Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 864186, 14 pages
doi:10.1155/2012/864186
Research Article
Uniqueness of Weak Solutions to an
Electrohydrodynamics Model
Yong Zhou1 and Jishan Fan2
1
2
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
Correspondence should be addressed to Yong Zhou, yzhoumath@zjnu.edu.cn
Received 11 October 2011; Accepted 3 March 2012
Academic Editor: Narcisa C. Apreutesei
Copyright q 2012 Y. Zhou and J. Fan. 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.
This paper studies uniqueness of weak solutions to an electrohydrodynamics model in Rd d 2, 3. When d 2, we prove a uniqueness without any condition on the velocity. For d 3, we
prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov
space.
1. Introduction
We consider the following model of electrokinetic fluid in Rd × 0, ∞ 1, 2:
∂t u u · ∇u ∇π − Δu Δφ∇φ,
1.1
div u 0,
∂t n u · ∇n ∇ · ∇n − n∇φ ,
∂t p u · ∇p ∇ · ∇p p∇φ ,
1.2
Δφ n − p,
u, n, p x, 0 u0 , n0 , p0 x,
1.3
1.4
1.5
x ∈ Rd d 2, 3.
1.6
The unknowns u, π, φ, n, and p denote the velocity, pressure, electric potential, anion
concentration, and cation concentration, respectively.
2
Abstract and Applied Analysis
Equations 1.3–1.5 are known as the electrochemical equations 3 or semiconductor
equations 4, 5, and electro-rheological systems 2, 6 when formally setting u 0. 1.1 and
1.2 are Navier-Stokes equations with the Lorentz force Δφ∇φ.
The uniqueness of weak solutions to the Navier-Stokes equations is still open. In 1962,
Serrin 7 gave the first uniqueness condition:
2 3
with 1,
u ∈ Lr 0, T ; Ls R3
r s
3 < s ≤ ∞.
1.7
Kozono and Taniuchi 8 proved the following uniqueness criterion:
u ∈ L2 0, T ; BMO R3 .
1.8
Here BMO denotes the functions of bounded mean oscillation. Ogawa and Taniuchi 9 obtained the uniqueness criterion:
0
∇u ∈ L Log L 0, T ; Ḃ∞,∞
R3
1.9
with
L Log L
0
0, T ; Ḃ∞,∞
:
T
f;
0
f 0 log e f 0 dt < ∞ .
Ḃ∞,∞
Ḃ∞,∞
1.10
Here it should be noted that Kozono et al. 10 proved that u is smooth if
0
R3 .
∇u ∈ L1 0, T ; Ḃ∞,∞
1.11
0
Here Ḃ∞,∞
is the homogeneous Besov space.
Kurokiba and Ogawa 4 considered the semiconductor equations 1.3–1.5 when
u 0 and proved that the existence and uniqueness of weak solutions with Lp initial data
n0 , p0 when p d/2d ≥ 3 and 1 < p < 2 d 2.
Note that the system 1.1–1.5 holds its form under the scaling u, π, φ, n, p →
uλ , πλ , φλ , nλ , pλ : λu, λ2 π, φ, λ2 n, λ2 pλ2 t, λx. Under this scaling, the space Lr 0, T ; Ls is invariant for u when 2/r d/s 1 and the space Lr 0, T ; Ls is invariant for n, p when
2/r d/s 2. Furthermore, Ld for u0 and Ld/2 for n0 , p0 are invariant spaces under this
scaling. Fan and Gao 11, Ryham 12, and Schmuck 13 proved the existence, uniqueness,
and regularity of global weak solutions to system 1.1–1.6 in a bounded domain Ω. When
Ω Rd , Jerome 14 established the first existence result in Kato’s semigroup framework.
Zhao et al. 15 obtained global well-posedness for small initial data in Besov spaces with
negative index.
The aim of this paper is to generalize the results of 4, 9. We will prove the following
results.
Abstract and Applied Analysis
3
Theorem 1.1. Let n0 , p0 ∈ L1 R2 ∩ L log LR2 , n0 , p0 ≥ 0 in R2 , n0 dx p0 dx, ∇φ0 ∈ L2 ,
and u0 ∈ L2 . Then there exists a unique weak solution u, n, p, φ to the problem 1.1–1.6 satisfying
n, p ∈ L∞ 0, T ; L1 ∩ L log L ∩ L2 0, T ; L2 ∩ L4/3 0, T ; W 1,4/3 , n, p ≥ 0 in R2 × 0, T ∂t n, ∂t p ∈ L4/3 0, T ; W −1,4/3 ,
∇φ ∈ L∞ 0, T ; L2 ∩ L2 0, T ; H 1 ∩ L4 0, T ; L4 ,
u ∈ L∞ 0, T ; L2 ∩ L2 0, T ; H 1 ∩ L4 0, T ; L4 ,
∂t u ∈ L4/3 0, T ; H −1
for any T > 0.
1.12
Remark 1.2. We can assume n0 − p0 ∈ H1 Hardy space and Δφ0 n0 − p0 gives ∇φ0 ∈ L2 R2 .
Theorem 1.3 d 3. Let n0 , p0 ∈ L3/2 , n0 , p0 ≥ 0 in R3 , n0 dx p0 dx, and u0 ∈ L2 .
Suppose that 1.9 holds true, then there exists a unique weak solution u, n, p, φ to the problem
1.1–1.6 satisfying
3/4 3/4 n ,p
∈ L∞ 0, T ; L2 ∩ L2 0, T ; H 1 , n, p ≥ 0 in R3 × 0, T ,
n, p ∈ L∞ 0, T ; L3/2 ∩ L5/2 0, T ; L5/2 ∩ L5/3 0, T ; W 1,5/3 ∩ L4 0, T ; L2 ,
∂t n, ∂t p ∈ L5/3 0, T ; W −1,3/2 ,
∇φ ∈ L∞ 0, T ; W 1,3/2 ∩ L5/2 0, T ; W 1,5/2 ,
∇φ ∈ L∞ 0, T ; L3 ∩ L5/2 0, T ; L15 ,
u ∈ L∞ 0, T ; L2 ∩ L2 0, T ; H 1 ,
∂t u ∈ L2 0, T ; W −1,3/2
1.13
for any T > 0.
Let ηj , j 0, ±1, ±2, ±3, . . ., be the Littlewood-Paley dyadic decomposition of unity that
−j ξ, and ∞
j ξ 1 except ξ 0. To fill the origin,
satisfies η ∈ C0∞ B2 \ B1/2 , ηj ξ η2
j−∞ η
∞
3
we put a smooth cut off ψ ∈ SR with ψξ
∈ C0 B1 such that
ψ ∞
ηj ξ 1.
1.14
j0
s
s
The homogeneous Besov space Ḃp,q
: {f ∈ S : f
Ḃp,q
< ∞} is introduced by the norm
⎞1/q
⎛
∞ q
js
f s : ⎝
2 ηj ∗ f p ⎠ ,
Ḃp,q
j−∞
L
1.15
for s ∈ R, 1 ≤ p, q ≤ ∞.
It is easy to prove the existence of weak solutions 14 and thus we omit the details
here; we only need to derive the estimates 1.12 and 1.13 and prove the uniqueness.
4
Abstract and Applied Analysis
2. Proof of Theorem 1.1
First, by the maximum principle, it is easy to prove that
2.1
n, p ≥ 0 in Rd × 0, ∞.
Testing 1.3 by 1 log n and testing 1.4 by 1 log p, respectively, using 1.2,
summing up the resulting equality, we obtain
T T √ 2 2
2
∇ n ∇ p dx dt Δφ dx dt
n log n p log p dx 4
0
0
n0 log n0 p0 log p0 dx.
2.2
Substracting 1.4 from 1.3, we see that
∂t n − p u · ∇ n − p ∇ · ∇ n − p − n p ∇φ .
2.3
Testing the above equation by −φ, using 1.5 and 2.1, we see that
1 d
2 dt
2
∇φ dx −
u · ∇Δφ · φ dx 2
Δφ dx n p ∇φ2 dx 0.
2.4
Whence
1 d
2 dt
2
∇φ dx uΔφ∇φ dx 2
Δφ dx n p ∇φ2 dx 0.
2.5
Testing 1.1 by u, using 1.2, we find that
1 d
2 dt
u dx 2
2
|∇u| dx 2.6
uΔφ∇φ dx.
Summing up 2.5 and 2.6, we get
1 d
2 dt
2
u ∇φ dx 2
2 2
|∇u|2 Δφ n p ∇φ dx 0,
2.7
whence
1
2
2
u ∇φ dx 2
T 0
2 2
1
|∇u| Δφ n p ∇φ dx dt ≤
2
2
2
u20 ∇φ0 dx.
2.8
Abstract and Applied Analysis
5
Integrating 1.3 and 1.4, we have
n dx n0 dx p0 dx 2.9
p dx.
Using the Gagliardo-Nirenberg inequality,
u
2L4 ≤ C
u
L2 ∇u
L2 ,
2.10
u
L4 0,T ;L4 ≤ C,
∇φ 4
≤ C,
L 0,T ;L4 n, p 2
≤ C.
L 0,T ;L2 2.11
we deduce that
2.12
2.13
√ √
√
√
Since ∇n 2∇ n · n, ∇ n ∈ L2 0, T ; L2 , n ∈ L4 0, T ; L4 , we easily infer that
∇n ∈ L4/3 0, T ; L4/3 ,
2.14
by the Hölder inequality. Similarly, we have
∇p ∈ L4/3 0, T ; L4/3 .
2.15
It is easy to show that
∂t n, ∂t p ∈ L4/3 0, T ; W −1,4/3 ,
∂t u ∈ L4/3 0, T ; H −1 .
2.16
Now we are in a position to prove the uniqueness. Let ui , πi , ni , pi , φi i 1, 2 be two weak
solutions to the problem 1.1–1.6. Also let us denote
u : u1 − u2 ,
π : π1 − π2 ,
n : n1 − n2 ,
p : p1 − p2 ,
φ : φ1 − φ2 .
2.17
We define N and P satisfying the following equations:
−ΔN N n in Rd × 0, ∞,
2.18
−ΔP P p in Rd × 0, ∞.
2.19
6
Abstract and Applied Analysis
It is easy to verify that
∂t n ∇ · u1 n un2 Δn − ∇ · n∇φ1 n2 ∇φ ,
2.20
2.21
∂t p ∇ · u1 p up2 Δp ∇ · p∇φ1 p2 ∇φ ,
∂t n − p ∇ · u1 n − p u n2 − p2 Δ n − p − ∇ · n p ∇φ1 n2 p2 ∇φ .
2.22
Testing 2.20 by N, we derive
1 d
2 dt
2
N |∇N| dx |∇N|2 |ΔN|2 dx
n∇φ1 · ∇N n2 ∇φ∇N u1 n∇N un2 ∇N dx : I1 I2 I3 I4 .
2
2.23
Using 2.10, 2.18 and 2.19, each term Ii i 1, 2, 3, 4 can be bounded as follows:
I1 ≤ ΔN
L2 ∇φ1 L4 ∇N
L4 N
L4 ∇φ1 L4 ∇N
L2
≤ C
ΔN
L2 ∇φ1 L4 ∇N
1/2
C∇φ1 L4 N
2H 1
ΔN
1/2
L2
L2
4
1
ΔN
2L2 C∇φ1 L4 ∇N
2L2 C∇φ1 L4 N
2H 1 ,
18
I2 ≤ n2 L2 ∇φL4 ∇N
L4
2
≤ n2 L2 ∇φL4 n2 L2 ∇N
2L4
≤ C
n2 L2 ∇φL2 ΔφL2 C
n2 L2 ∇N
L2 ΔN
L2
≤
≤
1
Δφ2 2 1 ΔN
2 2 C
n2 2 2 ∇φ2 2 ∇N
2 2 ,
L
L
L
L
L
18
18
I3 ≤ u1 L4 ΔN
L2 ∇N
L4
≤ C
u1 L4 ΔN
L2 ∇N
1/2
ΔN
1/2
L2
L2
≤
1
ΔN
2L2 C
u1 4L4 ∇N
2L2 ,
18
I4 ≤ n2 L2 u
L4 ∇N
L4
≤ n2 L2 u
2L4 n2 L2 ∇N
2L4
≤ C
n2 L2 u
L2 ∇u
L2 C
n2 L2 ∇N
L2 ΔN
L2
≤
1
1
∇u
2L2 ΔN
2L2 C
n2 2L2 u
2L2 ∇N
2L2 .
18
18
2.24
Abstract and Applied Analysis
7
Substituting these estimates into 2.23, we obtain
1 d
2 dt
1
N |∇N| dx |∇N|2 |ΔN|2 dx
2
2 4
≤ C ∇φ1 L4 n2 2L2 u1 4L4 1 N
2L2 ∇N
2L2 u
2L2 ∇φL2
2
2
2.25
1
Δφ2 2 1 ∇u
2 2 .
L
L
18
18
Similarly for the p-equation, we get
1 d
2 dt
2
1 2 2
∇p Δp dx
p2 ∇p dx 2
2
2 4 2
2
≤ C ∇φ1 L4 p2 L2 u1 4L4 1 pL2 ∇pL2 u
2L2 ∇φL2
2.26
1
Δφ2 2 1 ∇u
2 2 .
L
L
18
18
Testing 2.22 by −φ, using 1.5, we deduce that
1 d
2 dt
2
∇φ dx Δφ2 dx
2
− n p ∇φ1 ∇φ n2 p2 ∇φ u1 Δφ∇φ u n2 − p2 ∇φ dx
2.27
: J1 J2 J3 J4 .
Using 2.10, 2.18, and 2.19, each term Ji i 1, 2, 3, 4 can be bounded as follows:
J1 ≤ n pL2 ∇φ1 L4 ∇φL4
1/2 1/2
≤ C
ΔN
L2 ΔP L2 N
L2 P L2 ∇φ1 L4 ∇φL2 ΔφL2
≤
1
1
ΔN
2L2 ΔP 2L2 C
N
2L2 C
P 2L2
18
18
4 2
1 2
ΔφL2 C∇φ1 L4 ∇φL2 ,
18
J2 ≤ 0,
J3 ≤ u1 L4 ΔφL2 ∇φL4
8
Abstract and Applied Analysis
1/2 1/2
≤ C
u1 L4 ΔφL2 ∇φL2 ΔφL2
1
Δφ2 2 C
u1 4 4 ∇φ2 2 ,
L
L
18 L
J4 ≤ u
L4 n2 − p2 L2 ∇φ L4
2
≤ n2 p2 L2 u
2L4 n2 p2 L2 ∇φL4
≤ Cn2 p2 L2 u
L2 ∇u
L2 Cn2 p2 L2 ∇φL2 ΔφL2
≤
≤
2 2 1
1 2
∇u
2L2 ΔφL2 Cn2 p2 L2 u
2L2 ∇φL2 .
18
18
2.28
Substituting these estimates into 2.27, we have
1 d
2 dt
2
∇φ dx 1 Δφ2 dx
2
2
4
2
≤ C ∇φ1 L4 u1 4L4 n2 p2 L2 1 u
2L2 ∇φL2 N
2L2 P 2L2
1
1
1
ΔN
2L2 ΔP 2L2 ∇u
2L2 .
18
18
18
2.29
It is easy to find that u satisfies
∂t u u2 · ∇u u · ∇u1 ∇π − Δu Δφ∇φ1 Δφ2 ∇φ.
2.30
Testing this equation by u, using 1.2, we have
1 d
2 dt
u2 dx |∇u|2 dx Δφ∇φ1 u Δφ2 ∇φ · u − u · ∇u1 · u dx : 1 2 3 .
2.31
Using 2.10, each term i i 1, 2, 3 can be bounded as follows:
1 ≤ ΔφL2 ∇φ1 L4 u
L4
≤ CΔφL2 ∇φ1 L4 u
1/2
∇u
1/2
L2
L2
1
Δφ2 2 1 ∇u
2 2 C∇φ1 4 4 u
2 2 ,
L
L
L
L
18
18
2 ≤ Δφ2 L2 ∇φL4 u
L4
≤
Abstract and Applied Analysis
9
1/2 1/2
≤ CΔφ2 L2 ∇φL2 ΔφL2 u
1/2
∇u
1/2
L2
L2
1
Δφ2 2 1 ∇u
2 2 CΔφ2 2 2 u
2 2 ∇φ2 2 ,
L
L
L
L
L
18
18
3 ≤ u · ∇u · u1 dx
≤
≤ u
L4 ∇u
L2 u1 L4
≤ C
u
1/2
∇u
3/2
u1 L4
L2
L2
≤
1
∇u
2L2 C
u1 4L4 u
2L2 .
18
2.32
Substituting these estimates into 2.31, we have
1 d
2 dt
u dx 2
≤
|∇u|2 dx
1
Δφ2 2 C ∇φ1 4 4 Δφ2 2 2 u1 4 4 u
2 2 ∇φ2 2 .
L
L
L
L
L
L
9
2.33
Combining 2.25, 2.26, 2.29, and 2.33, using 2.8, 2.11, 2.12, 2.13, and the
Gronwall inequality, we conclude that
N P 0,
u 0,
∇φ 0,
2.34
and thus
n p 0.
2.35
This completes the proof.
3. Proof of Theorem 1.3
By the same calculations as that in 11, we can prove 1.13 and thus we omit the details
here.
Now we are in a position to prove the uniqueness. We still use the same notations as
that in Section 2, and similarly we get 2.23. But each term Ii i 1, 2, 3, 4 can be bounded
as follows:
I1 ≤ ΔN
L2 ∇N
L30/13 ∇φ1 L15 N
L6 ∇φ1 L3 ∇N
L2
∇φ1 15 C
∇N
2 2
≤ C
ΔN
L2 ∇N
4/5
ΔN
1/5
L
L
L2
L2
≤
5/2
1
ΔN
2L2 C∇φ1 L15 ∇N
2L2 C
∇N
2L2 ,
18
3.1
10
Abstract and Applied Analysis
by the Gagliardo-Nirenberg inequality,
,
ΔN
1/5
∇N
L30/13 ≤ C
∇N
4/5
L2
L2
I2 ≤ n2 L2 ∇φL6 ∇N
L3
≤ C
n2 L2 ΔφL2 ∇N
1/2
ΔN
1/2
L2
L2
3.2
≤ C
n2 L2 −ΔN N ΔP − P L2 ∇N
1/2
ΔN
1/2
L2
L2
≤
1
1
ΔN
2L2 ΔP 2L2 C
N
2L2 C
P 2L2 C
n2 4L2 ∇N
2L2
18
18
by the Gagliardo-Nirenberg inequality,
∇N
2L3 ≤ C
∇N
L2 ΔN
L2 ,
I3 u1 n∇N dx − u1 ΔN∇N dx.
3.3
Now we decompose u1 into three parts in the phase variable:
u1 M
ηj ∗ u1 j<−M
ηj ∗ u1 j−M
ηj ∗ u1
j>M
3.4
h
: u1 um
1 u1 .
Thus
I3 −
u1 ΔN∇N dx ∂i um
1 · ∂i N∇N dx −
uh1 ΔN∇N dx
i
3.5
: I31 I32 I33 .
Recalling the Bernstein inequality,
ηj ∗ u q ≤ C23j1/p−1/q ηj ∗ u p ,
L
L
1 ≤ p ≤ q ≤ ∞,
3.6
Abstract and Applied Analysis
11
the low-frequency part is estimated as
I31 ≤ ∇N
L6 ΔN
L2 u1 ≤ C
ΔN
2L2
j<−M
⎛
≤ C
ΔN
2L2 ⎝
L3
2j/2 ηj ∗ u1 L2
⎞1/2 ⎛
2j ⎠
⎞1/2
∞ 2
ηj ∗ u1 2 ⎠
⎝
L
3.7
j−∞
j<−M
≤ C2−M/2 ΔN
2L2 u1 L2
≤ C2−M/2 ΔN
2L2 .
The second term can be bounded as follows:
I32 ≤
i
∂i N
L2 ∇N
L2 ∂i um
1 L∞
≤ C
∇N
2L2 ∇um
1 L∞
≤
C
∇N
2L2
3.8
M ηj ∗ ∇u1 ∞
L
j−M
0
≤ CM
∇N
2L2 ∇u1 Ḃ∞,∞
.
On the other hand, the last term is simply bounded by the Hausdorff-Young inequality
as
I33 ≤ ΔN
L2 ∇N
L2 uh1 L∞
≤ ΔN
L2 ∇N
L2
−Δ−1/2 ηj−1 ηj ηj1 ∗ ηj ∗ −Δ1/2 u1 L∞
j>M
≤ C
ΔN
L2 ∇N
L2
2−j ηj ∗ −Δ1/2 u1 L∞
j>M
≤ C
ΔN
L2 ∇N
L2
0
2−j ∇u1 Ḃ∞,∞
j>M
0
≤ C2−M ΔN
L2 ∇N
L2 ∇u1 Ḃ∞,∞
≤ C2−M ∇N
2L2 ∇u1 2Ḃ∞,∞
C2−M ΔN
2L2 .
0
3.9
12
Abstract and Applied Analysis
0
≤ 1, we reach
Choosing M properly large so that C2−M/2 ≤ 1/36 and C2−M ∇u1 Ḃ∞,∞
I3 ≤
1
0
0
1 log e ∇u1 Ḃ∞,∞
,
ΔN
2L2 C
∇N
2L2 ∇u1 Ḃ∞,∞
8
I4 ≤ u
L6 n2 L2 ∇N
L3
3.10
≤ C
∇u
L2 n2 L2 ∇N
1/2
ΔN
1/2
L2
L2
≤
1
1
ΔN
2L2 ∇u
2L2 C
n
4L2 ∇N
2L2 .
18
18
Substituting the above estimates into 2.23, we obtain
1 d
2 dt
N 2 |∇N|2 dx 1
2
|∇N|2 |ΔN|2 dx
5/2
≤ C ∇φ1 L15 1 n2 4L2 N
2L2 P 2L2 ∇N
2L2
0
C
∇N
2L2 ∇u1 Ḃ∞,∞
1
1
ΔP 2L2 ∇u
2L2 .
18
18
0
1 log e ∇u1 Ḃ∞,∞
3.11
Similarly for the p-equation, we have
1 d
2 dt
P 2 |∇P |2 dx |∇P |2 |ΔP |2 dx
4 5/2
≤ C ∇φ1 L15 1 p2 L2 N
2L2 P 2L2 ∇P 2L2
0
C
∇P 2L2 ∇u1 Ḃ∞,∞
1
1
ΔN
2L2 ∇u
2L2 .
18
18
0
1 log e ∇u1 Ḃ∞,∞
3.12
As in Section 2, we still have 2.31. But each term i i 1, 2, 3 can be bounded as
follows:
1 ≤ ∇φ1 L15 ΔφL2 u
L30/13
≤ C∇φ1 L15 ΔN
L2 ΔP L2 N
L2 P L2 u
4/5
∇u
1/5
L2
L2
≤
1
1
ΔN
2L2 ΔP 2L2 C
N
2L2 C
P 2L2
18
18
5/2
1
∇u
2L2 C∇φ1 L15 u
2L2 ,
18
3.13
Abstract and Applied Analysis
13
by the Gagliardo-Nirenberg inequality,
,
u
L30/13 ≤ C
u
4/5
∇u
1/5
L2
L2
2 ≤ Δφ2 L2 ∇φL6 u
L3
≤ CΔφ2 L2 ΔφL2 u
1/2
∇u
1/2
L2
L2
≤ CΔφ2 L2 ΔN
L2 ΔP L2 N
L2 P L2 u
1/2
∇u
1/2
L2
L2
≤
3.14
1
1
ΔN
2L2 ΔP 2L2 C
N
2L2 C
P 2L2
18
18
4
1
∇u
2L2 CΔφ2 L2 u
2L2 ,
18
by the Gagliardo-Nirenberg inequality
u
2L3 ≤ C
u
L2 ∇u
L2 .
3.15
By the similar calculations as that of I3 , 3 can be bounded as follows:
3 ≤
1
0
0
1 log e ∇u1 Ḃ∞,∞
.
∇u
2L2 C
u
2L2 ∇u1 Ḃ∞,∞
18
3.16
Substituting the above estimates into 2.31, we have
1 d
2 dt
u2 dx ≤
1
2
|∇u|2 dx
1
1
ΔN
2L2 ΔP 2L2 C
N
2L2 C
P 2L2
9
9
5/2 4 ∇φ1 L15 Δφ2 L2 u
2L2
3.17
0
0
1 log e ∇u1 Ḃ∞,∞
.
C
u
2L2 ∇u1 Ḃ∞,∞
Combining 3.11, 3.12, and 3.17, using 1.13 and the Gronwall inequality, we
arrive at
N P 0,
u 0,
3.18
∇φ 0.
3.19
as thus
n p 0,
This completes the proof.
14
Abstract and Applied Analysis
Acknowledgments
The authors would like to thank the referee for careful reading and helpful suggestions.
This work is partially supported by Zhejiang Innovation Project Grant no. T200905, ZJNSF
Grant no. R6090109, and NSFC Grant no. 11171154.
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