# Non-conservative solutions of the Euler-$\alpha$ equations

Rajendra Beekie  Matthew Novack

###### Abstract

The Euler-$\alpha$ equations model the averaged motion of an ideal
incompressible fluid when filtering over spatial scales smaller than $\alpha$.
We show that there exists $\beta>1$ such that weak solutions to the two and
three dimensional Euler-$\alpha$ equations in the class
$C^{0}_{t}H^{\beta}_{x}$ are not unique and may not conserve the Hamiltonian
of the system, thus demonstrating flexibility in this regularity class. The
construction utilizes a Nash-style intermittent convex integration scheme. We
also formulate an appropriate version of the Onsager conjecture for
Euler-$\alpha$, postulating that the threshold between rigidity and
flexibility is the regularity class
$L^{3}_{t}B^{\nicefrac{{1}}{{3}}}_{3,\infty,x}$.

###### Contents

  1. 1 Introduction
    1. 1.1 Euler-$\alpha$ as a Hamiltonian System
    2. 1.2 Definition of Weak solution
    3. 1.3 Rigidity for weak solutions of Euler-$\alpha$
    4. 1.4 Results and Main Ideas
  2. 2 Convex Integration Scheme
  3. 3 Mollification
  4. 4 Linear Algebra
  5. 5 Technical Preliminaries
    1. 5.1 Decoupling
      1. 5.1.1 $L^{p}$ decoupling lemma
      2. 5.1.2 Decoupling for intersections of deformed intermittent pipes
    2. 5.2 Inverse Divergence
  6. 6 Construction of the Perturbation
    1. 6.1 Definition of and estimates on the perturbation
    2. 6.2 The new equation at level $q+1$
  7. 7 Stress Estimates
    1. 7.1 Oscillation Error
      1. 7.1.1 Type 1 oscillation errors
      2. 7.1.2 Type 2 oscillation errors
    2. 7.2 Nash Error
    3. 7.3 Transport Error
    4. 7.4 Parameters and inequalities
  8. A Proof of Lemma 1.5
  9. B Proof of Lemma 4.1

##  1 Introduction

We consider the Lagrangian averaged Euler equations (LAE-$\alpha$ or
Euler-$\alpha$) for the unknown velocity field $u$ and pressure $p$:

$\displaystyle\partial_{t}\left(u-\alpha^{2}\Delta
u\right)+u\cdot\nabla\left(u-\alpha^{2}\Delta
u\right)+\left(u-\alpha^{2}\Delta u\right)^{j}\nabla u^{j}+\nabla p$
$\displaystyle=0$ (1.1a) $\displaystyle\operatorname{\mathrm{div}}u$
$\displaystyle=0\,.$ (1.1b)

The parameter $\alpha>0$ corresponds to a _length scale_ , and we pose the
system for $(x,t)\in\mathbb{T}^{n}\times[0,T],n=2\text{ or }3\,,T>0$.
Euler-$\alpha$ was originally derived within an Euler-Poincaré variational
framework to model the averaged motion of an ideal incompressible fluid when
filtering over spatial scales smaller than $\alpha$
[HolmMarsdenJerroldRatiu98a, HolmMarsdenJerroldRatiu98b]. It was later proven
that, like the Euler equations, (1.1) have a deep geometric significance as
the geodesic equations with respect to an $H^{1}$ metric on the diffeomorphism
group of volume preserving flows [Arnold66, Shkoller98]. We also note that the
Euler-$\alpha$ equations are identical to the inviscid second-grade fluid
equations, where $\alpha$ now represents the elastic response of the fluid
(see [DunnFosdick74] and references therein for more on second-grade fluids).
The viscous variant of (1.1) is known as the Lagrangian averaged Navier-Stokes
equation (LANS-$\alpha$) or the viscous Camassa-Holm equation and has been
successfully used as a turbulence closure model [ChenFoiasHolmOlsonTiti98,
FoiasHolmTiti01, MohseniKosovicShkollerMarsden03, ChenHolmMargolinZhang99].

When posed on a compact Riemannian manifold of dimension $n$ without boundary,
local-wellposedness in $H^{s}$ for $s>\frac{n}{2}+1$ was proven in
[Shkoller98] (see also [Shkoller00, MarsdenRatiuShkoller00] for the case of
domains with boundary). The state of the global wellposedness question for
Euler-$\alpha$ is analogous to that of the classical Euler equations; in two
dimensions, global wellposedness of smooth solutions is known [busuioc99], but
in three dimensions, finite time singularity formation has not been ruled out.
It was shown in [HouLi06] that global existence may be phrased in terms of a
Beale-Kato-Majda type continuation criterion which requires that

$\int_{0}^{T}\|\nabla\times(-\Delta)^{-1}(1-\alpha^{2}\Delta)u(t)\|_{\text{BMO}}dt<\infty\,.$
(1.2)

Note that by setting $\alpha=0$ we formally recover the Euler equations after
redefining the pressure. In the case of Euclidean space, the convergence of
smooth Euler-$\alpha$ solutions to Euler solutions was demonstrated in
[LinshizTiti10] (see also [LFNLTitiZang15, BusuiocIftimieLFNL12,
BusuiocIftimie17, BusuiocIftimieLFNL16] for progress in the case of domains
with boundary).

###  1.1 Euler-$\alpha$ as a Hamiltonian System

Sufficiently smooth solutions to Euler-$\alpha$ satisfy the conservation law

$\frac{d}{dt}\mathcal{H}_{\alpha}:=\frac{d}{dt}\left(\|u\|_{L^{2}}^{2}+\alpha^{2}\|\nabla
u\|_{L^{2}}^{2}\right)=0\,.$ (1.3)

Beyond merely being a conserved quantity, $\mathcal{H}_{\alpha}$ is actually
the _Hamiltonian_ associated to the action function from which Euler-$\alpha$
can be derived . We follow [BSV16] to expand upon this point. In the Euler-
Poincaré framework, Euler-$\alpha$ arises as the geodesic equations associated
with the metric

$(u,w)_{\alpha}:=\int_{\mathbb{T}^{n}}(1-\alpha^{2}\Delta)u\cdot w\,dx\,.$
(1.4)

Moreover, the geodesic equations can be computed by identifying extrema of the
action function

$s(u)=\int_{\mathbb{T}^{n}}(1-\alpha^{2}\Delta)u\cdot
u\,dx=\int_{\mathbb{T}^{n}}|u|^{2}+\alpha^{2}|\nabla u|^{2}\,dx$ (1.5)

when considering variations that are Lie-advected and satisfy suitable
boundary conditions. Since solutions are extrema of the action, they formally
conserve the Hamiltonian

$\mathcal{H}_{\alpha}(t)=\int_{\mathbb{T}^{n}}(1-\alpha^{2}\Delta)u\cdot
u\,dx=\int_{\mathbb{T}^{n}}|u|^{2}+\alpha^{2}|\nabla u|^{2}\,dx\,.$ (1.6)

This defines a functional framework in which one may study wellposedness for
Euler-$\alpha$. A natural question then follows: _Do solutions of ( 1.1) which
live in the function space determined by (1.6) necessarily conserve the
Hamiltonian?_ We answer this question in the negative:

######  Theorem 1.1.

Let $n\geq 2$. Then there exists $\beta>1$ such that there exist weak
solutions of (1.1) in $C^{0}_{t}H_{x}^{\beta}(\mathbb{T}^{n})$ which do not
conserve $\mathcal{H}_{\alpha}$.

###  1.2 Definition of Weak solution

We first start by specifying the meaning of a weak solution of (1.1). Using
incompressibility, the identity

$\partial_{kk}u^{j}\partial_{l}u^{j}=\partial_{k}\left(\partial_{k}u^{j}\partial_{l}u^{j}\right)-\partial_{k}u^{j}\partial_{kl}u^{j}=\partial_{k}\left(\partial_{k}u^{j}\partial_{l}u^{j}\right)-\frac{1}{2}\partial_{l}(\partial_{k}u^{j}\partial_{k}u^{j})\,,$

and redefining the pressure, we can formally rewrite (1.1) as

$\displaystyle\partial_{t}\left(u-\alpha^{2}\Delta
u\right)+\operatorname{\mathrm{div}}(u\otimes\left(u-\alpha^{2}\Delta
u\right)-\alpha^{2}\nabla^{T}u\nabla u)+\nabla p$ $\displaystyle=0$ (1.7a)
$\displaystyle\operatorname{\mathrm{div}}u$ $\displaystyle=0\,,$ (1.7b)

or

$\displaystyle\partial_{t}\left(u-\alpha^{2}\Delta
u\right)^{l}+\partial_{k}\left(u^{k}\left(u^{l}-\alpha^{2}\Delta
u^{l}\right)-\alpha^{2}\partial_{k}u^{j}\partial_{l}u^{j}\right)+\partial_{l}p$
$\displaystyle=0$ (1.8a) $\displaystyle\partial_{l}u^{l}$ $\displaystyle=0\,$
(1.8b)

when written in coordinates. Multiplying (1.7) by a smooth, divergence-free
vector field $\varphi$, integrating in space and time, and integrating by
parts leads to

$\int_{\mathbb{R}}\int_{\mathbb{T}^{n}}\partial_{t}\varphi\cdot
u+\alpha^{2}\nabla\partial_{t}\varphi:\nabla
u+\nabla\varphi:\left(u\otimes\left(u-\alpha^{2}\Delta
u\right)-\alpha^{2}\nabla u^{T}\nabla u\right)dxdt=0\,.$ (1.9)

This motivates the following definition:

######  Definition 1.2.

A vector field $u\in C_{t}^{0}H_{x}^{1}$ is called a weak solution of (1.1) if
the vector field $u(\cdot,t)$ is weakly divergence free and has zero mean for
any $t$ and satisfies (1.9) for any divergence free test function $\varphi$.

######  Remark 1.3.

In order to interpret the term
$\int_{\mathbb{R}}\int_{\mathbb{T}^{n}}\nabla\varphi:u\otimes\Delta
u\,dx\,dt$, we use duality. Specifically, since $u\in H_{x}^{1}$,
$u\cdot\nabla\varphi\in H^{1}_{x}$ as well, and $\Delta u$ is well defined in
$H_{x}^{-1}$. Using the observation that

$(\Delta
u_{l},u\cdot\nabla\varphi_{l})_{H^{-1},H^{1}}=-\langle\partial_{i}u_{l},\partial_{i}(u\cdot\nabla\varphi_{l})\rangle_{L^{2},L^{2}}=-\langle\partial_{i}u_{l},\partial_{i}u_{k}\partial_{k}\varphi_{l}+u_{k}\partial_{k}\partial_{i}\varphi_{l}\rangle_{L^{2},L^{2}}\,,$

we have the following _equivalent_ weak formulation of (1.9):

$\int_{\mathbb{R}}\int_{\mathbb{T}^{n}}\partial_{t}\varphi\cdot
u+\alpha^{2}\nabla\partial_{t}\varphi:\nabla u+\nabla\varphi:\left(u\otimes
u-\alpha^{2}\nabla u^{T}\nabla u+\alpha^{2}\nabla u\nabla
u^{T}\right)+\alpha^{2}u\cdot\nabla\nabla\varphi:\nabla u^{T}dxdt=0\,.$ (1.10)

######  Remark 1.4.

For later use, it will also be convenient to utilize the following formulation
of the Euler-$\alpha$ equations, which in the smooth context a notion of
solution equivalent to the previous formulations:

$\displaystyle\partial_{t}\left(u-\alpha^{2}\Delta
u\right)+\operatorname{\mathrm{curl}}\left(u-\alpha^{2}\Delta u\right)\times
u+\nabla p$ $\displaystyle=0$ (1.11a)
$\displaystyle\operatorname{\mathrm{div}}u$ $\displaystyle=0\,.$ (1.11b)

###  1.3 Rigidity for weak solutions of Euler-$\alpha$

Inspired by Onsager’s conjecture for the classical Euler equations
[Onsager49], one naturally may ask at what level of regularity weak solutions
as defined in Definition 1.2 conserve $\mathcal{H}_{\alpha}$. To this end, we
have the following lemma (see Appendix A for the proof):

######  Lemma 1.5 (Energy Conservation).

Let $u$ be a mean-zero in space weak solution of Euler-$\alpha$ as defined by
definition 1.2 on $\mathbb{T}^{2}$ or $\mathbb{T}^{3}$, and let $s>1$. If
${u\in L^{3}\left([0,T);B_{3,\infty}^{s}\right)\cap
C^{0}\left([0,T);H^{1}\right)}$, then the quantity

$\mathcal{H}_{\alpha}(t)=\|u(t)\|_{L^{2}}^{2}+\alpha^{2}\|\nabla
u\|_{L^{2}}^{2}$

is constant in time for $t\in[0,T)$.

In light of Lemma 1.5 we make the following conjecture:

######  Conjecture 1.

Weak solutions of (1.1) obey the following dichotomy.

  1. 1.

For $s>1$, weak solutions in
$C_{t}^{0}H_{x}^{1}\cap{L_{t}^{3}}B_{3,\infty}^{s}$ conserve
$\mathcal{H}_{\alpha}$.

  2. 2.

For $s<1$, there exist weak solutions in
$C_{t}^{0}H_{x}^{1}\cap{L_{t}^{3}}B_{3,\infty}^{s}$ which do not conserve
$\mathcal{H}_{\alpha}$.

######  Remark 1.6.

Stated in terms of $L^{2}$-based Sobolev spaces, $H^{\nicefrac{{3}}{{2}}}$
embeds into the critical regularity class $B_{3,\infty}^{1}$ in three
dimensions.

###  1.4 Results and Main Ideas

We will conclude Theorem 1.1 from the following:

######  Theorem 1.7.

Let $u^{(1)}$ and $u^{(2)}$ be smooth solutions of (1.1) defined on $[0,T]$
with zero mean. Moreover, suppose that
$\mathcal{H}_{\alpha}(u^{1})\neq\mathcal{H}_{\alpha}(u^{2})$. There exists a
$\beta^{\prime}>1$ and a weak solution
$u\in{C^{0}}\left([0,T];H_{x}^{\beta^{\prime}}\right)$ of (1.1) such that

$u\equiv u^{(1)}\,\text{ on }\left[0,\frac{T}{3}\right]\quad\text{and}\quad
u\equiv u^{(2)}\,\text{ on }\left[\frac{2T}{3},T\right]\,.$ (1.12)

The method of proof for Theorem 1.7 is based on the iterative technique known
as convex integration. In the context of fluid equations, convex integration
schemes generally use a specially constructed stationary solution to the
equations as the basic building block of the iteration. In our iteration, we
use _intermittent Mikado flows_. Mikado flows were introduced in
[DaneriSzekelyhidi17] and featured crucially in the resolution of the Onsager
conjecture for nonconservative solutions [Isett2018]; see also [BDLSV17] for
an extension to dissipative solutions. The introduction of _intermittency_ to
convex integration schemes dates back to the work [BV19], which gave the first
example of non-uniqueness of weak solutions for the Navier-Stokes equations.
Intermittent Mikado flows were later introduced in [ModenaSZ17]. One may check
that “generic” Mikado tubes are not necessarily stationary solutions of (1.1),
but Mikado flows with a radial flow profile about the axis (as defined in
(6.8)) are in fact stationary solutions; see Proposition 6.1. For further
applications of and background on convex integration schemes, we refer to the
papers [cheskidov2020nonuniqueness, cheskidov2020sharp, IssetVicol15,
novack2020, BCV18, brue2020positive, Dai18nonunique, BV_EMS19, DLSZ19].

We emphasize that the need for intermittency arises from the combination of
the rigidity threshhold for $\mathcal{H}_{\alpha}$, and the notion of weak
solution we employ; we simultaneously need $\nabla u\in L^{2}$ to define the
weak formulation, but also $\nabla u\notin L^{3}$, so that
$\mathcal{H}_{\alpha}$ may vary in time. Thus our iteration should _not_
produce spatially homogeneous objects. We instead need solutions for which the
regularity threshold measured in $L^{p}$ depends fundamentally on $p$. This
dependence on $p$ of the regularity of both the building blocks and the final
solution is the main attribute of intermittent flows [BV19].

A key aspect of our construction is the composition of the high-frequency
Mikado flows with Lagrangian flow maps. This composition respects the
transport structure of (1.1) and has been employed in earlier convex
integration schemes [BDLISZ15, Isett12]. The _combination_ of intermittency
with Lagrangian flow maps presents particular challenges and was exploited for
the first time in the recent work [bmnv21]. It may in fact be possible to
close a convex integration scheme for (1.1) _without_ composing with
Lagrangian flow maps, perhaps using different building blocks such as
intermittent jets [BCV18]. However, this would require the building blocks to
be very intermittent. Utilization of such intermittent building blocks appears
to impede the scheme from reaching the $L^{3}$ rigidity threshold. We refer to
Remark 7.2, in which we make a simple computation with the _Nash error_ term,
to justify this assertion. Since we seek to build techniques which are as
robust as possible in the hopes of eventually proving an Onsager-type
conjecture for Euler-$\alpha$, we therefore use Lagrangian flow maps.

On the other hand, one difficulty created by the composition with Lagrangian
flow maps is the non-trivial interaction of building blocks arising from
different Lagrangian coordinate systems. We circumvent this difficulty by
expanding on an insight from [BBV19], in which the supports of separate
building blocks are concentrated around planes which intersect _orthogonally_
, in the sense that the intersection is a periodized line, and thus a lower-
dimensional set. Since the intersection of the building blocks is concentrated
around a lower dimensional set, one can use intermittency to make the error
terms manageable. The new difficulty not present in [BBV19] is that the
supports of our building blocks deform in time due to the composition with
Lagrangian flow maps. On a timescale which is commensurate with that
determined by $\left\|\nabla_{x}u\right\|_{L^{\infty}_{t,x}}^{-1}$, we show in
Proposition 5.3 that the intersection of deformed intermittent Mikado tubes
oriented around non-parallel axes is concentrated around a collection of
nearly-periodic _points_. Then since the intersection is concentrated around a
lower-dimensional set, intermittency may be used to make these errors small.
We remark that this technique allows us to treat the intersection of non-
parallel Mikado tubes in both two and three dimensions. In fact this is the
only part of the scheme which requires an adjustment in two dimensions; see
Remark 5.4 and Remark 7.3 for details.

There are other methodologies one could employ for preventing the intersection
of Mikado tubes arising from different Lagrangian coordinate systems. One
could try to appeal to the gluing technique [Isett2018], a placement strategy
predicated on a restriction of the timescale [dlk20], or a placement strategy
predicated on the degree of freedom offered by the sparseness of intermittent
Mikado tubes [bmnv21]. However, these methods either do not appear to work for
the Euler-$\alpha$ equations, or may work but would introduce error terms
which again appear to impede the scheme from reaching the sharp $L^{3}$
threshold. Furthermore, none of the aforementioned methods could treat the 2D
Euler-$\alpha$ equations, in which the usage of Mikado flows conflicts with
the fact that non-parallel lines must intersect in two dimensions. Therefore
we use Proposition 5.3, which functions perfectly well in our two- and three-
dimensional intermittent context. For additional commentary on this issue, we
refer to Remark 7.1.

##  2 Convex Integration Scheme

Our aim is to construct a sequence of solutions $(u_{q},{\mathring{R}}_{q})$
to the relaxed system of equations111The pressure $p_{q}$ is determined via
the incompressibility of $u_{q}$.

$\displaystyle\partial_{t}(u_{q}-\alpha^{2}\Delta
u_{q})^{l}+\partial_{k}\left(u_{q}^{k}(u_{q}^{l}-\alpha^{2}\Delta
u_{q}^{l})-\alpha^{2}\partial_{k}u_{q}^{j}\partial_{l}u_{q}^{j}\right)+\partial_{l}p$
$\displaystyle=\partial_{k}{\mathring{R}}_{q}^{kl}$ (2.1a)
$\displaystyle\partial_{l}u_{q}^{l}$ $\displaystyle=0\,,$ (2.1b)

where ${\mathring{R}}_{q}$ is a symmetric traceless matrix, and show that
${\mathring{R}}_{q}\to 0$. In order to quantify the convergence of
${\mathring{R}}_{q}$ and $u_{q}$, we use a frequency parameter $\lambda_{q}$
and an amplitude parameter $\delta_{q}$ defined as follows:

$\lambda_{q}=a^{b^{q}}\qquad\delta_{q}=\lambda_{q}^{-2\beta}\,,$ (2.2)

where $a,b\in\mathbb{N}$ are large and $\beta>1$. We will assume the following
inductive bounds on the sequence

$\displaystyle\left\|{\mathring{R}}_{q}\right\|_{L^{1}}$
$\displaystyle\leq{{\alpha}^{2}}\cdot{\mathcal{C}_{\mathring{R}}}\cdot\delta_{q+1}\lambda_{q+1}^{2}$
(2.3a) $\displaystyle\left\|\nabla u_{q}\right\|_{L^{2}}$ $\displaystyle\leq
1$ (2.3b) $\displaystyle\left\|\nabla^{3}_{t,x}u_{q}\right\|_{0}$
$\displaystyle\leq{\lambda_{q}^{4}}$ (2.3c) $\displaystyle r_{q}$
$\displaystyle={\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\Gamma}}\,.$
(2.3d)

where $\mathcal{C}_{\mathring{R}}$ is a dimensional constant and
$\Gamma\in(0,1)$. All norms in the above assumptions are Lebesgue norms in
space, measured uniformly in time; furthermore, we use
$\left\|\cdot\right\|_{k}$ to denote the spatial $C^{k}$ norm of functions,
measured uniformly in time. All parameters used above, as well as all other
parameters used throughout the paper, are defined in subsection 7.4.

Theorem 1.7 is a consequence of the following proposition:

######  Proposition 2.1.

There exists $\beta>1$, $b\in\mathbb{N}$, and $a_{0}\in\mathbb{N}$ such that
for any natural number $a\geq a_{0}$, the following holds. Suppose that
$(u_{q},{\mathring{R}}_{q})$ are given and satisfy (2.1) and (2.3). Then there
exists $(u_{q+1},{\mathring{R}}_{q+1})$ which satisfy (2.1) and (2.3) with $q$
replaced by $q+1$. Furthermore, $u_{q+1}-u_{q}$ satisfies the bounds

$\left\|u_{q+1}-u_{q}\right\|_{L^{2}}+\lambda_{q+1}^{-1}\left\|\nabla\left(u_{q+1}-u_{q}\right)\right\|_{L^{2}}+\lambda_{q+1}^{-2}\left\|\nabla^{2}\left(u_{q+1}-u_{q}\right)\right\|_{L^{2}}\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,.$
(2.4)

Finally, if the temporal support of ${\mathring{R}}_{q}$ satisfies

$\operatorname{\mathrm{supp}}_{t}{\mathring{R}}_{q}:=\\{t:|{\mathring{R}}_{q}(x,t)|\not\equiv
0\\}\subseteq[t_{1},t_{2}]\,,$ (2.5)

then

$\operatorname{\mathrm{supp}}_{t}{\mathring{R}}_{q+1}\cup\operatorname{\mathrm{supp}}_{t}(u_{q+1}-u_{q})\subset\left(t_{1}-\frac{1}{30\lambda_{q}},t_{2}+\frac{1}{30\lambda_{q}}\right)\,.$
(2.6)

Assuming Proposition 2.1, we now prove Theorem 1.7.

###### Proof of Theorem 1.7.

We proceed as in [BCV18]. Let $\eta$ be a smooth cutoff function with the
property that $\eta=1$ on $\left[0,\frac{2T}{5}\right]$ and $\eta=0$ on
$\left[\frac{3T}{5},T\right]$. Define

$u_{0}=\eta u^{(1)}+(1-\eta)u^{(2)}\,,$ (2.7)

where $u^{(1)},u^{(2)}$ are as in Theorem 1.7, and

$\displaystyle{\mathring{R}}_{0}=\mathcal{R}\partial_{t}\eta\left(u^{(1)}-u^{(2)}\right)$
$\displaystyle+\eta(1-\eta)\mathcal{R}\operatorname{\mathrm{div}}\left[u^{(1)}\otimes(u^{(2)}-\alpha^{2}\Delta
u^{(2)})+u^{(2)}\otimes(u^{(1)}-\alpha^{2}\Delta
u^{(1)})-\alpha^{2}\left((\nabla u^{(1)})^{T}\nabla u^{(2)}+(\nabla
u^{(2)})^{T}\nabla u^{(1)}\right)\,.\right]$

Above we have used the operator $\mathcal{R}$, which is defined in Proposition
5.6; note that the usage of ${\mathring{R}}$ is justified since the inputs
have zero mean. Then the pair $(u_{0},{\mathring{R}}_{0})$ solves the system
(2.1) for an appropriate choice of pressure $p_{0}$, and
$\operatorname{\mathrm{supp}}_{t}{\mathring{R}}_{0}$ is contained in
$[\nicefrac{{2T}}{{5}},\nicefrac{{3T}}{{5}}]$. We _assume_ for the moment that
$(u_{0},{\mathring{R}}_{0})$ satisfy (2.3); at the end of the proof, we shall
see that by a rescaling, this assumption may be imposed without loss of
generality. We now inductively apply Proposition 2.1 to construct a sequence
$\\{(u_{q},{\mathring{R}}_{q})\\}_{q=0}^{\infty}$. For
$\beta^{\prime}\in(1,\beta)$, we have

$\displaystyle\sum_{q\geq 0}\|u_{q+1}-u_{q}\|_{H^{\beta^{\prime}}}$
$\displaystyle\lesssim\sum_{q\geq
0}\|\nabla(u_{q+1}-u_{q})\|_{L^{2}}^{2-\beta^{\prime}}\|\nabla^{2}(u_{q+1}-u_{q})\|_{L^{2}}^{\beta^{\prime}-1}$
$\displaystyle\lesssim(\lambda_{q+1}\delta_{q+1}^{\frac{1}{2}})^{2-\beta^{\prime}}(\lambda_{q+1}^{2}\delta_{q+1}^{\frac{1}{2}})^{\beta^{\prime}-1}$
$\displaystyle\lesssim 1\,,$

where we have used (2.4) and that the $u_{q}$ have zero mean. This implies the
existence of a strong limit $u\in C^{0}_{t}H^{\beta^{\prime}}$. By (2.3),
${\mathring{R}}_{q}$ converges to zero strongly in $L^{1}$. Combining this
with the regularity on $u_{q}$ implies that the limit $u$ satisfies the weak
formulation (1.9). Note also that by the inductive assumption on the support
of the stress and the increment $u_{q+1}-u_{q}$, we can conclude that the
temporal support of $u-u_{0}$ is contained in the interval

$\left(\frac{2T}{5}-\frac{1}{30}\sum_{q\geq
0}\lambda_{q}^{-1}\,,\quad\frac{3T}{5}+\frac{1}{30}\sum_{q\geq
0}\lambda_{q}^{-1}\right)\subset\left(\frac{2T}{5}-\frac{1}{30(1-a^{-b})}\,,\quad\frac{3T}{5}+\frac{1}{30(1-a^{-b})}\right)\subset\left(\frac{T}{3},\frac{2T}{3}\right)$

where we have used that $b\geq 2$ to deduce that $a^{bq}\leq a^{b^{q}}$ (for
the first containment) and that $a$ is sufficiently large (for the second
containment). As a consequence of this and the definition of $u_{0}$, (1.12)
follows.

If $(u_{0},{\mathring{R}}_{0})$ do not satisfy (2.3), we note that (1.1) is
invariant under the rescaling $u(x,t)\to\zeta u(x,\zeta t)$ for
$\zeta\in\mathbb{R}$. Therefore, we can associate to $u^{(i)}$ defined on
$[0,T]$ a function $u_{\zeta}^{(i)}(x,t)=\zeta u^{(i)}(x,\zeta t)$ defined on
$[0,\zeta^{-1}T]$. By taking $\zeta$ sufficiently small depending on the
choice of $a$ in (2.2), we can ensure that the bounds in (2.3) are satisfied
for $u_{\zeta,0}:=\eta_{\zeta}u^{(1)}+(1-\eta_{\zeta})u^{(2)}$, where
$\eta_{\zeta}(t)=\eta(\zeta t)$, and the associated stress
${\mathring{R}}_{\zeta,0}$. We can then inductively apply Proposition 2.1 as
before to construct a weak solution $u_{\zeta}$ for (1.11) defined on
$[0,\zeta^{-1}T]$. Rescaling the solution to produce $u$ defined on $[0,T]$
allows us to conclude the theorem. ∎

The remainder of this paper will be devoted to proving Proposition 2.1.

##  3 Mollification

It will be necessary to mollify the equation in order to mitigate the loss of
derivatives characteristic of convex integration schemes. We define the
parameter $\ell$ in terms of $\lambda_{q}$ by

$\ell=\lambda_{q}^{-8}\,.$ (3.1)

######  Lemma 3.1 (Mollifying the equation).

Assuming we have a pair $(u_{q},{\mathring{R}}_{q})$ satisfying (2.1) and
(2.3), there exists a pair $(u_{\ell},{\mathring{R}}_{\ell})$ and a commutator
stress ${\mathring{R}}_{\textnormal{comm}}$ satisfying the new equation

$\displaystyle\partial_{t}(u_{\ell}-\alpha^{2}\Delta
u_{\ell})^{l}+\partial_{k}\left(u_{\ell}^{k}(u_{\ell}^{l}-\alpha^{2}\Delta
u_{\ell}^{l})-\alpha^{2}\partial_{k}u_{\ell}^{j}\partial_{l}u_{\ell}^{j}\right)+\partial_{l}p_{\ell}$
$\displaystyle=\partial_{k}({\mathring{R}}_{\ell}^{kl}+{\mathring{R}}_{\textnormal{comm}}^{kl})$
(3.2a) $\displaystyle\partial_{l}u_{\ell}^{l}$ $\displaystyle=0\,.$ (3.2b)

The new pair satisfies the inductive estimates (2.3) as well as the higher-
order estimates

${\left\|\nabla_{t,x}^{m}u_{\ell}\right\|_{L^{\infty}}\lesssim{\ell^{-\nicefrac{{3}}{{2}}-m}}}\,,\qquad\left\|\nabla_{t,x}^{m}{\mathring{R}}_{\ell}\right\|_{L^{\infty}}\lesssim{\ell^{-m-2}}\,,\qquad\left\|{\mathring{R}}_{\textnormal{comm}}\right\|_{L^{1}}\leq\lambda_{q}^{-1}\delta_{q+2}\lambda_{q+2}^{2}\,$
(3.3)

for $m\geq 0$.

###### Proof.

Let $\phi_{\ell}$ be a family of spatial Friedrichs mollifiers at scale
$\ell$, and let $\varphi_{\ell}$ be a family of temporal Friedrichs mollifiers
at scale $\ell$. Then defining

$\displaystyle u_{\ell}$ $\displaystyle:=(u_{q}*\phi_{\ell})*\varphi_{\ell}$
(3.4a) $\displaystyle{\mathring{R}}_{\ell}$
$\displaystyle:=({\mathring{R}}_{q}*\phi_{\ell})*\varphi_{\ell}\,,$ (3.4b)

we have that the mollified objects satisfy the equation

$\displaystyle\partial_{t}(u_{\ell}-\alpha^{2}\Delta
u_{\ell})^{l}+\partial_{k}\left(u_{\ell}^{k}(u_{\ell}^{l}-\alpha^{2}\Delta
u_{\ell}^{l})-\alpha^{2}\partial_{k}u_{\ell}^{j}\partial_{l}u_{\ell}^{j}\right)+\partial_{l}p_{\ell}$
$\displaystyle=\partial_{k}({\mathring{R}}_{\ell}^{kl}+R_{\text{comm}}^{kl})$
(3.5a) $\displaystyle\partial_{l}u_{\ell}^{l}$ $\displaystyle=0\,.$ (3.5b)

In the above display, $R_{\textnormal{comm}}^{kl}$ is a symmetric (but not yet
traceless) tensor given by the formula

$\displaystyle R_{\text{comm}}^{kl}$
$\displaystyle:=u_{\ell}^{k}u_{\ell}^{l}-((u_{q}^{k}u_{q}^{l})*\phi_{\ell})*\varphi_{\ell}-\alpha^{2}\left(\partial_{k}u_{\ell}^{j}\partial_{l}u_{\ell}^{j}-((\partial_{k}u_{q}^{j}\partial_{l}u_{q}^{j})*\phi_{\ell})*\varphi_{\ell}\right)$
$\displaystyle\qquad-\alpha^{2}\mathcal{R}^{kl}\operatorname{\mathrm{div}}\left(u_{\ell}\otimes\Delta
u_{\ell}-((u_{q}\otimes\Delta u_{q})*\phi_{\ell})*\varphi_{\ell}\right)\,.$
(3.6)

Using the double commutator estimate from [CET94] (see also Proposition E.1
from [BDLISZ15]) and (2.3), we may estimate the last term in $L^{1}$ by

$\displaystyle\|\mathcal{R}\operatorname{\mathrm{div}}(\left(u_{\ell}\otimes\Delta
u_{\ell}-((u_{q}\otimes\Delta
u_{q})*\phi_{\ell})*\varphi_{\ell}\right))\|_{L^{1}}$
$\displaystyle\lesssim\|\left(u_{\ell}\otimes\Delta
u_{\ell}-((u_{q}\otimes\Delta
u_{q})*\phi_{\ell})*\varphi_{\ell}\right)\|_{L^{p}}$
$\displaystyle\lesssim\|\left(u_{\ell}\otimes\Delta
u_{\ell}-((u_{q}\otimes\Delta
u_{q})*\phi_{\ell})*\varphi_{\ell}\right)\|_{L^{\infty}}$
$\displaystyle\lesssim\ell^{2}\|\nabla^{3}u_{\ell}\|_{L^{\infty}}\left\|\nabla
u_{\ell}\right\|_{L^{\infty}}$ $\displaystyle\lesssim\ell^{2}\lambda_{q}^{8}$
$\displaystyle{\leq\lambda_{q}^{-1}\alpha^{2}\cdot\mathcal{C}_{\mathring{R}}\cdot\delta_{q+2}\lambda_{q+2}^{2}}\,,$

where $p\in(1,\infty)$ is close to $1$, we used the bound for
$\mathcal{R}\operatorname{\mathrm{div}}$ from Proposition 5.6, and we appealed
to inequality (1). The terms on the first line of (3) obey similar bounds, and
we omit further details. Subtracting off the trace of
$R_{\textnormal{comm}}^{kl}$ and absorbing it into the pressure concludes the
proof of the estimate for ${\mathring{R}}_{\textnormal{comm}}$. The estimates
for $u_{\ell}$ and ${\mathring{R}}_{\ell}$ follow from Young’s inequality for
convolutions, and we omit further details. ∎

##  4 Linear Algebra

We begin with the linear algebra lemma which ensures that there are sets of
vectors $v$ for which the linear combinations of simple tensors $v\otimes v$
can be used to “span” a set of symmetric traceless matrices. The significance
of the quantity $3v\otimes v-\operatorname{\mathrm{Id}}$ in this Lemma will
become clear after Lemma 6.2. The proof can be found in Appendix B.

######  Lemma 4.1 (Linear Algebra).

Let $N\in\mathbb{N}$ be given. Then there exists a fixed positive constant
$\mathcal{C}_{\rm sum}>0$, a small positive number $\varepsilon>0$, and sets
of nine distinct vectors
$\mathcal{K}_{n}=\\{k_{i}^{n}\\}_{i=1}^{9}\subset\mathbb{S}^{2}\cap\mathbb{Q}^{3}$
indexed by $n\in\\{0,\dots,N\\}$ such that the following holds. First, if
$n_{1}\neq n_{2}$, then
$\mathcal{K}_{n_{1}}\cap\mathcal{K}_{n_{2}}=\emptyset$. Secondly, let
${\mathring{R}}$ be a smooth, symmetric, traceless matrix with
$\left\|{\mathring{R}}\right\|_{0}\leq{\varepsilon}$. Then there exist smooth,
strictly positive functions
$\left\\{c_{i}^{n}\left({\mathring{R}}\right)\right\\}_{i=1}^{9}$ depending on
${\mathring{R}}$ and $n$ such that for all $n\in\\{0,\dots,N\\}$,

$\displaystyle\sum_{i=1}^{9}\left(c_{i}^{n}\right)^{2}\left({\mathring{R}}\right)\left(3k_{i}^{n}\otimes
k_{i}^{n}-\textnormal{Id}\right)={\mathring{R}}\,,\qquad\sum_{i=1}^{9}\left(c_{i}^{n}\right)^{2}\left({\mathring{R}}\right)=\mathcal{C}_{\rm
sum}\,.$ (4.1)

##  5 Technical Preliminaries

###  5.1 Decoupling

####  5.1.1 $L^{p}$ decoupling lemma

The following lemma may be found in [BV19] as Lemma 3.7. Although it is stated
there for $\mathbb{T}^{3}$, the proof adapts easily to other dimensions (up to
adjustments of certain constants).

######  Lemma 5.1 ($L^{p}$ decoupling).

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

$\displaystyle\lambda^{{\mathbf{N}_{\textnormal{dec}}}+4}\leq\left(\frac{\mu}{2\pi\sqrt{3}}\right)^{{\mathbf{N}_{\textnormal{dec}}}}\,.$
(5.1)

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

$\displaystyle\max_{0\leq
N\leq{\mathbf{N}_{\textnormal{dec}}}+4}\lambda^{-N}\|D^{N}f\|_{L^{p}}\leq\mathcal{C}_{f}$
(5.2)

for a constant $\mathcal{C}_{f}>0$. 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 $\lambda$).

####  5.1.2 Decoupling for intersections of deformed intermittent pipes

Before stating the main decoupling result for products of deformed pipes in
Proposition 5.3, we recall the following lemma from [bmnv21]. The version
written below is nearly identical to Lemma 4.7 from subsection 4.2 of
[bmnv21]. The only difference is that the analogue in [bmnv21] of the
Lipschitz bound in (5.5) is quite sharp, whereas (5.5) is lossy. Since the
timescale $\tau$ has been shortened commensurately in this version of the
Lemma, the proof is identical in spirit, and we refer the reader to [bmnv21]
for more details.

######  Lemma 5.2 (Control on Axes, Support, and Spacing).

Consider a convex neighborhood of space $\Omega\subset\mathbb{T}^{3}$. Let $v$
be an incompressible velocity field, and define the flow $X(x,t)$ as the
solution to

$\displaystyle\partial_{t}X(x,t)$ $\displaystyle=v\left(X(x,t),t\right)$
(5.3a) $\displaystyle X|_{t=t_{0}}$ $\displaystyle=x\,,$ (5.3b)

and inverse $\Phi(x,t)=X^{-1}(x,t)$ as the solution to

$\displaystyle\partial_{t}\Phi+v\cdot\nabla\Phi$ $\displaystyle=0$ (5.4a)
$\displaystyle\Phi_{t=t_{0}}$ $\displaystyle=x\,.$ (5.4b)

Define $\Omega(t):=\\{x\in\mathbb{T}^{3}:\Phi(x,t)\in\Omega\\}=X(\Omega,t)$,
and let ${\tau=\ell^{3}}$ be fixed. Suppose that for
$(x,t)\in\Omega(t)\times[t_{0}-\tau,t_{0}+\tau]$,

$|\nabla v(x,t)|\lesssim{\ell^{-\frac{3}{2}}}\,$ (5.5)

where the implicit constant is independent of $q$. Let
$\mathbb{W}_{\lambda_{q+1},r,\xi}:\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ be
a set of straight pipe flows constructed as in Proposition 6.1 which are
$\frac{\mathbb{T}^{3}}{\lambda_{q+1}r}$-periodic for
$\frac{\lambda_{q}}{\lambda_{q+1}}\leq r\leq 1$ and concentrated around axes
$\\{A_{i}\\}_{i\in\mathcal{I}}$ oriented in the vector direction $\xi$ for
$\xi\in\mathcal{K}_{n}$. Then
$\mathbb{W}:=\mathbb{W}_{\lambda_{q+1},r,\xi}(\Phi(x,t)):\Omega(t)\times[t_{0}-\tau,t_{0}+\tau]$
satisfies the following conditions:

  1. 1.

If $x$ and $y$ belong to a particular axis $A_{i}\subset\Omega$, then

$\frac{X(x,t)-X(y,t)}{|X(x,t)-X(y,t)|}=\xi+\delta_{i}(x,y,t)$ (5.6)

where $|\delta_{i}(x,y,t)|<\ell^{{\nicefrac{{1}}{{3}}}}$.

  2. 2.

Let $x$ and $y$ belong to a particular axis $A_{i}\subset\Omega$. Denote the
length of the axis $A_{i}(t):=X(A_{i}\cap\Omega,t)$ in between $X(x,t)$ and
$X(y,t)$ by $L(x,y,t)$. Then

$L(x,y,t)\leq\left(1+\ell^{\nicefrac{{1}}{{3}}}\right)\left|x-y\right|.$ (5.7)

  3. 3.

The support of $\mathbb{W}$ is contained in a
$\mathcal{C}_{\xi}\lambda_{q+1}^{-1}$-neighborhood of

$\bigcup_{i}A_{i}(t)\,,$ (5.8)

where $\mathcal{C}_{\xi}$ is a dimensional constant depending only on $N$ and
the sets $\mathcal{K}_{n}$ from Lemma 4.1.

  4. 4.

$\mathbb{W}$ is “approximately periodic” in the sense that for distinct axes
$A_{i},A_{j}$ with $i\neq j$ and
$\textnormal{dist}(A_{i}\cap\Omega,A_{j}\cap\Omega)=d$,

$\left(1-\ell^{\nicefrac{{1}}{{3}}}\right)d\leq\textnormal{dist}\left(A_{i}(t),A_{j}(t)\right)\leq\left(1+\ell^{\nicefrac{{1}}{{3}}}\right)d.$
(5.9)

We now state and prove the main proposition which allows us to obtain a
decoupling-style estimate for the product of two pipe flows with equal
periodicity but oriented around axes with non-parallel tangent vectors.

######  Proposition 5.3 (Decoupling of Minimally Orthogonal Pipes).

Consider neighborhoods of space $\Omega^{1}$ and $\Omega^{2}$ satisfying the
assumptions of Lemma 5.2. Let $v$ be an incompressible velocity field, and
define

$\displaystyle\partial_{t}\Phi_{1}+v\cdot\nabla\Phi_{1}$ $\displaystyle=0$
(5.10a) $\displaystyle\Phi|_{t=t_{1}}$ $\displaystyle=x\,.$ (5.10b)

and

$\displaystyle\partial_{t}\Phi_{2}+v\cdot\nabla\Phi_{2}$ $\displaystyle=0$
(5.11a) $\displaystyle\Phi|_{t=t_{2}}$ $\displaystyle=x\,.$ (5.11b)

Define $\Omega^{1}(t)$ and $\Omega^{2}(t)$ as in Lemma 5.2, and assume that
$v$ satisfies the usual estimate (5.5) on $\Omega^{1}(t)\cup\Omega^{2}(t)$.
For $\xi_{1},\xi_{2}$ belonging to $\mathcal{K}_{n_{1}}$,
$\mathcal{K}_{n_{2}}$, with $n_{1}\neq n_{2}$, let
$\mathbb{W}^{1}_{\lambda_{q+1},r_{2},\xi_{1}}:\Omega^{1}\rightarrow\mathbb{R}^{3}$
and
$\mathbb{W}^{2}_{\lambda_{q+1},r_{2},\xi_{2}}:\Omega^{2}\rightarrow\mathbb{R}^{3}$
be pipe flows with _shared_ periodicity $(\lambda_{q+1}r_{2})^{-1}$ but
concentrated around axes with _distinct_ unit vector directions
$\xi_{1},\xi_{2}\in\mathbb{S}^{2}$ such that there exists $\epsilon_{0}>0$
satisfying

$|\langle\xi_{1},\xi_{2}\rangle|\leq
1-\epsilon_{0}\,,\qquad{\ell^{\nicefrac{{1}}{{3}}}}\leq\frac{\epsilon_{0}}{2}\,.$
(5.12)

Define
$\mathbb{W}^{1}:=\mathbb{W}^{1}_{\lambda_{q+1},r_{2},\xi_{1}}(\Phi_{1})$ and
$\mathbb{W}^{2}:=\mathbb{W}^{2}_{\lambda_{q+1},r_{2},\xi_{2}}(\Phi_{2})$ to be
the transported pipe flows with shifting spatial domains $\Omega^{1}(t)$ and
$\Omega^{2}(t)$, respectively. Set ${\tau=\ell^{3}}$ and assume that there
exists

$t_{0}\in[t_{1}-\tau,t_{1}+\tau]\cap[t_{2}-\tau,t_{2}+\tau]$

such that $\Omega^{1}(t_{0})\cap\Omega^{2}(t_{0})\neq\emptyset$. Set

$\Omega(t):=\Omega^{1}(t)\cap\Omega^{2}(t)\,.$

Let $f:\Omega(t)\times[t_{1}-\tau,t_{1}+\tau]\cap[t_{2}-\tau,t_{2}+\tau]$ be a
function such that

$\left\|D^{n}f\right\|_{L^{\infty}_{t}\left(L^{1}(\Omega(t))\right)}\leq{\mathcal{C}_{f}}\left(\lambda_{q+1}r_{1}\right)^{n}\,,\qquad
n\leq{\mathbf{N}_{\textnormal{dec}}}+4$ (5.13)

where $m_{1},m_{2},{\mathbf{N}_{\textnormal{dec}}}\in\mathbb{N}$ and
${\mathbf{N}_{\textnormal{dec}}},r_{1},r_{2}$ satisfy

$0\leq r_{1}\leq r_{2}\leq
1\,,\qquad\frac{r_{1}}{r_{2}}\leq\frac{2}{3}\,,\qquad(\lambda_{q+1}r_{1})^{{\mathbf{N}_{\textnormal{dec}}}+4}\leq\left(\lambda_{q+1}r_{2}\right)^{\mathbf{N}_{\textnormal{dec}}}\,.$
(5.14)

Furthermore, assume that for $i=1,2$ and $\tau\in[t_{i}-\tau,t_{i}+\tau]$,
$\Phi_{i}$ satisfies

${|\nabla\Phi_{i}-\operatorname{\mathrm{Id}}|\leq\ell\,,\qquad\|\nabla^{n}\Phi_{i}\|_{L^{\infty}}\leq\mathcal{C}_{\Phi}(\lambda_{q+1}r_{1})^{n-1}\,,\qquad
1\leq n\leq\max(m_{1},m_{2})\,.}$ (5.15)

Then for $t\in[t_{1}-\tau,t_{1}+\tau]\cap[t_{2}-\tau,t_{2}+\tau]$,

$\left\|f(t,\cdot)\nabla^{m_{1}}\mathbb{W}^{1}(t,\cdot)\otimes\nabla^{m_{2}}\mathbb{W}^{2}(t,\cdot)\right\|_{L^{1}(\Omega(t))}\lesssim{r_{2}\cdot\mathcal{C}_{f}\cdot\lambda_{q+1}^{m_{1}+m_{2}}}$
(5.16)

for a universal implicit constant depending on $\epsilon_{0}$, the sets
$\mathcal{K}_{n}$ for $n\in\\{0,...,N\\}$, and $\mathcal{C}_{\Phi}$, but
independent of $f$, $r_{1}$, $r_{2}$, or $\tau$.

######  Remark 5.4.

We note that the entire Proposition applies as long as the vector directions
$\xi_{1}$ and $\xi_{2}$ are minimally orthogonal in the sense of (5.12).
Furthermore, the scaling in two dimensions of (5.16) is identical, since
$L^{2}$-normalized objects have magnitude $r_{2}^{-\nicefrac{{1}}{{2}}}$ in
$L^{\infty}$, and the area of the support of an intersection of two orthogonal
intermittent pipe flows is $r_{2}^{2}$. The proof goes through _mutatis
mutandis_ after making these slight adjustments, and we omit the details. In
the case of the 2D Euler-$\alpha$ equations, Mikado flows may not be taken to
be disjoint. This lemma allows Theorem 1.1 to apply for the 2D Euler-$\alpha$
equations as well, since the all intersection terms (not just the ones arising
from pipes originating from different Lagrangian coordinate systems) may be
estimated using this Proposition.

###### Proof of Proposition 5.3.

Define the set of intersection points $I(t)$ at time $t$ for the flowed pipes
$\mathbb{W}^{1}$ and $\mathbb{W}^{2}$ by

$I(t):=\\{x:x\in\operatorname{\mathrm{supp}}\mathbb{W}^{1}(t)\cap\operatorname{\mathrm{supp}}\mathbb{W}^{2}(t)\\}\,.$
(5.17)

We will show that at time $t_{0}$,

$I(t_{0})\subset\bigcup_{i}B_{i}\,,$ (5.18)

where the $B_{i}$’s are pairwise disjoint balls of radius
$\mathcal{C}(\xi_{1},\xi_{2})\left(\lambda_{q+1}r_{2}\right)^{-1}$ for some
dimensional constant $\mathcal{C}(\xi_{1},\xi_{2})$. Furthermore, we will show
that

$\left|I(t)\cap
B_{i}\right|{\lesssim}_{\epsilon_{0},\mathcal{C}(\xi_{1},\xi_{2})}\lambda_{q+1}^{-3}$
(5.19)

for a universal constant which we emphasize depends only on $\epsilon_{0}$ and
$\mathcal{C}(\xi_{1},\xi_{2})$. After proving this, we will be able to follow
the proof of the usual decoupling lemma to prove that the desired $L^{1}$
estimate holds at $t_{0}$. Since the intersection points are transported by
the incompressible velocity field $v$, the same decoupling estimate will hold
for all times. One could equivalently rerun the argument for all $t$ for which
$\Omega_{1}(t)\cap\Omega_{2}(t)\neq\emptyset$.

Let $x\in I(t_{0})$. Then
$x\in\operatorname{\mathrm{supp}}\mathbb{W}^{1}(t_{0})\cap\operatorname{\mathrm{supp}}\mathbb{W}^{2}(t_{0})$,
and by (5.8) from Lemma 5.2, there exist $x_{1},x_{2}$ belonging to flowed
axes $A_{i_{1}}(t_{0})$, $A_{i_{2}}(t_{0})$, respectively, where we haved
mimicked the notation from (5.2) to define the flowed axes for
$\mathbb{W}^{1}$ and $\mathbb{W}^{2}$, such that

$|x_{1}-x|\lesssim\lambda_{q+1}^{-1},\quad|x_{2}-x|\lesssim\lambda_{q+1}^{-1}\,.$
(5.20)

The implicit constants in the above inequality depend only on
$\mathcal{C}_{\xi_{1}}$ and $\mathcal{C}_{\xi_{2}}$ but nothing else. By (5.9)
from Lemma 5.2 and the shared periodicity of $\mathbb{W}^{1}$ and
$\mathbb{W}^{2}$ to scale $(\lambda_{q+1}r_{2})^{-1}$, $A_{i_{1}}(t_{0})$ and
$A_{i_{2}}(t_{0})$ are the only axes of $\mathbb{W}^{1}(t)$ and
$\mathbb{W}^{2}(t)$, respectively, which are closer than a dimensional
constant $\rho_{1,2}^{-1}$ multiplied by $(\lambda_{q+1}r_{2})^{-1}$ to any of
the points $x$, $x_{1}$, or $x_{2}$. We therefore define $B_{x}$ by

$B_{x}:=\left\\{y\in\mathbb{T}^{3}:|x-y|\leq\left(\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-1}\right\\}.$
(5.21)

We emphasize that $\rho_{1,2}$ depends only on $\xi_{1}$ and $\xi_{2}$, or
equivalently on the sets $\mathcal{K}_{n}$ from Proposition 4.1. We also note
that after setting
$\mathcal{C}_{\xi}=\max\\{\mathcal{C}_{\xi_{1}},\mathcal{C}_{\xi_{2}}\\}$,
$I(t_{0})\cap B_{x}$ only contains points which are within
$\mathcal{C}_{\xi}\lambda_{q+1}^{-1}$ of either $A_{i_{1}}(t_{0})$ or
$A_{i_{2}}(t_{0})$ by (5.8).

We will now prove that

$\left|I(t_{0})\cap B_{x}\right|\lesssim_{\epsilon_{0}}\lambda_{q+1}^{-3}\,.$
(5.22)

Towards this end, consider the cones $C^{j}$ defined for $j=1,2$ by

$C^{j}:=\\{x\in\mathbb{T}^{3}:x=x_{j}+r\xi\,\textnormal{ for
}\,|\xi-\xi_{j}|\leq\ell^{\nicefrac{{1}}{{3}}},r\in\mathbb{R}\\}\,.$ (5.23)

By (5.6), these cones contain the axes $A_{i_{1}}(t_{0})$ and
$A_{i_{2}}(t_{0})$. Let

$\tilde{C}^{j}:=\\{x\in\mathbb{T}^{3}:|x-C^{j}|\leq\mathcal{C}_{\xi}\lambda_{q+1}^{-1}\\}.$
(5.24)

By (5.8) and the previous observation that only $A_{i_{1}}(t_{0})$ and
$A_{i_{2}}(t_{0})$ have non-empty intersection with $B_{x}$, $\tilde{C}^{j}$
contains the support of $\mathbb{W}^{j}\cap B_{x}$ for $j=1,2$. Thus to
provide an upper bound on the measure of $I(t_{0})\cap B_{x}$, it suffices to
estimate the measure of $\tilde{C}^{1}\cap\tilde{C}^{2}$.

Let $x\in\tilde{C}^{1}\cap\tilde{C}^{2}$. Then for $j=1,2$, there exist
$r_{j}\in\mathbb{R}$, $\xi_{j,x}\in\mathbb{S}^{2}$ such that
$|\xi_{j}-\xi_{j,x}|<\ell^{\nicefrac{{1}}{{3}}}$, and $y_{j}$ such that
$|y_{j}-x_{j}|\leq\mathcal{C}_{\xi}\lambda_{q+1}^{-1}$, with

$x=y_{1}+r_{1}\xi_{1,x}=y_{2}+r_{2}\xi_{2,x}\,.$ (5.25)

By (5.20), we have that then

$|y_{1}-y_{2}|\lesssim_{\mathcal{C}_{\xi}}\lambda_{q+1}^{-1}\,.$

As a consequence of (5.12),

$\displaystyle\lambda_{q+1}^{-2}$
$\displaystyle\gtrsim|r_{1}\xi_{1,x}-r_{2}\xi_{2,x}|^{2}$
$\displaystyle=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\langle\xi_{1,x},\xi_{2,x}\rangle$
$\displaystyle\geq
r_{1}^{2}+r_{2}^{2}-\left(1-\frac{\epsilon_{0}}{2}\right)\left(r_{1}^{2}+r_{2}^{2}\right)$
$\displaystyle\geq\frac{\epsilon_{0}}{2}\left(r_{1}^{2}+r_{2}^{2}\right)\,,$

which is only possible when

$r_{1}^{2}+r_{2}^{2}\lesssim\frac{1}{\epsilon_{0}}\lambda_{q+1}^{-2}\,.$

We can then brutally bound the volume of $\tilde{C}^{1}\cap\tilde{C}^{2}$ by
the volume of the union of two balls centered at $x_{j}$ whose radius is twice
the maximum value of $r_{j}$, yielding

$\left|\tilde{C}^{1}\cap\tilde{C}^{2}\right|\lesssim_{\epsilon_{0},\mathcal{C}_{\xi}}\lambda_{q+1}^{-3}.$
(5.26)

Since $I(t_{0})\cap B_{x}\subset\tilde{C}^{1}\cap\tilde{C}^{2}$, we have
verified (5.19) for $B_{x}$. Repeating the above steps by finding $x\in
I(t_{0})\setminus B_{x}$ and so forth verifies (5.18).

To demonstrate the desired decoupling estimate at $t_{0}$, consider

$\displaystyle\int_{\Omega(t_{0})}\left|f\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right|\,dx$
$\displaystyle=\int_{I(t_{0})}\left|f\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right|\,dx$
$\displaystyle\leq\sum_{i}\int_{B_{i}}\left|f\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right|\,dx\,.$
(5.27)

Letting $\bar{h}_{i}$ denote the mean of any function $h$ on $B_{i}$, for
$x\in B_{i}$, we can write that

$\displaystyle|f(x)|$ $\displaystyle=|(\bar{f}_{i}+f(x)-\bar{f}_{i})|$
$\displaystyle\leq|\bar{f}_{i}|+(2\rho_{1,2}\lambda_{q+1}r_{2})^{-1}\sup_{B_{i}}|Df|$

since the diameter of $B_{i}$ is $(2\rho_{1,2}\lambda_{q+1}r_{2})^{-1}$.
Iterating ${\mathbf{N}_{\textnormal{dec}}}$ times, we obtain the pointwise
estimate for $x\in B_{i}$

$|f(x)|\leq\sum_{m=0}^{{\mathbf{N}_{\textnormal{dec}}}-1}\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-m}\left|\overline{D^{m}f}_{i}\right|+\left(\lambda_{q+1}r_{2}\right)^{-{\mathbf{N}_{\textnormal{dec}}}}\left\|D^{{\mathbf{N}_{\textnormal{dec}}}}f\right\|_{L^{\infty}(\Omega(t_{0}))}\,.$

Multiplying by
$\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}$,
integrating, and summing over $i$, (5.27) is less than or equal to

$\displaystyle\sum_{i}\int_{B_{i}}$
$\displaystyle|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}|\left(\sum_{m=0}^{{\mathbf{N}_{\textnormal{dec}}}-1}\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-m}|\overline{D^{m}f}_{i}|\right)$
$\displaystyle\qquad+\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-{\mathbf{N}_{\textnormal{dec}}}}\left\|D^{{\mathbf{N}_{\textnormal{dec}}}}f\right\|_{L^{\infty}(\Omega(t_{0}))}\left\|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right\|_{L^{1}(\Omega(t_{0}))}$
$\displaystyle\leq\sum_{i}\int_{B_{i}}|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}|\left(\sum_{m=0}^{{\mathbf{N}_{\textnormal{dec}}}-1}\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-m}\frac{1}{|B_{i}|}\left\|D^{m}f_{i}\right\|_{L^{1}(B_{i})}\right)$
$\displaystyle\qquad+\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-{\mathbf{N}_{\textnormal{dec}}}}\left\|D^{{\mathbf{N}_{\textnormal{dec}}}}f\right\|_{L^{\infty}(\Omega(t_{0}))}\left\|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right\|_{L^{1}(\Omega(t_{0}))}.$
(5.28)

We note that by (6.2), (5.15), repeated application of the chain rule, and the
inequality $\lambda_{q+1}\gg\lambda_{q+1}r_{1}$, we have

$\|\nabla^{m_{i}}\mathbb{W}^{i}\|_{L^{\infty}}\lesssim
r_{2}^{-1}\lambda_{q+1}^{m_{i}}\,.$

Since

$\left|I(t_{0})\cap B_{i}\right|\lesssim\lambda_{q+1}^{-3}\,,$

we have

$\int_{B_{i}}|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}|\leq\left|I(t_{0})\cap
B_{i}\right|\left\|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right\|_{L^{\infty}(B_{i})}\lesssim
r_{2}^{-2}\lambda_{q+1}^{m_{1}+m_{2}-3}\,.$ (5.29)

Using the fact that there exist at most
$\approx\left(\lambda_{q+1}r_{2}\right)^{3}$ disjoint balls of radius
$(\rho_{1,2}\lambda_{q+1}r_{2})^{-1}$ in $\mathbb{T}^{3}$ (by volume
constraints), we have that using Sobolev embedding, (5.13), (5.14), and
(5.29), we can estimate the second term in (5.28) by

$\displaystyle\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-{\mathbf{N}_{\textnormal{dec}}}}$
$\displaystyle\left\|D^{{\mathbf{N}_{\textnormal{dec}}}}f\right\|_{L^{\infty}(\Omega(t_{0}))}\left\|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\right\|_{L^{1}(\Omega(t_{0}))}$
$\displaystyle\lesssim\left(\lambda_{q+1}r_{2}\right)^{-{\mathbf{N}_{\textnormal{dec}}}}\cdot\mathcal{C}_{f}\cdot\left(\lambda_{q+1}r_{1}\right)^{{\mathbf{N}_{\textnormal{dec}}}+4}\|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}\|_{L^{1}\left(\Omega(t_{0})\right)}$
$\displaystyle\leq
C(\epsilon_{0},\mathcal{K})\cdot{\mathcal{C}_{f}}\cdot\left(\lambda_{q+1}r_{2}\right)^{3}\lambda_{q+1}^{m_{1}+m_{2}-3}r_{2}^{-2}$
$\displaystyle\leq C(\epsilon_{0},\mathcal{K})\cdot{\mathcal{C}_{f}}\cdot
r_{2}\lambda_{q+1}^{m_{1}+m_{2}}\,.$

The constant in the last line depends only on $\epsilon_{0}$, the sets of
vector directions $\mathcal{K}_{n}$, and geometric quantities. Thus it remains
to estimate the first term. We may write that

$\displaystyle\sum_{i}\int_{B_{i}}$
$\displaystyle|\nabla^{m_{1}}\mathbb{W}^{1}\otimes\nabla^{m_{2}}\mathbb{W}^{2}|\left(\sum_{m=0}^{{\mathbf{N}_{\textnormal{dec}}}-1}\left(2\rho_{1,2}\lambda_{q+1}r_{2}\right)^{-m}\frac{1}{|B_{i}|}\left\|D^{m}f\right\|_{L^{1}(B_{i})}\right)$
$\displaystyle\leq
C(\epsilon_{0},\mathcal{K})\sum_{m=0}^{{\mathbf{N}_{\textnormal{dec}}}-1}\lambda_{q+1}^{m_{1}+m_{2}-3}r_{2}^{-2}\left(\lambda_{q+1}r_{2}\right)^{3}\left(\frac{\lambda_{q+1}r_{1}}{\lambda_{q+1}r_{2}}\right)^{m}{\mathcal{C}_{f}}$
$\displaystyle\leq
C(\mathcal{K},\epsilon_{0})\lambda_{q+1}^{m_{1}+m_{2}}r_{2}{\mathcal{C}_{f}}$

using again (5.13), (5.14), (5.21), and (5.29). Summing estimates for the two
terms proves (5.16) at the fixed time $t_{0}$.

To prove the estimate for all times $t$, first notice that by the
incompressibility of the flow, the $B_{i}$’s remain disjoint upon advection.
Furthermore, since $\mathbb{W}^{1}$ and $\mathbb{W}^{2}$ are constant along
the characteristics of $v$, the $L^{1}$ norm of their product on each $B_{i}$
remains constant as well. Thus, the only part of the argument which requires
adjustment is the estimate on the diameters of the flowed balls $B_{i}$, which
is easily seen to increase by a geometric factor on the timescale in question
after using the results of Lemma 5.2. ∎

###  5.2 Inverse Divergence

The inverse divergence we use in this paper must be flexible enough to handle
both intermittency and composition with diffeomorphisms. For these purposes,
we use an inverse divergence predicated on “differentation by parts” rather
than Fourier multipliers. The inverse divergence operator from [bmnv21] is
built upon this principle; in fact, the estimates proven there are much more
detailed than anything necessary in this paper. However, we reproduce
identically the “iterative step” from [bmnv21] but state a significantly
streamlined and simplified version of the estimates satisfied by the inverse
divergence operator. For the reader’s convenience, we include an outline of
the proof of the claimed estimates for the inverse divergence operator.

######  Proposition 5.5 (Inverse divergence iteration step).

Let $n\geq 2$ and fix two zero-mean $\mathbb{T}^{n}$-periodic functions
$\varrho$ and $\vartheta$ with $\varrho=\Delta\vartheta$. Let $\Phi$ be a
volume preserving transformation of $\mathbb{T}^{n}$ such that
$\left\|\nabla\Phi-\operatorname{\mathrm{Id}}\right\|_{L^{\infty}(\mathbb{T}^{n})}\leq\nicefrac{{1}}{{2}}$.
Define the matrix $A=(\nabla\Phi)^{-1}$. Given a smooth vector field $G^{i}$,
we have

$\displaystyle
G^{i}\varrho\circ\Phi=\partial_{m}{\mathring{R}}^{im}+\partial_{i}P+E^{i}\,,$
(5.30)

where the traceless symmetric stress $R^{im}$ is given by

$\displaystyle{\mathring{R}}^{im}$
$\displaystyle=\left(G^{i}A^{m}_{\ell}+G^{m}A^{i}_{\ell}-A^{i}_{k}A^{m}_{k}G^{p}\partial_{p}\Phi^{\ell}\right)(\partial_{\ell}\vartheta)\circ\Phi-P\delta_{im}\,,$
(5.31)

the pressure term is given by

$\displaystyle P$
$\displaystyle=\left(2G^{m}A^{m}_{\ell}-A^{m}_{k}A^{m}_{k}G^{p}\partial_{p}\Phi^{\ell}\right)(\partial_{\ell}\vartheta)\circ\Phi\,,$
(5.32)

and the error term $E^{i}$ is given by

$\displaystyle E^{i}$
$\displaystyle=\left(\partial_{m}\left(G^{p}A^{i}_{k}A^{m}_{k}-G^{m}A^{i}_{k}A^{p}_{k}\right)\partial_{p}\Phi^{\ell}-\partial_{m}G^{i}A^{m}_{\ell}\right)(\partial_{\ell}\vartheta)\circ\Phi\,.$
(5.33)

Before defining and collecting estimates on the primary inverse divergence
operator, we recall the Fourier-multiplier frequently used in convex
integration schemes, for example in [BDLISZ15].

######  Proposition 5.6 (Fourier-multiplier inverse divergence).

Let $v:\mathbb{T}^{3}\rightarrow\mathbb{T}^{3}$ be a $C^{\infty}$ vector
field. Define

$(\mathcal{R}v)^{ij}=\left(-\frac{1}{2}\Delta^{-2}\partial_{i}\partial_{j}\partial_{k}-\frac{1}{2}\Delta^{-1}\partial_{k}\delta_{ij}+\Delta^{-1}\partial_{i}\delta_{jk}+\Delta^{-1}\partial_{j}\delta_{ik}\right)\left(v^{k}-\fint_{\mathbb{T}^{3}}v^{k}\right)\,.$
(5.34)

Then
$\operatorname{\mathrm{div}}\left(\mathcal{R}v\right)=v-\fint_{\mathbb{T}^{3}}v$,
and the operator $\nabla\circ\mathcal{R}$ is a singular integral operator
which is bounded from $L^{p}(\mathbb{T}^{3})$ to itself for all $1<p<\infty$
(with $p$-dependent bounds).

With the iterative step and nonlocal inverse divergence operator in hand, we
can now construct the full inverse divergence operator and record and prove
the estimates satisfied by the output. For the purposes of the statement of
this Proposition, we use the notation

$\mathcal{M}\left(n,N,\lambda,\Lambda\right)=\lambda^{\min\\{n,N\\}}\Lambda^{\max\\{n-N,0\\}}\,.$
(5.35)

######  Proposition 5.7 (Inverse divergence with estimates).

Fix an incompressible vector field
$v(t,x):\mathbb{R}\times\mathbb{T}^{n}\rightarrow\mathbb{R}^{n}$ for $n=2,3$
and denote its material derivative by $D_{t}=\partial_{t}+v\cdot\nabla$. Let
$G:\mathbb{R}\times\mathbb{T}^{n}\rightarrow\mathbb{R}^{n}$ be a vector field,
and assume there exists a constant $\mathcal{C}_{G}>0$, $\lambda\geq 1$, and a
large integer ${\mathsf{d}}$ such that

$\displaystyle\left\|D^{N}G\right\|_{L^{1}(\mathbb{T}^{n})}\lesssim\mathcal{C}_{G}\lambda^{N}$
(5.36)

for all $N\leq 2{\mathsf{d}}$ with implicit constants which may depend on $N$
but not $q$. Let $\Phi$ be a volume preserving transformation of
$\mathbb{T}^{n}$ such that

$D_{t}\Phi=0\,\qquad\mbox{and}\qquad\left\|\nabla\Phi-\operatorname{\mathrm{Id}}\right\|_{L^{\infty}(\operatorname{\mathrm{supp}}G)}\leq\nicefrac{{1}}{{2}}\,.$
(5.37)

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 bounds

$\displaystyle\left\|D^{N+1}\Phi\right\|_{L^{\infty}(\operatorname{\mathrm{supp}}G)}+\left\|D^{N+1}\Phi^{-1}\right\|_{L^{\infty}(\operatorname{\mathrm{supp}}G)}+\left\|D^{N}\left(\nabla\Phi\right)^{-1}\right\|_{L^{\infty}(\operatorname{\mathrm{supp}}G)}$
$\displaystyle\lesssim\lambda^{N}\,$ (5.38)

for all $N\leq 2{\mathsf{d}}$, where the implicit constants may again depend
on $N$ but not $q$. Lastly, let
$\varrho,\vartheta\colon\mathbb{T}^{n}\to\mathbb{R}$ be two zero mean
functions with the following properties:

  1. 1.

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

  2. 2.

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

  3. 3.

There exist parameters $\Lambda\geq\zeta$ and $\mathcal{C}_{*}\geq 1$ such
that

$\displaystyle\left\|D^{N}\varrho\right\|_{L^{1}}\lesssim\mathcal{C}_{*}\Lambda^{N}\qquad\mbox{and}\qquad\left\|D^{N}\vartheta\right\|_{L^{1}}\lesssim\mathcal{C}_{*}\mathcal{M}\left(N,2{\mathsf{d}},\zeta,\Lambda\right)$
(5.39)

for all $N\leq 4{\mathsf{d}}$ except for the case $N=2{\mathsf{d}}$ when the
Calderón-Zygmund inequality fails. In this exceptional case, the second
inequality in (5.39) is allowed to be weaker by a factor of
$\Lambda^{\alpha}$, for an arbitrary $\alpha\in(0,1]$; that is, we only
require that
$\left\|D^{2{\mathsf{d}}}\vartheta\right\|_{L^{1}}\lesssim\mathcal{C}_{*}\Lambda^{\alpha}\zeta^{2{\mathsf{d}}}$.

If the parameters satisfy

$\displaystyle{\left(\frac{\lambda}{\zeta}\right)^{{\mathsf{d}}}\lambda^{4}\leq\zeta^{-1}\,,}$
(5.40)

then we have that

$\displaystyle G\;\varrho\circ\Phi$
$\displaystyle=\operatorname{\mathrm{div}}{\mathring{R}}+\nabla
P+\fint_{\mathbb{T}^{3}}G\varrho\circ\Phi\,.$ (5.41)

Furthermore, the traceless symmetric stress ${\mathring{R}}$ and pressure $P$
satisfy the bound

$\displaystyle\left\|{\mathring{R}}\right\|_{L^{1}}+\left\|P\right\|_{L^{1}}$
$\displaystyle\lesssim\mathcal{C}_{G}{\lambda^{4}}\mathcal{C}_{*}\zeta^{-1}\,,$
(5.42)

with an implicit constant which is independent of $G$, $\varrho$, or $\Phi$.

###### Proof.

The proof mirrors that of [bmnv21] in spirit. Since there is no need to
propagate any sharp derivative estimates, however, the proof is significantly
simpler. In fact there is no need to ensure that decoupling lemmas apply as in
[bmnv21], leading to the $\lambda^{4}$ Sobolev loss in (5.42). These
shortcomings are irrelevant for applications in this paper but could be
avoided by appealing to the more sophisticated estimates in [bmnv21]. In any
case, the strategy is to apply Proposition 5.5 to produce a sequence of
matrices ${\mathring{R}}_{(j)}$ and scalar functions $P_{(j)}$ for $0\leq
j\leq{\mathsf{d}}-1$ which each satisfy the estimates in (5.42). Every
iteration will produce a symmetric tensor ${\mathring{R}}_{(j)}$ and pressure
$P_{(j)}$ which are each smaller than $G\varrho\circ\Phi$ by at least
$\zeta^{-1}$ and an error term $E_{(j)}$ which has gained one factor of
$\zeta^{-1}\lambda$ relative to its predecessor; that is, we repeatedly
differentiate by parts to produce a symmetric tensor and an error term which
has “exchanged expensive derivatives for cheap derivatives.” We may halt this
exchanging process after reaching the threshold indicated in (5.40), at which
point we simply apply the usual Fourier multiplier inverse divergence and use
its trivial $L^{2}(\mathbb{T}^{n})\rightarrow L^{2}(\mathbb{T}^{n})$ bound.

To set notation let us define

$\varrho_{(0)}=\varrho\,,\qquad\varrho_{(k)}=\left(\zeta^{-2}\Delta\right)^{{\mathsf{d}}-k}\vartheta\quad\forall
1\leq k\leq{\mathsf{d}}\,.$ (5.43)

From these definitions, we have that

$\varrho_{(k-1)}=\zeta^{-2}\Delta\varrho_{(k)}\,.$ (5.44)

We begin by applying Proposition 5.5 with $\varrho=\varrho_{(0)}$ and
$\vartheta=\varrho_{(1)}$ so that $\varrho=\zeta^{-2}\Delta\varrho_{(1)}$.
From (5.37) we have that the assumption on $\Phi$ is satisfied, and so we
obtain

$G^{i}\varrho_{(0)}\circ\Phi=\partial_{m}{\mathring{R}}^{im}_{(0)}+\partial_{i}P_{(0)}+E^{i}_{(0)}\,.$

From (5.31) and (5.32), we have that both the symmetric traceless stress
${\mathring{R}}^{im}_{(0)}$ and the pressure $P^{i}_{(0)}$ are defined in
terms of the symmetric tensor

$\left(G^{i}A^{m}_{\ell}+G^{m}A^{i}_{\ell}-A^{i}_{k}A^{m}_{k}G^{p}\partial_{p}\Phi^{\ell}\right)\left(\zeta^{-2}\partial_{\ell}\varrho_{(1)}\right)\circ\Phi\,,$
(5.45)

and so it will suffice to estimate (5.45). From (5.38), (5.36), and the
Sobolev embedding $W^{n+1,1}(\mathbb{T}^{n})\hookrightarrow
L^{\infty}(\mathbb{T}^{n})$, we have that

$\left\|G^{i}A^{m}_{\ell}+G^{m}A^{i}_{\ell}-A^{i}_{k}A^{m}_{k}G^{p}\partial_{p}\Phi^{\ell}\right\|_{L^{\infty}}\lesssim\mathcal{C}_{G}{\lambda^{4}}\,.$

From item (1) and (5.39), we have that

$\left\|\zeta^{-2}\partial_{\ell}\varrho_{(1)}\circ\Phi\right\|_{L^{1}}=\left\|\zeta^{-2}\partial_{\ell}\varrho_{(1)}\right\|_{L^{1}}=\left\|\zeta^{-2}\zeta^{-2({\mathsf{d}}-1)}\partial_{\ell}\Delta^{{\mathsf{d}}-1}\vartheta\right\|_{L^{1}}\lesssim\zeta^{-1}\mathcal{C}_{*}\,.$

Thus we may bound the tensor in (5.45) in $L^{1}(\mathbb{T}^{n})$ by

$\mathcal{C}_{G}{\lambda^{4}}\zeta^{-1}\mathcal{C}_{*}$

as desired in (5.42). The error term $E^{i}_{(0)}$ is given from (5.33) by

$\displaystyle E^{i}_{(0)}$
$\displaystyle=\left(\partial_{m}\left(G^{p}A^{i}_{k}A^{m}_{k}-G^{m}A^{i}_{k}A^{p}_{k}\right)\partial_{p}\Phi^{\ell}-\partial_{m}G^{i}A^{m}_{\ell}\right)(\zeta^{-2}\partial_{\ell}\varrho_{(1)})\circ\Phi\,.$

Upon decomposition of the summation over $m$, $p$, $k$, and $\ell$, what
remains is a sum of terms, each of which is a vector field with components
indexed by $i$. Each of these vector fields is of the form

$G^{i}_{(1)}\zeta^{-2}\partial_{\ell}\varrho_{(1)}\circ\Phi\,,$

where $\ell\in\\{1,\dots,n\\}$. In the above expression, the vector field
$G^{i}_{(1)}$ is a single term of the form

$\partial_{m}\left(G^{p}A^{i}_{k}A^{m}_{k}-G^{m}A^{i}_{k}A^{p}_{k}\right)\partial_{p}\Phi^{\ell}-\partial_{m}G^{i}A^{m}_{\ell}\,,$

where $m,p,k,\ell\in\\{1,\dots,n\\}$. From (5.36) and (5.37), we have that

$\left\|D^{N}\left(\partial_{m}\left(G^{p}A^{i}_{k}A^{m}_{k}-G^{m}A^{i}_{k}A^{p}_{k}\right)\partial_{p}\Phi^{\ell}-\partial_{m}G^{i}A^{m}_{\ell}\right)\right\|_{L^{1}}\lesssim\mathcal{C}_{G}\lambda^{N+1}\,$

for all $N\leq 2{\mathsf{d}}-1$. However, the $L^{1}$ norm of
$\left(\zeta^{-2}\partial_{\ell}\varrho_{(1)}\circ\Phi\right)$ is
$\zeta^{-1}\mathcal{C}_{*}$ as calculated above. Hence we are in a scenario
similar to that in which we started, except we have “traded a cheap derivative
for an expensive derivative” - i.e. lost a factor of $\lambda$ but gained a
factor of $\zeta^{-1}$ when estimating the $L^{1}$ norm of $E_{(0)}^{i}$
relative to that of the original product $G\varrho\circ\Phi$. So applying
Proposition 5.7 with the new functions $G^{i}_{(1)}$ and
$\zeta^{-2}\partial_{\ell}\varrho_{(1)}\circ\Phi$ and estimating as before
will produce a new symmetric stress ${\mathring{R}}_{(1)}$, pressure
$P_{(1)}$, and error $E_{(1)}$. The bounds for these terms will differ from
those of their respective predecessors by a factor of $\lambda\zeta^{-1}$.
Repeating this process ${\mathsf{d}}$ times produces an explicitly computable
sum of the form

$\displaystyle
G^{i}\varrho\circ\Phi=\sum_{j=0}^{{\mathsf{d}}-1}\left(\partial_{m}{\mathring{R}}^{im}_{(j)}+\partial_{i}P_{(j)}\right)+E^{i}_{({\mathsf{d}}-1)}\,.$
(5.46)

The final error term $E_{({\mathsf{d}}-1)}^{i}$ has the same mean as the left-
hand size of (5.46) and has gained the factor of smallness
$\left(\lambda\zeta\right)^{{\mathsf{d}}-1}$ relative to $E_{(0)}$, which had
already gained one factor of $\lambda\zeta^{-1}$. Hence we may estimate the
size of $E^{i}_{({\mathsf{d}}-1)}$ in $L^{2}$ by

$\mathcal{C}_{G}\lambda^{4+{\mathsf{d}}}\mathcal{C}_{*}\zeta^{-{\mathsf{d}}}\,.$

Applying $\mathcal{R}$ to $\mathbb{P}_{\neq 0}E^{i}_{(d-1)}$, appealing to its
$L^{2}\hookrightarrow L^{2}$ bound, and utilizing (5.40) concludes the proof.
∎

##  6 Construction of the Perturbation

We next define the intermittent Mikado flows used in this paper. The following
Proposition is part of Proposition 4.3 from [bmnv21], to which we refer the
reader for a proof of everything except (2), as well as further properties
about intermittent Mikado flows which are not relevant to this paper.

######  Proposition 6.1 (Construction and Properties of Intermittent Pipe
Flows).

Given a vector $\xi$ belonging to the one of the sets of rational vectors
$\mathcal{K}_{n}\subset\mathbb{Q}^{3}$ from Lemma 4.1,
$r^{-1},\lambda\in\mathbb{N}$ with $\lambda r\in\mathbb{N}$, and large
integers $N$ and $d$, there exist vector fields
$\mathbb{W}_{\xi,\lambda,r}:\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ and
implicit constants depending on $N$ and $d$ but not $\lambda$ or $r$ such that
the following hold:

  1. 1.

There exists $\varrho:\mathbb{R}^{2}\rightarrow\mathbb{R}$ which is radially
symmetric and given by the iterated Laplacian $\Delta^{d}\vartheta=:\varrho$
of a radially symmetric potential
$\vartheta:\mathbb{R}^{2}\rightarrow\mathbb{R}$ with compact support in a ball
of radius $\frac{1}{4}$ such that the following holds. There exists
$\mathbb{U}_{\xi,\lambda,r}:\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ such that

$\displaystyle{\operatorname{\mathrm{curl}}\mathbb{U}_{\xi,\lambda,r}=\xi\lambda^{-2d}\Delta^{d}\left(\vartheta_{\xi,\lambda,r}\right)=\xi\varrho_{\xi,\lambda,r}=:\mathbb{W}_{\xi,\lambda,r}}\,,\qquad\xi\cdot\nabla\varrho_{\xi,\lambda,r}=\xi\cdot\nabla\vartheta_{\xi,\lambda,r}=0\,.$

  2. 2.

$\mathbb{W}_{\xi,\lambda,r}$ is a stationary, pressureless solution to the
Euler equations, i.e.

$\operatorname{\mathrm{div}}\mathbb{W}_{\xi,\lambda,r}=0,\qquad\operatorname{\mathrm{div}}\left(\mathbb{W}_{\xi,\lambda,r}\otimes\mathbb{W}_{\xi,\lambda,r}\right)=0\,.$

In addition, $\mathbb{W}_{\xi,\lambda,r}$ is a stationary solution to the
Euler-$\alpha$ equations with an explicitly computable pressure222The pressure
corresponding to a stationary solution of the Euler-$\alpha$ equations is not
uniquely determined unless a mean-zero condition is imposed. \- see (6.10).

  3. 3.

For all $n\leq N$,

${\left\|\nabla^{n}\vartheta_{\xi,\lambda,r}\right\|_{L^{p}(\mathbb{T}^{3})}\lesssim\lambda^{n}r^{\left(\frac{2}{p}-1\right)}},\qquad{\left\|\nabla^{n}\varrho_{\xi,\lambda,r}\right\|_{L^{p}(\mathbb{T}^{3})}\lesssim\lambda^{n}r^{\left(\frac{2}{p}-1\right)}}$
(6.1)

and

${\left\|\nabla^{n}\mathbb{U}_{\xi,\lambda,r}\right\|_{L^{p}(\mathbb{T}^{3})}\lesssim\lambda^{n-1}r^{\left(\frac{2}{p}-1\right)}},\qquad{\left\|\nabla^{n}\mathbb{W}_{\xi,\lambda,r}\right\|_{L^{p}(\mathbb{T}^{3})}\lesssim\lambda^{n}r^{\left(\frac{2}{p}-1\right)}}.$
(6.2)

Furthermore, each of the functions listed above is
$\frac{\mathbb{T}^{3}}{\lambda r}$-periodic.

  4. 4.

Let $\Phi:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{T}^{3}$ be the periodic
solution to the transport equation

$\displaystyle\partial_{t}\Phi+v\cdot\nabla\Phi$ $\displaystyle=0,$ (6.3a)
$\displaystyle\Phi_{t=t_{0}}$ $\displaystyle=x\,,$ (6.3b)

with a smooth, divergence-free, periodic velocity field $v$. Then

$\nabla\Phi^{-1}\cdot\left(\mathbb{W}_{\xi,\lambda,r}\circ\Phi\right)=\operatorname{\mathrm{curl}}\left(\nabla\Phi^{T}\cdot\left(\mathbb{U}_{\xi,\lambda,r}\circ\Phi\right)\right).$
(6.4)

###### Proof of Proposition 6.1.

We refer to [bmnv21] for the proofs of each item, save for the stationarity
with respect to the Euler-$\alpha$ equations. Due to the presence of the
Laplacian in the Euler-$\alpha$ equations, we must define the Mikado flows
using a radially symmetric flow profile. We shall define the objects
$\mathbb{U}$, $\mathbb{W}$, $\varrho$, and $\vartheta$ at unit scale and refer
to [bmnv21] for the concentration and periodizing. Thus, let
$h:\mathbb{R}^{+}\to\mathbb{R}$ be a compactly supported smooth function
satisfying

$\int_{0}^{\infty}h(r){r}\,dr=0\,,\qquad
h(r)=\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}\right)^{d}H(r)\,.$ (6.5)

Note that the second property asserts that in Cartesian coordinates, $h$ is
the iterated Laplacian of $H$, and so the second property implies the first.
Then define

$\varrho(y_{1},y_{2}):=h\left(\sqrt{y_{1}^{2}+y_{2}^{2}}\right)\,,\qquad\vartheta(y_{1},y_{2}):=H\left(\sqrt{y_{1}^{2}+y_{2}^{2}}\right)\,.$
(6.6)

Furthermore, define the rotated version of (6.6) by

$\varrho_{k}(x):=\varrho(k_{1}\cdot x,k_{2}\cdot x)\,,\quad
x\in\mathbb{R}^{3}\,,$ (6.7)

where $(k_{1},k_{2},k)$ forms a rational orthonormal basis of
$\mathbb{R}^{3}$. Define the radial Mikado flow associated to (6.7) as

$\mathbb{W}_{k}(x)=\varrho_{k}(x)k.$ (6.8)

We note that one may convert to cylindrical coordinates adapted to the support
of $\varrho_{k}$; after doing so, it is clear that $\Delta u^{j}\nabla u^{j}$
is curl-free. Since it is also mean-zero, it must be given by a pressure
gradient. It is easy to check that the other terms in (1.11) vanish for the
radial Mikado flow, or are equal to a pressure gradient, showing the
stationarity. For the sake of completeness, we still compute explicitly the
pressure associated with the radial flow profile, using the formulation
(1.11).

We have that

$\operatorname{\mathrm{div}}(\mathbb{W}_{k})=k\cdot\nabla\varrho_{k}=0\,.$

A stationary solution of (1.11) must satisfy

$\operatorname{\mathrm{curl}}\left(u-\alpha^{2}\Delta u\right)\times u+\nabla
p=0\,.$ (6.9)

We first note that

$\displaystyle\operatorname{\mathrm{curl}}_{x}\mathbb{W}_{k}\times\mathbb{W}_{k}$
$\displaystyle=\operatorname{\mathrm{curl}}_{x}(\varrho_{k}k)\times\varrho_{k}k$
$\displaystyle=(-\partial_{y_{1}}\varrho(k_{1}\cdot x,k_{2}\cdot
x)k_{2}+\partial_{y_{2}}\varrho(k_{1}\cdot x,k_{2}\cdot
x)k_{1})\times\varrho_{k}k$
$\displaystyle=-(k_{1}\partial_{y_{1}}\varrho(k_{1}\cdot x,k_{2}\cdot
x)\varrho_{k}+k_{2}\partial_{y_{2}}\varrho(k_{1}\cdot x,k_{2}\cdot
x)\varrho_{k})$ $\displaystyle=-\frac{1}{2}\nabla_{x}\varrho_{k}^{2}\,.$

To compute the pressure arising from the term

$\operatorname{\mathrm{curl}}\Delta\mathbb{W}_{k}\times\mathbb{W}_{k}\,,$

we will use polar coordinates adapted to the $k_{1},k_{2}$ plane. Hence
$r=\sqrt{(k_{1}\cdot x)^{2}+(k_{2}\cdot x)^{2}}$ will represent the distance
from the $k$ axis, and $\theta$ will measure rotation around the $k$ axis. We
also define $g(y)=\Delta_{y}\varrho(y)$ and $g_{k}(x)=g(k_{1}\cdot
x,k_{2}\cdot x)$ so that

$g_{k}(x)=g(k_{1}\cdot x,k_{2}\cdot
x)=\Delta_{x}(\varrho_{k}(x))=(\Delta_{y}\varrho)(k_{1}\cdot x,k_{2}\cdot x)$

Using these conventions we have

$\displaystyle\operatorname{\mathrm{curl}}\Delta$
$\displaystyle\mathbb{W}_{k}\times\mathbb{W}_{k}$
$\displaystyle=(-k_{2}\partial_{y_{1}}g(k_{1}\cdot x,k_{2}\cdot
x)+k_{1}\partial_{y_{2}}g(k_{1}\cdot x,k_{2}\cdot x))\times\varrho_{k}k$
$\displaystyle=-k_{1}\partial_{y_{1}}g(k_{1}\cdot x,k_{2}\cdot
x)\varrho_{k}-k_{2}\partial_{y_{2}}g(k_{1}\cdot x,k_{2}\cdot x)\varrho_{k}$
$\displaystyle=-k_{1}\cos\theta\partial_{r}\left(\partial_{r}^{2}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot
x)^{2}}\right)+\frac{1}{r}\partial_{r}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)\right)h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)$ $\displaystyle\quad-
k_{2}\sin\theta\partial_{r}\left(\partial_{r}^{2}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot
x)^{2}}\right)+\frac{1}{r}\partial_{r}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)\right)h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)$
$\displaystyle=-e_{r}\partial_{r}\left(\partial_{r}^{2}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot
x)^{2}}\right)+\frac{1}{r}\partial_{r}h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)\right)h\left(\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot x)^{2}}\right)\,,$

where we used that the Laplacian applied to radial functions is given as
$\partial_{r}^{2}+\frac{\partial_{r}}{r}$ to go from the second to the third
equality. Note that $h$ and its derivatives are evaluated at
$\sqrt{(k_{1}\cdot x)^{2}+(k_{2}\cdot x)^{2}}$. We can write this as

$-e_{r}\partial_{r}\left(\partial_{r}^{2}h+\frac{1}{r}\partial_{r}h\right)h=-\nabla_{x}\int_{0}^{\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot
x)^{2}}}\partial_{r}\left(\partial_{r}^{2}h+\frac{1}{r}\partial_{r}h\right)h\,dr\,.$

Therefore, the pressure is a radial function given by

$p\left(\sqrt{(k_{1}\cdot x)^{2}+(k_{2}\cdot
x)^{2}}\right)=\frac{\left(h\left(\sqrt{(k_{1}\cdot x)^{2}+(k_{2}\cdot
x)^{2}}\right)\right)^{2}}{2}-\alpha^{2}\int_{0}^{\sqrt{(k_{1}\cdot
x)^{2}+(k_{2}\cdot
x)^{2}}}\partial_{r}\left(\partial_{r}^{2}h+\frac{1}{r}\partial_{r}h\right)h\,dr\,.$
(6.10)

∎

######  Lemma 6.2 (Calculating the Averages).

Let $\xi$ be a rational unit direction vector, and let
$\mathbb{W}_{\xi}=\varrho_{\xi}\xi$ be the unit scale, un-concentrated Mikado
flow in the direction $\xi$ as in (6.8). Then if
$\left\|\nabla\varrho_{\xi}\right\|_{L^{2}(\mathbb{T}^{3})}^{2}=\mathcal{C}$,
we have that

$\displaystyle\int_{\mathbb{T}^{3}}\mathbb{W}_{\xi}^{k}\partial_{mm}\mathbb{W}_{\xi}^{\ell}+\partial_{k}\mathbb{W}_{\xi}^{j}\partial_{\ell}\mathbb{W}_{\xi}^{j}=\frac{\mathcal{C}}{2}\left(\delta^{k\ell}-3\xi^{k}\xi^{\ell}\right)\,.$
(6.11)

###### Proof.

Since $\xi$ is fixed throughout the proof, we simply write $\varrho$ rather
than $\varrho_{\xi}$ to lighten the notation. Then to calculate the first
integral, we write

$\displaystyle\int_{\mathbb{T}^{3}}\mathbb{W}_{\xi}^{k}\partial_{mm}\mathbb{W}_{\xi}^{\ell}\,dx$
$\displaystyle=\int_{\mathbb{T}^{3}}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\partial_{mm}\left(\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\right)\xi^{k}\xi^{\ell}\,dx$
$\displaystyle=-\xi^{k}\xi^{\ell}\int_{\mathbb{T}^{3}}\partial_{p}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\xi_{p}^{m}\partial_{n}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\xi_{n}^{m}\,dx$
$\displaystyle=-\xi^{k}\xi^{\ell}\int_{\mathbb{T}^{3}}\delta_{np}\partial_{p}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\partial_{n}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,dx$
$\displaystyle=-\xi^{k}\xi^{\ell}\int_{\mathbb{T}^{3}}\partial_{n}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\partial_{n}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,dx$
$\displaystyle:=-\mathcal{C}\xi^{k}\xi^{\ell}\,.$ (6.12)

Note that in the above calculation, $\partial_{mm}$ denotes partial
differentation computed with respect to the standard coordinate basis, while
$\partial_{p}\varrho$ denotes the derivative of $\varrho$ with respect to
$x\cdot\xi_{p}$. We have also used that

$\mathcal{C}=\left\|\nabla_{e_{1},e_{2},e_{3}}\varrho_{\xi}\right\|_{L^{2}(\mathbb{T}^{3})}^{2}=\int_{\mathbb{T}^{3}}\left((\partial_{1}\varrho)^{2}+(\partial_{2}\varrho)^{2}\right)(x\cdot\xi_{1},x\cdot\xi_{2})\,dx\,.$

For the second integral, we must calculate

$\int_{\mathbb{T}^{3}}\partial_{k}\mathbb{W}_{\xi}^{j}\partial_{\ell}\mathbb{W}_{\xi}^{j}=\int_{\mathbb{T}^{3}}\xi^{j}\,\partial_{n}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,\xi_{n}^{k}\,\xi^{j}\,\partial_{p}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,\xi_{p}^{\ell}\,.$

Since we are summing over both $n$ and $p$, we have four cases corresponding
to

$(n,p)\in\\{(1,1),(1,2),(2,1),(2,2)\\}\,.$

If $n=p=1$, then the expression is equal to

$\displaystyle\int_{\mathbb{T}^{3}}\xi^{j}\,\partial_{1}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,\xi_{1}^{k}\,\xi^{j}\,\partial_{1}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,\xi_{1}^{\ell}$
$\displaystyle=\left\|\partial_{1}\varrho\right\|_{L^{2}(\mathbb{T}^{3})}^{2}\xi_{1}^{k}\xi_{1}^{\ell}\,,$
(6.13)

and for $n=p=2$, we obtain

$\left\|\partial_{2}\varrho\right\|_{L^{2}(\mathbb{T}^{3})}^{2}\xi_{2}^{k}\xi_{2}^{\ell}\,.$
(6.14)

However, since $\mathbb{W}_{\xi}$ is in fact radial around its axis, we have
that by the assumption that
$\left\|\nabla\varrho\right\|_{L^{2}}^{2}=\mathcal{C}$,

$\left\|\partial_{1}\varrho\right\|_{L^{2}(\mathbb{T}^{3})}^{2}=\left\|\partial_{2}\varrho\right\|_{L^{2}(\mathbb{T}^{3})}^{2}=\frac{\mathcal{C}}{2}\,.$
(6.15)

Moving to the cases $n\neq p$, for example $n=1$ and $p=2$, the expression
becomes

$\displaystyle\int_{\mathbb{T}^{3}}\xi^{j}\,\partial_{1}\varrho(x\cdot\xi_{1}$
$\displaystyle,x\cdot\xi_{2})\,\xi_{1}^{k}\,\xi^{j}\,\partial_{2}\varrho(x\cdot\xi_{1},x\cdot\xi_{2})\,\xi_{2}^{\ell}\,.$
(6.16)

Consider a single cross section of the pipe with diameter $2d$; in cylindrical
coordinates, this is the set

$\\{z=z_{0},\,r\leq d,\,0\leq\theta\leq 2\pi\\}\,.$

The coordinate transformations for partial derivatives in cylindrical (or
polar) coordinates for functions which do not depend on $\theta$ are

$\partial_{1}\rightarrow\cos(\theta)\partial_{r},\qquad\partial_{2}\rightarrow\sin(\theta)\partial_{r}\,.$

Since $\varrho$ does not depend on $\theta$, the integrand over this cross-
section of pipe becomes

$\xi_{1}^{k}\xi_{2}^{\ell}\int_{0}^{2\pi}\int_{0}^{d}\cos(\theta)\partial_{r}\varrho(r)\sin(\theta)\partial_{r}\varrho(r)r\,dr\,d\theta=0\,,$

since $\cos(\theta)\sin(\theta)=\nicefrac{{1}}{{2}}\sin(2\theta)$ integrates
to $0$ from $0$ to $2\pi$, which shows that the entire term in (6.16)
vanishes.

Finally, we have that

$\xi\otimes\xi+\xi_{1}\otimes\xi_{1}+\xi_{2}\otimes\xi_{2}=\textnormal{Id},\qquad\xi_{1}^{k}\xi_{1}^{\ell}+\xi_{2}^{k}\xi_{2}^{\ell}=\textnormal{Id}-\xi^{k}\xi^{\ell}\,$

for any orthonormal basis $\\{\xi,\xi_{1},\xi_{2}\\}$ of $\mathbb{R}^{3}$.
Combining this with (6.12), (6.13), (6.14), and (6.15), we deduce the equality
in (6.11). ∎

###  6.1 Definition of and estimates on the perturbation

We define a compactly supported temporal cutoff function
$\eta:(-1,1)\rightarrow[0,1]$ which induces a $C^{\infty}$ partition of unity
of $\mathbb{R}$ according to

$\sum_{i\in\mathbb{Z}}\eta^{2}(\cdot-i)\equiv 1\,.$ (6.17)

To set the scale for our cutoffs we define a parameter $\tau_{q}$ as

$\tau_{q}:=\ell^{3}\,.$ (6.18)

Letting $t_{i}=i\tau_{q}$ for $i\in\mathbb{Z}$, we define the rescaled and
translated cutoff functions by

$\eta_{q,i}(t)=\begin{dcases}\eta\left(\tau_{q}^{-1}(t-t_{i})\right)&\operatorname{\mathrm{supp}}_{t}{\mathring{R}}_{q}\cap\left[t_{i}-\tau_{q},t_{i}+\tau_{q}\right]\neq\emptyset\\\
0&\textnormal{otherwise}\,.\end{dcases}$ (6.19)

We have that $\eta_{q,i}$ satisfies the properties

$\sum_{i\in\mathbb{Z}}\eta_{q,i}^{2}(t)\equiv 1\quad\forall
t\in\operatorname{\mathrm{supp}}_{t}{\mathring{R}}_{q}\,,\qquad\left|\partial_{t}^{m}\eta_{q,i}\right|\lesssim\tau_{q}^{-m}\,,\qquad\eta_{q,i}\eta_{q,i^{\prime}}\neq
0\implies|i-i^{\prime}|\leq 1\,.$ (6.20)

Define $\Phi_{q,i}$ to be the solution to the transport equation

$\displaystyle\left(\partial_{t}+u_{\ell}\cdot\nabla\right)\Phi_{q,i}$
$\displaystyle=0$ (6.21a) $\displaystyle\Phi_{q,i}(t_{i},x)$
$\displaystyle=x\,.$ (6.21b)

######  Lemma 6.3 (Deformation bounds).

The following estimates hold for the deformation maps $\Phi_{q,i}$:

  1. 1.

For each $t\in\mathbb{R}$,
$\Phi_{q,i}(t,\cdot):\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ is a
diffeomorphism, and we denote the (time-dependent) inverse map by $X_{q,i}$,
so that $\Phi_{q,i}(t,\cdot)\circ X_{q,i}(t,\cdot)=\operatorname{\mathrm{Id}}$
for all $t\in\mathbb{R}$.

  2. 2.

For all $t\in\operatorname{\mathrm{supp}}\eta_{q,i}$ and $N\geq 2$,

$\displaystyle\left\|\nabla\Phi_{q,i}(t)-\operatorname{\mathrm{Id}}\right\|_{0}\,,\left\|\nabla
X_{q,i}(t)-\operatorname{\mathrm{Id}}\right\|_{0}$ $\displaystyle\leq{\ell}$
(6.22a)
$\displaystyle\left\|\nabla^{N}\Phi_{q,i}(t)\right\|_{0}\,,\left\|\nabla^{N}X_{q,i}(t)\right\|_{0}$
$\displaystyle\lesssim\ell^{-(N-1)}$ (6.22b)
$\displaystyle\left\|\nabla^{N-2}\partial_{t}\Phi_{q,i}\right\|_{0}\,,\left\|\nabla^{N-2}\partial_{t}X_{q,i}\right\|_{0}$
$\displaystyle\lesssim\ell^{-(N+1)}\,.$ (6.22c)

###### Proof.

The fact that $\Phi_{q,i}$ is a diffeomorphism follows from the divergence-
free property of the vector field $u_{\ell}$, and then automatically
$\Phi_{q,i}^{-1}=X_{q,i}$ is a diffeomorphism as well. The proof of (6.22a)
for $\Phi(t)$ follows precisely the proof of Proposition D.1, (135) in
[BDLISZ15]. We use (2.3) and Young’s inequality to deduce that $\|\nabla
u_{\ell}\|_{0}\lesssim\ell^{-\frac{3}{2}}$. The choice of $\tau_{q}=\ell^{3}$
then implies that the deviation of $\nabla\Phi_{q,i}$ from the identity is
bounded by $\ell^{-\frac{3}{2}}\cdot\ell^{3}\leq\ell$. The same estimate for
$\nabla X$ follows from the inverse function theorem. The estimate (6.22b) for
$\Phi_{q,i}$ follows [BDLISZ15] again, while the same estimate for $X_{q,i}$
follows from the Fa’a di Bruno formula and the fact that $\nabla X_{q,i}$ is a
$C^{\infty}$ function of the entries of $\nabla\Phi_{q,i}$. The final estimate
in (6.22c) for $\Phi_{q,i}$ follows from writing
$\nabla^{N-2}\partial_{t}\Phi_{q,i}=\nabla^{N-2}\left(u_{\ell}\cdot\nabla\Phi_{q,i}\right)$
and utilizing the spatial derivative bounds on $u_{\ell}$ and $\Phi_{q,i}$ and
standard Hölder estimates for products. The bound for $\partial_{t}X_{q,i}$
follow from the fact that $\partial_{t}X_{q,i}(t,x)=u_{\ell}(X_{q,i}(t,x),t)$
and repeated application of the chain rule (or Fa’a di Bruno formula); see for
example Proposition A.1, estimate A.5 from [BDLSV17]. ∎

Define the cutoff function $\chi(z):[0,\infty)\rightarrow\mathbb{R}$ to be a
smooth function satisfying

$\chi(z)=\begin{dcases}1&0\leq z\leq 1\\\ z&z\geq 2\end{dcases}\,,\qquad z\leq
2\chi(z)\leq{4z}\quad\forall z\in(1,2)\,.$ (6.23)

Recall that in Lemma 4.1, we chose a value of $N\in\mathbb{N}$, which in turn
sets a value of $\varepsilon$. We choose $N=2$ and let $\varepsilon$ be as in
the statement of the Lemma. We define

$\rho_{q}(x,t)=\left(2\delta_{q+1}{\lambda_{q+1}^{2}}\varepsilon^{-1}\mathcal{C}_{\mathring{R}}\right)\cdot\chi\left((\mathcal{C}_{\mathring{R}}\delta_{q+1}{\lambda_{q+1}^{2}}\alpha^{2})^{-1}\left|{\mathring{R}}_{\ell}(x,t)\right|\right)\,.$
(6.24)

Simple calculations give that

$\frac{\left|\alpha^{-2}{\mathring{R}}_{\ell}\right|}{\rho_{q}(x,t)}\leq\varepsilon\,,\qquad\left\|\rho_{q}\right\|_{L^{p}}\leq\frac{3}{\varepsilon}\left(\mathcal{C}_{{\mathring{R}}}(8\pi^{3})^{\frac{1}{p}}\delta_{q+1}\lambda_{q+1}^{2}+\left\|\alpha^{-2}{\mathring{R}}_{\ell}\right\|_{L^{p}}\right)\,.$
(6.25)

For ease of notation, let us define

$R_{\ell}=\alpha^{-2}\frac{{\mathring{R}}_{\ell}}{\rho_{q}}\,.$ (6.26)

From (6.25) we have that $c_{k}^{n}\left(R_{\ell}\right)$ is well-defined for
$1\leq k\leq 9$, $n=0,1$, and $i\in\mathbb{Z}$. After referring to Proposition
6.1 to set the notations

$\mathbb{W}_{q+1,k}:=\mathbb{W}_{k,\lambda_{q+1},r_{q}}\,,\qquad\varrho_{q+1,k}:=\varrho_{k,\lambda_{q+1},r_{q}}\,,$
(6.27)

we may then define the principal part of the perturbation

$\displaystyle w_{q+1}^{(p)}$
$\displaystyle={\frac{1}{\lambda_{q+1}}}\sum_{i\in
2\mathbb{Z}}\sum_{k\in\mathcal{K}_{0}}c_{k}^{0}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\left(\nabla\Phi_{q,i}(x,t)\right)^{-1}\mathbb{W}_{q+1,k}\circ\Phi_{q,i}(x,t)$
$\displaystyle\qquad+{\frac{1}{\lambda_{q+1}}}\sum_{i\in
2\mathbb{Z}+1}\sum_{k\in\mathcal{K}_{1}}c_{k}^{1}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\left(\nabla\Phi_{q,i}(x,t)\right)^{-1}\mathbb{W}_{q+1,k}\circ\Phi_{q,i}(x,t)\,,$
(6.28)

where as indicated in (6.27), $\mathbb{W}_{q+1,k}$ is an intermittent Mikado
flow with parameter choices $\xi=k$, $\lambda=\lambda_{q+1}$,
$r=r_{q}=\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\Gamma}$. We may also
define the divergence corrector by

$\displaystyle w_{q+1}^{(c)}$
$\displaystyle={\frac{1}{\lambda_{q+1}}}\sum_{i\in
2\mathbb{Z}}\sum_{k\in\mathcal{K}_{0}}\nabla\left(c_{k}^{0}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\right)\times\left(\left(\nabla\Phi_{q,i}(x,t)\right)^{T}\
\mathbb{U}_{q+1,k}\circ\Phi_{q,i}(x,t)\right)$
$\displaystyle\qquad+{\frac{1}{\lambda_{q+1}}}\sum_{i\in
2\mathbb{Z}+1}\sum_{k\in\mathcal{K}_{1}}\nabla\left(c_{k}^{1}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\right)\times\left(\left(\nabla\Phi_{q,i}(x,t)\right)^{T}\mathbb{U}_{q+1,k}\circ\Phi_{q,i}(x,t)\right)\,.$

We then define the entire perturbation as

$\displaystyle w_{q+1}$ $\displaystyle=w_{q+1}^{(p)}+w_{q+1}^{(c)}$
$\displaystyle={\frac{1}{\lambda_{q+1}}}\sum_{i\in\mathbb{Z}}\sum_{k\in\mathcal{K}}\operatorname{\mathrm{curl}}\left(c_{k}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\left(\nabla\Phi_{q,i}(x,t)\right)^{T}\mathbb{U}_{q+1,k}\circ\Phi_{q,i}(x,t)\right)\,,$
(6.29)

where in a slight abuse of notation, we have condensed the separate sums over
even and odd integers into a single sum over _all_ integers, while suppressing
the superscript on $c_{k}$ and the subscript on $\mathcal{K}$. Notice that by
the definition of $\eta_{q,i}$ in (6.19) and the assumption on the temporal
support of ${\mathring{R}}_{q}$ from (2.5), we have that

$\operatorname{\mathrm{supp}}_{t}w_{q+1}\subset(t_{1}-2\tau_{q}^{-1},t_{2}+2\tau_{q}^{-1})\,.$
(6.30)

######  Lemma 6.4 (Estimates on the perturbation).

The perturbation $w_{q+1}$ as defined in (6.1) satisfies the following
estimates.

  1. 1.

For $j\geq 1$, we have that

$\displaystyle\left\|\rho\right\|_{0}\lesssim\delta_{q+1}\lambda_{q+1}^{2}\ell^{-3}\,,$
$\displaystyle\qquad\left\|\nabla_{x,t}^{j}\rho\right\|_{0}\lesssim\delta_{q+1}\lambda_{q+1}^{2}\ell^{-4j}$
(6.31a)
$\displaystyle\left\|\rho^{\nicefrac{{1}}{{2}}}\right\|_{0}\lesssim\delta_{q+1}^{\nicefrac{{1}}{{2}}}\
\lambda_{q+1}\ell^{-2}\,,$
$\displaystyle\qquad\left\|\nabla_{x,t}^{j}\rho^{\nicefrac{{1}}{{2}}}\right\|_{0}\lesssim\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}\ell^{-5j}\,.$
(6.31b)

  2. 2.

For $j\geq 1$, we have that

$\displaystyle\left\|c_{k}\left(R_{\ell}\right)\eta_{q,i}\rho^{\nicefrac{{1}}{{2}}}_{q}\right\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}\,,$
(6.32a)
$\displaystyle\left\|\nabla_{x,t}^{j}\left(c_{k}\left(R_{\ell}\right)\eta_{q,i}\rho^{\nicefrac{{1}}{{2}}}_{q}\right)\right\|_{0}$
$\displaystyle\lesssim\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}\ell^{-5j-2}\lesssim\ell^{-5j-2}\,.$
(6.32b)

  3. 3.

For $j\geq 0$, we have that

$\displaystyle\left\|w_{q+1}^{(p)}\right\|_{L^{2}}+\lambda_{q+1}^{-1}\left\|\nabla
w_{q+1}^{(p)}\right\|_{L^{2}}+\lambda_{q+1}^{-2}\left\|\nabla^{2}w_{q+1}^{(p)}\right\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}$ (6.33a)
$\displaystyle\left\|w_{q+1}^{(c)}\right\|_{L^{2}}+\lambda_{q+1}^{-1}\left\|\nabla
w_{q+1}^{(c)}\right\|_{L^{2}}+\lambda_{q+1}^{-2}\left\|\nabla^{2}w_{q+1}^{(c)}\right\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\left(\ell^{8}\lambda_{q+1}\right)^{-1}$
(6.33b)
$\displaystyle\left\|w_{q+1}^{(p)}\right\|_{\dot{C}^{j}}+\left\|w_{q+1}^{(c)}\right\|_{\dot{C}^{j}}$
$\displaystyle\leq\lambda_{q+1}^{j+2}\,.$ (6.33c)

###### Proof.

(6.31a) and (6.31b) follow from Young’s inequality, (2.3), and standard Hölder
estimates for compositions, cf. [BDLSV17] Appendix A.1. (6.32a) follows from
(2.3), (6.25), (6.20), the definition of $c_{j}(R_{\ell})$ as a smooth bounded
function, and a sufficiently small choice of $\mathcal{C}_{\mathring{R}}$.
(6.32b) follows from $\tau=\ell^{3}$, (6.31a), (6.31b), standard Hölder
estimates, and $\ell\leq\lambda_{q+1}^{2}\delta_{q+1}$, which is inequality
(2).

To prove (6.33a) we fix an arbitrary time
$t_{0}\in\operatorname{\mathrm{supp}}_{t}w_{q+1}$. At $t_{0}$, at most two of
the $\eta_{i}$ are nonzero, so it is enough to focus on bounding each term in
the sum over $i$ in (6.1) individually. We note that from (6.32b), (6.22a)
(6.22b), (6.2), the chain and product rule, and the ordering
$\ell^{-12}\ll\lambda_{q+1}$ from inequality (3), it is enough to consider the
case where the derivatives all land on $\mathbb{W}\circ\Phi_{q,i}$. Before we
can apply Lemma 5.1, we perform the change of variables $y:=\Phi_{q,i}(x)$:
doing so converts $\nabla^{j}\mathbb{W}\circ\Phi_{q,i}$ to
$\nabla^{j}\mathbb{W}$ which is a genuinely
$\left(\mathbb{T}/(\lambda_{q+1}r_{q})\right)^{3}$ periodic function. Then we
let $g=\nabla^{j}\mathbb{W}$ and $f$ be the product of $\lambda_{q+1}^{-1}$,
$c_{k}^{i\mod
2}\left(R_{\ell}\right)\eta_{q,i}(t)\rho^{\nicefrac{{1}}{{2}}}_{q}(x,t)\left(\nabla\Phi_{q,i}\right)^{-1}$
, $j$ copies of $\nabla\Phi_{q,i}$ (all composed with $X_{q,i}$) , and the
Jacobian $\det\nabla X_{q,i}(y)$. Using (6.32a), (6.32b), Lemma 6.3, and
Hölder estimates for compositions, we have that

$\ell^{8j}\|D^{j}f\|_{0}\leq{c}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,,$

where ${c}$ is sufficiently small enough to absorb any dimensional constants;
note that this is allowed because we have used an extra factor of $\ell$, and
we may assume $\mathcal{C}_{\mathring{R}}$ from (2.3) is sufficiently small.
From the parameter inequality (4), which states that

$\ell^{-8({\mathbf{N}_{\textnormal{dec}}}+4)}\leq\lambda_{q+1}^{{\mathbf{N}_{\textnormal{dec}}}(1-\Gamma-\frac{\Gamma}{b})}\,,$

we may thus apply Lemma 5.1 with $\mu=\lambda_{q+1}r_{q}$ and
$\lambda=\ell^{-8}$, producing (6.33a).

To prove $\eqref{eq:wc}$ we once again use the fact that it is enough to
consider all the derivatives landing on
$\mathbb{U}_{\lambda_{q+1},\xi,r_{q}}$, where $\mathbb{U}$ is the potential
defined in Proposition 6.1. In addition, by (6.32b) and (3) from Proposition
6.1 and a similar application of Lemma 5.1, the bound (6.33b) follows.
Finally, to prove (6.33c) we simply use (3) from Proposition 6.1 and
inequality (5), which implies that $r_{q}^{-1}\ll\lambda_{q+1}$. ∎

###  6.2 The new equation at level $q+1$

We define

$u_{q+1}:=u_{\ell}+w_{q+1}.$ (6.34)

Adding $w_{q+1}$ to (3.2a), we find that $u_{q+1}$ satisfies

$\displaystyle\partial_{t}(u_{q+1}-$ $\displaystyle\alpha^{2}\Delta
u_{q+1})^{l}+\partial_{k}\left(u_{q+1}^{k}(u_{q+1}^{l}-\alpha^{2}\Delta
u_{q+1}^{l})-\alpha^{2}\partial_{k}u_{q+1}^{j}\partial_{l}u_{q+1}^{j}\right)+\partial_{l}p_{\ell}$
$\displaystyle=\partial_{k}{\mathring{R}}_{\text{comm}}^{kl}$
$\displaystyle\quad+\partial_{k}\left({\mathring{R}}_{\ell}^{kl}+w_{q+1}^{k}\left(w_{q+1}^{l}-\alpha^{2}\Delta
w_{q+1}^{l}\right)-\alpha^{2}\partial_{k}w_{q+1}^{j}\partial_{l}w_{q+1}^{j}\right)$
(6.35) $\displaystyle\quad+\partial_{t}(w_{q+1}-\alpha^{2}\Delta
w_{q+1})^{l}+u_{\ell}\cdot\nabla(w_{q+1}-\alpha^{2}\Delta w_{q+1})^{l}$ (6.36)
$\displaystyle\quad+\partial_{k}\left(w_{q+1}^{k}\left(u_{\ell}^{l}-\alpha^{2}\Delta
u_{\ell}^{l}\right)-\alpha^{2}\partial_{k}u_{\ell}^{j}\partial_{l}w_{q+1}^{j}-\alpha^{2}\partial_{k}w_{q+1}^{j}\partial_{l}u_{\ell}^{j}\right)$
(6.37)
$\displaystyle=:\partial_{k}{\mathring{R}}_{\text{comm}}^{kl}+\partial_{k}{\mathring{R}}_{\text{osc}}^{kl}\,+\,\partial_{k}{\mathring{R}}_{\text{transport}}^{kl}\,+\,\partial_{k}{\mathring{R}}_{\text{Nash}}^{kl}\,.$
(6.38)

We do not need to address the commutator stress, and so we analyze the three
main error terms in the next sections.

##  7 Stress Estimates

With Proposition 5.5 and Proposition 5.3, we are now ready to estimate the
remaining stress terms from (6.35)-(6.37). In what follows we will use that
$\mathcal{R}\circ\operatorname{\mathrm{div}}$ is bounded on $L^{p}$ for
$p\in(1,\infty)$. In particular, $\mathcal{R}\circ\operatorname{\mathrm{div}}$
is _not_ bounded on $L^{1}$. To circumvent this we will choose a $p$ slightly
larger than 1 and bound the stress in $L^{p}$ by interpolating between $L^{1}$
and $L^{\infty}$. This will be accounted for by a parameter $\gamma$
satisfying $0<\gamma\ll 1$ appearing as a power on $\lambda_{q+1}$.

###  7.1 Oscillation Error

The oscillation error was given in (6.35) by

$\partial_{k}{\mathring{R}}_{\text{osc}}^{kl}:=\partial_{k}\left({\mathring{R}}_{\ell}^{kl}+w_{q+1}^{k}\left(w_{q+1}^{l}-\alpha^{2}\Delta
w_{q+1}^{l}\right)-\alpha^{2}\partial_{k}w_{q+1}^{j}\partial_{l}w_{q+1}^{j}\right)\,.$

Eschewing index notation for the time being, we can split this sum as

$\displaystyle\operatorname{\mathrm{div}}$ $\displaystyle\left(w_{q+1}\otimes
w_{q+1}-\alpha^{2}\left(w_{q+1}^{(c)}\otimes\Delta
w_{q+1}^{(c)}+w_{q+1}^{(c)}\otimes\Delta
w_{q+1}^{(p)}+w_{q+1}^{(p)}\otimes\Delta w_{q+1}^{(c)}\right)\right)$
$\displaystyle\quad-\alpha^{2}\operatorname{\mathrm{div}}\left(\left(\nabla
w_{q+1}^{(c)}\right)^{T}\nabla w_{q+1}^{(c)}+\left(\nabla
w_{q+1}^{(p)}\right)^{T}\nabla w_{q+1}^{(c)}+\left(\nabla
w_{q+1}^{(c)}\right)^{T}\nabla w_{q+1}^{(p)}\right)$
$\displaystyle\quad-\alpha^{2}\operatorname{\mathrm{div}}\left(w_{q+1}^{(p)}\otimes\Delta
w_{q+1}^{(p)}+\left(\nabla w_{q+1}^{(p)}\right)^{T}\nabla
w_{q+1}^{(p)}-\alpha^{-2}{\mathring{R}}_{\ell}\right)\,.$ (7.1)

We estimate now the first two lines of the above display, which are all of
lower order. We may bound $w_{q+1}\otimes w_{q+1}$ in $L^{1}$ by
$\delta_{q+1}$ after appealing to (6.33a) and (6.33b) and in $L^{\infty}$ by
$\lambda_{q+1}^{2}$. The subsequent terms on the first line, which we
abbreviate $R_{\textnormal{corr},1}$ may be bounded in $L^{1}$ by
$\delta_{q+1}\lambda_{q+1}^{2}\left(\ell^{8}\lambda_{q+1}\right)^{-1}$ after
appealing to the higher order bounds in (6.33a) and (6.33b), and in
$L^{\infty}$ by $\lambda_{q+1}^{6}$ using (6.33c). The terms on the second
line, which we abbreviate by $R_{\textnormal{corr},2}$, all obey analogous
$L^{1}$ and $L^{\infty}$ bounds compared to the terms in
$R_{\textnormal{corr},1}$. We obtain that

$\displaystyle\left\|\mathcal{R}\circ\operatorname{\mathrm{div}}\left(w_{q+1}\otimes
w_{q+1}+R_{\textnormal{corr},1}+R_{\textnormal{corr},2}\right)\right\|_{L^{1}}$
$\displaystyle=\left\|\mathcal{R}\circ\operatorname{\mathrm{div}}\left(\mathbb{P}_{\neq
0}\left(w_{q+1}\otimes
w_{q+1}+R_{\textnormal{corr},1}+R_{\textnormal{corr},2}\right)\right)\right\|_{L^{1}}$
$\displaystyle\lesssim\delta_{q+1}\lambda_{q+1}^{\gamma}+\delta_{q+1}\lambda_{q+1}^{2+\gamma}\left(\ell\lambda_{q+1}\right)^{-1}$
$\displaystyle\leq{\lambda_{q+2}^{2-\gamma}\delta_{q+2}}\,.$ (7.2)

Therefore these errors are symmetric traceless stresses of the proper
magnitude and may be absorbed into the new stress ${\mathring{R}}_{q+1}$. Note
that in the last line, we appealed to inequality (6).

For the remainder of this section, we focus on the analysis of the third line

$\alpha^{2}\operatorname{\mathrm{div}}\left(w_{q+1}^{(p)}\otimes\Delta
w_{q+1}^{(p)}+\left(\nabla w_{q+1}^{(p)}\right)^{T}\nabla
w_{q+1}^{(p)}-\alpha^{-2}{\mathring{R}}_{\ell}\right)$ (7.3)

from (7.1). Identical arguments to those which produced the bounds (6.33b)
imply that when the differential operators $\Delta$ or $\nabla$ from (7.3)
land on the low-frequency coefficient functions or flow maps in the definition
of $w_{q+1}^{(p)}$ in (6.1), the resulting terms may be bounded by (7.2). So
we may immediately absorb these terms into ${\mathring{R}}_{q+1}$ and focus on
the terms in which the differential operator falls on the high-frequency
objects.

At this point we revert back to index notation and index the tensors by
$\theta$ and $l$ rather than $k$ and $l$ (to avoid the overload of notation
with vector directions $k$) and ignore the prefactor $\alpha^{2}$ in front of
the last line of (7.3). Utilizing the notation from (6.27) and the identities

$\displaystyle\partial_{m}\left(\mathbb{W}_{q+1,k}\circ\Phi_{q,i}\right)$
$\displaystyle=k\partial_{n}\varrho_{q+1,k}(\Phi_{q,i})\partial_{m}\Phi_{q,i}^{n}$
$\displaystyle\partial_{mm}\left(\mathbb{W}_{q+1,k}\circ\Phi_{q,i}\right)$
$\displaystyle=k\partial_{np}\varrho_{q+1,k}(\Phi_{q,i})\partial_{m}\Phi_{q,i}^{n}\partial_{m}\Phi_{q,i}^{p}+k\partial_{n}\varrho_{q+1,k}(\Phi_{q,i})\partial_{mm}\Phi_{q,i}^{n}\,,$
(7.4)

we find that we must bound (after throwing away the lower order term from the
Laplacian in (7.4))

$\displaystyle\frac{\partial_{\theta}}{\lambda_{{q+1}}^{2}}\left(\sum_{\begin{subarray}{c}i,i^{\prime}\in\mathbb{Z}\\\
k\in\mathcal{K}_{i\,\textnormal{mod}\,2}\\\
k^{\prime}\in\mathcal{K}_{i^{\prime}\,\textnormal{mod}\,2}\end{subarray}}\left(c_{k}c_{k^{\prime}}\eta_{q,i}\eta_{q,i^{\prime}}\rho_{q}^{\nicefrac{{1}}{{2}}}\right)^{2}\left(\nabla\Phi_{q,i}\right)^{-1}_{\beta\theta}k^{\beta}\varrho_{q+1,k}(\Phi_{q,i})\left(\nabla\Phi_{q,i^{\prime}}\right)^{-1}_{jl}k^{j}\partial_{np}\varrho_{q+1,k^{\prime}}(\Phi_{q,i^{\prime}})\partial_{m}\Phi_{q,i^{\prime}}^{n}\partial_{m}\Phi_{q,i^{\prime}}^{p}\right)$
(7.5)
$\displaystyle+\frac{\partial_{\theta}}{\lambda_{{q+1}}^{2}}\left(\sum_{\begin{subarray}{c}i,i^{\prime}\in\mathbb{Z}\\\
k\in\mathcal{K}_{i\,\textnormal{mod}\,2}\\\
k^{\prime}\in\mathcal{K}_{i^{\prime}\,\textnormal{mod}\,2}\end{subarray}}\left(c_{k}c_{k^{\prime}}\eta_{q,i}\eta_{q,i^{\prime}}\rho_{q}^{\nicefrac{{1}}{{2}}}\right)^{2}\left(\nabla\Phi_{q,i}\right)^{-1}_{\beta
j}k^{\beta}\partial_{n}\varrho_{q+1,k}(\Phi_{q,i})\partial_{\theta}\Phi_{q,i}^{n}\left(\nabla\Phi_{q,i^{\prime}}\right)^{-1}_{mj}k^{m}\partial_{p}\varrho_{q+1,k^{\prime}}(\Phi_{q,i^{\prime}})\partial_{l}\Phi_{q,i^{\prime}}^{p}\right)\,$
(7.6)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\alpha^{-2}\partial_{\theta}{\mathring{R}}_{\ell}^{\theta
l}\,.$

By the properties of the time cutoffs $\eta_{q,i}$, the only nonzero terms in
the above sum occur when $|i-i^{\prime}|\leq 1$. The Type 1 oscillation error
is the standard oscillation error in most convex integration schemes and is
precisely the sum of
$\alpha^{-2}\partial_{\theta}{\mathring{R}}_{\ell}^{\theta l}$ and the terms
in the above sums in which $i=i^{\prime}$; for such terms, we have that all
terms besides those with $k=k^{\prime}$ vanish, since Mikado flows which
belong to the same set $\mathcal{K}_{i}$ but have different vector directions
may be taken to be disjoint.333In two dimensions, these error terms do _not_
vanish but may be analyzed using Proposition 5.3 in the same way as the terms
for which $i\neq i^{\prime}$. These terms will be analyzed using Proposition
5.7. The Type 2 oscillation errors are the terms in the above sums for which
$i\neq i^{\prime}$; these terms will be analyzed using Proposition 5.3.

####  7.1.1 Type 1 oscillation errors

We first analyze the terms in (7.5) for which $i=i^{\prime}$, which (after a
slight obfuscation of notation in which we denote the sets of vector
directions with $\mathcal{K}$) may be written as

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}\left(\sum_{i\in\mathbb{Z},k\in\mathcal{K}}c_{k}^{2}\eta_{q,i}^{2}\rho_{q}\left(\nabla\Phi_{q,i}\right)^{-1}_{\beta\theta}k^{\beta}\varrho_{q+1,k}(\Phi_{q,i})\left(\nabla\Phi_{q,i}\right)^{-1}_{jl}k^{j}\partial_{np}\varrho_{q+1,k}(\Phi_{q,i})\partial_{m}\Phi_{q,i}^{n}\partial_{m}\Phi_{q,i}^{p}\right)\,.$

In order to simplify notation further in the second term above, for the time
being we omit all subscripts $q$, $i$, and $k$ on the derivative matrices
$\nabla\Phi_{q,i}$ and flow profiles $\varrho_{q+1,k}$, and we will abbreviate
$c_{k}^{2}\eta_{q,i}^{2}\rho_{q}$ with simply $a^{2}$. With these conventions,
the above display may be rewritten as

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}\sum_{i,k}a^{2}$
$\displaystyle\bigg{[}k^{\theta}k^{l}\varrho(\Phi)\partial_{mm}\varrho(\Phi)$
(7.7)
$\displaystyle\quad+k^{\beta}\left((\nabla\Phi)_{\beta\theta}^{-1}-\delta_{\theta\beta}\right)k^{l}\varrho(\Phi)\partial_{mm}\varrho(\Phi)$
$\displaystyle\quad+k^{\beta}\left(\nabla\Phi\right)^{-1}_{\beta\theta}\left((\nabla\Phi)_{jl}^{-1}-\delta_{jl}\right)k^{j}\varrho(\Phi)\partial_{mm}\varrho(\Phi)$
$\displaystyle\quad+k^{\beta}\left(\nabla\Phi\right)^{-1}_{\beta\theta}(\nabla\Phi)_{jl}^{-1}k^{j}\left(\partial_{m}\Phi^{n}-\delta_{mn}\right)\varrho(\Phi)\partial_{nm}\varrho(\Phi)$
$\displaystyle\quad+k^{\beta}\left(\nabla\Phi\right)^{-1}_{\beta\theta}(\nabla\Phi)_{jl}^{-1}k^{j}\partial_{m}\Phi^{n}\left(\partial_{m}\Phi^{p}-\delta_{mp}\right)\varrho(\Phi)\partial_{np}\varrho(\Phi)\bigg{]}\,.$

The first term above will be kept and used to cancel the stress. We must
analyze the second through fifth terms. The arguments for each are quite
similar, and so we estimate the second and leave further details to the
reader. Ignoring the $\partial_{\theta}$ but holding onto the rescaling
$\lambda_{q+1}^{-2}$ for the time being, we recall (6.22a), (6.32a), (6.1)
with $\lambda=\lambda_{q+1}$, $p=2,\infty$ and
$r=\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\Gamma}\geq\lambda_{q+1}^{-1}$,
and (6.32b). Then applying Lemma 5.1 in the same way that produced the bounds
(6.33a), we may estimate the second term in $L^{1}$ by

$\displaystyle\lambda_{q+1}^{-2}\left\|a^{2}\left(\nabla\Phi^{-1}-\operatorname{\mathrm{Id}}\right)\varrho(\Phi)\partial_{mm}\varrho(\Phi)\right\|_{L^{1}}\lesssim\lambda_{q+1}^{-2}\left\|a^{2}\right\|_{L^{1}}\left\|\nabla\Phi^{-1}-\textnormal{Id}\right\|_{0}\left\|\varrho\Delta\varrho\right\|_{L^{1}}$
$\displaystyle\lesssim\ell\delta_{q+1}\lambda_{q+1}^{2}\,,$ (7.8)

and in $L^{\infty}$ by

$\displaystyle\left\|a^{2}\right\|_{0}\left\|\nabla\Phi-\textnormal{Id}\right\|_{0}\left\|\varrho\Delta\varrho\right\|_{0}\lesssim\ell^{-4}\ell\lambda_{q+1}^{4}\,.$
(7.9)

Applying $\mathcal{R}\circ\partial_{\theta}\circ\mathbb{P}_{\neq 0}$, we may
absorb this error term into ${\mathring{R}}_{q+1}$ after appealing to the
parameter inequality (7), which implies that

${\ell\delta_{q+1}\lambda_{q+1}^{2+\gamma}\leq\delta_{q+2}\lambda_{q+2}^{2-\gamma}\,.}$
(7.10)

Returning to (7.7), we may rewrite the first line as

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}\sum_{i,k}a^{2}$
$\displaystyle\left(k^{\theta}k^{l}\mathbb{P}_{\neq
0}\left(\varrho\partial_{mm}\varrho\right)(\Phi)+k^{\theta}k^{l}\mathbb{P}_{=0}\left(\varrho\partial_{mm}\varrho\right)\right)\,.$
(7.11)

The second portion of this term will be used to cancel the stress, and the
first part is an error itself. In order to analyze it, we have to telescope
back to a term which has the correct factors of $\nabla\Phi$ so that we again
have an approximately stationary solution to the equations. Thus we rewrite
the first term from the above display as

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}\sum_{i,k}a^{2}$
$\displaystyle\bigg{[}k^{\beta}\left(\delta_{\theta\beta}-(\nabla\Phi)^{-1}_{\beta\theta}\right)k^{l}\mathbb{P}_{\neq
0}\left(\varrho\partial_{mm}\varrho\right)(\Phi)$
$\displaystyle\quad+k^{\beta}(\nabla\Phi)^{-1}_{\beta\theta}k^{j}\left(\delta_{jl}-(\nabla\Phi)^{-1}_{jl}\right)\mathbb{P}_{\neq
0}\left(\varrho\partial_{mm}\varrho\right)(\Phi)$
$\displaystyle\quad+k^{\beta}(\nabla\Phi)^{-1}_{\beta\theta}k^{j}(\nabla\Phi)^{-1}_{jl}\mathbb{P}_{\neq
0}\left(\varrho\partial_{mm}\varrho\right)(\Phi)\bigg{]}\,.$ (7.12)

The first and second of these terms are estimated in precisely the same
fashion as the terms estimated in (7.8), and we omit further details. In the
third term, the product rule gives that $\partial_{\theta}$ lands either on
the low-frequency objects $a^{2}$ or $\nabla\Phi$ or the high-frequency, mean-
zero object $\mathbb{P}_{\neq 0}(\varrho\partial_{mm})$. When
$\partial_{\theta}$ lands on the low-frequency portion, we then must estimate

$\lambda_{q+1}^{-2}\sum_{i,k}\partial_{\theta}\left(a^{2}k^{\beta}(\nabla\Phi)_{\beta\theta}^{-1}k^{j}(\nabla\Phi)^{-1}_{jl}\right)\mathbb{P}_{\neq
0}\left(\varrho\partial_{mm}\varrho\right)(\Phi)\,.$ (7.13)

We apply the inverse divergence operator from Proposition 5.7 with the choices

$\displaystyle v=u_{\ell}\,,\qquad
G^{l}=\partial_{\theta}\left(a^{2}k^{\beta}(\nabla\Phi)_{\beta\theta}^{-1}k^{j}(\nabla\Phi)^{-1}_{jl}\right)\,,\qquad\mathcal{C}_{G}=\ell^{-9}\,,\qquad\lambda=\ell^{-5}\,,\qquad\Phi=\Phi_{q,i}\,,$
$\displaystyle\qquad\varrho=\mathbb{P}_{\neq
0}\left(\varrho_{q+1,k}\partial_{mm}\varrho_{q+1,k}\right)\,,\qquad\vartheta=\left(\lambda_{q+1}r_{q}\right)^{2{\mathsf{d}}}\Delta^{-{\mathsf{d}}}\varrho\,,$
$\displaystyle\qquad\zeta=\mu=(\lambda_{q+1}r_{q})\,,\qquad\Lambda=\lambda_{q+1}\,,\qquad{\mathcal{C}_{*}=\lambda_{q+1}^{2}}\,.$
(7.14)

Then (6.32b) shows that (5.36) is satisfied, (6.22a) shows that (5.37) is
satisfied, (6.22b) shows that (5.38) is satisfied, item (1) is satisfied by
the $\frac{\mathbb{T}^{3}}{\lambda_{q+1}r_{q}}$-periodicity guaranteed by item
(3) from Proposition 6.1, item (2) and (3) are satisfied by item (3) from
Proposition 6.1, and (5.40) is satisfied by the parameter inequality (9),
which states that

$\left(\frac{\ell^{-5}}{\lambda_{q+1}r_{q}}\right)^{\mathsf{d}}\ell^{-4}\leq(\lambda_{q+1}r_{q})^{-1}\,.$

Since we may assume that all terms inside the operator $\partial_{\theta}$ in
(7.12) have zero mean, and we shall see shortly that the complementary term to
(7.13) in which the $\partial_{\theta}$ lands on the high-frequency object
vanishes, the error term we are currently estimating may be taken to have zero
mean. Then applying (5.41) and (5.42), we have rewritten this error term as
the sum of the divergence of a traceless symmetric stress ${\mathring{R}}$ and
gradient of a pressure $P$ which obey the $L^{1}$ bounds (recall the factor of
$\lambda_{q+1}^{-2}$ in front of the expression in (7.13))

$\left\|{\mathring{R}}\right\|_{L^{1}}+\left\|P\right\|_{L^{1}}\lesssim\lambda_{q+1}^{-2}\mathcal{C}_{G}\lambda^{4}\mathcal{C}_{*}\zeta^{-1}=\lambda_{q+1}^{-2}\ell^{-9}\ell^{-20}\lambda_{q+1}^{2}\left(\lambda_{q+1}r_{q}\right)^{-1}\leq\delta_{q+2}\lambda_{q+2}^{2-\gamma}\,.$
(7.15)

In the last inequality, we have used inequality (11).

We now must calculate what happens in (7.12) when the $\partial_{\theta}$
lands on the high frequency portion. By the product rule, we have two terms,
which we may write out as

$\displaystyle
k^{\beta}(\nabla\Phi)^{-1}_{\beta\theta}k^{j}(\nabla\Phi)^{-1}_{jl}\mathbb{P}_{\neq
0}\left(\partial_{n}\varrho\partial_{mm}\varrho\right)(\Phi)\partial_{\theta}\Phi^{n}$
$\displaystyle=k^{n}k^{j}(\nabla\Phi)_{jl}^{-1}\partial_{n}\varrho(\Phi)\partial_{mm}\varrho(\Phi)=0\,.$

and

$\displaystyle
k^{\beta}(\nabla\Phi)^{-1}_{\beta\theta}k^{j}(\nabla\Phi)^{-1}_{jl}\mathbb{P}_{\neq
0}\left(\varrho\partial_{nmm}\varrho\right)(\Phi)\partial_{\theta}\Phi^{n}$
$\displaystyle=k^{n}k^{j}(\nabla\Phi)_{jl}^{-1}\varrho(\Phi)\partial_{nmm}\varrho(\Phi)=0\,,$

To conclude that these terms vanish, we have used that
$k^{n}\partial_{n}(D^{N}\varrho)=0$ for all $N\geq 0$, which follows from (1)
of Proposition 6.1. This concludes the analysis of the terms in (7.5), save
for the low frequency terms which we set aside in (7.11) to cancel the stress
later.

We now return to (7.6) and handle the terms for which $i=i^{\prime}$, which
after adopting the same abbreviations as in the analysis of (7.5) are written
as

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}$
$\displaystyle\left[\sum_{i,k}a^{2}(\nabla\Phi)^{-1}_{\beta
j}k^{\beta}\partial_{n}\varrho(\Phi)\partial_{\theta}\Phi^{n}(\nabla\Phi)^{-1}_{mj}k^{m}\partial_{p}\varrho(\Phi)\partial_{l}\Phi^{p}\right]$
$\displaystyle=\lambda_{q+1}^{-2}\partial_{\theta}\left[\sum_{i,k}a^{2}\partial_{\theta}\varrho(\Phi)\partial_{l}\varrho(\Phi)\right]+T\,,$

where $T$ is a sum of telescoping terms exactly analogous to those in (7.7)
and is estimated in exactly the same way, and we omit further details.
Focusing on the first term after the equality in the above display, we again
set aside the low-frequency portion to cancel the stress. Then the high-
frequency portion of this term is given by

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}\left(\sum_{i,k}a^{2}\mathbb{P}_{\neq
0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)(\Phi)\right)\,.$

Again by the product rule, the derivative may land on low-frequency objects as
in (7.13), and this term is fed into the inverse divergence exactly as before.
We omit further details, as the only differences are that the high-frequency
object $\mathbb{P}_{\neq 0}(\varrho\partial_{mm}\varrho)$ from (7.13) has been
replaced with $\mathbb{P}_{\neq
0}(\partial_{\theta}\varrho\partial_{l}\varrho)$, and a few indices have been
shifted in the definition of the low frequency object $G^{l}$ from (7.14).

The relatively more troublesome term arises when $\partial_{\theta}$ lands on
the high-frequency portion $\mathbb{P}_{\neq
0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)(\Phi)$. We analyze
this term as follows. Since radial Mikado flows are stationary solution of the
Euler-$\alpha$ equations (with pressure calculated in (6.10)), we have that

$\displaystyle\partial_{\theta}\left(\mathbb{P}_{\neq
0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)\right)=\partial_{l}P\,,$

where $P$ is a mean-zero pressure. We want to show that

$\displaystyle\partial_{\theta}\left(a^{2}T_{\theta
l}(\Phi)\right)=\partial_{l}\tilde{P}+\partial_{\theta}({\mathring{R}}_{\theta
l})$ (7.16)

where $T_{\theta l}:=\mathbb{P}_{\neq
0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)$. We now highlight
the properties of $a^{2}$ and $T_{\theta l}$ which are analogous to those in
(7.14), which later allow applications of the inverse divergence operator with
choices essentially identical to those of (7.14). The mean-zero tensor
$T_{\theta l}$ is periodic to scale $\left(\lambda_{q+1}r_{q}\right)^{-1}$ and
given by the iterated Laplacian of a potential (similar to the choices of
$\varrho$ and $\vartheta$ in the analysis of (7.13)), has derivatives which
cost $\Lambda=\lambda_{q+1}$, satisfies $\partial_{\theta}(T_{\theta
l})=\partial_{l}P$, and has $L^{1}$ norm comparable to $\lambda_{q+1}^{2}$
(just as in the choice for $\mathcal{C}_{*}$ from (7.13)). Note that by virtue
of the equality $\partial_{\theta}T_{\theta l}=\partial_{l}P$ and the $L^{1}$
norm of $T_{\theta l}$, we have that

$\displaystyle\left\|P\right\|_{L^{1}}=\left\|\Delta^{-1}\partial_{ll}P\right\|_{L^{1}}=\left\|\Delta^{-1}\partial_{l\theta}T_{\theta
l}\right\|_{L^{1}}\leq\lambda_{q+1}^{2+\gamma}\,.$ (7.17)

The diffeomorphism $\Phi=\Phi_{q,i}$ in (7.16) is the usual one, and the low-
frequency coefficient function $a^{2}$ has $L^{1}$ norm given by a power of
$\ell^{-1}$, and each subsequent derivative costs $\ell^{-5}$ (exactly like
the term $G^{l}$ in the analysis of (7.13)). The tensor
${\mathring{R}}_{\theta l}$ in (7.16) will be symmetric and traceless and,
from the assumptions on $a^{2}$ and $T_{\theta l}$, satisfy estimates which
allow it to be absorbed into ${\mathring{R}}_{q+1}$.

Feeding the term in (7.16) in which $\partial_{\theta}$ lands on the low
frequency function $a$ into the inverse divergence from Proposition 5.5 to
obtain an error term ${\mathring{R}}_{\textnormal{low}}^{\theta l}$ and
pressure $P_{\textnormal{low}}$, we have that

$\displaystyle\partial_{\theta}\left(a^{2}T_{\theta l}(\Phi)\right)$
$\displaystyle=\partial_{\theta}(a^{2})T_{\theta
l}(\Phi)+a^{2}\partial_{\theta}\left(T_{\theta l}(\Phi)\right)$
$\displaystyle=\partial_{\theta}{\mathring{R}}_{\textnormal{low}}^{\theta
l}+\partial_{l}P_{\textnormal{low}}+\fint_{\mathbb{T}^{3}}\partial_{\theta}(a^{2})T_{\theta
l}(\Phi)+a^{2}\partial_{m}T_{\theta l}(\Phi)\partial_{\theta}\Phi^{m}$
$\displaystyle=\partial_{\theta}{\mathring{R}}_{\textnormal{low}}^{\theta
l}+\partial_{l}P_{\textnormal{low}}+\fint_{\mathbb{T}^{3}}\partial_{\theta}(a^{2})T_{\theta
l}(\Phi)+a^{2}\partial_{m}T_{\theta
l}(\Phi)\left(\partial_{\theta}\Phi^{m}-\delta_{\theta
m}\right)+a^{2}\left(\partial_{\theta}T_{\theta l}\right)(\Phi)\,.$ (7.18)

The first two terms on the second line, which come from the application of the
inverse divergence, are estimated entirely analogously to how (7.13) was
estimated and satisfy analogous bounds due to the properties of $a$,
$T_{\theta l}$, and $\Phi$. The third term, which contains the mean, will
cancel with the means resulting from subsequent applications of the inverse
divergence to the remaining terms, since the left-hand side of (7.18) is zero
mean. Therefore, it suffices to show these two terms are sufficiently small.
The fourth term in the above display may be rewritten using the Piola identity
and differentiation by parts as

$\displaystyle a^{2}\partial_{m}T_{\theta
l}(\Phi)(\partial_{\theta}\Phi^{m}-\delta_{\theta m})$
$\displaystyle=a^{2}(\nabla\Phi)^{-1}_{mn}\partial_{n}\left(T_{\theta
l}(\Phi)\right)(\partial_{\theta}\Phi^{m}-\delta_{\theta m})$
$\displaystyle=\partial_{n}\left(a^{2}(\nabla\Phi)^{-1}_{mn}T_{\theta
l}(\Phi)(\partial_{\theta}\Phi^{m}-\delta_{\theta
m})\right)-\partial_{n}\left(a^{2}(\nabla\Phi)^{-1}_{mn}(\partial_{\theta}\Phi^{m}-\delta_{\theta
m})\right)T_{\theta l}(\Phi)\,.$ (7.19)

The first term in (7.19) is the divergence of a tensor indexed by $n$ and $l$
and satisfies bounds identical to those in (7.8), although it is not symmetric
and traceless. Applying the same methodology that converted the telescoping
term in (7.7) into a symmetric traceless stress, we see that this term may be
absorbed into ${\mathring{R}}_{q+1}$ with a bound given by (7.10). The second
term is of the exact same form as the term in (7.13) and may be fed into the
inverse divergence from Proposition 5.5 to produce bounds identical to those
in (7.15).

The final term on the second line of (7.18) may be rewritten using the Piola
identity and differentiation by parts as

$\displaystyle a^{2}\left(\partial_{\theta}T_{\theta l}\right)(\Phi)$
$\displaystyle=a^{2}(\partial_{l}P)(\Phi)$
$\displaystyle=a^{2}\partial_{m}\left((\nabla\Phi)^{-1}_{lm}P\circ\Phi\right)$
$\displaystyle=a^{2}\partial_{l}(P\circ\Phi)+a^{2}\partial_{m}\left(\left((\nabla\Phi)^{-1}_{lm}-\delta_{lm}\right)P\circ\Phi\right)$
$\displaystyle=\partial_{l}\left(a^{2}P\circ\Phi\right)-\partial_{l}a^{2}P\circ\Phi+\partial_{m}\left(a^{2}\left((\nabla\Phi)_{lm}^{-1}-\delta_{lm}\right)P\circ\Phi\right)-\partial_{m}a^{2}\left((\nabla\Phi)^{-1}_{lm}-\delta_{lm}\right)P\circ\Phi$
$\displaystyle=\partial_{l}\left(a^{2}P\circ\Phi\right)+\partial_{m}\left(a^{2}\left((\nabla\Phi)_{lm}^{-1}-\delta_{lm}\right)P\circ\Phi\right)-\partial_{m}a^{2}(\nabla\Phi)^{-1}_{lm}P\circ\Phi\,.$

The first term on the last line above is pressure. The second term is the
divergence of a tensor, but not a symmetric traceless one. We estimate it in
the same fashion as the previous telescoping terms, noting along the way that
the $L^{1}$ bound on $P$ costs an extra factor of $\lambda_{q+1}^{\gamma}$.
The final term goes into the inverse divergence from Proposition 5.5 and obeys
similar bounds as before, albeit with an extra $\lambda_{q+1}^{\gamma}$ loss
again due to the $L^{1}$ bound on $P$.

It remains to show that the low-frequency portions of the terms analyzed above
cancel the stress. We write that

$\displaystyle\lambda_{q+1}^{-2}\partial_{\theta}$
$\displaystyle\left(\sum_{i}\eta_{q,i}^{2}(t)\sum_{k}c_{k}^{2}(R_{\ell})\rho_{q}(x,t)\left(k^{\theta}k^{l}\mathbb{P}_{=0}\left(\varrho\partial_{mm}\varrho\right)+\mathbb{P}_{=0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)\right)\right)-\alpha^{-2}\partial_{\theta}{\mathring{R}}_{\ell}^{\theta
l}$
$\displaystyle=\lambda_{q+1}^{-2}\partial_{\theta}\left(\sum_{i}\eta_{q,i}^{2}(t)\sum_{k}c_{k}^{2}(R_{\ell})\rho_{q}(x,t)\left(k^{\theta}k^{l}\mathbb{P}_{=0}\left(\varrho\partial_{mm}\varrho\right)+\mathbb{P}_{=0}\left(\partial_{\theta}\varrho\partial_{l}\varrho\right)\right)\right)$
$\displaystyle\qquad-\sum_{i}\eta_{q,i}^{2}(t)\alpha^{-2}\partial_{\theta}{\mathring{R}}_{\ell}^{\theta
l}$
$\displaystyle=\partial_{\theta}\sum_{i}\eta_{q,i}^{2}(x,t)\left(\sum_{k}c_{k}^{2}(R_{\ell})\rho_{q}(x,t)\left(3k^{\theta}k^{l}-\delta_{\theta
l}\right)-\alpha^{-2}{\mathring{R}}_{\ell}^{\theta l}\right)$
$\displaystyle=\partial_{\theta}\sum_{i}\eta_{q,i}^{2}(x,t)\left(\rho_{q}(x,t)\left(\alpha^{-2}\frac{{\mathring{R}}_{\ell}^{\theta
l}}{\rho_{q}(x,t)}\right)-\alpha^{-2}{\mathring{R}}_{\ell}^{\theta l}\right)$
$\displaystyle=0\,.$

We have used Lemma 6.2 and the normalization
$\|\nabla\varrho_{q+1,k}\|_{L^{2}}=\lambda_{q+1}$, which is commensurate with
the estimates in Proposition 6.1.

####  7.1.2 Type 2 oscillation errors

From (7.5) and (7.6) and the ensuing discussion, the only non-zero terms for
which $i\neq i^{\prime}$ are those in which $|i-i^{\prime}|\leq 1$. As only
two time cutoffs are non-zero at any point in time, it therefore suffices to
analyze a single term of the form

$\mathcal{E}_{i}=a_{i}a_{i+1}\mathbb{W}_{q+1,k}\circ\Phi_{q,i}\otimes\Delta\left(\mathbb{W}_{q+1,k^{\prime}}\circ\Phi_{q,i+1}\right)+a_{i}a_{i+1}\nabla\left(\mathbb{W}_{q+1,k}\circ\Phi_{q,i}\right)^{T}\nabla\left(\mathbb{W}_{q+1,k^{\prime}}\circ\Phi_{q,i+1}\right)\,,$
(7.20)

where we have used the simpler notation from (7.4) for the Mikado flows and
the shorthand

$a_{i}:=a_{q,i,k}:=\lambda_{q+1}^{-1}c_{k}^{i\text{ mod
}2}\eta_{q,i}\rho^{\nicefrac{{1}}{{2}}}_{q,i}(\nabla\Phi_{q,i})^{-1}\,.$

To estimate (7.20), we will use Proposition 5.3 with the following choices:

$\displaystyle\mathbb{W}^{1}=\mathbb{W}_{q+1,k}\circ\Phi_{q,i}\,,\qquad\mathbb{W}^{2}=\mathbb{W}_{q+1,k^{\prime}}\circ\Phi_{q,i+1}\,,\qquad
v=u_{\ell}\,,\qquad\Phi_{1}=\Phi_{q,i}\,,\qquad\Phi_{2}=\Phi_{q,i+1}\,,$
$\displaystyle f=a_{i}a_{i+1}\,,\qquad
r_{1}=\frac{\ell^{-8}}{\lambda_{q+1}}\,,\qquad
r_{2}=r_{q}\,,\qquad{\mathcal{C}_{f}=\delta_{q+1}}\,.$

By the definition of $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ and Lemma 4.1, we
have that (5.12) is satisfied. From Lemma 6.3 we also have that (5.15) is
satisfied for an appropriate choice of $\mathcal{C}_{\Phi}$. We now just need
to check the bounds for $f$, which was defined as

$f=a_{i}a_{i+1}=\lambda_{q+1}^{-2}c_{k}^{0}c_{k^{\prime}}^{1}\eta_{q,i}\eta_{q,i+1}\rho_{q,i}^{\nicefrac{{1}}{{2}}}\rho_{q,i+1}^{\nicefrac{{1}}{{2}}}(\nabla\Phi_{q,i})^{-1}(\nabla\Phi_{q,i+1})^{-1}\,.$

By the product rule (6.32a), and (6.32b), we have that

$\|\nabla^{n}f\|_{L^{1}}\leq\delta_{q+1}\ell^{-8n}\,,$

and so (5.13) is satisfied. By the inequality (4), we have that

${\ell^{-8({\mathbf{N}_{\textnormal{dec}}}+4)}\leq\lambda_{q+1}^{{\mathbf{N}_{\textnormal{dec}}}(1-\Gamma-\frac{\Gamma}{b})}}\,,$

and so (5.14) is satisfied. Therefore, (5.16) yields that

$\|\mathcal{E}_{i}\|_{L^{1}}\lesssim\delta_{q+1}r_{q}\lambda_{q+1}^{2}\,.$
(7.21)

Since $\mathcal{E}_{i}$ is not necessarily traceless and symmetric, we define
the stress

$R_{\textnormal{osc},2}=\sum_{i}\mathcal{R}\circ\operatorname{\mathrm{div}}\mathcal{E}_{i}\,.$

As before, we may interpolate between the sharp $L^{1}$ bound in (7.21) and a
lossy $L^{\infty}$ bound to deduce that,

$\left\|R_{\textnormal{osc},2}\right\|_{L^{1}}\leq{\delta_{q+1}r_{q}\lambda_{q+1}^{2+\gamma}\leq\delta_{q+2}\lambda_{q+2}^{2-\gamma}\,,}$
(7.22)

which concludes the analysis of the Type 2 error terms. In the last line we
have used inequality (8).

######  Remark 7.1 (Different strategies for solving the “intersection
problem”).

Following [bmnv21] (see section 2.5.2 for heuristics, or Proposition 4.8 for a
precise statement), discretizing in space at scale $\lambda_{q}^{-1}$ and
using $\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\nicefrac{{3}}{{4}}}$
worth of intermittency ensures that pipes from different transport maps do not
intersect at all. The problem with this method is that it requires a large
amount of intermittency. This will not impede the maximization of the
regularity in $L^{2}$ of a solution, but will cause extremely lossy
$L^{\infty}$ estimates. So although this method prevents intersections
entirely and would suffice for our current purposes, it appears to impede the
successful resolution of the Onsager conjecture for the Euler-$\alpha$
equations.

A second option is that of De Lellis and Kwon from the paper [dlk20]. In this
paper, the authors restrict the timescale to be much shorter than the inverse
Lipschitz timescale. On this short timescale, they can also prevent
intersections entirely. However, the heuristic Nash error bounds are _already_
sharp when utilizing the inverse Lipschitz timescale; see Remark 7.2.
Shortening the timescale will damage these error terms and force the usage of
more intermittency, thus preventing $b$ from approaching $1$. Thus it appears
this technique also will not allow for a resolution of the Onsager conjecture
for the Euler-$\alpha$ equations.

A third option would be the gluing technique of Isett [Isett2018]. However,
the gluing technique as employed by Isett requires global Lipschitz bounds
which are not available in an intermittent convex integration scheme. Since
the usage of intermittency seems _necessary_ in order to obtain estimates for
the transport error which allow $\nabla u\in L^{2}$, and there is no clear way
to pass to the limit in the weak formulation of (2.1) unless $\nabla u\in
L^{2}_{t,x}$, the gluing technique does not appear to be consistent with a
convex integration scheme which produces a function satisfying a meaningful
weak formulation of the equations. Based on these heuristics, none of the
current methodologies for preventing intersections of deformed pipe flows will
allow for a resolution of the Onsager conjecture for Euler-$\alpha$. Since
Proposition 5.3 requires no minimum amount of intermittency to function, no
artificial restriction of the timescale, and no global Lipschitz bounds, it
appears to be more amenable to sharper regularity estimates which would allow
the scheme to reach the $L^{3}$-threshold.

###  7.2 Nash Error

Per (6.37), the Nash error is given by

$\partial_{\theta}{\mathring{R}}_{\text{Nash}}^{\theta
l}:=\partial_{\theta}\left(w_{q+1}^{\theta}\left(u_{\ell}^{l}-\alpha^{2}\Delta
u_{\ell}^{l}\right)-\alpha^{2}\partial_{\theta}u_{\ell}^{j}\partial_{l}w_{q+1}^{j}-\alpha^{2}\partial_{\theta}w_{q+1}^{j}\partial_{l}u_{\ell}^{j}\right)\,.$
(7.23)

It is clear that the most costly terms in this expression are the second and
third terms; more specifically, the instances in which the differential
operator $\partial_{\theta}$ falls on the principal part of
$\partial_{l}w_{q+1}^{j}$ or $\partial_{\theta}w_{q+1}^{j}$. For this reason
we begin by analyzing one such term. In the second term in (7.23), the most
expensive term is then

$\displaystyle\partial_{\theta}u_{\ell}^{j}\partial_{\theta}\partial_{l}\left(w_{q+1}^{(p)}\right)^{j}$
$\displaystyle=\partial_{\theta}u_{\ell}^{j}\partial_{\theta}\partial_{l}\left(\frac{1}{\lambda_{q+1}}\sum_{i\in\mathbb{Z}}\sum_{k\in\mathcal{K}}c_{k}(R_{\ell})\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(x,t)(\nabla\Phi_{q,i})_{\beta
j}^{-1}k^{\beta}\varrho_{q+1,k}(\Phi_{q,i})\right)\,.$

We simplify again to the case in which the differential operators
$\partial_{\theta}\partial_{l}$ fall on $\varrho_{q+1,k}$, which again will
clearly be the most costly error term from the above expression. Thus we are
tasked with bounding

$\displaystyle\partial_{\theta}u_{\ell}^{j}\left(\frac{1}{\lambda_{q+1}}\sum_{i\in\mathbb{Z}}\sum_{k\in\mathcal{K}}c_{k}(R_{\ell})\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(x,t)(\nabla\Phi_{q,i})_{\beta
j}^{-1}k^{\beta}\partial_{mn}\varrho_{q+1,k}(\Phi_{q,i})\partial_{\theta}\Phi_{q,i}^{m}\partial_{l}\Phi_{q,i}^{n}\right)\,.$
(7.24)

We shall apply Proposition 5.5 with the following choices, after fixing
choices of $m,n\in\\{1,2,3\\}$:

$\displaystyle v=u_{\ell}\,,\qquad
G^{l}=\partial_{\theta}u_{\ell}^{j}c_{k}(R_{\ell})\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(\nabla\Phi_{q,i})_{\beta
j}^{-1}k^{\beta}\partial_{\theta}\Phi_{q,i}^{m}\partial_{l}\Phi_{q,i}^{n}\,,\qquad\mathcal{C}_{G}={\ell^{-7}}$
$\displaystyle\lambda=\ell^{-5}\,,\qquad\Phi=\Phi_{q,i}\,,\qquad\varrho=\partial_{mn}\varrho_{q+1,k}\,,\qquad\vartheta=\partial_{mn}\vartheta_{k,\lambda_{q+1},r_{q}}\,,$
$\displaystyle\qquad\zeta=\Lambda=\lambda_{q+1}\,,\qquad\mathcal{C}_{*}={\lambda_{q+1}^{2}}r_{q}\,,\qquad\mu=\lambda_{q+1}r_{q}\,.$
(7.25)

Then from (6.32b), Lemma 6.3, and (3.3), we may bound $G^{l}$ by

$\left\|\partial_{\theta}u_{\ell}^{j}\right\|_{0}\left\|c_{k}\rho_{q}^{\nicefrac{{1}}{{2}}}\eta_{q,i}\right\|_{0}\left\|(\nabla\Phi_{q,i})^{-1}\right\|_{0}\left\|\nabla\Phi_{q,i}\right\|_{0}^{2}\lesssim\ell^{-7}\,.$
(7.26)

Furthermore, from (6.32b), the most expensive spatial derivatives arise when
differentiating $c_{k}\rho_{q}^{\nicefrac{{1}}{{2}}}\eta_{q,i}$, and cost
$\ell^{-5}$ from (6.32b). Thus we have satisfied (5.36) with our choices of
$\mathcal{C}_{G}$ and $\lambda$. The deformation bounds in (5.37) and (5.38)
follow from Lemma 6.3. The equality in (1) follows from applying
$\partial_{mn}$ to both sides of the first equality in (1) from Proposition
6.1. The desired periodicity in (2) follows as well from (3) from Proposition
6.1. Finally, since $\zeta=\Lambda=\lambda_{q+1}$, (5.39) follows also from
(3) from Proposition 6.1. The parameter inequality in (5.40) translates into

$\left(\frac{\ell^{-5}}{\lambda_{q+1}}\right)^{{\mathsf{d}}}\ell^{-20}\leq\lambda_{q+1}^{-1}\,,$

which is inequality (10). Applying (5.42) and recalling the factor
$\frac{1}{\lambda_{q+1}}$ in front of (7.24), we find that (7.24) satisfies
(5.41) for a traceless symmetric stress ${\mathring{R}}$ and pressure $P$
which satisfy the bound

$\left\|{\mathring{R}}\right\|_{L^{1}}+\left\|P\right\|_{L^{1}}\lesssim\lambda_{q+1}^{-1}\ell^{-7}\ell^{-20}\lambda_{q+1}^{2}r_{q}\lambda_{q+1}^{-1}\leq{\delta_{q+2}\lambda_{q+2}^{2-\gamma}}\,,$
(7.27)

which is a consequence of inequality (12). Finally, since the entirety of the
Nash error in (7.23) has mean zero since it is the divergence of a tensor, we
may ignore the mean in equality (5.41). Furthermore, since we have bounded a
term which had one spatial derivative landed on $u_{\ell}$ and two on the flow
profile in the definition of $w_{q+1}^{(p)}$, all terms in (7.23) are either
similar in structure and will satisfy an identical bound, or have _less_
derivatives landing on the flow profile and will thus satisfy better bounds.
We omit further details.

######  Remark 7.2 (Usage of intermittency and the $L^{3}$ rigidity
threshold).

To the authors knowledge, the 3D Euler equations provide the only known
example of a proven Onsager-type threshold between rigidity and flexibility.
In the proofs of flexibility from [Isett2018], [BDLSV17], estimates are made
in $L^{\infty}$ rather than $L^{3}$ since no intermittency is employed, and so
we compute the following $L^{\infty}$ bound for the Nash error term, which is
then written as

$\left\|\operatorname{\mathrm{div}}^{-1}\left[w_{q+1}^{k}\partial_{k}\left(u_{q}-\alpha\Delta
u_{q}\right)^{\ell}-\alpha\partial_{k}\left(\partial_{k}u_{q}^{j}\partial^{\ell}w_{q+1}^{j}+\partial_{k}w_{q+1}^{j}\partial^{\ell}u_{q}^{j}\right)\right]\right\|_{L^{\infty}}\,.$
(7.28)

The first part of this error term is quite benign since no derivatives have
landed on $w_{q+1}$. The second part of the error term is less benign, since
it is possible that all the derivatives can land on $w_{q+1}$. The worst case
scenario is that in which all the derivatives land on $w_{q+1}$. Assuming that
_no_ intermittency is being employed, we find that

$\displaystyle\lambda_{q+1}^{-1}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{2}$
$\displaystyle\leq\delta_{q+2}\lambda_{q+2}^{2}$
$\displaystyle\iff\delta_{q+2}^{-1}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\delta_{q}^{\nicefrac{{1}}{{2}}}$
$\displaystyle\leq\lambda_{q+2}^{2}\lambda_{q+1}^{-1}\lambda_{q}^{-1}$
$\displaystyle\iff 2\beta b^{2}-\beta b-\beta$ $\displaystyle\leq 2b^{2}-b-1$
$\displaystyle\iff\beta$ $\displaystyle\leq 1\,.$

We thus conclude that when estimating the Nash error, the apparent optimal
$L^{\infty}$ bound would not allow the final solution to enjoy $C^{1}$
regularity. Any intermittency would _weaken_ this $L^{\infty}$ estimate, and a
significant amount of intermittency would produce a bound which would be far
from $C^{1}$ regularity, and likely also far from the sharp $L^{3}$ threshold.

###  7.3 Transport Error

Per (6.36), the transport error is given by

$\displaystyle\partial_{\theta}{\mathring{R}}_{\text{transport}}^{\theta
l}:=\partial_{t}(w_{q+1}-\alpha^{2}\Delta
w_{q+1})^{l}+u_{\ell}\cdot\nabla(w_{q+1}-\alpha^{2}\Delta w_{q+1})^{l}\,.$
(7.29)

It is clear that the terms $(\partial_{t}+u_{\ell}\cdot\nabla)w_{q+1}$ and
$(\partial_{t}+u_{\ell}\cdot\nabla)(\Delta w_{q+1}^{(c)})$ are lower order
when compared with $(\partial_{t}+u_{\ell}\cdot\nabla)(\Delta w_{q+1}^{(p)})$,
and so we shall bound only the latter and omit further details. Thus we are
tasked with bounding

$\displaystyle\left(\partial_{t}+u_{\ell}\cdot\nabla\right)\left[\Delta\left(\frac{1}{\lambda_{q+1}}\sum_{i\in\mathbb{Z}}\sum_{k\in\mathcal{K}}c_{k}(R_{\ell})\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(x,t)(\nabla\Phi_{q,i})_{\beta
l}^{-1}k^{\beta}\varrho_{q+1,k}(\Phi_{q,i})\right)\right]\,.$

As usual, the terms in which $\Delta$ lands anywhere besides $\rho_{q+1,k}$
are lower order, and so we shall simply bound

$\displaystyle\left(\partial_{t}+u_{\ell}\cdot\nabla\right)\left(\frac{1}{\lambda_{q+1}}\sum_{i\in\mathbb{Z}}\sum_{k\in\mathcal{K}}c_{k}(R_{\ell})\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(x,t)(\nabla\Phi_{q,i})_{\beta
l}^{-1}k^{\beta}\partial_{mn}\varrho_{q+1,k}(\Phi_{q,i})\partial_{j}\Phi_{q,i}^{m}\partial_{j}\Phi_{q,i}^{n}\right)\,.$

After fixing choices of $m,n\in\\{1,2,3\\}$, we shall apply Proposition 5.7
with the following choices:

$\displaystyle v=u_{\ell}\,,\qquad
G^{l}=\left(\partial_{t}+u_{\ell}\cdot\nabla\right)\left(c_{k}\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}(\nabla\Phi_{q,i})_{\beta
l}^{-1}k^{\beta}\partial_{j}\Phi_{q,i}^{m}\partial_{j}\Phi_{q,i}^{n}\right)\,,\qquad\mathcal{C}_{G}={\ell^{-13}}\,,$
$\displaystyle{\lambda=\ell^{-5}}\,,\qquad\Phi=\Phi_{q,i}\,,\qquad\varrho=\partial_{mn}\varrho_{q+1,k}\,,\qquad\vartheta=\partial_{mn}\vartheta_{k,\lambda_{q+1},r_{q}}\,,$
$\displaystyle\zeta=\Lambda=\lambda_{q+1}\,,\qquad\mathcal{C}_{*}=\lambda_{q+1}^{2}r_{q}\,,\qquad\mu=\lambda_{q+1}r_{q}\,.$
(7.30)

In the above choices, we have used that the material derivative of any
function which has been precomposed with $\Phi_{q,i}$ is zero, and so the
material derivative falls on everything except
$\partial_{mn}\rho_{q+1,k}(\Phi_{q,i})$. To bound $G^{l}$, we split up the
material derivative into $\partial_{t}$ and $u_{\ell}\cdot\nabla$ and appeal
to (3.3), (6.32b) with $j=1$, (6.20), and Lemma 6.3 to bound $G^{l}$ by

$(1+\left\|u_{\ell}\right\|_{0})\left\|\nabla_{x,t}\left(c_{k}\eta_{q,i}\rho_{q}^{\nicefrac{{1}}{{2}}}\right)\right\|_{0}\left\|\nabla_{x,t}\left(\nabla\Phi_{q,i}^{-1}\otimes\nabla\Phi_{q,i}\otimes\nabla\Phi_{q,i}\right)\right\|_{0}\lesssim\ell^{-5}\cdot{\ell^{-7}}\cdot\ell^{-1}=\ell^{-13}\,.$

Appealing to the same estimates, we see that spatial derivatives on $G^{l}$
cost at most $\ell^{-5}$, and so (5.36) is satisfied. The rest of the choices
in (7.30) are _identical_ to those in (7.25), and so the rest of the
assumptions in Proposition 5.5 are satisfied. Thus (5.41) gives a traceless
symmetric stress ${\mathring{R}}$ and pressure $P$ satisfying bounds identical
to those in (7.27), save for 6 extra factors of $\ell^{-1}$ from the slightly
worse choice of $\mathcal{C}_{G}$. Since the highest order error terms
satisfies a bound which matches the desired estimate on
${\mathring{R}}_{q+1}$, it only remains to address the mean of the transport
error. From the definition of $u_{q+1}$ in (6.34) and the identification of
error terms immediately below (6.34), we see that

$\fint\left(\partial_{t}+u_{\ell}\cdot\nabla\right)\left(w_{q+1}-\alpha^{2}\Delta
w_{q+1}\right)=\fint\partial_{t}\left(u_{q+1}-\alpha^{2}\Delta
u_{q+1}\right)\,.$

Since $u_{q+1}$ was constructed as an explicit sum of curls of vector fields,
cf. (6.1), the right-hand side in the above equality has zero mean, and thus
the left-hand side does as well. Therefore we may ignore the terms arising