Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics
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Stability in three dimensions 59
Remarks
• As in the two-dimensional case, if the domain satisfies the Poincar´e
inequality, then the estimate becomes
(u − v)(t)
2
L
2
+ ν
t
0
∇(u − v)(t
′
)
2
L
2
dt
′
≤
u
0
− v
0
2
L
2
+
2
ν
t
0
(f − g)(t
′
)
2
V
′
σ
dt
′
exp
C
ν
3
t
0
∇u(t
′
)
4
L
2
dt
′
.
• The proof that such an L
4
([0,T]; V
σ
) solution u exists will be detailed in
Section 3.4 in the case of bounded domains, and in Section 3.5 in the case
without boundary.
Proof of Theorem 3.3 Thanks to Lemma 2.3, the fact that u belongs
to L
4
([0,T]; V
σ
) implies that
Q(u, u)
L
2
([0,T ];V
′
)
≤ Cu
1
2
L
∞
([0,T ];L
2
)
u
3
2
L
3
([0,T ];H
1
0
)
≤ CT
1
8
u
1
2
L
∞
([0,T ];L
2
)
u
3
2
L
4
([0,T ];H
1
0
)
. (3.3.2)
Hence the non-linear term Q(u, u) belongs to L
2
([0,T]; V
′
). Thus, exactly as in
the two-dimensional case, u is the solution of (ES
ν
) with initial data u
0
and
external force f + Q(u, u). Theorem 3.1 immediately implies that u belongs
to C([0,T]; H) and satisfies, for any (s, t) such that 0 ≤ s ≤ t,
1
2
u(t)
2
L
2
+ ν
t
s
∇u(t
′
)
2
L
2
dt
′
=
1
2
u(s)
2
L
2
+
t
s
f(t
′
),u(t
′
)dt
′
+
t
s
Q(u(t
′
),u(t
′
)),u(t
′
)dt
′
.
Using Lemma 2.3, we get the energy equality (3.3.1).
The method now used in the proof of the stability is important because we
shall follow its lines quite often in this book. As u and v are two Leray solutions,
we can write that
δ
ν
(t)
def
= (u − v)(t)
2
L
2
+2ν
t
0
∇(u − v)(t
′
)
2
L
2
dt
′
= u(t)
2
L
2
+2ν
t
0
∇u(t
′
)
2
L
2
dt
′
+ v(t)
2
L
2
+2ν
t
0
∇v(t
′
)
2
L
2
dt
′
− 2(u(t)|v(t))
L
2
− 4ν
t
0
(∇u(t
′
)|∇v(t
′
))
L
2
dt
′
≤u
0
2
L
2
+2
t
0
f(t
′
),u(t
′
)dt
′
+ v
0
2
L
2
+2
t
0
g(t
′
),v(t
′
)dt
′
− 2(u(t)|v(t))
L
2
− 4ν
t
0
(∇u(t
′
)|∇v(t
′
))
L
2
dt
′
. (3.3.3)
60 Stability of Navier–Stokes equations
Now the problem consists in evaluating the cross-product terms
(u(t)|v(t))
L
2
+2ν
t
0
(∇u(t
′
)|∇v(t
′
))
L
2
dt
′
−
t
0
g(t
′
),v(t
′
)dt
′
−
t
0
f(t
′
),u(t
′
)dt
′
.
Let us suppose for a moment that u and v are smooth in space and time; then
we can simply scalar-multiply by v the equation satisfied by u, and conversely
scalar-multiply by u the equation satisfied by v.Weget
(∂
t
u|v)
L
2
+ ν(∇u|∇v)
L
2
+(u ·∇u|v)
L
2
=(f|v)
L
2
and
(∂
t
v|u)
L
2
+ ν(∇v|∇u)
L
2
+(v ·∇v|u)
L
2
=(g|u)
L
2
;
hence summing both equalities yields
∂
t
(u(t)|v(t))
L
2
+2ν(∇u(t)|∇v(t))
L
2
− (f(t)|v(t))
L
2
− (g(t)|u(t))
L
2
+(u(t) ·∇u(t)|v(t))
L
2
+(v(t) ·∇v(t)|u(t))
L
2
=0.
After an easy algebraic computation we find that
(u(t) ·∇u(t)|v(t))
L
2
+(v(t) ·∇v(t)|u(t))
L
2
=((u − v)(t) ·∇(u − v)(t)|u(t))
L
2
,
hence after integration in time, we obtain
(u(t)|v(t))
L
2
+2ν
t
0
(∇u(t
′
)|∇v(t
′
))
2
L
2
dt
′
−
t
0
(g(t
′
)|v(t
′
))
L
2
dt
′
−
t
0
(f(t
′
)|u(t
′
))
L
2
dt
′
= −(u
0
|v
0
)
L
2
+
t
0
((u − v)(t
′
) ·∇(u − v)(t
′
)|u(t
′
))
L
2
dt
′
+
t
0
(f(t
′
)|(v − u)(t
′
)) dt
′
+
t
0
(g(t
′
)|(u − v)(t
′
)) dt
′
. (3.3.4)
Plugging (3.3.4) into (3.3.3) yields
(u − v)(t)
2
L
2
+2ν
t
0
∇(u − v)(t
′
)
2
L
2
dt
′
≤u
0
− v
0
2
L
2
+2
t
0
((f − g)|(u − v))(t
′
) dt
′
+2
t
0
((u − v) ·∇(u − v)|u)
L
2
(t
′
) dt
′
,
and the Gronwall lemma gives the smallness of δ
ν
. However unfortunately the
above computations make no sense if no smoothness in space and time is known
Stability in three dimensions 61
on u or on v (and in particular the final Gronwall argument does not seem pos-
sible to write correctly). So some precautions have to be taken in order to make
these computations valid, and to conclude the argument by the Gronwall lemma –
in particular we are going to see that the assumption that u ∈ L
4
([0,T]; V
σ
)is
enough to make the above computations valid.
Let us therefore proceed with the rigorous computations. Identity (3.3.3)
involves scalar products of u and v, which naturally lead to using the definition
of weak solutions choosing for instance u as a test function. Unfortunately, in
order to be admissible, test functions need to belong to C
1
(R
+
; V
σ
), so that
some preliminary smoothing in time of u is required. We shall use the following
approximation lemma and postpone its proof to the end of the proof of the
theorem.
Lemma 3.2 Let u be a Leray solution which belongs to L
4
([0,T]; V
σ
).A
sequence (u
k
)
k∈N
of C
1
(R
+
; V
σ
) exists such that
• the sequence (u
k
)
k∈N
tends to u in L
4
([0,T]; V
σ
) ∩ L
∞
([0,T]; H);
• for all k ∈ N, we have
∂
t
u
k
− ν∆u
k
= Q(u
k
, u
k
)+f + R
k
+ ∇p
k
(3.3.5)
with lim
k→∞
R
k
L
2
([0,T ];V
′
σ
)
=0.
The function u
k
belongs to C
1
(R
+
; V
σ
), so it can be used as a test function in
Definition 2.5. As v is a Leray solution, we have
B
k
(t)
def
=(v(t)|u
k
(t))
L
2
=(v(0)|u
k
(0))
L
2
− ν
t
0
(∇v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
g(t
′
), u
k
(t
′
)dt
′
+
t
0
(v(t
′
) ⊗ v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
v(t
′
),∂
t
u
k
(t
′
)dt
′
.
Thanks to (3.3.5), we get
B
k
(t)=(v(0)|u
k
(0))
L
2
− 2ν
t
0
(∇v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
g(t
′
), u
k
(t
′
)dt
′
+
t
0
f(t
′
),v(t
′
)dt
′
+
t
0
(v(t
′
) ⊗ v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
Q(u
k
(t
′
), u
k
(t
′
)),v(t
′
)dt
′
+
t
0
v(t
′
),R
k
(t
′
)dt
′
.
62 Stability of Navier–Stokes equations
Lemma 3.2 implies that lim
k→∞
B
k
(t)=(v(t)|u(t))
L
2
and that
lim
k→∞
(v(0)|u
k
(0))
L
2
− 2ν
t
0
(∇v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
v(t
′
),R
k
(t
′
)dt
′
+
t
0
g(t
′
), u
k
(t
′
)dt
′
is equal to
(v(0)|u(0))
L
2
− 2ν
t
0
(∇v(t
′
)|∇u(t
′
))
L
2
dt
′
+
t
0
g(t
′
),u(t
′
)dt
′
.
Thus stating
N
k
(t)
def
=
t
0
(v(t
′
) ⊗ v(t
′
)|∇u
k
(t
′
))
L
2
dt
′
+
t
0
Q(u
k
(t
′
), u
k
(t
′
)),v(t
′
)dt
′
,
we obtain
(v(t)|u(t))
L
2
=(v(0)|u(0))
L
2
− 2ν
t
0
(∇v(t
′
)|∇u(t
′
))
L
2
dt
′
+
t
0
g(t
′
),u(t
′
)dt
′
+
t
0
f(t
′
),v(t
′
)dt
′
+ lim
k→∞
N
k
(t).
Plugging this into (3.3.3) gives
δ
ν
(t)=u
0
− v
0
2
L
2
+2
t
0
(f − g)(t
′
), (u − v)(t
′
)dt
′
+ lim
k→∞
N
k
(t).
It remains to study the term N
k
(t). In order to do this, let us observe that,
for any vector field a and b in V
σ
,wehave(b ⊗ b|∇a)
L
2
= Q(b, b),a and thus
(b ⊗ b|∇a)
L
2
+ Q(a, a),b = Q(b, b),a + Q(a, a),b.
Using Lemma 2.3, we can write
Q(b, b),a + Q(a, a),b = Q(b, b),a−b + Q(a, a),b− a
= Q(b, a),a−b + Q(a, a),b− a.
Thus, it turns out that
Q(b, b),a + Q(a, a),b = Q(a − b, a),b−a
=((a − b) ⊗ a|∇(b − a))
L
2
.
Using the Gagliardo–Nirenberg inequality (see Corollary 1.2), we get for any a
and c in V
σ
,
|(c ⊗ a|∇c)
L
2
|≤Ca
L
6
c
L
3
∇c
L
2
≤ C∇a
L
2
c
1
2
L
2
∇c
3
2
L
2
. (3.3.6)
Stability in three dimensions 63
For almost every time t, the vector field v(t) belongs to V
σ
. It follows that for
all k ∈ N and t ≥ 0, taking a = u
k
(t
′
) and b = v(t
′
), t
′
∈ [0,t], we have
N
k
(t) ≤ C
t
0
∇u
k
(t
′
)
L
2
(u
k
− v)(t
′
)
1
2
L
2
∇(u
k
− v)(t
′
)
3
2
L
2
dt
′
.
Using Lemma 3.2, we know that
(∇u
k
(·)
L
2
)
k∈N
tends to ∇u(·)
L
2
in L
4
([0,T])
((u
k
− v)(t
′
)
L
2
)
k∈N
tends to (u − v)(·)
L
2
in L
∞
([0,T])
(∇(u
k
− v)(·)
L
2
)
k∈N
tends to ∇(u − v)(·)
L
2
in L
2
([0,T]).
Therefore, we have
lim
k→∞
N
k
(t) ≤
t
0
∇u(t
′
)
L
2
(u − v)(t
′
)
1
2
L
2
∇(u − v)(t
′
)
3
2
L
2
dt
′
.
We conclude that
δ
ν
(t) ≤u
0
− v
0
2
L
2
+2
t
0
(f − g)(t
′
)
V
′
σ
(u − v)(t
′
)
V
dt
′
+ C
t
0
∇u(t
′
)
L
2
∇(u − v)(t
′
)
3
2
L
2
(u − v)(t
′
)
1
2
L
2
dt
′
.
Using the convexity inequality (1.3.5) with θ =1/4 and θ =1/2, we obtain
(u − v)(t)
2
L
2
+ ν
t
0
∇(u − v)(t
′
)
2
L
2
dt
′
≤u
0
− v
0
2
L
2
+
2
ν
t
0
(f − g)(t
′
)
2
V
′
σ
dt
′
+
t
0
C
ν
3
∇u(t
′
)
4
L
2
+ ν
(u − v)(t
′
)
2
L
2
dt
′
.
Gronwall’s lemma allows us to conclude the proof of Theorem 3.3 provided we
prove Lemma 3.2.
Proof of Lemma 3.2 Thanks to inequality (3.3.2), Q(u, u) belongs
to L
2
([0,T]; V
′
). This implies that if in addition u is a Leray solution, then ∂
t
u
also belongs to L
2
([0,T]; V
′
). Lebesgue’s theorem together with Proposition 2.3,
page 40, yields
lim
k→∞
P
k
u − u
L
4
([0,T ];V
σ
)
= lim
k→∞
P
k
∂
t
u − ∂
t
u
L
2
([0,T ];V
′
σ
)
=0.
Then, as in Lemma 2.2, page 44, we can define by a standard regularization
procedure in time, a sequence (u
k
)
k∈N
in C
1
(R
+
, V
σ
) such that u
k
tends to u
64 Stability of Navier–Stokes equations
in L
4
([0,T]; V
σ
) ∩L
∞
([0,T], H). Moreover ∂
t
u
k
tends to ∂
t
u in L
2
([0,T]; V
′
) and
inequality (3.3.2) implies that
lim
k→∞
Q(u
k
, u
k
) − Q(u, u)
L
2
([0,T ];V
′
σ
)
=0.
Lemma 3.2 and thus Theorem 3.3 are now proved.
3.4 Stable solutions in a bounded domain
The purpose of this section is the proof of the existence of solutions of the
system (NS
ν
) which are L
4
in time with values in V
σ
in the case when the
domain Ω is bounded. In order to state (and prove) a sharp theorem, we shall
introduce intermediate spaces between the spaces V
′
σ
and V
σ
. Then, we shall
prove a global existence theorem for small data and then a theorem local in time
for large data.
3.4.1 Intermediate spaces
We shall define a family of intermediate spaces between the spaces V
′
σ
and V
σ
.
This can be done by abstract interpolation theory but we prefer to do it here in
an explicit way.
Definition 3.2 Let s be in [−1, 1]. We shall denote by V
s
σ
the space of
vector fields u in V
′
such that
u
2
V
s
σ
def
=
j∈N
µ
2s
j
u, e
j
2
< +∞.
Here (e
j
)
j∈N
denotes the Hilbert basis on H given by Theorem 2.1, page 34.
Theorem 2.1 implies that V
0
σ
= H and V
1
σ
= V
σ
. Moreover, it is obvious that,
when s is non-negative, V
s
σ
endowed with the norm ·
V
s
σ
is a Hilbert space.
The following proposition will be important in the following two paragraphs.
Proposition 3.1 The space V
1
2
σ
is embedded in L
3
and the space L
3
2
is
embedded in V
−
1
2
σ
.
Proof This proposition can be proved using abstract interpolation theory.
We prefer to present here a self-contained proof in the spirit of the proof of
Theorem 1.2. Let us consider a in V
1
2
σ
. Without loss of generality, we can assume
that a
V
1
2
σ
≤ 1. Let us define, for a positive real number Λ,
a
Λ
def
=
j/µ
j
<Λ
a, e
j
e
j
and b
Λ
def
= a − a
Λ
.
Stable solutions in a bounded domain 65
Using the fact that
{x ∈ Ω / |a(x)| > Λ}⊂{x ∈ Ω / |a
Λ
(x)| > Λ/2}∪{x ∈ Ω / |b
Λ
(x)| > Λ/2}
,
we can write
a
3
L
3
≤ 3
+∞
0
Λ
2
meas ({x ∈ Ω / |a
Λ
(x)| > Λ/2}) dΛ
+3
+∞
0
Λ
2
meas ({x ∈ Ω / |b
Λ
(x)| > Λ/2}) dΛ
≤ 3 × 2
6
+∞
0
Λ
−4
a
Λ
6
L
6
dΛ+3× 2
2
+∞
0
b
Λ
2
L
2
dΛ.
Thanks to Theorem 1.2, we have, by definition of the ·
V
s
σ
norm,
a
Λ
2
L
6
≤ Ca
Λ
2
V
σ
≤ C
j/µ
j
<Λ
µ
2
j
a, e
j
2
≤ CΛ
j/µ
j
<Λ
µ
j
a, e
j
2
≤ CΛ.
Thus we have
a
3
L
3
≤ C
+∞
0
Λ
−2
a
Λ
2
V
σ
dΛ+C
+∞
0
b
Λ
2
L
2
dΛ
≤ C
j∈N
+∞
µ
j
Λ
−2
µ
2
j
a, e
j
2
dΛ+C
j∈N
µ
j
0
a, e
j
2
dΛ
≤ C
j∈N
µ
j
a, e
j
2
≤ C.
This proves the first part of the proposition.
The second part is obtained by a duality argument. By definition, we have,
for any a in V
′
,
a
V
−
1
2
σ
= (µ
−
1
2
j
a, e
j
)
j∈N
ℓ
2
= sup
(α
j
)
j∈N
(α
j
)
j∈N
ℓ
2
≤1
j∈N
α
j
µ
−
1
2
j
a, e
j
. (3.4.1)
66 Stability of Navier–Stokes equations
The map L defined by
L
ℓ
2
→V
1
2
σ
(α
j
)
j∈N
→
j∈N
α
j
µ
−
1
2
j
e
j
is an onto isometry. Thus, thanks to (3.4.1), we have
a
V
−
1
2
σ
= sup
ϕ
V
1
2
σ
≤1
j∈N
(L
−1
ϕ)
j
µ
−
1
2
j
a, e
j
.
For any ϕ in V
σ
,wehave
j∈N
(L
−1
ϕ)
j
µ
−
1
2
j
a, e
j
= a, ϕ.
If we assume that a is in L
3
2
, we have, because ϕ is in L
3
,
a, ϕ =
Ω
a(x) · ϕ(x) dx.
H¨older’s inequality and the first part of Proposition 3.1 imply that
|a, ϕ| ≤ a
L
3
2
ϕ
L
3
≤ Ca
L
3
2
ϕ
V
1
2
σ
.
Thus we have
a
V
−
1
2
σ
≤ sup
ϕ
V
1
2
σ
≤1
a, ϕ
≤ Ca
L
3
2
.
This completes the proof of Proposition 3.1.
3.4.2 The well-posedness result
The aim of this paragraph is the proof of the following existence theorem with
data in V
1
2
σ
.
Theorem 3.4 If the initial data u
0
belongs to V
1
2
σ
and the bulk force f belongs
to L
2
loc
(R
+
; V
−
1
2
σ
), then a positive time T exists such that a solution u of (NS
ν
)
exists in L
4
([0,T]; V
σ
). This solution is unique and belongs to C([0,T]; V
1
2
σ
).
Moreover, a constant c exists (which can be chosen independently of the
domain Ω) such that, if
u
0
V
1
2
σ
+
1
ν
f
L
2
(R
+
;V
−
1
2
σ
)
≤ cν,
Stable solutions in a bounded domain 67
then the above solution is global.
Proof For the sake of simplicity, we shall ignore the bulk force in the proof.
Let us consider the sequence (u
k
)
k∈N
used in the proof of Leray’s theorem and
defined by the ordinary differential equation (NS
ν,k
), page 45. The point is to
prove that this sequence (u
k
)
k∈N
is bounded in L
4
([0,T]; V
σ
) for some positive T .
Let us recall the remark on page 56 which tells us that
u
k
=
k
j=0
U
j,k
(t)e
j
with
U
j,k
(t)
def
=(u
0
|e
j
)
L
2
e
−νµ
2
j
t
+
t
0
e
−νµ
2
j
(t−t
′
)
Q(u
k
(t
′
),u
k
(t
′
)),e
j
dt
′
. (3.4.2)
Using Proposition 3.1 we claim that for any vector field a and b in V
σ
and for
all j ∈ N,
P
j
div(a ⊗ b)
V
−
1
2
σ
= P
j
(a ·∇b)
V
−
1
2
σ
≤ Ca ·∇b
L
3
2
≤ Ca
L
6
∇b
L
2
.
Using Sobolev embeddings, we deduce that
P
j
div(a ⊗ b)
V
−
1
2
σ
≤ Ca
V
σ
b
V
σ
. (3.4.3)
By definition of the norm on V
−
1
2
σ
, we infer that for all k ∈ N,a(c
j,k
(t))
j∈N
exists such that
|Q(u
k
(t),u
k
(t)),e
j
| ≤ Cc
j,k
(t)µ
1
2
j
u
k
(t)
2
V
σ
(3.4.4)
with, for any t,
j∈N
c
2
j,k
(t) = 1. Plugging this inequality into (3.4.2), we get
|U
j,k
(t)|≤|(u
0
|e
j
)|e
−νµ
2
j
t
+ Cµ
1
2
j
t
0
e
−νµ
2
j
(t−t
′
)
c
j,k
(t
′
)u
k
(t
′
)
2
V
σ
dt
′
. (3.4.5)
Thanks to Young’s inequality f⋆g
L
4
≤f
L
4
3
g
L
2
, we have, for any
positive T ,
U
j,k
L
4
([0,T ])
≤|(u
0
|e
j
)|µ
−
1
2
j
#
1 − e
−4νµ
2
j
T
4ν
$
1
4
+
C
ν
3
4
µ
−1
j
#
T
0
c
2
j,k
(t)u
k
(t)
4
V
σ
dt
$
1
2
.
68 Stability of Navier–Stokes equations
Multiplying by µ
j
and taking the ℓ
2
norm gives
j∈N
µ
2
j
U
j,k
2
L
4
([0,T ])
1
2
≤
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
+
C
ν
3
4
j∈N
T
0
c
2
j,k
(t)u
k
(t)
4
V
σ
dt
1
2
.
Thanks to (3.4.4), we have
j∈N
µ
2
j
U
j,k
2
L
4
([0,T ])
1
2
≤
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
+
C
ν
3
4
u
k
2
L
4
([0,T ];V
σ
)
.
Now let us observe that, thanks to the Cauchy–Schwarz inequality, for any a
in ℓ
2
(L
4
[0,T]),
T
0
a
j
(t)
4
ℓ
2
(N)
dt =
T
0
j∈N
a
2
j
(t)
2
dt
=
j∈N,k∈N
T
0
a
2
j
(t)a
2
k
(t) dt
≤
j∈N,k∈N
a
j
2
L
4
([0,T ])
a
k
2
L
4
([0,T ])
≤
(a
j
L
4
([0,T ])
)
j∈N
4
ℓ
2
Let us notice that this is a particular case of the Minkowski inequality. Thus we
infer that
u
k
L
4
([0,T ];V
σ
)
≤
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
+
C
ν
3
4
u
k
2
L
4
([0,T ];V
σ
)
.
Let us define
T
k
def
= sup
T>0 / u
k
L
4
([0,T ];V
σ
)
≤
ν
3
4
2C
%
.