# On the vanishing viscosity limit of statistical solutions of the
incompressible Navier–Stokes equations

Ulrik Skre Fjordholm
Department of Mathematics
University of Oslo
Postboks 1053 Blindern, 0316 Oslo, Norway<EMAIL_ADDRESS>https://www.mn.uio.no/math/english/people/aca/ulriksf/ ,  Siddhartha Mishra
Seminar for Applied Mathematics (SAM)
ETH Zürich
HG G 57.2, Rämistrasse 101, Zürich, Switzerland<EMAIL_ADDRESS>http://www.sam.math.ethz.ch/ smishra and  Franziska Weber
Department of Mathematical Sciences
Carnegie Mellon University
5000 Forbes Avenue, Pittsburgh, PA 15213, USA<EMAIL_ADDRESS>
###### Abstract.

We study statistical solutions of the incompressible Navier–Stokes equation
and their vanishing viscosity limit. We show that a formulation using
correlation measures, which are probability measures accounting for spatial
correlations, and moment equations is equivalent to statistical solutions in
the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version
of Kolmogorov’s self-similarity at small scales hypothesis that allows for
intermittency corrections, we show that the limit is a statistical solution of
the incompressible Euler equations. To pass to the limit, we derive a
Kármán–Howarth–Monin relation for statistical solutions and combine it with
the weak scaling assumption and a compactness theorem for correlation
measures.

##  1\. Introduction

The motion of an incompressible viscous fluid can be described by the
Navier–Stokes equations

(1.1) $\begin{split}\partial_{t}u+\operatorname{div}\bigl{(}u\otimes
u\bigr{)}+\nabla p&=\varepsilon\Delta u\\\ \operatorname{div}u&=0\\\
u\big{|}_{t=0}&=u_{0},\end{split}$

where $u\colon D\to U\coloneqq\mathbb{R}^{d}$ is the fluid velocity and
$p\colon D\to\mathbb{R}$, the pressure, acting as a Lagrange multiplier to
enforce the divergence constraint $\operatorname{div}u=0$, and $u_{0}$ is the
initial condition. Here, we take the spatial domain $D$ to be the
$d$-dimensional torus $D\coloneqq\mathbb{T}^{d}$, and we denote the phase
space by $U\coloneqq\mathbb{R}^{d}$. The divergence is defined as
$\operatorname{div}u:=\sum_{i=1}^{d}\partial_{x_{i}}u^{i}$ and $\nabla
p:=(\partial_{x_{1}}p,\dots,\partial_{x_{d}}p)^{\top}$ is the spatial
gradient. The parameter $\varepsilon>0$ denotes the _viscosity_ and is
proportional to the reciprocal of the Reynolds number. It is well-known that
many flows of interest are characterized by high to very-high Reynolds
numbers. Hence, one is interested in studying what happens when viscosity
$\varepsilon$ equal to zero. In this formal limit $\varepsilon\to 0$, one
obtains the incompressible Euler equations, which are the prototypical models
for an ideal fluid.

The question of whether the $\varepsilon\to 0$ limit is a good approximation
of (1.1) is of great practical relevance and has received considerable
attention, both from a physical as well as mathematical point of view.
Furthermore, it plays an essential role in computational fluid dynamics, as
many numerical methods for the Euler equations, as well as large eddy
simulations (LES) for Navier-Stokes equations, can be viewed as
discretizations of (1.1) with $\varepsilon$ of the order of the discretization
parameter.

While in two spatial dimension, the convergence of a sequence of solutions
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ of (1.1) to a solution of the Euler
equations has been proved rigorously for many settings, see e.g., [14, 51, 13,
15], it turns out to be very challenging in 3D. Leray [46] proved in 1934 the
existence of “Leray–Hopf solutions” of (1.1), which are weak solutions of
(1.1), i.e., they satisfy (1.1) in the sense of distributions and in addition
an energy inequality of the form

(1.2) $\int_{D}|u(t,x)|^{2}dx+\varepsilon\int_{0}^{t}\int_{D}|\nabla
u(s,x)|^{2}dxds\leqslant\int_{D}|u_{0}(x)|^{2}dx,\quad t\geqslant 0.$

(Here and in the remainder, we suppress the dependence of $u^{\varepsilon}$ on
$\varepsilon$ for convenience and write just $u$.) Hence, if the initial data
$u_{0}$ lies in $L^{2}_{\operatorname{div}}(D;U)$, such solutions satisfy
$u\in
L^{\infty}\big{(}[0,T];L^{2}_{\operatorname{div}}(D;\mathbb{R}^{d})\big{)}\cap
L^{2}\big{(}[0,T];H^{1}_{\operatorname{div}}(D;\mathbb{R}^{d})\big{)}$. (Here,
$L^{2}_{\operatorname{div}}$ and $H^{1}_{\operatorname{div}}$ are the weakly
divergence free functions in $L^{2}$ and $H^{1}$, respectively.) However, as
$\varepsilon\to 0$, the $L^{2}$-bound on the gradient of $u$, that stems from
the energy inequality (1.2), no longer suffices for deriving sufficient
compactness of the sequence $\\{u\\}_{\varepsilon>0}$ in $L^{2}$—which would
be needed to pass to the limit in the nonlinear terms. It appears that, at
least as far as global solutions are concerned, there is currently no means of
gaining sufficient compactness through other conserved quantities or
bootstrapping; in fact, it is unclear if global solutions of higher regularity
than the one given by (1.2) exist in 3D. Closely related to this issue is the
lack of stability estimates, i.e., well-posedness of Leray-Hopf solutions [21,
44]. The main obstruction to better regularity or stability estimates is
caused by the nonlinear convective term $(u\cdot\nabla)u$. The role of the
nonlinear term and possible instabilities in the Leray-Hopf solutions are
often related to the issue of _turbulence_ in fluid flows.

The theory of mathematical turbulence was initiated in the 1930s and 1940s by
Taylor, Richardson, Kolmogorov and others, see [31] and references therein,
and has since influenced fluid mechanics, as well as atmospheric sciences and
plasma physics heavily. In his sequence of three papers [41, 40, 42], nowadays
referred to as K41, Kolmogorov took a probabilistic approach to turbulence and
formulated basic hypotheses about fluid flow at high Reynolds numbers and
derived predictions based on these. Many of these have later been confirmed by
experiments. The idea of studying equations (1.1) in a probabilistic setting
has since been taken up again in many works, in different frameworks, by
adding stochastic forcing terms to (1.1), see e.g. [25, 48, 10], or taking
uncertain or measure-valued initial data, e.g. [17]. In the latter case, the
solution of (1.1) may not be a function any more but instead a time-
parametrized probability measure on the phase space. Global existence of such
measure-valued solutions for incompressible flows has been shown in 3D, and
even the passage to the limit $\varepsilon\to 0$ can be made rigorous in this
case [17]. However, measure-valued solutions are generally not unique, which
can be shown by counterexample even in the case of Burgers’ equation [22].
Hence, measure-valued solutions are too broad a solution concept to resolve
the problem of non-uniqueness, and more information or constraints need to be
added.

To overcome this, in [23], it was suggested to take into account the (time)
evolution of all possible multi-point spatial correlations. Instead of a
single probability measure on the phase space $U$, such a _statistical
solution_ is a family of probability measures on the phase space and products
of the phase space $U^{k}$, for $k\in\mathbb{N}$, corresponding to the multi-
point correlations. Hence, one can interpret the solution as a measure-valued
solution augmented with information about higher order spatial correlations.
From a practical point of view, this approach is very natural, as often only
averaged quantities of interest of the fluid flow can be observed. Moreover,
it is also in line with Kolmogorov’s turbulence theory, as this theory studies
statistical properties of the fluid and makes predictions about these. The
system of equations that arises for the higher order correlations is also
known as the _Friedman–Keller infinite chain of moment equations_ [38, 57] and
finite closure relation for this infinite family of equations have been
studied for small and large Reynolds numbers in [33, 35, 34, 36].

An alternative point of view in this context, is to consider instead
probability measures on a space of suitable initial conditions; in the case of
(1.1) this would be $L^{2}_{\operatorname{div}}(D;U)$. Equation (1.1) is then
interpreted as a Liouville equation on an infinite dimensional function space
and the solution is a mapping assigning to each time $t$ a probability measure
on $L^{2}_{\operatorname{div}}(D;U)$. This setting was first considered by
Prodi [52] and later on extensively studied by Foiaş and collaborators [26,
27, 29, 28], see also [37]. A closely related notion of statistical solutions
was studied by Vishik and Fursikov [57]. Foiaş and his collaborators proved
existence of such solutions in 2D and 3D, uniqueness in 2D, and further
properties related to turbulence [28]. The relations between the Foiaş and
Prodi notion of statistical solutions and the Vishik-Fursikov version were
explored in [30, 8, 9]. The latter work etends the notion of statistical
solutions to other relevant PDEs in fluid mechanics.

Given this plurality of definitions of statistical solutions, it is natural to
examine, if and under what conditions, these solution concepts are equivalent.
_The first goal of this paper is to prove that both these concepts of
statistical solutions of the incompressible Navier-Stokes equations ( 1.1) are
equivalent_ as long as a statistical version of the of the energy inequality
(1.2) holds.

_The second and main goal of this paper is to investigate the vanishing
viscosity limit of the statistical solutions of incompressible Navier-Stokes
equations_. Under an _weak scaling_ assumption on the Navier-Stokes
statistical solutions, we will use compactness criteria, presented recently in
[24], to prove that vanishing viscosity limits of the statistical solutions of
Navier-Stokes equations are statistical solutions of the incompressible Euler
equations.

Our weak scaling assumption is a significantly weaker version of the scaling
hypothesis of Kolmogorov’s 1941 theory and allows for _intermittent_
corrections. Our main technical tool is a statisical version of the well-known
Kármán–Howarth–Monin relation [31, 16, 47], that relates the evolution of
2-point correlations to the longitudinal structure function $S_{\|}^{3}$,
which is, roughly speaking, defined as

$S_{\|}^{3}(|\ell|)\coloneqq\Bigl{\langle}\Bigl{(}(u(x+\ell)-u(x))\cdot\widehat{\ell}\Bigr{)}^{3}\Bigr{\rangle},\quad\widehat{\ell}\coloneqq\frac{\ell}{|\ell|}.$

Here $\langle\cdot\rangle$ is a suitable average of the flow.

Thus, by characterizing this vanishing viscosity limit, we establish a
rigorous relationship between the incompressible Navier-Stokes and Euler
equations, while accommodating physically observed facts about turbulent flows
in this description.

The remainder of this article is organized as follows: In Section 2, we
introduce the concept of correlation measures and in Section 3 we show the
equivalence of statistical solutions as introduced by Foiaş and Prodi with
families of correlation measures satisfying the Friedman–Keller chain of
moment equations. Then in Section 4, we consider the passage to the limit
$\varepsilon\to 0$ and conclude with an appendix with technical results.

##  2\. Correlation measures

In this section, we recall the definition of correlation measures and some
important properties of them from [23, 24]. We start by introducing the
necessary notation.

###  2.1. Notation

For $k\in\mathbb{N}$, $k\geqslant 1$, we denote the tensor products

$D^{k}=\underbrace{D\times\dots\times D}_{k\,\textrm{times}},\quad
U^{k}=\underbrace{U\otimes\dots\otimes U}_{k\,\textrm{times}}.$

If $X$ is a topological space then we let $\mathscr{B}(X)$ denote the Borel
$\sigma$-algebra on $X$, we let $\mathscr{M}(X)$ denote the set of signed
Radon measures on $(X,\mathscr{B}(X))$, and we let
$\mathscr{P}(X)\subset\mathscr{M}(X)$ denote the set of all probability
measures on $(X,\mathscr{B}(X))$, i.e. all $0\leqslant\mu\in\mathscr{M}(X)$
with $\mu(X)=1$ (see e.g. [3, 7, 39]). For $k\in\mathbb{N}$ and a multiindex
$\alpha\in\\{0,1\\}^{k}$ we write $|\alpha|=\alpha_{1}+\dots+\alpha_{k}$ and
$\bar{\alpha}=\mathbbm{1}-\alpha=(1-\alpha_{1},\dots,1-\alpha_{k})$, and we
let $x_{\alpha}$ be the vector of length $|\alpha|$ consisting of the elements
$x_{i}$ of $x$ for which $\alpha_{i}$ is non-zero. For a vector
$x=(x_{1},\dots,x_{k})$ we write
$\widehat{x}_{i}=(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{k})$. For a vector
$\xi=(\xi_{1},\dots,\xi_{k})$ we write
$|\xi^{\alpha}|=|\xi_{1}|^{\alpha_{1}}\cdots|\xi_{k}|^{\alpha_{k}}$ with the
convention $0^{0}=1$.

####  2.1.1. Carathéodory functions

If $E$ and $V$ are Euclidean spaces then a measurable function $g\colon
E\times V\to\mathbb{R}$ is called a _Carathéodory function_ if $\xi\mapsto
g(x,\xi)$ is continuous for a.e. $y\in E$ and $y\mapsto g(y,\xi)$ is
measurable for every $\xi\in V$ (see e.g. [1, Section 4.10]). Given
$k\in\mathbb{N}$ and a Carathéodory function $g=g(x,\xi)\colon D^{k}\times
U^{k}\to\mathbb{R}$ we define the functional $L_{g}\colon
L^{p}(D;U)\to\mathbb{R}$ by

(2.1)
$L_{g}(u)\coloneqq\int_{D^{k}}g(x_{1},\dots,x_{k},u(x_{1}),\dots,u(x_{k}))\,dx.$

(It is not obvious that $L_{g}$ is continuous, or even well-defined; see
[23].) We denote the set of Carathéodory functions depending on space and time
by $\mathcal{H}^{k}_{0}([0,T),D;U)\coloneqq L^{1}([0,T)\times
D^{k};C_{0}(U^{k}))$ and its dual space by
$\mathcal{H}^{k*}_{0}([0,T),D;U)\coloneqq L^{\infty}_{w}([0,T)\times
D^{k};\mathscr{M}(U^{k}))$ (see e.g. [5]).

In the following, we will focus on a specific type of Carathéodory functions.
In particular, for $p\geqslant 1$ we let $\mathcal{H}^{k,p}([0,T],D;U)$ denote
the space of Carathéodory functions $g\colon[0,T]\times D^{k}\times
U^{k}\to\mathbb{R}$ satisfying

(2.2)
$|g(t,x,\xi)|\leqslant\sum_{\alpha\in\\{0,1\\}^{k}}\varphi_{|\bar{\alpha}|}(t,x_{\bar{\alpha}})|\xi^{\alpha}|^{p}\qquad\forall\
x\in D^{k},\ \xi\in U^{k}$

for nonnegative functions $\varphi_{i}\in L^{\infty}([0,T];L^{1}(D^{i}))$,
$i=0,1,\dots,k$. We let
$\mathcal{H}^{k,p}_{1}([0,T],D;U)\subset\mathcal{H}^{k,p}([0,T],D;U)$ denote
the subspace of functions $g$ satisfying the local Lipschitz condition

(2.3) $\begin{split}\big{|}g(t,x,\zeta)-g(t,y,\xi)\big{|}\leqslant&\
\psi(t)\sum_{i=1}^{k}|\zeta_{i}-\xi_{i}|\max\big{(}|\xi_{i}|,|\zeta_{i}|\big{)}^{p-1}h(t,\widehat{x}_{i},\widehat{\xi}_{i})\\\
&+O(|x-y|)\widetilde{h}(t,x,\xi)\end{split}$

for every $x\in D^{k}$, $y\in B_{r}(x)$ for some $r>0$, for some nonnegative
$h\in\mathcal{H}^{k-1,p}([0,T],D;U)$ and $0\leqslant\psi(t)\in
L^{\infty}([0,T])$ and some $\widetilde{h}\in\mathcal{H}^{k,p}([0,T],D;U)$.
(Note that the term $\widetilde{h}$ was not present in [24, Definition 2.2],
but one can generalize the results of that paper to include such a term.)

We also denote for a parametrized probability measure $\nu^{k}\in
L^{\infty}_{w}([0,T)\times D^{k};\mathscr{M}(U^{k}))$ and a Carathéodory
function $g$ the pairing

$\bigl{\langle}\nu^{k},g\bigr{\rangle}_{\mathcal{H}^{k}}=\int_{D^{k}}\bigl{\langle}\nu^{k}_{t,x},g(t,x)\bigr{\rangle}\,dx$

(where
$\bigl{\langle}\nu^{k}_{x},g(t,x)\bigr{\rangle}=\int_{U^{k}}g(t,x,\xi)\,d\nu^{k}_{t,x}(\xi)$
is the usual duality pairing between Radon measures $\mathscr{M}(U^{k})$ and
continuous functions $C_{0}(U^{k})$).

###  2.2. Definitions

We are now in a position to define time-dependent correlation measures.

######  Definition 2.1.

A _time-dependent correlation measure_ is a collection
$\bm{\nu}=(\nu^{1},\nu^{2},\dots)$ of functions
$\nu^{k}\in\mathcal{H}^{k*}_{0}([0,T),D;U)$ such that

  1. (i)

$\nu^{k}_{t,x}\in\mathscr{P}(U^{k})$ for a.e. $(t,x)\in[0,T]\times D^{k}$, and
the map $x\mapsto\bigl{\langle}\nu^{k}_{t,x},f\bigr{\rangle}$ is measurable
for every $f\in C_{b}(U^{k})$ and almost every $t\in[0,T]$. (In other words,
$\nu^{k}_{t}$ is a Young measure from $D^{k}$ to $U^{k}$.)

  2. (ii)

$L^{p}$ integrability:

(2.4)
$\operatorname*{ess\,sup}_{t\in[0,T)}\left(\int_{D}\bigl{\langle}\nu^{1}_{t,x},|\xi|^{p}\bigr{\rangle}\,dx\right)^{1/p}\leqslant
c<+\infty$

  3. (iii)

Diagonal continuity (DC):

(2.5) $\int_{0}^{T^{\prime}}\omega_{r}^{p}\big{(}\nu^{2}_{t}\big{)}\,dt\to
0\qquad\text{ as }r\to 0\text{ for all }T^{\prime}\in(0,T),$

where

$\omega_{r}^{p}(\nu^{2}_{t})\coloneqq\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{r}(x)}\bigl{\langle}\nu^{2}_{t,x,y},|\xi_{1}-\xi_{2}|^{p}\bigr{\rangle}\,dy\,dx$

is called the _modulus of continuity_ of $\bm{\nu}$.

We denote the set of all time-dependent correlation measures by
$\mathcal{L}^{p}([0,T),D;U)$.

In [24] (and see [23] for a time-independent version), the following
equivalence between time-dependent correlation measures and parametrized
probability measures on $L^{p}(D)$ was proved:

######  Theorem 2.2.

For every time-dependent correlation measure
$\bm{\nu}\in\mathcal{L}^{p}([0,T),D;U)$ there is a unique (up to subsets of
$[0,T)$ of Lebesgue measure $0$) map
$\mu\colon[0,T)\to\mathscr{P}(L^{p}(D;U))$ such that

  1. (i)

the map

(2.6)
$t\mapsto\bigl{\langle}\mu_{t},L_{g}\bigr{\rangle}=\int_{L^{p}}\int_{D^{k}}g(x,u(x))\,dx\,d\mu_{t}(u)$

is measurable for all $g\in\mathcal{H}^{k}_{0}(D;U)$,

  2. (ii)

$\mu$ is $L^{p}$-bounded:

(2.7)
$\operatorname*{ess\,sup}_{t\in[0,T)}\int_{L^{p}}\|u\|_{L^{p}}^{p}\,d\mu_{t}(u)\leqslant
c^{p}<\infty$

  3. (iii)

$\mu$ is dual to $\bm{\nu}$: the identity

(2.8)
$\int_{D^{k}}\bigl{\langle}\nu_{t}^{k},g(x)\bigr{\rangle}\,dx=\int_{L^{p}}\int_{D^{k}}g(x,u(x))\,dx\,d\mu_{t}(u)$

holds for a.e. $t\in[0,T)$, every $g\in\mathcal{H}^{k}_{0}(D;U)$ and all
$k\in\mathbb{N}$.

Conversely, for every $\mu\colon[0,T)\to\mathscr{P}(L^{p}(D;U))$ satisfying
(i) and (ii), there is a unique correlation measure
$\bm{\nu}\in\mathcal{L}^{p}([0,T),D;U)$ satisfying (iii).

We also have the following “Compactness” Theorem for time-dependent
correlation measures [24, Theorem 2.21]

######  Theorem 2.3.

Let $\bm{\nu}_{n}\in\mathcal{L}^{p}([0,T),D;U)$ for $n=1,2,\dots$ be a
sequence of correlation measures such that

(2.9)
$\displaystyle\sup_{n\in\mathbb{N}}\operatorname*{ess\,sup}_{t\in[0,T)}\left(\int_{D}\bigl{\langle}\nu^{1}_{n;t,x},|\xi|^{p}\bigr{\rangle}\,dx\right)^{1/p}\leqslant
c<+\infty$ (2.10) $\displaystyle\lim_{r\to
0}\limsup_{n\to\infty}\int_{0}^{T^{\prime}}\omega_{r}^{p}\bigl{(}\nu^{2}_{n,t}\bigr{)}\,dt=0$

for some $c>0$ and all $T^{\prime}\in[0,T)$. Then there exists a subsequence
$(n_{j})_{j=1}^{\infty}$ and some $\bm{\nu}\in\mathcal{L}^{p}([0,T),D;U)$ such
that

  1. (i)

$\bm{\nu}_{n_{j}}\overset{*}{\rightharpoonup}\bm{\nu}$ as $j\to\infty$, that
is,
$\bigl{\langle}\nu^{k}_{n_{j}},g\bigr{\rangle}_{\mathcal{H}^{k}}\to\bigl{\langle}\nu^{k},g\bigr{\rangle}_{\mathcal{H}^{k}}$
for every $g\in\mathcal{H}^{k}_{0}([0,T),D;U)$ and every $k\in\mathbb{N}$

  2. (ii)

$\bigl{\langle}\nu^{1}_{t},|\xi|^{p}\bigr{\rangle}_{\mathcal{H}^{1}}\leqslant
c^{p}$ for a.e. $t\in[0,T)$

  3. (iii)

$\int_{0}^{T^{\prime}}\omega_{r}^{p}\big{(}\nu^{2}_{t}\big{)}\,dt\leqslant\liminf_{n\to\infty}\int_{0}^{T^{\prime}}\omega_{r}^{p}\big{(}\nu^{2}_{n,t}\big{)}\,dt$
for every $r>0$ and $T^{\prime}\in[0,T)$

  4. (iv)

for $k\in\mathbb{N}$, let $\varphi\in L^{1}_{\textrm{loc}}([0,T)\times D^{k})$
and $\kappa\in C(U^{k})$ be nonnegative, and let
$g(t,x,\xi)\coloneqq\varphi(t,x)\kappa(\xi)$. Then

(2.11)
$\bigl{\langle}\nu^{k},g\bigr{\rangle}_{\mathcal{H}^{k}}\leqslant\liminf_{j\to\infty}\bigl{\langle}\nu_{n_{j}}^{k},g\bigr{\rangle}_{\mathcal{H}^{k}}.$

  5. (v)

Assume moreover that $D\subset\mathbb{R}^{d}$ is compact, $T<\infty$ and that
$\bm{\nu}_{n}$ have uniformly bounded support, in the sense that

(2.12) $\|u\|_{L^{p}}\leqslant R\qquad\text{for $\mu^{n}_{t}$-a.e.\ }u\in
L^{p}(D;U)\text{ for every }n\in\mathbb{N},~{}a.e~{}t\in(0,T),$

with $\mu_{t}^{n}\in\mathscr{P}_{T}(L^{p}(D;U))$ being dual to $\bm{\nu}_{n}$,
then the following _observables converge strongly:_

(2.13)
$\lim_{j\to\infty}\int_{D^{k}}\left|\int_{0}^{T}\left(\bigl{\langle}\nu^{k}_{n_{j};t,x},g(t,x)\bigr{\rangle}-\bigl{\langle}\nu^{k}_{t,x},g(t,x)\bigr{\rangle}\right)\,dt\right|\,dx=0$

for every $g\in\mathcal{H}^{k,p}_{1}([0,T],D;U)$.

##  3\. Statistical solutions

The goal of this section is to show that the statistical solutions of
Navier–Stokes as introduced by Foiaş and Prodi [26, 27, 28, 29, 52] are
equivalent to families of correlation measures as introduced in [23] that
satisfy the Friedman–Keller system of moment equations. For the sake of
simplicity we will assume that the support of the initial measure $\mu_{0}$
lies in a bounded set $B\subset L^{2}(D;U)$, that is,

(3.1) $\text{supp}(\mu_{0})\subset B\subset L^{2}(D;U).$

###  3.1. The Leray projector

We recall first that the _Helmholtz–Leray projector_ , or simply _Leray
projector_ , is the linear map $\mathbb{P}\colon L^{2}(D;U)\to
L^{2}_{\operatorname{div}}(D;U)\coloneqq\\{v\in
L^{2}(D;U):\operatorname{div}v=0,\,\int vdx=0\\}$ that projects a vector field
$f\in L^{2}(D;U)$ to its divergence free component, that is
$f=\mathbb{P}f+\nabla\psi_{f}$ with $\operatorname{div}(\mathbb{P}f)=0$ and
$\mathbb{P}f,\nabla\psi_{f}\in L^{2}(D;U)$. One can show that $\nabla\psi_{f}$
is orthogonal in $L^{2}(D;U)$ to any function $u\in
L^{2}_{\operatorname{div}}(D;U)$,

$\int_{D}u\cdot\nabla\psi_{f}\,dx=0.$

For functions in the tensor product space $L^{2}(D^{k};U^{k})$ we let
$\mathbb{P}_{x_{i}}$ denote the Leray projector in the $i$th component, i.e.,
$\varphi=\mathbb{P}_{x_{i}}\varphi+\nabla_{x_{i}}\psi_{\varphi,i}$ where
$\operatorname{div}_{x_{i}}(\mathbb{P}_{x_{i}}\varphi)=0$, and
$\psi_{\varphi,i}\in L^{2}(D^{k};U^{k-1})$ with
$\nabla_{x_{i}}\psi_{\varphi,i}\in L^{2}(D^{k};U^{k})$.

###  3.2. Definitions

We will start by recalling the different definitions of statistical solutions
introduced in [28, 30, 23].

######  Definition 3.1 (Definition 3.2 in [23]).

Let $\varepsilon\geqslant 0$. The _Friedman–Keller system of moment equations_
, defined for time-dependent correlation measures
$\bm{\nu}\in\mathcal{L}^{2}([0,T),D;U)$, is the hierarchy of equations

(3.2)
$\begin{split}&\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\frac{\partial\varphi}{\partial
t}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\\\
&+\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx\\\
&+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\\\
=&\
-\varepsilon\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\Delta_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\end{split}$

for all $k\in\mathbb{N}$, for all $\varphi\in C^{2}_{c}([0,T)\times
D^{k};U^{k})$ with $\operatorname{div}_{x_{i}}\varphi=0$ for all
$i=1,\dots,k$, along with the divergence constraint

(3.3)
$\int_{D^{k}}\int_{U^{k}}\xi_{1}\otimes\dots\otimes\xi_{\ell}\otimes\alpha_{\ell+1}(\xi_{\ell+1})\otimes\dots\otimes\alpha_{k}(\xi_{k})\,d\nu_{t,x}^{k}(\xi)\cdot\nabla_{x_{1},\dots,x_{\ell}}\varphi(x)\,dx=0,$

where
$\nabla_{x_{1},\dots,x_{\ell}}=(\nabla_{x_{1}},\dots,\nabla_{x_{\ell}})^{\top}$,
$1\leqslant\ell\leqslant k\in\mathbb{N}$, for all $\varphi\in
H^{1}(D^{k};U^{k-\ell})$, $\alpha_{j}\in C(U;U)$, with $\alpha_{j}(v)\leqslant
C(1+|v|^{2})$ for all $j=1,\dots,k$.

If $\bm{\nu}$ solves the Friedman–Keller system of moments equations and in
addition satisfies the energy inequality

(3.4)
$\displaystyle\sum_{k=0}^{K}a_{k}\int_{D^{k}}\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{t,x}^{k}(\xi)\,dx$
$\displaystyle+2\varepsilon\sum_{k=0}^{K}a_{k}\sum_{i=1}^{k}\sum_{j=1}^{d}\lim_{h\rightarrow
0}\frac{1}{h^{2}}\int_{0}^{t}\\!\\!\int_{D^{k}}\\!\int_{U^{k+1}}\\!\\!\\!|\xi_{1}|^{2}\dots|\xi_{i}-\xi_{k+1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{t,(x,x_{i}+h\mathbf{e}_{j})}^{k+1}(\xi,\xi_{k+1})\,dx\,ds$
$\displaystyle\leqslant\sum_{k=0}^{K}a_{k}\int_{D^{k}}\\!\\!\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{0,x}^{k}(\xi)\,dx$

for all $K\in\mathbb{N}$ and $a_{k}\in\mathbb{R}$, $k=0,\dots,K$ such that
$p_{K}(s)=\sum_{k=0}^{K}a_{k}s^{k}$ is a nonnegative, nondecreasing polynomial
for $s\in[0,R]$ for $R$ sufficiently large related to the support of the
correlation measure (see (2.12)), then we call $\bm{\nu}$ a _Friedman–Keller
statistical solution_ of the Navier–Stokes (when $\varepsilon>0$) or Euler
(when $\varepsilon=0$) equations.

######  Remark 3.2.

Friedman–Keller statistical solutions are analogous to the definition of
statistical solutions for hyperbolic systems of conservation laws as
introduced in [23, 24] for compressible flows.

######  Remark 3.3.

By a standard argument for weak solutions to continuity equations, the map
$t\mapsto\langle\nu^{k}_{t,\cdot},\,\xi_{1}\otimes\cdots\otimes\xi_{k}\rangle$
is weakly continuous for every $k\in\mathbb{N}$; see e.g. [2, Remark 2.2].

######  Remark 3.4 (Formulation of (3.2) with non-divergence free test
functions).

Denoting $\mathbb{P}_{k}\coloneqq\mathbb{P}_{x_{1}}\dots\mathbb{P}_{x_{k}}$
(cf. Section 3.1), we can replace the divergence-free test function
$\varphi\in C^{2}_{c}([0,T]\times D^{k};U^{k})$ in (3.2) by
$\mathbb{P}_{k}\varphi$ for an arbitrary $\varphi\in C^{2}_{c}([0,T]\times
D^{k};U^{k})$. Using the fact that the Leray projection is self-adjoint and
that, by (3.3),

$\operatorname{div}_{x_{i}}\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k})\,d\nu_{t,x}^{k}(\xi)=0\qquad\forall\
i=1,\dots,k,\quad\text{a.e. }\,(t,x)\in[0,T]\times D^{k}$

we observe that

$\mathbb{P}_{k}\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k})\,d\nu_{t,x}^{k}(\xi)=\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k})\,d\nu_{t,x}^{k}(\xi),\quad\text{a.e.
}\,t\in[0,T],\,x\in D^{k}.$

Therefore,

$\displaystyle\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\frac{\partial\mathbb{P}_{k}\varphi}{\partial
t}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\mathbb{P}_{k}\left(\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k})\,d\nu_{t,x}^{k}(\xi)\right):\frac{\partial\varphi}{\partial
t}(t,x)\,dx\,dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k})\,d\nu_{t,x}^{k}(\xi):\frac{\partial\varphi}{\partial
t}(t,x)\,dx\,dt.$

Similarly,

$\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\mathbb{P}_{k}\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx=\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx,$

and

$\displaystyle\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\Delta_{x_{i}}\mathbb{P}_{k}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\Delta_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt,$

the last one being true due to the fact that the Laplacian and the Leray
projection commute on the torus. Hence, the weak formulation (3.2) can be
rewritten as

(3.5)
$\begin{split}&\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\frac{\partial\varphi}{\partial
t}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt+\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx\\\
&+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\\\
=&-\varepsilon\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\Delta_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\\\
&+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\nabla_{x_{i}}\psi_{\varphi,i}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt\end{split}$

where $\varphi\in C^{2}_{c}([0,T]\times D^{k};U^{k})$. The terms

(3.6)
$\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\nabla_{x_{i}}\psi_{\varphi,i}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$

correspond to the pressure in the deterministic setting.

To define statistical solutions in the sense of Foiaş and Prodi, we need to
introduce some notation. We denote by $L^{2}_{\operatorname{div}}(D;U)$ the
space of divergence free $L^{2}(D;U)$-vector fields and by
$H^{1}_{\operatorname{div}}(D;U)$ the space of divergence free functions in
$H^{1}(D;U)$ (these can be obtained as the closures of
$C^{\infty}(D;U)\cap\\{\operatorname{div}u=0\\}$ in $L^{2}(D;U)$ and
$H^{1}(D;U)$, respectively, with suitable integral conditions:

$\begin{split}L^{2}_{\operatorname{div}}(D;U)&=\left\\{u\in
L^{2}(\mathbb{T}^{d})\ :\ \operatorname{div}u=0,\
\int_{\mathbb{T}^{d}}u(x)\,dx=0\right\\},\\\
H^{1}_{\operatorname{div}}(D;U)&=\left\\{u\in H^{1}(\mathbb{T}^{d})\ :\
\operatorname{div}u=0,\ \int_{\mathbb{T}^{d}}u(x)\,dx=0\right\\},\end{split}$

for periodic boundary conditions. We denote the $L^{2}$-inner product by

$(u,v)=\int_{D}u(x)v(x)\,dx,$

and for $\varepsilon>0$ the $H^{1}$-inner product by

$a(u,v)=\varepsilon\sum_{i=1}^{d}\int_{D}\frac{\partial u}{\partial
x^{i}}\cdot\frac{\partial u}{\partial x^{i}}\,dx,$

Define the Stokes operator $A$ by

$\begin{split}Au&=-\mathbb{P}\Delta u,\quad\text{for all }u\in
D(A)=H^{1}_{\operatorname{div}}(D;U)\cap H^{2}(D;U),\\\
\varepsilon(Au,v)&=a(u,v),\quad\text{for all }u,v\in D(A^{1/2}),\end{split}$

where $\mathbb{P}$ is the Leray projector, and the skew-symmetric trilinear
form $b$ by

(3.7) $\begin{split}b(u,v,w)&\coloneqq\int_{D}(u\cdot\nabla)v\cdot
w\,dx=(B(u,v),w),\quad u,v,w\in D(A^{1/2})\\\ B(u)&\coloneqq
B(u,u).\end{split}$

We can then write the Navier–Stokes equations in the functional formulation:
Let $T>0$, $u_{0}\in L^{2}_{\operatorname{div}}(D;U)$, find $u\in
L^{\infty}([0,T];L^{2}_{\operatorname{div}}(D;U))\cap
L^{2}([0,T];H^{1}_{\operatorname{div}}(D;U))$ with
$u^{\prime}\coloneqq\frac{d}{dt}u\in L^{1}([0,T];D(A^{-1/2}))$ such that

(3.8) $u^{\prime}+\varepsilon Au+B(u)=0$

and $u(0)=u_{0}$ in a suitable sense. This corresponds to the weak formulation

(3.9) $\frac{d}{dt}(u,v)+a(u,v)+b(u,u,v)=0\qquad\text{for all }v\in
H^{1}_{\operatorname{div}}(D;U).$

If we denote

(3.10) $F(t,u)\coloneqq-\varepsilon Au-B(u),$

the functional formulation becomes

(3.11) $u^{\prime}(t)=F(t,u(t)).$

We need the following class of test functions:

######  Notation 3.5.

[28] Let $\mathscr{T}_{\text{cyl}}$ denote the class of cylindrical test
functions consisting of the real-valued functionals $\Phi=\Phi(u)$ that depend
on a finite number $k\in\mathbb{N}$ of components of $u$, that is,

$\Phi(u)=\varphi\bigl{(}(u,g_{1}),\dots,(u,g_{k})\bigr{)},$

where $\varphi\in C^{1}_{c}(\mathbb{R}^{k})$ and $g_{1},\dots,g_{k}\in
H^{1}(D;U)$. Let $\mathscr{T}_{\text{cyl}}^{0}$ denote the subset of such
functions which satisfy $g_{1},\dots,g_{k}\in
H^{1}_{\operatorname{div}}(D;U)$. We denote by $\Phi^{\prime}$ the
differential of $\Phi$ in $L^{2}_{\operatorname{div}}(D;U)$, which can be
expressed as

$\Phi^{\prime}(u)=\sum_{j=1}^{k}\partial_{j}\varphi\bigl{(}(u,g_{1}),\dots,(u,g_{k})\bigr{)}g_{j},$

where $\partial_{j}\varphi$ is the derivative of $\varphi$ with respect to its
$j$th component.

We can now define statistical solutions in the sense of Foiaş and Prodi. We
will use the definition as it stated in their newer work [30, Def. 3.2]:

######  Definition 3.6 (Foiaş–Prodi [28, 26, 29, 30]).

A family of probability measures $(\mu_{t})_{0\leqslant t\leqslant T}$ on
$L^{2}_{\operatorname{div}}(D;U)$ is a _Foiaş–Prodi statistical solution_ of
the Navier–Stokes equations on $L^{2}_{\operatorname{div}}(D;U)$ with initial
data $\mu_{0}$ if

  1. (a)

The function

(3.12) $t\mapsto\int_{L^{2}_{\operatorname{div}}}\varphi(u)\,d\mu_{t}(u),$

is measurable on $[0,T]$ for every $\varphi\in
C_{b}(L^{2}_{\operatorname{div}}(D;U))$;

  2. (b)

$\mu$ satisfies the weak formulation

(3.13)
$\int_{L^{2}_{\operatorname{div}}}\Phi(u)\,d\mu_{t}(u)=\int_{L^{2}_{\operatorname{div}}}\Phi(u)\,d\mu_{0}(u)+\int_{0}^{t}\int_{L^{2}_{\operatorname{div}}}(F(s,u),\Phi^{\prime}(u))\,d\mu_{s}(u)\,ds$

for all $t\in[0,T]$ and all cylindrical test functions
$\Phi\in\mathscr{T}_{\text{cyl}}^{0}$, where $F$ is given in (3.10).

  3. (c)

$\mu$ satisfies the strengthened mean energy inequality: For any $\psi\in
C^{1}(\mathbb{R},\mathbb{R})$ nonnegative, nondecreasing with bounded
derivative and $t\in[0,T]$, the inequality

(3.14)
$\int_{L^{2}_{\operatorname{div}}}\psi(\|u\|_{L^{2}(D)}^{2})\,d\mu_{t}(u)+2\varepsilon\int_{0}^{t}\int_{L^{2}_{\operatorname{div}}}\psi^{\prime}(\|u\|_{L^{2}(D)}^{2})|u|_{H^{1}(D)}^{2}\,d\mu_{s}(u)\\\
\leqslant\int_{L^{2}_{\operatorname{div}}}\psi(\|u\|_{L^{2}(D)}^{2})\,d\mu_{0}(u)$

holds.

  4. (d)

The function

(3.15)
$t\mapsto\int_{L^{2}_{\operatorname{div}}}\psi(\left\|u\right\|_{L^{2}(D)}^{2})\,d\mu_{t}(u)$

is continuous at $t=0$ from the right, for any function $\psi\in
C^{1}(\mathbb{R},\mathbb{R})$ nonnegative, nondecreasing with bounded
derivative.

######  Remark 3.7.

Note that, as a consequence of the energy inequality (3.14) for $\psi(s)=s$,
the function

$t\mapsto\int_{L^{2}_{\operatorname{div}}}\|u\|_{L^{2}(D)}^{2}\,d\mu_{t}(u),$

belongs to $L^{\infty}([0,T])$ and the function

(3.16)
$t\mapsto\int_{L^{2}_{\operatorname{div}}}|u|_{H^{1}(D)}^{2}\,d\mu_{t}(u),$

belongs to $L^{1}([0,T])$. Notice also that (3.13) implies that

$t\mapsto\int_{L^{2}_{\operatorname{div}}}\Phi(u)d\mu_{t}(u)$

for $\Phi(u)$ a cylindrical test function, is continuous since

$\int_{L^{2}_{\operatorname{div}}}(F(s,u),\Phi^{\prime}(u))\,d\mu_{t}(u)$

is locally integrable. Combining this fact with condition (c), conditionn (d)
follows directly.

###  3.3. Equivalence between the solution concepts

Next, we show that the Friedman–Keller statistical solutions in Definition 3.1
and the Foiaş–Prodi statistical solutions in Definition 3.6 are in fact the
same.

######  Theorem 3.8 (Foiaş–Temam statistical solutions satisfy the
Friedman–Keller system).

Let $\mu$ be a Foiaş–Prodi statistical solution such that the initial
condition $\mu_{0}$ has bounded support,

$\text{supp}(\mu_{0})\subset B\subset L^{2}_{\operatorname{div}}(D;U),\qquad
B\subset\big{\\{}u\in L^{2}(D;U)\ :\ \|u\|_{L^{2}(D)}\leqslant R\big{\\}}$

for some $R>0$. Then $\mu$ corresponds (cf. Theorem 2.2) to a correlation
measure $\bm{\nu}_{t}=(\nu^{1},\nu^{2},\dots)$ that is a statistical solution
in the Friedman–Keller sense (cf. Definition 3.1).

Conversely, we have:

######  Theorem 3.9 (Friedman–Keller solutions are Foiaş–Prodi statistical
solutions).

Let $\bm{\nu}=(\bm{\nu}_{t})_{0\leqslant t\leqslant T}$ be a Friedman–Keller
statistical solution of Navier–Stokes (cf. Definition 3.1) with bounded
support, i.e.,

(3.17)
$\int_{D^{k}}\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{t,x}^{k}(\xi)\,dx\leqslant
R^{k}<\infty,$

for some $0<R<\infty$, every $k\in\mathbb{N}$, and almost every $t\in[0,T]$.
Then $\bm{\nu}$ corresponds to a probability measure
$\mu=(\mu_{t})_{0\leqslant t\leqslant T}$ on a bounded set of
$L^{2}_{\operatorname{div}}(D;U)$ which is a Foiaş–Prodi statistical solution
of the Navier–Stokes equations (cf. Definition 3.6).

The proofs of these two results are given in Appendix A.

Foiaş et al. have shown existence of Foiaş–Prodi statistical solutions for the
(forced) Navier–Stokes equations, see e.g. [26, 27, 28]. Using these
equivalence theorems, this implies existence of statistical solutions via
correlation measures as in Definition 3.1.

######  Remark 3.10.

The equivalence theorems 3.8 and 3.9 are restricted to probability measures
with bounded support. It should be possible to extend these results to
probability measures having sufficiently fast decay near infinity; however,
the proofs would become significantly more technical. We have therefore
decided to restrict ourselves to probability measures with bounded support.

##  4\. Vanishing viscosity limit of statistical solutions of Navier–Stokes

The goal of this section is to pass to the inviscid limit $\varepsilon\to 0$
under the assumption of _weak statistical scaling_ (c.f. Section 4.3,
Assumption 1). We will first prove a rigorous result on the longitudinal third
order structure function

(4.1)
$S_{\|}^{3}(\tau,r)\coloneqq\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}\big{(}(u(x+rn)-u(x))\cdot
n\big{)}^{3}\,dxdS(n)\,d\mu_{t}(u)\,dt,$

and then relate it to the similarly defined second order structure function
using the weak scaling assumption. Together with weak statistical anisotropy,
this yields diagonal continuity of the correlation measures
$\bm{\nu}^{\varepsilon}$ that is needed to apply the compactness theorem 2.3
and pass to the limit. The proof of the scaling estimate for the third order
structure function (4.1) in Lemma 4.2 and 4.3 largely follows the proof of a
similar result for martingale solutions of stochastic Navier–Stokes equations
in [6]. To simplify notation, we will omit writing the dependence of
$\bm{\nu}$ and $\mu$ on $\varepsilon$ in the following sections.

###  4.1. Kármán–Howarth–Monin relation

The key to deriving an estimate on the behavior of the third order structure
function (4.1) is the so-called Kármán–Howarth–Monin (KHM) relation [16] that
describes the evolution of the second correlation marginal. Similar relations
have been derived before for various settings (stochastic, forced, etc.), see
[31, 49, 16, 47, 6, 20, 18]. For statistical solutions we derive:

######  Proposition 4.1.

Let $\bm{\nu}$ be a Friedman–Keller statistical solution of the Navier–Stokes
equations. Then the second correlation marginal $\nu^{2}$ satisfies the
Kármán–Howarth–Monin relation for correlation measures:

(4.2)
$\begin{split}&\quad\sum_{ij}\int_{D}\\!\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\,d\nu_{\tau,x,x+h}^{2}(\xi)\,dx\,\sigma^{ij}(h)\,dh-\sum_{ij}\int_{D}\\!\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\,\sigma^{ij}(h)\,dh\\\
&\quad+\frac{1}{2}\sum_{ijk}\int_{0}^{\tau}\\!\\!\int_{D}\\!\int_{D}\\!\int_{U^{2}}(\xi_{2}^{i}-\xi^{i}_{1})(\xi_{2}^{j}-\xi_{1}^{j})(\xi_{2}^{k}-\xi^{k}_{1})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,\partial_{h^{k}}\sigma^{ij}(h)\,dh\,dt\\\
&=-\varepsilon\sum_{ij}\int_{0}^{\tau}\\!\\!\int_{D}\\!\int_{D}\\!\int_{U^{2}}\\!\\!(\xi_{1}^{i}-\xi_{2}^{i})(\xi_{1}^{j}-\xi_{2}^{j})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,\Delta_{h}\sigma^{ij}(h)\,dh\,dt.\end{split}$

for any $\tau>0$, where $\sigma=(\sigma_{ij})_{i,j=1}^{3}$ is any smooth,
compactly supported, isotropic rank 2 tensor – that is, any $\sigma\in
C^{2}_{c}(\mathbb{R}^{d},\mathbb{R}^{d\times d})$ of the form

(4.3)
$\sigma(h)=\left(\omega_{1}(|h|)\mathbf{I}+\omega_{2}(|h|)\hat{h}\otimes\hat{h}\right)\qquad\text{where
}\hat{h}=\begin{cases}h/|h|&h\neq 0\\\ 0&h=0\end{cases}$

for $\omega_{1},\omega_{2}\in C^{2}_{c}(\mathbb{R})$.

###### Proof.

We consider equation (3.5) for $k=2$ with the test function
$\varphi(t,x,y)=\eta(t,y-x)$ (for simplicity replacing $x_{1}$ and $x_{2}$ by
$x$ and $y$ and writing in component form $\eta=(\eta^{ij})_{ij}$),

(4.4)
$\begin{split}&\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\frac{\partial\eta^{ij}}{\partial
t}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\\\
&\quad+\sum_{ij}\int_{D}\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\eta^{ij}(0,y-x)\,d\nu_{0,x,y}^{2}(\xi)\,dx\,dy\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{x^{k}}\eta^{ij}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{y^{k}}\eta^{ij}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\\\
&=-\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}(\Delta_{x}+\Delta_{y})\eta^{ij}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{x^{k}}\partial_{x^{i}}\psi^{j}_{\eta,1}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{y^{k}}\partial_{y^{j}}\psi^{i}_{\eta,2}(t,y-x)\,d\nu_{t,x,y}^{2}(\xi)\,dx\,dy\,dt\end{split}$

Since $\psi_{\eta,1}$ solves
$\Delta_{x}\psi_{\eta,1}=\operatorname{div}_{x}\eta$ and $\psi_{\eta,2}$
solves
$\Delta_{y}\psi_{\eta,2}=\operatorname{div}_{y}\eta=-\operatorname{div}_{x}\eta$,
we have $\psi_{\eta,1}=(\Delta_{x})^{-1}\operatorname{div}_{x}\eta$ and
$\psi_{\eta,2}=-(\Delta_{y})^{-1}\operatorname{div}_{x}\eta=-(\Delta_{x})^{-1}\operatorname{div}_{x}\eta$
and so $\psi_{\eta}\coloneqq\psi_{\eta,1}=-\psi_{\eta,2}$ (up to additive
constants). Using this and changing the integration variables to $x$ and
$h\coloneqq y-x$, we obtain

(4.5)
$\begin{split}&\sum_{ij}\int_{0}^{T}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\frac{\partial\eta^{ij}}{\partial
t}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ij}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\eta^{ij}(0,h)\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\,dh\\\
&\quad-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&=-2\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{h^{k}}\partial_{h^{i}}\psi^{j}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{h^{k}}\partial_{h^{j}}\psi^{i}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\end{split}$

where $\psi_{\eta}=\psi_{\eta,1}$. The cubic terms can be rewritten using the
following simple fact (which can also be found in Frisch [31, equation
(6.13)], and in a similar, weak form in [6]):

(4.6)
$\begin{split}&\quad-2\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&=\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt.\end{split}$

The proof of this is postponed to the end of this proof. Using this, we can
rewrite (4.5) as

(4.7)
$\begin{split}&\quad\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\frac{\partial\eta^{ij}}{\partial
t}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ij}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\eta^{ij}(0,h)\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\,dh\\\
&\quad-\frac{1}{2}\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&=-2\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{h^{k}}\partial_{h^{i}}\psi^{j}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{h^{k}}\partial_{h^{j}}\psi^{i}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt.\end{split}$

Since

$\displaystyle\quad\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{1}^{j}\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle=\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\int_{U}\\!\\!\xi_{1}^{i}\xi_{1}^{j}\,d\nu_{t,x}^{1}(\xi)\,dx\int_{D}\Delta_{h}\eta^{ij}(t,h)\,dh\,dt=0,$

we can rewrite

$\displaystyle\quad-2\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle=\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!(\xi_{1}^{i}-\xi_{2}^{i})(\xi_{1}^{j}-\xi_{2}^{j})\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt.$

Moreover, we have for symmetric, smooth and compactly supported rank 2 tensors
$\eta$ of the form

(4.8)
$\eta(t,h)=\omega_{1}(t,|h|)\mathbf{I}+\omega_{2}(t,|h|)\hat{h}\otimes\hat{h},$

with $\psi^{i}_{\eta}=-(\Delta_{h})^{-1}\operatorname{div}_{h}\eta^{i,\cdot}$,

(4.9)
$\begin{split}&\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{1}^{k}\xi_{2}^{j}\partial_{h^{k}}\partial_{h^{i}}\psi^{j}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt=0,\\\
&\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{h^{k}}\partial_{h^{j}}\psi^{i}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt=0,\end{split}$

whose proof is postponed to the end of this proof. Using this, (4.7) becomes

(4.10)
$\begin{split}&\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\frac{\partial\eta^{ij}}{\partial
t}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&\quad+\sum_{ij}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\eta^{ij}(0,h)\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\,dh\\\
&\quad-\frac{1}{2}\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
&=\varepsilon\sum_{ij}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!(\xi_{1}^{i}-\xi_{2}^{i})(\xi_{1}^{j}-\xi_{2}^{j})\Delta_{h}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt.\end{split}$

Let $\theta_{\delta}$ be a sequence of smooth, uniformly bounded functions
with the property that $\theta_{\delta}\to\mathbf{1}_{(0,\tau]}(t)$ for every
$t$ as $\delta\to 0$. If we now use a test function

(4.11) $\eta(t,h)=\sigma(h)\theta_{\delta}(t),$

where $\sigma$ is of the form (4.3), then we can use the weak continuity in
time of the moments
$\int_{U^{k}}\xi_{1}\otimes\dots\otimes\xi_{k}\,d\nu^{k}_{t,x}(\xi)$ to obtain
for any $\tau>0$, as $\delta\to 0$,

(4.12)
$\begin{split}&\quad-\sum_{ij}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\,d\nu_{\tau,x,x+h}^{2}(\xi)\,dx\sigma^{ij}(h)\,dh+\sum_{ij}\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!\xi_{1}^{i}\xi_{2}^{j}\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\sigma^{ij}(h)\,dh\\\
&\quad-\frac{1}{2}\sum_{ijk}\int_{0}^{\tau}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\partial_{h^{k}}\sigma^{ij}(h)\,dh\,dt\\\
&=\varepsilon\sum_{ij}\int_{0}^{\tau}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\\!\\!(\xi_{1}^{i}-\xi_{2}^{i})(\xi_{1}^{j}-\xi_{2}^{j})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\Delta_{h}\sigma^{ij}(h)\,dh\,dt.\qed\end{split}$

###### Proof of (4.6).

We expand the right hand side:

$\displaystyle\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle=\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}\xi_{1}^{j}\xi_{1}^{k}-\xi_{2}^{i}\xi_{2}^{j}\xi_{2}^{k})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,\partial_{h^{k}}\eta^{ij}(t,h)\,dh\,dt$
$\displaystyle\quad+\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{2}^{i}\xi_{2}^{j}\xi_{1}^{k}-\xi_{1}^{i}\xi_{1}^{j}\xi_{2}^{k})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle\quad-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}\xi_{2}^{j}\xi_{1}^{k}-\xi_{2}^{i}\xi_{1}^{j}\xi_{2}^{k})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle\quad-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{2}^{i}\xi_{1}^{j}\xi_{1}^{k}-\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt.$

The first term on the right hand side is zero since $\eta$ is compactly
supported (after changing the integration variable from $x$ to $x-h$ in one of
the terms). The second term on the right hand side vanishes using the
divergence constraint (3.3). Using that $\eta$ and $\nu$ are symmetric, the
last two terms are identical and so

$\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{1}^{i}-\xi^{i}_{2})(\xi_{1}^{j}-\xi_{2}^{j})(\xi_{1}^{k}-\xi^{k}_{2})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt\\\
=-2\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}(\xi_{2}^{i}\xi_{1}^{j}\xi_{1}^{k}-\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k})\partial_{h^{k}}\eta^{ij}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt,$

which proves the claim. ∎

###### Proof of (4.9).

We consider the second expression, assume $\eta$ is of the form

(4.13)
$\eta(t,h)=\omega_{1}(t,|h|)\mathbf{I}+\omega_{2}(t,|h|)\hat{h}\otimes\hat{h},$

where $\omega_{i}$, $i=1,2$ are compactly supported in the torus. Then using
that $\psi^{i}_{\eta}=-(\Delta_{h})^{-1}\operatorname{div}_{h}\eta^{i,\cdot}$
(the first term is treated in a similar way)

$\displaystyle E$
$\displaystyle\coloneqq-\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\partial_{h^{k}}\partial_{h^{j}}\psi^{i}_{\eta}(t,h)\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,dh\,dt$
$\displaystyle=\sum_{ijk\ell}\int_{0}^{T}\\!\\!\int_{D}\\!\\!\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\,\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\partial_{h^{\ell}}\eta^{i\ell}(t,h)\right)\,dh\,dt$
$\displaystyle=\sum_{ijk\ell}\int_{0}^{T}\\!\\!\int_{D}\\!\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\right)\partial_{h^{\ell}}\eta^{i\ell}(t,h)dh\,dt.$

We note that since $\omega_{i}$ have compact support, we can write in polar
coordinates

$\sum_{\ell}\partial_{h^{\ell}}\eta^{i\ell}(t,h)=\biggl{(}\underbrace{\omega^{\prime}_{1}(t,|h|)+\omega_{2}^{\prime}(t,|h|)+2\frac{\omega_{2}(t,|h|)}{|h|}}_{\eqqcolon\,G(t,|h|)}\biggr{)}\hat{h}^{i}$

and so

$\begin{split}E&=\sum_{ijk\ell}\int_{0}^{T}\\!\\!\int_{D}\\!\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\partial_{h^{\ell}}\eta^{i\ell}(t,h)\,dx\right)dh\,dt\\\
&=\sum_{ijk}\int_{0}^{T}\\!\\!\int_{D}\\!\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\int_{D}\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\right)G(t,|h|)\hat{h}^{i}\,dh\,dt\\\
&=\sum_{jk}\int_{0}^{T}\\!\\!\int_{0}^{\infty}\int_{|h|=r}\\!\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\int_{D}\\!\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\right)\hat{h}^{i}dS(h)\,G(t,r)\,dr\,dt\\\
&=\sum_{ijk}\int_{0}^{T}\\!\\!\int_{0}^{\infty}\int_{|h|\leqslant
r}\\!\partial_{h^{i}}\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\left(\int_{D}\\!\int_{U^{2}}\xi_{1}^{i}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\right)dh\,G(t,r)\,dr\,dt\\\
&=\sum_{jk}\int_{0}^{T}\\!\\!\int_{0}^{\infty}\int_{|h|\leqslant
r}\\!\partial_{h^{k}}\partial_{h^{j}}\Delta^{-1}\operatorname{div}_{h}\left(\int_{D}\\!\int_{U^{2}}\xi_{1}\xi_{2}^{j}\xi_{2}^{k}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\right)dh\,G(t,r)\,dr\,dt=0,\end{split}$

where we used the divergence theorem in the second to last identity and the
divergence constraint (3.3) for the last identity. ∎

###  4.2. Scaling of third order structure functions

Next, we use the KHM-relation (4.2) to derive a scaling relation for the
averaged third order structure function in terms of the measure $\mu_{t}$,

(4.14)
$S^{3}_{0}(\tau,r)=\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\fint_{\mathbb{S}^{2}}\int_{D}\big{|}u(x)-u(x+rn)\big{|}^{2}\big{(}u(x+rn)-u(x)\big{)}\cdot
n\,dx\,dS(n)\,d\mu_{t}(u)\,dt,$

which will be more convenient to work with for this purpose. We have:

######  Lemma 4.2.

Let $\mu_{t}$ be a Foiaş–Prodi statistical solution of the Navier–Stokes
equations (cf. Definition 3.6). Then

(4.15) $\left|\frac{S^{3}_{0}(\tau,r)}{r}\right|\leqslant 2E_{0},$

where $E_{0}$ is the initial energy,

(4.16)
$E_{0}\coloneqq\int_{L^{2}_{\operatorname{div}}(D;U)}\left\|u(x)\right\|^{2}_{L^{2}(D)}\,d\mu_{0}(u).$

###### Proof.

We take a test function of the form $\sigma(h)=\omega(|h|)\mathbf{I}$ in the
KHM-relation (4.2) with $\omega$ having compact support in $[0,0.5)$. A little
bit of algebra yields (denoting $\hat{h}_{k}=h^{k}/|h|$)

$\partial_{h^{k}}\omega(|h|)=\omega^{\prime}(|h|)\hat{h}^{k},$

and so (4.2) for this particular test function reads

(4.17)
$\begin{split}&\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}\cdot\xi_{2}\,d\nu_{\tau,x,x+h}^{2}(\xi)\,dx\omega(|h|)\,dh-\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}\cdot\xi_{2}\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\omega(|h|)\,dh\\\
&+\frac{1}{2}\int_{0}^{\tau}\\!\\!\int_{D}\int_{D}\\!\int_{U^{2}}|\xi_{1}-\xi_{2}|^{2}(\xi_{2}-\xi_{1})\cdot\hat{h}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\omega^{\prime}(|h|)\,dh\,dt\\\
=&-\varepsilon\int_{0}^{\tau}\\!\\!\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!|\xi_{1}-\xi_{2}|^{2}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\Delta_{h}\omega(|h|)\,dh\,dt.\end{split}$

In terms of the statistical solution $(\mu_{t})_{t>0}$ this is

(4.18)
$\begin{split}&\quad\int_{L^{2}_{\operatorname{div}}}\\!\int_{D}\int_{D}\\!\\!u(x)\cdot
u(x+h)\,dx\omega(|h|)\,dh\,d\mu_{\tau}(u)-\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!u(x)u(x+h)\,dx\,\omega(|h|)\,dh\,d\mu_{0}(u)\\\
&\quad+\frac{1}{2}\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!|u(x)-u(x+h)|^{2}(u(x+h)-u(x))\cdot\hat{h}\,\,dx\omega^{\prime}(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=-\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!|u(x)-u(x+h)|^{2}\,dx\Delta_{h}\omega(|h|)\,dh\,d\mu_{t}(u)\,dt.\end{split}$

The last term can also be written as

(4.19)
$\begin{split}&\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!|u(x)-u(x+h)|^{2}\,dx\Delta_{h}\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=2\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!\nabla_{h}u(x+h)\cdot(u(x)-u(x+h))\,dx\nabla_{h}\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=2\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!\nabla_{x}u(x+h)\cdot(u(x)-u(x+h))\,dx\nabla_{h}\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=2\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!\nabla_{x}u(x)\cdot(u(x-h)-u(x))\,dx\nabla_{h}\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=-2\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!\nabla_{x}u(x):\nabla_{h}u(x-h)\,dx\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\\\
&=2\varepsilon\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!\nabla_{x}u(x+h):\nabla_{x}u(x)\,dx\omega(|h|)\,dh\,d\mu_{t}(u)\,dt\end{split}$

Changing to spherical coordinates and using the definition of $S^{3}_{0}$,
(4.14), we obtain

$\displaystyle\frac{1}{2}\int_{0}^{\infty}S_{0}^{3}(\tau,r)r^{2}\omega^{\prime}(r)dr$
$\displaystyle=-2\varepsilon\int_{0}^{\infty}\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!\nabla_{x}u(x):\nabla_{x}u(x+rn)\,dxdS(n)\,d\mu_{t}(u)\,dtr^{2}\omega(r)dr$
$\displaystyle\quad-\int_{0}^{\infty}\int_{L^{2}_{\operatorname{div}}}\\!\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
u(x+rn)\,dxdS(n)\,d\mu_{\tau}(u)r^{2}\omega(r)dr$
$\displaystyle\quad+\int_{0}^{\infty}\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
u(x+rn)\,dxdS(n)\,d\mu_{0}(u)r^{2}\omega(r)dr.$

We denote

$\displaystyle m_{2}(\tau,r)$
$\displaystyle\coloneqq\int_{L^{2}_{\operatorname{div}}}\\!\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
u(x+rn)\,dxdS(n)\,d\mu_{\tau}(u)$ $\displaystyle v(\tau,r)$
$\displaystyle\coloneqq\int_{0}^{\tau}\\!\\!\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!\nabla_{x}u(x):\nabla_{x}u(x+rn)\,dxdS(n)\,d\mu_{t}(u)\,dt.$

Since $(\mu_{t})_{t>0}$ is supported on functions in
$L^{\infty}([0,\infty);L^{2}_{\operatorname{div}}(D;U))\cap
L^{2}([0,\infty);H^{1}_{\operatorname{div}}(D;U))$, $S_{0}^{3}$ is a
continuous function. Moreover, notice that due to the _a priori_ bounds
following from the energy inequality (3.14), both $m_{2}$ and $v$ are
uniformly bounded and continuous in $r$ and $\tau$ (for the continuity in
$\tau$ of the first quantity, one needs weak time continuity of the moments
which follows the fact that they satisfy the equations (4.2) where all the
terms are integrable). We obtain

$\displaystyle\frac{1}{2}\int_{0}^{\infty}S_{0}^{3}(\tau,r)r^{2}\omega^{\prime}(r)dr$
$\displaystyle=-2\varepsilon\int_{0}^{\infty}v(\tau,r)r^{2}\omega(r)dr$
$\displaystyle\quad-\int_{0}^{\infty}m_{2}(\tau,r)r^{2}\omega(r)dr+\int_{0}^{\infty}m_{2}(0,r)r^{2}\omega(r)dr,$

which is an ODE in the sense of distributions for $S_{0}^{3}(\tau,\cdot)$, and
because the right hand side is uniformly bounded and continuous, we can
consider it in the strong sense (note that boundary terms when integrating the
$S_{0}^{3}$ term by parts vanish):

$\frac{1}{r^{2}}\partial_{r}(r^{2}S_{0}^{3}(r))=4\varepsilon
v(\tau,r)+2m_{2}(\tau,r)-2m_{2}(0,r),$

or

$\frac{S_{0}^{3}(r)}{r}=\frac{2}{r^{3}}\int_{0}^{r}s^{2}\big{(}2\varepsilon
v(\tau,s)+m_{2}(\tau,s)-m_{2}(0,s)\big{)}\,ds.$

The energy inequality (3.14) and the Cauchy–Schwarz inequality imply that
$2\varepsilon v(\tau,s)$ and $m_{2}(\tau,s)$ are both bounded by $E_{0}$
(defined in (4.16)), uniformly in $\tau,s$. Hence,

(4.20)
$\left|\frac{S_{0}^{3}(r)}{r}\right|\leqslant\frac{2}{r}\int_{0}^{r}\left(\frac{s}{r}\right)^{2}\big{(}2\varepsilon|v(\tau,s)|+|m_{2}(\tau,s)|+|m_{2}(0,s)|\big{)}\,ds\leqslant
2E_{0}.$

(See also [6, Proposition 1.9] for a related result.) ∎

Using this lemma, we can derive a scaling relation for the averaged
longitudinal structure function $S^{3}_{\|}$, where

(4.21)
$S_{\|}^{p}(\tau,r)=\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}\big{(}(u(x+rn)-u(x))\cdot
n\big{)}^{p}\,dxdS(n)\,d\mu_{t}(u)\,dt.$

######  Lemma 4.3.

Let $\mu_{t}$ be a Foiaş–Prodi statistical solution of the Navier–Stokes
equations (cf. Definition 3.6). Then

(4.22) $\left|\frac{S^{3}_{\|}(\tau,r)}{r}\right|\leqslant 2E_{0},$

where $C>0$ is some constant independent of $\varepsilon$ and $E_{0}$ is the
initial energy,

(4.23)
$E_{0}\coloneqq\int_{L^{2}_{\operatorname{div}}(D;U)}\left\|u(x)\right\|^{2}_{L^{2}(D)}\,d\mu_{0}(u)$

###### Proof.

Again, we start with the KHM relation (4.2). This time we use the test
function $\sigma(h)=\omega(|h|)\hat{h}\otimes\hat{h}$ where $\omega\in
C_{c}^{\infty}(\mathbb{R})$ is an even function. We have

$\begin{split}\partial_{h^{k}}(\omega(|h|)\hat{h}^{i}\hat{h}^{j})&=\omega^{\prime}(|h|)\hat{h}^{i}\hat{h}^{j}\hat{h}^{k}+\frac{1}{|h|}\left(\delta_{ik}\hat{h}^{j}+\delta_{jk}\hat{h}^{i}-2\hat{h}^{i}\hat{h}^{j}\hat{h}^{k}\right)\omega(|h|)\\\
&=\left(\omega^{\prime}(|h|)-2\frac{\omega(|h|)}{|h|}\right)\hat{h}^{i}\hat{h}^{j}\hat{h}^{k}+\frac{1}{|h|}\left(\delta_{ik}\hat{h}^{j}+\delta_{jk}\hat{h}^{i}\right)\omega(|h|)\end{split}$

Therefore, (4.2) becomes

(4.24)
$\begin{split}&-\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}\cdot\hat{h}\,\xi_{2}\cdot\hat{h}\,d\nu_{\tau,x,x+h}^{2}(\xi)\,dx\omega(|h|)\,dh\\\
&+\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!\xi_{1}\cdot\hat{h}\,\xi_{2}\cdot\hat{h}\,\,d\nu_{0,x,x+h}^{2}(\xi)\,dx\omega(|h|)\,dh\\\
&-\frac{1}{2}\int_{0}^{\tau}\\!\\!\int_{D}\int_{D}\\!\int_{U^{2}}\big{(}(\xi_{2}-\xi_{1})\cdot\hat{h}\big{)}^{3}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\left(\omega^{\prime}(|h|)-2|h|^{-1}\omega(|h|)\right)\,dh\,dt\\\
&+\int_{0}^{\tau}\\!\\!\int_{D}\int_{D}\\!\int_{U^{2}}|\xi_{1}-\xi_{2}|^{2}(\xi_{1}-\xi_{2})\cdot\hat{h}\,d\nu_{t,x,x+h}^{2}(\xi)\,dx|h|^{-1}\omega(|h|)\,dh\,dt\\\
=&\
\varepsilon\sum_{ij}\int_{0}^{\tau}\\!\\!\int_{D}\int_{D}\\!\int_{U^{2}}\\!\\!(\xi_{1}^{i}-\xi_{2}^{i})(\xi_{1}^{j}-\xi_{2}^{j})\,d\nu_{t,x,x+h}^{2}(\xi)\,dx\Delta_{h}\sigma^{ij}(h)\,dh\,dt.\end{split}$

Again, in terms of $(\mu_{t})_{t>0}$, this means

(4.25)
$\begin{split}&\quad-\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!u(x)\cdot\hat{h}\,u(x+h)\cdot\hat{h}\,\,dx\omega(|h|)\,dh\,d\mu_{\tau}(u)\\\
&\quad+\int_{L^{2}_{\operatorname{div}}}\int_{D}\int_{D}\\!u(x)\cdot\hat{h}\,u(x+h)\cdot\hat{h}\,dx\omega(|h|)\,dh\,d\mu_{0}(u)\\\
&\quad-\frac{1}{2}\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}\big{(}(u(x+h)-u(x))\cdot\hat{h}\big{)}^{3}\,dx\,d\mu_{t}(u)\left(\omega^{\prime}(|h|)-2|h|^{-1}\omega(|h|)\right)\,dh\,dt\\\
&\quad+\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}|u(x)-u(x+h)|^{2}\big{(}u(x)-u(x+h)\big{)}\cdot\hat{h}\,dx\,d\mu_{t}(u)|h|^{-1}\omega(|h|)\,dh\,dt\\\
&=\varepsilon\sum_{ij}\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}\\!\big{(}u^{i}(x)-u^{i}(x+h)\big{)}\big{(}u^{j}(x)-u^{j}(x+h)\big{)}\,dx\,d\mu_{t}(u)\Delta_{h}\sigma^{ij}(h)\,dh\,dt.\end{split}$

Similar to the computation in (4.19), we have

$\displaystyle\quad\varepsilon\sum_{ij}\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}\\!\big{(}u^{i}(x)-u^{i}(x+h)\big{)}\big{(}u^{j}(x)-u^{j}(x+h)\big{)}\,dx\,d\mu_{t}(u)\Delta_{h}\sigma^{ij}(h)\,dh\,dt$
$\displaystyle=2\varepsilon\sum_{ij}\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}\\!\nabla_{x}u^{i}(x+h)\nabla_{x}u^{j}(x)\,dx\,d\mu_{t}(u)\sigma^{ij}(h)\,dh\,dt$
$\displaystyle=2\varepsilon\int_{0}^{\tau}\int_{D}\int_{L^{2}_{\operatorname{div}}}\\!\\!\int_{D}\\!\big{(}\nabla_{x}u(x+h)\cdot\hat{h}\big{)}\cdot\big{(}\nabla_{x}u(x)\cdot\hat{h}\big{)}\,dx\,d\mu_{t}(u)\omega(|h|)\,dh\,dt,$

so equation (4.25) becomes (after switching to polar coordinates)

(4.26)
$\begin{split}&\quad-\int_{0}^{\infty}\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
n\,u(x+rn)\cdot n\,\,dx\,dS(n)\,d\mu_{\tau}(u)r^{2}\omega(r)\,dr\\\
&\quad+\int_{0}^{\infty}\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
n\,u(x+rn)\cdot n\,dx\,dS(n)\,d\mu_{0}(u)r^{2}\omega(r)\,dr\\\
&\quad-\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}\big{(}\big{(}u(x+rn)-u(x)\big{)}\cdot
n\big{)}^{3}\,dx\,dS(n)\,d\mu_{t}(u)\,dt\left(r^{2}\omega^{\prime}(r)-2r\omega(r)\right)\,dr\\\
&\quad+\int_{0}^{\infty}\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}|u(x)-u(x+rn)|^{2}\big{(}u(x)-u(x+rn)\big{)}\cdot
n\,dx\,dS(n)\,d\mu_{t}(u)\,dtr\omega(r)\,dr\\\
&=2\varepsilon\int_{0}^{\infty}\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}\\!\big{(}\nabla_{x}u(x+rn)\cdot
n\big{)}\cdot\big{(}\nabla_{x}u(x)\cdot
n\big{)}\,dx\,dS(n)\,d\mu_{t}(u)\,dtr^{2}\omega(r)\,dr\end{split}$

Denote

$\displaystyle\widetilde{m}_{2}(\tau,r)$
$\displaystyle\coloneqq\int_{L^{2}_{\operatorname{div}}}\fint_{\mathbb{S}^{2}}\int_{D}\\!u(x)\cdot
n\,u(x+rn)\cdot n\,\,dx\,dS(n)\,d\mu_{\tau}(u)$
$\displaystyle\widetilde{v}(\tau,r)$
$\displaystyle\coloneqq\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}\\!\big{(}\nabla_{x}u(x+rn)\cdot
n\big{)}\cdot\big{(}\nabla_{x}u(x)\cdot n\big{)}\,dx\,dS(n)\,d\mu_{t}(u)\,dt.$

Writing
$\omega^{\prime}(r)-2r^{-1}\omega(r)=r^{2}\frac{d}{dr}\big{(}\omega(r)r^{-2}\big{)}$,
(4.26) becomes

$\displaystyle\quad\frac{1}{2}\int_{0}^{\infty}r^{4}S_{\|}^{3}(\tau,r)\partial_{r}\big{(}r^{-2}\omega(r)\big{)}\,dr+\int_{0}^{\infty}S_{0}^{3}(\tau,r)r\omega(r)\,dr$
$\displaystyle=-\int_{0}^{\infty}\widetilde{m}_{2}(\tau,r)r^{2}\omega(r)\,dr+\int_{0}^{\infty}\widetilde{m}_{2}(0,r)r^{2}\omega(r)\,dr-2\varepsilon\int_{0}^{\infty}\widetilde{v}(\tau,r)r^{2}\omega(r)\,dr.$

Again, we note that due to the estimates from the energy inequality (3.14),
$S^{3}_{\|}$, $\widetilde{m}$ and $\widetilde{v}$ are continuous and bounded
quantities in $\tau$ and $r$. And so we can consider this ODE in the sense of
distributions as an ODE in the strong sense,

$\partial_{r}\big{(}r^{4}S_{\|}^{3}(\tau,r)\big{)}=2r^{4}\left(\frac{S_{0}^{3}(\tau,r)}{r}+\widetilde{m}_{2}(\tau,r)-\widetilde{m}_{2}(0,r)+2\varepsilon\widetilde{v}(\tau,r)\right),$

or

(4.27)
$S_{\|}^{3}(\tau,r)=2\int_{0}^{r}\frac{s^{4}}{r^{4}}\left(\frac{S_{0}^{3}(\tau,s)}{s}+\widetilde{m}_{2}(\tau,s)-\widetilde{m}_{2}(0,s)+2\varepsilon\widetilde{v}(\tau,s)\right)\,ds.$

By the energy bound and Cauchy–Schwarz inequality, $\widetilde{m}_{2}$ and
$\varepsilon\widetilde{v}$ are uniformly bounded in $\varepsilon$ for all
$s,\tau>0$. Moreover from Lemma 4.2, we have that $s^{-1}S_{0}^{3}(s,\tau)$ is
uniformly bounded in $\varepsilon$ by $2E_{0}$. Hence,

(4.28) $\left|\frac{S_{\|}^{3}(\tau,r)}{r}\right|\leqslant 2E_{0}$

for some $C>0$ independent of $\varepsilon$. ∎

######  Remark 4.4.

Combining (4.20) and (4.28), we also obtain a uniform bound on
$r^{-1}S^{3}_{\perp}(\tau,r)$, where $S^{3}_{\perp}$ is the transversal
structure function

(4.29)
$S^{3}_{\perp}(\tau,r)\coloneqq\int_{0}^{\tau}\int_{L^{2}_{\operatorname{div}}}\\!\\!\fint_{\mathbb{S}^{2}}\int_{D}|\delta^{\perp}_{rn}u(x)|^{2}\delta_{rn}u(x)\cdot
n\,dx\,dS(n)\,d\mu_{t}(u)\,dt=S^{3}_{0}(\tau,r)-S^{3}_{\|}(\tau,r),$

where

(4.30)
$\delta_{h}u(x)=u(x+h)-u(x),\quad\delta^{\perp}_{h}u=(\mathbf{I}-\hat{h}\otimes\hat{h})\delta_{h}u,\quad\hat{h}\coloneqq\frac{h}{|h|}.$

None of these quantities has a sign and therefore the previously derived
bounds do not imply compactness without further assumptions.

###  4.3. Scaling assumption

In order to pass to the limit $\varepsilon\to 0$, we need to an additional
assumption about the behavior of structure functions. Specifically, we need

######  Assumption 1 (Weak statistical scaling).

For any $\varepsilon>0$, let $\mu^{\varepsilon}$ be a Foiaş–Prodi statistical
solution of the incompressible Navier–Stokes equations. We assume that for
$r\ll 1$, the second and third order longitudinal structure functions (4.21)
are related by

(4.31) $\left|S^{2}_{\|}(\tau,r)\right|\leqslant
C\left|S^{3}_{\|}(\tau,r)\right|^{\alpha},$

where $C$ is a constant independent of $\varepsilon$ and $\alpha>0$.

######  Remark 4.5 (Weak statistical scaling).

Assumption 1 is inspired by the following stronger scaling assumption often
encountered in turbulence theory: For any $p,q$ with $q\geqslant p\geqslant
1$, the $p$-th and $q$-th order longitudinal structure functions (4.21) are
related by

(4.32) $\left|S^{p}_{\|}(\tau,r)\right|\leqslant
C\left|S^{q}_{\|}(\tau,r)\right|^{\frac{\lambda(p)}{\lambda(q)}},$

where $C$ is a constant independent of $\varepsilon$ and $\lambda(p)>0$ for
$p\leqslant p_{0}$ where $3\leqslant p_{0}\in\mathbb{R}\cup\\{\infty\\}$. In
Kolmogorov’s 1941 (“K41”) theory [41, 40, 42], $\lambda(p)=p$. However, this
cannot be confirmed with physical experiments [4, 53]. Various physicists
therefore suggested intermittency corrections to account for the deviation
from Kolmogorov’s original theory, among others, Kolmogorov himself in 1962
[43] in his refined theory of turbulence, Frisch et al. the $\beta$-model
[32], as well as Novikov and Stewart [50]. Assumption (4.32) can also
accommodate the frequently used model by She and Leveque [54] who suggested

(4.33)
$\lambda(p)=\frac{p}{9}+2\left(1-\left(\frac{2}{3}\right)^{p/3}\right).$

######  Remark 4.6.

Combining the bound on the third order structure function in Lemma 4.3 with
Assumption 1, we obtain

$\left|S^{2}_{\|}(\tau,r)\right|\leqslant Cr^{\alpha}.$

We will combine Assumption 1 with the following lemma, whose proof is given in
Appendix C:

######  Lemma 4.7 (Weak anisotropy).

Let $\mu_{t}$ be a statistical solution of the Navier–Stokes equation. Then
$\mu$ satisfies

(4.34)
$\begin{split}&3\int_{0}^{T}\int_{D}\int_{L^{2}_{\operatorname{div}}(D;U)}\fint_{\partial
B_{r}(0)}(\delta_{rn}u\cdot n)^{2}dS(n)\,dx\,d\mu_{t}(u)\,dt\\\
&=\int_{0}^{T}\int_{D}\int_{L^{2}_{\operatorname{div}}(D;U)}\fint_{B_{r}(0)}|\delta_{\ell}u(x)|^{2}d\ell\,dx\,d\mu_{t}(u)\,dt.\end{split}$

Under Assumption 1, we obtain

(4.35)
$\int_{0}^{T}\int_{D}\int_{L^{2}_{\operatorname{div}}(D;U)}\fint_{B_{r}(0)}|\delta_{\ell}u(x)|^{2}d\ell\,dx\,d\mu_{t}(u)\,dt\leqslant
Cr^{\alpha}.$

Using the equivalence theorem 2.2, we can write this as

(4.36)
$\int_{0}^{T}\int_{D}\fint_{B_{r}(0)}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{2}\,d\nu_{x,x+y}^{2}(\xi)\,dy\,dx\,dt\leqslant
Cr^{\alpha}.$

and since this is uniform with respect to the viscosity coefficient
$\varepsilon$ (by the weak scaling assumption), it implies uniform diagonal
continuity of the sequence $\\{\bm{\nu}\\}_{\varepsilon>0}$.

###  4.4. Passage to the limit $\varepsilon\to 0$

Now we are in a position to prove our main result. We will keep track of the
superscript $\varepsilon$ again in order to distinguish between the
approximating sequence $\\{\bm{\nu}^{\varepsilon}\\}_{\varepsilon>0}$ and the
limiting measure $\bm{\nu}$ for $\varepsilon=0$.

######  Theorem 4.8.

Let $\\{\bm{\nu}^{\varepsilon}\\}_{\varepsilon>0}$ be a sequence of (either
Foiaş–Temam or Friedman–Keller) statistical solutions to the Navier–Stokes
equations with initial data $\mu_{0}$ with bounded support (cf. (3.1)). Assume
that $\bm{\nu}^{\varepsilon}$ all satisfy Assumption 1. Then, as
$\varepsilon\to 0$, $\bm{\nu}^{\varepsilon}$ converges (along a subsequence)
to a correlation measure $\bm{\nu}$ on $L^{2}$ with bounded support (cf.
(3.17))

(4.37)
$\int_{D^{k}}\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{t,x}^{k}(\xi)\,dx\leqslant
R^{k}<\infty,$

for some $0<R<\infty$, any $k\in\mathbb{N}$ and that satisfies the “inviscid
Friedman–Keller system”:

(4.38)
$\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\frac{\partial\varphi}{\partial
t}(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt+\int_{D^{k}}\\!\\!\int_{U^{k}}\\!\\!(\xi_{1}\otimes\cdots\otimes\xi_{k}):\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx\\\
+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\\!\\!\int_{U^{k}}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt=0$

for all $k\in\mathbb{N}$, for all $\varphi\in C^{2}_{c}([0,T]\times
D^{k};U^{k})$ with $\operatorname{div}_{x_{i}}\varphi(x)=0$, a.e. $x\in D^{k}$
for all $i=1,\dots,k$ and (corresponding to the divergence constraint)

(4.39)
$\int_{D^{k}}\int_{U^{k}}\xi_{1}\otimes\dots\otimes\xi_{\ell}\otimes\alpha_{\ell+1}(\xi_{\ell+1})\otimes\dots\otimes\alpha_{k}(\xi_{k})\,d\nu_{t,x}^{k}(\xi)\cdot\nabla_{x_{1},\dots,x_{\ell}}\psi(x)\,dx=0,$

where
$\nabla_{x_{1},\dots,x_{\ell}}=(\nabla_{x_{1}},\dots,\nabla_{x_{\ell}})^{\top}$,
$1\leqslant\ell\leqslant k\in\mathbb{N}$, for all $\psi\in
H^{1}(D^{k};U^{k-\ell})$, $\alpha_{j}\in C(U;U)$, $\alpha_{j}(v)\leqslant
C(1+|v|^{2})$ and $j=1,\dots,k$.

###### Proof.

From the condition on the initially bounded support (3.1) and the energy
inequality (3.4), we obtain that the sequence $\bm{\nu}^{\varepsilon}$
satisfies (2.9) for $p=2$ uniformly in $\varepsilon>0$. The reasoning of
Subsection 4.1, 4.2 and 4.3 resulting in (4.36) imply that
$\bm{\nu}^{\varepsilon}$ is uniformly diagonal continuous as in (2.10). Hence,
using Theorem 2.3, we obtain, up to subsequence, the existence of a limiting
correlation measure $\bm{\nu}\in\mathcal{L}^{2}([0,T),D;U)$. So it remains to
check whether $\bm{\nu}$ satisfies the equations (4.38) and (4.39). We note
that the functions

(4.40)
$\begin{split}g_{1}(t,x,\xi)&\coloneqq(\xi_{1}\otimes\cdots\otimes\xi_{k}):\frac{\partial\varphi}{\partial
t}(t,x),\\\
g_{2}(t,x,\xi)&\coloneqq\sum_{i=1}^{k}(\xi_{1}\otimes\cdots\otimes\xi_{k}):\Delta_{x_{i}}\varphi(t,x),\\\
g_{3}(t,x,\xi)&\coloneqq\sum_{i=1}^{k}(\xi_{1}\otimes\cdots\otimes(\xi_{i}\otimes\xi_{i})\otimes\cdots\xi_{k}):\nabla_{x_{i}}\varphi(t,x)\end{split}$

for $\varphi\in C^{2}_{c}([0,T]\times D^{k};U^{k})$, $1\leqslant\ell\leqslant
k\in\mathbb{N}$, are all functions in $\mathcal{H}^{k,p}_{1}([0,T],D;U)$.
Hence, we can pass to the limit in all the terms in the Friedman–Keller system
(3.2). The term that is multiplied by $\varepsilon$ vanishes because it is a
uniformly bounded in $\varepsilon>0$ quantity that is multiplied by
$\varepsilon$. For the divergence constraint (4.39), we note that the function

$g_{4}(t,x,\xi)\coloneqq\theta(t)\xi_{1}\otimes\dots\otimes\xi_{\ell}\otimes\alpha_{\ell+1}(\xi_{\ell+1})\otimes\dots\otimes\alpha_{k}(\xi_{k})\cdot\nabla_{x_{1},\dots,x_{\ell}}\psi(x),$

lies in $\mathcal{H}^{k,p}_{1}([0,T],D;U)$ for any $\theta\in
C^{\infty}_{c}((0,T))$, and $\psi\in H^{1}(D^{k};U^{k-\ell})$, and
$\alpha_{j}\in C(U;U)$ with $|\alpha_{j}(v)|\leqslant C(1+|v|^{2})$. Passing
$\varepsilon\to 0$ in $\langle\nu^{\varepsilon,k},\,g_{4}\rangle$ and using
that $\nu^{\varepsilon,k}$ satisfy the divergence constraint (3.3), we can
conclude, by the arbitrariness of $\theta$, that (4.39) holds for a.e.
$t\in[0,T]$. ∎

######  Remark 4.9.

By the equivalence theorem 2.2, we know that the limiting correlation measure
$\bm{\nu}$ corresponds to a parametrized measure
$\mu=(\mu_{t})_{t>0}:[0,T)\to\mathscr{P}(L^{2}_{\operatorname{div}}(D;U))$
that satisfies

(4.41)
$\int_{L^{2}(D)}\Phi(u)\,d\mu_{t}(u)=\int_{L^{2}(D)}\Phi(u)\,d\mu_{0}(u)+\int_{0}^{t}\int_{L^{2}(D)}\int_{D}(u(x)\otimes
u(x)):\nabla_{x}\Phi^{\prime}(u)(x)\,dx\,d\mu_{s}(u)\,ds$

for all cylindrical test functions $\Phi\in\mathscr{T}_{\text{cyl}}^{0}$ that
satisfy $g_{j}\in C^{2}(D)$, and the energy inequality

(4.42)
$\int_{L^{2}(D)}\|u\|_{L^{2}(D)}^{2}\,d\mu_{t}(u)\leqslant\int_{L^{2}(D)}\|u\|_{L^{2}(D)}^{2}\,d\mu_{0}(u),\quad\text{for
all }t\in[0,T].$

The proof of this fact follows along the lines of the proof of Theorem 3.9
while ignoring the terms involving $\varepsilon$ and not attempting to recover
$B(u)$ as it may be unbounded.

##  5\. Discussion

It is well-known that many incompressible fluid flows of interest are
characterized by very-high Reynolds number. Hence, a precise characterization
of the vanishing viscosity ($\varepsilon\to 0$) limit of the Navier-Stokes
equations (1.1) is of great interest. Formally, one would expect that the
vanishing viscosity limit of Navier-Stokes equations is related to the
incompressible Euler equations. However as mentioned in the introduction,
rigorous results in this direction are only available in two space dimensions,
even in the case of periodic boundary conditions. The key aim of this article
was to investigate the vanishing viscosity limit of the Navier-Stokes
equations, including in three space dimensions.

It is well known that fluid flows at high Reynolds numbers are characterized
by _turbulence_ , loosely speaking, marked by the presence of energy
containing eddies at smaller and smaller scales. This phenomenon is clearly
linked to the lack of compactness in the Leray-Hopf Navier-Stokes solutions as
well as their possible instabilities/non-uniqueness.

Hence, one needs to make further assumptions on the Leray-Hopf solutions that
can yield additional information and facilitate passage to the limit. One
avenue for making such assumptions, which are realistic and possibly observed
in experiments, comes from physical theories of turbulence. In particular,
Kolmogorov’s well-known K41 theory is based on several verifiable assumptions
on the underlying fluid flow and results in a precise characterization of
quantities such as structure functions and _energy spectra_.

In [12], Chen and Glimm relate the K41 energy spectra to compactness results
on the Leray-Hopf solutions, in appropriate Sobolev and Hölder spaces.
Consequently, under the assumption of the K41 energy spectrum, the authors
prove that the underlying Leray-Hopf solutions converge to _weak solutions_ of
the incompressible Euler equations as $\varepsilon\to 0$. However,
Kolmogorov’s derivation of the decay of energy spectra is based on a
_probabilistic charectization_ of the underlying fluid flow. In particular,
assumptions such as (statistical) homogeneity, isotropy and scaling, which
form the foundation of Kolmogorov’s theory, are too stringent if imposed at
the deterministic level, as done in [12]. Moreover, it is now well-established
that the strong scaling assumptions of Kolmogorov might not hold in real fluid
flows and _intermittent_ corrections are necessary. Hence, the applicability
of the assumptions and results of [12] can be questioned from this
perspective.

Nevertheless, the connection with Kolmogorov’s theories of turbulence and
their variants forms the basis of our work. We start with the realization that
a probabilistic description of the solutions of Navier-Stokes equations is
necessary to relate physical theories of turbulence to rigorous mathematical
statements. To this end, we focus on _statistical solutions_ of Navier-Stokes
equations. Two possible frameworks of such statistical solutions are
available, namely the _Foiaş-Prodi_ statistical solutions (see Definition 3.6)
and the _Friedman-Keller_ statistical solutions (see Definition 3.1), which is
based on the concept of correlation measures of [23]. We prove that both these
solution concepts are equivalent as long as a statistical version of the
energy inequality holds. This also allows us to prove the existence of
Friedmann-Keller statistical solutions of the incompressible Navier-Stokes
equations.

Then, we investigated the vanishing viscosity limit of the statistical
solutions of the incompressible Navier-Stokes equations. To this end, we
derived a suitable statistical version of the well-known Kármán-Howarth-Monin
relation and used it to prove precise rates for the asymptotic decay of a
averaged third-order structure function in Lemma 4.2. However, these estimates
do not suffice to pass to the $\varepsilon\to 0$ limit. To this end, we
assumed a _weak statistical scaling_ of the Navier-Stokes statistical
solutions (see Assumption 1). This assumption is a weaker version of
Kolmogorov’s scaling assumptions in his K41 theory. Moreover, it is consistent
with and incorporates different variants of scaling that are proposed in the
physics literature to explain intermittent corrections to Kolmogorov’s theory.
Under this assumption, we proved a weak anisotropy result and invoked
compactness results of [24] to rigorously prove that the statistical solutions
of the Navier-Stokes equations converge, in a suitable sense, to a statistical
solution of the incompressible Euler equations. Thus, we were able to
characterize the vanishing viscosity limit of the Navier-Stokes equations in a
relevant regime.

At this juncture, it is essential to point that that no assumption, other than
weak statistical scaling, is made in our results and all other estimates are
derived rigorously. This should be contrasted with the results of [12] where
the authors directly assume a decay of the energy spectrum for the weak
solutions of the Navier-Stokes equations. It is currently unclear if one can
relax the weak statistical scaling assumption or even if it holds for all
incompressible fluid flows. Experimental evidence strongly supports that this
assumption is verified in practice, see e.g., [4, 54, 53].

To the best of our knowledge, the only rigorous study of the vanishing
viscosity limit of the (Foiaş-Prodi) statistical solutions was carried out by
Chae in [11] where he proved that these statistical solutions converge to a
measure-valued solution of the incompressible Euler equations. In contrast, we
prove convergence to statistical solutions of the incompressible Euler
equations and recall that statistical solutions are much more informative than
measure-valued solutions as they also incorporate knowledge of all multi-point
correlations.

Finally, our characterization of the vanishing viscosity limit can be viewed
in connection to recent results in [45] where the authors proved convergence
of numerical spectral viscosity approximations to the statistical solutions of
the Euler equations under very similar weak scaling assumptions.

##  Appendix A Equivalence of different definitions of statistical solutions
for the incompressible Navier–Stokes equations

This appendix is devoted to the proof of Theorems 3.8 and 3.9. For
convenience, we restate the result:

######  Theorem A.1 (Foiaş–Prodi statistical solutions are Friedman–Keller
solutions).

Let $\mu$ be a Foiaş–Prodi statistical solution such that the initial
condition $\mu_{0}$ has bounded support, $\text{supp}(\mu_{0})\subset B\subset
L^{2}_{\operatorname{div}}(D;U)$, $B=\\{u\in
L^{2}(D;U)\,:\,\|u\|_{L^{2}(D)}\leqslant R\\}$ for some $0<R\in\mathbb{R}$
large enough. Then $\mu$ is a Friedmann–Keller statistical solution (cf.
Definition 3.1).

###### Proof.

It is shown in [26, Theorem 2, Section 3] that Foiaş–Prodi statistical
solutions with initial measure $\mu_{0}$ having bounded support in
$B_{L^{2}_{\operatorname{div}}(D;U)}(R)\coloneqq\\{u\in
L^{2}_{\operatorname{div}}(D;U):\|u\|_{L^{2}(D)}\leqslant R\\}$ have bounded
support for all times, i.e., $\text{supp}(\mu_{t})\subset
B_{L^{2}_{\operatorname{div}}(D;U)}(R)$. Therefore, we can assume that
$\mu_{t}$ has uniformly bounded support. This implies in particular that
$\mu_{t}$ have bounded moments:

(A.1)
$\int_{L^{2}_{\operatorname{div}}(D;U)}\|u\|_{L^{2}(D)}^{2k}\,d\mu_{t}(u)\leqslant
R^{2k}\quad\text{for a.e. }t\in[0,T].$

Moreover, by [26, Lemma 5, Section 3], we have that statistical solutions of
Navier–Stokes satisfy

(A.2)
$\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\left[-\partial_{t}\Phi(t,u)+a(u,\partial_{u}\Phi(t,u))+b(u,u,\partial_{u}\Phi(t,u))\right]\,d\mu_{t}(u)\,dt\\\
=\int_{L^{2}_{\operatorname{div}}(D;U)}\Phi(0,u)\,d\mu_{0}(u)$

for any test function $\Phi(t,u)$ that is Fréchet differentiable on
$[0,T]\times L^{2}_{\operatorname{div}}(D;U)$ with $\Phi(t,\cdot)=0$ near
$t=T$ and $|\partial_{u}\Phi(t,u)|\leqslant C$ and
$|\partial_{t}\Phi(t,u)|\leqslant C_{1}+C_{2}\|u\|_{L^{2}(D)}$ for all $u$ and
$t$ and some constants $C,C_{1},C_{2}$ (equation ($3.13_{\mathrm{I}}$) and
condition (3.8) in [26]). Therefore, we can choose test functions

$\Phi(u)=q((u,\varphi_{1}),\dots,(u,\varphi_{k}))\theta(t)$

for $q$ a polynomial on $\mathbb{R}^{k}$ and $\varphi_{j}\in
H^{1}_{\operatorname{div}}(D;U)\cap C^{2}_{c}(D,U)$ and $\theta\in
C^{1}_{c}([0,T))$, $j=1,\dots,k$ in (A.2), so that we get

$\int_{0}^{T}\\!\\!\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!-\theta^{\prime}(t)q((u,\varphi_{1}),\dots,(u,\varphi_{k}))d\mu_{t}(u)dt\\\
+\int_{0}^{T}\\!\\!\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!\Big{[}\theta(t)\sum_{i=1}^{k}\partial_{i}q((u,\varphi_{1}),\dots,(u,\varphi_{k}))\left[a(u,\varphi_{i})+b(u,u,\varphi_{i})\right]\Big{]}\,d\mu_{t}(u)\,dt\\\
=\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!\\!\\!\\!\theta(0)q((u,\varphi_{1}),\dots,(u,\varphi_{k}))\,d\mu_{0}(u),$

Note that we can integrate by parts in the terms involving $a(u,\varphi_{i})$
and $b(u,u,\varphi_{i})$, $i=1,\dots,k$, so that all the derivatives are on
the test functions $\varphi_{i}$, $i=1,\dots,k$ and $\theta$:

$\displaystyle 0=$
$\displaystyle\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!\theta^{\prime}(t)q((u,\varphi_{1}),\dots,(u,\varphi_{k}))\,d\mu_{t}(u)\,dt+\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!\theta(0)q((u,\varphi_{1}),\dots,(u,\varphi_{k}))\,d\mu_{0}(u)$
$\displaystyle\quad+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!\theta(t)\partial_{i}q((u,\varphi_{1}),\dots,(u,\varphi_{k}))$
$\displaystyle\hphantom{\quad+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\\!\\!}\times\int_{D}\left[\varepsilon
u(x_{i})\Delta_{x_{i}}\varphi_{i}(x_{i})+(u(x_{i})\otimes
u(x_{i})):\nabla_{x_{i}}\varphi_{i}(x_{i})\right]\,dx_{i}\,d\mu_{t}(u)\,dt$

Now take $q(s_{1},\dots,s_{k})=s_{1}\cdots s_{k}$, so that the last identity
becomes (denote $\,dx=\,dx_{1}\dots\,dx_{k}$)

$0=\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}\theta^{\prime}(t)u(x_{1})\cdot\varphi_{1}(x_{1})\cdots
u(x_{k})\cdot\varphi_{k}(x_{k})\,dx\,d\mu_{t}(u)\,dt\\\
+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}\theta(t)\big{(}u(x_{1})\cdot\varphi_{1}(x_{1})\cdots\varepsilon
u(x_{i})\cdot\Delta_{x_{i}}\varphi_{i}(x_{i})\cdots
u(x_{k})\cdot\varphi_{k}(x_{k})\big{)}\,dx\,d\mu_{t}(u)\,dt\\\
+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}\theta(t)\big{(}u(x_{1})\cdot\varphi_{1}(x_{1})\cdots(u(x_{i})\otimes
u(x_{i})):\nabla_{x_{i}}\varphi_{i}(x_{i})\cdots
u(x_{k})\cdot\varphi_{k}(x_{k})\big{)}\,dx\,d\mu_{t}(u)\,dt\\\
+\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}\theta(0)u(x_{1})\cdot\varphi_{1}(x_{1})\cdots
u(x_{k})\cdot\varphi_{k}(x_{k})\,dx\,d\mu_{0}(u),$

which is, denoting $x=(x_{1},\dots,x_{k})$ and
$\varphi(t,x)\coloneqq\theta(t)\varphi_{1}(x_{1})\otimes\dots\otimes\varphi_{k}(x_{k})$,
equivalent to

$\displaystyle 0=$ $\displaystyle\
\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}(u(x_{1})\otimes\dots\otimes
u(x_{k})):\partial_{t}\varphi(t,x)\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle+\varepsilon\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}(u(x_{1})\otimes\dots\otimes
u(x_{k})):\Delta_{x_{i}}\varphi(t,x)\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}(u(x_{1})\otimes\dots\otimes(u(x_{i})\otimes
u(x_{i}))\otimes\dots\otimes
u(x_{k})):\nabla_{x_{i}}\varphi(t,x)\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle+\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D^{k}}(u(x_{1})\otimes\dots\otimes
u(x_{k})):\varphi(0,x)\,dx\,d\mu_{0}(u),$

Since we assume that the support of $\mu_{0}$ is bounded (and therefore also
the support of $\mu_{t}$ for almost all $t\in[0,T]$), we can use a density
argument to conclude that the above identity holds for all $\varphi\in
L^{2}([0,T];(H^{1}_{\operatorname{div}}(D;U))^{k})\cap C^{2}_{c}([0,T]\times
D^{k};U^{k})$, that is, all $\varphi\in C^{2}_{c}([0,T]\times D^{k};U^{k})$
with $\operatorname{div}_{x_{i}}\varphi=0$ for all $i=1,\dots,k$. Observe next
that all the terms in the above identity have the form required to apply
Theorem 2.2 (recall that $\mu_{t}$ has bounded support). Therefore, by the
same theorem, there exists a unique correlation measure
$\bm{\nu}=(\bm{\nu}_{t})_{0\leqslant t\leqslant
T}=\big{(}(\nu^{1}_{t},\nu^{2}_{t},\dots)\big{)}_{0\leqslant t\leqslant T}$
corresponding to $\\{\mu_{t}\\}$ satisfying

$\displaystyle 0=$ $\displaystyle\
\int_{0}^{T}\\!\\!\int_{D^{k}}\int_{U^{k}}(\xi_{1}\otimes\dots\otimes\xi_{k}):\partial_{t}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$
$\displaystyle+\varepsilon\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\int_{U^{k}}(\xi_{1}\otimes\dots\otimes\xi_{k}):\Delta_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$
$\displaystyle+\sum_{i=1}^{k}\int_{0}^{T}\\!\\!\int_{D^{k}}\int_{U^{k}}(\xi_{1}\otimes\dots\otimes(\xi_{i}\otimes\xi_{i})\otimes\dots\otimes\xi_{k}):\nabla_{x_{i}}\varphi(t,x)\,d\nu_{t,x}^{k}(\xi)\,dx\,dt$
$\displaystyle+\int_{D^{k}}\int_{U^{k}}(\xi_{1}\otimes\dots\otimes\xi_{k}):\varphi(0,x)\,d\nu_{0,x}^{k}(\xi)\,dx,$

which is (3.2). By previous arguments, each measure $\mu_{t}$ has bounded
support (and in particular bounded moments), so we can take any nonnegative,
nondecreasing polynomial $\psi(s)=p_{K}(s)$, of degree $K$, for
$K\in\mathbb{N}$ in the strengthened energy inequality (3.14), so that we get

(A.3)
$\int_{L^{2}(D)}p_{K}(\|u\|_{L^{2}(D)}^{2})\,d\mu_{t}(u)+2\varepsilon\int_{0}^{t}\int_{L^{2}(D)}p_{K}^{\prime}(\|u\|_{L^{2}(D)}^{2})|u|_{H^{1}(D)}^{2}\,d\mu_{s}(u)\,ds\\\
\leqslant\int_{L^{2}(D)}p_{K}(\|u\|_{L^{2}(D)}^{2})\,d\mu_{0}(u)\qquad\text{for
a.e. }t\in[0,T].$

Thanks to the bounded support of $\mu_{t}$, all the moment terms have the form
required to apply Theorem 2.2, moreover, with Lemma B.3, we can rewrite the
gradient term such that we obtain for a.e. $t\in[0,T]$, the energy inequality
(3.4). Indeed, we have with Theorem 2.2

$\displaystyle\int_{L^{2}(D)}p_{K}(\|u\|_{L^{2}(D)}^{2})\,d\mu_{t}(u)$
$\displaystyle=\sum_{k=0}^{K}a_{k}\int_{L^{2}(D)}\|u\|_{L^{2}(D)}^{2k}\,d\mu_{t}(u)$
$\displaystyle=\sum_{k=0}^{K}a_{k}\int_{L^{2}(D)}\int_{D^{k}}|u(x_{1})|^{2}\dots|u(x_{k})|^{2}dx\,d\mu_{t}(u)$
$\displaystyle=\sum_{k=0}^{K}a_{k}\int_{D^{k}}\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}d\nu_{t,x}^{k}(\xi)dx.$

Similarly, with Lemma B.3, (via Lemma B.1)

$\displaystyle\int_{0}^{t}\int_{L^{2}(D)}p_{K}^{\prime}\big{(}\|u\|_{L^{2}(D)}^{2}\big{)}|u|_{H^{1}(D)}^{2}\,d\mu_{s}(u)\,ds$
$\displaystyle\qquad=\sum_{k=0}^{K}a_{k}k\int_{0}^{t}\int_{L^{2}(D)}\|u\|_{L^{2}(D)}^{2(k-1)}|u|_{H^{1}(D)}^{2}\,d\mu_{s}(u)\,ds$
$\displaystyle\qquad=\sum_{k=0}^{K}a_{k}k\int_{0}^{t}\int_{L^{2}(D)}\int_{D^{k}}|u(x_{1})|^{2}\dots|u(x_{k-1})|^{2}|\nabla
u(x_{k})|^{2}dx\,d\mu_{s}(u)\,ds$
$\displaystyle\qquad=\sum_{k=0}^{K}a_{k}\sum_{i=1}^{k}\int_{0}^{t}\int_{L^{2}(D)}\int_{D^{k}}|u(x_{1})|^{2}\dots|\nabla
u(x_{i})|^{2}\dots|u(x_{k})|^{2}dx\,d\mu_{s}(u)\,ds$
$\displaystyle\qquad=\sum_{k=0}^{K}a_{k}\sum_{i=1}^{k}\sum_{j=1}^{d}\lim_{h\to
0}\frac{1}{h^{2}}\int_{0}^{t}\int_{D^{k}}\int_{U^{k+1}}|\xi_{1}|^{2}\dots|\xi_{i}-\xi_{k+1}|^{2}\dots|\xi^{k}|^{2}d\nu^{k+1}_{t,(x,x_{i}+h\mathbf{e}_{j})}(\xi,\xi_{k+1})dx\,ds$

and a similar computation for the term involving $\mu_{0}$ yields (3.4). The
bounded support of $\mu_{0}$ implies, using the equivalence theorem [23,
Theorem 2.7],

$\int_{D^{k}}\\!\\!\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{0,x}^{k}(\xi)\,dx<\infty$

for any $k\in\mathbb{N}$ and thus the boundedness of all terms in (3.4). It
remains to show that each $\nu^{k}$ satisfies (3.3). We let $\varphi\in
C^{1}_{c}(D^{k};U^{k})$, $1\leqslant\ell\leqslant k$, $\alpha_{j}\in C(U;U)$,
with $\alpha_{j}(v)\leqslant C(1+|v|^{2})$, $j=\ell,\dots,k$ and compute using
the equivalence theorem 2.2,

$\begin{split}&\int_{D^{k}}\int_{U^{k}}\xi_{1}\otimes\dots\otimes\xi_{\ell}\otimes\alpha_{\ell+1}(\xi_{\ell+1})\otimes\dots\otimes\alpha_{k}(\xi_{k})\,d\nu_{t,x}^{k}(\xi)\cdot\nabla_{x_{1},\dots,x_{\ell}}\varphi(x)\,dx\\\
&\qquad=\int_{L^{2}(D)}\int_{D^{k}}\\!\\!\\!\begin{aligned}
&u(x_{1})\otimes\dots\otimes
u(x_{\ell})\otimes\alpha_{\ell+1}(u(x_{\ell+1}))\otimes\dots\otimes\alpha_{k}(u(x_{k}))\\\
&\cdot\nabla_{x_{1},\dots,x_{\ell}}\varphi(x)\,dx\,d\mu_{t}(u)\end{aligned}\\\
&\qquad=0,\end{split}$

since $\mu_{t}$ is supported on divergence free functions. This concludes the
proof. ∎

Let us show the reverse direction now, that is, that any correlation measure
that solves the Friedman–Keller system and satisfies in addition an energy
inequality, is a statistical solution of the Navier–Stokes equations in the
sense of Foiaş–Prodi.

######  Theorem A.2 (Friedman–Keller solutions are Foiaş–Prodi solutions).

Let $\bm{\nu}=(\bm{\nu}_{t})_{0\leqslant t\leqslant T}$ be a Friedman–Keller
statistical solution of Navier–Stokes (cf. Definition 3.1) with bounded
support, i.e.,

(A.4)
$\int_{D^{k}}\int_{U^{k}}|\xi_{1}|^{2}\dots|\xi_{k}|^{2}\,d\nu_{t,x}^{k}(\xi)\,dx\leqslant
R^{k}<\infty,$

for some $0<R<\infty$, every $k\in\mathbb{N}$, and almost every $t\in[0,T]$.
Then $\bm{\nu}$ corresponds to a probability measure
$\mu=(\mu_{t})_{0\leqslant t\leqslant T}$ on a bounded set of
$L^{2}_{\operatorname{div}}(D;U)$ which is a Foiaş–Prodi statistical solution
of the Navier–Stokes equations (cf. Definition 3.6).

###### Proof.

From the equivalence theorem 2.2, we obtain that $(\bm{\nu}_{t})_{0\leqslant
t\leq T}$ corresponds to a family of measures $(\mu_{t})_{0\leqslant
t\leqslant T}\subset\mathscr{P}(L^{2}(D;U))$ with bounded support, that is
$\text{supp}(\mu_{t})\subset\\{u\in L^{2}(D;U)\,:\,\|u\|_{L^{2}(D;U)}\leqslant
R\\}$ for almost every $t\in[0,T]$. Property (a) of Definition 3.6 – the fact
that (3.12) is measurable for all $\varphi\in C_{b}(L^{2})$ – follows from a
monotone class argument, which we include here. By property (i) of Theorem
2.2, the function

$t\mapsto\int_{L^{2}(D;U)}L_{f}(u)\,d\mu_{t}(u)=\int_{L^{2}(D;U)}\int_{D}f(x,u(x))\,dx\,d\mu_{t}(u)$

is measurable for every $f\in L^{1}(D^{k},C_{0}(U^{k}))$. Let $\mathbf{M}$ be
the collection of sets

$\mathbf{M}=\left\\{E\in\mathscr{B}(L^{2}(D;U))\text{ such that
}t\mapsto\int_{L^{2}(D;U)}\mathbbm{1}_{E}(u)\,d\mu_{t}(u)\text{ is
measurable}\right\\}.$

By the monotone convergence theorem, $\mathbf{M}$ is a monotone class, that
is, it is closed under (countable) unions of increasing sequences of sets and
intersections of decreasing sequences of sets. By the same argument as in the
proof of [23, Proposition 2.12], $\mathbf{M}$ contains the collection of
cylinder sets $\mathrm{Cyl}(L^{2})=\\{\text{all cylinder sets on
}L^{2}(D;U)\\}$, that is, all sets of the form $E=\big{\\{}u\in L^{2}\ :\
(\langle\varphi_{1},\,u\rangle,\dots,\langle\varphi_{n},\,u\rangle)\in
F\big{\\}}$ for some $n\in\mathbb{N}$, a Borel set $F\subset\mathbb{R}^{n}$
and $\varphi_{1},\dots,\varphi_{n}\in L^{2}(D;U)$. Since $\mathrm{Cyl}(L^{2})$
is an algebra which generates $\mathscr{B}(L^{2})$ (cf. e.g. [23, Appendix]),
it follows from the monotone class lemma that
$\mathbf{M}=\sigma(\mathrm{Cyl}(L^{2}))=\mathscr{B}(L^{2})$. Approximating an
arbitrary $\varphi\in C_{b}(L^{2}(D;U))$ by simple functions now gives the
desired conclusion.

We claim that $\mu_{t}$ is supported on $L^{2}_{\operatorname{div}}(D;U)$.
This follows from Lemma B.5: Since $\mu$ has bounded support on $L^{2}(D)$ we
may take $\varphi(\xi)=|\xi|^{2}$ in that lemma, which is continuous and
bounded on any compact subset of $\mathbb{R}$ and satisfies $\varphi(0)=0$.
Then by the lemma, for any $g\in H^{1}(D)$,

$\int_{L^{2}(D)}\left|\int_{D}u\cdot\nabla g\,dx\right|^{2}\,d\mu_{t}(u)=0.$

Hence, by Chebychev’s inequality,

$\mu_{t}\left(\left\\{u\in L^{2}(D):\,\int_{D}u\cdot\nabla g\,dx\neq
0\right\\}\right)=0.$

Since $g\in H^{1}(D;U)$ was arbitrary and $H^{1}(D;U)$ is separable, this
implies that $\mu_{t}$ is supported on $L^{2}$-functions that are weakly
divergence free, which is exactly the space $L^{2}_{\operatorname{div}}(D;U)$
(see e.g. [56, Section 1, Chapter 1]).

Next we claim that $\mu$ satisfies condition (c) in Definition 3.6. As
$\bm{\nu}$ is assumed to satisfy (3.4), we can apply [23, Theorem 2.7]
combined with Lemma B.4 to each of the terms and obtain that $\mu_{t}$
satisfies (A.3) for any nonnegative and nondecreasing polynomial
$p_{K}(s)=\sum_{k=0}^{K}a_{k}s^{k}$ on $[0,R]$. This implies in particular
that $\mu_{t}$ is supported on $H^{1}(D;U)$, for a.e. $t\in[0,T)$, and (3.16).
Now any differentiable nondecreasing function $\psi$ on a bounded interval can
be approximated by nondecreasing polynomials [55], from which (3.14) follows
after passing to the limit in a suitable polynomial approximation.
Specifically, for a given $\psi$, let $\\{p_{K_{n}}\\}_{n\in\mathbb{N}}$ be a
sequence of nonnegative, nondecreasing polynomials ($K_{n}\to\infty$)
satisfying

$\left\|p_{K_{n}}-\psi\right\|_{C^{1}([0,R])}\leqslant\frac{1}{n}.$

Then, for $t\geqslant 0$, by the compact support property of $\mu$ on $B$,

$\left|\int_{L^{2}(D)}\psi(\left\|u\right\|_{L^{2}(D)}^{2})d\mu_{t}(u)-\int_{L^{2}(D)}p_{K_{n}}(\left\|u\right\|^{2}_{L^{2}(D)})d\mu_{t}(u)\right|\leqslant\left\|p_{K_{n}}-\psi\right\|_{C^{1}([0,R])}\leqslant\frac{1}{n},$

and

$\displaystyle\left|\int_{0}^{t}\int_{L^{2}(D)}\psi^{\prime}(\left\|u\right\|_{L^{2}(D)}^{2})|u|^{2}_{H^{1}(D)}d\mu_{t}(u)ds-\int_{0}^{t}\int_{L^{2}(D)}p_{K_{n}}^{\prime}(\left\|u\right\|^{2}_{L^{2}(D)})|u|^{2}_{H^{1}(D)}d\mu_{t}(u)ds\right|$
$\displaystyle\qquad\leqslant\int_{0}^{t}\int_{L^{2}(D)}\left|\psi^{\prime}(\left\|u\right\|_{L^{2}(D)}^{2})-p_{K_{n}}^{\prime}(\left\|u\right\|^{2}_{L^{2}(D)})\right||u|^{2}_{H^{1}(D)}d\mu_{t}(u)ds$
$\displaystyle\qquad\leqslant\left\|p_{K_{n}}-\psi\right\|_{C^{1}([0,R])}\int_{0}^{t}\int_{L^{2}(D)}|u|^{2}_{H^{1}(D)}d\mu_{t}(u)ds$
$\displaystyle\qquad\leqslant\frac{C}{n},$

where the last inequality follows from (3.16).

It remains to show that the correlation measures satisfy the evolution
equation (3.13) for all $\Phi\in\mathscr{T}_{\text{cyl}}^{0}$. From this also
part (d) of the definition will follow (see Remark 3.7). Let
$\Phi(u)=\varphi((u,g_{1}),\dots,(u,g_{k}))$ be an arbitrary function in
$\mathscr{T}_{\text{cyl}}^{0}$ with $g_{j}\in H^{1}(D;U)$ and let
$\widetilde{R}=\max_{1\leqslant j\leqslant k}(\left\|g_{j}\right\|_{L^{2}})$.
Since $\varphi$ is continuously differentiable and bounded on
$B\coloneqq[-R\widetilde{R}-1,R\widetilde{R}+1]^{k}$, we can approximate it
arbitrarily well by polynomials thanks to the Weierstrass approximation
theorem. Let

$p_{n}(\zeta)\coloneqq\sum_{|\underline{j}|=0}^{N_{n}}\beta_{\underline{j}}\zeta_{1}^{j_{1}}\dots\zeta_{k}^{j_{k}},\quad
j_{i}\geqslant
0,\quad\underline{j}=(j_{1},\dots,j_{k}),\quad|\underline{j}|\coloneqq
j_{1}+\dots+j_{k},$

for some $N_{n}\in\mathbb{N}$ large enough, be approximations of $\varphi$
that satisfy

$\left\|\varphi-p_{n}\right\|_{C^{1}(B)}\leqslant\frac{1}{n},\quad
n\in\mathbb{N}.$

Let $\theta\in C_{c}^{1}((0,T))$ be an arbitrary compactly supported test
function. Since by the equivalence theorem, for any
$\underline{j}=(j_{1},\dots,j_{k})$, $j_{i}\geqslant 0$,
$|\underline{j}|=j_{1}+\dots+j_{k}$,

$\displaystyle\int_{0}^{T}\theta^{\prime}(t)\int_{D^{|\underline{j}|}}\left(\int_{U^{|\underline{j}|}}\xi_{1}\otimes\cdots\otimes\xi_{|\underline{j}|}\,d\nu_{t,x}^{|\underline{j}|}(\xi)\right)$
$\displaystyle:\Big{(}g_{1}(x_{1})\otimes\cdots\otimes
g_{1}(x_{j_{1}})\otimes\cdots\otimes
g_{k}(x_{|\underline{j}|-j_{k}+1})\otimes\cdots\otimes
g_{k}(x_{|\underline{j}|})\Big{)}\,dx\,dt$
$\displaystyle=\int_{0}^{T}\theta^{\prime}(t)\int_{L^{2}(D)}(u,g_{1})^{j_{1}}\dots(u,g_{k})^{j_{k}}\,d\mu_{t}(u)\,dt,$

and if $\widetilde{j}=j_{1}+\dots+j_{i-1}$, then

$\displaystyle\sum_{\ell=\widetilde{j}+1}^{\widetilde{j}+j_{i}}\int_{0}^{T}\\!\\!\\!\theta(t)\\!\int_{D^{|\underline{j}|}}\left(\int_{U^{|\underline{j}|}}\xi_{1}\otimes\cdots\otimes\xi_{|\underline{j}|}\,d\nu_{t,x}^{|\underline{j}|}(\xi)\right)$
$\displaystyle:\Big{(}g_{1}(x_{1})\otimes\cdots\otimes
g_{i}(x_{\widetilde{j}+1})\otimes\dots\otimes\Delta_{x_{\ell}}g_{i}(x_{\ell})\otimes\dots\otimes
g_{i}(x_{\widetilde{j}+j_{i}})\otimes\dots\otimes
g_{k}(x_{|\underline{j}|})\Big{)}\,dx\,dt$ $\displaystyle=$ $\displaystyle\
j_{i}\int_{0}^{T}\\!\\!\theta(t)\int_{L^{2}(D;U)}(u,g_{1})^{j_{1}}\cdots(u,g_{i})^{j_{i}-1}(u,\Delta
g_{i})\cdots(u,g_{k})^{j_{k}}\,d\mu_{t}(u)\,dt$ $\displaystyle=$
$\displaystyle\
j_{i}\int_{0}^{T}\\!\\!\theta(t)\int_{L^{2}(D;U)}(u,g_{1})^{j_{1}}\cdots(u,g_{i})^{j_{i}-1}(Au,g_{i})\cdots(u,g_{k})^{j_{k}}\,d\mu_{t}(u)\,dt,$

where the last equality follows because $g_{i}$ is divergence free because
$\mu_{t}$ is supported on $H^{1}(D;U)$-functions by the energy inequality
(3.14). Furthermore, we have

$\displaystyle\sum_{\ell=\widetilde{j}+1}^{\widetilde{j}+j_{i}}\int_{0}^{T}\\!\\!\\!\theta(t)\\!\int_{D^{|\underline{j}|}}\left(\int_{U^{|\underline{j}|}}\xi_{1}\otimes\cdots\otimes\xi_{\widetilde{j}+1}\otimes\cdots\otimes(\xi_{\ell}\otimes\xi_{\ell})\otimes\cdots\otimes\xi_{\widetilde{j}+j_{i}}\otimes\cdots\otimes\xi_{|\underline{j}|}\,d\nu_{t,x}^{|\underline{j}|}(\xi)\right)$
$\displaystyle:\Big{(}g_{1}(x_{1})\otimes\cdots\otimes\nabla_{x_{\ell}}g_{i}(x_{\ell})\otimes\cdots\otimes
g_{k}(x_{|\underline{j}|})\Big{)}\,dx\,dt$ $\displaystyle=$ $\displaystyle\
j_{i}\int_{0}^{T}\\!\\!\theta(t)\int_{L^{2}(D)}(u,g_{1})^{j_{1}}\cdots(u,g_{i})^{j_{i}-1}(u\otimes
u,\nabla g_{i})\cdots(u,g_{k})^{j_{k}}\,d\mu_{t}(u)\,dt$ $\displaystyle=$
$\displaystyle\
-j_{i}\int_{0}^{T}\\!\\!\theta(t)\int_{L^{2}(D)}(u,g_{1})^{j_{1}}\cdots(u,g_{i})^{j_{i}-1}(B(u),g_{i})\cdots(u,g_{k})^{j_{k}}\,d\mu_{t}(u)\,dt,$

again, the last equality following because $g_{i}$ is divergence free and
$\mu_{t}$ is supported on $H^{1}(D;U)$-functions for almost every $t\in[0,T]$.
We obtain, by combining the terms to the Friedman–Keller system, that if

$\Phi_{n}(u)\coloneqq p_{n}\big{(}(u,g_{1}),\dots,(u,g_{k})\big{)}$

then $\mu$ satisfies the equation

(A.5)
$\int_{0}^{T}\int_{L^{2}(D)}\theta^{\prime}(s)\Phi_{n}(u)\,d\mu_{s}(u)\,ds+\int_{0}^{T}\theta(s)\int_{L^{2}(D)}(F(s,u),\partial_{u}\Phi_{n}(u))\,d\mu_{s}(u)\,ds=0$

for every test function $\theta\in C_{c}^{1}((0,T))$. Next, note that

(A.6)
$\begin{split}&\left|\int_{L^{2}(D)}\Phi_{n}(u)\,d\mu_{t}(u)-\int_{L^{2}(D)}\Phi(u)\,d\mu_{t}(u)\right|=\left|\int_{L^{2}(D)}\left(\Phi_{n}(u)-\Phi(u)\right)\,d\mu_{t}(u)\right|\\\
&\quad\leqslant\int_{L^{2}(D)}\big{|}p_{n}\big{(}(u,g_{1}),\dots,(u,g_{k})\big{)}-\varphi\big{(}(u,g_{1}),\dots,(u,g_{k})\big{)}\big{|}\,d\mu_{t}(u)\\\
&\quad\leqslant\int_{L^{2}(D)}\sup_{\zeta\in
B}\left|p_{n}(\zeta)-\varphi(\zeta)\right|\,d\mu_{t}(u)\\\
&\quad=\left\|\varphi-
p_{n}\right\|_{C^{0}(B)}\leqslant\frac{1}{n},\end{split}$

because $\mu_{t}$ has support on $\\{u\in
L^{2}(D)\,:\,\|u\|_{L^{2}(D)}\leqslant R\\}$. Moreover,

(A.7)
$\begin{split}&\left|\int_{0}^{T}\theta(s)\int_{L^{2}(D)}\big{(}\partial_{u}\Phi_{n}(u),F(s,u)\big{)}\,d\mu_{s}(u)\,ds-\int_{0}^{T}\theta(s)\int_{L^{2}(D)}\big{(}\partial_{u}\Phi(u),F(s,u)\big{)}\,d\mu_{s}(u)\,ds\right|\\\
&\quad\leqslant\|\theta\|_{L^{\infty}}\int_{0}^{T}\int_{L^{2}(D)}\big{|}\big{(}\partial_{u}(\Phi(u)-\Phi_{n}(u)),F(s,u)\big{)}\big{|}\,d\mu_{s}(u)\,ds\\\
&\quad\leqslant\|\theta\|_{L^{\infty}}\sum_{j=1}^{k}\int_{0}^{T}\int_{L^{2}(D)}\big{|}\partial_{j}\varphi\big{(}(u,g_{1}),\dots,(u,g_{k})\big{)}-\partial_{j}p_{n}\big{(}(u,g_{1}),\dots,(u,g_{k})\big{)}\big{|}\big{|}(g_{j},F(s,u))\big{|}\,d\mu_{s}(u)\,ds\\\
&\quad\leqslant\|\theta\|_{L^{\infty}}\sum_{j=1}^{k}\sup_{B}\left|\partial_{j}(\varphi-
p_{n})\right|\int_{0}^{T}\int_{L^{2}(D)}\big{|}(g_{j},F(s,u))\big{|}\,d\mu_{s}(u)\,ds\\\
&\quad\leqslant\|\theta\|_{L^{\infty}}\left\|\varphi-
p_{n}\right\|_{C^{1}(B)}\sum_{j=1}^{k}\int_{0}^{T}\int_{L^{2}(D)}\big{|}(g_{j},F(s,u))\big{|}\,d\mu_{s}(u)\,ds\\\
&\quad\leqslant\frac{1}{2n}\|\theta\|_{L^{\infty}}\sum_{j=1}^{k}\left(\left\|g_{j}\right\|_{H^{1}(D)}^{2}+\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla
u\right\|_{L^{2}(D)}^{2}\,d\mu_{s}(u)\,ds\right)\end{split}$

where we used Young’s inequality for the last inequality. Thanks to (A.6) and
(A.7), we can pass to the limit $n\rightarrow\infty$ in (A.5) and obtain

(A.8)
$\int_{0}^{T}\theta^{\prime}(s)\int_{L^{2}(D)}\Phi(u)\,d\mu_{s}(u)\,ds+\int_{0}^{T}\theta(s)\int_{L^{2}(D)}(F(s,u),\partial_{u}\Phi(u))\,d\mu_{s}(u)\,ds=0$

for every $\theta\in C_{c}^{1}((0,T))$. It follows that the distributional
derivative of $\varphi(s)\coloneqq\int_{L^{2}(D)}\Phi(u)\,d\mu_{s}(u)$ is

$\frac{d}{ds}\varphi(s)=\int_{L^{2}(D)}(F(s,u),\partial_{u}\Phi(s,u))\,d\mu_{s}(u),$

and since the right-hand side lies in $L^{1}((0,T))$ we see that $\varphi$ is
absolutely continuous. Consequently, we can (through a standard approximation
procedure) insert $\theta=\mathbbm{1}_{(0,t)}$ in (A.8) and conclude that
(3.13) is true.

∎

##  Appendix B Auxiliary lemmata

######  Lemma B.1 (Representation of the gradient).

Let $(\mu_{t})_{0\leqslant t\leqslant T}$ be a measure on $L^{2}(D)$
satisfying

(B.1) $\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla
u\right\|_{L^{2}(D)}^{2}\,d\mu_{t}(u)\,dt\leqslant C<\infty.$

Define for $h\in\mathbb{R}$ the finite difference gradient $\nabla_{h}$ by

$\nabla_{h}f(x)=\left(D^{h}_{1}f(x),\dots,D^{h}_{d}f(x)\right),\quad
D^{h}_{j}f(x)=\frac{f(x+h\mathbf{e}_{j})-f(x)}{h},$

where $\mathbf{e}_{j}\in\mathbb{R}^{d}$ is the $j$th unit vector. Then we have
for any $V\subset\subset D$ (or in the case that $D$ is the torus also for
$V=D$)

(B.2) $\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla
u\right\|_{L^{2}(V)}^{2}\,d\mu_{t}(u)\,dt=\lim_{h\rightarrow
0}\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla_{h}u\right\|_{L^{2}(V)}^{2}\,d\mu_{t}(u)\,dt.$

###### Proof.

Consider a compact $V\subset\subset D$ and a smooth $u\in C^{\infty}(D)$.
Then, since we can write

(B.3) $D^{h}_{j}u(x)=\int_{0}^{1}\partial_{x^{j}}u(x+th\mathbf{e}_{j})\,dt,$

and hence

(B.4)
$|\nabla_{h}u(x)|\leqslant\sum_{j=1}^{d}\int_{0}^{1}|\partial_{x^{j}}u(x+th\mathbf{e}_{j})|\,dt\leqslant\sum_{j=1}^{d}\left\|\partial_{x^{j}}u\right\|_{L^{\infty}(V)}\leqslant
C\left\|\nabla u\right\|_{L^{\infty}(V)},$

for all $x\in V$ and $\int_{V}C\left\|\nabla
u\right\|_{L^{\infty}(V)}\,dx<\infty$, we can use the dominated convergence
theorem to obtain that

$\int_{V}|\nabla u|^{2}\,dx=\lim_{h\rightarrow
0}\int_{V}|\nabla_{h}u|^{2}\,dx.$

Moreover, by [19, Theorem 3, Section 5.8.2], we have for any $u\in H^{1}(D)$,

(B.5) $\left\|\nabla_{h}u\right\|_{L^{2}(V)}\leqslant C_{V}\left\|\nabla
u\right\|_{L^{2}(D)},$

for some constants $0\leqslant c_{V},C_{V}<\infty$ and any $V\subset\subset
D$. Since $C^{\infty}(D)$ is dense in $H^{1}(D)$, we can find a sequence of
functions $(u_{n})_{n\in\mathbb{N}}\subset C^{\infty}(D)$ such that

$\lim_{n\rightarrow\infty}\left\|u_{n}-u\right\|_{H^{1}(V)}=0,$

for any $V\subset D$ bounded. So fix $V\subset\subset D$ and an arbitrary
$\varepsilon>0$ and pick $N_{\varepsilon}\in\mathbb{N}$ large enough such that
for all $n\geqslant N_{\varepsilon}$,

$\left\|u-u_{n}\right\|_{H^{1}(V)}\leqslant\frac{1}{2}\frac{1}{1+C_{V}}\varepsilon,$

where $C_{V}$ is the constant in (B.5). Then for $n\geqslant N_{\varepsilon}$

$\displaystyle\left\|\nabla u-\nabla_{h}u\right\|_{L^{2}(V)}$
$\displaystyle\leqslant\left\|\nabla(u-u^{n})\right\|_{L^{2}(V)}+\left\|\nabla
u^{n}-\nabla_{h}u^{n}\right\|_{L^{2}(V)}+\left\|\nabla_{h}(u^{n}-u)\right\|_{L^{2}(V)}$
$\displaystyle\leqslant\left\|\nabla(u-u^{n})\right\|_{L^{2}(V)}+\left\|\nabla
u^{n}-\nabla_{h}u^{n}\right\|_{L^{2}(V)}+C_{V}\left\|\nabla(u^{n}-u)\right\|_{L^{2}(V)}$
$\displaystyle\leqslant\frac{\varepsilon}{2}+\left\|\nabla
u^{n}-\nabla_{h}u^{n}\right\|_{L^{2}(V)}.$

Now pick $h_{0}$ with $|h_{0}|$ small enough that

$\left\|\nabla
u^{n}-\nabla_{h}u^{n}\right\|_{L^{2}(V)}\leqslant\frac{\varepsilon}{2}$

for all $h$ with $|h|\leqslant|h_{0}|$. Then

$\left\|\nabla u-\nabla_{h}u\right\|_{L^{2}(V)}\leqslant\varepsilon$

for all $h$ with $|h|\leqslant|h_{0}|$. Since $\varepsilon>0$ was arbitrary,
we obtain

(B.6) $\lim_{h\rightarrow 0}\int_{V}|\nabla_{h}u|^{2}\,dx=\int_{V}|\nabla
u|^{2}\,dx.$

Combining (B.5) with (B.1), we can apply the dominated convergence theorem and
use (B.6) to pass to the limit:

$\begin{split}\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla
u\right\|_{L^{2}(V)}^{2}\,d\mu_{t}(u)\,dt&=\int_{0}^{T}\int_{L^{2}(D)}\lim_{h\rightarrow
0}\left\|\nabla_{h}u\right\|_{L^{2}(V)}^{2}\,d\mu_{t}(u)\,dt\\\
&=\lim_{h\rightarrow
0}\int_{0}^{T}\int_{L^{2}(D)}\left\|\nabla_{h}u\right\|_{L^{2}(V)}^{2}\,d\mu_{t}(u)\,dt.\end{split}$

∎

######  Corollary B.2.

Let $(\mu_{t})_{0\leqslant t\leqslant T}$ be a measure on $L^{q}(D)$. If
$\mu_{t}$ satisfies

$\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla
u\right\|_{L^{q}(D)}^{q}\,d\mu_{t}(u)\,dt\leqslant C<\infty,$

then for any $V\subset\subset D$ (or if $D$ is the torus also for $V=D$)

$\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla
u\right\|_{L^{q}(V)}^{q}\,d\mu_{t}(u)\,dt=\lim_{h\rightarrow
0}\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla_{h}u\right\|_{L^{q}(V)}^{q}\,d\mu_{t}(u)\,dt.$

######  Lemma B.3 (More representation of gradients).

Let $(\mu_{t})_{0\leqslant t\leqslant T}$ be a measure on $L^{q}(D)$, where
$D$ is the $d$-dimensional torus satisfying

(B.7) $\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla
u\right\|_{L^{q}(D)}^{q}\,d\mu_{t}(u)\,dt\leqslant C<\infty.$

Then we can represent the integral of the gradient in terms of correlation
measures as

(B.8) $\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla
u\right\|_{L^{q}(D)}^{q}\,d\mu_{t}(u)\,dt=\lim_{h\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\sum_{j=1}^{d}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu^{2}_{x,x+h\mathbf{e}_{j}}(\xi)\,dx\,dt.$

###### Proof.

From Lemma B.1, we know that we can represent the left hand side of (B.8) as
the limit as $h$ goes to zero of

$\frac{1}{h^{q}}\int_{0}^{T}\int_{L^{q}(D)}\int_{D}|u(x+h)-u(x)|^{q}\,dx\,d\mu_{t}(u)\,dt.$

Note that for arbitrary nonnegative functions $\varphi\in L^{1}(D)$ with
$\varphi_{\varepsilon}(x)\coloneqq\varepsilon^{-d}\varphi(x/\varepsilon)$, we
have that

$\int_{D}\int_{D}|f(x)-f(x-y)|^{q}\varphi_{\varepsilon}(y)\,dy\,dx\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}0,$

for any $f\in L^{q}(D)$, since shifts are continuous in $L^{p}$ for
$L^{p}$-functions. In particular, choosing
$\varphi(x)=\frac{1}{|B_{1}(0)|}\mathbbm{1}_{B_{1}(0)}(x)$ this yields

(B.9) $\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x)}|f(x)-f(y)|^{q}\,dy\,dx\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}0,$

and

$\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|f(x)-f(y)|^{q}\,dy\,dx\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}\int_{D}|f(x)-f(x+h\mathbf{e}_{j})|^{q}\,dx,$

for any fixed $h>0$ and $\mathbf{e}_{j}$ is the $j$th unit vector. We have for
any $\varepsilon>0$,

(B.10)
$\displaystyle\Biggl{|}\frac{1}{h}\left(\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\\!\\!\\!|u(x)-u(x+h\mathbf{e}_{j})|^{q}\,dx\,d\mu_{t}(u)\,dt\right)^{1/q}$
(B.11)
$\displaystyle\quad\qquad-\frac{1}{h}\left(\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|u(x)-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt\right)^{1/q}\Biggr{|}$
(B.12)
$\displaystyle\quad\leqslant\left(\frac{1}{h^{q}}\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|u(x+h\mathbf{e}_{j})-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt\right)^{1/q}$
(B.13)
$\displaystyle\quad=\left(\frac{1}{h^{q}}\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x)}|u(x)-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt\right)^{1/q}\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}0,$

by (B.9) and Lebesgue dominated convergence theorem. Hence

(B.14)
$\frac{1}{h^{q}}\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\\!\\!\\!|u(x)-u(x+h\mathbf{e}_{j})|^{q}\,dx\,d\mu_{t}(u)\,dt\\\
=\frac{1}{h^{q}}\lim_{\varepsilon\to
0}\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|u(x)-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt.$

Now we can apply [23, Theorem 2.7] to rewrite the last expression in terms of
the unique correlation measure $\bm{\nu}_{t}$ to which $\mu_{t}$ corresponds:

$\frac{1}{h^{q}}\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\\!\\!\\!\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|u(x)-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt\\\
=\frac{1}{h^{q}}\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\\!\\!\\!\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\,dt$

Because of the equivalence theorem [23, Theorem 2.7] and assumption (B.7)
combined with Corollary B.2, we have

$\lim_{\varepsilon\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\,dt<\infty,$

uniformly in $0<h<h_{0}$ for a small enough $h_{0}$. Since

$\displaystyle\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx$
$\displaystyle\qquad\leqslant C\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}\left(|\xi_{1}|^{q}+|\xi_{2}|^{q}\right)\,d\nu_{x,y}^{2}(\xi)\,dy\,dx$
$\displaystyle\stackrel{{\scriptstyle\text{consistency}}}{{=}}2C\int_{D}\int_{U}|\xi_{1}|^{q}\,d\nu_{x}^{1}(\xi_{1})\,dx\leqslant
C<\infty\quad\text{ for almost every }t,$

we can apply the dominated convergence theorem and pass the limit in
$\varepsilon$ inside:

(B.15) $\lim_{\varepsilon\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\,dt\\\
=\frac{1}{h^{q}}\int_{0}^{T}\lim_{\varepsilon\rightarrow
0}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\,dt$

Now fix $\varepsilon>0$, a ‘good’ $t\in[0,T]$, and note that for almost every
$x\in D$,

$\displaystyle\left|\left(\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\right)^{1/q}-\left(\int_{D}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,x+h\mathbf{e}_{j}}^{2}(\xi)\,dx\right)^{1/q}\right|$
$\displaystyle\qquad\stackrel{{\scriptstyle\text{consistency}}}{{=}}\Biggl{|}\left(\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{3}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y,x+h\mathbf{e}_{j}}^{3}(\xi)\,dy\,dx\right)^{1/q}$
$\displaystyle\qquad\hphantom{\stackrel{{\scriptstyle\text{consistency}}}{{=}}\Bigl{|}}-\left(\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{3}}|\xi_{1}-\xi_{3}|^{q}\,d\nu_{x,y,x+h\mathbf{e}_{j}}^{3}(\xi)\,dy\,dx\right)^{1/q}\Biggr{|}$
$\displaystyle\qquad\stackrel{{\scriptstyle\Delta\text{-ineq.}}}{{\leqslant}}\left(\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{3}}|\xi_{3}-\xi_{2}|^{q}\,d\nu_{x,y,x+h\mathbf{e}_{j}}^{3}(\xi)\,dy\,dx\right)^{1/q}$
$\displaystyle\qquad\stackrel{{\scriptstyle\text{consistency}}}{{=}}\left(\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{2}-\xi_{1}|^{q}\,d\nu_{y,x+h\mathbf{e}_{j}}^{2}(\xi)\,dy\,dx\right)^{1/q}$

The last term goes to zero by the diagonal continuity property of $\bm{\nu}$
(c.f. [23, Definition 2.5]). Hence, for almost every $t\in[0,T]$,

(B.16) $\lim_{\varepsilon\to 0}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx=\int_{D}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,x+h\mathbf{e}_{j}}^{2}(\xi)\,dx.$

Thus, combining (B.16) and (B.15),

$\displaystyle\int_{0}^{T}\int_{L^{q}(D)}\int_{D}\frac{|u(x)-u(x+h\mathbf{e}_{j})|^{q}}{h^{q}}\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle\qquad=\lim_{\varepsilon\to
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{L^{q}(D)}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}|u(x)-u(y)|^{q}\,dy\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle\qquad=\lim_{\varepsilon\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.31105pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.50664pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.4333pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-3.96664pt}}\\!\int_{B_{\varepsilon}(x+h\mathbf{e}_{j})}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,y}^{2}(\xi)\,dy\,dx\,dt$
$\displaystyle\qquad=\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,x+h\mathbf{e}_{j}}^{2}(\xi)\,dx\,dt.$

Since this last identity holds for any $h>0$, and the limit is bounded by
assumption (B.7), we can pass to the limit $h\to 0$ and obtain, using Lemma
B.1,

$\displaystyle\int_{0}^{T}\int_{L^{q}(D)}\int_{D}|\partial_{j}u|^{q}\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle=\lim_{h\rightarrow
0}\int_{0}^{T}\int_{L^{q}(D)}\int_{D}\frac{|u(x)-u(x+h\mathbf{e}_{j})|^{q}}{h^{q}}\,dx\,d\mu_{t}(u)\,dt$
$\displaystyle=\lim_{h\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu_{x,x+h\mathbf{e}_{j}}^{2}(\xi)\,dx\,dt.$

Summing over $j=1,\dots,d$, we obtain the claim. ∎

######  Lemma B.4 (Reverse direction of Lemma B.3).

Let $\bm{\nu}=(\bm{\nu}_{t})_{0\leqslant t\leqslant
T}=((\nu^{1}_{t},\nu_{t}^{2}\dots))_{0\leqslant t\leqslant T}$ be a
correlation measure as in [23, Definition 2.5] that satisfies

(B.17) $\lim_{h\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\sum_{j=1}^{d}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu^{2}_{x,x+h\mathbf{e}_{j}}(\xi)\,dx\,dt\leqslant
C<\infty.$

Let $D$ be the $d$-dimensional torus. Then $\bm{\nu}$ corresponds to a family
of probability measures $\mu=(\mu_{t})_{0\leqslant t\leqslant T}$ on
$L^{q}(D)$ that satisfy

(B.18) $\int_{0}^{T}\int_{L^{q}(D)}\left\|\nabla
u\right\|_{L^{q}(D)}^{q}\,d\mu_{t}(u)\,dt=\lim_{h\rightarrow
0}\frac{1}{h^{q}}\int_{0}^{T}\int_{D}\sum_{j=1}^{d}\int_{U^{2}}|\xi_{1}-\xi_{2}|^{q}\,d\nu^{2}_{x,x+h\mathbf{e}_{j}}(\xi)\,dx\,dt\leqslant\infty.$

###### Proof.

The proof follows from the proof of Lemma B.3 by going in the reverse
direction and checking that all the steps are equivalences. ∎

######  Lemma B.5.

Let $\Phi:H^{1}(D)\to\mathbb{R}$ be a function of the form

$\Phi(u)=\varphi\left((u,\nabla g_{1}),\dots,(u,\nabla g_{k})\right),$

where $\varphi:\mathbb{R}^{k}\to\mathbb{R}$ is bounded and continuous with
$\varphi(0)=0$ and $g_{i}\in H^{1}(D)$. Then

$\int_{L^{2}(D)}\Phi(u)\,d\mu_{t}(u)=0,$

for any $\mu$ supported on a bounded set of $L^{2}(D)$ corresponding to a
family of correlation measures $\bm{\nu}=(\nu_{1},\nu_{2},\dots)$ satisfying
(3.3), i.e.,

$\int_{D^{k}}\int_{U^{k}}\xi_{1}\otimes\dots\otimes\xi_{\ell}\otimes\alpha_{\ell+1}(\xi_{\ell+1})\otimes\dots\otimes\alpha_{k}(\xi_{k})\,d\nu_{t,x}^{k}(\xi)\cdot\nabla_{x_{1},\dots,x_{\ell}}\varphi(x)\,dx=0,$

where
$\nabla_{x_{1},\dots,x_{\ell}}=(\nabla_{x_{1}},\dots,\nabla_{x_{\ell}})^{\top}$,
$1\leqslant\ell\leqslant k\in\mathbb{N}$, for all $\varphi\in
H^{1}(D^{k};U^{k-\ell})$, $\alpha_{j}\in C(U;U)$, with $\alpha_{j}(v)\leqslant
C(1+|v|^{2})$ for all $j=1,\dots,k$.

###### Proof.

Since $\varphi$ is bounded and continuous, we can approximate it on every
compact subset $B\subset\mathbb{R}^{k}$ by polynomials $p_{n}$ such that

$\left\|\varphi-p_{n}\right\|_{C(B)}\leqslant\frac{1}{n},$

$n\in\mathbb{N}$. Since $\varphi(0)=0$, we may assume that $p_{n}(0)=0$ and
hence the constant term of the polynomial is zero. It is therefore of the form

$p_{n}(\zeta)=\sum_{|\underline{j}|=1}^{N_{n}}\beta_{\underline{j}}\zeta_{1}^{j_{1}}\dots\zeta_{k}^{j_{k}},$

where $N_{n}\in\mathbb{N}$ large enough, $\zeta=(\zeta_{1},\dots,\zeta_{k})$,
$\underline{j}=(j_{1},\dots,j_{k})$, $j_{i}\geqslant 0$, and
$|\underline{j}|=j_{1}+\dots+j_{k}$. Hence

$\int_{L^{2}(D)}p_{n}((u,\nabla g_{1}),\dots,(u,\nabla
g_{k}))\,d\mu_{t}(u)=\sum_{|\underline{j}|=1}^{N_{n}}\beta_{\underline{j}}\int_{L^{2}(D)}(u,\nabla
g_{1})^{j_{1}}\cdots(u,\nabla g_{k})^{j_{k}}\,d\mu_{t}(u).$

Consider one of the terms in the sum:

$\displaystyle\int_{L^{2}(D)}(u,\nabla g_{1})^{j_{1}}\cdots(u,\nabla
g_{k})^{j_{k}}\,d\mu_{t}(u)$
$\displaystyle=\int_{L^{2}(D)}\int_{D}u(x_{1})\cdot\nabla_{x_{1}}g_{1}(x_{1})\,dx_{1}\cdots\int_{D}u(x_{j_{1}})\cdot\nabla_{x_{j_{1}}}g_{1}(x_{j_{1}})\,dx_{j_{1}}$
$\displaystyle\qquad\cdot\int_{D}u(x_{j_{1}+1})\cdot\nabla_{x_{j_{1}+1}}g_{2}(x_{j_{1}+1})\,dx_{j_{1}+1}\cdots\int_{D}u(x_{|\underline{j}|})\cdot\nabla_{x_{|\underline{j}|}}g_{k}(x_{|\underline{j}|})dx_{|\underline{j}|}\,d\mu_{t}(u)$
$\displaystyle=\int_{D^{|\underline{j}|}}\int_{(\mathbb{R}^{N})^{|\underline{j}|}}\xi_{1}\otimes\dots\otimes\xi_{|\underline{j}|}\,d\nu^{|\underline{j}|}_{x}(\xi)\nabla_{x_{1}}g_{1}(x_{1})\dots\nabla_{x_{|\underline{j}|}}g_{k}(x_{|\underline{j}|})\,dx=0,$

where the last equality follows from (3.3). Hence

$\int_{L^{2}(D)}p_{n}((u,\nabla g_{1}),\dots,(u,\nabla
g_{k}))\,d\mu_{t}(u)=0.$

By the approximation property of the $p_{n}$, after passing $n\to\infty$, we
obtain the result for arbitrary continuous $\varphi$. ∎

##  Appendix C An identity by T. Drivas for second order structure functions

We prove the following lemma discovered by Theo Drivas (Lemma 4.7):

######  Lemma C.1 (Weak anisotropy).

Let $\mu_{t}$ be a statistical solution of the Navier–Stokes equation. Then
$\mu$ satisfies

(C.1)
$\begin{split}&3\int_{0}^{T}\int_{D}\int_{L^{2}_{\operatorname{div}}(D;U)}\fint_{\partial
B_{r}(0)}(\delta_{rn}u\cdot n)^{2}dS(n)\,dx\,d\mu_{t}(u)\,dt\\\
&=\int_{0}^{T}\int_{D}\int_{L^{2}_{\operatorname{div}}(D;U)}\fint_{B_{r}(0)}|\delta_{\ell}u(x)|^{2}d\ell\,dx\,d\mu_{t}(u)\,dt.\end{split}$

###### Proof.

We let $\omega\in C^{\infty}_{c}(\mathbb{R}^{d})$, $\omega\geqslant 0$,
compactly supported in $B_{1}(0)$ and
$\omega_{\eta}(y)\coloneqq\eta^{-d}\omega(y/\eta)$ for $\eta>0$ its rescaled
version. For some fixed $\ell\in\mathbb{R}^{3}$ and $x\in D$, we let
$\varphi(t,y)=\ell\omega_{\eta}(x-y)$. We use this as a test function in (3.5)
for $k=1$. Using the Einstein summation convention we then have

(C.2)
$\begin{split}&\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u(y)\cdot\ell\omega_{\eta}(x-y)\theta^{\prime}(t)\,dy\,d\mu_{t}(u)\,dt\\\
&-\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)u^{j}(y)\ell^{j}\partial_{y^{i}}\omega_{\eta}(x-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt\\\
&+\varepsilon\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)\ell^{i}\Delta_{y}\omega_{\eta}(x-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt\\\
&+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)u^{j}(y)\partial_{y^{i}}\partial_{y^{j}}\psi_{\omega}(x-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt=0.\end{split}$

where the last term comes from the fact that the test function is not
divergence free, c.f. Remark 3.4. We define
$u_{\eta}(x)\coloneqq\int_{D}u(y)\omega_{\eta}(x-y)\,dy$ and rewrite the above
in terms of $u_{\eta}$,

(C.3)
$\begin{split}&\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}u_{\eta}(x)\cdot\ell\theta^{\prime}(t)\,d\mu_{t}(u)\,dt+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}u^{i}_{\eta}(x)\partial_{x^{i}}u^{j}_{\eta}(x)\ell^{j}\,d\mu_{t}(u)\theta(t)\,dt\\\
&+\varepsilon\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\Delta_{x}u^{i}_{\eta}(x)\ell^{i}\theta(t)\,d\mu_{t}(u)\,dt\\\
=&-\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)u^{j}(y)\partial_{y^{i}}\partial_{y^{j}}\psi_{\omega}(x-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt\\\
&+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}[u^{i}_{\eta}(x)\partial_{x^{i}}u^{j}_{\eta}(x)-\partial_{x^{i}}((u^{i}u^{j})_{\eta}(x))]\ell^{j}\,d\mu_{t}(u)\theta(t)\,dt.\end{split}$

We subtract this identity from the same identity at $x+\ell$ (and use the
notation $\delta_{\ell}u=u(x+\ell)-u(x)$):

(C.4)
$\begin{split}&\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\delta_{\ell}u_{\eta}(x)\cdot\ell\theta^{\prime}(t)\,d\mu_{t}(u)\,dt\\\
&+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}[u^{i}_{\eta}(x+\ell)\partial_{x^{i}}u^{j}_{\eta}(x+\ell)-u^{i}_{\eta}(x)\partial_{x^{i}}u^{j}_{\eta}(x)]\ell^{j}\,d\mu_{t}(u)\theta(t)\,dt\\\
&+\varepsilon\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\Delta_{x}\delta_{\ell}u^{i}_{\eta}(x)\ell^{i}\theta(t)\,d\mu_{t}(u)\,dt\\\
=&-\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)u^{j}(y)\partial_{y^{i}}\partial_{y^{j}}\psi_{\omega}(x+\ell-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt\\\
&+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}\int_{D}u^{i}(y)u^{j}(y)\partial_{y^{i}}\partial_{y^{j}}\psi_{\omega}(x-y)\theta(t)\,dy\,d\mu_{t}(u)\,dt\\\
&+\int_{0}^{T}\int_{L^{2}_{\operatorname{div}}(D;U)}[u^{i}_{\eta}(x+\ell)\partial_{x^{i}}u^{j}_{\eta}(x+\ell)-\partial_{x^{i}}((u^{i}u^{j})_{\eta}(x+\ell))]\ell^{j}\,d\mu_{t}(u)\theta(t)\,dt\\\