Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems
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276 IV Steady Stokes Flow in Bounded Domains
u ∈ D
2,q
(R
n
+
), π ∈ D
1,q
(R
n
+
). (IV.5.17)
To fix the ideas, consider the case n = 3. From (IV. 5.16) and (IV.5.3)
2
II.1.8
it follows that
(D
2
3
u
3
+ c
3i
D
2
3
u
i
) ∈ L
q
(R
n
+
). (IV.5.18)
Moreover, integrating (IV.5.14) by parts and again employing (IV.5.16) one
can prove for all i = 1, 2, 3,
[(1 + a
33
)D
2
3
u
i
−(δ
3i
+ b
3i
)D
3
π] ∈ L
q
(R
n
+
). (IV.5.19)
Conditions (IV.5. 18) and (IV.5.19) then yield
(1 + a
33
) {[c
31
D
2
3
u
1
+ c
32
D
2
3
u
2
+ (1 + c
33
)D
2
3
u
3
− D
3
π [c
31
b
31
+ c
32
+ (1 + b
33
)(1 + c
33
)]} ∈ L
q
(R
n
+
).
(IV.5.20)
Because of (IV.5 .5) we can choose d so smal l that the coefficient of D
3
π is
strictly positive. From (IV.5.18) and (IV.5. 20) we thus derive
D
3
π ∈ L
q
(R
n
+
), (IV.5.21)
which, in view of (IV.5. 19), gives
D
2
3
u
i
∈ L
q
(R
n
+
). (IV.5.22)
From (IV.5.16), (IV.5.21), and (IV.5.22) we then conclude the validity of
(IV.5.17). On the other hand, from (IV.5.10), (IV.5.13) and (IV.5.17) we can
finall y assert
v ∈ W
2,q
(ω
0
), p ∈ W
1,q
(ω
0
),
which is what we wanted to show.
The results obtained so far can then be summarized in the following
Theorem IV.5.1 Let Ω be an arbitrary domain in R
n
, n ≥ 2, with a bound-
ary portio n σ of class C
m+2
, m ≥ 0. Let Ω
0
be any bounded subdomain of Ω
with ∂Ω
0
∩ ∂Ω = σ. Further, let
v ∈ W
1,q
(Ω
0
), p ∈ L
q
(Ω
0
), 1 < q < ∞,
be such that
(∇v, ∇ψ) = −hf , ψi + (p, ∇ · ψ), for al l ψ ∈ C
∞
0
(Ω
0
),
(v, ∇ ϕ) = 0 for all ϕ ∈ C
∞
0
(Ω
0
),
v = v
∗
at σ.
Then, if
f ∈ W
m,q
(Ω
0
), v
∗
∈ W
m+2−1/q,q
(σ) ,
we have
v ∈ W
m+2,q
(Ω
0
), p ∈ W
m+1,q
(Ω
0
),
for a ny Ω
0
satisfying:
IV.5 L
q
-Estimates Near the Boundary 277
(i) Ω
0
⊂ Ω
0
;
(ii) ∂Ω
0
∩∂Ω is a strictly interior subregion of σ.
Finally, the following estimate holds
kvk
m+2,q,Ω
0
+ kpk
m+1,q,Ω
0
≤ c
kf k
m,q,Ω
0
+ kv
∗
k
m+2−1/q,q(σ)
+ kvk
1,q ,Ω
0
+ kpk
q,Ω
0
,
(IV.5.23)
where c = c(m, n, q, Ω
0
, Ω
0
).
Remark IV.5.1 A consideration similar to that ma de in Remark IV.4.1 ap-
plies also to the estimate (IV.5.23).
From this theorem and Theorem IV.4.3 we thus obtain, in particular, the
following result concerning globa l regularity of q- generalized solutions.
Theorem IV.5.2 Let v be a q-generalized solution to the Stokes problem
(IV.0.1), (IV.0.2). Then if Ω is of class C
∞
, f ∈ C
∞
(Ω) and v
∗
∈ C
∞
(∂Ω),
3
it
follows that v, p ∈ C
∞
(Ω), where p is the pressure field associated to v by
Lemma IV.1.1.
As in the case of interior regularity, intermediate smoothness results can b e
obtained from Theorem IV.4.1, Theorem IV.5.1, and the embedding Theorem
II.3.4. We leave them to the reader as an exercise. Regularity in H¨older spaces
can b e obtained from the results of Section IV.7.
Exercise IV.5.1 (Agmon, Douglis, & Nirenberg 1959). Let Ω ⊂ R
n
have a smooth
boundary portion σ and let φ ∈ W
m−1/q,q
(σ) , m ≥ 1, 1 < q < ∞, be of compact
support in σ. Perform the change of variables (IV .5.2) for a sufficiently regular ζ,
and denote by
b
φ,
b
φ, bσ the transforms of φ and σ under this change, so that
b
φ can
be considered defined in the whole of R
n−1
and of compact support in bσ. Show the
existence of a constant c independent of φ such that
c
−1
kφk
m−1/q,q(σ)
≤ k
b
φk
m−1/q,q(R
n−1
)
≤ ckφk
m−1/q,q(σ)
.
Hint: Let Ω
1
= Ω ∩B, with ∂Ω
1
∩∂Ω = σ and B a ball centered at x
0
∈ σ. Denote
by
b
Ω
1
the transform of Ω
1
. By Theorem II.10.2, we may find v ∈ W
m,q
(R
n
+
), of
compact support in the closure of
b
Ω
1
and such that v =
b
φ at bσ, v = 0 at ∂
b
Ω
1
− bσ,
and, moreover,
kvk
m,q,R
n
+
≤ c
1
k
b
φk
m−1/q,q(R
n−1
)
.
If u is the transform of v under the inverse of (IV.5.2), one has u = φ at σ and
kvk
m,q,R
n
+
≤ c
1
kuk
m,q,Ω
1
≤ c
2
kvk
m,q,R
n
+
≤ c
3
c
1
k
b
φk
m−1/q,q(R
n−1
),
which proves the first inequality. The second one is proved analogously.
3
Namely, v
∗
is infinitely differentiable along the boundary.
278 IV Steady Stokes Flow in Bounded Domains
Exercise IV.5.2 Show that Theorem IV.5.1 also holds when ∇ · v = g 6≡ 0, g ∈
W
m+1,q
(Ω
0
), provided we add t he term
kgk
m+1,q,Ω
0
on the right-hand side of (IV. 5.20).
Exercise IV.5.3 Let Ω, σ and Ω
0
, and Ω
0
be as in Theorem IV.5.1. Let u ∈
W
1,q
(Ω
0
), q ∈ (1, ∞), satisfy the following condition
(∇u, ∇ψ) = −hf, ψi, for all ψ ∈ C
∞
0
(Ω
0
)
u = u
∗
at σ .
Show that, if f ∈ W
m,q
(Ω
0
), u
∗
∈ W
m+2−1/q,q
(σ) , m ≥ 0, then necessarily u ∈
W
m+2,q
(Ω
0
), and the following estimate holds
kuk
m+2,q,Ω
0
≤ c (kfk
m,q,Ω
0
+ ku
∗
k
m+2−1/q,q,(σ)
) ,
with c independent of u, f and u
∗
.
We conclude this section by giving an estimate near the bo undary involving
the L
q
-norm of ∇v and p. Specifically we have
Theorem IV.5.3 Let Ω be an arbitrary domain in R
n
, n ≥ 2, with a bound-
ary portion σ of class C
1
.
4
Let, further, Ω
0
, Ω
0
, v, and p be as in Theorem
IV.5. 1. Then if
f ∈ W
−1,q
0
(Ω
0
), v
∗
∈ W
1−1/q,q
(σ),
the following inequality holds:
kvk
1,q ,Ω
0
+ kpk
q,Ω
0
≤ c
kf k
−1,q,Ω
0
+ kv
∗
k
1−1/q,q(σ)
+ kvk
q,Ω
0
+ kpk
−1,q,Ω
0
.
(IV.5.24)
Proof. Transforming v and p into u and π, respectively, as befo re, we readily
obtain that u and π obey (IV.5.3 ), (IV.5.4) i n the weak form. Successively,
we apply the results of Theorem IV.3.3 to u, π and derive, in particular, that
they obey the inequality
|u|
1,q
+ kπk
q
≤ c
1
(|F |
−1,q
+ kgk
q
) . (IV.5.25)
Recalling (IV.5.4) and (IV.5.10) it is not difficult to show that
|F |
−1,q,R
n
+
≤ c
2
kfk
−1,q,ω
+ kpk
−1,q,ω
+ kvk
q,ω
+ a k∇uk
q,R
n
+
+ bkπk
q,R
n
+
kgk
q,R
n
+
≤ c
3
kvk
q,ω
+ ck∇uk
q,R
n
+
.
(IV.5.26)
Therefore, (IV.5.24) becomes a consequence of (IV.5.25), (IV. 5.26), and
(IV.5.10). ut
4
Notice that we only need σ of class C
1
. In fact the result can be extended to σ
Lipschitz but with “not too sharp” corners; see Galdi, Simader, & Sohr (1994).
IV.6 Existence, Uniqueness, and L
q
-Estimates in a Bounded Domain 279
Remark IV.5.2 A consideration similar to that ma de in Remark IV.4.2 ap-
plies also to the estimate (IV.5.24).
Exercise IV.5.4 Show that Theorem IV.5.3 continues t o hold if ∇ · v = g 6≡ 0,
where g ∈ L
q
(Ω
0
). The inequality (IV.5.21) is then modified by adding the term
kgk
q,Ω
0
on its right-hand side.
IV.6 Existence, Uniqueness, and L
q
-Esti mates in a
Bounded Domain
The i nterior and boundary inequaliti es demonstrated in Section IV.4 and Sec-
tion IV.5 allow us to derive L
q
-estimates for q-generalized solutions holding
for the whole of Ω, in the case where Ω is bounded and suitably regular.
Specifically, setting
kwk
k,q/ R
= inf
c∈R
kw + ck
k,q
(IV.6.1)
we have
Lemma IV.6.1 Let v be a q-generalized solution to the Stokes problem
(IV.0.1), (IV.0.2) in a bounded domain Ω of R
n
, n ≥ 2, of class C
m+2
, m ≥ 0,
corresponding to
f ∈ W
m,q
(Ω), v
∗
∈ W
m+2−1/q,q
(∂Ω).
Then,
v ∈ W
m+2,q
(Ω), p ∈ W
m+1,q
(Ω),
where p is the pressure field associated to v by Lemma IV.1.1. Moreover, the
following inequality holds:
kvk
m+2,q
+ kpk
m+1,q/ R
≤ c
kfk
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
(IV.6.2)
with c = c(m, n, q, Ω).
Proof. By covering Ω wi th a finite number of open balls, from Theorem IV.4.1
and Theorem IV.5.1 we deduce
v ∈ W
m+2,q
(Ω), p ∈ W
m+1,q
(Ω)
and the vali dity of the inequality
|v|
m+2,q
+ |p|
m+1,q
≤ c
1
kf k
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
+ kpk
q
+ kvk
1,q
.
We add to both sides of this inequal ity the L
q
-norms of all derivatives of v
[respectively, of p] up to the order m+1 [respectively, m] and employ Ehrling’s
inequality (II.5.20 ) to derive
280 IV Steady Stokes Flow in Bounded Domains
kvk
m+2,q
+ kpk
m+1,q
≤ c
2
kfk
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
+ kpk
q
+ kvk
q
.
(IV.6.3)
Clearly, (IV.6.3) remains unaffected if we replace p with p + c, for any c ∈ R.
Thus, taking the inf
c∈R
of both sides of this new inequality, we obtain
kvk
m+2,q
+ kpk
m+1,q/ R
≤ c
2
kfk
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
+kpk
q/ R
+ kvk
q
.
(IV.6.4)
It is easy to show that, provided the solution is unique, we can drop the last
two terms on the right-hand side of (IV.6.4). In fact, it is enough to show the
existence of a constant c
3
independent of the data and the particular solution
such that
kvk
q
+ kpk
q/ R
≤ c
3
kf k
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
. (IV.6.5)
If (IV.6.5) were not true, a sequence would exist such as
{v
k
} ⊂ W
m+2,q
(Ω), {p
k
} ⊂ W
m+1,q
(Ω),
with
kv
k
k
q
+ kp
k
k
q/ R
= 1, for all k ∈ N,
while the right-hand side of (IV.6.5) tends to zero. By (IV.6.3) we then have
kv
k
k
m+2,q
+ kp
k
k
m+1,q/ R
uniform ly bounded in k and therefore we may select a subsequence which, by
the com pactness result of Exercise II.5.8, converges strongly to limits
u ∈ W
1,q
(Ω), π ∈ L
q
(Ω),
respectively, with
kuk
q
+ kπk
q/ R
= 1.
However, this last relati on is easily contradicted. Actually, it is immediately
shown that u is a q-generalized solution to the Stokes problem in Ω cor-
responding to f ≡ v
∗
≡ 0 and so, by the uniqueness hypothesis, u ≡ 0,
π = const. The proof of the lemm a is therefore completed, once we have
shown the following result. ut
Lemma IV.6.2 Let Ω be a bounded, C
2
-smooth domain of R
n
. If v is a q-
weak solution to the Stokes problem (IV.0.1), (IV.0.2) corresponding to zero
data, then v ≡ 0, p ≡ const. a.e. in Ω, where p is the pressure field associated
to v by Lemma IV.1.1.
Proof. If q ≥ 2, by the uniqueness part of Theorem IV.1.1 we already know
that the previous statement is true (even assuming less smoothness on Ω). If
q < 2, from the first part o f Lemma IV.6.1 we have
IV.6 Existence, Uniqueness, and L
q
-Estimates in a Bounded Domain 281
v ∈ W
2,q
(Ω), p ∈ W
1,q
(Ω),
and so, by the embedding Theorem II.3.2, it follows that
v ∈ W
1,r
1
(Ω), p ∈ L
r
1
(Ω), r
1
= nq/(n − q).
Now, if r
1
≥ 2 we are finished; otherwise, by the first part of Lemma IV.6.1
we have
v ∈ W
2,r
1
(Ω), p ∈ W
1,r
1
(Ω),
and so, again by Theorem II.3.2, it follows that
v ∈ W
1,r
2
(Ω), p ∈ L
r
2
(Ω), r
2
= nq/(n − 2q) (> r
1
).
If r
2
≥ 2 the proof is achieved; otherwise,
v ∈ W
2,r
2
(Ω), p ∈ W
1,r
2
(Ω)
and we continue this procedure as many times as needed until we arrive to
show, after a finite number of steps,
v ∈ W
1,2
(Ω).
The lemma is therefore completely proved. ut
We now turn our attention to the question of existence of q-generalized
solutions. When q = 2 the answer is already furnished in (IV.1.1). In the
general case we argue as follows. Given
f ∈ W
m,q
(Ω), v
∗
∈ W
m+2−1/q,q
(∂Ω), 1 < q < ∞,
with
Z
∂Ω
v
∗
· n = 0, (IV. 6.6)
let us approximate them with sequences {f
k
}, {v
∗k
} of sufficiently smooth
functions. We can always assume
Z
∂Ω
v
∗k
·n = 0 .
1
1
Actually, let {v
k
} be a sequence of smooth functions tending to v
∗
in
W
2−1/q,q
(∂Ω), and let φ be a smooth function with
R
∂Ω
φ = 1. The sequence
v
∗k
= v
k
−φ
Z
∂Ω
v
k
· n, k ∈ N
is smooth, tends to v in W
2−1/q,q
(∂Ω), and satisfies
Z
∂Ω
v
∗k
· n = 0.
282 IV Steady Stokes Flow in Bounded Domains
Denote by
{v
k
}, {p
k
}
the corresponding solutions whose existence is ensured by Theorem IV.1.1.
From Lemma IV.6.1 we have for all k ∈ N:
v
k
∈ W
2,2
(Ω), p
k
∈ W
1,2
(Ω).
If n = 2, the embedding Theorem II. 3.4 tells us
v
k
∈ W
1,r
(Ω), p
k
∈ L
r
(Ω), for any r ∈ (1, ∞)
and so Lemma IV.6.1 ensures
v
k
∈ W
2,q
(Ω), p
k
∈ W
1,q
(Ω)
and estimate (IV.6.1) holds. We then let k → ∞ and use (IV.6.1) to obtain
for som e v ∈ W
2,q
(Ω), p ∈ W
1,q
(Ω)
v
k
→ v in W
2,q
(Ω),
p
k
→ p in W
1,q
(Ω).
Clearly, v, p solve a.e. the Stokes system (IV.0.1) corresponding to f, while
v equal s v
∗
at the boundary in the trace sense. For n > 2, we have
v
k
∈ W
1,r
(Ω), p
k
∈ L
r
(Ω), for any r ∈ (1, 2n/(n − 2 )).
Thus, if 2 < n ≤ 4, we again use Lemma IV.6.1 and Theorem II.3.4 to deduce
v ∈ W
2,q
(Ω), p ∈ W
1,q
(Ω).
We then proceed as in the case where n = 2. Fo r n > 4, by a double application
of Lemma IV.6. 1 and Theorem II.3.4 we have
v
k
∈ W
1,r
(Ω), p
k
∈ L
r
(Ω), for any r ∈ (1, 2n/(n −4))
and, by the same token, we recover existence if 4 < n ≤ 6, a nd so forth.
Existence of solutio ns for all 1 < q < ∞ a nd any space dimension can therefore
be fully established.
By means o f a simila r procedure, we may also show existence of q-weak
solutions corresponding to arbitrary
f ∈ W
−1,q
0
(Ω), v
∗
∈ W
1−1/q,q
(∂Ω), 1 < q < ∞,
with v
∗
satisfying (IV.6.3) and Ω of cla ss C
2
. In fact, if v is a q-weak solution,
from Theorem IV.4.4 and Theorem IV.5.3 we derive
kvk
1,q
+ kpk
q
≤ c
kfk
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+ kvk
q
+ kpk
−1,q
(IV.6.7)
IV.6 Existence, Uniqueness, and L
q
-Estimates in a Bounded Domain 283
where c = c(n, q, Ω) and p is the pressure field associated to v by Lemma
IV.1. 1. From (IV.1.3) it is apparent that the inequality just obtained remains
unaffected if we replace p with p + c, c ∈ R. We then recover
kvk
1,q
+ kpk
q/R
≤ c
kfk
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
+ kvk
q
+ kpk
−1,q/R
.
The last two terms on the right-hand side of this relation can be increased by
the data:
kvk
q
+ kpk
−1,q/R
≤ C
kfk
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
, (IV.6.8)
with C = C(q, n, Ω). This can be proved by the same contradiction argument
used to show (IV.6.5). In fact, if (IV.6.8) were not true, there would exist a
sequence of sol utions
{v
k
, p
k
} ⊂ W
1,q
(Ω) × (L
q
(Ω) / R)
with
kv
k
k
q
+ kp
k
k
−1,q/R
= 1, for all k ∈ N,
corresponding to data {f
k
, v
∗k
} converging to zero in the space W
−1,q
(Ω) ×
W
1−1/q,q
(∂Ω). However, by the compactness results of Theorem II. 5.3 and
Exercise II.5.8 , we find
{v, p} ∈ W
1,q
(Ω) × L
q
(Ω) (IV.6.9)
such that
v
k
converges to v weakly in W
1,q
(Ω), strongly in L
q
(Ω)
p
k
converges to p weakly in L
q
(Ω) / R, strongly in W
−1,q
(Ω).
Since v, p is a solution to the Stokes problem with f ≡ v
∗
≡ 0, by Lemma
IV.6. 2 it follows that, as k → ∞ ,
v ≡ 0, p = const, in Ω,
and therefore (IV.6.9) cannot hold. We then conclude the validity of the in-
equality
kvk
1,q
+ kpk
q
≤ c
kf k
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
.
By means of this relati on, we may argue as before to prove existence o f q-
generalized solutions.
The results shown so far in this section are collected in the following main
theorem.
Theorem IV.6.1 Let Ω be a bounded domain of R
n
, n ≥ 2. The following
properties hold.
284 IV Steady Stokes Flow in Bounded Domains
(a) Suppose Ω of class C
m+2
, m ≥ 0. Then, for any
f ∈ W
m,q
(Ω), v
∗
∈ W
m+2−1/q,q
(∂Ω) , 1 < q < ∞,
with
Z
∂Ω
v
∗
·n = 0,
there exists one and only one pair v, p
2
such that
(i) v ∈ W
m+2,q
(Ω), p ∈ W
m+1,q
(Ω);
(ii) v, p verify the Stokes system (IV.0.1) a.e. in Ω and v satisfies (IV.0.2)
in the trace sense.
In addition, this solution obeys the inequality
kvk
m+2,q
+ kpk
m+1,q/ R
≤ c
1
kf k
m,q
+ kv
∗
k
m+2−1/q,q(∂Ω)
, (IV.6.10)
where c
1
= c
1
(n, m, q, Ω).
(b) Suppose Ω of class C
2
. Then, for every
f ∈ W
−1,q
0
(Ω), v
∗
∈ W
1−1/q,q
(∂Ω) , 1 < q < ∞,
there exists one and only one q-generalized solution v to the Stokes prob-
lem (IV.0.1), (IV.0.2). This solution satisfies the inequality
kvk
1,q
+ kpk
q/ R
≤ c
2
kfk
−1,q
+ kv
∗
k
1−1/q,q(∂Ω)
, (IV.6.11)
where p is the pressure field associated to v by Lemma IV.1.1.
Exercise IV.6.1 Let u ∈ H
1
q
(Ω), 1 < q < ∞, with Ω a C
2
-smooth bounded
domain. Show that there exists c = c(n, q, Ω) such that
kuk
1,q
≤ c sup
ϕ ∈ H
1
q
0
(Ω), ϕ 6= 0
|(∇u, ∇ϕ)|
kϕk
1,q
0
ff
. (IV.6.12)
Hint: The map ϕ ∈ H
1
q
0
(Ω) ⊂ W
1,q
0
0
(Ω) → (∇u, ∇ϕ) defines a linear functional.
Therefore, by the Hahn-Banach Theorem II.1.7, there is f ∈ W
−1,q
0
(Ω) such that
hf , ϕi = (∇u, ∇ϕ), ϕ ∈ H
1
q
0
(Ω) and with kfk
−1,q
equal to the right-hand side of
(IV.6.12). Consider then the Stokes problem with v
∗
≡ 0 and f ≡ f and apply the
results of Theorem IV.6.1(b).
Exercise IV.6.2 Suppose v, p solves the Stokes problem (IV.0.1) with Ω ≡ B
R
,
and suppose also v ∈ W
m+2,q
(Ω), f ∈ W
m,q
(Ω), for some m ≥ 0, q ∈ (1, ∞), and
v
∗
≡ 0. Show that there exists a constant c i ndependent of R such that
|v|
m+2,q
≤ c kf k
m,q
.
2
p is determined up to a constant that may be fixed by requiring p
Ω
= 0. In such
a case, the term kpk
m+1,q/R
can be replaced by kpk
m+1,q
.
IV.6 Existence, Uniqueness, and L
q
-Estimates in a Bounded Domain 285
Exercise IV.6.3 Show that the first [respectively, second] part of Theorem IV.6.1
continues to hold if ∇ · v = g 6≡ 0, with g ∈ W
m+1,q
(Ω) [respectively, g ∈ L
q
(Ω)]
and
Z
Ω
g =
Z
∂Ω
v
∗
· n.
Inequality (IV.6.10) [respectively, (IV.6.11)] is t hen accordingly modified by adding
to its right-hand side the term
kgk
m+1,q
, [respectively, kgk
q
].
Hint: Use Exercise IV.4.2, Exercise IV.5.2, Exercise IV.4.3, and Exercise IV .5.4,
together with the reasonings employed to arrive at Theorem IV.6.1.
We end this section by proving a further useful estimate satisfied by
the pressure field p, in addition to those already provided by (IV.6.1 0) and
(IV.6.11). Specifically, we have
Theorem IV.6.2 Let Ω be a bounded domain of R
n
, n ≥ 2, of class C
2
,
and let (v, p) ∈ W
2,q
(Ω) × W
1,q
(Ω) be a solution to (IV.0.1), corresponding
to f ∈ H
q
(Ω), fo r some q ∈ (1, ∞).
3
Furthermore, we norm alize p by the
condition p
Ω
= 0. Then, given ε > 0, there exists c = c(Ω, n, q, r, ε) such that
kpk
q
≤ c |v|
1,q
+ ε kvk
2,r
,
for any r ∈ [ q(n−1)/n, q], if q > n/(n−1), and any r ∈ (1, q], if q ≤ n/(n−1).
Proof. We dot-multiply both sides of (IV.0.1) by ∇ϕ, ϕ ∈ W
1,q
0
(Ω), to obtain
Z
Ω
∆v · ∇ϕ =
Z
Ω
∇p · ∇ϕ , for all ϕ ∈ W
1,q
0
(Ω) . (IV.6.13)
We then choose ϕ as the solution to the following Neumann problem
∆ϕ = g −g
Ω
in Ω ,
∂ϕ
∂n
∂Ω
= 0 ,
Z
Ω
ϕ = 0 , (IV.6.14 )
where g ∈ L
q
0
(Ω). From a classical result of Agmon, Douglis & Nirenberg
(1959, §15), it follows that this problem has one and only one sol ution ϕ ∈
W
2,q
0
(Ω), which, in addi tion, satisfies the estimate
kϕk
2,q
0
≤ c kgk
q
0
, (IV.6.15)
for some c = c(Ω, q, n). Thus, integrating by parts on the right-hand side of
(IV.6.13), using (IV. 6.14), and the condition p
Ω
= 0, we obtain
Z
Ω
∆v · ∇ϕ = −
Z
Ω
p g . (IV.6.16)
3
In view of the Helmholtz–Weyl decomposition Theorem III.1.2, we may assume,
without loss, f ∈ H
q
(Ω), instead of f ∈ L
q
(Ω).