J. Nonlinear Sci. Appl. 4 (2011), no. 2, 126–129
The Journal of Nonlinear Science and Applications
http://www.tjnsa.com
A NEW REGULARITY CRITERION FOR THE
NAVIER-STOKES EQUATIONS
HU YUE1∗ AND WU-MING LI2
This paper is dedicated to Themistocles M. Rassias on the occasion of his sixtieth birthday
Communicated by Choonkil Park
Abstract. We study the incompressible Navier-Stokes equations in the entire
three-dimensional space. We prove that if ∂3 u3 ∈ Lst 1 Lrx1 and u1 , u2 ∈ Lst 2 Lrx2 ,
then the solution is regular. Here s21 + r31 ≤ 1, 3 ≤ r1 ≤ ∞, s22 + r32 ≤ 1 and
3 ≤ r2 ≤ ∞.
1. Introduction and preliminaries
In this paper, we consider Cauchy problem for the incompressible Navier-Stokes
equations
 ∂u
 ∂t + u · ∇u + ∇p = ν4u
(1.1)
∇·u = 0

u(x, 0) = u0 (x)
in which u = (u1 (x, t), u2 (x, t), u3 (x, t)) is the unknown velocity field, u0 =
(u1 (x, 0), u2 (x, 0), u3 (x, 0)) is the initial velocity field with ∇ · u0 = 0, and p(x, t)
is a scalar pressure. While ν is the kinematic viscosity coefficient, we will assume
that ν ≡ 1 for simplicity in this paper. Here we use the classical notations
∇u = (∂1 u, ∂2 u, ∂3 u), 4u =
3
X
∂i2 u,
∇·u=
3
X
i=1
i=1
∂i ui , u · ∇u =
3
X
uj ∂j u.
j=1
Date: Received: August 12, 2010; Revised: November 24, 2010.
∗
Corresponding author
c 2011 N.A.G.
°
2000 Mathematics Subject Classification. Primary 35Q30, 76D03.
Key words and phrases. Navier-Stokes equations; Leray-Hopf weak solution; Regularity.
126
REGULARITY CRITERION FOR NAVIER-STOKES EQUATIONS
127
In 1934, Leray proved the existence of a global weak solution
\
u ∈ L∞ (0, ∞; L2 (R3 )) L2 (0, ∞; H 1 (R3 )),
on the condition that u0 ∈ L2 and ∇ · u0 = 0. It is called the Leray-Hopf weak
solution which satisfies the energy inequality [1]. If ku(t)kH 1 is continuous, then
we say that u(t) is regular. Up to the present, the regularity and uniqueness of
the weak solutions is still an open problem. On the other hand, there are already
many criterions which ensure that the weak solution is regular [2, 3].
In this paper, we give a regularity criterion concerning one derivative of one
component of the velocity and other components of velocity.
2. Main results
Our main result can be stated as follows.
Theorem 2.1. Let
∂3 u3 ∈ Lst 1 Lrx1 , where
2
3
+
≤ 1, 3 ≤ r1 ≤ ∞,
s 1 r1
u1 , u2 ∈ Lst 2 Lrx2 , where
2
3
+
≤ 1, 3 ≤ r2 ≤ ∞.
s2 r2
and
(2.1)
Then u is regular.
At first, let us recall the definition of the Leray-Hopf weak solution.
Definition 2.2. (Leray-Hopf weak solution) If a measurable vector u satisfies
the following properties in 0 ≤ T ≤ ∞:
(1) u is weakly continuous in [0, T ) × L2 (R3 ),
(2) u verifies (1.1) in the sense of distribution
Z TZ
Z
∂Φ
(
+ u · ∇Φ + 4Φ) · udxdt +
u0 · Φ(x, 0)dx = 0,
0
R3 ∂t
R3
∀Φ ∈ C0∞ (R3 × [0, T )), ∇ · Φ = 0, Φ is a vector function,
(3) u satisfies the energy inequality
Z t
2
ku(·, t)kL2 + 2
k∇u(·, s)k2L2 ds ≤ ku0 k2L2 , 0 ≤ t ≤ T,
0
then u is called a Leray-Hopf weak solution of the Navier-Stokes equations.
We give three lemmas for the proof of the theorem.
Lemma 2.3. (The estimate for the pressure in Lq (R3 ))
kpkLq (t) ≤ Ckuk2L2q (t), 1 < q < ∞.
Proof. See [2].
¤
128
H. YUE, W.-M. LI
2
3
Lemma 2.4. (Interpolation inequality) Assume u ∈ L∞
t Lx (R × I) and ∇u ∈
2 2
3
s r
Lt Lx (R × I), where I is an open interval. Then u ∈ Lt Lx (R3 × I) for all r, s
such that
3
2 3
+ ≤
s r
2
and 2 ≤ r ≤ 6. Moreover,
(6−r)/2r
(3r−6)/2r
kukLst Lrx ≤ CkukL∞ L2x k∇ukL2 L2
t
t
x
.
Proof. See [3].
¤
By Lemmas 2.3 and 2.4, we get the following.
Lemma 2.5. (The estimate for the pressure in Lst Lrx (R3 × I))
2 3
kpkLst Lrx ≤ Cku0 k2L2 , + = 3, 1 < r ≤ 3.
s r
Proof. (Proof of Theorem 2.1) At first, let us multiply the equation for u3 of (1.1)
by |u3 |u3 and integrate over [0, T ] × R3 t ∈ (0, T )
Z tZ
Z tZ
Z tZ
∂u3
· |u3 |u3 −
4u3 · |u3 |u3 = −
u · ∇u3 · |u3 |u3
0
R3 ∂t
0
R3
0
R3
Z tZ
−
∂3 p · |u3 |u3
0
R3
I1 + I2 = I3 + I4 .
Calculating every term,
Z Z
∂|u3 |3
1
1
1
1 t
= ku3 k3L3 |t0 = ku(·, t)k3L3 − ku(·, 0)k3L3 ,
I1 =
3 0 R3 ∂t
3
3
3
Z
Z
Z
Z
3
3
X t
8
8X t
(∂i |u3 |3/2 )2 = k∇|u3 |3/2 k2L2t L2x ,
I2 = −
∂ii u3 |u3 |u3 =
9 i=1 0 R3
9
R3
i=1 0
3 Z tZ
3 Z Z
X
1X t
I3 = −
ui ∂i u3 |u3 |u3 =
∂i ui |u3 |3 = 0,
3
3
3
R
R
i=1 0
i=1 0
Z tZ
1
3/2
3/2
I4 =
p∂3 (|u3 |u3 ) ≤ CkpkLs2 Lr2 k∂3 u3 kLs1 Lr1 + ku3 k3L∞
3,
t Lx
x
x
t
t
6
0
R3
where
 1
+ 1 = 1


 s11 + s12 = 2
r1
r2
3
2 ≤ s1 ≤ ∞, 3 ≤ r1 ≤ ∞
 s21 + r31 = 1

 2
3
+ r2 = 3
1 ≤ s2 ≤ 2, 23 ≤ r2 ≤ 3.
s2
By Lemma 2.3, we get
(2.2)
kpkLst 2 Lrx2 ≤ Cku0 k2L2 ,
and so
1
3/2
(2.3)
I4 ≤ Cku0 k3L2 k∂3 u3 kLs1 Lr1 + ku3 k3L∞
3.
t Lx
x
t
6
REGULARITY CRITERION FOR NAVIER-STOKES EQUATIONS
129
Summarizing the above calculating of I1 , I2 , I3 and I4 , we get
3
3
3
3
1
8
ku3 (·, t)k3L3 + k∇|u3 | 2 k2L2t L2x ≤ Cku0 k3L2 k∂3 u3 kL2 s1 Lr1 + Cku0 kL2 2 k∇u0 kL2 2 .
x
t
6
9
By (2.1), we get
3
3 + k∇|u3 | 2 kL2 L2 ≤ C,
ku3 (·, t)kL∞
t Lx
t x
moreover,
3 2
ku3 (·, t)kL3t L9x ≤ Ck∇|u3 | 2 kL3 2 L2 ,
t
x
and so
3 + ku3 (·, t)kL3 L9 ≤ C,
ku3 (·, t)kL∞
t Lx
t x
i.e., u3 ∈ Lst Lrx , where 2s + 3r ≤ 1, 3 ≤ r ≤ 9, so the solution is regular making
use of the regularity criterion [4, 5]:
2 3
u ∈ Lst Lrx , where + ≤ 1, 3 ≤ r ≤ ∞,
s r
which completes the proof.
¤
Acknowledgements: This work was supported by NSFC Grant 10771052.
References
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193–248. 1
2. D. Chae and J. Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal.–TMA 46 (2001), 727–735. 1, 2
3. I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math.
Phys. 48 (2007), Art. ID 065203. 1, 2
4. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch.
Ration. Mech. Anal. 9 (1962), 187–195. 2
5. L. Escauriaza, G. Seregin and V. S̆verk, On backward uniqueness for parabolic equations,
Zap. Nauch. Seminarov POMI 288 (2002), 100–103. 2
1
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China.
E-mail address: huu3y2@163.com
2
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China.
E-mail address: liwum0626@126.com