Energy conservation and Onsager's conjecture for the Euler equations
arXiv:0704.0759v1  [math.AP]  5 Apr 2007
ENERGY CONSERVATION AND ONSAGER’S CONJECTURE
FOR THE EULER EQUATIONS
A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
ABSTRACT. Onsager conjectured that weak solutions of the Euler equa-
tions for incompressible fluids in R3 conserve energy only if they have a
certain minimal smoothness, (of order of 1/3 fractional derivatives) and
that they dissipate energy if they are rougher. In this paper we prove
that energy is conserved for velocities in the function space B
3,c(N)
show that this space is sharp in a natural sense. We phrase the energy
spectrum in terms of the Littlewood-Paley decomposition and show that
the energy flux is controlled by local interactions. This locality is shown
to hold also for the helicity flux; moreover, every weak solution of the
Euler equations that belongs to B
3,c(N)
conserves helicity. In contrast,
in two dimensions, the strong locality of the enstrophy holds only in the
ultraviolet range.
1. INTRODUCTION
The Euler equations for the motion of an incompressible inviscid fluid
+ (u · ∇)u = −∇p,
(2) ∇ · u = 0,
where u(x, t) denotes the d-dimensional velocity, p(x, t) denotes the pres-
sure, and x ∈ Rd. We mainly consider the case d = 3. When u(x, t) is a
classical solution, it follows directly that the total energyE(t) = 1
|u|2 dx
is conserved. However, conservation of energy may fail for weak solutions
(see Scheffer [25], Shnirelman [24]). This possibility has given rise to a
considerable body of literature and it is closely connected with statistical
theories of turbulence envisioned 60 years ago by Kolmogorov and On-
sager. For reviews see, for example, Eyink and Sreenivasan [14], Robert
[23], and Frisch [15].
Date: April 5, 2007.
2000 Mathematics Subject Classification. Primary: 76B03; Secondary: 76F02.
Key words and phrases. Euler equations, anomalous dissipation, energy flux, Onsager
conjecture, turbulence, Littlewood-Paley spectrum.
http://arxiv.org/abs/0704.0759v1
2 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
Onsager [22] conjectured that in 3-dimensional turbulent flows, energy
dissipation might exist even in the limit of vanishing viscosity. He sug-
gested that an appropriate mathematical description of turbulent flows (in
the inviscid limit) might be given by weak solutions of the Euler equations
that are not regular enough to conserve energy. According to this view, non-
conservation of energy in a turbulent flow might occur not only from vis-
cous dissipation, but also from lack of smoothness of the velocity. Specif-
ically, Onsager conjectured that weak solutions of the Euler equation with
Hölder continuity exponent h > 1/3 do conserve energy and that turbulent
or anomalous dissipation occurs when h ≤ 1/3. Eyink [12] proved energy
conservation under a stronger assumption. Subsequently, Constantin, E and
Titi [7] proved energy conservation for u in the Besov spaceBα3,∞, α > 1/3.
More recently the result was proved under a slightly weaker assumption by
Duchon and Robert [11].
In this paper we sharpen the result of [7]: we prove that energy is con-
served for velocities in the Besov space of tempered distributions B
3,p . In
fact we prove the result for velocities in the slightly larger spaceB
3,c(N)
Section 3). This is a space in which the “Hölder exponent” is exactly 1/3,
but the slightly better regularity is encoded in the summability condition.
The method of proof combines the approach of [7] in bounding the trilinear
term in (3) with a suitable choice of the test function for weak solutions in
terms of a Littlewood-Paley decomposition. Certain cancelations in the tri-
linear term become apparent using this decomposition. We observe that the
space B
3,c(N)
is sharp in the context of no anomalous dissipation. We give
an example of a divergence free vector field in B
3,∞ for which the energy
flux due to the trilinear term is bounded from below by a positive constant.
This construction follows ideas in [12]. However, because it is not a solu-
tion of the unforced Euler equation, the example does not prove that indeed
there exist unforced solutions to the Euler equation that live in B
3,∞ and
dissipate energy.
Experiments and numerical simulations indicate that for many turbu-
lent flows the energy dissipation rate appears to remain positive at large
Reynolds numbers. However, there are no known rigorous lower bounds
for slightly viscous Navier-Stokes equations. The existence of a weak so-
lution of Euler’s equation, with positive smoothness and that does not con-
serve energy remains an open question. For a discussion see, for example,
Duchon and Robert [11], Eyink [12], Shnirelman [25], Scheffer [24], de
Lellis and Szekelyhidi [10].
We note that the proof in Section 3 applied to Burger’s equation for 1-
dimensional compressible flow gives conservation of energy in B1/3
3,c(N)
ENERGY CONSERVATION 3
this case it is easy to show that conservation of energy can fail in B
which is the sharp space for shocks.
The Littlewood-Paley approach to the issue of energy conservation ver-
sus turbulent dissipation is mirrored in a study of a discrete dyadic model
for the forced Euler equations [4, 5]. By construction, all the interactions
in that model system are local and energy cascades strictly to higher wave
numbers. There is a unique fixed point which is an exponential global at-
tractor. Onsager’s conjecture is confirmed for the model in both directions,
i.e. solutions with bounded H5/6 norm satisfy the energy balance condition
and turbulent dissipation occurs for all solutions when the H5/6 norm be-
comes unbounded, which happens in finite time. The absence of anomalous
dissipation for inviscid shell models has been obtained in [8] in a space with
regularity logarithmically higher than 1/3.
In Section 3.2 we present the definition of the energy flux employed in
the paper. This is the flux of the Littlewood-Paley spectrum, ([6]) which
is a mathematically convenient variant of the physical concept of flux from
the turbulence literature. Our estimates employing the Littlewood-Paley de-
composition produce not only a sharpening of the conditions under which
there is no anomalous dissipation, but also provide detailed information
concerning the cascade of energy flux through frequency space. In sec-
tion 3.3. we prove that the energy flux through the sphere of radius κ is
controlled primarily by scales of order κ. Thus we give a mathematical
justification for the physical intuition underlying much of turbulence the-
ory, namely that the flux is controlled by local interactions (see, for exam-
ple, Kolmogorov [16] and also [13], where sufficient conditions for locality
were described). Our analysis makes precise an exponential decay of non-
local contributions to the flux that was conjectured by Kraichnan [17].
The energy is not the only scalar quantity that is conserved under evolu-
tion by classical solutions of the Euler equations. For 3-dimensional flows
the helicity is an important quantity related to the topological configura-
tions of vortex tubes (see, for example, Moffatt and Tsinober [21]). The
total helicity is conserved for smooth ideal flows. In Section 4 we observe
that the techniques used in Section 3 carry over exactly to considerations of
the helicity flux, i.e., there is locality for turbulent cascades of helicity and
every weak solution of the Euler equation that belongs to B
3,c(N)
conserves
helicity. This strengthens a recent result of Chae [2]. Once again our argu-
ment is sharp in the sense that a divergence free vector field in B
3,∞ can be
constructed to produce an example for which the helicity flux is bounded
from below by a positive constant.
4 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
An important property of smooth flows of an ideal fluid in two dimen-
sions is conservation of enstrophy (i.e. the L2 norm of the curl of the ve-
locity). In section 4.2 we apply the techniques of Section 3 to the weak
formulation of the Euler equations for velocity using a test function that
permits estimation of the enstrophy. We obtain the result that, unlike the
cases of the energy and the helicity, the locality in the enstrophy cascade is
strong only in the ultraviolet range. In the infrared range there are nonlo-
cal effects. Such ultraviolet locality was predicted by Kraichnan [18] and
agrees with numerical and experimental evidence. Furthermore, there are
arguments in the physical literature that hold that the enstrophy cascade is
not local in the infrared range. We present a concrete example that exhibits
this behavior.
In the final section of this paper, we study the bilinear term B(u, v). We
show that the trilinear map (u, v, w) → 〈B(u, v), w)〉 defined for smooth
vector fields in L3 has a unique continuous extension to {B1/2
18/7,2
}3 (and a
fortiori to {H5/6}3, which is the relevant space for the dyadic model prob-
lem referred to above). We present an example to show that this result is
optimal. We stress that the borderline space for energy conservation is much
rougher than the space of continuity for 〈B(u, v), w〉.
2. PRELIMINARIES
We will use the notation λq = 2
q (in some inverse length units). Let
B(0, r) denote the ball centered at 0 of radius r in Rd. We fix a nonnegative
radial function χ belonging to C∞0 (B(0, 1)) such that χ(ξ) = 1 for |ξ| ≤
1/2. We further define
(3) ϕ(ξ) = χ(λ−11 ξ)− χ(ξ).
Then the following is true
(4) χ(ξ) +
ϕ(λ−1q ξ) = 1,
(5) |p− q| ≥ 2 ⇒ Supp ϕ(λ−1q ·) ∩ Supp ϕ(λ−1p ·) = ∅.
We define a Littlewood-Paley decomposition. Let us denote by F the
Fourier transform on Rd. Let h, h̃, ∆q (q ≥ −1) be defined as follows:
ENERGY CONSERVATION 5
h = F−1ϕ and h̃ = F−1χ,
∆qu = F−1(ϕ(λ−1q ξ)Fu) = λdq
h(λqy)u(x− y)dy, q ≥ 0
∆−1u = F−1(χ(ξ)Fu) =
h̃(y)u(x− y)dy.
For Q ∈ N we define
(6) SQ =
Due to (3) we have
(7) SQu = F−1(χ(λ−1Q+1ξ)Fu).
Let us now recall the definition of inhomogeneous Besov spaces.
Definition 2.1. Let s be a real number, p and r two real numbers greater
than 1. Then
‖u‖Bsp,r
= ‖∆−1u‖Lp +
λsq‖∆qu‖Lp
ℓr(N)
is the inhomogeneous Besov norm.
Definition 2.2. Let s be a real number, p and r two real numbers greater
than 1. The inhomogeneous Besov space Bsp,r is the space of tempered
distributions u such that the norm ‖u‖Bsp,r is finite.
We refer to [3] and [19] for background on harmonic analysis in the con-
text of fluids. We will use the Bernstein inequalities
Lemma 2.3.
‖∆qu‖Lb ≤ λ
q ‖∆qu‖La for b ≥ a ≥ 1.
As a consiquence we have the following inclusions.
Corollary 2.4. If b ≥ a ≥ 1, then we have the following continuous em-
beddings
Bsa,r ⊂ B
B0a,2 ⊂ La
6 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
In particular, the following chain of inclusions will be used throughout
the text.
(8) H
6 (R3) ⊂ B
(R3) ⊂ B
(R3) ⊂ B
3,2(R
3. ENERGY FLUX AND LOCALITY
3.1. Weak solutions.
Definition 3.1. A function u is a weak solution of the Euler equations with
initial data u0 ∈ L2(Rd) if u ∈ Cw([0, T ];L2(Rd)), (the space of weakly
continuous functions) and for every ψ ∈ C1([0, T ];S(Rd)) with S(Rd) the
space of rapidly decaying functions, with ∇x · ψ = 0 and 0 ≤ t ≤ T , we
(u(t), ψ(t))− (u(0), ψ(0))−
(u(s), ∂sψ(s))ds =
b(u, ψ, u)(s)ds,
where
(u, v) =
u · vdx,
b(u, v, w) =
u · ∇v · w dx,
and ∇x · u(t) = 0 in the sense of distributions for every t ∈ [0, T ].
Clearly, (9) implies Lipschitz continuity of the maps t → (u(t), ψ) for
fixed test functions. By an approximation argument one can show that for
any weak solution u of the Euler equation, the relationship (9) holds for all
ψ that are smooth and localized in space, but only weakly Lipschitz in time.
This justifies the use of physical space mollifications of u as test functions
ψ. Because we do not have an existence theory of weak solutions, this is a
rather academic point.
3.2. Energy flux. For a divergence-free vector field u ∈ L2 we introduce
the Littlewood-Paley energy flux at wave number λQ by
(10) ΠQ =
Tr[SQ(u⊗ u) · ∇SQu]dx.
If u(t) is a weak solution to the Euler equation, then substituting the test
function ψ = S2Qu into the weak formulation of the Euler equation (9) we
obtain
(11) ΠQ(t) =
‖SQu(t)‖22.
ENERGY CONSERVATION 7
Let us introduce the following localization kernel
(12) K(q) =
q , q ≤ 0;
q , q > 0,
For a tempered distribution u in R3 we denote
dq = λ
q ‖∆qu‖3,(13)
d2 = {d2q}q≥−1.(14)
Proposition 3.2. The energy flux of a divergence-free vector field u ∈ L2
satisfies the following estimate
(15) |ΠQ| ≤ C(K ∗ d2)3/2(Q).
From (15) we immediately obtain
(16) lim sup
|ΠQ| ≤ lim sup
We define B
3,c(N)
to be the class of all tempered distributions u in R3 for
which
(17) lim
λ1/3q ‖∆qu‖3 = 0,
and hence dq → 0. We endow B1/33,c(N) with the norm inherited from B
Notice that the Besov spaces B
3,p for 1 ≤ p < ∞, and in particular B
are included in B
3,c(N)
As a consequence of (11) and (16) we obtain the following theorem.
Theorem 3.3. The total energy flux of any divergence-free vector field in
the class B
3,c(N)
∩ L2 vanishes. In particular, every weak solution to the
Euler equation that belongs to the class L3([0, T ];B
3,c(N)
) ∩Cw([0, T ];L2)
conserves energy.
Proof of Proposition 3.2. In the argument below all the inequalities should
be understood up to a constant multiple.
Following [7] we write
(18) SQ(u⊗ u) = rQ(u, u)− (u− SQ)⊗ (u− SQ) + SQu⊗ SQu,
where
rQ(u, u) =
hQ(y)(u(x− y)− u(x))⊗ (u(x− y)− u(x))dy,
h̃Q(y) = λ
Q+1h̃(λQ+1y).
8 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
After substituting (18) into (10) we find
Tr[rQ(u, u) · ∇SQu]dx(19)
Tr[(u− SQ)⊗ (u− SQ) · ∇SQu]dx.(20)
We can estimate the term in (19) using the Hölder inequality by
‖rQ(u, u)‖3/2‖∇SQu‖3,
whereas
(21) ‖rQ(u, u)‖3/2 ≤
∣∣∣h̃Q(y)
∣∣∣ ‖u(· − y)− u(·)‖23dy.
Let us now use Bernstein’s inequalities and Corollary 2.4 to estimate
‖u(· − y)− u(·)‖23 ≤
|y|2λ2q‖∆qu‖23 +
‖∆qu‖23(22)
Q |y|2
Q−q d
q + λ
q(23)
≤ (λ4/3Q |y|2 + λ
Q )(K ∗ d2)(Q).(24)
Collecting the obtained estimates we find
Tr[rQ(u, u) · ∇SQu]dx
≤ (K ∗ d2)(Q)
∣∣∣h̃Q(y)
∣∣∣λ4/3Q |y|2dy + λ
λ2q‖∆qu‖23
≤ (K ∗ d2)(Q)λ−2/3Q
λ4/3q d
≤ (K ∗ d2)3/2(Q)
Analogously we estimate the term in (20)
Tr[(u− SQ)⊗ (u− SQ) ·∆SQu]dx
≤ ‖u− SQu‖23‖∆SQu‖3
‖∆qu‖23
λ2q‖∆qu‖23
≤ (K ∗ d2)3/2(Q).
This finishes the proof.
ENERGY CONSERVATION 9
3.3. Energy flux through dyadic shells. Let us introduce the energy flux
through a sequence of dyadic shells between scales −1 ≤ Q0 < Q1 < ∞
as follows
(25) ΠQ0Q1 =
Tr[SQ0Q1(u⊗ u) · ∇SQ0Q1u] dx,
where
(26) SQ0Q1 =
Q0≤q≤Q1
∆q = SQ1 − SQ0.
We will show that similar to formula (15) the flux through dyadic shells
is essentially controlled by scales near the inner and outer radii. In fact it
almost follows from (15) in view of the following decomposition
S2Q0Q1 = (SQ1 − SQ0−1)
= S2Q1 + S
Q0−1 − 2SQ0−1SQ1
= S2Q1 + S
Q0−1 − 2SQ0−1
= S2Q1 − S
Q0−1 − 2SQ0−1(1− SQ0−1)
= S2Q1 − S
Q0−1 − 2∆Q0−1∆Q0 .
Therefore
(28) ΠQ0Q1 = ΠQ1 − ΠQ0−1 − 2
Tr[∆̄Q0(u⊗ u) · ∇∆̄Q0u] dx,
where
(29) ∆̄Q0(u) =
h̄Q0(y)u(x− y) dy,
and h̄Q0(x) = F−1
ϕ(λ−1Q0−1ξ)ϕ(λ
Note that the flux through a sequence of dyadic shells is equal to the
difference between the fluxes across the dyadic spheres on the boundary
plus a small error term that can be easily estimated. Indeed, let us rewrite
the tensor product term as follows
(30) ∆̄Q0(u⊗ u) = r̄Q0(u, u) + ∆̄Q0u⊗ u+ u⊗ ∆̄Q0u,
where
r̄Q(u, u) =
h̄Q(y)(u(x− y)− u(x))⊗ (u(x− y)− u(x)) dy.
10 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
Thus we have
Tr[∆̄Q0(u⊗ u) · ∇∆̄Q0u] dx =
Tr[r̄Q(u, u) · ∇∆̄Q0u] dx
∆̄Q0u · ∇u · ∆̄Q0u dx
We estimate the first integral as previously to obtain
Tr[r̄Q0(u, u) · ∇∆̄Q0u] dx
∣∣∣∣ ≤ dQ0(K ∗ d
2)(Q0).
As to the second integral we have
∆̄Q0u · ∇u · ∆̄Q0u dx
∣∣∣∣ =
∆̄Q0u · ∇SQ0u · ∆̄Q0u dx
≤ d2Q0(K ∗ d
2)1/2(Q0).
Applying these estimates to the flux (28) we arrive at the following con-
clusion.
Theorem 3.4. The energy flux through dyadic shells between wavenumbers
λQ0 and λQ1 is controlled primarily by the end-point scales. More precisely,
the following estimate holds
(33) |ΠQ0Q1| ≤ C(K ∗ d2)3/2(Q0) + C(K ∗ d2)3/2(Q1).
3.4. Construction of a divergence free vector field with non-vanishing
energy flux. In this section we give a construction of a divergence free
vector field in B
3,∞(R
3) for which the energy flux is bounded from below
by a positive constant. This suggests the sharpness ofB
3,c(N)
(R3) for energy
conservation. Our construction is based on Eyink’s example on a torus [12],
which we transform to R3 using a method described below.
Let χQ(ξ) = χ(λ
Q+1ξ). We define P
ξ for vectors ξ ∈ R3, ξ 6= 0 by
P⊥ξ v = v − |ξ|−2(v · ξ)ξ =
I− |ξ|−2(ξ ⊗ ξ)
for v ∈ C3 and we use v · w =
vjwj for v, w ∈ C3.
Lemma 3.5. Let Φk(x) be R3 – valued functions, such that
Ik :=
|ξ||FΦk(ξ)| dξ <∞.
ENERGY CONSERVATION 11
Let also Ψk(x) = P(e
ik·xΦk(x)) where P is the Leray projector onto the
space of divergence free vectors. Then
(34) sup
∣∣Ψk(x)− eik·x(P⊥k Φk)(x)
∣∣ ≤ 1
|k| ,
(35) sup
∣∣(S2QΨk)(x)− χ2Q(k)Ψk(x)
∣∣ ≤ c
(2π)3
where c is the the Lipschitz constant of χ(ξ)2.
Proof. First, note that for any k, ξ ∈ R3 and v ∈ C3 we have
(v · ξ)ξ
|ξ|2 +
(v · ξ)k
∣∣∣∣ ≤
|ξ| ξ +
|v||ξ + k|
|k| .
In addition, it follows that
(v · k)k
|k|2 +
(v · ξ)k
∣∣∣∣ =
|(v · (k + ξ))k|
≤ |v||ξ + k||k| .
Adding (36) and (37) we obtain
|P⊥ξ v − P⊥k v| =
(v · ξ)ξ
|ξ|2 −
(v · k)k
(v · ξ)ξ
|ξ|2 +
(v · ξ)k
∣∣∣∣ +
(v · k)k
|k|2 +
(v · ξ)k
≤ 2 |v||ξ + k||k| .
Using this inequality we can now derive the following estimate:
|Ψk(x)− eik·x(P⊥k Φk)(x)| = |F−1[P⊥ξ (FΦk)(ξ + k)− P⊥k (FΦk)(ξ + k)]|
(2π)3
|ξ + k|
|k| |(FΦk)(ξ + k)| dξ
= |k|−1 1
|ξ||(FΦk(ξ))| dξ.
12 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
Finally, we have
|(S2QΨk)(x)− χQ(k)2Ψk(x)| = |F−1[(χQ(ξ)2 − χQ(k)2)(FΨk)(ξ)]|
(2π)3
c|ξ + k|
|(FΦk)(ξ + k)| dξ
= λ−1Q+1
(2π)3
|ξ||(FΦk)(ξ)| dξ,
where c is the the Lipschitz constant of χ(ξ)2. This concludes the proof. �
Example illustrating the sharpness of Theorem 3.3. Now we proceed to
construct a divergence free vector field in B
3,∞(R
3) with non-vanishing
energy flux. Let U(k) be a vector field U : Z3 → C3 as in Eyink’s example
[12] with
U(λq, 0, 0) = iλ
q (0, 0,−1), U(−λq, 0, 0) = iλ−1/3q (0, 0, 1),
U(0, λq, 0) = iλ
q (1, 0, 1), U(0,−λq, 0) = iλ−1/3q (−1, 0,−1),
U(λq, λq, 0) = iλ
q (0, 0, 1), U(−λq,−λq, 0) = iλ−1/3q (0, 0,−1),
U(λq,−λq, 0) = iλ−1/3q (1, 1,−1), U(−λq, λq, 0) = iλ−1/3q (−1,−1, 1),
for all q ∈ N and zero otherwise. Denote ρ(x) = F−1χ(4ξ) and A =∫
ρ(x)3 dx. Since χ(ξ) is radial, ρ(x) is real. Moreover,
ρ(x)3 dx =
(2π)3
F(ρ2)Fρ dξ
(2π)6
χ(4η)χ(4(ξ − η))χ(4ξ) dηdξ > 0.
Now let
u(x) = P
U(k)eik·xρ(x).
Note that u ∈ B1/33,∞(R3). Our goal is to estimate the flux ΠQ for the vector
field u. Define
Φk = |k|1/3U(k)ρ(x) and Ψk(x) = P(eik·xΦk(x)).
Then clearly Φk(x) and Ψk(x) satisfy the conditions of Lemma 3.5, and we
(41) u(x) =
k∈Z3\{0}
|k|−1/3Ψk(x).
ENERGY CONSERVATION 13
Now note that
Ψk1 · ∇S2QΨk2 = Ψk1 · S2QP[∇(eik·xΦk2)]
= i(Ψk1 · k2)S2QΨk2 +Ψk1 · S2QP(eik2·x∇Φk2).
In addition, the following equality holds by construction:
(43) P⊥k Φk = Φk, ∀k ∈ Z3.
Define the annulusAQ = Z
3∩B(0, λQ+2)\B(0, λQ−1). Thanks to Lemma 3.5,
for any sequences k1(Q), k2(Q), k3(Q) ∈ AQ with k1 + k2 = k3, we have
(Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx = i
(Ψk1 · k2)S2QΨk2 ·Ψ∗k3 dx+O(λ
(eik1·xΦk1 · k2)χQ(k2)2eik2·xΦk2 · e−ik3·xΦ∗k3 dx+O(λ
= i(|k1||k2||k3|)1/3A(U(k1) · k2)χQ(k2)2U(k2) · U(k3)∗ +O(λ0Q).
On the other hand, since the Fourier transform of Ψk is supported in
B(k, 1/4), we have
(Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx = 0,
whenever k1 + k2 6= k3. In addition, due to locality of interactions in this
example, (44) also holds if Aq \ {k1, k2, k3} 6= ∅ for all q ∈ N. Finally,
(Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx+
(Ψk1 · ∇S2QΨk3) ·Ψ∗k2 dx = 0,
whenever k2 /∈ AQ and k3 /∈ AQ. Hence, the flux for u can be written as
(46) ΠQ = −
k1,k2,k3∈AQ
k1+k2+k3=0
(|k1||k2||k3|)−1/3
(Ψk1 · ∇S2QΨk2) ·Ψk3 dx.
Since the number of nonzero terms in the above sum is independent of Q,
we obtain
(47) ΠQ = AΠ̃Q +O(λ
where Π̃ is the flux for the vector field U , i.e.,
(48) Π̃Q := −
k1,k2,k3∈AQ
k1+k2+k3=0
i(U(k1) · k2)χQ(k2)2U(k2) · U(k3).
14 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
The flux Π̃Q has only the following non-zero terms (see [12] for details):
|k2|=λQ
|k3|=
i(U1(−k2 − k3) · k2)U2(k2) · U3(k3)(χQ(k2)2 − χQ(k3)2)
≥ 4(χ(1/2)2 − χ(1/
2)2),
|k2|=
|k3|=2λQ
i(U1(−k2 − k3) · k2)U2(k2) · U3(k3)(χQ(k2)2 − χQ(k3)2)
≥ 4(χ(1/
2)2 − χ(1)2).
Hence
Π̃Q ≥ 4(χ(1/2)2 − χ(1/
2)2 + χ(1/
2)2 − χ(1)2) = 4.
This together with (47) implies that
lim inf
ΠQ ≥ 4A.
4. OTHER CONSERVATION LAWS
In this section we apply similar techniques to derive optimal results con-
cerning the conservation of helicity in 3D and that of enstrophy in 2D for
weak solutions of the Euler equation. In the case of the helicity flux we
prove that simultaneous infrared and ultraviolet localization occurs, as for
the energy flux. However, the enstrophy flux exhibits strong localization
only in the ultraviolet region, and a partial localization in the infrared re-
gion. A possibility of such a type of localization was discussed in [18].
4.1. Helicity. For a divergence-free vector field u ∈ H1/2 with vorticity
ω = ∇ × u ∈ H−1/2 we define the helicity and truncated helicity flux as
follows
u · ω dx(49)
Tr [SQ(u⊗ u) · ∇SQω + SQ(u ∧ ω) · ∇SQu] dx,(50)
where u ∧ ω = u ⊗ ω − ω ⊗ u. Thus, if u was a solution to the Euler
equation, then HQ would be the time derivative of the Littlewood-Paley
helicity at frequency λQ,
SQu · SQω dx.
ENERGY CONSERVATION 15
Let us denote
bq = λ
q ‖∆qu‖3,(51)
b2 = {b2q}∞q=−1,(52)
T (q) =
q , q ≤ 0;
q , q > 0,
Proposition 4.1. The helicity flux of a divergence-free vector field u ∈ H1/2
satisfies the following estimate
(54) |HQ| ≤ C(T ∗ b2)3/2(Q).
Theorem 4.2. The total helicity flux of any divergence-free vector field in
the class B
3,c(N)
∩H1/2 vanishes, i.e.
(55) lim
HQ = 0.
Consequently, every weak solution to the Euler equation that belongs to the
class L3([0, T ];B
3,c(N)
) ∩ L∞([0, T ];H1/2) conserves helicity.
Proposition 4.1 and Theorem 4.2 are proved by direct analogy with the
proofs of Proposition 3.2 and Theorem 3.3.
Example illustrating the sharpness of Theorem 4.2. We can also construct
an example of a vector field in B
3) for which the helicity flux is
bounded from below by a positive constant. Indeed, let U(k) be a vector
field U : Z3 → C3 with
U(±λq, 0, 0) = λ−2/3q (0, 0,−1),
U(0,±λq, 0) = λ−2/3q (1, 0, 1),
U(±λq,±λq, 0) = λ−2/3q (0, 0, 1),
U(±λq,∓λq, 0) = λ−2/3q (1, 1,−1),
for all q ∈ N and zero otherwise. Denote ρ(x) = F−1χ(4ξ),A =
ρ(x)3 dx,
and let
(56) u(x) = P
U(k)eik·xρ(x).
Note that u ∈ B2/33,∞(R3). On the other hand, a computation similar to the
one in Section 3.4 yields
(57) lim inf
|HQ| ≥ 4A.
16 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
4.2. Enstrophy. We work with the case of a two dimensional fluid in this
section. In order to obtain an expression for the enstrophy flux one can use
the original weak formulation of the Euler equation for velocities (9) with
the test function chosen to be
(58) ψ = ∇⊥S2Qω.
Let us denote by ΩQ the expression resulting on the right hand side of (9):
(59) ΩQ =
SQ(u⊗ u) · ∇∇⊥SQω
Thus,
d‖SQω‖22
= ΩQ.
As before we write
rQ(u, u) · ∇∇⊥SQω
(u− SQu)⊗ (u− SQu) · ∇∇⊥SQω
Let us denote
cq = ‖∆qω‖3,(61)
c2 = {c2q}∞q=−1,(62)
W (q) =
λ2q, q ≤ 0;
λ−4q , q > 0,
We have the following estimate (absolute constants are omitted)
|ΩQ| ≤
∣∣∣h̃Q(y)
∣∣∣ (‖∇SQu‖23|y|2 + ‖(I − SQ)u‖23)‖∇2SQω‖3dy
+ ‖(I − SQ)u‖23‖∇2SQω‖3
λ−2Q ‖SQω‖23 +
λ−2q c
λ−2q c
≤ ‖SQω‖23
λ−4Q−qc
λ2Q−qc
λ−4Q−qc
≤ ‖SQω‖23(W ∗ c2)1/2(Q) + (W ∗ c2)3/2(Q)
Thus, we have proved the following proposition.
ENERGY CONSERVATION 17
FIGURE 1. Construction of the vector field illustrating in-
frared nonlocality.
Proposition 4.3. The enstrophy flux of a divergence-free vector field satis-
fies the following estimate up to multiplication by an absolute constant
(64) |ΩQ| ≤ ‖SQω‖23(W ∗ c2)1/2(Q) + (W ∗ c2)3/2(Q).
Consequently, every weak solution to the 2D Euler equation
with ω ∈ L3([0, T ];L3) conserves enstrophy.
Much stronger results concerning conservation of enstrophy are available
for the Euler equations ([13], [20]) and for the long time zero-viscosity limit
for damped and driven Navier-Stokes equations ([9]).
Example illustrating infrared nonlocality. We conclude this section with a
construction of a vector field for which the enstrophy cascade is nonlocal in
the infrared range. Let θq = arcsin(λq−Q−2) and
(65) U lq = (cos(θq),− sin(θq)), Uhq = (sin(θq), cos(θq)),
klq = λq(sin(θq), cos(θq)), k
λ2Q+2 − λ2q(cos(θq),− sin(θq)),
see Fig. 4.2 for the case q = Q. Denote ρ(x) = δh̃(δx),A =
ρ(x)3 dx =∫
h̃(x)3 dx. Note that A > 0 and is independent of δ. Now let
(67) ulq(x) = P[U
q sin(k
q · x)ρ(x)], uhq (x) = P[Uhq sin(khq · x)ρ(x)].
(68) uq(x) = u
q(x) + u
q (x)
18 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
for q = 0, . . . , Q, and
(69) uQ+1(x) = P[V sin(λQ+2x1)ρ(x)],
where V = (0, 1). Now define
(70) u(x) =
uq(x).
Our goal is to estimate the enstrophy flux for u. Since Fu is compactly
supported, the expression (59) is equivalent to
(71) ΩQ =
(u · ∇)S2Qω · ω dx.
It is easy to see that
(72) ΩQ ≥
(uhq · ∇)S2Q(∇⊥ · ulq)(∇⊥ · uQ+1) dx.
Using Lemma 3.5 we obtain
ΩQ ≥ A
|Uhq |λ2q|U lq|λQ+2|V |+O(δ)
= λQ+2‖∆Q+2u‖3
λ2q‖∆qu‖23 +O(δ),
which shows sharpness of (64) in the infrared range.
5. INEQUALITIES FOR THE NONLINEAR TERM
We take d = 3 and consider u, v ∈ B
3,2 with ∇ · u = 0 and wish to
examine the advective term
(74) B(u, v) = P(u · ∇v) = ΛH(u⊗ v)
where
(75) [H(u⊗ v)]i = Rj(ujvi) +Ri(RkRl(ukvl))
and P is the Leray-Hodge projector, Λ = (−∆) 12 is the Zygmund operator
and Rk = ∂kΛ
−1 are Riesz transforms.
Proposition 5.1. The bilinear advective term B(u, v) maps continuously
the space B
3,2 × B
3,2 to the space B
. More precisely, there exist
ENERGY CONSERVATION 19
bilinear continuous maps C(u, v), I(u, v) so that B(u, v) = C(u, v) +
I(u, v) and constants C such that, for all u, v ∈ B
3,2 with ∇ · u = 0,
(76) ‖C(u, v)‖
≤ C‖u‖
(77) ‖I(u, v)‖
≤ C‖u‖
hold. If u, v, w ∈ B
(78) |〈B(u, v), w〉| ≤ C‖u‖
holds. So the trilinear map (u, v, w) 7→ 〈B(u, v), w〉 defined for smooth
vector fields in L3 has a unique continuous extension to
and a
fortiori to
Proof. We use duality. We take w smooth (w ∈ B
) and take the scalar
product
〈B(u, v), w〉 =
B(u, v) · wdx
We write, in the spirit of the paraproduct of Bony ([1])
(79) ∆q(B(u, v)) = Cq(u, v) + Iq(u, v)
(80) Cq(u, v) =
p≥q−2, |p−p′|≤2
∆q(ΛH(∆pu,∆p′v))
Iq(u, v) =
[∆qΛH(Sq+j−2u,∆q+jv) + ∆qΛH(Sq+j−2v,∆q+ju)]
20 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
We estimate the contribution coming from the Cq(u, v):
|〈Cq(u, v), w〉|
|q−q′|≤1
p≥q−2, |p−p′|≤2
3∆pu‖L3‖Λ
3∆p′v‖L3‖∆q′w‖L3
|p−p′|≤2
3∆pu‖L3‖Λ
3∆p′v‖L3
q≤p+2,|q−q′|≤1
3∆q′w‖L3
|p−p′|≤2
‖Λ 13∆pu‖ L3‖Λ
3∆p′v‖L3
 ‖w‖
≤ C‖u‖
This shows that the bilinear map C(u, v) =
q≥−1Cq(u, v) maps continu-
ously
(82) |〈C(u, v), w〉| ≤ C‖u‖
The terms Iq(u, v) contribute
|〈Iq(u, v), w〉|
|j|≤2, |q−q′|≤1
λq‖Sq+j−2u‖
‖∆q+jv‖L3‖∆q′w‖
|j|≤2, |q−q′|≤1
λq‖Sq+j−2v‖
‖∆q+ju‖L3‖∆q′w‖
≤ C‖u‖
|j|≤2,|q−q′|≤1
q ‖∆q+jv‖L3λ
q ‖∆q′w‖
+C‖v‖
|j|≤2,|q−q′|≤1
q ‖∆q+ju‖L3λ
q ‖∆q′w‖
≤ C‖u‖
Here we used the fact that
‖Squ‖
≤ C‖u‖
ENERGY CONSERVATION 21
This last fact is proved easily:
‖Sq(u)‖
∥∥∥∥∥∥
|∆ju|2
∥∥∥∥∥∥
‖∆ju‖2
≤ C‖u‖
We used Minkowski’s inequality in L
4 in the penultimate inequality and
Bernstein’s inequality in the last. This proves that I maps continuously
3,2 × B
3,2 to B
The proof of (78) follows along the same lines. Because of Bernstein’s
inequalities, the inequality (82) for the trilinear term 〈C(u, v), w〉 is stronger
than (78). The estimate of I follows:
|〈Iq(u, v), w〉|
|j|≤2, |q−q′|≤1
λq‖Sq+j−2u‖
‖∆q+jv‖
‖∆q′w‖
|j|≤2, |q−q′|≤1
λq‖Sq+j−2v‖
‖∆q+ju‖
‖∆q′w‖
≤ C‖u‖
|j|≤2,|q−q′|≤1
q ‖∆q+jv‖
q ‖∆q′w‖
+C‖v‖
|j|≤2,|q−q′|≤1
q ‖∆q+ju‖
q ‖∆q′w‖
+ ‖v‖
This concludes the proof. ✷
The inequality (82) is not true for 〈B(u, v), w〉 and (78) is close to being
optimal:
Proposition 5.2. For any 0 ≤ s ≤ 1
, 1 < p < ∞, 2 < r ≤ ∞ there exist
functions u, v, w ∈ Bsp,r and smooth, rapidly decaying functions un, vn, wn,
such that limn→∞ un = u, limn→∞ vn = v, limn→∞wn = w hold in the
norm of Bsp,r and such that
〈B(un, vn), wn〉 = ∞
Proof. We start the construction with a divergence-free, smooth function
u such that Fu ∈ C∞0 (B(0, 14)) and
u31dx > 0. We select a direction
22 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
e = (1, 0, 0) and set Φ = (0, u1, 0). Then
(83) A :=
(u(x) · e)
∣∣P⊥e Φ(x)
∣∣2 dx > 0.
Next we consider the sequence aq =
so that (aq) ∈ ℓr(N) for r > 2, but
not for r = 2, and the functions
(84) vn =
q aqP [sin(λqe · x)Φ(x)]
(85) wn =
q aqP [cos(λqe · x)Φ(x)] .
Clearly, the limits v = limn→∞ vn and w = limn→∞wn exist in norm in
every Bsp,r with 0 ≤ s ≤ 12 , 1 < p < ∞ and r > 2. Manifestly, by
construction, u, vn and wn are divergence-free, and because their Fourier
transforms are in C∞0 , they are rapidly decaying functions. Clearly also
〈B(u, vn), wn〉 =
P(u · ∇vn)wndx =
(u · ∇vn) · wndx.
The terms corresponding to each q in
u · ∇vn =
(u(x) · e)aqλ
q P [cos(λqe · x)Φ(x)]
q u(x) · P [sin(λqe · x)∇Φ(x)]
and in (85) have Fourier transforms supported B(λqe,
) ∪ B(−λq, 12) and
respectively B(λqe,
) ∪ B(−λqe, 14). These are mutually disjoint sets for
distinct q and, consequently, the terms corresponding to different indices q
do not contribute to the integral
(u · ∇vn) · wndx. The terms from the
second sum in (86) form a convergent series. Therefore, using Lemma 3.5,
we obtain
(u · vn) · wn =
(u(x) · e) {P [cos(λqe · x)Φ(x)]}2 dx+O(1)
(u(x) · e)
∣∣P⊥e Φ(x)
∣∣2 dx+O(1)
A+O(1),
ENERGY CONSERVATION 23
which concludes the proof. �
ACKNOWLEDGMENT
The work of AC was partially supported by NSF PHY grant 0555324,
the work of PC by NSF DMS grant 0504213, the work of SF by NSF DMS
grant 0503768, and the work of RS by NSF DMS grant 0604050.
REFERENCES
[1] J-M Bony, Calcul symbolique et propagation des singularité pour leséquations aux
dérivées partielles non linéaires, Ann. Ecole Norm. Sup. 14 (1981) 209–246.
[2] D. Chae, Remarks on the helicity of the 3-D incompressible Euler equations, Comm.
Math. Phys. 240 (2003), 501–507.
[3] J-Y Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford Univ 1998.
[4] A. Cheskidov, S. Friedlander and N. Pavlović, An inviscid dyadic model of turbu-
lence: the fixed point and Onsager’s conjecture, Journal of Mathematical Physics, to
appear.
[5] A. Cheskidov, S. Friedlander and N. Pavlović, An inviscid dyadic model of turbu-
lence: the global attractor (with S. Friedlander and N. Pavlović), preprint.
[6] P. Constantin, The Littlewood-Paley spectrum in 2D turbulence, Theor. Comp. Fluid
Dyn.9 (1997), 183-189.
[7] P. Constantin, W. E, E. Titi, Onsager’s conjecture on the energy conservation for
solutions of Euler’s equation, Commun. Math. Phys. 165 (1994), 207–209.
[8] P. Constantin, B. Levant, E. Titi, Regularity of inviscid shell models of turbulence,
Physical Review E 75 1 (2007) 016305.
[9] P. Constantin, F. Ramos, Inviscid limit for damped and driven incompressible Navier-
Stokes equations in R2, Commun. Math. Phys., to appear (2007).
[10] C. De Lellis and L. Székelyhidi, The Euler equations as differential inclusion,
preprint.
[11] J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incom-
pressible Euler and Navier-Stokes equations, Nonlinearity 13 (2000), 249–255.
[12] G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier
analysis and local energy transfer, Phys. D 78 (1994), 222–240.
[13] G. L. Eyink, Locality of turbulent cascades, Phys. D 207 (2005), 91–116.
[14] G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbu-
lence, Rev. Mod. Phys. 78 (2006).
[15] U. Frisch, Turbulence. The legacy of A. N. Kolmogorov. Cambridge University Press,
Cambridge, 1995.
[16] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluids
at very large Reynolds numbers, Dokl. Akad. Nauk. SSSR 30 (1941), 301–305.
[17] R. H. Kraichnan, The structure of isotropic turbulence at very high Reynolds num-
bers, J. Fluid Mech. 5 (1959), 497–543.
[18] R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10
(1967), 1417-1423.
[19] P-G Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman
and Hall/CRC, Boca Raton, 2002.
[20] M. Lopes Filho, A. Mazzucato, H. Nussenzveig-Lopes, Weak solutions, renormalized
solutions and enstrophy defects in 2D turbulence, ARMA 179 (2006), 353-387.
24 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
[21] H. K. Moffatt and A. Tsinober, Helicity in laminar and turbulent flow, Ann. Rev. Fluid
Mech. 24 (1992), 281–312.
[22] L. Onsager, Statistical Hydrodynamics, Nuovo Cimento (Supplemento) 6 (1949),
279–287.
[23] R. Robert, Statistical Hydrodynamics ( Onsager revisited ), Handbook of Mathemat-
ical Fluid Dynamics, vol 2 ( 2003), 1–55. Ed. Friedlander and Serre. Elsevier.
[24] V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal.
3(4) (1993), 343–401.
[25] A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Comm.
Pure Appl. Math. 50 (1997), 1261–1286.
(A. Cheskidov) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN
ARBOR, MI 48109
E-mail address: acheskid@umich.edu
(P. Constantin) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO, CHICAGO,
IL 60637
E-mail address: const@cs.uchicago.edu
(S. Friedlander and R. Shvydkoy) DEPARTMENT OF MATHEMATICS, STAT. AND
COMP. SCI., UNIVERSITY OF ILLINOIS, CHICAGO, IL 60607
E-mail address: susan@math.northwestern.edu
E-mail address: shvydkoy@math.uic.edu
