# Preserving positive intermediate curvature

Tsz-Kiu Aaron Chow  Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA, 02139<EMAIL_ADDRESS>,  Florian Johne  Department of
Mathematics
Columbia University
New York, NY, 10027<EMAIL_ADDRESS>and  Jingbo Wan  Department of
Mathematics
Columbia University
New York, NY, 10027<EMAIL_ADDRESS>
###### Abstract.

Consider a compact manifold $N$ (with or without boundary) of dimension $n$.
Positive $m$-intermediate curvature interpolates between positive Ricci
curvature ($m=1$) and positive scalar curvature ($m=n-1$), and it is
obstructed on partial tori $N^{n}=M^{n-m}\times\mathbb{T}^{m}$. Given
Riemannian metrics $g,\bar{g}$ on $(N,\partial N)$ with positive
$m$-intermediate curvature and $m$-positive difference $h_{g}-h_{\bar{g}}$ of
second fundamental forms we show that there exists a smooth family of
Riemannian metrics with positive $m$-intermediate curvature interpolating
between $g$ and $\bar{g}$. Moreover, we apply this result to prove a non-
existence result for partial torical bands with positive $m$-intermediate
curvature and strictly $m$-convex boundaries.

##  1\. Introduction

The existence of Riemannian metrics with positive curvature implies
obstructions on the topology of closed manifolds: Manifolds with topology
$N^{n}=M^{n-1}\times\mathbb{S}^{1}$ do not admit metrics of positive Ricci
curvature by the Theorem of Bonnet–Myers, while manifolds with topology
$N^{n}=\mathbb{T}^{n}$ do not admit metrics of positive scalar curvature by
the resolution of the Geroch conjecture due to R. Schoen and S.-T. Yau [12]
(for $3\leq n\leq 7$ by using minimal hypersurfaces) and M. Gromov and H.-B.
Lawson [7] (by using spinors and the Atyah–Singer index theorem).

The above results yield obstructions for positive curvature on the the partial
tori $N^{n}=M^{n-m}\times\mathbb{T}^{m}$ for the limit cases $m=1$ and
$m=n-1$. Recently, S. Brendle, S. Hirsch and the second author introduced the
notion of positive intermediate curvature (see Section 2 for a precise
definition), which interpolates between positive Ricci curvature $(m=1)$ and
positive scalar curvature $(m=n-1)$. They obtained the following obstruction
result on partial tori:

######  Theorem 1.1 (Generalized Geroch conjecture, Theorem 1.5 in [1]).

Assume $n\leq 7$ and $1\leq m\leq n-1$. Let $N^{n}$ be a closed and orientable
manifold of dimension $n$, and suppose that there exists a closed and
orientable manifold $M^{n-m}$ and a map $F:N^{n}\rightarrow
M^{n-m}\times\mathbb{T}^{m}$ with non-zero degree. Then the manifold $N$ does
not admit a metric with positive $m$-intermediate curvature.

The associated rigidity question was studied by J. Chu, K-K. Kwong and M.-C.
Lee [5] in ambient dimension at most five. S. Chen [3] extended the
obstruction result to manifolds with arbitrary ends. Moreover, K. Xu [15]
showed sharpness of the result in [1] by constructing counterexamples for
dimensions $n>7$ and $3\leq m\leq n-3$. The above result was recently used by
M. L. Labbi [8] to compute the Riemann invariant of products of spheres and
tori.

In this work, we study the interaction of the internal geometry and the
boundary geometry for metrics of positive intermediate curvature. The
corresponding question for positive scalar curvature and mean curvature on the
boundary dates back to work by M. Gromov and H.-B. Lawson [6]. A similar
interaction appears in the proof of the positive mass theorem by R. Schoen and
S.-T. Yau [13] — planes with positive mean curvature act as barriers for
minimal hypersurfaces in the bulk region with positive scalar curvature.
Related is work by Y. Shi and L.-F. Tam [14], where they proved an estimate
for the integral of the mean curvature over the boundary in manifolds with
non-negative scalar curvature. P. Miao [9] proved a positive mass theorem on
manifolds with corners by performing a suitable intrinsic bending
construction. A pertubation argument, which made the boundary totally geodesic
while keeping the scalar curvature non-decreasing, allowed S. Brendle, F.C.
Marques and A. Neves [2] to construct counterexamples to the Min-Oo
conjecture.

Recently, the first author proved a general result [4, Main Theorem 1] on the
interaction of internal geometry and boundary geometry by suitably gluing
Riemannian metrics. The result applies to a wide range of positive curvature
conditions, for example to metrics with positive curvature operator, PIC 2,
PIC 1 (with convex boundary), positive isotropic curvature (with two-convex
boundary) and positive scalar curvature (with mean-convex boundary). For a
different approach to the gluing problem, see also the thesis by A.
Schlichting, [11].

The first result of our work extends the gluing result of the first author to
$m$-intermediate curvature with the natural condition of $m$-convexity on the
boundary:

######  Theorem 1.2 (Preserving positive $m$-intermediate curvature).

Suppose that $N^{n}$ is a compact smooth manifold with smooth boundary
$\partial N$ of dimension $\dim N=n$. Let $g,\tilde{g}$ be Riemannian metrics
on $N$, such that $g=\tilde{g}$ on the boundary $\partial N$.

Then there exists $\lambda_{0}>0$, a family of smooth Riemannian metrics
$\\{\hat{g}_{\lambda}\\}_{\lambda>\lambda_{0}}$, and a neighborhood $U$ of the
boundary $\partial N$, such that the metric $\hat{g}_{\lambda}$ agrees with
the metric $g$ outside of $U$, and the metric $\hat{g}_{\lambda}$ agrees with
the metric $\tilde{g}$ in a neighbourhood of $\partial N$. Additionally, we
have $\hat{g}_{\lambda}\rightarrow g$ as $\lambda\rightarrow\infty$ in
$C^{\alpha}$ for any $\alpha\in(0,1)$.

Moreover, let $1\leq m\leq n-1$. If

  1. (1)

the Riemannian manifolds $(N,g)$ and $(N,\tilde{g})$ have positive
$m$-intermediate curvature,

  2. (2)

the difference $h_{g}-h_{\tilde{g}}$ is strictly $m$-convex (i.e. strictly
$m$-positive),

then the Riemannian manifold $(N,\hat{g}_{\lambda})$ has positive
$m$-intermediate curvature for all $\lambda>\lambda_{0}$.

The main ingredient in the proof is Proposition 3.1 relating the cone of
positive $m$-intermediate curvature to the Kulkarni–Nomizu product of
$m$-convex symmetric two-tensors.

In the second part of the paper we consider Riemannian manifolds with
$m$-positive intermediate curvature and strictly $m$-convex boundary: Let us
first recall the doubling lemma by M. Gromov and H.-B. Lawson for manifolds
with positive scalar curvature and strictly mean convex boundaries.

######  Lemma 1.3 (Doubling of positive scalar curvature metrics, M. Gromov
and H.-B. Lawson [6]).

Suppose $(N,g)$ is an orientable compact smooth Riemannian manifold with
smooth boundary $\partial N$. Assume the metric $g$ has positive scalar
curvature and is strictly mean convex (i.e. $H_{\partial N}>0$) with respect
to the outward unit normal. Then the double of $N$ carries a metric of
positive scalar curvature.

This lemma (in conjunction with the nonexistence of positive scalar curvature
metrics on the torus) then implies the following obstruction to positive
scalar curvature on torical bands:

######  Theorem 1.4 (Boundaries of a torical band, M. Gromov and H.-B. Lawson,
[6]).

Consider the smooth manifold with boundary $N=\mathbb{T}^{n-1}\times[-1,1]$
and let $g$ be a Riemannian metric on $N$ with positive scalar curvature. Then
the boundary $\partial N$ cannot be strictly mean convex.

D. Räde extended the above result to a scalar curvature and mean curvature
comparison result on more general bands [10].

We extend the above result on scalar curvature, mean curvature and torical
bands to partially torical bands by proving a generalization of the doubling
lemma of M. Gromov and H.-B. Lawson.

######  Theorem 1.5 (Boundaries of a partially torical band).

Let $n\leq 7$ and $1\leq m\leq n-1$. Suppose $M^{n-m}$ is a closed orientable
manifold. Consider the smooth manifold with boundary
$N=M^{n-m}\times\mathbb{T}^{m-1}\times[-1,1]$ and a Riemannian metric $g$ with
positive $m$-intermediate curvature on $N$. Then the boundary $\partial N$
cannot be strictly $m$-convex.

The work is structured as follows: In Section 2 we introduce our notation and
recall the definition of intermediate curvature. In Section 3 we prove an
algebraic lemma connecting the cone of positive $m$-intermediate curvature and
$m$-convexity. In Section 4 and 5 we prove the gluing result and in Section 6
we perform the doubling constructions.

Acknowledgements:
The authors would like to thank Simon Brendle for discussions and
encouragement, and the anonymous referee for careful reading of the article.

##  2\. Preliminaries

Let $(V,\langle\cdot,\cdot\rangle)$ be a $n$-dimensional real inner product
space. The space of algebraic curvature tensors on $V$ denoted by $C_{B}(V)$
is given by multilinear maps $R:V\times V\times V\times
V\rightarrow\mathbb{R}$ with the symmetries of the curvature tensor, i.e.

$R(v_{1},v_{2},v_{3},v_{4})=-R(v_{2},v_{1},v_{3},v_{4})\;\text{and}\;R(v_{1},v_{2},v_{3},v_{4})=R(v_{3},v_{4},v_{1},v_{2})$

for all $v_{1},v_{2},v_{3},v_{4}\in V$, and satisfying the first Bianchi
identity, i.e.

$R(v_{1},v_{2},v_{3},v_{4})+R(v_{3},v_{1},v_{2},v_{4})+R(v_{2},v_{3},v_{1},v_{4})=0$

for all $v_{1},v_{2},v_{3},v_{4}\in V$.

We denote the space of symmetric bilinear maps $T:V\times
V\rightarrow\mathbb{R}$ by $\operatorname{Sym}^{2}V$. The Kulkarni–Nomizu
product ${\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}:\operatorname{Sym}^{2}V\times\operatorname{Sym}^{2}V\rightarrow C_{B}(V)$ is
defined by

$(S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}T)(v_{1},v_{2},v_{3},v_{4})=S(v_{1},v_{3})T(v_{2},v_{4})+S(v_{2},v_{4})T(v_{1},v_{3})-S(v_{1},v_{4})T(v_{2},v_{3})-S(v_{2},v_{3})T(v_{1},v_{4})$

for $v_{1},v_{2},v_{3},v_{4}\in V$.

Following Definition 1.1 in work of the S. Brendle, S. Hirsch and the second
author [1] we define the cone $\mathcal{C}_{m}(V)$ of non-negative
$m$-intermediate curvature in the space of algebraic curvature tensors by

$\displaystyle\mathcal{C}_{m}(V):=\left\\{T\in
C_{B}(V)\;\left|\;\sum_{p=1}^{m}\sum_{q=p+1}^{n}T(e_{p},e_{q},e_{p},e_{q})\geq
0\;\text{for all orthonormal
bases}\;\\{e_{i}\\}_{i=1}^{n}\;\text{of}\;V\right.\right\\}.$

For a Riemannian manifold $(N^{n},g)$ with boundary $\partial N$ we consider
its Levi-Civita connection $D$ and its Riemann curvature tensor
$\operatorname{Rm}_{N}$ given by the formula

$\operatorname{Rm}_{N}(X,Y,Z,W)=-g(D_{X}D_{Y}Z-D_{Y}D_{X}Z-D_{[X,Y]}Z,W)$

for vector fields $X,Y,Z,W\in\Gamma(TN)$.

The Riemannian manifold $(N^{n},g)$ has positive $m$-intermediate curvature,
if $\operatorname{Rm}_{N}(p)\in\operatorname{Int}(\mathcal{C}_{m})$ for all
$p\in N$ (compare with Definition 1.1 in [1]).

Let $\nu$ be the inward pointing unit normal vector field on the boundary
$\partial N$. The scalar-valued second fundamental form $h_{g}:T(\partial
N)\otimes T(\partial N)\rightarrow C^{\infty}(\partial N)$ of the boundary
$\partial N$ with respect to the Riemannian metric $g$ is defined by

$h_{g}(X,Y)=g(\nu,D_{X}Y)$

for $X,Y\in\Gamma(TN)$. With this convention the scalar-valued second
fundamental form is positive on the standard sphere
$\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ with respect to the inward pointing
unit normal vector $\nu=-x$.

We say that the boundary $\partial N$ is strictly $m$-convex (where $1\leq
m\leq n-1$), if the bilinear form $h_{g}(p)$ is $m$-positive for all
$p\in\partial N$, i.e. if $\lambda_{1}\leq\lambda_{2}\leq\dots\lambda_{n-1}$
denote the eigenvalues of $h_{g}(p)$, then $\lambda_{1}+\dots+\lambda_{m}>0$.
For $m=1$ we recover the notion of strict convexity, and for $m=n-1$ we
recover the notion of strict mean convexity.

##  3\. Connecting $m$-convexity and $m$-intermediate curvature

In this section we prove a lemma in linear algebra, which allows us to connect
the cone of positive $m$-intermediate curvature and strict $m$-convexity.

######  Proposition 3.1 ($m$-intermediate curvature cone and $m$-convexity).

Let $(V,\langle\cdot,\cdot\rangle)$ be a $n$-dimensional inner product space
and $S:V\times V\rightarrow\mathbb{R}$ be a symmetric bilinear form. Let
$W\subset V$ be a $(n-1)$-dimensional subspace, and let $\nu\in W^{\perp}$ be
a unit vector. Let $S|_{W}$ be the restriction of $S$ on $W$. Fix $1\leq m\leq
n-1$. Then the bilinear form $S|_{W}$ is $m$-positive, if and only if

$S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\in\operatorname{Int}(\mathcal{C}_{m}(V)).$

###### Proof.

Suppose that the bilinear form $S|_{W}$ is $m$-positive. We extend the
bilinear form $S|_{W}$ to a bilinear $T$ on $V$ by setting

$T(v,w):=S|_{W}(v^{\parallel},w^{\parallel})$

for $v,w\in V$. Here $v^{\parallel}$ denotes the orthogonal projection from
$V$ to $W$.

The assumption on the $m$-positivity of the bilinear form $S|_{W}$ on $W$
implies that the bilinear form $T$ is $(m+1)$-positive on $V$.

We first want to show that $T$ being $(m+1)$-positive implies
$T{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\in\operatorname{Int}(\mathcal{C}_{m}(V))$.

Let $\\{e_{1},\dots,e_{n}\\}$ be an orthonormal basis of $V$ with respect to
the inner product $\langle\cdot,\cdot\rangle$. We denote the components of the
vector $\nu$ with respect to this orthonormal basis by $a_{p}$, i.e.
$a_{p}=\langle\nu,e_{p}\rangle$.

We have

$\displaystyle\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})=a_{p}^{2}T(e_{q},e_{q})+a_{q}^{2}T(e_{p},e_{p})-2a_{p}a_{q}T(e_{p},e_{q})$

for $1\leq p,q\leq n$ by definition of the Kulkarni–Nomizu product. We observe
the identity

$\displaystyle 2\sum_{p=1}^{m}\sum_{q=p+1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})$ $\displaystyle=$
$\displaystyle\left(\sum_{p=1}^{n}\sum_{q=1}^{n}-\sum_{p=m+1}^{n}\sum_{q=m+1}^{n}\right)\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q}).$

We evaluate the first term in the above sum:

$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})$ $\displaystyle=$
$\displaystyle
2\sum_{p=1}^{n}a_{p}^{2}\sum_{q=1}^{n}T(e_{q},e_{q})-2T\left(\sum_{p=1}^{n}a_{p}e_{p},\sum_{q=1}^{n}a_{q}e_{q}\right)=2\operatorname{tr}_{V}(T)-2T(\nu,\nu)=2\operatorname{tr}_{V}(T).$

We evaluate the second term in the above sum:

$\displaystyle\sum_{p=m+1}^{n}\sum_{q=m+1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})$ $\displaystyle=$
$\displaystyle
2\sum_{p=m+1}^{n}a_{p}^{2}\sum_{q=m+1}^{n}T(e_{q},e_{q})-2T\left(\sum_{p=m+1}^{n}a_{p}e_{p},\sum_{q=m+1}^{n}a_{q}e_{q}\right).$

This implies

(1) $\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})$ $\displaystyle=$
$\displaystyle\operatorname{tr}_{V}(T)\sum_{p=1}^{m}a_{p}^{2}+\sum_{p=m+1}^{n}a_{p}^{2}\sum_{q=1}^{m}T(e_{q},e_{q})+T\left(\sum_{p=m+1}^{n}a_{p}e_{p},\sum_{q=m+1}^{n}a_{q}e_{q}\right).$

If $a_{p}=0$ for all $m+1\leq p\leq n$, then we deduce the estimate

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})=\operatorname{tr}_{V}(T)\sum_{p=1}^{m}a_{p}^{2}>0,$

since the bilinear form $T$ is $(m+1)$-positive by construction and hence
$n$-positive. Hence we deduce $T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\in\operatorname{Int}(\mathcal{C}_{m})$ in this case.

Now suppose that $a_{p}\neq 0$ for some $m+1\leq p\leq n$. We define the unit
vector

$w:=\left(\sum_{p=m+1}^{n}a_{p}^{2}\right)^{-\frac{1}{2}}\sum_{q=m+1}^{n}a_{q}e_{q}.$

With this definition we deduce from equation (1) the identity

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\left[T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\right](e_{p},e_{q},e_{p},e_{q})=\operatorname{tr}_{V}(T)\sum_{p=1}^{m}a_{p}^{2}+\sum_{p=m+1}^{n}a_{p}^{2}\left(\sum_{q=1}^{m}T(e_{q},e_{q})+T(w,w)\right).$

The first term involving the trace $\operatorname{tr}_{V}(T)$ is positive as
above. The term in the bracket is positive, since the bilinear form $T$ is
$(m+1)$-positive, and $w\perp\operatorname{span}\\{e_{1},\dots,e_{m}\\}$ by
construction. Hence the sum is positive and we deduce $T{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\in\operatorname{Int}(\mathcal{C}_{m}(V))$.

On the other hand, by the construction of $T$, the restriction of $S-T$ to the
subspace $W$ vanishes. Therefore, we may write

$S=T+\omega\otimes\nu^{b}+\nu^{b}\otimes\omega,$

where $\omega$ is a suitable 1-form. Note that

$(\omega\otimes\nu^{b}+\nu^{b}\otimes\omega){\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\ }(\nu^{b}\otimes\nu^{b})=0$

by symmetry. Hence,

$S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{b}\otimes\nu^{b})\in\operatorname{Int}(\mathcal{C}_{m}(V)).$

The other implication in the equivalence follows by taking the orthonormal
basis $\\{f_{1},\dots,f_{n-1},\nu\\}$ of the vector space $V$, where
$\\{f_{1},\dots,f_{n-1}\\}$ is an orthonormal basis of the subspace $W$. ∎

##  4\. Preserving Curvature Conditions

In this section, we assume that $g$ and $\tilde{g}$ are Riemannian metrics on
$N$ such that $g-\tilde{g}=0$ along $\partial N$. We describe our choice of
perturbation as in work of S. Brendle, F.C. Marques and A. Neves [2]. We fix a
neighborhood $U$ of the boundary $\partial N$ and a smooth boundary defining
function $\rho:N\to[0,\infty)$ by taking it to be the distance function from
the boundary $\partial N$ with respect to the metric $g$. Then we have
$|D\rho|_{g}=1$. Since $g-\tilde{g}=0$ along the boundary $\partial N$, we can
find a symmetric (0,2)-tensor $S$ such that $\tilde{g}=g+\rho S$ in a
neighborhood of $\partial N$ and $S=0$ outside $U$. The scalar-valued second
fundamental forms and the boundary defining function satisfy

$\displaystyle\frac{1}{2}S(X,Y)=h_{g}(X,Y)-h_{\tilde{g}}(X,Y),\;\text{and}\;D^{2}\rho(X,Y)=-h_{g}(X,Y).$

for all $X,Y\in\Gamma(T(\partial N))$. This implies that the identity

(2) $\displaystyle
h_{g}(X,Y)-h_{\tilde{g}}(X,Y)=\frac{1}{2}S(X,Y)=-D^{2}\rho(X,Y)-h_{\tilde{g}}(X,Y)$

holds on the boundary $\partial N$ for all $X,Y\in\Gamma(T(\partial N))$.

We choose a smooth cut-off function $\chi:[0,\infty)\to[0,1]$ with the
following properties (compare with [2, Lemma 17]):

  * •

$\chi(s)=s-\frac{1}{2}s^{2}$ for $s\in[0,\frac{1}{2}]$;

  * •

$\chi(s)$ is constant for $s\geq 1$;

  * •

$\chi^{\prime\prime}(s)<0$ for $s\in[0,1)$.

Moreover, we choose a smooth cut-off function $\beta:(-\infty,0]\to[0,1]$ such
that

  * •

$\beta(s)=\frac{1}{2}$ for $s\in[-1,0]$;

  * •

$\beta(s)=0$ for $s\in(-\infty,-2]$.

For $\lambda>0$ sufficiently large we define a smooth metric
$\hat{g}_{\lambda}$ on the manifold $N$ by the formula

(3)
$\displaystyle\hat{g}_{\lambda}=\begin{cases}g+\lambda^{-1}\chi(\lambda\rho)S\quad\quad\quad\quad&\text{for}\quad\rho\geq
e^{-\lambda^{2}}\\\
\tilde{g}-\lambda\rho^{2}\beta(\lambda^{-2}\log\rho)S\quad&\text{for}\quad\rho<e^{-\lambda^{2}}.\end{cases}$

In the sequel, we will show that $\hat{g}_{\lambda}$ preserves positive
$m$-intermediate curvature of $g$ and $\tilde{g}$ for sufficiently large
$\lambda>0$. Note that we have the identity $\hat{g}_{\lambda}=\tilde{g}$ in
the region $\\{\rho\leq e^{-2\lambda^{2}}\\}$ and $\hat{g}_{\lambda}=g$
outside the neighbourhood $U$. Moreover, from the construction it follows that
$\hat{g}_{\lambda}\rightarrow g$ as $\lambda\rightarrow\infty$ in $C^{\alpha}$
for any $\alpha\in(0,1)$.

We first derive a lower bound for the $m$-intermediate curvature of the metric
$\hat{g}_{\lambda}$. We first consider the region $\\{\rho\geq
e^{-\lambda^{2}}\\}$.

######  Proposition 4.1 (Curvature estimates in inner gluing region).

Suppose that $h_{g}-h_{\tilde{g}}$ is $m$-positive on the boundary $\partial
N$. Let $\epsilon>0$ be given. If $\lambda=\lambda(\epsilon,\chi)>0$ is
sufficiently large, then

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(x)(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}_{g}(x)(e_{p},e_{q},e_{p},e_{q}))\geq-\epsilon$

for any $\hat{g}_{\lambda}$-orthonormal frame $\\{e_{1},\dots,e_{n}\\}$ and
any $x\in N$ in the region $\\{\rho(x)\geq e^{-\lambda^{2}}\\}$.

###### Proof.

We fix a point $x\in N$ such that $\rho(x)\geq e^{-\lambda^{2}}$. Let
$\\{e_{1},\dots,e_{n}\\}$ be a geodesic normal frame around the point $x$ with
respect to the metric $\hat{g}_{\lambda}$. Let $\varphi$ be a two-form. We
write $\varphi=\sum_{i,j}\varphi^{ij}e_{i}\wedge e_{j}$ for coefficients
$\varphi^{ij}$, which are anti-symmetric in $i$ and $j$. In the following the
Einstein summation convention will be adopted freely. Since $\varphi$ is in
particular a (2,0)-tensor, $\varphi$ induces by the fundamental principle of
tensor calculus a linear map $[\varphi]:(T_{x}N)^{*}\to T_{x}N$ via the action
$[\varphi]w:=\varphi^{ij}w(e_{i})e_{j}$. Equation (5) in work of the first
author [4] yields the estimate

$\displaystyle\operatorname{Rm}_{\hat{g}_{\lambda}}(\varphi,\varphi)-\operatorname{Rm}_{g}(\varphi,\varphi)$
$\displaystyle\geq
2\lambda(-\chi^{\prime\prime}(\lambda\rho))S([\varphi]d\rho,[\varphi]d\rho)$
$\displaystyle\quad+\varphi^{ij}\varphi^{kl}\chi^{\prime}(\lambda\rho)(-2D_{i}D_{k}\rho\,S_{jl}-\frac{1}{2}\chi^{\prime}(\lambda\rho)|D\rho|_{\hat{g}_{\lambda}}^{2}S_{ik}S_{jl})$
$\displaystyle\quad-C\chi^{\prime}(\lambda\rho)|[\varphi]d\rho|-C\lambda^{-1},$

where $C>0$ is a positive constant independent of $\lambda$.

To pick up the sectional curvatures, we choose for $1\leq p,q\leq n$ the two-
form $\varphi_{pq}$ by specifying the components
$(\varphi_{pq})^{ij}=\delta_{p}^{i}\delta_{q}^{j}-\delta_{p}^{j}\delta_{q}^{i}$.
This implies the identities

$\displaystyle[\varphi_{pq}]d\rho=\nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p}\quad\text{and}\quad
4\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q})=\operatorname{Rm}(\varphi_{pq},\varphi_{pq}).$

Recall that $S$ is supported in $U$, so $D\rho$ and $D^{2}\rho$ are uniformly
bounded with respect to the metric $\hat{g}_{\lambda}$. Thus the above
estimate implies after summation the inequality

(4)
$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\lambda(-\chi^{\prime\prime}(\lambda\rho))\
\sum_{p=1}^{m}\sum_{q=p+1}^{n}S\Big{(}\nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p},\ \nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p}\Big{)}-C\chi^{\prime}(\lambda\rho)-C\lambda^{-1}$

in the region $\\{\rho\geq e^{-\lambda^{2}}\\}$. Here the constant $C$ is
independent of $\lambda$, but it does depend on $S,g,\rho,N$.

By our assumption on the scalar-valued second fundamental forms and
Proposition 3.1 we deduce

$S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(d\rho\otimes d\rho)\in\operatorname{Int}(\mathcal{C}_{m})$

(with respect to the metric $g$) at each point on $\partial N$. This allows us
to fix a small number $a>0$ such that

$S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(d\rho\otimes d\rho)-4a|\nabla\rho|^{2}\delta{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\ }\delta\in\mathcal{C}_{m}$

(with respect to the metric $g$) in a small neighborhood of $\partial N$ where
$\rho\geq e^{-\lambda^{2}}$. From the construction of the metric
$\hat{g}_{\lambda}$ we deduce

$S{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(d\rho\otimes d\rho)-2a|\nabla\rho|^{2}\delta{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\ }\delta\in\mathcal{C}_{m}$

(with respect to the metric $\hat{g}_{\lambda}$) in a small neighborhood of
$\partial N$ where $\rho\geq e^{-\lambda^{2}}$. Moreover, we observe that

$S\Big{(}\nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\ e_{p},\ \nabla_{p}\rho\
e_{q}-\nabla_{q}\rho\ e_{p}\Big{)}=S{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\ }(d\rho\otimes
d\rho)(e_{p},e_{q},e_{p},e_{q}).$

For a positive constant
$C(n,m):=\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\delta{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}\delta)(e_{p},e_{q},e_{p},e_{q})$ we observe the estimate

(5) $\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}S\Big{(}\nabla_{p}\rho\
e_{q}-\nabla_{q}\rho\ e_{p},\ \nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p}\Big{)}$
$\displaystyle=\sum_{p=1}^{m}\sum_{q=p+1}^{n}(S{\leavevmode\nobreak\
\wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\ }(d\rho\otimes
d\rho))(e_{p},e_{q},e_{p},e_{q})$ $\displaystyle\geq 2aC(n,m)|\nabla\rho|^{2}$

in a small neighborhood of the boundary $\partial N$ where $\rho\geq
e^{-\lambda^{2}}$. Combining the estimate (4) and the estimate (5) we obtain
the estimate

(6)
$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))$
$\displaystyle\geq$ $\displaystyle
aC(n,m)\lambda(-\chi^{\prime\prime}(\lambda\rho))|\nabla\rho|^{2}-C\chi^{\prime}(\lambda\rho)-C\lambda^{-1}$

in the region $\\{\rho\geq e^{-\lambda^{2}}\\}$.

We split the region into two sub-regions as follows. Let us fix a real number
$s_{0}\in[0,1)$ such that $C\chi^{\prime}(s_{0})<\frac{\epsilon}{2}$. By the
construction of the cut-off function $\chi$, we have $\inf_{0\leq s\leq
s_{0}}(-\chi^{\prime\prime}(s))>0$. This implies in the region
$\\{e^{-\lambda^{2}}\leq\rho<s_{0}\lambda^{-1}\\}$ the estimate

$\displaystyle\inf_{e^{-\lambda^{2}}\leq\rho<s_{0}\lambda^{-1}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))\right)$
$\displaystyle\geq$ $\displaystyle
aC(n,m)\lambda\,\inf_{e^{-\lambda^{2}}\leq\rho<s_{0}\lambda^{-1}}\left((-\chi^{\prime\prime}(\lambda\rho))|\nabla\rho|^{2}\right)-C-C\lambda^{-1}$
$\displaystyle\geq$ $\displaystyle aC(n,m)\lambda\,\inf_{0\leq s\leq
s_{0}}(-\chi^{\prime\prime}(s))-C-C\lambda^{-1}.$

Thus, we obtain

$\inf_{e^{-\lambda^{2}}\leq\rho<s_{0}\lambda^{-1}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))\right)\to\infty$

as $\lambda\to\infty$. Moreover, in the region $\\{\rho\geq
s_{0}\lambda^{-1}\\}$ we have

$\displaystyle\inf_{\rho\geq
s_{0}\lambda^{-1}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))\right)\geq-C\chi^{\prime}(s_{0})-C\lambda^{-1}.$

Since $C\chi^{\prime}(s_{0})<\frac{\epsilon}{2}$, it follows that

$\inf_{\rho\geq
s_{0}\lambda^{-1}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))\right)\geq-\epsilon$

if $\lambda>0$ is sufficiently large. Putting the above together, we conclude
that

$\inf_{\rho\geq
e^{-\lambda^{2}}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}(e_{p},e_{q},e_{p},e_{q}))\right)\geq-\epsilon$

if $\lambda>0$ is sufficiently large. This completes the proof of Proposition
4.1. ∎

We next consider the region $\\{\rho<e^{-\lambda^{2}}\\}$:

######  Proposition 4.2 (Curvature estimates in outer gluing region).

Suppose that $h_{g}-h_{\tilde{g}}$ is $m$-positive on the boundary $\partial
N$. Let $\epsilon>0$ be an arbitrary positive real number. If
$\lambda=\lambda(\epsilon,\beta)>0$ is sufficiently large, then

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(x)(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}_{\tilde{g}}(x)(e_{p},e_{q},e_{p},e_{q}))\geq-\epsilon$

for any $\hat{g}_{\lambda}$-orthonormal frame $\\{e_{1},\dots,e_{n}\\}$ and
any $x\in N$ in the region $\\{\rho(x)<e^{-\lambda^{2}}\\}$.

###### Proof.

In the region $\\{\rho<e^{-\lambda^{2}}\\}$, we have
$\hat{g}_{\lambda}=\tilde{g}+\tilde{h}_{\lambda}$, where $\tilde{h}_{\lambda}$
is defined by

$\displaystyle\tilde{h}_{\lambda}=-\lambda\rho^{2}\beta(\lambda^{-2}\log\rho)S.$

Let $\\{e_{1},\dots,e_{n}\\}$ be a geodesic normal frame around $x$ with
respect to the metric $\hat{g}_{\lambda}$. Equation (12) in work of the first
author [4] implies

(7)
$\displaystyle{}\operatorname{Rm}_{\hat{g}_{\lambda}}(\varphi,\varphi)-\operatorname{Rm}_{\tilde{g}}(\varphi,\varphi)\geq\
$ $\displaystyle
2\lambda\beta(\lambda^{-2}\log\rho)S([\varphi]d\rho,[\varphi]d\rho)-L\lambda^{-1}$

for any two-form $\varphi=\varphi^{ij}e_{i}\wedge e_{j}$ and a positive
constant $L>0$ independent of $\lambda$. Choosing for $1\leq p,q\leq n$ the
two-form $\varphi_{pq}$ by
$(\varphi_{pq})^{ij}=\delta_{p}^{i}\delta_{q}^{j}-\delta_{p}^{j}\delta_{q}^{i}$
in equation (7) and summing over $p,q$, we obtain

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}_{\tilde{g}}(e_{p},e_{q},e_{p},e_{q}))$
$\displaystyle\geq\frac{1}{2}\lambda\beta(\lambda^{-2}\log\rho)\sum_{p=1}^{m}\sum_{q=p+1}^{n}S\Big{(}\nabla_{p}\rho\
e_{q}-\nabla_{q}\rho\ e_{p},\ \nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p}\Big{)}-L\lambda^{-1}.$

Proceeding similarly as in the proof of Proposition 4.1, we have

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}S\Big{(}\nabla_{p}\rho\
e_{q}-\nabla_{q}\rho\ e_{p},\ \nabla_{p}\rho\ e_{q}-\nabla_{q}\rho\
e_{p}\Big{)}\geq 2a|\nabla\rho|_{\tilde{g}}^{2}.$

(with respect to the metric $\hat{g}_{\lambda}$) in a neighborhood of
$\partial N$ where $\rho<e^{-\lambda^{2}}$. This implies

(8)
$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}_{\tilde{g}}(e_{p},e_{q},e_{p},e_{q}))\geq
a\lambda\beta(\lambda^{-2}\log\rho)|\nabla\rho|^{2}-L\lambda^{-1}$

in the region $\\{\rho<e^{-\lambda^{2}}\\}$. Hence, if $\lambda>0$ is
sufficiently large, then we have

$\displaystyle\inf_{\rho<e^{-\lambda^{2}}}\left(\sum_{p=1}^{m}\sum_{q=p+1}^{n}(\operatorname{Rm}_{\hat{g}_{\lambda}}(e_{p},e_{q},e_{p},e_{q})-\operatorname{Rm}_{\tilde{g}}(e_{p},e_{q},e_{p},e_{q}))\right)\geq-\epsilon.$

From this, the assertion follows. ∎

Combining Proposition 4.1 and Proposition 4.2, we can summarize the results in
this section:

######  Corollary 4.3.

Suppose that $h_{g}-h_{\tilde{g}}$ is $m$-positive on the boundary $\partial
N$. Let $\epsilon>0$ be an arbitrary positive real number. If
$\lambda=\lambda(\epsilon,\chi,\beta)>0$ is sufficiently large, then we have
the pointwise inequality

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\hat{g}_{\lambda}}(x)(e_{p},e_{q},e_{p},e_{q})$
$\displaystyle\geq$
$\displaystyle\min\left\\{\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{g}(x)(e_{p},e_{q},e_{p},e_{q}),\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\tilde{g}}(x)(e_{p},e_{q},e_{p},e_{q})\right\\}-\epsilon$

for any $\hat{g}_{\lambda}$-orthonormal frame $\\{e_{1},\dots,e_{n}\\}$ and
any $x\in N$.

##  5\. Proof of Theorem 1.2 on preserving positive intermediate curvature

Suppose that $h_{g}-h_{\tilde{g}}$ is $m$-positive on the boundary $\partial
N$. Suppose also that $(N,g)$ and $(N,\tilde{g})$ have positive
$m$-intermediate curvature.

Fix a point $x\in N$ and let $\\{E_{1},\dots,E_{N}\\}$ be an orthonormal basis
of the tangent space $T_{x}N$ with respect to the Riemannian metric
$\hat{g}_{\lambda}$. We want to show that

$\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\hat{g}_{\lambda}}(E_{p},E_{q},E_{p},E_{q})>0.$

We divide the proof into two cases: In the first case $x\in N$ is in the inner
gluing region $\\{\rho\geq e^{-\lambda^{2}}\\}$; and in the second case $x\in
N$ is in the outer gluing region $\\{\rho<e^{-\lambda^{2}}\\}$.

For the first case, we have $g=\hat{g}_{\lambda}-h_{\lambda}$ where
$h_{\lambda}(x)=\lambda^{-1}\chi(\lambda\rho)S(x)$. Fix a point $x\in N$. We
evolve the orthonormal basis $\\{E_{1},\dots,E_{n}\\}$ in the tangent space
$T_{x}N$ by the linear ordinary differential equation

(9) $\displaystyle\begin{cases}\frac{d}{ds}E_{i}(s)&=\
\frac{1}{2}h_{\lambda}\circ E_{i}(s)\\\ E_{i}(0)&=\ E_{i}.\end{cases}$

Then we see that the basis $\\{E_{i}(s)\\}$ remains orthonormal with respect
to the metric $g_{s}=\hat{g}_{\lambda}-sh_{\lambda}$. We define
$e_{i}:=E_{i}(1)$. Then the basis $\\{e_{1},\dots,e_{n}\\}$ is orthonormal
with respect to the metric $g=\hat{g}_{\lambda}-h_{\lambda}$. It follows that

$\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{g}(e_{p},e_{q},e_{p},e_{q})>0.$

In view of Corollary 4.3, it suffices to show that

$\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{g}(E_{p},E_{q},E_{p},E_{q})>0.$

By writing $E_{i}(s)=A_{i}^{k}(s)E_{k}$, we observe that

$\displaystyle\frac{d}{ds}E_{i}(s)$
$\displaystyle=\frac{d}{ds}A_{i}^{k}(s)E_{k}=\frac{1}{2}h_{l}^{k}A_{i}^{l}(s)E_{k}=\frac{1}{2}\lambda^{-1}\chi(\lambda\rho)S_{l}^{k}A_{i}^{l}(s)E_{k}$

for any $s\in[0,1]$. This implies the estimate

$\displaystyle\Big{|}\frac{d}{ds}\operatorname{Rm}_{g}(E_{p}(s),E_{q}(s),E_{p}(s),E_{q}(s))\Big{|}\leq
C\lambda^{-1},$

where $C$ is a positive constant depending only on $N,g,\tilde{g}$ and $\chi$.
This implies that

$\displaystyle\Big{|}\operatorname{Rm}_{g}(E_{p},E_{q},E_{p},E_{q})-\operatorname{Rm}_{g}(e_{p},e_{q},e_{p},e_{q})\Big{|}$
$\displaystyle\leq\int_{0}^{1}\frac{C}{\lambda}d\tau\leq\frac{C}{\lambda}.$

Hence we have

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{g}(E_{p},E_{q},E_{p},E_{q})\geq\frac{1}{2}\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{g}(e_{p},e_{q},e_{p},e_{q})>0,$

if $\lambda>0$ is sufficiently large.

For the second case, we have
$\tilde{g}=\hat{g}_{\lambda}+\lambda\rho^{2}\beta(\lambda^{-2}\log\rho)S$.
Thus $h_{\lambda}=\lambda\rho^{2}\beta(\lambda^{-2}\log\rho)S$. Following the
same argument as in the first case and using the fact that

$\displaystyle|h_{\lambda}|\leq
C(g,\tilde{g},\beta)e^{-\lambda^{2}}\leq\frac{C(g,\tilde{g},\beta)}{\lambda}$

in the region $\\{\rho<e^{-\lambda^{2}}\\}$ for sufficiently large
$\lambda>0$, we also deduce

$\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\tilde{g}}(E_{p},E_{q},E_{p},E_{q})\geq\frac{1}{2}\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\tilde{g}}(\tilde{e}_{p},\tilde{e}_{q},\tilde{e}_{p},\tilde{e}_{q})>0$

for sufficiently large $\lambda>0$.

Combining the two cases together, Corollary 4.3 implies that

$\displaystyle\sum_{p=1}^{m}\sum_{q=p+1}^{n}\operatorname{Rm}_{\hat{g}_{\lambda}}(E_{p},E_{q},E_{p},E_{q})>0$

for sufficiently large $\lambda>0$. Since the point $x\in N$ and the
orthonormal basis $\\{E_{1},...,E_{n}\\}\subset T_{x}N$ are arbitrary, we
conclude that

$\operatorname{Rm}_{\hat{g}_{\lambda}}\in\operatorname{Int}(\mathcal{C}_{m})$

on $N$ for sufficiently large $\lambda>0$. This finishes the proof of the
theorem.

##  6\. Proof of Theorem 1.5 on boundaries of partially torical bands

In this section we prove Theorem 1.5 by using the strategy outlined by M.
Gromov and H.-B. Lawson. We prove a generalization of their doubling lemma (as
stated in Lemma 1.3).

Once we have established the doubling lemma Theorem 1.5 follows directly by
the generalization of the Geroch conjecture, Theorem 1.1.

######  Lemma 6.1 (Doubling for positive $m-$intermediate curvature).

Suppose $(N^{n},g)$ is an orientable compact smooth Riemannian manifold with
smooth boundary $\partial N$, such that the metric $g$ has positive
$m$-intermediate curvature, and the boundary $\partial N$ is strictly
$m$-convex (possibly finitely many connected components). Then the double of
$N$ carries a metric of positive $m$-intermediate curvature.

We mimic the doubling process by M. Gromov and H.-B. Lawson, see Theorem 5.7
in [6]. We will closely follow their notations and constructions. For
completeness we describe their construction in detail.

We first shrink the manifold $N$ a little bit while preserving its boundary
condition as follow: let $N_{1}=N\backslash C$ where $C$ is a thin collar of
the boundary $\partial N$ and $N_{1}$ is chosen so that $\partial N_{1}$ is
still strictly $m$-mean convex. We then consider the Riemannian product
$N\times I$ with Riemannian metric $g_{N}+dt\otimes dt$ and define
$D(N)=\\{p\in N\times I\mid\text{dist}(p,N_{1})=\epsilon\\}$, where
$0<\epsilon\ll 1$. Note that $D(N)$ is homeomorphic to the double of $N$.

Now we fix a point $x\in\partial N_{1}$, and let $\sigma$ be the geodesic
segment in $N_{1}$ emanating orthogonally from $\partial N_{1}$ at $x$. Then
the product $\sigma\times I$ will be totally geodesic in the product $N\times
I$.

Let $\mu_{1},...\mu_{n-1}$ be the principal curvatures of $\partial N_{1}$ at
$x$. By the construction of M. Gromov and H.-B. Lawson the principle
curvatures of $D(N)$ will be of the following form for a point $x_{\theta}$
corresponding to an angle $\theta$ (see Figure 8 in [6]):

(10)
$\displaystyle\lambda_{k}=(\mu_{k}+O(\epsilon))\cos\theta+O(\epsilon^{2})\;\text{for}\;k=1,\dots,n-1,\;\text{and}\;\lambda_{n}=\frac{1}{\epsilon}\cos\theta+O(\epsilon).$

As in the Figure 8 in [6], we have a natural polar coordinates describing
these $x_{\theta}$, let us denote $pr_{N}(x_{\theta})$ the projection of
$x_{\theta}$ to the corresponding point on $\partial N_{1}$. Since the
bilinear forms $h_{\partial N_{1}}$ at $pr_{N}(x_{\theta})$ and $h_{D(N)}$ at
$x_{\theta}$ are diagonalized simultaneously in this construction, we have the
following relation:

$\displaystyle h_{D(N)}\Big{|}_{x_{\theta}}=\left((\cos\theta)h_{\partial
N_{1}}\Big{|}_{pr_{N}(x_{\theta})}+O(\epsilon)\right)\oplus\left(\frac{1}{\epsilon}\cos\theta+O(\epsilon)\right)(\nu^{\flat}\otimes\nu^{\flat})\Big{|}_{pr_{N}(x_{\theta})}.$

We apply the Gauss equation to $D(N)$ as a submanifold of $N\times I$ at the
point $x_{\theta}$: We have for any orthonormal basis $\\{e_{i}\\}_{i=1}^{n}$
at the point $x_{\theta}$ the relation

(11)
$\displaystyle\text{Rm}^{D(N)}(e_{i},e_{j},e_{i},e_{j})=\text{Rm}^{N\times
I}(e_{i},e_{j},e_{i},e_{j})+h_{D(N)}(e_{i},e_{i})h_{D(N)}(e_{j},e_{j})-h_{D(N)}(e_{i},e_{j})^{2}.$

We note that the second fundamental form terms of $D(N)$ are related to the
second fundamental form terms of the boundary $\partial N_{1}$. This implies

$\displaystyle
h_{D(N)}(e_{i},e_{i})h_{D(x)}(e_{j},e_{j})-h_{D(N)}(e_{i},e_{j})^{2}\Big{|}_{x_{\theta}}$
$\displaystyle=$ $\displaystyle\left((\cos\theta)h_{\partial
N_{1}}(e_{i},e_{i})+\frac{1}{\epsilon}\cos\theta\cdot\nu^{\flat}(e_{i})^{2}+O(\epsilon)\right)\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle\cdot\left((\cos\theta)h_{\partial
N_{1}}(e_{j},e_{j})+\frac{1}{\epsilon}\cos\theta\cdot\nu^{\flat}(e_{j})^{2}+O(\epsilon)\right)\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle-\left((\cos\theta)h_{\partial
N_{1}}(e_{i},e_{j})+\frac{1}{\epsilon}\cos\theta\cdot\nu^{\flat}(e_{i})\nu^{\flat}(e_{j})+O(\epsilon)\right)^{2}\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle=$
$\displaystyle\frac{1}{\epsilon^{2}}\left(\cos^{2}\theta\nu^{\flat}(e_{i})^{2}\nu^{\flat}(e_{j})^{2}-\cos^{2}\theta\nu^{\flat}(e_{i})^{2}\nu^{\flat}(e_{j})^{2}\right)\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle+\frac{1}{\epsilon}\cos^{2}\theta\left(h_{\partial
N_{1}}(e_{i},e_{i})\nu^{\flat}(e_{j})^{2}+h_{\partial
N_{1}}(e_{j},e_{j})\nu^{\flat}(e_{i})^{2}-2h_{\partial
N_{1}}(e_{i},e_{j})\nu^{\flat}(e_{i})\nu^{\flat}(e_{j})\right)\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle+O(1)$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon}\cos^{2}\theta\left(h_{\partial
N_{1}}{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{\flat}\otimes\nu^{\flat})\right)(e_{i},e_{j},e_{i},e_{j})\Big{|}_{pr_{N}(x_{\theta})}+O(1),$

where the terms of order $O(\frac{1}{\epsilon^{2}})$ cancelled.

Summation of equation (11) yields the following identity for the
$m$-intermediate curvature:

$\displaystyle\sum\limits_{p=1}^{m}\sum\limits_{q=p+1}^{n}\text{Rm}^{D(N)}(e_{p},e_{q},e_{p},e_{q})\Big{|}_{x_{\theta}}$
$\displaystyle=$
$\displaystyle\sum\limits_{p=1}^{m}\sum\limits_{q=p+1}^{n}\left(\text{Rm}^{N\times
I}(e_{p},e_{q},e_{p},e_{q})+h_{D(N)}(e_{p},e_{p})h_{D(N)}(e_{q},e_{q})-h_{D(N)}(e_{p},e_{q})^{2}\right)\Big{|}_{x_{\theta}}$
$\displaystyle=$
$\displaystyle\sum\limits_{p=1}^{m}\sum\limits_{q=p+1}^{n}\text{Rm}^{N}(e_{p},e_{q},e_{p},e_{q})\Big{|}_{pr_{N}(x_{\theta})}$
$\displaystyle+\sum\limits_{p=1}^{m}\sum\limits_{q=p+1}^{n}\frac{1}{\epsilon}\cos^{2}\theta\left(h_{\partial
N_{1}}{\leavevmode\nobreak\ \wedge\\!\\!\\!\\!\\!\bigcirc\leavevmode\nobreak\
}(\nu^{\flat}\otimes\nu^{\flat})\right)(e_{p},e_{q},e_{p},e_{q})\Big{|}_{pr_{N}(x_{\theta})}+O(1)$

The result then follows from Proposition 3.1 by choosing $\epsilon>0$
sufficient small.

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# A Principle of Maximum Entropy
for the Navier-Stokes Equations

Gui-Qiang G. Chen  Mathematical Institute, University of Oxford, Oxford, OX2
6GG, UK<EMAIL_ADDRESS>,  James Glimm  Stony Brook University, Stony
Brook, NY 11794, and GlimmAnalytics LLC, USA<EMAIL_ADDRESS>and  Hamid
Said  Department of Mathematics, College of Science, Kuwait University, Safat
13060, Kuwait<EMAIL_ADDRESS>
(Date: August 28, 2024; Date: August 28, 2024)

###### Abstract.

A principle of maximum entropy is proposed in the context of viscous
incompressible flow in Eulerian coordinates. The relative entropy functional,
defined over the space of $L^{2}$ divergence-free velocity fields, is
maximized relative to alternate measures supported over the energy–enstrophy
surface. Since thermodynamic equilibrium distributions are characterized by
maximum entropy, connections are drawn with stationary statistical solutions
of the incompressible Navier-Stokes equations. Special emphasis is on the
correspondence with the final statistics described by Kolmogorov’s theory of
fully developed turbulence.

###### Key words and phrases:

Principle of maximum entropy, energy-enstrophy surface, Navier-Stokes
equations, fully developed turbulence.

###### 2020 Mathematics Subject Classification:

28D20, 76F02, 28C20, 76D05, 49S05, 35A15, 70G10, 35Q30, 37A50

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation for entropy maximization
    2. 1.2 Principle of maximum entropy: classical formulation
  2. 2 Entropy Maximization and the Physical Measure
    1. 2.1 Principle of maximum entropy: formulation for the Navier-Stokes equations
    2. 2.2 Main theorem for the principle of maximum entropy
  3. 3 Connections and Remarks

##  1\. Introduction

We are concerned with the reformulation of the principle of maximum entropy,
as established in the context of statistical mechanics and information theory,
and its applications to mathematical fluid dynamics. More specifically, the
Boltzmann–Gibbs entropy under investigation (cf. [9, 21]):

$S(f)=-\int f(u)\log f(u)\mathop{}\\!\mathrm{d}\eta(u)$ (1.1)

is maximized under appropriate constraints resulting from physical
considerations111In case of a discrete system in which probabilities
$f_{1},f_{2},\cdots,f_{k}$ are assigned to a random variable $Y$ taking values
$y_{1},y_{2},\cdots,y_{k}$, respectively, then the entropy is expressed as
$S=-\sum_{i}f_{i}\log f_{i}$.. From a mathematical standpoint, these
considerations impose constraints on the class of admissible velocity fields
$u$ and their associated probability distributions $f(u)$.

The principle of maximum entropy, introduced by E. Jaynes [13] in 1957,
revolves around making the most desirable choice, for given prior data. This
choice aims to maximize the entropy functional (1.1). In other words, as
expressed in [12], “[T]he [principle of maximum entropy] is the probability
assignment that is consistent with the available information but is maximally
noncommittal with regard to missing information.” From the standpoint of
statistical mechanics, it is seen as a method to determine the correct
probability distribution at the state of equilibrium, depending heavily on the
constraints imposed on the system, i.e., our knowledge of prior or given
information. As the principle of maximum entropy provides a rule for assigning
probabilities to data, it proves particularly useful in “filling gaps” between
otherwise scattered data. Therefore, maximum entropy modeling can be applied
to a wide range of problems. We refer to [14] for applications to several
areas including statistics, biology, and medicine.

The physics underlying our problem is fundamentally rooted in the first and
second laws of thermodynamics. The former signifies the conservation of energy
$E$, while the latter imposes a constraint on admissible processes in terms of
the production of entropy $S$. At a micro-level, where each micro-state is
equally probable, entropy can be expressed in terms of the Boltzmann factor
$k_{B}$:

$S=k_{B}\log(\Omega)$

where $\Omega$ is the number of states present at a given energy $E$. Equation
(1.1) can be understood as a generalization of the above formula for entropy
$S$ when the micro-states of the system may not be equally distributed.

This formulation of the second law in terms of a single entropy can be
rewritten in a perhaps more conventional manner. Introduce two entropies,
$S_{E}$ for the entropy of the energetic degrees of freedom which is related
to the viscous and turbulent dissipation of energy, and $S_{Z}$ for the
entropy of the vortical and enstropic degrees of freedom which is related to
the viscous and turbulent dissipation of enstrophy. We assume that both the
energetic and the enstrophic degrees of freedom are governed by a common
temperature, which can be fixed in the context of the second law of
thermodynamics. Consequently, the second law takes the form:

${\rm d}E=T{\rm d}S_{E}+T{\rm d}S_{Z}.$

The first of the terms on the right-hand side aligns with the familiar
expression from standard thermodynamics, a scenario that typically excludes
physics models featuring an enstropic degree of freedom. The inclusion of the
second term is necessitated by the much richer physical model under
consideration here.

In tandem with the modified or reformulated second law, we can express the
Legendre transforms and the definition of entropy as the logarithm of the
volume of a constant energy surface, now accounting for two energy sources.
Consequently, we represent

${\rm d}E={\rm d}E_{K}+{\rm d}E_{Z}$

as a summation of the kinetic and enstrophic energies, resulting in two terms
on the left-hand side of the second law. The kinetic energy $E_{K}$ aligns
with the conventional description in thermodynamics, while $E_{Z}$ is an
additional energy source specific to the effect that the entropy considered is
for the simplest of the fluid physics theories possible, namely incompressible
with no mixtures.

From a mathematical standpoint, the Boltzmann–Gibbs entropy (or its discrete
counterpart) is a special case of what is known as the _relative entropy_
first introduced by Kullback and Leibler in 1951 [16] (consult [15] for more
details). Consider a probability space $(\mathcal{H},\mathcal{B(H)})$ and two
measures $\eta$ and $\mu$ defined on it, where $\mathcal{B(H)}$ denotes the
space of Borel probability measures over space $\mathcal{H}$. Then the
relative entropy $S(\mu|\eta)$ (of $\mu$ with respect to $\eta$) is defined by

$\mathcal{S}(\mu|\eta):=\begin{cases}-\displaystyle\int_{\mathcal{H}}\dfrac{\mathop{}\\!\mathrm{d}\mu}{\mathop{}\\!\mathrm{d}\eta}\log(\dfrac{\mathop{}\\!\mathrm{d}\mu}{\mathop{}\\!\mathrm{d}\eta})\mathop{}\\!\mathrm{d}\eta(u)&\text{if
}\mu\ll\eta,\\\ -\infty&\mathrm{otherwise}.\end{cases}$ (1.2)

In the case where measure $\mu$ is absolutely continuous with respect to
$\eta$, the Radon-Nikodym derivative
$\frac{\mathop{}\\!\mathrm{d}\mu}{\mathop{}\\!\mathrm{d}\eta}$ is in
$L^{1}({\rm d}\eta)$. Once $\eta$ is chosen as the Lebesgue measure (when
$\mathcal{H}=\mathbb{R}^{n}$), then
$\frac{\mathop{}\\!\mathrm{d}\mu}{\mathop{}\\!\mathrm{d}\eta}$ can identified
with the probability distribution $f(u)$ in equation (1.1). The relative
entropy turns into a functional that is seen to be maximized over the space of
all probability measures, once we fix the _reference_ measure $\eta$.
Therefore, the maximum entropy problem consists of finding a probability
measure $\mu$ on $(\mathcal{H},\mathcal{B(H)})$, depending now on the choice
of measure $\eta$, such that the relative entropy is maximized under some
given information. In the probability space, $-\mathcal{S}(\mu|\eta)$ can be
perceived as a measure of “distance” between the maximum entropy measure $\mu$
and the reference measure $\eta$, or any two measures for that matter.

In the context of fluid flow, a theory for entropy maximization was first
established in [10], in which a well-defined probability measure acting as a
linear functional on the configuration space of the Euler and Navier-Stokes
equations, reduced to enforce particle interchange symmetries, was proved to
maximize the Boltzmann–Gibbs entropy relative to alternate solutions,
restricted to the constant energy surface. In this paper, we establish the
existence of a probability measure on the configuration space satisfying a
maximum entropy admissibility condition relative to alternate solutions of the
Navier-Stokes equations, yet with a restriction to the _energy-enstrophy_
surface. The significance of this restriction lies in its connection to the
pursuit of establishing a rigorous mathematical foundation for the
investigation of turbulent phenomena, a link that we briefly outline in the
following section.

###  1.1. Motivation for entropy maximization

The recent nonuniqueness result for the Navier-Stokes equations [2] suggest
that weak solutions may fall short from describing the complex nature of fluid
flow. On this basis, a probabilistic description of the flow can serve as an
alternate framework for studying the Navier-Stoke and Euler equations, as
opposed to classical weak solutions and their Euler limits. In fact, any
comprehensive exploration of turbulent flow phenomena must contend with some
notion of _averaging_ ; especially as it relates to formulating the Kolmogorov
theory of fully developed turbulence, widely considered to be valid by the
practitioners of the field. We refer the reader to [8] for a modern
comprehensive account on this topic.

In an endeavor to establish a mathematical theory of turbulence, Foias [5]
(see also [6]), inspired by earlier work of G. Prodi, initiated a program to
give, among other things, a rigorous formulation of the notion of _statistical
solutions_ for the Navier-Stokes equations and eventually derive the
predictions of Kolmogorov for the energy spectrum and the structure functions
[17, 18]. The Kolmogorov power laws are formulated in relation to the
_average_ (long time averages or ensemble averages) of certain physically
measurable quantities; hence, by constructing a family of time-dependent and
spatially homogeneous probability measures $\mu_{t}$ (over an appropriate
Hilbert space) that satisfy the Navier-Stokes equations in some averaged
sense222A closely related theory was introduced by M. Vishik and A. Furkisov
[20], where statistical solutions exist as probability measures on the set of
solution trajectories of the Navier-Stokes equations, e.g., $C^{\rm
w}_{t}L^{2}_{x}$. See also the work of Foias, Rosa, and Temam [7] for a
connection between these two notions of solutions.. Therefore, Foias, later
with Manley, Roas, and Temam [5, 6], was able to attach an unambiguous meaning
to the notion of an ensemble average333 In fact, Foias _et al._ proved the
convergence of long-time averages
$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\Phi(u(t))\mathop{}\\!\mathrm{d}t$
for a suitable class of functions $\Phi$ and the weak solutions $u$ of the
Navier-Stokes equations. See [6] for the precise statement and more details..
In the context of this program, one main assumption embedded in the derivation
of the Kolmogorov power laws is that the global energy _identity_ for the
Navier-Stokes equations (with normalized mass density and viscosity $\nu$):

$\dfrac{1}{2}\|u(t)\|^{2}+\nu\int_{0}^{t}\|\nabla
u(s)\|^{2}\mathop{}\\!\mathrm{d}s=\frac{1}{2}\|u_{0}\|^{2}$ (1.3)

is satisfied when integrated against a (spatially) homogeneous statistical
solution $\mu_{t}$ for all $t>0$. Here $u_{0}$ is taken to be the (given)
initial velocity of the fluid, _i.e._ , $u(x,0)\doteq u_{0}$, and $\|\cdot\|$
denotes the $L^{2}_{x}$-norm for simplicity. These solutions, which also
remain invariant under re-scaling, are called _self-similar_ homogeneous
statistical solutions to the Navier-Stokes equations.

Leray-Hopf solutions are known to satisfy the global energy _inequality_
(obtained by replacing “=” with “$\leq$” in relation (1.3)). Whether identity
(1.3) holds in three space dimensions remains one of the major open problems
for weak solutions of the Navier-Stokes equations. Incidentally, the same is
true for statistical solutions: only an averaged version of the global energy
inequality is known to hold.

In other words, these claims infer that a probabilistic description consistent
with Kolmogorov’s spectrum needs to be supported over the set

$G_{e}(t)\doteq\Big{\\{}u\,:\,\dfrac{1}{2}\|u(t)\|^{2}+\nu\int_{0}^{t}\|\nabla
u(s)\|^{2}\mathop{}\\!\mathrm{d}s=e(t)\Big{\\}}$ (1.4)

where the prescribed energy $e(t)$ is taken to be the energy profile for the
initial velocity. However, since the existence and uniqueness of such
solutions remain in question, energy $e(t)$ can be prescribed _a priori_ to
ensure that $G_{e}(t)$ is nonempty. Throughout this paper, we assume that, for
each $t\geq 0$, $G_{e}(t)$ is defined for any prescribed positive energy
$e(t)$; such a set $G_{e}(t)$ is called the _energy-enstrophy surface._

A similar challenge arises when constructing numerical solutions. It is known
that statistical solutions of the incompressible Euler equations and
multidimensional systems of conservation laws can be computed. However, due to
the lack of the well-posedness of these problems and the lack of convergence
of their numerical approximations, it is not known whether these solutions are
entropy or entropy maximizing solutions444Incidentally, as is mentioned in
[4], the lack of rigorous convergence of numerical schemes may be attributed
to the emergence of turbulent-like structures at finer scales as the mesh is
refined.. See [4, 19] and references therein for more details.

Next, we demonstrate how the principle of maximum entropy provides a
systematic approach for determining probability distributions in the presence
of constraints on the mean energy of the system.

###  1.2. Principle of maximum entropy: classical formulation

As indicated above, the principle of maximum entropy can be formulated in a
number of different ways depending on modeling considerations. The formulation
in the context of equilibrium statistical mechanics, which is presented here,
most resembles the application of the principle for the purposes of this work.
Consider the entropy functional (1.1) together with the two constraints:

$\int g(u)\mathop{}\\!\mathrm{d}\xi(u)=1,\qquad\int
g(u)\,\dfrac{m\,u^{2}}{2}\mathop{}\\!\mathrm{d}\xi(u)=\varepsilon,$ (1.5)

where $\dfrac{m\,u^{2}}{2}$ is the kinetic energy of a single particle as
measured in a fixed frame, $\varepsilon$ is the mean energy per particle which
is _a priori_ known, and measure $\xi$ is assumed to exist over the
configuration space consisting of all velocity fields555In the finite-
dimensional setting, $\xi$ is taken to be the Lebesgue measure, and the
configuration space consists of all possible velocity components of a single
particle: $u=(u_{1},u_{2},\cdots,u_{n})\in\mathbb{R}^{n}$.. The principle of
maximum entropy, then, asserts that the physical entropy achieves its maximum,
which we denote $g^{*}(u)$, for the constrained system. Hence we may apply the
methods of the calculus of variations to functional (1.1) subject to
constraints (1.5) to conclude that there exist two Lagrange multipliers
$\alpha$ and $\beta$ satisfying

$\int\dfrac{\delta L}{\delta
g}(g(u);\alpha,\beta)\,\phi(u)\mathop{}\\!\mathrm{d}\xi(u)\bigg{|}_{g=g*}=0$

for any test function $\phi(u)$, where

$L(g(u);\alpha,\beta)=-g(u)\log
g(u)+\alpha(1-g(u))+\beta(\varepsilon-g(u)\,\dfrac{m\,u^{2}}{2}).$

Therefore, we obtain

$-\int\phi(u)\Big{(}\big{(}1+\log
g^{*}(u)\big{)}+\alpha+\beta\dfrac{m\,u^{2}}{2}\Big{)}\mathop{}\\!\mathrm{d}\xi(u)=0.$
(1.6)

If the integrand is assumed to be sufficiently regular, then

$\big{(}1+\log g^{*}(u)\big{)}+\alpha+\beta\dfrac{m\,u^{2}}{2}=0$

for all $u$, so that

$g^{*}(u)=c_{\alpha}\exp(-\beta\frac{m\,u^{2}}{2}),$ (1.7)

where $c_{\alpha}=\exp(-(1+\alpha))$. The two constraints (1.5) can be used to
find parameters $\alpha$ and $\beta$ in terms of the known mean energy
$\varepsilon$. One can even go a step further and determine the Lagrange
multipliers in terms of more fundamental thermodynamics quantities by
employing the first law of thermodynamics [1]. In any case, we obtain the
well-known Maxwell-Boltzmann distribution describing the thermal equilibrium
velocity distribution through the principle of maximum entropy.

In a similar fashion, the maximum entropy probability measure $\mu_{t}$ (whose
support is $G_{e}(t)$) obtained for the problem of fluid flow should be
associated with a universal law of decay in the limit: $t\to\infty$. This is
indeed the case for self-similar homogeneous statistical solutions
constructed: the average kinetic energy scales like $\dfrac{M}{t}$ as
$t\to\infty$, where constant $M=M(\nu)$ is independent of any initial
conditions (cf. [6]).

As will be illustrated in §2, a version of the energy constraint similar to
(1.5) will be presented, but not explicitly: it is included in terms of the
support of the maximum entropy probability measure, that is to say over set
$G_{e}(t)$. This choice is motivated by the role that the global (averaged)
energy identity for self-similar homogeneous statistical solutions plays in
producing the predictions of Kolmogorov’s theory of turbulence.

##  2\. Entropy Maximization and the Physical Measure

In this section, we establish the principle of maximum entropy in the context
of incompressible flow. We begin first by setting out the notation and
formulating the problem.

###  2.1. Principle of maximum entropy: formulation for the Navier-Stokes
equations

The fixed time particle configuration space is the Hilbert space
${\mathcal{H}}$ of $L^{2}$ divergence-free velocity fields defined on a cube
$V\subset\mathbb{R}^{3}$ and satisfying periodic boundary conditions on the
boundary of $V$. For every $t\in[0,T]$ (fixed), define the Banach space
$X_{t}$ to be the space $L^{\infty}_{t}L^{2}_{x}\cap L^{2}_{t}H^{1}_{x}$
endowed with the norm:

$\|u\|_{t}^{2}\doteq\dfrac{1}{2}\|u(t)\|^{2}+\nu\int_{0}^{t}\|\nabla
u(s)\|^{2}{\rm d}s,$

where $\|\cdot\|$ denotes the $L^{2}_{x}$-norm. Throughout this section, given
a fixed $t\geq 0$, we fix a scalar $e=e(t)\geq 0$ to define the energy-
enstrophy surface:

$G_{e}(t)=\big{\\{}u\in X_{t}\,:\,\|u\|_{t}^{2}=e(t)\big{\\}}\qquad\mbox{for
each $t\in[0,T]$},$ (2.1)

and make the identification

$G_{e}(0)=\big{\\{}u_{0}\in\mathcal{H}(V)\,:\,\|u\|_{0}=\dfrac{1}{2}\|u_{0}\|^{2}=e(0)\big{\\}}.$
(2.2)

It is clear that $G_{e}(t)$ and $G_{e}(0)$ are non-empty. For each $t\geq 0$
and non-negative $e(t)$, a given physical probability measure $\eta_{e,t}$ is
supported over $G_{e}(t)$. For measures $\eta_{e,t}$ and $\mu_{e,t}$ belonging
the probability space $\mathcal{M}:=(\mathcal{H,B(H)})$, we define

$\mathcal{S}(\mu_{e,t}|\eta_{e,t}):=\begin{cases}-\displaystyle\int_{\mathcal{H}}\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\log(\dfrac{d\mu_{e,t}}{d\eta_{e,t}})\mathop{}\\!\mathrm{d}\eta_{e,t}(u)&\text{if
}\mu_{e,t}\ll\eta_{e,t},\\\ -\infty&\mathrm{otherwise},\end{cases}$ (2.3)

which is known as the _relative entropy_ functional of measure $\mu_{e,t}$
with respect to the reference probability measure $\eta_{e,t}$. Since
$\log(y)$ is a concave function and
$f_{e,t}(u):=\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}(u)}{\mathop{}\\!\mathrm{d}\eta_{e,t}(u)}\in
L^{1}(\eta_{e,t})$, we employ Jensen’s inequality to obtain

$\displaystyle\mathcal{S}(\mu_{e,t}|\eta_{e,t})=\mathcal{S}(f_{e,t})$
$\displaystyle=\int_{\mathcal{H}}\log(\frac{1}{f_{e,t}(u)})\mathop{}\\!\mathrm{d}\mu_{e,t}(u)\leq\log(\int_{\mathcal{H}}\frac{1}{f_{e,t}(u)}\mathop{}\\!\mathrm{d}\mu_{e,t}(u))$
$\displaystyle\leq\log(\int_{\mathcal{H}}\mathop{}\\!\mathrm{d}\eta_{e,t}(u))=\log(\eta_{e,t}(G_{e}(t)))=0.$

We postpone the discussion on the measure $\eta_{e,t}$ over the energy-
enstrophy to the next section.

Our problem consists of maximizing the relative entropy functional (2.3)
subject to some appropriate constraints: the solution obtained is a
probability distribution

$\int
f_{e,t}(u)\mathop{}\\!\mathrm{d}\eta_{e,t}(u)=\int\mathop{}\\!\mathrm{d}\eta_{e,t}(u)=1,$
(2.4)

and the support of the resulting measure must be supported on $G_{e}(t)$. For
the latter constraint, it will be shown that the fact that $\eta_{e,t}$ is
supported on $G_{e}(t)$ is sufficient to ensure the second constraint is
satisfied by the maximum entropy measure. We hence require a solution to the
following maximum problem:

$\sup\big{\\{}\mathcal{S}(f_{e,t})\,:\,\mu_{e,t}\in\mathcal{M}\,\,\mathrm{satisfying}\,\,\eqref{constr_1a}\big{\\}}$
(2.5)

for each fixed $t\in[0,T]$ and a priori prescribed $e(t)$.

###  2.2. Main theorem for the principle of maximum entropy

The following basic lemma is key in establishing our main result – Theorem 2.2
– below.

######  Lemma 2.1.

Fix $t\in[0,T]$ and $e(t)\geq 0$. Then the following statements hold:

  1. (i)

The relative entropy is non-positive: $\mathcal{S}(f_{e,t})\leq 0$ for all
$\mu_{e,t}\in\mathcal{M}$, where
$f_{e,t}=\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}$.

  2. (ii)

The relative entropy is concave in $\mu_{e,t}$: for
$\mu^{1}_{e,t},\mu^{2}_{e,t}\in\mathcal{M}$ so that
$\mathcal{S}(\mu^{1}_{e,t}|\eta_{e,t})$ and
$\mathcal{S}(\mu^{2}_{e,t}|\eta_{e,t})$ are finite, then

$\mathcal{S}(\mu_{e,t}|\eta_{e,t})\geq\alpha\mathcal{S}(\mu^{1}_{e,t}|\eta_{e,t})+(1-\alpha)\mathcal{S}(\mu^{2}_{e,t}|\eta_{e,t}),$

where $\mu_{e,t}=\alpha\mu_{e,t}^{1}+(1-\alpha)\mu_{e,t}^{2}$ and
$\alpha\in[0,1]$.

  3. (iii)

The total variation of $\mu_{e,t}$, defined by

$|\mu_{e,t}|=\displaystyle\sup_{\Pi}\sum_{i=1}^{m}\int_{A_{i}}{\rm
d}\mu_{e,t}(u),$

satisfies

$|\mu_{e,t}-\eta_{e,t}|^{2}\leq-2\mathcal{S}(\mu_{e,t}|\eta_{e,t}),$

where the supremum is taken over all finite partitions
$\Pi=\\{A_{1},A_{2},\cdots A_{m}\\}$ of $\mathcal{H}$.

  4. (iv)

Functional $\mathcal{S}(f_{e,t})$ is upper-semicontinuous: if
$|\mu_{e,t}^{n}-\mu_{e,t}|\to 0$ as $n\to\infty$, then

$\mathcal{S}(\mu_{e,t}|\eta_{e,t})\geq\limsup_{n}\mathcal{S}(\mu^{n}_{e,t}|\eta_{e,t}).$

Proof. Statement (i) has been proven in §2.1. The proofs of (iii)–(iv) can be
found in Ch.1 of [12].

A slight modification of (ii) is also stated in [12] but without explicit
proof, so we include the proof of (ii) for the sake of completeness. We begin
by invoking the inequality:

$\displaystyle\sum_{i=1}^{m}a_{i}\log(\dfrac{a_{i}}{b_{i}})\geq
a\log(\dfrac{a}{b}),$

where $a_{i},b_{i}$ are any non-negative numbers for all $i=1,2,...,m$, with
$a=\sum_{i}a_{i}$ and $b=\sum_{i}b_{i}$. The equality holds if and only if
$\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\cdots=\frac{a_{m}}{b_{m}}$.

Fix $\alpha\in[0,1]$, and let
$\mu_{e,t}=\alpha\mu_{e,t}^{1}+(1-\alpha)\mu_{e,t}^{2}$. Consider

$\displaystyle\alpha\dfrac{d\mu_{e,t}^{1}}{d\eta_{e,t}}\log(\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{1}}{\mathop{}\\!\mathrm{d}\eta_{e,t}})+(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{2}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\log(\dfrac{d\mu_{e,t}^{2}}{d\eta_{e,t}})$
$\displaystyle=\alpha\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{1}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\log\left(\dfrac{\alpha\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{1}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}{\alpha\dfrac{\mathop{}\\!\mathrm{d}\eta_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}\right)+(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{2}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\log\left(\dfrac{(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{2}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}{(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\eta_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}\right)$
$\displaystyle\geq\left(\alpha\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{1}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}+(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{2}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\right)\log\left(\dfrac{\alpha\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{1}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}+(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}^{2}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}{\alpha\dfrac{\mathop{}\\!\mathrm{d}\eta_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}+(1-\alpha)\dfrac{\mathop{}\\!\mathrm{d}\eta_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}}\right)$
$\displaystyle=\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}\log(\dfrac{\mathop{}\\!\mathrm{d}\mu_{e,t}}{\mathop{}\\!\mathrm{d}\eta_{e,t}}).$

Integrating with respect to $\eta_{e,t}$ gives the desired conclusion.

We now state and prove the main result of this paper.

######  Theorem 2.2.

The physical measure $\eta_{e,t}$ is a solution to the maximization problem
(2.5). That is, the physical measure maximizes the entropy functional relative
to all the comparison probability measures $\mu_{e,t}$, both restricted to the
global energy-enstrophy surface $G_{e}(t)$.

Proof. We divide the proof into two steps.

1\. Existence. We follow the argument in Ch.3 of [12]. Fix $t\in[0,T]$ and a
non-negative function $e(t)$. Consider the problem

$I_{\rm sup}=\sup_{\mu_{e,t}\in\mathcal{A}}\mathcal{S}(f_{e,t})\leq 0,$ (2.6)

where
$\mathcal{A}:=\\{\mu_{e,t}\in\mathcal{M}\,\,\mathrm{satisfying}\,\,\eqref{constr_1a}\\}$.
Let $\mu_{e,t}^{n}$ be a maximizing sequence for our problem, i.e.,
$\mu^{n}_{e,t}\in\mathcal{A}$, and $\mathcal{S}(f^{n}_{e,t})\to I_{\rm sup}$
as $n\to\infty$. It can be directly checked that

$\displaystyle\mathcal{S}(f^{m}_{e,t})+\mathcal{S}(f^{n}_{e,t})$
$\displaystyle=\mathcal{S}(\mu_{e,t}^{m}|\eta_{e,t})+\mathcal{S}(\mu_{e,t}^{n}|\eta_{e,t})$
$\displaystyle=2\mathcal{S}(\mu_{e,t}^{m,n}|\eta_{e,t})+\mathcal{S}(\mu_{e,t}^{m}|\mu_{e,t}^{m,n})+\mathcal{S}(\mu_{e,t}^{n}|\mu_{e,t}^{m,n}),$
(2.7)

where $\mu_{e,t}^{m,n}=\dfrac{1}{2}(\mu^{m}_{e,t}+\mu^{n}_{e,t})$. Because of
the linear nature of the constraint, we see that
$\mu_{e,t}^{m,n}\in\mathcal{A}$. The concavity of the relative entropy, Lemma
2.1, gives

$2\mathcal{S}(\mu_{e,t}^{m,n}|\eta_{e,t})\geq\mathcal{S}(\mu_{e,t}^{m}|\eta_{e,t})+\mathcal{S}(\mu_{e,t}^{n}|\eta_{e,t})$

which, together with the non-positivity of $\mathcal{S}(\cdot)$, implies that
$\lim_{m,n}\mathcal{S}(\mu_{e,t}^{m,n}|\eta_{e,t})=2I_{\rm sup}$ so that
$\lim_{m,n}\mathcal{S}(\mu_{e,t}^{m}|\mu_{e,t}^{m,n})=0=\lim_{m,n}\mathcal{S}(\mu_{e,t}^{n}|\mu_{e,t}^{m,n})$.
By Lemma 2.1(iii),

$\displaystyle|\mu_{e,t}^{m}-\mu_{e,t}^{n}|$
$\displaystyle\leq|\mu_{e,t}^{m}-\mu_{e,t}^{m,n}|+|\mu_{e,t}^{n}-\mu_{e,t}^{m,n}|$
$\displaystyle\leq\sqrt{-2\mathcal{S}(\mu_{e,t}^{m}|\mu_{e,t}^{m,n})}+\sqrt{-2\mathcal{S}(\mu_{e,t}^{n}|\mu_{e,t}^{m,n})}\to
0\qquad\,\,\mbox{as $m,n\to\infty$}.$ (2.8)

Then we conclude that there exists $\mu^{*}_{e,t}\in\mathcal{M}$ satisfying

$\lim_{n}|\mu_{e,t}^{n}-\mu^{*}_{e,t}|=0.$ (2.9)

Moreover, due again to the linearity of constraint (2.4), it readily follows
that $\mu^{*}_{e,t}\in\mathcal{A}$. Now, by the upper-semicontinuity of
$\mathcal{S}$, Lemma 2.1(iv), we conclude

$I_{\rm sup}=\mathcal{S}(\mu^{*}_{e,t}|\eta_{e,t}).$

2\. Euler-Lagrange equations. Since the constraint is linear in $f$, then, by
the standard methods of the Calculus of Variations, there exists a constant
$\alpha_{t}$ (independent of $u$) satisfying the Euler-Lagrange equations:

$\displaystyle-\int\Psi_{t}(u)\big{(}1+\log(f^{\ast}_{e,t}(u))\big{)}\,\mathop{}\\!\mathrm{d}\eta_{e,t}(u)-\alpha_{t}\int\Psi_{t}(u)\mathop{}\\!\mathrm{d}\eta_{e,t}(u)=0$
(2.10)

for any test function $\Psi_{t}(u)$. Then we have

$\displaystyle-\big{(}1+\log(f^{\ast}_{e,t}(u))\big{)}-\alpha_{t}=0.$ (2.11)

for $\eta_{e,t}$-almost every $u\in G_{e}(t)$. Thus,

$\displaystyle\log(f^{\ast}_{e,t}(u))=-(1+\alpha_{t}).$ (2.12)

or equivalently

$\displaystyle f^{\ast}_{e,t}(u)=\exp\\{-(1+\alpha_{t})\\}.$ (2.13)

That is, $f^{\ast}_{e,t}$ is constant in $u$. Since both $\mu_{e,t}^{\ast}$
and $\eta_{e,t}$ are probability measures, constraint (2.4) yields
$\alpha_{t}\equiv-1$, implying that $f^{\ast}_{e,t}\equiv 1$ or equivalently
$\mu_{e,t}^{\ast}=\eta_{e,t}$. In other words, the physical measure
$\eta_{e,t}$ achieves the maximum entropy under constraint (2.4).

Remark 2.1. The global energy-enstrophy surface $G_{e}(t)$ in (2.1) for
Theorem 2.2 can be replaced by the local energy-enstrophy surface: For any
given $e(x,t)>0$,

$\displaystyle\frac{1}{2}\int_{V}|u(x,t)|^{2}\psi(x,t)\mathop{}\\!\mathrm{d}x+\nu\int_{0}^{t}\int_{V}|\nabla
u|^{2}\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}\tau$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\int_{V}\big{\\{}|u|^{2}(\partial_{t}\psi+\nu\Delta\psi)+(|u|^{2}+2p)u\cdot\nabla\psi)\big{\\}}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}\tau=\int_{V}e(x,t)\psi(x,t)\mathop{}\\!\mathrm{d}x$
(2.14)

for any $\psi(x,\tau)\in C^{\infty}_{\rm c}(V\times[0,t])$. Then Theorem 2.2
can state as follows: The physical measure maximizes the entropy production
rate relative to all the comparison probability measures $\mu_{e,t}$, both
restricted to the local entropy-enstrophy surface (2.2).

Remark 2.2. A connection can be drawn to the classical results highlighted in
§1.2: at $t=0$, we set $\eta_{e,0}=g(u)\,\xi$, where $g(u)$ is given by
equation (1.7); this implies that
$\dfrac{\mathop{}\\!\mathrm{d}\mu^{*}_{e,0}}{\mathop{}\\!\mathrm{d}\xi}=g(u)\in
L^{1}({\rm d}\xi)$ in agreement with the classical principle of maximum
entropy. Furthermore, at $t=0$, the energy condition in (2.2) gives the
classical energy constraint (1.5)2, once we identify the mean energy with
$\int e(0)g(u)\mathop{}\\!\mathrm{d}\xi$.

##  3\. Connections and Remarks

In view of the previous work [10] on the principle of maximum entropy applied
to the problem of inviscid fluid flow, the current results outline a general
approach for studying the case of viscous flow under a common framework. As
such, it is clear that the standard energy class associated with the Navier-
Stokes equations provides the natural setting for expressing the total energy
$\|u\|^{2}_{t}$ of the fluid at each time $t$, and hence defining the energy-
enstrophy surface $G_{e}(t)$.

Equally clear is that having a complete theory of entropy maximization for
fluid flow is tied with specifying selection criteria for the reference
measure $\eta_{e,t}$. Whether or not such criteria are model dependent is open
to question. For instance, in the finite-dimensional setting, a natural
candidate for the reference measure is the Lebesgue measure (see [3, 10]) as
it is the only measure (up to a multiplicative constant) that possesses the
property of translation invariance; that is, in the velocity space, it remains
invariant under the Galilean transformation. Yet another candidate for the
reference measure is given by considering the statistics of fluid flows at a
long time; this is related to _stationary_ statistical solutions of the
Navier-Stokes equations and Kolmogorov’s theory of turbulence (cf. [6]).

We first recall the notion of the stationary statistical solutions of the
Naiver-Stokes equations. Assume that $\mu$ is a Borel probability measure over
space $\mathcal{H}$ with finite enstrophy, that is,

$\int_{\mathcal{H}}\|\nabla u\|^{2}\mathop{}\\!\mathrm{d}\mu(u)<\infty.$

Then the measure $\mu$ is said to be a stationary statistical solution of the
Naiver-Stokes equations if $\mu$ satisfies the following (stationary)
Liouville-type equation:

$\int_{\mathcal{H}}\int_{V}Q(u)\,\Phi^{\prime}(u)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}\mu(u)=0$
(3.1)

for all suitable test functions $\Phi$, and the energy inequality:

$\int_{\mathcal{H}}\int_{V}\left(\nu|\nabla u|^{2}-F\cdot
u\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}\mu(u)\leq 0,$ (3.2)

where $Q(u)=-\nu\Delta u+(u\cdot\nabla)u-F$.

Now from the perspective of thermodynamics, the flow relaxes, after
sufficiently long enough time, to its equilibrium state characterized by
maximum entropy; so we may assume that the statistics of this (final) state is
encoded by the stationary solution $\mu\doteq\mu_{\infty}$. On the other hand,
one of the fundamental postulates of Kolmogorov’s theory for fully developed
isotropic turbulence is the similarity hypothesis that the statistical
behavior of the flow must be determined by only two parameters: the (average)
energy dissipation rate $\overline{\epsilon}_{\nu}(t)$ and viscosity $\nu$
(see also [11]). Therefore, the average energy balance, in the absence of body
forces, for the case of non-zero energy dissipation rate (i.e., the average of
equation (1.3)) reads

$\dfrac{1}{2}\int_{\mathcal{H}}\|u\|^{2}\mathop{}\\!\mathrm{d}\mu_{t}(u)+\underbrace{\nu\int_{\mathcal{H}}\int_{0}^{t}\|\nabla
u(s)\|^{2}\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}\mu_{t}(u)}_{\overline{\epsilon}_{\nu}(t)}=\frac{1}{2}\int_{\mathcal{H}}\|u\|^{2}\mathop{}\\!\mathrm{d}\mu_{0}(u),$
(3.3)

where the probability measure $\mu_{t}$ is such that equation (3.3) is well-
defined (e.g. a time-dependent statistical solution). However, if we set
$\mu_{t}=\mu_{\infty}$, then the non-zero energy (finite) dissipation rate can
be maintained only if the kinetic energy is infinite, as seen by equation
(3.2). This nonphysical conclusion was pointed out first by E. Hopf [11] (see
also [6]). The resolution of this paradox, according to Foias, Manley, Rosa,
and Temam [5, 6, 7], lies instead in introducing a _family_ of probability
measures $\mu_{t}$ (parameterized by time $t\in[0,\infty)$)666More precisely,
for each homogeneous statistical solution of the Navier-Stokes equations
$\mu_{t}$ (with viscosity $\nu$), we define
$\mu_{t}=\mu^{\nu,\overline{\epsilon}_{\nu}(t)}$ for some
$\overline{\epsilon}_{\nu}(t)>0$., a feature which the physical measure
$\eta_{e,t}$ enjoys through the energy-enstrophy surface $G_{e}(t)$. These
solutions, called self-similar homogeneous statistical solutions, were
originally postulated for describing the decay of fully developed turbulence
according to Kolmogorov [17, 18], though to date the question of their
existence has not been resolved yet.

Meanwhile, our results suggest replacing the right-hand side of equation (3.3)
with $\overline{e}(t)$, a quantity that can ostensibly be identified with
$\int_{G_{e}(t)}e(t)\mathop{}\\!\mathrm{d}\eta_{e,t}$. Such a choice adds a
degree of freedom to the final statistics of the flow, now depending upon the
initial prescribed energy $e(t)$. In keeping with the above similarity
hypothesis, however, we can in principle prescribe the total energy $e(t)$ at
a high Reynolds number in terms of parameters
$(\overline{\epsilon}_{\nu},\nu)$ and wavenumber $\kappa$, removing the
ambiguity associated with the term:
$\int_{G_{e}(t)}e(t)\mathop{}\\!\mathrm{d}\eta_{e,t}$. Let
$\mathcal{E}(\kappa,t)$ be the _energy spectrum_ (in the Fourier space with
$|\mbox{\boldmath$\kappa$}|=\kappa$) associated with the average kinetic
energy. Then the average total energy can be expressed in terms of the energy
spectrum:

$\int_{0}^{\infty}\mathcal{E}(\kappa,t)\big{(}\dfrac{1}{2}+\nu\kappa^{2}\big{)}\mathop{}\\!\mathrm{d}\kappa.$
(3.4)

Equation (3.4) provides a natural alternative for our (average) total energy
$\overline{e}(t)$. In fact, the amount of total energy contained between
wavenumbers $\kappa_{1}$ and $\kappa_{2}$ at a given time $t$, _i.e._ ,

$\int_{\kappa_{1}}^{\kappa_{2}}\mathcal{E}(\kappa,t)\big{(}\dfrac{1}{2}+\nu\kappa^{2}\big{)}\mathop{}\\!\mathrm{d}\kappa$

can now be identified with the change in the total energy
$\overline{e}_{\kappa_{2}}(t)-\overline{e}_{\kappa_{1}}(t)$.

As $\nu\to 0$, or equivalently at _infinite_ Reynolds number, the implications
of Kolmogorov’s theory give

$\mathcal{E}(\kappa,t)\sim
C(t)\,\overline{\epsilon}(t)^{\frac{2}{3}}\kappa^{-\frac{5}{3}}$ (3.5)

for large enough $|\mbox{\boldmath$\kappa$}|=\kappa$ but still belonging to
the inertial range, where $C(t)$ is a positive dimensionless constant
independent of viscosity $\nu$, and $\lim_{\nu\to
0}\overline{\epsilon}_{\nu}(t)=\overline{\epsilon}(t)>0$. In both cases,
nevertheless, the explicit dependence on $e(t)$ can be removed in the final
statistics in favor of the energy dissipation rate and viscosity.

In view of interpreting the total energy $e(t)$ in terms of the energy
spectrum, one can conceive of the maximum entropy probability measure, that is
the reference probability measure, as assigning a probability for finding the
total energy carried by a certain wavenumber $\kappa$ to be
$\overline{e}_{\kappa}(t)$. When the wavenumber lies in the inertial range and
in the limit of infinite Reynolds number, this probability can be calculated
via the Kolmogorov energy spectrum (3.4).

Acknowledgments. Gui-Qiang G. Chen’s research is partially supported by the UK
Engineering and Physical Sciences Research Council Awards EP/L015811/1,
EP/V008854/1, and EP/V051121/1.

Declarations

The authors have no competing interests to declare that are relevant to the
content of this article. In addition, our manuscript has no associated data.

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