Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems
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476 VII Steady Oseen Fl ow in Exterior Domains
and whi ch at l arge distances has the asymptotic structure of the type proved
in Theorem VII.6.2 . Denote next by ψ a “cut-off” function that equals one in
Ω
R/2
and zero in Ω
ρ
, where R/2 > ρ > δ(Ω
c
). Putting u = ψv, π = ψp, from
(VII.0.3 ) it follows that u, π satisfies the following Oseen problem in R
n
:
∆u + R
∂u
∂x
1
= ∇π + RF
∇ · u = g,
(VII.7.1 )
where
RF = Rψf − R
∂ψ
∂x
1
v + (2∇ψ · v + ∆ψv − p∇ψ)
g = ∇ψ · v.
(VII.7.2 )
Employing Theorem VII.4.1 we deduce the existence of a solutio n w, τ to
(VII.7.1 ), (VII.7.2) satisfying, in particula r, the properties
w ∈
m
\
`=0
D
`+2,q
(R
n
), τ ∈
m
\
`=0
D
`+1,q
(R
n
), 1 < q < ∞
w ∈
m
\
`=0
D
`+1,s
1
(R
n
), s
1
=
(n + 1)q
n + 1 − q
, 1 < q < n + 1
w ∈
m
\
`=0
D
`,s
2
(R
n
) s
2
=
(n + 1)q
n + 1 − 2q
, 1 < q <
n + 1
2
(VII.7.3 )
together with inequalities (VII.4.30)–(VII.4.35). We then apply Theorem
VII.6.2 to w in the domain Ω
R/2
, which does not contain the support of
g. Because of (VII.7 .3)
3
, w satisfies assumption (ii) of that theorem and, con-
sequently, it has the asymptotic structure (VII.6.18 )
1
, from which we conclude
u ≡ w, π ≡ p+const; see Exercise VII.6.2. Recalling (VII.7.2) and that v = u,
p = π in Ω
R/2
, from (VII.4.34) and (VII.4.35) i t foll ows, in particular, that
for a ll ` ∈ [0, m] and all q ∈ (1, (n + 1)/2), the pair v, p obeys the inequality
R
2/(n+1)
|v|
`,s
2
,Ω
R/2
+ R
1/(n+1)
|v|
`+1,s
1
,Ω
R/2
+R
∂v
∂x
1
`,q,Ω
R/2
+ |v|
`+2,q,Ω
R/2
+ |p|
`+1,q,Ω
R/2
≤ c
1
(R|f|
`,q
+ (1 + R)|v|
`+1,q,Ω
R
+ |p|
`,q,Ω
R
) ,
(VII.7.4 )
where s
1
=
(n+1)q
n+1−q
, s
2
=
(n+1)q
n+1−2q
and, for n = 2,
VII.7 Existence, Uniqueness, and L
q
-Estimates in Exterior Domains 477
R|v
2
|
`,2q/(2−q),Ω
R/2
+ R|v
2
|
`+1,q,Ω
R/2
+ R
2/3
|v|
`,3q/(3−2q),Ω
R/2
+R
1/3
|v|
`+1,3q/(3−q), Ω
R/2
+ R
∂v
∂x
1
`,q,Ω
R/2
+|v|
`+2,q,Ω
R/2
+|p|
`+1,q,Ω
R/2
≤ c
1
(R|f|
`,q
+ (1 + R)|v|
`+1,q,Ω
R
+ |p|
`,q,Ω
R
) .
(VII.7.5 )
Let us next derive analo gous inequalities in Ω
R
. From (IV.6.3) we have
kvk
m+2,q,Ω
R
+ kpk
m+1,q,Ω
R
≤ c
2
(
Rkfk
m,q,Ω
R
+ kv
∗
k
m+2−1/q,q(∂Ω)
+ R
∂v
∂x
1
m,q,Ω
R
+kvk
m+2−1/q,q(∂B
R
)
+ kvk
q,Ω
R
+ kpk
q,Ω
R
)
,
(VII.7.6 )
where, as usual, the origin of coordinates has been taken in the interior of Ω
c
.
By the trace Theorem II.4.4 we have
kvk
m+2−1/q,q(∂B
R
)
≤ c
3
|v|
m+2,q,Ω
R/2
+ kvk
m+1,q,Ω
R
. (VII.7 .7)
Furthermore, by the embedding Theorem II.3.4,
kvk
m,s,Ω
R/2
+
m
X
`=0
|v|
`+1,s
1
,Ω
R/2
≤ c
4
kvk
m+2,q,Ω
R
(VII.7.8 )
and
kvk
m,2q/(2−q),Ω
R/2
≤ c
4
kvk
m+1,q,Ω
R
, if n = 2, (VII.7.9)
and so, col lecting (VII.7.4), (VII. 7.5)–(VII.7.9), we derive, in particular, for
some c = c(n, q, Ω, m) and all q ∈ (1, (n + 1)/2 )
a
1
kvk
m,s
2
,Ω
+ R
∂v
∂x
1
m,q,Ω
+
m
X
`=0
{a
2
|v|
`+1,s
1
,Ω
+ |v|
`+2,q,Ω
+ |p|
`+1,q,Ω
}
≤c
5
Rkfk
m,q,Ω
+kv
∗
k
m+2−1/q,q(∂Ω)
+(1 + R)kvk
m+1,q,Ω
R
+kpk
m,q,Ω
R
(VII.7.1 0)
and, if n = 2,
478 VII Steady Oseen Fl ow in Exterior Domains
R
kv
2
k
m,2q/(2−q),Ω
+ k∇v
2
k
m+1,q,Ω
+a
1
kvk
m,3q/(3−2q),Ω
+ R
∂v
∂x
1
m,q,Ω
+
m
X
`=0
a
2
|v|
`+1,3q/(3−q), Ω
+ |v|
`+2,q,Ω
+ |p|
`+1,q,Ω
≤c
5
Rkfk
m,q,Ω
+kv
∗
k
m+2−1/q,q(∂Ω)
+(1 + R)kvk
m+1,q,Ω
R
+kpk
m,q,Ω
R
(VII.7.1 1)
where
a
1
= min{1, R
2/(n+1)
}, a
2
= min{1, R
1/(n+1)
},
s
1
=
(n + 1)q
n + 1 − q
, s
2
=
(n + 1)q
n + 1 −2q
.
(VII.7.1 2)
By a repeated use of Ehrling’s inequality (see Exercise II.5.16), for all ε > 0
it follows that
kpk
m,q,Ω
R
≤ εkpk
m+1,q,Ω
R
+ c
6
kpk
q,Ω
R
(VII.7.1 3)
with c
6
= c
6
(ε, n, m, q, Ω
R
). In addi tion, possibly modifying p by a suitable
constant (which causes no loss of generality), from Lemma IV.4.1 we derive
kpk
q,Ω
R
≤ c
7
[(1 + R)kvk
1,q,Ω
R
+ Rkfk
q,Ω
]. (VII.7.14 )
Inequalities (VII.7 .10), (VII.7 .11), (VII.7.13), and (VII.7.14) then yi el d
a
1
kvk
m,s
2
,Ω
+ R
∂v
∂x
1
m,q,Ω
+
m
X
`=0
{a
2
|v|
`+1,s
1
,Ω
+ |v|
`+2,q,Ω
+ |p|
`+1,q,Ω
}
≤c
8
Rkfk
m,q,Ω
+kv
∗
k
m+2−1/q,q(∂Ω)
+(1 + R)kvk
m+1,q,Ω
R
(VII.7.1 5)
and, if n = 2,
R
kv
2
k
m,2q/(2−q),Ω
+ k∇v
2
k
m+1,q,Ω
+a
1
kvk
m,3q/(3−2q),Ω
+ R
∂v
∂x
1
m,q,Ω
+
m
X
`=0
a
2
|v|
`+1,3q/(3−q), Ω
+ |v|
`+2,q,Ω
+ |p|
`+1,q,Ω
≤c
8
Rkfk
m,q,Ω
+kv
∗
k
m+2−1/q,q(∂Ω)
+(1 + R)kvk
m+1,q,Ω
R
.
(VII.7.1 6)
VII.7 Existence, Uniqueness, and L
q
-Estimates in Exterior Domains 479
We now look for an inequality of the type
kvk
m+1,q,Ω
≤ c
9
Rkfk
m,q,Ω
+ kv
∗
k
m+2−1/q,q(∂Ω)
(VII.7.1 7)
for a suitable constant independent of v, f and v
∗
. The proof of (VII.7.17)
can be obtained, as in the case of the Stokes problem, by a contradiction
argument. Actually, admitting the invalidity of (VII. 7.17) means to assume
the existence of two sequences
{f
k
} ⊂ C
∞
0
(Ω), {v
∗k
} ⊂ W
m+2−1/q,q
(∂Ω)
such that, by denoting by {v
k
, p
k
} the corresponding solutions,
Rkf
k
k
m,q,Ω
+ kv
∗k
k
m+2−1/q,q(∂Ω)
≤ 1/k
kv
k
k
m+1,q,Ω
R
= 1.
(VII.7.1 8)
However, reasoning as in the proof of Lemma V.4.5, we can show that, as
k → ∞, v
k
converges to a solution v of the Oseen problem (VII.0.2), (VII.0.3)
with f ≡ v
∗
≡ v
∞
≡ 0. Furthermore, by (VII.7.15), it follows that v ∈ L
s
2
(Ω)
and therefore by Theorem VII.6.2 and Exercise VII.6.2 we have v ≡ 0. The
point to discuss now is that a priori the constant c depends also on R and,
consequently, we lose the dependence of inequality (VII.7.15) +
c
8
c
7
(VII .7.17)
on the Reynolds number R. We may thus wonder if, at least in some cases,
this undesired feature can be avoided. We shal l presently show that if n > 2
and q ∈ (1, n/2) we may take the constant c
9
independent of R ∈ (0, B] for
any positive, arbitrarily fixed B. Actually assume (VII. 7.17) does not hold,
then there exist sequences
{f
k
} ⊂ C
∞
0
(Ω), {v
∗k
} ⊂ W
m+2−1/q,q
(∂Ω) (VII.7.19)
and
{R
k
} ⊂ (0, B] (VII.7.20)
such that, denoting by {v
k
, p
k
} the solutions to the Oseen problems
∆v
k
+ R
k
∂v
k
∂x
1
− ∇p
k
= R
k
f
k
∇ · v
k
= 0
v
k
= v
∗k
at ∂Ω,
(VII.7.2 1)
the following condition holds
R
k
kf
k
k
m,q,Ω
+ kv
∗k
k
m+2−1/q,q(∂Ω)
≤ 1/k
kv
k
k
m+1,q,Ω
R
= 1.
(VII.7.2 2)
480 VII Steady Oseen Fl ow in Exterior Domains
In view of (VII.7.20) there is a subsequence, indi cated ag ain by {R
k
}, and a
number R ≥ 0 to which R
k
converges as k → ∞. Furthermore, by (VII.7 .6),
(VII.7.1 4), (VII.7.1 5), and (VII.7.18), for all k ∈ N we have for all fixed R
kv
k
k
1,q,Ω
R
+ kD
2
v
k
k
m,q
+ k∇p
k
k
m,q
≤ M (VII.7.23)
for some constant M independent of k. The results of Exercise II.6.2 on weak
compactness of
˙
D
m,q
-spaces along with Exercise II.5.8 on strong compactness
(on bounded domains) imply the exi stence of a subsequence, still denoted by
{v
k
, p
k
}, and of two functions v ∈ D
2,q
(Ω), p ∈ D
1,q
(Ω) such that as k → ∞
D
2
v
k
w
→ D
2
v, ∇p
k
w
→ ∇p, in L
q
(Ω)
v
k
→ v, in W
m+1,q
(Ω
R
).
(VII.7.2 4)
From (VII.7.22), (VII.7.21), and (VII.7.24) it immediately follows that v, p is
a solution to the homogeneous Oseen problem
∆v + R
∂v
∂x
1
− ∇p = 0
∇ · v = 0
v = 0 at ∂Ω,
(VII.7.2 5)
satisfying
kvk
m+1,q,Ω
R
= 1. (VII.7.26)
Let us now distinguish the following two cases: (i) R > 0, (ii) R = 0. In case
(i) from (VII.7.15) (written along the subsequences) and (VII.7.22) we obtain
v ∈ L
s
2
(Ω) (VII.7.27)
and, since v solves (VII.7.25), from Theorem VII.6.2 and Exercise VII.6.2 we
deduce v ≡ 0, contradicting (VII.7 .26). If the limiting value R is zero, we can
no longer deduce (VII.7.27). Nevertheless, if q ∈ (1, n/2), we can still deduce
that v b elongs to some space L
r
(Ω). Actually, by a double application of
inequality (II.6.22), and recalling that, fo r each fixed k, v
k
(x) and ∇v
k
(x)
tend to zero uniformly as |x| tends to infinity, from (VII.7.23) we have
kv
k
k
nq/(n−2q)
≤ ckD
2
v
k
k
q
≤ M
and therefore
v ∈ L
nq/(n−q)
(Ω).
Replacing this information into (VII.7.25) w ith R = 0 and using this time
Theorem V.3.2 and Theorem V.3.4 we conclude v ≡ 0, which contradicts
(VII.7.2 6).
Once (VI I .7.15) and (VII.7.17) have been established, we can prove the
following.
VII.7 Existence, Uniqueness, and L
q
-Estimates in Exterior Domains 481
Theorem VII.7.1 Let Ω be an exterior domain in R
n
of class C
m+2
, m ≥ 0.
Given
f ∈ W
m,q
(Ω), v
∗
∈ W
m+2−1/q,q
(∂Ω), 1 < q < (n + 1)/2 ,
there exists one and only one corresponding solution v, p to the Oseen problem
(VII.0.2 ), (VII.0.3) such that
v ∈ W
m,s
2
(Ω) ∩
(
m
\
`=0
D
`+1,s
1
(Ω) ∩ D
`+2,q
(Ω)
)
p ∈
m
\
`=0
D
`+1,q
(Ω)
with s
1
=
(n+1)q
n+1−q
, s
2
=
(n+1)q
n+1−2q
. If n = 2 , we al so have
v
2
∈ W
m,2q/(2−q)
(Ω) ∩
m
\
`=0
D
`+1,q
(Ω)
!
.
Moreover, v, p verify
a
1
kvk
m,s
2
+ R
∂v
∂x
1
m,q
+
m
X
`=0
{a
2
|v|
`+1,s
1
+ |v|
`+2,q
+ |p|
`+1,q
}
≤ c
Rkfk
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
(VII.7.2 8)
and, if n = 2,
R
kv
2
k
m,2q/(2−q)
+ k∇v
2
k
m+1,q
+a
1
kvk
m,3q/(3−2q)
+ R
∂v
∂x
1
m,q
+
m
X
`=0
a
2
|v|
`+1,3q/(3−q)
+ |v|
`+2,q
+ |p|
`+1,q
≤ c
Rkfk
m,q,
+kv
∗
k
m+2−1/q,q(∂Ω)
(VII.7.2 9)
with a
1
and a
2
given in (VI I.7.12). The constant c depends on m, q, n, Ω, and
R. However, if q ∈ (1, n/2) and R ∈ (0, B] for some B > 0, c depends solely
on m, q, n, Ω, and B.
Proof. The existence part, together with the validity of (VII.7.28), has been
already established for f ∈ C
∞
0
(Ω). However, from (VII.7.28 ) and from a
now standard density argument we can extend existence to all f ∈ W
m,q
(Ω).
Finally, uniqueness of solutions is most easily discussed if we take into account
that, indicating by u, π the difference between two solutions correspondi ng to
the same data, we have u ∈ L
s
2
(Ω). T herefore, Theorem VII.6.2 and Exercise
VII.6.2 ensure u ≡ 0. The proof is thus completed. ut
482 VII Steady Oseen Fl ow in Exterior Domains
Remark VI I.7.1 Theorem VII.7.1 leaves out the question of existence and
uniqueness for q > (n + 1)/2. However, by using a treatment analogous to
that employed in Section V.3 for the Stokes problem, one could show exis-
tence, uniqueness, and validity of corresponding L
q
-estimates in suitable quo-
tient spaces. We shall not treat this here. For related results f or q-generalized
solutions, we refer the reader to Remark VII.7.3.
Remark VI I.7.2 The validity of inequalities of the type (VII.7.29) with c
independent of R ∈ [0, B] is of fundamental importance for treating nonlin-
ear, plane-steady flow with nonzero velocity at infinity. However, because of
the Stokes paradox, one expects that the constant c in (VII.7.29) may become
unbounded as R approaches zero. On the other hand, if 1 < q < 6/5, in Sec-
tionXII.4, we shall prove the validity of an inequal ity weaker than (VII.7.29)
for a constant c which, in general, can be rendered independent of R fo r R
ranging in [0, B].
Exercise VII.7.1 Extend the results of Theorem VII.7.1 to the case ∇·v = g 6≡ 0,
with g a prescribed function from W
m+1,q
(Ω). Show further that, in this case, in-
equalities (VII.7.28) and (V II.7.29) are modified by adding the term (1+R)kgk
m+1,q
to its right-hand side.
Our subsequent objective is to extend Theorem VII.4.2 to the case of
an exterior do main Ω of class C
2
. We start with f ∈ C
∞
0
(Ω), and v
∗
∈
W
1−1/q,q
(∂Ω). As in Theorem VII.7.1, correspondi ng to these data there ex-
ists a sol ution v, p to (VII.0.2), (VII.0.3) with
v ∈ W
1,q
loc
(Ω) ∩ C
∞
(Ω), p ∈ L
q
loc
(Ω) ∩ C
∞
(Ω)
satisfying the asymptotic behavior of the typ e described in Theorem VII.7.2.
Furthermore, u ≡ ψv and π ≡ ψp satisfy problem (VII.7.1), (VII.7.2) in R
n
,
to which we apply the results stated in Theorem VII.4.2. We thus deduce the
existence of a solution w, τ to (VII.7.1), (VII.7.2) enjoying, in particular, the
properties
w ∈ D
1,q
(R
n
) ∩ L
s
1
(R
n
), τ ∈ L
q
(R
n
)
s
1
=
(n + 1)q
(n + 1 − q)
, 1 < q < n + 1
w
2
∈ L
q
(R
2
), if n = 2,
(VII.7.3 0)
together with the estimates
VII.7 Existence, Uniqueness, and L
q
-Estimates in Exterior Domains 483
R
1/(n+1)
kwk
s
1
+|w|
1,q
+ kτ k
q
≤ c
1
(R|F |
−1,q
+ R|g|
−1,q
+ kgk
q
) ,
(if n > 2)
Rkw
2
k
q
+ R
1/3
kwk
3q/(3−q)
+ |w|
1,q
+ kτk
q
≤ c
1
(R|F |
−1,q
+ R|g|
−1,q
+ kgk
q
) .
(if n = 2)
(VII.7.3 1)
As in Theorem VII.7.1, we prove w ≡ u, τ ≡ p.
1
Assuming q > n/(n −1) and
reasoning exactly as we did in Theorem V.5.1, we have
R|F |
−1,q
+R|g|
−1,q
+ kgk
q
≤ c
2
(R|f|
−1,q
+ (1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
)
(VII.7.3 2)
and so, recalling that u = v, π = p in Ω
R/2
, from (VII.7.30)–(VII.7.32 ) we
deduce if n > 2:
R
1/(n+1)
kvk
s
1
,Ω
R/2
+|v|
1,q,Ω
R/2
+ kpk
q,Ω
R/2
≤ c
3
(R|F |
−1,q
+ (1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
) ,
(VII.7.3 3)
and if n = 2:
Rkv
2
k
q,Ω
R/2
+ R
1/3
kvk
3q/(3−q),Ω
R/2
+ |v|
1,q,Ω
R/2
+ kpk
q,Ω
R/2
≤ c
3
(R|F |
−1,q
+ (1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
) .
(VII.7.3 4)
To obtain an estimate “near” the bounda ry, we apply inequality (IV.6.7), that
is,
kvk
1,q,Ω
R
+ kpk
q,Ω
R
≤ c
4
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+(1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
+kvk
1−1/q,q(∂B
R
)
,
(VII.7.3 5)
where we used the obvious inequality
kf k
−1,q,Ω
R
≤ |f |
−1,q
.
Employing the trace Theorem II.4.4 at the boundary term on ∂B
R
we find
kvk
1−1/q,q(∂B
R
)
≤ ckvk
1,q,Ω
R,2R
and so, by (VII.7.33), (VII.7.34), and (VII.7.35 ) it follows that
kvk
1,q ,Ω
R
+ kpk
q,Ω
R
≤ c
5
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+(1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
) .
(VII.7.3 6)
1
A priori, τ = p + const. but we can choose the constant up to which p is defined
in such a way that τ = p.
484 VII Steady Oseen Fl ow in Exterior Domains
Taking into account that, by Theorem II.3. 4,
kvk
s
1
≤ c
6
kvk
1,q,Ω
R/2
, (VII.7.37 )
combining (VII.7.33), (VII.7.36), and (VII.7.3 7) we conclude, for all q ∈
(n/(n − 1), n + 1), if n > 2:
a
2
kvk
s
1
,Ω
+ |v|
1,q,Ω
+ kpk
q,Ω
≤ c
7
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+ (1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
,
(VII.7.3 8)
and if n = 2:
Rkv
2
k
q,Ω
+ a
2
kvk
3q/(3−q),Ω
+ |v|
1,q,Ω
+ kpk
q,Ω
≤ c
7
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+ (1 + R)kvk
q,Ω
R
+ kpk
−1,q,Ω
R
,
(VII.7.3 9)
where a
2
is defined in (VII. 7.12). By a reasoning totally similar to that devel-
oped in the proof o f Theorem VII.7.1 we can prove the existence of a constant
c
8
independent of v, p, f, and v
∗
such that
kvk
q,Ω
R
+ kpk
−1,q,Ω
R
≤ c
8
{R|f |
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
}. (VII.7.40)
The constant c
8
will a lso depend, in general, on R. However, if q ∈ (n/(n −
1), n), by employing inequality (II.6.22), namely,
kvk
nq/(n−q)
≤ c|v|
1,q
,
one shows, as in Theorem VII .7.1, that c
8
can be chosen independent of R
ranging in (0, B], with an arbitrarily fixed B.
We are thus in a position to prove the foll owing.
Theorem VII.7.2 Let Ω be an exterior domai n in R
n
of class C
2
. Given
f ∈ D
−1,q
0
(Ω), v
∗
∈ W
1−1/q,q
(∂Ω), n/(n − 1) < q < n + 1 .
there exists one and only one q-g eneralized solution v to (VII.0.2)–(VII.0.3).
Furthermore, it holds that
v ∈ L
s
1
(Ω), s
1
=
(n + 1)q
(n + 1 − q)
,
p ∈ L
q
(Ω)
with p pressure field associated to v by Lemm a VII.1.1, and if n = 2,
v
2
∈ L
q
(Ω).
Finally, v and p obey the estimate
VII.7 Existence, Uniqueness, and L
q
-Estimates in Exterior Domains 485
a
2
kvk
s
1
+ |v|
1,q
+ |p|
q
≤ c
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
(VII.7.4 1)
and, if n = 2,
Rkv
2
k
q
+ a
2
kvk
3q/(3−q)
+ |v|
1,q
+ kpk
q
≤ c
R|f|
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
,
(VII.7.4 2)
where c = c(n, q, Ω, R) and a
2
is defined in (VII.7.12). If n > 2 and q ∈
(n/(n − 1), n), for R ∈ (0, B], with B a rbitrarily fixed number, c depends
solely on n, q, Ω, and B.
Proof. The existence part foll ows from (VII.7.38) and (VII.7.40) whenever f ∈
C
∞
0
(Ω). By a density a rgument, based on Theorem II.8. 1 and on inequality
(VII.7.4 1), we easily extend the result to all f ∈ D
−1,q
0
(Ω). Final ly, uniqueness
follows from Exercise VII.6. 2.
ut
Exercise VII.7.2 Extend the results of Theorem VII.7.2 to the case where ∇·v =
g 6≡ 0, with g a prescribed function from L
q
(Ω) ∩ D
−1,q
0
(Ω). Show further that, in
this case, inequality (VII.7.41) is modified by adding to its right-hand side the term
kgk
q
+ R|g|
−1,q
.
Remark VI I.7.3 For future reference, we would like to po int out that, as
far as existence goes, Theorem VII.7.2 admits a suitable extension to the case
q ∈ [n + 1, ∞), n ≥ 3. More precisely, assuming Ω as in that theorem, for any
given
f ∈ D
−1,q
0
(Ω), v
∗
∈ W
1−1/q,q
(∂Ω), n + 1 ≤ q < ∞.
there exists at least a solenoidal vector field v ∈ D
1,q
(Ω), with v = v
∗
at
∂Ω (in the sense of trace), and a scalar field p ∈ L
q
(Ω) satisfying (VII.1.2).
The proof of this result is sim ilar to that of Theorem V.5.1 (when q ≥ n)
and we shall sketch it here. Let (
e
v
i
, eπ
i
) be the solutions to the Oseen problem
(VII.0.3 ) with f ≡ 0 a nd v
∗
= −e
i
, i = 1, . . ., n. From Theorem VII.1.1,
Theorem VII.2.1 and Theorem VII.6.2, we deduce that these solutions exist,
are of class C
∞
(Ω) and, moreover, they have the asymptotic behavior given
in (VII.6.16). From this latter and (VII.3.49) we o bta in
e
h
i
:= (
e
v
i
+ e
i
) ∈
D
1,q
(Ω), q ∈ ((n + 1)/n, ∞), with
e
h
i
= 0 at ∂Ω, i = 1, . . ., n. Thus, from the
characterization given in Theorem II.7.6(ii), it follows that
e
h
i
∈ D
1,q
0
(Ω), q ∈
[n+1, ∞). As in Theorem V. 5.1, we easily show that the set {
e
h
i
, eπ
i
} is linearly
independent and its linear span constitutes an n-dimensional subspace,
e
S
q
, of
D
1,q
0
(Ω) ×L
q
(Ω), q ∈ [ n + 1 , ∞). Arguing as in the proof of Theorem VII.7.2,
corresponding to smo oth data f and v
∗
, we can find a solution (v, p) to the
Oseen problem sati sfy ing the following estimate, f or any (fixed) q ∈ [n+1, ∞):