Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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218 R. Farwig, H. Kozono and H. Sohr
3. The approximate system
There are several proofs of Theorem 1.3 in the special case when g =0,see
[13], [15], [16], [20], [22]. The proofs rest on three steps: (1) an approximation
procedure yielding a sequence of solutions, (u
m
), (2) an energy estimate for u
m
with bounds independent of m ∈ N to show that each u
m
exists on the maximal
interval of existence, [0,T), and (3) weak and strong convergence properties of a
suitable subsequence of {u
m
} to construct a weak solution of the Navier-Stokes
equation. One possibility in (1) is the use of the Yosida approximation yielding
the approximate system
u
t
− Δu +(J
m
u) ·∇u + ∇p = f, div u =0
together with u
|
∂Ω
=0,u(0) = u
0
on Ω × [0,T).
In our case (1.12) let v = v
m
, m ∈ N, be a weak solution of the approximate
perturbed Navier-Stokes system
v
t
− Δv +(J
m
v + E) ·∇(v + E)+∇˜p = f
1
, div v =0
v
|
∂Ω
=0,v(0) = v
0
(3.1)
where v
0
∈ L
2
σ
(Ω), f
1
=divF
1
, F
1
∈ L
2
0,T; L
2
(Ω)
,andE results from (1.4)
as a (very) weak solution satisfying (1.8). However, as already explained, we will
only need that the vector field E satisfies (1.8) without referring to (1.4). Then
v = v
m
∈ L
∞
loc
[0,T); L
2
σ
(Ω)
∩L
2
loc
[0,T); W
1,2
0
(Ω)
(3.2)
is called a (Leray-Hopf type) weak solution of (3.1) in Ω ×[0,T)iftherelation
−v,w
t
Ω,T
+ ∇v, ∇w
Ω,T
−(J
m
v + E)(v + E), ∇w
Ω,T
= v
0
,w(0)
Ω
−F
1
, ∇w
Ω,T
(3.3)
is satisfied for every w ∈ C
∞
0
([0,T; C
∞
0,σ
(Ω)) and the energy inequality
1
2
v(t)
2
2
+
t
0
∇v
2
2
dτ ≤
1
2
v
0
2
2
−
t
0
F
1
− (J
m
v + E)E,∇v
Ω
dτ,
(3.4)
0 ≤ t<T,holds.
Lemma 3.1. Let v
0
∈ L
2
σ
(Ω), f
1
=divF
1
, F
1
∈ L
2
(0,T; L
2
(Ω)),andE satisfying
(1.8) be given. If s = ∞, suppose the smallness assumption (1.11) on E as well.
Then there exists some T
= T
(v
0
,F
1
,E,m) ∈ (0, min(1,T)] such that (3.1) has
auniqueweaksolutionv = v
m
on Ω × (0,T
), i.e., v satisfies (3.2)–(3.4) with T
replaced by T
.
Proof. Assume that v = v
m
is a solution of (3.1) on Ω ×(0,T
), 0 <T
≤ 1. Hence
v is contained in the space
X
T
:= L
∞
0,T
; L
2
σ
(Ω)
∩L
2
0,T
; W
1,2
0
(Ω)
Global Leray-Hopf Weak Solutions 219
with
v
X
T
:= v
2,∞;T
+ A
1
2
v
2,2;T
< ∞.
Then we obtain for any 0 <T
≤ min(1,T), using H¨older’s inequality, the proper-
ties (2.1)–(2.3) and (2.5), the following estimates for (J
m
v + E)(v + E)withsome
constant c = c(Ω) > 0:
(J
m
v)v
2,2;T
≤J
m
v
6,4;T
v
3,4;T
≤ c A
1
2
J
m
v
2,4;T
v
X
T
≤ cmv
2,4;T
v
X
T
≤ cm(T
)
1/4
v
2
X
T
,
(J
m
v)E
2,2;T
≤J
m
v
4,4;T
E
4,4;T
≤ c J
m
v
6,4;T
E
4,4;T
≤ cm(T
)
1/4
v
X
T
E
4,4;T
,
Ev
2,2;T
≤E
q,s;T
v
(
1
2
−
1
q
)
−1
,(
1
2
−
1
s
)
−1
,T
≤ c E
q,s;T
v
X
T
.
Since EE
2,2;T
≤E
2
4,4;T
, we proved the estimate
(J
m
v + E)(v + E)
2,2;T
≤ cm(T
)
1/4
v
X
T
E
4,4;T
+ v
X
T
+ E
2
4,4;T
+ c E
q,s;T
v
X
T
.
(3.5)
Obviously, (3.5) also holds in the limit case s = ∞, q =3.
With the definition
ˆ
F
1
(v)=F
1
− (J
m
v + E)(v + E)
we write the system (3.1) in the form
v
t
− Δv + ∇p =div
ˆ
F
1
(v), div v =0
v =0on∂Ω,v(0) = v
0
.
Since v
0
∈ L
2
σ
(Ω) and
ˆ
F
1
(v) ∈ L
2
(0,T
; L
2
(Ω)), we apply classical L
2
-results
[20, Ch. IV] on weak solutions of the instationary Stokes system to get that v ∈
C
0
([0,T
); L
2
σ
(Ω)) and satisfies the fixed point relation
v = F
T
(v)inX
T
, (3.6)
where
F
T
(v)
(t)=e
−tA
v
0
−
t
0
A
1
2
e
−(t−τ)A
A
−
1
2
P
2
div
ˆ
F
1
(v)(τ) dτ ;
see [20, III.2.6] concerning the operator A
−
1
2
P
2
div. Moreover, v satisfies even an
energy equality for t ∈ [0,T
) instead of the energy inequality (3.4), and, by (3.5),
the energy estimate
F
T
(v)
X
T
≤ a v
2
X
T
+ b v
X
T
+ d (3.7)
220 R. Farwig, H. Kozono and H. Sohr
where
a = cm(T
)
1/4
,b= c E
q,s;T
+ cm(T
)
1/4
E
4,4;T
,
d = c
v
0
2
+ E
2
4,4;T
+ F
1
2,2;T
)
(3.8)
with constants c>0 independent of v, m and T
.
By analogy, we get for two elements v
1
,v
2
∈ X
T
the estimate
F
T
(v
1
) −F
T
(v
2
)
X
T
≤ cm(T
)
1/4
v
1
−v
2
X
T
v
1
X
T
+ v
2
X
T
+ E
4,4;T
+ c v
1
−v
2
X
T
E
q,s;T
≤v
1
− v
2
X
T
a (v
1
X
T
+ v
2
X
T
)+b
.
(3.9)
Up to now, to derive (3.7), (3.9), we considered a given solution v = v
m
∈ X
T
of
(3.1).
In the next step, we treat (3.6) as a fixed point problem in X
T
.Assuming
the smallness condition
4 ad+2b<1 (3.10)
we easily see that the quadratic equation y = ay
2
+ by + d has a minimal positive
root y
1
which also satisfies 2ay
1
+ b<1. Hence, under the assumption (3.10),
F
T
maps the closed ball B
T
= {v ∈ X
T
: v
X
T
≤ y
1
} into itself. Moreover,
(3.9), (3.10) imply that F
T
is a strict contradiction on B
T
.NowBanach’sFixed
Point Theorem yields the existence of a unique solution v = v
m
∈ B
T
of the
fixed point problem (3.6). This solution is a weak solution of the approximate
perturbed Navier-Stokes system (3.1). Moreover, v satisfies an energy identity and
v ∈ C
0
([0,T
); L
2
σ
(Ω)).
To satisfy the smallness assumption (3.10) (for fixed m ∈ N), it suffices in
view of (3.8) to choose T
∈ (0, min(1,T)) sufficiently small in the case that s<∞.
However, if s = ∞,lookingatthetermE
q,s;T
in (3.8) we have to assume that
E
C
0
([0,T );L
3
(Ω))
is sufficiently small.
Finally, we show that the solution just found, v = v
m
, which is unique in B
T
,
is even unique in X
T
.Indeed,consideranysolution˜v ∈ X
T
of (3.1). Then there
exists 0 <T
∗
≤ min(1,T
) such that ˜v
X
T
∗
≤ y
1
, and the estimate (3.9) with T
replaced by T
∗
∈ (0, min(1,T)) implies that v−˜v
X
T
∗
= F
T
∗
(v)−F
T
∗
(˜v)
X
T
∗
≤
v − ˜v
X
T
∗
(2ay
1
+ b). Since 2ay
1
+ b<4ad + b<1, we conclude that v =˜v on
[0,T
∗
]. When T
∗
<T
, we repeat this step finitely many times to see that v =˜v
on [0,T
].
To prove that the approximate solution v = v
m
does not only exist on an
interval [0,T
)whereT
= T
(m), but on [0,T), and to pass to the limit m →∞,
we need a global (in time) and uniform (in m ∈ N)energyestimateofv
m
.
Lemma 3.2. Let v = v
m
, m ∈ N, be a weak solution of the approximate perturbed
Navier-Stokes system (3.1) on some interval [0,T
) ⊆ [0,T) where v
0
∈ L
2
σ
(Ω),
Global Leray-Hopf Weak Solutions 221
f
1
=divF
1
, F
1
∈ L
2
(0,T; L
2
(Ω)), and let E satisfy (1.8).If2 <s<∞, 3 <q<
∞,thenv satisfies the energy estimate
v
2
2,∞;t
+ ∇v
2
2,2;t
≤
v
0
2
2
+4F
1
2
2,2;t
+4E
4
4,4;t
exp
c E
s
q,s;t
(3.11)
for all t ∈ [0,T
) where c = c(Ω,q) > 0 is a constant. If s = ∞, q =3,underthe
smallness assumption (1.11), v satisfies the energy estimate
v(t)
2
2
+ ∇v
2
2,2;t
≤v
0
2
2
+4E
4
4,4;t
+4F
1
2,2;t
. (3.12)
Proof. In view of the energy inequality (3.4) we have to estimate the crucial term
t
0
(J
m
v + E)E,∇v
Ω
dτ.ByH¨older’s inequality, (2.1)–(2.3) and (2.5), we get
"
"
"
"
t
0
(J
m
v)E,∇v
Ω
dτ
"
"
"
"
≤
t
0
J
m
v
(
1
2
−
1
q
)
−1
E
q
∇v
2
dτ
≤ c
t
0
J
m
v
α
2
∇J
m
v
1−α
2
E
q
∇v
2
dτ (3.13)
≤ c
t
0
v
α
2
E
q
∇v
2−α
2
dτ
where α =1−
3
q
=
2
s
, cf. (2.3). Hence, if 2 <s<∞, by Young’s inequality
"
"
t
0
(J
m
v)E,∇v
Ω
dτ
"
"
≤
1
8
∇v
2
2,2;t
+ c
t
0
v
2
2
E
s
q
dτ (3.14)
with a constant c = c(q,Ω) > 0. In the case s = ∞, q =3,wehaveα =0andget
"
"
"
"
t
0
(J
m
v)E,∇v
Ω
dτ
"
"
"
"
≤ c E
3,∞;t
∇v
2
2,2;t
. (3.15)
Moreover,
"
"
"
"
t
0
EE,∇v
Ω
dτ
"
"
"
"
≤
t
0
E
2
4
∇v
2
dτ ≤
1
8
∇v
2
2,2;t
+2E
4
4,4;t
;
(3.16)
the term
t
0
F, ∇v
Ω
dτ is treated similarly. Inserting these estimates into (3.4) we
are led to the estimate (2 <s<∞)
v(t)
2
2
+ ∇v
2
2,2;t
≤v
0
2
2
+4E
4
4,4;t
+4F
1
2
2,2;t
+ c
t
0
v
2
2
E
s
q
dτ.
(3.17)
Then Gronwall’s Lemma proves (3.11).
Togetasimilarresultwhens = ∞ we have to assume in view of (3.15) with
α =0thatc E
3,∞;t
≤
1
4
; then we immediately get (3.12).
222 R. Farwig, H. Kozono and H. Sohr
Lemma 3.3. Under the assumptions of Lemma 3.1 for every m ∈ N there exists a
unique weak solution v = v
m
of (3.1) on [0,T). This solution v ∈ C
0
([0,T); L
2
σ
(Ω))
satisfies in addition to the energy inequality (3.4) the strong energy identity
1
2
v(t)
2
2
+
t
t
0
∇v
2
2
dτ =
1
2
v(t
0
)
2
2
−
t
t
0
F
1
− (J
m
v + E)E, ∇v
Ω
dτ
(3.18)
for all t
0
∈ [0,T) and t
0
<t<T, and it holds
d
dt
1
2
v(t)
2
2
+ ∇v(t)
2
2
= −F
1
− (J
m
v + E)E,∇v
Ω
(t) (3.19)
in the sense of distributions on [0,T).
Proof. Let [0,T
∗
) ⊆ [0,T) be the largest interval of existence of v = v
m
,and
assume that T
∗
<T.Sincev ∈ C
0
([0,T
∗
); L
2
σ
(Ω)), we find 0 <T
0
<T
∗
arbitrarily
close to T
∗
with v(T
0
) ∈ L
2
σ
(Ω) which will be taken as initial value at T
0
in (3.1)
in order to extend v beyond T
0
. Since the length δ of the interval of existence
[T
0
,T
0
+ δ) of this unique extension depends only on v(T
0
)
2
and F
1
2,2;T
,
E
4,4;T
, E
q,s;T
by Lemma 3.1, we see that v can be extended beyond T
∗
in
contradiction with the assumption.
Since v = v
m
in Lemma 3.1 satisfies an energy identity instead of only an
energy inequality, v = v
m
will satisfy the strong energy equality (3.18) on [0,T).
Since both integrands in (3.18) are L
1
-functions, the corresponding integrals are
absolutely continuous in t; hence we get the differential identity (3.19) in the sense
of distributions.
4. Proof of Theorem 1.3
Let (v
m
) denote the sequence of approximate solutions on [0,T) constructed in
Section 3 and let 0 <T
≤ T be finite. By (3.11) we find a constant c>0such
that
v
m
2,∞;T
+ ∇v
m
2,2;T
≤ c for all m ∈ N. (4.1)
Hence there exists a subsequence of (v
m
) which for simplicity will again be denoted
by (v
m
) with the following properties:
There exists a vector field v ∈ L
∞
(0,T
; L
2
σ
(Ω)) ∩ L
2
(0,T
; H
0
(Ω)):
v
m
∗
v in L
∞
(0,T
; L
2
σ
(Ω)) (weakly-∗)
v
m
v in L
2
(0,T
; H
1
0
(Ω)) (weakly)
v
m
→ v in L
2
(0,T
; L
2
(Ω) (strongly)
v
m
(t) → v(t)inL
2
(Ω) for a.a. t ∈ [0,T)(strongly).
(4.2)
We note that the third property is based on compactness arguments just as
for the classical Navier-Stokes system and that the fourth property is a well-
known consequence of the strong convergence in L
2
(0,T
; L
2
(Ω)). Moreover, for
Global Leray-Hopf Weak Solutions 223
all t ∈ [0,T)
∇v
2,2;t
≤ lim inf
m→∞
∇v
m
2,2;t
,
v(t)
2
≤ lim inf
m→∞
v
m
(t)
2
.
(4.3)
By H¨older’s inequality, (4.1) and (4.2) we also conclude (after extracting a further
subsequence again denoted by (v
m
)) that
v
m
v in L
s
1
(0,T
; L
q
1
(Ω)),
2
s
1
+
3
q
1
=
3
2
, 2 ≤ s
1
,q
1
< ∞
v
m
v
m
vv in L
s
2
(0,T
; L
q
2
(Ω)),
2
s
2
+
3
q
2
=3, 1 ≤ s
1
,q
1
< ∞
v
m
·∇v
m
v·∇v in L
s
3
(0,T
; L
q
3
(Ω)),
2
s
3
+
3
q
3
=4, 1 ≤ s
1
,q
1
< ∞.
(4.4)
When passing to the limit in the weak formulation (3.3) only some terms in
(J
m
v
m
+E)(v
m
+E), ∇w
Ω,T
need a special consideration. Concerning (J
m
v
m
)v
m
we first note that
J
m
v
m
= J
m
(v
m
− v)+J
m
v → v in L
2
(0,T
; L
2
(Ω));
hence, as m →∞, by (4.4)
1
and H¨older’s inequality
(J
m
v
m
)v
m
− vv =(J
m
v
m
− v
m
)v
m
+ v
m
·(v
m
− v)+(v
m
−v)v0
in L
s
4
(0,T
; L
q
4
(Ω)),
2
s
4
+
3
q
4
=4, 1 ≤ s
4
,q
4
≤ 2.
(4.5)
Proceeding similarly with all other terms we prove that v is a weak solution of the
perturbed Navier-Stokes system satisfying Definition 1.2 (i),(ii).
It remains to show that v satisfies the energy inequality (1.13). To this aim
we consider the energy equality (3.18) for v
m
and those t
0
∈ [0,T)wherethe
strong convergence v
m
(t
0
) → v(t
0
)inL
2
(Ω) holds, see (4.2)
4
. By (4.3) and (4.2)
4
the first three terms in (3.18) pose no problems for m →∞. The same holds true
for the terms F,∇v
m
Ω
and EE,∇v
m
. To treat the remaining term we have to
prove that
t
t
0
(J
m
v
m
)E,∇v
m
Ω
dτ →
t
t
0
vE, ∇v
Ω
dτ (4.6)
as m →∞.
Since E ∈ L
s
(t
0
,t; L
q
(Ω)) and C
∞
0
((t
0
,t) ×Ω) is dense in L
s
(t
0
,t; L
q
(Ω)) for
1 ≤ s, q < ∞, it suffices to show (4.6) for any smooth
˜
E and that the sequence
((J
m
v
m
)∇v
m
) is bounded in L
s
(t
0
,t; L
q
(Ω)). Indeed, for
˜
E ∈ C
∞
0
((t
0
,t) × Ω)
t
t
0
(J
m
v
m
)
˜
E,∇v
m
Ω
−v
˜
E,∇v
Ω
)dτ
= −
t
t
0
(J
m
v
m
)v
m
−vv, ∇
˜
E
Ω
dτ → 0asm →∞
due to (4.5). Moreover,
(J
m
v
m
)∇v
m
q
,s
;t
≤∇v
m
2,2;t
J
m
v
m
(
1
q
−
1
2
)
−1
,(
1
s
−
1
2
)
−1
;t
224 R. Farwig, H. Kozono and H. Sohr
is uniformly bounded in m ∈ N by (4.1) and since 2(
1
s
−
1
2
)+3(
1
q
−
1
2
)=
3
2
where
2
s
+
3
q
=1, 2 ≤ s ≤∞, 3 ≤ q ≤∞. In the case E ∈ C
0
((0,T); L
3
(Ω))
the same argument applies; note that only for this argument we needed E ∈ C
0
instead of E ∈ L
∞
. Summarizing the previous ideas we prove (4.6).
Proof of Corollary 1.5. Since the proof is based on a differential inequality and
not on Gronwall’s Lemma applied to an integral inequality we have to consider
the sequence of approximate solutions (v
m
) first of all. By the differential equation
(3.19) for v = v
m
we get the estimate
1
2
d
dt
v
m
(t)
2
2
+ ∇v
m
(t)
2
2
≤ (F
1
2
+ E
2
4
)∇v
m
2
+ (J
m
v
m
)E,∇v
m
Ω
where the last term due to the assumption (1.15) can be estimated as follows:
|((J
m
v
m
)E,∇v
m
)|≤
1
4
∇J
m
v
m
2
∇v
m
2
≤
1
4
∇v
m
2
2
.
Then Young’s inequality and an absorption argument lead to the estimate
d
dt
v
m
(t)
2
2
+ ∇v
m
(t)
2
2
≤ 4(F
1
2
2
+ E
4
4
)(t) (4.7)
for a.a. t ≥ 0. Let μ
0
> 0 denote the smallest eigenvalue of the Dirichlet Laplacian
on Ω. Then ∇v
m
(t)
2
2
≥ μ
0
v
m
(t)
2
2
for a.a. t>0, and we get that
d
dt
v
m
(t)
2
2
+ μ
0
v
m
(t)
2
2
≤ 4(F
1
2
2
+ E
4
4
)(t).
This estimate together with the assumption (1.17) immediately implies that
v
m
(t)
2
2
≤ e
−μ
0
t
v
0
2
2
+ e
−μ
0
t
t
0
e
μ
0
τ
4(F
1
2
2
+ E
4
4
)dτ
≤ e
−μ
0
t
v
0
2
2
+
4
μ
0
(F
1
2
L
∞
(0,∞;L
2
(Ω))
+ E
4
L
∞
(0,∞;L
4
(Ω))
).
(4.8)
Hence v
m
is uniformly bounded on (0, ∞) with a bound independent of m ∈ N.
By the pointwise convergence property (4.2)
4
v(t) satisfies the same bound, first
of all for a.a. t ≥ 0, but due to its weak continuity property in L
2
(Ω) even for all
t ≥ 0.
Moreover, (4.8) applied to v yields a time T
0
= T
0
(v
0
2
) such that v(t)
2
2
is bounded by κ
∗
for all t ≥ T
0
where κ
∗
is chosen larger than the last two terms
in (4.8).
Finally, integrating (4.7) with respect to time and passing to the limit with
m →∞, we get the estimate (1.18).
5. Construction of the vector field E
To apply Theorem 1.3 and Corollary 1.5 and to find solutions u of the Navier-
Stokes system (1.1) in the form u = v + E we have to construct a suitable vector
field E solving (1.4); the solution should satisfy the assumptions (1.8) to apply
Theorem 1.3 and (1.15) to apply Corollary 1.5, respectively. First we consider very
Global Leray-Hopf Weak Solutions 225
weak solutions E of (1.8), see [2]–[5], for given suitable data g, E
0
and f
0
.Fortheir
definition we introduce the space of initial values, J
q,s
σ
(Ω), by
J
q,s
σ
(Ω) =
u
0
∈D(A
q
)
: u
0
J
q,s
σ
=
∞
0
A
q
e
−τA
q
(A
−1
q
P
q
u
0
)
s
q
dτ
1
s
< ∞
.
(5.1)
Here, D(A
q
) is equipped with the graph norm or equivalently with the norm
A
q
·
q
,andthetermA
−1
q
P
q
u
0
for u
0
∈D(A
q
)
denotes the unique element
u
∗
∈ L
q
σ
(Ω) such that A
−1
q
P
q
u, ϕ = u
∗
,ϕ = u, P
q
A
−1
q
ϕ for all ϕ ∈ L
q
σ
.
Proposition 5.1. Let Ω ⊂ R
3
be a bounded domain with ∂Ω ∈ C
1,1
,let0 <T ≤∞
and let 1 <s,q,r <∞ satisfy
1
3
+
1
q
≥
1
r
. Assume that f
0
=divF
0
,
F
0
∈ L
s
(0,T; L
r
(Ω)),g∈ L
s
(0,T; W
−
1
q
,q
(∂Ω)) (5.2)
such that g(t),N
∂Ω
=0for a.a. t ∈ (0,T) and let E
0
∈J
q,s
σ
(Ω). Then the Stokes
system (1.4) has a unique very weak solution
E ∈ L
s
(0,T; L
q
(Ω)) (5.3)
in the sense that for all test functions w ∈ C
1
0
([0,T); C
2
0,σ
(Ω))
−E,w
t
Ω,T
−E, Δw
Ω,T
= E
0
,w(0)
Ω
−F
0
, ∇w
Ω,T
−g, N ·∇w
∂Ω,T
div E =0 in Ω ×(0,T),E·N = g · N on ∂Ω × (0,T).
(5.4)
This solution satisfies the a priori estimate
E
q,s;T
≤ c (F
0
r,s;T
+ g
L
s
(0,T ;W
−
1
q
,q
(∂Ω))
+ u
0
J
q,s
σ
) (5.5)
with a constant c = c(q, r, s, Ω) > 0 independent of T and of the data.
For more details on very weak solutions we refer to [1]–[5]. See, e.g., [5, Chap-
ters 2.1, 2.3], for the well-definedness of all terms in (5.1) and (5.4); Proposition
5.1 is a special case of [5, Theorem 2.14] and a remark in [5, §1.3] on the extension
of results in [3], [4] where ∂Ω ∈ C
2,1
to the case ∂Ω ∈ C
1,1
. Note that Serrin’s
condition
2
s
+
3
q
= 1 is not needed in the linear theory. Moreover, s = ∞ is not
included in Proposition 5.1; hence the case s = ∞ will not be dealt with in the
next result.
Corollary 5.2. Let Ω ⊂ R
3
be a bounded domain with ∂Ω ∈ C
1,1
,let0 <T ≤∞
and let 1 <s,q,r <∞ satisfy
2
s
+
3
q
=1,
1
3
+
1
q
≥
1
r
. Assume that f
0
=divF
0
,
F
0
∈ L
s
(0,T; L
r
(Ω)) ∩ L
4
(0,T; L
12
7
(Ω)),
g ∈ L
s
(0,T; W
−
1
q
,q
(∂Ω)) ∩L
4
(0,T; W
−
1
4
,4
(∂Ω)),
E
0
∈J
q,s
σ
(Ω) ∩J
4,4
σ
(Ω)
(5.6)
226 R. Farwig, H. Kozono and H. Sohr
such that g(t),N
∂Ω
=0for a.a. t ∈ (0,T). Then the inhomogeneous Stokes
system (1.4) has a unique very weak solution E satisfying (1.8), i.e.,
E ∈ L
s
(0,T; L
q
(Ω)) ∩ L
4
(0,T; L
4
(Ω)), (5.7)
and the a priori estimate
E
q,s;T
+ E
4,4;T
≤ c (F
0
q,s;T
+ F
0
12
7
,4;T
+g
L
s
(0,T ;W
−
1
q
,q
(∂Ω))
+ g
L
4
(0,T ;W
−
1
4
,4
(∂Ω))
+u
0
J
q,s
σ
+ u
0
J
4,4
σ
)
(5.8)
holds with a constant c = c(q, r, s,Ω) > 0 independent of T .
Proof. We apply Proposition 5.1 with the exponents s, q, r and 4, 4,
12
7
.Sincethe
very weak solution E of (1.4) in [3, 4, 5] is constructed in a finite number of steps
where each of them yields the same result for s, q, r and for 4, 4,
12
7
, it is easily seen
that the unique solution E satisfies (5.7), (5.8).
Remark 5.3. (i) In the case 4 ≤ s ≤ 8, 4 ≤ q ≤ 6andT finite the L
s
(L
q
)-conditions
in (5.6) imply the L
4
(L
4
)-conditions; then (5.6)–(5.8) simplify considerably.
(ii) For the system (1.1) consider data f =divF , F ∈ L
2
(0,T; L
2
(Ω)),
u
0
∈ L
2
(Ω) and boundary data g as in (5.6)
3
satisfying g(t),N
∂Ω
=0fora.a.
t ∈ (0,T). Then solve (1.4) with data f
0
=0,E
0
=0andg to get a (unique) very
weak solution E satisfying (5.7) and the a priori estimate
E
q,s;T
+ E
4,4;T
≤ c
g
L
s
(0,T ;W
−
1
q
,q
(∂Ω))
+ g
L
4
(0,T ;W
−
1
4
,4
(∂Ω))
.
Next, by Theorem 1.3, we find a weak solution v of the perturbed Navier-Stokes
system (1.12) with data f
1
= f =divF , F
1
= F ,andv
0
= u
0
satisfying (1.13),
(1.14). Then u = v + E is a weak solution of (1.1) split into a weak and a very
weak part, v and E. It is an easy exercise to write down a corresponding energy
estimate for u in terms of u
0
, f and g only.
(iii) Assuming more regularity on the boundary data g better properties of
u = v + E can be achieved; we refer to [3, 4] and to the forthcoming paper [6] for
such results.
In the second part of this Section we consider the assumption (1.15) and
Corollary 1.5. Assume that the bounded domain Ω ⊂ R
3
with ∂Ω=
L
∪
j=0
Γ
j
∈ C
1,1
has boundary components Γ
0
,...,Γ
L
with Γ
0
being the “outer” boundary of Ω
and Γ
j
, 1 ≤ j ≤ L, being the boundary of “holes” Ω
j
. Further, let the boundary
data g with g(t) ∈ W
1
2
,2
(∂Ω) for a.a. t ∈ (0,T) satisfy the restricted flux condition
Γ
j
g(t) · Ndo =0, 0 ≤ j ≤ L. (5.9)
Then, due to a construction in [14], there exists a solenoidal extension E = E
ε
∈
W
1,2
(Ω) of g for a.a. t ∈ (0,T) satisfying (1.16) (for arbitrary but fixed ε>0and
Global Leray-Hopf Weak Solutions 227
for a.a. t). However, we do need also an estimate of E and E
t
in terms of g and
g
t
, respectively.
Proposition 5.4. Let Ω ⊂ R
3
be a bounded domain as above and let the boundary
function
g ∈ L
∞
(0, ∞; W
1
2
,2
(∂Ω)),g
t
∈ L
∞
(0, ∞; W
−
1
2
,2
(∂Ω)) (5.10)
satisfy the restricted flux condition (5.9). Then there exists an extension
E ∈ L
∞
(0, ∞; W
1,2
(Ω)),E
t
∈ L
∞
(0, ∞; W
−1,2
(Ω)) (5.11)
of g satisfying inequality (1.15)
2
and the a priori estimate
E
L
∞
(0,∞;W
1,2
(Ω))
≤ c g
L
∞
(0,∞;W
1
2
,2
(∂Ω))
E
t
L
∞
(0,∞;W
−1,2
(Ω))
≤ c g
t
L
∞
(0,∞;W
−
1
2
,2
(∂Ω))
(5.12)
with a constant c = c(Ω) > 0.
Proof. We follow the ideas of E. Hopf [14] as described in [8, 11] to find an extension
E of g written as the curl of a suitable vector potential and defined by a bounded
linear operator g → E.
Ignoring t ∈ (0,T) for a moment we consider g ∈ W
1/2,2
(∂Ω) satisfying the
restricted flux condition as in (5.9). Then we use the theory of very weak solutions,
see [3]–[5], to find a solution u
j
∈ L
2
(Ω
j
), 1 ≤ j ≤ L, of the stationary Stokes
system
−Δu
j
+ ∇p
j
=0, div u
j
=0inΩ
j
,u= g on ∂Ω
j
(5.13)
for each hole Ω
j
,1≤ j ≤ L. By definition
−(u
j
, Δw)
Ω
j
+ g,N ·∇w
∂Ω
j
=0 forallw ∈ C
2
0,σ
(Ω
j
)
div u
j
=0inΩ
j
,u
j
·N = g · N on ∂Ω
j
,
and [4, Theorem 3], yields the existence of a unique very weak solution u
j
satisfying
the a priori estimate
u
j
2,Ω
j
≤ c g
W
−1/2,2
(∂Ω
j
)
;
here the necessary compatibility condition g, N
∂Ω
j
= 0 is fulfilled due to (5.9)
for each j. By analogy, we find a very weak solution u
0
∈ W
1,2
(Ω) such that
−Δu
0
+ ∇p
0
=0, div u
0
=0inΩ,u= g on ∂Ω,
again taking into account (5.9), and get that
u
0
2,Ω
≤ c g
W
−1/2,2
(∂Ω)
.
Finally, we consider A = B
R
\(Ω ∪
:
L
j=1
Ω
j
) for a ball of radius R and center 0
such that Ω ⊂ B
R
, and find a unique very weak solution u
A
∈ W
1,2
(A)ofthe
Stokes system
−Δu
A
+ ∇p
A
=0, div u
A
=0inA, u
|
Γ
0
= g, u
|
∂B
R
=0
since g,N
Γ
0
= 0 by (5.9). Moreover, u
A
2,A
≤ c g
W
−1/2,2
(∂Ω)
.