fourierlargesymbols147

# Anomalous dissipation and Euler flows

Jan Burczak  Mathematisches Institut, Leipzig University, Augustusplatz 10,
04109 Leipzig, Germany and Max Planck Institute for Mathematics in the
Sciences, Inselstraße 22-26, 04103 Leipzig, Germany<EMAIL_ADDRESS>leipzig.de ,  László Székelyhidi, Jr  Max Planck Institute for Mathematics in
the Sciences, Inselstraße 22-26, 04103 Leipzig, Germany
<EMAIL_ADDRESS>and  Bian Wu  Max Planck Institute for Mathematics in
the Sciences, Inselstraße 22-26, 04103 Leipzig, Germany<EMAIL_ADDRESS>
###### Abstract.

We show anomalous dissipation of scalar fields advected by a (typical) weak
solution to the Euler equations with $C^{\frac{1}{3}^{-}}$ regularity in the
3D periodic setting.

LSz gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through GZ SZ 325/2-1.

##  1\. Introduction

We consider the Cauchy problem for the linear advection-diffusion equation

$\begin{split}\partial_{t}\rho_{\kappa}+u\cdot\nabla\rho_{\kappa}&=\kappa\Delta\rho_{\kappa}\\\
\rho_{\kappa}|_{t=0}&=\rho_{in},\end{split}$ (1.1)

on the 3-dimensional flat torus $\mathbb{T}^{3}$. The quantity $\rho_{\kappa}$
is a passive scalar and $\kappa>0$ represents molecular diffusivity. We are
interested in the case where the vectorfield $u=u(x,t)$ is a weak solution of
the incompressible Euler equations

$\begin{split}\partial_{t}u+u\cdot\nabla u+\nabla p&=0,\\\ \mathop{\rm
div}\nolimits u&=0.\end{split}$ (1.2)

We study the behavior of

$\kappa\int_{0}^{T}\|\nabla\rho_{\kappa}\|_{L^{2}}^{2}\,dt$ (1.3)

as $\kappa\to 0$. Our main result is the following rigorous verification of
the phenomenon of scalar anomalous dissipation induced by turbulent flows, in
the deterministic setting.

######  Theorem 1.1.

Let $0<\beta<1/3$. There exists a weak solution $u\in
C^{\beta}(\mathbb{T}^{3}\times[0,T])$ to (1.2) such that for every non-zero
initial datum $\rho_{in}\in H^{1}(\mathbb{T}^{3})$ with zero mean, the family
of unique solutions $\\{\rho_{\kappa}\\}_{\kappa>0}$ to (1.1) satisfies

$\limsup_{\kappa\to
0}\quad\kappa\int_{0}^{T}\|\nabla\rho_{\kappa}\|_{L^{2}}^{2}\,dt\geq
c\|\rho_{in}\|_{L^{2}}^{2},$ (1.4)

where $c>0$ depends only on
$\frac{\|\nabla\rho_{in}\|_{L^{2}(\mathbb{T}^{3})}}{\|\rho_{in}\|_{{}_{L^{2}(\mathbb{T}^{3})}}}$
and $T>0$. Moreover, any smooth solution of (1.2) can be uniformly
approximated by such weak solutions $u$.

The inequality (1.4), meaning that as molecular diffusivity $\kappa\to 0$, the
dissipation rate becomes independent of the diffusivity, is referred to as
anomalous dissipation, and is a highly nonlinear feature of the vectorfield
$u$. We remark also that if we drop the requirement that $u$ is a weak
solution of (1.2) and only require $u$ to be divergence-free, then the set of
such vectorfields is $C^{0}$-dense in the space of all time-dependent
divergence-free vectorfields. Also, the vectorfields we construct have the
stronger feature that anomalous dissipation happens on every time interval
contained in $[0,T]$. That is, given $u$ as in Theorem 1.1 and any
$\mathcal{I}\subset[0,T]$, the restriction of $u$ to $\mathcal{I}$ has the
same property, of course with a constant $c>0$ in (1.4) depending on
$\mathcal{I}$.

###  1.1. Context

####  1.1.1. Turbulence

In the 1949 paper [Obu49] titled ’Structure of the Temperature Field in
Turbulent Flow’ A.N. Obukhov writes

In other words, the turbulent motion inside a thermally heterogeneous medium
with gradients which are initially weak can contribute to the local gradients
of temperature, which are subsequently smoothed out by the action of molecular
heat conductivity.

This statement is precisely a prediction that little (’molecular’) diffusion
$\kappa\Delta$ present in (1.1) can be amplified by advection $u\cdot\nabla$
by a turbulent velocity field $u$. The responsible mechanism should be
transport of modes towards higher frequencies (smaller scales), where
eventually diffusivity of a molecular length-scale steps in. This prediction
has been further corroborated by phenomenological, experimental and numerical
arguments, see classical [Cor51], [Bat59], and more recent [SS10], [DSY05],
[SS00]. Furthermore, the expected amplification of diffusivity is supposed to
keep the quantity (1.3) bounded away from zero, a phenomenon referred to as
’anomalous dissipation’. Thus, it is strongly believed that anomalous
dissipation is a key feature of turbulence.

On the other hand, it is argued that turbulent flows are irregular. More
precisely, in his famous 1949 note on statistical hydrodynamics [Ons49], L.
Onsager conjectured that the threshold regularity for the validity of energy
conservation in the class of Hölder continuous weak solutions of the Euler
equations is the exponent $1/3$. Onsager’s interest in this issue came from an
effort to explain the primary mechanism of energy dissipation in turbulence,
one that prevails even in the absence of viscosity. Thus, the implicit
suggestion was that Hölder continuous weak solutions of the Euler equations
may be an appropriate mathematical description of turbulent flows in the
inviscid limit.

Putting these two pieces together, considering non-smooth velocity fields
advecting a scalar is of primary interest for turbulence.

There are several analytical models of scalar turbulence, including the 1968
Kraichnan model [Kra68] (seemed nowadays as rather incompatible, compare the
relevant discussion and references of [AV23]) and the more recent alternating
shear flows, discussed in more detail below.

####  1.1.2. Anomalous dissipation in PDEs

For any fixed initial datum, the energy solutions of (1.1) converge (along
subseqences) to a distributional solution of the transport equation
$\partial_{t}\rho+u\cdot\nabla\rho=0$. These are given by a uniquely defined
and measure preserving flow map, provided the drift is regular enough
(classically $u\in L^{1}(W^{1,\infty})$, but also $u\in L^{1}(W^{1,1})$
suffices [DL89]) and divergence-free. In such a case, anomalous dissipation is
excluded, since measure preservation implies in particular conservation of the
$L^{2}$ norm.

On the other hand, below the regularity threshold which required for a well-
posedness theory of the transport equation, there are examples of divergence
free velocity fields that give rise to anomalous dissipation. We mention the
recent [EL23], where $u\in C^{\infty}([0,T];C^{\alpha}(\mathbb{T}^{2}))$,
$\alpha<1$ (in fact it is merely logarithmically below Lipschitz regularity)
and earlier [DEIJ22], where $u\in L^{1}(C^{\alpha}),\alpha<1$. The
constructions of $u$ in the aforementioned articles are essentially based on
(i) shear flows alternating in time in a quasi self-similar fashion, which by
’stirring’ push towards smaller scales, where molecular diffusion takes over
(inspired by [Dep03, Pie94, Aiz78]) and on (ii) comparison with the inviscid
case [CZDE20a].

It is worth mentioning the recent papers [BDL23], [BCC+22], partially drawing
from [JY22, IJJ22], that manage to embed the above mentioned passive scalar
methodology into nonlinear fluid dynamics (Navier-Stokes equations), at the
cost of an arising forcing term.

A different approach based on _iterated quantitative homogenization_ has been
recently proposed by Armstrong-Vicol in [AV23]. Although both iterated
quantitative homogenization (see e.g. [NSX20]) and homogenization in the
present context of passive tracers in turbulent flows [MPP85] has been studied
previously, in [AV23] this idea was successfully combined with an infinite
iterative scheme, and has substantially inspired our work. The methodology of
[AV23] has several advantages over the previous ones: First of all, it
abandons the need of the aforementioned comparison with the inviscid case,
consequently becomes compatible with the existence of inertial range in
turbulence. Secondly, in works based on alternating shear flows focussing
towards a singular time, the anomalous dissipation actually occurs at only one
instant of time. This seems a major drawback in the context of turbulent flows
in light of the statistical time-invariance. Thirdly, as in classical
homogenization, the method allows for enhanced and anomalous dissipation of
arbitrary initial tracer fields, in contrast with previous constructions based
on convex integration [MS18, MS19, BCDL21, MS20], where the tracer is
constructed in parallel to the velocity field scale-by-scale.

The main step forward that we make compared with [AV23] is that our velocity
field solves incompressible Euler equation, thus providing the first
deterministic link on the PDE level between the Obukhov-Corrsin theory and
fluid mechanics. Indeed, to our knowledge this is the first result on
anomalous dissipation, where the advecting vectorfield is itself the solution
of a PDE. In general, we interpret iterative quantitative homogenization in
parallel with convex integration, with the basic principle in mind that whilst
convex integration is a form of ”inverse renormalization” strategy, iterative
quantitative homogenization can be seen as ”forward renormalization”.

As an interesting byproduct of our approach, which intertwines convex
integration techniques with iterated homogenization, we construct not just a
single velocity field exhibiting anomalous dissipation, but a $C^{0}$-dense
set of such fields. Moreover, in our approach we abandon the shear-flow
setting, giving us more flexibility in constructed such vectorfields.

####  1.1.3. Related topics: enhanced dissipation, mixing, and beyond

Within or above Lipschitz regularity for $u$, divergence-free drift can of
course also assist dissipation (while it cannot counteract it), but in a
weaker sense than anomalous dissipation (roughly, $T$ must grow at at rate
commensurate with regularity), which is referred to as enhanced dissipation.

The enhanced dissipation by a single shear flow, or a related simple ’lower
dimensional’ flow is well studied, with roots in the computations by Kelvin
and Kolmogorov [Kol34], compare [ABN22], [BCZ17], [CZEW20], [ZG23], [CCZW21]
and their references. In such cases, one needs to restrict initial datum to a
class not orthogonal to the shearing direction, but the setting usually allows
to extract precise decay rates. For a more general related approach see
[Vuk21]. The general condition in the Lipschitz case has been provided in the
seminal [CKRZ08] (albeit without clear rate, for an optimal example see
[ELM23]), whose results has been recently slightly strenghtened and reproved
by a different method in [Wei21].

Let us finally point out, without getting into details, natural connections
between enhanced dissipation and mixing (which is the phenomenon of inviscid-
case migration towards smaller scales caused by a vector field), see [CZDE20b]
and the recent survey [ZCIM23] with its references. There are also interesting
developments in the stochastic setting, see [BBPS22a], [BBPS22b] and [HPZZ23],
in the kinetic setting (Landau damping) cf [MV11], and in suppression of
chemotactic blowup, including recent beautiful [HKY23]. In this context we
mention the recent work [CMOW23], where optimal quantitative enhanced
dissipation estimates are obtained for random drift given by the Gaussian free
field - the strategy here, based on a renormalization strategy of computing
effective diffusivities scale-by-scale, is very much reminiscent of our point
of view.

###  1.2. Notation

We use $\mathbb{R}^{3}:=\mathbb{R}^{3\times 1}$. For two vectors
$v_{1},v_{2}\in\mathbb{R}^{3}$, we use $v_{1}^{T}v_{2}$ or $v_{1}\cdot v_{2}$
to denote their inner product. For an invertible matrix
$A\in\mathbb{R}^{3\times 3}$, we use $A^{T}$ to denote its transpose, $A^{-1}$
to denote its inverse and $A^{-T}$ to denote the inverse of its transpose. For
a vector-value function
$\chi=(\chi_{1},\chi_{2},\chi_{3})^{T}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$,
we use $\nabla\chi$ to denote
$(\nabla\chi_{1},\nabla\chi_{2},\nabla\chi_{3})^{T}$. We use
$\boldsymbol{\alpha}$ to denote the multiindex in partial derivative and we
use $|\boldsymbol{\alpha}|$ to denote its total order of differentiation.

##  2\. Outline of proof of Theorem 1.1

In this section we provide an overview of the key steps in our proof, and
postpone technical details to subsequent sections. As mentioned in the
introduction, our strategy involves a ”backward renormalization” in
constructing the velocity field in Section 2.1 (in the sense of an iteration
from large-scale to small-scale, $u_{q}\mapsto u_{q+1}\mapsto\dots$), and a
”forward renormalization” in constructing the passive tracer in Section 2.2
(in the sense of an iteration from small-scale to large-scale,
$\rho_{q+1}\mapsto\rho_{q}\mapsto\dots$).

We have tried to present the proofs in later sections in a self-contained
manner, because we believe these may be of independent interest.

###  2.1. Construction of the vectorfield $u$ \- Convex integration

The general scheme for producing Hölder continuous weak solutions of the Euler
equations (1.2) is by now well understood. One proceeds via an inductive
process on a sequence of approximate solutions $u_{q}$ and associated Reynolds
defect $\mathring{R}_{q}$ and pressure $p_{q}$, $q=0,1,2,\dots$, which satisfy
the _Euler-Reynolds system_

$\begin{split}\partial_{t}u_{q}+\mathop{\rm div}\nolimits(u_{q}\otimes
u_{q})+\nabla p_{q}&=\mathop{\rm div}\nolimits\mathring{R}_{q}\,,\\\
\mathop{\rm div}\nolimits u_{q}&=0\,,\end{split}$ (2.1)

with constraints

$\mathrm{tr\,}\mathring{R}_{q}(x,t)=0,\quad\int_{\mathbb{T}^{3}}u_{q}(x,t)\,dx=0,\quad\int_{\mathbb{T}^{3}}p_{q}(x,t)\,dx=0.$
(2.2)

We note in passing that these normalizations determine the pressure $p_{q}$
uniquely from $(u_{q},\mathring{R}_{q})$ so that one may speak of the pair
$(u_{q},\mathring{R}_{q})$ being a solution of (2.1).

####  2.1.1. Inductive assumptions

The induction process involves a set of _inductive estimates_. These estimates
are in terms of a frequency parameter $\lambda_{q}$ and amplitude
$\delta_{q}$, which are given by

$\lambda_{q}:=2\pi\lceil
a^{(b^{q})}\rceil,\quad\delta_{q}:=\lambda_{q}^{-2\beta}$ (2.3)

where $\lceil x\rceil$ denotes the smallest integer $n\geq x$, $a\gg 1$ is a
large parameter, $b>1$ is close to $1$ and $0<\beta<\nicefrac{{1}}{{3}}$ is
the exponent of Theorem 1.1. The parameters $a$ and $b$ are then related to
$\beta$. With these parameters the inductive estimates take the form111In
[BLSV19] an additional estimate is added for $\|u_{q}\|_{C^{0}}$, but it turns
out this can be avoided. The only place where a bound on $\|u_{q}\|_{C^{0}}$
was needed in [BLSV19] is in [BLSV19, Proposition 5.9], but as we show below,
the (much worse) bound induced by (2.5) for $n=1$ suffices to control the
relevant terms - see Proposition 6.3 and Lemma 6.15.

$\displaystyle\left\|\mathring{R}_{q}\right\|_{C^{0}}$
$\displaystyle\leq\delta_{q+1}\lambda_{q}^{-\gamma_{R}},$ (2.4)
$\displaystyle\left\|u_{q}\right\|_{C^{n}}$ $\displaystyle\leq
M\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{n}\quad\textrm{ for
}n=1,2,\dots,\bar{N},$ (2.5)
$\displaystyle\left|e(t)-\int_{\mathbb{T}^{3}}\left|u_{q}\right|^{2}\,dx-\bar{e}\delta_{q+1}\right|$
$\displaystyle\leq\delta_{q+1}\lambda_{q}^{-\gamma_{E}},$ (2.6)

where $\gamma_{R},\gamma_{E}>0$ are small parameters, $\bar{N}\in\mathbb{N}$
is a large parameter to be chosen suitably (depending on $\beta>0$ and $b>1$),
and $\bar{e}>0$, $M\geq 1$ are universal constants, which will be fixed
throughout the iteration and depend on the particular geometric form of the
perturbing building blocks - see they will be specified in Definition 6.10 and
6.14. We remark that in [BLSV19] only the case $\bar{N}=1$ is required in
(2.5) and (2.6) is slightly weaker. Moreover, in [BLSV19] a generic small
parameter $\alpha>0$ is used in place of $\gamma_{R},\gamma_{E}$, but for our
purposes we need to choose these small corrections more carefully.

####  2.1.2. Parameter choices

The inductive construction for passing from $u_{q}$ to $u_{q+1}$ involves
three steps: _mollification_ , _gluing_ and _perturbation_. In these steps two
more scales are introduced, an adjusted length-scale $\ell_{q}$ and an
adjusted time-scale $\tau_{q}$. In our case these will be defined as

$\ell_{q}:=\lambda_{q}^{-1-\gamma_{L}},\quad\tau_{q}:=\lambda_{q}^{-1+\beta-\gamma_{T}},$
(2.7)

where $\gamma_{L},\gamma_{T}$ are additional small parameters to be chosen
suitably in dependence of $\beta,b$. It is worth pointing out that, if one
were to set $\gamma_{L}=\gamma_{T}=0$, then $\ell_{q},\tau_{q}$ would be the
natural (dimensionally consistent) length- and time-scales induced by the
velocity field $u_{q}$ (c.f. (2.3) and (2.5)). Indeed, we can think of
$\delta_{q}^{\nicefrac{{1}}{{2}}}$ having physical dimension of velocity, i.e.
$LT^{-1}$, and $\lambda_{q}^{-1}$ having physical dimension of length $L$.

Moreover, setting $\gamma_{R}=0$ would be the consistent estimate in light of
the basic principle in convex integration, that the error $R_{q}$ is cancelled
by the new average stress
$\langle(u_{q+1}-u_{q})\otimes(u_{q+1}-u_{q})\rangle$.

In addition to these small parameters we will also use $\alpha>0$, as is also
done in [BLSV19], to take care of the lack of Schauder estimates in
$C^{0},C^{1},\dots$ spaces. Thus, in summary, we will use the following set of
additional parameters, which will all be chosen depending on $\beta,b$:

  * •

$\gamma_{R}\in(0,1)$: smallness of $\|\mathring{R}_{q}\|_{C^{0}}$ with respect
to $\|R_{q}\|_{C^{0}}$ ;

  * •

$\gamma_{L}\in(0,1)$: smallness of $\ell_{q}$ with respect to
$\lambda_{q}^{-1}$;

  * •

$\gamma_{T}\in(0,1)$: smallness of $\tau_{q}$ with respect to
$(\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q})^{-1}$;

  * •

$\gamma_{E}\in(0,1)$: smallness of energy gap with respect to
$\|R_{q}\|_{C^{0}}$;

  * •

$\alpha\in(0,1)$: Schauder exponent;

  * •

$\bar{N}\in\mathbb{N}$: number of derivatives in the induction.

With a suitable choice of these parameters we have the following analogue of
[BLSV19, Proposition 2.1]:

######  Proposition 2.1.

There exist universal constants $M\geq 1$, $\bar{e}>0$ with the following
property. Assume $0<\beta<\frac{1}{3}$ and

$1<b<\frac{1-\beta}{2\beta}\,.$ (2.8)

Further, assume that $\gamma_{T},\gamma_{R},\gamma_{E}>0$ satisfy

$\max\\{\gamma_{T}+b\gamma_{R},\gamma_{E}\\}<(b-1)\bigl{(}1-(2b+1)\beta\bigr{)}$
(2.9)

and let $e:[0,T]\to\mathbb{R}$ be a strictly positive smooth function. Then
there exists $\gamma_{L}>0$, $\bar{N}\in\mathbb{N}$ and $\alpha_{0}>0$
depending on $\beta,b,\gamma_{T},\gamma_{R},\gamma_{E}$ such that, for any
$0<\alpha<\alpha_{0}$, there exists $a_{0}\gg 1$ (in addition depending on
$e$), such that for any $a\geq a_{0}$ the following holds:

Let $(u_{q},\mathring{R}_{q})$ be a smooth solution of (2.1) satisfying the
estimates (2.4)–(2.6). Then there exists another solution
$(u_{q+1},\mathring{R}_{q+1})$ to (2.1) satisfying (2.4)–(2.6) with $q$
replaced by $q+1$, and we have

$\left\|u_{q+1}-u_{q}\right\|_{C^{0}}+\frac{1}{\lambda_{q+1}}\left\|u_{q+1}-u_{q}\right\|_{C^{1}}\leq
M\delta_{q+1}^{\nicefrac{{1}}{{2}}}.$ (2.10)

The new velocity field $u_{q+1}$ is obtained as

$u_{q+1}:=\bar{u}_{q}+w_{q+1},$ (2.11)

where $\bar{u}_{q}$ is constructed from $u_{q}$ via Isett’s gluing procedure
[Ise18], and $w_{q+1}$ is the new perturbation consisting of a deformed family
of Mikado flows. In Sections 2.1.3-2.1.4 below we now sketch the proof of
Proposition 2.1. The detailed proof, which is very much based on the proof in
[BLSV19], is deferred to Section 6.

####  2.1.3. Gluing procedure

The gluing procedure amounts to the following statement:

######  Proposition 2.2.

Within the setting of Proposition 2.1 we have the following statement. Let
$(u_{q},\mathring{R}_{q})$ be a smooth solution of (2.1) satisfying the
estimates (2.4)–(2.6). Then there exists another solution
$(\bar{u}_{q},\mathring{\bar{R}}_{q})$ to (2.1) such that

$\mathop{\rm
supp}\nolimits\mathring{\bar{R}}_{q}\subset\mathbb{T}^{3}\times\bigcup_{i\in\mathbb{N}}(i\tau_{q}+\tfrac{1}{3}\tau_{q},i\tau_{q}+\tfrac{2}{3}\tau_{q})$
(2.12)

and the following estimates hold for any $N\geq 0$:

$\displaystyle\|\bar{u}_{q}\|_{C^{N+1}}$
$\displaystyle\lesssim\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\ell_{q}^{-N}\,,$
(2.13) $\displaystyle\|\mathring{\bar{R}}_{q}\|_{C^{N+\alpha}}$
$\displaystyle\lesssim\mathring{\delta}_{q+1}\ell_{q}^{-N-2\alpha}\,,$ (2.14)
$\displaystyle\|(\partial_{t}+\bar{u}_{q}\cdot\nabla)\mathring{\bar{R}}_{q}\|_{C^{N+\alpha}}$
$\displaystyle\lesssim\tau_{q}^{-1}\mathring{\delta}_{q+1}\ell_{q}^{-N-2\alpha}\,,$
(2.15)
$\displaystyle\left|\int_{\mathbb{T}^{3}}|\bar{u}_{q}|^{2}-|u_{q}|^{2}\,dx\right|$
$\displaystyle\lesssim\mathring{\delta}_{q+1}\,.$ (2.16)

Moreover, the vector potentials of $u_{q}$ and $\bar{u}_{q}$ satisfy

$\|z_{q}-\bar{z}_{q}\|_{C^{\alpha}}\lesssim\tau_{q}\mathring{\delta}_{q+1}\ell_{q}^{-\alpha}\,.$
(2.17)

Here and in the sequel, the vector potential $z$ of a divergence-free velocity
field $u$ is given by the Biot-Savart operator on $\mathbb{T}^{3}$, defined as
$\mathcal{B}=(-\Delta)^{-1}\mathop{\rm curl}\nolimits$, so that
$z=\mathcal{B}u$ is the unique solution of

$\mathop{\rm div}\nolimits z=0\qquad\textrm{ and }\qquad\mathop{\rm
curl}\nolimits z=u\,,$ (2.18)

recalling that we assume zero spatial average $\int_{\mathbb{T}^{3}}u(x,t)dx$.

Proposition 2.2 is restated below as Corollary 6.7 and is a direct consequence
of Corollary 6.4 and Proposition 6.6.

####  2.1.4. The perturbation

The formula for the new perturbation $w_{q+1}$ uses Mikado flows, which we
recall here briefly. Mikado flows were introduced originally in [DS17] and
widely used since in applications of convex integration to fluid dynamics.

We fix a finite set $\Lambda\subset\mathbb{R}^{3}$ consisting of nonzero
vectors $\vec{k}\in\mathbb{R}^{3}$ with rational coordinates, and for each
$\vec{k}\in\Lambda$ let $\varphi_{\vec{k}}$ be a periodic function with the
properties

  * •

$\vec{k}\cdot\nabla\varphi_{\vec{k}}=0$,

  * •

For any $\vec{k}\neq\vec{k}^{\prime}\in\Lambda$ we have $\mathop{\rm
supp}\nolimits\varphi_{\vec{k}}\cap\mathop{\rm
supp}\nolimits\varphi_{\vec{k}^{\prime}}=\emptyset$,

  * •

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\varphi_{\vec{k}}(\xi)\,d\xi=0$ and
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}|\Delta\varphi_{\vec{k}}(\xi)|^{2}\,d\xi=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}|\nabla\varphi_{\vec{k}}(\xi)|^{2}\,d\xi=1$.

Next, let $\psi_{\vec{k}}=\Delta\varphi_{\vec{k}}$ and define

$W_{\vec{k}}(\xi)=\psi_{\vec{k}}(\xi)\vec{k},\quad
U_{\vec{k}}(\xi)=\vec{k}\times\nabla\varphi_{\vec{k}},$ (2.19)

so that222Here we use the vector calculus identity $\mathop{\rm
curl}\nolimits(F\times G)=F\mathop{\rm div}\nolimits G-G\mathop{\rm
div}\nolimits F+(G\cdot\nabla)F-(F\cdot\nabla)G$. $\mathop{\rm curl}\nolimits
U_{\vec{k}}=W_{\vec{k}}$ and, for any $a_{\vec{k}}$, $\vec{k}\in\Lambda$, the
vectorfield $W=\sum_{\vec{k}\in\Lambda}a_{\vec{k}}W_{\vec{k}}$ satisfies

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
W\,d\xi=0,\quad\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int W\otimes
W\,d\xi=\sum_{\vec{k}\in\Lambda}a_{\vec{k}}^{2}\vec{k}\otimes\vec{k}.$

Further, we define $H_{\vec{k}}=H_{\vec{k}}(\xi)$ to be the antisymmetric
zero-mean matrix with the property that, for any $v\in\mathbb{R}^{3}$ we have
$H_{\vec{k}}v=U_{\vec{k}}\times v$.

The following lemma (originating in the work of Nash [Nas54]) is crucial:

######  Lemma 2.3.

For any compact set $\mathcal{N}\subset\mathcal{S}^{3\times 3}_{+}$ there
exists a finite $\Lambda\subset\mathbb{Q}^{3}$ such that there exists smooth
functions $a_{\vec{k}}:\mathcal{N}\to\mathbb{R}_{+}$ with

$\sum_{\vec{k}\in\Lambda}a_{\vec{k}}^{2}(R)\vec{k}\otimes\vec{k}=R\quad\textrm{
for any }R\in\mathcal{N}.$ (2.20)

The corresponding vectorfield
$W=W(R,\xi)=\sum_{\vec{k}\in\Lambda}a_{\vec{k}}(R)W_{\vec{k}}(\xi)$ is called
a Mikado flow. In the following we will fix
$\mathcal{N}:=B_{1/2}(\mathrm{Id})$, the metric ball of radius $1/2$ around
the identity matrix in $\mathcal{S}^{3\times 3}_{+}$ and denote the
corresponding Mikado vectorfield by $W=W(R,\xi)$.

Following [BLSV19] the new perturbation is then defined as

$w_{q+1}=\frac{1}{\lambda_{q+1}}\mathop{\rm
curl}\nolimits\left[\sum_{i\in\mathbb{N}}\sum_{\vec{k}\in\Lambda}\eta_{i}\sigma_{q}^{\nicefrac{{1}}{{2}}}a_{\vec{k}}(\tilde{R}_{q,i})\nabla\Phi_{i}^{T}U_{\vec{k}}(\lambda_{q+1}\Phi_{i})\right]\,,$
(2.21)

where

  * •

$\eta_{i}=\eta_{i}(x,t)$, $i\in\mathbb{N}$, are smooth nonnegative cutoff
functions with pairwise disjoint supports such that

$\|\partial_{t}^{m}\nabla^{n}_{x}\eta_{i}\|_{L^{\infty}}\leq
C_{n,m}\tau_{q}^{-m},\quad\sum_{i}\eta_{i}^{2}(x,t)=\bar{\eta}^{2}(x,\tau_{q}^{-1}t)\,,$
(2.22)

where $\bar{\eta}=\bar{\eta}(x,t)$ is a (universal) function, $1$-periodic in
$t$, such that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\bar{\eta}^{2}(x,t)\,dx=c_{0},\quad\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0}^{1}\bar{\eta}^{2}(x,t)\,dt=c_{1}$

for some universal constants $c_{0},c_{1}>0$.

  * •

$\sigma_{q}=\sigma_{q}(t)$ is a positive scalar function with the property

$|\sigma_{q}(t)-c_{1}^{-1}\delta_{q+1}|\leq
C\delta_{q+1}\lambda_{q}^{-\min\\{\gamma_{E},\gamma_{R},(b-1)\beta\\}}\leq\tfrac{1}{3c_{1}}\delta_{q+1}$
(2.23)

  * •

The maps $\Phi_{i}=\Phi_{i}(x,t)$ are the volume-preserving diffeomorphisms
defined as the inverse flow map of the velocity field $\bar{u}_{q}$, which
satisfy for $(x,t)\in\mathop{\rm supp}\nolimits\eta_{i}$:

$\|\nabla\Phi_{i}(x,t)\|_{C^{n}}+\|(\nabla\Phi_{i})^{-1}(x,t)\|_{C^{n}}\leq
C_{n}\ell_{q}^{-n},\quad\|\nabla\Phi_{i}-\mathrm{Id}\|_{L^{\infty}}\leq
C\lambda_{q}^{-\gamma_{T}}\,.$ (2.24)

  * •

$\tilde{R}_{q,i}=\tilde{R}_{q,i}(x,t)$, $i\in\mathbb{N}$, satisfies

$\tilde{R}_{q,i}=\nabla\Phi_{i}(\mathrm{Id}-\sigma_{q}^{-1}\mathring{\bar{R}}_{q})\nabla\Phi_{i}^{T},\quad\|\tilde{R}_{q,i}\|_{C^{n}}\leq
C_{n}\ell_{q}^{-n}\,.$ (2.25)

These properties will be obtained in Section 6.3, where we will conclude the
proof of Proposition 2.1.

###  2.2. Enhanced dissipation via iterative stages

We define an additional $q$-dependent parameter which plays a key role in this
paper:

$\kappa_{q}:=\lambda_{q}^{-\theta},\quad\theta=\frac{2b}{b+1}(1+\beta)\,.$
(2.26)

Using (2.3) it is easy to verify the recursive identity

$\kappa_{q}=\frac{\delta_{q+1}}{\lambda_{q+1}^{2}\kappa_{q+1}}.$ (2.27)

In turn, this identity indicates that we can think of $\kappa_{q}$ has having
physical dimension of diffusion coefficient $L^{2}T^{-1}$.

We consider the sequence of advection-diffusion equations on
$\mathbb{T}^{3}\times[0,T]$:

$\begin{split}\partial_{t}\rho_{q}+u_{q}\cdot\nabla\rho_{q}&=\kappa_{q}\Delta\rho_{q}\,,\\\
\rho_{q}|_{t=0}&=\rho_{in}\,,\end{split}$ (2.28)

where $u_{q}$ is the sequence of velocity fields obtained in Section 2.1 via
Proposition 2.1. We are interested in comparing the cumulative dissipation for
subsequent values of $q$, given by

$D_{q}:=\kappa_{q}\int_{0}^{T}\|\nabla\rho_{q}\|_{L^{2}}^{2}\,dt=\tfrac{1}{2}(\|\rho_{in}\|_{L^{2}}^{2}-\|\rho_{q}(T)\|_{L^{2}}^{2}).$
(2.29)

Our main result is

######  Proposition 2.4.

Let $0<\beta<\frac{1}{3}$ and $b>1$ as in (2.8). Assume that
$\gamma_{L},\gamma_{T},\gamma_{R}\in(0,1)$ satisfy

$\gamma_{T}<\frac{b-1}{b+1}(1-(2b+1)\beta)<\gamma_{R}+\gamma_{T},$ (2.30)
$2\gamma_{L}<\frac{b-1}{b+1}(1+\beta).$ (2.31)

Then there exists $\gamma>0$ and $\alpha_{1}>0$ such that for all
$0<\alpha<\alpha_{1}$ there exists $\tilde{N}\in\mathbb{N}$ and
$q_{0}\in\mathbb{N}$ with the following property.

For any $q\geq q_{0}$ and any initial datum $\rho_{in}\in
L^{2}(\mathbb{T}^{3})$ with $\int_{\mathbb{T}^{3}}\rho_{in}\,dx=0$ such that

$\|\rho_{in}\|_{H^{n}}\leq\lambda_{q}^{n}\min\\{D_{q}^{\nicefrac{{1}}{{2}}},D_{q+1}^{\nicefrac{{1}}{{2}}}\\}\qquad\textrm{
for }1\leq n\leq\tilde{N},$ (2.32)

we have

$|D_{q+1}-D_{q}|\leq\tfrac{1}{2}\lambda_{q}^{-\gamma}D_{q}.$ (2.33)

###### Proof.

We start by fixing $\gamma,\alpha_{1}>0$. The exponent $\gamma>0$ is
determined by the inequalities

$\begin{split}\gamma<\tfrac{1}{4}\min\Bigl{\\{}\gamma_{T},\gamma_{R},\gamma_{E},(b-1)\beta,(b-1)\theta,\gamma_{T}+\gamma_{R}+(2b-1)\beta+1-\theta,\\\
\frac{b-1}{b+1}(1-(2b+1)\beta)-\gamma_{T}\Bigr{\\}}.\end{split}$ (2.34)

We remark that the last condition in the first line above can be satisfied
because of the right inequality in (2.30). In turn, $\alpha_{1}$ is then
chosen sufficiently small so that

$\alpha_{1}(1+\gamma_{L})+\gamma+\theta<\gamma_{T}+\gamma_{R}+(2b-1)\beta+1.$
(2.35)

We also need to choose

$\begin{split}\tilde{N}\geq\max\Bigl{\\{}(b+1)\left(\frac{2}{b-1}+\frac{\theta}{b}\right),\,1+\frac{2b(b-1)(1+\beta)}{\gamma_{T}(b+1)}\Bigr{\\}},\end{split}$
(2.36)

where the first lower bound is given by $N_{h}$ in (4.18) and the second lower
bound is given by $N$ to validate (A1) in section 5.

_Step 1: Preparation ( $\rho_{q+1}\leadsto\tilde{\rho}_{q+1}$)_

We start with the equation

$\begin{split}\partial_{t}\rho_{q+1}+u_{q+1}\cdot\nabla\rho_{q+1}&=\kappa_{q+1}\Delta\rho_{q+1}\,,\\\
\rho_{q+1}|_{t=0}&=\rho_{in}\,,\end{split}$ (2.37)

and recall the form $u_{q+1}=\bar{u}_{q}+w_{q+1}$ of the advecting field
$u_{q+1}$, where $w_{q+1}$ is given in (2.21). Then we make use of the linear
algebra identity $(Au)\times(Av)=\textrm{cof }A(u\times v)=\det
AA^{-T}(u\times v)$ for any $3\times 3$ matrix $A$ and vectors
$u,v\in\mathbb{R}^{3}$. From this we deduce that if $H$ is the antisymmetric
matrix such that $Hv=u\times v$ and $\det A=1$, then $\tilde{H}=A^{-1}HA^{-T}$
is the antisymmetric matrix such that $\tilde{H}v=(A^{T}u)\times v$. In
particular, using the antisymmetric matrix-valued functions $H_{\vec{k}}(\xi)$
introduced above in Section 2.1.4, we deduce

$(\nabla\Phi^{-1}_{i}H_{\vec{k}}\nabla\Phi^{-T}_{i})v=(\nabla\Phi_{i}^{T}U_{\vec{k}})\times
v.$

Therefore, using the identity $\mathop{\rm
div}\nolimits(z\times\nabla\rho)=(\mathop{\rm curl}\nolimits
z)\cdot\nabla\rho$, we can write equation (2.37) equivalently as

$\begin{split}\partial_{t}\rho_{q+1}+\bar{u}_{q}\cdot\nabla\rho_{q+1}&=\mathop{\rm
div}\nolimits A_{q+1}\nabla\rho_{q+1}\,,\\\
\rho_{q+1}|_{t=0}&=\rho_{in}\,,\end{split}$ (2.38)

where the elliptic matrix $A_{q+1}=A_{q+1}(x,t)$ is defined as

$A_{q+1}(x,t)=\kappa_{q+1}\mathrm{Id}+\frac{\eta_{i}(x,t)}{\lambda_{q+1}}\sum_{i}\nabla\Phi_{i}^{-1}(x,t)H^{(i)}(x,t,\lambda_{q+1}\Phi_{i})\nabla\Phi_{i}^{-T}(x,t)\,,$
(2.39)

and

$\displaystyle H^{(i)}(x,t,\xi)$
$\displaystyle=\sum_{\vec{k}\in\Lambda}\sigma_{q}^{\nicefrac{{1}}{{2}}}(t)a_{\vec{k}}(\tilde{R}_{q,i}(x,t))H_{\vec{k}}(\lambda_{q+1}\xi)\,.$

Let $(\tilde{\eta}_{i})_{i}$ be a partition of unity such that
$\tilde{\eta}_{i}\eta_{i}=\eta_{i}$, $\tilde{\eta}_{i}\eta_{j}=0$ for $j\neq
i$ and satisfying estimates of the same type as (2.22):

$\|\partial_{t}^{m}\nabla^{n}_{x}\tilde{\eta}_{i}\|_{L^{\infty}}\leq\tilde{C}_{n,m}\tau_{q}^{-m}.$
(2.40)

Define the elliptic matrix

$\tilde{A}_{q+1}(x,t)=\sum_{i}\tilde{\eta}_{i}(x,t)\nabla\Phi_{i}^{-1}(x,t)\left[\kappa_{q+1}\mathrm{Id}+\frac{\eta_{i}(x,t)}{\lambda_{q+1}}H^{(i)}(x,t,\lambda_{q+1}\Phi_{i}(x,t))\right]\nabla\Phi_{i}^{-T}(x,t)\,.$
(2.41)

The estimate (2.24) implies

$\|A_{q+1}-\tilde{A}_{q+1}\|_{L^{\infty}}\leq\kappa_{q+1}\sum_{i}\tilde{\eta}_{i}\|\nabla\Phi_{i}^{-1}\nabla\Phi_{i}^{-T}-\mathrm{Id}\|_{L^{\infty}}\leq
C\kappa_{q+1}\lambda_{q}^{-\gamma_{T}}.$ (2.42)

Therefore we can compare the cumulative dissipation in (2.38) with that in

$\begin{split}\partial_{t}\tilde{\rho}_{q+1}+\bar{u}_{q}\cdot\nabla\tilde{\rho}_{q+1}&=\mathop{\rm
div}\nolimits\tilde{A}_{q+1}\nabla\tilde{\rho}_{q+1}\,,\\\
\tilde{\rho}_{q+1}|_{t=0}&=\rho_{in}\,.\end{split}$ (2.43)

More precisely, assuming that $q_{0}$ is sufficiently large and $q\geq q_{0}$,
we can ensure
$C\lambda_{q}^{-\gamma_{T}}<\tfrac{1}{2}\lambda_{q}^{-\nicefrac{{\gamma_{T}}}{{2}}}$,
and then apply Proposition 3.3 with
$\varepsilon=\tfrac{1}{2}\lambda_{q}^{-\nicefrac{{\gamma_{T}}}{{2}}}<\tfrac{1}{2}$
to conclude

$\left|\|\rho_{q+1}(T)\|^{2}_{L^{2}}-\|\tilde{\rho}_{q+1}(T)\|_{L^{2}}^{2}\right|\lesssim\lambda_{q}^{-\nicefrac{{\gamma_{T}}}{{2}}}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\tilde{\rho}_{q+1}(T)\|_{L^{2}}^{2}\right|.$

We may also write this as

$\left|D_{q+1}-\tilde{D}_{q+1}\right|\lesssim\lambda_{q}^{-\nicefrac{{\gamma_{T}}}{{2}}}\tilde{D}_{q+1},$
(2.44)

where
$\tilde{D}_{q+1}=\tfrac{1}{2}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\tilde{\rho}_{q+1}(T)\|_{L^{2}}^{2}\right|$.

_Step 2: Spatial homogenization (
$\tilde{\rho}_{q+1}\leadsto\bar{\rho}_{q}^{(1)}$)_

In the second step, we use classical estimates in quantitative homogenization
and an explicit formula for the corrector, to replace (2.43) by the
homogenized equation

$\begin{split}\partial_{t}\bar{\rho}_{q}^{(1)}+\bar{u}_{q}\cdot\nabla\bar{\rho}^{(1)}_{q}&=\mathop{\rm
div}\nolimits\bar{A}_{q}\nabla\bar{\rho}^{(1)}_{q}\,,\\\
\bar{\rho}_{q}^{(1)}|_{t=0}&=\rho_{in}\,.\end{split}$ (2.45)

The homogenized elliptic coefficient is given, like in classical
homogenization theory, by

$\bar{A}_{q}(x,t)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\tilde{A}_{q+1}(x,t,\xi)\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}(x,t)\nabla\Phi_{i}^{T}(x,t)\nabla_{\xi}\chi_{i}^{T}(x,t,\xi)\Big{)}\,d\xi\,$

with corrector
$\chi_{i}:\mathbb{T}^{3}\times[0,T]\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$.
We will show below in Section 4 that, because of the special structure of the
oscillating vectorfield $w_{q+1}$, $\chi$ can be defined _explicitly_ by

$\chi_{i}(x,t,\xi)=-\frac{\sigma_{q}^{1/2}(t)}{\kappa_{q+1}\lambda_{q+1}}\nabla\Phi_{i}^{-1}(x,t)\sum_{\vec{k}}a_{\vec{k}}\big{(}\tilde{R}_{q,i}(x,t)\big{)}\varphi_{\vec{k}}(\xi)\vec{k}\,.$
(2.46)

Using the properties of Mikado flows in Section 2.1.4, this formula allows us
to obtain an explicit expression for $\bar{A}_{q+1}$. Indeed, since both
$H^{(i)}$ and $\chi_{i}$ satisfy $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int H^{(i)}\,d\xi=0$,
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\chi_{i}\,d\xi=0$, we have

$\bar{A}_{q}=\kappa_{q+1}\sum_{i}\tilde{\eta}_{i}\nabla\Phi_{i}^{-1}\nabla\Phi_{i}^{-T}+\frac{1}{\lambda_{q+1}}\sum_{i}\eta_{i}^{2}\nabla\Phi_{i}^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int H^{(i)}(\nabla_{\xi}\chi_{i})^{T}\,d\xi.$

On the other hand, using (2.46) and the definition of $H^{(i)}(x,t,\xi)$,

$\displaystyle H^{(i)}(\nabla_{\xi}\chi_{i})^{T}$
$\displaystyle=-\frac{\sigma_{q}}{\lambda_{q+1}\kappa_{q+1}}\sum_{\vec{k}}a_{\vec{k}}^{2}(\tilde{R}_{q,i})H_{\vec{k}}(\nabla\varphi_{\vec{k}}\otimes(\nabla\Phi_{i}^{-1}\vec{k}))$
$\displaystyle=\frac{\sigma_{q}}{\lambda_{q+1}\kappa_{q+1}}\sum_{\vec{k}}a_{\vec{k}}^{2}(\tilde{R}_{q,i})|\nabla\varphi_{\vec{k}}|^{2}(\vec{k}\otimes\vec{k})\nabla\Phi_{i}^{-T}\,.$

Here we used the definition of $H_{\vec{k}}(\xi)$ to deduce

$H_{\vec{k}}\nabla\varphi_{\vec{k}}=(\vec{k}\times\nabla\varphi_{\vec{k}})\times\nabla\varphi_{\vec{k}}=-|\nabla\varphi_{\vec{k}}|^{2}$

Using the normalization in the definition of Mikado flows in Section 2.1.4 as
well as Lemma 2.3, we conclude

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
H^{(i)}(\nabla_{\xi}\chi_{i})^{T}\,d\xi=\frac{\sigma_{q}}{\lambda_{q+1}\kappa_{q+1}}\tilde{R}_{q,i}\nabla\Phi_{i}^{-T}\,.$

Using (2.25) we deduce

$\bar{A}_{q}(x,t)=\sum_{i}\tilde{\eta}_{i}\kappa_{q+1}\nabla\Phi^{-1}_{i}\nabla\Phi_{i}^{-T}+\frac{\sigma_{q}}{\lambda_{q+1}^{2}\kappa_{q+1}}\sum_{i}\eta_{i}^{2}(\mathrm{Id}-\sigma_{q}^{-1}\mathring{\bar{R}}_{q})\,.$
(2.47)

Next, let

$\begin{split}\tilde{\kappa}_{q}(x,t)&=\kappa_{q+1}+c_{1}^{-1}\frac{\delta_{q+1}}{\lambda_{q+1}^{2}\kappa_{q+1}}\sum_{i}\eta_{i}^{2}(x,t)\\\
&=\kappa_{q+1}+c_{1}^{-1}\bar{\eta}^{2}(x,\tau_{q}^{-1}t)\kappa_{q},\end{split}$
(2.48)

where we have used (2.27) and (2.22). Then we compute

$\displaystyle\bar{A}_{q}-\tilde{\kappa}_{q}\mathrm{Id}$
$\displaystyle=\kappa_{q+1}\sum_{i}\tilde{\eta}_{i}(\nabla\Phi_{i}^{-1}\nabla\Phi_{i}^{-T}-\mathrm{Id})+$
$\displaystyle+\kappa_{q}\sum_{i}\eta_{i}^{2}(\delta_{q+1}^{-1}\sigma_{q}-c_{1}^{-1})\mathrm{Id}-\kappa_{q}\sum_{i}\eta_{i}^{2}\delta_{q+1}^{-1}\mathring{\bar{R}}_{q}.$

It then follows from (2.14), (2.23) and (2.24) that, for all $(x,t)$,

$|\bar{A}_{q}(x,t)-\tilde{\kappa}_{q}(x,t)\mathrm{Id}|\leq
C\tilde{\kappa}_{q}(x,t)\lambda_{q}^{-\min\\{\gamma_{T},\gamma_{R},\gamma_{E},(b-1)\beta\\}}\,,$
(2.49)

for some fixed constant $C$. Then, as in Step 1 we can choose $q_{0}$ is
sufficiently large to ensure that
$C\lambda_{q}^{-2\gamma}<\tfrac{1}{2}\lambda_{q}^{-\nicefrac{{\gamma}}{{2}}}$.
In particular, we may bound pointwise

$\frac{1}{2}\tilde{\kappa}_{q}\mathrm{Id}\leq\bar{A}_{q}\leq
2\tilde{\kappa}_{q}\mathrm{Id}.$ (2.50)

In Section 4, specifically Corollary 4.3, we compare the cumulative
dissipation of $\tilde{\rho}_{q+1}$ with that of $\bar{\rho}_{q}^{(1)}$. The
setting in Section 4 is based on validity of the inequality (2.50) as well as
(2.30), (2.31). Then, Corollary 4.3 and Remark 4.4 yields:

$\left|\|\tilde{\rho}_{q+1}(T)\|^{2}_{L^{2}}-\|\bar{\rho}_{q}^{(1)}(T)\|_{L^{2}}^{2}\right|\lesssim\tfrac{1}{2}\lambda_{q}^{-\gamma}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\bar{\rho}^{(1)}_{q}(T)\|_{L^{2}}^{2}\right|,$

or equivalently

$\left|\tilde{D}_{q+1}-D_{q}^{(1)}\right|\lesssim\lambda_{q}^{-\gamma}D_{q}^{(1)},$
(2.51)

where
$D_{q}^{(1)}=\tfrac{1}{2}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\bar{\rho}^{(1)}_{q}(T)\|_{L^{2}}^{2}\right|$.

_Step 3: Diagonal reduction (
$\bar{\rho}_{q}^{(1)}\leadsto\bar{\rho}_{q}^{(2)}$)_

Using once more (2.49) we now apply Proposition 3.3 with
$\varepsilon=\tfrac{1}{2}\lambda_{q}^{-\gamma}<\tfrac{1}{2}$ to conclude

$\left|D_{q}^{(2)}-D_{q}^{(1)}\right|\lesssim\lambda_{q}^{-\gamma}D_{q}^{(2)},$
(2.52)

where
$D_{q}^{(1)}=\tfrac{1}{2}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\bar{\rho}^{(1)}_{q}(T)\|_{L^{2}}^{2}\right|$
and $\bar{\rho}^{(2)}_{q}$ is the solution of

$\begin{split}\partial_{t}\bar{\rho}^{(2)}_{q}+\bar{u}_{q}\cdot\nabla\bar{\rho}^{(2)}_{q}&=\mathop{\rm
div}\nolimits\tilde{\kappa}_{q}\nabla\bar{\rho}^{(2)}_{q}\,,\\\
\bar{\rho}_{q}^{(2)}|_{t=0}&=\rho_{in}\,.\end{split}$ (2.53)

_Step 4: Time averaging ( $\bar{\rho}_{q}^{(2)}\leadsto\bar{\rho}_{q}^{(3)}$)_

The advection-diffusion equation (2.53) has two different characteristic time-
scales: the advective time-scale $\|\nabla\bar{u}_{q}\|_{L^{\infty}}^{-1}$ and
the time-scale $\tau_{q}$ given by the time-oscillatory behaviour of the
ellipticity coefficient $\tilde{\kappa}_{q}(x,t)$ given in (2.48), with the
relationship given by (2.7): $\|\nabla\bar{u}_{q}\|_{L^{\infty}}\tau_{q}\leq
M\lambda_{q}^{-\gamma_{T}}$. Let us now invoke Proposition 5.1 with the
following choices (on the right-hand side we choose parameters used in
Proposition 5.1 upon the current construction)

$\eta:=c_{1}^{-1}\bar{\eta}^{2},\quad\kappa_{0}:=\kappa_{q+1},\quad\kappa_{1}:=\kappa_{q},\quad\mu:=\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q},\quad\tau:=\tau_{q}.$

We remark that Proposition 5.1 requires assumptions (A1)-(A4). Control of the
cutoff-functions in (2.22) yields (A4), whereas (A2)-(A3) follow from
assumptions (2.30), (2.31), (2.13) as well as (2.7). (A1) gives the second
lower bound for $\tilde{N}$ in (2.36).

In this setting one can average over the faster time-scale $\tau_{q}$ and
obtain the estimate

$\left|\|\bar{\rho}^{(3)}_{q}(T)\|^{2}_{L^{2}}-\|\bar{\rho}^{(2)}_{q}(T)\|_{L^{2}}^{2}\right|\leq
C\lambda_{q}^{-\gamma_{T}}(D^{(2)}_{q}+D^{(3)}_{q}),$ (2.54)

where

$D^{(i)}_{q}=\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\bar{\rho}^{(i)}_{q}(T)\|_{L^{2}}^{2}\right|$

and $\bar{\rho}^{(3)}_{q}$ is the solution of

$\begin{split}\partial_{t}\bar{\rho}^{(3)}_{q}+\bar{u}_{q}\cdot\nabla\bar{\rho}^{(3)}_{q}&=(\kappa_{q+1}+\kappa_{q})\Delta\bar{\rho}^{(3)}_{q}\,,\\\
\bar{\rho}_{q}^{(3)}|_{t=0}&=\rho_{in}\,.\end{split}$ (2.55)

Choosing $q_{0}$ sufficiently large, we may ensure that
$C\lambda_{q}^{-\gamma_{T}}<\tfrac{1}{4}\lambda_{q}^{-\gamma}$ in (2.54), from
which we can then conclude

$\left|\frac{D_{q}^{(2)}}{D_{q}^{(3)}}-1\right|\leq\frac{1}{4}\lambda_{q}^{-\gamma}\left(1+\frac{D_{q}^{(2)}}{D_{q}^{(3)}}\right).$

Therefore we deduce

$\left|\frac{D_{q}^{(2)}}{D_{q}^{(3)}}-1\right|\leq\frac{1}{2}\lambda_{q}^{-\gamma}.$
(2.56)

_Step 5: Gluing estimate ( $\bar{\rho}_{q}^{(3)}\leadsto\rho_{q}$)_

Finally, we compare (2.55) to (2.28) by using the estimate (2.17). Indeed, we
can write (2.55) as

$\begin{split}\partial_{t}\bar{\rho}^{(3)}_{q}+u_{q}\cdot\nabla\bar{\rho}^{(3)}_{q}&=\mathop{\rm
div}\nolimits\left(\kappa_{q}\nabla\bar{\rho}^{(3)}_{q}+\kappa_{q+1}\nabla\bar{\rho}^{(3)}_{q}+(z_{q}-\bar{z}_{q})\times\nabla\bar{\rho}^{(3)}_{q}\right)\,,\\\
\bar{\rho}_{q}^{(3)}|_{t=0}&=\rho_{in}\,.\end{split}$

On the other hand (2.17) and our choice of parameters imply

$\|z_{q}-\bar{z}_{q}\|_{L^{\infty}}\lesssim\tau_{q}\delta_{q+1}\lambda_{q}^{-\gamma_{R}+\alpha(1+\gamma_{L})}\leq\kappa_{q}\lambda_{q}^{-2\gamma}\,,\quad\kappa_{q+1}=\lambda_{q}^{-(b-1)\theta}\kappa_{q}\leq\lambda_{q}^{-2\gamma}\kappa_{q}.$
(2.57)

A final application of Proposition 3.3, assuming $q_{0}$ is suffuciently large
to absorb the implicit constant, leads to

$\left|\|\bar{\rho}^{(3)}_{q}(T)\|^{2}_{L^{2}}-\|\rho_{q}(T)\|_{L^{2}}^{2}\right|\lesssim\lambda_{q}^{-2\gamma}\left|\|\rho_{in}\|^{2}_{L^{2}}-\|\rho_{q}(T)\|_{L^{2}}^{2}\right|,$

where $\rho_{q}$ is the solution of (2.28), equivalently

$\left|D_{q}^{(3)}-D_{q}\right|\lesssim\lambda_{q}^{-2\gamma}D_{q}.$ (2.58)

_Final estimate_

Overall, we see that after the five steps above, we achieve the estimate

$\frac{D_{q+1}}{D_{q}}=\frac{D_{q+1}}{\tilde{D}_{q+1}}\frac{\tilde{D}_{q+1}}{D_{q}^{(1)}}\frac{D_{q}^{(1)}}{D_{q}^{(2)}}\frac{D_{q}^{(2)}}{D_{q}^{(3)}}\frac{D_{q}^{(3)}}{D_{q}}\geq(1-C\lambda_{q}^{-\gamma})^{5}D_{q}\geq(1-\tfrac{1}{2}\lambda_{q}^{-\gamma/2})D_{q},$
(2.59)

where we have again assumed $q_{0}$ is suffuciently large to absorb the
constants and the exponent $5$ at the expense of decreasing the exponent from
$\gamma$ to $\gamma/2$. This concludes the proof of Proposition 2.4, with
$\gamma$ replaced by $\gamma/2$. ∎

###  2.3. Construction of the vectorfield $u$ \- h-principle

In Section 2.1 we detailed the main iteration scheme for producing Hölder-
continuous weak solutions of the Euler equations. What remains is to produce
an initial vectorfield and associated Reynolds tensor, which satisfies the
inductive assumptions (2.4)-(2.6) for _some_ $q\in\mathbb{N}$. To this end we
recall that a smooth strict subsolution to the Euler equations is a smooth
triple $(\bar{u},\bar{p},\bar{R})$ on $\mathbb{T}^{3}\times[0,T]$ solving the
Euler-Reynolds system (2.1) with the normalizations
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\bar{u}\,dx=0$,
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\bar{p}\,dx=0$, such that
$\bar{R}(x,t)>0$ is (uniformly) positive definite on
$\mathbb{T}^{3}\times[0,T]$. The energy associated with a subsolution is (c.f.
[DS17, DRS21, BLSV19])

$e(t)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}|\bar{u}|^{2}+\mathrm{tr\,}\bar{R}\,dx.$
(2.60)

We have the following statement, which is a variant of [DS17, Proposition
3.1], see also [BLSV19, Theorem 7.1] and [DRS21, Proposition 4.1].

######  Proposition 2.5.

Let $(\bar{u},\bar{p},\bar{R})$ be a smooth strict subsolution with energy
$e$. Let the parameters $M,\bar{e}$,
$\beta,b,\gamma_{T},\gamma_{R},\gamma_{E},\alpha_{0}>0$ and $a_{0}\gg 1$ be as
in Proposition 2.1, and $(\delta_{q},\lambda_{q})_{q\in\mathbb{N}}$ be as in
(2.3).

Then, there exists $q_{0}\in\mathbb{N}$ such that, for any $q\geq q_{0}$ there
exists $(u_{q},\mathring{R}_{q})$ solutions of (2.1) with (2.2) such that
(2.4)-(2.6) hold. Moreover,

$\displaystyle\|\bar{u}-u_{q}\|_{C^{0}}$
$\displaystyle\leq\bar{C}\|\bar{R}\|_{C^{0}}^{\nicefrac{{1}}{{2}}}\,,$ (2.61)
$\displaystyle\|\bar{z}-z_{q}\|_{C^{\alpha}}$
$\displaystyle\leq\bar{C}\|\bar{R}\|_{C^{\alpha}}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{-1+\alpha}\,,$
(2.62)

where the constant $\bar{C}$ only depends on $\bar{u}$ and on
$\delta:=\inf_{(x,t)}\frac{\min\\{\bar{R}\zeta\cdot\zeta:|\zeta|=1\\}}{\mathrm{tr\,}\bar{R}}$.

###### Proof.

We construct, following [DS17, DRS21], $u_{q}:=\bar{u}+w$, where $w$ is from
the perturbation step of Section 6, but without the need for time-cutoffs.
More precisely, we define

$\tilde{R}=\bar{R}-\tfrac{1}{3}\delta_{q+1}\mathrm{Id}.$

For $q$ sufficiently large, we ensure that

$\frac{1}{\mathrm{tr\,}\tilde{R}}\tilde{R}\geq\tfrac{1}{2}\delta\mathrm{Id}.$

Further, define $\Phi=\Phi(x,t)$ to be the inverse flow associated to the
vectorfield $\bar{u}$, i.e. such that
$(\partial_{t}+\bar{u}\cdot\nabla)\Phi=0$ and $\Phi(x,0)=x$. Since $\bar{u}\in
C^{\infty}(\mathbb{T}^{3}\times[0,T])$ with $\mathop{\rm
div}\nolimits\bar{u}=0$, for every $t$ the map $\Phi(\cdot,t)$ is a volume-
preserving diffeomorphisms, and there exists a constant $C$ such that

$C^{-1}\leq|\nabla\Phi|,|\nabla\Phi^{-1}|\leq C.$

Now we apply Lemma 2.3 with

$\mathcal{N}:=\left\\{R\in\mathcal{S}^{3\times
3}_{+}:\,R=\tfrac{1}{\mathrm{tr\,}S}ASA^{T}\textrm{ where
$C^{-1}\leq|A|,|A^{-1}|\leq C$ and
$S\geq\tfrac{1}{2}\delta\mathrm{tr\,}S\mathrm{Id}$}\right\\}\,,$

to obtain $\Lambda\subset\mathcal{Q}^{3}$ and $a_{\vec{k}}$ satisfying (2.20).
We set (c.f. (6.38))

$\displaystyle\tilde{S}=\frac{1}{\mathrm{tr\,}\tilde{R}}\nabla\Phi\tilde{R}\nabla\Phi^{T},$

and observe that $\tilde{S}(x,t)\in\mathcal{N}$. The new perturbation $w$ is
then defined, in analogy with (2.21), as

$w=\frac{1}{\lambda}\mathop{\rm
curl}\nolimits\left[\sum_{\vec{k}\in\Lambda}(\mathrm{tr\,}\tilde{R})^{\nicefrac{{1}}{{2}}}a_{\vec{k}}(\tilde{S})\nabla\Phi^{T}U_{\vec{k}}(\lambda\Phi)\right]\,.$
(2.63)

The associated Reynolds stress is (c.f. (6.54))

$\mathring{R}_{q}=\mathcal{R}\left[\partial_{t}w+\bar{u}\cdot\nabla
w+w\cdot\nabla\bar{u}+\mathop{\rm div}\nolimits(w\otimes w-\tilde{R})\right],$

where $\mathcal{R}$ is the inverse divergence operator on symmetric
$2$-tensors, introduced in [DLS13]. Since $\bar{R}-\tilde{R}$ is a constant
multiple of the identity, it easily follows, as in [DS17, BLSV19], that
$u_{q},\mathring{R}_{q}$ is a solution of (2.1). Moreover, the estimates in
[DS17, BLSV19] (see also Section 6.3) imply

$\displaystyle\|\mathring{R}_{q}\|_{C^{0}}$ $\displaystyle\leq
C\lambda^{-1+\alpha}\,,$ $\displaystyle\|w\|_{C^{n}}$ $\displaystyle\leq
C\|\bar{R}\|_{C^{0}}^{\nicefrac{{1}}{{2}}}\lambda^{n}\quad\textrm{ for all
}n\leq\bar{N}\,,$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}|u_{q}|^{2}-|\bar{u}|^{2}-\mathrm{tr\,}\tilde{R}\,dx\right|$
$\displaystyle\leq C\lambda^{-1+\alpha}\,.$

Here the constant $C$ depends on $(\bar{u},\bar{R})$ and on $\mathcal{N}$. It
remains to choose $\lambda$ and $q$. We fix an exponent $0<\gamma_{o}$ such
that

$\beta<\gamma_{o}\textrm{ and
}\alpha+\gamma_{o}<1-2b\beta-\max\\{\gamma_{E},\gamma_{R}\\}$

and define $\lambda=\lambda_{q}^{1-\gamma_{o}}$ ($q$ still to be fixed).
Observe that (2.8)-(2.9) guarantee the existence of such $\gamma_{o}$. Now,
validity of (2.4)-(2.6) follows from

$C\lambda^{-1+\alpha}\leq\delta_{q+1}\lambda_{q}^{-\max\\{\gamma_{E},\gamma_{R}\\}},\textrm{
as well as }C\lambda\leq
M\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q},\,\lambda\leq\lambda_{q}.$

But by our choice of exponent $\gamma_{o}$ these inequalities are satisfied
for $q$ sufficiently large. This concludes the proof. ∎

######  Remark 2.6.

An obvious way to apply Proposition 2.5 is to take $\bar{u}=0$, $\bar{p}=0$
and $\bar{R}=\tfrac{1}{3}e(t)\mathrm{Id}$. More generally, one can take any
smooth $\bar{u},\bar{p}$ which is a classical solution of the Euler equations
(1.2).

###  2.4. Proof of Theorem 1.1

###### Proof.

_Step 1. Choice of parameters._

Given $0<\beta<\tfrac{1}{3}$, we choose
$1<b<\min\\{\sqrt{\tfrac{3}{2}},\tfrac{1-\beta}{2\beta}\\}$, and then fix
$\gamma_{T},\gamma_{R},\gamma_{E}>0$ as

$\gamma_{E}=\gamma_{T}=\gamma_{R}=\frac{b-1}{b(b+1)}(1-(2b+1)\beta)\,.$ (2.64)

One easily verifies that then

$\displaystyle\gamma_{T}+b\gamma_{R}$ $\displaystyle<(b-1)(1-(2b+1)\beta)\,,$
(2.65) $\displaystyle\gamma_{T}$
$\displaystyle<\frac{b-1}{b+1}(1-(2b+1)\beta)\,,$ (2.66)
$\displaystyle\gamma_{R}+\gamma_{T}$
$\displaystyle>\frac{b-1}{b+1}(1-(2b+1)\beta)\,.$ (2.67)

Indeed, we have

$\gamma_{T}+b\gamma_{R}=\frac{b-1}{b}(1-(2b+1)\beta),\quad\gamma_{R}+\gamma_{T}=\frac{2}{b}\frac{b-1}{b+1}(1-(2b+1)\beta).$

Note that (2.65) is the requirement (2.9) in Proposition 2.1, whereas (2.66)
and (2.67) are the left and right inequalities in (2.30) in Proposition 2.4.
Therefore, with this choice of parameters both Propositions are applicable.

_Step 2. Construction of the vectorfield $u$._

Having fixed the parameters $\beta,b,\gamma_{T},\gamma_{R},\gamma_{E}$ in Step
1, as well as $\alpha_{0},\alpha_{1},a_{0}$ in Propositions 2.1 and 2.4, we
apply Proposition 2.5 to obtain $q_{0}\in\mathbb{N}$ and
$(u_{q_{0}},\mathring{R}_{q_{0}})$ satisfying the estimates (2.4)-(2.6). Then,
Proposition 2.1 applies inductively for any $q\geq q_{0}$ and yields a
sequence $\\{u_{q}\\}_{q\geq q_{0}}$. Arguing as in [BLSV19], we deduce that
$u_{q}\to u$ in $C([0,T];C^{\beta^{\prime}}(\mathbb{T}^{3}))$ for any
$\beta^{\prime}<\beta$, and moreover, that $u\in
C^{\beta^{\prime\prime}}([0,T]\times\mathbb{T}^{3})$ for any
$\beta^{\prime\prime}<\beta^{\prime}$. Since
$\beta^{\prime\prime}<\beta^{\prime}\beta<1/3$ was arbitrary in this argument,
by renaming $\beta^{\prime\prime}<\beta^{\prime}\beta$ we deduce the existence
of $u\in C^{\beta}([0,T]\times\mathbb{T}^{3})$ as in the statement of the
Theorem.

_Step 3. Macroscopic diffusion for mollified initial datum_

Next, we turn to the enhanced dissipation properties of the velocity field
$u$. First, we fix $\gamma>0$ and $\tilde{N}$ as in Proposition 2.4. Let
$\rho_{in}\in H^{1}(\mathbb{T}^{3})$ be a nonzero initial datum with
$\mathop{\rm div}\nolimits\rho_{in}=0$ and
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}^{3}}\rho_{in}\,dx=0$. We define the
length-scale

$\ell:=\frac{\|\rho_{in}\|_{L^{2}}}{\|\nabla\rho_{in}\|_{L^{2}}}.$

Define

$\tilde{\rho}_{in}:=\rho_{in}*\psi_{r\ell}\,,$

where $\psi$ is a standard symmetric mollifier (or one can use e.g. $\psi$
from Section 6.1) and $0<r<1$ is still to be fixed. Then (see e.g. Lemma 6.2)

$\displaystyle\|\tilde{\rho}_{in}-\rho_{in}\|_{L^{2}}$ $\displaystyle\leq
Cr\ell\|\nabla\rho_{in}\|_{L^{2}}\leq Cr\|\rho_{in}\|_{L^{2}}$ (2.68)
$\displaystyle\|\tilde{\rho}_{in}\|_{H^{n+1}}$ $\displaystyle\leq
C(r\ell)^{-n}\|\nabla\rho_{in}\|_{L^{2}}\leq
C(r\ell)^{-n}\ell^{-1}\|\rho_{in}\|_{L^{2}}\quad\textrm{ for all }0\leq
n\leq\tilde{N}.$ (2.69)

The small factor $r$ will be chosen below; for now we merely assume $r<1$ is
sufficiently small so that the first inequality becomes

$\|\tilde{\rho}_{in}-\rho_{in}\|_{L^{2}}\leq\tfrac{1}{2}\|\rho_{in}\|_{L^{2}}\,.$
(2.70)

Next, for any $q\geq q_{0}$ we consider the solution $\tilde{\rho}_{q}$ of the
equation

$\begin{split}\partial_{t}\tilde{\rho}_{q}+u_{q}\cdot\nabla\tilde{\rho}_{q}=&\kappa_{q}\Delta\tilde{\rho}_{q}\,,\\\
\tilde{\rho}_{q}|_{t=0}=&\tilde{\rho}_{in}\,.\end{split}$ (2.71)

By applying the Poincaré inequality, there exists $c_{0}>0$ such that

$\displaystyle\frac{d}{dt}\|\tilde{\rho}_{q}(t)\|^{2}_{L^{2}}dt\leq-2\kappa_{q}\|\nabla\tilde{\rho}_{q}(t)\|^{2}_{L^{2}}\leq-
c_{0}\kappa_{q}\|\tilde{\rho}_{q}(t)\|^{2}_{L^{2}},$ (2.72)

and therefore

$\begin{split}\kappa_{q}\int_{0}^{T}\|\nabla\tilde{\rho}_{q}\|^{2}_{L^{2}}dt=&\frac{1}{2}\left(\|\tilde{\rho}_{in}\|_{L^{2}}^{2}-\|\tilde{\rho}_{q}(T)\|_{L^{2}}^{2}\right)\\\
\geq&\frac{1}{2}\left(1-\exp(-c_{0}\kappa_{q}T)\right)\|\tilde{\rho}_{in}\|_{L^{2}}^{2}\\\
\geq&16c\kappa_{q}T\|\tilde{\rho}_{in}\|_{L^{2}}^{2}\geq
4c\kappa_{q}T\|\rho_{in}\|^{2}_{L^{2}}\end{split}$ (2.73)

for some universal constant $c$, provided $q$ is sufficiently large so that
$c_{0}\kappa_{q}T<1/2$. Here we also used (2.70).

_Step 4. The inertial range - choice of $q_{I}$ and $r$._

For some $q_{I}\geq q_{0}$ to be fixed, set

$r:=\lambda_{q_{I}}^{-\nicefrac{{b\theta}}{{2}}}.$ (2.74)

We claim that for sufficiently large $q_{I}$ the following inequalities hold:

$\displaystyle C\ell^{-1}(r\ell)^{-n}$
$\displaystyle\leq\lambda_{q}^{n+1}(cT\kappa_{q+1})^{\nicefrac{{1}}{{2}}}\textrm{
for all }n\geq 0\,,$ (2.75) $\displaystyle Cr^{2}$
$\displaystyle\leq\tfrac{1}{4}cT\kappa_{q_{I}}\,.$ (2.76)

Indeed, (2.75) is satisfied if

$(r\ell)^{-1}\leq\lambda_{q}\textrm{ and
}C\ell^{-1}\leq(cT)^{\nicefrac{{1}}{{2}}}\lambda_{q}^{1-\nicefrac{{b\vartheta}}{{2}}}\,,$
(2.77)

where we used (2.26). Comparing powers of $\lambda_{q}$ we then observe that
(2.75) is valid provided $\lambda_{q_{I}}^{b\theta-\theta}$ is sufficiently
large, whereas (2.76) is valid provided $\lambda_{q_{I}}^{1-b\theta/2}$ is
sufficiently large. Since by choice $b<\sqrt{\tfrac{3}{2}}$, from (2.26) it
follows $b\theta/2<1$. We conclude that with sufficiently large $q_{I}$ and
the choice (2.74), the inequalities (2.75)-(2.76) hold.

_Step 5. Enhanced dissipation in the inertial range._

Combining (2.75) with (2.73) and (2.69) we observe that for any $q\geq q_{I}$

$\|\tilde{\rho}_{in}\|_{H^{n}}\leq\lambda_{q}^{n}\left(\kappa_{q+1}\int_{0}^{T}\|\nabla\tilde{\rho}_{q+1}\|^{2}_{L^{2}}dt\right)^{\nicefrac{{1}}{{2}}}\textrm{
for any }1\leq n\leq\tilde{N}\,,$ (2.78)

i.e. condition (2.32) in Proposition 2.4 holds. Therefore, for any $q\geq
q_{I}$, we may apply Proposition 2.4 with initial data $\tilde{\rho}_{in}$, to
obtain

$\displaystyle(1-\tfrac{1}{2}\lambda_{q}^{-\gamma})\kappa_{q}\int_{0}^{T}\|\nabla\tilde{\rho}_{q}\|^{2}_{L^{2}}dt\leq\kappa_{q+1}\int_{0}^{T}\|\nabla\tilde{\rho}_{q+1}\|^{2}_{L^{2}}dt.$
(2.79)

Next, observe that there exists $\tilde{q}\in\mathbb{N}$ and $\tilde{c}>0$
(depending only on $\gamma>0$ and the choice of $a,b$ for defining the
sequence $(\lambda_{q})_{q}$), such that
$\prod_{q^{\prime}\geq\tilde{q}}(1-\tfrac{1}{2}\lambda_{q}^{\prime-\gamma})\geq
e^{-\tilde{c}\lambda_{\tilde{q}}^{-\gamma}}\geq\tfrac{1}{2}$. Consequently,
assuming $q_{I}\geq\tilde{q}$, we have

$\displaystyle\kappa_{q}\int_{0}^{T}\|\nabla\tilde{\rho}_{q}\|^{2}_{L^{2}}dt\geq\kappa_{q_{I}}\int_{0}^{T}\|\nabla\tilde{\rho}_{q_{I}}\|^{2}_{L^{2}}dt\prod_{q^{\prime}=q_{I}}^{q-1}(1-\tfrac{1}{2}\lambda_{q^{\prime}}^{-\gamma})\geq\tfrac{1}{2}\kappa_{q_{I}}\int_{0}^{T}\|\nabla\tilde{\rho}_{q_{I}}\|^{2}_{L^{2}}dt.$
(2.80)

We deduce, for any $q\geq q_{I}$

$\kappa_{q}\int_{0}^{T}\|\nabla\tilde{\rho}_{q}\|^{2}_{L^{2}}dt\geq
2c\kappa_{q_{I}}T\|\rho_{in}\|_{L^{2}}^{2}\,.$ (2.81)

_Step 6. dissipation in the molecular range._

Now, let us fix $\kappa=\kappa_{q_{M}}$ for some $q_{M}\geq q_{I}$, and
compare $\tilde{\rho}_{q_{M}}$ to the solution $\tilde{\rho}$ of

$\begin{split}\partial_{t}\tilde{\rho}+u\cdot\nabla\tilde{\rho}=&\kappa\Delta\tilde{\rho},\\\
\tilde{\rho}|_{t=0}=&\tilde{\rho}_{in}.\end{split}$ (2.82)

To this end we consider the vector potentials $z_{q_{M}},z$ of $u_{q_{M}},u$.
We have, for any $q$,

$\|z_{q+1}-z_{q}\|_{C^{0}}\leq\|z_{q+1}-\bar{z}_{q}\|_{C^{0}}+\|\bar{z}_{q}-z_{q}\|_{C^{0}},$

where $\bar{z}_{q}$ is the vector potential of $\bar{u}_{q}$ obtained in
Proposition 2.2. Using the same arguments as in Section 6 (see for instance
Proposition 6.16) it is easy to verify

$\|z_{q+1}-\bar{z}_{q}\|_{C^{0}}\lesssim\|\mathcal{R}w_{q+1}\|_{C^{\alpha}}\lesssim\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-1+\alpha}=\lambda_{q}^{-b(1+\beta-\alpha)}.$

Since $\theta-b(1+\beta)=-b\frac{(b-1)}{b+1}(1+\beta)>2\gamma$, we deduce
(assuming $\alpha$ sufficiently small)

$\|z_{q+1}-z_{q}\|_{C^{0}}\lesssim\kappa_{q}\lambda_{q}^{-2\gamma},$

where we used (2.57) for estimating $\bar{z}_{q}-z_{q}$. In particular we
obtain

$\|z-z_{q_{M}}\|_{C^{0}}\leq\sum_{q=q_{*}}^{\infty}\|z_{q+1}-z_{q}\|_{C^{0}}\lesssim\kappa_{q_{M}}\lambda_{q_{M}}^{-2\gamma}=\kappa\lambda_{q_{M}}^{-2\gamma}.$

Writing the equation (2.82) as

$\partial_{t}\tilde{\rho}+u_{q_{M}}\cdot\nabla\tilde{\rho}=\mathop{\rm
div}\nolimits(\kappa\nabla\tilde{\rho}+(z-z_{q_{M}})\times\nabla\tilde{\rho}),$

we may then apply Proposition 3.3 to deduce

$\displaystyle\kappa\int_{0}^{T}\|\nabla\tilde{\rho}\|^{2}_{L^{2}}dt\geq
c\kappa_{q_{I}}T\|\rho_{in}\|_{L^{2}}^{2}.$ (2.83)

_Step 7. Enhanced dissipation for original initial datum._

Finally, we compare $\tilde{\rho}$ to the solution $\rho$ of

$\begin{split}\partial_{t}\rho+u\cdot\nabla\rho=&\kappa\Delta\rho,\\\
\rho|_{t=0}=&\rho_{in}.\end{split}$ (2.84)

The basic energy estimate together with (2.68) gives

$\kappa\int_{0}^{T}\|\nabla\rho-\nabla\tilde{\rho}\|_{L^{2}}^{2}\,dt\leq\frac{1}{2}\|\rho_{in}-\tilde{\rho}_{in}\|_{L^{2}}\leq
Cr^{2}\|\rho_{in}\|_{L^{2}}^{2}\,.$

Consequently

$\displaystyle\kappa^{\nicefrac{{1}}{{2}}}\left(\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}\,dt\right)^{\nicefrac{{1}}{{2}}}$
$\displaystyle\geq\kappa^{\nicefrac{{1}}{{2}}}\left(\int_{0}^{T}\|\nabla\tilde{\rho}\|_{L^{2}}^{2}\,dt\right)^{\nicefrac{{1}}{{2}}}-C^{\nicefrac{{1}}{{2}}}r\|\rho_{in}\|_{L^{2}}$
$\displaystyle\geq\left[(c\kappa_{q_{I}}T)^{\nicefrac{{1}}{{2}}}-C^{\nicefrac{{1}}{{2}}}r\right]\|\rho_{in}\|_{L^{2}}$
$\displaystyle\geq\tfrac{1}{2}(c\kappa_{q_{I}}T)^{\nicefrac{{1}}{{2}}}\|\rho_{in}\|_{L^{2}},$

where we used (2.76).

Since this is true for $\kappa=\kappa_{q_{M}}$ for any $q_{M}\geq q_{I}$, we
deduce

$\limsup_{\kappa\to
0}\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}\,dt\geq\tfrac{1}{4}c\kappa_{q_{I}}T\|\rho_{in}\|_{L^{2}}^{2}.$

This concludes the proof.

∎

##  3\. Energy estimates

###  3.1. Estimates for advection-diffusion with Laplacian

Estimates in this section are needed for both spatial homogenisation and time
averaging. We consider the advection-diffusion equation

$\displaystyle\partial_{t}\rho+u\cdot\nabla\rho$
$\displaystyle=\kappa\Delta\rho\qquad\textrm{ on }\mathbb{T}^{3}\times[0,T]$
(3.1) $\displaystyle\rho|_{t=0}$ $\displaystyle=\rho_{in}$

with smooth initial datum. For short notation, in this section we use space-
time norms on time interval $[0,T]$, denoted by $L_{xt}$.

We assume the scales relationship

$\left(\frac{\|\nabla
u\|_{L^{\infty}}}{\kappa}\right)^{n}\geq\left(\frac{\|\nabla^{n}u\|_{L^{\infty}}}{\kappa}\right)^{\frac{2n}{n+1}}.$
(3.2)

######  Lemma 3.1.

Let $\kappa>0$ and $u\in C^{\infty}(\mathbb{T}^{d};\mathbb{R}^{d})$ be
divergence free. Assume (3.2). Then the advection-diffusion equation (3.1)
satisfies for any ${t\leq T}$

$\frac{1}{2}\|\rho(t)\|_{L^{2}}^{2}+\kappa\int_{0}^{t}\|\nabla\rho(s)\|_{L^{2}}^{2}\,ds=\frac{1}{2}\|\rho_{in}\|_{L^{2}}^{2},$

and, for any $n\geq 1$

$\sup_{t\leq
T}\|(\nabla^{n}\rho)(t)\|^{2}_{L^{2}}+\kappa\int_{0}^{T}\|\nabla^{n+1}\rho(s)\|_{L^{2}}^{2}ds\leq\|\nabla^{n}\rho_{in}\|^{2}_{L^{2}}+C_{n}\left(\frac{\|\nabla
u\|_{L^{\infty}}}{\kappa}\right)^{n}\kappa\int_{0}^{T}\|\nabla\rho(s)\|_{L^{2}}^{2}.$
(3.3)

###### Proof.

Apply $\partial^{\boldsymbol{\alpha}}$, where $|\boldsymbol{\alpha}|=n$, to
the equation and and test the result with $\partial^{\boldsymbol{\alpha}}$ to
get after integration in time

$\frac{1}{2}\sup_{t\leq
T}\|(\partial^{\boldsymbol{\alpha}}\rho)(t)\|^{2}_{L^{2}}+\kappa\|\nabla\partial^{\boldsymbol{\alpha}}\rho\|_{L_{xt}^{2}}^{2}\leq\frac{1}{2}\|\nabla^{n}\rho_{in}\|^{2}_{L^{2}}+\|[u\cdot\nabla,\partial^{\boldsymbol{\alpha}}]\rho\|_{L_{xt}^{2}}\|\partial^{\boldsymbol{\alpha}}\rho\|_{L_{xt}^{2}}.$

For the last term use the commutator estimate

$\|[u\cdot\nabla,\partial^{\boldsymbol{\alpha}}]f\|_{L^{2}}\lesssim_{n}\|\nabla^{n}u\|_{L^{\infty}}\|\nabla
f\|_{L^{2}}+\|\nabla u\|_{L^{\infty}}\|\nabla^{n}f\|_{L^{2}}$

(followed by Hölder’s inequality in time) to write

$\begin{split}\|[u\cdot\nabla,\partial^{\boldsymbol{\alpha}}]\rho\|_{L_{xt}^{2}}&\lesssim_{n}\|\nabla^{n}u\|_{L^{\infty}}\|\nabla\rho\|_{L_{xt}^{2}}\|\nabla^{n}\rho\|_{L_{xt}^{2}}+\|\nabla
u\|_{L^{\infty}}\|\nabla^{n}\rho\|^{2}_{L_{xt}^{2}}\\\
&\lesssim_{n}\|\nabla^{n}u\|_{L^{\infty}}\|\nabla\rho\|^{\frac{n+1}{n}}_{L_{xt}^{2}}\|\nabla^{n+1}\rho\|^{\frac{n-1}{n}}_{L_{xt}^{2}}+\|\nabla
u\|_{L^{\infty}}\|\nabla\rho\|^{\frac{2}{n}}_{L_{xt}^{2}}\|\nabla^{n+1}\rho\|^{\frac{2n-2}{n}}_{L_{xt}^{2}}.\end{split}$

Here, for the latter $\lesssim$ we use interpolation. Thus, summing over all
partial derivatives of order $n$ we have

$\begin{split}\frac{1}{2}\sup_{t\leq
T}\|(\nabla^{n}\rho)(t)\|^{2}_{L^{2}}+\kappa\|\nabla^{n+1}&\rho\|_{L_{xt}^{2}}^{2}\leq
C_{n}\kappa^{-1}\|\nabla^{n}u\|_{L^{\infty}}(\kappa^{\frac{1}{2}}\|\nabla\rho\|_{L_{xt}^{2}})^{\frac{n+1}{n}}(\kappa^{\frac{1}{2}}\|\nabla^{n+1}\rho\|_{L_{xt}^{2}})^{\frac{n-1}{n}}\\\
+C_{n}&\kappa^{-1}\|\nabla
u\|_{L^{\infty}}(\kappa^{\frac{1}{2}}\|\nabla\rho\|_{L_{xt}^{2}})^{\frac{2}{n}}(\kappa^{\frac{1}{2}}\|\nabla^{n+1}\rho\|_{L_{xt}^{2}})^{\frac{2n-2}{n}}+\frac{1}{2}\|\nabla^{n}\rho_{in}\|^{2}_{L^{2}}.\end{split}$

The above and Young’s inequality gives

$\sup_{t\leq
T}\|(\nabla^{n}\rho)(t)\|^{2}_{L^{2}}+\kappa\|\nabla^{n+1}\rho\|_{L_{xt}^{2}}^{2}\leq\|\nabla^{n}\rho_{in}\|^{2}_{L^{2}}+C_{n}\left(\left(\frac{\|\nabla
u\|_{L^{\infty}}}{\kappa}\right)^{n}+\left(\frac{\|\nabla^{n}u\|_{L^{\infty}}}{\kappa}\right)^{\frac{2n}{n+1}}\right)\kappa\|\nabla\rho\|_{L_{xt}^{2}}^{2},$

which with assumption (3.2) gives (3.3). ∎

We will need also the following immediate corollary for the forced advection-
diffusion equation

$\displaystyle\partial_{t}\rho+u\cdot\nabla\rho$
$\displaystyle=\kappa\Delta\rho+\mathop{\rm div}\nolimits f\qquad\textrm{ on
}\mathbb{T}^{3}\times[0,T]$ (3.4) $\displaystyle\rho|_{t=0}$
$\displaystyle=\rho_{in}$

with smooth initial datum and forcing.

######  Corollary 3.2.

Let $\kappa>0$ and $u\in C^{\infty}(\mathbb{T}^{d};\mathbb{R}^{d})$ be
divergence free. Assume (3.2). Then the forced advection-diffusion equation
(3.4) satisfies for any ${t\leq T}$

$\frac{1}{2}\|\rho(t)\|_{L^{2}}^{2}+\kappa\int_{0}^{t}\|\nabla\rho(s)\|_{L^{2}}^{2}\,ds=\frac{1}{2}\|\rho_{in}\|_{L^{2}}^{2}+\int_{0}^{t}\int
f\cdot\nabla\rho,$

and for any $n\geq 1$:

$\begin{split}&\sup_{t\leq
T}\|(\nabla^{n}\rho)(t)\|^{2}_{L^{2}}+\kappa\int_{0}^{T}\|\nabla^{n+1}\rho(s)\|_{L^{2}}^{2}ds\leq\\\
&\|\nabla^{n}\rho_{in}\|^{2}_{L^{2}}+C_{n}\left(\frac{\|\nabla
u\|_{L^{\infty}}}{\kappa}\right)^{n}\kappa\int_{0}^{T}\|\nabla\rho(s)\|_{L^{2}}^{2}+\sum_{|\boldsymbol{\alpha}|=n}\left|\int_{0}^{T}\int\partial^{\boldsymbol{\alpha}}f\cdot\nabla\partial^{\boldsymbol{\alpha}}\rho\right|.\end{split}$
(3.5)

Observe that we don’t use above norms for forcing, which will be important
later. In particular, this is why we have immediately integrated in time while
deriving estimate (3.3).

###  3.2. Estimates for advection-diffusion with an elliptic matrix

Estimates in this section are needed mainly in the space homogenisation
proposition. We will consider

$\displaystyle\partial_{t}\bar{\rho}+u\cdot\nabla\bar{\rho}$
$\displaystyle=\mathop{\rm div}\nolimits\bar{A}\nabla\bar{\rho}\qquad\textrm{
on }\mathbb{T}^{3}\times[0,T]$ (3.6) $\displaystyle\bar{\rho}|_{t=0}$
$\displaystyle=\rho_{in}$

with smooth initial datum and smooth matrix $\bar{A}$, or a difference of two
solutions $\bar{\rho}_{1}$ and $\bar{\rho}_{2}$ with respective matrices
$\bar{A}_{1}$, $\bar{A}_{2}$.

####  3.2.1. Comparison

First, we prove an estimate that allows to compare two solutions of (3.6) with
two ellipticity matrices.

######  Proposition 3.3 (Stability estimates).

Let $\varrho_{1}$ and $\varrho_{2}$ solve the following equations on
$\mathbb{T}^{3}\times[0,T]$

$\displaystyle\partial_{t}\varrho_{1}+u\cdot\nabla\varrho_{1}$
$\displaystyle=\mathop{\rm div}\nolimits(A_{1}\nabla\varrho_{1}),$
$\displaystyle\partial_{t}\varrho_{2}+u\cdot\nabla\varrho_{2}$
$\displaystyle=\mathop{\rm div}\nolimits(A_{2}\nabla\varrho_{2})$

with initial data $\varrho_{1}(0)=\varrho_{2}(0)=\rho_{\text{in}}\in
L^{2}(\mathbb{T}^{3})$ and uniformly elliptic symmetric matrices
$A_{1},A_{2}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$
satisfying for $\varepsilon\leq\frac{1}{2}$

$\displaystyle\big{|}(A_{1}-A_{2})\xi\cdot\zeta\big{|}\leq$
$\displaystyle\varepsilon(A_{1}\xi\cdot\xi)^{\frac{1}{2}}(A_{1}\zeta\cdot\zeta)^{\frac{1}{2}},\quad\text{for
any }(x,t)\in\mathbb{T}^{3}\times[0,T]\text{ and }\xi,\zeta\in\mathbb{R}^{3}.$
(3.7)

Let $\tilde{\varrho}:=\varrho_{1}-\varrho_{2}$, then we have

$\displaystyle|A_{1}\nabla\varrho_{2}\cdot\nabla\varrho_{2}|\leq$
$\displaystyle\;2A_{2}\nabla\varrho_{2}\cdot\nabla\varrho_{2},$ (3.8)
$\displaystyle\sup_{t\leq
T}\|\tilde{\varrho}(t)\|^{2}_{L^{2}}+\int_{0}^{T}\int
A_{1}\nabla\tilde{\varrho}\cdot\nabla\tilde{\varrho}dxdt\leq$
$\displaystyle\varepsilon^{2}\int_{0}^{T}\int
A_{1}\nabla\varrho_{2}\cdot\nabla\varrho_{2}dxdt,$ (3.9)
$\displaystyle\big{|}\|\varrho_{1}(t)\|_{L^{2}}^{2}-\|\varrho_{2}(t)\|_{L^{2}}^{2}\big{|}\leq$
$\displaystyle 9\varepsilon\int_{0}^{t}\int
A_{1}\nabla\varrho_{2}\cdot\nabla\varrho_{2}dxdt\leq
18\varepsilon\int_{0}^{t}\int
A_{2}\nabla\varrho_{2}\cdot\nabla\varrho_{2}dxdt$ (3.10)

for any $t\leq T$.

###### Proof.

The pointwise inequality (3.8) follows from (3.7), ellipticity, and
$\varepsilon\leq\frac{1}{2}$. Taking the difference of the two equations, we
have

$\displaystyle\partial_{t}\tilde{\varrho}+u\cdot\nabla\tilde{\varrho}$
$\displaystyle=\mathop{\rm
div}\nolimits\big{(}A_{1}\nabla\tilde{\varrho}\big{)}+\mathop{\rm
div}\nolimits\big{(}(A_{1}-A_{2})\nabla\varrho_{2}\big{)}.$ (3.11)

Testing with $\tilde{\varrho}$ and integrate by parts. For the last term in
(3.11), we use Young’s inequality to absorb the term $\tilde{\varrho}$ from
the first term on the right hand side. This gives (3.9).

From the equations for $\varrho_{1}$ and $\varrho_{2}$, we also can derive

$\partial_{t}\big{(}\tilde{\varrho}\varrho_{2}\big{)}=-\tilde{\varrho}u\cdot\nabla\varrho_{2}+\tilde{\varrho}\mathop{\rm
div}\nolimits(A_{2}\nabla\varrho_{2})\\\
-\varrho_{2}u\cdot\nabla\tilde{\varrho}+\varrho_{2}\mathop{\rm
div}\nolimits\big{(}A_{1}\nabla\tilde{\varrho}\big{)}+\varrho_{2}\mathop{\rm
div}\nolimits\big{(}(A_{1}-A_{2})\nabla\varrho_{2}\big{)}.$ (3.12)

Integrate in $\mathbb{T}^{3}\times[0,T]$, the first and the third terms on the
right hand side of (3.12) cancel. For the rest, we integrate by parts

$\begin{split}\left|\int\tilde{\varrho}(t)\varrho_{2}(t)dx\right|&\leq\left|\int_{0}^{t}\int(A_{1}+A_{2})\nabla\tilde{\varrho}\nabla\varrho_{2}dxds\right|+\left|\int_{0}^{t}\int(A_{1}-A_{2})\nabla\varrho_{2}\nabla\varrho_{2}dxds\right|\\\
&\leq 2\left|\int_{0}^{t}\int
A_{1}\nabla\tilde{\varrho}\nabla\varrho_{2}dxds\right|+2\left|\int_{0}^{t}\int(A_{1}-A_{2})\nabla\varrho_{2}\nabla\varrho_{2}dxds\right|.\end{split}$

We estimate the latter right-hand side term by $2\varepsilon\int_{0}^{t}\int
A_{1}\nabla\varrho_{2}\cdot\nabla\varrho_{2}dxds$ using the assumption. For
the first term we use
$A_{1}\xi\cdot\zeta\leq(A_{1}\xi\cdot\xi)^{\frac{1}{2}}(A_{1}\zeta\cdot\zeta)^{\frac{1}{2}}$
(Cauchy-Schwarz) and Young’s inequality that utilizes $\varepsilon^{2}$ of
(3.9), to get

$\left|\int\tilde{\varrho}(t)\varrho_{2}(t)dx\right|\leq
4\varepsilon\int_{0}^{t}\int
A_{1}\nabla\varrho_{2}\cdot\nabla\varrho_{2}dxds.$

Then the following fact concludes the proof of (3.10),

$\displaystyle\left|\|\varrho_{1}(t)\|_{L^{2}}^{2}-\|\varrho_{2}(t)\|_{L^{2}}^{2}\right|=\left|\int\tilde{\varrho}(t)\big{(}\varrho_{1}(t)+\varrho_{2}(t)\big{)}dx\right|=\|\tilde{\varrho}(t)\|^{2}_{L^{2}}+2\left|\int\tilde{\varrho}(t)\varrho_{2}(t)dx\right|.$

∎

####  3.2.2. General weighted estimate

We define

$\bar{D}=\int_{0}^{T}\|\bar{\kappa}^{\frac{1}{2}}\nabla\bar{\rho}(s)\|_{L^{2}}^{2}\,ds.$

Assume the following inequalities

$2\bar{\kappa}(x,t)\mathrm{Id}\geq\bar{A}(x,t)\geq\frac{\bar{\kappa}(x,t)}{2}\mathrm{Id}$
(3.13)

and

$\left\|\frac{D^{u}_{t}\bar{\kappa}}{\bar{\kappa}}\right\|_{L^{\infty}}\leq
C\tau^{-1}.$ (3.14)

Assume further that for $m\geq 1$

$\|\bar{\kappa}^{\frac{m-2}{2}}\nabla^{m}\bar{A}\|_{L^{\infty}}\leq
C(\tau^{-1})^{\frac{m}{2}}\qquad\|\bar{\kappa}^{\frac{m-2}{2}}\nabla^{m}\bar{\kappa}\|_{L^{\infty}}\lesssim(\tau^{-1})^{\frac{m}{2}}$
(3.15)

and

$\|\bar{\kappa}^{\frac{m-1}{2}}\nabla^{m}u\|_{L^{\infty}}\leq
C\tau^{-1}(\tau^{-1})^{\frac{m-1}{2}}.$ (3.16)

######  Lemma 3.4.

Let $u$ be divergence free. Assume the inequalities (3.13), (3.14), (3.16),
(3.15) hold. Then the general advection-diffusion equation (3.6) satisfies for
any ${t\leq T}$

$\frac{1}{2}\|\bar{\rho}(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|\bar{\kappa}^{\frac{1}{2}}\nabla\bar{\rho}(s)\|_{L^{2}}^{2}\,ds=\frac{1}{2}\|\rho_{in}\|_{L^{2}}^{2}$

and for any $n\geq 1$

$\sup_{t\leq
T}\|(\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho})(t)\|^{2}_{L^{2}}+\int_{0}^{T}\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}^{2}\lesssim(\tau^{-1})^{n}\bar{D}+\sum_{i=1}^{n}(\tau^{-1})^{n-i}\|(\bar{\kappa}^{i/2}\nabla^{i}\bar{\rho})_{in}\|^{2}_{L^{2}}.$
(3.17)

Further, for any $|\boldsymbol{\alpha}|=n$

$\int_{0}^{T}\|\bar{\kappa}^{\frac{n}{2}}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}\|^{2}_{L^{2}}\lesssim(\tau^{-1})^{n+1}\bar{D}+\sum_{i=1}^{n+1}\left(\tau^{-1}\right)^{{n+1-i}}\|(\bar{\kappa}^{i/2}\nabla^{i}\bar{\rho})_{in}\|^{2}_{L^{2}}$
(3.18)

The constants in $\lesssim$ depend on the constants in assumptions and $n$

The proof occupies the rest of this section.

###### Proof.

_Step 1: Preliminary $m$-th order estimate_.

Apply $\partial^{\boldsymbol{\alpha}}$, where $|\boldsymbol{\alpha}|=n$, to
the equation (3.6) and test the result with
$\bar{\kappa}^{n}\partial^{\boldsymbol{\alpha}}\bar{\rho}$

$\begin{split}\frac{1}{2}\frac{d}{dt}&\int|(\bar{\kappa}^{n/2}\partial^{\boldsymbol{\alpha}}\bar{\rho})(t)|^{2}+\int\bar{\kappa}^{n}\bar{A}(\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho})(\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho})=\frac{1}{2}\int\frac{D^{u}_{t}(\bar{\kappa}^{n})}{\bar{\kappa}^{n}}\bar{\kappa}^{n}|\partial^{\boldsymbol{\alpha}}\bar{\rho}|^{2}\\\
&-\sum_{{\boldsymbol{\beta}}+{\boldsymbol{\gamma}}={\boldsymbol{\alpha}},{\boldsymbol{\beta}}>0}c_{\boldsymbol{\beta}}\int\partial^{\boldsymbol{\beta}}u\cdot\partial^{\boldsymbol{\gamma}}\nabla\bar{\rho}(\partial^{\boldsymbol{\alpha}}\bar{\rho})\bar{\kappa}^{n}+\partial^{\boldsymbol{\beta}}\bar{A}\nabla\partial^{\boldsymbol{\gamma}}\bar{\rho}\nabla(\bar{\kappa}^{n}\partial^{\boldsymbol{\alpha}}\bar{\rho})+\int\bar{A}\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho}\nabla(\bar{\kappa}^{n})\partial^{\boldsymbol{\alpha}}\bar{\rho},\end{split}$
(3.19)

where $c_{\boldsymbol{\beta}}$ are binomial coefficients. Estimate four right-
hand side terms of (3.19), in order of their appearance. For the first one,
use
$\frac{D^{u}_{t}(\bar{\kappa}^{n})}{\bar{\kappa}^{n}}=n\frac{D^{u}_{t}(\bar{\kappa})}{\bar{\kappa}}$
and estimate in modulus

$\int\frac{D^{u}_{t}(\bar{\kappa}^{n})}{\bar{\kappa}^{n}}\bar{\kappa}^{n}|\partial^{\boldsymbol{\alpha}}\bar{\rho}|^{2}\leq
n\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}^{2}\left\|\frac{D^{u}_{t}\bar{\kappa}}{\bar{\kappa}}\right\|_{L^{\infty}}.$

For a single summand of the second one, estimate its modulus by distributing
the weights according to derivatives as follows

$\int\bar{\kappa}^{\frac{n-|\boldsymbol{\gamma}|-1}{2}}\partial^{\boldsymbol{\beta}}u\cdot(\bar{\kappa}^{\frac{|\boldsymbol{\gamma}|+1}{2}}\partial^{\boldsymbol{\gamma}}\nabla\bar{\rho})((\partial^{\boldsymbol{\alpha}}\bar{\rho})\bar{\kappa}^{n/2})\leq\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}\|\bar{\kappa}^{\frac{|\boldsymbol{\gamma}|+1}{2}}\nabla^{|\boldsymbol{\gamma}|+1}\bar{\rho}\|_{L^{2}}\left\|\bar{\kappa}^{\frac{n-|\boldsymbol{\gamma}|-1}{2}}\nabla^{n-|\boldsymbol{\gamma}|}u\right\|_{L^{\infty}}.$

For a single summand of the third one we estimate its modulus by

$\begin{split}\int&\partial^{\boldsymbol{\beta}}\bar{A}\nabla\partial^{\boldsymbol{\gamma}}\bar{\rho}\nabla(\bar{\kappa}^{n}\partial^{\boldsymbol{\alpha}}\bar{\rho})=n\int\partial^{\boldsymbol{\beta}}\bar{A}(\nabla\partial^{\boldsymbol{\gamma}}\bar{\rho})\bar{\kappa}^{n-1}\nabla\bar{\kappa}\partial^{\boldsymbol{\alpha}}\bar{\rho}+\int\partial^{\boldsymbol{\beta}}\bar{A}(\nabla\partial^{\boldsymbol{\gamma}}\bar{\rho})\bar{\kappa}^{n}\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho}\leq\\\
&n\|\bar{\kappa}^{\frac{n-|\boldsymbol{\gamma}|-2}{2}}\nabla^{n-|\boldsymbol{\gamma}|}\bar{A}\|_{L^{\infty}}\|\bar{\kappa}^{-\frac{1}{2}}\nabla\bar{\kappa}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{|\boldsymbol{\gamma}|+1}{2}}\nabla^{|\boldsymbol{\gamma}|+1}\bar{\rho}\|_{L^{2}}\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}\\\
+&\|\bar{\kappa}^{\frac{n-|\boldsymbol{\gamma}|-2}{2}}\nabla^{n-|\boldsymbol{\gamma}|}\bar{A}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{|\boldsymbol{\gamma}|+1}{2}}\nabla^{|\boldsymbol{\gamma}|+1}\bar{\rho}\|_{L^{2}}\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}.\end{split}$
(3.20)

For the fourth, last term of (3.19) we estimate it in modulus writing, thanks
to the upper bound (3.13)

$\int\bar{\kappa}^{n-1}\bar{A}\nabla\bar{\kappa}\partial^{\boldsymbol{\alpha}}\bar{\rho}\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho}\leq
2\int|\bar{\kappa}|^{n}|\nabla\bar{\kappa}||\partial^{\boldsymbol{\alpha}}\bar{\rho}|\nabla\partial^{\boldsymbol{\alpha}}\bar{\rho}|\leq
2\|\bar{\kappa}^{-\frac{1}{2}}\nabla\bar{\kappa}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}.$

Together, these estimates for four right-hand side terms of (3.19), absorbing
their terms $\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}$
by the dissipative part using Young’s inequality, after summing over all
multiindices of order $n$ yield

$\begin{split}\frac{1}{2}\frac{d}{dt}&\|(\bar{\kappa}^{n/2}\nabla^{n}\bar{\rho})(t)\|^{2}_{L^{2}}+\frac{1}{2}\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}^{2}\lesssim_{n}\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}^{2}\left\|\frac{D^{u}_{t}\bar{\kappa}}{\bar{\kappa}}\right\|_{L^{\infty}}\\\
&+\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}\sum_{j=0}^{n-1}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|_{L^{2}}\left\|\bar{\kappa}^{\frac{n-j-1}{2}}\nabla^{n-j}u\right\|_{L^{\infty}}\\\
&+\|\bar{\kappa}^{-\frac{1}{2}}\nabla\bar{\kappa}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|_{L^{2}}\sum_{j=0}^{n-1}\|\bar{\kappa}^{\frac{n-j-2}{2}}\nabla^{n-j}\bar{A}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|_{L^{2}}\\\
&+\sum_{j=0}^{n-1}\|\bar{\kappa}^{\frac{n-j-2}{2}}\nabla^{n-j}\bar{A}\|^{2}_{L^{\infty}}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|^{2}_{L^{2}}+\|\bar{\kappa}^{-\frac{1}{2}}\nabla\bar{\kappa}\|^{2}_{L^{\infty}}\|\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho}\|^{2}_{L^{2}}.\end{split}$
(3.21)

_Step 2: Plugging in scales assumptions._

Use the assumptions (3.14), (3.15), (3.16) for right-hand side of (3.21) to
obtain

$\frac{d}{dt}\|(\bar{\kappa}^{n/2}\nabla^{n}\bar{\rho})(t)\|^{2}_{L^{2}}+\|\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho}\|_{L^{2}}^{2}\lesssim_{n}\sum_{j=0}^{n-1}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|^{2}_{L^{2}}(\tau^{-1})^{n-j}$
(3.22)

which after integrating in time and writing
$P(i)=\int_{0}^{T}\|\bar{\kappa}^{\frac{i}{2}}\nabla^{i}\bar{\rho}\|_{L^{2}}^{2}$
yields for any $n\geq 1$

$\sup_{t\leq
T}\|(\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho})(t)\|^{2}_{L^{2}}+P(n+1)\lesssim_{n}\|(\bar{\kappa}^{\frac{n}{2}}\nabla^{n}\bar{\rho})_{in}\|^{2}_{L^{2}}+\sum_{j=0}^{n-1}P(j+1)(\tau^{-1})^{n-j}.$
(3.23)

_Step 3: Iterations._

Take (3.23) with $n=1$. Observing that $P(1)=\bar{D}$ we have

$\sup_{t\leq
T}\|(\bar{\kappa}^{1/2}\nabla\bar{\rho})(t)\|^{2}_{L^{2}}+\int_{0}^{T}\|\bar{\kappa}\nabla^{2}\bar{\rho}\|_{L^{2}}^{2}\,dt\lesssim\tau^{-1}\bar{D}+\|(\bar{\kappa}^{1/2}\nabla\bar{\rho})_{in}\|^{2}_{L^{2}}$
(3.24)

i.e. (3.17) with $n=1$, this allows to start induction. Assume (3.17) holds
for any $j\leq n$, in particular

$P(j+1)\lesssim_{n}(\tau^{-1})^{j}\bar{D}+\sum_{i=1}^{j}(\tau^{-1})^{j-i}\|(\bar{\kappa}^{i/2}\nabla^{i}\bar{\rho})_{in}\|^{2}_{L^{2}}.$

For $n+1$ we have via (3.23)

$\begin{split}P(n+2)&\lesssim_{n}\|(\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho})_{in}\|^{2}_{L^{2}}+\sum_{j=0}^{n}P(j+1)(\tau^{-1})^{n+1-j}\\\
&\lesssim_{n}\|(\bar{\kappa}^{\frac{n+1}{2}}\nabla^{n+1}\bar{\rho})_{in}\|^{2}_{L^{2}}+\sum_{j=0}^{n}\left((\tau^{-1})^{j}\bar{D}+\sum_{i=1}^{j}(\tau^{-1})^{j-i}\|(\bar{\kappa}^{i/2}\nabla^{i}\bar{\rho})_{in}\|^{2}_{L^{2}}\right)(\tau^{-1})^{n+1-j},\end{split}$

which gives (3.17) for $n+1$.

_Step 4: Transport estimate._

Apply $\partial^{\boldsymbol{\alpha}}$, where $|\boldsymbol{\alpha}|=n$, to
the equation (3.6)

$\begin{split}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}=-\sum_{{\boldsymbol{\beta}}+{\boldsymbol{\gamma}}={\boldsymbol{\alpha}},{\boldsymbol{\beta}}>0}c_{\boldsymbol{\beta}}\partial^{\boldsymbol{\beta}}u\cdot\partial^{\boldsymbol{\gamma}}\nabla\bar{\rho}+\mathop{\rm
div}\nolimits\sum_{{\boldsymbol{\beta}}+{\boldsymbol{\gamma}}={\boldsymbol{\alpha}}}c_{\boldsymbol{\beta}}\partial^{\boldsymbol{\beta}}\bar{A}\partial^{\boldsymbol{\gamma}}\nabla\bar{\rho}.\end{split}$

Multiply both sides with $\bar{\kappa}^{\frac{n}{2}}$, distribute the weights,
and take space-time $L^{2}$ norms on time interval $[0,T]$, denoted by
$L^{2}_{xt}$. These give

$\begin{split}\|\bar{\kappa}^{\frac{n}{2}}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}\|_{L^{2}_{xt}}&\lesssim_{n}\sum_{i+j=n,i>0}\|\bar{\kappa}^{\frac{i-1}{2}}\nabla^{i}u\|_{L^{\infty}}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|_{L^{2}_{xt}}\\\
&+\sum_{i+j=n}\|\bar{\kappa}^{\frac{i-1}{2}}\nabla^{i+1}\bar{A}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{j+1}{2}}\nabla^{j+1}\bar{\rho}\|_{L^{2}_{xt}}+\sum_{i+j=n}\|\bar{\kappa}^{\frac{i-2}{2}}\nabla^{i}\bar{A}\|_{L^{\infty}}\|\bar{\kappa}^{\frac{j+2}{2}}\nabla^{j+2}\bar{\rho}\|_{L^{2}_{xt}}.\end{split}$

Use the assumptions (3.15), (3.16)

$\|\bar{\kappa}^{\frac{n}{2}}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}\|_{L^{2}_{xt}}\lesssim_{n}\tau^{-1}\sum_{j=1}^{n+2}(\tau^{-1})^{\frac{n-j}{2}}\|\bar{\kappa}^{\frac{j}{2}}\nabla^{j}\bar{\rho}\|_{L^{2}_{xt}}.$

Squaring both sides above yields

$\int_{0}^{T}\|\bar{\kappa}^{\frac{n}{2}}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}\|^{2}_{L^{2}}\lesssim_{n}(\tau^{-1})^{2}\sum_{j=1}^{n+2}(\tau^{-1})^{n-j}P(j).$

(3.18) now follows from using (3.17) to control $P(j)$.

∎

##  4\. Spatial homogenisation

###  4.1. Setup

In this section, we use $\xi$ to denote the variable in the cell. For a
function $f:\mathbb{T}^{3}\rightarrow\mathbb{R}$ defined in the cell, i.e.
taking the variable $\xi$ as its argument, we use $\langle f\rangle$ to denote
its integral in $\mathbb{T}^{3}$. Without further specification, the integral
domain is $\mathbb{T}^{3}$ for the variables $x,\xi\in\mathbb{T}^{3}$. Let $g$
be a function taking arguments $(x,t,\xi)$, we use
$\|g(x,t,\cdot)\|_{L^{\infty}_{\xi}}$ to denote the supreme norm in $\xi$
variable. Note that $\|g\|_{L^{\infty}_{\xi}}$ is still a function in $(x,t)$.
The usual $L^{2}$ and $H^{-1}$ norms are in $x$ variable. The constant in
$\lesssim$ in this section depends on $N_{h}$ defined in (4.18).

In this section we consider the following advection-diffusion equation for
$\rho:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}$

$\begin{split}\partial_{t}\rho+u\cdot\nabla\rho=&\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla\rho\big{)},\\\
\rho|_{t=0}=&\rho_{\text{in}},\end{split}$ (4.1)

with elliptic tensors
$\tilde{A}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ and
$A:\mathbb{T}^{3}\times[0,T]\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3\times
3}$ given by

$\displaystyle\tilde{A}(x,t):=$ $\displaystyle
A\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)},$ (4.2) $\displaystyle A(x,t,\xi):=$
$\displaystyle\sum_{i}\tilde{\eta}_{i}(x,t)\nabla\Phi_{i}^{-1}(x,t)\Bigg{(}\kappa\mathrm{Id}+\frac{\eta_{i}(x,t)\sigma^{1/2}(t)}{\lambda}\sum_{\vec{k}}a_{\vec{k}}\big{(}\tilde{R}_{i}(x,t)\big{)}H_{\vec{k}}(\xi)\Bigg{)}\nabla\Phi_{i}^{-T}(x,t),$
(4.3)

where, compared to the setting in Step 2 of the proof of Proposition 2.4 we
drop the index $q$ for simplicity, so that

$\begin{split}\kappa=&\kappa_{q+1},\\\ \lambda=&\lambda_{q+1},\\\
\bar{\kappa}=&\tilde{\kappa}_{q},\\\
\tau=&\tau_{q},\end{split}\quad\quad\quad\begin{split}\ell=&\ell_{q},\\\
u=&\bar{u}_{q},\\\ \sigma=&\sigma_{q},\\\
\tilde{R}_{i}=&\tilde{R}_{q,i}.\end{split}$ (4.4)

and $\bar{u}_{q}$, $\Phi_{i}$, $\sigma_{q}$, $\eta_{i}$, $a_{\vec{k}}$,
$\tilde{R}_{q,i}$ are given in section 2.1.4. The partition of unity
$\tilde{\eta}_{i}$ is defined in (2.40). Here, we also clarify a notation
taking (4.2) as an example. We often use $(x,t,\lambda\Phi_{i}(x,t))$ as an
argument of some function, this function always involves $(\eta_{i})_{i}$ as a
family of cutoff functions with disjoint support. It means taking
$(x,t,\lambda\Phi_{i}(x,t))$ as the argument in the support of $\eta_{i}$ for
any $i$.

Our goal in this section is to show the solution $\rho$ to (4.1) homogenizes
to the solution $\bar{\rho}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}$ of
the following equation

$\begin{split}\partial_{t}\bar{\rho}+u\cdot\nabla\bar{\rho}=&\mathop{\rm
div}\nolimits\big{(}\bar{A}\nabla\bar{\rho}\big{)},\\\
\bar{\rho}|_{t=0}=&\rho_{\text{in}},\end{split}$ (4.5)

with elliptic tensor
$\bar{A}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ given by

$\displaystyle\bar{A}(x,t)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
A(x,\xi,t)\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}(x,t)\nabla\Phi_{i}^{T}(x,t)\nabla_{\xi}\chi_{i}^{T}(x,t,\xi)\Big{)}\,d\xi,$
(4.6)

and
$\chi_{i},\chi:\mathbb{T}^{3}\times[0,T]\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$
given by

$\displaystyle\chi_{i}(x,t,\xi)=$
$\displaystyle-\frac{\sigma^{1/2}(t)}{\kappa\lambda}\nabla\Phi_{i}^{-1}(x,t)\sum_{\vec{k}}a_{\vec{k}}\big{(}\tilde{R}_{i}(x,t)\big{)}\varphi_{\vec{k}}(\xi)\vec{k},$
(4.7) $\displaystyle\chi(x,t,\xi)=$
$\displaystyle\sum_{i}\chi_{i}(x,t,\xi)\eta_{i}(x,t).$ (4.8)

With the choice in (4.4), we collect the following facts:

$\displaystyle\|\nabla\Phi_{i}-\mathrm{Id}\|_{L^{\infty}}\leq$
$\displaystyle\lambda^{-2\gamma},$ (4.9)
$\displaystyle\tfrac{1}{2}\bar{\kappa}\leq\bar{A}$ $\displaystyle\leq
2\bar{\kappa},\quad\textrm{ where
}\bar{\kappa}=\kappa\left(1+\sum_{i}\frac{\sigma\eta_{i}^{2}}{\kappa^{2}\lambda^{2}}\right),$
(4.10)

and for $n=0,1$

$\displaystyle\|D_{t}^{n}\nabla_{x}^{m}\chi_{i}(x,t,\cdot)\|_{C^{1}_{\xi}}\lesssim$
$\displaystyle\frac{\sigma^{1/2}}{\kappa\lambda}\tau^{-n}\ell^{-m},$ (4.11)
$\displaystyle\|D_{t}^{n}\nabla_{x}^{m}\sigma\|_{L^{\infty}}\lesssim$
$\displaystyle\sigma\tau^{-n}\ell^{-m},$ (4.12)
$\displaystyle\|D_{t}^{n}\nabla_{x}^{m}\eta_{i}\|_{L^{\infty}}+\|D_{t}^{n}\nabla_{x}^{m}\tilde{\eta}_{i}\|_{L^{\infty}}+$
$\displaystyle\|D_{t}^{n}\nabla_{x}^{m}\nabla\Phi_{i}\|_{L^{\infty}}+\|D_{t}^{n}\nabla_{x}^{m}(a_{\vec{k}}\circ\tilde{R}_{i})\|_{L^{\infty}}\lesssim\tau^{-n}\ell^{-m}.$
(4.13)

Notice that from (2.30) and (2.31), and with $\gamma>0$ given in (2.34), we
have

$\displaystyle\tau\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\leq$ $\displaystyle
1,$ (4.14) $\displaystyle\tau\|\bar{\kappa}\|_{L^{\infty}}\lambda^{2}\geq$
$\displaystyle\lambda^{\frac{b-1}{b+1}},$ (4.15)
$\displaystyle\tau\|\bar{\kappa}\|_{L^{\infty}}\lambda^{\frac{2}{b}}\leq$
$\displaystyle\lambda^{-2\gamma},$ (4.16)
$\displaystyle\|\bar{\kappa}\|_{L^{\infty}}^{-1}\tau^{-1}\lambda^{-2}\leq\kappa^{-1}\tau^{-1}\lambda^{-2}<$
$\displaystyle\lambda^{-2\gamma}.$ (4.17)

Furthermore, we need to choose $N_{h}\geq 3$ sufficiently large such that

$\displaystyle N_{h}\geq(b+1)\left(\frac{2}{b-1}+\frac{\theta}{b}\right),$
(4.18)

then (4.18) and (4.15) give the following relation

$\displaystyle\lambda^{2}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}$
$\displaystyle\lesssim\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\lambda^{2}\big{)}^{N_{h}}.$
(4.19)

###  4.2. Quantitative estimates

The quantitative version of the above homogenisation process is given as
follows.

######  Proposition 4.1.

Given (4.4), the parameter setting in Proposition 2.1 and Proposition 2.4, let
$\rho$ be the solution to (4.1)-(4.3). Let $\bar{\rho}$ be the solution to
(4.5)-(4.7). Choose $N_{h}$ such that (4.18) holds. Define
$\tilde{\rho}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}$ such that

$\rho(x,t)=\bar{\rho}(x,t)+\frac{1}{\lambda}\chi\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}\cdot\nabla\bar{\rho}(x,t)+\tilde{\rho}(x,t).$
(4.20)

Then

$\begin{split}\|\tilde{\rho}(\cdot,T)\|^{2}_{L^{2}}+&\kappa\int_{0}^{T}\|\nabla\tilde{\rho}\|^{2}_{L^{2}}dt\lesssim\frac{1}{\lambda^{2}\kappa\tau}\mathcal{D}_{N_{h}},\\\
\mathcal{D}_{l}:=&\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\bar{\rho}\|^{2}_{L^{2}}dt+\sum_{i=1}^{l}\tau^{i}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{i}}{{2}}}\nabla^{i}\rho_{\text{in}}\|_{L^{2}}^{2}dt\quad\text{
for }l\in\mathbb{N}.\end{split}$ (4.21)

We also have the following corollaries

######  Corollary 4.2.

Let $\rho,\bar{\rho}$ and $\mathcal{D}_{l}$ be as in Proposition 4.1, then

$\begin{split}\|\rho(t)-\bar{\rho}(t)\|^{2}_{L^{2}}+&\kappa\int_{0}^{T}\|\nabla\rho-\nabla\bar{\rho}\|^{2}_{L^{2}}dt\lesssim\frac{1}{\lambda^{2}\kappa\tau}\mathcal{D}_{N_{h}}.\\\
\end{split}$ (4.22)

######  Corollary 4.3.

Let $\rho,\bar{\rho}$ and $\mathcal{D}_{l}$ be as in Proposition 4.1, then

$\begin{split}\Big{|}\|\rho(T)\|^{2}_{L^{2}}-\|\bar{\rho}(T)\|^{2}_{L^{2}}\Big{|}\lesssim\Big{(}\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}+\lambda^{-2\gamma}\Big{)}\mathcal{D}_{N_{h}}.\\\
\end{split}$ (4.23)

######  Remark 4.4.

If $\rho_{in}$ satisfies that, for any $i\in[0,N_{h}]$

$\displaystyle\|\nabla^{i}\rho_{in}\|_{L^{2}}^{2}\leq\lambda^{\frac{2i}{b}}\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt,$

then for $\mathcal{D}_{N_{h}}$, we have

$\begin{split}\mathcal{D}_{N_{h}}\leq&\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\bar{\rho}\|^{2}_{L^{2}}dt+\sum_{i=1}^{N_{h}}\tau^{i}\|\bar{\kappa}\|_{L^{\infty}}^{i}\lambda^{\frac{2i}{b}}\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt,\\\
\leq&{}_{\eqref{e:hom_para_assump:for_rem}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\bar{\rho}\|^{2}_{L^{2}}dt+\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt.\end{split}$
(4.24)

Therefore, combining (4.23), (4.24) and (4.17), we have

$\displaystyle\Big{|}\|\rho(T)\|^{2}_{L^{2}}-\|\bar{\rho}(T)\|^{2}_{L^{2}}\Big{|}\lesssim\lambda^{-\gamma}\min\left\\{\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\bar{\rho}\|^{2}_{L^{2}}dt,\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt\right\\}.$
(4.25)

The proofs of Proposition 4.1, Corollary 4.2 and Corollary 4.3 are given in
the end of this section. The $\tilde{\rho}$ in Proposition 4.1 is the
homogenization error term. To estimate this error term, we show $\tilde{\rho}$
satisfies an equation in the following lemma.

######  Lemma 4.5.

Let $\tilde{\rho}$ be as in (4.20), then $\tilde{\rho}$ satisfies

$\begin{split}\partial_{t}\tilde{\rho}&+u\cdot\nabla\tilde{\rho}-\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla\tilde{\rho}\big{)}=\mathop{\rm
div}\nolimits\big{(}\tilde{B}\nabla\bar{\rho}\big{)}\\\
&+\frac{1}{\lambda}\Big{(}\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla^{2}\bar{\rho}\chi\big{)}+\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla_{x}\chi^{T}\nabla\bar{\rho}\big{)}\Big{)}\\\
&-\frac{1}{\lambda}\Big{(}\chi\cdot
D_{t}\nabla\bar{\rho}\cdot+\nabla\bar{\rho}\cdot D_{t}\chi\Big{)}\end{split}$
(4.26)

with the matrices
$\tilde{B}:\mathbb{T}^{3}\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ and
$B:\mathbb{T}^{3}\times[0,T]\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3\times
3}$ given by

$\displaystyle\tilde{B}(x,t)=$ $\displaystyle
B\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)},$ (4.27) $\displaystyle B(x,t,\xi)=$
$\displaystyle
A\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\Big{)}-\Big{\langle}A\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\Big{)}\Big{\rangle},$
(4.28)

and $D_{t}\chi$ denotes the transport derivative in $x$, i.e.

$\displaystyle
D_{t}\chi=\big{(}\partial_{t}\chi+(u\cdot\nabla_{x})\chi\big{)}(x,t,\lambda\Phi_{i}).$

###### Proof of Lemma 4.5.

From the ansatz (4.20), omitting the argument
$\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}$, direct computations give

$\displaystyle\nabla\tilde{\rho}$
$\displaystyle=\nabla\rho-\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\Big{)}\nabla\bar{\rho}-\frac{1}{\lambda}\sum_{i}\Big{(}\nabla^{2}\bar{\rho}\chi_{i}\eta_{i}+\nabla_{x}(\chi_{i}\eta_{i})^{T}\nabla\bar{\rho}\Big{)},$
(4.29) $\displaystyle\partial_{t}\tilde{\rho}$
$\displaystyle=\partial_{t}\rho-\partial_{t}\bar{\rho}-\sum_{i}\eta_{i}\partial_{t}\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\nabla\bar{\rho}-\frac{1}{\lambda}\sum_{i}\Big{(}\partial_{t}\nabla\bar{\rho}\cdot\chi_{i}\eta_{i}+\nabla\bar{\rho}\cdot\partial_{t}(\chi_{i}\eta_{i})\Big{)}.$
(4.30)

Using $\partial_{t}\Phi_{i}+(u\cdot\nabla)\Phi_{i}=0$ and omitting the
argument $\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}$, we obtain

$\begin{split}\partial_{t}\tilde{\rho}+u\cdot\nabla\tilde{\rho}-\mathop{\rm
div}\nolimits\tilde{A}\nabla\tilde{\rho}=&\mathop{\rm
div}\nolimits\left(\left[A\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\Big{)}-\Big{\langle}A\Big{(}\mathrm{Id}+\sum_{i}\eta_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\Big{)}\Big{\rangle}\right]\nabla\bar{\rho}\right)\\\
&+\frac{1}{\lambda}\sum_{i}\Big{(}\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla^{2}\bar{\rho}\chi_{i}\eta_{i}\big{)}+\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla_{x}(\chi_{i}\eta_{i})^{T}\nabla\bar{\rho}\big{)}\Big{)}\\\
&-\frac{1}{\lambda}\sum_{i}\Big{(}\partial_{t}\nabla\bar{\rho}\chi_{i}\eta_{i}+\partial_{t}(\chi_{i}\eta_{i})\nabla\bar{\rho}\Big{)}-\frac{1}{\lambda}\sum_{i}\Big{(}u\nabla^{2}\bar{\rho}\chi_{i}\eta_{i}+u\nabla_{x}(\chi_{i}\eta_{i})^{T}\nabla\bar{\rho}\Big{)}\\\
=&\mathop{\rm
div}\nolimits\big{(}B\nabla\bar{\rho}\big{)}+\frac{1}{\lambda}\sum_{i}\Big{(}\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla^{2}\bar{\rho}\chi_{i}\eta_{i}\big{)}+\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla_{x}(\chi_{i}\eta_{i})^{T}\nabla\bar{\rho}\big{)}\Big{)}\\\
&-\frac{1}{\lambda}\sum_{i}\Big{(}\eta_{i}\chi_{i}\cdot
D_{t}\nabla\bar{\rho}\cdot+\nabla\bar{\rho}\cdot
D_{t}(\chi_{i}\eta_{i})\Big{)}.\end{split}$

∎

######  Lemma 4.6.

Let $\tilde{\rho}$ be as in (4.20). Then for any $N\in\mathbb{N}^{+}$,
$\tilde{\rho}$ satisfies

$\begin{split}\partial_{t}\tilde{\rho}&+u\cdot\nabla\tilde{\rho}-\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla\tilde{\rho}\big{)}=\frac{1}{\lambda}\Big{(}E_{1}+E_{2}+E_{3}+E_{4}+\sum_{l=1}^{N}F_{l}+G_{N}\Big{)}\end{split}$
(4.31)

with

$\displaystyle E_{1}=$ $\displaystyle-\sum_{i,j,l}\mathop{\rm
div}\nolimits\big{(}\nabla\partial_{j}\bar{\rho}\times(\nabla\Phi_{i}^{T}\tilde{c}^{(i)}_{j})\big{)}$
$\displaystyle E_{2}=$ $\displaystyle-\sum_{i,j,l}\mathop{\rm
div}\nolimits\big{(}\nabla_{x}\tilde{c}^{(i)}_{jl}\times\nabla\Phi_{i,l})\partial_{j}\bar{\rho}\big{)}$
$\displaystyle E_{3}=$ $\displaystyle\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla^{2}\bar{\rho}\chi\big{)}$ $\displaystyle
E_{4}=$ $\displaystyle\mathop{\rm
div}\nolimits\big{(}\tilde{A}\nabla_{x}\chi^{T}\nabla\bar{\rho}\big{)}$

and

$\displaystyle F_{1}=$ $\displaystyle 0,$ (4.32) $\displaystyle F_{l}=$
$\displaystyle\frac{1}{\lambda^{l-1}}\sum_{i,1\leq|\boldsymbol{\alpha}|\leq
l-1}\mathop{\rm
div}\nolimits\Big{(}f^{(l-1)}_{0,\boldsymbol{\alpha}}(x,t,\lambda\Phi_{i})D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}+f^{(l-1)}_{1,\boldsymbol{\alpha}}(x,t,\lambda\Phi_{i})\partial^{\boldsymbol{\alpha}}\bar{\rho}\Big{)},\text{
for }l\geq 2,$ (4.33) $\displaystyle G_{l}=$
$\displaystyle\frac{1}{\lambda^{l-1}}\sum_{i,1\leq|\boldsymbol{\alpha}|\leq
l}\Big{(}g^{(l)}_{0,\boldsymbol{\alpha}}(x,t,\lambda\Phi_{i})D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}+g^{(l)}_{1,\boldsymbol{\alpha}}(x,t,\lambda\Phi_{i})\partial^{\boldsymbol{\alpha}}\bar{\rho}\Big{)}$
(4.34)

where $\tilde{c}_{j}^{(i)}(x,t)=c_{j}^{(i)}(x,t,\lambda\Phi_{i}(x,t))$ and the
functions $c_{j}^{(i)}$, $f^{(l)}_{0,\boldsymbol{\alpha}}$,
$f^{(l)}_{1,\boldsymbol{\alpha}}$, $g^{(l)}_{0,\boldsymbol{\alpha}}$ and
$g^{(l)}_{1,\boldsymbol{\alpha}}$, taking arguments $(x,t,\xi)$, satisfy the
following estimates, for $n\in\\{0,1\\}$,

$\displaystyle\langle f^{(l)}_{0,\boldsymbol{\alpha}}\rangle=\langle
f^{(l)}_{1,\boldsymbol{\alpha}}\rangle$ $\displaystyle=\langle
g^{(l)}_{0,\boldsymbol{\alpha}}\rangle=\langle
g^{(l)}_{1,\boldsymbol{\alpha}}\rangle=0,$ (4.35)
$\displaystyle\|D_{t}^{n}\nabla_{x}^{m}c^{(i)}_{j}\|_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\kappa\Big{(}1+\frac{\sigma^{1/2}\eta_{i}}{\kappa\lambda}\Big{)}\frac{\sigma^{1/2}\eta_{i}}{\kappa\lambda}\tau^{-n}\ell^{-m},$
(4.36)
$\displaystyle\big{\|}D_{t}^{n}\nabla_{x}^{m}f^{(l)}_{0,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}+\big{\|}D_{t}^{n}\nabla_{x}^{m}g^{(l)}_{0,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\frac{\sigma^{1/2}\eta_{i}}{\kappa\lambda}\tau^{-n}\ell^{-(l-|\boldsymbol{\alpha}|+m)},$
(4.37)
$\displaystyle\big{\|}D_{t}^{n}\nabla_{x}^{m}f^{(l)}_{1,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}+\big{\|}D_{t}^{n}\nabla_{x}^{m}g^{(l)}_{1,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\frac{\sigma^{1/2}\eta_{i}}{\kappa\lambda}\tau^{-(n+1)}\ell^{-(l-|\boldsymbol{\alpha}|+m)}.$
(4.38)

###### Proof of Lemma 4.6.

Define $\chi_{ij}=\chi_{i}\cdot e_{j}$. Let $b_{j}=Be_{j}$ and
$\tilde{b}_{j}=\tilde{B}e_{j}$. Direct computations show

$\begin{split}b_{j}&(x,t,\xi)=\sum_{i}\eta_{i}b^{(i)}_{j}\\\
=&\sum_{i}\eta_{i}\nabla\Phi_{i}^{-1}\Bigg{[}\kappa\nabla_{\xi}\chi_{ij}+\frac{\sigma^{1/2}}{\lambda}\sum_{\vec{k}}a_{\vec{k}}(\tilde{R}_{i})\Big{(}H_{\vec{k}}\nabla\Phi_{i}^{-T}e_{j}+\eta_{i}H_{\vec{k}}\nabla_{\xi}\chi_{ij}-\eta_{i}\langle
H_{\vec{k}}\nabla_{\xi}\chi_{ij}\rangle\Big{)}\Bigg{]}.\end{split}$ (4.39)

We claim $\mathop{\rm div}\nolimits_{\xi}(\nabla\Phi_{i}b^{(i)}_{j})=0$.
Indeed, notice that

$\begin{split}\mathop{\rm
div}\nolimits_{\xi}\big{(}H_{\vec{k}}\nabla_{\xi}\chi_{ij}\big{)}=&\big{(}W_{\vec{k}}\cdot\nabla_{\xi}\big{)}\chi_{ij}=-\frac{\rho^{1/2}}{\kappa\lambda}\nabla\Phi_{i}^{-1}\sum_{\vec{k}}a_{\vec{k}}\big{(}\tilde{R}_{i}\big{)}\big{(}\psi_{\vec{k}}\vec{k}\cdot\nabla_{\xi}\big{)}\varphi_{\vec{k}}\vec{k}\cdot
e_{j}=0,\\\ \mathop{\rm
div}\nolimits_{\xi}\big{(}H_{\vec{k}}\nabla\Phi_{i}^{-T}e_{j}\big{)}=&W_{\vec{k}}\cdot\big{(}\nabla\Phi_{i}^{-T}e_{j}\big{)}=\psi_{\vec{k}}\big{(}\nabla\Phi_{i}^{-1}\vec{k}\big{)}\cdot
e_{j}.\end{split}$ (4.40)

Then plugging (4.40) and (4.7) into (4.39) gives $\mathop{\rm
div}\nolimits_{\xi}(\nabla\Phi_{i}b^{(i)}_{j})=0$.

Also notice that $\langle\nabla\Phi_{i}b^{(i)}_{j}\rangle=0$, hence we can
find a vector potential $c^{(i)}_{j}=c^{(i)}_{j}(x,t,\xi)$ so that
$\eta_{i}\nabla\Phi_{i}\,b^{(i)}_{j}=\nabla_{\xi}\times c^{(i)}_{j}$. Using
the fact $\det\nabla\Phi_{i}^{T}=1$ and omitting the argument
$\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}$, we have333Here, we use the formula
$(Av_{1})\times(Av_{2})=A^{-T}(v_{1}\times v_{2})\det A$ for
$v_{1},v_{2}\in\mathbb{R}^{3}$ and $A\in\mathbb{R}^{3\times 3}$ invertible.

$\begin{split}\frac{1}{\lambda}\nabla\times\big{(}\nabla\Phi_{i}^{T}c^{(i)}_{j}\big{)}&=\frac{1}{\lambda}\nabla\times\big{(}c^{(i)}_{jl}\nabla\Phi_{i,l}\big{)}=\frac{1}{\lambda}\nabla
c^{(i)}_{jl}\times\nabla\Phi_{i,l}\quad\text{(chain rule)}\\\
&=\frac{1}{\lambda}\nabla_{x}c^{(i)}_{jl}\times\nabla\Phi_{i,l}+\big{(}\nabla\Phi_{i}^{T}\nabla_{\xi}c^{(i)}_{jl}\big{)}\times\big{(}\nabla\Phi_{i}^{T}e_{l}\big{)}\\\
&=\frac{1}{\lambda}\nabla_{x}c^{(i)}_{jl}\times\nabla\Phi_{i,l}+\nabla\Phi_{i}^{-1}\big{(}\nabla_{\xi}\times
c^{(i)}_{j}\big{)}\quad\text{(linear transform for }\nabla\times).\end{split}$

Therefore, omitting the argument $\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}$, we
have

$\displaystyle b_{j}=$
$\displaystyle\sum_{i}\nabla\Phi_{i}^{-1}\nabla_{\xi}\times
c^{(i)}_{j}=\sum_{i}\Big{(}\frac{1}{\lambda}\nabla\times(\nabla\Phi_{i}^{T}c^{(i)}_{j})-\frac{1}{\lambda}\nabla_{x}c^{(i)}_{jl}\times\nabla\Phi_{i,l}\Big{)},$
$\displaystyle b_{j}\partial_{j}\bar{\rho}=$
$\displaystyle\frac{1}{\lambda}\nabla\times\big{(}\partial_{j}\bar{\rho}\nabla\Phi_{i}^{T}c^{(i)}_{j}\big{)}-\frac{1}{\lambda}\nabla\partial_{j}\bar{\rho}\times\big{(}\nabla\Phi_{i}^{T}c^{(i)}_{j}\big{)}-\frac{1}{\lambda}\big{(}\nabla_{x}c^{(i)}_{jl}\times\nabla\Phi_{i,l}\big{)}\partial_{j}\bar{\rho}.$

Let
$\tilde{c}^{(i)}_{j}(x,t)=c^{(i)}_{j}\big{(}x,t,\lambda\Phi_{i}(x,t)\big{)}$.
The estimate (4.36) follows from (4.11) by Schauder estimate. This gives
$E_{1}$ and $E_{2}$. $E_{3}$ and $E_{4}$ are directly from the second line of
(4.26).

Next, by induction, we show the last line of (4.26) gives
$\sum_{l=1}^{N}F_{l}+G_{N}$. When $l=1$, $|\boldsymbol{\alpha}|=1$, let
$g^{(1)}_{0,\boldsymbol{\alpha}}$ be given by $\chi$,
$g^{(1)}_{1,\boldsymbol{\alpha}}$ be given by $D_{t}\chi$ componentwise,
$F_{1}=0$. Using (4.11), we have (4.35) and the estimates (4.37) (4.38) for
$l=1$ by Schauder estimate. Assume (4.33), (4.34), (4.37) and (4.38) are true
for $l$. Using chain rule, for any function
$h(x,t,\xi):\mathbb{T}^{3}\times[0,1]\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$,
we have

$\begin{split}\frac{1}{\lambda}\mathop{\rm
div}\nolimits\bigl{(}\nabla\Phi_{i}^{-1}h(x,t,\lambda\Phi_{i})\bigr{)}&=(\mathop{\rm
div}\nolimits_{\xi}h)(x,t,\lambda\Phi_{i})+\frac{1}{\lambda}\mathop{\rm
div}\nolimits_{x}(\nabla\Phi_{i}^{-1}h)(x,t,\lambda\Phi_{i}).\end{split}$

Let $h_{0}=\mathop{\rm
div}\nolimits^{-1}_{\xi}g^{(l)}_{0,\boldsymbol{\alpha}}$ and
$h_{1}=\mathop{\rm div}\nolimits^{-1}_{\xi}g^{(l)}_{1,\boldsymbol{\alpha}}$,
then $\langle h_{0}\rangle=\langle h_{1}\rangle=0$. We can deduce (dropping
the arguments $(x,t,\lambda\Phi_{i})$)

$\begin{split}\frac{1}{\lambda}\mathop{\rm
div}\nolimits\big{(}\nabla\Phi_{i}^{-1}h_{0}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}+&\nabla\Phi_{i}^{-1}h_{1}\partial^{\boldsymbol{\alpha}}\bar{\rho}\big{)}=g^{(l)}_{0,\boldsymbol{\alpha}}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}+g^{(l)}_{1,\boldsymbol{\alpha}}\partial^{\boldsymbol{\alpha}}\bar{\rho}\\\
&+\frac{1}{\lambda}\mathop{\rm
div}\nolimits_{x}\big{(}\nabla\Phi_{i}^{-1}h_{0}\big{)}D_{t}\partial^{\boldsymbol{\alpha}}\bar{\rho}+\frac{1}{\lambda}\mathop{\rm
div}\nolimits_{x}\big{(}\nabla\Phi_{i}^{-1}h_{1}\big{)}\partial^{\boldsymbol{\alpha}}\bar{\rho}\\\
&+\frac{1}{\lambda}\nabla\Phi_{i}^{-1}h_{0}\cdot
D_{t}\partial^{\boldsymbol{\alpha}}\nabla\bar{\rho}+\frac{1}{\lambda}\nabla\Phi_{i}^{-1}h_{1}\cdot\partial^{\boldsymbol{\alpha}}\nabla\bar{\rho}.\end{split}$
(4.41)

Now let $f^{(l)}_{0,\boldsymbol{\alpha}}=\nabla\Phi_{i}^{-1}h_{0}$ and
$f^{(l)}_{1,\boldsymbol{\alpha}}=\nabla\Phi_{i}^{-1}h_{1}$. The second and the
third lines in (4.41) give the formulas for
$g^{(l+1)}_{0,\boldsymbol{\alpha}}$ and $g^{(l+1)}_{1,\boldsymbol{\alpha}}$.
Because $h_{0}$ and $h_{1}$ have zero mean in $\xi$,
$g^{(l+1)}_{0,\boldsymbol{\alpha}}$ and $g^{(l+1)}_{1,\boldsymbol{\alpha}}$
also have zero mean in $\xi$. The estimates (4.37) and (4.38) for $l+1$
directly follows from the form of $g^{(l+1)}_{0,\boldsymbol{\alpha}}$ and
$g^{(l+1)}_{1,\boldsymbol{\alpha}}$ and (4.11). ∎

###### Proof of Proposition 4.1 and Corollary 4.2.

Test (4.31) with $\tilde{\rho}$. By Hölder’s inequality and Young’s
inequality, we have

$\begin{split}\|\tilde{\rho}(\cdot,t)\|^{2}_{L^{2}}+\kappa\int_{0}^{T}\|\nabla\tilde{\rho}\|^{2}_{L^{2}}dt\lesssim&\frac{1}{\lambda^{2}\kappa}\bigg{(}\sum_{l=1}^{4}\int_{0}^{T}\|E_{l}\|^{2}_{H^{-1}}dt+\sum_{l=1}^{N}\int_{0}^{T}\|F_{l}\|^{2}_{H^{-1}}dt+\int_{0}^{T}\|G_{N}\|^{2}_{L^{2}}dt\bigg{)}\\\
&+\frac{1}{\lambda^{2}}\|(\chi\cdot\nabla\bar{\rho})(\cdot,0)\|^{2}_{L^{2}}\end{split}$
(4.42)

Recall the estimates for $\bar{\rho}$ from (3.17) and (3.18),

$\begin{split}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}\nabla^{m}\bar{\rho}\|_{L^{2}}^{2}dt\lesssim&\tau^{-(m-1)}\mathcal{D}_{m-1},\\\
\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}D_{t}\nabla^{m}\bar{\rho}\|_{L^{2}}^{2}dt\lesssim&\tau^{-(m+1)}\mathcal{D}_{m+1}.\\\
\end{split}$ (4.43)

Note that (4.36) gives

$\displaystyle\|\bar{\kappa}^{-1}D_{t}^{n}\nabla_{x}^{m}c^{(i)}_{j}\|_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\tau^{-n}\ell^{-m}.$ (4.44)

Also (4.11) and (4.3) give

$\displaystyle\|\bar{\kappa}^{-1}\tilde{A}D_{t}^{n}\nabla_{x}^{m}\chi\|_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\tau^{-n}\ell^{-m}.$ (4.45)

With (4.44) and (4.45), the estimates for $E_{1}$ and $E_{3}$ are the same.
The estimates for $E_{2}$ and $E_{4}$ are also the same. Here, we estimate
$E_{1}$ and $E_{2}$ as examples.

$\begin{split}\int_{0}^{T}\|E_{1}\|^{2}_{H^{-1}}dt\lesssim&\|\bar{\kappa}^{-1}c^{(i)}_{j}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}\nabla^{2}\bar{\rho}\|^{2}_{L^{2}}dt\\\
\lesssim&\tau^{-1}\mathcal{D}_{1},\\\
\int_{0}^{T}\|E_{2}\|^{2}_{H^{-1}}dt\lesssim&\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\|^{2}_{L^{\infty}}\|\bar{\kappa}^{-1}\nabla_{x}c^{(i)}_{j}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\bar{\rho}\|_{L^{2}}dt\\\
\lesssim&\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\mathcal{D}_{0}\lesssim_{\eqref{e:hom_para_assump:taukappaell}}\tau^{-1}\mathcal{D}_{0}.\end{split}$
(4.46)

Note that $\mathcal{D}_{1}$ is defined in (4.21).

Next, we estimate $F_{l}$ and $G_{N}$. In the rest of this proof, notice the
fact $|\nabla\Phi_{i}|<2$, contributing to a factor finally depending on
$N_{h}$. Note that from (4.37), (4.38) and (4.10) we have

$\begin{split}\big{\|}\bar{\kappa}^{-\nicefrac{{1}}{{2}}}f^{(l)}_{0,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}+\big{\|}\bar{\kappa}^{-\nicefrac{{1}}{{2}}}g^{(l)}_{0,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}&\lesssim\kappa^{-\nicefrac{{1}}{{2}}}\ell^{-(l-|\boldsymbol{\alpha}|)},\\\
\big{\|}\bar{\kappa}^{-\nicefrac{{1}}{{2}}}f^{(l)}_{1,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}+\big{\|}\bar{\kappa}^{-\nicefrac{{1}}{{2}}}g^{(l)}_{1,\boldsymbol{\alpha}}\big{\|}_{L^{\infty}_{\xi}}&\lesssim\kappa^{-\nicefrac{{1}}{{2}}}\tau^{-1}\ell^{-(l-|\boldsymbol{\alpha}|)}.\end{split}$
(4.47)

Note that $F_{1}=0$. For $F_{l}$ with $F\geq 2$, using (4.43) and (4.47), we
have

$\begin{split}\int_{0}^{T}\|F_{l}\|_{H^{-1}}^{2}dt\lesssim&\frac{1}{\lambda^{2(l-1)}}\sum_{m=1,|\boldsymbol{\alpha}|=m}^{l-1}\|\bar{\kappa}^{-(m-1)}\|_{L^{\infty}}\|\bar{\kappa}^{-\nicefrac{{1}}{{2}}}f^{(l-1)}_{0,\boldsymbol{\alpha}}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}D_{t}\nabla^{m}\bar{\rho}\|_{L^{2}}dt\\\
&+\|\bar{\kappa}^{-(m-1)}\|_{L^{\infty}}\|\bar{\kappa}^{-\nicefrac{{1}}{{2}}}f^{(l-1)}_{1,\boldsymbol{\alpha}}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}\nabla^{m}\bar{\rho}\|_{L^{2}}dt\\\
\lesssim&\frac{1}{\lambda^{2(l-1)}}\sum_{m=1}^{l-1}\|\bar{\kappa}\|_{L^{\infty}}^{-(m-1)}|\nabla\Phi_{i}|^{2(l-2)}\kappa^{-1}\ell^{-2(l-1-m)}\tau^{-(m+1)}\mathcal{D}_{m+1}\\\
\lesssim&(\lambda\ell)^{-2(l-1)}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\sum_{m=1}^{l-1}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\big{)}^{-m}\tau^{-1}\mathcal{D}_{m+1}\\\
(\eqref{e:hom_para_assump:taukappaell},m=l-1)\lesssim&(\lambda\ell)^{-2(l-1)}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\big{)}^{-(l-1)}\tau^{-1}\mathcal{D}_{l}\\\
\lesssim&\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\lambda^{2}\big{)}^{-(l-1)}\tau^{-1}\mathcal{D}_{l}\\\
(\eqref{e:hom_para_assump:kappataulambda},l=2)\lesssim&\kappa^{-1}\tau^{-1}\lambda^{-2}\tau^{-1}\mathcal{D}_{l}\\\
\lesssim&\tau^{-1}\mathcal{D}_{l}.\end{split}$ (4.48)

For $G_{N_{h}}$, using (4.43) and (4.47), we have

$\begin{split}\int_{0}^{T}\|G_{N_{h}}\|_{L^{2}}^{2}dt\lesssim&\frac{1}{\lambda^{2(N_{h}-1)}}\sum_{m=1,|\boldsymbol{\alpha}|=m}^{N_{h}}\|\bar{\kappa}^{-(m-1)}\|_{L^{\infty}}\|\bar{\kappa}^{-\nicefrac{{1}}{{2}}}g^{(N_{h})}_{0,\boldsymbol{\alpha}}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}D_{t}\nabla^{m}\bar{\rho}\|_{L^{2}}dt\\\
&+\|\bar{\kappa}^{-(m-1)}\|_{L^{\infty}}\|\bar{\kappa}^{-\nicefrac{{1}}{{2}}}g^{(N_{h})}_{1,\boldsymbol{\alpha}}\|^{2}_{L^{\infty}}\int_{0}^{T}\|\bar{\kappa}^{\nicefrac{{m}}{{2}}}\nabla^{m}\bar{\rho}\|_{L^{2}}dt\\\
\lesssim&\lambda^{-2(N_{h}-1)}\sum_{m=1}^{N_{h}}\|\bar{\kappa}\|_{L^{\infty}}^{-(m-1)}|\nabla\Phi_{i}|^{2(N_{h}-1)}\kappa^{-1}\ell^{-2(N_{h}-m)}\tau^{-(m+1)}\mathcal{D}_{m+1}\\\
\lesssim&(\lambda\ell)^{-2(N_{h}-1)}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\ell^{-2}\sum_{m=1}^{N_{h}}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\big{)}^{-m}\tau^{-1}\mathcal{D}_{m+1}\\\
(\eqref{e:hom_para_assump:taukappaell},m=N_{h})\lesssim&(\lambda\ell)^{-2(N_{h}-1)}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\ell^{-2}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\ell^{-2}\big{)}^{-N_{h}}\tau^{-1}\mathcal{D}_{N_{h}+1}\\\
\lesssim&\lambda^{2}\|\bar{\kappa}\|_{L^{\infty}}\kappa^{-1}\big{(}\tau\|\bar{\kappa}\|_{L^{\infty}}\lambda^{2}\big{)}^{-N_{h}}\tau^{-1}\mathcal{D}_{N_{h}+1}\\\
(\eqref{e:hom_para_assump:Nh_result})\lesssim&\tau^{-1}\mathcal{D}_{N_{h}+1}.\end{split}$
(4.49)

For the last term in (4.42) involving initial data, we have

$\displaystyle\|(\chi\nabla\bar{\rho})(\cdot,0)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\|\bar{\kappa}^{-\nicefrac{{1}}{{2}}}\chi\|_{L^{\infty}}^{2}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\rho_{\text{in}}\|_{L^{2}}^{2}\lesssim\kappa^{-1}\|\bar{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\rho_{\text{in}}\|_{L^{2}}^{2}\lesssim\kappa^{-1}\tau^{-1}\mathcal{D}_{0}.$
(4.50)

For the proof of Corollary 4.2, the only difference is that we need to
estimate the corrector as an error term, which is the same as (4.50). This
concludes our proof. ∎

###### Proof of Corollary 4.3.

To simplify the formulas, we introduce the following shorthand notation

$\displaystyle M(x,t,\xi):=$
$\displaystyle\mathrm{Id}+\sum_{i}\eta_{i}(x,t)\nabla\Phi_{i}^{T}(x,t)\nabla_{\xi}\chi_{i}^{T}\big{(}x,t,\xi\big{)}.$

We first claim the following estimates, omitting the argument
$(x,t,\lambda\Phi_{i})$

$\displaystyle\Big{|}\kappa\int_{0}^{T}\int\big{(}M^{T}M-\langle
M^{T}M\rangle\big{)}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt\Big{|}\lesssim$
$\displaystyle\Big{(}\frac{1}{\lambda\ell}+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\mathcal{D}_{1},$
(4.51)
$\displaystyle\frac{\kappa}{\lambda^{2}}\int_{0}^{T}\|\nabla^{2}\bar{\rho}\chi\|^{2}_{L^{2}}+\|\nabla_{x}\chi^{T}\nabla\bar{\rho}\|^{2}_{L^{2}}dt\lesssim$
$\displaystyle\frac{1}{\lambda^{2}\kappa\tau}\mathcal{D}_{1}.$ (4.52)

Direct computations give

$\displaystyle M^{T}M$
$\displaystyle=\mathrm{Id}+\sum_{i}\eta_{i}(\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}+\nabla_{\xi}\chi_{i}\nabla\Phi_{i})+\sum_{i}\eta_{i}^{2}\nabla_{\xi}\chi_{i}\nabla\Phi_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T},$
$\displaystyle M^{T}M-\langle M^{T}M\rangle$
$\displaystyle=\sum_{i}\eta_{i}(\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}+\nabla_{\xi}\chi_{i}\nabla\Phi_{i})+$
$\displaystyle+\sum_{i}\eta_{i}^{2}(\nabla_{\xi}\chi_{i}\nabla\Phi_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}-\langle\nabla_{\xi}\chi_{i}\nabla\Phi_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}\rangle).$

Furthermore, by using (4.7) and the properties of Mikado flows in Section
2.1.4 we compute

$\displaystyle\kappa\nabla_{\xi}\chi_{i}\nabla\Phi_{i}\nabla\Phi_{i}^{T}\nabla_{\xi}\chi_{i}^{T}$
$\displaystyle=\frac{\sigma}{\kappa\lambda^{2}}\sum_{\vec{k}}a_{\vec{k}}^{2}\nabla\Phi_{i}^{-1}(\vec{k}\otimes\nabla_{\xi}\varphi_{\vec{k}})\nabla\Phi_{i}\nabla\Phi_{i}^{T}(\nabla_{\xi}\varphi_{\vec{k}}\otimes\vec{k})\nabla\Phi_{i}^{-T}$
$\displaystyle=\frac{\sigma}{\kappa\lambda^{2}}\sum_{\vec{k}}a_{\vec{k}}^{2}\nabla\Phi_{i}^{-1}|\nabla\Phi_{i}^{T}\nabla_{\xi}\varphi_{\vec{k}}|^{2}\nabla\Phi_{i}^{-T}.$

In particular, comparing with the derivation of $\bar{A}$, we deduce that

$\displaystyle\big{|}\kappa\langle M^{T}M\rangle-\bar{A}\big{|}$
$\displaystyle\lesssim\sum_{i}\eta_{i}^{2}\|\nabla\Phi_{i}-\mathrm{Id}\|_{L^{\infty}}|\bar{A}|\lesssim\lambda^{-2\gamma}|\bar{A}|.$
(4.53)

Here we use (4.7), (4.9), (4.10) and the formula for homogenized matrix
(2.47).

Now we estimate the goal of this corollary. Using (4.29) and omitting
$(x,t,\lambda\Phi_{i})$, we have

$\displaystyle\kappa$
$\displaystyle\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt=\kappa\int_{0}^{T}\Big{\|}M\nabla\bar{\rho}+\frac{1}{\lambda}(\nabla^{2}\bar{\rho}\chi+\nabla_{x}\chi^{T}\nabla\bar{\rho})+\nabla\tilde{\rho}\Big{\|}_{L^{2}}^{2}dt$
$\displaystyle\leq$
$\displaystyle\Big{(}1+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\kappa\int_{0}^{T}\|M\nabla\bar{\rho}\|_{L^{2}}^{2}dt+C\big{(}1+\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}\big{)}\kappa\int_{0}^{T}\|\nabla\tilde{\rho}\|_{L^{2}}^{2}dt$
$\displaystyle+C\big{(}1+\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}\big{)}\frac{\kappa}{\lambda^{2}}\int_{0}^{T}\|\nabla^{2}\bar{\rho}\chi\|^{2}_{L^{2}}+\|\nabla_{x}\chi^{T}\nabla\bar{\rho}\|^{2}_{L^{2}}dt$
$\displaystyle\leq$
$\displaystyle\Big{(}1+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\kappa\int_{0}^{T}\int\big{(}M^{T}M-\langle
M^{T}M\rangle\big{)}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt+C\Big{(}1+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\int_{0}^{T}\int\bar{A}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt$
$\displaystyle+\Big{(}1+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\int_{0}^{T}\int\big{(}\kappa\langle
M^{T}M\rangle-\bar{A}\big{)}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt+C\big{(}1+\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}\big{)}\kappa\int_{0}^{T}\int_{0}^{T}\|\nabla\tilde{\rho}\|_{L^{2}}^{2}dt$
$\displaystyle+C\big{(}1+\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}\big{)}\frac{\kappa}{\lambda^{2}}\int_{0}^{T}\|\nabla^{2}\bar{\rho}\chi\|^{2}_{L^{2}}+\|\nabla_{x}\chi^{T}\nabla\bar{\rho}\|^{2}_{L^{2}}dt$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}+\frac{1}{\lambda\ell}+\lambda^{-2\gamma}\Big{)}\mathcal{D}_{N_{h}}+\Big{(}1+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\int_{0}^{T}\int\bar{A}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt.$

For the terms after the second $\lesssim$, we estimate them by (4.51), (4.53)
together with Proposition 3.3, Proposition 4.1 and (4.52) respectively. Then
we conclude

$\displaystyle\kappa\int_{0}^{T}\|\nabla\rho\|_{L^{2}}^{2}dt-\int_{0}^{T}\int\bar{A}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt\lesssim\Big{(}\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}+\frac{1}{\lambda\ell}+\lambda^{-2\gamma}\Big{)}\mathcal{D}_{N_{h}}.$

The proof of (4.52) is the same as the terms $E_{3}$ and $E_{4}$ in the proof
of Proposition 4.1, so we omit it here. Now we prove our claim (4.51). As in
the proof of Lemma 4.6, we could find a matrix potential
$\omega:=\sum_{i}\eta_{i}\omega^{(i)}$ taking arguments $(x,t,\xi)$ such that
for $n\in\\{0,1\\}$

$\displaystyle\big{(}M^{T}M-\langle M^{T}M\rangle\big{)}_{jl}=$
$\displaystyle\partial_{\xi_{1}}\omega_{jl}=\sum_{i}\eta_{i}\partial_{\xi_{1}}\omega_{jl}^{(i)},$
$\displaystyle\big{\|}D_{t}^{n}\nabla_{x}^{m}\omega_{j}\big{\|}_{L^{\infty}_{\xi}}\lesssim$
$\displaystyle\sum_{i}\Big{(}\frac{\sigma\eta_{i}^{2}}{\kappa^{2}\lambda^{2}}+\frac{\sigma^{1/2}\eta_{i}}{\kappa\lambda}\Big{)}\tau^{-n}\ell^{-m}$
$\displaystyle\lesssim$
$\displaystyle\sum_{i}\Big{(}1+\frac{\sigma\eta_{i}^{2}}{\kappa^{2}\lambda^{2}}\Big{)}\tau^{-n}\ell^{-m}\lesssim\bar{\kappa}\kappa^{-1}\tau^{-n}\ell^{-m}.$
(4.54)

Now by the chain rule,

$\displaystyle\nabla\big{(}\omega_{jl}^{(i)}(x,t,\lambda\Phi_{i})\big{)}$
$\displaystyle=(\nabla_{x}\omega_{jl}^{(i)})(x,t,\lambda\Phi_{i})+\lambda\nabla\Phi_{i}^{T}(\nabla_{\xi}\omega_{jl}^{(i)})(x,t,\lambda\Phi_{i}).$

This leads to

$\displaystyle(\partial_{\xi_{1}}\omega_{jl}^{(i)})(x,t,\lambda\Phi_{i})=\frac{1}{\lambda}(\nabla\Phi_{i}^{-T})_{1m}\partial_{m}\big{(}\omega_{jl}^{(i)}(x,t,\lambda\Phi_{i})\big{)}-\frac{1}{\lambda}(\nabla\Phi_{i}^{-T})_{1m}(\partial_{x_{m}}\omega_{jl}^{(i)})(x,t,\lambda\Phi_{i}).$

Omitting the argument $(x,t,\lambda\Phi_{i})$, consider the following
oscillation term and use integration by parts

$\begin{split}\kappa\int_{0}^{T}&\int\big{(}(M^{T}M)-\langle
M^{T}M\rangle\big{)}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt\\\
=&\frac{\kappa}{\lambda}\sum_{i,j,l,m}\Big{(}\int_{0}^{T}\int\eta_{i}(\nabla\Phi_{i}^{-T})_{1m}\partial_{m}\omega_{jl}^{(i)}\partial_{j}\bar{\rho}\partial_{l}\bar{\rho}dxdt-\int_{0}^{T}\int\eta_{i}(\nabla\Phi_{i}^{-T})_{1m}\partial_{x_{m}}\omega_{jl}^{(i)}\partial_{j}\bar{\rho}\partial_{l}\bar{\rho}dxdt\Big{)}\\\
=&\frac{\kappa}{\lambda}\sum_{i,j,l,m}\bigg{(}-\int_{0}^{T}\int\omega_{jl}^{(i)}\partial_{m}\Big{(}\eta_{i}(\nabla\Phi_{i}^{-T})_{1m}\partial_{j}\bar{\rho}\partial_{l}\bar{\rho}\Big{)}dxdt-\int_{0}^{T}\int\eta_{i}(\nabla\Phi_{i}^{-T})_{1m}\partial_{x_{m}}\omega_{jl}^{(i)}\partial_{j}\bar{\rho}\partial_{l}\bar{\rho}dxdt\bigg{)}.\end{split}$
(4.55)

Now, to estimate the terms above, in the following estimate, we use (4.54),
(4.13), Young’s inequality with the decomposition
$\bar{\kappa}|\nabla\bar{\rho}||\nabla^{2}\bar{\rho}|=\tau^{-\nicefrac{{1}}{{4}}}\bar{\kappa}^{\nicefrac{{1}}{{4}}}|\nabla\bar{\rho}|\cdot\tau^{\nicefrac{{1}}{{4}}}\bar{\kappa}^{\nicefrac{{3}}{{4}}}|\nabla^{2}\bar{\rho}|$.
Then we invoke energy estimates (4.43), and use (4.17),

$\displaystyle\kappa\int_{0}^{T}\int$ $\displaystyle\big{(}(M^{T}M)-\langle
M^{T}M\rangle\big{)}\nabla\bar{\rho}\cdot\nabla\bar{\rho}dxdt$
$\displaystyle\lesssim$
${}_{\eqref{c:hom_dissipation_eq7},\eqref{e:hom_assump:slow}}\frac{1}{\lambda\ell}\int_{0}^{T}\int\bar{\kappa}|\nabla\bar{\rho}|^{2}dxdt+\frac{1}{\lambda}\int_{0}^{T}\int\bar{\kappa}|\nabla\bar{\rho}||\nabla^{2}\bar{\rho}|dxdt$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{\lambda\ell}\int_{0}^{T}\int\bar{\kappa}|\nabla\bar{\rho}|^{2}dxdt+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\int_{0}^{T}\int\bar{\kappa}|\nabla\bar{\rho}|^{2}dxdt+\frac{\tau^{\nicefrac{{1}}{{2}}}}{\lambda\kappa^{\nicefrac{{1}}{{2}}}}\int_{0}^{T}\int|\bar{\kappa}\nabla^{2}\bar{\rho}|^{2}dxdt$
$\displaystyle\lesssim$
${}_{\eqref{p:hom_trho:p:e3}}\Big{(}\frac{1}{\lambda\ell}+\frac{1}{\lambda\kappa^{\nicefrac{{1}}{{2}}}\tau^{\nicefrac{{1}}{{2}}}}\Big{)}\mathcal{D}_{1}$

This concludes the proof. ∎

##  5\. Time averaging

###  5.1. Setup

Consider constants $\kappa_{1}>\kappa_{0}>0$, and the oscillatory
$\tilde{\kappa}=\tilde{\kappa}(x,t,\tau^{-1}t)$ of the form

$\tilde{\kappa}(x,t,\tau^{-1}t)=\kappa_{0}+\eta(x,t,\tau^{-1}t)\kappa_{1}\,,$
(5.1)

where $s\mapsto\eta(x,t,s)$ is $1$-periodic, nonnegative and smooth with

$\langle\eta\rangle:=\int_{0}^{1}\eta(x,t,s)\,ds=1\,.$

We define

$\kappa:=\kappa_{0}+\kappa_{1},\quad
g(x,t,s):=\frac{\kappa_{1}}{\kappa}(\eta(x,t,s)-1),\quad
g_{\tau}(x,t):=g(x,t,\tau^{-1}t),$ (5.2)

so that $\langle g\rangle=0$, $g$ is bounded, and we have the identity

$\tilde{\kappa}(x,t,\tau^{-1})=\kappa_{0}+\kappa_{1}\eta(x,t,\tau^{-1}t)=\kappa+\kappa
g_{\tau}(x,t).$ (5.3)

In this section we consider the solutions $\rho$ and $\bar{\rho}$ of the
following equations

$\begin{split}\partial_{t}\rho+u\cdot\nabla\rho&=\mathop{\rm
div}\nolimits\left(\tilde{\kappa}\nabla\rho\right)\\\
\rho|_{t=0}&=\rho_{in},\end{split}$ (5.4)

and

$\begin{split}\partial_{t}\bar{\rho}+u\cdot\nabla\bar{\rho}&=\mathop{\rm
div}\nolimits\left(\kappa\nabla\bar{\rho}\right)\\\
\bar{\rho}|_{t=0}&=\rho_{in}.\end{split}$ (5.5)

In view of definitions of $\kappa$, $\tilde{\kappa}$ (5.5) can be seen as the
time-averaged version of (5.4). Further, we define the respective total
dissipation

$\tilde{D}:=\int_{0}^{T}\|\tilde{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\rho\|_{L^{2}}^{2}\,dt\,,\quad
D:=\kappa\int_{0}^{T}\|\nabla\bar{\rho}\|_{L^{2}}^{2}\,dt.$ (5.6)

We make the following assumptions: there exists $\mu\geq\|\nabla u\|_{\infty}$
(we should think of $\mu^{-1}\sim\|\nabla u\|_{\infty}^{-1}$, being the
advection time-scale) such that

  1. (A1)

The fast time-scale is shorter than the advection time-scale, i.e.
$\tau\mu<1/2$. Moreover, there exists $N\in\mathbb{N}$ such that

$(\tau\mu)^{N-1}\kappa<\kappa_{0}.$ (5.7)

  2. (A2)

Control of higher-order spatial derivatives: for any $n\leq N$

$\|\nabla u\|_{C^{n}}^{2}\leq
C_{N}\left(\frac{\mu}{\kappa}\right)^{n}\mu^{2}.$ (5.8)

  3. (A3)

Bound on the initial data: for any $1\leq n\leq N$

$\|\rho_{in}\|_{H^{n}}^{2}\leq C_{N}\left(\frac{\mu}{\kappa}\right)^{n}D.$
(5.9)

  4. (A4)

Control of cutoff and its slow derivatives: for any $1\leq n\leq N$

$\|\nabla^{n}\eta\|_{L^{\infty}}\leq
C_{N}(\tau\mu)^{2}\left(\frac{\mu}{\kappa}\right)^{\frac{n}{2}},\qquad\|D_{t}\eta\|_{L^{\infty}}\leq
C_{N}\mu.$ (5.10)

######  Proposition 5.1.

Under the assumptions (A1), (A2), (A3), (A4) we have

$\|\rho(T)-\bar{\rho}(T)\|_{L^{2}}^{2}+\kappa\int_{0}^{T}\|\nabla\rho-\nabla\bar{\rho}\|^{2}_{L^{2}}\,dt\lesssim(\tau\mu)^{2}D.$
(5.11)

Moreover, the total dissipation satisfies

$|D-\tilde{D}|\lesssim(\tau\mu)(D+\tilde{D}).$ (5.12)

The implicit constants depend only on $N$ in asumptions.

###### Proof.

Starting with $\rho^{(0)}:=\bar{\rho}$ being the solution of (5.5), we
construct the solution $\rho$ of (5.4) by successive approximations
$\rho^{(i)}$, $i=1,2,\dots,N$, defined to be solutions of

$\begin{split}\partial_{t}\rho^{(i)}+u\cdot\nabla\rho^{(i)}&=\kappa\Delta\rho^{(i)}+\mathop{\rm
div}\nolimits\left(g_{\tau}\nabla\rho^{(i-1)}\right)\\\
\rho|_{t=0}&=\rho_{in}.\end{split}$ (5.13)

We will show below in Lemma 5.2 the improved energy bound444In Section 4 of
[AV23] a similar strategy was employed to deal with fast temporal
oscillations, albeit leading to an estimate with much more convolved right-
hand side and weaker improvement, compare Lemma 4.2 there.

$\|\rho^{(i)}(T)-\rho^{(i-1)}(T)\|_{L^{2}}^{2}+\kappa\int_{0}^{T}\|\nabla\rho^{(i)}(t)-\nabla\rho^{(i-1)}\|^{2}_{L^{2}}\,dt\leq
C_{N}(\tau\mu)^{2i}D.$

Setting

$\rho^{(error)}:=\rho-\rho^{(N)},$

and using (5.4), (5.13) and (5.3), we may write the equation for
$\rho^{(error)}$ as

$\begin{split}\partial_{t}\rho^{(error)}+u\cdot\nabla\rho^{(error)}&=\kappa\Delta\rho^{(error)}+\kappa\mathop{\rm
div}\nolimits(g_{\tau}\nabla\rho^{(error)})+\kappa\mathop{\rm
div}\nolimits(g_{\tau}(\nabla\rho^{(N)}-\nabla\rho^{(N-1)}))\,\\\
&=\kappa_{0}\Delta\rho^{(error)}+\bar{\kappa}\mathop{\rm
div}\nolimits(\eta\nabla\rho^{(error)})+\kappa\mathop{\rm
div}\nolimits(g_{\tau}(\nabla\rho^{(N)}-\nabla\rho^{(N-1)}))\,.\end{split}$

Since $\eta\geq 0$ and $|g|\leq 2$, the standard energy estimate and an
application of Young’s inequality yields

$\|\rho^{(error)}(T)\|_{L^{2}}^{2}+\kappa_{0}\int_{0}^{T}\|\nabla\rho^{(error)}\|_{L^{2}}^{2}\,dt\leq
C\frac{\kappa}{\kappa_{0}}\kappa\int_{0}^{T}\|\nabla\rho^{(N)}-\nabla\rho^{(N-1)}\|_{L^{2}}^{2}\,dt.$

Combining with the improved energy bound and (A1) we deduce

$\begin{split}\|\rho^{(error)}(T)\|_{L^{2}}^{2}+\kappa\int_{0}^{T}\|\nabla\rho^{(error)}\|_{L^{2}}^{2}\,dt&\leq
C_{N}\left(\frac{\kappa}{\kappa_{0}}\right)^{2}(\tau\mu)^{2N}D\\\ &\leq
C_{N}(\tau\mu)^{2}D\,.\end{split}$ (5.14)

Consequently,

$\displaystyle\|\rho(T)-\bar{\rho}(T)\|_{L^{2}}^{2}+\kappa\int_{0}^{T}\|\nabla\rho-\nabla\bar{\rho}\|^{2}_{L^{2}}$
$\displaystyle\,dt\leq
2\|\rho(T)-\rho^{(N)}(T)\|_{L^{2}}^{2}+2\kappa\int_{0}^{T}\|\nabla\rho-\nabla\rho^{(N)}\|^{2}_{L^{2}}\,dt+$
$\displaystyle+2^{N}\sum_{i=0}^{N-1}$
$\displaystyle\|\rho^{(i+1)}(T)-\rho^{(i)}(T)\|_{L^{2}}^{2}+\kappa\int_{0}^{T}\|\nabla\rho^{(i+1)}-\nabla\rho^{(i)}\|^{2}_{L^{2}}\,dt$
$\displaystyle\lesssim(\tau\mu)^{2}D\,,$

verifying (5.11).

Next, we consider the difference in total dissipation, and write

$\int_{0}^{T}\int_{\mathbb{T}^{3}}\kappa|\nabla\bar{\rho}|^{2}-\tilde{\kappa}|\nabla\rho|^{2}\,dx\,dt=\underbrace{\kappa\int_{0}^{T}\int_{\mathbb{T}^{3}}g_{\tau}|\nabla\bar{\rho}|^{2}\,dx\,dt}_{(I)}+\underbrace{\int_{0}^{T}\int_{\mathbb{T}^{3}}\tilde{\kappa}(|\nabla\bar{\rho}|^{2}-|\nabla\rho|^{2})\,dx\,dt}_{(II)}$

We show below (see (5.26)) the bound

$|(I)|\lesssim(\tau\mu)D,$

since, in the notation introduced below (c.f.(5.18)),
$I=B^{(0)}(\nabla\rho^{(0)},\nabla\rho^{(0)})$. Next, we use Young’s
inequality to estimate

$\displaystyle|(II)|$
$\displaystyle=\left|\int_{0}^{T}\int_{\mathbb{T}^{3}}\tilde{\kappa}(\nabla\rho+\nabla\bar{\rho})\cdot(\nabla\rho-\nabla\bar{\rho})\,dx\,dt\right|\,$
$\displaystyle\lesssim\varepsilon^{-1}\kappa\int_{0}^{T}\|\nabla\rho-\nabla\bar{\rho}\|_{L^{2}}^{2}\,dt+\varepsilon\left(\kappa\int_{0}^{T}\|\nabla\bar{\rho}\|_{L^{2}}^{2}\,dt+\int_{0}^{T}\|\tilde{\kappa}^{\nicefrac{{1}}{{2}}}\nabla\rho\|_{L^{2}}^{2}\,dt\right).$

Choosing $\varepsilon=\tau\mu$ and using (5.11) we deduce

$|(II)|\lesssim(\tau\mu)(D+\tilde{D}),$

verifying (5.12). ∎

The rest of this section is concerned with estimates on the successive
approximations (5.13). Since they may be of separate interest, we present them
independently from the specific choices made in the previous section. To this
end let us observe that the definition (5.2) of $g$ and of $\eta$ imply there
exists the fast time potential ie $G(x,t,s)=\int_{0}^{s}g(x,t,a)da$ such that

$\partial_{s}G(x,t,s)=g(x,t,s),\quad\text{ where }\quad
g_{\tau}(x,t)=g(x,t,\tau^{-1}t).$

Let us denote $g^{0}:=g,\;g^{l+1}:=G^{l}g,\;\partial_{s}G^{l}=g^{l}$. Since
$\frac{\bar{\kappa}}{\kappa}\leq 1$ and $\eta$ is bounded, assumption (A4) and
definitions yields that for $f=g^{l}\text{ or }G^{l}$ we have

$\begin{split}&\|f\|_{L^{\infty}}\lesssim
1,\qquad\|(D^{slow}_{t}f)\|_{L^{\infty}}\lesssim\mu,\\\ &\text{and for
}n>0\qquad\|\nabla^{n}f\|_{L^{\infty}}\lesssim(\tau\mu)^{2}\left(\frac{\mu}{\kappa}\right)^{\frac{n}{2}}\lesssim(\tau\mu)\left(\frac{\mu}{\kappa}\right)^{\frac{n}{2}}.\end{split}$
(5.15)

We will use usually the bound
$\|\nabla^{n}f\|_{L^{\infty}}\lesssim(\tau\mu)\left(\frac{\mu}{\kappa}\right)^{\frac{n}{2}}$,
only once the other is needed.

###  5.2. Notation

$D:=\kappa\int_{0}^{T}\|\nabla\rho_{0}(s)\|_{L^{2}}^{2}$ (5.16) $\VE
f\VE^{2}_{n}:=\sup_{t\leq
T}\|(\nabla^{n}f)(t)\|^{2}_{L^{2}}+\kappa\int_{0}^{T}\|\nabla^{n+1}f(s)\|_{L^{2}}^{2}ds+\left(\frac{\|\nabla
u\|_{L^{\infty}}}{\kappa}\right)^{n}\left(\kappa\int_{0}^{T}\|\nabla
f(s)\|_{L^{2}}^{2}ds\right)$ (5.17) $B^{l}(f,g):=\kappa\int_{0}^{T}\int
g^{l}_{\tau}f\cdot g,\quad\text{where}\quad
g^{0}:=g,\;g^{l+1}:=G^{l}g,\;\partial_{s}G^{l}=g^{l}$ (5.18)

In particular

$g^{l}=\tau D_{t}(G^{l}_{\tau})-\tau(D^{slow}_{t}G^{l})_{\tau}$ (5.19)

###  5.3. Commutators

We will need several estimates for commutators. Let $|\boldsymbol{\alpha}|=n$

$\|[g\cdot\nabla,\partial^{\boldsymbol{\alpha}}]f\|_{L^{2}}\lesssim\|\nabla^{n}g\|_{L^{\infty}}\|\nabla
f\|_{L^{2}}+\|\nabla g\|_{L^{\infty}}\|\nabla^{n}f\|_{L^{2}}$ (5.20)
$\|[\partial^{\boldsymbol{\alpha}},g]\nabla
f\|_{L^{2}}\lesssim\|\nabla^{n}g\|_{L^{\infty}}\|\nabla f\|_{L^{2}}+\|\nabla
g\|_{L^{\infty}}\|\nabla^{n}f\|_{L^{2}}$ (5.21)

###  5.4. Improved energy bound

The following is the key technical lemma of this section. Consider

$\begin{split}\partial_{t}\rho^{(i)}+u\cdot\nabla\rho^{(i)}&-\kappa\Delta\rho^{(i)}=\kappa\mathop{\rm
div}\nolimits\left(g_{\tau}\nabla\rho^{(i-1)}\right)\\\
\rho^{(i)}|_{t=0}&=0\;\text{ for
}i>0,\qquad\rho^{(0)}|_{t=0}=\rho_{in}\end{split}$ (5.22)

and $\rho^{(-1)}\equiv 0$.

######  Lemma 5.2.

Assume that there is $\partial_{s}G(x,t,s)=g(x,t,s)$ such that (5.15) holds
for both $f=g$, $f=G$. Assume (5.8), (5.9), (5.10), then the energy solutions
of (5.22) satisfy
(1) for $\rho^{(0)}$

$\|\rho^{(0)}(t)\|_{L^{2}}^{2}+\kappa\int_{0}^{t}\|\nabla\rho^{(0)}(s)\|_{L^{2}}^{2}\,ds=\|\rho_{in}\|_{L^{2}}^{2}$
(5.23)

and, for any $n\geq 1$

$\VE\rho^{(0)}\VE^{2}_{n}\lesssim_{n}\left(\frac{\mu}{\kappa}\right)^{n}D$
(5.24)

(2) for $\rho^{(i)}$, $i>0$

$\begin{split}\VE\rho^{(i)}\VE^{2}_{n}\lesssim_{n,i}(\tau\mu)^{2i}\left(\frac{\mu}{\kappa}\right)^{n}D\end{split}$
(5.25)

Furthermore

$\begin{split}&\sup_{|\boldsymbol{\alpha}|=n,|\boldsymbol{\beta}|=m}\left|B^{l}(\nabla\partial^{\boldsymbol{\alpha}}\rho^{(i)},\nabla\partial^{\boldsymbol{\beta}}\rho^{(i)})\right|\lesssim_{n,m,i}(\tau\mu)^{2i+1}\left(\frac{\mu}{\kappa}\right)^{\frac{n+m}{2}}D.\end{split}$
(5.26)

Note that the equation (5.13) with assumptions (A2)-(A4) falls within the
above lemma.

###### Proof.

We complete this proof in several steps.

_Step 1: Preliminary bound for the bilinear term._

Let $|\boldsymbol{\alpha}|=n$ and $|\boldsymbol{\beta}|=m$. It holds by