The Type 1 oscillation errors are defined in the three parameter regimes
${\widetilde{n}}=0$, $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, and
${\widetilde{n}}={n_{\rm max}}$. In the case ${\widetilde{n}}=0$, Type 1
oscillation errors stem from the term identified in (8.37), which we recall is

$\displaystyle(\mathcal{H}+\mathcal{R}^{*})\bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,0}}\Bigl{(}\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\Bigr{)}$
$\displaystyle\quad\qquad\qquad\qquad\times(\mathbb{W}_{\xi,q+1,0}\otimes\mathbb{W}_{\xi,q+1,0})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,0}}\Bigl{(}\sum\limits_{n=1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\Bigr{)}$
$\displaystyle\quad\qquad\qquad\qquad\times(\mathbb{W}_{\xi,q+1,0}^{\theta}\mathbb{W}_{\xi,q+1,0}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\bigg{)}.$
(8.81)

This sum is divided into the terms identified in (8.39), (8.40), (8.41),
(8.42), (8.44), and (8.45). The errors defined in (8.44) and (8.45) are
$\mathring{H}_{q,n,p}^{0}$ errors and will be corrected by later perturbations
$w_{q+1,n,p}$, while the others will be immediately absorbed into
$\mathring{R}_{q+1}^{0}$.

In the case $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, Type 1 oscillation
errors stem from the term identified in (8.53)

$\displaystyle\left(\mathcal{H}+\mathcal{R}^{*}\right)\Biggl{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\Bigl{(}\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\Bigr{)}$
$\displaystyle\quad\qquad\qquad\qquad\times(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\Bigl{(}\sum\limits_{n={\widetilde{n}}+1}^{{n_{\rm
max}}}\sum\limits_{p=1}^{p_{\rm
max}}\mathbb{P}_{[q,n,p]}+\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\Bigr{)}$
$\displaystyle\quad\qquad\qquad\qquad\times(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\Biggr{)}.$
(8.82)

This sum is divided into the terms identified in (8.57), (8.58), (8.59),
(8.60), (8.61), and (8.62). As before, the last two terms are
$\mathring{H}_{q,n,p}^{{\widetilde{n}}}$ errors and will be corrected by later
perturbations, while the others are absorbed into
$\mathring{R}_{q+1}^{\widetilde{n}}$.

In the case ${\widetilde{n}}={n_{\rm max}}$, Type 1 oscillation errors are
identified in (8.73) and (8.74) as

$\displaystyle{\left(\mathcal{H}+\mathcal{R}^{*}\right)\Bigg{(}\sum_{\xi,i,j,k,p,\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}\otimes\mathbb{W}_{\xi,q+1,{n_{\rm
max}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}}$
$\displaystyle\quad+{\sum_{\xi,i,j,k,p,\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}(\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}^{\theta}\mathbb{W}_{\xi,q+1,{n_{\rm
max}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\Bigg{)}}.$
(8.83)

These errors are completely absorbed into $\mathring{R}_{q+1}$.

To prove the desired estimates on these error terms, we will first analyze a
single term of the form

$\displaystyle(\mathcal{H}+\mathcal{R}^{*})\Bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad\qquad\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\alpha\theta}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\zeta\gamma}\Bigg{)}$
$\displaystyle\qquad=:\left(\mathcal{H}+\mathcal{R}^{*}\right)\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},n,p}\,.$
(8.84)

The estimates in Lemma 8.6 for this term on the support of a cutoff function
$\psi_{i,q}$ will depend on ${\widetilde{n}}$ and ${\widetilde{p}}$, which
range from $0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and
$1\leq{\widetilde{p}}\leq{p_{\rm max}}$, respectively, and $n$ and $p$, which
range from ${\widetilde{n}}+1\leq n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm
max}}$, with the additional endpoint case $n={n_{\rm max}}$, $p={p_{\rm
max}}+1$. We then use this general estimate to specify in Remark 8.7 how the
terms corresponding to various values of $n$, ${\widetilde{n}}$, $p$, and
${\widetilde{p}}$ are absorbed into either higher order stresses
$\mathring{H}_{q,n,p}^{\widetilde{n}}$ or
$\mathring{R}_{q+1}^{\widetilde{n}}$, and eventually $\mathring{R}_{q+1}$.

######  Lemma 8.6.

The terms $\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},n,p}$ defined in
(8.84) satisfy the following.

  1. (1)

For the special case $n={n_{\rm max}}$, $p={p_{\rm max}}+1$ and all
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$, $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$, as well as for all cases $0\leq{\widetilde{n}}<n\leq{n_{\rm max}}$,
$1\leq p,{\widetilde{p}}\leq{p_{\rm max}}$, the nonlocal portion of the
inverse divergence satisfies

$\left\|D^{N}D_{t,q}^{M}\left(\mathcal{R}^{*}\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},n,p}\right)\right\|_{L^{1}\left(\mathbb{T}^{3}\right)}\leq\frac{\delta_{q+2}}{\lambda_{q+1}}\lambda_{q+1}^{N}\tau_{q}^{-M}\,$
(8.85)

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

  2. (2)

For $n={n_{\rm max}}$, $p={p_{\rm max}}+1$, all
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$, the high frequency, local portion of the inverse divergence satisfies

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},{n_{\rm
max}},{p_{\rm
max}}+1}\right)\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\qquad\qquad\lesssim\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(8.86)

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

  3. (3)

For $0\leq{\widetilde{n}}<n\leq{n_{\rm max}}$ and $1\leq
p,{\widetilde{p}}\leq{p_{\rm max}}$, the medium frequency, local portion of
the inverse divergence satisfies

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},n,p}\right)\right\|_{L^{1}\mathrm{supp\,}\left(\psi_{i,q}\right)}$
$\displaystyle\lesssim\delta_{q+1,n,p}\lambda_{q,n,p}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(8.87)

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

######  Remark 8.7.

Note that after appealing to ${\widetilde{n}}\leq n-1$, (9.35), and (9.42),
(8.87) matches (7.15), (7.22), and (7.29), or equivalently (6.118). In
addition, after appealing again to ${\widetilde{n}}\leq n-1$, (9.35), and
(9.42), (8.85) and (8.86) are sufficient to meet (7.13), (7.20), and (7.27).

###### Proof of Lemma 8.6.

The first step is to use item (1) and (4.15) from Proposition 4.4 to rewrite
(8.84) as

$\displaystyle(\mathcal{H}+\mathcal{R}^{*})\Bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\nabla
a_{(\xi)}^{2}\nabla\Phi_{(i,k)}^{-1}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\otimes\mathbb{W}_{\xi,q+1,{\widetilde{n}}})(\Phi_{(i,k)})\nabla\Phi_{(i,k)}^{-T}$
$\displaystyle\qquad+\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}a_{(\xi)}^{2}(\nabla\Phi_{(i,k)}^{-1})_{\theta\alpha}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\theta}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{\gamma})(\Phi_{(i,k)})\partial_{\alpha}(\nabla\Phi_{(i,k)}^{-1})_{\gamma\kappa}\Bigg{)}$
$\displaystyle=\left(\mathcal{H}+\mathcal{R}^{*}\right)\Bigg{(}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right)(\Phi_{(i,k)})$
$\displaystyle\qquad\times\bigg{(}\partial_{\alpha}a_{(\xi)}^{2}\left(\nabla\Phi_{(i,k)}^{-1}\right)_{\gamma\kappa}\xi^{\theta}\xi^{\gamma}\left(\nabla\Phi_{(i,k)}^{-T}\right)_{\theta\alpha}+a_{(\xi)}^{2}\left(\nabla\Phi_{(i,k)}\right)^{-1})_{\theta\alpha}\xi^{\theta}\xi^{\gamma}\partial_{\alpha}\left(\nabla\Phi_{(i,k)}^{-1}\right)_{\gamma\kappa}\bigg{)}\Bigg{)}.$
(8.88)

Next, we must identify the functions and the values of the parameters which
will be used in the application of Proposition A.18, specifically Remark A.19.
We first address the bounds required in (A.66), (A.67), and (A.68), which we
can treat simultaneously for items (1), (2), and (3). Afterwards, we split the
proof into two parts. First, we set $n={n_{\rm max}}$, $p={p_{\rm max}}+1$ and
prove (8.85) for _only_ these specific values of $n$ and $p$, as we
simultaneously prove (8.86). Next, we consider $n<{n_{\rm max}}$ and prove
(8.85) in the remaining cases, as we simultaneously prove (8.87).

Returning to (A.66), 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 (9.60a), which gives that

$2{\mathsf{N}_{\rm
dec}}+4\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)-{\mathsf{d}}\rfloor\,.$ (8.89)

Denoting the $\kappa^{\textnormal{th}}$ component of the below vector field
$G$ by $G_{\kappa}$, we _fix_ a value of $(\xi,i,j,k,{\widetilde{p}},\vec{l})$
and set

$\displaystyle
G_{\kappa}=\partial_{\alpha}a_{(\xi)}^{2}\left(\nabla\Phi_{(i,k)}^{-1}\right)_{\gamma\kappa}\xi^{\theta}\xi^{\gamma}\left(\nabla\Phi_{(i,k)}^{-T}\right)_{\alpha\theta}+a_{(\xi)}^{2}\left(\nabla\Phi_{(i,k)}\right)^{-1})_{\alpha\theta}\xi^{\theta}\xi^{\gamma}\partial_{\alpha}\left(\nabla\Phi_{(i,k)}^{-1}\right)_{\kappa\gamma}\,.$
(8.90)

We now establish (A.66)–(A.68) with the parameter choices

$\displaystyle\mathcal{C}_{G}=|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma^{2j-3-{\mathsf{C}_{b}}}_{q+1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\,,$
(8.91)

$\lambda=\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}$,
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$, and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$. Applying Lemma 8.1 and estimate
(8.25) with $r=2$, $r_{2}=1$, $r_{1}=\infty$, and the bounds (6.113) and
(6.114), we see that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\Bigl{(}\partial_{\alpha}a_{(\xi)}^{2}\left(\nabla\Phi_{(i,k)}^{-1}\right)_{\gamma\kappa}\xi^{\theta}\xi^{\gamma}\left(\nabla\Phi_{(i,k)}^{-T}\right)_{\alpha\theta}\Bigr{)}\right\|_{L^{1}}$
$\displaystyle\lesssim|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma^{2j+5}_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}})^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\lesssim|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma^{2j-2-{\mathsf{C}_{b}}}_{q+1}$
$\displaystyle\qquad\times\Gamma_{q+1}^{-1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(8.92)

holds for all
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$. To achieve the last
inequality, we have used the definition of
$\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}$ in (9.34) and the definition of
$f_{q,{\widetilde{n}}}$ in (9.31) to rewrite

$\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\Gamma_{q+1}^{7+{\mathsf{C}_{b}}}=\Gamma_{q+1}^{-1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\,.$

For the second half of $G_{\kappa}$, we can appeal to (6.113) and (6.114), and
use that
$\widetilde{\lambda}_{q}\leq\lambda_{q,{\widetilde{n}},{\widetilde{p}}}$ for
all ${\widetilde{n}}$ and ${\widetilde{p}}$ 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

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

Combining these estimates shows that

$\displaystyle\left\|D^{N}D_{t,q}^{M}G_{\kappa}\right\|_{L^{1}}\lesssim\mathcal{C}_{G}\left(\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+3},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,$
(8.93)

for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor$, showing that (A.66) has been
satisfied.

We set the flow in Proposition A.18 as $\Phi=\Phi_{i,k}$, which by definition
satisfies $D_{t,q}\Phi_{i,k}=0$. Appealing to (6.109) and (6.112), we have
that (A.67) is satisfied. From (6.60), the choice of $\nu$ from earlier, and
(9.39), we have that $Dv=Dv_{\ell_{q}}$ satisfies the bound (A.68).

Proof of item (2) and of item (1) when $n={n_{\rm max}}$, $p={p_{\rm max}}+1$.
We first assume that ${\widetilde{n}}<{n_{\rm max}}$. In this case, we have
that the minimum frequency $\lambda_{q,{n_{\rm max}}+1,0}$ of
$\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm max}}+1\right]}$ is larger than the
minimum frequency $\lambda_{q,{\widetilde{n}}}$ of
$\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}$ from (9.28) and (9.22). We
therefore can discard $\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}$ from
(8.88) and with the goal of satisfying verifying (i)–(iii) of Proposition
A.18, we set

$\displaystyle\zeta=\lambda_{q,{n_{\rm max}}+1,0},$
$\displaystyle\qquad\mu=\lambda_{q,{\widetilde{n}}},\qquad\Lambda=\lambda_{q+1},$
(8.94)

and

$\displaystyle\varrho$ $\displaystyle=\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm
max}}+1\right]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right),$
(8.95a) $\displaystyle\vartheta$ $\displaystyle=\lambda_{q,{n_{\rm
max}}+1,0}^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm
max}}+1\right]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\,,$
(8.95b)

where we recall that $\varrho_{\xi,\lambda,r}$ is defined via Propositions 4.3
and 4.4. We then have immediately that

$\displaystyle\varrho$ $\displaystyle=\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm
max}}+1\right]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right)$
$\displaystyle=\lambda_{q,{n_{\rm
max}}+1,0}^{-2{\mathsf{d}}}\Delta^{\mathsf{d}}\lambda_{q,{n_{\rm
max}}+1,0}^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\left(\mathbb{P}_{\left[q,{n_{\rm
max}},{p_{\rm
max}}+1\right]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\right)$
$\displaystyle=\lambda_{q,{n_{\rm
max}}+1,0}^{-2{\mathsf{d}}}\Delta^{\mathsf{d}}\vartheta\,,$ (8.96)

and so (i) from Proposition A.18 is satisfied. By property (1) of Proposition
4.3, we have that the functions $\varrho$ and $\vartheta$ defined in (8.95)
are both periodic to scale
$\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}=\lambda_{q,{\widetilde{n}}}^{-1}$,
and so (ii) is satisfied. The estimates in (A.69) follow with
$\mathcal{C}_{*}=1$ from standard Littlewood-Paley arguments (see also the
discussion in part (b) of Remark A.21) and item (5) from Proposition 4.4. Note
that in the case $N=2{\mathsf{d}}$ in (A.69), the inequality is weakened by a
factor of $\lambda_{q+1}^{\alpha_{\mathsf{R}}}$, for an arbitrary
$\alpha_{\mathsf{R}}>0$; thus, (ii) is satisfied. At this stage let us fix a
value for this parameter $\alpha_{\mathsf{R}}$: we choose it to be
sufficiently small (with respect to $b$ and $\varepsilon_{\Gamma}$) to ensure
that the loss $\lambda_{q+1}^{\alpha_{\mathsf{R}}}$ may be absorbed by the
spare negative factor of $\Gamma_{q+1}$ in the definition of
$\mathcal{C}_{G}$, as is postulated in (9.53). From (9.19), (9.22), (9.26),
and (9.29), we have that

$\widetilde{\lambda}_{q}\leq\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\ll\lambda_{q,{\widetilde{n}}}\leq\lambda_{q,{n_{\rm
max}}+1,0}\leq\lambda_{q+1},$

and so (A.70) is satisfied. From (9.48) we have that

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

if ${\mathsf{N}_{\rm dec}}$ is chosen large enough, and so (A.71) is
satisfied. Applying the estimate (A.73) with $\alpha$ as in (9.53), recalling
the value for $\mathcal{C}_{G}$ in (8.91), using (6.19) and (6.147) with
$r_{1}=\infty$ and $r_{2}=1$, we obtain that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},{n_{\rm
max}},{p_{\rm
max}}+1}\right)\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\quad\lesssim\sum_{i^{\prime}=i-1}^{i+1}\sum_{\xi,j,k,\vec{l}}\Lambda^{\alpha_{\mathsf{R}}}|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma^{2j-3-{\mathsf{C}_{b}}}_{q+1}\Gamma_{q}^{{-\mathsf{C_{R}}}}$
$\displaystyle\qquad\qquad\qquad\times\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\mathcal{C}_{*}\zeta^{-1}\mathcal{M}\left(N,1,\zeta,\Lambda\right)\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)$
$\displaystyle\quad\lesssim\Gamma_{q+1}\Big{(}\Gamma_{q+1}^{-1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\big{(}f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\big{)}\Big{)}\lambda_{q,{n_{\rm
max}}+1,0}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\quad\lesssim\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,,$
(8.97)

for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$. In the last
inequality, we have used the parameter estimate (9.54), which directly implies

$\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\lambda_{q,{n_{\rm
max}}+1,0}^{-1}\leq\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\,.$
(8.98)

Then, after using (9.60c), which gives that for all ${\widetilde{n}}$ we have

$\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}\geq
3\mathsf{N}_{\textnormal{ind,v}},$ (8.99)

and thus the range of derivatives allowed in (8.97) is exactly as needed in
(8.86), thereby proving this bound.

Continuing to follow the parameter choices in Remark A.19, we set
$N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$, and as before
$N^{\sharp}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. From (9.60d), we have that the condition
$N_{\circ}\leq\nicefrac{{N^{\sharp}}}{{4}}$ is satisfied. The inequalities
(A.75) and (A.76) follow from the discussion in Remark A.19. The inequality in
(A.77) follows from (9.43), (9.55), the fact that
$\lambda=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\leq\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{p_{\rm
max}}}$, and $\zeta=\lambda_{q,{n_{\rm max}}+1,0}>\lambda_{q,{n_{\rm
max}}-1}\geq\lambda_{q,{\widetilde{n}}}$, as in the discussion in Remark A.19.
Having satisfied these assumptions, we may now appeal to estimate in (A.79),
which gives (8.85) for the case ${\widetilde{n}}<n={n_{\rm max}}$, $p={p_{\rm
max}}+1$, and any value of ${\widetilde{p}}$.

Recall we began this case by assuming that ${\widetilde{n}}<{n_{\rm max}}$. In
the case ${\widetilde{n}}={n_{\rm max}}$ and $1\leq{\widetilde{p}}\leq{p_{\rm
max}}$, we have from (9.22) and (9.29) that $\lambda_{q,{n_{\rm
max}}}>\lambda_{q,{n_{\rm max}}+1,0}$, and so

$\mathbb{P}_{\left[q,{n_{\rm max}},{p_{\rm
max}}+1\right]}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}=\mathbb{P}_{\geq\lambda_{q,{n_{\rm
max}}}}\,.$

Then we can set $\zeta=\mu=\lambda_{q,{n_{\rm max}}}$. The only change is that
(8.98) becomes stronger, since $\lambda_{q,{n_{\rm max}}}>\lambda_{q,{n_{\rm
max}}+1,0}$, and so the desired estimates follow by arguing as before. We omit
further details.

Proof of item (3) and of item (1) when $p\neq{p_{\rm max}}+1$ and
$n\leq{n_{\rm max}}$.  Note that in both of these cases we have
${\widetilde{n}}<n$. We first point that that we may assume that $n$ and $p$
are such that $\lambda_{q,{\widetilde{n}}}<\lambda_{q,n,p}$. If not, then
$\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}=0$, and so
the estimate is trivially satisfied. We then set

$\displaystyle\zeta=\max\left\\{\lambda_{q,{\widetilde{n}}},\lambda_{q,n,p-1}\right\\},$
$\displaystyle\qquad\mu=\lambda_{q,{\widetilde{n}}},\qquad\Lambda=\lambda_{q,n,p},$
(8.100)

and

$\displaystyle\varrho$
$\displaystyle=\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}\left(\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)^{2}\right),$
(8.101a) $\displaystyle\vartheta$
$\displaystyle=\zeta^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\,.$
(8.101b)

We then have from the discussion part (b) of Remark A.21 that

$\displaystyle\varrho$
$\displaystyle=\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}\left(\varrho_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}^{2}\right)$
$\displaystyle=\zeta^{-2{\mathsf{d}}}\Delta^{\mathsf{d}}\zeta^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\left(\mathbb{P}_{\geq\lambda_{q,{\widetilde{n}}}}\mathbb{P}_{[q,n,p]}\left(\varrho^{2}_{\xi,\lambda_{q+1},r_{q+1,{\widetilde{n}}}}\right)\right),$
$\displaystyle=\zeta^{-2{\mathsf{d}}}\Delta^{\mathsf{d}}\vartheta\,,$ (8.102)

and so (i) from Proposition A.18 is satisfied. By property (1) of Proposition
4.3, $\varrho$ and $\vartheta$ are both periodic to scale
$\left(\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{-1}=\lambda_{q,{\widetilde{n}}}^{-1}$,
and so (ii) is satisfied. The estimates in (A.69) follow with
$\mathcal{C}_{*}=1$ from the discussion in part (b) of Remark A.21. Note that
in the case $N=2{\mathsf{d}}$ in (A.69), the inequality is weakened by a
factor of $\lambda_{q+1}^{\alpha_{\mathsf{R}}}$, and so (ii) is satisfied.
Here we again use $\alpha_{\mathsf{R}}$ as in (9.53), so this loss will be
absorbed using a factor of $\Gamma_{q+1}$. From (9.19), (9.26), (9.29) and
(9.22), and the assumption that $\lambda_{q,{\widetilde{n}}}<\lambda_{q,n,p}$,
we have that

$\widetilde{\lambda}_{q}\leq\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\ll\lambda_{q,{\widetilde{n}}}\leq\max\left\\{\lambda_{q,{\widetilde{n}}},\lambda_{q,n,p-1}\right\\}\leq\lambda_{q,n,p},$

and so, since $\Lambda\leq\lambda_{q+1}$, (A.70) is satisfied. From (9.48) we
have that

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

and so (A.71) is satisfied. Applying the estimate (A.73) for the parameter
range in Remark A.19, recalling that (8.90) includes the indicator function of
$\mathrm{supp\,}\left(\psi_{i,q}\right)$, recalling the definition of
$\mathcal{C}_{G}$ in (8.91), using (6.19) and (6.147) with $r_{1}=\infty$ and
$r_{2}=1$, and using $\zeta^{-1}\leq\lambda_{q,n,p-1}^{-1}$, we have that

$\displaystyle\left\|D^{N}D_{t,q}^{M}\left(\mathcal{H}\mathcal{O}_{{\widetilde{n}},{\widetilde{p}},n,p}\right)\right\|_{L^{1}\left(\mathrm{supp\,}\psi_{i,q}\right)}$
$\displaystyle\qquad\lesssim\sum_{i^{\prime}=i-1}^{i+1}\sum_{\xi,j,k,\vec{l}}\Lambda^{\alpha_{\mathsf{R}}}|\mathrm{supp\,}(\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}})\bigr{|}\Gamma^{2j-3-{\mathsf{C}_{b}}}_{q+1}\Gamma_{q}^{{-\mathsf{C_{R}}}}$
$\displaystyle\qquad\qquad\times\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\mathcal{C}_{*}\zeta^{-1}\mathcal{M}\left(N,1,\zeta,\Lambda\right)\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)$
$\displaystyle\qquad\lesssim\Gamma_{q+1}\Gamma_{q+1}^{-1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\lambda_{q,n,p-1}^{-1}\lambda_{q,n,p}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\qquad\lesssim\delta_{q+1,n,p}\lambda_{q,n,p}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\,.$
(8.103)

In the last inequality, we have used that since $n<{\widetilde{n}}$, by (9.34)
we have

$\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{\widetilde{n}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\lambda_{q,n,p-1}^{-1}\leq\delta_{q+1,n,p}$
(8.104)

for all
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}$. Then after using
(9.61), which gives that for all ${\widetilde{n}}<n$ that

$\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5\right)\rfloor-{\mathsf{d}}\geq\mathsf{N}_{\rm
fin,n},$ (8.105)

we have achieved (8.87).

Continuing to follow the parameter choices in Remark A.19, we set
$N_{\circ}=M_{\circ}=3\mathsf{N}_{\textnormal{ind,v}}$, and as before
$N^{\sharp}=\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-5$. From (9.60d), we have that the condition
$N_{\circ}\leq\nicefrac{{N^{\sharp}}}{{4}}$ is satisfied. The inequalities
(A.75) and (A.76) follow from the discussion in Remark A.19. The inequality in
(A.77) follows from (9.55) and the fact that
$\lambda=\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}\leq\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{p_{\rm
max}}}$ and
$\zeta=\max\\{\lambda_{q,{\widetilde{n}}},\lambda_{q,n,p-1}\\}\geq\lambda_{q,{\widetilde{n}}}$.
We then achieve the concluded estimate in (A.79), which gives (8.85) for the
case $p\neq{p_{\rm max}}+1$, $n\leq{n_{\rm max}}$ and any values of
${\widetilde{n}}$, ${\widetilde{p}}$ with ${\widetilde{n}}<n$. ∎

###  8.7 Type 2 oscillation errors

In order to show that the Type 2 errors (previously identified in (8.43),
(8.55), (8.56), (8.71), (8.72)) vanish, we will apply Proposition 4.8 on the
support of a specific cutoff function

$\eta=\eta_{i,j,k,q,n,p,\vec{l}}=\psi_{i,q}\chi_{i,k,q}\overline{\chi}_{q,n,p}\omega_{i,j,q,n,p}\zeta_{i,q,k,n,\vec{l}}\,.$

Before we may apply the proposition, we first estimate in Lemma 8.8 the number
of cutoff functions $\eta^{*}$ which may overlap with $\eta$, with an eye
towards keeping track of all the pipes that we will have to dodge in order to
successfully place pipes on $\eta$. The next three Lemmas ((8.9)-(8.11)) are
technical in nature and are necessary in order to apply Lemma 4.7.
Specifically, we show that given $\eta$, $\eta^{*}$ and a fixed time $t^{*}$,
one may find a convex set which contains the intersection of the supports of
$\eta$ and $\eta^{*}$ at $t^{*}$. The time $t^{*}$ will be the time at which
the pipes on $\eta^{*}$ are _straight_ , and combined with the convexity,
Lemma 4.7 may be applied. The upshot of this is that the pipes belonging to
$\eta^{*}$ only undergo mild deformations on the support of $\eta$. This
allows us to finally apply Proposition 4.8 to place pipes on $\eta$ which
dodge all pipes originating from overlapping cutoff functions $\eta^{*}$. We
remark that since $\overline{\chi}_{q,n,p}$ depends only on $n$ and $p$, which
are indices already encoded in $\omega_{i,j,q,n,p}$, throughout this section
we will suppress the dependence of the cumulative cutoff function $\eta$ on
$\overline{\chi}_{q,n,p}$ (defined in (6.104)), as it does not affect any of
the estimates.

####  8.7.1 Preliminary estimates

######  Lemma 8.8 (Keeping Track of Overlap).

Given a cutoff function $\eta_{i,j,k,q,n,p,\vec{l}}$, consider the set of all
tuples $\left({i^{*}},{j^{*}},{k^{*}},{n^{*}},{p^{*}},\vec{l}^{*}\right)$ such
that the cutoff function
$\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{p^{*}},\vec{l}^{*}}$ satisfies:

  1. (1)

${n^{*}}\leq n$

  2. (2)

There exists $(x,t)$ such that

$\eta_{i,j,k,q,n,p,\vec{l}}(x,t)\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{p^{*}},\vec{l}^{*}}(x,t)\neq
0.$ (8.106)

Then the cardinality of the set of all such tuples is bounded above by
$\mathcal{C}_{\eta}\Gamma_{q+1}$, where the constant $\mathcal{C}_{\eta}$
depends only on ${n_{\rm max}}$, ${p_{\rm max}}$, ${j_{\rm max}}$, and
dimensional constants. In particular, due to (9.2), (9.3), and (6.129),
$\mathcal{C}_{\eta}$ is independent of $q$ and the values of the other
parameters indexing the cutoff functions.

###### Proof of Lemma 8.8.

Recall that the cutoff functions are defined by

$\eta_{i,j,k,q,n,p,\vec{l}}(x,t)=\psi_{i,q}(x,t)\chi_{i,k,q}(t)\overline{\chi}_{q,n,p}(t)\omega_{i,j,q,n,p}(x,t)\zeta_{i,q,k,n,\vec{l}}(x,t).$
(8.107)

As noted in the outline of this section, we will suppress the dependence on
$\overline{\chi}_{q,n,p}$, since the $n$ and $p$ indices are already accounted
for in $\omega_{i,j,q,n,p}$. The proof proceeds by first counting the number
of combinations $({i^{*}},{k^{*}})$ for which it is possible that there exists
$(x,t)$ such that

$\psi_{i,q}(x,t)\chi_{i,k,q}(t)\psi_{{i^{*}},q}(x,t)\chi_{{i^{*}},{k^{*}},q}(t)\neq
0.$ (8.108)

Next, for a given $({i^{*}},{k^{*}})$, we count the number of values of
$({j^{*}},{n^{*}},{p^{*}})$ such that there exists $(x,t)$ such that

$\omega_{i,j,q,n,p}(x,t)\omega_{{i^{*}},{j^{*}},q,{n^{*}},{p^{*}}}(x,t)\neq
0.$ (8.109)

Finally, for a given $({i^{*}},{k^{*}},{j^{*}},{n^{*}},{p^{*}})$, we count the
number of triples $(l^{*},w^{*},h^{*})$ such that ${n^{*}}\leq n$ and there
exists $(x,t)$ such that

$\zeta_{i,q,k,n,p,\vec{l}}(x,t)\zeta_{{i^{*}},q,{k^{*}},{n^{*}},{p^{*}},\vec{l}^{*}}(x,t)\neq
0.$ (8.110)

Recalling the definition of $\chi_{i,k,q}$ from (6.96) and (6.98), we see that
$\psi_{i,q}\chi_{i,{k^{*}},q}$ may have non-empty overlap with
$\psi_{i,q}\chi_{i,k,q}$ if and only if ${k^{*}}\in\\{k-1,k,k+1\\}$. Next,
from (6.19), we have that only $\psi_{i-1,q}$ and $\psi_{i+1,q}$ may overlap
with $\psi_{i,q}$. Now, let $(x,t)\in\mathrm{supp\,}\psi_{i,q}\chi_{i,k,q}$ be
given such that there exists $k_{i-1}$ such that

$\psi_{i,q}(x,t)\chi_{i,k,q}(t)\psi_{i-1,q}(x,t)\chi_{i-1,k_{i-1},q}(t)\neq
0.$

From the definition of $\chi_{i-1,k_{i-1},q}$, it is immediate that the
diameter of the support of $\chi_{i-1,k_{i-1},q}$ is _larger_ than the
diameter of the support of $\chi_{i,k,q}$. It follows that there can be at
most three values of ${k^{*}}$ (one of which is $k_{i-1}$) such that
$\chi_{i-1,{k^{*}},q}$ has non-empty overlap with $\chi_{i,k,q}$. Finally, let
$(x,t)\in\mathrm{supp\,}\psi_{i,q}\chi_{i,k,q}$ be given such that there
exists $k_{i+1}$ such that

$\psi_{i,q}(x,t)\chi_{i,k,q}(t)\psi_{i+1,q}(x,t)\chi_{i+1,k_{i+1},q}(t)\neq
0.$

From the definition of $\chi_{i+1,{k^{*}},q}$, there exists a constant
$\mathcal{C}_{\chi}$ depending on $\chi$ but not $i$, $q$, or ${k^{*}}$ such
that for all $|k^{\prime}|\geq\mathcal{C}_{\chi}\Gamma_{q+1}$

$\chi_{i+1,k_{i+1}+k^{\prime},q}(t)\chi_{i,k,q}(t)=0$

for all $t\in\mathbb{R}$. Therefore, the number of ${k^{*}}$ such that
$\chi_{i+1,{k^{*}},q}$ may have non-empty overlap with $\chi_{i,k,q}$ is no
more than $2\mathcal{C}_{\chi}\Gamma_{q+1}+1$. In summary, the number of pairs
$({i^{*}},{k^{*}})$ such that (8.108) holds for some $(x,t)$ is bounded above
by

$3+3+2\mathcal{C}_{\chi}\Gamma_{q+1}+1\leq 3\mathcal{C}_{\chi}\Gamma_{q+1}$
(8.111)

if $\lambda_{0}$ is sufficiently large, where the implicit constant is
independent of $q$ or any other parameters which index the cutoff functions.

Now let $({i^{*}},{k^{*}})$ be given such that
$\psi_{{i^{*}},q}\chi_{{i^{*}},{k^{*}},q}$ has nonempty overlap with
$\psi_{i,q}\chi_{i,k,q}$. Once values of ${n^{*}}$, ${p^{*}}$, and ${j^{*}}$
are chosen, these three parameters along with the value of ${i^{*}}$ uniquely
determine a stress cutoff function
$\omega_{{i^{*}},{j^{*}},q,{n^{*}},{p^{*}}}$. Since ${i^{*}}$ was fixed, we
may let ${j^{*}}$, ${n^{*}}$, and ${p^{*}}$ vary. Using that
${j^{*}}\leq{j_{\rm max}}\leq 4b/(\varepsilon_{\Gamma}(b-1))$ from (6.129),
${n^{*}}\leq{n_{\rm max}}$, ${p^{*}}\leq{p_{\rm max}}$ where ${n_{\rm max}}$,
and ${p_{\rm max}}$ are independent of $q$, the number of tuples
$({i^{*}},{k^{*}},{j^{*}},{n^{*}},{p^{*}})$ such that there exists $(x,t)$
with

$\psi_{i,q}(x,t)\chi_{i,k,q}(x,t)\omega_{i,j,q,n,p}(x,t)\psi_{{i^{*}},q}(x,t)\chi_{{i^{*}},{k^{*}},q}(x,t)\omega_{{i^{*}},{j^{*}},q,{n^{*}},{p^{*}}}(x,t)\neq
0$ (8.112)

is bounded by a dimensional constant multiplied by $\Gamma_{q+1}{n_{\rm
max}}{p_{\rm max}}4b/(\varepsilon_{\Gamma}(b-1))$.

Finally, fix a tuple $({i^{*}},{k^{*}},{j^{*}},{n^{*}},{p^{*}})$ such that
(8.112) holds at $(x,t)$. From (6.139), there exists
$\vec{l}^{*}=(l^{*},w^{*},h^{*})$ such that
$\zeta_{{i^{*}},q,{k^{*}},{n^{*}},\vec{l}^{*}}(x,t)\neq 0$. From (6.141),
(6.108), and the fact that ${n^{*}}\leq n$, there exists a dimensional
constant $\mathcal{C}_{\zeta}$ such at most $\mathcal{C}_{\zeta}$ of the
checkerboard cutoffs neighboring
$\zeta_{{i^{*}},q,{k^{*}},{n^{*}},\vec{l}^{*}}$ can intersect the support of
$\zeta_{i,q,k,n,\vec{l}}$. Since all Lagrangian trajectories originating at
$(x,t)$ follow the same velocity field $v_{\ell_{q}}$ and the checkerboard
cutoffs are precomposed with Lagrangian flows, this property is preserved in
time. Thus we have shown that for each tuple
$({i^{*}},{k^{*}},{j^{*}},{n^{*}},{p^{*}})$, the number of associated tuples
$(l^{*},w^{*},h^{*})$ such that
$\zeta_{{i^{*}},q,{k^{*}},{n^{*}},\vec{l}^{*}}$ can have nonempty intersection
with $\zeta_{i,q,k,n,\vec{l}}$ is bounded by a dimensional constant
independent of $q$.

Combining the preceding arguments, we obtain that the number of cutoff
functions $\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{p^{*}},\vec{l}^{*}}$ which
may overlap nontrivially with $\eta_{i,j,k,q,n,p,\vec{l}}$ is bounded by at
most a dimensional constant multiplied by $\Gamma_{q+1}{n_{\rm max}}{p_{\rm
max}}4b/(\varepsilon_{\Gamma}(b-1))$, finishing the proof. ∎

######  Lemma 8.9.

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

$|x-y|\geq\mathcal{C}_{*}\left(\Gamma_{q}\lambda_{q}\right)^{-1}.$ (8.113)

###### Proof Lemma 8.9.

Let $L(x,y)$ be the line segment connecting $x$ and $y$. From (6.36), we have
that for $z\in L(x,y)$ (in fact for all $z\in\mathbb{T}^{3}$),

$\left|\nabla\psi_{i,q}(z)\right|\lesssim\psi_{i,q}^{1-\frac{1}{\mathsf{N}_{\rm
fin}}}(z)\lambda_{q}\Gamma_{q}.$ (8.114)

Thus we can write

$\displaystyle\frac{1}{8}\leq\left|\psi_{i,q}^{2}(x,t)-\psi_{i,q}^{2}(y,t)\right|$
$\displaystyle\leq 2\left|\psi_{i,q}(x)-\psi_{i,q}(y)\right|$
$\displaystyle\leq
2\left|\int_{0}^{1}\nabla\psi_{i,q}(x+t(y-x))\cdot(y-x)\,dt\right|$
$\displaystyle\leq 2|x-y|\left\|\nabla\psi_{i,q}\right\|_{L^{\infty}}$
$\displaystyle\lesssim\Gamma_{q}\lambda_{q}|x-y|,$

and (8.113) follows. ∎

######  Lemma 8.10.

Consider cutoff functions

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

where $n^{*}\leq n$ and $\eta$ and $\eta^{*}$ overlap as in Lemma 8.8. Let
$t^{*}\in\mathrm{supp\,}\chi_{{i^{*}},{k^{*}},q}$ be given. Then there exists
a convex set $\Omega:=\Omega(\eta,\eta^{*},t^{*})$ with diameter
$\lambda_{q,n,0}^{-1}\Gamma_{q+1}$ such that

$\left(\mathrm{supp\,}\zeta_{i,k,q,n,\vec{l}}\cap\\{t=t^{*}\\}\right)\subset\Omega\subset\mathrm{supp\,}\psi_{i\pm,q}.$
(8.115)

###### Proof of Lemma 8.10.

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
6.24 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

$|X(x,t^{*})-y|\leq\lambda_{q,n,0}^{-1}\Gamma_{q+1}\leq\widetilde{\lambda}_{q}^{-1}<\mathcal{C}_{*}\widetilde{\lambda}_{q}^{-1}$

for $\mathcal{C}_{*}$ given in (8.113), where we have used the definitions of
$\lambda_{q,n,0}$ in (9.26), (9.27), and (9.28). Then from Lemma 8.9, it
cannot be the case that $\psi_{i^{\prime},q}^{2}(t^{*},y)\leq\frac{1}{8}$, and
so

$\displaystyle
y\in\mathrm{supp\,}\psi_{i^{\prime},q}\cap\\{t=t^{*}\\}\subset\mathrm{supp\,}\psi_{i\pm,q}\cap\\{t=t^{*}\\}\,.$
(8.116)

Since $y$ is arbitrary, we conclude that the ball of radius
$\Gamma_{q+1}\lambda_{q,n,0}^{-1}$ is contained in
$\mathrm{supp\,}\psi_{i\pm,q}\cap\\{t=t^{*}\\}$. We let
$\Omega(\eta,\eta^{*},t^{*})$ to be precisely this ball (hence a convex set).
Since $D_{t,q}\zeta_{i,k,q,n,\vec{l}}=0$ and
$(x,t_{0})\in\mathrm{supp\,}\zeta_{i,k,q,n,\vec{l}}$, we have that
$X(x,t^{*})\in\mathrm{supp\,}\zeta_{i,k,q,n,\vec{l}}\cap\\{t=t^{*}\\}$. Then,
recalling that the support of $\zeta_{i,k,q,n,\vec{l}}$ must obey the diameter
bound in (6.141) on the support of $\widetilde{\chi}_{i,k,q}$, which contains
the support of $\chi_{{i^{*}},{k^{*}},q}$ by (6.103), we conclude that

$\mathrm{supp\,}\zeta_{i,k,q,n,\vec{l}}\cap\\{t=t^{*}\\}\subset\Omega\,.$
(8.117)

Combining (8.116) and (8.117) concludes the proof of the lemma. ∎

######  Lemma 8.11.

As in Lemma 8.8, consider cutoff functions

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

Let $t^{*}\in\mathrm{supp\,}\chi_{{i^{*}},{k^{*}},q}$ be such that
$\Phi^{*}:=\Phi_{({i^{*}},{k^{*}})}$ is the identity at time $t^{*}$. Using
Lemma 8.10, define $\Omega:=\Omega(\eta,\eta^{*},t^{*})$. Define
$\Omega(t):=\Omega(\eta,\eta^{*},t^{*},t):=X(\Omega,t)$, where
$X(\cdot,t^{*})$ is the identity.

  1. (1)

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

$\mathrm{supp\,}\eta(\cdot,t)\subset\Omega(t)\subset\mathrm{supp\,}\psi_{i\pm,q}.$
(8.118)

  2. (2)

Let
$\mathbb{W}^{*}\circ\Phi^{*}:=\mathbb{W}_{\xi*,q+1,n*}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},\vec{l}^{*}}\circ\Phi_{({i^{*}},{k^{*}})}$
be an intermittent pipe flow supported on $\eta^{*}$. Then there exists a
geometric constant $\mathcal{C}_{\textnormal{pipe}}$ such that

$\left(\mathrm{supp\,}\mathbb{W}^{*}\circ\Phi^{*}\cap\\{t=t^{*}\\}\cap\Omega\right)\subset\bigcup_{n=1}^{N}S_{n},$

where the sets $S_{n}$ are cylinders concentrated around line segments $A_{n}$
for $n\in\\{1,...,N\\}$ with

$N\leq\mathcal{C}_{\textnormal{pipe}}\left(\frac{\lambda_{q,n}}{\lambda_{q,n,0}\Gamma_{q+1}^{-1}}\right)^{2}.$
(8.119)

  3. (3)

$\mathbb{W}^{*}\circ\Phi^{*}(\cdot,t)$ and the associated axes $A_{n}(t)$ and
sets $S_{n}(t)$ satisfy the conclusions of Lemma 4.7 on the set $\Omega(t)$
for $t\in\mathrm{supp\,}\chi_{i,k,q}$.

###### Proof of Lemma 8.11.

From the previous lemma, we have that for all $y\in\Omega$,
$\psi_{i\pm,q}^{2}(y,t^{*})\geq\nicefrac{{1}}{{8}}$. Applying Lemma 6.24, we
have that for all $t$ with
$|t-t^{*}|\leq\tau_{q}\Gamma_{q+1}^{-i+5+\mathsf{c_{0}}}$, the Lagrangian flow
originating from $(y,t^{*})$ has the property that

$\psi_{i\pm,q}^{2}(t,X(y,t))\geq\nicefrac{{1}}{{16}}\,.$ (8.120)

Recalling from (6.102) that the diameter of the support of
$\widetilde{\chi}_{{i^{*}},{k^{*}},q}$ is
$\tau_{q}\Gamma_{q+1}^{-{i^{*}}+\mathsf{c_{0}}}$ and that $i-1\leq{i^{*}}\leq
i+1$, we have that in particular the Lagrangian flow originating at
$(y,t^{*})$ satisfies (8.120) for all
$t\in\mathrm{supp\,}\widetilde{\chi}_{{i^{*}},{k^{*}},q}$. From (6.103),
(8.120) is then satisfied in particular for all
$t\in\mathrm{supp\,}\chi_{i,k,q}$, thus proving the second inclusion from
(8.118). To prove the first inclusion, we use (8.115), the definition of
$\Omega(t)$, and the equality $D_{t,q}\zeta_{i,k,q,n,\vec{l}}=0$ to deduce
that

$\mathrm{supp\,}\zeta_{i,k,q,n,\vec{l}}(\cdot,t)\subset\Omega(t),$

finishing the proof of (8.118).

To prove the second claim, recall that $\mathbb{W}^{*}\circ\Phi^{*}$ at
$t=t^{*}$ is periodic to scale $\lambda_{q,{n^{*}}}^{-1}$ for ${n^{*}}\leq n$,
and the diameter of $\Omega$ is $2\lambda_{q,n,0}^{-1}\Gamma_{q+1}$ (in fact
$\Omega$ is a ball). Considering the quotient of the respective diameters
squared, the claim then follows after absorbing the geometric constant
$n_{\xi}^{*}$ from Proposition 4.3 into $\mathcal{C}_{\textnormal{pipe}}$.

To see that we may apply Lemma 4.7, first note that $\Omega=\Omega(t^{*})$ is
convex by construction, and so the first assumption of Lemma 4.7 is met. We
choose $v=v_{\ell_{q}}$ and $X$ and $\Phi$ to be the associated backwards and
forwards flows originating from $t_{0}=t^{*}$. From (6.60), (8.118), and
(9.19), we have that for $t\in\mathrm{supp\,}\chi_{i,k,q}$ and
$x\in\Omega(t)$,

$\left|\nabla
v_{\ell_{q}}(x,t)\right|\lesssim\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}\Gamma_{q+1}^{i+2}=\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{i+7},$
(8.121)

and so (4.21) is satisfied with $C=i+7$. Recall again from (6.103) that
$\mathrm{supp\,}\widetilde{\chi}_{{i^{*}},{k^{*}},q}$ contains the support of
$\chi_{i,k,q}$, and that from (6.102) the support of
$\widetilde{\chi}_{{i^{*}},{k^{*}},q}$ has diameter
$\tau_{q}\Gamma_{q+1}^{-{i^{*}}+\mathsf{c_{0}}}$. We then use (9.39) and
(9.19) to write that for any
$t\in\mathrm{supp\,}\widetilde{\chi}_{{i^{*}},{k^{*}},q}$ we have

$\displaystyle|t-t^{*}|\leq\tau_{q}\Gamma_{q+1}^{-{i^{*}}+\mathsf{c_{0}}+1}$
$\displaystyle\leq\tau_{q}\Gamma_{q+1}^{-i+\mathsf{c_{0}}+2}$
$\displaystyle\leq\left(\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}\Gamma_{q+1}^{\mathsf{c_{0}}+6}\right)^{-1}\Gamma_{q+1}^{-i+\mathsf{c_{0}}+2}$
$\displaystyle=\left(\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{\mathsf{c_{0}}+11}\right)^{-1}\Gamma_{q+1}^{-i+\mathsf{c_{0}}+2}$
$\displaystyle\leq\left(\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{i+9}\right)^{-1},$

so that (4.20) is satisfied since $C+2=i+9$. We can now apply Lemma 4.7,
concluding the proof of the Lemma. ∎

####  8.7.2 Applying Proposition 4.8

######  Lemma 8.12.

The Type 2 oscillation errors vanish. More specifically,

  1. (1)

When ${\widetilde{n}}=0$, the Type 2 errors identified in (8.43) vanish.

  2. (2)

When $1\leq{\widetilde{n}}\leq{n_{\rm max}}-1$, the Type 2 errors identified
in (8.55) and (8.56) vanish.

  3. (3)

When ${\widetilde{n}}={n_{\rm max}}$, the Type 2 errors identified in (8.71)
and (8.72) vanish.

###### Proof of Lemma 8.12.

We first recall what the Type $2$ oscillation errors are. When
${\widetilde{n}}=0$, the errors identified in (8.43) can be written using
(8.30) as

$\mathcal{O}_{0,2}=\sum_{\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,0}\circ\Phi_{(i,k)}\right)\otimes\mathrm{curl\,}\left(a_{({\xi^{*}})}\nabla\Phi_{({i^{*}},{k^{*}})}^{T}\mathbb{U}_{{\xi^{*}},q+1,0}\circ\Phi_{({i^{*}},{k^{*}})}\right)\,,$
(8.122)

where the notation $\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}$ is defined in
(8.29) and denotes summation over all pairs of cutoff function indices for
which at least one parameter differs between the two pairs. When
$1\leq{\widetilde{n}}\leq{n_{\rm max}}$, the Type 2 errors identified in
(8.55) and (8.71) can be written as

$\displaystyle
2\sum_{n^{\prime}\leq{\widetilde{n}}-1}w_{q+1,{\widetilde{n}}}\otimes_{\textnormal{s}}w_{q+1,n^{\prime}}=2$
$\displaystyle\sum_{{n^{*}}\leq{\widetilde{n}}-1}\sum_{\xi,i,j,k,{\widetilde{p}},\vec{l}}\sum_{{\xi^{*}},{i^{*}},{j^{*}},{k^{*}},{p^{*}},\vec{l}^{*}}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)$
$\displaystyle\qquad\qquad\otimes_{\textnormal{s}}\mathrm{curl\,}\left(a_{({\xi^{*}})}\nabla\Phi_{({i^{*}},{k^{*}})}^{T}\mathbb{U}_{{\xi^{*}},q+1,{n^{*}}}\circ\Phi_{({i^{*}},{k^{*}})}\right)\,.$
(8.123)

When $1\leq{\widetilde{n}}\leq{n_{\rm max}}$, the Type 2 errors identified in
(8.56) and (8.72) can be written as

$\sum_{\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}}\mathrm{curl\,}\left(a_{(\xi)}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\right)\otimes\mathrm{curl\,}\left(a_{({\xi^{*}})}\nabla\Phi_{({i^{*}},{k^{*}})}^{T}\mathbb{U}_{{\xi^{*}},q+1,{\widetilde{n}}}\right)\,,$
(8.124)

where the notation $\neq\\{\xi,i,j,k,{\widetilde{p}},\vec{l}\\}$ has been
reused from (8.29). To show that the errors defined in (8.122), (8.7.2), and
(8.124) vanish, it suffices to show the following. For pairs of cutoff
functions $\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}$ and
$\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{p^{*}},\vec{l}^{*}}$ satisfying the
two conditions in Lemma 8.8, and vectors $\xi,{\xi^{*}}\in\Xi$,

$\displaystyle\mathrm{supp\,}\left(\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{i,j,k,{\widetilde{n}},{\widetilde{p}},\vec{l}}\circ\Phi_{(i,k)}\right)\cap\mathrm{supp\,}\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}$
$\displaystyle\qquad\qquad\qquad\cap\mathrm{supp\,}\left(\mathbb{W}_{{\xi^{*}},q+1,{n^{*}}}^{{i^{*}},{j^{*}},{k^{*}},{n^{*}},{p^{*}},\vec{l}^{*}}\circ\Phi_{({i^{*}},{k^{*}})}\right)\cap\mathrm{supp\,}\eta_{{i^{*}},{j^{*}},{k^{*}},q,{n^{*}},{p^{*}},\vec{l}^{*}}=\emptyset.$
(8.125)

The proof of this claim will proceed by fixing ${\widetilde{n}}$, using the
preliminary estimates, and applying Proposition 4.8.

Let ${\widetilde{n}}$ be fixed and assume that $w_{q+1,n^{\prime}}$ for
$n^{\prime}<{\widetilde{n}}$ has been defined (when ${\widetilde{n}}=0$, this
assumption is vacuous). In particular, placements have been chosen for all
intermittent pipe flows indexed by $n^{\prime}$. Now, consider all the cutoff
functions $\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}$ utilized at
stage ${\widetilde{n}}$. Since the parameters indexing the cutoff functions
are countable, we may choose _any_ ordering of the tuples
$(i,j,k,{\widetilde{p}},\vec{l})$ at level ${\widetilde{n}}$. Combined with an
ordering of the direction vectors $\xi\in\Xi$, we thus have an ordering of the
cutoff functions $\eta_{i,j,k,q,{\widetilde{n}},{\widetilde{p}},\vec{l}}$ and
the associated intermittent pipe flows
$\mathbb{W}_{\xi,q+1,{\widetilde{n}}}^{i,j,k,{\widetilde{n}},{\widetilde{p}},\vec{l}}\circ\Phi_{(i,k)}$.

To ease notation, we will abbreviate the 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 4.8
inductively on $z$ such that the following two conditions hold. Our goal is to
place the pipe flow $(\mathbb{W}\circ\Phi)_{z}$ such that

$\mathrm{supp\,}(\mathbb{W}\circ\Phi)_{z^{\prime}}\cap\mathrm{supp\,}(\mathbb{W}\circ\Phi)_{z}\cap\mathrm{supp\,}\eta_{z}=\emptyset\,,$
(8.126)

for all $z^{\prime}<z$, and such that

$\mathrm{supp\,}w_{q+1,n^{\prime}}\cap\mathrm{supp\,}(\mathbb{W}\circ\Phi)_{z}\cap\mathrm{supp\,}\eta_{z}=\emptyset\,,$
(8.127)

for all $n^{\prime}<{\widetilde{n}}$. The first condition shows that all Type
2 errors such as (8.122) and (8.124) which arise from two sets of pipes both
indexed by ${\widetilde{n}}$ vanish, while the second condition shows that the
Type 2 errors which arise from pipes indexed by $n^{\prime}<{\widetilde{n}}$
interacting with pipes indexed by ${\widetilde{n}}$ vanish, such as (8.7.2).

Throughout the rest of the proof, $z^{\prime}$ will only ever denote an
integer less than $z$ such that $\eta_{z}$ and $\eta_{z^{\prime}}$ overlap.
Although we have suppressed the indices, note that $\eta_{z^{\prime}}$ and
$\eta_{z}$ both correspond to the index ${\widetilde{n}}$. Conversely, let
$\eta_{z^{\prime\prime}}$ denote a generic cutoff function indexed by
$n^{\prime}$ which overlaps with $\eta_{z}$. By Lemma 8.8, there exists a
geometric constant $\mathcal{C}_{\eta}$ such that the number of cutoff
functions $\eta_{z^{\prime}}$ or $\eta_{z^{\prime\prime}}$ which overlap with
$\eta_{z}$ is bounded above by $\mathcal{C}_{\eta}\Gamma_{q+1}$. Let
$t_{z^{\prime}}\in\mathrm{supp\,}\chi_{i_{z^{\prime}},k_{z^{\prime}},q}$ be
the time for which $\Phi_{i_{z^{\prime}},k_{z^{\prime}},q}$ is the identity,
and let $\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}}\right)$ be the
convex set constructed in Lemma 8.10, where we have set
$t^{*}=t_{z^{\prime}}$. Let
$\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t\right)$ denote the
image of $\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}}\right)$ under
this flow, as defined in Lemma 8.11. We then have that the set

$\mathrm{supp\,}(\mathbb{W}\circ\Phi)_{z^{\prime}}\cap\mathrm{supp\,}\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}}\right)\cap\\{t=t_{z^{\prime}}\\}$
(8.128)

is contained in the union of sets $S_{n}^{z^{\prime}}$ concentrated around
axes $A_{n}^{z^{\prime}}$ for

$n\leq\mathcal{C}_{\textnormal{pipe}}\Gamma_{q+1}^{2}\frac{\lambda_{q,{\widetilde{n}}}^{2}}{\lambda_{q,{\widetilde{n}},0}^{2}}\,,$

and the flowed axes $A_{n}^{z^{\prime}}$ and pipes of
$(\mathbb{W}\circ\Phi)_{z^{\prime}}$ satisfy the conclusions of Lemma 4.7.
Furthermore, substituting $z^{\prime\prime}$ for $z^{\prime}$ in the preceding
discussion, all the analogous definitions and conclusions can be made for
cutoff functions $\eta_{z^{\prime\prime}}$ and pipe flows
$(\mathbb{W}\circ\Phi)_{z^{\prime\prime}}$.

We will apply Proposition 4.8 with the following choices. Let $t_{z}$ be the
time at which the flow map $\Phi_{i,k,q}$ corresponding to $\eta_{z}$ is the
identity. Set

$\Omega=\left(\bigcup_{z^{\prime}<z}\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t_{z}\right)\right)\bigcup\left(\bigcup_{n^{\prime}<{\widetilde{n}}}\Omega\left(\eta_{z},\eta_{z^{\prime\prime}},t_{z^{\prime\prime}},t_{z}\right)\right)$
(8.129)

and set

$\displaystyle
r_{1}=\Gamma_{q+1}^{-1}\frac{\lambda_{q,{\widetilde{n}},0}}{\lambda_{q+1}}=\begin{cases}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}-1}\cdot\frac{5}{6}}\Gamma_{q+1}^{-1}&\mbox{if
}{\widetilde{n}}\geq 2\\\
\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\frac{4}{5}}\Gamma_{q+1}^{-1}&\mbox{if
}{\widetilde{n}}=1\\\ \frac{\widetilde{\lambda}_{q}}{\lambda_{q+1}}&\mbox{if
}{\widetilde{n}}=0.\end{cases}$ (8.130)

We have used here the definitions of $\lambda_{q,{\widetilde{n}},0}$ given in
(9.27), (9.26), and (9.28). Note that by (8.118),
$\mathrm{supp\,}\eta_{z}(\cdot,t_{z})\subset\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t_{z}\right)$
for each $z^{\prime}<z$, with the analogous inclusion holding when
$z^{\prime}$ is replaced by $z^{\prime\prime}$. In particular, we have that
$\mathrm{supp\,}\eta_{z}(\cdot,t_{z})\subset\Omega$. Furthermore, we have
additionally from Lemma 8.11 that Lemma 4.7 may be applied on $\Omega(t)$ for
all $t\in\chi_{i,k,q}$. Thus, the diameter of
$\Omega(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t_{z})$ satisfies

$\textnormal{diam}\left(\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t_{z}\right)\right)\leq(1+\Gamma_{q+1}^{-1})\textnormal{diam}\left(\Omega(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}})\right)=2(1+\Gamma_{q+1}^{-1})\lambda_{q,{\widetilde{n}},0}^{-1}\Gamma_{q+1}$
(8.131)

Using that the diameter of the support of $\eta_{z}(\cdot,t_{z})$ is bounded
by a dimensional constant times $\lambda_{q,{\widetilde{n}},0}^{-1}$ from
(6.141) and recalling that
$\mathrm{supp\,}\eta_{z}(\cdot,t_{z})\subset\Omega\left(\eta_{z},\eta_{z^{\prime}},t_{z^{\prime}},t_{z}\right)$
with the analogous conclusion holding for $z^{\prime\prime}$, we have that

$\displaystyle\textnormal{diam}(\Omega)$ $\displaystyle\leq
4(1+\Gamma_{q+1}^{-1})\lambda_{q,{\widetilde{n}},0}^{-1}\Gamma_{q+1}+\Gamma_{q+1}\lambda_{q,{\widetilde{n}},0}^{-1}$
$\displaystyle\leq
6(1+\Gamma_{q+1}^{-1})\Gamma_{q+1}\left(\lambda_{q,{\widetilde{n}},0}\right)^{-1}$
$\displaystyle\leq 16(\lambda_{q+1}r_{1})^{-1}$

for each value of ${\widetilde{n}}$ from (8.130), and so (4.28) is satisfied.

Now set

$\mathcal{C}_{A}=\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1},\qquad
r_{2}=r_{q+1,n}\approx\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}},$

where above we have appealed to (9.23) and (9.25). By (8.119) and Lemma 8.8,
the total number of pipes contained in $\Omega$ is no more than

$\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}^{3}\frac{\lambda_{q,{\widetilde{n}}}^{2}}{\lambda_{q,{\widetilde{n}},1}^{2}}.$

Then we can write

$\displaystyle\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}^{3}\frac{\lambda_{q,{\widetilde{n}}}^{2}}{\lambda_{q,{\widetilde{n}},0}^{2}}=\mathcal{C}_{A}\frac{r_{2}^{2}}{r_{1}^{2}},$

and so (4.29) is satisfied. Furthermore, the assumptions on the axes and the
neighborhoods of the axes required by Proposition 4.8 follow from Lemma 8.11,
which allows us to appeal to the conclusions of Lemma 4.7. Finally, from
(9.58a), we have that for ${\widetilde{n}}\geq 2$,

$\displaystyle C_{*}\mathcal{C}_{A}r_{2}^{4}\leq
16C_{*}\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}\cdot
4}\leq\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}-1}\cdot\frac{5}{6}\cdot
3}\Gamma_{q+1}^{-3}=r_{1}^{3},$ (8.132)

showing that (4.31) is satisfied for ${\widetilde{n}}\geq 2$. In the cases
${\widetilde{n}}=0$ and ${\widetilde{n}}=1$, the desired inequalities follow
from (8.130) and (9.58b) and (9.58c), and so we have checked that (4.31) is
satisfied for all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$. Then from the
conclusion (4.32) of Proposition 4.8, we have that on the support of $\Omega$,
which in particular contains the support of $\eta_{z}(\cdot,t_{z})$ from
(8.118), we can choose the support of $(\mathbb{W}\circ\Phi)_{z}$ to be
disjoint from the support of $(\mathbb{W}\circ\Phi)_{z^{\prime}}$ and
$(\mathbb{W}\circ\Phi)_{z^{\prime\prime}}$ for all overlapping
$z^{\prime\prime}$ and $z^{\prime}$. Then since
$D_{t,q}(\mathbb{W}\circ\Phi)_{z}=D_{t,q}(\mathbb{W}\circ\Phi)_{z^{\prime}}=D_{t,q}(\mathbb{W}\circ\Phi)_{z^{\prime\prime}}=0$,
(8.126) and (8.127) are satisfied, concluding the proof. ∎

###  8.8 Divergence corrector errors

######  Lemma 8.13.

For all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$,
$1\leq{\widetilde{p}}\leq{p_{\rm max}}$, and $j\in\\{2,3\\}$, the divergence
corrector errors $\mathcal{O}_{{\widetilde{n}},1,j}$ satisfy

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

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

###### Proof of Lemma 8.13.

The divergence corrector errors are given in (8.31), (8.49), and (8.66). The
estimates for $j=\\{2,3\\}$ are each similar, and so we shall only prove the
case $j=2$. Thus we estimate

$\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\sum_{\xi,i^{\prime},j,k,{\widetilde{p}},\vec{l}}\left(\left(a_{(\xi)}\nabla\Phi_{(i^{\prime},k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i^{\prime},k)}\right)\otimes\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i^{\prime},k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i^{\prime},k)}\right)\right)\right)\right\|_{L^{1}}.$
(8.133)

Recall that $\xi$ takes only six distinct values and that $j\leq{j_{\rm
max}}$, ${\widetilde{p}}\leq{p_{\rm max}}$ are bounded independently of $q$.
Furthermore, on the support of $\psi_{i,q}$, only $\psi_{i-1,q}$,
$\psi_{i,q}$, and $\psi_{i+1,q}$ are non-zero from (6.19). As a result, only
time cutoffs $\chi_{i-1,k,q}$, $\chi_{i,k,q}$, and $\chi_{i+1,k,q}$ may be
non-zero. Since for each $i$ the $\chi_{i,k,q}$’s form a partition of unity in
time for which only two cutoff functions are non-zero at any fixed time, for
every time, the sum in (8.133) is a finite sum for which the number of non-
zero terms in the summand is bounded independently of $q$. Similarly, the sum
over $\vec{l}$ forms a partition of unity which only finitely many cutoff
functions overlap at any fixed point in space and time. Therefore we may
absorb the effects of $\xi$, $j$, $k$, ${\widetilde{p}}$, and $\vec{l}$ in the
implicit constant in the inequality.

Using Hölder’s inequality and estimates (8.16) and (8.17) from Corollary 8.2
with $r=2$, $r_{2}=1$, and $r_{1}=\infty$, we have that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9\right)\rfloor$,

$\displaystyle\sum_{\xi,i^{\prime},j,k,{\widetilde{p}},\vec{l}}\left\|\psi_{i,q}D^{k}D_{t,q}^{m}\left(\left(a_{(\xi)}\nabla\Phi_{(i^{\prime},k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i^{\prime},k)}\right)\otimes\left(\nabla
a_{(\xi)}\times\left(\nabla\Phi_{(i^{\prime},k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i^{\prime},k)}\right)\right)\right)\right\|_{L^{1}}$
$\displaystyle\qquad\lesssim\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)\frac{\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}{\lambda_{q+1}}$
$\displaystyle\qquad\lesssim\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\lambda_{q+1}^{k}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$

which proves the desired estimate after recalling that for all
${\widetilde{n}}$,

$\displaystyle\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9\right)\rfloor$
$\displaystyle\geq 3\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\frac{\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}{\lambda_{q+1}}$
$\displaystyle\leq\delta_{q+2}\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}$
$\displaystyle-\mathsf{c}_{\widetilde{\textnormal{n}}}+4$ $\displaystyle\leq
1\,,$

which follow from (9.60b), (9.34) and (9.54), and (9.42), respectively. ∎

###  8.9 Time support of perturbations and stresses

First, we prove (7.12). Indeed, appealing to (5.1), which defines
$\mathring{R}_{\ell_{q}}$ in terms of a mollifier applied to
$\mathring{R}_{q}$, (9.20), which defines the scale at which
$\mathring{R}_{q}$ is mollified, and (6.104), which ensures that the time
support of $w_{q+1,0}$ is only enlarged relative to the time support of
$\mathring{R}_{\ell_{q}}$ by
$2\left(\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{2}\right)^{-1}$,
we achieve (7.12). To prove (7.14) and (7.16), first note that application of
the inverse divergence operators $\mathcal{H}$ and $\mathcal{R}^{*}$
_commutes_ with multiplication by $\overline{\chi}_{q,n,p}$.383838This is
simple to check from the formula given in Proposition A.17 and the formula for
the standard nonlocal inverse divergence operator given in (A.100), both of
which involve operations which are purely spatial, such as differentiation and
application of Fourier multipliers. Then by the definition of
$\mathring{R}_{q+1}^{0}$ and $\mathring{H}_{q,n,p}^{0}$ in Section 8.3, we
achieve (7.14) and (7.16). Proving the inclusions in (7.19), (7.21), (7.23),
(7.26), (7.28), and (7.30), follows similarly from (6.104), the properties of
$\mathcal{H}$ and $\mathcal{R}^{*}$, and the definitions of
$\mathring{R}_{q+1}^{\widetilde{n}}$ and
$\mathring{H}_{q,n,p}^{\widetilde{n}}$ in Section 8.3. Finally, to see that
(7.4) follows from the inclusions already demonstrated, notice that the
threshold in (7.4) is weaker than any of the previous inclusions by a factor
of $\Gamma_{q+1}$, and so we may allow the time support of
$\mathring{R}_{q+1}^{\widetilde{n}}$ to expand slightly as ${\widetilde{n}}$
increases from $0$ to ${n_{\rm max}}$ while still meeting the desired
inclusion.

##  9 Parameters

The purpose of this section is to provide an exhaustive delineation of the
many parameters, inequalities, and notations which arise throughout the bulk
of the paper. In Section 9.1, we define the $q$-independent parameters _in
order_ , beginning with the regularity index $\beta$, and ending with the
number $a_{*}$, which will be used to absorb every implicit constant
throughout the paper. Then in Section 9.2, we define the parameters which
depend on $q$, as well as the parameters which depend in addition on $n$ and
$p$. The definitions of both the $q$-independent and $q$-dependent parameters
will appear rather arbitrary, but are justified in Section 9.3. This section
contains, in no particular order, consequences of the definitions made in the
previous two sections which are necessary to close the estimates in the proof.
Finally, Sections 9.4 and 9.5 contain the definitions of a few operators and
some notations that are used throughout the paper.

###  9.1 Definitions and hierarchy of the parameters

The parameters in our construction are chosen as follows:

  1. (i)

Choose an arbitrary regularity parameter
$\beta\in[\nicefrac{{1}}{{3}},\nicefrac{{1}}{{2}})$. In light of [11, 43],
there is no reason to consider the regime $\beta<\nicefrac{{1}}{{3}}$.

  2. (ii)

Choose $b\in(1,\nicefrac{{3}}{{2}})$ sufficiently small such that

$\displaystyle 2\beta b$ $\displaystyle<1\,.$ (9.1)

The heuristic reason for (9.1) is given by (2.4.1). Note that (9.1) and the
inequality $\beta<\nicefrac{{1}}{{2}}$ imply that
$\beta(2b+1)<\nicefrac{{3}}{{2}}$, which is a required inequality for the
heuristic estimate (2.22).

  3. (iii)

With $\beta$ and $b$ chosen, we may now designate a number of parameters:

    1. (a)

The parameter ${n_{\rm max}}$, which per Section 2.4.2 denotes the total
number of higher order stresses $\mathring{R}_{q,n}$ and thus primary
frequency divisions in between $\lambda_{q}$ and $\lambda_{q+1}$, is defined
as the smallest integer for which

$\displaystyle 1-2\beta b>\frac{5}{6}\left(\frac{4}{5}\right)^{{n_{\rm
max}}-1}\,.$ (9.2)

    2. (b)

The parameter ${p_{\rm max}}$, which per Section 2.4.2 denotes the total
number of subdivided components $\mathring{R}_{q,n,p}$ of a higher order
stress $\mathring{R}_{q,n}$ and thus secondary frequency divisions in between
$\lambda_{q}$ and $\lambda_{q+1}$, is defined as the smallest integer for
which

$\displaystyle\frac{1}{{p_{\rm max}}}<\frac{1-2\beta b}{10}\,.$ (9.3)

    3. (c)

The parameter ${\mathsf{C}_{b}}$ appearing in (3.21) is use to quantify the
$L^{1}$ norm of the velocity cutoff functions $\psi_{i,q}$. It is defined as

$\displaystyle{\mathsf{C}_{b}}=\frac{b+4}{b-1}\,.$ (9.4)

    4. (d)

The exponent $\mathsf{C_{R}}$ is used in order to define a small parameter in
the estimate for the Reynolds stress, cf. (3.15). This parameter is then used
in the proof to absorb geometric constants in the construction. It is defined
as

$\displaystyle\mathsf{C_{R}}=4b+1\,.$ (9.5)

  4. (iv)

The parameter $\mathsf{c_{0}}$, which is first introduced in (3.20) and
utilized in Sections 7 and 8 to control small losses in the sharp material
derivative estimates, is defined in terms of ${n_{\rm max}}$ as

$\displaystyle\mathsf{c_{0}}=4{n_{\rm max}}+5\,.$ (9.6)

  5. (v)

The parameter $\varepsilon_{\Gamma}>0$, which is used in (9.18) to quantify
the _finest_ frequency scale between $\lambda_{q}$ and $\lambda_{q+1}$
utilized throughout the scheme, is defined as the greatest real number for
which the following inequalities hold

$\displaystyle\varepsilon_{\Gamma}\Big{(}7+\mathsf{C_{R}}+{n_{\rm
max}}(8+{\mathsf{C}_{b}}))\Big{)}$ $\displaystyle<\frac{1-2\beta}{10}$ (9.7a)
$\displaystyle\varepsilon_{\Gamma}$
$\displaystyle<\frac{1}{100}\left(\frac{4}{5}\right)^{{n_{\rm max}}-1}$ (9.7b)
$\displaystyle\varepsilon_{\Gamma}$ $\displaystyle<\frac{b}{9(b-1)}$ (9.7c)
$\displaystyle 2b\varepsilon_{\Gamma}(\mathsf{c_{0}}+7)$
$\displaystyle<1-\beta\,.$ (9.7d)

  6. (vi)

The parameter $\alpha_{\mathsf{R}}>0$ from the $L^{1}$ loss of the inverse
divergence operator is now defined as

$\displaystyle\alpha_{\mathsf{R}}=\frac{\varepsilon_{\Gamma}(b-1)}{2b}\,.$
(9.8)

  7. (vii)

The parameters $\mathsf{N}_{\rm cut,t}$ and $\mathsf{N}_{\rm cut,x}$ are used
in Section 6 in order to define the velocity and stress cutoff functions.
$\mathsf{N}_{\rm cut,x}$ is the number of space derivatives which are embedded
into the definitions of these cutoff functions, while $\mathsf{N}_{\rm cut,t}$
is the number of material derivatives. See (6.6), (6.14), and (6.119). These
large parameters are chosen solely in terms of $b$ and $\varepsilon_{\Gamma}$
as

$\displaystyle\frac{1}{2}\mathsf{N}_{\rm cut,x}=\mathsf{N}_{\rm
cut,t}=\left\lceil\frac{3b}{\varepsilon_{\Gamma}(b-1)}+\frac{15b}{2}\right\rceil\,.$
(9.9)

  8. (viii)

The parameter $\mathsf{N}_{\textnormal{ind,t}}$, which is the number of sharp
material derivatives propagated on stresses and velocities in Sections 3
through 8, is chosen as the smallest integer for which we have

$\displaystyle\mathsf{N}_{\textnormal{ind,t}}=\left\lceil\frac{4}{\varepsilon_{\Gamma}(b-1)}\right\rceil\mathsf{N}_{\rm
cut,t}\,.$ (9.10)

  9. (ix)

The parameter $\mathsf{N}_{\textnormal{ind,v}}$, whose primary role is to
quantify the number of sharp space derivatives propagated on the velocity
increments and stresses, cf. (3.12) and (3.15), is chosen as the smallest
integer for which we have the bounds

$\displaystyle
4b\mathsf{N}_{\textnormal{ind,t}}+8+b(\mathsf{C_{R}}+3)\varepsilon_{\Gamma}(b-1)+2\beta(b^{3}-1)$
$\displaystyle<\varepsilon_{\Gamma}(b-1)\mathsf{N}_{\textnormal{ind,v}}\,.$
(9.11)

  10. (x)

The value of the decoupling parameter ${\mathsf{N}_{\rm dec}}$, which is used
in the $L^{p}$ decorrelation Lemma A.2, is chosen as the smallest integer for
which we have

$\displaystyle{\mathsf{N}_{\rm
dec}}\left(\frac{1}{30}\left(\frac{4}{5}\right)^{{n_{\rm
max}}}-\varepsilon_{\Gamma}\right)>\frac{4b}{b-1}\,.$ (9.12)

  11. (xi)

The value of the parameter ${\mathsf{d}}$, which in essence is used in the
inverse divergence operator of Proposition A.18 to count the order of a
parametrix expansion, is chosen as the smallest integer for which we have

$\displaystyle({\mathsf{d}}-1)\left(\frac{1}{30}\left(\frac{4}{5}\right)^{{n_{\rm
max}}}-\varepsilon_{\Gamma}\right)>\frac{(12\mathsf{N}_{\textnormal{ind,v}}+7)b}{b-1}\,.$
(9.13)

  12. (xii)

The value of $\mathsf{N}_{\rm fin}$, which is introduced in Section 3 and used
to quantify the highest order derivative estimates utilized throughout the
scheme is chosen as the smallest integer such that

$\displaystyle\frac{3}{2}\mathsf{N}_{\rm fin}>(2\mathsf{N}_{\rm
cut,t}+\mathsf{N}_{\rm
cut,x}+14\mathsf{N}_{\textnormal{ind,v}}+2{\mathsf{d}}+2{\mathsf{N}_{\rm
dec}}+12)2^{{n_{\rm max}}+1}\,.$ (9.14)

  13. (xiii)

Having chosen all the previous parameters in items (i)–(xii), there exits a
sufficiently large parameter $a_{*}\geq 1$, which depends on all the
parameters listed above (which recursively means that $a_{*}=a_{*}(\beta,b)$),
and which allows us to choose $a$ an arbitrary number in the interval
$[a_{*},\infty)$. While we do not give a formula for $a_{*}$ explicitly, it is
chosen so that $a_{*}^{(b-1)\varepsilon_{\Gamma}}$ is at least twice larger
than all the implicit constants in the $\lesssim$ symbols throughout the
paper; note that these constants only depend on the parameters in items
(i)–(xii) — never on $q$ — which justifies the existence of $a_{*}$.

Having made the choices in items (i)–(xiii) above, we are now ready to define
the $q$-dependent parameters which appear in the proof.

###  9.2 Definitions of the $q$-dependent parameters

####  9.2.1 Parameters which depend on $q$

For $q\geq 0$, we define the fundamental frequency parameter used in this
paper as

$\displaystyle\lambda_{q}$
$\displaystyle=2^{\big{\lceil}(b^{q})\log_{2}a\big{\rceil}}\,.$ (9.15)

Definition (9.15) gives that $\lambda_{q}$ is an integer power of $2$, and
that we have the bounds

$\displaystyle a^{(b^{q})}\leq\lambda_{q}\leq
2a^{(b^{q})}\qquad\mbox{and}\qquad\frac{1}{3}\lambda_{q}^{b}\leq\lambda_{q+1}\leq
2\lambda_{q}^{b}$ (9.16)

for all $q\geq 0$. Throughout the paper the above two inequalities are used by
putting the factors of $\nicefrac{{1}}{{3}}$ and $2$ into the implicit
constants of $\lesssim$ symbols. In terms of $\lambda_{q}$, the fundamental
amplitude parameter used in the paper is

$\displaystyle\delta_{q}$
$\displaystyle=\lambda_{1}^{(b+1)\beta}\lambda_{q}^{-2\beta}\,.$ (9.17)

In terms of the parameter $\varepsilon_{\Gamma}$ from (9.7), we introduce a
parameter which is used repeatedly throughout the paper to mean “a tiny power
of the frequency parameter”:

$\displaystyle\Gamma_{q+1}$
$\displaystyle=\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\varepsilon_{\Gamma}}\,.$
(9.18)

In order to cap off our derivative losses, we need to mollify in space and
time using the operators described in Section 9.4 below. This is done in terms
of the following space and time parameters:

$\displaystyle\widetilde{\lambda}_{q}$
$\displaystyle=\lambda_{q}\Gamma_{q+1}^{5}$ (9.19)
$\displaystyle\widetilde{\tau}_{q}^{-1}$
$\displaystyle=\tau_{q}^{-1}\widetilde{\lambda}_{q}^{3}\widetilde{\lambda}_{q+1}\,.$
(9.20)

While $\widetilde{\tau}_{q}$ is used for mollification and thus for rough
material derivative bounds, the fundamental time parameter used in the paper
for sharp material derivative bounds is

$\displaystyle\tau_{q}$
$\displaystyle=\left(\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}\Gamma_{q+1}^{\mathsf{c_{0}}+6}\right)^{-1}\,.$
(9.21)

Note that besides depending on the parameters introduced in (i)–(xiii), the
parameters introduced above only depend on $q$, but are independent of $n$ and
$p$.

####  9.2.2 Parameters which depend also on $n$ and $p$

The rest of the parameters depend on $n\in\\{0,\ldots,{n_{\rm max}}\\}$ and on
$p\in\\{0,\ldots,{p_{\rm max}}\\}$. We start by defining the frequency
parameter $\lambda_{q,n}$ and the intermittency parameter $r_{q+1,n}$ by

$\displaystyle\lambda_{q,n}$
$\displaystyle=2^{\left\lceil\left(\frac{4}{5}\right)^{n+1}\log_{2}\lambda_{q}+\left(1-\left(\frac{4}{5}\right)^{n+1}\right)\log_{2}\lambda_{q+1}\right\rceil}$
(9.22) $\displaystyle r_{q+1,n}$
$\displaystyle=\frac{\lambda_{q,n}}{\lambda_{q+1}}$ (9.23)

for $0\leq n\leq{n_{\rm max}}$. In particular, (9.22) shows that
$\lambda_{q+1}r_{q+1,n}$ is an integer power of $2$, and we have the bound

$\displaystyle\lambda_{q}^{\left(\frac{4}{5}\right)^{n+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n+1}}\leq\lambda_{q,n}\leq
2\lambda_{q}^{\left(\frac{4}{5}\right)^{n+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n+1}}\,,$
(9.24)

while (9.23) implies that $r_{q+1}^{-1}$ is an integer power of $2$, and we
have the estimates

$\displaystyle\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}}\leq
r_{q+1,n}\leq
2\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}}\,.$
(9.25)

As with (9.16) we absorb the factors of $2$ in (9.24) and (9.25) into the
implicit constants in $\lesssim$ symbols.

We also define the frequency parameters $\lambda_{q,n,p}$ by

$\displaystyle\lambda_{q,0,p}$
$\displaystyle=\Gamma_{q+1}\widetilde{\lambda}_{q}$ $\displaystyle n=0,0\leq
p\leq{p_{\rm max}}$ (9.26) $\displaystyle\lambda_{q,1,0}$
$\displaystyle=\lambda_{q}^{\frac{4}{5}}\lambda_{q+1}^{\frac{1}{5}}$
$\displaystyle n=1,p=0$ (9.27) $\displaystyle\lambda_{q,n,0}$
$\displaystyle=\lambda_{q}^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}$
$\displaystyle 2\leq n\leq{n_{\rm max}}+1$ (9.28)
$\displaystyle\lambda_{q,n,p}$
$\displaystyle=\lambda_{q,n,0}^{1-\nicefrac{{p}}{{{p_{\rm
max}}}}}\lambda_{q,n+1,0}^{\nicefrac{{p}}{{{p_{\rm max}}}}}$ $\displaystyle
1\leq n\leq{n_{\rm max}},0\leq p\leq{p_{\rm max}}.$ (9.29)

For $0\leq n\leq{n_{\rm max}}$, we define

$\displaystyle f_{q,0}$ $\displaystyle=1$ $\displaystyle n=0$ (9.30)
$\displaystyle f_{q,n}$
$\displaystyle=\left(\frac{\lambda_{q,n+1,0}}{\lambda_{q,n,0}}\right)^{\nicefrac{{1}}{{{p_{\rm
max}}}}}$ $\displaystyle 1\leq n\leq{n_{\rm max}}.$ (9.31)

We define $\delta_{q+1,0,p}$ by

$\displaystyle\delta_{q+1,0,1}$
$\displaystyle=\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}$ $\displaystyle p=1$
(9.32) $\displaystyle\delta_{q+1,0,p}$ $\displaystyle=0$ $\displaystyle 2\leq
p\leq{p_{\rm max}}.$ (9.33)

When $1\leq n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm max}}$, we define
$\delta_{q+1,n,p}$ by

$\delta_{q+1,n,p}=\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\cdot\left(\frac{\widetilde{\lambda}_{q}}{\lambda_{q,n,p-1}}\right)\cdot\prod_{n^{\prime}<n}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\,.$
(9.34)

We remark that by the definition of $\lambda_{q,1,0}$ given in (9.27), and
more generally $\lambda_{q,n,p}$ in (9.29), the fact that $n\geq 1$, and a
large choice of ${p_{\rm max}}$ which makes $f_{q,n}$ (defined in (9.31))
small, $\delta_{q+1,n,p}$ is significantly smaller than
$\Gamma_{q}^{-\mathsf{C_{R}}}\delta_{q+1}$.

For $1\leq n\leq{n_{\rm max}}$, we define $\mathsf{c}_{\textnormal{n}}$ in
terms of $\mathsf{c_{0}}$ by

$\displaystyle\mathsf{c}_{\textnormal{n}}$
$\displaystyle=\mathsf{c_{0}}-4n\,.$ (9.35)

For $n=0$, we set

$\mathsf{N}_{\textnormal{fin},0}=\frac{3}{2}\mathsf{N}_{\rm fin},$ (9.36)

while for $1\leq n\leq{n_{\rm max}}$, we define $\mathsf{N}_{\rm fin,n}$
inductively on $n$ by using (9.36) and the formula

$\mathsf{N}_{\rm
fin,n}=\left\lfloor\frac{1}{2}\left(\mathsf{N}_{\textnormal{fin},\textnormal{n}-1}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-6\right)-{\mathsf{d}}\right\rfloor\,.$ (9.37)

###  9.3 Inequalities and consequences of the parameter definitions

The definitions made in the previous two sections have the following
consequences, which will be used frequently throughout the paper.

Due to (9.15) we have that
$\Gamma_{q+1}\geq(\nicefrac{{1}}{{2}})^{b\varepsilon_{\Gamma}}\lambda_{q}^{(b-1)\varepsilon_{\Gamma}}\geq(\nicefrac{{1}}{{2}})^{b\varepsilon_{\Gamma}}\lambda_{0}^{(b-1)\varepsilon_{\Gamma}}\geq(\nicefrac{{1}}{{2}})a_{*}^{(b-1)\varepsilon_{\Gamma}}$.
As was already mentioned in item (xiii), we have chosen $a_{*}$ to be
sufficiently large so that $a_{*}^{(b-1)\varepsilon_{\Gamma}}$ is at least
twice larger than all the implicit constants appearing in all $\lesssim$
symbols throughout the paper. Therefore, for any $q\geq 0$, we may use a
single power of $\Gamma_{q+1}$ to absorb any implicit constant in the paper:
an inequality of the type $A\lesssim B$ may be rewritten as
$A\leq\Gamma_{q+1}B$.

From (9.18), (9.19), and (9.7c), we have that

$\displaystyle\Gamma_{q+1}^{4}\widetilde{\lambda}_{q}\leq\lambda_{q+1}\,.$
(9.38)

From the definition (9.21) of $\tau_{q}$ and (9.35), which gives that
$\mathsf{c}_{\textnormal{n}}$ is decreasing with respect to $n$, we have that
for all $0\leq n\leq{n_{\rm max}}$

$\displaystyle\Gamma_{q+1}^{\mathsf{c}_{\textnormal{n}}+6}\delta_{q}^{\nicefrac{{1}}{{2}}}{\widetilde{\lambda}_{q}}$
$\displaystyle\leq\tau_{q}^{-1}\,.$ (9.39)

Using the definitions (9.17), (9.18), (9.19), and (9.21), writing out
everything in terms of $\lambda_{q-1}$, and appealing to (9.7d), we have that

$\displaystyle\tau_{q-1}^{-1}\Gamma_{q+1}^{3+\mathsf{c_{0}}}$
$\displaystyle\leq\tau_{q}^{-1}$ (9.40)
$\displaystyle\tau_{q-1}^{-1}\Gamma_{q+1}$
$\displaystyle\leq\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\,.$ (9.41)

From the definitions (9.6) of $\mathsf{c_{0}}$ and (9.35) of
$\mathsf{c}_{\textnormal{n}}$, we have that for all $0\leq n\leq{n_{\rm
max}}$,

$-\mathsf{c}_{\textnormal{n}}+4\leq-1.$ (9.42)

From the definition of $\widetilde{\tau}_{q}$, it is immediate that

$\displaystyle\tau_{q}^{-1}{\widetilde{\lambda}_{q}^{4}}$
$\displaystyle\leq{\widetilde{\tau}_{q}^{-1}}\leq\tau_{q}^{-1}{\widetilde{\lambda}_{q}^{3}}{\widetilde{\lambda}_{q+1}}\,.$
(9.43)

From (9.7d), the assumption that $\beta\geq\nicefrac{{1}}{{3}}$, and the
assumption $b\leq\nicefrac{{3}}{{2}}$, we can write everything out in terms of
$\lambda_{q}$ to deduce that

$\displaystyle\tau_{q}^{-1}\Gamma_{q+1}^{9}$
$\displaystyle\leq\tau_{q+1}^{-1}\,.$ (9.44)

From the definitions (9.22) and (9.26)–(9.29), for all $0\leq n\leq{n_{\rm
max}}$ and $0\leq p\leq{p_{\rm max}}$ we have

$\frac{\lambda_{q,n,p}}{\lambda_{q,n}}\ll 1\,.$

More precisely, when $n=0$ we have that

$\displaystyle\frac{\Gamma_{q+1}\lambda_{q,n,p}}{\lambda_{q,n}}=\frac{\Gamma_{q+1}^{2}\widetilde{\lambda}_{q}}{\lambda_{q,0}}=\frac{\Gamma_{q+1}^{7}\lambda_{q}}{\lambda_{q,0}}=\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{-\frac{1}{5}+7\varepsilon_{\Gamma}}$
(9.45)

while for $n\geq 1$ it holds that

$\displaystyle\frac{\Gamma_{q+1}\lambda_{q,n,p}}{\lambda_{q,n}}\leq\frac{\Gamma_{q+1}\lambda_{q,n+1,0}}{\lambda_{q,n}}=\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{(\frac{4}{5})^{n}(\frac{4}{5}-\frac{5}{6})+\varepsilon_{\Gamma}}\leq\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{-\frac{1}{30}(\frac{4}{5})^{{n_{\rm
max}}}+\varepsilon_{\Gamma}}$ (9.46)

as it is clear that the quotient on the left hand side is largest when
$n={n_{\rm max}}$. Note that due to (9.2) we have
$\frac{1}{30}\left(\frac{4}{5}\right)^{{n_{\rm
max}}}-\varepsilon_{\Gamma}<\frac{1-2\beta
b}{30}-\varepsilon_{\Gamma}\leq\frac{1}{5}-7\varepsilon_{\Gamma}$; here we
also used that $\varepsilon_{\Gamma}\leq\frac{1}{36}$, which handily follows
from (9.7b). Combining (9.45) and (9.46) we thus arrive at

$\displaystyle\frac{\Gamma_{q+1}\lambda_{q,n,p}}{\lambda_{q,n}}\leq\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{-\frac{1}{30}(\frac{4}{5})^{{n_{\rm
max}}}+\varepsilon_{\Gamma}}\leq\left(2\lambda_{q}^{b-1}\right)^{-\frac{1}{30}(\frac{4}{5})^{{n_{\rm
max}}}+\varepsilon_{\Gamma}}$ (9.47)

for all $0\leq n\leq{n_{\rm max}}$ and $0\leq p\leq{p_{\rm max}}$. Combining
the above estimate with our choice of ${\mathsf{N}_{\rm dec}}$ in (9.12), we
thus arrive at

$\displaystyle\lambda_{q+1}^{4}\leq\left(\frac{\lambda_{q,{\widetilde{n}}}}{2\pi\sqrt{3}\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{\widetilde{p}}}}\right)^{{\mathsf{N}_{\rm
dec}}}\,.$ (9.48)

for all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$ and
$1\leq{\widetilde{p}}\leq{p_{\rm max}}$.

Next, we a list a few consequences of the fact that
$\mathsf{N}_{\textnormal{ind,v}}\gg\mathsf{N}_{\textnormal{ind,t}}$, as
specified in (9.11). First, we note from (9.43) that

$\displaystyle\widetilde{\tau}_{q-1}^{-1}\tau_{q-1}\leq\widetilde{\lambda}_{q-1}^{3}\widetilde{\lambda}_{q}\leq\lambda_{q}^{4}$
(9.49)

where in the second inequality we have used that
$\varepsilon_{\Gamma}\leq\frac{3}{20b}$. In turn, the above inequality
combined with (9.11) implies the following estimates, all of which are used
for the first time in Section 5:

$\displaystyle\lambda_{q-1}^{8}\Gamma_{q+1}^{1+\mathsf{C_{R}}}\frac{\delta_{q-1}}{\delta_{q+2}}\left(\widetilde{\tau}_{q-1}^{-1}\tau_{q-1}\right)^{\mathsf{N}_{\textnormal{ind,t}}}$
$\displaystyle\leq\Gamma_{q}^{\mathsf{N}_{\textnormal{ind,v}}-2}$ (9.50a)
$\displaystyle\widetilde{\lambda}_{q}^{2}\left(\tilde{\tau}_{q-1}^{-1}\tau_{q-1}\right)^{\mathsf{N}_{\textnormal{ind,t}}}$
$\displaystyle\leq\Gamma_{q+1}^{5\mathsf{N}_{\textnormal{ind,v}}}$ (9.50b)
$\displaystyle\lambda_{q-1}^{4}\delta_{q-1}^{\nicefrac{{1}}{{2}}}\Gamma_{q}^{2}\delta_{q}^{-\nicefrac{{1}}{{2}}}(\widetilde{\tau}_{q-1}^{-1}\tau_{q-1})^{\mathsf{N}_{\textnormal{ind,t}}}$
$\displaystyle\leq\Gamma_{q}^{\mathsf{N}_{\textnormal{ind,v}}}\,.$ (9.50c)

Next, as a consequence of our choice of $\mathsf{N}_{\rm cut,t}$ and
$\mathsf{N}_{\rm cut,x}$ in (9.9), we obtain the following bounds, which are
used in Section 6

$\displaystyle\widetilde{\lambda}_{q}^{\nicefrac{{3}}{{2}}}\Gamma_{q}^{-\mathsf{N}_{\rm
cut,t}}\leq\lambda_{q}^{3}\Gamma_{q}^{-\mathsf{N}_{\rm cut,t}}\leq 1\,.$
(9.51)

for all $q\geq 0$. The fact that $\mathsf{N}_{\textnormal{ind,t}}$ is taken to
be much larger than $\mathsf{N}_{\rm cut,t}$, as expressed in (9.10), implies
when combined with (9.49) the following bound, which is also used in Section
6:

$\displaystyle\left(\tau_{q}\widetilde{\tau}_{q}^{-1}\right)^{\mathsf{N}_{\rm
cut}}\leq\lambda_{q+1}^{4\mathsf{N}_{\rm
cut}}\leq\Gamma_{q+1}^{\mathsf{N}_{\textnormal{ind,t}}}$ (9.52)

for all $q\geq 1$.

The parameter $\alpha_{\mathsf{R}}$ is chosen in (9.8) in order to ensure the
inequality

$\lambda_{q+1}^{\alpha_{\mathsf{R}}}\leq\Gamma_{q+1}.$ (9.53)

for all $q\geq 0$. This fact is used in Section 8. Several other, much more
hideous, parameter inequalities are used in Section 8, and for the readers’
convenience we list them next. First, we claim that

$\Gamma_{q+1}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{n_{\rm
max}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\lambda_{q,{n_{\rm
max}}+1,0}^{-1}\leq\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\,.$
(9.54)

In order to verify the above bound, we appeal to to the choices made in (9.1),
(9.2), and (9.3), to the definitions (9.19), (9.27), (9.28), (9.31), and the
fact that ${\widetilde{n}}\leq{n_{\rm max}}$, to deduce that the left side of
(9.54) is bounded from above by

$\displaystyle\delta_{q+1}\Gamma_{q+1}^{6+{n_{\rm
max}}(8+{\mathsf{C}_{b}})}\frac{\lambda_{q}}{\lambda_{q,{n_{\rm
max}}+1,0}}\left(\frac{\lambda_{q,{n_{\rm
max}}+1,0}}{\lambda_{q,1,0}}\right)^{\frac{1}{{p_{\rm max}}}}$
$\displaystyle=\delta_{q+1}\Gamma_{q+1}^{6+{n_{\rm
max}}(8+{\mathsf{C}_{b}})}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(1-(\frac{4}{5})^{{n_{\rm
max}}}\frac{5}{6}\right)}\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\frac{1}{{p_{\rm
max}}}\left(\frac{4}{5}-(\frac{4}{5})^{{n_{\rm max}}}\frac{5}{6}\right)}$
$\displaystyle\leq\frac{\lambda_{q}\delta_{q+1}}{\lambda_{q+1}}\Gamma_{q+1}^{6+{n_{\rm
max}}(8+{\mathsf{C}_{b}})}\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{(1-2\beta
b)\frac{4}{5}}\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\frac{1-2\beta
b}{10}\frac{4}{5}}$
$\displaystyle\leq\left(\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\right)\frac{\lambda_{q}\delta_{q+1}}{\lambda_{q+1}\delta_{q+2}}\Gamma_{q+1}^{7+\mathsf{C_{R}}+{n_{\rm
max}}(8+{\mathsf{C}_{b}})}\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{(1-2\beta
b)\frac{22}{25}}$
$\displaystyle\leq\left(\Gamma_{q+1}^{-\mathsf{C_{R}}}\Gamma_{q+1}^{-1}\delta_{q+2}\right)\Gamma_{q+1}^{7+\mathsf{C_{R}}+{n_{\rm
max}}(8+{\mathsf{C}_{b}})}\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{-(1-2\beta
b)\frac{3}{25}}$

The proof of (9.54) is now completed by appealing to (9.7a), which ensures
that $\Gamma_{q+1}$ represents a sufficiently small power of
$\nicefrac{{\lambda_{q+1}}}{{\lambda_{q}}}$.

Next, we claim that due to our choice of ${\mathsf{d}}$, we have

$\Gamma_{q}^{-\mathsf{C_{R}}}\delta_{q+1}\widetilde{\lambda}_{q}\prod_{n^{\prime}\leq{n_{\rm
max}}}\left(f_{q,n^{\prime}}\Gamma_{q+1}^{8+{\mathsf{C}_{b}}}\right)\lambda_{q+1}\left(\frac{\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{p_{\rm
max}}}}{\lambda_{q,{\widetilde{n}}}}\right)^{{\mathsf{d}}-1}\left(\lambda_{q+1}^{4}\right)^{3\mathsf{N}_{\textnormal{ind,v}}}\leq\frac{\delta_{q+2}}{\lambda_{q+1}^{5}}.$
(9.55)

In order to verify the above bound we use the previously established estimate
(9.54) in conjunction with (9.47); after dropping the helpful factor of
$\Gamma_{q+1}^{-2-\mathsf{C_{R}}}$, we deduce that the left side of (9.55) is
bounded from above by

$\displaystyle\delta_{q+2}\lambda_{q,{n_{\rm
max}}+1,0}\lambda_{q+1}\left(\frac{\Gamma_{q+1}\lambda_{q,{\widetilde{n}},{p_{\rm
max}}}}{\lambda_{q,{\widetilde{n}}}}\right)^{{\mathsf{d}}-1}\left(\lambda_{q+1}^{4}\right)^{3\mathsf{N}_{\textnormal{ind,v}}}$
$\displaystyle\leq\frac{\delta_{q+2}}{\lambda_{q+1}^{5}}\lambda_{q+1}^{3}\left(2\lambda_{q}^{b-1}\right)^{-({\mathsf{d}}-1)\left(\frac{1}{30}(\frac{4}{5})^{{n_{\rm
max}}}-\varepsilon_{\Gamma}\right)}\lambda_{q+1}^{12\mathsf{N}_{\textnormal{ind,v}}}$

The choice of ${\mathsf{d}}$ in (9.13) shows that the above estimate directly
implies (9.55).

The amplitudes of the higher order corrections $w_{q+1,n,p}$ must meet the
inductive assumptions stated in (3.13). In order to meet the satisfactory
bound in Remark 8.3, from (9.32)–(9.34), we deduce the bound

$\delta_{q+1,{\widetilde{n}},{\widetilde{p}}}^{\nicefrac{{1}}{{2}}}\leq\Gamma_{q+1}^{-2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}.$
(9.56)

Indeed, the case ${\widetilde{n}}=0$ follows from the definition of
$\mathsf{C_{R}}$ in (9.5), while the case ${\widetilde{n}}\geq 1$ is a
consequence of the definition (9.34), which implies that
$\delta_{q,{\widetilde{n}},{\widetilde{p}}}\leq\delta_{q,0,1}$, for any
${\widetilde{n}}\geq 1$ and any ${\widetilde{p}}\geq 1$.

Another parameter inequality which is necessary to estimate the transport and
Nash errors in Sections 8.4 and 8.5, is

$\Gamma_{q+1}^{4+\frac{{\mathsf{C}_{b}}}{2}}\delta_{q+1,{\widetilde{n}},1}^{\frac{1}{2}}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}$
(9.57)

for all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$. When ${\widetilde{n}}=0$,
this inequality may be deduced by writing everything out in terms of
$\lambda_{q}$, appealing to the appropriate definitions, and then using that
$\beta<\nicefrac{{1}}{{2}}$ from item i, (9.1), (9.4), (9.5), (9.6), (9.7b),
after which one arrives at

$\displaystyle\varepsilon_{\Gamma}\left(4+\frac{b-4}{2}+\frac{1}{2}{-\mathsf{C_{R}}}+\mathsf{c_{0}}+12\right)+\beta(2b+1)<\frac{1}{100}+\frac{3}{2}<\frac{9}{5}.$

It is clear there is quite a bit of room in the above inequality, and
similarly, (9.57) becomes _most_ restrictive when ${\widetilde{n}}={n_{\rm
max}}$. In this case, one may again write everything out in terms of
$\lambda_{q}$, move everything to the left hand side, and appeal to the most
of the same referenced inequalities as before to see that

$\displaystyle\varepsilon_{\Gamma}\left(22+4{n_{\rm
max}}\right)+\beta(2b+1)-\frac{3}{2}$
$\displaystyle\leq\varepsilon_{\Gamma}\left(22+4{n_{\rm
max}}\right)+\beta-\frac{1}{2}<0\,,$

where in the last inequality we have instead appealed to (9.7a) rather than
(9.7b), proving (9.57) in the remaining cases $1\leq{\widetilde{n}}\leq{n_{\rm
max}}$.

Parameter inequalities which play a crucial role in showing that the
Oscillation 2 type errors vanish, see Section 8.7, are:

$\displaystyle
16C_{*}\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}\cdot
4}$
$\displaystyle<\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}-1}\cdot\frac{5}{6}\cdot
3}\Gamma_{q+1}^{-3}\,,\qquad\mbox{for}\qquad{\widetilde{n}}\geq 2\,,$ (9.58a)
$\displaystyle
16C_{*}\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\frac{4}{5}\cdot
4}$
$\displaystyle<\left(\frac{\widetilde{\lambda}_{q}}{\lambda_{q+1}}\right)^{3}\,\,,$
(9.58b) $\displaystyle
16C_{*}\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\Gamma_{q+1}^{4}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{2}\cdot
4}$
$\displaystyle<\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\frac{4}{5}\cdot
3}\,.$ (9.58c)

where $C_{*}$ is the geometric constant from Lemma 4.8–estimate (4.31),
$\mathcal{C}_{\textnormal{pipe}}$ is a geometric constant which appears in
Lemma 8.11–estimate (8.119), and $\mathcal{C}_{\eta}$ is the constant from
Lemma 8.8. In order to verify (9.58), we first note that
$C_{*}\mathcal{C}_{\textnormal{pipe}}\mathcal{C}_{\eta}\leq\Gamma_{q+1}$,
since $a_{*}$ was chosen to be sufficiently large. Inequality (9.58b) is then
an immediate consequence of the fact that $\nicefrac{{16}}{{5}}>3$. The bound
(9.58a) follows from

$\Gamma_{q+1}^{5}<\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\left(\frac{4}{5}\right)^{{n_{\rm
max}}-1}\left(\frac{64}{25}-\frac{5}{2}\right)}\leq\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\left(\frac{4}{5}\right)^{{\widetilde{n}}+1}\cdot
4-\left(\frac{4}{5}\right)^{{\widetilde{n}}-1}\cdot\frac{5}{6}\cdot 3}\,.$
(9.59)

The second inequality in the above display is a consequence of
${\widetilde{n}}\leq{n_{\rm max}}$, while the first one follows from (9.7b).
Finally, inequality (9.58c) is a consequence of the fact that
$\nicefrac{{64}}{{25}}-\nicefrac{{12}}{{5}}>\nicefrac{{64}}{{25}}-\nicefrac{{5}}{{2}}$
and the first inequality in (9.59), which bounds $\Gamma_{q+1}^{5}$.

We conclude this section by verifying a few inequalities concerning the
parameter $\mathsf{N}_{\rm fin,n}$, which counts the number of available
space-plus-material derivative for the residual stress $\mathring{R}_{q,n}$.
For all $0\leq n\leq{n_{\rm max}}$ we require that

$\displaystyle\mathsf{N}_{\textnormal{ind,t}},2{\mathsf{N}_{\rm dec}}+4$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm
cut,x}-5\right)\rfloor-{\mathsf{d}}\,,$ (9.60a) $\displaystyle
14\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9\,,$ (9.60b)
$\displaystyle 6\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm
cut,x}-6\right)\rfloor-{\mathsf{d}}\,,$ (9.60c) $\displaystyle
6\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{4}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm cut,x}-7\right)\rfloor\,.$
(9.60d)

for all $0\leq n\leq{n_{\rm max}}$. Additionally for
$0\leq{\widetilde{n}}<n\leq{n_{\rm max}}$, we require that

$\displaystyle\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-6\right)\rfloor-{\mathsf{d}}\geq\mathsf{N}_{\rm
fin,n}$ (9.61)

holds. The inequality (9.61) is a direct consequence of the recursive formula
(9.37) and of the fact that the sequence $\mathsf{N}_{\rm fin,n}$ is monotone
decreasing with respect to $n$. Using (9.36) and (9.37) one may show that

$\mathsf{N}_{\rm fin,n}\geq
2^{-n}\mathsf{N}_{\textnormal{fin},0}-(2{\mathsf{d}}+\mathsf{N}_{\rm
cut,t}+\mathsf{N}_{\rm cut,x}+8)\,.$

Noting that the bounds (9.60) are most restrictive for $n={n_{\rm max}}$, they
now readily follow from our choice (9.14).

###  9.4 Mollifiers and Fourier projectors

Let $\phi(\zeta):\mathbb{R}\rightarrow\mathbb{R}$ be a smooth, $C^{\infty}$
function compactly supported in the set $\\{\zeta:|\zeta|\leq 1\\}$ which in
addition satisfies

$\int\phi(\zeta)\,d\zeta=1,\qquad\int\phi(\zeta)\zeta^{n}=0\quad\forall
n=1,2,...,\mathsf{N}_{\textnormal{ind,v}}.$ (9.62)

Let $\widetilde{\phi}(x):\mathbb{R}^{3}\rightarrow\mathbb{R}$ be defined by
$\widetilde{\phi}(x)=\phi(|x|)$. For $\lambda,\mu\in\mathbb{R}$, define

$\phi_{\lambda}^{(x)}(x)={{\lambda}^{3}}\widetilde{\phi}\left(\lambda
x\right),\qquad\phi_{\mu}^{(t)}(t)=\mu\phi(\mu t).$ (9.63)

For $q\in\mathbb{N}$, we will define the spatial and temporal convolution
operators

$\mathcal{P}_{q,x}:=\phi_{\widetilde{\lambda}_{q}}^{(x)}\ast,\qquad\mathcal{P}_{q,t}:=\phi_{\widetilde{\tau}_{q-1}^{-1}}^{(t)}\ast,\qquad\mathcal{P}_{q,x,t}:=\mathcal{P}_{q,x}\mathcal{P}_{q,t}.$
(9.64)

We will use the notation $\mathbb{P}_{\leq\lambda}$ to denote the standard
(Littlewood-Paley) Fourier projection operators onto spatial frequencies which
are less than or equal to $\lambda$, $\mathbb{P}_{\geq\lambda}$ to denote the
standard Littlewood-Paley projection operators onto spatial frequencies which
are greater than or equal to $\lambda$, and the notation

$\mathbb{P}_{[\lambda_{1},\lambda_{2})}$

to denote the Fourier projection operator onto spatial frequencies $\xi$ such
that $\lambda_{1}\leq|\xi|<\lambda_{2}$. If $\lambda_{1}=\lambda_{2}$, we
adopt the convention that $\mathbb{P}_{[\lambda_{1},\lambda_{2})}f=0$ for any
$f$.

###  9.5 Notation

$\displaystyle\mathcal{M}\left(n,N,\lambda,\Lambda\right)$
$\displaystyle=\lambda^{\min\\{n,N\\}}\Lambda^{\max\\{n-N,0\\}}$
$\displaystyle a\otimes_{\mathrm{s}}b$ $\displaystyle=\frac{1}{2}(a\otimes
b+b\otimes a)$ (9.65) $\displaystyle a\,\mathring{\otimes}_{\rm s}\,b$
$\displaystyle=\frac{1}{2}(a\,\mathring{\otimes}\,b+b\,\mathring{\otimes}\,a)$
(9.66) $\displaystyle\mathrm{supp\,}_{t}f$
$\displaystyle=\overline{\\{t:f|_{\mathbb{T}^{3}\times\\{t\\}}\not\equiv
0\\}}$ (9.67)

We will use repeatedly the notation (noted in the introduction in (2.3) and
(2.4) and in Remark 3.2)

$\left\|f\right\|_{L^{p}}:=\left\|f\right\|_{L^{\infty}_{t}(L^{p}(\mathbb{T}^{3}))}\,.$
(9.68)

That is, all $L^{p}$ norms stand for $L^{p}$ norms in space, uniformly in
time. Similarly, when we wish to emphasize a set dependence on
$\Omega\subset\mathbb{R}\times\mathbb{T}^{3}$ of an $L^{p}$ norm, we write

$\left\|f\right\|_{L^{p}(\Omega)}:=\left\|{\mathbf{1}}_{\Omega}\;f\right\|_{L^{\infty}_{t}(L^{p}(\mathbb{T}^{3}))}\,.$
(9.69)

##  Appendix A Useful lemmas

This appendix contains a collection of auxiliary lemmas which are used
throughout the paper:

  * •

Section A.1 recalls the classical $C^{N}$ estimates for solutions of the
transport equation. This is for instance used in Section 6.4.

  * •

Section A.2 gives the detailed construction of the basic cutoff functions
$\widetilde{\psi}_{m,q}$ and $\psi_{m,q}$, which are used in Section 6 to
construct the velocity and the stress cutoff functions.

  * •

Section A.3 recalls the fundamental fact that the $L^{p}$ norm of the product
of a slowly oscillating function and a fast periodic function is essentially
bounded by the product of their $L^{p}$ norms.

  * •

Section A.4 contains a version of the Sobolev inequality which takes into
account the support of the velocity cutoff functions.

  * •

Section A.5 contains a number of consequences of the multivariate Faa di Bruno
formula. Most of the results here are used for bounding the space and material
derivatives of the cutoff functions in Section 6. We also present here, cf.
Lemma A.7, a version of the $L^{p}$ decorrelation lemma from Section A.3 in
which the fast periodic function is composed with a volume-preserving flow
map. Lemma A.7 plays a crucial role in estimating the $L^{2}$ norms of the
velocity increments in Section 8.2.

  * •

Sections A.6 and A.7 contain a number of lemmas which allow us to go back and
forth between information for (arbitrarily) high order derivative bounds in
Eulerian and Lagrangian variables. These lemmas concerning sums of operators
and commutators with material derivatives are frequently used throughout the
paper to overcome the fact that material derivatives and spatial/temporal
derivatives do not commute.

  * •

Section A.8 introduces in Proposition A.18 the inverse divergence operator
used in this paper. We call this operator “intermittency friendly” because it
is composed of a principal part which precisely maintains the spatial support
of the vector field it is applied to, plus a secondary part which is nonlocal,
but whose amplitude is incredibly small. It is here that the definition (4.10)
for the density of our pipe flows plays an important role, as the high order
${\mathsf{d}}$ of the Laplacian present in (4.10) allows us to perform a
parametric expansion which maintains (to leading order) the support of pipes,
and also takes into account deformations due to the flow map.

###  A.1 Transport estimates

We shall require the following estimates for smooth solutions of transport
equations. For proofs we refer the reader to [8, Appendix D].

######  Lemma A.1 (Transport Estimates).

Consider the transport equation

$\partial_{t}f+u\cdot\nabla f=g,\qquad\qquad f|_{t_{0}}=f_{0}$

where $f,g:\mathbb{T}^{n}\rightarrow\mathbb{R}$ and
$u:\mathbb{T}^{n}\rightarrow\mathbb{R}^{n}$ are smooth functions. Let $X$ be
the flow of $u$, defined by

$\frac{d}{dt}X=u(X,t),\qquad X(x,t_{0})=x,$

and let $\Phi$ be the inverse of the flow of $X$, which is the identity at
time $t_{0}$. Then the following hold:

  1. (1)

$\displaystyle\|f(t)\|_{C^{0}}\leq\|f_{0}\|_{C^{0}}+\int_{t_{0}}^{t}\|g(s)\|_{C^{0}}\,ds$

  2. (2)

$\displaystyle\|Df(t)\|_{C^{0}}\leq\|Df_{0}\|_{C^{0}}e^{(t-t_{0})\|Du\|_{C^{0}}}+\int_{t_{0}}^{t}e^{(t-s)\|Du\|_{C^{0}}}\|Dg(s)\|_{C^{0}}\,ds$

  3. (3)

For any $N\geq 2$, there exists a constant $C=C(N)$ such that

$\displaystyle\qquad\quad\|D^{N}f(t)\|_{C^{0}}\leq$
$\displaystyle\left(\|D^{N}f_{0}\|_{C^{0}}+C(t-t_{0})\|D^{n}u\|_{C^{0}}\|Df\|_{C^{0}}\right)e^{C(t-t_{0})\|Du\|_{C^{0}}}$
$\displaystyle\quad+\int_{t_{0}}^{t}e^{C(t-s)\|Du\|_{C^{0}}}\left(\|D^{N}g(s)\|_{C^{0}}+C(t-s)\|D^{N}u\|_{C^{0}}\|Dg(s)\|_{C^{0}}\right)\,ds$

  4. (4)

$\displaystyle\|D\Phi(t)-\mathrm{Id}\|_{C^{0}}\leq
e^{(t-t_{0})\|Du\|_{C^{0}}}-1\leq(t-t_{0})\|Du\|_{C^{0}}e^{(t-t_{0})\|Du\|_{C^{0}}}$

  5. (5)

For $N\geq 2$ and a constant $C=C(N)$,

$\|D^{N}\Phi(t)\|_{C^{0}}\leq
C(t-t_{0})\|D^{N}u\|_{C^{0}}e^{C(t-t_{0})\|Du\|_{C^{0}}}$

###  A.2 Proof of Lemma 6.2

We first consider the function

$\displaystyle f(x)=\begin{cases}0&\mbox{if }x\leq 0\\\
e^{-\frac{1}{x^{2}}}&\mbox{if }x>0.\end{cases}$ (A.1)

We claim that for all $0\leq N\leq\mathsf{N}_{\rm fin}$ and $x>0$,

$\frac{|D^{N}f(x)|}{(f(x))^{1-\frac{N}{\mathsf{N}_{\rm fin}}}}\lesssim 1.$
(A.2)

The proof of this is achieved in two steps; first, one can show by induction
that for all $0\leq N\leq\mathsf{N}_{\rm fin}$, there exist constants $K_{N}$
and $c_{k}$ for $0\leq k\leq K_{N}$ such that

$D^{N}\left(e^{-\frac{1}{x^{2}}}\right)=\sum\limits_{k=0}^{K_{N}}\frac{c_{k}}{x^{k}}e^{-\frac{1}{x^{2}}}.$
(A.3)

Next, one may also check that for any powers $p,q>0$,

$\lim_{x\rightarrow 0^{+}}e^{-\frac{q}{x^{2}}}\frac{1}{x^{p}}=0.$ (A.4)

Then for $1\leq N\leq\mathsf{N}_{\rm fin}$, we see that $0\leq
1-\frac{N}{\mathsf{N}_{\rm fin}}<1$, and so using (A.3), we have that the
left-hand side of (A.2) may be split into a finite linear combination of terms
of the form in (A.4), showing that (A.2) is valid.

We now glue together two versions of $f$ as follows with the goal of forming a
prototypical cutoff function $\psi$. First, let
$x_{0}=\sqrt{\frac{1}{\ln(2)}}$ so that $f(x_{0})=\frac{1}{2}$. Now consider
the function $\widetilde{f}(x)=f(2x_{0}-x)$, and set

$\displaystyle F(x)=\begin{cases}f(x)&\mbox{if }x\leq x_{0}\\\
1-f(2x_{0}-x)&\mbox{if }x>x_{0}.\end{cases}$ (A.5)

Then $F(x)$ is continuous everywhere, and $C^{\infty}$ everywhere except
$x_{0}$, where it is not necessarily differentiable. Furthermore, one can
check that by the definition of $F$ and (A.2), for all $0\leq
N\leq\mathsf{N}_{\rm fin}$,

$\frac{|D^{N}F(x)|}{(F(x))^{1-\frac{N}{\mathsf{N}_{\rm fin}}}}\lesssim 1\mbox{
for all
}0<x<x_{0},\qquad\frac{|D^{N}\left(1-(F(x))^{2}\right)^{\frac{1}{2}}|}{\left(1-(F(x))^{2}\right)^{\frac{1}{2}\left(1-\frac{N}{\mathsf{N}_{\rm
fin}}\right)}}\lesssim 1\mbox{ for all }x_{0}<x<2x_{0}.$ (A.6)

The latter inequality follows from noticing that for $x$ close to $2x_{0}$,

$\displaystyle\left(1-(F(x))^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\left((1+F(x))(1-F(x))\right)^{\frac{1}{2}}=\left(1+F(x)\right)^{\frac{1}{2}}\left(f(2x_{0}-x)\right)^{\frac{1}{2}}\,.$

Since multiplying by a smooth function strictly larger than $1$, rescaling $f$
by a fixed parameter, and raising $f$ to a positive power preserves the
estimate (A.2) up to implicit constants (in fact raising $f$ to a power is
equivalent to rescaling), (A.6) is verified.

Towards the goal of adjusting $F$ to be differentiable at $x_{0}$, let $E$ be
the set $\left(\frac{x_{0}}{2},\frac{3x_{0}}{2}\right)$, and let $\phi$ be a
compactly, $C^{\infty}$ mollifier such that the support of the mollified
characteristic function $\mathcal{X}_{E}\ast\phi(x)$ is contained in
$\left(\frac{x_{0}}{4},\frac{7x_{0}}{4}\right)$. Setting

$\psi(x)=\left(\mathcal{X}_{E}\ast\phi(x)\right)\phi\ast
F(x)+\left(1-\mathcal{X}_{E}\ast\phi(x)\right)F(x),$ (A.7)

one may check that $\psi$ is $C^{\infty}$ and has the following properties:

$\displaystyle\psi(x)$ $\displaystyle=0\mbox{ for }x\leq 0$ (A.8)
$\displaystyle 0<\psi(x)$ $\displaystyle<1\mbox{ for }0<x<2x_{0}$ (A.9)
$\displaystyle\psi(x)$ $\displaystyle=1\mbox{ for }x\geq 2x_{0}$ (A.10)
$\displaystyle\frac{|D^{N}\psi(x)|}{(\psi(x))^{1-\frac{N}{\mathsf{N}_{\rm
fin}}}}$ $\displaystyle\lesssim 1\mbox{ for all }0<x$ (A.11)
$\displaystyle\frac{|D^{N}\left(1-(\psi(x))^{2}\right)^{\frac{1}{2}}|}{\left(1-(\psi(x))^{2}\right)^{\frac{1}{2}\left(1-\frac{N}{\mathsf{N}_{\rm
fin}}\right)}}$ $\displaystyle\lesssim 1\mbox{ for all }0<x<2x_{0}.$ (A.12)

We can now build $\widetilde{\psi}_{m,q}$. By rescaling and translating $\psi$
and using (A.8)-(A.10), one can check that

$\widetilde{\psi}_{m,q}(x)=\psi\left(\frac{x-\Gamma_{q}^{2(m+1)}}{\frac{1}{2x_{0}}\left(\frac{1}{4}-1\right)\Gamma_{q}^{2(m+1)}}\right)$
(A.13)

satisfies all components of (1). Notice that this rescaling involves a factor
proportional to $\Gamma_{q}^{-2(m+1)}$. Then using (A.11) and the fact that
every derivative $\psi_{m,q}$ introduces another factor of
$\Gamma_{q}^{-2(m+1)}$, we have that (6.3) is satisfied.

We now outline how to construct $\psi_{m,q}(\Gamma_{q}^{-2(m+1)}y)$, which is
the first term in the series in (6.1) and will define $\psi_{m,q}(y)$. The
basic idea is that the region
$(\frac{1}{4}\Gamma_{q+1}^{2(m+1)},\Gamma_{q+1}^{2(m+1)})$ where
$\widetilde{\psi}_{m,q}$ decreases from $1$ to $0$ will be the region where
$\psi_{m,q}(\Gamma_{q+1}^{-2(m+1)}y)$ increases from $0$ to $1$, and
furthermore in order to satisfy (6.1), we have a formula for
$\psi_{m,q}(\Gamma_{q+1}^{-2(m+1)}y)$ for these $y$-values. Specifically, in
order to ensure (6.1) for
$y\in(\frac{1}{4}\Gamma_{q+1}^{2(m+1)},\Gamma_{q+1}^{2(m+1)})$, we define

$\psi_{m,q}^{2}\left(\Gamma_{q+1}^{-2(m+1)}y\right)=1-\widetilde{\psi}_{m,q}^{2}(y)$

in this range of $y$-values. Then by adjusting (A.12) to reflect the
rescalings present in the definition of $\widetilde{\psi}_{m,q}$ and
$\psi_{m,q}(\Gamma_{q}^{-2(m+1)}y)$, we have that for
$y\in\left(\frac{1}{4},1\right)$, $\psi_{m,q}$ is well-defined and (6.4)
holds. To define $\psi_{m,q}(\Gamma_{q}^{-2(m+1)}y)$ for
$y\in[\frac{1}{4}\Gamma_{q}^{4(m+1)},\Gamma_{q}^{4(m+1)}]$ and thus
$\psi_{m,q}\left(y\right)$ for
$y\in[\frac{1}{4}\Gamma_{q}^{2(m+1)},\Gamma_{q}^{2(m+1)}]$, we can use that
for $y\in[\frac{1}{4}\Gamma_{q}^{4(m+1)},\Gamma_{q}^{4(m+1)}]$, the rescaled
function $\psi_{m,q}(\Gamma_{q+1}^{-4(m+1)}y)$ (i.e. the term in (6.1) with
$i=2$) is now well-defined. Then we can set

$\psi_{m,q}^{2}\left(\Gamma_{q+1}^{-2(m+1)}y\right)=1-\psi^{2}_{m,q}\left(\Gamma_{q+1}^{-4(m+1)}y\right)$

so that $\psi_{m,q}$ is well-defined for
$y\in[\frac{1}{4}\Gamma_{q}^{2(m+1)},\Gamma_{q}^{2(m+1)}]$ and (6.1) holds in
this range of $y$-values. Appealing again to (A.11) and (A.12), we have that
(6.5) is satisfied in the claimed range of $y$-values. Finally, in the missing
interval $[1,\frac{1}{4}\Gamma_{q}^{2(m+1)}]$, we set $\psi_{m,q}\equiv 1$.
One can now check that (6.1) holds for all $y\geq 0$, and that (6.2) follows
from (1), (2), and (6.1), concluding the proof.

###  A.3 $L^{p}$ decorrelation

The following lemma may be found in [13, Lemma 3.7].

######  Lemma A.2 ($L^{p}$ de-correlation estimate).

Fix integers ${\mathsf{N}_{\rm dec}}\geq 1$, $\mu\geq\lambda\geq 1$ and assume
that these integers obey

$\displaystyle\lambda^{{\mathsf{N}_{\rm
dec}}+4}\leq\left(\frac{\mu}{2\pi\sqrt{3}}\right)^{{\mathsf{N}_{\rm dec}}}\,.$
(A.14)

Let $p\in\\{1,2\\}$, and let $f$ be a $\mathbb{T}^{3}$-periodic function such
that

$\displaystyle\max_{0\leq N\leq{\mathsf{N}_{\rm
dec}}+4}\lambda^{-N}\|D^{N}f\|_{L^{p}}\leq\mathcal{C}_{f}$ (A.15)

for a constant $\mathcal{C}_{f}>0$.393939For instance, if $f$ has frequency
support in the ball of radius $\lambda$ around the origin, we have that
$\mathcal{C}_{f}\approx\left\|f\right\|_{L^{p}}$. Then, for any
$(\mathbb{T}/\mu)^{3}$-periodic function $g$, we have that

$\displaystyle\|fg\|_{L^{p}}\lesssim\mathcal{C}_{f}\|g\|_{L^{p}}\,,$

where the implicit constant is universal (in particular, independent of $\mu$
and $\lambda$).

###  A.4 Sobolev inequality with cutoffs

######  Lemma A.3.

Let $0\leq\psi_{i}\leq 1$ be cutoff functions such that
$\psi_{i\pm}=(\psi_{i-1}^{2}+\psi_{i}^{2}+\psi_{i+1}^{2})^{\nicefrac{{1}}{{2}}}=1$
on $\mathrm{supp\,}(\psi_{i})$, and such that for some $\rho>0$ we have

$\displaystyle|D^{K}\psi_{i}(x)|\lesssim\psi_{i}^{1-K/\mathsf{N}_{\rm
fin}}(x)\rho^{K}$ (A.16)

for all $K\leq 4$. Fix parameters $p\in[1,\infty]$,
$0<\lambda\leq\widetilde{\lambda}$, $0<\mu_{i}\leq\widetilde{\mu}_{i}$,
$N_{x},N_{t}\geq 0$, and assume that the sequences $\\{\mu_{i}\\}_{i\geq 0}$
and $\\{\widetilde{\mu}_{i}\\}_{i\geq 0}$ are nondecreasing. Assume that there
exist $N_{*},M_{*}\geq 0$ such that the function
$f\colon\mathbb{T}^{3}\to\mathbb{R}$ obeys the estimate

$\displaystyle\left\|\psi_{i}D^{N}D_{t}^{M}f\right\|_{L^{p}}\lesssim\mathcal{C}_{f}\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu_{i},\widetilde{\mu}_{i}\right)$
(A.17)

for all $N\leq N_{*}$ and $M\leq M_{*}$. Then, we have that

$\displaystyle\left\|\psi_{i}^{2}D^{N}D_{t}^{M}f\right\|_{L^{\infty}}$
$\displaystyle\lesssim\mathcal{C}_{f}(\max\\{1,\rho,\widetilde{\lambda}\\})^{\nicefrac{{3}}{{p}}}\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu_{i},\widetilde{\mu}_{i}\right)$
(A.18a)
$\displaystyle\left\|D^{N}D_{t}^{M}f\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i})}$
$\displaystyle\lesssim\mathcal{C}_{f}(\max\\{1,\rho,\widetilde{\lambda}\\})^{\nicefrac{{3}}{{p}}}\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu_{i+1},\widetilde{\mu}_{i+1}\right)$
(A.18b)

for all $N\leq N_{*}-\lfloor\nicefrac{{3}}{{p}}\rfloor-1$ and $M\leq M_{*}$.

Lastly, if the inequality (A.17) holds for all $N+M\leq N_{\circ}$ for some
$N_{\circ}\geq 0$ (instead of $N\leq N_{*}$ and $M\leq M_{*}$), then the
bounds (A.18a) and (A.18b) hold for $N+M\leq
N_{\circ}-\lfloor\nicefrac{{3}}{{p}}\rfloor-1$.

###### Proof of Lemma A.3.

The proof uses that $\lfloor\nicefrac{{3}}{{p}}\rfloor+1>\nicefrac{{3}}{{p}}$
for all $p\in[1,\infty]$, and that $W^{s,p}\subset L^{\infty}$ for
$s>\nicefrac{{3}}{{p}}$. Moreover, the proof of (A.18a) is nearly identical to
that of (A.18b), and thus we only give the proof of (A.18b); moreover, for
simplicity we only give the proof for $p=2$, as all the other Lebesgue
exponents are treated in the same way. By Gagliardo-Nirenberg-Sobolev
interpolation we have

$\displaystyle\left\|D^{N}D_{t}^{M}f\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i})}$
$\displaystyle\leq\left\|\psi_{i\pm}^{2}D^{N}D_{t}^{M}f\right\|_{L^{\infty}(\mathbb{T}^{3})}$
$\displaystyle\lesssim\left\|\psi_{i\pm}^{2}D^{N}D_{t}^{M}f\right\|_{L^{2}(\mathbb{T}^{3})}^{\nicefrac{{1}}{{4}}}\left\|\psi_{i\pm}^{2}D^{N}D_{t}^{M}f\right\|_{\dot{H}^{2}(\mathbb{T}^{3})}^{\nicefrac{{3}}{{4}}}+\left\|\psi_{i\pm}^{2}D^{N}D_{t}^{M}f\right\|_{L^{2}(\mathbb{T}^{3})}\,.$

Using (A.16), (A.17), and the monotonicity of the $\mu_{i}$ and
$\widetilde{\mu}_{i}$, we obtain

$\displaystyle\left\|\psi_{i\pm}^{2}D^{N}D_{t}^{M}f\right\|_{\dot{H}^{2}(\mathbb{T}^{3})}$
$\displaystyle\lesssim\left\|\psi_{i\pm}D^{N+2}D_{t}^{M}f\right\|_{L^{2}}+\left\|D\psi_{i\pm}\right\|_{L^{\infty}}\left\|\psi_{i\pm}D^{N+1}D_{t}^{M}f\right\|_{L^{2}}+\left\|\frac{D^{2}(\psi_{i\pm}^{2})}{\psi_{i\pm}}\right\|_{L^{\infty}}\left\|\psi_{i\pm}D^{N}D_{t}^{M}f\right\|_{L^{2}}$
$\displaystyle\lesssim\left\|\psi_{i\pm}D^{N+2}D_{t}^{M}f\right\|_{L^{2}}+\rho\left\|\psi_{i\pm}D^{N+1}D_{t}^{M}f\right\|_{L^{2}}+\rho^{2}\left\|\psi_{i\pm}D^{N}D_{t}^{M}f\right\|_{L^{2}}$
$\displaystyle\lesssim(\max\\{\widetilde{\lambda},\rho\\})^{2}\mathcal{C}_{f}\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu_{i+1},\widetilde{\mu}_{i+1}\right)\,,$

for all $N\leq N_{*}-2$ and $M\leq M_{*}$. In the second inequality above we
have used that $|D^{2}(\psi_{i\pm}^{2})|\lesssim\rho^{2}\psi_{i\pm}(x)$, which
follows from (A.16). Combining the above two displays proves (A.18b).

Note that for $p=1$ we require that
$|D^{4}(\psi_{i\pm}^{2})|\lesssim\rho^{4}\psi_{i\pm}(x)$, which also follows
from (A.16) since $\mathsf{N}_{\rm fin}\geq 4$, and this is why we have
assumed this inequality to hold for all $K\leq 4$.

Lastly, assume that (A.17) holds for all $N+M\leq N_{\circ}$, and fix any
$N^{\prime},M^{\prime}\geq 0$ such that $N^{\prime}+M^{\prime}\leq
N_{\circ}-\lfloor\nicefrac{{3}}{{p}}\rfloor-1$. Let
$N_{*}=N^{\prime}+\lfloor\nicefrac{{3}}{{p}}\rfloor+1$ and $M_{*}=M^{\prime}$.
Then (A.17) gives a bound for
$\|\psi_{i}D^{N^{\prime\prime}}D_{t}^{M^{\prime\prime}}f\|_{L^{p}}$ for all
$N^{\prime\prime}\leq N_{*}$ and $M^{\prime\prime}\leq M_{*}$. The bounds
(A.18a) and (A.18b) thus give an estimate for
$\|\psi_{i}D^{N^{\prime}}D_{t}^{M^{\prime}}f\|_{L^{p}}$, which concludes the
proof. ∎

###  A.5 Consequences of the Faa di Bruno formula

We are using the following version of the multivariable Faa di Bruno formula
[25, Theorem 2.1]. Let $g=g(x_{1},\ldots,x_{d})=f(h(x_{1},\ldots,x_{d}))$,
where $f\colon\mathbb{R}^{m}\to\mathbb{R}$, and
$h\colon\mathbb{R}^{d}\to\mathbb{R}^{m}$ are $C^{n}$ smooth functions of their
respective variables. Let $\alpha\in{\mathbb{N}}_{0}^{d}$ be s.t.
$|\alpha|=n$, and let $\beta\in\mathbb{N}_{0}^{m}$ be such that
$1\leq|\beta|\leq n$. We then define

$\displaystyle p(\alpha,\beta)$
$\displaystyle=\Bigg{\\{}(k_{1},\ldots,k_{n};\ell_{1},\ldots,\ell_{n})\in(\mathbb{N}_{0}^{m})^{n}\times(\mathbb{N}_{0}^{d})^{n}\colon\exists
s\mbox{ with }1\leq s\leq n\mbox{ s.t. }$
$\displaystyle\qquad\qquad|k_{j}|,|\ell_{j}|>0\Leftrightarrow 1\leq j\leq
s,\,0\prec\ell_{1}\prec\ldots\prec\ell_{s},\sum_{j=1}^{s}k_{j}=\beta,\sum_{j=1}^{s}|k_{j}|\ell_{j}=\alpha\Bigg{\\}}.$
(A.19)

Then the multivariable Faa di Bruno formula states that we have the equality

$\displaystyle\partial^{\alpha}g(x)=\alpha!\sum_{|\beta|=1}^{n}(\partial^{\beta}f)(h(x))\sum_{p(\alpha,\beta)}\prod_{j=1}^{n}\frac{(\partial^{\ell_{j}}h(x))^{k_{j}}}{k_{j}!(\ell_{j}!)^{k_{j}}}.$
(A.20)

Note that in (A.19) we have that $k_{j}=0\in\mathbb{N}_{0}^{m}$ and
$\ell_{j}=0\in N_{0}^{d}$ for $j\geq s+1$. Therefore, we could write the sums
and products with $j\in\\{1,\ldots,s\\}$ as sums for $j\in\\{1,\ldots,n\\}$.
Keeping in mind this convention, we importantly note that in (A.20) we can
have $|\ell_{j}|=0$ only if $|k_{j}|=0$, and in this case the entire term in
the product is equal to $1$. That is, the product in (A.20) only goes from $1$
to $s$, and in this case $|\ell_{j}|\geq 1$ for $j\in\\{1,\ldots,s\\}$. This
fact will be used frequently.

For applications to cutoff functions we apply this formula for scalar
functions $h$, i.e. $m=1$, while for applications to the perturbation or
Reynolds stress sections we apply this formula for vector fields $h$, i.e.
$m=3$.

Since throughout this manuscript the number of derivatives that we need to
estimate is uniformly bounded (say by $\mathsf{N}_{\rm fin}$), we may ignore
the factorial terms in (A.20) and include them in the implicit constant of
$\lesssim$. Using this convention, we summarize in the following lemma a
useful consequence of the Faá di Bruno formula above.

######  Lemma A.4 (Faá di Bruno).

Fix $N\leq\mathsf{N}_{\rm fin}$. Let $\psi\colon[0,\infty)\to[0,1]$ be a
smooth function obeying

$\displaystyle|D^{B}\psi|\lesssim\Gamma_{\psi}^{-2B}\psi^{1-B/\mathsf{N}_{\rm
fin}}$ (A.21)

for all $B\leq N$, and some $\Gamma_{\psi}>0$. Let $\Gamma,\lambda,\Lambda>0$
and $N_{*}\leq N$. Furthermore, let
$h\colon\mathbb{T}^{3}\times\mathbb{R}\to\mathbb{R}$ and denote

$g(x)=\psi(\Gamma^{-2}h(x)).$

Assume the function $h$ obeys

$\displaystyle\left\|D^{B}h\right\|_{L^{\infty}(\mathrm{supp\,}g)}\lesssim\mathcal{C}_{h}\mathcal{M}\left(B,N_{*},\lambda,\Lambda\right)$
(A.22)

for all $B\leq N$, where the implicit constant is independent of
$\lambda,\Lambda,\Gamma,\mathcal{C}_{h}>0$. Then, we have that for all points
$(x,t)\in\mathrm{supp\,}h$, the bound

$\displaystyle\frac{|D^{N}g|}{g^{1-N/\mathsf{N}_{\rm
fin}}}\lesssim\mathcal{M}\left(N,N_{*},\lambda,\Lambda\right)\max\\{(\Gamma_{\psi}\Gamma)^{-2}\mathcal{C}_{h},(\Gamma_{\psi}\Gamma)^{-2N}\mathcal{C}_{h}^{N}\\}$
(A.23)

holds. If the $\psi^{1-B/\mathsf{N}_{\rm fin}}$ factor on the right side of
(A.21) is replaced by $1$, then the $g^{1-N/\mathsf{N}_{\rm fin}}$ factor on
the left side of (A.23) also has to be replaced by $1$.

###### Proof of Lemma A.4.

The goal is to apply (A.19)–(A.20) with $f(x)=\psi(\Gamma^{-2}x)$. For
$(x,t)\in\mathrm{supp\,}(g)$ we obtain from (3.9), (A.21), and (A.23) that

$\displaystyle\frac{|D^{N}g|}{g^{1-N/\mathsf{N}_{\rm fin}}}$
$\displaystyle\lesssim\sum_{B=1}^{N}\frac{|D^{B}\psi|}{\psi^{1-B/\mathsf{N}_{\rm
fin}}}\psi^{(N-B)/\mathsf{N}_{\rm
fin}}\Gamma^{-2B}\sum_{p(\alpha,B)}\prod_{j=1}^{n}\left\|\partial^{\ell_{j}}h\right\|_{L^{\infty}(\mathrm{supp\,}g)}^{k_{j}}$
$\displaystyle\lesssim\sum_{B=1}^{N}(\Gamma_{\psi}\Gamma)^{-2B}\sum_{p(\alpha,B)}\prod_{j=1}^{n}\left(\mathcal{C}_{h}\mathcal{M}\left(\ell_{j},N_{*},\lambda,\Lambda\right)\right)^{k_{j}}$
$\displaystyle\lesssim\sum_{B=1}^{N}(\Gamma_{\psi}\Gamma)^{-2B}\mathcal{C}_{h}^{B}\mathcal{M}\left(N,N_{*},\lambda,\Lambda\right)$

for any $1\leq B\leq N$. The conclusion of the lemma follows upon bounding the
geometric sum. ∎

Frequently in the paper, we need a version of Lemma A.4 which also deals with
mixed spatial and material derivatives. A convenient statement is:

######  Lemma A.5 (Mixed derivative Faá di Bruno).

Fix $N,M\in\mathbb{N}$ such that $N+M\leq\mathsf{N}_{\rm fin}$. Let
$\psi\colon[0,\infty)\to[0,1]$ be a smooth function obeying

$\displaystyle|D^{B}\psi|\lesssim\Gamma_{\psi}^{-2B}\psi^{1-B/\mathsf{N}_{\rm
fin}}$ (A.24)

for all $B\leq N$ and a constant $\Gamma_{\psi}>0$. Let $v$ be a fixed vector
field, and denote $D_{t}=\partial_{t}+v\cdot\nabla$, which is a scalar
differential operator. Let
$\Gamma,\lambda,\widetilde{\lambda},\mu,\widetilde{\mu}\geq 1$ and
$N_{x},N_{t}\leq N$. Furthermore, let
$h\colon\mathbb{T}^{3}\times\mathbb{R}\to\mathbb{R}$ and denote

$g(x,t)=\psi(\Gamma^{-2}h(x,t)).$

Assume the function $h$ obeys

$\displaystyle\left\|D^{N^{\prime}}D_{t}^{M^{\prime}}h\right\|_{L^{\infty}(\mathrm{supp\,}g)}\lesssim\mathcal{C}_{h}\mathcal{M}\left(N^{\prime},N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M^{\prime},N_{t},\mu,\widetilde{\mu}\right)$
(A.25)

for all $N^{\prime}\leq N$, and $M^{\prime}\leq M$, where the implicit
constant is independent of
$\lambda,\widetilde{\lambda},\mu,\widetilde{\mu},\Gamma$, and
$\mathcal{C}_{h}$. Then, we have that for all points
$(x,t)\in\mathrm{supp\,}h$, the bound

$\displaystyle\frac{|D^{N}D_{t}^{M}g|}{g^{1-(N+M)/\mathsf{N}_{\rm
fin}}}\lesssim\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu,\widetilde{\mu}\right)\max\left\\{(\Gamma_{\psi}\Gamma)^{-2}\mathcal{C}_{h},((\Gamma_{\psi}\Gamma)^{-2}\mathcal{C}_{h})^{N+M}\right\\}$
(A.26)

holds. If the $\psi^{1-B/\mathsf{N}_{\rm fin}}$ factor on the right side of
(A.24) is replaced by $1$, then the $g^{1-(N+M)/\mathsf{N}_{\rm fin}}$ factor
on the left side of (A.26) also has to be replaced by $1$.

###### Proof of Lemma A.5.

Let $X(a,t)$ be the flow induced by the vector field $v$, with initial
condition $X(a,t)=x$. Denote by $a=X^{-1}(x,t)$ the inverse of the map $X$. We
then note that

$\displaystyle D_{t}^{M}g(x,t)=\left(\partial_{t}^{M}((g\circ
X)(a,t))\right)|_{a=X^{-1}(x,t)}.$

We wish to apply the above with the function $g(x,t)=\psi(\Gamma^{-2}h(x,t))$.
We apply the Faa di Bruno formula (A.19)–(A.20) with the one dimensional
differential operator $\partial_{t}^{M}$ to the composition $g\circ X$, note
that $\partial_{t}^{\beta_{i}}(h(X(a,t),t))=(D_{t}^{\beta_{i}}h)(X(a,t),t)$,
and then evaluate the resulting expression at $a=X^{-1}(x,t)$, to obtain

$\displaystyle
D_{t}^{M}g(x,t)=M!\sum_{B=1}^{M}\Gamma^{-2B}\psi^{(B)}(\Gamma^{-2}h(x,t))\sum_{\begin{subarray}{c}\\{\kappa,\beta\in\mathbb{N}^{M}\colon\\\
|\kappa|=B,\kappa\cdot\beta=M\\}\end{subarray}}\prod_{i=1}^{M}\frac{\left((D_{t}^{\beta_{i}}h)(x,t)\right)^{\kappa_{i}}}{\kappa_{i}!(\beta_{i}!)^{\kappa_{i}}}.$

We now apply $D^{N}$ to the above expression, use the Leibniz rule, and then
appeal again to the Faa di Bruno formula (A.19)–(A.20), this time for spatial
derivatives. We obtain

$\displaystyle D^{N}D_{t}^{M}g(x,t)$
$\displaystyle=M!N!\sum_{B=1}^{M}\sum_{K=0}^{N}\sum_{B^{\prime}=0}^{K}\Gamma^{-2(B+B^{\prime})}\psi^{(B+B^{\prime})}(\Gamma^{-2}h(x,t))\sum_{p(K,B^{\prime})}\prod_{j=1}^{K}\frac{(D^{\ell_{j}}h(x,t))^{k_{j}}}{k_{j}!(\ell_{j}!)^{k_{j}}}$
$\displaystyle\quad\times\sum_{\begin{subarray}{c}\\{\alpha\in\mathbb{N}^{M}\colon\\\
|\alpha|=N-K\\}\end{subarray}}\sum_{\begin{subarray}{c}\\{\kappa,\beta\in\mathbb{N}^{M}\colon\\\
|\kappa|=B,\kappa\cdot\beta=M\\}\end{subarray}}\prod_{i=1}^{M}\frac{D^{\alpha_{i}}(((D_{t}^{\beta_{i}}h)(x,t))^{\kappa_{i}})}{\alpha_{i}!\kappa_{i}!(\beta_{i}!)^{\kappa_{i}}}.$
(A.27)

Upon dividing by $g^{1-(N+M)/\mathsf{N}_{\rm fin}}$ and noting that
$B+B^{\prime}\leq M+N$, from (A.24), identity (A.27), the Leibniz rule, and
assumption (A.25), we obtain

$\displaystyle\frac{|D^{N}D_{t}^{M}g|}{g^{1-(N+M)/\mathsf{N}_{\rm fin}}}$
$\displaystyle\lesssim\sum_{B=1}^{M}\sum_{K=0}^{N}\sum_{B^{\prime}=0}^{K}(\Gamma_{\psi}\Gamma)^{-2(B+B^{\prime})}\mathcal{C}_{h}^{B^{\prime}}\mathcal{M}\left(K,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{C}_{h}^{B}\mathcal{M}\left(N-K,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu,\widetilde{\mu}\right)$
$\displaystyle\lesssim\mathcal{M}\left(N,N_{x},\lambda,\widetilde{\lambda}\right)\mathcal{M}\left(M,N_{t},\mu,\widetilde{\mu}\right)\sum_{B=1}^{M}\sum_{B^{\prime}=0}^{N}(\Gamma_{\psi}\Gamma)^{-2(B+B^{\prime})}\mathcal{C}_{h}^{B^{\prime}+B}$

from which (A.26) follows by summing the geometric series. ∎

######  Lemma A.6.

Given a smooth function $f\colon\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}$,
suppose that for $\lambda\geq 1$ the vector field
$\Phi\colon\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}^{3}$ satisfies the
estimate

$\displaystyle\left\|D^{N+1}\Phi\right\|_{L^{\infty}(\mathrm{supp\,}f)}$
$\displaystyle\lesssim\lambda^{N}$ (A.28)

for $0\leq N\leq N_{*}$. Then for any $1\leq N\leq N_{*}$ we have

$\displaystyle\left|D^{N}\left(f\circ\Phi\right)(x,t)\right|\lesssim$
$\displaystyle\sum_{m=1}^{N}\lambda^{N-m}\left|(D^{m}f)\circ\Phi(x,t)\right|$
(A.29)

and thus trivially we obtain

$\displaystyle\left|D^{N}\left(f\circ\Phi\right)(x,t)\right|\lesssim$
$\displaystyle\sum_{m=0}^{N}\lambda^{N-m}\left|(D^{m}f)\circ\Phi(x,t)\right|.$

for any $0\leq N\leq N_{*}$.

###### Proof of Lemma A.6.

Applying (A.20), noting that $|\ell_{j}|=0$ implies $|k_{j}|=0$, and
assumption (A.28), we have that for any multi index
$\alpha\in\mathbb{N}_{0}^{3}$ with $|\alpha|=N$,

$\displaystyle\left|\partial^{\alpha}\left(f\circ\Phi\right)(x,t)\right|$
$\displaystyle\lesssim\sum_{|\beta|=1}^{N}\left|((\partial^{\beta}f)\circ\Phi)(x,t)\right|\prod_{j=1}^{N}\sum_{p(\alpha,\beta)}\left|\left(\partial^{\ell_{j}}\Phi(x,t)\right)^{k_{j}}\right|$
$\displaystyle\lesssim\sum_{|\beta|=1}^{N}\left|(\partial^{\beta}f)\circ\Phi\right|\prod_{j=1}^{N}\sum_{p(\alpha,\beta)}\lambda^{({\left|\ell_{j}\right|-1})|k_{j}|}$
$\displaystyle\lesssim\sum_{m=1}^{N}\lambda^{N-m}\left|(D^{m}f)\circ\Phi\right|$

by the definition (A.19). Thus we obtain (A.29). ∎

In order to estimate the perturbation in $L^{p}$ spaces as well as terms
appearing in the Reynolds stress we will need the following abstract lemma,
which follows from Lemmas A.2 and A.6.

######  Lemma A.7.

Let $p\in\\{1,2\\}$, and fix integers $N_{*}\geq M_{*}\geq{\mathsf{N}_{\rm
dec}}\geq 1$. Suppose $f\colon\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}$ and
let $\Phi\colon\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}^{3}$ be a vector
field advected by an incompressible velocity field $v$, i.e.
$D_{t}\Phi=(\partial_{t}+v\cdot\nabla)\Phi=0$. Denote by $\Phi^{-1}$ the
inverse of the flow $\Phi$, which is the identity at a time slice which
intersects the support of $f$. Assume that for some
$\lambda,\nu,\widetilde{\nu}\geq 1$ and $\mathcal{C}_{f}>0$ the functions $f$
satisfies the estimates

$\displaystyle\left\|D^{N}D_{t}^{M}f\right\|_{L^{p}}$
$\displaystyle\lesssim\mathcal{C}_{f}\lambda^{N}\mathcal{M}\left(M,N_{t},\nu,\widetilde{\nu}\right)$
(A.30)

for all $N\leq N_{*}$ and $M\leq M_{*}$, and that $\Phi$, and $\Phi^{-1}$ are
bounded as

$\displaystyle\left\|D^{N+1}\Phi\right\|_{L^{\infty}(\mathrm{supp\,}f)}$
$\displaystyle\lesssim\lambda^{N}$ (A.31)
$\displaystyle\left\|D^{N+1}\Phi^{-1}\right\|_{L^{\infty}(\mathrm{supp\,}f)}$
$\displaystyle\lesssim\lambda^{N}$ (A.32)

for all $N\leq N_{*}$. Lastly, suppose that $\varphi$ is
$(\mathbb{T}/\mu)^{3}$-periodic, and that there exist parameters
$\widetilde{\zeta}\geq\zeta\geq\mu$ and $\mathcal{C}_{\varphi}>0$ such that

$\displaystyle\left\|D^{N}\varphi\right\|_{L^{p}}\lesssim\mathcal{C}_{\varphi}\mathcal{M}\left(N,N_{x},\zeta,\widetilde{\zeta}\right)\,$
(A.33)

for all $0\leq N\leq N_{*}$. If the parameters

$\lambda\leq\mu\leq\zeta\leq\widetilde{\zeta}$

satisfy

$\displaystyle\widetilde{\zeta}^{4}\leq\left(\frac{\mu}{2\pi\sqrt{3}\lambda}\right)^{{\mathsf{N}_{\rm
dec}}}\,,$ (A.34)

and we have

$2{\mathsf{N}_{\rm dec}}+4\leq N_{*}\,,$ (A.35)

then the bound

$\displaystyle\left\|D^{N}D_{t}^{M}\left(f\;\varphi\circ\Phi\right)\right\|_{L^{p}}$
$\displaystyle\lesssim\mathcal{C}_{f}\mathcal{C}_{\varphi}\mathcal{M}\left(N,N_{x},\zeta,\widetilde{\zeta}\right)\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)$
(A.36)

holds for $N\leq N_{*}$ and $M\leq M_{*}$.

######  Remark A.8.

We emphasize that (A.36) holds for the same range of $N$ and $M$ that (A.30)
holds, as soon as $N_{*}$ is sufficiently large compared to ${\mathsf{N}_{\rm
dec}}$ so that (A.35) holds.

######  Remark A.9.

We note that if estimate (A.30) is known to hold for $N+M\leq N_{\circ}$ for
some $N_{\circ}\geq 2{\mathsf{N}_{\rm dec}}+4$ (instead of, for $N\leq N_{*}$
and $M\leq M_{*}$), and if the bounds (A.31)–(A.32) hold for all $N\leq
N_{\circ}$, then it follows from the below proof that the bound (A.36) holds
for $N+M\leq N_{\circ}$ and $M\leq N_{\circ}-2{\mathsf{N}_{\rm dec}}-4$. The
only modification required to the proof (given below) is that instead of
considering the cases $N^{\prime}\leq N_{*}-{\mathsf{N}_{\rm dec}}-4$ and
$N^{\prime}>N_{*}-{\mathsf{N}_{\rm dec}}-4$, we now have to split according to
$N^{\prime}+M\leq N_{\circ}-{\mathsf{N}_{\rm dec}}-4$ and
$N^{\prime}+M>N_{\circ}-{\mathsf{N}_{\rm dec}}-4$. In the second case we use
that $N-N^{\prime\prime}\geq N_{\circ}-M-{\mathsf{N}_{\rm
dec}}-4\geq{\mathsf{N}_{\rm dec}}$, which holds exactly because $M\leq
N_{\circ}-2{\mathsf{N}_{\rm dec}}-4$.

###### Proof of Lemma A.7.

Since $D_{t}\Phi=0$ we have $D_{t}^{M}(\varphi\circ\Phi)=0$. Using that
$\mathrm{div\,}v\equiv 0$, so that $\Phi$ and $\Phi^{-1}$ preserve volume, and
Lemma A.6, which we may apply due to (A.31), we have

$\displaystyle\left\|D^{N}D_{t}^{M}\left(f\;\varphi\circ\Phi\right)\right\|_{L^{p}}$
$\displaystyle\lesssim\sum_{N^{\prime}=0}^{N}\left\|D^{N^{\prime}}D_{t}^{M}f\;D^{N-N^{\prime}}\left(\varphi\circ\Phi\right)\right\|_{L^{p}}$
$\displaystyle\lesssim\sum_{N^{\prime}=0}^{N}\sum_{N^{\prime\prime}=0}^{N-N^{\prime}}\lambda^{N-N^{\prime}-N^{\prime\prime}}\left\|D^{N^{\prime}}D_{t}^{M}f\;(D^{N^{\prime\prime}}\varphi)\circ\Phi\right\|_{L^{p}}$