Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics
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The particular case of the Rossby operator in R
3
99
As the integral
(R
+
)
2
1
(t − s)
1
2
e
−νr
2
(t+s)
dtds
is finite, this yields the expected estimate in the case p = 1. To get the whole
interval p ∈ [1, +∞], we simply notice that due to the skew-symmetry of the
rotation operator, the L
∞
(R
+
; L
2
(R
3
)) norm of v is bounded, uniformly in ε.
As its frequencies are bounded by R, the same goes for the L
∞
(R
+
×R
3
) norm
since according to Lemma 1.2, page 26,
v
L
∞
(R
+
×R
3
)
≤ C
R
v
L
∞
(R
+
;L
2
(R
3
))
≤v
0
L
2
(R
3
)
.
Interpolation of that uniform bound with the L
1
(R
+
; L
∞
(R
3
)) estimate found
above yields estimate (5.2.7).
Now let us turn to the anisotropic estimate (5.2.8), which is obtained in a very
similar manner to the isotropic estimate above. If we denote, for any function g
defined on R
3
,byg its vertical Fourier transform
g(x
h
,ξ
3
)
def
=
R
e
−ix
3
ξ
3
g(x
h
,x
3
) dx
3
,
then we have of course
G
ε,±
ν
(τ)g
L
∞,2
x
h
,x
3
= C
G
ε,±
ν
(τ)g
L
∞,2
x
h
,ξ
3
,
where we have defined
G
ε,±
ν
(τ)g(x
h
,ξ
3
)
def
=
R
2
y
h
I
±
(ετ,τ,x
h
− y
h
,ξ
3
)g(y
h
,ξ
3
)dy
h
where I
±
is defined by (5.2.4). By Lemma 5.2 we have
G
ε,±
ν
t
ε
g
L
∞,2
x
h
,ξ
3
≤ C
r,R
ε
1
2
t
1
2
e
−ct
g
L
1,2
x
h
,ξ
3
.
We shall now be using exactly the same TT
∗
argument as in the isotropic case;
we reproduce it here for the reader’s convenience. We have
b
L
1
(R
+
;L
∞,2
x
h
,ξ
3
)
= sup
ϕ∈
B
R
+
×R
3
b(t, x
h
,ξ
3
) ϕ(t, x
h
,ξ
3
) dx
h
dξ
3
dt,
100 Dispersive cases
with
B =
'
ϕ ∈D(R
+
×R
3
), ϕ
L
∞
(R
+
;L
1,2
x
h
,ξ
3
)
≤ 1
(
. So we can write
G
def
=
G
ε,+
ν
t
ε
g
L
1
(R
+
;L
∞,2
x
h
,ξ
3
)
= sup
ϕ∈
B
R
+
×R
5
I
+
t,
t
ε
,x
h
− y
h
,ξ
3
g(y
h
,ξ
3
) ϕ(t, x
h
,ξ
3
) dtdx
h
dy
h
dξ
3
= sup
ϕ∈
B
R
+
×R
3
g(y
h
,ξ
3
)
×
R
2
I
+
t,
t
ε
,x
h
− y
h
,ξ
3
ϕ(t, x
h
,ξ
3
) dx
h
dtdy
h
dξ
3
.
We infer that
G ≤g
L
2
(R
3
)
sup
ϕ∈
B
R
+
ˇ
I
+
t,
t
ε
, ·,ξ
3
∗ ϕ(t, ·,ξ
3
) dt
L
2
(R
3
)
.
Similarly to the isotropic case, we have
R
+
ˇ
I
+
t,
t
ε
, ·,ξ
3
∗ ϕ(t, ·,ξ
3
) dt
2
L
2
≤ C
r,R
(R
+
)
2
ε
1
2
(t − s)
1
2
e
−νr
2
(t+s)
ϕ(t, ·)
L
1,2
x
h
,ξ
3
ϕ(s, ·)
L
1,2
x
h
,ξ
3
dtds
and the result follows as previously: to get the L
p
estimate in time for all p,
we now notice that the L
∞
(R
+
,L
∞,2
x
h
,x
3
) norm of the solution is bounded, again
by frequency localization and the fact that the rotation operator is an isometry
on L
2
(R
3
), and the corollary is proved.
5.3 Application to rotating fluids in R
3
In this section, we focus on the small Rossby number limit of solutions to the
incompressible Navier–Stokes–Coriolis system, namely
(NSC
ε
)
∂
t
u
ε
+ u
ε
·∇u
ε
− ν∆u
ε
+
e
3
∧ u
ε
ε
+ ∇p
ε
=0
div u
ε
=0
u
ε
|t=0
= u
0
.
We refer to the introduction of this book for physical discussions of the model.
In the sequel, we will focus on the viscous case ν>0. We have neglected the
presence of bulk forces to simplify the presentation. Let us introduce some nota-
tion: let P be the Leray projector onto divergence-free vector fields; the fact that
the operator P(e
3
∧ u
ε
) is skew-symmetric implies that if the initial velocity u
0
belongs to L
2
(R
3
), then we obtain a sequence (u
ε
)
ε>0
of Leray’s weak solutions,
uniformly bounded in the space L
∞
(R
+
; L
2
(R
3
)) ∩ L
2
(R
+
;
˙
H
1
(R
3
)). We leave
Application to rotating fluids in R
3
101
as an exercise to the reader, the adaptation of the proof of Theorem 2.3, page 42,
to obtain the existence of such solutions.
We now wish to analyze that system in the limit when ε goes to zero. In the
introduction (Part I), we claimed that the weak limit of u
ε
does not depend on
the vertical variable x
3
. The only element of L
2
(R
3
) which does not depend on x
3
is zero, so in order to get some more relevant results on the asymptotics of u
ε
,
we are going to study the existence (and convergence) of Leray-type solutions
for initial data of the type
u
j
0
=
u
j
0
(x
h
)+w
j
0
(x
h
,x
3
),j∈{1, 2, 3}, (5.3.1)
with div
h
u
h
0
= div w
0
=0.
Let us denote by
H(R
2
) the space of vector fields
u with three components
in L
2
(R
2
), such that ∂
1
u
1
+ ∂
2
u
2
= 0. We have, of course, for u in
H(R
2
),
u =(u
h
, 0) + u
3
(0, 0, 1)
with
u
h
in H(R
2
) and u
3
in L
2
(R
2
).
We will denote by (
NS) the two-dimensional Navier–Stokes equations, when
the velocity field has three components and not two:
(
NS)
∂
t
u + u
h
·∇
h
u − ν∆
h
u +(∇
h
p, 0)=0
div
h
u
h
=0
u
|t=0
= u
0
∈
H.
Our aim is to prove the convergence of the solutions of the rotating fluid equa-
tions (NSC
ε
) associated with data of the type (5.3.1) towards the solution
of (
NS). So we first need to define what a solution of (NS) is, and to prove
an existence theorem for that system; that will be done in Section 5.3.1 below.
Then we will investigate the existence and convergence of solutions to (NSC
ε
)
in a “Leray” framework (in Section 5.3.2) and we will discuss their stability and
global well-posedness in time in Section 5.3.3.
5.3.1 Study of the limit system
In this brief section we shall discuss the existence of solutions to system (
NS).
Definition 5.1 We shall say that a vector field u in the space
L
∞
loc
(R
+
;
H(R
2
)) ∩ L
2
loc
(R
+
; H
1
(R
2
))
is a weak solution of (
NS) with initial data u
0
in
H(R
2
) if and only if u
h
is
a solution of the two-dimensional Navier–Stokes equations in the sense of
Definition 2.5, page 42, and if for any function Ψ in C
1
(R
+
; H
1
(R
2
)),
u
3
(t), Ψ(t) = u
3
(0), Ψ(0) +
t
0
R
2
ν∇
h
u
3
·∇
h
Ψ − u
3
· ∂
t
Ψ
(t
′
,x
h
) dx
h
dt
′
.
102 Dispersive cases
We state without proof the following theorem, which is a trivial adaptation of
Theorem 3.2, page 56.
Theorem 5.4 Let
u
0
be a vector field in
H(R
2
). Then there is a unique solu-
tion
u to (NS) in the sense of Definition 5.1. Moreover, this solution belongs
to C(R
+
;
H(R
2
)) and satisfies the energy equality
1
2
u(t)
2
L
2
+ ν
t
0
∇
h
u(t
′
)
2
L
2
dt
′
=
1
2
u
0
2
L
2
.
5.3.2 Existence and convergence of solutions to the rotating-fluid equations
The goal of this section is to study the well-posedness of the rotating-fluid equa-
tions in the functional framework presented above. We will then investigate their
asymptotics as the parameter ε goes to zero.
Due to the skew-symmetry of the rotation operator, studying (NSC
ε
) in such
a framework means in fact studying three-dimensional perturbations of the two-
dimensional Navier–Stokes equations. Let us first establish the equation on such
a perturbed flow. We consider
u
0
in
H(R
2
) and the associate solution u of (NS)
given by Theorem 5.4. Let us now define w
def
= u −
u, where u is assumed to solve
the three-dimensional Navier–Stokes equation. Then we can write formally
(PNS
ν
)
∂
t
w + w ·∇w +
u ·∇w + w ·∇u − ν∆w = −∇p
div w =0.
The definition of a solution is simply a duplication of Definition 2.5, page 42.
Definition 5.2 Let w
0
be a vector field in H(R
3
). We shall say that w is
a weak solution of (PNS
ν
) with initial data w
0
if and only if w belongs to the
space L
∞
loc
(R
+
; H) ∩ L
2
loc
(R
+
; V
σ
) and for any function Ψ in C
1
(R
+
; V
σ
),
R
3
w(t, x) · Ψ(t, x) dx −
t
0
R
3
(w · ∂
t
Ψ+ν∇w : ∇Ψ) (t
′
,x) dxdt
′
−
t
0
R
3
(w ⊗ (
u + w)+u ⊗ w):∇Ψ(t
′
,x) dxdt
′
=
R
3
w
0
(x) · Ψ(0,x) dx,
where
u is the unique solution of (NS) associated with u
0
, given by
Theorem 5.4.
We have the following result.
Theorem 5.5 Let
u
0
and w
0
be two vector fields, respectively in
H(R
2
) and
in H(R
3
).Let
u be the unique solution of (NS) associated with u
0
, given by
Application to rotating fluids in R
3
103
Theorem 5.4. Then, there exists a global weak solution w to the system (PNS
ν
) in
the sense of Definition 5.2. Moreover, this solution satisfies the energy inequality
R
3
|w(t, x)|
2
dx + ν
t
0
R
3
|∇w(t
′
,x)|
2
dxdt
′
≤w
0
2
L
2
exp
C
ν
2
u
0
2
L
2
.
Proof We shall not give all the details of the proof here, as it is very similar to
the proof of the Leray Theorem 2.3, page 42. The only difference is the presence
of the additional vector field
u, which could cause some trouble, were it not for
the fact that it only depends on two variables and not on three. So the key to
the proof of that theorem is to establish some estimates on the product of two
vector fields, one of which only depends on two variables.
Let us therefore consider the sequence (w
k
)
k∈N
defined by
˙w
k
(t)=νP
k
∆w
k
(t)+F
k
(w
k
(t)) + G
k
(w
k
(t),
u(t)), (5.3.2)
where we recall that F
k
(a)=P
k
Q(a, a), and where we have defined
G
k
(a, b)=P
k
G(a, b)
def
= P
k
(Q(a, b)+Q(b, a)) . (5.3.3)
The only step of the proof of Theorem 2.3 we shall retrace here is the proof of
the analog of the energy bound (2.2.4), page 46. An integration by parts yields
1
2
d
dt
w
k
(t)
2
L
2
+ ν∇w
k
(t)
2
L
2
= −
R
3
(
u(t, x
h
) ·∇w
k
(t, x)) · w
k
(t, x) dx
−
R
3
(w
h
k
(t, x) ·∇
h
u(t, x
h
)) · w
k
(t, x) dx
and again
1
2
d
dt
w
k
(t)
2
L
2
+ ν∇w
k
(t)
2
L
2
= −
R
3
(w
h
k
(t, x) ·∇
h
u(t, x
h
)) · w
k
(t, x) dx.
Let us define
I
k
(t)
def
=
R
3
(w
h
k
(t, x) ·∇
h
u(t, x
h
)) · w
k
(t, x) dx.
We can write
|I
k
(t)|≤
R
w
k
(t, ·,x
3
)
2
L
4
(R
2
)
dx
3
∇
h
u(t, ·)
L
2
(R
2
)
and the Gagliardo–Nirenberg inequality (see Corollary 1.2), page 25 yields
|I
k
(t)|≤C∇
h
u(t)
L
2
(R
2
)
R
w
k
(t, ·,x
3
)
L
2
(R
2
)
∇
h
w
k
(t, ·,x
3
)
L
2
(R
2
)
dx
3
.
It is then simply a matter of using the Cauchy–Schwarz inequality to find that
|I
k
(t)|≤
ν
2
∇w
k
(t)
2
L
2
(R
3
)
+
C
ν
w
k
(t)
2
L
2
(R
3
)
∇
h
u(t)
2
L
2
(R
2
)
.
104 Dispersive cases
Going back to the estimate on w
k
we find that
d
dt
w
k
(t)
2
L
2
+ ν∇w
k
(t)
2
L
2
≤
C
ν
w
k
(t)
2
L
2
(R
3
)
∇
h
u(t)
2
L
2
(R
2
)
which by Gronwall’s lemma yields
w
k
(t)
2
L
2
+ ν
t
0
∇w
k
(t
′
)
2
L
2
dt
′
≤w
k
(0)
2
L
2
exp
C
ν
t
0
∇
h
u(t
′
)
2
L
2
dt
′
.
The energy estimate on
u enables us to infer finally that
w
k
(t)
2
L
2
+ ν
t
0
∇w
k
(t
′
)
2
L
2
dt
′
≤w
k
(0)
2
L
2
exp
C
ν
2
u
0
2
L
2
.
The end of the proof of Theorem 5.5 is then identical to the case treated in
Section 2.2, page 42, so we will not give any more detail here.
We are now ready to study the asymptotics of Leray solutions of (NSC
ε
).
We start by defining the Coriolis version, denoted by (PNSC
ε
) of the perturbed
system (PNS
ν
):
∂
t
w
ε
+ w
ε
·∇w
ε
+
u ·∇w
ε
+ w
ε
·∇u − ν∆w
ε
+
e
3
∧ w
ε
ε
= −
∇p
ε
ε
div w
ε
=0.
Let us note that
u belongs to the kernel of the Coriolis operator because e
3
∧
u
is a gradient.
Definition 5.3 Let w
0
be a vector field in H(R
3
), and let ε>0 be given.
We shall say that w
ε
is a weak solution of (PNSC
ε
) with initial data w
0
if
and only if w
ε
belongs to the space L
∞
loc
(R
+
; H) ∩L
2
loc
(R
+
; V
σ
) and for any
function Ψ in C
1
(R
+
; V
σ
),
R
3
w
ε
(t, x) · Ψ(t, x) dx −
t
0
R
3
(w
ε
· ∂
t
Ψ+ν∇w
ε
: ∇Ψ) (t
′
,x) dxdt
′
+
t
0
R
3
e
3
∧ w
ε
ε
· Ψ − (w
ε
⊗ (
u + w
ε
)+u ⊗ w
ε
):∇Ψ
(t
′
,x) dxdt
′
=
R
3
w
0
(x) · Ψ(0,x) dx,
where
u is the unique solution of (NS) associated with u
0
, given by
Theorem 5.4.
Theorem 5.6 Let u
0
and w
0
be two vector fields, respectively in
H(R
2
) and
in H(R
3
).Let
u be the unique solution of (NS) associated with u
0
, given by
Application to rotating fluids in R
3
105
Theorem 5.4. Then, there exists a global weak solution w
ε
to the system (PNSC
ε
),
satisfying
R
3
|w
ε
(t, x)|
2
dx + ν
t
0
R
3
|∇w
ε
(t
′
,x)|
2
dxdt
′
≤w
0
2
L
2
exp
C
ν
2
u
0
2
L
2
.
Moreover, for any q ∈]2, 6[, for any time T , we have
lim
ε→0
T
0
w
ε
(t)
2
L
q
(R
3
)
dt =0.
This means that one can solve the system (NSC
ε
) with initial data
u
0
+ w
0
, and
any such solution converges towards the unique solution
u of (NS) associated
with
u
0
.
Proof of Theorem 5.6 We shall omit the proof of the existence of w
ε
sat-
isfying the energy estimate, as it is identical to the proof of Theorem 5.5 due
to the skew-symmetry of the Coriolis operator. So what we must concentrate on
now is the proof of the convergence of w
ε
to zero. It is here that the Strichartz
estimates proved in Section 5.2 will be used. In order to use those estimates, we
have to get rid of high frequencies and low vertical frequencies. Let us define the
following operator
P
R
f
def
= χ
|D|
R
f, where χ ∈D(] − 2, 2[),χ(x)=1for|x|≤1.
Let us observe that, thanks to Sobolev embeddings (see Theorem 1.2, page 23)
and the energy estimate, we have, for any q ∈ [2, 6[,
w
ε
−P
R
w
ε
L
2
(R
+
;L
q
(R
3
))
≤ Cw
ε
−P
R
w
ε
L
2
(R
+
;
˙
H
3
(
1
2
−
1
q
)
)
≤ CR
−α
q
w
ε
L
2
(R
+
;
˙
H
1
)
≤ CR
−α
q
w
0
L
2
exp
C
ν
2
u
0
2
L
2
(R
2
)
(5.3.4)
with α
q
def
=(3/q) − (1/2)· Now let us define
χ
D
3
r
a
def
= F
−1
χ
ξ
3
r
a(ξ)
.
As the support of χ(D
3
/r)P
R
w
ε
is included in B
r,R
def
= {ξ ∈ B(0,R) / |ξ
3
|≤2r},
we have, thanks to Lemma 1.1, page 24,
χ
D
3
r
P
R
w
ε
(t)
L
∞
≤
#
B
r,R
dξ
|ξ|
2
$
1
2
χ
D
3
r
P
R
w
ε
(t)
˙
H
1
≤ Cr
1
2
log(1 + R
2
)
1
2
w
ε
(t)
˙
H
1
.
106 Dispersive cases
Then, by the energy estimate, we have
χ
D
3
r
P
R
w
ε
L
2
(R
+
;L
∞
)
≤ C
R
r
1
2
w
0
L
2
exp
C
ν
2
u
0
2
L
2
(R
2
)
. (5.3.5)
Let us define
P
r,R
f
def
=
Id −χ
D
3
r
P
R
f. (5.3.6)
The following lemma, which we admit for the moment, describes the dispersive
effects due to fast rotation.
Lemma 5.3 For any positive real numbers r, R and T , and for any q
in ]2, +∞[,
∀ε>0 , P
r,R
w
ε
L
2
([0,T ];L
q
(R
3
))
≤ Cε
1
8
(
1−
2
q
)
,
the constant C above depending on r, q, R, T ,
u
0
L
2
and w
0
L
2
but not on ε.
Together with inequalities (5.3.4) and (5.3.5), this lemma implies that, for any
positive r, R and T , for q ∈]2, 6[,
∀ε>0 , w
ε
L
2
([0,T ];L
q
)
≤ CR
−α
q
+ C
R
r
1
2
+ C
3
ε
1
8
(
1−
2
q
)
,
the constant C
3
above depending on r, R, T ,
u
0
L
2
and w
0
L
2
but not on ε.
We deduce that, for any positive r, R and T , for q ∈]2, 6[,
lim sup
ε→0
w
ε
L
2
([0,T ];L
q
)
≤ CR
−α
q
+ C
R
r
1
2
.
Passing to the limit when r tends to 0 and then when R tends to ∞ gives
Theorem 5.6, provided of course we prove Lemma 5.3.
Proof of Lemma 5.3 Thanks to Duhamel’s formula we have,
P
r,R
w
ε
(t)=
3
j=1
P
j
r,R
w
ε
(t) with
P
1
r,R
w
ε
(t)
def
= G
ε
ν
t
ε
P
r,R
w
0
,
P
2
r,R
w
ε
(t)
def
=
t
0
G
ε
ν
t − t
′
ε
P
r,R
Q(w
ε
(t
′
),w
ε
(t
′
)) dt
′
and
P
3
r,R
w
ε
(t)
def
=
t
0
G
ε
ν
t − t
′
ε
P
r,R
(Q(w
ε
(t
′
),
u(t
′
)) + Q(u(t
′
),w
ε
(t
′
))) dt
′
.
Theorem 5.3 implies
P
1
r,R
w
ε
L
2
(R
+
;L
∞
)
≤ C
r,R
ε
1
8
w
0
L
2
.
Application to rotating fluids in R
3
107
By interpolation with the energy bound, we infer that
P
1
r,R
w
ε
L
2
([0,T ];L
q
)
≤ C
r,R,T
ε
1
8
(
1−
2
q
)
w
0
L
2
exp
C
ν
2
u
0
2
L
2
(R
2
)
. (5.3.7)
Using again Theorem 5.3, we have
P
2
r,R
w
ε
L
2
([0,T ];L
∞
)
≤ C
r,R
ε
1
8
P
R
Q(w
ε
,w
ε
)
L
1
([0,T ];L
2
)
.
Lemma 1.2, page 26, together with the energy estimate implies that
P
R
Q(w
ε
,w
ε
)
L
1
([0,T ];L
2
)
≤ CRP
R
(w
ε
⊗ w
ε
)
L
1
([0,T ];L
2
)
≤ CR
1+
3
2
w
ε
⊗ w
ε
L
1
([0,T ];L
1
)
≤ C
R
T w
0
2
L
2
(R
3
)
exp
C
ν
2
u
0
2
L
2
(R
2
)
.
Thus we have
P
2
r,R
w
ε
L
2
([0,T ];L
∞
)
≤ C
r,R
Tε
1
8
w
0
2
L
2
(R
3
)
exp
C
ν
2
u
0
2
L
2
(R
2
)
.
By interpolation with the energy bound, we infer
P
2
r,R
w
ε
L
2
([0,T ];L
q
)
≤ C
r,R,T
ε
1
8
(
1−
2
q
)
w
0
2(1−
1
q
)
L
2
exp
C
ν
2
u
0
2
L
2
(R
2
)
. (5.3.8)
Still using Theorem 5.3, we have
P
3
r,R
w
ε
L
2
([0,T ];L
∞
)
≤ C
r,R
ε
1
8
P
R
(Q(w
ε
,
u)+Q(u, w
ε
))
L
1
([0,T ];L
2
)
. (5.3.9)
Lemma 1.2, page 26, implies that
P
R
Q(
u, w
ε
)
L
1
([0,T ];L
2
)
≤ CRP
R
(u ⊗ w
ε
)
L
1
([0,T ];L
2
)
.
We shall prove the following lemma, which is an anisotropic version of Lemma 1.2,
page 26.
Lemma 5.4 For any function f ∈ L
1,2
x
h
,x
3
, we have
P
R
f
L
2
(R
3
)
≤ CRf
L
1,2
x
h
,x
3
.
Proof Let x
3
∈ R be given. Then Lemma 1.2 in two space dimensions
implies that
P
R
f(·,x
3
)
L
2
(R
2
)
≤ CRf (·,x
3
)
L
1
(R
2
)
.
Taking the L
2
norm in x
3
therefore yields
P
R
f
L
2
(R
3
)
≤ CR
R
f(·,x
3
)
2
L
1
(R
2
)
dx
3
1
2
,
108 Dispersive cases
and the result follows from the fact that
R
f(·,x
3
)
2
L
1
(R
2
)
dx
3
≤f
2
L
1,2
x
h
,x
3
.
This lemma, together with H¨older’s estimate and the now classical energy
bound, implies that
P
R
Q(
u, w
ε
)
L
1
([0,T ];L
2
(R
3
))
≤ CR
2
u ⊗ w
ε
L
1
([0,T ];L
1,2
x
h
,x
3
)
≤ CR
2
u
L
2
([0,T ];L
2
(R
2
))
w
ε
L
2
([0,T ];L
2
(R
3
))
≤ C
R,T
u
0
L
2
(R
2
)
w
0
L
2
(R
3
)
× exp
C
ν
2
u
0
2
L
2
(R
2
)
.
Clearly, the term P
R
Q(w
ε
,
u)
L
1
([0,T ];L
2
(R
3
))
can be estimated in the same way.
Thus, inequality (5.3.9) becomes
P
3
r,R
w
ε
L
2
([0,T ];L
∞
)
≤ C
r,R,T
ε
1
8
u
0
L
2
w
0
L
2
exp
C
ν
2
u
0
2
L
2
.
By interpolation with the energy bound, we get
P
3
r,R
w
ε
L
2
([0,T ];L
q
)
≤ C
r,R,T
ε
1
8
(
1−
2
q
)
u
0
1−
2
q
L
2
w
0
L
2
exp
C
ν
2
u
0
2
L
2
.
Combining inequalities (5.3.7)–(5.3.9) concludes the proof of the lemma and thus
of Theorem 5.6.
5.3.3 Global well-posedness
In the same way as in Part II, once solutions have been obtained it is natural
to address the question of their stability. The stability arguments of Section 3.5,
page 72, still hold in the setting of rotating fluids, due as usual to the skew-
symmetry of the rotation operator. What we are interested in, therefore, is
proving the existence, in the framework set up in the introduction of this section,
of a solution in L
4
([0,T]; V
σ
) with uniform bounds. Actually we will do better
than that since we will be able to prove a result global in time, with no small-
ness assumption on the initial data. That will be of course due to the presence
of strong enough rotation in the equation.
The theorem we shall prove is the following.
Theorem 5.7 Let
u
0
and w
0
be two divergence-free vector fields, respectively,
in the spaces
H and H
1
2
(R
3
). Then a positive ε
0
exists such that for all ε ≤ ε
0
,
there is a unique global solution u
ε
to the system (NSC
ε
). More precisely, denot-
ing by
u the (unique) solution of (NS) associated with u
0
,byv
ε
F
the solution