# Non-conservative $H^{\nicefrac{{1}}{{2}}-}$ weak solutions of the
incompressible 3D Euler equations

Tristan Buckmaster Department of Mathematics, Princeton University, Princeton,
NJ 08544<EMAIL_ADDRESS> Nader Masmoudi NYUAD Research Institute,
New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE & Courant
Institute of Mathematical Sciences, New York University, New York, NY 10012,
<EMAIL_ADDRESS> Matthew Novack Courant Institute of Mathematical
Sciences, New York University, New York, NY 10012<EMAIL_ADDRESS> Vlad
Vicol Courant Institute of Mathematical Sciences, New York University, New
York, NY 10012<EMAIL_ADDRESS>
###### Abstract

For any positive regularity parameter $\beta<\nicefrac{{1}}{{2}}$, we
construct non-conservative weak solutions of the 3D incompressible Euler
equations which lie in $H^{\beta}$ uniformly in time. In particular, we
construct solutions which have an $L^{2}$-based regularity index _strictly
larger_ than $\nicefrac{{1}}{{3}}$, thus deviating from the
$H^{\nicefrac{{1}}{{3}}}$-regularity corresponding to the Kolmogorov-Obhukov
$\nicefrac{{5}}{{3}}$ power spectrum in the inertial range.

###### Contents

  1. 1 Introduction
    1. 1.1 Context and motivation
    2. 1.2 Ideas and difficulties
  2. 2 Outline of the convex integration scheme
    1. 2.1 A guide to the parameters
    2. 2.2 Inductive assumptions
    3. 2.3 Intermittent pipe flows
    4. 2.4 Higher order stresses
    5. 2.5 Cut-off functions
    6. 2.6 The perturbation
    7. 2.7 The Reynolds stress error and heuristic estimates
  3. 3 Inductive assumptions
    1. 3.1 General notations
    2. 3.2 Inductive estimates
    3. 3.3 Main inductive proposition
    4. 3.4 Proof of Theorem 1.1
  4. 4 Building blocks
    1. 4.1 A careful construction of intermittent pipe flows
    2. 4.2 Deformed pipe flows and curved axes
    3. 4.3 Placements via relative intermittency
  5. 5 Mollification
  6. 6 Cutoffs
    1. 6.1 Definition of the velocity cutoff functions
    2. 6.2 Properties of the velocity cutoff functions
    3. 6.3 Definition of the temporal cutoff functions
    4. 6.4 Estimates on flow maps
    5. 6.5 Stress estimates on the support of the new velocity cutoff functions
    6. 6.6 Definition of the stress cutoff functions
    7. 6.7 Properties of the stress cutoff functions
    8. 6.8 Definition and properties of the checkerboard cutoff functions
    9. 6.9 Definition of the cumulative cutoff function
  7. 7 From $q$ to $q+1$: breaking down the main inductive estimates
    1. 7.1 Induction on $q$
    2. 7.2 Notations
    3. 7.3 Induction on $\widetilde{n}$
  8. 8 Proving the main inductive estimates
    1. 8.1 Definition of $\mathring{R}_{q,{\widetilde{n}},{\widetilde{p}}}$ and $w_{q+1,{\widetilde{n}},{\widetilde{p}}}$
    2. 8.2 Estimates for $w_{q+1,{\widetilde{n}},{\widetilde{p}}}$
    3. 8.3 Identification of error terms
    4. 8.4 Transport errors
    5. 8.5 Nash errors
    6. 8.6 Type 1 oscillation errors
    7. 8.7 Type 2 oscillation errors
    8. 8.8 Divergence corrector errors
    9. 8.9 Time support of perturbations and stresses
  9. 9 Parameters
    1. 9.1 Definitions and hierarchy of the parameters
    2. 9.2 Definitions of the $q$-dependent parameters
    3. 9.3 Inequalities and consequences of the parameter definitions
    4. 9.4 Mollifiers and Fourier projectors
    5. 9.5 Notation
  10. A Useful lemmas
    1. A.1 Transport estimates
    2. A.2 Proof of Lemma 6.2
    3. A.3 $L^{p}$ decorrelation
    4. A.4 Sobolev inequality with cutoffs
    5. A.5 Consequences of the Faa di Bruno formula
    6. A.6 Bounds for sums and iterates of operators
    7. A.7 Commutators with material derivatives
    8. A.8 Intermittency-friendly inversion of the divergence

##  1 Introduction

We consider the homogeneous incompressible Euler equations

$\displaystyle\partial_{t}v+\mathrm{div\,}(v\otimes v)+\nabla p$
$\displaystyle=0$ (1.1a) $\displaystyle\mathrm{div\,}v$ $\displaystyle=0$
(1.1b)

for the unknown velocity vector field $v$ and scalar pressure field $p$, posed
on the the three dimensional box $\mathbb{T}^{3}=[-\pi,\pi]^{3}$ with periodic
boundary conditions. We consider weak solutions of (1.1), which may be defined
in the usual way for $v\in L^{2}_{t}L^{2}_{x}$.

We show that within the class of weak solutions of regularity
$C^{0}_{t}H^{\nicefrac{{1}}{{2}}-}_{x}$, the 3D Euler system (1.1) is
flexible.111Loosely speaking, we consider a system of partial differential
equations of physical origin to be flexible in a certain regularity class, if
at this regularity level the PDEs are not anymore predictive: there exist
infinitely many solutions, which behave in a non-physical way, in stark
contrast to the behavior of the PDE in the smooth category. We refer the
interested reader to the discussion in the surveys of De Lellis and
Székelyhidi Jr. [30, 32] which draw the analogy with the flexibility in
Gromov’s $h$-principle [40]. An example of this flexibility is provided by:

######  Theorem 1.1 (Main result).

Fix $\beta\in(0,\nicefrac{{1}}{{2}})$. For any divergence-free
$v_{\mathrm{start}},v_{\mathrm{end}}\in L^{2}(\mathbb{T}^{3})$ which have the
same mean, any $T>0$, and any $\epsilon>0$, there exists a weak solution $v\in
C([0,T];H^{\beta}(\mathbb{T}^{3}))$ to the 3D Euler equations (1.1) such that
$\left\|v(\cdot,0)-v_{\mathrm{start}}\right\|_{L^{2}(\mathbb{T}^{3})}\leq\epsilon$
and
$\left\|v(\cdot,T)-v_{\mathrm{end}}\right\|_{L^{2}(\mathbb{T}^{3})}\leq\epsilon$.

Since the vector field $v_{\mathrm{end}}$ may be chosen to have a much higher
(or much lower) kinetic energy than the vector field $v_{\mathrm{start}}$, the
above result shows the existence of infinitely many non-conservative weak
solutions of 3D Euler in the regularity class
$C^{0}_{t}H^{\nicefrac{{1}}{{2}}-}_{x}$. Theorem 1.1 further shows that the
set of so-called wild initial data is dense in the space of $L^{2}$ periodic
functions of given mean. The novelty of this result is that these weak
solutions have more than $\nicefrac{{1}}{{3}}$ regularity, when measured on a
$L^{2}_{x}$-based Banach scale.

######  Remark 1.2 (Corollaries of the proof).

We have chosen to state the flexibility of the 3D Euler equations as in
Theorem 1.1 because it is a simple way to exhibit weak solutions which are
non-conservative, leaving the entire emphasis of the proof on the regularity
class in which the weak solutions lie. Using by now standard approaches
encountered in convex integration constructions for the Euler equations, we
may alternatively establish the following variants of flexibility for (1.1)
within the class of $C^{0}_{t}H^{\nicefrac{{1}}{{2}}-}_{x}$ weak solutions:

  1. (a)

The proof of Theorem 1.1 also shows that: given any
$\beta<\nicefrac{{1}}{{2}}$, $T>0$, and $E>0$, there exists a weak solution
$v\in C(\mathbb{R},H^{\beta}(T^{3}))$ of the 3D Euler equations such that:
$\mathrm{supp\,}_{t}v\subset[-T,T]$, and
$\left\|v(\cdot,0)\right\|_{L^{2}}\geq E$. Such weak solutions are nontrivial
and have compact support in time, thereby implying the non-uniqueness of weak
solutions to (1.1) in the regularity class
$C^{0}_{t}H^{\nicefrac{{1}}{{2}}-}_{x}$. The argument is sketched in Remark
3.7 below.

  2. (b)

The proof of Theorem 1.1 may be modified to show that: given any
$\beta\in(0,\nicefrac{{1}}{{2}})$, and any $C^{\infty}$ smooth function
$e\colon[0,T]\to(0,\infty)$, there exists a weak solution $v\in
C^{0}([0,T];H^{\beta}(\mathbb{T}^{3}))$ of the 3D Euler equations, such that
$v(\cdot,t)$ has kinetic energy $e(t)$, for all $t\in[0,T]$. In particular,
the flexibility of 3D Euler in $C^{0}_{t}H^{\nicefrac{{1}}{{2}}-}_{x}$ may be
shown to also hold within the class of dissipative weak solutions, by choosing
$e$ to be a non-increasing function of time. This is further discussed in
Remark 3.8 below.

###  1.1 Context and motivation

Classical solutions of the Cauchy problem for the 3D Euler equations (1.1) are
known to exist, locally in time, for initial velocities which lie in
$C^{1,\alpha}$ for some $\alpha>0$ (see e.g. Lichtenstein [48]). These
solutions are unique, and they conserve (in time) the kinetic energy
${\mathcal{E}}(t)=\frac{1}{2}\int_{\mathbb{T}^{3}}|v(x,t)|^{2}dx$, giving two
manifestations of rigidity of the Euler equations within the class of smooth
solutions.

Motivated by hydrodynamic turbulence, it is natural to consider a much broader
class of solutions to the 3D Euler system; these are the distributional or
weak solutions of (1.1), which may be defined in the natural way as soon as
$v\in L^{2}_{t}L^{2}_{x}$, since (1.1) is in divergence form. Indeed, one of
the fundamental assumptions of Kolmogorov’s ’41 theory of turbulence [46] is
that in the infinite Reynolds number limit, turbulent solutions of the 3D
Navier-Stokes equations exhibit anomalous dissipation of kinetic energy; by
now, this is considered to be an experimental fact, see e.g. the book of
Frisch [39] for a detailed account. In particular, this anomalous dissipation
of energy necessitates that the family of Navier-Stokes solutions does not
remain uniformly bounded in the topology of $L^{3}_{t}B^{\alpha}_{3,\infty,x}$
for any $\alpha>\nicefrac{{1}}{{3}}$, as the Reynolds number diverges, as was
alluded to in the work of Onsager [57].222Onsager did not use the Besov norm
$\left\|v\right\|_{B^{\alpha}_{p,\infty}}=\left\|v\right\|_{L^{p}}+\sup_{|z|>0}|z|^{-\alpha}\left\|v(\cdot+z)-v(\cdot)\right\|_{L^{p}}$;
here we use this modern notation and the sharp version of this conclusion, cf.
Constantin, E, and Titi [22], Duchon and Robert [35], Drivas and Eyink [34].
Thus, in the infinite Reynolds number limit for turbulent solutions of 3D
Navier-Stokes, one expects the convergence to _weak_ solutions of 3D Euler,
not classical ones.

It turns out that even in the context of weak solutions, the 3D Euler
equations enjoy some conditional variants of rigidity. An example is the
classical weak-strong uniqueness property.333If $v$ is a strong solution of
the Cauchy problem for (1.1) with initial datum $v_{0}\in L^{2}$, and $w\in
L^{\infty}_{t}L^{2}_{x}$ is merely a weak solution of the Cauchy problem for
(1.1), which has the additional property that it its kinetic energy
${\mathcal{E}}(t)$ is less than the kinetic energy of $v_{0}$, for a.e. $t>0$,
then in fact $v\equiv w$. See e.g. the review [65] for a detailed account.
Another example is the question of whether weak solutions of the 3D Euler
equation conserve kinetic energy. This is the subject of the Onsager
conjecture [57], one of the most celebrated connections between
phenomenological theories in turbulence and the rigorous mathematical analysis
of the PDEs of fluid dynamics. For a detailed account we refer the reader to
the reviews [37, 21, 60, 30, 63, 32, 33, 12, 14] and mention here only a few
of the results in the Onsager program for 3D Euler.

Constantin, E, and Titi [22] established the rigid side of the Onsager
conjecture, which states that if a weak solution $v$ of (1.1) lies in
$L^{3}_{t}B^{\beta}_{3,\infty,x}$ for some $\beta>\nicefrac{{1}}{{3}}$, then
$v$ conserves its kinetic energy. The endpoint case
$\beta=\nicefrac{{1}}{{3}}$ was addressed by Cheskidov, Constantin,
Friedlander, and Shvydkoy [16], who established a criterion which is known to
be sharp in the context of 1D Burgers. By using the Bernstein inequality to
transfer information from $L^{2}_{x}$ into $L^{3}_{x}$ , the authors of [16]
also prove energy-rigidity for weak solutions based on a regularity condition
for an $L^{2}_{x}$ based scale: if $v\in L^{3}_{t}H^{\beta}_{x}$ with
$\beta>\nicefrac{{5}}{{6}}$, then $v$ conserves kinetic energy (see also the
work of Sulem and Frisch [62]). We emphasize the discrepancy between the
energy-rigidity threshold exponents $\nicefrac{{5}}{{6}}$ for the
$L^{2}$-based Sobolev scale, and $\nicefrac{{1}}{{3}}$ for $L^{p}$-based
regularity scales with $p\geq 3$.

The first flexibility results were obtained by Scheffer [58], who constructed
non-trivial weak solutions of the 2D Euler system, which lie in
$L^{2}_{t}L^{2}_{x}$ and have compact support in space and time. The existence
of infinitely many dissipative weak solutions to the Euler equations was first
proven by Shnirelman in [59], in the regularity class
$L^{\infty}_{t}L^{2}_{x}$. Inspired by the work [53] of Müller and Šverak for
Lipschitz differential inclusions, in [29] De Lellis and Székelyhidi Jr. have
constructed infinitely many dissipative weak solutions of (1.1) in the
regularity class $L^{\infty}_{t}L^{\infty}_{x}$ and have developed a
systematic program towards the resolution of the flexible of the Onsager
conjecture, using the technique of convex integration. Inspired by Nash’s
paradoxical constructions for the isometric embedding problem [54], the first
proof of flexibility of the 3D Euler system in a Hölder space was given by De
Lellis and Székelyhidi Jr. in the work [31]. This breakthrough or crossing of
the $L^{\infty}_{x}$ to $C^{0}_{x}$ barrier in convex integration for 3D Euler
[31] has in turn spurred a number of results [8, 6, 9, 27] which have used
finer properties of the Euler equations to increase the regularity of the wild
weak solutions being constructed. The flexible part of the Onsager conjecture
was finally resolved by Isett [43, 42] in the context of weak solutions with
compact support in time (see also the subsequent work by the first and last
authors with De Lellis and Székelyhidi Jr. [11] for dissipative weak
solutions), by showing that for any regularity parameter
$\beta<\nicefrac{{1}}{{3}}$, the 3D Euler system (1.1) is flexible in the
class of $C^{\beta}_{t,x}$ weak solutions. We refer the reader to the review
papers [30, 63, 32, 33, 12, 14] for more details concerning convex integration
constructions in fluid dynamics, and for open problems in this area.

Since the aforementioned convex integration constructions are spatially
homogenous, they yield weak solutions whose Hölder regularity index cannot be
taken to be larger than $\nicefrac{{1}}{{3}}$ (recall that weak solutions in
$L^{3}_{t}C^{\beta}_{x}$ with $\beta>\nicefrac{{1}}{{3}}$ must conserve
kinetic energy). However, the exponent $\nicefrac{{1}}{{3}}$ is not expected
to be a sharp threshold for energy-rigidity/flexibility if the weak solutions’
regularity is measured on an $L^{p}_{x}$-based Banach scale with $p<3$. This
expectation stems from the measured intermittent nature of turbulent flows,
see e.g. Frisch [39, Figure 8.8, page 132]. In broad terms, intermittency is
characterized as a deviation from the Kolmogorov ’41 scaling laws, which were
derived under the assumptions of homogeneity and isotropy (for a rigorous way
to measure this deviation, see Cheskidov and Shvydkoy [20]). A common
signature of intermittency is that for $p\neq 3$, the $p^{th}$ order structure
function444In analogy with $L^{p}$-based Besov spaces, absolute $p^{th}$ order
structure functions are typically defined as
$S_{p}(\ell)=\fint_{0}^{T}\fint_{\mathbb{T}^{3}}\fint_{{\mathbb{S}}^{2}}|v(x+\ell
z,t)-v(x,t)|^{p}dzdxdt$. The structure function exponents in Kolmogorov’s ’41
theory are then given by $\zeta_{p}=\limsup_{\ell\to 0^{+}}\frac{\log
S_{p}(\ell)}{\log(\epsilon\ell)}$, where $\epsilon>0$ is the postulated
anomalous dissipation rate in the infinite Reynolds number limit. Of course,
for any non-conservative weak solution we may define a positive number
$\epsilon=\fint_{0}^{T}|\frac{d}{dt}{\mathcal{E}}(t)|dt$ as a substitute for
Kolmogorov’s $\epsilon$, which allows one to define $\zeta_{p}$ accordingly.
exponents $\zeta_{p}$ deviate from the Kolmogorov-predicted values of
$\nicefrac{{p}}{{3}}$. We note that the regularity statement $v\in
C^{0}_{t}B^{s}_{p,\infty}$ corresponds to a structure function exponent
$\zeta_{p}=sp$; that is, Kolmogorov ’41 predicts that $s=\nicefrac{{1}}{{3}}$
for all $p$. The exponent $p=2$ plays a special role, as it allows one to
measure the intermittent nature of turbulent flows on the Fourier side as a
power-law decay of the energy spectrum. Throughout the last five decades, the
experimentally measured values of $\zeta_{2}$ (in the inertial range, for
viscous flows at very high Reynolds numbers) have been consistently observed
to exceed the Kolmogorov-predicted value of $\nicefrac{{2}}{{3}}$ [1, 50, 61,
45, 15, 44, 55], thus showing a steeper decay rate in the inertial range power
spectrum than the one predicted by the Kolmogorov-Obhukov $5/3$ law. Moreover,
in the mathematical literature, Constantin and Fefferman [23] and Constantin,
Nie, and Tanveer [24] have used the 3D Navier-Stokes equations to show that
the Kolmogorov ’41 prediction $\zeta_{2}=\nicefrac{{2}}{{3}}$ is only
consistent with a lower bound for $\zeta_{2}$, instead of an exact equality.

Prior to this work, it was not known whether the 3D Euler equation can sustain
weak solutions which have kinetic energy that is uniformly bounded in time but
not conserved, and which have spatial regularity equal to or exceeding
$H^{\nicefrac{{1}}{{3}}}_{x}$, corresponding to
$\zeta_{2}\geq\nicefrac{{2}}{{3}}$; see [12, Open Problem 5] and [14,
Conjecture 2.6]. Theorem 1.1 gives the first such existence result. The
solutions in Theorem 1.1 may be constructed to have second order structure
function exponent $\zeta_{2}$ an arbitrary number in $(0,1)$, showing that
(1.1) exhibits weak solutions which severely deviate from the Kolmogorov-
Obhukov $5/3$ power spectrum.

We note that in a recent work [18], Cheskidov and Luo established the
sharpness of the $L^{2}_{t}L^{\infty}_{x}$ endpoint of the Prodi-Serrin
criteria for the 3D Navier-Stokes equations, by constructing non-unique weak
(mild) solutions of these equations in $L^{p}_{t}L^{\infty}_{x}$, for any
$p<2$.555See also [19] for a proof that the space $C^{0}_{t}L^{p}_{x}$ is
critical for uniqueness at $p=2$, in two space dimensions. As noted in [18,
Theorem 1.10], their approach also applies to the 3D Euler equations, yielding
weak solutions that lie in $L^{1}_{t}C^{\beta}_{x}$ for any $\beta<1$, and
thus these weak solutions also have more than $\nicefrac{{1}}{{3}}$
regularity. The drawback is that the solutions constructed in [18] do not have
bounded (in time) kinetic energy, in contrast to Theorem 1.1, which yields
weak solutions with kinetic energy that is continuous in time.

Theorem 1.1 is proven by using an intermittent convex integration scheme,
which is necessary in order to reach beyond the $\nicefrac{{1}}{{3}}$
regularity exponent, uniformly in time. Intermittent convex integration
schemes have been introduced by the first and last authors in [13] in order to
prove the non-uniqueness of weak (mild) solutions of the 3D Navier-Stokes
equations with bounded kinetic energy, and then refined in collaboration with
Colombo [7] to construct solutions which have partial regularity in time.
Recently, intermittent convex integration techniques have been used
successfully to construct non-unique weak solutions for the transport equation
(cf. Modena and Székelyhidi Jr. [52, 51], Brué, Colombo, and De Lellis [5],
and Cheskidov and Luo [17]), the 2D Euler equations with vorticity in a
Lorentz space (cf. [4]), the stationary 4D Navier-Stokes equations (cf. Luo
[49]), the $\alpha$-Euler equations (cf. [3]), in the context of the MHD
equations (cf. Dai [26], the first and last authors with Beekie [2]), and the
effect of temporal intermittency has recently been studied by Cheskidov and
Luo [18]. We refer the reader to the reviews [12, 14] for further references,
and for a comparison between intermittent and homogenous convex integration.

When applied to three-dimensional nonlinear problems, intermittent convex
integration has insofar only been successful at producing weak solutions with
negligible spatial regularity indices, uniformly in time. As we explain in
Section 1.2, there is a fundamental obstruction to achieving high regularity:
in physical space, intermittency causes concentrations that results in the
formation of intermittent peaks, and to handle these peaks the existing
techniques have used an extremely large separation between the frequencies in
consecutive steps of the convex integration scheme.666This becomes less of an
issue when one considers the equations of fluid dynamics in very high space
dimensions, cf. Tao [64]. This paper is the first to successfully implement a
high-regularity (in $L^{2}$), spatially-intermittent, temporally-homogenous,
convex integration scheme in three space dimensions, and shows that for the 3D
Euler system any regularity exponent $\beta<\nicefrac{{1}}{{2}}$ may be
achieved.777It was known within the community (see Section 2.4.1 for a
detailed explanation) that there is a key obstruction to reaching a regularity
index in $L^{2}$ for a solution to the Euler equations larger than
$\nicefrac{{1}}{{2}}$ via convex integration. In fact, the techniques
developed in this paper are the backbone for the recent work [56] of the last
two authors, which gives an alternative, intermittent, proof of the Onsager
conjecture.

###  1.2 Ideas and difficulties

As alluded to in the previous paragraph, the main difficulty in reaching a
high regularity exponent for weak solutions of (1.1) is that the existing
intermittent convex integration schemes do not allow for consecutive frequency
parameters $\lambda_{q}$ and $\lambda_{q+1}$ to be close to each other. In
essence, this is because intermittency smears out the set of active
frequencies in the approximate solutions to the Euler system (instead of
concentric spheres, they are more akin to thick concentric annuli), and
several of the key estimates in the scheme require frequency separation to
achieve $L^{p}$-decoupling (see Section 2.4.1). Indeed, high regularity
exponents necessitate an almost geometric growth of frequencies
($\lambda_{q}=\lambda_{0}^{q}$), or at least a barely super-exponential growth
rate $\lambda_{q+1}=\lambda_{q}^{b}$ with $0<b-1\ll 1$ (in comparison, the
schemes in [13, 7] require $b\approx 10^{3}$). Essentially every new idea in
this paper is aimed either directly or indirectly at rectifying this issue:
how does one take advantage of intermittency, and at the same time keep the
frequency separation to be nearly geometric?

The building blocks used in the convex integration scheme are intermittent
pipe flows,888The moniker used in [27] and the rest of the literature for
these stationary solutions has been “Mikado flows”. However, we rely rather
heavily on the geometric properties of these solutions, such as orientation
and concentration around axes, and so to emphasize the tube-like nature of
these objects, we will frequently use the name “pipe flows”. which we describe
in Section 2.3. Due to their spatial concentration and their periodization
rate, quadratic interactions of these building blocks produce both the helpful
low frequency term which is used to cancel the previous Reynolds stress
$\mathring{R}_{q}$, and also a number of other errors which live at
intermediate frequencies. These errors are spread throughout the frequency
annulus with inner radius $\lambda_{q}$ and outer radius $\lambda_{q+1}$, and
may have size only slightly less than that of $\mathring{R}_{q}$. If left
untreated, these errors only allow for a very small regularity parameter
$\beta$. In order to increase the regularity index of our weak solutions, we
need to take full advantage of the frequency separation between the slow
frequency $\lambda_{q}$ and the fast frequency $\lambda_{q+1}$. As such, the
intermediate-frequency errors need to be further corrected via velocity
increments designed to push these residual stresses towards the frequency
sphere of radius $\lambda_{q+1}$. The quadratic interactions among these
higher-order velocity corrections themselves, and in principle also with the
old velocity increments, in turn create higher order Reynolds stresses, which
live again at intermediate frequencies (slightly higher than before), but
whose amplitude is slightly smaller than before. This process of adding higher
order velocity perturbations designed to cancel intermediate-frequency higher
order stresses has to be repeated many times until all the resulting errors
are either small, or they live at frequency $\approx\lambda_{q+1}$, and thus
are also small upon inverting the divergence. See Sections 2.4 and 2.6 for a
more thorough account of this iteration.

Throughout the process described in the above paragraph, we need to keep
adding velocity increments, while at the same time keeping the high-high-high
frequency interactions under control. The fundamental obstacle here is that
when composing the intermittent pipe flows with the Lagrangian flow of the
slow velocity field, the resulting deformations are not spatiotemporally
homogenous. In essence, the intermittent nature of the approximate velocity
fields implies that a sharp global control on their Lipschitz norm is
unavailable, thus precluding us from implementing a gluing technique as in
[42, 11]. Additionally, we are faced with the issue that pipe flows which were
added at different stages of the higher order correction process have
different periodization rates and different spatial concentration rates, and
may a-priori overlap. Our main idea here is to implement a placement technique
which uses the relative intermittency of pipe flows from previous or same
generations, in conjunction with a sharp bound on their local Lagrangian
deformation rate, to determine suitable spatial shifts for the placement of
new pipe flows so that they dodge all other bent pipes which live in a
restricted space-time region. This geometric placement technique is discussed
in Section 2.5.2.

A rigorous mathematical implementation of the heuristic ideas described in the
previous two paragraphs, which crucially allows us to slow down the frequency
growth to be almost geometric, requires extremely sharp information on all
higher order errors and their associated velocity increments. For instance, in
order to take advantage of the transport nature of the linearized Euler system
while mitigating the loss of derivatives issue which is characteristic of
convex integration schemes, we need to keep track of essentially _infinitely
many sharp material derivative estimates_ for all velocity increments and
stresses. Such estimates are naturally only attainable on a local inverse
Lipschitz timescale, which in turn necessitates keeping track of the precise
location in space of the peaks in the densities of the pipe flows, and
performing a frequency localization with respect to both the Eulerian and the
Lagrangian coordinates. In order to achieve this, we introduce carefully
designed cutoff functions, which are defined recursively for the velocity
increments (in order to keep track of overlapping pipe flows from different
stages of the iteration), and iteratively for the Reynolds stresses (in order
to keep track of the correct amplitude of the perturbation which needs to be
added to correct these stresses); see Section 2.5. The cutoff functions we
construct effectively play the role of a joint Eulerian-and-Lagrangian
Littlewood-Paley frequency decomposition, which in addition keeps track of
both the position in space and the amplitude of various objects (more akin to
a wavelet decomposition). The analysis of these cutoff functions requires
estimating very high order commutators between Lagrangian and Eulerian
derivatives which in great part are responsible for the length of this paper
(see Section 6 and Appendix A). Lastly, we mention an additional technical
complication: since the sharp control of the Lipschitz norm of the approximate
velocities in our scheme is local in space and time, we need to work with an
inverse divergence operator (e.g. for computing higher order stresses) which,
up to much lower order error terms, maintains the spatial support of the
vector fields that it is applied to. Additionally, we need to be able to
estimate an essentially infinite number of material derivatives applied to the
output of this inverse divergence operator. This issue is addressed in Section
A.8.

The rest of the paper is organized as follows. Section 2 contains an outline
of the convex integration scheme, in which we replace some of the actual
estimates and definitions appearing in the proof with heuristic ones in order
to highlight the new ideas at an intuitive level. The proof of Theorem 1.1 is
given in Section 3, assuming that a number of estimates hold true inductively
for the solutions of the Euler-Reynolds system at every step of the convex
integration iteration. The remainder of the paper is dedicated to showing that
the inductive bounds stated in Section 3.2 may indeed be propagated from step
$q$ to step $q+1$. Section 4 contains the construction of the intermittent
pipe flows used in this paper and describes the careful placement required to
show that these pipe flows do not overlap on a suitable space-time set. The
mollification step of the proof is performed in Section 5. Section 6 contains
the definitions of the cutoff functions used in the proof and establishes
their properties. Section 7 breaks down the the main inductive bounds from
Section 3.2 into components which take into account the higher order stresses
and perturbations. Section 8 then proves the constituent parts of the
inductive bounds outlined in the previous section. Section 9 carefully defines
the many parameters in the scheme, states the precise order in which they are
chosen, and lists a few consequences of their definitions. Finally, Appendix A
contains the analytical toolshed to which we appeal throughout the paper.

### Acknowledgements

T.B. was supported by the NSF grant DMS-1900149 and a Simons Foundation
Mathematical and Physical Sciences Collaborative Grant. N.M. was supported by
the NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research
Institute grant of the center SITE. V.V. was supported by the NSF grant CAREER
DMS-1911413.

##  2 Outline of the convex integration scheme

###  2.1 A guide to the parameters

In order to make sharp estimates throughout the scheme, we will require
numerous parameters. For the reader’s convenience, we have collected in this
section the heuristic definitions of all the parameters introduced in the
following sections of the outline. The parameters are listed in Section 2.1.1
in the order corresponding to their first appearance in the outline. We give
as well brief descriptions of the significance of each parameter.

####  2.1.1 Definitions

######  Definition 2.1 (Parameters Introduced in Section 1).

  1. (1)

$\beta$ \- The regularity exponent corresponding to a final solution $v\in
C\left(\mathbb{R};H^{\beta}(\mathbb{T}^{3})\right)$.

######  Definition 2.2 (Parameters Introduced in Section 2.2).

  1. (1)

$q$ \- The integer which represents the primary stages of the iterative convex
integration scheme.

  2. (2)

${\lambda_{q}=a^{(b^{q})}}$ \- The primary parameter used to quantify
frequencies. $a$ and $b$ will be chosen later, with $a\in\mathbb{R}_{+}$ being
a sufficiently large positive number and $b\in\mathbb{R}$ a real number
slightly larger than $1$.

  3. (3)

$\delta_{q}=\lambda_{q}^{-2\beta}$ \- The primary parameter used to quantify
amplitudes of stresses and perturbations.

  4. (4)

$\tau_{q}=(\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q})^{-1}$ \- The primary
parameter used to quantify the cost of a material derivative
$\partial_{t}+v_{q}\cdot\nabla$.999For technical reasons, $\tau_{q}^{-1}$ will
be chosen to be slightly shorter than $\delta_{q}^{\frac{1}{2}}\lambda_{q}$.
For the heuristic calculations, one may ignore this modification and simply
use $\tau_{q}^{-1}=\delta_{q}^{\frac{1}{2}}\lambda_{q}$.

######  Definition 2.3 (Parameters Introduced in Section 2.3).

  1. (1)

$n$ \- The primary parameter which will be used to divide up the frequencies
between $\lambda_{q}$ and $\lambda_{q+1}$ and which will take non-negative
integer values. The divisions will be used both for the frequencies of the
higher order stresses in Section 2.4 as well as the thickness of the
intermittent pipe flows used to correct the higher order stresses.

  2. (2)

${n_{\rm max}}$ \- A large integer which is fixed independently of $q$ and
which sets the largest allowable value of $n$.

  3. (3)

$\displaystyle{r_{q+1,n}=\left(\lambda_{q}\lambda_{q+1}^{-1}\right)^{\left(\frac{4}{5}\right)^{n+1}}}$
\- The parameter quantifying intermittency, or the thickness of a tube
periodized at unit scale for values of $n$ such that $0\leq n\leq{n_{\rm
max}}$.101010In particular, this choice gives
$r_{q+1,n+1}=r_{q+1,n}^{\frac{4}{5}}$. In our proof, the inequality
$r_{q+1,n}^{3}\ll r_{q+1,n+1}^{4}$ plays a crucial role. In order to absorb
$q$ independent constants, as well as to ensure that there is a sufficient gap
between these parameters to ensure decoupling, we have chosen to work with the
$\frac{4}{5}$ instead of the $\frac{3}{4}$ geometric scale.

  4. (4)

$\displaystyle{\lambda_{q,n}=\lambda_{q+1}r_{q+1,n}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n+1}}}$
\- The minimum frequency present in an intermittent pipe flow
$\mathbb{W}_{q+1,n}$. Equivalently, $\left(\lambda_{q+1}r_{q+1,n}\right)^{-1}$
is the scale to which $\mathbb{W}_{q+1,n}$ is periodized.

Figure 1: Schematic of the frequency parameters appearing in Definitions 2.2
and 2.4.

######  Definition 2.4 (Parameters Introduced in Section 2.4).

  1. (1)

For $2\leq n\leq{n_{\rm max}}$,
$\displaystyle\lambda_{q,n,0}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}$
is the minimum frequency present in the higher order stress
$\mathring{R}_{q,n}$. Conversely, $\lambda_{q,n+1,0}$ is the maximum frequency
present in $\mathring{R}_{q,n}$. When $n=0$, we set
$\lambda_{q,0,0}=\lambda_{q}$ to be the maximum frequency present in
$\mathring{R}_{q,0}=\mathring{R}_{q}$, and when $n=1$,
$\lambda_{q,1,0}=\lambda_{q,0}$ is the minimum frequency present in
$\mathring{R}_{q,1}$, while $\lambda_{q,2,0}$ is the maximum frequency.

  2. (2)

$p$ \- A secondary parameter which takes positive integer values and which
will be used to divide up the frequencies in between $\lambda_{q,n,0}$ and
$\lambda_{q,n+1,0}$, as well as the higher order stresses.

  3. (3)

${p_{\rm max}}$ \- A large integer, fixed independently of $q$, which is the
largest allowable value of $p$.

  4. (4)

$\displaystyle\lambda_{q,n,p}=\lambda_{q,n,0}^{1-\frac{p}{{p_{\rm
max}}}}\lambda_{q,n+1,0}^{\frac{p}{{p_{\rm max}}}}$ \- The maximum frequency
present in the higher order stress $\mathring{R}_{q,n,p}$ for $1\leq
n\leq{n_{\rm max}}$ and $1\leq p\leq{p_{\rm max}}$. Conversely,
$\lambda_{q,n,p-1}$ is the minimum frequency in $\mathring{R}_{q,n,p}$. When
$n=0$ and $p$ takes any value, we adopt the convention that
$\lambda_{q,0,p}=\lambda_{q}$.

  5. (5)

$\displaystyle f_{q,n}=\lambda_{q,n+1,0}^{\frac{1}{{p_{\rm
max}}}}\lambda_{q,n,0}^{-\frac{1}{{p_{\rm max}}}}$ \- The increment between
frequencies $\lambda_{q,n,p-1}$ and $\lambda_{q,n,p}$ for $n\geq 1$. We have
the equalities

$\lambda_{q,n,p}=\lambda_{q,n,0}f_{q,n}^{p},\qquad\lambda_{q,n+1,0}=\lambda_{q,n,0}f_{q,n}^{p_{\rm
max}}\,.$

For ease of notation, when $n=0$ we set $f_{q,n}=1$.

  6. (6)

For $n=0$ and $p=1$, $\delta_{q+1,0,1}:=\delta_{q+1}$ is the amplitude of
$\mathring{R}_{q}:=\mathring{R}_{q,0}$. For $n=0$ and $p\geq 2$,
$\delta_{q+1,0,p}=0$, since there are no higher order stresses at $n=0$. For
$n\geq 1$ and any value of $p$, the amplitude of $\mathring{R}_{q,n,p}$ is
given by

$\displaystyle\delta_{q+1,n,p}:=\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,p-1}}\cdot\prod_{n^{\prime}<n}f_{q,n^{\prime}}.$

One should view the product of $f_{q,n^{\prime}}$ terms as a negligible error,
which is justified by calculating

$\displaystyle\prod\limits_{0\leq n^{\prime}\leq{n_{\rm
max}}}f_{q,n^{\prime}}$ $\displaystyle=\left(\frac{\lambda_{q,{n_{\rm
max}}+1,0}}{\lambda_{q,1,0}}\right)^{\frac{1}{{p_{\rm
max}}}}\leq\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\frac{1}{{p_{\rm
max}}}}$ (2.1)

and assuming that ${p_{\rm max}}$ is large.

######  Definition 2.5 (Parameters Introduced in Section 2.5).

  1. (1)

$\varepsilon_{\Gamma}$ \- A very small positive number.

  2. (2)

$\displaystyle{\Gamma_{q+1}=\left(\lambda_{q+1}\lambda_{q}^{-1}\right)^{\varepsilon_{\Gamma}}}$
\- A parameter which will be used to quantify deviations in amplitude. In
particular, $\Gamma_{q}$ will be used to quantify amplitudes of both velocity
fields and (higher-order) stresses.

###  2.2 Inductive assumptions

For every non-negative integer $q$ we will construct a solution
$(v_{q},p_{q},\mathring{R}_{q})$ to the Euler-Reynolds system

$\displaystyle\partial_{t}v_{q}+\mathrm{div\,}(v_{q}\otimes v_{q})+\nabla
p_{q}$ $\displaystyle=\mathrm{div\,}\mathring{R}_{q}$ (2.2a)
$\displaystyle\mathrm{div\,}v_{q}$ $\displaystyle=0\,.$ (2.2b)

Here $\mathring{R}_{q}$ is assumed to be a trace-free symmetric matrix. The
relative size of the approximate solution $v_{q}$ and the Reynolds stress
error $\mathring{R}_{q}$ will be measured in terms of the frequency parameter
$\lambda_{q}$ and the amplitude parameter $\delta_{q}$, which are defined in
Definition 2.2. We will propagate the following basic inductive estimates on
$(v_{q},\mathring{R}_{q})$:111111By $\left\|v_{q}\right\|_{H^{1}}$, we
actually mean $\left\|v_{q}\right\|_{C^{0}_{t}H^{1}_{x}}$. Similarly,
$\|\mathring{R}_{q}\|_{L^{1}}$ stands for
$\|\mathring{R}_{q}\|_{C^{0}_{t}L^{1}_{x}}$. Unless stated explicitly
otherwise, all the norms used in this paper represent analogous uniform in
time estimates and will be abbreviated as such.

$\displaystyle\left\|v_{q}\right\|_{H^{1}}$
$\displaystyle\leq\delta_{q}^{\frac{1}{2}}\lambda_{q}$ (2.3)
$\displaystyle\|\mathring{R}_{q}\|_{L^{1}}$ $\displaystyle\leq\delta_{q+1}.$
(2.4)

We shall see later that in order to build solutions belonging to
$\dot{H}^{\beta}$ for $\beta$ approaching $\frac{1}{2}$, we must propagate
additional estimates on higher order material and spatial derivatives of both
$v_{q}$ and $\mathring{R}_{q}$ in $L^{2}$ and $L^{1}$, respectively. Roughly
speaking, every spatial derivative on either $v_{q}$ or $\mathring{R}_{q}$
costs a factor of $\lambda_{q}$. Additional material derivatives are more
delicate and will be discussed further in Section 2.5, but for the time being,
one may imagine that each material derivative
$D_{t,q}:=\partial_{t}+v_{q}\cdot\nabla$ on $v_{q}$ or $\mathring{R}_{q}$
costs a factor of $\tau_{q}^{-1}$.

###  2.3 Intermittent pipe flows

Pipe flows, both homogeneous and intermittent, have proven to be one of the
most useful components of many convex integration schemes. Homogeneous pipe
flows were introduced first by Daneri and Székelyhidi Jr. [27]. The
prototypical pipe flow in the $\vec{e}_{3}$ direction is constructed using a
smooth function $\rho:\mathbb{R}^{2}\rightarrow\mathbb{R}$ which is compactly
supported, for example in a ball of radius $1$ centered at the origin, and has
zero mean. Letting $\varrho:\mathbb{T}^{2}\rightarrow\mathbb{R}$ be the
$\mathbb{T}^{2}$-periodized version of $\rho$, the $\mathbb{T}^{3}$-periodic
pipe flow $\mathbb{W}:\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ is defined as

$\mathbb{W}(x_{1},x_{2},x_{3})=\varrho(x_{1},x_{3}){e}_{2}\,.$ (2.5)

It is immediate that $\mathbb{W}$ is divergence-free and a stationary solution
to the Euler equations. Pipe flows such as $\mathbb{W}$ have been used in
convex integration schemes which produce solutions in $L^{\infty}$-based
spaces [27, 43, 11]. At the $q^{\textnormal{th}}$ stage of the iteration, the
$\frac{\mathbb{T}^{3}}{\lambda_{q+1}}$-periodized pipe flow
$\mathbb{W}\left(\lambda_{q+1}\cdot\right)$ is used to construct the
perturbation.

By contrast, intermittent pipe flows are _not_ spatially homogeneous.
Intermittency in the context of convex integration schemes was introduced by
the first and last authors in [13] via _intermittent Beltrami flows_ , which
are defined via their Fourier support and may be likened to modified and
renormalized Dirichlet kernels. Intermittent pipe flows were introduced by
Modena and Székelyhidi Jr. in the context of the transport and transport-
diffusion equation [52] and have also been utilized for the higher dimensional
(at least four dimensional121212In three dimensions, intermittent pipe flows
are not sufficiently sparse to handle the error term arising from the
Laplacian. This issue was addressed by Colombo and the first and last authors
in [7] through the usage of _intermittent jets_ , and similar objects have
been used in subsequent papers as well (see work of Brue, Colombo, and De
Lellis [5], Cheskidov and Luo [17, 18]).) Navier-Stokes equations [49, 64].
The precise objects we use are defined in (4.10) in Proposition 4.4, but let
us briefly describe some of their important attributes. The intermittency is
quantified by the parameter $r_{q+1,n}\ll 1$. Let
$\rho_{r_{q+1,n}}\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by
$\rho_{r_{q+1,n}}(\cdot)=\rho\left(\frac{\cdot}{r_{q+1,n}}\right)$, and let
$\varrho_{r_{q+1,n}}$ be the $\mathbb{T}^{2}$-periodized version of
$\rho_{r_{q+1,n}}$. Thus one can see that $r_{q+1,n}$ describes the _thickness
of the pipes at unit scale._ In order to make the intermittent pipe flows of
unit size in $L^{2}(\mathbb{T}^{3})$, one must multiply by a factor of
$r_{q+1,n}^{-1}$, meaning that the Lebesgue norms of the resulting object
$\mathbb{W}_{r_{q+1,n}}$ scale like

$\left\|\mathbb{W}_{r_{q+1,n}}\right\|_{L^{p}(\mathbb{T}^{3})}\sim
r_{q+1,n}^{\frac{2}{p}-1}.$ (2.6)

Let $\mathbb{W}_{q+1,n}$ be the
$\frac{\mathbb{T}^{3}}{(r_{q+1,n}\lambda_{q+1})}$-periodic version of
$\mathbb{W}_{r_{q+1,n}}$. Notice that this implies that the thickness of the
pipes comprising $\mathbb{W}_{q+1,n}$ is of order $\lambda_{q+1}^{-1}$ for all
$n$, and that the Lebesgue norms of the periodized object $\mathbb{W}_{q+1,n}$
depend only on $r_{q+1,n}$. Per Definition 2.3, the thickness of the pipes
used in the perturbation at stage $q+1$ will be quantified by

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

This choice will be jusified upon calculation of the heuristic bounds.

Figure 2: A pipe flow $\mathbb{W}_{q+1,n}$ which is periodized to scale
$(\lambda_{q+1}r_{q+1,n})^{-1}=\lambda_{q,n}^{-1}$ is placed in a direction
parallel to the $e_{2}$ axis. Upon taking into account periodic shifts, we
note that there are $r_{q+1,n}^{-2}$ many options to place this pipe. This
degree of freedom will be used later, see e.g. Figure 7.

####  2.3.1 Lagrangian coordinates, intermittency, and placements

In order to achieve the optimum regularity $\beta$, we will define the pipe
flows which comprise the perturbation at stage $q+1$ in Lagrangian coordinates
corresponding to the velocity field $v_{q}$. Due to the inherent instability
of Lagrangian coordinates over timescales longer than that dictated by the
Lipschitz norm of the velocity field, there will be many sets of coordinates
used in different time intervals which are then patched together using a
partition of unity. This technique has been used frequently in recent convex
integration schemes, beginning with work of Isett [41], the first author, De
Lellis, and Székelyhidi Jr. [10], and Isett, the first author, De Lellis, and
Székelyhidi Jr. [8], but perhaps most notably in the proof of the Onsager
conjecture by Isett [43] and the subsequent strengthening to dissipative
solutions by the first and last authors, De Lellis, and Székelyhidi Jr. [11].

The proof of Onsager’s conjecture employs the gluing technique to prevent pipe
flows defined using different Lagrangian coordinate systems from overlapping.
The intermittent quality of our building blocks, and thus the approximate
solution $v_{q}$, appears to obstruct the successful implementation of the
gluing technique, since gluing requires a sharp control on the global
Lipschitz norm of the velocity field which will be unavailable. Thus, we
cannot use the gluing technique and must control in a different fashion the
possible interactions between two intermittent pipe flows defined using
different Lagrangian coordinate systems.

To control these interactions, we have introduced a _placement technique_ (cf.
Proposition 4.8) which is used to completely prevent all such interactions.
This placement technique is predicated on a simple observation about
intermittent pipe flows, which to our knowledge has not yet been used in any
convex integration schemes to date. When the diameter of the pipe at unit
scale is of size $r_{q+1,n}$, there are $(r_{q+1,n})^{-2}$ disjoint choices
for the support of pipe. These choices simply correspond to shifting the
intersection of the axis of the pipe in the plane which is perpendicular to
the axis, cf. Proposition 4.3. This degree of freedom is unaffected by
periodization and is depicted in Figure 2 for a
$\frac{\mathbb{T}^{3}}{\lambda_{q+1}r_{q+1,n}}$-periodic intermittent pipe
flow $\mathbb{W}_{q+1,n}$. We will exploit this degree of freedom to choose
placements for each set of pipes which _entirely avoid_ other sets of pipes on
small discretized regions of space-time. The space-time discretization is made
possible through the usage of cutoff functions which will be discussed in more
detail later in Section 2.5. We remark that De Lellis and Kwon [28] have
introduced a placement technique in the context of $C^{\alpha}$, globally
dissipative solutions to the 3D Euler equations which is predicated on
restricting the timescale of the Lagrangian coordinate systems to be
significantly shorter than the Lipschitz timescale. This restriction
significantly limits the regularity of the final solution and is thus not
suited for a intermittent scheme aimed at $H^{\frac{1}{2}-}$ regularity.

###  2.4 Higher order stresses

####  2.4.1 Regularity beyond $\nicefrac{{1}}{{3}}$

The resolution of the flexible side of the Onsager conjecture in [43] and [11]
mentioned previously shows that given some prescribed regularity index
$\beta\in(0,\frac{1}{3})$, one can construct dissipative weak solutions $u$ in
$C^{\beta}$. Conversely, following on partial work by Eyink [36], Constantin,
E, and Titi [22] have proven that conservation of energy in the Euler
equations requires only that $u\in
L^{3}_{t}\left(B^{\alpha}_{3,\infty}\right)$ for $\alpha>\nicefrac{{1}}{{3}}$.
This leaves open the possibility of building dissipative weak solutions with
more than $\frac{1}{3}$-many derivatives in $L^{p}\left(\mathbb{T}^{3}\right)$
(uniformly in time in our case) for $p<3$.

Let us present a heuristic estimate which indicates a regularity limit of
$H^{\frac{1}{2}}$ for solutions produced via convex integration schemes. For
this purpose, let us focus on one of the principal parts of the stress in an
intermittent convex integration schemes (for the familiar reader, this is part
of the oscillation error). The perturbations include a coefficient function
$a$ which depends on $\mathring{R}_{q}$ and thus for which derivatives cost
$\lambda_{q}$ and which has amplitude $\delta_{q+1}^{\nicefrac{{1}}{{2}}}$
(the square root of the amplitude of the stress). These coefficient functions
are multiplied by intermittent pipe flows $\mathbb{W}_{q+1,0}$ for which
derivatives cost $\lambda_{q+1}$ and which have unit size in $L^{2}$, but are
only periodized to scale $\left(\lambda_{q+1}r_{q+1,0}\right)^{-1}$. When the
divergence lands on the square of the coefficient function $a^{2}$ in the
nonlinear term, the resulting error term satisfies the estimate

$\displaystyle\left\|\mathrm{div\,}^{-1}\left(\nabla(a^{2})\mathbb{P}_{\neq
0}(\mathbb{W}_{q+1,0}\otimes\mathbb{W}_{q+1,0})\right)\right\|_{L^{1}}\leq\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q+1}r_{q+1,0}}.$
(2.7)

The numerator is the size of $\nabla(a^{2})$ in $L^{1}$, while the denominator
is the gain induced by inverting the divergence at $\lambda_{q+1}r_{q+1,0}$,
which is the minimum frequency of $\mathbb{P}_{\neq
0}(\mathbb{W}_{q+1,0}\otimes\mathbb{W}_{q+1,0})=\mathbb{W}_{q+1,0}\otimes\mathbb{W}_{q+1,0}-\fint_{\mathbb{T}^{3}}\mathbb{W}_{q+1,0}\otimes\mathbb{W}_{q+1,0}$.
Note that we have used implicitly that $\mathbb{W}_{q+1,0}$ has unit $L^{2}$
norm, and that by periodicity $\mathbb{P}_{\neq
0}(\mathbb{W}_{q+1,0}\otimes\mathbb{W}_{q+1,0})$ decouples from
$\nabla(a^{2})$. This error would be minimized when $r_{q+1,0}=1$, in which
case

$\displaystyle\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q+1}}<\delta_{q+2}$
$\displaystyle\iff\lambda_{q+1}^{-2\beta+\frac{1}{b}}<\lambda_{q+1}^{-2\beta
b+1}$ $\displaystyle\iff 2\beta b^{2}-2\beta b<b-1$ $\displaystyle\iff 2\beta
b(b-1)<b-1$ $\displaystyle\iff\beta<\frac{1}{2b}.$ (2.8)

Any intermittency parameter $r_{q+1,0}\ll 1$ would weaken this estimate since
the gain induced from inverting the divergence will only be
$\lambda_{q+1}r_{q+1,0}\ll\lambda_{q+1}$. On the other hand, we will see that
a small choice of $r_{q+1,0}$ _strengthens all other error terms_ , and
because of this, in our construction we will choose $r_{q+1,0}$ as in
Definition 2.3, item (3). One may refer to the blog post of Tao [64] for a
slightly different argument which reaches the same apparent regularity limit.
This apparent regularity limit is independent of dimension, and we believe
that the method in this paper can not be modified to yield weak solutions with
regularity $L^{\infty}_{t}W^{s,p}_{x}$ with $s>1/2$, for any $p\in[1,2]$.

The higher order stresses mentioned in Section 1.2 will compensate for the
losses incurred in this nonlinear error term when $r_{q+1,0}\ll 1$. As we
shall describe in the next section, we use the phrase “higher order stresses”
to describe errors which are higher in frequency and smaller in amplitude than
$\mathring{R}_{q}$, but not sufficiently small enough or at high enough
frequency to belong to $\mathring{R}_{q+1}$. Similarly, “higher order
perturbations” are used to correct the higher order stresses and thus increase
the extent to which an approximate solution solves the Euler equations.

####  2.4.2 Specifics of the higher order stresses

In convex integration schemes which measure regularity in $L^{\infty}$ (i.e.
using Hölder spaces $C^{\alpha}$), pipe flows interact through the
nonlinearity to produce low ($\approx\lambda_{q}$) and high
($\approx\lambda_{q+1}$) frequencies. We denote by $w_{q+1,0}$ the
perturbation designed to correct $\mathring{R}_{q}$. In the absence of
intermittency, the low frequencies from the self-interaction of $w_{q+1,0}$
cancel the Reynolds stress error $\mathring{R}_{q}$, and the high frequencies
are absorbed by the pressure up to an error small enough to be placed in
$\mathring{R}_{q+1}$. In an intermittent scheme, the self-interaction of the
intermittent pipe flows comprising $w_{q+1,0}$ produces low, intermediate, and
high frequencies. The low and high frequencies play a similar role as before.
However, the intermediate frequencies cannot be written as a gradient, nor are
small enough to be absorbed in $\mathring{R}_{q+1}$. This issue has limited
the available regularity on the final solution in many previous intermittent
convex integration schemes. In order to reach the threshold $H^{\frac{1}{2}}$,
we address this issue using higher order Reynolds stress errors
$\mathring{R}_{q,n}$ for $n=1,2,\dots,{{n_{\rm max}}}$, cf. Figure 3.

Figure 3: Adding the increment $w_{q+1,0}$ corrects the stress
$\mathring{R}_{q,0}=\mathring{R}_{q}$, but produces error terms which live at
frequencies that are intermediate between $\lambda_{q}$ and $\lambda_{q+1}$,
due to the intermittency of $w_{q+1,0}$. These new errors are sorted into
higher order stresses $\mathring{R}_{q,n}$ for $1\leq n\leq{n_{\rm max}}$, as
depicted above. The heights of the boxes corresponds to the amplitude of the
errors that will fall into them, while the frequency support of each box
increases from $\lambda_{q}$ for $\mathring{R}_{q,0}=\mathring{R}_{q}$, to
$\lambda_{q+1}$ for $\mathring{R}_{q+1}$.

After the addition of $w_{q+1,0}$ to correct $\mathring{R}_{q}$, which is
labeled in Figure 4 as $\mathring{R}_{q,0}$, low frequency error terms are
produced, which we divide into higher order stresses. To correct the error
term of this type at the _lowest_ frequency, which is labeled
$\mathring{R}_{q,1}$ in Figure 4, we add a sub-perturbation $w_{q+1,1}$. The
subsequent bins are lighter in color to emphasize that they are not yet full;
that is, there are more error terms which have yet to be constructed but will
be sorted into such bins. The emptying of the bins then proceeds inductively
on $n$, as we add higher order perturbations $w_{q+1,n}$, which are designed
to correct $\mathring{R}_{q,n}$. For $1\leq n\leq{n_{\rm max}}$, the frequency
support of $\mathring{R}_{q,n}$ is131313In reality, the higher order stresses
are not compactly supported in frequency. However, they will satisfy
derivative estimates to very high order which are characteristic of functions
with compact frequency support.

$\left\\{k\in{\mathbb{Z}}^{3}\colon\lambda_{q,n,0}\leq|k|<\lambda_{q,n+1,0}\right\\}.$
(2.9)

This division will be justified upon calculation of the heuristic bounds in
Section 2.7.

Figure 4: Adding $w_{q+1,n}$ to correct $\mathring{R}_{q,n}$ produces error
terms which are distributed among the Reynolds stresses
$\mathring{R}_{q,n^{\prime}}$ for $n+1\leq n^{\prime}\leq{n_{\rm max}}$.

Let us now explain the motivation for the division of $\mathring{R}_{q,n}$
into the further subcomponents $\mathring{R}_{q,n,p}$. Suppose that we add a
perturbation $w_{q+1,n}$ to correct $\mathring{R}_{q,n}$ for $n\geq 1$. The
amplitude of $w_{q+1,n}$ would depend on the amplitude of
$\mathring{R}_{q,n}$, which in turn depends on the gain induced by inverting
the divergence to produce $\mathring{R}_{q,n}$, which depends then on the
minimum frequency $\lambda_{q,n,0}$. However, derivatives on the low frequency
coefficient function used to define $w_{q+1,n}$ would depend on the maximum
frequency of $\mathring{R}_{q,n}$, which is $\lambda_{q,n+1,0}$. The (sharp-
eyed) reader may at this point object that the first derivative on the low-
frequency coefficient function $\nabla(a(\mathring{R}_{q,n}))$ should be
cheaper, since $\mathring{R}_{q,n}$ is obtained from inverting the divergence,
and taking the gradient of the cutoff function written above should thus
morally involve bounding a zero-order operator. However, constructing the low-
frequency coefficient function presents technical difficulties which prevent
us from taking advantage of this intuition. In fact, the failure of this
intuition is the sole reason for the introduction of the parameter $p$, as one
may see from the heuristic estimates later. In any case, increasing the
regularity $\beta$ of the final solution requires minimizing this gap between
the gain in amplitude provided by inverting the divergence and the cost of a
derivative, and so we subdivide $\mathring{R}_{q,n}$ into further components
$\mathring{R}_{q,n,p}$ for $1\leq p\leq p_{max}$.141414There are certainly a
multitude of ways to manage the bookkeeping for amplitudes and frequencies.
Using both $n$ and $p$ is convenient because then $n$ is the only index which
quantifies the rate of periodization. Both ${{n_{\rm max}}}$ and ${p_{\rm
max}}$ are fixed independently of $q$. Each component $\mathring{R}_{q,n,p}$
then will have frequency support in the set

$\left\\{k\in{\mathbb{Z}}^{3}\colon\lambda_{q,n,p-1}\leq|k|<\lambda_{q,n,p}\right\\}=\left\\{k\in{\mathbb{Z}}^{3}\colon\lambda_{q,n,0}f_{q,n}^{p-1}\leq|k|<\lambda_{q,n,0}f_{q,n}^{p}\right\\}.$
(2.10)

Notice that by the definition of $f_{q,n}$ in Definition 2.4, 2.10 defines a
partition of the frequencies in between $\lambda_{q,n,0}$ and
$\lambda_{q,n+1,0}$ for $1\leq p\leq{p_{\rm max}}$. Figure 5 depicts this
division, and we shall describe in the heuristic estimates how each
subcomponent $\mathring{R}_{q,n,p}$ is corrected by $w_{q+1,n,p}$, with all
resulting errors absorbed into either $\mathring{R}_{q+1}$ or
$\mathring{R}_{q,n^{\prime}}$ for $n^{\prime}>n$.

Figure 5: The higher order stress $\mathring{R}_{q,n}$ is decomposed into
components $\mathring{R}_{q,n,p}$, which increase in frequency and decrease in
amplitude as $p$ increases. We use the base of the red boxes to indicate
support in frequency, where frequency is increasing from left to right, and
the height to indicate amplitudes. Each subcomponent $\mathring{R}_{q,n,p}$ is
corrected by its own corresponding sub-perturbation $w_{q+1,n,p}$, which has a
commensurate frequency and amplitude.

Thus, the net effect of the higher order stresses is that one may take errors
for which the inverse divergence provides a weak estimate due to the presence
of relatively low frequencies and push them to higher frequencies for which
the inverse divergence estimate is stronger. We will repeat this process until
all errors are moved (almost) all the way to frequency $\lambda_{q+1}$, at
which point they are absorbed into $\mathring{R}_{q+1}$. Heuristically, this
means that in constructing the perturbation $w_{q+1}$ at stage $q$, we have
_eliminated_ all the higher order error terms which arise from self-
interactions of intermittent pipe flows, thus producing a solution $v_{q+1}$
to the Euler-Reynolds system at level $q+1$ which is _as close as possible_ to
a solution of the Euler equations. We point out that one side effect of the
higher order perturbations is that the total perturbation $w_{q+1}$ has
spatial support which is _not_ particularly sparse, since as $n$ increases the
perturbations $w_{q+1,n}$ become successively less intermittent and thus more
homogeneous. At the same time, the frequency support of our solution is also
not too sparse, since $b$ is close to $1$ and
$r_{q+1,0}=\left(\lambda_{q}\lambda_{q+1}^{-1}\right)^{\frac{4}{5}}$, so that
many of the frequencies between $\lambda_{q}$ and $\lambda_{q+1}$ are active.

###  2.5 Cut-off functions

####  2.5.1 Velocity and stress cut-offs

The concept of a turnover time, which is proportional to the inverse of the
gradient of the mean flow $v_{q}$, is crucial to the previous convex
integration schemes mentioned earlier which utilized Lagrangian coordinates.
Since the perturbation is expected to be roughly flowed by the mean flow
$v_{q}$, the turnover time determines a timescale on which the perturbation is
expected to undergo significant deformations. An important property of pipe
flows, first noted by Daneri and Székelyhidi Jr. in [27] and utilized
crucially by Isett [43] towards the proof of Onsager’s conjecture, is that the
length of time for which pipe flows written in Lagrangian coordinates remain
approximately stationary solutions to Euler depends only on the Lipschitz norm
of the transport velocity $v_{q}$ and not the Lipschitz norms of the original
(undeformed) pipe flow. However, the timescale under which pipe flows
transported by an intermittent velocity field remain coherent is space-time
dependent, in contrast to previous convex integration schemes in which the
timescale was uniform across $\mathbb{R}\times\mathbb{T}^{3}$. As such, we
will need to introduce space-time cut-offs $\psi_{i,q}$ in order to determine
the local turnover time. In particular, the cut-off $\psi_{i,q}$ will be
defined such that

$\displaystyle\left\|\nabla
v_{q}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q})}\lesssim\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{i}:=\tau_{q}^{-1}\Gamma_{q+1}^{i}\,.$
(2.11)

With such cut-offs defined, we then define in addition a family of temporal
cut-offs $\chi_{i,k,q}$ which will be used to restrict the timespan of the
intermittent pipe flows in terms of the local turnover. Each cut-off function
$\chi_{i,k,q}$ will have temporal support contained in an interval of length

$\tau_{q}\Gamma_{q+1}^{-i}.$ (2.12)

It should be noted that we will design the cut-offs so that we can deduce much
more on its support than (2.11). Since the material derivative
$D_{t,q}:=\partial_{t}+v_{q}\cdot\nabla$ will play an important role, we will
require estimates involving material derivatives $D_{t,q}^{N}$ of very high
order.151515The loss of material derivative in the transport error means that
to produce solutions with regularity approaching $\dot{H}^{\frac{1}{2}}$, we
have to propagate material derivative estimates of arbitrarily high order on
the stress. We expect the cost of a material derivative to be related to the
turnover time, which itself is local in nature. As such, high order material
derivative estimates will be done on the support of the cut-off functions and
will be of the form

$\left\|\psi_{i,q}D_{t,q}^{N}\mathring{R}_{q,n,p}\right\|_{L^{r}}\,.$

In addition to the family of cut-offs $\psi_{i,q}$ and $\chi_{i,k,q}$, we will
also require stress cut-offs $\omega_{i,j,q,n,p}$ which determine the local
size of the Reynolds stress errors $\mathring{R}_{q,n,p}$; in particular
$\omega_{i,j,q,n,p}$ will be defined such that

$\displaystyle\left\|\nabla^{M}\mathring{R}_{q,n,p}\right\|_{L^{\infty}(\mathrm{supp\,}\omega_{i,j,q,n,p})}\leq\delta_{q+1,n,p}\Gamma_{q+1}^{2j}\lambda_{q,n,p}^{M}\,.$
(2.13)

Previous intermittent convex integration schemes have managed to successfully
cancel intermittent stress terms with much simpler stress cutoff functions
than the ones we use. However, mitigating the loss of spatial derivative in
the oscillation error means that we have to propagate sharp spatial derivative
estimates of arbitrarily high order on the stress in order to produce
solutions with regularity approaching $\dot{H}^{\frac{1}{2}}$. Due to this
requirement, we then have to estimate the _second_ derivative (and higher) of
the stress cutoff function

$\left\|\nabla^{2}\left(\omega^{2}\left(\mathring{R}_{q,n,p}\right)\right)\right\|_{L^{1}}\,,$

which in turn necessitates bounding the local $L^{2}$ norm of
$\nabla\mathring{R}_{q,n,p}$ due to the term

$\left\|\left(\nabla^{2}(\omega^{2})\right)\left(\mathring{R}_{q,n,p}\right)\;\left|\nabla\mathring{R}_{q,n,p}\right|^{2}\right\|_{L^{1}}\,.$

Given inductive estimates about the derivatives of $\mathring{R}_{q}$ _only
in_ $L^{1}$ which have not been upgraded to $L^{p}$ for $p>1$, this term will
obey a fatally weak estimate, which is why we must estimate
$\mathring{R}_{q,n,p}$ in $L^{\infty}$ as in (2.13).

####  2.5.2 Checkerboard cut-offs

As mentioned in the discussion of intermittent pipe flows, we must prevent
pipes originating from different Lagrangian coordinate systems from
intersecting. The first step is to reduce the complexity of this problem by
restricting the size of the spatial domain on which intersections must be
prevented. Towards this end, consider the maximum frequency of the original
stress $\mathring{R}_{q}=\mathring{R}_{q,0}$, or any of the higher order
stresses $\mathring{R}_{q,n}$ for $n\geq 1$. We may write these frequencies as
$\lambda_{q+1}r_{1}$ for $\lambda_{q}\lambda_{q+1}^{-1}\leq r_{1}<1$. We then
decompose $\mathring{R}_{q,n}$ using a checkerboard partition of unity
comprised of bump functions which follow the flow of $v_{q}$ and have support
of diameter $\left(\lambda_{q+1}r_{1}\right)^{-1}$. These two properties
ensure that we have _preserved_ the derivative bounds on $\mathring{R}_{q,n}$.
Thus, we fix the set $\Omega$ to be the support of an individual checkerboard
cutoff function in this partition of unity at a fixed time, cf. (4.28).

Suppose furthermore that $\Omega$ is inhabited by disjoint sets of deformed
intermittent pipe flows which are periodized to spatial scales no finer than
$\left(\lambda_{q+1}r_{2}\right)^{-1}$ for $0<r_{1}<r_{2}<1$. In practice,
$r_{2}$ will be $r_{q+1,n}$, where $r_{q+1,n}$ is the amount of intermittency
used in the pipes which comprise the perturbation $w_{q+1,n}$ which is used to
correct $\mathring{R}_{q,n}$. The pipes which already inhabit $\Omega$ may
first be from previous generations of perturbations $w_{q+1,n^{\prime}}$ for
$n^{\prime}<n$, in which case they are periodized to spatial scales much
broader than $\left(\lambda_{q+1}r_{2}\right)^{-1}$, or from an overlapping
checkerboard cutoff function used to decompose $\mathring{R}_{q,n}$ on which a
placement of pipes periodized to spatial scale
$\left(\lambda_{q+1}r_{2}\right)^{-1}$ has already been chosen. In either
case, these pipes will have been deformed by the velocity field $v_{q}$ on the
time-scale given by the inverse of the local Lipschitz norm. We represent the
support of these deformed pipe flows in terms of axes
$\\{A_{i}\\}_{i\in\mathcal{I}}$ around which the pipes
$\\{P_{i}\\}_{i\in\mathcal{I}}$ are concentrated to thickness
$\lambda_{q+1}^{-1}$ (recall from Section 2.3 that all intermittent pipe flows
used in our scheme have this thickness).

We will now explain that one may choose a new set of (straight, i.e. not
deformed) intermittent pipe flows $\mathbb{W}_{r_{2},\lambda_{q+1}}$
periodized to scale $\left(\lambda_{q+1}r_{2}\right)^{-1}$ which are disjoint
from each deformed pipe $P_{i}$ and _on the support of_ $\Omega$ and _under
appropriate restrictions on $r_{1}$ and $r_{2}$_. Heuristically, this task
becomes easier when $r_{2}$ is smaller, since this means both that we have
more choices of placement for the new set, and there are less pipes $P_{i}$
inhabiting $\Omega$. Conversely, this task becomes more difficult when $r_{1}$
is smaller, since then $\Omega$ is larger and will contain more pipes $P_{i}$.
We assume throughout that the deformations of the $P_{i}$’s are mild enough to
preserve the expected length, curvature, and spacing bounds between
neighboring pipes that arise from writing pipes in Lagrangian coordinates and
flowing for a length of time which is strictly less than the inverse of the
Lipschitz norm of the velocity field.

First, we can estimate the cardinality of the set $\mathcal{I}$ (which indexes
the axes $A_{i}$ and pipes $P_{i}$) from above by $r_{2}^{2}r_{1}^{-2}$. To
understand this bound, first note that if we had _straight_ pipes $P_{i}$
periodized to scale $\left(\lambda_{q+1}r_{2}\right)^{-1}$ inhabiting a _cube_
of side length $\left(\lambda_{q+1}r_{1}\right)^{-1}$, this bound would hold.
Using the fact that our deformed pipes obey similar length, curvature, and
spacing bounds as straight pipes and that our set $\Omega$ can be considered
as a subset of a cube with side length proportional to
$\left(\lambda_{q+1}r_{1}\right)^{-1}$, the same bound will hold up to
dimensional constants. Secondly, by the intermittency of the desired set of
new pipes, we have $r_{2}^{-2}$ choices for the placement of the new set, as
indicated in Figure 2.

To finish the argument, we must estimate how many of these $r_{2}^{-2}$
choices would lead to non-empty intersections between the new pipes and any
$P_{i}$. To calculate this bound, we will imagine the placement of the new set
of straight pipes as occurring on a two-dimensional plane which is
perpendicular to the axes of the pipes. After projecting each $P_{i}$ onto
this two-dimensional plane, our task is to choose the intersection points of
the new pipes with the plane so that the new pipes do not intersect the
shadows of the $P_{i}$’s.

Figure 6: In the figure on the left we display $\mathbb{T}^{3}$, in which we
have: in green, a set of pipe flows (old generation, very sparse) that were
deformed by $v_{q}$; in blue, the support of a cutoff function
$\zeta_{q,i,k,n,\vec{l}}$, whose diameter is
$\approx(\lambda_{q+1}r_{1})^{-1}$. Due to the sparseness, very few (if any!)
of these green pipes intersect the blue region. The figure on the right
further zooms into the blue region, to emphasize its contents. On the support
of $\zeta_{q,i,k,n,\vec{l}}$ we have displayed two sets of deformed pipe
flows, in pink and orange. These pipes flows were deformed also by $v_{q}$,
from a nearby time at which they were straight and periodic at scale
$(\lambda_{q+1}r_{2})^{-1}$. At the current time, at which the above figure is
considered, these pipe flows aren’t quite periodic anymore, but they are
close. The question now is: can we place a straight pipe flow, periodic at
scale $(\lambda_{q+1}r_{2})^{-1}$, whose axis is orthogonal to the front face
of the blue box (pictured in black), and which does not intersect any of the
existing pipes in this region? To see that this is possible, in Figure 7 we
estimate the area of shadows on this face of the cube. Figure 7: As mentioned
in the caption of Figure 6, we consider the image on the right and take the
projection of all pipes present in the blue box (green, pink, orange), onto
the front face of the cube (parallel to the $e_{3}-e_{1}$ plane). Because
these existing pipes were bent by $v_{q}$, the shadow does not consist of
straight lines, and in fact the projections can overlap. By estimating the
area of this projection, we see that if $r_{2}^{4}\ll r_{1}^{3}$ then there is
enough room left to insert a new pipe flow with orientation axis $e_{2}$
(represented by the black disks in the above figure), which will not intersect
any of the projections of the existing pipes, and thus not intersect the
existing pipes themselves.

Given one of the deformed pipes $P_{i}$, since its thickness is
$\lambda_{q+1}^{-1}$ and its length inside $\Omega$ is proportional to the
diameter of $\Omega$, specifically $\left(\lambda_{q+1}r_{1}\right)^{-1}$, we
may cover the shadow of $P_{i}$ on the plane with $\approx r_{1}^{-1}$ many
balls of diameter $\lambda_{q+1}^{-1}$. Covering all the $P_{i}$’s thus
requires $\approx r_{2}^{2}r_{1}^{-2}\cdot r_{1}^{-1}$ balls of diameter
$\lambda_{q+1}^{-1}$. Now, imagine the intersection of the new set of pipes
with the plane. Each choice of placement defines this intersection as
essentially a set of balls of diameter $\approx\lambda_{q+1}^{-1}$ equally
spaced at distance $\left(\lambda_{q+1}r_{2}\right)^{-1}$. The intermittency
ensures that there are $r_{2}^{-2}$ disjoint choices of placement, i.e.
$r_{2}^{-2}$ disjoint sets of balls which represent the intersection of a
particularly placed new set of pipes with the plane. As long as

$r_{2}^{2}r_{1}^{-2}\cdot r_{1}^{-1}\ll r_{2}^{-2}\quad\iff r_{2}^{4}\ll
r_{1}^{3}$

there must exist at least one choice of placement which does not produce _any_
intersections between $\mathbb{W}_{r_{2},\lambda_{q+1}}$ and the $P_{i}$’s.
Notice that if $r_{1}$ is too small or if $r_{2}$ is too large, this
inequality will not be satisfied, thus validating our previous heuristics
about $r_{1}$ and $r_{2}$.

To obey the relative intermittency inequality between $r_{1}$ and $r_{2}$
derived above for placements of new intermittent pipes on sets of a certain
diameter, we will utilize cutoff functions

$\zeta_{q,i,k,n,\vec{l}}$

which are defined using a variety of parameters. The index $q$ describes the
stage of the convex integration scheme, while $i$ and $k$ refer to the
velocity and temporal cutoffs defined above. The parameter $n$ corresponds to
a higher order stress $\mathring{R}_{q,n}$ and refers to its minimum frequency
$\lambda_{q,n,0}$, quantifying the value of $(\lambda_{q+1}r_{1})^{-1}$ and
the diameter of the support as described earlier. The parameter
$\vec{l}=(l,w,h)$ depends on $q$ and $n$ and provides an enumeration of the
(three-dimensional) checkerboard covering $\mathbb{T}^{3}$ at scale
$\left(\lambda_{q,n,0}\right)^{-1}$. On the support of one of these
checkerboard cutoff functions, we can inductively place pipes periodized to
scale $\left(\lambda_{q+1}r_{2}\right)^{-1}=\lambda_{q,n}^{-1}$ which are
disjoint. The checkerboard cutoff functions and the pipes themselves all
follow the same velocity field, and so ensuring the disjointness at a single
time slice is sufficient.

####  2.5.3 Cumulative cut-off function

Finally, the variety of cut-offs described above will be combined into the
family of cut-offs

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

which have timespans of $\tau_{q}\Gamma_{q+1}^{-i}$ and $L^{2}$ norms

$\left\|\eta_{i,j,k,q,n,p,\vec{l}}\right\|_{L^{2}}\lesssim\Gamma^{-\frac{i}{2}}_{q+1}\cdot\Gamma^{-\frac{j}{2}}_{q+1}$
(2.14)

We will also require a cut-off $\eta_{i\pm,j\pm,k\pm,q,n,p,\vec{l}}$ which is
defined to be $1$ on the support of $\eta_{i,j,k,q,n,\vec{l}}$ and satisfies
the estimate

$\left\|\eta_{i\pm,j\pm,k\pm,q,n,\vec{l}}\right\|_{L^{2}}\lesssim\Gamma^{-\frac{i}{2}}_{q+1}\cdot\Gamma^{-\frac{j}{2}}_{q+1}.$
(2.15)

We remark that (2.14) and (2.15) are only heuristics (see Lemma 6.41 for the
precise estimate). Designing the cut-offs turned out to be for the authors
perhaps the most significant technical challenge of the paper. Their
definition will be inductive and estimates involving them will involve several
layers of induction.

###  2.6 The perturbation

The intermittent pipe flows of Section 2.3, the higher order stresses of
Section 2.4, and the cut-off functions of Section 2.5 provide the key
ingredients in the construction of the perturbation

${w_{q+1}:=\sum_{n=0}^{n_{\rm max}}\sum_{p=1}^{p_{\rm
max}}w_{q+1,n,p}}:=\sum_{n=0}^{n_{\rm max}}w_{q+1,n}.$

In the above double sum, we will adopt the convention that $w_{q+1,0,p}=0$
unless $p=1$ to streamline notation. Let us emphasize that $w_{q+1}$ is
constructed _inductively_ on $n$ for the following reason. Each perturbation
$w_{q+1,n}=\sum_{p=1}^{p_{\rm max}}w_{q+1,n,p}$ will contribute error terms to
all higher order stresses $\mathring{R}_{q,{\widetilde{n}},p}$ for
${\widetilde{n}}>n$ and $1\leq p\leq{p_{\rm max}}$, and so
$\mathring{R}_{q,{\widetilde{n}}}=\sum_{p=1}^{p_{\rm
max}}\mathring{R}_{q,{\widetilde{n}},p}$ is not a well-defined object until
each $w_{q+1,n^{\prime}}$ has been constructed for all $n^{\prime}<n$. For the
purposes of the following heuristics, we will abbreviate the cutoff functions
by $a_{n,p}$, and ignore summation over many of the indexes which parametrize
the cutoff functions, as they are not necessary to understand the heuristic
estimates. We will freely use the heuristic that the cutoff functions allow us
to use the $L^{\infty}_{t}H^{1}_{x}$ norm of $v_{q}$ to control terms (usually
related to the turnover time) which previously required global Lipschitz
bounds on $v_{q}$.

Let $\Phi_{q,k}:\mathbb{R}\times\mathbb{T}^{3}\rightarrow\mathbb{T}^{3}$ be
the solution to the transport equation

$\partial_{t}\Phi_{q,k}+v_{q}\cdot\nabla\Phi_{q,k}=0$

with initial data given to be the identity at time $t_{k}=k\tau_{q}$. We
mention that this definition is _purely_ heuristic, since as mentioned
previously, the Lagrangian coordinate systems will have to be indexed by
another parameter which encodes the fact that $\nabla v_{q}$ is spatially
inhomogeneous.161616The actual transport maps used in the proof are defined in
Definition 6.26. For the time being let us ignore this issue. Each map
$\Phi_{q,k}$ has an effective timespan
$\tau_{q}=(\delta_{q}^{\frac{1}{2}}\lambda_{q})^{-1}$, at which point one
resets the coordinates and defines a new transport map $\Phi_{q,k+1}$ starting
from the identity. Let $\mathbb{W}_{q+1,n}$ denote the pipe flow with
intermittency $r_{q+1,n}$ periodized to scale
$\left(\lambda_{q+1}r_{q+1,n}\right)^{-1}$. The perturbation $w_{q+1,n,p}$ is
then defined heuristically by

$\displaystyle w_{q+1,n,p}(x,t)$
$\displaystyle=\sum_{k}a_{n,p}\left(\mathring{R}_{q,n,p}(x,t)\right)\left(\nabla\Phi_{{q,k}}(x,t)\right)^{-1}(x,t)\mathbb{W}_{q+1,n}(\Phi_{q,k}(x,t)).$

We have adopted the convention that
$\mathring{R}_{q}=\mathring{R}_{q,0}=\mathring{R}_{q,0,1}$ and
$\mathring{R}_{q,0,p}=0$ if $p\geq 2$. Composing with $\Phi_{q,k}$ adapts the
pipe flows to the Lagrangian coordinate system associated to $v_{q}$ so that
$(\nabla\Phi_{q,k})^{-1}\mathbb{W}_{q+1,n}(\Phi_{q,k})$ is Lie-advected and
remains divergence-free to leading order. The perturbation $w_{q+1,n,p}$ has
the following properties:

  1. (1)

The thickness (at unit scale) of the pipes on which $w_{q+1,n,p}$ is supported
depends only on $q$ and $n$ and is quantified by

$r_{q+1,n}=\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}}.$
(2.16)

Thus, the perturbations become less intermittent as $n$ increases, since the
thickness of the pipes (periodized at unit scale) becomes larger as $n$
increases. Notice that the maximum frequency of $\mathring{R}_{q,n,p}$ is
$\lambda_{q,n,p}$ for $n\geq 1$ per (2.10), and $\lambda_{q}$ for $n=0$, while
the minimum frequency of the intermittent pipe flow $\mathbb{W}_{q+1,n}$ used
to construct $w_{q+1,n,p}$ is $\lambda_{q,n}$. Referring back to Definition
2.3 and Definition 2.4, we have that for $1\leq n\leq{n_{\rm max}}$ and $1\leq
p\leq{p_{\rm max}}$,

$\lambda_{q,n,p}=\lambda_{q,n,0}^{1-\frac{p}{{p_{\rm
max}}}}\lambda_{q,n+1,0}^{\frac{p}{{p_{\rm
max}}}}\leq\lambda_{q,n+1,0}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n}\cdot\frac{5}{6}}\ll\lambda_{q}^{\left(\frac{4}{5}\right)^{n+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n+1}}=\lambda_{q,n},$

which ensures that the low frequency portion of $w_{q+1,n,p}$ decouples from
the high frequency intermittent pipe flow $\mathbb{W}_{q+1,n}$. For $n=0$, the
maximum frequency of $\mathring{R}_{q,0}=\mathring{R}_{q}$ is $\lambda_{q}$,
which is much less than $\lambda_{q,0}$ per Definition 2.3.

  2. (2)

The $L^{2}$ size of $w_{q+1,n,p}$ is equal to the square root of the $L^{1}$
norm of $\mathring{R}_{q,n,p}$, which in turn depends on the minimum frequency
of $\mathring{R}_{q,n,p}$ and will be $\delta_{q+1,n,p}$, where we define
$\delta_{q+1,0,p}=\delta_{q+1}$. For $n\geq 1$ and $1\leq p\leq{p_{\rm max}}$,
we have from Definition 2.5 that

$\displaystyle\delta_{q+1,n,p}=\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,p-1}}\prod_{n^{\prime}<n}f_{q,n^{\prime}}.$

  3. (3)

For $n\geq 1$, derivatives on the low frequency coefficient function of
$w_{q+1,n,p}$ cost the maximum frequency of $\mathring{R}_{q,n,p}$, which is
$\lambda_{q,n,p}$. For $n=0$, $\mathring{R}_{q,0}=\mathring{R}_{q}$, so that
each spatial derivative on the coefficient function of $w_{q+1,0}$ costs
$\lambda_{q}$.

  4. (4)

The transport error and Nash error created by the addition of $w_{q+1,n,p}$
are small enough to be absorbed into $\mathring{R}_{q+1}$ for every $n$ .

  5. (5)

Per Definition 2.3, the oscillation error which results from $w_{q+1,n,p}$
interacting with itself has minimum frequency

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

###  2.7 The Reynolds stress error and heuristic estimates

Note that since the relation (2.2) is linear in the Reynolds stress, replacing
$q$ with $q+1$, the right hand side can be split into three components:

$\begin{split}\mathrm{div\,}(w_{q+1}\otimes w_{q+1}+\mathring{R}_{q})\\\
\partial_{t}w_{q+1}+v_{q}\cdot\nabla w_{q+1}\\\ w_{q+1}\cdot\nabla
v_{q}~{},\end{split}$ (2.17)

which we call the _oscillation error_ , _transport error_ and _Nash error_
respectively.

####  2.7.1 Type 1 oscillation error

In this section, we sketch the heuristic estimates which justify the following
principle: the low frequency, high amplitude errors arising from the self
interaction of an intermittent pipe flow can be transferred to higher
frequencies and smaller amplitudes through the higher order stresses and
perturbations. We shall show that the following estimates are self-consistent
and allow for the constructions of solutions approaching the regularity
threshold $\dot{H}^{\frac{1}{2}}$:

$\left\|\nabla^{M}\mathring{R}_{q}\right\|_{L^{1}}\leq\delta_{q+1}\lambda_{q}^{M}$
(2.18)
$\left\|\nabla^{M}\mathring{R}_{q,n,p}\right\|_{L^{1}}\leq\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,p-1}}\prod_{n^{\prime}<n}f_{q,n^{\prime}}\lambda_{q,n,p}^{M}=\delta_{q+1,n,p}\lambda_{q,n,p}^{M}\,.$
(2.19)

The higher order stress $\mathring{R}_{q,n,p}$ is defined using the spatial
Littlewood-Paley projection operator

$\mathbb{P}_{[q,n,p]}:=\mathbb{P}_{\left[\lambda_{q,n,p-1},\lambda_{q,n,p}\right)}=\mathbb{P}_{\geq\lambda_{q,n,p-1}}\mathbb{P}_{<\lambda_{q,n,p}},$

which projects onto the frequencies from (2.10). We define
$\mathring{R}_{q,n,p}$ as follows:

$\mathring{R}_{q,n,p}:=\sum_{n^{\prime}<n}\sum\limits_{p^{\prime}=1}^{p_{\rm
max}}\mathrm{div\,}^{-1}\left(\nabla\left(a_{n^{\prime},p^{\prime}}^{2}(\mathring{R}_{q,n^{\prime},p^{\prime}})\nabla\Phi_{q,k}^{-1}\otimes\nabla\Phi_{q,k}^{-T}\right):\left(\mathbb{P}_{[q,n,p]}\left(\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}\right)\right)(\Phi_{q,k})\right).$
(2.20)

We pause here to point out an important consequence of this definition. Let
$n^{\prime}$ be fixed, and consider the right side of the above equality.
Then, due to the periodicity of $\mathbb{W}_{q+1,n^{\prime}}$ at scale
$(\lambda_{q+1}r_{q+1,n^{\prime}})^{-1}$ we have171717We denote by
$\mathbb{P}_{\neq 0}$ the operator which subtracts from a function its mean in
space.

$\displaystyle\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}$
$\displaystyle=\mathbb{P}_{=0}\left(\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}\right)+\mathbb{P}_{\neq
0}\left(\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}\right)$
$\displaystyle=\mathbb{P}_{=0}\left(\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}\right)+\mathbb{P}_{\geq\lambda_{q+1}r_{q+1,n^{\prime}}}\left(\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}\right).$

For $n^{\prime}\geq 1$, we have that

$\lambda_{q+1}r_{q+1,n^{\prime}}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n^{\prime}+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n^{\prime}+1}}\gg\lambda_{q}^{\left(\frac{4}{5}\right)^{n^{\prime}}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n^{\prime}}\cdot\frac{5}{6}}=\lambda_{q,n^{\prime}+1,0}=\lambda_{q,n^{\prime},{p_{\rm
max}}},$

where $\lambda_{q,n^{\prime}+1,0}$ is the minimum frequency of
$\mathring{R}_{q,n^{\prime}+1}=\sum_{p^{\prime}=0}^{p_{\rm
max}}\mathring{R}_{q,n^{\prime}+1,p^{\prime}}$, while for $n^{\prime}=0$ we
have that

$\lambda_{q+1}r_{q+1,0}=\lambda_{q,1}=\lambda_{q}^{\left(\frac{4}{5}\right)}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)}=\lambda_{q,1,0},$

which is the minimum frequency of $\mathring{R}_{q,1}$. Therefore, we have
shown that the error terms arising from _all_ non-zero modes of
$\mathbb{W}_{q+1,n^{\prime}}\otimes\mathbb{W}_{q+1,n^{\prime}}$ are accounted
for in the higher order stresses $\mathring{R}_{q,{\widetilde{n}}}$ for
${\widetilde{n}}>n^{\prime}$. Thus, the higher order stresses created by the
interaction of $w_{q+1,n^{\prime}}$ will be absorbed into higher order
stresses with _strictly larger_ values of $n$.

Now assuming that $\mathring{R}_{q,n^{\prime},p^{\prime}}$ and
$w_{q+1,n^{\prime},p^{\prime}}$ are well-defined for all $n^{\prime}<n$ and
$1\leq p^{\prime}\leq{p_{\rm max}}$ and using the heuristic estimates from the
previous section for $w_{q+1,n^{\prime},p^{\prime}}$, we can estimate the
component of $\mathring{R}_{q,n,p}$ coming from
$w_{q+1,n^{\prime},p^{\prime}}$ by recalling (2.20) and writing

$\displaystyle\left\|\mathring{R}_{q,n,p}\right\|_{L^{1}}$
$\displaystyle\leq\sum_{n^{\prime}<n}\frac{\delta_{q+1,n^{\prime},p^{\prime}}\lambda_{q,n^{\prime},p^{\prime}}}{\lambda_{q,n,p-1}}$
$\displaystyle=\sum_{n^{\prime}<n}\frac{\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n^{\prime},p^{\prime}-1}}\prod_{n^{{}^{\prime\prime}}<n^{\prime}}f_{q,n^{{}^{\prime\prime}}}{\lambda_{q,n^{\prime},p^{\prime}}}}{\lambda_{q,n,p-1}}$
$\displaystyle\leq\sum_{n^{\prime}<n}\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,p-1}}\prod_{n^{{}^{\prime\prime}}\leq
n^{\prime}}f_{q,n^{{}^{\prime\prime}}}$
$\displaystyle\lesssim\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,p-1}}\prod_{n^{{}^{\prime\prime}}<n}f_{q,n^{{}^{\prime\prime}}}=\delta_{q+1,n,p}\,.$

The denominator comes from the gain induced by the combination of the inverse
divergence and the Littlewood-Paley projector $\mathbb{P}_{[q,n,p]}$. The
numerator is the amplitude of
$\nabla|a_{n^{\prime},p^{\prime}}(\mathring{R}_{q,n^{\prime},p^{\prime}})|^{2}$,
computed using the chain rule and the assumption (2.19) on
$\nabla\mathring{R}_{q,n^{\prime},p^{\prime}}$. We have used that the $L^{2}$
norm of $\mathbb{W}_{q+1,n^{\prime}}$ is normalized to unit size. Any
derivatives on $\mathring{R}_{q,n,p}$ will cost $\lambda_{q,n,p}$, which is
the maximum frequency in the Littlewood-Paley projector
$\mathbb{P}_{[q,n,p]}$. Thus, all terms which will land in
$\mathring{R}_{q,n,p}$ will satisfy the correct estimates given that
$\mathring{R}_{q,n^{\prime},p^{\prime}}$ satisfies the correct estimates for
$n^{\prime}<n$ and $1\leq p^{\prime}\leq{p_{\rm max}}$. Since
$\mathring{R}_{q}=:\mathring{R}_{q,0}$ satisfies the inductive assumptions, we
can initiate this iteration at level $n=0$ while satisfying (2.18).

Now that $\mathring{R}_{q,n,p}$ satisfies the appropriate estimates, we can
correct it with a perturbation $w_{q+1,n,p}$ as described in the previous
section. As before, since $\mathbb{W}_{q+1,n}$ has minimum frequency

$\lambda_{q,n}=\lambda_{q+1}r_{q+1,n}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n+1}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n+1}}\gg\lambda_{q}^{\left(\frac{4}{5}\right)^{n}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n}\cdot\frac{5}{6}}=\lambda_{q,n+1,0}\,,$

and the minimum frequency in $\mathring{R}_{q,n+1}$ is $\lambda_{q,n+1,0}$,
every error term resulting from the self interaction of $w_{q+1,n,p}$ will be
absorbed into higher order stresses $\mathring{R}_{q,{\widetilde{n}}}$ for
${\widetilde{n}}>n$. Therefore, we can induct on $n$ to add a sequence of
perturbations $w_{q+1,n}=\sum_{p=1}^{p_{\rm max}}w_{q+1,n,p}$ such that all
nonlinear error terms are canceled by subsequent perturbations. Upon reaching
${n_{\rm max}}$ and recalling (2.1), we can estimate the final nonlinear error
term by

$\displaystyle\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q+1}r_{q+1,{n_{\rm
max}}}}\prod_{n^{{}^{\prime}}<{n_{\rm
max}}}f_{q,n^{{}^{\prime}}}\leq\delta_{q+2}$
$\displaystyle\impliedby\delta_{q+1}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{1-\left(\frac{4}{5}\right)^{{n_{\rm
max}}+1}-\frac{1}{{p_{\rm max}}}}\leq\delta_{q+2}$
$\displaystyle\iff\lambda_{q+1}^{-2\beta}\lambda_{q+1}^{\left(\frac{1}{b}-1\right)\left(1-\left(\frac{4}{5}\right)^{{n_{\rm
max}}+1}-\frac{1}{{p_{\rm max}}}\right)}\leq\lambda_{q+1}^{-2\beta b}$
$\displaystyle\iff 2\beta
b(b-1)\leq(b-1)\left(1-\left(\frac{4}{5}\right)^{{n_{\rm
max}}+1}-\frac{1}{{p_{\rm max}}}\right)$
$\displaystyle\iff\beta\leq\frac{1}{2b}\left(1-\left(\frac{4}{5}\right)^{{n_{\rm
max}}+1}-\frac{1}{{p_{\rm max}}}\right).$

Choosing $b$ to be close to $1$ and ${n_{\rm max}}$ and ${p_{\rm max}}$
sufficiently large shows that these error terms are commensurate with
$\dot{H}^{\frac{1}{2}-}$ regularity.

####  2.7.2 Type 2 oscillation error

We now consider the second type of oscillation error, which would arise as a
result of two distinct pipes intersecting and thus serves no purpose in the
cancellation of stresses. Beginning with
$\mathring{R}_{q}=\mathring{R}_{q,0}$, we have that every derivative on
$\mathring{R}_{q,0}$ costs $\lambda_{q}$. Therefore, we may decompose
$\mathring{R}_{q,0}$ using a checkerboard partition of unity at scale
$\lambda_{q}^{-1}$. Referring back to the discussion of the checkerboard
cutoff functions, this sets the value of $r_{1}$ to be
$\lambda_{q}\lambda_{q+1}^{-1}$. Now, suppose that on a single square of this
checkerboard, we have placed a set of intermittent pipe flows
$\mathbb{W}_{q+1,0}$ which are periodized to scale
$\left(\lambda_{q+1}r_{q+1,0}\right)^{-1}$. After flowing the pipes and the
checkerboard square by $v_{q}$ for a short length of time181818The length of
time is equal to the local Lipschitz norm of $v_{q}$ on the support of the
cutoff $\psi_{i,q}$, given by the time-cutoff hidden in $a_{n,p}$., we must
place a new set of pipes $\mathbb{W}^{\prime}_{q+1,0}$ which are disjoint from
the flowed pipes $\mathbb{W}_{q+1,0}$. Given the choice of $r_{1}$, this will
be possible provided that

$\displaystyle r_{q+1,0}=r_{2}\ll r_{1}^{\frac{3}{4}}\,.$ (2.21)

Thus, the _minimum_ amount of intermittency needed to successfully place
disjoint sets of intermittent pipes is
$\left(\lambda_{q}\lambda_{q+1}^{-1}\right)^{\frac{3}{4}}$. Per Definition
2.3, our choice of $r_{q+1,0}$ is
$\left(\lambda_{q}\lambda_{q+1}^{-1}\right)^{\frac{4}{5}}$, which is then
sufficiently small.

Let us now assume that we have successfully corrected
$\mathring{R}_{q,n^{\prime}}$ for $n^{\prime}<n$, and that we wish to correct
$\mathring{R}_{q,n}=\sum_{p=1}^{p_{\rm max}}\mathring{R}_{q,n,p}$ with a
perturbation $w_{q+1,n}=\sum_{p=1}^{p_{\rm max}}w_{q+1,n,p}$. First, we recall
that

$\left\|\nabla^{M}\mathring{R}_{q,n,p}\right\|_{L^{1}}\lesssim\delta_{q+1,n,p}\lambda_{q,n,p}^{M}.$

Therefore, we can multiply $\mathring{R}_{q,n,p}$ by a checkerboard partition
of unity at scale $\lambda_{q,n,0}^{-1}\gg\lambda_{q,n,p}^{-1}$ while
preserving these bounds. We must choose values of $r_{1}$ and $r_{2}$, as in
Section 2.5.2. Since for $n\geq 2$

$\lambda_{q+1}r_{1}=\lambda_{q,n,0}=\lambda_{q}^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}=\lambda_{q+1}\cdot\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}},$

and for $n=1$

$\lambda_{q,1,0}=\lambda_{q}^{\frac{4}{5}}\lambda_{q+1}^{\frac{1}{5}}\gg\lambda_{q+1}\cdot\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{1-1}\cdot\frac{5}{6}},$

we have that for all $n\geq 1$

$r_{1}\geq\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}.$

Recall that $\mathring{R}_{q,n,p}$ will be corrected by $w_{q+1,n,p}$, which
is constructed using intermittent pipe flows $\mathbb{W}_{q+1,n}$ with
intermittency

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

Thus in order to succeed in placing pipes $\mathbb{W}_{q+1,n}$ which avoid
both previous generations of pipes, which are periodized to scales rougher
than $\mathbb{W}_{q+1,n}$, and pipes from the same generation on overlapping
cutoff functions, we must ensure that

$\displaystyle r_{2}$ $\displaystyle\ll r_{1}^{\frac{3}{4}}$
$\displaystyle\iff\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}}$
$\displaystyle\ll\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}\cdot\frac{3}{4}}$
$\displaystyle\iff\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}\cdot\frac{3}{4}$
$\displaystyle<\left(\frac{4}{5}\right)^{n+1}$ $\displaystyle\iff\frac{1}{2}$
$\displaystyle<\left(\frac{4}{5}\right)^{3}=\frac{64}{125}.$

So our choice of $r_{q+1,n}$ is sufficient to ensure that we can successfully
place intermittent pipe flows when constructing $w_{q+1,n,p}$ which are
disjoint from all other pipe flows from either previous generations
($n^{\prime}<n$) or the same generation (the same value of $n$).

####  2.7.3 Nash and transport errors

The heuristic for the Nash and transport errors is that our choice of
$r_{q+1,n}$ provides much more intermittency than is needed to ensure that
linear errors arising from $w_{q+1,n,p}$ can be absorbed into
$\mathring{R}_{q+1}$.191919One may verify that in three dimensions, the
minimum amount of intermittency needed to absorb the Nash and transport errors
arising from $w_{q+1,0}$ into $\mathring{R}_{q+1}$ at regularity approaching
$\dot{H}^{\frac{1}{2}}$ is
$r_{q+1,0}=\lambda_{q}^{\frac{1}{2}}\lambda_{q+1}^{-\frac{1}{2}}$. In general,
one can further verify that given errors supported at frequency
$\lambda_{q}^{\alpha}\lambda_{q+1}^{1-\alpha}$, one could correct them using
intermittent pipe flows with minimum frequency
$\lambda_{q}^{\frac{\alpha}{2}}\lambda_{q+1}^{1-\frac{\alpha}{2}}$ while
absorbing the resulting Nash and transport errors into $\mathring{R}_{q+1}$.
One should compare this with (2.21), which shows that the placement technique
requires more intermittency, which at level $n=0$ corresponds to
$\lambda_{q}^{\frac{3}{4}}\lambda_{q+1}^{-\frac{3}{4}}$. In other words, the
Type 2 oscillation errors required much more intermittency than the Nash and
transport errors will.

Let us start with the Nash error arising from the addition of $w_{q+1,0,1}$,
which is designed to correct $\mathring{R}_{q}$. Using decoupling, the cost of
a derivative on $\mathbb{W}_{q+1,0}$ being $\lambda_{q+1}$ (so that inverting
the divergence gains a factor of $\lambda_{q+1}$), the size of $\nabla v_{q}$
in $L^{2}$, and the $L^{1}$ size of $\mathbb{W}_{q+1,n}$ being $r_{q+1,0}$,
the size of this error is

$\displaystyle\frac{1}{\lambda_{q+1}}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}r_{q+1,0}$
$\displaystyle=\frac{1}{\lambda_{q+1}}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)}.$

This is (much) less than $\delta_{q+2}$ since

$\displaystyle\frac{\delta_{q+1}^{\nicefrac{{1}}{{2}}}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\nicefrac{{3}}{{2}}}}{\lambda_{q+1}^{\nicefrac{{3}}{{2}}}}\leq\delta_{q+2}$
$\displaystyle\iff\lambda_{q+1}^{-\beta}\lambda_{q+1}^{-\frac{\beta}{b}}\lambda_{q+1}^{\frac{1}{b}\cdot\frac{3}{2}}\lambda_{q+1}^{-\frac{3}{2}}\leq\lambda_{q+1}^{-2\beta
b}$ $\displaystyle\iff 2\beta b^{2}-\beta b-\beta\leq(b-1)\cdot\frac{3}{2}$
$\displaystyle\iff\beta(2b+1)(b-1)\leq(b-1)\cdot\frac{3}{2}.$ (2.22)

Choosing $b$ close to $1$ will make this error commensurate with
$\dot{H}^{\frac{1}{2}-}$ regularity.

Let us now estimate the Nash error arising from the addition of $w_{q+1,n,p}$
for $n\geq 2$, given by

$\left\|\mathrm{div\,}^{-1}\left(\left(a_{n,p}\nabla\Phi_{q,k}^{-1}\mathbb{W}_{q+1,n}(\Phi_{q,k})\right)\cdot\nabla
v_{q}\right)\right\|_{L^{1}}.$

Using again decoupling, the cost of a derivative on $\mathbb{W}_{q+1,n}$ being
$\lambda_{q+1}$ (so that inverting the divergence gains a factor of
$\lambda_{q+1}$), the size of $\nabla v_{q}$ in $L^{2}$, the $L^{1}$ size of
$\mathbb{W}_{q+1,n}$ being $r_{q+1,n}$, and (2.1), we have that for $n\geq 2$,
the size of this error is

$\displaystyle\frac{1}{\lambda_{q+1}}\cdot\delta_{q+1,n,p}^{\frac{1}{2}}r_{q+1,n}\cdot\delta_{q}^{\frac{1}{2}}\lambda_{q}$
$\displaystyle\leq\frac{1}{\lambda_{q+1}}\cdot\delta_{q+1,n,1}^{\frac{1}{2}}r_{q+1,n}\cdot\delta_{q}^{\frac{1}{2}}\lambda_{q}$
$\displaystyle=\frac{1}{\lambda_{q+1}}\left(\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q,n,0}}\right)^{\frac{1}{2}}\left(\prod_{n^{\prime}<n}f_{q,n^{\prime}}\right)^{\frac{1}{2}}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}}\delta_{q}^{\frac{1}{2}}\lambda_{q}$
$\displaystyle\leq\frac{1}{\lambda_{q+1}}\left(\frac{\delta_{q+1}\lambda_{q}}{\lambda_{q}^{\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}\lambda_{q+1}^{1-\left(\frac{4}{5}\right)^{n-1}\cdot\frac{5}{6}}}\right)^{\frac{1}{2}}\left(\frac{\lambda_{q}}{\lambda_{q+1}}\right)^{\left(\frac{4}{5}\right)^{n+1}-\frac{1}{2{p_{\rm
max}}}}\delta_{q}^{\frac{1}{2}}\lambda_{q}\,.$

Since

$\frac{1}{2{p_{\rm
max}}}+\frac{1}{2}\cdot\frac{5}{6}\cdot\left(\frac{4}{5}\right)^{n-1}<\left(\frac{4}{5}\right)^{n+1}$

independently of $n\geq 2$ if ${p_{\rm max}}$ is sufficiently large, the Nash
error will be smaller than $\delta_{q+2}$ based on the preceding estimates.
Furthermore, one may check that
$\delta_{q+1,1,1}^{\frac{1}{2}}r_{q+1,1}<\delta_{q+1,2,1}^{\frac{1}{2}}r_{q+1,2},$
so that the Nash error arising from the addition of $w_{q+1,1,p}$ is also
satisfactorily small for all $p$.

Now let us consider the transport error. The size of the transport error
arising from the addition of $w_{q+1,n,p}$ is

$\displaystyle\left\|\mathrm{div\,}^{-1}\left((D_{t,q}a_{n,p})\nabla\Phi_{q,l}^{-1}\mathbb{W}_{q+1,n}\right)\right\|_{L^{1}}$
$\displaystyle\leq\frac{1}{\lambda_{q+1}}\tau_{q}^{-1}\delta_{q+1,n,p}^{\frac{1}{2}}r_{q+1,n}$
$\displaystyle=\frac{1}{\lambda_{q+1}}\cdot\delta_{q+1,n,p}^{\frac{1}{2}}r_{q+1,n}\cdot\delta_{q}^{\frac{1}{2}}\lambda_{q}.$
(2.23)

Thus, the transport error is the same size as the Nash error and is
sufficiently small to be put into $\mathring{R}_{q+1}$.

##  3 Inductive assumptions

While in Section 2 we have outlined in broad terms the main steps in the proof
of Theorem 1.1, along with the heuristics for some of the choices we have made
in our proof, starting with the current section, we work with precise
estimates.

In Section 3.1 we introduce some of the notation used in the proof, such as
the Euler-Reynolds system, the mollified velocity, velocity increments,
material/directional derivatives, our notation for geometric upper bounds with
tho different bases, and our notation for $\left\|\cdot\right\|_{L^{p}}$.

In Section 3.2 we introduce the principal amplitude and frequency parameters
used in proof (the precise definitions and the order of choosing these
parameters is detailed in Section 9.1). Next, in Sections 3.2.1 and 3.2.2 we
state the primary inductive assumptions for the velocity, velocity increments,
and Reynolds stress. These primary estimates hold on the support of previous
generation velocity cutoff functions, which are inductively assumed to satisfy
a number of properties, listed in Section 3.2.3. Lastly, in Section 3.2.4 we
list a number of bounds for the velocity increments and mollified velocities,
which involve all possible combinations of space and material derivatives, up
to a certain order. These bounds are technical in nature, and should be
ignored at a first reading; their sole purpose is to allow us to bound
commutators between $D^{n}$ and $D_{t,q}^{m}$ for very high values of $n$ and
$m$.

In conclusion, in Section 3.4 we show that if we are able to propagate the
previously stated inductive estimates from step $q$ to step $q+1$, for every
$q\geq 0$, then Theorem 1.1 follows. At the end of the section we discuss the
modifications to the proof which would be necessary in order to obtain other
types of flexibility statements.

###  3.1 General notations

As is standard in convex integration schemes for the Euler system [29], we
introduce the Euler-Reynolds system for the unknowns
$(v_{q},\mathring{R}_{q})$:

$\displaystyle\partial_{t}v_{q}+\mathrm{div\,}(v_{q}\otimes v_{q})+\nabla
p_{q}$ $\displaystyle=\mathrm{div\,}\mathring{R}_{q}$ (3.1a)
$\displaystyle\mathrm{div\,}v_{q}$ $\displaystyle=0.$ (3.1b)

Here and throughout the paper, the pressure $p_{q}$ is uniquely defined by
solving $\Delta
p_{q}=\mathrm{div\,}\mathrm{div\,}(\mathring{R}_{q}-v_{q}\otimes v_{q})$, with
$\int_{\mathbb{T}^{3}}p_{q}dx=0$.

In order to avoid the usual derivative-loss issue in convex integration
schemes, for $q\geq 0$ we use the space-time mollification operator defined in
Section 9.4 – equation (9.64), to smoothen out the velocity field $v_{q}$ as:

$\displaystyle v_{\ell_{q}}:=\mathcal{P}_{q,x,t}v_{q}\,.$ (3.2)

In particular, we note that spatial mollification is performed at scale
$\widetilde{\lambda}_{q}^{-1}$ (which is just slightly smaller than
$\lambda_{q}^{-1}$), while temporal mollification is at scale
$\widetilde{\tau}_{q-1}$ (which is a lot smaller than $\tau_{q-1}$).

Next, for all $q\geq 1$, define

$\displaystyle w_{q}:=v_{q}-v_{\ell_{q-1}},\qquad
u_{q}:=v_{\ell_{q}}-v_{\ell_{q-1}}.$ (3.3)

For consistency of notation, define $w_{0}=v_{0}$ and $u_{0}=v_{\ell_{0}}$.
Note that

$\displaystyle
u_{q}=\mathcal{P}_{q,x,t}w_{q}+(\mathcal{P}_{q,x,t}v_{\ell_{q-1}}-v_{\ell_{q-1}})$
(3.4)

so that we may morally think that $u_{q}=w_{q}+$ a small error term (the
smallness of this error term will be ensured by choosing a mollifier with a
large number of vanishing moments, cf. (9.62)).

We use the following notation for the material derivative corresponding to the
vector field $v_{\ell_{q}}$:

$\displaystyle D_{t,q}:=\partial_{t}+v_{\ell_{q}}\cdot\nabla.$ (3.5)

With this notation, we have that

$\displaystyle D_{t,q}=D_{t,q-1}+u_{q}\cdot\nabla.$ (3.6)

We also introduce the directional derivatives

$\displaystyle D_{q}:=u_{q}\cdot\nabla$ (3.7)

which allow us to transfer information between $D_{t,q-1}$ and $D_{t,q}$ via
$D_{t,q}=D_{t,q-1}+D_{q}$.

######  Remark 3.1 (Geometric upper bounds with two bases).

If for a sequence of numbers $\\{a_{n}\\}_{n\geq 0}$, and for two parameters
$0<\lambda<\Lambda$ we have the bounds

$\displaystyle a_{n}$ $\displaystyle\leq\lambda^{n},\quad\mbox{for all}\quad
n\leq N_{*}$ $\displaystyle a_{n}$
$\displaystyle\leq\lambda^{N_{*}}\Lambda^{n-N_{*}}\quad\mbox{for all}\quad
n>N_{*},$

for some $N_{*}\geq 1$, we will abbreviate these bounds as

$\displaystyle a_{n}\leq\mathcal{M}\left(n,N_{*},\lambda,\Lambda\right)\,,$

where we define

$\displaystyle\mathcal{M}\left(n,N_{*},\lambda,\Lambda\right):=\lambda^{\min\\{n,N_{*}\\}}\Lambda^{\max\\{n-N_{*},0\\}}$
(3.8)

for all $n\geq 0$. The first entry of
$\mathcal{M}\left(\cdot,\cdot,\lambda,\Lambda\right)$ measures the index in
the sequence (typically number of derivatives considered) and the second entry
determines the index after which the base of the geometric bound changes from
$\lambda$ to $\Lambda$. This notation has the following consequence, which
will be used throughout the paper: if $1\leq\lambda\leq\Lambda$, then

$\displaystyle\mathcal{M}\left(a,N_{*},\lambda,\Lambda\right)\mathcal{M}\left(b,N_{*},\lambda,\Lambda\right)\leq\mathcal{M}\left(a+b,N_{*},\lambda,\Lambda\right).$
(3.9)

When either $a$ or $b$ are larger than $N_{*}$ the above inequality creates a
loss; for $a+b\leq N_{*}$, it is an equality.

######  Remark 3.2 (Norms are uniform in time).

Throughout this section, and the remainder of the paper, in order to
abbreviate notation we shall use the notation $\left\|f\right\|_{L^{p}}$ to
denote $\left\|f\right\|_{L^{\infty}_{t}(L^{p}(\mathbb{T}^{3}))}$. That is,
all $L^{p}$ norms stand for $L^{p}$ norms in space, uniformly in time.
Similarly, when we wish to emphasize a set dependence of an $L^{p}$ norm, we
write $\left\|f\right\|_{L^{p}(\Omega)}$, for some space-time set
$\Omega\subset\mathbb{R}\times\mathbb{T}^{3}$, to stand for
$\left\|{\mathbf{1}}_{\Omega}\;f\right\|_{L^{\infty}_{t}(L^{p}(\mathbb{T}^{3}))}$.

###  3.2 Inductive estimates

The proof is based on propagating estimates for solutions
$(v_{q},\mathring{R}_{q})$ of the Euler-Reynolds system (3.1), inductively for
$q\geq 0$. In order to state these bounds, we first need to fix a number of
parameters in terms of which these inductive estimates are stated. We start by
picking a regularity exponent
$\beta\in(\nicefrac{{1}}{{3}},\nicefrac{{1}}{{2}})$, and a super-exponential
rate parameter $b\in(1,\nicefrac{{3}}{{2}})$ such that $2\beta b<1$. In terms
of this choice of $\beta$ and $b$, a number of additional parameters (${n_{\rm
max}},\ldots\mathsf{N}_{\rm fin}$) are fixed, whose precise definition is
summarized for convenience in items (iii)–(xii) of Section 9.1. Note that at
this point the parameter $a_{*}(\beta,b)$ from item (xiii) in Section 9.1 is
not yet fixed. With this choice, we then introduce the fundamental
$q$-dependent frequency and amplitude parameters from Section 9.2. We state
here for convenience the main $q$-dependent parameters defined in (9.15),
(9.17), (9.18), and (9.21):

$\displaystyle\lambda_{q}=2^{\big{\lceil}{(b^{q})\log_{2}a}\big{\rceil}}\approx\lambda_{q-1}^{b}\,,\qquad$
$\displaystyle\delta_{q}=\lambda_{1}^{\beta(b+1)}\lambda_{q}^{-2\beta}\,,$
(3.10a)
$\displaystyle\tau_{q}^{-1}=\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\Gamma_{q+1}^{\mathsf{c_{0}}+11}\,,\qquad$
$\displaystyle\Gamma_{q+1}=\left(\frac{\lambda_{q+1}}{\lambda_{q}}\right)^{\varepsilon_{\Gamma}}\approx\lambda_{q}^{(b-1)\varepsilon_{\Gamma}}\,,$
(3.10b)

where the constant $\mathsf{c_{0}}$ is defined by (9.6). The $\approx$ symbols
in (3.10) mean that the left side of the $\approx$ symbol lies between two
(universal) constant multiples of the right side (see e.g. (9.16)).

######  Remark 3.3 (Usage of the symbol $\lesssim$ and choice of $a_{*}$).

Throughout the subsequent sections, we will make frequent use of the symbol
$\lesssim$. We emphasize that any implicit constants indicated by $\lesssim$
are only allowed to depend on the parameters defined in Section 9.1, items
(i)–(xii). The implicit constants in $\lesssim$ are however always independent
of the parameters $a$ and $q$, which appear in (3.10). This allows us at the
end of the proof, cf. item (xiii) in Section 9.1 to choose $a_{*}(\beta,b)$ to
be sufficiently large so that for all $a\geq a_{*}(\beta,b)$ and all $q\geq
0$, the parameter $\Gamma_{q+1}$ appearing in (3.10) is larger than all the
implicit constants in $\lesssim$ symbols encountered throughout the paper.
That is, upon choosing $a_{*}$ sufficiently large, any inequality of the type
$A\lesssim B$ which appears in this manuscript, may be rewritten as
$A\leq\Gamma_{q+1}B$, for any $q\geq 0$.

In order to state the inductive assumptions we use four large integers,
defined precisely in Section 9.1. For the moment it is just important to note
that these fixed parameters are independent of $q$ and that they satisfy the
ordering

$\displaystyle 1\ll\mathsf{N}_{\rm
cut,t}\ll\mathsf{N}_{\textnormal{ind,t}}\ll\mathsf{N}_{\textnormal{ind,v}}\ll\mathsf{N}_{\rm
fin}\,.$ (3.11)

The precise definitions of these integers, and the meaning of the $\ll$
symbols in (3.11), are given in (9.9), (9.10), (9.11), and (9.14). Roughly
speaking, the role of these parameters is as follows:

  * •

$\mathsf{N}_{\rm cut,t}$ is the number of sharp material derivatives which are
built into the velocity and stress cutoff functions.

  * •

$\mathsf{N}_{\textnormal{ind,t}}$ is the number of sharp material derivatives
propagated for velocities and stresses.

  * •

$\mathsf{N}_{\textnormal{ind,v}}$ is used to quantify the number of (lossy)
higher order space and time derivatives for velocities and stresses.

  * •

$\mathsf{N}_{\rm fin}$ is used to quantify the highest order derivatives
appearing in the proof.

Next, we state the inductive assumptions for the velocity increments and
stresses at various levels $q\geq 0$. Throughout the section we frequently
refer to the notation $\mathcal{M}\left(n,N_{*},\lambda,\Lambda\right)$ from
(3.8).

####  3.2.1 Primary inductive assumption for velocity increments

We make $L^{2}$ inductive assumptions for
$u_{q^{\prime}}=v_{\ell_{q^{\prime}}}-v_{\ell_{q^{\prime}-1}}$ at levels
$q^{\prime}$ strictly below $q$. For all $0\leq q^{\prime}\leq q-1$ we assume
that

$\displaystyle\left\|\psi_{i,q^{\prime}-1}D^{n}D^{m}_{t,q^{\prime}-1}u_{q^{\prime}}\right\|_{L^{2}}\leq\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}}\mathcal{M}\left(n,2\mathsf{N}_{\textnormal{ind,v}},\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}}^{i}\tau_{q^{\prime}-1}^{-1},\widetilde{\tau}_{q^{\prime}-1}^{-1}\right)$
(3.12)

holds for all $n+m\leq\mathsf{N}_{\rm fin}$.

At level $q$, we assume that the velocity increment $w_{q}$ satisfies

$\displaystyle\left\|\psi_{i,q-1}D^{n}D^{m}_{t,q-1}w_{q}\right\|_{L^{2}}\leq\Gamma_{q}^{-1}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{n}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q}^{i-1}\tau_{q-1}^{-1},\Gamma_{q}^{-1}\widetilde{\tau}_{q-1}^{-1}\right)$
(3.13)

for $n,m\leq 7\mathsf{N}_{\textnormal{ind,v}}$. Moreover, recalling from
(9.67) that $\mathrm{supp\,}_{t}f$ denotes the temporal support of a function
$f$, we inductively assume that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{q-1})\subset[T_{1},T_{2}]\quad\Rightarrow\quad\mathrm{supp\,}_{t}(w_{q})\subset\left[T_{1}-(\lambda_{q-1}\delta_{q-1}^{\nicefrac{{1}}{{2}}})^{-1},T_{2}+(\lambda_{q-1}\delta_{q-1}^{\nicefrac{{1}}{{2}}})^{-1}\right]\,.$
(3.14)

####  3.2.2 Inductive assumption for the stress

For the Reynolds stress $\mathring{R}_{q}$, we make $L^{1}$ inductive
assumptions

$\displaystyle\left\|\psi_{i,q-1}D^{n}D_{t,q-1}^{m}\mathring{R}_{q}\right\|_{L^{1}}\leq\Gamma_{q}^{-\mathsf{C_{R}}}\delta_{q+1}\lambda_{q}^{n}\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q}^{i+1}\tau_{q-1}^{-1},\Gamma_{q}^{-1}\widetilde{\tau}_{q-1}^{-1}\right)$
(3.15)

for all $0\leq n,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$.

####  3.2.3 Inductive assumption for the previous generation velocity cutoff
functions

More assumptions are needed in relation to the previous velocity perturbations
and old cutoffs functions. First, we assume that the velocity cutoff functions
form a partition of unity for $q^{\prime}\leq q-1$:

$\displaystyle\sum_{i\geq 0}\psi_{i,q^{\prime}}^{2}\equiv
1,\qquad\mbox{and}\qquad\psi_{i,q^{\prime}}\psi_{i^{\prime},q^{\prime}}=0\quad\textnormal{for}\quad|i-i^{\prime}|\geq
2.$ (3.16)

Second, we assume that there exists an ${i_{\rm max}}={i_{\rm max}}(q)>0$,
which is bounded uniformly in $q$ as

$\displaystyle\frac{b+1}{b-1}\leq{i_{\rm
max}}(q)\leq\frac{4}{\varepsilon_{\Gamma}(b-1)}\,,$ (3.17)

such that

$\displaystyle\psi_{i,q^{\prime}}$ $\displaystyle\equiv 0\quad\mbox{for
all}\quad i>{i_{\rm
max}}(q^{\prime})\,,\qquad\mbox{and}\qquad\Gamma_{q^{\prime}+1}^{{i_{\rm
max}}(q^{\prime})}\leq\lambda_{q^{\prime}}^{\nicefrac{{5}}{{3}}}\,,$ (3.18)

for all $q^{\prime}\leq q-1$. For all $0\leq q^{\prime}\leq q-1$ and $0\leq
i\leq{i_{\rm max}}$ we assume the following pointwise derivative bounds for
the cutoff functions $\psi_{i,q^{\prime}}$. For mixed space and material
derivatives (recall the notation from (3.5)) we assume that

$\displaystyle\frac{{\mathbf{1}}_{\mathrm{supp\,}\psi_{i,q^{\prime}}}}{\psi_{i,q^{\prime}}^{1-(K+M)/\mathsf{N}_{\rm
fin}}}\left|\left(\prod_{l=1}^{k}D^{\alpha_{l}}D_{t,q^{\prime}-1}^{\beta_{l}}\right)\psi_{i,q^{\prime}}\right|$
$\displaystyle\qquad\lesssim\mathcal{M}\left(K,\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\Gamma_{q^{\prime}}\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}}-\mathsf{N}_{\rm
cut,t},\Gamma_{q^{\prime}+1}^{i+3}\tau_{q^{\prime}-1}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.19)

for $K,M,k\geq 0$ with $0\leq K+M\leq\mathsf{N}_{\rm fin}$, where
$\alpha,\beta\in{\mathbb{N}}^{k}$ are such that $|\alpha|=K$ and $|\beta|=M$.
Lastly, we consider mixtures of space, material, and directional derivatives
(recall the notation from (3.7)). Then with $K,M,\alpha,\beta$ and $k$ as
above, and with $N\geq 0$, we assume that

$\displaystyle\frac{{\mathbf{1}}_{\mathrm{supp\,}\psi_{i,q^{\prime}}}}{\psi_{i,q^{\prime}}^{1-(N+K+M)/\mathsf{N}_{\rm
fin}}}\left|D^{N}\left(\prod_{l=1}^{k}D_{q^{\prime}}^{\alpha_{l}}D_{t,q^{\prime}-1}^{\beta_{l}}\right)\psi_{i,q^{\prime}}\right|$
$\displaystyle\qquad\lesssim\mathcal{M}\left(N,\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\Gamma_{q^{\prime}}\widetilde{\lambda}_{q^{\prime}}\right)(\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1})^{K}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}}-\mathsf{N}_{\rm
cut,t},\Gamma_{q^{\prime}+1}^{i+3}\tau_{q^{\prime}-1}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.20)

as long as $0\leq N+K+M\leq\mathsf{N}_{\rm fin}$.

In addition to the above pointwise estimates for the cutoff functions
$\psi_{i,q^{\prime}}$, we also assume that we have a good $L^{1}$ control.
More precisely, we postulate that

$\displaystyle\left\|\psi_{i,q^{\prime}}\right\|_{L^{1}}\lesssim\Gamma_{q^{\prime}+1}^{-2i+{\mathsf{C}_{b}}}\qquad\mbox{where}\qquad{\mathsf{C}_{b}}=\frac{4+b}{b-1}$
(3.21)

holds for $0\leq q^{\prime}\leq q-1$ and all $0\leq i\leq{i_{\rm
max}}(q^{\prime})$.

####  3.2.4 Secondary inductive assumptions for velocities

Next, for $0\leq q^{\prime}\leq q-1$, $0\leq i\leq{i_{\rm max}}$, $k\geq 1$,
$K,M\geq 0$, $\alpha,\beta\in\mathbb{N}^{k}$ with $|\alpha|=K$ and
$|\beta|=M$, we assume that the following mixed space-and-material derivative
bounds hold

$\displaystyle\left\|\Big{(}\prod_{i=1}^{k}D^{\alpha_{i}}D_{t,q^{\prime}-1}^{\beta_{i}}\Big{)}u_{q^{\prime}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q^{\prime}})}$
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}})\mathcal{M}\left(K,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i+3}\tau_{q^{\prime}-1}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.22)

for $K+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$,

$\displaystyle\left\|\Big{(}\prod_{i=1}^{k}D^{\alpha_{i}}D_{t,q^{\prime}}^{\beta_{i}}\Big{)}Dv_{\ell_{q^{\prime}}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q^{\prime}})}$
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}}\widetilde{\lambda}_{q^{\prime}})\mathcal{M}\left(K,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.23)

for $K+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}$, and

$\displaystyle\left\|\Big{(}\prod_{i=1}^{k}D^{\alpha_{i}}D_{t,q^{\prime}}^{\beta_{i}}\Big{)}v_{\ell_{q^{\prime}}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q^{\prime}})}$
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}}\lambda_{q^{\prime}}^{2})\mathcal{M}\left(K,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.24)

for $K+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$. Additionally, for
$N\geq 0$ we postulate that mixed space-material-directional derivatives
satisfy

$\displaystyle\left\|D^{N}\Big{(}\prod_{i=1}^{k}D_{q^{\prime}}^{\alpha_{i}}D_{t,q^{\prime}-1}^{\beta_{i}}\Big{)}u_{q^{\prime}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q^{\prime}})}$
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}})^{K+1}\mathcal{M}\left(N+K,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i+3}\tau_{q^{\prime}-1}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.25a)
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}})\mathcal{M}\left(N,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)(\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1})^{K}\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i+3}\tau_{q^{\prime}-1}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.25b)

whenever $N+K+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$.

######  Remark 3.4.

Identity (A.39) shows that (3.25b) automatically implies the bound

$\displaystyle\left\|D^{N}D_{t,q^{\prime}}^{M}u_{q^{\prime}}\right\|_{L^{\infty}(\mathrm{supp\,}\psi_{i,q^{\prime}})}$
$\displaystyle\qquad\lesssim(\Gamma_{q^{\prime}+1}^{i+1}\delta_{q^{\prime}}^{\nicefrac{{1}}{{2}}})\mathcal{M}\left(N,2\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}},\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.26)

for all $N+M\leq\nicefrac{{3\mathsf{N}_{\rm fin}}}{{2}}+1$. To see this, we
take $B=D_{t,q^{\prime}-1}$ and $A=D_{q^{\prime}}$, so that
$A+B=D_{t,q^{\prime}}$. The estimate (3.26) now is a consequence of identity
(A.39) and the parameter inequalities
$\Gamma_{q^{\prime}+1}^{\mathsf{c_{0}}+3}\tau_{q^{\prime}-1}^{-1}\leq\tau_{q^{\prime}}^{-1}$
(which follows from (9.40)) and
$\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}+1}\tau_{q^{\prime}}^{-1}\leq\widetilde{\tau}_{q^{\prime}}^{-1}$
(which is a consequence of (3.18) and (9.43)). In a similar fashion, the bound
(3.20) and identity (A.39) imply that

$\displaystyle\frac{{\mathbf{1}}_{\mathrm{supp\,}\psi_{i,q^{\prime}}}}{\psi_{i,q^{\prime}}^{1-(N+M)/\mathsf{N}_{\rm
fin}}}\left|D^{N}D_{t,q^{\prime}}^{M}\psi_{i,q^{\prime}}\right|$
$\displaystyle\qquad\lesssim\mathcal{M}\left(N,\mathsf{N}_{\textnormal{ind,v}},\Gamma_{q^{\prime}}\lambda_{q^{\prime}},\Gamma_{q^{\prime}}\widetilde{\lambda}_{q^{\prime}}\right)\mathcal{M}\left(M,\mathsf{N}_{\textnormal{ind,t}}-\mathsf{N}_{\rm
cut,t},\Gamma_{q^{\prime}+1}^{i-\mathsf{c_{0}}}\tau_{q^{\prime}}^{-1},\Gamma_{q^{\prime}+1}^{-1}\widetilde{\tau}_{q^{\prime}}^{-1}\right)$
(3.27)

for all $N+M\leq\mathsf{N}_{\rm fin}$. Indeed, the above estimates follow from
the same parameter inequalities mentioned above, and from identity (A.39) with
$A=D_{q^{\prime}}$ and $B=D_{t,q^{\prime}-1}$.

######  Remark 3.5.

The inductive assumptions for the velocities given in Sections 3.2.1 and
3.2.4, with the definition of the mollifier operator ${\mathcal{P}}_{q,x,t}$
in Section 9.4, imply that the new velocity field $v_{q}=w_{q}+v_{\ell_{q-1}}$
is very close to its mollification $v_{\ell_{q}}$, uniformly in space and
time. That is, we have

$\displaystyle\left\|D^{n}D_{t,q-1}^{m}(v_{\ell_{q}}-v_{q})\right\|_{L^{\infty}}\leq\lambda_{q}^{-2}\delta_{q}^{\nicefrac{{1}}{{2}}}\mathcal{M}\left(n,2\mathsf{N}_{\textnormal{ind,v}},\lambda_{q},\widetilde{\lambda}_{q}\right)\mathcal{M}\left(m,\mathsf{N}_{\textnormal{ind,t}},\tau_{q-1}^{-1}\Gamma_{q}^{i-1},\tilde{\tau}_{q-1}^{-1}\Gamma_{q}^{-1}\right)$
(3.28)

for all $n,m\leq 3\mathsf{N}_{\textnormal{ind,v}}$. The proof of the above
bound is given in Lemma 5.1, cf. estimate (5.4).

###  3.3 Main inductive proposition

The main inductive proposition, which propagates the inductive estimates in
Section 3.2 from step $q$ to step $q+1$, is as follows.

######  Proposition 3.6.

Fix $\beta\in[\nicefrac{{1}}{{3}},\nicefrac{{1}}{{2}})$ and choose
$b\in(1,\nicefrac{{1}}{{2\beta}})$. Solely in terms of $\beta$ and $b$, define
the parameters ${n_{\rm max}}$, ${\mathsf{C}_{b}}$, $\mathsf{C_{R}}$,
$\mathsf{c_{0}}$, $\varepsilon_{\Gamma}$, $\alpha_{\mathsf{R}}$,
$\mathsf{N}_{\rm cut,t}$, $\mathsf{N}_{\rm cut,x}$,
$\mathsf{N}_{\textnormal{ind,t}}$, $\mathsf{N}_{\textnormal{ind,v}}$,
${\mathsf{N}_{\rm dec}}$, ${\mathsf{d}}$, and $\mathsf{N}_{\rm fin}$, by the
definitions in Section 9.1, items (i)–(xii). Then, there exists a sufficiently
large $a_{*}=a_{*}(\beta,b)\geq 1$, such that for any $a\geq a_{*}$, the
following statement holds for any $q\geq 0$. Given a velocity field $v_{q}$
which solves the Euler-Reynolds system with stress $\mathring{R}_{q}$, define
$v_{\ell_{q}},w_{q}$, and $u_{q}$ via (3.2)–(3.3). Assume that
$\\{u_{q^{\prime}}\\}_{q^{\prime}=0}^{q-1}$ satisfies (3.12), $w_{q}$ obeys
(3.13)–(3.14), $\mathring{R}_{q}$ satisfies (3.15), and that for every
$q^{\prime}\leq q-1$ there exists a partition of unity
$\\{\psi_{i,q^{\prime}}\\}_{i\geq 0}$ such that properties (3.16)–(3.18) and
estimates (3.19)–(3.25) hold. Then, there exists a velocity field $v_{q+1}$, a
stress $\mathring{R}_{q+1}$, and a partition of unity $\\{\psi_{i,q}\\}_{q\geq
0}$, such that $v_{q+1}$ solves the Euler-Reynolds system with stress
$\mathring{R}_{q+1}$, $u_{q}$ satisfies (3.12) for $q^{\prime}\mapsto q$,
$w_{q+1}$ obeys (3.13)–(3.14) for $q\mapsto q+1$, $\mathring{R}_{q+1}$
satisfies (3.15) for $q\mapsto q+1$, and the $\psi_{i,q}$ are such that
(3.16)–(3.25) hold when $q^{\prime}\mapsto q$.

###  3.4 Proof of Theorem 1.1

Choose the parameters $\beta,b,\ldots,a_{*}$, as described in Section 9.1, and
assume that with these parameter choices, and for any $a\geq a_{*}$, we are
able to propagate the inductive bounds claimed in Sections 3.2.1–3.2.4 from
step $q$ to step $q+1$, for all $q\geq 0$; this is achieved in Sections 6–8.
We next show that if $a\geq a_{*}$ is chosen sufficiently large, depending
additionally on the $v_{\mathrm{start}}$, $v_{\mathrm{end}}$, $T>0$, and
$\epsilon>0$ from the statement of Theorem 1.1, then the inductive assumptions
imply Theorem 1.1.

Without loss of generality, assume that
$\int_{\mathbb{T}^{3}}v_{\mathrm{start}}(x)dx=\int_{\mathbb{T}^{3}}v_{\mathrm{end}}(x)dx=0$.
Since these functions lie in $L^{2}(\mathbb{T}^{3})$, there exists $R>0$ such
that upon defining

$\displaystyle v_{0}^{(1)}:=\mathbb{P}_{\leq
R}v_{\mathrm{start}}\,,\qquad\mbox{and}\qquad v_{0}^{(2)}:=\mathbb{P}_{\leq
R}v_{\mathrm{end}}\,,$

where $\mathbb{P}_{\leq R}$ denotes the Fourier truncation operator to
frequencies $|\xi|\leq R$, we have that

$\displaystyle\|v_{0}^{(1)}-v_{\mathrm{start}}\|_{L^{2}(\mathbb{T}^{3})}+\|v_{0}^{(2)}-v_{\mathrm{end}}\|_{L^{2}(\mathbb{T}^{3})}\leq\frac{\epsilon}{2}\,.$
(3.29)

Note that $v_{0}^{(1)},v_{0}^{(2)}\in C^{\infty}(\mathbb{T}^{3})$, and thus by
the classical local well-posedness theory plus propagation of regularity (see
Foias, Frisch, and Temam [38]), there exists $T_{0}>0$ and unique strong
solutions $v^{(1)}\in C^{\infty}((-T_{0},T_{0})\times\mathbb{T}^{3})$ and
$v^{(2)}\in C^{\infty}((T-T_{0},T+T_{0})\times\mathbb{T}^{3})$ of the 3D Euler
system (1.1), such that $v^{(1)}(x,0)=v_{0}^{(1)}(x)$ and
$v^{(2)}(x,T)=v_{0}^{(2)}(x)$. Without loss of generality, we may take
$T_{0}\leq\nicefrac{{T}}{{2}}$.

Next, let $\varphi\colon[0,T]\to[0,1]$ be a non-increasing $C^{\infty}$ smooth
function such that $\varphi\equiv 1$ on $[0,\nicefrac{{T_{0}}}{{2}}]$ and
$\varphi\equiv 0$ on $[T_{0},T]$. Define the $C^{\infty}$-smooth function

$\displaystyle v_{0}(x,t):=\varphi(t)v^{(1)}(x,t)+\varphi(T-t)v^{(2)}(x,t)\,.$
(3.30)

On $[0,T]$, $v_{0}$ solves the Euler-Reynolds system (3.1) for a suitable zero
mean scalar pressure $p_{0}$, with the $C^{\infty}$-smooth stress
$\mathring{R}_{0}$ defined by

$\displaystyle\mathring{R}_{0}(x,t)$
$\displaystyle:=(\partial_{t}\varphi)(t)\mathcal{R}v^{(1)}(x,t)-(\partial_{t}\varphi)(T-t)\mathcal{R}v^{(2)}(x,t)$
$\displaystyle\quad+\varphi(t)(\varphi(t)-1)(v^{(1)}\mathring{\otimes}v^{(1)})(x,t)+\varphi(T-t)(\varphi(T-t)-1)(v^{(2)}\mathring{\otimes}v^{(2)})(x,t)\,,$
(3.31)

where $\mathcal{R}$ is the classical nonlocal inverse-divergence operator (see
(A.100) for the definition). From the above definition and the fact that
$\varphi\equiv 1$ on $[0,\nicefrac{{T_{0}}}{{2}}]$, we deduce that

$\displaystyle\mathrm{supp\,}_{t}(\mathring{R}_{0})\subset[\nicefrac{{T_{0}}}{{2}},T-\nicefrac{{T_{0}}}{{2}}]\,.$
(3.32)

This fact will be needed towards the end of the proof.

For consistency of notation, we also define $v_{-1}=v_{\ell_{-1}}=u_{-1}=0$,
so that $v_{0}=w_{0}$ holds by (3.3). For the velocity cutoffs, we let
$\psi_{0,-1}=1$ and $\psi_{i,-1}=0$ for all $i\geq 1$. It is then immediate to
check that the $\\{\psi_{i,-1}\\}_{i\geq 0}$ satisfy the inductive assumptions
(3.16)–(3.21), for $q^{\prime}=-1$, with the derivative bounds (3.19) and
(3.20) being empty statements for $K+M\geq 1$, respectively when $N+K+M\geq
1$. Moreover, the bounds (3.12) and (3.22)–(3.25b) hold for $q^{\prime}=-1$
since the left side of these inequalities vanishes identically. Lastly, the
assumption (3.14) is empty since there is no $\mathring{R}_{-1}$ stress to
speak of.

It thus remains to verify that the pair $(v_{0},\mathring{R}_{0})$ defined in
(3.30)–(3.31) satisfies the estimates (3.13) and (3.15), where by the above
choices we have $D_{t,-1}=\partial_{t}$. Note that the parameter
$\mathsf{N}_{\textnormal{ind,v}}$ was already chosen; thus, we have that

$\displaystyle C_{\mathrm{datum}}$ $\displaystyle:=\max_{0\leq n,m\leq
7\mathsf{N}_{\textnormal{ind,v}}}\left\|D^{n}\partial_{t}^{m}v_{0}\right\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}+\max_{0\leq
n,m\leq
3\mathsf{N}_{\textnormal{ind,v}}}\left\|D^{n}\partial_{t}^{m}\mathring{R}_{0}\right\|_{L^{\infty}(0,T;L^{1}(\mathbb{T}^{3}))}<\infty\,.$
(3.33)

Note that $C_{\mathrm{datum}}$ only depends on $v_{\mathrm{start}}$,
$v_{\mathrm{end}}$, the cutoff frequency $R>0$, the choice of the cutoff
function $\varphi$, on $T>0$, and on the parameter
$\mathsf{N}_{\textnormal{ind,v}}$. In particular, $C_{\mathrm{datum}}$ does
not depend on the parameter $a$, which is the base of the exponential defining
$\lambda_{q}$ in (3.10). Defining
$\tau_{-1}=\Gamma_{0}^{-1}=\lambda_{0}^{-\varepsilon_{\Gamma}}$ and
$\widetilde{\tau}_{-1}=\Gamma_{0}^{-3}=\lambda_{0}^{-3\varepsilon_{\Gamma}}$
(these parameters are never used again), and that $\lambda_{0}\geq a\geq
a_{*}\geq 1$, we thus have that (3.13) and (3.15) hold if we ensure that

$\displaystyle
C_{\mathrm{datum}}\leq\Gamma_{0}^{-1}\delta_{0}^{\nicefrac{{1}}{{2}}}\qquad\mbox{and}\qquad
C_{\mathrm{datum}}\leq\Gamma_{0}^{{-\mathsf{C_{R}}}}\delta_{1}\,.$ (3.34)

Using that $\varepsilon_{\Gamma}$ is sufficiently small with respect to
$\beta$ and $b$, we have that
$\Gamma_{0}^{-1}\delta_{0}^{\nicefrac{{1}}{{2}}}=\lambda_{0}^{-\varepsilon_{\Gamma}}\lambda_{1}^{(b+1)\beta/2}\lambda_{0}^{-\beta}\geq(\lambda_{1}\lambda_{0}^{-1})^{(b+1)\beta/2}\geq(a^{b-1}/2)^{\beta}$.
Also, by using that $\varepsilon_{\Gamma}$ is chosen to be sufficiently small
with respect to $\beta$ and $b$, we have that
$\Gamma_{0}^{{-\mathsf{C_{R}}}}\delta_{1}=\lambda_{0}^{(4b+1)\varepsilon_{\Gamma}}\lambda_{1}^{(b-1)\beta}\geq(\lambda_{1}\lambda_{0}^{-1})^{(b-1)\beta}\geq(a^{b-1}/2)^{(b-1)\beta}$.
Thus, if in addition to $a\geq a_{*}$, as specified by item (xiii) in Section
9.1, if we choose $a\geq a_{*}$ to be sufficiently large in terms of $\beta,b$
and the constant $C_{\mathrm{datum}}$ from (3.33), in order to ensure that

$a^{(b-1)^{2}\beta}\geq 4C_{\mathrm{datum}}\,,$

then the condition (3.34) is satisfied. We make this choice of $a$, and thus
all the estimates claimed in Sections 3.2.1–3.2.4 hold true for the base step
in the induction, the case $q=0$.

Proceeding inductively, these estimates thus hold true for all $q\geq 0$. This
allows us to define a function $v\in
C^{0}(0,T;H^{\beta^{\prime}}(\mathbb{T}^{3}))$ for any $\beta^{\prime}<\beta$
via the absolutely convergent series202020We may equivalently define
$v=\lim_{q\to\infty}v_{q}=\lim_{q\to\infty}w_{q}+\sum_{q^{\prime}=0}^{q-1}u_{q^{\prime}}=\sum_{q^{\prime}\geq
0}u_{q^{\prime}}$. We choose to work with (3.35) because it highlights the
dependence on $v_{0}$.

$\displaystyle v=\lim_{q\to\infty}v_{q}=v_{0}+\sum_{q\geq
0}(v_{q+1}-v_{q})=v_{0}+\sum_{q\geq
0}\left(w_{q+1}+(v_{\ell_{q}}-v_{q})\right)\,,$ (3.35)

where we recall the notation (3.2) and (3.3). Indeed, by (3.13), (3.16), and
interpolation, we have that $\left\|w_{q}\right\|_{H^{\beta^{\prime}}}\leq
2\Gamma_{q}^{-1}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\beta^{\prime}}=2\Gamma_{q}^{-1}\lambda_{1}^{\nicefrac{{(b+1)\beta}}{{2}}}\lambda_{q}^{-(\beta-\beta^{\prime})}$
which is summable for $q\geq 0$ whenever $\beta^{\prime}<\beta$. By appealing
to the bound (3.28), we furthermore obtain that
$\left\|v_{\ell_{q}}-v_{q}\right\|_{H^{\beta^{\prime}}}\lesssim\lambda_{q}^{-2}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{\beta^{\prime}}\lesssim\lambda_{1}^{\nicefrac{{(b+1)\beta}}{{2}}}\lambda_{q}^{-2-(\beta-\beta^{\prime})}$,
which is again summable over $q\geq 0$. This justifies the definition of $v$
in (3.35), and the fact that $v\in
C^{0}(0,T;H^{\beta^{\prime}}(\mathbb{T}^{3}))$ for any $\beta^{\prime}<\beta$.
Finally, we note that by additionally appealing to (3.15), which yields
$\|\mathring{R}_{q}\|_{L^{1}}\lesssim\Gamma_{q}^{{-\mathsf{C_{R}}}}\delta_{q+1}\to
0$ as $q\to\infty$, in view of (3.1) the function $v$ defined in (3.35) is a
weak solution of the Euler equations on $[0,T]$.

In order to complete the proof, we return to (3.35) and note that due to