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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
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Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
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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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