$\displaystyle\bigl{\|}D^{{\boldsymbol{\alpha}}}D_{t,q}^{{\boldsymbol{\beta}}}\mathring{R}_{q,n}\bigr{\|}_{L^{\infty}(\mathrm{supp\,}\psi_{i,q}\omega_{i,j,q,n})}\lesssim\Gamma_{q+1}^{2(j+1)}\delta_{q+1,n}(\Gamma_{q+1}\lambda_{q,n})^{|{\boldsymbol{\alpha}}|}\mathcal{M}\left(|{\boldsymbol{\beta}}|,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}+2}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\tilde{\tau}_{q}^{-1}\right)$
(5.28)

for all $|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|\leq\mathsf{N}_{\rm
fin,n}-4$.

The next Lemma provides estimate on the maximum value of $j$ for which
$\psi_{i,q}\omega_{i,j,q,n}$ may be non-zero. While the proof is similar in
spirit to [5, Lemma 6.35], we include the proof since propagating sharp
$L^{\infty}$ estimates of the stress is one of the crucial new ideas in this
paper.

######  Lemma 5.14 (Maximal $j$ index in the stress cutoffs).

Fix $q\geq 0$ and $0\leq n\leq n_{\rm max}$. There exists a ${j_{\rm
max}}={j_{\rm max}}(q,n)\geq 1$, which is bounded as

$\displaystyle{j_{\rm
max}}(q,n)\leq\frac{1}{2}\left(2+\frac{{\mathsf{C_{u}}}+3}{b}+\frac{2\beta
b^{2}}{\varepsilon_{\Gamma}(b-1)}\right)\,,$ (5.29)

such that for any $0\leq i\leq{i_{\rm max}}(q)$, we have

$\displaystyle\psi_{i,q}\,\omega_{i,j,q,n}\equiv 0\qquad\mbox{for all}\qquad
j>j_{\rm max}.$

Moreover, assuming that $a=\lambda_{0}$ is sufficiently large, we have the
bound

$\displaystyle\Gamma_{q+1}^{2j_{\rm
max}(q,n)}\leq\Gamma_{q}^{{\mathsf{C_{u}}}}\delta_{q+1,n}^{-1}\Gamma_{q+1}^{14\Upsilon(n)+3}\,.$
(5.30)

###### Proof of Lemma 5.14.

We define $j_{\rm max}$ by

$\displaystyle{j_{\rm max}}={j_{\rm
max}}(q,n)=\frac{1}{2}\left\lceil\frac{\log(\Gamma_{q}^{{\mathsf{C_{u}}}}\delta_{q+1,n}^{-1})\Gamma_{q+1}^{14\Upsilon(n)+2}}{\log(\Gamma_{q+1})}\right\rceil.$
(5.31)

To see that $j_{\rm max}$ may be bounded independently of $q$ and $n$, we note
that $\delta_{q+1,n}^{-1}\leq\delta_{q+2}^{-1}$, and thus

$\displaystyle 2j_{\rm max}\leq
1+\frac{{\mathsf{C_{u}}}}{b}+\frac{\log(\delta_{q+2}^{-1})}{\log(\Gamma_{q+1})}+14\Upsilon(n)+2\to
3+\frac{{\mathsf{C_{u}}}}{b}+\frac{2\beta
b^{2}}{\varepsilon_{\Gamma}(b-1)}+14\Upsilon(n)\quad\mbox{as}\quad
q\to\infty\,.$

Thus, assuming that $a=\lambda_{0}$ is sufficiently large, we obtain that

$\displaystyle 2{j_{\rm max}}(q,n)\leq
4+\frac{{\mathsf{C_{u}}}}{b}+\frac{2\beta
b^{2}}{\varepsilon_{\Gamma}(b-1)}+14\Upsilon({n_{\rm max}})$ (5.32)

for all $q\geq 0$ and $0\leq n\leq{n_{\rm max}}$.

To conclude the proof of the Lemma, let $j>j_{\rm max}$, as defined in (5.31),
and assume by contradiction that there exists a point
$(x,t)\in\mathrm{supp\,}(\psi_{i,q}\omega_{i,j,q,n})\neq\emptyset$. In
particular, $j\geq 1$. Then, by (5.24)–(5.25) and the pigeonhole principle, we
see that there exists $0\leq k\leq\mathsf{N}_{\rm cut,x}$ and $0\leq
m\leq\mathsf{N}_{\rm cut,t}$ such that

$\displaystyle|D^{k}D_{t,q}^{m}\mathring{R}_{q,n}(x,t)|\geq\frac{\Gamma_{q+1}^{2j}}{\sqrt{8\mathsf{N}_{\rm
cut,x}\mathsf{N}_{\rm
cut,t}}}\delta_{q+1,n}(\Gamma_{q+1}\lambda_{q,n})^{k}(\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}+2}\tau_{q}^{-1})^{m}.$

On the other hand, from (5.20b) and (5.23), we have that

$\displaystyle|D^{k}D_{t,q}^{m}\mathring{R}_{q,n}(x,t)|\leq\Gamma_{q}^{{\mathsf{C_{u}}}+1}\Gamma_{q+1}^{14\Upsilon(n)}\Gamma_{q+1}\lambda_{q,n}^{k}(\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}}\tau_{q}^{-1})^{m}.$

The above two estimates imply that

$\displaystyle\Gamma_{q+1}^{2j}\leq\Gamma_{q}^{{\mathsf{C_{u}}}+1}\Gamma_{q+1}^{14\Upsilon(n)}\sqrt{8\mathsf{N}_{\rm
cut,x}\mathsf{N}_{\rm
cut,t}}\delta_{q+1,n}^{-1}\leq\Gamma_{q}^{{\mathsf{C_{u}}}+2}\Gamma_{q+1}^{14\Upsilon(n)}\delta_{q+1,n}^{-1},$

which contradicts the fact that $j>j_{\rm max}$, as defined in (5.31). ∎

The following two lemmas correspond to [5, Lemmas 6.36 and 6.38],
respectively. As with Corollary 5.13, the method of proof applies _mutatis
mutandis_ after dropping the unnecessary subscript $p$. We therefore refer the
reader to [5] for further details.

######  Lemma 5.15 (Derivative bounds for the stress cutoffs).

For $q\geq 0$, $0\leq n\leq{n_{\rm max}}$, $0\leq i\leq{i_{\rm max}}$, and
$0\leq j\leq{j_{\rm max}}$, we have that

$\displaystyle\frac{{\bf
1}_{\mathrm{supp\,}\psi_{i,q}}|D^{N}D_{t,q}^{M}\omega_{i,j,q,n}|}{\omega_{i,j,q,n}^{1-(N+M)/\mathsf{N}_{\rm
fin}}}\lesssim(\Gamma_{q+1}\lambda_{q,n})^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}+3}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\tilde{\tau}_{q}^{-1}\right)$
(5.33)

for all $N+M\leq\mathsf{N}_{\rm fin,n}-\mathsf{N}_{\rm cut,x}-\mathsf{N}_{\rm
cut,t}-4$.

######  Lemma 5.16 ($L^{r}$ norm of the stress cutoffs).

Let $q\geq 0$ and define
$\psi_{i\pm,q}=\left(\psi_{i-1,q}^{2}+\psi_{i,q}^{2}+\psi_{i+1,q}^{2}\right)^{\nicefrac{{1}}{{2}}}$.
Then for $r\geq 1$ we have that

$\displaystyle\left\|\omega_{i,j,q,n}\right\|_{L^{r}\left(\mathrm{supp\,}\psi_{i\pm,q}\right)}\lesssim\Gamma_{q+1}^{-\nicefrac{{2j}}{{r}}}$
(5.34)

holds for all $0\leq i\leq{i_{\rm max}}$, $0\leq j\leq{j_{\rm max}}$, and
$0\leq n\leq{n_{\rm max}}$. The implicit constant is independent of $i,j,q,n$.

###  5.4. Anisotropic checkerboard cutoff functions

We construct anisotropic checkerboard cutoff functions which are well-suited
for intermittent pipe flows with axes parallel to $e_{3}$. The construction
for general $\xi\in\Xi$ follows by rotation. Consider a partition of
$\mathbb{T}^{3}$ into the rectangular prisms defined using

$\left\\{(x_{1},x_{2},x_{3})\in\mathbb{T}^{3}\,:\,0\leq
x_{1},x_{2}\leq\mathcal{C}_{\Gamma}\Gamma_{q+1}\left(\lambda_{q+1}r_{q+1,n}\right)^{-1},\,0\leq
x_{3}\leq 2\pi\lambda_{q,n}^{-1}\right\\}$ (5.35)

and its translations by

$\bigl{(}l_{1}\mathcal{C}_{\Gamma}\Gamma_{q+1}(\lambda_{q+1}r_{q+1,n})^{-1},l_{2}\mathcal{C}_{\Gamma}\Gamma_{q+1}(\lambda_{q+1}r_{q+1,n})^{-1},l_{3}2\pi\lambda_{q,n}^{-1}\bigr{)}\,$

for

$l_{1},l_{2}\in\\{0,\dots,\nicefrac{{\mathcal{C}_{\Gamma}}}{{2\pi}}\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,n}-1\\}\,,\qquad
l_{3}\in\\{0,\dots,\lambda_{q,n}-1\\}\,,$

where $\mathcal{C}_{\Gamma}\geq 1$ ensures that the prisms evenly partition
$[-\pi,\pi]^{3}$ and is bounded above independently of $q$. Index these prisms
by integer triples $\vec{l}=(l_{1},l_{2},l_{3})$. Let
$\mathcal{X}_{q,n,e_{3},\vec{l}}$ be a $C^{\infty}$ partition of unity adapted
to this checkerboard of anisotropic rectangular prisms which satisfies

$\sum_{\vec{l}}\bigl{(}\mathcal{X}_{q,n,e_{3},\vec{l}}\bigr{)}^{2}=1$ (5.36)

for any $q$ and $n$. Specifically, we impose that spatial derivatives applied
to cutoffs belonging to this partition of unity cost
$\approx\Gamma_{q+1}(\lambda_{q+1}r_{q+1,n})^{-1}$ in the $x_{1}$ and $x_{2}$
directions, and $\approx\lambda_{q,n}^{-1}$ in the $x_{3}$ direction, so that

$\bigl{\|}\partial_{1}^{M_{1}}\partial_{2}^{M_{2}}\partial_{3}^{M}\chi_{q,n,e_{3},\vec{l}}\bigr{\|}_{L^{\infty}}\lesssim\left(\lambda_{q+1}r_{q+1,n}\Gamma_{q+1}^{-1}\right)^{M_{1}+M_{2}}\lambda_{q,{\widetilde{n}}}^{M}\,$

for $M_{1},M_{2},M\leq 3\mathsf{N}_{\rm fin}$. Furthermore, for
$\vec{l},\vec{l}^{*}$ such that

$|l_{1}-l_{1}^{*}|\geq 2,\qquad|l_{2}-l_{2}^{*}|\geq
2,\qquad|l_{3}-l_{3}^{*}|\geq 2,$

we impose that

$\mathcal{X}_{q,n,e_{3},\vec{l}}\;\mathcal{X}_{q,n,e_{3},\vec{l}^{*}}=0\,.$

Incorporating rotations into the above construction, we may similarly produce
cutoff functions $\mathcal{X}_{q,n,\xi,\vec{l}}$ satisfying analogous
properties for $\xi\in\Xi$. Note that if
$\\{\xi,\xi^{\prime},\xi^{\prime\prime}\\}$ forms an orthonormal basis for
$\mathbb{R}^{3}$, then

$\bigl{\|}\left(\xi^{\prime}\cdot\nabla\right)^{M_{1}}\left(\xi^{\prime\prime}\cdot\nabla\right)^{M_{2}}(\xi\cdot\nabla)^{M}\chi_{q,n,\xi,\vec{l}}\bigr{\|}_{L^{\infty}}\lesssim\left(\lambda_{q+1}r_{q+1,n}\Gamma_{q+1}^{-1}\right)^{M_{1}+M_{2}}\lambda_{q,{\widetilde{n}}}^{M}\,.$
(5.37)

######  Definition 5.17 (Anisotropic checkerboard cutoff function).

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

$\zeta_{q,i,k,n,{\xi},\vec{l}}\,(x,t)=\mathcal{X}_{q,n,{\xi},\vec{l}}\left(\Phi_{i,k,q}(x,t)\right).$
(5.38)

These cutoff functions satisfy properties which we enumerate in the following
lemma.

######  Lemma 5.18.

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

  1. (1)

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

  2. (2)

For each $t\in\mathbb{R}$ and all $x=(x_{1},x_{2},x_{3})\in\mathbb{T}^{3}$,

$\sum_{\vec{l}}\bigl{(}\zeta_{q,i,k,n,{\xi},\vec{l}}\,(x,t)\bigr{)}^{2}=1\,.$
(5.39)

  3. (3)

Let $A=(\nabla\Phi_{i,k,q})^{-1}$. Then we have the spatial derivative
estimate

$\displaystyle\bigl{\|}D^{N_{1}}D_{t,q}^{M}({\xi^{\ell}A_{\ell}^{j}\partial_{j}})^{N_{2}}\zeta_{q,i,k,n,\xi,\vec{l}}\bigr{\|}_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\widetilde{\chi}_{i,k,q}\right)}$
$\displaystyle\lesssim\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,n}\right)^{N_{1}}\lambda_{q,n}^{N_{2}}$
$\displaystyle\qquad\qquad\times\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c}_{0}}\tau_{q}^{-1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(5.40)

for all $N_{1}+N_{2}+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$.

  4. (4)

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

$\textnormal{diam}(\mathrm{supp\,}(\zeta_{q,i,k,n,\xi,\vec{l}}\,(\cdot,t)))\lesssim\lambda_{q,n}^{-1}\,.$
(5.41)

###### Proof of Lemma 5.18.

The proof of (1) is immediate from (5.38). (5.39) follows from (1) and (5.36).
To verify (3), the only nontrivial calculations are those including the
differential operator $(\xi^{\ell}A_{\ell}^{j}\partial_{j})$. Using the
Leibniz rule, the contraction

$\xi^{\ell}A_{\ell}^{j}\partial_{j}\zeta_{q,i,k,n,\xi,\vec{l}}=\xi^{\ell}A_{\ell}^{j}(\partial_{m}\mathcal{X}_{q,n,\xi,\vec{l}})(\Phi_{i,k,q})\partial_{j}\Phi_{i,k,q}^{m}=\xi^{m}(\partial_{m}\mathcal{X}_{q,n,\xi,\vec{l}})(\Phi_{i,k,q})\,,$

(5.37), and (5.19g) gives the desired estimate. The proof of (5.41) follows
from the construction of $\mathcal{X}_{q,n,\xi,\vec{l}}$ and the Lipschitz
bound obeyed by $\nabla v_{\ell_{q}}$ on the support of $\psi_{i,q}$; see for
example (3.16). ∎

###  5.5. Definition of the cumulative cutoff function

Finally, combining the cutoff functions defined in Definition 5.5,
(5.25)–(5.26), and (5.15), we define the cumulative cutoff function by

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

Since the values of $q$ and $n$ are clear from the context and the values of
$\xi$ and $\vec{l}$ are irrelevant in many arguments, we may abbreviate the
above using any of

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

It follows from (2.11) at level $q$, (5.27), (5.13), and (5.39) that for every
${(q,n,\xi)}$ fixed, we have

$\displaystyle\sum_{i,j\geq
0}\sum_{k\in\mathbb{Z}}\sum_{\vec{l}}\eta_{i,j,k,q,n,\xi,\vec{l}}\,^{2}(x,t)=1\,.$
(5.42)

The sum in $i$ goes up to $i_{\rm max}$ (defined in (5.12)), while the sum in
$j$ goes up to $j_{\rm max}$ (defined in (5.31)).

We conclude this section with support estimates on the cumulative cutoff
functions $\eta_{i,j,k,q,n,\xi,\vec{l}}$.

######  Lemma 5.19.

For $r_{1},r_{2}\in[1,\infty]$ with $\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$ and
any $0\leq i\leq{i_{\rm max}}$, $0\leq j\leq{j_{\rm max}}$, and $\xi\in\Xi$,
we have that

$\sum_{\vec{l}}\left|\mathrm{supp\,}(\eta_{i,j,k,q,n,\xi,\vec{l}})\right|\lesssim\Gamma_{q+1}^{\frac{-2i+{\mathsf{C}_{b}}}{r_{1}}+\frac{-2j}{r_{2}}+2}\,.$
(5.43)

###### Proof of Lemma 5.19.

From (2.17) at level $q$ and (5.34), we have that for each fixed time $t$,

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

Using the fact that $\\{\eta_{q,i,k,n,\xi,\vec{l}}\\}_{\vec{l}}$ forms a
partition of unity from (5.39) and $\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$ gives
the desired estimate. ∎

##  6\. Inductive propositions

###  6.1. Induction on $q$

The main claim of this section is an induction on $q$. Notice that the
estimates in this proposition match the inductive assumptions (2.9) and (2.10)
at level $q+1$.

######  Proposition 6.1 (Inductive Step on $q$).

Given the velocity field $v_{\ell_{q}}$ which solves the Euler-Reynolds system
with stress $\mathring{R}_{\ell_{q}}+\mathring{R}_{q}^{\textnormal{comm}}$,
where $v_{\ell_{q}}$, $\mathring{R}_{\ell_{q}}$, and
$\mathring{R}_{q}^{\textnormal{comm}}$ satisfy the conclusions of Lemma 4.1 in
addition to (2.8a)–(2.21b), there exist $v_{q+1}=v_{\ell_{q}}+w_{q+1}$ and
$\mathring{R}_{q+1}$ which satisfy the following:

  1. (1)

$v_{q+1}$ solves the Euler-Reynolds system with stress $\mathring{R}_{q+1}$.

  2. (2)

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

$\displaystyle\left\|\psi_{i,q}D^{k}D_{t,q}^{m}w_{q+1}\right\|_{L^{2}}$
$\displaystyle\leq\Gamma_{q+1}^{-1}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(6.1a)
$\displaystyle\left\|D^{k}D_{t,q}^{m}w_{q+1}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\Theta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(6.1b)

  3. (3)

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

$\displaystyle\bigl{\|}\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}\bigr{\|}_{L^{1}}$
$\displaystyle\leq\Gamma_{q+1}^{-\mathsf{C_{R}}}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(6.2a)
$\displaystyle\bigl{\|}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}\bigr{\|}_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+2}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(6.2b)

###  6.2. Notations

The proof of Proposition 6.1 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 $\displaystyle
w_{q+1,{\widetilde{n}}}$. We shall employ the notation:

  1. (1)

${\widetilde{n}}$ \- An integer taking values $0\leq{\widetilde{n}}\leq{n_{\rm
max}}$ over which induction is performed, indexing the component
$w_{q+1,{\widetilde{n}}}$ of the velocity increment $w_{q+1}$. 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)

$n$ \- 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)

${\mathring{H}}_{q,n}^{n^{\prime}}$ \- The component of $\mathring{R}_{q,n}$
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)

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

$\displaystyle\mathbb{P}_{[q,n]}=\begin{cases}\mathbb{P}_{\left[\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1},\lambda_{q,1}\right)}&\mbox{if
}n=1\,,\\\ \mathbb{P}_{\left[\lambda_{q,n-1},\lambda_{q,n}\right)}&\mbox{if
}2\leq n\leq{n_{\rm max}}\,,\\\ \mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}&\mbox{if }n={n_{\rm max}}+1\,,\end{cases}$ (6.3)

where $\mathbb{P}_{[\lambda_{1},\lambda_{2})}$ is defined in Remark 3.4 as
$\mathbb{P}_{\geq\lambda_{1}}\mathbb{P}_{<\lambda_{2}}$. Errors which include
the frequency projector $\mathbb{P}_{[q,{n_{\rm max}}+1]}$ will be small
enough to be absorbed into $\mathring{R}_{q+1}$. We note that if
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, then from (2.7b), any
$\frac{\mathbb{T}^{3}}{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}$-periodic
function satisfies

$\displaystyle f=\mathchoice{{\vbox{\hbox{$\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+1}r_{q+1,{\widetilde{n}}}}f=\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+\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}f\,.$ (6.4)

  5. (5)

In order to later deduce a useful refinement of (6.4), we set

$\displaystyle r({\widetilde{n}})=\begin{cases}0&\mbox{if
}{\widetilde{n}}=0\,,\\\ \frac{{n_{\rm max}}+{\widetilde{n}}}{2}&\mbox{if
}1\leq{\widetilde{n}}\leq{n_{\rm max}}-1\,.\end{cases}$ (6.5)

  6. (6)

In order to keep track of small losses related to the process of building a
stress $\mathring{R}_{q,{\widetilde{n}}}$, corrector
$w_{q+1,{\widetilde{n}}}$, and new stresses $\mathring{R}_{q,n}$ for
$n>{\widetilde{n}}$, we define

$\displaystyle\Upsilon(n)=\begin{cases}0&\mbox{if }n=0\,,\\\ 1&\mbox{if }1\leq
n\leq\frac{{n_{\rm max}}}{2}\\\ k&\mbox{if }\frac{2^{k-1}-1}{2^{k-1}}{n_{\rm
max}}<n\leq\frac{2^{k}-1}{2^{k}}{n_{\rm max}}\\\ 2+\lceil\log_{2}({n_{\rm
max}})\rceil&\mbox{if }n={n_{\rm max}}\,.\end{cases}$ (6.6)

$\Upsilon({\widetilde{n}})$ gives an upper bound on the number of steps in the
induction on ${\widetilde{n}}$ it takes to produce the _entire_ error term
$\mathring{R}_{q,{\widetilde{n}}}$. A consequence of (6.5) and (6.6) is that

$\displaystyle
n>r({\widetilde{n}})\qquad\implies\qquad\Upsilon(n)\geq\Upsilon({\widetilde{n}})+1\,.$
(6.7)

To prove this, first consider the case $n={n_{\rm max}}$. Then for all
$0\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, we have that
$r({\widetilde{n}})<{n_{\rm max}}$, and so (6.7) should hold for all
${\widetilde{n}}<{n_{\rm max}}$. Since ${\widetilde{n}}<{n_{\rm max}}$, there
exists a minimum value of $k$, say $k_{\widetilde{n}}$, such that
${\widetilde{n}}\leq{n_{\rm max}}-\frac{{n_{\rm
max}}}{2^{k_{\widetilde{n}}}}$, which implies that
$\Upsilon({\widetilde{n}})\leq k_{\widetilde{n}}$. For
$k=\lceil\log_{2}({n_{\rm max}})\rceil+2$, however, we have that ${n_{\rm
max}}-\frac{{n_{\rm max}}}{2^{k-1}}\geq{n_{\rm max}}-\frac{1}{2}$, and so it
must be the case that $k_{\widetilde{n}}\leq\lceil\log_{2}({n_{\rm
max}})\rceil+1$, which proves (6.7) in the case $n={n_{\rm max}}$, and shows
that

$\Upsilon(n)\leq 2+\lceil\log_{2}({n_{\rm max}})\rceil\qquad\forall
n\leq{n_{\rm max}}\,.$ (6.8)

To prove (6.7) in the remaining cases, note that if ${\widetilde{n}}=0$, then
$n>r(0)\implies n\geq 1$ and so (6.7) holds. If ${\widetilde{n}}=1$, then
$n>\frac{{n_{\rm max}}+1}{2}$, and again (6.7) holds. Finally, if
$2\leq{\widetilde{n}}\leq{n_{\rm max}}-1$ and $\Upsilon({\widetilde{n}})=k$,
then

$\displaystyle n$ $\displaystyle>\frac{{n_{\rm
max}}+{\widetilde{n}}}{2}>\frac{{n_{\rm
max}}+\frac{2^{k-1}-1}{2^{k-1}}{{n_{\rm
max}}}}{2}=\frac{2^{k}-1}{2^{k}}{n_{\rm
max}}\qquad\implies\qquad\Upsilon(n)\geq k+1\,.$

  7. (7)

$\mathring{R}_{q+1}^{\widetilde{n}}$ \- For any
$0\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, this is 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

$\mathring{R}_{q+1}^{{\widetilde{n}}+1}=\mathring{R}_{q+1}^{{\widetilde{n}}}+\left(\textnormal{errors
coming from }w_{q+1,{\widetilde{n}}+1}\textnormal{ that also go into
}\mathring{R}_{q+1}\right)\,.$ (6.9)

We adopt the convention that $\mathring{R}_{q+1}^{-1}=0$.

  8. (8)

We adopt the convention that $\sum_{n=0}^{-1}f(n)\equiv\sum_{n={n_{\rm
max}}+1}^{{n_{\rm max}}}f(n)\equiv 0$ denotes an _empty_ summation.

###  6.3. Induction on $\widetilde{n}$

We split the verification of Proposition 6.1 using a sub-inductive procedure
on the parameter ${\widetilde{n}}$. Note that summing (6.11a)-(6.12b) over
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, appealing to (8.43) and (8.48), and
using the extra factor of $\Gamma_{q+1}^{-1}$ to kill implicit constants, we
have matched the desired bounds in (6.1a)-(6.2b).

######  Proposition 6.2 (Induction on ${\widetilde{n}}$: From
${\widetilde{n}}-1$ to ${\widetilde{n}}$ for $0\leq{\widetilde{n}}\leq{n_{\rm
max}}$).

Under the assumptions of Proposition 6.1 and Lemma 4.1, we let
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$ be given, and let
$v_{q,{\widetilde{n}}-1}=v_{\ell_{q}}+\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}w_{q+1,n^{\prime}}$,
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$, and
$\mathring{H}_{q,n}^{n^{\prime}}$ be given for $0\leq
n^{\prime}\leq{\widetilde{n}}-1$ and ${\widetilde{n}}\leq n\leq{n_{\rm max}}$,
such that the following are satisfied:

  1. (1)

${v_{q,{\widetilde{n}}-1}}$ solves the Euler-Reynolds system with stress

$\displaystyle\mathbf{1}_{\\{{\widetilde{n}}=0\\}}\mathring{R}_{\ell_{q}}+\mathring{R}_{q+1}^{{\widetilde{n}}-1}+\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\sum\limits_{n>r(n^{\prime})}^{{n_{\rm
max}}}\mathring{H}_{q,n}^{n^{\prime}}+\mathring{R}_{q}^{\textnormal{comm}}\,.$
(6.10)

  2. (2)

For all
$k+m\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}^{\prime}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9$ and $0\leq
n^{\prime}\leq{\widetilde{n}}-1$,

$\displaystyle\left\|D^{k}D_{t,q}^{m}w_{q+1,n^{\prime}}\right\|_{L^{2}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,n^{\prime}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{3}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}^{\prime}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(6.11a)
$\displaystyle\left\|D^{k}D_{t,q}^{m}w_{q+1,n^{\prime}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon(n^{\prime})+\frac{7}{2}}r_{q+1,n^{\prime}}^{-1}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}^{\prime}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(6.11b)

  3. (3)

For all $k,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$ and
$1\leq{\widetilde{n}}\leq{n_{\rm max}}$,

$\displaystyle\bigl{\|}\psi_{i,q}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}^{{\widetilde{n}}-1}\bigr{\|}_{L^{1}}$
$\displaystyle\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(6.12a)
$\displaystyle\bigl{\|}D^{k}D_{t,q}^{m}\mathring{R}_{q+1}^{{\widetilde{n}}-1}\bigr{\|}_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i+1}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(6.12b)

  4. (4)

For $0\leq n^{\prime}\leq{\widetilde{n}}-1$, $r(n^{\prime})<n\leq{n_{\rm
max}}$, and all $k+m\leq\mathsf{N}_{\rm fin,n}$,

$\displaystyle\bigl{\|}D^{k}D_{t,q}^{m}\mathring{H}_{q,n}^{n^{\prime}}\bigr{\|}_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,n}\lambda_{q,n}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,,$
(6.13a)
$\displaystyle\bigl{\|}D^{k}D_{t,q}^{m}\mathring{H}_{q,n}^{n^{\prime}}\bigr{\|}_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon(n)}\lambda_{q,n}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(6.13b)

Then if $0\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, there exists
$w_{q+1,{\widetilde{n}}}$, $\mathring{R}_{q+1}^{{\widetilde{n}}}$, and
$\mathring{H}_{q,n}^{n^{\prime}}$ for $0\leq n^{\prime}\leq{\widetilde{n}}$,
such that (6.10)–(6.13b) are satisfied with ${\widetilde{n}}-1$ replaced with
${\widetilde{n}}$. If ${\widetilde{n}}={n_{\rm max}}$, then there exists
$w_{q+1,{n_{\rm max}}}$ and $\mathring{R}_{q+1}$ such that
$v_{q+1}:=v_{q,{n_{\rm max}}-1}+w_{q+1,{n_{\rm max}}}$ solves the Euler-
Reynolds system with stress $\mathring{R}_{q+1}$, and $v_{q+1}$, $w_{q+1}$,
and $\mathring{R}_{q+1}$ satisfy conclusions (6.1a)–(6.2b) from Proposition
6.1.

##  7\. Proving the main inductive estimates

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

In this section we define the stresses $\mathring{R}_{q,{\widetilde{n}}}$ and
the perturbations $w_{q+1,{\widetilde{n}}}$ used to correct them. For
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, we define

$\mathring{R}_{q,{\widetilde{n}}}=\mathbf{1}_{\\{{\widetilde{n}}=0\\}}\mathring{R}_{\ell_{q}}+\sum_{0\leq
n^{\prime}\leq{\widetilde{n}}-1}\mathring{H}_{q,{\widetilde{n}}}^{n^{\prime}}\,.$
(7.1)

In Subsection 7.3, we will show that
$\mathring{H}^{n^{\prime}}_{q,{\widetilde{n}}}$ is zero in certain parameter
regimes, although for the moment this is irrelevant. Now for any fixed values
of ${\widetilde{n}}$, $i$, $j$, and $k$, we may define

$R_{q,{\widetilde{n}},j,i,k}=\nabla\Phi_{(i,k)}\left(\delta_{q+1,{\widetilde{n}}}\Gamma^{2j+4}_{q+1}\mathrm{Id}-\mathring{R}_{q,{\widetilde{n}}}\right)\nabla\Phi_{(i,k)}^{T}\,.$
(7.2)

Let $\xi\in\Xi$ be a vector from Proposition 3.1. For all $\xi\in\Xi$, we
define the coefficient function $a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}$ by

$a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}:=a_{\xi,i,j,k,q,{\widetilde{n}}}:=a_{(\xi)}=\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}\,\gamma_{\xi}\left(\frac{R_{q,{\widetilde{n}},j,i,k}}{\delta_{q+1,{\widetilde{n}}}\Gamma^{2j+4}_{q+1}}\right)\,.$
(7.3)

From Corollary 5.13, we see that on the support of $\eta_{(i,j,k)}$ we have
$|\mathring{R}_{q,{\widetilde{n}}}|\lesssim\Gamma_{q+1}^{2j+2}\delta_{q+1,{\widetilde{n}}}$,
and thus by estimate (5.19a) from Corollary 5.10, we have that

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

once $\lambda_{0}$ is sufficiently large. Thus we may apply Proposition 3.1.

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

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

where we have used Proposition 3.3 (with $\lambda=\lambda_{q+1}$ and
$r=r_{q+1,{\widetilde{n}}}$) and the shorthand notation

$\displaystyle\mathbb{W}_{\xi,q+1,{\widetilde{n}}}=\mathbb{W}^{(i,j,k,{\widetilde{n}},\vec{l})}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}=:\mathbb{W}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{s}\,.$
(7.4)

The superscript $s=(i,j,k,{\widetilde{n}},\vec{l})$ indicates the placement of
the intermittent pipe flow
$\mathbb{W}^{i,j,k,{\widetilde{n}},\vec{l}}_{\xi,q+1,{\widetilde{n}}}$ (cf.
(2) from Proposition 3.3), which depends on $i$, $j$, $k$, ${\widetilde{n}}$,
and $\vec{l}$ and is only relevant in Section 7.5. To ease notation, we will
suppress the superscript except in Section 7.5. We will also adopt the same
notational conventions for the potentials
$\mathbb{U}_{\xi,q+1,{\widetilde{n}}}$. Furthermore, (3.8) from Proposition
3.3 gives that we can now write the principal part of the first term of the
perturbation as

$w_{q+1,{\widetilde{n}}}^{(p)}=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}a_{(\xi)}\mathrm{curl\,}\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right):=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}\,.$
(7.5)

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

$w_{q+1,{\widetilde{n}}}^{(c)}=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}w_{(\xi)}^{(c)}$
(7.6)

so that

$w_{q+1,{\widetilde{n}}}=w_{q+1,{\widetilde{n}}}^{(p)}+w_{q+1,{\widetilde{n}}}^{(c)}=\sum_{i,j,k}\sum_{\vec{l}}\sum_{\xi}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\,.$
(7.7)

###  7.2. Estimates for $w_{q+1,{\widetilde{n}}}$

In this section, we verify (6.11a) and (6.11b). 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 7.1.

For $N,N^{\prime},M$ with $N^{\prime}=0,1$ and
$N+N^{\prime}+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-4$, and $r,r_{1},r_{2}\in[1,\infty]$ with
$\frac{1}{r_{1}}+\frac{1}{r_{2}}=1$, we have the following estimate.

$\displaystyle\bigl{\|}D^{N}D_{t,q}^{M}(\xi^{\ell}A_{\ell}^{j}\partial_{j})^{N^{\prime}}a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}\bigr{\|}_{L^{r}}$
$\displaystyle\lesssim|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|^{\nicefrac{{1}}{{r}}}\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{j+2}\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{N}$
$\displaystyle\qquad\qquad\times\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}}}\right)^{N^{\prime}}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$
(7.8)

In the case that $r=\infty$, the above estimate gives that

$\displaystyle\bigl{\|}D^{N}D_{t,q}^{M}(\xi^{\ell}A_{\ell}^{j}\partial_{j})^{N^{\prime}}a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}\bigr{\|}_{L^{\infty}}$
$\displaystyle\lesssim\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon({\widetilde{n}})+\frac{7}{2}}\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{N}$
$\displaystyle\qquad\qquad\times\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}}}\right)^{N^{\prime}}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.9)

###### Proof of Lemma 7.1.

We first compute (7.8) for the case $r=\infty$. Recalling estimate (5.28), we
have that for all $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

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

From Corollary 5.10, 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 definition (7.2), for
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}R_{q,{\widetilde{n}},j,i,k}\right\|_{L^{\infty}(\mathrm{supp\,}\eta_{(i,j,k)})}\lesssim\delta_{q+1,{\widetilde{n}}}\Gamma^{2j+4}_{q+1}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.10)

The above estimates allow us to apply [5, Lemma A.5] with $N=N^{\prime}$,
$M=M^{\prime}$ so that $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,
$\psi=\gamma_{\xi,}$, $\Gamma_{\psi}=1$, $v=v_{\ell_{q}}$, $D_{t}=D_{t,q}$,
$h(x,t)=R_{q,{\widetilde{n}},j,i,k}(x,t)$,
$C_{h}=\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}=\Gamma^{2}$,
$\lambda=\widetilde{\lambda}=\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$,
$\mu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+2}$,
$\widetilde{\mu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, and
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$. We obtain that for all
$N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-4$,

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

From the above bound, definition (7.3), the Leibniz rule, estimate (2.23) at
level $q$ in conjunction with (8.35), (5.19g), (5.16), (5.33), and (3), we
obtain that for
$N+N^{\prime}+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,x}-\mathsf{N}_{\rm cut,t}-4$,

$\displaystyle\bigl{\|}D^{N}D_{t,q}^{M}(\xi^{\ell}A_{\ell}^{j}\partial_{j})^{N^{\prime}}a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}\bigr{\|}_{L^{\infty}}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{j+2}(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}})^{N}$
$\displaystyle\qquad\qquad\times(\Gamma_{q+1}\lambda_{q,{\widetilde{n}}})^{N^{\prime}}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

Then, using (5.30) the above bound becomes (7.9). When $r\neq\infty$, we use
$\left\|f\right\|_{L^{r}}\leq\left\|f\right\|_{L^{\infty}}|\\{\mathrm{supp\,}f\\}|^{\nicefrac{{1}}{{r}}}$
and the demonstrated bound for $r=\infty$ to obtain (7.8) for the full range
of $r$. ∎

An immediate consequence of Lemma 7.1 is that we have estimates for the
velocity increments themselves. These are summarized in the following
corollary. The proofs for $r\neq\infty$ are analogous to those from [5,
Corollary 8.2] and therefore use Lemma A.1. We only note that the gap between
the spatial derivative cost of $a_{(\xi)}$
($\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$ from Lemma 7.1) and
the minimum frequency of $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$
($\lambda_{q+1}r_{q+1,{\widetilde{n}}}$ from (7.4) and Proposition 3.3) is now
only $\Gamma_{q+1}$, and so we need the inequality (8.37) in order to satisfy
(A.2). The assumption (A.1) follows from (8.58a). The estimates for $r=\infty$
follow directly from (7.9) and (3.6).

######  Corollary 7.2.

For $N+M\leq\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-8$ and
$(r,r_{1},r_{2})\in\left\\{(1,2,2),(2,\infty,1)\right\\}$, for $w_{(\xi)}$ we
have the estimates

$\displaystyle\left\|D^{N}D_{t,q}^{M}w_{(\xi)}\right\|_{L^{r}}$
$\displaystyle\lesssim|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|^{\nicefrac{{1}}{{r}}}\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}{\left(r_{q+1,{\widetilde{n}}}\right)}^{\frac{2}{r}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.11a) $\displaystyle\left\|D^{N}D_{t,q}^{M}w_{(\xi)}\right\|_{L^{\infty}}$
$\displaystyle\lesssim
r_{q+1,{\widetilde{n}}}^{-1}\lambda_{q+1}^{N}\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon({\widetilde{n}})+\frac{7}{2}}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.11b)

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 that

$\displaystyle\bigl{\|}D^{N}D_{t,q}^{M}w_{(\xi)}^{(c)}\bigr{\|}_{L^{r}}$
$\displaystyle\lesssim\frac{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}{\lambda_{q+1}}|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|^{\nicefrac{{1}}{{r}}}\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma^{j+2}_{q+1}{\left(r_{q+1,{\widetilde{n}}}\right)}^{\frac{2}{r}-1}$
$\displaystyle\qquad\qquad\qquad\times\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.12a)
$\displaystyle\bigl{\|}D^{N}D_{t,q}^{M}w_{(\xi)}^{(c)}\bigr{\|}_{L^{\infty}}$
$\displaystyle\lesssim
r_{q+1,{\widetilde{n}}}^{-1}\frac{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}{\lambda_{q+1}}\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon({\widetilde{n}})+\frac{7}{2}}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.12b)

######  Remark 7.3.

Note that the above estimates verify the bounds (6.11a) and (6.11b) after
summing on $(i,j,k,{\widetilde{n}},\xi,\vec{l})$ and using (5.43) with
$r_{1}=\infty$ and $r_{2}=2$. Then from (7.5)–(7.7), (7.11a)–(7.12b), and the
parameter inequalities (8.25), (8.43), and (8.48), the bounds (6.1a) and
(6.1b) follow after using the extra factor of $\Gamma_{q+1}^{-1}$ to absorb
implicit constants.

###  7.3. Identification of error terms

Recall that $v_{q,{\widetilde{n}}-1}$ is divergence-free and is a solution to
the Euler-Reynolds system with stress given in (6.10). Now using the
definition of $\mathring{R}_{q,{\widetilde{n}}}$ from (7.1) for
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, we add $w_{q+1,{\widetilde{n}}}$ as
defined in (7.7), we have that
$v_{q,{\widetilde{n}}}:=v_{q,{\widetilde{n}}-1}+w_{q+1,{\widetilde{n}}}$
solves

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}}+\mathrm{div\,}\left(v_{q,{\widetilde{n}}}\otimes
v_{q,{\widetilde{n}}}\right)+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)+\mathrm{div\,}\Biggl{(}\sum\limits_{n^{\prime}=0}^{{\widetilde{n}}-1}\sum\limits_{n>r(n^{\prime})}^{{n_{\rm
max}}}\mathring{H}_{q,n}^{n^{\prime}}\Biggr{)}+\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}$
$\displaystyle\quad+D_{t,q}w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}+2\sum_{n^{\prime}\leq{\widetilde{n}}-1}\mathrm{div\,}\left(w_{q+1,n^{\prime}}\otimes_{\rm
s}w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle\quad+\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}+\mathring{R}_{q,{\widetilde{n}}}\right).$ (7.13)

Here we use the notation $a\otimes_{\rm s}b=\frac{1}{2}(a\otimes b+b\otimes
a)$. The first term on the right hand side is
$\mathring{R}_{q+1}^{{\widetilde{n}}-1}$, which for ${\widetilde{n}}\geq 1$
satisfies the same estimates as $\mathring{R}_{q+1}^{\widetilde{n}}$ by
(6.12a) and will thus be absorbed into $\mathring{R}_{q+1}^{{\widetilde{n}}}$.
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 $\mathring{H}_{q,n}^{\widetilde{n}}$,
matches (6.10) at level ${\widetilde{n}}$ (i.e. replacing every instance of
${\widetilde{n}}-1$ with ${\widetilde{n}}$). We apply the inverse divergence
operators from Proposition A.2 to the transport and Nash errors to obtain

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

and these errors are absorbed into $\mathring{R}_{q+1}^{\widetilde{n}}$ or the
new pressure. We will show in Section 7.5 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

$2\sum_{0\leq n^{\prime}\leq{\widetilde{n}}-1}w_{q+1,n^{\prime}}\otimes_{\rm
s}w_{q+1,{\widetilde{n}}}=0\,.$ (7.14)

So to verify (6.10) at level ${\widetilde{n}}$, only the analysis of last line
of the right-hand side of (7.13) remains.

For a fixed ${\widetilde{n}}$, throughout this section we will consider sums
over indices $(\xi,i,j,k,\vec{l})$, where the direction vector $\xi$ takes on
one of the finitely many values in Proposition 3.3, $0\leq i\leq{i_{\rm
max}}(q)$ indexes the velocity cutoffs, $0\leq j\leq{j_{\rm
max}}(q,{\widetilde{n}})$ indexes the stress cutoffs, the parameter
$k\in\mathbb{Z}$ indexes the time cutoffs defined in (5.15), and lastly,
$\vec{l}\in\mathbb{N}^{3}_{0}$ indexes the checkerboard cutoffs from
Definition 5.17. For brevity of notation, we denote sums over such indexes as

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

Moreover, we shall denote as

$\sum_{\neq\\{\xi,i,j,k,\vec{l}\\}}$

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

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

We may now write out the self-interaction of $w_{q+1,{\widetilde{n}}}$ as

$\displaystyle\mathrm{div\,}\left(w_{q+1,{\widetilde{n}}}\otimes
w_{q+1,{\widetilde{n}}}\right)$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\left(\mathrm{curl\,}\bigl{(}a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\bigr{)}\otimes\mathrm{curl\,}\bigl{(}a_{(\xi)}\nabla\Phi_{i,k}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\bigr{)}\right)$
$\displaystyle+\sum_{\neq\\{\xi,i,j,k,\vec{l}\\}}\mathrm{div\,}\left(\mathrm{curl\,}\bigl{(}a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\bigr{)}\otimes\mathrm{curl\,}\bigl{(}a_{({\xi^{*}})}\nabla\Phi_{({i^{*}},{k^{*}})}^{T}\mathbb{U}_{{\xi^{*}},q+1,{\widetilde{n}}}\bigr{)}\right)$
$\displaystyle=:\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1}+\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},2}.$
(7.15)

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

$\mathcal{O}_{{\widetilde{n}},2}=0\,.$ (7.16)

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

$\displaystyle\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1}$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\left(\bigl{(}a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\otimes\bigl{(}a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\right)$
$\displaystyle\qquad+2\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\left(\bigl{(}a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\otimes_{\rm
s}\bigl{(}\nabla
a_{(\xi)}\times\bigl{(}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\bigr{)}\right)$
$\displaystyle\qquad+\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\left(\bigl{(}\nabla
a_{(\xi)}\times\bigl{(}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\bigr{)}\otimes\bigl{(}\nabla
a_{(\xi)}\times\bigl{(}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\bigr{)}\bigr{)}\right)$
$\displaystyle:=\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}+\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}\right).$
(7.17)

The last two of these terms are divergence corrector errors and will therefore
be absorbed into $\mathring{R}_{q+1}^{\widetilde{n}}$ and estimated in Section
7.6. So the only terms which we have yet to identify from (7.13) are
$\mathcal{O}_{{\widetilde{n}},1,1}$ and $\mathring{R}_{q,{\widetilde{n}}}$.

Recall cf. (7.4) that $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$ is periodized to
scale $\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}$. Using (6.4),
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}}}}+\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}\left(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\right)\,.$

Using (4) and (3.9) from Proposition 3.3 in combination with the above
identity, and the convention that $\bullet$ denotes the unspecified components
of a vector field, we then split $\mathcal{O}_{{\widetilde{n}},1,1}$ as

$\displaystyle\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}\right)$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\left(a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}\right)$
$\displaystyle\quad+\sum_{\xi,i,j,k,\vec{l}}\mathrm{div\,}\bigg{(}a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle=\mathrm{div\,}\sum_{\xi,i,j,k,\vec{l}}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}\eta_{(i,j,k)}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{\widetilde{n}},j,i,k}}{\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\quad+\sum_{\xi,i,j,k,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\quad+\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\sum_{n={\widetilde{n}}+1}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\,.$
(7.18)

By (3.1) from Proposition 3.1, identity (7.2), and (5.39), we obtain that

$\displaystyle\sum_{i,j,k,\xi}\sum_{\vec{l}}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}^{2}\,\gamma_{\xi}^{2}\left(\frac{R_{q,{\widetilde{n}},j,i,k}}{\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad=\sum_{i,j,k,\xi}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}\psi_{i,q}^{2}\omega_{i,j,q,{\widetilde{n}}}^{2}\chi_{i,k,q}^{2}\gamma_{\xi}^{2}\left(\frac{R_{q,{\widetilde{n}},j,i,k}}{\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}}\right)\nabla\Phi_{(i,k)}^{-1}\left(\xi\otimes\xi\right)\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad=\sum_{i,j,k}\psi_{i,q}^{2}\omega_{i,j,q,{\widetilde{n}}}^{2}\chi_{i,k,q}^{2}\left(\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}\mathrm{Id}-\mathring{R}_{q,{\widetilde{n}}}\right)$
$\displaystyle\qquad=-\mathring{R}_{q,{\widetilde{n}}}+\mathrm{Id}\biggl{(}\sum_{\;i,j,k}\psi_{i,q}^{2}\omega_{i,j,q,{\widetilde{n}}}^{2}\chi_{i,k,q}^{2}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{2j+4}\biggr{)}\,,$
(7.19)

where in the last equality we have appealed to the fact that
$\eta^{2}_{i,j,k}$ forms a partition of unity, cf. (5.42). The second term on
the right hand side of (7.19) is a pressure term.

Returning to the second and third lines in (7.18), we first note that when
${\widetilde{n}}=0$, (2.7b) gives that
$\lambda_{q+1}r_{q+1,0}=\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q,n}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{-2}=\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}$.
Then from (6.3) and (7.4), for all $1\leq n\leq{n_{\rm max}}+1$, we deduce
that
$\mathbb{P}_{[q,n]}\left(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0}\right)\neq
0$. Conversely, when $1\leq{\widetilde{n}}\leq{n_{\rm max}}$, for all
$n\geq{\widetilde{n}}+1$ such that
$\lambda_{q,n}<\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{-2}$,
i.e. such that the maximal frequency of $\mathbb{P}_{[q,n]}$ is less than the
minimal frequency of $\mathbb{P}_{\neq
0}\left(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\right)$,
we have that
$\mathbb{P}_{[q,n]}\left(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\right)=0$.
Using (2.7a), we write that

$\displaystyle\qquad\lambda_{q}^{\frac{1}{2}-\frac{n}{2({n_{\rm
max}}+1)}}\lambda_{q+1}^{\frac{1}{2}+\frac{n}{2({n_{\rm
max}}+1)}}<\lambda_{q+1}^{\frac{1}{2}}\lambda_{q}^{\frac{1}{4}-\frac{{\widetilde{n}}}{4({n_{\rm
max}}+1)}}\lambda_{q+1}^{\frac{1}{4}+\frac{{\widetilde{n}}}{4({n_{\rm
max}}+1)}}\Gamma_{q+1}^{-2}$ $\displaystyle\Leftrightarrow$
$\displaystyle\qquad\lambda_{q}^{\frac{1}{4}+\frac{{\widetilde{n}}-2n}{4({n_{\rm
max}}+1)}}<\lambda_{q+1}^{\frac{1}{4}+\frac{{\widetilde{n}}-2n}{4({n_{\rm
max}}+1)}}\Gamma_{q+1}^{-2}$ $\displaystyle\Leftrightarrow$
$\displaystyle\qquad
2\varepsilon_{\Gamma}<\frac{1}{4}+\frac{{\widetilde{n}}-2n}{4({n_{\rm
max}}+1)}$ $\displaystyle\Leftrightarrow$ $\displaystyle\qquad
8\varepsilon_{\Gamma}({n_{\rm max}}+1)<{n_{\rm max}}+1+{\widetilde{n}}-2n$
$\displaystyle\Leftarrow$ $\displaystyle\qquad n\leq\frac{{n_{\rm
max}}+{\widetilde{n}}}{2}\,,\qquad\frac{1}{2}-4\varepsilon_{\Gamma}({n_{\rm
max}}+1)>0\,.$ (7.20)

The second inequality in the last line follows from (8.6a). Based on (7.20)
and (6.5), we apply Proposition A.2 in the parameter regimes
${\widetilde{n}}=0,1\leq n\leq{n_{\rm max}}$ and
$1\leq{\widetilde{n}}\leq{n_{\rm max}}-1,r({\widetilde{n}})=\frac{{n_{\rm
max}}+{\widetilde{n}}}{2}<n\leq{n_{\rm max}}$ to define

$\displaystyle\mathring{H}_{q,n}^{\widetilde{n}}:=\mathcal{H}\bigg{(}\sum_{\xi,i,j,k}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{[q,n]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad\qquad+\sum_{\xi,i,j,k}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\mathbb{P}_{[q,n]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\bigg{)}\,.$
(7.21)

The terms from (7.18) with $\mathbb{P}_{[q,{n_{\rm max}}+1]}$ will be absorbed
into $\mathring{R}_{q+1}^{\widetilde{n}}$. We will show shortly that the terms
$\mathring{H}_{q,n}^{\widetilde{n}}$ in (7.21) are precisely the terms needed
to make (7.13) match (6.10) at level ${\widetilde{n}}$.

Recall from (6.9) 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 (6.10), we return to (7.13)
and use (7.14), (7.15), (7.17), (7.18), (7.19), (7.20), (6.5), and (7.21) to
write

$\displaystyle\partial_{t}v_{q,{\widetilde{n}}}+\mathrm{div\,}\left(v_{q,{\widetilde{n}}}\otimes
v_{q,{\widetilde{n}}}\right)+\nabla p_{q,{\widetilde{n}}-1}$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\biggl{(}\sum\limits_{\;n^{\prime}=0}^{{\widetilde{n}}-1}\sum\limits_{n>r(n^{\prime})}^{{n_{\rm
max}}}\mathring{H}_{q,n}^{n^{\prime}}\biggr{)}+\mathrm{div\,}\left(\mathring{R}_{q+1}^{{\widetilde{n}}-1}\right)$
$\displaystyle\quad+D_{t,q}w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}+\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}\right)+\mathrm{div\,}\left(\mathcal{O}_{{\widetilde{n}},1,1}+\mathring{R}_{q,{\widetilde{n}}}\right)$
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\biggl{(}\sum\limits_{\;n^{\prime}=0}^{{\widetilde{n}}-1}\sum\limits_{n>r(n^{\prime})}^{{n_{\rm
max}}}\mathring{H}_{q,n}^{n^{\prime}}\biggr{)}$
$\displaystyle\quad+\mathrm{div\,}\bigg{(}\mathring{R}_{q+1}^{{\widetilde{n}}-1}+(\mathcal{H}+\mathcal{R}^{*})\left(D_{t,q}w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)+\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}\bigg{)}+\nabla\pi$
$\displaystyle\quad+\mathrm{div\,}\bigg{[}\left(\mathcal{H}+\mathcal{R}^{*}\right)\bigg{(}\sum_{\xi,i,j,k,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\sum_{n>r({\widetilde{n}})}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle\qquad\qquad+(\mathcal{H}+\mathcal{R}^{*})\bigg{(}\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\sum_{n>r({\widetilde{n}})}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\bigg{)}\bigg{]}$
(7.22)
$\displaystyle=\mathrm{div\,}\mathring{R}_{q}^{\textnormal{comm}}+\mathrm{div\,}\biggl{(}\sum\limits_{\;n^{\prime}=0}^{{\widetilde{n}}}\sum\limits_{n>r(n^{\prime})}^{{n_{\rm
max}}}\mathring{H}_{q,n}^{n^{\prime}}\biggr{)}+\mathrm{div\,}\mathring{R}_{q+1}^{{\widetilde{n}}}+\nabla\pi\,,$
(7.23)

where

$\displaystyle\mathring{R}_{q+1}^{{\widetilde{n}}}$
$\displaystyle=\mathring{R}_{q+1}^{{\widetilde{n}}-1}+(\mathcal{H}+\mathcal{R}^{*})\left(D_{t,q}w_{q+1,{\widetilde{n}}}+w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)+(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1,2}+\mathcal{O}_{{\widetilde{n}},1,3}+\pi\mathrm{Id}$
$\displaystyle\quad+\mathcal{H}\bigg{(}\sum_{\xi,i,j,k,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{[q,{n_{\rm
max}}+1]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle\quad+\mathcal{H}\bigg{(}\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\mathbb{P}_{[q,{n_{\rm
max}}+1]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\bigg{)}$
$\displaystyle\quad+\mathcal{R}^{*}\bigg{(}\sum_{\xi,i,j,k,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\sum_{n>r({\widetilde{n}})}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}\otimes\mathbb{W})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}\bigg{)}$
$\displaystyle\quad+\mathcal{R}^{*}\bigg{(}\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\sum_{n>r({\widetilde{n}})}^{{n_{\rm
max}}+1}\mathbb{P}_{[q,n]}(\mathbb{W}^{\theta}\mathbb{W}^{\gamma})_{\xi,q+1,{\widetilde{n}}}(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\bigg{)}\,.$
(7.24)

We emphasize that to obtain (7.23), we have used that the Type 2 oscillation
errors from (7.14) and (7.16) will be shown to vanish. The equality (7.23)
completes the proof of (6.10) at level ${\widetilde{n}}$.

###  7.4. Type 1 oscillation errors

Recall from (7.23) that there are two main categories of Type 1 oscillation
errors which arise from the addition of $w_{q+1,{\widetilde{n}}}$: the higher
order stresses $H_{q,n}^{{\widetilde{n}}}$, which are defined and non-
vanishing in (7.21) in the parameter regimes ${\widetilde{n}}=0,1\leq
n\leq{n_{\rm max}}$ and $1\leq{\widetilde{n}}<{n_{\rm
max}},r({\widetilde{n}})<n\leq{n_{\rm max}}$, and the portions of
$\mathring{R}_{q+1}^{{\widetilde{n}}}$, which are defined in the last four
lines of (7.24). To estimate these error terms, we will first analyze a single
term of the form

$\displaystyle\left(\mathcal{H}+\mathcal{R}^{*}\right)\Bigg{(}\sum_{\xi,i,j,k,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{[q,n]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad\qquad+\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta}^{\alpha}\mathbb{P}_{[q,n]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma}^{\bullet}\Bigg{)}$
$\displaystyle\qquad=:\mathcal{O}_{n,{\widetilde{n}}}+\mathcal{O}_{n,{\widetilde{n}}}^{*}\,,$
(7.25)

where $\bullet$ refers to the unspecified components of a vector field, and
superscripts on $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}$ refer to components of
vectors over which summation is performed. In the above display, we allow
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and $r({\widetilde{n}})<n\leq{n_{\rm
max}}+1$, thus including both $H_{q,n}^{{\widetilde{n}}}$ from (7.21) and all
Type 1 error terms in (7.24).

######  Lemma 7.4.

The terms $\mathcal{O}_{n,{\widetilde{n}}}$ and
$\mathcal{O}_{n,{\widetilde{n}}}^{*}$ defined in (7.25) satisfy the following
estimates.

  1. (1)

For all error terms $\mathcal{O}_{n,{\widetilde{n}}}^{*}$, which are the
outputs of $\mathcal{R}^{*}$, we have for all $N,M\leq
3\mathsf{N}_{\textnormal{ind,v}}$ that

$\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}^{*}\right\|_{L^{\infty}}\leq{\delta_{q+2}}\lambda_{q+1}^{N-1}\tau_{q}^{-M}\,.$
(7.26)

  2. (2)

For $0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and $n={n_{\rm max}}+1$, the high
frequency, local part of the Type 1 errors satisfies

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{{n_{\rm
max}}+1,{\widetilde{n}}}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.27a) $\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{{n_{\rm
max}}+1,{\widetilde{n}}}\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.27b)

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

  3. (3)

For $0\leq{\widetilde{n}}<{n_{\rm max}}$ and $r({\widetilde{n}})<n\leq{n_{\rm
max}}$, the medium frequency, local part of the Type 1 errors satisfies

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}\right\|_{L^{1}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\delta_{q+1,n}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.28a)
$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon(n)}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.28b)

for all $N+M\leq\mathsf{N}_{\rm fin,n}$.

######  Remark 7.5.

In order to verify (6.13a) for $n^{\prime}={\widetilde{n}}$ and
$r({\widetilde{n}})<n\leq{n_{\rm max}}$, we first note that
$\mathcal{O}_{n,{\widetilde{n}}}=\mathring{H}_{q,n}^{{\widetilde{n}}}$, and
the inequality
$\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}\leq\Gamma_{q+1}^{i-\mathsf{c}_{\textnormal{n}}}\,,$
holds from ${\widetilde{n}}\leq n-1$ and (8.27). Then (7.28a) provides the
desired bound. (6.13b) follows similarly from (7.28b). The bound in (6.12a)
follows from (7.26) and (7.27a), since
$\mathsf{c}_{\widetilde{\textnormal{n}}}\geq 4$ from (8.5) and (8.27). The
bound in (6.12b) follows from (7.26) and (7.27b). Lastly, when
${\widetilde{n}}={n_{\rm max}}$, and hence $n={n_{\rm max}}+1$, (7.26),
(7.27a), and (7.27b) match (6.2a) and (6.2b).

###### Proof of Lemma 7.4.

We use (1) from Proposition 3.3 and the notation $A=(\nabla\Phi)^{-1}$ to
rewrite (7.25) as

$\displaystyle\qquad+\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}^{2}(A_{(i,k)})_{\theta}^{\alpha}\mathbb{P}_{[q,n]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(A_{(i,k)})_{\gamma}^{\bullet}\Bigg{)}$
$\displaystyle\left(\mathcal{H}+\mathcal{R}^{*}\right)\biggl{(}\sum_{\;\xi,i,j,k,\vec{l}}\mathbb{P}_{[q,n]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right)(\Phi_{(i,k)})\xi^{\theta}\xi^{\gamma}\left(A_{(i,k)}\right)_{\theta}^{\alpha}\left(\partial_{\alpha}a_{(\xi)}^{2}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}+a_{(\xi)}^{2}\partial_{\alpha}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}\right)\biggr{)}\,.$

Next, we must identify the functions and the values of the parameters which
will be used in the application of Proposition A.2. We first address the
bounds required in (A.4), (A.5), and (A.6), which we can treat simultaneously
for items (1), (2), and (3). Afterwards, we split the proof into two parts.
First, we set $n={n_{\rm max}}+1$ and prove (7.26), (7.27a), and (7.27b) for
any value of ${\widetilde{n}}$. Next, we consider
$0\leq{\widetilde{n}}<{n_{\rm max}}$ and $r({\widetilde{n}})<n\leq{n_{\rm
max}}$ and prove (7.26) in the remaining cases, as we simultaneously prove
(7.28a) and (7.28b).

Returning to (A.4), we will verify that this inequality holds with
$v=v_{\ell_{q}}$, $D_{t}=D_{t,q}=\partial_{t}+v_{\ell_{q}}\cdot\nabla$, and
$\displaystyle N_{*}=M_{*}=\lfloor\nicefrac{{N^{\sharp}}}{{2}}\rfloor$, where
$N^{\sharp}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. In order to verify the assumption
$N_{*}-{\mathsf{d}}\geq 2{\mathsf{N}_{\rm dec}}+4$, we use that
${\mathsf{N}_{\rm dec}}$ and ${\mathsf{d}}$ satisfy (8.58a). We fix values of
$(i,j,k,{\widetilde{n}},\xi,\vec{l})$ and set

$\displaystyle
G^{\bullet}=\xi^{\theta}\xi^{\gamma}\left(A_{(i,k)}\right)_{\theta}^{\alpha}\left(\partial_{\alpha}a_{(\xi)}^{2}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}+a_{(\xi)}^{2}\partial_{\alpha}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}\right)\,.$
(7.29)

Note crucially that the differential operator falling on
$a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}^{2}$ in the first term is precisely
$\xi^{\theta}\left(A_{(i,k)}\right)_{\theta}^{\alpha}\partial_{\alpha}$, which
from (3) and (7.8) will obey a good bound. We now establish (A.4)–(A.6) with
the parameter choices

$\displaystyle\mathcal{C}_{G,1}$
$\displaystyle=|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{2j+5}\,,\qquad\qquad\mathcal{C}_{G,\infty}=\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+8}\lambda_{q,{\widetilde{n}}}\,,$
(7.30)

$\lambda=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$.

To establish an $L^{1}$ bound for the first term from (7.29), we appeal to
Lemma 7.1, estimate (7.8) with $N^{\prime}=1$, and (5.19g) to deduce that

$\displaystyle\Biggl{\|}D^{N}D_{t,q}^{M}\left(\xi^{\theta}\left(A_{(i,k)}\right)_{\theta}^{\alpha}\partial_{\alpha}a_{(\xi)}^{2}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}\xi^{\gamma}\right)\Biggr{\|}_{L^{1}}$
$\displaystyle\qquad\lesssim|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{2j+5}\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.31)

holds for all
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$. It is precisely at this point
that we have used that the differential operator
$\xi^{\theta}(A_{(i,k)})_{\theta}^{\alpha}\partial_{\alpha}$ costs only
$\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$. For the $L^{\infty}$ bound on the
same term, we argue similarly except we apply estimate (7.9) to obtain

$\displaystyle\Biggl{\|}D^{N}D_{t,q}^{M}\left(\xi^{\theta}\left(A_{(i,k)}\right)_{\theta}^{\alpha}\partial_{\alpha}a_{(\xi)}^{2}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}\xi^{\gamma}\right)\Biggr{\|}_{L^{\infty}}$
$\displaystyle\qquad\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+8}\lambda_{q,{\widetilde{n}}}\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.32)

For the second term from (7.29), we can appeal to (5.19g) and use that
$\widetilde{\lambda}_{q}\leq\lambda_{q,{\widetilde{n}}}$ for all
${\widetilde{n}}$ to deduce that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$, we have

$\displaystyle\left\|D^{N}D_{t,q}^{M}\partial_{\alpha}\left(A_{(i,k)}\right)_{\gamma}^{\bullet}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q}\widetilde{\chi}_{i,k,q})}\lesssim\lambda_{q,{\widetilde{n}}}^{N+1}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c_{0}}+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right).$

Combining this with Lemma 7.1, estimate (7.8) in the case $p=1$ and (7.9) in
the case $p=\infty$ produces identical bounds as for the first term and in the
range
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$. Adding both estimates together
shows that (A.4) has been satisfied for both $p=1,\infty$.

We set the flow in Proposition A.2 as $\Phi=\Phi_{i,k}$, which by definition
satisfies $D_{t,q}\Phi_{i,k}=0$. Appealing to (5.19b) and (5.19e), we have
that (A.5) is satisfied. From (2.19) at level $q$, which follows from
Proposition 5.6, the choice of $\nu$ from earlier, and (8.30), we have that
$Dv=Dv_{\ell_{q}}$ satisfies the bound (A.6).

Proof of items (1) and (2) for $0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and
$n={n_{\rm max}}+1$.  We first assume that ${\widetilde{n}}<{n_{\rm max}}$.
With the goal of verifying (i)–(iii) of Proposition A.2, we choose
$\zeta,\mu,\Lambda,\rho$ and $\varrho$ as

$\displaystyle\zeta=\lambda_{q,{n_{\rm max}}}\,,\qquad\mu$
$\displaystyle=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\,,\qquad\Lambda=\lambda_{q+1}\,,$
$\displaystyle\varrho=\mathbb{P}_{\left[q,{n_{\rm
max}}+1\right]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right)\,,$
$\displaystyle\qquad\vartheta=\lambda_{q,{n_{\rm
max}}}^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\mathbb{P}_{\left[q,{n_{\rm
max}}+1\right]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\,,$
(7.33)

where we recall that $\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}$ is
defined in Propositions 3.2 and 3.3. We then have by definition that (i) from
Proposition A.2 is satisfied. By property (1) of Proposition 3.2, we have that
the functions $\varrho$ and $\vartheta$ defined in (7.33) are both periodic to
scale $\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}$, and so (ii) is
satisfied. In the case $p=1$, the estimates in (A.7) follow with
$\mathcal{C}_{*,1}=1$ from standard Littlewood-Paley arguments (see also the
discussion in part (b) of [5, Remark (A.21)]) and item (5) from Proposition
3.3. In the case $p=\infty$, the estimates follow from Lemma 3.5, (3.11b) with
the choices $\mathcal{C}_{*,\infty}=r_{q+1,{\widetilde{n}}}^{-2}$,
$\lambda_{1}=\lambda_{q,{n_{\rm
max}}},\lambda_{2}=\infty,\lambda=\lambda_{q+1},r=r_{q+1,{\widetilde{n}}}$. We
recall from (8.36) the choice of $\alpha=\varepsilon_{\Gamma}\frac{b-1}{b}$,
so that the loss $\lambda_{q+1}^{\alpha}$ gives exactly a loss of
$\Gamma_{q+1}$. From (8.20), (8.24), and the temporary assumption that
${\widetilde{n}}<{n_{\rm max}}$, we have that

$\widetilde{\lambda}_{q}\ll\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}\ll\lambda_{q+1}r_{q+1,{\widetilde{n}}}\leq\lambda_{q,{n_{\rm
max}}}\leq\lambda_{q+1},$

and so (A.8) is satisfied. From (8.37) we have that

$\lambda_{q+1}^{4}\leq\left(\frac{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}{2\pi\sqrt{3}\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}}\right)^{\mathsf{N}_{\rm
dec}}=\left(\frac{\Gamma_{q+1}}{2\pi\sqrt{3}}\right)^{\mathsf{N}_{\rm
dec}}\,,$

and so (A.9) is satisfied. Applying the estimate (A.11) for $p=1$ with
$\alpha$ as in (8.36), recalling the value for $\mathcal{C}_{G,1}$ in (7.30),
summing over $i$ and using (2.11) at level $q$, summing over $j,k,\xi$,
summing over $\vec{l}$ and using (5.43) with $r_{1}=\infty$ and $r_{2}=2$, and
appealing to (8.38) and (8.40), we obtain that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\lambda_{q+1}^{N}\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{8}\lambda_{q,{n_{\rm
max}}}^{-1}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.34)

Applying the same steps but in the case $p=\infty$ and using the parameter
inequality (8.45) yields the bound

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+9}\lambda_{q,{\widetilde{n}}}r_{q+1,{\widetilde{n}}}^{-2}\lambda_{q,{n_{\rm
max}}}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.35)

in the same range of $N,M$. The proof is complete after using (8.58c), which
gives that the range of derivatives allowed in (7.34) and (7.35) is as much as
is needed in (7.27a).

Following the parameter choices in [5, Remark A.19], we set
$N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$, and
$N^{\sharp}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. From (8.58d), we have that the condition
$N_{\circ}\leq\nicefrac{{N^{\sharp}}}{{4}}$ is satisfied. The inequalities
(A.13) and (A.14) follow from the discussion in [5, Remark A.19]. The
inequality in (A.15) follows from the choices
$\lambda=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$,
$\zeta=\lambda_{q,{n_{\rm
max}}}\geq\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$, (8.32), and
(8.50). Having satisfied these assumptions, we may now appeal to estimate
(A.17) for $p=\infty$ and sum over all parameters $(i,j,k,\xi,\vec{l})$. Since
$\vec{l}$ takes at most $\lambda_{q+1}^{3}$ values, $i$, and $j$ are bounded
independently of $q$, and $k$ corresponds to a partition of unity in time, we
obtain (7.26) for the case ${\widetilde{n}}<{n_{\rm max}}$ and $n={n_{\rm
max}}+1$.

Recall that we began this case with the temporary assumption that
${\widetilde{n}}<{n_{\rm max}}$. In the case ${\widetilde{n}}={n_{\rm max}}$,
we have from (8.24) that $\lambda_{q+1}r_{q+1,{n_{\rm
max}}}>\lambda_{q,{n_{\rm max}}}$. Then we can set
$\zeta=\mu=\lambda_{q+1}r_{q+1,{n_{\rm max}}}$ and substitute
$\mathbb{P}_{\geq\lambda_{q+1}r_{q+1,{\widetilde{n}}}}$ for
$\mathbb{P}_{[q,{n_{\rm max}}]}$. The only change is that (7.34) and (7.35)
become stronger, since $\lambda_{q,{n_{\rm max}}}<\lambda_{q+1}r_{q+1,{n_{\rm
max}}}$, and so the desired estimates follow by arguing as before. We omit
further details.

Proof of item (3) and of item (1) when $0\leq{\widetilde{n}}<{n_{\rm max}}$
and $r({\widetilde{n}})<n\leq{n_{\rm max}}$.  We set

$\displaystyle\zeta$
$\displaystyle=\begin{dcases}\max\left\\{\lambda_{q+1}r_{q+1,{\widetilde{n}}},\lambda_{q,n-1}\right\\}&\mbox{if}\quad
2\leq n\leq{n_{\rm max}}\\\
\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}&\mbox{if}\quad
n=1\,,\end{dcases}\qquad\mu=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\,,\qquad\Lambda=\lambda_{q,n}\,,$
(7.36)

and

$\displaystyle\varrho$
$\displaystyle=\mathbb{P}_{[q,n]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right)\,,\qquad\vartheta=\zeta^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\mathbb{P}_{[q,n]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\,.$

We then have by definition that (i) from Proposition A.2 is satisfied. By
property (1) of Proposition 3.2, $\varrho$ and $\vartheta$ are both periodic
to scale $\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}$, and so (ii)
is satisfied. The estimates in (A.7) follow with $\mathcal{C}_{*,1}=1$ in the
case $p=1$ as before. In the case $p=\infty$, we appeal to Lemma 3.5, (3.11b)
with $\lambda_{1}=\zeta$, $\lambda_{2}=\lambda_{q,n}=\Lambda<\lambda_{q+1}$,
and $r=r_{q+1,{\widetilde{n}}}$ to deduce that (A.7) holds with
$\mathcal{C}_{*,\infty}=\left(\frac{\lambda_{q,n}}{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}\right)^{2}$.
We again set $\alpha$ as in (8.36). From (8.24) and the condition that
$r({\widetilde{n}})<n$, we have that if $n\neq 1$, then

$\widetilde{\lambda}_{q}\leq\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}\ll\lambda_{q+1}r_{q+1,{\widetilde{n}}}\leq\max\left\\{\lambda_{q+1}r_{q+1,{\widetilde{n}}},\lambda_{q,n-1}\right\\}\leq\lambda_{q,n}\,,$

and so (A.8) is satisfied if $n\neq 1$. If $n=1$, then it must be the case
that ${\widetilde{n}}=0$, and so

$\widetilde{\lambda}_{q}\leq\lambda_{q+1}r_{q+1,0}\Gamma_{q+1}^{-1}\ll\lambda_{q+1}r_{q+1,0}\leq\lambda_{q,1}\,.$

From (8.37), the inequality $\lambda_{q,n}\leq\lambda_{q+1}$, and the choices
of $\mu$ and $\lambda$, we have that (A.9) is satisfied.

We now use the definition of $\mathcal{C}_{G,p}$ in (7.30) and apply the
estimate (A.11). In the case that $p=1$ and $n=1$, then we must have
${\widetilde{n}}=0$, and so for all
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$, we sum over
$(i,j,k,\xi,\vec{l})$ as before and obtain that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{0,1}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\Gamma_{q+1}^{9}\bigl{(}\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}\bigr{)}^{-1}\lambda_{q,1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{0}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\delta_{q+1,1}\lambda_{q,1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{0}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.37)

The inequality in the last line follows immediately from the definitions in
(8.26). Alternatively, if $n>1$, then $\zeta^{-1}\leq\lambda_{q,n-1}^{-1}$
from (7.36), and so if
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{{\widetilde{n}},n}\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{8}\lambda_{q,n-1}^{-1}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\delta_{q+1,n}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.38)

In the last inequality, we have used (8.42). After using (8.59), which gives
$\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}\geq\mathsf{N}_{\rm
fin,n}$ for all ${\widetilde{n}}<n$, we have achieved (7.28a).

In the case that $p=\infty$ and $n=1$, then we must have that
${\widetilde{n}}=0$, and so

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{0,1}\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q}^{\mathsf{C_{u}}}\Gamma_{q+1}^{9}\lambda_{q,0}\left(\frac{\lambda_{q,1}}{\lambda_{q+1}r_{q+1,0}}\right)^{2}\left(\lambda_{q+1}\lambda_{q}\Gamma_{q+1}^{2}\right)^{-\nicefrac{{1}}{{2}}}$
$\displaystyle\qquad\qquad\times\lambda_{q,1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{0}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\Gamma_{q+1}^{9}\Gamma_{q}^{{\mathsf{C_{u}}}}\lambda_{q,1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{0}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.39)

To achieve the last line, we have appealed to (8.24) and the inequality
$\frac{\lambda_{q,1}^{2}}{\lambda_{q+1}^{\nicefrac{{3}}{{2}}}\lambda_{q}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}}<1$,
which is immediate from a large choice of ${n_{\rm max}}$. In the case that
$p=\infty$ and $n\geq 2$, we have that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\mathcal{O}_{n,{\widetilde{n}}}\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+9}\lambda_{q,{\widetilde{n}}}\left(\frac{\lambda_{q,n}}{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}\right)^{2}\lambda_{q,n-1}^{-1}$
$\displaystyle\qquad\qquad\times\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+13}\frac{\lambda_{q,n}^{2}}{\lambda_{q+1}\lambda_{q,n-1}}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon(n)}\lambda_{q,n}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.40)

To achieve the second inequality, we have used (8.24). To achieve the third
inequality, we have used (8.39) and (6.7). The estimates above are again valid
in the range
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$, which from (8.59)
completes the proof of (7.28b).

Following again the parameter choices in [5, Remark (A.19)], we set
$N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$, and
$N^{\sharp}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. From (8.58d), we have that the condition
$N_{\circ}\leq\nicefrac{{N^{\sharp}}}{{4}}$ is satisfied. The inequalities
(A.13) and (A.14) follow from the discussion in [5, Remark (A.19)]. The
inequality in (A.15) follows from the choices
$\lambda=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$,
$\zeta\geq\lambda_{q+1}r_{q+1,{\widetilde{n}}}$, (8.32), and (8.50). We then
achieve the concluded estimate in (A.17), which after summing as before gives
(7.26) in the remaining cases $0\leq{\widetilde{n}}<{n_{\rm max}}$,
$r({\widetilde{n}})<n\leq{n_{\rm max}}$. ∎

###  7.5. Type 2 oscillation errors

In order to show that the Type 2 errors identified in (7.14) and (7.15)
vanish, we will apply Proposition 3.8 on the support of a specific cutoff
function

$\eta=\eta_{i,j,k,q,n,\xi,\vec{l}}=\psi_{i,q}\chi_{i,k,q}\omega_{i,j,q,n}\zeta_{q,i,k,n,\xi,\vec{l}}\,$

in order to place pipes parallel to $\xi$ on $\mathrm{supp\,}\eta$. We first
collect several preliminary estimates in the first subsubsection, mainly with
the goal of verifying assumption (3) from Proposition 3.8, before applying
Proposition 3.8 in the second.

####  7.5.1. Preliminary estimates

######  Lemma 7.6 (Keeping Track of Overlap).

For every tuple $(i,j,k,n)$, define the index set $\mathcal{I}$ as

$\mathcal{I}=\mathcal{I}(i,j,k,n)=\bigl{\\{}({i^{*}},{j^{*}},q,{n^{*}})\colon{n^{*}}\leq
n,\;\psi_{i,q}\omega_{i,j,q,n}\chi_{i,k,q}\psi_{{i^{*}},q}\omega_{{i^{*}},{j^{*}},q,{n^{*}}}\chi_{{i^{*}},{k^{*}},q}\not\equiv
0\bigr{\\}}\,.$

Then, the cardinality of $\mathcal{I}$ is bounded above by
$\mathcal{C}_{\eta}\Gamma_{q+1}$, where $\mathcal{C}_{\eta}$ depends only on
${n_{\rm max}}$, ${j_{\rm max}}$, and dimensional constants. In particular,
$C_{\eta}$ is independent of $q$.

###### Proof of Lemma 7.6.

The proof proceeds similarly to the proof of [5, Lemma 8.6]. In fact it is
somewhat simpler, since the parameter $p$ (see [5, equation (9.3)]) is no
longer part of the scheme, and we are not considering the checkerboard cutoffs
$\zeta_{q,i,k,n,\xi,\vec{l}}$ yet, but will only incorporate them later. We
thus give only an idea of the proof. Once $i$ is fixed, we first note that
$\psi_{i,q}$ may only overlap with $\psi_{i+1,q}$ and $\psi_{i-1,q}$ from
(2.11) at level $q$. The factor of $\Gamma_{q+1}$ in the upper bound for the
cardinality of $\mathcal{I}$ comes from the fact that the timescale of the
$\chi_{i+1,{k^{*}},q}$’s on the support of $\psi_{i+1,q}$ is faster by a
factor of $\Gamma_{q+1}$ than the timescale of the $\chi_{i,k,q}$’s on the
support of $\psi_{i,q}$. Considering then values of $j$ and $n$ introduces a
dependence on ${j_{\rm max}}$ and ${n_{\rm max}}$ which is nevertheless
independent of $q$. ∎

######  Lemma 7.7.

Let $(x,t),(y,t)\in\mathrm{supp\,}\psi_{i,q}$ be such that
$\psi_{i,q}^{2}(x,t)\geq\nicefrac{{1}}{{4}}$ and
$\psi_{i,q}^{2}(y,t)\leq\nicefrac{{1}}{{8}}$. Then there exists a geometric
constant $\mathcal{C}_{\ast}>1$ such that

$|x-y|\geq\mathcal{C}_{*}\left(\Gamma_{q}\lambda_{q}\right)^{-1}.$ (7.41)

For the proof of Lemma 7.7, we refer to [5, Lemma 8.7].

######  Lemma 7.8.

Consider cutoff functions

$\displaystyle\eta$
$\displaystyle:=\eta_{i,j,k,q,n,\xi,\vec{l}}=\psi_{i,q}\chi_{i,k,q}\omega_{i,j,q,n}\zeta_{q,i,k,n,\xi,\vec{l}},$
$\displaystyle\eta^{*}$
$\displaystyle:=\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{\xi^{*}},\vec{l}^{*}}=\psi_{{i^{*}},q}\chi_{{i^{*}},{k^{*}},q}\omega_{{i^{*}},{j^{*}},q,{n^{*}}}\zeta_{q,{i^{*}},{k^{*}},{n^{*}},{\xi^{*}},\vec{l}^{*}},$

where $({i^{*}},{j^{*}},{k^{*}},{n^{*}})\in\mathcal{I}(i,j,k,n)$, as defined
in Lemma 7.6. Let $t^{*}\in\mathrm{supp\,}\chi_{{i^{*}},{k^{*}},q}$ be given.
Assume furthermore that $\eta\eta^{*}\not\equiv 0$, which implies that
$\zeta_{q,i,k,n,\xi,\vec{l}}\,\zeta_{q,{i^{*}},{k^{*}},{n^{*}},{\xi^{*}},\vec{l}^{*}}\not\equiv
0$. Then there exists a convex set
$\Omega:=\Omega(\eta,\eta^{*},t^{*})\subset\mathbb{T}^{3}$ with diameter
$\lambda_{q,n}^{-1}\Gamma_{q+1}$ such that

$\bigl{(}\mathrm{supp\,}\zeta_{q,i,k,n,\xi,\vec{l}}\cap\\{t=t^{*}\\}\bigr{)}\subset\Omega\subset\mathrm{supp\,}\psi_{i\pm,q}\,.$

###### Proof of Lemma 7.8.

Let $(x,t_{0})\in\mathrm{supp\,}\left(\eta\eta^{*}\right)$. Then there exists
$i^{\prime}\in\\{i-1,i,i+1\\}$ such that
$\psi_{i^{\prime},q}^{2}(x,t_{0})\geq\frac{1}{2}$. Consider the flow $X(x,t)$
originating from $(x,t_{0})$. Then for any $t$ such that
$|t-t_{0}|\leq\tau_{q}\Gamma_{q+1}^{-i+5+\mathsf{c_{0}}}$, we can apply Lemma
5.8 to deduce that $\psi_{i^{\prime},q}^{2}(t,X(x,t))\geq\frac{1}{4}$. By the
definition of $\chi_{{i^{*}},{k^{*}},q}$, the fact that
${i^{*}}\in\\{i-1,i,i+1\\}$, the existence of
$(x,t_{0})\in\mathrm{supp\,}(\chi_{i,k,q}\chi_{{i^{*}},{k^{*}},q})$, and the
fact that $t^{*}\in\mathrm{supp\,}\chi_{{i^{*}},{k^{*}},q}$, we in particular
deduce that $\psi_{i^{\prime},q}^{2}(t^{*},X(x,t^{*}))\geq\frac{1}{4}$. Now,
let $y$ be such that

$|X(x,t^{*})-y|\leq\lambda_{q,n}^{-1}\Gamma_{q+1}\leq\widetilde{\lambda}_{q}^{-1}<\mathcal{C}_{*}(\Gamma_{q}\lambda_{q})^{-1}$

for $\mathcal{C}_{*}$ given in (7.41), where we have used the definition of
$\lambda_{q,n}$ in (8.23). Then from Lemma 7.7, it cannot be the case that
$\psi_{i^{\prime},q}^{2}(y,t^{*})\leq\frac{1}{8}$, and so

$\displaystyle
y\in\mathrm{supp\,}\psi_{i^{\prime},q}\cap\\{t=t^{*}\\}\subset\mathrm{supp\,}\psi_{i\pm,q}\cap\\{t=t^{*}\\}\,.$
(7.42)

Since $y$ is arbitrary, we conclude that the ball of radius
$\Gamma_{q+1}\lambda_{q,n}^{-1}$ is contained in
$\mathrm{supp\,}\psi_{i\pm,q}\cap\\{t=t^{*}\\}$. We let
$\Omega(\eta,\eta^{*},t^{*})$ to be precisely this ball. Since
$D_{t,q}\zeta_{q,i,k,n,\xi,\vec{l}}=0$ and
$(x,t_{0})\in\mathrm{supp\,}\zeta_{q,i,k,n,\xi,\vec{l}}$, we have that
$X(x,t^{*})\in\mathrm{supp\,}\zeta_{q,i,k,n,\xi,\vec{l}}\cap\\{t=t^{*}\\}$.
Then, recalling that the support of $\zeta_{q,i,k,n,\xi,\vec{l}}$ must obey
the diameter bound in (5.41) on the support of $\widetilde{\chi}_{i,k,q}$,
which contains the support of $\chi_{{i^{*}},{k^{*}},q}$ by (5.18), we
conclude that

$\mathrm{supp\,}\zeta_{q,i,k,n,\xi,\vec{l}}\cap\\{t=t^{*}\\}\subset\Omega\,.$
(7.43)

Combining (7.42) and (7.43) concludes the proof of the lemma. ∎

######  Lemma 7.9.

As in Lemma 7.8, consider cutoff functions $\eta$ and $\eta^{*}$ satisfying
the conditions from Lemma 7.6 and the assumption $\eta\eta^{*}\not\equiv 0$.
Let $t^{*}\in\mathrm{supp\,}\chi_{{i^{*}},{k^{*}},q}$ be such that
$\Phi^{*}:=\Phi_{({i^{*}},{k^{*}})}$ is the identity at time $t^{*}$. Using
Lemma 7.8, define $\Omega:=\Omega(\eta,\eta^{*},t^{*})$. Define
$\Omega(t):=\Omega(\eta,\eta^{*},t^{*},t):=X^{*}(\Omega,t)$, where $X^{*}$ is
the inverse of $\Phi^{*}$. Then the following conclusions hold.

  1. (1)

For $t\in\mathrm{supp\,}\chi_{i,k,q}$,

$\mathrm{supp\,}\eta(\cdot,t)\subset\Omega(t)\subset\mathrm{supp\,}\psi_{i\pm,q}\,.$

  2. (2)

Let
$\mathbb{W}^{*}\circ\Phi^{*}:=\mathbb{W}_{\xi*,q+1,n^{*}}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},\vec{l}^{*}}\circ\Phi_{({i^{*}},{k^{*}})}$
be the intermittent pipe flow supported on $\eta^{*}$. Then
$\mathbb{W}^{*}\circ\Phi^{*}$ satisfies the conclusion of Lemma 3.7 on the set
$\Omega(t)$ for $t\in\mathrm{supp\,}\chi_{i,k,q}$.

  3. (3)

For $\mathcal{I}=\mathcal{I}(i,j,k,n)$ defined as in Lemma 7.6, we denote

$\mathsf{P}:=\bigcup_{\mathcal{I}}\left(\mathrm{supp\,}\left(\psi_{{i^{*}},q}\omega_{{i^{*}},{j^{*}},q,{n^{*}}}\right)\bigcap\left(\bigcup_{\vec{l}^{*},{\xi^{*}}}\mathrm{supp\,}\left(\zeta_{q,{i^{*}},{k^{*}},{n^{*}},{\xi^{*}},\vec{l}^{*}}\mathbb{W}_{{\xi^{*}},q+1,{n^{*}}}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},\vec{l}}\circ\Phi_{({i^{*}},{k^{*}})}\right)\right)\right)\,,$
(7.44)

which is precisely the union of the supports of all pipes living on cutoff
functions indexed by tuples belonging to $\mathcal{I}$, which are however
_not_ restricted to the support of their corresponding time cutoffs
$\chi_{{i^{*}},{k^{*}},q}$. Then there exists $\mathcal{C}_{P}$ such that for
any convex set $\Omega^{\prime}\subset\mathbb{T}^{3}$ with
$\textnormal{diam}(\Omega^{\prime})\leq(\lambda_{q+1}r_{q+1,n})^{-1}$ and any
$t\in\mathrm{supp\,}\chi_{i,k,q}$, the set
$\mathsf{P}\cap\left(\\{t\\}\times\Omega^{\prime}\right)$ consists of at most
$\mathcal{C}_{P}\Gamma_{q+1}$ segments of deformed pipes of length
$(\lambda_{q+1}r_{q+1,n})^{-1}$.

######  Remark 7.10.

The third item simply asserts that at stage $n$, there exists a geometric
constant $\mathcal{C}_{P}$ such that in any
$(\nicefrac{{\mathbb{T}}}{{\lambda_{q+1}r_{q+1,n}}})^{3}$-periodic cell of
diameter approximately $(\lambda_{q+1}r_{q+1,n})^{-1}$, there exist at most
$\mathcal{C}_{P}\Gamma_{q+1}$ segments of deformed pipes of length
$(\lambda_{q+1}r_{q+1,n})^{-1}$. This will later allow us to apply Proposition
3.8. The factor of $\Gamma_{q+1}$ comes from the fact that overlapping time
cutoffs $\chi_{i,k,q}$ and $\chi_{i+1,k^{\prime},q}$ have timescales which
differ by a factor of $\Gamma_{q+1}$, and that we have _not_ restricted
$\mathbb{W}_{{\xi^{*}},q+1,{n^{*}}}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},\vec{l}^{*}}$
to the support of its corresponding time cutoff $\chi_{{i^{*}},{k^{*}},q}$.
Notice also that since choosing a shift moves a segment of pipe inside a
$(\nicefrac{{\mathbb{T}}}{{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}})^{3}$-periodic
cell but does not increase the _number_ of such segments, the conclusion in
(3) is independent of the choice of placement. We may thus appeal to it in the
next subsection in order to choose a placement.

###### Proof of Lemma 7.9.

The statement and proof are quite similar to the proof of [5, Lemma 8.9], and
we refer there for the proof of the first two claims. The only difference is
contained in the third claim above, since we have rephrased the way in which
we count the number of deformed segments of pipe comprising
$\mathbb{W}^{*}\circ\Phi^{*}$ which may overlap with $\mathrm{supp\,}\eta$. We
remind the reader that a single “segment of deformed pipe” consists of the
support of $\mathbb{W}^{*}\circ\Phi^{*}$ restricted to a single (deformed)
$(\nicefrac{{\mathbb{T}^{3}}}{{\lambda_{q+1}r_{q+1,n}}})^{-1}$-periodic cell.
Then to prove the third claim, we first fix a tuple
$({i^{*}},{j^{*}},{k^{*}},{n^{*}})\in\mathcal{I}$ and note that in any convex
set $\Omega^{\prime}$ of diameter at most $(\lambda_{q+1}r_{q+1,n})^{-1}$, the
conclusions of Lemma 3.7 and the construction of the checkerboard cutoff
functions implies that there exist at most finitely many
$\zeta_{q,{i^{*}},{k^{*}},{n^{*}},{\xi^{*}},\vec{l}^{*}}$ such that

$\psi_{{i^{*}},q}\omega_{{i^{*}},{j^{*}},q,{n^{*}}}\chi_{{i^{*}},{k^{*}},q}\zeta_{q,{i^{*}},{k^{*}},{n^{*}},{\xi^{*}},\vec{l}^{*}}\not\equiv
0\,.$

From the construction of $\mathbb{W}^{*}\circ\Phi^{*}$ in Proposition 3.3 and
the fact that $\mathbb{W}^{*}\circ\Phi^{*}$ satisfies the conclusions of Lemma
3.7 on $\mathrm{supp\,}\chi_{i,k,q}$, we then have that taking the union over
just $\vec{l}^{*}$ and ${\xi^{*}}$ in (7.44) allows for the desired conclusion
with a $q$-independent constant. Then applying Lemma (7.7) and taking the
union over the $C_{\eta}\Gamma_{q+1}$ many tuples in $\mathcal{I}$ then
provides the conclusion with a new constant $\mathcal{C}_{P}$ multiplied by
$\Gamma_{q+1}$. ∎

####  7.5.2. Applying Proposition 3.8

######  Lemma 7.11.

The Type 2 oscillation errors identified in (7.14) and (7.15) vanish.

###### Proof of Lemma 7.11.

To show that the errors defined in (7.14) and (7.15) vanish, it suffices to
show the following: for any pairs of cutoff functions
$\eta=\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}$ and
$\eta^{*}=\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{\xi^{*}},\vec{l}^{*}}$
where $({i^{*}},{j^{*}},{n^{*}},{k^{*}})\in\mathcal{I}(i,j,k,n)$, we have that

$\displaystyle\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}\;\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{\xi^{*}},\vec{l}^{*}}\;\Bigl{(}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{i,j,k,{\widetilde{n}},\vec{l}}\circ\Phi_{(i,k)}\otimes\mathbb{W}_{{\xi^{*}},q+1,{n^{*}}}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},\vec{l}^{*}}\circ\Phi_{({i^{*}},{k^{*}})}\Bigr{)}\equiv
0\,.$ (7.45)

The proof of this claim will proceed by fixing ${\widetilde{n}}$, using the
preliminary estimates, and applying Proposition 3.8.

Now, consider all cutoff functions
$\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}$ utilized at stage
${\widetilde{n}}$. We may choose an ordering of the tuples
$(i,j,k,\xi,\vec{l})$ at level ${\widetilde{n}}$, which automatically provides
orderings for the cutoff functions
$\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}}$ and associated pipe flows
$\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{i,j,k,{\widetilde{n}},\vec{l}}\circ\Phi_{(i,k)}$.
To lighten the notation, we will abbreviate the newly ordered cutoff functions
as $\eta_{z}$ and the associated intermittent pipe flows as
$(\mathbb{W}\circ\Phi)_{z}$, where $z\in\mathbb{N}$ corresponds to the
ordering. We will apply Proposition 3.8 inductively on $z\in\mathbb{N}$,
according to the chosen ordering, so that (7.45) holds.

Fix $\eta_{z}$, and fix the associated index set
$\mathcal{I}(z)=\mathcal{I}(i,j,k,{\widetilde{n}})$. Since we are proving
(7.45) iteratively, we only need to consider the elements
$z^{\prime}\in\mathcal{I}(z)$ such that ${n^{*}}<{\widetilde{n}}$, and
$\widehat{z}\in\mathcal{I}(z)$ such that ${n^{*}}={\widetilde{n}}$ and
$\widehat{z}<z$, according to the aforementioned ordering.

We will apply Proposition 3.8 with the following choices. First, we recall
that at the time $t_{z}$ at which $\Phi_{z}$ is the identity, the cutoff
function $\eta_{z}$ contains a checkerboard cutoff function $\zeta_{z}$ which
from (5.35) is adapted to a rectangular prism of dimensions
$2\pi\lambda_{q,{\widetilde{n}}}^{-1}$ in the direction of $\xi_{z}$, and
$\mathcal{C}_{\Gamma}\Gamma_{q+1}(\lambda_{q+1}r_{q+1,{\widetilde{n}}})^{-1}$
in the directions perpendicular to $\xi_{z}$. Thus we can bound the dimensions
of the support of the anistropic checkerboard cutoff by
$4\pi\lambda_{q,{\widetilde{n}}}^{-1}$ and
$2\mathcal{C}_{\Gamma}\Gamma_{q+1}(\lambda_{q+1}r_{q+1,{\widetilde{n}}})^{-1}$,
and we thus set

$\Omega=\mathrm{supp\,}\zeta_{z}\cap\\{t=t_{z}\\}\,,\qquad
r_{1}=\frac{\lambda_{q,{\widetilde{n}}}}{4\pi\lambda_{q+1}}\,,\qquad
r_{2}=r_{q+1,{\widetilde{n}}}\,,\qquad\mathcal{C}_{\Omega}=2\mathcal{C}_{\Gamma}.$

Recalling item 3 from Lemma 7.9, we choose the support of
$(\mathbb{W}\circ\Phi)_{z}|_{t=t_{z}}$ to have empty intersection with

$\mathsf{P}\cap\Omega\,,\qquad\mathsf{P}\textnormal{ as defined in
}\eqref{eq:counting:pipelets}\,,$ (7.46)

and so by definition $\mathsf{P}$ satisfies item 3 from Proposition 3.8. Thus
it remains to check (3.21). From the definition of $r_{q+1,{\widetilde{n}}}$
in (8.24), we have that

$\displaystyle
C_{*}\mathcal{C}_{\Omega}^{2}\mathcal{C}_{P}\Gamma_{q+1}^{3}r_{2}^{2}=C_{*}\mathcal{C}_{\Omega}^{2}\mathcal{C}_{P}\Gamma_{q+1}^{3}r_{q+1,{\widetilde{n}}}^{2}\lesssim
C_{*}\mathcal{C}_{\Omega}^{2}\mathcal{C}_{P}\Gamma_{q+1}^{3}\frac{\lambda_{q,{\widetilde{n}}}}{\lambda_{q+1}}\Gamma_{q+1}^{-4}<r_{1}$
(7.47)

if $a$ is chosen sufficiently large so that $\Gamma_{q+1}^{-1}$ can absorb the
constants $C_{*}$, $\mathcal{C}_{\Omega}^{2}$, $\mathcal{C}_{P}$ and the
implicit constant, all of which are bounded independently of $q$. Therefore
(3.21) is satisfied, and we may apply Proposition 3.8 to choose a placement
for $\mathbb{W}_{z}$ which has empty intersection with $\mathsf{P}$ at time
$t=t_{z}$. This shows that at time $t=t_{z}$, $\mathbb{W}_{z}$ has empty
intersection with _all_ previously existing pipes which may be non-zero at
_any_ time $t\in\mathrm{supp\,}\chi_{i,k,q}$ but have been flowed to time
$t=t_{z}$. Finally, since
$D_{t,q}(\mathbb{W}\circ\Phi)_{z}=D_{t,q}(\mathbb{W}\circ\Phi)_{z^{\prime}}=D_{t,q}(\mathbb{W}\circ\Phi)_{\widehat{z}}$,
and $\mathsf{P}$ has been constructed to contain _all_ pipes which are non-
zero at _any_ time $t\in\mathrm{supp\,}\chi_{i,k,q}$, (7.45) is satisfied for
all $t\in\mathrm{supp\,}\chi_{z}$, concluding the proof. ∎

###  7.6. Divergence corrector errors

In this subsection we estimate the stresses arising from the incompresibility
correctors, namely

$\mathrm{div\,}\bigl{(}w_{q+1,{\widetilde{n}}}^{(p)}\otimes
w_{q+1,{\widetilde{n}}}^{(c)}+w_{q+1,{\widetilde{n}}}^{(c)}\otimes
w_{q+1,{\widetilde{n}}}^{(p)}+w_{q+1,{\widetilde{n}}}^{(c)}\otimes
w_{q+1,{\widetilde{n}}}^{(c)}\bigr{)}\,.$

######  Lemma 7.12.

For all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$, the divergence corrector
errors
$(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1,2}$
and $\mathcal{O}_{{\widetilde{n}},1,3}$ identified in (7.17) and the first
line of (7.24), satisfy the bounds

$\displaystyle\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\bigl{(}\mathcal{O}_{{\widetilde{n}},1,3}+(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1,2}\bigr{)}\right\|_{L^{1}}\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.48a)
$\displaystyle\left\|D^{k}D_{t,q}^{m}\bigl{(}\mathcal{O}_{{\widetilde{n}},1,3}+(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1,2}\bigr{)}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}\tau_{q}^{-1},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.48b)

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

###### Proof of Lemma 7.12.

We first present the estimates for the stress
$\mathcal{O}_{{\widetilde{n}},1,3}=w_{q+1,{\widetilde{n}}}^{(c)}\otimes
w_{q+1,{\widetilde{n}}}^{(c)}$, which is also given explicitly by the last
line in (7.17). By the Leibniz rule, the estimate (7.12a) with
$(r,r_{1},r_{2})=(2,\infty,1)$, and the fact that
$\mathrm{supp\,}\psi_{i,q}\cap\mathrm{supp\,}\eta_{(i^{\prime},j^{\prime},k^{\prime})}\neq\emptyset$
if and only if $|i^{\prime}-i|\leq 1$, it follows that

$\displaystyle\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\mathcal{O}_{{\widetilde{n}},1,3}\right\|_{L^{1}}\lesssim
r_{q+1,{\widetilde{n}}}^{2}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{6}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

The bound (7.48a) for $\mathcal{O}_{{\widetilde{n}},1,3}$ now follows from the
parameter inequality (8.52). Similarly, from (7.12b) it follows that

$\displaystyle\left\|D^{k}D_{t,q}^{m}\mathcal{O}_{{\widetilde{n}},1,3}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+7}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

The bound (7.48b) for $\mathcal{O}_{{\widetilde{n}},1,3}$ then follows from
the inequality (8.54).

It thus remains to estimate
$(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}\mathcal{O}_{{\widetilde{n}},1,2}=(\mathcal{H}+\mathcal{R}^{*})\mathrm{div\,}(w_{q+1,{\widetilde{n}}}^{(p)}\otimes
w_{q+1,{\widetilde{n}}}^{(c)}+w_{q+1,{\widetilde{n}}}^{(c)}\otimes
w_{q+1,{\widetilde{n}}}^{(p)})$. Using the second line of (7.17), we have

$\displaystyle\mathrm{div\,}\bigl{(}w_{q+1,{\widetilde{n}}}^{(p)}\otimes
w_{q+1,{\widetilde{n}}}^{(c)}+w_{q+1,{\widetilde{n}}}^{(c)}\otimes
w_{q+1,{\widetilde{n}}}^{(p)}\bigr{)}^{\bullet}$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\partial_{m}\bigl{(}a_{(\xi)}\varrho_{(\xi)}\circ\Phi_{(i,k)}\xi^{\ell}\bigl{(}A_{\ell}^{m}\epsilon_{\bullet
pr}+A_{\ell}^{\bullet}\epsilon_{mpr}\bigr{)}\partial_{p}a_{(\xi)}\partial_{r}\Phi_{(i,k)}^{s}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\circ\Phi_{(i,k)}\bigr{)}$
(7.49)

where $\epsilon_{i_{1}i_{2}i_{3}}$ is the Levi-Civita alternating tensor, we
implicitly contract the repeated indices $\ell,m,p,r,s$, and the $\bullet$
refers to the indices of the vectors on either side of the above display. The
subtle point is that if the derivative in $\nabla a_{(\xi)}$ is not in a good
direction, cf. Lemma 5.18, one seemingly obtains the wrong bound. As such we
use that $\\{\xi,\xi^{\prime},\xi^{\prime\prime}\\}$ is an orthonormal basis
associated with the direction vector $\xi$ with
$\xi\times\xi^{\prime}=\xi^{\prime\prime}$, and so
$\xi^{n}\xi^{\ell}+(\xi^{\prime})^{n}(\xi^{\prime})^{\ell}+(\xi^{\prime\prime})^{n}(\xi^{\prime\prime})^{\ell}=\delta^{n\ell}$,
and decompose

$\displaystyle\partial_{p}a_{(\xi)}$
$\displaystyle=\underbrace{\partial_{p}\Phi_{(i,k)}^{n}\xi^{n}\xi^{\ell}A_{\ell}^{j}\partial_{j}a_{(\xi)}}_{=:\partial_{p}a_{(\xi)}^{\rm
good}}+\underbrace{\partial_{p}\Phi_{(i,k)}^{n}(\xi^{\prime})^{n}(\xi^{\prime})^{\ell}A_{\ell}^{j}\partial_{j}a_{(\xi)}+\partial_{p}\Phi_{(i,k)}^{n}(\xi^{\prime\prime})^{n}(\xi^{\prime\prime})^{\ell}A_{\ell}^{j}\partial_{j}a_{(\xi)}}_{=:\partial_{p}a_{(\xi)}^{\rm
bad}}\,,$ (7.50)

where we have also set $A=A_{(i,k)}=(\nabla\Phi_{(i,k)})^{-1}$. Using this
decomposition, we note that from Lemma 7.1, the derivative of $a_{(\xi)}$ in
the “good” term costs a factor of $\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$,
whereas the derivatives landing on $a_{(\xi)}$ in the “bad” terms cost a
factor of
$\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}\gg\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$.

In view of (7.8) and (7.9), we leave the part of (7.49) which contains $\nabla
a_{(\xi)}^{\rm good}$ in divergence form and simply move the resulting
symmetric stress

$\displaystyle(\mathcal{O}_{{\widetilde{n}},1,2}^{\rm
good})^{m\bullet}:=\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}\varrho_{(\xi)}\circ\Phi_{(i,k)}\xi^{\ell}\bigl{(}A_{\ell}^{m}\epsilon_{\bullet
pr}+A_{\ell}^{\bullet}\epsilon_{mpr}\bigr{)}\partial_{p}a_{(\xi)}^{\rm
good}\partial_{r}\Phi_{(i,k)}^{s}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\circ\Phi_{(i,k)}\,,$
(7.51)

in $\mathring{R}_{q+1}^{{\widetilde{n}}}$, up to removing a trace term which
is thrown into the pressure. This good part of
$\mathcal{O}_{{\widetilde{n}},1,2}$ obeys the same $L^{1}$ and $L^{\infty}$
bounds as $\mathcal{O}_{{\widetilde{n}},1,3}$ above. To see this, we apply the
$L^{1}$ de-correlation estimate from Lemma A.1, for $p=1$,
$f=a_{(\xi)}\xi^{\ell}(A_{\ell}^{m}\epsilon_{\bullet
pr}+A_{\ell}^{\bullet}\epsilon_{mpr})\partial_{p}a_{(\xi)}^{\rm
good}\partial_{r}\Phi_{(i,k)}^{s}$, $\Phi=\Phi_{(i,k)}$, $v=v_{\ell_{q}}$, and
$\varphi=\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}$. In light of
Proposition 3.3, Corollary 5.10, estimate (7.8), and definition (7.50), we
have that the assumptions of Lemma A.1 hold with the parameter choices
$\mathcal{C}_{f}=\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{7}\lambda_{q,{\widetilde{n}}}$,
$\lambda=\Gamma_{q+1}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}$,
$\nu=\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}\tau_{q}^{-1}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$,
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\mu=\lambda_{q+1}r_{q+1,{\widetilde{n}}}=\Gamma_{q+1}\lambda$,
$\mathcal{C}_{\varphi}=\lambda_{q+1}^{-1}$,
$\zeta=\widetilde{\zeta}=\lambda_{q+1}$, $N_{x}=0$, and
$N_{\circ}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. By (8.58a) we have that $N_{\circ}\geq
2{\mathsf{N}_{\rm dec}}+4$, and by (8.37) we have that
$\lambda_{q+1}^{4}\leq(\Gamma_{q+1}(2\pi\sqrt{3})^{-1})^{{\mathsf{N}_{\rm
dec}}}$, and so condition (A.2) is verified. Thus, from (A.3) we deduce the
$L^{1}$ estimate

$\displaystyle\bigl{\|}\psi_{i,q}D^{k}D_{t,q}^{m}\mathcal{O}_{{\widetilde{n}},1,2}^{\rm
good}\bigr{\|}_{L^{1}}\lesssim\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{7}\lambda_{q,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

The bound (7.48a) for $\mathcal{O}_{{\widetilde{n}},1,2}^{\rm good}$ now
follows from the parameter inequality (8.53), and the fact that
$\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm
cut,x}-5\geq\max\\{2\mathsf{N}_{\textnormal{ind,t}}+4+3\mathsf{N}_{\textnormal{ind,v}},6\mathsf{N}_{\textnormal{ind,v}}\\}$,
which is a consequence of (8.58a) and (8.58c). Similarly, from Proposition
3.3, Corollary 5.10, estimate (7.9), and definition (7.50), we have the
$L^{\infty}$ estimate

$\displaystyle\bigl{\|}D^{k}D_{t,q}^{m}\mathcal{O}_{{\widetilde{n}},1,2}^{\rm
good}\bigr{\|}_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}\lesssim
r_{q+1,{\widetilde{n}}}^{-2}\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+8}\lambda_{q,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$

The bound (7.48b) for $\mathcal{O}_{{\widetilde{n}},1,2}^{\rm good}$ then
follows from the parameter inequality (8.47).

Returning to (7.49), it remains to consider the bad part, coming from the
second term in (7.50), namely

$\displaystyle\sum_{\xi,i,j,k,\vec{l}}\partial_{m}\left(a_{(\xi)}\varrho_{(\xi)}\circ\Phi_{(i,k)}\xi^{\ell}\bigl{(}A_{\ell}^{m}\epsilon_{\bullet
pr}+A_{\ell}^{\bullet}\epsilon_{mpr}\bigr{)}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\circ\Phi_{(i,k)}\right)=\mathbf{V}_{1}^{\bullet}+\mathbf{V}_{2}^{\bullet}$
(7.52)

where $\mathbf{V}_{1}$ corresponds to the term containing
$A_{\ell}^{m}\epsilon_{\bullet pr}$, and $\mathbf{V}_{2}$ corresponds to the
term containing $A_{\ell}^{\bullet}\epsilon_{mpr}$. When we distribute the
$\partial_{m}$ derivative in (7.52), we need to be careful that the derivative
does not land on the fast (at frequency $\lambda_{q+1}$) object
$\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}$.

Let us first handle $\mathbf{V}_{1}$. For this purpose, note that

$\displaystyle\xi^{\ell}A_{\ell}^{m}\partial_{m}\left((\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\circ\Phi_{(i,k)}\right)$
$\displaystyle=\xi^{\ell}A_{\ell}^{m}\partial_{m}\Phi_{(i,k)}^{r}\left(\partial_{r}(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\right)\circ\Phi_{(i,k)}$
$\displaystyle=\left(\xi^{\ell}\partial_{\ell}(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\right)\circ\Phi_{(i,k)}$
$\displaystyle=0$

because $\xi\cdot\nabla$ annihilates both $\varrho_{(\xi)}$ and
$\mathbb{U}_{\xi,q+1,{\widetilde{n}}}$, from (3.4). Thus, by (7.52), the term
$\mathbf{V}_{1}^{\bullet}$ becomes

$\displaystyle\mathbf{V}_{1}^{\bullet}$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\partial_{m}\bigl{(}a_{(\xi)}\xi^{\ell}A_{\ell}^{m}\epsilon_{\bullet
pr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\bigr{)}\left(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\right)\circ\Phi_{(i,k)}$
(7.53)

Notice that by the Piola identity, we have
$\partial_{m}(a_{(\xi)}\xi^{\ell}A_{\ell}^{m}\epsilon_{\bullet
pr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s})=\xi^{\ell}A_{\ell}^{m}\partial_{m}(a_{(\xi)}\epsilon_{\bullet
pr}\partial_{p}a_{(\xi)}^{\rm bad}\partial_{r}\Phi_{(i,k)}^{s})$, and so the
slow objects contain a derivative that costs the good factor of
$\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$, and a derivative that costs the bad
factor of $\lambda_{q+1}r_{q+1,{\widetilde{n}}}$. We then apply the inverse
divergence operator $\mathcal{H}+\mathcal{R}^{*}$ from Proposition A.2, with
the following choices: $p=1$,
$G=\xi^{\ell}A_{\ell}^{m}\partial_{m}(a_{(\xi)}\epsilon_{\bullet
pr}\partial_{p}a_{(\xi)}^{\rm bad}\partial_{r}\Phi_{(i,k)}^{s})$,
$\varrho=\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}$,
$\Phi=\Phi_{(i,k)}$, $v=v_{\ell_{q}}$, and
$N_{*}=M_{*}=\lfloor\frac{1}{2}(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5)\rfloor$. By (2.19), Corollary 5.10, and
estimate (7.8), assumption (A.4) holds for
$\mathcal{C}_{G}=\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{7}(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}})\lambda_{q,{\widetilde{n}}}$,
$\lambda=\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}$,
$N_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}$,
and $\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, while
assumptions (A.5)–(A.6) hold with $\lambda^{\prime}=\widetilde{\lambda}_{q}$.
From Proposition 3.3 and standard Littlewood-Paley analysis, upon letting
$\zeta=\mu=\lambda_{q+1}r_{q+1,{\widetilde{n}}}$,
$\vartheta=(\zeta^{-2}\Delta)^{-{\mathsf{d}}}(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})$,
$\Lambda=\lambda_{q+1}$, $\mathcal{C}_{*}=\lambda_{q+1}^{-1}$, and $\alpha$ as
in (8.36), we have that condition (A.7) is satisfied. With these chosen
parameters, the condition (A.8) trivially holds, while condition (A.9) is
equivalent to
$\lambda_{q+1}^{4}\leq(\Gamma_{q+1}(2\pi\sqrt{3})^{-1})^{{\mathsf{N}_{\rm
dec}}}$, which in this case holds due to (8.37). Conditions (A.13)–(A.14) are
verified for $N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$ and
$\mathcal{C}_{v}=\Gamma_{q+1}^{{i_{\rm
max}}+1}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{2}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}}\Theta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{2}$,
in view of (2.1), (2.13), and (2.20), and (8.20)–(8.21). Lastly, the
inequality (A.15) holds because ${\mathsf{d}}$ is taken to be sufficiently
large to ensure (8.51). From (A.11) and (A.17) we deduce the $L^{1}$ bound

$\displaystyle\left\|\psi_{i,q}D^{k}D_{t,q}^{m}(\mathcal{H}+\mathcal{R}^{*})\mathbf{V}_{1}\right\|_{L^{1}}$
$\displaystyle\quad\lesssim\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{8}\frac{(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}})\lambda_{q,{\widetilde{n}}}}{(\lambda_{q+1}r_{q+1,{\widetilde{n}}})\lambda_{q+1}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(7.54)

Since
$\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{7}\lambda_{q,{\widetilde{n}}}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}$
– see (8.53), and
$\lfloor\frac{1}{2}(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5)\rfloor-{\mathsf{d}}\geq
3\mathsf{N}_{\textnormal{ind,v}}$ – see (8.58c), the above bound is consistent
with (7.48a).

The $L^{\infty}$ estimate is obtained similarly. We again apply Proposition
A.2 with the only parameters that change being: $p=\infty$,
$\mathcal{C}_{G}=\Gamma_{q}^{\mathsf{C_{u}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+8}\lambda_{q,{\widetilde{n}}}(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}$)
– see (A.14), and
$\mathcal{C}_{*}=r_{q+1,{\widetilde{n}}}^{-2}\lambda_{q+1}^{-1}$ – see (3.5)
and (3.6). From (A.11) and (A.17) we obtain

$\displaystyle\left\|D^{k}D_{t,q}^{m}(\mathcal{H}+\mathcal{R}^{*})\mathbf{V}_{1}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\quad\lesssim\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+9}\frac{(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}})\lambda_{q,{\widetilde{n}}}r_{q+1,{\widetilde{n}}}^{-2}}{(\lambda_{q+1}r_{q+1,{\widetilde{n}}})\lambda_{q+1}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,,$
(7.55)

Since
$\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+8}\lambda_{q,{\widetilde{n}}}r_{q+1,{\widetilde{n}}}^{-2}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}$,
see (8.47), the above bound is consistent with (7.48b).

It remains to consider the term $\mathbf{V}_{2}$ in (7.52). We distribute the
$\partial_{m}$ derivative on either the slow or the fast objects and decompose

$\displaystyle\mathbf{V}_{2}^{\bullet}$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\partial_{m}\left(a_{(\xi)}\varrho_{(\xi)}\circ\Phi_{(i,k)}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\circ\Phi_{(i,k)}\right)$
$\displaystyle=\sum_{\xi,i,j,k,\vec{l}}\Bigl{(}\partial_{m}\bigl{(}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{r}\Phi_{(i,k)}^{s}\bigr{)}a_{(\xi)}\partial_{p}a_{(\xi)}^{\rm
bad}+\partial_{m}a_{(\xi)}^{\rm
good}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \
-a_{(\xi)}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{m}(\partial_{p}a_{(\xi)}^{\rm
good})\partial_{r}\Phi_{(i,k)}^{s}\Bigr{)}\left(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s}\right)\circ\Phi_{(i,k)}$
$\displaystyle+\sum_{\xi,i,j,k,\vec{l}}a_{(\xi)}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\partial_{m}\left((\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\circ\Phi_{(i,k)}\right)\,.$
(7.56)

In the second equality above we have used the identities
$\epsilon_{mpr}\partial_{m}(\partial_{p}a_{(\xi)}^{\rm
bad})=-\epsilon_{mpr}\partial_{m}(\partial_{p}a_{(\xi)}^{\rm good})$, and that
$\epsilon_{mpr}\partial_{m}a_{(\xi)}^{\rm bad}\partial_{p}a_{(\xi)}^{\rm
bad}=0$. We first consider the terms in which the $\partial_{m}$ has not
landed on functions related to pipe densities. Similarly to the definition of
$\mathbf{V}_{1}$ in (7.53), the slow functions in each term contain a
derivative that costs the good factor of
$\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}$, and a derivative that costs the bad
factor of $\lambda_{q+1}r_{q+1,{\widetilde{n}}}$. As such, when applying
$\mathcal{H}+\mathcal{R}^{*}$ to the last line of (7.56), the resulting stress
obeys exactly the same estimates as (7.54) and (7.55).

Finally, we are left to consider the term on the last line of (7.56), in which
the $\partial_{m}$ derivative lands on the fast objects, at frequency
$\lambda_{q+1}$. The key observation is that this term is in fact equal to
$0$! To see this cancellation, we recall the identification of
$\partial_{p}a_{(\xi)}^{\rm bad}$ in (7.50), and we recall from (3.3) that
$\mathbb{U}_{\xi,q+1,{\widetilde{n}}}=-\xi^{\prime}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime\prime}+\xi^{\prime\prime}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime}$.
With these identities, we have

$\displaystyle
a_{(\xi)}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\partial_{m}\left((\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\circ\Phi_{(i,k)}\right)$
$\displaystyle\qquad=a_{(\xi)}\xi^{\ell}A_{\ell}^{\bullet}\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\partial_{m}\Phi^{n}\partial_{n}(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\circ\Phi_{(i,k)}\,.$

Note that from (7.50), that $\partial_{p}a_{(\xi)}^{\rm bad}$ contains either
a factor of $\partial_{p}\Phi_{(i,k)}^{k}\xi^{\prime}_{k}$ or a factor of
$\partial_{p}\Phi_{(i,k)}^{k}\xi^{\prime\prime}_{k}$. From (3.4), we also have
that

$\displaystyle\partial_{r}\Phi_{(i,k)}^{s}\partial_{m}\Phi^{n}\partial_{n}(\varrho_{(\xi)}\mathbb{U}^{s}_{\xi,q+1,{\widetilde{n}}})$
$\displaystyle=-\partial_{r}\Phi_{(i,k)}^{s}\xi^{\prime}_{s}\partial_{m}\Phi^{n}\xi^{\prime}_{n}\left((\xi^{\prime}\cdot\nabla)\left(\varrho_{(\xi)}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime\prime}\right)\right)\circ\Phi_{(i,k)}$
$\displaystyle\qquad+\partial_{r}\Phi_{(i,k)}^{s}\xi^{\prime\prime}_{s}\partial_{m}\Phi^{n}\xi^{\prime}_{n}\left((\xi^{\prime}\cdot\nabla)\left(\varrho_{(\xi)}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime}\right)\right)\circ\Phi_{(i,k)}$
$\displaystyle\qquad-\partial_{r}\Phi_{(i,k)}^{s}\xi^{\prime}_{s}\partial_{m}\Phi^{n}\xi_{n}^{\prime\prime}\left((\xi^{\prime\prime}\cdot\nabla)\left(\varrho_{(\xi)}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime\prime}\right)\right)\circ\Phi_{(i,k)}$
$\displaystyle\qquad+\partial_{r}\Phi_{(i,k)}^{s}\xi^{\prime\prime}_{s}\partial_{m}\Phi^{n}\xi_{n}^{\prime\prime}\left((\xi^{\prime}\cdot\nabla)\left(\varrho_{(\xi)}\varphi_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{\prime\prime}\right)\right)\circ\Phi_{(i,k)}\,.$

Thus, the expression $\epsilon_{mpr}\partial_{p}a_{(\xi)}^{\rm
bad}\partial_{r}\Phi_{(i,k)}^{s}\partial_{m}\Phi^{n}\partial_{n}(\varrho_{(\xi)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}^{s})\circ\Phi_{(i,k)}$
equals the sum of eight terms, each of which is of the type

$\displaystyle\epsilon_{mpr}\partial_{p}\Phi_{(i,k)}^{k}\xi^{(1)}_{k}\partial_{r}\Phi_{(i,k)}^{s}\xi^{(2)}_{s}\partial_{m}\Phi^{n}\xi^{(3)}_{n}\times(\mbox{product
of fast pipe densities or fast cutoffs})\circ\Phi_{(i,k)}$

where
$(\xi^{(1)},\xi^{(2)},\xi^{(3)})\in\\{\xi^{\prime},\xi^{\prime\prime}\\}^{3}$.
Since in each of these eight terms, at least two of the vectors in the tuple
$(\xi^{(1)},\xi^{(2)},\xi^{(3)})$ are equal to each other, either to
$\xi^{\prime}$ or $\xi^{\prime\prime}$, by the skew symmetry of the Levi-
Civita symbol, we must have

$\displaystyle\epsilon_{mpr}\partial_{p}\Phi_{(i,k)}^{k}\xi^{(1)}_{k}\partial_{r}\Phi_{(i,k)}^{s}\xi^{(2)}_{s}\partial_{m}\Phi^{n}\xi^{(3)}_{n}=0\,.$

This proves that the last term on the right side of (7.56) is indeed equal to
$0$, concluding the proof. ∎

###  7.7. Transport errors

######  Lemma 7.13.

For all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$, the transport error satisfies
the following estimates for $N,M\leq 3\mathsf{N}_{\textnormal{ind,v}}$:

$\displaystyle\left\|\psi_{i,q}D^{N}D_{t,q}^{M}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(D_{t,q}w_{q+1,{\widetilde{n}}}\right)\right)\right\|_{L^{1}}$
$\displaystyle\lesssim\delta_{q+2}\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.57a)
$\displaystyle\left\|D^{k}D_{t,q}^{m}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)(D_{t,q}w_{q+1,{\widetilde{n}}})\right)\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(7.57b)

###### Proof of Lemma 7.13.

Recall from the first line of (7.24) that the transport error is given by
$\mathcal{H}+\mathcal{R}^{*}$ applied to $D_{t,q}w_{q+1,{\widetilde{n}}}$,
which we further expand as

$\displaystyle D_{t,q}w_{q+1,{\widetilde{n}}}$
$\displaystyle=D_{t,q}\biggl{(}\sum_{\;i,j,k,\vec{l},\xi}\mathrm{curl\,}\left(a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\biggr{)}$
$\displaystyle=\sum_{i,j,k,\vec{l},\xi}D_{t,q}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\right)\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}+\sum_{i,j,k,\vec{l},\xi}\left(D_{t,q}\nabla
a_{(\xi)}\right)\times\left(\nabla\Phi_{(i,k)}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
$\displaystyle\qquad+\sum_{i,j,k,\vec{l},\xi}\nabla
a_{(\xi)}\times\left(\left(D_{t,q}\nabla\Phi_{(i,k)}\right)\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
(7.58)

Since the second two terms contain the corrector defined in (7.6), and the
bounds for the corrector in (7.12a) 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.2, recall that the inverse divergence of (7.58) needs to be
estimated on the support of a cutoff $\psi_{i,q}$ in order to verify (7.57a)
and (7.57b). Recall that for all ${\widetilde{n}}$,
$D_{t,q}w_{q+1,{\widetilde{n}}}$ has zero mean. Thus, although each individual
term in the final equality in (7.58) 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.16).

We will apply Proposition A.2 to the first term with the following choices.
Let $p\in\\{1,\infty\\}$. 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 (8.58a). We define

$G=D_{t,q}(a_{(\xi)}\nabla\Phi_{(i,k)}^{-1})\xi\,,$

with $\lambda=\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$. In order to
obtain the value of the amplitude constant $\mathcal{C}_{G}$, which now
depends on $p$, when $p=1$ we use (7.8) with $r=1$ and (5.19g), while when
$p=\infty$ we use (7.9) and (5.19g), obtaining

$\displaystyle\mathcal{C}_{G,1}$
$\displaystyle=|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},\xi,\vec{l}})|\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{j+2}\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3}\,,$
(7.59a) $\displaystyle\mathcal{C}_{G,\infty}$
$\displaystyle=\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon({\widetilde{n}})+\frac{7}{2}}\tau_{q}^{-1}\Gamma_{q+1}^{{i_{\rm
max}}-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}$
$\displaystyle\leq\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{{\mathsf{C_{u}}}+7\Upsilon({\widetilde{n}})+9-\mathsf{c}_{\widetilde{\textnormal{n}}}}\tau_{q}^{-1}\Theta_{q}^{\nicefrac{{1}}{{2}}}\delta_{q}^{-\nicefrac{{1}}{{2}}}\leq\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{{\mathsf{C_{u}}}+7\Upsilon({n_{\rm
max}})+20+\mathsf{c}_{0}-\mathsf{c}_{\widetilde{\textnormal{n}}}}\Theta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\,.$
(7.59b)

Note that we have used (8.33), (5.9), and (8.22) to simplify the above
expressions. We have that

$\displaystyle\|D^{N}D_{t,q}^{M}G\|_{L^{p}}$
$\displaystyle\lesssim\mathcal{C}_{G,p}\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}}-1,\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim\mathcal{C}_{G,p}\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{-1}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$
(7.60)

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 (8.35), and so
(A.4) is satisfied. We set $\Phi=\Phi_{i,k}$ and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$. Appealing as usual to Corollary
5.10 and (2.19) with $q^{\prime}=q$, which is valid from Proposition 5.6, we
have that (A.5) and (A.6) are satisfied.

Referring to (1) from Proposition 3.3, 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
$\mu=\lambda_{q+1}r_{q+1,{\widetilde{n}}}$ and referring to (2) from
Proposition 3.3, we have that (ii) is satisfied. Setting
$\Lambda=\zeta=\lambda_{q+1}$,
$C_{*,p}=r_{q+1,{\widetilde{n}}}^{\frac{2}{p}-1}$, $\alpha$ as in (8.36), and
referring to (3.5) and (3.6) from Proposition 3.3, we have that (A.7) is
satisfied. (A.8) is immediate from the definitions. Referring to (8.37), we
have that (A.9) is satisfied.

After summing on $(i,j,k,{\widetilde{n}},\xi,\vec{l})$, using (2.11) at level
$q$, and (5.43) with $r_{1}=r_{2}=2$, we conclude from (A.11) that for $p=1$
and
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$,

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\left(D_{t,q}w_{q+1,{\widetilde{n}}}\right)\right)\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\quad\lesssim\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{{\mathsf{C}_{b}}+9-\mathsf{c}_{\widetilde{\textnormal{n}}}}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\quad\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.61)

after also using (8.55) and (8.33). From (8.58c), these bounds are valid for
all $N,M\leq 3\mathsf{N}_{\textnormal{ind,v}}$. Similarly, for $p=\infty$, we
have

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\left(D_{t,q}w_{q+1,{\widetilde{n}}}\right)\right)\right\|_{L^{\infty}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\quad\lesssim\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{{\mathsf{C_{u}}}+7\Upsilon({n_{\rm