DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Volume 26, Number 4, April 2010
doi:10.3934/dcds.2010.26.1141
pp. 1141–1151
SUFFICIENT CONDITIONS FOR THE REGULARITY TO THE
3D NAVIER–STOKES EQUATIONS
Chongsheng Cao
Department of Mathematics
Florida International University
Miami, FL 33199, USA
Abstract. In this paper we consider the three–dimensional Navier–Stokes
equations subject to periodic boundary conditions or in the whole space. We
provide sufficient conditions, in terms of one direction derivative of the velocity
field, namely, uz , for the regularity of strong solutions to the three-dimensional
Navier–Stokes equations.
1. Introduction. The three-dimensional Navier–Stokes equations (NSE) of viscous incompressible fluid read:
∂u
− ν∆u + (u · ∇)u + ∇p = 0,
(1)
∂t
∇ · u = 0,
(2)
u(x, y, z, 0) = u0 (x, y, z),
(3)
where u, the velocity field, and p, the pressure, are the unknowns, and ν > 0, the
viscosity, is given. We equip the system (1)–(3) with periodic boundary conditions
with period 1. Namely,
u(x + 1, y, z, t) = u(x, y + 1, z, t) = u(x, y, z + 1, t) = u(x, y, z, t).
(4)
Because of the periodic boundary conditions we consider here the base domain
Ω = (0, 1)3 . We emphasize, however, that one can apply our proof almost line by
line to obtain same results for the three–dimensional Navier–Stokes equations in
the whole space R3 .
We denote by Lq (Ω) and H m (Ω) the usual Lq −Lebesgue and Sobolev spaces,
respectively (cf. [1]). We denote by
Z
1q
q
kφkq =
|φ| dxdydz
,
for every φ ∈ Lq (Ω).
(5)
Ω
We set
V
=
{φ : the 3D vector valued trigonometric polynomials with ∇ · φ = 0} ,
which will form the space of test functions. Let H and V be the closure spaces
of V in L2 (Ω) under L2 −topology, and in H 1 (Ω) under H 1 −topology, respectively.
2000 Mathematics Subject Classification. Primary: 35Q35, 65M70.
Key words and phrases. Three-dimensional Navier–Stokes equations, Regularity criterion,
global regularity.
The author is supported by NSF grant no. DMS-0709228.
1141
1142
CHONGSHENG CAO
Let u0 ∈ H, we say u is a Leray–Hopf weak solution to the system (1)–(3) on the
interval [0, T ] with initial value u0 if u satisfies
(1) u ∈ Cw ([0, T ], H) ∩ L2 ([0, T ], V ), and ∂t u ∈ L1 ([0, T ], V ′ ), where V ′ is the
dual space of V ;
(2) the weak formulation of the NSE:
Z
Z
(u · φ)(t) dxdydz − (u · φ)(t0 ) dxdydz
Ω
Ω
Z tZ
[u · (φt + ν∆φ)] dx ds
=
t0
+
Ω
Z tZ
t0
[(u · ∇)φ] · u dxdydz,
Ω
for every φ ∈ V, and almost every t, t0 ∈ [0, T ];
(3) the energy inequality:
Z t
ku(t)k22 + ν
k∇u(s)k22 ds ≤ ku(t0 )k22 ,
(6)
t0
for every t and almost every t0 .
Moreover, if u0 ∈ V , a weak solution is called strong solution of (1)–(3) on [0, T ] if,
in addition, it satisfies
u ∈ C([0, T ], V ) ∩ L2 ([0, T ], H 2 (Ω)), and ∂t u ∈ L2 ([0, T ], H).
In this case one also has energy equality in (6) instead of and inequality.
It is well known that the 2D Navier–Stokes equations have a unique weak and
strong solutions which exist globally in time (cf., for example, [6], [16], [24], [27],
[28]). In the 3D case, the weak solutions are known to exist globally in time.
But, the uniqueness, regularity, and continuous dependence on initial data for weak
solutions are still open problems.
Since Prodi [21], and Serrin in [23] provided a sufficient conditions for the global
regularity of the 3D Navier–Stokes equations, many articles were dedicated for this
subjects (for details see, for example, the survey papers [18], [29] and references
therein). Most recently, there has been some progress along these lines (see, for
example, [3], [8], [10], [11], [13], [25], [26], and references therein) which states,
roughly speaking, that a strong solution u exists on the time interval [0, T ] for as
long as
2 3
u ∈ Lp ([0, T ], Lq (Ω)), with
+ ≤ 1, for q ≥ 3,
(7)
p q
or as long as
2 3
∇u ∈ Ls ([0, T ], Lr (Ω)), with
+ ≤ 2, for r ≥ 3/2.
(8)
s r
In particular, in case that r = 2 and s = 4 in (8), namely,
∇u ∈ L4 ([0, T ], L2(Ω)),
(9)
then u is a strong solution on [0, T ].
Moreover, many sufficient regularity conditions were established in terms of only
partial components of the velocity field, or partial components of gradient of velocity
field of the 3D NSE (cf. e.g., [2], [4], [5], [12], [14], [15], [19], [20], [30]). In this
paper, we provide improved sufficient conditions, in terms of only one direction
REGULARITY TO THE 3D NAVIER–STOKES EQUATIONS
1143
derivative of the velocity field, that guarantee the global regularity of the 3D NSE.
Specifically, if u0 ∈ V , and if for some T > 0 we have
uz ∈ Lβ ([0, T ], Lα(Ω));
with α >
27
3
2
, β ≥ 1, and + ≤ 2,
16
α β
(10)
where u is a weak solution with the initial datum u0 , then u is a strong solution
of the 3D Navier–Stokes equations which exists on the interval [0, T ]. Notice that
we have improved the condition in [15], which requires α ≥ 9/4. Moreover, our
condition (10) implies that we only need (9) to be true on one direction derivative
of the velocity field in order to get the regularity of the 3D NSE.
For convenience, we recall the following three-dimensional Sobolev and Ladyzhenskaya inequalities (see, e.g., [1], [6], [9], [17]). There exists a positive constant Cr
such that
6−r
kψkr ≤ Cr kψk22r (k∂x ψk2 + kψk2 )
r−2
2r
(k∂y ψk2 + kψk2 )
r−2
2r
(k∂z ψk2 + kψk2 )
r−2
2r
3(r−2)
6−r
≤ Cr kψk22r kψkH 12r(Ω) ,
(11)
for every ψ ∈ H 1 (Ω), 2 ≤ r ≤ 6. And also,
rq+6q
2r−6q
(2+q)r
kψkr ≤ Cr kψk2 + kψk3q
k∆ψk2(2+q)r ,
(12)
for every ψ ∈ H 2 (Ω), 3q ≤ r. More generally, one has
2
1
3
kφkL3q ≤ C (kφz kq + kφkq ) 3 kφkH
1,
(13)
and
2
1
5
5
kφkL5q ≤ Ckφ2 kH
.
1 (kφz kq + kφkq )
(14)
We have not found a reference to the above two inequalities (13) and (14). We
present a proof here. It is clear that for every q ≥ 1, we get
Z 1
Z 1
3q+2
3q+2
3q
|φ(x, y, x)| 2 ≤ C
|φ(x, y, z)| 2 dx +
|φ(x, y, z)| 2 |φx (x, y, z)| dx ,
0
|φ(x, y, x)|
3q+2
2
≤C
0
1
Z
3q+2
2
|φ(x, y, z)|
dy +
0
3q−2
|φ(x, y, x)|
≤C
Z
1
Z
0
1
3q−2
|φ(x, y, z)|
Z
dz +
0
1
3q
|φ(x, y, z)| 2 |φy (x, y, z)| dy ,
3(q−1)
|φ(x, y, z)|
|φz (x, y, z)| dz .
0
As a result, we obtain
3q
|φ(x, y, z)|
≤C
Z
1
|φ(x, y, z)|
3q+2
2
dx +
0
×
Z
1
Z
1
0
1
|φ(x, y, z)|
3q
2
0
|φ(x, y, z)|
3q+2
2
dy +
0
×
Z
Z
1
Z
1
3q
2
|φ(x, y, z)|
3q−2
|φ(x, y, z)|
dz +
0
3(q−1)
|φ(x, y, z)|
21
|φx (x, y, z)| dx
|φy (x, y, z)| dy
0
12
|φz (x, y, z)| dz
21
.
1144
CHONGSHENG CAO
Thus, by Cauchy–Schwarz inequality, we get
Z
21
Z
Z
3q+2
3q
3q
2
2
|φ(x, y, z)| dxdydz ≤ C
|φ(x, y, z)|
dxdydz +
|φ| |φx | dxdydz
Ω
Ω
×
Ω
Z
|φ(x, y, z)|
Z
|φ(x, y, z)|3q−2 dxdydz +
3q+2
2
dxdydz +
Ω
×
|φ|
Z
|φ|3(q−1) |φz | dxdydz
3q
2
|φy | dxdydz
Ω
Ω
12
Z
Ω
3q
4
3q
4
1
2
3(q−1))
2
1
≤ kφk3q (kφk2 + kφx k2 ) kφk3q (kφk2 + kφy k2 ) 2 kφk3q
12
1
(kφkq + kφz kq ) 2 .
Therefore,
1
1
1
kφk3q ≤ C kφx k23 kφy k23 kφz kq3 ,
and (13) follows. Similarly, for every q ≥ 1, we get
Z 1
Z 1
5q+4
5q+4
5q
2
2
2
2
|φ(x, y, x)|
≤C
|φ(x, y, z)|
dx +
|φ(x, y, z)| |(φ )x (x, y, z)| dx ,
0
|φ(x, y, x)|
5q+4
2
≤C
0
1
Z
|φ(x, y, z)|
5q+4
2
dy +
0
|φ(x, y, x)|5q−4 ≤ C
Z
As a result, we obtain
Z
|φ(x, y, z)|5q ≤ C
1
|φ(x, y, z)|5q−4 dz +
0
1
|φ(x, y, z)|
5q+4
2
dx +
0
×
Z
1
Z
1
1
|φ(x, y, z)|
|φ(x, y, z)|
5q+4
2
dy +
Z
1
Z
1
|(φ )y (x, y, z)| dy ,
1
0
|φ(x, y, z)|5(q−1) |φz (x, y, z)| dz ,
1
|φ(x, y, z)|
5q
2
5q
2
|φ(x, y, z)|
12
|(φ )x (x, y, z)| dx
2
|φ(x, y, z)|
dz +
0
2
|(φ )y (x, y, z)| dy
0
5q−4
2
0
Z
Z
5q
2
0
0
×
Z
5(q−1)
|φ(x, y, z)|
12
|φz (x, y, z)| dz
0
12
.
Thus, by Cauchy–Schwarz inequality, we get
Z
12
Z
Z
5q+4
5q
5q
2
2
2
|φ(x, y, z)| dxdydz ≤ C
|φ|
dxdydz +
|φ| |(φ )x | dxdydz
Ω
Ω
×
Z
|φ|
Z
|φ|5q−4 dxdydz +
5q+4
2
dxdydz +
Ω
×
2
≤ kφk5q kφ k2 + k(φ )x k2
5(q−1)
×kφk5q 2
Z
|φ|5(q−1) |φz | dxdydz
2
|(φ )y | dxdydz
Ω
2
21
12
|φ|
5q
2
Ω
Ω
5q
4
Ω
Z
5q
4
kφk5q kφ2 k2 + k(φ2 )y k2
1
(kφkq + kφz kq ) 2 .
12
12
Therefore,
kφk5q ≤ C kφ2 k2 + k(φ2 )x k2
and (14) follows.
15
kφ2 k2 + k(φ2 )y k2
15
1
(kφkq + kφz k2 ) 5 ,
REGULARITY TO THE 3D NAVIER–STOKES EQUATIONS
1145
2. The main result. In this section we will prove our main result, which states
that the strong solution to system (1)–(3) exists on the interval [0, T ] provided the
assumption (10) on uz holds.
Theorem 2.1. Let u0 ∈ V, and let u be a Leray–Hopf weak solution to system (1)–
(3) with the initial value u0 . Let T > 0, and suppose that uz satisfies the condition
(10), namely,
Z T
27
3
2
kuz (s)kβα ds ≤ M, with α ≥
, 1 ≤ β < ∞, and + ≤ 2. (15)
16
α
β
0
Then u is a strong solution of system (1)–(3) on the interval [0, T ]. Moreover, it is
the only weak solution on [0, T ] with the initial datum u0 .
Proof. It is clear that we only need to prove this theorem under α3 + β2 = 2, i.e.
2α
. Also, in [15] the author has proved this theorem when α ≥ 9/4. Therefore,
β = 2α−3
from now on, we only consider
27/16 ≤ α ≤ 9/4,
β=
2α
.
2α − 3
(16)
By standard procedure for the 3D Navier–Stokes equations with periodic boundary
condition (see, e.g., [6], [7], [18], [22], [24], [27], [28]) one can show that there
exists a global in time Leray–Hopf weak solution to the system (1)–(3), if u0 ∈ H.
Furthermore, one can show the short time existence of a unique strong solution if
u0 ∈ V . In addition, this strong solution is the only weak solution, with the initial
datum u0 , on the maximal interval of existence of the strong solution.
Suppose that u is the strong solution with initial value u0 ∈ V such that u ∈
C([0, T ∗ ), V )∩L2 ([0, T ∗ ), H 2 (Ω)), where [0, T ∗ ) is the maximal interval of existence
of the unique strong solution. If T ∗ ≥ T then there is nothing to prove. If, on the
other hand, T ∗ < T our strategy is to show that the H 1 norm of this strong solution
is bounded on [0, T ∗ ), provided condition (10) is valid. As a result [0, T ∗ ) cannot
be a maximal interval of existence and consequently T ∗ ≥ T . Which will conclude
our proof.
From now on we focus on the strong solution, u, on its maximal interval of existence [0, T ∗ ), where we assume that T ∗ < T. As we have observed earlier the strong
solution u will also be the only weak solution on the interval [0, T ∗ ). Therefore, by
the energy inequality (6) for Leray–Hopf weak solutions we have (see, for example,
[6], [27] or [28] for details)
Z t
2
ku(t)k2 + ν
k∇u(s)k22 ds ≤ K1 ,
(17)
0
where
K1 = ku0 k22 .
Moreover, by (16) and (17), we get
( Rt
Z t
ku(s)kβ2 ds 27/16 ≤ α ≤ 2
0
β
3(α−2)
6−α
ku(s)kα ds ≤
Rt
2α−3
2α−3
0
ku(s)k
ds 2 ≤ α ≤ 9/4
ku(s)k
2
H1
0
β/2
≤ CK1
,
(18)
(19)
(20)
1146
CHONGSHENG CAO
For convenience , from now on, we will rewrite the Navier–Stokes equations as
∂v
− ν∆h v − νvzz + (v · ∇h )v + wvz + ∇h p = 0,
∂t
∂w
− ν∆h w − νwzz + v · ∇h w + wwz + pz = 0,
∂t
∇h · v + wz = 0,
(21)
(22)
(23)
where v = (v1 , v2 ) is the horizontal velocity field components and w is the vertical
velocity component. Namely, u = (v, w). We set ∇h = (∂x , ∂y ) to be the horizontal
gradient operator and ∆h = ∂x2 + ∂y2 the horizontal Laplacian, respectively. Next,
let us show that the H 1 norm of the strong solution u is bounded on interval [0, T ∗ ).
Denote by
X(t) = ku(t)k22 + k∇v(t)k22 + kuz (t)k22 + kw(t)k44 ,
Z t
Y (t) =
k∆v(s)k22 + k∇uz (s)k22 + k∇(w2 )(s)k22 ds,
0
Z t
2α
V (t) =
1 + ku(s)k22 + k∇u(s)k22 + (kukα + kuz (s)kα ) 2α−3
0
× k∇v(s)k22 + kuz (s)k22 + kw2 (s)k22 ds,
(24)
(25)
(26)
where α is as in (10).
2.1. L4 estimate. Taking the inner product of the equation (22) with w3 in L2 (Ω),
we get
1 dkw2 k22
3ν
+
k∇(w2 )k22
4 dt
4
Z
3
=−
(u · ∇w) w dxdydz −
Ω
Z
pz w3 dxdydz.
(27)
Ω
By integration by parts, and the incompressibility condition (2) we get
Z
(u · ∇w) w3 dxdydz = 0.
(28)
Ω
By the standard procedure one can get
−∆pz = 2∇ · (∇ · (u uz )).
Let us denote by
p1 = 2(−∆)−1 [∇ · (∇ · (v uz ))] ,
(29)
p2 = 2(−∆)−1 [∇ · (∇ · (w uz ))] .
(30)
and
Moreover, by Hölder’s inequality and (11), we obtain
Z
Z
Z
pz w3 dxdydz ≤ p1 w3 dxdydz + p2 w3 dxdydz Ω
Ω
≤ Ckp1 k
6α(3α−1)
2α2 +23α−18
≤ Ckv uz k
≤ Ckvk
kw3 k
6α(3α−1)
2α2 +23α−18
6α(3α−1)
(α+4)(2α−3)
Ω
6α(3α−1)
16α2 −29α+18
kw3 k
3
4α kw k
+ kp2 k α+3
6α(3α−1)
16α2 −29α+18
kuz kα kw3 k
4α
3(α−1)
3
4α kw k
4α
+ kw uz k α+3
3(α−1)
6α(3α−1)
16α2 −29α+18
3
4α kuz kα kw k
4α
+ kwk α−1
. (31)
3(α−1)
REGULARITY TO THE 3D NAVIER–STOKES EQUATIONS
1147
Let us estimate the above term by term. First, by (11), we get
3
4α kuz kα kw k
4α
kwk α−1
= kwk44α kuz kα
3(α−1)
α−1
= kw2 k22α kuz kα
α−1
2α−3
3
≤ C kw2 k22 + kw2 k2 α k∇w2 k2α kuz kα .
Next, applying (12) with r =
kvk
6α(3α−1)
(α+4)(2α−3)
6α(3α−1)
(α+4)(2α−3)
and q = α, we obtain
5α−6
5−2α
3α−1
≤ C kvk2 + kvk3α
k∆vk23α−1 .
By interpolation inequality we get
2(16α−27)
kw3 k
6α(3α−1)
16α2 −29α+18
5(9−α)
3(3α−1)
.
= kwk3 18α(3α−1) ≤ Ckwk4 3(3α−1) kwk5α
16α2 −29α+18
By (13), we reach
5α−6
2(5α−6)
5α−6
3α−1
kvk3α
≤ C (kvk2 + k∇vk2 ) 3(3α−1) (kvkα + kvz kα ) 3(3α−1) ,
and by (14), we obtain
5(9−α)
3(3α−1)
kwk5α
≤ C kw2 k2 + k∇w2 k2
Thus,
3ν
1 dkw2 k22
+
k∇(w2 )k22
4 dt
4
2(9−α)
3(3α−1)
9−α
(kwkα + kwz kα ) 3(3α−1) .
2(5α−6)
5α−6
5−2α
≤ C kvk2 + (kvk2 + k∇vk2 ) 3(3α−1) (kvkα + kvz kα ) 3(3α−1) k∆vk23α−1
16α−27
2(9−α)
9−α
× kuz kα kw2 k23(3α−1) kw2 k2 + k∇w2 k2 3(3α−1) (kwkα + kwz kα ) 3(3α−1)
2α−3
3
+C kw2 k22 + kw2 k2 α k∇w2 k2α kuz kα
14α−9
6+8α
≤ C kvk2 (kukα + kuz kα ) 3(3α−1) kw2 k23(3α−1)
2(9−α)
16α−27
6+8α
+kvk2 (kukα + kuz kα ) 3(3α−1) kw2 k23(3α−1) k∇w2 k23(3α−1)
2(5α−6)
13α
5−2α
13α
3(3α−1)
5−2α
3α−1
14α−9
3(3α−1)
+kvkH
(kukα + kuz kα ) 3(3α−1) k∆vk23α−1 kw2 k23(3α−1)
1
2(5α−6)
3(3α−1)
H1
2
16α−27
3(3α−1)
+kvk
(kukα + kuz kα )
k∆vk2
kw k2
2α−3
3
+C kw2 k22 kuz kα + kw2 k2 α k∇w2 k2α kuz kα .
2
2(9−α)
3(3α−1)
k∇w k2
Thanks to Gronwall’s inequality and (17), (15) and notations (24)–(26), we obtain
Z t
2
2
kw (t)k2 + ν
k∇(w2 )(s)k22 ds
(32)
0
26α−39
33−8α
2α−3
3
≤ kw2 (t = 0)k22 + C 1 + V (t) + V (T ) 6(3α−1) Y (t) 6(3α−1) + V (t) 2α Y (t) 2α .
1148
CHONGSHENG CAO
2.2. H 1 estimates. Taking the inner product of the equation (21) with −∆v and
the equation (1) with −uzz in L2 , and using the fact that the Stokes operator is
the same as the Laplacian operator under periodic boundary conditions, we obtain
1 d(k∇vk22 + kuz k22 )
+ ν(k∆vk22 + k∇uz k)
2 Z
dt
Z
=
=
[(u · ∇)v] · ∆v dxdydz +
ZΩ
∇h p∆v dxdydz +
Ω
[(v · ∇h )v · ∆h v + (v · ∇h )v · vzz
Z
+
[(u · ∇)u] · uzz dxdydz.
Ω
Z
[(u · ∇)u] · uzz dxdydz
Z
+ wvz · ∆v] dxdydz +
∇h p∆v dxdydz
Ω
Ω
Ω
By integration by parts, we get
Z
Z
(v · ∇h )v · ∆h v dxdydz =
wz |∇h v|2 + ∂x v2 ∂y v1 − ∂x v1 ∂y v2 dxdydz (33)
Ω
Ω
By integration by parts in the above we have
Z
Z
((v · ∇h )v + wvz ) · vzz dxdydz = − ((vz · ∇h )v + wz vz ) · vz dxdydz,
Ω
(34)
Ω
and
Z
Z
∇h p∆v dxdydz = −
∆p (∇h · v) dxdydz
Ω
Ω
Z
= − (2(∂x v1 )2 + 2(∂y v2 )2 + 2∂x v2 ∂y v1 + 2∂x v1 ∂y v2 + 2∇h w · vz ) wz dxdydz
ZΩ
= − (2(∂x v1 )2 + 2(∂y v2 )2 + 2∂x v2 ∂y v1 + 2∂x v1 ∂y v2 ) wz dxdydz
ZΩ
+
2w(∇h vz wz + vz ∇h wz ) dxdydz.
(35)
Ω
As a result of all of the above, and Hölder inequality, we get
1 d(k∇vk22 + kuz k22 )
+ ν(k∆vk22 + k∇uz k)
2 Z
dt
≤C
|uz | |∇v|2 + |w| |uz | |∇2 v| + |w| |uz | |∇uz | dxdydz
Ω
≤ C kuz kα k∇vk22α + kwk 2α(7α−3) kuz k 2α(7α−3) (k∆vk2 + k∇uz k2 ) .
6α2 +6−11α
α−1
α2 +8α−6
By (11)–(14), we obtain
h
i
2α−3
3
k∇vk22α ≤ C k∇vk22 + k∇vk2 α k∆vk2α ,
α−1
6(2α−3)
5(3−α)
7α−3
≤ Ckwk4 7α−3 kwk5α
3(2α−3)
2(3−α)
2(3−α)
3−α
7α−3
2
2 7α−3
2 7α−3
≤ Ckw k2
kw k2
+ k∇w k2
(kwkα + kwz kα ) 7α−3 ,
4α−3
3α
7α−3
7α−3
kuz k 2α(7α−3) ≤ C kuz k2 + kuz kα k∇uz k2
.
kwk
2α(7α−3)
6α2 +6−11α
α2 +8α−6
REGULARITY TO THE 3D NAVIER–STOKES EQUATIONS
1149
Therefore, we get
1 d(k∇vk22 + kuz k22 )
+ ν(k∆vk22 + k∇uz k)
2
dt
h
i
2α−3
3
≤ C k∇vk22 + k∇vk2 α k∆vk2α kuz kα
3(2α−3)
2(3−α)
2(3−α)
3−α
7α−3
2
2 7α−3
2 7α−3
+Ckw k2
kw k2
+ k∇w k2
(kwkα + kwz kα ) 7α−3
4α−3
3α
7α−3
7α−3
(k∆vk2 + k∇uz k2 )
× kuz k2 + kuz kα k∇uz k2
h
i
2α−3
3
≤ C k∇vk22 kuz kα + k∇vk2 α k∆vk2α kuz kα +
4α
3−α
+C (k∆vk2 + k∇uz k2 ) kuz k2 (kukα + kuz kα ) 7α−3 kw2 k27α−3
3α
3α
(36)
(37)
4α
+ (kukα + kuz kα ) 7α−3 k∇uz k27α−3 kw2 k27α−3
3(2α−3)
3−α
2(3−α)
+kuz k2 (kukα + kuz kα ) 7α−3 kw2 k2 7α−3 k∇w2 k27α−3
3α
3α
3(2α−3)
2(3−α)
+ (kukα + kuz kα ) 7α−3 k∇uz k27α−3 kw2 k2 7α−3 k∇w2 k27α−3
.
Thanks to Gronwall’s inequality and (17), we obtain
Z t
k∇v(t)k22 + kuz (t)k22 + ν
(k∆v(s)k22 + k∇uz (s)k) ds
(38)
0
3(2α−3)
3+8α
2α−3
3
≤ k∇u(t = 0)k22 + C 1 + V (t) + V (t) 2α Y (t) 2α + V (t) 2(7α−3) Y (t) 2(7α−3) ,
where X(t), Y (t) and V (t) are as in (24)–(26), respectively. Therefore, by (32) and
(38), we reach
X(t) + Y (t) ≤ ku(t = 0)k2H 1 + kw2 (t = 0)k22 + C(1 + V (t))
(39)
3(2α−3)
26α−39
33−8α
3+8α
2α−3
3
+C V (t) 6(3α−1) Y (t) 6(3α−1) + V (t) 2α Y (t) 2α + V (t) 2(7α−3) Y (t) 2(7α−3) .
Thanks to Young’s inequality, we get
X(t) + Y (t) ≤ k∇u0 k22 + ku0 k44 + C(K1 , α, t) + CV (t),
(40)
where C(K1 , α, t) is a constant depended on K1 , α, t. Therefore, by Gronwall’s
inequality we obtain
X(t) + Y (t) ≤ C k∇u0 k22 + ku0 k44 + C(K1 , α, t) eC(1+K1 +M) ,
(41)
for all t ∈ [0, T ∗ ), where M is as in (15). Therefore, the k∇vk2 and kuz k2 are for all
t ∈ [0, T ∗ ). Following the previous results (e.g., [4], [14]), we have that the H 1 norm
of the strong solution u is bounded on the maximal interval of existence [0, T ∗ ),
which leads to a contradiction. This completes the proof of Theorem 2.1.
Acknowledgments. This work was supported in part by the NSF grants no. DMS0709228 and the Florida International University Foundation.
1150
CHONGSHENG CAO
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Received December 2008; revised February 2009.
E-mail address: [email protected]