2,q}D^{k}D_{t,q}^{m}\mathring{R}_{q,n,p}\right\|_{L^{1}}$
$\displaystyle\lesssim 16\Gamma_{q+1}^{-2j}\sum_{k=0}^{\mathsf{N}_{\rm
cut,x}}\sum_{m=0}^{\mathsf{N}_{\rm cut,t}}\Gamma_{q+1}^{-k}$
$\displaystyle\lesssim\Gamma_{q+1}^{-2j}$

where the implicit constant only depends on $\mathsf{N}_{\rm cut,t}$. The
proof is concluded since the $L^{1}$ norm of a function with range in $[0,1]$
is bounded by the measure of its support. ∎

###  6.8 Definition and properties of the checkerboard cutoff functions

For $0\leq n\leq{n_{\rm max}}$, consider all the
$\frac{\mathbb{T}^{3}}{\lambda_{q,n,0}}$-periodic cells contained in
$\mathbb{T}^{3}$, of which there are $\lambda_{q,n,0}^{3}$. Index these cells
by integer triples $\vec{l}=(l,w,h)$ for
$l,w,h\in\\{0,...,\lambda_{q,n,0}-1\\}$. Let $\mathcal{X}_{q,n,\vec{l}}$ be a
partition of unity adapted to this checkerboard of periodic cells which
satisfies

$\sum_{\vec{l}=(l,w,h)}\left(\mathcal{X}_{q,n,\vec{l}}\right)^{2}=1$ (6.136)

for any $q$ and $n$. Furthermore, for
$\vec{l}=(l,w,h),\vec{l}^{*}=(l^{*},w^{*},h^{*})\in\\{0,...,\lambda_{q,n,0}-1\\}^{3}$
such that

$|l-l^{*}|\geq 2,\qquad|w-w^{*}|\geq 2,\qquad|h-h^{*}|\geq 2,$

we impose that

$\mathcal{X}_{q,n,\vec{l}}\mathcal{X}_{q,n,\vec{l}^{*}}=0.$ (6.137)

######  Definition 6.39 (Checkerboard Cutoff Function).

Given $q$, $0\leq n\leq{n_{\rm max}}$, $i\leq{i_{\rm max}}$, and
$k\in\mathbb{Z}$, we define

$\zeta_{q,i,k,n,\vec{l}}(x,t)=\mathcal{X}_{q,n,\vec{l}}\left(\Phi_{i,k,q}(x,t)\right).$
(6.138)

######  Lemma 6.40.

The cutoff functions $\left\\{\zeta_{q,i,k,n,\vec{l}}\right\\}_{\vec{l}}$
satisfy the following properties:

  1. (1)

The material derivative $D_{t,q}\left(\zeta_{q,i,k,n,\vec{l}}\right)$
vanishes.

  2. (2)

For each $t\in\mathbb{R}$ and all $x\in\mathbb{T}^{3}$,

$\sum_{\vec{l}=(l,w,h)}\left(\zeta_{q,i,k,n,\vec{l}}(x,t)\right)^{2}=1.$
(6.139)

  3. (3)

We have the spatial derivative estimate for all
$m\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$

$\left\|D^{m}\zeta_{q,i,k,n,\vec{l}}\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\widetilde{\chi}_{i,k,q}\right)}\lesssim\lambda_{q,n,0}^{m}.$
(6.140)

  4. (4)

There exists an implicit dimensional constant independent of $q$, $n$, $k$,
$i$, and $\vec{l}$ such that for all
$(x,t)\in\mathrm{supp\,}\psi_{i,q}\widetilde{\chi}_{i,k,q}$,

$\textnormal{diam}\left(\mathrm{supp\,}\left(\zeta_{q,i,k,n,\vec{l}}(\cdot,t)\right)\right)\lesssim\left(\lambda_{q,n,0}\right)^{-1}.$
(6.141)

###### Proof of Lemma 6.40.

The proof of (1) is immediate given that $\zeta_{q,i,k,n,\vec{l}}$ is
precomposed with the flow map $\Phi_{i,k,q}$. (6.139) follows from (1),
(6.136), and the fact that for each $t\in\mathbb{R}$, $\Phi_{i,k,q}(t,\cdot)$
is a diffeomorphism of $\mathbb{T}^{3}$. The spatial derivative estimate in
(6.140) follows from Lemma A.4, (6.109), and the parameter definitions in
(9.19), (9.26), and (9.29). The property in (6.141) follows from the
construction of the $\mathcal{X}_{q,n,\vec{l}}$ functions (which can be taken
simply as a dilation by a factor of $\lambda_{q,n,1}$ of a $q$-independent
partition of unity on $\mathbb{R}^{3}$) and (6.108). ∎

###  6.9 Definition of the cumulative cutoff function

Finally, combining the cutoff functions defined in Definition 6.6,
(6.120)–(6.121), and (6.96), we define the cumulative cutoff function by

$\displaystyle{\eta_{i,j,k,q,n,p,\vec{l}}(x,t)=\psi_{i,q}(x,t)\omega_{i,j,q,n,p}(x,t)\chi_{i,k,q}(t)\overline{\chi}_{q,n,p}(t)\zeta_{q,i,k,n,\vec{l}}(x,t).}$

Since the values of $q$ and $n$ are clear from the context, the values in
$\vec{l}$ are irrelevant in many arguments, and the time cutoffs
$\overline{\chi}_{q,n,p}$ are only used in Section 8.9, we may abbreviate the
above using any of

$\displaystyle\eta_{i,j,k,q,n,p,\vec{l}}\,(x,t)=\eta_{i,j,k,q,n,p}(x,t)=\eta_{(i,j,k)}(x,t)=\psi_{(i)}(x,t)\omega_{(i,j)}(x,t)\chi_{(i,k)}(t)\zeta_{(i,k)}(x,t).$

It follows from Lemma 6.8, (6.122), (6.94), and (6.139) that for every
${(q,n,p)}$ fixed, we have a partition of unity

$\displaystyle\sum_{i,j\geq
0}\sum_{k\in\mathbb{Z}}\sum_{\vec{l}}\eta_{i,j,k,q,n,p,\vec{l}}^{2}\,(x,t)=1.$
(6.142)

The sum in $i$ goes up to $i_{\mathrm{max}}$ (defined in (6.53)), while the
sum in $j$ goes up to $j_{\mathrm{max}}$ (defined in (6.128)). In analogy with
$\psi_{i\pm,q}$, we define

$\omega_{(i,j\pm)}(x,t):=\left(\omega_{(i,j-1)}^{2}(x,t)+\omega_{(i,j)}^{2}(x,t)+\omega_{(i,j+1)}^{2}(x,t)\right)^{\frac{1}{2}},$
(6.143)

which are cutoffs with the property that

$\omega_{(i,j\pm)}\equiv 1\textnormal{ on }\mathrm{supp\,}{(\omega_{(i,j)})}.$
(6.144)

We then define

$\eta_{(i\pm,j\pm,k,\pm)}(x,t):=\psi_{i\pm,q}(x,t)\omega_{(i,j\pm)}(x,t)\widetilde{\chi}_{i,k,q}(t)\zeta_{q,i,k,n,\vec{l}}(x,t),$
(6.145)

which are cutoffs with the property that

$\eta_{(i,\pm,j\pm,k\pm)}\equiv\zeta_{q,i,k,n,\vec{l}}\quad\textnormal{ on
}\mathrm{supp\,}{\left(\psi_{(i)}\omega_{(i,j)}\chi_{(i,k)}\right)}.$ (6.146)

We conclude this section with estimates on the $L^{p}$ norms of the cumulative
cutoff function $\eta_{(i,j,k)}$.

######  Lemma 6.41.

For $r_{1},r_{2}\in[1,\infty]$ with $\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$ we
have

$\sum_{\vec{l}}\left|\mathrm{supp\,}(\eta_{i,j,k,q,n,p,\vec{l}})\right|\lesssim\Gamma_{q+1}^{-2\left(\frac{i}{r_{1}}+\frac{j}{r_{2}}\right)+\frac{{\mathsf{C}_{b}}}{r_{1}}+2}$
(6.147)

###### Proof of Lemma 6.41.

Applying Lemma 6.23, Lemma 6.38, Hölder’s inequality, and interpolating yields

$\displaystyle\left|\mathrm{supp\,}(\psi_{i,q})\cap\mathrm{supp\,}(\omega_{i,j,q,n,p})\right|$
$\displaystyle\leq\left\|\psi_{i\pm,q}\omega_{(i,j\pm)}\right\|_{L^{1}}$
$\displaystyle\leq\left\|\psi_{i\pm,q}\right\|_{L^{r_{1}}}\left\|\omega_{(i,j\pm)}\right\|_{L^{r_{2}}}$
$\displaystyle\lesssim\Gamma_{q+1}^{-\frac{2(i-1)-{\mathsf{C}_{b}}}{r_{1}}-\frac{2(j-1)}{r_{2}}}.$

Using $\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$ and (6.139), which gives that the
$\zeta_{q,i,k,n,\vec{l}}$ form a partition of unity, yields (6.147). ∎

##  7 From $q$ to $q+1$: breaking down the main inductive estimates

The overarching goal of this section is to state several propositions which
decompose the verification of the main inductive assumptions (3.13) and (3.14)
for the perturbation $w_{q+1}$ and (3.15) for the stress $\mathring{R}_{q+1}$
into digestible components. We remind the reader, cf. Remark 6.1, that the
rest of the inductive estimates stated in Section 3.2.3 are proven in Section
6. We begin in Section 7.1 with Proposition 7.1, which simply translates the
main inductive assumptions into statements phrased at level $q+1$. At this
point, we then introduce in Section 7.2 a handful of notations which will be
necessary in order to state the propositions which form the constituent parts
of the proof of Proposition 7.1. The next three propositions (7.3, 7.4, and
7.5) are described and presented in Section 7.3. They are significantly more
detailed than Proposition 7.1, as they contain the precise estimates that will
be propagated throughout the construction and cancellation of the higher order
stresses $\mathring{R}_{q,{\widetilde{n}}}$. These three propositions will be
verified in Section 8.

###  7.1 Induction on $q$

The main claim of this section is an induction on $q$.

######  Proposition 7.1 (Inductive Step on $q$).

Given $v_{\ell_{q}}$, $\mathring{R}_{\ell_{q}}$, and
$\mathring{R}_{q}^{\textnormal{comm}}$ satisfying the Euler-Reynolds system

$\displaystyle\partial_{t}v_{\ell_{q}}+\mathrm{div\,}(v_{\ell_{q}}\otimes
v_{\ell_{q}})+\nabla p_{\ell_{q}}$
$\displaystyle=\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
(7.1a) $\displaystyle\mathrm{div\,}v_{\ell_{q}}$ $\displaystyle=0$ (7.1b)

with $v_{\ell_{q}}$, $\mathring{R}_{\ell_{q}}$, and
$\mathring{R}_{q}^{\textnormal{comm}}$ satisfying the conclusions of Lemma 5.1
in addition to (3.12)-(3.25b), there exist $v_{q+1}=v_{\ell_{q}}+w_{q+1}$ and
$\mathring{R}_{q+1}$ which satisfy the following:

  1. (1)

$v_{q+1}$ and $\mathring{R}_{q+1}$ solve the Euler-Reynolds system

$\displaystyle\partial_{t}v_{q+1}+\mathrm{div\,}(v_{q+1}\otimes
v_{q+1})+\nabla p_{q+1}$ $\displaystyle=\mathring{R}_{q+1}$ (7.2a)
$\displaystyle\mathrm{div\,}v_{q+1}$ $\displaystyle=0.$ (7.2b)

  2. (2)

For all $k,m\leq 7\mathsf{N}_{\textnormal{ind,v}}$,

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}w_{q+1}\right\|_{L^{2}}\leq\Gamma_{q+1}^{-1}\delta_{q+1}^{\frac{1}{2}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.3)

Furthermore, we have that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q})\subset[T_{1},T_{2}]\quad\Rightarrow\quad\mathrm{supp\,}_{t}(w_{q+1})\subset\left[T_{1}-(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}})^{-1},T_{2}+(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}})^{-1}\right]\,.$
(7.4)

  3. (3)

For all $k,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$,

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}\right\|_{L^{1}}\leq\Gamma_{q+1}^{-\mathsf{C_{R}}}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.5)

######  Remark 7.2.

In achieving the conclusions (7.2), (7.3), and (7.5), we have verified the
inductive assumptions (3.13)-(3.15) at level $q+1$. The inductive assumption
(3.12) at levels $q^{\prime}<q+1$ follows from Lemma (5.1). The proof of
Proposition 7.1 will entail many estimates which are much more detailed than
(7.3) and (7.5), but for the time being we record only the basic estimates
which are direct translations of (3.13)-(3.15) at level $q+1$.

###  7.2 Notations

The proof of Proposition 7.1 will be achieved through an induction with
respect to $\widetilde{n}$, where $0\leq{\widetilde{n}}\leq{n_{\rm max}}$
corresponds to the addition of the perturbation $\displaystyle
w_{q+1,{\widetilde{n}}}=\sum\limits_{{\widetilde{p}}=1}^{{p_{\rm
max}}}w_{q+1,{\widetilde{n}},{\widetilde{p}}}$. The addition of each
perturbation $w_{q+1,{\widetilde{n}}}$ will move the minimum effective
frequency present in the stress terms to $\lambda_{q,{\widetilde{n}}+1,0}$.
This induction on ${\widetilde{n}}$ requires three sub-propositions; the base
case ${\widetilde{n}}=0$, the inductive step from ${\widetilde{n}}-1$ to
${\widetilde{n}}$ for ${\widetilde{n}}\leq{n_{\rm max}}-1$, and the final step
from ${n_{\rm max}}-1$ to ${n_{\rm max}}$. Throughout these propositions, we
shall employ the following notations.

  1. (1)

$\boldsymbol{{\widetilde{n}}}$ \- An integer taking values
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$ over which induction is performed. At
every step in the induction, we add another component
$w_{q+1,{\widetilde{n}}}$ of the final perturbation

$w_{q+1}=\sum\limits_{{\widetilde{n}}=0}^{n_{\rm
max}}\sum\limits_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,{\widetilde{n}},{\widetilde{p}}}.$

We emphasize that the use of ${\widetilde{n}}$ at various points in statements
and estimates means that we are _currently_ working on the inductive step at
level ${\widetilde{n}}$.

  2. (2)

$\boldsymbol{n}$ \- An integer taking values $1\leq n\leq{n_{\rm max}}$ which
correspond to the higher order stresses $\mathring{R}_{q,n}$. Occasionally, we
shall use the notation $\mathring{R}_{q,0}=\mathring{R}_{\ell_{q}}$ to
streamline an argument. We emphasize that $n$ will be used at various points
in statements and estimates to reference higher order objects in addition to
those at level ${\widetilde{n}}$, and so will satisfy the inequality
${\widetilde{n}}\leq n$.

  3. (3)

$\boldsymbol{{\mathring{H}}_{q,n,p}^{n^{\prime}}}$ \- The component of
$\mathring{R}_{q,n,p}$ originating from an error term produced by the addition
of $w_{q+1,n^{\prime}}$. The parameter $n^{\prime}$ will always be a
_subsidiary_ parameter used to reference objects created at or _below_ the
level ${\widetilde{n}}$ that we are currently working on, and so will satisfy
$n^{\prime}\leq{\widetilde{n}}$.

  4. (4)

$\boldsymbol{\mathbb{P}_{[q,n,p]}}$ \- We use the spatial Littlewood-Paley
projectors $\mathbb{P}_{[q,n,p]}$ defined by

$\displaystyle\mathbb{P}_{[q,n,p]}=\begin{cases}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}},{p_{\rm max}}}}&\mbox{if }n={n_{\rm max}},p={p_{\rm max}}+1\\\
\mathbb{P}_{\left[\lambda_{q,n,p-1},\lambda_{q,n,p}\right)}&\mbox{if }1\leq
n\leq{n_{\rm max}},1\leq p\leq{p_{\rm max}}\end{cases}$ (7.6)

where $\mathbb{P}_{[\lambda_{1},\lambda_{2})}$ is defined in Section 9.4 as
$\mathbb{P}_{\geq\lambda_{1}}\mathbb{P}_{<\lambda_{2}}$. Note that for
$n={n_{\rm max}}$ and $p={p_{\rm max}}+1$, $\mathbb{P}_{[q,{n_{\rm
max}},{p_{\rm max}}+1]}$ projects onto _all_ frequencies larger than
$\lambda_{q,{n_{\rm max}},{p_{\rm max}}}=\lambda_{q,{n_{\rm max}}+1,0}$.
Errors which include the frequency projector $\mathbb{P}_{[q,{n_{\rm
max}},{p_{\rm max}}+1]}$ will be small enough to be absorbed into
$\mathring{R}_{q+1}$.

We shall frequently utilize sums of Littlewood-Paley projectors
$\mathbb{P}_{[q,n,p]}$ to decompose products of intermittent pipe flows
periodized to scale $\lambda_{q,{\widetilde{n}}}^{-1}$. These sums will be
written in terms of three parameters - $n$, $p$, and ${\widetilde{n}}$. As a
consequence of (7.6), (9.29), (9.23), and (9.22), we have that
$\lambda_{q,{\widetilde{n}}+1,0}\leq\lambda_{q,{\widetilde{n}}}$ for
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, so that

$\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}=\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}+1,0}}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}=\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}.$
(7.7)

A consequence of (7.7) is that for
$\frac{\mathbb{T}^{3}}{\lambda_{q,{\widetilde{n}}}}$-periodic
functions292929We note that in the second equality in (7.8), such functions do
not have active frequencies in between $\lambda_{q,{\widetilde{n}}+1,0}$ and
$\lambda_{q,{\widetilde{n}}}$. where $0\leq{\widetilde{n}}\leq{n_{\rm max}}$,

$\displaystyle f$ $\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}f+\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}f$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}f+\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)f.$ (7.8)

These equalities will be useful in the calculations in Section 8.3, and we
will recall their significance when we estimate the Type 1 errors in Section
8.6.

  5. (5)

$\boldsymbol{\mathring{R}_{q+1}^{\widetilde{n}}}$ \- Any stress term which
satisfies the estimates required of $\mathring{R}_{q+1}$ and which has already
been estimated at the ${\widetilde{n}}^{th}$ stage of the induction; that is,
error terms arising from the addition of $w_{q+1,n^{\prime}}$ for
$n^{\prime}\leq{\widetilde{n}}$. We _exclude_
$\mathring{R}_{q}^{\textnormal{comm}}$ from
$\mathring{R}_{q+1}^{\widetilde{n}}$, only absorbing it at the very end when
we define $\mathring{R}_{q+1}$. Thus

$\mathring{R}_{q+1}^{{\widetilde{n}}+1}=\mathring{R}_{q+1}^{{\widetilde{n}}}+\left(\textnormal{errors
coming from }w_{q+1,{\widetilde{n}}+1}\textnormal{ that also go into
}\mathring{R}_{q+1}\right)\,.$ (7.9)

###  7.3 Induction on $\widetilde{n}$

The first proposition asserts that there exists a perturbation $w_{q+1,0}$
which we add to $v_{\ell_{q}}$ so that $v_{q,0}:=v_{\ell_{q}}+w_{q+1,0}$
satisfies the following. First, $v_{q,0}$ solves the Euler-Reynolds system
with a righthand side consisting of stresses $\mathring{R}_{q+1}^{0}$ and
$\mathring{H}_{q,n,p}^{0}$ which belong respectively to $\mathring{R}_{q+1}$
and $\mathring{R}_{q,n,p}$ for $1\leq n\leq{n_{\rm max}}$ and $1\leq
p\leq{p_{\rm max}}$. Secondly, $w_{q+1,0}$ satisfies estimates which in
particular imply the inductive assumptions required of the velocity
perturbation $w_{q+1}$ in (7.3).303030This is checked in Remark 8.3. Thirdly,
$\mathring{R}_{q+1}^{0}$ satisfies the estimates required of
$\mathring{R}_{q+1}$ in the inductive assumption (6.118) (with an extra factor
of smallness). Finally, each $\mathring{H}_{q,n,p}^{0}$ satisfies the
inductive assumptions required of $\mathring{R}_{q,n,p}$ in (6.118).

######  Proposition 7.3 (Induction on ${\widetilde{n}}$: The Base Case
${\widetilde{n}}=0$).

Under the assumptions of Proposition 7.1 (equivalently the conclusions of
Lemma 5.1), there exist $\displaystyle
w_{q+1,0}=\sum\limits_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,0,p}=w_{q+1,0,1}$, $\mathring{R}_{q+1}^{0}$, and
$\mathring{H}_{q,n,p}^{0}$ for $1\leq n\leq{n_{\rm max}}$ and $1\leq
p\leq{p_{\rm max}}$ such that the following hold.

  1. (1)

$v_{q,0}:=v_{\ell_{q}}+w_{q+1,0}$ solves

$\displaystyle\partial_{t}v_{q,0}+\mathrm{div\,}(v_{q,0}\otimes
v_{q,0})+\nabla p_{q,0}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{0}\right)+\mathrm{div\,}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{{p_{\rm
max}}}\mathring{H}_{q,n,p}^{0}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
(7.10a) $\displaystyle\mathrm{div\,}v_{q,0}$ $\displaystyle=0.$ (7.10b)

  2. (2)

For all $k+m\leq\mathsf{N}_{\textnormal{fin},0}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$ and
$1\leq{\widetilde{p}}\leq{p_{\rm max}}$ (although only $w_{q+1,0,1}$ is non-
zero)

$\displaystyle\left\|D^{k}D_{t,q}^{m}w_{q+1,0,{\widetilde{p}}}\right\|_{L^{2}\left(\mathrm{supp\,}\psi_{i,q}\right)}\lesssim\delta_{q+1,0,{\widetilde{p}}}^{\frac{1}{2}}\Gamma_{q+1}^{3+\frac{{\mathsf{C}_{b}}}{2}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c_{0}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.11)

Furthermore, we have that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q})\subset[T_{1},T_{2}]\quad\Rightarrow\quad\mathrm{supp\,}_{t}(w_{q+1,0,{\widetilde{p}}})\subset\left[T_{1}-(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1},T_{2}+(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1}\right]\,.$
(7.12)

  3. (3)

For all $k,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$,

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}^{0}\right\|_{L^{1}}\lesssim\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.13)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{R}_{q+1}^{0}\subseteq\mathrm{supp\,}_{t}w_{q+1,0}\,.$
(7.14)

  4. (4)

For all $k+m\leq\mathsf{N}_{\rm fin,n}$ and $1\leq n\leq{n_{\rm max}}$, $1\leq
p\leq{p_{\rm max}}$,

$\left\|D^{k}D_{t,q}^{m}\mathring{H}_{q,n,p}^{0}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}\lesssim\delta_{q+1,n,p}\lambda_{q,n,p}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.15)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{H}_{q,n,p}^{0}\subseteq\mathrm{supp\,}_{t}w_{q+1,0}\,.$
(7.16)

The second proposition assumes that perturbations $w_{q+1,n^{\prime}}$ have
been added for $n^{\prime}\leq{\widetilde{n}}-1$ while satisfying four
criteria. First,
$v_{q,{\widetilde{n}}-1}=v_{\ell_{q}}+\sum\limits_{n^{\prime}\leq{\widetilde{n}}-1}w_{q+1,n^{\prime}}$
solves an Euler-Reynolds system with stresses
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$ and
$\mathring{H}_{q,n,p}^{n^{\prime}}$. Secondly, the perturbations
$w_{q+1,n^{\prime}}$ satisfy the inductive assumptions required of $w_{q+1}$
in (7.3) for $n^{\prime}\leq{\widetilde{n}}-1$. Thirdly,
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$ satisfies the inductive assumption
(7.5) at level $q+1$. Finally, $\mathring{H}_{q,n,p}^{n^{\prime}}$ satisfies
the assumption (6.118) in the parameter regime ${\widetilde{n}}\leq
n\leq{n_{\rm max}}$, $n^{\prime}\leq{\widetilde{n}}-1$, $1\leq p\leq{p_{\rm
max}}$. The conclusion of the proposition replaces each ${\widetilde{n}}-1$ in
the assumptions with ${\widetilde{n}}$.

######  Proposition 7.4 (Induction on ${\widetilde{n}}$: From
${\widetilde{n}}-1$ to ${\widetilde{n}}$ for $1\leq{\widetilde{n}}\leq{n_{\rm
max}}-1$).

Let $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$ be given, and let

$v_{q,{\widetilde{n}}-1}:=v_{\ell_{q}}+\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}w_{q+1,n^{\prime}}=v_{\ell_{q}}+\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\sum\limits_{p^{\prime}=1}^{p_{\rm
max}}w_{q+1,n^{\prime},p^{\prime}},$

$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$, and
$\mathring{H}_{q,n,p}^{n^{\prime}}$ be given for
$n^{\prime}\leq{\widetilde{n}}-1$, ${\widetilde{n}}\leq n\leq{n_{\rm max}}$
and $1\leq p,p^{\prime}\leq{p_{\rm max}}$ such that the following are
satisfied.

  1. (1)

${v_{q,{\widetilde{n}}-1}}$ solves:

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}-1}$
$\displaystyle+\mathrm{div\,}(v_{q,{\widetilde{n}}-1}\otimes
v_{q,{\widetilde{n}}-1})+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)+\mathrm{div\,}\left(\displaystyle\sum\limits_{n={\widetilde{n}}}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
(7.17a) $\displaystyle\mathrm{div\,}v_{q,{\widetilde{n}}-1}$
$\displaystyle=0\,.$ (7.17b)

  2. (2)

For all
$k+m\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}^{\prime}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$,
$n^{\prime}\leq{\widetilde{n}}-1$, and $1\leq p^{\prime}\leq{p_{\rm max}}$,

$\left\|D^{k}D_{t,q}^{m}w_{q+1,n^{\prime},p^{\prime}}\right\|_{L^{2}\left(\mathrm{supp\,}\psi_{i,q}\right)}\lesssim\delta_{q+1,n^{\prime},p^{\prime}}^{\frac{1}{2}}\Gamma_{q+1}^{3+\frac{{\mathsf{C}_{b}}}{2}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}^{\prime}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.18)

Furthermore, we have that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q,n^{\prime},p^{\prime}})$
$\displaystyle\subset[T_{1,n^{\prime},p^{\prime}},T_{2,n^{\prime},p^{\prime}}]$
$\displaystyle\Rightarrow\mathrm{supp\,}_{t}(w_{q+1,n^{\prime},p^{\prime}})\subset\left[T_{1,n^{\prime},p^{\prime}}-(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1},T_{2,n^{\prime},p^{\prime}}+(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1}\right]\,.$
(7.19)

  3. (3)

For all $k,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$,

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right\|_{L^{1}}\lesssim\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right).$
(7.20)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{R}_{q+1}^{{\widetilde{n}}-1}\subseteq\bigcup_{n^{\prime}\leq{\widetilde{n}}-1}\mathrm{supp\,}_{t}w_{q+1,n^{\prime}}\,.$
(7.21)

  4. (4)

For all $k+m\leq\mathsf{N}_{\rm fin,n}$, ${\widetilde{n}}\leq n\leq{n_{\rm
max}}$, $n^{\prime}\leq{\widetilde{n}}-1$, and $1\leq p\leq{p_{\rm max}}$,

$\left\|D^{k}D_{t,q}^{m}\mathring{H}_{q,n,p}^{n^{\prime}}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}\lesssim\delta_{q+1,n,p}\lambda_{q,n,p}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.22)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{H}_{q,n,p}^{n^{\prime}}\subseteq\mathrm{supp\,}_{t}w_{q+1,n^{\prime}}\,.$
(7.23)

Then there exists $w_{q+1,{\widetilde{n}}}$ such that (1)-(4) are satisfied
with ${\widetilde{n}}-1$ replaced with ${\widetilde{n}}$.

The final proposition considers the case $\widetilde{n}={n_{\rm max}}$ and
shows that, under assumptions analogous to those in Proposition 7.4, there
exists $w_{q+1,{n_{\rm max}}}$ such that all remaining errors after the
addition of $w_{q+1,{n_{\rm max}}}$ can be absorbed into $\mathring{R}_{q+1}$,
thus verifying the conclusions of Proposition 7.1.

######  Proposition 7.5 (Induction on ${\widetilde{n}}$: The Final Case
${\widetilde{n}}={n_{\rm max}}$).

Let

$v_{q,{n_{\rm max}}-1}:=v_{\ell_{q}}+\sum\limits_{n^{\prime}=0}^{{n_{\rm
max}}-1}w_{q+1,n^{\prime}}=v_{\ell_{q}}+\sum_{n^{\prime}=0}^{{n_{\rm
max}}-1}\sum_{p^{\prime}=1}^{p_{\rm max}}w_{q+1,n^{\prime},p^{\prime}}$

$\mathring{R}_{q+1}^{{n_{\rm max}}-1}$, and $\mathring{H}_{q,{n_{\rm
max}},p}^{n^{\prime}}$ be given for $n^{\prime}\leq{n_{\rm max}}-1$ and $1\leq
p,p^{\prime}\leq{p_{\rm max}}$ such that the following are satisfied.

  1. (1)

${v_{q,{n_{\rm max}}-1}}$ solves:

$\displaystyle\partial_{t}v_{q,{n_{\rm max}}-1}$
$\displaystyle+\mathrm{div\,}(v_{q,{n_{\rm max}}-1}\otimes v_{q,{n_{\rm
max}}-1})+\nabla p_{q,{n_{\rm max}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right)+\mathrm{div\,}\left(\displaystyle\sum\limits_{n^{\prime}=0}^{{n_{\rm
max}}-1}\sum\limits_{p=1}^{p_{\rm max}}\mathring{H}^{n^{\prime}}_{q,{n_{\rm
max}},p}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$ (7.24a)
$\displaystyle\mathrm{div\,}v_{q,{n_{\rm max}}-1}$ $\displaystyle=0\,.$
(7.24b)

  2. (2)

For all
$k+m\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}^{\prime}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$,
$n^{\prime}\leq{n_{\rm max}}-1$, and $1\leq p^{\prime}\leq{p_{\rm max}}$,

$\left\|D^{k}D_{t,q}^{m}w_{q+1,n^{\prime},p^{\prime}}\right\|_{L^{2}\left(\mathrm{supp\,}\psi_{i,q}\right)}\lesssim\delta_{q+1,n^{\prime},p^{\prime}}^{\frac{1}{2}}\Gamma_{q+1}^{3+\frac{{\mathsf{C}_{b}}}{2}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}^{\prime}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.25)

Furthermore, we have that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q,n^{\prime},p^{\prime}})$
$\displaystyle\subset[T_{1,n^{\prime},p^{\prime}},T_{2,n^{\prime},p^{\prime}}]$
$\displaystyle\Rightarrow\quad\mathrm{supp\,}_{t}(w_{q+1,n^{\prime},p^{\prime}})\subset\left[T_{1,n^{\prime},p^{\prime}}-(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1},T_{2,n^{\prime},p^{\prime}}+(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1})^{-1}\right]\,.$
(7.26)

  3. (3)

For all $k,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$,

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right\|_{L^{1}}\lesssim\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right).$
(7.27)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\subseteq\bigcup_{n^{\prime}\leq{n_{\rm
max}}-1}\mathrm{supp\,}_{t}w_{q+1,n^{\prime}}\,.$ (7.28)

  4. (4)

For all
$k+m\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}_{\textnormal{max}}}$,
$n^{\prime}\leq{n_{\rm max}}-1$, and $1\leq p\leq{p_{\rm max}}$

$\displaystyle\left\|D^{k}D_{t,q}^{m}\mathring{H}_{q,{n_{\rm
max}},p}^{n^{\prime}}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,{n_{\rm max}},p}\lambda_{q,{n_{\rm
max}},p}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}_{\textnormal{max}}}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.29)

Furthermore, we have that

$\mathrm{supp\,}_{t}\mathring{H}_{q,n,p}^{n^{\prime}}\subseteq\mathrm{supp\,}_{t}w_{q+1,n^{\prime}}\,.$
(7.30)

Then there exists $w_{q+1,{n_{\rm max}}}$ and $\mathring{R}_{q+1}$ such that
$v_{q+1}:=v_{q,{n_{\rm max}}-1}+w_{q+1,{n_{\rm max}}}$ and
$\mathring{R}_{q+1}$ satisfy conclusions (7.2), (7.3), (7.4), and (7.5) from
Proposition 7.1.

##  8 Proving the main inductive estimates

Because the proofs of Propositions 7.3, 7.4, and 7.5 will be comprised of
multiple arguments with many similarities, we divide up the proofs of the
Propositions into sections corresponding to these arguments.313131This
organization of proof avoids having to alternate between the definitions of
$w_{q+1,{\widetilde{n}},{\widetilde{p}}}$ and
$\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$ for all
$1\leq{\widetilde{n}}\leq{n_{\rm max}}$ and $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$. We judge that it is wiser to define all the perturbations
simultaneously under the assumptions of Propositions 7.3, 7.4, and 7.5.
Namely, we assume that each $\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$
exists and satisfies the enumerated properties, some of which may not be
verified until later. First, we define
$\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$ and
$w_{q+1,{\widetilde{n}},{\widetilde{p}}}$ in Section 8.1 for each
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$. Then, Section 8.2 collects estimates on
$w_{q+1,{\widetilde{n}},{\widetilde{p}}}$, thus verifying (7.11) and (7.12),
(7.18) and (7.19), and (7.25) and (7.26) at levels ${\widetilde{n}}=0$,
$1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, and ${\widetilde{n}}={n_{\rm
max}}$, respectively. Next, in Section 8.3 we separate out the different types
of error terms and write down the Euler-Reynolds system satisfied by
$v_{q,{\widetilde{n}}}$, which verifies (7.10), (7.17), and (7.24), again at
the respective values of ${\widetilde{n}}$.

The error estimates are then divided into five sections. We first estimate the
transport and Nash errors in Sections 8.4 and 8.5. The next section estimates
the Type 1 oscillation errors (notated with
$\mathring{H}_{q,n,p}^{\widetilde{n}}$), which are obtained via Littlewood-
Paley projectors $\mathbb{P}_{[q,n,p]}$. In the parameter regime $1\leq
n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm max}}$, Type 1 oscillation errors
will satisfy the estimates (7.15), (7.22), and (7.29) at respective parameter
values ${\widetilde{n}}=0$, $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, and
${\widetilde{n}}={n_{\rm max}}$. Type 1 oscillation errors obtained from
$\mathbb{P}_{[q,{n_{\rm max}},{p_{\rm max}}+1]}$ have a sufficiently high
minimum frequency (from (7.6) specifically $\lambda_{q,{n_{\rm max}}+1,0}$,
which by a large choice of ${n_{\rm max}}$ is very close to $\lambda_{q+1}$)
to be absorbed into $\mathring{R}_{q+1}$. Then in Section 8.7, we use
Proposition 4.8 to show that on the support of a checkerboard cutoff function,
Type 2 oscillation errors vanish. The divergence corrector errors are
estimated in Sections 8.8. The divergence corrector, Nash, and transport
errors will always be absorbed into $\mathring{R}_{q+1}$ and thus must again
satisfy one of (7.13), (7.20), or (7.27). Finally, the conclusions (7.12),
(7.14), (7.16), (7.19), (7.21), (7.23), (7.26), (7.28), and (7.30), concerning
the time support will be verified in Section 8.9.

###  8.1 Definition of $\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$ and
$w_{q+1,{\widetilde{n}},{\widetilde{p}}}$

In this section we construct the perturbations $w_{q+1,{\widetilde{n}}}$.
Before doing so, we recall the significance of each parameter used to define
the perturbations.

  1. (a)

$\xi$ is the vector direction of the axis of the pipe

  2. (b)

$i$ quantifies the amplitude of the velocity field $v_{\ell_{q}}$ along which
the pipe will be flowed

  3. (c)

$j$ quantifies the amplitude of the Reynolds stress

  4. (d)

$k$ describes which time cut-off $\chi_{i,k,q}$ is active

  5. (e)

$q+1$ is the stage of the overall convex integration scheme

  6. (f)

${\widetilde{n}}$ and ${\widetilde{p}}$ signify which higher order stress
$\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$ is being corrected, and
${\widetilde{n}}$ also denotes the intermittency parameter
$r_{q+1,{\widetilde{n}}}$

  7. (g)

$\vec{l}=(l,w,h)$ is used to index the checkerboard cutoff functions. Recall
that the admissible values of $l$, $w$, and $h$ range from $0$ to
$\lambda_{q,{\widetilde{n}},0}-1$ and thus depend on ${\widetilde{n}}$.

####  8.1.1 The case ${\widetilde{n}}=0$

To define $\displaystyle w_{q+1,0}=\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,0,p}=w_{q+1,0,1}$, we recall the notation
$\mathring{R}_{\ell_{q}}=\mathring{R}_{q,0}$ and set

$R_{q,0,1,j,i,k}=\nabla\Phi_{(i,k)}\left(\delta_{q+1,0,1}\Gamma^{2j+4}_{q+1}\mathrm{Id}-\mathring{R}_{q,0}\right)\nabla\Phi_{(i,k)}^{T}.$
(8.1)

For ${\widetilde{p}}\geq 2$, we set $R_{q,0,{\widetilde{p}},j,i,k}=0$. Fix
values of $i$, $j$, and $k$. Let $\xi\in\Xi$ be a vector from Proposition 4.1.
For all $\xi\in\Xi$, we define the coefficient function
$a_{\xi,i,j,k,q,0,{\widetilde{p}},\vec{l}}$ by

$a_{\xi,i,j,k,q,0,{\widetilde{p}},\vec{l}}:=a_{\xi,i,j,k,q,0,{\widetilde{p}}}:=a_{(\xi)}=\delta_{q+1,0,{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\eta_{i,j,k,q,0,{\widetilde{p}},\vec{l}}\gamma_{\xi}\left(\frac{R_{q,0,{\widetilde{p}},j,i,k}}{\delta_{q+1,0,{\widetilde{p}}}\Gamma^{2j+4}_{q+1}}\right)\,.$
(8.2)

From Lemma 6.31, we see that on the support of $\eta_{(i,j,k)}$ we have
$|\mathring{R}_{q,0,{\widetilde{p}}}|\leq\Gamma_{q+1}^{2j+2}\delta_{q+1,0,{\widetilde{p}}}$,
and thus by estimate (6.108) from Corollary 6.27, for ${\widetilde{p}}=1$ we
have that

$\left|\frac{R_{q,0,{\widetilde{p}},j,i,k}}{\delta_{q+1,0,{\widetilde{p}}}\Gamma^{2j+4}_{q+1}}-\mathrm{Id}\right|\leq\Gamma_{q+1}^{-1}<\frac{1}{2}$

once $\lambda_{0}$ is sufficiently large. Thus we may apply Proposition 4.1.

The coefficient function $a_{(\xi)}$ is then multiplied by an intermittent
pipe flow

$\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,0}\circ\Phi_{(i,k)},$

where we have used the objects defined in Proposition 4.4 and the shorthand
notation

$\displaystyle\mathbb{W}_{\xi,q+1,0}=\mathbb{W}^{(i,j,k,0,\vec{l})}_{\xi,q+1,0}=\mathbb{W}_{\xi,q+1,0}^{s}=\mathbb{W}^{s}_{\xi,\lambda_{q+1},r_{q+1,0}}.$
(8.3)

The superscript $s=(i,j,k,0,\vec{l})$ indicates the placement of the
intermittent pipe flow $\mathbb{W}^{i,j,k,0,p,\vec{l}}_{\xi,q+1,0}$ (cf. (2)
from Proposition 4.4), which depends on $i$, $j$, $k$, ${\widetilde{n}}=0$,
and $\vec{l}$ and is only relevant in Section 8.7.323232Note that for
${\widetilde{p}}\geq 2$, $\delta_{q+1,0,{\widetilde{p}}}=0$, so there is no
need for the placement to depend on ${\widetilde{p}}$ in this case, as
$w_{q+1,0,{\widetilde{p}}}$ will uniformly vanish. To ease notation, we will
suppress the superscript except in Section 8.7. Furthermore, item 1 from
Proposition 4.4 gives that

$\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,0}\circ\Phi_{(i,k)}=\mathrm{curl\,}\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right).$

We can now write the principal part of the first term of the perturbation as

$w_{q+1,0}^{(p)}=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}a_{(\xi)}\mathrm{curl\,}\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right):=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}.$
(8.4)

The notation $w_{(\xi)}$ implicitly encodes all indices and thus will be a
useful shorthand for the principal part of the perturbation. To make the
perturbation divergence free, we add

$w_{q+1,0}^{(c)}=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}^{(c)}$
(8.5)

so that

$w_{q+1,0}=w_{q+1,0}^{(p)}+w_{q+1,0}^{(c)}=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)$
(8.6)

is divergence-free and mean-zero.

####  8.1.2 The case $1\leq{\widetilde{n}}\leq{n_{\rm max}}$

With $w_{q+1,0}$ constructed, we construct $\displaystyle
w_{q+1,{\widetilde{n}}}=\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,{\widetilde{n}},{\widetilde{p}}}$ for
$1\leq{\widetilde{n}}\leq{n_{\rm max}}$. For $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$, we define

$\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}=\sum_{n^{\prime}\leq{\widetilde{n}}-1}\mathring{H}_{q,{\widetilde{n}},{\widetilde{p}}}^{n^{\prime}}.$
(8.7)

With this definition in hand, we set

$R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}=\nabla\Phi_{(i,k)}\left(\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1}\mathrm{Id}-\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right)\nabla\Phi_{(i,k)}^{T},$
(8.8)

and define the coefficient function
$a_{\xi,i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}$ by

$a_{\xi,i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}=a_{\xi,i,j,k,q,{\widetilde{n}},{\widetilde{p}}}=a_{(\xi)}=\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}\gamma_{\xi}\left(\frac{R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}}{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1}}\right)\,.$
(8.9)

By Lemma 6.31 and Corollary 6.27 as before,
$R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}/(\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1})$
lies in the domain of $\gamma_{\xi}$, as soon as $\lambda_{0}$ is sufficiently
large (similarly to the display below (8.2)). The coefficient function is
multiplied by an intermittent pipe flow

$\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}=\mathrm{curl\,}\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right),$

where we have used the shorthand notation

$\displaystyle\mathbb{W}_{\xi,q+1,{\widetilde{n}}}=\mathbb{W}^{i,j,k,{\widetilde{n}},{\widetilde{p}},\vec{l}}_{\xi,q+1,{\widetilde{n}}}=\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{s}=\mathbb{W}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{s}\,.$
(8.10)

As before, the superscript $s=(i,j,k,{\widetilde{n}},{\widetilde{p}},\vec{l})$
refers to the placement of the pipe, depends on $i$, $j$, $k$,
${\widetilde{n}}$, ${\widetilde{p}}$, and $\vec{l}$, and will be chosen in
Section 8.7. Thus the principal part of the perturbation is defined by

$w_{q+1,{\widetilde{n}},{\widetilde{p}}}^{(p)}=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}a_{(\xi)}\mathrm{curl\,}\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}.$
(8.11)

As before, we add a corrector

$w_{q+1,{\widetilde{n}},{\widetilde{p}}}^{(c)}=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}^{(c)},$
(8.12)

producing the divergence-free perturbation

$\displaystyle w_{q+1,{\widetilde{n}}}=\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,{\widetilde{n}},{\widetilde{p}}}$
$\displaystyle=\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\left(w_{q+1,{\widetilde{n}},{\widetilde{p}}}^{(p)}+w_{q+1,{\widetilde{n}},{\widetilde{p}}}^{(c)}\right)$
$\displaystyle=\sum_{i,j,k,{\widetilde{p}}}\sum_{\vec{l}}\sum_{\xi}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\,.$
(8.13)

###  8.2 Estimates for $w_{q+1,{\widetilde{n}},{\widetilde{p}}}$

In this section, we verify (7.11), (7.18), and (7.25). We first estimate the
$L^{r}$ norms of the coefficient functions $a_{(\xi)}$. We have consolidated
the proofs for each value of ${\widetilde{n}}$ into the following lemma.

######  Lemma 8.1.

For $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-4$, $r\geq 1$, and $r_{1},r_{2}\in[1,\infty]$
with $\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$, we have

$\displaystyle\left\|D^{N}D_{t,q}^{M}a_{\xi,i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}\right\|_{L^{r}}$
$\displaystyle\qquad\lesssim\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(8.14)

###### Proof of Lemma 8.1.

We begin by considering the case $r=\infty$. The general case $r\geq 1$ will
then follow from the size of the support of $a_{(\xi)}$. Recalling estimate
(6.125), we have that for all
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$

From Corollary 6.27, we have that for all $N+M\leq\nicefrac{{3\mathsf{N}_{\rm
fin}}}{{2}}$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}D\Phi_{(i,k)}\right\|_{L^{\infty}(\mathrm{supp\,}(\psi_{i,q}\chi_{i,k,q}))}$
$\displaystyle\leq\widetilde{\lambda}_{q}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c_{0}}}\tau_{q}^{-1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$

Thus from the Leibniz rule and the definitions (8.8), (8.1), for
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}$
$\displaystyle\qquad\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(8.15)

The above estimates allow us to apply Lemma A.5 with $N=N^{\prime}$,
$M=M^{\prime}$ so that $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,
$\psi=\gamma_{\xi,}$ (which is allowable since by Proposition 4.1 we have that
$D^{B}\gamma_{\xi}$ is bounded uniformly with respect to $q$, and we have
checked in Section 8.1 that the argument of $\gamma_{\xi}$ remains strictly
within a ball of radius $\varepsilon$ of the identity), $\Gamma_{\psi}=1$,
$v=v_{\ell_{q}}$, $D_{t}=D_{t,q}$,
$h(x,t)=R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}(x,t)$,
$C_{h}=\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}=\Gamma^{2}$,
$\lambda=\widetilde{\lambda}=\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}$,
$\mu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2}$,
$\widetilde{\mu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, and
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$. We obtain that for all
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\gamma_{\xi}\left(\frac{R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}}{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1}}\right)\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}$
$\displaystyle\lesssim\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

From the above bound, definitions (8.2) and (8.9), the Leibniz rule, estimates
(6.84), (6.97), (6.131), and Lemma 6.40, we obtain that for
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-4$,333333The limit on the number of derivatives
comes from (6.131) and (8.15). The sharp cost of a material derivative comes
from (6.131).

$\displaystyle\left\|D^{N}D_{t,q}^{M}a_{(\xi)}\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\sum_{\begin{subarray}{c}N^{\prime}+N^{\prime\prime}=N,\\\
M^{\prime}+M^{\prime\prime}=M\end{subarray}}\left\|D^{N^{\prime}}D_{t,q}^{M^{\prime}}\eta_{(i,j,k)}\right\|_{L^{\infty}}\left\|D^{N^{\prime\prime}}D_{t,q}^{M^{\prime\prime}}\gamma_{\xi}\left(\frac{R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}}{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma^{2j+4}_{q+1}}\right)\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\sum_{\begin{subarray}{c}N^{\prime}+N^{\prime\prime}=N,\\\
M^{\prime}+M^{\prime\prime}=M\end{subarray}}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N^{\prime}}\mathcal{M}\left(M^{\prime},\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N^{\prime\prime}}\mathcal{M}\left(M^{\prime\prime},\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

This concludes the proof of (8.14) when $r=\infty$. For general $r\geq 1$, we
just note that
$\mathrm{supp\,}(a_{(\xi)})\subseteq\mathrm{supp\,}(\eta_{i,j,k,q,n,p,\vec{l}})$.
∎

An immediate consequence of Lemma 8.1 is that we have estimates for the
velocity increments themselves. These are summarized in the following
corollary.

######  Corollary 8.2.

For $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-8$ we have the estimate

$\displaystyle\left\|D^{N}D_{t,q}^{M}w_{(\xi)}\right\|_{L^{r}}$
$\displaystyle\lesssim\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}{\left(r_{q+1,{\widetilde{n}}}\right)}^{\nicefrac{{2}}{{r}}-1}$
$\displaystyle\qquad\qquad\qquad\times\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(8.16)

For $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$ and
$(r,r_{1},r_{2})\in\left\\{(1,2,2),(2,\infty,1)\right\\}$, we have the
estimates

$\displaystyle\left\|D^{N}D_{t,q}^{M}w_{(\xi)}^{(c)}\right\|_{L^{r}}$
$\displaystyle\lesssim\frac{\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}{\lambda_{q+1}}\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}{\left(r_{q+1,{\widetilde{n}}}\right)}^{\nicefrac{{2}}{{r}}-1}$
$\displaystyle\qquad\qquad\qquad\times\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(8.17)
$\displaystyle\left\|D^{N}D_{t,q}^{M}w_{q+1,{\widetilde{n}},{\widetilde{p}}}\right\|_{L^{r}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{\frac{-2i+{\mathsf{C}_{b}}}{r_{1}r}+2+\frac{2}{r}}_{q+1}{\left(r_{q+1,{\widetilde{n}}}\right)}^{\nicefrac{{2}}{{r}}-1}$
$\displaystyle\qquad\qquad\qquad\times\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(8.18)

Finally, we have that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q})\subset[T_{1},T_{2}]\quad\Rightarrow\quad\mathrm{supp\,}_{t}(w_{q+1,{\widetilde{n}},{\widetilde{p}}})\subset\left[T_{1}-(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}})^{-1},T_{2}+(\lambda_{q}\delta_{q}^{\nicefrac{{1}}{{2}}})^{-1}\right]\,.$
(8.19)

######  Remark 8.3.

By choosing $r=2$, $r_{2}=1$, and $r_{1}=\infty$ in (8.18) and recalling that
(9.56) and (9.60b) give that

$\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\leq\Gamma_{q+1}^{-2}\delta_{q+1}^{\nicefrac{{1}}{{2}}},\qquad\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9\geq
14\mathsf{N}_{\textnormal{ind,v}},$

we may sum over ${\widetilde{n}}$ and ${\widetilde{p}}$ in (8.18) and use the
extra negative factor of $\Gamma_{q+1}$ to absorb any implicit constants.
Finally, from (9.42), we have that the cost of a sharp material derivative in
(8.18) is sufficient to meet the bounds in (7.3). Then we have verified
(7.11), (7.18), and (7.25) at levels ${\widetilde{n}}=0$,
$1\leq{\widetilde{n}}<{n_{\rm max}}$, and ${\widetilde{n}}={n_{\rm max}}$,
respectively, and (7.3).

###### Proof of Corollary 8.2.

Recalling the definition of $w_{(\xi)}$ from (8.4) and (8.13), we aim to prove
the first estimate by applying Remark A.9, with
$f=a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}$,
$\mathcal{C}_{f}=\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}$,
$\Phi=\Phi_{(i,k)}$, $v=v_{\ell_{q}}$,
$\lambda=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}$,
$\zeta=\widetilde{\zeta}=\lambda_{q+1}$,
$\mathcal{C}_{\varphi}=r_{q+1,{\widetilde{n}}}^{\nicefrac{{2}}{{r}}-1}$,
$\mu=\lambda_{q,{\widetilde{n}}}=\lambda_{q+1}r_{q+1,{\widetilde{n}}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$,
$g=\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$,
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$, and
$N_{\circ}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-4$. From (8.14) and Corollary 6.27, we have that
for $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-4$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}a_{(\xi)}\right\|_{L^{r}}\lesssim\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(8.20)
$\displaystyle\left\|D^{N}D_{t,q}^{M}(D\Phi_{(i,k)})^{-1}\right\|_{L^{\infty}(\mathrm{supp\,}(\psi_{i,q}\widetilde{\chi}_{i,k,q}))}\leq\widetilde{\lambda}_{q}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c_{0}}}\tau_{q}^{-1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$
(8.21)
$\displaystyle{\left\|D^{N}\Phi_{(i,k)}\right\|_{L^{\infty}(\mathrm{supp\,}(\psi_{i,q}\widetilde{\chi}_{i,k,q}))}}\lesssim\Gamma_{q+1}^{-1}\widetilde{\lambda}_{q}^{N-1}$
(8.22)
$\displaystyle\left\|D^{N}\Phi^{-1}_{(i,k)}\right\|_{L^{\infty}(\mathrm{supp\,}(\psi_{i,q}\widetilde{\chi}_{i,k,q}))}\lesssim\Gamma_{q+1}^{-1}\widetilde{\lambda}_{q}^{N-1}$
(8.23)

showing that (A.30), (A.31), and (A.32) are satisfied. Recall that
$\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$ is periodic to scale

$\displaystyle{\lambda_{q,{\widetilde{n}}}^{-1}=\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}=\left(\lambda_{q}^{\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}}\right)^{-1}}.$

By (9.48) and (9.60a), we have that for all $q$, ${\widetilde{n}}$, and
${\widetilde{p}}$,

$\displaystyle\lambda_{q+1}^{4}\leq\left(\frac{\lambda_{q,{\widetilde{n}}}}{2\pi\sqrt{3}\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}\right)^{{\mathsf{N}_{\rm
dec}}},\qquad 2{\mathsf{N}_{\rm
dec}}+4\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$ (8.24)

and so the assumptions (A.34) and (A.35) from Lemma A.5 are satisfied. From
the estimates in Proposition 4.4, we have that (A.33) is satisfied with
$\zeta=\widetilde{\zeta}=\lambda_{q+1}$. We may thus apply Lemma A.7, Remark
A.9 to obtain that for both choices of $(r,r_{1},r_{2})$ and
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-8$,

$\displaystyle\left\|D^{N}\left(D_{t,q}^{M}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\right)\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right\|_{L^{r}}$
$\displaystyle\lesssim\sum_{m=0}^{N}\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N-m}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\left\|D^{m}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\right\|_{L^{r}}$
$\displaystyle\lesssim\sum_{m=0}^{N}\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N-m}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\lambda_{q+1}^{m}\left(r_{q+1,n}\right)^{\nicefrac{{2}}{{r}}-1}$
$\displaystyle\lesssim\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\lambda_{q+1}^{N}\left(r_{q+1,n}\right)^{\nicefrac{{2}}{{r}}-1}\,.$

Here we have used that
${\lambda_{q+1}\geq\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}$
for all $0\leq n\leq{n_{\rm max}}$ and $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$, and thus we have proven (8.16).

The argument for the corrector is similar, save for the fact that $D_{t,q}$
will land on $\nabla a_{(\xi)}$, and so we require an extra commutator
estimate from Lemma A.14, specifically Remark A.15. Note that
$D_{t,q}\Phi_{(i,k)}=0$ gives

$\displaystyle D_{t,q}^{M}w_{(\xi)}^{(c)}$
$\displaystyle=D_{t,q}^{M}\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle=\sum_{M^{\prime}+M^{\prime\prime}=M}c(M^{\prime},M)\left(D_{t,q}^{M^{\prime}}\nabla
a_{(\xi)}\right)\times\left(\left(D_{t,q}^{M^{\prime\prime}}\nabla\Phi_{(i,k)}^{T}\right)\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right).$

Using (6.60) and (8.20) shows that (A.50) and (A.51) are satisfied with
$f=\nabla a_{(\xi)}$,

$\mathcal{C}_{f}=\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}},$

$\mathcal{C}_{v}=\delta_{q}^{\frac{1}{2}}\Gamma_{q+1}^{i+1}$,
$\lambda_{v}=\widetilde{\lambda}_{v}=\widetilde{\lambda}_{q}$,
$\mu_{v}=\Gamma_{q+1}^{i-\mathsf{c_{0}}}\tau_{q}^{-1}$,
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\widetilde{\mu}_{v}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$,
$\lambda_{f}=\widetilde{\lambda}_{f}=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}$,
$\mu_{f}=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3}$,
and $\widetilde{\mu}_{f}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$. Applying
Lemma A.14 (estimate (A.54)) as before, we obtain that for
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\nabla
a_{(\xi)}\right\|_{L^{r}}\lesssim\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}})^{N+1}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(8.25)

In view of (8.21) and (8.24), we may apply Lemma A.7, Remark A.9, and the
estimates from Proposition 4.4 to obtain that for
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)\right\|_{L^{r}}$
$\displaystyle\
\lesssim\sum_{m=0}^{N}\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}^{N-m}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\left\|D^{m}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\right\|_{L^{r}}$
$\displaystyle\lesssim\lambda_{q+1}^{m-1}\sum_{m=0}^{N}\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}^{N-m}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\left(r_{q+1,n}\right)^{\frac{2}{r}-1}$
$\displaystyle\lesssim\frac{\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}{\lambda_{q+1}}\lambda_{q+1}^{N}\bigl{|}\mathrm{supp\,}(\eta_{(i,j,k)})\bigr{|}^{\frac{1}{r}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\left(r_{q+1,n}\right)^{\nicefrac{{2}}{{r}}-1}\,,$
(8.26)

proving (8.17).

The final estimate (8.18) follows from the first two after recalling that
$\psi_{i,q}$ may overlap with $\psi_{i+1,q}$, so that on the support of
$\psi_{i,q}$, we will have to appeal to (8.14) at level $i+1$. Then, we sum
over $\vec{l}$ and appeal to the bound (6.147). Next, we may sum on $j$, index
which we recall from Lemma 6.35 is bounded independently of $q$, and
${\widetilde{p}}$, $k$. The powers of $\Gamma_{q+1}^{j}$ cancel out since
$rr_{2}=1$. Next, we sum over ${\widetilde{p}}$, which is bounded
independently of $q$, and recall that the parameter $k$, although not bounded
independently of $q$, corresponds to a partition of unity, so that the number
of cutoff functions which may overlap at any fixed point _is_ finite and
bounded independently of $q$. ∎

###  8.3 Identification of error terms

In this section, we identify the error terms arising from the addition of
$\displaystyle w_{q+1,{\widetilde{n}}}=\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}w_{q+1,{\widetilde{n}},{\widetilde{p}}}$. After doing so, we can write
down the Euler-Reynolds system satisfied by $v_{q,{\widetilde{n}}}$, in turn
verifying at level ${\widetilde{n}}$ the conclusions (7.10), (7.17), and
(7.24) of Propositions 7.3, 7.4, and 7.5, respectively.

####  8.3.1 The case ${\widetilde{n}}=0$

By the inductive assumption of Proposition 7.3, we have that
$\mathrm{div\,}v_{\ell_{q}}=0$, and

$\partial_{t}{v_{\ell_{q}}}+\mathrm{div\,}(v_{\ell_{q}}\otimes
v_{\ell_{q}})+\nabla
p_{\ell_{q}}=\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}.$

Adding $w_{q+1,0}$ as defined in (8.6), we obtain that
$v_{q,0}:=v_{\ell_{q}}+w_{q+1,0}$ solves

$\displaystyle\partial_{t}v_{q,0}+\mathrm{div\,}(v_{q,0}\otimes
v_{q,0})+\nabla p_{\ell_{q}}$
$\displaystyle=(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,0}+w_{q+1,0}\cdot\nabla
v_{\ell_{q}}+\mathrm{div\,}\left(w_{q+1,0}\otimes
w_{q+1,0}\right)+\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle:=\mathcal{T}_{0}+\mathcal{N}_{0}+\mathcal{O}_{0}+\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}.$
(8.27)

For a fixed ${\widetilde{n}}$, throughout this section we will consider sums
over indices

$\displaystyle(\xi,i,j,k,\widetilde{p},\vec{l})$

where the direction vector $\xi$ takes on one of the finitely many values in
Proposition 4.4, $0\leq i\leq{i_{\rm max}}(q)$ indexes the velocity cutoffs
(there are finitely many such values, cf. (6.50)), $0\leq j\leq{j_{\rm
max}}(q,{\widetilde{n}},{\widetilde{p}})$ indexes the stress cutoffs (there
are finitely many such values, cf. (6.129)), the parameter $k$ indexes the
time cutoffs defined in (6.96) (the number of values of $k$ is $q$-dependent,
but this is irrelevant because they form a partition of unity cf. (6.94)), the
parameter $1\leq\widetilde{p}\leq{p_{\rm max}}$ indexes which component of
$\mathring{R}_{q+1,{\widetilde{n}},{\widetilde{p}}}$ we are working with
(there are finitely many such values, cf. (9.3)), and lastly, $\vec{l}$
indexes the checkerboard cutoffs from Definition 6.39 (again, the number of
such indexes is $q$-dependent, but this is acceptable because they form a
partition of unity cf. (6.139)). For brevity of notation, we denote sums over
such indexes as

$\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\,.$

Moreover, we shall denote as

$\sum_{\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}}$ (8.28)

the double-summation over indexes $(\xi,i,j,k,\widetilde{p},\vec{l})$ and
$({\xi^{*}},{i^{*}},{j^{*}},{k^{*}},{p^{*}},\vec{l}^{*})$ which belong to the
set

$\left\\{(\xi,i,j,k,{\widetilde{p}},\vec{l},{\xi^{*}}),({i^{*}},{j^{*}},{k^{*}},{p^{*}},\vec{l}^{*}):\xi\neq{\xi^{*}}\lor
i\neq{i^{*}}\lor j\neq{j^{*}}\lor
k\neq{k^{*}}\lor{\widetilde{p}}\neq{p^{*}}\lor\vec{l}\neq\vec{l}^{*}\right\\},$
(8.29)

although we remind the reader that at the current stage ${\widetilde{n}}=0$,
the sum over ${\widetilde{p}}$ is superfluous since $w_{q+1,0}=w_{q+1,0,1}$.
For the sake of consistency between $w_{q+1,0}$ and $w_{q+1,{\widetilde{n}}}$
for $1\leq{\widetilde{n}}\leq{n_{\rm max}}$, we shall include the index
${\widetilde{p}}$ throughout this section. Expanding out the oscillation error
$\mathcal{O}_{0}$, we have that

$\displaystyle\mathcal{O}_{0}$
$\displaystyle=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\otimes\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle+\sum_{\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\otimes\mathrm{curl\,}\left(a_{({\xi^{*}})}\nabla\Phi_{({i^{*}},{k^{*}})}^{T}\mathbb{U}_{{\xi^{*}},q+1,0}\circ\Phi_{({i^{*}},{k^{*}})}\right)\right)$
$\displaystyle:=\mathrm{div\,}\mathcal{O}_{0,1}+\mathrm{div\,}\mathcal{O}_{0,2}.$
(8.30)

In Section 8.7, we will show that $\mathcal{O}_{0,2}$ is a Type 2 oscillation
error so that

$\mathcal{O}_{0,2}=0.$

Recalling identity (4.14) and the notation (9.65), we further split
$\mathcal{O}_{0,1}$ as

$\displaystyle\mathrm{div\,}\mathcal{O}_{0,1}=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\otimes\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle\qquad+2\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\otimes_{\mathrm{s}}\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\right)\otimes\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle:=\mathrm{div\,}\left(\mathcal{O}_{0,1,1}+\mathcal{O}_{0,1,2}+\mathcal{O}_{0,1,3}\right).$
(8.31)

Aside from $\mathcal{O}_{0,1,1}$, each of these terms is a divergence
corrector error and will therefore be estimated in Section 8.8.

Recall by Propositions 4.3, 4.4, and by (8.3) that $\mathbb{W}_{\xi,q+1,0}$ is
periodized to scale
$\left(\lambda_{q+1}r_{q+1,0}\right)^{-1}=\lambda_{q,0}^{-1}$. Using the
definition of $\mathbb{P}_{[q,n,p]}$ and (7.8), we have that343434The case
${\widetilde{n}}=0$ is exceptional in the sense that the minimum frequency of
$\mathbb{P}_{\geq\lambda_{q,0}}$ and the minimum frequency of
$\mathbb{P}_{[q,1,0]}$ are in fact both equal to
$\lambda_{q,0}=\lambda_{q,1,0}=\lambda_{q}^{\frac{4}{5}}\lambda_{q+1}^{\frac{1}{5}}$
from (9.27) and (9.22). For the sake of consistency with the
${\widetilde{n}}\geq 1$ cases, we will include the superfluous
$\mathbb{P}_{\geq\lambda_{q,0}}$ in the calculations in this section.

$\displaystyle\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}{\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0}}$
$\displaystyle+\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)\left(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0}\right)\,.$

Combining this observation with identity (4.15) from Proposition 4.4, and with
the definition of the $a_{(\xi)}$ in (8.2), we further split
$\mathcal{O}_{0,1,1}$ as

$\displaystyle\mathrm{div\,}\left(\mathcal{O}_{0,1,1}\right)=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0}(\Phi_{(i,k)})\right)\nabla\Phi_{(i,k)}^{-T}\right)$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\bigg{(}a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,0}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle=\mathrm{div\,}\sum_{i,j,k,{\widetilde{p}},\vec{l}}\sum_{\xi}\delta_{q+1,0,{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,0,{\widetilde{p}},j,i,k}}{\delta_{q+1,0,{\widetilde{p}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\times\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,0}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}$
$\displaystyle\qquad\times\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,0}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\,.$
(8.32)

By Proposition 4.1, equation (4.1), and the definition (8.1), we may rewrite
the first term on the right side of the above display as

$\displaystyle\mathrm{div\,}\sum_{i,j,k,{\widetilde{p}},\vec{l}}\sum_{\xi}\delta_{q+1,0,{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,0,{\widetilde{p}},j,i,k}}{\delta_{q+1,0,{\widetilde{p}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad=\mathrm{div\,}\sum_{i,j,k,\vec{l}}\eta_{(i,j,k)}^{2}\left(\delta_{q+1,0,1}\Gamma_{q+1}^{2j+4}\mathrm{Id}-\mathring{R}_{\ell_{q}}\right)$
$\displaystyle\qquad=-\mathrm{div\,}\sum_{i,j,k,\vec{l}}\eta^{2}_{(i,j,k)}\mathring{R}_{\ell_{q}}+\nabla\left(\sum_{i,j,k,\vec{l}}\eta_{(i,j,k)}^{2}\delta_{q+1,0,1}\Gamma_{q+1}^{2j+4}\right)$
$\displaystyle\qquad:=-\mathrm{div\,}\left(\mathring{R}_{\ell_{q}}\right)+\nabla\pi$
(8.33)

In the last equality of the above display we have used the fact that by
(6.142) we have

$\mathring{R}_{\ell_{q}}=\sum_{i,j,k,\vec{l}}\eta_{(i,j,k)}^{2}\mathring{R}_{\ell_{q}}\,.$
(8.34)

We apply Proposition A.18 to the remaining two terms from (8.32) to define for
$1\leq n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm max}}$ 353535Recall that
$\mathcal{H}$ is the local portion of the inverse divergence operator. The
pressure and the nonlocal portion will be accounted for shortly. We will check
in Section 8.6 that these errors are of the form required by the inverse
divergence operator as well as check the associated estimates.

$\displaystyle\mathring{H}_{q,n,p}^{0}$
$\displaystyle:=\mathcal{H}\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,0}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,0}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,0}^{\theta}\mathbb{W}_{\xi,q+1,0}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}.$
(8.35)

The last terms from (8.32) with $\mathbb{P}_{[q,{n_{\rm max}},{p_{\rm
max}}+1]}$ will be absorbed into $\mathring{R}_{q+1}$, whereas the terms in
(8.35) correspond to the error terms in (7.15).

Before amalgamating the preceding calculations, we pause to calculate the
means of various terms to which the inverse divergence operator from
Proposition A.18 will be applied. Examining the equality

$\partial_{t}v_{q,0}+\mathrm{div\,}\left(v_{q,0}\otimes v_{q,0}\right)+\nabla
p_{\ell_{q}}=\mathcal{T}_{0}+\mathcal{N}_{0}+\mathcal{O}_{0}+\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
(8.36)

and recalling the definitions of $\mathcal{T}_{0}$, $\mathcal{N}_{0}$, and
$\mathcal{O}_{0}$, we see immediately that every term can be written as the
divergence of a tensor except for $\partial_{t}v_{q,0}$ and $\mathcal{T}_{0}$.
Note however that $v_{q,0}=v_{\ell_{q}}+w_{q+1,0}$, that
$\int_{\mathbb{T}^{3}}\partial_{t}v_{\ell_{q}}=0$ (by integrating in space
(5.2)), and that $w_{q+1,0}$ is the curl of a vector field (cf. (8.13)). This
shows that $\int_{\mathbb{T}^{3}}\partial_{t}v_{q,0}=0$, and thus
$\int_{\mathbb{T}^{3}}\mathcal{T}_{0}=0$ as well. Therefore, we may use (A.72)
and (A.78) write

$\mathcal{T}_{0}=\mathrm{div\,}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)\mathcal{T}_{0}\right)+\nabla
P.$

We can now combine the calculations of (8.27), (8.30), (8.31), (8.32), (8.33)
(8.34), and (8.35) and let the notation $\nabla\pi$ change from line to line
to incorporate all the pressure terms to write that

$\displaystyle\partial_{t}v_{q,0}+\mathrm{div\,}\left(v_{q,0}\otimes
v_{q,0}\right)+\nabla p_{\ell_{q}}$
$\displaystyle=\mathcal{T}_{0}+\mathcal{N}_{0}+\mathcal{O}_{0}+\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle=\mathcal{T}_{0}+\mathcal{N}_{0}+\mathrm{div\,}\left(\mathcal{O}_{0,1}\right)+\mathrm{div\,}\left(\mathcal{O}_{0,2}\right)+\mathrm{div\,}\mathring{R}_{\ell_{q}}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle=\mathcal{T}_{0}+\mathcal{N}_{0}+\mathrm{div\,}\left(\mathring{R}_{\ell_{q}}+\mathcal{O}_{0,1,1}\right)+\mathrm{div\,}\left(\mathcal{O}_{0,1,2}+\mathcal{O}_{0,1,3}+\mathcal{O}_{0,2}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle=\mathcal{T}_{0}+\mathcal{N}_{0}+\nabla\pi$
$\displaystyle\quad+\mathrm{div\,}(\mathcal{H}+\mathcal{R}^{*})\bigg{[}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\quad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,0}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{]}$
(8.37)
$\displaystyle\qquad+\mathrm{div\,}\left(\mathcal{O}_{0,1,2}+\mathcal{O}_{0,1,3}+\mathcal{O}_{0,2}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle=\nabla\pi+\mathrm{div\,}\bigg{[}\underbrace{\left(\mathcal{H}+\mathcal{R}^{*}\right)(\mathcal{T}_{0})}_{\textnormal{transport}}+\underbrace{\left(\mathcal{H}+\mathcal{R}^{*}\right)(\mathcal{N}_{0})}_{\textnormal{Nash}}+\mathring{R}_{q}^{\textnormal{comm}}$
(8.38)
$\displaystyle\qquad+\underbrace{\left(\mathcal{H}+\mathcal{R}^{*}\right)\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{[q,{n_{\rm max}},{p_{\rm
max}}+1]}(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{0}}$ (8.39)
$\displaystyle\qquad\qquad+\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{[q,{n_{\rm
max}},{p_{\rm
max}}+1]}(\mathbb{W}_{\xi,q+1,0}^{\theta}\mathbb{W}_{\xi,q+1,0}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{0}}\bigg{)}$ (8.40)
$\displaystyle\quad+\underbrace{\mathcal{R}^{*}\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{0}}$ (8.41)
$\displaystyle\qquad+\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}_{\xi,q+1,0}^{\theta}\mathbb{W}_{\xi,q+1,0}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{0}}\bigg{)}$ (8.42)
$\displaystyle\qquad\qquad+\underbrace{\mathcal{O}_{0,1,2}+\mathcal{O}_{0,1,3}}_{\textnormal{divergence
corrector}}+\underbrace{\mathcal{O}_{0,2}}_{\textnormal{Type 2}}\bigg{]}$
(8.43)
$\displaystyle\quad+\mathrm{div\,}\mathcal{H}\bigg{(}\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - }\mathring{H}_{q,n,p}^{0}}$ (8.44)
$\displaystyle\qquad+\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,0}}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}_{\xi,q+1,0}^{\theta}\mathbb{W}_{\xi,q+1,0}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}}_{\textnormal{Type
1 - }\mathring{H}_{q,n,p}^{0}}\bigg{)}$ (8.45)
$\displaystyle:=\mathrm{div\,}(\mathring{R}_{q+1}^{0})+\mathrm{div\,}\left(\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{{p_{\rm
max}}}\mathring{H}_{q,n,p}^{0}\right)+\nabla\pi+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}},$

thus verifying (7.10) from Proposition 7.3 after condensing the labeled terms
into $\mathring{R}_{q+1}^{0}$ and using (8.35) on the pieces labeled
$\mathring{H}_{q,n,p}^{0}$.

####  8.3.2 The case $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$

From (7.17), we assume that $v_{q,{\widetilde{n}}-1}$ is divergence-free and
is a solution to

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}-1}+$
$\displaystyle\mathrm{div\,}\left(v_{q,{\widetilde{n}}-1}\otimes
v_{q,{\widetilde{n}}-1}\right)+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)+\mathrm{div\,}\left(\sum\limits_{n={\widetilde{n}}}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}\,.$

Now using the definition of $\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$
from (8.7) and adding $w_{q+1,{\widetilde{n}}}$ as defined in (8.13), we have
that
$v_{q,{\widetilde{n}}}:=v_{q,{\widetilde{n}}-1}+w_{q+1,{\widetilde{n}}}=v_{\ell_{q}}+\sum_{0\leq
n^{\prime}\leq{\widetilde{n}}-1}w_{q+1,n^{\prime}}+w_{q+1,{\widetilde{n}}}$
solves

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}}+$
$\displaystyle\mathrm{div\,}\left(v_{q,{\widetilde{n}}}\otimes
v_{q,{\widetilde{n}}}\right)+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)+\mathrm{div\,}\left(\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle\quad+(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}$
$\displaystyle\quad+\sum_{n^{\prime}\leq{\widetilde{n}}-1}\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle\quad+\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}+\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right).$ (8.46)

The first term on the right hand side is
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$, which satisfies the same estimates
as $\mathring{R}_{q+1}^{\widetilde{n}}$ by (7.20) and will thus be absorbed
into $\mathring{R}_{q+1}^{{\widetilde{n}}}$ (these estimates do not change in
${\widetilde{n}}$ save for implicit constants). The second term, save for the
fact that the sum is over $n^{\prime}\leq{\widetilde{n}}-1$ rather than
$n^{\prime}\leq{\widetilde{n}}$ and is therefore missing the terms
$\mathring{H}_{q,n,p}^{\widetilde{n}}$, matches (7.17) at level
${\widetilde{n}}$ (i.e. replacing every instance of ${\widetilde{n}}-1$ with
${\widetilde{n}}$). As before, we apply the inverse divergence operators from
Proposition A.18 to the transport and Nash errors to obtain

$(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}=\mathrm{div\,}\left((\mathcal{H}+\mathcal{R}^{*})\left((\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)\right)+\nabla\pi,$

and these errors are absorbed into $\mathring{R}_{q+1}^{\widetilde{n}}$ or the
new pressure. We will show in Section 8.7 that the interaction of
$w_{q+1,{\widetilde{n}}}$ with previous terms $w_{q+1,n^{\prime}}$ is a Type 2
oscillation error so that

$\sum_{n^{\prime}\leq{\widetilde{n}}-1}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes
w_{q+1,{\widetilde{n}}}\right)=0.$ (8.47)

So to verify (7.17) at level ${\widetilde{n}}$, only the analysis of

$\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}+\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right)$

remains. Reusing the notations from (8.28)363636In a slight abuse of notation,
notice that the admissible values of $\vec{l}$ have changed, since these
parameters describe the checkerboard cutoff functions at scale
$\lambda_{q,{\widetilde{n}},1}^{-1}$ and thus depend on ${\widetilde{n}}$. and
writing out the self-interaction of $w_{q+1,{\widetilde{n}}}$ yields

$\displaystyle\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\right)\otimes\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{i,k}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\right)\right)$
$\displaystyle+\sum_{\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\right)\otimes\mathrm{curl\,}\left(a_{(\xi^{\prime})}\nabla\Phi_{(i^{\prime},k^{\prime})}^{T}\mathbb{U}_{\xi^{\prime},q+1,{\widetilde{n}}}\right)\right)$
$\displaystyle:=\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1}+\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},2}.$
(8.48)

As before, we will show that $\mathcal{O}_{{\widetilde{n}},2}$ is a Type 2
oscillation error so that

$\mathcal{O}_{{\widetilde{n}},2}=0.$

Splitting $\mathcal{O}_{{\widetilde{n}},1}$ gives

$\displaystyle\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1}=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\otimes\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle\qquad+2\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\otimes_{\mathrm{s}}\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)\otimes\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle:=\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}+\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}\right).$
(8.49)

The last two of these terms are again divergence corrector errors and will
therefore be absorbed into $\mathring{R}_{q+1}^{\widetilde{n}}$ and estimated
in Section 8.8. So the only terms remaining from (8.46) are
$\mathcal{O}_{{\widetilde{n}},1,1}$ and $\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$, which are analyzed in
a fashion similar to the ${\widetilde{n}}=0$ case, save for the fact that
summation over ${\widetilde{p}}$ is now crucial.

Recall cf. (8.10) that $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$ is periodized to
scale
$\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}=\lambda_{q,{\widetilde{n}}}^{-1}$.
Using (7.8), we have that

$\displaystyle\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}{\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}}$
$\displaystyle+\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)\left(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\right).$

Combining this division with identity (4.15) from Proposition 4.4, we further
split $\mathcal{O}_{{\widetilde{n}},1,1}$ as

$\displaystyle\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}\right)=\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\left(a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\right)\nabla\Phi_{(i,k)}^{-T}\right)$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathrm{div\,}\bigg{(}a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle=\mathrm{div\,}\sum_{i,j,k,{\widetilde{p}},\vec{l}}\sum_{\xi}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}}{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}$
$\displaystyle\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\,.$
(8.50)

By Proposition 4.1, equation (4.1), and identity (8.8), we obtain that

$\displaystyle\mathrm{div\,}\sum_{i,j,k,{\widetilde{p}},\vec{l}}\sum_{\xi}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{\widetilde{n}},{\widetilde{p}},j,i,k}}{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad=\mathrm{div\,}\sum_{i,j,k,{\widetilde{p}},\vec{l}}\eta_{(i,j,k)}^{2}\left(\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\mathrm{Id}-\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right)$
$\displaystyle\qquad=-\mathrm{div\,}\sum_{i,j,k,\vec{l}}\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\eta^{2}_{(i,j,k)}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}+\nabla\left(\sum_{i,j,k,\vec{l}}\eta_{(i,j,k)}^{2}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{2j+4}\right)$
$\displaystyle\qquad:=-\mathrm{div\,}\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}+\nabla\pi\,,$ (8.51)

where in the last equality we have appealed to (6.142). We can finally apply
Proposition A.18 to the remaining terms in (8.50) for ${\widetilde{n}}+1\leq
n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm max}}$ to define

$\displaystyle\mathring{H}_{q,n,p}^{\widetilde{n}}:=\mathcal{H}\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad+\sum_{\xi,i,j,k,{\widetilde{p}}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}.$
(8.52)

As before, the terms from (8.50) with $\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm max}}+1\right]}$ will be absorbed into
$\mathring{R}_{q+1}^{\widetilde{n}}$. We will show shortly that the terms
$\mathring{H}_{q,n,p}^{\widetilde{n}}$ in (8.52) are precisely the terms
needed to make (8.46) match (7.17) at level ${\widetilde{n}}$. As before, any
nonlocal inverse divergence terms will be absorbed into
$\mathring{R}_{q+1}^{\widetilde{n}}$.

Recall from (7.9) that $\mathring{R}_{q+1}^{\widetilde{n}}$ will include
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$ in addition to error terms arising
from the addition of $w_{q+1,{\widetilde{n}}}$ which are small enough to be
absorbed in $\mathring{R}_{q+1}$. Then to check (7.17), we return to (8.46)
and use (8.48), (8.49), (8.50), (8.51), and (8.52) to write

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}}+\mathrm{div\,}\left(v_{q,{\widetilde{n}}}\otimes
v_{q,{\widetilde{n}}}\right)+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\left(\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle\quad+(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}$
$\displaystyle\quad+\sum_{n^{\prime}\leq{\widetilde{n}}-1}\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle\quad+\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}+\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right)$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\left(\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{\widetilde{n}}-1}+(\mathcal{H}+\mathcal{R}^{*})\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle\qquad+(\mathcal{H}+\mathcal{R}^{*})\left(w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)+\sum_{n^{\prime}\leq{\widetilde{n}}-1}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes
w_{q+1,{\widetilde{n}}}\right)\bigg{)}$
$\displaystyle\qquad+\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}+\mathcal{O}_{{\widetilde{n}},2}\right)+\nabla\pi+\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}+\sum_{{\widetilde{p}}=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}\right)$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\left(\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}\right)+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{\widetilde{n}}-1}+(\mathcal{H}+\mathcal{R}^{*})\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle\quad+(\mathcal{H}+\mathcal{R}^{*})\left(w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)+\sum_{n^{\prime}\leq{\widetilde{n}}-1}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes
w_{q+1,{\widetilde{n}}}\right)\bigg{)}$
$\displaystyle\quad+\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}+\mathcal{O}_{{\widetilde{n}},2}\right)+\nabla\pi$
$\displaystyle\quad+\mathrm{div\,}\bigg{[}\left(\mathcal{H}+\mathcal{R}^{*}\right)\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}$
$\displaystyle\qquad\qquad\qquad\times\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}\bigg{]}$
(8.53)
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\left({\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\mathring{H}_{q,n,p}^{n^{\prime}}}\right)+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{\widetilde{n}}-1}+\underbrace{(\mathcal{H}+\mathcal{R}^{*})\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)}_{\textnormal{transport}}$ (8.54)
$\displaystyle\qquad+\underbrace{(\mathcal{H}+\mathcal{R}^{*})\left(w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)}_{\textnormal{Nash}}+\underbrace{\sum_{n^{\prime}\leq{\widetilde{n}}-1}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes
w_{q+1,{\widetilde{n}}}\right)}_{\textnormal{Type 2}}\bigg{)}$ (8.55)
$\displaystyle\qquad+\mathrm{div\,}\bigg{(}\underbrace{\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}}_{\textnormal{divergence
corrector}}+\underbrace{\mathcal{O}_{{\widetilde{n}},2}}_{\textnormal{Type
2}}\bigg{)}+\nabla\pi$ (8.56)
$\displaystyle\qquad+\mathrm{div\,}\bigg{[}\underbrace{(\mathcal{H}+\mathcal{R}^{*})\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{[q,{n_{\rm max}},{p_{\rm
max}}+1]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{\widetilde{n}}}$ (8.57)
$\displaystyle\qquad\qquad+\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{[q,{n_{\rm
max}},{p_{\rm
max}}+1]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{\widetilde{n}}}$ (8.58)
$\displaystyle+\mathcal{R}^{*}\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}$
$\displaystyle\qquad\qquad\times\underbrace{\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{\widetilde{n}}}$ (8.59)
$\displaystyle+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}$
$\displaystyle\qquad\qquad\times\underbrace{\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}}_{\textnormal{Type
1 - part of }\mathring{R}_{q+1}^{\widetilde{n}}}\bigg{)}\bigg{]}$ (8.60)
$\displaystyle\quad+\mathrm{div\,}\underbrace{\mathcal{H}\bigg{[}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type
1 - }\mathring{H}_{q,n,p}^{\widetilde{n}}}$ (8.61)
$\displaystyle\qquad+\underbrace{\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\left(\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum_{p=1}^{{p_{\rm
max}}}\mathbb{P}_{[q,n,p]}\right)(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{]}}_{\textnormal{Type
1 - }\mathring{H}_{q,n,p}^{\widetilde{n}}}$ (8.62)
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\mathring{R}_{q+1}^{{\widetilde{n}}}+\mathrm{div\,}\sum\limits_{n={\widetilde{n}}+1}^{n_{\rm
max}}\sum_{p=1}^{p_{\rm
max}}\sum\limits_{n^{\prime}=0}^{\widetilde{n}}\mathring{H}_{q,n,p}^{n^{\prime}}+\nabla\pi,$

which concludes the proof after identifying the first seven lines (save for
the double sum of $\mathring{H}_{q,n}^{n^{\prime}}$ terms) of the second to
last equality as $\mathring{R}_{q+1}^{\widetilde{n}}$ and using (8.52) to
incorporate the eighth and ninth lines into the new double sum of
$\mathring{H}_{q,n}^{n^{\prime}}$ terms. Note that we have implicitly used in
the above equalities that
$\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)w_{q+1,{\widetilde{n}}}$ has
zero mean, which can be deduced in the same fashion as for the case
${\widetilde{n}}=0$.

####  8.3.3 The case ${\widetilde{n}}={n_{\rm max}}$

From (7.24), we assume that $v_{q,{n_{\rm max}}-1}$ is divergence-free and is
a solution to

$\displaystyle\partial_{t}v_{q,{n_{\rm max}}-1}+$
$\displaystyle\mathrm{div\,}\left(v_{q,{n_{\rm max}}-1}\otimes v_{q,{n_{\rm
max}}-1}\right)+\nabla p_{q,{n_{\rm max}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right)+\mathrm{div\,}\left(\sum\limits_{n^{\prime}=0}^{{n_{\rm
max}}-1}\sum_{p=1}^{p_{\rm max}}\mathring{H}_{q,{n_{\rm
max}},p}^{n^{\prime}}\right)+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}\,.$

Now using the definition of $\mathring{R}_{q,{n_{\rm max}},p}$ from (8.7) and
adding $w_{q+1,{n_{\rm max}}}$ as defined in (8.13), we have that
$v_{q+1}:=v_{q,{n_{\rm max}}-1}+w_{q+1,{n_{\rm max}}}$ solves

$\displaystyle\partial_{t}v_{q+1}+$
$\displaystyle\mathrm{div\,}\left(v_{q+1}\otimes v_{q+1}\right)+\nabla
p_{q,{n_{\rm max}}-1}$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\left(\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right)+(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{n_{\rm
max}}}+w_{q+1,{n_{\rm max}}}\cdot\nabla v_{\ell_{q}}$
$\displaystyle\quad+\sum_{n^{\prime}\leq{n_{\rm
max}}-1}\mathrm{div\,}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{n_{\rm max}}}\right)$
$\displaystyle\quad+\mathrm{div\,}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,{n_{\rm max}}}+\sum_{p=1}^{p_{\rm max}}\mathring{R}_{q,{n_{\rm
max}},p}\right).$ (8.63)

We absorb the term $\mathrm{div\,}\left(\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right)$ into $\mathring{R}_{q+1}$ immediately. We will then show that
up to a pressure term,

$\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)w_{q+1,{n_{\rm
max}}}\right),\qquad\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(w_{q+1,{n_{\rm
max}}}\cdot\nabla v_{\ell_{q}}\right)$

can be absorbed into $\mathring{R}_{q+1}$ in Sections 8.4 and 8.5,
respectively. We will be show in 8.7 that the interaction of $w_{q+1,{n_{\rm
max}}}$ with previous perturbations $w_{q+1,n^{\prime}}$ will satisfy

$\sum_{n^{\prime}\leq{n_{\rm max}}-1}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{n_{\rm max}}}\right)=0.$
(8.64)

Thus it remains to analyze

$\mathrm{div\,}\left(w_{q+1,{n_{\rm max}}}\otimes w_{q+1,{n_{\rm
max}}}+\sum_{p=1}^{p_{\rm max}}\mathring{R}_{q,{n_{\rm max}}}\right)$

from (8.63). Reusing the notations from (8.28)–(8.29), we can write out the
self-interaction of $w_{q+1,{n_{\rm max}}}$ as

$\displaystyle\mathrm{div\,}$ $\displaystyle\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,{n_{\rm max}}}\right)$
$\displaystyle=\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\right)\otimes\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{i,k}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\right)\right)$
$\displaystyle\qquad+\sum_{\neq\\{\xi,i,j,k,p,\vec{l}\\}}\mathrm{div\,}\left(\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\right)\otimes\mathrm{curl\,}\left(a_{(\xi^{\prime})}\nabla\Phi_{(i^{\prime},k^{\prime})}^{T}\mathbb{U}_{\xi^{\prime},q+1,{n_{\rm
max}}}\right)\right)$ $\displaystyle:=\mathrm{div\,}\mathcal{O}_{{n_{\rm
max}},1}+\mathrm{div\,}\mathcal{O}_{{n_{\rm max}},2}.$ (8.65)

As before, we will show in Section 8.7 that $\mathcal{O}_{{n_{\rm max}},2}$ is
a Type 2 oscillation error and so

$\mathcal{O}_{{n_{\rm max}},2}=0.$

Splitting $\mathcal{O}_{{n_{\rm max}},1}$ gives

$\displaystyle\mathrm{div\,}\mathcal{O}_{{n_{\rm
max}},1}=\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\otimes\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle\qquad+2\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\otimes_{\mathrm{s}}\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle\qquad+\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\right)\otimes\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{n_{\rm
max}}}\circ\Phi_{(i,k)}\right)\right)\right)$
$\displaystyle:=\mathrm{div\,}\left(\mathcal{O}_{{n_{\rm
max}},1,1}+\mathcal{O}_{{n_{\rm max}},1,2}+\mathcal{O}_{{n_{\rm
max}},1,3}\right).$ (8.66)

The last two of these three terms are again divergence corrector errors and
will therefore be absorbed into $\mathring{R}_{q+1}$ and estimated in Section
8.8.

Recall cf. (8.3) that $\mathbb{W}_{\xi,q+1,{n_{\rm max}}}$ is periodized to
scale $\left(\lambda_{q+1}r_{q+1,{n_{\rm max}}}\right)^{-1}=\lambda_{q,{n_{\rm
max}}}^{-1}$. Combining this observation with (4.15) from Proposition 4.4 and
(7.8), we further split $\mathcal{O}_{{n_{\rm max}},1,1}$ as373737In this
case, $\mathbb{P}_{\geq\lambda_{q,{n_{\rm max}}}}$ has a greater minimum
frequency than $\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm max}}+1\right]}$,
cf. (9.28), (9.22), and (7.6). For the sake of consistency, we write
$\mathbb{P}_{\geq\lambda_{q,{n_{\rm max}}}}\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm max}}+1\right]}$ throughout this section.

$\displaystyle\mathrm{div\,}\left(\mathcal{O}_{{n_{\rm
max}},1,1}\right)=\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\otimes\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}(\Phi_{(i,k)})\right)\nabla\Phi_{(i,k)}^{-T}\right)$
$\displaystyle\quad+\sum_{\xi,i,j,k,p,\vec{l}}\mathrm{div\,}\left(a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\otimes\mathbb{W}_{\xi,q+1,{n_{\rm
max}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\right)$
$\displaystyle=\mathrm{div\,}\sum_{i,j,k,p,\vec{l}}\sum_{\xi}\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{n_{\rm
max}},p,j,i,k}}{\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\quad+\sum_{\xi,i,j,k,p,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\otimes\mathbb{W}_{\xi,q+1,{n_{\rm
max}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\quad+\sum_{\xi,i,j,k,p,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}^{\theta}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}.$
(8.67)

By (4.1) from Proposition 4.1 and (8.8), we obtain that

$\displaystyle\mathrm{div\,}\sum_{i,j,k,p,\vec{l}}\sum_{\xi}\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{n_{\rm
max}},p,j,i,k}}{\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad=\mathrm{div\,}\sum_{i,j,k,p,\vec{l}}\eta_{(i,j,k)}^{2}\left(\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}\mathrm{Id}-\mathring{R}_{q,{n_{\rm
max}},p}\right)$
$\displaystyle\qquad=-\mathrm{div\,}\sum_{i,j,k,\vec{l}}\sum_{p=1}^{p_{\rm
max}}\eta^{2}_{(i,j,k)}\mathring{R}_{q,{n_{\rm
max}},p}+\nabla\left(\sum_{i,j,k,p,\vec{l}}\eta_{(i,j,k)}^{2}\delta_{q+1,{n_{\rm
max}},p}\Gamma_{q+1}^{2j+4}\right)$
$\displaystyle\qquad:=-\mathrm{div\,}\sum_{p=1}^{p_{\rm
max}}\mathring{R}_{q,{n_{\rm max}},p}+\nabla\pi\,,$ (8.68)

where in the last line we have used (6.142). We can apply Proposition A.18 to
the remaining two terms in (8.67) to produce the terms

$\displaystyle\left(\mathcal{H}+\mathcal{R}^{*}\right)\bigg{(}\sum_{\xi,i,j,k,p,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{n_{\rm
max}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad+\sum_{\xi,i,j,k,p,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{n_{\rm
max}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)},$
(8.69)

which will be absorbed into $\mathring{R}_{q+1}$ and estimated in Section 8.6.

Before combining the previous steps, we remind the reader that at this point,
$\mathring{R}_{q+1}$ will be fully defined, and will include
$\mathring{R}_{q+1}^{{n_{\rm max}}-1}$, all the error terms arising from the
addition of $w_{q+1,{n_{\rm max}}}$, and
$\mathring{R}_{q}^{\textnormal{comm}}$. Then from (8.63), (8.64), (8.65),
(8.66), (8.67), (8.68), and (8.69), we can finally write that

$\displaystyle\partial_{t}$ $\displaystyle
v_{q+1}+\mathrm{div\,}\left(v_{q+1}\otimes v_{q+1}\right)+\nabla p_{q,{n_{\rm
max}}-1}$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\left(\mathring{R}_{q+1}^{{n_{\rm
max}}-1}\right)+(\partial_{t}+v_{\ell_{q}}\cdot\nabla)w_{q+1,{n_{\rm
max}}}+w_{q+1,{n_{\rm max}}}\cdot\nabla v_{\ell_{q}}$
$\displaystyle\quad+\sum_{n^{\prime}\leq{n_{\rm
max}}-1}\mathrm{div\,}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{n_{\rm max}}}\right)$
$\displaystyle\quad+\mathrm{div\,}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,{n_{\rm max}}}+\sum_{p=1}^{p_{\rm max}}\mathring{R}_{q,{n_{\rm
max}},p}\right)$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{n_{\rm
max}}-1}+\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(\partial_{t}w_{q+1,{n_{\rm
max}}}+v_{\ell_{q}}\cdot\nabla w_{q+1,{n_{\rm max}}}\right)$
$\displaystyle\qquad+(\mathcal{H}+\mathcal{R}^{*})\left(w_{q+1,{n_{\rm
max}}}\cdot\nabla v_{\ell_{q}}\right)+\sum_{n^{\prime}\leq{n_{\rm
max}}-1}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{n_{\rm
max}}}\right)\bigg{)}$
$\displaystyle\qquad+\mathrm{div\,}\left(\mathcal{O}_{{n_{\rm
max}},1,1}+\mathcal{O}_{{n_{\rm max}},1,2}+\mathcal{O}_{{n_{\rm
max}},1,3}+\mathcal{O}_{{n_{\rm max}},2}+\sum_{p=1}^{p_{\rm
max}}\mathring{R}_{q,{n_{\rm max}},p}\right)+\nabla\pi$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{n_{\rm
max}}-1}+\underbrace{\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(\partial_{t}w_{q+1,{n_{\rm
max}}}+v_{\ell_{q}}\cdot\nabla w_{q+1,{n_{\rm
max}}}\right)}_{\textnormal{transport}}$ (8.70)
$\displaystyle\qquad+\underbrace{(\mathcal{H}+\mathcal{R}^{*})\left(w_{q+1,{n_{\rm
max}}}\cdot\nabla
v_{\ell_{q}}\right)}_{\textnormal{Nash}}+\underbrace{\sum_{n^{\prime}\leq{n_{\rm
max}}-1}\left(w_{q+1,{n_{\rm max}}}\otimes
w_{q+1,n^{\prime}}+w_{q+1,n^{\prime}}\otimes w_{q+1,{n_{\rm
max}}}\right)}_{\textnormal{Type 2}}\bigg{)}$ (8.71)
$\displaystyle\qquad+\mathrm{div\,}\bigg{(}\underbrace{\mathcal{O}_{{n_{\rm
max}},1,2}+\mathcal{O}_{{n_{\rm max}},1,3}s}_{\textnormal{divergence
corrector}}+\underbrace{\mathcal{O}_{{n_{\rm max}},2}}_{\textnormal{Type
2}}\bigg{)}+\nabla\pi$ (8.72)
$\displaystyle+\mathrm{div\,}\underbrace{\left(\mathcal{H}+\mathcal{R}^{*}\right)\bigg{(}\sum_{\xi,i,j,k,p,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{n_{\rm
max}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}_{\textnormal{Type 1}}$ (8.73)
$\displaystyle\qquad\qquad+\underbrace{\sum_{\xi,i,j,k,p}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{n_{\rm
max}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}}_{\textnormal{Type
1}}$ (8.74) $\displaystyle=\mathrm{div\,}(\mathring{R}_{q+1})+\nabla\pi,$

concluding the proof after again noting that
$\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)w_{q+1,{\widetilde{n}}}$ has
zero mean.

###  8.4 Transport errors

######  Lemma 8.4.

For all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$, the transport errors satisfy

$D_{t,q}w_{q+1,{\widetilde{n}}}=\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}=\mathrm{div\,}\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)+\nabla p_{\widetilde{n}}$

with the estimates

$\displaystyle\left\|\psi_{i,q}D^{N}D_{t,q}^{M}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)\right)\right\|_{L^{1}}$
$\displaystyle\qquad\qquad\lesssim\delta_{q+2}\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$

for all $N,M\leq 3\mathsf{N}_{\textnormal{ind,v}}$.

###### Proof of Lemma 8.4.

The transport errors are given in (8.38), (8.54), and (8.70). Writing out the
transport error, we have that

$\displaystyle\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)w_{q+1,{\widetilde{n}}}$
$\displaystyle=\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)\left(\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}\mathrm{curl\,}\left(a_{\xi,i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)$
$\displaystyle=\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\right)\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}$
$\displaystyle\qquad+\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}\left(\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)\nabla
a_{(\xi)}\right)\times\left(\nabla\Phi_{(i,k)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
$\displaystyle\qquad+\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}\nabla
a_{(\xi)}\times\left(\left(\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)\nabla\Phi_{(i,k)}\right)\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
(8.75)

Due to the fact that the second two terms arise from the addition of the
corrector defined in (8.5) and (8.12), and the fact that the bounds for the
corrector in (8.17) are stronger than that of the principal part of the
perturbation, we shall completely estimate only the first term and simply
indicate the set-up for the second and third. Before applying Proposition
A.18, recall that the inverse divergence of (8.75) needs to be estimated on
the support of a cutoff $\psi_{i,q}$ in order to verify (7.13), and (7.20),
and (7.27). Recall from the identification of the error terms (cf. (8.36) and
the subsequent argument) that for all ${\widetilde{n}}$,
$\left(\partial_{t}+v_{\ell_{q}}\cdot\nabla\right)w_{q+1,{\widetilde{n}}}$ has
zero mean. Thus, although each individual term in the final equality in (8.75)
may not have zero mean, we can safely apply $\mathcal{H}$ and
$\mathcal{R}^{*}$ to each term and estimate the outputs while ignoring the
last term in (A.78).

We will apply Proposition A.18, specifically Remark A.19, to each summand in
the first term on the right side of (8.75), with the following choices. We set
$v=v_{\ell_{q}}$, and $D_{t}=D_{t,q}=\partial_{t}+v_{\ell_{q}}\cdot\nabla$ as
usual. We set
$N_{*}=M_{*}=\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$, with ${\mathsf{N}_{\rm dec}}$
and ${\mathsf{d}}$ satisfying (9.60a). We define

$G=(\partial_{t}+v_{\ell_{q}}\cdot\nabla)(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1})\xi,$

with $\lambda=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, and

$\mathcal{C}_{G}=\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\delta_{q+1,{\widetilde{n}},1}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+j+5}\tau_{q}^{-1}\,,$

which is the correct amplitude in view of (8.14) with $r=1$ and
$r_{1}=r_{2}=2$, and (6.114). Thus, we have that

$\displaystyle\left\|D^{N}D_{t,q}^{M}G\right\|_{L^{1}}\lesssim\mathcal{C}_{G}\left(\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}}-1,\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$
(8.76)

for all
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$ after using (9.42) and (9.52),
and so (A.66) is satisfied. We set $\Phi=\Phi_{i,k}$ and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$. Appealing as usual to Corollary
6.27 and (6.60), we have that (A.67) and (A.68) are satisfied.

Referring to (1) from Proposition 4.4, we set
$\varrho=\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}$ and
$\vartheta=\vartheta_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}$. Setting
$\zeta=\lambda_{q+1}$, we have that (i) is satisfied. Setting
$\mu=\lambda_{q+1}r_{q+1,{\widetilde{n}}}=\lambda_{q,{\widetilde{n}}}$ and
referring to (2) from Proposition 4.4, we have that (ii) is satisfied. Setting
$\Lambda=\zeta=\lambda_{q+1}$ and $C_{*}=r_{q+1,{\widetilde{n}}}$ and
referring to (4.11) and (4.12) from Proposition 4.4, we have that (A.69) is
satisfied. (A.70) is immediate from the definitions. Referring to (9.48), we
have that (A.71) is satisfied. Thus, we conclude from (A.73) with
$\alpha_{\mathsf{R}}$ as in (9.53), that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\left((\partial_{t}+v_{\ell_{q}}\cdot\nabla)(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1})\xi\right)\right)\right\|_{L^{1}}=\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\left(G\varrho\circ\Phi\right)\right)\right\|_{L^{1}}$
$\displaystyle\qquad\lesssim\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\delta_{q+1,{\widetilde{n}},1}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+j+6}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,,$

after appealing to (9.42). From (9.60c), these bounds are valid for all
$N,M\leq 3\mathsf{N}_{\textnormal{ind,v}}$. The bound obtained above is next
summed over $(i,j,k,{\widetilde{p}},{\widetilde{n}},\vec{l})$. First, we treat
the sum over $\vec{l}$. By noting that (6.147) with $r_{1}=2$ and $r_{2}=2$,
and (9.42) imply

$\sum_{\vec{l}}\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+j+6}\leq\Gamma_{q+1}^{-2\left(\frac{i}{2}+\frac{j}{2}\right)+\frac{{\mathsf{C}_{b}}}{2}+2}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+j+6}=\Gamma_{q+1}^{\frac{{\mathsf{C}_{b}}}{2}+3}\,,$

we conclude that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\left(\partial_{t}w_{q+1,{\widetilde{n}}}+v_{\ell_{q}}\cdot\nabla
w_{q+1,{\widetilde{n}}}\right)\right)\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\qquad\lesssim\sum_{i^{\prime}=i-1}^{i+1}\sum_{j,k,{\widetilde{p}},\xi}\Gamma_{q+1}^{\frac{{\mathsf{C}_{b}}}{2}+3}\delta_{q+1,{\widetilde{n}},1}^{\nicefrac{{1}}{{2}}}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i^{\prime}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\qquad\lesssim\Gamma_{q+1}^{4+\frac{{\mathsf{C}_{b}}}{2}}\delta_{q+1,{\widetilde{n}},1}^{\frac{1}{2}}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\qquad\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(8.77)

after also using (9.57).

To finish the proof for the first term in (8.75), we must provide a matching
estimate for the $\mathcal{R}^{*}$ portion. Following again the parameter
choices in Remark A.19, we set
$N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$. As in the argument
from Lemma 8.6, we have that (A.75), (A.76), and (A.77) are satisfied, this
time with $\zeta=\lambda_{q+1}$. Thus we achieve the estimate in (A.79).
Summing over $\vec{l}$ loses a factor less than $\lambda_{q+1}^{3}$, while
summing over the other indices costs a constant independent of $q$. This
completes the estimate for the first term from (8.75).

For the second and third terms, we explain how to identify $G$ and $\varrho$
in order to give an idea of how to obtain similar estimates. Using 1 from
Proposition 4.4 and the vector calculus identity
$\mathrm{curl\,}\mathrm{curl\,}=\nabla\mathrm{div\,}-\Delta$, we obtain that

$\mathbb{U}_{\xi,q+1,{\widetilde{n}}}=\mathrm{curl\,}\left(\xi\lambda_{q+1}^{-2{\mathsf{d}}}\Delta^{{\mathsf{d}}-1}\left(\vartheta_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\right)=\lambda_{q+1}^{-2{\mathsf{d}}}\xi\times\nabla\left(\Delta^{{\mathsf{d}}-1}\left(\vartheta_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\right).$
(8.78)

With a little massaging, one can now rewrite the second and third terms in
(8.75) in the form $G\varrho\circ\Phi_{(i,k)}$. Since both terms have traded a
spatial derivative on $\mathbb{U}_{\xi,q+1,{\widetilde{n}}}$ for a spatial
derivative on $a_{(\xi)}$, inducing a gain, one can easily show that the
estimates for these terms will be even stronger than those for the first term.
Notice that we have set
$N_{*}=M_{*}=\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-7\right)\rfloor$ since we have lost a spatial
derivative on $a_{(\xi)}$. We omit the rest of the details. ∎

###  8.5 Nash errors

######  Lemma 8.5.

For all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$, the Nash errors satisfy

$w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}=\mathrm{div\,}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)+\nabla p_{\widetilde{n}}$

with

$\displaystyle\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)\right\|_{L^{1}}\lesssim\delta_{q+2}\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$

for all $N,M\leq 3\mathsf{N}_{\textnormal{ind,v}}$.

###### Proof of Lemma 8.5.

The estimates are similar to those in Lemma 8.4. Writing out the Nash error,
we have that

$\displaystyle w_{q+1,{\widetilde{n}}}\cdot\nabla v_{\ell_{q}}$
$\displaystyle=\sum_{i-1\leq i^{\prime}\leq
i+1}\sum_{j,k,{\widetilde{p}},\vec{l},\xi}\mathrm{curl\,}\left(a_{\xi,i,j,k,q,{\widetilde{n}}}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
$\displaystyle=\left(\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}\nabla
a_{(\xi)}\times\left(\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\right)\cdot\nabla
v_{\ell_{q}}$
$\displaystyle\qquad+\left(\sum_{i,j,k,{\widetilde{p}},\vec{l},\xi}a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\cdot\nabla
v_{\ell_{q}}.$ (8.79)

Due to the fact that the first term arises from the addition of the corrector
defined in (8.5) and (8.12), and the fact that the bounds for the corrector in
(8.17) are stronger than that of the principal part of the perturbation, we
shall completely estimate only the second term and simply indicate the set-up
for the first. Before applying Proposition A.18, recall that the inverse
divergence of (8.75) needs to be estimated on the support of a cutoff
$\psi_{i,q}$ in order to verify (7.5), (7.13), and (7.20). Note that the Nash
error can be written as $\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\cdot
v_{\ell_{q}}\right)$ and so has zero mean. Thus, although each individual term
in the final equality in (8.79) may not have zero mean, we can safely apply
$\mathcal{H}$ and $\mathcal{R}^{*}$ to each term and estimate the outputs
while ignoring the last term in (A.78).

We will apply Proposition A.18 to the second term with the following choices.
We set $v=v_{\ell_{q}}$, and
$D_{t}=D_{t,q}=\partial_{t}+v_{\ell_{q}}\cdot\nabla$ as usual. We set
$N_{*}=M_{*}=\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,x}-\mathsf{N}_{\rm cut,t}-4\right)\rfloor$, with ${\mathsf{N}_{\rm dec}}$
and ${\mathsf{d}}$ satisfying (9.60a). We define

$G=a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\xi\cdot\nabla v_{\ell_{q}}$

and set

$\mathcal{C}_{G}=\bigl{|}\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\delta_{q+1,{\widetilde{n}},1}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+j+5}\tau_{q}^{-1}\,,$

$\lambda=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$, and
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$. From (8.14) with
$r=1$ and $r_{1}=r_{2}=2$, (6.114), and (6.60), we have that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,x}-\mathsf{N}_{\rm cut,t}-4\right)\rfloor$

$\displaystyle\left\|D^{N}D_{t,q}^{M}G\right\|_{L^{1}}\lesssim\mathcal{C}_{G}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$
(8.80)

and so (A.66) is satisfied. Note that we have used (9.39) when converting the
$\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}$ to a
$\tau_{q}^{-1}$. Setting $\Phi=\Phi_{(i,k)}$ and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$, we have that (A.67) and (A.68) are
satisfied as usual. The choices of $\varrho$, $\vartheta$, $\zeta$, $\mu$,
$\Lambda$, and $\mathcal{C}_{*}$ are identical to those of the transport error
(both terms contain $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}$),
and so we have that (i)-(ii), (A.69), (A.70), and (A.71) are satisfied as
well. Since the bound (8.80) is identical to that of (8.76), we obtain an
estimate identical to (8.77). The argument for the $\mathcal{R}^{*}$ portion
follows analogously to that for the first term from the transport error.
Finally, after using (8.78) again, one may obtain similar estimates for the
first term in (8.79), concluding the proof. ∎

###  8.6 Type 1 oscillation errors