Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 856896, 15 pages
doi:10.1155/2011/856896
Research Article
Lr -Lp Stability of the Incompressible Flows with
Nonzero Far-Field Velocity
Jaiok Roh
Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea
Correspondence should be addressed to Jaiok Roh, joroh@dreamwiz.com
Received 16 June 2011; Accepted 15 September 2011
Academic Editor: Ibrahim Sadek
Copyright q 2011 Jaiok Roh. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the stability of stationary solutions w for the exterior Navier-Stokes flows with a nonzero constant velocity u∞ at infinity. For u∞ 0 with nonzero stationary solution w, Chen 1993,
Kozono and Ogawa 1994, and Borchers and Miyakawa 1995 have studied the temporal stability
in Lp spaces for 1 < p and obtained good stability decay rates. For the spatial direction, we recently
0, Heywood 1970, 1972 and Masuda 1975 have studied the
obtained some results. For u∞ /
temporal stability in L2 space. Shibata 1999 and Enomoto and Shibata 2005 have studied the
temporal stability in Lp spaces for p ≥ 3. Then, Bae and Roh recently improved Enomoto and
Shibata’s results in some sense. In this paper, we improve Bae and Roh’s result in the spaces Lp for
p > 1 and obtain Lr -Lp stability as Kozono and Ogawa and Borchers and Miyakawa obtained for
u∞ 0.
1. Introduction
The motion of nonstationary flow of an incompressible viscous fluid past an isolated rigid
body is formulated by the following initial boundary value problem of the Navier-Stokes
equations:
∂
u − Δu u · ∇u ∇p f,
∂t
u|t0 u0 ,
u|∂Ω 0,
∇ · u 0 in Ω × 0, ∞,
lim ux, t u∞ ,
1.1
|x| → ∞
where Ω is an exterior domain in Rn with a smooth boundary ∂Ω, and u∞ denotes a given
constant vector describing the velocity of the fluid at infinity. In this paper, we consider a
nonzero constant u∞ . The physical model of the exterior Navier-Stokes equations with a
nonzero constant u∞ can be considered as the motion of water in the sea when a boat is
2
Abstract and Applied Analysis
moving with the speed −u∞ , while the one with zero constant u∞ can be considered when a
0, while, with u∞ 0, many
boat is stopped. There are few known results for the case u∞ /
results were obtained for the temporal decay and weighted estimates of solutions of 1.1
refer 1–12.
Now, we set u u∞ v in 1.1 and have
∂
v − Δv u∞ · ∇v v · ∇v ∇p1 f,
∂t
v|t0 u0 − u∞ ,
v|∂Ω −u∞ ,
∇ · v 0 in Ω × 0, ∞,
lim vx, t 0.
1.2
|x| → ∞
Consider the following linear problem:
∂
u − Δu u∞ · ∇u ∇p 0,
∂t
u|t0 u0 ,
u|∂Ω 0,
∇·u0
in Ω × 0, ∞,
lim ux, t 0,
1.3
|x| → ∞
which is referred to as the Oseen equations; see 13.
In order to formulate the problem 1.3, Enomoto and Shibata 14 used the Helmholtz
decomposition:
Lp Ωn Jp Ω ⊕ Gp Ω,
1.4
where 1 < p < ∞,
Lp Ωn u u1 , . . . , un : uj ∈ Lp Ω, j 1, . . . , n ,
∞
u u1 , . . . , un ∈ C0∞ Ωn : ∇ · u 0 in Ω ,
C0,σ
∞
Jp Ω the completion of C0,σ
Ω, in Lp Ωn ,
Gp Ω ∇π ∈ Lp Ωn : π ∈ Lp,loc Ω .
1.5
The Helmholtz decomposition of Lp Ωn was proved by Fujiwara and Morimoto 15,
Miyakawa 16, and Simader and Sohr 17. Let P be a continuous projection from Lp Ωn
onto Jp Ωn .
By applying P into 1.3 and setting Ou∞ P −Δ u∞ · ∇, one has
ut Ou∞ u 0,
for t > 0, u0 u0 ,
1.6
where the domain of Ou∞ is given by
Dp Ou∞ u ∈ Jp Ω ∩ Wp2 Ωn : u|∂Ω 0 .
1.7
Abstract and Applied Analysis
3
Then, Enomoto and Shibata 14 proved that Ou∞ generates an analytic semigroup {T t}t≥0
which is called the Oseen semigroup one can also refer to 16, 18 and obtained the following properties.
Proposition 1.1. Let σ0 > 0 and assume that |u∞ | ≤ σ0 . Let 1 ≤ r ≤ q ≤ ∞, then
T ta
Lq Ω ≤ Cr,q,σ0 t−3/21/r−1/q a
Lr Ω ,
t > 0,
1.8
where r, q / 1, 1 and ∞, ∞,
∇T ta
Lq Ω ≤ Cr,q,σ0 t−3/21/r−1/q−1/2 a
Lr Ω ,
t > 0,
1.9
where 1 ≤ r ≤ q ≤ 3 and r, q /
1, 1.
The main purpose of this paper is to discuss the temporal stability of stationary solution w of the nonlinear Navier-Stokes equation 1.2. One can note that w satisfies the following equations:
−Δw u∞ · ∇w w · ∇w ∇p2 f,
w|∂Ω −u∞ ,
∇ · w 0,
lim wx 0.
1.10
|x| → ∞
For suitable f, Shibata 19 proved that, for any given 0 < δ < 1/4, there exists such
that if 0 < |u∞ | ≤ , then one has
w
p,2 |
w
|δ p2 p,1 ≤ |u∞ |β ,
1.11
where
u
p,m ∂m u
Lp Ω ,
|
u
|δ sup1 |x|1 su∞ xδ |ux| sup1 |x|3/2 1 su∞ x1/2δ |∇ux|,
x∈Ω
x∈Ω
su∞ x |x| − xT ·
u∞
|u∞ |
1.12
δ < β < 1 − δ.
Throughout this paper, we assume that f satisfies the assumption in Shibata 19. Now, we
consider the polar coordinate system
y1 r cos θ,
y2 r sin θ cos φ,
y3 r sin θ sin φ,
1.13
for 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, and 0 ≤ r < ∞. Let S be an orthogonal matrix such that Su∞ |u∞ |1, 0, 0T and put sy |y| − y1 . By a change of variable y Sx,
|x| y r,
su∞ x s y r1 − cos θ.
1.14
4
Abstract and Applied Analysis
See Shibata 19 for the detail. Now, by using the above change of variable, we can see easily
that w satisfies
w
L3/1δ1 Ω w
L3/1−δ2 Ω ∇w
L3/2δ1 Ω ∇w
L3/2−δ2 Ω ≤ C|u∞ |1/2 ,
1.15
for small δ1 , δ2 , where C is independent on u∞ .
One can also refer to 20 for more general cases of the existence and regularity of
stationary Navier-Stokes equations.
For the stability of stationary solutions w, by setting u v − w and p p1 − p2 for v, p1 ,
w, p2 in 1.2 and 1.10, we have the following equations in Ω:
∂
u − Δuu∞ · ∇uu · ∇ww · ∇u u · ∇u∇p 0,
∂t
∇ · u 0,
ux, 0 u0 x
ux, t 0 for x ∈ ∂Ω,
for x ∈ Ω,
1.16
lim ux, t 0.
|x| → ∞
Here, in fact, the initial data should be u0 − u∞ − w, but for our convenience, we denote by u0
for u0 − u∞ − w if there is no confusion.
First, Heywood 21, 22 and Masuda 23 have studied the temporal stability in L2
space. Shibata 19 proved that there exists small such that if 0 < |u∞ | ≤ and u0 3 ≤ ,
then a unique solution ux, t of 1.16 has the following properties: for any 3 < p < ∞,
√
u3,0,t up,μp,t ∇u3,1/2,t ≤ ,
lim ut − u0 3 up,μp,t ∇u3,1/2,t 0,
1.17
t→0
where
zp,ρ,t sup sρ zs, ·
p ,
0<s<t
1
3
.
μ p −
2 2p
1.18
After that, Enomoto and Shibata 14 considered the stability for arbitrary u∞ by
deleting the smallness condition of |u∞ |. But in this case, all constants in their results depend
on σ0 when |u∞ | ≤ σ0 . Also, they assumed the existence of stationary solution w with
w
L3/1δ1 Ω w
L3/1−δ2 Ω ∇w
L3/2δ1 Ω ∇w
L3/2−δ2 Ω ≤ α,
1.19
for small δ1 , δ2 and α. Then, as a result, they proved 1.16 has a unique solution ux, t with
lim ut − u0 3 t1/2 ut
L∞ ∇ut
L3 0,
t→0
ut
ut o t−1/2−3/2p , for any 3 ≤ p ≤ ∞,
∇ut
3 o t−1/2
as t → ∞ when u0 is small enough in the space L3 Ω.
1.20
Abstract and Applied Analysis
5
Also, Bae and Roh 24 improved Enomoto-Shibata’s result in some sense. But their
result is limited in the space Lp for 3/2 < p, while we consider all 1 < p. Moreover, their result
depends on s and r, while ours only depends on r, where w ∈ Ls and u0 ∈ Lr . Also, their
optimal decay rate is 2/3 δ, while ours is 3/2 δ.
Now, in the next main Theorem, we settle the temporal stability of stationary solutions
for the Navier-Stokes equations with a nonzero constant vector at infinity. The idea of the
proof is initiated by Kato 25 for w 0 and a very well-known method. Also, for w /
0 with
u∞ 0, Kozono and Ogawa 12 also used similar method.
Theorem 1.2. There exists small p, q, r such that if 0 < |u∞ | ≤ and u0 L3 Ω < , then a unique
solution ux, t of 1.16 has the following properties:
ut
Lp Ω ≤ C t−3/21/r−1/p u0 r
for 1 < r < p ≤ ∞, t > 0,
∇ut
Lq Ω ≤ C t−3/21/r−1/q−1/2 u0 r
1.21
for 1 < r < q ≤ 3, t > 0,
where u0 ∈ L3 Ω ∩ Lr Ω.
2. Proof of Main Theorem
First, we consider the following linear problem:
∂
u − Δu u∞ · ∇u w · ∇u u · ∇w ∇p 0,
∂t
2.1
∇ · u 0,
u|t0 u0 ,
u|∂Ω 0,
lim ux, t 0.
|x| → ∞
By applying Helmholtz-Leray projection P and setting
Lu P −Δu u∞ · ∇u w · ∇u u · ∇w
Ou∞ u P w · ∇u u · ∇w,
2.2
we have
ut Lu 0,
for t > 0, u0 u0 .
2.3
And we note that the domain of L is
Dp L Dp Ou∞ u ∈ Jp Ω ∩ Wp2 Ωn |u|∂Ω 0 .
2.4
6
Abstract and Applied Analysis
Let St be a semigroup generated by the linear operator L, then, by Duharmel’s
Principle, a solution ux, t of 2.1 can be written as in the following integral form,
ux, t Stu0 T tu0 t
2.5
T t − τP w · ∇u u · ∇w dτ,
0
where T t is an analytic semigroup generated by the Oseen operator Ou∞ .
Lemma 2.1. Let u0 ∈ L3 Ω ∩ Lr Ω for 1 < r < 3, then there exists a small p, q, r such that if
|u∞ | ≤ and u0 L3 Ω < , then a solution ux, t represented by 2.5 satisfies 1 < p ≤ ∞ with
1/r − 1/p < 2/3,
ut
Lp Ω Stu0 Lp Ω ≤ C t−3/21/r−1/p u0 Lr Ω ,
2.6
t > 0,
and for 1 < q ≤ 3 with 1/r − 1/q < 1/3,
∇ut
Lq Ω ∇Stu0 Lq Ω ≤ C t−3/21/r−1/q−1/2 u0 Lr Ω ,
t > 0.
2.7
Proof. Before we prove Lemma 2.1 note from 1.15 that we have
w
L3/1δ1 Ω w
L3/1−δ2 Ω ∇w
L3/2δ1 Ω ∇w
L3/2−δ2 Ω ≤ C|u∞ |1/2 ,
2.8
for small δ1 , δ2 > 0. In fact, by straight calculations, we can choose any δ1 , δ2 ≤ 3/16.
Step 1. Let 3 < p ≤ ∞ with 1/3 ≤ 1/r − 1/p < 2/3 and 3/2 < q ≤ 3 with 1/r − 1/q < 1/3. We
consider the following iteration method to obtain our estimates:
uk1 t T tu0 t
2.9
T t − τP w · ∇uk uk · ∇wdτ.
0
We let 1/q − 1/p 1/3 and
Mpk sup tn/21/r−1/p uk t ,
p
t∈0,∞
Nqk sup tn/21/r−1/q1/2 ∇uk t .
q
t∈0,∞
2.10
If t ≥ 2, then by Proposition 1.1, for small δ1 , δ2 > 0, we have
t
0
T t − τP w · ∇uk uk · ∇w
p dτ
≤C
t−1
t − τ
−n/21/r1 −1/p
0
t−1
0
t − τ
w · ∇uk r1 dτ −n/21/r1 −1/p
t
t−1
uk · ∇w
r1 dτ t − τ−n/21/r2 −1/p w · ∇uk r2 dτ
t
t−1
t − τ
−n/21/r2 −1/p
uk · ∇w
r2 dτ
Abstract and Applied Analysis
t−1
≤
C|u∞ |Nqk
C|u∞ |Mpk
t − τ
−1−δ1 /2 −3/21/r−1/q−1/2
τ
dτ 7
t
0
t − τ−1δ2 /2 τ −3/21/r−1/q−1/2 dτ
t−1
t−1
t − τ
−1−δ1 /2 −3/21/r−1/p
τ
dτ 0
t
t − τ
−1δ2 /2 −3/21/r−1/p
τ
dτ
t−1
≤ C|u∞ | Mpk Nqk t−3/21/r−1/p ,
2.11
where 1/r1 1/p 2/3 δ1 /3 and 1/r2 1/p 2/3 − δ2 /3. If 0 < t < 2, then we have
t
0
T t − τP w · ∇uk uk · ∇w
p dτ
≤C
t
t − τ
−n/21/r3 −1/p
0
w · ∇uk r3 dτ t
t − τ
−n/21/r3 −1/p
0
uk · ∇w
r3 dτ
≤ C|u∞ | Mpk Nqk t−3/21/r−1/p ,
2.12
where 1/r3 1/p 2/3 − δ2 /3. So, we obtain
uk1 t
p ≤ Ct−n/21/r−1/p u0 r C|u∞ |t−3/21/r−1/p Mpk Nqk ,
∀t > 0,
2.13
which implies
Mpk1 ≤ C
u0 r C|u∞ | Mpk Nqk .
2.14
Similarly, we obtain for t ≥ 2,
∇uk1 t
q ≤ ∇T tu0 q t
0
∇T t − τP w · ∇uk uk · ∇w
q dτ
≤ Ct−n/21/r−1/q−1/2 u0 r C|u∞ |Nqk
C|u∞ |Nqk
C|u∞ |Mpk
t
t−1
t − τ−1−δ1 /2 τ −3/21/r−1/q−1/2 dτ
0
t − τ−1δ2 /3 τ −3/21/r−1/q−1/2 dτ
t−1
t
0
t − τ−n/21/r4 −1/q−1/2 τ −3/21/r−1/p dτ
≤ Ct−n/21/r−1/q−1/2 u0 r C|u∞ |t−3/21/r−1/q−1/2 Mpk Nqk ,
2.15
8
Abstract and Applied Analysis
where 1/r4 2/3 1/p 1/3 1/q. Also, for 0 < t < 2, we have
t
0
∇T t − τP w · ∇uk uk · ∇w
q dτ
≤ C|u∞ |
Mpk
Nqk
t
−3/21/r−1/q−1/2δ2 /2
≤ C|u∞ |
Mpk
Nqk
2.16
t
−3/21/r−1/q−1/2
.
Therefore, we get
Mpk1 Nqk1 ≤ C
u0 r C|u∞ | Mpk Nqk .
2.17
So if C|u∞ | < 1 the constant C is bounded as |u∞ | goes to zero, so we can make C|u∞ | < 1 by
choosing small u∞ , then we have some K such that
Mpk1 Nqk1 < K,
2.18
for all k. Hence, by taking the limit, we complete the proof.
Step 2. Now, we want to prove 1 < r < p ≤ 3. For this case, we choose 3/2 < q ≤ 3 and p1 > 3
such that
1 1 1
− < ,
r q 3
1
1
2
−
< .
r p1 3
2.19
Then, we have
ut
p ≤ T tu0 p t
0
T t − τP w · ∇u u · ∇w
p dτ
≤ Ct−3/21/r−1/p u0 r C
C
t
0
≤ C t
t
0
t − τ−3/21/r1 −1/p w
3 ∇u
q dτ
2.20
t − τ−3/21/r2 −1/p u
p1 ∇w
3/2 dτ
−3/21/r−1/p
u0 r ,
where 1/r1 1/3 1/q and 1/r2 1/p1 2/3. One can note that 1/r1 − 1/p < 2/3 and
1/r2 − 1/p < 2/3.
Step 3. Now, we want to prove 1 < r < q ≤ 3/2. For this case, we choose 3/2 < q1 ≤ 3 and
p > 3 such that
1 1
1
−
< ,
r q1 3
1 1 2
− < .
r p 3
2.21
Abstract and Applied Analysis
9
Similar to Step 2, we have
∇ut
q ≤ ∇T tu0 q t
0
∇T t − τP w · ∇u u · ∇w
q dτ
−3/21/r−1/q−1/2
≤ Ct
C
t
0
u0 r C
t
0
t − τ−3/21/r1 −1/q−1/2 w
3 ∇u
q1 dτ
2.22
t − τ−3/21/r2 −1/q−1/2 u
p ∇w
3/2 dτ
−3/21/r−1/q−1/2
≤ Ct
u0 r ,
where 1/r1 1/3 1/q1 and 1/r2 1/p 2/3. One can note that 1/r1 − 1/q < 1/3 and
1/r2 − 1/q < 1/3.
Step 4. At last, we want to prove 3 < p < ∞ with 1/r − 1/p < 1/3. In this case, we can do
easily, by interpolation inequality, Steps 1 and 2.
Therefore, we complete the proof by Steps 1–4.
Now, by applying the Helmholtz-Leray projection P into 1.16, we can obtain
ut Lu P u · ∇u 0,
for t > 0, u0 u0 ,
2.23
where
Lu P −Δu u∞ · ∇u w · ∇u u · ∇w
Ou∞ u P w · ∇u u · ∇w,
Dp L Dp Ou∞ u ∈ Jp Ω ∩ Wp2 Ωn |u|∂Ω 0 .
2.24
One can note from of 14, Lemma 2.6 that for 1 < p < ∞ and u ∈ Dp L Dp Ou∞ ,
u
W 2,p Ω ≤ Cp Ou∞ u
p u
p .
2.25
w · ∇u u · ∇w
p ≤ w
∞ ∇w
∞ u
W 2,p Ω
≤ |u∞ |
u
W 2,p Ω ≤ Cp |u∞ | Ou∞ u
p u
p .
2.26
Also, from 1.11, we have
Since the linear operator Ou∞ generates an analytic semigroup T t refer to 14, 19, we
obtain an analytic semigroup St generated by the linear operator L if |u∞ | is small enough.
The proof is from perturbation theory of analytic semigroup refer to 26, Theorem 2.4, page
499.
10
Abstract and Applied Analysis
Remark 2.2. In Lemma 2.1, by the property of a semigroup, we can remove the conditions
1/r − 1/p < 2/3 for ut
Lp Ω and 1/r − 1/p < 1/3 for ∇ut
Lp Ω , because we have ux, t
Stu0 St/2St/2u0 .
Now, we are in the position to prove Theorem 1.2. For the proof, we consider a solution
ux, t 1.16 as the limit of the following usual iteration method:
uk1 t Stu0 −
t
St − τP uk · ∇uk dτ.
2.27
0
Here, we will prove by a similar method with the proof of Lemma 2.1. One can note
that we will prove without Remark 2.2.
Step 1. We prove that, for any p > 3, we have
∇ut
3 < Ct−1/2 ,
ut
p < Ct−1/23/2p ,
∀t > 0.
2.28
Let
Mpk sup t1/2−3/2p uk t ,
p
t∈0,∞
for p > 3,
N3k sup t1/2 ∇uk t .
2.29
3
t∈0,∞
By Lemma 2.1 and 2.27, we obtain
uk1 t
p ≤ Ct
−1/23/2p
u0 3 C
t
0
t − τ−1/2 uk t
p ∇uk t
3 dτ
≤ Ct−1/23/2p u0 3 CMpk N3k
t
t − τ−1/2 τ −1/23/2p τ −1/2 dτ
2.30
0
≤ t−1/23/2p C
u0 3 CMpk N3k ,
which implies
Mpk1 ≤ C
u0 3 CMpk N3k .
2.31
Similarly, we have
∇uk1 t
3 ≤ Ct−1/2 u0 3 C
t
0
t − τ−3/2p−1/2 uk t
p ∇uk t
3 dτ ≤ t−1/2 C
u0 3 CMpk N3k ,
2.32
Abstract and Applied Analysis
11
which implies
N3k1 ≤ C
u0 3 CMpk N3k .
2.33
2
Mpk1 N3k1 < C
u0 3 C Mpk N3k .
2.34
Hence, we have
Now, we have a sequence of the form
2.35
xk1 ≤ α βxk2 ,
and we know that such sequence satisfies
1/2
1 − 1 − 4αβ
1
<
,
xk ≤ M ≡
2β
2β
if α <
1
.
4β
2.36
Therefore, by recurrence estimates, smallness of u0 3 implies
Mpk1 N3k1 < K,
2.37
for some constant K. Finally, we obtain
∇ut
3 < Ct−1/2 ,
ut
p < Ct−1/23/2p ,
∀t > 0.
2.38
Step 2. We prove that if 3/2 < p with 1/r − 1/p < 1/3 and u0 ∈ Lr Ω ∩ L3 Ω, then we have
ut
p ≤ Ct−3/21/r−1/p ,
∀t > 0.
2.39
Let
Mp sup t3/21/r−1/p ut
p .
t∈0,∞
2.40
From estimates of Step 1, one can note that we have
∇ut
3 ≤ Ct−1/2 u0 3 ,
∀t > 0.
2.41
12
Abstract and Applied Analysis
So, we have
ut
p ≤ Ct
−3/21/r−1/p
≤ Ct
−n/21/r−1/p
−n/21/r−1/p
≤t
u0 r C
t
0
t − τ−n/21/r8 −1/p ut
p ∇ut
3 dτ
u0 r C
u0 3
t
t − τ−1/2 τ −n/21/r−1/p τ −1/2 dτ
2.42
0
C
u0 r C
u0 3 Mp ,
which implies
Mp < C
u0 r C
u0 3 Mp ,
2.43
where 1/r8 1/3 1/p.
Hence, we complete the proof with C
u0 3 < 1.
Step 3. We prove that if 3/2 < q ≤ 3 with 1/r − 1/q < 1/3 and u0 ∈ Lr Ω ∩ L3 Ω, then we
have
∇ut
q ≤ Ct−3/21/r−1/q−1/2 ,
∀t > 0.
2.44
Let
Nq sup tn/21/r−1/q1/2 ∇ut
q .
t∈0,∞
2.45
We choose some p1 ≈ 3 with p1 > 3 such that
∇u
q ≤ Ct−n/21/r−1/q−1/2 u0 r C
≤ Ct
−n/21/r−1/q−1/2
−n/21/r−1/q−1/2
≤t
t
0
t − τ−n/21/r7 −1/q−1/2 u
p1 ∇u
q dτ
u0 r C
u0 3 Nq
t
t − τ−1/2−3/2p τ −1/23/2p1 τ −n/21/r−1/q−1/2 dτ
0
C
u0 r C
u0 3 Nq .
2.46
So we complete the proof with C
u0 3 < 1.
Step 4. We prove that if 1 < r < p < ∞, 1 < r < 3, and u0 ∈ Lr Ω ∩ L3 Ω, then we have
ut
p ≤ Ct−3/21/r−1/p ,
∀t > 0.
2.47
Case 1 let p > 3/2. Since we proved for 1/r − 1/p < 1/3 in Step 2, we can assume that
1/3 ≤ 1/r − 1/p. One notes that we can rewrite solutions ut in the form
ut S
t
t
t
u
−
St − τP u · ∇udτ.
2
2
t/2
2.48
Abstract and Applied Analysis
13
For any r > 1, we choose l > 3/2 such that 1/r − 1/l < 1/3 and 1/l − 1/p < 2/3. Also, for any
1 < r < p ≤ ∞ with 1 < r < 3, we choose s1 > 3 and 3/2 < s2 < 3 such that
1 1
1
−
< ,
r s2 3
1
1
1 2
− < .
s1 s2 p 3
2.49
Then, by Steps 1–3, we have
t
t C
u
t − τ−3/21/s−1/p u · ∇u
s dτ
ut
p ≤ Ct−3/21/l−1/p 2 t/2
l
≤ Ct−3/21/r−1/p u0 r C
u0 r
≤ Ct−3/21/r−1/p u0 r ,
t
t − τ−3/21/s−1/p τ −1/2−3/21/r−1/s2 τ −1/23/2s1 dτ
t/2
∀t > 0.
2.50
Case 2 let 1 < p ≤ 3/2. By Step 1–3, we have
ut
p ≤ Ct−3/21/r−1/p u0 r C
t
0
t − τ−3/21/s−1/p u · ∇u
s dτ
≤ Ct−3/21/r−1/p u0 r C
u0 r
≤ Ct−3/21/r−1/p u0 r ,
t
t − τ−3/21/s−1/p τ −3/21/r−1/s1 τ −1/2 dτ
2.51
0
∀t > 0,
where s1 > 3/2, 1/r − 1/s1 < 1/3, 1/s 1/s1 1/3.
Step 5. We prove that if 1 < r < q ≤ 3 and u0 ∈ Lr Ω ∩ L3 Ω, then
∇ut
q ≤ Ct−3/21/r−1/q−1/2 .
2.52
Case 1 let 3/2 < q ≤ 3. Since we proved 1/r − 1/q < 1/3 in Step 3, we can assume that
1/3 ≤ 1/r − 1/q. Now, we choose l > 3/2 such that 1/r − 1/l < 1/3 and 1/l − 1/q < 1/3. We
also can have s1 > 3 and 3/2 < s2 < 3 with
1
1
1
,
s s1 s2
1 1
1
−
< ,
r s2 3
1 1 1
− < .
s q 3
2.53
14
Abstract and Applied Analysis
So, by Step 1–4, we obtain
t
t C
u
t − τ−3/21/s−1/q−1/2 ut
s1 ∇ut
s2 dτ
∇ut
q ≤ Ct−3/21/l−1/q−1/2 2 t/2
l
≤ Ct−3/21/r−1/q−1/2 u0 r C
u0 r
×
t
t − τ−3/21/s−1/q−1/2 τ −1/23/2s1 τ −3/21/r−1/s2 −1/2 dτ
t/2
≤ Ct−3/21/r−1/q−1/2 u0 r .
2.54
Case 2 Let 1 < q ≤ 3/2. By Step 1-Step 3, we have
∇ut
q ≤ Ct−3/21/r−1/q−1/2 u0 r C
−3/21/r−1/q−1/2
≤ Ct
t
0
t − τ−3/21/s−1/q−1/2 u · ∇u
s dτ
u0 r C
u0 r
≤ Ct−3/21/r−1/q−1/2 u0 r ,
t
t − τ−3/21/s−1/q−1/2 τ −3/21/r−1/s1 τ −1/2 dτ
0
∀t > 0,
2.55
where s1 > 3/2, 1/r − 1/s1 < 1/3, 1/s 1/s1 1/3, and 1/s − 1/q < 1/3.
Therefore, by Step 1–5, we complete the proof of Theorem 1.2.
Acknowledgment
The author would like to thank Professor Hyeong-Ohk Bae for valuable comments. The
author also thanks deeply Professors Cheng He and Lizhen Wang for the preprint 27
“Weighted Lp -Estimates for Stokes Flow in Rn , with Applications to the Non-Stationary
Navier-Stokes Flow” which motivates the idea of the proof. This research was supported by
Basic Science Research Program through the National Research Foundation of Korea NRF
funded by the Ministry of Education, Science and Technology 2010-0023386.
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