Comptes Rendus Mathématique

Asaf Nachmias

Yuval Peres

Comptes Rendus
Mathématique
Asaf Nachmias and Yuval Peres
Every recurrent network has a potential tending to infinity
Volume 363 (2025), p. 793-798
Online since: 4 July 2025
https://doi.org/10.5802/crmath.758
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 793-798
https://doi.org/10.5802/crmath.758
Research article / Article de recherche
Probability theory / Probabilités
Every recurrent network has a potential
tending to infinity
Tout réseau récurrent a un potentiel tendant vers l’infini
Asaf Nachmias a and Yuval Peres b
a Department of Mathematical Sciences, Tel Aviv University, Israel
b Beijing Institute of Mathematical Sciences and Applications, Beijing, China
E-mails: asafnach@tauex.tau.ac.il, yperes@gmail.com
Abstract. A rooted network consists of a connected, locally finite graph G , equipped with edge conductances
and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1
at o, and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted
network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and
Nakai in the settings of Euclidean domains and Riemannian surfaces.
Résumé. Un réseau enraciné est un graphe G localement fini et connexe, doté de conductances sur les arêtes
et d’un sommet distingué o. Un potentiel est une fonction sur les sommets de G qui est positive, s’annule
en o, a un laplacien 1 en o et est harmonique en tous les autres sommets. Nous prouvons que tout réseau
enraciné, récurrent et infini admet un potentiel tendant vers l’infini. Il s’agit d’un analogue de théorèmes
classiques de Evans et Nakai dans le cadre des domaines euclidiens et des surfaces riemanniennes.
Funding. This work was supported by ERC consolidator grant 101001124 (UniversalMap) as well as ISF
grants 1294/19 and 898/23.
Manuscript received 27 February 2025, revised and accepted 6 May 2025.
1. Introduction
A rooted network (G ,c,o) consists of a connected, locally finite graph G = (V ,E), endowed with
positive edge weights c : E → (0,∞) and a distinguished vertex o ∈V . The corresponding network
random walk with transition probabilities p(x, y) = cx y /cx is reversible, with reversing measure
cx =∑z∼x cxz . Define the network Laplacian of a function f : V →R by
∆ f (v) = ∑
x∼v
cv x
(
f (x)− f (v)).
A potential on (G ,c,o) is a function h : V → [0,∞) that is harmonic off o (i.e., ∆h(v) = 0 for
every vertex v ̸= o), normalized so that h(o) = 0 and ∆h(o) = 1. In every infinite rooted network
there is at least one potential, see Claim 5. Our focus is on recurrent networks.
We say that a function h : V → [0,∞) tends to infinity and write h → ∞, if for all M > 0 the
set
{
v ∈ V : h(v) ≤ M} is finite. Observe that on Z, the potentials h1(n) = max(n,0) for n ∈ Z and
h2(n) = max(−n,0) do not tend to infinity, but any non-trivial convex combination of them does.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
794 Asaf Nachmias and Yuval Peres
On Z2, on the other hand, it is well known that there is a unique potential h with h(v) =Θ(log|v |)
as v → ∞, see [6, Sections 12 and 31]. Recently there has been renewed interest in recurrent
potential theory, motivated by close connections to unimodular random networks and uniform
spanning trees in work of Berestycki, van Engelenburg and Hutchcroft, see [1,2].
Theorem 1. On any infinite recurrent rooted network there exists a potential which tends to
infinity.
The earliest theorem of this type is due to G. C. Evans [3] who showed that every compact
polar set K in Rd supports a probability measure such that its Newtonian potential is infinite
on K . Using Evans’ approach, Nakai [4] proved that every parabolic noncompact Riemannian
surface admits a function which is harmonic off a compact set and tends to infinity (i.e., its level
sets are compact). In Nakai’s proof, the set K is replaced by the Stone–Čech boundary of the
surface. Nakai’s argument can be adapted to prove Theorem 1, but the use of the Stone–Čech
compactification makes this argument non-constructive. Here we present a more constructive
probabilistic approach, using the minimax theorem.
Remark 2. Theorem 1 implies that if a rooted network admits a unique potential, then it must
tend to infinity. This answers a question raised following [1, Proposition 5.1].
Remark 3. It is well known that the existence of a potential tending to infinity implies recurrence.
Indeed, let h be a potential, let {X t } be the network random walk and consider the hitting time
τo = min
{
n ≥ 0 : Xn = o
}
. Then h(Xn∧τo ) is a nonnegative martingale, whence it almost surely
converges; this contradicts transience if h →∞.
2. Preliminaries
2.1. Dipoles and potentials
The Green kernel Go(x, y) of the random walk {Xn}n≥0 killed at o is defined by
Go(x, y) :=
∞∑
n=0
Px
(
Xn = y, τo > n
)
.
The corresponding Green density,
go(x, y) = Go(x, y)
cy
,
is symmetric by reversibility. A dipole from o to another vertex y in a network (G ,c) is a function
f : V → [0,∞), for which f (o) = 0 and
∀ v ∈V ∆ f (v) = 1o(v)−1y (v).
Claim 4. In a recurrent network, the Green density go( · , y) for the killed random walk is a dipole
from o to y. Moreover, supy go(x, y) <∞ for each x ∈V .
Proof. First observe that go( · , y) is harmonic off {o, y} and satisfies go(o, y) = 0. Second,
∆go( · , y)(y) =−1, since∑
x∼y
cx y
(
go(y, y)−go(x, y)
)=Go(y, y)− ∑
x∼y
cx y
cy
Go(x, y) = 1. (1)
Also,
∆go( · , y)(o) =
∑
x∼o
cox go(x, y) =
∑
x∼o
Go(y, x)
cox
cx
= 1, (2)
where the last equality is obtained by summing, over n ≥ 0, the identity∑
x∼o
Py
(
Xn = x, τo > n
)cox
cx
=Py
(
Xn+1 = o, τo > n
)=Py (τo = n +1)
and using recurrence. Thus, go( · , y) is a dipole.
Asaf Nachmias and Yuval Peres 795
Lastly, the identity (2) implies that go(x, y) ≤ c−1ox for every x ∼ o. More generally, since go( · , y)
is superharmonic off o, for any x ̸= o we have go(x, y) ≤ ∏ℓi=1 c−1xi−1xi , where x0, . . . , xℓ is a simple
path from o to x. □
Claim 5. The space of potentials on a recurrent infinite rooted (G ,c,o) is a nonempty, compact,
convex subset of RV endowed with the product topology.
Proof. By Claim 4, for any sequence of vertices yn →∞, Cantor diagonalization yields a subse-
quential pointwise limit of go( · , yn), and any such limit is a potential. Next, for any potential h
we have h(x) ≤ ∏ℓi=1 c−1xi−1xi , where x0, . . . , xℓ is a simple path from o to x, by the argument in the
proof of Claim 4, so the set of potentials is a closed subset of a compact product space. Finally,
convexity is obvious. □
2.2. The Doob transform determined by a potential
Any potential h defines a Markov chain on the state space Sh =
{
v : h(v) > 0} via the Doob h-
transform that has transition probabilities
ph(x, y) = p(x, y)h(y)
h(x)
.
The corresponding Markov chain {Yn}n≥0 is called the h-process. Write Phµ for the law of the h-
process with initial distribution µ and Ehµ for the corresponding expectation.
For any finite sequenceγ= (γ1, . . . ,γk ) in V , we writePγ1 (γ) :=
∏k
i=2 p(γi−1,γi ). Ifγ is contained
in Sh , we define P
h
γ1
(γ) similarly using ph(γi−1,γi ). We have the telescoping product
Phγ1 (γ) =Pγ1 (γ)
h(γk )
h(γ1)
. (3)
We will use the initial distribution µh defined by
µh(v) = cov h(v) for all v ∼ o. (4)
Observe that
Phµh (γ) = coγ1 h(γ1)Phγ1 (γ) = coγ1Pγ1 (γ)h(γk ) = coPo(oγ)h(γk ). (5)
There is a standard relation between the Green kernels of the h-process and the network
random walk killed at o.
Claim 6. Let h be a potential on a recurrent rooted network (G ,c,o). Then the h-process {Yn}n≥0
is transient. Moreover, its Green kernel Gh(x, y) =∑k≥0Phx (Yk = y), defined for x, y ∈ Sh , satisfies
Gh(x, y) =Go(x, y) h(y)
h(x)
, (6)
and for all v ∈ Sh
Gh(µh , v) = h(v)cv , (7)
where Gh(µ, v) :=∑x µ(x)Gh(x, v).
Proof. Transience will follow once we show Gh(x, y) <∞. To prove (6), we sum (3) over all paths
from x to y that avoid o. Note that by harmonicity of h off o, any such path is contained in Sh .
Next, (7) follows from (6) by
Gh(µh , y) =
∑
v
cov h(v)Gh(v, y) = h(y)
∑
v
cov Go(v, y) = h(y)cy ,
using (2) and the symmetry of go( · , · ) in the last step. □
796 Asaf Nachmias and Yuval Peres
Claim 7. Let h be a potential for a recurrent rooted network (G ,c,o) and consider the h-pro-
cess {Yn} started at v ∈ Sh . Then for every M > 0,
Phv
(
h(Yℓ) > M
)−→ 1 as ℓ→∞.
This was also observed in [2, Lemma 5].
Proof. Recall that {Xn} is the network random walk. By (3), we have
Phv
(
h(Yℓ) ≤ M
)= Ev [h(Xℓ)
h(v)
1{h(Xℓ)≤M and τo≥ℓ}
]
≤ M
h(v)
Pv (τo ≥ ℓ),
which tends to 0 as ℓ→∞ by recurrence. □
3. Proof of main theorem
Let d( · , · ) denote graph distance. For any integer r > 0, we set
B(o,r ) = {w : d(o, w) < r } and ∂B(o,r ) = {w : d(o, w) = r },
Next we state a formally weaker version of Theorem 1, where we can choose a different potential
for each radius, and show that it implies the theorem.
Lemma 8. LetΨ be the set of potentials on a recurrent infinite rooted network (G ,c,o). Then
M(r ) := max
ψ∈Ψ
min
v∈∂B(o,r )
ψ(v) −→∞ as r →∞. (8)
Remark 9. For every integer r > 0 and ψ ∈ Ψ, the minimum of ψ on V \ B(o,r ) is attained on
∂B(o,r ). Indeed, let {X t } be the network random walk started at w ̸∈ B(o,r ) and let τr be the
hitting time of ∂B(o,r ). Then ψ(X t∧τr ) is a positive martingale, so by recurrence and optional
stopping,
ψ(w) ≥ Ew
[
ψ(Xτr )
]≥ min
v∈∂B(o,r )
ψ(v).
Consequently, M(r ) = maxψ∈Ψminw∈V \B(o,r )ψ(w) is non-decreasing in r .
Proof of Theorem 1 given Lemma 8. For each integer r , let ψr be a potential where the maxi-
mum in the definition (8) of M(r ) is attained. For every n ≥ 1, choose rn such that M(rn) ≥ n2n .
By Claim 5,
h :=
∞∑
n=1
2−nψrn
is a potential. Next, if w ∉ B(o,rn), then h(w) ≥ 2−nψrn (w) ≥ n. Thus, for each n, the set{
v ∈V : h(v) < n} is finite. □
3.1. Preliminary lemmas
The following lemma shows that if the dipoles go( · , vn) converge to a potential h, then the
network random walks conditioned on τvn < τo converge weakly to the h-process. We actually
need this for dipole mixtures. Given a probability measure η on V , write
go( · ,η) :=
∑
v
η(v)go( · , v).
Lemma 10. Let (G ,c,o) be a recurrent rooted network. Let rn →∞ be a sequence of integers and for
each n let ηrn be a probability measure supported on ∂B(o,rn). Assume that the sequence go( · ,ηrn )
converges pointwise to a potential h as n → ∞. Then for any finite path γ in Sh starting at a
neighbor of o, we have ∑
v
ηrn (v)Po
(
oγ
∣∣τv < τ+o )−→Phµh (γ) as n →∞. (9)
Asaf Nachmias and Yuval Peres 797
Proof. First note that for v, w different from o, we have go(w, v) = Pw (τv < τo)go(v, v) and by
path reversal,
coPo(τv < τ+o ) = cvPv (τo < τ+v ) = cv Go(v, v)−1 = go(v, v)−1.
Therefore, if γ= (γ1, . . . ,γℓ) does not contain the vertex v , then
Po
(
oγ
∣∣τv < τ+o )=Po(oγ)Pγℓ (τv < τo)
Po(τv < τ+o )
= coPo(oγ)go(γℓ, v). (10)
Integrating this with respect to ηrn gives∑
v
ηrn (v)Po
(
oγ
∣∣τv < τ+o )= coPo(oγ)go(γℓ,ηrn ) −→ coPo(oγ)h(γℓ),
as n →∞ by our hypothesis. This yields the desired result by (5). □
Let h be a potential and set TM = inf
{
t > 0 : h(X t ) ≥ M
}
where {X t } is the network random
walk. Optional stopping for the nonnegative martingale h(X t∧τo ) gives Pv (TM < τo) ≤ h(v)/M .
Careful path reversal and conditioning yields the following.
Lemma 11. Let h be a potential for a recurrent rooted network (G ,c,o) and let v ̸= o be a vertex.
Then the network random walk {X t } satisfies
Po
(
h(Xℓ) ≥ M
∣∣τv < τ+o )≤ h(v)M ,
for all M > 0 and integer 0 < ℓ< d(o, v), where τ+o = min
{
t ≥ 1 : X t = o
}
.
Proof. We may assume that h(v) < M . Since ℓ< d(o, v) we have
Po
(
h(Xℓ) ≥ M
∣∣τv < τ+o )≤Po(TM < τv ∣∣τv < τ+o ). (11)
Recall that coPo(τv < τ+o ) = cvPv (τo < τ+v ) by path reversal. Similarly,
coPo(TM < τv < τ+o ) = cvPv (TM < τo < τ+v ),
so Po
(
TM < τv
∣∣τv < τ+o )=Pv (TM < τo ∣∣τo < τ+v ). (12)
Consider the random walk started at v and write L = max{t < τo : X t = v}. Then XL , . . . , Xτo is
distributed as a random walk started at v , conditioned on τo < τ+v , and stopped at o. Hence
Pv
(
TM < τo
∣∣τo < τ+v )=Pv (∃ s ∈ (L,τo) : h(Xs ) ≥ M)≤Pv (TM < τo). (13)
To bound the last probability we note that St = h(X t∧τo ) is a non-negative martingale, so by
optional stopping
h(v) = Ev S0 ≥ Ev STM ≥ MPv (TM < τo).
Combining this with (11), (12) and (13) concludes the proof. □
3.2. Proof of Lemma 8
Recall that Ψ denotes the set of potentials on (G ,c,o). Given an integer r > 0, let Υr be the
compact space of probability measures on ∂B(o,r ). Observe that for every ψ : V →R
min
v∈∂B(o,r )
ψ(v) = min
η∈Υr
∑
v
η(v)ψ(v). (14)
Next, we will apply Sion’s minimax theorem [5] to the function fr : Ψ×Υr →R, defined by
fr (ψ,η) =
∑
v∈∂B(o,r )
η(v)ψ(v).
As noted in Claim 5, the setΨ is a convex subset of the linear space RV endowed with the product
topology. Additionally, it is clear that fr is continuous and bilinear and that Υr is a compact
798 Asaf Nachmias and Yuval Peres
convex subset of the linear topological space R∂B(o,r ). Hence the assumptions of Sion’s theorem
hold, and we obtain, using (14), that M(r ) defined in (8) satisfies
M(r ) = max
ψ∈Ψ
min
η∈Υr
fr (ψ,η) = min
η∈Υr
max
ψ∈Ψ
fr (ψ,η). (15)
For each integer r > 0, let ηr ∈ Υr denote a minimizer of the right-hand side of (15). By
Claim 4, there exists a sequence rn → ∞ such that go( · ,ηrn ) converges pointwise to a function
h : V → [0,∞). Clearly h is a potential.
Fix m > 0. By Claim 7, there exists an integer ℓ such that Phµh
(
h(Yℓ) ≥ 2m
)≥ 1/2. We now apply
Lemma 10 and sum (9) over all paths γ= (γ1, . . . ,γℓ) ⊂ Sh such that h(γℓ) ≥ 2m. We infer that
lim
n→∞
∑
v
ηrn (v)Po
(
h(Xℓ) ≥ 2m
∣∣τv < τ+o )=Phµh (h(Yℓ) ≥ 2m)≥ 1/2.
Lemma 11 states that if ℓ< rn and v ∈ ∂B(o,rn), then
Po
(
h(Xℓ) ≥ 2m
∣∣τv < τ+o )≤ h(v)2m .
Combining the last two inequalities yields that
liminf
n→∞ frn (h,ηrn ) = liminfn→∞
∑
v
ηrn (v)h(v) ≥ m.
In conjunction with (15), this gives that liminfn M(rn) ≥ m for every m > 0. Since M(r ) is non-
decreasing (Remark 9), it must tend to infinity as r →∞. □
Acknowledgments
We thank Omer Angel, Ori Gurel-Gurevich and Fedja Nazarov for helpful discussions.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] N. Berestycki and D. van Engelenburg, “Harnack inequality and one-endedness of UST on
reversible random graphs”, Probab. Theory Relat. Fields 188 (2024), no. 1-2, pp. 487–548.
[2] D. van Engelenburg and T. Hutchcroft, “The number of ends in the uniform spanning tree
for recurrent unimodular random graphs”, Ann. Probab. 52 (2024), no. 6, pp. 2079–2103.
[3] G. C. Evans, “Potentials and positively infinite singularities of harmonic functions”,
Monatsh. Math. Phys. 43 (1936), no. 1, pp. 419–424.
[4] M. Nakai, “On Evans potential”, Proc. Japan Acad. 38 (1962), pp. 624–629.
[5] M. Sion, “On general minimax theorems”, Pac. J. Math. 8 (1958), pp. 171–176.
[6] F. Spitzer, Principles of random walk, Second edition, Graduate Texts in Mathematics,
Springer, 1976, pp. xiii+408.


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Comptes Rendus
Mathématique
Jinlu Li, Yanghai Yu and Weipeng Zhu
Anomalous dissipation for the d-dimensional Navier–Stokes equations
Volume 363 (2025), p. 345-351
Online since: 20 May 2025
https://doi.org/10.5802/crmath.709
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 345-351
https://doi.org/10.5802/crmath.709
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
Anomalous dissipation for the d-dimensional
Navier–Stokes equations
Dissipation anomale pour les équations de
Navier–Stokes à d dimensions
Jinlu Li a, Yanghai Yu ∗,b andWeipeng Zhu c
a School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China
b School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
c School of Mathematics, Foshan University, Foshan, Guangdong 528000, China
E-mails: lijinlu@gnnu.edu.cn (J. Li), yuyanghai214@sina.com (Y. Yu), mathzwp2010@163.com (W. Zhu)
Abstract. The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier–
Stokes equations in the whole space:{
∂t uε+uε ·∇uε−ε∆uε+∇pε = 0,
divuε = 0.
We aim to present a simple rigorous example of initial data which generates the corresponding solutions of
the Navier–Stokes equations for which the dissipation rate of the kinetic energy is bounded away from zero.
Résumé. L’objectif de cet article est d’étudier la limite de disparition de la viscosité pour les équations de
Navier–Stokes à d dimensions dans l’espace entier :{
∂t uε+uε ·∇uε−ε∆uε+∇pε = 0,
divuε = 0.
Nous visons à présenter des exemples simples et rigoureux de données initiales qui génèrent les solutions
correspondantes des équations de Navier–Stokes pour lesquelles le taux de dissipation de l’énergie cinétique
est borné loin de zéro.
Keywords. Navier–Stokes equations, inviscid limit.
Mots-clés. Équations de Navier–Stokes, limite inviscide.
2020 Mathematics Subject Classification. 35Q35, 76D09.
Funding. J. Li is supported by the National Natural Science Foundation of China (12161004), Train-
ing Program for Academic and Technical Leaders of Major Disciplines in Ganpo Juncai Support Pro-
gram (20232BCJ23009). Y. Yu is supported by the National Natural Science Foundation of China (12101011).
W. Zhu is supported by the National Natural Science Foundation of China (12201118) and Guangdong Basic
and Applied Basic Research Foundation (2021A1515111018).
Manuscript received 29 May 2024, revised 7 November 2024, accepted 5 December 2024.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
346 Jinlu Li, Yanghai Yu and Weipeng Zhu
1. Introduction
One of the most important laws in the study of developed turbulence is Kolmogorov’s 4/5-law,
and in the derivation of this law, a significant assumption is the zeroth law (see Frisch [10] for
example). The zeroth law of turbulence states that, in the limit of zero viscosity, the rate of kinetic
energy dissipation for solutions to the incompressible Navier–Stokes equations is non-vanishing.
This is one of the central ansatz of Kolmogorov’s 1941 theory [18]. In this paper we consider the
vanishing viscosity limit for the d-dimensional incompressible Navier–Stokes equations
∂t uε+uε ·∇uε−ε∆uε+∇pε = f ε, (t , x) ∈R+×Rd ,
divuε = 0, (t , x) ∈R+×Rd ,
uε(t = 0) = uε0, x ∈Rd ,
(NS)
where ε> 0 is the kinematic viscosity, uε : [0,∞)×Rd →Rd is the velocity of the fluid, pε : [0,∞)×
Rd → R is the pressure and f ε : [0,∞) ×Rd → R is an external body force. Assuming that the
solution is sufficiently smooth and the initial data uε0 satisfy a uniform L
2 bound (in other words
the total kinetic energy at the initial time is bounded).
Since uε is divergence free, taking the dot product of the equation with uε and integrating
over Rd , it is immediate to see that the energy balance:
1
2
d
dt
∥∥uε(t )∥∥2L2(Rd ) =−ε∥∥∇uε(t )∥∥2L2(Rd ) +∫
Rd
f ε ·uεdx. (1)
The second term appearing on the right-hand side is the total work of the force f ε, while the first
term is the energy dissipation rate due to the viscosity of the fluid. Throughout the paper, we
assume that the external body force f ε = 0. From (1) we see that the L2 energy decay of solutions
is governed by
1
2
∥∥uε(t )∥∥2L2(Rd ) − 12∥u0∥2L2(Rd ) =−ε
∫ t
0
∥∥∇uε(τ)∥∥2L2(Rd ) dτ.
A fundamental postulate of Kolmogorov’s 1941 theory of fully developed turbulence [16–18],
called the zeroth law of turbulence, is that the anomalous dissipation of the kinetic energy holds,
namely the following inequality
liminf
ε→0+
{
ε
∫ t
0
∥∥∇uε(τ)∥∥2L2(Rd ) dτ}> 0
is valid for finite time t .
As for passive scalars, experimental and numerical observations of hydrodynamic turbulence
suggest that, in the limit of vanishing viscosity, the rate of kinetic energy dissipation becomes
nonzero [15,19–21], i.e. there exists η > 0 independent of ε such that, in turbulent regimes, a
family of Leray–Hopf solutions {uε}ε>0 satisfies
ε
∫ T
0
∫
Rd
∣∣∇uε(t , x)∣∣2 dx dt ≥ η> 0.
This phenomenon of anomalous dissipation is so fundamental to our modern understanding
of turbulence that it is often termed the “zeroth law”. Laboratory experiments and numerical
simulations of turbulence both confirm the above zeroth law [4,9,15,22]. However, this heuristic
phenomenon is difficult to capture mathematically, and only a few rigorous results are known.
Recently, in [8], Drivas et al. investigated the anomalous scalar dissipation for fluid velocities with
Hölder regularity and gave sufficient conditions for anomalous dissipation in terms of the mixing
rates of the advecting flow. Bruè and De Lellis [3] established the anomalous dissipation for the
forced Navier–Stokes equations. Jeong and Yoneda [12,13] considered the anomalous dissipation
for the 3D (NS) with zero external force in the framework of 2+ 12 -dimensional flows. We would
like to mention that all the results mentioned above are obtained on the Torus case. For more
Jinlu Li, Yanghai Yu and Weipeng Zhu 347
results on the anomalous dissipation for the Navier–Stokes equations, we refer to see [2,5–7,14]
and the refereed references.
To the best of our knowledge, there are no known anomalous dissipation for (NS) in the whole
space (even in R2) where it is rigorously proved. Keeping this classical physical phenomenon in
mind, in this paper, we compare the Navier–Stokes flow and the heat flow with the same initial
data, and deduce anomalous dissipation in the vanishing viscosity limit of (NS) under some
particular conditions on the initial data. Our main result is the following.
Theorem 1. Let d ≥ 2. There exists a sequence of C∞-smooth initial data uεn0 on Rd with
εn = 2−2n satisfying a uniform L2 bound such that there are families of smooth solutions uεn (t , x) ∈
C∞
(
[0,1]×Rd ) to the zero forced (NS) with the property that
εn
∫ 1
0
∥∥∇uεn (t , x)∥∥2L2(Rd ) dt >∼ ∥uεn0 ∥L2(Rd ). (2)
In particular, we have
liminf
n→∞
{
εn
∫ 1
0
∥∥∇uεn (t , x)∥∥2L2(Rd ) dt}≥ η, (3)
for some absolute constant η> 0.
Remark 2. This result suggests that one possible mechanism for anomalous dissipation would
be a cascade towards high-frequencies, whose intensity matches the scaling laws of the vanishing
viscosity (more than half of the initial energy gets transferred in time T = 1, by non-linear effects,
towards modes of frequency higher than 1/
p
εT ). The construction of a family of flows with that
property as ε→ 0 remains an open question. Here, the transfer is artificially ensured by choosing
initial data with a shifting Fourier support.
Now let us say some words on the key idea of proof of Theorem 1. We compare the Navier–
Stokes equation and the heat equation with the same initial data, and show that the solution U
to the associated heat equation displays anomalous dissipation, i.e.,
εn
∫ 1
0
∥∥∇Un(τ)∥∥2L2(Rd ) dτ≈ ∫ 1
0
∥∥eεnτ∆uεn0 ∥∥2L2(Rd ) dτ> 0. (4)
The main difficulty in the proof of Theorem 1 is to construct the initial data satisfying the
following:
• the L2-norm of initial data is bounded uniformly with respect to the viscosity;
• the difference of the Navier–Stokes flow and the heat flow is sufficiently small.
To achieve the above two points, taking advantage of the scaling in the whole space, we choose
suitably the initial data whose frequency is supported in the annulus
{
ξ : |ξ|2 ∼ ε−1n
}
. This
naturally gives that ∫ 1
0
∥∥eεnτ∆uεn0 ∥∥2L2(Rd ) dτ≈ ∥uεn0 ∥2L2(Rd ). (5)
Lastly, we would like to emphasize that we have to cancel the effect of the advection term. The
key observation is that, based on our construction of initial data, the advection term U ·∇U can
be seen as uεn0 ·∇uεn0 , which is sufficiently small.
Notation. For X a Banach space and I ⊂R, we denote by C (I ; X ) the set of continuous functions
on I with values in X . Sometimes we will denote Lp (0,T ; X ) by LpT X . The symbol A
<∼ (>∼)B means
that there is a uniform positive “harmless” constant C independent of A and B such that
A ≤ (≥)CB,
and we sometimes use the notation A ≈ B means that A <∼ B and B <∼ A. Let us recall that for all
u ∈ S ′, the Fourier transform Fu is defined by Fu(ξ) = ∫Rd e−ix·ξu(x)dx for any ξ ∈ Rd . Choose
a radial, non-negative, smooth function ϑ : Rd 7→ [0,1] such that suppϑ ⊂ B(0,4/3) and ϑ(ξ) ≡ 1
348 Jinlu Li, Yanghai Yu and Weipeng Zhu
for |ξ| ≤ 3/4. Setting ϕ(ξ) := ϑ(ξ/2)−ϑ(ξ), then we deduce that suppϕ⊂ {ξ ∈Rd : 3/4 ≤ |ξ| ≤ 8/3}
and ϕ(ξ) ≡ 1 for 4/3 ≤ |ξ| ≤ 3/2. For more details see [1]. The nonhomogeneous dyadic block is
defined by
∀ u ∈S ′(Rd ), ∆ j u =

0, if j ≤−2,
ϑ(D)u if j =−1,
ϕ(2− j D)u, if j ≥ 0,
where the pseudo-differential operator is defined by σ(D) : u →F−1(σFu).
Finally we recall the definition of the nonhomogeneous Besov spaces
B sp,r (R
d ) :=
{
f ∈S ′(Rd ) : ∥ f ∥B sp,r (Rd ) :=
∥∥2 j s∥∆ j f ∥Lpx (Rd )∥∥ℓr ( j≥−1) <∞}.
2. Proof of Theorem 1
We divide the proof of Theorem 1 into three steps. Now, by the heat flow, we shall mean the
unique L2 solution U = eεt∆uε0(x) of 
∂tU −ε∆U = 0,
divU = 0,
U (0, x) = uε0(x).
(6)
From (NS) with f ε = 0 and (6), we deduce that the difference v := uε−U satisfies the equation
with zero initial data
∂t v + v ·∇v −ε∆v +∇pε =−v ·∇U −U ·∇v −U ·∇U . (7)
Step 1: Estimation of difference. Multiplying equation (7) by v and integrating in space, then
using the fact that v is divergence and the Cauchy–Schwarz inequality we conclude
∥v∥2
L2(Rd )
+ε
∫ t
0
∥∇v∥2
L2(Rd )
dτ≤ 2
∫ t
0
(∥∇U∥L∞(Rd ) +1)∥v∥2L2(Rd ) dτ+∫ t0 ∥U ·∇U∥2L2(Rd ) dτ,
which implies
∥v∥2
L2(Rd )
+ε
∫ t
0
∥∇v∥2
L2(Rd )
dτ≤ exp
(∫ t
0
2
(∥∇U∥L∞(Rd ) +1)dτ)∫ t
0
∥U∥2
L2(Rd )
∥∇U∥2
L∞(Rd ) dτ.
Hence, noting that uε =U + v , we have
ε
∫ t
0
∥∇uε∥2
L2(Rd )
dτ− ε
2
∫ t
0
∥∇U∥2
L2(Rd )
dτ
≥−ε
∫ t
0
∥∇v∥2
L2(Rd )
dτ
≥−exp
(∫ t
0
(∥∇U∥L∞(Rd ) +1)dτ)∫ t
0
∥U∥2
L2(Rd )
∥∇U∥2
L∞(Rd ) dτ.
(8)
Step 2: Construction of initial data. We need to introduce smooth, radial cut-off functions to
localize the frequency region. Let φ̂ ∈C ∞0 (R) be an even, real-valued and non-negative function
on R that satisfies
φ̂(ξ) =
{
1, if |ξ| ≤ 14 ,
0, if |ξ| ≥ 12 .
Remark 3. It is not difficult to see that
∥φ∥L2(R) ≈ 1 and ∥φ∥L∞(R) ≈ 1.
Jinlu Li, Yanghai Yu and Weipeng Zhu 349
From now on, we set εn := 2−2n and define the initial data uεn0 by
uεn0 (x) := 2−dn−n
(−∂2 fn(x), ∂1 fn(x), 0, . . . , 0),
where
fn(x) := cos
(
17
12
2n x1
) d∏
i=1
φ(εn xi ).
Then we have the following
Lemma 4. We have
∥uεn0 ∥L2(Rd ) ≈ ∥φ∥dL2(R) and ∥u
εn
0 ∥B 1∞,1(Rd ) ≈ 2
(1−d)n∥φ∥dL∞(R).
Proof. Due to the simple fact that cosθ = 12 (e−iθ+e iθ), we deduce easily that
F ( fn) = 22nd−1
∫
R
(
e−i2
2n x1
(
ξ1+ 1712 2n
)
+e−i22n x1
(
ξ1− 1712 2n
))
φ(x1)dx1
d∏
i=2
φ̂(22nξi )
= 22nd−1
[
φ̂
(
22n
(
ξ1 + 17
12
2n
))
+ φ̂
(
22n
(
ξ1 − 17
12
2n
))] d∏
i=2
φ̂(22nξi ),
which implies that
suppF (un0 ) = suppF ( fn) ⊂
{
ξ ∈Rd : 3324 2n ≤ |ξ| ≤ 3524 2n
}
. (9)
Recalling that ϕ≡ 1 for 43 ≤ |ξ| ≤ 32 , we have
∆ j (u
εn
0 ) =
{
uεn0 , if j = n,
0, if j ̸= n.
Thus, for any σ ∈R, we deduce that
∥uεn0 ∥Bσp,r (Rd ) = 2
nσ∥uεn0 ∥Lp (Rd ) = 2n(σ−d−1)
(∥∂1 fn∥Lp (Rd ) +∥∂2 fn∥Lp (Rd )),
which completes the proof of Lemma 4. □
Step 3: Completion of the proof. Easy computations show that for t ∈ [0,Tεn ) (for more details
see [11]) ∥∥uεn (t )∥∥B 1∞,1(Rd ) ≤CeC V(t )
(
∥uεn0 ∥B 1∞,1(Rd ) +
∫ t
0
e−C V(τ)
∥∥∇pεn (τ)∥∥B 1∞,1(Rd ) dτ
)
≤CeC V(t )
(
∥uεn0 ∥B 1∞,1(Rd ) +C
∫ t
0
e−C V(τ)
∥∥uεn (τ)∥∥2B 1∞,1(Rd ) dτ
)
,
where V(t ) := ∫ t0 ∥∥uεn (τ)∥∥B 1∞,1(Rd ) dτ. Solving the above inequality, we deduce that
∥∥uεn (t )∥∥B 1∞,1(Rd ) ≤ ∥u
εn
0 ∥B 1∞,1(Rd )
1−C t∥uεn0 ∥B 1∞,1(Rd )
whenever t < 1
C∥uεn0 ∥B 1∞,1(Rd )
.
By Lemma 4, one has ∥uεn0 ∥B 1∞,1(Rd ) ≈ 2
(1−d)n , which tells us that there exists Tεn ≥ 1 such that (NS)
has a unique solution uεn in C
(
[0,1]; H∞
)
.
Noticing that Un(t , x) = eεn t∆uεn0 (x), we obtain that for some uniform positive constant C
∥Un∥L∞T (L2(Rd )) ≤ ∥u
εn
0 ∥L2(Rd ) ≤C
and ∥∇Un∥L∞T (L∞(Rd )) ≤ ∥∇u
εn
0 ∥L∞(Rd ) ≤C∥uεn0 ∥B 1∞,1(Rd ) ≤C 2
(1−d)n ,
350 Jinlu Li, Yanghai Yu and Weipeng Zhu
where we have used the classical Lp -Lp estimate: ∥eεt∆ f ∥Lp ≤ ∥ f ∥Lp with 1 ≤ p ≤ ∞, the
embedding B 0∞,1 ,→ L∞, and Lemma 4 in the last step. Thus
exp
(∫ t
0
2
(∥∇Un∥L∞(Rd ) +1)dτ)∫ t
0
∥Un∥2L2(Rd )∥∇Un∥
2
L∞(Rd ) dτ≤C 2
2(1−d)n .
Then, coming back to (8), we conclude
εn
∫ 1
0
∥∇Un∥2L2(Rd ) dτ≥
εn
2
∫ 1
0
∥∇Un∥2L2(Rd ) dτ−C 2
2(1−d)n . (10)
Due to the support condition (9) and the Plancherel’s identity, it holds that
εn
∫ 1
0
∥∥∇Un(τ)∥∥2L2(Rd ) dτ≈ ∫ 1
0
∥∥eεnτ∆uεn0 ∥∥2L2(Rd ) dτ≈ ∥uεn0 ∥2L2(Rd ) ≈ ∥φ∥2dL2(R).
Inserting the above into (10) gives (2)–(3). Thus we complete the proof of Theorem 1.
Acknowledgments
We would like to thank the anonymous reviewer for the valuable and detailed comments which
greatly improved the presentation of this paper.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
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