Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics
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Study of the limit system with anisotropic viscosity 149
and we are going to show that each of those two terms can be estimated in
the same way. In the following computations we shall use the notation of the
above sections. The computations will be quite similar to Section 6.4, apart
from the fact that one needs to take into account the anisotropy of the problem
here (since the Laplacian is anisotropic). The first term in (6.6.7), where Q
h
appears, is a finite sum of terms of the form
(k,n)∈Z
6
k∈K(n)
u
j
osc
(k)(k
h
− n
h
)(k
3
− n
3
)u
j
′
osc
(n − k)
+ k
3
u
j
osc
(k)(k
h
− n
h
)u
j
′
osc
(n − k)
n
3
u
j
′′
osc
(n), (6.6.8)
where j, j
′
and j
′′
are in {1, 2, 3}. We have denoted K
n
def
= ∪
σ
K
σ
n
with nota-
tion (6.2.5). Now we are going to show that (∂
3
Q
v
(u
osc
,u
osc
)|∂
3
u
osc
)
L
2
can be
written in the same way. We have
∂
3
Q
v
(u
osc
,u
osc
)
∂
3
u
osc
L
2
= − lim
ε→0
#
∂
3
#
L
t
ε
u
osc
3
∂
3
L
t
ε
u
osc
$
∂
3
L
t
ε
u
osc
$
L
2
and using the fact that L
t
ε
u
osc
is divergence-free, we can write on the
one hand
#
∂
3
L
t
ε
u
osc
3
∂
3
L
t
ε
u
osc
∂
3
L
t
ε
u
osc
$
L
2
= −
#
div
h
L
t
ε
u
osc
h
∂
3
L
t
ε
u
osc
∂
3
L
t
ε
u
osc
$
L
2
whereas, on the other hand, an integration by parts implies that
#
L
t
ε
u
osc
3
∂
2
3
L
t
ε
u
osc
∂
3
L
t
ε
u
osc
$
L
2
= −
1
2
#
∂
3
L
t
ε
u
osc
3
∂
3
L
t
ε
u
osc
∂
3
L
t
ε
u
osc
$
L
2
.
So finally using once again the fact that L
t
ε
u
osc
is divergence-free we get
∂
3
Q
v
(u
osc
,u
osc
)
∂
3
u
osc
L
2
=
1
2
lim
ε→0
#
div
h
L
t
ε
u
osc
h
∂
3
L
t
ε
u
osc
∂
3
L
t
ε
u
osc
$
L
2
.
150 The periodic case
So we are back to the formulation (6.6.8). The following lemma shows how to
estimate a sum of the type (6.6.8).
Lemma 6.6 There is a constant C such that the following estimate holds: for
all vector fields a, b, c of the form 7.0.5, we have
(k,n)∈Z
6
k∈K
n
|a(k)
b(n − k)c(n)|≤C
n∈Z
3
(1 + |n
h
|
2
)
1
2
|a(n)|
2
1
2
×
n∈Z
3
|
b(n)|
2
1
2
n∈Z
3
(1 + |n
h
|
2
)
1
2
|c(n)|
2
1
2
.
Remark This result is similar to Lemma 6.2, only the spaces appearing on
the right-hand side are anisotropic-type Sobolev spaces. Contrary to the case of
Lemma 6.2 and in order to avoid any additional complication, we shall not prove
that result in any more generality (with a general “resonant set” and with more
general Sobolev norms on the right-hand side).
Before proving Lemma 6.6, let us finish the proof of Proposition 6.5. We use
Lemma 6.6 twice, once with a = u
osc
, b = ∂
3
∇
h
u
osc
and c = ∂
3
u
osc
, and once
with a = ∂
3
u
osc
, b = ∇
h
u
osc
and c = ∂
3
u
osc
.Weget
|(∂
3
Q(u
osc
,u
osc
)|∂
3
u
osc
)
L
2
|
≤
∂
3
u
osc
L
2
+ |∇
h
|
1
2
∂
3
u
osc
L
2
2
∇
h
u
osc
L
2
+
u
osc
L
2
+ |∇
h
|
1
2
u
osc
L
2
∇
h
∂
3
u
osc
L
2
×
∂
3
u
osc
L
2
+ |∇
h
|
1
2
∂
3
u
osc
L
2
.
An easy computation using the interpolation inequality
|∇
h
|
1
2
u
osc
(t)
2
L
2
≤u
osc
(t)
L
2
∇
h
u
osc
(t)
L
2
, (6.6.9)
yields the following estimate:
|(∂
3
Q(u
osc
,u
osc
)|∂
3
u
osc
)
L
2
|
≤
ν
4
∇
h
∂
3
u
osc
2
L
2
+ ∂
3
u
osc
2
L
2
∇
h
u
osc
L
2
+
C
ν
∂
3
u
osc
2
L
2
(∇
h
u
osc
2
L
2
+ u
osc
2
L
2
)+
C
ν
u
osc
2
L
2
∂
3
|∇
h
|
1
2
u
osc
2
L
2
+
C
ν
∂
3
u
osc
2
L
2
|∇
h
|
1
2
u
osc
2
L
2
+
C
ν
∂
3
|∇
h
|
1
2
u
osc
2
L
2
|∇
h
|
1
2
u
osc
2
L
2
.
Study of the limit system with anisotropic viscosity 151
Using (6.6.9) once again yields
|(∂
3
Q(u
osc
,u
osc
)|∂
3
u
osc
)
L
2
|
≤
ν
2
∇
h
∂
3
u
osc
2
L
2
+ ∂
3
u
osc
2
L
2
∇
h
u
osc
L
2
+
C
ν
∂
3
u
osc
2
L
2
∇
h
u
osc
2
L
2
+ u
osc
2
L
2
+
C
ν
2
∂
3
u
osc
2
L
2
u
osc
2
L
2
.
Going back to (6.6.3), we have finally
d
dt
∂
3
u
osc
(t)
2
L
2
+ ∇
h
∂
3
u
osc
(t)
2
L
2
≤∂
3
u
osc
2
L
2
∇
h
u
osc
L
2
+
C
ν
(∇
h
u
osc
2
L
2
+ u
osc
2
L
2
)
+
C
ν
3
∂
3
u
osc
2
L
2
u
osc
2
L
2
.
The result follows from the L
2
energy estimate (6.6.3) on u
osc
thanks to the
Gronwall lemma.
To end the proof of Proposition 6.5, we still need to prove (6.6.2). This result
relies on some refined analysis of diophantine equations, and is beyond the scope
of this book. We will therefore not prove the result, but refer to [49] where the
proof can be found. Proposition 6.5 is proved.
Proof of Lemma 6.6 This is, as in the case of Proposition 6.2, due to the
special structure of the quadratic form Q. We recall to this end that if n and k
h
are fixed, then there are at most eight integers k
3
such that k is in the resonant
set K
n
. So we can write, using the Cauchy–Schwarz inequality,
(k,n)∈Z
6
k∈K
n
|a(k)
b(n − k)c(n)|≤C
(k
h
,n)∈Z
5
|c(n)|
#
k
3
∈Z
|a(k)|
2
|
b(n − k)|
2
$
1
2
which again by a Cauchy–Schwarz inequality yields
(k,n)∈Z
6
k∈K
n
|a(k)
b(n − k)c(n)|≤C
(k
h
,n
h
)∈Z
4
#
n
3
∈Z
|c(n)|
2
$
1
2
×
(k
3
,n
3
)∈Z
2
|a(k)|
2
|
b(n − k)|
2
1
2
.
152 The periodic case
We deduce that
(k,n)∈Z
6
k∈K
n
|a(k)
b(n − k)c(n)|
≤ C
(k
h
,n
h
)∈Z
4
#
n
3
∈Z
|c(n)|
2
$
1
2
#
k
3
|a(k)|
2
$
1
2
#
ℓ
3
|
b(n
h
− k
h
,ℓ
3
)|
2
$
1
2
.
Two-dimensional product rules enable us to write, for all two-dimensional vector
fields A, B and C
(k
h
,n
h
)∈Z
4
|
A(n
h
)
B(n
h
− k
h
)
C(k
h
)|
≤ C
n
h
∈Z
2
(1 + |n
h
|
2
)
1
2
|
A(n
h
)|
2
1
2
×
n
h
∈Z
2
|
B(n
h
)|
2
1
2
n
h
∈Z
2
(1 + |n
h
|
2
)
1
2
|
C(n
h
)|
2
1
2
so the result follows, taking
A(n
h
)=
&
n
3
∈Z
|a(n)|
2
1
2
and similarly for B
and C. Lemma 6.6 is proved.
Note that in particular Lemma 6.6 implies the following result.
Proposition 6.6 The quadratic form Q satisfies, for any divergence-free vector
field a,
Q(a, a)
H
−1,0
≤ Ca
L
2
a
H
1,0
+ ∂
3
a
L
2
a
1
2
L
2
a
1
2
H
1,0
.
In particular Q is bilinear continuous from
L
∞
([0,T]; H
0,1
) × L
2
([0,T]; H
1,0
) into L
2
([0,T]; H
−1,0
).
Proof As in (6.6.5) and (6.6.6) let us decompose
Q(a, a)=Q
h
(a, a)+Q
v
(a, a).
Let b ∈ H
1,0
be given. The divergence-free condition on a implies that
(Q(a, a)|b) ≤
k∈K
σ
n
|a(k)||a(n − k)||n
h
||
b(n)| +
k∈K
σ
n
|a(k)||a(n − k)|n
3
|
b(n)|.
Let us start with the first term. We can write
|(Q
h
(a, a)|b)|≤
k∈K
σ
n
|a(k)||a(n − k)||n
h
||
b(n)|,
Study of the limit system with anisotropic viscosity 153
so Lemma 6.6 implies that
|(Q
h
(a, a)|b)|≤C
n∈Z
3
(1 + |n
h
|
2
)
1
2
|a(n)|
2
n∈Z
3
(1 + |n
h
|
2
)|
b(n)|
2
1
2
.
By the Gagliardo–Nirenberg inequality we get
|(Q(a, a)|b)|≤Ca
L
2
a
H
1,0
b
H
1,0
.
For Q
v
we write
|(Q
v
(a, a)|b)|≤
k∈K
σ
n
k
3
|a(k)||a(n − k)||
b(n)|
+
k∈K
σ
n
|a(k)||n
3
− k
3
|a(n − k)||
b(n)|,
and Lemma 6.6 along with the Gagliardo–Nirenberg inequality implies that
|(Q
v
(a, a)|b)|≤C∂
3
a
L
2
a
1
2
L
2
a
1
2
H
1,0
b
1
2
L
2
b
1
2
H
1,0
,
and Proposition 6.6 is proved.
7
Ekman boundary layers for rotating fluids
In this chapter, we investigate the problem of rapidly rotating viscous fluids
between two horizontal plates with Dirichlet boundary conditions. We present
the model with so-called “turbulent” viscosity. More precisely, we shall study the
limit when ε tends to 0 of the system
(NSC
ε
)
∂
t
u
ε
+ div(u
ε
⊗ u
ε
) − ν∆
h
u
ε
− βε∂
2
3
u
ε
+
e
3
∧ u
ε
ε
= −∇p
ε
div u
ε
=0
u
ε
|∂Ω
=0
u
ε
|t=0
= u
ε
0
,
where Ω = Ω
h
×]0, 1[: here Ω
h
will be the torus T
2
or the whole plane R
2
.We
shall use, as in the previous chapters, the following notation: if u is a vector
field on Ω we state u =(u
h
,u
3
). In all that follows, we shall assume that on
the boundary ∂Ω, u
ε
0
· n = u
ε,3
0
= 0, and that div u
ε
0
= 0. The condition u
3
0
=0
on the boundary implies the following fact: for any vector field u ∈H(Ω), the
function ∂
3
u
3
is L
2
(]0, 1[) with respect to the variable x
3
with values in H
−1
(Ω
h
)
due to the divergence-free condition. So by integration, we get
u
3
(x
h
, 1) − u
3
(x
h
, 0) = −
1
0
div
h
u
h
(x
h
,x
3
)dx
3
=0. (7.0.1)
So the vertical mean value of the horizontal part of the vector field is divergence-
free as a vector field on Ω
h
.
We proved in Chapter 2 that a weak solution u
ε
exists such that if
E
ε
t
(v)
def
= v(t)
2
L
2
+2ν
t
0
∇v(t
′
)
2
L
2
dt
′
+2βε
t
0
∂
3
v(t
′
)
2
L
2
dt
′
, (7.0.2)
then we have
E
ε
t
(u
ε
) ≤u
0
2
L
2
. (7.0.3)
It follows that the family (u
ε
)
ε>0
has some weak limit points, for instance in the
space L
2
loc
(R
+
×Ω), and as seen in the introduction in Part I they do not depend
on the vertical variable. Let us recall the arguments. Let
u be in the weak closure
of u
ε
. Obviously, the term
∂
t
u
ε
+ div(u
ε
⊗ u
ε
) − ν∆
h
u
ε
− βε∂
2
3
u
ε
156 Ekman boundary layers for rotating fluids
is, up to an extraction of a subsequence, convergent in D
′
(R
+
×Ω). Thus this
implies that
e
3
∧
u = −∇p
which can be written
−
u
2
u
1
0
=
−∂
1
p
−∂
2
p
−∂
3
p
,
from which we deduce that ∂
3
u
h
= 0 and div
h
u
h
=0.Asu is a divergence-free
vector field on the three-dimensional domain Ω, then ∂
3
u
3
=0.
In the following we will also define
E
T
(v)
def
= sup
t∈[0,T ]
v(t)
2
L
2
(Ω)
+2ν
T
0
∇
h
v(t)
2
L
2
(Ω)
dt.
The aim of this chapter is to study the asymptotics of the family (u
ε
)
ε>0
as ε
goes to zero. The easiest situation occurs when the initial data do not depend
on the vertical variable x
3
(which corresponds to the so-called “well-prepared”
case). The precise theorem in the well-prepared case is the following. It will be
proved in Section 7.3, using a fundamental linear lemma stated and proved in
Section 7.1. Let us first introduce space with anisotropic regularity.
Definition 7.1 We shall denote by H
1,0
(Ω) (or by H
1,0
when no confu-
sion is possible) the space of L
2
functions u on Ω such that ∇
h
u also belongs
to L
2
(Ω).
Theorem 7.1 Let (u
ε
)
ε>0
be a family of weak solutions of (NSC
ε
) associated
with a family of initial data u
ε
0
∈H(Ω) such that
lim
ε→0
u
ε
0
=(
u
h
0
, 0) in H(Ω),
where
u
h
0
is a horizontal vector field in H(Ω
h
). Denoting by u the global solution
of the two-dimensional Navier–Stokes-type equations on Ω
h
(NSE
ν,β
)
∂
t
u + div
h
(u ⊗ u) − ν∆
h
u +
2β u = −∇
h
p
div
h
u =0
u
|t=0
= u
h
0
,
then u
ε
goes to (
u, 0) as ε goes to zero, in the space
L
∞
loc
(R
+
; L
2
(Ω)) ∩ L
2
loc
(R
+
; H
1,0
).
Remark We will omit the proof that there is a unique global solution to the
system (NSE
ν,β
) with initial data
u
h
0
in H(Ω
h
), as it is exactly the same proof as
Ekman boundary layers for rotating fluids 157
for the two-dimensional Navier–Stokes equations studied in Part II, Section 2.2
and Chapter 3. In particular the solution
u belongs to the space
C
0
b
(R
+
, H(Ω
h
)) ∩ L
2
(R
+
, V
σ
(Ω
h
)),
and for all t ≥ 0,
1
2
u(t)
2
L
2
+ ν
t
0
∇
u(t
′
)
2
L
2
dt
′
+
2β
t
0
u(t
′
)
2
L
2
dt
′
=
1
2
u
0
2
L
2
. (7.0.4)
We shall investigate the more delicate case when the initial data u
0
do actu-
ally depend on the vertical variable x
3
(the so-called “ill-prepared” case) in
Section 7.4, where we shall compute explicitly approximate solutions for the
linear problem. The study of the full non-linear problem depends strongly on
the domain Ω. In the case when Ω = R
2
×]0, 1[, the result is the following,
proved in Section 7.6.
Theorem 7.2 Let u
0
be a vector field in H(R
2
×]0, 1[). Let us consider a fam-
ily (u
ε
)
ε>0
of weak solutions of (NSC
ε
) associated with the initial data u
0
.
Denoting by
u the global solution of the two-dimensional Navier–Stokes-type
equations (NSE
ν,β
) with initial data
u
|t=0
(x
h
)
def
=
1
0
u
h
0
(x
h
,x
3
)dx
3
,
we have, for any positive time T and any compact subset K of R
2
×]0, 1[,
lim
ε→0
[0,T ]×K
|u
ε
(t, x) − (u(t, x
h
), 0)|
2
dx
h
dx
3
=0.
The key point of the proof of this theorem is that the dispersive phenomena
studied in Chapter 5 are not affected by the Ekman boundary layer.
When the domain Ω
h
is periodic, the result is different, due to the absence of
dispersion; in particular the methods of Chapter 6 will be used here to deal with
the periodic horizontal boundary conditions. We will therefore be using some
notation of the purely periodic case: in the coming theorem, L is the filtering
operator of Chapter 6, and Q is the quadratic form defined in Proposition 6.1,
page 124. Before stating the result let us define the periodic setting: the horizontal
torus will be defined as
T
2
def
=
2
)
j=1
R
/a
j
Z
,
where the a
j
’s are positive real numbers. We will see that solving the problem
in Ω with Dirichlet boundary conditions corresponds to considering vector fields
having the following symmetries:
u(x
h
,x
3
)=(u
h
(x
h
, −x
3
), −u
3
(x
h
, −x
3
)).
158 Ekman boundary layers for rotating fluids
These symmetry properties are clearly preserved by the rotating-fluid equations.
Such vector fields will be decomposed in the following way:
u(x)=
k∈N
3
u
k,h
e
iπk
h
·x
h
cos(k
3
πx
3
)
−
i
k
3
π
k
h
· u
k,h
e
iπk
h
·x
h
sin(k
3
πx
3
)
. (7.0.5)
This form is very similar to the decomposition of periodic vector fields used in
Chapter 6, and in particular the Fourier components are
(u
k,1
,u
k,2
,u
k,3
) with u
k,3
= −
i
k
3
π
k
h
· u
k,h
.
Comparing the vertical Fourier transform defined in (7.0.5) with the definition
of the classical Fourier transform (6.1.1), page 117, shows that with the notation
of Chapter 6 we have a
3
= 2, so as in Chapter 6 we will define
k
def
=
k
1
a
1
,
k
2
a
2
,
k
3
a
3
with a
3
=2.
The theorem proved in Section 7.7 is the following. We have denoted by H
0,1
the space of functions f ∈ L
2
(Ω) such that ∂
3
f is in L
2
(Ω), and H
0
the space of
vector fields in H with vanishing horizontal mean.
Theorem 7.3 A non-negative, continuous operator E on H
0
∩H
0,1
(Ω) exists (it
is defined by (7.4.39) below) such that the following results hold. Let us consider
initial data u
0
in H
0
∩ H
0,1
, and let u be the associated solution of the system
(NSE
ν,E
)
∂
t
u − ν∆
h
u + E u + Q(u, u)=0
div u =0
u
|t=0
= u
0
given by Proposition 6.5, page 145. If T
2
satisfies condition (A) in the sense of
Definition 6.2, then the following result is true. Let (u
ε
)
ε>0
be a family of weak
solutions of (NSC
ε
) associated with the initial data u
0
. Then for all T>0,
lim
ε→0
E
T
u
ε
−L
t
ε
u
=0.
Note that the structure of the limiting equation in the periodic case is very
different from the case of the whole space. The filtering operator acts like the
identity on two-dimensional vector fields, and that is why it does not appear in
the statement of Theorem 7.3, as the three-dimensional vector fields disappear
due to dispersion. In the same way, the operator E acts like
√
2β Id on two-
dimensional vector fields. In the case of T
2
×]0, 1[, the above theorem generalizes
Theorem 7.1. Here three-dimensional vector fields remain in the limit, since u is
no longer two dimensional. We recall (see Section 6.6, page 144) that the fact
that the periodic box satisfies condition (A) ensures that the horizontal mean
of the solution of the limit system is preserved–avanishing horizontal mean is
needed to construct the boundary layers in Section 7.4.