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##  Appendix A List of acronyms

## Glossary

## Acronyms

2dFGRS

    2-degree Field Galaxy Redshift Survey
2PCF

    2-point correlation function
3PCF

    3-point correlation function
AA

    alpha angle
ACT

    Atacama Cosmology Telescope
ADS

    Airbus Defence and Space
AGN

    active galactic nucleus
ALMA

    Atacama Large Millimeter/Submillimetre Array
AM

    abundance matching
AOCS

    attitude and orbit-control system
AP

    Alcock-Paczynski
APE

    absolute pointing error
BAO

    baryon acoustic oscillation
BBN

    big bang nucleosynthesis
BFE

    brighter-fatter effect
BG

    blue grism
BNT

    Bernardeau–Nishimichi–Taruya
BOSS

    Baryon Oscillation Spectroscopic Survey
BPT

    Baldwin–Phillips–Terlevich
CCD

    charge-coupled device
CDPU

    Control and Data Processing Unit
CFHT

    Canada-France-Hawaii Telescope
CIB

    cosmic infrared background
CMB

    cosmic microwave background
CMD

    colour-magnitude diagram
CPC

    completeness and purity calibration
CPU

    central processing unit
CTI

    charge-transfer inefficiency
CU

    calibration unit
CXB

    cosmic X-ray background
DECam

    Dark Energy Camera
DES

    Dark Energy Survey
DESI

    Dark Energy Spectroscopic Instrument experiment
DI

    direct integration
DPU

    data-processing unit
DR

    data release
DR1

    first data release
DTCP

    daily telemetry communication period
DUNE

    Dark Universe Explorer
EAF

    Euclid Auxiliary Field
EC

    Euclid Consortium
EDF-F

    Euclid Deep Field Fornax
EDF-N

    Euclid Deep Field North
EDF-S

    Euclid Deep Field South
EDF

    Euclid Deep Field
EDS

    Euclid Deep Survey
EFS

    Euclid Flagship Simulation
EFT

    effective field theory
EGC

    extragalactic globular cluster
EoR

    epoch of reionisation
ERO

    Early Release Observations
ESA

    European Space Agency
ESAC

    European Space Astronomy Centre
ESOC

    European Science Operations Centre
ESOP

    early science operations
EUDF

    Euclid Ultra-Deep Field
EW

    equivalent width
EWS

    Euclid Wide Survey
FFT

    fast Fourier transform
FGS

    fine-guidance sensor
FITS

    flexible image transport system
FKP

    Feldman–Kaiser–Peacock
FoM

    figure of merit
FoV

    field of view
FPA

    focal-plane array
FWHM

    full width at half maximum
G3L

    galaxy-galaxy-galaxy lensing
GBTDS

    Galactic Bulge Time Domain Survey
GC

    globular cluster
GF

    Gaussian fit
GGL

    galaxy-galaxy lensing
GP

    Gaussian process
GR

    general relativity
HGA

    high-gain antenna
HOD

    halo-occupation distribution
HOS

    higher-order statistics
HSC

    Hyper Suprime-Cam
HST

     Hubble Space Telescope
IA

    intrinsic alignment
ICRS

    International Celestial Reference System
IGM

    intergalactic medium
ISW

    integrated Sachs–Wolfe
JWST

    James Webb Space Telescope
KiDS

    Kilo-Degree Survey
KS

    Kaiser–Squires
kSZ

    kinetic Sunyaev–Zeldovich
LED

    light-emitting diode
LF

    luminosity function
LoS

    line of sight
LRG

    luminous red galaxy
LSB

    low-surface brightness
LSS

    large-scale structure
LSST

    Legacy Survey of Space and Time
M2M

    M2 mechanism
MACC

    multi-accumulate
MAMBO

    Mocks with Abundance Matching in BOlogna
MCMC

    Markov chain Monte Carlo
MLI

    multi-layer thermal insulation
MOC

    mission operation centre
MS

    main sequence
NEP

    north ecliptic pole
NFW

    Navarro–Frenk–White
NIR

    near-infrared
NIRCAM

    Near-InfraRed Camera
NISP

    Near Infrared Spectrometer and Photometer
NLA

    nonlinear linear alignment
Pan-STARRS

    Panchromatic Survey Telescope and Rapid Response System
PDC

    phase-diversity calibration
PDF

    probability density function
PLM

    payload module
pRF

    probabilistic random forest
PRNU

    pixel-response non-uniformity
PSF

    point spread function
PTC

    photon-transfer curve
PV

    performance-verification
QE

    quantum efficiency
RG

    red grism
RMS

    root mean square
RoI

    region of interest
ROS

    reference observing sequence
RPE

    relative pointing error
RSD

    redshift-space distortion
RSU

    readout shutter unit
S/N

    signal-to-noise ratio
SAA

    Solar aspect angle
SBF

    surface brightness fluctuation
SDC

    Science Data Centre
SDSS

    Sloan Digital Sky Survey
SED

    spectral energy distribution
SEF

    single-epoch frame
SEP

    South Ecliptic Pole
SFE

    surface figure error
SFR

    star-formation rate
SGS

    Science Ground Segment
SHMR

    stellar mass-halo mass relation
SiC

    silicon carbide
SMBH

    supermassive black hole
SMF

    stellar mass function
SNe

    supernovae
SO

    Simons Observatory
SOC

    Science Operations Centre
SOM

    self-organising map
SPACE

    Spectroscopic All-Sky Cosmic Explorer
SPT

    South Pole Telescope
SPV

    Science Performance Verification
SSO

    Solar System object
STOP

    structural thermal optical performance
SVM

    service module
SZ

    Sunyaev–Zeldovich
TCM

    transfer correction manoeuvre
TDE

    tidal disruption event
tSZ

    thermal Sunyaev–Zeldovich
UCD

    ultra-cool dwarf
UNIONS

    Ultraviolet Near Infrared Optical Northern Survey
VIPERS

    VIMOS Public Extragalactic Redshift Survey
VISTA

    Visible and Infrared Survey Telescope for Astronomy
WFC3

    Wide Field Camera 3
WISHES

    Wide Imaging with Subaru-Hyper Suprime-Cam Euclid Sky
WL

    weak lensing
ZP

    zero point
$z$PDF

    redshift probability density function


max}})+21+\mathsf{c}_{0}-\mathsf{c}_{\widetilde{\textnormal{n}}}}\Theta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}r_{q+1,{\widetilde{n}}}^{-1}\lambda_{q+1}^{-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
$\displaystyle\quad\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+5},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right)$
(7.62)

after also using (8.27) and (8.56).

To conclude the proof, we must still estimate the nonlocal ($\mathcal{R}^{*}$)
portion of the inverse divergence, and the error terms coming from the
divergence correctors. These error terms, however, obey stronger estimates
than the bounds in (7.61) and (7.62), and so we refer to the proof of [5,
Lemma 8.12] for further details. ∎

###  7.8. Nash errors

######  Lemma 7.14.

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

$\displaystyle\left\|\psi_{i,q}D^{N}D_{t,q}^{M}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)\left(w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}}\right)\right)\right\|_{L^{1}}$
$\displaystyle\lesssim\delta_{q+2}\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)$
(7.63a)
$\displaystyle\left\|D^{k}D_{t,q}^{m}\left(\left(\mathcal{H}+\mathcal{R}^{*}\right)(w_{q+1,{\widetilde{n}}}\cdot\nabla
v_{\ell_{q}})\right)\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}$
$\displaystyle\lesssim\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\lambda_{q+1}^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4},\Gamma_{q+1}^{-1}\widetilde{\tau}_{q}^{-1}\right)\,.$
(7.63b)

###### Proof of Lemma 7.14.

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

$\displaystyle w_{q+1,{\widetilde{n}}}\cdot\nabla v_{\ell_{q}}$
$\displaystyle=\sum_{i,j,k,\vec{l},\xi}\mathrm{curl\,}\left(a_{\xi,i,j,k,q,{\widetilde{n}},\vec{l}}\nabla\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)\cdot\nabla
v_{\ell_{q}}$ $\displaystyle=\Biggl{(}\sum_{i,j,k,\vec{l},\xi}\nabla
a_{(\xi)}\times\left(\Phi_{(i,k)}^{T}\mathbb{U}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\right)+\sum_{i,j,k,\vec{l},\xi}a_{(\xi)}\nabla\Phi_{(i,k)}^{-1}\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}\Biggr{)}\cdot\nabla
v_{\ell_{q}}\,.$ (7.64)

Due to the fact that the first term arises from the addition of the corrector
defined in (7.6), and the fact that the bounds for the corrector in (7.12a)
are stronger than that of the principal part of the perturbation, we shall
only consider the second term. Note that the Nash error can be written as
$\mathrm{div\,}(w_{q+1,{\widetilde{n}}}\otimes v_{\ell_{q}})$ and so has zero
mean. Thus, although each individual term in the final equality in (7.64) may
not have zero mean, we can safely apply $\mathcal{H}$ and $\mathcal{R}^{*}$ to
each term and estimate the outputs while ignoring the last term in (A.16).

We will apply Proposition A.2 to the second term with the following choices.
We set $v=v_{\ell_{q}}$, and
$D_{t}=D_{t,q}=\partial_{t}+v_{\ell_{q}}\cdot\nabla$ as usual. We set
$N_{*}=M_{*}=\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,x}-\mathsf{N}_{\rm cut,t}-4\right)\rfloor$, with ${\mathsf{N}_{\rm dec}}$
and ${\mathsf{d}}$ satisfying (8.58a). We define

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

and set $\mathcal{C}_{G,1}$, $\mathcal{C}_{G,\infty}$ to be equal to the
quantities in (7.59),
$\lambda=\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}$,
$\nu=\tau_{q}^{-1}\Gamma_{q+1}^{i-\mathsf{c}_{\widetilde{\textnormal{n}}}+4}$,
$M_{t}=\mathsf{N}_{\textnormal{ind,t}}$, and
$\widetilde{\nu}=\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}$. Note that these
choices match exactly the choices from the estimates on the transport error.
From (7.8) with $r=1$ and $r_{1}=r_{2}=2$, (5.19g), and (2.19) at level $q$,
we have that for
$N,M\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,x}-\mathsf{N}_{\rm cut,t}-4\right)\rfloor$

$\displaystyle\left\|D^{N}D_{t,q}^{M}G\right\|_{L^{1}}$
$\displaystyle\lesssim\mathcal{C}_{G,p}\left(\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}\right)^{N}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\tau_{q}^{-1}\Gamma_{q+1}^{i+1},\widetilde{\tau}_{q}^{-1}\Gamma_{q+1}^{-1}\right),$
(7.65)

and so (A.4) is satisfied. Note that we have used (8.30) when converting the
$\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}$ coming from (2.19)
at level $q$ to a $\tau_{q}^{-1}$. Setting $\Phi=\Phi_{(i,k)}$ and
$\lambda^{\prime}=\widetilde{\lambda}_{q}$, we have that (A.5) and (A.6) are
satisfied as usual. The choices of $\varrho$, $\vartheta$, $\zeta$, $\mu$,
$\Lambda$, and $\mathcal{C}_{*}$ are identical to those of the transport error
(both terms contain $\mathbb{W}_{\xi,q+1,{\widetilde{n}}}\circ\Phi_{(i,k)}$),
and so we have that (i)-(ii), (A.7), (A.8), and (A.9) are satisfied as well.
Since the bound (7.65) is identical to that of (7.60), we obtain an estimate
identical to (7.61) in the case $p=1$. The case $p=\infty$ and the estimates
for the $\mathcal{R}^{*}$ portion follows analogously to that for the first
term from the transport error. We omit further details. ∎

##  8\. Parameters

The purpose of the first subsection is to 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 8.2, we define the parameters
which depend on $q$, as well as the parameters which depend in addition on
$n$. Section 8.3 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.

###  8.1. Definitions and hierarchy of the parameters

The parameters in our construction are chosen as:

  1. (i)

Choose an $L^{2}$ regularity index
$\beta\in[\nicefrac{{1}}{{3}},\nicefrac{{1}}{{2}})$; in light of [4, 36],
there is no reason to take $\beta<\nicefrac{{1}}{{3}}$.

  2. (ii)

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

$\displaystyle 2\beta b$ $\displaystyle<1\,.$ (8.1)

  3. (iii)

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

    1. (a)

The parameter ${n_{\rm max}}$, which denotes the total number of higher order
stresses $\mathring{R}_{q,n}$, is defined as the smallest integer such that

$\displaystyle\frac{2}{{n_{\rm max}}+1}$ $\displaystyle<\frac{(b-1)^{2}}{2b}$
(8.2a) $\displaystyle 2\beta b+\frac{3+\lceil\log_{2}{n_{\rm
max}}\rceil}{2({n_{\rm max}}+1)}$ $\displaystyle<\frac{1}{2}+\frac{{n_{\rm
max}}}{2({n_{\rm max}}+1)}\,.$ (8.2b)

Notice that the second inequality is possible since $2\beta b<1$.

    2. (b)

The parameter ${\mathsf{C}_{b}}$ appearing in (2.17) to quantify
$\left\|\psi_{i,q}\right\|_{L^{1}}$ is defined as

$\displaystyle{\mathsf{C}_{b}}=\frac{b+4}{b-1}\,.$ (8.3)

    3. (c)

The exponent $\mathsf{C_{R}}$ is a small parameter used to estimate the
Reynolds stress, cf. (2.10a), and then absorb geometric constants in the
construction. It is defined as

$\displaystyle\mathsf{C_{R}}=10b+1\,.$ (8.4)

  4. (iv)

The parameter $\mathsf{c_{0}}$, which is first introduced in (2.16) and
utilized in Sections 6 and 7 to control small losses in the sharp material
derivative estimates, is defined in terms of ${n_{\rm max}}$ as

$\displaystyle\mathsf{c_{0}}=4{n_{\rm max}}+5\,.$ (8.5)

  5. (v)

The parameter $\varepsilon_{\Gamma}>0$, which is used in (8.19) to quantify
the _finest_ frequency scale between $\lambda_{q}$ and $\lambda_{q+1}$
utilized throughout the scheme, is defined as any real number such that

$\displaystyle\varepsilon_{\Gamma}300({n_{\rm max}}+1)(\lceil\log_{2}{n_{\rm
max}}\rceil)$ $\displaystyle<b-1$ (8.6a) $\displaystyle
7\varepsilon_{\Gamma}(2+\lceil\log_{2}{n_{\rm max}}\rceil+22+4{n_{\rm max}})$
$\displaystyle<\frac{b-1}{2b}-\frac{3}{2(b-1)({n_{\rm max}}+1)}$ (8.6b)
$\displaystyle\varepsilon_{\Gamma}\left(5+(2+\lceil\log_{2}{n_{\rm
max}}\rceil)(9+{\mathsf{C}_{b}})\right)$
$\displaystyle<\frac{1}{2}-\frac{2+\lceil\log_{2}{n_{\rm max}}\rceil}{{n_{\rm
max}}}$ (8.6c)
$\displaystyle\varepsilon_{\Gamma}\left(\mathsf{C_{R}}\left(\frac{b-1}{b}\right)+15+9(3+\lceil\log_{2}({n_{\rm
max}})\rceil)\right)$ $\displaystyle<\frac{1}{2}\left(1+\frac{{n_{\rm
max}}}{{n_{\rm max}}+1}\right)-2\beta b-\frac{3+\lceil\log_{2}{n_{\rm
max}}\rceil}{2({n_{\rm max}}+1)}\,$ (8.6d)
$\displaystyle\varepsilon_{\Gamma}\left(\frac{1}{2}{\mathsf{C}_{b}}+\mathsf{c}_{0}+10+\frac{1}{2}\mathsf{C_{R}}\right)$
$\displaystyle<1-2\beta b$ (8.6e)
$\displaystyle\varepsilon_{\Gamma}\left(7+\mathsf{C_{R}}+{n_{\rm
max}}(8+{\mathsf{C}_{b}})\right)$ $\displaystyle<\frac{1-2\beta}{10}$ (8.6f)
$\displaystyle 2b\varepsilon_{\Gamma}(\mathsf{c}_{0}+7)<1-\beta\,.$ (8.6g)

We note that the right-hand side of (8.6b) is positive from (8.2a) and the
right-hand sides of (8.6c) and (8.6d) are positive from (8.2b).

  6. (vi)

The parameter ${\mathsf{C_{u}}}$ is defined as

${\mathsf{C_{u}}}=\frac{1}{\varepsilon_{\Gamma}(b-1)({n_{\rm max}}+1)}\,.$
(8.7)

  7. (vii)

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

$\displaystyle\alpha=\frac{\varepsilon_{\Gamma}(b-1)}{2b}\,.$ (8.8)

  8. (viii)

The parameters $\mathsf{N}_{\rm cut,t}$ and $\mathsf{N}_{\rm cut,x}$ are used
in Section 5 in order to define the velocity and stress cutoff functions; see
(5.3), (5.7), and (5.24). These large integers are chosen solely in terms of
$b$ and $\varepsilon_{\Gamma}$ as

$\displaystyle\frac{1}{2}\mathsf{N}_{\rm cut,x}=\mathsf{N}_{\rm
cut,t}=\left\lceil\frac{3b}{\varepsilon_{\Gamma}(b-1)}+\frac{15b}{2}\right\rceil\,.$
(8.9)

  9. (ix)

The parameter $\mathsf{N}_{\textnormal{ind,t}}$, which is the number of sharp
material derivatives propagated on stresses and velocities in Sections 2
through 7, is chosen as the smallest integer for which we have

$\displaystyle\mathsf{N}_{\textnormal{ind,t}}=\left\lceil\frac{4}{\varepsilon_{\Gamma}(b-1)}\right\rceil\mathsf{N}_{\rm
cut,t}\,.$ (8.10)

  10. (x)

The parameter $\mathsf{N}_{\textnormal{ind,v}}$, whose primary role is to
quantify the number of sharp spatial derivatives propagated on the velocity
increments and stresses, cf. (2.8a) and (2.10a), is chosen as the smallest
integer for which we have the bound

$\displaystyle
4b\mathsf{N}_{\textnormal{ind,t}}+8+b(\mathsf{C_{R}}+3)\varepsilon_{\Gamma}(b-1)+2\beta(b^{3}-1)$
$\displaystyle<\varepsilon_{\Gamma}(b-1)\mathsf{N}_{\textnormal{ind,v}}\,.$
(8.11)

  11. (xi)

The value of the decoupling parameter ${\mathsf{N}_{\rm dec}}$, which is used
in the $L^{p}$ decorrellation conditions (A.2) and (A.9), is chosen as the
smallest integer for which

$\displaystyle{\mathsf{N}_{\rm dec}}>\frac{8b}{(b-1)\varepsilon_{\Gamma}}\,.$
(8.12)

  12. (xii)

The parameter ${\mathsf{d}}$, which is used in the inverse divergence operator
of Proposition A.2 to count the order of a parametrix expansion, is chosen as
the smallest integer for which we have

$\displaystyle({\mathsf{d}}-1)\varepsilon_{\Gamma}(b-1)$
$\displaystyle>b(6+13\mathsf{N}_{\textnormal{ind,v}})+2\beta
b^{2}+(2+\lceil\log_{2}{n_{\rm max}}\rceil)\left(\frac{b-1}{2({n_{\rm
max}}+1)}+\varepsilon_{\Gamma}(b-1)(9+{\mathsf{C}_{b}})\right)\,.$ (8.13)

  13. (xiii)

The value of $\mathsf{N}_{\rm fin}$, which is introduced in Section 2 and used
to quantify the highest order derivative estimates utilized throughout the
scheme is chosen as the smallest integer such that

$\displaystyle\frac{3}{2}\mathsf{N}_{\rm fin}>(2\mathsf{N}_{\rm
cut,t}+\mathsf{N}_{\rm
cut,x}+14\mathsf{N}_{\textnormal{ind,v}}+2{\mathsf{d}}+2{\mathsf{N}_{\rm
dec}}+12)2^{{n_{\rm max}}+1}\,.$ (8.14)

  14. (xiv)

Having chosen all the previous parameters in items (i)–(xiii), there exists 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)–(xiii) — never on $q$ — which justifies the existence of $a_{*}$.

###  8.2. Definitions of the $q$-dependent parameters

####  8.2.1. Parameters which depend only on $q$

For $q\geq 0$, we define the fundamental frequency parameter as

$\displaystyle\lambda_{q}$
$\displaystyle=2^{\big{\lceil}(b^{q})\log_{2}a\big{\rceil}}\,.$ (8.15)

Definition (8.15) gives that $\lambda_{q}$ is an integer power of $2$, and
that we have the bounds

$\displaystyle a^{(b^{q})}\leq\lambda_{q}\leq
2a^{(b^{q})}\qquad\mbox{and}\qquad\frac{1}{3}\lambda_{q}^{b}\leq\lambda_{q+1}\leq
2\lambda_{q}^{b}$ (8.16)

for all $q\geq 0$. Throughout the paper, if there exists a universal constant
$C>0$ such that $C^{-1}A\leq B\leq CA$, we say that $A\approx B$. In
particular, the above reads $\lambda_{q}\approx a^{(b^{q})}$ and
$\lambda_{q+1}\approx\lambda_{q}^{b}$. It will be convenient to denote the
quotient of two consecutive frequency parameters by

$\displaystyle\Theta_{q+1}=\lambda_{q+1}\lambda_{q}^{-1}\approx\lambda_{q}^{b-1}\,.$
(8.17)

The fundamental amplitude parameter is defined in terms of $\lambda_{q}$ by

$\displaystyle\delta_{q}$
$\displaystyle=\lambda_{1}^{(b+1)\beta}\lambda_{q}^{-2\beta}\,.$ (8.18)

We now introduce a parameter which is defined in terms of the parameter
$\varepsilon_{\Gamma}$ from (8.6) and used repeatedly to mean “a tiny power of
the frequency parameter”:

$\displaystyle\Gamma_{q+1}$
$\displaystyle=\Theta_{q+1}^{\varepsilon_{\Gamma}}\,.$ (8.19)

In order to cap off our derivative losses, we need to mollify in space and
time using the operators described in Section 4. This is done in terms of the
following space and time parameters:

$\displaystyle\widetilde{\lambda}_{q}$
$\displaystyle=\lambda_{q}\Gamma_{q+1}^{5}$ (8.20)
$\displaystyle\widetilde{\tau}_{q}^{-1}$
$\displaystyle=\tau_{q}^{-1}\widetilde{\lambda}_{q}^{3}\widetilde{\lambda}_{q+1}\,.$
(8.21)

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

$\displaystyle\tau_{q}$
$\displaystyle=\bigl{(}\delta_{q}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q}\Gamma_{q+1}^{\mathsf{c_{0}}+6}\bigr{)}^{-1}\,.$
(8.22)

Note that besides depending on the parameters introduced in (i)–(xiv), the
parameters introduced above only depend on $q$, but are independent of $n$. We
note that the definitions of the parameters listed so far in this subsection
have not been changed from the definitions used in [5].

####  8.2.2. Parameters which depend on $q$ and $n$

The rest of the parameters depend on both $q$ and $n$. We start by defining
the frequency parameter $\lambda_{q,n}$ and the intermittency parameter
$r_{q+1,n}$ by

$\displaystyle\lambda_{q,n}$
$\displaystyle=\begin{cases}2^{\lceil(1+6(b-1)\varepsilon_{\Gamma})\log_{2}\lambda_{q}\rceil},&n=0\\\
2^{\lceil(\frac{1}{2}-\frac{n}{2({n_{\rm
max}}+1)})\log_{2}\lambda_{q}+(\frac{1}{2}+\frac{n}{2({n_{\rm
max}}+1)})\log_{2}\lambda_{q+1}\rceil},&1\leq n\leq{n_{\rm
max}}\end{cases}\,,$ (8.23) $\displaystyle r_{q+1,n}$
$\displaystyle=\lambda_{q+1}^{-1}2^{\lceil\frac{1}{2}\log_{2}\lambda_{q,n}+\frac{1}{2}\log_{2}\lambda_{q+1}-2\log_{2}\Gamma_{q+1}\rceil}$
(8.24)

for $0\leq n\leq{n_{\rm max}}$. In particular, (8.23) shows that
$\lambda_{q,n}$ is a power of $2$, with
$\lambda_{q,0}\approx\lambda_{q}\Gamma_{q+1}^{6}$ and
$\lambda_{q,n}\approx\lambda_{q}^{\frac{1}{2}-\frac{n}{2({n_{\rm
max}}+1)}}\lambda_{q+1}^{\frac{1}{2}+\frac{n}{2({n_{\rm max}}+1)}}$ for $1\leq
n\leq{n_{\rm max}}$. Similarly, (8.24) shows that $\lambda_{q+1}r_{q+1,n}$ is
an integer power of $2$, and we have
$\lambda_{q+1}r_{q+1,{\widetilde{n}}}\approx\lambda_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{-2}$.
A consequence of these approximations are the inequalities

$r_{q+1,{\widetilde{n}}}^{-2}\leq
2\frac{\lambda_{q+1}}{\lambda_{q}}=2\Theta_{q+1}\,,\qquad
r_{q+1,{\widetilde{n}}}^{2}\leq
2\Gamma_{q+1}^{-4}\frac{\lambda_{q,{\widetilde{n}}}}{\lambda_{q+1}}\,.$ (8.25)

We recall from (2.7c) that the stresses $\mathring{R}_{q,n}$ for $0\leq
n\leq{n_{\rm max}}$ will be measured in terms of

$\displaystyle\delta_{q+1,n}=\begin{cases}\delta_{q+1}\Gamma_{q}^{-\mathsf{C_{R}}},&n=0\\\
\displaystyle\delta_{q+1,0}\frac{\widetilde{\lambda}_{q}}{\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{\nicefrac{{1}}{{2}}}}\Gamma_{q+1}^{9},&n=1\\\
\displaystyle\delta_{q+1,0}\frac{\widetilde{\lambda}_{q}}{\lambda_{q,n-1}}\Gamma_{q+1}^{8}\left(\Theta_{q+1}^{\frac{1}{2({n_{\rm
max}}+1)}}\Gamma_{q+1}^{9}\right)^{\Upsilon(n)},&2\leq n\leq{n_{\rm
max}}\,.\end{cases}$ (8.26)

The function $\Upsilon(n)$ is defined in (6.6) to quantify the number of steps
required to produce $\mathring{R}_{q,n}$. As each step accumulates negligible
losses, which correspond to the quantity in parentheses above, one may adhere
to the heuristic that $\delta_{q+1,n}$ is roughly speaking equal to
$\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n}}$. We remark that each of the
parameters defined so far in this subsubsection has a new definition compared
to that of [5].

Conversely, the following three parameters remain unchanged when compared to
[5]. For $1\leq n\leq{n_{\rm max}}$, we define $\mathsf{c}_{\textnormal{n}}$
in terms of $\mathsf{c_{0}}$ by

$\displaystyle\mathsf{c}_{\textnormal{n}}$
$\displaystyle=\mathsf{c_{0}}-4n\,.$ (8.27)

For $n=0$, we set

$\mathsf{N}_{\textnormal{fin},0}=\frac{3}{2}\mathsf{N}_{\rm fin},$ (8.28)

while for $1\leq n\leq{n_{\rm max}}$, we define $\mathsf{N}_{\rm fin,n}$
inductively on $n$ by using (8.28) and the formula

$\mathsf{N}_{\rm
fin,n}=\left\lfloor\frac{1}{2}\left(\mathsf{N}_{\textnormal{fin},\textnormal{n}-1}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-6\right)-{\mathsf{d}}\right\rfloor\,.$ (8.29)

###  8.3. Inequalities and consequences of the parameter definitions

Due to (8.15) 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 (xiv), 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 the definition (8.22) of $\tau_{q}$ and (8.27), which gives that
$\mathsf{c}_{\textnormal{n}}$ is decreasing with respect to $n$, we have that
for all $0\leq n\leq{n_{\rm max}}$,

$\displaystyle\Gamma_{q+1}^{\mathsf{c}_{\textnormal{n}}+6}\delta_{q}^{\nicefrac{{1}}{{2}}}{\widetilde{\lambda}_{q}}$
$\displaystyle\leq\tau_{q}^{-1}\,.$ (8.30)

Using the definitions (8.18), (8.19), (8.20), and (8.22), and writing out
everything in terms of $\lambda_{q-1}$, we have

$\displaystyle\tau_{q-1}^{-1}\Gamma_{q+1}^{3+\mathsf{c_{0}}}$
$\displaystyle\leq\tau_{q}^{-1}\,.$ (8.31)

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

$\displaystyle\tau_{q}^{-1}{\widetilde{\lambda}_{q}^{4}}$
$\displaystyle\leq{\widetilde{\tau}_{q}^{-1}}\leq\tau_{q}^{-1}{\widetilde{\lambda}_{q}^{3}}{\widetilde{\lambda}_{q+1}}\,.$
(8.32)

From the definitions (8.5) of $\mathsf{c_{0}}$ and (8.27) of
$\mathsf{c}_{\textnormal{n}}$, we have that for all $0\leq n\leq{n_{\rm
max}}$,

$-\mathsf{c}_{\textnormal{n}}+4\leq-1.$ (8.33)

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 (8.11). First, we note from (8.32) that

$\displaystyle\widetilde{\tau}_{q-1}^{-1}\tau_{q-1}\leq\widetilde{\lambda}_{q-1}^{3}\widetilde{\lambda}_{q}\leq\lambda_{q}^{4}$
(8.34)

where in the second inequality we have used that
$\varepsilon_{\Gamma}\leq\frac{3}{20b}$.

The fact that $\mathsf{N}_{\textnormal{ind,t}}$ is taken to be much larger
than $\mathsf{N}_{\rm cut,t}$, as expressed in (8.10), implies when combined
with (8.34) the following bound, which is also used in Section 5:

$\displaystyle\left(\tau_{q}\widetilde{\tau}_{q}^{-1}\right)^{\mathsf{N}_{\rm
cut}}\leq\Gamma_{q+1}^{\mathsf{N}_{\textnormal{ind,t}}}$ (8.35)

for all $q\geq 1$. The parameter $\alpha$ in (8.8) is chosen as such in order
to ensure that

$\lambda_{q+1}^{\alpha}\approx\Gamma_{q+1}.$ (8.36)

for all $q\geq 0$. We note that the previous seven inequalities only involve
parameters which have not changed when compared to [5].

Next, we list a number of parameter inequalities which are not the same as
those in [5]. Our choice of ${\mathsf{N}_{\rm dec}}$ in (8.12) and the
assumption that $a$ is chosen sufficiently large so that
$\Gamma_{q+1}^{\nicefrac{{1}}{{2}}}>2\pi\sqrt{3}$ yields

$\displaystyle\lambda_{q+1}^{4}\leq\left(\frac{\Gamma_{q+1}}{2\pi\sqrt{3}}\right)^{{\mathsf{N}_{\rm
dec}}}\qquad\impliedby\qquad\frac{8b}{(b-1)\varepsilon_{\Gamma}}<{\mathsf{N}_{\rm
dec}}\,.$ (8.37)

We need a number of new inequalities to manage the Type 1 oscillation errors.
The first of these is

$\displaystyle\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{8}\leq\begin{cases}\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\Gamma_{q+1}^{9}\left(\Theta_{q+1}^{\frac{1}{2({n_{\rm
max}}+1)}}\Gamma_{q+1}^{9}\right)^{{\widetilde{n}})}&\mbox{if}\quad{\widetilde{n}}=0,1\\\
\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\Gamma_{q+1}^{9}\left(\Theta_{q+1}^{\frac{1}{2({n_{\rm
max}}+1)}}\Gamma_{q+1}^{9}\right)^{\Upsilon({\widetilde{n}})+1}&\mbox{if}\quad
2\leq{\widetilde{n}}\leq{n_{\rm max}}\end{cases}\,.$ (8.38)

If ${\widetilde{n}}=0$, then the inequality follows from (8.26), (8.20),
(8.23), and (6.6). If ${\widetilde{n}}=1$, the inequality follows from the
aforementioned inequalities and the equality

$\frac{\lambda_{q,1}}{\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{\nicefrac{{1}}{{2}}}}=\frac{\lambda_{q,{\widetilde{n}}}}{\lambda_{q,{\widetilde{n}}-1}}=\frac{\lambda_{q+1}}{\lambda_{q,{n_{\rm
max}}}}=\Theta_{q+1}^{\frac{1}{2({n_{\rm max}}+1)}}\,,$ (8.39)

which holds for $2\leq{\widetilde{n}}\leq{n_{\rm max}}$. Finally, if
$2\leq{\widetilde{n}}\leq{n_{\rm max}}$, we use the aforementioned
inequalities in conjunction with (8.39). Next, we claim that for all
$0\leq{\widetilde{n}}\leq{n_{\rm max}}$,

$\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\widetilde{\lambda}_{q}\Gamma_{q+1}^{9}\left(\Theta_{q+1}^{\frac{1}{2({n_{\rm
max}}+1)}}\Gamma_{q+1}^{9}\right)^{\Upsilon({\widetilde{n}})+1}\lambda_{q,{n_{\rm
max}}}^{-1}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\,.$ (8.40)

The above inequality is a consequence of (6.8) and

$\displaystyle 2\beta
b(b-1)+(b-1)\varepsilon_{\Gamma}\left({-\mathsf{C_{R}}}\left(\frac{1}{b}-1\right)+15+9(3+\lceil\log_{2}({n_{\rm
max}})\rceil)\right)+\left(3+\lceil\log_{2}{n_{\rm
max}}\rceil\right)\frac{b-1}{2({n_{\rm max}}+1)}$
$\displaystyle\qquad\qquad<\frac{b-1}{2}+\frac{{n_{\rm max}}}{2({n_{\rm
max}}+1)}(b-1)\,,$ (8.41)

which in turn follows from (8.2b) and (8.6d). Finally, we claim that for $n$
such that $n>r({\widetilde{n}})$, as defined in (6.5), and $n>2$,

$\displaystyle\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\Gamma_{q+1}^{9}\lambda_{q,n-1}^{-1}\leq\delta_{q+1,n}\,.$
(8.42)

If ${\widetilde{n}}=0,1$, the inequality follows from the definitions of
$\lambda_{q,0}$ and $\lambda_{q,1}$ in (8.23), the definition of the
$\delta_{q+1,n}$’s in (8.26), and (6.6), which guarantees that
$\Upsilon(n)\geq 1$ for $n\geq 2$. In the case
$2\leq{\widetilde{n}}\leq{n_{\rm max}}$, the inequality follows from the
aforementioned inequalities combined with (8.39) and the fact that for
$n>r({\widetilde{n}})$, (6.7) gives that
$\Upsilon(n)\geq\Upsilon({\widetilde{n}})+1$.

The amplitudes of the higher order corrections $w_{q+1,n,p}$ must meet the
inductive assumptions stated in (2.9a). Towards this end, we claim that for
all $0\leq{\widetilde{n}}\leq{n_{\rm max}}$,

$\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\Gamma_{q+1}^{5}\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,.$
(8.43)

Indeed, the case ${\widetilde{n}}=0$ follows from the definition of
$\mathsf{C_{R}}$ in (8.4), while the case ${\widetilde{n}}\geq 1$ is a
consequence of the definition (8.26) and the inequality

$\varepsilon_{\Gamma}(b-1)\left(5+(2+\lceil\log_{2}{n_{\rm
max}}\rceil)(9+{\mathsf{C}_{b}})\right)+(b-1)\frac{2+\lceil\log_{2}{n_{\rm
max}}\rceil}{{n_{\rm max}}}<\frac{b-1}{2}\,,$ (8.44)

which in turn is a consequence of (8.2b) and (8.6c).

We will also need that

$\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+13}\lambda_{q,{\widetilde{n}}}r_{q+1,{\widetilde{n}}}^{-2}\lambda_{q,{n_{\rm
max}}}^{-1}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-2}\,.$ (8.45)

The above inequality is a consequence of (2.7a), (8.39), and

$\varepsilon_{\Gamma}\left(\frac{{\mathsf{C_{u}}}}{b}+14\left(2+\lceil\log_{2}{n_{\rm
max}}\rceil\right)+20-{\mathsf{C_{u}}}\right)+\frac{1}{2({n_{\rm
max}}+1)}<0\,,$ (8.46)

which holds due to the choice of ${\mathsf{C_{u}}}$ in (8.7) and (8.6a). The
inequality (8.45) then immediately implies that

$\displaystyle\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+13}\lambda_{q,{\widetilde{n}}}r_{q+1,{\widetilde{n}}}^{-2}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-2}\,.$
(8.47)

We claim now that ${\mathsf{C_{u}}}$ satisfies

$\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{7\Upsilon({\widetilde{n}})+\nicefrac{{7}}{{2}}}r_{q+1,{\widetilde{n}}}^{-1}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-2}\Theta_{q+1}^{\nicefrac{{1}}{{2}}}\,.$
(8.48)

We may verify this by using (8.25), the definition of ${\mathsf{C_{u}}}$ in
(8.7), and the inequalities

$\frac{{\mathsf{C_{u}}}}{2b}+7(2+\lceil\log_{2}{n_{\rm
max}}\rceil)+4\leq{\mathsf{C_{u}}}-2\,,\quad\impliedby\quad
1-\frac{1}{2b}>\varepsilon_{\Gamma}(b-1)({n_{\rm
max}}+1)(7\lceil\log_{2}{n_{\rm max}}\rceil+20)\,,$ (8.49)

the second of which follows from (8.6a).

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

$\widetilde{\lambda}_{q}\lambda_{q+1}\left(\Theta_{q+1}^{\frac{1}{2({n_{\rm
max}}+1)}}\Gamma_{q+1}^{9+{\mathsf{C}_{b}}}\right)^{\Upsilon({\widetilde{n}})}\lambda_{q+1}\left(\frac{\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}}{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}\right)^{{\mathsf{d}}-1}\left(\lambda_{q+1}^{4}\right)^{3\mathsf{N}_{\textnormal{ind,v}}}\leq\frac{\delta_{q+2}}{\lambda_{q+1}^{10}}\,,$
(8.50)

and

$(\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{{\mathsf{C}_{b}}+5}r_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\lambda_{q+1})\left(\frac{\Gamma_{q+1}^{-1}\lambda_{q+1}r_{q+1,{\widetilde{n}}}}{\lambda_{q+1}r_{q+1,{\widetilde{n}}}}\right)^{({\mathsf{d}}-1)}\left(\lambda_{q+1}^{4}\right)^{3\mathsf{N}_{\textnormal{ind,v}}}\leq\frac{\delta_{q+2}}{\lambda_{q+1}^{10}}\,.$
(8.51)

The bound (8.50) follows from (8.13) and (8.34), while (8.51) follows from
(8.50) and the parameter inequality
$\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{{\mathsf{C}_{b}}+5}r_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}}\lambda_{q+1}\leq\widetilde{\lambda}_{q}\lambda_{q+1}^{2}$
.

For estimating the stresses emerging from the divergence correctors, we shall
need the bound

$\displaystyle
r_{q+1,{\widetilde{n}}}^{2}\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{13}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\,,$
(8.52)

which follows from (8.25), (8.38), and (8.40) and implies that

$\displaystyle\delta_{q+1,{\widetilde{n}}}\Gamma_{q+1}^{9}\lambda_{q,{\widetilde{n}}}\lambda_{q+1}^{-1}=r_{q+1,{\widetilde{n}}}^{2}\Gamma_{q+1}^{13}\delta_{q+1,{\widetilde{n}}}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\,.$
(8.53)

We furthermore need that

$\displaystyle\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({\widetilde{n}})+7}\leq\Gamma_{q}^{{\mathsf{C_{u}}}}\Gamma_{q+1}^{14\Upsilon({n_{\rm
max}})+7}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\,,$ (8.54)

which in turn follows from
${\mathsf{C_{u}}}\geq\frac{b}{b-1}(8+14\Upsilon({n_{\rm max}}))$, which is a
consequence of (8.6a) and (6.8).

In order to estimate the transport and Nash errors in $L^{1}$ in Sections 7.7
and 7.8, we claim that

$\Gamma_{q+1}^{{{\mathsf{C}_{b}}}+4}\delta_{q+1,{\widetilde{n}}}^{\nicefrac{{1}}{{2}}}\tau_{q}^{-1}r_{q+1,{\widetilde{n}}}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2}\,.$
(8.55)

In order to verify (8.55), we note that by (8.18), (8.19), (8.22), (8.24),
(8.26), the definition of ${\mathsf{C}_{b}}$ in (8.3), and the previously
established parameter inequalities (8.38) and (8.40), the left side of (8.55)
is bounded from above by

$\displaystyle\Gamma_{q+1}^{{\mathsf{C}_{b}}+\mathsf{c}_{0}+13}(\delta_{q+1,{\widetilde{n}}}\lambda_{q,{\widetilde{n}}})^{\nicefrac{{1}}{{2}}}(\delta_{q}\lambda_{q})^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-\nicefrac{{3}}{{2}}}$
$\displaystyle\qquad\leq\Gamma_{q+1}^{{\mathsf{C}_{b}}+\mathsf{c}_{0}+13}(\Gamma_{q+1}^{-{\mathsf{C}_{b}}-9{-\mathsf{C_{R}}}}\lambda_{q+1}\delta_{q+2})^{\nicefrac{{1}}{{2}}}(\delta_{q+2}\lambda_{q+1}\lambda_{q+1}^{2\beta
b-1}\lambda_{q}^{1-2\beta})^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-\nicefrac{{3}}{{2}}}$
$\displaystyle\qquad\leq\Gamma_{q+1}^{\frac{1}{2}{\mathsf{C}_{b}}+\mathsf{c}_{0}+10+\frac{1}{2}\mathsf{C_{R}}}\lambda_{q}^{(\beta
b+\beta-1)(b-1)}(\Gamma_{q+1}^{{-\mathsf{C_{R}}}-1}\delta_{q+2})\,.$

Thus, (8.55) holds since
$\varepsilon_{\Gamma}(\frac{1}{2}{\mathsf{C}_{b}}+\mathsf{c}_{0}+10+\frac{1}{2}\mathsf{C_{R}})+2\beta
b<1$, in view of (8.1) and (8.6e). To estimate the transport and Nash errors
in $L^{\infty}$, we finally need that

$\Gamma_{q}^{\frac{{\mathsf{C_{u}}}}{2}}\Gamma_{q+1}^{{\mathsf{C_{u}}}+7\Upsilon({n_{\rm
max}})+21+4{\widetilde{n}}}\Theta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}r_{q+1,{\widetilde{n}}}^{-1}\lambda_{q+1}^{-1}\leq\Gamma_{q+1}^{{\mathsf{C_{u}}}-1}\,,$
(8.56)

which follows from the definition of ${\mathsf{C_{u}}}$ in (8.7), (8.25), and
(8.6b).

In Remark 2.7, have have used that

$\lim_{(\beta,b)\to(\nicefrac{{1}}{{2}}-,1+)}\frac{2\beta
b}{(b-1)({\mathsf{C_{u}}}\varepsilon_{\Gamma}+\nicefrac{{1}}{{2}})}+2\rightarrow\infty$
(8.57)

which is a consequence of the choice of ${\mathsf{C_{u}}}$ in (8.7).

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 $\mathring{R}_{q,n}$.
This verification is the same as in [5, Section 9.3]. For all $0\leq
n\leq{n_{\rm max}}$ we require that

$\displaystyle\mathsf{N}_{\textnormal{ind,t}},2{\mathsf{N}_{\rm dec}}+4$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm
cut,x}-5\right)\rfloor-{\mathsf{d}}\,,$ (8.58a) $\displaystyle
14\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\mathsf{N}_{\textnormal{fin},\textnormal{n}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-2{\mathsf{N}_{\rm dec}}-9\,,$ (8.58b)
$\displaystyle 6\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm
cut,x}-6\right)\rfloor-{\mathsf{d}}\,,$ (8.58c) $\displaystyle
6\mathsf{N}_{\textnormal{ind,v}}$
$\displaystyle\leq\lfloor\nicefrac{{1}}{{4}}\left(\mathsf{N}_{\rm
fin,n}-\mathsf{N}_{\rm cut,t}-\mathsf{N}_{\rm cut,x}-7\right)\rfloor\,.$
(8.58d)

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

$\displaystyle\lfloor\nicefrac{{1}}{{2}}\left(\mathsf{N}_{\rm{fin,}{\widetilde{n}}}-\mathsf{N}_{\rm
cut,t}-\mathsf{N}_{\rm cut,x}-6\right)\rfloor-{\mathsf{d}}\geq\mathsf{N}_{\rm
fin,n}$ (8.59)

holds. The inequality (8.59) is a direct consequence of the recursive formula
(8.29) and of the fact that the sequence $\mathsf{N}_{\rm fin,n}$ is monotone
decreasing with respect to $n$. Using (8.28) and (8.29) 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 (8.58) are most restrictive for $n={n_{\rm max}}$, they
now readily follow from (8.14).

##  Appendix A Auxiliary lemmas

###  A.1. $L^{p}$ decorrelation

In order to estimate the perturbation in $L^{p}$ spaces as well as terms
appearing in the Reynolds stress we will need a combination of [5, Lemma A.7]
and [5, Remark A.9], which we recall next.

######  Lemma A.1 ($L^{p}$ decorrelation with flows).

Let $p\in\\{1,2\\}$, and fix integers $N_{\circ}\geq{\mathsf{N}_{\rm dec}}\geq
1$. 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

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

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

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

for all $N\leq N_{\circ}$. 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.1)

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

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

satisfy

$\displaystyle\widetilde{\zeta}^{4}\bigl{(}2\pi\sqrt{3}\lambda\mu^{-1}\bigr{)}^{{\mathsf{N}_{\rm
dec}}}\leq 1\,,$ (A.2)

and we have

$2{\mathsf{N}_{\rm dec}}+4\leq N_{\circ}\,,$

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.3)

holds for $N+M\leq N_{\circ}$ and $M\leq N_{\circ}-2{\mathsf{N}_{\rm dec}}-4$.

###  A.2. Inversion of the divergence

Given a vector field $G^{i}$, a zero mean periodic function $\varrho$ and an
incompressible flow $\Phi$, our goal in this section is to write
$G^{i}(x)\varrho(\Phi(x))$ as the divergence of a symmetric tensor. For this
purpose, we use [5, Proposition A.18].

######  Proposition A.2 (Intermittency-friendly inverse divergence).

Fix an incompressible vector field $v$ and denote its material derivative by
$D_{t}=\partial_{t}+v\cdot\nabla$. Fix integers $N_{*}\geq M_{*}\geq 1$. Also
fix ${\mathsf{N}_{\rm dec}},{\mathsf{d}}\geq 1$ such that
$N_{*}-{\mathsf{d}}\geq 2{\mathsf{N}_{\rm dec}}+4$, and $p\in\\{1,\infty\\}$.

Let $G$ be a vector field and assume there exists a constant
$\mathcal{C}_{G}>0$ and parameters $\lambda,\nu\geq 1$ such that

$\displaystyle\left\|D^{N}D_{t}^{M}G\right\|_{L^{p}}\lesssim\mathcal{C}_{G}\lambda^{N}\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)$
(A.4)

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

Let $\Phi$ be a volume preserving transformation of $\mathbb{T}^{3}$, such
that

$D_{t}\Phi=0\,\qquad\mbox{and}\qquad\left\|\nabla\Phi-\mathrm{Id}\right\|_{L^{\infty}(\mathrm{supp\,}G)}\leq\nicefrac{{1}}{{2}}\,.$

Denote by $\Phi^{-1}$ the inverse of the flow $\Phi$, which is the identity at
a time slice which intersects the support of $G$. Assume that the velocity
field $v$ and the flow functions $\Phi$ and $\Phi^{-1}$ satisfy the following
bounds

$\displaystyle\left\|D^{N+1}\Phi\right\|_{L^{\infty}(\mathrm{supp\,}G)}+\left\|D^{N+1}\Phi^{-1}\right\|_{L^{\infty}(\mathrm{supp\,}G)}$
$\displaystyle\lesssim\lambda^{\prime N}$ (A.5)
$\displaystyle\left\|D^{N}D_{t}^{M}Dv\right\|_{L^{\infty}(\mathrm{supp\,}G)}$
$\displaystyle\lesssim\nu\lambda^{\prime
N}\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)\,,$ (A.6)

for all $N\leq N_{*}$, $M\leq M_{*}$, and some $\lambda^{\prime}>0$.

Lastly, let $\varrho,\vartheta\colon\mathbb{T}^{3}\to\mathbb{R}$ be two zero
mean functions with the following properties:

  1. (i)

there exists ${\mathsf{d}}\geq 1$ and a parameter $\zeta\geq 1$ such that
$\varrho(x)=\zeta^{-2{\mathsf{d}}}\Delta^{\mathsf{d}}\vartheta(x)$

  2. (ii)

there exists a parameter $\mu\geq 1$ such that $\varrho$ and $\vartheta$ are
$(\nicefrac{{\mathbb{T}}}{{\mu}})^{3}$-periodic

  3. (iii)

there exists parameters $\Lambda\geq\zeta$, $\mathcal{C}_{*}\geq 1$, and
$\alpha\in(0,1]$, such that

$\displaystyle\left\|D^{N}\vartheta\right\|_{L^{p}}\lesssim\mathcal{C}_{*}\Lambda^{\alpha}\mathcal{M}\left(N,2{\mathsf{d}},\zeta,\Lambda\right)$
(A.7)

for all $0\leq N\leq\mathsf{N}_{\rm fin}$.

If the above parameters satisfy

$\displaystyle\lambda^{\prime}\leq\lambda\ll\mu\leq\zeta\leq\Lambda\,,$ (A.8)

where by $\ll$ in (A.8) we mean that

$\displaystyle\Lambda^{4}\bigl{(}2\pi\sqrt{3}\lambda\mu^{-1}\bigr{)}^{{\mathsf{N}_{\rm
dec}}}\leq 1\,,$ (A.9)

then, we have that

$\displaystyle G\;\varrho\circ\Phi$
$\displaystyle=\mathrm{div\,}\mathring{R}+\nabla
P+E=\mathrm{div\,}\left(\mathcal{H}\left(G\varrho\circ\Phi\right)\right)+\nabla
P+E.$ (A.10)

where the traceless symmetric stress $\mathcal{H}(G\varrho\circ\Phi)$ and the
scalar pressure $P$ are supported in $\mathrm{supp\,}G$, and for any fixed
$\alpha\in(0,1)$ they satisfy

$\displaystyle\left\|D^{N}D_{t}^{M}\mathcal{H}\left(G\varrho\circ\Phi\right)\right\|_{L^{p}}+\left\|D^{N}D_{t}^{M}P\right\|_{L^{p}}$
$\displaystyle\lesssim\mathcal{C}_{G}\mathcal{C}_{*}\zeta^{-1}\Lambda^{\alpha}\mathcal{M}\left(N,1,\zeta,\Lambda\right)\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)$
(A.11)

for all $N\leq N_{*}-{\mathsf{d}}$ and $M\leq M_{*}$. The implicit constants
depend on $N,M,\alpha$ but not $G$, $\varrho$, or $\Phi$. Lastly, for $N\leq
N_{*}-{\mathsf{d}}$ and $M\leq M_{*}$ the error term $E$ in (A.10) satisfies

$\displaystyle\left\|D^{N}D_{t}^{M}E\right\|_{L^{p}}\lesssim\mathcal{C}_{G}\mathcal{C}_{*}\lambda^{\mathsf{d}}\zeta^{-{\mathsf{d}}}\Lambda^{\alpha+N}\mathcal{M}\left(M,M_{t},\nu,\widetilde{\nu}\right)\,.$
(A.12)

We emphasize that the range of $M$ in (A.11) and (A.12) is exactly the same as
the one in (A.4), while the range of permissible values for $N$ shrank from
$N_{*}$ to $N_{*}-{\mathsf{d}}$.

Lastly, let $N_{\circ},M_{\circ}$ be integers such that $1\leq M_{\circ}\leq
N_{\circ}\leq M_{*}/2$. Assume that in addition to the bound (A.6) we have the
following global lossy estimates

$\displaystyle\left\|D^{N}\partial_{t}^{M}v\right\|_{L^{\infty}(\mathbb{T}^{3})}\lesssim\mathcal{C}_{v}\widetilde{\lambda}_{q}^{N}\widetilde{\tau}_{q}^{-M}$
(A.13)

for all $M\leq M_{\circ}$ and $N+M\leq N_{\circ}+M_{\circ}$, where

$\displaystyle\mathcal{C}_{v}\widetilde{\lambda}_{q}\lesssim\widetilde{\tau}_{q}^{-1},\qquad\mbox{and}\qquad\lambda^{\prime}\leq\widetilde{\lambda}_{q}\leq\Lambda\leq\lambda_{q+1}\,.$
(A.14)

If ${\mathsf{d}}$ is chosen large enough so that

$\displaystyle\mathcal{C}_{G}\mathcal{C}_{*}\Lambda\left(\lambda\zeta^{-1}\right)^{{\mathsf{d}}-1}\left(1+\tau_{q}\max\\{\widetilde{\tau}_{q}^{-1},\widetilde{\nu},\mathcal{C}_{v}\Lambda\\}\right)^{M_{\circ}}\leq\delta_{q+2}\lambda_{q+1}^{-10}\,,$
(A.15)

then we may write

$\displaystyle
E=\mathrm{div\,}\left(\mathcal{R}^{*}(G\varrho\circ\Phi)\right)+\fint_{\mathbb{T}^{3}}G\varrho\circ\Phi
dx\,,$ (A.16)

where $\mathcal{R}^{*}(G\varrho\circ\Phi)$ is a traceless symmetric stress
which satisfies

$\displaystyle\left\|D^{N}D_{t}^{M}\mathcal{R}^{*}\left(G\varrho\circ\Phi\right)\right\|_{L^{p}}\leq\delta_{q+2}\lambda_{q+1}^{N-10}\tau_{q}^{-M}$
(A.17)

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

For $p=1$, Proposition A.2 is taken as is from [5, Proposition A.17]. For the
proof of Proposition A.2 in the case $p=\infty$, the proof of [5, Proposition
A.17] applies mutatis mutandis, after replacing each instance of an $L^{p}$
bound for $p\neq\infty$ in the proof with an $L^{\infty}$ bound. In fact, the
condition (A.9) involving the parameter ${\mathsf{N}_{\rm dec}}$ is
technically irrelevant in the case $p=\infty$, since $L^{\infty}$ is an
algebra.

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