Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 232, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
REMARKS ON REGULARITY CRITERIA FOR THE 3D
NAVIER-STOKES EQUATIONS
RUIYING WEI, YIN LI
Abstract. In this article, we study the regularity criteria for the 3D NavierStokes equations involving derivatives of the partial components of the velocity.
It is proved that if ∇h u
e belongs to Triebel-Lizorkin space, ∇u3 or u3 belongs
to Morrey-Campanato space, then the solution remains smooth on [0, T ].
1. Introduction
This article is devoted to the Cauchy problem for the following incompressible
3D Navier-Stokes equation:
ut + (u · ∇)u + ∇p = 4u,
div u = 0,
x ∈ R3 , t > 0
x ∈ R3 , t > 0
(1.1)
with initial data
u(x, 0) = u0 ,
x ∈ R3 ,
(1.2)
where u = (u1 (x, t), u2 (x, t), u3 (x, t)) and p = p(x, t) denote the unknown velocity vector and the unknown scalar pressure, respectively. In the last century,
Leray [11] and Hopf [8] proved the global existence of a weak solution u(x, t) ∈
L∞ (0, ∞; L2 (R3 )) ∩ L2 (0, ∞; H 1 (R3 )) to (1.1)-(1.3) for any given initial datum
u0 (x) ∈ L2 (R3 ). However, whether or not such a weak solution is regular and
unique is still a challenging open problem. From that time on, different criteria for
regularity of the weak solutions has been proposed.
The classical Prodi-Serrin conditions (see [16, 18, 19]) say that if
2 3
+ = 1, 3 ≤ s ≤ ∞,
t
s
then the solution is smooth. Similar results is showed by Beirão da Veiga [1]
involving the velocity gradient growth condition:
2 3
3
∇u ∈ Lt (0, T ; Ls (R3 )),
+ = 2,
< s ≤ ∞.
t
s
2
Actually, whether the weak solution is smooth when a part of the velocity components is involved. As for this direction, later on, criteria just for one velocity
u ∈ Lt (0, T ; Ls (R3 )),
2000 Mathematics Subject Classification. 35Q35, 76D05.
Key words and phrases. Regularity criteria; Triebel-Lizorkin space; Morrey-Campanato space.
c
2014
Texas State University - San Marcos.
Submitted June 23, 2014. Published October 29, 2014.
1
2
R. WEI, Y. LI
EJDE-2014/232
component appeared. The first result in this direction is due to Neustupa et al [15]
(see also Zhou [21]), where the authors showed that if
2 3
1
u3 ∈ Lt (0, T ; Ls (R3 )),
+ = , s ∈ (6, ∞],
t
s
2
then the solution is smooth. A similar result, for the gradient of one velocity
component, is independently due to Zhou [22] and Pokorný [17]. In [22], Zhou
proved that if
2 3
3
∇u3 ∈ Lt (0, T ; Ls (R3 )),
+ = , 3 ≤ s < ∞.
t
s
2
then the solution is smooth on [0, T ]. This result is extended by Zhou and Pokorný
[26]; that is,
2 3
23
∇u3 ∈ Lt (0, T ; Ls (R3 )),
+ =
, 2 ≤ s ≤ 3.
t
s
12
Further criteria, including several components of the velocity gradient, pressure
or other quantities, can be found, here we just list some. Zhou and Pokorný [25]
proved the regularity condition
3
1
10
2 3
+ = + , s>
.
u3 ∈ Lt (0, T ; Ls (R3 )),
t
s
4 2s
3
And in [10], Jia and Zhou proved that if a weak solution u satisfies one of the
following two conditions:
10
u3 ∈ L∞ (0, T ; L 3 (R3 ));
∇u3 ∈ L∞ (0, T ; L30/19 (R3 )),
then u is regular on [0, T ]. Dong and Zhang [5] proved that if the horizontal
derivatives of the two velocity components
Z T
k∇h ũ(s)kḂ 0 ds < ∞,
0
∞,∞
then the solution keeps smoothness up to time T , where ũ = (u1 , u2 , 0), and ∇h ũ =
(∂1 ũ, ∂2 ũ, 0). For other kinds of regularity criteria, see [2, 6, 7, 9, 23, 24, 28, 29, 30]
and the references cited therein.
Throughout this paper C will denote a generic
R positive constantRwhich can vary
from line to line. For simplicity, we shall use f (x) dx to denote R3 f (x) dx, use
k · kp to denote k · kLp .
The purpose of this article is to improve and extend above known regularity
criterion of weak solution for the equations (1.1), (1.2) to the Triebel-Lizorkin space
and Morrey-Campanato spaces. The main results of this paper read:
Theorem 1.1. Assume that u0 ∈ H 1 (R3 ) with div u0 = 0. u(x,t) is the corresponding weak solution to (1.1) and (1.2) on [0, T ). If additionally
Z T
2 3
3
(1.3)
k∇h u
e(·, t)kpḞ 0 dt < ∞, with + = 2, < q ≤ ∞,
p q
2
q, 2 q
0
3
then the solution remains smooth on [0, T ].
Theorem 1.2. Assume that u0 ∈ H 1 (R3 ) with div u0 = 0. u(x, t) is the corresponding weak solution to (1.1) and (1.2) on [0, T ). If additionally
Z T
8
3
k∇u3 (·, t)k 3(2−r)
(1.4)
dt < ∞, with 0 < r ≤ 1, 2 ≤ p ≤ ,
p, 3
r
Ṁ
r
0
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REMARKS ON REGULARITY
3
then the solution remains smooth on [0, T ].
Theorem 1.3. Assume that u0 ∈ H 1 (R3 ) with div u0 = 0. u(x,t) is the corresponding weak solution to (1.1) and (1.2) on [0, T ). If additionally
Z T
8
3
3
ku3 (·, t)k 3−4r
dt < ∞, with 0 < r < , 2 ≤ p ≤ ,
(1.5)
p, 3
r
Ṁ
4
r
0
then the solution remains smooth on [0, T ].
0
Remark 1.4. Noticing that the classical Riesz transformation is bounded in Ḃ∞,∞
,
if we take q = ∞ in Theorem 1.1, then the classical Beal-Kato-Majda criterion for
the Navier-Stokes equations is obtained; that is, if
Z T
k∇h u
e(·, t)kḂ 0 dt < ∞
∞,∞
0
then the solution remains smooth on [0, T ].
Remark 1.5. Since (it is proved in [12, 13])
Lq (R3 ) = Ṁ q,q (R3 ) ⊂ Ṁ p,q (R3 ),
1 < p ≤ q < ∞,
3
3
3
3
3
L r (R3 ) ⊂ Ṁ p, r (R3 ) ⊂ Ẋr (R3 ) ⊂ Ṁ 2, r (R3 ), 2 < p ≤ , 0 < r < ,
r
2
the result of Theorem 1.2 is an improved version of [27, Theorem 2]. Also we obtain,
if
Z T
8
3
3−4r
dt < ∞, with 0 < r < ,
ku3 (·, t)kẊ
r
4
0
then the solution remains smooth on [0, T ].
2. Preliminaries
In this section, we shall introduce the Littlewood-Paley decomposition theory,
and then give some definitions of the homogeneous Besov space, homogeneous
Triebel-Lizorkin space, Morrey-Campanato space and multiplier space as well as
some relate spaces used throughout this paper. Before this, let us first recall the
weak solutions of (1.1)-(1.3):
Let u0 ∈ L2 (R3 ) with ∇ · u0 = 0, a measurable R3 -valued vector u is said to be
a weak solution of (1.1)-(1.3) if the following conditions hold:
(1) u(x, t) ∈ L∞ (0, ∞; L2 (R3 )) ∩ L2 (0, ∞; H 1 (R3 ));
(2) u solves (1.1)-(1.2) in the sense of distributions;
(3) the energy inequality holds; i.e,
Z t
kuk22 + 2
k4u(·, τ )k22 dτ ≤ ku0 k22 , 0 ≤ t ≤ T.
0
Let us choose a nonnegative radial function ϕ
∈ C ∞ (R3 ) be supported in the
P∞
3
8
3
−l
annulus {ξ ∈ R : 4 ≤ |ξ| ≤ 3 }, such that
l=−∞ ϕ(2 ξ) = 1, ∀ξ 6= 0. For
0
3
f ∈ S (R ), the frequency projection operators 4l is defined as
4l f = F −1 (ϕ(2−j ·)) ∗ f,
where F −1 (g) is the inverse Fourier transform of g. The formal decomposition
∞
X
f=
4l f.
(2.1)
l=−∞
4
R. WEI, Y. LI
EJDE-2014/232
is called the homogeneous Littlewood-Paley decomposition. Noticing
l+1
X
4l f =
4j (4l f )
j=l−1
and using the Young inequality, we have the following class Bernstein inequality:
Lemma 2.1 ([3]). Let α ∈ N , then for all 1 ≤ p ≤ q ≤ ∞, sup|α|=k k∂ α 4l f kq ≤
1
1
C2lk+3l( p − q ) k4l f kp . and C is a constant independent of f, l.
s
For s ∈ R and (p, q) ∈ [1, ∞] × [1, ∞], the homogeneous Besov space Ḃp,q
is
defined by
s
Ṡp,q
= {f ∈ Z 0 (R3 ) : kf kṠ s < ∞},
p,q
where
kf kḂ s =
p,q
0

 P
jsq
k4j f (·)kqp
j∈z 2
1/q
sup
js
j∈z 2 k4j f (·)kp ,
,
1 ≤ q < ∞,
q = ∞.
3
and Z (R ) denote the dual space of
Z 0 (R3 ) = {f ∈ S(R3 ) : Dα fˆ(0) = 0. ∀a ∈ N 3 }.
On the other hand, for s ∈ R, (p, q) ∈ [1, ∞) × [1, ∞], and for s ∈ R, p = q = ∞,
the homogenous Triebel-Lizorkin space is defined as
kf kḞ s
p,q
s
Ḟp,q
= {f ∈ Z 0 (R3 ) : kf kḞ s < ∞},
p,q
( P
jsq
q 1/q
k( j∈z 2 |4j f (·)| ) kp , 1 ≤ q < ∞,
=
k supj∈z (2js |4j f (·)|)kp ,
q = ∞.
Notice that by Minkowski inequality, we have the following two imbedding relations:
s
s
Ḃp,q
⊂ Ḟp,q
,
q ≤ p;
s
s
Ḟp,q
⊂ Ḃp,q
,
p ≤ q.
and the following two inclusions:
s
s
Ḣ s = Ḃ2,2
= Ḟ2,2
,
0
0
L∞ ⊂ Ḟ∞,∞
= Ḃ∞,∞
.
We refer to [20] for more properties.
For 1 < q ≤ p < ∞, the homogeneous Morrey-Campanato space in R3 is
3
3
Ṁ p,q = f ∈ Lqloc (R3 ); kf kṀ p,q = sup sup R p − q kf kq(B(x,R)) < ∞ ,
x∈R3 R>0
For 1 ≤ p0 ≤ q 0 < ∞, we define the homogeneous space
n
X
0 0
0
0
Ṅ p ,q = f ∈ Lq |f =
gk , where gk ∈ Lqcomp (R3 ) and
k∈N
X
3( 10 − 10 )
dk p q kgk kq0
o
< ∞, where dk = diam(supp gk ) < ∞ .
k∈N
For 0 < α < 3/2, we say that a function belongs to the multiplier spaces M (Ḣ α , L2 )
if it maps, by pointwise multiplication, Ḣ α to L2 :
Ẋα := M (Ḣ α , L2 ) := f ∈ S 0 ; kf · gkL2 ≤ CkgkḢ α , ∀g ∈ Ḣ α .
EJDE-2014/232
REMARKS ON REGULARITY
5
Here, Ḣ α is the homogeneous Sobolev space of order α,
Z
1/2
Ḣ α = f ∈ L1loc ; kf kL2 ≡
|ξ|2α |û(ξ)|2
<∞ .
R3
p
where L (1 ≤ p ≤ ∞) is the Lebsgue space endowed with norm k · kp .
Lemma 2.2 ([4, 12]). Let 1 ≤ p0 ≤ q 0 < ∞, and p, q such that p1 + p10 = 1, 1q + q10 = 1.
0
0
Then Ṁ p,q is the dual space of Ṅ p ,q .
Lemma 2.3 ([4, 7, 12]). Let 1 < p0 ≤ q 0 < 2, m ≥ 2, and p1 + p10 = 1. Denote
n
α = − n2 + np + m
∈ (0, 1] Then there exists a constant C > 0, such that for any
u ∈ Lm (Rn ), v ∈ Ḣ α (Rn ),
ku · vkṄ p0 ,q0 ≤ CkukLm kvkḢ α .
Lemma 2.4 ([13]). For 0 ≤ r ≤ 23 , let the space M (Ḃ2r,1 → L2 ) be the space of
functions which are locally square integrable on R3 and such that pointwise multiplication with these functions maps boundedly the Besov space Ḃ2r,1 (R3 ) to L2 (R3 ).
The norm in M (Ḃ2r,1 → L2 ) is given by the operator norm of pointwise multiplication:
kf kM (Ḃ r,1 →L2 ) = sup{kf gk2 : kgkḂ r,1 ≤ 1}.
2
2
3
Then, f belongs to M (Ḃ2r,1 → L2 ) if and only if f belongs to Ṁ 2, r (with equivalence
of norms).
3. The proof of main results
Proof of Theorem 1.1. Multiplying (1.1)1 by −4u, integrating by parts, noting
that ∇ · u = 0, we have
Z
Z
Z
1 d
|∇u|2 dx + |4u|2 dx = [(u · 5)u] · 4u dx =: I.
(3.1)
2 dt
Next we estimate the right-hand side of (3.1), with the help of integration by parts
and −∂3 u3 = ∂1 u1 + ∂2 u2 , one shows that
Z
3
X
I=−
∂k ui ∂i uj ∂k uj dx
i,j,k=1
=−
2 X
3 Z
X
i,j=1 k=1
−
2 Z
X
2 Z
X
Z
3 Z
X
∂k u3 ∂3 u3 ∂k u3 dx
k=1
∂3 ui ∂i u3 ∂3 u3 dx −
i=1
≤C
∂k ui ∂i u3 ∂k u3 dx
i,k=1
∂k u3 ∂3 uj ∂k uj dx −
j,k=1
−
2 Z
X
∂k ui ∂i uj ∂k uj dx −
2 Z
X
∂3 u3 ∂3 uj ∂3 uj dx
j=1
|∇h u
e||∇u|2 dx.
Thus, the above inequality implies
Z
1 d
2
2
k∇uk2 + k4uk2 ≤ C |∇h u
e||∇u|2 dx.
2 dt
(3.2)
6
R. WEI, Y. LI
EJDE-2014/232
Using the Littlewood-Paley decomposition (2.1), ∇h u
e can be written as
∇h u
e=
X
N
X
4j (∇h u
e) +
j<−N
4j (∇h u
e) +
j=−N
X
4j (∇h u
e).
j>N
where N is a positive integer to be chosen later. Substituting this into (3.2), one
obtains
1 d
k∇uk22 + k4uk22
2 dt
N Z
X Z
X
≤C
|4j (∇h u
e)||∇u|2 dx + C
|4j (∇h u
e)||∇u|2 dx
j<−N
+C
j=−N
XZ
(3.3)
|4j (∇h u
e)||∇u|2 dx
j>N
=: K1 + K2 + K3 .
For Ki (i = 1, 2, 3), we now give the estimates one by one. For K1 , using the Hölder
inequality, the Young inequality and Lemma 2.1, it follows that
X
K1 ≤ C
k4j (∇h u
e)k∞ k∇uk22
j<−N
≤C
X
≤C
X
23j/2 k4j (∇h u
e)k2 k∇uk22
j<−N
(3.4)
2
3j
1/2 X
j<−N
≤ C2
−3N/2
k4j (∇h u
e)k22
1/2
k∇uk22
j<−N
k∇uk32 .
s
Where in the last inequality, we use the fact that for all s ∈ R, Ḣ s = Ḃ2,2
.
For K2 , by the Hölder inequality and the Young inequality, one has
Z X
N
K2 = C
|4j (∇h u
e)||∇u|2 dx
j=−N
≤ CN
2q−3
2q
Z X
N
|4j (∇h u
e)|2q/3
3/(2q)
|∇u|2 dx
j=−N
≤ CN
≤ CN
2q−3
2q
k∇h u
ekḞ 0
k∇uk22q
k∇h u
ekḞ 0
k∇uk2
2q
q,
3
2q−3
2q
q,
≤
(3.5)
q−1
2q−3
q
2q
3
2q
1
k4uk22 + CN k∇h u
ekḞ2q−3
k∇uk22 .
0
2q
2
q,
3
where we used the interpolation inequality
3
kuks ≤ Ckuk2s
for 2 ≤ s ≤ 6.
− 12
3
3/q
k4uk2
−3
2
s
kukḢ
,
1
EJDE-2014/232
REMARKS ON REGULARITY
7
Finally, using Hölder inequality and Lemma 2.1, K3 can be estimated as
XZ
|4j (∇h u
e)||∇u|2 dx
K3 = C
j>N
≤C
X
k4j (∇h u
e)k3 k∇uk23
j>N
≤C
X
j
(3.6)
2 2 k4j (∇h u
e)k2 k∇uk23
j>N
X
≤ C(
j>N
≤ C2
X
2−j )1/2 (
22j k4j (∇h u
e)k22 )1/2 k∇uk2 k4uk2
j>N
−N/2
k∇uk2 k4uk22 .
Substituting (3.4), (3.5) and (3.6) in (3.3), we obtain
d
k∇uk22 + k4uk22
dt
2q
3
k∇uk22 + C2−N/2 k∇uk2 k4uk22 .
≤ C2− 2 N k∇uk32 + CN k∇h u
ekḞ2q−3
0
q,
(3.7)
2q
3
Now we choose N such that C2−N/2 k∇uk2 ≤ 12 ; that is
N≥
ln(k∇uk22 + e) + ln C
+ 2.
ln 2
Thus (3.7) implies
d
k∇uk22 ≤ C + Ck∇h u
ekpḞ 0 ln(k∇uk22 + e)k∇uk22 .
2q
dt
q,
3
Taking the Gronwall inequality into consideration, we obtain
RT
Z T
k∇h u
ekp 0
(τ )dτ i
h
0
Ḟ 2q
p
2
q,
3
ln(k∇uk2 + e) ≤ C 1 +
k∇h u
ekḞ 0 (τ )dτ · e
.
0
q,
2q
3
The proof of Theorem 1.1 is complete under the condition (1.3).
Proof of Theorem 1.2. Multiplying (1.1)1 by −4h u, integrating by parts, noting
that ∇ · u = 0, we have
Z
Z
Z
1 d
|∇h u|2 dx + |∇∇h u|2 dx = [(u · 5)u] · 4h u dx =: J.
(3.8)
2 dt
Next we estimate the right-hand side of (3.8), with the help of integration by parts
and −∂3 u3 = ∂1 u1 + ∂2 u2 , one shows that
3 X
2 Z
X
J =−
∂k ui ∂i uj ∂k uj dx
i,j=1 k=1
=−
Z
2
X
∂k ui ∂i uj ∂k uj dx −
i,j,k=1
−
3 X
2 Z
X
2 Z
X
i,k=1
∂k u3 ∂3 uj ∂k uj dx
j=1 k=1
=: J1 + J2 + J3 .
∂k ui ∂i u3 ∂k u3 dx
(3.9)
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R. WEI, Y. LI
EJDE-2014/232
For J2 and J3 , we obtain
Z
|J2 + J3 | ≤ C
|∇u3 ||∇h u||∇u| dx.
(3.10)
J1 is a sum of eight terms, using the fact −∂3 u3 = ∂1 u1 + ∂2 u2 , we can estimate it
as
Z
J1 = − (∂1 u1 + ∂2 u2 )[(∂1 u1 )2 − ∂1 u1 ∂2 u2 + (∂2 u2 )2 ] dx
Z
− (∂1 u1 + ∂2 u2 )[(∂2 u1 )2 + ∂1 u2 ∂2 u1 + (∂1 u2 )2 ] dx
Z
= ∂3 u3 [(∂1 u1 )2 − ∂1 u1 ∂2 u2 + (∂2 u2 )2 + (∂2 u1 )2 + ∂1 u2 ∂2 u1 + (∂1 u2 )2 ] dx
Z
≤ C |∇u3 ||∇h u||∇u| dx.
(3.11)
Substituting the estimates (3.9)-(3.11) in (3.8), we obtain
Z
1 d
2
2
k∇h uk2 + k∇∇h uk2 ≤ C |∇u3 ||∇h u||∇u| dx =: L.
2 dt
(3.12)
when 2 < p ≤ 3r , using Lemmas 2.2 and 2.3, and the Young inequality, we obtain
L ≤ Ck∇u3 k
≤ Ck∇u3 k
≤ Ck∇u3 k
≤
3
Ṁ p, r
k|∇u| · |∇h u|k
Ṅ
p0 ,
3
3−r
3
k∇h ukḢ r k∇uk2
3
k∇ukL2 k∇h uk1−r
k∇∇h ukr2
2
Ṁ p, r
Ṁ p, r
(3.13)
2
1
k∇∇h uk22 + Ck∇u3 k 2−rp, 3 k∇uk22 .
Ṁ r
2
where we used the inequality
kf kḢ r
Z
ˆ
= k|ξ| f k2 = ( |ξ|2r |fˆ|2r |fˆ|2−2r dξ)1/2 ≤ kf k1−r
k∇f kr2 ,
2
r
with 0 < r ≤ 1.
In the case p = 2, using Hölders inequality, Lemma 2.4, and the Young inequality,
we can estimate L as
L ≤ Ck|∇u3 | · |∇h u|k2 k∇uk2
≤ Ck∇u3 k
3
k∇h ukḂ r,1 k∇uk2
3
k∇h uk1−r
k∇∇h ukr2 k∇uk2
2
Ṁ 2, r
≤ Ck∇u3 k
Ṁ 2, r
≤
2
(3.14)
2
1
k∇∇h uk22 + Ck∇u3 k 2−r2, 3 k∇uk22 .
Ṁ r
2
where we used the following interpolation inequality [14]: for 0 ≤ r ≤ 1, kf kḂ r,1 ≤
2
kf k1−r
k∇f kr2 .
2
Now, gathering (3.13) and (3.14) together and substituting into (3.12), we obtain
2
d
k∇h uk22 + k∇∇h uk22 ≤ Ck∇u3 k 2−rp, 3 k∇uk22 .
Ṁ r
dt
(3.15)
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REMARKS ON REGULARITY
9
Multiplying (1.1)1 by −4u, integrating by parts, noting that ∇ · u = 0, we have
(see [25])
Z
1 d
k∇uk22 + k4uk22 = [(u · 5)u] · 4u dx
2 dt
Z
≤ C |∇h u||∇u|2 dx
≤ Ck∇h uk2 k∇uk24
1/2
3
≤ Ck∇h uk2 k∇uk2 k∇uk62
1/2
1/2
≤ Ck∇h uk2 k∇uk2 k∇∇h uk2 k4uk2 .
Integrating, with respect to t, yields
Z t
1
k∇u(t)k22 +
k4u(τ )k22 dτ
2
0
Z t
1
k∇u(τ )k22 dτ )1/4
≤ k∇u0 k22 + C sup k∇h u(τ )k2 (
2
0≤τ ≤t
0
Z t
1/2 Z t
1/4
2
×
k∇∇h u(τ )k2 dτ
k4u(τ )k22 dτ
.
0
(3.16)
0
Substituting (3.15) in (3.16), using Hölders inequality and the Young inequality,
we obtain
Z t
1
2
k∇u(t)k2 +
k4u(τ )k22 dτ
2
0
Z t
Z t
1/4
2
1
≤ k∇u0 k22 + (C + C
k∇u3 k 2−rp, 3 k∇u(τ )k22 dτ )
k4u(τ )k22 dτ
Ṁ r
2
0
0
Z t
4/3
2
3
1/2
≤C +C
k∇u3 k 2−rp, 3 k∇u(τ )k22 k∇u(τ )k2 dτ
Ṁ r
0
Z
1 t
(3.17)
k4u(τ )k22 dτ
+
2 0
Z t
Z t
1/3
8
2
2
≤C +C
k∇u3 k 3(2−r)
k∇u(τ
)k
dτ
k∇u(τ
)k
dτ
3
2
2
Ṁ p, r
0
0
Z t
1
k4u(τ )k22 dτ
+
2 0
Z t
Z
8
1 t
2
≤C +C
k∇u3 k 3(2−r)
k∇u(τ
)k
dτ
+
k4u(τ )k22 dτ.
2
p, 3
r
Ṁ
2
0
0
Absorbing the last term into the left hand side, applying the Gronwall inequality
and combining with the standard continuation argument, we conclude that the solu8
3
tions u can be extended beyond t = T provided that ∇u3 ∈ L 3(2−r) (0, T ; Ṁ p, r (R3 )).
This completes the proof of Theorem 1.2.
Proof of Theorem 1.3. We start from (3.9), we can estimate J2 and J3 as
Z
|J2 + J3 | ≤ C |u3 ||∇u||∇∇h u| dx.
(3.18)
10
R. WEI, Y. LI
EJDE-2014/232
From (3.11), we find that
Z
J1 ≤ C
|u3 ||∇u||∇∇h u| dx.
Combining (3.9),(3.17), (3.18) with (3.8), we obtain
Z
1 d
k∇h uk22 + k∇∇h uk22 ≤ C |u3 ||∇u||∇∇h u| dx =: V.
2 dt
(3.19)
(3.20)
When 2 < p ≤ 3r , similarly as in the proof of L in (3.13), we obtain
V ≤ Cku3 k
3
≤ Cku3 k
3
k∇ukḢ r k∇∇h uk2
≤ Cku3 k
3
k∇∇h ukL2 k∇uk1−r
k4ukr2
2
Ṁ p, r
Ṁ p, r
Ṁ p, r
k|∇u| · |∇∇h u|k
Ṅ
p0 ,
3
3−r
(3.21)
1
2(1−r)
k∇∇h uk22 + Cku3 k2 p, 3 k∇uk2
k4uk2r
2 .
Ṁ r
2
When p = 2, similarly as in the proof of L in (3.14), we have
≤
V ≤ Ck|u3 | · |∇u|k2 k∇∇h uk2
≤ Cku3 k
3
Ṁ 2, r
≤ Cku3 k
3
Ṁ 2, r
k∇ukḂ r,1 k∇∇h uk2
2
k∇uk1−r
k4ukr2 k∇∇h uk2
2
(3.22)
1
2(1−r)
k∇∇h uk22 + Cku3 k2 2, 3 k∇uk2
k4uk2r
2 .
Ṁ r
2
Substituting (3.21) and (3.22) in (3.20), we find that
d
2(1−r)
k∇h uk22 + k∇∇h uk22 ≤ Cku3 k2 2, 3 k∇uk2
k4uk2r
(3.23)
2 .
Ṁ r
dt
Substituting (3.23) in (3.16), we obtain
Z t
1
k∇u(t)k22 +
k4u(τ )k22 dτ
2
0
Z t
1
2(1−r)
2
≤ k∇u0 k2 + C + C
ku3 k2 p, 3 k∇u(τ )k2
k4u(τ )k2r
dτ
2
Ṁ r
2
0
Z t
1/4
×
k4u(τ )k22 dτ
0
Z
4/3
Z t
1 t
2(1−r)
2
≤C+
k4u(τ )k2 dτ + C
k4u(τ )k2r
dτ
ku3 k2 p, 3 k∇u(τ )k2
2
Ṁ r
4 0
0
Z t
Z t
4r/3
1
≤C+
k4u(τ )k22 dτ + C
k4u(τ )k22 dτ
4 0
0
Z t
4(1−r)
2
3
×
ku3 k 1−rp, 3 k∇u(τ )k22 dτ
≤
Ṁ
0
r
Z t
4(1−r)
3−4r
1
2
1
3−4r
k4u(τ )k22 dτ + C
ku3 k 1−rp, 3 k∇u(τ )k22(1−r) k∇u(τ )k22(1−r) dτ
r
Ṁ
2 0
0
Z
1
3−4r
Z t
Z t
8
1 t
3−4r
2
2
≤C+
k4u(τ )k2 dτ + C
ku3 k p, 3 k∇u(τ )k2 dτ
k∇u(τ )k22 dτ
Ṁ r
2 0
0
0
Z t
Z t
8
1
2
≤C+
k4u(τ )k22 dτ + C
ku3 k 3−4r
3 k∇u(τ )k2 dτ.
Ṁ p, r
2 0
0
Z
≤C+
t
EJDE-2014/232
REMARKS ON REGULARITY
11
By a similar argument as in the proof of Theorem 1.2, provided that u3 ∈
8
3
L 3−4r (0, T ; Ṁ p, r (R3 )), we complete the proof of Theorem 1.3.
Acknowledgements. Ruiying Wei and Yin Li would like to express sincere gratitude to Professor Zheng-an Yao of Sun Yat-sen University for enthusiastic guidance and constant encouragement. This work is partially supported by Guangdong Provincial culture of seedling of China (no. 2013LYM0081), and Guangdong
Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University.
References
[1] H. Beirão da Veiga; A new regularity class for the Navier-Stokes equations in Rn , Chinese
Ann. Math. Ser. B, 16 (1995) 407-412.
[2] C. Cao, E. Titi; Global regularity criterion for the 3D Navier-Stokes equations involving one
entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011) 919-932.
[3] J. Chemin; Perfect Incompressible Fluids, Oxford University Press, New York, 1998.
[4] S. Dubois; Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations, J. Diff.
Eqns., 189 (2003) 99-147.
[5] B. Dong, Z. Zhang; The BKM criterion for the 3D Navier-Stokes equations via two velocity
components, Nonlinear Anal., Real World Applications, 11 (2010) 2415-2421.
[6] D. Fang, C. Qian; The regularity criterion for 3D Navier-Stokes equations involving one
velocity gradient component, Nonlinear Anal., 78 (2013) 86-103
[7] J. Fan, S. Jiang, G. Ni; On regularity criteria for the n-dimensional Navier-Stokes equations
in terms of the pressure, J. Diff. Eqns., 244 (2008) 2963-2979.
[8] E. Hopf; Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math.
Nachr. 4 (1951) 213-231.
[9] X. Jia, Z. Jiang; An anisotropic reuglarity criterion for the Navier-Stokes equations, Commun. Pure Appl. Anal., 12 (2013) 1299-1306.
[10] X. Jia, Y. Zhou; Remarks on regularity criteria for the Navier- Stokes equations via one
velocity component. Nonlinear Anal. Real World Appl., 15 (2014), 239-245.
[11] J. Leray; Sur le mouvement d’un liquide visqueux emplissant l’espace Acta Math. 63 (1934)
183-248.
[12] P. Lemarié-Rieusset; Recent Developments in the Navier-Stokes Problem, Chapman and Hall,
London. 2002.
[13] P. Lemarié-Rieusset; The Navier-Stokes equations in the critical Morrey-Campanato space,
Rev. Mat. Iberoam., 23 (2007) 897-930.
[14] S. Machihara, T. Ozawa; Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc.,
131 (2003) 1553-1556.
[15] J. Neustupa, A. Novotný P. Penel; An interior regularity of a weak solution to the NavierStokes equations in dependence on one component of velocity, Topics in mathematical fluid
mechanics, Quad. Mat., 10 (2002) 163-183.
[16] T. Ohyama; Interior regularity of weak solutions of the time-dependent Navier-Stokes equation, Proc. Japan Acad., 36 (1960) 273-277.
[17] M. Pokorný; On the result of He concerning the smoothness of solutions to the Navier-Stokes
equations, Electron. J. Diff. Eqns., 11 (2003) 1-8.
[18] G. Prodi; Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl.,
48 (1959), 173-182.
[19] J. Serrin; The initial value problem for the Navier-Stokes equations, Nonlinear Problems,
Proc. Symposium, Madison, Wisconsin, University of Wisconsin Press, Madison, Wisconsin,
(1963) 69-98.
[20] H. Triebel; Interpolation Theory, Function Spaces, Differential Operators, North Holland,
Amsterdam, New-York, Oxford, 1978.
[21] Y. Zhou; A new regularity criterion for weak solutions to the Navier-Stokes equations. J.
Math. Pures Appl. 84 (2005) 1496-1514.
[22] Y. Zhou; A new regularity criterion for the Navier-Stokes equations in terms of the gradient
of one velocity component, Methods Appl. Anal., 9 (2002) 563-578.
12
R. WEI, Y. LI
EJDE-2014/232
[23] Y. Zhou, S. Gala; Regularity criteria for the solutions to the 3D MHD equations in the
multiplier space. Z. Angew. Math. Phys., 61 (2010) 193-199.
[24] Y. Zhou, S. Gala; Regularity criteria in terms of the pressure for the Navier-Stokes equations
in the critical Morrey-Campanato space. Z. Anal. Anwend., 30 (2011) 83-93.
[25] Y. Zhou, M. Pokorný; On the regularity of the solutions of the Navier-Stokes equations via
one velocity component, Nonlinearity, 23 (2010) 1097-1107.
[26] Y. Zhou, M. Pokorný; On a regularity criterion for the Navier-Stokes equations involving
gradient of one velocity component, J. Math. Phys., 50 (2009) 123514.
[27] Z. Zhang; A remark on the regularity criterion for the 3D Navier-Stokes equations involving
the gradient of one velocity component, J. Math. Anal. Appl., 414 (2014) 472-479.
[28] Z. Zhang; A Serrin-type reuglarity criterion for the Navier-Stokes equations via one velocity
component, Commun. Pure Appl. Anal., 12 (2013) 117-124.
[29] Z. Zhang, F. Alzahrani, T. Hayat, Y. Zhou; Two new regularity criteria for the Navier-Stokes
equations via two entries of the velocity Hessian tensor. Appl. Math. Lett., 37 (2014) 124-130.
[30] Z. Zhang, Z. Yao, P. Li, C. Guo, M. Lu; Two new regularity criteria for the 3D Navier-Stokes
equations via two entries of the velocity gradient, Acta Appl. Math., 123 (2013) 43-52.
Ruiying Wei
School of Mathematics and Information Science, Shaoguan University, Shaoguan,
Guangdong 512005, China.
Department of Mathematics, Sun Yat-sen University , Guangzhou, Guangdong 510275,
China
E-mail address: weiruiying521@163.com
Yin Li
School of Mathematics and Information Science, Shaoguan University, Shaoguan,
Guangdong 512005, China.
Department of Mathematics, Sun Yat-sen University , Guangzhou, Guangdong 510275,
China
E-mail address: liyin2009521@163.com