Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics
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Application to rotating fluids in R
3
109
of (VC
ε
) with initial data w
0
(with f =0), and defining w
ε
def
= u
ε
−
u, we have
w
ε
∈ C
0
b
(R
+
; H
1
2
(R
3
)) and ∇w
ε
∈ L
2
(R
+
; H
1
2
(R
3
)),
w
ε
− v
ε
F
→ 0 in L
∞
(R
+
;
˙
H
1
2
(R
3
))
and ∇(w
ε
− v
ε
F
) → 0 in L
2
(R
+
;
˙
H
1
2
(R
3
))
as ε goes to zero.
Proof Let us start by proving the uniqueness of such solutions. In order to do
so, it is enough to prove the uniqueness of w
ε
solving(PNSC
ε
). So let us consider
two vector fields w
ε
1
and w
ε
2
solving (PNSC
ε
), with the same initial data w
0
in H
1
2
and the same
u in L
∞
(R
+
;
H) ∩L
2
(R
+
;
˙
H
1
) solution of (NS). As in Section 3.5,
page 72, we are going to use results on the time-dependent Stokes system. If w
ε
denotes the difference w
ε
def
= w
ε
1
−w
ε
2
, then w
ε
solves the time-dependent Stokes
system with force
Q(w
ε
1
,w
ε
1
) − Q(w
ε
2
,w
ε
2
)+G(
u, w
ε
),
using notation (5.3.3). As seen in Chapter 3, page 53, for any vector field a
which belongs to L
4
([0,T]; V
σ
), Q(a, a) is in the space L
2
([0,T]; V
′
σ
), so the
term Q(w
ε
1
,w
ε
1
) − Q(w
ε
2
,w
ε
2
) is dealt with exactly as for the three-dimensional
stability result (Theorem 3.3, page 58– it is in fact even easier here since both w
ε
1
and w
ε
2
are supposed to be in L
4
([0,T]; V
σ
)). So the only term we must control
is G(
u, w
ε
). Let us prove that it is an element of L
2
([0,T]; V
′
σ
). In order to do
so, it is enough to prove that
u ⊗ w
ε
is in L
2
([0,T]; L
2
). But we have, by the
Gagliardo–Nirenberg inequality (1.3.4), page 25,
u ⊗ w
ε
2
L
2
(R
3
)
=
R
(
u ⊗ w
ε
)(·,x
3
)
2
L
2
(R
2
)
dx
3
≤
u
2
L
4
(R
2
)
R
w
ε
(·,x
3
)
2
L
4
(R
2
)
dx
3
≤ C
u
L
2
(R
2
)
∇u
L
2
(R
2
)
w
ε
L
2
(R
3
)
∇w
ε
L
2
(R
3
)
,
so the result follows; for G(
u, w
ε
)tobeinL
2
([0,T]; V
′
σ
) it is in fact enough to
suppose that w
ε
belongs to L
∞
([0,T]; H) ∩L
2
([0,T]; V
σ
), contrary to the purely
three-dimensional case where a L
4
([0,T]; V
σ
) bound is required.
Let us now prove the global existence of w
ε
as stated in the theorem. We
can immediately note that it is hopeless to try to prove the global existence
of w
ε
simply by considering
˙
H
1
2
estimates on (5.3.2): by skew-symmetry, the
rotation would immediately disappear and unless the initial data w
0
are small,
it is impossible to find an estimate global in time using the methods introduced
in Section 3.5. So the idea is to subtract from (5.3.2) the solution v
ε
F
of (VC
ε
),
which we know goes to zero (at least for low frequencies) by the Strichartz
estimates. We will then be led to solving a system of the type (5.3.2) on the
difference w
ε
k
− v
ε
F
, which will have small data and small source terms. The
110 Dispersive cases
methods of Section 3.5 should then enable us to find the expected result, up to the
additional difficulty consisting in coping with the interaction of two-dimensional
and three-dimensional vector fields.
Let us be more precise. We can introduce v
ε
, the solution of (VC
ε
) associated
with the initial data P
r,R
w
0
where P
r,R
is the cut-off operator defined in (5.3.6).
Notice that P
r,R
w
0
converges towards w
0
in H
1
2
as r goes to zero and R goes to
infinity. So the proof of the fact that w
ε
−v
ε
can be made arbitrarily small, in
the function spaces given in Theorem 5.7, implies that w
ε
−v
ε
F
can also be made
arbitrarily small in those same spaces; therefore, from now on we shall restrict
our attention to v
ε
, and we are ready to consider the following approximate
system:
˙
δ
ε
k
(t)=−
e
3
∧ P
k
δ
ε
k
(t)
ε
+ νP
k
∆δ
ε
k
(t)+F
k
(δ
ε
k
(t)) + F
k
(v
ε
(t))
+ G
k
(δ
ε
k
(t),
u(t)+v
ε
(t)) + G
k
(u(t),v
ε
(t)),
where F
k
and G
k
were defined in (5.3.3). The initial data for δ
ε
k
are
δ
ε
k|t=0
= P
k
(Id −P
r,R
)w
0
.
We omit in the notation of δ
ε
k
and v
ε
the dependence on r and R, although that
fact will of course be crucial in the estimates. We notice that δ
ε
k
remains bounded
in L
2
(R
3
) and is defined as a smooth solution for any time t. The point is now
to prove a global bound in
˙
H
1
2
(R
3
), as well as the fact that δ
ε
k
can be made
arbitrarily small. Indeed w
ε
k
def
= v
ε
+ δ
ε
k
solves
∂
t
w
ε
k
+ P
k
(w
ε
k
·∇w
ε
k
+
u ·∇w
ε
k
+ w
ε
k
·∇u) − νP
k
∆w
ε
k
+
P
k
(e
3
∧ w
ε
k
)
ε
=0,
with initial data P
k
P
r,R
w
0
.
The main step in the proof of the theorem reduces to the following
proposition.
Proposition 5.1 For any positive η, one can find three positive real num-
bers r
0
, R
0
and ε
0
such that for any k ∈ N and for any ε<ε
0
, we have
sup
t≥0
δ
ε
k
(t)
2
˙
H
1
2
(R
3
)
+
ν
2
R
+
∇δ
ε
k
(t
′
)
2
˙
H
1
2
(R
3
)
dt
′
≤ η.
We leave the reader to complete the proof of Theorem 5.7 using that proposition,
since it consists simply in copying the end of the proof of Theorem 3.5 page 73.
Proof of Proposition 5.1 Let us consider from now on three positive real
numbers η<1, r and R. We define the time T
k
(depending of course on those
three numbers), as the biggest time t for which
sup
t
′
≤t
δ
ε
k
(t
′
)
2
˙
H
1
2
+
ν
2
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
≤ η.
Application to rotating fluids in R
3
111
Thanks to the Lebesgue theorem, r and R can be chosen in order to have
δ
ε
k
(0)
2
˙
H
1
2
≤(Id −P
r,R
)w
0
2
˙
H
1
2
≤ η
2
, (5.3.10)
which implies in particular that T
k
is positive for each k, since η
2
<η.
Let us now write an energy estimate in
˙
H
1
2
(R
3
)onδ
ε
k
.Wehave
1
2
δ
ε
k
(t)
2
˙
H
1
2
+ ν
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
≤
1
2
δ
ε
k
(0)
2
˙
H
1
2
+
t
0
(F
k
(δ
ε
k
(t
′
))|δ
ε
k
(t
′
))
˙
H
1
2
dt
′
+
t
0
(F
ε
k
(t
′
)|δ
ε
k
(t
′
))
˙
H
1
2
dt
′
where
F
ε
k
def
= G
k
(δ
ε
k
,
u + v
ε
)+G
k
(
u, v
ε
)+F
k
(v
ε
).
By (3.5.2) we have
F
k
(δ
ε
k
)
˙
H
−
1
2
≤ C∇δ
ε
k
2
L
2
≤ Cδ
ε
k
˙
H
1
2
∇δ
ε
k
˙
H
1
2
,
so we infer that
1
2
δ
ε
k
(t)
2
˙
H
1
2
+ ν
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
≤
1
2
δ
ε
k
(0)
2
˙
H
1
2
+ Cδ
ε
k
(t)
˙
H
1
2
∇δ
ε
k
(t)
2
˙
H
1
2
+
t
0
(F
ε
k
(t
′
)|δ
ε
k
(t
′
))
˙
H
1
2
dt
′
.
Now let us suppose that η ≤ (1/2C)
,
where C is the constant appearing in the
estimate above. Then by (5.3.10) we have, for all times t ≤ T
k
,
δ
ε
k
(t)
2
˙
H
1
2
+ ν
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
≤ η
2
+
t
0
(F
ε
k
(t
′
)|δ
ε
k
(t
′
))
˙
H
1
2
dt
′
. (5.3.11)
We have the following estimate for F
ε
k
; we postpone the proof to the end of this
section.
Proposition 5.2 With the previous choices of η, r
0
and R
0
, there is a ε
0
> 0
and a family of functions (f
ε
)
ε≤ε
0
, uniformly bounded in L
1
(R
+
) by a constant
depending on
u
0
L
2
and on w
0
˙
H
1
2
, such that, for all t ≤ T
k
, we can write,
for any ε ≤ ε
0
,
t
0
(F
ε
k
|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
2
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
t
0
f
ε
(t
′
)δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+ η
2
·
Let us end the proof of Proposition 5.1. Applying Gronwall’s lemma to the
estimate (5.3.11) and using Proposition 5.2, yields
δ
ε
k
(t)
2
˙
H
1
2
(R
3
)
+
ν
2
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
(R
3
)
dt
′
≤ 2η
2
exp f
ε
L
1
(R
+
)
.
112 Dispersive cases
It is therefore now a matter of choosing η small enough so that
2η
2
exp
sup
ε≤ε
0
f
ε
L
1
(R
+
)
≤
η
2
,
to infer that T
k
=+∞; Proposition 5.1 is proved.
Proof of Proposition 5.2 Recalling that
F
ε
k
def
= G
k
(δ
ε
k
,v
ε
)+G
k
(δ
ε
k
,
u)+G
k
(v
ε
, u)+F
k
(v
ε
),
we have four terms to estimate. First of all we can write, again using (3.5.2),
(G
k
(δ
ε
k
,v
ε
)|δ
ε
k
)
˙
H
1
2
≤ C∇v
ε
L
2
∇δ
ε
k
L
2
∇δ
ε
k
˙
H
1
2
so
t
0
(G
k
(δ
ε
k
,v
ε
)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
3
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇v
ε
(t
′
)
4
L
2
dt
′
. (5.3.12)
Then to conclude we just need to notice that
R
+
∇v
ε
4
L
2
dt
′
≤
C
ν
w
0
4
˙
H
1
2
. (5.3.13)
The second term is more delicate: it is here that we have to deal with interactions
between two-dimensional and three-dimensional vector fields. This is done by the
following lemma.
Lemma 5.5 A constant C exists such that for all vector fields a and b
|(a|b)
˙
H
1
2
|≤Ca
L
2
(R
x
3
;L
4
3
(R
2
))
∇b
L
2
(R
x
3
;
˙
H
1
2
(R
2
))
.
Proof By definition of the scalar product on
˙
H
1
2
,wehave
(a|b)
˙
H
1
2
=(2π)
−d
R
3
|ξ|a(ξ)
ˇ
b(ξ) dξ
=(2π)
−d
R
3
|ξ
h
|
−
1
2
a(ξ)|ξ
h
|
1
2
|ξ|
ˇ
b(ξ) dξ.
We have denoted
ˇ
b = F(b(−·)). By the Cauchy–Schwarz inequality, we get
|(a|b)
˙
H
1
2
|≤(2π)
−d
R
3
|ξ
h
|
−1
|a(ξ)|
2
dξ
1
2
R
3
|ξ
h
||F(∇b)(ξ)|
2
dξ
1
2
.
Using the Fourier–Plancherel theorem in R
x
3
we infer
(a|b)
˙
H
1
2
≤ (2π)
−d
R
a(·,x
3
)
2
˙
H
−
1
2
(R
2
)
dx
3
1
2
R
∇b(·,x
3
)
2
˙
H
1
2
(R
2
)
dx
3
1
2
.
Application to rotating fluids in R
3
113
Thanks to the dual Sobolev estimate stated in Corollary 1.1, page 25, we have
∀x
3
∈ R , a(·,x
3
)
2
˙
H
−
1
2
(R
2
)
≤ Ca(·,x
3
)
2
L
4
3
(R
2
)
.
Lemma 5.5 is proved.
As for any function a in
˙
H
1
2
(R
3
),
a
L
2
(R
x
3
;
˙
H
1
2
(R
2
))
≤ Ca
˙
H
1
2
(R
3
)
,
Lemma 5.5 enables to find
t
0
(G
k
(δ
ε
k
,
u)|δ
ε
k
)
˙
H
1
2
dt
′
≤
t
0
δ
ε
k
(t
′
) ·∇
u(t
′
)
L
2
(R
x
3
;L
4
3
(R
2
))
+
u(t
′
) ·∇δ
ε
k
(t
′
)
L
2
(R
x
3
;L
4
3
(R
2
))
∇δ
ε
k
(t
′
)
˙
H
1
2
dt
′
. (5.3.14)
On the one hand, we have by H¨older’s inequality
u ·∇δ
ε
k
L
2
(R
x
3
;L
4
3
(R
2
))
≤
u
L
4
(R
2
)
∇δ
ε
k
L
2
(R
3
)
,
hence using the Gagliardo–Nirenberg inequality (1.3.4), page 25,
t
0
u(t
′
) ·∇δ
ε
k
(t
′
)
L
2
(R
x
3
;L
4
3
(R
2
))
∇δ
ε
k
(t
′
)
˙
H
1
2
dt
′
≤
ν
16
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
3
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
u(t
′
)
4
L
4
(R
2
)
dt
′
. (5.3.15)
On the other hand, we have
δ
ε
k
·∇
u
L
2
(R
x
3
;L
4
3
(R
2
))
≤∇
u
L
2
(R
2
)
δ
ε
k
˙
H
1
2
(R
3
)
.
Thus we infer
t
0
δ
ε
k
(t
′
) ·∇
u(t
′
)
L
2
(R
x
3
;L
4
3
(R
2
))
∇δ
ε
k
(t
′
)
˙
H
1
2
dt
′
≤
ν
16
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇
u(t
′
)
2
L
2
(R
2
)
dt
′
. (5.3.16)
Plugging (5.3.15) and (5.3.16) into (5.3.14) yields finally
t
0
(G
k
(δ
ε
k
,
u)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
C
ν
2
u(t
′
)
4
˙
H
1
2
+ ∇
u(t
′
)
2
L
2
dt
′
,
and the result follows from the fact that
R
+
u(t)
4
˙
H
1
2
+ ∇
u(t)
2
L
2
dt ≤
C
ν
u
0
2
L
2
1
ν
3
u
0
2
L
2
+1
. (5.3.17)
114 Dispersive cases
Next let us estimate the term containing G
k
(v
ε
,
u). We have
(G
k
(v
ε
,
u)|δ
ε
k
)
˙
H
1
2
≤v
ε
·∇
u + u ·∇v
ε
L
2
∇δ
ε
k
L
2
≤
v
ε
L
∞,2
x
h
,x
3
∇
u
L
2
(R
2
)
+ Rv
ε
L
∞,2
x
h
,x
3
u
L
2
(R
2
)
∇δ
ε
k
L
2
,
so
(G
k
(v
ε
,
u)|δ
ε
k
)
˙
H
1
2
≤∇δ
ε
k
2
L
2
∇
u
L
2
+ v
ε
L
∞,2
x
h
,x
3
+ Cv
ε
2
L
∞,2
x
h
,x
3
∇
u
L
2
+ C
R
v
ε
L
∞,2
x
h
,x
3
u
2
L
2
.
Finally
t
0
(G
k
(v
ε
,
u)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇
u(t
′
)
2
L
2
+ v
ε
(t
′
)
2
L
∞,2
x
h
,x
3
dt
′
+
t
0
v
ε
(t
′
)
L
∞,2
x
h
,x
3
C
R
u(t
′
)
2
L
2
+ Cv
ε
(t
′
)
L
∞,2
x
h
,x
3
∇u(t
′
)
L
2
dt
′
.
But using the a priori bounds on
u along with the Strichartz estimates on v
ε
we have
t
0
v
ε
(t
′
)
L
∞,2
x
h
,x
3
C
R
u(t
′
)
2
L
2
+ Cv
ε
(t
′
)
L
∞,2
x
h
,x
3
∇
u(t
′
)
L
2
dt
′
≤ C
r,R
w
0
˙
H
1
2
u
0
L
2
ε
1
4
u
0
L
2
+ ε
1
8
w
0
˙
H
1
2
.
So finally, we get
t
0
(G
k
(v
ε
,
u)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇
u(t
′
)
2
L
2
+ v
ε
(t
′
)
2
L
∞,2
x
h
,x
3
dt
′
+ C
r,R
ε
1
8
w
0
˙
H
1
2
u
0
L
2
ε
1
8
u
0
L
2
+1
. (5.3.18)
Let us notice that
R
+
∇
u(t
′
)
2
L
2
+ v
ε
(t
′
)
2
L
∞,2
x
h
,x
3
dt
′
≤
C
ν
u
0
2
L
2
+ C
r,R
ε
1
4
w
0
2
˙
H
1
2
, (5.3.19)
and that for ε small enough,
C
r,R
ε
1
8
w
0
˙
H
1
2
u
0
L
2
ε
1
8
u
0
L
2
+1
≤
η
2
2
·
The third term is therefore correctly estimated.
Application to rotating fluids in R
3
115
The last term is very easy to estimate: we simply write
(F
k
(v
ε
)|δ
ε
k
)
˙
H
1
2
≤ Cv
ε
L
∞
∇v
ε
L
2
∇δ
ε
k
L
2
to find finally
t
0
(F
k
(v
ε
)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇v
ε
(t
′
)
2
˙
H
1
2
dt
′
+ C
t
0
v
ε
(t
′
)
2
L
∞
v
ε
(t
′
)
˙
H
1
2
dt
′
.
Since
t
0
v
ε
(t
′
)
2
L
∞
v
ε
(t
′
)
˙
H
1
2
dt
′
≤
C
r,R
ν
ε
1
4
w
0
2
˙
H
1
2
we find
t
0
(F
k
(v
ε
)|δ
ε
k
)
˙
H
1
2
dt
′
≤
ν
8
t
0
∇δ
ε
k
(t
′
)
2
˙
H
1
2
dt
′
+
C
ν
t
0
δ
ε
k
(t
′
)
2
˙
H
1
2
∇v
ε
(t
′
)
2
˙
H
1
2
dt
′
+
C
r,R
ν
ε
1
4
w
0
2
˙
H
1
2
. (5.3.20)
Finally we use the fact that
R
+
∇v
ε
(t
′
)
2
˙
H
1
2
dt
′
≤
C
ν
w
0
2
˙
H
1
2
, (5.3.21)
and that for ε small enough,
C
r,R
ν
ε
1
4
w
0
2
˙
H
1
2
≤
η
2
2
·
Putting together estimates (5.3.12), (5.3.17), (5.3.18) and (5.3.20) yields the
result. We notice in particular that
f
ε
(t)=C
ν
∇v
ε
(t)
4
L
2
+ v
ε
(t)
2
L
∞,2
x
h
,x
3
+
u(t)
4
˙
H
1
2
(R
2
)
+ ∇
u(t)
2
L
2
(R
2
)
so as computed in (5.3.13), (5.3.17), (5.3.19), (5.3.21) we have
t
0
|f
ε
(t
′
)| dt
′
≤ C
ν
w
0
2
˙
H
1
2
+ w
0
4
˙
H
1
2
+
u
0
2
L
2
+
u
0
4
L
2
+ C
r,R
ε
1
4
w
0
2
˙
H
1
2
.
Proposition 5.2 is proved.
6
The periodic case
6.1 Setting of the problem, and statement of the main result
This chapter deals with the rotating-fluid equations (NSC) in a purely periodic
setting. Let a
1
, a
2
and a
3
be three positive real numbers and define the
periodic box
T
3
def
=
3
)
j=1
R
/a
j
Z
unlike the previous chapter where T
3
denoted the unit periodic box in R
3
.In
this chapter the size of the box will have some importance in the analysis.
The system we shall study is the following.
(NSC
ε
T
)
∂
t
u
ε
− ν∆u
ε
+ P(u
ε
·∇u
ε
)+
1
ε
P(e
3
∧ u
ε
)=0in T
3
u
ε
|t=0
= u
ε
0
with div u
ε
0
=0,
where P is the Leray projector onto the space of divergence-free vector fields.
Let us recall its definition in Fourier variables. As we are in a periodic setting,
the Fourier variables are discrete variables n =(n
1
,n
2
,n
3
) ∈ Z
3
and we will
denote throughout this chapter
n =(n
1
, n
2
, n
3
) with n
j
def
=
n
j
a
j
,
j ∈{1, 2, 3}.
Then the Fourier transform of any function h is
∀ n ∈ Z
3
,
h(n)=Fh(n)
def
=
T
3
e
−2iπn·x
h(x) dx, (6.1.1)
and n ·x is the scalar product of (n
1
, n
2
, n
3
)by(x
1
,x
2
,x
3
). The expression of P
in Fourier variables is the following:
P(n)=Id−
1
|n|
2
n
2
1
n
1
n
2
n
1
n
3
n
2
n
1
n
2
2
n
2
n
3
n
3
n
1
n
3
n
2
n
2
3
,
where Id denotes the identity matrix in Fourier space.
118 The periodic case
Let us also recall the definition of Sobolev spaces on T
3
. We shall say that a
function h is in H
s
(T
3
)if
h
H
s
(T
3
)
def
=
n∈Z
3
(1 + |n|
2
)
s
|
h(n)|
2
1
2
< +∞.
Similarly h is in the homogeneous Sobolev space
˙
H
s
(T
3
)if
h
˙
H
s
(T
3
)
def
=
n∈Z
3
|n|
2s
|
h(n)|
2
1
2
< +∞.
A vector field is in H
s
(T
3
) (respectively in
˙
H
s
(T
3
)) if each of its components is.
All the vector fields considered in this chapter will be supposed to be mean-free,
and one can notice that for such vector fields, homogeneous and inhomogeneous
spaces coincide on T
3
.
Our goal is to study the behavior of the solutions of (NSC
ε
T
) as the Rossby
number ε goes to zero. As in the R
3
case studied in the previous chapter, we will
be concerned with the existence and the convergence of both weak and strong
solutions.
Let us start by discussing the case of weak solutions. We have the following
definition.
Definition 6.1 We shall say that u is a weak solution of (NSC
ε
T
) with
initial data u
0
in H(T
3
) if and only if u belongs to the space
C(R
+
; V
′
) ∩ L
∞
loc
(R
+
; H) ∩ L
2
loc
(R
+
; V),
and for any function Ψ in C
1
(R
+
; V
σ
),
T
3
u(t, x) · Ψ(t, x) dx −
t
0
T
3
u · ∂
t
Ψ+
t
0
T
3
ν∇u : ∇Ψ
(t
′
,x) dxdt
′
+
t
0
T
3
e
3
∧ u
ε
· Ψ − u ⊗ u : ∇Ψ
(t
′
,x) dxdt
′
=
T
3
u
0
(x) · Ψ(0,x) dx.
Let us recall the definition of the Coriolis operator L:
Lw
def
= P(e
3
∧ w).
As L is skew-symmetric and commutes with derivatives, (NSC
ε
T
) satisfies the
same energy estimates as the three-dimensional Navier–Stokes equations in