Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics
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Stable solutions in a bounded domain 69
As u
k
belongs to C
1
(R
+
; P
k
V
σ
), the supremum T
k
is positive for all k. We have,
for all T ∈ [0,T
k
],
u
k
L
4
([0,T ];V
σ
)
≤ 2
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
. (3.4.6)
In the case of small data, it is enough to observe that, for any positive T ,
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
≤
1
ν
1
2
u
0
2
V
1
2
σ
.
Thus, if u
0
V
1
2
σ
≤
ν
8C
, we have, for any T smaller than T
k
,
u
k
L
4
([0,T ];V
σ
)
≤
ν
3
4
4C
·
This implies that T
k
=+∞ and that
u
k
L
4
(R
+
;V
σ
)
≤
2
ν
1
4
u
0
V
1
2
σ
.
In the case of large data, let us define the smallest integer j
0
such that
j>j
0
µ
j
(u
0
|e
j
)
2
1
2
≤
ν
16C
· (3.4.7)
Then, using the fact that 1 − e
−x
≤ x for all non-negative x, we can write for
all T<T
k
,
U
1
(T )
def
=
j∈N
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
≤
ν
3
4
16C
+
j≤j
0
µ
j
(u
0
|e
j
)
2
#
1 − e
−4νµ
2
j
T
ν
$
1
2
1
2
≤
ν
3
4
16C
+ µ
j
0
√
2 T
1
4
u
0
L
2
.
Thus, stating
T
u
0
def
=
#
ν
3
4
16
√
2Cµ
j
0
u
0
L
2
$
4
,
70 Stability of Navier–Stokes equations
we have, for any positive T less than min{T
k
; T
u
0
},
u
k
L
4
([0,T );V
σ
)
≤
ν
3
4
2C
·
Thus, for all k, T
k
≥ T
u
0
. This implies that (u
k
)
k∈N
is a bounded sequence
of L
4
([0,T]; V
σ
). We infer that a Leray solution u of (NS
ν
) exists such that u
belongs to L
4
([0,T]; V
σ
). Thanks to Theorem 3.3, this solution is unique on [0,T],
continuous from [0,T]intoH and satisfies the energy equality on [0,T].
The only thing we have to prove now is the continuity of u from [0,T]
into V
1
2
σ
.Asu belongs to L
4
([0,T]; V
σ
), we infer from (3.4.3) that Q(u, u)is
in L
2
([0,T]; V
−
1
2
σ
). Using Theorem 3.1, we get
(u(t)|e
j
)=(u
0
|e
j
)e
−νµ
2
j
t
+
t
0
e
−νµ
2
j
(t−t
′
)
Q(u(t
′
),u(t
′
)),e
j
dt
′
.
Using (3.4.3) again, we infer by definition of the norm on V
−
1
2
σ
that there exists
a sequence (c
j
(t))
j∈N
, such that
|(u(t)|e
j
)|≤|(u
0
|e
j
)|e
−νµ
2
j
t
+ Cµ
1
2
j
t
0
e
−νµ
2
j
(t−t
′
)
c
j
(t
′
)u(t
′
)
2
V
σ
dt
′
with
&
j∈N
c
2
j
(t) = 1. Using the Cauchy–Schwarz inequality, we have
(u(t)|e
j
)
L
∞
([0,T ])
≤|(u
0
|e
j
)| +
C
ν
1
2
µ
−
1
2
j
#
T
0
c
2
j
(t)u(t)
4
V
σ
dt
$
1
2
.
Multiplying by µ
1
2
j
and taking the ℓ
2
norm gives
U
2
(T )
def
=
j∈N
µ
j
(u(·)|e
j
)
2
L
∞
([0,T ])
1
2
≤
√
2u
0
V
1
2
σ
+
√
2
C
ν
1
2
j∈N
T
0
c
2
j
(t)u(t)
4
V
σ
dt
1
2
≤
√
2u
0
V
1
2
σ
+
C
ν
1
2
u
2
L
4
([0,T ];V
σ
)
.
This gives that u is in L
∞
([0,T]; V
1
2
σ
). In fact, it will imply continuity using the
following argument. Let η be any positive number. An integer j
0
exists such that
j>j
0
µ
j
(u(·)|e
j
)
2
L
∞
([0,T ])
1
2
<
η
2
·
Stable solutions in a bounded domain 71
Now, it turns out that for all (t
1
,t
2
) ∈ [0,T]
2
, one has
u(t
1
)−u(t
2
)
V
1
2
σ
≤
j>j
0
µ
j
(u(·)|e
j
)
2
L
∞
([0,T ])
1
2
+
j≤j
0
µ
j
(u(t
1
)−u(t
2
)|e
j
)
2
1
2
≤
η
2
+ µ
1
2
j
0
u(t
1
) − u(t
2
)
L
2
.
Theorem 3.1 tells us that u is continuous from [0,T]intoH. Thus the whole
Theorem 3.4 is proved.
3.4.3 Some remarks about stable solutions
In this paragraph, we shall assume that the bulk force f is identically 0. We
shall establish some results about the maximal existence time of the solution
constructed in the preceding paragraph.
Proposition 3.2 Let us assume that the initial data u
0
belongs to V
σ
. Then
the maximal time of existence T
⋆
of the solution u in the space
C([0,T
⋆
[; V
1
2
σ
) ∩ L
4
loc
([0,T
⋆
[; V
σ
)
satisfies
T
⋆
≥
cν
3
∇u
0
4
L
2
·
Proof Thanks to (3.4.6), the maximal time of existence T
⋆
is bounded from
below by T such that
4
j
µ
j
(u
0
|e
j
)
2
#
1 − e
−νµ
2
j
T
ν
$
1
2
≤ cν
3
2
.
As 1 − e
−νµ
2
j
T
≤ νµ
2
j
T , we infer that
4
j
µ
j
(u
0
|e
j
)
2
#
1 − e
−νµ
2
j
T
ν
$
1
2
≤ 4T
1
2
j
µ
2
j
(u
0
|e
j
)
2
≤ 4T
1
2
u
0
2
V
σ
.
This proves the proposition.
From this proposition, we infer the following corollary.
Corollary 3.1 Let T
⋆
be the maximal time of existence for a solution u of the
system (NS
ν
) in the space C([0,T
⋆
); V
1
2
σ
) ∩ L
4
loc
([0,T
⋆
); V
σ
).IfT
⋆
is finite, then
T
⋆
0
∇u(t)
4
L
2
dt =+∞ and T
⋆
≤
c
ν
5
u
0
4
L
2
.
72 Stability of Navier–Stokes equations
Proof For almost every t, u(t) belongs to V
σ
. Then, thanks to the above pro-
position, the maximal time of existence of the solution starting at time t, which
is of course T
⋆
− t, satisfies
T
⋆
− t ≥
cν
3
∇u(t)
4
L
2
·
This can be written as
∇u(t)
4
L
2
≥
cν
3
T
⋆
− t
·
This gives the first part of the corollary. Taking the square root of the above
inequality gives, thanks to the energy estimate,
cν
5
2
T
⋆
0
dt
(T
⋆
− t)
1
2
≤
1
2
u
0
2
L
2
.
The corollary is proved.
3.5 Stable solutions in a domain without boundary
The case of two dimensions was dealt with in Section 3.2 in complete general-
ity. The purpose of this section reduces to the study of well-posedness of the
incompressible Navier–Stokes equations in the whole space R
3
or in a periodic
box T
3
.
The basic remark about the level of regularity which is necessary to get
uniqueness and stability is related to the scaling of the Navier–Stokes equation.
If u is a solution of the Navier–Stokes equations in the whole space R
d
, then the
vector field u
λ
defined by
u
λ
(t, x)
def
= λu(λ
2
t, λx)
is also a solution with initial data λu
0
(λ·). It turns out that non-linear estimates
will be based in this section on scaling invariant norms. It is very easy to check
that in two dimensions, the energy norm
u(t)
2
L
2
+2ν
t
0
∇u(t
′
)
2
L
2
dt
′
is scaling invariant. This is not the case in three dimensions. In three dimensions,
the analogous scaling invariant norm is
u(t)
2
˙
H
1
2
+2ν
t
0
∇u(t
′
)
2
˙
H
1
2
dt
′
. (3.5.1)
Unfortunately, no conservation or global a priori control is known about that
quantity.
Stable solutions in a domain without boundary 73
Theorem 3.5 Let u
0
be a divergence-free vector field in H
1
2
and let f be an
external force in L
2
loc
(R
+
;
˙
H
−
1
2
). There is a positive time T such that there is a
solution of (NS
ν
) satisfying
u ∈ C([0,T]; H
1
2
) and ∇u ∈ L
2
([0,T]; H
1
2
).
Moreover a constant c exists such that
u
0
˙
H
1
2
+
1
ν
f
L
2
(R
+
;
˙
H
−
1
2
)
≤ cν =⇒ T = ∞,
in which case one also has lim
t→+∞
u(t)
˙
H
1
2
=0.
Remark The convexity inequality on Sobolev norms implies that
C([0,T];
˙
H
1
2
) ∩ L
2
([0,T];
˙
H
3
2
) ⊂ L
4
([0,T];
˙
H
1
).
Thus the solutions constructed by the above theorem are stable.
Proof of Theorem 3.5
Step 1: The case of small data
We shall start by examining the case when the initial data u
0
are small in
˙
H
1
2
.
We shall of course be using the sequence of approximate solutions (u
k
)
k∈N
intro-
duced in Subsection 2.2.1 as solutions of the system (NS
ν,k
) defined on page 45.
Let us write the energy estimate in the space
˙
H
1
2
. Taking the
˙
H
1
2
scalar product
of (NS
ν,k
) with u
k
, it turns out, due to the divergence-free condition, that
d
dt
u
k
(t)
2
˙
H
1
2
+2ν∇u
k
(t)
2
˙
H
1
2
=2(Q(u
k
(t),u
k
(t))|u
k
(t))
˙
H
1
2
.
By definition of the scalar product on
˙
H
1
2
,weget
(Q(u
k
(t),u
k
(t))|u
k
(t))
˙
H
1
2
≤Q(u
k
(t),u
k
(t))
˙
H
−
1
2
∇u
k
(t)
˙
H
1
2
.
The Sobolev embeddings proved in Theorem 1.2, page 23, and Corollary 1.1,
page 25, imply that
Q(a, b)
˙
H
−
1
2
≤ Ca ·∇b
L
3
2
≤ Ca
L
6
∇b
L
2
≤ C∇a
L
2
∇b
L
2
. (3.5.2)
By the interpolation inequality between
˙
H
1
2
and
˙
H
3
2
, we infer
d
dt
u
k
(t)
2
˙
H
1
2
+2ν∇u
k
(t)
2
˙
H
1
2
≤ Cu
k
(t)
˙
H
1
2
∇u
k
(t)
2
˙
H
1
2
. (3.5.3)
A quick examination of that inequality shows that it is of little use when the
norm u
k
(t)
˙
H
1
2
is large. On the other hand, it is very good when that norm
is small enough. This is a typical phenomenon of global existence theorems for
74 Stability of Navier–Stokes equations
small data. From now on we suppose that u
k
(0)
˙
H
1
2
≤ cν; let us define T
k
in [0, +∞]by
T
k
def
= sup{t/ ∀t
′
≤ t, u
k
(t
′
)
˙
H
1
2
≤ cν} with c
def
=
1
C
,
the constant C being that in (3.5.3) By estimate (3.5.3), we have
d
dt
u
k
(t)
2
˙
H
1
2
|t=0
< 0.
Thus T
k
is positive and there is some positive t
k
such that
∀t ∈]0,t
k
] , u
k
(t)
˙
H
1
2
<cν.
Moreover, still by inequality (3.5.3), the function u
k
(t)
˙
H
1
2
decreases on the
interval [0,T
k
[. So for any t ∈ [t
k
,T
k
[wehaveu
k
(t)
˙
H
1
2
<cν. Hence T
k
=+∞.
We deduce directly from inequality (3.5.3) that
∀t ≥ 0 ,
d
dt
u
k
(t)
2
˙
H
1
2
+ ν∇u
k
(t)
2
˙
H
1
2
≤ 0.
By integration it follows that for any real number t,wehave
u
k
(t)
2
˙
H
1
2
+ ν
t
0
∇u
k
(t
′
)
2
˙
H
1
2
dt
′
≤u
k
(0)
2
˙
H
1
2
.
Extracting a subsequence which converges weakly towards a Leray solution u,
the above estimate implies that this Leray solution u satisfies
u ∈ L
∞
(R
+
; H
1
2
) and ∇u ∈ L
2
(R
+
; H
1
2
).
Now let us prove that the solution u goes to zero for large times, in
˙
H
1
2
:
this result is simply due to the fact that as u is a Leray solution it is in the
space L
∞
(R
+
; L
2
) ∩ L
2
(R
+
,
˙
H
1
), hence by interpolation in L
4
(R
+
;
˙
H
1
2
). It fol-
lows that for any η>0 one can find a time T
η
such that u(T
η
, ·)
˙
H
1
2
≤ η, which
yields the result recalling that the
˙
H
1
2
norm decreases in time.
Step 2: The case of large data
Let us now consider the case of large data. We start by decomposing the initial
data into a high-frequency part and a low-frequency part. Let k
0
be a positive
real number which will be chosen later on and let us consider u
L
the solution of
the evolution Stokes problem
(ES
ν
)
∂
t
u
L
− ν∆u
L
= −∇p
div u
L
=0
u
L
|t=0
= P
k
0
u
0
.
Stable solutions in a domain without boundary 75
Stating w
k
def
= u
k
−u
L
, it is obvious after very elementary computations that w
k
is the solution of the evolution Stokes problem with initial data P
k
(u
0
−P
k
0
u
0
)
and external force
g
k
def
= P
k
(Q(w
k
,w
k
)+Q(w
k
,u
L
)+Q(u
L
,w
k
)+Q(u
L
,u
L
)) .
Using again an
˙
H
1
2
energy estimate, we get that
d
dt
w
k
(t)
2
˙
H
1
2
+2ν∇w
k
(t)
2
˙
H
1
2
=2(g
k
(t)|w
k
(t))
˙
H
1
2
≤g
k
(t)
˙
H
−
1
2
∇w
k
(t)
˙
H
1
2
.
Using estimate (3.5.2), we get that
g
k
(t)
˙
H
−
1
2
≤ C
∇w
k
(t)
2
L
2
+ ∇u
L
(t)
2
L
2
.
Then by an interpolation inequality between
˙
H
1
2
and
˙
H
3
2
, we deduce that
2(g
k
(t)|w
k
(t))
˙
H
1
2
≤
C
1
w
k
(t)
˙
H
1
2
+
ν
2
∇w
k
(t)
2
˙
H
1
2
+
C
2
ν
∇u
L
(t)
4
L
2
.
Thus we get
d
dt
w
k
(t)
2
˙
H
1
2
+
3
2
ν∇w
k
(t)
2
˙
H
1
2
≤ C
1
w
k
(t)
˙
H
1
2
∇w
k
(t)
2
˙
H
1
2
+
C
2
ν
∇u
L
(t)
4
L
2
.
Now let us prove that w
k
, which can be made arbitrarily small initially, will
remain so for a long enough time. More precisely, let us define T
k
by
T
k
def
= sup
t/ t
′
≤ t, w
k
(t
′
)
˙
H
1
2
≤
ν
2C
1
·
Let us choose k
0
the smallest integer such that
u
0
− P
k
0
u
0
˙
H
1
2
≤
ν
4C
1
· (3.5.4)
Let us prove that a positive time T exists such that for any integer k,we
have T
k
≥ T . For any time t smaller than or equal to T
k
,weget
d
dt
w
k
(t)
2
˙
H
1
2
+ ν∇w
k
(t)
2
˙
H
1
2
≤
C
2
ν
∇u
L
(t)
4
L
2
.
By time integration, we infer, for any t ≤ T
k
,
w
k
(t)
2
˙
H
1
2
+ ν
t
0
∇w
k
(t
′
)
2
˙
H
1
2
dt
′
≤
ν
4C
1
2
+
C
ν
t
0
∇u
L
(t
′
)
4
L
2
dt
′
. (3.5.5)
As u
L
belongs to P
k
0
H,wehave
∇u
L
(t)
4
L
2
≤ k
2
0
u
L
(t)
4
L
2
≤ k
2
0
u
0
4
L
2
76 Stability of Navier–Stokes equations
Thus, thanks to (3.5.4), we get that
w
k
(t)
2
˙
H
1
2
+ ν
t
0
∇w
k
(t
′
)
2
˙
H
1
2
dt
′
≤
ν
4C
1
2
+
C
ν
tk
2
0
u
0
4
L
2
.
Now let us state
T
def
=
ν
3
16C
2
1
C
1
k
2
0
u
0
4
L
2
· (3.5.6)
Then for any k we get T
k
≥ T .
Similarly to the case of small data, one can extract from (w
k
)
k∈N
a sub-
sequence which converges weakly towards u − u
L
, where u is a Leray solution.
By the above estimate on w
k
, u satisfies
u ∈ L
∞
([0,T]; H
1
2
) and ∇u ∈ L
2
([0,T]; H
1
2
).
To end the proof of the theorem, it remains therefore to prove that u belongs
to C([0,T];
˙
H
1
2
). In order to do so, we recall that the above bounds on u imply in
particular that u ∈ L
4
([0,T]; H
1
). Hence u is stable in the sense of Theorem 3.3,
page 58, and we can write, as noted in formula (3.1.3),
u(t, x)=(2π)
−d
R
d
e
ix·ξ
u(t, ξ) dξ with
u(t, ξ)
def
= e
−ν|ξ|
2
t
u
0
(ξ)+
t
0
e
−ν|ξ|
2
(t−t
′
)
FQ(u(t
′
),u(t
′
))(ξ) dt
′
.
Note that we have supposed here that the domain is R
3
, but the computations are
identical in the case of T
3
, simply replacing everywhere the integrals in ξ ∈ R
3
by sums on k ∈ Z
3
. We leave the details to the reader. Let us state the following
proposition, which we will prove at the end of this section.
Proposition 3.3 If v is the solution of the Stokes evolution system (ES
ν
), with
an initial data v
0
in H
1
2
and an external force in L
2
loc
(R
+
; H
−
1
2
), then
R
3
|ξ|v(·,ξ)
2
L
∞
([0,T ])
dξ ≤v
0
˙
H
1
2
+
1
2ν
1
2
f
L
2
([0,T ];
˙
H
−
1
2
)
.
Let us recall that
Q(u, u)
˙
H
−
1
2
≤ C∇u
2
L
2
.
Thus we may apply Proposition 3.3. This implies directly the fact that u is
continuous in time on [0,T] with values in
˙
H
1
2
. Indeed let η be any posit-
ive number. One can find, according to Proposition 3.3, a positive integer N
0
such that
|ξ|≥N
0
|ξ|u(·,ξ)
2
L
∞
([0,T ])
dξ <
η
2
·
Blow-up condition and propagation of regularity 77
Now consider t
1
and t
2
in the time interval [0,T]. We have
u(t
1
) − u(t
2
)
˙
H
1
2
≤
#
|ξ|≤N
0
|ξ||u(t
1
,ξ) − u(t
2
,ξ)|
2
dξ
$
1
2
+
#
|ξ|≥N
0
|ξ|u(·,ξ)
2
L
∞
([0,T ])
$
1
2
≤ N
1
2
0
u(t
1
, ·) − u(t
2
, ·)
L
2
+
η
2
·
But we know from Theorem 3.3, page 58, that u is continuous in time with values
in L
2
, so the result follows, up to the proof of Proposition 3.3.
Proof of Proposition 3.3 We have
|ξ|
1
2
|u(t, ξ)|≤|ξ|
1
2
|u
0
(ξ)| +
t
0
e
−ν|ξ|
2
(t−t
′
)
|ξ|
1
2
f(t
′
,ξ) dt
′
.
Young’s inequality enables us to infer that
|ξ|
1
2
|u(t, ξ)|
L
∞
([0,T ])
≤|ξ|
1
2
|u
0
(ξ)| +
1
2ν
1
2
|ξ|
−
1
2
f(t, ξ)
L
2
([0,T ])
.
Taking the L
2
norm gives
R
d
|ξ|v(·,ξ)
2
L
∞
([0,T ])
dξ
1
2
≤
R
d
|ξ||u
0
(ξ)|
2
dξ
1
2
+
1
2ν
1
2
#
T
0
R
d
|ξ|
−1
|
f(t, ξ)|
2
dtdξ
$
1
2
.
The proposition follows.
3.6 Blow-up condition and propagation of regularity
The aim of the first subsection is a version of Proposition 3.2 and Corollary 3.1
for the case of domains without boundary. The purpose of the second subsection
is the proof of the propagation of regularity result in the case of dimension two
which will be useful in Chapter 6 in the periodic case.
3.6.1 Blow-up condition
Proposition 3.4 Let u
0
be in H
1
(R
3
). Then the maximal time of existence
T
⋆
of the solution u in C([0,T
⋆
[; H
1
2
) ∩ L
2
loc
([0,T
⋆
[;
˙
H
3
2
) satisfies
T
⋆
≥
cν
3
∇u
0
4
L
2
·
78 Stability of Navier–Stokes equations
Proof If u
0
is in H
1
,wehave
u
0
− P
k
0
u
0
2
˙
H
1
2
≤
|ξ|≥k
0
|ξ||u
0
(ξ)|
2
dξ
≤ k
−1
0
|ξ|
2
|u
0
(ξ)|
2
dξ
≤ k
−1
0
∇u
0
2
L
2
.
Thus choosing k
0
=(4C
1
∇u
0
2
L
2
)/ν
2
ensures (3.5.4). Now, estimate (3.5.5)
gives, thanks to the conservation of the H
1
norm by the heat flow,
w
k
(t)
2
˙
H
1
2
+ ν
t
0
∇w
k
(t
′
)
2
˙
H
1
2
dt
′
≤
ν
4C
1
2
+
C
ν
t∇u
0
4
L
2
.
Proposition 3.4 is now proved.
We can now state an analog of Corollary 3.1, the proof of which is left to the
reader as an exercise.
Corollary 3.2 Let T
⋆
be the maximal time of existence for a solution of the
system (NS
ν
) in the space C([0,T
⋆
); H
1
2
) ∩L
4
loc
([0,T
⋆
); H
1
).IfT
⋆
is finite then
T
⋆
0
∇v(t)
4
L
2
dt =
T
⋆
0
∇u(t)
2
˙
H
1
2
dt =+∞ and T
⋆
≤
c
ν
5
u
0
4
L
2
.
This property will have the following useful application (see in particular the
scheme of the proof of the forthcoming Theorem 6.2, page 119).
Theorem 3.6 Two real numbers c and C exist which satisfy the following prop-
erty. Let u be the solution of (NS
ν
) in T
3
associated with initial data u
0
in H
1
2
and an external force f in L
2
(R
+
; H
−
1
2
). Let us assume that u is global and that
u
2
1
2
def
= sup
t≥0
u(t)
2
H
1
2
+2ν
t
0
∇u(t
′
)
2
H
1
2
dt
′
< +∞.
Then, for any v
0
in H
1
2
and any g in L
2
(R
+
; H
−
1
2
) such that
v
0
− u
0
2
H
1
2
+
4
ν
f − g
2
L
2
(R
+
;H
−
1
2
)
≤ cν exp
−
C
ν
4
u
4
1
2
,
the solution for (NS)
ν
associated with v
0
and g is global and belongs to the
space E
1
2
.
Proof Let us consider the maximal solution v given by Theorem 3.5. It belongs
to the space C([0,T
⋆
[; H
1
2
) ∩ L
2
loc
([0,T
⋆
[; H
3
2
). Let us state w
def
= v − u.Itis
solution of
∂
t
w − ν∆w = Q(w, w)+Q(u, w)+Q(w, u)+h
div w =0
w
|t=0
= v
0
− u
0