Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 369571, 15 pages
doi:10.1155/2011/369571
Research Article
Asymptotic Behavior of the Navier-Stokes
Equations 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 18 June 2011; Revised 7 October 2011; Accepted 17 October 2011
Academic Editor: Weiqing Ren
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.
Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the
temporal stability has been studied by Heywood 1970, 1972 and Masuda 1975 in L2 space and
by Shibata 1999 and Enomoto-Shibata 2005 in Lp spaces for p ≥ 3. However, their results did
not include enough information to find the spatial decay. So, Bae-Roh 2010 improved EnomotoShibata’s results in some sense and estimated the spatial decay even though their results are
limited. In this paper, we will prove temporal decay with a weighted function by using Lr − Lp
decay estimates obtained by Roh 2011. Bae-Roh 2010 proved the temporal rate becomes slower
by 1 σ/2 if a weighted function is |x|σ for 0 < σ < 1/2. In this paper, we prove that the
temporal decay becomes slower by σ, where 0 < σ < 3/2 if a weighted function is |x|σ . For the
proof, we deduce an integral representation of the solution and then establish the temporal decay
estimates of weighted Lp -norm of solutions. This method was first initiated by He and Xin 2000
and developed by Bae and Jin 2006, 2007, 2008.
1. Introduction
When a boat is sailing with a constant velocity u∞ , we may think that the water is flowing
around the fixed boat with opposite velocity −u∞ like the water flow around an island. As we
have seen, behind the boat the motion of the water is significantly different from other areas,
which is called the wake. 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,
∇·u0
in Ω × 0, ∞,
lim ux, t u∞ ,
|x| → ∞
1.1
2
Abstract and Applied Analysis
where Ω is an exterior domain in R3 with a smooth boundary ∂Ω and u∞ denotes a given
constant vector describing the velocity of the fluid at infinity. For u∞ 0, the temporal decay
and weighted estimates for solutions of 1.1 have been studied in 1–13
.
In this paper, we consider a nonzero constant u∞ . We set u u∞ v in 1.1 and have
∂
v − Δv u∞ · ∇v v · ∇v ∇p1 f,
∂t
∇ · v 0,
v|t0 u0 − u∞ ,
v|∂Ω −u∞ ,
in Ω × 0, ∞,
1.2
lim vx, t 0.
|x| → ∞
Consider the following linear equations of 1.2:
∂
u − Δu u∞ · ∇u ∇p 0,
∂t
u|t0 u0 ,
∇ · u 0 in Ω × 0, ∞,
u|∂Ω 0,
lim ux, t 0,
1.3
|x| → ∞
which is referred to as the Oseen equations; see 14
.
In order to formulate the problem 1.3, Enomoto and Shibata 15
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-Morimoto 16
, Miyakawa
17
, and Simader-Sohr 18
. 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 15
proved that Ou∞ generates an analytic semigroup {T t}t≥0
which is called the Oseen semigroup one can also refer to 17, 19
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.
By using Proposition 1.1, Bae-Jin 20
considered the spatial stability of stationary
solution w of 1.3 and obtained the following: if |x|u0 , u0 ∈ Lr Ω with ∇ · u0 0, then
for any t > 0,
|x|ut
p ≤ Ct−3/21/r−1/p |x|u0 Lr Ω C|u∞ |t−3/21/r−1/p1 u0 Lr Ω ,
1.10
where p ≥ 3 and 1 < r < 3.
And, for the nonstationary Navier-Stokes equations, we discuss the stability of stationary solution w of the nonlinear Navier-Stokes equation 1.2, and w satisfies the following
equations:
−Δw u∞ · ∇w w · ∇w ∇p2 f,
w|∂Ω −u∞ ,
∇ · w 0,
lim wx 0.
1.11
|x| → ∞
For suitable f, Shibata 21
proved that for any given 0 < δ < 1/4 there exists such
that if 0 < |u∞ | ≤ , then one has
w
L3/1δ1 Ω w
L3/1−δ2 Ω ∇w
L3/2δ1 Ω ∇w
L3/2−δ2 Ω ≤ C|u∞ |1/2 ,
1.12
for small δ1 , δ2 , where C is independent of u∞ .
By setting u v − w and p p1 − p2 for v, p1 , w, p2 in 1.2 and 1.11, we have the
following equations in Ω:
∂
u − Δu u∞ · ∇u u · ∇w w · ∇u u · ∇u ∇p 0,
∂t
∇ · ut, x 0,
ux, t 0
ux, 0 u0 x
for x ∈ ∂Ω,
for x ∈ Ω,
lim ux, t 0.
|x| → ∞
1.13
4
Abstract and Applied Analysis
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. Heywood 22, 23
, Masuda 24
, Shibata 21
, EnomotoShibata 15
, Bae-Roh 25
, and Roh 26
have studied the temporal decay for solutions of
1.13, and we have the followings in 26
.
Proposition 1.2. There exists small p, q, r such that if 0 < |u∞ | ≤ , and u0 L3 Ω < , then a
unique solution ux, t of 1.13 has
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
for 1 < r < q ≤ 3, t > 0,
1.14
where u0 ∈ L3 Ω ∩ Lr Ω.
Now, we are in the position to introduce our main theorems which are the weighted
stability of stationary solution w.
Theorem 1.3. Let 1 < r < p < ∞ and 1/r − 1/p > 2/3. Then there exists small p, r such that if
0 < |u∞ | ≤ , u0 L3 Ω < , |x|u0 ∈ L3r/3−2r Ω, and ∇ · u0 0, then the solution ux, t of 1.13
satisfies
|x|ut
Lp Ω ≤ C t−3/21/r−1/p1 u0 r ,
∀t > 0,
1.15
where u0 ∈ L3 Ω ∩ Lr Ω.
Remark 1.4. In Theorem 1.3, the assumption |x|u0 ∈ L3r/3−2r Ω is for simple calculations.
We also can obtain a similar result where |x|u0 ∈ Lr Ω. For the proof we have to consider
delay solution ut = ut t0 . Then we can follow the method in Bae and Roh 4
.
Theorem 1.5. Let 1/r − 1/p > 2σ/3 for 1 < σ < 3/2 and 1 < r < p < ∞. Then there exists small
p, r such that if 0 < |u∞ | ≤ , u0 L3 Ω < , |x|σ u0 ∈ L3r/3−2r Ω, and ∇ · u0 0, then the
solution ux, t of 1.13 satisfies
|x|σ ut
Lp Ω ≤ C t−3/21/r−1/pσ u0 r ,
∀t ≥ 1,
1.16
where u0 ∈ L3 Ω ∩ Lr Ω.
Remark 1.6. For the exterior Navier-Stokes flows with u∞ 0, temporal decay rate with
0, we
weight function |x|σ becomes slower by σ/2; refer to 1–4, 8, 13
. However, for u∞ /
found out from Theorems 1.3 and 1.5 that temporal decay rate with weight function |x|σ
becomes slower by σ for 0 ≤ σ < 3/2. In fact, Bae and Roh 25
concluded that it becomes
slower by 1 σ/2 for 0 < σ < 1/2. Hence, our decay rate is little faster than the one in Bae
and Roh 25
for 0 < σ < 1/2.
One of the difficulties for the exterior Navier-Stokes equations is dealing with the
boundary of Ω because a pressure representation in terms of velocity is not a simple problem.
So to remove the pressure term, we adapt an indirect method by taking a weight function φ
Abstract and Applied Analysis
5
vanishing near the boundary. This astonied method for exterior problem was initiated by He
and Xin 27
and then developed by Bae and Jin 1, 2, 4, 20
.
2. Proof of Main Theorems
In this section, we will prove the weighted stability of stationary solutions of the NavierStokes equations with nonzero far-field velocity. We first consider |x| for a weight function
and then |x|σ for σ < 3/2. Our method can be applied to the Oseen equations. As a result, we
note that we can improve the result of Bae-Jin 1
by the same method.
2.1. Proof of Theorem 1.3
We define φR x |x|χ|x|1 − χ|x|/R for large R > 0, where χ is a nonnegative cutoff
function with χ ∈ C∞ 0, ∞, χs 0 for s ≤ 1, and χs 1 for s ≥ 2. When there is no
confusion, we use the same notation φ instead of φR for convenience.
As in 1
, we set
vx ≡
R3
N x − y φ y ∇ × u y dy,
2.1
where N is the fundamental function of −Δ, that is, N Nx − y 1/4π|x − y|. By the
definition of v, we have −Δv φ∇ × u. Moreover,
∇×v
Ω
N x − y ∇ × φ∇ × u y dy φu R0 ,
2.2
where
R0 : ∇N ∗ u · ∇φ − ∇ × N ∗ ∇φ × u .
2.3
We first estimate ∇ × vt
p and then obtain the estimate of φut
p |x|ut
p .
Now, we consider the fundamental solutions for the nonstationary Oseen equations,
written as
Vti x V i x, t Γt xei ∇
∂
N ∗ Γt x,
∂xi
2.4
where Γt x Γx, t 4πt−3/2 e−|x−tu∞ | /4t refer to 15, 28
. In fact, Γ is a translation in the
2
direction of x by tu∞ of the heat kernel Kx, t 4πt−3/2 e−|x| /4t , that is, Γx, t Kx −
i
tu∞ , t. Set ωt x ωi x, t N ∗ Γt xei , i 1, 2, 3, where ei is the standard unit vector of
which the ith term is 1. Then, we have
2
∇ × ∇ × ωi −Δωi ∇ div ωi V i .
2.5
6
Abstract and Applied Analysis
Hence, we have the identity
∇y × φ y ∇y × ωi x − y, t − τ φ y V i x − y, t − τ Ri1 x, y, t − τ ,
2.6
Ri1 x, y, t − τ ∇φ y × ∇y × ωi x − y, t − τ .
2.7
where
From straightforward calculations we have that for 1 ≤ s ≤ ∞,
β
∂ Γt−τ ≤ ct − τ−3/21−1/s−|β|/2 .
s
2.8
One might note that we may sometimes use V i s ≤ Γt 3s/3s < ct−13/2s instead
of V s ≤ Γt s < ct−3/21−1/s because of technical reason. By the definition of V i , both
inequalities hold for any s ≥ 3/2. We multiply 1.13 by ∇y × φy∇y × ωi x − y, t − τ
and integrate over Ω × 0, t − , and then we have
i
t− ∂u
− Δy u u∞ · ∇y u w · ∇u u · ∇w u · ∇u
Ω ∂τ
· ∇y × φ y ∇y × ωi x − y, t − τ dy dτ
0
−
t− 0
Ω
2.9
∇p y · ∇y × φ y ∇y × ωi x − y, t − τ dy dτ 0.
We finally get the following integral representation for ∇ × v refer to 2, 3
for the
detail:
∇x × vi ∇x × v0 ∗ Γt
t −
u · ∂τ Δy u∞ · ∇y R1 x, y, t − τ dy dτ
0
−
0
−
Ω
t 0
−
Ω
t Ω
t 0
−
Ω
u · Ri2 x, y, t − τ dy dτ
u · V i x − y, t − τ u∞ · ∇y φ y dy dτ
wk u uk w · ∂yk φ y V i x − y, t − τ ∂yk Ri1 x, y, t − τ dy dτ
t− 0
Ω
uk u · ∂yk φ y V i x − y, t − τ ∂yk Ri1 x, y, t − τ dy dτ
I II III IV V V I,
2.10
Abstract and Applied Analysis
7
where
Ri2 x, y, t − τ 2 ∇y φ y · ∇y V i x − y, t − τ Δy φ y V i x − y, t − τ .
2.11
Applying Young’s convolution and the Calderon-Zygmund inequalities, we obtain
I
p ∇ × v0 ∗ Γt p ≤ u0 φ ∗ Γt p ∇N ∗ u0 ∇φ
∗ Γt p
≤ u0 φ
3r/3−2r Γt 3pr/5pr3r−3p ∇N ∗ u0 ∇φ
3r/3−2r Γt 3pr/5pr3r−3p
≤ Ct−3/21/r−1/p1 φu0 3r/3−2r Ct−3/21/r−1/p1 u0 r ,
2.12
∀t > 0,
if φu0 ∈ L3r/3−2r and u0 ∈ Lr .
And II is bounded by as follows:
II
p ≤ c
t
0
i i u
s1 ∇2 φ ∂k ∇ × ωt−τ
u
s3 ∇Δφ3 ∇ × ωt−τ
∞
s2
s4
i |u∞ ||x|−1 u ∇ × ωt−τ
dτ
s5
2.13
s6
II1 II2 II3 ,
where 1/s1 1/s2 1 1/p, 1/s3 1/s4 1 1/p − 1/3 and 1/s5 1/s6 1 1/p.
We have
II1 ≤ C
u0 r
t
0
τ −3/21/r−1/s1 t − τ−3/21−1/s2 dτ ≤ C
u0 r t−3/21/r−1/p1 ,
∀t > 0,
2.14
where 1/r − 1/s1 < 2/3 and s2 < 3. Also, we obtain
II2 ≤ C
u0 r
t
0
τ −3/21/r−1/s3 t − τ−13/2s4 dτ ≤ C
u0 r t−3/21/r−1/p1 ,
∀t > 0,
2.15
where 1/r − 1/s3 < 2/3. Finally, we get
II3 ≤ C
t
0
t
i ∇u
s5 ∇ × ωt−τ dτ ≤ C
u0 r τ −3/21/r−1/s5 −1/2 t − τ−13/2s6 dτ
≤ C
u0 r t−3/21/r−1/p1 ,
s6
0
2.16
∀t > 0,
where 1/r − 1/s5 < 1/3. Hence, for any t > 0, we have
I
p II
p ≤ Ct−3/21/r−1/p1 φu0 3r/3−2r C
u0 r t−3/21/r−1/p1 .
2.17
8
Abstract and Applied Analysis
Also, we obtain
t
III
p ≤
u∂j φ ∗ ∂j V i p uΔφ ∗ V i p dτ
0
t
≤
0
i
u
s ∂j V i ps/pss−p u
s1 ∇2 φ
∞ ∂k ∇ × ωt−τ
s2 dτ
≤ C
u0 r t−3/21/r−1/p1 ,
2.18
∀t > 0,
where 1/s1 1/s2 1 1/p, 1/r − 1/s1 < 2/3 and s2 < 3. In the above calculation, we
used ∂V i t
q ≤ t−3/21−1/q instead of ∂V i t
q ≤ t−3/21−1/q−1/2 because of simplicity of
calculations.
And we have
t
IV p ≤ c|u∞ |
0
u
s7 ∇φ∞ V i dτ ≤ C
u0 r t−3/21/r−1/p1 ,
s8
∀t > 0,
2.19
where 1/s7 1/s8 1 1/p, 1/r − 1/s7 < 2/3 and s8 < 3.
Next, for V , we have
V −
t 0
Ω
wk u uk w ·
∂yk φ y V i x − y, t − τ
φ y ∂yk V i x − y, t − τ ∂yk Ri1 x − y, t − τ dy dτ
2.20
≤ V1 V2 V3 .
We get
V1 p ≤ c
t
0
u
r1 w
3 ∇φ∞ V i dτ ≤ C
u0 r t−3/21/r−1/p1 ,
r2
∀t > 0,
2.21
where 1/r1 1/r2 2/3 1/p, 1/r − 1/r1 < 2/3 and r2 < 3. In the above calculation, we used
V i t
q ≤ t−3/21−1/q1/2 instead of V i t
q ≤ t−3/21−1/q because of simplicity of calculations.
Since |x|w
∞ < C see 21
, we have
V2 p ≤ c
t
0
u
r3 φw∞ ∇V i dτ ≤ C
u0 r t−3/21/r−1/p1 ,
r4
∀t > 0,
2.22
where 1/r3 1/r4 1 1/p, 1/r − 1/r3 < 2/3 and r4 < 3. In the above calculation, we
used ∂V i t
q ≤ t−3/21−1/q instead of ∂V i t
q ≤ t−3/21−1/q−1/2 because of simplicity of
calculations.
Abstract and Applied Analysis
9
Next, for any t > 0, we have
V3 p ≤
t
0
u
r5 w
3 ∇2 φ ∇ × ωi u
r7 w
∞ ∇φ∞ ∂k ∇ × ωi dτ
∞
≤ u0 r t
−3/21/r−1/p1
r6
r8
2.23
,
where 1/r5 1/r6 1/r7 1/r8 2/3 1/p, 1/r − 1/r5 < 2/3, 1/r − 1/r7 < 2/3 and r8 < 3.
Hence, we have
V p ≤ C
u0 r t−3/21/r−1/p1 ,
∀t > 0.
2.24
Consider V Ias follows:
VI −
t 0
Ω
uk u ·
∂yk φ y V i x − y, t − τ
φ y ∂yk V i x − y, t − τ ∂yk Ri1 x − y, t − τ dy dτ
2.25
≤ V I1 V I 2 V I 3 .
We have, for any t > 0,
V I1 p ≤
t
0
u
s1 u
s2 ∇φ∞ V i dτ ≤ C2 u0 r t−3/21/r−1/p1 ,
s3
2.26
where 1/s1 1/s2 1/s3 1 1/p, 1/r − 1/s1 1/s2 < 1/3, and ut
s2 ≤ ct−1/23/2s2 . In
the above calculation, we used V i t
q ≤ t−3/21−1/q1/2 instead of V i t
q ≤ t−3/21−1/q
because of technical reason.
Similar to V I1 , we get
V I3 p ≤
t
0
u
r1 u
r2 ∇2 φ ∇ × ωi u
s1 u
s2 ∇φ∞ ∂k ∇ × ωi dτ
∞
≤ C
u0 r t
−3/21/r−1/p1
,
r3
s3
2.27
∀t > 0,
where 1/r1 1/r2 1/r3 1 1/p 1/s1 1/s2 1/s3 , 1/r − 1/r1 1/r2 < 1/3, ut
r2 ≤
ct−1/23/2r2 , and 1/r − 1/s1 1/s2 < 1/3, ut
s2 ≤ ct−1/23/2s2 .
Note that
φut ≤ ∇ × vt
∇N ∗ ut∇φ ≤ ∇ × vt
ut
p
p
3p/32p
p
p
≤ ∇ × vt
p C
u0 r t−3/21/r−1/p1 ,
∀t > 0.
2.28
10
Abstract and Applied Analysis
√
Since for t > 0, ut
9 ≤
V I2 p ≤
t
i
φuτ uτ
9 ∇V t − τ
p
9/8
0
≤
t−1/3 , from Shibata 21
, we have
t
0
∇ × vτ
p τ
−1/3
t − τ
−2/3
dτ
2.29
dτ C
u0 r t
−3/21/r−1/p1
.
Hence, we have
V I
p ≤ t
0
∇ × vτ
p τ −1/3 t − τ−2/3 dτ C
u0 r t−3/21/r−1/p1 .
2.30
Thus, by 2.17–2.19, 2.24, 2.28, and 2.30, for all t > 0, we obtain
∇ × vt
p ≤ t
0
∇ × vτ
p τ −1/3 t − τ−2/3 dτ C
u0 r t−3/21/r−1/p1 .
2.31
Now, we use the following lemma refer to 25
.
Lemma 2.1. Let a function St satisfy the inequality, for some α < 2/3,
−α
St ≤ ct
ε
t
Sττ −1/3 t − τ−2/3 dτ
2.32
∀t > 0.
0
One also assumes that
lim t
t → 0
−ε
t
τ −1/3 Sτdτ 0.
2.33
0
Then, there is ε0 so that if ε ≤ ε0 , then one has
St ≤ ct−α
2.34
for some c independent of t.
Since
∇ × vt
p ≤ φutp ∇N ∗ ut∇φp ≤ R
ut
p ut
3p/3p
≤ CR
u0 3 t−1/23/2p C2 u0 3 t−3/21/3−1/p1/2 ,
∀t > 0,
2.35
Abstract and Applied Analysis
11
condition 2.33 satisfies
lim t
−ε
t
t → 0
τ
−1/3
0
∇ × vτ
p dτ lim t
−ε
t → 0
t
0
τ −1/3 CR
u0 3 τ −1/23/2p C2 u0 3 τ 3/2p dτ
lim CR
u0 3 τ −1/33/2p C2 u0 3 τ −13/2p 0,
t → 0
2.36
for < 1/3 3/2p.
So, by Lemma 2.1, we have
∇ × vt
p ≤ C
u0 r t−3/21/r−1/p1 .
2.37
Hence, by 2.28, for any t > 0, we have
φut ≤ C
u0 t−3/21/r−1/p1 ,
r
p
2.38
and by taking R → ∞, we complete the proof of Theorem 1.3.
2.2. Proof of Theorem 1.5
By using the results in previous section, for any 0 < α < 1, we have small β > 0 such that
α
1−α |x| ut ≤ |x|α uα u
3/α−3β
s
1/1−α−β
α 1−α
≤ Ct−3/21/r−α−3β/3α1 Ct−3/21/r−1−α−β/1−α
≤ Ct−3/21/r−1/sα ,
2.39
where 1 − 2α/3 − 2β 1/s.
Now, in this section, we consider φx |x|σ χ|x|, where 1 < σ < 3/2.
Similar to previous section, for I
p , II1 , and II2 , we obtain the same decay rate with
previous section. And for any t > 0, we have
II3 ≤ C
t σ−2 |x| 0
≤ C
u0 r
i ×
ω
u
∇
s1
t−τ dτ
2
3/2−σ
t
0
τ −3/21/r−1/s1 t − τ
s2
−13/2s2
2.40
2
dτ ≤ C
u0 r t−3/21/r−1/p3/2−2−σ /2 ,
12
Abstract and Applied Analysis
where 1/s1 1/s2 1 1/p and 1/r − 1/s1 < 2/3. Also, for III
p , we obtain
t III
p ≤ u∂j φ ∗ ∂j V i uΔφ ∗ V i dτ
p
0
≤
p
t σ−1 |x| u ∂j V i s
0
ps/pss−p
i u
s1 ∇2 φ ∂k ∇ × ωt−τ
dτ
∞
≤ C
u0 r t−3/21/r−1/pσ−1/2 C
u0 r t−3/21/r−1/p1 ,
2.41
s2
∀t > 0,
where 1/s1 1/s2 1 1/p, 1/r − 1/s1 < 2/3 and s2 < 3.
Next, we have
t IV p ≤ c|u∞ | |x|σ−1 u V i dτ ≤ C|u∞ |
u0 r t−3/21/r−1/pσ ,
s1
0
∀t > 0,
s2
2.42
where 1/s1 1/s2 1 1/p, s2 < 3, and 1/r − 1/s1 < 2σ/3. Also, since |x|w
∞ < C, we get
V1 p ≤ c
t
0
u
r1 |x|σ−1 w V i dτ ≤ C
u0 r t−3/21/r−1/p1 ,
∞
r2
∀t > 0,
2.43
where 1/r1 1/r2 1 1/p, 1/r − 1/r1 < 2/3, and r2 < 3.
And we obtain
V2 p ≤ c
t σ−1 |x| u |x|w
∞ ∇V i dτ ≤ C
u0 r t−3/21/r−1/pσ−1/2 ,
r3
0
r4
∀t > 0,
2.44
where 1/r3 1/r4 1 1/p, 1/r − 1/r3 < 2σ/3, and r4 < 3/2.
Next, for any t > 0, we have
V3 p ≤
t
0
u
r5 w
3 ∇2 φ ∇ × ωi |x|σ−1 u w
3 ∂k ∇ × ωi dτ
∞
r6
r7
r8
2.45
≤ C
u0 r t−3/21/r−1/p1 C
u0 r t−3/21/r−1/pσ−1/2 ,
where 1/r5 1/r6 1/r7 1/r8 2/3 1/p, 1/r − 1/r5 < 2/3, 1/r − 1/r7 < 2σ/3, and r8 < 3.
Hence, we have
V p ≤ C
u0 r t−3/21/r−1/p1 C
u0 r t−3/21/r−1/pσ−1/2 ,
∀t > 0.
2.46
Abstract and Applied Analysis
13
Consider V I as follows:
VI −
t Ω
0
uk u ·
∂yk φ y V i x − y, t − τ
φ y ∂yk V i x − y, t − τ ∂yk Ri1 x − y, t − τ dy dτ
2.47
≤ V I1 V I 2 V I 3 .
We have, for any t > 0,
t V I1 p ≤ |x|σ−1 u u
s2 V i dτ ≤ C
u0 r t−3/21/r−1/pσ−1/2 ,
s1
0
s3
2.48
where 1/s1 1/s2 1/s3 1 1/p, s3 < 3, 1/r − 1/s1 < 2σ/3, and ut
s2 ≤ ct−1/23/2s2 .
Similar to V I1 , we get
V I3 p ≤
t
0
u
r1 u
r2 ∇2 φ ∇ × ωi |x|σ−1 u u
s2 ∂k ∇ × ωi dτ
∞
≤ C
u0 r t
r3
−3/21/r−1/p1/2
C
u0 r t
s1
−3/21/r−1/pσ
s3
,
2.49
∀t > 0,
where 1/r1 1/r2 1/r3 11/p 1/s1 1/s2 1/s3 , 1/r−1/r1 < 2/3, 1/r−1/r1 1/r2 < 1/3,
1/r − 1/s1 < 2σ/3, 1/r − 1/s2 < 2/3, 1/r − 1/s1 1/s2 < 2σ − 1/3, ut
s2 ≤ ct−1/23/2s2 ,
and ut
r2 ≤ ct−1/23/2r2 . In the above calculation, we used V i t
q ≤ t−3/21−1/q 1/2 instead
of V i t
q ≤ t−3/21−1/q because of technical reason. Now, we have
V I2 p ≤
t
0
|x|uτ
s1 |x|σ−1 uτ ∇V i t − τ dτ
≤ C
u0 r t
s2
−3/21/r−1/pσ1/2−3/2r1
2.50
s3
,
where 1/s1 1/s2 1/s3 1 1/p, s2 < 3/2, |x|σ−1 uτ
s2 < Ct−3/21/r1 −1/s2 σ−1 , and r1 <
3 ≈ 3.
So, we obtain
φut ≤ ∇ × vt
∇N ∗ ut∇φ ≤ ∇ × vt
|x|σ−1 ut
p
p
p
p
3p/3p
≤ ∇ × vt
p C
u0 r t−3/21/r−1/pσ−1/2 ,
≤ C
u0 r t−3/21/r−1/pσ ,
which completes the proof.
∀t ≥ 1,
∀t > 0
2.51
14
Abstract and Applied Analysis
Acknowledgments
The author would like to express her appreciation to Professors Hyeong-Ohk Bae and Bum Ja
Jin for valuable comments. The author would like to thank the referee for helpful comments.
This research was supported by the 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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