Calc. Var. (2023) 62:219
https://doi.org/10.1007/s00526-023-02558-8 Calculus of Variations

On energy conservation for the hydrostatic Euler equations:
an Onsager conjecture

Daniel W. Boutros1 · Simon Markfelder2 · Edriss S. Titi1,3,4

Received: 6 May 2023 / Accepted: 25 July 2023 / Published online: 25 August 2023
© The Author(s) 2023

Abstract
Onsager’s conjecture, which relates the conservation of energy to the regularity of weak
solutions of the Euler equations, was completely resolved in recent years. In this work, we
pursue an analogue of Onsager’s conjecture in the context of the hydrostatic Euler equations
(also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In
this case the relevant conserved quantity is the horizontal kinetic energy. We first consider
the standard notion of weak solution which is commonly used in the literature. We show that
if the horizontal velocity (u, v) is sufficiently regular then the horizontal kinetic energy is
conserved. Interestingly, the spatial Hölder regularity exponent which is sufficient for energy
conservation in the context of the hydrostatic Euler equations is 1

2 and hence larger than
the corresponding regularity exponent for the Euler equations (which is 1

3 ). This is due to
the anisotropic regularity of the velocity field: Unlike the Euler equations, in the case of
the hydrostatic Euler equations the vertical velocity w is one degree spatially less regular
with respect to the horizontal variables, compared to the horizontal velocity (u, v). Since
the standard notion of weak solution is not able to deal with this anisotropy properly, we
introduce two new notions of weak solutions for which the vertical part of the nonlinearity

Communicated by L. Szekelyhidi.

B Daniel W. Boutros
dwb42@cam.ac.uk

Simon Markfelder
simon.markfelder@gmail.com

Edriss S. Titi
titi@math.tamu.edu ; Edriss.Titi@maths.cam.ac.uk ; edriss.titi@weizmann.ac.il

1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge
CB3 0WA, UK

2 Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany

3 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

4 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100
Rehovot, Israel

123
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URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-42it0mxsoodg8

http://crossmark.crossref.org/dialog/?doi=10.1007/s00526-023-02558-8&domain=pdf
http://orcid.org/0000-0003-1921-2102
http://orcid.org/0000-0002-0011-7597
http://orcid.org/0000-0002-5004-1746


219 Page 2 of 40 D. W. Boutros et al.

is interpreted as a paraproduct. We finally prove several sufficient conditions for such weak
solutions to conserve energy.

Mathematics Subject Classification 76B99 (primary) · 35Q35 · 35D30 (secondary)

1 Introduction

In this paper, we study the three-dimensional hydrostatic Euler equations with rotation (also
known as the reduced inviscid primitive equations of planetary scale oceanic and atmospheric
dynamics, i.e. without coupling to temperature), which are given [17]

∂t u + ∇ · (uu) − �v + ∂x p = 0, (1.1)

∂tv + ∇ · (vu) + �u + ∂y p = 0, (1.2)

∂z p = 0, (1.3)

∇ · u = 0. (1.4)

Here u = (u, v, w) is the velocity field, p is the pressure and � ∈ R is the Coriolis param-
eter. We will also denote by uH = (u, v) the horizontal velocity vector field for notational
convenience.

It is already apparent from equation (1.4) that the vertical velocity w will have one degree
less spatial regularity in x and y than the horizontal velocities u and v. The fact that the
regularity of the velocity vector field is anisotropic will keep appearing when we study
regularity criteria which are sufficient for energy conservation.

In terms of the domain, the physical boundary-value problem for the system (1.1)–(1.4)
is usually studied in a three-dimensional channel

M = {
(x, y, z) ∈ R

3 : 0 ≤ z ≤ L, (x, y) ∈ R
2}

subject to the no-normal flow boundary condition in the vertical direction

w(x, y, 0, t) = w(x, y, L, t) = 0. (1.5)

Furthermore, u, v, w and p are periodic of period 2L in x and y. We consider the above
system with the initial conditions

u(x, y, z, 0) = u0(x, y, z), v(x, y, z, 0) = v0(x, y, z). (1.6)

System (1.1)–(1.6) is ill-posed in any Sobolev space [36], while it is locally well-posed in
the space of analytic functions [31, 45].

For classical (analytic) solutions of the equations in the channel, we can extend the solution
in the z-direction by extending the vertical velocity w to be odd in z and the horizontal
velocities u and v to be even in z with periodic domain [−L, L]. The odd extension of w is
made possible by the no-normal flow boundary condition (1.5).

For classical solutions, these symmetry conditions are invariant under the equations [31,
36].Without loss of generality we take L = 1

2 whichmakes the fundamental periodic domain
to be the flat unit torus T

3. Therefore for classical solutions, studying the physical boundary-
value problem in the channel with the no-normal flow boundary conditions is equivalent to
posing the problem on the torus subject to the symmetry conditions given above for u and p.

For this reason, throughout the paper, we will study weak solutions of the primitive equa-
tions on the torus with periodic boundary conditions and subject to the symmetry conditions.

123



On energy conservation for the hydrostatic Euler equations... Page 3 of 40 219

Notably, if such a weak solution satisfies suitable regularity assumptions such that one can
make sense of the trace when z = 0, L , then the odd extension of w actually satisfies the
original physical boundary condition (1.5). Consequently, such a weak solution solves the
original boundary value problem in M . In all the cases we consider in this paper, the weak
solutions have sufficient regularity to interpret boundary condition (1.5) in a trace sense.

One can observe that the horizontal kinetic energy

E(t) =
∫

T3

(
|u(x, t)|2 + |v(x, t)|2

)
dx

is a formally conserved quantity (i.e. it is conserved by classical solutions). From now on,
we will refer to this conserved quantity as the kinetic energy or simply the energy.

In this paper, we will show, loosely speaking, that the kinetic energy of weak solutions is
conserved if u and v are Hölder continuous with exponent bigger than 1

2 (we do not assume
this for w). This is in contrast to the corresponding result for the Euler equations, where the
critical exponent is 1

3 (for u, v and w). One might plausibly explain that the Hölder exponent
being higher in this case is due to the lower spatial regularity of the vertical velocity w.

The primitive equations model the planetary scale dynamics of the ocean and the atmo-
sphere. These equations can be derived as an asymptotic limit of the small aspect ratio (the
ratio of height to depth in a fluid) from the Rayleigh–Bénard (Boussinesq) system. These
equations have been the subject of a lot of attention over the years. So the overview that will
be given below will necessary be incomplete in summarising all the relevant work.

The primitive equationswere first introduced in [63]. The viscous primitive equationswere
studied in amathematical context for the first time in [51–53] (inwhich the global existence of
weak solutions was established). The short time existence and uniqueness of strong solutions
was established in [33] (see also [70]). The global existence of strong solutions was then
proven in [17] (see also [42]), with results for different boundary conditions in [47, 48]. In
[35] the case with less regular initial data was investigated using a semigroup method. The
uniqueness of weak solutions was discussed in [10, 11, 40, 41, 44, 50, 57].

The inviscid primitive equations have also been the subject of many works. The linear ill-
posedness in Sobolev spaces for the inviscid casewithout rotation (i.e.� = 0)was established
in [62]. Subsequently, the nonlinear ill-posedness was proven in [34]. Then it was shown in
[15, 72] that smooth solutions to the inviscid primitive equations without rotation can develop
a finite-time singularity. These results were extended to the case with rotation in [36]. Then
in [31] it was proved that the lifespan of solutions can be prolonged by fast rotation for
“well-prepared” initial data. The papers [8, 9, 31, 32, 43, 45, 46, 56] investigated the local
well-posedness for analytic initial data or initial data with a monotonicity assumption.

The nonuniqueness (and global existence) of weak solutions with initial data in L∞ for
the inviscid primitive equations was established in [19]. The paper [19] also showed the
non-conservation of energy. To the knowledge of the authors, it is the only paper that uses
convex integration for these equations. It is then natural to consider an analogue of Onsager’s
conjecture for energy conservation in the hydrostatic Euler equations, which is the subject
matter of this contribution.

Onsager’s conjecture was first posed in [60] for the Euler equations. It states that if a
weak solution u of the Euler equations belongs to L3((0, T );C0,α(T3)) with α > 1

3 then the
spatial L2 norm of u is conserved. In this context, the L2 norm of the velocity field u has the
physical interpretation of kinetic energy. The second half of the conjecture states that there
exist weak solutions of the Euler equations for α < 1

3 which do not conserve energy. Yet, it

123



219 Page 4 of 40 D. W. Boutros et al.

was also shown in [4] that there exist weak solutions that only have L2 spatial regularity and
which do conserve energy.

The importance of Onsager’s conjecture was emphasised in [29], which gave a proof of
the first half of the conjecture under a slightly more strict assumption thanC0,α , for α > 1

3 . A
complete proof of the first half was then given in [20]. A different approach was introduced
in [28] to prove the result, by deriving an equation of local energy balance with a defect term
(which describes the energy flux at the vanishing limit of small scales) to capture a possible
lack of regularity. The first half of the conjecture in the presence of wall boundaries was
proven in [5] (and references therein).

The second half of Onsager’s conjecture has also been proven by now. The first result
in this direction was by [66], which proved that there exist nontrivial weak solutions of the
Euler equation with compact support in time (which as a result do not conserve energy).
The first construction of a dissipative solution of the Euler equations was given by [67].
Subsequently techniques from convex integration were applied to the Euler equations to
prove the existence of non-energy conserving solutions in C0,1/3− in [39], after gradual
improvement in a sequence of papers [12–14, 21–25, 37, 38] (and see references therein).

It is natural to ask whether one can consider analogues of Onsager’s conjecture for related
PDEs. In [2] the authors consider the analogue of the first half of Onsager’s conjecture and
establish the universality of the Onsager exponent 1

3 for conservation of entropy as well
as other companion laws in a general class of conservation laws (see also [3]). The paper
[6] looked at the Euler-α model, while [7] investigated several subgrid-scale α-models of
turbulence.

The goal of this paper is to establish a formulation of the Onsager conjecture for energy
conservation of weak solutions to the hydrostatic Euler equations, in particular we prove
results of the following type (where Bs

p,q(T
3) is a Besov space, see Definition 2.4). The

concept of a weak solution for the hydrostatic Euler equations will be made precise later.

Theorem Letu beaweak solution of the hydrostaticEuler equations (which assumes thatw ∈
L2((0, T ); L2(T3))) and (u, v) ∈ L4((0, T ); Bα

4,∞(T3)) with α > 1
2 . Then the horizontal

kinetic energy is conserved, in particular

‖u(t1, ·)‖2L2 + ‖v(t1, ·)‖2L2 = ‖u(t2, ·)‖2L2 + ‖v(t2, ·)‖2L2 ,

for almost every t1, t2 ∈ (0, T ).

This theorem suggests that the analogue of theOnsager exponent (i.e. the threshold for energy
conservation) for these equations is 1

2 . Throughout the paper the term ‘Onsager exponent’
will denote the regularity threshold for a weak solution to conserve energy. Depending on
the context, this can either refer to a Besov, Hölder or Sobolev exponent.

This result is in contrast to the Onsager exponent for the incompressible Euler equations
which is 1

3 . See Sect. 7 for a discussion of this increase of the Onsager exponent from
1
3 to

1
2 .

The proof of this result will use techniques from [28]. In particular we will establish
an equation that describes the time evolution of the conserved quantity (referred to as the
equation of local energy balance) which contains a ‘defect term’ (which can be interpreted
as an energy flux at the vanishing limit of small scales or dissipation term). This defect
term captures the roughness in the weak solution which causes the anomalous dissipation of
energy and will be zero if the underlying weak solution is sufficiently regular.

We observe that there are two ways to approach the issue of the decrease in regularity of
w, as well as the nonlocality and anisotropic regularity imposed by equation (1.4).

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On energy conservation for the hydrostatic Euler equations... Page 5 of 40 219

1. Either one starts by imposing a regularity assumption on w and then proceeds to derive a
sufficient condition for energy conservation of weak solutions in terms of the horizontal
velocities.

2. Or one imposes sufficient regularity assumptions on the horizontal velocities such that
both the vertical velocity has sufficient regularity for a weak solution to be well-defined,
and the horizontal velocities are sufficiently regular for the energy to be conserved.

We will consider both these approaches in this paper. The first approach will be considered
in Sects. 2–4, while the second approach will be investigated in Sects. 5 and 6.

Throughout this paper, we will consider three types of weak solutions to the hydrostatic
Euler equations. We will refer to these notions as type I, II or III weak solutions. For later
reference, we already mention roughly what the differences between these notions are:

• A type I weak solution (see Definition 2.1) is the ‘canonical’ weak solution to the hydro-
static Euler equations. For such a weak solution it is assumed that w ∈ L2(T3 × (0, T ))

and u, v ∈ L∞((0, T ); L2(T3)).
• For a type II weak solution (see Definition 4.16) it is assumed thatw ∈ L2((0, T ); L2(T;

B−s
2,∞(T2))) and u, v ∈ L∞((0, T ); L2(T3)) ∩ L2((0, T ); L2(T; Bσ ′

2,∞(T2))) for σ ′ > s

(here L2(T; Bσ ′
2,∞(T2)) means L2(T) regularity in the vertical direction and Bσ ′

2,∞(T2)

regularity in the horizontal directions), where 0 < s ≤ 1
2 .

• For a type IIIweak solution (seeDefinition4.1)weassume thatw ∈ L2((0, T ); B−s
2,∞(T3))

and u, v ∈ L∞((0, T ); L2(T3)) ∩ L2((0, T ); Bσ ′
2,∞(T3)) for 0 < s ≤ 1

2 and s < σ ′.
Note that a difficulty in defining weak solutions where w is just a functional (i.e. type II
and III weak solutions) is to make sense of the products uw and vw as distributions. This is
done by using Bony’s paradifferential calculus. Furthermore, let us point out that boundary
condition (1.5) is imposed by using trace theorems. In the case of type II and type III weak
solutions the boundary condition will hold as a functional in a negative Besov space. We
refer to Remark 7.1 below for a comment on the relation of these notions of weak solution
to turbulence.

We now give an outline of the paper. In Sect. 2 we establish the equation of local energy
balance for weak solutions of type I. We then introduce a criterion that ensures that the defect
term, in the local energy balance, is zero. In Sect. 3 we investigate the adequate regularity
assumption necessary for the defect term to be zero, which then implies that the underlying
weak solution conserves energy. In other words, the goal is to interpret the criterion derived
in Sect. 2. Such a sufficient condition for type I weak solutions (in terms of Besov spaces) is
stated in the theorem above, which will be proven in Sect. 3.

Inspired by the quasilinear Lipschitz condition used in proving global existence and
uniqueness for the 2D Euler equations (see [54, 55, 73] for further details), we will sub-
sequently consider a logarithmic Hölder space. We will show that functions of this kind also
yield a zero defect term and solutions with such regularity conserve energy. The idea of this
example is to show that the additional decay requirement of the mollified defect term (on
top of the Lipschitz decay) to ensure conservation of energy is only very weak. Such a result
was already proven for the case of the Euler equations in [18].

We will also show that in order to ensure energy conservation u and v can have a lower
Hölder exponent if w is Hölder continuous. Roughly speaking, if we assume that w(·, t) ∈
C0,β(T3) then if u(·, t), v(·, t) ∈ C0,α(T3) with α ≥ β > 0 and α > 1

2 − 1
2β there is

conservation of energy. In particular, if u, v and w are all Hölder continuous with exponent
bigger than 1

3 the energy is conserved (i.e. the ‘original’ Onsager conjecture is also true,
which is consistent with the general result for conservation laws in [2]).

123



219 Page 6 of 40 D. W. Boutros et al.

We continue Sect. 3 by proving that it is possible to have different ‘horizontal’ and ‘ver-
tical’ Onsager exponents. This is caused by the fact that the hydrostatic Euler equations are
anisotropic in terms of regularity of the velocity field. We will show that if u and v are
Hölder continuous with exponent α in the vertical and β in the horizontal directions (where
2α + β > 2, α > 1

3 and β > 2
3 ) then the energy is conserved.

In Sect. 4 we will consider type II and type III weak solutions, namely when w is allowed
to be functional that belongs to a negative Besov space. We follow the same procedure as
we did for type I weak solutions and prove an analogue of Onsager’s conjecture for types II
and III weak solutions. We again establish an equation of local energy balance with a defect
term and find sufficient criteria for the vanishing of the defect term.

For type III weak solutions we show that if w(·, t) ∈ B−s
3,∞(T3) for 0 < s < 1

3 then

the the condition (u, v) ∈ L3((0, T ); B1/2+s/2+
3,∞ (T3)) is sufficient for the conservation of

energy. We also show that the Onsager exponent cannot be higher than 2
3 , for any notion of

weak solution. Indeed, if (u, v) ∈ L3((0, T ); B2/3+
3,∞ (T3)) then a type III weak solution of

the hydrostatic Euler equations conserves energy, regardless of any regularity assumption
on the vertical velocity w. This is because the regularity assumption on u and v ensures via
equation (1.4) thatw has enough regularity to make sense of the equation and also for energy
to be conserved.

The arguments for type II weak solutions are the same as for type III so therefore we only
state the results without proof.

Finally, in Sects. 5 and 6 we formulate sufficient criteria for energy conservation solely in
terms of regularity assumptions on the horizontal velocities u and v. In Sect. 5 we formulate
criteria in terms of Sobolev spaces, while in Sect. 6 we formulate a criterion in terms of
Besov spaces.

In Sect. 7 we conclude and give a plausible explanation for the increase in the Onsager
exponent from 1

3 for the Euler equations to 1
2 for the hydrostatic Euler equations with more

details. In the conclusion we also provide an overview of the different regularity criteria that
ensure conservation of energy. In particular, there seems to be a ‘family’ of Onsager con-
jectures for the hydrostatic Euler equations, which is due to the aforementioned anisotropic
regularity of the velocity field.

In Appendix A we state a few standard estimates for Besov norms. Appendix B gives an
overview of the techniques from Bony’s paradifferential calculus that are used in Sect. 4.
Finally, we note that many results in this paper could also have been proven by using the
commutator estimates approach as presented in [5, 20]. In Appendix C we give an example
of a proof done by using commutator estimates, namely the proof of Proposition 3.8.

2 The equation of local energy balance

In this section, we establish the equation of local energy balance. We first define a notion of
weak solutions (type I) to the hydrostatic Euler equations as in [19].

Definition 2.1 A type I weak solution of the hydrostatic Euler equations on the domain
T
3 × (0, T ) is given by a velocity field u = (u, v, w) : T

3 × (0, T ) → R
3 and a pressure

p : T
3 × (0, T ) → R such that:

• u ∈ L2(T3 × (0, T )), u, v ∈ L∞((0, T ); L2(T3)) and p ∈ L∞((0, T ); L1(T3)).

123



On energy conservation for the hydrostatic Euler equations... Page 7 of 40 219

• For all φ1, φ2 ∈ D(T3 × (0, T ); R) the following equations hold
∫ T

0

∫

T3
u∂tφ1dx dt +

∫ T

0

∫

T3
uu · ∇φ1dx dt +

∫ T

0

∫

T3
�vφ1dx dt (2.1)

+
∫ T

0

∫

T3
p∂xφ1dx dt = 0,

∫ T

0

∫

T3
v∂tφ2dx dt +

∫ T

0

∫

T3
vu · ∇φ2dx dt −

∫ T

0

∫

T3
�uφ2dx dt (2.2)

+
∫ T

0

∫

T3
p∂yφ2dx dt = 0.

• For all φ3 ∈ D(T3 × (0, T ); R), the following equation holds for the pressure
∫ T

0

∫

T3
p∂zφ3dxdt = 0. (2.3)

• For all φ4 ∈ D(T3 × (0, T ); R) the incompressibility condition holds, i.e.
∫ T

0

∫

T3
u · ∇φ4dx dt = 0. (2.4)

• It holds that w(x, y, 0, t) = w(x, y, 1, t) = 0 (we are considering the unit torus) for
almost all (x, y) ∈ T

2 and t ∈ (0, T ). This ensures that the symmetry conditions are
obeyed, namely that uH and p are even in z and that w is odd in z. We will explain in
Remark 2.2 why this requirement makes sense.

Remark 2.2 If one wants to define the notion of a type I weak solution to the boundary-value
problem for the hydrostatic Euler equations in the three-dimensional channel M , one has to
ensure that the no-normal flow boundary conditions (1.5) in the vertical direction are obeyed
(i.e.w(x, y, 0, t) = w(x, y, L, t) = 0). Equivalently, if the physical problem is posed on the
unit torus T

3 one has to ensure that the symmetry conditions are obeyed, which again means
that the boundary conditions (1.5) have to be imposed.

For type I weak solutions we know that u(·, t) ∈ L2(M) as well as that ∇ · u(·, t) =
0 ∈ L2(M). Then by the generalised trace theorem (see [68]) we know that

(
u ·n(·, t))|∂M∈

H−1/2(∂M). In particular, on the top and bottom of the channel u ·n = ±w, so the condition
w|∂M= 0 (recall that the channel has no boundary in the horizontal directions) makes sense
as an equation in H−1/2(∂M).

By the same reasoning, we can impose the condition w(x, y, 0, t) = 0 on the torus. Once
again, this condition makes sense as a trace. Then a weak solution of hydrostatic Euler on
the torus obeying this condition can be viewed as a solution of the physical boundary-value
problem on the channel because one can do an even extension in the horizontal directions
and an even/odd extension in the vertical direction. For this reason, we will only consider the
problem on the torus here. But the results can be proven by the same methods in the case of
the channel.

We will make one remark about the pressure for later reference. One can write down the
following equation for the pressure (if � = 0, otherwise an extra term appears which does
not significantly affect the regularity of the pressure)

�H p = −(∇H ⊗ ∇H ) :
∫

T

(
uH ⊗ uH

)
dz, (2.5)

123



219 Page 8 of 40 D. W. Boutros et al.

where we recall that uH = (u, v) is a two-dimensional vector and �H is the Laplacian in x
and y. The solution is uniquely determined by the requirement that

∫
T2 pdxdy = 0.

We will now show that we can weaken the regularity requirements on the test functions
φ1 and φ2 in the definition of a type I weak solution (Definition 2.1). We will need this result
later to prove an equation of local energy balance.

Lemma 2.3 Equations (2.1) and (2.2) still hold for test functions φ1, φ2 ∈ W 1,1
0 ((0, T );

L2(T3)) ∩ L2((0, T ); H3(T3)).

Proof Let ϕ ∈ W 1,1
0 ((0, T ); L2(T3)) ∩ L2((0, T ); H3(T3)) be arbitrary. Then there exists

a sequence {ϕn}∞n=1 ⊂ D(T3 × (0, T )) such that ϕn → ϕ in W 1,1
0 ((0, T ); L2(T3)) ∩

L2((0, T ); H3(T3)).
Now note that equations (2.1) and (2.2) hold for any ϕn , since they lie in D(T3 × (0, T )).

We observe that u∂tϕn → u∂tϕ in L1((0, T ); L1(T3)) as n → ∞ and therefore
∫ T

0

∫

T3
u∂tϕndx dt

n→∞−−−→
∫ T

0

∫

T3
u∂tϕdx dt .

Similarly, one can see that uu · ∇ϕn → uu · ∇ϕ in L1((0, T ); L1(T3)) as n → ∞, which
means that

∫ T

0

∫

T3
uu · ∇ϕndx dt

n→∞−−−→
∫ T

0

∫

T3
uu · ∇ϕdx dt dt .

Recall that we have made a separate regularity assumption on the pressure (p ∈ L∞((0, T );
L1(T3))). We observe that p∂xϕn → p∂xϕ in L2((0, T ); L1(T3)) as n → ∞. Thus it holds
that

∫ T

0

∫

T3
p∂xϕndx dt

n→∞−−−→
∫ T

0

∫

T3
p∂xϕdx dt dt .

The convergence of the other terms works exactly the same way and hence we conclude the
proof. 
�
We will also fix some notation. Let ϕ ∈ C∞

c (R3; R) be a radial standard C∞
c mollifier with

the property that
∫
R3 ϕdx = 1. We define

ϕε(x) := 1

ε3
ϕ

(
x

ε

)
.

Moreover, we introduce the notation

uε := u ∗ ϕε.

Throughout the paper we will be using the Einstein summation convention.
We also recall the definition of Besov spaces. Note that there are several equivalent defi-

nitions of Besov spaces, see, e.g., [1, 49] for more details.

Definition 2.4 For 1 ≤ p, q ≤ ∞ and s > 0, we say that a measurable function u : T
n → R

belongs to the Besov space Bs
p,q(T

n) if

‖u‖Bs
p,q

:= ‖u‖L p + |u|Bs
p,q

< ∞.

Here we have defined the Besov seminorm to be (for 1 ≤ q < ∞)

|u|Bs
p,q

:=
( ∫

Rn
‖��s�+1

h u‖qL p
dh

|h|n+sq

)1/q

.

123



On energy conservation for the hydrostatic Euler equations... Page 9 of 40 219

For the case q = ∞, we define the alternative seminorm

|u|Bs
p,∞ := sup

h∈Rn\{0}
1

|h|s ‖��s�+1
h u‖L p . (2.6)

Note that �·� has been used here to denote the integer part. We have also introduced the
difference quotients through the following two equations

�1
hu(x) := u(x + h) − u(x),

�m
h u(x) := �h

(
�m−1

h u(x)
)

, m ≥ 2,

for x ∈ T
n and h ∈ R

n .

The equation of local energy balance will be stated in the following theorem.

Theorem 2.5 Let u be a type I weak solution of the hydrostatic Euler equations such that

u, v ∈ L4 (
(0, T ); L4(T3)

)
. (2.7)

Then the following equation of local energy balance (which holds in the sense of distributions
with the space of test functions D(T3 × (0, T ))) is satisfied

∂t (u
2 + v2) + ∇ ·

[
(u2 + v2)u

]
+ 2∇ · (pu) + 1

2
D(u) = 0. (2.8)

In the above we have introduced the defect term

D(u)(x, t) := lim
ε→0

∫

R3

[
∇ϕε(ξ) · δu(ξ ; x, t) (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2)

]
dξ, (2.9)

where

δu(ξ ; x, t) := u(x + ξ, t) − u(x, t).

The limit in equation (2.9) is independent of the choice of mollifier ϕε and the convergence
is in the space W−1,1((0, T );W−1,1(T3)).

Proof We first mollify equations (1.1) and (1.2) in space (with ϕε) and obtain that

0 = ∂t u
ε + ∇ · (uu)ε − �vε + ∂x p

ε, (2.10)

0 = ∂tv
ε + ∇ · (vu)ε + �uε + ∂y p

ε . (2.11)

These equations hold pointwise. We observe that uε, vε ∈ L∞((0, T );C∞(T3)). We note
that the terms in the system (2.10)–(2.11) have the following regularities: ∇ · (uu)ε,∇ ·
(vu)ε ∈ L2((0, T );C∞(T3)), �vε,�uε ∈ L∞((0, T );C∞(T3)) and ∂x pε, ∂y pε ∈
L∞((0, T );C∞(T3)) (and similarly for the terms in equation (2.11)). From this we con-
clude that

∂t u
ε, ∂tv

ε ∈ L2 (
(0, T );C∞(T3)

)
.

As a result, we obtain that

uε, vε ∈ H1 (
(0, T );C∞(T3)

) ∩ L∞((0, T );C∞ (
T
3)

)

⊂ W 1,1((0, T ); L2(T3)) ∩ L2((0, T ); H3(T3)).

123



219 Page 10 of 40 D. W. Boutros et al.

Therefore, we multiply uε and vε by a function ψ ∈ D(T3 × (0, T ); R) and consider ψuε

and ψvε as the test functions in our weak formulation for type I weak solutions, which is
allowed by Lemma 2.3. Adding the equations together gives

∫ T

0

∫

T3

[
u∂t (u

εψ) + v∂t (v
εψ) + uu · ∇(uεψ) + vu · ∇(vεψ) + �vuεψ − �uvεψ

+ p∂x (u
εψ) + p∂y(v

εψ)

]
dxdt = 0. (2.12)

Now we multiply equations (2.10) and (2.11) by ψu and ψv respectively (they therefore
hold almost everywhere), integrate them in space and time and subtract them from equation
(2.12). This yields

∫ T

0

∫

T3

[
u∂t (u

εψ) − uψ∂t u
ε + v∂t (v

εψ) − vψ∂tv
ε + uu · ∇(uεψ) − ψu∇ · (uu)ε

+ vu · ∇(vεψ) − ψv∇ · (vu)ε + p∂x (u
εψ) − uψ∂x p

ε + p∂y(v
εψ) − vψ∂y p

ε

]
dxdt = 0.

Now we deal with the different terms in turn, we observe by the Leibniz rule that (since
uε, vε and ψ are weakly differentiable in time)

∫ T

0

∫

T3

[
u∂t (u

εψ) − uψ∂t u
ε + v∂t (v

εψ) − vψ∂tv
ε

]
dxdt =

∫ T

0

∫

T3

[
(uuε + vvε)∂tψ

]
dxdt

= 〈−∂t (uu
ε + vvε), ψ〉,

where the brackets 〈·, ·〉 denote the distributional action. Note that we have used the fact that
ψ is C∞ to be able to justify the time derivative of the products uεψ and vεψ satisfies the
Leibniz rule.

Now we look at the pressure terms, we recall that ∂z pε = 0, as well as equation (2.3).
Then we rewrite the pressure terms as follows

∫ T

0

∫

T3

[
p∂x (u

εψ) − uψ∂x p
ε + p∂y(v

εψ) − vψ∂y p
ε

]
dxdt

=
∫ T

0

∫

T3

[
p∂x (u

εψ) + p∂y(v
εψ) + p∂z(w

εψ) − uψ∂x p
ε − vψ∂y p

ε − wψ∂z p
ε

]
dxdt

=
∫ T

0

∫

T3

[
puε · ∇ψ + pεu · ∇ψ

]
dxdt = 〈−∇ · (puε + pεu), ψ〉,

where we have used the incompressibility ∇ · uε = 0 condition in the above. Finally, we
look at the advective terms. We first introduce a defect term and calculate it to be

Dε(u) :=
∫

R3
dξ

[
∇ϕε(ξ) · δu(ξ ; x, t)

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

) ]
= −∇ ·

[
(u2 + v2)u

]ε

+ u · ∇(u2 + v2)ε + 2u∇ · (uu)ε + 2v∇ · (vu)ε − 2uu · ∇uε − 2vu · ∇vε − (u2 + v2)∇ · uε

= −∇ ·
[
(u2 + v2)u

]ε

+ u · ∇(u2 + v2)ε + 2u∇ · (uu)ε + 2v∇ · (vu)ε − 2uu · ∇uε − 2vu · ∇vε.

123



On energy conservation for the hydrostatic Euler equations... Page 11 of 40 219

Subsequently, we can write the advective terms as follows

∫ T

0

∫

T3

[
uu · ∇ (

uεψ
) − ψu∇ · (uu)ε + vu · ∇(vεψ) − ψv∇ · (vu)ε

]
dxdt

=
∫ T

0

∫

T3

[
ψuu · ∇uε − ψu∇ · (uu)ε + ψvu · ∇vε − ψv∇ · (vu)ε + (

uuε + vvε
)
u · ∇ψ

]
dxdt

=
∫ T

0

∫

T3

[
− 1

2
ψDε(u) − 1

2
ψ∇ ·

[ (
u2 + v2

)
u
]ε

+ 1

2
ψu · ∇

(
u2 + v2

)ε + (
uuε + vvε

)
u · ∇ψ

]
dxdt

=
∫ T

0

∫

T3

[
− 1

2
ψDε(u) + 1

2

[ (
u2 + v2

)
u
]ε

· ∇ψ − 1

2

(
u2 + v2

)ε
u · ∇ψ + (

uuε + vvε
)
u · ∇ψ

]
dxdt

=
〈
− 1

2
Dε(u) − ∇ · ((

uuε + vvε
)
u
) + 1

2
∇ ·

((
u2 + v2

)εu − ( (
u2 + v2

)
u
)ε

)
, ψ

〉
.

Once again we have used the incompressibility condition here. Then we end up with the
following distributional equation (again for all ψ ∈ D(T3 × (0, T )))

〈
∂t (uu

ε + vvε) + ∇ · (puε + pεu) + 1

2
Dε(u) + ∇ · ((uuε + vvε)u)

+ 1

2
∇ ·

((
(u2 + v2)u

)ε − (
u2 + v2

)εu
)

, ψ

〉
= 0. (2.13)

Nowwe consider the convergence of the different terms as ε → 0.We first observe that since
u, v ∈ L∞((0, T ); L2(T3)) it holds that

uuε + vvε ε→0−−→ u2 + v2 in L∞ (
(0, T ); L1(T3)

)
.

Now we recall the assumption that u, v ∈ L4((0, T ); L4(T3)). By equations (1.3) and (2.5),
one has p ∈ L2((0, T ); L2(T3)). Therefore,

puε + pεu
ε→0−−→ 2pu in L1 (

(0, T ); L1(T3)
)
.

Now again by assumption (2.7) it holds that

uuε + vvε ε→0−−→ u2 + v2 in L2 (
(0, T ); L2(T3)

)
.

In particular one has (uuε + vvε)u
ε→0−−→ (u2 + v2)u in L1((0, T ); L1(T3)). From this we

conclude that
(
(u2 + v2)u

)ε − (
u2 + v2

)εu
ε→0−−→ 0 in L1 (

(0, T ); L1 (
T
3)) .

Finally, we need to consider the convergence of the defect term Dε as ε → 0. We first
observe that Dε(u) is well-defined as a function in L1((0, T ); L1(T3)) for any ε > 0 due to
the following bound

‖Dε(u)(·, t)‖L1 ≤ Cε‖δu(·, t)‖L2

(
‖δu(·, t)‖2L4 + ‖δv(·, t)‖2L4

)
.

In the aboveCε is a constantwhich depends on ε > 0.We already saw that δu(|δu|2+|δv|2) ∈
L1((0, T ); L1(T3)), therefore Dε(u) ∈ L1((0, T ); L1(T3)) for any ε > 0.

Regarding the limit ε → 0, by virtue of equation (2.13) one has

Dε(u) = −2∂t (uu
ε + vvε) − 2∇ · (puε + pεu) − 2∇ · ((uuε + vvε)u)

−∇ ·
((

(u2 + v2)u
)ε − (

u2 + v2
)εu

)
.

123



219 Page 12 of 40 D. W. Boutros et al.

In the above we have proven that the right-hand side converges inW−1,1((0, T );W−1,1(T3))

and is independent of the choice of mollifier. We call its limit D(u). As a result,

D(u) = −2
(
∂t (u

2 + v2) + ∇ · (2pu) + ∇ · ((u2 + v2)u)
)
,

which holds in the sense of distributions, that is
〈
∂t (u

2 + v2) + ∇ · (2pu) + 1

2
D(u) + ∇ · ((u2 + v2)u), ψ

〉
= 0,

for every test function ψ ∈ D(T3 × (0, T )). 
�
It should be mentioned that it is possible to include the viscous terms when deriving the
equation of local energy balance, but this is not needed for our purposes here. Now we
introduce a criterion which ensures that the defect term, D(u), is zero. Note that the term
D(u) can be physically interpreted as an energy flux at the limit of vanishing small spatial
scales, i.e. as ε → 0, and hence is the source of dissipation of energy in the case of not
smooth enough type I weak solutions.

Proposition 2.6 Let u be a type I weak solution of the hydrostatic Euler equations and
assume that u, v ∈ L4((0, T ); L4(T3)). Let C ∈ L1(0, T ) and assume that σ ∈ L∞

loc(R) is
a nonnegative real-valued function with the property that σ(|ξ |) → 0 as |ξ | → 0. Suppose
u satisfies the inequality

∫

T3
|δu(ξ ; x, t)|

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx ≤ C(t)|ξ |σ(|ξ |). (2.14)

Then D(u) = 0.

Proof Observe that

|Dε(u)| = 1

2

∣∣∣∣

∫

R3
∇ϕε(ξ) · δu(ξ ; x, t)

[
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

]
dξ

∣∣∣∣

≤
∫

R3
|∇ϕε(ξ)||δu(ξ ; x, t)|

[
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

]
dξ.

Integrating this inequality over T
3 × (0, T ) gives us that

∫ T

0

∫

T3
|Dε(u)|dxdt ≤

∫ T

0
dt

∫

R3
|∇ϕε(ξ)|

∫

T3
|δu(ξ ; x, t)| (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2) dxdξ

≤
∫ T

0
C(t)dt

∫

R3

1

ε3
|∇ξ ϕ(ξ/ε)||ξ |σ(|ξ |)dξ.

Note thatwe are allowed to useFubini’s theorembecause the integrand is absolutely integrable
(as u(·, t) ∈ L2(T3) and u(·, t), v(·, t) ∈ L4(T3)). Now by using that

∫ T
0 C(t)dt < ∞ and

the change of variable ξ = εz, we find that
∫ T

0

∫

T3
|Dε(u)|dxdt �

∫

T3
|∇ξ ϕ(z)||ξ |σ(|ξ |)dz =

∫

T3
|∇zϕ(z)||z|σ(ε|z|)dz.

Thanks to the assumption that σ(|ξ |) → 0 as |ξ | → 0, we find that Dε(u) → 0, in
L1(T3 × (0, T )) as ε → 0 by the Lebesgue dominated convergence theorem. Therefore
D(u) = 0. 
�
The goal of the next section is to interpret criterion (2.14), i.e. by finding the biggest function
space for which bound (2.14) holds.

123



On energy conservation for the hydrostatic Euler equations... Page 13 of 40 219

3 Proof of conservation of energy

We will prove several different sufficient conditions for (2.14) to hold. We will then prove
that if condition (2.14) is satisfied, conservation of energy holds. In particular, one can make
different regularity assumptions for w due to the aforementioned anisotropy of the velocity
field. In this section, we consider type I weak solutions (cf. Definition 2.1), i.e. we assume
that w ∈ L2((0, T ); L2(T3)).

First we will find a suitable Besov space such that condition (2.14) is satisfied. Later in
this section we will introduce the logarithmic Hölder space in order to show that even if
the solution is in a slightly weaker function space the function σ in equation (2.14) is still
o(1) as |ξ | → 0. We will also show that the Onsager exponent (the regularity threshold for
energy conservation) can be lowered if w is Hölder continuous. Finally, we will show that
there can be a different Onsager exponent in the vertical direction compared to the horizontal
directions.

Criterion (2.14) seems to dictate the condition u(·, t), v(·, t) ∈ Bα
3,∞(T3) with α > 1

3 ,
because it is possible to bound |δu| by

|δu| ≤ |δu| + |δv| + |δw|.

The first two terms in this equation multiplied by the term (|δu|2 + |δv|2) in equation (2.14)
give us products of the form |δu|3, |δv|3, |δu||δv|2 and |δv||δu|2. Because we want to treat
the horizontal velocities on an equal footing, we want to impose the same conditions on
the horizontal velocities. The terms in Dε(u) with only horizontal velocities should have
sufficient regularity such that the function σ in equation (2.14) is o(1) as |ξ | → 0. This
seems to necessitate the assumption u(·, t), v(·, t) ∈ Bα

3,∞(T3) with α > 1
3 .

These conditions alone do not ensure energy conservation, because in that case one needs
to assume that w(·, t) ∈ Bα

3,∞(T3) with α > 1
3 as well. That would be a very strong

requirement, because implicitly this imposes conditions on u and v via equation (1.4). These
conditions cannot be inferred directly however. Instead we need a stronger condition on u
and v, in order to not need any assumption on w (except for it being in L2((0, T ); L2(T3)),
which is required in the definition of a type I weak solution).

3.1 Sufficient condition for energy conservation

Wenowprove a propositionwhich gives a sufficient condition for (2.14) to be true,which then
implies conservation of energy. This result coincides with the theorem from the introduction.

Proposition 3.1 (Conservation of energy) Let u be a type I weak solution of the hydrostatic
Euler equations and assume that u, v ∈ L4((0, T ); Bα

4,∞(T3)) with α > 1
2 . Then D(u) = 0

(where D(u) was defined in equation (2.9)), which implies that the weak solution conserves
energy. That is,

‖u(t1, ·)‖2L2 + ‖v(t1, ·)‖2L2 = ‖u(t2, ·)‖2L2 + ‖v(t2, ·)‖2L2 , for a.e. t1, t2 ∈ (0, T ). (3.1)

Proof By the definition of a type I weak solution for the hydrostatic Euler equations (Def-
inition 2.1) it is assumed that u ∈ L2((0, T ); L2(T3)). Because u and v are in addition
assumed to be in L4((0, T ); L4(T3)) then by Theorem 2.5 the defect term D(u) exists and is
well-defined. Moreover, the equation of local energy balance holds. By using the additional

123



219 Page 14 of 40 D. W. Boutros et al.

regularity assumptions that u(·, t), v(·, t) ∈ Bα
4,∞(T3) with α > 1

2 we get that
∫

T3
|δu(ξ ; x, t)|

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx

≤ ‖δu(ξ ; ·, t)‖L2
(‖δu(ξ ; ·, t)‖2L4 + ‖δv(ξ ; ·, t)‖2L4

)

≤ 2‖u(·, t)‖L2 |ξ |2α
(∥∥∥∥

δu(ξ ; ·, t)
|ξ |α

∥∥∥∥

2

L4
+

∥∥∥∥
δv(ξ ; ·, t)

|ξ |α
∥∥∥∥

2

L4

)

≤ 2‖u(·, t)‖L2 |ξ |2α
(

‖u(·, t)‖2Bα
4,∞

+ ‖v(·, t)‖2Bα
4,∞

)
.

Because 2α > 1, we can take (in the notation of Proposition 2.6)

σ(|ξ |) := |ξ |2α−1,

which indeed goes to zero as |ξ | tends to zero. Finally, we observe that

C(t) = ‖u(·, t)‖L2

(
‖u(·, t)‖2Bα

4,∞
+ ‖v(·, t)‖2Bα

4,∞

)
∈ L1(0, T ),

since ‖u(·, t)‖L2 ∈ L2(0, T ) and ‖u(·, t)‖Bα
4,∞ , ‖v(·, t)‖Bα

4,∞ ∈ L4(0, T ). Therefore the
conditions of Proposition 2.6 are satisfied and as a consequence it follows that D(u) = 0.

Now according to equation (2.8) we have

∂t (u
2 + v2) + ∇ ·

[
(u2 + v2 + 2p)u

]
= 0,

which holds in the sense of distributions. Hence for any ψ ∈ D(T3 × (0, T ); R) we have
that

∫ T

0

∫

T3
(u2 + v2)∂tψdxdt = −

∫ T

0

∫

T3
∇ψ ·

((
u2 + v2 + 2p

)
u
)
dxdt .

Let 0 < t1 < t2 < T and choose the following test function

ψ(t) =
∫ t

0

(
φε(t

′ − t1) − φε(t
′ − t2)

)
dt ′.

Here φ is a C∞
c (R; R) mollifier such that

∫
R

φ(t)dt = 1, suppφ ⊂ (−1, 1) and φε(t) =
ε−1φ(t/ε). Note that ψ is zero for t ∈ (0, t1 − ε) ∪ (t2 + ε, T ) (i.e. suppψ ⊂ (0, T )), for ε

sufficiently small. The derivative of this test function is given by

∂tψ(t) = φε(t − t1) − φε(t − t2).

Becauseψ has no spatial dependence, it follows that the spatial gradient vanishes. As a result
we get that

∫ t1+ε

t1−ε

∫

T3
(u2 + v2)φε(t − t1)dxdt =

∫ t2+ε

t2−ε

∫

T3
(u2 + v2)φε(t − t2)dxdt .

Then by using the Lebesgue differentiation theorem in the limit ε → 0 [30, 71] we can
conclude that

∫

T3

(
u2(x, t1) + v2(x, t1)

)
dx =

∫

T3

(
u2(x, t2) + v2(x, t2)

)
dx,

for almost every t1, t2 ∈ (0, T ), which is equation (3.1). 
�

123



On energy conservation for the hydrostatic Euler equations... Page 15 of 40 219

It now becomes plausible why the Onsager exponent is 1
2 and not 1

3 , the required regularity
in order to ensure that the function σ in criterion (2.14) is o(1) has to be distributed over two
terms in the product instead of three (the latter being the case for the Euler equations, see
[28] for more details). This will be discussed further in Sect. 7.

3.2 Energy conservation for log-Hölder regularity

Note that it is also possible to satisfy criterion (2.14) by imposing a different regularity
condition on the solution. To this end, we introduce another function space.

Definition 3.2 Let U be a bounded and closed set. The logarithmic Hölder space C0,γ
log (U )

with 0 < γ < 1 consists of all continuous functions (i.e. in C0(U )) for which the following
seminorm is finite

|u|
C0,γ
log

:= sup
x,y∈U ,x �=y

|u(x) − u(y)|
|x − y|γ (1 + log−(|x − y|)). (3.2)

It is straightforward to see that C0,γ
log (T3) ⊂ C0,γ (T3). One can also show that C0,γ

log (T3) is a
Banach space (see [26]). We now prove that type I weak solutions with logarithmic Hölder
continuity with exponent 1

2 have a defect term which is zero and hence conserve energy.

Proposition 3.3 Suppose u is a type I weak solution of the hydrostatic Euler equations and
suppose that u, v ∈ L4((0, T );C0,1/2

log (T3)). Then D(u) = 0, which then implies the conser-
vation of energy (3.1).

Proof The proof is almost the same as the proof of Proposition 3.1. The only difference being
that we now have the inequality (for |ξ | < 1)

‖δu(ξ ; ·, t)‖L2
(‖δu(ξ ; ·, t)‖2L4 + ‖δv(ξ ; ·, t)‖2L4

)

≤ 2‖u(·, t)‖L2 |ξ | 1
(
1 − log(|ξ |))2

(
‖u(·, t)‖2

C0,1/2
log

+ ‖v(·, t)‖2
C0,1/2
log

)
.

This means that in the notation of Proposition 2.6 we have that (again for |ξ | < 1)

σ(|ξ |) := 1
(
1 − log(|ξ |))2

,

which clearly goes to zero as |ξ | → 0. Therefore the conditions of Proposition 2.6 are satisfied
and the defect term is zero. The proof that the solution conserves energy is analogous to the
proof of Proposition 3.1. 
�
Remark 3.4 Note that C0,1/2

log (T3) �⊂ Bα
4,∞(T3) with α > 1

2 . Therefore the above proposition
is not contained within Proposition 3.1.

Remark 3.5 We observe that the log-Hölder spaces can also be used for the original Onsager
conjecture for the Euler equations. We recall that in [28] it was proven that

∫

T3
|δu(ξ ; x, t)|3dx ≤ C(t)|ξ |σ(|ξ |), (3.3)

ensures that the defect term is zero (in the equation of local energy balance for a weak solution
of the Euler equations) Here we used the same notation as in Proposition 3.1, where u is

123



219 Page 16 of 40 D. W. Boutros et al.

a weak solution of the three-dimensional Euler equations. Now we observe that under the
assumption that u(·, t) ∈ C0,1/3

log (T3) condition (3.3) is satisfied with the following choice
for σ

σ(|ξ |) := 1
(
1 − log(|ξ |))3

, |ξ | < 1.

It is clear that σ(|ξ |) → 0 as |ξ | → 0 and therefore there is conservation of energy. Note that
also in this case u(·, t) ∈ C0,1/3

log (T3) is a different condition from u(·, t) ∈ Bα
3,∞(T3) with

α > 1
3 . Therefore log-Hölder spaces can be applied to prove the original Onsager conjecture.

It is also possible to consider even slower decay, e.g. a power of a logarithm. In addition, one
can also consider logarithmic Besov spaces. A result of this type for the Euler equations was
proven in [18].

The consideration of logarithmic Hölder spaces was inspired by the proof of global exis-
tence for the 2D Euler equations with the vorticity being bounded in L∞. In that case a
quasilinear Lipschitz condition is used of the form [54, 55, 73]

|u(x) − u(y)| ≤ Kϕ(|x − y|),
where we have defined

ϕ(r) :=
{
r(1 − log r), r < 1,

1, r ≥ 1.

In order to generalise this condition we have introduced the concept of logarithmic Hölder
continuity.

3.3 Hölder regularity for the vertical velocity

Next we consider w to be Hölder continuous with some exponent β. In that case the Onsager
exponent (for u and v) can be lower, as we will see in the next proposition.

Proposition 3.6 Letu be a type I weak solution of the hydrostatic Euler equations and assume
that u ∈ L p1((0, T ); Bβ

p2,∞(T3)) with β > 0 and 1 ≤ p1, p2 ≤ ∞. Moreover, assume that
u, v ∈ Lq1((0, T ); Bα

q2,∞(T3)) with 2p′
1 ≤ q1, 2p′

2 ≤ q2 (where p′
1, p

′
2 are the Hölder

conjugates of p1 and p2) and 2α > 1 − β (with α ≥ β). Then D(u) = 0, which implies
conservation of energy (3.1).

Proof One can obtain the following estimates
∫

T3
|δu(ξ ; x, t)|

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx

≤ ‖δu(ξ ; ·, t)‖L p2

(
‖δu(ξ ; ·, t)‖2

L2p′2
+ ‖δv(ξ ; ·, t)‖2

L2p′2

)

≤ |ξ |2α+β ‖δu(ξ ; ·, t)‖L p2

|ξ |β

⎛

⎝
‖δu(ξ ; ·, t)‖2

L2p′2
|ξ |2α +

‖δv(ξ ; ·, t)‖2
L2p′2

|ξ |2α

⎞

⎠

≤ |ξ |2α+β‖u(·, t)‖
Bβ
p2,∞

(
‖u(·, t)‖2Bα

2p′2,∞
+ ‖v(·, t)‖2Bα

2p′2,∞

)
.

123



On energy conservation for the hydrostatic Euler equations... Page 17 of 40 219

We have used the assumed Besov regularity ofw in estimating the term
‖δu(ξ ;·,t)‖L p2

|ξ |β . We can

now takeσ(|ξ |) := |ξ |2α+β−1 which goes to zero as |ξ | → 0 since 2α+β > 1 by assumption.
It is also the case that ‖u(·, t)‖

Bβ
p2,∞

(‖u(·, t)‖2Bα

2p′2,∞
+ ‖v(·, t)‖2Bα

2p′2,∞
) ∈ L1(0, T ). We can

then apply Proposition 2.6 to conclude that D(u) = 0. By analogous reasoning to the proof
of Proposition 3.1 we conclude that the solution conserves energy. 
�
Remark 3.7 In particular, Proposition 3.6 implies that energy is conserved as soon asu, v, w ∈
L3((0, T ); Bα

3,∞(T3)) for α > 1
3 . This means that the conservation part of the ‘original’

Onsager conjecture still holds for the inviscid primitive equations. This is consistent with the
general result for conservation laws in [2].

3.4 Anisotropic sufficient conditions for energy conservation

As was mentioned before, the hydrostatic Euler equations are anisotropic in terms of regu-
larity. The vertical velocity w has a lower regularity than the horizontal velocities u and v.
However, one can observe from equation (1.4) that w is one degree less regular in the hori-
zontal variables but one degree more regular in the z-variable. We will use this observation
in the next proposition to show that there are ‘vertical’ and ‘horizontal’ Onsager exponents.
To this end, we introduce the following norm (for 0 < α < β < 1)

‖ f ‖
Bα
3,∞

(
T;Bβ

3,∞(T2)
) := ‖ f ‖L3(T3) + sup

ξ∈R3\{0}

‖δ f (· ; ξ)‖L3(T3)

|ξ |α + sup
ξh∈R2\{0}

‖δ f (· ; (ξh, 0))‖L3(T3)

|ξ |β .

(3.4)

A function being in the space Bα
3,∞(T; Bβ

3,∞(T2)) means that it is Besov regular in the
z-direction with exponent α, while it has regularity β in the horizontal directions.

Proposition 3.8 (Horizontal and vertical Onsager exponents) Let u be a type I weak solution
of the hydrostatic Euler equations such that u, v ∈ L3((0, T ); Bα

3,∞(T; Bβ
3,∞(T2))) with

β > 2
3 , α > 1

3 and 2α + β > 2. Under these assumptions it holds that D(u) = 0. In
particular, the weak solution conserves energy (3.1).

Proof First we will restrict to the case β > 1, as otherwise α > 1
2 and then the result

was proven already in Proposition 3.1. By recalling equation (1.4), we find that ∂zw ∈
L3((0, T ); Bα

3,∞(T; Bβ−1
3,∞ (T2))). This then implies thatw ∈ L3((0, T ); Bα+1

3,∞ (T; Bβ−1
3,∞ (T2)))

(by using the Lebesgue differentation theorem). So w has Bα+1
3,∞ regularity in the z-direction

and Bβ−1
3,∞ regularity in the horizontal directions. We take any ξ = (ξh, ξz) ∈ R

3 with the

horizontal increment ξh ∈ R
2 and the vertical increment ξz ∈ R. From this we obtain that

δu(ξ ; x, t) = δu(ξz; x + ξh, t) + δu(ξh; x, t).
We introduce the notation δuh := δu(ξh; x, t) and δuz := δu(ξz; x + ξh, t). We only look at
the ‘vertical’ part of the defect term

∫

T3
|δw(ξ ; x, t)|

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx,

since the other part
∫

T3

(
|δu(ξ ; x, t)| + |δv(ξ ; x, t)|

)(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx

123



219 Page 18 of 40 D. W. Boutros et al.

automatically satisfies the conditions of Proposition 2.6 if α, β > 1
3 . For the ‘vertical’ part

of the defect term, we can derive the estimates
∫

T3
|δw(ξ ; x, t)|

(
|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2

)
dx

≤ 2
∫

T3
(|δwh | + |δwz |)

(
|δuh |2 + |δuz |2 + |δvh |2 + |δvz |2

)
dx

≤ C
(|ξ |β+1 + |ξ |3 + |ξ |2α+β−1 + |ξ |2α+1) ‖w‖

Bα+1
3,∞

(
T;Bβ−1

3,∞ (T2)
)

·
(

‖u‖2
Bα
3,∞

(
T;Bβ

3,∞(T2)
) + ‖v‖2

Bα
3,∞(T;Bβ

3,∞(T2))

)
.

By assumption, we have that β +1 > 1, 2α+β −1 > 1 and 2α+1 > 1. Moreover, we know
that C(t) = ‖w‖

Bα+1
3,∞ (T;Bβ−1

3,∞ (T2))
(‖u‖2

Bα
3,∞(T;Bβ

3,∞(T2))
+ ‖v‖2

Bα
3,∞(T;Bβ

3,∞(T2))
) ∈ L1(0, T ).

Therefore by Proposition 2.6 it follows that D(u) = 0. By a similar argument as in the proof
of Proposition 3.1 we conclude that the solution conserves energy. 
�

Notice that this proposition is not implied by Proposition 3.1 in the range 1
3 < α < 1

2 .
As was mentioned before, if α > 1

2 then this proposition is merely a weaker result than
Proposition 3.1. We can conclude from Proposition 3.8 that the Onsager exponents seem to
be ‘direction-dependent’, as can be expected from the fact that the regularity of the different
components of the velocity field in the hydrostatic Euler equations is anisotropic. In other
words, we can weaken the regularity requirement in the vertical direction at the expense of
a stronger regularity requirement in the horizontal directions.

4 Negative Besov regularity for the vertical velocity

In the previous sections, in particular in Definition 2.1 of type I weak solutions, we assumed
that w ∈ L2((0, T ); L2(T3)). It is difficult to find an equivalent regularity condition to this
assumption solely in terms of u and v. Implicity, w ∈ L2((0, T ); L2(T3)) implies some
regularity of the horizontal divergence of (u, v).

In order toweaken the assumptions onw, in this sectionwe introduce two different notions
of ‘veryweak’ solutions of the hydrostatic Euler equations. In this context, ‘veryweak’means
that eitherw(·, t) ∈ B−s

2,∞(T3) (type III weak solution) orw(·, z, t) ∈ B−s
2,∞(T2) (type IIweak

solution) for s > 0. Now because we need to make sense of the equation and in particular
the products uw and vw, we are obliged to make stronger regularity assumptions on u and
v compared to Definition 2.1. To this end, we will use techniques from paradifferential
calculus. A brief overview of these techniques that will be used in this paper are presented
in Appendix B, for the convenience of the reader.

In this section, we will mainly focus on type III weak solutions. We will prove sufficient
criteria for energy conservation for such solutions. The results will then be stated without
proof for type II weak solutions, as they are identical in nature.

4.1 Weak solutions with lower regularity on the vertical velocity

In order to define the notion of a type III weak solution, we let w be a functional on Besov
functions.We need to prove that the products uw and vw also have negative Besov regularity,
for this we use the paraproduct estimates.

123



On energy conservation for the hydrostatic Euler equations... Page 19 of 40 219

We will find that for type III weak solutions, the Onsager exponent seems to increase. We
begin by introducing the notion of a type III weak solution.

Definition 4.1 We call a pair of a velocity field u = (u, v, w) : T
3 × (0, T ) → R

3 and a
pressure p : T

3 × (0, T ) → R a type III weak solution of the hydrostatic Euler equations
with regularity parameter s (where s > 0) if it satisfies the following conditions:

• w ∈ L2((0, T ); B−s
2,∞(T3)), u, v ∈ L∞((0, T ); L2(T3)) ∩ L2((0, T ); Bσ ′

2,∞(T3)) for

some σ ′ > s and p ∈ L∞((0, T ); L1(T3)).
• For all φ1, φ2 ∈ D(T3 × (0, T ); R) the following equations hold

∫ T

0

∫

T3
u∂tφ1dx dt +

∫ T

0
〈uu,∇φ1〉dt +

∫ T

0

∫

T3
�vφ1dx dt

+
∫ T

0

∫

T3
p∂xφ1dx dt = 0, (4.1)

∫ T

0

∫

T3
v∂tφ2dx dt +

∫ T

0
〈vu, ∇φ2〉dt −

∫ T

0

∫

T3
�uφ2dx dt

+
∫ T

0

∫

T3
p∂yφ2dx dt = 0. (4.2)

Note that here 〈·, ·〉 denotes the spatial duality bracket between D′(T3) and D(T3).
• For all φ3 ∈ D(T3 × (0, T ); R) it holds that

∫ T

0

∫

T3
p∂zφ3dxdt = 0. (4.3)

• The velocity field u is divergence-free, i.e. for all φ4 ∈ D(T3 × (0, T ); R) we have that
∫ T

0
〈u,∇φ4〉dt = 0. (4.4)

• It must hold that for almost every t ∈ (0, T ) that

w(·, 0, t) = w(·, 1, t) = 0 in Bs−1
2,∞(T2). (4.5)

Remark 4.2 We first have to show that this definition makes sense, in particular we must be
able to show that the products uw(·, t) and vw(·, t) (which appear in equations (4.1) and
(4.2)) are distributions, i.e. are elements of the space D′(T3). To this end, we use Bony’s
paradifferential calculus. By using the regularities of u, v and w as stated in Definition 4.1
and applying Lemma B.3 we get that uw, vw ∈ L1((0, T ); B−s

1,∞(T3)) and hence they are
distributions.

In addition, we obtain that the duality brackets betweenD′(T3) andD(T3) are equivalent
to the duality brackets between B−s

1,∞(T3) and Bs∞,1(T
3), i.e.

〈uu,∇φ1〉 = 〈uu,∇φ1〉B−s
1,∞×Bs∞,1

, 〈vu, ∇φ1〉 = 〈vu, ∇φ1〉B−s
1,∞×Bs∞,1

.

In particular, we conclude that equations (1.1) and (1.2) not only hold in the sense of distri-
butions, but in fact they hold in the space W−1,1((0, T ); B−s−1

1,∞ (T3)).

The no-normal flow boundary condition in equation (4.5) will be made sense of by using a
trace theorem. In order to do so, we recall the following lemma from [69, Chapter 3, Lemma
1.1] or [64, Lemma 1.31].

123



219 Page 20 of 40 D. W. Boutros et al.

Lemma 4.3 Let X be a Banach space. Assume that f , g ∈ L1((a, b); X). Then the following
two statements are equivalent:

1. The function f is equal almost everywhere to the primitive function of g, i.e.

f (z) = ξ +
∫ z

0
g(z′)dz′, ξ ∈ X , for almost all z ∈ [a, b].

2. For all test functions φ5 ∈ D(a, b) it holds that
∫ b

a
f (z′)φ′

5(z
′)dz′ = −

∫ b

a
g(z′)φ5(z

′)dz′.

In particular, both statements imply that f is almost everywhere equal to a continuous
function from [a, b] to X.

By using this lemma we show that the boundary condition in (4.5) is well-defined.

Remark 4.4 Firstly, by using Lemma 4.3 with f = w, g = −∇H · uH , X = Bs−1
2,∞(T2),

[a, b] = [0, 1] and keeping t fixed, we get that w(·, z, t) is continuous as a function of the
z-variable (since Bs−1

2,∞(T2) is a Banach space). This subsequently means that the no-normal
boundary condition (1.5) for the vertical velocity is obeyed as an equation in the space
Bs−1
2,∞(T2). In particular, we are able to conclude from this that we can extend a solution on

the torus to a solution in the channel.

Remark 4.5 Toestablish an equation of local energybalance,wewill need tomake sense of the
products u2w, v2w and pw as distributions. To this endwe have to require in addition thatw ∈
L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bσ ′
3,∞(T3)) for σ ′ > s. Again by Bony’s parad-

ifferential calculus (specificallyLemmaB.3)wefind that u2, v2 ∈ L3/2((0, T ); Bσ ′
3/2,∞(T3)).

Another application of Lemma B.3 then yields that u2w, v2w ∈ L1((0, T ); B−s
1,∞(T3)), thus

they belong to D′(T3 × (0, T ); R). As a result, the action on D(T3 × (0, T )) makes sense
and the equation of local energy balance will hold in the sense of distributions.

Regarding the pressure, we observe from equations (1.3) and (2.5) by using ellip-
tic regularity results that p ∈ L3/2((0, T );W σ ′−ε,3/2(T3)) (because u2, v2, uv ∈
L3/2((0, T ); Bσ ′

3/2,∞(T3)), as was mentioned before), where ε > 0 can be chosen sufficiently

small such that σ ′ − ε > s. Therefore the pressure is more regular than L∞((0, T ); L1(T3)),
as was assumed in the definition of a type III weak solution. Then by proceeding as before we
can show that pu ∈ L1((0, T ); B−s

1,∞(T3)), which will be needed to establish the equation
of local energy balance.

Remark 4.6 Note that in the case s > 1
2 the regularity requirement w(·, t) ∈ B−s

2,∞(T3)

already follows from the regularity requirement on u and v. Indeed from u(·, t), v(·, t) ∈
Bσ ′
2,∞(T3) (for someσ ′ > s) and equation (4.4)wededuce that ∂zw(·, t) ∈ L2(T; Bσ ′−1

2,∞ (T2)).

Then we find thatw(·, t) ∈ L2(T; Bσ ′−1
2,∞ (T2)). This can be seen by applying Lemma 4.3 and

using the no-normal flow boundary condition imposed in Definition 4.1.
Then we observe that w(·, t) ∈ Bσ ′−1

2,∞ (T3) ⊂ B−s
2,∞(T3). The inclusion holds because

σ ′ − 1 > s − 1 > − 1
2 > −s. Equivalently, if u is a type III weak solution with regularity

parameter s ∈ ( 12 , 1), then u is in fact a type III weak solution with smaller regularity
parameter 1 − s.

Naturally, if s ≥ 1 then ∂zw(·, t) will lie in L2(T3). Then by applying Lemma 4.3 we find
that w(·, t) ∈ L2(T3), which means that we are considering a type I weak solution.

123



On energy conservation for the hydrostatic Euler equations... Page 21 of 40 219

Now we show that we can expand the space of test functions, as it was done in Lemma 2.3.

Lemma 4.7 Letu be a type III weak solution of the hydrostatic Euler equationswith regularity
parameter s, such that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bσ ′
3,∞(T3)). Then

the weak formulation in equations (4.1) and (4.2) still holds for test functions in φ1, φ2 ∈
W 1,1

0 ((0, T ); L2(T3)) ∩ L3((0, T ); H3+s(T3)).

Proof The proof works the same way as the proof for Lemma 2.3. We take an arbitrary
ϕ ∈ W 1,1

0 ((0, T ); L2(T3)) ∩ L3((0, T ); H3+s(T3)). Then we know that there exists a

sequence {ϕn}∞n=1 ⊂ D(T3 × (0, T )) such that ϕn → ϕ in W 1,1
0 ((0, T ); L2(T3)) ∩

L3((0, T ); H3+s(T3)) as n → ∞.
The only convergence that we need to show that differs from the proof of Lemma 2.3

is for the integral
∫ T
0 〈uu,∇ϕn〉B−s

1,∞×Bs∞,1
dt (and the same integral for v). We observe that

∇ϕn
n→∞−−−→ ∇ϕ in L3((0, T ); H2+s(T3)) ⊂ L3((0, T ); Bs∞,1(T

3)). This means that we can
prove the convergence

∣∣∣∣

∫ T

0
〈uu,∇ϕn〉B−s

1,∞×Bs∞,1
dt −

∫ T

0
〈uu, ∇ϕ〉B−s

1,∞×Bs∞,1
dt

∣∣∣∣

=
∣∣∣∣

∫ T

0
〈uu,∇ϕ − ∇ϕn〉B−s

1,∞×Bs∞,1
dt

∣∣∣∣

≤
∫ T

0

(‖uu‖B−s
1,∞

‖∇ϕ − ∇ϕn‖Bs∞,1

)
dt ≤ ‖uu‖

L3/2
t

((
B−s
1,∞

)

x

)‖ϕ − ϕn‖L3
t

((
Bs+1

∞,1

)

x

)

n→∞−−−→ 0.

Here we have used that uu ∈ L3/2((0, T ); B−s
1,∞(T3)) which follows from the regularity

assumptions on u, v, w and Lemma B.3. Therefore the final inequality follows, which con-
cludes the proof. 
�

4.2 The equation of local energy balance

Now that we have introduced the notion of a type III weak solution and extended the space
of test functions, we can establish an equation of local energy balance.

Theorem 4.8 Let u be a type III weak solution of the hydrostatic Euler equations with regu-
larity parameter s, such that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bσ ′
3,∞(T3)).

Then for all ψ ∈ D(T3 × (0, T )) the following equation of local energy balance is satisfied
∫ T

0

[ ∫

T3

(
u2∂tψ + v2∂tψ

)
dx +

〈
(u2 + v2 + 2p)u,∇ψ

〉

B−s
1,∞×Bs∞,1

− 1

2

〈
D(u), ψ

〉

B−s
1,∞×Bs∞,1

]
dt = 0. (4.6)

Here D(u) denotes the element in W−1,1((0, T ); B−s−1
1,∞ (T3)) given by

D(u) := lim
ε→0

∫

R3
dξ

[
∇ϕε(ξ) · δu(ξ ; x, t) (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2)

]
.

The limit D(u) is independent of the choice of mollifier.

123



219 Page 22 of 40 D. W. Boutros et al.

Remark 4.9 Notice that in particular, since L∞((0, T ); Bs+1(T3)) ⊂ D(T3 × (0, T ); R),
equation (4.6) holds in the sense of distributions (the duality brackets in this equation are
equivalent to the distributional action betweenD′(T3×(0, T )) andD(T3×(0, T ))). However,
we show in fact that the equation holds in the space W−1,1((0, T ); B−s−1

1,∞ (T3)).

Proof Throughout the proof we will write duality brackets with explicit subscripts to make
clear in which spaces the elements lie as to make the proof more understandable. Any duality
bracket in this proof is equivalent to a distributional duality bracket, as ψ is a test function.

We first mollify equations (1.1) and (1.2) in space with ϕε and obtain

0 = ∂t u
ε + ∇ · (uu)ε − �vε + ∂x p

ε, (4.7)

0 = ∂tv
ε + ∇ · (vu)ε + �uε + ∂y p

ε . (4.8)

It holds that uε, vε ∈ L∞((0, T );C∞(T3)). We observe that (uu)ε, (vu)ε ∈ L1((0, T );
C∞(T3)) and pε ∈ L∞((0, T );C∞(T3)), from this we can conclude that ∂t uε, ∂tv

ε ∈
L1((0, T );C∞(T3)), which implies that uε, vε ∈ W 1,1((0, T );C∞(T3)) ∩ L∞((0, T );
C∞(T3)).

As a result, we are able to apply Lemma 4.7 and use ψuε and ψvε as test functions in the
weak formulation in Definition 4.1 (for ψ ∈ D(T3 × (0, T ); R)). Adding the equations for
u and v together gives that

∫ T

0

[ ∫

T3

(
u∂t (u

εψ) + v∂t (v
εψ) + �vuεψ − �uvεψ + p∂x (u

εψ) + p∂y(v
εψ)

)
dx

+ 〈uu,∇(uεψ)〉B−s
1,∞×Bs∞,1

+ 〈vu, ∇(vεψ)〉B−s
1,∞×Bs∞,1

]
dt = 0.

By multiplying Eqs. (4.7) and (4.8) by uψ respectively vψ and subtracting them from the
previous equation, we get that

∫ T

0

[ ∫

T3

(
u∂t (u

εψ) − uψ∂t u
ε + v∂t (v

εψ) − vψ∂tv
ε + p∂x (u

εψ) − uψ∂x p
ε

+ p∂y(v
εψ) − vψ∂y p

ε − ψu∇ · (uu)ε − ψv∇ · (vu)ε
)
dx + 〈uu, ∇(uεψ)〉B−s

1,∞×Bs∞,1

+ 〈vu, ∇(vεψ)〉B−s
1,∞×Bs∞,1

]
dt

=
∫ T

0

[ ∫

T3

(
u∂t (u

εψ) − uψ∂t u
ε + v∂t (v

εψ) − vψ∂tv
ε + p∂x (u

εψ) − uψ∂x p
ε

+ p∂y(v
εψ) − vψ∂y p

ε

)
dx +

〈
uu · ∇uε − u∇ · (uu)ε + vu · ∇vε − v∇ · (vu)ε, ψ

〉

B−s
1,∞×Bs∞,1

+
〈
(uuε + vvε)u, ∇ψ

〉

B−s
1,∞×Bs∞,1

]
dt = 0.

We will mainly discuss the advective terms here, as the derivation of the equation of energy
for the other terms proceeds in the same way as Theorem 2.5. To handle them, we introduce
a defect term

Dε(u) :=
∫

R3
dξ

[
∇ϕε(ξ) · δu(ξ ; x, t) (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2)

]

= −∇ ·
[
(u2 + v2)u

]ε

+ u · ∇(u2 + v2)ε + 2u∇ · (uu)ε + 2v∇ · (vu)ε

123



On energy conservation for the hydrostatic Euler equations... Page 23 of 40 219

− 2uu · ∇uε − 2vu · ∇vε.

Notice that the equation holds in L1((0, T ); B−s
1,∞(T3)), in particular in D′(T3 × (0, T ); R).

Now we can write the advective terms as follows
∫ T

0

〈
uu · ∇uε − u∇ · (uu)ε + vu · ∇vε − v∇ · (vu)ε, ψ

〉

B−s
1,∞×Bs∞,1

dt

=
∫ T

0

〈
− 1

2
Dε(u) − ∇ ·

[
(u2 + v2)u

]ε

+ u · ∇(u2 + v2)ε, ψ

〉

B−s
1,∞×Bs∞,1

dt

=
∫ T

0

〈
− 1

2
Dε(u), ψ

〉

B−s
1,∞×Bs∞,1

+
〈[

(u2 + v2)u
]ε

− u(u2 + v2)ε,∇ψ

〉

B−s
1,∞×Bs∞,1

dt .

We observe that

[
(u2+v2)u

]ε

−u(u2+v2)ε → 0 in B−s
1,∞(T3) by the paraproduct estimates

(see Remark 4.5 andAppendix B for more details). In particular, we have that (again for some
σ ′ > s, by the assumptions on u, v, w and Lemma B.3)

∥∥∥∥

[
(u2 + v2)u

]ε

− u(u2 + v2)ε
∥∥∥∥
B−s
1,∞

≤
∥∥∥∥

[
(u2 + v2)u

]ε

− u(u2 + v2)

∥∥∥∥
B−s
1,∞

+
∥∥∥∥(u2 + v2)u − u(u2 + v2)ε

∥∥∥∥
B−s
1,∞

≤
∥∥∥∥

[
(u2 + v2)u

]ε

− u(u2 + v2)

∥∥∥∥
B−s
1,∞

+ ‖(u2 + v2) − (u2 + v2)ε‖
Bσ ′
3/2,∞

‖u‖B−s
3,∞

ε→0−−−→ 0.

Finally, we need to prove that the defect term makes sense. First we claim that
〈
Dε(u), ψ

〉

B−s
1,∞×Bs∞,1

=
∫

R3

[
∇ϕε(ξ)〈δu (|δu|2 + |δv|2) , ψ〉B−s

1,∞×Bs∞,1

]
dξ. (4.9)

This can be shown as follows. For the mollification (in space) uη of u, we have by Fubini’s
theorem

〈 ∫

R3
dξ ∇ϕε(ξ) · δuη

(|δu|2 + |δv|2) , ψ

〉

B−s
1,∞×Bs∞,1

=
∫

R3
dξ

[
∇ϕε(ξ)〈δuη

(|δu|2 + |δv|2) , ψ〉B−s
1,∞×Bs∞,1

]
.

Now we observe that

−
∫

R3
dξ

[
∇ϕε(ξ)〈δuη

(|δu|2 + |δv|2) , ψ〉B−s
1,∞×Bs∞,1

]

+
∫

R3
dξ

[
∇ϕε(ξ)〈δu (|δu|2 + |δv|2) , ψ〉B−s

1,∞×Bs∞,1

]

=
∫

R3
dξ

[
∇ϕε(ξ)〈δ(u − uη)

(|δu|2 + |δv|2) , ψ〉B−s
1,∞×Bs∞,1

]

≤ C‖u − uη‖B−s
3,∞

(
‖u‖2

Bσ ′
3,∞

+ ‖v‖2
Bσ ′
3,∞

)
‖ψ‖Bs∞,1

dt
η→0−−−→ 0.

123



219 Page 24 of 40 D. W. Boutros et al.

Moreover, it is straightforward to verify that
〈
Dε(u) −

∫

R3
dξ ∇ϕε(ξ) · δuη

(|δu|2 + |δv|2) , ψ

〉

B−s
1,∞×Bs∞,1

≤ C‖u − uη‖B−s
3,∞

(
‖u‖2

Bσ ′
3,∞

+ ‖v‖2
Bσ ′
3,∞

)
‖ψ‖Bs∞,1

η→0−−−→ 0.

Therefore claim (4.9) is true. Hence we have

〈Dε(u), ψ〉B−s
1,∞×Bs∞,1

=
∫

R3
dξ

[
∇ϕε(ξ)〈δu (|δu|2 + |δv|2) , ψ〉B−s

1,∞×Bs∞,1

]

≤ C‖ψ‖Bs∞,1
‖u‖B−s

3,∞

(
‖u‖2

Bσ ′
3,∞

+ ‖v‖2
Bσ ′
3,∞

)
< ∞.

In particular, Dε(u) makes sense as an element in L1((0, T ); B−s
1,∞(T3)). The convergence

as ε → 0 follows by looking at the equation we established (for fixed ε > 0)

Dε(u) = −2∂t (uu
ε + vvε) − 2∇ · (puε + pεu) − 2∇ · (

(uuε + vvε)u
)

−∇ ·
(( (

u2 + v2
)
u
)ε − (

u2 + v2
)εu

)
.

As ε → 0 the right-hand side converges in W−1,1((0, T ); B−1−s
1,∞ (T3)) which means that

Dε(u) converges in the same space. Since the limit of the right-hand side is independent
of the choice of mollifier, hence the limit D(u) is independent of the choice of mollifier.
Therefore we conclude that the derived equation of local energy balance holds in the sense
of distributions. 
�
Remark 4.10 In order to make sense of the hydrostatic Euler equations for a type III weak
solution, the assumptions w ∈ L2((0, T ); B−s

2,∞(T3)) and u, v ∈ L2((0, T ); Bσ ′
2,∞(T3)) for

some σ ′ > s are sufficient. To derive an equation of local energy balance however, the
assumption that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bσ ′
3,∞(T3)) is necessary.

Alternatively we could requirew ∈ L2((0, T ); B−s
2,∞(T3)) and u, v ∈ L4((0, T ); Bσ ′

4,∞(T3))

in order to establish an equation of local energy balance. Propositions 4.11, 4.12, 4.15 and
Theorem 4.14 then hold with straightforward modifications.

4.3 Proof of conservation of energy

Now we prove an estimate on the defect term like in Proposition 2.6.

Proposition 4.11 Letu be a type III weak solution of the hydrostatic Euler equations with reg-
ularity parameter s such that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bσ ′
3,∞(T3)).

Assume that for all ψ ∈ D(T3 × (0, T )) it holds that
∣∣∣∣〈δu

(|δu|2 + |δv|2) , ψ〉B−s
1,∞×Bs∞,1

∣∣∣∣ ≤ C(t)|ξ |σ(|ξ |)‖ψ‖Bs∞,1
, (4.10)

where C ∈ L1(0, T ) and σ ∈ L∞
loc(R) with the property that σ(|ξ |) → 0 as |ξ | → 0. Then

D(u) = 0, which implies that energy is conserved.

Proof Under this assumption, we estimate the defect term as follows (by using identity (4.9))

123



On energy conservation for the hydrostatic Euler equations... Page 25 of 40 219

∣∣∣∣

∫ T

0

〈
− 1

2
Dε(u), ψ

〉

B−s
1,∞×Bs∞,1

dt

∣∣∣∣

=
∣∣∣∣

∫ T

0

∫

R3
∇ϕε(ξ)〈δu

(
|δu|2 + |δv|2

)
, ψ〉B−s

1,∞×Bs∞,1
dξdt

∣∣∣∣

≤
∫ T

0

∫

R3
|∇ϕε(ξ)|

∣∣∣∣〈δu
(
|δu|2 + |δv|2

)
, ψ〉B−s

1,∞×Bs∞,1

∣∣∣∣dξdt

≤
∫ T

0
C(t)‖ψ‖Bs∞,1

dt
∫

R3
|∇ϕε(ξ)||ξ |σ(|ξ |)dξ

≤
∫ T

0
C(t)‖ψ‖Bs∞,1

dt
∫

R3
|∇zϕ(z)||z|σ(ε|z|)dz,

where in the last line we made a change of variable ξ = εz (like was done in Proposition
2.6). Using the dominated convergence theorem and the assumption on σ we deduce that the
right-hand side converges to 0 as ε → 0. Therefore we conclude that D(u) = 0. 
�
Next we give sufficient conditions that ensure that the defect term is zero, in particular such
that condition (4.10) is satisfied.

Proposition 4.12 Assume that u is a type III weak solution of the hydrostatic Euler equa-
tions with regularity parameter s, and that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈
L3((0, T ); Bα

3,∞(T3)) with α > 1
2 + s

2 . Then D(u) = 0 in the sense of distributions. This
then implies that the weak solution conserves energy.

In order to prove Proposition 4.12 we need the following lemma.

Lemma 4.13 Let 1 ≤ p ≤ ∞, s, ε > 0 with s + ε < 1. Then for all u ∈ Bs+ε
p,∞ and

ξ ∈ R
3\{0} we have

‖δu‖Bs
p,∞ � |ξ |ε‖u‖Bs+ε

p,∞ .

Proof First we observe that

‖δu‖L p = |ξ |ε 1

|ξ |ε ‖δu‖L p ≤ |ξ |ε sup
h∈R3\{0}

1

|h|ε ‖δhu‖L p ≤ |ξ |ε‖u‖Bε
p,∞ � |ξ |ε‖u‖Bs+ε

p,∞ .

Hence it remains to estimate the Besov seminorm. Due to the assumption s + ε < 1, this
seminorm only contains a first-order increment. We obtain

|δu|Bs
p,∞ = sup

h∈R3\{0}
1

|h|s ‖δξ δhu‖L p = |ξ |ε sup
h∈R3\{0}

1

|h|s |ξ |ε ‖δξ δhu‖L p

≤ |ξ |ε sup
h∈R3\{0},|h|≤|ξ |

1

|h|s |ξ |ε ‖δξ δhu‖L p + |ξ |ε sup
h∈R3\{0},|h|≥|ξ |

1

|h|s |ξ |ε ‖δξ δhu‖L p

≤ |ξ |ε sup
h∈R3\{0},|h|≤|ξ |

1

|h|s+ε
‖δξ δhu‖L p + |ξ |ε sup

h∈R3\{0},|h|≥|ξ |
1

|ξ |s+ε
‖δξ δhu‖L p

≤ 4|ξ |ε‖u‖Bs+ε
p,∞ .


�

123



219 Page 26 of 40 D. W. Boutros et al.

Proof of Proposition 4.12 By Proposition 4.11 we have to prove bound (4.10). By assumption
we have 2α−1−s > 0, thuswe can fix θ > 0 sufficiently small such that 2α−1−s−3θ > 0.
Using Lemma B.3 we obtain

∣∣∣∣〈δu
(|δu|2 + |δv|2) , ψ〉B−s

1,∞×Bs∞,1

∣∣∣∣

≤ ‖ψ‖Bs∞,1
‖δu (|δu|2 + |δv|2)‖B−s

1,∞

≤ ‖ψ‖Bs∞,1
‖δu‖B−s

3,∞

∥∥|δu|2 + |δv|2∥∥Bs+θ
3/2,∞

≤ ‖ψ‖Bs∞,1
‖δu‖B−s

3,∞

(
‖δu‖Bθ

3,∞
‖δu‖Bs+2θ

3,∞
+ ‖δv‖Bθ

3,∞
‖δv‖Bs+2θ

3,∞

)
.

Now in order to apply Lemma 4.13 we observe that we may assume that s ≤ 1
2 , see

Remark 4.6, and that α < 1. So we deduce from Lemma 4.13 that for all 0 �= ξ ∈ R
3

‖δu‖Bθ
3,∞

≤ |ξ |α−θ‖u‖Bα
3,∞ , ‖δu‖Bs+2θ

3,∞
≤ |ξ |α−2θ−s‖u‖Bα

3,∞ ,

and similarly for δv. We therefore conclude that
∣∣∣∣〈δu

(|δu|2 + |δv|2) , ψ〉B−s
1,∞×Bs∞,1

∣∣∣∣ ≤ |ξ |2α−s−3θ‖ψ‖Bs∞,1
‖δu‖B−s

3,∞

(
‖u‖2Bα

3,∞
+ ‖v‖2Bα

3,∞

)
.

By assumption it holds that ‖δu‖B−s
3,∞

(‖u‖2Bα
3,∞

+ ‖v‖2Bα
3,∞

) ∈ L1(0, T ), moreover we can

take σ(|ξ |) := |ξ |2α−1−s−3θ , which satisfies the assumptions of Proposition 4.11. Therefore
by Proposition 4.11 we conclude that D(u) = 0. 
�

Finally, we are able to prove conservation of energy under sufficient regularity analogously
to Proposition 3.1.

Theorem 4.14 Let u be a type III weak solution of the hydrostatic Euler equations with reg-
ularity parameter s, and that w ∈ L3((0, T ); B−s

3,∞(T3)) and u, v ∈ L3((0, T ); Bα
3,∞(T3))

with α > 1
2 + s

2 . Then there is conservation of energy. In other words, it holds that

‖u(t1, ·)‖2L2 + ‖v(t1, ·)‖2L2 = ‖u(t2, ·)‖2L2 + ‖v(t2, ·)‖2L2 , (4.11)

for almost all t1, t2 ∈ (0, T ).

Proof By Proposition 4.12 we know that D(u) = 0 under the regularity assumptions. There-
fore we have the following equation of local energy balance

∫ T

0

[ ∫

T3

(
u2∂tψ + v2∂tψ

)
dx +

〈
(u2 + v2 + 2p)u,∇ψ

〉

B−s
1,∞×Bs∞,1

]
dt = 0.

We conclude proceeding exactly as in the proof of Proposition 3.1. 
�
We observe that the there is an upper bound on the Onsager exponent, namely 2

3 . We will
show this in the following proposition.

Proposition 4.15 Let u be a type III weak solution of the hydrostatic Euler equations with
(arbitrary) regularity parameter s such that u, v ∈ L3((0, T ); Bα

3,∞(T3)) with α > 2
3 . Then

conservation of energy (as in equation (4.11)) holds.

123



On energy conservation for the hydrostatic Euler equations... Page 27 of 40 219

Proof We prove that the sufficient condition for energy conservation of Proposition 4.11
is satisfied. By equation (1.4) and Lemma 4.3 we know that w ∈ L3((0, T ); Bα−1

3,∞ (T3)).

As α − 1 > − 1
3 , this implies that w ∈ L3((0, T ); B−1/3

3,∞ (T3)) and u is a type III weak

solution with regularity parameter s = 1/3. Now because 1
2 + s

2 = 2
3 < α, the claim follows

immediately from Theorem 4.14. 
�
One can infer from this result that the notion of a type III weak solution will only have

implications on the Onsager conjecture in the range 0 < s < 1
3 , because if s ≥ 1

3 Theorem
4.14 follows from Proposition 4.15.

4.4 Type II weak solutions

In this section, we introduce type II weak solutions and state sufficient conditions for such
solutions to conserve energy.

Definition 4.16 A type II weak solution consists of a velocity field u = (u, v, w) : T
3 ×

(0, T ) → R
3 as well as a pressure p : T

3 × (0, T ) → R together with a regularity parameter
0 < s < 1

2 , if the following conditions hold:

• w ∈ L2((0, T ); L2(T; B−s
2,∞(T2))), u, v ∈ L∞((0, T ); L2(T3)) ∩ L2((0, T ); L2(T;

Bσ ′
2,∞(T2))) for some σ ′ > s and where L2(T) refers to the regularity in the vertical

direction.
• For all φ1, φ2 ∈ D(T3 × (0, T ); R) it holds that

∫ T

0

∫

T3
u∂tφ1dx dt +

∫ T

0

∫

T

〈uu, ∇φ1〉dz dt +
∫ T

0

∫

T3
�vφ1dx dt

+
∫ T

0

∫

T3
p∂xφ1dx dt = 0, (4.12)

∫ T

0

∫

T3
v∂tφ2dx dt +

∫ T

0

∫

T

〈vu, ∇φ2〉dz dt −
∫ T

0

∫

T3
�uφ2dx dt

+
∫ T

0

∫

T3
p∂yφ2dx dt = 0. (4.13)

The brackets 〈·, ·〉 refer to the distributional duality between D′(T2) and D(T2).
• For all φ3 ∈ D(T3 × (0, T )) it holds that

∫ T

0

∫

T3
p∂zφ3dxdt = 0. (4.14)

• The velocity field u is divergence-free, in particular for all φ4 ∈ D(T3 × (0, T )) it holds
that

∫ T

0

∫

T

〈u, ∇φ4〉dzdt = 0. (4.15)

• Finally, for almost all t ∈ (0, T ) it holds that

w(·, 0, t) = w(·, 1, t) = 0 in B−s
2,∞(T2). (4.16)

In a similar fashion as it was argued in Remark 4.2, one can prove that uw(·, z, t), vw(·, z, t)
∈ B−s

1,∞(T2). We skip the details here. One should observe that the vertical velocity is

123



219 Page 28 of 40 D. W. Boutros et al.

more regular for type II weak solutions compared to type III weak solutions, whereas the
opposite holds for the horizontal velocities. Hence, neither notion is stronger than the other
if 0 < s < 1

2 . In the case s > 1
2 , type III weak solutions are also type II weak solutions.

Finally, we observe that boundary condition (4.16) can be made sense of in a similar fashion
as it was argued in Remark 4.4.

Next we consider sufficient conditions for energy conservation. We only state the results
and omit the proofs, as they are very similar in nature as for the type III weak solutions, see
Lemma 4.7, Theorem 4.8, Propositions 4.11 and 4.12, Theorem 4.14 and Proposition 4.15.
We first state the equation of local energy balance.

Theorem 4.17 Let u be a type II weak solution of the hydrostatic Euler equations
with regularity parameter s, such that w ∈ L3((0, T ); L3(T; B−s

3,∞(T2))) and u, v ∈
L3((0, T ); L3(T; Bσ ′

3,∞(T3))) for some σ ′ > s. Then for all ψ ∈ D(T3 × (0, T )) the follow-
ing equation of local energy balance is satisfied

∫ T

0

[ ∫

T3

(
u2∂tψ + v2∂tψ

)
dx +

∫

T

〈
(u2 + v2 + 2p)u,∇ψ

〉

B−s
1,∞×Bs∞,1

dz

− 1

2

∫

T

〈
D(u), ψ

〉

B−s
1,∞×Bs∞,1

dz

]
dt = 0.

The defect term D(u) is given by

D(u) := lim
ε→0

∫

R3
dξ

[
∇ϕε(ξ) · δu(ξ ; x, t) (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2)

]
,

and is independent of the choice of mollifier.

Remark 4.18 Note that the duality brackets are now on the two-dimensional torus T
2, while

for the type III weak solutions they were three-dimensional.

The usual sufficient condition for the defect term to be zero is given below.

Lemma 4.19 Let u be a type II weak solution of the hydrostatic Euler equations
with regularity parameter s, such that w ∈ L3((0, T ); L3(T; B−s

3,∞(T2))) and u, v ∈
L3((0, T ); L3(T; Bσ ′

3,∞(T3))) for some σ ′ > s. Moreover, assume that the following inequal-

ity holds for all ψ ∈ D(T3 × (0, T ))

∣∣∣∣

∫ 1

0
〈δu (|δu|2 + |δv|2) , ψ〉B−s

1,∞×Bs∞,1
dz

∣∣∣∣ ≤ C(t)|ξ |σ(|ξ |)
∫ 1

0
‖ψ‖Bs∞,1(T

2)dz,

where C ∈ L1(0, T ) and σ ∈ L∞
loc(R) has the property that σ(|ξ |) → 0 as |ξ | → 0. Then

D(u) = 0, which in turn implies conservation of energy.

Finally, we can state the sufficient condition for energy conservation for type II weak solu-
tions.

Proposition 4.20 Let u be a type II weak solution of the hydrostatic Euler equations
with regularity parameter s such that w ∈ L3((0, T ); L3(T; B−s

3,∞(T2))) and u, v ∈
L3((0, T ); Bα

3,∞(T; Bβ
3,∞(T2))) with α > 1

2 and β > 1
2 + s

2 , then the solution conserves
energy. In particular, equation (4.11) holds.

123



On energy conservation for the hydrostatic Euler equations... Page 29 of 40 219

Like in the case of type III weak solution, we have a result which is analogous to Proposition
4.15.

Proposition 4.21 Let u be a type II weak solution of the hydrostatic Euler equations with the
property that u, v ∈ L3((0, T ); Bα

3,∞(T; Bβ
3,∞(T2))) with α > 1

2 and β > 2
3 (note that this

space can be defined similarly to equation (3.4)). Then the weak solution conserves energy.

5 Sufficient conditions in terms of Sobolev spaces

In the next two sections, we give sufficient conditions for type I weak solutions (or at least
with similar regularity requirements to type I) to conserve energy solely in terms of the
horizontal velocities. In the previous sections, we started with a regularity assumption on the
vertical velocity and then derived sufficient conditions for energy conservation in terms of
the horizontal velocities (with sometimes additional assumptions on the vertical velocity).
However, these assumptions are not independent of each other by the anisotropic regularity
and the nonlocality imposed by (1.4).

The fact that the assumption that w ∈ L2(T3 × (0, T )) induces implicit assumptions on
the horizontal velocities can be seen as follows. The fact that the vertical velocity w has L2

spatial regularity means that either ∂xu and ∂yv are in L2((0, T ); L2(T3)) or their potential
lack of regularity cancels in a particular way that is unclear at this point. In order to gain a
better understanding of this issue wewant to look at the required regularity assumption on the
horizontal velocities u and v such that a regularity assumption on w is no longer necessary
for defining a weak solution or proving conservation of energy.

In this section we will prove such sufficient conditions in terms of Sobolev spaces and
Lebesgue spaces. These results complement our earlier results. As was shown in Propositions
2.6 and 3.1, condition (2.14) is sufficient for a type I weak solution withw ∈ L2(T3×(0, T ))

to conserve energy. However, if we would like to avoid any explicit assumptions on w, two
separate conditions on u and v are necessary:

1. The horizontal velocities u and v need to have sufficient Hölder regularity with a given
exponent such that the function σ in equation (2.14) is o(1). This automatically implies
that their Hölder exponent must be bigger than 1

2 .
2. The vertical velocity w needs to satisfy some integrability condition in order for the

inequality (2.14) to hold, because δu needs to be bounded in some L p space. Note that
we want to proceed here as we did for type I weak solutions.

From this reasoning we can conclude that we can either assume that u, v are in a sufficiently
regular Sobolev space such that both conditions are satisfied (by use of theSobolev embedding
theorem), or we have to make two separate assumptions.

Proposition 5.1 Let u, v ∈ L3((0, T );C0,β(T3) ∩ W 1,p(T3)) with β > 1
2 and p > 1 or

alternatively u, v ∈ L3((0, T );W 1,p(T3)) with p > 6. Moreover, assume that
∫
T

∇H ·
uHdz = 0 and that together with the associated w (via equation (1.4)) the velocity field u
satisfies the weak formulation in equations (2.1)–(2.3). Then D(u) = 0. In particular, the
weak solution conserves energy (see equation (3.1)).

Proof Wefirst observe that the second case reduces to the first case by the Sobolev embedding
theorem, therefore we will only consider the first case. We first show that the defect term is
bounded. Now since u(·, t), v(·, t) ∈ W 1,p(T3), we know that ∂xu(·, t)+∂yv(·, t) ∈ L p(T3)

123



219 Page 30 of 40 D. W. Boutros et al.

and by virtue of equation (1.4) and Lemma 4.3 it follows that w ∈ L3((0, T ); L p(T3)) with
p > 1. Using Hölder’s inequality we obtain that

‖δu(ξ ; ·, t)(|δu(ξ ; ·, t)|2 + |δv(ξ ; ·, t)|2)‖L1

≤ ‖δu(ξ ; ·, t)‖L p‖|δu(ξ ; ·, t)|2 + |δv(ξ ; ·, t)|2‖L p/(p−1)

≤ ‖δu(ξ ; ·, t)‖L p

(
‖δu(ξ ; ·, t)‖2L2p/(p−1) + ‖δv(ξ ; ·, t)‖2L2p/(p−1)

)
.

Observe that u(·, t), v(·, t) ∈ L2p/(p−1)(T3) because of the fact that u(·, t), v(·, t) ∈
C0,β(T3) with β > 1

2 .
In addition we see that

‖δu(·, t)‖L p

(
‖δu(·, t)‖2L2p/(p−1) + ‖δv(·, t)‖2L2p/(p−1)

)
∈ L1(0, T ),

because we assumed that u, v and w have L3 temporal regularity.
Finally we need to check condition (2.14). We see that

‖δu(ξ ; ·, t)(|δu(ξ ; ·, t)|2 + |δv(ξ ; ·, t)|2)‖L1

� |ξ |2β‖δu(ξ ; ·, t)‖L p ·
(∥∥∥∥

δu

|ξ |β (ξ ; ·, t)
∥∥∥∥

2

L2p/(p−1)
+

∥∥∥∥
δv

|ξ |β (ξ ; ·, t)
∥∥∥∥

2

L2p/(p−1)

)

� 4|ξ |2β‖u(·, t)‖L p‖1‖2L2p/(p−1)

(‖u(·, t)‖2C0,β + ‖v(·, t)‖2C0,β

)
.

Thus by Proposition 2.6 D(u) = 0. Now energy conservation can be proven analogously to
the proof of Proposition 3.1. 
�

6 Sufficient conditions in terms of Besov spaces

As was already mentioned in the previous section, the regularity assumption on w contains
(by means of equation (1.4)) implicit assumptions on the horizontal velocities u and v. In
this section we prove a criterion in terms of Besov spaces which is similar to the one in
Proposition 5.1.

Proposition 6.1 Assume that u, v ∈ L3((0, T ); Bs
9
4 ,∞(T3)) with s > 1 and

∫
T

∇H · uHdz =
0, such that they (together with the associated w via equation (1.4)) satisfy the weak formu-
lation given in equations (2.1)–(2.3). Then D(u) = 0, which implies that energy is conserved
(see equation (3.1)).

Proof We first recall the following property of Besov spaces [1, 49]

‖∂α f ‖
Bs−|α|
p,q

≤ ‖ f ‖Bs
p,q

, |α| < s. (6.1)

This implies that ∂zw and hence w ∈ L3((0, T ); Bs−1
9
4 ,∞(T3)) by equation (1.4) and by apply-

ing Lemma 4.3, in particular we obtain that w ∈ L3((0, T ); L9/4(T3)). Subsequently by a
Besov embedding [49] we can conclude that

Bs
9/4,∞(T3) ⊂ Bs−1/2

18/5,∞(T3), s > 1.

Therefore u, v ∈ L3((0, T ); Br
18/5,∞(T3)) with r > 1

2 . Now we estimate the defect term
given in equation (2.9) and use Proposition 2.6 to prove it is zero. By usingHölder’s inequality

123



On energy conservation for the hydrostatic Euler equations... Page 31 of 40 219

we can derive that
∫

T3
|δu(ξ ; x, t)| (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2) dx

≤ ‖δu(ξ ; ·, t)‖L9/4
(‖δu(ξ ; ·, t)‖2L18/5 + ‖δv(ξ ; ·, t)‖2L18/5

)
.

We then get

‖δu(ξ ; ·, t)‖L9/4
(‖δu(ξ ; ·, t)‖2L18/5 + ‖δv(ξ ; ·, t)‖2L18/5

)

≤ 2‖u(·, t)‖L9/4 |ξ |2r
(∥∥∥∥

δu(ξ ; ·, t)
|ξ |r

∥∥∥∥

2

L18/5
+

∥∥∥∥
δv(ξ ; ·, t)

|ξ |r
∥∥∥∥

2

L18/5

)

≤ 2‖u(·, t)‖Bs−1
9/4,∞

|ξ |2r
(
‖u(·, t)‖2Br

18/5,∞
+ ‖v(·, t)‖2Br

18/5,∞

)
.

Hence condition (2.14) is satisfied. Finally, we need to prove the regularity in time. From the
derived conclusions that u, v ∈ L3((0, T ); Br

18/5,∞(T3)) (with r > 1
2 ) and w ∈ L3((0, T );

Bs−1
9/4,∞(T3)) (and s > 1) we observe that

‖u(·, t)‖Bs−1
9/4,∞

(
‖u(·, t)‖2Br

18/5,∞
+ ‖v(·, t)‖2Br

18/5,∞

)
∈ L1(0, T ).

Thus we conclude by Proposition 2.6 that D(u) = 0. Then by a similar argument to Propo-
sition 3.1 we find that the energy is conserved. 
�
Remark 6.2 From the Besov embedding theorem [49] we can derive that

Bs
9/4,∞(T3) ⊂ Bt

4,∞(T3)

where s > 1 and t > 7
12 . It then obviously follows that the same embedding is true if t > 1

2 .
Therefore the result derived in Proposition 3.1 is stronger than the results from this section,
but for Proposition 3.1 we needed a separate regularity assumption on w.

Remark 6.3 Proposition 6.1 is in the context of type I weak solutions. It is also possible to
consider sufficient regularity conditions to both define a weak solution and prove energy
conservation in the context of type II and III weak solutions. This was discussed to some
extent in Remark 4.6 and Proposition 4.15.

7 Conclusion

In this paper,we have proven that theOnsager exponent of the hydrostatic Euler equations is at
most 12 (for type I weak solutions). This result differs from the original Onsager conjecture for
the Euler equations, where it has been established that the threshold for energy conservation
is 1

3 . As it was mentioned before, a technical explanation for this increase can be given by
looking again at the defect term

D(u)(x, t) := 1

2
lim
ε→0

∫

R3

[
∇ϕε(ξ) · δu(ξ ; x, t) (|δu(ξ ; x, t)|2 + |δv(ξ ; x, t)|2)

]
dξ.

It is clear from Proposition 2.6 that for the expression to be zero, the term σ in equation
(2.14) has to be o(1). The convergence to zero of σ as |ξ | → 0 can be very slight however.
This was illustrated by the consideration of the logarithmic Hölder space in Definition 3.2
and the proof that the defect term is zero in Proposition 3.3.

123



219 Page 32 of 40 D. W. Boutros et al.

The difference with the defect term for the Euler equations, as stated in [28], is that in the
latter case there is a cubic term |δu|3 and that the vertical velocity has the same regularity
as the horizontal velocities in that case. This means that for the Euler equations the Besov
regularity necessary for σ to be o(1) can be equally distributed among the three terms in the
product |δu|3, leading to an Onsager exponent of 1

3 .
For the hydrostatic Euler equations, due to the anistropic nature of the equations, the

required Besov regularity (for σ to be o(1) in equation (2.14)) must be distributed on the
term |δuH |2, which leads to an Onsager exponent of 1

2 .
We would also like to summarise the different results presented in this paper. Unlike

the Euler equations, the hydrostatic Euler equations seem to have a ‘family’ of Onsager
conjectures. We now give an overview of these:

1. We first consider type I weak solutions (cf. Definition 2.1), i.e. we assume that w ∈
L2((0, T ); L2(T3)). Then there are several subcases:

• Without any further assumptions, the Onsager exponent is 1
2 , cf. Proposition 3.1.

• Another sufficient condition for energy conservation is that u, v ∈ L4((0, T );
C1/2
log (T3)), cf. Proposition 3.3.

• If we assume that w is Besov regular with exponent β, then the Onsager exponent
for the horizontal velocities is 1

2 − 1
2β, cf. Proposition 3.6.• If we assume that the horizontal velocities u and v are Besov regular with exponent

α in the z-direction (where 1
3 < α < 1

2 ) and we make no additional assumption on
w, then the Onsager exponent in the horizontal directions is 2 − 2α, cf. Proposition
3.8.

2. Under the assumption that w ∈ L2((0, T ); B−s
2,∞(T3)) (for s ≤ 1

2 ), we can make sense
of the equation (with additional regularity assumptions on u and v). This is in the context
of type III weak solutions, as introduced in Definition 4.1. In the range 0 < s ≤ 1

3 , the
Onsager exponent is 1

2 + s
2 (cf. Proposition 4.12), where the regularity exponents refer

to L3-based Besov spaces. For s ≥ 1
3 , the Onsager exponent is

2
3 (cf. Proposition 4.15).

3. In Sect. 4 we also considered the type II weak solutions, for whichw ∈ L2((0, T ); L2(T;
B−s
2,∞(T2))), see Definition 4.16. For such solutions we proved that they conserve energy

if w ∈ L3((0, T ); L3(T; B−s
3,∞(T3))) and u, v ∈ L3((0, T ); Bα

3,∞(T; Bβ
3,∞(T2))) with

α > 1
2 and β > 1

2 + s
2 , cf. Proposition 4.20. Moreover, we proved that if α > 1

2 and
β > 2

3 the solution conserves energy, regardless of assumptions on w (cf. Proposition
4.21).

4. Finally, there is the case where we do not make explicit regularity assumptions on w

(neither for making sense of the equation nor for proving energy conservation) but only
on the horizontal velocities u and v. Then we considered two subcases in Sects. 5 and 6:

• In terms of Sobolev and Hölder spaces, we proved that u, v ∈ L3((0, T );C0,β(T3)∩
W 1,p(T3)) with p > 1 and β > 1

2 , or alternatively u, v ∈ L3((0, T );W 1,p(T3))

with p > 6 suffices to define a w such that (u, v, w) is a type I weak solution which
conserves energy (cf. Proposition 5.1).

• Moreover, we proved that if u, v ∈ L3((0, T ); Bs
9/4,∞(T3)) with s > 1 then we can

define w such that (u, v, w) is a type I weak solution which conserves energy (cf.
Proposition 6.1).

Note that the result of Proposition 4.15 is related to the results of Sects. 5 and 6.
Namely, that for u, v ∈ L3((0, T ); Bα

3,∞(T3)) with α > 2
3 it is implied that w ∈

123



On energy conservation for the hydrostatic Euler equations... Page 33 of 40 219

L3((0, T ); B−1/3
3,∞ (T3)) such that (u, v, w) is a type III weak solution with regularity

parameter 1
3 which conserves energy, under the condition that the associated w is peri-

odic and satisfies the symmetry conditions.

The reason we have considered these different cases is that the vertical velocity w no
longer has its own dynamical equation but is constrained by the incompressibility instead,
therefore the equations are anisotropic and nonlocal. In order to establish an equation of local
energy balance, a regularity assumption on w is necessary. One needs the vertical velocity to
have sufficient regularity to make sense of a weak solution but on the other hand regularity
assumptions onw impose implicit regularity assumptions on u and v. This leads us to consider
different types of weak solutions to the hydrostatic Euler equations.

Remark 7.1 In [27, Theorem 2] a theorem for the incompressible Navier-Stokes equations
was established that describes the absence of anomalous dissipation in the vanishing viscosity
limit under sufficient regularity assumptions. If one would assume sufficient regularity for
the horizontal velocities uniform in viscosity to rule out anomalous dissipation in the viscous
primitive equations, we expect the limiting solution of the inviscid primitive equations to be
a type III weak solution rather than a type I weak solution.

This is due to the lack of an a priori bound of the L2(T3) norm of the vertical velocity w.
In particular, if one assumes solutions of the viscous primitive equations to satisfy uniform
regularity bounds in viscosity, say u, v ∈ L3((0, T ); Bα

3,∞(T3)) for 1
2 < α < 1, then

by the incompressibility condition the vertical velocity w only has uniform regularity in
L3((0, T ); Bα−1

3,∞ (T3)). Therefore we expect that the type III notion of weak solution has
more implications on turbulence than the type I notion.

As was noted in the introduction, for classical solutions of the inviscid primitive equations
the physical boundary-value problem in the channelwith no-normal flowboundary conditions
is equivalent to the problem posed on the unit torus with the symmetry assumptions

w odd in z, u, v and p even in z. (7.1)

In the case of the viscous primitive equations, for strong solutions the problem posed on the
torus with the aforementioned symmetry assumptions is equivalent to the physical boundary-
value problem in the channel with no-permeability and stress-free boundary conditions (see
[16])

w(x, y, z, t)|z=0,L= ∂zu(x, y, z, t)|z=0,L= ∂zv(x, y, z, t)|z=0,L= 0. (7.2)

For this reason, even if one wishes to formulate sufficient conditions for the absence of
anomalous dissipation in the viscous primitive equations, it is possible to consider the problem
on the toruswith symmetry conditions insteadof in the channelwith physical boundaries (as in
this case the boundary is ‘virtual’ in some sense). Finally, we note that in the case of Dirichlet
boundary conditions, studying sufficient conditions to rule out anomalous dissipation for the
viscous primitive equations becomes more subtle due to the appearance of boundary layers.

There are mainly two ways to establish sufficient conditions for energy conservation.
One either starts with a regularity assumption on w to properly define a weak solution and
then derive sufficient conditions for u and v to ensure conservation of energy. Alternatively,
one can impose sufficient conditions on u and v to define a weak solution and then obtain
sufficient conditions for energy conservation. This range of possibilities leads to a ‘family’
of Onsager conjectures, which was summarised above and is the content of this contribution.

123



219 Page 34 of 40 D. W. Boutros et al.

Remark 7.2 We observe that it is possible to even extend this ‘family’ of Onsager conjectures
by combining the different ideas we used in this paper. For example, it is possible to derive
the horizontal and vertical Onsager exponents for type III weak solutions, similar to what
was done in Proposition 3.8 for type I weak solutions of the hydrostatic Euler equations.

Acknowledgements The authors would like to thank the anonymous referee for the thorough reading of
the paper as well as the useful comments and suggestions. D.W.B. acknowledges support from the Cam-
bridge Trust, the Cantab Capital Institute for Mathematics of Information and the Hendrik Muller fund. S.M.
acknowledges support from the Alexander von Humboldt foundation. E.S.T. acknowledges the partial support
by the Simons Foundation Award No. 663281 granted to the Institute of Mathematics of the Polish Academy
of Sciences for the years 2021–2023. The authors would like to thank the Isaac Newton Institute for Math-
ematical Sciences, Cambridge, for support and hospitality during the programme “Mathematical aspects of
turbulence: where do we stand?” where part of this work was undertaken and supported by EPSRC grant
no. EP/K032208/1.

Declarations

Conflict of interest The authors declare that they have no competing interests. Data sharing is not applicable
to this article as no datasets were generated or analysed during the current study.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix A: Essential estimates regarding the Besov norm

The following estimates are used several times in this paper.

Lemma A.1 For any 1 ≤ p, q, q1, q2 ≤ ∞, α ∈ R, θ > 0 the following estimates hold

‖ f ‖B0
p,∞ � ‖ f ‖L p � ‖ f ‖B0

p,1
,

‖ f ‖Bα
p,q1

� ‖ f ‖Bα+θ
p,q2

,

‖ f ‖Bα
p,q1

� ‖ f ‖Bα
p,q2

, if q1 ≥ q2

‖∂i f ‖Bα−1
p,q

� ‖ f ‖Bα
p,q

.

For the proof we refer to [65] Propositions 2.1, 2.2, 2.3 and Theorem 2.2.

Appendix B: Review of paradifferential calculus

We briefly recall some basic notions of paradifferential calculus, further details can be found
in [1, 58, 59]. We first introduce some basic notions of Littlewood-Paley theory, which can
again by found in [1]. Let ρ be a smooth function with support in the annulus of radii 3

4 and

123

http://creativecommons.org/licenses/by/4.0/


On energy conservation for the hydrostatic Euler equations... Page 35 of 40 219

8
3 . We let {ρ j }∞j=−1 be a dyadic partition of unity, i.e.

ρ0(ξ) = ρ(ξ), ρ j (ξ) = ρ(2− j ξ) for j = 1, 2, . . . , ρ−1(ξ) = 1 −
∞∑

j=0

ρ j (ξ).

Then the Littlewood-Paley blocks are defined as follows for f ∈ S ′(T3)

�̂ j f (ξ) = ρ j (ξ) f̂ (ξ), j = −1, 0, . . . .

Bony’s decomposition of a product is formally given by

f g = T f g + Tg f + R( f , g).

Here T f g and Tg f are called the paraproducts which are given by

T f g =
∞∑

j=−1

j−2∑

i=−1

�i f � j g, Tg f =
∞∑

j=−1

j−2∑

i=−1

�i g� j f .

The resonance term R( f , g) is given by

R( f , g) =
∑

|k− j |≤1

�k f � j g.

For the paraproducts we have the following estimates.

Lemma B.1 (Lemma 2.1 in [61]) Let α, β ∈ R and 1 ≤ p, p1, p2, q, q1, q2 ≤ ∞ with

1

p
= 1

p1
+ 1

p2
,

1

q
= 1

q1
+ 1

q2
.

• For any f ∈ L p1(T3) and g ∈ Bβ
p2,q(T

3) we have that

‖T f (g)‖Bβ
p,q

� ‖ f ‖L p1 ‖g‖Bβ
p2,q

.

• If α < 0 then for any f ∈ Bα
p1,q1(T

3) and g ∈ Bβ
p2,q2(T

3) we have

‖T f (g)‖Bα+β
p,q

� ‖ f ‖Bα
p1,q1

‖g‖
Bβ
p2,q2

.

For the resonance term the following estimate holds.

Lemma B.2 (Theorem 2.85 in [1]) Let α, β ∈ R and 1 ≤ p, p1, p2, q, q1, q2 ≤ ∞ with

1

p
= 1

p1
+ 1

p2
,

1

q
= 1

q1
+ 1

q2
.

• If α + β > 0 then we have for any f ∈ Bα
p1,q1(T

3) and g ∈ Bβ
p2,q2(T

3)

‖R( f , g)‖
Bα+β
p,q

� ‖ f ‖Bα
p1,q1

‖g‖
Bβ
p2,q2

.

• If α + β = 0 and q = 1 then we have for any f ∈ Bα
p1,q1(T

3) and g ∈ Bβ
p2,q2(T

3)

‖R( f , g)‖B0
p,∞ � ‖ f ‖Bα

p1,q1
‖g‖

Bβ
p2,q2

.

For further details and proofs we refer to [1, 61].
Combining Lemmas B.1 and B.2 leads to the following result.

123



219 Page 36 of 40 D. W. Boutros et al.

Lemma B.3 Let α < β, β + α > 0, θ > 0, and 1 ≤ p1, p2, p, q1, q2, q ≤ ∞ with

1

p1
+ 1

p2
= 1

p
.

• For any f ∈ Bα
p1,q1(T

3) and g ∈ Bβ
p2,q2(T

3) we have that

‖ f g‖Bα
p,q1

� ‖ f ‖Bα
p1,q1

‖g‖
Bβ
p2,q2

.

• If α > 0 we have for any f ∈ Bα
p1,q(T

3) and g ∈ Bα
p2,q(T

3)

‖ f g‖Bα
p,q

� ‖ f ‖Bα
p1,q

‖g‖Bα
p2,q

.

• If α > 0 we have for any f ∈ Bθ
p1,q1(T

3) ∩ Bα+θ
p2,q2(T

3)

‖ f 2‖Bα
p,q

� ‖ f ‖Bθ
p1,q1

‖ f ‖Bα+θ
p2,q2

.

Proof The estimates claimed in Lemma B.3 are a simple consequence of Lemmas B.1 and
B.2. We only show the third bullet point in more detail as this is the most important estimate
needed for this paper. The other estimates can be proven similarly.

By Bony’s decomposition we may write

‖ f 2‖Bα
p,q

≤ 2‖T f f ‖Bα
p,q

+ ‖R( f , f )‖Bα
p,q

.

Lemma B.1 yields together with Lemma A.1

‖T f f ‖Bα
p,q

� ‖ f ‖L p1 ‖ f ‖Bα
p2,q

� ‖ f ‖B0
p1,1

‖ f ‖Bα
p2,q

� ‖ f ‖Bθ
p1,q1

‖ f ‖Bα+θ
p2,q2

.

Moreover we find by using Lemmas B.2 and A.1

‖R( f , f )‖Bα
p,q

� ‖ f ‖B0
p1,∞‖ f ‖Bα

p2,q
� ‖ f ‖Bθ

p1,q1
‖ f ‖Bα+θ

p2,q2
.


�

Appendix C: Proof of Proposition 3.8 by using commutator estimates

The results in this paper were proven by using an equation of local energy balance and
subsequently showing that the defect term in this equation is zero. Most of the results could
also have been proven by using commutator estimates (of the type first introduced in [20]).

In this section, we will present a proof of Proposition 3.8 to illustrate how one can prove
these results by using commutator estimates.We repeat the statement of Proposition 3.8 here.

Proposition C.1 (Horizontal and vertical Onsager exponents) Let u be a type I weak solution
of the hydrostatic Euler equations such that u, v ∈ L3((0, T ); Bα

3,∞(T;
Bβ
3,∞(T2))) with β > 2

3 , α > 1
3 and 2α + β > 2. We recall that the regularity assumption

means that u and v have Bα
3,∞(T) regularity in the z-direction, while they have Bβ

3,∞(T2)

regularity in the horizontal directions. Under these assumptions the weak solution conserves
energy.

Proof We restrict to the caseβ > 1, as otherwise by the assumptions of the theoremwewould
need α > 1

2 . Then both α and β are bigger than 1
2 and the conclusion of the proposition was

already proven in Proposition 3.1.

123



On energy conservation for the hydrostatic Euler equations... Page 37 of 40 219

We take φ1 = (uε)ε and φ2 = (vε)ε in the weak formulation in Definition 2.1, transfer
one of the mollifications and add the equations together. This gives

∫ T

0

∫

T3
uε∂t u

εdx dt +
∫ T

0

∫

T3
vε∂tv

εdx dt +
∫ T

0

∫

T3
(uu)ε · ∇uεdx dt

+
∫ T

0

∫

T3
(vu)ε · ∇vεdx dt −

∫ T

0

∫

T3
�uεvεdx dt +

∫ T

0

∫

T3
�vεuεdx dt

+
∫ T

0

∫

T3
pε∂xu

εdx dt +
∫ T

0

∫

T3
pε∂yv

εdx dt = 0.

We observe that

−
∫ T

0

∫

T3
�uεvεdx dt +

∫ T

0

∫

T3
�vεuεdx dt = 0.

Regarding the pressure terms, we obtain
∫ T

0

∫

T3
pε∂xu

εdx dt +
∫ T

0

∫

T3
pε∂yv

εdx dt

=
∫ T

0

∫

T3
pε∂xu

εdx dt +
∫ T

0

∫

T3
pε∂yv

εdx dt +
∫ T

0

∫

T3
pε∂zw

εdx dt = 0.

Concerning the advective terms, we only present the details in handling the integral∫ T
0

∫
T3(uu)ε · ∇uεdx dt , as the derivation for the other advective term proceeds in a similar

fashion. First we look at the first and second terms of the divergence, we get that

∫ T

0

∫

T3
(u2)ε∂xu

εdx dt +
∫ T

0

∫

T3
(uv)ε∂yu

εdx dt

=
∫ T

0

∫

T3

[
uεuε +

∫

R3

(
ϕε(y)(δu(−y; x, t) ⊗ δu(−y; x, t)))dy − (u − uε) ⊗ (u − uε)

]

· ∂xu
εdx dt

+
∫ T

0

∫

T3

[
uεvε +

∫

R3

(
ϕε(y)δu(−y; x, t) ⊗ δv(−y; x, t))dy − (u − uε) ⊗ (v − vε)

]

· ∂yu
εdx dt .

In the above we have used commutator estimates as introduced in [5, 20]. Then we do the
estimates

∫ T

0

∫

T3

[ ∫

R3

(
ϕε(y)(δu(−y; x, t) ⊗ δu(−y; x, t)))dy − (u − uε) ⊗ (u − uε)

]
∂xu

εdx dt

+
∫ T

0

∫

T3

[ ∫

R3

(
ϕε(y)δu(−y; x, t) ⊗ δv(−y; x, t))dy − (u − uε) ⊗ (v − vε)

]
∂yu

εdx dt

≤
[
ε2α‖u‖2

Bα
z (Bβ

h )
+ ε2α‖u‖

Bα
z (Bβ

h )
‖v‖

Bα
z (Bβ

h )

]
εα−1‖u‖

Bα
z (Bβ

h )

≤ ε3α−1‖u‖2
Bα
z (Bβ

h )

[
‖u‖

Bα
z (Bβ

h )
+ ‖v‖

Bα
z (Bβ

h )

]
.

Recall that we assumed that α > 1
3 , therefore these terms go to zero as ε → 0.

123



219 Page 38 of 40 D. W. Boutros et al.

Subsequently, we look at the third component. By using a commutator estimate, we see
that
∫ T

0

∫

T3
(uw)ε∂zu

εdx dt =
∫ T

0

∫

T3

[
uεwε +

∫

R3

(
ϕε(y)(δu(−y; x, t) ⊗ δw(−y; x, t)))dy

− (u − uε) ⊗ (w − wε)

]
∂zu

εdx dt .

We now make the estimate

∫ T

0

∫

T3

[ ∫

R3

(
ϕε(y)(δu(−y; x, t) ⊗ δw(−y; x, t)))dy − (u −