Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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528 J. Pr¨uss and G. Simonett
mean value theorem follows
L
−1
b
K
b
(z)
0
E(a
1
)
≤L
−1
b
K
b
(z) − L
−1
b
K
b
(0)
0
E(a
1
)
+ L
−1
b
K
b
(0)
0
E(a
1
)
≤ r
1
/2+L
−1
b
K
b
(0)
0
E(a
1
)
,z∈
0
B
E(a
1
)
(0,r
1
).
Here we observe that the norm of L
−1
b
K
b
(0) = L
−1
b
(K(z
∗
) − (0,f
∗
d
,g
∗
,g
∗
h
)) in
0
E(a
1
) can be made as small as we wish by choosing a
1
small enough. We may
assume that a
1
was already chosen so that L
−1
b
K
b
(0)
0
E(a
1
)
≤ r
1
/2.
(v) We have shown in (iv) that the mapping
L
−1
b
K
b
:
0
B
E(a
1
)
(0,r
1
) →
0
B
E(a
1
)
(0,r
1
)
is a contraction. By the contraction mapping theorem L
−1
b
K
b
has a unique fixed
point ˆz ∈
0
B
E(a
1
)
(0,r
1
) ⊂
0
E(a
1
) and it follows immediately from (4.10)–(4.11)
that ˆz+z
∗
is the (unique) solution of the nonlinear problem (2.1) in
0
B
E(a
1
)
(z
∗
,r
1
).
Setting t
0
= a
1
gives the assertion in part (a) of the Theorem.
(vi) The proof that (u, π, q, h) is analytic in space and time proceeds exactly in
the same way as in steps (vi)–(vii) of the proof of Theorem 6.3 in [28], with the
only difference that here g
∗
h
= g
∗
h,λ,ν
= 0, and that the operator D
ν
in formula
(6.30) of [28] is to be replaced by D
λ,ν
, defined by
D
λ,ν
h := (λb
λ,ν
−ν|∇h),b
λ,ν
(t, x):=b(λt, x + tν). (4.25)
We note that D
1,0
=(b|∇·). In the same way as in [25, Lemma 8.2] one obtains
that
[(λ, ν) → b
λ,ν
]:(1−δ, 1+δ) × R
n
→ F
4
(a) (4.26)
is real analytic. The remaining arguments are now the same as in [28], and this
completes the proof of Theorem 4.2.
Proof of Theorem 1.1. Clearly, the compatibility conditions of Theorem 1.1 are
satisfied if and only if (4.6) is satisfied. Moreover, the smallness-boundedness con-
dition of Theorem 1.1 is equivalent to (4.7), where we have slightly abused notation
by using the same symbol for u
0
and its transformed version Θ
∗
h
0
u
0
.
Theorem 4.2 yields a unique solution (v, w, π, [π],h) ∈ E(t
0
) which satisfies
the additional regularity properties listed in part (b) of the theorem. Setting
(u, q)(t, x, y)=(v, w,π)(t, x, y − h(t, x)), (t, x, y) ∈O,
we then conclude that (u, q) ∈ C
ω
(O, R
n+2
)and[q] ∈ C
ω
(M). The regularity
properties listed in Remark 1.2(a) are implied by Proposition 5.1(a),(c). Finally,
since π(t, x, y) is defined for every (t, x, y) ∈O, we can conclude that
q(t, ·) ∈
˙
H
1
p
(Ω(t)) ⊂ UC (Ω(t))
for every t ∈ (0,t
0
).
Two-phase Navier-Stokes Equations 529
5. Appendix
In this section we state and prove some technical results that were used above.
Proposition 5.1. Suppose p>n+3. Then the following embeddings hold:
(a) E
1
(a) → BC(J; W
2−2/p
p
(
˙
R
n+1
, R
n+1
)) → BC(J; BC
1
(
˙
R
n+1
, R
n+1
)) and
there is a constant C
0
= C
0
(p) such that
u
0
BC(J ;W
2−2/p
p
)
+ u
0
BC(J ;BC
1
)
≤ C
0
u
0
E
1
(a)
for all u ∈
0
E
1
(a) and all a ∈ (0, ∞).
(b) E
3
(a) → BC(J; BC(R
n
)) and there exists a constant C
0
= C
0
(p) such that
g
0
BC(J;BC)
≤ C
0
g
0
E
3
(a)
for all g ∈
0
E
3
(a) and all a ∈ (0, ∞).
(c) F
4
(a) → BC(J; W
1
p
(R
n
)) ∩ BC(J; BC
1
(R
n
)) and there exists a constant
C
0
= C
0
(p) such that
g
0
BC(J;W
1
p
)
+ g
0
BC(J;BC
1
)
≤ C
0
g
0
F
4
(a)
for all g ∈
0
F
4
(a) and all a ∈ (0, ∞).
(d) E
4
(a) → BC
1
(J; BC
1
(R
n
)) ∩ BC(J; BC
2
(R
n
)) and there exists a constant
C
0
= C
0
(p) such that
h
0
BC
1
(J;BC
1
)
+ h
0
BC(J ;BC
2
)
≤ C
0
h
0
E
4
(a)
for all h ∈
0
E
4
(a) and all a ∈ (0, ∞).
(e) ∂
j
∈L(E
4
(a), E
3
(a)) ∩L(E
4
(a), F
4
(a)) for j =1,...,n. Moreover, for every
given a
0
> 0 there is a constant C
0
= C
0
(a
0
,p) such that
∂
j
h
E
3
(a)
+ ∂
j
h
F
4
(a)
≤ C
0
h
E
4
(a)
for all h ∈ E
4
(a) and all a ∈ (0,a
0
].
Proof. We refer to [25, Proposition 6.2] for a proof of (a)–(b). The assertion in
(c) can established in the same way, using that F
4
(a) → BC(J; W
2−3/p
p
(R
n
)), see
[19, Remark 5.3(d)]. In order to show that the embedding constant in (d) does not
depend on a ∈ (0,a
0
] we define an extension operator in the following way: for
h ∈
0
BC
1
([0,a]; X), with X an arbitrary Banach space, we first set
˜
h(t):=0for
t ≤ 0, so that
˜
h ∈ BC
1
((−∞,a]; X), and then define
(Eh)(t):=
h(t)if0≤ t ≤ a,
3
˜
h(2a − t) − 2
˜
h(3a −2t)ifa ≤ t.
(5.1)
A moment of reflection shows that Eh ∈
0
BC
1
([0, ∞); X), and that Eh is an ex-
tension of h. It is evident that the norm of E :
0
BC
1
([0,a]; X) →
0
BC
1
([0, ∞); X)
is independent of a ∈ [0,a
0
]. The assertion follows now by the same arguments as
in the proof of [25, Proposition 6.2].
530 J. Pr¨uss and G. Simonett
Let a
0
> 0 be fixed. In order to establish part (e) it suffices to show that there is
a constant C
0
= C(a
0
,p,r) such that
g
W
r
p
([0,a];X)
≤ C
0
g
H
1
p
([0,a];X)
,a∈ (0,a], (5.2)
where X is an arbitrary Banach space and r ∈ [0, 1]. This follows from Hardy’s
inequality as follows: for r ∈ (0, 1) fixed we have
1
2
g
p
W
r
p
([0,a];X)
=
a
0
a
s
g(t) − g(s)
p
X
(t − s)
1+rp
dt ds
=
a
0
a−s
0
g(s + τ) − g(s)
p
X
τ
1+rp
dτ ds
≤
a
0
a−s
0
1
τ
1+rp
τ
0
∂g(s + σ)
X
dσ
p
dτ ds
≤ c(r, p)
a
0
a−s
0
1
τ
1−(1−r)p
∂g(s + τ)
p
X
dτ ds
= c(r, p)
a
0
1
τ
1−(1−r)p
a−τ
0
∂g(s + τ)
p
X
ds dτ
≤ c(r, p)
a
0
1
τ
1−(1−r)p
dτ ∂g
p
L
p
([0,a];X)
where ∂g is the derivative of g, and this readily yields (5.2).
Our next result will be important in order to derive estimates for the nonlin-
earities in (2.1).
Lemma 5.2. Suppose p>n+3.Leta
0
∈ (0, ∞) be given. Then
(a) E
3
(a) is a multiplication algebra and we have the following estimate
g
1
g
2
E
3
(a)
≤ (g
1
∞
+ g
1
E
3
(a)
)(g
2
∞
+ g
2
E
3
(a)
) (5.3)
for all (g
1
,g
2
) ∈ E
3
(a) × E
3
(a) and all a>0.
(b) There exists a constant C
0
= C
0
(a
0
,p) such that
g
1
g
2
0
E
3
(a)
≤ C
0
(g
1
∞
+ g
1
E
3
(a)
)g
2
0
E
3
(a)
(5.4)
for all (g
1
,g
2
) ∈ E
3
(a) ×
0
E
3
(a) and all a ∈ (0,a
0
].
(c) There exists a constant C
0
= C
0
(a
0
,p) such that
g∂
j
h
0
E
3
(a)
≤ C
0
g
E
3
(a)
h
0
E
4
(a)
,j=1,...,n, (5.5)
for all (g, h) ∈ E
3
(a) ×
0
E
4
(a) and a ∈ (0,a
0
].
(d) Suppose (g, ψ) ∈ E
3
(a) × E
3
(a) and let β(t, x):=
1+ψ
2
(t, x).Then
g
β
k
∈
E
3
(a) for k ∈ N and the following estimate holds
#
#
#
#
g
β
k
#
#
#
#
E
3
(a)
≤ (1 + ψ
E
3
(a)
)
k
(g
∞
+ g
E
3
(a)
). (5.6)
Two-phase Navier-Stokes Equations 531
Proof. The assertions in (a)–(b) follow from (the proof of) Proposition 6.6.(ii) and
(iv) in [25].
(c) To economize our notation we set r =1/2 −1/2p and θ =1−1/p.
Suppose that (g, h) ∈ E
3
(a) ×
0
E
4
(a). We first observe that
g∂
j
h
W
r
p
(J;L
p
)
≤
g
L
p
(J;L
p
)
+ g
W
r
p
(J;L
p
∂
j
h
0
BC(J;L
∞
)
+
a
0
a
0
g(s)(∂
j
h(t) − ∂
j
h(s))
p
L
p
dt ds
|t − s|
1+rp
1/p
.
Using H¨older’s inequality, and the fact that (1 − r − 1/p)=r>0, we obtain the
estimate
a
0
a
0
g(s)(∂
j
h(t) − ∂
j
h(s))
p
L
p
dt ds
|t −s|
1+rp
≤
a
0
a
0
g(s)
p
L
∞
"
"
"
"
t
s
∂
t
∂
j
h(τ)
L
p
dτ
"
"
"
"
p
dt ds
|t − s|
1+rp
≤
a
0
g(s)
p
L
∞
a
0
dt
|t − s|
1−(1−r−1/p)p
ds
a
0
∂
t
∂
j
h(τ)
p
L
p
dτ
≤ C
0
(a
0
,p)g
p
L
p
(J;L
∞
)
∂
t
∂
j
h
p
L
p
(J;L
p
)
(5.7)
for a ∈ (0,a
0
]. Hence we conclude that
g∂
j
h
0
W
r
p
(J;L
p
)
≤ C
0
g
W
r
p
(J;L
p
)
∂
j
h
0
BC(J;L
∞
)
+ ∂
t
∂
j
h
L
p
(J;L
p
)
≤ C
0
g
E
3
(a)
h
0
E
4
(a)
(5.8)
uniformly in a ∈ (0,a
0
]. It is easy to verify that
g∂
j
h
L
p
(J;W
θ
p
)
≤g
L
p
(J;W
θ
p
)
∂
j
h
0
BC(J;L
∞
)
+ g
L
p
(J;L
∞
)
∂
j
h
0
BC(J;W
θ
p
)
≤ C
0
g
E
3
(a)
h
0
E
4
(a)
.
(5.9)
Combining the estimates (5.8)–(5.9) yields (5.5).
(d) As in the proof of Proposition 6.6.(v) in [25] we obtain
g/β
W
r
p
(J;L
p
)
≤1/β
∞
(g
L
p
(J;L
p
)
+ g
W
r
p
(J;L
p
)
)+g
∞
1/β
W
r
p
(J;L
p
)
≤ (1 + 1/β
W
r
p
(J;L
p
)
)(g
∞
+ g
W
r
p
(J;L
p
)
).
Thus it remains to estimate the term 1/β
W
r
p
(J;L
p
)
.Usingthatβ
2
(t, x)−β
2
(s, x)=
ψ
2
(t, x) − ψ
2
(s, x) one easily verifies that
"
"
"
"
1
β(s, x)
−
1
β(t, x)
"
"
"
"
=
"
"
"
"
β
2
(t, x) −β
2
(s, x)
β(s, x)β(t, x)(β(t, x)+β(s, x))
"
"
"
"
≤|ψ(t, x) −ψ(s, x)|
and this yields 1/β
W
r
p
(J;L
p
)
≤ψ
W
r
p
(J;L
p
)
.Consequently,
g/β
W
r
p
(J;L
p
)
≤ (1 + ψ
W
r
p
(J:L
p
)
)(g
∞
+ g
W
r
p
(J;L
p
)
).
532 J. Pr¨uss and G. Simonett
A similar argument shows that
g/β
L
p
(J;W
θ
p
)
≤ (1 + ψ
L
p
(J;W
θ
p
)
)(g
∞
+ g
L
p
(J;W
θ
p
)
).
Combining the last two estimates gives (5.6) for k = 1. The general case then
follows by induction.
Corollary 5.3. Suppose p>n+3.Leta
0
∈ (0, ∞) and k ∈ N with k ≥ 1 be given.
(a) There exists a constant C
0
= C
0
(a
0
,p,k) such that
(g
1
···g
k
)¯g
0
E
3
(a)
≤ C
0
k
F
i=1
g
i
∞
+ g
i
E
3
(a)
¯g
0
E
3
(a)
for all functions g
i
∈ E
3
(a), 1 ≤ i ≤ k, ¯g ∈
0
E
3
(a),andalla ∈ (0,a
0
].
(b) There exists a constant C
0
= C
0
(a
0
,p,k) such that
g(∂
1
h ···∂
k
h∂
j
¯
h)
0
E
3
(a)
≤ C
0
∇h
k
∞
+h
E
4
(a)
∇h
k−1
∞
+ ∇h
k
BC(J ;W
1−1/p
p
)
g
E
3
(a)
¯
h
0
E
4
(a)
≤ C
0
∇h
k
BC(J ;BC
1
)
+ h
E
4
(a)
∇h
k−1
∞
g
E
3
(a)
¯
h
0
E
4
(a)
for h ∈ E
4
(a),
¯
h ∈
0
E
4
(a), a ∈ (0,a
0
], 1 ≤ j ≤ n,and
i
∈{1,...n} with
i =1,...,k.
Proof. (a) follows from (5.4) by iteration.
(b) The first line in (5.8) shows that
g(∂
1
h ···∂
k
h∂
j
¯
h)
W
r
p
(J;L
p
)
≤ C
0
g
W
r
p
(J;L
p
)
(∂
1
h ···∂
k
h∂
j
¯
h
0
BC(J;L
∞
)
+ ∂
t
(∂
1
h ···∂
k
h∂
j
¯
h)
L
p
(J;L
∞
)
).
Next we note that
∂
1
h ···∂
k
h(∂
t
∂
j
¯
h)
L
p
(J;L
∞
)
≤∇h
k
∞
∂
t
∂
j
¯
h
L
p
(J;L
∞
)
,
and
∂
1
h···(∂
t
∂
i
h)···∂
k
h∂
j
¯
h
L
p
(J;L
∞
)
≤∂
t
∂
i
h
L
p
(J;L
∞
)
∇h
k−1
∞
∂
j
¯
h
0
BC(J;L
∞
)
.
Proposition 6.1(d) now implies the assertion for ·
W
r
p
(J;L
p
)
. On the other hand
we have by (5.9) for θ =1−1/p
g(∂
1
h ···∂
k
h∂
j
¯
h)
L
p
(J;W
θ
p
)
≤g
L
p
(J;W
θ
p
)
∂
1
h ···∂
k
h∂
j
¯
h
∞
+ g
L
p
(J;L
∞
)
∂
1
h ···∂
k
h∂
j
¯
h
0
BC(J ;W
θ
p
)
≤ C
0
g
E
3
(a)
∇h
k
∞
+ ∇h
k
BC(J;W
1−1/p
p
)
h
0
E
4
(a)
since W
θ
p
(R
n
) is a multiplication algebra. The last inequality then follows from
Sobolev’s embedding theorem.
Two-phase Navier-Stokes Equations 533
Remark 5.4. It can be shown that the estimate in (5.5) can be improved as follows:
For every a
0
∈ (0, ∞) there is a constant C
0
= C
0
(a
0
,p) > 0andaconstant
θ = θ(p) > 0 such that
g∂
j
h
0
E
3
(a)
≤ C
0
a
θ
g
E
3
(a)
h
0
E
4
(a)
holds for all (g,h) ∈ E
3
(a) ×
0
E
4
(a)anda ∈ (0,a
0
]. In the same way, the constant
C
0
in Corollary 5.3(b) can be replaced by C
0
a
θ
.
Lemma 5.5. Suppose p>n+3.Leta
0
∈ (0, ∞) be given. Then
(a) F
4
(a) is a multiplication algebra and we have the estimate
g
1
g
2
F
4
(a)
≤ C
a
g
1
F
4
(a)
g
2
F
4
(a)
for all (g
1
,g
2
) ∈ F
4
(a) × F
4
(a), where the constant C
a
depends on a.
(b) There exists a constant C
0
= C
0
(a
0
,p) such that
g
1
g
2
0
F
4
(a)
≤ C
0
(g
1
∞
+ g
1
F
4
(a)
)g
2
0
F
4
(a)
(5.10)
for all (g
1
,g
2
) ∈ F
4
(a) ×
0
F
4
(a) and all a ∈ (0,a
0
].
(c) There exists a constant C
0
= C
0
(a
0
,p) such that
g∂
j
h
0
F
4
(a)
≤ C
0
(g
∞
+ g
F
4
(a)
)h
0
E
4
(a)
,j=1,...,n, (5.11)
for all (g, h) ∈ F
4
(a) ×
0
E
4
(a) and a ∈ (0,a
0
].
Proof. Here we equip F
4
(a) with the (equivalent) norm
g
F
4
(a)
= g
W
1−1/2p
p
(J;L
p
)
+
n
!
i=1
∂
i
g
L
p
(J;W
1−1/p
p
)
. (5.12)
(a) This follows from Proposition 5.1(c) by similar arguments as in the proof of
Proposition 6.6(ii) and (iv) in [25].
(b) It follows from part (a) and Proposition 5.1(c) that
g
1
g
2
0
W
r
p
(J;L
p
)
≤ C
0
(g
1
∞
+g
1
W
r
p
(J;L
p
)
)g
2
0
F
4
(a)
, (g
1
,g
2
) ∈ F
4
(a)×
0
F
4
(a)
where r =1− 1/2p. Next, observe that again by Proposition 5.1(c)
(∂
i
g
1
)g
2
L
p
(J;W
θ
p
)
≤∂
i
g
1
L
p
(J;W
θ
p
)
g
2
0
BC(J;L
∞
)
+ ∂
i
g
1
L
p
(J;L
∞
)
g
2
0
BC(J;W
θ
p
)
≤ C
0
g
1
F
4
(a)
g
2
0
F
4
(a)
where θ =1−1/p.Moreover,
g
1
∂
i
g
2
L
p
(J;W
θ
p
)
≤g
1
L
p
(J;W
θ
p
)
∂
i
g
2
0
BC(J ;L
∞
)
+ g
1
∞
∂
i
g
2
L
p
(J;W
θ
p
)
≤ C
0
(g
1
∞
+ g
1
F
4
(a)
)g
2
0
F
4
(a)
.
The estimates above in conjunction with (5.12) yields (5.10).
(c) follows from (b) by setting g
2
= ∂
j
h and from Proposition 5.1(e), which cer-
tainly is also true for
0
E
4
(a).
534 J. Pr¨uss and G. Simonett
Proof of Proposition 4.1. It follows as in the proof of [28, Proposition 6.2] that
N
b
∈ C
ω
(E(a), F(a)), and moreover, that DH
b
(z) ∈L(
0
E(a),
0
F(a)) for z ∈ E(a).
It thus remains to prove the estimates stated in the proposition.
Without always writing this explicitly, all the estimates derived below will
be uniform in a ∈ (0,a
0
], for a
0
> 0 fixed. Moreover, all estimates will be uniform
for (¯u, ¯π, ¯q,
¯
h) ∈
0
E(a).
(i) Without changing notation we consider here the extension of h from R
n
to
R
n+1
defined by h(t, x, y)=h(t, x)for(x, y) ∈ R
n
× R and t ∈ J.Withthis
interpretation we have
∂h
∞,J×R
n+1
= ∂h
∞,J×R
n
,h∈ E(a),∂∈{∂
j
, Δ,∂
t
},
(5.13)
where ·
∞,U
denotes the sup-norm for the set U ⊂ J × R
n+1
. Next we observe
that
BC(J; BC(R
n+1
)) · L
p
(J; L
p
(R
n+1
)) → L
p
(J; L
p
(R
n+1
)),
BC(J; L
p
(R
n+1
)) · L
p
(J; BC(R
n+1
)) → L
p
(J; L
p
(R
n+1
)),
BC(J; BC(R
n+1
)) · BC(J; BC(R
n+1
)) → BC(J; BC(R
n+1
)),
(5.14)
that is, multiplication is continuous and bilinear in the indicated function spaces
(with norm equal to 1).
LetusfirstconsiderthetermF
1
(u, h):=|∇h|
2
∂
2
y
u appearing in the definition
of F .ItsFr´echet derivative at (u, h)isgivenby
DF
1
(u, h)[¯u,
¯
h]=|∇h|
2
∂
2
y
¯u +2(∇h|∇
¯
h)∂
2
y
u.
Suppose (¯u,
¯
h) ∈
0
E
1
(a) ×
0
E
4
(a). From (5.13), the first and third line in (5.14)
and Proposition 5.1(d) follows the estimate
DF
1
(u, h)[¯u,
¯
h]
0
F(a)
≤ C
0
∇h
∞
(∇h
∞
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
for all (u, h) ∈ E
1
(a) × E
4
(a). It is important to note that the constant C
0
does
not depend on the length of the interval J =(0,a)fora ∈ (0,a
0
].
Next, let us take a closer look at the term F
2
(u, h):=Δh∂
y
u in the definition
of F .TheFr´echet derivative is DF
2
(u, h)[¯u,
¯
h]=Δh∂
y
¯u +Δ
¯
h∂
y
u. We infer from
(5.13), the first and second line in (5.14), and Proposition 5.1 that
DF
2
(u, h)[¯u,
¯
h]
0
F(a)
≤ (Δh
L
p
(J;L
∞
)
+ ∂
y
u
L
p
(J;L
p
)
)·
(∂
y
¯u
0
BC(J;L
p
)
+ Δ
¯
h
0
BC(J;L
∞
)
)
≤ C
0
(h
E
4
(a)
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
for all (u, h) ∈ E
1
(a) × E
4
(a).
The derivative of F
3
(u, h):=(u|∇h)∂
y
u,where∇h := (∇h, 0), is given by
DF
3
(u, h)[¯u,
¯
h]=(¯u|∇h)∂
y
u +(u|∇h)∂
y
¯u +(u|∇
¯
h)∂
y
u
Two-phase Navier-Stokes Equations 535
and it follows from (5.13)–(5.14) and Proposition 5.1(a), (d) that there is a constant
C
0
> 0 such that
DF
3
(u, h)[¯u,
¯
h]
0
F(a)
≤ C
0
(∇h
∞
+ u
∞
)u
E
1
(a)
(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
for all (u, h) ∈ E
1
(a) × E
4
(a).
Let us also consider the term F
4
(u, h):=∂
t
h∂
y
u. Observing that
DF
4
(u, h)[¯u,
¯
h]=∂
t
h∂
y
¯u + ∂
t
¯
h∂
y
u,
that ∂
t
: E
4
(a) → F
4
(a) is linear and continuous and
F
4
(a) → L
p
(J; BC
1
(R
n
)) ∩ BC(J; BC
1
(R
n
))
(5.15)
we conclude from (5.13)–(5.15) and Proposition 5.1(a), (c) that there is a constant
C
0
= C
0
(a
0
) such that
DF
4
(u, h)[¯u,
¯
h]
0
F(a)
≤ (∂
t
h
L
p
(J;L
∞
)
+ ∂
y
u
L
p
(J;L
p
)
)
(∂
y
¯u
0
BC(J;L
p
)
+ ∂
t
¯
h
0
BC(J;L
∞
)
)
≤ C
0
(h
E
4
(a)
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
for all (u, h) ∈ E
1
(a) × E
4
(a).
The derivative of F
5
(π, h):=∂
y
π∇h is given by
DF
5
(π, h)[¯π,
¯
h]=∂
y
¯π∇h + ∂
y
π∇
¯
h.
It follows from (5.13)–(5.14) and Proposition 5.1(d) that there exists C
0
such that
DF
5
(π, h)[¯π,
¯
h]
0
F(a)
≤ (∇h
∞
+ ∂
y
π
L
p
(J;L
p
)
)
(∂
y
¯π
L
p
(J;L
p
)
+ ∇h
0
BC(J;L
∞
)
)
≤ C
0
(∇h
∞
+ π
E
2
(a)
)(¯π
0
E
2
(a)
+
¯
h
0
E
4
(a)
)
for all (π, h) ∈ E
2
(a) × E
4
(a). The remaining terms in the definition of F can be
analyzed in the same way. Summarizing we have shown that there is a constant
C
0
such that
DF(u, π, h)[¯u, ¯π,
¯
h]
0
F
1
(a)
≤C
0
5
∇h
∞
+ ∇h
2
∞
+ (u, π, h)
E(a)
+(∇h
∞
+ u
∞
)u
E
1
(a)
6
(¯u, ¯π,
¯
h)
0
E(a)
(5.16)
for all (u, π, h) ∈ E(a)andalla ∈ (0,a
0
].
(ii) We will now consider the nonlinear function F
d
(u, h)=(∇h|∂
y
v), stemming
from the transformed divergence. Since h(x, y):=h(x) does not depend on y we
have
F
d
(u, h)=(∇h|∂
y
u)=∂
y
(∇h|u). (5.17)
We note that
∂
y
∈L
H
1
p
(J; L
p
(R
n+1
)),H
1
p
(J; H
−1
p
(R
n+1
))
. (5.18)
536 J. Pr¨uss and G. Simonett
The norm of this linear mapping does not depend on the length of the interval
J =[0,a]. It is easy to see that multiplication is continuous in the following
function spaces:
H
1
p
(J; BC(R
n+1
)) ·H
1
p
(J; L
p
(R
n+1
)) → H
1
p
(J; L
p
(R
n+1
))
BC(J; BC
1
(
˙
R
n+1
)) · L
p
(J; H
1
p
(
˙
R
n+1
)) → L
p
(J; H
1
p
(
˙
R
n+1
)).
(5.19)
The derivative of F
d
at (u, h) ∈ E
1
(a) × E
4
(a)isgivenby
DF
d
(u, h)[¯u,
¯
h]=(∇h|∂
y
¯u)+(∇
¯
h|∂
y
u)=∂
y
((∇h|¯u)+(∇
¯
h|u)).
We want to derive a uniform estimate for DF
d
(u, h)[¯u,
¯
h] which does not depend
on the length of the interval J =[0,a]. We conclude from (5.13)–(5.15) that
(∇h|¯u)
0
H
1
p
(J;L
p
)
∼(∇h|¯u)
L
p
(J;L
p
)
+ (∂
t
∇h|¯u)
L
p
(J;L
p
)
+ (∇h|∂
t
¯u)
L
p
(J;L
p
)
≤∇h
∞
¯u
L
p
(J;L
p
)
+ ∂
t
∇h
L
p
(J;L
∞
)
¯u
0
BC(J;L
p
)
+ ∇h
∞
∂
t
¯u
L
p
(J;L
p
)
≤ C
0
(∇h
∞
+ h
E
4
(a)
)¯u
0
H
1
p
(J;L
p
)
.
Similar arguments also yield (∇
¯
h|u)
0
H
1
p
(J;L
p
)
≤ C
0
u
H
1
p
(J;L
p
)
¯
h
0
E
4
(a)
. These
estimates in combination with (5.18) show that there is a constant C
0
such that
(∇h|∂
y
¯u)+(∂
y
u|∇
¯
h)
0
H
1
p
(J;H
−1
p
)
≤ C
0
(∇h
∞
+ h
E
4
(a)
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
for all (u, h) ∈ E
1
(a) × E
4
(a), where C
0
is uniform in a ∈ (0,a
0
]. Observing that
(∇h|∂
y
¯u)
L
p
(J;L
p
)
+Σ
n+1
j=1
(∂
j
∇h|∂
y
¯u)+(∇h|∂
j
∂
y
¯u)
L
p
(J;L
p
)
defines an equivalent norm for (∇h|∂
y
¯u)
L
p
(J;H
1
p
)
, we infer once more from (5.13)–
(5.14) and Proposition 5.1 that
(∇h|∂
y
¯u)
L
p
(J;H
1
p
)
≤ C
0
(h
E
4
(a)
+ ∇h
∞
)¯u
0
E
1
(a)
(∇
¯
h|∂
y
u)
L
p
(J;H
1
p
)
≤ C
0
u
L
p
(J;H
2
p
)
¯
h
0
E
4
(a)
.
Summarizing we have shown that there exists a constant C
0
such that
DF
d
(u, h)[¯u,
¯
h]
0
F
2
(a)
≤ C
0
(∇h
∞
+ h
E
4
(a)
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
(5.20)
for all (u, h) ∈ E
1
(a) × E
4
(a)anda ∈ (0,a
0
].
(iii) We remind that
[[ μ∂
i
·]] ∈L
H
1
p
(J; L
p
(
˙
R
n+1
)) ∩ L
p
(J; H
2
p
(
˙
R
n+1
)), E
3
(a)
(5.21)
where [[μ∂
i
u]] denotes the jump of the quantity μ∂
i
u with u a generic function from
˙
R
n+1
→ R,andwhere∂
i
= ∂
x
i
for i =1,...,n and ∂
n+1
= ∂
y
.
The mapping G(u, q, h) is made up of terms of the form
[[ μ∂
i
u
k
]] ∂
j
h, [[ μ∂
i
u
k
]] ∂
j
h∂
l
h, q∂
j
h, Δh∂
j
h, G
κ
(h),G
κ
(h)∂
j
h
Two-phase Navier-Stokes Equations 537
where u
k
denotes the kth component of a function u ∈ E
1
(a). It follows from
Lemma 5.2(a) and (5.21) that the mappings
(u, h) → [[ μ∂
i
u
k
]] ∂
j
h, [[ μ∂
i
u
k
]] ∂
j
h∂
l
h : E
1
(a) × E
4
(a) → E
3
(a),
(q, h) → q∂
j
h : E
3
(a) × E
4
(a) → E
3
(a),h→ Δh∂
j
h : E
4
(a) → E
3
(a)
are multilinear and continuous.
LetusnowtakeacloserlookatthetermG
1
(u, h):=[[μ∂
i
u
k
]] ∂
j
h.ItsFr´echet
derivative is given by
DG
1
(u, h)[¯u,
¯
h]=∂
j
h[[ μ∂
i
¯u
k
]] + [[ μ∂
i
u
k
]] ∂
j
¯
h.
Setting g
1
= ∂
j
h and g
2
:= [[μ∂
i
¯u
k
]] we obtain from (5.4) and (5.21) the estimate
∂
j
h[[ μ∂
i
¯u
k
]]
0
E
3
(a)
≤ C
0
(∇h
∞
+ ∇h
E
3
(a)
)¯u
0
E
1
(a)
.
On the other hand, setting g := [[μ∂
i
u
k
]] we conclude from (5.5) and (5.21) that
[[ μ∂
i
u
k
]] ∂
j
¯
h
0
E
3
(a)
≤ C
0
u
E
1
(a)
¯
h
0
E
4
(a)
.
Consequently,
DG
1
(u, h)[¯u,
¯
h]
0
E
3
(a)
≤ C
0
(∇h
∞
+∇h
E
3
(a)
+ u
E
1
(a)
)(¯u
0
E
1
(a)
+
¯
h
0
E
4
(a)
)
(5.22)
for all (u, h) ∈ E
1
(a) × E
4
(a), and all a ∈ (0,a
0
].
Given (u, h) ∈ E
1
(a)×E
4
(a)letG
2
(u, h):=[[μ∂
i
u
k
]] ∂
j
h∂
l
h.TheFr´echet derivative
of G
2
is given by
DG
2
(u, h)[¯u,
¯
h]=∂
j
h∂
l
h[[ μ∂
i
¯u
k
]] + [[ μ∂
i
u
k
]] ∂
j
h∂
j
¯
h +[[μ∂
i
u
k
]] ∂
l
h∂
j
¯
h.
From Corollary 5.3(a),(b) and (5.21) follows that there is a constant C
0
such that
DG
2
(u, h)[¯u,
¯
h]
0
E
3
(a)
≤ C
0
(∇h
∞
+ h
E
4
(a)
)
2
¯u
0
E
1
(a)
+ C
0
∇h
BC(J;BC
1
)
+ h
E
4
(a)
u
E
1
(a)
¯
h
0
E
4
(a)
)
(5.23)
for all (u, h) ∈ E
1
(a) × E
4
(a)andalla ∈ (0,a
0
].
The terms G
3
(q, h):=q∂
j
h and G
4
(h):=Δh∂
j
h canbeanalyzedinthesameway
as the term G
1
, yielding the following estimates
DG
3
(q, h)[¯q,
¯
h]
0
E
3
(a)
≤ C
0
(∇h
∞
+ ∇h
E
3
(a)
+ q
E
3
(a)
)(¯q
0
E
3
(a)
+
¯
h
0
E
4
(a)
)
(5.24)
as well as
DG
4
(h)
¯
h
0
E
3
(a)
≤ C
0
(∇h
∞
+ ∇h
E
3
(a)
+ ∇
2
h
E
3
(a)
)
¯
h
0
E
4
(a)
.
(5.25)
Let us now consider the term
G
5
(h)=
1
(1 + β)β
(∂
j
h)
2
Δh, β(t, x):=
1+|∇h(t, x)|
2
,