(\lambda).$

Instead,

$-\mathbf{P}_{\neq
0}\Bigl{(}a_{\xi}^{2}\partial_{t}\eta_{\xi}^{2}\frac{\xi}{\mu}\Bigr{)}+P_{H}\mathbf{P}_{\neq
0}\Bigl{(}a_{\xi}^{2}\eta_{\xi}^{2}\frac{\xi}{\mu}\Bigr{)}=-(I-P_{H})\Bigl{(}\mathbf{P}_{\neq
0}\partial_{t}(a_{\xi}^{2}\eta_{\xi}^{2})\frac{\xi}{\mu}\Bigr{)}+\mathbf{P}_{\neq
0}\Bigl{(}\frac{\xi}{\mu}\partial_{t}(a_{\xi}^{2})\eta_{\xi}^{2}\Bigr{)},$

where the first term goes into the pressure thanks to
$I-P_{H}=\nabla\Delta^{-1}\operatorname{div}$, while the second one can be
seen as part of $R_{q+1}$, and

$\bigl{\|}\mathcal{R}\mathbf{P}_{\neq
0}\bigl{(}\tfrac{\xi}{\mu}\partial_{t}(a_{\xi}^{2})\eta_{\xi}^{2}\bigr{)}\bigr{\|}_{L^{\infty}_{t}L^{1}_{x}}\lesssim
C_{a}(5)\frac{1}{\mu}\log^{2}(\lambda).$

####  4.3.4. Conclusion of the proof

Summing up all the contributions to $R_{q+1}$, we end up with

(4.23) $\displaystyle|R_{q+1}|_{L^{\infty}_{t}L^{1}_{x}}\lesssim$
$\displaystyle C_{a}(1)\big{(}\frac{1}{\lambda}+\sigma
r+\frac{r^{\frac{3}{2}}}{\mu}\big{)}\big{(}\frac{1}{\lambda}+\sigma
r+\frac{r^{\frac{3}{2}}}{\mu}+\delta_{q+1}^{\frac{1}{2}}+(\lambda\sigma)^{-\frac{1}{2}}\big{)}$
$\displaystyle+C_{a}(0)r^{\frac{3}{2}-\frac{3}{s}}+C_{a}(1)\sigma\mu
r^{-\frac{1}{2}}\log(\lambda)+C_{a}(2)\big{(}\frac{\lambda}{r}\big{)}^{\frac{3}{2}}\log^{4-\beta}(\lambda)$
$\displaystyle+C_{a}(2)\frac{(\lambda\sigma
r)^{\frac{3}{2}}}{\mu}\log(\lambda)+C_{a}(5)\frac{\log^{3}(\lambda\sigma)}{\lambda\sigma}\log^{2}(\lambda)$
$\displaystyle+C_{a}(5)\log^{5}(\lambda)\sigma
r+C_{a}(5)\frac{\log^{2}(\lambda)}{\mu}.$

Since the values of $C_{a}(N)$ do not depend on the parameter $\lambda$ (see
Lemma 4.7), then we can choose the parameters $s$, $\sigma$, $\mu$ and $r$
such that Eq. 4.5 hold and $\|R_{q+1}\|_{L^{\infty}_{t}L^{1}_{x}}$ is smaller
than $\delta_{q+2}$ as long as $\lambda$ is large enough. Therefore, consider

$s=\frac{3}{2},\quad\sigma=\frac{1}{\lambda}\log^{2}(\lambda)\log^{4}(\log(\lambda)),\quad
r=\frac{\lambda}{\log^{y}(\lambda)},\quad\mu=\lambda^{\frac{3}{2}}\log^{z}(\lambda),$

where $y>2$ and $z>-\frac{3}{2}y$ in order to satisfy Eq. 4.5. In particular,
by this choice, the RHS of Eq. 4.23 tends to zero as long as $\lambda$
diverges to $\infty$ if the following inequalities hold

$\displaystyle z>-\frac{3}{2}y,\qquad z+\frac{y}{2}<-3,\qquad y>7,$
$\displaystyle\frac{3}{2}y+4-\beta<0,\qquad\frac{3}{2}y+z>4.$

It is possible to find a pair $(y,z)$ that satisfies the previous inequality
if and only if $\beta>\frac{29}{2}$. In order to conclude the proof of Lemma
2.3, we need to show that Eq. 2.2 and Eq. 2.3 are fully satisfied by our
construction and our choice of the parameters. In particular, the control on
the temporal support follows directly from Eq. 4.18, while the estimates on
$\|v_{q+1}-v_{q}\|_{L^{\infty}_{t}L^{2}_{x}}$ and
$\||\nabla|^{\frac{3}{2}}T_{m}(v_{q+1}-v_{q})\|_{L^{\infty}_{t}L^{1}_{x}}$ are
due to Lemma 4.9 and 4.14.

This concludes the proof of the _Iteration Lemma_ (Lemma 2.3).

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